A Theory of Fairness and Social Welfare Marc Fleurbaey and François Maniquet December 2009

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Contents Introduction

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I

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Basics

1 A contribution to welfare economics 1.1 Introduction . . . . . . . . . . . . . . . . . . 1.2 The theory of fair allocation . . . . . . . . . 1.3 Arrovian social choice theory . . . . . . . . 1.4 Welfarism . . . . . . . . . . . . . . . . . . . 1.5 Multidimensional inequality . . . . . . . . . 1.6 Bergson-Samuelson social welfare functions 1.7 Cost-benefit analysis . . . . . . . . . . . . . 1.8 Money-metric utilities . . . . . . . . . . . . 1.9 Conclusion . . . . . . . . . . . . . . . . . .

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2 Efficiency versus equality 2.1 Introduction . . . . . . . . . . . . . . . 2.2 Resource equality vs Pareto efficiency 2.3 Accommodating equality . . . . . . . . 2.4 Conclusion . . . . . . . . . . . . . . .

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23 23 24 28 37

3 Priority to the worst-off 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 3.2 Absolute priority to the worst-off . . . . . . . . . . 3.3 From Nested-Contour Transfer to absolute priority 3.4 From Equal-Split Transfer to absolute priority . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . .

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4 The 4.1 4.2 4.3 4.4

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informational basis of social orderings Introduction . . . . . . . . . . . . . . . . . . The feasible set . . . . . . . . . . . . . . . . Information on preferences . . . . . . . . . . Allocation rules as social ordering functions iii

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CONTENTS 4.5

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Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Distribution

5 Fair 5.1 5.2 5.3 5.4 5.5

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distribution of divisible goods: Introduction . . . . . . . . . . . . . From equal split to Ω-Equivalence A Walrasian SOF . . . . . . . . . . Second-best applications . . . . . . Conclusion . . . . . . . . . . . . .

6 Specific domains 6.1 Introduction . . . . . . . 6.2 Expected utility . . . . . 6.3 Non-convex preferences 6.4 Homothetic preferences 6.5 Indivisibles . . . . . . . 6.6 Conclusion . . . . . . .

68

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73 73 74 77 87 91

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95 . 95 . 95 . 97 . 101 . 104 . 109

7 Extensions 7.1 Introduction . . . . . . . . . . 7.2 Rationalizing allocation rules 7.3 Cross-profiles SOFs . . . . . . 7.4 Legitimate endowments . . . 7.5 Functionings . . . . . . . . . 7.6 Conclusion . . . . . . . . . .

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III

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Production

8 Public good 8.1 Introduction . . . . . . . 8.2 The model . . . . . . . . 8.3 Three axioms . . . . . . 8.4 Two solutions . . . . . . 8.5 Pure public good . . . . 8.6 Second-best applications 8.7 Conclusion . . . . . . .

111 111 112 115 121 124 126

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133 133 134 137 141 152 154 158

9 Private good 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . 9.2 A simple model with private rights on the input . 9.3 Assessing reforms . . . . . . . . . . . . . . . . . . 9.4 A third-best analysis . . . . . . . . . . . . . . . . 9.5 A general model of production and allocation . . 9.6 Conclusion . . . . . . . . . . . . . . . . . . . . .

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161 161 162 164 165 168 171

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CONTENTS

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10 Unequall skills 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Compensation versus responsibility: ethical dilemmas 10.4 Weakening the basic axioms . . . . . . . . . . . . . . . 10.5 Compromise axioms . . . . . . . . . . . . . . . . . . . 10.6 From transfer to priority axioms . . . . . . . . . . . . 10.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . .

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173 173 175 176 181 187 189 194

11 Income taxation 11.1 Introduction . . . . . . . . . . . . . . . . . . . . 11.2 The model . . . . . . . . . . . . . . . . . . . . . 11.3 Assessing reforms . . . . . . . . . . . . . . . . . 11.4 Optimal taxation schemes in a simple economy 11.5 Optimal taxation schemes . . . . . . . . . . . . 11.6 Conclusion . . . . . . . . . . . . . . . . . . . .

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197 197 198 201 211 215 225

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12 Conclusions A Proofs A.1 Chapter A.2 Chapter A.3 Chapter A.4 Chapter A.5 Chapter Bibliography

227 3: Priority to the worst-off . . . . . . 5: Fair distribution of divisible goods 6: Specific domains . . . . . . . . . . 7: Extensions . . . . . . . . . . . . . 10: Unequal Skills . . . . . . . . . .

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231 231 234 253 262 266 269

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CONTENTS

Preface

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viii

PREFACE

Introduction The evaluation of allocations of resources, distributions of well-being, and public policies is a pervasive need in economics and a frequent activity of economists. But this is also generally considered a difficult and hazardous exercise. The danger is not just the risk of mixing value judgments and factual assessments. It is also, perhaps primarily, the risk of getting lost in the morass of controversies and impossibility theorems of the field that one can broadly call normative economics. However, the development of this field in the last century has been impressive, and has provided very powerful analytical tools. In particular, the theory of social choice and the theory of fair allocation have separately proposed an array of concepts and methods which are promising. In this book we put the concepts of these two theories to use and propose a general theory of social criteria for economic allocation problems. In a nutshell, then, this book studies the elaboration of criteria for the evaluation of social and economic situations, and the application of such criteria to the search for optimal public policies. Several broad objectives are assigned to the criteria developed in our analysis. Firstly, the criteria should be sufficiently comprehensive so that when they declare a situation to be preferable to another, there is a sense in which this evaluation takes account of all relevant considerations and is not merely a judgment that the considered situation is better in only one respect. More specifically, the idea is that the criteria should incorporate principles of efficiency as well as principles of equity. Restricting attention to efficiency only, or to equity only, would not provide very useful criteria in our opinion.1 A related tenet of this requirement of comprehensiveness is that the criteria must be individualistic in the sense of taking account of every individual situation in its own right, and of giving due consideration to every individual’s perspective on his own position. Criteria that directly rely on global quantities of the population without grounding this on an assessment of every individual situation are therefore excluded from the outset, although, obviously, it will be considered quite valuable when, from a properly individualistic criterion, we will sometimes be able to derive simple criteria based on global data. Secondly, the criteria should be fine-grained enough to be useful in most contexts of public decision-making and institutional design. Criteria that simply point to the optimal solution in some special context of implementation are not 1 See,

e.g., Arrow (1963) for a criticism of pure efficiency criteria.

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INTRODUCTION

sufficient from this standpoint.2 We are looking for sufficiently precise rankings of all the options which may be on the agenda in various economic and political contexts. This may be slightly more than what is really needed in practice, because decision-makers only need to know what the optimal option is in their particular context, not how to rank all the suboptimal options or the infeasible options. But given the great variety of possible contexts, and in particular having in mind that the job of public decision-makers is mostly to evaluate imperfect reforms in characteristically suboptimal situations, we think that the most convenient kind of criterion is a fine-grained ranking of all options that may be on the agenda in some possible context. Finally, because we think of applications in public economics, the criteria must be relevantly applicable to the evaluation of social and economic situations, that is, primarily, the evaluation of conflicts of interests between individuals in the allocation of economic resources. We believe that for such applications, general abstract criteria will not sufficiently take account of the ethically relevant features of individuals’ interests. More precisely, requirements of fairness are sometimes general, but also often particular to the situation at hand. Fairness in the distribution of unproduced commodities is not the same as fairness in production, and fairness in the production of a private good is not the same as for a public good. As a consequence, the main focus here is on economic models of resource allocation, rather than on abstract models of political or collective decision, although some words will be said on the latter. One defining feature of our analysis throughout is that individual situations are not described by interpersonally comparable measures of well-being but only by ordinal non-comparable preferences. Moreover, we will put special emphasis on cases when preferences are heterogeneous. Some words of explanation are needed about this particular feature of our approach. Following Robbins and Samuelson,3 an important stream of economic analysis has adopted the view that utility cannot easily be compared across individuals, and that interpersonal utility comparisons always involve value judgments with little or no objective basis. This is a controversial view, as it may be thought that mental states like happiness are amenable to some kind of objective measurement, or are likely to become so in the near future. It is true, nonetheless, that individuals’ various goals in life, and their corresponding achievements, are essentially incommensurable, so that measuring satisfaction (as distinct from simple mental states like happiness) in a meaningful and interpersonally comparable way is often considered problematic. Other possible motivations for eschewing interpersonally comparable measures of satisfaction must be mentioned. Many authors4 have argued that for the evaluation of social and economic allocations, a focus on subjective or mental states is not appropriate. They argue that 2 In a more philosophical context, Sen (2009) emphasizes that, in view of the imperfections of the societies analysts and decision-makers are grappling with, it is important to rank the social alternatives instead of merely delineating a perfectly just but unattainable society as is done in many theories of justice. 3 Robbins (1932), Samuelson (1947). 4 Rawls (1982), Dworkin (2000), Sen (1992).

xi social justice primarily deals with the distribution of resources and means of flourishing (including personal characteristics that may be registered as internal resources) rather than the distribution of subjective satisfaction. The way in which individuals obtain degrees of satisfaction out of given amounts of resources or capabilities should, in this view, be considered a matter of personal responsibility. In summary, our approach in terms of ordinal non-comparable preferences is compatible with several important ethical views, not just with the positivist “no-bridge” argument that has been influential in economics. Obviously, however, being able to devise social criteria on the sole basis of data which can be extracted from observable demand behavior, or even less as we shall see, is indeed a practical advantage. The topic of this book can, then, be more technically described as the aggregation of conflicting individual preferences into a consistent social ranking. Admittedly, we are a far cry from achieving the ultimate goal of devising a set of criteria for the evaluation of general social and economic situations, where by “general” we mean a complete description of societies in all dimensions and all details. What we provide here is a series of analyses for simple contexts depicted by tractable models. Here is an outline of the contents. The first chapter maps out the field of welfare economics and social choice theory and explains how our approach relates to, and supplements, the existing approaches. Social choice theory has been dubbed the “science of the impossible” and we explain in particular why in our view there is a lot of room for interesting possibilities. It is now well known that obtaining possibility results has to do with the information that is used by the criteria. The fourth chapter examines the informational basis of our approach with greater detail, after some general results of the approach have been presented in the second and third chapters. Our focus in the first part of the book is the canonical model of distribution of unproduced goods. This model is simple but useful as a basic tool for the analysis of multi-dimensional problems. Some of the results we obtain with it are recurrent in all contexts. Moreover, it is sufficiently abstract to be versatile, and some results can be easily transposed to other contexts, for instance when goods are replaced by functionings in the description of individual situations. The second and third chapters present two basic results that are pervasive in our approach, as they come up in some way or other in all economic models that have been studied so far. The first basic result is a conflict between the idea of reducing resource inequalities across individuals and the Pareto principle. This efficiency-equality tension is not totally obvious when one has in mind the prominent place, in the theory of fair allocation, of the fully efficient and fair Walrasian equilibrium with equal budgets. It appears, however, that this nice conjunction in the first-best context does not extend well to the ranking of all allocations. The second basic result is that, under reasonable restrictions, the combination of the Pareto principle and some mild requirements which impose a minimal inequality aversion (namely, it must be positive, or even simply non-negative) force the social criteria to actually have an infinite aversion to inequality, as in the maximin criterion. The literature on social welfare con-

xii

INTRODUCTION

tains justifications of the maximin and the leximin criteria which involve rather strong egalitarian requirements, in the one-dimensional context when individual well-being is measured by an interpersonally comparable index of income or utility. The different justification we obtain here hinges on the multi-dimensional context of multiple goods being allocated among individuals. The second part of the book examines the particular social rankings that can be defined for the model of distribution of unproduced goods. It considers in turn the case of divisible goods and the case of indivisibles. The third part introduces production, for the relatively simple case when one output is produced with one input such as labor. We do, however, examine in detail the case of unequal skills, which is particularly relevant for applications to public economics. As alluded to above, the main value of defining fine-grained rankings of all allocations is the possibility of giving policy advice under any restriction of the set of feasible allocations. A particularly relevant context of application is provided by incentive constraints which arise when the public authority has imperfect information about individual characteristics. We show in particular how the social rankings obtained can be used for the evaluation of income tax schedules, when the population is heterogeneous both in skill levels and in preferences about leisure and consumption, and such characteristics are private knowledge. This study of production deals with what is technically described as the production of a private good, but we also examine the problem of production of a public good, which is also very relevant to public economics. An example of application to public good funding in the second-best context (i.e., when individuals may misrepresent their willingness to pay for the public good) is provided. In the second and third parts we adopt the same methodology, which consists in defining efficiency and equity requirements and determining what kind of social rankings satisfy these requirements. Once a social ranking is obtained, it can be used for the evaluation of public policies and in the last part we particularly focus on the translation of the ranking of allocations into a ranking of policies, for standard tax-and-transfer instruments. As acknowledged above, this work is not yet at the stage of proposing social criteria for a general model of the economy with production of several private and public goods, with various inputs and jobs, and externalities. We do hope however that this book will be an encouragement for research in this direction. We end this introduction with a caveat. This is a work in normative economics, where we derive social criteria from basic ethical principles and apply them to policy issues. We consider that the role of the economist in this kind of analysis is to establish the link between value judgments and policy conclusions, not to use the authority of expertise in order to promote personal prejudice. As an illustration of this stance, we often end up considering different criteria that rely on alternative ethical principles. We do not endorse each and every criterion that is proposed here, which would be inconsistent, and we refrain, as much as we can, from expressing definite preferences when several criteria are on the table. Of course, we exercised some judgment in the selection of the basic principles, retaining (or focusing on) those that appear reasonable for current

xiii prevailing views. All in all, we find support in Samuelson’s defense of welfare economics: “It is a legitimate exercise of economic analysis to examine the consequences of various value judgments, whether or not they are shared by the theorist” (1947, p. 220).

How to read this book. The bulk of the argument in this book does not require any special mathematical competence and our hope is that it is accessible to most economists. However, economic allocations are complex objects and our proofs often involve the examination of several different allocations which differ from each other in all sorts of ways. As a consequence, many of the long and tedious proofs of our results have been relegated to Appendix A, in which case the main text only contains an intuitive explanation of the logic of the argument. Among other things, the index lists all of the axioms that are used in the search of social criteria, so as to make it easy to locate their first appearance in the book. General notations. The set of real numbers (resp., non-negative, positive real numbers) is R (resp., R+ , R++ ) the set of natural (resp., relative, positive) integers is N (resp., Z, Z++ ), the set of rational numbers (resp., positive rational numbers) is Q (resp., Q++ ). Vector inequalities are denoted ≥, >, À . Weak (resp., strict) set inclusion is denoted ⊆ (resp., Ã). The (Minkowski) addition of sets is defined as A + B = {x | ∃ (a, b) ∈ A × B, x = a + b} . An ordering is a reflexive and transitive binary relation on a set. The subset of maximal elements of a set A for an ordering R is denoted max|R A.

xiv

INTRODUCTION

Part I

Basics

1

3 This first part is an introduction to the approach developed in this book. Its first aim is to help the reader understand where the approach lies in the broad field of normative economics, and the first chapter reviews the various subdomains of the field in order to highlight the main differences. Its main point is to show that our approach is the only one that combines various features that are scattered in various classical approaches. Fundamentally, this approach opens a working space at the intersection of social choice theory and fair allocation theory: like the former, it constructs rankings of all possible alternatives; like the latter, it involves fairness principles about resource allocation rather than interpersonal comparisons of utility; like both theories, it puts the Pareto principle, the ideal of respecting individual preferences, first in the order of priorities. The expression “fair social choice” is often used as a name for the approach.5 The second and third chapters introduce general results that appear to be common to all models that have been studied so far. These results are striking and somewhat counterintuitive. The first is that there is a tension between the Pareto principle and the deceptively simple idea that an agent who has more in all dimensions than another, e.g., who consumes more of all commodities, is necessarily better-off and should transfer some of his surplus to the other agent. The idea that “having more in all dimensions” is an obvious situation of advantage has been flagged by Sen (1985, 1992) as a partial but robust solution to the problem of indexing well-being in a multidimensional context. This seems indeed very natural, but our approach, because it involves respecting individual preferences, implies that it must sometimes happen that the better-off agent is the one who has less in all dimensions. As we will explain, this is in fact much less counterintuitive than it seems when individual preferences are taken into account. The second result is that, once again because of the multidimensional context and the respect of individual preferences, a minimal degree of preference for resource equality implies that one must actually give absolute priority to the worst-off. This is surprising because it appears easy to use utility functions with a certain degree of concavity in order to obtain a finite preference for equality in resources. But it turns out that this is not easy at all. The construction of such utility functions is so difficult and informationally demanding that the only “simple” approach consists in adopting the absolute priority for the worstoff. As will be illustrated several times in this book, this does not mean that this approach is only for radicals. Giving absolute priority for the worst-off is compatible with many possible ways of identifying the worst-off. As we will show, even free-market libertarians can see their ideas reflected in particular social criteria developed along this vein. The fourth and last chapter of this part may be skipped by the reader who is more interested in applications than in theoretical underpinnings. It examines the informational requirements of our approach. In the theory of social choice, following Sen (1970), it has become classical to analyze the problem of finding possibility results as a problem about the information that is used in the con5 It

is more transparent but less entertaining than the alternative name “welfair economics”.

4 struction of social preferences. This is justified and in Chapter 4 we show how our possibility results are linked to the fact that our social preferences involve certain kinds of interpersonal comparisons. However, they do not perform interpersonal comparisons of utility but instead compare resource bundles or, more precisely, indifference curves (or sets, in more than two dimensions). We also examine in that chapter other aspects of information that are important in understanding the approach, such as the fact that social preferences may depend in a limited way on the feasible set.

Chapter 1

A contribution to welfare economics 1.1

Introduction

Social welfare and fairness are concepts with a venerable history in economic theory. In this chapter, we would like to situate the approach that is developed in this book within the field of welfare economics. The simplest way to do this is to compare it to the various existing approaches in the field. For each of them, we will list the main features which are shared with our approach, and explain why we keep them. We will also present the main differences and we will explain and justify why we have to, or choose to, depart from those classical approaches. We hope that this short overview will clarify how our undertaking can contribute to the development of some of these subfields. This discussion will rely on a simple example. Assume that a positive quantity of several divisible private goods has to be distributed to a population of agents, each of whom has personal preferences over his own consumption.1 We will discuss each classical subfield of welfare economics, as well as our approach, in this simple framework. The preferences are assumed to be well-behaved and self-centered (i.e., without consumption externalities). What we call an economy is a population with a profile of preferences and a social endowment to be distributed. Formally, we consider that there are goods (with ≥ 2). The social endowment of goods is denoted Ω ∈ R++ . The population is a non-empty finite subset N. Its cardinality is denoted |N |. Each agent i in N has a preference relation Ri which is a complete ordering over bundles zi belonging to her consumption set X = R+ . For two bundles x, y ∈ X, we write x Ri y to denote that agent i is at least as well off at x as at y. The corresponding strict preference and indifference rela1 This canonical model has a long tradition in welfare economics. In the more recent literature, it is examined in Arrow (1963), Kolm (1972), Varian (1974), Moulin and Thomson (1988), and Moulin (1990, 1991), among many others.

5

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CHAPTER 1. A CONTRIBUTION TO WELFARE ECONOMICS

tions are denoted Pi and Ii , respectively. We restrict attention to preferences which are continuous, monotonic (i.e., for two bundles x, y ∈ X, if x ≥ y, then x Ri y and if x À y, then x Pi y), and convex. Let R denote the set of such preferences. An economy is denoted E = (RN , Ω), where RN = (Ri )i∈N is the profile of preferences for the whole population. Let E denote the class of all economies satisfying the above conditions. An allocation is a list of bundles zN = (zi )i∈N ∈ X N . It is feasible if X zi ≤ Ω. i∈N

We denote the set of feasible allocations for E by Z(E). As we will see in the next sections, each subfield of welfare economics would address different questions in this model, and offer specific ways of solving them.

1.2

The theory of fair allocation

The theory of fair allocation (for a survey, see Thomson 2010), pioneered by Kolm (1966, 1971) and Varian (1974) looks for ways of allocating resources which are efficient, in the sense of Pareto, and fair. It turns out that, typically, fairness does not receive a unique interpretation in resource allocation models. As a consequence, the theory aims at identifying all possible ways of capturing intuitions of fairness, defining axioms that encapsulate these intuitions, and looking at allocation rules which satisfy the axioms. An allocation rule is a correspondence S that associates to each economy E, in a domain D ⊆ E, a subset S(E) of the feasible allocations. The two allocation rules that have received the most attention are the Egalitarian Walrasian and the Egalitarian-Equivalent rules. The first one, which we denote S EW , selects all the allocations arising as competitive equilibrium allocations from an equal division of the social endowment of goods. The individual budget delineated by the initial endowment ω i ∈ X and market prices p ∈ R+ is defined as B(ω i , p) = {zi ∈ X | pzi ≤ pω i }. Allocation rule Egalitarian Walrasian S EW For all E = (RN , Ω) ∈ E, ½ µ ¶¾ Ω S EW (E) = zN ∈ Z(E) | ∃p ∈ R+ , ∀i ∈ N, zi ∈ max|Ri B ,p . |N | A feasible allocation zN for an economy E ∈ E is (Pareto) efficient if there 0 is no feasible zN such that zi0 Ri zi for all i ∈ N and zi0 Pi zi for some i ∈ N. Let P (E) denote the set of efficient allocations for E. We can now define the second allocation rule. It selects all the Pareto efficient allocations having the property that each agent is indifferent between her bundle and a fraction of the social endowment, the same for all agents.

1.2. THE THEORY OF FAIR ALLOCATION

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Allocation rule Egalitarian Equivalent S EE For all E = (RN , Ω) ∈ E, S EE (E) = {zN ∈ P (E) | ∃λ ∈ R+ , ∀i ∈ N, zi Ii λΩ}. Figure 1.1 illustrates these allocation rules in the Edgeworth box. We have N = {1, 2}, and preferences R1 , R2 are represented by two indifference curves for each agent. The two allocations represented in the figure are selected by the W E allocation rules defined above: zN = (z1W , z2W ) ∈ S EW (E) and zN = (z1E , z2E ) ∈ EE S (E). good 2 µ R1 ¡ ¾6 ¡¡ good 1

O2 ´ ´ ´ ´ E E E ´ p ´ szN = (z1 , z2 ) ± ´ ´ λΩ ´ s s ´ W W W zN = (z1 , z2 ) 3 ´ ´ ´ s ( Ω2 , Ω2 ) ´ ´ s´ + ´ λΩ ´ ´ ´ ¡ ´ ¡ ´ ª ¡ ´ ´ R2 ´ ´ -good 1 ? O1 good 2 Figure 1.1: Illustration of two major allocation rules.

Our theory has a lot in common with the theory of fair allocation, and we consider that we are mainly contributing to that theory. First, the (sometimes implicit) central ethical objective on which the theory is grounded is that of resource equality. In the simple model we use in this chapter, as well as in any other resource allocation model studied from that fairness point of view, if allocating goods equally were always possible and compatible with Pareto efficiency, then no other solution would be looked for. As explained in the introduction, we believe equality of resources can be a key objective of a theory of fairness and social welfare, following Rawls and other philosophers’ argument that social justice is a matter of allocating resources rather than subjective satisfaction or happiness. Our work can be seen as an application of this approach in economic theory, to the limited extent that the models we study in this book are very partial descriptions of societies and agents’ preferences are a crude description of their conceptions of a good life. On the other hand, our simple models will already be sufficiently rich to prove that the seemingly simple notion of equal-

8

CHAPTER 1. A CONTRIBUTION TO WELFARE ECONOMICS

ity of resources can receive several interpretations, all of which axiomatically justified. The main difference between the theory of fair allocation in its current shape and our contribution is that a solution to a resource allocation problem in the former is an allocation rule, whereas in the latter it is a social ordering function. Let us explain this key point. When one studies allocation rules, the objective is limited to identifying the optimal allocations for each economy in a given domain, optimality being defined by a combination of efficiency and fairness axioms. In our approach, we look for social ordering functions, hereafter SOFs, which specify, for each economy, a complete ranking of the corresponding allocations. Formally, a social ordering (for economy E = (RN , Ω)) is a complete ordering over the set X N of allocations. A SOF R associates every economy 0 E in some domain D ⊆ E with a social ordering R(E). For zN , zN ∈ X N , we 0 0 write zN R(E) zN to denote that allocation zN is at least as good as zN in E. The corresponding strict social preference and social indifference relations are denoted P(E) and I(E), respectively. Let us illustrate this notion. The following example of a SOF also relies on the concept of “egalitarian-equivalence” and applies the leximin criterion to specific individual indices. The general definition of the leximin criterion is the following. For two vectors of real numbers aN , a0N , one says that aN is better than a0N for the leximin criterion, which will be denoted here by aN ≥lex a0N , when the smallest component of aN is not lower than the smallest component of a0N , and if they are equal, the second smallest component is not lower, and so forth. The specific individual indices to which the leximin criterion is applied by the SOF are defined as follows. They evaluate every agent’s bundle by the fraction of Ω to which this agent is indifferent.2 Indices of this sort are actually standard “utility” representations of individual preferences.3 Formally, let us define the function uΩ (zi , Ri ) by the condition zi Ii uΩ (zi , Ri )Ω. We propose to call it a “Ω-equivalent” utility function, as it measures the proportional share of Ω which would give agent i the same satisfaction as with zi . Figure 1.2 illustrates this notion. When the leximin criterion is applied to Ω-equivalent utilities, one obtains the Ω-Equivalent Leximin SOF: Social ordering function Ω-Equivalent Leximin (RΩlex ) 0 For all E = (RN , Ω) ∈ E, zN , zN ∈ XN ,

0 ⇔ (uΩ (zi , Ri ))i∈N ≥lex (uΩ (zi0 , Ri ))i∈N . zN RΩlex (E) zN

2 This “egalitarian-equivalent” SOF was introduced by Pazner and Schmeidler (1978), and, with an approach that is closer to ours, by Pazner (1979). The general idea of egalitarianequivalence can be traced back at least to Kolm (1966), who attributes it to Lange (1936). 3 See e.g. Debreu (1959), Kannai (1970).

1.2. THE THEORY OF FAIR ALLOCATION

9

sΩ ¡ ¡ ¡ ¡ ¡ ¡ ¡ s uΩ (zi , Ri )Ω¡ ¡ ¡ szi ¡ ¡ ¡ ¡ good 1 0

good 2 6

Ri

Figure 1.2: Computation of uΩ (zi , Ri ). Of course, an allocation rule can be viewed as a simple social ordering function, where the optimal allocations are socially strictly preferred to the nonoptimal ones, and where all allocations are socially indifferent in each of these two classes. Such a SOF is obviously too coarse for many applications in which the fully optimal allocations cannot be obtained. Moreover, it violates the Weak Pareto requirement according to which an allocation is strictly better than another one as soon as each agent strictly prefers her bundle in the former allocation to the one in the latter. The SOFs we study in this book, like the Ω-Equivalent Leximin, all satisfy that requirement. Axiom Weak Pareto 0 For all E = (RN , Ω) ∈ D, and zN , zN ∈ X N , if zi Pi zi0 for all i ∈ N , then 0 zN P(E) zN . We think that SOFs may play a crucial role in the study of social fairness, once we take the implementation issue into account. Indeed, it is often the case that the set of allocations among which the public decision-maker has to choose is so narrow that it does not contain any of the optimal allocations. Such constraints may come from asymmetries of information and, hence, incentive considerations (associated to the revelation of preferences). They may also be associated to the very nature of the problem. This is the case, for instance, if a status quo exists and the solution needs to be looked for in a neighborhood of it, or if the tools which the policy-maker can resort to are limited, for instance, to linear taxation. Having a complete ranking of the allocations and maximizing it always leads to a well-defined solution (provided the set of implementable allocations is compact) no matter which constraints turn out to be the binding ones. Let us assume, for instance, that the social choice has to be made among the set of allocations containing equal division and the allocations that can be

10

CHAPTER 1. A CONTRIBUTION TO WELFARE ECONOMICS

obtained from it through trade at a fixed price of p. Figure 1.3 illustrates, E Ωlex in a two-agent economy, the allocation (z E (both 1 , z 2 ) that maximizes R E agents are indifferent between the bundle z i they get and λΩ, whereas all other allocations in the p trade line assign bundles which at least one agent finds worse than λΩ). p good 2 µ ¾6 good 1

O2 E E s(z 1 , z 2 )

λΩ ´ s + ´ ´ ´ ´ ª ¡ ´ ´ ´ R2 ´ ´ ´ O1

´ ´ ´ ´ ´ ´ ´ ´ λΩ ´ s ´ 3 ´ ´ s µ ¡ ´ R1

-good 1 ? good 2

Figure 1.3: Best allocation for RΩlex when only trade at fixed price p is possible. In the following chapters, we will give examples of applications of SOF maximisation processes in the framework of second-best theory (that is, when the only constraint is that preferences or skills are private information of the agents). Let us complete this section by pointing out another key aspect of the theory of fair allocation that is retained here, namely, its informational basis. Consider for instance RΩlex . It applies the leximin criterion to Ω-equivalent utilities. These “utilities”, however, are mere indexes representing ordinal and noncomparable preferences Ri . They should be interpreted as indexes of resource value (they are expressed in fractions of a resource bundle), not as measures of subjective satisfaction. To make this clear, imagine that instead of starting our analysis at the level of preferences, we had introduced exogenous utility functions (measuring subjective satisfaction, for instance) ui : X → R and defined the set U of all utility functions representing continuous, monotonic and convex preferences. In this alternative framework, an economy would be a list E = (uN , Ω) ∈ U N × R+ . In such a setting, our approach would simply ignore the properties of such utility functions other than the underlying preferences, i.e., would seek SOFs which satisfy the following axiom. Axiom Ordinalism and Non-Comparability 0 For all E = (uN , Ω), E 0 = (u0N , Ω) ∈ D, and zN , zN ∈ X N , if for all i ∈ N

1.3. ARROVIAN SOCIAL CHOICE THEORY

11

there exists an increasing function gi : R → R such that for all x ∈ X, u0i (x) = 0 0 gi (ui (x)), then zN R(E) zN ⇐⇒ zN R(E 0 ) zN . Ordinalism follows from the fact that any utility function can be replaced by any strictly increasing transformation of it, so that only the ranking of bundles matters. Non comparability follows from the fact that those transformations of utility functions can differ among agents (we may have as many gi functions as agents) so that no common meaning can be attributed to utility levels or utility differences, in particular. Under Ordinalism and Non-Comparability we do not lose any generality by defining economies directly in terms of preferences Ri .

1.3

Arrovian social choice theory

Contrary to the theory of fair allocation, Arrovian social choice looks for finegrained rankings of the allocations. As explained above, our approach displays this feature as well. The main difference between Arrovian social choice and our approach comes from the axioms we impose. The key axiom in the Arrovian tradition is the following independence requirement.4 A SOF R satisfies this requirement if and only if the ranking between two allocations only depends on the individual preferences about these two allocations. Axiom Arrow Independence 0 0 For all E = (RN , Ω), E 0 = (RN , Ω0 ) ∈ D, and zN , zN ∈ X N , if Ri and Ri0 0 0 0 coincide on {zi , zi } for all i ∈ N , then zN R(E) zN ⇐⇒ zN R(E 0 ) zN . We illustrate the consequences of imposing Arrow Independence in Figure 1.4. By monotonicity of preferences, agent 1 always prefers z10 to z1 , and agent 0 2 z2 to z20 . By Arrow Independence, that implies that zN = (z1 , z2 ) and zN = 0 0 (z1 , z2 ) need to be ordered the same way independently of the preferences. Now, in the economy E = (RN , Ω), zN is a competitive allocation from equal division, 0 i.e., zN ∈ S EW (E), whereas zN is an inefficient allocation. In the economy 0 0 0 E = (RN , Ω), on the other hand, zN ∈ S EW (E) and zN is inefficient. It may 0 0 look reasonable to use this information to rank zN above zN in E, and zN above 0 zN in E . Arrow Independence prevents us from doing so. The key objective of Arrovian social choice is to combine Arrow Independence with Pareto axioms. If the domain D over which the SOF is defined contains, for instance, all the profiles of continuous, monotonic and convex preferences, Arrow Independence and Weak Pareto lead to dictatorship (Bordes and Le Breton 1989). Dictatorship means that there is an individual who imposes his strict preferences Pi to the social ordering for all profiles in the domain. More precisely: fixing N and Ω, there is an individual i ∈ N such that for all ¡ ¢N 0 0 ∈ R+ \ {0} , zi Pi zi0 ⇒ zN P (E) zN . This is, obviously, RN ∈ RN , all zN , zN 4 It was called “Independence of Irrelevant Alternatives” by Arrow (1951), a normative name that we do not retain as it is controversial whether the concerned alternatives are indeed irrelevant or not.

12

CHAPTER 1. A CONTRIBUTION TO WELFARE ECONOMICS µ R1 ¡ good 2 ¡ µ ¾6 ¡¡ good 1

(z1 , z2 )

0 0 s(z1 , z2 )

s R20

O1

O2

¡ ¡ ¡ ª R2

R10

(Ω, Ω) s2 2

-good 1 ? good 2

0 0 Figure 1.4: Indep. of Irrelevant Alternatives requires zN R(E)zN ⇔ zN R(E 0 )zN .

in contradiction with resource equality as this agent’s monotonic preferences will then be satisfied at the expense of all the others. The literature on Arrovian social choice in economic domains (that is, domains where trade or production opportunities are explicitly modelled and preferences are typically restricted to be self-centered and satisfy conditions of continuity, monotonicity and convexity) has focussed on identifying the domains of economies over which Arrow’s negative result is reproduced. The general picture is that the domain restrictions consistent with the nature of the allocation problems are not sufficient to escape Arrow’s impossibility, or, when Arrow’s impossibility is avoided, to be consistent with any minimal objective of resource equality. We depart from Arrovian social choice by shifting the emphasis from Arrow Independence to resource equality properties. We agree that saving on the information needed to make social judgements is important, but we do not accept to make it more important than allocating resources fairly. Our strategy, later in the book, will therefore be to look at possible weakenings of Arrow Independence compatible with our objective of resource equality. This discussion gives us the opportunity to make the following clarification point. We believe that any attempt to give axiomatic foundations to welfare economics refers, explicitly or implicitly, to a hierarchy of the norms which are taken into account. The hierarchy adopted in this book is the following: the norm which is given priority is efficiency; none of the SOFs studied here would recommend a Pareto inferior allocation. Our second norm is resource equality. As it will be clear in the following chapters, many axioms capture the ideal of resource equality, and one ultimately has to make a selection among them.

1.4. WELFARISM

13

Our third norm is informational simplicity or parsimony, and the weakenings of Arrow Independence belong to this category. Consequently, our strategy will always be to begin with a discussion of resource equality axioms, in order to check whether they are compatible with Pareto axioms, and, when necessary, to identify the trade-offs between them. We do not claim that the hierarchy adopted here is the only reasonable one, but we hope it is sufficiently reasonable to serve as a useful guide for the development of an attractive approach. To conclude this section, let us mention an important consequence of the fact that we give priority to resource equality axioms over informational parsimony axioms: we allow our ranking of the allocation to depend on the available resources (which is captured by the fact that the SOFs have E = (RN , Ω) as their argument), whereas the Arrovian approach to social choice excludes this possibility, making a radical distinction between the objectives (the satisfaction of the population) and the constraints (the feasible set of allocations). Again, it would be better if, at the end, we obtained SOFs that do not depend on this kind of information (and, sometimes, this will be achieved), but we are not ready to drop our resource equality objective because of such informational parsimony considerations.

1.4

Welfarism

Welfarism is the social ethics according to which social welfare comes from an appropriate aggregation of individual welfare indices. If one understands the notion of “welfare index” in a broad sense, few social criteria fall out of this category. In particular, the RΩlex SOF is welfarist as it applies the leximin criterion to welfare indices corresponding to Ω-equivalent utilities. Most of the SOFs analyzed in this book are welfarist in a similar way. An important branch of social choice theory has sprung out of Arrow’s impossibility by introducing exogenous interpersonally comparable utility functions. It has explained the links between certain types of aggregation and certain types of interpersonal comparisons. The sum-utilitarian criterion, for instance, requires that welfare indices be cardinally measurable and that, at least, the unit of welfare measure be comparable. An egalitarian criterion such as the leximin, on the other hand, requires the utility levels to be comparable. In comparison, our approach satisfies Ordinalism and Non Comparability, as explained in the previous section, and, instead of taking welfare indices as exogenous data, it proposes to construct welfare indices out of ordinal and non comparable preferences. Moreover, the particular welfare indices used in the various SOFs studied in this book will be derived from specific fairness axioms, thereby obtaining an ethical justification. In this way, the interpersonal comparisons performed in our approach embody specific value judgments in an explicit and transparent way. In summary, for a broad interpretation of welfarism, our approach is welfarist but, rather than leaving the definition of welfare to an outside analysis, it offers specific proposals about how to define and measure the individual welfare

14

CHAPTER 1. A CONTRIBUTION TO WELFARE ECONOMICS

indices. As soon as individual welfare is, as is often the case, defined in a more specific sense that is directly tied to a subjective notion of well-being such as the level of satisfaction or happiness, it is clear that our approach should then be branded as non-welfarist. The individual indices that serve in the SOFs of this book are indices of resources (or resource value) rather than of subjective well-being. Two opposite objections might be raised at this point. One could object to our approach that our emphasis on resource equality should lead us to drop preferences altogether. Our reply is that we want to allocate resources to agents to allow them to follow their view of the good life and a better satisfaction of one agent’s preferences means that she has moved closer to what she thinks is a good life. Consequently, there is no reason to waste individuals’ opportunities to achieve their goals, and this is precisely why Pareto efficiency should be kept. The opposite objection is that by focusing on resource equality our approach falls prey to “resource fetishism” (Sen 1985) and neglects the inequalities that individuals may endure in their abilities to make use of resources to their own benefit. The answer to this objection, which takes inspiration from Dworkin’s theory, is that it is an easy extension of our approach to take account of personal abilities as internal resources over which individuals may have preferences as well, jointly with external resources. This extension will not be studied in this book but has been the topic of another book (Fleurbaey 2008). Finally, let us mention some further differences between most welfarist theories and our approach. First, our theory is less ambitious, as we only aim at giving recommendations for specific allocation problems, whereas welfarist solutions are generally presented as readily applicable to any kind of social choice problem. Second, as we will prove at length in the next chapters, it turns out that not all aggregation processes are compatible with our approach. More specifically, we will have to reach the conclusion that, when combining the axioms we are interested in, only the leximin aggregation method (or, at least, some aggregation method compatible with the maximin) is acceptable.

1.5

Multidimensional inequality

In the model that serves as the illustrative tool of this chapter, there are quantities of goods to be allocated among agents. In the special case where = 1, we are left with a situation where all preferences coincide: more is better than less. This is the classical model of the theory of inequality measurement. This theory aims at providing complete rankings over the set of distribution vectors, and reducing inequality is identical to allocating resources equally. The classical theory of inequality measurement has been recently extended to take multiattribute situations into account, that is, to build orderings over allocations of several goods based on the ideal of resource equality. Our approach has exactly the same objective. The only, but essential, difference regards the way preferences are taken into account. What is key in the theory of SOFs is that individual preferences are allowed

1.5. MULTIDIMENSIONAL INEQUALITY

15

to differ among agents, and we impose Pareto axioms. From that point of view, we need to distinguish between two approaches to multidimensional inequality measurement. The first approach, which we call the dominance approach, assumes that there exists a function measuring how quantities of the good to allocate are transformed into utility, and this function is the same for all individuals (e.g., Atkinson and Bourguignon 1982). The second approach does not use any utility function, and we examine it later on. In the dominance approach, the utility function which is used to evaluate quantities of resources can be interpreted in two ways. In one interpretation, it represents individual preferences, which must then be assumed to be identical. In another interpretation, it consists of an index chosen by the evaluator, and reflecting the evaluator’s ethical choice about, for instance, how an increase in some resources allocated to a relatively poor agent trades off against a decrease in some other resource allocated to a relatively richer agent. The key feature of the dominance approach is that an allocation of resources, 0 zN , is said to be better than another one, zN , if the sum of the individual utilities at the former one is larger than the corresponding sum at the latter one independently of which utility function is chosen to evaluate the individual bundles. More precisely, for a given set U of admissible utility functions, an allocation is said to dominate another if its sum of utilities is greater for each and every utility function in that set. Axiom Dominance w.r.t. U 0 For all E = (RN , Ω) ∈ D, and zN , zN ∈ XN , X X 0 ⇐⇒ ∀U ∈ U, U (zi ) ≥ U (zi0 ). zN R(E) zN i∈N

i∈N

An immediate consequence of this definition is that the resulting ordering will be partial. In the one-dimensional context, it coincides with the generalized Lorenz criterion due to Shorrocks (1983) when U is the set of increasing concave functions. Another major difference with our approach is the fact that the same index is used to evaluate the bundle of all the agents, whereas we take as a starting point that individuals may have different preferences and that these preferences should be respected. In the second approach to inequality measurement, no index of individual welfare appears at all. It is assumed that more of any good is better, but how agents trade off relatively more of one good against relatively more of another good is not taken into account. The emphasis is on how unequally each good is distributed and how the inequalities in the distribution of each good should be aggregated. One branch of this approach seeks direct generalizations of the Lorenz criterion without considering additive social welfare functions.5 Another branch constructs inequality indices. For instance, consider the following multidimensional generalization of the Gini coefficient (Gajdos and Weymark 2005). Two allocations are compared in the following way. For each allocation, first, 5 See,

e.g., Kolm (1977), Koshevoy and Mosler (2007).

16

CHAPTER 1. A CONTRIBUTION TO WELFARE ECONOMICS

the Gini mean6 of the distribution of each good is computed separately, and, second, the geometric mean of these means is computed. The allocation with the largest geometric mean is socially preferred to the other. To define this SOF formally, we need to introduce the following notation. For E = (RN , Ω) ∈ D, zN ∈ X N and k ∈ {1, . . . , }, let zek ∈ RN + denote the permutation vector of (zik )i∈N where the components are in decreasing order: ze1k ≥ ze2k ≥ . . . ≥ ze|N|k . Social ordering function Multi-dimensional Gini (RMG ) 0 For all E = (RN , Ω) ∈ E, zN , zN ∈ XN , 0 zN RMG (E) zN

⎞1 ⎛ ⎞1 |N | |N | X Y X ⎝ 1 ⎝ 1 ⇔ (2i − 1)e zik ⎠ ≥ (2i − 1)ze0 ik ⎠ 2 2 |N | |N | i=1 i=1 k=1 k=1 Y

⎛

Contrary to the results that can be derived from the dominance approach, this SOF provides us with a complete ranking of the allocations. Another advantage of this approach is that it limits the relevant information regarding individual preferences. The RMG SOF, for instance, satisfies Arrow Independence. On the other hand, the immediate disadvantage is that it violates the Pareto principle. Consequently, the approach is justified only in contexts where no information on preferences can be obtained, or when the evaluator has good reasons to disregard the information that can be obtained. In conclusion, the main contribution of our approach to the study of multidimensional inequality is to put individual preferences at the heart of the theory and to study inequality under a strict application of the Pareto principle. On the other hand, we share with this literature the key concern of resource equality, and, as it will become clear in the very next chapters, we borrow from this literature some crucial properties of inequality aversion.

1.6

Bergson-Samuelson social welfare functions

Arrovian social choice theory has developed mainly as an attempt to make sense of the social welfare functions introduced by Bergson (1938) and further discussed, clarified and vindicated by Samuelson (1947). Conceptually, a Bergson-Samuelson social welfare function is a representation tool for a SOF that satisfies the Pareto principle. Consequently, it can be written as a social welfare function of individual utilities as follows. Axiom Bergson-Samuelson Social Welfare Function For all E = (RN , Ω) ∈ D, there exists a function W : RN → R and, for each i ∈ N , a utility function Ui : ¡X → R representing for all ¢ ¡ Ri such that ¢ 0 0 zN , zN ∈ X N , zN R(E) zN ⇐⇒ W (Ui (zi ))i∈N ≥ W (Ui (zi0 ))i∈N . 6 The Gini mean of a distribution is the weighted arithmetic mean where the weights are determined by the rank of the corresponding component in the distribution of that good among agents.

1.6. BERGSON-SAMUELSON SOCIAL WELFARE FUNCTIONS

17

An important feature of this axiom is that the functions W and Ui can be picked ¡ arbitrarily ¢ provided the ordering obtained with the composite function W (Ui (zi ))i∈N corresponds to R(E). The function W is not a fixed social welfare function meant to be applied to exogenous and arbitrary utilities. As emphasized by Samuelson: “There are an infinity of equally good indicators... which can be ¯ = W (U1 , ..., Un ), and used. Thus, if one of these is written as W if we were to change from one set of cardinal indexes of individual utility to another set (U10 , ..., Un0 ), we should simply change the form of the function W so as to leave all social decisions invariant.” (1947, p. 228, with adapted notation). This makes it possible for a SOF that is representable in this way to satisfy Ordinalism and Non-Comparability, as Bergson and Samuelson were keen to insist. There has been a controversy on such a possibility a few decades ago, which was triggered by the mistaken belief of some commentators that the W function was meant to be fixed and that the utility functions were exogenous.7 Not many of the SOFs we define and study in this book can be represented by Bergson-Samuelson functions, but this is only because they involve the leximin criterion, which is not representable by a function. For instance, a maximin variant of the Ω-Equivalent Leximin SOF defined in Section 1.2 is representable by a Bergson-Samuelson social welfare function, with W as the min function, and, in E = (RN , Ω) ∈ D, Ui (zi ) = uΩ (zi , Ri ). In this light, our work can be viewed as a contribution to the Bergson-Samuelson tradition. The relationship between our approach and the Bergson-Samuelson tradition is more than a formal one. The original objective of Bergson, indeed, was to construct objects that could be used to evaluate societies from a fairness point of view: “The optimum income distribution... is not determined by an empirical comparison of marginal social welfare per dollar among different households. Rather, it is determined by the rule of equity, which itself defines social welfare in the sphere of income distribution” (1938, p. 66). Unfortunately, neither Bergson nor Samuelson provided hints at what the “rule of equity” could be. We believe that the Bergson-Samuelson tradition can be developed in three ways. First, it is not sufficient to find examples of Bergson-Samuelson functions (as Samuelson 1977 did, for instance), it is also important to provide an axiomatic basis for them in order to elucidate their ethical underpinnings. Second, because of Samuelson’s reluctance to consider cross-economy axioms such as Arrow Independence, it is generally believed that the Bergson-Samuelson approach is a “single-profile” approach. But SOFs that are representable by Bergson-Samuelson functions can satisfy cross-economy 7A

detailed study of this controversy is made in Fleurbaey and Mongin (2005).

18

CHAPTER 1. A CONTRIBUTION TO WELFARE ECONOMICS

robustness properties. In particular, there is a specific weakening of Arrow’s independence property that is needed to obtain reasonable Bergson-Samuelson SOFs. This point has already been made in the literature (by Hansson (1973), Mayston (1974) and Pazner (1979)), but is not well known. Third, properly justified Bergson-Samuelson functions can be used to reach original policy conclusions in public economics. While public economics is often made with not very precise social welfare functions, we believe that more precise conclusions can be obtained with the more specific social criteria that emerge from a sufficiently rich axiomatic analysis. In this book we propose several second-best analyses and devote one full chapter to the optimal income taxation problem. It should be clear now why we see our approach as part of welfare economics, and we do hope to contribute to the revival of a field that has received too little attention in the last decades.

1.7

Cost-benefit analysis

Cost-benefit analysis is one of the oldest branch of welfare economics. Its aim is to evaluate social policy and it refers specifically to willingness-to-pay in a market economy. One can distinguish three variants of cost-benefit analysis. The oldest variant is based on Marshallian consumer surplus and producer surplus in partial equilibrium analysis. This has been refined for the general equilibrium approach by replacing the Marshallian surplus by compensating and equivalent variations or the related Kaldor-Hicks-Scitovsky compensation tests. Finally, most specialists now advocate using something like a Bergson-Samuelson social welfare function in order to avoid the problems raised by the older approaches. The old approaches, however, remain in use in much of the applied work done in evaluation units of governments throughout the world, as well as in branches of normative economics such as industrial policy and international economic policy. As the Bergson-Samuelson approach has already been discussed in the previous section, let us focus here on compensating and equivalent variations. In the context of our example, the typical question would be to assess the change from a market equilibrium (zN , p) corresponding to endowments ω N , to another 0 market equilibrium (zN , p0 ) corresponding to endowments ω 0N . This reveals an immediate difference with the objective of a SOF: here, the ranking of allocations need not be complete. The compensating variation for agent i ∈ N is defined by p0 ω 0i − min {p0 x | x Ri zi } , which can be interpreted as the amount agent i would be willing to pay in the final state in order to move from zi to zi0 . Her equivalent variation is defined by min {px | x Ri zi0 } − pω i , corresponding to the payment the agent would accept, in the initial state, in order to forgo the move from zi to zi0 .

1.7. COST-BENEFIT ANALYSIS

19

The change is socially acceptable, in that approach, if over all agents the sum of the compensating (or equivalent) variations is positive. Informally, a reform is implemented if there exists a way for those who gain to strictly compensate those who lose, so as to leave no one worse off than in the initial allocation. This is a kind of utilitarian point of view where gains and losses are measured and compared in monetary terms, and where the essential feature is that compensation be possible, should it be implemented or not. There are two classical problems with this approach.8 First, such criteria are not transitive. This occurs because the reference prices which serve to evaluate the changes depend on the pair of allocations to be compared. Second, it is ethically dubious to consider that a virtual compensation of losers can justify a reform. Suppose the losers are mostly among the poorest of the population, while the gainers are among the richest. It then seems shocking to declare the change to be good simply because a compensation is possible, even if the compensation will never be implemented. Our approach seeks to remedy these two problems, first by constructing transitive orderings, and second by putting enough inequality aversion to avoid sacrificing the worse-off in this way. What our approach has in common with cost-benefit analysis, however, is the focus on resource valuation as opposed to subjective utility. In both approaches ordinal non-comparable preferences are the main informational ingredient about individual welfare. To illustrate how the cost-benefit criteria can be minimally changed to come close to our approach, observe that the sum of compensating or equivalent variations evaluates a particular allocation zN by computing X

i∈N

min {p∗ x | x Ri zi } ,

where p∗ is the reference price vector (either the final or the initial prices, for compensating or equivalent variations, respectively). As we will see in Chapter 5, an interesting ordering in our approach consists in evaluating an allocation by the formula max min min {px | x Ri zi } .

p:pΩ=1 i∈N

In this formula we see that the sum over all i ∈ N is replaced by the maximin criterion, while the moving reference price is replaced by a reference price that is specific to the allocation but does not depend on which other allocation the allocation zN is compared to. These two changes alleviate the problems that plague the sum of compensating or equivalent variations. In a sense our approach can be viewed as proposing a particular refinement or extension of cost-benefit analysis. 8 Criticisms of compensating or equivalent variations, or of the related compensation tests, can be found in Arrow (1963), Boadway and Bruce (1984), Blackorby and Donaldson (1990), among many other references.

20

1.8

CHAPTER 1. A CONTRIBUTION TO WELFARE ECONOMICS

Money-metric utilities

For a fixed reference price p∗ , the expression mi (zi ) = min {p∗ x | x Ri zi } is a utility representation of agent i’s preferences, a “money-metric” utility. Samuelson and Swamy (1974) and Samuelson (1974) have proposed to use this number as the key ingredient of a quantity index that is convenient to track individual welfare accurately, in contrast to standard indexes such as the Laspeyres, Paasche or Fisher quantity indexes which do not respect individual preferences in general. Once this number is put on the table, it is tempting to incorporate it in a social welfare function in order to evaluate allocations: W (m1 (z1 ), . . . , mn (zn )). This has not been proposed by Samuelson, who was particularly adamant against summing up money-metric utilities across individuals, but the idea has been promoted by many authors, in particular by Deaton and Muellbauer (1980) and King (1983). As should be clear by now, our own approach is very close to this idea, although it encompasses many variants of it. This idea has attracted a lot of criticism and its popularity quickly declined. Three main critiques seem to have been influential. The first critique is that the equivalent income does not incorporate sufficient information about subjective welfare, as it only depends on ordinal non-comparable preferences. “A variation of one’s intensities of pleasure or welfare cannot, therefore, find any reflection in this numbering system as long as the ordering remains unchanged.” (Sen 1979, p. 11) This criticism may have gained momentum from the fact that moneymetric utilities have often been presented as a “monetary measure of subjective welfare”, clearly an oxymoron. It is much more sensible to defend the notion as depicting “the budget constraints to which the agents are submitted to” (Deaton and Muellbauer 1980, p. 225), which is well in line with our own focus on resource equality. A second critique is that a social welfare function W whose arguments are money-metric utilities mi (zi ) may fail to be quasi-concave in commodity consumptions zN , even when individual preferences are convex and W is quasiconcave (Blackorby and Donaldson, 1988). This appears incompatible with a minimal preference for equality. For instance, consider a two-agent population in which both individuals have the same direct utility function over two-commodity bundles (x, y): Ui (x, y) = min {x, y + 1} . Take reference prices (1, 1) and consider the allocation in which both individuals consume the bundle (1, 1). Their money-metric utility is then equal to 1. Now introduce some inequality, letting individual 1 consume (1 − δ, 1) and individual 2 consume (1 + δ, 1) for a small δ > 0. Then individual 1’s money-metric utility is 1 − δ while individual 2’s money-metric utility is 1 + 2δ. For δ small enough, any differentiable and symmetric social welfare function bearing on money-metric utilities will declare this

1.9. CONCLUSION

21

change to be an improvement:

dW = −

∂W ∂W δ+ 2δ > 0 ∂m1 ∂m2

∂W ∂W = ∂m , as should be the case for a symmetrical social welfare function if ∂m 1 2 because the two individuals are equally well-off in the initial situation. The problem disappears only if one take the maximin or the leximin criterion instead of a social welfare function with finite inequality aversion. We will see in Chapter 3 how this observation can be generalized and, instead of undermining the approach, can serve to justify the maximin or the leximin as aggregation criteria.

The third criticism, developed by Roberts (1980) and Slesnick (1991), is that the mi function depends on the reference vector p∗ , and that, if one wants to be able to perform interpersonal comparisons or social evaluations that are independent of the reference, severe restrictions are required (e.g., identical homothetic preferences). When individual preferences are different from one individual to another, with crossing indifference curves, the mere ranking of individuals simply cannot be independent of p∗ . What we argue in our approach is that references need not be arbitrary but can be selected on the ground of specific fairness considerations. In this light, it appears unduly demanding, perhaps even ethically inappropriate, to seek evaluations which are independent of the reference. In conclusion, we believe that none of the criticisms raised against moneymetric utilities is decisive. There seems to be an interesting conceptual area that is jointly close to the theory of fair allocation, social choice theory, and various branches of welfare economics such as the Bergson-Samuelson approach, costbenefit analysis and money-metric utilities. Our thesis in this book is that this area is more fertile than has been considered.

1.9

Conclusion

This quick general view of the main fields of welfare economics has identified four key items that allow us to compare our approach with those fields: whether the objective defines a complete and fine-grained ordering, whether possibly different preferences are taken into account and respected, whether the emphasis is put on resource equality, and whether the informational basis of the social judgments satisfy the ordinalism non-comparability requirement. How each field behaves with respect to these items is summarized in the following table. A question mark in a cell means that the approach leaves this issue unspecified.

22

CHAPTER 1. A CONTRIBUTION TO WELFARE ECONOMICS Table 1.1: Comparison of the main approaches Resource Ordinal SOF Pareto equality non-comp. Fair allocation − + + + Arrovian social choice + + − + Welfarism (broad) + ? ? ? Welfarism (subjective) + + − − Dominance (incomplete) − + + Multidim. ineq. indices + − + + Bergson-Samuelson + + ? + Cost-benefit Fair SOFs (this book)

(incomplete + & intransitive)

+

+

−

+

+

+

These comparisons allow us to characterize our approach as follows: we study social ordering functions that are consistent with the Pareto principle and satisfy fairness properties, while sticking to ordinal and non-comparable information on individual preferences. The underlying view of a just society is one where resources are allocated among agents in such a way that the value they attach to bundles of resources is equalized. With this work, we especially hope to convey two key messages: • Welfare economics is not caught in a dilemma between impossibility and interpersonal comparisons of subjective utility. • Considerations of fairness (interpreting justice in terms of resource equality) are useful to address second-best policy problems.

Chapter 2

Efficiency versus equality 2.1

Introduction

The evaluation of allocations of resources in a given context should depend on the characteristics of the context, in particular the preferences of the population. This is why the object of our study is a social ordering function, which specifies, for each economy in an admissible domain, a complete ranking of the corresponding allocations. As announced in the introduction, we will study SOFs in distribution models, with divisible or indivisible goods, as well as in production models of a private or a public good. This study will be developed in Parts 2 and 3 below. In all this work, fairness is interpreted as resource equality, but there are different ways of specifying this notion, in particular in relation to the specific features of the environment, so that the SOFs which end up being selected do not have much in common. Some partial results, however, turn out to be general in the sense that they have their counterpart in each of the studied models. In this first part, we present and discuss these general and basic results. They provide two insights into the possibilities and limitations surrounding the construction of SOFs. The first concerns the way in which resource equality requirements need to be defined in order to be compatible with basic efficiency principles. The second lesson concerns the degree of inequality aversion which is compatible with efficiency and informational simplicity requirements. The model we rely upon in order to present these general results is the canonical “fair division” model, which was introduced in the previous chapter and is also the topic of a more specific analysis of SOFs in Part II. The fact that this model is rather abstract but at the same time contains the main ingredients of economic allocation models that pertain to our analysis (multiple goods, heterogeneous individual preferences) makes it quite suitable for the presentation of general findings on SOFs. The topic of this chapter is the way in which resource equality requirements may clash with the Pareto principle when they are defined without sufficient 23

24

CHAPTER 2. EFFICIENCY VERSUS EQUALITY

precaution. In particular, requiring that a transfer of resources from agent j to agent k be a social improvement, whenever j is assigned more of all commodities than k, turns out to be incompatible with standard versions of the Pareto principle. This means that “consuming more of all commodities than another agent” is not sufficient for an agent to be unambiguously identified as better-off than the other agent in the ethically appropriate sense. In this chapter we present this result formally (Section 2) and we analyze several ways of modifying the equality requirement so as to recover compatibility with the Pareto principle (Section 3).

2.2

Resource equality vs Pareto efficiency

The idea of resource equality has a well established tradition in the theory of fair allocation, where it finds a variety of expressions: equality of budgets, no-envy (no agent should prefer another’s bundle), egalitarian-equivalence (every agent should be indifferent to a reference bundle), lower bounds (e.g., every agent should be at least as well-off as at equal split), solidarity (every agent should be affected in the same direction when general parameters such as resources or population size change), etc.1 Here we deal with orderings, so that the idea of the “optimality” of resource equality has to be translated into the “betterness” of inequality reduction. This naturally leads us to consider the Pigou-Dalton transfer principle2 as a starting point. The transfer principle is standardly formulated in the one-dimensional framework of income distributions. It then says that a transfer from an agent to another with lower income reduces inequality, or increases social welfare, provided it does not reverse their ranking, or at least provided the transfer does not exceed the initial gap between them. We have to adapt this to our multi-dimensional framework with multiple goods. The version of the Pigou-Dalton principle that we retain focuses on transfers of positive bundles of resources from an agent to another, such that, even after the transfer, the recipient consumes less of every good than the donor. More precisely, the following axiom says that if a transfer of positive quantities of each good is made from j to k, the other agents being unaffected, and after the transfer j still consumes more of every good than k, the final allocation is at least as good as the initial allocation. Axiom Transfer 0 For all E = (RN , Ω) ∈ D, and zN , zN ∈ X N , if there exist j, k ∈ N, and ∆ ∈ R++ such that zj − ∆ = zj0 À zk0 = zk + ∆ 0 R(E) zN . and for all i 6= j, k, zi = zi0 , then zN 1 For 2 Due

recent surveys, see Moulin and Thomson (1997), Thomson (2001). to Pigou (1912), Dalton (1920).

2.2. RESOURCE EQUALITY VS PARETO EFFICIENCY

25

In the literature on multidimensional inequality, it is common to consider a larger class of Pigou-Dalton transfers defined by the formula zj0 = zj − λ (zj − zk ) , zk0 = zk + λ (zj − zk ) , for λ ∈ [0, 1]. No requirement is made about zj > zk , so that these complex Pigou-Dalton transfers may go in opposite directions for different dimensions. For instance, in a two-good economy, if zj = (1, 3) and zk = (3, 1), a complex Pigou-Dalton transfer, in which i is a recipient in the first dimension and a donor in the second, may yield zj0 = zk0 = (2, 2). But as it may happen that zj Pj zj0 and zk Pk zk0 , such transfers may go against unanimous individual preferences. See Figure 2.1(a) for illustration. The clash with the Pareto axioms is then immediate. good 2 6

szi @ Rs @ zi0 = zj0 @ I z @s j

0

good 1

good 2 6

0

µ ¡ szk ¡

0 szk

szj ¡ ª s¡ zj0

(a) (b) Figure 2.1: Pigou-Dalton transfers.

good 1

This kind of problem can be avoided if, as in the axiom of Transfer defined above, the transfer is performed only between two agents who are ranked in the same way in all dimensions, i.e., when one consumes more of every good than the other, and the transfer involves a positive amount of every good going from the richer to the poorer. In that case the agent who receives the transfer is bound to benefit from it, while the donor is bound to have a reduced satisfaction (see Figure 2.1(b)). The transfer cannot then directly go against unanimous preferences. Unfortunately, the Pigou-Dalton principle as embodied in the axiom of Transfer does conflict with efficiency. Theorem 2.1 On the domain E, no SOF satisfies Weak Pareto and Transfer. The simple proof is illustrated in Figure 2.2. Four allocations are represented in a two-agent economy. Agent j is “richer” than agent k in allocations 1 and 2, while the converse is true in the other allocations. Weak Pareto requires 1 4 3 2 2 1 that zN P (E) zN and zN P (E) zN whereas Transfer requires zN P (E) zN and 4 3 zN P (E) zN , which creates a cycle. There exists a single-profile version of this theorem.3 Namely, in any economy where not all agents have the same preferences, it is impossible to construct 3 The

Weak Pareto and Transfer conditions are defined for SOFs, not for social orderings

26

CHAPTER 2. EFFICIENCY VERSUS EQUALITY good 2 6

Rj zj2

zk1

s s zk2

s

szj1

zj4 s s

s zk4

zsk3

Rk

zj3

good 1

0

Figure 2.2: Proof of Theorem 2.1. a social ordering satisfying Weak Pareto and Transfer.4 It should be emphasized here that all the (positive and negative) results of this book involving transfer axioms would remain valid if the transfer axioms combined the requirement that zj0 dominate zk0 with the additional restriction that ∆ should be a certain fraction λ ∈ (0, 1/2) of zj − zk : zj0 = zj − λ (zj − zk ) À zk0 = zk + λ (zj − zk ) . There is a similar problem with a transfer axiom that restricts attention to reducing inequality of budgets at unchanged prices.5 It is indeed very tempting to think that when agents have unequal budgets and this inequality is reduced (without altering market prices), then the situation is improved. But things are, again, not so simple. This provides a way to slightly strengthen Theorem 2.1. One can observe that Theorem 2.1 still holds when Transfer is replaced by the logically weaker axiom stating that: For all E = (RN , Ω) ∈ D, and 0 zN , zN ∈ X N , if there exist j, k ∈ N, and ∆ ∈ R++ , p ∈ R++ such that zj ∈ max|Rj B (zj , p) , zk ∈ max|Rk B (zk , p) , ¡ ¢ zj0 ∈ max|Rj B zj0 , p , zk0 ∈ max|Rk B (zk0 , p) , zj − ∆ = zj0 À zk0 = zk + ∆

in a particular economy. But we can directly understand them as conditions on the social ordering constructed for economy E by considering SOFs defined on the singleton domain D = {E}. 4 See Fleurbaey and Trannoy (2003). Brun and Tungodden (2004) prove a similar singleprofile result with a different transfer condition, which involves zk0 ≥ zj > zj0 > zk , implying that the receiver gets more than the donor. 5 This is due to Gibbard (1979).

2.2. RESOURCE EQUALITY VS PARETO EFFICIENCY

27

0 and for all i 6= j, k, zi = zi0 , then zN R(E) zN . In other words, even assuming additionally that agents choose their bundles in budgets and that the transfer does not alter prices, the idea that reducing inequality is good may conflict with efficiency. The proof is a simple adaptation of Figure 2.2 (one just has to bend the indifference curves so as to equalize the 1 2 marginal rates of substitutions of j and k in allocations zN and zN as well as in 3 4 zN and zN ). This first set of results is undoubtedly bad news, as the axioms involved seem so basic. This questions the intuition that when an agent is above another in all dimensions she is undoubtedly better-off. There is a paradox here. Any monotonic preference relation values a greater bundle more, so that when an agent has more of every good than another, all agents are unanimous in considering that she has a better bundle. But, as shown in Figure 2.2, all agents may also be indifferent between this allocation and another in which they all consider that, because she now has less of every good, her bundle is less valuable. There is a tension between different facets of the idea of respecting unanimous preferences. There are two possible ways out of the impossibility. One can weaken or forget the efficiency requirements, or adopt weaker versions of the transfer principle. We follow the latter route, because we find it worthwhile to explore the extent to which fairness considerations can be taken into account while retaining the Pareto principle. As it turns out, a lot can be done along this vein and this provides the material for this book. The alternative route might also be worth exploring, but it has a lesser priority in our opinion because in the simple settings of distribution and production that we study in this model, the Pareto principle seems hard to relinquish. A criterion violating this principle will sometimes, maybe often, lead to selecting inefficient allocations, and such situations are inherently unstable because agents can find other allocations that make every agent better-off. Moreover, whereas it is relatively easy to formulate weaker versions of the transfer principle, as done in the next section, it seems harder to weaken the Pareto axioms without abandoning the Pareto principle altogether.6 In this chapter we lay the stress on the conflict between the Pareto principle and the transfer principle, but it is worth noting that the problem does not come from the inequality aversion embodied in the transfer principle. Rather, it comes from the fact that the Transfer axiom is not sufficiently sensitive to the agents’ preferences. One can have similar difficulties with axioms reflecting only a principle of impartiality, for instance. Consider a two-agent economy. Out of impartiality, one might consider requiring

(z1 , z2 ) I(E) (z2 , z1 ), but this obviously clashes with the Pareto principle because it may be that, for instance, (z1 , z2 ) is unanimously preferred to (z2 , z1 ). 6 See,

however, Sprumont (2006).

