Microeconomic Theory for the Social Sciences Takashi Hayashi August 30, 2015

Preface This book covers microeconomic theory at the level of intermediate/advanced undergraduates, but I also intend it to be an introduction for those with other intellectual backgrounds, who do not necessarily agree to what so-called ”mainstream economists” say, but at least feel it OK to know how they think and see things.

Stance of this book I tried to give thorough explanations of definitions and assumptions which the theory is based upon. Also I tried to give thorough accounts of motivations and reservations behind the theory. Professional work in economic theory is presented as a sequence of definitions, assumptions and their implications. Its result is presented as a theorem, which is a statement in the form ”If A is true, then B is true.” It is vital for theorists to share the understanding of what assumptions the present theory is relying on, because there is no conclusion without assumption. If you think you are free from any assumption, it is either that you don’t know what assumption your argument is relying on or that you know it and you are hiding it. Introductory teaching of economics, on the other hand, tends to omit giving thorough explanations of underlying assumptions and reservations. There is a good educational reason to do so, because teachers don’t want to make their students bored before getting into ”useful” stuff. This causes a danger, however, that learners do not care about the underlying assumptions and the logical process of how the assumptions lead to the conclusion. As a result, learners quite often abuse theory by applying it to situations in which its assumption does not hold, or criticize theory on the ground that its conclusion is wrong again by applying it to situations in which its assumption does not hold. My aim is to help the readers to get able to draw a precise line between what economic theory says (or can say) and what it does not (or cannot). I hope this is helpful for economics majors as well as those from other disciplines. I am more than happy if my attempt is successful.

i

ii

On the mixture and order of formal presentation, verbal discussion and examples To achieve the above goal at the level of intermediate/advanced undergraduate textbook, I tried to find the best mixture and order of formal presentation of the model, verbal discussions on its motivation and reservations, and explanations through examples. The mixture and order vary across the topics. When I find it helpful to put emphasis on formal presentation as I believe it conveys the point more vividly (and it does not let you behind), I do so. When I find that verbal discussion gives you a richer understanding or helps you to avoid confusion which I expect may happen in the first-look at the formal model, I put discussions before or along with it. When I find it helpful to go over an example first when I introduce a new notion rather than presenting a formal definition or giving wordy discussions, I do so.

On the degree of generality and simplification To achieve the above goal, I tried to find the best level of generality and simplification at which the theory is presented. I would say there are two kinds of simplification. One is made when we can make the model more general and complex in order to get closer to the reality and it is feasible at a methodological level, but it does not essentially change the argument or it is irrelevant to the issue you are looking at. This is called simplification without loss of generality. For example, in many places in the book I assume that there are just two goods in the economy. This of course does not mean that there are really only two goods in the world, but simply means that in order to understand the point it is enough to think of just two goods. What we mean by ”without loss of generality” depends naturally on the type of audiences and their interests. First possibility is that professional researchers are not content with the simplification since it abstracts away what they think is important in the professional works, while it should be allowed for certain educational purposes. There is an opposite possibility: theorists are quite often content with simplified illustration and want to concentrate on the main point, when they see that its extension to more complicated settings is straightforward, while learners or readers from outside of the discipline rather feel that relying on simplified illustration is a cheat. I take the latter case more seriously, and present certain topics at a higher level of generality when I believe it helps understanding more effectively. The other kind of simplification is of course due to the limitation of our analytical ability. In such cases I put discussions on what are abstracted away and what the present theory is missing, and I hope it helps you to go beyond.

iii

On terminologies Microeconomics is related to the society, and because of this the words used there naturally overlap real-life wordings. This is actually dangerous. For example, what do you imagine from the words such as ”utility,” ”perfect competition,” and ”efficiency?” If you imagine some kind of ”substance” from the word ”utility,” that’s wrong. If you imagine a situation like everybody killing each other from the word ”perfect competition,” that’s wrong. If you imagine a one-dimensional criterion which ranks between all social alternatives from the word ”efficiency,” that’s wrong. In economics these words are given precise boundaries in the form of definitions, and I relegate them to the corresponding chapters. What I like to say to you here is that you should wipe away what you imagine from the usual life usages of those words. Unfortunately, the terminologies like above are accepted ones, and it may be embarrassing if I make up new ones, so I decided to accept most of them by putting adequate discussions when I introduce them. Nevertheless, I decided to use unconventional terminologies in some cases when I’m afraid using the accepted terminology causes a serious misunderstanding. I hope it doesn’t embarrass you too much.

On mathematical expositions I use mathematical exposition as long as it is easier than the verbal one. Economic theorists use mathematics not because they like to mystify but because it is the easiest way to share understandings precisely. It is the easiest way to precisely share definitions, assumptions and the process of deriving conclusions from them, and to avoid confusions and errors which often happen in the arguments by natural languages. Of course what we mean by easier will depend on the audiences. I guess it is hard in the beginning, but I bet you will see it much easier as you proceed. Also I tried to give explanations to mathematical notions in a self-contained matter as much as possible when they are necessary for reading ahead.

He or She Because of the nature of the subject, I use third-person singular pronoun repeatedly. It is always a problem for economic theorists if we should use he or she. There isn’t a ”gender-neutral” third-person singular pronoun in English (neither in my mother language) which refers to an ”abstract individual,” and it will be embarrassing if I make up such one. So I have to make a choice. I sometimes use ”she” in research papers, but given that I’m a male this might be somewhat artificial. So I decided to use ”he.” I wish this doesn’t make my texts look sexist.

iv

Acknowledgements This book is largely based on my book published in Japanese (1st edition in 2007, 2nd in 2013). I like to thank Minerva Shobo publisher and Mr. Kentaro Horikawa for their cooperation which enabled the publication of that book. Most of the materials are originally based on my lecture notes given at the University of Texas at Austin. Some topics are based on my lecture notes given at the University of Glasgow.

v

Mathematical Notation I’m not actually using serious mathematics and the most of difficulties you might face will be simply due to unfamiliar notations, which I use for the purpose of concision. Here I give brief description of them. • x ∈ X is read as ”x belongs to X” or ”x belonging to X.” • {x ∈ X : f (x)} denotes the set consisting of x belonging to X which satisfies proposition f (x). • ∀ is universal quantifier. ”∀x; f (x)” is read as ”every x satisfies proposition f (x),” and ”∀x ∈ X; f (x)” is read as ”every x belonging to X satisfies proposition f (x).” • ∃ is existential quantifier. ”∃x; f (x)” is read as ”there exists at least one x which satisfies proposition f (x),” and ”∃x ∈ X; f (x)” is read as ”there exists at least one x belonging to X which satisfies proposition f (x).” • =⇒ is the symbol of implication. For two propositions A, B, ”A =⇒ B” is read as ”If A is true, then B is true.” • ⇐⇒ is the symbol of logical equivalence. For two propositions A, B, ”A ⇐⇒ B” is read as ”A is true if and only if B is true.” • R denotes the set of real numbers. • R+ denotes the set of non-negative real numbers. • R++ denotes the set of positive real numbers. • Rn denotes the set of n-dimensional vectors of real numbers. Its element is for example denoted by x = (x1 , · · · , xn ), where its i-th coordinate is xi . • Rn+ = {x ∈ Rn : xi ≥ 0, i = 1, · · · , n} denotes the set of n-dimensional non-negative vectors. • Rn++ = {x ∈ Rn : xi > 0, i = 1, · · · , n} denotes the set of n-dimensional positive vectors. ∏n • i=1 Ai denotes the product of n sets A1 , · · · , An , that is, A1 × · · · × An .

Contents I

Individual Preference and Choice

1

1 On the concept of ”rationality” in economics 2 Choice objects and choice opportunities 2.1 Description of choice objects . . . . . . . 2.2 Opportunity sets . . . . . . . . . . . . . 2.3 Consumption set . . . . . . . . . . . . . 2.4 Budget constraint . . . . . . . . . . . . . 2.5 Exercises . . . . . . . . . . . . . . . . .

2

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

11 11 12 13 16 21

3 Preference 3.1 Preference relation . . . . . . . . . . . . . . . . . . . . . 3.2 Preference over consumptions . . . . . . . . . . . . . . . 3.3 Marginal rate of substitution . . . . . . . . . . . . . . . 3.4 Smooth preferences . . . . . . . . . . . . . . . . . . . . . 3.5 Convexity and diminishing marginal rate of substitution 3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

23 23 25 31 33 34 35

. . . . .

. . . . .

4 So-called utility function 4.1 ”Utility” representation of preference . . . . 4.2 Marginal utility . . . . . . . . . . . . . . . . 4.3 Describing marginal rate of substitution by utilities . . . . . . . . . . . . . . . . . . . . 4.4 Ordinal utility and cardinal utility . . . . . 4.5 Exercises . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . . . . . . . . . means of . . . . . . . . . . . . . . . . . .

5 Choice and demand 5.1 Maximal elements for preference . . . . . . . . . . . . 5.2 Smooth consumption choice . . . . . . . . . . . . . . . 5.3 The case of perfect substitution . . . . . . . . . . . . . 5.4 The case of perfect complementarity . . . . . . . . . . 5.5 Demand function . . . . . . . . . . . . . . . . . . . . . 5.6 Consumption choice and demand in exchange economy 5.7 Describing choice as utility maximization . . . . . . .

vi

. . . . . . . . . . . . marginal . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

36 36 40 48 51 53 54 54 57 61 63 63 64 65

CONTENTS 5.8 5.9

vii

Expenditure minimization and compensated demand . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Demand analysis 6.1 Normal and inferior goods . . . . . . . . . 6.2 Ordinary and Giffen goods . . . . . . . . . 6.3 Gross substitutes and gross complements . 6.4 Elasticity of demand . . . . . . . . . . . . 6.5 Substitution effect and income effect . . . 6.6 Income evaluation of welfare change . . . 6.7 Exercises . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

66 70 71 71 72 73 75 77 81 88

. . . . . . .

7 Willingness to pay and consumer surplus 7.1 Naive utility argument . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The assumption of no income effect . . . . . . . . . . . . . . . . . 7.3 Marginal willingness to pay as marginal rate of substitution . . . 7.4 No income effect and inverse demand function . . . . . . . . . . . 7.5 Compensated variation, equivalent variation and consumer surplus 8 Intertemporal choice 8.1 Intertemporal choice and intertemporal budget constraint 8.2 How to deal with inflation . . . . . . . . . . . . . . . . . . 8.3 Discounted present value of streams . . . . . . . . . . . . 8.4 Preference over consumption streams . . . . . . . . . . . . 8.5 Intertemporal consumption choice . . . . . . . . . . . . . 8.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

89 89 90 95 97 99

. . . . . .

. . . . . .

. . . . . .

101 101 102 103 105 113 115

9 Choice under risk 9.1 Risk and uncertainty . . . . . . . . . . . . . . . . . . . . . . . 9.2 Risk attitude . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Expected utility representation: an experimental construction 9.4 Expected utility representation: the formulation . . . . . . . 9.5 Axiomatic characterization of expected utility representation 9.6 ”Cardinal” properties of vNM indices . . . . . . . . . . . . . 9.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Violation of the expected utility theory . . . . . . . . . . . . . 9.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

116 116 116 117 119 120 122 125 129 133

10 Revealed preference

. . . . . .

134

II Perfectly Competitive and Complete Market with Complete Information 139 11 Perfectly competitive and complete market with complete information 140 11.1 Perfect competition . . . . . . . . . . . . . . . . . . . . . . . . . . 140

CONTENTS

viii

11.2 Complete market . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 11.3 Complete information . . . . . . . . . . . . . . . . . . . . . . . . 146 12 Competitive equilibrium in exchange economies 12.1 Exchange economy . . . . . . . . . . . . . . . . . 12.2 Competitive equilibrium . . . . . . . . . . . . . . 12.3 Interest rate in borrowing-lending economies . . . 12.4 Security exchange and security price . . . . . . . 12.5 Exercise . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

148 148 149 152 157 164

13 Efficiency of competitive allocation 13.1 Pareto improvement and Pareto efficiency . . . 13.2 Efficiency of competitive equilibrium allocation 13.3 Important remarks on Pareto efficiency . . . . . 13.4 Exercises . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

165 165 169 171 172

14 Production technology 14.1 1-input/1-output case . . . . . . . . . . . . . . . . . . . . . . . . 14.2 2-input/1-output case . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

173 173 175 180

. . . .

15 Profit maximization and cost minimization 181 15.1 Profit maximization when output price and input prices are given 181 15.2 Cost minimization when input prices are given . . . . . . . . . . 187 15.3 Long-run and short-run . . . . . . . . . . . . . . . . . . . . . . . 190 16 Cost curve and supply 193 16.1 Average cost and marginal cost . . . . . . . . . . . . . . . . . . . 193 16.2 Profit maximization under perfect competition . . . . . . . . . . 195 16.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 17 Competitive equilibrium in production economies 200 17.1 Private ownership economy . . . . . . . . . . . . . . . . . . . . . 200 17.2 The representative consumer/producer model . . . . . . . . . . . 206 17.3 Interest rate determination in an intertemporal production economy211 17.4 Efficiency of competitive equilibria . . . . . . . . . . . . . . . . . 213 17.5 Socialist calculation debate . . . . . . . . . . . . . . . . . . . . . 220 18 Partial equilibrium analysis 18.1 Competitive partial equilibrium . . . . . . . . . . . . . . . . . . . 18.2 Pareto efficiency and maximal surplus . . . . . . . . . . . . . . . 18.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

222 223 224 228

CONTENTS

III

ix

Imperfect competition and game theory

19 Monopoly 19.1 Monopoly equilibrium . . . . . . . . . . . . . 19.2 Pareto inefficiency of monopoly equilibrium . 19.3 Price discrimination and monopolistic surplus 19.4 Exercises . . . . . . . . . . . . . . . . . . . .

229 . . . .

. . . .

. . . .

230 231 233 234 241

20 Basic game theory I: normal-form games 20.1 Description of strategic interdependence: normal-form games 20.2 Dominant strategy . . . . . . . . . . . . . . . . . . . . . . . . 20.3 Iterated elimination of dominated strategies . . . . . . . . . . 20.4 Rationalizable strategies . . . . . . . . . . . . . . . . . . . . . 20.5 Nash equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 20.6 Mixed strategies . . . . . . . . . . . . . . . . . . . . . . . . . 20.7 Refinement of Nash equilibria . . . . . . . . . . . . . . . . . . 20.8 How should we think of multiple equilibria? . . . . . . . . . . 20.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

243 244 247 248 251 253 260 265 268 271

21 Basic game theory II: extensive-form games 21.1 Description of strategic interdependence: extensive-form games 21.2 Subgame-perfect Nash equilibrium . . . . . . . . . . . . . . . . 21.3 Extensive-form games with imperfect information . . . . . . . . 21.4 Bargaining game . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Repeated games and sustainable cooperation . . . . . . . . . . 21.6 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

272 272 273 278 280 283 286

22 Oligopoly 22.1 Simultaneous quantity setting (Cournot competition) 22.2 Sequential quantity setting: Stackelberg competition 22.3 Simultaneous price setting: Bertand competition . . 22.4 Sequential price setting . . . . . . . . . . . . . . . . 22.5 Convergence to perfect competition . . . . . . . . . . 22.6 Collusion . . . . . . . . . . . . . . . . . . . . . . . . 22.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

287 288 291 294 299 302 304 306

IV

. . . . . . . . . . . . extraction . . . . . .

. . . . . . .

. . . . . . .

. . . .

. . . . . . .

. . . .

. . . . . . .

. . . . . . .

. . . . . . .

Economic Analysis with Incomplete Information 308

23 Basic game theory III: games with incomplete 23.1 Bayesian game and Bayesian Nash equilibrium 23.2 On the common prior assumption . . . . . . . . 23.3 Exercises . . . . . . . . . . . . . . . . . . . . .

information 309 . . . . . . . . . . 309 . . . . . . . . . . 315 . . . . . . . . . . 316

CONTENTS

x

24 Auction 24.1 Prominent auction formats . . . . . . . . . . . 24.2 Information, timeline and the natures of values 24.3 Preferences . . . . . . . . . . . . . . . . . . . . 24.4 First-price auction . . . . . . . . . . . . . . . . 24.5 Second-price auction . . . . . . . . . . . . . . . 24.6 The revenue equivalence theorem . . . . . . . . 24.7 Exercises . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

317 317 318 319 319 323 325 326

25 Trade with incomplete information 25.1 Adverse selection . . . . . . . . . . 25.2 Moral hazard . . . . . . . . . . . . 25.3 Signaling . . . . . . . . . . . . . . 25.4 Speculative trade . . . . . . . . . . 25.5 Exercises . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

327 327 331 335 338 342

V

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

Market Failure and Normative Economic Analysis 344

26 Externality 345 26.1 Market failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 26.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 27 Public goods and the free-rider problem 27.1 Public goods . . . . . . . . . . . . . . . . . . 27.2 Efficiency criterion: the Samuelson condition 27.3 The case of quasi-linear preferences . . . . . . 27.4 The free-rider problem . . . . . . . . . . . . . 27.5 Strategy-proof mechanism . . . . . . . . . . . 27.6 Exercises . . . . . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

353 353 353 357 359 360 362

28 Indivisibility and heterogeneity 363 28.1 Allocation of indivisible objects . . . . . . . . . . . . . . . . . . . 364 28.2 Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 28.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 29 Efficiency, welfare comparison and fairness 29.1 The Kaldor/Hicks criteria . . . . . . . . . . 29.2 Fair allocation in exchange economies . . . 29.3 Fairness in production economies . . . . . . 29.4 Exercises . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

371 371 377 380 382

30 Aggregation of preferences and social choice 30.1 Motivations from welfare economics and political science 30.2 Axioms for aggregation of preferences . . . . . . . . . . 30.3 Arrow’s theorem . . . . . . . . . . . . . . . . . . . . . . 30.4 May’s theorem . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

383 383 386 389 389

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

CONTENTS

xi

30.5 Borda rule again . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 30.6 Domain restriction and single-peaked preferences . . . . . . . . . 391 30.7 Proof of Arrow’s theorem . . . . . . . . . . . . . . . . . . . . . . 393 31 Implementability of social choice objectives 31.1 Social choice function and mechanism . . . . . . . . . . . 31.2 Implementation in dominant strategy equilibrium . . . . . 31.3 Implementation in Nash equilibrium and allowing multiple libria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Appendix: Proof of the Gibbard-Satterthwaite theorem .

. . . . . . . . equi. . . . . . . .

397 397 398 401 404

Postscripts

407

Solutions to the exercises

424

Part I

Individual Preference and Choice

1

Chapter 1

On the concept of ”rationality” in economics Economics is very often criticized of making unrealistic assumptions. The most common criticism will be about ”rationality,” saying that real human beings are not rational as assumed in economics. I agree to some of them eventually (probably not in the way the readers expect), but let me give some clarifications before I proceed, since the word ”rationality” is broad — in daily life the word ”rational” or ”irrational” has been used even as a convenient rhetoric to justify and praise or criticize and dis somebody’s choice or action while maintaining the appearance of being valueneutral. It is obviously a hard problem to summarize the notion of rationality in economics so that everybody can agree. Let me try, however, to summarize what I understand is consistently underling economic theory. I would say ”rationality” in economics refers to that 1. an individual has certain consistent subjective criterion of value (called preference); 2. he takes all the relevant contingencies into account and perceive them correctly; 3. he goes through ”logically correct” reasonings; and 4. he fulfills the criterion up to the maximum. That is, the notion of ”rationality” here is purely a formal one at an individual level. As far as the above conditions are met we have to say that one is ”rational” even if he is a vicious killer. At the same time, this notion of rationality does not presume that one is ”selfish,” and does not exclude altruism to be a component of individual’s subjective criterion at all, as far as it remains to be consistent. 2

CHAPTER 1. ”RATIONALITY”

3

Put differently, some action being ”rational” for an individual and its being socially desirable are different issues. Of course this clarified notion of rationality again faces criticisms, saying that real human beings are not consistent or knowledgeable or precise or smart as described above. It is an ”idealized” principle which is impossible if you take it literally. To borrow an outdated analogy (it’s a shame but I can come up with only this one), this is analogous to how physics starts its first-step argument by assuming vacuum and no friction. There is no vacuum or frictionless situation in reality, but such assumption helps us to build the first several laws in classical mechanics, which kicks us upstairs so that we can understand more realistic situations. I view that in social sciences it is not only helpful to start with such idealized assumption but rather necessary, in that we cannot see or understand ”reality as it is” without standing on such ”baseline.” Reality is of course different from the baseline, but it can be understood only by seeing how it is different or distant from the baseline. A natural question arising here is what is the postulate for a good choice of such baseline. A most extreme form of ”positivist” view says that it doesn’t have to have anything to do with reality and it should be the simplest assumption under the simplest setting which can derive predictions consistent with real phenomena as many as possible, and it is rather better as it is more unrealistic. It says for example that it is meaningless to test whether individuals are really solving their maximization problems rationally, and what is important is that their behaviors are explained ”as if” they are solving maximization problems rationally. I don’t take this view, however, because it does not say anything about the necessity of particular assumptions, as there may be several equally simple principles which can explain the reality in the ”as if” way. Why do we have to choose the above assumptions over the others? Also, economists have a task to do welfare analysis and provide normative arguments, which critically depend on how much individuals are responsible for their rationality. If an economist takes the above ”pure positivist” view he should not be able to draw any normative implication from his positive analysis. If he does so it must be a deception. A typical deception is that in ”positive” analysis one describes individuals’ choices ”as if” they are acting rationally and in its normative implication he switches the interpretation implicitly so that the individuals are indeed rational and responsible for their choices. Another view about rationality often invoked is an evolutionary story, which says that if one is not rational he would die or perish either in the social or biological sense, meaning that there is little to lose by assuming that those who are living (that is, who have survived by now) are rational.

CHAPTER 1. ”RATIONALITY”

4

I don’t take this view either, since it says at most that certain characteristics make one the ”fittest” and more likely to survive under certain environment. We cannot draw any normative implication from this either, while application of evolutionary arguments to social sciences often falls in the confusion that such characteristics are desirable and those with such characteristics should be dominant in the society, even in a modern civilized society — it is in the beginning strange to insist that, since if some people are really the fittest they would have been already dominant before saying ”should.” I would say, the primary role of economics is to provide a consistent and meaningful understanding of commensurability between different individual values each of which is solid, stable and deliberate, and how to realize such commensuration. It is not to provide an explanation or prediction of ”behavior in general,” nor to grab an organic formation of value sentiments in the society ”as a whole.” The choice of baseline should serve this objective, and certain abstraction is necessary in order that we can clearly see solid, stable and deliberate individual values. Note that such abstraction is purely a formal one, in the sense that we identify time horizons, spaces and contingencies over which the notion of solid, stable and deliberate individual values makes sense, by abstracting away certain ranges of idiosyncratic details of choice situations. It is not selecting a particular content of social life over another, such as selecting ”economic rationality” and ”abstracting away” the other ones such as political, social and cultural. Of course, in this sense, we should note that economics has an ”imperialistic ambition” which tries to apply its methodology to any aspect of social life that possesses the same formal structure. It is natural that you wonder if such rationality approach works (or should work). So let me briefly explain the nature of the approach, what types of abstraction are carried out there, and list challenges to it. To illustrate, let x denote input, which is an observable external element given to the individual (such as constraint or situation), and let y denote output, which is his observed behavior. If we take the most extreme stance of socalled behavioralism, then we consider only a functional relationship which holds between input and output. Denote such functional relationship by f , then the relation between input and output is denoted by y = f (x). In the empirical side, given data consisting of pairs of input and output (x1 , y2 ), (x2 , y2 ), · · · , (xn , yn ),

CHAPTER 1. ”RATIONALITY”

5

a purely behavioralistic research would look for f which meets y1

= f (x1 )

y2

= f (x2 ) .. .

yn

= f (xn ),

but it does not impose any other features on it, particularly the ones about internal mental state. On the other hand, the ”rationality” approach in economics considers the functional relationship y = g(θ, x), where the parameter θ describes preference (corresponding to 1 in the above) and g describes the maximization behavior (corresponding to 2 to 4 in the above). There the right-hand-side g(θ, x) refers the choice which achieves preference θ up to the maximum under given external condition x. In the empirical side, given data consisting of pairs of input and output (x1 , y2 ), (x2 , y2 ), · · · , (xn , yn ), if you can ”back up” parameter θ which meets y1 y2

= g(θ, x1 ) = g(θ, x2 ) .. .

yn

= g(θ, xn )

then we say that this behavioral data is ”rationalizable.” Again, ”rationalize” here does not mean justification. This θ is called a structural parameter, and supposed to be unchanged across different external conditions. In policy arguments, we use θ ”backed up” from data and then predict the effect of input on output and evaluate if such change is desirable, which is called counter-factual analysis. This is the format of theory, empirics and policy recommendation in the ”rationality” approach. Economics is ”behavioralistic” in the sense that it considers only the behaviorgoverning parameter θ which can be ”backed up” from observed choices. That’s why the research of subjective well-being based on self-report is not taken to be a ”standard” research in economics. On the other hand, economics is ”not behavioralistic” in the sense that it is already interested in ”value” and has certain prior tendency or preconception to pick the ”rational” choice model (that is, g) among many possible models which can explain the given data.

CHAPTER 1. ”RATIONALITY”

6

By the way, I put quotation marks on ”logically correct” in Postulate 3 in the above formulation. This obviously suggests that I don’t want to be seen as naively believing that ”logic” is single and universal. Yes, the ways of reasonings used in this book are limited to the standard formal logic and probability theory. However, legal reasonings such as argumentum e contrario and analogy are not quite a logical operation in the sense of formal logic, but they may be a convincing ”logic” indeed. There may be ”rational” choices based on ”logically correct” reasonings which are different from the standard formal logic, and it is one of the jobs of theorists to investigate the consequences of being ”rational” in such sense. The ”rationality” approach has always been facing criticisms. The first thing one can think of in order to absorb the criticisms is to introduce ”noise” or ”unobserved heterogeneity,” denoted ε here, and extend the above rationality model like y = g(θ, x) + ε When ”noise” ε is zero ”on average” we can say that the individual is ”rational on average.” This is exactly the sense in which I wrote ”individually solid and stable” above.1 Now, how should we think when a deviation from ”rationality” looks systematic and cannot be explained by ”errors” or ”noise?” Let me bring up some issues which I like to share before proceeding. Issue 1: Preference is not fixed, but it rather changes over time. In order to deal with this issue we need to select the most relevant time horizon to the given problem. For, in my view, most of the cases which are regarded as showing ”irrationality” or ”inconsistency” of human choices are simply due to the observer’s misspecification of the relevant time horizon. For example, even when somebody ate pasta for lunch yesterday and hamburger for lunch today we won’t say his choice over foods is inconsistent, since we know that the problem of what to eat today is not identical to or separate from the problem of what to eat on another day. Much less when we spend our lunch budget on weakly or monthly basis. That is, what we observe here is not a contradiction like pasta > hamburger,

hamburger > pasta

but a revealed ranking (pasta, hamburger) > (pasta, pasta), (hamburger, hamburger), (hamburger, pasta), 1 Of course it is no more than an analogy at this point since we don’t yet have the definition of ”+.”

CHAPTER 1. ”RATIONALITY”

7

where (pasta, hamburger) denotes the choice of eating pasta on the first day and hamburger on the second day (although we cannot specify the ranking among the three on the right-hand-side from this observation alone). It is natural to wonder, ”As you set the time horizon longer you can absorb variation of choices across periods into the length of the time horizon. Doesn’t it mean that by taking time horizon arbitrarily long we can explain anything as ”rational” choice generated by a ”fixed” preference over arbitrarily long objects?” In other words, if we take it literally that life is just once anything is ”rational” since there is only one sample. This is ultimately a problem for the outside observer, who judges how long the time horizon should be taken so that observed choice data are seen as a ”repetition of some complete problem” and it is meaningful to think of consistency and inconsistency across samples. For example, you can think of proportions of kinds of lunch meals during one month, and see the data as a repetition of such monthly summary. What if the life is not a repetition of an identical problem, and like a ”whole life” only one dynamic choice problem is given to an individual and we can observe just one sample of his life path? In statistical treatment of such dynamic choice problems, we usually consider that people are ”ex-ante identical” and take different life paths because of different inputs, which are observable, and unobservable noises. Now, how should we think if we see inconsistencies of choices even after selecting the time horizon as adequately as possible? Let us think of the following example. Example 1.1 You have a choice of starting cocaine or not. There are three possible paths: • A: Start it and quit it later. • B: Start it and continue. • C: Don’t take it at all. As you have not started cocaine yet and you are curious, your preference over the paths is A>C>B However, because of the nature of addiction, once you start cocaine you become a different person, literally, and the preference of your new personality is B>A That is, there are two different ”selves,” before and after taking cocaine, who have contradicting preferences.

CHAPTER 1. ”RATIONALITY”

8

In such cases one preference cannot simply determine the choice. There are at least two ways of choice. One is that the current self makes choice by (mistakenly) believes that he can control his future selves. It is called naive decision. In the above example, the naive decision is to start taking cocaine, intending to stop later, and does not actually stop it later. The other one is that the current self foresees how future selves behave, and makes choice by taking how future selves behave as a given constraint. It is called sophisticated decision. In the above example the sophisticated decision is not to take cocaine at all, given that the future self cannot stop taking cocaine once he starts it. The cocaine example may be a bit too extreme, but this type of problem often occurs in choice with habit formation. Let us think of one more example. Example 1.2 Consider the following two choice problems. Problem A A1: Receiving 1000 dollars after one year. A2: Receiving 1050 dollars after one year and one week. Problem B B1: Receiving 1000 dollars now. B2: Receiving 1050 dollars after one week. The example as presented like above is somewhat misleading since you can save money, so assume that you have to spend the money immediately after receiving. Then, (in more carefully designed experiments) the pair of choices like A2 and B1 is frequently observed. What’s the problem with this? Suppose you choose A2 in A and B1 in B. Then you would choose to sign a contract to receive 1050 dollars after one year and one week, rather than a contract to 1000 dollars after one year. After one year, you will regret, since you like to receive 1000 dollars immediately rather than to wait for one more week. Thus, there is a conflict between current self and self after one year. How do economists think when such ”successive selves” are present? Mostly we then take an individual as a ”society” consisting of different ”selves,” and game theory or social choice theory to such micro-society. Although, we should be careful about the applicability of these theories to the micro-society, since ”successive selves” are not totally different persons from each other. I will come to this issue in the last part of the postscripts.

Issue 2: It is untrue that an individual chooses the best available thing for him. There are cases in which he gives up selfish choice, due to certain social reasons.

CHAPTER 1. ”RATIONALITY”

9

Consider the following example. Suppose you receive 100 dollars from somebody, and you can either spend all of that for yourself or give half of that to your brother. If you care only about your consumption in the current period you will spend all for your self, but many of you may give half of that to your brother. We can explain this in two ways. 1. The apparent departure from ”rationality” is again due to misspecification of time horizon and relevant contingencies. There is nothing wrong in that a rational action causes loss in the short-run whereas it realizes gains in the long run. Also, there is nothing wrong in that a rational action causes loss under particular state, while it is profitable in expectation from the ex-ante viewpoint. Insurance is a typical example. In such a way, we can explain mutual help as a collection of individuals’ selfish behaviors in the long run or under uncertainty. 2. Altruism and care for social status are nothing but a part of preference. It appears that an individual is not choosing what he likes because the outside observer is mis-specifying his preference or captures it only partially. In such cases, an individual compares between satisfaction of is his ”selfish” motive and satisfaction of his altruistic motive and his taste for social status, and after ”weighing” he makes the total decision. By going through the above ways we can extend the standard choice theory, often borrowing helps of game theory to be covered in the later part of the book. I would say that economics puts priority on the first way, since allowing the second way without discipline may lead to ”anything goes.” In any case, from the viewpoint of ”rationality” as summarized above it is not essential whether an individual is ”selfish” or not. Issue 3: Individual’s choice criterion does not exist independently of choice situation. In the above model, the structural parameter θ is supposed to exist prior to and independently of x being given. That is, preference is supposed to exist independently of choice situation. The ”rationality” approach then considers that behavior is a function of preference and choice situation. It is shown by many experimental studies, however, that individual’s choice criterion depends on how choice opportunities are given. The following example is due to Benartzi and Thaler [3]. Example 1.3 Consider coin flipping and the following two choice problems. Problem 1: Split 100 dollars between two securities below.

CHAPTER 1. ”RATIONALITY”

10

Security A: It doubles the allocated money if it shows head, and it becomes junk if tail. Security B: It doubles the allocated money if it shows tail, and it becomes junk if head. Problem 2: Split 100 dollars between two securities as below. Security A: It doubles the allocated money if it shows head, and it becomes junk if tail. Security C: It returns the allocated money as it is regardless of the coin-flip outcome. Since head and tails are equally likely it is natural to split money equally between A and B in Problem 1. However, it is reported that they tend to split money between A and C in Problem 2 as well. This is strange from ”rational” viewpoint, since splitting money equally between A and B is nothing but buying Security C, and there is no point in allocating money to A in Problem 2. Such deviation from rational choice, which is inconsistent but non-random and has certain tendency, is called an anomaly. How does a ”die-hard” rational choice theorist handle this problem? He would go one step back, and take an ex-ante viewpoint. In the above example, he would consider probability distribution of security choice problems to be faced by the individual, and takes ex-ante evaluation of such distribution, by means of expectation calculation. The last criticism will be, 4: Humans are not ”choosing.” I would say, if humans are not ”choosing,” all what we can say is ”it is what it is” and we cannot talk about ”value” or what is ”good” or ”bad” or ”better” or ”worse.” I know I’m taking the order of causality upside-down, and it must be absurd to say that the nature has to work so that the rationality approach allows us to talk about ”values.” But the problem is obviously beyond my intellect. Let me stop here for now. I know there are many undiscussed problems, but I think it is better to share them after digesting the main body of this book. I will come back to this at the end of the postscript.

Chapter 2

Choice objects and choice opportunities 2.1

Description of choice objects

The method (not substantive contents conveyed by it) covered in this book can be applied to general kinds of spheres of society, not only to individual and material consumptions. It allows for example that one’s consumption may affect other ones’ consumptions (externality), and that there may be a good which many people can use at the same time (public good). Also it is not limited to material consumption but can be applied to non-material kinds of social actions such as political or social or cultural choice. In the beginning, the set of choice objects X is just an abstract set. For example, the set of choice objects in US presidential election is let’s say X = {Obama, Clinton, Romney, Palin, McCain, · · · }, and the set of choice objects in the problem of which school to attend is let’s say X = {Univ. A, Univ. B, Univ. C, Univ. D, Univ. E, · · · }, and the set of choice objects in the problem of which company to work for is let’s say X = {Co. A, Co. B, Co. C, Co. D, Co. E, · · · }. I guess the readers wonder here. ”How can we choose from them even when not all of them can be the presidential candidates?” ”How can we choose from them even when not all of those schools make offers to me?” ”How can we choose from them even when not all of those companies make offers to me?” Here I take X to be the set of all the conceivable and potentially available objects, putting it aside which ones are actually available to choose.

11

CHAPTER 2. OBJECTS AND OPPORTUNITIES

12

The first step in microeconomics is to write down the right set of choice objects according to the interest. Consider again the example X = {Co.A, Co.B, Co.C, Co.D, Co.E, · · · }. Here it is implicitly assumed that salary is already a fixed component of each company’s feature. However, one can consider that salary is an explicit variable as well, then the set of choice objects is X = {Co.A, Co.B, Co.C, Co.D, Co.E, · · · } × R+ , where R+ denotes the non-negative half line. Its element is for example (Co.D, m), which means working for Company D for salary m. Also, one can consider that which city to work in is also an explicit variable as well. Then the set of choice objects is X = {Co.A, Co.B, Co.C, Co.D, Co.E, · · · } × R+ × {City α, City β, City γ, · · · }, and its element is for example (Co.D, m, γ), which means working for Company D for salary m and living in City γ. Also, consider what to eat for lunch then the set of choice objects is X = {pizza, humburger, pasta, sandwitch, fish and chips, · · · } but if you are talking not just about lunch for today lunch but also about lunch for tomorrow, the right description is X

=

{pizza, hamburger, pasta, sandwich, fish and chips, · · · } ×{pizza, hamburger, pasta, sandwich, fish and chips, · · · },

and its element is for example (sandwich, pizza), which says ”eating sandwich today and pizza tomorrow.”

2.2

Opportunity sets

As explained above, the set of choice objects X consists of all the potentially available ones. However, in actual choice opportunities we are given only a subset of it. Let us call it an opportunity set. Denote it let’s say by B, then it must satisfy B ⊂ X and B ̸= ∅. In the example of school choice, given the set of all schools X = {Univ. A, Univ. B, Univ. C, Univ. D, Univ. E, · · · }, the set of schools one can be admitted to is let’s say B = {Univ. C, Univ. E, Univ. J}.

CHAPTER 2. OBJECTS AND OPPORTUNITIES

13

In the example choosing which company to work for, given the set of all companies X = {Co. A, Co. B, Co. C, Co. D, Co. E, · · · }. the set of companies from which one can get an offer is let’s say B = {Co. A, Co. K, Co. M, Co. Q}. Here let B denote the family of opportunity sets which are institutionally possible. The simplest form of B will consist of all the non-empty subsets of X, but it is not always the case institutionally. For example, in US presidential election because no more than one candidate can run from one party we cannot have a choice opportunity like B = {Obama, Clinton}.

2.3

Consumption set

So far, the set of choice object X or choice objects x and y can be anything. However, in the first half of this book we consider individual consumptions mostly, as far as we are concerned with market theory. In this context the set of choice objects, which consists of all the potentially possible combination of consumptions, is called consumption set. Again, because consumers are constrained by their incomes not all of them are always available to choose. I need to explain consumption set first, however.

2.3.1

Standard consumption set

To simplify the explanation, we mostly assume that there are just two goods. Of course this does not mean that there are really only two goods in the world, and it is simply that the two-good illustration is enough for understanding of the contents covered in this book. Here let me call them Good 1 and Good 2. Also, we mostly assume that each good is homogeneous and divisible. Like gasoline, we consider that this 1 gallon of it and that 1 gallon of it are identical and we can buy it in arbitrarily fine quantities such as 1.367... gallons. Of course actual accounting does now allow this but let us consider that we can do something like this as closely as possible. On the other hand, this house and that house are typically different. They are heterogeneous. Also typically we cannot buy 0.47 units of house. It is indivisible. We will consider heterogeneous goods and indivisible goods in the next section and in a later chapter. As we assume each of the two goods is homogeneous and divisible, the consumption set is given as the non-negative quadrant of the 2-dimensional plane R2+ . Its element x = (x1 , x2 ) is called consumption vector. When the consumer is receiving x = (x1 , x2 ) it means he is receiving x1 units of Good 1 and x2 units of Good 2 (see Figure 2.1). For example, when Good 1 is gasoline and

CHAPTER 2. OBJECTS AND OPPORTUNITIES

14

Good 2 6

x2

r x

x1

- Good 1

Figure 2.1: 2-dimensional consumption set

Good 2 is water, consumption vector x = (x1 , x2 ) refers to x1 units of gasoline and x2 units of water. We assume that consumption of each good is non-negative, but depending on interpretation it is possible to think of negative consumption, and I will come to it when it is necessary to consider.

2.3.2

Indivisible goods

There are variations in how to describe heterogeneity and indivisibility, but here let me pick a simple illustration: Good 1 is indivisible but homogeneous and Good 2 is homogeneous and divisible. I will take heterogeneity and indivisibility more seriously in Chapter 28. Good 1, which is homogeneous but indivisible, allows only integer amounts of consumption. Good 2 is the same as before. Then the consumption set is Z+ × R+ . See Figure 2.2, where the first-coordinate consists only of integer values. For example, x = (x1 , x2 ) is in the consumption set because x1 = 3 is an integer. On the other hand, y = (y1 , y2 ) is not in the consumption set because y1 is not an integer.

2.3.3

Labor and leisure

In the above formulation of consumption set we have assumed that unless the consumer is constrained by his budget we may consider arbitrarily large amounts of consumptions. This will be inadequate for the case of labor and leisure, as one cannot work more than 24 hours a day in the beginning. Therefore, when we analyze the choice of labor and leisure we assume that available hours per period are limited in the outset, after excluding minimal necessary hours for subsistence such as sleeping hours.

CHAPTER 2. OBJECTS AND OPPORTUNITIES

15

Good 2 6 ry

rx

- Good 1 Figure 2.2: Good 1 is indivisible

In economics we mostly take the description that the consumer chooses leisure hours out of his disposable hours, and out the rest to labor (though it doesn’t have to be). For simplicity let the disposable hours be 1. On the other hand, let us assume that there is just one consumption good or that all the consumption goods have been aggregated into one. This is enough for understanding tradeoffs between consumption and leisure. Then the set of possible combinations of leisure and consumption is [0, 1]×R+ as depicted in Figure 2.3. When its element (l, c) is given it means that the consumer puts l into leisure and 1 − l into labor, and consumes c units of the consumption good. Of course one can think that labor and leisure are an indivisible. Right, in reality one can be either employed or unemployed rather than he can choose between working 35 hours or 40 hours per week. Also, where to work will matter as well. I will come to such problem of indivisibility and heterogeneity in Chapter 28.

2.3.4

Consumption over time

What is important in economics is that even if goods are materially the same they are treated as different goods if they are to be consumed at different time periods and different contingencies. For example, gasoline to be consumed today and gasoline to be consumed tomorrow are different goods. Saving is an action to buy future consumptions by means of selling current consumptions. The simplest model of such intertemporal consumption is 2-period model. Assume that there are just two periods, Period 1 and Period 2, and there is just one material good in each period. It might look too simple, but this is enough for the understanding and it can be extended to many periods. Then the consumption set is the non-negative quadrant R2+ . That is, when

CHAPTER 2. OBJECTS AND OPPORTUNITIES

16

Consumption 6

r (l, c)

- Leisure Figure 2.3: Leisure/labor and consumption

a consumption vector x = (x1 , x2 ) is given to the consumer it means the he consumes x1 units in Period 1 and x2 units in Period 2. We call this a consumption stream. It may be too short to be called a ”stream” but we adopt this word in order to emphasize the relevance of time.

2.3.5

Consumption under uncertainty: state-contingent consumption

Same argument holds for uncertainty as well. For example, 1 gallon of gasoline when Republicans win the US presidential election is a different good than one gallon of gasoline when Democrats win. If you have to make some investment decision before the election your choice = bet is described in the form of statecontingent consumption. To simplify, focus on the case that there are just two possible states of the world, like ”Republicans or Democrats” and ”hot summer or cold summer.” Call the first one State 1 and the second one State 2, and there is just one material good at each state. It might look too simple again, but this is enough for the understanding and it can be extended to many states. Then the set of state-contingent consumption vectors is described by the nonnegative quadrant R2+ . That is, when a vector of state-contingent consumption x = (x1 , x2 ) is given it means that the consumer receives x1 units of consumption at State 1 and x2 units st State 2.

2.4

Budget constraint

Consumption set describes all possible combinations of consumptions which are potentially available for the consumer. Not all of them are actually affordable, however, and he is constrained by his budget according to this income and

CHAPTER 2. OBJECTS AND OPPORTUNITIES

17

Good 2 6

w/p2 ry p1 x1 + p2 x2 = w w/p1

- Good 1

Figure 2.4: Budget constraint

prices of the goods. That is, the set of consumptions affordable under the budget constraint is a subset of the consumption set. Budget constraint in the standard form Let us think of the simplest case, which I call standard form. When a pair of prices (called price vector) p = (p1 , p2 ) and income w are given, any affordable combination of consumption x = (x1 , x2 ) must satisfy p1 x1 + p2 x2 ≦ w. As the left-hand-side is expenditure and the right-hand-side is income, the above inequality says that expenditure should not exceed income. Graphically speaking, any affordable consumption vector cannot go outside of the triangle depicted in Figure 2.4. For example, consumption vector y = (y1 , y2 ) is not affordable. Remark 2.1 One may naturally ask, ”where does the price p = (p1 , p2 ) come from, and how is it determined?” Please delay this question until the chapters on market. Here I’m just talking about how consumers respond to given prices. Given a price vector p = (p1 , p2 ) and income w, denote the set of consumption vectors satisfying the budget constraint by B(p, w) = {x ∈ R2+ : p1 x1 + p2 x2 ≦ w} This is called budget set. Graphically, B(p, w) corresponds to the area surrounded by the triangle as in Figure 2.4. Its upper-left face is called budget line. When the consumer spends all his income his consumption vector must line on the budget line. Budget line is described by the equality p1 x1 + p2 x2 = w,

CHAPTER 2. OBJECTS AND OPPORTUNITIES

18

which is called budget equation. When a consumption vector is strictly below the budget line it means the consumer is not spending all his income. Remark 2.2 In most cases we consider that our consumer spends all his income and chosen consumption vector lies on the budget line, and that the budget constraint is met with equality. One may say, ”that’s wrong, it excludes saving.” This does not take future consumption into account as a different good. In economics goods to be consumed at different time periods are taken to be different goods. As you take all the relevant time periods into account it is without loss of generality to assume that the consumer spends all his income in the end. One may ask, ”what about bequest?” Yes, this should be counted as different a consumption good. By the way, let us consider that prices of all goods and income are both doubled, that is, consider a change from (p, w) = (p1 , p2 , w) to (2p, 2w) = (2p1 , 2p2 , 2w). How does the budget constraint change? You will immediately see that it does not make any difference. The new budget constraint is 2p1 x1 + 2p2 x2 ≦ 2w, but as you divide both sides by 2 it is the same as the original budget constraint p1 x1 + p2 x2 ≦ w. In general, for all positive number λ > 0 we have B(λp, λw) = B(p, w). Opportunity cost What you have to give up when you choose something is called its opportunity cost. As the slope of budget line is − pp12 , when you like to increase the amount of Good 1 you have to give up pp12 units of Good 2. Thus opportunity cost of extra 1 unit of Good 1 is pp12 units of Good 2. Likewise, opportunity cost of extra 1 unit of Good 2 is pp21 units of Good 1.

2.4.1

Budget constraint in exchange economy

In the above budget constraint in the standard form I did not specify the source of income w. Income may have many sources in reality, such as sales of goods, wage, returns from assets, dividend from firm shares, and so on. Here let me consider the simplest one, income in an exchange economy. In an exchange economy each consumer brings her initial endowment e = (e1 , e2 ) to the market. Then he either sells Good 1 and buys Good 2, or sells Good 2 and buys Good 1, or sells or buys nothing. Given a price vector p = (p1 , p2 ), his income is the market valuation of his initial endowment p1 e1 + p2 e2 .

CHAPTER 2. OBJECTS AND OPPORTUNITIES

19

Good 2 6

rx re p1 x1 + p2 x2 = p1 e1 + p2 e2 - Good 1 Figure 2.5: Budget constraint in an exchange economy

Hence the budget constraint is p1 x1 + p2 x2 ≦ p1 e1 + p2 e2 . Here we assume that initial e is fixed, and only p = (p1 , p2 ) is variable. Then denote the set of consumption vectors meeting the budget constraint by B(p) = {x ∈ R2+ : p1 x1 + p2 x2 ≦ p1 e1 + p2 e2 }. Note that here once price p is given income is determined as well. Likewise, the budget line is described by p1 x1 + p2 x2 = p1 e1 + p2 e2 Note that the initial endowment point is always in the budget line as in Figure 2.5. If the consumer chooses a point like x on the budget line which is left to the endowment point as in Figure 2.5, that is, if it holds x 1 < e1 ,

x2 > e 2

then he is selling Good 1 and buying Good 2. Similarly for the opposite direction. Numeraire By the way, it is immediate to see that the budget constraint p1 x1 + p2 x2 ≦ p1 e1 + p2 e2 is equivalent to p1 p1 x1 + x2 ≦ e1 + e2 . p2 p2

CHAPTER 2. OBJECTS AND OPPORTUNITIES

20

That is, in an exchange economy only relative price does matter, not its absolute level. Therefore it is OK to normalize the price of some good equal to 1. Such good is called numeraire. Any good can be a numeraire, but here let’s say it is Good 2, and let p denote the relative price of Good 1 for Good 2, then the budget constraint it. px1 + x2 ≦ pe1 + e2 .

2.4.2

Labor and consumption

The labor-consumption model can be treated as a special case of the model of exchange economy. Here the consumer’s initial endowment is 1 unit of disposable hours and e unit of the consumption good. In the consumption space it is (1, e). Let q denote wage and p denote the price of the consumption good. If the consumer puts l unit hours into leisure he works for 1 − l unit hours. Then he earns income from labor q(1 − l). His income consists of this and the market value of his initial holding of the consumption good e, which is pe. Therefore his consumption c must follow the constraint pc ≦ q(1 − l) + pe. Note that this is equivalent to ql + pc ≦ q + pe, in which the right-hand-side is the market value of initial endowment (1, e). As only relative price does matter in exchange economies, here let’s take the consumption good to be the numeraire and divide the both sides of the above by p, then we get q q l + c ≦ + e. p p Here

q p

is the wage measure by the consumption good, which is the real wage.

Here if the consumer wants to increase 1 extra unit of leisure then he has to give up pq units of consumption. Thus, the opportunity cost of extra 1 unit of leisure is pq units of the consumption good.

2.4.3

Saving and borrowing

Let me repeat that even if goods are materially the same they are treated as different goods if they are to be consumed at different time periods. We describe this by the two-period model, and let me introduce budget constraint here. In the two-period model initial endowment is interpreted as earning stream. That is, when the consumer has initial endowment e = (e1 , e2 ) it means that he earns e1 units of the consumption good in the current period and e2 units in the future period. It might be too short to be called a ”stream,” but let me go with this.

CHAPTER 2. OBJECTS AND OPPORTUNITIES

21

Denote the pure interest rate by r. Assume there is no inflation, as I will come to it in Chapter 9. Now suppose one consumes x1 units of the consumption good in the current period. Then he saves e1 − x1 units, which can be negative and in that case he is borrowing. Return from saving (or repayment for borrowing) to come in the future period is obtained by multiplying the gross interest rate 1 + r to e1 − x1 , which is (1 + r)(e1 − x1 ). Thus the upper limit of consumption in the future period consists of this plus earning in the future e2 , hence it is e2 + (1 + r)(e1 − x1 ). Therefore consumption in the future period x2 has to obey x2 ≦ e2 + (1 + r)(e1 − x1 ). By rewriting this we obtain (1 + r)x1 + x2 ≦ (1 + r)e1 + e2 . Here the right-hand-side is the amount of consumption in the future period which is obtained when the consumer saves all the earning in the current period. As it is the lifetime earning measured by future consumption it is called future value of lifetime earning. Note that this is a special case of the budget constraint in standard form, where future consumption is taken to be the numeraire, that is, p1 = 1 + r and p2 = 1. On the other hand, as we divide both sides of the above by 1 + r we obtain 1 1 x2 ≦ e1 + e2 . 1+r 1+r Here the right-hand-side is the amount of current consumption one can obtain if he borrows up to limit against his lifetime income. As it is the lifetime earning measured by current consumption it is called present value of lifetime earning Again note that this is a special case of the budget constraint in standard form, where current consumption is taken to be the numeraire, that is, p1 = 1 1 and p2 = 1+r . Future value of lifetime earning corresponds to the x2 -intercept of the budget line, and present value of it corresponds to the x1 -intercept of the budget line. In either formulation, the (absolute value of) slope of the budget line is pp12 = 1 + r, which means that gross interest rate is the relative price of current consumption for future consumption. In other words, gross interest rate is the opportunity cost of 1 extra unit of current consumption measured by future consumption, as the consumer has to give up 1 + r units of future consumption as he gets one extra unit of current consumption. x1 +

2.5

Exercises

Exercise 1 You have 120 units of income. Price of Good 1 is 4, that of Good 2 is 3.

CHAPTER 2. OBJECTS AND OPPORTUNITIES

22

(i) Write down the budget constraint. (ii) What is the relative price of Good 1 for Good 2? (iii) Suppose Good 1 is taxed 20% per price and Good 2 is taxed 0.5 per unit. Then write down the new budget constraint.

Chapter 3

Preference 3.1

Preference relation

Preference relation describes an individual’s subjective ranking over choice objects. It is denoted by ≿, ≻, ∼. To help understanding you may take an analogy to inequality and equality symbols ≧, >, =, while this analogy is not quite right as seen below. Let X be the set of choice objects, which is at this point abstract and it may consist of anything. Then, given choice objects x, y ∈ X, the relation x≿y is read as ”x is at least as good as y for the individual.” Likewise, x≻y is read as ”x is better than y for the individual.” Also, x∼y is read as ”x is as good as y for the individual” or ”the individual is indifferent between x and y.” We will need ≿ only, under the completeness condition introduced below, because x ≻ y may be defined by ”y ≿ x is not true,” and x ∼ y may be defined by ”both x ≿ y and y ≿ x are true.” Above I wrote that the analogy to ≧, >, = is not quite right. This is because two difference objects can be equally preferable. That is, the relation x ∼ y can be true for two different objects x and y. On the other hand, the equality relation x = y can be true only when x and y are an identical object. Throughout the book we assume that individual preference relation satisfies the following two properties. One is 23

CHAPTER 3. PREFERENCE

24

Completeness:For every x, y ∈ X, it holds at least one of x ≿ y and y ≿ x. Note that this allows both x ≿ y and y ≿ x hold, and we have x ∼ y then. Completeness says that the individual can compare any two alternatives. It might sound obvious, but it is not. Consider the following example. Example 3.1 Mr. S has double selves. Call his one self S1 and the other S2. S1 has preference ≿1 and S2 has ≿2 , each of which satisfies completeness. Given these, Mr. S forms unified preference ≿ by x ≿ y holds if and only if both x ≿1 y and x ≿2 y hold. Each of his selves has complete preference, but the preference of his unified self fails to satisfy completeness. When x ≿1 y but y ≻2 x, the unified self cannot determine which one is better or if they are equally preferable. Similarly for the case that y ≻1 x and x ≿2 y. The other assumption is transitivity Transitivity: If x ≿ y and y ≿ z, then x ≿ z This requires that preference does not cycle, which means that it is consistent. The following example shows violation of it. Example 3.2 Mr. N has triple selves. Call his selves N1, N2 and N3, respectively. N1 has preference ≿1 , N2 had preference ≿2 , and N3 has preference ≿3 , each of which satisfies transitivity. Given these, Mr. N forms his unified preference ≿ by majority voting among the three selves. Even if each of his selves satisfies transitivity, the preference of his unified self fails to satisfy it. For example, consider a profile x ≻1 y ≻1 z z ≻2 x ≻2 y y ≻3 z ≻3 x

Then x beats y by 2 vs. 1, y beats z by 2 vs. 1, and z beats x by 2 vs. 1, and it violates transitivity. As I acknowledged in Chapter 1 I will proceed with assuming that at least individual preferences satisfy completeness and transitivity. Of course, as suggested by the above examples if we attempt to aggregate individual preferences into a ”social preference” it is non-obvious if that can satisfy completeness and transitivity (see Chapter 30 for details).

CHAPTER 3. PREFERENCE

25

Good 2 6 A rx B

C - Good 1

Figure 3.1: An indifference curve

3.2 3.2.1

Preference over consumptions Indifference curves

Below, choice object x is not an abstract point but refers to a 2-dimensional consumption vector x = (x1 , x2 ). In the 2-dimensional consumption space, you can can think of preference just like you think of level curves of a mountain. For example, the set of consumption vectors which are better than x = (x1 , x2 ) is described by area A in Figure 3.1. In other words, it is the area ”higher than” x. Likewise, the set of consumption vectors worse than x is described by area B. It is the area ”lower than” x. And, the border between A and B consists of consumption vectors which are equally preferable to x. We call this an indifference curve. It is just like a level curve of a mountain. When you draw an indifference curve passing through another consumption vector y = (y1 , y2 ) which is preferred to x, it looks like in Figure 3.2. This is a level curve which is passing above x. Likewise, we can draw a series of indifference curves like in Figure 3.2. Because we can actually draw indefinitely many indifference curves what I drew in the figure is only a part of that. Here I took an analogy of level curves of a mountain, but in our case there are no numbers which describe the ”height” of the mountain. This is because indifference curves describe only a relative ordering about which one is better or worse. We can know if something is better than another for a given consumer, but cannot know ”how much she is happy,” and such quantitative statement does not have economic content. Therefore our level curves, i.e., indifference curves, do not accompany numbers signifying heights. Notice that under Transitivity indifference curves do not cross. Suppose

CHAPTER 3. PREFERENCE

26

Good 2 6

ry rx

- Good 1 Figure 3.2: Indifference curves

indifference curve C and C ′ cross as in Figure 3.3, and denote the intersection by x. Then, since y in the figure is above x across C we have y ≻ x. Likewise, since z is above y across C ′ we have z ≻ y. However, since x is above z across C we have x ≻ z, which leads to a cycle and contradicts Transitivity. Let me give you some examples of preference. The simplest one consists of parallel and straight indifference curves as in Figure 3.4. Here the two goods are said to be perfect substitutes of each other. Here the slope of indifference curves being −3 in the graph means (x1 , x2 ) ∼ (x1 + t, x2 − 3t) for any t. In other words, the consumer is willing to give up 3 units of Good 2 per one extra unit of Good 1. Thus the slope of indifference curves express the consumer’s subjective rate of exchange between two goods. We call this marginal rate of substitution of Good 2 for Good 1, while its more general definition will be given later. Next example consists of L-shaped indifference curves located parallel along an upward-sloping straight line passing through the origin. Here the two goods are said to be perfect complements of each other. In this graph the L-shaped indifference curves are located parallel along the line x1 = 2x2 . This means the consumer sticks to some fixed proportion between Good 1 and Good 2, which is 2:1 here, and any extras have no value for him. So for example when he originally has (8, 4), receives extra 6 units of Good 1 and ends up with (14, 4), because the extra 6 units of Good 1 have no value we have (8, 4) ∼ (14, 4). Likewise, when we add 5 units of Good 2 to (8, 4) so as to obtain (8, 9), again because the extra 5 units of Good 2 have no value and thus we have (8, 4) ∼ (8, 9). In this book I refer to perfect substitution and perfect complementarity as extreme cases mostly. More flexible preferences will be between the two.

CHAPTER 3. PREFERENCE

27

Good 2 6 r z rx ry

C′ C - Good 1

Figure 3.3: Crossing indifference curves

Good 2 6

1 −3 ? - Good 1 Figure 3.4: Perfect substitution

CHAPTER 3. PREFERENCE

28

Good 2 6

9

4

8

14

- Good 1

Figure 3.5: Perfect complementarity

3.2.2

Monotonicity

Next I introduce two assumptions which are natural for preferences over consumptions. One is monotonicity, which says more is better. Strong Monotonicity: For any x = (x1 , x2 ) and y = (y1 , y2 ), if x1 ≧ y1 , x2 ≧ y2 and if at least one of these inequalities are strict, then x ≻ y. This means the consumer is better off when the consumptions of both goods increase or the consumption of one good increases while that of the other stays the same. Therefore our ”mountain” does not have a peak, and extends upward to the north-east direction. Under Monotonicity, the set of consumption vectors better than x contains the quadrant of north-east direction and the indifference curves are always downward-sloping (see Figure 3.6). Of course you can think of preferences which violate monotonicity. Consider for example that the consumer dislikes some commodity, which is a bad for him, then his preference violates monotonicity. Also it is violated when the consumer gets ”full” and consumption more than that makes him sick. Monotonicity is pretty innocuous, however. If there is a harmful commodity one can trade it for a negative price and it is equivalent to trading the ”right to put that away” for a positive price, in which monotonicity is taken to hold with regard to such right. What about the case of becoming ”full?” It is a matter of how long we take one period to be. If we take it to be short we may have a case that the consumer becomes full, but it we take to be sufficiently long then the consumer’s preference satisfies the property that more is better. Now, as I put the word ”strong” in the above definition it suggests that there is a weaker definition. Weak Monotonicity: For all x = (x1 , x2 ), y = (y1 , y2 ), if x1 > y1 and x2 > y2 , then x ≻ y.

CHAPTER 3. PREFERENCE

29

Good 2 6

x

r

- Good 1 Figure 3.6: Monotonicity

This says it is better if you increase the amounts of both goods, and it leaves the possibility that you don’t get strictly better off when you increase the amount of just one good. To illustrate the difference, consider the case of perfect complementarity. When you increase the amounts of both Good 1 and Good 2 at (x1 , x2 ) and obtain (y1 , y2 ), we have (y1 , y2 ) ≻ (x1 , x2 ) and weak monotonicity is met. On the other hand, when we increase the amount of Good 1 only, let’s say by t, as we just move along the same indifference curve we have (x1 + t, x2 ) ∼ (x1 , x2 ), which says strong monotonicity fails. Likewise, when we increase the amount of Good 2 only, let’s say by t, as we just move along the same indifference curve we have (x1 , x2 + s) ∼ (x1 , x2 ), which again says strong monotonicity fails.

3.2.3

Convexity

The other assumption is convexity, which says taking middle is better. Strict Convexity: For any x = (x1 , x2 ) and y = (y1 , y2 ), if x ∼ y then for all 0 < λ < 1 it holds λx + (1 − λ)y ≻ x ∼ y, where λx + (1 − λ)y = (λx1 + (1 − λ)y1 , λx2 + (1 − λ)y2 ). Given two equally preferable consumption vectors x and y, consider any point in the middle, λx + (1 − λ)y. Then convexity says such middle point is better than the two extreme points (seet Figure 3.7). Convexity looks somewhat artificial as compared to monotonicity. It is a very natural assumption, however, in the context of intertemporal consumption and consumption under uncertainty. In the setting of intertemporal consumption, taking middle corresponds to reducing fluctuation of consumption between periods. For example, while consumption stream (10, 0) refers to consuming 10

CHAPTER 3. PREFERENCE

30

Good 2 6 rx rλx + (1 − λ)y

ry - Good 1 Figure 3.7: Convexity

now and 0 in the future, (0, 10) refers to consuming 0 now and 10 in the future, the midpoint (5, 5) refers to consuming 5 both now and in the future. It is thus natural to prefer the midpoint when the consumer dislikes fluctuation over time.1 In the setting of consumption under uncertainty, taking middle corresponds to hedging uncertainty. For example, while state-contingent vector (10, 0) refers to consuming 10 if Republicans win and 0 if Democrats win, (0, 10) refers to consuming 0 if Republicans win and 10 if Democrats win, the mid point (5, 5) refers to consuming 5 regardless of the election outcome. It is thus natural to prefer the midpoint when the consumer dislikes uncertainty. Now, as I put the word ”strict” in the above definition it suggests that there is a weaker definition. Weak Convexity: For any x = (x1 , x2 ) and y = (y1 , y2 ), if x ∼ y then for all 0 < λ < 1 it holds λx + (1 − λ)y ≿ x ∼ y. This means that taking middle of any two equally preferable points does not make the consumer worse off. The difference here is that the consumer may not get strictly better off. To illustrate, consider the case of perfect substitution. Because indifference curves are straight here, any point between any two equally preferable points is again equally preferable to those, which fails to satisfy strict convexity while the weak one is met. 1 We need to be a bit more careful, since typically a consumer is not indifferent between 10 units to be received now and 10 units to be received in the future, since he is normally impatient. I will come to the issue of impatience in Chapter 8, and let us pretend here that the consumer is patient and 10 units now and 10 units in the future are equally valuable.

CHAPTER 3. PREFERENCE

3.3

31

Marginal rate of substitution

Marginal rate of substitution is the subjective rate of exchange between goods. It refers to relative importance of a good which is measured in terms of another good. Consider the following question. How important is Good 1 for you as compared to Good 2? Note that I’m not asking ”How important is Good 1 for you?,” which is not an economically meaningful question, as importance of something can be measured only by something else. In other words, value of something for one can be revealed only by telling how much of something else he can sacrifice. In any case, let me restate the above question as follows: In order to get extra one unit of Good 1 how many units of Good 2 can you give up? Answer to this question is called marginal rate of substitution of Good 2 for Good 1, which is the subjective relative value of Good 1 measured by means of Good 2. Of course one can define ”marginal rate of Good 1 for Good 2”=”the amounts of Good 1 which can be given up in order to get extra one unit of Good 2,” and it is the inverse of the above one. In order to avoid confusion, however, we adopt the first one throughout the book, where Good 1 is what is measured and Good 2 is what measures. Hence we omit the part ”of Good 2 for Good 1” hereafter. To illustrate, look again at the case of perfect substitution as depicted in Figure 3.4. Here the slope of indifference curves is −3, which means that (x1 , x2 ) and (x1 + t, x2 − 3t) are equally preferable. In other words the consumer is will to sacrifice up to 3t units of Good 2 in order to get t extra units of Good 1. That is, the slope corresponds to how many units of Good 2 one can give up in order to get 1 extra unit of Good 1. Thus, the marginal rate of substitution of Good 2 for Good 1 is given by the (absolute value) of the slope of the indifference curves. In this example it is 3. Now, why do we use the term ”marginal” rate of substitution, not just ”rate of substitution?” This is because indifference curves in general are not straight or parallel, and it rather looks like in Figure 3.8 typically. Here the slope of indifference curves varies across points. Therefore we need to look at local slope of the indifference curves, and marginal rate substitution is defined by the absolute value of such local slope. Suppose we are at point x = (x1 , x2 ) on an indifference curve as in Figure 3.8. Now suppose we add a ”slight amount” of Good 1. Denote this ”slight amount” by ∆x1 . Let ∆x2 the amount of Good 2 the consumer can give up in order of get extra ∆x1 of Good 1. Then (x1 , x2 ) and (x1 + ∆x1 , x2 + ∆x2 )

CHAPTER 3. PREFERENCE

32

Good 2 6

∆x1 (x1 , x2 ) r ∆x2 r?(x + ∆x , x + ∆x ) 1 1 2 2

- Good 1 Figure 3.8: Marginal rate of substitution

must be on the same indifference curve. While this indifference curve is not straight, it is taken to be straight ”locally” when the change in consumption is very small. This local slope is equal to the slope of the tangent line to the 2 indifference curve at (x1 , x2 ). Denote this by ∆x ∆x1 . This is the amount of Good 2 one can give up in order to get 1 extra unit of Good 1, in the local sense. Recall that the indifference curves are downward sloping as preference satisfies monotonicity. Thus, when ∆x1 is positive ∆x2 is negative. Hence the local 2 slope of any indifference curve ∆x ∆x1 is negative. Because we are interested in the absolute value of it, the marginal rate of substitution at x is given by ∆x2 = − ∆x2 . M RS(x) = ∆x1 ∆x1 Here I put (x) after M RS because I like to emphasize the fact that marginal rate of substitution, which is the local slope of an indifference curve, varies across points. That is, M RS(x) is a function of x. Now let us describe marginal rate of substitution for several examples of preference. Example 3.3 Perfect substitution: Consider preference exhibiting perfect substitution, where the (absolute value of) slope of indifference curves is α. Then we have M RS(x) = α for all x = (x1 , x2 ). Note that α = 3 in the first example. Example 3.4 Cobb-Douglass preference: MRS of Good 2 for Good 1 takes the following form. x2 M RS(x) = α x1

CHAPTER 3. PREFERENCE

33

Here MRS is disproportional to x1 and proportional to x2 . Here indifference curves are (generalized) hyperbola with x1 -axis and x2 -axis being its asymptotes. It looks odd, but this preference fits empirical data quite well. I’ll come back to the detail properties of this preference later. Example 3.5 Perfect complementarity: MRS may not be given to such preference in the above-noted sense, because indifference curves have kinks. However, one may give MRS in a generalize sense like below. When the line of locus of kinks is described by x1 /x2 = α, ”marginal rate of substitution” of Good 2 for Good 1 is M RS(x) = ∞ = indefinite = 0

if x1 /x2 < α if x1 /x2 = α if x1 /x2 > α

That is, when Good 1 is scarce compared to the determined rate α it cannot be compensated by any larger amount of Good 2, hence MRS is infinity. On the other hand, when Good 1 is abundant compared to the determined rate α it is simply valueless and the consumer does not want to sacrifice any amount of Good 2 for that, hence MRS is 0. Finally, MRS is not uniquely determined at kinked points.

3.4

Smooth preferences

Let me define a class of ”well-behaved” preferences, which will be relevant in later chapters. Definition 3.1 Preference is said to be smooth if it satisfies strong monotonicity and strict convexity on the positive quadrant R2++ , and additionally that 1. any indifference in the positive quadrant is ”smooth” in the sense that it has no ”kink.” 2. any indifference in the positive quadrant tends to be flat as it gets closer to the x1 -axis, and tends to be vertical as it gets closer to the x2 -axis, and it does not hit any of the two axis. In the words of marginal rate of substitution, this is restated as 1. M RS(x) is uniquely determined at any point x in the positive quadrant. 2. M RS(x) converges to infinity as x1 tends to 0, and converges to 0 as x2 tends to 0.

CHAPTER 3. PREFERENCE

34

Good 2 6 r? r-? - Good 1 Figure 3.9: Diminishing MRS

Condition 2 says that when Good 1 is indefinitely scarce it becomes indefinitely valuable compared to Good 2 and the amount of Good 2 the consumer can give up in order to get extra one unit of Good 1 becomes indefinitely large, and also says that when Good 2 is indefinitely scarce it becomes indefinitely valuable compared to Good 1 and the amount of Good 2 the consumer can give up in order to get extra one unit of Good 1 becomes indefinitely small. That is, marginal rate of substitution varies flexibly between 0 and infinity, like in Figure 3.8 and 3.9 For example, preference exhibiting perfect complementarity between two goods has kinks on their indifference curves and therefor is not smooth. Also, preference exhibiting perfect substitution between two goods satisfies condition 1 because its indifference curves are parallel and straight, which have no kinks, while it fails to meet condition 2 because the indifference curves hit both axis. In contract, Cobb-Douglass preference is smooth.

3.5

Convexity and diminishing marginal rate of substitution

Let us rethink the meaning of convexity in terms of marginal rate of substitution. Indifference curves generated by convex preference are steeper as Good 1 quantity is smaller, and flatter as Good 1 quantity is larger, as in Figure 3.9. Equivalently, marginal rate of substitution M RS(x) is larger as x1 is smaller, and smaller as x1 is larger. That is, when Good 1 is scarce the amount of Good 2 one can give for one extra unit of it is larger, and when Good 1 is abundant the amount of Good 2 one can give for one extra unit of it is smaller. This is called the law of diminishing marginal rate of substitution. I’ll come tho this in relation to so called ”the law of diminishing marginal utility.”

CHAPTER 3. PREFERENCE

3.6

35

Exercises

Exercise 2 Let Good 1 be consumption good at Period 1, and Good 2 be consumption good at Period 2. (i) Describe by means of indifference curves the preference of a consumer who cares only about consumption at Period 1. (ii) Describe by means of indifference curves the preference of a consumer exhibiting perfect substitution between consumptions at two periods, such that he cares more about the current consumption. (iii) Describe by means of indifference curves the preference of a consumer exhibiting perfect substitution between consumptions at two periods, such that he cares more about the future consumption. Exercise 3 Let Good 1 be the consumption good available at State 1, and Good 2 be the one available at State 2. (i) Suppose the probability of State 1 2/3, and describe by means of indifference curves the preference of a consumer who cares only about the expected value of his consumption. (ii) Describe by means of indifference curves the preference of a consumer who cares only about the worst case, meaning the case with the lower amount of consumption.

Chapter 4

So-called utility function 4.1

”Utility” representation of preference

There are many statements in economics which can be obtained by directly tracking the properties of preferences. Not only that, ”in principle” every statement in economics is obtained by doing this. However, it is customary to represent preferences by means of numerical functions in order to make analysis more operational. In the previous chapter I wrote that the ”mountain” described by a series of indifference curves is not assigned numbers describing its height, but I’m here saying that we will assign such numbers for our convenience. Why do we do this? I have two answers. 1. Once a preference is given its representation, we can describe actor’s ”rational” behavior of choosing the most preferred one from the set of available options as the maximization of a function under constraints. We know much about the mathematics of maximization of functions (I suppose), and I’m saying we should make better use of it. 2. Representing preferences by means of functions allows us to derive optimal choice in a way that we can carry out econometric analysis of it. By means of comparing the solution with data we can estimate the functional form and its parameters. Then we can predict and evaluate the effect of policy changes by using the estimated functional form and parameters. This is called counter-factual analysis. In economics we call this ”utility function.” This might give an impression of saying that there exists a ”substance” called ”utility,” however. Of course this reflects the history that economists once believed naively that such ”substance” exists, but I use the word utility representation as consistently as possible in order to emphasize the modern standpoint. As I will repeat persistently below, utility representation is only an ”analyst’s choice” to describe the ranking of which option is better than another, by means

36

CHAPTER 4. SO-CALLED UTILITY FUNCTION

37

of relative values, and we should not read any economic content in such numbers or be aware that there must be certain ”faith” in order to do so. Definition 4.1 Function u : X → R is said to be a utility representation of preference ≿ if for all x, y ∈ X it holds that x ≿ y implies u(x) ≧ u(y) x ≻ y implies u(x) > u(y) x ∼ y implies u(x) = u(y). In other words, it says that the assigned numbers are consistent with the given preference. For example, suppose that the preference ≿ is given by x ≻ y ∼ z ≻ w. Let us assign numbers to the above alternatives consistently with the ranking. Denote the number assigned to each alternative by u(x), u(y), u(z), u(w), respectively. Then a set of numbers consistent with the ranking is for example u(x) = 3, u(y) = u(z) = 2, u(w) = 1. This is one utility representation of the preference. However, for the preference x ≻ y ∼ z ≻ w the above one is not the only utility representation. For example let us double the assigned numbers and obtain u∗ (x) = 6, u∗ (y) = u∗ (z) = 4, u∗ (w) = 2. This is also a utility representation of preference x ≻ y ∼ z ≻ w. Now, if we represent the preference by u∗ instead of u is our consumer’s ”happiness doubled?” That’s nonsense. They are simply representations of the same preference. Likewise, let us add 4 uniformly to the first set of numbers: u∗∗ (x) = 7, u∗∗ (y) = u∗∗ (z) = 6, u∗∗ (w) = 5 This is also a utility representation of preference x ≻ y ∼ z ≻ w. Now, if we represent the preference by u∗∗ instead of u is our consumer ”4 units happier?” That’s nonsense again. They are simply representations of the same preference. There is no meanings like ”4 units happier” Actually, any set of number is fine as far as it is consistent with the ranking. For example u ˆ(x) = 5, u ˆ(y) = u ˆ(z) = 2, u ˆ(w) = −3 is fine and u ˜(x) = −2, u ˜(y) = u ˜(z) = −5, u ˜(w) = −8

CHAPTER 4. SO-CALLED UTILITY FUNCTION

38

is fine as well. More generally, for any monotone transformation we have the following result. Any function from real numbers to real numbers f is said to be monotone transformation if f (u) > f (v) whenever u > v. That is, it is any function such that its graph is upward-sloping. Theorem 4.1 Suppose that function u is a utility representation of preference ≿. Then for any monotone transformation the function defined by u ˆ(x) ≡ f (u(x)) is also a utility representation of ≿. Proof. If x ≻ y, because uis a utility representation we have u(x) > u(y). Since f is a monotone transformation, f (u(x)) > f (u(y)). If x ∼ y, because u is a utility representation we have u(x) = u(y). Hence we have f (u(x)) = f (u(y)). If x ≿ y it holds either x ≻ y or x ∼ y, from the above we have either f (u(x)) > f (u(y)) or f (u(x)) = f (u(y)). Therefore it holds f (u(x)) ≧ f (u(y)). That is, once there is a utility representation to a given preference relation we can cook up arbitrarily many representations for the same preference by taking arbitrary monotone transformation. Utility representation has meaning only as a representation of preference ordering and it has no quantitative meaning. Thus is called ordinal utility. Let me restate this point with regard to preference over a consumption space. To illustrate consider the case of perfect substitution, in which the slope of indifference curve is −α, that is marginal rate of substitution is always constant and equal to α. The simplest representation of it will be like u(x) = αx1 + x2 . Indeed, if we take ”utility level” u ¯ any consumption vector (x1 , x2 ) yielding this satisfies αx1 + x2 = u ¯, which describes a straight line with slope −α. This is not the only representation, of course. For example, if we double the above representation (that is, by transforming via f (u) = 2u) we obtain u ˆ(x) = 2αx1 + 2x2 . Then the above noted indifference curve is described by 2αx1 + 2x2 = 2¯ u Is our consumer’s happiness doubled? No. Both u and u ˆ are no more than representations of the same preference. It is immediate to see that the indifference

CHAPTER 4. SO-CALLED UTILITY FUNCTION

39

curve described by αx1 + x2 = u ¯ and the one described by 2αx1 + 2x2 = 2¯ u are the same. That is, the ”utility level” u ¯ here is no more than an index referring to ”which indifference curve” we are looking at. Likewise, we can think of transformation such as f (u) = eu . Because this is a monotone transformation the function u ˜(x) = eαx1 +x2 is also a representation of the same preference, and the above-noted indifference curve is described by eαx1 +x2 = eu¯ . Again it is immediate to see that the indifference curve described by αx1 +x2 = u ¯ and the one described by eαx1 +x2 = eu¯ are the same. And of course for any monotone transformation f the function f (αx1 +x2 ) is also a utility representation of the above preference. It is therefore no more than mathematical convenience which representation we use among many equivalent ones. We can use eαx1 +x2 , we can use ln(αx1 + x2 ), but in any case they are representations of the same preference and it does not make any difference in results of the analysis. So we can just use a mathematically convenient one, which is, say, u(x) = αx1 + x2 . Next let us give representations to preferences exhibiting perfect complementarity. When we have parallel L-shaped indifference curves along a straight line x2 = αx1 , the preference is represented let’s say by u(x) = min{αx1 , x2 }. Here min{A, B} refers to the smaller number out of A and B.1 That means any extra amount deviating from the fixed proportion has no value. It is of course not the only utility representation, and we can represent this preference in other functions such as emin{αx1 ,x2 } and ln(min{αx1 , x2 }). Because they represent the same preference and it does not make any difference which one to use, however, we typically use the simplest one, u(x) = min{αx1 , x2 }. Finally we give representations to Cobb-Douglas preference. It typical representation is u(x) = xa1 xb2 I will show later why this represents Cobb-Douglas preference which is pinned down by marginal rate of substitution M RS(x) = α xx12 . An indifference curve generated by this preference is described by xa1 xb2 = u ¯, 1 For

example, min{3, 5} = 3, min{6, 2} = 2, and min{4, 4} = 4.

CHAPTER 4. SO-CALLED UTILITY FUNCTION

40

which is a generalization of hyperbola and has x1 -axis and x2 -axis as its asymptotes. Again, this is of course not the utility representation. We can let’s say take log transformation and obtain u ˆ(x) = ln(xa1 xb2 ) = a ln x1 + b ln x2 , which is a representation of the same preference. It is immediate to see that the indifference curve described by xa1 xb2 = u ¯ and the one described by a ln x1 + b ln x2 = ln u ¯ are the same. We can obtain arbitrarily many representations of Cobb-Douglass preference by taking arbitrary monotone transformation, but the above two are the typical ones. Sometimes the first one is easier to handle and sometimes the second is easier.

4.2

Marginal utility

Just like values of utility representation have no quantitative or economic meaning, so-called marginal utility has no quantitative or economic meaning either. Why do I introduce it, nevertheless? It is a technical concept which helps us to describe marginal rate of substitution in an operational manner, but that is the only role. Economists tend to use the word ”marginal utility” as if it is a substantive concept, however, when its relation to marginal rate of substitution is straightforward. We should keep that in mind.

4.2.1

One-good case

To illustrate, let me start with the case that there is just one good and preference satisfies monotonicity, which simply says more of it is better. Then its representation u(x) can be any monotone increasing function, that is, any function with upward-sloping graph. Let me start with the simplest example of monotone increasing function, which is a function with a linear graph, u(x) = ax + b with a > 0. Now let me ask in this representation how much ”utility” increases as the consumer gains extra one unit of the good, knowing that such question has no economic content. It is immediate to see that the answer is a, slope of the graph. Now, let us consider another representation u b(x) = ln(ax + b). This time its graph is not linear, so we have to look at local slopes of the graph. Suppose we are currently at x and we increase the amount of the good ”slightly.”

CHAPTER 4. SO-CALLED UTILITY FUNCTION

41

Denote this ”slight” amount by ∆x. When ∆x is very sufficiently small we can take the graph to be linear, locally, and its slope is u b(x + ∆x) − u b(x) . ∆x Now make this ”slight” amount ∆x infinitesimally small, then we obtain the derivative of u b(x), which is db u(x) u b(x + ∆x) − u b(x) = lim . ∆x→0 dx ∆x It is the marginal utility of the good at x. In this representation, because the derivative of log is fraction it is db u(x) d a = ln(ax + b) = . dx dx ax + b Consider one more representation u e(x) = eax+b . Marginal utility which corresponds to this representation is de u(x) d ax+b = e = aeax+b . dx dx Now you we that marginal utility depends on which representation to consider. This means that marginal utility itself does not have any economic content. Although u, u b and u e represent the same preference the marginal utilities derived from these respectively are different. This difference does not make any difference in consumer behavior, and we cannot have different analytical results depending on which utility representation or corresponding marginal utility to consider. In introductory books you might have seen so-called ”the law of diminishing marginal utility.” It says for example ”the first-sip of beer is very tasty, the second one is good but less tasty than the first, and the third is not much...” However, the law of diminishing marginal utility does not have any economic content. Look at marginal utilities of the above representations u, u b and u e, respectively. The marginal utility coming from u b(x) = ln(ax + b), which is db u(x)/dx = a/(ax + b), decreases as x increases. However, the marginal utility coming from u(x) = ax + b is du(x)/dx = a and it is constant, and the marginal utility coming from u e(x) = eax+b is de u(x)/dx = aeax+b and it is rather increasing as x increases. Since whether it’s true or not depends on which representation to consider we should say that the law of diminishing marginal utility does not have any economic content. Of course the statement like ”the first sip of beer...” captures some aspect of our preference correctly, but it is not what should be called ”the law of diminishing marginal utility.” It should be called the law of diminishing marginal

CHAPTER 4. SO-CALLED UTILITY FUNCTION

42

rate of substitution instead, in the sense that the first sip of beer if more precious and the consumer is willing to sacrifice more units of the other goods, and as he has more units beer he is willing to give up less units of the other goods. To understand this, however, we need to think of the case of 2 goods at least.

4.2.2

2-good case

When there are two or more goods, one has to obtain marginal utility of each good. That is, in the two-good case, marginal utility of Good 1 corresponds to how much ”utility” increases as the consumer gets extra one unit of Good 1 as he keeps the amount of Good 2 to be constant. Likewise, marginal utility of Good 2 corresponds to how much ”utility” increases as the consumer gets extra one unit of Good 2 as he keeps the amount of Good 1 to be constant. Of course, we should note that a phrase like ”how much utility increases” has no economic content. Let me illustrate with the case of perfect substitution. Of course there are arbitrarily many equivalent representation of the same preference, so I’m taking the simplest one, which is a linear function u(x) = ax1 + bx2 . Here it is easy to see how much utility increases as we add 1 unit of Good 1 alone, which is a, the coefficient on x1 . This is the marginal utility of Good 1 in the given representation. Likewise, it is easy to see how much utility increases as we add 1 unit of Good 2 alone, which is b, the coefficient on x2 . This is the marginal utility of Good 2 in the given representation. Graphically, this is illustrated as follows. The ”mountain” given by representation u(x) = ax1 + bx2 consist of a plane as in Figure 4.1. Marginal utility of Good 1 corresponds to the slope of this ”mountain” toward the east direction, that is, in the direction along the x1 -axis. Likewise, marginal utility of Good 2 corresponds to the slope of this ”mountain” toward the north direction, that is, in the direction along the x2 -axis. Since this mountain is a straight plane, the slope toward east is a everywhere and the slope toward north is b everywhere. Of course one can take another representation of the same preference, however, for example u b(x) = ln(ax1 + bx2 ). The ”mountain” given by this representation looks like Figure 4.2. Because u and u b are representations of the same preference they induce the same series of indifference curves. That is, when you look at the ”mountain” given by u and the ”mountain” given by u b from above you will see the same series of ”level curves” as in Figure 4.3. Let me repeat that only how the level curves look like should matter economically. Put this in mind, and look at the mountain given by general utility representation u. This time the mountain is not necessarily straight but its slope

CHAPTER 4. SO-CALLED UTILITY FUNCTION

”Utility” 6

 * Good 2       PP PPP PP PP PP P q Good 1 Figure 4.1: u(x) = ax1 + bx2

”Utility” 6

 * Good 2      PP PP  PP PP PP P P q Good 1 Figure 4.2: u b(x) = ln(ax1 + bx2 )

43

CHAPTER 4. SO-CALLED UTILITY FUNCTION

44

Good 2 6

- Good 1 Figure 4.3: Preference represented by u(x) = ax1 + bx2 , u b(x) = ln(ax1 + bx2 )

varies across points, therefore we need to look at its local slope. See Figure 4.4. Suppose we are now at x = (x1 , x2 ), and consider ”local slope toward east.” To see that, keep the amount of Good 2 x2 to be constant and consider adding a ”slight amount” of Good 1. Let ∆x1 denote this ”slight amount.” That is, we move from (x1 , x2 ) to (x1 + ∆x1 , x2 ) along the x1 -axis. Then the ”local slope toward east” is given by u(x1 + ∆x1 , x2 ) − u(x1 , x2 ) . ∆x1 Now as we make this ”slight amount” ∆x1 tend to be indefinitely small we obtain so-called partial derivative of u by x1 at x = (x1 , x2 ), ∂u(x) u(x1 + ∆x1 , x2 ) − u(x1 , x2 ) = lim ∆x1 →0 ∂x1 ∆x1 This is the marginal utility of Good 1 obtained for representation u at x = (x1 , x2 ). Consider for example u b(x) = ln(ax1 +bx2 ), then the marginal utility of Good 1 is a ∂ ln(ax1 + bx2 ) = ∂x1 ax1 + bx2 Here the partial derivative of a function by x1 is obtained by taking x2 to be constant and taking the derivative of it as if it is a single-variable function of x1 . Likewise, consider the ”local slope toward north” at x = (x1 , x2 ). See Figure 4.5. Now keep the Good 1 quantity x1 to be constant and consider adding a ”slight amount” of Good 2. Let ∆x2 denote this ”slight amount.” That is, we are moving from (x1 , x2 ) to (x1 , x2 + ∆x2 ) along the x2 -axis. Then the ”local

CHAPTER 4. SO-CALLED UTILITY FUNCTION

45

”Utility” 6 ∆u P q6 P ∆x1  * Good 2    x 2  PP P q P PPP ∆x1 PP x1 PP PP P q Good 1 Figure 4.4: Marginal utility of Good 1

slope toward north” is given by u(x1 , x2 + ∆x2 ) − u(x1 , x2 ) . ∆x2 Now as we make this ”slight amount” ∆x2 tend to be indefinitely small we obtain so-called partial derivative of u by x2 at x = (x1 , x2 ), ∂u(x) u b(x1 , x2 + ∆x2 ) − u(x1 , x2 ) = lim ∆x2 →0 ∂x2 ∆x2 This is the marginal utility of Good 1 obtained for representation u at x = (x1 , x2 ). Consider for example u b(x) = ln(ax1 +bx2 ), then the marginal utility of Good 2 is ∂ ln(ax1 + bx2 ) b = ∂x2 ax1 + bx2 Here the partial derivative of a function by x2 is obtained by taking x1 to be constant and taking the derivative of it as if it is a single-variable function of x2 . Likewise, another one such as u e(x) = eax1 +bx2 represents the same preference as u(x) = ax1 +bx2 does. For this representation, marginal utility of Good 1 at x = (x1 , x2 ) is ∂e u(x) ∂eax1 +bx2 = = aeax1 +bx2 ∂x1 ∂x1

CHAPTER 4. SO-CALLED UTILITY FUNCTION

46

”Utility” 6 ∆u 6 *   ∆x2  * Good 2    x 2  PP *   ∆x2 PPP PP x1 PP PP P q Good 1 Figure 4.5: Marginal utility of Good 2

and that of Good 2 is ∂eax1 +bx2 ∂e u(x) = = beax1 +bx2 . ∂x2 ∂x2 Now let me emphasize again that marginal utility depends on which representation of a given preference to consider, and that marginal utility itself has no economic content. All of u, u b and u e are representations of the same preference, and they induce the same series of indifference curves. Marginal utilities derived of those representations differ, while as these representations correspond to the same preference this difference does not make any difference in consumer choice. Therefore so-called ”the law of diminishing marginal utility” has no economic content again. To illustrate, look at the marginal utility of Good 1 for each of u, u b and u e. In representation u b(x) = ln(ax1 + bx2 ), marginal utility of u b(x) a Good 1 ∂∂x = , which is decreasing in x1 and exhibits diminishing ax1 +bx2 1 marginal utility. However, in representation u(x) = ax1 + bx2 , marginal utility of Good 1 is ∂u(x) ∂x1 = a, which is constant in x1 . Moreover, in representation u e(x) u e(x) = eax1 +bx2 , marginal utility of Good 1 is ∂∂x = aeax1 +bx2 , which is 1 increasing in x1 . When different representations of the same preference lead to different conclusions about a proposition, it means that such proposition has no economic content.

Now let me illustrate this again by taking Cobb-Douglas preference. While it can have arbitrarily many representation, a simple one will be like u(x) = xa1 xb2 .

CHAPTER 4. SO-CALLED UTILITY FUNCTION

47

For this representation u, marginal utility of Good 1 at x = (x1 , x2 ) is ∂u(x) ∂xa1 xb2 = = axa−1 xb2 1 ∂x1 ∂x1 and that of Good 2 is ∂u(x) ∂xa1 xb2 = = bxa1 xb−1 2 . ∂x2 ∂x2 Marginal utility of Good 1 is obtained by taking x2 to be constant taking the derivative as if it is a single-variable function of x1 . Likewise, marginal utility of Good 2 is obtained by taking x1 to be constant taking the derivative as if it is a single-variable function of x2 . Of course one can think of difference representations. For example, by taking log transformation of the above we get u b(x) = a ln x1 + b ln x2 , which represents the same preference as the above one does. For representation u b, marginal utility of Good 1 at (x1 , x2 ) is ∂b u(x) ∂(a ln x1 + b ln x2 ) a = = ∂x1 ∂x1 x1 and that of Good 2 is ∂b u(x) ∂(a ln x1 + b ln x2 ) b = = . ∂x2 ∂x2 x2 Marginal utility of Good 1 is obtained by taking x2 to be constant and taking the derivative as if it is a single-variable function of x1 . Likewise, marginal utility of Good 2 is obtained by taking x1 to be constant and taking the derivative as if it is a single-variable function of x2 . Again, while u and u b represent the same preference they lead to different values of marginal utilities. Let me verify again that the ”law of diminishing marginal utility” has no economic content. For example, in representation u(x) = xa1 xb2 let us consider three cases a = 13 , b = 23 , a = 1, b = 2 and a = 2, b = 4. Then each case delivers utility representations 1

2

x13 x23 x1 x22 x21 x42

CHAPTER 4. SO-CALLED UTILITY FUNCTION

48

respectively. Note that they represent the same preference and yield the same series of indifference curves. However, for each of them marginal utility of Good 1 at x = (x1 , x2 ) is 1

2

∂x13 x23 1 −2 2 = x1 3 x23 ∂x1 3 ∂x1 x22 = x22 ∂x1 ∂x21 x42 = 2x1 x42 ∂x1 respectively, and the first one is decreasing, second is constant, third is increasing in x1 , respectively. Again, when different representations of the same preference lead to different conclusions about a proposition, it means that such proposition has no economic content.

4.3

Describing marginal rate of substitution by means of marginal utilities

This books takes the stance that the concept of marginal utility helps us to describe marginal rate of substitution in an operational manner, but it has no more role than that. In other words, only marginal rate of substitution has economic content and marginal utility itself has no such content. Now how can we describe marginal rate of substitution by means of marginal utilities? Consider that we are at x = (x1 , x2 ) as in Figure 3.8, and ask how much of Good 2 the consumer can give up in order to get a ”slight amount” of Good 1. Denote this ”slight amount” of Good 1 by ∆x1 , and let ∆x2 denote the amount of Good 2 he is willing to give up. Then two points (x1 , x2 ) and (x1 + ∆x1 , x2 + ∆x2 ) must be on the same indifference curve. Because indifference curves are downward-sloping, when ∆x1 is positive ∆x2 must be negative. Then let ∆u denote the change from utility at u(x1 , x2 ) at (x1 , x2 ) to utility u(x1 + ∆x1 , x2 + ∆x2 ) at (x1 + ∆x1 , x2 + ∆x2 ). Here because the ”local slope toward the east” is ∂u(x) ∂x1 , the change of ”height” made when we move to the ”east” by ∆x1 is cause the ”local slope toward the north”

is ∂u(x) ∂x2 , the ∂u(x) ∂x2 ∆x2 .

∂u(x) ∂x1 ∆x1 .

Also, be-

change of ”height” made

when we move to the ”north” by ∆x2 is Therefore, the total change of ”height” made when we move from x = (x1 , x2 ) to the east by ∆x1 and to the north by ∆x2 is ∆u =

∂u(x) ∂u(x) ∆x1 + ∆x2 ∂x1 ∂x2

This is because the ”mountain” is taken to be locally straight.

CHAPTER 4. SO-CALLED UTILITY FUNCTION

49

Now recall that we are moving along the same indifference curve and the change of ”utility” must be kept zero (See Figure 3.8). Therefore, from ∆u = 0 we have ∂u(x) ∂u(x) 0= ∆x1 + ∆x2 . ∂x1 ∂x2 By rearranging this equality we obtain −

∆x2 = ∆x1

∂u(x) ∂x1 ∂u(x) ∂x2

Notice that the left-hand-side above is the marginal rate of substitution, the absolute value of the local slope of the indifference curve. Recall that since indifference curves are downward-sloping, when ∆x1 is positive ∆x2 is negative, and that’s why we are putting the minus sign. Summing up, we obtain ∂u(x) ∂x1 ∂u(x) ∂x2

M RS(x) =

.

This is the description of marginal rate of substitution by means of marginal utilities. Let me describe marginal rate of substitution for the previous examples. First one is the case of perfect substitution. From the utility representation in linear function u(x) = ax1 + bx2 we have M RS(x) =

∂u(x) ∂x1 ∂u(x) ∂x2

=

a , b

which will be immediate to see. From another representation u b(x) = ln(ax1 + bx2 ) we obtain \ M RS(x) =

∂u b(x) ∂x1 ∂u b(x) ∂x2

=

a ax1 +bx2 b ax1 +bx2

=

a . b

Also, from another one u e(x) = eax1 +bx2 we have ^ M RS(x) =

∂u e(x) ∂x1 ∂u e(x) ∂x2

=

aeax1 +bx2 a = . beax1 +bx2 b

We obtain the same marginal rate of substitution. This is not a coincidence. As I will give a general proof below, since the above u, u b, u e represent the same preference they must give the same marginal rate of substitution.

CHAPTER 4. SO-CALLED UTILITY FUNCTION

50

Another example. Let us describe marginal rate of substitution given by Cobb-Douglas preference. From the power function form u(x) = xa1 xb2 we obtain M RS(x) =

∂u(x) ∂x1 ∂u(x) ∂x2

=

axa−1 xb2 ax2 1 = b−1 a bx1 bx1 x2

From its log transformation u b(x) = a ln x1 + b ln x2 we obtain \ M RS(x) =

∂u b(x) ∂x1 ∂u b(x) ∂x2

=

a x1 b x2

=

ax2 . bx1

Again they are the same. One more example. For a general utility representation u(x) consider doubling it so as to obtain u b(x) = 2u(x). Because the second one represents the same preference as the first one does, since it is just doubling the scale of utility representation, they must give the same marginal rate of substitution. To see this, denote the marginal rate of substitution derived form the first \ one by M RS(x), and denote the rate derived from the second one by M RS(x). From the above formula we have M RS(x) =

∂u(x) ∂x1 ∂u(x) ∂x2

.

\ M RS(x) =

∂u b(x) ∂x1 ∂u b(x) ∂x2

,

On the other hand, we have

but since u b(x) = 2u(x) we have \ M RS(x) =

∂2u(x) ∂x1 ∂2u(x) ∂x2

=

2 ∂u(x) ∂x1 2 ∂u(x) ∂x2

and as 2s in the numerator and the denominator are canceled out we obtain \ M RS(x) = \ which implies M RS(x) = M RS(x).

∂u(x) ∂x1 ∂u(x) ∂x2

,

CHAPTER 4. SO-CALLED UTILITY FUNCTION

51

While marginal utilities have no economic content, why does marginal rate of substitution described by them have economic content? This is because marginal rate of substitution is given as the ratio between marginal utilities, where the scales of marginal utilities are canceled out across the numerator and the denominator. Now we have the following general claim. Theorem 4.2 For any monotone transformation f both u(x) and u b(x) = f (u(x)) give the same marginal rate of substitution. Proof. Let M RS(x) denote the marginal rate of substitution derived from \ u(x), and let M RS(x) denote the marginal rate of substitution derived from u b(x). Marginal utility of Good 1 in representation u b(x) is ∂b u(x) ∂f (u(x)) ∂u(x) = = f ′ (u(x)) , ∂x1 ∂x1 ∂x1 from the formula for partial derivative of composite function. Likewise, marginal utility of Good 2 in representation u b(x) is ∂f (u(x)) ∂u(x) ∂b u(x) = = f ′ (u(x)) ∂x2 ∂x2 ∂x2 Therefore, \ M RS(x) =

∂u b(x) ∂x1 ∂u b(x) ∂x2

=

f ′ (u(x)) ∂u(x) ∂x1 f ′ (u(x)) ∂u(x) ∂x1

=

∂u(x) ∂x1 ∂u(x) ∂x1

= M RS(x).

Utility representation obtained by taking any monotone transformation of any given representation represent the same preference as the original one does. The above theorem states that marginal rate of substitution is invariant to taking any monotone transformation, which means it is independent of which representation to consider for a given preference.

4.4

Ordinal utility and cardinal utility

Numerical representation is said to be ordinal if it is just a representation of ranking and has no quantitative meaning. On the other hand, it is said to be cardinal if has quantitative meaning. Unless noted particularly, this book adopts the standpoint of ordinal utility throughout, which says that the concept of”utility” has no meaning other than a representation of preference ordering. Hence any statement like ”if I buy this my happiness is doubled” has no meaning, and much less ”interpersonal comparison of utility” does, which allows statements like ”I’m happier than

CHAPTER 4. SO-CALLED UTILITY FUNCTION

52

you” and ”she is 1.3 times happier than he is.” From the standpoint of ordinal utility, the only cardinal notion, which has quantitative meanings, is marginal rate of substitution. On the other hand, there is a standpoint of cardinal utility, asserting that utilities do have quantitative meanings. Without a ”faith,” it is hard to get convinced with the standpoint of cardinal utility, in particular the assertion that utilities are comparable across individuals. However, it is true that this appeals to our intuition. One may say for example, ”if we cannot compare utilities across individuals, can’t we compare between the utility from extra 10 dollars for a poor and the utility from extra 10 dollars for a rich? That’s absurd!” I somehow understand such claims, but I think it should be justified (if we want to) in a richer model with richer dimensions and descriptions of time and uncertainty which are implicit and fixed in our current argument. Having said that, I will proceed with the standpoint of ordinal utility. While this book maintains the standpoint of ordinal utility throughout, in later chapters the readers may wonder that I’m using some notions of cardinal utility, particularly in the chapters about quasi-linear preferences, discounted utility and expected utility. Let me emphasize I’m not. If it looks as if I’m using some notions of cardinal utility, it will be because certain type of preference allows a class of ”natural” forms of representations, which is a subset of all the possible utility representations of it, and I restrict attention to such ”natural” ones. For example, if a preference relation allows representation in the form u(x) = v1 (x2 ) + v2 (x2 ) it is said to be additively separable. When we argue about quasi-linear preferences, discounted utility and expected utility, v1 and v2 are considered to have certain quantitative meanings as far as we restrict attention to the class of additively separable representations Of course, even if a preference has additively separable representations it is not the only class of representations of it. For example we can take exponential transformation and obtain u b(x) = eu(x) = ev1 (x2 )+v2 (x2 ) = ev1 (x1 ) ev2 (x2 ) , and let vb1 (x1 ) = ev1 (x1 ) , vb2 (x2 ) = ev2 (x1 ) . Then obtain a ”multiplicatively separable” representation u b(x) = vb1 (x1 )b v2 (x2 ). Also we can take any arbitrary monotone transformation f and obtain another representation u e(x) = f (u(x)), which is in general not additively separable.

CHAPTER 4. SO-CALLED UTILITY FUNCTION

53

With this point kept in mind, economists choose the explanation that the class of additive representations is the ”natural” one and interprets v1 to be the ”utility of Good 1” and v2 to be the ”utility of Good 2” and that the individual cares about the ”sum” of the two, and call v1 and v2 ”cardinal utility.” We should note, however, that such ”additions” and ”quantitative comparisons” of ”utilities” are limited to those within a given individual. Therefore, it still does not get into any kind of interpersonal comparability of utility, which requires a different level of ”faith” as said above.

4.5

Exercises

Exercise 4 Let Good 1 be consumption good at Period 1, and Good 2 be consumption good at Period 2. (i) Give a representation to the preference of a consumer who cares only about consumption at Period 1. (ii) Give a representation to the preference of a consumer exhibiting perfect substitution between consumptions at two periods, such that he cares more about the current consumption. (iii) Give a representation to the preference of a consumer exhibiting perfect substitution between consumptions at two periods, such that he cares more about the future consumption. Exercise 5 Let Good 1 be the consumption good available at State 1, and Good 2 be the one available at State 2. (i) Suppose the probability of State 1 2/3, and give a representation to the preference of a consumer who cares only about the expected value of his consumption. (ii) Give a representation to the preference of a consumer who cares only about the worst case, meaning the case with the lower amount of consumption. 2

3

Exercise 6 Consider preference represented by u(x) = x15 x25 . (i) Find marginal utility of Good 1 under the representation. (ii) Find marginal utility of Good 2 under the representation. (iii) Find the marginal rate of substitution of Good 2 for Good 1 (iv) Redo the above for another representation v(x1 , x2 ) = 2 ln x1 + 3 ln x2 .

Chapter 5

Choice and demand 5.1

Maximal elements for preference

As before, let X the set of all the potentially available choice alternatives, which is the consumption set in the context of choosing consumptions. Let ≿ denote preference relation defined over X. Now suppose that an opportunity set B is a subset of X. Then the ”best” choice in B for a given individual is defined as a maximal element in it according to his preference. Definition 5.1 Say that x∗ ∈ B is a maximal element in B for preference relation ≿ if there is no x ∈ B such that x ≻ x∗ . When the preference satisfies completeness and transitivity, this is equivalent to saying that ”it is at least as good as any element in B.” That is, one can also say that x∗ ∈ B is a maximal element in B for preference relation ≿ if x∗ ≿ x for all x ∈ B. Now let us apply this to consumption choice under budget constraint. Definition 5.2 Given price vector p = (p1 , p2 ) and income w, consumption vector x∗ = (x∗1 , x∗2 ) ∈ B(p, w) is said to be a maximal element in budget set B(p, w) if there is no x ∈ B(p, w) such that x ≻ x∗ . Let me illustrate this in Figure 5.1. Pick any point in the budget set, say x. Is the maximal? No, because we can find a point in the budget set which is above the indifference curve passing through x, such as y. Then how about y? It is not maximal, because we can find a point in the budget set which is above the indifference curve passing through y, such as z. And so on. In the end, the consumer will choose a point like x∗ . There is no point in the budget set which is above the indifference curve passing through x∗ . Thus x∗ is a maximal element in the given budget set.

54

CHAPTER 5. CHOICE AND DEMAND

55

Good 2 6

ry

∗ rx

x

r

rz - Good 1

Figure 5.1: Maximal element

Now I depicted the maximal element x∗ on the budget line, but if preference does not meet monotonicity and allows to get ”full” at some point then at the consumer might not spend all his income at his optimal choice. However, we can show that under monotonicity any maximal element must be on the budget line. Proposition 5.1 If preference satisfies (weak) monotonicity then any maximal element in the budget set must be on the budget line. Proof. Suppose that there is a maximal element which is not on the budget line, at which the consumer does not spend all his income. Denote it by x = (x1 , x2 ), then we have p1 x1 +p2 x2 < w. Then we can take y = (y1 , y2 ) such that y1 > x1 , y2 > x2 and p1 y1 + p2 y2 ≦ w. From the weak monotonicity of preference we have y ≻ x, which contradicts to x being maximal in the budget set. Since monotonicity of preference is an innocuous assumption after making appropriate interpretation as discussed before, we consider the case that any maximal element is on the budget equation throughout. Now look at Figure 5.1 again, then you might notice that there is just one maximal element here, which is x∗ . In general, however, maximal elements may not be just one. For example, when preference which is weakly convex but not strictly convex then we may have a situation like Figure 5.2, in which all the points on the flat part of the indifference curve coinciding with the budget line are maximal elements. Also, if preference even fails be weakly convex then as in Figure 5.3 all the points mutually distant to each other are maximal. However, we can show that maximal element is always unique when preference is strictly convex.

CHAPTER 5. CHOICE AND DEMAND

56

Good 2 6

- Good 1 Figure 5.2: Weakly convex preference

Good 2 6

r r r

- Good 1

Figure 5.3: Non-convex preference

CHAPTER 5. CHOICE AND DEMAND

57

Proposition 5.2 If preference is strictly convex then maximal element in any budget set is uniquely determined. Proof. Suppose there are two distinct maximal elements in the given budget set, denoted by x = (x1 , x2 ) and y = (y1 , y2 ), respectively. Since x is maximal we have x ≿ y. Likewise, since y is maximal as well we have y ≿ x. Therefore we have x ∼ y. Now pick let’s say 21 x + 12 y, then it is in the budget set and from strict convexity leads to 12 x + 12 y ≻ x ∼ y. This contradicts to x and y being maximal. In this book we mostly assume strict convexity and consider the case that maximal element is uniquely determined. Exceptional but important cases are perfect substitution and perfect complementarity, in which weak convexity is met but not strict convexity. I will come to this later.

5.2

Smooth consumption choice

There are two more points I like you to notice about the maximal element as depicted in Figure 5.1. One is that it is not on either edge of the budget line and the other is that at the maximal element the indifference curve passing through it is tangent to the budget line. When the maximal element is strictly between the endpoints and both of quantities of Good 1 and Good 2 are positive it is said to be an interior solution. On the other hand, when the maximal element is on one of the edges as in Figure 5.4 it is called a corner solution. Thus, let me say that the consumption choice meets tangency condition when the corresponding indifference curve is tangent to the budget line at the maximal element. The tangency condition may not hold even when the solution is in interior, when the corresponding indifference curve has a ”kink” as in Figure 5.5. Let me say that the consumption choice is smooth when it is an interior solution and meets the tangency condition. Proposition 5.3 When preference is smooth consumption choice is smooth. Let me omit the rigorous proof, but the idea is as follows. Consider the case that the corresponding indifference curve is steeper than the budget line, like at x as depicted in Figure 5.6. Then the local slope of the indifference curve is greater than the slope of the budget line. This means that the marginal rate of substitution of Good 2 for Good 1 is greater than the relative price of Good 1 for Good 2. That is, we have M RS(x) >

p1 p2

CHAPTER 5. CHOICE AND DEMAND

58

Good 2 6

r

- Good 1

Figure 5.4: Corner solution

Good 2 6

r - Good 1 Figure 5.5: Indifference curves with kinks

CHAPTER 5. CHOICE AND DEMAND

59

Recall that M RS(x) is the amount of Good 2 the consumer is willing to give up in order to get one extra unit of Good 1. On the other hand, pp12 is the amount of Good 2 he has to give up in order to get one extra unit of Good 1, that is, the opportunity cost of one extra unit of Good 1 measure by Good 2. In the current situation the amount of sacrifice he is willing to make in order to get extra one unit of Good 1 is greater than the amount of sacrifice he has to pay. Thus he will increase the amount of Good 1 by sacrificing the consumption of Good 2. Consider the opposite case as well. Consider the case that the corresponding indifference curve is flatter than the budget line, like at x′ as depicted in Figure 5.6. Then the local slope of the indifference curve is smaller than the slope of the budget line. This means that the marginal rate of substitution of Good 2 for Good 1 is smaller than the relative price of Good 1 for Good 2. That is, we have p1 M RS(x′ ) < p2 In the current situation the amount of sacrifice he is willing to make in order to get extra one unit of Good 1 is smaller than the amount of sacrifice he has to pay. Or, equivalently speaking by taking the inverse in the above inequality, the amount of Good 1 he is willing to make in order to get extra one unit of Good 2 is greater than the amount of Good 1 he has to sacrifice. Thus he will increase the amount of Good 2 by sacrificing the consumption of Good 1. When the corresponding indifference curve is tangent to the budget line, like at x∗ as depicted in Figure 5.6, the local slope of the indifference curve is equal to the slope of the budget line. That is, we have equality M RS(x∗ ) =

p1 . p2

Here the amount of sacrifice he is willing to make in order to get extra one unit of Good 1 is equal to the amount of sacrifice he has to pay. If he moves to upper-left on the budget line he leads to the first situation in which the amount of Good 1 is too small, and if he moves to lower-right on the budget line he leads to the second situation in which the amount of Good 1 is too large. Therefore, the optimal consumption is determined to be the point on the budget line at which the marginal rate of substitution and relative price are equalized. As marginal rate of substitution moves between 0 and infinity monotonically and continuously under smoothness of preference, there is a unique point x∗ with x∗1 , x∗2 > 0 which satisfies the above equality. Thus, smooth consumption is determined by the tangency condition and the maximal element x∗ satisfies the equation M RS(x∗ ) =

p1 . p2

Also, since x∗ is on the budget line it satisfies the budget equation p1 x∗1 + p2 x∗2 = w.

CHAPTER 5. CHOICE AND DEMAND

60

Good 2 6

rx ∗ rx ′ rx

- Good 1

Figure 5.6: Tangency condition

By solving the above two equations with two unknowns we can find x∗ = (x∗1 , x∗2 ).

5.2.1

An example of smooth consumption: the case of Cobb-Douglas preference

Let us derive optimal consumption from the tangency condition in a concrete example. Let us consider Cobb-Douglas preference, which is represented in the form u(x) = a ln x1 + b ln x2 . First let us find the marginal rate of substitution. We already showed that marginal rate of substitution is given as the ration between marginal utilities, that is, M RS(x) =

∂u(x) ∂x1 ∂u(x) ∂x2

∂u(x) ∂x1

For the above representation, we have marginal rate of substitution is M RS(x) =

a x1 b x2

=

=

. a x1

and

∂u(x) ∂x2

=

b x2 .

Hence the

ax2 . bx1

Therefore at the maximal element x∗ = (x∗1 , x∗2 ) it holds ax∗2 p1 = . ∗ bx1 p2 By rewriting this we obtain a linear relationship x∗2 = budget equation, then we have p1 x∗1 + p2 x∗2 = p1 x∗1 + p2

bp1 ∗ ap2 x1 .

Plug this into the

(a + b)p1 ∗ bp1 ∗ x = x1 = w. ap2 1 a

CHAPTER 5. CHOICE AND DEMAND

61

By solving this for x∗1 we obtain x∗1 =

w a · a + b p1

By plugging this into the linear relationship x∗2 = x∗2 =

bp1 ∗ ap2 x1

we obtain

b w · . a + b p2

Consumption choice given by Cobb-Douglas preference is that first the consumer splits his income between Good 1 and Good 2 at the proportion a versus b and then divides the income allocated to each good by its price. This proportionality is observed in data in a quite robust manner, and this is the reason why Cobb-Douglas preference (or its generalization) is frequently used in application.

5.3

The case of perfect substitution

Next let us consider consumption choice given by preferences exhibiting perfect substitution, which is an example of multiple optima and corner solutions. Here represent the preference exhibiting perfect substitution let’s say by u(x) = ax1 + bx2 , then the marginal rate of substitution is ab , which is constant everywhere. Case 1 — When pp12 < ab : See Figure 5.7. Here the budget line is flatter than the indifference curves, that is, the marginal rate of substitution is always greater than the relative price of Good 1 for Good 2. Therefore the consumer is always willing to increase the amount of Good 1 by sacrificing the amount of Good 2. Because the marginal rate of substitution is constant this does not stop until we reach the lower-right endpoint of the budget line. Therefore the maximal element is w x∗1 = , x∗2 = 0 p1 Case 2 — When pp12 < ab : See Figure 5.8. Here the budget line is steeper than the indifference curves, that is, the marginal rate of substitution is always smaller than the relative price of Good 1 for Good 2. Therefore the consumer is always willing to increase the amount of Good 2 by sacrificing the amount of Good 1. Because the marginal rate of substitution is constant this does not stop until we reach the upper-left endpoint of the budget line. Therefore the maximal element is w x∗1 = 0, x∗2 = p2 Case 3 — pp12 = ab : See 5.9. Here the budget line coincides with one of the indifference curves. Therefore all the points on the budget line are equally preferable. Also, any point on the budget line the marginal rate of substitution is equal to the relative price of Good 1 for Good 2. Therefore all the points on the budget line are maximal elements.

CHAPTER 5. CHOICE AND DEMAND

62

Good 2 6

r

- Good 1

Figure 5.7: Perfect substitution: Case 1

Good 2 6

r

- Good 1 Figure 5.8: Perfect substitution: Case 2

CHAPTER 5. CHOICE AND DEMAND

63

Good 2 6

- Good 1 Figure 5.9: Perfect substitution: Case 3

5.4

The case of perfect complementarity

Finally let us consider consumption choice given by preferences exhibiting perfect complementarity, which is an example of interior solution to which the tangency condition does not apply. Here represent the preference exhibiting perfect complementarity let’s say by u(x) = min{ xa1 , xb2 }. The series of indifference curves given by this preference are L-shaped and parallel along the line of locus of kinks xx12 = ab . Therefore the maximal element is obtained as the intersection of the line of locus of kinks xx21 = ab and the budget line p1 x1 + p2 x2 = w, as depicted in Figure 5.10. By solving these two equations we obtain x∗1 =

aw , ap1 + bp2

x∗2 =

bw . ap1 + bp2

Here the tangency condition does not apply because the corresponding indifference curve is kinked at the maximal element and we cannot take the marginal rate of substitution at this point.1

5.5

Demand function

Hereafter, given price p = (p1 , p2 ) and income w denote the maximal element in B(p, w) by x(p, w) = (x1 (p, w), x2 (p, w)) and call it demand function I restate like this because I want to emphasize that consumption choice may vary as the price-income pair (p, w) varies. 1 This may fall in a ”generalized” version of the tangency condition when we ”generalize” the notion of derivative, though.

CHAPTER 5. CHOICE AND DEMAND

64

Good 2 6

- Good 1 Figure 5.10: Perfect complementarity

For example, the demand function generated by Cobb-Douglas preference represented in the form u(x) = a ln x1 + b ln x2 is ( ) a w b w x(p, w) = (x1 (p, w), x2 (p, w)) = · , · . a + b p1 a + b p2 In order to be able to define the demand ”function” the maximal element has to be unique, but what if there are multiple optima? For example, consider preference exhibiting perfect substitution represented in the form u(x) = ax1 + bx2 . Then what would correspond to ”demand function” is  w if pp12 < ab  ( p1 , 0) x(p, w) = all the points on the budget line if pp12 = ab   (0, pw2 ) if pp12 > ab This maps each point to a set, not from point to point. It will be better to call this ”correspondence.” So more generally it is better to call the mapping demand correspondence rather than demand function, but as we mostly consider the case that the maximal element is uniquely determined given each price-income pair, we will call it demand function.

5.6

Consumption choice and demand in exchange economy

So far we have not specified the source of income w, but here let us consider consumption choice and demand in an exchange economy in which each consumer’s income is given as the market value of his initial endowment.

CHAPTER 5. CHOICE AND DEMAND

65

Given a price vector p = (p1 , p2 ), the income of a consumer who has initial endowment e = (e1 , e2 ) is p1 e1 + p2 e2 . Therefore the budget constraint is p1 x2 + p2 x2 ≦ p1 e1 + p2 e2 Taking the initial endowment e to be fixed, the budget set is denoted by B(p), where B(p) = {x ∈ R2+ : p1 x2 + p2 x2 ≦ p1 e1 + p2 e2 }. Here we obtain the demand function denoted by x(p) = (x1 (p), x2 (p)), as the maximal element in B(p) for each given p. It is obtained by replacing w by p1 e1 + p2 e2 in the previous argument. For example, the demand function given by Cobb-Douglas preference represented in the form u(x) = a ln x1 + b ln x2 is ( ) a p1 e1 + p2 e2 b p1 e1 + p2 e2 (x1 (p), x2 (p)) = · , · . a+b p1 a+b p2 Here if demand (x1 (p), x2 (p)) is in the left of the initial endowment point, that is, x1 (p) < e1 , x2 (p) > e2 then the consumer is buying Good 2 by selling Good 1 under given price p = (p1 , p2 ). Similarly for the opposite case.

5.7

Describing choice as utility maximization

Recall that x∗ ∈ B is a maximal element in B according to preference ≿ if x∗ ≿ x for all x ∈ B. If we describe this by means of utility representation u it is u(x∗ ) ≧ u(x) for all x ∈ B. That is, a maximal element according to a given preference is described as a maximizer of its representation. Since the maximized value of utility representation depends on opportunity set B we write this as v(B) = max u(x) x∈B

and call it an indirect utility function. As I have been repeatedly emphasizing, utility function itself has no quantitative or economic meaning. Consider taking any monotone transformation of u and maximizing u b = f (u) to look for x∗ such that u b(x∗ ) ≧ u b(x) for all x ∈ B,

CHAPTER 5. CHOICE AND DEMAND

66

then such x∗ must be the same. Recall that in consumption choice under budget constraint given price p = (p1 , p2 ) and income w, a consumption vector in the budget set x∗ = (x∗1 , x∗2 ) ∈ B(p, w) is said to be a maximal element in B(p, w) according to ≿ if x∗ ≿ x for all x ∈ B(p, w). If we describe this by means of utility representation u it is u(x∗ ) ≧ u(x) for all x ∈ B(p, w). That is, maximal element x∗ here is described as a maximizer of u(x) in the budget set B(p, w). This can be written also as a maximization problem max u(x) x

subject to

p1 x1 + p2 x2 ≦ w.

Because the maximized value depends on the price-income pair (p, w) we write it as v(p, w) = max u(x) x:p1 x1 +p2 x2 ≦w

and call it an indirect utility function. Again, consider taking any monotone transformation of u and consider the maximization problem for u b = f (u) as max u b(x) x

subject to

p1 x1 + p2 x2 ≦ w

then the solution must be the same. Consider for example Cobb-Douglas preference, then we must obtain the same solution whether we maximize u(x) = xa1 xb2 or we maximize u b(x) = a ln x1 + b ln x2 . In the above I first introduced the notion of maximal element without using utility maximization, because the description of utility maximization is only for convenience to the analyst. Having said that, hereafter we will borrow the description by utility maximization in order to be operational.

5.8

Expenditure minimization and compensated demand

Here I introduce a ”dual” of optimal choice under budget constraint, which is expenditure minimization. It is to minimize the expenditure for consumption as far as it satisfies a given ”utility level.”

CHAPTER 5. CHOICE AND DEMAND

67

Denote an arbitrary ”utility level” by u. Now under the constraint that obtained utility must be at least as large as u, that is, u(x) ≧ u consider minimizing the expenditure for x = (x1 , x2 ), which is p1 x2 + p2 x2 . That is, the problem is written as min p1 x2 + p2 x2 x

subject to

u(x) ≧ u.

This is somehow an uncomfortable formulation because it takes a particular utility representation as given. However, later I show that it is without loss of generality to formulate the problem with a particular representation, as different formulations obtained from different utility representations are suitably translatable to each other. Solution to an expenditure minimization problem is obtained as follows. See Figure 5.11. Here does consumption vector z = (z1 , z2 ) satisfying u(z) = u minimize the expenditure? To see that draw a straight line passing through z with slope − pp21 , then it is described by the formula p1 x1 + p2 x2 = p1 z1 + p2 z2 and it is the line consisting of consumption vectors which yield the same expenditure under p = (p1 , p2 ) as z does. Let me call this an iso-expenditure line. Here we can find a point on the given indifference curve which is below the iso-expenditure line passing through z, let’s say y = (y1 , y2 ). Hence z is not minimizing the expenditure. What about y then? To see that, again draw the iso-expenditure line passing through y, which is described by p1 x1 + p2 x2 = p1 y1 + p2 y2 . Again we can find a point on the given indifference curve which is below the iso-expenditure line passing through y, let’s say x∗ = (x∗1 , x∗2 ). Hence y is not minimizing the expenditure. Now draw the iso-expenditure line passing through x∗ described by p1 x1 + p2 x2 = p1 x∗1 + p2 x∗2 . Notice that this iso-expenditure line is tangent to the given indifference curve at x∗ . Hence there is no point on the indifference curve which is below the isoexpenditure line passing through x∗ . Thus x∗ is minimizing the expenditure.

CHAPTER 5. CHOICE AND DEMAND

68

Good 2 6

r x∗

ry

rz - Good 1

Figure 5.11: Expenditure minimization

Since the iso-expenditure line is tangent to the given indifference curve at the expenditure-minimizing point x∗ , the local slope of the indifference curve and the slope of the iso-expenditure line are equal. That is, marginal rate of substitution and the relative price of Good 1 for Good 2, which is pp12 , are equalized. Thus we obtain an equation M RS(x∗ ) =

p1 . p2

On the other hand, we are following the constraint that consumption at least satisfies the given ”utility level” u. Since it is enough to satisfy the minimally necessary level of utility here, we obtain another equation u(x∗ ) = u By solving these two equations we obtain the expenditure-minimizing point. This is called compensated demand under p = (p1 , p2 ) given ”utility level” u. Let us do this with Cobb-Douglas preference which is represented by u(x) = a ln x1 + b ln x2 . As obtained in the previous arguments, marginal rate of substitution here is M RS(x) =

ax2 . bx1

Therefore at the expenditure-minimizing point x∗ = (x∗1 , x∗2 ) it holds ax∗2 p1 = . bx∗1 p2

CHAPTER 5. CHOICE AND DEMAND

69

On the other hand, the other equation u(x∗ ) = u leads to a ln x∗1 + b ln x∗2 = u. By solving the two equations we obtain x∗1 x∗2

= (bp1 )− a+b (ap2 ) a+b e a+b a a u = (bp1 ) a+b (ap2 )− a+b e a+b b

b

u

Hereafter, denote the compensated demand given price p = (p1 , p2 ) and ”utility level” u by h(p, u) = (h1 (p, u), h2 (p, u)), and call it compensated demand function. Also, denote the minimized expenditure by e(p, u) = p1 h1 (p, u) + p2 h2 (p, u), and call it expenditure function. For example, Cobb-Douglas preference represented by u(x) = a ln x1 +b ln x2 yields the compensated demand function h1 (p, u) =

(bp1 )− a+b (ap2 ) a+b e a+b b

b

u

h2 (p, u) = (bp1 ) a+b (ap2 )− a+b e a+b a

a

u

and the expenditure function [ ] a b a a b b u e(p, u) = a− a+b b a+b + a a+b b− a+b p1a+b p2a+b e a+b . Expenditure-minimizing point is utility-maximizing given the price when the minimized expenditure is given as the income. Thus it holds =

x1 (p, e(p, u))

h2 (p, u) =

h1 (p, u)

x2 (p, e(p, u))

Also, utility-maximizing point is minimizing the expenditure given the price in order to satisfy the same level of utility as it yields. Thus it holds x1 (p, w) x2 (p, w)

= =

h1 (p, u(x(p, w))) h2 (p, u(x(p, w)))

This relationship is called duality. Now let me come back to the problem of taking ”utility level” as given. Here I show that it is without loss of generality to formulate compensated demand function and expenditure function with a particular representation, as different formulations obtained from different utility representations are suitably translatable to each other. Proof follows directly from the definitions.

CHAPTER 5. CHOICE AND DEMAND

70

Proposition 5.4 Let h(p, u) and e(p, u) denote the compensated demand function and the expenditure function defined for utility representation u(x), respectively. Let f be any monotone transformation, and denote the compensated demand function and the expenditure function defined for representation v(x) = f (u(x)). Then it holds b h1 (p, v) = b h2 (p, v) = eb(p, v) =

5.9

h1 (p, f −1 (v)) h2 (p, f −1 (v)) e(p, f −1 (v)).

Exercises 2

3

Exercise 7 Consider preference represented by u(x) = x15 x25 . (i) Solve for the demand function and the indirect utility function. (ii) Solve for the compensated demand function and the expenditure function.

Chapter 6

Demand analysis In this chapter let us consider how demand responds to price and income changes, and welfare evaluation of such changes.

6.1

Normal and inferior goods

We can classify goods according to how demands for them respond to income change. Definition 6.1 A good is said to be normal if higher (lower) income leads to larger (smaller) demand for it. A good is said to be inferior if higher (lower) income leads to smaller (larger) demand for it. More mathematically speaking, let’s say Good 1 is normal if for ”slight amount” of extra income ∆w it holds x1 (p, w + ∆w) − x1 (p, w) > 0 Now by making this ”slight amount” ∆w tend to be infinitesimally small, the condition means that the partial derivative of demand function for Good 1 by w is positive, that is, ∂x1 (p, w) > 0. ∂w Likewise, the condition for inferior good is that the inequality above is reversed. For example, consider the demand function generate by Cobb-Douglas preference represented by u(x) = a ln x1 + b ln x2 , x1 (p, w) =

w a · , a + b p1

x2 (p, w) =

b w · a + b p2

Since income w appears in the numerator in the formula for x1 (p, w), and similarly for x2 (p, w), when it increase the demand for both good increase. Hence both Good 1 and 2 are normal. 71

CHAPTER 6. DEMAND ANALYSIS

72

Good 2 6 w+∆w p2

r

w p2

r w p1

- Good 1

w+∆w p1

Figure 6.1: Good 1 is inferior

It sounds puzzling that when one has more income he consumes less amount of some good, but it is well-possible under standard preferences satisfying completeness, transitivity monotonicity and convexity. See for example Figure 6.1, in which as income increase from w to w+∆w the demand for Good 1 decreases. This is intuitive is such that you consume it only when you are poor, and do not consume it when you are rich. Note, however, that not all goods can be inferior simultaneously (why?). Let me give an example of demand function allowing the existence of inferior good. a w(1 − e−w ) b we−w x1 (p, w) = · , x2 (p, w) = · a+b p1 a+b p2 When you take the derivative of demand function for Good 1 by income you get ∂x1 (p, w) a 1 − e−w + we−w = · > 0, ∂w a+b p1 which implies Good 1 is always normal. On the other hand, when you take the derivative of demand function for Good 2 by income you get ∂x2 (p, w) a (1 − w)e−w = · > 0, ∂w a+b pw which is negative when w > 1, meaning inferior there.

6.2

Ordinary and Giffen goods

Next let us consider how demand for each good responds to own price change. That is, we consider how demand for Good 1 changes as the price of Good 1 changes, and how demand for Good 2 changes as the price of Good 2 changes.

CHAPTER 6. DEMAND ANALYSIS

73

Definition 6.2 A good is said to be ordinary if higher (lower) price of it leads to smaller (larger) demand for it. A good is said to be a Giffen good if higher (lower) price of it leads to larger (smaller) demand for it. More mathematically speaking, let’s say Good 1 is ordinary if for ”slight amount” of own price increase ∆p1 it holds x1 (p1 + ∆p1 , p2 , w) − x1 (p1 , p2 , w) < 0 Now by making this ”slight amount” ∆p1 tend to be infinitesimally small, the condition means that the partial derivative of demand function for Good 1 by p1 is negative, that is, ∂x1 (p, w) <0 ∂p1 Likewise, the condition for Giffen good is that the inequality above is reversed. Consider again the previous example of demand function generated by CobbDouglas preference. Since p1 appears in the denominator in the formula for x1 (p, w) and p2 appears in the denominator in the formula for x2 (p, w), demand for each good decreases as its own price increases. Hence both Good 1 and 2 are ordinary. It is again puzzling that demand for a good increases as its own price increases, but it is possible under standard preferences satisfying completeness, transitivity, monotonicity and convexity. See Figure 6.2 for example. Here as the price increase from p1 to p1 + ∆p1 the demand for Good 1 increases. Later it is shown that the Giffen good phenomenon is explained as the case that the ”inferiority” of an inferior good is ”very severe.” To reach there we need some more concepts to go over.

6.3

Gross substitutes and gross complements

Next let us consider how demand for each good responds to price change of another good. Definition 6.3 A good is said to be a gross substitute of another one if the demand for it increases as the price of the second increases. A good is said to be a gross complement of another one if the demand for it decreases as the price of the second increases. More mathematically speaking, let’s say Good 1 is a gross substitute of Good 2 if for ”slight amount” of own price increase ∆p2 it holds x1 (p1 , p2 + ∆p2 , w) − x1 (p1 , p2 , w) > 0

CHAPTER 6. DEMAND ANALYSIS

74

Good 2 6 w p2

r

r w p1 +∆p1

w p1

- Good 1

Figure 6.2: Good 1 is Giffen

Now by making this ”slight amount” ∆p2 tend to be infinitesimally small, the condition means that the partial derivative of demand function for Good 1 by p2 is positive, that is, ∂x1 (p, w) >0 ∂p2 Likewise, the condition for gross substitute is that the inequality above is reversed. Let me relegate the explanation of what ”gross” means to the section on substitution and income effect. Here let me just give you an intuition. For example, suppose Good 1 is chicken and Good 2 is beef. I guess in most kinds of cuisine you take either chicken or beef, and not both. Then as the beef price is higher consumers will switch from buying beef to buying chicken. Thus chicken is a gross substitute of beef. On the other hand, suppose Good 1 is game soft and Good 2 is game hardware.1 One cannot enjoy game software without the hardware, and the hardware alone is meaningless when there is no software accompanied. Now as the hardware price is higher its consumption will be lower, and it leads to lower consumption of the software. Hence game software is a gross complement of game hardware. Here we consider three examples of demand functions. First, let us con√ √ sider preference represented by u(x) = a x1 + b x2 . Let us derive the demand function, as it is the first time to see this one. Since this preference is smooth we can use the tangency condition. As the marginal utility of Good 1 is √ √ √ √ ∂(a x1 +b x2 ) ∂(a x1 +b x2 ) a = 2√x1 and that of Good 2 is = 2√bx2 , the marginal ∂x1 ∂x2

1 Is

this example being outdated after the rise of smartphone?

CHAPTER 6. DEMAND ANALYSIS rate of substitution is

Since M RS(x) =

p1 p2

√a 2 x1 √b 2 x2

75

√ a x2 = √ b x1

holds at optimal choice, we have √ a x2 p1 √ = b x1 p2 b2 p2

Solve this for x2 , then we obtain x2 = a2 p12 . Plug this into the budget equation 2 p1 x1 + p2 x2 = w and solve for x1 . Plug this again into the relationship between x2 and x1 , then we obtain x1 . Thus we obtain x1 (p, w) =

(

a2 w

p1 a2 + b2 ·

p1 p2

),

x2 (p, w) =

(

b2 w

p2 a2 ·

p2 p1

+ b2

).

In this demand function p2 appears in the denominator of a term appearing in the denominator in the formula for x1 (p, w). Hence if Good 2 becomes more expensive the demand for Good 1 increases. Likewise, p1 appears in the denominator of a term appearing in the denominator in the formula for x2 (p, w). Hence if Good 1 becomes more expensive the demand for Good 2 increases. Therefore, Good 1 and 2 are a gross substitute of each other. Next let us consider preference exhibiting perfect complementarity, which is represented by u(x) = min{ xa1 , xb2 }. As derived in the previous chapter the demand function is x1 (p, w) =

aw , ap1 + bp2

x2 (p, w) =

bw . ap1 + bp2

Since p2 is in the denominator in the formula for x1 (p, w), as Good 2 becomes more expensive the demand for Good 1 decreases. Likewise, since p1 is in the denominator in the formula for x2 (p, w), as Good 1 becomes more expensive the demand for Good 2 decreases. Hence Good 1 and 2 are a gross complement of each other. Also, let’s go back to the demand function given by Cobb-Douglas preference, then p2 is not in the formula for x1 (p, I). Hence the demand for Good 1 is not affected by the price of Good 2. Likewise, p1 is not in the formula for x2 (p, I). Hence the demand for Good 2 is not affected by the price of Good 1.

6.4

Elasticity of demand

By the way, we have to adjust units in order to compare how demands are sensitive to income and price change across goods, otherwise such comparison will be meaningless. In economics we use a concept called elasticity.

CHAPTER 6. DEMAND ANALYSIS

76

Elasticity of demand for some good to the change in some variable is the percentage change in quantity of the good demanded in response to one percent change in the variable. To illustrate let me continue to focus on Good 1. Then income elasticity of demand for Good 1 is the percentage change in quantity of Good 1 demanded in response to one percent change in income, which is given by e1,w =

∆x1 x1 ∆w w

=

∆x1 w ∆w x1

One may also write it with derivative notation like ∂x1 w . ∂w x1

e1,w =

For example, when demand for Good 1 increases from 100 to 110 as income increases from 500 to 600 the income elasticity is 110−100 100 600−500 500

=

0.1 = 0.5. 0.2

Likewise, price elasticity of demand for Good 1 is the percentage change in quantity of Good 1 demanded in response to one percent change in its price, which is given by ∆x1 ∆x1 p1 x1 e1,p1 = ∆p = 1 ∆p1 x1 p 1

One may also write it with derivative notation like ∂x1 p1 . ∂p1 x1

e1,p1 =

For example, when demand for Good 1 decreases from 100 to 90 as its price rises from 20 to 30 the price elasticity is 90−100 100 30−20 20

=

0.1 = −0.2 0.5

By the way, when the (absolute value of) price elasticity is smaller than 1 the good is said to be a necessity and when it is greater than 1 it is said to be a luxury. Likewise, cross price elasticity of demand for Good 1 against Good 2 is the percentage change in quantity of Good 1 demanded in response to one percent change in the price of Good 2, which is given by e1,p2 =

∆x1 x1 ∆p2 p2

=

∆x1 p2 ∆p2 x1

CHAPTER 6. DEMAND ANALYSIS

77

One may also write it with derivative notation like e1,p2 =

∂x1 p2 ∂p2 x1

For example, when demand for Good 1 increases from 100 to 120 as the price of Good 2 rises from 20 to 22 the cross price elasticity is 120−100 100 22−20 20

=

0.2 =2 0.1

You can define the corresponding definitions for Good 2 similarly.

6.5

Substitution effect and income effect

Now let me come back to the reason why I put the word ”gross” on substitute and complement in the previous definition. First of all, price change of a good must have two effects, one is the change of relative price for other goods and the other is the change of real income. When some good becomes more expensive, first it becomes relatively more expensive compared to other goods (rise of relative price) and second the quantity of it you can buy within your income decreases (fall of real income). The first effect is called substitution effect and the second is called income effect. ”Gross” demand change now means the whole change without making a distinction between the two. See Figure 6.3. Let w be the income, let p = (p1 , p2 ) denote the price pair before price change, and let x = (x1 , x2 ) denote the consumption choice before the price change. Suppose the price of Good 1 goes up from p1 to p1 + ∆p1 . Hence the price pair after the change is (p1 + ∆p1 , p2 ). Then the new budget w 1 line passes through ( p1 +∆p , 0) and (0, pw2 ), and its slope is − p1 +∆p . Note that p2 1 the x2 -intercept has not changed. Denote the demand after the price change by x′ = (x′1 , x′2 ). Gross demand change for Good 1 is x′1 − x1 and for Good 2 it is x′2 − x2 . Now in order to separate between substitution effect and income effect, draw 1 a line with slope − p1 +∆p which is tangent to the indifference curve passing p2 through x, the demand before the price change. 1 Let x b denote the point at which the line with slope − p1 +∆p is tangent to the p2 corresponding indifference curve. Then it is the expenditure-minimizing point which yields the same welfare level as x does. That is, x b is the compensated demand under (p1 + ∆p1 , p2 ) corresponding to the welfare level given by x. Since the change from the original demand x to the compensated demand x b is made by suitably compensating income so that the consumer can maintain the same welfare level after the price change, it reflects only the change of relative price. Therefore the move from x to x b, that is x b − x = (b x1 − x1 , x b2 − x2 ), is the substitution effect of the price change.

CHAPTER 6. DEMAND ANALYSIS

78

Good 2 6 I p2

x′

r

b rx rx

I p1 +∆p1

I p1

- Good 1

Figure 6.3: Substitution and income effect of the price change of Good 1

1 On the other hand, since the line with slope − p1 +∆p which passes through x b p2 is parallel to the budget line after the price change, the move from x b to x′ reflects b2 ) b1 , x′2 − x only the change of real income. Therefore, the vector x′ − x b = (x′1 − x is the income effect of the price change. Notice that the substitution effect of price increase of Good 1 is always negative (non-positive, precisely) on Good 1 and positive (non-negative, precisely) on Good 2. Since indifference curves are downward-sloping, the increase of relative price of Good 1 moves the point meeting the tangency condition to the upper-left direction. Therefore the increase of Good 1 price always decrease the compensated demand for Good 1 and increases the compensated demand for Good 2. More generally, we have

Proposition 6.1 When the price of some good goes up, its substitution effect on itself is non-positive and that on the other goods is non-negative. Let me explain this analytically. From the duality relationship stated in the last chapter the compensated demand for Good 1 h1 (p, u) is given by h1 (p, u) = x1 (p, e(p, u)). By taking the partial derivative of this with regard to p1 we obtain ∂h1 ∂x1 ∂x1 ∂e = + ∂p1 ∂p1 ∂w ∂p1 From the definition of expenditure function e(p, u) = p1 h1 (p, u) + p2 h2 (p, u)

CHAPTER 6. DEMAND ANALYSIS

79

we have ∂e ∂p1

∂ (p1 h1 + p2 h2 ) ∂p1 ∂h1 ∂h2 = h1 + p1 + p2 ∂p1 ∂p1 ( ) ∂h2 p1 ∂h1 = h1 + p2 + p2 ∂p1 ∂p1 =

Since the tangency condition M RS(x) = pp12 is met at x(p, e(p, u)) = x we can rewrite the above formula into ( ) ∂e ∂h1 ∂h2 = h1 + p2 M RS(x) + ∂p1 ∂p1 ∂p1 Now notice that (M RS(x), 1) is the normal vector to the corresponding indifference curve at x, since (1, −M RS(x)) is the tangent vector of the curve at x from the definition of marginal rate of substitution. Because the compensated demand( moves along ) this indifference curve, we see that vector (M RS(x), 1) and ∂h1 ∂h2 vector ∂p1 , ∂p1 are orthogonal to each other. Hence the inner product of the 1 two is zero, that is, M RS(x) ∂h ∂p1 +

∂h2 ∂p1

∂e ∂p1

= 0. Thus we obtain = h1 .

This is called Shepard’s lemma. From Shepard’s lemma we obtain ∂h1 ∂x1 ∂x1 = + h1 . ∂p1 ∂p1 ∂w By rearranging this we get ∂h1 ∂x1 ∂x1 = − h1 , ∂p1 ∂p1 ∂w ∂h1 1 in which ∂x ∂p1 explains the gross effect of price change, ∂p1 explains the sub1 stitution effect, and − ∂x ∂w h1 explains the income effect. This formula is called Slutsky equation Likewise, we obtain Slutsky equations for the other combinations

∂x2 ∂p1 ∂x1 ∂p2 ∂x2 ∂p2

= = =

∂h2 ∂x2 − h1 ∂p1 ∂w ∂h1 ∂x1 − h2 ∂p2 ∂w ∂h2 ∂x2 − h2 ∂p2 ∂w

CHAPTER 6. DEMAND ANALYSIS

80

Good 2 6 w p2

b rx rx

x′

r w p1 +∆p1

w p1

- Good 1

Figure 6.4: Substitution and income effect on a Giffen good

Example 6.1 (Giffen good): As depicted in Figure 6.4, consider the case that Good is a Giffen good, that is, demand for it is larger despite its price goes up. As stated above, the substitution effect on Good 1 is always negative (at least b1 | > |b x1 − x1 | shows in the weak sense), that is, x b1 − x1 ≦ 0. However, |x′1 − x that the income effect is large enough to overcome the substitution effect. Here Good 1 is an inferior good, which exhibits negative income effect in the sense that as income is smaller it is consumed more. Moreover, since the negativity of income effect is so severe that it overcomes the substitution effect and leads to the gross demand increase. Thus, Giffen good is characterized as an inferior good such that its negativity of income effect is severe enough to overcome the substitution effect to itself. Example 6.2 (Gross complement): I wrote that substitution effect of price increase of some good on another good is always greater than or equal to zero. That is, in a ”pure” sense every good is a substitute of any other good. How can it be consistent with the existence of ”gross” complements? Reconsider the example of preference exhibiting perfect complementarity. As depicted in Figure 6.5, consider that the price of Good 1 goes up from p1 to p1 + ∆p1 . Then the compensated demand under (p1 + ∆p1 , p2 ) which gives the same welfare level as x does is x itself. That is, the substitution effect is zero and the gross demand change consists only of income effect. More generally, one is a gross complement of another if income effect of price increase of the other is greater than the substitution effect of it, and one is a gross substitute of another otherwise.

CHAPTER 6. DEMAND ANALYSIS

81

Good 2 6 w p2

rx



b rx = x

w p1 +∆p1

w p1

- Good 1

Figure 6.5: Substitution and income effect under perfect complementarity

6.6

Income evaluation of welfare change

Consumer may gain or lose from price changes. If the government should compensate consumer’s loss due to price change or tax consumer’s gain due to price change in a lump-sum manner, how much should it be? Here we consider how to evaluate such gain and loss in terms of income.

6.6.1

Compensate variation and equivalent variation

There are two views about this. One is Under the price after the change, if we are to maintain the given consumer’s welfare level the same as before, how much of income should be compensated or taken away? Such value is called compensated variation. Compensated variation is defined as follows. Let p = (p1 , p2 ) be the price before the change and let p′ = (p′1 , p′2 ) be the one after the change. Denote the income by w. Then the minimal income which is necessary for purchasing consumption after the price change in order to maintain the same welfare level as before, u(x(p, w)), is given using the expenditure function in the form e(p′ , u(x(p, w))). Hence the compensated variation is the difference between this and the original income, CV = e(p′ , u(x(p, w))) − w Using indirect utility function it is given in the form v(p′ , w + CV ) = v(p, w)

CHAPTER 6. DEMAND ANALYSIS

82

Let me illustrate for the case that the price of Good 1 increases by ∆p1 , using Figure 6.6 Let x = x(p, w) denote the demand under the price before the change p = (p1 , p2 ), and let x′ = x(p′ , w) denote the demand under the price after the change p′ = (p1 + ∆p1 , p). Then compensated demand is x b = h(p′ , u(x(p, w))), which minimizes expenditure under the price after the change as far as it yields the same welfare level as before. Then the budget line corresponding to the minimal necessary income is the dotted line passing through x b, and the income corresponding to this is e(p′ , u(x(p, w))) = w + CV . That is, this budget line is obtained by shifting the budget line after the price change to the above vertically by CV /p2 (or to the right horizontally by CV /(p1 + ∆p1 )). The other one is Under the current price, if we are to maintain the given consumer’s welfare level the same as after the potential price change, how much of income should be taken away or compensated? Such value is called equivalent variation. Again, let p = (p1 , p2 ) be the current price and let p′ = (p′1 , p′2 ) be the one after the potential change. Denote the income by w. Then the minimal income which is necessary for purchasing consumption under the current price in order to maintain the same welfare level as after the potential price change, u(x(p′ , w)), is given using the expenditure function in the form e(p, u(x(p′ , w))) Hence the equivalent variation is the difference between the original income and this, EV = w − e(p, u(x(p′ , w))) Using indirect utility function it is given in the form v(p, w − EV ) = v(p′ , w). Let me illustrate for the case that the price of Good 1 increases by ∆p1 , using Figure 6.6. Let x = x(p, w) denote the demand under the current price p = (p1 , p2 ), and let x′ = x(p′ , w) denote the demand under the price after the potential change p′ = (p1 + ∆p1 , p). Then compensated demand is x e = h(p, u(x(p′ , w))), which minimizes expenditure under the current price as far as it yields the same welfare level as after the potential price change. Then the budget line corresponding to the minimal necessary income is the dotted line passing through x e, and the income corresponding to this is e(p, u(x(p′ , w))) = w + EV . Here EV is negative, and this budget line is obtained by shifting the budget line under the current price to the below vertically by |EV |/p2 (or to the left horizontally by |EV |/p1 ). Let us compare between compensated variation and equivalent variation, again for the case that the price of Good 1 increase from p1 to p1 + ∆p1 . Note

CHAPTER 6. DEMAND ANALYSIS

83

that CV

= e(p′ , u(x(p, w))) − w = e((p1 + ∆p1 , p2 ), u(x(p1 , p2 , w))) − e((p1 , p2 ), u(x(p1 , p2 , w)))

From Shepard’s lemma ∂e , = h1 ∂p1 since integral of derivative is the difference between values of the original function at two end-points, we obtain CV

= e((p1 + ∆p1 , p2 ), u(x(p1 , p2 , w))) − e((p1 , p2 ), u(x(p1 , p2 , w))) ∫ p1 +∆p1 = h1 (a, p2 , u(x(p1 , p2 , w)))da p1

Since it holds u(x(p1 + ∆p1 , p2 , w)) < u(x(a, p2 , w)) < u(x(p1 , p2 , w)) at all p1 < a < p1 + ∆p1 and it holds h1 (a, p2 , u(x(a, p2 , w)) = x1 (a, p2 , w) because of duality, we have ∫ p1 +∆p1 CV = h1 (a, p2 , u(x(p1 , p2 , w)))da ∫ ≥

p1 p1 +∆p1

h1 (a, p2 , u(x(a, p2 , w)))da p1 ∫ p1 +∆p1

=

x1 (a, p2 , w)da p1

where the right-hand-side is the change of consumer surplus to be defined below. In Figure 6.7, the compensated demand curve h1 passing through the pair of consumption and price of Good 1, (x1 , p1 ) is above the demand curve X1 . The compensated variation is given as the area surrounded by horizontal lines p1 and p1 + ∆p1 , the compensated demand curve h1 and the vertical axis. This is smaller than the change in consumer surplus, which is given by horizontal lines p1 and p1 + ∆p1 , the demand curve X1 and the vertical axis. On the other hand, we have EV

= I − e(p, u(x(p′ , w))) = e((p1 + ∆p1 , p2 ), u(x(p1 + ∆p1 , p2 , w))) −e((p1 , p2 ), u(x(p1 + ∆p1 , p2 , w)))

Again from Shepard’s lemma we obtain ∫ p1 +∆p1 EV = h1 (a, p2 , u(x(p1 + ∆p1 , p2 , w)))da p1

CHAPTER 6. DEMAND ANALYSIS

84

Good 2 I+CV p2

6

I p2 I+EV p2

x′

r

b rx x er

rx

I p1 +∆p1

I p1

- Good 1

Figure 6.6: Compensated and equivalent variation by price change of Good 1

Since it holds u(x(p1 + ∆p1 , p2 , w)) < u(x(a, p2 , w)) < u(x(p1 , p2 , w)) for all p1 < a < p1 + ∆p1 and it holds h1 (a, p2 , u(x(a, p2 , w)))) = x1 (a, p2 , w) because of duality, we have ∫ p1 +∆p1 EV = h1 (a, p2 , u(x(p1 + ∆p1 , p2 , w)))da p1 p1 +∆p1

∫ ≤

h1 (a, p2 , u(x(a, p2 , w)))da p1 ∫ p1 +∆p1

=

x1 (a, p2 , w)da p1

where the right-hand-side is the change of consumer surplus to be defined below. In Figure 6.7, the compensated demand curve h′1 passing through the pair of consumption and price of Good 1, (x′1 , p1 + ∆p1 ) is below the demand curve X1 . The equivalent variation is given as the area surrounded by horizontal lines p1 and p1 + ∆p1 , the compensated demand curve h′1 and the vertical axis. This is smaller than the change in consumer surplus, which is given by horizontal lines p1 and p1 + ∆p1 , the demand curve X1 and the vertical axis. Now denote the change in consumer surplus by ∆CS, then we have EV ≤ ∆CS ≤ CV. Note that for price drops we obtain the reverse relationship.

6.6.2

Inverse demand and consumer surplus

In order to calculate compensated variation and equivalent variation we need to know the compensated demand function, which is not directly observable

CHAPTER 6. DEMAND ANALYSIS

85

Good 1 price h′1 h1 6 r

p1 + ∆p1

r

p1

x′1

x1

- Good 1

Figure 6.7: Compensated variation, equivalent variation and change in consumer surplus

and we need to know the entire preference in order to know it. We are given only a demand function at best as data in practice, however. Now how can we evaluate welfare change in terms of income given only a demand function. Again, consider the price change of Good 1. Given demand function x(p, w) = (x1 (p, w), x2 (p, w)) fix income w and p2 the price of Good 2, solve the equation x1 = x1 (p1 , p2 , w) for p1 , and denote the solution by p1 = p1 (x1 ) This is called inverse demand function. Plot this on the plane as in Figure 6.8, where we take quantity of Good 1 on the horizontal axis and its price on the vertical axis. For example, Cobb-Douglas preference represented by u(x) = a ln x1 +b ln x2 yields the demand function x1 (p, w) =

w a · , a + b p1

x2 (p, w) =

where the inverse demand function for Good 1 is p1 (x1 ) =

a w · . a + b x1

b w · , a + b p2

CHAPTER 6. DEMAND ANALYSIS

86

Good 1 price 6

- Good 1 Figure 6.8: Inverse demand function

This inverse demand function describes apparent willingness to pay for the given good, that is, the amount of income the consumer is willing to give up in order to have an extra unit of consumption of it. I’ll come back later to explain why it is an ”apparent” one. Let me first illustrate this using integers, then p1 (1) is the willingness to pay for the first unit, p1 (2) is the willingness to pay for the second unit, and so on, and p1 (k) is the willingness to pay for the k-th unit. Given price p1 , the consumer buys an extra unit as far as p1 (k) > p1 , since willingness to pay is greater than the price, and decreases the consumption when p1 (k) < p1 , since willingness to pay is smaller than the price. Thus optimal consumption is x1 such that p1 (x1 ) = p1 . Since the consumer is willing to pay p1 (1) while he has to pay p1 , he has gained p1 (1) − p1 from the first unit of purchase in the net sense. Since the consumer is willing to pay p1 (2) while he has to pay p1 , he has gained p1 (2) − p1 from the first unit of purchase in the net sense. By adding this up to x1 we have an accumulated net gain x1 ∑

p1 (k) − px1

k=1

Now consider making the grid finer rather than taking integers, we have an integral formula ∫ x1 p1 (z)dz − px1 0

This is called consumer surplus, and it describes apparent amount of net gain from trade measured in terms of income. It is the area surrounded by the inverse demand, horizontal line p1 and the vertical axis in Figure 6.9. Now consider form example that the price of Good 1 increase from p1 to p1 +

CHAPTER 6. DEMAND ANALYSIS

87

Good 1 price 6

r

p1

x1

- Good 1

Figure 6.9: Consumer surplus

∆p1 . Then the consumer surplus after the price change is the area surrounded by the inverse demand curve, horizontal line p1 + ∆p1 and the vertical axis in Figure 6.10. Hence the change in consumer surplus due to the price change is the area surrounded by two horizontal lines p1 +∆p1 and p1 , the inverse demand curve and the vertical axis in Figure 6.10. By rotating Figure 6.10 by 90-degrees, we see the integral description of the change in consumer surplus ∫ p1 +∆p1 ∆CS = x1 (r)dr, p1

which is counted as a loss. Now I come back to why this is an ”apparent” description of willingness to pay. Note that the definition of consumer surplus does not distinguish between substitution effect and income effect. Because of this the change in consumer surplus does not in general coincide with either compensated variation or equivalent variation. Thus, for example, on the price increase of Good 1 we have the relationship EV ≤ ∆CS ≤ CV, which means that the change in consumer surplus is too little as a proxy of compensated variation, and too much as a proxy of equivalent variation. More concretely, if the government should compensate the loss due to price increase the compensation is too little if it is substituted by the change in consumer surplus. Also, if the consumer should pay in order to stop the policy leading to the price increase the payment is too much if it is substituted by the change in consumer surplus. These three criteria coincide only under the assumption of no income effect, which I will discuss in the next chapter.

CHAPTER 6. DEMAND ANALYSIS

88

Good 1 price 6 r

p1 + ∆p1

r

p1

x1

- Good 1

Figure 6.10: Change in consumer surplus

6.7

Exercises

Exercise 8 Consider the demand function generated by preference represented √ √ in the form u(x) = a x1 + b x2 x1 (p, I) =

a2 p2 I , + b2 p21

a2 p1 p2

x2 (p, I) =

b2 p1 I + b2 p1 p2

a2 p22

On the demand for Good 1, find is income elasticity, price elasticity and cross price elasticity to Good 1. 2

3

Exercise 9 Consider preference represented by u(x) = x15 x25 . On the price change from p = (p1 , p2 ) to p′ = (p1 +∆p1 , p2 ), find its compensated variation, equivalent variation and the change in consumer surplus.

Chapter 7

Willingness to pay and consumer surplus Cost-benefit analysis will be the heart of introductory micro, and well-known (and probably notorious) even to non-economists. What do we mean by benefit, however, while cost is more or less clear?1

7.1

Naive utility argument

You might have read the following argument in introductory books, or not. In any case, it is what economists once believed seriously, but it does not have any economic content. So I like you to read this thinking of which part is flawed. 1. Denote the utility of q units of consumption by U (q). 2. Given price of a given good p, the consumer’s surplus, which is the net utility obtained by subtracting expenditure pq from the gross utility U (q), is U (q) − pq The consumer maximizes this. 3. At the utility-maximizing point, we have equality of marginal utility and price, U ′ (q) = p. That is, the net utility is maximized when incremental utility from incremental consumption of the given good is equal to the incremental cost of it.

1 I’ll

come to cost in the chapters on production

89

CHAPTER 7. WILLINGNESS TO PAY

90

4. Take q on the horizontal axis and p on the vertical axis, and plot the above maximization point, then we obtain a demand curve p = U ′ (q).

Now, what’s wrong? At least the following points should be considered. 1. In the above argument utility is treated as if it is a monetary evaluation of consumption. However, utility function is no more than a representation of a given preference ranking and has it has no quantitative meaning. How can we measure it in terms of money? 2. In the argument the obtained demand is determined only by the relationship between ”marginal utility” and price. However, in the previous chapters we saw that demands in general depend on income. Where has the income gone?

7.2

The assumption of no income effect

Due to the above difficulties certain some people say that the consumer surplus analysis is dead. From this standpoint, consumer surplus can be taught in introductory courses only as a ”convenience” to motivate learners and it should be forgotten once they go up to higher levels. Nevertheless the concept of benefit (how much the consumer is willing to pay for the given good) and consumer’s surplus (how much he has gained fro trade) is very intuitive. In this chapter I argue how this concept can be defined consistently in the framework of ordinal utility. We should note that the argument below is ”difficult”, however. It is ”difficult,” in the sense that it can be applied only to a very local and specific phase of an economy which can be ”isolated” from the rest, and cannot be extended to the economy as a whole. Therefore modern economics applies the concept of benefit and consumer surplus when we can ”isolate” a local and specific phase and calls it partial equilibrium analysis. On the other hand, directly treating an economy as a whole without doing such isolation is called general equilibrium analysis. How can such ”isolation” work? Market of a good under consideration of partial equilibrium analysis is like a ”boat floating on the ocean.” Since the boat is negligibly small compared to the ocean, one can ignore its effect on the ocean, and can focus on the movement of the boat itself. That is, when the market for the commodity under consideration is very small compared to the entire economy its behavior will not affect the entire economy and the behavior of the markets for ”the other commodities” will be unchanged at least in the approximate sense.

CHAPTER 7. WILLINGNESS TO PAY

91

Good 2 6

x1

x2

- Good 1

r x

Figure 7.1: Consumption space R+ × R

When we read any discourse on economy we should be very careful about whether it is a partial equilibrium argument or not. In reading a partial equilibrium argument, we should ask if the above-noted isolation is successful, or otherwise we cannot ignore ”general equilibrium effects” and have to look at the entire economy ”as a whole.” By the way, after reading the explanations below you might think, ”what’s the difference from the explanation in the beginning?” No, although the ”established form” looks quite the same and everything looks parallel, it is categorically different entirely. I’ll come to this again. Consider the following two-good model. Here Good 1 is the good we are focusing on, and Good 2 is income transfer to be spent over the other goods. This income transfer is allowed to take negative values as well. When it is negative it means that the consumer is decreasing income to be spent over the other goods, that is, paying. Therefore the consumption space is not the negative quadrant like before, but it is the right half of the plain R+ × R. For example, when x = (x1 , x2 ) is given as in Figure 7.1 the consumer is losing |x2 | units of income. The assumption behind this is that the market for Good 1 is very small compared to the entire economy, so that the consumer’s income is sufficiently large compared to Good 1 and taken to be unlimited in the local sense, and only its relative increase or decrease matters. That is, as in Figure 7.2 the consumption space R+ × R is obtained by magnifying the portion where the ”background income” to be allocated to the other commodities is already sufficiently large. Then consumer’s preference is supposed to be quasi-linear in Good 2, the

CHAPTER 7. WILLINGNESS TO PAY

92 Change in income 6

Income 6  

- Good 1

- Good 1 Figure 7.2: Change in income when the background income is sufficiently large

income transfer. Definition 7.1 The preference is said to be quasi-linear if the indifference curves are parallel along the vertical axis. Indifference curves for a quasi-linear preference are depicted as in Figure 7.3. Indifference curves being parallel along the Good-2-axis means that marginal rate of substitution is independent of the quantity of Good 2. That is, the amount of income the consumer is willing to give up in order to get one extra unit of Good 1 is independent of how much income he is holding (see the dotted line in Figure 7.3). It is that there is no income effect on Good 1. When is the no income effect assumption valid? This argument dates back to Marshall, who thought that when the commodity under consideration is negligibly small compared to the entire set of commodities the income effect on it is negligible. For example, if you don’t buy a small thing such one can of cola for 2 dollars it is not because you can’t pay 2 dollars or paying 2 dollars seriously affects consumption of the other goods, but because you judge that it does not deserve 2 dollars. Marshall thought that in such situations in which income effect is negligible consumption is determined only by comparison between willingness to pay and price. 2 2 Of course Marshall did not lead to handle this formally. It was done first by Vives [37], who considered an increasing set of commodities and shows that when the number of commodity tends to be arbitrarily large and income tends to be large as well at the same rate, each single commodity becomes negligibly small compared to the entire set income effect on it converges to zero.

CHAPTER 7. WILLINGNESS TO PAY

93

Good 1 6

- Good 1

Figure 7.3: Quasi-linear preference

The no income effect assumption is important in making welfare judgments. If you mistakenly assume that income effect is negligible while it actually matters, when the consumer’s apparent willingness to pay for the good is for example low you cannot distinguish whether it is because he does not care for it or it is because of the actual income effect. Since the indifference curves are parallel along the Good 2 axis, one can pick one of them and let it represent the whole preference. As in Figure 7.4 pick the difference curve passing through the origin and look at a point on it with Good 1 quantity denoted by x1 . On a given indifference curve if Good 1 quantity is determined it automatically determines the quantity of Good 2, and it is represented as a function of x1 . Let it be denoted by −v(x1 ) Here v(x1 ) is the largest possible amount of Good 2 the consumer is willing to give up in order to get x1 units of Good 1, which meets the condition (x1 , −v(x1 )) ∼ (0, 0) Thus v(x1 ) is understood to be the willingness to pay for x1 units of Good 1. Under the assumption that there is no income effect on Good 1 this willingness to pay does not depend on income and depends only on x1 . When the consumer is indifferent between bundle x = (x1 , x2 ) and another bundle consisting only of Good 2, denoted (0, w2 ), that is, when x ∼ (0, w2 ),

CHAPTER 7. WILLINGNESS TO PAY

94

Good 1 6

x1

- Good 1

−v(x1 )

Figure 7.4: Willingness to pay

let us call w2 the consumer surplus generated by x, in the sense that it is the evaluation of net gain from consumption in terms of income. To find the consumer surplus, draw the indifference curve passing through x = (x1 , x2 ) and look at its intercept with the vertical axis, (0, w2 ), as in Figure 7.5. Since the indifference curves are parallel along the vertical axis we have w2 = v(x1 ) + x2 , that is, it holds x ∼ (0, v(x1 ) + x2 ) Hence the consumer surplus gained from x is v(x1 ) + x2 . Now, pick any x = (x1 , x2 ) and y = (y1 , y2 ). Since x ∼ (0, v(x1 ) + x2 ), y ∼ (0, v(y1 ) + y2 ), we have x ≿ y if and only if x2 + v(x1 ) ≧ y2 + v(y1 ). Thus consumer surplus v(x1 ) + x2 is a representation of the quasi-linear preference. Notice, however, that any monotone transformation of a representation of a given preference is again a representation of the same preference, for arbitrary monotone transformation f the function f (v(x1 )+x2 ) represents the above quasi-linear preference. Thus we obtain the proposition below. Proposition 7.1 When a preference is quasi-linear with respect to Good 2 it is represented in the form u(x) = f (v(x1 ) + x2 ) where f is arbitrary monotone transformation.

CHAPTER 7. WILLINGNESS TO PAY

95

Good 2 6 w2 = v(x1 ) + x2

x2 x1

- Good 1

−v(x1 )

Figure 7.5: Consumer surplus

Notice that while the whole utility representation u has no quantitative meaning since f is arbitrary, the consumer surplus v(x1 ) + x2 inside f has certain quantitative meaning, in that it is interpreted to be a measure of gains from trade in terms of income. While there can be arbitrarily many utility representations for a given preference, the function describing one’s willingness to pay, that is v, is uniquely determined when the preference is quasi-linear in income. This v is a component of utility representation, not a utility representation by itself. To emphasize the distinction let me call v the willingness to pay function for a given consumer. Willingness to pay has economic content under the assumption of quasi-linearity within the framework of partial equilibrium analysis, while the whole utility representation again has no quantitative meaning. Historically, the willingness to pay function was believed to be ”the” utility function and this made people believe that utility function has a quantitative meaning. This is a confusion, however, while nowadays teachers very often make use of this confusion between them in introductory courses for ”educational purpose.”

7.3

Marginal willingness to pay as marginal rate of substitution

Let us look more into the implication of quasi-linearity. Proposition 7.2 When preference is quasi-linear and represented in the form f (v(x1 ) + x2 ), the marginal rate of substitution of Good 2 (income transfer) for

CHAPTER 7. WILLINGNESS TO PAY

96

Good 1 (the good under consideration) at x = (x1 , x2 ) is given by M RS(x) = v ′ (x1 ) Proof. Given representation f (v(x1 ) + x2 ), marginal utility of Good 1 is ∂f (v(x1 ) + x2 ) = f ′ (v(x1 ) + x2 )v ′ (x1 ) ∂x1 On the other hand, marginal utility of Good 2, income transfer to be spent over the other goods, is ∂f (v(x1 ) + x2 ) = f ′ (v(x1 ) + x2 ) ∂x2 Hence the marginal rate of substitution is M RS(x) =

∂f (v(x1 )+x2 ) ∂x1 ∂f (v(x1 )+x2 ) ∂x2

=

f ′ (v(x1 ) + x2 )v ′ (x1 ) = v ′ (x1 ) f ′ (v(x1 ) + x2 )

The above result says two things. 1. Marginal rate of substitution is independent of monotone transformation f. 2. The marginal rate of substitution depends only on x1 and independent of x2 . 1 is the same as before. Generally, marginal rate of substitution is independent of which representation to pick for a given preference. 2 is specific to quasi-linearity. Recall that marginal rate of substitution of Good 1 for Good 2 is the amount of Good 2 one is willing to give up in order to have an extra one unit of Good 1, which is exactly the notion of willingness to pay in the current context. In general, marginal rate of substitution depends on both x1 and x2 , which means willingness to pay is not determined independently of income. Under the assumption of quasi-linearity, however, willingness to pay for Good 1 is independent of the amount of Good 2 held and determined by x1 alone. Thus, v ′ (x1 ) is understood to be the amount of income the consumer is willing to give up in order to get an extra one unit of the good under consideration, that is, marginal willingness to pay. ”What’s the difference from the argument in the beginning of the chapter?” you might say. There is a difference at a categorical level. There may be arbitrarily many utility representations for a given preference, since one can take arbitrary monotone transformation of them. Marginal utility has no quantitative meaning either. On the other hand, willingness to pay and marginal willingness to pay have quantitative meanings under the assumption of quasi-linearity, or equivalently speaking, the assumption of no income effect.

CHAPTER 7. WILLINGNESS TO PAY

7.4

97

No income effect and inverse demand function

Let us now look at consumption choice when preference is quasi-linear. Let p be the relative price of Good 1 for Good 2 (that is income transfer), and I be the income (while income would not matter since no income effect is assumed here). Thus, we consider the maximization problem max f (v(x1 ) + x2 ) x

subject to px1 + x2 = w Because we have x2 = −px1 + w from the budget constraint the above problem is equivalent to max f (v(x1 ) − px1 + w), x1

which is the problem of maximizing consumer surplus since w is just a constant added. As before, apply the tangency condition assuming that the willingness to pay function v is ”smooth.” That is, the consumption choice is determined by the equality between marginal rate of substitution and relative price M RS(x1 , x2 ) = p1 ′ p2 . Because M RS(x1 , x2 ) = v (x1 ) holds under quasi-linearity and we are p1 taking p2 = p as the normalization we have v ′ (x1 ) = p Notice that here the consumption of Good 1 is determined by just one equation independently of income I. Hence by solving this one equation for the one unknown x1 we obtain the demand function for Good 1 x1 (p), which is independent of income holding and depends only on the relative price for income. Let us look more into the tangency condition that marginal rate of substitution is equal to the relative price v ′ (x1 ) = p The meaning is the same as before, but it has a clearer looking because of quasi-linearity. For example, when v ′ (x1 ) > p, because the marginal willingness to pay is greater than the price the consumer will buy more. On the other hand, when v ′ (x1 ) < p,

CHAPTER 7. WILLINGNESS TO PAY

98

Good 2 6

b rx r x′

rx - Good 1

Figure 7.6: Substitution and income effect in quasi-linear preference

because the marginal willingness to pay is smaller than the price he will rather reduce the quantity to buy. In the end the consumption choice of Good 1 will be the point at which the equality holds. Also the tangency condition is saying that the price such that the consumer continues to buy Good 1 up to x1 units is v ′ (x1 ). That is, the marginal willingness to pay is interpreted to be the inverse demand function in which the Good 1 quantity x1 is the independent variable. That is, the inverse function of the demand function x1 (p), which is denoted by p(x1 ), is given by p(x1 ) = v ′ (x1 ). Now let us reconfirm that the income effect on Good 1 is zero. Assuming that the price of Good 2 is normalized to 1, consider that the price of Good 1 goes up from p to p′ as in Figure 7.7, in which the consumption moves from x = (x1 , x2 ) to x′ = (x′1 , x′2 ). Let x b = (b x1 , x b2 ) be the compensated demand under the price after the change which yields the same welfare level as x does. Because the indifference curves are parallel along the vertical axis, x b is precisely in the above of x′ , hence the income effect on Good 1, that is x′1 − x b1 , is zero. Let us go over an example. Suppose that the willingness to pay function is √ v(x1 ) = x1 . Consumption of Good 1 is determined by the tangency condition v ′ (x1 ) = p. Since v ′ (x1 ) = 2√1x1 here, the condition leads to 1 √ =p 2 x1

CHAPTER 7. WILLINGNESS TO PAY

99

Hence the inverse demand function is 1 p(x1 ) = √ , 2 x1 and by solving the for x1 we obtain the demand function x1 (p) =

1 . 4p2

Since consumer surplus is v(x1 (p)) − px1 (p), by plugging the demand function 1 into this we obtain 4p .

7.5

Compensated variation, equivalent variation and consumer surplus

As indicated in the previous chapter, under the assumption of no income effect the three measures of welfare change, change in consumer surplus, compensated variation and equivalent variation, coincide. Go back to the Slutsky equation ∂h1 ∂x1 ∂x1 = + h1 ∂p1 ∂p1 ∂w When income effect is zero, we have

∂x1 ∂w

= 0 and the equation becomes

∂h1 ∂x1 = , ∂p1 ∂p1 which means that the demand curve and compensated demand curve coincide. In Figure 6.7, it means that all three curves collapse into one curve. Hence the three criteria coincide under the assumption of no income effect.. Let me explain this using Figure 7.7. Assuming that the price of Good 2 is normalized to 1, consider that the price of Good 1 goes up from p to p + ∆p. Let x denote the demand before the price change, x′ denote the demand after the price change, x b denote the compensated demand under p + ∆p which yields the same welfare level as x does, and x e denote the compensated demand under p which yields the same welfare level as x′ does. Here the compensated variation is x b2 −x′2 and the equivalent variation is x2 − x e2 , and the change in consumer surplus is the difference between the intercepts of the two indifference curves with the vertical axis. Because the indifference curves are parallel along the vertical axis, these three coincide.

CHAPTER 7. WILLINGNESS TO PAY

100

Good 2 6

b rx r x′

rx r x e

- Good 1

Figure 7.7: Compensated and equivalent variation in quasi-linear preference

Chapter 8

Intertemporal choice 8.1

Intertemporal choice and intertemporal budget constraint

Let me repeat that even if goods are materially the same they are treated as different goods if they are to be consumed at different time periods. Here saving is understood to be selling current consumption and buying future consumption, and borrowing is understood to be buying current consumption and selling future consumption. In the book I describe this by the two-period model. For simplicity let us assume that there are just two periods and there is just one consumption good available at each period. This is enough for our purpose while it can be extended so as to allow many periods and many good at each period. Thus the consumption set for a given consumer is R2+ . Its element, typically denoted by x = (x1 , x2 ) is called a consumption stream, where x1 refers to consumption at Period 1 and x2 refers to consumption at Period 2. In the two-period model one’s initial endowment is interpreted to be his earning stream. That is, when a consumer has his initial endowment e = (e1 , e2 ) it means that he earns e1 units of consumption good at Period 1 and e2 units of consumption good at Period 2. Let r denote pure interest rate. Assume there is no inflation, as I will come to it in the next section. Now, suppose one consumes x1 units in this period then he is saving e1 − x1 units of consumption good available today. This can be negative, as he is borrowing. Return from saving to be received in the next period is obtained by multiplying gross interest rate 1 + r to e1 − x1 , that is, (1 + r)(e1 − x1 ). The disposable income in the next period is obtained by adding this (1 + r)(e1 − x1 ) to the earning in the next period e2 , that is, e2 + (1 + r)(e1 − x1 ). Therefore consumption in the next period, denoted x2 obeys the constraint x2 ≦ e2 + (1 + r)(e1 − x1 ). 101

CHAPTER 8. INTERTEMPORAL CHOICE

102

By rearranging this we obtain (1 + r)x1 + x2 ≦ (1 + r)e1 + e2 , in which the right-hand-side is the amount of consumption in the future period which you can get when you consume all the earning in the current period. Since it is the lifetime income measure in terms of consumption in the future, it is called the future value of lifetime income. Notice that this lifetime budget constraint is a special case of the standard form of budget constraint, where we take future consumption to be the numeraire and take p1 = 1 + r and p2 = 1. On the other hand, by dividing both sides in the above formula by gross interest rate 1 + r we obtain x1 +

1 1 x2 ≦ e1 + e2 . 1+r 1+r

Here the right-hand-side is the amount of consumption good in the current period which you can get when you borrow against all of your future earnings and spend all your lifetime income on current consumption. Since it is the lifetime income measured in terms of current consumption it is called the present value of lifetime income. Notice again that this is a special case of the standard form of budget constraint, in which we take current consumption to be 1 numeraire and take p1 = 1 and p2 = 1+r . Future value of lifetime income corresponds to the x2 -intercept of the budget line and present value of lifetime income corresponds to the x1 -intercept. In any case the slope of budget line is pp12 = 1 + r in its absolute value, hence gross interest rate is the relative price of current consumption for future consumption. That is, as you increase one unit of current consumption you have to give up 1 + r units of future consumption. In other words, it is the opportunity cost of extra one unit of current consumption which is measured in terms of future consumption.

8.2

How to deal with inflation

Now let us see how the intertemporal budget constraint looks like under inflation. Denote the inflation rate by π, and normalize the price level of consumption good in the current period equal to 1, then the price level of consumption good in the future period is 1 + π. Here r is now nominal interest rate. Notice that under inflation there is difference between nominal and real. Suppose you consume x1 in the current period then you save e1 − x1 in terms of current consumption good. Then return from saving is obtained by multiplying gross interest rate 1 + r to saving e1 − x1 , that is, (1 + r)(e1 − x1 ). Since saving is made in terms of current consumption good it is not inflated in the future period. On the other hand, as you earn e2 units of consumption good in the future period its value is inflated and (1 + π)e2 is counted into the disposable income in the future period. Summing up the disposable income in

CHAPTER 8. INTERTEMPORAL CHOICE

103

the future period is (1 + π)e2 + (1 + r)(e1 − x1 ). Expenditure on consumption x2 in the future period is inflated as well and it is (1 + π)x2 . Thus we have the constraint (1 + π)x2 ≦ (1 + π)e2 + (1 + r)(e1 − x1 ) Divide both sides by 1 + π then we obtain x 2 ≦ e2 +

1+r (e1 − x1 ), 1+π

Now consider

1+r 1+π to be the real gross interest rate then actually nothing changes. That is, define real (pure) interest rate ρ by 1+r =1+ρ 1+π Then the budget constraint is rewritten into the form x2 ≦ e2 + (1 + ρ)(e1 − x1 ) and we can rewrite this into the future value form of budget constraint (1 + ρ)x1 + x2 ≦ (1 + ρ)e1 + e2 or into the present value form of budget constraint x1 +

1 1 x2 ≦ e1 + e2 . 1+ρ 1+ρ

Therefore, when there is inflation intertemporal budget constraint is determined by real interest rate. Since our consumer is assumed to be ”rational” throughout the book he cares only about ”real.” Thus, unless mentioned specifically the interest rate r is hereafter taken to be the real one, which is already adjusted to inflation.

8.3

Discounted present value of streams

Here let us look more into discounted present value. Consider an asset which yields e1 units of consumption good in the current period and e2 units of consumption good in the future period. Given that the real interest rate is r, how much of consumption good in the current period is it worth? The answer is its present discounted value e1 +

1 e2 . 1+r

CHAPTER 8. INTERTEMPORAL CHOICE

104

1 That is, holding this asset and holding e1 + 1+r e2 units of consumption good in the current period are equivalent. This is because if you take e1 out of 1 1 e1 + 1+r e2 units of consumption good in the current period and save 1+r e2 1 then in the future period you receive (1 + r) · 1+r e2 = e2 units of consumption good. Thus you can mimic the same consumption stream generated by the this asset. Suppose the relative price of this asset for the consumption good in the current period, denoted P , is smaller than its discounted present value, that is,

P < e1 +

1 e2 . 1+r

Then what happens? Consider that a consumer borrows P units of consumption good at the market interest rate and buys this asset for P , then he can generate ”something from nothing,” as he can generate a consumption stream (x1 , x2 ) satisfying 1 1 x1 + x2 ≦ e1 + e2 − P 1+r 1+r without having anything in the beginning. Conversely, suppose the relative price of this asset for the consumption good in the current period is larger than its discounted present value, that is, P > e1 +

1 e2 . 1+r

The one can get this asset from somebody telling that he pays e1 in the current period and e2 in the future period, and sells it for P , then he can generate ”something from nothing” again, as he can generate a consumption stream (x1 , x2 ) satisfying ( ) 1 1 x1 + x2 ≦ P − e1 + e2 . 1+r 1+r Such action of generating ”something from nothing” is called arbitrage. When there is an arbitrage opportunity anybody will try to exploit it unless he is ”stupid.” In the first case then there will be an excess demand for the asset and the asset price will go up, and in the second there will be an excess supply of it and its price will do down. In the end the arbitrage opportunity will disappear and there will be no free lunch, and there the asset price will be equal to the discounted present value of its earning stream. 1 We can extend the discounted present value formula to three or more periods. Given that real interest rate per period is r the discounted present value of stream (e1 , e2 , e3 , · · · , eT ) is ( )2 )T −1 )t−1 ( ( T ∑ 1 1 1 1 e1 + e2 + e3 + · · · + eT = et , 1+r 1+r 1+r 1+r t=1 1 Practically, there may be may people who are ”stupid,” or even when they recognize the arbitrage opportunity they may not be able to exploit because of barriers such as transaction fees. If that is the case arbitrage opportunities may not disappear.

CHAPTER 8. INTERTEMPORAL CHOICE

105

where the receipt/payment one period later is discounted once, the receipt/payment two periods later is discounted twice, and so on, ....., and the receipt/payment T − 1 periods later is discounted T − 1 times. W can think of T being infinity, then the discounted present value of stream (e1 , e2 , e3 , · · · ) is e1 +

1 e2 + 1+r

(

1 1+r

)2 e3 + · · · =

∞ ∑ t=1

( et

1 1+r

)t−1 .

Of course we die on some day, but we are not sure exactly when we die. In order to think of such open-ended situation, infinity is a good ”approximation.” Also, if we think of monthly or weekly consumption-savings let’s say 30 years is an extremely long period and we can take it to be essentially infinite. Also we can consider that the interest rate varies over time, while it has been assume to be constant in th above. Let r1 denote the interest rate between Period 1 and Period 2, r2 denote that between Period 2 and Period 3, and so on, then the discounted present value of stream (e1 , e2 , e3 , · · · ) is given by ∞ t−1 ∑ ∏ 1 1 1 1 e1 + e2 + · e3 + · · · = , et 1 + r1 1 + r1 1 + r2 1 + rk t=1 k=0

where r0 = 0.

8.4

Preference over consumption streams

8.4.1

Impatience and intertemporal substitution

Let me introduce two appealing properties of preference over consumption streams. Consider the following problem. Which one do you like better, (10, 2) or (2, 10)? For serious experimental supports you may consult a book on experimental economics such as ....., here most readers will choose (10, 2). That is, if you have sacrifice either of current and future consumptions you will choose to sacrifice the latter. In other words, people discount evaluation of future consumptions compared the current ones. This is called impatience. In other words, in order to make consumers give up current consumptions by compensating by means of future consumptions you need to pay certain premium. Here is another problem. Which one do you like better, (10, 2) or (6, 6)? Now most readers will choose (6, 6) (for serious experimental supports see for example .....). That is, you will like to smooth your consumption stream by reducing fluctuations over time . In other words, there is certain level of complementarity between current and future consumptions.

CHAPTER 8. INTERTEMPORAL CHOICE

8.4.2

106

Discounted utility representation

In order to handle the above two properties in an operational manner, we restrict attention to a class of preferences with more specific form. Definition 8.1 Preference is said to have discounted utility representation if it is represented in the form u(x) = f (v(x1 ) + βv(x2 )), where I would call v a periodwise evaluation function and β > 0 is a number called discount factor, and f is any monotone transformation. Recall that as before the whole utility representation u(x) is ordinal and has no quantitative meanings, and that f can be any monotone transformation. It is usual in practice, however, to ignore f and simply write u(x) = v(x1 ) + βv(x2 ). There is a problem in how we should call function v. Some people simply call it ”utility function,” but this is inconsistent with out standpoint of ordinal utility, since in our definition utility function is no more than a representation of ranking and has no quantitative meaning, and any monotone transformation of it represents the same ranking, while here the function v has some quantitative meanings as explained below. Some people call it ”period utility function,” but I still like to avoid it for the same reason. The following result shows the role of periodwise evaluation function. Proposition 8.1 Preference represented in the discounted utility form with periodwise evaluation function v and discount factor β induces marginal rate of substitution at x = (x1 , x2 ) in the form M RS(x) =

v ′ (x1 ) . βv ′ (x2 )

Proof. Given a discounted utility representation f (v(x1 ) + βv(x2 )), where f can be any monotone transformation, marginal utility of Good 1 (consumption in Period 1) is ∂f (v(x1 ) + βv(x2 )) = f ′ (v(x1 ) + βv(x2 ))v ′ (x1 ) ∂x1 On the other hand, marginal utility of Good 2 (consumption in Period 2) is ∂f (v(x1 ) + βv(x2 )) = f ′ (v(x1 ) + βv(x2 ))βv ′ (x2 ) ∂x2 Therefore the marginal rate of substitution of Good 2 for Good 1 is ∂f (v(x1 )+βv(x2 )) ∂x1 ∂f (v(x1 )+βv(x2 )) ∂x1

=

f ′ (v(x1 ) + βv(x2 ))v ′ (x1 ) v ′ (x1 ) = . ′ ′ f (v(x1 ) + βv(x2 ))βv (x2 ) βv ′ (x2 )

CHAPTER 8. INTERTEMPORAL CHOICE

107

Thus, while the whole representation u has no quantitative meaning the function v and discount factor β have quantitative meanings, as far as we restrict attention to the class of discounted utility representations, as discussed in Section 4.4. The function v explains how the consumer is willing to substitute between consumptions at different periods, by means of its ”curvature.” As the graph of v is ”curvier” the corresponding indifference curves over consumption streams are more convex toward the origin, which means that the consumer dislikes fluctuation of consumptions over time, that is, substitutability between periods is smaller. As the graph of v is ”more straight” the corresponding indifference curves over consumption streams are more straight, which means that the consumer tends to accept fluctuation of consumptions over time, that is, substitutability between periods is higher. Consider two consumers with preferences represented in discounted utility forms, one’s periodwise evaluation function is v(z) = z and the other’s periodwise evaluation function is vb(z) = ln z, where they have the same discount factor β. First one’s preference is represented by u(x) = x1 + βx2 , which exhibits perfect substitution between consumptions at different periods. On the other hand, second one’s preference is represented by u b(x) = ln x1 + β ln x2 , which is a Cobb-Douglas preference. See Figure 8.1 and consider (c, c), the consumption stream in which one consumes c in both periods. From the above result, as far as preference allows discounted utility representation the marginal rate of substitution at (c, c) is v ′ (c) 1 M RS(c, c) = βv ′ (c) = β . Hence both indifference curves have the same slope at (c, c). However, since the first one exhibits perfect substitution the consumer stays indifferent after he increases ∆c units of current consumption and decreases β1 ∆c units of future consumption. Thus he is willing to accept fluctuation of consumption over time. On the other hand, since the second one exhibits an indifference curve which is bending across the 45-degree line, and therefore dislikes fluctuation. While taking any monotone transformation of the whole representation u does not change the ranking it represents, take monotone transformation of v (log transformation in the above example) in general changes the preference and indifference curves it describes. Hence v has certain quantitative meanings and it is cardinal, in a limited sense as explained below. Unfortunately, it is an accepted terminology to call v ”utility function” despite the difference that u is ordinal but v is cardinal within the class of discounted utility representations. We should keep this in mind as we proceed.

CHAPTER 8. INTERTEMPORAL CHOICE

108

Period 2 6

r ∆c

c

−∆c/β ? - Period 1

c

Figure 8.1: Intertemporal substitution

We should note, however, the periodwise evaluation function has quantitative meanings only in its ”curvature”, not in its absolute level or scale. Hence it has no meaning like ”how much the consumer is happy or happier.” This thing is the same as before. While curvature of a function may change by taking monotone transformation in general, it is invariant under taking any positive affine transformation (adding constants, multiplying positive constants). Therefore, whenever you take any positive affine transformation of a given periodwise evaluation function it describes the preference in the discounted utility form. For example, when transform v(z) = ln z into ve(z) = a ln z + b it does not change the series of indifference curves. This will be clear from ve(x1 ) + βe v (x2 ) = (a ln x1 + b) + β(a ln x2 + b) = =

a(ln x1 + β ln x2 ) + b a(v(x1 ) + βv(x2 )) + b

In general we have the following proposition Proposition 8.2 Suppose v is a periodwise evaluation function which describes some preference in the discounted utility form together with some discount factor β. Then for ant constants a, b with a > 0, av + b is also a periodwise evaluation function which describes the same preference in the discounted utility form together with the discount factor β. Proof. Since v is a periodwise evaluation function which describes the given preference in the discounted utility form together with some discount factor β, for any monotone transformation f the function f (v(x1 ) + βv(x2 ))

CHAPTER 8. INTERTEMPORAL CHOICE

109

is a representation of the given preference. Then the right-hand-side of f (av(x2 ) + b + β(av(x2 ) + b)) = f (a(v(x1 ) + βv(x2 )) + b) also represents the same preference because f (av +b) is a monotone transformation. Hence the discounted utility form in the left-hand-side above represents the same preference. Discount factor is the weight on future consumption, which means it is larger as the given consumer is more patient. It will be realistic to assume that the discount factor is between 0 and 1, though, since most consumers will not put more weight on future consumptions than on current consumptions.

8.4.3

Extension to many periods

We can extend the argument on discounted utility preferences to many-period cases. Suppose there are T periods then the consumption space is RT+ , which consists of consumption streams over T periods, say denoted by x = (x1 , x2 , · · · , xT ). Let ≿ denote the preference over T -period consumption streams, which falls in the extended class of discounted utility preferences. Then it is represented by u : RT+ → R, that is, we have the relation x ≿ y ⇐⇒ u(x) ≧ u(y) for all x, y ∈ RT+ , where u has the form ( T ) ∑ t−1 u(x) = f β v(xt ) . t=1

We can even think of T being infinite. Of course we die on some day, but we are not sure exactly when we die. In order to think of such open-ended situation, infinity is a good ”approximation.” Also, if we think of monthly or weekly consumption-savings let’s say 30 years is an extremely long period and we can take it to be essentially infinite. Problem of time consistency Potential problem in the many-period extension is consistency of intertemporal choice. In the two-period model the consumer’s saving decision is made just once, where the saving decision in Period 1 automatically determines the level of consumption in Period 2 basically. 2 However, when there are three periods or more, there consumer faces tradeoffs not only between ”today and tomorrow,” but also between ”tomorrow and the day after tomorrow,” and so on, and he faces the problem of ”reoptimization.” To illustrate, let me give one stupid example. 2 This may leave a problem of ”reoptimization” in Period 2 about which good to consume more or less, but we are rather focusing on intertemporal trade-offs between consumption levels at different periods.

CHAPTER 8. INTERTEMPORAL CHOICE

110

Example 8.1 (Cake-eating problem): There is one unit of cake. It is good to eat for three days. Consider here a preference saying ”I like to eat twice just today, but like to eat equal amounts per day after that.” Such individual will make a consumption plan (0.5, 0.25, 0.25), saying ”I eat 0.5 today, 0.25 tomorrow and 0.25 the day after tomorrow.” Now, suppose he ate 0.5 on the first day and the second day comes, how does he eat the remaining 0.5 unit of cake then? As he has a particular kind of impatience that he likes to eat twice on the current day, it will be natural to assume that he has the same kind of impatience on the second day as well. Then the amount he eats on the second day is now more than 0.25, which violates the ex-ante plan made on the first day. The example will be sharper when it consists of four periods. Then the consumption plan made in the first period will be (0.4, 0.2, 0.2, 0.2). However, once the individual consumes 0.4 on the first day, he will reoptimize on the second day how to eat the remaining 0.6 units of cake, and he leads to a new consumption plan (0.3, 0.15, 0.15), which is against the ex-ante plan made on the first day. Preference, or more precisely a time-series of preferences, is said to be timeinconsistent if ”future selves” do not follow the ex-ante plan made by the ”initial self.” Economics models consumer’s intertemporal consumption path as a solution to a ”planning problem.” When the consumer has time-inconsistency problems in his ”selves,” however, we cannot model his choice simply as a solution to a planning problem, as we discussed in Chapter 1. Throughout the book we mostly assume that any time series of preferences is time-consistent. To formalize the notion of time consistency, let me start with saying that the individual’s self Period 1, self at Period 2, self at Period 3,..... are ”potentially” different, and formalize that there is no inconsistency between them. Denote the individual’s preference at Period 1 by ≿1 , which is defined over consumption streams from Period 1 to Period T . That is, when the self at Period 1 ranks consumption stream (x1 , x2 , · · · , xT ) over consumption stream (y1 , y2 , · · · , yT ), we write (x1 , x2 , · · · , xT ) ≿1 (y1 , y2 , · · · , yT ). Likewise, denote the individual’s preference at Period 2 by ≿2 , which is defined over consumption streams from Period 2 to Period T . That is, when the self at Period 2 ranks consumption stream (x2 , x3 , · · · , xT ) over consumption stream (y2 , y3 , · · · , yT ), we write (x2 , x3 , · · · , xT ) ≿2 (y2 , y3 , · · · , yT ). In general, denote the individual’s preference at Period t by ≿t , which is defined over consumption streams from Period t to Period T . That is, when the self at Period t ranks consumption stream (xt , xt+1 , · · · , xT ) over consumption stream (yt , yt+1 , · · · , yT ), we write (xt , xt+1 , · · · , xT ) ≿t (yt , yt+1 , · · · , yT ).