28

CHAPTER 2. EFFICIENCY VERSUS EQUALITY

The conflict with Pareto would remain even if one restricted the application of this indifference requirement to the case in which z1 À z2 , so that (z1 , z2 ) cannot be unanimously preferred to (z2 , z1 ). In Figure 2.3, one sees that Weak Pareto implies that (z1 , z2 ) P(E) (z3 , z4 ) and (z4 , z3 ) P(E) (z2 , z1 ). It is therefore impossible to also have (z1 , z2 ) I(E) (z2 , z1 ) and (z3 , z4 ) I(E) (z4 , z3 ), even though z1 À z2 and z3 À z4 . R1

good 2 6 R2 z2

s

sz1

s

z4

0

zs3 good 1

Figure 2.3: Impartiality vs Pareto.

2.3

Accommodating equality

In this section, we present two main ways to escape the impossibility by weakening Transfer. The first one consists in restricting application of the transfer principle to particular regions of the space of goods. Take the well known principle that in fair division no agent should receive a bundle that is worse, for her preferences, than an equal split of the available resources (Steinhaus 1948, Moulin 1996). This suggests applying the transfer principle to cases where the relatively rich agent is one who gets a bundle she strictly prefers to equal split, and the relatively poor agent prefers equal split to the bundle she gets. It is clear that the ranking of the agents according to this criterion cannot be affected as one moves along indifferent curves, which allows us to escape the impossibility. We will actually consider a simpler restriction, namely, focusing on cases where bundles physically dominate or are dominated by equal split. In this way the resulting requirement does no longer need to refer to preferences. See Figure 2.4 for an illustration. We can now formulate the axiom saying that if a transfer of positive quantities of each good is made from j to k, all other agents being unaffected, and after the transfer j still consumes more than his per capita share while k still consumes less, then the after-transfer allocation is at least as good as the initial allocation.

2.3. ACCOMMODATING EQUALITY

29

good 2 6

Ω/2 s

s0 ¡ zk µ s¡ zk

z sj ¡ ª s¡ zj0

0

sΩ

good 1

Figure 2.4: Equal Split Transfer. Axiom Equal-Split Transfer 0 For all E = (RN , Ω) ∈ D, and zN , zN ∈ X N , if there exist j, k ∈ N, and ∆ ∈ R+ such that Ω zj − ∆ = zj0 À À zk0 = zk + ∆ |N |

0 and for all i 6= j, k, zi = zi0 , then zN R(E) zN .

A related way of carving out a region of the space where the transfer principle is applied consists in referring to bundles that are proportional to the social endowment. Let us say that an allocation zN ∈ X N is proportional for E = (RN , Ω) if for all i ∈ N, zi = λi Ω for some λi ∈ R+ . We denote the set of proportional allocations for E by Pr(E). Proportional allocations delineate a simple setting in which all interpersonal comparisons, as well as transfers between agents, can be conceived directly in terms of fractions of the social endowment. Again, restricting the application of the transfer principle to proportional allocations prevents the ranking of agents into richer and poorer to vary (technically, it prevents us from using Pareto Indifference to move along the indifference surfaces and apply the axiom twice). Again, no information on preferences is necessary to formulate the axiom. Figure 2.5 illustrates this kind of transfer. In a nutshell, the following axiom is identical to Transfer except that it restricts attention to the case in which all individual bundles in the initial and final allocations are proportional to Ω. Axiom Proportional-Allocations Transfer 0 For all E = (RN , Ω) ∈ D, and zN , zN ∈ Pr(E), if there exist j, k ∈ N, and ∆ ∈ R++ such that zj − ∆ = zj0 À zk0 = zk + ∆ 0 R(E) zN . and for all i 6= j, k, zi = zi0 , then zN

30

CHAPTER 2. EFFICIENCY VERSUS EQUALITY sΩ

good 2 6

sz ¡ j ª s¡ 0 s 0 zj z µ k ¡ sz ¡ k

0

good 1

Figure 2.5: Proportional Allocations Transfer. One can of course generalize this approach and consider allocations in which bundles are proportional to any arbitrary reference bundle. The argument underlying Theorem 2.1, however, indicates that it is hopeless to simultaneously satisfy two transfer axioms referring to two different (non-proportional) reference bundles. Only one direction of proportionality can be chosen. Why, then, focus on the direction of the social endowment Ω rather than any other reference bundle? The same kind of argument as in Theorem 2.1, again, shows that any other direction would entail a conflict with Equal-Split Transfer. Even more dramatically perhaps, any other direction would make it impossible to satisfy the following basic axiom jointly with the transfer axiom. The following axiom says that when the simple equal split of Ω is efficient, then it should be among the allocations that are optimal for R(E) in Z(E). Axiom Equal-Split Selection For all E = (RN , Ω) ∈ D, if (Ω/ |N | , . . . , Ω/ |N |) ∈ P (E), then (Ω/ |N | , . . . , Ω/ |N |) ∈ max|R(E) Z(E). Another way of escaping the impossibility is by restricting application of the transfer principle to pairs of agents having the same preferences. Technically, this prevents any crossing of indifference curves as in the above figures. Ethically, the fact that agents have the same preferences guarantees not only that both prefer the bundle with more of every good, but also that this unanimous (among the two concerned agents) ranking of bundles extends to all Pareto-indifferent allocations. There is then no doubt at all about who has a more or a less valuable position. Contrary to the above axioms, the preferences of the agents now play a role. The following axiom, then, is identical to Transfer but for the fact that the agents j and k between which the transfer takes place must identical preferences.

2.3. ACCOMMODATING EQUALITY

31

Axiom Transfer among Equals 0 For all E = (RN , Ω) ∈ D, and zN , zN ∈ X N , if there exist j, k ∈ N such that Rj = Rk and ∆ ∈ R++ such that zj − ∆ = zj0 À zk0 = zk + ∆, 0 R(E) zN . and for all i 6= j, k, zi = zi0 , then zN

The justification of this axiom can also refer to the concept of equity as “no-envy”, a key notion in the theory of fair allocation.7 Agent k envies agent j 0 in zN whenever zj Pk zk . At the allocations zN and zN considered in the above axiom, k envies j and moreover, as they have identical preferences, k would envy j at any Pareto-indifferent allocation. The no-envy concept is a selection concept adapted to allocation rules and does not easily apply to SOFs in the multi-dimensional context, for the same sort of reason as the Pigou-Dalton transfer. Indeed, when indifference curves cross as in Figure 2.6 (compare with Figure 2.2), an allocation in which agent 1 k envies agent j, such as zN in the figure, can be Pareto equivalent to another 2 allocation, such as zN , in which j envies k. It is then difficult to decide who, among j and k, should be given priority on the basis of the no-envy concept. In particular, it is incompatible with Pareto axioms to require an allocation to be preferred to another when they differ in terms of the number or presence of envy occurrences.8 good 2 6

Rj zj1 1 szk

s

zj2

0

s

2 szk Rk

good 1 Figure 2.6: No-envy vs Pareto.

The situation is not problematic, however, when indifference curves do not cross, as in Figure 2.7. In such a case, when an agent envies another, this remains 7 The seminal references are Kolm (1972), Varian (1974). For a recent survey see Arnsperger (1994). 8 See Tadenuma (2002) and Suzumura (1981a,b).

32

CHAPTER 2. EFFICIENCY VERSUS EQUALITY

true for all other allocations which are Pareto-equivalent to this one. This noncrossing property can be obtained when preferences are not exactly identical and this suggests another axiom of transfer, which is stronger than Transfer among Equals. Instead of requiring that the preferences of j and k be identical, it simply requires that their indifference curves through their bundles in zN 0 and zN are nested and thereby do not cross. The condition that indifference curves of the donor and recipient do not cross after (and, as a result, before) the transfer can be formally written as having an empty intersection of the former’s upper contour and the latter’s lower contour sets. See Figure 2.7. good 2 6 U (zj0 , Rj )

L(zk0 , Rk )

µ ¡ sz ¡ k

szk

z sj ¡ ª s¡ zj0

sΩ

0

0

good 1

Figure 2.7: Nested-Contour Transfer. The (closed) upper and lower contour sets will be denoted U (zi , Ri ) = {x ∈ X | x Ri zi } , L (zi , Ri ) = {x ∈ X | zi Ri x} . The following axiom then says that a transfer from j to k, the other agents being unaffected, such that after the transfer j’s upper contour set is still disjoint from k’s lower contour set, produces an allocation that is at least as good as the initial allocation. Axiom Nested-Contour Transfer 0 For all E = (RN , Ω) ∈ D, and zN , zN ∈ X N , if there exist j, k ∈ N, and ∆ ∈ R++ such that zj − ∆ = zj0 À zk0 = zk + ∆, U (zj0 , Rj ) ∩ L(zk0 , Rk ) = ∅

0 and for all i 6= j, k, zi = zi0 , then zN R(E) zN .

2.3. ACCOMMODATING EQUALITY

33

A third way of weakening P Transfer may be considered. Let an allocation zN be called balanced when i∈N zi = Ω, and let B (E) denote the set of balanced allocations in economy E. One sees that in Figure 2.2, it cannot be the case that both z 1 (or z 2 ) and z 3 (or z 4 ) are feasible and balanced. Restricting the requirement to balanced allocations may therefore look as another way out of the impossibility, but it is not, if the economy has at least three agents. Consider the following axiom, which is identical to Transfer but for the fact that the initial and final allocations must be balanced. Axiom Balanced-Allocations Transfer 0 For all E = (RN , Ω) ∈ D, and zN , zN ∈ B (E), if there exist j, k ∈ N, and ∆ ∈ R++ such that zj − ∆ = zj0 À zk0 = zk + ∆ 0 and for all i 6= j, k, zi = zi0 , then zN R(E) zN .

One then obtains the following result. Let E 3 denote the subdomain of E containing the economies with a population of size |N | ≥ 3. Theorem 2.2 On the domain E 3 , no SOF satisfies Weak Pareto and BalancedAllocations Transfer. 1 5 3 2 P(E) zN and zN P(E) zN Figure 2.8 illustrates the argument. One sees that zN by Weak Pareto. On the other hand, Balanced-Allocations Transfer implies 2 1 4 3 5 4 zN R(E) zN , zN R(E) zN and zN R(E) zN , which produces a cycle.

good 2 6

Ri 3 4 szi = zi s 5

zi Rj

zs j1 zk2 s s 2 z 1 s zj k

0

z3 zk5 s s k Rk s s zk4 s zj4 = zj5

zj3

zi1 =s zi2 good 1

Figure 2.8: Proof of Theorem 2.2. We conclude this analysis in the following statement that the relevant transfer axioms are not only separately, but jointly compatible with Weak Pareto and even with the stronger axiom of Strong Pareto. Strong Pareto requires an allocation to be weakly preferred to another whenever all agents weakly prefer

34

CHAPTER 2. EFFICIENCY VERSUS EQUALITY

it. It also requires strict preference as soon as at least one agent displays strict preference. Axiom Strong Pareto 0 For all E = (RN , Ω) ∈ D, and zN , zN ∈ X N , if zi Ri zi0 for all i ∈ N , then 0 0 0 zN R(E) zN ; if, in addition, zi Pi zi for some i ∈ N , then zN P(E) zN . Theorem 2.3 On the domain E, the Ω-Equivalent Leximin RΩlex satisfies Strong Pareto, Equal-Split Transfer, Proportional-Allocations Transfer, EqualSplit Selection, Transfer among Equals and Nested-Contour Transfer. Note that the Ω-Equivalent Leximin RΩlex satisfies the transfer axioms in a strong way, as the post-transfer allocation is deemed strictly better than the pre-transfer allocation. Finally, it is worthwhile comparing RΩlex to two other SOFs which will serve as useful examples in this book. Let us first define these two SOFs. The first one is another kind of Ω-Equivalent SOF.9 It relies on the same valuation of individual bundles, but applies the Nash-product social welfare function instead of the leximin criterion. This is less strongly egalitarian.10 Social ordering function Ω-Equivalent Nash (RΩNash ) 0 For all E = (RN , Ω) ∈ E, zN , zN ∈ XN , 0 zN RΩNash (E) zN ⇔

Y

i∈N

uΩ (zi , Ri ) ≥

Y

uΩ (zi0 , Ri ).

i∈N

The other example orders allocations with respect to the coefficient of resource utilization. This coefficient measures the level of efficiency of an allocation.11 It is equal to the scalar λ ∈ [0, 1] such that a fraction λ of the resources would have been just enough to provide the same welfare level to all agents, that is, if, in an economy with resources limited to a fraction λ of the current resources, there exists a Pareto-efficient allocation which is Pareto-indifferent to 9 It is inspired from a family of SOFs proposed by Eisenberg (1961) and Milleron (1970). These authors are less specific about the utility indices to be used, because they focus on the case of homothetic individual preferences. In their particular context, any choice of a homogeneous utility representation of individual preferences yields the same social ordering. See Chapter 6 for a study of this particular subdomain. 1 0 As it is defined here, this SOF is not good at evaluating allocations in which z I 0 for some j j j ∈ N, since i∈N uΩ (zi , Ri ) = 0 for such allocations, no matter what happens to agents i 6= j. One can, for instance, refine the definition in the following way. Let D(zN ) = {i ∈ N | zi Ii 0} . Then: 0 0 zN RΩN a sh (E) zN ⇔ |D(zN )| < D(zN ) ⎡

1 1 The

⎢ 0 or ⎣|D(zN )| = D(zN ) and

i∈D(z / N)

uΩ (zi , Ri ) ≥

0 ) i∈D(z / N

coefficient of resource utilization was introduced by Debreu (1951).

⎤

⎥ uΩ (zi0 , Ri )⎦ .

2.3. ACCOMMODATING EQUALITY

35

the current allocation (i.e., all agents are indifferent between the two).12 Let P (E) denote the set of Pareto-efficient allocations for economy E. Social ordering function Resource Utilization (RRU ) 0 For all E = (RN , Ω) ∈ E, zN ,¡zN ∈ X N¢ , and λ, λ0 ∈ R+ such that there exist ∗ ∗0 zN ∈ P (RN , λΩ) and zN ∈ P RN , λ0 Ω with zi Ii zi∗ and zi0 Ii zi0∗ for all i ∈ N , 0 ⇔ λ ≥ λ0 . zN RRU (E) zN

The computation of λ is illustrated in Figure 2.9 with the Edgeworth box. Computing λ is equivalent to moving agent 2’s origin O2 down the ray O1 O2 until agent 2’s indifference curve (which moves down correspondingly) is tangent to agent 1’s. Once this is obtained, the new origin for agent 2 represents precisely λΩ. O s2 # #Ω R1 # # s # # λΩ # # # # # # # R2 # # # # # # # # # szN # # # ? O1 ¾6

Figure 2.9: RRU in the Edgeworth box. All the Pareto-efficient allocations are considered socially equivalent by this ordering function, and they are socially preferred to any inefficient allocation. Obviously, this SOF only incorporates efficiency considerations and disregards distributional issues. An interesting feature of the Resource Utilization SOF is that, contrary to RΩlex , it does not involve the intermediate computation of specific utility representations of individual preferences. However, it can 1 2 When one ranks all allocations, including the infeasible, one will of course have λ > 1 for some of them.

36

CHAPTER 2. EFFICIENCY VERSUS EQUALITY

be defined in terms of sums of money-metric utilities, for endogenous prices. Indeed, it ranks allocations in the same way as the expression X max min {px | x Ri zi } . p:pΩ=1

i∈N

∗ ∈ P (RN , λΩ), with p such that pΩ = 1 This fact is proved as follows. Let zN ∗ 13 a supporting price vector for zN , and let zN ∈ X N be Pareto-indifferent to ∗ ∗ ∗ zP N . One has min {px | x Ri zi } = min {px | x Ri zi } = pzi for all i ∈ N and ∗ i zi = λΩ, so that

X

i∈N

min {px | x Ri zi } = p

X

zi∗ = λ.

i

For any arbitrary price vector p0 such that p0 Ω = 1, any i ∈ N, one has min {p0 x | x Ri zi } = min {p0 x | x Ri zi∗ } ≤ p0 zi∗ , so that

X

i∈N

min {p0 x | x Ri zi } ≤ p0

X

zi∗ = λ,

i

implying in turn that λ = max

p:pΩ=1

X

i∈N

min {px | x Ri zi } .

The relationship between efficiency and the sum of money-metric utility functions was critically commented upon by Samuelson with his characteristically sharp style: “Whatever the merits of the money-metric utility concept developed here, a warning must be given against its misuse. Since money can be added across people, those obsessed by Pareto-optimality in welfare economics as against interpersonal equity may feel tempted to add money-metric utilities across people and think that there is ethical warrant for maximizing the resulting sum. That would be an illogical perversion, and any such temptation should be resisted” (1974, p. 1266)

Let us now briefly examine what axioms, among those introduced so far, are satisfied by the three SOFs RRU , RΩlex , and RΩNash . As Table 2.1 shows, unsurprisingly RRU has a bad record with respect to transfer axioms. More surprisingly perhaps, RΩNash does not perform very well either in terms of resource equality. It does satisfy Proportional Allocations Transfer, but not the others, not even Equal-Split Selection (which is satisfied by RRU ). This suggests that a strong aversion to inequality may, paradoxically, be necessary to satisfy minimally egalitarian requirements. This will be the topic of the next chapter. 1 3 The vector p supports allocation z N when for all i ∈ N, zi is a best bundle for Ri in the set {x ∈ X | px ≤ pzi } .

2.4. CONCLUSION

37

Table 2.1: Properties satisfied by the three SOFs Strong Pareto Transfer Equal-Split Transfer Proportional-Allocations Transfer Equal-Split Selection Transfer among Equals Nested-Contour Transfer

2.4

RRU + − − − + − −

RΩlex + − + + + + +

RΩNash + − − + − − −

Conclusion

All the fairness requirements defined in this chapter work as follows. By comparing the bundles of resources agents get, it is possible to identify pairs of agents such that one is relatively richer than the other. Then, a transfer of resources from the richer to the poorer is a strict (or weak) social improvement. Such requirements are compatible with efficiency axioms only if there is an unambiguous way to identify who is relatively rich and who is relatively poor. This may require to look at the preferences of the agents, as in Transfer among Equals or Nested-Contour Transfer, or to relate the bundles that agents receive to the total amount of available resources, as in Equal-Split Transfer or Proportional-Allocations Transfer. It is already clear from the above how a SOF can be defined without comparing individual utilities. The SOFs considered here compare bundles of resources, or more precisely indifference sets, and this is the additional information which makes it possible to stick to ordinal non-comparable information about preferences. Theorem 2.1, gives a warning against hasty comparisons of bundles when one wants the SOF to obey the Pareto principle. It is not too surprising that the Pareto principle reduces the scope for immediate comparisons of objective situations. One may find it frustrating that this undermines what Sen (1992) dubbed the “intersection approach”, namely, the idea that when an agent’s bundle is unanimously considered better than another’s bundle by the population’s preferences, this should be endorsed by the social criterion by giving priority to the agent with the dominated bundle. Our results imply this intuitive idea is deceptive. This is, however, much less paradoxical than it seems at first glance. On the contrary, one may even argue that it makes perfect sense. When the Pareto principle is respected, what matters is not simply the amount of resources agents receive, but also how they value them compared to alternative bundles. Formally, the Pareto principle requires us to disregard bundles of resources and focus exclusively on the agents’ indifference sets. When the indifference curves of two agents cross, it is not obvious who is the better-off, even though one of them may consume more of every commodity than the other. This is not unreasonable at all, especially if one thinks of more concrete settings. Those who consume more and work less are not necessarily better-off if they find their work

38

CHAPTER 2. EFFICIENCY VERSUS EQUALITY

much more unpleasant, those who have a higher income and less sickness may be actually worse-off because they find it harder to cope with their health problems, and so on. In conclusion, it is possible to rely on ordinal non-comparable individual preferences to define consistent resource-egalitarian and Paretian social criteria, but it is important to realize that this may require taking account of preferences in a somewhat comprehensive way.

Chapter 3

Priority to the worst-off 3.1

Introduction

Chapter 2 has described the tension between efficiency and resource equality. From this, one could worry that imposing the Pareto principle will restrict the scope of egalitarian requirements excessively. This chapter dispels such worries. On the contrary, the combination of the egalitarian axioms proposed in Chapter 2 with Pareto requirements has strong egalitarian consequences. More precisely, a small aversion to inequality may, when combined with Pareto axioms, justify a strong aversion to inequality, under some circumstances. A preliminary intuition for this fact can be obtained by observing that an agent may be indifferent between making a small donation in one region of the space and a big donation in another region, as illustrated in Figure 3.1. good 2 6

zk1

s

zj2 s

2 szk

z1 sj

zsj3

zk3

z4 ss k

z4 sj good 1

0

Figure 3.1: Justifying a leaky-bucket transfer ¢ ¡ ¢ ¡ In this figure, allocation zj2 , zk2 is obtained from allocation zj1 , zk1 by mak39

40

CHAPTER 3. PRIORITY TO THE WORST-OFF

ing a transfer from j to k, and this transfer is balanced in the sense that what j gives equals what But, for both agents, this is equivalent to go¡ ¢ k¡ receives. ¢ ing from zj3 , zk3 to zj4 , zk4 , where the transfer made between j and k is a “leaky-bucket” transfer in which the recipient gets less than is given by the donor. Now, consider the situation in Figure 3.2. In this case, it is impos¡ depicted ¢ sible to compare (zj , zk ) to zj0 , zk0 by reference to a single balanced transfer. A leaky-bucket transfer appears necessary here. Now, the transfer axioms introduced in the previous chapter, by themselves, only justify balanced transfers. In this chapter, nonetheless, we show that such axioms may actually justify all “leaky-bucket” transfers, in the sense that they may imply that the SOF must display an infinite aversion to inequality. This holds when, in addition to Pareto axioms, rather innocuous additional requirements of independence and separability (to be defined below) are imposed on the SOF. good 2 6 szj

zk s 0

zk0

s

zs j0 good 1

Figure 3.2: A leaky-bucket transfer. The chapter is structured as follows. The next section examines how to combine the axiom of Transfer among Equals with the Pareto principle, showing how an infinite inequality aversion is hard to avoid, while Sections 3 and 4 repeat the exercise for Nested-Contour Transfer and Equal-Split Transfer.

3.2

Absolute priority to the worst-off

Consider the following strengthening of Transfer among Equals. Instead of considering that some amount of resources is transferred from j to k, we consider that some amount of resources is taken out of j’s bundle and some other, possibly much smaller, amount of resources is added to k’s bundle (as in Figure 3.2). If we still require the resulting allocation to be socially as least as good as the initial one, this means that we give absolute priority to the worse-off, accepting

3.2. ABSOLUTE PRIORITY TO THE WORST-OFF

41

any possible loss of resources in the transfer.1 This axiom is called “Priority among Equals”, but the word “Priority” here is a shortcut for “absolute priority for the worse-off in the relevant pair of agents”.2 We will later define similar “priority” axioms, with the same convention. Axiom Priority among Equals 0 For all E = (RN , Ω) ∈ D, and zN , zN ∈ X N , if there exist j, k ∈ N such that Rj = Rk , zj À zj0 À zk0 À zk ,

0 R(E) zN . and for all i 6= j, k, zi = zi0 , then zN

Incidentally, if a SOF satisfies Strong Pareto and Priority among Equals, then it must consider that the post-transfer allocation is strictly better than the pre-transfer allocation, and not just at least as good. Indeed, consider an 00 allocation zN such that zj À zj0 À zj00 À zk0 À zk00 À zk , 00 R(E) zN and by and for all i 6= j, k, zi00 = zi . By Priority among Equals, zN 0 00 0 Strong Pareto, zN P(E) zN , implying zN P(E) zN by transitivity. We now present a particular situation in which, combined with Strong Pareto, Transfer among Equals may justify the leaky-bucket transfer depicted in Figure 3.2 by a sequence of transfers that are all balanced. In fact we will make use of Pareto Indifference, a weaker axiom than Strong Pareto that says that two allocations are equally good if all agents are indifferent between the bundles they consume at these two allocations.

Axiom Pareto Indifference 0 For all E = (RN , Ω) ∈ D, and zN , zN ∈ X N , if zi Ii zi0 for all i ∈ N , then 0 zN I(E) zN . Consider Figure 3.3 in which, compared to Figure 3.2, some additional indifference curves have been drawn. Since there is no crossing of indifference curves in this figure, it is possible that these curves belong to one and the same preference ordering, and we will indeed assume that agents j and k have the same preferences, so that Transfer among Equals can be applied. 2 1 In virtue of Transfer among Equals, indeed, one sees that zN P(E) zN and 4 3 1 2 3 zN P(E) zN . But Pareto Indifference also requires zN I(E) zN , zN I(E) zN and 4 0 zN I(E) zN . Putting together these relations, by transitivity one concludes that 1 A similar “equity” condition was introduced by Hammond (1976) for the context of utilities: it says that if uj > u0j ≥ u0k > uk , and u0i = ui for all i 6= j, k, then (u0i )i∈N is at least as good as (ui )i∈N . 2 Parfit (1995) introduced the word “priority” in order to cover the whole span of degrees of priority, from very low to infinite. His main intention was to distinguish views that focus on the worst-off out of a concern for the badly-off from those that are inspired by aversion to inequality per se. We do not dwell on this distinction here, and our transfer (and priority) axioms can be motivated from both standpoints.

42

CHAPTER 3. PRIORITY TO THE WORST-OFF zj2 s

good 2 6

s

zk1

zs k2

zk s

zk0

s

0

zs j1 zs j

zs j0

z3 zj4 s s j s zk3 s z 4 k

good 1

Figure 3.3: From Transfer among Equals to absolute priority. 0 zN P(E) zN . The leaky-bucket transfer has been proved equivalent to a sequence of balanced transfers and Pareto-equivalent changes and is thereby vindicated. This reasoning crucially depends on the particular shape of the intermediate indifference curves (the thin curves) drawn in Figure 3.3. For other configurations it would generally be impossible to produce a leaky-bucket transfer out of a sequence of balanced transfers and Pareto-equivalent changes. However, it 0 would be cumbersome if the ranking of two allocations such as zN , zN had to depend on other indifference curves than those observed at these allocations. The independence axiom that we now introduce requires the ranking to disregard such “irrelevant” indifference curves. More precisely, it says that the social ranking of two allocations should remain unaffected by a change in individual preferences which does not modify the agents’ indifference sets at the bundles they receive in these two allocations.3 Let the indifference set of i at zi be defined as I (zi , Ri ) = {x ∈ X | x Ii zi }.

Axiom Unchanged-Contour Independence 0 0 , Ω) ∈ D, and zN , zN ∈ X N , if for all i ∈ N, For all E = (RN , Ω), E 0 = (RN I(zi , Ri0 ) = I(zi , Ri ) and I(zi0 , Ri0 ) = I(zi0 , Ri ), 0 0 then zN R(E) zN ⇔ zN R(E 0 ) zN .

This axiom is weaker than Arrow Independence because it makes it possible to use more information about individual preferences than pairwise preferences about zi , zi0 . Chapter 4 will contain a detailed discussion of independence axioms in social choice. For the time being, let us observe that Unchanged-Contour Independence is, unlike Arrow Independence, satisfied by a wide range of social 3 This kind of property was introduced by Hansson (1973) in the abstract voting model, and was adapted to the context of fair division by Pazner (1979).

3.2. ABSOLUTE PRIORITY TO THE WORST-OFF

43

criteria. The three SOFs presented as examples in the previous chapters (RΩlex , RΩNash , RRU ) do satisfy it. It is also satisfied by cost-benefit criteria based on sums of compensating or equivalent variations.4 It is satisfied by the fair allocation solutions5 when they are redefined as SOFs. For instance, consider the SOF which simply ranks all envy-free and efficient allocations above all others, or the SOF which ranks all egalitarian Walrasian allocations above all others. Both satisfy Unchanged-Contour Independence. The fact that this axiom is widely satisfied does not justify it, but suggests that it is connected to basic principles. Simplicity is one such basic principle. Unchanged-Contour Independence makes the evaluation of allocations simpler and things could be very complex otherwise. Apart from simplicity, there is also the question of relevance, in relation to the agents’ responsibility for their preferences. What matters for the evaluation of an allocation, arguably, is what agents think of the bundle they have, how it compares to other bundles. It does not seem to matter so much how, in detail, agents rank bundles which are worse than theirs, or better than theirs, apart from the fact that the latter are above the former. When agents experience changes of preferences within their lower or their upper contour sets, without affecting the content of these sets themselves, one may say that there is no reason why this should be a public matter justifying a reallocation of resources. It seems plausible that it should remain a private matter. For instance, in a market economy, when the indifference curve does not change, the agent’s choice in her budget set does not change either. This may be related to the idea that agents are to some extent responsible for their preferences. Under Unchanged-Contour Independence, such a strongly egalitarian requirement as Priority among Equals turns out to be implied by Transfer among Equals and Pareto Indifference. We can now state the result. Note that, unless this is absolutely obvious, we also check at the end of the proofs that the axioms are necessary for the result, in the sense that counterexamples can be found if any of them is dropped. Theorem 3.1 On the domain E, if a SOF satisfies Pareto Indifference,6 Transfer among Equals and Unchanged-Contour Independence, then it satisfies Priority among Equals. Proof. The argument is a direct development of the explanations given about Figure 3.3. Let R satisfy Pareto Indifference, Transfer among Equals and 0 Unchanged-Contour Independence. Let E = (RN , Ω) ∈ E, zN , zN ∈ X N and 0 0 j, k ∈ N be such that Rj = Rk , zj À zj À zk À zk , and for all i 6= j, k, zi = zi0 . First case: there exist x ∈ U (zj , Rj ), x0 ∈ L(zj0 , Rj ) such that x ≯ x0 . Let 0 Rj = Rk0 ∈ R, zj1 , zj2 , zj3 , zj4 , zk1 , zk2 , zk3 , zk4 ∈ X, ∆ ∈ R++ be constructed in such 4 See

Section 1.7. Section 1.2. 6 There is a variant of Theorem 3.1, Lemma A.1 proved in the appendix, which involves Weak Pareto instead of Pareto Indifference. It does not imply Priority among Equals, but a variant of it in which “for all i 6= j, k, zi = zi0 ” is replaced by “for all i 6= j, k, zi0 Pi zi ”. Similar variants can also be constructed for the next results of this chapter. 5 See

44

CHAPTER 3. PRIORITY TO THE WORST-OFF

a way that for i ∈ {j, k} , I(zi , Ri0 ) = I(zi , Ri ), I(zi0 , Ri0 ) = I(zi0 , Ri ), zi1 Ii0 zi , zi3 Ii0 zi2 , zi0 Ii0 zi4 , and zj2

= zj1 − ∆ À zk2 = zk1 + ∆,

zj4

= zj3 − ∆ À zk4 = zk3 + ∆.

This construction is illustrated in Figure 3.3. The thick curves represent indifference curves for Ri = Rj as well as for Rj0 = Rk0 , while the thin curves are indifference curves for Rj0 = Rk0 . ¡¡ ¢ ¢ Let E 0 = RN\{j,k} , Rj0 , Rk0 , Ω ∈ E. By Pareto Indifference, (zN\{j,k} , zj1 , zk1 ) I(E 0 ) zN .

By Transfer among Equals, (zN \{j,k} , zj2 , zk2 ) R(E 0 ) (zN \{j,k} , zj1 , zk1 ). By Pareto Indifference, (zN\{j,k} , zj3 , zk3 ) I(E 0 ) (zN\{j,k} , zj2 , zk2 ). By Transfer among Equals, (zN \{j,k} , zj4 , zk4 ) R(E 0 ) (zN \{j,k} , zj3 , zk3 ). 0 0 I(E 0 ) (zN\{j,k} , zj4 , zk4 ). By transitivity, zN R(E 0 ) zN . By Pareto Indifference, zN 0 By Unchanged-Contour Independence, zN R(E) zN . Second case: there are no x ∈ U (zj , Rj ), x0 ∈ L(zj0 , Rj ) such that x ≯ x0 . Then let zj∗ , zk∗ ∈ X be such that zj À zj∗ À zj0 and zk0 À zk∗ À zk , and such that there exist x ∈ U (zj , Rj ), x∗ ∈ L(zj∗ , Rj ) such that x ≯ x∗ , as well as x∗∗ ∈ U (zj∗ , Rj ), x0 ∈ L(zj0 , Rj ) such that x∗∗ ≯ x0 . By the argument of the first case, ¡ ¢ ¢ ¡ 0 one shows that zN \{j,k} , zj∗ , zk∗ R(E) zN and that zN R(E) zN\{j,k} , zj∗ , zk∗ . 0 By transitivity, zN R(E) zN . Finally, let us check that every axiom is necessary for the conclusion of the theorem. 1) Drop Pareto Indifference. A counterexample is Rpsum , definedPas follows, 0 for p ∈ R++ . For all E = (RN , Ω) ∈ E, zN Rpsum (E) zN iff i∈N pzi ≥ P a given 0 i∈N pzi . Ωsum 2) Drop Transfer among Equals. Take , defined P R Pas follows. For all 0 E = (RN , Ω) ∈ E, zN RΩsum (E) zN iff i∈N uΩ (zi , Ri ) ≥ i∈N uΩ (zi0 , Ri ). 3) Drop Unchanged-Contour Independence. Let E IL denote the class of economies where agents have identical linear preferences, and for every linear R, let u(., R) be a linear representation of R. Consider RIL defined as follows. It

3.3. FROM NESTED-CONTOUR TRANSFER TO ABSOLUTE PRIORITY45 0 coincides with RΩlex P on E \E IL , while for all E = (RN , Ω) ∈ E IL , zN RIL (E) zN P 0 iff i∈N u(zi , Ri ) ≥ i∈N u(zi , Ri ).

Transfer among Equals is, alone, compatible with a zero inequality aversion. For instance, it is satisfied by the (non-Paretian) P SOF which ranks allocations according to the value of total consumption, p0 i zi , for some fixed price vector p0 . The above result then produces absolute inequality aversion from axioms which are each, separately, compatible with no aversion to inequality at all! This kind of result draws much from the multi-dimensional framework of this model, and has no counterpart in the theory of social choice or the theory of inequality measurement dealing with one-dimensional measures of well-being. As alluded to in Section 1.8, this result generalizes the observation made by Blackorby and Donaldson (1988) that a differentiable increasing and symmetric social welfare function bearing on money-metric utilities may fail to be quasi-concave in zN . By construction, such a social welfare function satisfies Unchanged-Contour Independence and Strong Pareto. If Rj = Rk and zj À zk , permuting zj and zk gives an equivalent allocation by symmetry, and a convex combination of the permuted allocation and the initial allocation amounts to making a transfer from j to k. Under quasi-concavity, this post-transfer allocation should be at least as good as the initial one, implying that Transfer among Equals would be satisfied as well. But as such a social welfare function does not give absolute priority to the worst-off, by our result it cannot satisfy Transfer among Equals and therefore it cannot be quasi-concave in zN .

3.3

From Nested-Contour Transfer to absolute priority

As we saw in Section 2.3, Nested-Contour Transfer is stronger than Transfer among Equals because it allows transfers between agents with non-crossing indifference curves even when they have different preferences. Consider the following strengthening of Nested-Contour Transfer, which gives absolute priority to the worse-off by allowing leaky-bucket transfers. Axiom Nested-Contour Priority 0 For all E = (RN , Ω) ∈ D, and zN , zN ∈ X N , if there exist j, k ∈ N such that zj À zj0 À zk0 À zk ,

U (zj0 , Rj ) ∩ L(zk0 , Rk ) = ∅ 0 and for all i 6= j, k, zi = zi0 , then zN R(E) zN .

When, at two allocations, two agents have non-crossing indifference curves (i.e., four non-crossing curves altogether), if one does not know the rest of their indifference maps one must consider it possible that they actually have identical preferences throughout. As a consequence, under Unchanged-Contour Independence which forces to consider all possibilities concerning the rest of

46

CHAPTER 3. PRIORITY TO THE WORST-OFF

the indifference map, a transfer between agents with nested indifference curves must be treated like a transfer between agents with identical preferences. Under Unchanged-Contour Independence, therefore, Priority among Equals implies Nested-Contour Priority (similarly, Transfer among Equals implies NestedContour Transfer). One can then immediately state the following corollary to Theorem 3.1. Corollary 3.1 On the domain E, if a SOF satisfies Pareto Indifference, Transfer among Equals and Unchanged-Contour Independence, then it satisfies NestedContour Priority. There is another way in which Nested-Contour Transfer, combined with Strong Pareto, can imply an infinite inequality aversion. Consider the situation depicted in Figure 3.4. In addition to agents j and k, some other agent l is considered, with two of her indifference curves. These three agents may have different preferences, but it is important that no crossing occurs between the curves of the figure. good 2 6

zl2 zk0 zk s 0

z1 ss l

s

zj0

zl3

s

s

szj

4 szl

good 1

Figure 3.4: From Nested-Contour Transfer to absolute priority. The bundles represented on the figure are such that zl1 − zl2 =

¢ 1 0 1¡ (zk − zk ) and zl4 − zl3 = zj − zj0 . 4 4

This is important because it makes the following reasoning possible. Start 1 allocation ¡from ¡ 1 ¢ 2 (zj , zk , zl ). By Nested-Contour Transfer, the allocation ¢ (z3j , zk + 1 2 2 zl − zl , zl ) is better. It is Pareto equivalent to¡ (zj , zk + z − z l l ¡ 1 , z2l ¢). By ¢ 4 Nested-Contour Transfer again, the allocation (zj − zl4 − zl3 , zk + ¡ 4zl − 3z¢l , zl ) is the previous one. It is Pareto equivalent to (zj − zl − zl , zk + ¡ 1better2 ¢than zl − zl , zl1 ). Notice that in this last allocation, agent l is back at zl1 , while the net operation between j and k is a leaky-bucket transfer, because zl4 − zl3 À zl1 − zl2 .

3.3. FROM NESTED-CONTOUR TRANSFER TO ABSOLUTE PRIORITY47 Now, repeat this sequence of transfers and Pareto-equivalent changes three more times. One then ends up with allocation ¡ ¡ ¢ ¢ (zj − 4 zl4 − zl3 , zk + 4 zl1 − zl2 , zl1 ),

which is equal to (zj0 , zk0 , zl1 ). In this configuration, then, we are able to deduce from Nested-Contour Transfer and Strong Pareto that (zj0 , zk0 , zl1 ) is better than (zj , zk , zl1 ). A sequence of balanced transfers and Pareto-equivalent changes, once again, eventually produces a leaky-bucket transfer. In this case, what is important is the presence of agent l with appropriate indifference curves in the relevant area between zj0 and zk0 . Now, should the evaluation of the change from (zj , zk ) to (zj0 , zk0 ) depend on the presence of a third agent with special preferences? This is not necessarily unreasonable, as we will see below, but it violates the principle of separability, according to which a change of allocation concerning a subpopulation should not depend on the rest of the population which is not affected. This principle of separability can be formalized in the following axiom which states that modifying the parameters describing an unconcerned agent (i.e., her bundle and/or her preferences) does not modify the social ordering of the two allocations under consideration. Axiom Separability 0 N For all E = (RN , Ω) ∈ D, and zN , zN is i ∈ N such that zi = zi0 , ¡ ∈ X , if there ¢ 0 0 0 then for all Ri ∈ R such that E = ( RN \{i} , Ri , Ω) ∈ D, zi00 ∈ X, ³ ´ ¡ ¢ 0 0 00 zN R(E) zN ⇔ zN \{i} , zi00 R(E 0 ) zN , z \{i} i .

There is a similar axiom in welfare economics7 and in social choice,8 requiring that agents who are indifferent between two alternatives do not influence social preferences over these alternatives. The separability principle can be related to the principle of subsidiarity saying that unconcerned agents should not influence a particular decision. As Fleming colorfully put it, “in considering policies affecting inhabitants of this planet we do not feel hampered by our ignorance regarding states of mind which prevail among the inhabitants of Mars.” (1952, p. 372)

A similar result to Corollary 3.1 can be obtained with Separability instead of Unchanged-Contour Independence. As is not surprising from the preceding discussion around Figure 3.4, it requires a minimum number of three agents in the population. Theorem 3.2 On the domain E 3 , if a SOF satisfies Pareto Indifference, NestedContour Transfer and Separability, then it satisfies Nested-Contour Priority. 7 See 8 Key

in particular Fleming (1952). references are Sen (1970) and d’Aspremont and Gevers (1977).

48

CHAPTER 3. PRIORITY TO THE WORST-OFF

Proof. The argument relies directly on the reasoning made around Figure 3.4. Let R satisfy Pareto Indifference, Nested-Contour Transfer and Separability. 0 Let E = (RN , Ω) ∈ E 3 , zN , zN ∈ X N , j, k ∈ N be such that zj À zj0 À zk0 À zk , 0 0 U (zj , Rj ) ∩ L(zk , Rk ) = ∅, and for all i 6= j, k, zi = zi0 . Let ∆j = zj − zj0 and ∆k = zk0 − zk . Let l ∈ N, l 6= j, k, Rl0 ∈ R, zl1 , zl2 , zl3 , zl4 ∈ X, and m ∈ Z++ be defined ∆ in such a way that zl1 Il0 zl4 , zl2 Il0 zl3 , zl1 = zl2 + ∆mk , zl4 = zl3 + mj , U (zl2 , Rl0 ) ∩ 0 0 1 0 0 t 0 L(zk , Rk ) = ∅, U (zj , Rj )∩L(zl , Rl ) = ∅, and zj À zl À zk for all t ∈ {1, 2, 3, 4}. This construction ¡¡ is illustrated ¢ ¢ in Figure 3.4 with m = 4. Let E 0 = RN\{l} , Rl0 , Ω ∈ E. By Nested-Contour Transfer, ¶ µ ¡ ¢ ∆k 2 zN \{k,l} , zk + , zl R(E 0 ) zN\{l} , zl1 . m By Pareto Indifference, µ ¶ ¶ µ ∆k 3 ∆k 2 0 I (E ) zN \{k,l} , zk + zN\{k,l} , zk + ,z ,z . m l m l

By Nested-Contour Transfer, µ ¶ ¶ µ ∆j ∆k 4 ∆k 3 0 R (E ) zN \{k,l} , zk + zN \{j,k,l} , zj − , zk + ,z ,z . m m l m l

By Pareto Indifference, µ ¶ ¶ µ ∆j ∆k 1 ∆j ∆k 4 zN\{j,k,l} , zj − , zk + , zl I (E 0 ) zN \{j,k,l} , zj − , zk + , zl . m m m m To sum up, using transitivity, µ ¶ ¡ ¢ ∆j ∆k 1 zN\{j,k,l} , zj − , zk + , zl R (E 0 ) zN\{l} , zl1 . m m

Repeating the argument m times and using transitivity again, ¡ ¢ ¢ ¡ zN \{j,k,l} , zj − ∆j , zk + ∆k , zl1 R (E 0 ) zN\{l} , zl1 .

0 , zj − ∆j = zj0 , zk + ∆k = zk0 , one obtains Since zN \{j,k,l} = zN\{j,k,l} ´ ³ ¡ ¢ 0 , zl1 R (E 0 ) zN \{l} , zl1 . zN\{l}

0 R(E) zN . By Separability, zN Finally, let us check that every axiom is necessary for the conclusion of the theorem. The examples below are defined in the proof of Theorem 3.1. 1) Drop Pareto Indifference. Take Rpsum . 2) Drop Nested-Contour Transfer. Take RΩsum . 3) Drop Separability. Take RIL .

There is no counterpart of this result for Transfer among Equals, as one can find SOFs satisfying Strong Pareto, Transfer among Equals and Separability but not Priority among Equals.9 9 Here is an example. For any economy E = (R , Ω), partition N into I(E) and J(E), N where I(E) is the subset of agents with linear preferences and J(E) is the complement. If

3.4. FROM EQUAL-SPLIT TRANSFER TO ABSOLUTE PRIORITY

3.4

49

From Equal-Split Transfer to absolute priority

In a similar way as for Transfer among Equals and Nested-Contour Transfer, we can strengthen the Equal-Split Transfer axiom by requiring that a transfer given to an agent consuming strictly less than equal split be a social improvement, whatever the amount of resources drawn from a donor who gets strictly more than equal split (see Figure 3.5). sΩ

good 2 6

Ω/2 s

s0 ¡ zk µ s¡ zk

sz ¡ j ¡ ¡ ª s¡ zj0

good 1

0

Figure 3.5: Equal-Split Priority.

Axiom Equal-Split Priority 0 For all E = (RN , Ω) ∈ D, and zN , zN ∈ X N , if there exist j, k ∈ N such that zj À zj0 À

Ω À zk0 À zk |N |

0 and for all i 6= j, k, zi = zi0 , then zN R(E) zN .

Theorem 3.3 On the domain E, if a SOF satisfies Strong Pareto, Equal-Split Transfer and Unchanged-Contour Independence, then it satisfies Equal-Split Priority. The logic of the proof, which is relegated to the appendix, is similar to the proof of Theorem 3.1. But the constraint that the bundle Ω/ |N | be in between the bundles of the donor and recipient makes a construction as in Figure 3.5 impossible in general in one step (the thin indifference curve at zj2 , zj3 , in that figure, cannot in general be close to the thick indifference curves at zj2 , zj3 when I(E) = N, the SOF evaluates an allocation by the sum the leximin criterion to the vector

uΩ (zj , Rj ) +

i∈N

uΩ (zi , Ri ) . Otherwise, it applies

i∈I(E) uΩ (zi , Ri )

j∈J(E)

.

50

CHAPTER 3. PRIORITY TO THE WORST-OFF

these points are required to dominate Ω/ |N |). The argument then requires a few additional steps. Unchanged-Contour Independence cannot be replaced by Separability in Theorem 3.3, because one can find SOFs satisfying Strong Pareto, Equal-Split Transfer and Separability but not Equal-Split Priority (e.g., utilitarian SOFs that add up utilities computed in such a way that marginal utility for bundles below Ω/ |N | is always greater than some fixed value, whereas marginal utility for bundles above Ω/ |N | is always smaller than this value). There is, however, a similar result involving a stronger separability axiom. This axiom is defined as follows. It states that when an agent has the same bundle in two allocations, the ranking of these two allocations should remain the same if this agent were simply absent from the economy. Axiom Separation 0 For all E = (RN , Ω) ∈ D with |N | ≥ 2, and zN , zN ∈ X N , if there is i ∈ N such 0 that zi = zi , then 0 0 zN R(E) zN ⇔ zN \{i} R(RN \{i} , Ω) zN\{i} .

We also need to introduce a Replication axiom, requiring that in a replicated economy, that is, an economy obtained by “cloning” each agent a given number of times, and multiplying the available resources by the same number, replicated allocations should be ranked exactly as in the initial economy. Some notation is needed to make this statement precise. If r is a positive integer, an economy E 0 = (RN 0 , Ω0 ) is a r-replica of E = (RN , Ω) if Ω¯0 = rΩ¯ and there exists a mapping γ : N 0 → N such that for all i ∈ N, ¯γ −1 (i)¯ = r and for all j ∈ γ −1 (i), Rj = Ri . This implies, in particular, that |N 0 | = r |N | . We use a similar terminology for allocations. Thus, if E 0 = (RN 0 , Ω0 ) is a r-replica of 0 E = (RN , Ω) and zN ∈ X N , the allocation zN 0 ∈ X N denotes the r-replica of zN , i.e., the allocation such that for all i ∈ N and all j ∈ γ −1 (i), zj0 = zi . Axiom Replication 0 ∈ X N , and r ∈ Z++ , if E 0 = (RN 0 , Ω0 ) ∈ D is For all E = (RN , Ω) ∈ D, zN , zN 0 0 a r-replica of E, then zN R (E) zN ⇔ zN 0 R (E 0 ) zN 0. Theorem 3.4 On the domain E, if a SOF satisfies Pareto Indifference, EqualSplit Transfer, Separation and Replication, then it satisfies Equal-Split Priority. The basic idea of the proof, provided in the appendix, is the same as for Theorem 3.2, but there is a difficulty due to the fact that the added agent (agent l in the proof of Theorem 3.2) must now be below Ω/ |N | when he receives a transfer from agent j, and above Ω/ |N | when he donates to agent k. It is impossible, with monotonic preferences, to move him from one situation to the other by Pareto Indifference. This is why Replication comes into play. By Lemma 4.1 proved in Chapter 4, Separation and Replication together imply that the SOF is insensitive to multiplying Ω by a positive rational number q.

3.5. CONCLUSION

51

Then one can locate qΩ wherever suitable to make the application of Equal-Split Transfer legitimate. There is no result of this sort with Proportional-Allocations Transfer, as this axiom is satisfied by RΩNash , which also satisfies Strong Pareto, UnchangedContour Independence, Separation and Replication. The fact that ProportionalAllocations Transfer does not imply a strong inequality aversion can be understood by noting that it has a purely one-dimensional content, applying to allocations with bundles on a given line. The results of this chapter rely on the multi-dimensional setting in which the other transfer axioms operate.

3.5

Conclusion

The main conclusion of this chapter is that as soon as one accepts axioms limiting the information that can be used in order to compare two allocations, such as Unchanged-Contour Independence or Separability, one is led to an extreme form of egalitarianism. Increasing the amount of resources allocated to the worst-off is always good for society, no matter how much is taken from the better-off agents. This conclusion will be reproduced in the subsequent chapters. Almost all the SOFs we will define and characterize are of the maximin or leximin type. As explained in this chapter, this results from our willingness to take agents’ preferences into account and at the same time to seek some mild degree of resource equality. This confirms and generalizes the observation made at the end of the previous chapter. The Resource Utilization SOF RRU involves the summation operator and is concerned only with efficiency. The Ω-Equivalent Nash SOF RΩNash relies on the product but fails to satisfy the most basic equity axioms. In this first series of examples, only the “Rawlsian” Ω-Equivalent Leximin SOF RΩlex combines efficiency and equity concerns in a satisfactory way. We now know that, in general, studying fairness in (multi-dimensional) models where both resources and preferences matter yields social criteria which are necessarily of the maximin or leximin type. Not all results in this book, however, will deal with such extreme criteria and we will see that certain “moderate” SOFs such as RΩNash have interesting properties in particular contexts. But, more importantly, we will also see that the class of leximin SOFs contain a great variety of social criteria, because there remains a wide array of possibilities about the way in which individual indices of resources are computed. Our approach appears sufficiently flexible to incorporate various notions of needs, rights and personal responsibilities and liabilities, which may substantially affect the distributional conclusions based on such criteria. Under the appropriate assumptions about how individual situations are evaluated, it is even possible for a maximin SOF to advocate laisser-faire policies (see in particular Chapters 7 and 10). Therefore we do not think that the apparently restrictive conclusions of these first two chapters are bad news. It is, on the contrary, quite helpful to know the

52

CHAPTER 3. PRIORITY TO THE WORST-OFF

answer to the question: “How should individual indices be aggregated?” The crucial question that remains to be addressed, and on which policy conclusions will dramatically depend, is: “How should individual indices be defined?”

Chapter 4

The informational basis of social orderings 4.1

Introduction

This chapter seeks to clarify some features of SOFs which may appear intriguing to the specialist of social choice theory or of fair allocation theory. The reader who is more interested in the applications of the approach may skip this chapter. Three issues are examined, which all have to do with the informational basis of SOFs. The first issue is connected to the fact that we define SOFs as functions R(RN , Ω) instead of functions R(RN ). It may appear strange that the ranking of allocations should vary as a function of the available resources, as if an ethical objective could depend on feasibility constraints. Section 2 explains why this dependence is important in order to obtain some results, although it is not essential to the notion of SOF in general. Then Section 3 examines how our theory relates to the theory of social choice in economic environments. We have already mentioned in Chapter 3 that our SOFs satisfy a weaker axiom of independence, namely, Unchanged-Contour Independence, than Arrow’s famous Independence of Irrelevant Alternatives. Relaxing Arrow’s independence axiom is the key ingredient that enables us to obtain possibility results. In Section 3 we examine how the possibility results are affected when one varies the quantity of information that is used, through various axioms of independence. Finally, Section 4 compares the theory of SOFs to the theory of fair allocation, in which allocation rules are the subject instead of SOFs. We argue that the difference between the two objects is thinner than usually thought, and that the informational basis is similar in the two theories. 53

54CHAPTER 4. THE INFORMATIONAL BASIS OF SOCIAL ORDERINGS

4.2

The feasible set

Many of the SOFs studied in this book depend on Ω, and therefore do depend on (some of) the feasibility constraints bearing on the economy. This may seem somehow in contradiction with the celebrated Arrow Program of social choice theory, which allegedly consists of two separate steps. The first step constructs a social preference ordering for every profile of individual preference orderings. The second step derives a social choice function that selects a subset of alternatives in terms of the optimization of social preferences within every possible set of feasible social alternatives. The first step, then, is meant to determine the uniform social objective before the set of feasible social alternatives is revealed. The second step is meant to determine the rational social choice after the set of feasible social alternatives is revealed. In Section 5.4 second-best applications will be presented, in which the best allocations will be determined under feasibility and incentive-compatibility constraints. Such constraints delineate a smaller set of attainable allocations than the feasible set Z(E), and the precise set of attainable allocations may vary depending on preferences and endowments. These second-best applications partly respect the Arrow program because the same SOF is applied independently of the particular set of attainable allocations that is at hand. But a more purist application of the Arrow Program would exclude any consideration of feasibility from the definition of social preferences. It seems to us that this purist approach can be simply accommodated in our framework by requiring SOFs to satisfy the following axiom, which says that the social ordering should depend only on RN and not on anything else, specifically Ω. It is called “Independence of Feasible Set”:1 Axiom Independence of Feasible Set 0 0 For all E = (RN , Ω), E 0 = (RN , Ω0 ) ∈ D, and zN , zN ∈ X N , zN R(E) zN ⇔ 0 0 zN R(E ) zN . The analysis of Chapter 2 showed that egalitarian axioms formulated without restriction to particular regions of the consumption set conflict with Pareto axiom. Some of the restrictions, such as those referring to equal split, involve the social endowment Ω. This observation suggests that imposing Independence of Feasible Set entails some costs with respect to the fairness properties of the SOF, in particular those relating to equal split. For instance, none of the three SOFs of Table 2.1 satisfies Independence of Feasible Set. This does not mean that a SOF satisfying Independence of Feasible Set cannot have some good properties. For instance, consider the Ω0 -Equivalent Leximin SOF which relies on a fixed reference bundle Ω0 for any economy. This SOF satisfies Strong Pareto, Nested-Contour Priority (and therefore Nested-Contour Transfer, Transfer among Equals, Priority among Equals) and the secondary axioms 1 Note, however, that Ω may not be the only variable which directly determines the actual feasible agenda in some contexts (e.g., incentive-compatibility constraints or pure political constraints may also play a role).

4.2. THE FEASIBLE SET

55

(Separation and Replication). But it does not satisfy the equal split axioms and related axioms bearing on proportional allocations. Actually, we have the following impossibility. Theorem 4.1 On the domain E, no SOF satisfies Weak Pareto, Equal-Split Selection and Independence of Feasible Set. Proof. Let N = {1, 2} and E = (RN , Ω), E 0 = (RN , Ω0 ) ∈ E. This is il1 2 lustrated in Figure 4.1, which displays two allocations zN , zN , in addition to ∗ 0∗ two equal-split allocations zN , zN , which are efficient in E and E 0 , respectively. By Independence of Feasible Set, there is a social ordering R over X N such ∗ 1 that R = R(E) = R(E 0 ). By Equal-Split Selection, zN R zN . By Weak Pareto, 2 ∗ 0∗ 2 0∗ 1 zN R zN . By Equal-Split Selection, zN R zN . By transitivity, zN R zN . But by 1 0∗ Weak Pareto, zN P zN , a contradiction. s ¢Ω ¢ ¢ R1 ¢ ¢ ¢sz11 ¢ z1∗ = z2∗ ¢s ©sΩ0 © 1 z2 s¢ © z22 s ©© ¢ © ¢ s© R2 © ¢ © s 0∗ 0∗ © z = z 2 1 2 ¢ © z1 ¢©©© ¢ © good 1 0

good 2 6

Figure 4.1: Proof of Theorem 4.1. By the same argument, one shows that no SOF can satisfy Weak Pareto, Independence of Feasible Set and select egalitarian Walrasian allocations (in the sense that max|R(E) Z(E) ∩ S EW (E) 6= ∅) in every economy E ∈ E. The same holds about egalitarian-equivalent allocations S EE (E).2 Such impossibilities suggest that Independence of Feasible Set is not so uncontroversial as it appears at first glance. In our framework we restrict attention to Ω ∈ R++ with positive quantities of all goods. But it may enhance intuition about this issue to momentarily extend the analysis to the case in which some goods may be absent. Consider the problem of distributing bread and water to a given population. When there is no water, a particular ranking of allocations of bread will be formed. According to Independence of Feasible Set, this ranking should be retained even if water became available. This is questionable, for 2 See

Fleurbaey and Maniquet (2008).

56CHAPTER 4. THE INFORMATIONAL BASIS OF SOCIAL ORDERINGS the following reason. In absence of water, presumably some simple egalitarian ranking would seem reasonable for the allocation of bread. But when water is available, the allocation of bread could legitimately take account of how much individuals are willing to substitute water for bread. The problem becomes acute, indeed, under Pareto axioms. For simplicity, consider a population with two agents. Assume for instance that, for the allocations of bread (good 1) only, in absence of water, giving 10 to agent 1 and 8 to agent 2 is better than 12 and 6, respectively: ((10, 0) , (8, 0)) R(E) ((12, 0) , (6, 0)) .

(4.1)

Now suppose that individual preferences are such that (10, 0) I1 (0, 6) , (12, 0) I1 (0, 8) , (6, 0) I1 (0, 10) , (8, 0) I1 (0, 12) . By Independence of Feasible Set, the above ranking of allocations of bread should be retained even when water is available. Now, in view of the agents’ preferences, Pareto Indifference and (4.1) entail that ((0, 6) , (0, 12)) R(E) ((0, 8) , (0, 10)) . By Independence of Feasible Set, this ranking of allocations of water should be retained even when there is no bread. This shows that Independence of Feasible Set is very restrictive in such a context. It prevents social preferences from taking account of the relative scarcity of goods, and from focusing on the appropriate parts of individual preferences. For instance, if one thinks that an egalitarian Walrasian allocation is a good social objective, it makes little sense to look for social preferences that are independent of the relative scarcity of goods, as individual situations have to be evaluated in terms of budgets, and the relative prices of goods will depend on total supply. We must emphasize, however, that the SOFs studied in this book are generally insensitive to multiplication of Ω by a scalar. Let us encapsulate this idea as an axiom stating that a proportional expansion of the social endowment has no effect on the ranking of any pair of allocations. Axiom Independence of Proportional Expansion 0 For all E = (RN , Ω), E 0 = (RN , λΩ) ∈ D with λ > 1, and zN , zN ∈ XN , 0 0 0 zN R(E) zN ⇔ zN R(E ) zN . In particular, the social orderings obtained with any of the SOFs from Table 2.1 satisfy this axiom. According to the above discussion, it is not the feasibility set per se which appears relevant, but the relative scarcity of goods as reflected in the direction of Ω. The size of Ω itself does not seem to matter so much in this argument. The idea that the SOF should be insensitive to multiplications of Ω by a scalar can actually be related to some of the axioms already introduced, namely,

4.2. THE FEASIBLE SET

57

Separation and Replication. These axioms together imply that the social ordering should be independent of multiplication of Ω by a rational number. Lemma 4.1 If a SOF R satifies Separation and Replication, then for all E = (RN , Ω) ∈ E and all rational numbers q ∈ Q++ , R(RN , qΩ) = R(E). Proof. Let R be defined on E. Step 1. First we claim that if R satisfies Separation and Replication, it 0 ∈ XN , r ∈ satisfies the following property: For all E = (RN , Ω) ∈ E, and zN , zN 0 0 0 Z++ , if E = (RN , Ω) ∈ E is such that RN is a r−replica of RN , then 0 0 zN R (E) zN ⇔ zN 0 R (E 0 ) zN 0.

Let E = (RN , Ω) ∈ E. Let RN 0 be a r−replica of RN . Consequently, RN 0 can be decomposed into (RN 1 , RN 2 , . . . , RN r ) such that RN s is a 1-replica of RN for each s ∈ {1, . . . , r} . By Replication, 0 0 zN R (E) zN ⇔ zN s R (RN s , Ω) zN s. 0

0 1 2 r N 0 Let zN be defined by: zN 0 , zN 0 , zN 0 , . . . , zN 0 ∈ X s = zN for all s ∈ {1, . . . , r} , t 0 t and zN s = zN if s ≤ t whereas zN s = zN if s > t, for t ∈ {1, . . . , r} . That is, 0 zN 0 1 zN 0 2 zN 0

r zN 0

= (zN , ..., zN ) , 0 = (zN , zN , ..., zN ) , 0 0 = (zN , zN , zN , ..., zN ) , .. . 0 0 = (zN , ..., zN ).

0 r 0 One has zN 0 = zN 0 and zN 0 = zN 0 . By Separation, for all s ∈ {1, . . . , r} , s−1 0 s 0 zN 0 R (E ) zN 0 ⇔ zN s R (RN s , Ω) zN s .

Applying this last relationship s − t times, we get, for 0 ≤ t < s ≤ r, t 0 s 0 zN 0 R (E ) zN 0 ⇔ zN R (E) zN .

In particular, for t = 0 and s = r, we get the desired result. 0 Step 2. Now, let us complete the proof. Let E = (RN , Ω) ∈ E. Let zN , zN ∈ N X , and p, q ∈ Z++ . Let RN p be a p−replica of RN . By Replication, 0 0 zN R (E) zN ⇔ zN p R (RN p , pΩ) zN p.

By Step 1, 0 0 zN p R (RN p , pΩ) zN p ⇔ zN R (RN , pΩ) zN .

58CHAPTER 4. THE INFORMATIONAL BASIS OF SOCIAL ORDERINGS Let RN q be a q−replica of RN . By the claim above, again, 0 0 ⇔ zN q R (RN q , pΩ) zN zN R (RN , pΩ) zN q.

Finally, by Replication, µ ¶ p 0 0 RN , Ω zN . zN q R (RN q , pΩ) zN q ⇔ zN R q Gathering the relationships, we obtain the desired outcome. We conclude this section with a discussion of our general approach, in relation to feasibility. As explained in Chapter 1, the social orderings we study rank all allocations in X N , the feasible as well as the infeasible. This is justified because the relevant feasibility constraints (resources, incentives, politics) may vary depending on the context, and in particular this allows us to formulate and discuss the Independence of Feasible Set axiom and similar axioms in a convenient way. It is, however, possible to modify the framework a little and restrict the social orderings to bear only on Z(E) which is, in any given economy E, and more precisely for any given Ω, the greatest feasible set under any circumstance. There is not much to be gained by this restriction, as there is no difficulty of defining social orderings that rank infeasible allocations as well. But, from the technical point of view, this restriction is interesting because it reduces the set of allocations which can be constructed in the proofs of characterization theorems. This raises the (non purely technical) question of whether other kinds of SOFs would then become admissible. We doubt that this is the case. Indeed, even if a SOF only ranks allocations in Z(E), one can formulate a variant of Independence of Proportional Expansion applying to feasible allocations and stating that the multiplication of Ω by a scalar greater than one should not alter the initial ranking over the set Z(E). If this axiom is added to the axioms considered so far, then all the allocations considered in the proofs of our results can be rendered feasible by multiplying Ω by a sufficiently large λ. Then all the proofs go through and the results remain valid. This implies that the new SOFs obtained in a framework where only Z(E) is ordered would violate this axiom and be very sensitive not only to the direction of Ω but also to its size. It is not easy to find arguments that would justify this. It is worth noting another interesting property of RΩlex . In order to rank two allocations z, z 0 from Z(E), one only needs to know the Ω-equivalent utilities, i.e., the bundles λi Ω and λ0i Ω such that for all i, λi Ω Ii zi and λ0i Ω Ii zi0 . When z, z 0 ∈ Z(E), necessarily λi , λ0i ≤ 1, implying that for all i, λi Ω and λ0i Ω are “feasible” in the sense that they belong to feasible allocations (even if the allocations (λi Ω)i∈N and (λ0i Ω)i∈N themselves may not be feasible). As a consequence, in order to rank z, z 0 , it is then sufficient to know individual preferences over feasible bundles. RΩlex is independent of individual preferences over infeasible bundles. This property is also satisfied by RΩNash and RRU but is not satisfied by all the SOFs studied in this book–in particular it is not

4.3. INFORMATION ON PREFERENCES

59

satisfied by the Walrasian SOF introduced in the next chapter. Related ideas of independence of preferences are discussed in more detail in the next section.