CHAPTER 8. INTERTEMPORAL CHOICE

111

Thus we have a sequence of preferences (≿1 , ≿2 , · · · , ≿T ). Now time-consistency means that decision made one’s self at a previous period is not overturned by himself in a later period, that is, ”he does not change his mind.” In the relationship between his self at Period 1 and self at Period 2, it says (c1 , x2 , · · · , xT ) ≿1 (c1 , y2 , · · · , yT ) ⇐⇒ (x2 , · · · , xT ) ≿2 (y2 , · · · , yT ) holds for any c1 , x2 , · · · , xT , y2 , · · · , yT . Here the left-hand-side in the above condition says that from the viewpoint of self at Period 1 a consumption plan stating at Period 2 denoted by (x2 , · · · , xT ) is at least as good as another such one (y2 , · · · , yT ), given that the current consumption is c1 . The right-hand-side condition says that from the view point of self at Period 2 now the consumption plan starting at that period (x2 , · · · , xT ) is at least as good as such one (y2 , · · · , yT ). The meaning of this will be clearer when you think of its violation, let’s say (c1 , x2 , · · · , xT ) ≿1 (c1 , y2 , · · · , yT ) and (x2 , · · · , xT ) ≺2 (y2 , · · · , yT ). Then even when self at Period 1 decides to follow the plan (x2 , · · · , xT ) from the next period, self at Period 2 does not follow it since another plan (y2 , · · · , yT ) is better for him. Now let me finish the general definition. Definition 8.2 A sequence of preferences (≿1 , ≿2 , · · · , ≿T ) is said to be timeconsistent if for all t = 1, 2, · · · , T − 1 and ct , xt+1 , · · · , xT , yt+1 , · · · , yT it holds (ct , xt+1 , · · · , xT ) ≿t (ct , yt+1 , · · · , yT ) ⇐⇒ (xt+1 , · · · , xT ) ≿t+1 (yt+1 , · · · , yT ) A class of intertemporal preferences which is standard in practice takes the following form (after ignoring monotone transformation without loss of gener-

CHAPTER 8. INTERTEMPORAL CHOICE

112

ality), u1 (x1 , · · · , xT ) =

T ∑

v(xτ )β τ −1

τ =1

u2 (x2 , · · · , xT ) =

T ∑

v(xτ )β τ −2

τ =2

.. . ut (xt , · · · , xT )

=

uT (xT )

.. . =

T ∑

vi (xτ )β τ −t

τ =t

v(xT )

To illustrate, let me write down the preference at Period 1 without using the summation symbol, u1 (x1 , · · · , xT ) = v(x1 ) + v(x2 )β + v(x3 )β 2 + · · · + v(xT )β T −1 Here we see that we evaluation consumption at each period by the same function v and take the sum of them after discounting by β, where we don’t discount the current consumption, discount once for the next period, discount twice for two periods later, and so on. Since we use the same function v and discount factor β over time we call this a series of stationary discounted utility preferences. Let us confirm that any series of stationary discounted utility preferences is time-consistent. Suppose it holds at Period 1 that (c1 , x2 , · · · , xT ) ≿1 (c1 , y2 , · · · , yT ). Under stationary discounted utility preference this is equivalent to



v(c1 ) + v(x2 )β + v(x3 )β 2 + · · · + v(xT )β T −1 v(c1 ) + v(y2 )β + v(y3 )β 2 + · · · + v(yT )β T −1

Notice that this is equivalent to v(x2 ) + v(x3 )β + · · · + v(xT )β T −2 ≥

v(y2 ) + v(y3 )β + · · · + v(yT )β T −2 .

Under stationary discounted utility preference this is equivalent to (x2 , · · · , xT ) ≿2 (y2 , · · · , yT ) In general, suppose it holds at Period t that (ct , xt+1 , · · · , xT ) ≿t (ct , yt+1 , · · · , yT ).

CHAPTER 8. INTERTEMPORAL CHOICE

113

Under stationary discounted utility preference this is equivalent to v(ct ) + v(xt+1 )β + v(xt+2 )β 2 + · · · + v(xT )β T −t v(ct ) + v(yt+1 )β + v(yt+2 )β 2 + · · · + v(yT )β T −t



Notice that this is equivalent to v(xt+1 ) + v(xt+2 )β + · · · + v(xT )β T −t−1 ≥

v(yt+1 ) + v(yt+2 )β + · · · + v(yT )β T −t−1 .

Under stationary discounted utility preference this is equivalent to (xt+1 , · · · , xT ) ≿t+1 (yt+1 , · · · , yT ). We should notice that this is not the only class of series of preferences which are time-consistent. For example, a series of preferences like • Period 1: I like to eat twice today, and like to eat equal amounts everyday from tomorrow. • Every period after Period 1: I like to eat equal amounts everyday. is time-consistent. However, this example is very unlikely empirically, because if one has the above kind of impatience in the first period it is natural to say that he has the same type of impatience in future periods as well, rather than such impatience disappears from the next period. Under the assumption that a given individual has the same type of impatience every period, a time-consistent class of series of preferences is limited basically to the class of stationary discounted utility preferences.3

8.5

Intertemporal consumption choice

Now let us look into intertemporal consumption choice made by a consumer with discounted utility preference. The consumer takes market interest rate r as given, and obeys the intertemporal budget constraint. Here let me use the present-value form x1 +

x2 e2 ≦ e1 + . 1+r 1+r

As before, let us focus on smooth consumption choice which is determined by the tangency condition. 3 Here the word ”basically” suggests that an additional assumption is needed. It is about the property that the stationary discounted utility form is written the form of ”summation,” but I omit this since it is quite a technical argument.

CHAPTER 8. INTERTEMPORAL CHOICE

114

Recall that the general form of tangency condition is M RS(x) = pp12 . Since the relative price of current consumption for future consumption is the gross interest rate here we have pp12 = 1 + r. Hence the tangency condition is now M RS(x) =

v ′ (x1 ) =1+r βv ′ (x2 )

Combine this with the intertemporal budget equation x1 +

e2 x2 = e1 + 1+r 1+r

then we can obtain the demand for current consumption and future consumption respectively, x1 (r), x2 (r) Thus, saving (or borrowing) in the current period is given e1 − x1 (r)

Lets us go over an example. Assume that the periodwise evaluation function is v(z) = ln z. Then the discounted utility representation is u(x) = ln x1 + β ln x2 , which describes Cobb-Douglas preference. Hence the marginal rate of substitux2 tion is βx and the tangency condition is 1 x2 = 1 + r. βx1 Solve the above for x2 , then we obtain x2 = (1+r)βx1 Plug this into the budget equation and solve for x1 , and again plug it into the relationship between x2 and x1 in order to obtain x2 , then we get ( ) 1 1 β x1 (r) = e1 + e2 , x2 (r) = ((1 + r)e1 + e2 ) 1+β 1+r 1+β and saving (or borrowing) is 1 1 β e1 − · e2 . 1+β 1+β 1+r We can see the following properties from the above result, while they can be seen more generally, 1. As interest rate r is higher (lower), assuming the other elements stay the same, the saving is larger (smaller).

CHAPTER 8. INTERTEMPORAL CHOICE

115

2. As current earning e1 is larger(smaller), assuming the other elements stay the same, the saving is larger (smaller). 3. As future earning e2 is larger (smaller), assuming the other elements stay the same, the saving is smaller (larger). 4. Compare between individuals with the same earning streams and the same periodwise evaluation function. Then one with higher (lower) β saves more (less).

8.6

Exercises

Exercise 10 (i) When the pure interest rate is 4%, what is the present value of an asset which yields 300 in the current period and 400 in the next period? (ii) When the pure interest rate is 10%, what is the present value of an asset which yields 200 in the current period, 300 in the next period and 500 two periods later? (iii) When the pure interest rate is 6%, what is the present value of an asset which yields 300 every period? Exercise 11 Consider a two-period consumption-saving problem. Consumer’s preference is represented by u(x) = ln x1 + 0.95 ln x2 , his earning stream is (40, 30) and the pure interest rate is 4%. How much does he save in Period 1?

Chapter 9

Choice under risk 9.1

Risk and uncertainty

Uncertainty in a broad sense can be classified into two specific notions. Risk refers to situations in which probability distributions over outcomes are given as objects, such as in coin flipping or throwing a die, or as in game theory such that players make choice over probability distributions over strategies. On the other hand, uncertainty in the genuine sense refers to situations in which decision makers are not given such objects and have to have some subjective beliefs by them selves. We will focus on risk in this book.

9.2

Risk attitude

When you make choice under risk, what do you care about? The first thing will be the expected value of return. But is that all? Let us think of the following examples. Example 9.1 Consider two options. One is a bet such that you flip a coin and you 100 dollars if you have head and nothing if you have tail. The other is to receive 50 dollars for sure. If you care only about expected return you will be indifferent between the two, and strictly prefer the first one if the sure amount in the second is slightly less, such as 49. This is not realistic, however. Example 9.2 (St. Petersburg paradox): You can flip a coin repeatedly until you have tail. If you have head k times before having tail, you receive 2k dollars. What is the expected return of this gamble? Since the probability that you have head in the first k flips and tail in the

116

CHAPTER 9. CHOICE UNDER RISK k + 1-th flip is

( 1 )k 2

·

1 2

=

( 1 )k+1 2

, the expected value of the return is

∞ ( )k+1 ∑ 1 k=0

2

117

k

2 =

∞ ∑ 1 k=0

2

=∞

If you care only about expected value of returns you are willing to pay arbitrary amount in order to attend this gamble, let’s say one million. But this is not realistic. The above example suggest that we need to take decision makers’ risk attitudes into account. Risk attitudes can be classified into three types, although we can think of mixtures of them. 1. Risk-averse: ”If I can receive a sure amount which is equal to the expected return of a given gamble, I will take the sure receipt.” 2. Risk-loving: ”If I can receive a sure amount which is equal to the expected return of a given gamble, I will rather take the gamble.” 3. Risk-neutral: ”I care only about expected returns.”

9.3

Expected utility representation: an experimental construction

To capture risk attitudes quantitatively, let us do the following operations. To simplify the illustration let us assume that possible returns are from 0 to 100. Our task is to assign a number to arbitrary number x which is between 0 and 100. Let v(x) be the number to be assigned to x. That is, we are going to construct a function v defined over numbers between 0 and 100. First let us set v(0) = 0 and v(100) = 1. This is just for a normalization purpose and whatever is actually fine, which I will explain later. Now consider a bet (100; 0.5, 0; 0.5), which means you receive 100 with probability 0.5 and 0 with probability 0.5. If the decision maker is risk neutral he is already indifferent between (100; 0.5, 0; 0.5) and sure receipt of 50. Let us suppose, however, that the decision maker is risk-averse and prefers the sure 50 to the bet. Then you can ask, ”what if the sure receipt is 49?” ”what about 48?” ”what about 47?” and so on. At some point the decision maker is indifferent between the sure receipt and the bet, and switches to take the bet if the sure receipt is less than that. Such value of sure receipt is called the certainty equivalent of the bet for him. While the certainty equivalent of (100; 0.5, 0; 0.5) for a risk-neutral decision maker is the expected return 50, since we are talking about a risk averse decision maker the certainty equivalent of this bet for him is below 50. To illustrate let’s say that it is 25. In order to be consistent with the value of certainty equivalent, we assign v(25) so that v(25) = 0.5v(100) + 0.5v(0) = 0.5.

CHAPTER 9. CHOICE UNDER RISK

118

16

0.6 0.5

25 36

50 60

100

Figure 9.1: vNM index

Now what about another bet (100; 0.6, 0; 0.4)? Again if the decision maker is risk-neutral the certainty equivalent of this bet for hims is the expected return which is 60. However, again because we are talking about a risk-averse decision maker, we say the certainty equivalent is lower than 60, say 36. In order to be consistent with the value of certainty equivalent, we assign v(36) so that v(36) = 0.6v(100) + 0.4v(0) = 0.6. By repeating this argument in the end we obtain a function v : [0, 100] → [0, 1] which exhibits a graph like Figure 9.1. Let us call this function von-Neumann/Morgenstern index (vNM index), after the names of the founders of the theory. When the decision maker is risk-averse the vNM index obtained exhibits a graph which is convex to the top. On the other hand, if the decision maker is risk-neutral the certainty equivalent of any bet is equal to its expected return and the obtained vNM index is a straight line (it corresponds to the dotted line connecting (0, 0) and (100, 1)). Once we obtain the vNM index of a given decision maker, we can find the certainty equivalent of any bet for him. For example, the certainty equivalent of (25; 0.5, 81; 0.5) is z such that v(z) = 0.5v(25) + 0.5v(81). In order √ to fit the above numerical example one can take the vNM index v(z) = z/10 (actually I made the numerical examples so that it is the case, in order to make the explanation simpler...).√Hence the certainty equivalent of √ √ bet (25; 0.5, 81; 0.5) is given by z such that z/10 = 0.5 25/10 + 0.5 81/10, and we obtain z = 49 by solving it. Note that the expected return of the bet is 53 on the other hand. √ Now, when the vNM is v(z) = z/10 as we obtain as an example, what is the certainty equivalent of the St. Petersburg gamble? This is described by the

CHAPTER 9. CHOICE UNDER RISK equation





∑ z = 10

k=0

119

( )k+1 √ k 1 2 . 2 10

In contrast to the previous case, the right -hand-side of the above equation is finite, since √ )k ∞ ( ∞ ( )k+1 √ k ∑ 1 ∑ 1 1 2 2+ 2 √ = . = 2 10 20 20 2 k=0

Hence the certainty equivalent is z = solves the paradox.

9.4

k=0 √ 3+2 2 , 2

which is a finite number. Thus it

Expected utility representation: the formulation

Expected utility represents the decision maker’s preference over bets. Thus we first need to formalize ”bets.” A bet is given as a probability distribution over outcomes, which is called a lottery. Let Z be the set of outcomes. In typical application we have Z = R+ which consists of possible values of receipts. A lottery is a list of possible outcomes and their probabilities. For example, a lottery denoted by p = (x1 ; p1 , x2 , p2 , · · · , xn ; pn ) gives outcome xk ∈ Z with probability pk for each k = 1, · · · , n, where n is the number of possible outcomes in lottery Since lottery is a probability distribution over outcomes we have ∑p. n to have k=1 pk = 1. Let ∆(Z) denote the set of such lotteries over Z. Note that we can mix two lotteries in ∆(Z) and make a so-called compound lottery again as an element of ∆(Z). For example, given p = (x1 ; p1 , · · · , xn , pn ) and q = (y1 ; q1 , · · · , ym ; qm ), consider a lottery which gives ”p with probability λ で p and q with probability 1 − λ.” In this compound lottery the probability of outcome xk is λpk for each k = 1, · · · , n, and the probability of outcome yj is (1 − λ)qj for each j = 1, · · · , m. Hence the compound lotter is given by λp + (1 − λ)q = (x1 ; λp1 , · · · , xn ; λpn , y1 ; (1 − λ)q1 , · · · , ym ; (1 − λ)qm ), where duplicated outcomes are suitably recounted. For example, the compound lottery made of p = (A; 0.5, B; 0.3, C; 0.2) and q = (C; 0.7, D; 0.3) with proportion 0.4 : 0.6 is 0.4p + 0.6q = (A; 0.2, B; 0.12, C; 0.5, D; 0.18)

The decision maker has preference ≿ over ∆(Z). For example, the relation p≿q

CHAPTER 9. CHOICE UNDER RISK

120

for p, q ∈ ∆(Z) states that lottery p is at least as good as q. A function u : ∆(Z) → R is said to represent ≿ if it holds p ≿ q ⇐⇒ u(p) ≿ u(q). We restrict attention to a class of preferences which allow representation with the following structure, not just general representations. Definition 9.1 Say that preference over lottery ≿ allows expected utility representation if there a vNM index v : Z → R such that the preference is represented in the form ( n ) ∑ u(p) = f v(xk )pk k=1

for any p = (x1 ; p1 , · · · , xn ; pn ), where f is an arbitrary monotone transformation. As before, the entire representation u(p) is ordinal and therefore has no quantitative meaning, since f is an arbitrary monotone ∑ntransformation. In practice, however, we often ignore f and write u(p) = k=1 v(xk )pk if no confusion arises.

9.5

Axiomatic characterization of expected utility representation

Not all preferences over lotteries allow expected utility representations. It is a class of preferences which satisfy certain conditions. Here are the conditions 1. Preference ≿ is complete and transitive. Also, p ≻ q holds for some p, q. 2. Mixture Continuity: Given any sequence λν converging to λ, if λν p + (1 − λν )q ≿ r holds for all ν, then λp + (1 − λ)q ≿ r. If λν p + (1 − λν )q ≾ r holds for all ν and λν converges to λ, then it holds λp + (1 − λ)q ≾ r. 3. Mixture Independence: For all p, q, r and 0 < λ < 1, if p ≿ q then λp + (1 − λ)r ≿ λq + (1 − λ)r. Completeness and transitivity are just like before. The latter part of 1 excludes the case that the decision maker is indifferent between any lotteries, since otherwise the argument is trivial. Mixture Continuity states that the preference ”does not jump” with regard to change in risks. It will be easier to understand the meaning by thinking of

CHAPTER 9. CHOICE UNDER RISK

λ p≿q

=⇒

121

p

λ ≿

r 1−λ

r

q

r 1−λ

r

Figure 9.2: Mixture Independence its violation. For example, consider a decision maker saying ”I never like to get infected with mad cow disease. As far as the probability of infection is positive it is the same as being infected.” This preference is discontinuous. Let D denote disease and S denote safety, and let (D; p1 , S; p2 ) denote a lottery which yields infection with probability p1 and safety with probability p2 . Now consider a sequence of lotteries ( ) 1 1 D; , S; 1 − ν = 1, 2, 3, · · · ν ν Since this decision maker takes any( positive probability of infection to be the ) same as being infected, we have D; ν1 , S; 1 − ν1 ∼ (D; 1, S; 0) for all ν = 1, 2, 3, · · · . However, as ν → ∞ we have ν1 → 0, hence the above sequence of lotteries converges to (D; 0, S; 1), which is the perfect safety. Here the continuity condition requires that (D; 0, S; 1) ∼ (D; 1, S; 0) holds in the limit, but his preference says (D; 0, S; 1) ≻ (D; 1, S; 0). Thus Mixture Continuity is violated. This suggests that the expected utility theory is suitable for describing decision criteria which accept certain risks when the reward is sufficiently large, but not suitable for describing criteria which require ”perfect safety” at the sacrifice of any rewards. Mixture Independence has the following interpretation. Here the mixture lottery λp + (1 − λ)r is interpreted as a two-stage lottery which yields ”p with probability λ and r with probability 1 − λ.” Thus the difference between the ”lottery which yields p with probability λ and r with probability 1 − λ” and the ”lottery which yields q with probability λ and r with probability 1 − λ” lies only between p and q. The independence condition is illustrated as in Figure 9.2. Now suppose the ranking between p and q and that between λp+(1−λ)r and λq + (1 − λ)r contradict with each other, let’s say that p ≻ q and λp + (1 − λ)r ≺ λq + (1 − λ)r. Then, despite that this decision maker ranks the ”lottery which yields q with probability λ and r with probability 1 − λ” over the ”lottery which yields p with probability λ and r with probability 1 − λ,” after the first stage

CHAPTER 9. CHOICE UNDER RISK

122

which realizes q and p respectively he would say, ”wait, I like to receive p rather than q.” Such phenomenon is called dynamic inconsistency Mixture Independence is interpreted to require that such inconsistency does not arise. Thus it is appealing at least as a normative ”rationality” requirement which imposes dynamic consistency. The independence condition is often violated in experiments, however. I will come to this later. Below is called the expected utility representation theorem, but I would relegate its proof to an advanced book such as Mas-Colell, Whinston and Green [21]. Theorem 9.1 Preference over lotteries ≿ satisfies the above three conditions if and only if there exists a vNM index v such that the preference is represented in the form p = (x1 ; p1 , · · · , xn ; pn ) ( n ) ∑ u(p) = f v(xk )pk k=1

for any lottery p = (x1 ; p1 , · · · , xn ; pn ), where f is an arbitrary monotone transformation.

9.6

”Cardinal” properties of vNM indices

While utility representation u has no quantitative meaning as stated before, vNM index v has certain quantitative meaning, since the curvature of v describes the decision maker’s √ risk attitude. For example, vNM indices v(z) = z and vb(z) = z are both monotone increasing, but the preference described by the former is risk averse while the one described by the latter is risk neutral. To illustrate, find √ equivalents √ √ certainty of (0; 0.5, 100; 0.5) for each case, then the former yields z = 0.5 0 + 0.5 100, which implies the certainty equivalent is 25, while the latter yields z = 0.5 · 0 + 0.5 · 100, which implies the certainty equivalent is 50. When you take any monotone transformation of the whole representation u it again represents the same preference. However, if you take an arbitrary monotone transformation of a vNM index v, which is square root in the above casse, it in general changes the preference which it describes. Hence v has certain quantitative meaning, while its meaning has certain limitation as discussed below. Again, let me emphasize that while v is cardinal the entire representation u is ordinal because any monotone transformation of it represents the same preference. That’s why I choose not to call v a utility function but call it a vNM index in order to avoid confusion. Note, however that a vNM index has its meaning only in its curvature and its value itself has no quantitative meaning. That is, it has no

CHAPTER 9. CHOICE UNDER RISK

123

meaning like how much happy the decision maker is. This is the same as before. Curvature of a function is changed in general if you take an arbitrary monotone transformation of the function, but it does not change under any affine transformation, which consists of multiplying positive constants and adding constants. Thus the preference to be described by a given vNM index and certainty equivalents for the preference are unchanged √ under affine transformations. √ For example, if you use ve(z) = a z + b instead of v(z) = z, the certainty equivalent of any bet is unchanged. Let z denote the certainty equivalent of a binary √ bet p = (x1 ; p1 , x2 ; p2 ) for the preference described by vNM index v(z) = z in the expected utility form, then it satisfies √ √ √ z = p1 x1 + p2 x2 Notice that this z satisfies √ √ √ √ √ a z + b = p1 (a x1 + b) + p2 (a x2 + b) = a(p1 x1 + p2 x2 ) + b √ as well, which implies it is the certainty equivalent obtained from ve(z) = a z+b. Converse is true as well In general you can say the following. Proposition 9.1 Suppose v is a vNM index describing some preference in the expected utility form. The for any constants a, b with a > 0, av + b describes the same preference in the expected utility form. Also, if two vNM indices describe the same preference over lotteries in the expected utility form, then there exist constant a, b with a > 0 such that vb = av + b. Proof. ⇐= part: Since v is a vNM index which describes the given preference in the expected utility form, for any monotone transformation f ( n ) ∑ v(xk )pk f k=1

is a representation of the preference. Then, since ( n ) ( ( n ) ) ∑ ∑ f (av(xk ) + b)pk = f a v(xk )pk + b k=1

k

and the right-hand-side is a representation of the same preference because f (av+ b) is a monotone transformation. Hence the left-hand-side of the above also represents the same preference. =⇒ part: Let v and vb are vNM indices which describe the same preference in the expected utility form. Now pick z, z such that δz ≻ δz (otherwise the claim is trivial if there don’t exist such z, z), where δz denotes the lottery which yields z with probability 1. Let vb(z) − vb(z)) a= v(z) − v(z)

CHAPTER 9. CHOICE UNDER RISK b=

124

v(z)b v (z) − v(z)b v (z) v(z) − v(z)

Then for any z there exists λz ∈ [0, 1] such that δz ∼ λδz + (1 − λ)δz . Existence and uniqueness of such λz follows from Mixture Continuity and Mixture Independence, but I omit the proof. Now from the expected utility representation theorem we have v(z) = λz v(z) + (1 − λz )v(z) vb(z) = λz vb(z) + (1 − λz )b v (z) From the first formula we obtain λz =

v(z) − v(z) , v(z) − v(z)

and by plugging this into the second formula we get ( ) v(z) − v(z) v(z) − v(z) vb(z) = vb(z) + 1 − vb(z) v(z) − v(z) v(z) − v(z) vb(z) − vb(z) v(z)b v (z) − v(z)b v (z)) = v(z) + v(z) − v(z) v(z) − v(z) = av(z) + b

We can characterize the decision maker’s attitude by means of the shape of the graph of the vNM index. 1. A decision maker is risk-neutral when the vNM index describing his preference in the expected utility form is affine, that is, when v(z) = az+b. You can see the by comparing between bet (x1 ; λ, x2 ; 1−λ) and its expected return λx1 + (1 − λ)x2 . Since v is affine, we have λv(x1 ) + (1 − λ)v(x2 ) =

λ(ax1 + b) + (1 − λ)(ax2 + b)

= a(λx1 + (1 − λ)x2 ) + b = v(λx1 + (1 − λ)x2 ) which implies the decision maker is indifferent between the bet and sure receipt of its expected return. 2. A decision maker is risk-averse when the vNM index describing his preference in the expected utility form is concave, that is, when it holds λv(x1 ) + (1 − λ)v(x2 ) < v(λx1 + (1 − λ)x2 ). This means that given a bet (x1 ; λ, x2 ; 1 − λ) and its expected return λx1 + (1 − λ)x2 the decision maker chooses the latter to be received for sure.

CHAPTER 9. CHOICE UNDER RISK

125

3. A decision maker is risk-loving when the vNM index describing his preference in the expected utility form is convex, that is, when it holds λv(x1 ) + (1 − λ)v(x2 ) > v(λx1 + (1 − λ)x2 ). This means that given a bet (x1 ; λ, x2 ; 1 − λ) and its expected return λx1 + (1 − λ)x2 the decision maker chooses the bet. You can think of which of the following ones correspond to risk aversion or risk loving. v(z) = ln z v(z) = z 2 v(z) = ez v(z) = −e−z Of course one may think of a vNM index which exhibits risk aversion (concavity) in some region and risk loving (convexity) in some region, but let me omit it in this book.

9.7

Applications

In the applications below we assume there any bet can have at most two outcomes and also that the probabilities are fixed. That is, we look at the expected utility representation applied to binary bets u(x1 ; π, x2 ; 1 − π) = πv(x1 ) + (1 − π)v(x2 ), where we fix π, 1 − π and vary x = (x1 , x2 ) only. Thus, simplify the notation to u(x) = πv(x1 ) + (1 − π)v(x2 )

9.7.1

Insurance purchase

Example 9.3 The decision maker has initial income w, his risk attitude is described by vNM index v. If an accident happens to his he loses L, where the accident probability is π. There is an insurance available for purchase, and one dollar of expense on it pays R dollars of income, where R > 1. How much does he spend on the insurance? Let t denote the income he spends on the insurance. Then, if an accident happens he ends up with final income w − L − t + Rt = w − L + (R − 1)t; if no accident happens he ends up with final income w − t.

CHAPTER 9. CHOICE UNDER RISK

126

When we take the expected return it is π(w − L + (R − 1)t) + (1 − π)(w − t) = w − πL + (πR − 1)t. Thus if the decision maker is risk-neutral he buys the insurance as much as possible when πR > 1 (meaning that he bets on the accident), buys no insurance at all when πR < 1, and leads to total indifference when πR = 1. Now let’s suppose the decision maker is risk averse where his risk attitude is described by vNM index v. Then the expected utility evaluation of insurance purchase is πv(w − L + (R − 1)t) + (1 − π)v(w − t) Thus the best amount of expenditure on the insurance is obtained by maximizing the above function. For simplicity let us assume interior solution and take the first-order condition that the derivative being equal to zero, π(R − 1)v ′ (w − L + (R − 1)t) − (1 − π)v ′ (w − t) = 0 By rearranging the above we obtain v ′ (w − L + (R − 1)t) 1−π 1 = · v ′ (w − t) π R−1

(∗)

We find the optima expenditure on the insurance by solving equation (∗). For example, if v(z) = ln z the solution is t=

πR − 1 1−π w+ L. R−1 R−1

Now, suppose 1 (∗∗) π Then the right-hand-side in (∗) is 1 and hence the left-hand-side must be 1 as well, which implies that the final receipt is riskless in the sense that remains the same whether an accident happens or not. This is called perfect insurance. L Under perfect insurance the expenditure on the insurance is t = R , and the L decision maker ends up with w − R regardless of whether an accident happens or not. While the condition (∗∗) is only an ad hoc condition here, but it will be shown in Chapter 12 that it holds as a consequence of market equilibrium under certain condition. R=

9.7.2

Portfolio choice 1

Example 9.4 The decision maker has income w, and his risk attitude is described by vNM index v(x). There are two assets available. One is safe, and pays R per one unit of investment. The other is risky, which pays R per one unit of investment with probability π and pays R per one unit with probability 1 − π, where R > R > R. Let me call the first case State 1 and the second State 2.

CHAPTER 9. CHOICE UNDER RISK

127

Let t denote the investment on the safe asset, where 0 ≦ t ≦ w. Then, if State 1 happens the decision maker ends up with final income Rt+R(w−t) = Rw − (R − R)t if State 2 happens the decision maker ends up with final income Rt+R(w−t) = Rw + (R − R)t When we take the expected return it is (πR + (1 − π)R)w + (R − (πR + (1 − π)R))t Thus if the decision maker is risk-neutral he puts all his initial income on the risky asset when (πR + (1 − π)R) > R, puts all on the safe asset when (πR + (1 − π)R) < R and anything is optimal when (πR + (1 − π)R) = R. Now let’s suppose the decision maker is risk averse where his risk attitude is described by vNM index v. Then the expected utility evaluation of his investment is πv(Rw − (R − R)t) + (1 − π)v(Rw + (R − R)t) Thus the best amount of expenditure on the insurance is obtained by maximizing the above function. Whether optimality is simply given by the ”derivative=zero” condition depends on short-sale constraints. In general, if you solve the ”derivative=zero” −π(R − R)v ′ (Rw − (R − R)t) + (1 − π)(R − R)v ′ (Rw + (R − R)t) = 0 there is a possibility that t is negative or greater than w. When short-sale of the assets is allowed it is OK, but if the 0 ≦ t ≦ w is imposed we may have a corner solution.

9.7.3

Portfolio choice 2: choosing state-contingent consumptions

Remember that even if goods are materially the same they are treated as different goods if they are to be delivered at different contingencies. For example, 1 gallon of gasoline when Republicans win the US presidential election is a different good than one gallon of gasoline when Democrats win. If you have to make some investment decision before the election your choice=bet is described in the form of state-contingent consumption. Again, to illustrate let’s restrict attention the cases that there are just two scenarios, such as ”Republicans or Democrats” or ”hot summer or cool summer.” Let me call one State 1, the other State 2, while the analysis is easily extended to the cases of many states. Then, a vector of state-contingent consumption denoted by x = (x1 , x2 ) says that the decision maker receives x1 units of consumption good if State 1 happens and x2 units if State 2 happens.

CHAPTER 9. CHOICE UNDER RISK

128

This is nothing but a special case of the two-good model we studied in the previous chapters, where Good 1 refers to consumption contingent on State 1 and Good 2 refers to consumption contingent on State 2. State-contingent consumption has the following interpretation: Good 1 = a security which gives one unit of consumption per one unit of it if State 1 occurs and is a junk if State 2 occurs; Good 2 = a security which is a junk if State 1 occurs and gives one unit of consumption per one unit of it if State 2 occurs. Such security is called Arrow security. Any security is given as a combination of Arrow securities. To illustrate, go back to the previous example. There holding one unit of a safe asset which yields R units of consumption for sure is equivalent to holding R units of Arrow security 1 and R units of Arrow security 2. Also, holding one unit of a risk asset which yields R units of consumption at State 1 and R units of consumption at State 2 is equivalent to holding R units of Arrow security 1 and R units of Arrow security 2. Note, however, that this interpretation assumes that short-sales are allowed for such real securities as far as the holdings of Arrow securities are non-negative. To illustrate, see Figure 9.3. When you put all initial income w on the safe asset you get the upper-left end point of the thick segment on the 45-degree line, which gives you the holding of Arrow securities (Rw, Rw). When you put all initial income on the risky asset you get the lower-right end point of the thick segment which gives you the holding of Arrow securities (Rw, Rw). However, without allowing short-selling you cannot obtain every point on the standard budget line in the sense introduced before. For example, any point on the dotted part lower-right than (Rw, Rw) can be obtained only by short-selling the safe asset, and any point on the dotted part upper-right than (Rw, Rw) can be obtained only by short-selling the risky asset. Thus directly handling the exchange of Arrow securities on the budget line in the standard sense requires that such short-selling is allowed. Let us assume that the decision maker’s preference is represented in the expected utility form where the vNM index is denoted by v. Let π denote the probability that State 1 occurs, where State 2 occurs with probability 1 − π. Then his preference induced over state-contingent consumptions is represented in the form u(x) = πv(x1 ) + (1 − π)v(x2 ) Notice that here the marginal rate of substitution of Good 2 for Good 1 is given by πv ′ (x1 ) M RS(x) = . (1 − π)v ′ (x2 ) Let e = (e1 , e2 ) denote the decision maker’s initial holding of the securities. In a security exchange market, given a vector of security prices p = (p1 , p2 ), he

CHAPTER 9. CHOICE UNDER RISK

129

Security 2 6

- Security 1 Figure 9.3: Short-sale constraints

is demand for securities x(p) = (x1 (p), x2 (p)) is determined by solving max πv(x1 ) + (1 − π)v(x2 ) x

subject to

p1 x1 + p2 x2 = p1 e1 + p2 e2

where his income is given by the market value of his initial security holdings. When the vNM index is smooth, the optimal choice is given by the tangency condition that marginal rate of substitution is equal to the relative price. Then we have πv ′ (x1 ) p1 = (1 − π)v ′ (x2 ) p2 Combine this with the budget constraint p1 x1 + p2 x2 = p1 e1 + p2 e2 and solve the equations for x = (x1 , x2 ) then we obtain the security demand. For example, when the vNM index is v(z) = ln z, his preference over statecontingent consumptions are represented by u(x) = π ln x1 + (1 − π) ln x2 This is nothing but Cobb-Douglass preference, the demand for state-contingent consumptions is obtained by applying the previous result, which yields x1 (p) =

9.8 9.8.1

π(p1 e1 + p2 e2 ) p1

x2 (p) =

(1 − π)(p1 e1 + p2 e2 ) . p2

Violation of the expected utility theory Violation of the independence condition

As noted above, the mixture independence condition is often violated in experiments. Consider the following example.

CHAPTER 9. CHOICE UNDER RISK

130

Problem A: Which one to choose, A1=(100; 0.9, 0; 0.1) or A2=(90; 1)? Problem B: Which one to choose, B1=(100; 0.45, 0; 0.55) or B2=(90; 0.5, 0; 0.5)? In experiments many subjects choose A2 in Problem A while they choose B1 in Problem B. Notice, however, that (10000; 0.45, 0; 0.55) = 0.5(10000; 0.9, 0; 0.1) + 0.5(0; 1) (9000; 0.5, 0; 0.5) = 0.5(9000; 1) + 0.5(0; 1) Thus if they follow the independence condition one chooses A2 in Problem A must choose B2 in Problem B. Thus the combination A2-B1 is a violation of the independence condition. This is called Allais paradox. Where does the paradox come from? One can raise so-called certainty effect. Note that the expected utility theory imposes evaluation of probabilities to be ”linear,” while it allows non-linearity of evaluation of returns. In terms of representation, the functional form ( n ) ∑ v(xk )pk u(p) = f k=1

is linear in probability pk of each outcome, whereas v is allowed to be non-linear in outcomes. That is, increase of probability of a given outcome from 0 to 0.1, that from 0.1 to 0.2, that from 0.2 to 0.3, and so on, and that from 0.9 to 1, are all evaluated equally. The paradox is explained by allowing that the increase of probability to 1 has a more significant effect than that around a lower probability value. Having something for sure has a special effect. Also, there is a normative argument that the independence condition should be violated. It says that when there are two equally preferable alternatives we should flip a coin, for the sake of fairness, rather than choosing one from them arbitrarily. Recall that under the independence condition when two alternatives are equally preferable flipping a coin is at best equally preferable to them, and it does give a strong recommendation to randomize. Thus, if the society takes the position to respect ex-ante notion of fariness the social decision must violated the independence condition. However, it is impossible to ”just” give up or weaken the independence condition. For, as we already saw in the the explanation of the independence condition that it is equivalent to dynamic consistency, any violation of the the condition leads to a dynamic inconsistency problem, in that future selves do not follow the will of the current self. To illustrate the point consider the following example due to Machina [19].

CHAPTER 9. CHOICE UNDER RISK

131

Example 9.5 Mom has a single indivisible item — a ”treat” — which she can give to either daughter Abigail or son Benjamin. Assume that she is indifferent between Abigail getting the treat and Benjamin getting the treat, and strongly prefers either of these outcomes to the case where neither child gets it. However, in a violation of the precepts of expected utility theory, Mom strictly prefers a coin flip over either of these sure outcomes, and in particular, strictly prefers 1/2: 1/2 to any other pair of probabilities. This random allocation procedure would be straightforward, except that Benjie, who cut his teeth on Raiffa’s classic Decision Analysis, behaves as follows: Before the coin is flipped, he requests a confirmation from Mom that, yes, she does strictly prefer a 50:50 lottery over giving the treat to Abigail. He gets her to put this in writing. Had he won the flip, he would have claimed the treat. As it turns out, he loses the flip. But as Mom is about to give the treat to Abigail, he reminds Mom of her ”preference for flipping a coin over giving it to Abigail (producing her signed statement), and demands that she flip again. What would your Mom do if you tried to pull a stunt like this? She would undoubtedly say ”You had your chance!” and refuse to flip the coin again. This is precisely what Mom does. Each of Mom’s claim and Benjamin’s claim amounts to a problem. If we accept Benjamin’s claim and flip coin again, we run into a problem of dynamic inconsistency, since Abigail winning the item with probability 1/2 × 1/2 = 1/4 and Benjamin winning with probability 1/2 + 1/2 × 1/2 = 3/4, which is unfair in any sense from the ex-ante viewpoint. Abigail will say the same thing when she loses. So Mom will have to flip coins forever. If we accept Mom’s claim, we have to go outside of the standard notion of ”rationality” called consequentialism which says out decision should not be affected by ”bygones.” Here Mom’s claim ”you had a change” brings up what Benjamin (and Abigail) could have got if the first he won the flip, which is nothing but a ”bygone.”

9.8.2

Timing of resolution of risk

There is another but related implicit assumption behind the equivalence between Mixture Independence and dynamic consistency. It is that the decision maker is indifferent to timing of resolution of risk so that only the probability distributions over final outcomes matter.1 To illustrate, let us think of the following example. 1 Depending

on context it is also called consequentialism.

CHAPTER 9. CHOICE UNDER RISK

132

1. Flip a coin twice, and receive 100 dollars if both flips are head, 50 if the first is head and the second flip is tail, 30 if the first flip is tail and the second is head, 0 if both flips are tail. 2. Throw a four-face die, and receive 100 dollars if the face is 1, 50 if it is 2, 30 if 3, 0 if 4. Since each outcome occurs with quarter probability in both gambles, they induce the same probability distribution over outcomes. However, they are different when the decision maker cares about timing of resolution of risk, and even more when the second coin flip is made after certain time. Recall the explanation of Mixture Independence. There was actually a ”cheat.” It is the assumption that the two stage lottery which gives p with probability λ and q with probability 1 − λ and the compound lottery λp + (1 − λ)q are equivalent. However, the latter by definition is a one-stage lottery λp + (1 − λ)q = (x1 ; λp1 , · · · , xn ; λpn , y1 ; (1 − λ)q1 , · · · , ym ; (1 − λ)qm ) If the decision maker cares about timing of resolution of risk they may not be equivalent. Then we need to think of ”lotteries over lotteries,” ”lotteries over lotteries over lotteries,” and so on, so that they are treated differently across the layers of timings. It is the idea of recursive utility theory due to Kreps and Porteus [15].

9.8.3

Dependence on reference points

Let us think more about the assumption that only probability distributions over final outcomes matter. Consider the following example. QA: When 50 dollars are initially given, which one do you choose? A1= receiving 50 more dollars with probability 50% and nothing with probability 50% A2= receiving 25 more dollars for sure QB: When 100 dollars are initially given, which one do you choose? B1= losing 50 dollars with probability 50% and nothing with probability 50% B2= losing 25 dollars for sure It is observed in experiments using similar numbers that there are many subjects who choose A2 in QA and B1 in QB. The two choice problems are equivalent, however, under the assumption that only probability distributions over final outcomes matter, as both A1 and B1 induce the same lottery (100; 0.5, 0; 0.5) and A2 and B2 induce (75; 1). The above example shows that the decision makers tend to be risk averse when the risk is about how much to gain and tend to be risk loving when the

CHAPTER 9. CHOICE UNDER RISK

133

risk is about how much to lose, in the sense that they rather choose gambling than losing something for sure. This is called loss aversion, and one of the key components of the prospect theory due to Tversky and Kahnemann [34].

9.9

Exercises

Exercise 12 √There is a consumer whose risk attitude is described by vNM index v(z) = z. (i) For him, what is the certainty equivalent of a bet which yields 256 with probability 0.4 and 81 with probability 0.6? (ii) For him, the certainty equivalent of a bet which yields 361 if it rains and 64 if it is sunny is 225. What is the probability of rain? Exercise 13 There is a consumer whose risk attitude is described by vNM index v(z) = ln z. There are two states of the world, State 1 and State 2, which the probability of State 1 is 0.6. There are two assets, A and B. Asset A’s gross return rate is 1.2 at State 1 and 0.9 at State 2. Asset B’s gross return rate is 0.8 at State 1 and 1.5 at State 2. Given that his initial income is 100, how much does he invest on each asset?

Chapter 10

Revealed preference Until the previous chapter we have assumed a priori that each individual has his preference and chooses best alternatives according to it. It is of course a natural response to wonder, however, if that’s true. You cannot open your brain physically, however, in order to show your preference or maximization process directly as biological objects or substances or structures or processes.1 So here we take the standpoint to consider if observed choices can be explained consistently as the maximization some preference, rather than thinking if we can find preferences as physical entities. This is called the revealed preference approach. The revealed preference approach starts with observed choice data. Here the data is taken to be a list of pairs, each of which consists of a set of available alternatives called an opportunity set, say denoted by B, and a subset of it denoted by φ(B) consisting of alternatives chosen from B. Let X be the set of all the potentially available alternatives, which is assumed to be finite for simplicity. We assume that an opportunity set can be any nonempty subset of X. Thus the family of all the possible opportunity sets is given by B = {B : B ⊂ X, B ̸= ∅} Given an opportunity set B ∈ B, let φ(B) denote the set of alternatives chosen from it. Here we allow that φ(B) may consist of several elements, that is, we leave ties as they are and do not get into how ties are broken. Thus φ(B) is a nonempty subset of B In other words, φ is a mapping from B into itself with the property that φ(B) ⊂ B for all B ∈ B. Thus it is called a choice mapping. Any observed data is given as a choice mapping. 1 I’m of aware that there are such line of researches in recent decades, but let me take a classical and conservative standpoint here.

134

CHAPTER 10. REVEALED PREFERENCE

135

Definition 10.1 Choice mapping φ is said to be rationalizable if there exists a complete and transitive preference relation ≿ over X such that it holds φ(B) = {x ∈ B : x ≿ y for all y ∈ B} for all B ∈ B. That is, rationalizability says the exists a preference ordering ≿ such that φ(B) is equal to the set of maximal elements in B according to ≿ for all B ∈ B. Of course this ”rationalize” has nothing to do with justifying things, as discussed in Chapter 1. Now, what properties does a rationalizable choice mapping satisfy? I list two properties. Condition 10.1 (Contraction): For all B, C ∈ B with B ⊂ C and any x ∈ B, if x ∈ φ(C) then x ∈ φ(B). This condition says that what is chosen from a larger set must be chosen from any smaller set containing it. Plainly speaking, a world champion must be a state champion in his country. It is easy to see that Contraction is a necessary condition for rationalizability. If all players in the world are ranked according to some world ranking then if one is No.1 in south America he must be No.1 in his country, let’s say in Paraguay. The next condition is Condition 10.2 (Expansion): For all B, C ∈ B with B ⊂ C and any x, y ∈ B, if x, y ∈ φ(B) and x ∈ φ(C) then y ∈ φ(C). This condition is about ties. It says that when several alternatives are chosen from a smaller set, if one of them is chosen from a larger set as well the others chosen from the smaller set must be chosen there as well. Plainly speaking, if there are multiple champions in a country with ties and if one of them is a world champion then the other champions in the country must be world champions with ties as well. It is easy to see as well that Expansion is a necessary condition for rationalizability. If all players in the world are ranked according to some world ranking then if there are several No.1 players in Paraguay with ties and if one of them is No.1 in south America then the other No.1 players in Paraguay must be No.1 in south America with ties. Theorem 10.1 Choice mapping φ satisfies Contraction and Expansion if and only if it is rationalizable. Proof. (”If” part): Given a preference relation ≿, define a choice mapping φ≿ by φ≿ (B) = {x ∈ B : x ≿ y for all y ∈ B}

CHAPTER 10. REVEALED PREFERENCE

136

and I show that this satisfies Contraction and Expansion. Contraction: Pick any B, C with B ⊂ C and x ∈ B, and assume x ∈ φ≿ (C). Then since x ≿ y for all y ∈ C, we have x ≿ y for all y ∈ B, which implies x ∈ φ≿ (B). Expansion: Pick any B, C with B ⊂ C and x, y ∈ B, and assume x, y ∈ φ≿ (B) and x ∈ φ≿ (C). Since x, y are maximal elements in B with respect to ≿, it holds x ≿ y and y ≿ x, which implies x ∼ y. Since x is a maximal element in C with regard to ≿, it holds x ≿ z for all z ∈ C. Because of x ∼ y and transitivity of ≿, it holds y ≿ z for all z ∈ C, which implies y is also a maximal element in C with respect to ≿. Thus we have y ∈ φ≿ (C). (”Only if” part): Given a choice mapping φ, define a preference relation ≿φ by x ≿φ y ⇐⇒ x ∈ φ({x, y}) That is, we say choice reveals that x is at least as good as y when x is chosen from the pair x and y (this does not exclude that y is chosen there as well). First we show that ≿φ is complete and transitive. Completeness: Since φ({x, y}) is nonempty, it holds at least either x ∈ φ({x, y}) or y ∈ φ({x, y}). In the first case we obtain x ≿φ y, and in the latter case we obtain y ≿φ x (both may hold). Transitivity: Suppose x ≿φ y and y ≿φ z, and for a proof by contradiction suppose x ̸≿ z. By definition, this means x ∈ φ({x, y}), y ∈ φ({y, z}) and x∈ / φ({x, z}). Now we consider three cases for φ({x, y, z}). Case 1: Suppose x ∈ φ({x, y, z}), then by Contraction we have x ∈ φ({x, z}), which leads to a contradiction. Case 2: Suppose y ∈ φ({x, y, z}), then by Contraction we have y ∈ φ({x, y}). Since now x, y ∈ φ({x, y}), by Expansion we have x ∈ φ({x, y, z}), which reduces to Case 1. Case 3: Suppose z ∈ φ({x, y, z}), then by Contraction we have z ∈ φ({y, z}). Since now y, z ∈ φ({y, z}), by Expansion we have y ∈ φ({x, y, z}), which reduces to Case 2. Finally, I show that φ is indeed generated by ≿φ , that is, φ(B) = {x ∈ B : x ≿φ y for all y ∈ B} for all B. ⊂-direction: Let x ∈ φ(B). Then by Contraction it holds x ∈ φ({x, y}) for all y ∈ B, which means x ≿φ y. ⊃-direction: Let x ∈ B be such that x ≿φ y for all y ∈ B. Since φ(B)is a nonempty set there is x′ ∈ B such that x′ ∈ φ(B). Since the proof is already done when such x′ is equal to x, we assume x′ ̸= x. Then by Contraction it holds x′ ∈ φ({x, x′ }), but by assumption we have x ≿φ x′ , which implies x, x′ ∈ φ({x, x′ }). Now by Expansion we get x ∈ φ(B).

CHAPTER 10. REVEALED PREFERENCE

137

What kind of choices are not rationalizable in the above sense? There are still certain order in such choices. Let us think of the following example. Example 10.1 (Minimax regret): Consider choice under uncertainty with two states of the world, denoted by s1 and s2 . Consider the following statecontingent receipt of prize, where for example y = (1, 5) denotes 1 unit if s1 occurs and 5 units if s2 occurs. x y z

s1 2 1 5

s2 2 5 1

Here the minimization of maximal regret is done as follows. 1. For each possible state, imagine hypothetically that you could have made choice after knowing that state happened. In this example, if you could have chosen after seeing s1 you would choose z and would get 5 units, and if you could have chosen after seeing s2 you would choose y and would get 5 units. This is called ex-post optimum. 2. For each alternative, for each state take the difference between its outcome and the ex-post optimum at that state. Let us call it anticipated ex-post regret. Suppose you choose for example y, then if s1 happens the ex-post regret is 5 − 1 = 4, and if s2 happens his choice was right and the ex-post regret is 0. 3. For each alternative, take the maximum of the anticipated ex-post regrets across states. Consider for example y again, then the anticipated ex-post regret of choosing it is 4 at s1 0 at s2 , hence the maximal anticipated ex-post regret is 4. 4. Choose the alternative which minimizes the maximal anticipated ex-post regret. In the table below x minimizes the maximal regret and it is chosen from {x, y, z}.

x y z ex-post maximum

outcome s1 2 1 5 5

s2 2 5 1 5

regret s1 3 4 0

maximal regret s2 3 0 4

3 4 4

The minimax regret choice violates Contraction (it violates Expansion too, but I omit it here). For example, if you drop z and consider a binary choice {x, y}, then the table becomes

CHAPTER 10. REVEALED PREFERENCE

x y ex-post maximum

outcome s1 2 1 2

s2 2 5 5

regret s1 0 1

138 maximal regret s2 3 0

3 1

and now y is chosen. Thus the minimax regret choice does not allow explanation by the maximization of a single preference relation, but you see that violation may come from such choice rule which has certain order.

Part II

Perfectly Competitive and Complete Market with Complete Information

139

Chapter 11

Perfectly competitive and complete market with complete information The title will sound scary. This is actually the simplest kind of situation taught in introductory courses, however, in which price is determined so that demand matches supply. Economics typically launches its first-step arguments by assuming these three as a ”baseline.” I restate it in such way because I want to clarify the underlying assumption behind it. In this chapter I will explain and discuss these three conditions.

11.1

Perfect competition

First component of the ”baseline” is perfect competition.

11.1.1

Its definition and meaning

What do you imagine from the word ”competition?” Do you imagine a situation like everybody killing each other? To my understanding, whether one is for or against economic competition is largely affected by how the word ”competition” sounds to him or her, that is, whether he or she feels heroism in this word in positive way or negative way. It has nothing to do with what economics is talking about. I like you to forget about this sound. Let me start with giving a bare-bone tedious definition. Definition 11.1 Market is said to be perfectly competitive if every market participant takes the market price as given.

140

CHAPTER 11. MARKET

141

Now what does it mean? Substantively speaking, it says the market is populated with a large number of competitors, so that every market participant is negligibly small compare to the entire economy, and he cannot affect the market price by himself alone, and has to take it as given. This is also called the assumption of price-taking. Each consumer takes the market price as given, and has no power to affect that by himself alone, and decides consumption under the budget constraint. The term ”demand” refers to such passive decision. Each firm takes the market price as given, and has no power to affect that by itself alone, and determines production under that. The term ”supply” refers to such passive decision. In other words, nobody has market power.

11.1.2

Validity of the perfect competition assumption: criticisms

Let me list criticisms to the assumption of perfect competition as far as I can come up with. 1: Who is setting the price? The assumption of price taker at least literally says that every consumer and every producer are just passively responding to the price, saying ”I would by how much when the price is blah blah,” and ”I would sell how much when the price is blah blah” and so on. I you take this literally, nobody would be setting the price. It is as if the price is falling from the heaven. Who is setting this price. ”Market sets the price” is not an answer. I don’t want an explanation with ”invisible hand,” I want an explanation with ”visible hand.” As far as we take the assumption of perfect competition literally we have no choice but interpreting that the price is falling from the heaven. Leon Walras, one of the founders of the modern market theory, called such hypothetical agent (meant by ”heaven” here) auctioneer, and consider a priceadjustment process called tatonnement: the auctioneer announces price, the each market participant returns his demand or supply as a response to the announced price; if demand does not match supply the auctioneer revises the price, and so on, and eventually he announces the right price so that demand matches supply; trades are supposed to be done only after such adjustment is completed. There are markets with auctioneer, although it would not exactly coincide with what Walras meant, such as the markets for agricultural products, and security markets from which he got the inspiration. However, in most cases the sellers simply set their prices and buyers just buy or not, or the two sides negotiate in order to determine the price. Let us concede for moments and accept the existence of auctioneer. But still a problem remains.

CHAPTER 11. MARKET

142

2: It may be OK for consumers, but produces cannot be just passively responding to the price announced by the auctioneer. A large producer would rather take price as a function of its supply, instead that it returns its supply as a function of price. Such producer would rather try to manipulate price in in favor of it. Assuming the existence of auctioneer itself does not guarantee that market participants passively respond to price. When there are large market participants which can manipulate price we say that the market is imperfectly competitive, and that they have market power. When the market is perfectly competitive such large participants will strategically behave and try to outwit each other. This may the ”competitive” market many of you might imagine, but in economics we call it ”imperfect” competition. The above two are criticism to the assumption of price-taking. Let us concede again for moments and accept this. Even after this, there still remains a problem. 3: Even if the assumption of price-taking is met it is a different question if we reach competitive equilibrium. Demand and supply may remain unmatched. Even if we assume the existence of auctioneer and consider the tatonnement process as described it if a different question if the process lead the auctioneer announce the right price so that demand matches supply. This problem applies to the case of imperfect competition as well, but let me first explain this in the context of perfect competition.

11.1.3

Validity of the perfect competition assumption: response

As is discussed in Chapter 1, the most extreme form of ”positivist” view says that an assumption doesn’t have to have anything to do with reality and it should be the simplest assumption under the simplest setting which can derive predictions consistent with real phenomena as many as possible, and it is rather better as such it is more unrealistic. It says that the validity of assumption is not in the assumption itself and it is in the validity and richness of conclusions it derives. Again, I don’t take this view, since the validity of an assumption itself does matter since it makes difference in welfare analysis and normative interpretation. So let me try to give explanations of when it is OK to assume perfect competition, as much as possible. Basically, the argument is As there is a large number of market participants, each participant is negligibly small compared to the entire economy, and has to take certain market price as given, even when he is acting to set individual prices by himself.

CHAPTER 11. MARKET

143

Let me emphasize that when the market consists of a small number of large traders instead they have in general market powers and the condition of perfect competition fails. Then we need a theory of imperfect competition, which is covered in Part 3. The exact process of how imperfect competition converges to perfect competition is covered Part 3, but let me give a brief explanation of it. First let us resolve Problem 2. Let me assume that there is an auctioneer, as I will come to the case without that in the next paragraph. In general, a market participant with market power takes the effect of his decision on the market price into account. However, when the number of market participants is large the effect of an individual participant’s change in his quantity on the market price is almost zero. Thus, each market participant alone has to take the market price as given, which is determined by the mass behavior of a large number of participants. Now what about Problem 1? In order to clear this we have to dispense with the auctioneer. Consider the simplest case that each seller sets the price of his commodity. Then can each seller freely set his price? No. When there is a large number of competitors if you set the price too high you will lose demand and will lose profit. Thus even though each seller is setting his price by himself he has to take certain market price as given, which is set by the mass of a large number of sellers. There are two ways to argue, though. One is to consider that the number of actual market participants tends to be large, the other is that the number of actual participants does not have to be large but the market is open to any potential entrants and potentially there are indefinitely many entrants. In either case, the word ”competitive” means that since there are indefinitely many actual or potential competitors in the market no market participant cannot dominate the market by himself alone, and has to respond to the mass tendency in a passive manner. Finally, what about Problem 3? First thing we can think of is the tatonnement process as explained above. However, the tatonnement argument assumes that all trades are done only after the adjustment process finishes and reaches an equilibrium. This is not a realistic explanation either, as well as the assumption of auctioneer, at least as far as we take it literally. This necessitates to think if and how the market reaches competitive equilibrium through decentralized behaviors of market participants without any centralized adjustment. When there is a large number of market participants each one is negligibly small and has to take the mass behavior as given. Let us be content with, as it is already explained above. The problem here is if such mass behavior indeed leads to a competitive equilibrium. Since we do not rely on the auctioneer story, we have to thin of a situation in which the mass of market participants set their prices and quantities in a decentralized manner. However, in order that such decentralized mass behavior indeed leads to an equilibrium in the market, as far as such game is taken literally, each market

CHAPTER 11. MARKET

144

participants has to have a right prediction of how the others set prices and quantities. This seems to require each participant a terribly high level of rationality. — This point applies to the case of imperfect competition as well. There will be two views about this. One is The market equilibrium theory is a loose association of particular models, in which incomplete strategic reasoning and incomplete adjustment mutually complement each other. It is a ”detail-free” argument, saying that regardless of how particular models work the situation overall falls in a competitive equilibrium. It is therefore rather counter-productive to bring up a model of ”perfect adjustment” or ”perfect reasoning” and dismiss the equilibrium theory on the ground that these are unrealistic. This sounds cheating somehow, but it is understandable because the frameworks of our recognition are limited we have to make an ”economical” choice. The other is of course to continue to pursue a theoretical foundation of competitive equilibrium, or the notion of equilibrium in general. This is still an open question, but it seems to be commonly accepted that the key is learning and imitation. We have to give up the assumption that people trades only after complete adjustment is done, and that people perfectly read each other’s mind and lead to an equilibrium in a timeless manner. But it says the market will be able to form an equilibrium situation through the process of repeating actual trades, which may or may not be an equilibrium one, and learning the mass behavior and taking that into accounts. In the experimental literature it is known that repeated transactions converge to competitive equilibrium pretty quickly (see for example Joyce [5], Smith [32]). But the problem is that we don’t know yet why.

11.2

Complete market

Second component of the ”baseline” is market completeness. Definition 11.2 Market is said to be complete if every good is exchangeable with every other good. What do I mean? To understand this it is better to see the examples of market incompleteness. • When milk is not allowed to trade and everybody has to consumer only milk which he has home we cannot exchange between any good with milk. In this sense the market is incomplete. • Consider that there is a constraint on borrowing and you cannot borrow an amount despite you have enough lifetime income to pay back. For example, suppose you have 1 million of life time income in the present

CHAPTER 11. MARKET

145

value, but you cannot borrow this amount from the bank. Then you cannot buy current consumption by means of selling future consumption in the fully flexible manner. In this sense the market, in particular the market for intertemporal trading, is incomplete. • Consider that there is only a safe asset which pays constant return regardless of uncertainty. This may sound good. But if your earning is lower at some state (State 1, say, Republicans winning) and higher at another state (State 2, say, Democrats winning) you would like to hedge risk by means of transferring income from State 2 to State 1. You can do this if there is another asset, which is risky, and pays higher return than the safe asset at State 2 and lower return than the safe asset at State 1. Then you can transfer income from State 2 to State 1 by means of buying the risky asset and (short-)selling the safe asset. You cannot do this when there is only a safe asset or generally just one asset. In this sense the asset market is incomplete. • A consumption or production activity is said to have an externality if there is no market for it and it is not taken into account in consumption and production decision in the markets. A typical example is pollution. The polluter does not take the social effect of pollution into account in its consumption or production decision and the other economic agents cannot stop it. If there is a market for the ”right to prevent pollution,” people can pay for it in order to enjoy cleaner environment. Or, if there is a market for the ”right to do the activity with pollution” the polluter may pay for it and the other people may receive the payment. But the current type of market incompleteness says that there is no such market. As a result, resource allocation in the market may be inefficient even when it is perfectly competitive. This is called market failure. What is important in the above definition is that market prices do not take it into account. Even when an activity directly affects other economic agents it is not called externality when it is priced and traded in markets, such as service. • Complete autarky in which no trade is allowed is the most extreme form of market incompleteness. Completeness is critical for efficiency of allocation: when the market is incomplete even when it is perfectly competitive the resulting market outcome may be generally inefficient, in the sense that there is another allocation which makes everybody better off. We start with complete markets nevertheless, because it is the best way to find out and understand precisely what types of market incompleteness are significant.

CHAPTER 11. MARKET

11.3

146

Complete information

Third component of the ”baseline” is information completeness. Definition 11.3 Market is said to be with complete information if all the market participants share the same information. Again, to understand this it is better to see the examples of information incompleteness. • Buyers may not know the quality of products to buy, while the sellers know. • Sellers may not know how much buyers are willing to pay for the product, while each buyer knows his willingness to pay. • When several firms compete for the same set of buyers, each firm may not know the other firms’ production technology and cost structure. • When several buyers are competing for the same item, each buyer may not know how much the other buyers are willing to pay for it. • Some traders may have good information about a stock and other traders may have bad information about it. and so on. Nevertheless we start with markets with complete information, again. There are two reasons. One is that again it is the best way to find out and understand precisely what types of information incompleteness are significant. The other is that when the market consists of a large number of participants again asymmetrically distributed informations are aggregated into the market price so that the price conveys information to all the participants. The second argument is related to so-called efficient market hypothesis, which states that non-random predictable components of future asset values are quickly aggregated into the current asset prices so that the future asset prices consist only of random components which cannot be systematically predictable. In other words, you cannot outwit the market. This appears to be a contradiction if it is taken literally — if price tells us everything, why do we need to do research? The key here is that the very fact that each market participant tries to outwit each other is the driving force which leads the market to the situation such that nobody can outfit anybody, through the bidding behaviors being aggregated into the price. In the literature of auction, which handles situation in which bidders don’t know each others’ valuations of the item, it has been shown that when the number of bidders is large the auction outcome converges to an efficient one which is obtained under complete information. This is consistent with the fact that at an idiosyncratic level there are individuals who outwit the market by fortune. One will not be able to fully exploit

CHAPTER 11. MARKET

147

such fortune if he does not act rationally, but the fortune cannot come in a systematically on average basis either, and he cannot systematically outwit the market.

Apologies became long. Bracketing these apologies, let us see how the baseline model of market works in the next several chapters.

Chapter 12

Competitive equilibrium in exchange economies 12.1

Exchange economy

Although we are interested in analyzing production economies in the end, it is actually better to start with exchange economies, in order to understand the nature of gains from trade and the role of prices. That is, we consider an economy with a fixed supply which is the sum of initial holdings brought by the individuals. I will come back to production economy in later chapters. There are n consumers in the economy. Each consumer i = 1, · · · , n has his initial endowment ei = (ei1 , ei2 ) and brings it to the market. Here ei1 denotes the amount of Good 1 initially held by i, and ei2 denotes the amount of Good 2 initially held by i. Also, consumer i’s consumption vector is for example denoted by xi = (xi1 , xi2 ), where xi1 refers to i’s consumption of Good 1 and xi2 refers to i’s consumption of Good 2. Since we have many market participants and multiple goods now, we operate subscripts in this way. Also, let ≿i denote each i’s preference respectively. For example, when xi = (xi1 , xi2 ) is at least as good as yi = (yi1 , yi2 ) for i we write xi ≿i yi . Now any allocation x = (x1 , · · · , xn ) must satisfy the feasibility condition. First, the sum of consumptions of Good 1 across all consumers cannot exceed the sum of initial holdings of it, hence it must hold n ∑ i=1

xi1 ≤

n ∑

ei1 .

i=1

Because we are excluding the case of wasting resource without loss of generality, we assume that this constraint is met with equality. Similarly for Good 2. Thus, 148

CHAPTER 12. COMPETITIVE EXCHANGE ECONOMIES

149

any feasible allocation x = (x1 , · · · , xn ) obeys the constraint n ∑ i=1 n ∑ i=1

12.1.1

xi1 = xi2 =

n ∑ i=1 n ∑

ei1 ei2

i=1

Edgeworth box

To illustrate, let us consider that there are just two consumers, A and B. There must be a large number of small participants, however, in order that the assumption of perfect competition makes sense. The assumption of two consumer seems to contradict to that apparently when it is interpreted literally. Therefore you should imagine a large number of consumers behind these two, since the assumption is only for making the illustration easier to follow. Now, when we draw the two consumers’ consumption choices in separate twodimensional graphs it is hard to see if the feasibility condition is met. So we draw two consumers’ consumptions in one diagram, which is called Edgeworth box. First, draw A’s consumption set and B’s one respectively, and depict their initial endowment points on them respectively. Denote A’s initial endowment by eA = (eA1 , eA2 ) and B’s one by eB = (eB1 , eB2 ). Next, rotate B’s consumption set and paste it onto A’s one so that B’s initial endowment point coincides with A’s one. Then you get a rectangular-shaped diagram as in Figure 12.1, such that its horizontal length is equal to the total amount of Good 1 available, eA1 + eB1 , and its vertical length is equal to the total amount of Good 2 available, eA2 +eB2 . Denote the point at which the two endowment points coincide by e = (eA , eB ), which is seen as A’s initial endowment point when it is seen from A’s origin, and seen as B’s initial endowment point when it is seen from B’s origin. Now pick any point in the box, denoted let’s say by x = (xA , xB ), then the sum of its horizontal coordinates across A and B is equal to the vertical length of the box, and the sum of its horizontal coordinates across A and B is equal to the vertical length of the box. Thus we have xA1 + xB1 = eA1 + eB1 and xA2 + xB2 = eA2 + eB2 , which is nothing but the feasibility condition. That is, any feasible allocation is described as a point in this box diagram. Also, the budget line passing through the initial endowment point is seen as A’s one when it is seen from A’s origin and seen as B’s one when it is seen from B’s origin.

12.2

Competitive equilibrium

Now, under the assumption of price-taking each consumer i = 1, · · · , n responds to given price p = (p1 , p2 ) by returning his demand xi (p) = (xi1 (p), xi2 (p)),

CHAPTER 12. COMPETITIVE EXCHANGE ECONOMIES

150

A’s Good 2 6 B’s Good1

xA2

eB1

xB1

rx

xB2 re

eA2

OA

OB

eA1

xA1

eB2 - A’s Good 1 ? B’s Good 2

Figure 12.1: Edgeworth box

where the initial endowments are taken to be fixed and omitted from the notation. Then the prices are determined so that demand matches supply (again, assuming that you accepted my apologies in the previous chapter...). Denote such price vector by p∗ = (p∗1 , p2∗ ) n ∑ i=1 n ∑ i=1

xi1 (p∗ ) = xi2 (p∗ ) =

n ∑ i=1 n ∑

ei1 , ei2 ,

i=1

Such p∗ is called competitive equilibrium price, and the resulting allocation (x1 (p∗ ), · · · , xn (p∗ )) is called competitive equilibrium allocation. In the Edgeworth box as in Figure 12.2, the equilibrium condition is saying that A’s demand point and B’s demand point coincide. Since A is choosing his optimal consumption given the price, his corresponding indifference curve IA is tangent to the budget line. Likewise, since B is choosing his optimal consumption given the price, his corresponding indifference curve IB is tangent to the budget line. Thus, A and B’s indifference curves passing through the equilibrium allocation are tangent to the budget line, which is seen as A’s one when seen from A’s origin and seen as B’s one when seen from B’s origin. Under the assumption of smooth preferences, since marginal rate of substitution is equal to the relative price at each consumer’s optimal consumption, in competitive equilibrium it holds M RSi (xi (p∗ )) = for every i = 1, · · · , n.

p∗1 p∗2

CHAPTER 12. COMPETITIVE EXCHANGE ECONOMIES A’s Good 2

151

IA

6 B’s Good1

OB re

IB ∗ rx

OA

12.2.1

- A’s Good 1 ? B’s Good 2 Figure 12.2: Competitive equilibrium

Competitive equilibrium under Cobb-Douglas preferences

Let us solve for competitive equilibrium using a specific class of preferences, Cobb-Douglas preferences. Assume that each i’s preference is represented in the form ui (xi ) = λi ln xi1 + θi ln xi2 .

(i) We already know that the demand function generated by Cobb-Douglas preference for each i is given by xi1 (p) = xi2 (p) =

p1 ei1 + p2 ei2 , p1 p1 ei1 + p2 ei2 (1 − αi ) , p2 αi

where αi denote the relative weight on Good 1 defined by αi =

λi . λi + θi

Note that the above form is simplified to ) ( p2 xi1 (p) = αi ei1 + ei2 , p1 ( ) p1 xi2 (p) = (1 − αi ) ei1 + ei2 , p2 from which you see that only a relative price should matter.

CHAPTER 12. COMPETITIVE EXCHANGE ECONOMIES

152

(ii) Since there are just two goods here, when the market for one good is balanced the one for the other is automatically balanced. Thus it suffices to look at the condition on Good 1, ( ) ∑ n n ∑ p2 αi ei1 + ei2 = ei1 p1 i=1 i=1 By solving this we obtain

∑n αi ei2 p∗1 = ∑n i=1 p∗2 (1 − αi )ei1 i=1 p∗

Note that we can only find the ratio between p1∗ and it is sufficient. You 2 can see this by seeing that when p∗ = (p∗1 , p∗2 ) is an equilibrium price vector its double 2p∗ = (2p∗1 , 2p∗2 ) is also an equilibrium price vector because it does not change anybody’s budget constraint. That is, only relative prices matter. p∗ (iii) By plugging the equilibrium relative price p∗1 in to each consumer’s 2 demand function we obtain each i’s consumption in equilibrium ) ( ∑n j=1 (1 − αj )ej1 ∑n ei2 xi1 = αi ei1 + j=1 αj ej2 ( ∑n ) j=1 αj ej2 xi2 = (1 − αi ) ∑n ei1 + ei2 j=1 (1 − αj )ej1

12.3

Interest rate in borrowing-lending economies

12.3.1

Lifetime budget constraint and intertemporal competitive equilibrium

Let us apply the model of competitive market to an intertemporal economy. First we consider a borrowing-lending economy with no production. We will come to the case including production later. Consider the two-period model, in which Good 1 corresponds to consumption in Period 1 and Good 2 corresponds to consumption in Period 2. Denote consumer i’s earning stream by ei = (ei1 , ei2 ). That is, consumer i earns ei1 units of consumption good in Period 1 and ei2 units in Period 2. His consumption stream denoted let’s say by xi = (xi1 , xi2 ) means that he consumes xi1 in Period 1 and xi2 units in Period 2. Given interest rate r, suppose i consumes xi1 units in Period 1. Then he is saving ei1 − xi1 in Period 1, which is borrowing if it is negative. In Period 2, his disposable income comes from two things, one is his earning ei2 and the other is return from saving (1+r)(ei1 −xi1 ), which is debt repayment if it is negative. Since he dies in the end of Period 2, his consumption in Period 2, xi2 , must be equal to this disposable income in Period 2. Hence we obtain xi2 = ei2 + (1 + r)(ei1 − xi1 ).

CHAPTER 12. COMPETITIVE EXCHANGE ECONOMIES

153

By rearranging the above formula, we obtain the lifetime budget constraint in the present-value form xi2 ei2 xi1 + = ei1 + 1+r 1+r Thus, consumer i’s consumption behavior is described as a solution to max vi (xi1 ) + βi vi (xi2 ) xi

ei2 xi2 = ei1 + 1+r 1+r and we obtain his demand for consumption at each period denoted by (xi1 (r), xi2 (r)). This is nothing but a special case of the market model above, where the price of current consumption is normalized to 1 and the price of future consumption is taken to be the inverse of gross interest rate, that is, subject to

xi1 +

p1 = 1, p2 =

1 p1 , and =1+r 1+r p2

The interest rate in competitive equilibrium r∗ is determined so that demand matches supply in current consumption and future consumption respectively (assuming that you accepted my apologies in the previous chapter), n ∑



xi1 (r ) =

i=1 n ∑

n ∑

ei1

i=1

xi2 (r∗ ) =

i=1

n ∑

ei2

i=1

In other words, the interest rate is determined so that borrowing demand matches lending supply. Then, if xi1 (r∗ ) − ei1 > 0, xi2 (r∗ ) − ei2 < 0 it means consumer i is borrowing in Period 1, and if xi1 (r∗ ) − ei1 < 0,

xi2 (r∗ ) − ei2 > 0

it means he is saving and lending in Period 1. Now for example, assume that the periodwise evaluation function is vi (z) = ln z for all i then it is a special case of Cobb-Douglas preference applied over 1 for each i. By applying the previous consumption streams, where αi = 1+β i result we obtain the equilibrium interest rate ∑n 1 p∗1 i=1 1+βi ei2 ∗ 1 + r = ∗ = ∑n βi p2 i=1 1+βi ei1 From this we see the following comparative statics result (while this holds for more general class of preferences).

CHAPTER 12. COMPETITIVE EXCHANGE ECONOMIES

154

1. As earnings in the current period are larger the equilibrium interest is lower. 2. As earnings in the future period are larger the equilibrium interest is higher.

12.3.2

Ricardian equivalence

Consider that government expenditure is g1 units in terms of consumption good at Period 1 and g2 units in Period 2. Suppose they are given and consider how to finance them by a combination of issuing bond and taxation, where the market interest rate r is given. Below, everything is measured in terms of consumption good in the corresponding period. Let d1 denote the amount of debt at the end of Period 1, let h1 denote tax revenue at Period 1 and h2 for Period 2. Then we have g1 = d1 + h1 Since this economy ends at the end of Period 2 all debts must be repaid then. Hence we have g2 + (1 + r)d1 = h2 The we can derive the consolidated budget constraint of the government in the form g2 h2 g1 + = h1 + r r Denote consumer i’s tax payment in Period 1 by hi1 and that in Period 2 by hi2 . Also, let bi1 denotes the amount of purchase of bond in Period 1, and ai1 denote the amount of purchase of other financial assets. Then in Period 1 his consumption xi1 must satisfy the constraint xi1 + ai1 + bi1 + hi1 = ei1 In Period 2, his consumption xi2 must satisfy xi2 + hi2 = ei2 + (1 + r)(ai1 + bi1 ). Then by the same argument as before, we obtain the consumer’s lifetime budget constraint ( ) xi2 ei2 hi2 xi1 + = ei1 + − hi1 + 1+r 1+r 1+r This means that this lifetime consumption path depends only on the present value of his lifetime earning and the present value of the government’s taxation schedule on him. Now consider that there is a representative consumer, then his lifetime budget constraint is ( ) ( ) e2 h2 e2 g2 x2 = e1 + − h1 + = e1 + − g1 + , x1 + 1+r 1+r 1+r 1+r 1+r

CHAPTER 12. COMPETITIVE EXCHANGE ECONOMIES

155

which depends only on the government’s expenditure schedule, and it is not affected by how the government mixes between taxation and debt issuance. Remark 12.1 We should note that the government policy on taxation and debt still matters when distributional concerns matter, particularly when it is an intergenerational one. Also note that the above equivalence relies on the assumption of complete market, which I come to in the next section.

12.3.3

Liquidity constraint and market incompleteness

Go back to the lifetime budget constraint for consumer i, xi1 +

xi2 ei2 = ei1 + 1+r 1+r

What is implicit here is that the consumer obeys no liquidity constraint. ei2 Note that present value of his lifetime income is ei1 + 1+r . This would mean ei2 that when he can borrow against his lifetime income he can buy ei1 + 1+r units of current consumption good. However, even if the present value of your lifetime income is certainly 1 million it is a different question if you can borrow 1 million from a bank. If your borrowing ability or ability of financing does not catch up with your earning ability, you are said to obey liquidity constraint. If consumer i obeys borrowing constraint how does his temporal budget constraint look like? Let bi be the upper limit for borrowing by i which is imposed for some reason. Thus we have an additional constraint ei1 − xi1 ≥ −bi . Summing up, i’s lifetime budget constraint with borrowing limit is ei1 − xi1 xi2 xi1 + 1+r

≥ −bi = ei1 +

ei2 1+r

Note that the first constraint can be written xi1 ≤ ei1 + bi as well, which means that his current consumption cannot exceed the sum of current earning plus the upper limit of borrowing. Figure 12.3 depicts the case that the borrowing constraint does not bind and it does not affect the consumer’s intertemporal consumption. This is the case ei2 when ei1 + bi ≥ ei1 + 1+r . Figure 12.4 depicts the case that the borrowing constraint binds and it affects the consumer’s intertemporal consumption. This is the case when ei1 + bi ≤ ei2 ei1 + 1+r . The assumption of complete market says that one can exchange between any two goods. In the intertemporal context this means one can freely exchange between current consumption and future consumption. The absence of liquidity constraint thus guarantees market completeness.

CHAPTER 12. COMPETITIVE EXCHANGE ECONOMIES

Period 2 6

r ei ei1 + bi

- Period 1

Figure 12.3: Non-binding case

Period 2 6 r ei

ei1 + bi

- Period 1

Figure 12.4: Binding case

156

CHAPTER 12. COMPETITIVE EXCHANGE ECONOMIES

157

However, when there exists liquidity constraint and it binds the complete market assumption fails, and one cannot freely exchange between current consumption and future consumption. Then, marginal rates of substitution of future consumption for current consumption may not be equalized across individuals in competitive equilibrium. This causes inefficiency, as we will show in the next chapter that equalization of MRS is necessary for efficiency. Under liquidity constraints there may be a consumer who wants to borrow more and has enough earning for repayment indeed but cannot borrow, and a consumer who likes to save more but nobody can borrow from him because of the constraint despite he is willing to. Liquidity constraint is thus a friction as it is. There might be a role for such friction, however, if there is a dimension of bounded rationality such self-control problem. It is known that when there are conflicts between current self and future selves as illustrated in the chapter on intertemporal choice the borrowing constraint has a role of ”commitment device” in the sense that it prevents the current self from over-borrowing, which makes both the current self and future selves better off. See Laibson [17] for more details.

12.4

Security exchange and security price

12.4.1

Risk-sharing through security exchange

Next let us apply the model of competitive market to security exchange. As as before, we assume for simplicity that there are just two states. Let π denote the probability of State 1, and 1−π be the probability of State 2. In this application Good 1 is taken to be the consumption good to be delivered if State 1 occurs and Good 2 is taken to be the one to be delivered if State 2 occurs. As discussed before, such specification of goods are interpreted as follows Good 1 = a security which promises to deliver 1 unit of consumption per unit of it if State 1 occurs, and is a junk if State 2 occurs. Good 1 = a security which is a junk if State 1 occurs, and promises to deliver 1 unit of consumption per unit of it if State 2 occurs. Such securities are called Arrow securities. Now consumer i’s initial holding of ei = (ei1 , ei2 ) means that he earns ei1 units of consumption good id State 1 occurs, and ei2 units if State 2 occurs. Under the security interpretation, it means he is initially holding ei1 units of Arrow security 1 and ei2 units of Arrow security 2. Consider for example that the uncertainty is about presidential election in US. Let State 1 refers to the winning of Republicans and State 2 refers to the winning of Democrats. For simplicity, let me assume that A is entirely on the Republican side and B is entirely on the Democrat side, and the one in the winner-side receives all of resources falling from somewhere, let’s say 10 units of consumption good. Then A’s initial holding is eA = (10, 0), and B’s one is

CHAPTER 12. COMPETITIVE EXCHANGE ECONOMIES

158

eB = (0, 10). In terms of securities, A holds 10 units of ”Republicans stock” and 0 units of ”Democrats stock,” whereas B holds 0 units of ”Republicans stock” and 10 units of ”Democrats stock.” If there is no exchange the result is simply that A gets entire 10 units if Republicans win and B gets entire 10 units if Democrats win. Is that reasonable? No, since people are more or less risk-averse in general they will try to exchange the risks. For example, each of them can receive 5 units regardless of the election outcome, by means of a trade that A sells 5 units of the ”Republicans stock” and buys 5 units of the ”Democrats stock” whereas B buys 5 units of the ”Republicans stock” and sells 5 units the ”Democrats stock.” This example is a symmetric one, but it is not symmetric in general. If Republicans are likely to win A will be unwilling to sell the ”Republicans stock” and it will be more expensive relative to ”Democrats stock.” Let us see how the security prices are determined based on the competitive market model. Again, when a vector of i’s state-contingent consumption by xi = (xi1 , xi2 ) is given it means i consumes xi1 units if State 1 occurs and xi2 units if State 2 occurs. Given a vector of security prices p = (p1 , p2 ), i chooses hid demand for statecontingent consumptions (or Arrow securities, equivalently speaking) (xi1 (p), xi2 (p)) so that it solves max πvi (xi1 ) + (1 − π)vi (xi2 ) xi

subject to

p1 xi1 + p2 xi2 = p1 ei1 + p2 ei2

Here the budget constraint states that the market value of portfolio after the exchange is equal to the market of value of the initial portfolio. Then, the competitive equilibrium security price vector p∗ = (p∗1 , p∗2 ) is determined so that demand for each security matches its supply. (assuming that you accepted my apologies in the previous chapter), that is, n ∑

xi1 (p∗ ) =

i=1 n ∑

ei1

i=1

xi2 (p∗ ) =

i=1

If we have

n ∑

xi1 (p∗ ) − ei1 < 0,

n ∑

ei2

i=1

xi2 (p∗ ) − ei2 > 0

individual i is selling Arrow security 1 and buying Arrow security 2 in equilibrium, and similarly for the opposite case. Now for example, assume that the vNM index is vi (z) = ln z for all i then it is a special case of Cobb-Douglas preference applied over consumption streams, where αi = π for all i. By applying the previous result we obtain the relative price of Security 1 for Security 2 in equilibrium, ∑n π i=1 ei2 p∗1 ∑n . = p∗2 (1 − π) i=1 ei1

CHAPTER 12. COMPETITIVE EXCHANGE ECONOMIES

159

From the above formula we see that 1. As the probability of State 1 is higher the relative price of Arrow security 1 is higher. Similarly for State 2 and Arrow security 2. 2. As the resources available at State is more the relative price of Arrow security 1 is lower. Similarly for State 2 and Arrow security 2. Point 2 might be counterintuitive. However, you can understood that by seeing that when the resources at State 1 are scarce people need to prepare for that by means of buying Arrow security 1 and it makes its relative price higher.

12.4.2

The case of no aggregate risk

Let us consider the simplest case of so-called no aggregate risk. It is the case where the total amount of resources in the economy is riskless, and the uncertainty is only about who will get how much, which is described by n ∑

ei1 =

i=1

n ∑

ei2 .

i=1

Imagine a situation such that some fixed amount of resources is falling from the heaven and there is an uncertainty about who catches how much. It is realistic in some situation, however. For example, the number of car accident is quite constant every year in the entire society, while there is of course an uncertainty about who hits a crash. Recall that marginal rate of substitution given by a preference having expected utility representation takes the form M RSi (xi ) =

vi′ (xi1 ) (1 − π)vi′ (xi2 )

π for each i. Note that this is equal to 1−π when xi1 = xi2 , that is, when his consumption is riskless. We call this 45-degree line the certainty line since consumption is riskless on it. Now in the case of no-aggregate risk, as depicted in Figure 12.5 the Edgeworth box is exactly a square and the certainty line for A coincides with that for B. On the common certainty line, we have

M RSi (xi ) =

π 1−π

for each i. In competitive equilibrium all consumers’ MRSs are equalized through the relative price of Good 1 for Good 2, the above condition, and the above condition implies that it is met when all the consumers’ consumptions are riskless. Thus we can take the equilibrium relative price of Arrow security 1 for Arrow security 2 to be π p∗1 . = p∗2 1−π

CHAPTER 12. COMPETITIVE EXCHANGE ECONOMIES

160

A’s Good 2 6 B’s Good1

IA re

IB

OB xr ∗

-A’s Good 1 ? B’s Good 2 Figure 12.5: Competitive equilibrium under no aggregate risk OA

As in Figure 12.5, the equilibrium allocation is obtained as the intersection π of the common certainty line and the line with slope − 1−π passing through the initial endowment point. There all consumers receive riskless consumptions whether State 1 or 2 occurs. This is called perfect insurance, and it holds under the no aggregate risk condition regardless of particular functional forms of vNM indices as far as they exhibit risk aversion.

12.4.3

Risk-sharing between a risk-neutral agent and a risk-averse individual

Next consider risk sharing between a risk-neutral agent and a risk-averse individual. This situation appears very often in applied analysis. Here the risk-neutral agent is seen to be a large body which pools a large number of independent idiosyncratic risks so that in aggregate its performance is statistically certain thanks to the law of large numbers, such as financial institution. Let A be the small individual and B be the large body. Maintain the assumption that there are two states of the world, Stare 1 and State 2. Let π be the probability of State 1, then the probability of State 2 is 1 − π. Let me depict the situation as in Figure 12.6. At e the initial endowment point B is holding a large and riskless wealth, and A is facing a risk of earning which is small from B’s viewpoint. Hence trade between them is limited to occur in a small region such as the little Edgeworth box taken around A’s origin, in which B’s origin ′ OB is suitably adjusted. Notice that curves are straight, locally. Therefore B’s indifference curves are almost straight within the smaller Edgeworth box. Now recall that along B’s π certainty line the slopes of B’s indifference curves are always 1−π . Therefore within the small Edgeworth box B’s indifference curves are straight and parallel, π . In this sense, an expected utility their slopes are constant and equal to 1−π

CHAPTER 12. COMPETITIVE EXCHANGE ECONOMIES

161

A’d Good 2 B’s Good 1



OB

6

IB 

r e

′ OB

- A’s Good 1 OA

? B’s Good 2 Figure 12.6: Risk sharing between a large body and a small individual ?

preference can be seen to be risk-neutral when the relevant risk is small. Now let us magnify the small Edgeworth box in Figure 12.6 then we obtain an Edgeworth box looking like Figure 12.7. Since B’s indifference curves are π straight and parallel, their constant slope 1−π determines the relative price of consumption at State 1 for consumption at State 2. Given this, A’s optimal π consumption is always on his certainty line on which A’s MRS equals to 1−π . Since B is indifferent between any consumptions which yield the same expected amount of consumption, the competitive equilibrium allocation comes to x∗ , π where the straight line with slope 1−π is the budget line and at the same time B’s corresponding indifference curve. Thus, in competitive equilibrium the risk-neutral agent takes the entire risk and the risk-averse individual is guaranteed to receive riskless consumption.

12.4.4

Incomplete security markets

The above argument presumes that consumer can freely transfer his resources across states by means of trading securities. This is what the complete market assumption says in the current context. The security market is said to be incomplete when consumers cannot freely transfer their resources across states. To illustrate, consider that there are three states of the world, where con-

CHAPTER 12. COMPETITIVE EXCHANGE ECONOMIES

162

A’s Good 2 6 B’s Good 1  IB

IA

OB xr ∗ r e

-A’s Good 1 ? B’s Good 2 Figure 12.7: Risk charing between risk-averse and risk-neutral agents OA

sumer i’s state-contingent consumption is denoted by xi = (xi1 , xi2 , xi3 ) and his state-contingent earning is denoted by ei = (ei1 , ei2 , ei3 ). On the other hand, there are only two securities available to trade despite there are three states. Suppose let’s say that one unit of Security 1 pays one unit of consumption if State 1 occurs and nothing otherwise, and one unit of Security 2 pays one unit of consumption if State 2 occurs and nothing otherwise. Notice that there is no security which helps you to move resources to and from State 3. Let p1 denote the price of Security 1 and p2 denote the price of Security 2. Let zi1 denote consumer i’s holding of Security 1 and zi2 denote his holding of Security 2, where negative holdings are allowed which means short-selling. Then his portfolio choice zi = (zi1 , zi2 ) has to obey the budget constraint p1 zi1 + p2 zi2 = 0 where the right-hand-side is zero since the initial income is zero. If State 1 occurs he receives his earning ei1 plus return from Security 1 (or payment if he was short-selling it) zi1 . Hence his consumption is then xi1 = ei1 + zi1 Likewise, if State 2 occurs he receives his earning ei2 plus return from Security 2 (or payment if he was short-selling it) zi2 . Hence his consumption is then xi2 = ei1 + zi2 On the other hand, if State 3 occurs he simply receives his earning ei3 and nothing else. Hence his consumption is then xi3 = ei3

CHAPTER 12. COMPETITIVE EXCHANGE ECONOMIES

163

Summing up, his budget constraint is p1 zi1 + p2 zi2

= 0

xi1 xi2 xi3

= ei1 + zi1 = ei1 + zi2 = ei3

Notice that by replacing zi1 and zi2 in the first equation by those in the second and third the above equations reduce to p1 xi1 + p2 xi2 xi3

= =

p1 ei1 + p2 ei2 ei3

This does not fall in the corresponding budget constraint under the complete market assumption. In order that the assumption of complete market is met, there has to be another security. Let’s say that one unit of Security 3 pays one unit of consumption if State 3 occurs and nothing otherwise. Again let p1 denote the price of Security 1, p2 denote the price of Security 2 and p3 denote the price of Security 3. Let zi1 denote consumer i’s holding of Security 1, zi2 denote his holding of Security 2 and zi3 denote his holding of Security 3, where negative holdings are allowed which means short-selling. Then his portfolio choice zi = (zi1 , zi2 , zi3 ) has to obey the budget constraint p1 zi1 + p2 zi2 + p3 zi3 = 0 where the right-hand-side is zero since the initial income is zero. By the similar argument as above we obtain p1 zi1 + p2 zi2 + p3 zi3 xi1

= 0 = ei1 + zi1

xi2 xi3

= ei1 + zi2 = ei3 + zi3

Now this reduces to the standard budget constraint for three goods, p1 xi1 + p2 xi2 + p3 xi3

= p1 ei1 + p2 ei2 + p3 ei3

in which one can freely exchange between Good 1 and Good 2, Good 2 and Good 3, Good 3 and Good 1. When the security market is incomplete the consumers cannot share risks in an efficient manner (while I’ll come to the definition of efficiency in the next chapter). For example consider the simplest case that there are two states of the world but there it no security, or just one which doesn’t help at all for the diversification purpose. Then the only thing consumers can do is to eat their

CHAPTER 12. COMPETITIVE EXCHANGE ECONOMIES

164

earnings without any trades across states, which is typically inefficient from the risk-hedging viewpoint. A reader might think it is better to have more kinds of securities available to trade in order to help risk-hedging. This answer is in general NO! It is always better to go from nothing to some, but it is known that everyone may lose when more securities get available to trade in addition to some existing ones. See Hart [12] if you are interested.

12.5

Exercise

Exercise 14 Consider an exchange economy consisting of n consumers, where √ √ each i = 1, · · · , n has preference represented by ui (xi ) = ai xi1 + bi xi2 and has initial holding ei = (ei1 , ei2 ). Find the equation met by the competitive p∗ equilibrium relative price p1∗ . You don’t have to be able to solve the equation, 2 though. Exercise 15 Consider a security exchange market with two states, where the probability of State 1 is 3/4 and the probability of State 2 is 1/4. There are two consumers, A and B, whose preferences fall in the expected utility theory. A”s state-contingent earning is (2, 6), which means he earns 2 if State 1 occurs and 6 if State 2 occurs. Likewise, B’s state contingent earning is (8, 4). Find the relative price of Arrow security 1 for Arrow security 2 in competitive equilibrium and the resulting allocation of state-contingent consumptions.

Chapter 13

Efficiency of competitive allocation 13.1

Pareto improvement and Pareto efficiency

Now, is the competitive market (if it really exists) a ”good” way of resource allocation? Of course it depends on what we mean by ”good.” It is at least known that competitive equilibrium allocation satisfies so-called ”Pareto efficiency.” Some books call it ”Pareto optimality,” but for the reason I will state later it is far from what we imagine from the word ”optimality,” throughout this book we use the term ”Pareto efficiency.” Let me start with the two-person case. See Figure 13.1 and look at point x = (xA , xB ). Here IA (resp. IB ) is the indifference curve passing through xA (resp. xB ). Is this a ”good” allocation? It is not desirable in the following sense. Here you can take a point let’s say y = (yA , yB ) from the lens-shaped region surrounded by IA and IB . Since yA is above A’s indifference curve passing through xA , yB is above B’s indifference curve passing through xB from B’s viewpoint, allocation y is better than x for both A and B. Since both say that y is better than x, there will be no reason to object to it.1 Also, when you take a point in the boundary of the lens such as z = (zA , zB ), this makes A better off without hurting B, because B is indifferent between zB and xB and A strictly prefers zA to xA . Since we are not hurting B here there will be no reason to object to rank z over x socially.2 Such change which makes everybody better off or makes somebody better off without hurting anybody else is called Pareto improvement. Formally, Definition 13.1 An allocation y = (y1 , · · · , yn ) is a Pareto improvement of 1 I’m

aware of the standpoint saying ”No, there is a reason to object to it.” I’m aware of the standpoint saying that there IS a reason to object to it.