4.3

Information on preferences

In Chapter 3, we introduced Unchanged-Contour Independence, an axiom limiting the relevant information about preferences to the indifference sets at the allocations under consideration. The main SOFs studied in this book satisfy this axiom, which therefore appears perfectly compatible with efficiency and with fairness requirements. This is a good property because, as was explained in Section 3.2, it substantially simplifies the evaluation of allocations, and also because it goes well with the idea that individual preferences are a personal responsibility and should not be relied upon more than needed. Now, can we use even less information about preferences without sacrificing too much about efficiency or fairness? The theory of social choice famously started with the much stronger Arrow Independence, as recalled in Section 1.3. Arrow’s theorem proves that this condition forces one to either abandon efficiency (Weak Pareto) or to abandon impartiality between agents. If one abandons efficiency, one can for instance use an index U0 , independent of RN , in a SOF R defined by: 0 zN R(E) zN ⇔ (U0 (zi ))i∈N ≥lex (U0 (zi0 ))i∈N .

If one keeps Weak Pareto, the theorem says that one has to drop impartiality in a severe way, because there must be a “dictator ”, namely, an agent i0 such 0 0 that for all E = (RN , Ω) ∈ D, and zN , zN ∈ (X \ {0})N , one has zN P(E) zN 0 whenever zi0 Pi0 zi0 . A proof of this statement for the present model can be found in Bordes and Le Breton (1989). For instance, a SOF R satisfying Arrow Independence and Strong Pareto can be defined in the following way: 0 zN I(E) zN ⇔ zi Ii zi0 for all i ∈ N, 0 zN P(E) zN ⇔ zi Pi zi0 for some i ∈ N and zj Ij zj0 for all j < i.

In this SOF, there is a sequence of dictators such agent i’s strict preference decides only when the agents of higher authority (taken to be j < i in this example) are indifferent. It is transparent that Arrow Independence is logically stronger than UnchangedContour Independence, and Arrow’s theorem shows that it is too strong because it requires too much sacrifice either on efficiency or on fairness. It is in fact very intuitive that this condition is not compelling. For instance, in axioms like Transfer among Equals or Nested-Contour Transfer, one needs to check that indifference curves do not cross for the transfer to be recommended, and one cannot do so if the only information usable, as required by Arrow Independence, is that the recipient would like to obtain the transfer while the donor would rather not donate.

60CHAPTER 4. THE INFORMATIONAL BASIS OF SOCIAL ORDERINGS At this stage, a natural question is to ask what happens when the information about preferences that can be used in the evaluation is intermediate between pairwise preferences (as with Arrow Independence) and indifference sets (as with Unchanged-Contour Independence). Starting from Arrow Independence, let us introduce additional information. A first step consists in introducing information about marginal rates of substitution. As we do not assume differentiability, we will rely on supporting cones defined as follows: C(zi , Ri ) is the cone of price ˚ (zi , Ri ) denote vectors that support the upper contour set for Ri at zi . Letting U the interior of the upper contour set U (zi , Ri ) , one defines: ˚ (zi , Ri ) , pq > pzi }. C(zi , Ri ) = {p ∈ R | ∀q ∈ U As preferences Ri ∈ R are assumed to be monotonic, one has C(zi , Ri ) ⊆ R+ . The following axiom says that the ranking of two allocations should not be affected by a change of individual preferences if individual preferences on the corresponding bundles remain unchanged, with unchanged supporting cones (in particular, with unchanged marginal rates of substitution for the case of differentiable preferences). Axiom Unchanged-Cone Independence 0 0 For all E = (RN , Ω), E 0 = (RN , Ω) ∈ D, and zN , zN ∈ X N , if for all i ∈ N, Ri 0 0 and Ri agree on {zi , zi } and C(zi , Ri ) = C(zi , Ri0 ), C(zi0 , Ri ) = C(zi0 , Ri0 ), 0 0 ⇔ zN R(E 0 ) zN . then zN R(E) zN

As it turns out, this condition entails the same dictatorial result as Arrow Independence. The proof of this is tedious3 but we can illustrate the problem by focusing on Anonymity, which is a stronger requirement of impartiality than non-dictatorship, but still very compelling. It requires the social ordering to be invariant ¡ when ¢ preferences and bundles are permuted across agents. Let zπ(N ) denote zπ(i) i∈N . Axiom Anonymity 0 ∈ X N , all permutations π : N → N , For all E = (RN , Ω) ∈ D, all zN , zN 0 0 zN R(E) zN ⇔ zπ(N ) R(Rπ(N ) , Ω) zπ(N ) .

The proof that Anonymity and Weak Pareto are incompatible under some independence condition has a typical structure. First, observe that Anonymity implies the following axiom saying that permuting the bundles of two agents with identical preferences yields a new allocation which is just as good as the initial allocation. 3 See

Fleurbaey et al. (2005a).

4.3. INFORMATION ON PREFERENCES

61

Axiom Anonymity among Equals 0 For all E = (RN , Ω) ∈ D, all j, k ∈ N, and zN , zN ∈ X N , if Rj = Rk and 0 0 0 0 (zj , zk ) = (zk , zj ) while zi = zi for all i 6= j, k, then zN I(E) zN . Then suppose one can find a two-agent economy E = ((R1 , R2 ), Ω) and an allocation zN = (a, b) with a > b such that given the information about preferences at (a, b) and at (b, a) that can be used under the independence axiom, one could as well assume that the agents have the same preferences. More precisely, this sentence means the following: There is R0 such that, under the independence axiom, R(E) and R((R0 , R0 ), Ω) must agree on how to rank (a, b) and (b, a). Observe that as a > b, individual preferences over such bundles are always the same when they are strictly monotonic. Therefore, when preferences are strictly monotonic, applying the independence condition simply requires checking some extra condition such as, in the case of Unchanged-Cone Independence, equality of supporting cones at a and b for R1 , R2 and R0 . In the case of Arrow Independence no extra condition needs to be checked. Now, by Anonymity among Equals, (a, b) I((R0 , R0 ), Ω) (b, a). Therefore, by independence, (a, b) I(E) (b, a). 0 Introduce another allocation zN = (a0 , b0 ), in some other region of X, such 0 0 that a À b and, again, such that given the information about preferences at (a0 , b0 ) and at (b0 , a0 ) that can be used under the independence axiom, one could as well assume that the agents have the same preferences. By the same reasoning as above, this implies that (a0 , b0 ) I(E) (b0 , a0 ). 0 The reason zN was said to belong to some other region of X is that one needs to assume the following pattern of preferences: a0 P1 a P1 b P1 b0 and a P2 a0 P2 b0 P2 b. This requires that there is no domination between bundles of zN and bundles of 0 zN . Under this preference profile, Weak Pareto implies that (a0 , b0 ) P(E) (a, b) and (b, a) P(E) (b0 , a0 ). Recapitulating, one has (a, b) I(E) (b, a) P(E) (b0 , a0 ) I(E) (a0 , b0 ) P(E) (a, b), which is impossible. Figure 4.2 illustrates a possible profile of preferences that produces this outcome when Unchanged-Cone Independence (this also works for Arrow Independence) is considered and requires equal supporting cones between R1 and R2 at each of the four bundles. Let us write down the result. Theorem 4.2 On the domain E, no SOF satisfies Weak Pareto, Anonymity and Unchanged-Cone Independence. It is transparent that the same impossibility result would be obtained with 0 an even weaker independence condition requiring to have RN and RN coincide on some vicinity of fixed size surrounding the bundles, i.e., Bε (x) = {q ∈ X | kq − xk ≤ ε} ,

62CHAPTER 4. THE INFORMATIONAL BASIS OF SOCIAL ORDERINGS

good 2 6 a0 s A A b0 s A A

0

good 2 6 0s a A A b0 s A A

R1

HHs b

HHs a good 1

R2

HHs b

0

(a)

HHs a good 1

(b) Figure 4.2: Individual preferences.

where k·k denotes the Euclidean norm and ε > 0.4 The bundles a, b, a0 , b0 must then be located sufficiently far from each other so that the regions Bε (a), Bε (b), Bε (a0 ), Bε (b0 ) do not overlap and contain portions of indifference sets that can be connected in various ways so as to have the pattern of preferences required for the proof. Note that this new independence condition is not logically stronger than Unchanged-Contour Independence because it involves information about other indifference sets than the indifference sets containing the four bundles. Another direction of weakening Arrow Independence is obtained by considering the sets of allocations that one can obtain from the allocations by certain operations. Two examples will be presented here. The first one is reminiscent of the condition mentioned at the end of the previous section and referring to 0 0 feasible bundles. For a given pair of allocations zN , zN , let F (zN , zN ) denote 0 the set of bundles that can be obtained from resources used in zN or zN . In the following expression, a ∨ b = (max {ak , bk })k . 0 ) F (zN , zN

=

(

q∈X|q≤

X

i∈N

zi ∨

X

i∈N

zi0

)

.

To fix ideas, let us write down the corresponding axiom. It says that the ranking of two allocations is unchanged when individual preferences remain unchanged over all bundles which could be distributed to an individual with resources 0 sufficient to make both zN and zN feasible. Axiom Unchanged-Fset Independence 0 0 For all E = (RN , Ω), E 0 = (RN , Ω) ∈ D, and zN , zN ∈ X N , if for all i ∈ N, Ri 0 0 0 0 and Ri agree on F (zN , zN ), then zN R(E) zN ⇔ zN R(E 0 ) zN . 4 While Arrow’s dictatorship result is preserved with Unchanged-Cone Independence, it no longer holds true for this new axiom. See Fleurbaey et al. (2005a) for details.

4.3. INFORMATION ON PREFERENCES

63

This axiom is not strictly weaker than the previous ones because information about supporting cones or vicinities may involve bundles which do not belong 0 to F (zN , zN ). Nonetheless it should seem rather weak because it involves potentially large sets over which information about preferences is relevant. But in fact the dictatorship result still holds with this axiom. Observe that Figure 4.2 also illustrates how to apply this axiom in a very simple way in order to prove a variant of Theorem 4.2: When (a, b), and therefore (b, a), belong to an axis the feasible bundles involve only one good. An even weaker axiom consists in considering the whole subspace of bundles 0 spanned by zN and zN . Let !) ( Ã X X 0 0 . S(zN , zN ) = q ∈ X | ∃λ > 0, q ≤ λ zi + zi i∈N

This definition requires qk = 0 whenever

P

i∈N

zik +

i∈N

P

i∈N

0 zik = 0.

Axiom Unchanged-Subspace Independence 0 0 For all E = (RN , Ω), E 0 = (RN , Ω) ∈ D, and zN , zN ∈ X N , if for all i ∈ N, Ri 0 0 0 0 and Ri agree on S(zN , zN ), then zN R(E) zN ⇔ zN R(E 0 ) zN . This axiom is quite natural: It requires ignoring individual preferences over 0 in order to rank these two allocacommodities which are not used in zN , zN tions. Arrow himself, trying to defend Arrow Independence, ended up defending something that is closer to Unchanged-Subspace Independence than to Arrow Independence: ‘Suppose that there are just two commodities, bread and wine. A distribution, deemed equitable by all, is arranged, with the winelovers getting more wine and less bread than the abstainers. Suppose now that all the wine is destroyed. Are the wine-lovers entitled, because of that fact, to more than an equal share of bread? The answer is, of course, a value judgment. My own feeling is that tastes for unattainable alternatives should have nothing to do with the decision among the attainable ones; desires in conflict with reality are not entitled to consideration.’ (Arrow 1951, p. 73) With this axiom the dictatorship result no longer holds, but Theorem 4.2 is immediately extended, and once again Figure 4.2 provides the relevant illustration with the two axes playing the role of subspaces. It may be worth emphasizing that in these cases, when the dictatorship implication no longer holds, the SOFs that satisfy the properties are still substantially dictatorial. This can be viewed when one defines a dictator as follows: For a given population N, agent i ∈ N is a dictator on (D, Y ) if for all RN ∈ D, 0 0 all zN , zN ∈ Y, zN P(E) zN whenever zi Pi zi0 . With this definition, the larger D and Y for a given dictator, the more dictatorial the rule. Note for instance that, letting E N denote the subset of E corresponding to population N, Weak Pareto

64CHAPTER 4. THE INFORMATIONAL BASIS OF SOCIAL ORDERINGS ¡ N N¢ and Arrow Independence do not ³ ´ imply the existence of a dictator on E , X but only on E N , (X \ {0})N .5

Let a subset of commodities K ⊆ {1, ..., } be called sufficient for Ri if for xi ∈ X, there is yi ∈ X such that yik = 0 for all k ∈ / K and yi Ri xi . In other words, K is sufficient if the agent can always reach any arbitrarily high level of satisfaction with the consumption of commodities from K only. Let R∗ denote the subset of R containing strictly monotonic preferences and E ∗ , E ∗N the corresponding subdomains. Fix the population N (with |N | ≥ 2) and let ∗ DN denote the set of preference profiles for which a strict subset of {1, ..., } (possibly specific to the profile) is sufficient for all i ∈ N, and Y N denote the set of allocations which do not use all commodities: n o |N | ∗ DN = RN ∈ (R∗ ) | ∃K Ã {1, ..., } , ∀i ∈ N, K is sufficient , ( ) X N YN = zN ∈ X | zi ∈ / R++ . i

Observe that on Figure 4.2, the preferences could be such that every commodity is sufficient, because the indifference curves meet each axis, and the allocations under consideration do not use all commodities. Fleurbaey and Tadenuma (2007, Th. ³1) give the following result showing ´ N N that even though there is no dictator on E , (X \ {0}) , there is still a lot of dictatorship under Unchanged-Subspace Independence. Theorem 4.3 If a SOF defined on E ∗ satisfies Weak Pareto and UnchangedSubspace Independence, then´ for all N with |N | ≥ 2 there is i ∈ N who is a ³ ¡ ¢ N ∗ dictator on DN , (X \ {0}) and on E ∗N , Y N .

A similar result relative to independence defined with respect to vicinities around the bundles has yet to be established. The negative results in this section all have a common feature. They involve an independence axiom which forces the evaluation to rely only on information about preferences that is local in a certain sense. This is quite clear for Arrow Independence or Unchanged-Cone Independence. It is also true for UnchangedFset Independence and Unchanged-Subspace Independence, even though these axioms accept information about preferences over whole sets or subspaces, because the particular sets or subspaces under consideration are specific to the allocations under consideration. In contrast, Unchanged-Contour Independence, for which possibility results are obtained, makes it possible to use information about the indifference set that is arbitrarily extended in the commodity space.

5 Consider the SOF such that for every N there are i, j ∈ N such that for all E = (R , Ω), N 0 ∈ X |N | , z P(E) z 0 if z > z 0 = 0; if either z = z 0 = 0 or z , z 0 > 0, z R(E) z 0 all zN , zN j j j j N N j j N N

iff zi Ri zi0 . For this SOF, for every N, there i who is a dictator on E N , (X \ {0})N on E N , X N . It satisfies Weak Pareto and Arrow Independence.

but not

4.4. ALLOCATION RULES AS SOCIAL ORDERING FUNCTIONS

65

Recall, however, that SOFs such that RΩlex do not need more information than preferences about bundles x ≤ Ω in order to rank feasible allocations. In conclusion, the lesson that the construction of attractive SOFs requires “non-local” information must not be exaggerated. The problem becomes more serious for SOFs which satisfy Independence of Feasible Set. If one wants to construct a SOF that depends only on preferences and not at all on the available resources, then the above results imply that one must accept the possibility that the ranking of two particular allocations may depend on properties of the individual indifference sets at these allocations which are arbitrarily remote from the bundles consumed in these allocations. For instance, in a two-good economy, the fixed-reference SOF RΩ0 lex , for Ω0 = (1, 1), will always check the location of indifference curves at bundles on the 45◦ line in order to rank allocations, even when these allocations contain only one good, or contain thousands of units more in one commodity than in the other. We consider that this observation reinforces the view that Independence of Feasible Set is not a compelling requirement.

4.4

Allocation rules as social ordering functions

The requirement of “non-local” information about preferences vanishes when the Pareto axiom is weakened, as shown in this section. The topic of this section is not, however, the weakening of Pareto axioms but the comparison of the SOFs studied in this book with the allocation rules which one finds in the theory of fair allocation. The theory of fair allocation has often been contrasted with the theory of social choice because the latter has mostly delivered impossibility theorems whereas the former contains many positive results in the form of characterizations of nice allocation rules. An intuitive but deceptive explanation for this contrast is that the theory of social choice seeks orderings of all allocations whereas the theory of fair allocation is satisfied with the selection of a small subset of optimal allocations. This explanation is not satisfactory, in fact, because a selection of a subset of allocations can be formally associated with a complete ordering classifying all allocations into two tiers: the optimal allocations and the rest. In order to understand why the theory of fair allocation succeeds where the theory of social choice fails, one must confront the allocation rules and the complete orderings they generate to the axioms of social choice. The only plausible explanation is that some of the axioms of the impossibility theorems of social choice are not satisfied by the allocation rules. Recall that an allocation rule S associates every economy E in its domain with a non-empty subset S(E) ⊆ Z(E). The two-tier SOF generated by S can 0 be denoted RS and is defined as follows: zN RS (E) zN if and only if zN ∈ S(E) 0 0 or zN ∈ / S(E). One therefore has zN PS (E) zN if and only if zN ∈ S(E) and 0 zN ∈ / S(E). The two prominent allocation rules of the theory of fair allocation are, as recalled in Section 1.2, the egalitarian Walrasian rule S EW , which selects the competitive equilibria in which all agents have equal budgets, and the egalitarian-equivalent rule S EE , which selects the allocations zN such that

66CHAPTER 4. THE INFORMATIONAL BASIS OF SOCIAL ORDERINGS zN ∈ P (E) and there is λ such that for all i ∈ N, zi Ii λΩ. There is a variant of this rule which is based on a fixed Ω0 that does not depend on E. Let us confront these SOFs to the core impossibility theorem of social choice theory, namely, Arrow’s impossibility. This theorem has three ingredients: Weak Pareto, Arrow Independence, and non-dictatorship. The SOFs RS EW and RS EE are obviously not dictatorial, and actually no two-tier SOF can be dictatorial, because the dictator imposes his strict preferences over the SOF, and this implies that the SOF must have more than two tiers. Therefore the theory of fair allocation satisfies the non-dictatorship requirement automatically. For a similar reason, however, no two-tier SOF can satisfy Weak Pareto. From such a SOF one can at most require that the allocations it selects in Z(E) be Pareto-efficient. This requirement can be formulated as an axiom for any kind of SOF: Axiom Pareto Efficiency For all E ∈ D, max|R(E) Z(E) ⊆ P (E). This weakening of Pareto reflects the fact that allocation rules only select a subset of allocations and do not seek to rank all allocations in a more precise way. Arrow Independence can be satisfied by a two-tier SOF, but is not satisfied by those associated with the prominent allocation rules of the theory of fair allocation. In Section 1.3 this has been illustrated for the egalitarian Walrasian allocation rule.6 A similar illustration can be made for the egalitarianequivalent allocation rule. More generally, no reasonable allocation rule will satisfy Arrow Independence. Indeed, on the subdomain E ∗ of economies with strictly monotonic preferences, Wilson’s theorem (as extended by Bordes and Le Breton 1989 to economic environments) states that a SOF satisfying Arrow Independence must be either dictatorial, or anti-dictatorial (which means always going against the strict preferences of the “anti-dictator”), or constant (i.e., independent of the profile of preferences). As a two-tier SOF cannot be dictatorial or anti-dictatorial, for the reasons explained above, it must be constant.7 From this observation, one derives the following conclusion. Theorem 4.4 On the domain E ∗ , there exists a unique two-tier SOF satisfying Pareto Efficiency, Anonymity and Arrow Independence. It selects all the allocations such that one agent consumes Ω. On the domain E, no two-tier SOF satisfies Pareto Efficiency and Arrow Independence. 6 See Figure 1.4, which displays two allocations, z 0 N and zN , in a two-agent economy; by 0 are the same Arrow Independence, the fact that individual preferences about zN and zN in both profiles implies that social preferences should be identical for the two cases; but 0 is not, so that with the profile RN , zN is an equal-budget competitive equilibrium and zN 0 ; the reverse occurs with the profile R0 . zN PS W (E) zN N 7 Fleurbaey et al. (2005b) provide a simple direct proof of this fact.

4.4. ALLOCATION RULES AS SOCIAL ORDERING FUNCTIONS

67

Proof. The first result comes from the fact that the allocations in which one agent consumes Ω are the only allocations of Z(E) which are efficient for all profiles RN in E ∗ . By Anonymity, all of them must be selected. The second result comes from the fact that no allocation of Z(E) is efficient 0 for all profiles RN in E. Let zN ∈ max|R(E) Z(E) and zN ∈ / max|R(E) Z(E) 0 0 0 0 such that for all i ∈ N, zi ¿ zi or zi À zi . Let E = (RN , Ω) be such that 0 zN ∈ / P (E 0 ). Then one can no longer have zN P(E) zN . It is, however, required 0 by Arrow Independence because for all i ∈ N, Ri and Ri0 agree on {zN , zN }. Hence a contradiction. This result shows that Arrow Independence cannot be satisfied by a reasonable allocation rule, and that the fact that allocation rules correspond to two-tier SOFs is not a sufficient explanation for the success of the theory of fair allocation in terms of positive results. The key condition of social choice that has to be weakened in order to obtain positive results for allocation rules is, as for any kind of SOF, Arrow Independence. There is, however, a grain of truth in the idea that allocation rules are easier to obtain than more fine-grained SOFs. They are indeed less informationally demanding thanks to the weakening of Pareto requirements into Pareto Efficiency. Indeed, weakening the Pareto requirement in this fashion is sufficient to turn the negative results of the previous section into positive ones. Theorem 4.5 On the domain E, there exists a SOF satisfying Pareto Efficiency, Anonymity and either Unchanged-Cone Independence or UnchangedFset Independence (and therefore Unchanged-Subspace Independence). In fact, as far as fairness is concerned, much more than Anonymity can be satisfied. The example that proves the first part of the result is the egalitarian Walrasian allocation rule, viewed as a two-tier SOF. It satisfies Pareto Efficiency because Walrasian equilibria are efficient. It is clearly anonymous. In order to check that it satisfies Unchanged-Cone Independence, observe that when preferences change but the supporting cones at zN remain the same, then whether zN is or is not an egalitarian Walrasian equilibrium cannot be changed. As this determines the position of zN in the two-tier ranking defined by this SOF, Unchanged-Cone Independence is satisfied. This positive result, however, becomes an impossibility again if one adds to the list of axioms the requirement that the selection made by the SOF be essentially single-valued, i.e., that the allocations selected by the SOF for any given economy are equivalent by Pareto Indifference.8 The second part of the result is exemplified by the Pazner-Schmeidler allocation rule, viewed as a two-tier SOF. Pareto Efficiency and Anonymity are again obvious. Unchanged Fset-Independence is satisfied for the following reason. If 0 F (zN , zN ) ⊇ {q ∈ X | q ≤ Ω} , a change of preferences outside this set cannot 0 change whether zN or zN is efficient and egalitarian-equivalent and therefore 0 cannot change their ranking by this SOF. If F (zN , zN ) + {q ∈ X | q ≤ Ω} they 8 See

Fleurbaey et al. (2005b).

68CHAPTER 4. THE INFORMATIONAL BASIS OF SOCIAL ORDERINGS are not in P (E) and there is no change in preferences that can move them to P (E) so that they are deemed equivalent by this SOF independently of the preference profile RN . Observe that this second example generates an essentially single-valued selection, so that the problem encountered in this respect with Unchanged-Cone Independence does not occur with Unchanged-Fset Independence. In conclusion, abandoning Arrow Independence is indispensable to obtain possibility results in the theory of fair allocation just as in the theory of SOFs developed in this book. It it true, however, that allocation rules are informationally less demanding than fully Paretian SOFs, as they only require local information about preferences.

4.5

Conclusion

In order to avoid an Arrovian type of impossibility, our theory of SOFs must make room for a richer informational basis. In order to rank two allocations, one must know not just the population preferences over these two allocations, but some substantial parts of the agents’ indifference sets at these two allocations. This additional information is still only about ordinal non-comparable preferences, but it makes it possible to “compare” the agents’ situations by comparing their indifference sets. If i’s indifference set is everywhere above j’s set, we can conclude that i is better-off than j. When the indifference sets cross, who is better-off depends on the relative position of the indifference sets in the relevant portion of the consumption set. It is therefore not surprising that we have found it impossible to obtain efficient and anonymous SOFs on the sole basis of strictly local information such as pairwise preferences or supporting cones at the bundles under consideration. By focusing on a very limited information about indifference sets, the theory of social choice could not obtain positive results. Moreover, we have also seen that if one wants the SOFs to be not only anonymous and egalitarian but to satisfy fairness properties associated to the equal-split allocation, it is necessary to make the ranking of allocations depend on the available resources Ω. This is another important extension of the informational basis compared to social choice theory. In contrast, the difference between the SOFs studied in this book and the allocation rules of the theory of fair allocation is less important than is often thought, because both objects provide complete orderings that rely on more information than allowed by Arrow Independence. Allocation rules, however, provide only two-tier orderings and because of this cannot fully satisfy the standard Pareto axioms. This makes it possible to obtain fair allocation rules with local information about indifference sets, such as supporting cones. In other words, allocation rules are slightly less demanding, in terms of information about preferences, than fully Paretian SOFs. But this difference appears to be of second order compared to their basic similarity regarding the need to relax Arrow Independence and also the need to take account of Ω.

Part II

Distribution

69

71 After a first part that has introduced the approach, its general features and its relationship with the rest of welfare economics, we are now ready to start the real work, namely, constructing social criteria for the evaluation of allocations. This second part is about the same model that served as a workhorse for the generalities of the first part. Even though the division problem is not the most exciting when one is eager to say something about the pressing social problems of the real world, this model is very convenient in order to understand the basic concepts. The two subapproaches that are highlighted in the next chapter appear to haunt all the models that have been examined so far, and they reflect the classic divide, in the theory of fair allocation (and even in the earlier theory of index numbers), between egalitarian-equivalence and the Walrasian approach. The former evaluates individual situations by looking at specific parts of indifference curves that are located in well-chosen parts of the consumption set, whereas the latter compares individuals in terms of budget sets that are either their actual budget sets, or hypothetical budget sets that are suitably related to the general configuration of the allocation under consideration. These two ways of analyzing individual situations in the context of social evaluation are so basic and natural that one may suspect that any conceivable approach that similarly ignores utility information and focuses on indifference curves and resource bundles must in some way derive from one of them. The two other chapters of this part explore how the approach developed in this book can be specialized to specific domains of preferences, specific goods, or extended in order to tackle broader issues than the main ones studied here. Specific domains sometimes yield surprising results. For instance, when non-convex preferences are allowed the egalitarian-equivalence approach can be defended as the best extension of the Walrasian approach. When the smaller domain of homethetic preferences is considered, the Walrasian approach widens and certain criteria with moderate inequality aversion appear interesting. When indivisible goods are considered, the egalitarian-equivalence and the Walrasian approaches seem to come closer if not merge. It is also in this part that we will illustrate the possibility to construct libertarian social orderings that sanctify property rights, or, in another part of the ideological spectrum, social orderings that suggest an interesting solution to the indexing problem for the capabilities approach proposed by Sen. We will also see how to go beyond the standard social choice exercise of ranking social alternatives for a given population with fixed preferences, and consider the problem of ranking social alternatives for populations with different preferences, which is quite relevant in the context of comparisons of living standards.

72

Chapter 5

Fair distribution of divisible goods: two approaches 5.1

Introduction

This chapter and the next one pursue the analysis of the model that already provided the framework of Part I. Our aim now is to obtain precise conclusions about how to rank allocations. We already know from Chapter 3 that a good deal of egalitarianism will necessarily be part of the picture, but it remains to determine how individual situations should be measured and compared to each other. Among a variety of options, two main social orderings are singled out below. One has already been introduced in Chapter 1, namely the ΩEquivalent Leximin SOF. But another one also appears salient, and is more closely connected to the market mechanism. The choice between one or the other of these two different SOFs can be made easier, we hope, with the results of this chapter that show how they derive from different ethical principles. In the next section, a first series of results highlight the Ω-Equivalent Leximin SOF on the basis of axioms focusing on the equal split of the available resources. In Section 3, the alternative approach of a “Walrasian” kind of social preferences is introduced and motivated on the basis of axioms that relate to responsibility and neutrality (with respect to individual preferences) and to efficiency. Section 4 shows how these SOFs can be used for the selection of allocations in the “second-best” context in which the policy-maker does not know individuals’ characteristics but has information about the statistical distribution of characteristics. Section 5 concludes with comments on the differences between the two SOFs analyzed here. 73

74

5.2

CHAPTER 5. FAIR DISTRIBUTION: TWO APPROACHES

From equal split to Ω-Equivalence

In Chapter 2 we have seen that the Pareto principle is compatible with egalitarian axioms which can be restricted in two ways, namely, by restricting the region of the consumption set where the transfer principle applies (e.g., proportional allocations), or by restricting the type of agents to which it applies (e.g., agents with identical preferences). The following result shows the typical consequences of combining the two kinds of egalitarian axioms, together with Pareto and independence axioms. Recall that uΩ (zi , Ri ) denotes the Ω-equivalent utility. Theorem 5.1 On the domain E, if a SOF satisfies Strong Pareto, Transfer among Equals, Proportional-Allocations Transfer and Unchanged-Contour In0 dependence, then for all E = (RN , Ω) ∈ E and zN , zN ∈ XN , 0 min uΩ (zi , Ri ) > min uΩ (zi0 , Ri ) ⇒ zN P(E) zN . i∈N

i∈N

In other words, the SOF then exhibits a “maximin property” with respect to Ω-equivalent utilities: if the smallest utility is greater in one allocation, this allocation is preferred. The Ω-Equivalent Leximin SOF satisfies this property, but is not the only one. The proof makes use of Theorem 3.1 as a lemma.1 Let us give an intuitive description of the argument (the full proof is in the appendix). Notice that when min uΩ (zi , Ri ) > min uΩ (zi0 , Ri ), i∈N

i∈N

1 2 zN , zN

one can find ∈ Pr(Ω) (i.e., allocations with bundles proportional to Ω) such that for some i0 ∈ N and for all j 6= i0 , ½ ¾ µ ¶ zj1 À max max uΩ (zi , Ri ), max uΩ (zi0 , Ri ) Ω ≥ min uΩ (zi , Ri ) Ω À i∈N

i∈N

i∈N

zj2

=

zi20

À zi10 À uΩ (zi00 , Ri0 )Ω.

See Figure 5.1 (on which (maxi∈N uΩ (zi , Ri )) Ω has been arbitrarily put above (maxi∈N uΩ (zi0 , Ri )) Ω –the order does not matter). As a matter of fact, the number of agents in N \ {i0 } does not matter much for the logic of the argument, so that one can grasp the basic idea by focusing on the simple case of a two-agent population {j, i0 }. The argument is a reductio 0 ad absurdum. Suppose that zN R(E) zN . 0 Let Rj be such that I(zj , Rj0 ) = I(zj , Rj ), I(zj0 , Rj0 ) = I(zj0 , Rj ) and © ª U (zj1 , Rj0 ) = x ∈ X | x ≥ zj1 . ¡¡ ¢ ¢ See Figure 5.1. Let E 0 = Ri0 , Rj0 , Ω . By Unchanged-Contour Indepen0 1 0 2 R(E 0 ) zN . By Strong Pareto, zN P(E 0 ) zN and zN P(E 0 ) zN , so that dence, zN 1 0 2 by transitivity, zN P(E ) zN . 1 Theorem 3.1 involves Pareto Indifference, which is why Strong Pareto is invoked in Theorem 5.1. But the result is also true with Weak Pareto, as explained in the appendix.

5.2. FROM EQUAL SPLIT TO Ω-EQUIVALENCE Rj0

good 2 6

Rj00 zj3 s s zi1

zi30

0

s

75

sΩ zj1

s s s (maxi uΩ (zi , Ri ))Ω (maxi uΩ (zi0 , Ri ))Ω

s (mini uΩ (zi , Ri ))Ω

s2 zi0 = zj2

0 s uΩ (zi00 , Ri0 ))Ω

good 1

Figure 5.1: Proof of Theorem 5.1. 3 Let zN ∈ Pr(Ω) be such that

zj1 À zj3 À zj2 À zi30 À zi10 and 00 Let RN

¢ ¢ 1¡ 3 1¡ 2 zi0 + zj3 ¿ zi0 + zj2 = zi20 . 2 2 00 00 be such that Rj = Ri0 and such that for all i ∈ N, I(Ri00 , zi1 ) = I(Ri0 , zi1 ) and I(Ri00 , zi2 ) = I(Ri0 , zi2 )

and

© ª U (Rj00 , zj3 ) = x ∈ X | x ≥ zj3 .

00 See Figure 5.1. Let E 00 = (RN , Ω) . By Unchanged-Contour Independence, 1 00 2 3 1 3 2 zN P(E ) zN . By Theorem 3.1, zN P(E 00 ) zN . By transitivity, zN P(E 00 ) zN . 3 3 2 2 2 But as zi0 + zj ¿ zi0 + zj and zN is egalitarian, there exists an allocation 4 4 2 zN ∈ Pr(Ω) such that zN ¿ zN and for some ∆ ∈ R++ ,

zj3 − ∆ = zj4 À zi40 = zi30 + ∆. 4 3 2 4 R(E 00 ) zN . By Strong Pareto, zN P(E 00 ) zN . By Proportional-Allocations Transfer, zN 2 00 3 By transitivity, zN P(E ) zN . This yields a contradiction. Thus, the proof of this result is, in its structure, quite simple. It provides a useful illustration of the basic argument underlying many of our results below. Besides, in this particular theorem, the direction of proportionality referred to in Proportional-Allocations Transfer can be anything, so that this kind of result can be adapted to support any SOF that involves a ray of reference not necessarily containing Ω. It is even possible to take one axis as the reference ray, if one restricts the domain to preferences such that all indifference sets intersect

76

CHAPTER 5. FAIR DISTRIBUTION: TWO APPROACHES

this axis. The meaning of Proportional-Allocations Transfer in this case is that allocations in which this particular good only is distributed can be submitted to the transfer principle independently of the agents’ preferences. However, we have argued in Chapter 2 that the ray containing Ω is attractive when one takes the equal split allocation as a benchmark. It is also worth noting that in the proof above, Proportional Allocations Transfer is not used in its full force. The only consequence of it which is really 2 used is that an egalitarian allocation proportional to Ω (zN in the proof) is better than another proportional but non-egalitarian allocation using less total 3 resources (zN ). This is essentially an “equal-split” requirement. We now turn to another result which gives the prominent role to an equal split axiom, and indeed the interesting feature of this result is that it does not involve Transfer among Equals. The only egalitarian axiom refers to equal split. Theorem 5.2 On the domain E, if a SOF satisfies Strong Pareto, EqualSplit Transfer, Separation and Replication, then for all E = (RN , Ω) ∈ E and 0 zN , zN ∈ XN , 0 min uΩ (zi , Ri ) > min uΩ (zi0 , Ri ) ⇒ zN P(E) zN . i∈N

i∈N

The appendix presents and proves a stronger result which involves Weak Pareto and a weakening of Separation. Here we focus on the above statement, for which an intuitive argument is easier. First, by Theorem 3.4 (involving Pareto Indifference, Equal-Split Transfer, Separation and Replication), the SOF must satisfy Equal-Split Priority. Second, by Lemma 4.1 (involving Separation and Replication), it is insensitive to multiplications of Ω by a rational number. Consider the two-agent case {i0 , j} and assume that min uΩ (zi , Ri ) > min uΩ (zi0 , Ri ). i∈N

i∈N

0 Invoking Strong Pareto, we can focus on the special case when zN , zN ∈ Pr(Ω) and zj0 À zj À zi0 À zi00 .

1 2 , zN (As explained after Theorem 5.1, in the other cases one can always find zN 2 1 0 satisfying these conditions and such that zN P(E) zN and zN P(E) zN by Strong 2 1 0 Pareto, so that if one can prove that zN R(E) zN , then by transitivity, zN P(E) zN .) 0 Take E = (RN , qΩ), where q is a rational number such that

zj0 À zj À qΩ À zi0 À zi00 . 0 By Equal-Split Priority, one has zN R(E 0 ) zN . By Separation and Replication 0 (in virtue of Lemma 4.1), this implies zN R(E) zN . When there are more than two agents, one can similarly end up with a situation in which for all j 6= i0 ,

zj1 À zj2 À qΩ À zi20 À zi10 ,

5.3. A WALRASIAN SOF

77

2 1 0 and by Strong Pareto, zN P(E) zN and zN P(E) zN . A repeated application 2 0 1 of Equal-Split Priority yields zN R(E ) zN . Separation and Replication imply 2 1 zN R(E) zN and the desired conclusion is obtained by transitivity. These results (Theorems 5.1-5.2) strongly point in the direction of RΩlex . But they fail to uniquely characterize it, because there are other SOFs which satisfy the axioms of the theorems. The existence of reasonable axioms that would force 0 us to have R = RΩlex seems doubtful. Consider the following SOF, RΩΩ lex . 0 0 Take any arbitrary Ω0 that is not proportional to Ω. Let zN RΩΩ lex (E) zN whenΩlex 0 Ωlex 0 Ω0 lex 0 (E) zN , or zN I (E) zN and zN R (E) zN . This SOF ever either zN P satisfies all the axioms of the above theorems. It is also anonymous, in the sense that it does not give any biased consideration to any particular agent’s situation (permuting the agents’ preferences and their bundles does not alter the evaluation of the allocations).2 In view of this fact, we leave it open what kind of additional ethical principle, in this model, would single out the leximin SOF.

5.3

A Walrasian SOF

The previous section has provided justifications for the egalitarian-equivalent approach to the definition of social orderings. But this is not the only approach, as recalled in Chapter 1. In the theory of fair allocation, another important contender is the Walrasian approach, which relies on the idea that agents should be given fair endowments and left free to trade these endowments at competitive prices. The allocation rule S EW (E), defined in Section 1.2, selects the subset of egalitarian Walrasian allocations for E. Certainly, by referring to such competitive allocations the Walrasian approach sounds quite natural to those who live in market economies, but it remains to pin down the ethical justifications for this alternative view. The theory of fair allocation actually provides many arguments for it, most of which can be related to the idea that preferences should be respected in a neutral way. Take for instance the heraldic no-envy condition (saying that the selected allocation zN should be such that for all i, j ∈ N, zi Ri zj ). It is satisfied when for every i ∈ N, zi is the best bundle for Ri in some common set containing all the bundles. A situation in which agents are able to choose their own bundle in some common set of options may indeed be taken to reflect such neutrality about preferences. As is well known, an egalitarian Walrasian allocation (i.e., a competitive equilibrium with identical endowments) is always envy-free, and conversely, in large economies with sufficient diversity of preferences, envy-free efficient allocations are egalitarian Walrasian.3 In Chapter 2 we have already discussed how to make use of the no-envy concept in the definition of requirements for SOFs, and Nested-Contour Transfer 2 Anonymity is in fact implied by Replication as we formulated it, since an economy where preferences are permuted across agents is a 1-replica of the original economy. 3 For the former point, see Kolm (1972), and for the latter, Varian (1976) and Champsaur and Laroque (1981).

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CHAPTER 5. FAIR DISTRIBUTION: TWO APPROACHES

was in particular connected to this idea. Here is another axiom which, in the most immediate fashion, translates the no-envy requirement, as it is usually defined for allocation rules which select a first-best allocation in the feasible set Z(E), into a requirement for SOFs. It simply says that the first-best allocation in Z(E), when it is selected by the SOF, should satisfy the no-envy condition: Axiom Envy-Free Selection For all E = (RN , Ω) ∈ D and zN ∈ max|R(E) Z (E), for all i, j ∈ N, zi Ri zj . Another axiom which has been used for the justification of Walrasian allocations is the monotonicity requirement saying that when an allocation is selected, it should remain so when the individual upper contour sets at this allocation shrink, meaning that the corresponding bundles go up in individual preferences relative to other bundles.4 Here the idea may be that the social decision should not be too sensitive to the agents’ preferences, and this can be motivated by responsibility and neutrality concerns (as well as incentive concerns). Indeed, if agents are responsible for their preferences, and if we do not want to give them a treatment that is biased in favor of particular types of preferences, the case in which the bundle they are already granted goes up in their preferences seems a natural case in which no redistribution of resources is needed. Axiom Selection Monotonicity 0 For all E = (RN , Ω), E 0 = (RN , Ω) ∈ D, and zN ∈ max|R(E) Z (E), if for all 0 i ∈ N, U (zi , Ri ) ⊆ U (zi , Ri ), then zN ∈ max|R(E 0 ) Z (E 0 ) . Selection Monotonicity5 entails Envy-Free Selection when one considers a Paretian SOF which also satisfies Transfer among Equals. Under Strong Pareto it actually implies the egalitarian Walrasian way of obtaining envy-free allocations. Theorem 5.3 On the domain E, any SOF satisfying Selection Monotonicity, Weak Pareto and Transfer among Equals also satisfies Envy-Free Selection. If it also satisfies Strong Pareto, then for all E = (RN , Ω) ∈ E, S EW (E) ⊆ max|R(E) Z (E) . Proof. First part. Let E = (RN , Ω) ∈ E and zN ∈ max|R(E) Z (E) . Suppose there exist j, k ∈ N such that zj Pk zk . Then there exists R0 such that 4 See

Gevers (1986). This is closely related to Maskin Monotonicity (Maskin 1999). that these axioms are formulated about the “first-best” selection in Z(E) and not in terms of selection in any subset A ⊂ X |N | . This seems indeed essentially impossible. For instance, an axiom saying that the selection in any subset A should be envy-free whenever A contains envy-free allocations is incompatible with Pareto requirements because it may be that all envy-free allocations are Pareto inefficient in A. Even the weaker condition that the selection in any subset A should be envy-free whenever A contains envy-free efficient allocations is incompatible with Pareto Indifference because envy-free efficient allocations may be Pareto indifferent to allocations with envy. Similar concerns can be formulated about Selection Monotonicity, because it is somehow stronger than Envy-Free Selection in an egalitarian context, as shown in the next theorem. 5 Notice

5.3. A WALRASIAN SOF

79

U (zj , R0 ) ⊆ U (zj , Rj ) and U (zk , R0 ) = U (zk , Rk ), implying zj P0 zk (see Figure 5.2). good 2 6 Rk Rj

R0 = Rj0 = Rk0 z sj zj3s

3 szk

zk 0

s

zk2s sz 1

zj1 s s zj2

k

good 1

Figure 5.2: Proof of Theorem 5.3. 0 0 Let E 0 = (RN , Ω), with RN ∈ R being defined by Rj0 = Rk0 = R0 and = Ri for all i 6= j, k. By Selection Monotonicity, zN ∈ max|R(E 0 ) Z (E 0 ) . In 1 2 3 , zN ∈ X N , zN ∈ Z(E 0 ), ∆ ∈ addition, one can choose R0 so that there exist zN 1 0 3 0 2 2 1 R++ such that for all i ∈ N, zi Pi zi and zi Pi zi ; zj = zj − ∆ À zk1 + ∆ = zk2 ; and for all i 6= j, k, zi2 = zi1 (see Figure 5.2). 1 3 2 P(E 0 ) zN and zN P(E 0 ) zN . By Transfer among Equals, By Weak Pareto, zN 2 0 1 3 0 3 zN R(E ) zN . By transitivity, zN P(E ) zN . Since zN ∈ Z(E 0 ), this contradicts 0 zN ∈ max|R(E 0 ) Z (E ) . Second part. Consider E = (RN , Ω) ∈ E, and zN ∈ S EW (e). Let p be a price vector supporting6 zN , and Rp be defined by:

Ri0

∀x, x0 ∈ X, x Rp x0 ⇔ px ≥ px0 .

¡ ¢ ∗ ∈ max|R(Ep ) Z (Ep ) . By the first Let Ep = (Rp )i∈N , Ω ∈ E, and consider zN ∗ ∗ part, one must have zi Ip zj for all i, j ∈ N (when preferences are identical, envy-freeness implies that agents have bundles on the same indifference curve). ∗ By Strong Pareto, zN ∈ P (Ep ), so that zi∗ Ip zi for all i ∈ N, so that Strong ∗ Pareto implies zN R(E) zN , so that zN ∈ max|R(Ep ) Z (Ep ) . As a consequence, by Selection Monotonicity, zN ∈ max|R(E) Z (E) . Among the three SOFs introduced in Chapters 1 and 2 (namely, RRU , RΩlex , ΩNash R ), only RRU satisfies Selection Monotonicity, and none of them satisfies Envy-Free Selection. We now present a SOF which satisfies both axioms. It relies on the maximin criterion applied to money-metric utilities (defined in Section 1.8), but with the pecular feature that the reference price that serves to 6 That

is, for all i ∈ N, zi is a best bundle for Ri in the set {x ∈ X | px ≤ pzi } .

80

CHAPTER 5. FAIR DISTRIBUTION: TWO APPROACHES

measure such utilities is specific to every allocation, and is computed so as to obtain the most favorable maximin evaluation for the contemplated allocation. Social ordering function Egalitarian Walrasian (REW ) 0 For all E = (RN , Ω) ∈ E, zN , zN ∈ XN , 0 ⇔ max min min {px | x Ri zi } ≥ max min min {px | x Ri zi0 } zN REW (E) zN p:pΩ=1 i∈N

p:pΩ=1 i∈N

To make the discussion of money-metric utilities easier, let up (zi , Ri ) = min {px | x Ri zi } and ΠΩ = {p | pΩ = 1} . Note that when p ∈ ΠΩ , up (zi , Ri ) = min {px/pΩ | x Ri zi } . sΩ ¡ ¡ ¡ ¡ zs 1 ¡ ¡ ¡ s s λΩ ¡ z3 ¡ ¡ zs 2 ¡ ¡ µp ¡ ¡ ¡ good 1 0

good 2 6

Figure 5.3: Evaluation of an allocation by REW . The definition of REW may look complicated, but there is a simple geometrical way of evaluating an allocation with it. It suffices to look for the smallest point on the ray of Ω which belongs to the convex hull of the union of the individual upper contour sets at the contemplated allocation. Figure 5.3 illustrates this. On the figure, the relevant point for evaluation is λΩ, and in this example one has, for the corresponding p ∈ ΠΩ , λ = up (z1 , R1 ) = up (z2 , R2 ) < up (z3 , R3 ). It is clear from the figure that for any other p0 ∈ ΠΩ , λ > up0 (z1 , R1 ) or λ > up0 (z2 , R2 ), so that λ = max min up (zi , Ri ). p∈ΠΩ i∈{1,2,3}

E,

The name of this SOF comes from the observation that for all E = (RN , Ω) ∈ max|REW (E) Z(E) = S EW (E).

5.3. A WALRASIAN SOF

81

In other words, REW always selects, in Z(E), each and every egalitarian Walrasian allocation, and no other allocation. This can be proved as follows. For any zN ∈ EW (E), with supporting price p ∈ ΠΩ , one has up (zi , Ri ) = pzi = p

Ω 1 = . |N | |N |

Conversely, for any zN ∈ Z(E), any p ∈ ΠΩ , if up (zi , Ri ) ≥ 1/ |N | for all i ∈ N, this means that pzi ≥ pΩ/ |N | for all i ∈ N, which, in view of the fact that P i∈N zi ≤ Ω, is only possible if pzi = pΩ/ |N | and zi ∈ max|Ri B (Ω/ |N | , p) for all i ∈ N . This in turn entails that zN ∈ S EW (E), with supporting price p (and also that up (zi , Ri ) = 1/ |N | for all i ∈ N ). As a consequence, one has zN ∈ S EW (E) if and only if maxp∈ΠΩ mini∈N up (zi , Ri ) = 1/ |N | and for all zN ∈ Z(E) \ S EW (E), maxp∈ΠΩ mini∈N up (zi , Ri ) < 1/ |N | . Observe that REW bears some similarity with RRU , as the latter evaluates allocations by computing the expression max

p∈ΠΩ

X

up (zi , Ri ),

i∈N

in which, compared to REW , the summation operator has replaced the minimum. Let us now focus on REW . In addition to the above two axioms (Selection Monotonicity and Envy-Free Selection), it also satisfies Weak Pareto, Pareto Indifference, Transfer among Equals and Nested-Contour Transfer, Equal-Split Selection, Unchanged-Contour Independence, and Replication. Interestingly, like RRU it does not satisfy any of the separability axioms. More specifically, there is a strong incompatibility between separability axioms and Selection Monotonicity or Envy-Free Selection, as can be seen from the following result. Theorem 5.4 On the domain E, no SOF satisfies Strong Pareto, Envy-Free Selection and Separability. Proof. Let R satisfy the axioms. Let N = {1, 2, . . . , n} and let E = (RN , Ω) ∈ E be such that for all i ∈ N \ {1, 2}, Ri = Rp (defined in the proof of Th. 5.3). Figure 5.4 illustrates ¡ the proof. Ω ¢ We claim that z1∗ , z2∗ , Ω n , . . . , n ∈ max|R(E) Z (E) . Let zN ∈ Z (E) be such that for all i, j ∈ N, zi Ri zj . By Envy-Free Selection, it must be the case that for all i, j ∈ N \ {1, 2}, zi Ii zj . It is impossible to have Ω n P3 z3 as it would imply either z1 P3 z3 or z2 P3 z3 , violating Envy-Free Selection. It is impossible 0 to have z3 P3 Ω n , too. Indeed, this implies that for some bundle z3 ∈ X such 0 0 0 that z3 I3 z3 , either z3 P1 z1 or z3 P2 z2 , and, if n is sufficiently large, there is an allocation that is Pareto indifferent to zN and is such that agent 3 receives z30 , in contradiction to Envy-Free Selection. Therefore zi Ip Ω n for all i ∈ N \ {1, 2}. or pz = 6 pΩ For the same reason one cannot have pz1 6= p Ω 2 n n.

82

CHAPTER 5. FAIR DISTRIBUTION: TWO APPROACHES good 2 6

zs 2∗

zs 2∗∗ Ω n

p ¸¢ 3p0 ¢´´ ¢s ´

zs 1∗∗

0

z1∗ s good 1

Figure 5.4: Proof of Theorem 5.4. ¢ ¡ Ω Therefore, by Strong Pareto, z1∗ , z2∗ , Ω n , . . . , n ∈ max|R(E) Z (E) . In particular, we have µ ¶ µ ¶ Ω Ω Ω Ω z1∗ , z2∗ , , . . . , P (E) z1∗∗ , z2∗∗ , , . . . , . n n n n 0 , Ω) ∈ E be such that R1 = R10 and R2 = R20 and for all i ∈ Let E 0 = (RN 0 N \ {1, 2}, Ri = Rp0 . By a similar argument to the previous paragraph, we can prove that µ µ ¶ ¶ Ω Ω ∗∗ ∗∗ Ω 0 ∗ ∗ Ω z1 , z2 , , . . . , P (E ) z1 , z2 , , . . . , , n n n n violating Separability. Nonetheless, REW does satisfy a separability axiom that takes account of the egalitarian context of the analysis. This axiom restricts the application of Separation to agents who are not only unconcerned but are also obviously better-off than others in the allocations under consideration, because each of them receives a better bundle in the two allocations than another agent with the same preferences. It is only in that case that the ranking of the two allocations is required not to be affected by their absence.

Axiom Well-Off Separation 0 For all E = (RN , Ω) ∈ D with |N | ≥ 2, and zN , zN ∈ X N , if there are i, j ∈ N such that Ri = Rj , zi Pj zj , zi0 Pj zj0 , zi = zi0 , then 0 0 ⇔ zN \{i} R(RN \{i} , Ω) zN\{i} . zN R(E) zN

This axiom is also obviously satisfied by RΩlex and RΩNash which already satisfy Separation.7 We are now able to give an argument to the effect that REW is a key player in this game. 7 It

is also satisfied by RΩm in , the maximin variant of RΩlex defined before the statement

5.3. A WALRASIAN SOF

83

Theorem 5.5 On the domain E, if a SOF satisfies Strong Pareto, Transfer among Equals, Selection Monotonicity, Unchanged-Contour Independence, 0 Well-Off Separation and Replication, then for all E = (RN , Ω) ∈ E and zN , zN ∈ N X , 0 0 zN PEW (E) zN ⇒ zN P(E) zN . The main idea of the proof (provided in the appendix) is the following. As0 0 sume that zN PEW (E) zN and that, contrary to the desired conclusion, zN R(E) zN . 0 0 One then constructs an economy E = (RN 0 , mΩ), where the population of agents is modified by Well-Off Separation and Replication, and allocations 1 2 1 0 2 0 zN 0 and zN 0 , such that zN 0 P(E ) zN 0 as a consequence of zN R(E) zN , Strong Pareto, Weak Transfer among Equals and Unchanged-Contour Independence. 2 0 1 But these allocations are constructed so as to have zN 0 ∈ EW (E ) and zN 0 ∈ 0 Z(E ), which, by Strong Pareto, Weak Transfer among Equals, Selection Monotonic2 0 2 0 1 ity and Theorem 5.3, implies zN 0 ∈ max|R(E 0 ) Z(E ) and therefore zN R(E ) zN . This is a contradiction.8 The fact that REW bears some similarity with RRU may suggest that it reflects a greater concern for efficiency than the (Ω-equivalent) SOFs such as RΩlex and RΩNash . This impression is vindicated by the next result. It relies on a weak variant of Proportional Allocations Transfer that gives a preference to efficient allocations over wasteful allocations. More precisely, it says that if a proportional allocation zN is P efficient in the economy with total endowment 0 equal to its total consumption i∈N zi , whereas P zN 0is comparatively wasteful 1 in the sense that its average consumption |N| i∈N zi is below zi for all i ∈ N, 0 then zN is at least as good as zN . Axiom Proportional-Efficient Dominance P 0 ∈ X N , if zN ∈ P (RN , i∈N zi ) For all E = (RN , Ω) ∈ D, and zN ∈ P r(E), zN and for all i ∈ N 1 X 0 zi À zi , |N | i∈N

then

0 . zN R(E) zN

In this axiom, the premiss that for all i ∈ N, zi is above the average bundle 0 of zN implies that when zN is less egalitarian (i.e., there are low zi ), it remains 0 better than zN only if the latter is sufficiently wasteful (i.e., its mean is below the lowest zi ). In this way, this axiom does not give an absolute priority to efficiency concerns and reckons with some trade-off between efficiency and equality. When 0 there is more inequality in zN , the waste in zN has to be greater in order to vindicate the requirement that zN is (weakly) better. of Theorem A.1 in the appendix. It is not satisfied by RRU , which appears to be strongly non separable. 8 It may be noted that the full force of Strong Pareto is not used in the proof of Theorem 5.5 (and the proof of Theorem 5.3). Only Weak Pareto and Pareto Indifference are used, and we suspect that Pareto Indifference is not necessary for Theorem 5.5 to hold (the second part of Theorem 5.3, however, is not true in absence of Pareto Indifference).

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CHAPTER 5. FAIR DISTRIBUTION: TWO APPROACHES

In a one-dimensional setting ( = 1), when this configuration occurs, that is, when a distribution has its entire support above the mean of another, necessarily there is generalized Lorenz dominance, a very strong argument in favor of the former distribution.9 Therefore the above axiom is quite weak in terms of egalitarianism. Theorem 5.6 On the domain E, if a SOF satisfies Weak Pareto, Transfer among Equals, Proportional Efficient Dominance, Unchanged-Contour Indepen0 dence and Replication, then for all E = (RN , Ω) ∈ E and zN , zN ∈ XN , 0 0 zN PEW (E) zN ⇒ zN P(E) zN .

The argument in the proof is not very different, in its basic structure, from the proof of Theorem 5.1, but it is worth explaining how Proportional Efficient Dominance and Replication play their part here. Consider allocations 0 zN , zN in Figure 5.5. Imagine that, contrary to the desired result, one has good 2 6

sz1

sz10 Ω *

sz2 0 sz2

0

good 1

0 Figure 5.5: zN PEW (E) zN . 0 zN R(E) zN . Recall that by Unchanged-Contour Independence, other indifference curves could be anything. Therefore, by a combined use of Weak Pareto, Transfer among Equals, Unchanged-Contour Independence and Lemma A.1 (see 2 fn 6 in Section 3.2), one can show that allocation zN as shown in Figure 5.6, is 0 2 worse than zN . By transitivity, zN R(E) zN . 9 Generalized Lorenz dominance of (x , ..., x ) over (y , ..., y ) means that, for all k = n n 1 1 1, ..., n, ki=1 x(i) ≥ ki=1 y(i) , where x(i) is the component of (x1 , ..., xn ) which is at the ith rank by increasing order: x(1) ≤ ... ≤ x(n) . By a variant of the Hardy-Littlewood-Polya (1952) theorem, it is equivalent to having W (x1 , ..., xn ) ≥ W (y1 , ..., yn ) for all increasing and quasin 1 concave functions W. If for all i = 1, ..., n, xi > n i=1 yi , necessarily one has generalized Lorenz dominance of (x1 , ..., xn ) over (y1 , ..., yn ) because one then has, for all k = 1, ..., n, k

x(i) > i=1

k n

n i=1

k

yi ≥

y(i) . i=1

5.3. A WALRASIAN SOF

85

good 2 6

sz1

sz10

z12

s z22

s

sz2

s z20

0

Ω *

good 1

2 Figure 5.6: zN P(E) zN .

Now, by Replication, every agent can be given an arbitrarily large (but equal) number m − 1 of clones without altering the comparison, so that in the 0 0 2 1 replicated economy E 0 = (RN 0 , mΩ), zN 0 R(E ) zN 0 . Let zN 0 be an allocation 0 such that one agent of each sort is just better-off than in zN 0 , while all her 2 1 0 0 m − 1 clones are given z2 . By Weak Pareto, zN 0 R(E ) zN 0 . By transitivity, 1 0 2 zN 0 R(E ) zN 0 . Then, because indifference curves are nested, one can refer to Unchanged1 Contour Independence and notice that every clone who receives z22 in zN 0 could have the same preferences as any of the initial agents. Therefore, by Lemma 1 A.1, the clones at z22 in zN 0 can be brought back near the indifference curves 1 1 at z1 or z2 (dotted curves in the figure) in arbitrary proportions. This yields a 3 0 new allocation zN 0 which is just above the indifference curves at zN 0 . Then, by 3 2 transitivity, one has zN 0 preferred to zN 0 . 3 But if the bundles in zN 0 are well located, and the reallocation of clones among initial agents is well apportioned, one can obtain 1 X 3 zi ¿ z12 . m |N | 0 i∈N

2 In addition, zN 0 is efficient in the economy with endowment equal to its total 2 consumption. By Proportional Efficient Dominance, then, one should have zN 0 3 weakly better than zN 0 , and this yields a contradiction. The detailed proof is in the appendix. This last result suggests that REW is more sensitive than RΩlex to the efficiency or lack of efficiency of allocations (in particular proportional allocations) but, obviously, this does not make REW satisfy Strong Pareto, and one may wonder how to refine REW so as to remedy this deficiency. A natural option is to use the leximin criterion applied to (up (zi , Ri ))i∈N , for p ∈ arg lexp∈ΠΩ mini∈N up (zi , Ri ) (meaning that p is chosen also with the leximin criterion). Let us call it REW lex for further reference.

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One drawback of REW lex is that it satisfies Well-Off Separation only on the subdomain of differentiable preferences, because, when indifference curves of badly-off agents have cusps, a well-off agent may influence the computation of p, no matter how great his share is. A lexicographic combination of REW 0 0 0 and RΩlex (i.e., zN R(E) zN iff either zN PEW (E) zN , or zN IEW (E) zN and Ωlex 0 zN R (E) zN ) is better in this respect. REW has been called the Egalitarian Walrasian SOF in reference to the fact that it selects S EW (E) for all E ∈ E. Actually, a whole family of SOFs, with an arbitrarily low degree of inequality aversion, have this feature. Let W be any social welfare function (defined over vectors xN ∈ RN + ) which is minimally Paretian and inequality averse in the following sense: The maximum of W (xN ) P under the constraint x ≤ c is attained only by the egalitarian vector i i∈N xN such that x = c/ |N | for all i. A typical example of such a function is i P W (xN ) = i∈N ϕ (xi ) , with a strictly concave ϕ, but the family of functions W satisfying the above property is much larger. We have the following simple result. Theorem 5.7 Let R be defined on the domain E by: for all E = (RN , Ω) ∈ E 0 and zN , zN ∈ XN , 0 zN R(E) zN ⇔ max W ((up (zi , Ri ))i∈N ) ≥ max W ((up (zi0 , Ri ))i∈N ) . p∈ΠΩ

Then, for all E ∈ E,

p∈ΠΩ

max|R(E) Z(E) = S EW (E).

Proof. Let E = (RN , Ω) ∈ E, zN ∈ S EW (E), with supporting price p∗ ∈ ΠΩ , 0 and zN ∈ Z(E). Let p0 ∈ ΠΩ maximize W ((up (zi0 , Ri ))i∈N ). For all i ∈ N, 0 up0 (zi , Ri ) ≤ p0 zi0 . Therefore one has X

i∈N

up0 (zi0 , Ri ) ≤

X

i∈N

p0 zi0 ≤

X

p∗ zi = 1,

i∈N

and moreover p∗ zi = up∗ (zi , Ri ) = 1/ |N | for all i ∈ N. By the property of W, ¡ ¢ ¡ ¢ W (up0 (zi0 , Ri ))i∈N ≤ W (up∗ (zi , Ri ))i∈N . Since

¡ ¡ ¢ ¢ W (up0 (zi0 , Ri ))i∈N = max W (up (zi0 , Ri ))i∈N p∈ΠΩ

and

¡ ¡ ¢ ¢ W (up∗ (zi , Ri ))i∈N ≤ max W (up (zi , Ri ))i∈N , p∈ΠΩ

one obtains

¡ ¡ ¢ ¢ max W (up (zi0 , Ri ))i∈N ≤ max W (up (zi , Ri ))i∈N .

p∈ΠΩ

p∈ΠΩ

5.4. SECOND-BEST APPLICATIONS

87

0 with a strict inequality if zN ∈ / S EW (E) because in that case either (up0 (zi0 , Ri ))i∈N P is unequal or i∈N up0 (zi , Ri ) < 1, implying

¡ ¢ W (up0 (zi0 , Ri ))i∈N < W ((up∗ (zi , Ri ))i∈N ) .

Among the whole family depicted in Theorem 5.7, however, only one of them satisfies Transfer among Equals, namely REW .

5.4

Second-best applications

The problem of distributing unproduced commodities is not the main focus of public economics and this is quite understandable, given the importance of production and income in taxation issues. We can nonetheless use this simple framework to illustrate how SOFs can be relied upon in the evaluation of allocations under incentive constraints. The general idea is the following. Once a SOF is chosen, and a given economy E is the context of decision, the policy-maker can determine the best policy, provided she has the appropriate information about E and the social ordering uses the same kind of information. To be more specific, the informational context in which we focus is the standard setting for taxation theory. The policy-maker knows the statistical distribution of the agents’ characteristics, without knowing the exact profile. For instance, in a two-agent economy, she knows that the profile is either RN = π (R, R0 ) or the permuted RN = (R0 , R), but no more. Let E = (RN , Ω) and π π E = (RN , Ω). This lack of information is not too bad if the social ordering is itself anonymous. Because the policy-maker can then propose a menu of bundles {z, z 0 } to the agents, knowing in advance that the agent with R will choose z while the agent with R0 will choose z 0 . If the social ordering is anonymous, it π does not matter at all whether zN = (z, z 0 ) is obtained in E or zN = (z 0 , z) is π obtained in E , because if zN is the best allocation in some subset for R(E), π then zN is also the best in the corresponding permuted subset for R(E π ). (With a non-anonymous social ordering, things are more difficult for the policy-maker if, in order to favor an agent i over another agent j, it is important to know what i’s and j’s true preferences are.) Nonetheless, as is well documented in taxation theory, the lack of precise knowledge of the profile reduces the set of feasible allocations, because it makes it impossible to give precise agents with particular (unobservable) characteristics a menu that is less favorable than others’. Since the purpose of this section is only to provide a simple illustration of the general method, we focus on the two-agent case,10 with the advantage of being able to use the Edgeworth box for graphical representations. Let E = (RN , ω N ) with N = {1, 2} . Agents have private endowments, but we will suppose that these endowments are not particularly legitimate, so that the only morally relevant information about 1 0 The following analysis also applies when there is a continuum of agents with two homogeneous groups of equal size.

88

CHAPTER 5. FAIR DISTRIBUTION: TWO APPROACHES

P resources is the total amount Ω = i∈N ω i and the relevant SOF is either RΩlex or REW . We suppose that the policy-maker does not observe endowments ω i , preferences Ri , or final consumption zi , but only the agents’ net trades zi − ω i . By anonymity it is undesirable to give some agents with a particular name (which is observable) a less favorable menu than to others’, and incentive compatibility, as explained above, makes it also impossible to discriminate among agents with different unobservable characteristics. As a consequence, the relevant set ˆ of allocations among which the policy-maker can choose is the set Z(E) defined by: ˆ Z(E) = {zN ∈ Z(E) | ∀i, j ∈ N, zi Ri ω i + zj − ω j or ω i + zj − ω j ∈ / X} . / X corresponds to the simple situation when i The condition ω i + zj − ω j ∈ cannot envy j’s trade because it is impossible for himself (it would entail negative consumption for some goods). The first part of the incentive constraint defining this set can also be written ω i + (zi − ω i ) Ri ω i + (zj − ω j ) , which makes it more transparent that no agent should envy another’s net trade. This condition is satisfied in a given allocation zN if and only if it can be obtained by letting agents choose in a menu of net trades which contains at least (zi − ω i )i∈N . The restriction to two agents is helpful in order to simplify the incentive constraint. If z1 + z2 ≤ ω 1 + ω 2 , then for i 6= j, zj − ω j ≤ − (zi − ω i ), so that zi Ri ω i − (zi − ω i ) ≥ ω i + zj − ω j . In other words, if for all i ∈ {1, 2} , zi Ri 2ω i − zi ,

(5.1)

then necessarily zi Ri ω i +zj −ω j for all i, j ∈ {1, 2} . Conversely, if an allocation is balanced (i.e., z1 + z2 = ω 1 + ω 2 ) and is incentive-compatible, then it satisfies (5.1). Incidentally, it can be shown that wasting resources would not help in any way achieving a better allocation, for a SOF satisfying Weak or Strong Pareto, so that we can focus on balanced allocations. The relevant incentive constraint is then (5.1) in the two-agent case. This constraint can be given a simple geometric representation in terms of an adaptation of the “Kolm curve” (Kolm 1972 defines it for ω i = Ω/ |N |): K(ω i , Ri ) = {x ∈ X | x Ii 2ω i − x} . This set contains ω i and is symmetric with respect to it, because x ∈ K(ω i , Ri ) ⇔ 2ω i − x ∈ K(ω i , Ri ),

5.4. SECOND-BEST APPLICATIONS

89

and this combined with x Ii 2ω i − x also implies that the marginal rates of substitution for this set at ω i are the same as for agent i’s indifference set I(ω i , Ri ). The incentive constraint then boils down to zi being in K(ω i , Ri ) or above it. Figure 5.7 illustrates how this produces a simple graphical representation in the Edgeworth box. The dotted curves are some of the agents’ indifference curves. The set K(ω 1 , R1 ) is represented in the figure by the curve AB, while K(ω 2 , R2 ) is depicted by DE. The line segments BC and EF are the rest of the frontier of the set of incentive-compatible allocations, and they correspond to the case in which ω i + zj − ω j ∈ / X, which forms part of the definition of ˆ Z(E). This set, in the figure, is therefore represented by the area CBωEF G. Notice that the line segments BC and EF themselves do not belong to the set of incentive-compatible allocations, which therefore turns out not to be closed in this example.

G6 ¾

F

O2 # # # R1 # # # # # E # # # s# # Ω/2 R2 # # contract curve ∗ # szN # # # B # C # # sω # # # D A ? O1 Figure 5.7: Incentive-compatibility in the Edgeworth box. The thick curve in the figure represents the subset of allocations which are ˆ Pareto efficient in Z(E), or second-best efficient allocations for short. As in the figure, this subset typically contains part of the contract curve and part of ˆ the two Kolm curves.11 It never contains allocations in the interior of Z(E) which are not part of the contract curve, because for every such allocation ˆ one finds other allocations in Z(E) which are better for both agents. The 1 1 The subset of a particular Kolm curve which contains second-best efficient allocations is not necessarily connected.

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CHAPTER 5. FAIR DISTRIBUTION: TWO APPROACHES

endpoints are the allocations which are the best for one agent under the incentive constraint of the other. Rigorously speaking, BC does not belong to this set, but is thickened because it contains allocations which are Pareto efficient in the ˆ closure of Z(E). Such allocations are almost incentive-compatible, as there are incentive-compatible allocations arbitrarily close to any of them. An additional constraint which may or may not be added to the analysis is the individual rationality constraint: ∀i ∈ N, x Ri ω i .

(5.2)

In this model there is no deep reason to impose it, but the political context may make it relevant (a poll tax may seem unacceptable). In the particular context of the two-agent economy, it turns out that (5.2) implies (5.1). Indeed, convexity of preferences entails that if 2ω i − x Pi x, then (x + 2ω i − x) /2 = ω i Pi x. Therefore, in this special case, individual rationality preempts incentive constraints. The extra-thick part of the contract curve illustrates this in the figure. The best policy, in this context, consists in choosing the point in the thick (or extra-thick) curve which is the best for the social ordering R(E). When R(E) = RΩlex (E), in the example of Figure 5.7, the best incentive-compatible ∗ allocation is zN because the corresponding indifference curve for agent 1 cuts the ray of Ω just above Ω/2 whereas agent 2’s indifference curve intersects the ∗ ray below the DE curve and therefore farther away from Ω/2. Notice that zN is not Pareto efficient in Z(E). In order to improve agent 1’s situation, agent 2 is induced by public policy to trade more than would be efficient. The best incentive-compatible and individually rational allocation is the highest point on the extra-thick curve. Determining the best point for REW (E) seems less easy graphically. We conclude this section with a few observations. First, moving from one point to the other on the thick curve always makes one agent better-off and the other worse-off. This is obvious on the contract curve but it is also true on the relevant parts of the Kolm curves, because, as can be seen from the construction of the Kolm curve, for any agent i, the farther from ω i a point is on K(ω i , Ri ), the worse for Ri . Second, it may be that the best point on the thick curve belongs to a line segment like BC, so that this point is not incentive-compatible. This implies that, in such a case, there is no best incentive-compatible allocation for R(E), but at the same time that it is possible to find an incentive-compatible allocation arbitrarily close to this point. The non-existence of a best incentive-compatible allocation, in this configuration, is therefore not really problematic in practice. A third, more exotic, observation is that the second-best allocation for RΩlex (E) or REW (E) may turn out to be better than the competitive equilibrium for the agent who has the greater endowment, and worse than the competitive equilibrium for the agent with the smaller endowment. In other words, the best policy may consist in taxing the poor at the benefit of the rich. Figure 5.8 illustrates this possibility for RΩlex (E). The Kolm curve for agent 1 is AB,

5.5. CONCLUSION

91

for agent 2 it is DE (extended by EF in order to take account of the second part of the incentive constraint). The set of incentive-compatible allocations is AωDG, the set of Pareto efficient allocations among incentive-compatible allocations is the thick curve. G6 ¾

A

contract curve

# # #

D

O2 # # # ∗ # s zN # # c s zN # # R 1 # s # #ω B # s #Ω/2 # # # # R2 # # # F # E # ?

O1 Figure 5.8: The richer is the worse-off.

In this example, agent 1 is better endowed than agent 2, but at the comc his indifference curve cuts the ray of Ω closer to Ω/2 petitive equilibrium zN than agent 2’s indifference curve. Therefore the optimal second-best allocation for RΩlex (E) is better for agent 1 and worse for agent 2. In this example it is also the optimal first-best allocation. For REW (E) a similar paradox may occur simply because, as is well known from the “transfer paradox”, starting from the egalitarian Walrasian allocation, it may happen that a transfer of endowment from an agent to the other produces a new equilibrium which is better than the egalitarian Walrasian allocation for the donor and worse for the recipient.

5.5

Conclusion

The Ω-Equivalent Leximin SOF RΩlex on one side and the Egalitarian Walrasian SOF REW on the other provide two different ways of evaluating individual situations and allocations. The latter gives more attention to efficiency (Proportional Efficient Dominance), responsibility and neutrality with respect to preferences (Envy-Free Selection, Selection Monotonicity), whereas the former

92

CHAPTER 5. FAIR DISTRIBUTION: TWO APPROACHES

is more favorable to equal split allocations (Proportional-Allocations Transfer, Equal-Split Transfer) and satisfies more separability properties (Separation and weaker axioms). Table 5.1 recalls the various properties satisfied by these two SOFs. When several properties are logically related, we only retain the strongest that is satisfied in similar fashion to the others. Table 5.1: Properties of the two SOFs Strong Pareto Nested-Contour Priority Equal-Split Priority Prop.-Allocations Transfer Equal-Split Selection Separation Well-Off Separation Envy-Free Selection Selection Monotonicity Prop.-Efficient Dominance

RΩlex + + + + + + + -

REW + + + + + +

The fact that the Ω-Equivalent SOFs focus more on equal split has an interesting consequence. They cater more to the preferences of agents who particularly like bundles proportional to Ω. This “bias” can be seen in the following thought experiment. Suppose that agent i’s preferences change from Ri to Ri0 , such that for all λ > 0, U (λΩ, Ri0 ) ⊂ U (λΩ, Ri ), which means that bundles proportional to Ω go up in i’s ranking. Then, in any allocation zN , necessarily uΩ (zi , Ri0 ) ≤ uΩ (zi , Ri ), implying that agent i’s degree of priority in the egalitarian social evaluation can only go up if it changes. See Figure 5.9 for an illustration. In the figure, agent i at zi gets a lower Ω-equivalent utility with preferences that exhibit a stronger complementarity between goods, with an optimal proportion corresponding to the direction of the social endowment Ω. If a change of allocation is considered, the interests of agents with such preferences are therefore given priority, by RΩlex , over those of agents with more substitutability in their preferences. This slight bias must not, however, be interpreted as a perfectionist view trying to impose “good” preferences on the population. The RΩlex satisfies Strong Pareto and therefore never goes against the agents’ actual preferences. Let us compare this with the Egalitarian Walrasian SOF REW . It actually also favors some preferences. With this SOF, however, there is no absolute kind of preferences that are likely to put an agent in higher priority, but it is a matter of relative situation in the population profile. Roughly speaking, being more eccentric, i.e., having preferences more different from the rest of the population, is more likely to put an agent in the category of the worst-off. For instance, in Figure 5.3, one may consider agent 1 as eccentric compared to the others and one sees that this agent is indeed considered to be one of the worst-off in this example. The fact that REW satisfies properties reflecting

5.5. CONCLUSION

93 sΩ ¡ ¡ ¡ ¡ ¡ ¡ s¡uΩ (zi , Ri ) ¡ s Ri0 uΩ (zi , Ri0 ) ¡ s ¡ zi ¡ ¡ Ri ¡ ¡ good 1 0

good 2 6

Figure 5.9: Preference changes and RΩlex . ideas of responsibility and neutrality with respect to preferences, in particular Envy-Free Selection and Selection Monotonicity, is therefore in agreement with this observation that there is no special type of preferences, defined absolutely, that it particularly favors. The two ethical approaches presented in this chapter illustrate our claim in the introduction to the book that this study is about clarifying value judgments and their implications rather than proposing a unique kind of social criteria. We have just identified a first ethical divide. It separates social preferences focusing on special kinds of allocations (equal split, here) and therefore favoring agents whose preferences are oriented toward such allocations, from social preferences displaying a greater concern for responsibility and neutrality and a greater preference for efficient allocations. Other important divides will appear in the sequel, for different contexts.

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CHAPTER 5. FAIR DISTRIBUTION: TWO APPROACHES

Chapter 6

Specific domains 6.1

Introduction

The results of the previous chapter suggest that RΩlex and REW are salient SOFs for the problem of distribution of unproduced commodities. The characterization results have highlighted the ethical underpinnings of such criteria. The analysis of the previous chapter focused on the canonical framework and the basic concepts, and we now turn to a series of extensions of the analysis to different frameworks or different concepts. First, we focus on different domains of preference profiles. Section 6.2 is about a particular domain restriction, namely, when preferences correspond to expected utility maximization and the various goods are contingent amounts of money in different states of nature. In Section 6.3 we examine what happens when the domain of profiles of individual preferences is enlarged so as to admit non-convex preferences. This is more than a technical issue because non-convexities in preferences are very likely to occur in many contexts. In this larger domain the results single out RΩlex as the outstanding solution. Section 6.4 is devoted to the special case of homothetic preferences. In this smaller domain, the RΩNash SOF, introduced in Section 2.3 and based on the product of Ω-equivalent utilities, becomes interesting for the Walrasian perspective. Section 6.5 is devoted to the distribution of indivisible goods in the presence of an additional divisible good like money. Indivisibilities affect the description of allocations, the domain of preferences, many of the axioms and also the definition of solutions.

6.2

Expected utility

In this section, we examine a specific domain restriction, which corresponds to the case of individual preferences based on expected utility maximization. Suppose that the goods are in fact contingent goods corresponding to the consumption of a single physical good in distinct states of nature, so that 95

96

CHAPTER 6. SPECIFIC DOMAINS

zis ∈ R+ is agent i’s consumption in state s. Agent i’s preferences Ri are now ex ante preferences over contingent bundles zi ∈ R+ . In order to avoid any ethical problem linked to the fact that agents’ preferences in such a setting may reflect imperfect beliefs and irrational behavior, we will restrict attention to the case in which agents are expected utility maximizers (with state-independent utilities) and when the probabilities of the various states of nature are objective (or at least the subjective probabilities of the agents are identical). Agent i’s expected utility is written EUi (zi ) =

X

π s Ui (zis ),

s=1

where π is the vector of probabilities and Ui : R+ → R is i’s Bernoulli utility function. Let E R denote this restricted domain of economies. In Section 4.2, the Ω0 -Equivalent Leximin SOF RΩ0 lex , which relies on a fixed reference bundle Ω0 and therefore satisfies Independence of Feasible Set, has been chastised for failing to satisfy Equal-Split Selection. But there are cases when a fixed reference Ω0 offers itself as a natural benchmark, and in such cases a condition like Equal-Split Selection may appear less compelling. This seems to occur here. In the current setting, indeed, Equal-Split Selection loses its appeal because even when Ω/ |N | is efficient, it may happen that agents suffer from the risk in Ω/ |N | differently. Checking efficiency of equal split may not be enough, then, if one wants to evaluate individual situations accurately. The only case in which the evaluation may disregard individual risk aversions, or so it seems, is obtained with “certainty” allocations (i.e., allocations containing only bundles zi such that zis = zis0 for all s, s0 ∈ {1, . . . , }). All bundles in such allocations are proportional to each other, so that this corresponds to a case of proportional allocations. Moreover, the marginal rate of substitution at a certainty bundle is equal to the ratio of probabilities for all preferences based on the expected utility. This implies that every certainty allocation is, in this restricted domain, Pareto efficient with respect to all possible redistributions (including risky ones) of the total quantity consumed in this allocation. Therefore, it appears that applying the transfer principle to such allocations, independently of individual preferences, is quite sensible. For further reference, let Xc denote the subset of certainty bundles. This suggests that in this particular framework, the relevant ray of reference is not the ray to Ω but the certainty ray (i.e., the ray of certainty bundles). Let 1 = (1, . . . , 1) ∈ R+ . Let c(zi , Ri ) denote the “certainty-equivalent utility”, i.e., the real number such that zi Ii c(zi , Ri )1 . The R1 lex SOF assesses individual situations by the certainty-equivalent utilities and applies the leximin criterion. The restriction of the domain, however, questions the salience of this SOF. Arguments used in the proofs of the characterization results such as Theorem 5.1 no longer work on this domain.1 However, the following variant of Theorem 5.1, 1 Whether the theorem itself still holds is doubtful, but quite hard to ascertain. In particular, Unchanged-Contour Independence loses some but not all power on this domain, because

6.3. NON-CONVEX PREFERENCES

97

suggests that R1 lex is not a bad candidate. This result relies on an adaptation of Proportional Allocations Transfer which focuses on certainty allocations: Axiom Certainty Transfer 0 ∈ XcN , if there exist j, k ∈ N, and For all E = (RN , Ω) ∈ D, and zN , zN ∆ ∈ R++ such that zj − ∆ = zj0 À zk0 = zk + ∆ 0 R(E) zN . and for all i 6= j, k, zi = zi0 , then zN

Theorem 6.1 On the domain E R , if a SOF satisfies Weak Pareto, NestedContour Transfer, Certainty Transfer and Separation, then for all E = (RN , Ω) ∈ 0 E R and zN , zN ∈ XN , 0 min c(zi , Ri ) > min c(zi0 , Ri ) ⇒ zN P(E) zN . i∈N

i∈N

The proof is quite simple. It consists in introducing clones of all j 6= i0 (where, as usual, i0 has the smallest c(zi0 , Ri )) with same preferences and bundles between c(zi00 , Ri0 )1 and (mini∈N c(zi , Ri )) 1 . By a variant of Theorem 3.2 (recall that this theorem derives Nested-Contour Priority from Nested-Contour Transfer, Pareto Indifference and Separability), one can move the clones up by a small amount while the agents j 6= i0 are moved down. Then Certainty Transfer (and Weak Pareto) implies that the resulting allocation cannot actually be better than zN . See the appendix for details.

6.3

Non-convex preferences

It is well known that the competitive mechanism may break down in non-convex economies. This difficulty has echoes in the theory of fairness, for instance in the possible non-existence of envy-free and efficient allocations, or even of efficient allocations in which no individual bundle strictly dominates another.2 The Egalitarian Walrasian SOF REW partly remedies this difficulty. It selects Walrasian allocations with equal budgets whenever they exist, even in non-convex economies, and is well defined in all economies, including the nonconvex. One could then imagine that in non-convex economies it provides, by its first-best selection, an interesting generalization of the concept of Walrasian equilibrium. Unfortunately, the fact that REW is able to select an allocation in non-convex economies does not mean that it always select a satisfactory allocation, or more generally that it still yields appealing social preferences in this context. The following example illustrates the problem. Figure 6.1 displays an economy with two agents, where the first-best optimal allocation for REW may be, for some preferences,3 as in the figure. It is quite unequal and this knowing two indifference curves of an agent leaves some leeway about the rest of the indifference map. We leave this for future research. 2 See e.g. Varian (1974), Maniquet (1999). 3 There are some conditions about the other indifference curves for this statement to be true. The details are omitted here.

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CHAPTER 6. SPECIFIC DOMAINS

seems rather unjustified when one looks at the agents’ indifference curves. In particular, the indifference curve of agent 2 strongly dominates that of agent 1. In contrast, RΩlex always avoids such gross inequalities and, at the minimum, always guarantees that, at the first-best optimal allocation, all agents are at least as well-off as at the equal split allocation. good 2 6 R2

Ω ©s © ©© s© R1 © z2 s © s ©Ω/2 © z1 ©© © © © good 1 0 Figure 6.1: A first-best allocation for REW . The next result examines what happens to social preferences when, in the wider domain allowing for non-convex preferences, one requires the SOF to satisfy Equal-Split Selection (saying that whenever the equal-split allocation is efficient, it should be ranked among the best of the feasible allocations). Recall that, in the convex domain E, both RΩlex and REW satisfy this axiom. Let E ∗ denote the domain of economies E = (RN , Ω) such that for all i ∈ N, Ri is a continuous, monotonic, but not necessarily convex, ordering. In E ∗ , REW does no longer satisfy Equal-Split Selection, but one may hope to define a variant of REW for non-convex economies which satisfies this condition. The following theorem dashes such hope, and suggests that, paradoxically enough, RΩlex might be the best extension of REW for the wider domain.4 In the list below, Equal-Split Selection is the only axiom that is not satisfied by REW on E ∗. Theorem 6.2 On the domain E ∗ , if a SOF satisfies Weak Pareto, Transfer among Equals, Equal-Split Selection, Unchanged-Contour Independence and Independence of Proportional Expansion, then for all E = (RN , Ω) ∈ E ∗ and 0 zN , zN ∈ XN , 0 . min uΩ (zi , Ri ) > min uΩ (zi0 , Ri ) ⇒ zN P(E) zN i∈N

i∈N

4 Fleurbaey and Maniquet (1999b), in the context of production economies, have obtained similar results with allocation rules: an egalitarian-equivalent rule may be, on a non-convex domain, the only one that satisfies weak versions of axioms satisfied by a Walrasian rule on a convex domain.

6.3. NON-CONVEX PREFERENCES

99

In order to see how this is obtained, recall the proof of Theorem 5.1 as intuitively explained in Section 5.2. Let us reproduce the relevant part of the corresponding figure (Figure 5.1), but with indifference curves so as to be able to address the efficiency issue (see Figure 6.2–for clarity, agent j’s indifference curves are thick and i0 ’s are thin). By the same reasoning as in Section 5.2, 2 3 3 2 one arrives at allocations zN , zN such that zN P(E 00 ) zN in economy E 00 with 00 00 preferences RN = (Ri0 , Rj ). The end of the proof, however, is different. The 2 in order to arrive at an allocation which argument goes by worsening on zN 2 is egalitarian like zN but is moreover efficient (after some suitable expansion 3 3 or reduction of Ω), and uses still more resources than zN (so that zN is not efficient). This produces a contradiction with Equal-Split Selection. Ri0

good 2 6

Ri0 0

µΩ ¡ ¡ zj3 Rj00 s¡ s ¡ zi40 ¡ s @ I 5 @ 4 zj H zj ¡ Rj00 s @ s j¡ H s 1 3 zi20 = zj2 = zi50 s (z + zj3 ) ¡ Ri0 2 i0 ¡ 6 ¡ ¡ ¡ zi60 = zj6 Rj000 s zi30 ¡ ¡ ¡ good 1 0 Figure 6.2: Proof of Theorem 6.2.

4 In more detail, this last step proceeds as follows. First, let zN be such that

zi20 À zi40 À zj4 À

¢ 1¡ 3 z + zj3 , 2 i0

and let Rj000 be such that I(zj2 , Rj000 ) = I(zj2 , Rj00 ), I(zj3 , Rj000 ) = I(zj3 , Rj00 ) and U (zi40 , Ri0 ) ∩ L(zj4 , Rj000 ) = ∅, that is, the indifference curves at zi40 and zj4 do not meet or cross. The existence of such Rj000 is guaranteed to be possible only if the domain admits non-convex preferences. On Figure 6.2, in particular, U (zj4 , Rj000 ) is not convex. ¡¡ ¢ ¢ 3 2 By Unchanged-Contour Independence, zN P(E 000 ) zN for E 000 = Ri0 , Rj000 , Ω . 2 4 3 4 P(E 000 ) zN so that, by transitivity, zN P(E 000 ) zN . By Weak Pareto zN 0 3 0 3 4 0 4 Let Ri0 be such that I(zi0 , Ri0 ) = I(zi0 , Ri0 ), I(zi0 , Ri0 ) = I(zi0 , Ri0 ) and © ª U (zi20 , Ri0 0 ) = x ∈ X | x ≥ zi20 .

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CHAPTER 6. SPECIFIC DOMAINS

3 4 By Independence, zN P(E 0 ) zN for E 0 =

zj4 À zj5 À (See Figure 6.2.) Since

¡¡ 0 ¢ ¢ Ri0 , Rj000 , Ω . Let zi50 = zi20 and

¢ 1¡ 3 zi0 + zj3 . 2

zi20 À zi40 À zj4 À zj5 ,

then by Theorem 3.1, or more precisely by its variant Lemma A.15 (which re4 5 3 5 mains valid on the wider domain E ∗ ), zN P(E 0 ) zN . By transitivity, zN P(E 0 ) zN . 6 Finally, let zN be such that zj5 À zi60 = zj6 À

¢ 1¡ 3 zi0 + zj3 2

and let Ri000 be such that I(zi30 , Ri000 ) = I(zi30 , Ri0 0 ), I(zi50 , Ri000 ) = I(zi50 , Ri0 0 ) and © ª U (zi60 , Ri0 0 ) = x ∈ X | x ≥ zi60 .

¡¡ ¢ ¢ 3 5 P(E 00 ) zN for E 00 = Ri000 , Rj000 , Ω and by Weak Pareto, By Independence, zN 5 6 3 6 P(E 00 ) zN , so that by transitivity zN P(E 00 ) zN . This raises a contradiction zN 6 with Equal-Split Selection, because zN (after expansion or reduction of Ω and invoking Independence of Proportional Expansion) is an efficient equal split 3 allocation and zN is not efficient (as it wastes quantities). This argument illustrates how much more constraining the combination of Weak Pareto, Transfer among Equals and Unchanged-Contour Independence is in the wider domain E ∗ admitting non-convex preferences. good 2 6

0

good 2 6

(a)

good 1

0

good 1

(b) Figure 6.3: A hypothetical worst-off agent.

The result of this section can be related to the following simple observation, illustrated in Figure 6.3. Take a given allocation zN and imagine adding a hypothetical agent who would be unambigously worse-off than every agent of the actual population, in the sense that his indifference curve would lie below the indifference curve of everybody else. Or, in more rigorous terms, his upper 5 It

is mentioned in fn 6 of Section 3.2.

6.4. HOMOTHETIC PREFERENCES

101

contour set would contain all the agents’ upper contour sets. Now look for the smallest contour set (with respect to inclusion) having this property (this amounts to seeking the most favorable situation for the hypothetical worst-off agent). If one looks for such a contour set among convex preferences, then one obtains a contour set that coincides with the convex hull of the union of all agents’ upper contour sets (Fig. 6.3a). This is the object that serves to evaluate an allocation with REW . If, in contrast, one looks for such a contour set among convex and non-convex preferences, then one simply obtains a contour set that coincides with the union of all agents’ upper contour sets (Fig. 6.3b). This can serve to evaluate an allocation with RΩlex or, rather, its maximin variant RΩmin which evaluates allocations computing mini∈N uΩ (zi , Ri ) . In other words, the same operation which leads to computing REW (E) in a convex domain produces RΩmin (E) in the wider domain.

6.4

Homothetic preferences

The subdomain of profiles in which all individual preferences are homothetic offers a different outlook for some of the issues dealt with in the previous chapters. Let E H ⊂ E denote the subset of economies E = (RN , Ω) such that for all i ∈ N, zi , zi0 ∈ X, α ∈ R++ , zi Ri zi0 ⇔ αzi Ri αzi0 . Homotheticity entails two interesting properties for Ω-equivalent utilities and money-metric utilities. First, for all zi , zi0 ∈ X, Ri ∈ R, Ω, Ω0 ∈ R++ , uΩ (zi , Ri ) uΩ0 (zi , Ri ) = . 0 uΩ (zi , Ri ) uΩ0 (zi0 , Ri ) Second, for all zi , zi0 ∈ X, Ri ∈ R, Ω, p ∈ R++ , uΩ (zi0 , Ri ) up (zi0 , Ri ) = . uΩ (zi , Ri ) up (zi , Ri ) On this restricted domain, RΩNash satisfies a larger set of properties than on E. Recall that this SOF is defined as follows: Y Y 0 zN RΩNash (E) zN ⇔ uΩ (zi , Ri ) ≥ uΩ (zi0 , Ri ). i∈N

i∈N

Most remarkable is the fact6 that for all E ∈ E H one has max|RΩN a s h (E) Z(E) = S EW (E). In other words, on this subdomain RΩNash selects the egalitarian Walrasian allocations, like REW or related SOFs highlighted in Theorem 5.7. 6 This

is due to Eisenberg (1961).

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CHAPTER 6. SPECIFIC DOMAINS

Here is a simple proof of this fact.7 Let E = (RN , Ω) ∈ E H , zN ∈ S EW (E), 0 0 with supporting price p ∈ ΠΩ , and zN ∈ Z(E). Since zN ∈ Z(E), one has X

i∈N

up (zi0 , Ri ) ≤

X

up (zi , Ri ) = 1.

i∈N

The geometric mean is below the arithmetic mean except in case of equality. Note that up (zi , Ri ) = 1/ |N | for all i ∈ N. Therefore, "

Y

# |N1 |

up (zi0 , Ri )

i∈N

# |N1 | " Y 1 X 1 ≤ up (zi0 , Ri ) ≤ up (zi , Ri ) , (6.1) = |N | |N | i∈N

which implies

i∈N

Y up (z 0 , Ri ) i ≤ 1. up (zi , Ri )

i∈N

By homotheticity of preferences, for all i ∈ N, up (zi0 , Ri ) uΩ (zi0 , Ri ) = , uΩ (zi , Ri ) up (zi , Ri ) so that Y

i∈N

Y

up (zi0 , Ri ) up (zi , Ri ) i∈N " #" # Y Y up (z 0 , Ri ) Y i = uΩ (zi , Ri ) uΩ (zi , Ri ), ≤ up (zi , Ri )

uΩ (zi0 , Ri ) =

uΩ (zi , Ri )

i∈N

i∈N

i∈N

0 0 . If zN ∈ / S EW (E), then the first inequality in (6.1) is i.e., zN RΩNash (E 0 ) zN ΩNash 0 0 strict and zN P (E ) zN . This completes the proof. The fact that RΩNash selects egalitarian Walrasian allocations implies that it satisfies Selection Monotonicity on E H . Another interesting property that RΩNash satisfies on E H (but not on E) is Independence of Feasible Set. This is somewhat surprising as Ω appears in the definition of this SOF. 0 (RN , Ω0 ) ∈ E H , This fact is shown as follows. Consider E Q = (RN , Ω), E =Q 0 N ΩNash 0 0 zN , zN ∈ X such that zN R (E) zN . If i∈N uΩ (zQ i , Ri ) or i∈N uΩ (zi , Ri ) Q 0 equals zero, the same holds for i∈N uΩ0 (zi , Ri ) or i∈N uΩ0 (zi , Ri ), respec0 tively, and therefore zN RΩNash (E 0 ) zN as well. We now focus on the case in which both products are positive. 0 means that The fact that zN RΩNash (E) zN

Y

i∈N 7 This

uΩ (zi , Ri ) ≥

Y

uΩ (zi0 , Ri )

i∈N

is a simplified version of Eisenberg’s proof, which proves a more general statement. Milleron (1970) gives a simple analytical proof for the case of differentiable preferences.

6.4. HOMOTHETIC PREFERENCES and is equivalent to

103

Y uΩ (zi , Ri ) ≥ 1. uΩ (zi0 , Ri )

i∈N

By homotheticity of preferences, for all i ∈ N,

uΩ0 (zi , Ri ) uΩ (zi , Ri ) = . uΩ (zi0 , Ri ) uΩ0 (zi0 , Ri ) Therefore

Y uΩ0 (zi , Ri ) Y uΩ (zi , Ri ) ≥1⇔ ≥ 1, 0 uΩ (zi , Ri ) uΩ0 (zi0 , Ri )

i∈N

i∈N

0 , as was to be proved. the latter meaning that zN RΩNash (E 0 ) zN Another feature which gives some advantage to RΩNash over REW is that it satisfies Separation (on the whole domain) for allocations zN in which every zi Pi 0 for all i ∈ N . This indicates that the conflict between the Walrasian approach and separability almost vanishes in the homothetic domain. Let us call Restricted Separation this restricted version of Separation. In summary, the RΩNash SOF, on the E H domain, selects S EW (E) for all E, satisfies Independence of Feasible Set, and is separable for most allocations. Are there other interesting SOFs satisfying these properties? The following result gives a negative answer and therefore singles out RΩNash as outstanding in this context.

Theorem 6.3 On the domain E H , if a SOF R satisfies Pareto Indifference, Independence of Feasible Set, Restricted Separation, Continuity and if max|R(E) Z(E) = S EW (E) for all E, then R coincides with RΩNash . The intuition for this result is the following. The requirements imply that R(E) is representable by X W ((ui (zi ))i∈N ) = ϕi (ui (zi )) , i∈N

where ui is equal to uΩ (., Ri ) and is homogeneous when Ri is homothetic. Assuming differentiability (for the purpose of this intuitive explanation), the first-order condition for maximization of W over Z(E) implies that for all k = 1, . . . , , ∂ui ϕ0i (ui (zi )) (zi ) = λk (Ω) , ∂zik where λ (Ω) is the vector of Lagrange multipliers for the resource constraint P EW (E), one has λ (Ω) zi = i∈N zi ≤ Ω. Since the maximizer zN belongs to S λ (Ω) Ω/ |N | for all i ∈ N. Besides, ui being homogeneous implies λ (Ω) zi =

X

k=1

ϕ0i (ui (zi ))

∂ui (zi ) zik = ϕ0i (ui (zi )) ui (zi ) . ∂zik

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CHAPTER 6. SPECIFIC DOMAINS

Summarizing, one has, for all Ω ∈ R++ , all zN ∈ S EW (RN , Ω), all i ∈ N, ϕ0i (ui (zi )) ui (zi ) = λ (Ω) Ω/ |N | . Considering the possibility that preferences may be different for some agents then allows to conclude that ϕ0i (ui ) ui is actually constant (see the appendix for the precise argument). By integration, one obtains that ϕi is logarithmic. One should not, however, conclude from this section that RΩNash is unquestionably the best SOF in the subdomain of homothetic preferences. It remains true on this subdomain that, contrary to RΩlex and REW , RΩNash does not satisfy Transfer among Equals and that, contrary to RΩlex , it does not satisfy Equal-Split Transfer. Actually, these two properties are violated by every mem0 0 ber of the class of SOFs defined by: for all zN , zN ∈ X N , zN RΩα (E) zN if and only if 1 X 1 X 1−α (uΩ (zi , Ri ))1−α ≥ (uΩ (zi0 , Ri )) , 1−α 1−α i∈N

i∈N

where α ≥ 0, α 6= 1 is a given parameter. By taking the limit when α → 1, RΩNash can be considered a member of this class. The violation of Transfer among Equals and Equal-Split Transfer can be shown easily in a twogood, two-agent economy by considering Leontief preferences RL represented by min {zi1 , zi2 } . For Ω = (3, 3) , one obtains, for ε ∈ [0, 1), uΩ ((1, 1) , RL ) uΩ ((1 + ε, 2) , RL ) uΩ ((5 − ε, 4) , RL ) uΩ ((5, 5) , RL )

= = = =

1/3, (1 + ε) /3, 4/3, 5/3.

One has (5, 5)−(5 − ε, 4) = (1 + ε, 2)−(1, 1) , uΩ ((5, 5) , RL )−uΩ ((5 − ε, 4) , RL ) = 1/3, and uΩ ((1 + ε, 2) , RL ) − uΩ ((1, 1) , RL ) = ε/3. For every α there is ε small enough such that ((1, 1) , (5, 5)) PΩα (E) ((1 + ε, 2) , (5 − ε, 4)) , in contradiction to both transfer axioms. In other words, an absolute priority to the worst-off remains attractive in this subdomain. Interestingly, this is obtained in spite of the results of Chapter 3 no longer being true (in particular, Unchanged-Contour Independence becomes vacuous in this domain).

6.5

Indivisibles

Many division problems of interest involve goods, or objects, which are not perfectly divisible. Let us study, in this section, how SOFs can be defined and axiomatized in economies with indivisibilities. Note that, contrary to the previous sections of this chapter, this is not about a specific domain of preferences, but about a different technology of resource sharing. A great variety of models with indivisibles can be studied. Here we concentrate on the allocation of desirable objects having the property that each agent may consume at most one of these objects. Let us think of apartments in a housing complex, seats at a

6.5. INDIVISIBLES

105

concert, parking lots, tasks in a board of directors, etc. Again, preferences may differ among agents. We assume that monetary compensations are allowed for those who do not receive any objects or who receive an object they would be willing to exchange for another object. The question we have to address is then: What could be an equitable way of assigning objects among those agents, and how should the compensations be computed? Let us begin by defining the model formally. There is an infinite set A of objects. In specific economies, an agent may be assigned either an available object from A or no object at all. In the latter case, we say that this agent receives the “null object”, which is denoted ν. Let A∗ ≡ A ∪ {ν}. The resources available in an economy are described by a finite set of objects A ⊂ A. We assume that there are at least two agents and that there is at least one object to assign but never more objects than agents, that is, #N ≥ 2 and 1 ≤ #A ≤ #N . Preferences are defined over bundles zi = (ai , mi ) ∈ A∗ × R. We assume that preferences are continuous and strictly monotone with respect to money. We also assume that all objects are desirable and their value is always finite, that is, for all (ai , mi ) ∈ A × R, (ai , mi ) Pi (ν, mi ), and there exists m0i ∈ R, such that (ai , mi ) Ii (ν, m0i ). Let Rind and E ind respectively denote the set of all such preferences and the set of all such economies. In the preceding chapters and sections, SOFs were required to be able to rank feasible and unfeasible allocations. An allocation was declared unfeasible as soon as the sum of what agents were allocated was not lower than the available amounts of goods. The basic reason of that requirement was that unfeasible allocations may become feasible after operations that are associated with axioms, notably the removal of agents implied by Separation. It is not immediate to determine the set of allocations that a SOF should be required to rank in the current model. By assumption, each object comes in exactly one unit and will never be consumed by more than one agent. On the other hand, the total amount of money associated to an allocation may be strictly positive, so that this allocation is not feasible as long as money does not come from outside the economy, but the arrival of new agents is likely to make this allocation become feasible. Nevertheless, given the simple structure of the model, we can restrict ourselves to allocations that do not require money from outside. This assumption makes some reasoning longer but does not change the nature of the result. It simply shows that in this context, no proof requires to construct unfeasible allocations. Consequently, we require SOFs to rank allocations zN for an economy E = (RN , A) ∈ E ind such that all objects assigned in zN come from A, no two agents are assigned the same “real” object (there is no limit on the set ofPagents who receive the null object), and no money is required from outside ( i∈N mi ≤ 0). Let Z(E) denote this set. Let us begin our study by adapting the two major SOFs encountered in the previous chapters to the current setting. There is no immediate way of defining bundles that are “proportional” to the available set of objects. On the other hand, given that there is only one divisible good, money, the relevant well-being indices must be measured in money. One possible solution consists in defining a SOF, Rmlex , that applies the leximin criterion to indices um (zi , Ri ) defined

106

CHAPTER 6. SPECIFIC DOMAINS

as the amount of money that leaves the agent indifferent between her assigned bundle and receiving that amount of money and no object, that is, um (zi , Ri ) = m ⇔ zi Ii (ν, m). Social ordering function m-Equivalent Leximin (Rmlex ) 0 For all E = (RN , A) ∈ E ind , zN , zN ∈ Z(E), 0 ⇔ (um (zi , Ri ))i∈N ≥lex (um (zi0 , Ri ))i∈N . zN Rmlex (E) zN

The egalitarian Walrasian SOF is easier to adapt to this model than the egalitarian-egalitarian SOF. The price of the divisible good can be fixed at 1, so that we focus on price vectors p ∈ RA + , and, abusing notation, we let the money value of a bundle zi = (ai , mi ) be defined as pzi pzi

= pai + mi if ai ∈ A, and = mi if ai = ν.

Thus, the SOF REW is defined as follows. Social ordering function Egalitarian Walrasian (REW ) 0 For all E = (RN , A) ∈ E ind , zN , zN ∈ Z(E), 0 ⇔ max min min {px | x Ri zi } ≥ max min min {px | x Ri zi0 } . zN REW (E) zN i∈N p∈RA +

i∈N p∈RA +

Let us observe that in this model, for any fixed p ∈ RA + , the money metric utility associated to a pair (zi , Ri ) is the quantity of money which, given price vector p, enables this agent to buy an object (maybe the “null” object) and reach the same satisfaction level as at zi . So, clearly, min {px | x Ri zi } ≤ um (zi , Ri ), because (ν, um (zi , Ri )) Ri zi and the value of the bundle (ν, um (zi , Ri )) is um (zi , Ri ). Moreover, when the price of the real objects is high enough, she indeed chooses not to buy any of them and reaches a satisfaction level of um (zi , Ri ), that is, max min {px | x Ri zi } = um (zi , Ri ).

p∈RA +

This reasoning extends to groups of agents. For a fixed p, one has min min {px | x Ri zi } ≤ min um (zi , Ri ), i∈N

i∈N

and for sufficiently high prices of the real objects the two terms are equal, so that max min min {px | x Ri zi } = min um (zi , Ri ). i∈N p∈RA +

i∈N

This fact is illustrated in Figure 6.4. The economy is composed of the set N = {j, k, } of agents, and the set A = {a, b} of objects. On the right part of the figure, the money metric utility of bundle z is computed, given the price

6.5. INDIVISIBLES

107

vector p0 = (p0a , p0b ) and preferences R . The thin curve depicts the monetary value of each of the three bundles that are on the same indifference as z . We can deduce that the minimum quantity of money that is necessary to help agent reach the same well-being as at z corresponds to her buying good a. In the left part of the figure, the price vector is now endogenous, and it is computed to maximize the money metric utility of agent j, as agent j appears to be the poorest one. Price vector p = (pa , pb ) is an example of a maximizing price vector.

b

a

ν

z

-Rj z

pb }|

z{ -Rk

pa zs}| }|{ zj

}| {

{

p0 z}|{a

p0 z sz }| b { -R

-

-

s szk Q k o S Q S Q S min {p0 x | x Ri z } S min maxp∈RA i∈{j,k, } min {px | x Ri zi } + Figure 6.4: The equivalence between the REW and Rmlex SOFs.

To sum up, the above discussion has shown that the two SOFs defined so far are actually almost identical in the current model, as Rmlex is a refinement of REW . More formally, we have proven the following lemma. 0 Lemma 6.1 For all E = (RN , A) ∈ E ind , zN , zN ∈ Z(E), 0 0 zN PEW zN ⇒ zN Pmlex zN .

We can therefore expect Rmlex to satisfy a long list of axioms. It is easily seen that it satisfies Strong Pareto, Separation, and Independence of Feasible Set. Let us add two axioms to this list. First, this SOF is independent of changes in preferences over unfeasible bundles. Indeed, if only objects a and b are currently available but preferences over object c change, then the ranking of the allocations we have to rank does not change, as an agent’s money-metric utility of the feasible bundles remains unaffected. Let us note that, in the division model of Chapter 5 (with divisible commodities), RΩlex shares this property with Rmlex , whereas REW does not. Axiom Independence of Preferences over Unfeasible Bundles 0 0 For all E = (RN , A), E 0 = (RN , A) ∈ D, and zN , zN ∈ X N , if for all a, b ∈

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CHAPTER 6. SPECIFIC DOMAINS

A ∪ {ν} and m, m0 ∈ R, ∀ i ∈ N : (a, m) Ri (b, m0 ) ⇔ (a, m) Ri0 (b, m0 ), 0 0 then zN R(E) zN ⇔ zN R(E 0 ) zN .

The second additional axiom that turns out to be satisfied by Rmlex is the following Consistency axiom. It belongs to the family of axioms like Separability and Separation. It requires that the ranking of two allocations be unaffected by the withdrawal from society of an agent consuming the same bundle in the two allocations, providing this agent withdraws from society by bringing her bundle with her, with the consequence that the set of available objects shrinks. Formally, Axiom Consistency 0 For all E = (RN , A) ∈ D with |N | ≥ 3, and zN , zN ∈ X N , if there is i ∈ N such 0 that zi = zi = (ai , mi ), then 0 0 zN R(E) zN ⇒ zN\{i} R(RN \{i} , A \ {ai }) zN \{i} ,

and 0 0 zN P(E) zN ⇒ zN\{i} P(RN\{i} , A \ {ai }) zN \{i} ,

Not many SOFs defined up to now satisfy this axiom. On the one hand, in the division problem of an amount Ω of divisible goods, our two main SOFs fail to satisfy Consistency, and, in both cases, it comes from the fact that the withdrawal of an agent typically changes the aggregate endowment, and both RΩlex and REW are sensitive to such changes. On the other hand, some SOFs that do not depend on the aggregate endowment do satisfy this axiom, like RΩ0 lex (which consists in applying the leximin criterion to indices that are built by reference to the fraction of some fixed amount of resources Ω0 that leaves the agent indifferent with her current bundle), defined in Chapter 4 (Section 4.2), and R1 lex , the certainty equivalent leximin SOF, defined in Section 6.2 above (which consists in applying the leximin aggregator to indices that are built by reference to the certain amount of money leaving the agent indifferent with her current possibly uncertain bundle). The combination of Strong Pareto, Independence of Preferences over Unfeasible Bundles and Consistency force us to focus our attention on SOFs that aggregate individual utility levels measured according to the um representation of preferences. Indeed, only preferences over feasible bundles matter, but Consistency, and the withdrawal of agents consuming the current available objects, force us to focus on the bundles that would remain feasible after some agents leave the economy. These bundles are the ones involving the “null” object. As the set of bundles involving the “null” object can be used to build one and only one numerical representation of the preferences, it is not surprising that this representation becomes the one that ends up being used. If we add the fairness requirements of Transfer among Equals and Anonymity among Equals, then only Rmlex satisfies the axioms.

6.6. CONCLUSION

109

Theorem 6.4 On the domain E ind , a SOF R satisfies Strong Pareto, Independence of Preferences over Infeasible Bundles, Consistency, Transfer among Equals, and Anonymity among Equals if and only if it coincides with Rmlex . We could replace Independence of Preferences over Infeasible Bundles and Consistency by Independence of Feasible Set and Separation. Let us insist that Rmlex is the only SOF that satisfies the listed axioms, whereas we typically failed in the previous chapters and sections to single out the leximin from a set of maximin SOFs. That is of course related to the simple structure of the current model, and the fact that the set of bundles that are feasible in each and every economy (the set of bundles composed of money and the “null” objects) is one dimensional. There are many other models involving indivisible goods where fairness questions arise: models where agents can consume more than one object, models where objects are not desirable, matching models, etc. This section suggests that it could be a fruitful exercise to study SOFs in these other models as well.

6.6

Conclusion

Let us briefly take stock of the main insights from this chapter. Although Chapter 2 gave warnings about the necessary caution in the formulation of principles of equality of resources, the subsequent chapters have provided positive results and highlighted two broad approaches. The first, “equivalent”, approach refers to simple benchmark allocations (equal split) and evaluates other allocations by considering benchmark allocations to which every agent is indifferent. The second, “Walrasian”, approach refers to similar benchmark allocations but considers them as defining potential endowments used for market trades. These two approaches remain prominent in the extensions studied in this chapter, although some advantage seems to be given to the former (equivalence) when preferences may be non-convex, when one is looking for an evaluation that is independent of the vector of total resources (as becomes possible with homothetic preferences), or when a particular region of the consumption set offers itself as a natural anchor for the evaluation of bundles, as in the context of risk. This advantage, depending on the context, may or may not compensate for another difference which was described at the end of Chapter 5 and which one may consider as a slight disadvantage, namely the fact that the egalitarian-equivalent SOFs are less neutral with respect to individual preferences. The fact that the two approaches converge in the model of indivisibles studied in the last section is a useful reminder that in some simple contexts, certain ethical dilemmas simply vanish.

110

CHAPTER 6. SPECIFIC DOMAINS

Chapter 7

Extensions 7.1

Introduction

This chapter considers extensions of the analysis in three directions. In Section 7.2, we study how to proceed if one starts with a given allocation rule and would like to extend it into a reasonable ranking of all allocations, that is, a SOF that satisfies, at least, Pareto Indifference and Weak Pareto. That is, we show how one can construct a SOF that is specifically devoted to rationalizing the selections made by any given allocation rule. It will be no surprise that under some mild restrictions, the SOFs RΩlex and REW are indeed the natural candidates if one seeks to rationalize the allocation rules S EE and S EW , respectively. Then, in Section 7.3 we study how the analysis can be extended in order to deal with the evaluation of allocations in different economies. The practice of social evaluation is indeed seldom limited to the comparison of alternative allocations for a given population and economists are often asked to compare allocations of resources involving different populations, such as international comparisons of standards of living or assessment of national growth, inequalities and social welfare over long time periods. We show how such problems require extending the standard social choice approach, and how the SOFs introduced here can be refined to this purpose. Section 7.4 touches on the case in which, contrary to an implicit basic ethical assumption made so far, the agents have private endowments of resources which are considered legitimate. In other words, suppose that the total resources Ω have already been distributed in some fair way, although there may remain some opportunities for mutually beneficial trades among agents. How should one evaluate reallocations of resources following such trades? This section, in particular, shows how the egalitarian SOF REW can be generalized in order to rationalize any Walrasian equilibrium, not just the egalitarian equilibrium. In Section 7.5 we explore the possibility to incorporate dimensions of wellbeing which are not consumptions of resources but correspond to various “functionings” such as health or education. Such functionings may be the direct 111

112

CHAPTER 7. EXTENSIONS

objects of individuals preferences, along material consumption, although they do not fit the current framework because they cannot be transferred across individuals, or because it seems meaningless to add them up to compute total or average amounts. This section shows that the introduction of such additional dimensions of well-being in our approach is possible and once again illustrates the usefulness of the equivalence methodology epitomized by RΩlex .

7.2

Rationalizing allocation rules

In this section, we study how SOFs can be constructed when desirable allocation rules have been identified, and one would like to extend them into fine-grained social orderings. In other words, given an allocation rule S defined on a domain D, we look for a SOF R which rationalizes it in the sense that the allocations in S(E) must always be top ranked by the social ordering R(E). More precisely, let us say that R rationalizes S if for all E ∈ D, S(E) = max|R(E) Z(E). One can consider S(E) ⊆ max|R(E) Z(E) and S(E) ⊇ max|R(E) Z(E) as forms of partial rationalization.1 As explained in Section 4.4, an allocation rule can always be rationalized by a two-tier SOF where selected allocations are socially indifferent to each other and form the first class, and non-selected allocations are also socially indifferent to each other and form the second class. But the resulting social orderings typically fail to satisfy basic properties such as Weak Pareto. Let us introduce two general methods of constructing SOFs from allocation rules. They are both based on a real-valued function computing what could be called the value of an allocation depending on the parameters of the economy and on the allocation rule S to be rationalized. Then, the corresponding SOF is derived from the principle that an allocation is socially preferred if its value is greater. According to the first method, the value of an allocation zN for an economy E = (RN , Ω) is given by the highest real number λ satisfying the property that 0 an S-optimal allocation zN that is Pareto-inferior to zN exists in the economy (RN , λΩ). Formally, this amounts to computing, for all E = (RN , Ω) ∈ D, 0 ∈ S(RN , λΩ) s.t. zi Ri zi0 , ∀ i ∈ N }. VS∗ (zN , E) = sup{λ ∈ R+ | ∃zN

One then derives the corresponding SOF R∗S as follows: 0 0 ⇔ V ∗ (zN , E) ≥ V ∗ (zN , E). zN R∗S (e) zN

The second value function is similar to the first one, except that the supremum is no longer defined with respect to economies (RN , λΩ) but to economies 1 The limitations of partial rationalization are similar to the limitations of partial implementation (on which, see Thomson 1996). It is easy to satisfy S(E) ⊆ max|R(E) Z(E) when R is widely indifferent. And S(E) ⊇ max|R(E) Z(E) is compatible with the correspondence E →→ max|R(E) Z(E) failing to satisfy important properties of S (such as Anonymity).

7.2. RATIONALIZING ALLOCATION RULES

113

0 0 (RN , λΩ) where the choice of RN is simply restricted by the condition that agent i’s indifference sets at zi be the same at Ri as at Ri0 : 0 0 0 ∈ RN , ∃zN ∈ S(RN , λΩ) VS∗∗ (zN , E) = sup{λ ∈ R+ | ∃RN 0 s.t. ∀ i ∈ N, zi Ri zi and I(zi , Ri ) = I(zi0 , Ri0 )}. 2 The corresponding SOF is denoted R∗∗ S . ∗ ∗∗ The value of VS (zN , E) and VS (zN , E) is finite when S is such that for λ great enough, no allocation in S(RN , λΩ) is Pareto inferior to zN . This is always true for instance when S satisfies Pareto efficiency. The following result examines the Pareto and rationalization properties of R∗S and R∗∗ S . 0 ∈ X N , if for all i ∈ N, zi Ri zi0 , Theorem 7.1 1-i) For all E ∈ D, all zN , zN ∗ 0 ∗∗ 0 then zN RS (E) zN and zN RS (E) zN . 1-ii) If S is upper and lower-hemicontinuous3 with respect to Ω and for all E ∈ D, VS∗ (zN , E) < +∞ for all zN ∈ X N , then R∗S satisfies Weak Pareto. 2) If S satisfies Pareto efficiency, then for all E ∈ D,

S(E) ⊆ max|R∗ (E) Z(E) ∩ max|R∗∗ (E) Z(E). S

S

Assume moreover that S is upper-hemicontinuous with respect to Ω and satisfies 0 the following property: For all E ∈ D, all zN ∈ S(E), zN ∈ Z(E), if for all i ∈ N, 0 0 ∗ zi Ri zi , then zN ∈ S(E). Then RS rationalizes S. An immediate consequence of (1-i) is that R∗S and R∗∗ S satisfy Pareto Indifference. Some parts of this theorem are direct consequences of the definitions. For instance, when for all i ∈ N, zi Ri zi0 , necessarily {λ

∈ R+ | ∃qN ∈ S(RN , λΩ) s.t. zi0 Ri qi , ∀ i ∈ N } ⊆ {λ ∈ R+ | ∃qN ∈ S(RN , λΩ) s.t. zi Ri qi , ∀ i ∈ N },

0 0 , E) and therefore zN R∗S (E) zN . A similar which implies VS∗ (zN , E) ≥ VS∗ (zN ∗∗ observation applies to RS (E). It is also easy to see why when S is Pareto efficient, one has VS∗ (zN , E) ≤ 1 for all zN ∈ Z(E). Indeed, if one had VS∗ (zN , E) > 1, by definition there would 0 exist λ > 1 and zN ∈ S(RN , λΩ) such that zi Ri zi0 for all i ∈ N. This is impossible because zN ∈ Z(E) while, as S is Pareto efficient, one would then 0 have zN ∈ P (RN , λΩ). Now, for all zN ∈ S(E), necessarily VS∗ (zN , E) ≥ 1, 2 It is important to define V ∗∗ in terms of sup rather than max because the maximum is S often not defined. This problem, however, vanishes with the alternative, equivalent definition: 0 ∈ S(R0N , λΩ) VS∗∗ (zN , E) = sup{λ ∈ R+ | ∃R0N ∈ RN , ∃zN

s.t. ∀ i ∈ N, U(zi , Ri ) ⊆ U (zi0 , Ri0 )}.

3 A correspondence f is upper-hemicontinuous if x → x, y → y, and y ∈ f (x ) for all n n n n n implies y ∈ f (x). It is lower-hemicontinuous if for all x, all xn → x, all y ∈ f (x), there is yn → y such that yn ∈ f (xn ) for all n.

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which implies that VS∗ (zN , E) = 1 by the previous fact. Therefore, one must 0 0 always have zN R∗S (E) zN when zN ∈ S(E) and zN ∈ Z(E). ∗ ∗∗ The theorem is more precise about RS than about R∗∗ S . Although RS typically satisfies Weak Pareto as well, this involves some continuity condition with respect to the preference profile and will not be studied in detail here. One motivation for introducing R∗∗ S is that it always satisfies Unchanged-Contour Independence because VS∗∗ (zN , E) only depends on the indifference sets at zN . In some sense, the value functions VS∗ (zN , E) and VS∗∗ (zN , E), which refer to fractions of Ω, generalize Debreu’s coefficient of resource utilization. Actually, Debreu’s coefficient corresponds to the value computed by either method for the RU Pareto rule P (E). That is, when S = P, R∗S = R∗∗ . When S = S EE , S = R 0 N ∗ 0 one observes that for all E ∈ E, all zN , zN ∈ X , zN RS (E) zN if and only if min uΩ (zi , Ri ) ≥ min uΩ (zi0 , Ri ) , i∈N

i∈N

R∗S

may satisfy Unchanged-Contour Indepenwhich, incidentally, shows that dence even when VS∗ (zN , E) does not depend only on the indifference sets at 0 EW 0 zN . When S = S EW , zN R∗∗ (E) zN . As a conseS (E) zN if and only if zN R ∗ EE ∗∗ EW quence, RS EE rationalizes S and RS EW rationalizes S . Note that when S = S EE , one has VS∗∗ (zN , E) = 1 for all zN ∈ S EW (E). This is because, for any allocation zN ∈ S EW (E) one can construct indifferences curves that are just below I(zi , Ri ) and cross at a bundle proportional to Ω. EE Therefore, unlike R∗S EE , R∗∗ S EE is an eclectic SOF which includes the S EW optimal and the S -optimal allocations in its top rank: for all E ∈ D, S EE (E) ∪ S EW (E) ⊆ max|R∗∗

S EE

(E)

Z(E).

Let us now illustrate further the two methods by considering other allocation rules. Consider the allocation rule that selects all Pareto-efficient allocations such that every individual is at least as well-off as at the equal-split allocation: Allocation rule Equal Split Efficient S ESE For all E = (RN , Ω) ∈ E, S ESE (E) = {zN ∈ P (E) | ∀i ∈ N, zi Ri Ω/ |N |} . Let E = (RN , Ω) and take any allocation zN ∈ Z(E). Let λ be such that 0 there is zN ∈ P (RN , λΩ) Pareto-indifferent to zN . If for all i ∈ N, zi Ri λΩ/ |N | , 0 then zN ∈ S ESE (RN , λΩ) and therefore VS∗ESE (zN ) = VS∗∗ ESE (zN ) = λ. Therefore, the corresponding SOF coincides with RRU for allocations that are sufficiently egalitarian or inefficient. In contrast, if there is i ∈ N such that λΩ/ |N | Pi zi , then VS∗ESE (zN ) = VS∗∗ ESE (zN ) = |N | mini∈N uΩ (zi , Ri ) . This is 0 0 0 0 proved as follows. Let RN and zN be such that for some α ≥ 0, zN ∈ P (RN , αΩ) and for all i ∈ N, U (zi , Ri ) ⊆ U (zi0 , Ri0 ) . If for all i ∈ N, uΩ (zi0 , Ri0 ) = minj∈N uΩ (zj , Rj ) , then α/ |N | ≤ mini∈N uΩ (zi , Ri ) . If for all i ∈ N, U (zi0 , Ri0 ) = U (zi , Ri ) , then α = λ and therefore α/ |N | > mini∈N uΩ (zi , Ri ) . By continu0 0 ity, there is α such that zN ∈ P (RN , αΩ), for all i ∈ N, U (zi , Ri ) ⊆ U (zi0 , Ri0 ) ,

7.3. CROSS-PROFILES SOFS

115

0 0 and α/ |N | = mini∈N uΩ (zi , Ri ) . One then has zN ∈ S ESE (RN , αΩ), which ∗∗ 0 implies that VS ESE (zN ) ≥ |N | mini∈N uΩ (zi , Ri ) . Imposing RN = RN does not alter this reasoning and therefore VS∗ESE (zN ) ≥ |N | mini∈N uΩ (zi , Ri ) . As one obviously has VS∗ESE (zN ), VS∗∗ ESE (zN ) ≤ |N | mini∈N uΩ (zi , Ri ) , equality is obtained. In other words, the corresponding SOF coincides with REE for allocations that are sufficiently inegalitarian and efficient. In conclusion, one has RU R∗S ESE = R∗∗ and with REE on two S ESE and this SOF coincides with R different subsets that partition the set Z(E). Let us now turn to another allocation rule, namely, the allocation rule that selects the efficient and envy-free allocations:

Allocation rule Envy-free Efficient S EF E For all E = (RN , Ω) ∈ E, S EF E (E) = {zN ∈ P (E) | ∀i, j ∈ N, zi Ri zj } . 0 ∈ S EF E (E) that is ParetoLet E = (RN , Ω) and zN ∈ Z(E). If there is zN ∗ ∗∗ indifferent to zN , then VS EF E (zN ) = VS EF E (zN ) = 1. If zN is not efficient 0 and for some λ < 1 there is zN ∈ P (RN , λΩ) that is Pareto-indifferent to ∗ ∗∗ 0 zN , then VS EF E (zN ), VS EF E (zN ) ≤ λ. If zN is efficient but there is no zN ∈ EF E ∗ S (E) that is Pareto-indifferent to zN , then necessarily VS EF E (zN ) < 1. However, in this case it is possible to have VS∗∗ EF E (zN ) = 1, as illustrated in Figure 7.1. In the figure, by expanding the upper contour set of agent i, one 0 renders zN Pareto-indifferent to an allocation zN which is then envy-free and ∗ efficient. In conclusion, RS EF E rationalizes the allocation rule that contains S EF E and the allocations that are Pareto-indifferent to the allocations of S EF E . This is the smallest allocation rule, with respect to inclusion, that contains S EF E and always treats any pair of Pareto-indifferent allocations in the same way, either selecting both or rejecting both.4 In contrast, R∗∗ S EF E rationalizes a larger allocation rule. These examples have shown that the rationalization of allocation rules is a complex exercise for which there is no unique recipe. The SOFs R∗S and R∗∗ S appear to be prominent candidates but the choice between them may depend on the particular allocation rule under consideration. In a sense, these results support the view that a direct analysis of SOFs is worthwhile and cannot be replaced by the study of allocation rules and their rationalizations.

7.3

Cross-profiles SOFs

The SOFs that are the focus of this book rank all allocations for any given profile of preferences. They provide useful guidelines for the evaluation of public policies, for any given population. But they cannot serve for the comparisons 4 It is easy to refine R∗ so as to rationalize S EF E exactly. It suffices to say that if S EF E 0 is strictly better than z 0 if z I∗ 0 0 zN N S EF E (E) zN and zN is envy-free while zN is not. The N resulting SOF violates Pareto Indifference.