2 Again,

165

CHAPTER 13. EFFICIENCY

166

A’s Good 2 - IA 6 B’s Good1 rx ? IB

OB

rz rq ry

- A’s Good 1 ? B’s Good 2 Figure 13.1: Pareto improvement and Pareto efficiency OA

x = (x1 , · · · , xn ) if it holds for all i = 1 · · · , n, and

yi ≿i xi yi ≻i xi

for at least one i. Some books adopt the definition of Pareto improvement saying ”yi ≻ xi for all i = 1, · · · , n.” If an allocation is a Pareto improvement of another in this sense it is a Pareto improvement in our definition too, for if you are making all people better off it implies you are making at somebody without hurting anybody else. The converse may not be true as it says, however. Hence we are adopting a weaker definition for the improvement.3 This difference does not make a difference in the economy we are talking about now, however, in which we can vary quantities of goods in a continuous manner. For example, when yA ≻A xA and yB ∼B xB we can move slight amounts (∆x1 , ∆x2 ) from A to B so that B is strictly better off while A is still being strictly better off than xA , that is, we have (yA1 − ∆x1 , yA2 − ∆x2 ) ≻A xA and (yB1 + ∆x1 , yB2 + ∆x2 ) ≻B xB . Go back to Figure 13.1. Is y a ”good” allocation then? Again draw the corresponding indifference curves passing through y = (yA , yB ) wee that we can pick a point such as q = (qA , qB ) from the lens-shaped area, which is a Pareto improvement of y. Now what about q? When you draw the corresponding indifference curves this time we don’t have a lens-shaped area any longer, which means given q there is no feasible allocation which makes somebody better off without hurting anybody else. Then we say that allocation q is Paretoefficient. 3 It

means the definition of efficiency introduced later is stronger.

CHAPTER 13. EFFICIENCY

167

Now let me state the definition of feasible allocation again. Definition 13.2 Given a social sum of endowment (e1 , e2 ) = ( say that an allocation x = (x1 , · · · , xn ) is feasible if n ∑

∑n

i=1 ei1 ,

∑n

i=1 ei2 ),

xi1 = e1

i=1 n ∑

xi2 = e2

i=1

Definition 13.3 An allocation x is said to be Pareto-efficient when there is no feasible allocation which is a Pareto-improvement of it. In other words, an allocation is Pareto efficient when it is impossible to make anybody better off without hurting anybody else. When no confusion arises I would sometimes write ”efficiency” or ”efficient” instead of ”Pareto-efficiency” or ”Pareto-efficient.” When preferences are smooth and we restrict attention to allocations strictly inside the box (which are called interior allocations), having ”no lens-shaped area” is characterized by that the corresponding indifference curves of A and B are tangent to each other. That is, the local slope of A’s corresponding indifference curve is equal to the local slope of B’s one. Thus, Pareto efficiency is characterized by the condition that the marginal rates of substitution for A and B are equalized. In other words, at an efficient allocation subjective values of a good for another one are equal across individuals at margin. This is true not only for the two-person case. The following holds in general. Proposition 13.1 Suppose preferences are smooth. Then, an interior allocation x = (x1 , · · · , xn ) is Pareto efficient if and only if M RSi (xi ) = M RSj (xj ) for all i and j. Proof. (Efficiency =⇒ Equalization of MRSs): Pick any i, j and suppose M RSi (xi ) > M RSj (xj ) Recall that marginal rate of substitution of Good 2 for Good 1 is the amount of Good 2 one is willing to give up in order to get an extra one unit of Good 1. Since i’s such amount is larger than j’s such amount, one can pick p in between, such that M RSi (xi ) > p > M RSj (xj ). Now consider moving a ”slight amount” of Good 1, denoted ∆x1 , from j to i and moving a ”slight amount” of Good 2 given by p∆x1 from i to j. Then we can make both i and j better off while

CHAPTER 13. EFFICIENCY

168

keeping the welfare level of anybody else the same. For, i’s amount of Good 2 he has to give up is smaller than the amount of it he is willing to give up, and j’s amount of Good 2 he receives is larger than the amount of it he likes to receive at least. Hence there is a room for a Pareto improvement of x, which means x is not Pareto efficient. (Equalization of MRS =⇒ Efficienfy): Suppose MRSs are equalized at x, that is, we have M RSi (xi ) = µ > 0, i = 1, · · · , n Now suppose x is not Pareto efficient. Then there is an another feasible allocation ((x11 + ∆x11 , x12 + ∆x12 ), · · · , (xn1 + ∆xn1 , xn2 + ∆xn2 )) such that (xi1 + ∆xi1 , xi2 + ∆xi2 ) ≿i xi for all i = 1 · · · , n and (xi1 + ∆xi1 , xi2 + ∆xi2 ) ≻i xi ∑n ∑n for at least one i where i=1 ∆xi1 = i=1 ∆xi2 = 0. When preferences are smooth, the consumption (xi1 + ∆xi1 , xi2 + ∆xi2 ) is above the tangent line passing through xi for all i = 1 · · · , n, we have ∆xi2 ≥ −M RSi (xi )∆xi1 Also, since (xi1 + ∆xi1 , xi2 + ∆xi2 ) is strictly above the tangent line passing through xi for at least one i, we have ∆xi2 > −M RSi (xi )∆xi1 for such i. Denote the equalized value of MRSs by µ, then we have M RSi (xi ) = µ for all i = 1 · · · , n. Then by adding up the above inequalities we obtain n ∑ i=1

∆xi2 > −µ

n ∑

∆xi2 .

i=1

However, this contradicts to the assumed condition 0.

∑n i=1

∆xi1 =

∑n i=1

∆xi2 =

Notice here that there may be arbitrarily many Pareto-efficient allocations. As illustrated in Figure 13.2, we can draw arbitrarily many pairs indifference curves which are tangent to each other. We can actually draw a continuous curve by depicting points at which such tangency holds. In the current setting in which the goods are continuously divisible, there is actually a continuum of Pareto-efficient allocations.

CHAPTER 13. EFFICIENCY

169

A’s Good 2 - IA 6 B’s Good1

OB r

? IB

r r

- A’s Good 1 ? B’s Good 2 Figure 13.2: Set of Pareto efficient allocations OA

13.2

Efficiency of competitive equilibrium allocation

We see that competitive equilibrium allocation is Pareto-efficient when the preferences are smooth and it ends up to be an interior allocation, by seeing that MRSs of all consumers are equal to the relative price, that is, M RSi (xi ) = M RSj (xj ) =

p1 p2

for all i and j. We see that in the Edgeworth box diagram by seeing that the corresponding indifference curves of A and B are tangent to each other and to the budget line at the equilibrium allocation point. We can actually show that Pareto efficiency of competitive equilibrium allocation without relying on the assumption of smooth preference and its being interior. The following is called the first fundamental theorem of welfare economics. Theorem 13.1 Suppose the individuals’ preferences are monotone at least in the weak sense, then competitive equilibrium allocation is Pareto-efficient. Proof. Let p∗ = (p∗1 , p∗2 ) the equilibrium price, and let x∗ = (x∗1 , · · · , x∗n ) denote the equilibrium allocation. Suppose x∗ is not Pareto-efficient, then there is a feasible allocation x = (x1 , · · · , xn ) such that xi ≿i x∗i for all i = 1 · · · , n, and

xi ≻i x∗i

CHAPTER 13. EFFICIENCY

170

∑n ∑n ∑n ∑n ∗ ∗ for at least ∑ one i, where i=1 xi1 = i=1 xi1 = i=1 ei1 and i=1 xi2 = ∑ n n i=1 xi2 = i=1 ei2 hold because of feasibility. Recall that for each i his consumption in equilibrium x∗i is his optimal choice given the budget constraint p∗1 xi1 +p∗2 xi2 ≤ p∗1 ei1 +p∗2 ei2 . We obtain the following lemma because of this. Lemma 13.1 When xi ≿i x∗i it holds p∗1 xi1 + p∗2 xi2 ≥ p∗1 ei1 + p∗2 ei2 Proof. Suppose p∗1 xi1 + p∗2 xi2 < p∗1 ei1 + p∗2 ei2 , then we can sightly increase the quantities of Good 1 and Good 2, by ∆x1 > 0 and ∆x2 > 0 respectively, so that the budget constraint is still met by p∗1 (xi1 +∆x1 )+p∗2 (xi2 +∆x2 ) < p∗1 ei1 +p∗2 ei2 . By the weak monotonicity of preference we have (xi1 + ∆x1 , xi2 + ∆x2 ) ≻i xi . By transitivity of preference we get (xi1 + ∆x1 , xi2 + ∆x2 ) ≻i x∗i , but this contradicts to the assumption that x∗i is an optimal choice for i under the given budget constraint. Recall that we have xi ≻i x∗i for at least one i, but since x∗i is an optimal choice for i under the given budget constraint the strictly better consumption xi must have not been affordable to him. Hence it must hold p∗1 xi1 + p∗2 xi2 > p∗1 ei1 + p∗2 ei2 for such i. By adding up the inequalities above we obtain p1

n ∑ i=1

xi1 + p2

n ∑ i=1

xi2 > p1

n ∑

ei1 + p2

i=1

but this contradicts to the feasibility condition ∑ n i=1 ei2 .

n ∑

ei2 ,

i=1

∑n i=1

xi1 =

∑n

i=1 ei1

and

∑n i=1

xi2 =

The name ”first fundamental theorem” suggests that there is second one. It says that any Pareto-efficient allocation can be obtained through competitive equilibrium after suitable redistribution of income. Let x = (x1 , · · · , xn ) be any Pareto-efficient allocation. Since MRSs are equalized under efficiency, we have M RSi (xi ) = M RSj (xj ) for all i and j. This equalized MRS is taken to be the ”targeted price” to be obtained in competitive equilibrium. Thus one can take the ”targeted price” (p1 , p2 ) such that p1 = M RSi (xi ) p2 for all i.

CHAPTER 13. EFFICIENCY

171

In order to obtain this allocation in competitive equilibrium we need to redistribute income so that i’s income after the redistribution is p1 xi1 + p2 xi2 . Let p x + p2 xi2 ∑n 1 i1 ∑n λi = p1 j=1 xj1 + p2 j=1 xj2 be i’s wealth proportion compared to the entire economy. Here we can redistribute initial endowments by ei1 = λi e1 , ei2 = λi e2 . Then xi is an optimal choice for i under the budget constraint p1 xi1 + p2 xi2 = p1 ei1 + p2 ei2 , since MRS is equal to the relative price. Thus x = (x1 , · · · , xn ) is obtained as a competitive equilibrium allocation starting from initial endowments e = (e1 , · · · , en ) with equilibrium price (p1 , p2 ). This result does not depend on smoothness of preferences, but I relegate the proof to an advanced book such as Mas-Colell, Whinston and Green [21]. Theorem 13.2 (Second fundamental theorem of welfare economics): Assume that preferences are convex and monotone. Then any Pareto-efficient allocation is obtained in competitive equilibrium staring from suitably redistributed initial endowments.

13.3

Important remarks on Pareto efficiency

I give two very important remarks. One is already mentioned. 1: There are arbitrarily many Pareto-efficient allocations Again, as in Figure 13.2 we can draw arbitrarily many pairs of indifference curves of A and B which are tangent to each other. By depicting such points at which such tangency occurs we obtain a continuous curves. Thus we have a continuum of Pareto-efficient points. This is due to that the relation of Pareto-improvement does not satisfy completeness. For example, if A prefers xA to yA and B prefers yB to xB then none of x = (xA , xB ) and y = (yA , yB ) is a Pareto-improvement of the other. In particular, the Pareto criterion cannot rank between efficient allocations (because any efficient allocation is such that it cannot be improved upon!). That’s why I do not adopt the term ”Pareto optimality,” which will give a wrong impression that it determines a complete ranking which leads to a single solution. Also, notice that B’s origin at which A gets all the resources and B gets nothing is Pareto-efficient. Since the tangency argument does not apply on the boundary of the box I show this directly. Since A is receiving all the resources

CHAPTER 13. EFFICIENCY

172

it is impossible to make him better off. If you wan to make B better off on the other hand, you have to take resources from A, either Good 1 or 2 or both, which has to hurt A. Thus it is impossible to make A better of without hurting B and it is impossible to make B better of without hurting A, meaning that the allocation is Pareto-efficient. Likewise, A’s origin at which B gets all the resources and A gets nothing is Pereto-efficient. This also suggests the next point. 2: Efficiency is orthogonal to fairness Of course it will depend of the definition of ”fairness,” but whatever definition you adopt it will be obviously ”unfair” if A gets all the resources and similarly if B gets all, while such allocations meet the criterion of Pareto efficiency. We should not have a misunderstanding like ”pursuing efficiency sacrifices fairness.” What is said here is just that efficiency is silent about any notion of fairness, that, is they are orthogonal to each other. It is thus a different question if efficiency and fairness are compatible each other when any notion of fairness is proposed. See Chapter 29 for the details. We should not underrate the importance of Pareto efficiency, however. It has been a central issue in economics since its beginning that despite there are opportunities in which everybody gains from trades the society sometimes cannot fully utilize them.

13.4

Exercises

Exercise 16 There are two consumers, A and B, whose preferences are represented by uA (xA ) = λA ln xA1 + µA ln xA2 , uB (xB ) = λB ln xB1 + µB ln xB2 , respectively. Let e1 denote the total amount of Good 1 and e2 denote the total amount of Good 2. Find the set of Pareto efficient allocations.

Chapter 14

Production technology Let me start talking about production economy. Here I start with production technology.

14.1

1-input/1-output case

First let us consider the simplest case, in which there is one input good and one output good. Denote the quantity of input by x and denote the quantity of output by y, then production technology is described by a production function which relates between x and y in the form y = f (x) This simply means that y = f (x) units of output are produced from x units of input. The minimally necessary property of production function will be that f (0) = 0 That is, if nothing is input nothing is output. Also, it is necessary that x > x′ =⇒ f (x) ≥ f (x′ ) That is, when you increase input the output does not decrease (it is possible that the output does not increase, but at least it does not decrease). Below you will see some mathematical analogies between production function and utility function. They are conceptually different, however, since utility function is only a representation ranking and its value has no quantitative meanings, but value of production function is physical quantity of the output good and has quantitative meanings. One can classify production technology roughly into three. Production technology is said to exhibit 173

CHAPTER 14. PRODUCTION TECHNOLOGY y

6

174

Increasing returns

Constant returns Decreasing returns

-x Figure 14.1: Production function

1. constant returns to scale if it holds f (tx) = tf (x) for t > 1; 2. decreasing returns to scale if it holds f (tx) < tf (x) for t > 1; 3. increasing returns to scale if it holds f (tx) > tf (x) for t > 1. Consider for example the case of t = 2, that is, the case that we double the input. Then, constant returns says that the output is doubled exactly, decreasing returns says that the output is less than the double, and increasing returns says that the output is more than the double. You can think of combinations of them, of course. For example, you can think of a technology which exhibits increasing returns to scale when input is small and exhibits decreasing returns to scale when input is large. In the one input case, production function has the linear form f (x) = ax when it exhibits constant returns to scale, has its graph convex to the top (meaning it is a concave function) when it exhibits decreasing returns to scale, and has its graph convex to the bottom (meaning it is a convex function) when it exhibits decreasing returns to scale. See Figure 14.1. Next let me introduce the notion of marginal product, which is the amount of additional output obtained as you add one extra unit of input. Consider for example that the production function is linear and given by f (x) = ax. Then the marginal product is nothing but the slope coefficient a. In general, however, we have to look at local slope since returns to scale may not be constant and may decrease or increase. When the production function is ”smooth” then marginal product is given by the derivative of the production function. That is, the marginal product at input level x is given by f ′ (x) We say that production technology exhibits

CHAPTER 14. PRODUCTION TECHNOLOGY

175

1. constant marginal product if f ′ (x) is constant over x; 2. decreasing marginal product if f ′ (x) is decreasing in x; 3. increasing marginal product if f ′ (x) is increasing in x; Since how marginal product changes is given by f ′′ (x), which is obtained by taking the derivative again, constancy of marginal product is characterized by f ′′ (x) = 0, decreasing marginal product is characterized by f ′′ (x) < 0,and increasing marginal product is characterized by f ′′ (x) > 0. In the one-input case, we have 1. constant returns to scale = constant marginal product 2. decreasing returns to scale = decreasing marginal product 3. increasing returns to scale = increasing marginal product Note that this is not true when there are more than one inputs.

14.2

2-input/1-output case

14.2.1

Production function

Now we consider the case of two inputs and one output. While this may be extended to the case of many inputs the two-input illustration is enough for our purpose. Here a combination of inputs is denoted by a vector x = (x1 , x2 ), which means the amount of Input 1 is x1 units and the amount of Input 2 is x2 units. The amount of output is denoted by y as before. Then production technology is described by a production function which relates between y and x y = f (x) Graph of a production function may be depicted as in Figure 14.2. Here a curve obtained by connecting point which yield the same output level is called an isoquant curve, which resembles a level curve of a mountain. Such treatment of production function looks mathematically analogous to the treatment of utility function, but they are conceptually different. While utility function is no more than a representation of ranking and its value has no quantitative meaning, the value of production function is a measurable quantity of output which has quantitative meaning. While in the analysis of preferences only indifference curves have economic contents, in the analysis of production values of production function and marginal products have economic contents as well as the shapes of isoquant curves. Let me give you some example of production function.

CHAPTER 14. PRODUCTION TECHNOLOGY

176

y 6

 * x2      PP PPP PP PP PP P qx 1 Figure 14.2: Production function and isoquant curves

Example 14.1 (Linear production function): Consider a technology such that one unit of Input 1 yields a units output and one unit of Input 2 yields b units of output. Such technology is described by the production function f (x) = ax1 + bx2 Then its graph forms a straight mountain such that its isoquant curves are parallel straight lines. Note that having isoquant curves to be parallel straight does not mean that the production is linear. For example, a production function √ f (x) = ax1 + bx2 exhibits parallel and straight isoquant curves, but the ”mountain” becomes flatter and flatter as you go to the north-east direction. Example 14.2 (Leontief production function): Consider a technology such that one unit of output requires the combination of a units of Input 1 and b units of Input 2. Such technology is described by the production function {x x } 1 2 f (x) = min , a b Then its graph forms a straight mountain such that its isoquant curves are parallel and L-shaped. Again note that having isoquant curves to be parallel and L-shaped does not mean that the production is Leontief. For example, a production function √ {x x } 2 1 , f (x) = min a b

CHAPTER 14. PRODUCTION TECHNOLOGY

177

exhibits parallel and L-shaped isoquant curves, but the ”mountain” becomes flatter and flatter as you go to the north-east direction. Example 14.3 (Cobb-Douglas production function): It is given by f (x) = Axa1 xb2 I will come to the details of this production function below.

14.2.2

Returns to scale

We say that production technology exhibits 1. constant marginal product if f ′ (x) is constant over x; 2. decreasing marginal product if f ′ (x) is decreasing in x; 3. increasing marginal product if f ′ (x) is increasing in x; where tx = (tx1 , tx2 ). Consider for example the case of t = 2, that is, consider tha we double the quantities of both inputs. Then, constant returns says that the output is doubled exactly, decreasing returns says that the output is less than the double, and increasing returns says that the output is more than the double. Since a(tx1 ) + b(tx2 ) = t(ax1 + bx2 ), the linear production function exhibits constant returns to scale. Also, since { } {x x } tx1 tx2 1 2 min , = t min , a b a b the Leotief production function exhibits constant returns to scale. Finally, since A(tx1 )a (tx2 )b = ta+b Axa1 xb2 , the Cobb-Douglass production function exhibits constant returns to scale if a + b = 1, decreasing returns to scale if a + b < 1, and increasing returns to scale if a + b > 1.

14.2.3

Marginal product

Again let me introduce the notion of marginal product, which is the amount of additional output obtained as you add one extra unit of input. However, because there more than one input here, we have to restate precisely that marginal production of one input is the amount of additional output obtained as you add one extra unit of it while the amount of the other inputs stay the same.

CHAPTER 14. PRODUCTION TECHNOLOGY

178

In the case of linear production, the simplest case described by f (x) = ax1 + bx2 , the marginal product of Input 1 is the coefficient a and the marginal product of Input 2 is the coefficient b. Put differently, the slope of the ”mountain” toward the east direction is a and its slope toward the north direction is b. In general, however, the slopes of the ”mountain” are not constant and therefore we need to look ar local slopes. Given a combination of inputs x = (x1 , x2 ), consider adding a ”slight” amount of Input 1, denoted by ∆x1 , while the amount of Input 2 is kept the same. Then the increase of output per one additional unit of Input 1 is f (x1 + ∆x1 , x2 ) − f (x1 , x2 ) ∆x1 Now as you make this ”slight” amount ∆x1 tend to zero, we get marginal product of Input 1 measure at x, which is given as the partial derivative of the production function f (x) in x1 . That is, the marginal product of Input 1 measured at x is defined by ∂f (x1 , x2 ) ∂x1 Likewise, marginal product of Input 2 measured at x is defined by ∂f (x1 , x2 ) ∂x2

Consider for example the Cobb-Douglass production function f (x) = Axa1 xb2 Then marginal products of Input 1 and 2 are given respectively by ∂(Axa1 xb2 ) = Aaxa−1 xb2 , 1 ∂x1 ∂(Axa1 xb2 ) = Abxa1 xb−1 2 ∂x2 In the former, we treat x2 as a fixed constant and takes the derivative as if it is a single-variable function of x1 . In the latter, we treat x1 as a fixed constant and takes the derivative as if it is a single-variable function of x2 .

14.2.4

Technological rate of substitution

Now consider the following question.

CHAPTER 14. PRODUCTION TECHNOLOGY

179

Given that the amount of output to produce is fixed, if I have an extra one unit of Input 1, how many units of Input 2 can I dispense with? This is equivalent to asking Given that the amount of output to produce is fixed, if I lose one unit of Input 1, how many units of Input 2 do I need to supplement in order to maintain the given output level? In any case, it measures how much Input 1 is relatively productive compared to Input 2. For example, when Input 1 is capital and Input is labor it will measures how much labor can be dispensed with when you have an extra one unit of capital in order to achieve the given level of output. Such measure is given by the (absolute value of the) slope of an indifference surface. Let us look at the simplest case of linear production f (x) = ax1 + bx2 Then, any combination of inputs x = (x1 , x2 ) which yields a given level of output y is on the isoquant curve described by y = ax1 + bx2 , which is a straight line in this simplest example. By solving this for x2 , we obtain x2 = yb − ab · x1 , which implies the absolute value of the slope of the isoquant curve is ab . Thus as you increase one unit of Input 1 you can dispense with ab units of Input 2 in order to maintain the given output level y. Or, it means that when you lose one unit of Input 1 for some reason you need to add ab units of Input 2 in order to maintain the given level of output y. This is called technological rate of substitution of Input 2 for Input 1. One can think of the reverse definition by switching between Input 1 and 2, but I will fix the order between the two throughout the book for definiteness. In general, the slope of an isoquant curve is not constant. Hence we need to look at local slope of it. Given a combination of inputs x = (x1 , x2 ), the technological rate of substitution measure there, denoted T RS(x), is the amount of Input 2 you can dispense with when you add a ”slight” amount of Input 1, which is given by the local slope of the given isoquant curve. Thus we have T RS(x) = −

∆x2 ∆x1

Since slope is negative as the isoquant curve is downward-sloping, we put the 2 minus sign to ∆x ∆x1 in order to take its absolute value. Technological rate of substitution is described as the ratio between marginal products. Notice that a ”mountain” surface is curvy in general but in the local sense it is seen to be straight. Thus we have the local relationship change of height = slope toward the east × move to the east + slope toward the north × move to the north

CHAPTER 14. PRODUCTION TECHNOLOGY

180

Hence as we the combination of inputs from x = (x1 , x2 ) to (x1 +∆x1 , x2 +∆x2 ) the change in the output level ∆y is given locally by ∂f (x) ∂f (x) ∆x1 + ∆x2 , ∂x1 ∂x1

∆y =

Recall that we are moving along the given isoquant curve on which the change in the output level ∆y is kept to be zero, hence we have 0=

∂f (x) ∂f (x) ∆x1 + ∆x2 . ∂x1 ∂x1

By rearranging this we obtain −

∆x2 = ∆x1

∂f (x) ∂x1 ∂f (x) ∂x2

Since the left-hand-side above is nothing but the technological rate of substitution, we obtain T RS(x) =

∂f (x) ∂x1 ∂f (x) ∂x2

.

Consider for example the Cobb-Douglas production function f (x) = Axa1 xb2 Then marginal products are ∂f (x) = Aaxa−1 xb2 , 1 ∂x1 ∂f (x) = Abxa1 xb−1 2 ∂x2 Hence we have T RS(x) =

14.3

∂f (x) ∂x1 ∂f (x) ∂x2

=

Aaxa−1 xb2 ax2 1 = . b−1 a bx1 Abx1 x2

Exercises 1

1

Exercise 17 Suppose the production function is f (x) = x13 x25 . (i) Find the marginal product of each input. (ii) Fin the technical rate of substitution of Input 2 for Input 1.

Chapter 15

Profit maximization and cost minimization 15.1

Profit maximization when output price and input prices are given

In perfectly competitive markets, firms are suppose to maximize their profit, taking the prices as given.

15.1.1

One-input/one-output case

Again let me start with the case of one input and one output in order to convey the point first. Let p denote the given output price and q denote the given input price. Then profit earned when the firm inputs x units and produces y units of output and sells it is py − qx Since the production follows y = f (x), we obtain the profit as a one-variable function of x, pf (x) − qx The firm maximizes this in the range x ≥ 0. I would point out four assumptions behind this modeling (maybe more?). First one is 1: the firm is a ”small” participant in both output market and input markets, decides production activity taking the prices in those markets as given, assuming that all its products are sold. The assumption of price takes, as before, says that each market participant is very small compared to the entire economy and cannot manipulate the market 181

CHAPTER 15. PROFIT MAXIMIZATION AND COST MINIMIZATION182 price by himself alone, and has to take it as given. Now what do I mean by ”assuming that all its products are sold?” You might say, ”how can you assume that all what you produce are sold?” I would say that in a perfectly competitive economy such problem is to be solved by the market, not by individual firms. Because perfectly competitive market is such that there are large number of small participants, selling order and buying order of each participant alone has only a negligible effect on the entire market. Hence each participant is supposed to make his selling and buying order without worrying about the case of leftover or shortage. This suggests that when the market is not perfectly competitive we need to think of the possibilities of leftover and shortage. Second one is 2: The firm is free from liquidity constraint. Look again at the profit maximization problem max pf (x) − qx x≥0

Notice that there no constraint in this problem other than x ≥ 0. Production activity costs qx. Where does this firm finance from? This model has an implicit assumption on account processing that the firm can lump together the revenue pf (x) and the cost qx. This requires that 1. Revenue is correctly estimated. 2. One can finance any amount of necessary cost as far as the estimated revenue allows. In order to say that a firm can pay cost wx since it generates revenue pf (x), correct information about revenue and production technology behind it must be shared by all the market participants. This is what Point 1 says. Even if the firm can generate certain amount of revenue for sure and it is known among all the market participants it is a different question if the firm can fully finance the necessary cost. For example, even if the firm is known to generate 8 millions of revenue by spending 5 millions of cost, because of institutional constraints this firm may mot be allowed to lump together the cost and the revenue and might be required to pay 5 millions in advance. A firm is said to subject to liquidity constraint if it cannot finance cost which deserves the revenue which it can generate. This modeling assumes that firms are free from such liquidity constraints. The third assumption is related to the first and second. 3: The case of increasing returns to scale is excluded

CHAPTER 15. PROFIT MAXIMIZATION AND COST MINIMIZATION183 Since the profit maximization problem above has no constraint, when the technology exhibits increasing returns to scale it has no ”solution.” Consider for example a production function f (x) = Ax2 . Then the profit given p, w is pAx2 − wx, but you can make this arbitrarily large by making x arbitrarily large. This tells us that profit maximization in a competitive market is not consistent with the technology exhibiting increasing returns to scale. That is, when there are firms with technology exhibiting increasing returns to scale, such market cannot be competitive. Why? When there is a firm having an increasingreturns-to-scale technology, such firm will lead to have a market power by making its scale larger and raises its production efficiency. In the end it will dominate the market as a monopoly firm or one of oligopoly firms. Thus, in terms of technology, the perfect competition assumption will be said to presuppose that the market consists of a large number of firms each of which is small and has technology exhibiting decreasing or constant returns to scale. The last assumption is 4: Firm’s production decision follows that of shareholders, and there is a unanimous agreement among the shareholders that the firm’s profit should be maximized. We are treating a somewhat classic kind of firm. To my knowledge, the origin of corporation dates back to ”The Age of Discovery.” In that age, explorers collect shareholders to their project at each time of exploration, and the project dissolves after finishing the exploration and pays dividends to the shareholders. The firm in the current model rather resembles to it. If you take the model literally, one firm is one project, and when the project is finished it pays dividends to the shareholders and dissolves. In such situation, no shareholder will object to maximizing the profit generated by such specific project. Modern corporation as ”ongoing concern” may be viewed as a sequence of such projects each of which is set up, carried out and dissolved. If there is no conflict between those projects there will be no disagreement between shareholders to maximize profit in each project. However, when this ”ongoing concern” has different projects which may conflict with each other in particular across periods, since the composition of shareholders will change in general over time there may be disagreements between shareholders about ”which profit” to maximize. Apologies became too long. Let me get into the analysis of profit maximization. First let me consider the case that the production function f (x) is smooth and exhibits decreasing returns to scale. Then the graphs of revenue pf (x) and cost qx are depicted as in Figure 15.1, and the profit is maximized at the point at which the local slope of the revenue curve is equal to the slope of the cost line. In other words, by taking the derivative of profit by x and obtain the

CHAPTER 15. PROFIT MAXIMIZATION AND COST MINIMIZATION184 Profit 6 qx p(f x)

-x Figure 15.1: Profit maximization

first-order-condition

pf ′ (x) − q = 0

By solving this we obtain 1. factor demand function: x(p, q) 2. supply function: y(p, q) = f (x(p, q)) Consider for example f (x) = Axa , where 0 < a < 1. Then by taking the derivative of pAxa − qx by x and equate that to zero, we obtain pAxa−1 − q = 0. 1 ( ) 1−a By solving this for x, we obtain x(p, q) = pA . By plugging this into the q a ( ) 1 1−a production function we obtain y(p, q) = A 1−a pq . Let me give you a tricky but important example, the case of constant returns to scale, f (x) = Ax. Since the profit is pAx − qx = (pA − q)x, the profit maximizing quantity of input x(p, q) is   if pA < q 0 x(p, q) = any value if pA = q   none, or infinity if pA > q When pA < q any positive production yields deficits, hence inaction is profitmaximizing. When pA = q any level of production yields zero profit, any level is ”equally optimal.” When pA > q the firm can earn arbitrarily large profit by making the production level arbitrarily large, which diverges to infinity. This sounds tricky, but when this firm is seen as a ”representative firm” which is an aggregation of a large number of firms this is actually a good description of their ”collective behavior,” under the condition of free and costless entry and exist.

CHAPTER 15. PROFIT MAXIMIZATION AND COST MINIMIZATION185 For, when pA > q the industry is profitable, and under the condition of free entry arbitrarily many firms will enter the market in order to exploit this profit opportunity, and it will not stop until the profit goes down to zero. Also, when pA < q the industry is unprofitable, and under the condition of free exit all firms will exit from the market and it will not stop as far as the profit is negative.

15.1.2

Two-input/one-output case

This will be extended to the case of more than two inputs, but the current specification is enough for our purpose. Let p denote the given price of output, let q = (q1 , q2 ) denote the pair of given input prices. Then the profit earned by producing and (selling) y units of output from combination of inputs x = (x1 , x2 ) bought in the input markets is py − q1 x1 − q2 x2 Since technology follows y = f (x), it reduces to pf (x) − q1 x1 − q2 x2 Having said the same apologies I made in the one-input case, I say that the firm maximizes its profit by choosing x. When the production function is ”smooth” and exhibits decreasing returns to scale, the profit-maximizing combination of inputs is determined by taking the partial derivative of pf (x) − q1 x1 − q2 x2 by x1 and x2 respectively, and equate them to zeros. That is, the profit maximization condition is ∂f (x) = q1 ∂x1 ∂f (x) p = q2 ∂x2 p

(1) (2)

By solving these equations we obtain 1. Factor demand function: x(p, q) = (x1 (p, q), x2 (p, q)) 2. Supply function: y(p, q) = f (x(p, q)) Let us derive one condition which will be helpful later. By dividing (1) by (2) in the above we obtain ∂f (x) ∂x1 ∂f (x) ∂x2

=

q1 q2

CHAPTER 15. PROFIT MAXIMIZATION AND COST MINIMIZATION186 Since the left-hand-side is the technological rate of substitution, we get q1 T RS(x) = q2 This says that at the profit-maximizing point the technological rate of substitution is equal to the relative price between inputs. Note that the converse may not be true. Even when the technological rate of substitution is equal to the relative price between inputs the combination of inputs is not necessarily maximizing the profit, because the condition says only about the ”proportion” between inputs and it is silent about the levels of them. I will come to this point again in the section on cost minimization. Let us go over an example. Consider the Cobb-Douglass production function f (x) = Axa1 xb2 which exhibits decreasing returns to scale, where 0 < a + b < 1. Then the profit maximization condition is paAx1a−1 xb2 pbAxa1 xb−1 2

= =

q1 q2

As stated above, divide the first formula by the second one then we obtain q1 ax2 bx1 = q2 . Solve this equation for x2 then we get a linear relationship between bq1 the inputs, x2 = aq · x1 . Plug this into the first formula above and solve it for 2 x1 , then we get factor demand for Input 1. By plugging this now into the linear relationship we obtain x2 . After algebra, we then obtain 1−b b ( ) 1−(a+b) ( ) 1−(a+b) 1 a b x1 (p, q) = (Ap) 1−(a+b) q1 q2 1−a a ( ) 1−(a+b) ( ) 1−(a+b) 1 a b x2 (p, q) = (Ap) 1−(a+b) q1 q2

Plug these into the production function, then we obtain the supply function a b ( ) 1−(a+b) ( ) 1−(a+b) a+b 1 a b y(p, q) = A 1−(a+b) p 1−(a+b) q1 q2 What about the Cobb-Douglass production function with constant returns to scale? That is, suppose a + b = 1 f (x) = Axa1 xb2 , where we replace b by 1 − a. When we take the analogue of the previous argument, the profit maximization condition would be paAxa−1 x1−a 1 2 a −a p(1 − a)Ax1 x2

= q1 = q2

Then by dividing the first formula by the second we obtain the equalization of technological rate of substitution to the relative price between the inputs ax2 q1 = (1 − a)x1 q2

CHAPTER 15. PROFIT MAXIMIZATION AND COST MINIMIZATION187 It’s the same as before up to here, but once you plug this into the first formula you get pAaa (1 − a)1−a q11−a q2a−1 = q1 in which x1 , the quantity of Input 1 you are trying to find, has disappeared. w1a q21−a Also, when p is strictly greater or smaller than Aaa (1−a) 1−a the above condition is not met first of all. What’s happening here? This is actually the case similar to what we had for the one-input case. The ”solution” is either zero or infinity or indeterminate, where the difference is that we have to have a right ”proportion” between inputs while the level of output is indeterminate. ax2 To see this, plug in the above condition (1−a)x = qq12 into the original profit 1 formula pAxa1 x1−a −q1 x1 −q2 x2 . Then after some rearranging the profit formula 2 reduces to ( q1 ) pAaa−1 (1 − a)1−a q11−a q2a−1 − x1 . a Now consider the following three cases q a q 1−a

1 2 1. When p < Aaa (1−a) 1−a , the coefficient on x1 in the above profit formula is negative. Hence any positive level of production activity yields deficits. Hence the profit-maximizing choice is x1 = 0. Likewise, we get x2 = 0. Thus, inaction is profit-maximizing.

q a q 1−a

1 2 2. When p = Aaa (1−a) 1−a , the coefficient on x1 in the above profit formula is zero. Hence any level of activity is ”equally profit-maximizing” and yields zero profit. Since the two inputs must be combined with the proportion q1 ax2 (1−a)x1 = q2 , the solution is

Any (x1 , x2 ) satisfying

ax2 (1−a)x1

=

q1 q2 .

q a q 1−a

1 2 3. When p > Aaa (1−a) 1−a , the coefficient on x1 in the above profit formula is positive. Hence the firm can earn arbitrarily large profit by making the level of production activity arbitrarily higher. Thus there is no solution or I would say the solution is x1 = ∞ and x2 = ∞.

15.2

Cost minimization when input prices are given

Let us look at another level of production decision, which asks to solve the following problem. Given input prices q = (q1 , q2 ), what combination of inputs x = (x1 , x2 ) minimizes the cost to produce a given level of output y?

CHAPTER 15. PROFIT MAXIMIZATION AND COST MINIMIZATION188 This is called cost minimization problem. When a production activity is maximizing the profit it must be minimizing the cost to produce that level of output. The converse is not true, however, because minimizing cost to produce a given level of output is silent about what level of output maximizes the profit. Cost minimization is a concept which is more broadly applicable, however, because it requires only that the firm is price-taking in the input markets. Recall that profit maximization in a competitive market requires the presumption that the firm is price-taking in both output market and input markets. Hence the firm is assumed to be a small participants in both output market and input markets. On the other hand, cost minimization is applicable to the cases where the firm is not necessarily price-taking in the output market or its technology exhibits increasing returns to scale, as far as it is price-taking in the input markets. Now let us formalize the cost minimization problem. Given a pair of input prices q = (q1 , q2 ), the cost of combination of inputs x = (x1 , x2 ) is q1 x2 + q2 x2 Since we obey the constraint that we have to clear a given level of output y, the combination of inputs x must satisfy f (x) = y Thus, the firm varies x in order to minimize the cost under such constraint. Summing up, the cost minimization problem is written by min q1 x2 + q2 x2 x

subject to

f (x) = y

Notice that in contrast to the profit maximization problem in competitive markets this is a problem with a constraint. Hence it always has a solution even when the technology exhibits increasing returns to scale. How do we find the cost-minimizing point? See Figure 15.2. First, we draw the isoquant curve which corresponds to the given level of output y. Now pick an arbitrary point on the curve, let’s say x = (x1 , x2 ). Is it cost-minimizing? To see if it is, draw the line passing through x with slope − qq12 . Since any all the point on the line yield the same cost, let us call it an iso-cost line. Then, since we can pick a point on the isoquant curve which is strictly below the current iso-cost line, let’s say x′ = (x′1 , x′2 ). Then, since q1 x1 + q2 x2 > q1 x′1 + q2 x′2 , the point x is not cost-minimizing. Likewise, any point on the isoquant curve is not cost minimizing as far as the corresponding iso-cost line crosses the curve. Thus, the cost-minimizing point must be the point at which the corresponding iso-cost line is tangent to

CHAPTER 15. PROFIT MAXIMIZATION AND COST MINIMIZATION189 x2 6

r x∗

′ rx

rx - x1

Figure 15.2: Cost minimization

the isoquant curve, like x∗ = (x∗1 , x∗2 ) in the figure. When the isoquant curve is smooth it means that the local slope of the curve is equal to the slope of the iso-cost line, that is the technological rate of substitution is equal to the relative price between the inputs. Hence the cost minimization condition is T RS(x) =

q1 q2

Since this condition is implied by the profit maximization condition, we now confirm that when the firm is maximizing its profit it is always minimizing the cost. The converse is not true, as I repeat. By combining the above condition with f (x) = y, we obtain the conditional factor demand function x1 (q, y),

x2 (q, y)

I put ”conditional” because the problem is conditional on the level of output y. Also, by plugging this into the cost formula q1 x1 + q2 x2 , we obtain the cost function C(q, y) = q1 x1 (w, y) + q2 x2 (w, y)

Let us go over an example of the Cobb-Douglass production function f (x) = Axa1 xb2 . Since we already derives its technological rate of substitution before, 2 which is T RS(x) = ax bx1 , cost cost minimization condition is q1 ax2 = bx1 q2 By solving this for x2 we obtain the linear relationship between the inputs bq1 x2 = aq · x1 . By plugging this into the constraint y = Axa1 xb2 and solve it for 2

CHAPTER 15. PROFIT MAXIMIZATION AND COST MINIMIZATION190 x1 に, then we obtain b b ( ) a+b ( )− a+b 1 ( y ) a+b a b x1 (q, y) = A q1 q2

Then by plugging this into the linear relationship formula we obtain a a ( )− a+b ( ) a+b 1 ( y ) a+b a b x2 (q, y) = A q1 q2 By plugging the above conditional factor demand to the cost formula we obtain the cost function { a } ( ) a+b 1 b ( y ) a+b ( a ) a+b a b b C(q, y) = + q1a+b q2a+b A b a Notice that in contrast to the profit maximization problem when both output price and input prices are given, the solution here always exists and is equal as far as a + b > 0.

15.3

Long-run and short-run

Here ”long-run” means that the firm can vary its inputs in a fully flexible manner. On the other hand, ”short-run” means that the firm cannot change the level of some inputs. For example, once you build a factory you cannot immediately expand it or scrap it for certain periods. In such cases we have to take some inputs to be fixed quantities. Short-run profit maximization For illustration, let me assume that the quantity of Input 2 is fixed in the shortrun and the firm can vary the quantity of Input 1 only. Denote the fixed quantity of Input 2 by x ¯2 . Then the short-run profit maximization problem is max pf (x1 , x ¯ 2 ) − q1 x 1 − q2 x ¯2 x1

Since there the firm has just one variable input, this is a maximization problem with one variable. Hence the short-run profit maximization problem is pM P1 (x1 , x ¯ 2 ) = q1 . By solving this we obtain the short-run factor demand function for Input 1 x1 = x1 (p, q, x ¯2 ) By plugging this into the production function we obtain the short-run supply function y(p, q, x ¯2 ) = f (x(p, q, x ¯2 ))

CHAPTER 15. PROFIT MAXIMIZATION AND COST MINIMIZATION191

15.3.1

Short-run cost minimization

Now consider cost minimization in the short-run. As before, consider that the quantity of Input 2 is fixed to be x ¯2 . Then the cost minimization problem given output level y is min q1 x2 + q2 x ¯2 x1

subject to

f (x1 , x ¯2 ) = y

However, this problem is trivial, since y and x ¯2 are given there is just one x1 which satisfies f (x1 , x ¯2 ) = y. Hence in this case the short-run conditional factor demand function x1 x2

= xS1 (q, x ¯2 , y) = x ¯2

is automatically determined, and the short-run cost function is given by C S (q, x ¯2 , y) =

¯2 , y) + q2 x ¯2 q1 xS1 (q, x

Notice that the first term in the above short-run cost function is variable as output y varies, but the second term is just a constant. The cost term which is variable as the output level varies is called variable cost, and the constant cost which does not vary with the output level is called fixed cost. Denote the variable cost term by V C(y) and the fixed term by F C, and omit w, x ¯2 from the notation as they are given throughout, we can write the cost function as C S (y) = V C(y) + F C

Consider Cobb-Douglass production function f (x) = Axa1 xb2 for example. −b

¯b2 = y we have xS1 = A− a x Then, from the constraint Axa1 x ¯2 a y a , and the short-run cost function is given by 1

−b

1

C S (y) = q1 A− a x ¯ 2 a y a + q2 x ¯2 1

1

In order to think of the case that short-run cost minimization problem is not trivial, consider the case of three inputs. Now suppose that the quantity of Input 3 is fixed to be x ¯3 , and Input 1 and 2 are variable. Then the short-run cost minimization problem is min q1 x2 + q2 x2 + q3 x ¯3

x1 ,x2

subject to

f (x1 , x2 , x ¯3 ) = y

Here the minimization condition is, like before, given by the equality of the technical rate of substitution to the relative price between Input 1 and 2.

CHAPTER 15. PROFIT MAXIMIZATION AND COST MINIMIZATION192 1

1

Exercise 18 Production function is f (x) = x13 x25 . (i) Given output price p and input prices q = (q1 , q2 ), solve the profit maximization problem and find the factor demand functions x1 (p, q), x2 (p, q) and the supply function y(p, q). (ii) Given output level y and input prices q = (q1 , q2 ), solve the cost minimization problem and find the conditional factor demand function x1 (y, q), x2 (y, q) and cost function C(q, y).

Chapter 16

Cost curve and supply 16.1

Average cost and marginal cost

In the last chapter we derived the short-run cost function C(y) = V C(y) + F C as a consequence of short-run cost minimization. Here V C(y) denotes the variable cost and F C denotes the fixed cost. Let me define some derivative concepts. First, cost per one unit of production is called average cost and defined by AC(y) =

C(y) y

Average cost is decomposed into two parts, as AC(y) =

C(y) y

= =

V C(y) F C + y y AV C(y) + AF C(y)

where AV C(y) is called average variable cost, which is variable cost per one unit of production, and AF C(y) is called average fixed cost, which is fixed cost per one unit of production. It is obvious from the definition that average fixed cost falls as the firm produces more. Average cost is relevant to firm’s entry/exit decision, and average variable cost is relevant to firm’s shut down/operation decision. Also, in order to think of how much to produce we need to look at incremental cost for incremental unit of production rather than average costs. This is called marginal cost.

193

CHAPTER 16. COST CURVE AND SUPPLY

194

The marginal cost of one extra unit of production added to a given level of production y is M C(y) = C ′ (y) = lim

∆y→0

C(y + ∆y) − C(y) ∆y

Note that since the fixed cost part disappears after taking the derivative fixed cost does not affect marginal cost. When the marginal cost curve is upward-sloping we say that the technology exhibits increasing marginal cost. This says that as you produce more the additional cost tends to be more expensive, which corresponds to technology exhibiting decreasing returns to scale (with regard to variable inputs). When the marginal cost curve is downward-sloping we say that the technology exhibits decreasing marginal cost. This says that as you produce more the additional cost tends to be cheaper, which corresponds to technology exhibiting increasing returns to scale (with regard to variable inputs). When the marginal cost curve is straight we say that the technology exhibits constant marginal cost. This says that as you produce more the additional cost tends to be cheaper, which corresponds to technology exhibiting constant returns to scale (with regard to variable inputs). When you plot average cost, average variable cost and marginal cost it looks like Figure 16.1. Here the marginal cost curve is crossing the bottom of the average cost curve and the bottom of the average variable cost curve respectively. This is not just a coincidence, since ( )′ ( ) C(y) C ′ (y)y − C(y) 1 C(y) ′ AC ′ (y) = = = C (y) − y y2 y y 1 = (M C(y) − AC(y)) y and wit holds AC ′ (y) = 0 if and only M C(y) = AC(y). Similarly it holds AV C ′ (y) = 0 if and only if M C(y) = AV C(y). Let us go over two examples. Example 16.1 A short-run cost function C(y) = y 2 + 1 can come for example 1

1

from cost minimization by a firm with production function f (x1 , x2 ) = x12 x22 under input prices w1 = w2 = 1 and the short-run restriction x ¯2 = 1. Given this cost function we have the following (see Figure 16.2). 1. Variable cost V C(y) = y 2 2. Fixed cost F C = 1 3. Average cost AC(y) =

y 2 +1 y

=y+

4. Average variable cost AV C(y) =

1 y

V C(y) y

=

y2 y

=y

CHAPTER 16. COST CURVE AND SUPPLY AC 6

195

MC

AVC

-y Figure 16.1: Cost curves

5. Average fixed cost AF C(y) =

FC y

=

1 y

6. Marginal cost M C(y) = C ′ (y) = (y 2 + 1)′ = 2y. Since its derivative is C ′′ (y) = 2 > 0 it exhibits increasing marginal cost. Example 16.2 When the short-run cost function is given by C(y) = y 3 −2y 2 + 3y + 4 we have the following (see Figure 16.3). 1. Variable cost V C(y) = y 3 − 2y 2 + 3y 2. Fixed cost F C = 4 3. Average cost AC(y) =

y 3 −2y 2 +3y+1 y

4. Average variable cost AV C(y) = 5. Average fixed cost AF C(y) =

FC y

= y 2 − 2y + 3 +

V C(y) y

=

=

y 3 −2y 2 +3y y

4 y

= y 2 − 2y + 3

1 y

6. Marginal cost M C(y) = C ′ (y) = (y 3 − 2y 2 + 3y + 1)′ = 3y 2 − 4y + 3. Since its derivative is C ′′ (y) = 6y − 4, it exhbits decreasing marginal cost when 0 ≤ y ≤ 32 and increasing marginal cost when 23 ≤ y.

16.2

Profit maximization under perfect competition

Here let me describe firm’s profit maximization behavior in a perfectly competitive market in which there is a large number of market participants so that each one is small and has to take the market price as given.

CHAPTER 16. COST CURVE AND SUPPLY

AC 6

196

MC

2 AVC

-y

1

Figure 16.2: C(y) = y 2 + 1

AC 6

MC

5

AVC 2

1

2

-y

Figure 16.3: C(y) = y 3 − 2y 2 + 3y + 4

CHAPTER 16. COST CURVE AND SUPPLY

197

Denote the given price of output by p, then the profit maximization problem is formulated by max py − C(y) y≥0

By solving this we obtain the supply function y = S(p)

Depending on the shape of the cost function the condition ”derivative=zero” does not always determine the firm behavior. It is possible that when the output price is too low it may be better to shut down the operation and supply notion. Hence we consider the profit maximization choice case by case. 1. When the output price p is below the bottom of the AVC curve you cannot offset variable cost no matter what quantity you produce, so it is better not to produce at all. Such threshold is called the shut-down point. 2. When the price is above the shut-down point basically the ”derivative=zero” condition determines the profit-maximization choice, although we need some qualification as discussed below. Then by taking the derivative of the profit formula by y and equating it to zero we obtain p = M C(y) This equation may have several solutions, but at the profit maximization point the MC curve must be upward-sloping, since at the maximization point it must be that when you produce a bit less it is too cheap and when you produce a bit more it is too expensive. When the MC is downwardsloping then p = M C(y) is actually saying that the profit is minimized locally. 3. Note that when the price is below the AC curve, called the break-even point, you cannot offset fixed cost, hence in the long-run you should downsize the body making the fixed cost or scrap it entirely and exit from the market. Let’s go back to the first example C(y) = y 2 + 1 and consider the profit maximization problem max py − (y 2 + 1) y

Then, 1. Average variable cost AV C(y) = y has its minimal value to be equal to zero. Hence no worry about shut-down decision. 2. Marginal cost is M C(y) = 2y. Hence p = M C(y) yields S(p) = 21 p.

CHAPTER 16. COST CURVE AND SUPPLY 3. Average cost is AC(y) = y + p < 2 the firm should scrap.

1 y

198

and its minimal value is 2. Hence when

Hence the short-run supply function is S(p) =

1 p for every p > 0 2

What about the second example C(y) = y 3 − 2y 2 + 3y + 4? 1. Average variable cost is AV C(y) = y −2yy +3y = y 2 − 2y + 3 = (y − 1)2 + 2, which has a U-shape as depicted in Figure 16.3, and its bottom is 2. Hence when p < 2 the firm shuts down the operation. 3

2

2. When p ≥ 2 the marginal cost condition p = M C(y) applies. When you √ solve p = 3y 2 − 4y + 3 you get y = 2± 33p−4 , but since the MC curve must √ be upwa-dsloping at the maximizing point we have S(p) = 2+ 33p−5 . 3. Average cost is AC(y) = y 2 − 2y + 3 + y4 and its bottom is 5. Hence when p < 5 the firm should scrap. Hence the short-run supply function is { 0, √ S(p) = 2+ 3p−5 3

when p < 2 , when p ≥ 2

Let’s consider two more examples. What if the cost function is C(y) = √ y + 2? Here the average variable cost is AV C(y) = √1y , which is downwardsloping and gets closer to zero as y is large, so you don’t have to worry about 1 the shut-down decision. However, the marginal cost M C(y) = 2√ y exhibits a downward-sloping curve and cannot meet the profit maximizing condition not matter what the output price is. Hence there is no profit-maximizing point. Notice that such cost structure is incompatible with the assumption of pricetaking, since this firm can produce the product at cheaper cost per unit as it produces more, and such firm will in the end dominate the market and will have a market power, which cannot be price-taking. Since perfectly competitive markets refer to markets with ”a large number of small participants,” the analysis of firms with decreasing marginal cost should be handled by the models of monopoly and oligopoly, or by the model of government regulation. Finally, consider the case of constant marginal cost C(y) = 2y. 1. Average variable cost is AV C(y) = 2y y = 2, which exhibits a horizontal straight AVC curve. Hence when p < 2 the firm should shut down. 2. When p = 2 any output level is maximizing the profit.

CHAPTER 16. COST CURVE AND SUPPLY

199

3. When p > 2 there is no solution or the supply is infinity. Summing up, we have  when p < 2  0, any non-negative number, when p = 2 S(p) =  no solution, ot infinity, when p > 2 This looks tricky, but again when this firm is seen as a ”representative firm” of a large number of firms this is a good description of their ”mass behavior,” under the condition of free and costless entry and exist. For, when p > 2 the industry is profitable, and under the condition of free entry arbitrarily many firms will enter the market in order to exploit this profit opportunity, and it will not stop until the profit goes down to zero. Also, when p < 2 the industry is unprofitable, and under the condition of free exit all firms will exit from the market and it will not stop as far as the profit is negative. When p = 2, all the firms are on the knife-edge of break even and indifferent between staying and exiting.

16.3

Exercises

Exercise 19 Given a short-run cost function C(y) = y 3 − 4y 2 + 7y + 18, find (i) the average cost AC(y) (ii) the average variable cost AV C(y) (iii) the marginal cost M C(y) (iv) the break-even point (v) the shut-down point (vi) the short-run supply function

Chapter 17

Competitive equilibrium in production economies 17.1

Private ownership economy

Here we consider a model of production economy so-called private ownership economy. There are n consumers and m firms. The key here is that the firms are owned ultimately by the consumers = shareholders and their profits are paid to the consumers as dividends. Also, the ownership structure of each firm is fixed and remains unchanged. This might be a too ”classic” view to understand modern corporations in which ownership and management are separated and the composition of shareholders change over time. So I like you to read ahead by taking this model as a benchmark. Now, each consumer i = 1, · · · , n is characterized by his 1. initial holding of goods: ei = (ei1 , ei2 ), 2. ownership of the firms θi = (θi1 , · · · , θim ) 3. preference: ≿i Also, consumer i’s consumption is typically denoted by xi = (xi1 , xi2 ). Consumer i’s ownership over firm k refers to the percentage right of receiving the firm’s profit as dividend, that is θik denotes the proportion of firm k’s profit which i can receive. Since firms are owned ultimately by the consumers in this model, it holds n ∑ θik = 1 i=1

for all k = 1, · · · , m. To describe production in the current setting we take a little different notation. Production activity of firm k is typically denoted by yk = (yk1 , yk2 ), which 200

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

201

Good 2 6 ry

- Good 1 r y′

Figure 17.1: Transformation curve

means the firm produces yk1 units of Good 1 and yk2 units of Good 2. Since it is technologically impossible to have positive output from nothing, it must be either that yk1 < 0 and yk2 > 0 or that yk1 > 0 and yk2 < 0. In the first case Good 1 is taken to be the input and Good 2 is taken to be the output whereas in the second case Good 1 is taken to be the output and Good 2 is taken to be the input. That is, input is treated as negative output in this notation, and factor demand is treated as negative supply. We adopt this notation since it is more unified and actually convenient in formulating our arguments at least in this chapter. Firm k’s production activity yk = (yk1 , yk2 ) follows its technological constraint, which is described by an equation Tk (yk ) = 0 This function Tk is called firm k’s transformation function, which is a more general form of production function. When you depict the above equation on the two-dimensional plane as in Figure 17.1), we obtain a transformation curve. Transformation curves always pass through the origin. Also, it is impossible to have both yk1 > 0 and yk2 > 0. In general, it is possible that one produces Good 1 from Good 2 and Good 2 from Good 1. For example, at y in Figure 17.1 Good 2 is produced from Good 1, and at y ′ Good 1 is produced from Good 2. Technology which is described by production function is a special case of this, in the sense that the roles of input and output are fixed and there is no reverse production. For example, when Good 1 is always input and Good 2 is always output then technology described by ”output = f (input)” is written by means of transformation function in the form yk2 − fk (−yk1 ) = 0,

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

202

Good 2 6

- Good 1

Figure 17.2: No reverse production

and the corresponding transformation curve looks like Figure 17.2. In the technology described by a transformation function, what corresponds to marginal product in production function is marginal rate of transformation. It is the local slope of the transformation curve, which is given by ∆yk2 M RTk (yk ) = ∆yk1 See Figure 17.3. Its interpretation is the same as marginal product, given that the negative amount in production is taken to be the input and the positive amount is taken to be the output. Then, a list (x, y) consisting of consumption activities x = (x1 , · · · , xn ) and production activities y = (y1 , · · · , ym ) is said to be feasible if n ∑ i=1 n ∑

xi1 xi2

= =

n ∑ i=1 n ∑

ei1 + ei2 +

m ∑ k=1 m ∑

yk1 yk2

i=1

i=1

Tk (yk ) =

0 for all k = 1, · · · , m

k=1

Here the first and second formula mean that total consumption must be equal to the total amount of initial endowment plus total amount of output (including the negative amount, which is factor demand). The third formula means that production activities must obey the technological constraints.

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

203

Good 2 6 y ∆y2 6  r ∆y1 - Good 1

Figure 17.3: Marginal rate of transformation

17.1.1

Firms’ profit Maximization

In a perfectly competitive market each firm maximizes its profit by taking the price p = (p1 , p2 ) as given. It is formulated by max p1 yk1 + p2 yk2 yk

subject to Tk (yk ) = 0 for each k. How is the profit-maximizing production activity determined then? See Figure 17.4. It will be immediate that the origin (0, 0) provides zero profit. Then what about yk = (yk1 , yk2 ) on the transformation curve? When you draw a line passing through it with slope − pp12 it passes above the origin, hence it is generating positive profit. Let me call this line an iso-profit line, since all the point on it yield the same profit. It is not maximal, however, since you can find a point on the transformation ′ ′ curve such as yk′ = (yk1 , yk2 ). When you draw the iso-profit line passing through ′ yk it is above the previous one, which means yk′ is generating more profit. Thus yk is not profit-maximizing. Having said that, the profit maximization point must be such that the isoprofit line passing through it is tangent to the transformation curve, such as ∗ ∗ yk∗ = (yk1 , yk2 ) in Figure 17.4. This means that the local slope of the transformation curve is equal to the slope of the corresponding iso-profit line. Thus, the marginal rate of transformation must be equal to the relative price between the goods. Thus the profit maximization condition is M RTk (yk ) =

p1 p2

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

204

Good 2 6 ry

∗ ′ ry yr

- Good 1

Figure 17.4: Profit maximization

Combining this with Tk (yk ) = 0 and solve for yk , then we obtain yk (p) = (yk1 (p), yk2 (p)). While one of them is positive and the other is negative, or one of them of supply and the other is factor demand, we call the pair supply function altogether, since we treat factor demand as negative supply. When we plug in the supply function to the profit formula we obtain firm k’s maximal profit as a function of price, πk (p) = p1 yk1 (p) + p2 yk2 (p) This is called a profit function.

17.1.2

Consumers’ choice

As stated above, in the model of private ownership economy the maximized profit are distributed to the shareholders who own the firms ultimately, which means consumer sovereignty and at the same time shareholder sovereignty. Given price p = (p1 , p2 ), consumer i’s income consists of the market value of his initial holding plus dividends from the firms, while his income in the exchange economy consists only of the first one. That is, it is given by p1 ei1 + p2 ei2 +

m ∑

θik πk (p)

k=1

Consumer i receives θik percentage proportion of firm k’s profit πk (p), which is θ∑ ik πk (p). By adding up across k = 1, · · · , m, he receives income from dividends m k=1 θik πk (p).

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

205

Thus, consumer i chooses his optimal consumption under the budget constraint m ∑ p1 xi1 + p2 xi2 = p1 ei1 + p2 ei2 + θik πk (p) k=1

Denote the demand function by xi (p) = (xi1 (p), xi2 (p)). Again, note here that consumer makes consumption decision by taking dividend payments into account.

17.1.3

Competitive equilibrium

Consumers and firms respond to a given price p = (p1 , p2 ) by means of demand and supply xi (p) =

i = 1, · · · , n

(xi1 (p), xi2 (p)),

k = 1, · · · , m

yk (p) = (yk1 (p), yk2 (p)),

Then, competitive equilibrium price is the price at which demand matches supply, that is, it is the price p∗ = (p∗1 , p∗2 ) such that n ∑

xi1 (p∗ ) =

n ∑

i=1

i=1

n ∑

n ∑

xi2 (p∗ ) =

i=1

ei1 + ei2 +

i=1

m ∑ k=1 m ∑

yk1 (p∗ ) yk2 (p∗ )

k=1

In competitive equilibrium, since each consumer is making optimal consumption under the budget constrain we have equality between the marginal rate of substitution and the relative price M RSi (x∗i ) =

p∗1 p∗2

for every i. Since each firm is maximizing its profit, we have equality between the marginal rate of transformation and the relative price M RTk (yk∗ ) =

p∗1 p∗2

for every k, Summing up, we have M RSi (x∗i ) = M RTk (yk∗ ) = for all i = 1, · · · , n and k = 1, · · · , m.

p∗1 p∗2

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

17.2

206

The representative consumer/producer model

Here let us consider the case in which there is just one representative consumer and there is just one representative firm. It is called representative agent model. Since the assumption of competitive market requires substantially that there is a large number of small traders we should imagine a large number of consumer and produces behind the representative consumer and producer respectively. The representative consumer is characterized by 1. initial holding of goods: e = (e1 , e2 ), 2. ownership of the representative firm 3. preference ≿ and the representative firm is characterized by the transformation curve T (y) = 0 Note that here the representative consumer has the 100% ownership of the representative firm. A combination of consumption activity and production activity (x, y) is said to be feasible if x1 = e1 + y1 x2 = e2 + y2 T (y) = 0

(1) (2) (3)

Depict (e1 + y1 , e2 + y2 ) for each y = (y1 , y2 ) on the transformation curve T (y) = 0. This is obtained by shifting the transformation curve parallel by e = (e1 , e2 ) (see Figure 17.5). Take the intersection between this shifted curve and the non-negative quadrant, then it is the set of consumption activities provided by the production side. This is called the production possibility frontier. By doing this we can see consumption and production in one diagram. The representative firm maximizes its profit given p = (p1 , p2 ), which is formulated as max p1 y1 + p2 y2 y

subject to T (y) = 0 The profit maximization condition is given by the equality between the marginal rate of transformation and the relative price M RT (y) =

p1 p2

By combining this with T (y) = 0 we obtain the supply function y(p) = (y1 (p), y2 (p))

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

207

Good 2 6

re - Good 1

Figure 17.5: Production possibility frontier

and the profit function π(p) = p1 y1 (p) + p2 y2 (p). Since the representative consumer is the 100%-shareholder of the representative firm his budget constraint is p1 x1 + p2 x2 = p1 e1 + p2 e2 + π(p) Note again that the consumer makes consumption decision by taking the dividend payment from the firm into account. Thus, as in Figure 17.6 his budget line is shifted from the one passing through e = (e1 , e2 ) by the amount corresponding to the dividend, hence his budget set is expanded than in the case of no production. Also, his budget line is tangent to the production possibility frontier at (e1 + y1 (p), e2 + y2 (p)), the consumption point which is provided by the profit-maximizing production activity. The representative consumer makes consumption decision under this budget constraint, and we obtain the demand function x(p) = (x1 (p), x2 (p)). For example, when the corresponding indifference curve is as in Figure 17.6 and the demand is given by x = (x1 , x2 ) there, since it does not coincide with the consumption provided by the production side (e1 + y1 (p), e2 + y2 (p)) it is not in competitive equilibrium. Thus, competitive equilibrium price is p∗ = (p∗1 , p∗2 ) such that x1 (p∗ ) =

e1 + y1 (p∗ )



e2 + y2 (p∗ )

x2 (p ) =

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

208

Good 2 6 y∗ + e r x r re - Good 1

Figure 17.6: Budget constraint in production economy

In other words, it is the price at which the consumption demanded by the representative consumer coincides with the consumption provided by the profitmaximizing production activity. In Figure 17.7, the left-hand-side and the righthand-side of the above equality comes to the common point on the production possibility frontier. Suppose both the representative consumer’s indifference curves and the representative firm’s transformation curve are smooth. Since the consumer is choosing optimal consumption under the budget constraint the marginal rate of substitution is equal to the relative price, hence it holds M RS(x∗ ) =

p∗1 . p∗2

Since the firm is maximizing its profit the marginal rate of transformation is equal to the relative price, hence it holds M RT (y ∗ ) =

p∗1 p∗2

Summing up it holds M RS(x∗ ) = M RT (y ∗ ) =

p∗1 p∗2

In Figure 17.7, this means that the indifference curve and the production possibility frontier are tangent to the budget line/iso-profit line at the common point.

17.2.1

Example 1: decreasing returns

Let us go over an example. Fix Good 1 to be the input and Good 2 to be the output.

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

209

Good 2 6 r

x∗ = y ∗ + e

re - Good 1

Figure 17.7: Competitive equilibrium in production economy

Assume that the representative consumer’s preference is represented in the Cobb-Douglas form u(x) = λ ln x1 + θ ln x2 and his initial holding is e = (e1 , 0). That is, he holds some amount of the input good initially but not the output good at all. On the other hand, the representative firm’s technology exhibits decreasing returns to scale and the transformation function is given by √ T (y) = y2 − A −y1 , where y1 ≤ 0. The firm’s profit maximization by taking the market price as given yields supply function A2 p2 A2 p2 y1 (p) = − 22 , y2 (p) = 4p1 2p1 and profit function π(p) =

A2 p22 4p1

On the other hand, by taking it into account that the representative conA2 p2 sumer’s income is p1 e1 + p2 e2 + π(p1 , p2 ) = p1 e1 + 4p12 , we can derive the demand function generated by Cobb-Douglass preference ) ( ) ( A2 p2 A2 p22 p1 e1 + , x1 (p) = α e1 + , x2 (p) = (1 − α) 4p21 p2 4p1 where α =

λ λ+θ .

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

210

Then the relative price of Good 1 for Good 2 in competitive equilibrium is given by √ p∗1 A 1+α = p∗2 2 (1 − α)e1 and the production activity in the equilibrium is √ 1−α 1−α ∗ ∗ y1 = − e 1 , y2 = A e1 1+α 1+α and the consumption activity in the equilibrium is √ 2α 1−α ∗ ∗ x1 = e1 , x 2 = A e1 1+α 1+α

17.2.2

Example 2: constant returns

Next let us see the case of constant returns to scale. Maintain the assumption that Good 1 is the input and Good 2 is the output. As explained before, this gives us a reasonable description of ”collective behavior” of firms in the situation that there are large numbers of potential firms and free entries and exists are allowed. The representative consumer is again assume to have preference represented byu(x) = λ ln x1 + θ ln x2 and an initial holding e = (e1 , 0). On the other hand, production technology for the representative firm exhibits constant returns to scale, and it is described by a linear transformation function T (y) = y2 + Ay1 , where y1 ≤ 0. That is, the marginal rate of transformation is constant and equal to A. When the technology exhibits constant returns to scale the supply is given to be a ”correspondence” rather than a function. Here the supply correspondence is given by  when pp21 < A  (−∞, ∞), any (y1 , y2 ) with y2 + Ay1 = 0, when pp12 = A y(p) =  (0, 0) when pp12 > A That is, (i) when the relative price of input is cheaper compared to the marginal rate of substitution the representative firm can earn arbitrarily large profit by making the production scale arbitrarily larger; (ii) it chooses inaction in the opposite case, and (iii) anything level of production is profit-maximizing and the maximized profit is zero in the case of equality. The profit function is then { ∞, when pp21 < A π(p) = 0 when pp21 ≥ A

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

211

Since it cannot happen in equilibrium that the production level diverges to infinity, we restrict attention to the case that pp21 ≥ A, where the maximized profit is zero. Then the consumer receives no profit dividend and his income is p1 e1 . Hence his demand generated by the Cobb-Douglass preference is p1 e1 p1 e1 x1 (p) = α = αe1 , x2 (p) = (1 − α) p1 p2 where α =

λ λ+θ .

Since there is always a positive and finite demand, in equilibrium there must be a positive and finite supply. Hence it must be that the relative price in equilibrium is p∗1 =A p∗2 Thus, when technology exhibits constant returns to scale the equilibrium relative price is determined by the production technology alone (which is not true in general). Here the production activity in equilibrium is y1∗ = −(1 − α)e1 ,

y2∗ = (1 − α)Ae1

which is profit maximizing since any production level is equally profit maximizp∗ ing under p∗1 = A, where the maximized profit is zero. On the other hand, the 2 consumption activity in equilibrium is x∗1 = αe1 ,

17.3

x∗2 = (1 − α)Ae1 .

Interest rate determination in an intertemporal production economy

Let us apply the model of competitive production economy to an economy with intertemporal production. Consider the two-period setting as before, where Good 1 refers to the consumption good available at Period 1 and Good 2 refers to the consumption good available at Period 2. Denote the representative consumer’s earning stream by e = (e1 , e2 ). His consumption stream is generically denoted by x = (x1 , x2 ), where he consumes x1 at Period 1 and x2 units at Period 2. The representative consumer produces the consumption good available at Period 2 from the consumption good available at Period 1. Denote its generic production activity by y = (y1 , y2 ), then it holds y1 ≤ 0 and y2 ≥ 0. Given interest rate r the representative firm maximizes its profit measures in present value: 1 max y1 + y2 y 1+r

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

212

subject to T (y) = 0 Denote the maximized profit by π(r). The representative consumer’s problem is given by max v(x1 ) + βv(x2 ) x

x2 e2 = e1 + + π(r). 1+r 1+r Note that he takes the present value of the firm’s profit into his income in the present-value form. This is nothing but a special case of the previous model, in which the prices are written in the form subject to

x1 +

p1 = 1, p2 =

1 p1 , and = 1 + r. 1+r p2

Now the interest rate in competitive equilibrium is r∗ such that x1 (r∗ ) = e1 + y1 (r∗ ) x2 (r∗ ) = e2 + y2 (r∗ ). For example, when the period-wise evaluation function is given by v(z) = ln z and the earning stream is e = (e1 , 0) we can apply the previous example in which 1 α = 1+β . When the technology exhibits decreasing returns to scale and the transformation function is √ T (y) = y2 − A −y1 , by applying Example 1 above we obtain A p∗ 1 + r = ∗1 = p2 2 ∗

√( ) 2 1 1+ β e1

On the other hand, when the technology exhibits constant returns to scale and the transformation function is T (y) = y2 + Ay1 by applying Example 2 above we obtain 1 + r∗ =

p∗1 =A p∗2

Now, there was once a historically long debate about the source of interest rate. When I summarize the main views they are basically:

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

213

1. ”Interest is the reflection of intertemporal productivity.” 2. ”Interest is generated from the difference between current earning and future earning.” 3. ”Interest is a reward for patience.” This debate was very active in the eighteenth and nineteenth centuries, but from the current viewpoint each of them has a point but none of them is correct by itself alone, as the problem is a quantitative question about which one is relatively stronger or weaker. It is not more than saying that equilibrium price is determined not only by demand alone or by supply alone, and depends on both, since interest rate is nothing but the relative price of current consumption for future consumption. In the case of constant returns to scale the interest rate depends only on the production technology, but note that it is a limit situation.

17.4

Efficiency of competitive equilibria

Competitive equilibrium allocation in production economy is Pareto efficient, which is a natural generalization of efficiency in exchange economy. Let us recall the definition of Pareto improvement. Definition 17.1 An allocation of consumption (x′1 , · · · , x′n ) is said to be a Pareto improvement of (x1 , · · · , xn ) if it holds x′i ≿i xi for all i = 1 · · · , n and

x′i ≻i xi

for at least one i. Net let us redefine feasible∑consumption ∑n allocation in the setting of producn tion economy Let (e1 , e2 ) = ( i=1 ei1 , i=1 ei2 ) denote the social sum of initial endowments. Definition 17.2 An allocation of consumption (x1 , · · · , xn ) is said to be feasible if there exists (y1 , · · · , yn ) such that n ∑ i=1 n ∑

xi1 xi2

= e1 + = e2 +

i=1

Tk (yk1 , yk2 )

m ∑ k=1 m ∑

yk1 yk2

k=1

= 0

k = 1, · · · , m

Again here is the definition of Pareto efficiency. Note that for welfare comparison itself only consumption allocation should matter.

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

214

Definition 17.3 A feasible allocation of consumption (x1 , · · · , xn ) is said to be Pareto efficient if no other feasible allocation can be a Pareto improvement of it. Recall that in exchange economies Pareto efficiency is characterized by equalization of marginal rates of substitution across consumers. In production economy we have additional condition, equalization of marginal rates of transformation across firms and equalization of marginal rates of substitution and marginal rates of transformation between consumers and firms. Proposition 17.1 Suppose each consumer has smooth preference and each firm’s production is described by smooth transformation function, then an interior consumption allocation x = (x1 , · · · , xn ) is Pareto-efficient if and only if there is a production activity y = (y1 , · · · , ym ) which makes the consumption allocation feasible such that it holds M RSi (xi ) = M RTk (yk ) for all i = 1, · · · , n and k = 1, · · · , m. Proof. (Efficiency =⇒ Equalization of MRSs and MRTs): Equalization of MRSs saying that M RSi (xi ) = M RSj (xj ) holds for all i, j follows from the same reason as before, otherwise the consumption allocation is not Pareto efficient even in allocating the given amount of resource produced by the given production activity. Next we show that M RTk (yk ) = M RTl (yl ) holds for all k, l. Let me first state the intuition. When the marginal rates of substitution are not equalized across firms there exists a firm with relatively higher productivity and another firm with relatively lower productivity. Then by moving some units input from the firm with lower productivity to the firm with higher productivity we can produce more output from the same amount of input, which makes everybody better off, which means the current production activity is not Pareto efficient. Now suppose for example M RTk (yk ) > M RTl (yl ) Recall that when Good 1 is input and Good 2 is output the marginal rate of transformation is the extra amount of Good 2 produced by adding extra one unit of Good 1 as input; when Good 1 is output and Good 2 is input the marginal rate of transformation is the extra amount of Good 1 produced by adding extra

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

215

one unit of Good 2 as input. Without loss of generality, assume here that Good 1 is input and Good 2 is output. Under the current interpretation the above inequality means that marginal product in Firm k is greater than that in Firm l. Then consider moving a slight amount Good 1 for input denoted by ∆y1 from Firm l to k. Since the transformation curves are locally straight Firm k can do the production activity (yk1 − ∆y1 , yk2 + M RTk (yk )∆y1 ) and Firm l can do the production activity (yl1 + ∆y1 , yl2 − M RTl (yl )∆y1 ) By summing up the two production activities we have (yk1 + yl1 , yk2 + yl2 + (M RTk (yk ) − M RTl (yl ))∆y1 ) which means that we have extra (M RTk (yk ) − M RTl (yl ))∆y1 units of Good 2 produced from the same yk1 + yl1 units of Good 1 as input. By suitably distributing this extra units of Good 2 among the consumers we can make improve everybody’s welfare status. Thus there is a room for Pareto improvements and the current resource allocation is not Pareto efficient. Finally let me show that M RSi (xi ) = M RTk (yk ) hold for all i, k. Suppose for example M RSi (xi ) > M RTk (yk ) Again, let me assume without loss of generality that Good 1 is input and Good 2 is output. Then the above inequality means that the amount of Good 2 which consumer i is willing to give up in order to get one extra unit of Good 1 is greater than the amount of Good 2 which Firm k can produce from one unit of Good 1. This suggests that we can make i better off without hurting anybody else, by reducing the production activity of Firm k, moving its input Good 1 to him and let him compensate the reduction of production activity suitably. Now consider reducing Firm k’s input by ”slight” amount ∆y1 , then since the transformation curve is locally straight we have its production activity becomes (yk1 + ∆y1 , yk2 − M RTk (yk )∆y1 ). Note that yk1 < 0, which means adding ∆y1 > 0 means reduction of input. Now give ∆y1 units of Good 1 to Consumer i and make him pay M RTk (yk )∆y1 units of Good 2 to Firm k for compensation. Since his indifference curve is locally straight we have (xi1 + ∆y1 , xi2 − M RSi (xi )∆y1 ) ∼i xi

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

216

Since we have M RSi (xi ) > M RTk (yk ) by assumption, we obtain (xi1 + ∆y1 , xi2 − M RTk (yk )∆y1 ) ≻i xi Thus, the above change makes Consumer i better off without hurting anybody else, which means that the current resource allocation is Pareto-inefficient. Similar argument works for the case of underproduction. (Equalization of MRS and MRT =⇒ Efficiency): Suppose MRSs and MRTs are equalized at x and y, that is, we have M RSi (xi ) = M RTk (yk ) = µ > 0,

i = 1, · · · , n

Now suppose x is not Pareto efficient. Then there is an another feasible allocation ((x11 + ∆x11 , x12 + ∆x12 ), · · · , (xn1 + ∆xn1 , xn2 + ∆xn2 )) with production activity ((y11 + ∆y11 , y12 + ∆y12 ), · · · , (ym1 + ∆ym1 , ym2 + ∆ym2 )) such that (xi1 + ∆xi1 , xi2 + ∆xi2 ) ≿i xi for all i = 1 · · · , n and (xi1 + ∆xi1 , xi2 + ∆xi2 ) ≻i xi for at least one i, where n ∑ i=1 n ∑

∆xi1 ∆xi2

= =

i=1

m ∑ k=1 m ∑

∆yk1 ∆yk2

k=1

When preferences are smooth, the consumption (xi1 + ∆xi1 , xi2 + ∆xi2 ) is above the tangent line passing through xi for all i = 1 · · · , n, we have ∆xi2 ≥ −M RSi (xi )∆xi1 Also, since (xi1 + ∆xi1 , xi2 + ∆xi2 ) is strictly above the tangent line passing through xi for at least one i, we have ∆xi2 > −M RSi (xi )∆xi1 for such i. Denote the equalized value of MRSs by µ, then we have M RSi (xi ) = µ for all i = 1 · · · , n. Then by adding up the above inequalities we obtain n ∑ i=1

∆xi2 > −µ

n ∑ i=1

∆xi2 .

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

217

When transformation functions are are smooth, the production activity (yk1 + ∆yk1 , yk2 + ∆yk2 ) is below the tangent line passing through yk for all k = 1 · · · , m, we have ∆yk2 ≤ −M RTk (yk )∆yk1 Denote the equalized value of MRTs by µ, then we have M RSk (yk ) = µ for all k = 1 · · · , m. Then by adding up the above inequalities we obtain m ∑

∆yk2 ≤ −µ

k=1

Since

∑n i=1

∆xi1 =

∑m k=1

i=1

∆yk1 .

k=1

∆yk1 and n ∑

m ∑

∑n i=1

∆xi2 ≤ −µ

∆xi2 = n ∑

∑m k=1

∆yk2 this implies

∆xi2 ,

i=1

which is a contradiction to the previous inequality. Thus, in economies with smooth preferences and production technologies Pareto efficiency is characterized by the equality of marginal rates of substitution and marginal rates of transformation. Since in competitive equilibrium it holds M RSi (x∗i ) = M RTk (yk∗ ) =

p∗1 p∗2

p∗

i = 1, · · · , n for all k = 1, · · · , m where p1∗ is the equilibrium relative price 2 of Good 1 for Good 2, MRSs and MRTs are equalized. Hence competitive equilibrium allocation is Pareto-efficient. One can show Pareto-efficiency of competitive equilibrium allocation without assuming smoothness of preferences or technologies, though. We have the following result in general. Theorem 17.1 (First fundamental theorem of welfare economics) Suppose the individuals’ preferences are monotone at least in the weak sense, then competitive equilibrium allocation is Pareto-efficient. Proof. Let p∗ = (p∗1 , p∗2 ) denote the equilibrium price and let x∗ = (x∗1 , · · · , x∗n ) ∗ and y ∗ = (y1∗ , · · · , ym ) denote the consumption allocation and production activity in equilibrium respectively. Now suppose x∗ is not Pareto-efficient, then there exists another feasible consumption allocation and corresponding production activity x = (x1 , · · · , xn ) and y = (y1 , · · · , ym ) such that xi ≿i x∗i holds for all i = 1 · · · , n and

xi ≻i x∗i

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

218

holds for at least one i, where the feasibility condition n ∑ i=1 n ∑

xi1

=

xi2

=

i=1

n ∑ i=1 n ∑

ei1 + ei2 +

i=1

m ∑ k=1 m ∑

yk1 yk2

k=1

is met. Recall that for each i his consumption x∗i is optimal for him under the ∑k budget constraint p∗1 xi1 + p∗2 xi2 ≤ p∗1 ei1 + p∗2 ei2 + k=1 θik πk (p∗ ). Just like in the argument on exchange economies we have the following lemma. Lemma 17.1 xi1 ≿i x∗i implies p∗1 xi1 + p∗2 xi2 ≥ p∗1 ei1 + p∗2 ei2 +

m ∑

θik πk (p∗ )

k=1

Also, notice that for at least one i it holds xi ≻i x∗i . Since x∗i is again already optimal for i under his budget constraint anything strictly better than that must not have been affordable. Hence we have p∗1 xi1 + p∗2 xi2 > p∗1 ei1 + p∗2 ei2 +

m ∑

θik πk (p∗ )

k=1

By summing up the above inequalities we obtain p1

n ∑

xi1 + p2

i=1

n ∑

xi2 > p1

i=1

n ∑

ei1 + p2

i=1

n ∑

ei2 +

i=1

m ∑

πk (p∗ )

k=1

On the other hand, since each firm k is maximizing its profit at yk∗ respectively, it holds ∗ ∗ πk (p∗ ) = p∗1 yk1 + p∗2 yk2 ≥ p∗1 yk1 + p∗2 yk2 By summing up this across firms we obtain ( n ) ( n ) n n m m ∑ ∑ ∑ ∑ ∑ ∑ p1 xi1 + p2 xi2 > p1 ei1 + yk1 + p2 ei2 + yk2 , i=1

i=1

i=1

i=1

k=1

but this contradicts to the feasibility condition n ∑ i=1 n ∑ i=1

xi1 xi2

= =

n ∑ i=1 n ∑ i=1

ei1 + ei2 +

m ∑ k=1 m ∑ k=1

yk1 yk2

k=1

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

219

Conversely, in production economies as well any Pareto-efficient allocation can be obtained through competitive equilibrium after suitable redistribution of income. Let me explain this assuming smooth preferences and technologies. Pick any Pereto-efficient allocation x = (x1 , · · · , xn ) with corresponding production activity y = (y1 , · · · , yn ). Since MRSs and MRTs are equalized at efficient allocation it holds M RSi (xi ) = M RTk (yk ) This equalized rate is taken to be the ”targeted price” to be obtained in competitive equilibrium. Thus one can take the ”targeted price” (p1 , p2 ) such that p1 = M RSi (xi ) = M RTk (yk ) p2 for all i and k. For each producer k, since the marginal rate of transformation is equal to the relative price, under the convexity assumption yk solves the profit maximization problem max p1 yk1 + p2 yk2 yk

subject to T (yk ) = 0 Let πk (p) denote the maximized profit. Now for each consumer i we need to given him income equal to p1 xi1 +p2 xi2 . Let the proportion of his income to the entire society be given by λi =

p1

p1 xi1 + p2 xi2 ∑n j=1 xj1 + p2 j=1 xj2

∑n

Then, give him initial resources let’s say by ei1 = λi e1 , ei2 = λi e2 and ownership of the firms let’s say by θik = λi for each k = 1, · · · , m. Then, xi is consumer i’s optimal consumption under the budget constraint p1 xi1 + p2 xi2 = p1 ei1 + p2 ei2 +

m ∑

θik πk (p)

k=1

This result does not depend on the smoothness of preferences and technologies.

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

220

Theorem 17.2 (Second fundamental theorem of welfare economics) Assume convexity and monotonicity of preferences and convexity of technologies. Then any Pareto-efficient allocation x = (x1 , · · · , xn ) can be obtained through competitive equilibrium after suitably redistributing initial holdings e = (e1 , · · · , en ) and ownership of firms θ = (θ1 , · · · , θn ). Pareto efficiency per se does not mention how it should be achieved, it may be achieved potentially by a centralized economy in which the centralized government orders consumption allocations and production activities. The above two theorem state, however, that efficiency is achieved through each consumer’s selfish choice and each firm’s profit maximization. This is called decentralization.

17.5

Socialist calculation debate

You might think here, however, like this: OK, what is essential here is to obtain an efficient consumption allocation, which meets some additional normative properties if wanted. Why do we need to rely on market? Why do we need to depend on firms’ profit maximization? Why not doing it in a planning economy? Since there is no guarantee that the perfect competition condition holds in private-ownership economies as a matter of fact (less likely indeed, given the actual presence of monopoly for example), a centralized planning economy is rather reliable, isn’t it? This question is very understandable, and was indeed raised by economists advocating socialism such as Oskar Lange. A naive criticism to socialist planning in the outset was that it cannot achieve efficient allocation because there no price system to convey information about demand and supply. The socialists’ claim got more sophisticated and became like Price can be introduced not only in private ownership economies but also in planning economies as ”reference price.” As the planning agency plays the role of both perfectly competitive firms and the auctioneer it can set the right price which supports efficient allocation, and it is more reliable than the price adjustment operation as in the private ownership economies in which the perfect competition condition is likely to fail. This line of thought is called market socialism. Now we cannot distinguish between private ownership economy and market socialist economy, as far as we just look at the model of competitive market as it is in its ”established form.” Thus the issue shifts to which one can indeed ”reach” efficient allocations supported by competitive equilibria, or which one is the more effective way to reach there.

CHAPTER 17. COMPETITIVE PRODUCTION ECONOMIES

221

Economists such as Hayek raised a criticism to market socialism that the central planning agency does not have an incentive to set the right price. In private ownership economies, when the price is apart from the right one supporting efficient allocation, for example when the relative price of some good is too high, the firms driven by profit maximization increase the supply or or there will be an entry to the market for it in order to grab this profit opportunity, so that its price falls down to the level supporting efficiency. On the other hand, they say the planning agency does not have any such motives. Although, those economists did not provide formal models explaining why private ownership economies do better job in collecting demand-supply information more efficiently (in a different sense from allocative efficiency). It was much later that theoretical models encompassing such issues are presented by economists.

Chapter 18

Partial equilibrium analysis Although it was in a simplified two-good setting for illustration, we have worked in the previous chapters on the model in which demands match supplies for multiple goods simultaneously. To emphasize such simultaneity we call it general equilibrium model. It is hard to analyze the behavior of general equilibrium ”by hand,” which is particularly the case when we need to deal with more goods. Thus it is hard to analyze the effects of policies such as taxation, subsidy and regulation in the way that we can see their implications to the allocations of ALL goods simultaneously.1 Thus in this chapter we focus on looking at a market of one output good, by isolating it from the rests of the economy which are assumed to be fixed. Such analysis is called partial equilibrium analysis. But how can we ”isolate” one from the rests? If some change occurs in the market of a good we are looking at it will necessarily leads to changes in the rests of the economy, which will in turn leads to an effect to the market under the analysis. Hence the partial equilibrium analysis must be limited to situations in which such isolation works. Partial equilibrium analysis assumes: 1. on the consumer side, there is no income effect on the good under analysis; 2. on the producer side, inputs are summarized into an income term, provided that cost minimization is carried out by taking input prices as given. The assumption of no income effect on the consumer side essentially states that the market for it is sufficiently small compared to the entire economy so that increase of decrease of expenditure on it has almost no effect on the income to be spent on the rests of goods. 1 Although it is quite popular nowadays to solve the general equilibrium models by computation, thanks to the technological progress.

222

CHAPTER 18. PARTIAL EQUILIBRIUM ANALYSIS

223

The assumption of cost minimization on the producer side states that the firms are taking input prices as given, meaning that they are small buyers in the input markets while may be big and have market powers in the output market, which is essentially presuming that the market for it is sufficiently small compared to the entire economy. The market for the good under partial equilibrium analysis is thus like a ”boat floating on the ocean.” Since the boat is negligibly small compared to the ocean, one can ignore its effect on the ocean, and can focus on the movement of the boat itself. When the market for the good under consideration is very small compared to the entire economy its behavior will not affect the entire economy and the behavior of the markets for ”the other commodities” will remain unchanged at least in the approximate sense.

18.1

Competitive partial equilibrium

Hereafter let Good 1 be the object of partial equilibrium analysis and let Good 2 be the income transfer to be spent on the rest of goods. The amount of Good 2 can be either positive or negative, and it is taken to be a payment when it is negative. Fix the price of Good 2 equal to one, that is, take the normalization p2 = 1. Then denote the price of Good 1 by p instead of p1 . On the production side, Good 1 is produced as output whereas Good 2 is used as input. I apologize for switching the roles between Good 1 and 2 from before, but here I like to put priority on describing consumer’s willingness to pay for Good 1, which is marginal rate of substitution of Good 2 for Good 1. From the assumption of no income effect we assume that consumers’ preferences are linear in Good 2, that is, each consumer i = 1, · · · , n has preference represented in the form ui (xi1 , xi2 ) = f (vi (xi1 ) + xi2 ) Since consumption Good 1 does not depend on income under the assumption of no income effect, without loss of generality assume that initial income is zero, and consider that consumer i obeys his budget constraint pxi1 + xi2 = 0 That is, the income transfer here is xi2 = −pxi1 , which is the payment for the purchase of Good 1. As seen before, consumption of Good 1 is independent of income and it is determined by the condition of equality between marginal willingness to pay and (relative) price vi′ (xi1 ) = p From this we obtain inverse demand function pi (xi1 ) = vi′ (xi1 )

CHAPTER 18. PARTIAL EQUILIBRIUM ANALYSIS

224

and demand function xi1 (p) = (vi′ )−1 (p). Then the aggregate demand function is given by x1 (p) =

n ∑

xi1 (p),

i=1

and by taking its inverse we obtain the aggregate inverse demand function p(x1 ). It is helpful later to take an understanding that the buyer who is willing to buy the ”x1 -th” unit of the good is willing to pay p(x1 ). Such ”x1 -th” consumer is said to be the marginal consumer at x1 . On the other hand, each firm k = 1, · · · , m has technology which is summarized in the form of cost function Ck (yk1 ), where yk1 denotes its output level of Good 1. In a competitive market it solves max pyk1 − Ck (yk1 ) yk1

Here we assume that the shut-down condition does not bind, and the equality between marginal cost and price holds, M Ck (yk1 ) = p From this we obtain the inverse supply function pk (yk1 ) = M Ck (yk1 ) and supply function yk1 (p) = (M Ck )−1 (p). Likewise, we obtain the aggregate supply function m ∑ y1 (p) = yi1 (p) k=1

Then competitive partial equilibrium in which demand matches supply in the market for Good 1 is determined by x1 (p∗ ) =

n ∑

xi1 (p∗ ) =

i=1

18.2

m ∑

yk1 (p∗ ) = y1 (p∗ )

k=1

Pareto efficiency and maximal surplus

In order to talk about efficiency let rewrite down the definition of feasible allocation in the setting of partial equilibrium. Definition 18.1 Consumption allocation x = (x1 , · · · , xn ) is said to be feasible if there exists (y11 , · · · , ym1 ) such that n ∑ i=1 n ∑ i=1

xi1 xi2

=

m ∑

yk1

k=1 m ∑

= −

k=1

Ck (yk1 )

CHAPTER 18. PARTIAL EQUILIBRIUM ANALYSIS

225

Note that the second equality says that total payment by the consumers equals to the total cost for production. In the framework of partial equilibrium Pareto efficiency is equivalent to maximizing social surplus. Sometimes this is confused with the idea of ”the greatest happiness of the greatest number.” Since we are maintaining the viewpoint of ordinal utility we cannot simply take the sum of utilities across individuals, since we cannot compare them without bringing in a particular faith. On the other hand, willingness to pay and consumer surplus have quantitative meanings and it has an economic meaning to take the sum of them. Indeed, as is discussed below maximizing social surplus has totally silent about how much surplus each consumer should gain. Proposition 18.1 Consumption allocation x = (x1 , · · · , xn ) is Pareto-efficient if and only if (x11 , · · · , xn1 ) is given by the solution to max x,y

n ∑

vi (xi1 ) −

i=1

m ∑

Ck (yk1 )

k=1 n ∑

subject to

m ∑

xi1 =

i=1

yk1

k=1

′ ′ ) , · · · , ym1 Proof. ”Only there exists (x′11 , · · · , x′n1 ) and (y11 ∑Suppose ∑n if”′ Part: m ′ such that i=1 xi1 = k=1 yk1 and n ∑

vi (x′i1 ) −

i=1

Let

( S=

n ∑

m ∑

′ Ck (yk1 )>



n ∑

i=1

vi (xi1 ) −

i=1

k=1

vi (x′i1 )

n ∑

) vi (xi1 )

( −

i=1

m ∑

Ck (yk1 )

k=1

m ∑

′ Ck (yk1 )



k=1

m ∑

) Ck (yk1 )

k=1

then by assumption we ∑nhave S > 0. Divide this among all individuals so that si > 0 for each i and i=1 si = S. Now for each i = 1, · · · , n let

Then, from

∑n

n ∑ i=1

i=1

x′i2

xi2

x′i2 = xi2 + vi (xi1 ) − vi (x′i1 ) + si ∑m = − k=1 Ck (yk1 ) and the definition of S it holds

=

n ∑

xi2 +

i=1 m ∑

= − = −

k=1 m ∑ k=1

n ∑

vi (xi1 ) −

i=1

Ck (yk1 ) +

vi (x′i1 ) +

i=1 n ∑ i=1

′ Ck (yk1 )

n ∑

vi (xi1 ) −

n ∑

si

i=1 n ∑ i=1

vi (x′i1 ) + S

CHAPTER 18. PARTIAL EQUILIBRIUM ANALYSIS

226

which implies x′ = ((x′11 , x′12 ), · · · , (x′n1 , x′n2 )) is feasible. The for all i = 1, · · · , n it holds vi (x′i1 ) + x′i2 = vi (xi1 ) + xi2 + si > vi (xi1 ) + xi2 Therefore x′ is a Pareto improvement of x. Hence x is not Pareto efficient. ”If” Part: Suppose x = (x1 , · · · , xn ) is not Pareto efficient. Then there is a feasible allocation (x′11 , · · · , x′n1 ) such that vi (x′i1 ) + x′i2 ≥ vi (xi1 ) + xi2 for all i and

vi (x′i1 ) + x′i2 > vi (xi1 ) + xi2

for at least one i. Then by summing up the inequalities we obtain n ∑

{vi (x′i1 ) + x′i2 } >

i=1

n ∑

{vi (xi1 ) + xi2 }

i=1

∑m ∑n ∑m ∑n ′ ) By feasibility we have i=1 xi2 = − k=1 Ck (yk1 ) and i=1 x′i2 = − k=1 Ck (yk1 for the corresponding productions. Hence we have n ∑ i=1

vi (x′i1 ) −

m ∑

′ Ck (yk1 )>

k=1

n ∑

vi (xi1 ) −

i=1

m ∑

Ck (yk1 )

k=1

Remark 18.1 In the framework of partial equilibrium analysis, Pareto efficiency pins down allocation of Good 1 basically uniquely through maximization of social surplus, but we should notice that it is totally silent about how the maximized surplus (i.e., Good 2) should be distributed among individuals. Any distribution of maximized social surplus is efficient, indeed. How we should distributed social surplus is a question which is orthogonal to the notion of efficiency. To clarify the above point, consider an exchange economy between two individuals. Consumer A has initial holding of Good 1 denoted by eA1 , and consumer B has eB1 . Without loss of generality let the initial holdings of Good 2 be eA2 = eB2 = 0. Then feasible allocation (xA , xB ) must satisfy xA1 + xB1 xA2 + xB2

= eA1 + eA2 = 0

Then the set of Pareto-efficient allocations is depicted as in Figure 18.1, which is a vertical line. This is because when equality of marginal rates of substitution between A and B holds it holds at any point vertically above or below, since indifference curves are parallel along the vertical axis due to the

CHAPTER 18. PARTIAL EQUILIBRIUM ANALYSIS xA2 6

xB1

re

227

- xA1

? xB2 Figure 18.1: Set of efficient allocations under quasi-linear preferences

assumption of no income effect. Thus, while the allocation of Good 1 is uniquely determined allocation of Good 2 is totally undetermined. When willingness to pay and cost functions are smooth, Pareto efficiency is characterized by equalization of marginal willingness to pay and marginal cost. Proposition 18.2 Consumption allocation x∑ = (x1 , · · · , ∑ xn )is Pareto-efficient n m if and only if there exists (y11 , · · · , ym1 ) with i=1 xi1 = k=1 yk1 such that vi′ (xi1 ) = M Ck (yk1 ) holds for all i = 1, · · · , n and k = 1, · · · , m. Proof is similar to the one for the equivalence between efficiency and equalization of MRS and MRT, or it follows directly from the first-order condition for maximizing the social surplus. In competitive equilibrium marginal willingness to pay of all consumers and marginal cost of all firms are equal to the price, that is, vi′ (xi1 ) = M Ck (yk1 ) = p holds for all i = 1, · · · , n and k = 1, · · · , m. Thus it is Pareto-efficient. Here the sum of consumers’ surplus is given by n ∑ i=1

(v(xi1 ) − pxi1 ) =

n ∑ i=1

v(xi1 ) − p

n ∑ i=1

xi1

CHAPTER 18. PARTIAL EQUILIBRIUM ANALYSIS

228

On the other hand, the sum of producers’ surplus is given by m ∑

(pyk1 − Ck (yk1 ))

k=1

Note that since the firms are ultimately owned by individuals the producers’ surplus is in the end paid back to them, while it remains to be a problem who should receive how much. The point here is that in a competitive market maximization of social surplus is achieved through each individual’s pursue of maximizing his consumer surplus and each firm’s maximizing producer surplus (i.e., profit) in a decentralized manner.