116

CHAPTER 7. EXTENSIONS good 2 6 zi0

s

¡ ¡ ¡zj s ¡ ¡ ¡ ¡ 0

sΩ ¡ ¡ ¡ szi ¡ Ri ¡ ¡ ¡

0 szj Rj

good 1

Figure 7.1: VS∗∗ EF E (zN ) = 1 even if zN is not Pareto-indifferent to an allocation of S EF E (E).

of allocations across time and space, such as comparisons of the situations in a country at different periods, or comparisons of situations in different countries. Both kinds of comparisons are commonly performed by policy-makers who analyze growth statistics, GDP and human development rankings, and they arouse great interest in public debates. The SOFs analyzed so far are not relevant for such comparisons because the populations involved in these comparisons are not the same, and in particular their preferences are typically different. It would be quite useful to be able to define social preferences over allocations that are given to different populations, and this requires an extension of the above analysis. Another motivation for such an extension has to do with the evaluation of allocation procedures. Suppose we want to compare two different institutions that organize the distribution of resources in different ways. Such institutions may be games (such as divide-and-choose mechanisms), competitive equilibria, or second-best policies in which the policy-maker makes use of some information about the statistical distribution of characteristics in the population. The comparison of different procedures is made difficult by the fact that a given procedure may perform well for certain profiles and badly for other profiles. But just in order to record this, one must be able to compare allocations across profiles. Otherwise one cannot even say whether the situation produced by the procedure is more or less satisfactory in various profiles. Whatever the motivation, it should be clear by now that cross-profile comparisons of allocations would be a valuable tool for policy analysis. We do not devote a lot of space to this issue but show in this section that the required extension is not out of reach. Let us first extend the notion of SOF into the notion of a cross-profile social ¯ is now defined simply as a popordering function (CPSOF). An economy E ulation N ⊂ Z++ such that |N | < ∞ and a vector of resources Ω ∈ R++ :

7.3. CROSS-PROFILES SOFS

117

¯ = (N, Ω). Let E denote this domain of economies. A CPSOF R ¯ maps any E ¯ ¯ ¯ E of its domain D into a complete ordering R(E) over “allocations-profiles” (zN , RN ) ∈ X N × RN . The axioms defined up to now can be directly applied to this new setting, observing that they only express requirements about comparisons over pairs 0 (zN , RN ), (zN , RN ) with a same profile RN . For instance, Weak Pareto, without any change in content, now reads: Axiom Weak Pareto ¯ = (N, Ω) ∈ D, and zN , z 0 ∈ X N , RN ∈ RN , if zi Pi z 0 for all i ∈ N , For all E i N ¯ (z 0 , RN ). ¯ E) then (zN , RN ) P( N

The axiomatic analysis of the previous chapters, therefore, remains valid in 0 , RN ) this new setting and yields results about how to rank pairs (zN , RN ), (zN with the same preference profile. It remains to see how to obtain conclusions 0 0 about how to rank pairs (zN , RN ), (zN , RN ) with different preference profiles. In order to do this, the only axiom that we will extend into a requirement over changes of preferences is Unchanged-Contour Independence. In its extended version, it will say that the ranking of two allocations-profiles should only depend on indifference curves at each of these allocations-profiles. This is a very natural extension of Unchanged-Contour Independence in this context, because the motivation is the same as for the original condition, namely, the idea that the evaluation of an allocation in a given profile should only depend on the agents’ indifference curves at this allocation. Axiom Cross-Profile Independence ¯ = (N, Ω) ∈ D, zN , z 0 ∈ X N , RN , R0 , R ˆN , R ˆ 0 ∈ RN , if for all i ∈ N, For all E N N N ˆ i , zi ) = I(Ri , zi ) and I(R ˆ i0 , zi0 ) = I(Ri0 , zi0 ), I(R

¯ (z 0 , R0 ) ⇔ (zN , R ˆ N ) R( ¯ (z 0 , R ˆ 0 ). ¯ E) ¯ E) then (zN , RN ) R( N N N N It is not very difficult to imagine how to extend the definition of the SOFs studied in the previous chapter into CPSOFs. Since these SOFs evaluate an allocation in terms of fraction(s) of Ω (the smallest Ω-equivalent utilities, lexicographically, for RΩlex , the smallest pΩ-equivalent utility for REW ), one can simply proceed similarly with the comparison of such fractions of Ω for different allocations, independently of changes of preferences. This yields the following CPSOFs. ¯ Ωlex ) Cross-Profile SOF Ω-Equivalent Leximin (R 0 N 0 ¯ For all E = (N, Ω) ∈ E, all zN , zN ∈ X , RN , RN ∈ RN ,

0 0 ¯ (zN ¯ Ωlex (E) , RN ) ⇔ (uΩ (zi , Ri ))i∈N ≥lex (uΩ (zi0 , Ri0 ))i∈N . (zN , RN ) R

¯ EW ) Cross-Profile SOF Egalitarian Walrasian (R 0 N 0 ¯ For all E = (N, Ω) ∈ E, all zN , zN ∈ X , RN , RN ∈ RN ,

0 0 ¯ (zN ¯ EW (E) , RN ) ⇔ max min up (zi , Ri ) ≥ max min up (zi0 , Ri0 ). (zN , RN ) R p∈ΠΩ i∈N

p∈ΠΩ i∈N

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CHAPTER 7. EXTENSIONS

One must of course wonder whether there are other reasonable extensions of RΩlex and REW in this context. The answer is negative, as stated in the following result. ¯ coincides with RΩlex (resp. REW ) on singleTheorem 7.2 If a CPSOF R ¯ =R ¯ Ωlex except profile rankings and satisfies Cross-Profile Independence, then R 0 0 possibly over pairs (zN , RN ) , (zN , RN ) such that for all i, j ∈ N, uΩ (zi , Ri ) = ¯ =R ¯ EW ). uΩ (zj , Rj ) = uΩ (zi0 , Ri ) = uΩ (zj0 , Rj ) (resp. R In other words, the only new condition that is needed to extend SOFs into CPSOFs is Cross-Profile Independence, and in this result it is the only requirement involving comparisons of allocation-profiles for different preferences. In order to understand how this axiom operates in this extension, it is important ¯ (zN , R0 ) whento see that it is equivalent to the property that (zN , RN ) ¯ I(E) N 0 ever I(zi , Ri ) = I(zi , Ri ) for all i ∈ N. It is obvious that this property implies Cross-Profile Independence. In order to prove the converse, it suffices to apply 0 the definition of Cross-Profile Independence to the particular case zN = zN , 0 0 ˆ ˆ RN = RN and RN = RN . The axiom then concludes 0 0 ¯ (zN , RN ¯ (zN , RN ), ¯ E) ¯ E) ) ⇔ (zN , RN ) R( (zN , RN ) R(

¯ (zN , R0 ) . I(E) which is equivalent to (zN , RN ) ¯ N As an illustration of how the extension from a SOF to a CPSOF is made with Cross-Profile Independence, consider a two-agent economy {i, j} in which uΩ (zi0 , Ri0 ) = uΩ (zj , Rj ) < uΩ (zi , Ri ) = uΩ (zj0 , Rj0 ). ¯ coincides with RΩlex on single-profile One has to show that, if the CPSOF R ¯ (z 0 , R0 ) . Let z0 = uΩ (zj , Rj )Ω, rankings, then one must have (zN , RN ) ¯ I(E) N N z1 = uΩ (zi , Ri )Ω. The reasoning is illustrated in Figure 7.2. Let Rja be such that I(zj , Rja ) = I(zj , Rj ) and U (z1 , Rja ) = {x ∈ X | x ≥ z1 } . ¢¢ ¡ ¡ ¯ zN , Ri , Ra . By applicaBy Cross-Profile Independence, (zN , RN ) ¯ I(E) j tion of RΩlex , ¡ ¡ ¡ ¢¢ ¡ ¢¢ ¯ (z0 , z1 ) , Ri , Rja . zN , Ri , Rja ¯ I(E) Let Ria be such that I(z0 , Ria ) = I(z0 , Ri ) and

U (z1 , Ria ) = {x ∈ X | x ≥ z1 } . By Cross-Profile Independence, ¡ ¡ ¢¢ ¡ ¢¢ ¡ ¯ (z0 , z1 ) , Ria , Rja . I(E) (z0 , z1 ) , Ri , Rja ¯

By application of RΩlex , ¡ ¡ ¢¢ ¡ ¢¢ ¡ ¯ (z1 , z0 ) , Ria , Rja . I(E) (z0 , z1 ) , Ria , Rja ¯

7.3. CROSS-PROFILES SOFS

119

good 2 6 ­­ ­ ­ ­ s ­ zi ­ Ria ,Rja ,Rib ,Rjb s ­ z1 ­ ­ Ri ­ s ­ zj0 Rj0 ­ ­ ­ ­ ­ zi0 s ­ ­ s ­ zj z0­ s Rj ,Rja ­ ­ Ri ,Ria ­ 0 Ri ,Rib ­ 0 Rj ,Rjb ­ ­ ­ ­ good 1 0 Figure 7.2: Illustration of the extension argument. Let Rib be such that I(z1 , Rib ) = I(z1 , Ria ) and I(z0 , Rib ) = I(z0 , Ri0 ). By Cross-Profile Independence, ¡ ¡ ¡ ¢¢ ¡ ¢¢ ¯ (z1 , z0 ) , Rib , Rja . (z1 , z0 ) , Ria , Rja ¯ I(E) By application of RΩlex , ¡ ¡ ¡ ¢¢ ¡ ¢¢ ¯ (z0 , z1 ) , Rib , Rja . (z1 , z0 ) , Rib , Rja ¯ I(E)

Let Rjb be such that I(z1 , Rjb ) = I(z1 , Rja ) and I(z0 , Rjb ) = I(z0 , Rj0 ). Observe that I(z1 , Rib ) = I(z1 , Rjb ) = I(z1 , Ria ) = I(z1 , Rja ) By Cross-Profile Independence, ¡ ¡ ¡ ¢¢ ¡ ¢¢ ¯ (z0 , z1 ) , Ri0 , Rjb . (z0 , z1 ) , Rib , Rja ¯ I(E)

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CHAPTER 7. EXTENSIONS

By application of RΩlex , ¡ ¡ ¡ ¢¢ ¡ ¢¢ ¯ (z1 , z0 ) , Ri0 , Rjb . (z0 , z1 ) , Ri0 , Rjb ¯ I(E) By Cross-Profile Independence, ¡ ¡ ¡ ¢¢ ¡ ¢¢ ¯ (z1 , z0 ) , Ri0 , Rj0 (z1 , z0 ) , Ri0 , Rjb ¯ I(E)

By application of RΩlex , ¡ ¡ ¢¢ 0 0 ¯ (zN (z0 , z1 ) , Ri0 , Rj0 ¯ , RN ). I(E)

Now, one can apply transitivity over this chain of allocation-profiles, and con¯ (z 0 , R0 ) . clude that (zN , RN ) ¯ I(E) N N We complete this section with the observation that further extension of the concepts is possible. So far we have only extended the notion of SOF so as to cover the possibility of having different preferences in a given population N for a given amount of resources Ω. This is appropriate for the examination of procedures, in terms of how well they perform depending on the profile of preferences. But in the perspective of international or intertemporal comparisons of allocations, which typically involve different populations and different resources, one would actually like to produce a complete ordering R over “allocationseconomies” (zN , RN , Ω). Extending the analysis to different populations N raises the difficult issue of the optimal size of the population.5 There are, however, some contexts in which one simply wants to be neutral about size. For instance, in a comparison between countries, it makes sense to seek such neutrality and to be interested in the distribution of well-being independently of the size of the population. When this is the case, the extension of the above concepts is quite easy. For instance, the definition of RΩlex can be extended as follows: Ωlex

(zN , RN , Ω) R

0 0 0 0 (zN 0 , RN 0 , Ω) ⇔ (uΩ (zi , Ri ))i∈N ≥lex (uΩ (zi , Ri ))i∈N 0 ,

where the leximin for vectors of different dimensions is defined as the application of the standard leximin to replicas of the vectors which have the same dimension. For instance, the vector (2,5,7) can be compared to the vector (2,5) by comparing their replicas (2,2,5,5,7,7) and (2,2,2,5,5,5). This extension can be justified by relying in particular on a new axiom requiring the ranking to be indifferent to replications (i.e., (zN , RN , Ω) I (zN 0 , RN 0 , Ω0 ) whenever (RN 0 , Ω0 ) is a replica of (RN , Ω)). Extending the analysis to resources of different size is also easy, provided that the vectors of available resources are proportional to each other. It is a simple matter of applying Independence of Proportional Expansion to singleprofile rankings. The extended definition of RΩlex , for instance, would then become: (zN , RN , Ω) R 5 On

Ωlex

0 0 0 (zN ⇔ (uΩ (zi , Ri ))i∈N ≥lex (uΩ (zi0 , Ri0 ))i∈N 0 0 , RN 0 , Ω ) ⇔ (uΩ0 (zi , Ri ))i∈N ≥lex (uΩ0 (zi0 , Ri0 ))i∈N 0 .

this topic, see Broome (2004) and Blackorby et al. (2005).

7.4. LEGITIMATE ENDOWMENTS

121

Extending the analysis to vectors of resources which are not proportional to each other, by contrast, is much more problematic, because SOFs like RΩlex and REW do not satisfy Independence of Feasible Set. This problem has already been addressed in Section 4.2 and there is little to add here. Nonetheless, it is worth emphasizing that the extension into an ordering R is quite simple for a SOF that does satisfy Independence of Feasible Set. The analysis of this section applies just as well, for instance, to the extension of Ω0 -Equivalent SOFs which Ω0 lex

refer to a fixed bundle Ω0 . One can then define the ordering R (zN , RN , Ω) R

7.4

Ω0 lex

as follows:

0 0 0 0 0 (zN 0 , RN 0 , Ω ) ⇔ (uΩ0 (zi , Ri ))i∈N ≥lex (uΩ0 (zi , Ri ))i∈N 0 .

Legitimate endowments

Throughout this book we generally assume that agents’ endowments are not a legitimate source of inequality, adopting a “pure” view of fairness and equality. In this section, however, we show that the approach is flexible enough to accommodate quite different perspectives. Suppose that the agents have personal initial endowments of resources and that, for some reason, such endowments are considered legitimate, so much so that the purpose of social policy is no longer to redistribute toward more equality, but simply to achieve a proper sharing of the advantage obtained by agents over and above their legitimate initial position. Let ω i ∈ X denote agent i’s endowment, assuming that X ω i = Ω. i∈N

The framework is now slightly different, with an economy being defined as E = (RN , ω N ). The domain is now denoted E E and contains economies E = (RN , ω N ) such that RN ∈ R|N | and for all i ∈ N, ω i ∈ R++ . Notice that positive endowment of every good is required, so as to avoid situations in which an agent considers his endowment to be so bad that any multiple of his endowment would still be worse than his current consumption zi . The adaptation of the SOFs studied above to this different setting is actually quite simple. Recall that the central concepts are the Ω-equivalent utility function uΩ (zi , Ri ) defined by zi Ii uΩ (zi , Ri )Ω and the money-metric utility function up (zi , Ri ) computed at prices p such that pΩ = 1 : up (zi , Ri ) = min {px | x Ri zi } = min {px/pΩ | x Ri zi } . The reference to Ω in these formula may in fact be interpreted as pointing to the premise that the appropriate endowment of agents in the fairness ideal should be the equal split Ω/ |N |. If the current endowment ω i is actually the legitimate

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one, then one simply has to replace Ω in the formula by ω i , so as to obtain new functions: uωi (zi , Ri ) such that zi Ii uωi (zi , Ri )ω i , up (zi , Ri )/pω i = min {px/pωi | x Ri zi } . (Replacing Ω by ω i instead of |N | ω i involves a change of scale which has no consequence–priority is given here to simplicity of the formulae.) Applying the same definitions of SOFs as above to these new functions yields the following two SOFs in particular. Social ordering function ω-Equivalent Leximin (Rωlex ) 0 For all E = (RN , ω N ) ∈ E E , zN , zN ∈ XN , 0 ⇔ (uωi (zi , Ri ))i∈N ≥lex (uωi (zi0 , Ri ))i∈N . zN Rωlex (E) zN

n o P Let ∆ = p ∈ R+ | k=1 pk = 1 .

Social ordering function Walrasian (RW ) 0 For all E = (RN , ω N ) ∈ E E , zN , zN ∈ XN , 0 ⇔ max min up (zi , Ri )/pω i ≥ max min up (zi0 , Ri )/pω i . zN RW (E) zN p∈∆ i∈N

p∈∆ i∈N

Again, one can interpret the first one as being more focused on securing that for every i ∈ N, zi Ri ω i , and hence tends to favor the agents whose preferences are oriented toward the combination of goods featuring in their endowment. In contrast, the other one, which selects the subset of Walrasian allocations in Z(E), is more sensitive to the degree of efficiency of allocations and is more neutral regarding individual preferences. The fact that, for all E ∈ E E , max|RW (E) Z(E) coincides with the set of ∗ Walrasian equilibria to be fully understood. Let zN be a Walrasian P deserves ∗ ∗ ∗ equilibrium, i.e., i∈N zi = Ω and for some p ∈ ∆ , for all i ∈ N, zi ∈ ∗ max|Ri B (ω i , p ) . This implies that for all i ∈ N, up∗ (zi∗ , Ri ) = p∗ ω i . Take any vector p ∈ ∆ and any allocation zN ∈ Z(E). Necessarily,6 Ω∈

X

U (zi , Ri )

i∈N

6 Recall from the introduction that the sum of sets used in the expression is defined as follows: ⎫ ⎧ ⎬ ⎨ U (zi , Ri ) = x ∈ X | ∃xN ∈ U (zi , Ri ), x = xi . ⎭ ⎩ i∈N

i∈N

i∈N

7.4. LEGITIMATE ENDOWMENTS and

X

i∈N

so that

(

123

up (zi , Ri ) = min pq | q ∈ X

i∈N

X

i∈N

)

U (zi , Ri ) ,

up (zi , Ri ) ≤ pΩ.

P Recalling that Ω = i∈N ω i , this implies that for some i ∈ N, up (zi , Ri ) ≤ pω i and also that it is impossible to have up (zi , Ri ) ≥ pω i for all i ∈ N with a strict inequality for some i. As a consequence, max min up (zi∗ , Ri )/pω i = 1 ≥ max min up,ωi (zi , Ri )/pω i ,

p∈∆ i∈N

p∈∆ i∈N

∗ or equivalently, zN RW (E) zN . Now, suppose that for all i ∈ N,

up (zi , Ri )/pω i = 1. This implies that

X

up (zi , Ri ) = pΩ,

i∈N

and therefore that zN is Pareto-efficient, with supporting price vector p. One then has zi ∈ max|Ri B (ω i , p) for all i ∈ N, i.e., zN is a Walrasian equilibrium associated to the price vector p. It is worth emphasizing that RW does not require any redistribution and rationalizes the competitive equilibrium even though it is based on the maximin criterion. This illustrates our claim of Section 3.5 that even a libertarian policy can be accommodated in our approach. It all depends on how individual situations are measured and compared. The money-metric utilities up (zi , Ri )/pω i take account of legitimate endowments and compute the fraction of one’s endowment which would, with possibility of trade, yield the current level of satisfaction. Putting priority on the agents for whom this fraction is the smallest is a natural way to rationalize the competitive approach. We mention a variant of the Walrasian SOF. In the definition, one could replace the expression up (zi , Ri )/pω i by the expression up (zi , Ri ) − pω i , which computes the difference between minimal expenditure and the value of the endowment at price p. Like RW , the new SOF obtained in this way also selects Walrasian allocations. One drawback of this variant is that, for the ranking of suboptimal allocations, it is not independent on the choice of price normalization (i.e., p ∈ ∆ or otherwise) used in the computation of maxp mini∈N [up (zi , Ri ) − pω i ] . But its attraction lies in the fact that it appears very natural to compute the difference between the “equivalent value” of zi , i.e., up (zi , Ri ), and the value of ω i . This measures how much the agent gains or loses with respect to an ordinary Walrasian situation (at hypothetical prices

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p). This can easily be connected to standard notions of cost-benefit analysis, as it has been explained in Section 1.7. An additive variant of Rωlex can also be conceived. Consider the function uωi ,Ω (zi , Ri ) such that zi Ii ω i + uωi ,Ω (zi , Ri )Ω. Applying the leximin criterion to such utilities amounts to comparing individual situations in terms of fractions of Ω that are added to or subtracted from individual endowments. When legitimate endowments are the equal split bundle Ω/ |N | , this SOF coincides with RΩlex , just like Rωlex . We do not explore the axiomatic analysis of SOFs for this framework, as our purpose here is simply to hint at the possibility to deal with legitimate endowments. A detailed study of this setting would require more than a section.

7.5

Functionings

In most of this book we focus on allocations of resources. In the next part leisure will come to the front stage. This is a rather special kind of resource as it is not perfectly transferable across individuals. In this section, more broadly, we want to examine the possibility to extend the SOF approach to settings in which, beyond material consumption of goods and services, there are various “functionings” which matter to individuals, such as health, education, safety, social status, and so on. Some of these functionings may be purely personal while others may involve interactions between individuals. The notion of functionings has been popularized by A. Sen7 and, in the most general sense, covers all “doings and beings” that pertain to an individual life. The central part of the SOF approach is to find ways to compare indifference sets across individuals. The RΩlex SOF compares individual situations in terms of Ω-equivalent utilities, while the REW evaluates them in terms of moneymetric utilities–in both cases, indifference sets are the relevant informational basis of interpersonal comparisons. In this light, there is no particular difficulty in extending the analysis to non-material functionings, provided that individual preferences are well-defined over these functionings alongside material consumption. Suppose that for each i ∈ N, Ri bears on pairs (zi , fi ) , where zi ∈ X ⊂ R+ is, as previously, a vector of consumptions and fi ∈ F ⊂ Rm is a vector of functionings. Preferences are assumed to be continuous in (zi , fi ) and monotonic and convex in zi . For simplicity, let us assume that feasible functioning vectors in any given economy depend only on the available resources Ω. An important observation is that, even if some functionings may reflect social interactions, it remains sensible to evaluate individual situations in terms of what happens to the individual rather than in holistic terms. Taking account of social interactions does not require abandoning the basic principle that individuals are the primary units of evaluation. 7 See,

e.g., Sen (1985, 1992).

7.5. FUNCTIONINGS

125

In this enlarged framework, there is no particular difficulty in formulating axioms like the various Pareto requirements, Unchanged-Contour Independence, Separability, and so on. The transfer axioms are less obvious to adapt because transfers of functionings across individuals may appear less natural than transfers of resources. Fortunately, important parts of our analysis and results remain valid even if one restricts transfers to resources only. For instance, the axiom of Transfer among Equals can be rewritten as follows. The idea is to restrict the transfer axiom not only to cases of equal preferences, but also to cases in which the two individuals have the same functioning vectors,too. Axiom Transfer among Equals 0 0 For all E = (RN , Ω) ∈ D, and (zN , fN ) , (zN , fN ) ∈ X N × F N , if there exist j, k ∈ N such that Rj = Rk and ∆ ∈ R++ such that zj − ∆ = zj0 À zk0 = zk + ∆, fj = fj0 = fk0 = fk ,

0 and for all i 6= j, k, (zi , fi ) = (zi0 , fi0 ), then zN R(E) zN .

It is then an easy exercise to adapt Theorem 3.1 and show that Priority among Equals follows from Transfer among Equals combined with Pareto Indifference and Unchanged-Contour Independence. As a matter of fact, by Pareto Indifference the restriction fj = fj0 = fk0 = fk does not bite much and one 0 can therefore show that zN R(E) zN when the two individuals ¡ ¢ affected by the change, j and k, are such that Rj = Rk and (zj , fj ) Pj zj0 , fj0 Pj (zk0 , fk ) Pk (zk , fk ) . In other words, absolute priority is given to the individual whose indifference set is the lower. The reference to equal split, which plays a central role in the fair distribution model as it underlies the definition of RΩlex and REW , now becomes more problematic. If some individuals are affected by inequalities in functionings, seeking to equalize their material consumption may run against the most basic sense of fairness. However, suppose that the ideal of equality of resources remains sensible when all individuals enjoy the same level of functionings, at least at a particular reference level f ∗ . Then the analysis is easily adapted because by Pareto Indifference every allocation (zN , fN ) in a certain subset is ∗ ∗ Pareto equivalent to an allocation (zN , fN ) in which fi∗ = f ∗ for all i ∈ N. For such allocations (zN , fN ), one can then deduce a reasonable ranking from the ∗ application of RΩlex or REW to the “equivalent” allocations of goods zN . Ωlex Ωf ∗ lex into a R SOF which comMore precisely, this means extending R pares individuals in terms of Ωf ∗ -equivalent utilities uΩf ∗ ((zi , fi ) , Ri ) defined by: (zi , fi ) Ii (uΩf ∗ ((zi , fi ) , Ri )Ω, f ∗ ) . ∗

Similarly, one extends REW into a Rf EW SOF which relies on the computation of max min {px | (x, f ∗ ) Ri (zi , fi )} . p∈ΠΩ i∈N

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CHAPTER 7. EXTENSIONS

It is a consequence of the basic tension between transfer principles and the Pareto principle identified in Theorem 2.1 that this extension must rely on a single reference vector f ∗ and cannot involve a greater number of vectors f ∗ . This raises a new ethical problem, namely, the choice of the vector f ∗ . This problem is not, however, totally new because the Ω-equivalent utilities underlying RΩlex also rely on a reference bundle Ω. What triggers the choice of this particular bundle in our analysis? It derives from axioms like EqualSplit Transfer or Proportional Allocations Transfer, which say that a certain type of equality of resources is desirable even between individuals with different preferences. The same kind of considerations apply here. The choice of f ∗ must be dictated by the fact that it is when all individuals enjoy fi = f ∗ that the ideal of equality of material consumption is sensible. As an example, consider health. When all individuals have the same level of health, is it desirable to equalize their levels of consumption? Not necessarily. In case of bad health, some of them may need a greater compensation if health is more important according to their preferences than according to others’ preferences. Now, if all enjoy a perfect health, it seems quite sensible to seek to equalize their material consumption, and it would sound somewhat incongruous to consider that those who care less about health should consume more than the others. Such a view imposes good health as the proper reference level for health because it is only when everyone enjoys good health that individual preferences over health-consumption trade-offs can be disregarded. Not all dimensions of functionings provide such an obvious reference, but at least it should be clear that the need for a particular f ∗ is not an insurmountable obstacle for the application of this approach.8

7.6

Conclusion

Our study of the distribution of unproduced goods is now coming to a close. In this chapter we have hinted at more radical generalizations of the approach, which are relevant beyond the distribution problem and have been discussed here for convenience. The notion of cross-profile SOF is essential for the comparison of social welfare across countries or over periods of time and should become an important object of study for social choice theory in addition to the traditional SOF. The idea of taking account of legitimate endowments is one example of the possibility to vary the set of personal characteristics for which the agents are held responsible and which are therefore not subject to redistribution or compensation by other goods. Delineating this “sphere of responsibility” is an important ethical problem which has far-reaching consequences over redistributive policies. The importance of this issue will appear again in the next part. We have stressed in the introduction that each particular context requires 8 Fleurbaey and Gaulier (2009) apply this approach in order to make international comparisons of living standards that take account of leisure, life expectancy, household size, and the risk of losing one’s job.

7.6. CONCLUSION

127

a specific analysis because fairness principles are typically special to the form of redistribution, the kind of resources and dimensions of well-being, and the way in which resources are produced. The next part will tackle the specific questions pertaining to production, which are essential for applications to income taxation. We have nonetheless devoted many chapters to the distribution of unproduced goods because this setting provides the basic elements in which key concepts can be introduced without undue complication. In particular, the last section of this chapter has shown that extensions to non-material aspects of well-being are easy to conceive in principle (if not in practice) because the basis of the SOF approach is the possibility to compare individual situations in terms of indifference sets. Precise ways of making interpersonal comparisons of indifference sets may depend a lot on concrete aspects of the context (as shown by the example of health), but the key assumption that individuals have well defined preferences over the relevant dimensions of their own well-being is all that is needed at the most basic level.

128

CHAPTER 7. EXTENSIONS

Part III

Production

129

131 How does the study of the fair division problem carried out in Part II generalize to problems involving production? This is the topic of this Part. More precisely, we consider now that the economy is endowed with a stock of private goods that can be used to produce other commodities, which can be public or private goods. In Chapter 8 and the beginning of Chapter 9, the initial stock of private goods is legitimately owned by the agents themselves, and the problem boils down to identifying fair ways of jointly using the production technology. In Chapter 8, the commodity that can be produced is a public good, whereas it is a private one in Chapter 9. The general picture that will emerge from these chapters is, first, that the specific features of the problems involving production allow us to define several specific fairness conditions, which, in turn, yield specific social ordering functions. We take it as an illustration that the approach we propose is quite flexible and takes advantage of the richness of a model to offer richer possible definitions of what is fair. Secondly, the developments in these chapters will also confirm some key conclusions of the previous chapters, and, in particular, will confirm that our axiomatic analysis ends up justifying social ordering functions of the maximin type. In the last section of Chapter 9, we will drop the assumption that agents are the legitimate owners of some stock of private goods, and we will replace it with the assumption that all agents have an identical claim on the resources of the economy. Dropping any restriction on the number and the nature (public versus private) of the commodities, we will axiomatize a social ordering function inspired by what we axiomatized in the previous chapters. The lesson of this last exercise is that the approach presented in this book can be used to compute general indices of well-being. In Chapter 10, we carry on with the assumption that agents are not the legitimate owner of the inputs of the economy, but we formalize it in a completely different way. The story that best fits that model is that inputs are quantities of labor, the individuals having unequal productivities, and it is considered that they should not bear the consequences of their unequal endowments in productive skills. This chapter is a central one in the development of our approach, as it shows that our objective of resource equality can be adapted to situations involving internal resources (here, the productive skills). We then try to define our social objective by considering that external resources should be allocated unequally to compensate the agents suffering from a poor endowment in internal resources. The resulting model will be richer than the ones studied before, and the richest one in the whole book. Consequently, a long list of plausible fairness properties will appear, and a large part of the work will consist in identifying the trade-offs between fairness axioms. All the second-best applications associated with that model will be gathered in Chapter 11, the last chapter of this part. Indeed, given that the model is a canonical one to study the taxation of earnings, we will devote an entire chapter to studying the consequences of our social preferences on the shape of the tax function. It will also be a good opportunity to illustrate the difference between our second-best applications and the typical public economics approach

132 to second-best allocation in terms of utilitarian social objectives. In particular, some taxation policies will receive a strong support from our analysis, whereas they are hard to justify from a classical utilitarian viewpoint. That will conclude Part III.

Chapter 8

Public good 8.1

Introduction

Assume agents share a technology in common. The technology can be used to transform quantities of a private good that is currently owned by the agents into a public good (that is, the consumption of the good is non-rival: let us think of a software jointly developed by several firms, a public facility produced by members of a community, transportation infrastructures, etc.). How much should each agent be asked to contribute to the production, and how much should be produced? Also, assuming it is possible to exclude some agents from the consumption of the public good (let us think of hours of TV programmes, or access to a public facility), should that possibility be used, and if so, how much of the public good should each agent be allowed to consume? These are the questions we raise in this chapter.1 Even if the model we study here remains simple (there is one private good and one public good), the new ingredient we introduce, that is, the production technology, will turn out to sufficiently enrich the model so as to enable us to study new fairness properties. Two axioms are introduced below. The first one is based on the comparison of what an agent gets and what she would get if she were alone in the economy, and could use the production technology by herself. When the good to produce is a public good, being alone in the economy is a bad situation, actually the worse situation one can think of. The stand alone situation sets an intuitive (lower) benchmark. An obvious inequitable situation would be one in which one agent is worse off than at the benchmark whereas one other agent is better off. We study below the condition saying that in such situations, a transfer of resource from the latter agent to the former agent should be a social improvement. We call this condition Stand-Alone Transfer. The second equity condition we introduce is based on the idea that each agent should contribute to covering the production cost, or, at least, so should 1 This chapter is largely inspired by Maniquet and Sprumont (2004) and (2005), and the second-best application draws from Maniquet (2007).

133

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CHAPTER 8. PUBLIC GOOD

do each agent who consumes the produced good. Consequently, if two agents consume the same quantity of the produced good, but one of them contributes negatively (namely, she receives some amount of the input from the others) whereas the other one contributes positively, then a transfer from the former to the latter should be a social improvement. We call this condition Free-Lunch Transfer. We complement these two equity conditions with a third one reminiscent of the Equal-Split Transfer condition introduced in Chapter 2. It turns out that Stand-Alone Transfer and Equal-Split Transfer are easy to combine, whereas either one is incompatible with Free-Lunch Transfer. These three equity conditions force us to focus our attention on two social ordering functions. Unsurprisingly, they both are of the maximin type. This confirms one of the main lessons which could be drawn from Parts I and II: only an infinite inequality aversion is compatible with weak equity conditions and robustness conditions. One of the social ordering functions which ends up being characterized is quite similar to the Ω-Equivalent Leximin of Part II, with the social endowment Ω replaced by the cost function: the individual well-being levels are now calibrated with respect to the welfare an agent would reach if she were given free access to “fractions” of the cost function. The second social ordering function uses another way of calibrating individual well-being levels, that is, one now has to look at the quantity of the produced good, which, when consumed for free (that is, with a zero contribution) leaves an agent indifferent to her assigned bundle. Again, the ordering of allocations works by maximizing the minimal calibrated level of well-being. After having presented the model (Section 2), the equity axioms (Section 3) and the two prominent SOFs (Section 4), we show in Section 5 how the results extend to the case where the produced commodity is a pure public good. Then, we develop in Section 6 simple second-best applications where we compare the optimal choice associated with our two SOFs when exclusion is possible and when it is not.

8.2

The model

The new ingredient in this chapter is that one desirable public good is not initially available, but can be produced from one private good. The cost function C : R+ → R+ is strictly increasing, strictly convex, C(0) = 0, and limg→∞ C(g)/g = ∞.2 The set of all cost functions satisfying these conditions is denoted by C. All agents are assumed to be endowed with some quantity of the private good. Moreover, we consider that the quantity they will eventually be asked to contribute is small compared with their endowment. Consequently, we 2 The last assumption guarantees that, independently of the number and preferences of the agents, the set of feasible allocations that Pareto dominate the no activity allocation (that is, the allocation in which all agents consume their endowment and no public good is produced) is compact.

8.2. THE MODEL

135

do not need to specify the quantity of private good they begin with, and we only measure their contribution without imposing an upper bound on it. The implicit assumption is that the distribution of wealth before production is fair. This assumption allows us to disentangle two issues: wealth redistribution and sharing production costs. This chapter is only concerned with the latter. Bundles are pairs zi = (ti , gi ) ∈ R × R+ , where ti denotes agent i’s contribution to the production cost, and gi her consumption of the public good. Individual preferences are assumed to be continuous, strictly decreasing in the private good contribution level ti , strictly increasing in the produced good consumption level gi , and convex. Let R denote the set of preferences satisfying these conditions. An economy is now a list E = (RN , C), where N is a non-empty finite subset, RN ∈ RN , and C ∈ C. Let E denote the set of such economies, and D the domain of definition of the SOF to be constructed. As far as the allocations to be ranked by the SOF are concerned, we will restrict our attention to allocations, which we call admissible, where no agent obviously contributes too little, or too much, that is, there exists a production level g such that, for each i ∈ N, (0, g) Ri zi Ri (0, 0). The first part of the admissibility condition says that no agent prefers her bundle to the opportunity of consuming an arbitrarily large quantity of the produced good for free. The second part means that everyone agrees to participate (not participating would mean not contributing but not consuming the produced good either), that is, everyone gets a non-negative share of the surplus generated by the production. We denote the set of admissible allocations by A(RN ), and the set of admissible bundles for agent i ∈ N by Ai (Ri ) . An allocation zN is feasible for an economy E = (RN , C) if and only if X ti ≥ C (max{gi }i∈N ) . i∈N

Let Z(E) denote the set of feasible allocations. This definition of feasibility assumes that different agents may consume different levels of the public good and therefore that exclusion is possible. Let us note that an allocation with exclusion can never be Pareto optimal when preferences are strictly monotonic with respect to the public good, because, starting from an allocation with exclusion, it costs nothing to let every agent consume the full quantity of the public good. On the other hand, it has been proven that exclusion may alleviate the free-rider problem, that is, when one takes incentive compatibility constraints into account, threatening to exclude an agent from consuming fractions of the available quantity of the public good may help design the incentives properly (see Moulin 1994). Consequently, the set of allocations among which the policymaker will be likely to have to choose may contain allocations with exclusion. We come back to this issue in our second-best applications of Section 6. Now, we study the construction of SOFs over admissible allocations in economies satisfying the above definitions. This is illustrated in Figure 8.1,

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CHAPTER 8. PUBLIC GOOD

where two allocations are described in a two-agent economy. Allocation (z1 , z2 ) is such that agent 1 consumes strictly more of the public good than agent 2, who is, therefore, partially excluded. Allocation (z10 , z20 ) has no exclusion, but agent 1 contributes negatively, that is, receives a strictly positive amount of the private good (in addition to the amount she owns at the beginning). How to rank these two allocations? That is the question we answer in this chapter. C t6 s z1 z20

z10

s

s

R1

R2 z2 s

g

Figure 8.1: A two-agent public-good economy, and two allocations. Several axioms introduced in Part I are easily adapted to the current setting. The Pareto axioms do not need to be rewritten. Nor do the axioms of Unchanged-Contour Independence or Separability, as they do not involve any reference to the social resources. Even some fairness axioms, like Transfer among Equals or Nested Transfer are readily adapted. On the other hand, Equal-Split Transfer or Proportional Allocations Transfer were specific to the fair division problem. Maybe they could yield similar axioms in the current model but this needs a careful discussion which we develop in the next section. The only robustness condition which involves changes in the resources is Replication. When an economy is replicated, it must hold that an allocation is feasible in the initial economy if and only if the replicated allocation is feasible in the replicated economy. This requirement yields the following definition of the replica of a cost function. The production cost of a given amount of the public good is r times more expensive in the r-replica of the economy: Cr is a

8.3. THREE AXIOMS

137

r-replica of C if for all g ∈ R+ , Cr (g) = rC (g) . Consequently, E 0 = (RN 0 , Cr ) is a r-replica of E = (RN , C) if Cr is a r-replica of C, and, as before, there exists a mapping γ : N 0 → N such that for all i ∈ N, |γ −1 (i)| = r and for all j ∈ γ −1 (i), Rj = Ri .

8.3

Three axioms

Should agents consuming the same level of the public good contribute the same amount of the private good? One may think of it as an evident fairness property: resource equalisation requires that we ask agent to contribute identically. That would give us the following axiom. A transfer of private good between two agents is a weak social improvement if they both have the same quantity of the public good and the beneficiary of the transfer contributes strictly more than the other, before and after the transfer. Axiom Transfer 0 For all E = (RN , C) ∈ D, zN , zN ∈ A (RN ) , ∆ ∈ R++ , and j, k ∈ N such that 0 0 gj = gj = gk = gk , if tk − ∆ = t0k > t0j = tj + ∆ 0 R(E) zN . and for all i 6= j, k, zi0 = zi , then zN

As we know from Chapter 2, for any sufficiently rich domain D such an axiom conflicts with Pareto axioms. Does that mean that we have to forgo the general ideal of resource equality? As a matter of fact, the public good case offers a nice illustration of why an axiom such as Transfer is not compelling. Indeed, assume agent k, in the comparison of the two allocations contemplated in the axiom, has a high willingness to pay for the public good, whereas j has a low willingness to pay, and assume further that the production level of the public good is large, in better harmony with k’s demand than j’s demand. Given that they both consume the same quantity of the public good, which, for Pareto efficiency reasons, should be good news, it seems natural to request a greater contribution from k than from j. Conversely, if k has a lower willingness to pay but the produced quantity is small, closer to k’s aspiration, again it may be fair to ask k to pay more, and compensate j for the fact that her consumption of the public good is low. For these reasons, equalizing resources physically is not the most appropriate fairness objective. That is why we have to define axioms that are logically weaker than Transfer, and that embody a more compelling notion of fairness. The first two fairness axioms we introduce in this section follow the idea that individual resource distribution should be given some bounds, or more precisely, that the value agents give to resources, as a function of their preferences, should be given some bounds. Imposing lower and upper bounds is consistent with the idea that, on the one hand, all agents should be provided with a “safety net”

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and, on the other hand, no one should reach an excessive well-being level at the expense of others.3 Define the stand-alone satisfaction level of an agent as the level she would enjoy at an efficient allocation if she were the only agent in the economy. In the non-rival environment we are considering, being alone is not desirable because the cost of producing the public good cannot be shared with others. The standalone satisfaction level should be an agent’s lower welfare bound–this fairness principle was introduced by Moulin (1987). Somehow similarly, we may define the unanimity satisfaction level of an agent as the level she would enjoy at an efficient allocation if the others had the same preferences as hers and everyone were treated equally. In public good environments, disagreeing constitutes a social burden. Indeed, if all agents actually have identical preferences, they are all able to enjoy their unanimity satisfaction level. But as soon as preferences differ, no feasible allocation provides their unanimity satisfaction level to all agents. The unanimity satisfaction level should be an agent’s upper welfare bound–this fairness principle was introduced by Sprumont (1998). We apply these bounds in our context as follows. Suppose that an agent enjoys an “insufficient” satisfaction level, i.e., she is strictly worse off than at her stand-alone level. Then a private good transfer to her from an agent strictly better off than his own stand-alone level should be viewed as a social improvement. This is the Stand-Alone Transfer axiom. Likewise, suppose that an agent enjoys an “excessive” satisfaction level, i.e., she is strictly better off than at her unanimity level. Then a private good transfer from her to an agent who is strictly worse off than at his own unanimity level is a social improvement. That is the Equal-Split Transfer axiom. Both axioms express a form of welfare inequality aversion, but, again, a rather limited one. Both axioms can be related to the Equal-Split Transfer condition of Chapters 2 and 5. Indeed, in the fair division model of Part I, equal split can be justified from two different viewpoints: on the one hand, it is a natural safety net, and, on the other hand, it corresponds to the bundle an agent would consume if all other agents had the same preferences as hers. When a decision about how much to produce of a public good has to be taken, those two ideas give different axioms, but two axioms that, as it is proved below, turn out to be compatible with each other. Formally, if Ri ∈ R and C ∈ C, by a slight abuse of notation, we let max|Ri C denote the unique4 bundle zi = (ti , gi ) in max|Ri {(t, g)|t ≥ C(g)}. Our axioms read as follows. Note that a transfer of private good corresponds to a “transfer” of contribution in the opposite direction. Axiom Stand-Alone Transfer 0 For all E = (RN , C) ∈ D, zN , zN ∈ A (RN ) , ∆ ∈ R++ , and j, k ∈ N such that 3 The introduction of welfare lower and upper bounds in the literature on fairness is largely due to Moulin. See in particular Moulin (1991) and Moulin (1992). 4 Uniqueness follows from strict convexity of C.

8.3. THREE AXIOMS

139

gj = gj0 , gk = gk0 , if max|Rk C Pk zk0 = (tk − ∆, gk ) Pk zk and zj Pj zj0 = (tj + ∆, gj ) Pj max|Rj C 0 and for all i 6= j, k, zi0 = zi , then zN R(E) zN .

Axiom Equal-Split Transfer 0 For all E = (RN , C) ∈ D, zN , zN ∈ A (RN ) , ∆ ∈ R++ , and j, k ∈ N such that 0 0 gj = gj , gk = gk , if max|Rk (C/ |N |) Pk zk0 = (tk − ∆, gk ) Pk zk and zj Pj zj0 = (tj + ∆, gj ) Pj max|Rj (C/ |N |) 0 and for all i 6= j, k, zi0 = zi , then zN R(E) zN .

Actually, the close relationship between the two ideas is confirmed formally by Lemma 8.1 below, which requires a technical step that we develop now. For the sake of completeness, let us state the conditions of Separation and Replication in the current context. Separation requires that if an agent has the same bundle in two allocations, then removing this agent from the economy does not alter the social ranking between the two allocations. Axiom Separation 0 For all E = (RN , C) ∈ D with |N | ≥ 2, and zN , zN ∈ A (RN ) , if there is i ∈ N 0 such that zi = zi , then 0 0 ⇔ zN \{i} R(RN\{i} , C) zN zN R(E) zN \{i} .

Replication requires that in a replicated economy, the social ranking between replicas of allocations coincides with the social ranking of the original allocations in the original economy. Axiom Replication 0 For all E = (RN , C) ∈ D, zN , zN ∈ A (RN ), and r ∈ Z++ , if E 0 = (RN 0 , Cr ) ∈ D 0 0 is a r-replica of E, then zN R (E) zN ⇔ zN 0 R (E 0 ) zN 0. As we know from Lemma 4.1, combining Separation and Replication leads to some independence of the feasible set in the sense that multiplying or reducing the available resources should not affect social rankings. In the current context, any SOF satisfying Separation and Replication ranks allocations in the same way if the cost function is C or if it is qC, for any rational number q > 0. These irrelevant multiplication operations allow us to change the premisses of Stand-Alone Transfer into the premisses of Equal-Split Transfer, so that the two axioms become equivalent.

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Lemma 8.1 On the domain E, if a social ordering function satisfies Separation and Replication, then it satisfies Stand-Alone Transfer if and only if it satisfies Equal-Split Transfer. The proof mimics that of Lemma 4.1 and is omitted. Thanks to the possibility of multiplying the cost function by any positive rational number, implied by the combination of Separation and Replication, one can also deduce the following lemma. Lemma 8.2 On the domain E, if a social ordering function satisfies Separation, Replication and either Stand-Alone Transfer or Equal-Split Transfer, then it satisfies Nested-Contour Transfer (and, consequently, Transfer among Equals). 0 Proof. Let E = (RN , C) ∈ E, zN , zN ∈ A (RN ) be such that for ∆ ∈ R++ , and 0 0 j, k ∈ N such that gj = gj , gk = gk ,

t0j = tj + ∆,

t0k = tk − ∆,

U (zj0 , Rj ) ∩ L(zk0 , Rk ) = ∅

and for all i 6= j, k, zi = zi0 . As U (zj0 , Rj ) ∩ L(zk0 , Rk ) = ∅, there is q ∈ Q++ such that max|Rk qC Pk zk0 0 and zj0 Pj max|Rj qC. Therefore, by Stand-Alone Transfer, zN R(RN , qC) zN . As the ranking of allocations is unchanged when the cost function is rescaled 0 in this way, one obtains zN R(E) zN . This proves Nested-Contour Transfer. Transfer among Equals is implied by Nested-Contour Transfer (it corresponds to Nested-Contour Transfer restricted to the case in which Rj = Rk .) Finally, we turn to our third axiom, Free-Lunch Transfer. Consider a profile RN and an allocation at which two agents, j and k, consume the same quantity of the public good. Suppose that k’s private good contribution is positive but j’s contribution is negative. Since j enjoys a “free lunch”, a transfer of private good from j to k that does not reverse the signs of their contributions should be deemed to increase social welfare. The axiom reads as follows. A transfer of private good between two agents is a weak social improvement if they both have the same quantity of the public good and the beneficiary has a strictly positive contribution whereas the donor has a strictly negative contribution, before and after the transfer. Axiom Free-Lunch Transfer 0 For all E = (RN , C) ∈ D, zN , zN ∈ A (RN ) , ∆ ∈ R+ , and j, k ∈ N such that 0 0 gj = gj = gk = gk , if tj + ∆ = t0j < 0 < t0i = ti − ∆ 0 and for all i 6= j, k, zi0 = zi , then zN R(E) zN .

As it might be expected by now, no SOF satisfies Weak Pareto, FreeLunch Transfer and either Stand-Alone Transfer or Equal-Split Transfer. Figure 8.2 shows an economy E = (RN , C), N = {1, 2} , and four allocations

8.4. TWO SOLUTIONS

141

0 00 000 zN , zN , zN , zN ∈ A (RN ) such that i) g1 = g10 = g20 = g2 , g100 = g1000 = g2000 = g200 , 0 000 0 0 000 ii) t1 + t2 = t1 + t02 , t001 + t002 = t000 1 + t2 , iii) t1 < t1 ≤ 0 ≤ t2 < t2 , t1 < 00 00 000 000 00 00 000 t1 ≤ 0 ≤ t2 < t2 , and iv) max|R1 C P1 z1 P1 z1 and z2 P2 z2 P2 max|R2 C. 0 000 By Free-Lunch Transfer, zN R(E) zN , and by Stand-Alone Transfer, zN R(E) 00 00 0 000 zN . By Weak Pareto, zN P(E) zN and zN P(E) zN , which creates a cycle. If the graph of C is reinterpreted as that of C/2, then Figure 8.2 illustrates the incompatibility between Free-Lunch Transfer and Equal-Split Transfer.

t6 C or C/2

z2000

z200 @ @ R @ R2

s s

z s2 s0 z2

@ @ R @ R1

z10 s s z1 00 z1 s

g

sz1000

Figure 8.2: Free-Lunch Transfer versus Stand-Alone (or Equal-Split) Transfer.

8.4

Two solutions

The axioms introduced in the previous section force us to focus on two specific social ordering functions. Both SOFs give absolute priority to the worst-off, according to specific well-being indexes. The first index, a utility representation of the preferences, works as follows. The cost function is divided by the real number that would leave the agent indifferent between her actual bundle and the best bundle she could get if she were alone in the economy and had free access to that rescaled cost function. The associated well-being level is this number. Formally, for each Ri ∈ R, zi ∈ Ai (Ri ), and C ∈ C, uC (zi , Ri ) = a ⇔ zi Ii max|Ri C/a

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CHAPTER 8. PUBLIC GOOD

if such an a exists, and 0 otherwise.5 We call the numerical representation uC (·, Ri ) agent i’s cost-equivalent utility function. By definition of this representation, agent i is indifferent between receiving bundle zi or maximizing Ri subject to the cost function C/uC (zi , Ri ). This is illustrated on Figure 8.3. t6

C

zi

s

C/2 @ R@ i @ @ R @

s zi0

y

Figure 8.3: The cost-equivalent representation of Ri : uC (zi , Ri ) = 1 and uC (zi0 , Ri ) = 2. The C-Equivalent Leximin function RClex works by applying the leximin criterion to the cost-equivalent utility levels of the agents at the allocations we compare. Social ordering function C-Equivalent Leximin (RClex ) 0 For all E = (RN , Ω) ∈ E, zN , zN ∈ A (RN ), 0 ⇔ (uC (zi , Ri ))i∈N ≥lex (uC (zi0 , Ri ))i∈N . zN RClex (E) zN 5 The latter case occurs only if z I (0, 0) and the agent’s marginal rate of substitution in i i the (g, t) space is infinite at (0, 0).

8.4. TWO SOLUTIONS

143

We now turn to the second SOF. The associated utility representation of the preferences is the quantity of public good that, if received for free, would leave the agent indifferent between what she actually consumes and that quantity. Formally, for each Ri ∈ R and zi ∈ Ai (Ri ), there is a unique level of the public good, gi0 ∈ R+ , such that zi Ii (0, gi0 ). We may therefore define the numerical welfare representation function ug (., Ri ) : Ai (Ri ) → R+ by letting ug (zi , Ri ) = gi0 ⇔ zi Ii (0, gi0 ). We call the numerical representation ug (·, Ri ) agent i’s output-equivalent utility function. By definition of this representation, agent i is indifferent between receiving the bundle zi or consuming ug (zi , Ri ) of the public good for free. The g-Equivalent Leximin function Rglex applies the leximin criterion to the output-equivalent utility levels of the agents at the allocations to be compared. Social ordering function g-Equivalent Leximin (Rglex ) 0 For E = (RN , Ω) ∈ E, zN , zN ∈ A (RN ), 0 ⇔ (ug (zi , Ri ))i∈N ≥lex (ug (zi0 , Ri ))i∈N . zN Rglex (E) zN

The following table summarizes the relative merits of either solution. Table 8.1: Properties of the two SOFs RClex Strong Pareto + Tranfer among Equals + Nested-Contour Transfer + Stand-Alone Transfer + Equal-Split Transfer + Free-Lunch Transfer Unchanged-Contour Independence + Separation + Replication + Independence of the Feasible Set -

Rglex + + + + + + + +

As suggested by the table, there is a trade-off between Independence of the Feasible Set (in the current model, this axiom requires the ranking to be independent of changes in the cost function) and the fairness axioms of Stand-Alone Transfer and Equal-Split Transfer, under Pareto requirements. This trade-off is illustrated in Figure 8.4. If the cost function is C (resp., C/2), then agent 1 (resp., agent 2) is enjoying an insufficient (resp., excessive) well-being level according to Stand-Alone Transfer (resp., Equal-Split Trans0 fer) and zN R (RN , C) zN . If the cost function is C 0 , the situation of the two 000 00 agents appears reversed and one obtains zN R (RN , C 0 ) zN . Independence of 000 00 the Feasible Set then implies zN R (RN , C) zN . But Weak Pareto imposes that 00 0 000 zN P (RN , C) zN and zN P (RN , C) zN , which creates a cycle.

144

CHAPTER 8. PUBLIC GOOD t6

C 0 or C 0 /2

C or C/2

s z200 sz2000

zs1000 s z100 @ R1 R @

z0 s2 s z2

sz1 s0 z1 g

@ R2 R @

Figure 8.4: Incompatibility between Independence of the Feasible Set and either Stand-Alone Transfer or Equal-Split Transfer. The two SOFs above give absolute priority to the worst-off. The reason why the combination of robustness conditions and either Stand-Alone Transfer or Equal-Split Transfer yields a C-Equivalent Leximin function is quite similar to what we obtained with the Ω-Equivalent Leximin in Chapter 5. Indeed, as for Equal-Split Transfer in Part I, the axioms of Stand-Alone Transfer or Equal-Split Transfer can be strengthened into “priority” axioms when the social rankings are required to be independent of changes in preferences or of changes in the number of agents. Let us define two axioms of Stand-Alone Priority and Equal-Split Priority that convey an infinite aversion to inequality. Stand-Alone Priority is similar to Stand-Alone Transfer, except that the beneficiary of the transfer can get an arbitrarily small quantity of the private good, and the donor may be removed an arbitrarily large amount of the private good. Axiom Stand-Alone Priority 0 For E = (RN , C) ∈ D, zN , zN ∈ A (RN ), and j, k ∈ N such that gj = gj0 , gk = 0 gk , if max|Rk C Pk zk0 Pk zk and zj Pj zj0 Pj max|Rj C 0 and for all i 6= j, k, zi0 = zi , then zN R(E) zN .

As for Equal-Split Priority, it is similar to Equal-Split Transfer, except that now the beneficiary of the transfer can get an arbitrarily small quantity of the

8.4. TWO SOLUTIONS

145

private good, and the donor may lose an arbitrarily large amount of the private good. Axiom Equal-Split Priority 0 For E = (RN , C) ∈ D, zN , zN ∈ A (RN ), and j, k ∈ N such that gj0 = gj , gk0 = gk , if max|Rk (C/ |N |) Pk zk0 Pk zk and zj Pj zj0 Pj max|Rj (C/ |N |) 0 R(E) zN . and for all i 6= j, k, zi0 = zi , then zN

We then have the following. Lemma 8.3 a) On E, if a SOF R satisfies Pareto Indifference, Separation, Replication and Stand-Alone Transfer (or Equal-Split Transfer), then it satisfies Stand-Alone Priority and Equal-Split Priority. b) On E, if a SOF R satisfies Pareto Indifference, Unchanged-Contour Independence and Stand-Alone Transfer (resp., Equal-Split Transfer), then it satisfies Stand-Alone Priority (resp., Equal-Split Priority). 0 0 Proof. a) Let E = (RN , C) ∈ D, zN = (tN , gN ), zN = (t0N , gN ) ∈ A (RN ), and 0 0 j, k ∈ N be such that gj = gj , gk = gk ,

max|Rk C Pk zk0 Pk zk and zj Pj zj0 Pj max|Rj C, 0 R(E) zN . Let R0 , and for all i 6= j, k, zi0 = zi . We want to show that zN a b c d b c 0 a z0 , z0 , z0 , z0 be such that z0 I0 z0 P0 max|R0 C, zj P0 z0 I0 z0d , U (z0a , R0 ) = U (z0a , Rj ), g0a = g0b , g0c = g0d , (t0j − tj )/(tc0 − td0 ) = (tk − t0k )/(tb0 − ta0 ) ∈ N. Let E 0 = ((Rj , Rk , R0 ), C) and E 00 = ((Rj , Rk , R0 ), qC) where q ∈ Q++ is chosen so that max|R0 qC P0 z0a and zj0 Pj max|Rj qC. Let ∆1 = tb0 − ta0 and ∆2 = tc0 − td0 . This configuration is illustrated in Figure 8.5, which is similar to Figure 3.4. By Stand-Alone Transfer, ¡ ¢ zj , zk − (∆1 , 0) , z0b R(E 0 ) (zj , zk , z0a ) .

By Pareto Indifference,

¢ ¡ (zj , zk − (∆1 , 0) , z0c ) I(E 0 ) zj , zk − (∆1 , 0) , z0b .

By Stand-Alone Transfer, ¢ ¡ zj + (∆2 , 0) , zk − (∆1 , 0) , z0d R(E 00 ) (zj , zk − (∆1 , 0) , z0c ) .

By Pareto Indifference,

¡ ¢ (zj + (∆2 , 0) , zk − (∆1 , 0) , z0a ) I(E 00 ) zj + (∆2 , 0) , zk − (∆1 , 0) , z0d .

By Separation and Replication, one has R(E 00 ) = R(E 0 ). Therefore, wrapping up the previous lines, one obtains by transitivity: (zj + (∆2 , 0) , zk − (∆1 , 0) , z0a ) R(E 0 ) (zj , zk , z0a ) .

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CHAPTER 8. PUBLIC GOOD

C t6

zk s rr s zk0

z0b ss z0a

z0c s

d s z0

qC s zj0 r r

s zj

g

Figure 8.5: From Stand-Alone (or Equal-Split) Transfer to Stand-Alone (and Equal-Split) Priority. Repeating the argument (t0j − tj )/(tc0 − td0 ) times (Figure 8.5 illustrates the case in which three steps are needed), one eventually obtains: ¡ 0 0 a¢ zj , zk , z0 R(E 0 ) (zj , zk , z0a ) .

0 By Separation, this is equivalent to zN R(E) zN . It is an immediate extension of Lemma 8.1 that under Separation and Replication, Stand-Alone Priority and Equal-Split Priority are equivalent. b) The proof is essentially identical to that of Th. 3.1 and its basic line is illustrated in Figure 8.6. 0 By Unchanged-Contour Independence, one has zN R(E) zN if and only if 0 0 0 0 zN R(E ) zN , where E = (RN , C) is such that the indifference curves for j, k are as in the figure. By Pareto Indifference, ¡ a a ¡ ¢ ¢ zj , zk , zN \{j,k} I(E 0 ) zj , zk , zN \{j,k} .

By Stand-Alone Transfer, ¡ b b ¡ ¢ ¢ zj , zk , zN \{j,k} R(E 0 ) zja , zka , zN \{j,k} . By Pareto-Indifference, ¡ c c ¡ ¢ ¢ zj , zk , zN\{j,k} I(E 0 ) zjb , zkb , zN \{j,k} .

8.4. TWO SOLUTIONS

147

C or C/|N |

t6

zka s

zkb

s

s zkc s zkd

zk s

s zk0

s

zjc

zj0

s sd zj

s zj zjb

g

s sa zj

Figure 8.6: From Stand-Alone (resp., Equal-Split) Transfer to Stand-Alone (resp., Equal-Split) Priority. By Stand-Alone Transfer, ¡ ¢ ¢ ¡ d d zj , zk , zN\{j,k} R(E 0 ) zjc , zkc , zN\{j,k} . By Pareto-Indifference, ¡ 0 0 ¡ ¢ ¢ zj , zk , zN\{j,k} I(E 0 ) zjd , zkd , zN \{j,k} .

0 By transitivity, and taking account of the fact that zN \{j,k} = zN \{j,k} , one 0 obtains zN R(E 0 ) zN .

Regarding Free-Lunch Transfer, similar results are obtained. Separability (which is weaker than Separation) is sufficient to yield absolute priority to the worst off on the subdomain E 3 ⊆ E of economies with |N | ≥ 3. First, we define Free-Lunch Priority. It is identical to Free-Lunch Transfer, except that the beneficiary of the transfer can get an arbitrarily small quantity of the private good, and the donor may give away an arbitrarily large amount of the private good. Axiom Free-Lunch Priority 0 For all E = (RN , C) ∈ D, zN , zN ∈ A (RN ), and j, k ∈ N such that gj0 = gj =

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CHAPTER 8. PUBLIC GOOD

gk0 = gk , if

tj < t0j < 0 < t0i < ti

0 and for all i 6= j, k, zi0 = zi , then zN R(E) zN .

Lemma 8.4 a) On E 3 , if a SOF R satisfies Pareto Indifference, Free-Lunch Transfer and Separability, then it satisfies Free-Lunch Priority. b) On E, if a SOF R satisfies Strong Pareto, Free-Lunch Transfer and UnchangedContour Independence, then it satisfies Free Lunch Priority. Proof. a) The proof is very similar to that of Lemma 8.3 (a). We only provide the illustration in Figure 8.7. The thin indifference curves belong to a third agent l.

t6

zk s

s s r 0 r zk s s s

s zj0 s zj

s s s

g

r r

s

Figure 8.7: (a) From Free-Lunch Transfer to Free-Lunch Priority. b) The logic of the proof is basically the same as for Lemma 8.3 (b), but the constraint to have equal quantities of public good makes it impossible, in some cases, to construct a figure like Figure 8.6. Figure 8.8 illustrates the additional steps. 0 0 We want to prove that zN R(E) zN . Suppose not, i.e., zN P(E) zN . 0 0 By Unchanged-Contour Independence, one then has zN P(E ) zN , where 0 0 E 0 = (RN , C) is such that RN\{j} = RN \{j} , I(zj , Rj0 ) = I(zj , Rj ), I(zj0 , Rj0 ) = I(zj0 , Rj ), and I(zj∗ , Rj0 ) is the thin curve of Figure 8.8 that contains zj∗ .

8.4. TWO SOLUTIONS

149

t6

zk s

s zk0

s zj0 zj s s zj∗

g

Rj00 @ R Rj0 @

Figure 8.8: (b) From Free-Lunch Transfer to Free-Lunch Priority. ¡ ¢ 0 By Strong Pareto, zj∗ , zN\{j} P(E 0 ) zN . By Unchanged-Contour Indepen¡ ∗ ¢ 00 0 00 dence, one then has zj , zN \{j} P(E ) zN , where E 00 = (RN , C) is such that 00 ∗ 00 ∗ 0 0 00 0 RN \{j} = RN \{j} , I(zj , Rj ) = I(zj , Rj ), I(zj , Rj ) = I(zj , Rj ), and I(zj , Rj00 ) is the thin curve of Figure 8.8 that contains zj . It is then obvious how to complete the proof, following the steps of Lemma 8.3 (b). One can see on Figure 8.8 that it was impossible to construct suitable intermediate indifference curves in-between I(zj , Rj ) and I(zj0 , Rj ), because agent k’s indifference curves make it impossible to make a transfer backed by Free-Lunch Transfer on the extreme left side of the graph. This problem would vanish, however, if non convex preferences were allowed. We are now ready to state the main results of this section. Theorem 8.1 On E, if a SOF R satisfies Strong Pareto, Separation, Replication and either Stand-Alone Transfer or Equal-Split Transfer, then for all 0 E = (RN , C) and all zN , zN ∈ A(RN ), 0 . min uC (zi , Ri ) > min uC (zi0 , Ri ) ⇒ zN P(E) zN i∈N

i∈N

0 ∈ A(RN ) be such that mini∈N uC (zi , Ri ) > Proof. Let E = (RN , C) and zN , zN mini∈N uC (zi0 , Ri ).

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CHAPTER 8. PUBLIC GOOD

0 As in the proofs of Chapter 5, one first finds zˆN , zˆN ∈ A(RN ) such that by 0 0 Strong Pareto zN P(E) zˆN , zˆN P(E) zN , and such that there is i0 ∈ N for which for all j 6= i0 ,

zj0 , Rj ) > uC (ˆ zj , Rj ) > uC (ˆ zi0 , Ri0 ) > uC (ˆ zi00 , Ri0 ). uC (ˆ zi0 , Ri0 ), minj6=i0 uC (ˆ zj , Rj )) . One thereThere is q ∈ Q++ such that 1/q ∈ (uC (ˆ fore has max|Ri qC Pi0 zˆi0 and for all j 6= i0 , zˆj Pj max|Rj qC. Let zˆi10 , ..., zˆin−2 0 0 be such that zi0 , Ri0 ) > uC (ˆ zin−2 , Ri0 ) > · · · > uC (ˆ zi10 , Ri0 ) > uC (ˆ zi00 , Ri0 ). uC (ˆ 0 Let the n −1 agents other than i0 be numbered 1, ..., n−1. By n −1 applications of Stand-Alone Priority (permitted by Lemma 8.3), one obtains ¡ ¢ 0 0 , zˆi10 R(RN , qC) zˆN , zˆ1 , zˆ20 , ..., zˆn−1 ¡ ¢ ¢ ¡ 0 0 2 0 , zˆi10 , zˆ1 , zˆ2 , zˆ3 , ..., zˆn−1 , zˆi0 R(RN , qC) zˆ1 , zˆ20 , ..., zˆn−1 .. . ¡ ¡ ¢ ¢ 0 0 0 0 zˆ1 , zˆ2 , ..., zˆn−2 , zˆn−1 , zˆin−2 , zˆn−2 , zˆn−1 , zˆin−3 R(RN , qC) zˆ1 , zˆ2 , ..., zˆn−3 , 0 0 ¡ ¢ n−2 0 (ˆ z1 , ..., zˆn−1 , zˆi0 ) R(RN , qC) zˆ1 , zˆ2 , ..., zˆn−2 , zˆn−1 , zˆi0 ,

0 . implying by transitivity that zˆN R(RN , qC) zˆN 0 By Separation and Replication, this implies zˆN R(RN , C) zˆN . By transitiv0 ity, zN P(E) zN . By Lemma 8.1, Stand-Alone Transfer can be replaced by Equal-Split Transfer.

Theorem 8.2 On E (resp., E 3 ), if a SOF R satisfies Strong Pareto, Free-Lunch Transfer and Unchanged-Contour Independence (resp., Separability), then for 0 all E = (RN , C) and all zN , zN ∈ A(RN ), 0 . min ug (zi , Ri ) > min ug (zi0 , Ri ) ⇒ zN P(R) zN i∈N

i∈N

Proof. By part (b) (resp., part (a)) of Lemma 8.4, Free-Lunch Priority must 0 ∈ A(RN ) and be satisfied. Let us note that when for two allocations zN , zN two agents j, k, one has ug (zj0 , Rj ) > ug (zj , Rj ) > ug (zk , Rk ) > ug (zk0 , Rk ), while zi0 = zi for all i 6= j, k, then Pareto Indifference and Free-Lunch Priority 0 imply that zN R(E) zN . This is not completely obvious because I(zj0 , Rj ) may have a vertical asymptotic line above ug (zj , Rj ), which makes it impossible to apply Free-Lunch Priority in one blow, and requires the intervention of Pareto Indifference. Figure 8.9 illustrates the problem and its solution. As preferences are continuous and strictly monotonic, it is always possible to follow a finite

8.4. TWO SOLUTIONS

151

t6

zk0 s

s s zj zk

s zj0

g

Figure 8.9: Application of Free-Lunch Priority in the presence of vertical asymptotic lines. sequence of vertical line segments and indifference curves as in the figure. This corresponds to a sequence of applications of Free-Lunch Priority and Pareto Indifference, respectively. We can now prove the theorem. Let E = (RN , C) ∈ E (resp., E 3 ) and zN , 0 0 zN ∈ A(RN ) be such that mini∈N ug (zi , Ri ) > mini∈N ug (zi0 , Ri ). Let zˆN , zˆN ∈ 0 0 A(RN ) be such that by Strong Pareto zN P(E) zˆN , zˆN P(E) zN , and such that there is i0 ∈ N for which for all j 6= i0 , zj0 , Rj ) > ug (ˆ zj , Rj ) > ug (ˆ zi0 , Ri0 ) > ug (ˆ zi00 , Ri0 ). ug (ˆ

By n −1 applications of Free-Lunch Priority, as justified above, one obtains that 0 0 . By transitivity, zN P(E) zN . zˆN R(E) zˆN The latter theorem is somehow surprising, as very few axioms lead to sharp consequences. Indeed, the axioms complement each other quite efficiently, as, even if Strong Pareto, Separability and Unchanged-Contour Independence are reasonably weak axioms (at least in the sense that the large majority of SOFs we study in this book satisfy these axioms), combining them with Free-Lunch Transfer forces us to focus on output-equivalent utility levels, and to aggregate them using a criterion such as the leximin, that respects the strict-preference part of the maximin criterion.

152

8.5

CHAPTER 8. PUBLIC GOOD

Pure public good

The SOFs characterized above rank allocations where agents may consume different quantities of the public good. We argued in Section 3 that this is useful for making second-best social decisions. However, exclusion may be altogether impossible (or exceedingly costly): the non-rival good may be a pure public good. In this case, the set of admissible allocations should be redefined, as mentioned above (see Section 8.2), to incorporate the constraint that all agents consume the same quantity of the good. The axioms are rewritten without any further correction. Note that Separability and Separation become much weaker, as they only apply when comparing 0 allocations zN , zN that both involve the same level of public good. Consequently, Pareto Indifference plays a new and crucial role. Indeed, the comparison between two allocations that are associated to two different public good levels is always Pareto equivalent to the comparison of two allocations associated with the same produced level of public good. Therefore, all arguments involving Separability or Separation can be recovered by combining either axiom with Pareto Indifference. All the results presented in Section 4, therefore, generalize to the pure public good case, with a slight complication for part (b) of Lemma 8.3, in which Strong Pareto must replace Pareto Indifference, for the same reason as it already does so in part (b) of Lemma 8.4 in which, by definition of Free-Lunch Transfer, the quantities of public good consumed by the two agents must be identical. That both characterization results hold true is, of course, good news, but one may object that the model is no longer appropriate for the pure public good case. Indeed, we have assumed that contributions can be negative. This is legitimate if we think that a transfer could compensate an excluded agent but when exclusion is altogether impossible, we may simply restrict our attention to allocations where contributions are all non-negative. As a consequence, Free-Lunch Transfer is no longer applicable to the model. On the other hand, a small variant of it– dealing with the case in which one agent contributes a strictly positive amount while another agent does not contribute at all–can be formulated, and results in a characterization of a g-Equivalent maximin function.6 On the other hand, all our results regarding Stand-Alone Transfer and EqualSplit Transfer remain valid, as none of our arguments uses allocations involving negative contributions. Consequently, the C-Equivalent Leximin function is justified in exactly the same way as before. This discussion leads us naturally towards a solution that, surprisingly, is completely absent from the discussion so far: the Lindahl correspondence. The egalitarian Lindahl allocation rule is defined as follows.It involves individualized prices for the public good and the equal sharing of the profit of the production sector. Allocation rule Egalitarian Lindahl S EL For all E = (RN , C) ∈ E, S EL (E) is the set of allocations zN ∈ Z(E) such that 6 See

Fleurbaey and Sprumont (2009) for the details.

8.5. PURE PUBLIC GOOD

153

|N |

for some p ∈ R+ ,P (i) g maximizes g i∈N pi − C(g); ¡ P ¢ ª © (ii) for all i ∈ N, zi ∈ max|Ri (t, g) | gpi = t + g i∈N pi − C(g) / |N | .