18.3

Exercises

Exercise 20 Let Good 1 be the object of partial equilibrium analysis and Good 2 to be the income transfer to be spent on the other goods. Assume that each consumer i = 1, · · · , n has quasi-linear preference represented by ui (x) = √ ai xi1 +xi2 . Each firm k = 1, · · · , m is characterized by cost function Ck (yk ) = ck yk2 . Find the competitive equilibrium price.

Part III

Imperfect competition and game theory

229

Chapter 19

Monopoly Recall that perfect competition refers to situations in which the markets consist of a large number of small participants, each of which is negligibly small compared to the entire economy and has to take the market price as give which he cannot manipulate by himself alone. This assumption is pretty innocuous for the demand side, but it will be a problem for the producer side. There are many instances in which the markets consist of a small number of large firms. In such cases, sellers must be aware that their actions directly and indirectly affect the market price. We call such situation imperfect competition, in which certain market participants have market powers. Analysis of imperfect competition is mostly carried out in the framework of partial equilibrium analysis. For, the concept of imperfect competition and the concept of general equilibrium are actually very hard to be compatible with each other. First, when a given firm has a market power and be able to manipulate the market price, it may use this power either to maximize its profit or to manipulate the market price in favor of its shareholders’ interests in their consumptions, which are not necessarily compatible each other. This conflict leaves it ambiguous what the objective of a firm is. Second, while in the general equilibrium analysis only relative prices should matter and the analysis does not depend on how to normalize prices (which one to set equal to 1), price to be manipulated by firms in imperfect competition in the market for a particular output is an absolute one, or in other words it is a particular kind of relative price obtained by taking ”the price of income” equal to 1. Third point is a mathematical one, but it is that the profit functions to be used in general equilibrium analysis is not well-behaved (technically speaking, not concave) and cannot guarantee the existence of general equilibrium. Having said that, I explain imperfect competition in the framework of partial equilibrium analysis. I start with monopoly, the most extreme case of imperfect competition. 230

CHAPTER 19. MONOPOLY

19.1

231

Monopoly equilibrium

In monopoly, there is not an essential difference between quantity setting and price setting, since the monopolist firm wants to set the price equal to some particular value it can do that by manipulating the quantity in order that the resulting price is equal to that, and vice versa. Hence without loss of generality I start with a quantity setting argument. As discussed above, I adopt the partial equilibrium framework in which the demand side is summarized in the form of an inverse demand function p(x) and the monopolist firm is described by a cost function C(y). For comparison, first let me pretend that this firm behaves as a representative producer in a perfectly competitive market, then the firm takes the output price as a constant p which it cannot manipulate by itself alone. Hence the profit maximization problem is max py − C(y) y

and the profit maximization condition is p = M C(y) Since p = p(y) in competitive equilibrium, we obtain the competitive output level y CE by p(y CE ) = M C(y CE ). When the firm is a monopolist, on the other hand, it must be aware that if it provides y units of output the resulting price per unit is p(y). Hence the revenue considered by the monopolist firm is R(y) = p(y)y Since the cost is given by C(y), the monopolist’s profit is R(y) − C(y) The monopolist firm maximizes this by manipulating y. Thus the profit maximization problem for the monopolist firm is max R(y) − C(y) y

Here the firm’s marginal revenue denoted by M R(y), which is the additional revenue obtained from selling one extra unit of output, is given by the derivative of R(y), that is, M R(y) = R′ (y). Note that when you plot the marginal revenue curve it must be below the inverse demand curve as depicted in Figure 19.1. This follows from ′

M R(y) = R′ (y) = (p(y)y) = p(y) + p′ (y)y

CHAPTER 19. MONOPOLY

232

6

M C(y)

r

pM

r

pCE

p(y)

M R(y) r yM

y CE

-y

Figure 19.1: Monopoly equilibrium

and that the inverse demand curve is downward-sloping, meaning p′ (y) < 0, which implies M R(y) < p(y). Since the marginal cost is M C(y) = C ′ (y) as before, the profit-maximization condition is given by the equality between marginal revenue and marginal cost M R(y M ) = M C(y M ) Let us call such y M monopoly equilibrium. Then the monopoly price is given by pM = p(y M ). Because the marginal revenue curve is below the inverse demand curve, it holds y M < y CE and therefore pM > pCE . That is the output level is lower and the price is higher in monopoly equilibrium when it is compared to competitive equilibrium. To illustrate, consider the case of linear inverse demand and constant marginal cost. The inverse demand function is given by p(x) = a − bx, then the revenue takes the form R(y) = p(y)y = ay − by 2 and the marginal revenue is M R(y) = a − 2by On the other hand, the cost function is given by C(y) = cy, in which the marginal cost is M C(y) = c If this firm behaves as if it is the representative firm in a perfectly competitive market, its supply function is flat and given by p = c. Hence the competitive equilibrium is a−c pCE = c, y CE = b

CHAPTER 19. MONOPOLY

233

You can verify that the consumer surplus here is

(a−c)2 2b

and the producer surplus

(a−c)2 2b .

is zero, which implies the social surplus is In monopoly equilibrium on the other hand, you can verify from M R(y M ) = M C(y M ) that the monopoly quantity is yM =

a−c 2b

and the monopoly price is pM = p(y M ) =

19.2

a+c . 2

Pareto inefficiency of monopoly equilibrium

Since M R(y M ) = M C(y M ) holds in monopoly equilibrium y M and since the marginal revenue curve is below the inverse demand curve which means p(y M ) > M R(y M ), we obtain p(y M ) > M C(y M ) Thus in monopoly equilibrium there is a discrepancy between consumers’ marginal willingness to pay and the producer’s marginal cost, while they are equalized in competitive equilibrium. Now at quantity y M , there is a consumer who is willing to pay p(y M ) for one extra unit of the output good (he is called the marginal consumer at y M ), while the cost of producing one extra unit there is M C(y M ). Hence if the monopolist sells a ”slight” amount ∆yto this marginal consumer at price p such that p(y M ) > p > M C(y M ), then the marginal consumer gains additional surplus (p(y M ) − p)∆y and the monopolist gains additional surplus (p − M C(y M ))∆y. Thus at monopoly equilibrium there is a room that both consumers and the monopolist gain extra surplus. Since the extra surplus earned by the monopolist is paid back the shareholders who are consumers, by doing this we can make certain consumers without hurting anybody else, which is a Pareto improvement. Thus, allocation in monopoly equilibrium is Pareto-inefficient. While there is a consumer who are willing to pay more than the cost of additional production he cannot buy one. In this sense the monopolistic market is losing the opportunity to gain additional surplus equal to (p(y M )−M C(y M ))∆y. This marginal dead-weight-loss remains positive as far as y M ≤ y ≤ y CE , which becomes zero only when y reaches y CE . Hence the total or cumulative dead-weight-loss is obtained by adding up the marginal dead-weight-loss by ∫

y CE

(p(y) − M C(y)) dy, yM

CHAPTER 19. MONOPOLY

234

In Figure 19.1, the total dead-weight-loss is given by the area surrounded by (y CE , pCE ), (y M , M C(y M )) and (y M , pM ). There may be different answers to why monopoly is bad (if it is indeed bad). One may say it is bad because it is unfair. One may raise other ethical reasons. Here we can say that monopoly is bad without resorting to fairness, since the economy there is losing the opportunity of mutual gains from trade.

19.3

Price discrimination and monopolistic surplus extraction

One might argue, however, that monopoly itself is not the culprit of inefficiency, and the culprit is the constraint that the monopolist cannot exercise price discrimination so that it charges differently across different consumers and different units of purchase. What do I mean? I wrote in the previous explanation that the Pareto improvement at y M is done by trading at a lower price p such that p(y M ) > p > M C(y M ). However, if price discrimination is not allowed the monopolist has to sell every unit at this lower price p, which is not acceptable for it. Inefficiency due to such constraint may be recovered when the monopolist can charge higher amount to those with higher willingess to pay and lower amount to those with lower willingness to pay, so that the economy can fully exploit the opportunity of gains from trade. Then the monopolist can extract social surplus as much as possible, and if such extracted surplus is suitably redistributed among people we can make everybody better off. Of course, if this ”suitable” redistribution of extracted surplus is not guaranteed price discrimination may be regarded as undesirable from certain social viewpoints, and that’s why it is sometimes made illegal. So here I keep myself silent about how the extracted surplus should be distributed among people, and focus on how we can fully exploit the potential opportunity of gains from trade.

19.3.1

First-degree (or perfect) price discrimination

Let me start with the most extreme case in order to convey the point. That is, consider that the monopolist can charge differently across different consumers, consumer by consumer, and differently across units, unit by unit. Discrete case illustration To illustrate, consider that the are two types of consumer A and B, each of which has population NA and NB respectively, where NA < 2NB . Assume for

CHAPTER 19. MONOPOLY

235

simplicity that everybody can consume at most two units, and { 10, for the 1st unit A’s willingness to pay = 5, for the 2nd unit { 8, for the 1st unit B’s willingness to pay = 3, for the 2nd unit Consumers buy when the price is lower than willingness to pay, and do not buy when the willingness to pay is higher than willingness to pay. We assume without loss of generality that consumers buy when the price is equal to willingness to pay, because the monopolist can slightly lower the price so that consumers buy for sure while it does not make much difference for the seller side. We assume that the marginal cost for the monopolist is constant and given by M C = 4, that is, each unit of production of output costs 4 always. For comparison, first let me pretend that the monopolist firm behaves as if it is a representative firm in a competitive market. As the competitive equilibrium price equals to the marginal cost it is 4. Then Type A buys 2 units and Type B buys 1 unit. Type A consumer’s surplus is 10+5−4×2 = 7, Type B consumer’s surplus is 8 − 4 = 4. Since the producer surplus is zero here, the total social surplus is 7NA + 4NB . If no price discrimination is allowed for the monopolist, it has to charge equally across different consumers and different units. The natural candidates are as follows. 1. Charge 10 per unit: The profit is 10NA − 4NA = 6NA 2. Charge 8 per unit: The profit is 8(NA + NB ) − 4(NA + NB ) = 4NA + 4NB 3. Charge 5 per unit: Th profit is 5(2NA + NB ) − 4(2NA + NB ) = 2NA + NB 4. Charge 3 per unit: The profit is negative since 3 is below the marginal cost. Under the assumption NA < 2NB , the second one is profit maximizing and the maximized profit is 4NA + 4NB . If the monopolist knows every consumer’s willingness to pay, it can make the following offer: ”If you are a Type A consumer please pay 10 for the 1st unit and 5 for the second unit. If you are a Type B consumer please pay 8 for the 1st unit and please do not buy the 2nd unit.” Or Then Type A consumer buys 2 units and Type B consumer buys 1 units, and the profit is 10NA + 5NA + 8NB − 4 × (2NA + NB ) = 7NA + 4NB which is equal to the maximal social surplus. The consumer surplus is zero on the other hand, Hence the entire social surplus is extracted by the monopolist. This is called first-degree price discrimination, which is also called perfect price discrimination because of the complete extraction property.

CHAPTER 19. MONOPOLY

236

Continuous case Now consider the case of continuous quantity. The consumer side is summarized by an aggregate inverse demand function p(x), which says ”the marginal consumer at ”x-th unit” is willing to pay p(x).” Here the first-degree price discrimination says ”if you are the marginal consumer at ”x-th unit” please pay p(x),” and stops selling at yb such that p(b y ) = M C(b y) which is the surplus maximizing output level, that is, yb = y CE . Since each consumer pays his willingness to pay for each unit the consumer surplus is zero, and the entire maximal social surplus is extracted by the monopolist, which is given by ∫ yCE p(x)dx − C(y CE ). 0

Graphically, the maximized social surplus is the area surrounded by the inverse demand curve, the marginal cost curve and the vertical axis in Figure 19.1. The problem of misreporting The first-degree price discrimination is hardly observed in reality, however. The reason is that willingness to pay is in general each consumer’s private information which is not verifiable by others. Even if the others know one’s willingness to pay it is a different question if it is verifiable. There is a distinction between knowing something and being able to verify it. Go back to the leading example. There Type A consumer is supposed to pay more than Type B and receives no consumer surplus. This gives a Type A consumer an incentive to misreport his consumer type and mimic to be a Type B consumer. He would say, ”No, I’m not Type A but Type B.” Since the seller cannot verify buyer’s type, he cannot counter-argue against it and has to accept the claim.

19.3.2

Second-degree price discrimination

This leads us to think of charging differently across different units, unit by unit, while the same payment schedule must apply to all buyers. For, the seller can track and verify how many units a given consumer is purchasing. This is called second-degree price discrimination. Discrete case illustration Let me explain using the same numerical example as before. The natural candidates are as follows. Note that any consumer cannot buy the second unit without buying the first unit. 1. Charge 8 for each unit (profit-maximizing price when no price discrimination is allowed): The profit is 4NA + 4NB .

CHAPTER 19. MONOPOLY

237

2. Charge 10 for the 1st unit, 5 for the 2nd unit: The profit is 10NA + 5NA − 4 × 2NA = 7NA 3. Charge 8 for the 1st unit, 3 for the 2nd unit: The charge for the 2nd unit is obviously unprofitable. 4. Charge 10 for the 1st unit, 3 for the 2nd unit: The charge for the 2nd unit is obviously unprofitable. 5. Charge 8 for the 1st unit, 5 for the 2nd unit: The profit is 8(NA + NB ) + 5NA − 4(2NA + NB ) = 5NA + 4NB Under the assumption NA < 2NB the last one is profit maximizing and the profit 5NA + 4NB . This is lower than the profit earned if the first-degree price discrimination is possible, but it is higher than the profit earned when no price discrimination is allowed. The second-degree price discrimination does not make consumers fully reveal their willingness to pay, but it induces each consumer to self-select his type so that a heavy user buys more and a light user buys less. Charging system for for mobile phones is one such example. Continuous case In the setting of continuous quantity the second-degree price discrimination is also called non-linear pricing. This is because while the total charge is linear in pursed quantity when the price per unit is constant it is non-linear when the price per unit varies unit by unit. In Figure 19.2, the graph of total charge is linear when the price per unit is constant. In general, non-linear pricing can yield any curve, but how to choose such curve in order to maximize the profit is hard (technically speaking it is maximization in an infinite-dimensional space). So practically we restrict attention to piece-wise linear payment schedule in which price per unit is constant up to some quantity and shifts to another constant after than, and so on.

19.3.3

Third-degree price discrimination

On the other hand, when consumer types are physically verifiable to some extent the seller may group the consumers and charge differently across the groups. Student discount and senior discount are such examples. Sellers can ask customers to show their student IDs, and also can ask them to show their ID cards for age verification. However, this time the seller cannot charge differently between different units, unit by unit. This is called third-degree price discrimination. Sellers offer student discounts not because they are sympathetic to students. If they charge uniformly students will not buy when their willingness to pay is lower than that of business persons. Given that, it can be better for the monopolist to sell the good at cheaper price to students rather than losing them. However, if a business person can mimic to be a student such attempt simply

CHAPTER 19. MONOPOLY

238

total pay 6

-

quantity

Figure 19.2: Non-linear pricing

leads to selling the good uniformly at the cheaper price. Thus it is important here that a business person cannot mimic to be a student. Third-degree price discrimination is also known as multi-market monopoly. When several markets are geographically and institutionally separated so that consumers in one market cannot mimic to be a consumer in another market, the monopolist can price differently between the different markets. Discrete case illustration Let us consider the previous example again. Suppose the monopolist can verify between Type A and Type B, and can charge differently between the groups. However, each unit must be sold at the same price to consumers in a given group. Then the natural candidates are as follows. 1. Charge 8 per unit to both A and B (profit-maximizing price when no price discrimination is allowed): The profit is 4NA + 4NB 2. Charge 10 per unit to A and Charge 8 per unit to B: The profit is 10NA + 8NB − 4(NA + NB ) = 6NA + 4NB 3. Charge 10 per unit to A and Charge 3 per unit to B: Sales to B is obviously unprofitable. 4. Charge 5 per unit to A and charge 8 to B: The profit is 5 × 2NA + 8NB − 4(2NA + NB ) = 2NA + 4NB 5. Charge 5 per unit to A and charge 3 to B: Sales to B is obviously unprofitable. Here the second one is profit maximizing. The maximized profit is 6NA + 4NB , which is not as much as 7NA + 4NB which is earned if the first-degree

CHAPTER 19. MONOPOLY

239

discrimination is possible, but it is larger than the profit earned when no price discrimination is possible. Remember that third-degree price discrimination is possible only for verifiable types. For example, suppose there is another type of consumer C with population NC and the seller cannot distinguish between B and C. Then the monopolist can price-discriminate between Group A and Group BC, but cannot price-discriminate between Type B and Type C. Continuous case Now consider the case of continuous quantity. Suppose the monopolist can group consumers into two, A and B. The aggregate inverse demand in Market A is given by pA (xA ) and that in Market B is given by pB (xB ). Thus, when the monopolist provides yA units to Market A and yB to Market B the profit one unit is sold at price pA (yA )in Market A and at price pB (yB ) in Market B. Since the total cost is C(yA + yB ) the profit for the monopolist is pA (yA )yA + pB (yB )yB − C(yA + yB ) The monopolist varies both yA and yB in order to maximize its profit. The profit maximization condition is then M RA (yA ) = M C(yA + yB ) M RB (yB ) = M C(yA + yB ) Let us go over an example of linear demand and constant marginal cost. Suppose demand functions in the two markets are given by xA (p) = 100 − p xB (p) = 100 − 2p respectively. Then the inverse demand functions the two markets are pA (xA ) = 100 − xA 1 pB (xB ) = 50 − xB 2 respectively. Suppose the cost function is C(y) = 20y, that is, the marginal cost is constant and given by M C = 20. When no price discrimination is allowed the monopolist faces a single market, in which the aggregate demand function is x(p) = xA (p) + xB (p) = 200 − 3p, and its inverse demand function is p(x) =

200 3

− 13 y.

CHAPTER 19. MONOPOLY

240

2 Since marginal revenue in this single market is M R(y) = 200 3 − 3 y, the 200 2 profit maximizing condition is 3 − 3 y = 20, which implies y = 70. Thus the resulting price in the single market is p(70) = 130 3 and the monopolist’s profit 1 is 4900 = 1633 . 3 3 On the other hand, when third-degree price discrimination is allowed the monopolist solves ( ) 1 max (100 − yA )yA + 50 − yB yB − 20(yA + yB ) yA ,yB 2

Then the profit maximization condition is 100 − 2yA = 20 50 − yB = 20 Hence the monopolist provides yA = 40 to Market A and yB = 30 to Market B. The resulting price in Market A is pA = 60 and that in Market B is pB = 35, which are different. Now the monopolist’s profit is 60 × 40 + 35 × 30 − 20 × (40 + 30) = 2050, which is greater than that earned when no price discrimination is allowed.

19.3.4

Two-part tariff

Consider the case of linear inverse demand p(x) = a − bx and constant marginal cost c, where the linearity of inverse demand is not essential here but the constancy of marginal cost is. Then the competitive equilibrium quantity is y CE = a−c b , which is Pareto 2

efficient, and the price is y CE = c. Here the maximal social surplus is (a−c) 2b , all of which is taken by the consumer side. On the other hand, in monopoly equilibrium without price discrimination we already saw that there is an efficiency loss. Price discrimination is already discussed as a method to recover the efficiency loss, but there is another way here: charge the entire social surplus as an ”entry fee” and charge the marginal cost for each unit of purchase. In the above 2 example the entry fee is (a−c) and the charge per unit is c. By doing this the 2b monopolist can extract the entire maximal social surplus. The monopolist of course needs to know every consumer’s inverse demand function and the marginal cost has to be constant in order that the two-part tariff is carried out precisely. It is doable in a milder manner, however, and it quite often used in practice. A familiar examples are warehouse club and sports gym.

19.3.5

Bundling

I explain this through an example. Microsoft has two products, Word and Excel. We assume for simplicity that it can produce copies of them at no cost. Also we assume it has a technology to prevent piracy.

CHAPTER 19. MONOPOLY

241

There are two equally populated groups of consumers, A and B. Consumers in each group has willingness to pay for each of Word and Excel as in the following table. Like before we assume without loss of generality that consumer buys one when its price is exactly equal to his willingness to pay. A B

Word 12 8

Excel 8 12

Suppose Microsoft price each of Word and Excel. When the price of Word is 12 only consumer A buys it and the profit is 12, when it is 8 both A and B buy and the profit if 16. Hence it is profit-maximizing to set the price of Word equal to 8. Likewise, it is profit-maximizing to set the price of Excel equal to 8. Then the total profit is 16 × 2 = 32. Now consider that Microsoft can bundle Word and Excel into ”Office” and sell it. For simplicity assume that willingness to pay for Office is equal to the sum of willingness to pay for Word and willingness to pay for Excel. Then both A and B are willing to pay 20 for Office. Thus, by selling Office for price 20 instead of selling Word and Excel separately Microsoft can earn profit 20 × 2 = 40, which is greater than the above. Let us proceed one step further to consider that Microsoft Word and Excel separately as well as Office. Then each consumer chooses either of (i) buying Word only; (ii) buying Excel only; (iii) buying Office; (iv) buying both Word and Excel separately, and (v) nothing. Suppose there are four equally populated groups of consumers A,B,C and D. Their willingness to pay is as follows. A B C D

Word 12 8 15 0

Excel 8 12 0 15

Office 20 20 15 15

Here A and B are ”modest” consumers and C and D are ”extreme” consumers. Then the profit-maximizing choice is to let A and B buy Office, let C buy Word only and let D buy Excel only, which is to set the price of Word equal to 15, the price of Excel equal to 15 and the price of Word equal to 20. The problem is more complicated when willingness to pay for Office is not equal to the sum of willingness to pay for Word and willingness to pay for Excel, since if the price of office is too high consumers may buy both Word and Excel separately. Solution in such cases would require more sophisticated technique of combinatorial optimization.

19.4

Exercises

Exercise 21 The market inverse demand function is given by p(y) = 120 − 2y and the monopolist firm has cost function c(y) = 0.5y 2 .

CHAPTER 19. MONOPOLY

242

(i) Suppose this firm behaves as a price-taker, and find the quantity, price consumer surplus, producer surplus and social surplus in competitive equilibrium. (ii) Find the quantity, price consumer surplus, producer surplus, social surplus and dead weight loss in monopoly equilibrium. (iii) Described the first-degree price discrimination which achieves full surplus extraction here. Exercise 22 There is a firm being the monopolist in two markets. Its cost function is c(y) = 0.5y 2 . The market inverse demand function in Market A and B are respectively pA (yA ) = 90 − yA and pB (yB ) = 120 − 2yB . (i) Suppose the firm cannot price-discriminate and also behaves as a price-taker. Find the quantity and price in competitive equilibrium. (ii) Suppose the firm cannot price-discriminate. Find the quantity and price in monopoly equilibrium. (iii) Suppose the firm can price-discriminate. Find the quantity and price in monopoly equilibrium.

Chapter 20

Basic game theory I: normal-form games This book is not intended to be a textbook on game theory itself, but I would like to cover it as far as it is useful for the understanding of microeconomic theory. The word ”game” may sound pretty trivializing itself, and the theory is indeed often caricaturized irrelevantly because of this name. ”Game” here, however, refers to a situation of strategic interdependence, in which the consequence of one’s action depends not only on his own action but also on others’s actions, and vice versa for the others. It is even everything in our social life. In the analysis of perfect competitive markets, each market participant is suppose to take the market price as given and responds to it just passively, so we didn’t have to think of interaction between the participants explicitly. However, under imperfect competition a small number of large participants manipulate the market price while interacting with each other, which requires game-theoretic analysis. Also, even in analyzing perfectly competitive markets if you might like to describe the process of price formation in an explicit manner (since if you take the assumption of price-taking literally nobody is taking the role of price-setting). Thus if you want to explain thing not by ”invisible hands” but by ”visible hands” you need to explicitly describe trading behaviors by means of game theory. There are two ways of describing strategic interdependence. One is called normal form game, which is about simultaneous decisions by players, and the other is called extensive-form game, in which there are first mover, second mover and so on and they make decisions sequentially.1 1 As is discussed later, however, one can describe sequential decisions by means of normalform game, assuming that they simultaneously choose ”plans” on their course of actions and commit to them.

243

CHAPTER 20. BASIC GAME THEORY I

20.1

244

Description of strategic interdependence: normal-form games

Let me start with the simplest case that there are two players. There a normalform game is described in the form a payoff matrix. Let me explain through an example. Example 20.1 Market entry: There are two firms, A and B. They choose whether to enter the market or not, respectively. Payoffs are given in the following table, B Entry Non-entry Entry 5, 5 10, 0 A Non-entry 0, 10 0, 0 where the number in the left in each cell refers to A’s payoff and that in the right in each cell refers to B’s payoff. The table is read as If both A and B enter each of them gets 5. If A enters and B does not enter A gets 10 and B gets 0. If A does not enter and B enters A gets 0 and B gets 10. If neither A or B enters each of them gets 0. Now let me formalize this in a more general manner. A normal-form game consists of a set of players, strategy sets and payoff functions. The set of players is a finite set I = {1, · · · , n}. For each player i = 1, · · · , n, let Si denote the set of his∏strategies, where its generic element is denoted let’s say by si ∈ Si . Let n S = i=1 Si denote the set of all the combinations of all the players’ strategies, where its element denoted by s = (s1 , · · · , sn ) is called a strategy profile. Given a strategy profile s = (s1 , · · · , sn ), the payoff received by player i is denoted by vi (s). Since this is defined for all s ∈ S, player i’s payoff is described by a function vi : S → R, which is called payoff function for player i. Let us apply this formalization to the leading example. The set of players is I = {A, B}. The strategy sets are SA = {E, N } and SB = {E, N } respectively, where E denotes Entry and N denotes Non-entry. The payoff functions are given by vA (E, E) = vA (E, N ) = vA (N, E) =

5, 10, 0,

vA (N, N ) =

0

CHAPTER 20. BASIC GAME THEORY I

245

and vB (E, E) =

5,

vB (E, N ) = vB (N, E) =

0, 10,

vB (N, N ) =

0

where the first argument in the functions is A’s strategy and the second if B’s strategy. I guess the notion of payoff function may not be fully convincing to you at this point, but I will come to this after explaining one more example. Example 20.2 Prisoners’ dilemma Two gangs are arrested for a minor crime of which they are already convicted. They are suspected to have committed a serious crime, however. The prosecutor offers the following legal deal: if one confess while the other does not the one who confessed is free and the one who did not confess is prisoned for 10 years. If both confess each of them is prisoned for 5 years. If neither confesses each of them is prisoned for 1 year for the crimed they are already convicted of. For illustration, let me count payoffs by the negatives of years in prison (it doesn’t have to be, though, as I’ll explain in the next subsection), then the payoff matrix is B C N C −5, −5 0, −10 A N −10, 0 −1, −1 where C refers to confess and N refers to not to confess. Let us apply this formalization to the leading example. The set of players is I = {A, B}. The strategy sets are SA = {C, N } and SB = {C, N } respectively. The payoff functions are given by vA (C, C) =

−5

vA (C, N ) = 0 vA (N, C) = −10 vA (N, N ) =

−1

vB (C, C)

−5

and =

vB (C, N ) = −10 vB (N, C) = 0 vB (N, N ) = −1

CHAPTER 20. BASIC GAME THEORY I

20.1.1

246

On payoff functions

Let me now get into the detailed arguments on the payoff functions. Throughout this book I’m maintaining the standpoint that utility representation of preference is only an ordinal notion and it has no quantitative meanings. In the above specifications, however, I had assigned particular numbers to strategy profiles. It may be OK in the first example in which the firms’ payoff are described by their profits, but how can we describe individuals’ payoffs numerically like in the second example? In order to establish the precise definition of payoff function game theory borrows expected utility theory as introduced in Chapter 9. As is discussed later, strategies taken in games may be in general stochastic. Thus we consider the set of probability distributions over the set of strategy profiles and apply the expected utility theory there. That is, each player i has preference ≿i over the set of probability distributions over the set of strategy profiles, denoted by ∆(S), and it is represented in the form ( ) ( ) ∑ ∑ p ≿i q ⇐⇒ f vi (s)ps ≥ f vi (s)qs s∈S

s∈S

for p, q ∈ ∆(S), where vi is the vNM index which described player’s risk attitude and f is an arbitrary monotone transformation. The vNM index vi which forms the expected utility representation here is the payoff function what are talking about now. Thus we should understand that numbers appear in payoff matrices are already adjusted to the players’ risk attitudes. vMM index is ”cardinal” in the sense that it forms representation of preference in the expectation form, in which we take summation operations over the values of the index. I has been already discussed, however, that only the ”curvature” of the index has quantitative meanings and absolute amount of ”utility change” or absolute level of ”utility” have no meaning. Thus, if vi a vNM index which forms an expected utility representation for some preference, its any affine formation is a vNM index which forms an expected utility representation for the same preference. For example, in the Prisoners’ dilemma B A

C N

C −5, −5 −10, 0

N 0, −10 −1, −1

consider that we multiply 2 to each of A’s payoff and add 8 to it uniformly, and multiply 3 to each of B’s payoff and subtract 5 from it uniformly, then we

CHAPTER 20. BASIC GAME THEORY I

247

obtain B A

C N

C −2, −20 −12, −5

N 8, −35 6, −8

which describes the same game as the above. It is merely for simplicity that we use the first one instead of the second. Note again that the overall representation of preference allows arbitrary monotone transformation (denoted f here), which is consistent with the standpoint that representation of preference is ordinal.

20.2

Dominant strategy

What is the most unambiguous course of action in normal-form games, if it exists? In the first example there is an unambiguous answer fortunately (I made the example so that it is the case, though), for, Entry is optimal for A no matter what B chooses, and Entry is optimal for B no matter what A chooses When there is a strategy which is optimal no matter what opponents do, it will be unambiguously chosen. When a strategy is optimal for a given player no matter what the others to it is called a dominant strategy for him. In the above example Entry is a dominant strategy for A and also Entry is a dominant strategy for B. When all players have their dominant strategies respectively, the strategy profile consisting of them is called a dominant strategy equilibrium. More formally, Definition 20.1 s∗i ∈ Si is a dominant strategy for i if vi (s∗i , s−i ) ≥ vi (si , s−i ) holds for all si ∈ Si and all s−i ∈ S−i . A strategy profile (s∗1 , · · · , s∗n ) is said to be a dominant strategy equilibrium if for each i his strategy s∗i is a dominant strategy for him. Surprisingly maybe, the prisoners’ dilemma is a very example of game with a dominant strategy equilibrium. If your opponent confesses it is best for you to confess in order not to bear the whole responsibility. If your opponent does not confess if it best for you to confess since it lets you be free. In any case, no matter what your opponent does it is bet for you to confess. Since this holds for the opponent as well we have a dominant strategy equilibrium (C, C). Although it is unanimously better for both that both choose not to confess, this does not realize. What is interesting about the prisoners’ dilemma is that the consequence of each individual acting ”rationally” may be undesirable for

CHAPTER 20. BASIC GAME THEORY I

248

everybody, and that it happens in most the ”obvious” equilibrium situation, namely the dominant strategy equilibrium. Of course, however, dominant strategy does not always exist. Example 20.3 Market entry 2: Modify the previous example on market entry so that if both enter both lose, like B A

Entry Non-entry

Entry −2, −2 0, 10

Non-entry 10, 0 0, 0

Now there is no dominant strategy. If B is going to enter it is best for A to stay out, and if B is going to stay out it is best for A to enter. Similarly for B. Thus there is no strategy which is optimal for a given player no matter what the opponent does.

20.3

Iterated elimination of dominated strategies

What can we think of next when there is no dominant strategy? Let us consider the following question. Suppose each player is ”rational” in the sense that they can process given information in a logically correct manner and that he maximizes his expected utility under the given information. Then how much can players narrow down the strategies? Let me start by explaining with an example.

A

X Y Z

F 3, 0 4, 3 2, −2

B G 4, 1 6, 2 5, 0

H 2, 5 3, 1 8, −1

We see in the above table that strategy X is worse than Y from A’s viewpoint no matter what B chooses. Then we say that X is strictly dominated by Y for A. A strictly dominated strategy can never be an optimal choice no matter what the opponent does. Hence, if A1-1: A knows the payoff matrix correctly, A1-2: A is ”rational” in the sense explained above, X is never chosen by A. Now, if it holds

CHAPTER 20. BASIC GAME THEORY I

249

B1-1: B knows the payoff matrix correctly, B1-2: B knows A1-1 and A1-2 then B knows ”A never chooses X.” Thus B eliminates X from the possibility. The game after the elimination is now

A

Y Z

F 4, 3 2, −2

B G 6, 2 5, 0

H 3, 1 8, −1

In the game after the elimination, we see that H is worse than G for B, no matter what A chooses (except when A chooses X, which is the case already eliminated). That is, H is strictly dominated by G for B. Thus, if B1-3: B is ”rational” in the sense explained above, H is never chosen by B. Now, if it holds A2: A knows B1-1, B1-2 and B1-3, in addition to A1-1 and A2-2, then A knows ”B knows ”A never chooses X,” and because of this B never chooses H.” Thus A eliminates H from the possibility. The game after the elimination is now B A

Y Z

F 4, 3 2, −2

G 6, 2 5, 0

In the game after the elimination, we see that Z is worse than Y for A, no matter what B chooses (except when B chooses H, which has been eliminated because X had been eliminated). That is, Z is strictly dominated by Y for A. Thus, if A is ”rational” in the sense explained above, Z is never chosen by A. Now, if it holds B2: B knows A2, in addition to B1-1, B1-2, B1-3, then B knows ” A knows ”B knows ”A never chooses X,” and because of this B never chooses H,” and because of this A never chooses Z.” Thus B eliminates Z from the possibility. The game after the elimination is now B F G A Y 4, 3 6, 2 In the game after the elimination, we see that F is optimal for B. As a result, the only possibility is (F, G). This is called iterated eliminated of dominated strategies. Formally, it is defined as follows.

CHAPTER 20. BASIC GAME THEORY I

250

1. For each i, set Si0 = Si . 2. Then, for each i if it holds between si , s′i ∈ Si0 that ui (s′i , s−i ) > ui (si , s−i ) ∏ 0 for all s−i ∈ j̸=i Sj0 ≡ S−i , say that si is strictly dominated by s′i . 1 Thus, let Si denote the set of i’s strategies in Si0 which are not strictly dominated by anything else in Si0 . That is, Si1

0 = {si ∈ Si0 : ∄s′i ∈ Si0 , ∀s−i ∈ S−i , ui (s′i , s−i ) > ui (si , s−i )} 0 ′ 0 0 = {si ∈ Si : ∀si ∈ Si , ∃s−i ∈ S−i , ui (si , s−i ) ≧ ui (s′i , s−i )}

3. Inductively, given (S1k , · · · , Snk ) for general k, for each i let Sik+1

k = {si ∈ Si0 : ∄s′i ∈ Sik , ∀s−i ∈ S−i , ui (s′i , s−i ) > ui (si , s−i )} k = {si ∈ Sik : ∀s′i ∈ Sik , ∃s−i ∈ S−i , ui (si , s−i ) ≧ ui (s′i , s−i )}

4. Repeat this If this process leads to a unique strategy profile we say that the game is dominance-solvable. However, it is not in general the case that the iterated elimination leads to a unique strategy profile. Here is an example. B

A

X Y Z W

F 3, 0 6, 1 2, 3 0, 2

G 1, 1 −1, 2 0, 1 2, 4

H 4, 2 2, 0 1, −2 3, 1

I 2, 1 0, 5 −1, 4 1, 0

Apply the iterated elimination of dominated strategies, then it follows: 1. Z is strictly dominated by X for A, hence eliminated; 2. F is strictly dominated by G for B, hence eliminated; 3. Y is strictly dominated by W for A, hence eliminated; 4. I is strictly dominated by H for B, hence eliminated. but we cannot eliminate further. Thus {X, W } is the set A’s strategies and {G, H} is the set of B’s strategies which survive after the iterated elimination.

CHAPTER 20. BASIC GAME THEORY I

251

Elimination by weak dominance? Consider the following game.

A

X Y

F 1, 1 0, 0

B G 2, 0 2, 2

From A’s viewpoint here, X is strictly better than Y if B chooses F, and A and Y are equally preferable if B chooses G. Because of the tie in the latter case there is not strict dominance. However, A does not lose anything by choosing X over Y, and X is even a dominant strategy for him in the case. Here we say that Y is weakly dominated by X. We cannot immediately say that a weakly dominated strategy should be eliminated, for it could be an optimal choice ”with ties” depending on opponent’s strategy. In the above example, if B chooses G then X and Y are equally desirable for A. Taking a weakly dominated strategy is ”unlikely,” however, and under additional requirements we can eliminate it. I will come to this in the section on equilibrium refinement.

20.4

Rationalizable strategies

Consider the following example.

A

X Y Z

F 3, 2 0, 2 2, 4

B G 0, 0 5, 3 2, 0

H 3, 5 0, 0 2, 0

Here there is no strict dominance relation between any two strategies from neither player’s viewpoint. However, strategy Z cannot be an optimal choice for A no matter what B chooses. Hence we may eliminate it for the same reason as before, since it is never used. Then the game reduces to

A

X Y

F 3, 2 0, 2

B G 0, 0 5, 3

H 3, 5 0, 0

Here there is no strict dominance relation between any remaining strategies from neither player’s viewpoint. However, strategy F cannot be an optimal choice for B no matter what A chooses. Hence we may eliminate it for the same reason as before, since it is never used.

CHAPTER 20. BASIC GAME THEORY I

252

Then the game reduces to B X Y

A

G 0, 0 5, 3

H 3, 5 0, 0

When a strategy survives after such iterated elimination it is called rationalizable. Here is the formal definition of the elimination process. 1. For each i, set Sei0 = Si . 2. The, for each i, let Sei1 the set of strategies which is optimal for i for some ∏ 0 . If a strategy is not in this subset opponents’ strategies in j̸=i Sej0 ≡ Se−i it cannot be optimal against any opponents’ strategies. That is, Sei1

0 = {si ∈ Sei0 : ∃s−i ∈ Se−i , ∀s′i ∈ Sei0 , ui (si , s−i ) ≧ ui (s′i , s−i )}

3. Inductively, given (Se1k , · · · , Senk ) for general k, let Seik+1

=

k {si ∈ Seik : ∃s−i ∈ Se−i , ∀s′i ∈ Seik , ui (si , s−i ) ≧ ui (s′i , s−i )}

for each i. 4. Repeat this. A strategy is said to be rationalizable if it survives the above iterated eliminations. Here, ”rationalizable” just means that there is a reason to use it, and has nothing to do with other meanings. Let me go over one more example. B

A

X Y Z W

P 4, 4 2, 0 3, 1 3, 2

Q 4, 3 5, 4 4, 2 2, 3

R 1, 3 2, 5 3, 2 1, 3

S 1, 0 2, 0 2, 3 3, 4

Apply the rationalizability argument, then it follows: 1. Q is never a best response for B, hence eliminated; 2. Y is never a best response for A, hence eliminated; 3. R is never a best response for B, hence eliminated; 4. Z is never a best response for A, hence eliminated. but we cannot eliminate further. Thus {X, W } is the set A’s rationalizable strategies and {P, S} is the set of B’s rationalizable strategies.

CHAPTER 20. BASIC GAME THEORY I

20.5

253

Nash equilibrium

Given that dominant strategy does not always exist and iterated elimination of dominated strategies does not lead to a unique strategy profile and neither rationalizability does, what strategies should we think are chosen? In game theory, a solution concept called Nash equilibrium is taken to be the most standard one. In two-player games, Nash equilibrium refers to a strategy profile such that A’s strategy is optimal for him given B’s strategy, and B’s strategy is optimal for him given A’s strategy. In other words, it is a situation such that ”I do this because you do that, and you do that because I do this.” Compared this to the definition of dominant strategy equilibrium A’s strategy is optimal for him no matter what B does, and B’s strategy is optimal for him no matter what A does. Then you might notice a kind of ”jump” or ”circularity” in the definition of Nash equilibrium. Let me first finish the formal definition of Nash equilibrium. Definition 20.2 A strategy profile s∗ = (s∗1 , · · · , s∗n ) is said to be a Nash equilibrium if it holds vi (s∗i , s∗−i ) ≥ vi (si , s∗−i ) for all i and si ∈ Si . One can restate the definition of Nash equilibrium in the following way. For each player i, given a profile of the other players’ strategies s−i , let BRi (s−i ) denote the set of strategies which are optimal for i against s−i , which is called best response. We take a set in general rather than a point because there may be multiple optima. Then a strategy profile s∗ = (s∗1 , · · · , s∗n ) is said to be a Nash equilibrium if it holds s∗i ∈ BRi (s∗−i ) for all i. Now let us find Nash equilibria in specific examples. Example 20.4 Market entry 2: Consider the version of market entry game in which both lose when both enter, B A

Entry Non-entry

Entry −2, −2 0, 10

Non-entry 10, 0 0, 0

CHAPTER 20. BASIC GAME THEORY I

254

First let me first look at A’s best response. Suppose B enters then the best response for A is not to enter, hence BRA (E) = {N }. Suppose B does not enter then the best response for A is to enter, hence BRA (N ) = {E}. Do the same exercise for B, then we obtain BRB (E) = {N },BRB (N ) = {E} である. Since it is not immediate to see which strategy profile satisfies the condition s∗A ∈ BRA (s∗B ), s∗A ∈ BRA (s∗B ), let us do the following exercise. For each player and each possible opponent strategy, draw lines under the payoffs given by the best responses. For example, since A’s best response when B enters is not to enter, draw a line under A’s payoff 0 in the lower-left cell (0, 10) which corresponds to the strategy profile (N, E). — Be careful not to do it in the reverse way. Then we obtain B A

Entry Non-entry

Entry −2, −2 0, 10

Non-entry 10, 0 0, 0

Likewise, since A’s best response when B does not enter is to enter, draw a line under A’s payoff 10 in the upper-right cell (10, 0) which corresponds to the strategy profile (E, N ). Then we obtain

A

Entry Non-entry

Entry -2, −2 0, 10

B Non-entry 10, 0 0, 0

Do the same exercise for B for all possible strategies by A, then we obtain

A

Entry Non-entry

Entry -2, -2 0, 10

B Non-entry 10, 0 0, 0

In Nash equilibrium each player’s strategy must be a best response against each other’s strategy, hence it corresponds to a cell in which both players’ payoffs are underlined. Thus in the current example there are two Nash equilibria, (E, N ) and (N, E). In the first equilibrium, ”because A enters B does not enter, and because B does not enter A enters.” In the second equilibrium, ”because B enters A does not enter, and because A does not enter B enters.” This example also shows that there may be multiple Nash equilibria. Here it says only that ”either player concedes,” and does not tells us anything about which player concedes. I will come to the problem of multiple equilibria later. Let us go over three more examples for practice.

CHAPTER 20. BASIC GAME THEORY I

255

Example 20.5 Consider the game below. B

A

X Y Z

F 2, 4 3, 1 1, 7

G 3, 5 2, 0 4, 8

H 8, 3 5, −1 7, 9

Go over the underlying exercise, then we obtain B

A

X Y Z

F 2, 4 3, 1 1, 7

G 3, 5 2, 0 4, 8

H 8, 3 5, −1 7, 9

Thus Nash equilibrium is (Y, F ). Example 20.6 There are two firms A and B, which produce an identical product at constant and identical marginal cost. Denote the identical and constant marginal cost by c. They simultaneously decide their selling prices respectively, which is called Bertrand competition explained more in details in the chapter on oligopoly. Each firm can set one of five levels of price including the marginal cost itself, c < p1 < p2 < p3 < p4 . Then if one’s price is higher than the opponent one it will get nothing, if lower it receives all the demands, and if equal it receives half of the demands. Payoffs are let’s say given as follows. B

A

c p1 p2 p3 p4

c 0, 0 0, 0 0, 0 0, 0 0, 0

p1 0, 0 4, 4 0, 8 0, 8 0, 8

p2 0, 0 8, 0 6, 6 0, 12 0, 12

p3 0, 0 8, 0 12, 0 10, 10 0, 20

p4 0, 0 8, 0 12, 0 20, 0 15,15

Go over the underlying exercise, then we obtain B

A

c p1 p2 p3 p4

c 0, 0 0, 0 0, 0 0, 0 0, 0

p1 0, 0 4, 4 0, 8 0, 8 0, 8

p2 0, 0 8, 0 6, 6 0, 12 0, 12

p3 0, 0 8, 0 12, 0 10, 10 0, 20

p4 0, 0 8, 0 12, 0 20, 0 15,15

Thus there are two Nash equilibria, (c, c) and (p1 , p1 ). It looks better for both that they cooperate and play (p4 , p4 ), but they don’t do so in equilibrium. For,

CHAPTER 20. BASIC GAME THEORY I

256

if your opponent is setting high price then it is better for you to slight undercut the price and get all the demands than to set the same price. Your opponent will do the same thing as well. Thus they have to undercut prices down to either p1 or c. Example 20.7 Battle of sexes: Boy A and Girl B are a couple, and their problem where to go for a date. There are two places, one is boxing and the other is opera. Since the main objective is dating, if they go to different places they get nothing. If they go to the same place, A prefers boxing to opera and B prefers opera to boxing. Such situation can be described by a payoff matrix like below. B Boxing Opera Boxing 2, 1 0, 0 A 0, 0 1, 2 Opera Go over the underlying exercise, then we obtain B A

Boxing 2, 1 0, 0

Boxing Opera

Opera 0, 0 1, 2

Thus there are two Nash equilibria, (Boxing, Boxing) and (Opera, Opera). Again we have the multiple equilibria problem, since Nash equilibrium tells us only that they go to the same place and nothing about where they go.

20.5.1

Nash equilibrium and dominant strategy equilibrium

The following claim will be immediate. Proposition 20.1 If a strategy profile is a dominant strategy equilibrium then it is a Nash equilibrium. Indeed, when we go over the underling exercise for the prisoners’ dilemma we obtain B A

20.5.2

Confess Not to Confess

Confess −5, −5 -10, −1

Not to Confess −1, -10 −2, −2

Nash equilibrium, iterated elimination of dominated strategies and rationalizability

Proposition 20.2 If iterated elimination of dominated strategies leads to a unique strategy profile then it is Nash equilibrium.

CHAPTER 20. BASIC GAME THEORY I

257

Proof. Let s∗ = (s∗1 , · · · , s∗n ) be the unique strategy profile that survives the iterated elimination. Suppose it is not a Nash equilibrium then there exists i and si ∈ Si such that ui (si , s∗−i ) > ui (s∗i , s∗−i ). Since si has been eliminated as a dominated strategy in a previous round let’s say k, there is s′i ∈ Sik such that ui (s′i , s−i ) > ui (si , s−i ) k for all s−i ∈ S−i . ∗ k Since s−i has survived the elimination it must be that s∗−i ∈ S−i . Hence we obtain ui (s′i , s∗−i ) > ui (si , s∗−i )

If s′i = s∗i here, then it is a contradiction to the first inequality. Thus it must be that s′i ̸= s∗i . However, since s′i must have been eliminated in the round after ′ k, say k ′ , by the similar argument as above there is s′′i ∈ Sik such that ui (s′′i , s∗−i ) > ui (si , s∗−i ) Since Si is finite, as we repeat the above argument we get in the end that ui (s∗i , s∗−i ) > ui (si , s∗−i ), which is a contradiction to the first inequality. Proposition 20.3 Strategies taken in Nash equilibrium survive after iterated elimination of dominated strategies. Proof. Proof is by induction. Let sast = (s∗1 , · · · , s∗n ) be any Nash equilibrium strategy profile. 1. Suppose s∗i ∈ / Si1 for some i, it means there is si ∈ Si0 = Si such that ui (si , s−i ) > ui (s∗i , s−i ) 0 = S−i . From the definition of Nash equilibrium, however, for all s−i ∈ S−i ∗ si is an optimal choice for i given s∗−i ∈ S−i , which is a contradiction. Hence s∗i ∈ Si1 for all i.

2. Assume s∗i ∈ Sik is true for all i until k, and suppose s∗i ∈ / Sik+1 for some k i. This means there is si ∈ Si such that ui (si , s−i ) > ui (s∗i , s−i ) k for all s−i ∈ S−i . Combine this with the induction assumption that s∗j ∈ k Sj for all j, we obtain

ui (si , s∗−i ) > ui (s∗i , s∗−i ), which contradicts to the assumption that s∗ is a Nash equilibrium strategy profile. Hence we have s∗i ∈ Sik+1 for all i.

CHAPTER 20. BASIC GAME THEORY I

258

Go back to a previous example. B

A

X Y Z W

F 3, 0 6, 1 2, 3 0, 2

G 1, 1 −1, 2 0, 1 2, 4

H 4, 2 2, 0 1, −2 3, 1

I 2, 1 0, 5 −1, 4 1, 0

As we saw there, {X, W } is the set of A’s strategies and {G, H} is the set of B’s strategies which survive the iterated elimination respectively. Now there are two Nash equilibria, (X, H) and (W, G). B

A

X Y Z W

F 3, 0 6, 1 2, 3 0, 2

G 1, 1 −1, 2 0, 1 2, 4

H 4, 2 2, 0 1, −2 3, 1

I 2, 1 0, 5 −1, 4 1, 0

Similar results hold between Nash equilibrium strategies and rationalizable strategies. Proposition 20.4 If there is only one strategy profile which is rationalizable, then it is Nash equilibrium. Proof. Let s∗ = (s∗1 , · · · , s∗n ) be the unique strategy profile that is rationalizable. Suppose it is not a Nash equilibrium then there exists i and si ∈ Si such that ui (si , s∗−i ) > ui (s∗i , s∗−i ). Since si has been eliminated as an un-rationalizable in a previous round let’s k say k, for all s−i ∈ Se−i there is s′i ∈ Seik such that ui (s′i , s−i ) > ui (si , s−i ) k Since s∗−i is rationalizable it must be that s∗−i ∈ Se−i . Hence we obtain

ui (s′i , s∗−i ) > ui (si , s∗−i ) If s′i = s∗i here, then it is a contradiction to the first inequality. Thus it must be that s′i ̸= s∗i . However, since s′i must have been eliminated in the round after ′ k, say k ′ , by the similar argument as above there is s′′i ∈ Sik such that ui (s′′i , s∗−i ) > ui (si , s∗−i ) Since Si is finite, as we repeat the above argument we get in the end that ui (s∗i , s∗−i ) > ui (si , s∗−i ), which is a contradiction to the first inequality.

CHAPTER 20. BASIC GAME THEORY I

259

Proposition 20.5 Strategies taken in Nash equilibrium are rationalizable. Proof. Proof is by induction. Let sast = (s∗1 , · · · , s∗n ) be any Nash equilibrium strategy profile. 0 1. Suppose s∗i ∈ / Sei1 for some i, then it means that for all s−i ∈ Se−i = S−i there exists si ∈ Sei0 = Si such that

ui (si , s−i ) > ui (s∗i , s−i ) From the definition of Nash equilibrium, however, s∗i is an optimal choice for i given s∗−i ∈ S−i , which is a contradiction. Hence s∗i ∈ Si1 for all i. 2. Assume that s∗i ∈ Seik is true for all i until k, and suppose s∗i ∈ / Seik+1 for some i k This means that for all s−i ∈ Se−i there is si ∈ Seik such that ui (si , s−i ) > ui (s∗i , s−i ). Combine this with the induction assumption that s∗j ∈ Sejk for all j, we obtain ui (si , s∗−i ) > ui (s∗i , s∗−i ), which contradicts to the assumption that s∗ is a Nash equilibrium strategy profile. Hence we have s∗i ∈ Seik+1 for all i.

20.5.3

Why is Nash equilibrium played?

Let us go back to the definition of Nash equilibrium illustrated for the two-layer case, A’s strategy is optimal for him given B’s strategy, and B’s strategy is optimal for him given A’s strategy. In other words, it is a situation such that ”A does this because B does that, and B does that because A does this.” However, in order to say ”because B does that,” it must be that A is correctly predicting ”B does that,” and also in order to say ”because A does his,” it must be that B is correctly predicting ”B does this.” This ”correct prediction” cannot be reached just by the iterated elimination of dominated strategies or by the rationalizability argument. It is known that when they lead to a unique strategy profile it is Nash equilibrium, but in general they narrow down to a unique strategy profile. Thus there is a ”leap” from there to Nash equilibrium which must based on mutual ”correct prediction” about each others’ choices.

CHAPTER 20. BASIC GAME THEORY I

260

Since even the iterated elimination of dominated strategies or the rationalizability argument seems to require pretty high ability of logical reasonings, such further ”leap” toward Nash equilibrium may seem to require that players are ”super-rational.” One might argue in contrary, ”No, Nash equilibrium looks like requiring super-rationality because you are looking at the situation in a too static manner. If you look at the situation from a dynamic viewpoint, you will see that players learn to play Nash equilibrium through learning and imitations without having such super-rationality.” From this viewpoint players may not have correct predictions about each other’s actions initially, but as the game is played repeatedly they learn about each other and gradually form correct predictions about each other’s actions, which converges to Nash equilibrium in a long-run. See for example Kalai and Lehler [14] as one such theoretical result.

20.6

Mixed strategies

Does Nash equilibrium always exist, by the way? It is easy to find an example in which it does not exist. Rock-scissors-paper is a representative example. Here it is impossible that both players are taking actions each which is optimal against each other. Let us consider an even simpler example. Example 20.8 Marching pennies: A and B simultaneously show their coins. A wins if their faces match and B if they don’t. For simplicity, let me assume that if you win you get 1 and if you lose you lose 1. Then the payoff matrix is B A

Head Tail

Head 1, −1 −1, 1

Tail −1, 1 1, −1

It is easy to see that there is no Nash equilibrium. You can see this by doing the underlining exercise, which results in B A

Head Tail

Head 1, −1 −1, 1

Tail −1, 1 1, −1

However, Nash equilibrium is guaranteed to exist in an extended set of strategies. By ”extended” I mean that players are allowed to be randomize strategies. That is, we consider a strategy like ”showing Head with probability 0.3 and showing Tail with probability 0.7.” Such randomized strategy is called a mixed strategy. On the other hand, a strategy like before such as ”showing

CHAPTER 20. BASIC GAME THEORY I

261

Head” is called a pure strategy. Note, however, that pure strategy is a special case of mixed strategy, because ”showing Head” is nothing but ”showing Head with probability 1 and showing Tail with probability How should we interpret mixed strategies? We can think of two interpretations. One is literal, which says we literally randomize actions. For example, if you want to implement a mixed strategy ”showing Head with probability 0.3 and showing Tail with probability 0.7,” bring cards numbered from 1 to 10, and show Head if you draw one of numbers 1 to 3 and show Tail if you draw one of numbers from 4 to 10. For the other interpretation, imagine that players are randomly drawn and matched from a large population. Consider for example which side of the road you walk. Then a mixed strategy like ”walking Left with probability 0.3 and walking Right with probability 0.7” is interpreted that 30% of people you encounter walk Left and 70% of people you encounter walk Right. The second interpretation takes mixed strategies as such collective behaviors. The following result is known. If you are curious about its proof see an advanced textbook such as Mas-Colell, Whinston and Green [21]. Theorem 20.1 When the set of strategies are finite, Nash equilibrium always exists in mixed strategies. Now how do we find mixed-strategy Nash equilibria? You can do it by extending the best response argument to mixed strategies. Let me illustrate it using the example of matching pennies. Let pA denote the probability that A shows Head and let pB denote the probability that B shows Head. Then, A’s expected utility given a combination of mixed strategies (pA , pB ) is uA (pA , pB ) =

pA pB − pA (1 − pB ) − (1 − pA )pB + (1 − pA )(1 − pB )

= (4pB − 2)pA + 1 − 2pB Since the above function is linear in pA , its graph is a straight line either upwardsloping or downward-sloping or flat when pA is taken on the horizontal axis, where only the sign of 4pB − 2 matters. Thus, A’s best response to pB is  when pB < 0.5  {0}, [0, 1] , when pB = 0.5 BRA (pB ) =  {1}, when pB > 0.5. When pB < 0.5 the coefficient on A’s own probability of Head pA given by 2(2pB − 1) is negative. Hence A’s expected utility is linearly decreasing in pA . Since pA moves between 0 and 1, the maximal expected utility is obtained at the left end-point, which is pA = 0. When pB > 0.5 the coefficient on A’s own probability of Head pA given by 2(2pB − 1) is positive. Hence A’s

CHAPTER 20. BASIC GAME THEORY I

262

expected utility is linearly increasing in pA . Since pA moves between 0 and 1, the maximal expected utility is obtained at the right end-point, which is pA = 1. When pB = 0.5 the coefficient on A’s own probability of Head pA given by 2(2pB − 1) is zero. Hence A’s expected utility is constant in pA , and any point in the interval [0, 1] is optimal. Thus, entire interval [0, 1] is the best response. Note that best response is not a function from a point to a point but a ”correspondence” which maps each point to a set in general. Likewise, B’s expected utility given a combination of mixed strategies (pA , pB ) is EUB (pA , pB )

= −pA pB + pA (1 − pB ) + (1 − pA )pB − (1 − pA )(1 − pB ) = (2 − 4pA )pB − 1 + 2pA

By the similar argument as above, B’s best response to pA is  when pA < 0.5  {1}, [0, 1] , when pA = 0.5 BRB (pA ) =  {0}, when pA > 0.5. Now, the best responses of the two players are depicted as in Figure 20.1.2 In a mixed-strategy Nash equilibrium, the combination of each player’s probability to show Head (p∗A , p∗B ) satisfies p∗A



BRA (p∗B )

p∗B



BRB (p∗A )

That is, it is the point at which the two best response graphs coincide. Here the intersection is (p∗A , p∗B ) = (0.5, 0.5). Thus the mixed-strategy Nash equilibrium is ((Head 0.5, Tail 0.5), (Head 0.5, Tail 0.5)). Let us go over two more examples. Consider a version of market entry game we saw before, B A

Entry Non-entry

Entry −2, −2 0, 10

Non-entry 10, 0 0, 0

We already know that there are two pure-strategy Nash equilibria, (Entry,Nonentry) and (Non-entry,Entry). There is one more when we allow mixed strategies, however. A’s expected utility given a combination of mixed strategies (pA , pB ) is uA (pA , pB ) = 2 This

(10 − 12pB )pA

is the left-facing swastika, not the right-facing one.

CHAPTER 20. BASIC GAME THEORY I

263

pB 16

BRB BRA

0.5

0

- pA 1

0.5

Figure 20.1: Best responses in matching pennies

By the similar argument as before, A’s best response to pB is  when pB < 5/6  {1}, [0, 1] , when pB = 5/6 BRA (pB ) =  {0}, when pB > 5/6. Likewise, B’s expected utility given a combination of mixed strategies (pA , pB ) is EUB (pA , pB ) =

(10 − 12pA )pB

By the similar argument as before, B’s best response to pA is  when pA < 5/6  {1}, [0, 1] , when pA = 5/6 BRB (pA ) =  {0}, when pA > 5/6. The two players’ best responses are depicted as in Figure 20.2, and their graphs cross at three points. Thus we there are three Nash equilibria ((Entry 1, Non-entry 0), (Entry 0, Non-entry 1)) ((Entry 0, Non-entry 1), (Entry 1, Non-entry 0)) (( ) ( )) 5 1 5 1 Entry , Non-entry , Entry , Non-entry 6 6 6 6 two of which are pure-strategy Nash equilibria which have been already obtained. The nice thing of this method is that you obtain all equilibria at once. Consider the battle of sexes, B A

Boxing Opera

Boxing 2, 1 0, 0

Opera 0, 0 1, 2

CHAPTER 20. BASIC GAME THEORY I

264

pB 16

BRB

5/6

BRA

0

5/6

- pA 1

Figure 20.2: Best responses in the entry game

We already know that there are two pure-strategy Nash equilibria, (Boxing,Boxing) and (Opera,Opera). There is one more when we allow mixed strategies, however. A’s expected utility given a combination of mixed strategies (pA , pB ) is uA (pA , pB ) =

(3pB − 1)pA − pB + 1

By the similar argument as before, A’s best response to pB is  when pB < 1/3  {0}, [0, 1] , when pB = 1/3 BRA (pB ) =  {1}, when pB > 1/3. Likewise, B’s expected utility given a combination of mixed strategies (pA , pB ) is EUB (pA , pB ) =

(3pA − 2)pB − 2pA + 2

By the similar argument as before, B’s best response to pA is  when pA < 2/3  {0}, [0, 1] , when pA = 2/3 BRB (pA ) =  {1}, when pA > 2/3. である. The two players’ best responses are depicted as in Figure 20.3, and their graphs cross at three points. Thus we there are three Nash equilibria

((Boxing 0, Opera 1), (Boxing 0, Opera 1)) ((Boxing 1, Opera 0), (Boxing 1, Opera 0)) (( ) ( )) 2 1 1 2 Boxing , Opera , Boxing , Opera 3 3 3 3

CHAPTER 20. BASIC GAME THEORY I

265

pB 16

BRB

BRA 1/3

0

- pA 1

2/3

Figure 20.3: Best responses in the battle of sexes

20.7

Refinement of Nash equilibria

Reconsider the example which I raised in order to say that a weakly dominated strategy should not necessarily be eliminated.

A

X Y

F 1, 1 0, 0

B G 2, 0 2, 2

First let us find all Nash equilibria in this game. Let pA denote the probability that A chooses X, and let pB denote the probability that B chooses F. Then, A’s expected utility given (pA , pB ) is uA (pA , pB ) = =

pA pB + 2pA (1 − pB ) + 2(1 − pA )(1 − pB ) pA pB − 2pB + 2

Hence A’s best response to pB is { [0, 1] , when pB = 0 BRA (pB ) = {1}, when pB > 0. On the other hand, B’s expected utility given (pA , pB ) is uB (pA , pB ) = pA pB + 2(1 − pA )(1 − pB ) = (3pA − 2)pB − 2pA + 2 Hence B’s best response to pA is  when 0 ≤ pA < 2/3  {0}, [0, 1] , when pA = 2/3 BRB (pA ) =  {1}, when 2/3 ≤ pA ≤ 1.

CHAPTER 20. BASIC GAME THEORY I

266

pB 16

BRB

BRA

0

2/3

- pA 1

Figure 20.4: Best response including a weakly dominated strategy

When we depict the best responses we obtain Figure 20.4, The set of Nash equilibria consists of points at which the two graphs intersect, which is actually a continuum. It consists of two components, {(pA , pB ) : 0 ≤ pA ≤ 2/3, pB = 0} and (pA , pB ) = (1, 1) Thus, Y is a weakly dominated strategy, but it can be played in Nash equilibria. However, it is optimal for A to choose Y only when B chooses G for sure. It is an unreliable choice, given that there might be an error in the opponent’s choice. This leads us to think of a game in which each of A and B chooses any action at least with probability ε whether he wants it or not. This is called a perturbed game Recall that A’s expected utility is uA (pA , pB ) =

pA pB − 2pB + 2

Since pB is always positive under the perturbation, A makes pA equal to the maximal possible value under the perturbation 1 − ε. Thus, A’s best response under the perturbation is. BRA (pB ) = {1 − ε},

ε ≤ pB ≤ 1 − ε.

Recall that B’s expected utility is uB (pA , pB ) = (3pA − 2)pB − 2pA + 2

CHAPTER 20. BASIC GAME THEORY I

267

pB 6 1−ε

BRB

BRA

ε ε

2/3

- pA 1−ε

Figure 20.5: Best responses in the perturbed game

hence B’s best response under the perturbation is  when ε ≤ pA < 2/3  {ε}, [ε, 1 − ε] , when pA = 2/3 BRB (pA ) =  {1 − ε}, when 2/3 ≤ pA ≤ 1 − ε. When we depict the best responses of the two we obtain Figure 20.5. Let (pεA , pεB ) denote corresponding choice probabilities of X and F respectively in Nash equilibrium in the perturbed game, then it is given by the condition pεA



BRA (pεB )

pεB



BRB (pεA )

Since (pεA , pεB ) = (1 − ε, 1 − ε) is the unique intersection of the two graphs, there is just one Nash equilibrium in the perturbed game ((X; 1−ε, Y ; ε), (F ; 1− ε, G; ε)). Now let us make the perturbation ε tend to zero, then Nash equilibrium in the perturbed game converges to ((X; 1, Y ; 0), (F ; 1, G; 0)) This is called trembling-hand perfect equilibrium A weakly dominated strategy can never be optimal under perturbation, and it is chosen with the lowest possible probability ε. Thus, as ε converges to zero the probability that it is chosen converges to zero. Hence it is never chosen in trembling-hand perfect equilibrium. Summing up, we obtain Proposition 20.6 Weakly dominated strategy is never taken in tremblinghand-perfect equilibrium.

CHAPTER 20. BASIC GAME THEORY I

268

Since it is known that mixed-strategy Nash equilibrium always exists in any given game and since it converges in a mathematically suitable sense as ε converges to zero, trembling-hand perfect equilibrium always exists. Theorem 20.2 When there are finitely many strategies, trembling-hand perfect equilibrium always exists. By the way, in the above example, the trembling-hand perfect equilibrium yields payoff (1, 1), which is unanimously worse than (2, 2) yielded by a Nash equilibrium eliminated because of non-robustness. This suggests that whether an equilibrium is robust to errors and whether its consequence is social desirable are mutually orthogonal questions.

20.8

How should we think of multiple equilibria?

Which one is played when there are multiple Nash equilibria? The argument of equilibrium refinement excludes ”unlikely” equilibria by eliminating non-robust choices such as weakly dominate strategies. It leaves many of multiple equilibria problems unsolved, however. Consider for example a version of market entry game B Entry Non-entry

A

Entry −2, −2 0, 10

Non-entry 10, 0 0, 0

Here are three Nash equilibria, (E, N ), (N, E) and ((E and all of them are trembling-hand perfect. Also, in the Battle of Sexes game

5 1 5 1 6 , N 6 ), (E 6 , N 6 )),

B A

Boxing Opera

Boxing 2, 1 0, 0

Opera 0, 0 1, 2

there are thee Nash equilibria, (Box, Box), (Ope, Ope) and ((Box 32 , Ope 13 ), (Box 13 , Ope 32 )), and all of them are trembling-hand perfect. There is a literature called equilibrium selection, which proposes a criterion for selecting equilibrium positively rather than tries to eliminate ”unlikely” equilibrium as in equilibrium refinement. While the equilibrium refinement literature resorts to each player’s individual and intellectual rationality wanting strategic to choice be robust to certain kinds of errors, the equilibrium selection literature has a flavor of bringing in factors other than individual and intellectual rationality from outside. Consider the following example.

CHAPTER 20. BASIC GAME THEORY I

269

Example 20.9 Two players A and B simultaneously announce integers from 1 to 100, respectively. If they match each player receives that number times 100 dollars. Otherwise they get nothing. There are 100 pure-strategy Nash equilibria, (1, 1), (2, 2), · · · , (99, 99), (100, 100), but it is unambiguous to say that (100, 100) is the most desirable one for both unanimously and most ”natural” one to play. When we can rank between equilibria in the unanimous way, the unanimously best one will be played. Such equilibrium selection criterion is called payoff dominance. It is rather hard to observed games which possess this type of ”naturality,” however, and it is not a generally applicable criterion. For example, we don’t see unanimous ranking between equilibria in the market entry game and the battle of sexes game in the above. Also, even when the payoff dominance condition applies it may not be always appealing. Here is an example. Example 20.10 There are two hunters, A and B. Each of them can hunt rabbit alone, but they can hunt stag if they cooperate. Each of them cannot hunt stag alone, however. Each of them chooses between going to hunt stag and going to hunt rabbit. If both of them go to hunt stag they cooperate to get a stag, and each of them gets payoff 10. If A goes to hunt stag and B goes to hunt rabbit, A gets nothing because he cannot hunt stag alone while B gets payoff 6 since he can hunt rabbit alone. If both go to hunt rabbit each of them gets payoff 6 since each of them can hunt rabbit alone. Let me emphasize that payoffs are already adjusted to their risk attitudes. Thus payoff matrix is

A

Stag Rabbit

Stag 10, 10 6, 0

B Rabbit 0, 6 6, 6

There are three Nash equilibria, (S, S), (R, R) and ((S 53 , R 52 ), (S 35 , R 52 )), and all of them are trembling-hand perfect. According to payoff dominance (S, S) is played, but it is a ”risky choice” to play this equilibrium in that sense that it depends critically on that the opponent surely follows the same equilibrium play. Since it is unclear which pure strategy equilibrium the opponent will follow let us say that he takes each action with even chance. Then if you go to hunt stag your expected utility is 10 × 0.5 = 5. On the other hand, if you go to hunt rabbit you get 6 no matter what the opponent does. Thus we may say that going to hunt rabbit is the safe choice and that (R, R) is the ”safe” equilibrium. This criterion is called risk dominance.

CHAPTER 20. BASIC GAME THEORY I

270

Which one is more appealing, payoff dominance or risk dominance? So far it is known that risk dominance is stronger than payoff dominance (see for example cite a relevant paper or book). However, risk dominance does not apply to the market entry game and the battle of sexes, whereas payoff dominance neither. Consider next the following example Example 20.11 Two players A and B simultaneously announce integers from 1 to 100, respectively. If they match each player receives 100 dollars. Otherwise they get nothing. There are 100 pure-strategy Nash equilibria, (1, 1), (2, 2), · · · , (99, 99), (100, 100), and in contrary to one of the previous examples payoffs are the same in all equilibria. Hence neither payoff dominance or risk dominance applies. Not all equilibria are equally plausible, though. There are certain numbers which immediately come up in our mind, while it is hard to say that such thing is necessarily uniquely determined. For example one can think of (1, 1), (50, 50) and (100, 100), let’s say. Such thing which comes up with many people’s mind and is popular as a device for coordination because of that is called a focal point. Rendezvous is a typical example of such coordination, and people usually meet at a popular spot which is easy to come up with. Of course there does not always exist a focal point, and it depends on specific nature of each game. Thus it is not a criterion which applies to general games. Having said that, it seems that the theories of equilibrium refinement and equilibrium selection are in deadlock, while there may be a breakthrough in future, who knows. If you take game theory to be a ”predictive science” about human behaviors it is a bad news, or at least an obstacle. On the other hand, if you think game theory is a ”language” for ”understanding” social phenomena it is not actually a bad thing. When we observe different social phenomena between different societies and different groups we are tempted to specify a ”fundamental condition” which causes the difference. However, the multiplicity of equilibria tells us that entirely different social phenomena can come from an identical fundamental condition. It’s a ”chicken and egg” argument. Even when the fundamental condition (payoff matrix) is identical, we can have multiple chicken and egg stories, such as ”because A goes to Boxing B goes to Boxing, and because B goes to Boxing A goes to Boxing” and ”because A goes to Opera B goes to Opera, and because B goes to Opera A goes to Opera.” In this sense it is sometimes rather helpful for understanding diversities of our society.

CHAPTER 20. BASIC GAME THEORY I

20.9

271

Exercises

Exercise 23 Consider the game below. B

A

X Y Z V W

F 0, 2 2, 1 1, 3 −2, 1 3, 4

G 3, 1 0, 3 2, 0 4, −1 1, 3

H 0, 2 2, 4 1, 1 −1, 3 3, 0

I 5, 3 1, 1 3, 2 4, 1 2, 2

(1) Find the set of strategies which survive the iterated elimination of dominated strategies. (2) Find the set of rationalizable strategies. (3) Find all pure-strategy Nash equilibria. Exercise 24 Consider a game in which players simultaneously choose integers from 0 to 100 respectively, and one wins 100 dollars if his number is closest to the half of the averages of the chosen numbers. Find the set of rationalizable strategies. Exercise 25 There are three players A,B, and C. A chooses between A and Y, B chooses between F and G, and C chooses between K and L. The payoff matrix when A chooses X is in the left below, and the one when A chooses Y is in the right, where in each cell the the number in the lest if A’s payoff, the one in the middle is B’s payoff and the one in the right is C’s payoff. sA = X B

C F G

K 2, 1, 4 7, 2, 1

L 4, 3, 8 2, 1, 3

sA = Y B

C F G

K 5, 1, 8 4, 9, 3

L 1, 3, 2 3, 4, 5

Find all pure-strategy Nash equilibria. Exercise 26 Find all Nash equilibria in the game below, in pure or mixed strategies. B F G X 2, 2 5, 7 A Y 4, 6 2, 4

Chapter 21

Basic game theory II: extensive-form games 21.1

Description of strategic interdependence: extensive-form games

Extensive-form games deal with sequential decisions with turns, which are described by game trees. It will be better to start with an example. Example 21.1 Market entry: There are two firms, A and B. A is a potential entrant to the market and B is an incumbent monopoly firm. First, A decides whether to enter (E) the market or not (N). Then B decides whether to fight (F) or compromise (C) after seeing A’s action. A

E A

B q

q

B

F

−20

−5

C

10

10

0

20

N

Payoffs are explained as follows. If A does not enter A receives payoff 0 and B receives the monopoly profit 20. If A enters and B fights A loses 20 and B loses 5 as well. If A enters and B compromises each receives profit 10. We say that the above extensive-form game is with perfect information because the second-mover can monitor the first-mover’s action. First let us look into this extensive-form game by representing it by a normalform game. It is called a normal-form expression of extensive-form game. 272

CHAPTER 21. BASIC GAME THEORY II

273

Then the payoff matrix is given by B A

E N

F −20, −5 0, 20

C 10, 10 0, 20

This normal-form game has two (pure-strategy) Nash equilibria. One is (E, C) and the other is (N, F). The latter Nash equilibrium is unrealistic, however.The story behind it is that the entrant refrains from entry because of the incumbent’s threat saying ”if you enter I will fight.” However, once entry is done the decision problem for the incumbent is A B q

B

F

−20

−5

C

10

10

and its normal-form expression is B A

E

F −20, −5

C 10, 10

Such game which follows after preceding actions is called a subgame. In the above subgame after entry the incumbent firm never fights since it is simply a waste of resource. Therefore, the threat saying ”if you enter I will fight” is not credible. Such threat which is never carried out is called an empty threat.

21.2

Subgame-perfect Nash equilibrium

In order to exclude such equilibria based on empty threats, we narrow down Nash equilibria as follows. Definition 21.1 A strategy profile in a game is said to be a subgame-perfect Nash equilibrium if it induces a Nash equilibrium in every its subgame. It is already a Nash equilibrium because the entire game itself is counted as a subgame of it. From its name, subgame-perfect Nash equilibrium is a Nash equilibrium, for the entire game itself is counted as a subgame of it. Converse is not true. For example, since Fight is no longer optimal for B in the game following A’s entry (which is B’s one-person game), Fight is not a Nash equilibrium in the subgame. Hence (N, F) is not subgame-perfect.

CHAPTER 21. BASIC GAME THEORY II

274

In extensive-games with finite rounds with perfect information, subgameperfect Nash equilibrium is found by backward induction. Backward induction tells us here that we start with solving the subgame after A’s entry. Here it is optimal for B to choose Compromise. Next we look at A’s decision, in which A is supposed to foresee what B does in the game after A’s entry. Under the foresight that B chooses Compromise there, A’s payoff is 10 if he chooses Entry and 0 if he chooses Non-entry. Thus it is optimal for A to choose Entry. Thus the backward induction yields (E, C), which is the subgame-perfect Nash equilibrium in this game. Example 21.2 Market entry 2: Here A the entrant chooses whether to enter (E) or not to enter (N). Then B the incumbent chooses whether to take ”aggressive” pricing strategy (A) or ”conservative” pricing strategy (C), after seeing A’s action. A

E A

B q

q N

q B

B

A

−20

−5

C

10

10

A

0

5

C

0

20

Like before, let me start with the normal-form expression. Notice that here the number of B’s strategies is not two, but four. It is not simply ”aggressive” and ”conservative,” but it should be ”aggressive whether there is an entry or not (denoted AA),” ”aggressive if there is entry, conservative if there is no entry (AC),” ”conservative if there is entry, aggressive if there is no entry (CA),” and ”conservative whether there is an entry or not (CC).” In extensive-form games there is a distinction between a strategy and an action. In stead, it is a list of actions conditional on histories, such as ”I will be conservative if there is entry, aggressive if there is no entry.” To understand, imagine for example that this game is played online and the two players submit their programs to the mediator beforehand and the mediator runs the submitted programs. Then, what A needs to submit is simply the object name of an action (E or N), but what B needs to submit is an entire code saying for example ”output C if input is E; output A if input is N.” Now here is the payoff matrix for the normal-form expression B A

E N

AA −20, −5 0, 5

AC −20, −5 0, 20

CA 10, 10 0, 5

CC 10, 10 0, 20

CHAPTER 21. BASIC GAME THEORY II

275

There are three pure-strategy Nash equilibria in the above normal-form game: (E, CA) (E, CC) (N, AC) The third Nash equilibrium is not subgame-perfect, which is the case of empty threat. Also, the first one is not subgame-perfect. Here the incumbent firm takes an aggression action if there is no entry, but this is not an optimal choice ”if there is no entry.” Thus the subgame-perfect Nash equilibrium is the second one, (E, CC) Let us verify this by backward induction. If A enters B chooses Conservative which yields payoff 10 over Aggressive which yields payoff −5. If A does not enter B chooses Conservative which yields payoff 20 over Aggressive which yields payoff 5. Next we look at A’s decision, in which A is supposed to foresee what B does after A’s choice. Under the foresight that B chooses Conservative after A chooses Entry and that B chooses Conservative after A chooses Non-entry, A’s payoff is 10 if he chooses Entry and 20 if he chooses Non-entry. Thus it is optimal for A to choose Entry. Thus the backward induction yields (E, CC), which is the subgame-perfect Nash equilibrium in this game. Note that here ”(Entry, Conservative)” is not a right description of strategies while it is right as a description of observed path of actions. You might wonder, ”why do we care if the incumbent is taking optimal action when there is no entry, despite that the entrant is entering?” It does matter if the second-mover is taking optimal action at any node even if such node is not reached. To illustrate, consider the following extensive-form game.

X A

B q

q Y

q B

A

B

O

8

4

P

3

7

Q

2

6

R

100

5

CHAPTER 21. BASIC GAME THEORY II

276

The subgame-perfect Nash equilibrium strategy profile in the above game is (X, (O if X, Q if Y) Here ”Q if F” is a necessary description, for if B chooses R when A chooses Y for some reason A the first-mover changes his mind and chooses Y. Thus, even if the ”Q if F” is never realized it does affect the first-mover’s decision making. Let us go over another example. Example 21.3 There is 100 dollars on the table, and to be split between A and B. First A proposes how to split the 100 dollars. Here we assume there are three possible proposals, X1=”A receives 99 and B receives 1,” X2=”A receives 50 and B receives 50” and X1=”A receives 1 and B receives 99.” Then B says either Yes (Y) or No (N). If B says Yes A’s proposal goes through and if No all the money disappear. This is represented by an extensive-form in the following way.

q X1 q

X2

B q

Y

A 99

B 1

N

0

0

Y

50

50

N

0

0

Y

1

99

N

0

0

A X3

q B

The normal-form expression of the above game is below, where for example ”N Y Y ” denotes B’s strategy ”No if X1, Yes if X2 and Yes if X3,” and similarly for the other ones. B

A

X1 X2 X3

YYY 99, 1 50, 50 1, 99

YYN 99, 1 50, 50 0, 0

YNY 99, 1 0, 0 1, 99

YNN 99, 1 0, 0 0, 0

NYY 0, 0 50, 50 1, 99

NYN 0, 0 50, 50 0, 0

NNY 0, 0 0, 0 1, 99

NNN 0, 0 0, 0 0, 0

This normal-form expression has seven pure-strategy Nash equilibria, (X1, Y Y Y ), (X1, Y Y N ), (X1, Y N Y ), (X1, Y N N ), (X2, N Y Y ), (X2, N Y N ) and (X3, N N Y ). Only (X1, Y Y Y ) is subgame-perfect here, however. For, it is always better for B to accept the offer rather than rejecting it so that entire entire money disappear, even when A is making a greedy offer such as X1. Since A foresees it, he proposes X1.

CHAPTER 21. BASIC GAME THEORY II

Ar C S A B

Br S

1 1

0 21

C

Ar C S 20 20

Br S 19 40

C

277

Ar S

C

Br

C

A 58 B 58

S

39 39

38 59

Figure 21.1: Centipede game

In experiments it has been observed that when the first mover makes a greedy proposal the second mover rejects it despite that it is profitable to accept it. How should we interpret this? One explanation is that the above numbers are no more than material payoffs given by the experimenter, and they do not have to be the payoffs perceived by the subjects. When it is the case, payoffs perceived by the subjects are something which cannot be controlled by the experimenter and hence have to be estimated. Also, such perception severely depends on how experiments are carried out (e.g., whether it is face-to-face or not). Let us go over one more example. Example 21.4 The game gas six turns. First A chooses whether to continue the game (C) or to stop the game (S). If A continues then B chooses whether to continue or to stop the game. If B continues then A chooses whether to continue or to stop the game, and so on. As in Figure 21.1 each of A and B has at most three turns. Payoffs are as listed in the Figure. From the shape of the game tree we call it a centipede game. The normal-form expression of the above is below, where A’s strategy let’s say SCS says ”choose S in the first turn, C in the third turn and S in the fifth turn,” and similarly for the other ones. B

A

CCC CCS CSC CSS SCC SCS SSC SSS

CCC 58, 58 39, 39 20, 20 20, 20 1, 1 1, 1 1, 1 1, 1

CCS 38, 59 39, 39 20, 20 20, 20 1, 1 1, 1 1, 1 1, 1

CSC 19, 40 19, 40 20, 20 20, 20 1, 1 1, 1 1, 1 1, 1

CSS 19, 40 19, 40 20, 20 20, 20 1, 1 1, 1 1, 1 1, 1

SCC 0, 21 0, 21 0, 21 0, 21 1, 1 1, 1 1, 1 1, 1

SCS 0, 21 0, 21 0, 21 0, 21 1, 1 1, 1 1, 1 1, 1

SSC 0, 21 0, 21 0, 21 0, 21 1, 1 1, 1 1, 1 1, 1

SSS 0, 21 0, 21 0, 21 0, 21 1, 1 1, 1 1, 1 1, 1

CHAPTER 21. BASIC GAME THEORY II

278

You might wonder why we need to think of what A would do in the third and fifth turn even if he stops the game in the first turn. We need to think, however, about ”though I stop in the first turn, but what should I do in the third and fifth turn if I mistakenly continues the game for some reason?” Here all the strategy profiles corresponding to the sixteen cells in the lowerright are Nash equilibria. However, only (SSS, SSS) is subgame-perfect. For, backward induction leads to: 1. B stops the game if the sixth turn comes to him; 2. since A foresees it he stops the game if the fifth turn comes to him; 3. since B foresees it he stops the game if the fourth turn comes to him; 4. since A foresees it he stops the game if the third turn comes to him; 5. since B foresees it he stops the game if the second turn comes to him; 6. since A foresees it he stops the game if the first turn.

21.3

Extensive-form games with imperfect information

So far we have assumed that the succeeding players can observe the actions taken by the preceding players. This assumption is called perfect information. On the other hand, the game is called an extensive-form game with imperfect information when the succeeding players cannot necessarily observe the preceding players’ actions. Figure 21.2 is such an example. Here the second mover cannot monitor the action taken by the first mover when he makes an action. Thus, this game does not have a subgame which is strictly smaller than that, or the entire game itself is the only subgame of it. Therefore its subgame-perfect Nash equilibrium and its Nash equilibrium are the same. Thus in its normal-form expression B A

X Y

F 100, 30 80, 10

G 60, 10 180, 5

we find that its Nash equilibrium is (X, F ), which is also subgame-perfect vacuously. Now suppose that the game as in Figure 21.2 has actually a preceding stage in which B chooses whether to play this game or not. Thus let us consider an extensive-form game as in Figure 21.3. Here B’s action E is interpreted as entry and N is interpreted as non-entry. Then let NG for example denote B’s strategy such that he does not enter but

CHAPTER 21. BASIC GAME THEORY II

279

A B 100 30 Bq

F

X

G

Y

F

Aq q B

G

60

10

80

10

180 5

Figure 21.2: Imperfect information game 1

A B 100 30 Bq

F

X

G

Y

F

Aq E B q

q B

G

60

10

80

10

180 5

N 200 40 Figure 21.3: Imperfect information game 2

CHAPTER 21. BASIC GAME THEORY II

280

if he enters he chooses G in the game after the entry. Then the normal-form expression of this extensive-form game is given by

A

X Y

EF 100, 30 80, 10

EG 60, 10 180, 5

B NF 200, 40 200, 40

NG 200, 40 200, 40

Here this normal-form expression has four pure-strategy Nash equilibria, (X, N F ), (X, N G), (Y, N F ) and (Y, N G). We can still apply the backward induction argument in a generalized sense here, however. Consider that B has already chosen action E for any reason and the game after the entry as in Figure 21.2 is already here. Then Nash equilibrium in the game after the entry is (X, F ), which yields payoffs (100, 30). Now consider that B is choosing between E and F. Since he foresees that the consequence of choosing E is (100, 30) and that of choosing N is (200, 40), he chooses N over E. Hence the subgame-perfect Nash equilibrium is (X, N F ). Again let me emphasize that even when B chooses not to enter and the game after entry is not played we need to describe what they will do in there, since B needs to reason about the consequence of choosing entry when he decides whether to enter or not.

21.4

Bargaining game

Let us reconsider the previous example of proposal and acceptance/reject. This time let me assume that the unit to split between A and B is 1, and any proposal with continuous value is allowed.

21.4.1

One-period bargaining

First consider the one-period case like before, where A proposes and B either accepts or rejects. Denote A’s receipt by x, and denote an arbitrary proposal by (x, 1 − x), where x can be any number from 0 to 1. If B accepts then A’s offer goes through, and if he rejects all the money disappear. As we solve this extensive-form game by backward induction, it is optimal for B to accept any proposal 0 ≦ x ≦ 1. — It is strictly better for B to accept the proposal when 0 ≦ x < 1. When x = 1 he is indifferent between accepting and rejecting, so we can take that accepting it optimal with ties. Since A expects then that any proposal is accepted he makes the greediest proposal x = 1.

21.4.2

Two-period bargaining with alternate offers

Next consider that the two players alternately make proposals over two periods. They bargain in the following procedure:

CHAPTER 21. BASIC GAME THEORY II

281

1. At Period 1, A gives a proposal to B. Denote the proposed amount to be given to A by x1 , then the proposed allocation is (x1 , 1 − x1 ). B either accepts or rejects the proposal. If B accepts then A’s proposal goes through, and if he rejects they go to Period 2. 2. At Period 2, B gives a proposal to A. Denote the proposed amount to be given to A by x2 , then the proposed allocation is (x2 , 1 − x2 ). A either accepts or rejects the proposal. If A accepts then B’s proposal goes through, and if he rejects all the money disappear. Here we consider that waiting is not for free. That is, when they do not reach agreement in Period 1 and go to Period 2 then the outcomes to received at Period 2 are discounted from the viewpoint of Period 1 by a common discount factor 0 < β < 1. As we solve this extensive-form game by backward induction, from the previous argument A accepts any proposal 0 ≦ x2 ≦ 1. Since B expects that any proposal is accepted he makes the greediest proposal (0, 1). Given this look into B’s acceptance/rejection decision at Period 1. If B accepts A’s proposal (x1 , 1 − x1 ) he receives 1 − x1 . On the other hand, if B rejects A’s proposal they go to Period 2 in which B proposes (0, 1) and A accepts it. Since B’s payoff 1 to be received at Period 2 is seen to be β from the viewpoint of Period 1, B accepts A’s proposal as far as 1 − x1 ≧ β, that is, x1 ≦ 1−β, and rejects when x1 > 1−β. Note that since B is indifferent between accepting and rejecting, we can take that accepting it optimal with ties. Since A expects then that any x1 ≦ 1 − β and rejects if x1 > 1 − β, he makes the greediest proposal x1 = 1 − β, which yields (x1 , 1 − x1 ) = (1 − β, β).

21.4.3

Multi-period bargaining with alternate offers

Now let us extend the above argument to general even number of periods, 2T . They bargain in the following procedure: 1. At Period 1, A gives a proposal to B. Denote the proposed amount to be given to A by x1 , then the proposed allocation is (x1 , 1 − x1 ). B either accepts or rejects the proposal. If B accepts then A’s proposal goes through, and if he rejects they go to Period 2. 2. At Period 2, B gives a proposal to A. Denote the proposed amount to be given to A by x2 , then the proposed allocation is (x2 , 1 − x2 ). A either accepts or rejects the proposal. If A accepts then B’s proposal goes through, and if he rejects they go to Period 3. 3. After that, if they have not reached agreement by the previous period, A makes a proposal on odd periods and B does on even periods. 4. If they come to Period 2T and they do not reach agreement the money disappear.

CHAPTER 21. BASIC GAME THEORY II

282

As we solve this extensive-form game by backward induction, from the previous argument A accepts any proposal 0 ≦ x2T ≦ 1. Since B expects that any proposal is accepted he makes the greediest proposal (x2T , 1 − x2T ) = (0, 1). Given this look into B’s acceptance/rejection decision at Period 2T − 1. If B accepts A’s proposal (x2T −1 , 1 − x2T −1 ) he receives 1 − x2T −1 . On the other hand, if B rejects A’s proposal they go to Period 2T in which B proposes (0, 1) and A accepts it. Since B’s payoff 1 to be received at Period 2T is seen to be β from the viewpoint of Period 2T − 1, B accepts A’s proposal as far as 1 − x2T −1 ≧ β, that is, x2T −1 ≦ 1 − β, and rejects when x2T −1 > 1 − β. Note that since B is indifferent between accepting and rejecting when x2T −1 = 1 − β, we can take that accepting it is optimal with ties. Since A expects this he makes the greediest proposal x2T −1 = 1 − β, which yields (x2T −1 , 1 − x2T −1 ) = (1 − β, β). Given this look into A’s acceptance/rejection decision at Period 2T − 2. If A accepts B’s proposal (x2T −2 , 1 − x2T −2 ) he receives x2T −2 . On the other hand, if A rejects B’s proposal they go to Period 2T − 1 in which A proposes (1 − β, β) and B accepts it. Since A’s payoff 1 − β to be received at Period 2T − 1 is seen to be β(1 − β) from the viewpoint of Period 2T − 2, A accepts B’s proposal as far as x2T −2 ≧ β − β 2 , and rejects when x2T −2 < β − β 2 . Note that since A is indifferent between accepting and rejecting when x2T −2 = β − β 2 , we can take that accepting it is optimal with ties. Since B expects this he makes the greediest proposal x2T −2 = β − β 2 , which yields (x2T −2 , 1 − x2T −2 ) = (β − β 2 , 1 − β + β 2 ). Given this look into B’s acceptance/rejection decision at Period 2T − 3. If B accepts A’s proposal (x2T −3 , 1 − x2T −3 ) he receives 1 − x2T −3 . On the other hand, if B rejects A’s proposal they go to Period 2T − 2 in which B proposes (β − β 2 , 1 − β + β 2 ) and A accepts it. Since B’s payoff 1 − β + β 2 to be received at Period 2T − 2 is seen to be β(1 − β + β 2 ) from the viewpoint of Period 2T − 3, B accepts A’s proposal as far as 1 − x2T −3 ≧ β − β 2 + β 3 , that is, x2T −3 ≦ 1−β+β 2 −β 3 and rejects when x2T −3 > 1−β+β 2 −β 3 . Note that since B is indifferent between accepting and rejecting when x2T −3 = 1 − β + β 2 − β 3 , we can take that accepting it is optimal with ties. Since A expects this he makes the greediest proposal x2T −3 = 1−β +β 2 −β 3 , which yields (x2T −3 , 1 − x2T −3 ) = (1 − β + β 2 − β 3 , β − β 2 + β 3 ). By repeating this, we obtain that in Period 2 B proposes (x2 , 1 − x2 ) = (β − β 2 + · · · + β 2T −3 − β 2T −2 , 1 − β + β 2 − · · · − β 2T −3 + β 2T −2 )

CHAPTER 21. BASIC GAME THEORY II

283

and A accepts it, and in Period 1 A proposes (x1 , 1 − x1 ) = (1 − β + β 2 − β 3 + · · · + β 2T −2 − β 2T −1 , β − β 2 + β 3 − · · · − β 2T −2 + β 2T −1 ) and B accepts it.