What is its relationship with the SOFs or the axioms discussed so far? The above discussion, actually, has implicitly made clear that it is hard to justify the Lindahl approach to public goods in the framework of SOFs. There are three reasons for that, all of which are illustrated in Figure 8.10. Allocation z L = (z1L , z2L ) is an egalitarian Lindahl allocation (π denotes the profit of the firm producing the public good, and pi (i = 1, 2) denotes the individualized price paid by agent i). Any SOF consistent with Lindahlian ethics would recommend this allocation as the socially preferred among the feasible ones. t6

C

z2L s s

s z1L

p2 π 2

π

p1 p1 + p2

@ @ R @

g

R1 = R2

Figure 8.10: Any SOF consistent with the Egalitarian Lindahl correspondence violates Transfer among Equals, Stand Alone Transfer, Equal Split Transfer and Free Lunch Transfer. The first problem comes from the fact that z1L 6= z2L , whereas agents 1 and

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CHAPTER 8. PUBLIC GOOD

2 are identical. Consequently, any such SOF would violate Transfer among Equals. The second problem comes from the fact that uC (z2L , R2 ) < 1, that is, agent 2 ends up worse off than if she were alone in the economy: Stand-Alone Transfer has to be violated as well. The third problem is that agent 1 enjoys a free lunch and is strictly better off than if both agents had the same bundle. Consequently, any Lindalh SOF would violate Free Lunch Transfer and Equal Split Transfer. For all those reasons, we have to conclude that no desirable SOF would recommend us to single out the Lindahl allocations as the best ones. This shows the limits of the Lindahl correspondence, and it also sheds doubts on the common wisdom that the Walrasian correspondence is “extended” into the Lindahl correspondence when there are public goods.

8.6

Second-best applications

In this section, we illustrate how the SOFs axiomatized in the previous sections can be used to identify second-best allocations. Before that, and in order to evaluate the second-best allocations, let us begin by illustrating the first-best allocations in a two-agent economy. In Figure 8.11, allocation (z1C , z2C ) (resp., (z1g , z2g )) represents the optimal feasible allocation for the C-Equivalent Leximin function RClex (resp., the gEquivalent Leximin function Rglex ). They are associated in the graph to the same production level of the public good, but there is nothing essential in that (it only allows us to simplify the figure). What is essential is that agent 1, the low willingness to pay agent, strictly prefers z1g over z1C and the opposite preference holds for agent 2. At a first-best allocation, Rglex favors the agent with lower willingness to pay, whereas RClex favors the agent with greater willingness to pay. Let us now move towards second-best applications, by concentrating first on the pure public good case. In this case, given that all agents will consume the same quantity of the produced good, the incentive constraint (that is, the noenvy constraint) boils down to requiring that all contributions are equal, too. It is immediate, indeed, to check that if one type of agents has to pay less, all agents have a strict incentive to pretend that they are of that particular type, and the allocation is not incentive compatible. Therefore,the set of allocations among which the policy-maker can choose is the set Zˆ (E) which has the following very simple structure: for all E = (RN , C) ∈ E, Zˆ (E) = {zN ∈ A (RN ) | ∀i, j ∈ N, zi = zj } . Let us assume that we have a finite number of (types of) agents, and let us further assume that the agents’ preferences satisfy the Spence-Mirrlees singlecrossing property, i.e., an indifference curve of an agent crosses an indifference curve of another agent at most once. More precisely, we assume that for any pair of agents, either they have identical preferences or the intersection of two of their indifference curves (one for each agent) contains at most one point. A

8.6. SECOND-BEST APPLICATIONS

155

t6

C

C uC (ziC ,Ri )

z2g s @ @ @ R @ R2 s z2C C z1 s sP P

@ @ R @

sg z1

ug (zig , Ri )

g

R1

Figure 8.11: Optimal feasible allocations for Rglex and RClex in a two-agent economy. consequence of this property is that agents can be ordered according to their preferences over the public good. Indeed, if there exists a cost function such that an agent’s optimal bundle involves a lower production level than another agent’s optimal bundle, then, by this single-crossing property, it is also the case for any other cost function. Therefore, we do not lose any generality by assuming that N = {1, 2, . . . , n}, and that agent 1 has the lowest preference for the public good, agent 2 the second lowest, etc. and agent n has the largest preference for the public good. For all i ∈ N, let gi∗ ∈ R+ be the demand in the public good of agent i under the proviso that the production cost is divided equally, that is µ ¶ C (gi∗ ) ∗ max|Ri (C/n) = , gi . n The Spence-Mirrlees single-crossing property implies that g1∗ ≤ g2∗ ≤ . . . ≤ gn∗ .

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CHAPTER 8. PUBLIC GOOD

Let g glex and g Clex be the levels of production of the public good at the second-best optimal allocations obtained when the policy-maker maximizes the g-Equivalent Leximin SOF and the C-Equivalent Leximin SOF, respectively. We have the following result: g glex = g1∗ ≤ g Clex . This can be proved using the following figure. Let us begin by noting that if all agents consume some bundle (t, g) such that t ≥ 0, then the agent having the lowest output-equivalent utility level is agent 1. Indeed, as she has the lowest preference for the good, she also has the lowest willingness to pay for the consumption of the quantity g, so that the quantity of the public good that leaves her indifferent to (t, g) is the lowest. t6

C/n

s

C(g1 ) n

ug (z1 , R1 )

µ ¡

s

g1

R @ R3 R @ R2 R @ R1

g

Figure 8.12: Second-best provision of a pure public good. ´ ³ ∗ C(y1 ) ∗ , y When all agents consume 1 , one sees in Figure 8.12 that the agent n with the lowest output-equivalent utility level, agent 1, is precisely the agent maximizing at that bundle. Now, given that agent 1 is maximizing (over all allocations in Zˆ (E)) at that bundle, changing the level of production of the public good will necessarily decrease her welfare (under the constraint that the cost must be divided equally). Consequently, the allocation where all agents consume that bundle is the second-best optimal for the g-Equivalent Leximin function.

8.6. SECOND-BEST APPLICATIONS

157

Now, it is also clear that the cost-equivalent utility level of agent 1 is equal to at that bundle, whereas the utility level of all other agents is lower (strictly lower if they have different preferences, as in Figure 8.12). As a consequence, whenever one agent has a strictly higher preference for the public good than agent 1, increasing the level of production slightly will necessarily increase the well-being of the agent with the lowest cost-equivalent utility level. This proves the claim. Let us assume, now, that exclusion is possible: some agents may be prevented from consuming the whole available quantity of the public good, and, therefore, may be asked to pay a lower contribution. Is it good news? For whom? Let us show that the answers to these questions depend on the SOF we use. If the policy-maker wishes to maximize the g-Equivalent Leximin SOF Rglex , then the possibility of exclusion is never bad news for the low demand agent, in the sense that she may never be worse off at the second-best optimal allocation under exclusion than without exclusion. On the other hand, it may in some cases be bad news for the high demand agents. Figure 8.13 illustrates these two facts. Allocation (z1 , z1 ) is the optimal second-best allocation for Rglex in absence of exclusion. Let us observe, as we showed above, that the g-equivalent welfare level of agent 1 is lower than that of agent 2. Therefore, if some new welfare opportunities appear, agent 1 should benefit from it and this proves the first point. But let us observe that this allocation is also first-best Pareto optimal (we can deduce from the fact that both agents maximize on C2 that the Samuelson conditions are satisfied). Any other first or second-best allocation that assigns a strictly higher welfare level to agent 1 must assign a strictly lower welfare level to agent 2. One such allocation is illustrated in the figure. Let us observe that allocation (z10 , z20 ), which assigns to agent 1 a strictly larger welfare level than at z , is incentive compatible. Consequently, whether or not (z10 , z20 ) is second-best optimal, agent 1 will strictly prefer the second-best bundle she is assigned when exclusion is possible to the one she is assigned in the absence of exclusion. Thus, in this case exclusion strictly benefits the agent with a low willingness to pay, at the expense of the agent with a higher willingness to pay. If the policy-maker wishes to maximize the C-Equivalent Leximin SOF RClex , then, with two agents, the possibility of exclusion always leads to a Pareto improvement. This comes from the fact that if an allocation is optimal for RClex , then the C-equivalent welfare levels are equalized among the two agents.7 Indeed, agents have continuous single-peaked preferences over bundles b in Z(E) and the peak corresponds to a welfare level of 12 . There necessarily b exists an allocation of Z(E) that equalizes the C-equivalent welfare levels, as illustrated in Figure 8.14. It is the second best allocation recommended by RClex if exclusion is not possible. 1 |N|

7 It is easy to see that if there are more than two (types of) agents, then, at a RClex second-best optimal allocation without exclusion, the C-equivalent welfare levels of the lowest and the highest willingness to pay agents are equalized, and all the other agents’ welfare levels are strictly larger.

158

CHAPTER 8. PUBLIC GOOD t6

C

z20 s z1

s0 z1

s

@ R@ C/2 2 @ @ @ R @ @ R @ R1

g

Figure 8.13: If the planner maximises Rglex , then a high willingness to pay agent can be worse off at the second-best optimum when exclusion is possible than when it is not possible, whereas a low willingness to pay agent is always better off if exclusion is possible. If, now, exclusion is possible, so that some new welfare opportunities appear, then the minimal welfare level cannot decrease, which means that the new optimal allocation must Pareto dominate the one without exclusion: exclusion benefits both agents.8

8.7

Conclusion

The first lesson which should be drawn from this chapter is that models involving production confirm the main ideas developed in exchange models: fairness 8 If there are more than 2 agents, then the agents with the lowest and the largest willingness to pay will gain, but it may be the case that other agents lose.

8.7. CONCLUSION

159

t6

C

C uC (z ,Ri )

C 2

@ R @ R2

s z g

@ R @ R1

Figure 8.14: Optimal second-best allocation for RClex .

axioms need to be defined carefully in order to avoid incompatibilities with efficiency axioms; fairness axioms of the transfer type lead to infinite inequality aversion axioms when the SOF is also required to obey Pareto efficiency and cross-economy robustness axioms. We have also observed that the specificity of the production model, that is, the fact that the goods involved do not have the same status (one is a private good contribution and the other is a public good that nobody owns in strictly positive quantity before production takes place), allows us to define new axioms and axiomatize new solutions. Indeed, Free-Lunch Transfer is technically grounded on the fact that contributions can be negative, whereas consuming a negative quantity of the produced good does not make much sense. It is somehow comparable with what we observed in Section 6.2, in the allocation problem of uncertain quantities, where the set of certainty bundles ended up playing a specific role. In that case, as in this chapter, one advantage is that the resulting SOF satisfies Independence of the Feasible Set.

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CHAPTER 8. PUBLIC GOOD

Another lesson is that it is hard to justify SOFs that would rationalize the Lindahl correspondence. This first-best solution has received so many justifications parallel to the ones received by the Walrasian correspondence that one might have conjectured that it would have received some justification in the SOF framework. Nor did other well-known solutions to the public good first-best problem receive proper justification in our approach (such as the ratio equilibrium from Kaneko and its generalization by Mas-Colell). The public good case seems to be one in which only SOFs inspired by the general idea of egalitarian equivalence can be axiomatically justified. We will see below that if all goods are private, then, even in the presence of production, Walrasian allocations can also receive a strong justification.

Chapter 9

Private good 9.1

Introduction

In this chapter, we continue our analysis of production economies, under the assumption that the produced good is a private good. In the first sections of the chapter, we will stick to the assumption that agents privately own some endowment of the input. The most natural interpretation is that the inputs are agents’ labor contributions and the output is their income. The resulting model is not much different from the model of the previous chapter. Consequently, we could replicate the axiomatic study we carried out in Chapter 8, but we will put the emphasis on other issues. As the model under study is a pure private good model, we can expect to see the (equal income) Walrasian allocations play a role again. We will, indeed, see them back into the picture, but we will focus here on a different justification. In Chapter 5, we showed how a Walrasian SOF could be axiomatically derived. The existence of a production technology does not prevent us at all from deriving a similar SOF in the current model. The adaptation of the axiomatic analysis of Chapter 5 will not be examined here, however, as it is rather straightforward. Instead of following that direction, here we propose a derivation of the Walrasian allocations from an implementation point of view. That is, our focus will be on a specific SOF, the P -Equivalent Leximin SOF (P will denote the production set), that is reminiscent in the current model of the Ω-Equivalent Leximin SOF of Part II and the C-Equivalent Leximin SOF of the previous chapter. Our result will be that a policy-maker who wishes to maximize that SOF under incentive compatibility constraints and under the restriction to allocations that can be decentralized through linear consumption functions has to choose the Walrasian allocations. In the last section of the chapter, we drop the assumption that some endowments are privately owned by the agents. Moreover, we do not impose any restriction on the (finite) number of goods, nor on the nature (private or public) of the goods that are produced. We show that the generalization of the 161

162

CHAPTER 9. PRIVATE GOOD

Ω-Equivalent Leximin SOF to that model is immediate. That proves that the approach developed in this book is not confined to simple models, and can be used to build indices of well-being in more general contexts.

9.2

A simple model with private rights on the input

Let us consider that there are two goods in the economy: labor, denoted , and a private consumption good, denoted c. There is a set N of agents, and each agent i ∈ N has continuous and convex preferences Ri over ( i , ci ) bundles, which are decreasing in i (working is painful) and increasing in ci (the consumption good is always desirable). The consumption set is X = R2+ . There is a production set P ⊂ R2+ that describes what consumption level can be produced with any given labor contribution, that is, ( , c) ∈ P means that a total labor contribution of is sufficient to produce the level c of consumption. Let ∂P denote the upper boundary of P . The set P is assumed to be closed, strictly convex and such that (0, 0) ∈ ∂P . An economy is now a list E = (RN , P ) and we denote by E the set of all economies satisfying the above restrictions. An allocation zN = ( N , cN ) ∈ X N is feasible if the sum of labor contributions is sufficient, given the technology, to produce the sum of the consumptions: Ã ! X X ci ∈ P. i, i∈N

i∈N

Let Z(E) denote the set of feasible allocations for E. The SOFs we will be interested in, as in the previous chapter, only rank a subset of the feasible allocations, which we call “admissible” allocations and denote by A (E), the precise definition of which is given below. The first fairness axiom introduced here is reminiscent of the Stand-Alone Transfer axiom from the previous chapter. Remember that the Stand-Alone welfare level of an agent is the highest satisfaction level she can obtain when being the only one to use the technology. Again, we consider the satisfaction level associated to such a bundle1 max|Ri P as a threshold: if one agent is strictly above her Stand-Alone level, whereas another agent is strictly below, we view the former agent as strictly richer than the latter, and a transfer between the two increases social welfare. Our axiom reads as follows. A transfer of consumption good between two agents is a weak social improvement if the beneficiary has bundles yielding satisfaction levels below her stand-alone level, and the donor has bundles yielding satisfaction levels above her stand-alone level, before and after the transfer. Axiom Stand-Alone Transfer 0 For all E = (RN , P ) ∈ D, zN = ( N , cN ), zN =( 1 Uniqueness

follows from strict convexity of P.

0 0 N , cN )

∈ A (RN ) , ∆ ∈ R+ ,

9.2. A SIMPLE MODEL WITH PRIVATE RIGHTS ON THE INPUT

163

and j, k ∈ N , if max|Rk P Pk zk0 = ( k , ck + ∆) Pk zk and zj Pj zj0 = ( j , cj − ∆) Pj max|Rj P 0 P(E) zN . and for all i 6= j, k : zi0 = zi , then zN 0 . Let us note that if zN is feasible, then so is zN The inquiry into Egalitarian Walrasian SOFs in Chapter 5 could be conducted here as well. Since it would mainly consist in adapting results of Section 5.3 to the current framework, we omit it here. For the sake of completeness, let us mention that the Egalitarian Walrasian ranking of allocations is not compatible with Stand-Alone Transfer. An Egalitarian Walrasian allocation, indeed, can be such that one agent enjoys a strictly higher satisfaction level than her stand-alone level. This occurs when this agent has extreme preferences and either does not work and consumes his share of the profit of the production sector, or works a lot at a high wage rate when the others’ greater aversion to labor keeps the productivity high. When, in any of these two ways, an agent is better off than at her stand-alone level, necessarily another agent is strictly worse off. A transfer between them should be a weak social improvement, according to Stand-Alone Transfer, whereas for a SOF rationalizing the Egalitarian Walrasian allocation rule, the Egalitarian Walrasian allocations should be socially preferred to any other feasible allocation. Therefore not only the Egalitarian Walrasian SOF, but any SOF rationalizing the Egalitarian Walrasian allocation rule, must violate Stand-Alone Transfer. This is actually true even if we weaken our requirement that Walrasian allocations be selected to the requirement that envy-free allocations be selected. The Envy-Free Selection axiom that was introduced in Section 5.3 is immediately adapted to the current setting.

Axiom Envy-Free Selection For all E = (RN , P ) ∈ D and zN ∈ max|R(E) Z (E), for all i, j ∈ N, zi Ri zj . The following theorem summarizes the above discussion. Theorem 9.1 No SOF satisfies Strong Pareto, Envy-Free Selection and StandAlone Transfer. Proof. By Strong Pareto and Envy-Free Selection, the best allocations are envy-free and efficient allocations. For an economy with many agents and sufficient diversity of preferences, the envy-free and efficient allocations are close to the Egalitarian Walrasian allocations. If all Egalitarian Walrasian allocations have an agent above her stand alone level, and if the envy-free and efficient allocations are sufficiently close to the Egalitarian Walrasian allocations, one obtains the desired contradiction. The impossibility depicted in this theorem holds for sufficiently large economies. For economies with two agents, the P -Equivalent Leximin SOF defined below satisfies the three axioms. We will proceed, in this chapter, by making two applications of SOFs in the evaluation of policies. We choose to make these applications by using the

164

CHAPTER 9. PRIVATE GOOD

SOF that is similar in this private-good context to the C-Equivalent Leximin SOF defined in the previous chapter. By analogy, we call the new SOF the P -Equivalent Leximin SOF. It consists in applying the leximin to vectors of utility representations that are constructed in this way: the utility level that is associated to one bundle is equal to the parameter by which the production set needs to be rescaled so that the agent is indifferent between that bundle and being alone and free to use the rescaled production set. Formally, for a production set P and a positive real number a, the production set aP is defined by: µ ¶ c ( , c) ∈ aP ⇔ , ∈ P. a a For each i ∈ N , Ri ∈ R, let Ai (Ri ) denote the set of bundles for which the utility representation we now construct is well-defined. For each E = (RN , P ) ∈ E, uP (zi , Ri ) = a ⇔ zi Ii max|Ri aP. Sets Ai (Ri ) are precisely those for which such a number a exists. The set A(E) of admissible allocations is the set of allocations composed of admissible bundles. Let us note that if P satisfies the conditions: (i) for each ( , c) ∈ P , there exists a sufficiently large positive a such that (a , ac) ∈ / P , and (ii) for each ( , c) ∈ / P , there exists a sufficiently small positive a such that (a , ac) ∈ P , then Ai (Ri ) = {( i , ci ) ∈ X|( i , ci ) Ri (0, 0)}. We call the numerical representation uP (·, Ri ) agent i’s P -equivalent utility function. The P -Equivalent Leximin SOF RP lex works by applying the leximin criterion to the P -equivalent utility levels of the agents at the allocations we compare. Social ordering function P-Equivalent Leximin (RP lex ) 0 For E = (RN , P ) ∈ E, zN , zN ∈ A (RN ), 0 ⇔ (uP (zi , Ri ))i∈N ≥lex (uP (zi0 , Ri ))i∈N . zN RP lex (E) zN

As it is clear from the definition of this SOF, it fails to satisfy Independence of the Feasible Set (in the same way as the Ω-leximin and the C-leximin SOFs of Chapters 5 and 8 also do). Otherwise, in addition to Stand-Alone Transfer, it satisfies Strong Pareto, Transfer among Equals, Nested Transfer, UnchangedContour Independence, and Separation. It also satisfies the axioms we could define in the current model by analogy to Equal-Split Transfer and Replication. A characterisation result in the same vein as Theorem 8.1 above can be proven. We do not enter into the details here.

9.3

Assessing reforms

We now propose two applications of the P -Equivalent Leximin SOF. In this section, we show how to use it to assess reforms under information asymmetry.

9.4. A THIRD-BEST ANALYSIS

165

More precisely, let us assume that the set of agents is fixed, N = {1, . . . , n}, and let us assume that the economy E = (RN , P ) ∈ E is such that the policy-maker knows the distribution of preferences but does not know which agent has which preferences, that is, she knows RN up to a permutation. As in the previous chapters, she has to restrict her attention to incentivecompatible allocations zN ∈ A (RN ), that is, to allocations satisfying the noenvy, or self-selection, condition: for all i, j ∈ N, zi Ri zj . If this condition is satisfied, then it cannot be the case that one agent has a lower labor contribution and a higher consumption level than another agent. Consequently, there exists a non-decreasing consumption function φ : R+ → R+ such that for all i ∈ N : ci = φ( i ). We raise the following question: assume a society is applying some φ function. If the policy-maker is interested in reforming this consumption function, how should she modify it, that is, which agent should be paid more and which agent should be paid less? We show how to answer this question graphically. The key step consists in identifying which agent has the lowest P -equivalent welfare level. The only information that is needed to evaluate the welfare level of an agent is her indifference curve through the bundle she chooses. Given that the chosen bundles are observable, and given that, when the incentive compatibility constraints are satisfied, the agents reveal their preferences, the policy-maker, who knows the list of preferences in the economy is able to compute the welfare levels of the agents. Let us illustrate this reasoning in Figure 9.1. To make it simple, it represents a two-agent economy. Given consumption function φ, agent 1 chooses z1 and agent 2 z2 . Then, the policy-maker is able to identify that the agent having chosen z1 has a higher utility level than the other agent according to the P Equivalent measure. A reform of φ should then increase the consumption level around z2 and decrease the one around z2 . Indeed, given that the SOF we are interested in has a maximin shape, identifying the worst-off agent according to the constructed well-being measure is all that is needed to identify the direction of a strict social improvement. As a result, assessing reform is much easier than if the aggregator was, e.g., utilitarian, which would force us to measure the impact of the reform on the well-being measure of each agent.

9.4

A third-best analysis

Our second application restricts the set of feasible consumption functions even further. Let us assume, indeed, that only linear consumption functions can be used, i.e., there should exist some π ∈ R and ω ∈ R+ such that for all i ∈ N : φ( i ) = π + ω i . In such a framework, we try to identify the values of π and ω that maximize the P -Equivalent Leximin SOF. One of the possible consumption functions is the Egalitarian Walrasian one, that is, the one where ω is the competitive wage of that economy, while π is the per capita competitive profit of the firm operating the technology. Formally, φEW is an Egalitarian Walrasian consumption

166

CHAPTER 9. PRIVATE GOOD c

6

P R2

R1

I @

I @

z2

z1 s

uP (z1 , R1 )P

s

uP (z2 , R2 )P

φ

-

Figure 9.1: How to assess a reform by using the P -equivalent leximin SOF: given a consumption function φ, one computes utility indices uP (z1 , R1 ) and uP (z2 , R2 ). EW function and zN is an Egalitarian Walrasian allocation if there exist πEW and EW ω such that each agent maximizes her preferences over the budget set defined by profit π EW and wage ω EW , the resulting production plan is feasible and maximizes the total profit of the firm:

∀ i ∈ N,

ziEW ∈ max|Ri {( , c) ∈ X | c ≤ πEW + ω EW }, Ã ! X X ci ∈ P, i, i∈N

∀ ( 0 , c0 ) ∈ P,

i∈N

c0 ≤ nπ EW + ω EW 0 .

Let us note that the last property is equivalent to ∀ ( 0 , c0 ) ∈

1 P, n

c0 ≤ π EW + ω EW 0 .

This property will be used in the graphical representation below. We know that the allocation emerging from this consumption function is Pareto efficient in the set of allocations emerging from linear consumption functions (as it is Pareto efficient within the set of all allocations), but there are

9.4. A THIRD-BEST ANALYSIS

167

other linear consumption functions that yield allocations that are Pareto undominated. Moreover, in view of the above lemma, there is no clear connection between the P -Equivalent Leximin SOF and the Egalitarian Walrasian allocations. Yet we can show that the optimal linear consumption function is the Walrasian one in economies satisfying the following richness property about preferences. At any feasible allocation emerging from linear consumption function, there should exist at least one agent consuming a bundle that is average in the sense that it would still be feasible that all agents had chosen that precise bundle. Formally, the domain DA ⊂ E gathers all economies such that for all zN ∈ Z(E), if there exist π ∈ R and ω ∈ R+ such that for all i ∈ N : zi ∈ max|Ri {( , c) ∈ X|c ≤ π + ω }, then there exist one agent j ∈ N such that nzj ∈ P . This is a very demanding condition. In particular, it requires that, at all Egalitarian Walrasian allocations, there be one agent whose bundle is exactly the average of all other bundles. In large economies, it is consistent with the French proverb that “all tastes are in nature”, so that any kind of preferences is represented by at least one agent. Of course, the condition can also hold in finite economies (we can think, for instance, of economies where preferences are such that labor supplies do not depend on wages and one agent’s labor supply is equal to the average of all labor supplies). We are then able to derive the following result. Theorem 9.2 Let E = (RN , P ) ∈ DA . If a policy-maker maximizes the P Equivalent Leximin function RP lex and only linear consumption functions can be used, then she chooses the Egalitarian Walrasian consumption function. Again, the proof can be graphical. In Figure 9.2, a Walrasian allocation in a EW three-agent economy is drawn, zN . Let us observe that agent 2 is an average agent in the sense defined above: her labor contribution is precisely the third of the total labor supply. Let us assume that the economy belongs to the DA domain. The key argument is that the two properties of this allocation (its being Egalitarian Walrasian and agent 2 being an average agent) imply that bundle z2EW is precisely agent 2’s best bundle over the production set that one obtains by dividing P by n, as illustrated in the figure. That means that agent 2’s P -equivalent welfare level is precisely n1 , and that she is the worst-off agent EW in terms of P -equivalent welfare levels at zN . A Now, by the definition of D , for any other allocation that emerges from a linear consumption function, there is one agent, choosing a bundle that lies in the set n1 P , so that the reduction of P to which her indifference curve is tangent is strictly smaller than n1 P . Consequently, the P -equivalent welfare level of that EW agent is strictly below n1 , proving that zN is optimal. The reason why the Egalitarian Walrasian allocations turn out to be justified by a maximizing exercise of the P -equivalent SOF under the constraints we imposed can be explained as follows. First, the Egalitarian Walrasian allocations

168

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6 P R3 R2

I @

I @

R1 I @

ω EW

z1EW πEW

s

s

s z3EW 1 3P

z2EW -

Figure 9.2: The Egalitarian Walrasian allocation (z1EW , z2EW , z3EW ) is optimal for the P -equivalent leximin SOF among linear consumption functions. have attractive incentive properties as they are envy-free. Second, they have a simple structure (they are linear) and we have, by assumption, restricted our attention to those kind of allocations. Third, the Egalitarian Walrasian allocations turn out to be reasonably fair, even if fairness is evaluated by using the P -leximin SOF, which is the surprising feature of the above result.

9.5

A general model of production and allocation

This section contains the most general model that is studied in this book. There is a finite but arbitrarily large number of private goods that can be used as inputs or outputs, and an arbitrarily large number of public goods, that can be produced by using the private goods. The axiomatic analysis that we propose is grounded on generalizations of axioms previously introduced, and the main result of the section is an axiomatic characterization of a SOF that generalizes the Ω-Equivalent, the C-Equivalent and the P -Equivalent Leximin SOFs that have been studied in previous chapters and sections. The lesson from this section is that the approach presented in this book can be used to construct indices of

9.5. A GENERAL MODEL OF PRODUCTION AND ALLOCATION

169

social welfare in a general context. We assume that there is a set L of private goods and a set M of public goods. An agent’s consumption set is X = RL∪M and a typical consumption + bundle is denoted zi = (ci , gi ) . An economy is now defined as a quadruple E = (RN , Ω, V, Q) ∈ E, where N ⊂ N stands for the finite set of agents, RN ∈ RN stands for the preferences of these agents, Ω ∈ RL + is the social endowment in M private goods, V ⊂ RL is the private good production set and Q ⊂ RL × R+ is the public good production set. For vectors belonging to V or Q, negative components stand for inputs. We assume that (i) for all N ⊂ N, all i ∈ N, all Ri ∈ R, Ri is continuous, convex and monotonic in each (private and public) good, (ii) V is closed, strictly convex, comprehensive (if v ∈ V, then v 0 ∈ V for all v 0 ≤ v), and such that 0 ∈ ∂V, where 0 ∈ RL is the null vector and ∂V denotes the upper boundary of set V, and (iii) Q is closed, strictly convex, comprehensive and such that 0 ∈ ∂Q. Let E denote the set of economies satisfying those assumptions. Let¡ E = (RN ,¢Ω, V, Q) ∈ E. An allocation is a list zN ∈ X N . An allocation zN = (ci , gi )i∈N ∈ X N is feasible for E if there exists v ∈ V and q ∈ Q such that X ci ≤ Ω + v + qL , i∈N

max gik i∈N

≤ qk , ∀ k ∈ M.

Let Z (E) denote the set of feasible allocations for E. Let r ∈ R+ . If production possibilities are multiplied by r, we obtain the production sets rV and rQ, defined by: ½ ¾ 1 rV = v ∈ RL | v ∈ V , r µ ¶ ¾ ½ 1 | , q ∈ Q . rQ = q ∈ RL × RM q L M + r Note that for r > 1, rV is bigger than V, thanks to the convexity of V, whereas rQ is smaller than Q. This definition of rQ is motivated by the use of the Replication axiom in the analysis below. Indeed, an allocation is feasible with n agents and resources Ω, V, Q if and only if it is feasible with kn agents and resources kΩ, kV, kQ. In particular, the kn agents will consume the same quantity of public goods qM as the initial n agents with k times as many inputs qL . Finally, we denote by max|Ri (Ω, V, Q) the best bundle, according to Ri , among all bundles which can be produced and consumed given resources Ω, V, Q. We are now equipped to construct the numerical representation of the preferences which will be used in the SOF we will axiomatize. Let E = (RN , Ω, V, Q) ∈ E. Each Ri admits a unique numerical representation uΩV Q (·, Ri ) defined over a set of¡ “admissible ¢ bundles" ARi V Q , which is formally defined below, such that uΩV Q 0L∪M , Ri = 0 and for all r ∈ R++ , uΩV Q (zi , Ri ) = r ⇔ zi Ii max|Ri (rΩ, rV, rQ) .

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We call uΩV Q (zi , Ri ) agent i’s resource-equivalent welfare level at bundle zi . The set ARi V Q is the set of admissible bundles for agent i, that is, the subset of X where uΩV Q is well-defined. That ARi V Q is a strict subset of X follows from the fact that even if the set of bundles obtainable with resources rΩ, rV, rQ grows as r grows unboundedly, the welfare level of one agent freely maximizing over these resources may be bounded. This is the case, for instance, if agent i is interested only in the public goods, because with rΩ, rV, rQ the production of public goods is bounded for all r at the level that is producible with Ω, V, Q. What remains to be done is the rewriting of the axioms and the statement of the result. Axiom Stand-Alone Transfer 0 0 For all E = (RN , Ω, V, Q) ∈ E, zN = (cN , gN ), zN = (c0N , gN ) ∈ X N , ∆ ∈ RL +, and j, k ∈ N , if max|Rk (Ω, V, Q) Pk zk0 = (ck + ∆, gk ) Pk zk and zj Pj zj0 = (cj − ∆, gj ) Pj max|Rj (Ω, V, Q) , 0 and for all i 6= j, k, zi0 = zi , then zN R(E) zN .

Axiom Equal-Split Transfer 0 0 = (c0N , gN ) ∈ X N , ∆ ∈ RL For all E = (RN , Ω, V, Q) ∈ E, zN = (cN , gN ), zN +, and j, k ∈ N , if µ ¶ 1 1 1 max|Rk Ω, V, Q Pk zk0 = (ck + ∆, gk ) Pk zk and |N | |N | |N | µ ¶ 1 1 1 0 zj Pj zj = (cj − ∆, gj ) Pj max|Rj Ω, V, Q , |N | |N | |N | 0 R(E) zN . and for all i 6= j, k, zi0 = zi , then zN

Separation is defined exactly as in the other models above. Replicas of profiles of preferences or allocations are also defined as above. Axiom Replication 0 For all E = (RN , Ω, V, Q) ∈ E, zN , zN ∈ X N , and r ∈ Z++ , if E 0 = (RN 0 , Ω, V, Q) ∈ 0 E is such that RN 0 is a r-replica of RN , zN 0 , zN 0 are the corresponding r-replicas 0 0 0 of zN and zN respectively, Ω = rΩ, V = rV and Q0 = rQ, then 0 0 zN R (E) zN ⇔ zN 0 R (E 0 ) zN 0.

Theorem 9.3 If a SOF R satisfies Strong Pareto, Separation, Replication and either Stand-Alone Transfer or Equal-Split Transfer, then for all E = 0 ∈ XN , (RN , Ω, V, Q) ∈ E and all zN , zN 0 min uΩV Q (zi , Ri ) > min uΩV Q (zi0 , Ri ) ⇒ zN P(E) zN . i∈N

i∈N

The proof is an immediate adaptation of the proof of Theorem 8.1.

9.6. CONCLUSION

9.6

171

Conclusion

Let us sum up the contents of this chapter. First, on the axiomatic side, we have briefly examined how to extend the previous results to models of production of private goods. In Section 9.5, we have shown how to axiomatize a SOF in a general model of production and allocation of private and public goods. Second, on the application side, we have illustrated two different uses of SOFs. First, we have seen how to use RP lex in order to assess reforms. Starting with an arbitrary incentive-compatible allocation, it is sufficient to identify the bundle chosen by the agent having the lowest P -equivalent welfare level and to increase the assigned consumption level in the neighborhood of that bundle. We also observed how the remarkable simplicity of the Egalitarian Walrasian allocations makes it possible to single them out as the linear allocations that maximize the RP lex SOF. The objective of the SOF approach presented in this book is to build indices of social welfare based on fairness considerations and to apply these indices in the design of social institutions. Fairness has been interpreted here as equality of resources, and the first two chapters of this part have shown that equality of resources can receive precise axiomatic definitions independently of whether goods are public or private, and whether goods are available or have to be produced. One key assumption that was imposed in this chapter, as well as in all previous chapters of the book, is that the only relevant information about agents are their preferences. In the next chapters, we introduce the possibility that agents differ in terms of some internal resources, and those resources are assumed to be non-tranferable. Resource equality, therefore, requires distributing external resources in an unequal way in order to compensate the agents with lower internal resources. That will be the final topic of the book.

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Chapter 10

Unequall skills 10.1

Introduction

In this chapter, we present and study a new model, which is a variant of the private good production model of the previous chapter. Compared with that model, we assume that the production set is linear and that the agents may have different production skills. Consequently, the agents are characterized by two parameters: their preferences and their skill. The natural interpretation of the model, which has been introduced in the literature a long time ago, from different viewpoints, by Mirrlees (1971), and Pazner and Schmeidler (1974), is that the input is the labor time of the agents and their skill is the wage rate at which they are able to find a job. This is the canonical setting in which labor income taxation issues can be addressed. In the next chapter, we will show how some of the SOFs we derive here can be used to design optimal income tax schemes. Productive skills are assumed to be fixed and independent of the labor time. This amounts to assuming that agents use a linear technology. In view of such linearity, the laisser-faire allocations, i.e., the allocations at which agents choose their labor time and consume the share of the production that corresponds to their share in the contribution, are natural candidates to qualify as equitable allocations. Indeed, each agent freely chooses her preferred bundle and there is no externality among agents. One can argue, however, that these allocations are inequitable, in particular if differences in skills come (even partly) from inherited features which cannot be attributed to the agents’ responsibility, or if agents could be more productive but are constrained by the unavailability of jobs with higher wage rates.1 In this chapter, we endorse the viewpoint that, in the presence of unequal skills, laisser-faire is unfair, and we study the fairness objectives of 1) neutralizing 1 It may be, however, that agents’ skills are also partly the outcome of previous personal choices about investment in human capital. Symmetrically, one may argue that agents are only partially responsible for their preferences over consumption and leisure.

173

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CHAPTER 10. UNEQUALL SKILLS

the consequences of differential skills, and 2) being neutral with respect to the preferences, that is, not correcting for different choices of labor time. We believe these two objectives capture some basic ingredients of the classical debate about labor income taxation. Some people argue that differences in productive skills call for redistribution, whereas others claim that hardworking agents should benefit from their hard work. The SOFs we will propose below show possible ways out of the dilemma, and the study of income tax in the following chapter shows how tax schemes can be related to particular ways of finding a compromise between the two major fairness objectives defined in the previous paragraph. Let us consider these two objectives more closely. To capture the idea that differential skills should not entail unequal individual outcomes, let us consider allocating resources between two agents who have the same preferences. As they only differ with respect to their skills, society should help them reach the same outcome, in terms of welfare, for both of them. We call it the objective of compensation. To capture the idea that the various preferences displayed in the population should not be treated differently, i.e., that the agents should not receive any differential amount of resources on the pure basis that they have “good” or “bad” preferences, let us consider allocating resources between two agents who have the same skill. As they only differ with respect to their preferences, it is natural to consider that society should submit them to the same treatment in terms of redistribution. This implies that ideally, agents with identical skills should be submitted to the same lump sum transfer that does not depend on their preferences and their choice of labor. In such a situation, two agents with identical skills have the same linear budget set with a slope equal to their skill. If that is achieved, when an agent changes his preferences, one additional labor time unit results in an increased consumption by exactly the corresponding additional amount of production and society is not concerned with the agent’s change in preferences. We call it the objective of responsibility.2 The main lesson, from the point of view of fairnesss, of this chapter, is that there exists a deep conflict between these two objectives. One cannot simultaneously say that welfare should not depend on skills when agents have identical preferences, and that transfers should not depend on preferences when agents have identical skills. Given the richness of the model, we will be able to propose different axioms of either compensation or responsibility, and check for their mutual compatibility problems. The results of our study will be the characterizations of families of SOFs which offer reasonable compromises between the two objectives, that is, which satisfy combinations of appropriate weakenings of our basic requirements. A lot of fair allocation problems involve agents having different needs, talents, handicaps, etc., that is, untransferable attributes that determine how successful they may be in transforming external resources into personal outcomes. 2 This is called “liberal reward” in Fleurbaey (2008), in which two different conceptions of the implications of responsibility for reward schemes are distinguished, the “liberal” and the “utilitarian”. We focus here on the former.

10.2. THE MODEL

175

A widespread ethical view on those matters is that external resources need to be allocated or reallocated among these agents in a differentiated way so as to counterbalance such differences in personal attributes.3 We conceived this chapter and the following one in order to illustrate how to address these questions with the help of SOFs.

10.2

The model

The model of this chapter (and the next one) is a variant of the model we studied in Sections 9.2-9.4. Individual bundles are composed of a quantity of labor time ( ) and a level of consumption (c). We further add the restriction that labor time is bounded above by a “full time” level ( ≤ 1). The new and key ingredient of the current analysis is the agent’s production skill si ≥ 0 enabling her to produce the quantity si i of consumption good with labor time i . We further assume that si ∈ S = [smin , +∞), where smin denotes the lowest possible production skill in society, or, in some applications, the lowest wage rate at which agents can find jobs. Compared to the previous chapter, this amounts to assuming that the production function is fixed and linear, and it can be normalized as f ( ) = for all ≤ 0, so that we can forget about f in the definition of economies. An economy is now a list E = (sN , RN ), and the domain of all such economies is E. An allocation zN = (zi )i∈N ∈ X N is feasible in economy E = (sN , RN ) ∈ E if X X ci ≤ si i . i∈N

i∈N

Let Z(E) denote the set of feasible allocations for E. The following terminology will prove useful. For si ∈ R+ and zi = ( i , ci ) ∈ X, let B(si , zi ) ⊂ X denote the budget set obtained with skill si and such that zi is on the budget frontier: B(si , zi ) = {( 0i , c0i ) ∈ X | c0i − si

0 i

≤ ci − si i }.

In the special case where si = 0 and ci = 0, we adopt the convention that B(si , zi ) = {( 0i , c0i ) ∈ X | c0i = 0,

0 i

≥

i }.

For si ∈ R+ , Ri ∈ R and zi = ( i , ci ) ∈ X, let IB(si , Ri , zi ) ⊂ X denote the implicit budget at bundle zi for any agent with characteristics (si , Ri ), that is, the budget set with slope si having the property that zi is indifferent for Ri to the preferred bundle in that budget set: IB(si , Ri , zi ) = B(si , zi0 ) for any zi0 such that zi0 Ii zi and zi0 ∈ max|Ri B(si , zi0 ). By strict monotonicity of preferences, this definition is unambiguous. Notice that bundle z need not belong to the implicit budget. Figure 10.1 illustrates

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c

6 Ri

H Y HH zi •

(si )

B(si , zi )

• IB(si , Ri , zi ) (si ) 1 Figure 10.1: Budgets and implicit budgets. these definitions. In the figures of this chapter, we adopt the convention that the slopes of budget lines are in parentheses below the line. This new model does not require much rewriting of the axioms studied in the previous chapters. For instance, agents are now equals if they have the same preferences and skill. Also, Independence of the Feasible Set amounts to requiring independence of the vector of skills.

10.3

Compensation versus responsibility: ethical dilemmas

Let us now derive the basic axioms from the above definition of our two ethical objectives. Recall that, according to the objective of compensation, differential skills should not entail unequal individual outcomes. Consider two agents who have the same preferences but different skills. If one agent has a larger 3 This

broader class of compensation problems is the subject of Fleurbaey (2008).

10.3. COMPENSATION VERSUS RESPONSIBILITY

177

consumption whereas they have the same labor time, one can argue that the above ethical goal is not fully satisfied and that, other things equal, it would be a social improvement (or at least not a worsening) to transfer an amount of the consumtion good from the richer to the poorer agent. The resulting axiom is logically stronger than Transfer among Equals. Axiom Equal-Preferences Transfer 0 For all E = (sN , RN ) ∈ D, and zN = ( N , cN ) , zN = ( 0N , c0N ) ∈ X N , if there exist j, k ∈ N such that Rj = Rk , and ∆ ∈ R++ such that j = k = 0j = 0k , cj − ∆ = c0j > c0k = ck + ∆, 0 R(E) zN . and for all i 6= j, k, zi = zi0 , then zN

Let us now turn to the ethical objective of responsibility. It implies that the various preferences displayed in the population should not be treated differently, that is, the agents should not receive any differential amount of resources on the pure basis that they have “good” or “bad” preferences. Therefore, agents with identical skills should ideally receive equal lump-sum transfers and then be left free to choose their preferred bundle in the same budget set with a slope equal to their skill. Therefore, the next axiom requires that a transfer of resources between two agents having the same skill and having received unequal lump-sum transfers should be a strict social improvement. Before stating the axiom, we need to make the following formal point. Let us assume that agent j receives a lump-sum transfer and therefore maximizes her preferences over a budget with a slope equal to her skill, sj . If zj = ( j , cj ) is her chosen bundle, then cj − sj j , which may be negative, is equal to the lump-sum transfer she received and is a simple measure of the value of her budget. This measure allows us to introduce the notion of a “budget transfer” (corresponding to a transfer in lump-sum grants). If we move from an allocation ¡ ¢ where zj = ( j , cj ) and zk = ( k , ck ) to another allocation where zj0 = 0j , c0j and zk0 = ( 0k , c0k ) , and if both agents maximize their welfare over budgets of slopes sj = sk , ¡then we say ¢ that there has been a transfer of budget from j to k if (cj − sj j ) − c0j − s0j 0j = (c0k − s0k 0k ) − (ck − sk k ) . Our main responsibility requirement can be phrased as follows: a budget transfer from an agent having a larger budget to another agent having the same skill should be viewed as a (weak) social improvement. Axiom Equal-Skill Transfer 0 For all E = (sN , RN ) ∈ E, zN = ( N , cN ) , zN = ( 0N , c0N ) ∈ X N ¡ , ∆ ∈ R¢++ , if there exist j, k ∈ N such that sj = sk , (cj − sj j ) − ∆ = c0j − s0j 0j > (c0k − s0k 0k ) = (ck − sk k ) + ∆, zj ∈ max|Rj B(sj , zj ),

zj0 ∈ max|Rj B(sj , zj0 ),

zk ∈ max|Rk B(sk , zk ),

zk0 ∈ max|Rk B(sk , zk0 ),

0 R(E) zN . and for all i ∈ N, i 6= j, k, zi = zi0 , then zN

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The first basic result of this chapter states that Equal Preferences Transfer and Equal Skill Transfer are incompatible with each other: it is impossible to guarantee at the same time that agents having identical preferences will be treated independently of their skills, while agents with identical skills will be treated, in terms of budgets, independently of their preferences.4 Theorem 10.1 On the domain E, no SOF satisfies Weak Pareto, Equal-Preferences Transfer and Equal Skill Transfer. The exact proof is left to the reader, but we provide a hint by illustrating the basic structure of the problem. Let us look at Figure 10.2. Several budgets are depicted, associated to either skill s or s0 , with s < s0 , and several indifference curves are drawn, representing preferences R and R0 . Let us consider the four agent economy E = ((s, s, s0 , s0 ), (R, R0 , R, R0 )). Applying Equal-Preferences Transfer twice, first to the pair of agents (s, R) and (s0 , R), and then to the pair (s, R0 ) and (s0 , R0 ), we reach the conclusion that (z b , z¯c , z c , z¯b ) R(E) (z a , z¯d , z d , z¯a ). Applying Equal-Skill Transfer twice, first to the pair of agents (s, R) and (s, R0 ), and then to the pair (s0 , R) and (s0 , R0 ), we reach the conclusion that (z a , z¯d , z d , z¯a ) R(E) (z b , z¯c , z c , z¯b ), which is exactly the symmetric conclusion. Observe that this would yield a contradiction if the social preferences for the after-transfer allocations had to be strict. This simple illustration clearly shows the tension between the two ethical objectives. Agent (s, R), for instance, when consuming za , is considered relatively well-off when compared with agent (s0 , R) (who has the same preferences), according to the objective of compensation, but she is considered relatively disadvantaged when compared with agent (s, R0 ) (who has the same skill), according to the objective of responsibility. This turns out to be the case for each of the four agents of that economy, with the consequence that the successive application of the axioms leads to a cycle. The conflict between the two objectives is deep. We even get an impossibility to combine them when we remove the egalitarian objective embedded in our two axioms, and when we replace them with anonymity axioms. Equal-Preferences Anonymity requires that permuting the bundles of two agents having the same preferences should not affect social welfare. Note that this axiom is not logically related to Equal-Preferences Transfer. Axiom Equal-Preferences Anonymity 0 For all E = (sN , RN ) ∈ E, zN , zN ∈ X N , j, k ∈ N , if Rj = Rk , zj = zk0 and zk = 0 0 zj , and for all i ∈ N such that i 6= j, k, zi = zi0 , then zN I(E) zN . Equal-Skill Anonymity requires that permuting the budgets (lump-sum grants) of two agents having the same skill, when they are precisely, before and after the permutation, consuming their best bundles in their respective budgets, should not affect social welfare. 4 This result, and much of this chapter, draws from Fleurbaey and Maniquet (2005). There has been a long literature on the compensation responsibility dilemma, surveyed in Fleurbaey and Maniquet (1999) and Fleurbaey (2008).

10.3. COMPENSATION VERSUS RESPONSIBILITY c

R

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

179

⎫ ¡ ¡ ⎪ ¡ ⎪ ⎪ ¡ ⎪ ¡ ⎪ ¡ a ¡ ⎪ ¡ z¯ r ⎪ ¡ ⎪ ⎬ ¡ ¡ ¡ b ¡ R0 ¡ ¡ z ¯ ¡ r ⎪ ⎪ ¡ ¡ ¡ ⎪ ¡ ⎪ ⎪ ¡ ¡ ¡ ⎪ ⎪ ¡ ⎪ ⎭ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ÃÃÃÃà ¡ r c Ãà Ãá á ¡ á z¯ á à Ãà ¡ ¡ Ãà ¡ à ÃÃà r à à à à ¡ ¡ ¡ à à à ¡ ÃÃà z¯d ÃÃà ¡ Ãà ÃÃà Ãá ¡Ãá à ÃÃà à à à à ¡ ¡ ¡ à à à à ÃÃà ÃÃá ¡ ¡ à áà ÃÃà ¡ à ÃÃà à à à à ¡ ÃÃà Ãá ¡ (s) à à ¡ à à ÃÃà ÃÃà ¡ ¡ ¡ÃÃÃÃà ÃÃà ¡ ¡ ¡ Ãà à ¡Ã à ¡ rà Ãá à Ãà à a ¡ ¡ à z Ãá á á ¡ ¡ à à r à r c ¡ Ãà ¡ ¡ z b ¡ z ¡ r¡ ¡ ¡ ¡d ¡ ¡ ¡ z¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ (s0 ) ¡ ¡ ¡ ¡ 0 1 6

Figure 10.2: No SOF satisfies Weak Pareto, Equal-Preferences Transfer and Equal-Skill Transfer. Axiom Equal-Skill Anonymity 0 For all E = (sN , RN ) ∈ E, zN , zN ∈ X N , j, k ∈ N , if sj = sk , B(sj , zj ) = 0 0 B(sk , zk ), B(sj , zj ) = B(sk , zk ), and zj ∈ max|Rj B(sj , zj ),

zj0 ∈ max|Rj B(sj , zj0 ),

zk ∈ max|Rk B(sk , zk ),

zk0 ∈ max|Rk B(sk , zk0 ),

0 I(E) zN . and for all i ∈ N, i 6= j, k, zi = zi0 , then zN

The second result of this chapter, unfortunately, confirms the difficulty, revealed by the theorem above, of combining the objectives of compensation and responsibility. Theorem 10.2 On the domain E, no SOF satisfies Weak Pareto, Equal Preferences Anonymity and Equal Skill Anonymity.

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CHAPTER 10. UNEQUALL SKILLS

Again, the proof can be graphical, involving an economy with two skill levels, two preferences E = ((s, s, s0 , s0 ), (R, R0 , R, R0 )). We construct a cycle of allocations in Figure 10.3. Let us start with allocation (z b , z¯f , z e , z¯d ). By Weak Pareto, (z a , z¯e , z d , z¯c ) is a strictly preferred allocation. By EqualPreferences Anonymity, it is indifferent to (z d , z¯e , z a , z¯c ), which, by Equal-Skill Anonymity, is indifferent to (z d , z¯e , z f , z¯b ), itself indifferent to (z d , z¯b , z f , z¯e ), by Equal-Preferences Anonymity. By Weak Pareto, (z c , z¯a , z e , z¯d ) is strictly preferred, but it turns out to be indifferent to the allocation we started with, (z b , z¯f , z e , z¯d ), by application of Equal-Skill Anonymity.

# # r # ⎫ # z¯b ⎪ # ⎪ ⎪ ⎪ # ⎪ ⎪ # ⎪ ⎪ ⎪ # ⎪ ⎪ # ⎪ r za # ⎧ ( ⎪ ( ( ⎪ # ( # ( ⎬ ( ⎪ ( ⎪ ( r # ( ⎪ # ( ( ⎪ ( c R0 ⎪ # z¯ #((((( ⎪ ⎪ ⎪ ⎪ # ⎪ ( #( ⎪ ⎪ ⎪ ⎪ r ((( # r ⎪ (( ⎪ # ( ⎪ ⎪ ( ( r ⎪ b ⎪ # ( ⎪ # ( d ⎪ z ( ⎪ a ⎪ z ¯ ⎪ # z ¯ ⎪ # ⎪ ⎪ ⎪ ⎪ # ⎪ ⎪ # ⎪ ⎪ r ⎭ ⎪ # # ⎪ e ⎪ z ¯ # ⎪ # ⎪ ⎪ # ⎪ # ⎪ (( ⎨ # # (((( ( # ( # ( R (s) r ( # ((( ⎪ # ⎪ f ( ⎪ #(( ( z ¯ ( # ⎪ ( ⎪ (( # ⎪ ⎪ (((( ⎪ ( r # ( ⎪ ( ⎪ (( c r ⎪ # z ⎪ ⎪ # ⎪ zd r e ⎪ ⎪ r # z ⎪ ⎪ ⎪ # f ⎪ ⎪ # z ⎪ ⎪ ⎪ # ⎪ ⎪ ⎪ #0 ⎪ ⎩ # (s ) # # # # 0 1 c

6

Figure 10.3: No SOF satisfies Weak Pareto, Equal-Preferences Anonymity and Equal-Skill Anonymity.

10.4. WEAKENING THE BASIC AXIOMS

10.4

181

Weakening the basic axioms

One possible way out of the above negative results consists in weakening the axioms. In this section, we study examples of such weakened axioms. The first weakenings follow the idea that, in some economies, we know perfectly well what the optimal allocations should look like. Let us first apply this idea to the compensation objective. In an economy where all agents have the same preferences, the only optimal allocations should be the Pareto-efficient ones at which all agents are assigned bundles they deem equivalent. Indeed, no agent finds her bundle worse than that of any other agent, so that final outcomes can be claimed to have been equalized. If we apply the same idea to the responsibility objective, we look at economies where all agents have the same skill. In those economies, inequalities are justified, provided they only come from different choices. Consequently, letting agents maximize in the same budget set is optimal, and this is what laisser-faire prescribes. This leads us to the following two axioms. Axiom Equal-Welfare Selection For all E = (sN , RN ) ∈ D, zN ∈ max|R(E) Z(E), if for all i, j ∈ N , Ri = Rj , then for all i, j ∈ N , zi Ii zj . Axiom Laisser-Faire Selection For all E = (sN , RN ) ∈ D, zN ∈ max|R(E) Z(E), if for all i, j ∈ N , si = sj , then for all i ∈ N , zi ∈ max|Ri B(si , (0, 0)). Equal-Welfare Selection is logically weaker than Equal-Preferences Transfer, while Laisser-Faire Selection is logically weaker than Equal-Skill Transfer. Surprisingly, the duality between skills and preferences stops here. More precisely, we will show that, although it is easy to combine Laisser-Faire Selection with Equal-Preferences Transfer, the dual result does not hold (i.e., combining EqualWelfare Selection with Equal-Skill Transfer is harder). We begin by exploring the former possibility. First, we define the skill equivalent well-being index. It is the skill level that leaves the agent indifferent between her current bundle and being free to choose her labor time at that skill level. Formally, for zi ∈ X, and Ri ∈ R, us (zi , Ri ) = u ⇔ zi Ii max|Ri B(u, (0, 0)). That well-being index is well defined only over bundles zi ∈ X such that zi Ri (0, 0). For E = (sN , RN ) ∈ E, let the set of admissible allocations A(E) be defined as A(E) = {zN ∈ Z(E) | zi Ri (0, 0) ∀ i ∈ N }. The following SOF is defined only over those allocations.

Social ordering function s-Equivalent Leximin (Rslex ) 0 For all E = (sN , RN ) ∈ E, zN , zN ∈ A(E), 0 zN Rslex (E) zN ⇔ (us (zi , Ri ))i∈N ≥lex (us (zi0 , Ri ))i∈N .

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This SOF is illustrated in Figure 10.4, representing a similar economy to the one used in the proof of the impossibility theorems 10.1 and 10.2. There are two skill levels, s and s0 , and two preferences, R and R0 and the four agents have all the possible combinations of skill levels and preferences, so that s1 = s2 = s < s3 = s4 = s0 , R1 = R3 = R and R2 = R4 = R0 . In allocation (z1 , z2 , z3 , z4 ), agent 1, the low skill, low willingness to work, agent, enjoys a larger satisfaction level than agent 3 who has the same preferences but a higher skill. On the other hand, among the high willingness to work agents, the high skill one, agent 4, is assigned a better bundle than agent 2. According to the us well-being index, agent 2 is the worst-off agent, and agent 1 the best-off. At that allocation, Rslex would recommend to redistribute from 1 and 4 to 2 and 3. 6 c (us (z1 , R1 ))

z4 •

(us (z4 , R4 )) z2 •

• z1

(us (z2 , R2 )) z3 •

@ I @ (us (z3 , R3 ))

0

1 Figure 10.4: The s-Equivalent Leximin function.

The preferred allocations for this social ordering function, among feasible allocations, are such that all the agents’ skill-equivalent well-being indices are equal. This corresponds to the Equal-Wage Equivalent allocation rule studied by Fleurbaey and Maniquet (1999a).

10.4. WEAKENING THE BASIC AXIOMS

183

Theorem 10.3 On the domain E, the s-Equivalent Leximin function Rslex satisfies Strong Pareto, Equal-Preferences Transfer, Equal-Preferences Anonymity, Laisser-Faire Selection, Unchanged-Contour Independence, Separation, Replication and Independence of the Feasible Set. Conversely, if a SOF R satisfies Strong Pareto, Equal-Preferences Transfer, Laisser-Faire Selection, UnchangedContour Independence, and Separation, then for all E = (sN , RN ) ∈ E, all 0 zN , zN ∈ XN , 0 min us (zi , Ri ) > min us (zi0 , Ri ) ⇒ zN P(E) zN . i∈N

i∈N

The proof makes use of the fact that Strong Pareto, Equal-Preferences Transfer, and Unchanged-Contour Independence imply an absolute priority for the worse off (in the form of Equal-Preferences Priority), as explained in Section 10.6 below. The other key element of the proof is to justify a preference for zN 0 over zN when for some j, k ∈ N, us (zj0 , Rj ) > us (zj , Rj ) > us (zk , Rk ) > us (zk0 , Rk ), while for i 6= j, k, zi0 = zi . This is done in the following way. First, introduce two agents a and b (by Separation) with skill equal to s0 ∈ (us (zk , Rk ), us (zj , Rj )) , and such that Ra = Rj , Rb = Rk . Let za , zb be the bundles these two agents choose from the “laisser-faire” budget set B(s0 , (0, 0)). Observe that zj Pj za and zb Pk zk . By Unchanged-Contour Independence and Equal-Preferences Priority, agent a’s satisfaction can be raised a little while j is pulled down from zj0 to zj , and agent b’s satisfaction can be lowered a little while k is moved from zk0 to a better bundle zk00 below zk . By Separation, one can focus on changes affecting a and b separately from the rest, and conclude that by Laisser-Faire Selection, it would be at least as good to bring a and b back to their laisser-faire bundles. By Separation again, one can remove a and b from the economy and conclude that moving j and k from zj0 , zk0 to zj , zk00 is a weak improvement. By Strong Pareto, moving them to zj , zk is a strict improvement. Once this argument is made, it is easy to reach the conclusion of the theorem, in a similar fashion as it has been done in the characterizations of Chapter 5. On the other hand, if we try to combine Equal-Welfare Selection with EqualSkill Transfer, which is the dual to the combination of Laisser-Faire Selection and Equal-Preferences Transfer, we face the following difficulty. Theorem 10.4 On the domain E, no SOF satisfies Weak Pareto, Equal-Skill Transfer, Equal-Welfare Selection, and Separability. Proof. The economy is E = ((s, s0 , s, s0 ), (R, R, R0 , R0 )) but we will also consider the variants E 0 = ((s, s0 , s, s0 ), (R, R, R, R)) and E 00 = ((s, s0 , s, s0 ), (R0 , R0 , R0 , R0 )). The proof is illustrated in Figure 10.5. By Weak Pareto¡ and Equal-Welfare Selection, the unique best allocation in ¢ Z(E 0 ) for R(E 0 ) is z a , z b , z a , z b . Therefore, ¡ ¢ ¡ a b a b¢ z , z , z , z P(E 0 ) z¯a , z¯b , z a , z b .

184

CHAPTER 10. UNEQUALL SKILLS

c

⎫ ¡ ¡ ⎪ ¡ ⎪ ⎪ ¡ ⎪ ¡ ⎪ ⎪ ¡ ⎪ ¡ ⎪ ⎬ ¡ ¡ R0 ¡ ¡ d ¡ ⎪ ⎪ z ¯ ¡ ¡ ⎪ r ¡ ⎪ ¡ ¡ ⎪ ¡ ⎪ ⎪ ¡ d ⎪ ⎭ z ¡ ¡ ¡ r ¡ ¡ ¡ ¡ ¡ ¡ÃÃà ¡Ãà ¡ ¡ r à à ÃÃà ¡z c ¡ ¡ á á à ¡ ¡ ÃÃÃà à à ¡ r à Ãà à ¡Ã ¡ ¡ à à à à ¡ Ãà à z¯c à à à à ¡ ¡ ¡ Ãà ¡ÃÃÃà à à ¡ ¡ Ãà ¡ ÃÃà á ÃÃá ¡ ¡ ¡ à à ÃÃà ÃÃà ¡ ¡ ¡ ¡ ÃÃà ÃÃà à à ¡ ¡ ¡ (s) Ãà ¡ ÃÃà ¡Ãà ¡Ãà ¡ à ¡ à ÃÃà à à à à ¡ ¡ ¡ Ãà ¡ÃÃÃà à ¡ ¡ÃÃÃà ¡ à à r à ¡ à à à ¡Ã ¡ rà Ãà Ãà ¡z b ¡ ÃÃà ¡ ¡ z¯a ¡Ãà à ¡ r Ãà ¡ ¡ ¡(s0 ) rà Ãà z¯b a ¡ z¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ 0 1

R

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

6

Figure 10.5: No SOF satisfies Weak Pareto, Equal-Skill Transfer, Equal-Welfare Selection, and Separability. By Separability, this is equivalent to ¡ a b c d¢ ¡ ¢ z , z , z , z P(E) z¯a , z¯b , z c , z d .

00 00 By ¡ c a dsimilar ¢reasoning, as the unique best allocation in Z(E ) for R(E ) is c d z , z , z , z , one deduces that ¡ a b c d¢ ¡ ¢ z¯ , z¯ , z , z P(E) z¯a , z¯b , z¯c , z¯d .

By transitivity,

¡ a b c d¢ ¡ ¢ z , z , z , z P(E) z¯a , z¯b , z¯c , z¯d .

By Equal-Skill Transfer, ¡ a b c d¢ ¡ ¢ z¯ , z , z¯ , z R(E) z a , z b , z c , z d

and

¡ a b c d¢ ¡ ¢ z¯ , z¯ , z¯ , z¯ R(E) z¯a , z b , z¯c , z d ,

10.4. WEAKENING THE BASIC AXIOMS

185

implying by transitivity: ¡ a b c d¢ ¡ ¢ z¯ , z¯ , z¯ , z¯ R(E) z a , z b , z c , z d . One therefore has a contradiction.

A similar impossibility would be reached if Equal-Skill Transfer were replaced with Equal-Skill Anonymity. Note that the impossibility involves Separability and vanishes in absence of this axiom. The axioms of Strong Pareto, Equal-Skill Transfer, and Equal-Welfare Selection are satisfied by any SOF that applies the leximin criterion to indexes of well-being defined as max {uRN (z) | z ∈ IB(si , Ri , zi )} , where uRN is an increasing continuous utility function that depends on the profile RN and satisfies the property that when there is R0 such that for all i ∈ N, Ri = R0 , then uRN is a representation of R0 . We now study another possible weakening of Equal-Skill Transfer, proposed by Valletta (2009) in a slightly different context. Let us define the following partial order on the set R of preferences. For Ri , Ri0 ∈ R, we say that Ri is at least as industrious as Ri0 , noted Ri &M I Ri0 , if and only if for every possible budget set, the best bundle of Ri involves at least as much labor (and, therefore, at least as much consumption) as that of Ri0 . Formally, Ri &M I Ri0 ⇔ ∀s ∈ S, x ∈ X, max|Ri B(s, x) ≥ max|R0 B(s, x). i

The next axiom requires to apply Equal-Skill Transfer to pairs of agents such that the relatively richer agent is also more industrious. That axiom is consistent with the idea that consumption inequality, among two agents having the same skill, should be bounded above by what the difference in labor time justifies, that is, if sj = sk , Rj &M I Rk , and cj − ck > sj ( j − k ), then j has a relatively too large consumption level and a budget transfer from j to k is desirable. What the axiom does not require, contrary to Equal-Skill Transfer, is that the optimal difference in consumption be exactly the one related to the difference in labor time: cj − ck = sj ( j − k ). Consequently, the axiom also has a small compensation flavour, and that is why it is compatible with Equal-Preferences Transfer. Axiom &MI -Equal-Skill Transfer 0 For all E = (sN , RN ) ∈ E, zN = ( N , cN ) , zN = ( 0N , c0N ) ∈ X N , ∆ ∈ R++ , MI if there exist j, k ∈ N such that s = s , R Rk , (cj − sj j ) − ∆ = j k j & ¡ 0 ¢ cj − s0j 0j > (c0k − s0k 0k ) = (ck − sk k ) + ∆, zj ∈ max|Rj B(sj , zj ),

zj0 ∈ max|Rj B(sj , zj0 ),

zk ∈ max|Rk B(sk , zk ),

zk0 ∈ max|Rk B(sk , zk0 ),

0 and for all i ∈ N, i 6= j, k, zi = zi0 , then zN R(E) zN .

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CHAPTER 10. UNEQUALL SKILLS

The following SOF satisfies Equal Preferences Transfer and &M I -Equal-Skill Transfer. It works by applying the leximin to the following smin -equivalent well-being index. It corresponds to the lump-sum transfer that leaves the agent indifferent between her current bundle and being free to choose her labor time in a budget set defined by this lump-sum transfer and the minimal skill level. Formally, for zi ∈ X, and Ri ∈ R, usmin (zi , Ri ) = u ⇔ zi Ii max|Ri B(smin , (0, u)). Social ordering function smin -Equivalent Leximin (Rsmin lex ) 0 For all E = (sN , RN ) ∈ E, zN , zN ∈ XN , 0 ⇔ (usmin (zi , Ri ))i∈N ≥lex (usmin (zi0 , Ri ))i∈N . zN Rsmin lex (E) zN

This SOF is illustrated in Figure 10.6, using the same economy and the same allocation as in Figure 10.4. As with Rslex , agent 1 is considered better-off than agent 3, and agent 4 than agent 2, but so would consider any SOF satisfying Equal-Preferences Transfer. The interesting observation is that agent 1 is now considered worse-off than agent 4 and agent 3 worse-off than agent 2. A social welfare improving reform would now ask agent 4 to contribute more and agent 1 less than in the previous case. Compared to Rslex , Rsmin lex is less favorable to the hardworking agents.

c

6

z4 •

z2 • usmin (z4 , R4 ) usmin (z1 , R1 ) usmin (z2 , R2 )

z1 • • z3

usmin (z3 , R3 ) 0

1

Figure 10.6: The smin -Equivalent Leximin function. We have the following axiomatization result, the proof of which is in the appendix.

10.5. COMPROMISE AXIOMS

187

Theorem 10.5 On the domain E, the smin -Equivalent Leximin function Rslex satisfies Strong Pareto, Equal-Preferences Transfer, Equal-Preferences Anonymity, &M I -Equal-Skill Transfer, Unchanged-Contour Independence, Separation, Replication and Independence of the Feasible Set. Conversely, if a SOF R satisfies Strong Pareto, Equal-Preferences Transfer, &M I -Equal-Skill Transfer, UnchangedContour Independence, and Separation, then for all E = (sN , RN ) ∈ E, all 0 zN , zN ∈ XN , 0 . min usmin (zi , Ri ) > min usmin (zi0 , Ri ) ⇒ zN P(E) zN i∈N

i∈N

The proof is similar to that of Theorem 10.3 and again relies on the fact that Equal-Preferences Priority follows from Strong Pareto, Equal-Preferences Transfer, and Unchanged-Contour Independence (see Section 10.6). One also 0 proves that there must be a preference for zN over zN when for some j, k ∈ N, usmin (zj0 , Rj ) > usmin (zj , Rj ) > usmin (zk , Rk ) > usmin (zk0 , Rk ), while for i 6= j, k, zi0 = zi . This is done in the following way. First, introduce two agents a and b (by Separation) with skill equal to smin , and with smin equivalent utilities between usmin (zj , Rj ) and usmin (zk0 , Rk ). Their preferences are constructed so that the indifference curve of a (resp., b) does not cross I(zj , Rj ) (resp., I(zk , Rk )). Therefore, by Unchanged-Contour Independence and Equal-Preferences Priority, agent a’s satisfaction can be raised a little while j is pulled down from zj0 to zj , and agent b’s satisfaction can be lowered a little while k is moved from zk0 to a better bundle zk00 below zk . Moreover their preferences must be such that for any budget with slope smin or more, a chooses to work full time (this is obtained if a has linear preferences with a slope of indifference curves slightly less than smin ). Therefore his preferences dominate those of b for &M I , making it possible to invoke &M I -Equal-Skill Transfer and bring a and b back to their initial bundles. By Separation they can then be removed from the economy, and in conclusion one obtains that moving j and k from zj0 , zk0 to zj , zk00 is acceptable. By Strong Pareto, moving them to zj , zk is then a strict improvement.

10.5

Compromise axioms

Another way of avoiding the negative result of Section 10.3 is by defining axioms that combine the objectives of compensation for low skill and responsibility for preferences. We give an example of such an axiom in this section.5 This axiom is consistent with Kolm’s idea that members of a society should decide on a fraction ˜ of individual labor time such that incomes should be equalized among individuals working ˜ (see, e. g., Kolm 1996, 2004). More precisely, it requires that if two agents freely choose a labor time of ˜ but do not enjoy the same consumption level, then a transfer of consumption between them is a strict 5 This

section is drawn from Fleurbaey and Maniquet (2009).

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CHAPTER 10. UNEQUALL SKILLS

social improvement, provided they still end up freely choosing a labor time of ˜ after the transfer. The compensation content of this axiom comes from the fact that the two agents may have different skills, so that the larger consumption may originate in one agent having a higher skill. The axiom prevents the high-skilled agent from reaching a larger consumption level than the low-skilled one. The responsibility content of the axiom comes from the fact that it restricts its attention to agents having some particular preferences. Among such agents, i.e., among agents choosing a labor time of ˜, the objective of the axiom is to equalize consumption, so that, if the two agents have the same skill, a budget transfer between them is a social improvement. Axiom ˜-Labor Transfer: 0 For all E = (sN , RN ) ∈ E, zN , zN ∈ X N , j, k ∈ N , if 0 0 zj > zj > zk > zk , and

j

=

0 j

zj ∈ max|Rj B(sj , zj ),

zj0 ∈ max|Rj B(sj , zj0 ),

zk ∈ max|Rk B(sk , zk ),

zk0 ∈ max|Rk B(sk , zk0 ),

=

k

=

0 k

= ˜,

0 and for all i 6= j, k, zi = zi0 , then zN R(e) zN .

The key observation of this section is that there are SOFs that satisfy ˜Labor Transfer and Equal-Preferences Transfer or ˜-Labor Transfer and EqualSkill Transfer. The SOFs that satisfy ˜-Labor Transfer and Equal-Preferences Transfer are of the egalitarian equivalent kind, and are somehow similar to the SOFs already defined in this chapter. A prominent example is the leximin SOF that measures individual well-being by the intersection of an agent’s indifference curve with the vertical line of equation = ˜. We concentrate in this section on the SOFs that satisfy ˜-Labor Transfer and Equal-Skill Transfer. They are of the Walrasian kind, but like egalitarianequivalent SOFs they also work by applying the leximin criterion to some index of well-being. Let us define this index. The novelty is that it depends on the skill of the agent. The level of well-being associated to a bundle is defined with respect to the budget that leaves the agent indifferent between this bundle and freely choosing in that budget the slope of which is now the real skill of the agent. Once that budget has been identified, the value of the well-being index is the level of consumption that corresponds to a labor time of ˜ for that budget. Formally, for zi ∈ X, and Ri ∈ R, u ˜(zi , si , Ri ) = u ⇔ zi Ii max|Ri B(si , ( ˜, u)). That well-being index is non-negative only over bundles zi ∈ X such that zi Ri max|Ri B(si , ( ˜, 0)). For E = (sN , RN ) ∈ E, let the set of admissible ˜

allocations A (E) be defined as ˜

A (E) = {zN ∈ Z(E) | ∀ i ∈ N, zi Ri max|Ri B(si , ( ˜, 0))}.

10.6. FROM TRANSFER TO PRIORITY AXIOMS

189

The following SOF is defined only over those allocations.6 ˜ Social ordering function ˜-Egalitarian Walrasian Leximin (R EW ) ˜ 0 ∈ A (E), For all E = (sN , RN ) ∈ E, zN , zN

zN R

˜EW

¡ ¢ ¢ ¡ 0 (E) zN ⇔ u ˜(zi , si , Ri ) i∈N ≥lex u ˜(zi0 , si , Ri ) i∈N

This SOF is illustrated in Figure 10.7, using the same economy and the same allocation as before. The well-being index is now computed by drawing implicit budgets having the same slope as the real skill of the agents. The different values of the index are read on the vertical axis corresponding to a labor time of ˜, which is chosen in the graph to be equal to 0.6. We can see that the social evaluation of the agents’ positions is drastically different from the previous ones. Agent 4 is now considered the worst-off: the income she would earn by choosing a labor time of ˜ in her implicit budget is too low. A social welfare improving reform would benefit her. The other striking feature of the figure is that agent 1 is now considered worse-off than agent 3. ˜ That shows that R EW benefits the hardworking high-skilled agents and the lazy low-skilled ones. This conclusion depends on the value of ˜ and is valid only for intermediate values. For ˜ sufficient low, the worse-off agents would become 3 and 4, the two high-skilled agents, whereas for large values of ˜, the worse-off would become 1 and 2, the two low-skilled agents. ˜ The R EW SOF satisfies Strong Pareto, Equal-Skill Transfer, Equal-Skill Anonymity, ˜-Labor Transfer, Unchanged-Contour Independence, Separation, and Replication. On the other hand, it does not satisfy Independence of the Feasible Set, as the well-being index that it uses depends on the skills of the agents, so that changing the skill profile affects the social ranking. Importantly, many other SOFs satisfy the same list of axioms, and they do not all have the leximin or any maximin type. We develop this issue in the next section.

10.6

From transfer to priority axioms

All the SOFs that we have studied in this chapter up to now are leximin SOFs in a particular index of well-being. This is not a surprise, of course, given all the previous results, and given, especially, the discussion of Chapter 3. In this section, nonetheless, we look carefully at those results about infinite inequality aversion, as they are not all derived from the same combinations of axioms. Combining Equal-Preferences Transfer with Pareto axioms and either UnchangedContour Independence or Separation unsurprisingly leads to an axiom of EqualPreferences Priority. But a similar statement does not hold with Equal-Skill Transfer. There are SOFs, indeed, that do satisfy Strong Pareto, Equal Skill Transfer, strong robustness requirements, and that are not of the leximin type. 6 Allowing for negative values of the index, it is easy to extend this SOF over all X N . We do not do it here because the characterization of this SOF that is proposed in Section 10.6 works only for the admissible allocations.

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CHAPTER 10. UNEQUALL SKILLS 6

c

z4 •

•u ˜(z3 , s3 , R3 ) • u ˜(z1 , s1 , R1 ) z1

•

•

z2 •

u ˜(z2 , s2 , R2 )

• u ˜(z4 , s4 , R4 )

• z3

0

1

˜ = 0.6 Figure 10.7: The ˜-Egalitarian Walrasian leximin.

Here is an example. Let α ≤ 1, α 6= 0. The SOF R defined by: for all ˜ 0 ∈ A (E), E ∈ E, zN , zN 1X 1X 0 zN R(E) zN ⇔ (u ˜(zi , Ri ))α ≥ (u ˜(zi0 , Ri ))α , α α i∈N

i∈N

satisfies Strong Pareto, Equal-Skill Transfer, Equal-Skill Anonymity, UnchangedContour Independence, Separability, and Replication. Would it be promising to study non-leximin SOFs such as this one? This is doubtful. Indeed, for all values of α, the SOF defined in the theorem violates the simple Transfer among Equals axiom (recall that agents are equals in this model if they have both the same preferences and the same skill), and so would do any SOF satisfying Pareto Indifference, Equal-Skill Transfer and UnchangedContour Independence, as long as they do not satisfy Equal-Skill Priority. This is what we prove now. Axiom Transfer among Equals For all E = (sN , RN ) ∈ D, and zN = (

0 N , cN ) , zN

=(

0 0 N , cN )

∈ X |N | , j, k ∈ N ,

10.6. FROM TRANSFER TO PRIORITY AXIOMS ∆ ∈ R++ , if sj = sk and Rj = Rk ,

j

=

k

=

0 j

=

0 k,

191

and

cj − ∆ = c0j > c0k = ck + ∆, 0 and for all i 6= j, k, zi = zi0 , then zN R(E) zN .

Axiom Equal-Skill Priority 0 0 0 N For all E = (sN , RN ) ∈ E, zN = ( N , cN ) , zN ¡=0 ( N ,0 cN ¢) ∈ X0 , if0 there 0 exist j, k ∈ N such that sj = sk , (cj − sj j ) > cj − sj j > (ck − sk 0k ) > (ck − sk k ), zj ∈ max|Rj B(sj , zj ),

zj0 ∈ max|Rj B(sj , zj0 ),

zk ∈ max|Rk B(sk , zk ),

zk0 ∈ max|Rk B(sk , zk0 ),

0 and for all i 6= j, k, zi = zi0 , then zN R(E) zN .

Lemma 10.1 On the domain E, if a SOF R satisfies Pareto Indifference, Transfer among Equals, Equal-Skill Transfer, Unchanged-Contour Independence, and Separation, then it satisfies Equal-Skill Priority.7 Proof. The proof, which bears some similarity with the proof of Theorem 10.5, is illustrated in Figure 10.8 which features four agents: j, k such that sj = sk , and a, b such that (sa , Ra ) = (sj , Rj ) and (sb , Rb ) = (sk , Rk ). First, with the same reasoning as for Theorem 3.1, one sees that Pareto Indifference, Transfer among Equals and Unchanged-Contour Independence imply Priority among Equals: Axiom Priority among Equals 0 = ( 0N , c0N ) ∈ X |N| , j, k ∈ N , For all E = (sN , RN ) ∈ D, and zN = ( N , cN ) , zN 0 ∆ ∈ R++ , if sj = sk and Rj = Rk , j = k = j = 0k , and cj > c0j > c0k > ck , 0 and for all i 6= j, k, zi = zi0 , then zN R(E) zN .

Then, one can apply Prority among Equals between j and a and between k and b (see the figure) in order to derive that ¡ 0 0 0 0¢ zj , zk , za , zb R((sj , sk , sj , sk ) , (Rj , Rk , Rj , Rk )) (zj , zk , za , zb ) .

By Equal-Skill Transfer applied to a and b, ¡ 0 0 ¡ ¢ ¢ zj , zk , za , zb R((sj , sk , sj , sk ) , (Rj , Rk , Rj , Rk )) zj0 , zk0 , za0 , zb0 .

7 Separation can be weakened into Separability in the above lemma if there are at least four agents in the economy. Also, Unchanged-Contour Independence can be dropped altogether if Transfer among Equals is strengthened into Nested-Contour Transfer.

192 c

Rk

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

CHAPTER 10. UNEQUALL SKILLS 6

ÃÃà zj à s Ãà à à ÃÃà ÃÃà à à ÃÃà ÃÃà à zj0 ÃÃÃÃà à s Ãà à à à à à Ãà Ãà à 0 à à ÃÃà za à s ÃÃÃÃà ÃÃà à à à à s Ãà à ÃÃà ÃÃà ÃÃà ÃÃÃÃÃÃÃà za à à à à à Ãà ÃÃà à à à ÃÃà ÃÃà ÃÃÃÃÃÃÃà à à à à à à Ãà ÃÃÃà Ãà ÃÃà ÃÃà à ÃÃà ÃÃÃÃÃÃÃà à ÃÃà ÃÃÃÃÃà ÃÃà ÃÃà à à à à ÃÃà zbs ÃÃÃÃÃÃÃÃà ÃÃà ÃÃà à à Ãà à à ÃÃà à à à à à à s0 (sj = sk ) Ãà ÃÃà ÃÃà zb ÃÃà ÃÃà à à à à à sà ÃÃà Ãà ÃÃà zk0 à à à s ÃÃà Ãà zk 0

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

Rj

-

1

Figure 10.8: Pareto Indifference, Transfer among Equals, Equal-Skill Transfer, Unchanged-Contour Independence, and Separation imply Equal-Skill Priority. By transitivity, ¡ 0 0 ¢ zj , zk , za , zb R((sj , sk , sj , sk ) , (Rj , Rk , Rj , Rk )) (zj , zk , za , zb ) .

By Separation,

¡ 0 0¢ zj , zk R((sj , sk ) , (Rj , Rk )) (zj , zk ) .

By Separation again, reintroducing the other agents i 6= j, k for whom zi0 = zi , 0 one obtains zN R(E) zN . Equal-Skill Transfer seemed to be compatible with a non-infinite rate of inequality aversion, but the addition of the basic requirement of Transfer among Equals forces us to reintroduce an infinite inequality aversion. The above lemma confirms that only Leximin types of SOFs are left when one tries to combine axioms of efficiency, fairness and robustness. Indeed, we get the following result.

10.6. FROM TRANSFER TO PRIORITY AXIOMS

193

Theorem 10.6 On the domain E, if a SOF R satisfies Strong Pareto, Transfer among Equals, Equal-Skill Transfer, ˜-Labor Transfer, Unchanged-Contour Independence, and Separation, then it satisfies the following maximin property: 0 for all E = (sN , RN ) ∈ E, all zN , zN ∈ XN , 0 . min u ˜(zi , si , Ri ) > min u ˜(zi0 , si , Ri ) ⇒ zN P(E) zN i∈N

i∈N

We simply illustrate the key part of the easy proof in Figure 10.9. It consists in showing that if for two agents j, k ∈ N, u ˜(zj0 , sj , Rj ) > u ˜(zj , sj , Rj ) > u ˜(zk , sk , Rk ) > u ˜(zk0 , sk , Rk ), 0 . while for i 6= j, k, zi0 = zi , the allocation zN is better than zN

c

6

³³ zj0 ³³³ s ³³ ³³ ³ ³³ ³³ ³³ ³ ³ zj s ³³³ ³ ³³ ³³ ³³ ³³ ³³ ³ ³ ³³³³ ³ ³ ³ ³³ ³³ ³³ ³³ ³³³³³ ³ ³ (sj ) ³³ ³³ ³³ ³³ za ³³³³³³ ³ s ³³ ³³ ³³ s 0³ ³³³³ ³ ³ ³ z ³³³³³ 0 a ÃÃà ÃÃà zb à s ÃÃÃÃà ÃÃà ³³ à à à à s ÃÃÃÃÃÃà ÃÃà à ÃÃà ÃÃÃÃÃÃÃÃà zbÃà à à ÃÃà à à à à à à à à (sk ) Ãà ÃÃÃà ÃÃÃà ÃÃà ÃÃà ÃÃÃÃÃÃÃà à à szà ÃÃà Ãà ÃÃà 00 s k à à zk à s ÃÃà Ãà 0 zk 0

˜

-

1

Figure 10.9: Illustration of the proof of Theorem 10.6. Introduce agents a, b such that sa = sj and sb = sk . By Lemma 10.1, we know that we can apply Equal-Skill Priority. Doing so between j and a, as well as between k and b, one sees that ¡ ¢ (zj , zk00 , za , zb ) R((sj , sk , sj , sk ) , (Rj , Rk , Ra , Rb )) zj0 , zk0 , za0 , zb0 .

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CHAPTER 10. UNEQUALL SKILLS

By ˜-Labor Transfer applied to a and b, (zj , zk00 , za0 , zb0 ) R((sj , sk , sj , sk ) , (Rj , Rk , Ra , Rb )) (zj , zk00 , za , zb ) . By transitivity, ¡ ¢ (zj , zk00 , za0 , zb0 ) R((sj , sk , sj , sk ) , (Rj , Rk , Ra , Rb )) zj0 , zk0 , za0 , zb0 .

By Separation,

By Strong Pareto,

¡ ¢ (zj , zk00 ) R((sj , sk ) , (Rj , Rk )) zj0 , zk0 .

so that

¡ ¢ (zj , zk ) P((sj , sk ) , (Rj , Rk )) zj0 , zk00 , ¢ ¡ (zj , zk ) P((sj , sk ) , (Rj , Rk )) zj0 , zk0 .

By Separation again, reintroducing the other agents (with unchanged bundles 0 0 in zN and zN ), one obtains zN P(E) zN .

10.7

Conclusion

Several families of SOF’s end up being characterized in this chapter. The following table summarizes the relative merits of each solution. Table 10.1: Properties of the SOFs Strong Pareto Transfer among Equals Equal-Preferences Transfer Equal-Preferences Selection Equal-Skill Transfer Laisser-Faire Selection &M I -Equal-Skill Transfer ˜-Labor Transfer Unchanged-Contour Independence Separation Replication Independence of the Feasible Set

Rslex + + + + − + − − + + + +

Rsmin lex + + + + − − + − + + + +

R

˜EW

+ + − − + + + + + + + −

Again, this chapter has illustrated what can be expected from the SOF approach: identify the possible trade-offs between fairness axioms capturing ethical views on the resource allocation problem at hand, and deduce the SOF that satisfies some relevant set of fairness properties, together with efficiency and robustness axioms. The main ethical conflict, in the model we have studied here, is the one between compensation for low skills and responsibility for individual preferences. We saw that those conflicting axioms end up justifying quite

10.7. CONCLUSION

195

different SOFs. The Rslex SOF favors hardworking agents whereas the Rsmin lex ˜ SOF favors lazy agents and they both favor low-skilled agents. The R EW SOF, for intermediate values of ˜, favors hardworking high-skilled or lazy low-skilled agents. Given that the current model is the canonical model in which labor income taxation is discussed, we will investigate it further to identify which income tax scheme may be compatible with the maximization process of the SOFs defined here. This is the topic of the next chapter.

196

CHAPTER 10. UNEQUALL SKILLS

Chapter 11

Income taxation 11.1

Introduction

The ultimate objective of any approach to welfare economics is to provide criteria for the evaluation of social policies. In this chapter, we show how the SOFs axiomatized in the previous chapter engender such criteria. Once a policy-maker has chosen the SOF capturing her ethical preferences, she typically faces information constraints that prevent her from being able to implement the allocations that are optimal according to her SOF. The information constraints may be of different sorts, and we show here how the SOFs we have defined adjust to these different informational structures. We study two informational structures. In some applications, we assume that labor time and earnings are observable, but skills are not. Agents may then choose to work at a lower wage rate than their skill, if it is in their interest to do so. In other applications, we assume that labor time is no longer observable, only earnings are. Under either set of assumptions, the policy-maker has to identify the optimal allocation among the ones that are compatible with the agents’ incentives to hide their private information. Our SOF approach turns out to accommodate these different structures easily. This comes from two central properties, which we have already insisted on. The first property is that the SOFs that satisfy the axioms we impose are of the leximin type. Consequently, social welfare is appropriately measured by the well-being index of the worst-off agents. It turns out to be easily measured. The second property is the informational simplicity of the well-being indices we use. They rely on ordinal information on individual preferences, so that agents’ choices reveal the information that is needed to compute these indices. As is well known, restricting one’s attention to incentive-compatible allocations is equivalent to studying income tax schemes. The applications we present here will turn into identifying properties that optimal tax schemes should satisfy. Here is a preview of the main results of the chapter. First, we will prove that SOFs satisfying Equal Preferences Transfer, the 197

198

CHAPTER 11. INCOME TAXATION

most demanding compensation axiom, call for more redistribution than the SOFs satisfying Equal Skill Transfer, the most demanding responsibility axiom. Second, we will prove that all the SOFs highlighted here call for non-positive marginal rates of taxation on low incomes. That is the most important result of ˜ the chapter. Both Rsmin lex and R EW , in spite of the considerable differences in the axioms that justify them, call for a zero marginal rate of taxation, whereas ˜ Rslex calls for negative marginal tax rates. The third conclusion is that R EW , in spite of its egalitarian flavour, may call for low redistribution. In Section 2, we define the model and study the incentive compatibility constraints. In Section 3, we derive the simple criteria that can be used to evaluate current income tax schemes and identify who should benefit in priority from a reform. In Section 4, we study some characteristics of the optimal tax schemes in simple economies composed of four types of agents. In Section 5, we turn to the characteristics of optimal tax schemes in general economies.

11.2

The model

We keep the same model as in the previous chapter. An economy is a list E = (sN , RN ) ∈ E where for all i ∈ N , si ∈ S = [smin , smax ] denotes i’s production skill and Ri her preferences over labor time-consumption bundles zi = ( i , ci ) ∈ X = [0, 1] × R+ . The policy-maker has social preferences represented by a SOF R defined over all allocations. If information were complete, she could simply maximize R over the set of feasible allocations. We assume in this chapter that information is not complete. Skills and preferences are private information of the agents. The policy-maker knows the different elements of sN and RN but does not know who is who. To state it differently, she only knows the statistical distribution of types in the economy. Consequently, she has to restrict her attention to incentive-compatible allocations. The contents of the set of incentive-compatible allocations depend on what the policy-maker is able to observe. We consider two different informational contexts in turn. In the first context, the policy-maker observes labor time and pre-tax income for each agent. Given that the policy-maker does not observe the agent’s skill, the agent can choose to work at a wage rate that is lower than her skill, if it is in her interest. We denote by wi ≤ si the wage rate at which agent i ∈ N actually works. An allocation zN ∈ X N is incentive compatible if and only if there exists a list of individual wage rates wN ∈ S N at which agents work such that no agent envies the bundle of any other agent working at a wage rate she could earn: for all i, j ∈ N, si ≥ wj ⇒ ( i , ci ) Ri ( j , cj ).

(11.1)

b We denote by Z(E) the set of feasible incentive-compatible allocations for economy E when the policy-maker observes labor times and earnings. Allocations

11.2. THE MODEL

199

b that are Pareto undominated in Z(E) are called second-best efficient for E when labor is observable. b There is a very convenient way of describing Z(E). As the policy-maker observes i and wi i , she can infer wi . As a result, she can offer a tax function on labor time, τ w : [0, 1] → R, that is specific to each value of w. Agent i ∈ N will then choose wi and ( i , ci ) maximizing her satisfaction subject to the constraint that wi ci When ci = wi

i

≤ si ≤ wi

i

− τ wi ( i ).

− τ wi ( i ) for all i ∈ N, the allocation is feasible if and only if X τ wi ( i ) ≥ 0. i∈N

If the inequality is strict, we say that the allocation generates a budget surplus. Every incentive-compatible allocation can be obtained by such a menu of tax functions. Moreover, as stated in the following lemma, every incentivecompatible allocation can be obtained in such a way that every i ∈ N works at wi = si and is therefore submitted to the tax function τ si . Intuitively, this is explained as follows. If an allocation is incentive compatible, there exists, for each class of agents with a given skill s, an opportunity set in the ( , c) space that contains the best bundle of all agents having an equal skill or a lower skill than theirs and that lies everywhere below their indifference curves.1 The upper frontier of such sets can be used to construct suitable tax functions τ s . Lemma 11.1 Let E = (sN , RN ) ∈ E. A feasible allocation zN = ( N , cN ) for E is incentive compatible when the policy-maker observes labor time and earnings, that is, satisfies (11.1), if and only if for all w ∈ S there exists a tax function τ w : [0, w] → R such that, for all i ∈ N : (i) ci = i si − τ si ( i ), (ii) for all w ≤ si , ∈ [0, 1] : ( i , ci ) Ri ( , w − τ w ( )). b We Proof. Only if: Let E = (sN , RN ) ∈ E. Let zN = ( N , cN ) ∈ Z(E). need to construct a menu of τ w satisfying the conditions of the lemma. Let Nw = {i ∈ N | si ≥ w}. For w ∈ S, let τ w be such that the graph of fw ( ) = w − τ w ( ) in the ( , c) space is the lower envelope of the indifference curves of all individuals j ∈ N such that sj ≥ w, that is, τ w ( ) = sup{t ∈ R | ∃ i ∈ Nw , ( , w − t) Ri ( i , ci )}. To prove condition (i), we need to prove that for all i ∈ N , τ si ( i ) = si i − ci . When t = si i −ci , one has ( i , si i − t) = ( i , ci ), implying ( i , si i −t) Ri ( i , ci ) and therefore si 1 This

i

− ci ∈ {t ∈ R | ∃ j ∈ Nsi , ( i , si

i

− t) Rj ( j , cj )},

equivalence between the no-envy property and the existence of a common opportunity set has been mentioned in the early literature on fairness.

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so that necessarily, τ si ( i ) ≥ si i − ci . Suppose that τ si ( i ) > si i − ci . By definition, there is j ∈ Nsi such that ( i , si i − τ si ( i )) Rj ( j , cj ) and as ci > si i − τ si ( i ), by monotonicity one has ( i , ci ) Pj ( j , cj ). But this is precisely what incentive compatibility excludes. We prove condition (ii) in two steps. First, by construction, no part of the graph of fsi lies in the strict upper contour set of Ri at ( i , ci ), so that ( i , ci ) is among the best bundles for i if she chooses to work at si . Second, we need to show that it is in her interest to work at si . For all w < w0 ∈ S, Nw ⊇ Nw0 , so that for all ∈ [0, 1], τ w ( ) ≥ τ w0 ( ) (because τ w ( ) is the supremum of the union of more sets than τ w0 ( )). Choosing a lower w than si only decreases (with respect to inclusion) the budget set of labor time-consumption bundles from which to choose. If: Let E = (sN , RN ) ∈ E. Let zN = ( N , cN ) be obtained from a menu of tax functions τ w satisfying the two conditions of the lemma. Let i, j ∈ N be such that si ≤ sj . By condition (i), ci = si i − τ si ( i ). By condition (ii), ( j , cj ) Rj ( i , si i − τ si ( i )), the desired outcome. The consequence of this lemma is that we can restrict our attention to allocations such that each agent i chooses to work at her true skill level and freely chooses her labor time given how her pre-tax income will be taxed by function τ si . Let us now model the second informational context that is considered in this chapter. The policy-maker only observes earnings, i.e., the pre-tax incomes wi i . Note that, in this context, it is always best for agent i ∈ N to earn any given gross income by working at her maximal wage rate wi = si , as that minimizes his labor time for a fixed level of consumption. We can, therefore, focus on allocations zN and simply assume that wi = si for all i ∈ N. Also observe that by working sj j /si , agent i obtains the same earnings as agent j. An allocation zN ∈ X N is incentive compatible if and only if no agent envies the earnings-consumption bundle of any other agent: for all i, j ∈ N, µ ¶ sj , c (11.2) si ≥ sj j ⇒ ( i , ci ) Ri j j . si e We denote by Z(E) the set of feasible incentive-compatible allocations for economy E when the policy-maker only observes earnings. Allocations that are e Pareto undominated in Z(E) are called second-best efficient for E when labor is unobservable. It will prove useful to concentrate on bundles described by earnings rather than labor time. Let i ∈ N . Let , 0 ∈ [0, 1], y, y 0 ∈ R+ be such that y = si and y 0 = si 0 . As si is fixed, we can slightly abuse notation and write (y, c) Ri (y 0 , c0 ) to denote ( , c) Ri ( 0 , c0 ). e Again, there is a very convenient way of describing Z(E). As the policymaker observes yi = si i , she can propose a tax function on earnings τ : R+ → R. Agent i ∈ N will then choose (yi , ci ) maximizing her satisfaction subject to the constraint ci ≤ yi − τ (yi ).

11.3. ASSESSING REFORMS

201

When ci = yi − τ (yi ) for all i ∈ N, the allocation is feasible if and only if X τ (yi ) ≥ 0. i∈N

Again, if the inequality is strict, the tax function is said to generate a budget surplus. We now claim that any incentive-compatible allocation can be generated by such a tax function. The intuition for this result is identical to that of the previous one. Again, the fact that no-envy is equivalent to the free choice of agents in an equal opportunity set is essential. Lemma 11.2 Let E = (sN , RN ) ∈ E. A feasible allocation zN = ( N , cN ) for E is incentive compatible when the policy-maker only observes earnings, i.e., satisfies (11.2), if and only if there exists a tax function τ : R+ → R such that, letting yi = si i , for all i ∈ N : (i) ci = yi − τ (yi ), (ii) for all y ∈ [0, si ] , (yi , ci ) Ri (y, y − τ (y)) . The proof is very similar to the proof of the previous lemma and is omitted. In this chapter, we study the properties of the choice of a policy-maker when she maximises one of the SOFs we axiomatized in the previous chapter, and when she faces one of the informational contexts we just described. For the sake of simplicity, given that skill is always assumed to be unobservable and earnings to be observable, we characterize the two informational structures by specifying only whether labor time is observable or not. As incentive-compatible allocations can all be described in terms of tax functions, the second-best optimality exercise we carry on here is equivalent to evaluating tax functions. This justifies the title of the chapter.

11.3

Assessing reforms

Our first second-best application deals with the evaluation of reforms. More precisely, let us assume that there is a status quo taxation scheme, and moreover, let us assume that it is not optimal for the SOF the policy-maker is interested in. If the status quo has to be slightly changed, where should it be changed, that is, who should pay slightly more tax, and who should pay less? We study this question here, in each of the two informational contexts identified in the previous section. This application will again highlight the advantage of using SOFs built on the leximin aggregator. Indeed, identifying who should benefit from the reform is identical to identifying who is the worst-off according to the particular wellbeing index embodied in the SOF. Let us start with the case of observable labor time. Following Lemma 11.1, we assume that there exists a menu of tax functions satisfying the conditions listed above. Figure 11.1 illustrates such a menu in a three-skill economy, with

202 c

CHAPTER 11. INCOME TAXATION 6

(s3 )

(s2 ) τ s2 τ s3

τ s1 (s1 )

0

1

Figure 11.1: A menu of tax functions {τ s } when labor time is observable. s1 < s2 < s3 . Thin lines represent the opportunity sets agents would face in the absence of any taxation. The τ s functions are represented through the consumption functions si i − τ si ( i ). We assume that all the points along these curves are relevant, that is, each point of these curves hits the indifference curve of at least one agent.2 Comparing the curves, we can see that all the low-skilled agents are subsidized (they end up with a higher consumption than without taxation), whereas middle and high-skilled agents are subsidized only if their labor time is low. Figure 11.2 illustrates how to identify the worst-off agents when the policymaker’s preferences correspond to Rslex or Rsmin lex . Let us consider Rslex first. The ray of slope s represented in the figure is the highest such ray that is below the graph of all curves. It is tangent to the consumption function associated to τ s1 at s = 1. One can deduce that the indifference curve of agents choosing a labor time of s is tangent to the ray of slope s. Their skill-equivalent well-being index is, therefore, equal to s. By a simple revealed preferences argument, the graph shows that all other agents have a skill-equivalent index at least as large as s. That proves that, for Rslex , low-skilled agents working full time are, in this example, the ones that should benefit from a reform. It need not be the 2 In the case of a finite set of agents, this is achieved by assuming that the graph of the consumption function associated with τ s functions is the lower envelope of the indifference curves of all agents with that skill.

11.3. ASSESSING REFORMS

203

case, in general, that hard-working low-skilled agents are the worst-off agents according to Rslex . The key parameter is the average consumption per unit of labor, (si i − τ si ( i )) / i . c

6

τ s2 τ s3

•

τ s1

(s1 ) • t 0

(s) smin

s

=1

Figure 11.2: Evaluation of tax functions {τ s } according to Rslex or Rsmin lex . Let us now turn to Rsmin lex . In Figure 11.2, we claim that the worst-off agents according to Rsmin lex are the low-skilled agents choosing to work smin . Indeed, by a similar argument as above, the indifference curve of agents choosing that labor time is tangent to the budget of slope smin corresponding to a lump sum subsidy of t. Moreover, the implicit budget of slope smin of any agent choosing a different labor time is above the budget of that slope represented in the figure. That proves the claim. More generally, the worst-off agent is the low-skilled agent receiving the lowest subsidy, or equivalently, “paying” the largest tax, τ smin ( ). Let us study the generality of two properties of the above example. First, both Rslex and Rsmin lex force us to conclude that worst-off agents are those with minimal skill. This property is, actually, fully general. The incentivecompatibility constraints, indeed, protect higher-skilled agents and provide them, as proven in Lemma 11.1, with larger opportunity sets. It is therefore impossible that a higher-skilled agent ends up with a lower s-equivalent or smin -equivalent well-being level than the worst-off among the low-skilled agents. Second, the labor time of the worst-off agents according to Rslex is larger than according to Rsmin lex . This comes from the fact that t, in the graph,

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CHAPTER 11. INCOME TAXATION

is positive, so that the slope of the ray that is used to identify the worst-off according to Rslex is steeper than smin . It is easy to figure out that in the case t = 0, the two lines would coincide, so that the worst-off agents would be the same. From this observation, we can conclude that if τ smin is such that all the low-skilled agents are subsidized, then a policy-maker maximizing Rslex will focus on low-skilled agents working at least as much as those on which a policymaker maximizing Rsmin lex would focus. If τ smin is such that some low-skilled agents pay a strictly positive tax, then the conclusion is reversed. ˜ Figure 11.3 illustrates how to identify the worst-off agents according to R EW ∗ ∗∗ for two values of ˜, ˜ < ˜ . First, identifying which agents, in each skill subgroup, is worse-off, does not depend on ˜. Among low-skilled agents, the implicit budget of the worst-off is the same as when we apply Rsmin lex . In this example, they are the agents choosing a labor time of smin = ˜∗ . Among higher˜ skilled agents, however, the implicit budgets are different, as R EW recommends to use the agents’ actual skills to evaluate their well-being. In the graphical example, among s2 and s3 agents the worst-off agents are those working full time. c

6

(s3 )

•

τ s2

τ s3

τ s1

• •

(s2 )

(s1 )

• • • 0

˜∗

˜∗∗

1

Figure 11.3: Evaluation of tax functions {τ s } according to R

˜EW

.

Second, we have to compare the well-being of the worst-off across skill subgroups. The consequence of using implicit budgets of different slopes to evaluate the well-being of agents of different skills is drastic. With a low value of ˜, such as ˜∗ , the worst-off agents in the economy are the high-skilled agents working

11.3. ASSESSING REFORMS

205

full time. Their ˜-equivalent well-being index is ci − s3

i

+ s3 ˜∗ = s3

i

− τ s3 ( i ) − s3

i

+ s3 ˜∗ = s3 ˜∗ − τ s3 ( i ) = s3 ˜∗ − τ s3 (1).

˜∗

The R EW SOF recommends in this example that the reform benefit the agents with the largest income. With a higher value of ˜, such as ˜∗∗ , the ranking of the worst-off agents from each subgroup is different: low-skilled agents turn out to be worst-off. Their ˜-equivalent well-being index is now ci −s1 i +s1 ˜∗∗ = s1 i −τ s1 ( i )−s1 i +s1 ˜∗∗ = s1 ˜∗∗ −τ s1 ( i ) = s1 ˜∗∗ −τ s1 (

smin ).

The reform should then benefit low-skilled agents choosing a labor time of smin . ˜ Three general features of reforms consistent with R EW can be derived from this example. First, it need no longer be the case that the worst-off agents in the economy are to be found among the minimal-skilled agents. It may even be the case that they are to be found among the highest-skilled agents. Second, an increase in the reference value ˜ benefits the low-skilled agents: the larger ˜, the lower the skill of the subgroup of agents on which the reform should focus. Third, the simple computation to make in order to identify the agents who should benefit in priority from the reform is to look for the lowest s ˜ − τ s ( ) among all s and among all . Let us now turn to the assumption of non-observable labor time. Following Lemma 11.2, we assume that there is an income tax function τ , and we imagine that it represents the status quo from which a reform could depart. We need to identify the worst-off agents according to each of our SOFs. Figure 11.4 illustrates a tax function in a similar three-skill economy as above. The consumption function that is depicted has equation y − τ (y), so that the tax corresponding to a level of pre-tax income is the difference between the line of slope 1 (that is, the no taxation line) and the consumption level associated with that particular pre-tax income. Thanks to the incentivecompatibility constraints embedded in the figure, there is no loss of generality in assuming that this consumption function f (y) = y − τ (y) is non-decreasing. An earnings-consumption bundle in a strictly decreasing segment, indeed, could never be chosen by a rational agent, as she could get a larger consumption with a lower labor time. In this example, all low-skilled agents receive a subsidy (they pay a negative tax). Among middle-skilled agents (resp., high-skilled agents), those having a pre-tax income lower than y ∗ , that is, those working less than y ∗ /s2 (resp., y ∗ /s3 ), receive a subsidy, and the others pay a tax. As above, we assume that all the points along the curve are relevant, that is, each point of the curve hits the indifference curve of at least one agent. Moreover, we assume that for each si , each point of the curve for incomes lower than si hits the indifference curve of at least one agent having this skill. Figure 11.5 illustrates how to identify the worst-off agents when the policymaker’s objectives correspond to Rslex . We need to distinguish two steps. First, the worst-off agents need to be identified in each skill subgroup. This is done

206 c

CHAPTER 11. INCOME TAXATION 6

(1)

τ •

0

s1

y∗

s2

s3

y

Figure 11.4: A tax function τ s when labor time is unobservable.

in the same way as under the assumption of an observable labor time. In this example (and for the sake of clarity), it turns out that the worst-off among agents of each subgroup are the ones working full time. Second, we need to compare the s-equivalent well-being levels among the worst-off of each subgroup. In the graphical example, for each subgroup j ∈ {1, 2, 3}, that level is given by cj /sj . These numbers are not immediately comparable, as the denominators are expressed in terms of pre-tax income instead of labor time. There is an easy way of comparing them, which involves rescaling them so as to express them in terms of a common labor time unit. This is illustrated in the figure. The ray of slope cj /sj for middle and high-skilled agents is rescaled so as to fit the way labor time is measured ³ for´low-skilled agents. ³We are ´ left with s1 s1 comparing the quantities c1 /s1 , c2 / s2 s2 = c2 /s1 , and c3 / s3 s3 = c3 /s1 . Consequently, the simple observation that c1 ≤ c2 ≤ c3 leads us to conclude that the worst-off agents among the low-skilled ones are also the worst-off in the entire population. Again, higher-skilled agents are protected by the incentivecompatibility constraints and can never end up enjoying a lower well-being level than the worst-off among the low-skilled agents. In this example, therefore, the policy-maker should reform the tax system so as to decrease the tax (increase the subsidy) of the low-skilled agents working full time. More generally, those who should benefit from a tax reform can be found by minimizing (y − τ (y)) /y, or simply maximizing τ (y)/y, among the

11.3. ASSESSING REFORMS

207

low-skilled agents, i.e., among the pre-tax income levels y ≤ smin . c

6

(1)

c3

τ

c2

c1

0

s1

s2

s3

y

Figure 11.5: Evaluation of tax function τ according to Rslex . Figure 11.6 illustrates how to identify the worst-off agents when the policymaker’s preferences correspond to Rsmin lex . It is again convenient to distinguish two steps. First, the worst-off agents need to be identified in each skill subgroup. This is done in the same way as under the assumption of an observable labor time, and as in the case of Rslex in the previous paragraphs, the worst-off among agents of each subgroup, in this example, are the ones working full time. Second, we need to compare the smin -equivalent well-being levels among the worst-off agents from each subgroup. In each subgroup j ∈ {1, 2, 3}, that level is given by cj − smin j = cj − smin , as j = 1 for the worst-off agents. These operations are illustrated in the figure in the following way: earnings-consumption bundles are first rescaled so that they fit the low-skill consumption set. Then the three parallel lines correspond to the subtraction of s1 = smin , and we get t1 , t2 and t3 . Again, non-decreasingness of the consumption function guarantees that c1 ≤ c2 ≤ c3 , which, in turn, is necessary and sufficient to get t1 ≤ t2 ≤ t3 . Again, the worst-off in the population must be found among low-skilled agents. In this example, therefore, the policy-maker should reform the tax system so as to decrease the tax (increase the subsidy) of the low-skilled agents working full time. More generally, the beneficiary of a desirable tax reform can be found by minimizing c − smin = y − τ (y) − smin (y/smin ) = −τ (y), or simply maximizing

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τ (y), among pre-tax income levels y ≤ smin . c

6

(1)

c3

τ

c2 t3 c1 t2

t1 0

s1

s2

s3

y

Figure 11.6: Evaluation of tax function τ according to Rsmin lex . In our example, both Rslex and Rsmin lex force us to look at the shape of the tax function on low incomes. This is a fully general property, directly driven from the combination of the incentive-compatibility constraints and the fact that the well-being index we use does not depend on the agents’ actual skills. The two SOFs also force us in the examples given here to look at low-skilled agents working full time. This, however, is not general. It may even be the case that Rslex and Rsmin lex force us to look at agents with different labor time. Again, the key condition is whether all low-skilled agents receive a subsidy. In this case, Rslex focuses on agents working at least as much as those on which Rsmin lex focuses. If some low-skilled agents pay a positive tax, then the opposite relation holds. ˜ Let us now turn to R EW and show how it can be used to assess reforms. Figure 11.7 illustrates how to identify the worst-off agents when the policy˜ maker’s preferences correspond to R EW . Again, in a first step, the worst-off agents need to be identified in each skill subgroup. Among the minimal-skilled agents, we need to construct implicit budgets that have a slope of smin . This is done in the same way as for Rsmin lex . In our example, the agents working full time (or being indifferent between their current bundle and working full time) have the lowest implicit budget. Should an agent, facing this budget, choose a labor time of ˜ , she would enjoy a

11.3. ASSESSING REFORMS c

209

6

(1)

τ

cs˜min cy˜ 0

• • ˜smin ˜y smin

y

smax y

Figure 11.7: Evaluation of tax function τ according to R

˜EW

.

consumption level of cs˜min = smin ˜ − τ (smin ). This quantity measures the ˜equivalent well-being index of the worst-off agents in the subgroup of minimalskilled agents. For any y ∈ [smin , smax ] such that there are agents with a skill level equal to y, we can compute the corresponding well-being level in a similar way, as exemplified in the figure. Among agents having a skill level equal to y, those working full time (or being indifferent between their current bundle and working ˜ full time) are the worst-off. When evaluated according to R EW , the implicit budget of agents of skill y working full time has a slope of y. That is the key difference with the previous SOFs. In the (y, c)-space, such a budget, represented from y, has a slope of 1. Should an agent, facing this budget, choose a labor time of ˜ , she would enjoy a consumption level of cy˜ = ˜y − τ (y). This quantity measures the ˜-equivalent well-being index of the worst-off agent in the subgroup of y-skilled agents. Remember that it need not be the case that worst-off agents are those (indifferent to) working full time. In our example, it follows from the fact that the slope of the consumption function is lower than 1, which corresponds to a positive marginal tax rate. This is clearly the most interesting case, but it is not the only one. The most general result is that, among agents having³a skill level ´ equal ˜ to s, the worst-off have a well-being index equal to miny∈[0,s] s − τ (y) . It is now easy to define the criterion that needs to be used to find the worst-off among

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´ ³ the whole population: agents i ∈ N having the lowest miny∈[0,si ] ˜si − τ (y) need to benefit in priority from a reform. As it is clear from Figure 11.7, higher-skilled agents may have a lower wellbeing index than minimal-skilled agents: cy˜ < cs˜min . This conclusion is similar to the one we reached under the assumption that labor time is observable: while our two SOFs grounded on the axiom of Equal Preferences Transfer (i.e., Rslex and Rsmin lex ) force us to find the worst-off agents among the low-skilled, the ˜ R EW family of SOFs, grounded on the axiom of Equal Skill Transfer, may lead us to claim that some higher-skilled agents pay too much income tax. Let us, again, observe that the latter conclusion depends on the value of ˜. In the example, ˜ is rather low (around 1/3). For larger values of ˜, the well-being index values will change, and the ranking may change. For sufficiently large values of ˜, the low-skilled agents will be the worst-off. Let us now summarize the results obtained in this section about the evaluation of reforms using the SOFs characterized in the previous chapter. • As a function of the SOF one is interested in, and as a function of the informational structure one faces, the worst-off agents are identified by minimizing the following criteria. Rslex Rsmin lex R

˜EW

observable labor min ∈[0,1] −τ smin ( ) ( ) −τ min ∈[0,1] smin ´ ³ mins∈S, ∈[0,1] ˜s − τ s ( )

unobservable labor miny∈[0,smin ] −τ (y) miny∈[0,smin ] −τy(y) ³ ´ mins∈S,y∈[0,s] ˜s − τ (y)

• Independently of the informational structure, both Rslex and Rsmin lex , that is, our SOFs satisfying Equal Preferences Transfer, recommend to ˜ focus on low-skilled agents. On the contrary, R EW , satisfying Equal Skill Transfer, may recommend that the reform benefit higher-skilled agents. • Among low-skilled agents, Rslex recommends that the reform benefit agents who work at least as much as those who should benefit from a reform based on Rsmin lex if all low-skilled agents are currently subsidized (and conversely in the opposite case). This section has illustrated that our SOFs can be used to derive simple criteria ready to be used to evaluate reforms. The exercise consists of identifying the agents who should benefit, in priority, from the reform. This exercise is made particularly easy by the fact that our SOFs are of the leximin type. Those who should benefit from a reform are those currently experiencing the lowest wellbeing index. Thanks to the constraints imposed by incentive compatibility, they are easily identified by analyzing the shape of the tax functions, which is a striking fact for the practical implementation of such criteria. No information about the distribution of characteristics of the population is needed, knowing the tax rules is enough.

11.4. OPTIMAL TAXATION SCHEMES IN A SIMPLE ECONOMY

11.4

211

Optimal taxation schemes in a simple economy

The classical literature on optimal taxation has mostly focused on the design of the optimal tax scheme. A taxation scheme is optimal when it maximizes social preferences under incentive compatibility constraints. In this section and the next one, we derive some properties of the optimal tax schemes when social preferences are captured by the SOFs we characterized in Chapter 10, that is, ˜ Rslex , Rsmin lex and R EW . In this section, we begin our exploration with simple economies composed of four types of agents, similar to the economies we used in the previous chapter to illustrate the three SOFs we are interested in. The study of a more general model will be tackled in the next section. There are two possible skills, s1 < s2 , and two possible preferences in the ( , c) space, R1 and R2 such that R1 exhibit a lower willingness to work than R2 (that is, facing identical opportunity sets, a R1 agent would always choose a lower labor time).3 Studying the optimal menu of tax functions when labor time is observable does not bring more insights in this simple framework than in the general model. We postpone the study of this case to the next section. We therefore assume that labor time is not observable, so that the tax can only depend on earned income, yi . We have four types of agents, and their preferences in the (y, c) space can be denoted R11 , R12 , R21 , and R22 , where the first index refers to skill and the second one to preferences. In the graphical illustrations we assume that the four subgroups have the same size. In order to make the problem of incentive compatibility more interesting, we assume that low-skilled high-willingness-to-work agents cannot be distinguished from high-skilled low-willingness-to-work ones. That is, for any bundles (y, ), (y 0 , 0 ) that are affordable to both types (that is, such that y, y 0 ≤ s1 ), (y, ) R12 (y 0 , 0 ) ⇔ (y, ) R21 (y 0 , 0 ). An allocation in that economy is a list (zij = (yij , cij ))i,j∈{1,2} . Figure 11.8 illustrates two incentive-compatible allocations where z12 6= z21 . In the lefthand part of the figure, the two bundles are different, but agents of type 21 are indifferent between them. In the right-hand part, they strictly prefer the bundle they are assigned. The possibility of having z12 6= z21 comes from our assumption of an upper bound on labor time. In the classical literature, there is no such bound (and, therefore, nothing like a full-time job). To simplify the exposition and to be able to compare our results with the literature, we add the assumption that z12 = z21 . Optimal allocations for Rslex and Rsmin lex are illustrated in Figure 11.9. In the left hand part of the figure, the allocation (z11 , z12 , z21 , z22 ) , with z12 = z21 , is optimal for Rslex . The argument can be developed geometrically. First, the 3 That corresponds to the Spence-Mirrlees single crossing condition in the (l, c) space among agents with the same skill.

212

c

CHAPTER 11. INCOME TAXATION

6

¡(1) ¡ ¡ r ¡rz z22 21 ¡ r ¡ ¡z z11 r ¡ 12 ¡ ¡ ¡ ¡ ¡ s1 s2 y 0

c

6

¡(1) ¡ r ¡ rz 22 ¡z21 ¡ r ¡ ¡z12 z11 r ¡ ¡ ¡ ¡ ¡ ¡ s2 y s1 0

Figure 11.8: Incentive compatible allocations in a simple economy where z12 6= z21 .

indifference curves of both types of low-skilled agents are tangent to the ray of slope s. This implies that they have the same well-being index, and it is equal to s. A ray of slope s ss12 is also drawn in the figure. It represents the implicit budget agents of types 21 and 22 would face, should they enjoy a well-being of s as well. We see in the figure that both types of agents strictly prefer their bundle to choosing in that budget. Their well-being is therefore greater than s. The represented allocation does not succeed in equalizing the well-being indices of all the agents. One would like to further increase the well-being of types 11 and 12, at the expense of the others. But it is impossible to decrease the welfare of type 21, as it is indistinguishable from type 12 (any deviation would violate the incentive compatibility constraint). It is also impossible to decrease the welfare of the agents of type 22. Indeed, as they are indifferent between z12 and z22 , any decrease in their welfare would give them the incentive to claim they are of type 12. The incentive-compatibility constraint prevents us from improving the fate of the low-skilled agents. If such an allocation is second-best efficient–in the figure, it is–, then it is optimal. In the right-hand part of Figure 11.9, the allocation (z11 , z12 , z21 , z22 ) , with z12 = z21 , is optimal for Rsmin lex . Again, the indices are equalized among lowskilled agents, which is illustrated by the fact that both indifference curves are tangent to the same line of slope 1, which is equivalent to a slope of smin = s1 in their ( , c)-space. A similar implicit budget for the high-skilled agents is also represented in the figure, with a slope of s1 /s2 . As in the analysis relative to Rslex in the previous paragraph, we observe that types 21 and 22 are strictly better off than if they were given the opportunity to choose from that budget. Their well-being index is therefore strictly larger than the low-skilled agents’. We also observe that all incentive-compatibility constraints are binding, which, again, proves that if this allocation is second-best efficient–in the figure, it is–,

11.4. OPTIMAL TAXATION SCHEMES IN A SIMPLE ECONOMY c

c

6

z12 r

¡(1) ¡ ¡ z r ¡ 22 (s) ¡ ¡ (s ss12 ) ¡

¡ ¡ z11 ¡ r ¡ ¡ ¡ s1 0

s2

y

213

6

¡(1) ¡ ¡ ¡ ¡ z22 r ¡ ¡¡ ©© © ¡¡ © s1 z12 r ¡¡ ©© ( s2 ) © ¡© z11 r ¡© ¡ ¡ © ¡ © ¡© ¡ s2 y s1 0

Figure 11.9: Optimal second-best allocations according to Rslex , left, and Rsmin lex , right. then it is optimal. In Figure 11.10, the allocation (z11 , z12 , z21 , z22 ) , with z12 = z21 , is optimal ˜ for R EW , for the represented value of ˜. Both low-skilled agents maximize over a budget of slope 1, so that, again, their well-being index is equalized. The value of their well-being index is the consumption level they could afford by having a labor time of ˜ over that budget, i.e., by consuming ˜s1 , represented in the figure. c

6

(1) z22 ¡ ¡ ¡ s ¡¡ ¡ ¡ ¡ ¡ z12s ¡¡ ¡ ¡¡ ¡ ¡¡ ¡ s s¡ ¡ z11 s ¡ ¡¡ ¡ ¡¡ ¡ ¡ ¡ s2 y 0 ˜s1 ˜s2 s1 Figure 11.10: Optimal second-best allocations according to R

˜EW

.

Agents of type 22 are consuming z22 , a bundle they strictly prefer to z12 = z21 . The incentive compatibility constraint preventing agents of type 22 to claim

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they are of type 12 is not binding. That does not mean, though, that the wellbeing of type 12 should be increased. Indeed, agents of type 22 have exactly the same well-being. This is computed by considering the implicit budget to which agents 22 are indifferent, represented in the figure. Given that budget (of slope 1), the consumption level ˜s2 these agents would obtain by working ˜ is exactly the same as the low-skilled agents’ level. Any further redistribution from highskilled to low-skilled agents would therefore decrease the minimal well-being index in the population. Let us even observe that this allocation is first-best efficient (all slopes of the indifference curves are equal to 1 in the (y, c)-space). This completes the proof that this allocation is optimal. A larger value of ˜ would have increased the level of redistribution from agents of type 22 to low-skilled agents, with the consequence that the incentivecompatibility constraint would end up being binding. A lower value of ˜ would have decreased redistribution even more. At the limit, when ˜ = 0, the pol˜ icy of no redistribution, i.e., laisser-faire, would be optimal for R EW . This observation confirms that SOFs that are based on the leximin criterion do not necessarily call for a large redistribution. Having identified the properties of optimal second-best allocations for our SOFs, we can deduce the following results. • The incentive compatibility constraints of high-skilled agents are always binding when the policy-maker maximizes a SOF satisfying Equal Preferences Transfer. This follows from the fact that absolute priority is given to low-skilled agents. This property does not necessarily hold if the policymaker maximizes a SOF satisfying Equal Skill Transfer. • Independently of which SOF is maximized, the taxation is never progressive among low-skilled agents. Indeed, the amount transferred to lowskilled hardworking agents, type 12 agents, is always at least as large as the amount transferred to low-skilled low-willingness-to-work agents, type 11 agents.4 Moreover, in the case of Rslex , it is strictly larger. This result will be confirmed in the next section. • Finally, let us insist that, even if our social preferences are of the leximin type, they may lead to a very limited redistribution, as exemplified by the ˜ R EW SOF with a small ˜. The intuition for this result is the following ˜ one. The well-being index associated with the R EW SOF is equalized ˜ among agents when all agents working have the same consumption level, independently of their skill. For lower values of ˜, the redistribution necessary to equalize that consumption level among agents working ˜ is also lower. At the limit, if the policy-maker wishes to equalise the consumption of all agents choosing not to work, then no redistribution is necessary. Boadway et al. (2002) study the same two-skill two-preferences model under the assumptions that preferences are quasi-linear in leisure and that there is no 4 Recall that the transfers are measured by the vertical distance between the bundles agents are assigned and the no tax 45 ◦ line.

11.5. OPTIMAL TAXATION SCHEMES

215

upper bound on the labor time. They study utilitarian social preferences, that is, they equip each preferences R1 and R2 with a numerical representation u1 and u2 , linear in consumption, and social welfare is defined as a sum of individual utility levels. They show that when social preferences are utilitarian, the optimal allocations can be associated with a list of binding incentive-compatibility constraints very different from what we obtain here. In particular, it may be the case that redistribution benefits the agents of type 22. This cannot occur here because these agents always have the greatest index of well-being at the laisser-faire allocation, and we apply the maximin criterion. This proves that allocations maximizing one of the social preferences discussed in this paper form a strict subset of the second-best efficient allocations. Closer to what we develop here, they study the consequences of a family of utilitarian social preferences which give absolute priority to low-skilled agents in the sense that the weight assigned to the others’ utility is zero, whereas the weights given to the utilities of agents with different preferences, among the low-skilled, may vary. Boadway et al. then also reach the conclusion that redistribution may be regressive among low-skilled agents.

11.5

Optimal taxation schemes

In this section we turn to the general model. All results presented here are drawn from Fleurbaey and Maniquet (2006, 2007, 2009). Formal proofs of the results can be found in these papers. We focus here on intuitive presentations. There are two main differences between the SOFs we study in this section and the typical social preferences that are used in the classical literature on optimal taxation. The first difference comes from the aggregator. All our SOFs are of the leximin (or maximin) type, whereas the literature has concentrated on utilitarian SOFs.5 The second difference comes from the indices that are aggregated. Our SOFs aggregate indices of well-being that are axiomatically derived from some notion of equality of resources. The classical utilitarian SOFs aggregate utility levels coming from some exogenous utility function. In the latter case, the most frequent assumption is that all agents have the same utility function (and, therefore, the same preferences). One of the main achievements of the classical literature has been to provide us with the exact formula of the optimal tax scheme under various assumptions. As it is clear, now, a key feature of our taxation model is that agents differ both in their skill and in their preferences. This double heterogeneity has a direct consequence on our objective: it becomes almost impossible to derive the formula of the optimal tax scheme analytically. Optimal tax schemes have to be derived from the maximization of the criteria we defined in Section 11.3 under the proper incentive compatibility constraints. As we insisted above, these criteria are simple. On the other hand, characterizing the set of incentivecompatible allocations so as to be able to find the best allocation is an extremely 5 There are, however, important studies of the maximin criterion in Atkinson (1973), Boadway and Jacquet (2008), among others.

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difficult task. This explains why some of our results will be stated in the format: if this or that tax scheme is second-best efficient, then it is optimal. On the other hand, the tax schemes we will be able to point out have several properties that can be related to our objective of fairness. Identifying these properties is a key goal of second-best applications of our SOFs approach. The main properties we will identify will be a zero or negative marginal tax rate on low incomes.6 ˜ Our first formal result is that an optimal allocation for R EW or Rsmin lex can be obtained by a menu {τ s } such that the individuals with the lowest skill face a zero marginal tax. With Rslex (E), the average marginal tax for the low-skilled agents must be non-positive. Theorem 11.1 Assume labor time is observable. Let E = (sN , RN ) ∈ E. ˜ Every second-best optimal allocation for Rsmin lex (E) or R EW (E) can be obtained by a menu {τ s } such that τ smin is a non-positive constant-valued function. Every second-best optimal allocation for Rslex (E) can be obtained by a menu {τ s } such that τ smin satisfies τ smin ( ) ≥ τ smin (1) for all ∈ [0, 1]. The intuition for this result goes as follows. Let zN be an optimal allocation for Rsmin lex (E). Following the proof of Lemma 11.1, one can construct the menu {τ s } that implements this allocation in such a way that higher-skilled agents get better budget sets, i.e., for all ∈ [0, 1], all s > s0 , s − τ s ( ) ≥ s0 − τ s0 ( ) , and every agent i picks his bundle from the budget defined by τ si . Let b denote the lowest well-being level in the population at the optimal allocation. Let us change the tax menu by simply replacing τ smin with τ 0smin , defined by: τ 0smin ( ) = −b, for all ∈ [0, 1]. Observe that τ 0smin satisfies the requirement of the theorem. We claim that for all low-skilled agents i, ci = smin i + b, so that zN is still obtained with the new menu and the desired conclusion of the theorem is satisfied. First note that it is impossible that some low-skilled agent j has cj < smin j +b because this would imply that his well-being level is less than b, contradicting the assumption the lowest level is b. Now, suppose that there is some lowskilled agent j such that cj > smin j + b. Under the new menu there is a new al0 location zN in which this agent “pays” τ 0smin ( 0j ) = −b > smin j − cj = τ smin ( j ). 0 0 As τ smin ( i ) = −b ≥ τ smin ( i ) for all low-skilled i, and zi0 = zi for all higherskilled agents, the new menu generates a budget surplus. Note that at the new 0 allocation zN the lowest well-being level is still b, and it is now attained by all 0 low-skilled agents. One can redistribute the budget surplus of zN and obtain 00 another feasible and incentive-compatible allocation zN that strictly Pareto0 00 dominates zN . The lowest well-being level at zN is greater than b, which contra6 In recent developments of optimal taxation theory (Choné and Laroque 2005, Saez ????, Kaplow 2009), the assumption of identical preferences has been dropped, and some consequences of the heterogeneity in agents’ preferences on the shape of optimal tax schemes have been identified. In the models focusing on participation (the agents choosing only whether they work full time or do not work at all), it has appeared that a negative marginal tax rate may be second-best efficient (whereas the key result of Mirrlees’ (1971) seminal paper was that it had to be positive or zero).

11.5. OPTIMAL TAXATION SCHEMES

217

dicts the assumption that zN is optimal. This proves the claim of the previous paragraph. ˜ The reasoning is identical if one looks at R EW (E). If c denotes the lowest well-being level at the optimal allocation, then the new tax function τ 0smin one should look at is τ 0smin ( ) = −(c − smin ˜), for all ∈ [0, 1]. Apart from that slight change, the proof is similar. The reasoning is very similar for Rslex (E). If s denotes the lowest wellbeing level at the optimal allocation, the new tax function τ 0smin is defined by τ 0smin ( ) = max {τ smin ( ) , τ smin (1)} . Theorem 11.1 is a key result of this chapter. We need to remember that ˜ the axioms that justify Rsmin lex and Rslex on the one hand, and R EW on the other hand, are drastically different. The former SOFs satisfy Equal Preferences Transfer and, consequently, are definitely on the compensation side of the dilemma that we have studied in the previous chapter. The latter SOF satisfies Equal Skill Transfer and, consequently, is on the responsibility side of the dilemma. In spite of these opposing properties, second-best optimality, when labor time is observable, yields tax functions that exhibit a remarkable common feature: the marginal tax rate should be zero (or, on average, no greater than zero) on the earnings of low-skilled agents. ˜ Under the assumption of observable labor time, R EW (E) turns out to be easier to study than the other SOFs. The next result is our first characterization of an optimal second-best allocation. c

6

6 ¿ τ sk ¿ (s ) ¿ k ¿ , τ sj ¿ , (sj ) ¿ , © ¿ ,©© (si ) τ si © ¿,, ©»»»» τ © ¿ smin , © » » » (smin ) , © » ¿ » »»» »»» » b 0 1 ˜ = 0.5 Figure 11.11: Optimal second-best allocations according to R time is observable.

˜EW

when labor

Theorem 11.2 Assume labor time is observable. Let E = (sN , RN ) ∈ E be such that for all i ∈ N, there is j ∈ N such that Rj = Ri and sj = smin , and let b ≥ 0. For each s ∈ S, let τ s be defined by:

218

CHAPTER 11. INCOME TAXATION

τ s( ) =

½

(s − smin ) − b if ≤ ˜ (s − smin ) ˜ − b if ≥ ˜.

If an allocation obtained with the menu {τ s } is second-best efficient, then it is ˜ second-best optimal for R EW (E). Figure 11.11 represents a tax menu satisfying the definition given in the theorem. By construction, all low-skilled agents have a ˜-equivalent well-being level of b + smin ˜. All higher-skilled agents working more than ˜ have the same well-being level, as their actual budget coincides with their implicit budget for all labor time weakly greater than ˜. The allocation generated by this menu of tax functions has the property that all agents working more than ˜ must be considered as equally bad-off as the low-skilled agents. The only agents with a well-being level larger than b + smin ˜ are among the higher-skilled agents choosing to work less than ˜. Let us assume that the allocation zN generated by {τ s } is second-best efficient. The intuition of the proof that it must be optimal is the following. Suppose that there exists another feasible and incentive-compatible allocation 0 zN with a lowest well-being level that is greater than b+smin ˜.. As zN is second0 best efficient, necessarily some agents are worse-off in zN . Given the observation we made in the previous paragraph, such agents cannot be low-skilled agents or agents choosing to work more than ˜, because their well-being level is the lowest at zN . Consequently, any agent that is worse-off in the second allocation is a higher-skilled agent working less than ˜. Here is where the assumption of the theorem about preferences enters into play. At least one low-skilled agent has the same preferences as that higherskilled agent, makes the same choice (from a smaller budget) at zN and, con0 sequently, must also be worse-off in zN than in zN . But this contradicts the previous paragraph, and concludes the argument. This result may seem to have limited scope because it is generally unlikely that the menu {τ s } as defined in the proposition generates a second-best efficient allocation. But one can safely conjecture that if the allocation obtained with this menu is not too inefficient, then the optimal tax menu is close to {τ s } . Note that for ˜ = 0, this menu corresponds to the laissez-faire policy (one must then have b = 0), which yields an efficient allocation and is indeed optimal for R0EW (E). The likelihood that the optimal menu is close to {τ s } therefore increases when ˜ is smaller. In practice, a menu like {τ s } is easy to enforce (assuming that labor time or wage rates are observable), and one can then proceed to check if it generates large inefficiencies. We now turn our attention to the second informational structure, i.e., we now assume that labor time is no longer observable. When labor time is not observable, the optimal tax scheme is complex to analyze when there is no restriction on the distribution of skills and preferences. For the next results of this section, we need restrictions on the distribution of types in the economy.