21.5

Repeated games and sustainable cooperation

Let us come back to the prisoner’s dilemma

A

N C

B N v0 , v0 v− , v+

C v+ , v− vc , vc

where N stands for non-cooperation, C stands for cooperation, and v+ > vc > vo > v− . — Beware that I switched the notation between N and C from the original version of the prisoners’ dilemma. Here the unique Nash equilibrium is (N,N), which is even a dominant strategy equilibrium. That is, they try to cheat each other and lead together to an outcome which is worse for both. On the other hand, in our usual life we seem to be sustaining cooperation (more or less) instead of cheating each other. Where does the difference come from? The difference comes from how to take the time horizon. Note that the above game is for just once. Since the game is played just once, even when you cheat your opponent you are not punished unless there is third party who punishes. Consider on the other hand that this game played repeatedly indefinitely many times. Then a player who cheats and destroys the cooperation may be punished privately the opponents in the future by means of non-cooperative actions. Thus each player may choose to sustain cooperation even when there is no third party which enforces it, if such punishment is a credible threat. This is the basic idea of the repeated game theory. Hereafter we consider that a game is played repeatedly infinitely many times. It is hard to take this assumption literally as we die sometime, but since we don’t know when we die and we don’t know when the repetition ends, the assumption of infinite repetition is a reasonable description of such an open ended situation in which there is always the possibility that you are punished privately by your opponent or partner in the future. Here it is important that there is no terminal date, for, if the repetition is just for finite periods we can apply the backward induction, implying that neither

CHAPTER 21. BASIC GAME THEORY II

284

cooperates on the last day, implying that neither cooperates on the second last day, implying that neither cooperates on the third last day, on so on, implying neither cooperates on the first day. In repeated games we need to consider how the players evaluate streams of payoffs rather than payoffs on a given day. Here we assume that the players care about the discounted present values of payoff streams. Suppose player A receives vA1 in Period 1, vA2 in Period 2, and so on, and vAt at Period t in general, his payoff streams is denoted by (vA1 , vA2 , vA3 , · · · ). Let βA denote A’s discount factor which measures how A is patient, then the discounted present value of (vA1 , vA2 , vA3 , · · · ) is ∞ ∑

t−1 vAt βA

t=1

Similarly for B. Given B’s discount factor βB , the discounted present value of payoff stream (vB1 , vB2 , vB3 , · · · ) is ∞ ∑

t−1 vBt βB

t=1

Here we proceed by assuming that βA = βB = β, though. Now let us see how cooperation can be sustained in repeated games. The following strategy is a prominent one, called trigger strategy: Play C in Period 1. After this: (1) if both have played C in all the periods before, play C; (2) Otherwise, that is, if any of them chose N even once before, play N. Here I explain trigger strategy specifically because it is the simplest and intuitive one, and not because it is the ”optimal” way of punishing. We already know that in situations of strategic independence there is no strategy in general which is optimal regardless of opponent’s strategy. It is true here as well. In general, how you should punish your opponent depends on how your opponent punishes you. Indeed, it is known that there are infinitely many ways to sustain cooperation. A bit more precisely, for any pair of payoffs which is better than the noncooperative outcome for both there is a corresponding subgame-perfect Nash equilibrium which realizes it. This type of assertion is called folk theorem. The word ”folk” originally meant that it had been known informally for a long time, but nowadays a ”folk theorem” in this literature stands for a characterization of the range of payoffs attainable by sustainable cooperations in a given environment under study. Having said that, here I show Proposition 21.1 The pair of trigger strategies is a subgame-perfect Nash equilibrium,

CHAPTER 21. BASIC GAME THEORY II

285

That is, given that B is taking the trigger strategy it is optimal for A to follow the trigger strategy as well after any history, and vice versa. In the following argument I assume so-called one-shot deviation principle, though, saying that in order to check subgame-perfection of a given strategy profile it is enough to check if the prescribed action is optimal for each player at each time supposing that all the player (including the specified player himself) follow the given strategy profile in the future. In the current case the principle says it is enough to see check if playing C is optimal for let’s say A when cooperation has been sustained and if playing N is optimal for him when cooperation has been broken, supposing that both A and B follow the trigger strategy in the future. More specifically, it means we don’t have to consider for example A’s deviation from the trigger strategy such that he plays N when cooperation has been sustained and plays C after cooperation has already been broken, which form a ”double deviation” of the trigger strategy. For the proof of the one-shot deviation principle see for example Osborne and Rubinstein [24]. Here we look at A without loss of generality. 1. When cooperation has been broken before: Given that B is taking the trigger strategy, he chooses N forever including the current period. Also, given that A is following the trigger strategy from the next period he chooses N forever from the next period. Then, A gets only v− by choosing C and he gets vo at best by choosing N in this period, where this choice does not change that he gets vo forever from the next period. Hence it is optimal for him to choose N. 2. When cooperation has been sustained so far (including the first period): If A chooses N he gets v+ since B is choosing C in the current period. From Case 1, both play C forever from the next period and A gets vo at every period in the future. Hence the payoff stream he obtains is (v+ , vo , vo , · · · ), βvo and its discounted present value is v+ + 1−β . On the other hand, if A plays C he gets vc since B is choosing C in the current period. Given that B is taking the trigger strategy and that A is following the trigger strategy from the next period, A gets vc at every period in the future. Hence the payoff stream he obtains is (vc , vc , · · · ), vc and its discounted present value is 1−β . −vc βvo vc Therefore, when 1−β ≥ v+ + 1−β , that is, when β ≥ vv++ −v , it is optimal for o each player to follow the trigger strategy given that the opponent is taking the trigger strategy. + −vc The condition β ≥ vv+ −vo says that both players are sufficiently patient. 1 like firms, in which r When the players’ discount factor is given by β = 1+r denotes pure interest rate, the condition says that the interest rate is sufficiently low. In any case, when the players are not myopic but values long-terms gains

CHAPTER 21. BASIC GAME THEORY II

286

cooperation can be sustained by individuals’ selfish choices without bringing in any outside authority.

21.6

Exercise

Exercise 27 Provide the normal-form expression to the extensive-form game below, and find all pure-strategy Nash equilibria there. Find the subgameperfect Nash equilibrium then.

B q

X A

q Y

q B

A

B

F

0

3

G

60

5

F

9

12

G

5

10

Exercise 28 A and B split a land in the following procedure. First A cuts the land into two pieces. Then B takes one of the two, and A receives the remaining one. Suppose the area of the land is simply 1 and both A and B cares only about the area they receive respectively. Then what is the subgame-perfect equilibrium here? Exercise 29 Find the subgame perfect Nash equilibria (may or may not be unique) in the game below. A B 100 30 Bq

F

X

G

Y

F

Aq E B q

q B

G

60

10

80

10

180 60

N 200 40

Chapter 22

Oligopoly In monopoly there is only one firm which has market power. Now we consider oligopoly in which there are several firms that have market power. I start with duopoly, the case of two firms, and later extend the argument to the case of more firms. When there are several firms the consequence of one’s action depends not only on it but also on others’ actions. For example, Firm A’s profit depends not only on its action but also on Firm B’s action, and vice versa. Hence A needs to read what B will choose and B needs to read what A will choose. Such situation is called strategic interdependence. Discussion on strategic interaction in general is relegated to the chapters on game theory, and we focus on its implication in oligopoly markets. Models of oligopoly are classified by • what firms compete by • timing of their decision According to the first category we think of two kinds in this chapter, one is quantity and the other is price. According to the second category, we think of two cases, one in which the firms simultaneously choose their quantities or prices, the other in which there is first mover and second mover. Thus we will cover four kinds of competitions 1. simultaneous quantity setting 2. sequential quantity setting 3. simultaneous price setting 4. sequential price setting Again, we adopt the partial equilibrium framework because of the reason I discussed in the chapter on monopoly.

287

CHAPTER 22. OLIGOPOLY

22.1

288

Simultaneous quantity setting (Cournot competition)

First, let us consider that the firm simultaneously choose how much to provide to the market. This is called Cournot competition after the name of economists who first analyzed this. Consider a market in which there is certain market maker or auctioneer who sets the price so that a given provided units are sold. The demand side is summarized in the form of inverse demand function p(y). That is, when y units of output are provided to the market one unit of it is sold for p(y), Denote firm A’s cost function by, CA , that is, when firm A produces yA units it cost is CA (yA ). Likewise, let CB denote firm B’s cost function. Suppose A provides yA units and B provides yB units then the resulting market price is p(yA + yB ), A’s profit is p(yA + yB )yA − CA (yA ), and B’s profit is p(yA + yB )yB − CB (yB ) The two firms simultaneously choose their quantities. Here we assume that they play Nash equilibrium. Here is how to find Nash equilibrium. First, find each player’s best response to any possible combination of the other players’ strategies. In the current setting, given any B’s quantity choice yB , A solves its profit maximization problem max p(yA + yB )yA − CA (yA ) yA

Denote the solution to the above problem by BRA (yB ), which is A’s best response to yB . Likewise, given any A’s quantity choice yA , B solves its profit maximization problem max p(yA + yB )yB − CB (yB ) yB

Denote the solution to the above problem by BRB (yA ), which is B’s best response to yA . ∗ ∗ In Nash equilibrium (yA , yB ), each firm’s choice is a best response to each other’s choice, hence we have ∗ ∗ yA = BRA (yB ) ∗ ∗ yB = BRB (yA )

CHAPTER 22. OLIGOPOLY

289

We have now two equations with two unknowns, which we can solve. Let us do this is a simple example of linear inverse demand and constant marginal cost. The inverse demand is given by p(y) = a − by, A’s cost function is CA (yA ) = cA yA , and B’s cost function is CB (yB ) = cB yB . First let us solve for A’s best response. Given an arbitrary level of B’s quantity yB , solve A’s profit maximization problem max [a − b(yA + yB )] yA − cA yA yA

The first-order condition is a − 2byA − byB − cA = 0 By solving for yA as a function of yB we obtain BRA (yB ) =

a − cA − byB 2b

Notice that as the opponent’s quantity is large the own optimal quantity is smaller, and as the opponent’s quantity is smaller the own optimal quantity is larger. When the opponent is aggressive and provides more the market price is lower, the firm should be more conservative. When the opponent is conservative and provides more the market price is maintained higher, the firm should be more aggressive and take advantage of it. This is called strategic substitutes Now let us solve for B’s best response. Given an arbitrary level of A’s quantity yA , solve B’s profit maximization problem max [a − b(yA + yB )] yB − cB yB yB

Likewise, the first-order condition is a − byA − 2byB − cB = 0 By solving for yB as a function of yA we obtain BRB (yA ) =

a − cB − byA 2b

Plot the two players’ best response functions as in Figure 22.1. Then in ∗ ∗ Nash equilibrium (yA , yB ) we have ∗ yA =

∗ a − cA − byB , 2b

∗ yB =

∗ a − cB − byA 2b

By solving these two inequalities with two unknowns we obtain ∗ yA =

a − 2cA + cB 3b

∗ yB =

a − 2cB + cA 3b

CHAPTER 22. OLIGOPOLY

290

yB a−cA 6 b

BRA a−cB 2b ∗ yB

r BRB

0

∗ a−cA yA 2b

a−cB b

- yA

Figure 22.1: Best responses in Cournot competition

For later use, let me calculate the total quantity provided ∗ ∗ = + yB yA

2a − cA − cB , 3b

the resulting market price ∗ ∗ p(yA + yB )=

a + cA + cB , 3

and the two firms’ profits A’s profit =

(a − 2cA + cB )2 , 9b

B’s profit =

(a − 2cB + cA )2 . 9b

By the way, Nash equilibrium in Cournot competition played by two firms can be reached by the rationalizability criterion alone. See Figure 22.2. A Here the yA -intercept of A’s best response curve BRA is a−c 2b . That is, any quantity larger than this cannot be optimal for A no matter what yB is. Hence A A never chooses quantity larger than a−c 2b . Then, let y B denote the yB -coordinate of the intersection of the vertical ( ) A dotted line coming from a−c 2b , 0 and B’s best response curve BRB . Notice that quantity larger than y B can be optimal only when A chooses quantity larger a−cA A than a−c 2b . Since A never chooses quantity larger than 2b , any quantity larger than y B can never be optimal for B. B Likewise, the yB -intercept of B’s best response curve BRB is a−c 2b . That is, any quantity larger than this cannot be optimal for B no matter what yA is. B Hence B never chooses quantity larger than a−c 2b . Then, let y A denote the yA -coordinate of the intersection of the dotted hor( ) B izontal line coming from 0, a−c and A’s best response curve BRA . Notice 2b

CHAPTER 22. OLIGOPOLY

291

yB a−cA 6 b

BRA a−cB 2b

r

yB 0

BRB yA

a−cA 2b

a−cB b

- yA

Figure 22.2: Rationalizability in Cournot competition

that quantity larger than y A can be optimal only when B chooses quantity larger a−cB B than a−c 2b . Since B never chooses quantity larger than 2b , any quantity larger than y A can never be optimal cor A. [ Summing] up, the set of A’s quantity levels which A has a reason to choose A is y A , a−c , and the set of B’s quantity levels which B has a reason to choose 2b [ ] B is y B , a−c . 2b By repeat this argument, then he set of A’s quantity levels which A has a ∗ , and the set of B’s quantity reason to choose converges to the single point yA ∗ levels which B has a reason to choose converges to the single point yB .

22.2

Sequential quantity setting: Stackelberg competition

Next consider the case that there is first-mover and second-mover. Let A be the first mover and B be the second mover, and consider that B can set its quantity after seeing A’s quantity. This is called Stackelberg competition. Except for timing Stackelberg competition takes the same setting as Cournot competition. Consider a market in which there is certain market maker or auctioneer who sets the price so that a given provided units are sold. The demand side is summarized in the form of inverse demand function p(y). That is, when y units of output are provided to the market one unit of it is sold for p(y), Denote firm A’s cost function by, CA , that is, when firm A produces yA units it cost is CA (yA ). Likewise, let CB denote firm B’s cost function. Suppose A provides yA units and B provides yB units then the resulting market price is p(yA + yB ),

CHAPTER 22. OLIGOPOLY

292

A’s profit is p(yA + yB )yA − CA (yA ), and B’s profit is p(yA + yB )yB − CB (yB ) In the simultaneous-move case A’s strategy was A’s quantity yA and B’s strategy was B’s quantity yB , which means a strategy is simply a quantity, or a strategy is simply an action. However, in sequential setting there is generally a distinction between a strategy and an action. In stead, it is a list of actions conditional histories. For the first mover A it remains the same that his strategy is a quantity itself. For the second mover B, his strategy takes the form ′ If A provides yA units then I will provide yB units. If he provides yA units I ′ ′′ ′′ will provide yB . If he provides yA units I will provide yB units, .....

which is a list of actions contingent on all the possible actions by A. That is, whereas A’s strategy is simply a quantity yA the second mover B’s strategy is a function. Denote such function generically by fB , which is a list saying ”if A provides yA I will provide fB (yA ).” B will choose between such functions. Let FB denote the set of all functions mapping from A’s quantity into B’s quantity, where fB is taken to be its genetic element, then B’s set of strategies is FB . Notice that such function resembles the best response function as used in explaining Nash equilibrium, although how to find an optimal one of such function is the same as how to find a best response function. While best response function in the simultaneous move case is no more a construction in players’ mind, here B’s strategy fB is a choice object by itself. Here we look for a subgame-perfect Nash equilibrium strategy profile, which ∗ is denoted by (yA , fB∗ ). We can find it by backward induction which works as follows. 1. For each possible yA , the second mover B solves max p(yA + yB )yB − CB (yB ) yB

Denote B’s profit maximizing quantity conditional on yA by fB∗ (yA ). Then fB∗ is B’s optimal strategy as a list of actions contingent on A’s quantity. I’m putting ∗ in order to emphasize it is optimal for B. 2. Since the first mover is smart enough to foresee what B will choose conditional on his action, he foresees that if he chooses yA B will choose fB∗ (yA ). Based on such foresight, A solves max p(yA + fB∗ (yA ))yA − CA (yA ) yA

CHAPTER 22. OLIGOPOLY

293

Notice that here B’s quantity yB has been replaced by fB∗ (yA ). This ∗ is a one-variable maximization problem, and A’s optimal strategy yA is obtained by solving this. ∗ Plug A’s strategy as quantity yA into B’s strategy as a function fB∗ , we ∗ ∗ ∗ ∗ obtain B’s quantity choice fB (yA ). Here the sequence of quantities (yA , fB∗ (yA )) is called an equilibrium path. Again, don’t confuse between equilibrium path and equilibrium strategy profile.

Let us find the backward induction solution in the example of linear inverse demand and constant marginal cost. The inverse demand function is given by p(y) = a−by, A’s cost function is given by CA (yA ) = cA yA and B’s cost function is given by CB (yB ) = cB yB . Because it is ”backward” induction, first we solve the second mover B’s problem. For each possible yA , solve B’s profit maximization problem max [a − b(yA + yB )] yB − cB yB yB

Since the first-order condition is a − byA − 2byB − cB = 0 B’s optimal strategy is obtained as a function fB∗ given by fB∗ (yA ) =

a − cB − byA 2b

Since A is smart enough to foresee B’s choice conditional on his action, A knows if he chooses yA B will choose fB∗ (yA ). Hence A’s profit maximization problem after replacing yB by fB∗ (yA ) is

=

max [a − b(yA + fB∗ (yA ))] yA − cA yA yA [ ( )] a − cB − byA max a − b yA + yA − cA yA yA 2b

By taking the first-order condition we obtain ∗ yA =

a − 2cA + cB 2b

Summing up, the subgame-perfect Nash equilibrium strategy profile is ∗ yA

=

fB∗ (yA ) =

a − 2cA + cB 2b a − cB − byA for each yA 2b

As a result, the quantities provided are ∗ yA =

a − 2cA + cB , 2b

∗ fB∗ (yA )=

a − 3cB + 2cA 4b

CHAPTER 22. OLIGOPOLY

294

Again, don’t be confuse between strategy profile and a path of quantities. The total quantity provided as a result is ∗ ∗ yA + fB∗ (yA )=

3a − 2cA − cB 4b

and the market price there is ∗ ∗ + fB∗ (yA )) = p(yA

a + 2cA + cB 4

Hence the profits of A and B there are A’s profit =

(a − 2cA + cB )2 , 8b

B’s profit =

(a − 3cB + 2cA )2 16b

Recall that A’s profit in the simultaneous move case under the same demand2 A +cB ) cost assumption was (a−2c9b , which implies that in sequential quantity setting the first mover has an advantage compared to the simultaneous move quantity setting. This is because the first mover can drive out the second mover by providing its quantity aggressively so that the the second mover has to be cautious in order not to destroy the market price any longer. Of course, this does not mean that in any sequential decisions the first mover has an advantage. Imagine for example the rock-scissors-paper game played sequentially.

22.3

Simultaneous price setting: Bertand competition

Next let us consider the oligopoly situation in which the firms simultaneously set their prices. This is called Bertand competition. We will focus on the case of two firms.

22.3.1

No product differentiation

First let us consider that the two firms sell identical products. The case of identical marginal costs Also assume that the two firms have the same and constant marginal cost c. Denote the market demand function by x(p). Denote A’s price by pA and B’s price by pB , then the consequences are classified as follows: 1. When pA < pB , A takes all the demand x(pA ). 2. When pA > pB , B takes all the demand x(pB ).

CHAPTER 22. OLIGOPOLY

295

3. When pA = pB , they split the demand by half.1 Now consider that the two firm play Nash equilibrium. In this setting Nash equilibrium is unique and the equilibrium strategy profile (p∗A , p∗B ) is p∗A = p∗B = c That is, both firms set the same price equal to the marginal cost and each of them earns zero profit. First, let me show that it is a Nash equilibrium. Given that pB = c, A can catch all the demand when he sets pA < c, but it only generates deficit. Hence he does not have a profitable deviation by cutting the price further. Also, when A sets pA < c he only loses the demand and his profit is again zero, hence he does not have a profitable deviation by raising the price. Similarly for B. Since neither player has a profitable deviation it is a Nash equilibrium. Now let me show that it is the only Nash equilibrium. Suppose let’s say A is setting pA > c. Then, (i) if the current situation is pB ≥ pA then B can set p′B < pB with c < p′B < pA so that he catches all the demand and earns positive profit; (ii) if the current situation is pB < pA then B can set p′B > pB with c < p′B < pA so that he still catches all the demand and earns higher profit. Similarly for B. Hence in Nash equilibrium both firm must set their prices equal to c. The case of different marginal costs The previous argument heavily relies on the assumption that they have the same marginal cost. What if they are different. Denote A’s marginal cost, which is still assumed to be constant, by cA , and B’s marginal cost by cB . Without loss of generality assume that cA < cB . Then the only Nash equilibrium, if exists, is (pA , pB ) = (cB , cB ). That is, both set the prices equal to the higher cost. To see why, consider the following cases: 1. If pA , pB < cA , since the firm with lower price (or both firms if they set the same price) is earning negative profit it can get zero profit by setting higher price. 2. If pA > pB ≥ cB , Firm B can increase its profit by raising its price slightly. 3. If pB > pA ≥ cA , Firm A can increase its profit by raising its price slightly. 4. If pB ≤ pA and pB < cB , since Firm B is earning negative profit it can get zero profit by setting higher price. 5. If pA = pB > cB , one of them which is not receiving all the demands can increase its profit by raising its price slightly. 1 In the current case this does not matter while it does in some cases. In any case, let me fix this throughout the chapter.

CHAPTER 22. OLIGOPOLY

296

It depends on the tie-breaking rule if Nash equilibrium exists here. In the current rule that they split demand equally, (pA , pB ) = (cB , cB ) is not a Nash equilibrium, since A can slightly lower its price and can grab the whole demand, and earns more profit. To resolve the problem, maintain the tie-breaking rule but consider that there is a discrete grid for price setting, let’s say an integer grid, and assume that cA and cB are integer as well, and cB − cA ≥ 2. Then Nash equilibrium exists but uniqueness no longer holds, though, since all (pA , pB ) = (cA , cA +1), (cA +1, cA +2), · · · , (cB , cB −1), (cB −1, cB ), (cB , cB + 1) are Nash, since A is setting the highest price as far as it can grab the whole demand, and B is earning zero profit and can grab the demand only by earning negative or zero profit.

22.3.2

The case with product differentiation

Let us think of the case with product differentiation. Here even if one firm sets higher price than the other does it does not immediately lose the demand. Let pA denote A’s price and pB denote B’s price. The market demand function for A’s product is given by xA (pA , pB ) and that for B’s product is given by xB (pA , pB ). Let CA denote A’s cost function and CB denote B’s cost function. Then A’s profit is given by pA xA (pA , pB ) − CA (xA (pA , pB )), and B’s one is given by pB xB (pA , pB ) − CB (xB (pA , pB )) Again we consider that the two firms play Nash equilibrium in the game of simultaneously setting their prices. Let me explain how to find Nash equilibrium in the present context. Given any possible price pB taken by B, consider A’s profit maximization problem max pA xA (pA , pB ) − CA (xA (pA , pB )) pA

Denote its solution by BRA (pB ), which forms a best response function as pB is variable. Likewise, given any possible price pA taken by A, consider B’s profit maximization problem max pB xB (pA , pB ) − CB (xB (pA , pB )) pB

Denote its solution by BRB (pA ), which forms a best response function as pA is variable.

CHAPTER 22. OLIGOPOLY

297

pB 6

BRA BRB

p∗B

r

dB +cB eBB 2eBB

0

dA +cA eAA 2eAA

p∗A

- pA

Figure 22.3: Bertrand competition with product differentiation

In Nash equilibrium (p∗A , p∗B ) each firm’s price-setting choice is a best response to each other’s one. Thus we have p∗A = BRA (p∗B ) p∗B = BRB (p∗A ) Let us look for Nash equilibrium in the example of linear demand and linear costs. Demand for each firm’s product is xA (pA , pB ) = dA − eAA pA + eAB pB for A and xB (pA , pB ) = dB − eBB pB + eBA pA for B. A’s cost function is CA (yA ) = cA yA and B’s cost function is CB (yB ) = cB yB . First let us find A’s best response. Given any possible pB , solve A’s profit maximization problem max pA [dA − eAA pA + eAB pB ] − cA [dA − eAA pA + eAB pB ] pA

Since the first-order condition is dA − 2eAA pA + eAB pB + cA eAA = 0 A’s best response is BRA (pB ) =

dA + cA eAA + eAB pB 2eAA

In contrast to the case of quantity setting, as depicted in Figure 22.3 it is best response to set higher price as the opponent is setting higher price. This is called strategic complementarity. Likewise, given any possible pA , solve B’s profit maximization problem max pB [dB − eBB pB + eBA pA ] − cB [dB − eBB pB + eBA pA ] pB

CHAPTER 22. OLIGOPOLY

298

Since the first-order condition is dB − 2eBB pB + eBA pA + cB eBB = 0 B’s best response is BRB (pA ) =

dB + cB eBB + eBA pA 2eBB

See Figure 22.3 again. Since in Nash equilibrium (p∗A , p∗B ) we have p∗A =

dA + cA eAA + eAB p∗B , 2eAA

p∗B =

dB + cB eBB + eBA p∗A 2eBB

By solving the above equations we obtain p∗A =

dA + cA eAA + 2eAA −

eAB (dB +cB eBB ) 2eBB eAB eBA 2eBB

p∗B =

dB + cB eBB + 2eBB −

eBA (dA +cA eAA ) 2eAA eBA eAB 2eAA

which is the intersection of the two best response curves as in Figure 22.3. Nash equilibrium in Bertand competition can be reached by the rationalizability criterion alone. See Figure 22.4. In this argument let us assume that there is an upper bound for prices they can set, denoted by p. This looks like an ad hoc assumption but it is reasonable: here A’s best response becomes arbitrarily large as B’s prices tends to be arbitrarily higher, and vice versa, but this is rather an artifact of linear demand function which is assumed just for simplicity of calculation —– the demand for A’s product xA (pA , pB ) = dA − eAA pA + eAB pB tends to be arbitrarily large as pB tends to be arbitrarily higher here, but it won’t be the case in reality. A eAA Here the pA -intercept of A’s best response curve BRA is dA +c . That is, 2eAA any price lower than this cannot be optimal for A no matter what pB is. Hence A eAA A never chooses price lower than dA +c . 2eAA Let peA = BRA (pB ). Then, any A’s price higher than this can never be optimal since B does not set its price higher than pB . Hence A never chooses price higher than peA . B eBB Likewise, here the pB -intercept of B’s best response curve BRB is dB +c . 2eBB That is, any price lower than this cannot be optimal for B no matter what pA B eBB is. Hence B never chooses price lower than dB +c . 2eBB Let peB = BRA (pA ). Then, any B’s price higher than this can never be optimal since A does not set its price higher than pA . Hence B never chooses price higher than peB . ] the set of A’s price levels which A has a reason to choose is [ Summing up, dA +cA eAA , p e A , and the set of B’s price levels which B has a reason to choose [ 2eAA ] B eBB is dB +c , p e . B 2eBB By repeat this argument, then he set of A’s price levels which A has a reason to choose converges to the single point p∗A , and the set of B’s price levels which B has a reason to choose converges to the single point p∗B .

CHAPTER 22. OLIGOPOLY

299

pB pB 6 peB

BRA BRB

p∗B

r

dB +cB eBB 2eBB

0

dA +cA eAA 2eAA

p∗A peA

- pA pA

Figure 22.4: Rationaizability in Bertrand competition

22.4

Sequential price setting

22.4.1

The case of no product differentiation

Let us first assume that products are homogeneous and the two firms have the same and constant marginal cost c. Denote the market demand function by x(p). Denote A’s price by pA and B’s price by pB , then the consequences are classified as follows: 1. When pA < pB , A takes all the demand x(pA ). 2. When pA > pB , B takes all the demand x(pB ). 3. When pA = pB , they split the demand by half. Now suppose A sets its price first and B sets its price after seeing A’s price. Then, if pA > c, B can grab the whole demand by setting slightly lower price so that pA > pB > c. However, B can make pB close to pA as much as possible but the profit suddenly drops when pB is exactly equal to pA . Hence B does not always have optimal choice in the precise sense. So like before let us consider that there is a discrete grid for price setting, let’s say an integer grid, and assume that cA and cB are integer as well. Then for all pA ≥ c + 2 the optimal choice of B is pB = pA − 1. For all pA ≤ c − 1 the optimal choice of B is any pB ≥ c. When pA = c + 1 the optimal choice of B is pB = c + 1. When pA = c the optimal choice of B is any pB ≥ c. Since A receives zero profit except when pA = c + 1 and pB = c + 1, the

CHAPTER 22. OLIGOPOLY

300

unique subgame-perfect Nash equilibrium is p∗A

=

fB∗ (pA ) =

22.4.2

c+1   any ≥ c, when c + 1, when  pA − 11, when

pA ≤ c pA ≤ c + 1 pA ≥ c + 2

The case with product differentiation

Now consider the case with product differentiation. Again let A’s price by pA and B’s by pB . The market demand function for A’s product is given by xA (pA , pB ) and that for B’s product is given by xB (pA , pB ). Let CA denote A’s cost function and CB denote B’s one. Then A’s profit is given by pA xA (pA , pB ) − CA (xA (pA , pB )), and B’s one is given by pB xB (pA , pB ) − CB (xB (pA , pB )) Here let us assume that A sets its price first and B sets its price after seeing A’s price, and that they play subgame-perfect Nash equilibrium. Denote the equilibrium strategy profile by (p∗A , fB∗ ), then is is obtained by backward induction which works as follows. 1. For each possible pA , the second mover B solves its profit maximization problem max pB xB (pA , pB ) − CB (xB (pA , pB )) pB

Denote B’s profit maximizing price conditional on pA by fB∗ (pA ). Then fB∗ is B’s optimal strategy as a list of actions contingent on A’s price. I’m putting ∗ in order to emphasize it is optimal for B. 2. Since the first mover is smart enough to foresee what B will choose conditional on his action, he foresees that if he chooses pA B will choose fB∗ (pA ). Based on such foresight, A solves max pA xA (pA , fB∗ (pA )) − CA (xA (pA , fB∗ (pA ))), pA

Notice that here B’s price pB has been replaced by fB∗ (pA ). This is a onevariable maximization problem, and A’s optimal strategy p∗A is obtained by solving this. Plug A’s strategy as price p∗A into B’s strategy as a function fB∗ , we obtain B’s price choice fB∗ (p∗A ). Here the sequence of prices (p∗A , fB∗ (p∗A )) is called

CHAPTER 22. OLIGOPOLY

301

an equilibrium path. Again, don’t confuse between equilibrium path and equilibrium strategy profile. Let us look for the backward induction solution in the example of linear demand and linear costs. Demand for each firm’s product is xA (pA , pB ) = dA − eAA pA + eAB pB for A and xB (pA , pB ) = dB − eBB pB + eBA pA for B. A’s cost function is CA (yA ) = cA yA and B’s cost function is CB (yB ) = cB yB . Because it is ”backward” induction, first we solve the second mover B’s problem. For each possible pA , solve B’s profit maximization problem max pB [dB − eBB pB + eBA pA ] − cB [dB − eBB pB + eBA pA ] pB

Since the first-order condition is dB − 2eBB pB + eBA pA + cB eBB = 0 B’s optimal strategy is obtained as a function fB∗ given by fB∗ (pA ) =

dB + cB eBB + eBA pA 2eBB

Since A is smart enough to foresee B’s choice conditional on his action, A knows if he chooses pA B will choose fB∗ (pA ). Hence A’s profit maximization problem after replacing pB by fB∗ (pA ) is max pA [dA − eAA pA + eAB fB∗ (pA )] − cA [dA − eAA pA + eAB fB∗ (pA )] pA [ ] dB + cB eBB + eBA pA = max pA dA − eAA pA + eAB pA 2eBB [ ] dB + cB eBB + eBA pA −cA dA − eAA pA + eAB 2eBB By taking the first-order condition we obtain ( B eBB dA + eAB dB +c + cA eAA − 2e BB ( ) p∗A = eBA 2 eAA − eAB 2eBB

eAB eBA 2eBB

)

To see if the first-mover has advantage or disadvantage, consider the case that xA (pA , pB ) = 10 − pA + pB , xB (pA , pB ) = 10 − pB + pA and cA = cB = 2. Then in the simultaneous move case the prices are p∗A = p∗B = 12 and their profits are πA = πB = 100

CHAPTER 22. OLIGOPOLY

302

On the other hand, in the sequential move case the resulting prices are p∗A = 17, p∗B = f ∗ (p∗A ) = 14.5 and their profits are πA = 112.5, πB = 156.25 which shows that the second mover has an advantage (you can should this in a more general manner, of course).

22.5

Convergence to perfect competition

Recall the definition of perfectly competitive market the market consists of a large number of participants so that each one is negligibly small compare to the entire economy and has to take the market price as given. Now is it really true when the number of participants tends to be very large? That is, does imperfect competition converge to perfect competition then? If the answer is yes, that gives a strategic foundation of the perfect competition assumption.

22.5.1

Convergence of Cournot competition to perfect competition

Suppose there are n firms, and each firm k = 1, · · · , n is given its cost function Ck . Denote the inverse demand function by p(y). Then, given a profile of supplied quantities of n firms denoted by (y1 , · · · , yn ) the resulting market price is   n ∑ p yj  j=1

Now, when n is a very large number the total quantity provided y=

n ∑

yj .

j=1

is almost unaffected by the change of individual quantity yk of any given firm k. Then each individual firm k has to take the market price p(y) as a constant (hence rewritten by p) which it cannot manipulate by itself alone. Thus, in the approximate sense the profit maximization problem of each firm k is max pyk − Ck (yk ) yk

which yields the maximization condition p = M Ck (yk ). This is nothing but the firm behavior in a perfectly competitive market.

CHAPTER 22. OLIGOPOLY

303

Let me verify the above claim with a simpler example. Assume linear inverse demand function p(y) = a − by. Consider that there are n identical firms with a constant marginal cost c. Note that in a perfectly competitive market the equilibrium quantity follows from a − by = c, which is y CE = a−c b , and that the competitive equilibrium price of output is pCE = c. Now pick one firm arbitrarily, and denote its quantity by y, whereas the sum of the other firms’ quantities is denoted by Y . Then this firm’s profit maximization problem is max(a − b(y + Y ))y − cy y

Since the first-order condition is a − 2by − bY − c = 0 the firm’s best response against the total quantity of the others is BR(Y ) =

a − c − bY 2b

Now, consider that all the other firms behave just like this generic firm. Although it is a symmetric environment in which all the firms have the same cost structure, it is not necessary that they behave in a symmetric manner, so the assumption of symmetry of behavior is an additional assumption. This is called symmetric Nash equilibrium. In a symmetric Nash equilibrium when there are n firms, denote (y n , · · · , y n ), since all the other firms consequently provide the same quantity as the above generic firm does, we have Y = (n − 1)y n . Hence we it holds yn =

a − c − bY a − c − b(n − 1)y n = 2b 2b

which yields yn =

a−c (n + 1)b

and the resulting output price pn = a − bny n = a −

n(a − c) n+1

Now as we take n to be arbitrarily large, we have yn ny n pn

a−c → 0, (n + 1)b a−c a−c → , = (1 + 1/n)b b a−c = a− →c 1 + 1/n =

CHAPTER 22. OLIGOPOLY

304

Here each individual firm’s supply tends to be negligibly small, the total quantity ny n converges to that in the competitive equilibrium, and the price converges to the competitive equilibrium price level, that is, the marginal cost. However, the model of the quantity-setting competition relies on the existence of auctioneer. We need to think of price-setting competition ala Bertrand if we want to get a strategic foundation of perfect competition assumption without relying on auctioneer.

22.5.2

Convergence of Bertand competition to perfect competition

We know that when firms have identical and constant marginal costs competitive outcome is realized under Bertrand competition just with two firms, which set their prices equal to the marginal cost in Nash equilibrium. This argument relies heavily on the assumption that marginal costs are constant and identical across firms. However, it is known that each firm has to behave as a price-taker when there are indefinitely many potential firms with decreasing returns to scale and when the freedom of entry is guaranteed. For, if existing firms are earning profits with their prices being higher than the marginal cost there are arbitrarily many firms which can produce the good at cheaper cost and can set lower prices. Thus if the freedom of entry is guaranteed even the existing firms have to lower their prices to the marginal cost In the sense that it does not need to assume the existence of auctioneer it may be more convincing than the convergence argument for Cournot competition.

22.6

Collusion

22.6.1

Maximizing the joint profit

Let us now consider the firms form a cartel and manipulate quantities or prices cooperatively, so that they maximize their joint profit and share it nicely. To illustrate, assume linear inverse demand p(y) = a − by and also assume that the two firms have constant and identical marginal cost given by c.—– If they have different marginal cost, let the more efficient firm produce everything and let the less efficient firm do nothing and receive a reward for ”doing nothing.” Suppose they set quantities non-cooperatively in Cournot competition the Nash equilibrium quantities are ∗ yA =

a−c , 3b

∗ qB =

a−c 3b

and the resulting output price is ∗ ∗ p(yA + yB )=

a + 2c 3

CHAPTER 22. OLIGOPOLY

305

which yields profits A’s profit =

(a − c)2 9b

B’s profit =

(a − c)2 9b

On the other hand, what if they cooperatively manipulate the total quantity y = yA + yB in order to maximize the joint profit (a − by)y − cy where they behave together as a monopolist? Here the joint monopolist’s profit maximization yields a−c yb = . 2b This is smaller than the total quantity in the Cournot competition, and results in a higher price a+c p(b y) = 2 which yields the joint profit (a − c)2 4b This is greater than the sum of profits in the Cournot competition. So we can think of the following cartel arrangement. 1. Each of A and B produces half of yb, that is,

a−c 4b .

2. Each of A and B receives half of the maximized joint profit Then each of them earns higher profit than competition.

(a−c)2 9b

(a−c)2 8b .

which is earned in the Cournot

Such collusion can be arranged by means of cooperative price setting as well. For example, both can set the monopoly price p(b y) =

a+c 2

and share the demand and profit equally.

22.6.2

Do they keep the promise?

By forming a cartel the two firm can earn higher profits. But do they keep the promise? ∗ What if let’s say A breaks the promise and produces yA = a−c 3b , the quantity y b in Nash equilibrium instead of 2 ? Then the resulting market price is ) ( 5a + 7c yb ∗ = p yA + 2 12

CHAPTER 22. OLIGOPOLY

306

and their profits become A’s profit =

5(a − c)2 36b

B’s profit =

5(a − c)2 48b

Hence A earns higher profit than when it keeps the promise. The intuition is clear. The cartel arrangement here is that the two firms pay effort to reduce their quantities in order to keep the market price higher. Then, when B is keeping the promise and paying effort to keep the market price higher it is actually better for A to take advantage of B’s effort to produce more under the price which is relatively higher thanks to B. That’s the same thing for B, though. Now consider that A and B choose between the cartel quantity and the Counrot quantity respectively. Then the payoff matrix becomes as follows. B ∗ yB

A

∗ yA y b 2

2

2

(a−c) (a−c) 9b 2 , 9b 5(a−c) 5(a−c)2 , 48b 36b

y b 2

5(a−c)2 5(a−c) 36b 2 , 48b (a−c) (a−c)2 , 8b 8b 2

From the above table we see that it is always better fro A to provide the Cournot quantity, ans similarly for B. Thus the upper-left cell will occur. However, this outcome is unanimously worse for A and B than the cartel outcome. It’s a prisoners’ dilemma situation. This remains the case in the Bertrand price competition as well. Even when they promise to set the monopoly price together it is better for each one to break the promise and set lower price. However, when the competition is repeated indefinitely many times forever, as we saw in the section of repeated games cartel can be sustained when the firms are sufficiently patient, which means in this context the interest rate is sufficiently low.

22.7

Exercises

Exercise 30 Market inverse demand is given by が p(y) = 100 − 2y. There are two firms A and B, which have the same and constant marginal cost 4. (i) Find Nash equilibrium in the simultaneous quantity setting competition. (ii) Find subgame-perfect Nash equilibrium in the sequential quantity setting competition, in which A moves first and B moves second. Exercise 31 Consider the case with product differentiation, in which demand functions for A’s and B’s products are given by xA (pA , pB ) = 50 − 2pA + pB and xB (pA , pB ) = 50 − 2pB + pA , respectively. Assume that the two firms have the same and constant marginal cost. (i) Find Nash equilibrium in the simultaneous

CHAPTER 22. OLIGOPOLY

307

price setting competition. (ii) Find subgame-perfect Nash equilibrium in the sequential price setting competition, in which A moves first and B moves second.

Part IV

Economic Analysis with Incomplete Information

308

Chapter 23

Basic game theory III: games with incomplete information Until the previous chapter I have assumed that all the players or market participants have the same information. This assumption is called complete information. It means for example that firms know each other’s cost function, sellers know buyers’ willingness to pay, buyers’ know the qualities of the items being sold, bidders know each other’s valuation of the item, and so on. In many interesting situations different economic agents have different informations. They do not always know what information each other has, and can only infer that in a probabilistic manner. Such situation is called incomplete information. This chapter covers games with incomplete information, and the next chapter covers auction, one of its prominent applications.

23.1

Bayesian game and Bayesian Nash equilibrium

Let me start with an example. Example 23.1 A and B decide whether to enter the market or not, respectively. Here B may be either string or weak, but A cannot know which type B is. However, A knows that B’s two types are equally likely ex ante. Now if B is strong the game is

A

E N

B = Strong E N -2, 5 10, 0 0, 10 0, 0 309

CHAPTER 23. BASIC GAME THEORY III

310

and if B is weak the game is

E N

A

B = W eak E N 5, -2 10, 0 0, 10 0, 0

Since B knows his type he can condition his action on his type. Thus B’s strategy takes the form for example ”enter if he is strong and do not enter if he is weak.” The game extended in such a way is called a Bayesian game, and Nash equilibrium in a Bayesian game is called Bayesian Nash equilibrium. Since A cannot know B’s type when he chooses his action he cannot condition his action on B’s type. Thus A just chooses between entry and non-entry in an unconditional way. Now let me denote B’s strategy for example ”enter if he is strong and do not enter if he is weak” by ”EN” in short and similarly for the other ones. Then the payoffs in the Bayesian game are given by ex-ante expected values of payoffs in the original games. Hence the payoff matrix in the Bayesian game is

A

E N

EE 1.5, 1.5 0, 10

EN 4, 2.5 0, 5

B NE 7.5, -1 0, 5

NN 10, 0 0, 0

To illustrate, let us consider for example that A’s strategy is E and B’s strategy is EN. Since B’s types are equally likely here, payoffs (−2, 5) resulting from both firms’ entries when B is strong occur with probability half and payoffs (10, 0) resulting from only A’s entry when B is weak occur with probability half. Thus, A’s ex-ante expected payoff is −2 × 0.5 + 10 × 0.5 = 4 and B’s one is 5 × 0.5 + 0 × 0.5 = 2.5. Similarly for the other cells. By solving this game we obtain Bayesian Nash equilibrium (E,EN). Let me consider a some more general case, in which the ex-ante probability that B is strong is denoted by p, where 0 < p < 1. Then the payoff matrix of the Bayesian game is

A

E N

EE 5 − 7p, −2 + 7p 0, 10

B EN NE 10 − 12p, 5p 5 + 5p, −2 + 2p 0, 10p 0, 10 − 10p

NN 10, 0 0, 0

Notice that EN strictly dominates NN since 5p > 0 and 10p > 0. Also, EE strictly dominates NE since −2 + 7p > −2 + 2p and 10 > 10 − 10p. Thus by eliminating strictly dominated strategies we obtain B A

E N

EE 5 − 7p, −2 + 7p 0, 10

EN 10 − 12p, 5p 0, 10p

CHAPTER 23. BASIC GAME THEORY III

311

Since it holds −2 + 7p < 5p and 10 > 10p, if there exists a pure-strategy Nash equilibrium it must be either (E,EN) or (N,EE). When 10 − 12p ≧ 0 (E,EN) can be an equilibrium, and when 5 − 7p ≦ 0 (N,EE) can be an equilibrium. Summing up, we have When 0 < p < 5/7 (E, EN) is the unique equilibrium. When 5/7 ≦ p ≦ 5/6 both (E, EN) and (N, EE) are equilibria When 5/6 < p < 1 (N, EE) is the unique equilibrium. Here (E, EN) refers to an equilibrium in which B enters only when he is strong and A enters, in which B is ”timid.” This case can be an equilibrium when B’s probability of being strong is sufficiently low, that is, when p ≦ 5/6. On the other hand, (N,EE) refers to an equilibrium in which B enters no matter what his type is and A does not enter, in which B is ”bluffing.” This case can be an equilibrium when B’s probability of being strong is sufficiently high, that is, when p ≧ 5/7. In the intersection the two cases 5/7 ≦ p ≦ 5/6 there are two Bayesian Nash equilibria, one in which B is timid and the other in which B is bluffing. Thus we run into the multiple equilibria problem again. Example 23.2 Now suppose this time that both A and B have possibilities of being strong or weak. Assume for simlicity that the types of A and B are independently and identically distributed (I.I.D.), where the probability of being strong is 2/3. Payoff matrix for each combination of types of A and B is given by

A=S

A=W

E N

E -2, -2 0, 10

B=S N 10, 0 0, 0

E N

E -8, 5 0, 10

B=S N 10, 0 0, 0

A=S

A=W

B=W N 10, 0 0, 0

E N

E 5, -8 0, 10

E N

E -5, -5 0, 10

B=W N 10, 0 0, 0

Now each of A and B can condition his action on his type (but not on the opponent’s one as before), and the payoff matrix of the Bayesian game is

A

EE EN NE NN

EE −19/9, −19/9 2/9, 2/3 −13/9, 65/9 0, 10

B EN NE 2/3, 2/9 65/9, −13/9 4/3, 4/3 50/9, −2/3 −2/3, 50/9 5/9, 5/9 0, 20/3 0, 10/3

Here the Bayesian Nash equilibrium is (EN, EN).

NN 10, 0 20/3, 0 10/3, 0 0, 0

CHAPTER 23. BASIC GAME THEORY III

312

On the other hand, when the probability of being strong is 1/3 the payoff matrix for the Bayesian game is

A

EE EN NE NN

EE −28/9, −28/9 8/9, 14/3 −4, 20/9 0, 10

B EN NE 14/3, 8/9 20/9, −4 2, 2 20/9, 8/3 8/3, 20/9 0, 0 0, 10/3 0, 20/3

NN 10, 0 10/3, 0 20/3, 0 0, 0

in which there are two Bayesian Nash equilibria (EE, EN ),

(EN, EE)

Here in each equilibrium one is bluffing and the other is timid. There emerges the multiple equilibrium problem again. Now let me give you the formal definitions of games with incomplete information, corresponding Bayesian games and Bayesian Nash equilibria. A game with incomplete information consists of a set of players, sets of types, a common prior distribution, sets of strategies and payoff functions. The set of players I = {1, · · · , n} is a finite set. The set of types of each player i = 1,∏ · · · , n is n denoted by Θi . The set of all the players’ type profiles is given by Θ = i=1 Θi , where its element is generically denoted by θ = (θ1 , · · · , θn ). In the last example, the sets of types A and B are given by ΘA = {S, W } and ΘB = {S, W }, respectively. Let p denote an ex-ante probability distribution over Θ denoted by p, which is commonly held by all the players as their probabilistic beliefs about Θ. This is called the common prior assumption. This says From the ex-ante viewpoint all the players share the same belief, and any difference between their beliefs in the meantime arises only from the difference of informations they receive. In other words, this excludes a stubborn attitude such that one believes what he believes no matter what information he receives and no matter what others believe. I will explain later why we need to make such assumption. Let Si denote the set of strategies of player ∏n i = 1, · · · , n, where its generic element is denoted by si ∈ Si . Let S = i=1 Si denote the set of strategy profiles, where its generic element is denoted by s = (s1 , · · · , sn ). Let vi (θ, s) denote player i’s payoff realized when the type profile is θ and the strategy profile is s. That is, player i’s payoff function is given by vi : Θ×S → R. The Bayesian game corresponding to a given game with incomplete information consists of the set of players, the sets of Bayesian strategies and ex-ante expected payoff functions. The set of players is I just as before. A Bayesian strategy of each player is a function from the set of his types to the set of his strategies. Let Σi denote the set of all functions from Θi to Si , then it is the set of i’s Bayesian strategies. Let σi denote its generic element, then he chooses

CHAPTER 23. BASIC GAME THEORY III

313

∏n strategy σi (θi ) if his type is θi . Let Σ = i=1 Σi denote the set of Bayesian strategy profiles, where its generic element is denoted by σ = (σ1 , · · · , σn ). Also, let σ(θ) = (σ1 (θ1 ), · · · , σn (θn )) denote the strategy profile induced by a given Bayesian strategy profile σ when the realized type profile is θ. It is also denoted in the form σ(θ) = (σi (θi ), σ−i (θ−i )) for any given i. Given a Bayesian strategy profile σ = (σ1 , · · · , σn ), player i’s ex-ante expected payoff is given by ∑ vi (θ, σ(θ))p(θ). θ∈Θ

Here is the definition of Bayesian Nash equilibrium.1 Definition 23.1 A profile of Bayesian strategies σ b = (b σ1 , · · · , σ bn ) is said to be Bayesian Nash equilibrium if for all i and σi ∈ Σi it holds ∑ ∑ vi (θ, σ bi (θi ), σ b−i (θ−i ))p(θ) ≧ vi (θ, σi (θi ), σ b−i (θ−i ))p(θ) θ∈Θ

θ∈Θ

The next proposition says that Bayesian Nash equilibrium defined above in which every player is optimizing his Bayesian strategy from the ex-ante viewpoint is equivalent to an interim situation in which each player optimizes his action given that he knows his type only. For example, it states the equivalence between the two conditions, • ”A enters if A is strong and does not enter if A is weak” is optimal for A from the ex-ante view point. • It is optimal for A to enter once A knows he strong and it is optimal for A not to enter once A knows he weak . This allows us to find Bayesian Nash equilibrium in either of two ways, one based on ex-ante optimality of Bayesian strategies and the other based on interim optimality of them. In auction games to be covered later the second way is actually easier.2

1 When types are continuous quantities such as willingness to pay, the common prior distribution is a probability measure or probability distribution which is mathematically suitably defined, and the definition of equilibrium is that for all i and σi ∈ Σi it holds ∫ ∫ vi (θ, σ bi (θi ), σ b−i (θ−i ))dp(θ) ≧ vi (θ, σi (θi ), σ b−i (θ−i ))dp(θ) Θ

Θ

2 When types are continuous quantities the equivalent condition is stated as: for all i, θi ∈ Θi and si ∈ Si it holds ∫ vi (θi , θ−i , σ bi (θi ), σ b−i (θ−i ))dp(θ−i |θi ) Θ−i

∫ ≧

Θ−i

vi (θi , θ−i , si , σ b−i (θ−i ))dp(θ−i |θi )

CHAPTER 23. BASIC GAME THEORY III

314

Proposition 23.1 A profile of Bayesian strategies σ b = (b σ1 , · · · , σ bn ) is Bayesian Nash equilibrium if and only if for all i, θi ∈ Θi and si ∈ Si it holds ∑ vi (θi , θ−i , σ bi (θi ), σ b−i (θ−i ))p(θ−i |θi ) θ−i ∈Θ−i





vi (θi , θ−i , si , σ b−i (θ−i ))p(θ−i |θi ),

θ−i ∈Θ−i

where p(θ−i |θi ) denotes the probability of θ−i conditional on θi . Proof. Suppose σ b is a Bayesian Nash equilibrium, and suppose that for some i there exists θi ∈ Θi and si ∈ Si such that ∑ vi (θi , θ−i , si , σ b−i (θ−i ))p(θ−i |θi ) θ−i ∈Θ−i

>



vi (θi , θ−i , σ bi (θi ), σ b−i (θ−i ))p(θ−i |θi )

θ−i ∈Θ−i

Here define σi by σi (θi ) = si and σi (θi′ ) = σ bi (θi′ ) for all θi′ ̸= θi . Then it holds ∑ vi (θi , θ−i , σi (θi ), σ b−i (θ−i ))p(θ) θ∈Θ

=





vi (θi , θ−i , σi (θi ), σ b−i (θ−i ))p(θ−i |θi )p(θi )

θi θ−i ∈Θ−i

>





vi (θi , θ−i , σ bi (θi ), σ b−i (θ−i ))p(θ−i |θi )p(θi )

θi θ−i ∈Θ−i

=



vi (θi , θ−i , σ bi (θi ), σ b−i (θ−i ))p(θ)

θ∈Θ

which contradicts to the assumption that σ b is a Bayesian Nash equilibrium. To show the converse suppose that for all i, θi ∈ Θi and si ∈ Si it holds ∑ vi (θi , θ−i , σ bi (θi ), σ b−i (θ−i ))p(θ−i |θi ) θ−i ∈Θ−i





vi (θi , θ−i , si , σ b−i (θ−i ))p(θ−i |θi )

θ−i ∈Θ−i

Then, since for all σi ∈ Σi and θi ∈ Θi it holds ∑ vi (θi , θ−i , σ bi (θi ), σ b−i (θ−i ))p(θ−i |θi ) θ−i ∈Θ−i





θ−i ∈Θ−i

vi (θi , θ−i , σi (θi ), σ b−i (θ−i ))p(θ−i |θi )

CHAPTER 23. BASIC GAME THEORY III

315

it holds ∑ θ∈Θ

=



vi (θi , θ−i , σ bi (θi ), σ b−i (θ−i ))p(θ) ∑

vi (θi , θ−i , σ bi (θi ), σ b−i (θ−i ))p(θ−i |θi )p(θi )

θi θ−i ∈Θ−i







vi (θi , θ−i , σi (θi ), σ b−i (θ−i ))p(θ−i |θi )p(θi )

θi θ−i ∈Θ−i

=



vi (θi , θ−i , σi (θi ), σ b−i (θ−i ))p(θ)

θ∈Θ

23.2

On the common prior assumption

The assumption that all the players share an identical beliefs over the set of type profiles sounds absurd. However, we cannot just drop it. Suppose for example that A and B have different beliefs. Now, if A is ”rational” in the sense that he is ”not stubborn,” he must think of how B’s belief is like, and as a result A must have a belief about B’s belief. Likewise, if B is ”rational” in the sense that he is ”not stubborn,” he must think of how A’s belief is like, and as a result B must have a belief about A’s belief. Suppose that these ”second-order” beliefs held by A and B respectively are different. Then, if A is ”rational” in the sense that he is ”not stubborn,” he must think of how B’s second-order belief is like, and as a result A must have a third-order belief about B’s second-order belief. Likewise, if B is ”rational” in the sense that he is ”not stubborn,” he must think of how A’s second-order belief is like, and as a result B must have a third-order belief about A’s second-order belief. Suppose that these third-order beliefs held by A and B respectively are different. Then, if A is ”rational” in the sense that he is ”not stubborn,” he must think of how B’s third-order belief is like, and as a result A must have a fourth-order belief about B’s third-order belief. Likewise, if B is ”rational” in the sense that he is ”not stubborn,” he must think of how A’s third-order belief is like, and as a result B must have a fourth-order belief about A’s third-order belief. And so on. If you want to drop the common prior assumption and do not want the model to fall into an ad hoc one, you have to consider a set of types which contains all the above-mentioned infinite stairways as its element. Such set is called universal type space.

CHAPTER 23. BASIC GAME THEORY III

23.3

316

Exercises

Exercise 32 The game played by A and B is with probability p the prisoners’ dilemma in the left p and with probability 1 − p the game in the right in which cooperation is a dominant strategy. However, only B knows which game they are facing. B A

C N

C 10, 10 12, −1

N −1, 12 0, 0

C N

A

C 10, 10 0, 5

B N 5, 0 0, 0

Write down the Bayesian game and find Bayesian Nash equilibrium assuming 0 < p < 1. Exercise 33 Consider the game in which A is a potential buyer and B is a potential seller of a stock. With probability p the stock is a good one and will go up by 10, and with probability 1 − p it is a bad one will go down by 10.

A

Buy Not to Buy

Good Sell Not to Sell 10, 0 0, 10 0, 10 0, 10

A

Buy Not to Buy

Bad Sell Not to Sell −10, 0 0, −10 0, −10 0, −10

Write down the corresponding Bayesian game and find all the Bayesian Nash equilibria assuming that 0 < p < 1.

Chapter 24

Auction I focus on the case that a single item is being sold and buyers bid. Like procurement auctions we can consider that the sellers bid, but it can be treated by flipping the direction in the arguments below. If the seller knows buyers’ willingness to pay he simply points to the buyer with the highest willingness to pay and charge him the price equal to his willingness to pay. However, in genera the seller does not know the buyers’ willingness to pay. How can the seller make the buyers reveal their willingness to pay? Let them bid. If the bidding mechanism is a nice one the bids will nicely reveal their willingness to pay. But what auction format it better?

24.1

Prominent auction formats

Here let me give you well-known auction formats. 1. First-price auction: Bidders submit their bids in sealed envelopes. The auctioneers opens the bids. The highest bidder wins the item and pays his bid. 2. Second-price auction: Bidders submit their bids in sealed envelopes. The auctioneers opens the bids. The highest bidder wins the item but pays the second-highest bid. 3. English auction: It is a dynamic process. The auctioneer starts from sufficiently lower price and gradually raises it. In the meantime bidders give up and leave the room. When the last bidder but one gives up the remaining one wins the auction and pays the current price. 4. Dutch auction: It is a dynamic process. Contrary to English auction, the auctioneer starts from sufficiently high price and gradually lowers it. When there is the first bidder who announces to buy he wins and pays the current price.

317

CHAPTER 24. AUCTION

24.2

318

Information, timeline and the natures of values

In general, not only that the seller doesn’t know bidders’ willingness to pay, but also each bidder does not know how much the other bidders are willing to pay. Also, it is often the case that an individual bidder does not know even his willingness to pay since the value of the item is uncertain to him. To understand, consider that the auction is conducted in the following timeline. 1. Bidders are collected. At this point no bidder knows even his own valuation, and they only know the prior probabilistic distribution of values. This is called the ex-ante stage. 2. The bidders go to the viewing event or go to do research, and receive certain information (called signal in this literature). However, each bidder does not know what signals the other bidders have received. This is called interim stage. 3. The bidders bid. After the winner receives the item its actual value is realized. It is called the ex-post stage. The natures of values are classified as follows. 1. Independent private values: It’s just like what’s said by the proverb ”there is no accounting for taste.” Each bidder knows his valuation of the item at the interim stage, since it is just a matter of his own taste. Also, the prior distributions of individual bidders values are independent in the ex-ante stage, that is, how my value is likely to be higher or lower is not correlated with how you value is likely to be higher or lower. 2. Common value: When the value a stock for example becomes 100 dollars in the future it is 100 dollars for everybody. Here the difference of bidders’ valuations can come only from the difference between informations they receive at the interim stage. Another example is bidding for an oil field. Here all bidders’ interest is solely in how much oil it contains, and the difference of their valuations can come only from the difference of informations received in the interim stage about how much oil is contained. Here each bidder does not necessarily know the true value, and may learn about it from others valuations. I focus on the case of independent private values, since the common value case is actually harder and beyond the level of this book. Under the assumption of independent private values second-price auction and English auction are equivalent in the sense that the second highest bid in the second-price auction is equal to the price at which the last bidder but one gives up in English auction. Also, under expected utility theory first-price auction and Dutch auction are known to be equivalent in the sense that the

CHAPTER 24. AUCTION

319

highest bid in the first-price auction is equal to the price at which the winning bidder announces to buy. Thus we focus on the first-price auction and the second-price auction.

24.3

Preferences

Before getting into the details of auction formats let me first describe bidders’ preferences. Since winning and losing are uncertain in general and also payment or any income transfer is uncertain as well we need to consider bidders’ preferences over risky prospects. Consider that there are n bidders. We assume that each bidder’s preference is quasi-linear in income and each one is risk-neutral. The quasi-linearity assumption means that there is no income effect on the item to be sold. That is, we assume both of no income effect and risk neutrality. Let vi denote i’s willingness to pay. Let pi = ((xi1 , ai1 ); pi1 , (xi2 , ai2 ); pi2 , · · · , (xim , aim ); pim ) denote a lottery over the pairs of the item and income transfer, which says i receives xi1 units of the item (which is either 0 or 1) and ai1 units of income (payment if it is negative) with probability pi1 , xi2 units of the item and ai2 units of income with probability pi1 , and so on, and xim units of the item and aim units of income with probability pim . Then his preference over such lotteries is assumed to be represented in the form ui (pi ) =

m ∑

pik {vi xik + aik }

k=1

To illustrate, consider that bidder i pays ti if he wins and pays nothing if he loses, then he is facing a lottery pi = (1, −ti ; πi , 0, 0; 1 − πi ) where πi denotes his winning probability, and its expected utility evaluation is ui (pi ) = (vi − ti )πi There are some cruel auction formants in which you have to pay some even when you lose. Suppose that there bidder i pays ti1 if he wins and pays ti2 if he loses, then he is facing a lottery pi = (1, −ti1 ; πi , 0, −ti2 ; 1 − πi ) where πi denotes his winning probability, and its expected utility evaluation is ui (pi ) = (vi − ti1 )πi − ti2 (1 − πi )

24.4

First-price auction

Probably one of the most popular auction formats. It appears the seller expect more revenue in the first-price auction since the winner pays the highest bid. But this is not immediate.

CHAPTER 24. AUCTION

320

6 vi − maxj̸=i bj r r r maxj̸=i bj vi

- bi

Figure 24.1: Discontinuous payoff

24.4.1

The case of complete information and discrete bids

Let me start with the case of complete information, in the sense that bidders know how strong or weak each other bidder is, which the seller still doesn’t know the bidders’ willingness to pay. Problem of dealing with complete information and continuous bids It is hard to deal with the case of incomplete information in the setting of continuous bids. See Figure 24.1. Pick bidder i and let maxj̸=i bj denote the highest bids of the others. Since i’s value vi is greater than this he should win, but how much should he bid? In order to win his bid bi must be greater than maxj̸=i bj , and his net gain vi − bi is larger as his bid is closer to it. However, once his bid bi is exactly equal to maxj̸=i bj he loses the sure win and has to obey some tie-breaking rule, so that his expected utility suddenly drops down to somewhere between 0 and vi − maxj̸=i bj . Also, when his bid bi is below the highest bid of the others maxj̸=i bj now he gets nothing. Thus his maximization problem has no solution. There are two ways to think of this problem. One is to assume that bids are discrete, so that you don’t have any such discontinuity problem. I do this just below. The other is to introduce ”random noise” which smoothes out the discontinuity, which is the case of incomplete information I cover in the next subsection. Now consider the discrete-bid case. To illustrate, assume that there are two bidders and five possible values, 10, 20, 30, 40 and 50. Also, assume let’s say that A’s willingness to pay is 40 and B’s willingness to pay is 20. Tie-breaking

CHAPTER 24. AUCTION

321

is done by a coin-flip. Then the payoff matrix of this auction game is

A

10 20 30 40 50

10 30 10 2 , 2 20, 0 10, 0 0, 0 −10, 0

20 0, 0 20 0 2 , 2 10, 0 0, 0 −10, 0

B 30 0, −10 0, −10 10 −10 2 , 2 0, 0 −10, 0

40 0, −20 0, −20 0, −20 0 −20 2, 2 −10, 0

50 0, −30 0, −30 0, −30 0, −30 −10 −30 2 , 2

Let me give some explanations of where the above numbers come from. Consider for example the case where A bids 20 and B bids 10. Then A wins and pays 20. Since A’s willingness to pay is 40 his net gain is 40 − 20 = 20. B loses on the other hand he receives or pays nothing, hence his net gain is zero. Consider for another example the case where both bid 30. Since the tie is broken by coin-flip and both bidders are assumed to be risk-neutral, A’s expected utility is 1 1 10 1 1 10 2 × (40 − 30) + 2 × 0 = 2 and B’s expected utility is 2 × (20 − 30) + 2 × 0 = − 2 There are three pure-strategy Nash equilibria, (20, 10), (20, 20) and (30, 20). Notice that whichever equilibrium is played A is bidding strictly lower than his willingness to pay. Here A is the stronger bidder and he is aware of it, so he tries to lower his payment by bidding lower as far as it does not risk his winning to much.

24.4.2

The case of incomplete information

Let us now consider the case of incomplete information, in which each bidder knows only his value and does not know others’ values, which he knows only in the form of prior probability distribution. Now consider that there are n bidders who have preferences quasi-linear in income and are risk-neutral. Let vi denote i’s willingness to pay. Assume that each bidders’ valuations are distributed independently and each one is drawn from an identical distribution, which is called IID (independently and identically distributed) in the context of probability and statistics. Let F denote the cumulative distribution function for the identical distribution and f denote its density function. Note that given F and a fixed number v the probability that any drawn number is less than or equal to v is F (v). Note also that F and f have the relationship F ′ (v) = f (v). Denote bidder i’s value by vi . Denote the profile of bids other than i’s by b−i = (b1 , · · · , bi−1 , bi+1 , · · · , bn ), then the entire profile of biddings is denoted by (bi , b−i ). Then bidder i’s net gain is { vi − bi , if bi > maxj̸=i bj Ui (bi , b−i ) = 0, if bi < maxj̸=i bj We ignore the case of ties since its probability is zero in the setting of continuous bids. Here we consider a situation such that

CHAPTER 24. AUCTION

322

all the bidder know how each bidder conditions his bid on his value while nobody knows every other bidder’e realized value, and the way how one conditions his bid on his value is identical across bidders. This is called symmetric Bayesian Nash equilibrium. Since the concept of Bayesian Nash equilibrium itself does not imply that the bidding behaviors are symmetric even when the distributions of values are symmetric, we are imposing the symmetry of bidding behavior as an additional assumption. Denote the common function (called bidding function) by β : [0, 1] → R, where β(v) denotes one’s bid when his value is v. Suppose now that all the bidders except i with his value vi are following the bidding function β. Then bidder i’s expected utility of bidding bi is (vi − bi ) × Probability of winning since if he wins his net gain is his value minus his pay and if he loses he gets nothing and pays nothing. As the probability of winning is the probability that all the other bidders’ bids are lower than bi , it is P rob(maxj̸=i β(vj ) < bi ). Since it is equal to the probability that all the other bidders’ values are lower than β −1 (bi ), the value which yields bid bi under the bidding function β, it is F (β −1 (bi ))n−1 . Thus i’s expected utility is given by (vi − bi )F (β −1 (bi ))n−1 Take the first-order condition for maximization, then we have −F (β −1 (bi ))n−1 + (vi − bi )

(n − 1)F (β −1 (bi ))n−2 f (β −1 (bi )) = 0. β ′ (β −1 (bi ))

Here, in symmetric Bayesian Nash equilibrium, it is eventually optimal for i to follow β when all the other bidders’ are following β. This by replacing bi by β(vi ) we obtain the condition −F (vi )n−1 + (vi − β(vi ))

(n − 1)F (vi )n−2 f (vi ) =0 β ′ (vi )

By rearranging the formula we have β ′ (vi )F (vi )n−1 + β(vi )(n − 1)F (vi )n−2 f (vi ) = (n − 1)vi F (vi )n−2 f (vi ), Since the left-hand-side in the above is the derivative of β(vi )F (vi )n−1 we have d β(vi )F (vi )n−1 = (n − 1)vi F (vi )n−2 f (vi ). dvi By integrating the both sides above we obtain ∫ vi β(vi )F (vi )n−1 = (n − 1)vF (v)n−2 f (v)dv, 0

CHAPTER 24. AUCTION

323

which leads to ∫ vi β(vi ) =

0

] [ (n − 1)vF (v)n−2 f (v)dv = E max vj | max vj < vi j̸=i j̸=i F (vi )n−1

Here the right-hand-side is the expected value of the highest bid of the others conditional on the event that i’s bid is the highest. For example, when the distribution is uniform, that is, when F (v) = v and f (v) = 1, the above solution reduces to β(vi ) =

24.5

n−1 vi n

Second-price auction

Second-price sounds tricky, but it has actually a nice property. Let me explain this by an example first. Take the same numerical example as before in which A’s willingness to pay is 40 and B’s is 20, then the payoff matrix for the secondprice auction game is

A

10 20 30 40 50

10 30 10 2 , 2 30, 0 30, 0 30, 0 30, 0

20 0, 10 20 0 2 , 2 20, 0 20, 0 20, 0

B 30 0, 10 0, 0 10 −10 2 , 2 10, 0 10, 0

40 0, 10 0, 0 0, −10 0 −20 2, 2 0, 0

50 0, 10 0, 0 0, −10 0, −20 −10 −30 2 , 2

The difference from the previous one is that here if you win you pay the second highest bid (which is the opponent’s bid in the present illustration with two bidders). Thus in second-price auction the incentive to lower payment after winning is eliminated in the beginning, by conceding the payment down to the second-highest bid. Consider for example that A bids 20 and Bi bids 10, then A wins and pays 10, not 20. Since A’s willingness to pay is 40 his net gain is 40 − 10 = 30. It is immediate that B’s net gain is 0 then. Also, when both bid 30 let’s say the winner pays 30, since the first-price and the second-price coincide. Since both are risk-neutral A’s expected utility is 21 × (40 − 30) + 12 × 0 = 10 2 and B’s expected utility is 21 × (20 − 30) + 12 × 0 = − 10 . 2 You can see that there are many Nash equilibria because of lots of ties in payoffs, but there is an obvious one. Notice that bidding 40 is always optimal for A and bidding 20 is always optimal for B respectively, no matter what the opponent bids. Thus (40, 20) is a dominant strategy equilibrium here, in which each bidder bids his willingness to pay. This property holds generally in second-price auction. Consider that there are n. Denote bidder i’s willingness to pay by vi and his bid by bi . Also, denote

CHAPTER 24. AUCTION

324

6 vi − maxj̸=i bj r r r maxj̸=i bj vi

- bi

Figure 24.2: Case 1

the profile of bids other than i’s by b−i = (b1 , · · · , bi−1 , bi+1 , · · · , bn ), then the entire profile of bids is denoted by (bi , b−i ). Then bidder i’s net gain is { 0, when bi < maxj̸=i bj Ui (bi , b−i ) = vi − maxj̸=i bj , when bi > maxj̸=i bj Again ignore the case of ties in the present setting of continuous bids. Notice that one’s own bid does not appear in the term of his net gain as given above, and it affects only whether he wins or not. Thus any bidder cannot manipulate his payment by means of bidding either higher or lower than his willingness to pay, which means he does not lose anything by bidding his willingness to pay as it is. Thus we obtain the following result. Theorem 24.1 In second-price auctions it is always a dominant strategy to bid one’s willingness to pay. Proof. Case 1: When vi > maxj̸=i bj bidder i should win. Then the relation between his bid bi and his net gain is depicted as in Figure 24.2. There is a continuum of optimal choices here, but since the graph is flat the bidder does not lose anything by bidding vi . Case 2: When vi < maxj̸=i bj bidder i should not win. Then teh relation between his bid bi and his net gain is depicted as in Figure 24.3. There is a continuum of optimal choices here, but since the graph is flat the bidder does not lose anything by bidding vi .

Weakness to collusion Second-price auction has a weakness in the sense that it may be manipulated by collusion. Recall that in the previous example A bids 40 and B bids 20 in the dominant strategy equilibrium, where A’s net gain is 40 − 20 = 20 and B’s net gain is 0. If A and B can communicate, however, A can ask B to bid 10 instead of 20 by offering to pay him let’s say 5. Since B knows he cannot win

CHAPTER 24. AUCTION

325

6

r r - bi vi maxj̸=i bj vi − maxj̸=i bj r

Figure 24.3: Case 2

the auction anyway it is better for him to accept the offer. A gains as well, since even after paying 5 his net gain is 40 − 10 − 5 = 25, which is greater than 20.

24.6

The revenue equivalence theorem

Now, which auction format is good from the viewpoint of the seller? Consider the following timeline. 1. Bidders are collected. At this point no bidder knows even his own valuation, and they only know the prior probabilistic distribution of valuations. 2. Auction format is chosen. 3. The bidders go to the viewing event, in which each one realizes his valuation. However, each bidder does not know the realized values for the others. 4. The bidders bid. More realistically, bidders may decide whether to bid after setting the auction format or seeing his value, but let me omit it as it is an advanced topic. Thus, at the point of choosing an auction format the seller wants to maximize the ex-ante expected revenue since he is given only the prior distribution over the bidders’ valuations. In this setting the following result is know, which is called revenue equivalence theorem Theorem 24.2 Assume the following. 1. Risk-neural bidders 2. Independent private values Then, any auction format such that the bidder with the highest valuation always wins the item in Bayesian Nash equilibrium yields the same ex-ante expected revenue.

CHAPTER 24. AUCTION

326

Since the bidder with the highest valuation always wins in all of first-price, second-price, English and Dutch auctions they yield the same expected revenue ex-ante. One can also think of all-pay auction in which bidders pay their bids regardless of winning or losing and the highest bidder wins the item, third-price auction in which the highest bidder wins and pays the third highest bid, and many others. All these, however, maintains the property that the bidder with the highest valuation always wins they yield the same expected revenue ex-ante. The revenue equivalence theorem plays the role of benchmark in the auction literature, in the sense that it suggests which auction format is more favorable when we depart from its assumptions. It is known that when the bidders are risk-averse the first-price auction yields higher expected revenue than the second-price auction. When a riskaverse bidder knows he is strong he likes to make his winning sure more than a risk-neutral bidder does, and willing to bid higher. On the other hand, it holds regardless of risk attitudes that bidding own willingness to pay is always a dominant strategy in the second-price auction. Also, it is known that in the common-value case English auction yields higher expected revenue than the second-price auction. This is because English auction is a dynamic process and bidders learn about the common value as the auction is in progress. It is more likely to happen that even when a bidder initially plans to give up earlier but as he sees other bidders are still remaining in the room he learns that the value of the item was actually higher than his initial expectation, updates his valuations stays longer as well.

24.7

Exercises

Exercise 34 There are n bidders, who have quasi-linear preferences over consumptionincome pairs and are risk-neutral. Their willingness to pay are independently and identically distributed, where the cumulative distribution function is denoted by F and the density is denoted by f . Then find the bidding function for Bayesian Nash equilibrium in the all-pay auction game.

Chapter 25

Trade with incomplete information Many kinds of information such as quality of goods, ability of workers and consumers’ preferences or willingness to pay are private information which are not observable or verifiable by other market participants. Trades which are smoothly done under complete information are likely to fail under such types of incomplete information. We can roughly think of two kinds of such situations, one is adverse selection and the other is moral hazard.

25.1

Adverse selection

Adverse selection refers to a situation in which one side of the market cannot observe or verify the types of the other side of the market.

25.1.1

Market for a ”lemon”

”Lemon” is a slang in used car markets which refers to a defective car. There are 100 sellers in the used car markets each of which has one used car. Out of 100 sellers 50 own cars with good quality and 50 own cars with bad quality. The owners of good ones are willing to sell when they are paid 10000 dollars, the owners of bad ones are willing to sell when they are paid 3000 dollars. There are buyers, who are willing to pay 11000 dollars for a good one and 3500 for a bad one. For simplicity assume that all the market participants are risk-neutral. Now let me proceed by the assuming the following. Assumption on information 1: All the market participants know the above numbers. 2: But the buyers cannot observe or verify the quality of an individual car. You can know its quality only after driving several weeks or so. When 327

CHAPTER 25. TRADE WITH INCOMPLETE INFORMATION

328

you realize the car is defective and return to the seller claiming for some compensation the seller will say ”that’ll be because you wore it out,” and you don’t have any counter-evidence against it. 3: Also the owner of good one cannot differentiate their cars from the bad ones, for the owners of bad owns can claim the same thing. In such a situation how are the resulting trading pattern will be? If the buyers could distinguish between the good ones and the bad ones they are simply traded as different commodities. Then a good car will be traded for the price between 10000 and 11000 and a bad car will be traded for the price between 3000 and 3500. Since the buyers cannot observe or verify the qualities, however, they have to accept a price common between the good ones and the bad ones. Denote such price by p. Below we think step by step. 1. When 11000 < p: Since the price is higher even than the willingness to pay for the good ones nobody buys. 2. When 10000 ≤ p ≤ 11000: In this price range both types are provided to the market. The buyers know this. That is, they know 50 cars in the market are good and 50 are bad, hence they know if they buy they draw a good one with probability half and a bad one with probability half. Hence each buyer’s expected utility of buying is 0.5 × 110 + 0.5 × 35 − p = 72.5 − p from the assumption f risk neutrality. Notice that under the present assumption 10000 ≤ p ≤ 11000 it is negative. Hence nobody buys. 3. When 3500 < p < 10000: Since the price is lower than the reserve price for the good ones their owners do not sell. Thus only the bad ones are provided to the market. The buyers know this. That is, they know that all the cars in the market are bad. Hence each buyer’s expected utility of buying is 35 − p, which s negative under the current assumption 3500 < p < 10000. Hence nobody buys. 4. When p < 3000: Since the price is lower even that the reserve price for the bad ones nobody sells. 5. The remaining case is 3000 ≤ p ≤ 3500: Only the bad ones are provided to the market. The buyers know that all the cars in the market are bad, but they are willing to buy. Thus only the bad cars are traded.

CHAPTER 25. TRADE WITH INCOMPLETE INFORMATION

329

So-called adverse selection refers to such situation in which items with good quality are driven out of the market and only bad ones remain in the market and be traded. Whether is it the only case or not is a quantitative question, so let me go over the argument in a little more general model. The proportion of sellers owning good ones is denoted by π, then the proportion of sellers having bad ones is 1 − π. The reserve price for a good one is denoted by vH , that for a bad one is denoted by vL , where vH > vL . Also, willingness to pay for a good one is denoted by wH , and that for a bad one is wL , where wH > wL . Let us focus on the case that vH ≤ wH and vL ≤ wL , since if vH > wH or vL > wL trades are not made even under complete information. Again all the traders are assumed to be risk-neutral. Case 1: When πwH + (1 − π)wL ≥ vH , there are two kinds of equilibria. One is such that the price is in the range πwH + (1 − π)wL ≥ p ≥ vH and both of good ones and bad ones are traded. Here both types are provided and the buyers accept the risk of drawing a bad one with probability 1 − π, since the gambling is sufficiently attractive. The other is such that the price is in the range w L ≥ p ≥ vL and only the bad ones are traded, which is the adverse selection situation in the original example. There exist multiple equilibria here, in the sense that even though they come from the same fundamental in one equilibrium higher price promotes the items with higher quality to be provided and this substantiates buyers’ optimistic expectations in a self-fulfilling manner, and in the other equilibrium lower price prevents the items with higher quality from being provided and this substantiates buyers’ pessimistic expectations in a self-fulfilling manner. Case 2: When πwH + (1 − π)wL < vH the only price range which can support equilibrium is w L ≥ p ≥ vL in which only the bad ones are traded, which is the original adverse selection situation.

25.1.2

Insurance market

Suppose that the proportion of smokers out of potential buyers of an insurance is λ and the proportion of non-smokers is 1 − λ. Each potential buyer has initial

CHAPTER 25. TRADE WITH INCOMPLETE INFORMATION

330

income denoted by w. Each smoker dies of dies of lung cancer with probability πS and each non-smoker dies of lung cancer with probability πN (forget about other causes of death). If one dies his dependent loses income L, where L < w. On the other hand the insurance company can issue life insurance such that the insuree’s dependent can receive the entire loss L. Also, the insurees are their dependents are risk-averse and their vNM indices are given by v(z) = ln z. Assume that the insurance company is risk-neutral. Again, as before assume the following. Assumption on information 1: All the market participants know the above numbers. 2: But the insurance company cannot observe or verify if each individual is smoking or not (though I heard it is technically possible nowadays). 3: Also a non-smoker cannot differentiate himself from the smokers, for the smokers can claim that they are not smoking. Here we assume that an insuree and his dependent are considered as one. How does the insurance premium look like? If the insurance company can observe and verify if a given insuree is smoking or not, it can charge different insurance premia between smokers and nonsmokers. In order to make the point clearer let us first think of this case. Let pS denote the insurance premium for smokers and pN denote that for non-smokers. 1. In order that a smoker buys the insurance his expected utility of buying must be at least as large as his expected utility of not buying, that is, it has to be met ln(w − pS ) ≥ (1 − πS ) ln w + πS ln(w − L) Here the left-hand-side is ln(w−pS ) because the insurance is a full-coverage one, which guarantees the insuree’s final income w − pS . On the other hand, in order that the insurance company sells the insurance its expected profit must be non-negative, that is, pS − πS L ≥ 0 has to be met. Hence the price range in which an insurance contract can be made between a smoker and the insurance company is w − w1−πS (w − L)πS ≥ pS ≥ πS L For notational simplicity, let w − w1−πS (w − L)πS = vS . 2. Likewise, the price range in which an insurance contract can be made between a non-smoker and the insurance company is w − w1−πN (w − L)πN ≥ pN ≥ πN L Again for notational simplicity, let w − w1−πN (w − L)πN = vN .

CHAPTER 25. TRADE WITH INCOMPLETE INFORMATION

331

Now let’s consider that the insurance company cannot observe or verify if a given buyer is smoking or not. Then the company has to face the insurance premium common across smoker and non-smokers, which is denoted by p. We can think of two cases. Case 1: When vN ≥ (λπS + (1 − λ)πN )L there are two kinds of equilibria. One is such that the insurance premium is in the range vN ≥ p ≥ (λπS + (1 − λ)πN )L and both of smokers and non-smokers can buy the insurance. Here both types are buying the insurance and the insurance company accepts the risk of drawing a smoker one with probability π, since the gambling is sufficiently attractive. The other equilibrium is such that the insurance premium is in the range vS ≥ p ≥ πS L and only smokers buy the insurance. There exist multiple equilibria here, in which both types are willing to buy the insurance since the premium is sufficiently cheap, and the insurance company accepts the risk taking since the proportion of the smokers is sufficiently low. Case 2: When vN < (λπS + (1 − λ)πN )L the only price range which can support equilibrium is vS ≥ p ≥ πS L in which only smokers buy the insurance, which is the adverse selection situation.

25.2

Moral hazard

Moral hazard refers to a situation in which one side of the market cannot observe actions taken by the other side of the market. It originated from insurance industry and refers to a situation in which insurees neglect to protect their assets when they are fully insured.

25.2.1

Insurance market

Consider a market for car insurance. Assume that the driver is risk-averse and his risk attitude is described by a vNM index v(z) = ln z for simplicity. The insurance company is assumed to be risk neutral. The driver’s initial income is denoted by w. If the driver drives carefully the accident probability is π1 , and otherwise it is π2 , where 0 < π1 < π2 < 1. If he hits an accident he loses L.

CHAPTER 25. TRADE WITH INCOMPLETE INFORMATION

332

Being careful is not for free, it needs certain cost. Such cost given by c > 0 in the monetary term. It doesn’t have much quantitative meaning. The whole point is that it is not free. Assume that c is positive but sufficiently small, π1 is positive but sufficiently small, π2 is sufficiently large but less than 1. Then as is discussed in the chapter on competitive market the efficient risk-sharing is such that the risk-neutral agent, the insurance company here, takes all the risk, and the risk-averse agent receives riskless consumption, where the driver drives carefully because the cost of doing to is sufficiently cheap. Thus, as a benchmark consider an insurance with complete coverage, which pays L if the driver has an accident. Now consider, however, that the insurance company cannot observe or verify if the driver was driving carefully. We assume the following. Assumption on information 1: Both parties know the above setting and numbers. 2: However, the insurance company cannot observe or verify if an individual driver was driving carefully. Here the insurance company cannot make the premium or payment conditional on if the insuree was driving carefully. Thus, consider the unconditional insurance premium p, where the insurance is assumed to be full-coverage paying L upon accident. 1. Suppose the driver is not insured. Then his expected utility when he drives carefully is (1 − π1 ) ln(w − c) + π1 ln(w − L − c) On the other hand his expected utility when he does not drive carefully is (1 − π2 ) ln w + π1 ln(w − L) Since π1 is sufficiently low and π2 is sufficiently high, the driver drives carefully when he is not insured. 2. Suppose the driver is insured. Then, given that the insurance is fullcoverage his expected utility when he drives carefully is ln(w − c − p) On the other hand, his expected utility when he does not drive carefully is ln(w − p) since the insurance is full-coverage. Thus it is a waste of effort for the drive to drive carefully under the full-coverage insurance (unless he dies of the accident).

CHAPTER 25. TRADE WITH INCOMPLETE INFORMATION

333

3. Hence the drive buys the insurance when ln(w − p) ≥ (1 − π1 ) ln(w − c) + π1 ln(w − L − c) By solving the above inequality we obtain p ≤ w − (w − c)1−π1 (w − L − c)π1 4. The insurance company knows that the driver neglects to drive carefully when he is insured. However, knowing something is different from being able to verify it, and the company cannot prove if the drive neglected to be careful for individual cases. Thus the company has pay when the driver claim for it after accident regardless of whether he was driving carefully or not. Hence the insurance company estimates the accident probability higher equal to π2 , and its expected profit is p − π2 L Hence the company sells the insurance when p ≥ π2 L 5. When π1 is sufficiently small π2 is sufficiently high we have w − (w − c)1−π1 (w − L − c)π1 < π2 L, implying that there does not exist an insurance premium which meets the above two conditions. Hence the insurance contract does not make. This shows that under moral hazard a full-coverage insurance is impossible, meaning that the insuree has to accept some risk. Now, how does the insurance contract look like in order that the insuree pays effort on careful driving? Denote the payout by R. 1. As before, the drive pays effort on careful driving when he is not insured, and his expected utility is then (1 − π1 ) ln(w − c) + π1 ln(w − L − c) 2. The condition such that the drive pays effort on careful driving (called an incentive constraint) is (1 − π1 ) ln(w − c − p) + π1 ln(w − L + R − c − p) ≥ (1 − π2 ) ln(w − p) + π2 ln(w − L + R − p) This condition cannot be met unless R
CHAPTER 25. TRADE WITH INCOMPLETE INFORMATION

334

3. The condition that the drives chooses to buy the insurance (called a participation constraint) is (1 − π1 ) ln(w − c − p) + π1 ln(w − L + R − c − p) ≥ (1 − π1 ) ln(w − c) + π1 ln(w − L − c) 4. Given that the above constraints are met the insurance company can assume that the driver pays effort and the accident probability is π1 , hence the condition for the company to sell the insurance is p − π1 R ≥ 0

25.2.2

Reward contract

Consider that there is one employer and one employee. There is a project such that its outcome (revenue) is affected by the level of effort by the employee, in the sense that the probability of good outcome is higher (but not 1) when he pays effort and low (but not 0) when he does not pay effort. Assume that the employee is risk-averse and his risk attitude is described by a vNM index v(z) where z denotes final income. The employer is assumed to be risk neutral. If the employee pays effort the probability of good outcome is πH , and otherwise it is πL , where 0 < πL < πH < 1. If the outcome is good the employer receives revenue RG and if it is bad he receives RB . Effort is not for free. The cost of effort given by c > 0 in the monetary term. Assume that c is positive but sufficiently small compared to its effect on expected increase of revenue, πH is sufficiently large but not 1, πL is sufficiently small but not 0. Then as is discussed in the chapter on competitive market the efficient risk-sharing is such that the risk-neutral agent, the employer here, bears all the risk, and the risk-averse agent receives riskless reward, where the employee pays effor because the cost of doing to is sufficiently cheap. However, the riskless reward is impossible under moral hazard. The employer cannot observe or verify how the employee is working. Then, 1. Since the level of effort is not observable or verifiable the employer cannot make pay according to the employee’s effort. 2. On the other hand, when a fixed reward payment is made regardless of outcome the employee will not pay effort. When there is even a small uncertainty about outcomes in particular the employer cannot identify the employee’s effort level from the outcome. One can always say, ”I tired hard but unfortunately the outcome wasn’t good.” In such a situation the employer has condition the reward payment on the outcome.

CHAPTER 25. TRADE WITH INCOMPLETE INFORMATION

335

Let wG denote the reward payment for good outcome and wB for bad outcome. Then the incentive condition for the employee to pay effort is πH v(wG − c) + (1 − πH )v(wB − c) ≥ πL v(wG ) + (1 − πL )v(wB ) Let w be the income which the employee receives when he does not sign the contract and works elsewhere. Then the participation condition so that the employee signs the contract by his choice is πH v(wG − c) + (1 − πH )v(wB − c) ≥ v(w)

25.2.3

The principal-agent problem

Now consider that the insurance company or employer try to design a contract in order to maximize the expected profit. This is called a principal-agent problem, where the principal offers a contract to the agent but cannot monitor or verify the agent’s action. Typical examples are that the principals are an insurance company, an employer and a shareholder of a firm, and the agents are a driver, an employee and a manager, respectively. In general it is not clear if inducing the agent to pay effort is the best thing, since it might be too expensive to do so and it might be better to get content with lower effort of the agent. Hence the problem takes the following form. Let e denote the agent’s effort lever, which is assumed here for simplicity either H or L. Let ce denote the cost of making effort level e, which is measure in terms of income. There are two possible outcomes, good or bad. Let RG denote the principal’s revenue when the outcome is good, and RB be the revenue when the outcome is bad. Let πe be the probability of good outcome when the agent’s effort level is e. The principal is risk-neutral and cares for the expected profit. The agent is risk-averse and his risk attitude is described by the vNM index v(z), where z denotes the final income. Also, let v denote the level of vNM index given to the agent when he does not sign the contract and goes somewhere else. Then the principal-agent problem is formulated as max πe (RG − wH ) + (1 − πe )(RB − wB )

wG ,wB ,e

subject to πe v(wG − ce ) + (1 − πe )v(wB − ce )

≥ πe′ v(wG − ce′ ) + (1 − πe′ )v(wB − ce′ )

πe v(wG − ce ) + (1 − πe )v(wB − ce )

for all e′ = H, L ≥ v

where the first constraint is the incentive constraint and the second is the participation constraint.

25.3

Signaling

Go back to the adverse selection problem.

CHAPTER 25. TRADE WITH INCOMPLETE INFORMATION

25.3.1

336

Education as a signaling

Consider two types of workers, high and low in terms of productivity. Let a1 denote the marginal productivity of a high-type worker and a2 denote that of a low-type, where a1 > a2 . The proportion of high is known to be π. Hence the prior expectation of worker’s expected marginal productivity is πa1 + (1 − π)a2 . When the labor market is perfectly competitive and if the employers can observe and verify individual workers’ types the high-type worker is paid a1 and the low-type is paid a2 . However, when the employers cannot observe or verify individual workers’ types we run into the adverse selection problem. Now consider that the workers can send signals to the employers. Here let us consider an academic degree of a given level for the signal. Receiving the signal from a given employer, the employers pay the conditional expectation of his productivity. Let c1 be the cost acquiring the degree for the high type which is measure in income, and and that c2 be that for the low type, where c1 < c2 . This means that it is easier for the high type to acquire the degree. Here the cost is not just direct monetary cost, but rather about how painful it is. The assumption c1 < c2 says that it is relatively more painful for the low type to acquire the degree. We can think of three cases Case 1: Consider that only the high type acquires the degree. This is called a separating equilibrium. Then the employers know that the type of a given worker is high if he as the degree, and low otherwise. Hence they pay a1 to him if he has the degree and a2 otherwise. In order that this is the case it has to be optimal for the high-type worker to acquire the degree and it has to be optimal for the low-type worker not to acquire the degree. The first condition is a1 − c1 ≥ a2 , where the left-hand-side is the net gain of the high-type worker when he acquires the degree and the right-hand-side his net gain when he doesn’t. The second condition is a1 − c2 < a2 , where the left-hand-side is the net gain of the low-type worker when he acquires the degree and the right-hand-side his net gain when he doesn’t. Summing up, the separating equilibrium occurs when c1 ≤ a1 − a2 < c2 . Case 2: Consider that both types acquire the degree. This is called a pooling equilibrium. Then the employers receives no new information when they receive the signal, hence the expected marginal productivity of any individual worker who has the degree is πa1 + (1 − π)a2 .

CHAPTER 25. TRADE WITH INCOMPLETE INFORMATION

337

A problem occurs when they see a worker who does not have the degree. Since all the workers acquire the degree here, the existence of a worker without the degree is supposed to be ”impossible.” What would the employers believe when an impossible thing had happened? Since the standard Bayes rule does not apply here, their belief after seeing the impossible is taken to be another parameter. Let µ denote the employers’ belief that a worker without the degree is high type. Then the expected marginal productivity is µa1 + (1 − µ)a2 . Here it has to be the case that it is optimal for both types to acquire the degree. For the high type the required condition is πa1 + (1 − π)a2 − c1 ≥ µa1 + (1 − µ)a2 where the left-hand-side is the net gain of the high-type worker when he acquires the degree and the right-hand-side his net gain when he doesn’t. For the low type the required condition is πa1 + (1 − π)a2 − c2 ≥ µa1 + (1 − µ)a2 where the left-hand-side is the net gain of the low-type worker when he acquires the degree and the right-hand-side his net gain when he doesn’t. From the above two inequalities the pooling equilibrium to occurs when µ≤π−

c1 a1 − a2

and

c2 . a1 − a2 Since c1 < c2 the condition reduces to µ≤π−

µ≤π−

c2 . a1 − a2

The condition says that when the employers see a worker without the degree (which is impossible) they believe that the worker is less likely to be the high type.1 Case 3: Consider that neither type acquires the degree. This is called an another kind of pooling equilibrium. Then the employers receives no new information when they receive the signal of no degree, hence the expected marginal productivity of any individual worker who does not have the degree is πa1 + (1 − π)a2 . A problem occurs when they see a worker who does have the degree. Since no worker acquires the degree here, the existence of a worker having the degree is supposed to be”impossible.” Again, what would the employers believe when an impossible thing had happened? 1 Technically,

such equilibrium notion is called perfect Bayesian equilibrium.

CHAPTER 25. TRADE WITH INCOMPLETE INFORMATION

338

Since the standard Bayes rule does not apply here either, their belief after seeing the impossible is taken to be another parameter again. Let ν denote the employers’ belief that a worker having the degree is high type. Then the expected marginal productivity is νa1 + (1 − ν)a2 . Here it has to be the case that it is optimal for both types not to acquire the degree. For the high type the required condition is νa1 + (1 − ν)a2 − c1 < πa1 + (1 − π)a2 where the left-hand-side is the net gain of the high-type worker when he acquires the degree and the right-hand-side his net gain when he doesn’t. For the low type the required condition is νa1 + (1 − ν)a2 − c2 < πa1 + (1 − π)a2 where the left-hand-side is the net gain of the low-type worker when he acquires the degree and the right-hand-side his net gain when he doesn’t. From the above two inequalities the pooling equilibrium to occurs when ν <π+

c1 a1 − a2

ν <π+

c2 . a1 − a2

and

Since c1 < c2 the condition reduces to ν ≤π+

c1 . a1 − a2

The condition says that when the employers see a worker having the degree (which is impossible) they believe that the worker is less likely to be the high type. Notice that here the effort paid for the signaling purpose has nothing to do with enhancing productivity. It is a waste. In Japan, a new university graduate who seeks a job at an entry level is often supposed to submit his CV in handwriting. No correction marker is allowed. Here the employers are testing if a job candidate is ”crazy” enough (and revealing that they are wanting such ”crazy” workers). Of course I’m not saying that education is totally a waste of resources. But I would say education certainly has such an aspect.

25.4

Speculative trade

So far we have seen that information incompleteness prevents trades. Is it possible that incompleteness rather promotes trade?

CHAPTER 25. TRADE WITH INCOMPLETE INFORMATION

339

Under complete information, trades occur because of distributions of preferences and initial holdings, such as I have what you like and you have what I like. Now, consider that such factors have been eliminated and ask if trades can happen solely because of difference of information. That is, ask if speculative trades are possible. It sounds easy. When I receive information that the price of some stock is likely go up and you receive information that it is likely to go down, I might want to buy and you might want to sell. However, if I am ”rational” in the sense that I am ”not stubborn” and do ”not have baseless confidence,” I should get surprised when I encounter a trader who is willing to sell the stock: ”why is this guy willing to sell?” And if I am ”rational” in the sense that I can ”process information in the logically correct manner,” I should update my prediction about the stock price based on the probability theory, taking the emergence of such seller (you) into account. Similarly for you. If you are ”rational” in the sense that you are ”not stubborn” and do ”not have baseless confidence,” you should get surprised when you encounter a trader who is willing to buy the stock: ”why is this guy willing to buy?” And if you are ”rational” in the sense that you can ”process information in the logically correct manner,” you should update you prediction about the stock price based on the probability theory, taking the emergence of such buyer (me) into account. Can such ”rational” individuals ”agree to disagree” so that they trade? Let me explain using the following example, which is an economic version of so-called ”muddy children’s puzzle.” First let me explain the muddy children’s puzzle. Two children are playing in a muddy playground. Their faces may become muddy, but each child cannot see if his face is muddy or not, while he can see if the other child’s face is muddy or not. Now let’s say that both children’s faces are muddy. Here comes the teacher and said, ”At least one of your faces is muddy. If you see that your face is muddy go to the bathroom immediately, and otherwise stay here.” The two children stared at each other for a while, and as soon as each confirmed that the opponent was staying they immediately went to the bathroom. What happened here? When they are told by the teacher that at least one of their faces is muddy they cannot figure out if just one of them is muddy or both are muddy. So each child cannot see if the opponent’s face alone is muddy or his face is muddy as well. Thus each of them stays. However, once each of them sees that the opponent is staying he reasons, ”if my face is not muddy the opponent would have gone to the bathroom immediately, since he would have realized that his face alone is muddy when the teacher said that at least one of our faces is muddy.”

CHAPTER 25. TRADE WITH INCOMPLETE INFORMATION

340

An economic version of the above is as follows, which is an adaptation of the argument by Geanakoplos and Sebenius [8]. Example 25.1 A is holding a stock and B is a potential buyer. A receives information about the price of the stock, which is either of P1 , P2 , P3 , with equal probability. B receives information about the price of the stock, which is either of Q1 , Q2 , Q3 , with equal probability. Assume that the probability distributions of informations to be received by A and B respectively are independent. Also, A cannot observe what information B has received and B cannot observe what information A has received. The table below describes how the received informations are related to the stock price. P1 P2 P3

Q1 0.7 0.3 0.8

Q2 0.6 0.55 0.1

Q3 0.4 0.6 0.45

Here let’s say the number in the cell (P1 , Q1 ) shows that the stock price rises with probability 0.7 and falls with probability 0.3 if P1 and Q1 are true. Similarly for the other combinations. Assume that both traders are risk neutral, without loss of generality. Then A likes to hold the stock if the probability of rise is greater than 0.5 and likes to sell if it below 0.5. B likes to buy the stock if the probability of rise is greater than 0.5 and likes not to if it below 0.5. Now suppose A has received P2 and B has received Q3 . Now there is a mediator who asks the two questions in the following procedure. Each trader’s response is observed by the opponent. Step 1: Ask A, ”Would you like to sell?” If the response is NO, stop. If the response is YES, go to Step 2. Step 2: Ask B, ”Would you like to buy?” If the response is NO, stop. If the response is YES, go to Step 3. Step 3: Ask A, ”Would you still like to sell?” If the response is NO, stop. If the response is YES, go to Step 4. Step 4: Ask B, ”Would you still like to buy?” If the response is NO, stop. If the response is YES, go to Step 5. Repeat as far as they say YES. As you repeat this, after finite rounds either of them says NO. Why? Let us see this step by step. Initially, A knows P2 is true but he does not know what B knows. Also B knows Q3 but does not know what A knows. Hence knowledge commonly known to them is only a trivial one, ”either P1 or P2 or P3 is true, and Q1 or Q2 or Q3 is true.” Let me denote this

CHAPTER 25. TRADE WITH INCOMPLETE INFORMATION

341

trivial knowledge by CK0 = {P1 , P2 , P3 } × {Q1 , Q2 , Q3 }, where CK0 stands for common knowledge at the initial point. In Step 1, since A knows that P2 is true but knows only that either Q1 or Q2 or Q3 is true with equal probability, the probability of price rise conditional on his current information is P rob(up|{P2 } × {Q1 , Q2 , Q3 }) =

1.45 0.3 + 0.55 + 0.6 = < 0.5 3 3

Since A believes the stock price is likely to fall he says YES. Observing this response, B knows that P1 is not true. For, if P1 were true A would have said NO, because the conditional probability of price rise is then P rob(up|{P1 } × {Q1 , Q2 , Q3 }) =

0.7 + 0.6 + 0.4 1.7 = > 0.5. 3 3

But A said YES. Therefore B knows P1 is not true. After Step 1 it is known to both A and B that P1 is not true, the common knowledge is updated into CK1 = {P2 , P3 } × {Q1 , Q2 , Q3 }. In Step 2, since B knows that Q3 is true but knows only that either P2 or P3 is true with equal probability, the probability of price rise conditional on his current information is P rob(up|{P2 , P3 } × {Q3 }) =

0.6 + 0.45 1.05 = > 0.5 2 2

Since B believes the stock price is likely to rise he says YES. Observing this response, A knows that Q2 is not true. For, if Q2 were true B would have said NO, because the conditional probability of price rise is then 0.55 + 0.1 0.65 = < 0.5 2 2

P rob(up|{P2 , P3 } × {Q2 }}) =

But B said YES. Therefore A knows Q2 is not true. After Step 2 it is known to both A and B that Q2 is not true, the common knowledge is updated into CK2 = {P2 , P3 } × {Q1 , Q3 }. In Step 3, since A knows that P2 is true but knows only that either Q1 or Q3 is true with equal probability, the probability of price rise conditional on his current information is P rob(up|{P2 } × {Q1 , Q3 }) =

0.3 + 0.6 0.9 = < 0.5 2 2

Since A still believes the stock price is likely to fall he says YES. Observing this response, B knows that P3 is not true. For, if P3 were true A would have said NO, because the conditional probability of price rise is then P rob(up|{P3 } × {Q1 , Q3 }) =

1.25 0.8 + 0.45 = > 0.5 2 2

But A said YES. Therefore B knows P3 is not true.

CHAPTER 25. TRADE WITH INCOMPLETE INFORMATION

342

After Step 3 it is known to both A and B that P3 is not true, the common knowledge is updated into CK3 = {P2 } × {Q1 , Q3 }. In Step 4, Since B knows that Q3 is true and also that P2 is true, the probability of price rise conditional on his current information is P rob(up|{P2 } × {Q3 }) = 0.6 > 0.5 Since B knows the stock price is likely to rise he says YES. Observing this response, A knows that Q1 is not true. For, if Q1 were true B would have said NO, because the conditional probability of price rise is then P rob(up|{P2 } × {Q1 }}) = 0.3 < 0.5 But B said YES. Therefore A knows Q1 is not true. After Step 4 it is known to both A and B that Q1 is not true, the common knowledge is updated into CK4 = {P2 } × {Q3 }. Now both know the truth. Now in Step 5, since A knows that P2 is true and that Q3 is true, the probability of price rise conditional on his current information is P rob(up|{P2 } × {Q3 }) = 0.6 > 0.5 Since A knows the stock price is likely to rise he says NO. Thus, it is impossible for ”rational” agents to ”agree to disagree.” You may have a concern about the assumption here that each agent responds YES or NO sincerely at each step, while in a more realistic trading situation they may lie and also the trade makes when both say YES simultaneously. It is known, however, that it is impossible for them to agree to disagree even under such situations. See Geanakoplos [7] for more details. This result relies on the assumption of common-prior, which in the current example says that both A and B know that the probability distributions over P1 , P2 , P3 and Q1 , Q2 , Q3 are uniform and independent. As I discussed in the chapter of games with incomplete information, however, we cannot just drop or relax it.

25.5

Exercises

Exercise 35 There are 100 people who want to sell used cars and 100 people who want to buy a used car. Assume they are risk-neutral. 75 cars are ‘plum’ and 25 cars are ‘lemon.’ The owners of a plum is willing to part with it for 4000. The owners of a lemon is willing to part with it for 1000. The buyers are willing to pay 5000 for a plum and 1500 for a lemon. When the buyers cannot verify whether a given car is a plum or a lemon, they have to accept a common price. Describe all the trading patterns that can arise in this setting.

CHAPTER 25. TRADE WITH INCOMPLETE INFORMATION

343

Exercise 36 There is one employer and one employee. The employee has two choices of efforts, high or low. When she makes high effort, it generates a good outcome with probability 0.95 and a bad outcome with probability 0.05. When she makes low effort, it generates a good outcome with probability 0.2 and a bad outcome with probability 0.8. The cost of high effort for the employee is 3, and the cost of low effort is 0. Assume that the employee is risk-neutral. The employer cannot observe or verify whether the employee made high effort or not, although she knows the cost of high effort for the employee. She can pay different wages for different outcomes. Denote the wage for good outcome by wG , the wage for bad outcome by wB . To induce the employee make high effort, what condition do wG and wB have to satisfy? Exercise 37 A is holding a stock. B is a potential buyer of the stock. A receives some news that may be related to the stock price. She receives either of P1 , P2 , P3 , with equal probability. B receives another type of news that may be related to the stock price. She receives either of Q1 , Q2 , with equal probability. The probability distributions of the news they receive respectively are independent. Also, A cannot see what B receives and B cannot see what A receives. The table below shows how the stock price is related to the news. P1 P2 P3

Q1 0.55 0.3 0.6

Q2 0.4 0.9 0.3

For example, the number in the cell (P1 , Q2 ) tells that if you know both P3 and Q2 then you know that the stock price goes up with probability 0.3 and goes down with probability 0.7. Assume that both individuals are risk neutral. That is, A wants to sell the stock if the probability of its price going up is smaller than 0.5 based on what she knows (and does not want to sell if it is greater than 0.5). B wants to buy the stock if the probability of its price going up is greater than 0.5 based on what she knows (and does not want to buy if it is smaller than 0.5). Now suppose that A receives P1 and B receives Q1 . You successively ask questions to the two individuals in the following way. Every answer is immediately observable to the opponent. Step 1: You ask A, ‘would you like to sell the stock?’ Step 2: If A says Yes in Step 1, you ask B, ‘would you like to buy the stock?’ Step 3: If B says Yes in Step 2, you ask A, ‘would you still like to sell the stock?’ Step 4: If A says Yes in Step 3, you ask B, ‘would you still like to buy the stock?’ And so on. You continue this until one of them says No. Show that after finite rounds one of them ends up with saying No. Describe the logical process of how.

Part V

Market Failure and Normative Economic Analysis

344

Chapter 26

Externality Consumption or production activity is said to have an externality if its effect is not taken into account in the determination of market price. For example, when there is a consumption good such that its consumption harms or annoys others the consumer himself would not take that into account, the others cannot stop it unless they have a legal right to do so, hence the price of the consumption good does not take the harm or annoyance into account. For example, even if we know that consumption activity harms ourselves through transmission of CO2 each consumer need not take this into account in doing his consumption activity and we cannot to force each consumer to do so either. Or, even when there is a production activity which causes social costs (such as pollution) the corresponding firm does not take it into account in its profit maximization decision, and the other economic agents cannot stop it unless they have a legal right to do so, hence hence the prices of the output and inputs do not take such social costs into account. As a result, resource allocation in the market may be inefficient even when it is perfectly competitive. This is called market failure. What is important in the above definition is that market prices does not take it into account. Even when an activity directly affects other economic agents it is not called an externality when it is priced and traded in markets, such as service.

26.1

Market failure

To illustrate I will focus on the case that one’s consumption activity causes to another one. Externality between firms and between firms and consumers can be dealt in a similar way. Also for the illustration purpose let me assume the quasi-linear environment, in which Good 1 is the consumption good which causes externality and Good 2 is income transfer to be spent on the other goods. Each consumer i cares not only about his consumption of Good 1 xi1 and his income transfer xi2 but also others’ 345

CHAPTER 26. EXTERNALITY

346

consumptions of Good 1 denoted by x−i1 = (x11 , · · · , xi−1,1 , xi+1,1 , · · · , xin1 ). Let us consider for simplicity again that his preference is assumed to be represented in the form   ∑ xj1  + xi2 ui (xi1 , x−i1 , xi2 ) = vi (xi1 ) + ei  j̸=i

where v(i (xi1 ) is i’s ) benefit from his own consumption xi1 , which he can control, ∑ and ei j̸=i xj1 is the benefit (or cost if it is negative) caused by the others’ consumptions of Good 1. Let me call the first one internal benefit and the latter external benefit (or external cost if negative). Remark 26.1 One might think that a situation in which externality matters is typically the situation in which income effect does matter. True, but here I like to extract the problem of externality alone, and the assumption of no income effect is just for the convenience of doing it. Denote the relative price of Good 1 for Good 2 (income transfer) by p, then each consumer i solves   ∑ max vi (xi1 ) + ei  xj1  − pxi1 xi1

j̸=i

Note that in this individual choice others’ consumptions (xj1 )j̸=i are taken as given. That is, the above maximization problem is equivalent to the maximization problem in which only the internal benefit is taken into account: max vi (xi1 ) − pxi1 xi1

Therefore condition for optimal consumption in which only the internal benefit is taken into account is vi′ (xi1 ) = p By solving this we obtain i’s demand xi1 (p). On the other had, the representative firm solves its profit maximization problem max py − C(y) y

Its maximization condition is p = M C(y) as before. By solving this we obtain the supply y(p). In a competitive market the price of output is determined so that demand matches supply. That is, the competitive equilibrium price p∗ is such that n ∑ i=1

xi1 (p∗ ) = y(p∗ )

CHAPTER 26. EXTERNALITY

347

Let (x∗11 , x∗n1 ) denote the allocation of Good 1 in the competitive equilibrium. Then, since internal marginal benefit of each individual and marginal cost are all equal, that is, since it holds   n ∑ p∗ = vi′ (x∗i1 ) = M C  x∗j1  j=1

for all i = 1, · · · , n, the allocation is maximizing the difference between the sum of internal benefits and the producer surplus ( n ) n ∑ ∑ vi (xi1 ) − C xi1 i=1

i=1

This is not Pareto-efficient in general, since it does not maximize the social surplus    ( n ) n   ∑ ∑ ∑ xi1 vi1 (xi1 ) + ei  xj1  − C   i=1

i=1

j̸=i

which takes external effects into account. Let (b x11 , · · · , x bn1 ) denote the allocation of Good 1 which maximizes the social surplus, then it satisfies     n ∑ ∑ x bj1  x bj1  = M C  vi′ (b xi1 ) + e′i  j̸=i

j=1

When e′i is positive for i, that is, when there is positive externality ∑each n bi1 of ∑ consumptions (b x11 , · · · , x bn1 ) which maxiof consumptions, the sum i=1 x n mize the social surplus is greater than i=1 x∗i1 what is given by the competitive equilibrium. This means that consumption in competitive equilibrium is too little compared to the Pareto efficient level. On the other hand, when e′i is negative for each ∑n i, that is, when there is negative externality of consumptions, the sum bi1 of ∑ consumptions i=1 x n (b x11 , · · · , x bn1 ) which maximize the social surplus is smaller than i=1 x∗i1 what is given by the competitive equilibrium. This means that consumption in competitive equilibrium is too much compared to the Pareto efficient level.

26.2

Solutions

The market failure as illustrated above suggests that some intervention is necessary. To illustrate, let me further simplify the setting as below. There are two consumers, A and B. A prefers consumption of Good 1, but B dislikes it. Also, B dislikes that A consumes Good 1, while A does not care about

CHAPTER 26. EXTERNALITY

348

B’s consumption of Good 1. Without loss of generality, let me assume B never consumes Good 1 by himself. Thus, preferences of A and B are represented by uA (xA1 , xA2 ) = uB (xA1 , xB2 ) =

vA (xA1 ) + xA2 −lB (xA1 ) + xB2

Here vA denotes A’s internal benefit from his own consumption of Good 1, which ′ > 0. On the other hand, lB denotes B’s loss due to is positive and satisfies vA ′ (xA1 ) > 0. A’s consumption of Good 1, where lB ≥ 0 and lB First let us look at the outcome of competitive market. Since B never consumes Good 1 we only look at A’s consumption. A’s individually optimal consumption solves max vA (xA1 ) − pxA1 xA1

The individual optimal condition is then v ′ (xA1 ) = p. By solving this we obtain A’s demand xA1 (p). The representative firm as before solves max py − C(y) y

and its profit maximization condition is p = M C(y) as before. By solving this we obtain the supply function y(p). The competitive equilibrium price p∗ is determined by xA1 (p∗ ) = y(p∗ ). Here A’s consumption x∗A1 maximizes vA1 (xA1 ) − C(xA1 ) in which only A’s internal benefit is taken into account, but does not maximize the social surplus vA (xA1 ) − lB (xA1 ) − C(xA1 ) in which B’s loss is taken into account as well.

26.2.1

Rationing

First thing we can think of as the government’s policy is to maximize the social surplus max {vA (xA1 ) − lB (xA1 ) − C(xA1 )} xA1

directly and enforce the solution x bA1 . You will see that it is hard. The government needs to know A’s benefit vA and B’s lB as well as the production cost C. It is hard to know the first two, in particular, as A will try to overstate the benefit and B will try to overstate the loss. Also, it is quite politically infeasible that the government enforces the levels of individual consumptions.

CHAPTER 26. EXTERNALITY

26.2.2

349

Pigovian tax

Taxation will be a more realistic policy. The idea is due to Pigou. Here the tax rate which supports surplus maximization is B’s marginal loss at the surplus-maximizing point. Again let x bA1 be the surplus-maximizing level of A’s consumption of Good 1. Then consider tax rate t given by ′ (b xA1 ) t = lB

Given t, A solves his optimization problem max vA (xA1 ) − (p + t)xA1 xA1

and the optimization condition is now ′ vA (xA1 ) = p + t

Producer’s profit maximization condition is M C(y) = p as before. The representative firm as before solves max py − C(y) y

and its profit maximization condition is p = M C(y) as before. By solving this we obtain the supply function y(p). In competitive equilibrium under taxation x∗A1 with equilibrium price de′ noted by p∗ , since we have vA (x∗A1 ) = p∗ + t and M C(x∗A1 ) = p∗ it holds v ′ (x∗A1 ) − M C(x∗A1 ) = t Since this means

′ v ′ (x∗A1 ) − M C(x∗A1 ) = lB (b xA1 )

the equilibrium consumption x∗A1 coincides with x bA1 which maximizes the social surplus. However, in order to set the right tax rate (if you want to do it completely precisely) we need to know vA , lB and C again.

26.2.3

Internalization of externality: crating a right and trading it

Another idea is to create a ”right” to consume the good or a ”right” to stop the consumption and allow people to trade it. Case 1: When B is given the right First, let us consider the case that the government creates a ”right to consume the good” and gives it to B. Then A can consume one unit of the good only by buying one unit of the right and exercising it. A does not have to exercise

CHAPTER 26. EXTERNALITY

350

the right which he has bought from B, but here it is innocuous to assume he does. Let r denote the price of one unit of the right and p denote the price of the good. Let us look how many units of the right A demands. A’s demand for the right zA is determined by solving max vA (zA ) − rzA − pzA zA

where rzA is the cost of purchasing the right and pzA is the cost of exercising the right. Then the optimization condition is given by ′ vA (zA ) = p + r

By solving this we obtain A’s demand for the right zA (p, r), which is also A’s demand for the good in the output market. On the other hand, B’s supply of the right zB is determined by solving max −lB (zB ) + rzB zB

where rzB is the revenue from selling the right. Then the optimization condition is given by ′ lB (zB ) = r By solving this we obtain B’s supply of the demand zB (r). The representative firm as before solves max py − C(y) y

and its profit maximization condition is p = M C(y) as before. By solving this we obtain the supply function y(p). Here the competitive equilibrium condition consists of two equations, one that demand matches supply in the market for the right and the other that demand matches supply in the market for the right. Hence the equilibrium price vector (p∗ , r∗ ) is such that zA (p∗ , r∗ ) = zA (p∗ , r∗ ) =

zB (r∗ ) y(p∗ )

Let z ∗ denote the amount of the right traded in the equilibrium, which is also the amount of the good traded, then it satisfies ′ v ′ (z ∗ ) = p∗ + r∗ , lB (z ∗ ), M C(z ∗ ) = p∗

implying

v ′ (z ∗ ) − l′ (z ∗ ) − M C(z ∗ ) = 0

Now we see that z ∗ is the efficient level. Here A pays r∗ z ∗ units of income to B.

CHAPTER 26. EXTERNALITY

351

Case 2: When A is given the right Next, let us think of the case that the government creates a ”right to stop the consumption of the good” and gives it to A. Then B can stop one unit of A’s consumption the good only by buying one unit of the right from A and exercising it. B does not have to exercise the right which he has bought from A, but here it is innocuous to assume he does. Let w denote the price of one unit of the right and p denote the price of the good. Also, let x denote the amount of the good which A consumes when noting stops it. Let us look at A’s supply of the right denoted by sA . It is determined by solving max vA (x − sA ) + wsA − p(x − sA ) sA

where wsA is the revenue of selling the right and p(x − sA ) is the cost of consumption. Then the optimization condition is given by ′ vA (x − sA ) = w + p

By solving this we obtain A’s supply of the right sA (p, w) and A’s demand for the good x − sA (p, w). On the other hand, B’s demand for the right sB is determined by solving max −lB (x − sB ) − wsB sB

where wsB is the cost of purchasing the right to stop A’s consumption of the good. Then the optimization condition is ′ (x − sB ) = w lB

By solving this we obtain B’s demand for the right sB (w). The representative firm as before solves max py − C(y) y

and its profit maximization condition is p = M C(y) as before. By solving this we obtain the supply function y(p). Here the competitive equilibrium condition consists of two equations, one that demand matches supply in the market for the right and the other that demand matches supply in the market for the right. Hence the equilibrium price vector (p∗ , w∗ ) is such that sB (w∗ ) = x − sA (p∗ , w∗ ) =

sA (p∗ , w∗ ) y(p∗ )

Denote the amount of the right traded in the equilibrium by s∗ , then it satisfies ′ v ′ (x − s∗ ) = p∗ + w∗ , lB (x − s∗ ), M C(x − s∗ ) = p∗

This implies

v ′ (x − s∗ ) − l′ (x − s∗ ) − M C(x − s∗ ) = 0

CHAPTER 26. EXTERNALITY

352

and we see that x − s∗ is in the efficient level. Hence the amounts of consumption the good are the same in the two cases, z ∗ = x − s∗ = x bA1 and it does not depend on to which side a right is given. This result is called Coase’s theorem. This should not be interpreted that economic consequences are the same regardless of which side is given a right, since the distributions of the maximized social surplus are different. In case 1, A is paying r∗ z ∗ units of income to B. In case 2, on the other hand, B is paying w∗ s∗ units of income to A. Even though who should be given a right is neutral to efficient allocation of the good (Good 1 which is under the analysis), it is not neutral when distribution of income (Good 2 in the current illustration). Also, we should note that the ”neutrality” result depends heavily on the assumption of no income effect as well as the assumption of frictionless markets.

Chapter 27

Public goods and the free-rider problem 27.1

Public goods

A good with the following two properties are called a public good 1. Non-rivalry: One’s using it does not prevent anybody from using it. For example, one’s using a road does not prevent anybody from using it (here we abstract away the case of congestion). On the other hand, when one uses or consumes a private good in the standard sense nobody else cannot use or consume the identical object. 2. Non-excludability: Everybody can use it and we cannot exclude any person from using it. For example, everybody can use an open road and cannot be exclude for the reason that he is not paying for it. On the other hand, toll road meets the condition of non-rivalry but fails to meet non-excludability, since it has a toll gate so that one who does not pay for it cannot enter. Some adopt the definition only with non-rivalry, but in this chapter I focus on public goods with both non-rivalry and non-excludability.

27.2

Efficiency criterion: the Samuelson condition

To illustrate, consider that there one private good and one public good. Denote the quantity of the public good by g, and consumer i’s private consumption by xi . Then consumer i’s consumption space is the non-negative quadrant of the two-dimensional space R2+ , in which the public good is taken to be Good 1 and the private good is taken to be Good 2. Let (g, xi ) denote a combination of the public good and the private good for consumer i. 353

CHAPTER 27. PUBLIC GOODS

354

Let ≿i denote i’s preference over pair of the public good and the private good. For example when it holds (g, xi ) ≿i (g ′ , x′i ) it means that i weakly prefers (g, xi ) to (g ′ , x′i ). Let ui (g, xi ) denote a utility representation of i’s preference ≿i . Here, let us consider how much of the private good each consumer is willing to sacrifice in order to increase one extra unit of the public good. It is given by the slope of a given indifference curve, where Good 1 is taken to be the public good and Good 2 is taken to be the private good. Given that current combination is (g, xi ), let ∆xi denote the amount of the private good which consumer i is willing to change in order to increase one extra extra ∆g units of the public good. Then the amount of the private good he is willing to give up in order to increase one extra unit of the public good is approximately ∆xi ∆g where the sign of absolute value is put because ∆xi is negative. As we make ∆g tend to zero, we obtain marginal rate of substitution of the private good for the public good at (g, xi ) dxi M RSi (g, xi ) = dg which corresponds to the slope of the tangent line to the given indifference curve at (g, xi ). As we did for the case of two private goods, marginal rate of substitution is described as the ratio between marginal utilities (it is the same as before that (g,xi ) denote the marginal utilities themselves have no economic content). Let ∂ui∂g (g,xi ) marginal utility of the public good at (g, xi ) and let ∂ui∂x denote the marginal i utility of the private good at (g, xi ), then the marginal rate of substitution of the private good for the public good is given by

M RSi (g, xi ) =

∂ui (g,xi ) ∂g ∂ui (g,xi ) ∂xi

Next let us describe the production technology, which is summarized in the form of cost function. Let C(g) denote the amount of the private good needed in order to produce g units of the public good. Denote the marginal cost by M C(g), and assume that is is increasing. Finally, let ei denote i’s initial holding of the private good. Then a feasible allocation (g, x1 , · · · , xn ) must satisfy n ∑ i=1

xi + C(g) =

n ∑ i=1

ei

CHAPTER 27. PUBLIC GOODS

355

Let us consider Pareto-efficient allocation of the public good and the private good. First let us define efficiency. An allocation (g ′ , x′1 , · · · , x′n ) is said to a Pareto improvement of an allocation (g, x1 , · · · , xn ) if (g ′ , x′i ) ≿i (g, xi ) holds for all i and (g ′ , x′i ) ≻i (g, xi ) holds for at least one i A feasible allocation (g, x1 , · · · , xn )is said to be Pareto efficient if no feasible allocation can be a Pareto improvement of it. The following result states that efficiency requires the sum of willingness to sacrifice the private good across individuals equals to the marginal cost. Theorem 27.1 An (interior) allocation (g, x1 , · · · , xn ) is Pareto efficient if and only if n ∑ M RSi (g, xi ) = M C(g) i=1

Proof. ”Only part: Suppose the equality does not hold. ∑if” n Suppose i=1 M RSi (g, xi ) > M C(g). This is the case that more public good can be produced at cheaper cost. Since the cost function is locally linear the cost of producing extra ∆g units of the public good is M C(g)∆g. Let ∆xi denote the amount of private good to be sacrificed by each i, then we must have ∑ n i=1 ∆xi = M C(g)∆g. From the above inequality we can divide M C(g)∆g so that ∆xi < M RSi (g, xi )∆g holds for all i. Since the amount one needs to sacrifice is smaller than the amount he is willing to sacrifice, each i prefers (g + ∆g, xi − ∆xi ) to (g, xi ). Hence (g + ∆g, x1 − ∆x1 , · · · , xn − ∆xn ) is a Pareto-improvement of (g, x1 , · · · , xn ). ∑n Suppose i=1 M RSi (g, xi ) < M C(g) on the other hand. This is the case that the current level of public good is too costly. Hence by reducing the level of the public good and suitably distributing the dispensed cost back to the consumers we can make everybody better off. Since the cost function is locally linear the cost dispensed by reducing ∆g units of the public good is M C(g)∆g. Let ∆xi denote ∑n the amount of private good to be received by each i, then we must have i=1 ∆xi = M C(g)∆g. From the above inequality we can divide M C(g)∆g so that ∆xi > M RSi (g, xi )∆g holds for all i. Since the amount one receives is more than the amount one demands for the compensation of reduction, each i prefers (g + ∆g, xi + ∆xi ) to (g, xi ). Hence (g − ∆g, x1 + ∆x1 , · · · , xn + ∆xn ) is a Pareto-improvement of (g, x1 , · · · , xn ). ∑n ”If” part: Suppose i=1 M RSi (g, xi ) = M C(g) holds and (g, x1 , · · · , xn ) is not Pareto efficient. Then there is an allocation (g + ∆g, x1 + ∆x1 , · · · , xn + ∆xn ) with

n n ∑ ∑ (xi + ∆xi ) + C(g + ∆g) = ei i=1

i=1

CHAPTER 27. PUBLIC GOODS

356

which is a Pareto improvement of (g, x1 , · · · , xn ). Combined with n ∑ i=1

we obtain

n ∑

xi + C(g) =

n ∑

ei .

i=1

∆xi + C(g + ∆g) − C(g) = 0.

i=1

From the assumption of increasing marginal cost it follows C(g + ∆g) − C(g) ≥ M C(g)∆g Hence we have

Since

∑n i=1

n ∑

∆xi + M C(g)∆g ≤ 0.

i=1

M RSi (g, xi ) = M C(g), this implies n ∑ (∆xi + M RSi (g, xi )∆g) ≤ 0. i=1

On the other hand, since (g + ∆g, x1 + ∆x1 , · · · , xn + ∆xn ) is a Pareto improvement it holds (g + ∆g, xi + ∆xi ) ≿i (g, xi ) for all i and (g + ∆g, xi + ∆xi ) ≻i (g, xi ) for at least one i. From the property of MRS under convex preference the above implies ∆xi + M RSi (g, xi )∆g ≥ 0 for all i and ∆xi + M RSi (g, xi )∆g > 0 for at least one i. Summing up the above we obtain n ∑ (∆xi + M RSi (g, xi )∆g) > 0, i=1

which is a contradiction to the previous inequality. The above equality is called the Samuelson condition.

CHAPTER 27. PUBLIC GOODS

27.3

357

The case of quasi-linear preferences

As a special case let us consider that consumers’ preferences are quasi-linear as income effect is negligible. Each i is assumed to have preference ≿i represented in the form ui (g, xi ) = vi (g) + xi , where xi may be either positive or negative as before, and vi (g) is the amount of the private good i is willing to sacrifice in order to have g units of the public good. Note that here g needs not be continuous but it could be discrete. Since there is no income effect, assume ei = 0 without loss of generality, then the feasibility condition reduces to n ∑

xi + C(g) = 0

i=1

In the quasi-linear environment Pareto efficiency is characterized as the maximization of social surplus. Note that it determines only the level of public good and it says nothing about how the allocation of the private goods should be. That is, it is totally silent about who should pay how much. Proposition 27.1 Allocation (g, x1 , · · · , xn ) is Pareto efficient if and only if g maximizes n ∑ vi (g) − C(g) i=1

∑n ′ ′ ′ Proof. ”Only if” part: Suppose there i=1 vi (g ) − C(g ) > ∑n exists g with ∑n C(g). Then, by taking i=1 xi + C(g) = 0 into account we obtain i=1 vi (g) ∑− n C(g ′ ) < i=1 (vi (g ′ ) − vi (g) − xi ). In order to cover the cost C(g ′ ) we need to make ∑ each i contribute −x′i units of the private good, and they must satisfy n ′ C(g ) = i=1 (−x′i ). From the above inequality it is possible to divide C(g ′ ) so ′ that −xi < vi (g ′ )−vi (g)−xi holds for all i Thus we obtain vi (g ′ )+x′i > vi (g)+xi for all i, which is a Pareto improvement. ”If” part: Suppose (g, x1 , · · · , xn ) is not Pareto efficient, then there exists a feasible allocation (g ′ , x′1 , · · · , x′n ) such that vi (g ′ ) + x′i ≥ vi (g) + xi for all i and vi (g ′ ) + x′i > vi (g) + xi for at least one i. Then by summing up the equalities we obtain n n ∑ ∑ {vi (g ′ ) + x′i } > {vi (g) + xi } i=1

From the feasibility condition C(g) = − the above inequality reduces to n ∑ i=1

i=1

∑n

vi (g ′ ) − C(g ′ ) >

i=1 n ∑ i=1

xi and C(g ′ ) = −

vi (g) − C(g).

∑n i=1

x′i . Hence

CHAPTER 27. PUBLIC GOODS

27.3.1

358

Continuous case

Suppose the amount of public good is continuous and preference/cost are smooth. Then marginal rate of substitution reduces to marginal willingness to pay, so that we have M RSi (g, xi ) = vi′ (g). Thus the Sanuelson condition reduces to the following Theorem 27.2 In quasi-linear environments (g, x1 , · · · , xn ) is Pareto-efficient if and only if n ∑ vi′ (g) = M C(g) i=1

holds.

27.3.2

Discrete case

Provision of public good is quite often discrete, and quite often it is just provided or not. For simplicity, assume g is either 0 or 1. For the private good still assume that its quantity is continuous. Again (g, xi ) denotes a combination of the public good and the private good for i. Maintain the assumption of quasi-linear preference, which in this case reduces to the representation ui (g, xi ) = vi g + xi When g = 1 it is ui (1, xi ) = vi + xi , and when g = 0 it is ui (0, xi ) = xi . That is, vi is i’s willingness to pay for the public good. Let C be the cost of the public good. Then a feasible allocation must satisfy (g, x1 , · · · , xn ) n ∑ xi + Cg = 0 i=1

Recall that in the quasi-linear environment the equivalence between Pareto efficiency and maximization of social surplus is true regardless of the public good being continuous or discrete. Thus we obtain Theorem 27.3 In the quasi-linear environment with discrete public good, Pareto efficiency is characterized by n ∑

vi ≥ C

=⇒ g = 1

vi ≤ C

=⇒ g = 0

i=1 n ∑ i=1

That is, when the sum of willingness to pay exceeds the cost it is efficient to provide the public good and otherwise no. Again, this condition is totally silent about who should pay how much.

CHAPTER 27. PUBLIC GOODS

27.4

359

The free-rider problem

So far we have put aside the problem of who should pay how much. Now let’s think about it. The problem is very hard, since it is likely that each individual tries to keep his contribution as low as possible hoping that the others would contribute, and as everybody does so and nothing is contributed as a result. This is called free-rider problem. As is shown in the next section, it is known that efficiency of allocation has to be given up partially in order to resolve the problem. Here let me explain two examples in order to make the point clear. Example 27.1 Town A and B are planing to build a library jointly. Once a library is built anybody in either town can use it. If one of the two towns contributes 5 millions a small library is build and each town’s benefit evaluated in terms of income is 3 millions. If both towns contribute 5 millions respectively a large library is built and each town’s benefit is 7 millions. If just one of the two towns contributes and a small library is built the social surplus is 3 × 2 − 5 = 1. If both towns contribute and a large library is built the social surplus is 7 × 2 − 5 × 2 = 4. Hence the surplus-maximizing solution is that both towns contribute. Is this implementable? Consider let’s say that they decide whether to contribute respectively. Then the payoff matrix is

A

C N

C 2, 2 3, -2

B N -2, 3 0, 0

where C refers to contributing and N refers to not contributing. Then the unique Nash equilibrium is (N.N). Since each town neglects contributing hoping to free-ride on the other town, as a result none of them contributes and nothing is done. This is a typical prisoners’ dilemma situation. Let us consider another example. Example 27.2 (From Moulin [23]): A and B are facing the problem of whether to build a public facility and which should pay how much. For simplicity assume that the cost is 1. Consider the following mechanism. 1. Let each of A and B report his willingness to pay. Let vA (resp. vB ) denote the willingness to pay reported by A (resp. B), which may or may not be the true one. 2. If vA +vB ≥ 1 the project is undertaken and A pays If vA + vB < 1 nothing is done.

vA vA +vB ,

B pays

vB vA +vB .

CHAPTER 27. PUBLIC GOODS

360

Can it be a Nash equilibrium in this mechanism that each reports his true willingness to pay? The answer is NO. Let’s say A’s willingness to pay of 1.2 and B’s is 0.4. Then, if A is reporting vA = 1.2 the best thing for B is to report vB = 0, so that A pays the whole cost and B pays nothing. Also, if B is reporting vB = 0.4 the best thing for A is to report vA = 0.6, so that he needs to pay only the minimal necessary amount in order to make the project. Thus they do not report true willingness to pay in Nash equilibrium. There are many Nash equilibria in this game: pair of reports (vA , vB ) is Nash equilibrium when vA + vB = 1, vA ≥ 0.6, vB ≤ 0.4. The second example shows that unless you design a mechanism nicely it may be manipulated by misreporting preferences. Now, how can we let people report their true preferences by their choices?

27.5

Strategy-proof mechanism

In order to resolve the free-rider problem we have to exclude the case that one can gain from misreporting. Here let us consider a mechanism such that it is a dominant strategy to report one’s true preference/willingness to pay. The requirement that it is always optimal for everybody to report truthfully not matter what the others say might be too strong. Indeed it leads to an impossibility result that we have to give up efficiency at least partially. I will come to mechanisms with milder requirements in the last chapter. Let me introduce Vickrey-Clarke-Groves mechanism, which is a prominent one of strategy-proof mechanisms. Let G denote the set of possible choices of public good provision. In the first example, since the choice is whether to build a big one or a small one or nothing we can write G = {0, 1, 2}, 0 refers to nothing, 1 refers to small and 2 refers to big. In the second example the choice is just whether to under take or not we can write G = {0, 1}. When the public good provision is continuous we would write G = R+ . Given g ∈ G, denote its cost by C(g). We assume that individuals’ preferences are quasi-linear, and each i’s preference over pairs of the level of public good provision and the amount of income transfer to him is represented in the form ui (g, xi ) = vi (g) + xi Here vi (g) is i’s willingness to pay for the level of public good provision g, which is a private information. Given this, the VCG mechanism (actually its special case called the pivotal mechanism) is defined as follows.

CHAPTER 27. PUBLIC GOODS

361

1. Let each i = 1, · · · , n report his function of willingness to pay vi , which may or may not be the true one. 2. Given the list of reported functions v = (v1 , · · · , vn ), solve maxg∈G

n ∑

vi (g) − C(g)

i=1

Denote its solution by g(v). 3. Make each i = 1, · · · , n pay the following amount:   ∑ 1 n − 1 ti (v) = C(g(v)) + max  vj (g) − C(g) g∈G n n j̸=i   ∑ n − 1 − vj (g(v)) − C(g(v)) n j̸=i

It will be easy to see the term n1 C(g(v)). It is equal division of the cost. How should the term     ∑ ∑ n−1 n−1 vj (g(v)) − max  vj (g) − C(g) −  C(g(v)) g∈G n n j̸=i

j̸=i

be understood? It is a kind of tax imposed on changing public decision. Here the term   ∑ n − 1 max  vj (g) − G(g) g∈G n j̸=i

is the maximal surplus for those other than i and ∑

vj (g(v)) −

j̸=i

n−1 C(g(v)) n

is the actual surplus for those other than i. Thus the difference between them is viewed as the loss for those other than i which is caused by i’s report. In this mechanism i is required to pay this amount in addition to the equal division of the direct cost. This is called Clarke tax. Note that this Clarke tax is not paid to other individuals in the society but paid to a third party outside of the society. In other words, you have to throw that amount of income in to the garbage! When we take the sum of payments we have n ∑ i=1

ti (v) ≥ C(g(v))

CHAPTER 27. PUBLIC GOODS

362

which ∑n does not in general hold with equality, and the society has to throw i=1 ti (v) − C(g(v)) units of income to the outside. It is somehow the social cost of acquiring right information. Can we avoid such cost? The answer is known to be NO. See for example the corresponding chapter in Mas-Colell, Whinston and Green [21]. Let us solve for the Clarke tax in the second example. As before, let (vA (0), vA (1)) = (0, 1.2), (vA (0), vB (1)) = (0, 0.6) be the true willingness to pay by A and B, respectively. 1. Since v1 (1) + vB (1) − C(1) = 1.2 + 0.6 − 1 > 0 the public good provision is g(vA , vB ) = 1. 2. The payments are { 1 tA (vA , vA ) = · 1 + max 0.4 − 2 { 1 tB (vA , vB ) = · 1 + max 1.2 − 2

} ( 1 · 1, 0 − 0.4 − 2 } ( 1 · 1, 0 − 1.2 − 2

) 1 · 1 = 0.6 2 ) 1 · 1 = 0.5 2

We see that the sum of payments exceeds 1 and A is paying 0.1 units of Clarke tax to a third party.

27.6

Exercises

Exercise 38 The problem is where to locate a public facility on a line. Assume for simplicity that the construction cost is free or already paid. Assume that each individual i has quasi-linear preference over pairs of location and income transfer represented in the form ui (g, xi ) = −(vi − g)2 + xi , g denote the location and xi denotes income transfer to i, and vi denotes his ideal location. Find the location and Clarke tax given by the pivotal mechanism.

Chapter 28

Indivisibility and heterogeneity This chapter covers allocation of indivisible and heterogeneous objects. If there is a divisible and homogeneous good besides the indivisible and heterogeneous objects we can used it as the mean of payment and we can use some kind of auction mechanism. Here I’m talking about situation in which there is no such good, or even if it exists we are not allowed to pay by it either for ethical or institutional reasons. Let me give you one example, which sounds extreme but already quite actual. Suppose there are two patients A and B who need kidney transplantations. A has a brother who is willing to donate one of his kidneys but unfortunately his blood type does not match A’s one. B has a sister who is willing to donate one of her kidneys but unfortunately her blood type does not match B’s one. So neither of A or B can get transplantation as they are. However, what if A’s donor’s blood type matches B’s one and B’s donor’s blood type matches A’s one? If A and B swap their donors both can have kidney transplantations. Here using money or goods for payment is now allowed ethically or institutionally. Of course it is not ethically obvious either if we should allow swapping donors. One may say, ”what’s wrong with this, as they both gain and hurt nobody else.” Another may say, ”Even if both gain and hurt nobody else, is that all fine?” It is an open question, but in any case it is admitted in some states in US, and put in practice. Anyhow, here I will argue how to obtain a nice allocation when such kind of exchange is allowed.

363

CHAPTER 28. INDIVISIBILITY AND HETEROGENEITY

28.1

364

Allocation of indivisible objects

Organ transplantation is a somewhat extreme example, but allocation of indivisible objects is in many places in our life, such as which house to live, in which organization to work, which position to take in a given organization, and so on. Here let us restrict attention to a model in which each individual can get at exactly one unit. Example 28.1 There are 6 individuals. Each i initially holds an object denoted by ei . Each consumer has preference over the objects e1 , e2 , · · · , e6 , and they are given as follows. 1 e3 e5 e2 e4 e1 e6

2 e3 e4 e1 e5 e6 e2

3 e5 e4 e1 e6 e2 e3

4 e5 e2 e4 e6 e3 e1

5 e1 e3 e2 e5 e4 e6

6 e1 e2 e3 e6 e5 e4

Now consider an allocation such as (e4 , e6 , e5 , e2 , e3 , e1 ) Is this a ”good” allocation? First, since it holds e4 ≻1 e1 , e6 ≻2 e2 , e5 ≻3 e3 , e2 ≻4 e4 , e3 ≻5 e5 , e1 ≻6 e6 nobody gets worse off than his initial holding, which is the property called individual rationality. Thus nobody gets hurt by attending the trading arrangement. Next let us see that this allocation is Pareto-efficient. Since 3 and 6 are already receiving their best objects we cannot move what they get any longer. Next, 4 and 5 are receiving their second best objects, but their best ones are already taken by 3 and 6 respectively, we cannot move what they get any longer. Now between 1 and 2 it makes 1 worse off if we move e6 from 2 to 1. Since it is thus impossible to make a Pareto improvement, this allocation is Paretoefficient. Individual rationality and Pareto efficiency are not enough here, however, since we can think of the following kind of thing: 1 prefers e2 to e4 , and 2 prefers e1 to e6 , which means that it is better for 1 and 2 to escape from the trading arrangement and exchange between them alone. In such case we say that 1 and 2 block the allocation (e4 , e6 , e5 , e2 , e3 , e1 ). An allocation which can be blocked by some group is not desirable, and also it is not sustainable since the blocking group will deviate from it as a matter of fact.

CHAPTER 28. INDIVISIBILITY AND HETEROGENEITY

365

Thus we need to consider a condition which stronger than the conjunction of individual rationality and Pareto efficiency. Say that an allocation is a core allocation if it is not blocked by any group. Core allocation is individually rational, since otherwise it is blocked by a single individual who prefers holding his initial endowment than accepting the trade. Also, core allocation is Pareto-efficient since otherwise it is blocked by the group of all individuals, since retrading can make somebody happier without hurting anybody else. Fortunately, in this setting core allocation exists and also it is uniquely determined (see Shapley-Scarf [31]). Moreover, the core allocation is found very easily by the top-trading-cycle algorithm. Step 1: Each individual points to his best object. When the arrows of pointing form a cycle, the individuals forming the cycle trade according to the arrow directions and leave with the objects. Step 2: Each remaining individual points to his best object among the remaining ones. When the arrows of pointing form a cycle, the individuals forming the cycle trade according to the arrow directions and leave with the objects. ··· Step k: Each remaining individual points to his best object among the remaining ones. When the arrows of pointing form a cycle, the individuals forming the cycle trade according to the arrow directions and leave with the objects. ··· Repeat the above procedure until all the individuals leave, then it ends after finite rounds. Let us do it using the above example. Step 1: When each individual points to his best object we have 1 → e3 , 2 → e3 , 3 → e5 , 4 → e5 , 5 → e1 , 6 → e1 There is one cycle 1 → 3 → 5 → 1, hence 1 receives e3 , 3 receives e5 , 5 receives e1 and they leave. Step 2: After Step 1, three individuals 2,4,6 are remaining. When each of them points to his best object among the remaining ones we have 2 → e4 , 4 → e2 , 6 → e2 There is one cycle 2 → 4 → 2, hence 2 receives e4 , 4 receives e2 and they leave.

CHAPTER 28. INDIVISIBILITY AND HETEROGENEITY

366

Step 3: Only 6 remains after Step 2. Then he points his object and we have a self-cycle 6 → e6 Thus 6 receives e6 and leaves. END: The outcome is (e3 , e4 , e5 , e2 , e1 , e6 ) There is another nice thing of the core allocation here. Consider a mechanism in which each individual submits his preference and the above algorithm is run based on the submitted preferences. It is known that it is always a dominant strategy for every individual to submit his true preference in this mechanism. In other words, it is impossible to gain by telling a lie in this mechanism. Since each individual can try his best object according to his submitted preference goes down as the algorithm proceeds there is no point in trying with less preferable one first by telling a lie.

28.2

Matching

Placement of persons is a typical example of indivisibility and heterogeneity. It appears in many places such as employment, school admission and marriage. In contrast to the allocation of objects here what are allocated have their own preferences. Thus both employers and workers have preferences over the opposite side respectively, both students and schools have preferences over the opposite side respectively, and both men and women have preferences over the opposite side respectively. Hereafter let me proceed the illustration using marriage between men and women. Example 28.2 There are four men and four women. Men are denoted by m1 , m2 , m3 , m4 and women are denoted by w1 , w2 , w3 , w4 , respectively. Each individual’s preference over the opposite side is given as follows. m1 w4 w2 w3 w1

m2 w2 w1 w3 w4

m3 w4 w2 w1 w3

m4 w2 w3 w1 w4

w1 m4 m2 m3 m1

w2 m1 m3 m2 m4

w3 m2 m1 m4 m3

w4 m2 m3 m1 m4

Let us consider a matching let’s say (m1 , w1 ), (m2 , w2 ), (m3 , w3 ), (m4 , w4 ) Is this a reasonable matching? The answer is NO in the following sense: Look at m1 and w4 . Here m1 likes w4 better than his current match w1 . Also, w4 likes m1 better than her current match m4 . Hence they have an incentive to

CHAPTER 28. INDIVISIBILITY AND HETEROGENEITY

367

”run away” from the current matching arrangement. Here we way that m1 w4 block the current matching. Likewise, m4 and w1 block the current matching. A matching which is blocked will not last long. Or, even though it does not last long, since we have to go through a costly and painful process in order to dissolve a relationship once it is formed, it is better to avoid such unfortunate situation beforehand. Thus we consider the following concept. Definition 28.1 A matching is said to be stable if there is no blocking pair. Stable matching exists. It is not uniquely determined, however. The set of stable matching is ”segment-like,” though, and each of its two extremes corresponds to the most favorable one for men which all men unanimously prefer and the most favorable one for women which all women unanimously prefer, respectively. These two are called men-optimal matching and women-optimal matching Each of the two extreme stable matchings are found by the deferredacceptance algorithm. There are two ways to run this algorithm, one in which men propose and the other in which women propose. The men-proposing version obtains the men-optimal stable matching and the women-proposing version obtains the women-optimal stable matching. Let me first explain the men-proposing version. Step 1: (a) Each man proposes to his best woman. (b) Each woman picks the best man among those who propose to her (if any), keeps him and reject all the other proposers. Step 2: (a) Each man who got rejected in the previous step proposes to his second best woman. (b) Each woman picks the best man among those who propose to her (if any) and the man she has kept from the previous step (if any), keep him and reject all the other proposers. ···

Step k: (a) Each man who got rejected in the previous step proposes to his best woman among those to whom he has not yet proposed. (b) Each woman picks the best man among those who propose to her (if any) and the man she has kept from the previous step (if any), keep him and reject all the other proposers. ··· Repeat this and stop when there is no men competing for a woman.

CHAPTER 28. INDIVISIBILITY AND HETEROGENEITY

368

Let us do it using the above example. Step 1: (a) m1 , m3 propose to w4 . m2 , m4 proposes to w2 . (b) w2 keeps m2 and rejects m4 . w4 keeps m3 and rejects m1 . Step 2: (a) m1 proposes to w2 . m4 proposes to w3 . (b) w2 keeps m1 and rejects m2 . w3 keeps m4 . w4 keeps m3 . Step 3: (a) m2 proposes to w1 . (b) w1 keeps m2 . w2 keeps m1 . w3 keeps m4 . w4 keeps m3 . Thus the men-optimal stable matching is (m1 , w2 ), (m2 , w1 ), (m3 , w4 ), (m4 , w3 ). On the other hand, the women-proposing version works as follows. Step 1: (a) w1 proposes to m4 . w2 proposes to m1 . w3 , w4 propose to m2 . (b) m1 keeps w2 . m2 keeps w3 and rejects w4 . m4 keeps w1 . Step 2: (a) w4 proposes to m3 . (b) m1 keeps w2 . m2 keeps w3 . m3 keeps w4 . m4 keeps w1 . Thus the women-optimal stable matching is (m1 , w2 ), (m2 , w3 ), (m3 , w4 ), (m4 , w1 ). It might sound counterintuitive that the men-proposing version leads to menoptimal and the women-proposing version leads to women-optimal. It indeed sounds depressing for men that as the algorithm proceeds men are in general rejected and go down to their second-best, third-best, and so on. This is due to an psychological impression of ”propose” and ”being rejected,” and it is actually advantageous for men that men propose, because each man can try from his first-best woman. Or, the men-proposing version might look better for women since women are getting better men as the algorithm proceeds, but in general the algorithm may end before a woman sees a good man to propose to her. Thus the men-proposing version is actually disadvantageous for women. Indeed in the above example m2 likes w1 better than w3 and m4 likes w3 better than w1 . Also, w1 likes m4 better than m2 and w3 likes m2 better than m4 . In order to avoid such confusion, imagine that they submit their preferences to a third party and this third party runs the algorithm and do the process of ”proposal” and ”rejection” on behalf of them, and indeed the practical operation of the algorithm works so. When you think of such mechanism that each individual submits his/her preference over the opposite side and the mediator runs the algorithm the following properties are known.

CHAPTER 28. INDIVISIBILITY AND HETEROGENEITY

369

• When we adopt the men-proposing version it is always a dominant strategy for each man to submit his true preference. • When we adopt the women-proposing version it is always a dominant strategy for each woman to submit her true preference. This is because for example in the men-proposing version each man can try from his best woman and lying and proposing to a less preferable woman simply creates the possibility that he is accepted by a ”wrong” woman. Unfortunately, there is no mechanism which selects a stable matching such that it is always a dominant strategy for both every man and every woman to submit his/her true preference. However, when the preferences of one side are public information such as schools’ priority rankings over student characteristics only private information on the other side such as students’ preferences would matter. In such cases one can use the deferred acceptance algorithm in which the side with private information proposes, and then it is always a dominant strategy for everybody on the proposing side to submit his true preference.

28.3

Exercises

Exercise 39 There 6 individuals. Denote the initial endowment for each i = 1, 2, 3, 4, 5, 6 by ei . Their preferences are described as below. 1 e2 e5 e4 e1 e3 e6

2 e6 e3 e1 e5 e2 e4

3 e4 e3 e2 e5 e6 e1

4 e6 e5 e1 e4 e3 e2

5 e2 e5 e1 e4 e6 e3

6 e5 e3 e6 e1 e4 e2

Find core allocation using the top-trading-cycle algorithm. Exercise 40 There are 4 workers and 4 firms. Denote the workers by w1 , w2 , w3 , w4 , and firms by f1 , f2 , f3 , f4 . Each worker can work for at most one firm, and each firm can hire at most one firm. Workers’ preferences over firms and firms’ rankings over workers are described as below. w1 f2 f1 f3 f4

w2 f1 f3 f2 f4

w3 f1 f2 f4 f3

w4 f2 f3 f4 f1

f1 w1 w2 w3 w4

f2 w3 w4 w1 w2

f3 w1 w4 w2 w3

f4 w3 w1 w4 w2

(i) Find the worker-optimal stable matching by using the worker-proposing deferred-acceptance algorithm.

CHAPTER 28. INDIVISIBILITY AND HETEROGENEITY

370

(ii) Find the firm-optimal stable matching by using the firm-proposing deferredacceptance algorithm. Exercise 41 There are 6 students and 3 schools. Students are denoted by i1 , i2 , i3 , i4 , i5 , i6 , and schools are denoted by s1 , s2 , s3 . Each student can enroll to at most one school, and each school has 2 seats. Students’ preferences over schools and schools’ priority rankings over students are as follows. i1 s3 s2 s1

i2 s3 s2 s1

i3 s2 s3 s1

i4 s2 s1 s3

i5 s2 s3 s1

i6 s3 s1 s2

s1 i3 i1 i6 i5 i4 i2

s2 i2 i1 i4 i5 i3 i6

s3 i3 i4 i5 i6 i1 i2

(i) Run the so-called Boston mechanism, assuming that each student submits his preference truthfully. The Boston mechanism works as follows: First, each student applies to his first-best school, and each school admits students according to its priority ranking as far as its capacity allows. Second, each student rejected in the previous step applies to his second-best school, but it the seats there are already full by the previous step he is automatically rejected. Repeat this. Show that in this mechanism students can gain by misreporting their preferences. (ii) Run the student-proposing deferred-acceptance algorithm.

Chapter 29

Efficiency, welfare comparison and fairness I have used Pareto efficiency as a criterion for welfare judgment in many places in this book, but I also emphasized that the Pareto principle alone is silent about whether a change in economic activity is desirable when it does not improve all individuals welfare and about who should gain and who should lose, and how much. Also I emphasized that Pareto efficiency has nothing to do with notion of fairness in any sense. Thus I like to spend one chapter on what are often referred to as welfare criteria which can say some change is better even when it is not a Pareto improvement, and on some discussions on fairness in resource allocation.

29.1

The Kaldor/Hicks criteria

Change in economic activity does not always make all individuals better off. Can we think of a criterion which supports a change even in such situations? The so-called Kaldor criterion says that a change should be accepted if we can make everybody better off by reallocating the allocation obtained by the change than in the allocation before the change. It basically says, ”making the pie is better.” Formally it says Definition 29.1 An allocation y = (y1 , · · · , yn ) is a Kaldor-improvement of x = (x1 , · · · , xn ) if there exists an allocation y ′ = (y1′ , · · · , yn′ ) with n ∑

′ yi1

=

i=1 n ∑ i=1

n ∑

yi1

i=1 ′ yi2

=

n ∑ i=1

371

yi2

CHAPTER 29. WELFARE COMPARISON AND FAIRNESS

372

Good 2 r yB H 6 ′ H jr yB xr B

′ yA r Y r H H yA

r xA

- Good 1

Figure 29.1: Kaldor improvement

such that it holds for all i and

yi′ ≿i xi yi′ ≻i xi

for at least one i. It is obvious that if y is a Pareto-improvement of x it is a Kaldor-improvement of x. See Figure 29.1, in which there are two consumers A and B. Then allocation ′ ′ (yA , yB ) is a Kaldor-improvement of (xA , xB ), since we can obtain (yA , yB ) by reallocating (yA , yB ), which is a Pareto-improvement of (xA , xB ). Note that ′ ′ vectors yA − yA and yB − yB are exactly opposite of each other. Or, one can explain this by using so-called utility possibility frontiers. Fix a representation of A’s preference uA and also a representation of B’s preference uB . Given a vector of aggregate resources available to the society e = (e1 , e2 ), let I(e) = {(uA (xA ), uB (xB )) : xA1 + xB1 = e1 , xA2 + xB2 = e2 } be the set of pairs of A’s utility and B’s utility which are obtained by allocating e. Of course, this is only for describing trade-offs between A’s gain and B’s gain and utility numbers themselves have no quantitative meanings. See Figure 29.2, in which two utility possibility frontiers are drawn, I(e) obtained from e and I(e′ ) obtained from e′ . Then y on I(e′ ) makes a Kaldorimprovement of x on I(e) since we can pick y ′ on I(e′ ) which is in the upper-right of x. Let me state two problems of the Kaldor criterion. One is,

CHAPTER 29. WELFARE COMPARISON AND FAIRNESS

373

B 6 I(e)

r y

ry ′ xr I(e′ ) - A

Figure 29.2: Kaldor improvement

It says a change should be accepted when we can make everybody better off if we suitably reallocate the allocation obtained by the change. Why not just doing such reallocation? If the reallocation is indeed done it’s simply a Pareto-improvement, isn’t it? The definition of Kaldor-improvement only says ”if we suitably reallocate the allocation,” and it does not require that such reallocation is indeed done. Why should one get convinced by such unwarranted story of potential reallocation when he is in fact losing because of the change? If the reallocation is left undone such criterion is deceptive, and if the allocation is indeed done we just need the Pareto criterion and it is just redundant. The other problem is that an allocation which Kaldor-improves upon another may be Kaldor-improved upon by the latter. See Figure 29.3, in which (yA , yB ) is a Kaldor-improvement of (xA , xB ) through the potential realloca′ ′ tion to (yA , yB ), and (xA , xB ) is a Kaldor-improvement of (yA , yB ) through the potential reallocation to (x′A , x′B ). Hence the Kaldor-criterion cannot rank properly between allocations in general. One can explain this by using the utility possibility frontiers. See Figure 29.4, in which two utility possibility frontiers are drawn, I(e) obtained from e and I(e′ ) obtained from e′ . Then y on I(e′ ) makes a Kaldor-improvement of x on I(e) since we can pick y ′ on I(e′ ) which is in the upper-right of x. However, x makes a Kaldor improvement of y as well, since we can pick x′ on I(e) which is in the upper-right of y. Let me introduce a criterion which is the ”complement” of the Kaldor criterion. The so-called Hicks criterion says that a change should be accepted if

CHAPTER 29. WELFARE COMPARISON AND FAIRNESS

Good 2 r yB H 6

′ H jr yB

xr B r ′ xB ′ yA rH Y Hr

yA ′ rxA r xA - Good 1

Figure 29.3: Mutual Kaldor improvements

B 6 I(e)

′ x r

r y

ry ′ xr I(e′ ) - A

Figure 29.4: Mutual Kaldor improvements

374

CHAPTER 29. WELFARE COMPARISON AND FAIRNESS

375

B 6 I(e)

x r ry I(e′ ) - A Figure 29.5: Hicks improvement

we cannot make everybody better off by reallocating the allocation before the change than in the allocation obtained by the change. In another words, one is a Hicks-improvement of another if the latter is not a Kaldor-improvement of the former. Formally, Definition 29.2 An allocation y = (y1 , · · · , yn ) is a Hicks-improvement of x = (x1 , · · · , xn ) if there exists no allocation x′ = (y1′ , · · · , yn′ ) with n ∑

x′i1 =

i=1 n ∑ i=1

such that it holds for all i and

n ∑

xi1

i=1

x′i2 =

n ∑

xi2

i=1

x′i ≿i yi x′i ≻i yi

for at least one i. Let me explain this using utility possibility frontiers. See Figure 29.5. There we can never go to the upper-right of y on I(e′ ) by reallocating x on I(e). Hence y is a Hicks improvement of x.

CHAPTER 29. WELFARE COMPARISON AND FAIRNESS

376

The same comments as above apply to the Hicks criterion. Besides the ethical issue, an allocation which Hicks-improves upon another may be Hicksimproved upon by the latter. See Figure 29.5 again. There we can never go to the upper-right of y on I(e′ ) by reallocating x on I(e). Hence y is a Hicks improvement of x. However, it is also the case that we can never go to the upperright of x on I(e) by reallocating y on I(e′ ). Hence x is a Hicks improvement of y. Since the Kaldor criterion and Hicks criterion are the ”complement” of each other, if we impose both we can avoid the problem that two allocations dominate each other under the Kaldor criterion alone and under the Hicks criterion alone respectively. The idea is due to Scitovsky. Definition 29.3 An allocation y = (y1 , · · · , yn ) is a Scitovsky-improvement or of x = (x1 , · · · , xn ) if y is both a Kaldor-improvement and a Hicks-improvement of x. If one is a Hicks-improvement of another it means the latter is not a Kaldorimprovement of the former. Hence it is always the case that if one is a Scitovskyimprovement of another the latter is not a Scitovsky-improvement of the former. However, the ranking by the Scitovsky-improvement may be intransitive, that is, it may have a cycle. See Figure 29.6. There y is a Scitovsky-improvement of x, z is a Scitovsky-improvement of y, w is a Scitovsky-improvement of z, but x is a Scitovsky-improvement of w. It is called the Gorman paradox. So the Scitovsky-improvement doesn’t help, unfortunately.1 The above problems of inconsistency do not occur in the quasi-linear environment in which there is no income effect, while the ethical problem I discussed above still remains. There the Kaldor criterion and the Hicks criterion coincide and the comparisons are determined by the amount of social surplus, that is, ”making the pie is better.” Proposition 29.1 Assume the quasi-linear environment. Then, if y = (y1 , · · · , yn ) and x = (x1 , · · · , xn ) satisfy n ∑ i=1

{vi (yi1 ) + yi2 } >

n ∑

{vi (xi1 ) + xi2 }

i=1

y is both a Kaldor-improvement and a Hicks-improvement of x.

1 Samuelson considered a weakening of the condition that one allocation is better than another when the entire utility possibility frontier given by the former is above the entire utility possibility frontier by the latter. Let me call this Samuelson improvement. This leads to a cycle again when it is combined with the Pareto criterion, however. The same example works. In Figure 29.6, y is a Pareto-improvement of x, z is a Samuelson-improvement of y, w is a Pareto-improvement of z, but x is a Samuelson-improvement of w.

CHAPTER 29. WELFARE COMPARISON AND FAIRNESS

377

B 6

r x

ry

zr

w r - A

Figure 29.6: Gorman paradox ∑n ∑n Proof. Let w = i=1 {vi (yi1 ) + yi2 } − i=1 {vi (xi1 ) + xi2 } > 0. Then it is ′ enough to set let’s say yi1 = yi1 and ′ yi2 = vi (xi1 ) + xi2 − vi (yi1 ) +

w n

for all i. Again, let me emphasize that it is still totally silent about how we should cut ”the pie,” that is, how we should distribute the maximized surplus. Not only that. If we want to rank between allocation in order to satisfy completeness, the above result implies that we should be indifferent between any distributions of surplus. Thus it excludes any fairness or equity concerns.

29.2

Fair allocation in exchange economies

Now let me come to fairness What is fair or not depends on who should be responsible for what and how much. For example, even when one receives more consumption that another we cannot immediately say that it is unfair or fair, as it depends on how much he is responsible for or entitled to his ability, skill or his knowledge or personality. Here as a benchmark let us consider a ”primitive” situation in which nobody is responsible for anything. In the context of resource allocation, consider an economy without production in which people’s initial holdings are given solely by lucks. Imagine that a fixed amount of resource falls from the heaven so that how would grab how much is totally random. Since initial holdings here are purely

CHAPTER 29. WELFARE COMPARISON AND FAIRNESS

378

due to lucks, and nobody is responsible for it, hence it has no information. In such a primitive situation, only the total amount of resources should be the relevant information. On the other hand, we maintain the assumption that each person is responsible for his preference, that is, everybody is responsible for how he is. In the two-good illustration, initial holding of each i = 1, · · · , n denoted by ei = (ei1 , ei2 ) is coming only from luck and he has no responsibility for it. ∑nThus what ∑n is relevant is the sum of initial endowments E = (E1 , E2 ) = ( i=1 ei1 , i=1 ei2 ) only. Then a feasible allocation is any x = (x1 , · · · , xn ) such that n ∑ i=1 n ∑

xi1

=

E1

xi2

=

E2

i=1

Now what should be a ”fair” allocation.

29.2.1

Equal division

First, we can think of equal division, ( ) E E1 E2 xi = = , , i = 1, · · · , n n n n This doesn’t seem to be a clever solution, however, as people’s preferences are diverse. It may be a kind of solution when nobody is responsible even for his preference, though, while we are maintaining the the assumption that everybody is responsible for his preference.

29.2.2

Equal utility?

What about the idea that people should be ”equally happy,” while they don’t have to consume the same thing? That is, given utility representation ui (xi ) for each i = 1, · · · , n, consider that ui (xi ) = uj (xj ) should hold for all i, j. This needs a ”faith” of interpersonal comparison of utility, however, and we have to go outside of the framework of ordinal utility, in which utility representation is only an ordinal representation of preference ranking, it has no quantitative meanings or it is not comparable across individuals. We don’t even have a definition of ”equally happy” in a well-grounded sense. When ui (xi ) is a representation of i’s preference ≿i , let’s say its double u bi (xi ) = 2ui (xi ) is also a representation of the same preference, and we cannot

CHAPTER 29. WELFARE COMPARISON AND FAIRNESS

379

A’s Good 2 - IA

 16 B’s Good

OB

x∗∗ r ? IB

xr rE/2

rx∗ r x∗∗∗

- A’s Good 1 ? B’s Good 2 Figure 29.7: Allocations with envy and no envy OA

say which one is the ”right” representation. Also, when uj (xj ) is a represen1 tation of j’s preference ≿j , let’s say its cubic root u bj (xj ) = uj (xj ) 3 is also a representation of the same preference, and we cannot say which one is the ”right” representation. Here the condition of ”fairness” implied by ui (xi ) = uj (xj ) does not agree with the condition of ”fairness” implied by u bi (xi ) = u bj (xj ), and as far as we don’t have a particular ”faith” we cannot judge which one is the ”right” condition for ”equal happiness.”

29.2.3

Fairness as absence of envy

Now, is it possible to define a notion of fairness based only on preference relations, without bringing in a faith of interpersonal comparison of utility? Here let me introduce the idea of fairness as absence of envy. First let us define envy. Say that i envied j given allocation x = (x1 , · · · , xn ) if i prefers j’s consumption to his one, that is, x j ≻i x i holds. Since we are talking about a primitive situation in which nobody is responsible for anything, this is a ”justifiable” envy and we should avoid it. Then an allocation is said to be envy-free if nobody envies anybody there. See Figure 29.7. Here at x, A envies B and B does not envy A. At x∗ , A does not envy B but B envies A. At x∗∗ , both envy each other. And, at x∗∗∗ there is no envy. The equal division allocation is obviously envy-free, but it is not the only envy-free allocation.

CHAPTER 29. WELFARE COMPARISON AND FAIRNESS

29.2.4

380

Are efficiency and fairness compatible?

The answer is YES when goods are continuously divisible. Indeed, allocation in competitive equilibrium from equal division is Pareto efficient and envy-free. Competitive equilibrium from equal ( Edivision ) is such that all individuals are 1 E2 initially given the equal division E and they exchange in a competn = n , n itive market so as to obtain their final consumptions. Since competitive equilibrium allocation is Pareto-efficient for arbitrary initial holdings, the allocation in competitive equilibrium from equal division is Pareto-efficient. It is envy-free, since all the individuals have the same initial holding and hence face the same budget constraint. Thus anybody could buy what anybody else is consuming. Nevertheless he has chosen what he has chosen, it is cause he likes his one better than others’ ones. Hence there is no room for envy here. The answer is NO in general, however, when goods are indivisible. Consider the simplest case that there is just one indivisible object (which everybody wants) and nothing else. Since you cannot divide it if somebody receives it everybody else envies him. Hence the only way to avoid envy is to throw it into the garbage, but it is obviously inefficient. One can think of let’s say dividing the time to use the object when it is durable, though. Or, one can think of drawing a lottery in order to decide who should get it. Since probabilities are divisible, efficiency and envy-freeness are compatible with each other at least in the ex-ante sense, while ex-post they are obviously incompatible. Or, if there is another good which is continuously divisible we can use it as a mean of compensation so that the two notions are compatible. In any case, you will see that divisibility of goods or objects are crucial for the compatibility.

29.3

Fairness in production economies

One can we extend the above notion of fairness to production economies if 1. all individuals have an identical level of skill 2. production technology is not attributed to anybody or anybody’s contribution. When all individuals have the same skill, any inequality of their earning from labor reflects only how much somebody is more willing to work or less willing to work, which is purely a matter of taste. In such case we can extend the notion of envy-freeness as it is. Let (ci , li ) be denote i’s consumption and leisure, where 1 − li is taken to be his working hours. Then i is said to envy j given ((c1 , l1 ), · · · , (cn , ln )) if (cj , lj ) ≻i (ci , li )

CHAPTER 29. WELFARE COMPARISON AND FAIRNESS

381

Then say that an allocation ((c1 , l1 ), · · · , (cn , ln )) is envy-free if nobody envies anybody there. Consider now running a competitive market after dividing the initial resources equally and dividing ownerships of the firms equally, where all individuals are assumed to have equal amounts of time. Then the resulting allocation of consumption-leisure is Pareto-efficient and envy-free. Now what if individuals have different levels of skill? Now it depends on whether an individual is responsible for the level of his skill. First, let us consider that nobody is responsible for the level of his skill. Then we can adopt the same definition of envy and envy-freeness as above. However, there is a conflict between efficiency and such notion of fairness at a conceptual level. Here is an example due to Pazner and Schmeidler 1974 [25]. There are two individuals, A and B. Their preferences over pairs of consumption and leisure are represented as follows. uA (xA , lA ) = 1.1xA + lA uB (xB , lB ) = 2xB + lB Each of them is given 1 unit of time, which can be used for leisure or leisure. A produces 1 unit of the consumption good per 1 unit of labor time and B produces 0.1 units of the good per 1 unit of labor time. Then, Pareto efficiency implies that A should spend the entire 1 unit of time on labor and B should spend 0 unit of time on labor. That is, efficiency implies lA = 0 and lB = 1. Thus the total amount of the consumption good is 1. In order that A does not envy B, we have to have 1.1xA ≥ 1.1xB + 1 Together with xA + xB = 1 this implies xA ≥ that B does not envy A we have to have

21 22 .

On the other hand, in order

2xB + 1 ≥ 2xA Again together with xA + xB = 1 this implies xA ≤ 34 , which is incompatible with the previous inequality. Intuitively, A prefers more leisure compared to consumption than B does and B is willing to work more than A does. However, A is skillful and B has almost no skill. This leads A to envy B on the ground that B is enjoying more leisure what A likes, and leads B to envy A on the ground that A is enjoying more consumption what B likes. This suggests that we have to at least partially give up one of efficiency and envy-freeness as defined above. Next, let us consider that each individual is responsible for his skill. Assume that each individual can produce the consumption good with a constant marginal productivity of labor which is independent of the others’ labors. Let

CHAPTER 29. WELFARE COMPARISON AND FAIRNESS

382

Ai denote i’s marginal productivity of labor, then if i has li units of leisure time he spends 1 − li on labor and produces Ai (1 − li ) units of the consumption good. Now, given allocation ((c1 , l1 ), · · · , (cn , ln )) consider if i has a ”justifiable envy” against j. Here i is receiving (ci , li ) and j is receiving (cj , lj ). Note that as j works 1 − lj units of time he produces Aj (1 − lj ). Since i’s marginal productivity is Ai , in order to produce the same amount of the A (1−l ) consumption good i needs to spend j Ai j units of time on labor, implying that his leisure would be 1 −

Aj (1−lj ) . Ai

Therefore, if an individual is responsible ( ) A (1−l ) for his skill the ”right” comparison here is between (ci , li ) and cj , 1 − j Ai j , not between (ci , li ) and (cj , lj ). Thus, say that i ∗-envies j given ((c1 , l1 ), · · · , (cn , ln )) if ( ) Aj (1 − lj ) cj , 1 − ≻i (ci , li ) Ai Say that an allocation is ∗-envy-free if nobody ∗-envies anybody there. The ∗-envy-freeness is compatible with Pareto efficiency. For example, when each consumer i receives (Ai (1 − li ), li ) and chooses his best li freely it is Pareto efficient and ∗-envy-free. Of course, let me add that the right answer should reflect solely whether an individual is responsible for his skill or not, and it should not be that we pick one definition of fairness of another on the ground that it is fitting well to the notion of efficiency.

29.4

Exercises

Exercise 42 There are two individuals, A and B. In the partial equilibrium framework in which Good 2 is taken to be the numeraire, suppose their preferences are represented in the quasi-linear form uA (xA ) =

ln xA1 + xA2

uB (xB ) =

2 ln xB1 + xB2

There is no production, and the total amount of Good 1 is 12, the total of Good 2 is taken to be 0 without loss of generality. (i) Find the set of Pareto-efficient allocations. (ii) Find the set of Pareto-efficient and envy-free allocations.

Chapter 30

Aggregation of preferences and social choice 30.1

Motivations from welfare economics and political science

The argument in the last chapter suggests that we have to have a serious theory about who should gain or lose and how much when a change in resource allocation or state of the society in general from one to another cannot make everybody better off. The problem of the modern economics is rather that it is not even utilitarian. What do I mean? If you are a classical (and naive) utilitarian who believes interpersonal comparability of utilities and the law of diminishing marginal utilities, you would claim you can justify income redistribution from the rich to the poor on the ground that the marginal utility of wealth for the poor is greater than that for the rich. Since such claims need to bring in some ”faith,” it would be a natural thing to do for ”scientific” economists to try to dispense with such things. The socalled new welfare economics which advocates the criteria due to Pareto, Kaldor, Hicks and Scitovsky as illustrated in the last chapter is viewed as an attempt to dispense with the concept of interpersonally comparable cardinal utility. As we saw above it was a failure, in the sense that it cannot in general induce a consistent ranking. Or, even in the special case with no income effect in which the rankings can be consistent it cannot accommodate with any equity concern. This leads us to start with thinking of a complete and transitive ranking over all the possible social alternatives, and make it clear how making a consistent ranking requires to us to make a judgment about who should gain or lose and how much.

383

CHAPTER 30. AGGREGATION OF PREFERENCES

384

Bergson and later Samuelson considered a numerical-valued function defined over all the possible allocations. Since it is numerical-valued it is in the outset supposed to rank between any allocations in a complete and transitive manner. Suppose there are n individuals. For later arguments, let us take a wider scope. We consider an abstract set of social alternatives denoted by X. Then a social welfare function is given by W : X → R. Here X can involve for example political choices as well as resource allocations. In order to be consistent with the Pareto criterion, say that if everybody is indifferent between two social alternatives the entire society is indifferent between them as well. Let ≿i denote individual i’s preference relation over X, where i = 1, · · · , n. Then the condition states that x ∼i y =⇒ W (x) = W (y) hold for all x, y ∈ X. For each i = 1, · · · , n and given his preference relation ≿i , fix one of its representations let’s say denoted by ui . Given a list of such fixed representations u1 , · · · , un , the social welfare function reduces to a form W (x) = F (u1 (x), · · · , un (x)), where the function F is supposed to describe our ethical judgment about how we should evaluate individuals’ welfare. For example, we can think of 1. Benthamite social welfare function W (x) =

n ∑

αi ui (x)

i=1

in which F is additive. 2. Nash social welfare function W (x) =

n ∏

ui (x)αi

i=1

in which F is multiplicative. 3. Rawlsian social welfare function W (x) =

min ui (x)

i=1,··· ,n

which cares about the ”most unhappy” individual. Of course you will have a suspicion that the above argument relies on the concept of interpersonal comparability of utility, since in the above we are taking of their sum, product and minimum. What guarantees such operations?

CHAPTER 30. AGGREGATION OF PREFERENCES

385

We saw that in the partial equilibrium environment efficient allocation maximizes the sum of surplus. Historically and educationally, this seems to have caused a confusion with the idea of ”the greatest happiness of the greatest number.” However, surplus is measurable through marginal rate of substitution and does not require the notion of cardinal utility. Thus, taking the sum of surplus is consistent with the framework of ordinal utility theory. Indeed, maximization of the sum of surplus is totally silent about how the maximized surplus should be distributed. The above social welfare function approach seems to need a departure from the ordinal utility framework. But how exactly should the departure described? Moreover, it is not clear for us how F indeed describes the ethical views expressed in the social welfare function. For example, consider the simplest case of two individuals and consider a social welfare function in the form W (x) = uA (x) + uB (x) Should it be interpreted as treating the two individuals equally since it puts 1 vs. 1 weights to A and B respectively? You see that it is nonsense, as we can take another pair of representations u eA (x) = 2uA (x) and u eB (x) = 0.5uB (x) then we have W (x) = 0.5e uA (x) + 2e uB (x) which is now interpreted as giving 0.5 vs. 2 weights to A and B respectively. Also, as we take another pair of representations u eA (x) = euA (x) and u eB (x) = uB (x) e we have W (x) = ln u eA (x) + ln u eB (x) = ln u eA (x)e uB (x), which is equivalent to taking a Nash social welfare function with respect to u eA and u eB . Such ambiguities motivate us to take an axiomatic approach. It starts with a set of axioms each of which clearly expresses a normatively meaningful view by itself, instead of playing with a functional form. Another motivation comes from political science, which has a long literature of formal analysis of voting dating back to Condorcet. There the interest is how of if voting nicely aggregates people’s diverse opinions in order to make consistent social decisions. The theory of social welfare function established by Arrow stands more directly on this line. The subsequent part of the chapter follows Arrow’s argument and its sequels, in which the set of social alternatives are abstract. Also, we assume that X is a finite set, for the sake of simplicity. Such apparent difference of set-up seems to have given Samuelson a room for an excuse that Arrow’s negative result on aggregation does not affect the new welfare economics. However, similar negative results have been obtained in the setting of resource allocation. So without loss of generality we focus on the case that X is abstract and finite.

CHAPTER 30. AGGREGATION OF PREFERENCES

30.2

Axioms for aggregation of preferences

30.2.1

Completeness

386

It is known that the Pareto principle or unanimity rule alone cannot rank between any two social alternatives. Thus it fails to satisfy completeness, our first postulate, which says that we can always rank between (including indifference) any two alternatives. Let us confirm this with an example. Example 30.1 There are two individuals, A and B. There are six social alternatives x, y, z, u, v, w. Preferences of A and B are as follows. A y z u w x v

B w x v z y u

Since both A and B rank w over x, we see that w is a Pareto-improvement of x, hence x is not Pareto efficient. By similar argument we see that the set of Pareto-efficient alternatives are y, z, w. However, the Pareto principle alone cannot rank among y, z, w. For example, since A ranks y over z and B ranks z over y, we cannot rank between y and z here.

30.2.2

Transitivity

Another and even more familiar rule is majority rule. It says the society ranks x over y if majority of people rank x over y.— Here let me assume there are odd number of people and everybody has a strict preference in the sense that he is not indifferent between any two alternatives. However, the majority rule has the following problem, called voting paradox Example 30.2 There are there individuals, A,B and C. There are three social alternatives, x, y, z. Preferences of A,B and C are given as follows. A x y z

B z x y

C y z x

Then, the majority rule ranks, x over y, y over z but z over x, which forms a cycle and violates transitivity.

CHAPTER 30. AGGREGATION OF PREFERENCES

387

What’s wrong with violating transitivity? Although we cannot decide which one is the best when it is violated, we can make a choice using a tournament for example. However, the tournament method has the following problem under intransitivity. Example 30.3 Given the previous setting consider the following two tournament schedules. Schedule 1: First x is matched to y and the winner is then matched to z. Schedule 2: First y is matched to z and the winner is then matched to x. In Schedule 1, first x beats y and then it is beaten by z, hence the final winner is z. In Schedule 2, first y beats z and then it is beaten by x, hence the final winner is x. This example shows that the tournament method based on the majority rule is manipulable by manipulating the schedule.

30.2.3

Independence of irrelevant alternatives

Now let us come back to the idea of social welfare function. Since it is numericalvalued, the social ranking given by a social welfare function is complete and transitive. What’s the problem then? To illustrate, let us think of a specific example called Borda rule. Again assume strict preferences. It works as follows: Let m be the number of alternatives. For individual i, let (xi1 , xi2 , · · · , xim ) denote the list of the alternatives in the descending order, in which xi1 is his best and xim is his worst. Assign m to xi1 , m − 1 to xi2 , and so on, and assign 1 to xim . Then for each alternative take the sum of the assigned numbers across individuals. Such scoring is viewed as a social welfare function W (x) =

n ∑

ui (x)

i=1

in which ui (x) = m − k + 1 with x being i’s k-th best alternative. That is, this rule fixes a class of representations in which everybody’s highest utility is m and the lowest is 1, and the width of the grid of utility is 1. Let us do it with the following example. A x y z q

B y x q z

C z y x q

Given the above preference profile, A assigns 4, 3, 2, 1 to x, y, z, u respectively, and similarly for B and C. Then, the score of x is 4 + 3 + 2 = 9. By repeating

CHAPTER 30. AGGREGATION OF PREFERENCES

388

this we obtain W (x) = 9, W (y) = 10, W (z) = 7, W (q) = 4 and y is given the largest score. Now what’s any problem? To see that suppose A’s preference is different and it is A∗ as follows (where the preferences of B and C remain the same), A∗ x z q y

B y x q z

C z y x q

Then the scores are W (x) = 9, W (y) = 7, W (z) = 7, W (q) = 5 and x is the largest score. The problem here is that in both cases the rankings between x and y are A = A∗ x y

B y x

C y x

which remain unchanged across the cases, but in the former case y wins and in the latter case x wins. Thus, here the ranking between x and y depend not only on how individuals rank between just these two but also on how they rank z and q together with those as well. The requirement that how the society ranks between x and y should depend only on how individuals rank between x and y and should be independent of how they rank other alternatives (such as z and q) is called independence of irrelevant alternatives or IIA. Note that this is an ”inter-profile” axiom, in the sense that it is a property connecting across different preference profiles. IIA says in other words that in constructing the aggregate preference we should take individual preferences in a purely ordinal way and should not read any cardinal meaning in them. To see this, think of why Borda rule violates IIA. Borda rule not only takes how one ranks between x and y but also ”how much” he likes x over y let’s say through the comparisons with z and q. Here such cardinal information is taken by means of setting all individuals’s highest scores and lowest scores to be the same and the width of the grid of scores to be the same as well and constant. Why is IIA, or the requirement that preference aggregation should take a purely ordinal standpoint, important? See the first case in which y is the top element. A won’t like it. Then if he reports A∗ instead of A he can manipulate

CHAPTER 30. AGGREGATION OF PREFERENCES

389

the voting outcome from y to x. Thus Borda rule can be manipulated by misreporting preference. In general, if the society attempts to take cardinal information or ”intensity” of preferences it can be manipulated by exaggerating information about intensities. In the next chapter, it is shown that IIA is equivalent to the immunity to manipulation by strategic misreporting of preferences. That is, IIA is indispensable when you want the aggregation rule to be immune to manipulations.

30.3

Arrow’s theorem

Now consider a preference aggregation rule which satisfies the three axioms, while the more formal presentation is relegated to the appendix. 1. Order: The aggregate preference satisfies completeness and transitivity; 2. Pareto: If everybody ranks x over y the aggregate preference ranks x over y. 3. Independence of Irrelevant Alternatives (IIA): How the aggregate preference ranks between x and y depends only on how the individuals rank between x and y and not on how they rank any other alternatives. Does there exist a preference aggregation rule which satisfies the above axioms, and what does it look like? Arrow [1] gave a negative answer. Theorem 30.1 (Arrow’s theorem): When there are at least three individuals and at least three social alternatives, the only preference aggregation rule satisfying the above axiom is dictatorship, in the sense that there is one individual and the aggregate preference always follows his preference. Given our intuition that dictatorship is obviously bad, the above result suggests that we need to give up or weaken at least one of them the axioms. The proof is relegated to the appendix, but there you’ll see that the IIA axiom is repeatedly and heavily used. When you see how the IIA axiom is used there you will see it so thoughtless to draw an interpretation line ”the impossibility of democracy” based on Arrow’s theorem.

30.4

May’s theorem

When there are only two alternatives we don’t have to worry about violating transitivity. IIA is also met, vacuously. Hence the majority rule works, let’s say. Not only that, it is known to be the only rule which satisfies quite appealing axioms.

CHAPTER 30. AGGREGATION OF PREFERENCES

390

Theorem 30.2 (May’s theorem): Suppose there are just two social alternatives and odd number of individuals. Then the only preference aggregation rule satisfying the following axioms is the majority rule. 1. Anonymity: Every individual is treated equally. 2. Neutrality: The two alternatives are treated equally. 3. Monotonicity: Suppose x is winning under the current voting profile. Then if some people switch their voting from y to x, the winner is again x. Similarly for the opposite case.

30.5

Borda rule again

If you give up IIA you can allow the use of the Borda rule and other ruled based on social welfare functions. Violation of IIA leads to manipulability, though. However, the problem of manipulability may not be serious when the society is so large that the room for each individual to manipulate the outcome by himself alone tends to be negligibly small. This is an empirical question. Borda rule is characterized by the following set of axioms, in the setting in which the set of individuals is variable (see Young [39] for example). Theorem 30.3 The only rule satisfying the five axioms below is the Borda rule. 1. Order: The aggregate preference is complete and transitive. 2. Neutrality: All the alternatives are treated equally. 3. Monotonicity: When the group consist of a singe individual the aggregate preference is just his preference. 4. Combination: Suppose the aggregated preference of group A ranks x at least as good as y and the aggregate preference of group B, which does not overlap with A, ranks x at least as good as y. Then the aggregate preference of the group made by merging A and B again ranks x at least as good as y. The same assertion holds also when we replace ”at least as good as” by ”over.” 5. Cancellation: If the number of people who rank x over y is equal to the number of people who rank y over x then the aggregate preference take them to be equally preferable.

CHAPTER 30. AGGREGATION OF PREFERENCES

30.6

391

Domain restriction and single-peaked preferences

I did not write explicitly in the above explanation, but there is an implicit assumption in Arrow’s theorem. It is that the social welfare function has to take any combinatorially possible preferences into account. This assumption is called Unrestricted Domain. It will better to explain using an example. There are three alternatives, Left, Center and Right, which are denoted by L, C and R. Combinatorially, there are 3 × 2 × 1 = 6 preferences as follows. P1 L C R

P2 L R C

P3 C L R

P4 C R L

P5 R L C

P6 R C L

Look at P2 and P5. Here P2 ranks Left to be the best but ranks Right over Center. It is possible combinatorially, but we would say it is ”unlikely.” Likewise, P5 ranks Right to be the best but ranks Left over Center. It is possible combinatorially as well, but we would say it is ”unlikely.” Arrow’s theorem is assuming that even such preferences should be taken into account. However, when we are allowed to exclude such ”unlikely” preferences and narrow down to a restricted domain, Arrow’s theorem does not necessarily hold. A prominent case of restricted domain is single-peakedness. Assume that social alternatives are ordered in one dimension. Then preference is said to be single-peaked if there is a bliss point (peak) and any alternative gets worse as it goes far to the left from the peak and also as it goes far to the right from the peak. This includes the cases that the peak is the left endpoint or the right endpoint. In the above example, single-peaked preferences are limited to four, P1 L C R

P3 C L R

P4 C R L

P6 R C L

The majority rule works in the domain of single-peaked preferences. In the above example let’s say there are 5 individuals with preference P1, 4 with P3, 3 with P4, and 7 with P6. Then the majority rule yields • L vs.C: C wins by 5 vs.14 • C vs.R: C wins by 12 vs.7 • L vs.R: R wins by 9 vs.10

CHAPTER 30. AGGREGATION OF PREFERENCES

392

and we have a transitive ranking C ≻ R ≻ L. Let us look at another example. Social alternatives are numbers from 0 to 100. Consider choosing the rate of income tax for example. Of course everybody prefers low tax rate nominally, but here consider that the effects of income redistribution are taken into account, meaning that rich people typically prefer low rate and poor people prefer high rate. There are 25 individuals with single-peaked preferences, and the distribution of peaks is as follows. Peak #

G1 0 2

G2 10 3

G3 30 3

G4 50 4

G5 70 3

G6 90 6

G7 100 4

Group G2 for example consist of three individuals who prefer 10 the best. In contrast to the previous example, the above table describes only peaks. Hence individuals in the same group may have different preferences. For example, some in G2 may prefer 0 to 30 but some other may prefer 30 to 0. In any case, however, such difference does not affect the outcome of majority voting. It is because of the following. Out of these seven groups the median is G5, which is the group needed for making the majority from either side. Since the sum G1, G2, G3 and G4 is 12 and they cannot form the majority by themselves alone, and needs G5. Likewise, since the sum of G6 and G7 is 10 and they also need G5 in oder to form the majority. In the majority rule the peak for G5, which is 70, is chosen. When you compare between 70 and 50, while G1-4 prefer 50 since G5-7 prefer 70 the majority prefer 70 to 50. Similarly for the comparison between 70 and 30,10,0, respectively. When you compare between 70 and 90, while G6-7 prefer 90 since G1-5 prefer 70 the majority prefer 70 to 90. Similarly for the comparison between 70 and 100. Thus, 70 beats everything else in the majority rule. In general, we have the following result. Theorem 30.4 (Median Voter Theorem): When individuals have singlepeaked preferences the majority rule selects the median of the peaks of preferences. Note that median is different from mean. For example, in the distribution of peaks below the mean is almost 50 but the median is 100. Peak #

30.6.1

G1 0 50

G2 10 0

G3 30 0

G4 50 0

G5 70 0

G6 90 0

G7 100 51

Downsian electoral competition

There are two candidate running for the office. Policy is a point on the real line, and each candidate commits to the policy he announces. Voters’ preferences are

CHAPTER 30. AGGREGATION OF PREFERENCES

393

assumed to be single-peaked. Also, assume that the candidate are interested only in winning the office. This is called Downsian model. Now consider a game that the two candidate simultaneously announce their policies, in which tie-breaking is made in certain way. What do they announce in Nash equilibrium? The answer is that both announce the median peak, since if one announce something other than the median the opponent can beat him for sure by setting the policy to the median. Of course it is questionable if this answer matches the reality. It is easy to raise some points. 1. It is not realistic that candidates are interested only in winning the office, and it will be natural so consider that they are motivated by their own policies. 2. It is not realistic that the candidate know the voters’ preferences precisely. It is natural to say that there are certain informational asymmetries. 3. The candidate may not be able to commit to their policy announcements. 4. Policy space is not one-dimensional. and so on. I would say that the Downsian model is a benchmark, although, from which many recent researches have started so as to encompass the abovementioned issues.

30.7

Proof of Arrow’s theorem

Here the proof basically follows Austen-Smith and Banks [2]. Let X denote the set of social alternatives which is assumed to be finite. Let R denote the set of all the complete and transitive preference relations over X. Later we may consider in general a subset of R as the set of preference relations for which the preference aggregation rule is defined, since we might think of additional restrictions to preferences to be considered. Let D ⊂ R denote such set of preference relations that are to be considered. Then, any preference profile is let’s say denoted by ≿= (≿1 , · · · , ≿n ) ∈ Dn . Then a preference aggregation rule is a function R : Dn → R, which maps an arbitrary preference profile into a complete and transitive ranking over X. When a preference profile ≿= (≿1 , · · · , ≿n ) ∈ Dn is given, the aggregate preference is denoted by R(≿), where for any x, y xR(≿)y is read as the aggregate preference puts x to be at least as good as y, and xR(≿)y and not yR(≿)x

CHAPTER 30. AGGREGATION OF PREFERENCES

394

is read as the aggregate preference puts x over y, and we adopt a short-hand notation for this xP (≿)y Let P denote the subset of R, which consists of all the preference rankings which are never indifferent between any two alternatives. Let us call such preferences strict preferences. Now using the formal notations given above let us write down the four axiom and the theorem statement. Unrestricted Domain: D ⊃ P Order: For all ≿= (≿1 , · · · , ≿n ) ∈ Dn and x, y ∈ X, R(≿) is complete and transitive. — While this is already a part of the definition of the aggregation rule, since it takes value in R. Pareto: For all ≿= (≿1 , · · · , ≿n ) ∈ Dn and x, y ∈ X, if x ≻i y holds for all i = 1, · · · , n then xP (≿)y. Independence of Irrelevant Alternatives (IIA): For all ≿= (≿1 , · · · , ≿n ), ≿′ = (≿′1 , · · · , ≿′n ) ∈ Dn and x, y ∈ X, if x ≿i y ⇐⇒ x ≿′i y holds for all i = 1, · · · , n, then xR(≿)y ⇐⇒ xR(≿′ )y Theorem 30.5 (Arrow’s theorem): Suppose there are at least three individuals and at least three alternatives. Then, if R : Dn → R satisfies Unrestricted Domain, Order, Pareto and IIA there exists some i such that it holds x ≿i y ⇐⇒ xR(≿)y for all ≿= (≿1 , · · · , ≿n ) ∈ Dn and x, y ∈ X. We define the following concepts for the proof. Definition 30.1 A group is said to be semi-decisive for x over y if the aggregate preference ranks x over y whenever all the individuals in the group prefer x to y and all the others prefer y to x. A group is said to be semi-decisive for x over y if the aggregate preference ranks x over y whenever all the individuals in the group prefer x to y. Lemma 30.1 If a group is semi-decisive for x over y, for all z it is decisive for x over z.

CHAPTER 30. AGGREGATION OF PREFERENCES

395

Proof. Suppose group G is semi-decisive for x over y. Consider any preference profile ≿= (≿1 , · · · , ≿n ) such that x ≻i z holds for all i ∈ G. Suppose first that z ̸= x, y, and consider any preference profile ≿′ = (≿′1 , · · · , ≿′n ) such that x ≻′i y ≻′i z y ≻′j x, y ≻′j z

for all i ∈ G, for all j ∈ /G

Then, since G is semi-decisive for x over y we have xP (≿′ )y. Also, since everybody ranks y over z we have yP (≿′ )z from Pareto. By Transitivity, we have xP (≿′ )z. Since only the restriction on ≿′ above is that x ≻′i z for all i ∈ G and there is no restriction on how the individuals other than G rank between x and z, we can take ≿′ in the above argument so that x ≿′i z ⇐⇒ x ≿i z holds for all i = 1, · · · , n. Hence by IIA we obtain xP (≿)z. Thus we have shown that G is decisive for x over z for any z ̸= x, y. When G is decisive for x over z it immediately implies G is semi-decisive for x over z. Hence by switching between y and z in the above argument we can show that G is decisive for x over y. Lemma 30.2 If a group is semi-decisive for x over y, for all z it is decisive for y over z. Proof. Suppose group G is semi-decisive for x over y. Consider any preference profile ≿= (≿1 , · · · , ≿n ) such that y ≻i z holds for all i ∈ G. Suppose first that z ̸= x, y, and consider any preference profile ≿′ = (≿′1 , · · · , ≿′n ) such that y ≻′i x ≻′i z z

≻′j

x, y

≻′j

z

for all i ∈ G, for all j ∈ /G

Then, from Lemma 30.1 G is decisive for x over z, implying xP (≿′ )z. Also, since everybody ranks y over x we have yP (≿′ )x from Pareto. By Transitivity, we have yP (≿′ )z. Since only the restriction on ≿′ above is that y ≻′i z for all i ∈ G and there is no restriction on how the individuals other than G rank between y and z, we can take ≿′ in the above argument so that y ≿′i z ⇐⇒ y ≿i z holds for all i = 1, · · · , n. Hence by IIA we obtain yP (≿)z. Thus we have shown that G is decisive for y over z for any z ̸= x, y. When G is decisive for y over z it immediately implies G is semi-decisive for y over z. Hence by applying Lemma 30.1 with taking x = y, y = z and z = x we can show that G is decisive for y over x.

CHAPTER 30. AGGREGATION OF PREFERENCES

396

Lemma 30.3 If a group is semi-decisive for x over y, for all v, w it is decisive for v over w. Proof. Suppose group G is semi-decisive for x over y. The case of v = x and w = y and the case of v = y and w = x have been already shown. Thus, let v, w ̸= x, y. Then from Lemma 30.1 G is decisive for x over v. Then, since G is semi-decisive for x over v, by applying Lemma 30.2 by replacing y by v we can show that G is decisive for v over w. Proof of the theorem. From Lemma 30.3, the proof is done if we can show that there is an individual who is decisive for some x over some y. It is clear from Lemma 30.3 that such individual is only one if exists. From Pareto, the whole society is semi-decisive for any alternative over any other alternative. For x, y, let λ(x, y) be the minimal size of groups which are semi-decisive for x over y. Then let λ = min λ(x, y) (x,y)

The remaining is to show that λ = 1. Suppose λ > 1 and let G be a group with size λ such that for some x, y it is semi-decisive for x over y. Pick any i ∈ G, and consider any preference profile ≿= (≿1 , · · · , ≿n ) such that x ≻i y ≻i z z ≻j x ≻j y y ≻k z ≻k x

for all j ∈ G \ {i} for all k ∈ /G

where G \ {i} denote the group of people in G except for i. Then, since G is semi-decisive for x over y we have xP (≿)y. Suppose now zP (≿)y, then it implies G \ {i} is semi-decisive for z over y, which contradicts to the minimality of G. Hence we have yR(≿)z. By Transitivity we have xP (≿)z. However, this means that i is semi-decisive for x over z, which again contradicts to the minimality of G.

Chapter 31

Implementability of social choice objectives If you are a ”benevolent” policy maker, you would like to satisfy ”people’s voice” as much as possible. However, the argument in the previous chapter shows that it is a non-obvious problem to aggregate ”people’s voices” into ”people’s voice.” In this chapter, on the other hand, I shift the interest to how to design a mechanism so that ”people’s voices” are transmitted without lying. It will make me look bad to say that ”people” may lie. However, we already saw in the chapter on public good that a badly designed mechanism induces people to lie about the benefits from a project. Normative welfare criteria or fairness are meaningful only when the reported informations are true. It is empty if you boast of achieving some wonderful normative criterion by carrying out a policy which is based on fake informations. Hence this chapter considers the possibility that people may lie in order to manipulate social choice, and what form of social choice is implementable under such possibility.

31.1

Social choice function and mechanism

Denote the set of social outcomes by X, which is assumed to be finite. Let R denote the set of all complete and transitive preference relations over X. Denote the set of preference relations over which social choice functions are defined by D ⊂ R, since we sometimes restrict attention to a subset of preferences with additional properties. Then a social profile of preferences is generically denoted by ≿= (≿1 , · · · , ≿n ) ∈ Dn . A social choice function f : Dn → X is a function which maps any preference profile into an outcome. That is, when a social preference profile is ≿= (≿1 , · · · , ≿n ) ∈ Dn the outcome assigned by the function is f (≿). Definition 31.1 Let the set of social outcomes X be given. Then a mech397

CHAPTER 31. IMPLEMENTATION

398

anism Γ = (S1 , · · · , Sn , g) consists of sets of messages of the individuals S1 , · · · , Sn , where their product is denoted by S = S1 × · · · × Sn , and an outcome function g : S → X. Here I use the word ”message” instead of ”strategy” what I used in explaining the game theory. The reason is that the sets S1 , · · · , Sn themselves are separated from the set of social outcomes X, and they affect social outcomes only through the outcome function g which is designed by the planner. That is, messages themselves have no physical effects on anybody. In game theory as a positive theory, strategy sets consist of all the physical actions which affect the society, including violence let’s say, and it is an analyst’s problem to specify them. On the other hand, the sets of messages are to be set by the planner. Therefore, we might need to worry about the possibility that the individuals take physical actions outside the mechanism, which may affect the social outcomes. It is an open question, though. Direct mechanism is such each individual submits his preference as is message, which is an important kind of mechanism. That is, the direct mechanism is such that Si = D for all i. Then, a social choice function f : Dn → X itself is identified as a direct mechanism.

31.2

Implementation in dominant strategy equilibrium

First, let us think of implementation such that the message to be sent by each individual is optimal for him no matter what messages the others send. In other words, it is to design a mechanism so that nobody needs to worry about strategic interdependence. This is the most demand kind of requirement. In such type of implementation, each individual has to be able to make decision without knowing the other individuals’ preferences. Thus, each individual’s strategy on how to send his message depends only on his preference and not on the others’ ones. Definition 31.2 Given a mechanism Γ = (S1 , · · · , Sn , g), individual i’s privateinformation-dependent message is a function σi : D → Si . That is, we are think of a game with incomplete information in which a strategy take the form like if my true preference is ≿i I will send message σi (≿i ). Given a profile of preferences of those other than i denoted by ≿−i = (≿j )j̸=i , let σ−i (≿−i ) = (σj (≿j ))j̸=i denote the profile of messages to be sent by those other than i.

CHAPTER 31. IMPLEMENTATION

399

Definition 31.3 Given mechanism Γ = (S1 , · · · , Sn , g), a profile of privateinformation-dependent messages σ ∗ = (σ1∗ , · · · , σn∗ ) forms a dominant strategy equilibrium if for all ≿= (≿1 , · · · , ≿n ) ∈ Dn and for all i = 1, · · · , n と si ∈ Si it holds g(σi (≿i ), σ−i (≿−i )) ≿i g(si , σ−i (≿−i )) Definition 31.4 Say that a social choice function f : Dn → X is implementable in dominant strategy equilibrium if there exists a mechanism Γ = (S1 , · · · , Sn , g) and a profile of private-information-dependent messages σ ∗ = (σ1∗ , · · · , σn∗ ) forming a dominant strategy equilibrium such that for all ≿= (≿1 , · · · , ≿n ) ∈ Dn it holds g(σ ∗ (≿)) = f (≿). Definition 31.5 Say that a social choice function f : Dn → X is truthfully implementable in dominant strategy equilibrium if for all ≿= (≿1 , · · · , ≿n ) ∈ Dn and for all i = 1, · · · , n and all ≿′i ∈ D it holds f (≿i , ≿−i ) ≿i f (≿′i , ≿−i ). Let us prove a result called the revelation principle. This states that if a social choice function is implementable in dominant strategy equilibrium it is always truthfully implementable in dominant strategy equilibrium, which means that it if without loss of generality to focus on the direct mechanism. Theorem 31.1 (Revelation principle for implementation in dominant strategy equilibrium) If a social choice function f : Dn → X is implementable in dominant strategy equilibrium then it is truthfully implementable in dominant strategy equilibrium. Proof. Let Γ = (S1 , · · · , Sn , g) be a mechanism for which f is implementable in dominant strategy equilibrium, and let σ ∗ = (σ1∗ , · · · , σn∗ ) denote a profile of private-information-dependent messages which form a dominant strategy equilibrium. By the definition of implementability in dominant strategy equilibrium, for all ≿= (≿1 , · · · , ≿n ) ∈ Dn it holds g(σ ∗ (≿)) = f (≿), and for all i = 1, · · · , n and for all si ∈ Si it holds ∗ ∗ g(σi∗ (≿i ), σ−i (≿−i )) ≿i g(si , σ−i (≿−i ))

Hence for all i = 1, · · · , n and all ≿′i ∈ D, by taking si = σi∗ (≿′i ) in the above, we have ∗ ∗ g(σi∗ (≿i ), σ−i (≿−i )) ≿i g(σi∗ (≿′i ), σ−i (≿−i )) Since g(σ ∗ (≿)) = f (≿) holds for all ≿, we obtain that f (≿i , ≿−i ) ≿i f (≿′i , ≿−i ) holds for all i = 1, · · · , n and all ≿′i ∈ D.

CHAPTER 31. IMPLEMENTATION

400

We already saw the difficulty of implementability in dominant strategy equilibrium in the chapter on public goods. Here I introduce the Gibbard-Satterthwaite theorem in the abstract setting (Gibbard [10], Satterthwaite [30]). Definition 31.6 Social choice function f : Dn → X is が weakly Pareto efficient if for all ≿= (≿1 , · · · , ≿n ) ∈ Dn there is no y ∈ X such that y ≻i f (≿) holds for all i = 1, · · · , n. ”Weakly” means that it leaves the possibility that we can make somebody strictly better off while keeping others indifferent. In this sense it is a weak requirement. It doesn’t really matter here, since we put an additional restriction (which is rather technical) that one is never indifferent between any two alternatives. Let P be the subset of R consisting of preferences which are never indifferent between any two alternatives. Such preferences are called strict preferences. In the domain of strict preferences D = P we have the following result. Theorem 31.2 (Gibbard-Satterthwait theorem): If a social choice function f : P n → X is truthfully implementable in dominant strategy equilibrium and satisfies weak Pareto efficiency then it is dictatorial. That is, there is some i such that for all ≿= (≿1 , · · · , ≿n ) ∈ Dn and for all x ∈ X it holds f (x) ≿i x. How should we interpret this negative result? In general, the implementation problem is harder when it is easier to say whatever you want, and it is easier to say whatever you want when informational incompleteness is severer. Since implementation in dominant strategy equilibrium (via the direct mechanism) requires that it is always optimal for everybody to report his preference truthfully no matter what the others say, it is the hardest thing to do. Thus, the range of what can be implemented becomes larger as the individuals know more each other, like in the Bayesian situation in which they con’t know actual realization of others’ preferences but they know the ex-ante probability distribution of those. For implementability in Bayesian Nash equilibrium, see the corresponding chapter in Mas-Colell, Whinston and Green [21]. Also, it is harder to implement social choice objectives in abstract settings without relying on concrete natures of the objects to be handled, as you have to take logically possible but less likely preferences seriously and it makes easier to say whatever you want. However, when concrete natures of the objects to be handled are known it is harder to say whatever you want, and this makes it easier to implement social choice objectives. For example, in the problem of allocating indivisible objects in which everybody can get just one item, we already know that the core allocation is weakly Pareto efficient and obtained vby the top-trading-cycle mechanism in which it is a dominant strategy for everybody to submit his true preference. Also, when individual preferences are single-peaked it is known that selecting the median of peaks of reported preference is truthfully implementable in dominant strategy equilibrium.

CHAPTER 31. IMPLEMENTATION

31.3

401

Implementation in Nash equilibrium and allowing multiple equilibria

Nash equilibrium assumes complete information among the players, meaning that they know each other’s preferences and characters. What do I mean by adopting such equilibrium concept when the issue is that we don’t know people’s preferences? Here I’m talking about a situation in which ”people” know each other’s characters but the policy maker is isolated from them and has no information about that. To understand, imagine a society like classroom, in which students know each other’s true characters but the teacher is ignorant of that. In this sense implementation in Nash equilibrium deals with asymmetry of information between ”people” and and the policy maker. As an implementability condition it is less reliable than implementability in dominant strategy equilibrium or Bayesian-Nash equilibrium, since it requires people to have more knowledge about each other and more rationality. On the other hand, if people’s mutual knowledge and rationality are reliable, utilizing that may broaden the possibility of implementation. Unfortunately, even for a social choice function which is implementable in Nash equilibrium we have the same result as the Gibbard-Satterthwaite theorem. However, if we allow a set of outcomes to be given to a given profile of preferences, that is, if we consider a social choice correspondence instead of a function, we may escape from dictatorship, by allowing implementation in multiple equilibria. Definition 31.7 Given mechanism Γ = (S1 , · · · , Sn , g) and a preference profile ≿= (≿1 , · · · , ≿n ) ∈ Rn , message profile s = (s1 , · · · , sn ) is said to be a Nash equilibrium if it holds g(si , s−i ) ≿i g(s′i , s−i ) for all i = 1, · · · , n and s′i ∈ Si . Also, let N E(≿, Γ) denote the set of Nash equilibrium message profiles given Γ and ≿. Definition 31.8 Social choice correspondence F : Dn ↠ X is said to be Nash implementable if there exists a mechanism Γ = (S1 , · · · , Sn , g) such that for all ≿= (≿1 , · · · , ≿n ) ∈ Dn it holds g(N E(≿, Γ)) = F (≿), where g(N E(≿, Γ)) = {g(s) : s ∈ N E(≿, Γ)}. It is known that the following monotonicity condition is necessary for Nash implementability.

CHAPTER 31. IMPLEMENTATION

402

Definition 31.9 Social choice correspondence F : Dn ↠ X is said to be monotonic if for all ≿= (≿1 , · · · , ≿n ) ∈ Dn , x ∈ F (≿) and for all ≿′ = (≿′1 , · · · , ≿′n ) ∈ Dn such that x ≿i y =⇒ x ≿′i y holds for all i = 1, · · · , n and y ∈ X, it holds x ∈ F (≿′ ). Example 31.1 The correspondence which maps each preference profile to the set of weakly Pareto-efficient outcomes given that is monotonic. Given an arbitrary preference profile ≿= (≿1 , · · · , ≿n ) denote the set of weakly Pareto-efficient outcomes under it is denoted by P (≿). Pick any x ∈ P (≿), and consider any ≿′ = (≿′1 , · · · , ≿′n ) such that x ≿i y =⇒ x ≿′i y holds for all i = 1, · · · , n and y ∈ X. Now suppose x is not weakly Pareto-efficient under ≿′ , that is, there exists y such that y ≻′i x for all i = 1, · · · , n. Then from the above assumption (by taking the contrapositive) we have y ≻i x for all i = 1, · · · , n, which contradicts to x being weakly Pareto-efficient. Thus we obtain x ∈ P (≿′ ). Theorem 31.3 If social choice correspondence F : Dn ↠ X is Nash implementable then it is monotonic. Proof. Pick any ≿= (≿1 , · · · , ≿n ) ∈ Dn and x ∈ F (≿). Pick any ≿′ = (≿′1 , · · · , ≿′n ) ∈ Dn such that for all i = 1, · · · , n and y ∈ X it holds x ≿i y =⇒ x ≿′i y. Take s ∈ N E(≿, Γ) such that x = g(s), then for all i = 1, · · · , n and s′i ∈ Si it holds g(si , s−i ) ≿i g(s′i , s−i ) By assumption we have g(si , s−i ) ≿′i g(s′i , s−i ), Since i = 1, · · · , n and s′i ∈ Si are arbitrary, we obtain s ∈ N E(≿′ , Γ), implying x ∈ F (≿′ ). It is shown by Maskin [22] that together with the following condition monotonicity is sufficient for Nash implementability. Definition 31.10 Social choice correspondence F : Dn ↠ X is said to allow no veto power if for all ≿ and x ∈ X such that x is maximal element for n − 1 individuals it holds x ∈ F (≿) regardless of the remaining one’s preference.

CHAPTER 31. IMPLEMENTATION

403

Theorem 31.4 Assume n ≥ 3 and D = R. Then if social choice correspondence F : Dn ↠ X satisfies monotonicity and no veto power it is Nash implementable. Proof. We follow the proof by Repullo [28]. For each individual i = 1, · · · , n let the set of his messages be Si = Rn × X × N. That is, each i announces a list of all the individuals’ preferences, a social outcome and a natural number. Denote the list of all the individuals’ preferences which i submits by ≿i = (≿i1 , · · · , ≿in ), let the social outcome he submits by xi , and the number he submits by k i , then his message is denoted by si = (≿i , xi , k i ). Outcome function g : S → X is defined as follows. Given any message profile s = ((≿1 , x1 , k 1 ), · · · , (≿n , xn , k n )), 1. if all but one individual, denoted i, report the same message (≿, x, k) and if x ∈ F (≿), then let g(s) = xi if x ≿i xi , based on the preference profile ≿ reported by those n − 1 individuals, and let g(s) = x otherwise. 2. Otherwise, pick i who reported the largest number and let g(s) = xi , while tie-breaking is carried out based on certain priority ranking determined beforehand. First we show F (≿) ⊂ g(N E(≿, Γ)). Pick any ≿ and x ∈ F (≿). Consider a message profile such that all the individuals submit the same message (≿, x, 1). Then for any individual Rule 1 applies if he deviates, implying that he cannot change x by deviation, or can change it only to xi such that x ≿i xi , which is unprofitable. Hence this message profile is a Nash equilibrium, and implements x. Next we show g(N E(≿, Γ)) ⊂ F (≿). Let s ∈ N E(≿, Γ). Case 1: Suppose all the individuals report the same message (≿′ , x, k) and that x ∈ F (≿′ ). Pick an arbitrary i and consider a message s′i = (≿′ , y, k) with y being an arbitrary outcome satisfying x ≿′i y. Then from Rule 1 in the definition of the outcome function we have g(s′i , s−i ) = y. Since s ∈ N E(≿, Γ) we have x ≿i y. Hence by monotonicity it follows x ∈ F (≿). Case 2: Suppose all the individuals report the same message (≿′ , x, k) and that x∈ / F (≿′ ). Pick an arbitrary i and consider a message s′i = (≿′ , y, k ′ ) with y being an arbitrary outcome and k ′ > k. Then from Rule 2 in the definition of the outcome function we have g(s′i , s−i ) = y. Since s ∈ N E(≿, Γ) we have x ≿i y. Hence x is a maximal element according to ≿i . Since i was arbitrary, from the no-veto-power condition it follows x ∈ F (≿). Case 3: Suppose si ̸= sj for some i, j. Without loss of generality, let s1 ̸= s2 .

CHAPTER 31. IMPLEMENTATION

404

Pick arbitrary i ̸= 1, 2, y and k ′ > maxj̸=i k j , and consider a message of i given by s′i = (≿′ , y, k ′ ). Then from Rule 2 we obtain g(s′i , s−i ) = y. Since s ∈ N E(≿, Γ) it follows x ≿i y. Therefore x is a maximal element for i with respect to ≿i . Since either si ̸= s1 or si ̸= s2 has to hold without loss of generality let si ̸= s1 . Pick arbitrary y and k ′ > maxjne2 k j , and consider a message of 2 given by s′2 = (≿′ , y, k ′ ). Then from Rule 2 we obtain g(s′2 , s−2 ) = y. Since s ∈ N E(≿, Γ) it follows x ≿2 y. Therefore x is a maximal element for 2 with respect to ≿2 . Thus, x is a maximal element for all i but 1 with respect to ≿i . Therefore, from the no-veto-power condition it follows x ∈ F (≿).

31.4

Appendix: Proof of the Gibbard-Satterthwaite theorem

Here we follow the proof in Mas-Colell, Whinston and Green [21]. Definition 31.11 Social choice function f : Dn → X is said to be monotonic if for all ≿= (≿1 , · · · , ≿n ) and ≿′ = (≿′1 , · · · , ≿′n ) ∈ Dn such that for all i = 1, · · · , n and y ∈ X it holds f (≿) ≿i y =⇒ f (≿) ≿′i y, it holds f (≿′ ) = f (≿). Lemma 31.1 If a social choice function f : P n → X is truthfully implementable in dominant strategy equilibrium then it is monotonic. Proof. Consider ≿= (≿1 , · · · , ≿n ), ≿′ = (≿′1 , · · · , ≿′n ) ∈ P n such that for any i = 1, · · · , n and y ∈ X it holds f (≿) ≿i y =⇒ f (≿) ≿′i y. Then from truthful implementability we have f (≿1 , ≿2 , · · · , ≿n ) ≿1 f (≿′1 , ≿2 , · · · , ≿n ) From the assumption, it implies f (≿1 , ≿2 , · · · , ≿n ) ≿′1 f (≿′1 , ≿2 , · · · , ≿n ) On the other hand, from truthful implementability again we have f (≿′1 , ≿2 , · · · , ≿n ) ≿′1 f (≿1 , ≿2 , · · · , ≿n ) Since any ≿1 ∈ P in the domain of strict preferences cannot be indifferent between two distinct alternatives, we have f (≿′1 , ≿2 , · · · , ≿n ) = f (≿1 , ≿2 , · · · , ≿n ). By doing this for Individual 2, we obtain f (≿′1 , ≿′2 , · · · , ≿n ) = f (≿1 , ≿2 , · · · , ≿n ). By repeating the argument, we obtain the conclusion.

CHAPTER 31. IMPLEMENTATION

405

Definition 31.12 Given a preference profile ≿= (≿1 , · · · , ≿n ) and a set of alternatives X ′ ⊂ X, say that preference profile ≿′ = (≿′1 , · · · , ≿′n ) is a X ′ majorization of ≿ if ≿′ agrees with ≿ over X ′ and everybody prefers any alternative in X ′ over any alternative outside of X ′ . Lemma 31.2 Suppose social choice function f : Dn → X satisfies monotonicity and weak Pareto efficiency. Then, for any preference profile ≿ and any set of alternatives X ′ ⊂ X, for any preference profiles ≿′ = (≿′1 , · · · , ≿′n ) and ≿′′ = (≿′′1 , · · · , ≿′′n ) which are X ′ -majorization of ≿, it holds f (≿′ ) = f (≿′′ ). Proof. From weak Pareto efficiency we have f (≿′ ) ∈ X ′ . Now pick any y ∈ X and any i, and suppose f (≿′ ) ≿′i y. Then, if y ∈ X ′ since ≿′ and ≿′′ agree over X ′ we have f (≿′ ) ≿′′ y. Also, if y∈ / X ′ since it is inferior to any alternative in X ′ under ≿′′ we have f (≿′ ) ≻′′ y. Therefore the assumption of monotonicity is met between ≿′ and ≿′′ , leading to the conclusion. Definition 31.13 Given a social choice function f : Dn → X, define a preference aggregation rule generated by f , denoted Rf : Dn → R, as follows: for all ≿∈ Dn and x, y ∈ X, let xRf (≿)y either if x = y or if x = f (≿′ ) holds for some ≿′ being a {x, y}-majorization of ≿. From the above lemma this definition does not depend on how to take a {x, y}majorization of ≿. Lemma 31.3 For all ≿∈ Dn , Rf (≿)is a complete and transitive strict preference. Proof. Completeness: For any ≿ and x, y, take any its {x, y}- majorization, denoted by ≿′ , then from weak Pareto efficiency we have f (≿′ ) ∈ {x, y}. Thus either xRf (≿)y or yRf (≿)y holds. Strict preference: Moreover, since we cannot have both x = f (≿′ ) and y = f (≿′ ), we cannot have both xRf (≿)y and yRf (≿)y. Transitivity: Pick any ≿ and x, y, z, and suppose xRf (≿)y and yRf (≿)z. Take any {x, y, z}-majorization of ≿, denoted by ≿′′ , then from weak Pareto efficiency it follows f (≿′′ ) ∈ {x, y, z}. Suppose f (≿′′ ) = y. Take a {x, y}-majorization of ≿, denoted by ≿′ . Then since ≿′′ is a {x, y}-majorization of ≿ as well it follows from Lemma 31.2 that f (≿′ ) = f (≿′′ ) = y. However, xRf (≿)y on the other hand means x = f (≿′ ), a contradiction. Hence f (≿′′ ) ̸= y. Likewise we obtain f (≿′′ ) ̸= z, implying f (≿′′ ) = x. Now, pick any {x, z}-majorization of ≿, denoted by ≿′ . Since ≿′′ is a {x, z}majorization of ≿ as well it follows from Lemma 31.2 that f (≿′ ) = f (≿′′ ) = x, which implies xRf (≿)z. Lemma 31.4 Rf satisfies the Pareto principle.

CHAPTER 31. IMPLEMENTATION

406

Proof. Pick any ≿ and x, y, and suppose x ≻i y for all i. Take any {x, y}majorization of ≿, denoted by ≿′ , then from the weak Pareto efficiency it follows f (≿′ )x, which implies xRf (≿)y. Since Rf (≿) does not allow indifference, we obtain xPf (≿)y. Lemma 31.5 Rf satisfies Independence of Irrelevant Alternatives. Proof. Pick any ≿ and x, y, and take any ≿′ which agrees with ≿ over {x, y}. Take any {x, y}-majorization of ≿, denoted by ≿′′ . Without loss of generality, let f (≿′′ ) = x, meaning xRf (≿)y. Since ≿′′ is a {x, y}-majorization of ≿′ as well, xRf (≿)y and xRf (≿′ )y are equivalent. Lemma 31.6 If a social choice function f : P n → X satisfies monotonicity and weak Pareto efficiency it is dictatorial. Proof. From Arrow’s theorem, the preference aggregation rule Rf : Dn → R generated by f is dictatorial. Let i∗ be the dictator there, and for any ≿ let x denote the maximal element for ≿∗i , then for all y ̸= x it holds xRf (≿)y. Suppose now that f (≿) ̸= x, then by taking any {x, f (≿)}-majorization of ≿, denoted by ≿′ , we obtain f (≿′ ) = f (≿), which means f (≿)Rf (≿)x, a contradiction.

Postscripts There are many issues which I could not cover in this book, partly because it is intended to be an intermediate textbook, partly because of the limitation on my knowledge and ability. Here I list such issues and raise relevant books and articles, in order to help you to go to the next step. The choice is subjective and by no means exhaustive. First, let me refer to an advanced textbook which should be read after this. Mas-Colell, Whinston and Green (MWG),“Microeconomic Theory” [21] This has been the most popular textbook at graduate level in the last two decades. It’s a fat book even in the American standard, but it is the most comprehensive one.

Game theory General textbook on game theory I would recommend is Osborne and Rubinstein (OR) [24] or Fudenberg and Tirole (FT) [6] — it’s pretty much a matter of taste which one you choose, OR is more concise and FT may be more exhaustive. Also, FT covers more about incomplete information while OR focuses more on complete information. Although, game theory chapters of MWG may be enough to cover the first-year graduate course materials. Brief illustrations of specific fields of game theory and recommended readings follow. Equilibrium refinement As discussed in the text there may be many Nash equilibria in a game, and some of them are unlikely. For example, Nash equilibrium allows that a weakly dominated strategy is played, which will be unrealistic as you worry that the opponents may not precisely play the strategies to which such dominated strategy is optimal. Also, Nash equilibria in the normal-form expression of an extensiveform game may not be subgame-perfect. The theory of equilibrium refinement proposes concepts on robustness of equilibria to various kinds of errors or perturbations, and attempts to narrow 407

POSTSCRIPTS

408

down the set of equilibria based on them. After reading the corresponding chapters in OR or FT, you might want to go into a comprehensive book on this literature such as Van Damme [35]. Equilibrium selection The theory of equilibrium selection has a similar objective as the theory of equilibrium refinement, in the sense that both propose criteria to narrow down the set of equilibria, but it has a flavor more to bring in the selection criteria from ”outside” of the purely formal arguments on players’ optimization in games. Focal point is one such notion. Also, risk dominance as explained in the text is bringing ”risk attitudes” which is somewhat outside of the description of the game, since the standard kind of risk attitude is already taken into account in the description of payoffs. The most reputable book in the literature is Harsanyi and Selten [11]. Epistemic game theory Epistemic game theory attempts to make it precise what level of rationality indeed allows us to play a course of actions such as Nash equilibrum or other solutions. In the first chapter on game theory I gave a crude explanation on the relationship between iterated elimination of dominated strategy (or rationalizability) and common knowledge of rationality. In the chapter on incomplete information, I gave a crude explanation of what are behind the common prior assumption and any possible departure from it. Also I covered some remarkable implication of common knowledge under the common prior assumption, such as the impossibility of agreeing to disagreeing and the impossibility of speculative trades. In order to go further we need a precise definition of knowledge and common knowledge in order to handle this. You can start with the chapters on knowledge in OR and FT. Repeated games In the text I gave an illustration of how cooperation is sustained when the game is repeatedly played indefinitely many times, by means of a pair of trigger strategies. This argument is generalized into so-called folk theorem, where ”folk” means it had been informally known for a long time: any payoff profile being Pareto superior to the equilibrium payoffs in the one-shot setting is sustainable by means of a strategy profile which consists of a more sophisticated version of the trigger strategy. The basic argument as illustrated in the text assumes so-called perfect monitoring, meaning that every player observes and remembers all of what all players did before. Also it assumes complete information, meaning that every player knows all the players’ characteristics. The literature proceeds by considering imperfect monitoring or/and incomplete information.

POSTSCRIPTS

409

You can start with the chapters on repeated games in MWG, OR and FT. Then you might want to proceed to a reputable advanced textbook such as Mailath and Samuelson [20]. Evolutionary games One can interpret equilibrium in games as a consequence of imitation or learning process with trial and error at a population level, which is called evolutionary dynamic, rather than a consequence of deductive reasonings by individuals. Then equilibrium is thought to be a course of action which is stable to ”invasions.” Evolutionary game theory, which was originated in the field of mathematical biology, analyzes population dynamics of replication and characterize equilibria that are stable to invasions. Since it was imported to economics, economicsbased game theorists have worked on evolutionary dynamics which are closer to human responses and learning processes rather than direct biological processes. Such notions of stability are also related to equilibrium refinement and equilibrium selection. For the basics of evolutionary game theory you can consult Weibull [38]. Also, Vega-Redondo [36] puts more emphasis on implications to economic behavior. Cooperative games The games covered in the book are called non-cooperative games. On the other hand, there is a literature called cooperative game theory. It starts with describing what are attainable for any possible coalition, rather than starting with describing strategies, proposes solution concepts, and provides (axiomatic) characterizations. The key notion there is coalitional stability. When a coalition is to receive an outcome which is worse than what they can achieve by themselves, they will block the current proposal. A stable outcome is such that no coalition can block it. The literature also considers additional distributional conditions and obtains sharper solutions. There are two interpretations of such cooperative concepts. One is descriptive, in the sense that if an outcome is blocked or faces certain kind of objection by a coalition it will not last, although this is somehow a ”detail-free” argument as it does not provide explicit descriptions of strategies and equilibrium course of actions about how a coalition blocks the current proposal. The other interpretation is that it is a normative goal. Even though an unstable outcome does not last in the long run it takes time to dissolve and it is very costly — once you get married you cannot easily get divorced even if it was a wrong one. Thus it is desirable to look for a stable outcome beforehand, in order to avoid such tragedies, by designing mechanism nicely.

POSTSCRIPTS

410

Mechanism design In general, the party with private information is called agent and the party which cannot observe the agent’s private information (either his type or action) is called principal. For example, in the moral hazard problem the principal is the employer and the agent is the employee, and in the auction problem the seller is the principal and the bidders are the agents. Since the principal cannot observe or verify the agents’ private information he has to design a mechanism so that it is profitable for them to reveal their private information by their choices. This constraint is called incentive constraint. In addition to the incentive constraint, being subject to the participation constraint that it is profitable for agents to sign the contract rather than opting out (otherwise the contract doesn’t make), the principal designs the mechanism in order to maximize his payoff. This is what contract theory is about. An important subject to which the mechanism design approach is relevant is auction. Here the seller is to design the auction format in order to maximize his expected revenue. Since the seller does not know the bidders’ willingness to pay, he has to design auction so that bids reveal willingness to pay nicely. For this direction you can start with the corresponding chapters in MWG and FT. Then you might move onto reputable books such as Bolton and Dewatripont [4] on contract theory and Krishna [16] on auction theory. Mechanism design approach is adopted not only in analyzing the principals’ profit maximization but also in looking for a solution in the problems in which the principal is interpreted to be a planner whose objective is to achieve given normative requirements. Recall for example that efficient level of provision of pubic good is characterized by the Samuelson condition if individuals’ preferences are known. However, there is a gap between this and how to implement it, for the policy maker, the principal here, does not know those preferences. Hence he has to know people’s preferences either directly or indirectly, but there is no guarantee that people truthfully report their preferences. Thus it is necessary to design a game in which people choose to report true preferences by their choices, which is the incentive constraint. Pivotal mechanism is one such example, but we already saw that we have to give up efficient resource allocation at least partially there. Mechanism design theory in this direction takes incentive constraint as the basic condition and investigates if efficiency and other normative postulates are implementable, and how the implementing mechanisms look like. For this direction of mechanism design, I recommend to read the corresponding chapters in MWG, OR and FT. Mechanism design theory has also a role to complement the standard market theory.

POSTSCRIPTS

411

As is discussed in the book the theory of competitive market leaves it unspecified who sets the prices in what procedure. In other words, it is a theory about situations in which whoever sets the prices in whatever procedure the resulting prices have to fall in certain values. However, as we saw the section on social calculation debate such model does not have a formal distinction from the model in which a central planner plays the roles of both competitive firms and auctioneer. This necessitates to provide an explicit description of setting prices either as players’ strategies or a part of the rule of the game, that is, as a ”visible hand” instead of an ”invisible hand.” It necessitates to investigate what type of mechanism indeed implements competitive equilibrium allocation or efficient allocation, and which one is more ”informationally efficient” if if there exist several such things.

Political economics It is often said, ”economists very often lament that politicians do not choose ‘right’ economic policy, but isn’t it naive to say so without thing about why such policy is not chosen in the arena of politics?” Of course this is a problem. Political Economics (not Political Economy) is hence a popular research program now, which approaches to the interaction between political process and economic dynamics from positive viewpoints rather than normative. Persson and Tabellini [26] is a representative textbook in this direction.

General equilibrium theory and incomplete markets This book covered the analysis of saving and investment, determination of interest rates and asset prices, with the two-period or two-state model. This is ready to be extended to the cases of many periods and many states, and such extended theory is called dynamic general equilibrium (DGE) model or dynamic stochastic general equilibrium (DSGE) mode, which is one of the main frameworks of current macroeconomics. — I’m tempted to say that it is a unifying framework, but I wouldn’t, because of the reasons stated later. There are misunderstanding like ”since general equilibrium theory assumes the law of one price it cannot deal with price fluctuation or business cycle over time.” As explained in this book, however, even if goods are materially the same they are treated as different goods if they are to be delivered at different periods or contingencies. Thus in the extended model it is rather natural that prices differ across time periods and contingencies. However, such extension to multiple periods and multiple states of the world generates problems as well.

POSTSCRIPTS

412

Recall that borrowing is an action to buy current consumptions by means of selling future consumptions. When we extend the model to multiple periods it means we are assuming for example that you can buy current consumptions by means of selling consumptions ten years later. This apparently unimaginable action is actually more realistic than you may think, since we are doing this indirectly through renewing debt period by period. However, it is also true that borrowing-lending markets may not be that extensive and smooth. For example, even if you have 1 millions of lifetime income in the present value it is a different question if you can borrow 1 millions in the financial market. When your borrowing ability does not catch up with your earning you are said to be subject to a liquidity constraint. When you are subject to a liquidity constraint you may not buy current consumptions by means of selling consumptions ten years later. Also, the standard market theory assumes unconditionally that signed contracts are just carried out, or that it is costless to enforce signed contracts. This will not always apply to debt contracts, especially for sovereign debts, where it is costly for the lenders to do collect the debts, and they sometimes can collect only a part of those. It is called limited enforcement. In such cases you may not buy current consumptions by means of selling consumptions ten years later, since you cannot commit to have your consumptions ten years later to be collected for payment. The property that every good can be exchanged for any other good is called market completeness. The markets covered in this book are all complete, except for some examples I explained. When the market participants are subject to liquidity constraints the completeness does not however. Also, with regard to the case of limited enforcement of contracts we would say that the markets are endogenously incomplete. The problem of market incompleteness emerges also when the model is extended to the case of multiple states of the world. Assuming market completeness under uncertainty means we are assuming for example that we can buy gasoline available if Democrats win by means of selling gasoline available if Republicans win. This apparently unimaginable action is actually more realistic than you may think, since we are doing this indirectly through buying and selling securities: we are buying gasoline available upon Democrats’s winning and selling gasoline available upon Republicans’ winning, by means of buying securities positively related to Democrats’ winning and selling securities positively related to Republican’s winning. It is also true, however, that when there are many states of the world there may be no market or no security corresponding to such states. In such cases you cannot buy securities related to state X or sell securities related to state Y. When it is the case we say that the security markets are incomplete.

POSTSCRIPTS

413

When markets are incomplete equilibrium allocation there is not Pareto efficient even under perfect competition. The intuition is that there do not exist markets for some goods despite some people want to buy them or sell them. In the context of uncertainty it means that there are not enough devices for risk-hedging. This leads us to think about how much efficient competitive equilibrium allocation is then. It is known that competitive equilibrium allocation is Pareto efficient within the given constraint on exchangeability. This is called constrained Pareto efficiency. Thus Pareto efficiency of competitive equilibrium allocations in complete markets is viewed as a special case of this property. A ”neoclassical” intuition would then tells us that people get better off when markets become ”more complete,” in particular when new securities become available to trade. This intuition is actually wrong. Some people may get hurt when markets for new securities are opened. Not only that, Hart [12] gives an example that opening markets hurt everybody. Thus, competitive market achieves Pareto efficiency under given degree of exchangeability but it does not simply possess the property that more degrees of exchangeability is better. This leads us to investigate how we should innovate financial technologies so that everybody gets better off. Another problem arising due to market incompleteness is about objective and ownership of firms. When markets are complete the notion of profit is determined uniquely, hence there is a unanimous agreement among the shareholders of any firm that it should maximize the profit. However, the notion of ”profit” is not unique when markets are incomplete. While maximizing short-run profit period by period and maximizing long-run profit overall coincide when markets are complete, they do not in general when markets are incomplete. Thus there may be a conflict even among shareholders about ”which profit” should be maximized. This leads us to attempt to model entrepreneurs’ behavior and the process of determination of dividend policy within the firm explicitly. As an introduction to dynamic general equilibrium theory and theory of incomplete markets I would recommend the corresponding chapters in MWG. Then you can proceed to for example Ljungqvist and Sargent [18].

Search and frictions The models of markets covered in this book, whether they are perfectly competitive or imperfectly competitive, deal with how allocations of goods are determined, but stay silent about how objects are indeed exchanged. That is, as far as we take the model literally it means that determination of price and contract signing on the allocation of goods are carried out on the first day and they simply commit to it, where actual exchange is no more than an execution of an already signed contract. It assumes such a frictionless world,

POSTSCRIPTS

414

although it is an accepted first step in any field to start with a frictionless setting. There are two things which are necessarily lost by such modeling of frictionless world, money and unemployment. I used the words like dollars in this book, but it refers to just one of the three functions of money normally defined as 1. unit of account 2. store of value 3. medium of exchange That is, I have talked only about 1 at most. This is the view that money is no more than a ”proxy” of real goods and the gap between how goods are allocated and how they are exchanged is no more than a matter of appearance. In other words, in such model money is no more than a slip showing that the owner has enough goods or earning which correspond to the value of the good to purchase, or that one allows to spend certain portion of income on the good to purchase, and exchange by means of money is no more than a transaction of slips carried out in order to execute already signed contracts. In such a world, there no essential role for money. Also, in such modeling of the world unemployment is no more than a ”voluntary choice,” in that people are simply deciding whether to work or not given the market wages (which might be true to some extent). On the other hand, search theory takes the gap between how goods are allocated and how objects are exchanged seriously. If exchange of objects has to be done without money, it can occur only when ”double coincidence of wants” happens, but there is no guarantee that such thing happens nicely, meaning that search cost has to be taken seriously. Likewise, there is no guarantee that job-seekers and potential employers can meet nicely meaning that search cost has to be taken seriously here as well. In such situations, money has a significant role since it reduces search cost. Also the presence of search cost sometimes overturns the conclusion by the frictionless models. As an introduction I would recommend the corresponding chapters in Ljungqvist and Sargent [18].

So-called bounded rationality I have put ”so-called” of course because I worry that we are not communicating with each other as each one uses own definition of ”rationality” in an indefinite way without making it precise.

POSTSCRIPTS

415

Although there had been always criticisms to the assumption of rationality, serious researches of so-called bounded rationality emerged only after 90’s. My understanding of the reason for this is that the modeling alternatives to perfect rationality had tended to trace behavioral phenomena only as a totality which is driven automatically, and had left it unclear what departures from rationality we are making. Resent studies rather start with specifying ”minimal departure” from the assumption of rationality, and try to find its implication. For general introduction I would recommend Rubinstein [27]. Now let me come back to the four criteria of ””rationality” in economics” which I summarized in Chapter 1, saying 1. an individual has certain consistent subjective criterion of value (called preference); 2. he takes all the relevant contingencies into account and perceive them correctly; 3. he goes through ”logically correct” reasonings; and 4. he fulfills the criterion up to the maximum. It won’t be obvious if the departure from Criterion 1, lack of consistency such as time inconsistency, should be called ”lack of rationality” by itself. Lack of consistency itself may arise even when the individual is fully self-aware of it.1 When lack of consistency arises because of unawareness it would rather be a departure from Criterion 2. In any case, I would include the departure from Criterion 1 to ”bounded rationality.” Departure from Criterion 2 is important particularly under uncertainty. As is explained above, when we extend the standard theory of market to uncertainty we are thinking for example of ”gasoline available if Republicans win” and ”gasoline available if Democrats win.” However, could we trade for example ”gasoline available if 911 happens?” Not only that such thing was not available to trade it would not have been in many people’s mind. Not having something in mind is different from being aware of it and believing it cannot happen, that is, putting zero subjective probability on it. Unawareness is different from uncertainty. Departure from Criterion 3 is for example that people can not or do not do ”right” revision of probability distributions, as in the well-known problem of false positive. Let’s say there is a disease which occurs to 1 person in 10000. There is a test for this disease, which shows positive response with 90% probability when the subject indeed has the disease, but also shows positive 1 Although ”empirically” such individual is typically unaware of such inconsistency and gets aware of it only afterward.

POSTSCRIPTS

416

response with 5% probability when he does not have the disease (which is the false positive case). Now, given that one gets a positive response, what is the probability that he indeed as the disease? ”Correct answer” follows from Bayes rule, as

= = = ≈

P (Disease|Positive) P (Disease and Positive) P (Positive) P (Disease)P (Positive|Disease) P (Disease)P (Positive|Disease) + P (Non-disease)P (Positive|Non-disease) 0.0001 × 0.9 0.0001 × 0.9 + 0.9999 × 0.05 0.0018

It is typical, however, that even professionally trained medics answer numbers like 90%, when they answer only by intuition. Departure from Criterion 4 refers to that even though the decision maker has full knowledge and information and can go through ”correct” logic, but he does not have enough computational capacity to derive conclusion that are to be deduced from given information and logical rule. Consider the following example Question C: Which one do you choose? C1: Receive 100 dollars if the 100000000th decimal of π is even, and nothing otherwise. C1: Receive 100 dollars if the 100000000th decimal of π is odd, and nothing otherwise. and suppose you know the infinite series formula for π. Under the assumption that once necessary knowledge and information is given one can deduce the conclusion which is logically implied by these, you should be able to give a clear answer to the above problem. However, knowing the principle and rule of computation and having the capacity to carry out actual computation are different issues. So far I raised various departures from ”rationality” in both theoretical and empirical senses. However, let me emphasize that from the viewpoint of economic theory we cannot ”just” give up or weaken the assumption of rationality as noted above. What do I mean?

POSTSCRIPTS

417

First, departures from ”rationality” must generate various conflicts within an individual, which necessitates to think of how such conflicts are or should be resolved ”rationally.” An argument which does not contain or suffer from such conflicts in itself is no more than a transparent model of ”mere stupid,” which is of no value as theoretical understanding even if it is a ”truth” about human behavior. Being a ”truth” and being of theoretical interest are different. There does exist a ”trivial truth.” Note, however, that this applies to the rationality approach as well. For example, in the experimental literature it is known that actual bids in auction experiments are significantly higher than the theoretical prediction. It is transparent if you say that is because bidders like to win, which is ”trivially rational.” Also, in the literature of political behavior it has been an open question why people go to vote by paying certain cost despite that the probability of any voter’s vote being pivotal is almost zero. It is transparent if you say that is because voters like to vote, which is again ”trivially rational.” Now let me come back to the issue. The time inconsistency problem necessitates to think of a contradiction of why the current self makes a plan which has to be overturned by his future selves. If a man just has a wrong belief that he can control his future selves, and just makes a plan such as ”I quit smoking tomorrow” ”I start dieting tomorrow” everyday, and just fails to execute it, and nevertheless he just continues to have the same belief no matter how many times he fails, such transparent explanation of behavior is not of theoretical interest even if it is a ”truth.” Similarly for the issue of ignorance. If a man is just ignorant, after knowing what he did not know before he will cancel his previous plan and rebuild a new one. But then, it again necessitates to think of a contradiction of why the current self makes a plan which has to be overturned by his future selves, even though it is true that we don’t know what we don’t know and we cannot take it into account what we don’t know. However, it is not obvious how we can go beyond the trivial claim like ”we don’t know what we don’t know,” since it is not obvious how we can formalize a statement such as ”we don’t know what we don’t know but we know that we don’t know something” because if we could formalize it that means we already ”know” this ”something” what we don’t know. Also, one may say that since human thinking is not following the Bayes rule we can use non-Baysian updating in order to model human decision making. You cannot just do it, again, for it is known that following Bayes’ rule in updating probability distributions and being dynamically consistent are equivalent (see Ghirardato [9]). Consider for example that there are equal numbers of Red, Blue and Green balls in a box, each of which is thus drawn with 1/3 prior probability. Suppose

POSTSCRIPTS

418

now that the posterior probability of Red after knowing that the drawn ball is not Green is 2/5, instead of 1/2 derived from the Bayes rule. Then, assuming risk neutrality for simplicity, despite that the decision maker chooses ”90 dollars if Red is Drawn” over ”75 dollars if Blue is drawn” since 9000 × 13 = 3000 > 2500 = 7500 × 13 , after knowing that the ball drawn is not Green he changes his mind since 9000 × 52 = 3600 < 4500 = 7500 × 35 , meaning that himself ex-post does not follow the plan made by himself ex-ante. Again there emerges a contradiction of why the current self makes a plan which has to be overturned by his future selves. The second problem is that if we attempt to describe behaviors of boundedly rational agent as ”problem solving” such problem looks more difficult than the problem being solved by a rational agent. For example, a ”problem” being solved by multiple selves looks more complicated than a ”problem” being solved by a consistent individual. Also, a ”problem” with constraints on the decision maker’s ability of reasoning and computation looks more complicated (since there are more constraints!) than a ”problem” for the decision maker with unlimited ability of reasoning and computation Because of this, the models of ”satisficing” tend to be an apparently more sophisticated model of optimal stopping problem in which the decision maker ”optimally” decides when to stop searching. I guess I’m confusing or misunderstanding about the notion of ”solving” at some point, in the sense let’s say that there are various levels of ”solving” and I am confusing between them. I guess such problems are to be resolved by borrowing helps from neighboring disciplines such as computer sciences.2 The third problem is that under bounded rationality it is not any longer clear what is better or worse, even for a single individual, and nevertheless economics has to do welfare analysis as long as it is economics. From the standpoint of ”rationality” what is better or worse for an individual is simply what is revealed from his choice data. However, under departures from Criterion 1 above there may be disagreements among multiple selves about what is ”good.” Under departures from Criterion 2, an individual does not have knowledge and information enough to judge what is ”good” for him. Under departures from Criteria 3 and 4 due to bias or limitation of capacity in reasoning and computation one may not be able to draw a judgment about if something is ”good” or ”bad.” This leads us to ask the following question: Should it be now allowed that an external authority intervenes the judgment on what is ”good” for an individual? If so, to what extent? In recent years, Thaler and Sunstein [33] advocate a concept what they call ”libertarian paternalism.” This says, given that human choices depend heavily 2 Salant [29] recently shows that if a decision rule which uses smaller computational amounts than the fully rational one has to obey a framing effect. This proves that I’m indeed confusing.

POSTSCRIPTS

419

on how choice frames are given, which is called framing effects, we should induce people’s choice to ”better” ones, by means of manipulating frames. For example, in the enrollment to 401k-type pension plan it is known that the enrollment rate is significantly higher when the default choice is enrollment and one has to sign up when he likes to opt out, than when the default choice is non-enrollment and one has to sign up when he like to enroll. Libertarian paternalism says the former type of framing should be given. It is a paternalism in the sense that an authority intervenes to manipulate framing, but they say it is still libertarian in the sense that it is not enforcing a particular alternative and it is still the individual who chooses. It is nothing but an external authority’s subjective judgment, however, that the object toward which the individual’s choice is induced is ”good,” although the judgment has been relatively straightforward in the existing applications of libertarian paternalism so far, such as health choices with costs being given constants.

Final words I guess it is now the good time to summarize (if I may) the mode of thinking underlying the rationality approach. From the style of the arguments demonstrated in the book, you will notice the following mode of thinking: 1. Take things in a symmetric manner. 2. If you have to break symmetry, imagine an infinite hierarchy behind the apparent asymmetry. The symmetry principle tells you that if you have a reason to do something the other will do as well, and the level of thinking or knowledge you reach must have been reached by the others as well, and if your reasoning is disturbed by some noise so will be for the others. The principle is sometimes criticized of blinding ourselves to real asymmetries in the society. However, it is effective in restraining the temptations to easily introduce asymmetries in an ad hoc manner. Like informational asymmetries and bounded rationalities, sometimes we have to depart from the symmetry principle. The principle of infinite hierarchy, however, tells us that even if you have more information than the others you face uncertainty at a deeper level, with regard to how you believe about how the others believe about the world, and so on. Also it tells us that even if you are smarter than the others you may be fooled at another level. The arguments in bounded rationality will tempt you to think that you can outwit the market consistently, while it is impossible when everybody is rational. However, even if you know well about some dimension of irrationality and know well about how to manipulate people based on that, you may be actually manipulated at another dimension. Nobody will be free from that.

POSTSCRIPTS

420

I bet you got frustrated, ”Why are you still sticking to the rationality approach, knowing that it goes nowhere? You seem to be just toying with what you don’t believe, or toying with impossibilities. Why are you so cynical? How can it be anything other than an intellectual decadence?” If I’m allowed to use cheap rhetoric, I would say that enduring such contradictions and tensions within is the only way to escape from the dualism of alienated mechanical application of a theory and uncritically accepting or reacting to ”reality as it is.”

Bibliography [1] Arrow, Kenneth J. Social choice and individual values. Vol. 12. Yale university press, 2012. [2] David Austen-Smith and Jeffrey S. Banks, Positive Political Theory I: Collective Preference, University of Michigan Press, Ann Arbor, 1999. [3] Shlomo Benartzi and Richard H. Thaler, Naive Diversification Strategies in Defined Contribution Saving Plans, American Economic Review 91 (2001), 79-98. [4] Bolton, Patrick, and Mathias Dewatripont. Contract theory. MIT press, 2005. [5] Patrick Joyce, The Walrasian tˆ atonnement mechanism and information, RAND Journal of Economics, Vol. 15, No. 3 (1984), pp. 416-425. [6] Fudenberg, Drew, and Jean Tirole. ”Game theory. 1991.” (1991). [7] John Geanakoplos, Common knowledge, in Handbook of Game Theory with Economic Applications Volume 2 (1994), 1437-1496. [8] John Geanakoplos and James Sebenius, Don’t Bet On It: A Note on Contingent Agreements with Asymmetric Information, Journal of American Statistical Association (1983), 78(382): 224-226. [9] P. Ghirardato, Revisiting Savage in a conditional world, Economic Theory, Vol. 20 (2002), pp. 83-92. [10] Gibbard, Allan. ”Manipulation of voting schemes: a general result.” Econometrica: journal of the Econometric Society (1973): 587-601. [11] Harsanyi, John C., and Reinhard Selten. ”A general theory of equilibrium selection in games.” MIT Press Books 1 (1988). [12] Oliver Hart, On the optimality of equilibrium when the market structure is incomplete, Journal of Economic Theory 11 (1975), 418-443. [13] Geoffrey A. Jehle and Philip J. Reny, Advanced Microeconomic Theory, Prentice Hall; 2nd edition, 2000. 421

BIBLIOGRAPHY

422

[14] Ehud Kalai and Ehud Lehrer, Rational learning leads to Nash equilibrium, Econometrica Vol. 61, No. 5, 1993, pp. 1019-1045. [15] David M. Kreps and Evan L Porteus, Temporal Resolution of Uncertainty and Dynamic Choice Theory, Econometrica, vol. 46 (1978), pages 185-200. [16] Krishna, Vijay. Auction theory. Academic press, 2009. [17] David Laibson, ”Golden eggs and hyperbolic discounting, Quarterly Journal of Economics 112.2 (1997): 443-478. [18] Lars Ljungqvist and Thomas J. Sargent, Recursive Macroeconomic Theory, The MIT Press; 2nd edition, 2004. [19] Mark Machina, Dynamic consistency and non-expected utility models of choice under uncertainty, Journal of Economic Literature, 28 (1989), 16221668. [20] Mailath, George J., and Larry Samuelson. ”Repeated games and reputations: long-run relationships.” OUP Catalogue (2006). [21] Andrew Mas-Colell, Michael Whinston and Jerry Green, Microeconomic Theory, Oxford University Press, 1995. [22] Maskin, Eric. ”Nash equilibrium and welfare optimality*.” The Review of Economic Studies 66.1 (1999): 23-38. [23] Herve Moulin, Axioms of Cooperative Decision Making, Cambridge University Press, 1991. [24] Martin J. Osborne and Ariel Rubinstein, A Course in Game Theory, MIT Press, 1994. [25] Pazner, E., and D. Schmeidler, 1974. A difficulty in the concept of equity. Review of Economic Studies 41, 441-443. [26] Torsten Persson and Guido E. Tabellini, Political Economics: Explaining Economic Policy, MIT Press, 2000. [27] Ariel Rubinstein, Modeling Bounded Rationality, MIT Press, 1998. [28] Rafael Repullo, A simple proof of Maskin’s theorem on Nash implementation, Social Choice and Welfare, (1987), 4, 39-41. [29] Salant, Yuval. ”Procedural analysis of choice rules with applications to bounded rationality.” The American Economic Review 101.2 (2011): 724748. [30] Satterthwaite, Mark Allen. ”Strategy-proofness and Arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions.” Journal of economic theory 10.2 (1975): 187-217.

BIBLIOGRAPHY

423

[31] Shapley, Lloyd and Scarf, Herbert, On cores and indivisibility, Journal of Mathematical Economics, vol. 1 (1974), pages 23-37. [32] Vernon L. Smith, Experimental auction markets and the Walrasian hypothesis, Journal of Political Economy, Vol. 73 (1965), pp. 387-393. [33] Richard H. Thaler and Cass R. Sunstein, Nudge: Improving Decisions About Health, Wealth, and Happiness, Yale University Press, 2008. [34] Kahneman, Daniel, and Amos Tversky, Prospect Theory: An Analysis of Decision under Risk, Econometrica, 47 (1979), 263-291. [35] Van Damme, Eric. Stability and perfection of Nash equilibria. Springer, 1991. [36] Vega-Redondo, Fernando. Evolution, games, and economic behaviour. Oxford University Press, 1996. [37] Xavier Vives, Small income effects: A Marshallian theory of consumer surplus and downward sloping demand, Review of Economic Studies 54 (1) (1987) 87-103. [38] Weibull, Jorgen W. Evolutionary game theory. MIT press, 1997. [39] H. Peyton Young, An axiomatization of Borda’s rule, Journal of Economic Theory, 1974, vol. 9, issue 1, pages 43-52

Solutions to the exercises Answer 1 (i) 4x1 + 3x2 ≤ 120. (ii) obtain 4.8x1 + 3.5x2 ≤ 120.

4 . 3

(iii) From 4 × 1.2 = 4.8 and 3 + 0.5 = 3.5, we

Exercise 43 Let Good 1 be consumption good at Period 1, and Good 2 be consumption good at Period 2. (i) Describe by means of indifference curves the preference of a consumer who cares only about consumption at Period 1. (ii) Describe by means of indifference curves the preference of a consumer exhibiting perfect substitution between consumptions at two periods, such that he cares more about the current consumption. (iii) Describe by means of indifference curves the preference of a consumer exhibiting perfect substitution between consumptions at two periods, such that he cares more about the future consumption. Answer 2 Take consumption in Period 1 on the horizontal axis and consumption in Period 2 on the vertical axis. Then, (i) Vertical and parallel straight lines. (ii) Downward-sloping parallel straight lines, which are steeper than negative 45-degree lines. (iii) Downward-sloping parallel straight lines, which are flatter than negative 45-degree lines. Answer 3 Take consumption at State 1 on the horizontal axis and consumption at State 2 on the vertical axis. Then, (i) This is the case of perfect substitution, Indifference curves are downward-sloping parallel straight lines, with the absolute value of slope being (2/3)/(1/3) = 2. (ii) This is the case of perfect complementarity. Indifference curves are parallel Lshaped, aligned along the 45-degree line x1 = x2 . Answer 4 Let x1 denote consumption in Period 1 and x2 denote consumption in Period 2. (i) u(x) = x1 or its arbitrary monotone transformation. (ii) u(x) = ax1 + bx2 with a > b > 0, or its arbitrary monotone transformation. (iii) u(x) = ax1 + bx2 with b > a > 0, or its arbitrary monotone transformation. Answer 5 Let x1 denote consumption at State 1 and x2 denote consumption at State 2.

424

SOLUTIONS TO THE EXERCISES

425

(i) u(x) = 23 x1 + 13 x2 , or its arbitrary monotone transformation. (ii) u(x) = min{x1 , x2 }, or its arbitrary monotone transformation. 2 3 −3 3 ∂ x 5 x 5 = 25 x1 5 x25 . ∂x1 1 2 3 2 −2 2 ∂ x 5 x 5 = 35 x15 x2 5 . ∂x2 1 2

Answer 6 (i) (ii) (iii)

M RS(x) =

2

3

2

3

∂ x5 x5 ∂x1 1 2

3 3 2 −5 5 x x2 5 1 2 2 3 5 −5 x x 5 1 2

=

∂ x5 x5 ∂x2 1 2

(iv) (i) ∂x∂ 1 (2 ln x1 + 3 ln x2 ) = (ii) ∂x∂ 2 (2 ln x1 + 3 ln x2 ) = x32 . (iii)

2x2 . 3x1

2 . x1

∂ (2 ln x1 ∂x1 ∂ (2 ln x1 ∂x2

M RS(x) =

=

+ 3 ln x2 ) + 3 ln x2 )

2/x1 2x2 = . 3/x2 3x1

=

Answer 7 (i) From the tangency condition M RS(x) = pp21 it holds 1 · x1 . By plugging this into the budget equation we have x2 = 3p 2p2 p1 x1 + p2 x2 = p1 x1 + p2 ·

2x2 3x1

=

p1 , p2

implying

3p1 5p1 · x1 = · x1 = I 3p2 2

2I Thus we obtain x1 (p, I) = 5p . By plugging this into the previous formula we obtain 1 3I x2 (p, I) = 5p1 . By putting these into the utility representation we obtain the indirect 2

−2

3

−3

utility function v(p, I) = 2 5 3 5 5−1 p1 5 p2 5 I. (ii) From the tangency condition we obtain x2 = utility representation we have 2

3

2

(

u = x15 x25 = x15 3

3

3p1 · x1 2p2

−3

3p1 2p2

)3

· x1 . By plugging this into the

(

5

=

3p1 2p2

)3 5

x1

3

Thus we obtain h1 (p, u) = 3− 5 2 5 p1 5 p25 u. By plugging this to the previous formula 2

2

2 5

−2 5

we obtain h2 (p, u) = 3 5 2− 5 p1 p2 u. By putting this into the expenditure formula we ( 3 3 ) 2 3 2 2 obtain the expenditure function e(p, u) = 3− 5 2 5 + 3 5 2− 5 p15 p25 u. Answer 8 e1,I

=

e1,p1

=

e1,p2

=

1 a2 p2 + 2b2 p1 a2 p2 + b2 p1 b2 p 1 a2 p2 + b2 p1

Answer 9 From the previous one, demand function is x1 (p, I) = 3I , 5p1

2 5

3 5

indirect utility function is v(p, I) = 2 3 5 ) 2 3 ( 3 3 2 2 is e(p, u) = 3− 5 2 5 + 3 5 2− 5 p15 p25 u.

−1

−2 5

−3 5

2I 5p1

and x2 (p, I) =

p1 p2 I, and expenditure function

SOLUTIONS TO THE EXERCISES

426 ((

Hence ′

CV = e(p , v(p, I)) − I =

p1 + ∆p1 p1

( ′

EV = I − e(p, v(p , I)) = ∫

p1 +∆p1

∆CS = p1

( 1−

)

)2 5

−1 I

p1 p1 + ∆p1

)2 ) 5

I

2I p1 + ∆p1 2I dq = ln 5q 5 p1

1 · 400 ≈ 684.6 Answer 10 (i) 300 + 1.04 1 1 (ii) 200 + · 500 ≈ 885.95 · 300 + 2 1.1 1.1 ∑ 300 300 (iii) ∞ = 1.06 · 300 = 5300 t=1 1.06t−1 = 1− 1 0.06 1.06

Answer 11 Since marginal utility of current consumption is x11 and marginal utility of future consumption is 0.95 , MRS of future consumption for current consumption is x2 x2 M RS(x) = 0.95x . 1 x2 From the tangency condition M RS(x) = 1+r we obtain 0.9x = 1+r = 1.04, implying 1 x2 = 1.04 · 0.95x1 . 1 1 By plugging this into the lifetime budget equation x1 + 1.04 x2 = 40 + 1.04 · 30 = 68.85 and solve for x1 , then we obtain x1 = 35.3 and x2 = 34.88.Saving is thus 40−35.3 = 4.7. √ √ Answer 12 (i) Let z denote the certainty equivalent. Then from z = 0.4 256 + √ 0.6 81 we obtain z = 139.24. √ √ √ (ii) Let π denote the probability of rain. Then from 225 = π 361 + (1 − π) 64 we obtain π = 7/11. Answer 13 Let t denote the investment on A. Then the final income at State 1 is 1.2t + 0.8(100 − t) = 80 + 0.4t, and that at State 2 is 0.9t + 1.5(100 − t) = 150 − 0.6t. Hence the expected utility is 0.6 ln(80 + 0.4t) + 0.4 ln(150 − 0.6t) By taking the first order condition through taking the derivative by t we obtain 0.6 ·

0.4 −0.6 + 0.4 · =0 80 + 0.4t 150 − 0.6t

By solving the above we obtain t = 70. Hence the investment on A is 70 and that on B is 30. Answer 14 MRS of Good 2 for Good 1 in consumer i is a √ √i ai xi2 2 x M RSi (xi ) = b i1 = √ bi xi1 √i 2 x i2

From the tangency condition M RSi (xi ) = xi2 =

p1 p2

we obtain

b2i p21 · xi1 a2i p22

SOLUTIONS TO THE EXERCISES

427

By plugging this into the budget equation p1 xi1 + p2 xi2 = p1 ei1 + p2 ei2 and solve for xi1 , then we obtain the demand function xi1 (p) =

p1 e p2 i1 p1 p2

+

+ ei2

a2 i b2 i

,

p2 1 p2 2

·

xi2 (p) =

b2i p21 · a2i p22

p1 e p2 i1 p1 p2

+

+ ei2

a2 i b2 i

·

p2 1 p2 2

When the market for Good 1 is balance Good ∑ so it for ∑ 2, hence it is enough to see ∗ the equilibrium condition for Good 1, n i=1 xi1 (p ) = i ei1 . It is p∗ 1 ei1 p∗ 2

n ∑ i=1

p∗ 1 p∗ 2

+

a2 i b2 i

n + ei2 ∑ ei1 ( ∗ )2 = p i=1 · p1∗ 2

Answer 15 Since state-contingent earnings satisfy the condition of no aggregate risk, 3/4 the equilibrium relative price is pp21 = 1/4 = 3. Let t denote A’s purchase of Arrow security 1, then A’s state-contingent consumption is (2 + t, 6 − 3t). Since perfect insurance holds under the no aggregate risk condition, it holds 2 + t = 6 − 3t, implying t = 1. Hence A’s state-contingent consumption is (3, 3), and B’s one is (7, 7). Answer 16 Marginal rate of substitution of Good 2 for Good 1 is M RSA = in A and M RSB =

λB xB2 µB xB1

λA xA2 µA xA1

in B. Hence equalization og MRS leads to λA xA2 λB xB2 = (1 − λA )xA1 (1 − λB )xB1

Combine this with the feasibility condition xA1 + xB1 = e1 , xA2 + xB2 = e2 , then we obtain λB (e2 − xA2 ) λA xA2 = µA xA1 µB (e1 − xA1 ) Hence the set of Pareto-efficient allocations is { } λB (e2 − xA2 ) λA xA2 (xA , xB ) : = , xA1 + xB1 = e1 , xA2 + xB2 = e2 µA xA1 µB (e1 − xA1 ) −2

1

Answer 17 (i) M P1 (x) = 13 · x1 3 x25 , M P2 (x) = P1 (x1 ,x2 ) 2 = 5x . (ii) T RS(x) = M M P2 (x1 ,x2 ) 3x1

1 5

1

−4

· x13 x2 5 .

−2

1

Answer 18 From the previous question we have M P1 (x1 , x2 ) = 13 ·x1 3 x25 , M P2 (x1 , x2 ) = 1

−4

P1 (x1 ,x2 ) 2 · x13 x2 5 and T RS(x1 , x2 ) = M = 5x . M P2 (x1 ,x2 ) 3x1 (i) From the profit maximization condition pM P1 = w1 and pM P2 = w2 , we have 1 5

1 − 23 15 · x x2 3 1 1 −4 1 p · x13 x2 5 5 p

From the above two we obtain

5x2 3x1

=

w1 , w2

x2 =

=

w1

=

w2

implying

3w1 · x1 . 5w2

SOLUTIONS TO THE EXERCISES

428

By plugging this into the first equation we have )1 ( 5 1 − 2 3w1 p · x1 3 · x1 = w1 . 3 5w2 By solving this for x1 we obtain −12/7

x1 (p, w) = 3−12/7 5−3/7 w1

−3/7 15/7

w2

p

By plugging this into the above formula we obtain −5/7 −10/7 15/7 p2 p

x2 (p, w) = 3−5/7 5−10/7 p1

By plugging these into the production function we obtain 1

1

−5/7

y(p, w) = x13 x25 = 3−5/7 5−3/7 w1

−3/7 10/7

w2

(ii) From the tangency condition T RS(x) = w1 /w2 it holds x2 = 1

p

5x2 3x1

=

w1 , w2

hence we have

3w1 · x1 . 5w2

1

By plugging this into y = x13 x25 we obtain ( )1 1 5 3w1 3 · x1 y = x1 5w2 By solving this we obtain −3/8

x1 (y, w) = 3−3/8 53/8 w1

3/8

w2 y 15/8

By plugging this into the previous formula we obtain −5/8 15/8

x2 (y, w) = 35/8 5−5/8 w1 w2 5/8

y

By plugging this into the cost formula we obtain C(y, w) = w1 x1 + w2 x2 = (3−3/8 53/8 + 35/8 5−5/8 )w1 w2 y 15/8 5/8

3 2 C(y) = y −4y y+7y+9 = y 2 y 3 2 V C(y) = y −4yy +7y = y 2 − 4y + 7. y 3 2 ′ 2

Answer 19 (i) AC(y) =

− 4y + 7 +

3/8

18 . y

(ii) AV C(y) = (iii) M C(y) = (y − 4y + 7y + 9) = 3y − 8y + 7. (iv) Since AC ′ (y) = 2y − 4 − y182 becomes 0 at y = 3, the minimum of average cost is AC(3) = 10. Hence the break-even point is p = 10. (v) Since AV C(y) = y 2 − 4y + 7 = (y − 2)2 + 3 the minimum of average variable cost is 3. Hence the shut-down point is p = 3. (vi) Shut down and produce nothing when p < 3. When p ≥ 3 from p = M C(y) we √ have p = 3y 2 −8y+7. There are two solutions to this quadratic equation, y = 4± 33p−5 , but since the marginal cost √ curve must be upward-sloping at the profit maximization point we have S(p) = 4+ 33p−5 .. Summing up, we obtain { 0, √ when p < 3 S(p) = 4+ 3p−5 , when p ≥ 3 3

SOLUTIONS TO THE EXERCISES

429

Answer 20 Let p denote the relative price of Good 1 for Good 2. Then consumer i’s optimal consumption of Good 1 is determined by M RSi (xi ) = 2√1xi1 = p alone here a2

because of quasi-linearity, and by solving this we obtain xi1 (p) = 4pi2 . On the other hand, firm k’s profit maximization is determined by M Ck (yk ) = 2ck yk = p, and by solving this we obtain yk1 (p) = 2cpk . The equilibrium condition is m n ∑ ∑ p a2i = 2 4p 2c k i=1 k=1

and by solving this for p we obtain ( ∑ )1 3 n 2 i=1 ai ∑m 1 p = 2 k=1 ck ∗

Answer 21 (i) Since M C(y) = y the supply is determined by p = y. In competitive equilibrium, from 120 − 2y CE = y CE the quantity is y CE = 40 and the price is pCE = 120 − 2y CE = 40. Consumer surplus is the area of the triangle spanned by (0, 120), (40, 40) and (0, 40), which is 1600. Producer surplus is the area of the triangle spanned by (0, 0), (40, 40) and (0, 40), which is 800. Hence the social surplus is 1600 + 800 = 2400. (ii) From R(y) = (120 − 2y)y, the monopolist’s marginal revenue is M R(y) = 120 − 4y. In monopoly equilibrium, from M R(y M ) = M C(y M ) we have 120 − 4y M = y M , implying y M = 24. The monopoly price is p(y M ) = 120 − 2y M = 72. Consumer surplus is the area of the triangle spanned by (0, 120), (24, 72) and (0, 72), which is 576. Producer surplus is the area of the trapezoid spanned by (0, 0), (24, 24) (24, 72), (0, 72), which is 1440. Hence the social surplus is 576+1440 = 2016 and the deadweight loss is 2400 − 2016 = 384. (iii) As far as y ≤ 40, charge 120 − 2y for the y-th unit. Sell no more than that. Answer 22 (i) Since demand function in A is xA (p) = 90−p and that in B is xB (p) = 60−0.5p, the aggregate market demand function is x(p) = xA (p)+xB (p) = 150−1.5p, implying that the aggregate market inverse demand function is p(y) = 100 − 32 y. Since marginal cost is M C(y) = y, in competitive equilibrium it follows 100 − 2 CE y = y CE , hence the quantity is y CE = 60 and the price is pCE = 100− 32 y CE = 60. 3 (ii) Since the revenue is R(y) = (100 − 23 y)y the monopolist’s marginal revenue is M R(y) = 100 − 34 y, in monopoly equilibrium it follows from M R(y M ) = M C(y M ) that 100 − 43 y M = y M , implying that the quantity is y M = 300 ≈ 42.86 and the price 7 is p(y M ) = 100 − 23 y M = 500 ≈ 71.43. 7 (iii) Since revenue in A is RA (yA ) = (90−yA )yA marginal revenue there is M RA (yA ) = 90 − 2yA . Since revenue in B is RB (yB ) = (120 − 2yB )yB marginal revenue there is M RB (yB ) = 120 − 4yB . Since marginal cost is M C(yA + yB ) = yA + yB the profit maximization condition is 90 − 2yA

=

yA + yB

120 − 4yB

=

yA + yB

By solving the above we obtain yA = 66.43, pB = 570 ≈ 81.43. 7

165 7

≈ 23.57, yB =

135 7

≈ 19.29 and pA =

465 7



SOLUTIONS TO THE EXERCISES

430

Answer 23 (1) 1. For A, Y is strictly dominated by W. Hence eliminate Y. 2. For B, G is strictly dominated by F. Hence eliminate G. 3. For A, V is strictly dominated by X. Hence eliminate V. 4. For B, H is strictly dominated by I. Hence eliminate H. We cannot eliminate any further. Hence {X, Z, W } is the set of A’s strategies that survive the elimination, {F, I} is the set of B’s strategies that survive the elimination. (2) 1. For A, Y and Z can never be a best response, hence they are eliminated. 2. For B, G can never be a best response, hence it is eliminated. 3. For A, V can never be a best response, hence it is eliminated. 4. For B, H can never be a best response, hence it is eliminated. We cannot eliminate any further. Hence {X, Z, W } is the set of A’s rationalizable strategies, {F, I} is the set of B’s rationaizable strategies. (3) There are two pure-strategy Nash equilibria, (X, I) and (W, F ). Answer 24 Since nobody can announce any number greater than 100 it is impossible that the half of the average of announced numbers is greater than 50. Hence any number greater than 50 can never be a best response. Thus, nobody announces any number greater than 50, therefore it is impossible that the half of the average of announced numbers is greater than 25. Hence any number greater than 25 can never be a best response. Thus, nobody announces any number greater than 25, therefore it is impossible that the half of the average of announced numbers is greater than 12,5. And so on. By repeating this argument, only 0 is the rationalizable strategy for anybody. Answer 25 To explain As best response, consider for example that B chooses F and C chooses K. Then compare between A’s payoff in the upper-left cell in both matrices. If A chooses X he gets 2, if chooses Y he gets 5, hence the best response is Y and underline 5. Do the similar things for A, and for B and C do the known underlining exercises. Then we obtain sA = X B

F G

K 2, 1, 4 7, 2, 1

C L 4, 3, 8 2, 1, 3

sA = Y B

F G

K 5, 1, 8 4, 9, 3

C L 1, 3, 2 3, 4, 5

Thus, there are two pure-strategy Nash equilibria, (X, F, L) and (Y, G, L). Answer 26 Since A’s expected payoff given (pA , pB ) is uA (pA , pB )

=

(3 − 5pB )pA + 2pB + 2

A’s best response to pB is   {1}, [0, 1] , BRA (pB ) =  {0},

when when when

pB < 3/5 pB = 3/5 pB > 3/5.

SOLUTIONS TO THE EXERCISES

431

Likewise, since B’s expected payoff given (pA , pB ) is EUB (pA , pB )

=

(3 − 7pA )pB + 3pA + 4

B’s best response to pA is   {1}, [0, 1] , BRB (pA ) =  {0},

when when when

pA < 3/7 pA = 3/7 pA > 3/7.

Hence there are three Nash equilibria, ((X; 1, Y ; 0), (F ; 0, G; 1)) ((X; 0, Y ; 1), (F ; 1, G; 0)) (( ) ( )) 3 4 3 2 X; , Y ; , F ; , G; 7 7 5 5 Answer 27 Let GF denote B’s strategy that he chooses G if A chooses X and chooses F if A chooses Y, and similarly for the other cases. Then we obtain the normal-form expression B FF FG GF GG X 0, 3 0, 3 60, 5 60, 5 A Y 9, 12 5, 10 9, 12 5, 10 There are three pure-strategy Nash equilibria, (X, GF ), (X, GG) and (Y, F F ), while only (X, GF ) is subgame-perfect. Answer 28 A cuts the land equally. B always picks the larger one, while it doesn’t matter if the land is cut equally. Answer 29 There are two pure-strategy Nash equilibria in the subgame after B’s choosing E, (X, F ) and (Y, G). If (X, F ) is played in the subgame B chooses N, hence the subgame-perfect equilibrium is (X, N F ). On the other hand, if (Y, G) is played in the subgame B chooses E, hence the subgame-perfect equilibrium is (Y, EG). Thus there are two subgame-perfect equilibria. Answer 30 (i) Given arbitrary yB , A’s best response is determined by max(100 − 2(yA + yB) ))yA − 4qA yA

=

2 max 96yA − 2yA − 2yB yA yA

The first-order condition is 96 − 4yA − 2yB = 0, implying that A’s best response is BRA (yB ) =

48 − yB 2

SOLUTIONS TO THE EXERCISES

432

Likewise, B’s best response is BRB (yA ) =

48 − yA 2

In Nash equilibrium it holds ∗ = yA

48 − yB 48 − yA ∗ , yB = 2 2

∗ ∗ Hence we obtain yA = yB = 16. (ii) B’s optimal choice given yA is solved in the same way as above, at least mathematically. Then B’s strategy is a function fB given by

fB (yA ) =

48 − yA 2

Note that B’s strategy is the entire function. Then A solves

= =

max(100 − 2(yA + fB (yA ))yA − 4yA qA ( ( )) 48 − yA max 100 − 2 yA + yA − 4yA qA 2 2 max 48yA − yA qA

∗ ∗ In the equilibrium strategy yA from the first-order condition it holds 48 − 2yA = 0, ∗ implying yA = 24.

Answer 31 (i) Given arbitrary pB , A solves max pA [50 − 2pA + pB ] − 4 [50 − 2pA + pB ] pA

Since the first-order condition is 58 − 4pA + pB = 0 A’s best response is BRA (pB ) =

58 + pB 4

Likewise, B’s best response is BRB (pA ) =

58 + pA 4

In nash equilibrium (p∗A , p∗B ) it holds p∗A = By solving this we obtain

58 + p∗B , 4

p∗B =

58 + p∗A 4

p∗A = p∗B = 11.6

(ii) B’s optimal choice given pA is solved in the same way as above, at least mathematically. Then B’s strategy is a function fB given by fB (pA ) =

58 + pA 4

SOLUTIONS TO THE EXERCISES

433

Note that B’s strategy is the entire function. Then A solves

= =

max pA [50 − 2pA + fB (pA )] − 4 [50 − 2pA + fB (pA )] pA [ ] [ ] 58 + pA 58 + pA max pA 50 − 2pA + − 4 50 − 2pA + pA 4 4 7 2 143 pA − 200 max − pA + qA 4 2

In the equilibrium strategy p∗A from the first-order condition have p∗A = 143 . 7

7 ∗ p 2 A

+

143 2

= 0 we

Answer 32 The corresponding Bayesian game is B A

C N

CC 10, 10 12p, 5 − 6p

CN 5 + 5p, 10p 12p, −p

NC 10 − 11p, 10 + 2p 0, 5 − 5p

NN 5 − 6p, 12p 0, 0

and as far as 0 < p < 1 NC is the dominant strategy for B. Hence Bayesian-Nash equilibrium is (C, N C) when p ≤ 10/11 and (N, N C) when p ≥ 10/11. Answer 33 The corresponding Bayesian game is

A

B N

S/G, S/B 20p − 10, 0 0, 20p − 10

S/G, N/B 10p, −10 + 10p 0, 20p − 10

B N/G, S/B −10 + 10p, 10p 0, 20p − 10

N/G, N/B 0, 20p − 10 0, 20p − 10

Note that since 0 < p < 1 B’s only best response to A’s Buying is (N/G, S/B), and anything is optimal for B when A is not buying. There are two BNE for all 0 < p < 1, (N, (N/G, S/B)) and (N, (N/G, N/B)). When 0 < p < 1/2 there is one more BNE, (N, (S/G, S/B)). Answer 34 Let vi denote bidder i’s willingness to pay. Let b−i = (b1 , · · · , bi−1 , bi+1 , · · · , bn ) denote a profile of biddings except for i’s. Then an entire bidding profile is denoted by (bi , b−i ). Then bidder i’s payoff in the all-pay auction game is { vi − bi , if bi > maxj̸=i bj Ui (bi , b−i ) = −bi , if bi < maxj̸=i bj Ignore the case of ties since it is of probability zero here. Denote the bidding function in Symmetric Bayesian-Nash equilibrium by β : [0, 1] → R. Suppose all bidders other than i are following β. Then if i bids bi his expected payoff is vi P rob(max β(vj ) < bi ) − bi = vi F (β −1 (bi ))n−1 − bi j̸=i

Then the first-order condition is vi

(n − 1)F (β −1 (bi ))n−2 f (β −1 (bi )) −1=0 β ′ (β −1 (bi ))

SOLUTIONS TO THE EXERCISES

434

In the symmetric Bayesian-Nash equilibrium we can replace bi by β(vi ), which leads to (n − 1)F (vi )n−2 f (vi ) vi −1=0 β ′ (vi ) By arranging the above formula we obtain β ′ (vi ) = vi (n − 1)F (vi )n−2 f (vi ) By integrating both wides in the above we obtain ∫ vi (n − 1)vF (v)n−2 f (v)dv. β(vi ) = 0

Answer 35 There are two kinds of equilibria here. If p ≥ 4000 both types are provided to the market. Then the buyers’ expected utility of buying is 5000 × 0.75 + 1500 × 0.25 = 4125 > 4100. Hence they buy when 4000 ≤ p ≤ 4125, and both types are traded. The other kind of equilibrium is that only the low type ones are traded under 1000 ≤ p ≤ 1500. Answer 36 If the worker chooses high effort his expected utility is 0.95(wG − 3) + 0.05(wB − 3) If he chooses low effort his expected utlity is 0.2wG + 0.8wB Hence the required condition is 0.95(wG − 3) + 0.05(wB − 3) ≥ 0.2wG + 0.8wB which is simplified to wG − wB ≥ 4 Answer 37 Initial common knowledge is CK0 = (P1 ∪ P2 ∪ P3 ) × (Q1 ∪ Q2 ). = 0.475 < 0.5. Step 1: A says YES because prob(up|P1 × (Q1 ∪ Q2 )) = 0.55+0.4 2 Then B knows P2 is not true, since if P2 were true then A would have said NO because prob(up|P2 × (Q1 ∪ Q2 )) = 0.3+0.9 = 0.6 > 0.5. 2 Common knowledge after Step 1 is CK1 = (P1 ∪ P3 ) × (Q1 ∪ Q2 ). Step 2: B says YES because prob(up|(P1 ∪ P3 ) × Q1 ) = 0.55+0.6 = 0.525 > 0.5. 2 Then A knows Q2 is not true, since if Q2 were true B would have said NO because prob(up|(P1 ∪ P3 ) × Q2 ) = 0.4+0.3 = 0.35 < 0.5. 2 Common knowledge after Step 2 is CK2 = (P1 ∪ P3 ) × Q1 . Step 3: A says NO because prob(up|P1 × Q1 ) = 0.55 > 0.5. Answer 38 Location is determined by max − g

n ∑ (vi − g)2 i=1

SOLUTIONS TO THE EXERCISES Since the first-order condition is

∑n i=1

435

2(vi − g) = 0 we have

g(v) =

n 1∑ vi n i=1

Since surplus maximization for the those other than i is determined similarly, Clarke tax to be paid by i is ∑ ∑ (vj − g(v))2 (vj − g)2 − ti (v) = max g

=

∑ j̸=i

2 ( )2 n ∑ ∑ ∑ 1 1 vj − vj − vk  − vk n−1 n



=

j̸=i

j̸=i



−

k̸=i

j̸=i

2

k=1

n 1 ∑ 1∑  vk − vk n−1 n k̸=i

k=1

Answer 39 Round 1: From 1 → e2 , 2 → e6 , 3 → e4 , 4 → e6 , 5 → e2 , 6 → e5 , there is a cycle 2 → 6 → 5 → 2. Hence 2 gets e6 , 6 gets e5 and 5 gets e2 and leave. Round 2: From 1 → e4 , 3 → e4 , 4 → e1 , there is a cycle 1 → 4 → 1. Hence 1 gets e4 , 4 gets e1 and leave. Round 3: From 3 → e3 , 3 gets e3 . Summing up, the core allocation is (e4 , e6 , e3 , e1 , e2 , e5 ). Answer 40 (i) Round 1: w1 applies for f2 , w2 applies for f1 , w3 applies for f1 , and w4 applies for f2 . f1 keeps w2 and rejects w3 . f2 keeps w4 and rejects w1 . Round 2: w1 applies for f1 and w3 applies for f2 . f1 keeps w1 and rejects w2 . f2 keeps w3 and rejects w4 . Round 3: w2 applies for f3 and w4 applies foe f3 . f1 keeps w1 , f2 keeps w3 , f3 keeps w4 and rejects w2 . Round 4: w2 applies for f2 . f1 keeps w1 , f2 keeps w3 and rejects w2 , and f3 keeps w4 . Round 5: w2 applies for f4 . f1 keeps w1 , f2 keeps w3 , f3 keeps w4 , and f4 keeps w2 . Summing up, we obtain (w1 , f1 ), (w2 , f4 ), (w3 , f2 ), (w4 , f3 ) (ii) Round 1: f1 proposes to w1 , f2 proposes to w3 , f3 proposes to w1 , and f4 proposes to w3 . w1 keeps f1 and rejects f3 . w3 keeps f2 and rejects f4 . Round 2: f3 proposes to w4 , and f4 proposes to w1 . w1 keeps f1 and rejects f4 , w3 keeps f2 , and w4 keeps f3 . Round 3: f4 proposes to w4 . w1 keeps f1 , w3 keeps f2 , w4 keeps f3 and rejects f4 . Round 4: f4 proposes to w2 . w1 keeps f1 , w2 keeps f4 , w3 keeps f2 , and w4 keeps f3 . Summing up, we obtain (w1 , f1 ), (w2 , f4 ), (w3 , f2 ), (w4 , f3 )

SOLUTIONS TO THE EXERCISES

436

Answer 41 (i) Round 1: i1 applies foe s3 , i2 applies for s3 , i3 applies for s2 , i4 applies for s2 , i5 applies for s2 , and i6 applies for s3 . s2 admits i4 , i5 , and s3 admits i1 , i6 . Round 2: i2 applies for s2 , but the seats are already full and gets rejected. i3 applies for s3 , but the the seats are already full and gets rejected. Round 3: i2 and i3 apply for s1 , and get admitted. Here for example i2 can report s2 ≻ s3 ≻ s1 instead of his true preference, given that the others are reporting truthfully, then he is admitted s2 , which is better for him than s1 . (ii) Round 1: i1 applies for s3 , i2 applies for s3 , i3 applies for s2 , i4 applies for s2 , i5 applies for s2 , and i6 applies for s3 . s2 keeps i4 , i5 and rejects i3 . s3 keeps i1 , i6 and rejects i2 . Round 1: i2 applies for s2 , and i3 applies for s3 . s2 keeps i4 , i2 and rejects i5 . s3 keeps i6 , i3 and rejects i1 . Round 3: i1 applies for s2 , and i5 applies for s3 . s2 keeps i2 , i1 and rejects i4 . s3 keeps i3 , i5 and rejects i6 . Round 4: i4 applies for s1 , and i6 applies for s1 . s1 keeps i4 , i6 , s2 keeps i2 , i1 , and s3 keeps i3 , i5 . Summing up, we obtain i1 − s2 , i2 − s2 , i3 − s3 , i4 − s1 , i5 − s3 , i6 − s1 . Answer 42 (i) Since MRSs are equalized at Pareto-efficient allocations, we have M RSA (xA ) =

2 1 = = M RSB (xB ), xA1 xB1

which implies xB1 = 2xA1 . Combine this with the resource constraint xA1 + xB2 = 12, then we obtain xA1 = 4, xB1 = 8. Since the efficiency condition is silent about allocation of Good 2 here, the set of efficient allocations is {(xA , xB ) : xA1 = 4, xB1 = 8, xA2 + xB2 = 0} (ii) Since the allocation of Good 1 is xA1 = 4, xB1 = 8 from efficiency, the condition that A does not envy B is ln 4 + xA2 ≥ ln 8 + xB2 and the condition that B does not envy A is 2 ln 8 + xB2 ≥ 2 ln 4 + xA2 Combine these with the constraint xA2 + xB2 = 0, we obtain 12 ln 2 ≤ xA2 ≤ ln 2. Hence the set of efficient and envy-free allocations is } { 1 (xA , xB ) : xA1 = 4, xB1 = 8, xA2 + xB2 = 0, ln 2 ≤ xA2 ≤ ln 2 2

Microeconomic Theory for the Social Sciences

Aug 30, 2015 - To achieve the above goal, I tried to find the best level of generality and sim- plification at which ..... 12.4 Security exchange and security price . . . . . . . . . . . . . . . . ...... and the set of choice objects in the problem of which company to work for is let's say .... house and that house are typically different. They are ...

3MB Sizes 19 Downloads 146 Views

Recommend Documents

Microeconomic Theory
12/5; 12/7 - Asymmetric Information and Moral Hazards. • 12/7: Reading response 3 due. • 12/7: Quiz 11. • 12/10: MEL HW 14 due. • Ch. 19/20. • A (1970). FINAL. • SECTION 01: Monday, December 18, 9:00 am. • SECTION 06: Tuesday, December

Microeconomic Theory
Main Course Goal: The ultimate goal of this course is to prepare the applied Master's student for additional courses in applied microeconomics at the Master's and PhD levels. Learning Objectives: This course has .... Guidance requires selection of ap