11.5. OPTIMAL TAXATION SCHEMES

219

The intuitive meaning of these restrictions is that pre-tax income is not too informative a signal of one’s type. More precisely, the idea is that over an interval of income [0, s], it is impossible, by only looking at preferences restricted to that interval of income, to identify agents with greater productivity than s and distinguish them, on the basis of their preferences, from agents with productivity s. The first assumption applies this idea to the interval [0, smin ] and requires that for any agent, there exists a low-skilled agent who has the same preferences in (y, c) space over the relevant range. Assumption For all i ∈ N , there is j ∈ N such that sj = smin and for all (y, c), (y 0 , c0 ) ∈ [0, smin ] × R+ : µ ¶ ¶ µ 0 ¶ µ µ 0 ¶ y y 0 y y 0 , c Rj ,c ⇔ , c Ri ,c . sj sj si si The second restriction is more demanding and extends the idea to all relevant intervals. It requires that for any agent and any skill lower than his, there exists another agent with that lower skill who is indistinguishable in the relevant range. Let S(E) denote the set of skills in the current profile: S(E) = {s ∈ S | ∃i ∈ N, si = s} . Assumption For all i ∈ N , all s ∈ S(E) such that s < si , there is j ∈ N such that sj = s and for all (y, c), (y 0 , c0 ) ∈ [0, s] × R+ : ¶ ¶ µ 0 ¶ µ µ 0 ¶ µ y 0 y y 0 y , c Rj ,c ⇔ , c Ri ,c . sj sj si si These assumptions are a generalization of the restriction made in the previous section where agents of types 12 and 21 could not be distinguished. Let us insist that even if they are restrictive, we only impose them in order to exclude the implausible case where fiscal authorities can design the tax for some interval [y, y 0 ] (and, in particular, apply very high tax rates over that interval) knowing that only agents having an ability to earn much greater incomes would ever end up with an income in this interval. Our first result is reminiscent of Theorem 11.1. Even when labor time is ˜ unobservable, both Rsmin lex and R EW recommend to assign the same subsidy to all agents earning less than smin . Theorem 11.3 Assume labor time is unobservable. Let E = (sN , RN ) ∈ E satisfy Assumption 1. Every second-best optimal allocation for Rsmin lex (E) or ˜ R EW (E) can be obtained by a tax function τ that is constant over [0, smin ]. The intuition for that result is similar to the intuition of Theorem 11.1. Let zN be an optimal allocation with lowest well-being level b. Let us consider a tax function like τ represented in Figure 11.12, constructed so that the budget set

220

CHAPTER 11. INCOME TAXATION

is the lower envelope of the indifference curves of the population. In the figure it does not satisfy the property stated in the theorem. By Assumption 1, the part of the budget set corresponding to y ∈ [0, smin ] is the lower envelope of the indifference curves of the low-skilled agents. Therefore b equals the lowest value of −τ (y) for y ∈ [0, smin ] . Let us now consider the new tax function τ 0 and the resulting allocation 0 zN . It consists in applying a constant amount of subsidy, b, to all income levels y such that −τ (y) ≥ b. In the figure, this affects all incomes lower than y 0 . Beyond y 0 , τ and τ 0 coincide. This rise in taxation keeps the minimal wellbeing level identical among agents earning less than smin , and it does not affect that of agents earning more than y 0 (observe that those agents choose the same earnings under τ as under τ 0 ). The agents earning between smin and y 0 in zN 0 0 may be worse-off in zN , but the lowest well-being level in zN , over the whole population, is still b. If zN can be obtained with τ 0 , the conclusion of the theorem is satisfied. If zN cannot be obtained with τ 0 , this means that the tax paid by at least 0 one agent has increased in zN . As, by construction, no agent has benefited from a tax reduction, the allocation generated by τ 0 generates a budget surplus. By redistributing that surplus to all agents (which can be done by slightly translating τ 0 upwards), we can obtain a new allocation that strictly Pareto dominates the previous one, thereby strictly increasing its lowest well-being level above b, and contradicting the fact that the initial allocation zN was optimal. Our last three results are partial characterizations of the optimal allocations according to our three SOFs. Let us now examine the optimal allocations for Rsmin lex . The following theorem is almost a corollary of the previous one. Theorem 11.4 Assume labor time is unobservable. Let E = (sN , RN ) ∈ E satisfy Assumption 1. A tax function maximizing Rsmin lex (E) can be computed by minimizing τ (smin ) under the constraints that τ (y) = τ (smin ), for all y ≤ smin τ (y) ≥ τ (smin ), for all y > smin . The first constraint is a repetition of the zero marginal tax result of the previous theorem. The second constraint follows from the construction given above. Theorem 11.4 is illustrated in Fig. 11.13. In the earnings-consumption space, computing the optimal tax amounts to maximizing the height of the point (smin , smin − τ (smin )) under the constraint that the income function y − τ (y) must be located in the area delineated by the thick lines. Theorem 11.5 Assume labor time is unobservable. Let E = (sN , RN ) ∈ E satisfy Assumption 1. A tax function maximizing Rslex can be computed by maximizing the net income of the hardworking poor, smin − τ (smin ), under the

11.5. OPTIMAL TAXATION SCHEMES c

221

6

(1)

τ = τ0 •

τ τ0

b 0

smin

smax y

y0

Figure 11.12: Optimal allocations for Rsmin lex and R rate on low incomes.

˜EW

have a zero marginal

constraints that τ (smin ) τ (y) for all y ∈ (0, smin ], ≤ y smin τ (y) ≥ τ (smin ) for all y, τ (0) ≤ 0. The three constraints mean, respectively, that the average tax rate on low incomes is always lower than at smin , that the tax (subsidy) is the smallest (largest) at smin , and that the tax (subsidy) is nonpositive (nonnegative) at 0. Theorem 11.5 is illustrated in Figure 11.14. From the point (smin , smin − τ (smin )) one can construct the hatched area delimited by an upper line of slope 1 and a lower boundary made of the ray to the origin (on the left) and a flat line (on the right). Now, Theorem 11.5 says that computing the optimal tax may, without welfare loss, be done by maximizing the second coordinate of the point (smin , smin − τ (smin )) under the constraint that the income function y − τ (y) is located in the corresponding hatched area. The intuition of this theorem parallels that of the previous one. First, the lower bound on the amount of taxes, or, as illustrated in the figure, the upper bound on the consumption function, follows the same logic as the one illustrated in Figure 11.12: if the tax scheme does not satisfy this lower bound, then it is

222

CHAPTER 11. INCOME TAXATION

c

6

(1)

smin − τ (smin )

0

smin

y

Figure 11.13: Optimal second-best allocations according to Rsmin lex when labor time is unobservable. possible to construct another tax scheme that has this property and generates an allocation with the same associated lowest level of well-being. Second, the upper bound on the amount of taxes, or the lower bound on the consumption function, follows from the criterion that should be maximized. As proven in the section on reforms, above, we should try to maximize the average rate of subsidy among agents earning less than smin . That amounts to maximizing the consumption level of hardworking poor agents, under the constraint that no poorer agent gets a lower average rate of subsidy, as stated in the theorem. This result does not say that every optimal tax must satisfy these constraints, but it says, quite relevantly for the social policy-maker, that there is no problem, i.e. no welfare loss, in restricting attention to taxes satisfying those constraints, when looking for the optimal allocation. This result shows how the social preferences defined in this paper lead to focusing on the hardworking poor, who should get, in the optimal allocation, the greatest absolute amount of subsidy, among the whole population. However, the taxes computed for those with a lower income than smin also matter, as those agents must obtain at least as great a rate of subsidy as the hardworking poor. Our last result is a partial characterization of the optimal tax function ac˜ cording to R EW . Let ˜ ∈ [0, 1]. Let us define the following tax function: (i) for all y ∈ [0, smin ] , τ (y) = τ (0) ≤ 0; (ii) for all s, s0 ∈ S such that s < s0 and s < si < s0 for no i ∈ N, all y ∈ [s, s0 ] , τ (y) = min{τ (0) + (s − smin ) ˜ + (y − s) , τ (0) + (s0 − smin ) ˜}.

A tax function of this kind will be called a ˜-type tax. This formula calls for some explanations. The tax function is piece-wise linear. The segment on low

11.5. OPTIMAL TAXATION SCHEMES

c

223

6 (1)

smin − τ (smin )

0

smin

y

Figure 11.14: Optimal second-best allocations according to Rslex when labor time is unobservable. with a fixed subsidy −τ (0). Then comes a segment, £incomes1 ¤[0, smin ] is constant, smin , y , for some y 1 between smin and the next element s1 of S, where the rate of taxation is a hundred percent. Of course, no individual is expected to earn an income in this interval. The next segment covers the interval [y 1 , s1 ], and has a zero marginal tax rate. Then, the function continues with successive pairs of intervals, one with a hundred percent of marginal tax and the other with a zero marginal tax. The key feature is that the points (s, τ (s)), for s ∈ S (E) , are aligned, and the slope of the line is precisely ˜, that is, for all s, s0 ∈ S (E) , τ (s) − τ (s0 ) = ˜. s − s0 When S (E) is a large set with elements spread over the interval [smin , smax ], the tax function is therefore approximately a flat tax (constant marginal tax rate of ˜), except for the [0, smin ] interval where it is constant. The corresponding budget set delineated by y − τ (y) is illustrated in Figure 11.15, where the indifference curves of five individuals are also depicted. Here comes our last result. Theorem 11.6 Assume labor time is unobservable. Let E = (sN , RN ) ∈ E satisfy Restriction 2. If an allocation obtained with a ˜-type tax is second-best ˜ efficient, then it is second-best optimal for R EW (E). The proof goes in two steps. First, we identify the worst-off agents among agents of the same skill subgroup. As it is often the case, they will be those working full time, or, more generally, those having the largest labor time in their subgroup. Second, we compare the well-being level of the agents who are

224

c

CHAPTER 11. INCOME TAXATION

y − τ (y) ¡ ¡ • ¡z5 ¡ ¡ ¡ ¡ •¡ z ¡ 4 ¡ • ¡ ¡ z3 ¡¡ ¡ ¡ •z ¡ ¡ ¡ 2 ¡ ¡ ¡ ¡ ¡ ¡ z1 • ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ s ¡ 1 smax y 6 6min 6 s 6 s2 0 smin ˜ s1 ˜ s2 ˜ smax ˜ 6

Figure 11.15: Optimal second-best allocations according to R time is unobservable.

˜EW

when labor

the worst-off in their subgroup. The ˜-type tax scheme is constructed in the precise way that equalizes the minimal well-being across skill subgroups. Let us go through each step with more detail. An interesting property of a ˜-type tax function τ is that for all i ∈ N, if i chooses a bundle that is on the segment with slope 1 (zero marginal tax) just below si , then u ˜(zi , si , Ri ) = −τ (0) + smin ˜. Let us illustrate this fact in the case of individual 4, assuming that s4 = s2 . From the second term in the definition of τ , one has τ (y4 ) = τ (0) + (s2 − smin ) ˜, Bundle z4 is optimal for R4 in the budget set for which this level of tax is lump-sum. Therefore one simply has u ˜(z4 , s4 , R4 ) = −τ (y4 ) + s2 ˜ = −τ (0) + smin ˜, which achieves the proof. Now, if we assume that agent 4 has a larger skill than s2 , such as smax , then her implicit budget is larger than that of agents with the same skill, like agent 5. Agent 4, in that case, would have a larger implicit budget than agent 5, and, therefore, a larger well-being level. Now, by Assumption 2, there exists an agent of skill s2 who is undistinguishable from agent 4. The well-being of that agent is

11.6. CONCLUSION

225

−τ (0) + smin ˜. That proves that there necessarily exists an agent of skill s2 with a well-being level of −τ (0) + smin ˜, the lowest well-being level among agents of that skill. The second step consists of checking that the quantity −τ (0) + smin ˜ is independent of si . This tax function therefore equalizes u ˜ across individuals who work full time or just below full time. This equality can be seen in Figure 11.15, where the vertical segments at smin ˜, s1 ˜, s2 ˜ and smax ˜ between the horizontal axis and the lines of slope 1 corresponding to each of these values of s all have the same length. Finally, suppose that there exists another feasible and incentive-compatible 0 allocation zN with a greater minimal u ˜. As zN is second-best efficient, some agents must be worse-off. As zN is optimal, such agents must have a greater well-being in zN than −τ (0) + smin ˜. However, by Assumption 2, for each agent of this sort there is another agent with lower skill and a well-being level in zN at −τ (0) + smin ˜. This has been illustrated two paragraphs above with agent 0 these agents must also be worse-off, falling below −τ (0) + smin ˜. This 4. In zN contradiction proves that zN is optimal.

11.6

Conclusion

Chapters 10 and 11 together illustrate how the SOF approach developed in this book can be used to evaluate social policies. The first step consists in defining axioms capturing principles of fairness in the economic situations under scrutiny. There may be conflicts between these axioms, or between fairness and efficiency axioms. Once the lists of compatible axioms are identified, cross-economy robustness axioms can be added to characterize precise objectives a policy-maker should try to maximize. The second step deals with the implementation of these objectives when the policy-maker faces informational contraints. In this chapter, incentive-compatibility constraints are such that we can restrict our attention to tax schemes. The criteria we eventually obtain, consequently, directly bear on the shape of the tax functions. The applications in this chapter show that the only information one needs in order to evaluate and compare tax schemes are the relevant tax functions, i.e., the relationship between labor time or earnings on the one hand and consumption on the other. That information is easy to gather, it basically involves identifying the relevant tax legislation. As we emphasized already, it is not necessary in such evaluation to know the distribution of skills and preferences. Policy-makers, with this approach, obtain evaluation criteria that are readily applicable. We made the assumption, in this chapter, that the minimal skill is strictly positive. In our model, the skill refers to the wage rate at which you are able to find a job. In situations of high unemployment, one can argue that the skill of some agents is actually zero. Under such an assumption, the criterion corresponding to either Rslex or Rsmin lex boils down to maximizing the minimal ˜ consumption level. If the policy-maker is interested in maximizing R EW , then

226

CHAPTER 11. INCOME TAXATION

the zero marginal rate of taxation result disappear but the objective is not necessarily to maximize the lowest consumption level. As the last theorem of this chapter may suggest, linear taxation (at a rate of (1 − ˜)) is likely to be optimal.7 In almost all countries, one agent’s labor income tax does depend on more than her earnings. The typical additional argument is the size, or, more generally, the composition of the household. There may be other arguments as well, such as the working status of the person, or her age. The result of this chapter can still be applied in these cases, by considering each set of agents in the same categories in turn. The results presented here do not allow us, however, to draw conclusions about the relative tax agents should pay across different categories. This problem requires a richer model. Our last comment is about the difference between leximin and maximin type of SOFs. In the previous chapters, we saw how the combination of efficiency, fairness and robustness axioms forces us to adopt SOFs that maximize the minimal level of some well-being index, but it was frequently the case that, among all such SOFs, the leximin was impossible to single out axiomatically. Indeed, the results only said that an allocation with a greater lowest level of well-being than another allocation was strictly better, remaining silent about pairs of allocations with the same lowest level of well-being. In this chapter, for the sake of simplicity, we have always considered that the policy-maker had a leximin type of SOF. We would like to stress, now, that all the results we obtained in this chapter would be obtained with any kind of SOF displaying a strict preference for increasing the lowest well-being level, because this is the only element of social preferences that mattered in our proofs. In other words, the results we obtained in the axiomatic derivations of SOFs prove sufficient for second-best applications.

7 See

Fleurbaey and Maniquet (2009) for details.

Chapter 12

Conclusions This book has presented a theory of fairness and social welfare. By fairness, we meant that economic justice is a question of fair resource allocation.1 In order to be precise about what a fair resource allocation can be, we need to define the economic ingredients of the situation, that is, the available goods to allocate, the nature of the goods (whether they are private of public, divisible or not, in known or uncertain quantities, etc), the available technology to produce new goods, and the internal and external resources owned by the agents. What is fair, then, depends on the context, and any analysis begins by identifying fairness requirements that capture basic ethical principles in that context. We have surveyed a variety of contexts in this book. Even if they all turned out to have specificities, a general picture emerges. First, fairness involves the equalization of something. Second, that thing must be related to how agents value being given access to such and such quantity of resources. To sum up, our undertaking is consistent with the notion that a society is just if and only if it equalizes the subjective value that agents give to the entire bundle of internal and external resources they are given access to. Our theory refers to social welfare. Defining social welfare requires to aggregate the welfare of its members. If the objective is to evaluate social policy, then it is necessary to be able to claim that a given decline in the welfare of one agent is compensated, from society’s point of view, by an increase in the welfare of another agent. The common wisdom among the vast majority of economists on that question is that this process must involve interpersonal utility comparison. The whole book comes in opposition to that wisdom. Consistently with our notion of fairness, we replace utility comparison with the comparison of resource bundles, or, more accurately, with the comparison of indifference curves. We end up with an approach to policy evaluation that comprises two steps. In the first step, SOFs are defined. They arise from the combination of efficiency, fairness and robustness axioms. Let us recall that there is a hierarchy in this list of values. Efficiency is our central value. In all contexts we have been studying, 1 Recall

from Section 7.5 how the approach can be generalized to functionings.

227

228

CHAPTER 12. CONCLUSIONS

we do not see any reason to recommend a policy if another policy exists that leads to higher welfare for everybody. Fairness is our second value. Chapter 2 made clear that there may be conflicts between efficiency and fairness. Our strategy is then to weaken fairness requirements until they capture basic, sensible and maybe context-specific ethical objectives which are compatible with efficiency requirements. As made clear throughout the book, different fairness requirements may also conflict with each other. That is of course not particular to this approach and that should not be a surprise to anyone. Doesn’t economics keep telling us that our desires cannot be all satisfied simultaneously? When there are several conflicting requirements of fairness, there will be several conflicting ways of evaluating social policies. Our role is then to highlight the ethical choices that a policy-maker needs to make before proceeding to policy evaluation. Once the ethical dilemmas are identified, robustness axioms may enter the picture. They follow the general requirement that solutions to similar problems need also to be similar. Formally, they capture the general intuition that social preference for one allocation over another should be independent of changes in some parameters of the model, when these parameters are judged irrelevant. As proven in Parts 2 and 3 of the book, the addition of robustness axioms to efficiency and fairness axioms typically results in the selection of a very limited number of acceptable SOFs. That is how the first step of our approach ends, namely, with the definition of acceptable SOFs among which a policy-maker should choose as a function of her ethical position. This book has mainly be about defining acceptable SOFs. The most striking result about it is the proofs that only SOFs giving absolute priority to the worstoff are acceptable. The basic reasons are developed in Chapter 3. SOFs may apply to a variety of individual well-being indices. When one index is chosen, however, the primary concern of the policy-maker should be to increase the welfare of the agents being given access to the bundles of resources associated with the lowest value of that well-being index. Our axiomatic study of SOFs also revealed that there are two major families of acceptable SOFs. That was first exposed in Chapter 5, but is has been confirmed afterwards. The first family comprises egalitarian-equivalent SOFs and the second gathers Walrasian SOFs. In the former family, the relevant indices of well-being are constructed by only using individual agents’ characteristics. In the latter family, the relevant indices are often constructed by also taking account of the prices that would be needed to decentralize an efficient allocation that bears some relationship with the allocation at which the value of the indices are computed. This implies a greater interdependence between the evaluations of well-being for different individuals. Then comes the second step. Equipped with social preferences, the policymaker face a series of constraints. The typical ones are incentive-compatibility constraints. As the policy-maker does not know the agents’ preferences, and as preferences are a key ingredient in the definition of well-being indices (because we care about efficiency), she has to limit herself to policies that give the proper incentives to agents to reveal their private information about their pref-

229 erences. The policy-maker may face additional constraints as well. For instance, it may be the case that a status quo exists and that the only politically feasible allocations are in a neighborhood of the status quo. Independently of the constraints, policy evaluation works by maximizing the appropriate SOF under the constraints the policy-maker faces. As we saw throughout the book (in Chapters 5, 8, 9, and 11), such a maximization process boils down to identifying the bundle of resources chosen by the agent turning out to have the lowest well-being index value. Even if it is typically impossible to identify the well-being of all agents, agents’ choices in general reveal enough information to identify the worst-off. Consequently, policy evaluation is possible and clearcut recommendations can be made. Several recommendations have emerged, indeed, at several places in the book. In the production of public good, for instance, one of the two acceptable SOFs recommends to produce the public good at the level of its minimal demand, as shown in Section 8.6. In the labor income taxation problem, several SOFs based on different lists of axioms recommend to apply a zero marginal tax rate on incomes below the lowest income corresponding to a full-time job, as shown in Section 11.5. The next step in the development of this “fair social choice” approach to policy evaluation is to apply it to new problems, especially those in which governments are classically expected to play a role. Such a development is already in progress, especially in the field of health care provision and financing. Fleurbaey (2005) and Valletta (2008, 2009) study variants of our labor income taxation model where the internal resource is one’s health status, over which agents have preferences. The provision and financing of public education is another obvious candidate for the study of SOFs. Retirement benefits, family allowances and capital income taxation could also be profitably studied using SOFs. One of the assumptions that we have imposed throughout the book without questioning it is the assumption that agents are rational in the classical sense, i.e., their preferences are complete and transitive, and they are self-centered. We believe that this assumption is reasonable in the simple models we have studied. However, maybe the biggest current challenge to normative economics is to take account of the fact that agents behave in ways that may depart from what these assumptions capture. We mainly think of the fact that individual behavior may depend on what the others do. In the labor income taxation model, for instance, we have assumed that agents choose their labor time by simply looking at their own budget set. It is clear, however, that the labor time and living standards of others influence many people’s labor supply. For instance, some people are willing to work full-time, no matter how many hours this precisely means. Studying SOFs when agents have other-regarding preferences is another part of the future development of this approach. By offering a comprehensive presentation of the achievements of the theory so far, by discussing its foundations and by explaining how it can be used, we hope to have paved the way for these further developments.

230

CHAPTER 12. CONCLUSIONS

Appendix A

Proofs A.1

Chapter 3: Priority to the worst-off

Proof of Theorem 3.3. Let R satisfy Strong Pareto, Equal-Split Transfer 0 and Unchanged-Contour Independence. Let E = (RN , Ω) ∈ E, zN , zN ∈ XN , 0 0 j, k ∈ N be such that zj À zj À Ω/ |N | À zk À zk and for all i 6= j, k, zi = zi0 . 0 Assume, by way of contradiction, that zN P(E) zN . Pick an arbitrary bundle + 0 00 1 2 3 4 1 2 3 zj À zj and let Rj , Rj ∈ R and zj , zj , zj , zj , zk , zk , zk ∈ X and ∆ ∈ R++ be such that: © ª U (zj+ , Rj0 ) = U (zj+ , Rj00 ) = x ∈ X | x ≥ zj+ , © ª U (zj4 , Rj00 ) = x ∈ X | x ≥ zj4 , I(zj0 , Rj0 ) = I(zj0 , Rj00 ) = I(zj0 , Rj ), I(zj , Rj0 ) = I(zj , Rj ), zj2 Ij00 zj1 , zj4 Ij00 zj3 , zj1 = zj+ − ∆, zj3 = zj2 − ∆, zj4 = zj0 + ∆,

zk1 = zk + ∆, zk2 = zk1 + ∆, zk3 = zk2 + ∆ ¿ zk0 . This construction is illustrated in Figure A.1, where the indifference curves are those of Rj00 . Observe that necessarily zj1 , zj2 , zj3 , zj4 À Ω/ |N | À zk1 , zk2 , zk3 . ¢ ¢ ¢ ¢ ¡¡ ¡¡ Let E 0 = RN\{j} , Rj0 , Ω , E 00 = RN \{j} , Rj00 , Ω ∈ E. By Unchanged0 Contour Independence, zN P(E 0 ) zN . By Strong Pareto, (zN\{j} , zj+ ) P(E 0 ) zN . 0 . By Unchanged-Contour Independence, By transitivity, (zN\{j} , zj+ ) P(E 0 ) zN + 00 0 (zN \{j} , zj ) P(E ) zN . By Equal-Split Transfer, (zN\{j,k} , zj1 , zk1 ) R(E 00 ) (zN \{j} , zj+ ). By Pareto Indifference (entailed by Strong Pareto), (zN\{j,k} , zj2 , zk1 ) I(E 00 ) (zN \{j,k} , zj1 , zk1 ). 231

232

APPENDIX A. PROOFS good 2 6 zj1

0

zk0 2 ss zk s 3 s zk zk s zk1

z4 zj0 s s j sΩ

zj+ s s

zj2 ss

zj3

|N|

good 1

Figure A.1: Proof of Theorem 3.3. By Equal-Split Transfer, (zN\{j,k} , zj3 , zk2 )R(E 00 )(zN \{j,k} , zj2 , zk1 ). By Pareto Indifference, (zN \{j,k} , zj4 , zk2 )I(E 00 )(zN \{j,k} , zj3 , zk2 ). By Equal-Split Transfer, (zN\{j,k} , zj0 , zk3 )R(E 00 )(zN\{j,k} , zj4 , zk2 ). 0 0 By Strong Pareto, zN P(E 00 ) (zN\{j,k} , zj0 , zk3 ). By transitivity, zN P (E 00 ) (zN \{j} , zj+ ), a contradiction. Finally, let us check that every axiom is necessary for the conclusion of the theorem. The examples below are defined in the proof of Theorem 3.1. 1) Drop Strong Pareto. Take Rpsum . 2) Drop Equal-Split Transfer. Take RΩsum . 3) Drop Unchanged-Contour Independence. Take RIL .

Proof of Theorem 3.4. Let R satisfy Pareto Indifference, Equal-Split Trans0 fer, Separation and Replication. Let E = (RN , Ω) ∈ E, zN , zN ∈ X N , j, k ∈ N 0 0 be such that zj À zj À Ω/ |N | À zk À zk , and for all i 6= j, k, zi = zi0 . Let ∆j = zj − zj0 and ∆k = zk0 − zk . Let R0 ∈ R, r, m ∈ Z++ , z01 , z02 , z03 , z04 ∈ X be such that Ω Ω À z01 I0 z04 P0 z02 I0 z03 À À zk0 , |N | + 1/r |N | + 2/r z01 = z02 + ∆mk , and z04 = z03 +

∆j m.

The construction is illustrated in Figure A.2.

A.1. CHAPTER 3: PRIORITY TO THE WORST-OFF

good 2 6

Ω |N |

z02

z1 ss 0

Ω |N |+2/r s

0

s0 s zk ¡ zk s¡∆k

233

sz ¡ j ¡ ¡∆j 0 zj s¡ s

s|N |+1/r Ω

z03

s

4 sz0

good 1

Figure A.2: Proof of Theorem 3.4. Let E 0 = (RN 0 , rΩ) ∈ E be a r-replica of E. Let E 00 = (RN 0 , R0 , R0 , rΩ) , E = (RN 0 , R0 , rΩ) ∈ E. By Equal-Split Transfer, ¶ µ ¡ ¢ ∆k 2 1 zN 0 \{k} , zk + , z , z R (E 00 ) zN 0 , z01 , z01 . m 0 0 000

By Pareto Indifference, µ ¶ ¶ µ ∆k 3 1 ∆k 2 1 zN 0 \{k} , zk + , z0 , z0 I (E 00 ) zN 0 \{k} , zk + , z0 , z0 . m m By transitivity,

By Separation,

µ ¶ ¡ ¢ ∆k 3 1 zN 0 \{k} , zk + , z , z R (E 00 ) zN 0 , z01 , z01 . m 0 0 µ

zN 0 \{k} , zk +

¶ ¡ ¢ ∆k 3 , z0 R (E 000 ) zN 0 , z01 . m

By Equal-Split Transfer, µ ¶ ¶ µ ∆j ∆k 4 ∆k 3 000 zN 0 \{j,k} , zj − , zk + , z R (E ) zN 0 \{k} , zk + ,z . m m 0 m 0 By Pareto Indifference, µ ¶ ¶ µ ∆j ∆k 1 ∆j ∆k 4 zN 0 \{j,k} , zj − , zk + , z0 I (E 000 ) zN 0 \{j,k} , zj − , zk + , z0 . m m m m

234

APPENDIX A. PROOFS

By transitivity, µ ¶ ¡ ¢ ∆j ∆k 1 , zk + , z0 R (E 000 ) zN 0 , z01 . zN 0 \{j,k} , zj − m m By Separation,

µ ¶ ∆j ∆k zN 0 \{j,k} , zj − , zk + I (E 0 ) zN 0 . m m Repeating this argument m − 1 times, we obtain ¢ ¡ zN 0 \{j,k} , zj0 , zk0 R (E 0 ) zN 0 .

Now, repeating the whole argument r − 1 times, we obtain ⎞ ⎛

⎜ 0 0 0 0⎟ 0 ⎝zN \{j,k} , zj , ..., zj , zk , ..., zk ⎠ R (E ) zN 0 . | {z } | {z } | {z } (r times)

(r)

(r)

0 0 Recall that zN \{j,k} = zN \{j,k} . By Replication, zN R (E) zN . Finally, let us check that every axiom is necessary for the conclusion of the theorem. The first three examples below are defined in the proof of Theorem 3.1. 1) Drop Pareto Indifference. Take Rpsum . 2) Drop Equal-Split Transfer. Take RΩsum . 3) Drop Separation. Take RIL . 4) Drop Replication. A counter-example has yet to be found.

A.2

Chapter 5: Fair distribution of divisible goods

We first prove a variant of Th. 3.1. Lemma A.1 On the domain E, if a SOF satisfies Weak Pareto, Transfer among Equals and Unchanged-Contour Independence, then for all E = (RN , Ω) ∈ E, 0 all zN , zN ∈ X N , if there exist j, k ∈ N such that Rj = Rk , zj À zj0 À zk0 À zk , 0 P(E) zN . and for all i 6= j, k, zi0 Pi zi , then zN

Proof. Let R satisfy Weak Pareto, Transfer among Equals and Unchanged0 Contour Independence. Let E = (RN , Ω) ∈ E, zN , zN ∈ X N and j, k ∈ N 0 0 be such that Rj = Rk , zj À zj À zk À zk , and for all i 6= j, k, zi0 Pi zi . Let ε ∈ R++ be such that for all i 6= j, k, zi0 Pi zi + ε.

A.2. CHAPTER 5: FAIR DISTRIBUTION OF DIVISIBLE GOODS

235

First case: there exist x ∈ U (zj , Rj ), x0 ∈ L(zj0 , Rj ) such that x ≯ x0 . Let = Rk0 ∈ R, zj1 , zj2 , zj3 , zj4 , zk1 , zk2 , zk3 , zk4 ∈ X, ∆ ∈ R++ be constructed in such a way that for i ∈ {j, k} ,

Rj0

I(zi , Ri0 ) = I(zi , Ri ), I(zi0 , Ri0 ) = I(zi0 , Ri ), zi1 Pi0 zi , zi3 Pi0 zi2 , zi0 Pi0 zi4 , and zj2 zj4

= zj1 − ∆ À zk2 = zk1 + ∆, = zj3 − ∆ À zk4 = zk3 + ∆.

This construction of that illustrated in Figure 3.3. ¢ ¢ ¡¡ is a slight modification Let E 0 = RN \{j,k} , Rj0 , Rk0 , Ω ∈ E. By Weak Pareto, µ ¶ 1 zN\{j,k} + ε, zj1 , zk1 P(E 0 ) zN . 3

By Transfer among Equals, µ µ ¶ ¶ 1 1 zN \{j,k} + ε, zj2 , zk2 R(E 0 ) zN\{j,k} + ε, zj1 , zk1 . 2 2 By Weak Pareto, ¡ ¢ zN \{j,k} + ε, zj3 , zk3 P(E 0 )

µ ¶ 1 zN \{j,k} + ε, zj2 , zk2 . 2

By Transfer among Equals, ¡ ¡ ¢ ¢ zN \{j,k} + ε, zj4 , zk4 R(E 0 ) zN\{j,k} + ε, zj3 , zk3 . By Weak Pareto,

¢ ¡ 0 zN P(E 0 ) zN\{j,k} + ε, zj4 , zk4 .

0 0 P(E 0 ) zN . By Unchanged-Contour Independence, zN P(E) zN . By transitivity, zN 0 0 Second case: there are no x ∈ U (zj , Rj ), x ∈ L(zj , Rj ) such that x ≯ x0 . Then let zj∗ , zk∗ , zj∗∗ , zk∗∗ , ∈ X be such that zj À zj∗∗ À zj∗ À zj0 and zk0 À zk∗∗ À zk∗ À zk , and such that there exist x ∈ U (zj , Rj ), x∗ ∈ L(zj∗ , Rj ) such that x ≯ x∗ , as well as x∗∗ ∈ U (zj∗∗ , Rj ), x0 ∈ L(zj0 , Rj ) such that x∗∗ ≯ x0 . By the argument of the first case, one shows that µ ¶ 1 zN \{j,k} + ε, zj∗ , zk∗ P(E) zN 2

and that

¡ ¢ 0 zN P(E) zN\{j,k} + ε, zj∗∗ , zk∗∗ .

236

APPENDIX A. PROOFS

By Weak Pareto, ¡ ¢ zN \{j,k} + ε, zj∗∗ , zk∗∗ P(E)

¶ µ 1 ∗ ∗ zN \{j,k} + ε, zj , zk . 2

0 By transitivity, zN P(E) zN .

Proof of Theorem 5.1. We prove the result with Weak Pareto instead 0 of Strong Pareto. Let E = (RN , Ω) ∈ E and zN , zN ∈ X N be such that 0 mini∈N uΩ (zi , Ri ) > mini∈N uΩ (zi , Ri ). Assume that, contrary to the result, 0 one has zN R(E) zN . Let i0 ∈ N be such that uΩ (zi00 , Ri0 ) = mini∈N uΩ (zi0 , Ri ). 1 2 Let zN , zN ∈ Pr(Ω) be such that for all i, j ∈ N, ¶ µ ¶ µ 0 2 2 min uΩ (zi , Ri ) Ω ¿ zi = zj ¿ min uΩ (zi , Ri ) Ω, i∈N

i∈N

for all i ∈ N \ {i0 } , zi1 Pi zi , zi1 Pi zi0 , and uΩ (zi00 , Ri0 )Ω ¿ zi10 ¿ zi20 . 0 Let RN ∈ R be such that for all i ∈ N,

I(Ri0 , zi ) = I(Ri , zi ) and I(Ri0 , zi0 ) = I(Ri , zi0 ) and for all i ∈ N \ {i0 } ,

© ª U (Ri0 , zi1 ) = x ∈ X | x ≥ zi1 .

0 0 , Ω) . By Unchanged-Contour Independence, zN R(E 0 ) zN . By Let E 0 = (RN 1 0 0 0 2 1 2 Weak Pareto, zN P(E ) zN and zN P(E ) zN , so that by transitivity, zN P(E 0 ) zN . 3 Let zN ∈ Pr(Ω) be such that for all i ∈ N \ {i0 } ,

zi10 ¿ zi30 ¿ zi20 = zi2 ¿ zi3 ¿ zi1 , and

1 X 2 1 X 3 zi ¿ zi = zi20 . |N | |N | i∈N

i∈N

Let ε ∈ R++ be such that for all i ∈ N \{i0 } , εnΩ ¿ zi3 −zi2 and εnΩ ¿ zi30 −zi10 . 00 Let RN ∈ R be such that for all i ∈ N, I(Ri00 , zi1 ) = I(Ri0 , zi1 ) and I(Ri00 , zi2 ) = I(Ri0 , zi2 ) and for all i ∈ N \ {i0 } ,

© ª U (Ri00 , zi2 + εΩ) = x ∈ X | x ≥ zi2 + εΩ .

00 1 2 Let E 00 = (RN , Ω) . By Unchanged-Contour Independence, zN P(E 00 ) zN . 3 00 1 Moreover, one has zN P(E ) zN . This is proved as follows. Let n = |N | and h1i hn−1i be defined s be a bijection from {1, ..., n − 1} to N \ {i0 } . Let zN , ..., zN

A.2. CHAPTER 5: FAIR DISTRIBUTION OF DIVISIBLE GOODS

237

hti

as follows. For every t = 1, ..., n − 1, zi = zi2 + tεΩ if i = s(k) for k ≤ t; hti otherwise zi = zi1 + tεΩ. The number ε has been chosen so that for all i ∈ N, hn−1i ¿ zi3 . The following table summarizes the construction. zi i = s(1) zi2 + εΩ zi2 + 2εΩ

h1i

z z h2i .. .

i = s(2) zi1 + εΩ zi2 + 2εΩ

···

z hn−1i zi2 + (n − 1)εΩ zi2 + (n − 1)εΩ

i = s(n − 1) zi1 + εΩ zi1 + 2εΩ

i0 1 zi0 + εΩ zi10 + 2εΩ

zi2 + (n − 1)εΩ zi10 + (n − 1)εΩ

Table A.1 Let

h1i RN

∈ R be such that for all i ∈ N, h1i

h1i

h1i

h1i

I(Ri , zi1 ) = I(Ri00 , zi1 ) and I(Ri , zi ) = I(Ri00 , zi ) h1i

h1i

h1i

h1i

00 and Ri0 = Rs(1) . The latter is possible because U (Rs(1) , zs(1) )∩L(Ri000 , zi0 ) = ∅ ³ ´ h1i (i.e., indifference curves do not cross). By Lemma A.1, for E h1i = RN , Ω , h1i

h1i

1 1 zN P(E h1i ) zN . By Unchanged-Contour Independence, zN P(E 00 ) zN . hti

For t = 2, ..., n − 1, let RN ∈ R be such that for all i ∈ N, hti

ht−1i

I(Ri , zi hti

ht−1i

) = I(Ri00 , zi

hti

hti

hti

) and I(Ri , zi ) = I(Ri00 , zi )

hti

and Ri0 = Rs(t) . A similar argument to the previous reasoning leads to the hti

ht−1i

conclusion that zN P(E 00 ) zN

00

. hn−1i

3 1 P(E ) zN . By transitivity, zN P(E 00 ) zN , as was to By Weak Pareto, be proved. 3 2 4 3 By transitivity again, zN P(E 00 ) zN . Let zN ∈ Pr(Ω) be obtained from zN by (non-leaky) transfers, so that for all i ∈ N, zi4 ¿ zi2 . By Proportional4 3 4 2 Allocations Transfer, zN R(E 00 ) zN . By transitivity, zN P(E 00 ) zN . But by Weak 2 4 Pareto, zN P(E 00 ) zN , a contradiction. This completes the proof. In addition, we show that if any of the conditions is relaxed, one finds a SOF which does not obey the conclusion of the theorem. 1) Drop Weak Pareto. Take Rpsum . 2) Drop Transfer among Equals. Take RΩsum . 0 3) Drop Proportional-Allocations Transfer. Take RΩ lex for Ω0 not proportional to Ω. 4) Drop Unchanged-Contour Independence. Let RIL . 3 zN

We present a variant of Theorem 5.2 which involves a weak version of Separation. The following axiom is logically implied by the combination of Separation and Strong Pareto. It relies on the intuition that it would seem strange to tell an agent that his own ranking of two allocations would be accepted at the social

238

APPENDIX A. PROOFS

level only if he were not there.1 Axiom Opponent Separation 0 For all E = (RN , Ω) ∈ D, and zN , zN ∈ X N , if there is i ∈ N such that zi0 Pi zi , then 0 0 zN R(E) zN ⇒ zN \{i} R(RN \{i} , Ω) zN\{i} . A valuable feature of the list of axioms in the following theorem (as well as in the variant of Theorem 5.1 proved above) is that they are all satisfied by the Ω-Equivalent Maximin SOF RΩmin introduced in Section 6.3 and defined 0 0 by: for all E = (RN , Ω) ∈ E and zN , zN ∈ X N , zN RΩmin (E) zN if and only if 0 mini∈N uΩ (zi , Ri ) ≥ mini∈N uΩ (zi , Ri ). This SOF does not satisfy Strong Pareto and Separation, but it does satisfy Weak Pareto (as well as Pareto Indifference) and Opponent Separation. Theorem A.1 On the domain E, if a SOF satisfies Weak Pareto, Equal-Split Transfer, Opponent Separation and Replication, then for all E = (RN , Ω) ∈ E 0 and zN , zN ∈ XN , 0 . min uΩ (zi , Ri ) > min uΩ (zi0 , Ri ) ⇒ zN P(E) zN i∈N

i∈N

Its proof involves the following lemma, which is a variant of Theorem 3.4. Lemma A.2 On the domain E, if a SOF satisfies Weak Pareto, Equal-Split Transfer, Opponent Separation and Replication, then it satisfies: For all E = 0 (RN , Ω) ∈ D, and zN , zN ∈ X N , if for all i ∈ N, zi0 Pi Ω/ |N | and there exists 0 j ∈ N such that Ω/ |N | Pj zj , then zN P(E) zN . Proof of Lemma A.2. Let R satisfy Weak Pareto, Equal Split Transfer, Opponent Separation and Replication. Step 1. The structure of this step is similar to the proof of Theorem 3.4, to which the reader may refer for a first intuition. Let E = (RN , Ω) ∈ E, 0 zN , zN ∈ X N , j, k ∈ N be such that zj À zj0 À

Ω À zk0 À zk , |N |

and for all i 6= j, k, zi0 Pi zi . Let ∆j = zj − Ω/ |N | and ∆k = (zk0 − zk ) /8. We 0 0 prove that zN P(E) zN . Assume, by way of contradiction, that zN R(E) zN . 1 2 3 4 Let R0 ∈ R, r, m ∈ Z++ , ε ∈ R++ , z0 , z0 , z0 , z0 ∈ X be such that for all t = 1, ..., 4, Ω Ω À z0t + 2ε À z0t À À zk0 , |N | + 1/r |N | + 2/r z01 P0 z04 + ε P0 z03 + ε P0 z02 + 2ε,

1 This is reminiscent of the “no-show paradox” (Brams and Fishburn 1983) in voting theory, which occurs when an agent is better-off abstaining from voting than expressing her true preferences.

A.2. CHAPTER 5: FAIR DISTRIBUTION OF DIVISIBLE GOODS z01 = z02 +

239

∆k ∆j , z04 = z03 + , m m

zj0 − Ω/ |N | ∆k , ε¿ , 3rm rm and for all i 6= j, k, zi0 Pi zi + 3rmε. Let zN + ε denote (zi + ε)i∈N . Let E 0 = (RN 0 , rΩ) ∈ E be a r-replica of E. By Replication Invariance, 0 0 0 00 zN 0 R (E 0 ) zN = (RN 0 , R0 , R0 , rΩ) , 0 . By Weak Pareto, zN 0 +ε P (E ) zN 0 . Let E 000 E = (RN 0 , R0 , rΩ) ∈ E. By Opponent Separation, ¢ ¡ 0 ¢ ¡ 1 1 zN 0 + ε, z01 + ε, z01 + ε P (E 00 ) zN 0 , z0 , z0 . ε¿

Since

z02 + ε À

rΩ ∆k À zk0 À zk + + ε, r |N | + 2 m

one can apply Equal-Split Transfer and obtain ¶ µ ¡ ¢ ∆k + ε, z02 + ε, z01 + ε R (E 00 ) zN 0 + ε, z01 + ε, z01 + ε . zN 0 \{k} + ε, zk + m By Weak Pareto, ¶ µ ∆k zN 0 \{k} + 2ε, zk + + 2ε, z03 + ε, z01 + 2ε m µ ¶ ∆k P (E 00 ) zN 0 \{k} + ε, zk + + ε, z02 + ε, z01 + ε . m One has z02 + 2ε À

rΩ ∆k À zk0 À zk + 2 + 2ε, r |N | + 2 m

so that by Equal-Split Transfer, ¶ µ ∆k 3 2 + 2ε, z0 + ε, z0 + 2ε zN 0 \{k} + 2ε, zk + 2 m µ ¶ ∆k 00 3 1 R (E ) zN 0 \{k} + 2ε, zk + + 2ε, z0 + ε, z0 + 2ε m By transitivity, µ ¶ ∆k zN 0 \{k} + 2ε, zk + 2 + 2ε, z03 + ε, z02 + 2ε m ¡ 0 ¢ 00 1 1 P (E ) zN 0 , z0 , z0 .

By Opponent Separation, ¶ µ ¡ 0 ¢ ∆k 1 + 2ε, z03 + ε R (E 000 ) zN zN 0 \{k} + 2ε, zk + 2 0 , z0 . m

240

APPENDIX A. PROOFS

One has zj −

∆j rΩ À À z04 + ε m r |N | + 1

and by Equal-Split Transfer, ¶ µ ∆j ∆k zN 0 \{j,k} + 2ε, zj − + 2ε, zk + 2 + 2ε, z04 + ε m m µ ¶ ∆k R (E 000 ) zN 0 \{k} + 2ε, zk + 2 + 2ε, z03 + ε . m By Weak Pareto, ¶ µ ∆j ∆k 1 + 3ε, zk + 2 + 3ε, z0 + ε zN 0 \{j,k} + 3ε, zj − m m µ ¶ ∆j ∆k 000 4 P (E ) zN 0 \{j,k} + 2ε, zj − + 2ε, zk + 2 + 2ε, z0 + ε . m m By transitivity, ¶ µ ¡ 0 ¢ ∆j ∆k 1 zN 0 \{j,k} + 3ε, zj − + 3ε, zk + 2 + 3ε, z01 + ε P (E 000 ) zN 0 , z0 m m

Since

z02 + ε À

rΩ ∆k À zk0 À zk + 3 + 3ε, r |N | + 1 m

by Equal-Split Transfer one has ¶ µ ∆j ∆k + 3ε, zk + 3 + 3ε, z02 + ε zN 0 \{j,k} + 3ε, zj − m m µ ¶ ∆ ∆k j R (E 000 ) zN 0 \{j,k} + 3ε, zj − + 3ε, zk + 2 + 3ε, z01 + ε m m By transitivity, ¶ µ ¡ 0 ¢ ∆j ∆k 2 1 zN 0 \{j,k} + 3ε, zj − + 3ε, zk + 3 + 3ε, z0 + ε P (E 000 ) zN 0 , z0 . m m By Opponent Separation, µ ¶ ∆j ∆k 0 zN 0 \{j,k} + 3ε, zj − + 3ε, zk + 3 + 3ε R (E 0 ) zN 0. m m

Repeating this argument m − 1 times, we obtain ¢ ¡ 0 zN 0 \{j,k} + 3mε, zj − ∆j + 3mε, zk + 3∆k + 3mε R (E 0 ) zN 0.

This repetition of the argument requires, for the application of Equal-Split Transfer, to have, for all t = 1, ..., m, zk0 À zk + 3t

∆k ∆j rΩ + 3tε, zj − t À . m m r |N | + 1

A.2. CHAPTER 5: FAIR DISTRIBUTION OF DIVISIBLE GOODS

241

This is guaranteed by the fact that zk0 À zk + 6∆k À zk + 3∆k + 3mε, zj − ∆j =

Ω rΩ À . |N | r |N | + 1

Now, repeating the whole argument r − 1 times, we obtain ⎛ ⎜ ⎜ ⎝

zN \{j,k} + 3rmε, {z } | (r times)

zj − ∆j + 3mε, zj − ∆j + 6mε, ..., zj − ∆j + 3rmε, zk + 3∆k + 3mε, zk + 3∆k + 6mε, ..., zk + 3∆k + 3rmε

⎞

⎟ 0 ⎟ R (E 0 ) zN 0. ⎠

which contradicts Weak Pareto because zj0 zk0

Ω + 3rmε = zj − ∆j + 3rmε, |N | À zk + 6∆k À zk + 3∆k + 3rmε, À

and for all i ∈ N \ {j, k} , zi0 Pi zi + 3rmε. 0 Step 2. Let E = (RN , Ω) ∈ E, zN , zN ∈ X N be such that for all i ∈ N, 0 1 2 zi Pi Ω/ |N | and there exists j ∈ N such that Ω/ |N | Pj zj . There exist zN , zN 1 0 2 1 2 such that for all i ∈ N, zi Pi zi , zi Pi zi , and for all i ∈ N \ {j}, zi À zi À Ω/ |N | À zj2 À zj1 . 1 By Weak Pareto, zN P (E) zN . By a repeated application of step 1 (at each application, some i 6= j is moved down from zi1 to zi2 while up from ¡ j is moved ¢ zj1 + t∆ to zj1 + (t + 1) ∆, for t = 0, ..., |N | − 2 and ∆ = zj2 − zj1 / (|N | − 1)), 2 1 0 2 0 P (E) zN . By Weak Pareto, zN P (E) zN . By transitivity, zN P (E) zN . zN 0 Proof of Theorem A.1. Let E = (RN , Ω) ∈ E and zN , zN ∈ X N be such 0 0 that mini∈N uΩ (zi , Ri ) > mini∈N uΩ (zi , Ri ). We have to prove that zN P(E) zN . 0 c Let L = {i ∈ N | zi Pi zi } and L = N \ L. By construction L 6= ∅. If 0 Lc = ∅, one immediately gets zN P(E) zN by Weak Pareto. We now consider c the subcase L 6= ∅. Let l = |L| . 0 Step 1: Let n = |N |. We first prove that zN P(E) zN when mini∈N uΩ (zi0 , Ri ) < 1/(n−l). Let m, q ∈ Z++ be such that q > n, m/q < 1/(n−l), m/q 6= uΩ (zi0 , Ri ) for all i ∈ N and

min uΩ (zi , Ri ) > i∈N

zi4

m > min uΩ (zi0 , Ri ). i∈N q

1 2 3 4 , zN , zN , zN be such that for all i ∈ N, zi1 À zi À zi2 À zi3 and Let zN 0 À zi , and

m m Ω Pi zi4 ⇔ Ω Pi zi0 q q 4 zi Pi zi ⇔ zi Pi zi0 m zi3 Pi Ω for all i ∈ N . q

242

APPENDIX A. PROOFS

good 2 6 µR2 ¡ ¡ ¡

µR2 ¡ ¡

µΩ ¡ ¡ ¡ s 4 ¡ z21 s z2 ¡ s 0 ¡ z 2 s ¡ s z2 s 2 ¡ z z23 2 ¡ ¡ ¡ ¡ ¡ ¡ ¡ sz11 ¡ ¡ z12 ssz1 s s¡ 3 ¡m z 1 q Ω µR1 ¡ ¡ ¡ ¡ ¡ ¡ z4 ¡ s1 ¡ µR1 ¡ ¡ s ¡ 0 ¡ z1 ¡ ¡ ¡ good 1 Figure A.3: Proof of Theorem A.1.

The construction is illustrated in Figure A.3. 0 Case 1 : m/q ≤ ⎛1/n. ⎛ Let E = (RN ⎞0 , mΩ)⎞be a m-replica of E. Choose any

⎜⎜ ⎟ ⎟ i0 ∈ L. Let E0 = ⎝⎝RN 0 , Ri0 , ..., Ri0 ⎠ , mΩ⎠ . The average endowment in E0 | {z } q−mn

is

m q Ω.

By Lemma A.2,

¢ ¡ 3 0 zN 0 , zi30 , ..., zi30 P(E0 ) (zN 0 , zi0 , ..., zi0 ) .

3 0 0 0 3 By Opponent Separation, zN 0 R(E ) zN 0 . By Weak Pareto, zN 0 P(E ) zN 0 and 0 0 therefore zN 0 P(E 0 ) zN . By Replication, z P(E) z . 0 N N

A.2. CHAPTER 5: FAIR DISTRIBUTION OF DIVISIBLE GOODS

243

Case 2 : 1/n < m/q < 1/(n − l). Let m0 ∈ Z++ be a multiple of m such that m0 >

m n q min 1 −

l m q (n

o, − l), m n − 1 q

and let q 0 = m0 q/m. The previous inequality is equivalent to l + m0 (n − l) < q 0 < (m0 − 1)l + m0 (n − l). Let E 0 = (RN 0 , m0 Ω) be a m0 -replica of E. Let N (i) ⊂ N 0 be the set of agents who are “clones” of i (including i). Due to the above inequalities,P there exists a list of subsets KN such that for all i ∈ L, ∅ 6= Ki à N (i) and i∈L |Ki | + m0 (n − l) = q 0 . Let (N1 , N2 , N3 ) be a partition of N 0 defined by: N1 = ∪i∈L Ki , N2 = ∪i∈L N (i) \ Ki , N3 = ∪i∈Lc N (i). To make things clearer, we define allocations with tables, so that, for instance, the following table N1 zi1

N2 zi4

N3 zi2

means that for all i ∈ N, j ∈ N (i), if i ∈ L and j ∈ Ki , then j consumes zi1 ; if i ∈ L and j ∈ N (i) \ Ki , j consumes zi4 ; if i ∈ / L, then j consumes zi2 . Let a zN 0 = b zN 0 =

N1 zi4 zi3

N2 zi3 zi2

N3 zi4 zi2 0

m Let E 00 = (RN1 ∪N3 , m0 Ω). The average endowment in E 00 is m q 0 Ω = q Ω. For m a all j ∈ N1 ∪ N3 , zjb Pj m q Ω, whereas for some j ∈ N1 , q Ω Pj zj . Therefore, b 00 a by Lemma A.2, zN1 ∪N3 P(E ) zN1 ∪N3 . Moreover, one has, for all i ∈ N, all b 0 a j ∈ N2 ∩ N (i), zjb Pj zka , so that, by Opponent Separation, zN 0 P(E ) zN 0 . By a 0 0 0 b 0 Weak Pareto, zN 0 P(E ) zN 0 and zN 0 P(E ) zN 0 . By transitivity, zN 0 P(E ) zN 0 . 0 By Replication, zN P(E) zN . Step 2: We now extend the result to the more general case mini∈N uΩ (zi0 , Ri ) < 1. Let the population of set Lc be numbered i1 , ..., in−l . We construct allocak tions zN , for k = 1, ..., n − l. Let j ∈ L, λ0 ∈ (0, 1] be such that uΩ (zj0 , Rj ) = mini∈N uΩ (zi0 , Ri ) < λ0 < mini∈N uΩ (zi , Ri ). 1 The allocation zN is chosen so that

λ0 Ω Pj zj1 Pj zj0 , zi11 Pi1 zi01 , for all i ∈ Lc , i 6= i1 , zi Pi zi1 Pi λ0 Ω, for all i ∈ L \ {j}, zi Pi zi1 Pi zi0 and zi1 Pi λ0 Ω. One therefore has: uΩ (zj1 , Rj ) = mini∈N uΩ (zi1 , Ri ) < 1. For all i 6= i1 , zi Pi zi1 , whereas zi11 Pi1 zi01 Ri1 zi1 . Denoting l1 = |{i ∈ N | zi Pi zi1 }|, one then has l1 =

244

APPENDIX A. PROOFS

n − 1, and mini∈N uΩ (zi1 , Ri ) < 1/(n − l1 ) = 1. Moreover, mini∈N uΩ (zi1 , Ri ) < mini∈N uΩ (zi , Ri ). By a direct application of step 1 to the pair of allocations 1 1 zN , zN , one concludes that zN P(E) zN . k Similarly, for k = 2, ..., n − l, construct zN such that zjk−1 Pj zjk Pj zj0 , zikk Pik zi0k , for i = 1, ..., ik−1 , zik−1 Pi zik Pi zi0 , for i = ik+1 , ..., in−l , zik−1 Pi zik Pi λ0 Ω, for all i ∈ L \ {j}, zik−1 Pi zik Pi zi0 and zik Pi λ0 Ω. This implies uΩ (zjk , Rj ) = mini∈N uΩ (zik , Ri ). Necessarily uΩ (zjk , Rj ) < 1. De¯ ¯ noting lk = ¯{i ∈ N | zik−1 Pi zik }¯ , one has lk = n − 1, and mini∈N uΩ (zik , Ri ) < 1/(n − lk ) = 1. Moreover, min uΩ (zik , Ri ) = uΩ (zjk , Rj ) < uΩ (zjk−1 , Rj ) = min uΩ (zik−1 , Ri ). i∈N

i∈N

k−1 k P(E) zN . Finally, by construction, for all By a direct application of step 1, zN n−l n−l 0 0 i ∈ N, zi Pi zi , so that by Weak Pareto, zN P(E) zN . By transitivity, one 0 concludes that zN P(E) zN . Step 3: We extend again to the general case. Let m0 be such that mini∈N uΩ (zi0 , Ri ) < 0 m , and let E 0 = (RN 0 , m0 Ω) be a m0 -replica of E. One has mini∈N 0 uΩ0 (zi0 , Ri ) < 1. By application of Step 2, one must have zN 0 P(E 0 ) zN 0 . By Replication, 0 zN P(E) zN . This completes the proof. In addition, we show that each axiom is necessary. 1) Drop Weak Pareto. Take Rpsum . 2) Drop Equal-Split Transfer. Take RΩsum . 3) Drop Opponent Separation. Consider´ Rλ defined as follows. Let Λ : ³ R+ → X be defined by: Λ(λ) = Ωk |N |k−1 λk . This defines a monotonic k=1,...,

path which contains 0 and Ω/|N but is not a ray from the origin. Then 0 for any zN , zN , λN , λ0N such that for all i ∈ N, zi Ii Λ(λi ) and zi0 Ii Λ(λ0i ), let λ 0 zN R (E) zN iff mini∈N λi ≥ mini∈N λ0i . 4) Drop Replication: Consider R defined as follows. For any E = (RN , Ω), let A(E) = {zN | ∀i ∈ N, zi Ri Ω} and B(E) = X |N | \ A(E). Then R = RΩsum 0 over A(E), R = RΩmin over B(E), and zN P(E) zN whenever zN ∈ A(E), 0 zN ∈ B(E).

Lemma A.3 On the domain E, if a SOF satisfies Strong Pareto and Priority 0 among Equals, then for all E = (RN , Ω) ∈ E, and zN , zN ∈ X |N | , if for all i ∈ N 0 such that zi Pi zi , there is j ∈ N such that Rj = Ri and zi Pi zi0 Pi zj0 Pi zj , 0 then zN R(E) zN .

Proof. It derives directly from repeated application of the conditions.

A.2. CHAPTER 5: FAIR DISTRIBUTION OF DIVISIBLE GOODS

245

Lemma A.4 Let Ω ∈ R++ , b1 , ..., bn ∈ Z++ , and x1 , ..., xn ∈ R+ . If n X

1 Pn

i=1 bi i=1

bi xi ¿ Ω,

then for all p ∈ ΠΩ , there exist y1 , ..., yn ∈ R+ such that: ∀i ∈ {1, ..., n}, yi ≤ xi or yi À xi , ∀i ∈ {1, ..., n}, pyi = 1, n X 1 Pn bi yi = Ω. i=1 bi i=1

Proof. Let J = {i | pxi ≥ 1} and K = {i | pxi < 1}. As pΩ = 1, necessarily K 6= ∅. For all i ∈ J, define yi = px1 i xi . By construction, xi ≥ yi for all i ∈ J, so that " # X X 1 0 Ω = Pn bi yi + bi xi ¿ Ω. i=1 bi i∈J i∈K

Now, for all i ∈ K, let

yi = xi +

(1 − pxi ) (Ω − Ω0 ) . (1 − pΩ0 )

One has yi À xi and pyi = 1 for all i ∈ K. Moreover, one computes 1 Pn

n X

i=1 bi i=1

which completes the proof. Proof of Theorem 5.5. that

bi yi = Ω,

0 Let E = (RN , Ω) ∈ E and zN , zN ∈ X N be such 0 zN R(E) zN

EW

0 (E) zN .

in spite of zN P m, q ∈ Z++ such that

Let pz ∈ arg maxp∈ΠΩ mini∈N up (zi , Ri ). There exist

min upz0 (zi0 , Ri ) < i∈N

(A.1)

m < min upz (zi , Ri ). i∈N q

(A.2)

0 By Strong Pareto, zN R(E) zN implies that there exists i ∈ N such that zi0 Ri zi , EW 0 and zN P (E) zN implies that there exists i ∈ N such that zi Pi zi0 . Step 1: Construction of bundles and preferences (see Figure A.4). Let K = {i ∈ N | upz0 (zi0 , Ri ) = min upz0 (zi0 , Ri ) or zi0 Ri zi }. i∈N

P |N | In view of (A.2), there exist weights (ai )i∈N ∈ R++ with i∈N ai = 1, and an allocation zbN ∈ X |N | , such that: for all i ∈ K, zbi Ii zi0 ; for all i ∈ N \ K, zbi Ii zi ; P m a z b i∈N i i ¿ q Ω.

246 good 2 6

APPENDIX A. PROOFS

R10

µ C s z1 ¡ C¡ q C q qq C µΩ ¡ R3 q Cs C R1 ¡ R30 ¡ C z12 C R10 µ ¡ µ ¡ C ¡ C¡ ¡ C ¡ C ¡ C R0 C ¡ ¡ C 3 C ¡ ¡ sz3 µ C ¡ C¡ ¡ qq C q C ¡ q q C sq C ¡ sq q 2C C ¡ z3 z31 C C ¡ C C ¡ C p C ¡ R1 : ¡ » C» C 0 µ R1 ¡ C ¡ C ¡ C¡ C sm ¡ qq q Ω C C ¡ q ¡ z0 s C ¡ Cs 1 PPs 1 z 1 C ¡ x C PP 0 PP C R2 C PP¡ C R2 ¡PP ££± p0 C ¡ qq PP ¡ Cµ C sq ¡ PP µ ¡ C PP C¡ ¡ z21 R2 PqCP sz2 C ¡ 0 PP qq µ R2 ¡ C q PP C ¡ ¡ PP z20 Csq C ¡ PsP ¡ 2 P C ¡ z2 C ¡PP C P C ¡ good 1 Figure A.4: Proof of Theorem 5.5.

1+ 1++ 1+++ 1 , zN , zN , zN in X |N| , arbitrarily close to zbN , Therefore, there exist zN |N | and (ni )i∈N ∈ Z++ such that:

P

1

i∈N

ni

X

i∈N

ni zi1+++ ¿

m Ω q

(A.3)

and such that for all i ∈ N,

zi Pi zi0

zi1 Pi zi0 , ⇒ zi Pi zi1+++ ,

(A.4)

A.2. CHAPTER 5: FAIR DISTRIBUTION OF DIVISIBLE GOODS and for all i ∈ N \ K, zi1 Pi q for all q such that pz q ≤ pz m q Ω = 2 |N | such that for all i ∈ N, Lemma A.3 one can find zN ∈ X

S

pz zi2 = m q zi2 ≤P zi1 or zi2 À zi1 1 m 2 i∈N ni zi = q Ω. ni

m q .

247 Then by

(A.5)

i∈N

0 . Consider i such that zi0 Ri zi (agent 2 We now construct a new profile RN in Figure A.4–as noted above, some such i must exist). By (A.4), zi1 Pi zi , so that ½ ¾ ¡ 1 ¢ m U zi , Ri ⊂ U (zi , Ri ) ⊂ x ∈ X | pz x > , q ¡ ¢ ¡ ¢ One can then find Ri0 such that I zi1 , Ri = I zi1 , Ri0 , I (zi , Ri ) = I (zi , Ri0 ) and µ ¶ ½ ¾ m m I Ω, Ri0 = x ∈ X | pz x = . q q Now consider i such that zi Pi zi0 (agent 1 in Figure A.4–as noted above, some such i must exist). Then, by (A.4), zi Pi zi1 . If pz zi1 < m/q, then, by (A.5), zi2 À zi1 . In addition, for all x ∈ X¡ such ¢that p¡z x = m/q, one has zi Pi x. It ¢ is then easy to find ³Ri0 such ´that I zi1 , Ri = I zi1 , Ri0 , I (zi , Ri ) = I (zi , Ri0 )

and zi2 ∈ max|R0 B i

m q Ω, pz

. If, on the other hand, pz zi1 ≥ m/q (agent 3 in

Figure A.4), then, by (A.5), zi2 ≤ zi1 . Recall that by construction, zi1 Pi q for all q such that pz q ≤ m/q. In that case, one can also find Ri0 satisfying the above conditions. 0 Summing up, one can find RN such that for all i ∈ N, ¡ 1 ¢ ¡ ¢ I zi , Ri = I zi1 , Ri0 , I (zi , Ri ) = I (zi , Ri0 ) , (A.6) µ ¶ m zi2 ∈ max|R0 B (A.7) Ω, pz . i q and such that for some j ∗ with zj1∗ Pj ∗ zj ∗ µ ¶ ½ ¾ m m I Ω, Rj0 ∗ = x ∈ X | pz x = . q q Note that, by (A.2), (A.6) and (A.7), for all i ∈ N one has: zi Pi0 zi2 ,

zi Pi zi1+++

(A.8)

zi Pi0 zi1+++ . ∗

and also: iff By (A.5), one can find x ∈ X and ν ∈ Z++ such that m pz x∗ = q # " X m 1 2 ∗ zi + νx = Ω. |N | + ν q i∈N

(A.9)

248

APPENDIX A. PROOFS

Similarly, by (A.3) one can then choose μ, s ∈ Z++ such that X q ni = s > |N | + ν + μ m i∈N # " X 1 X 1++ m zi + νzj1++ +μ ni zi1+++ ¿ Ω. ∗ s q i∈N

(A.10)

i∈N

By (A.5) and (A.9), one then necessarily has: # " X m 1 X 2 ∗ 2 zi + νx + μ ni zi = Ω. s q i∈N

(A.11)

i∈N

ª © Let L = i ∈ N | zi Pi zi1 and Lc = N \ L. Notice that one also has L = ª © i ∈ N | zi Pi0 zi1+++ . Indeed, by (A.4) and (A.6), zi Pi zi1 zi1 Ri zi

⇒ zi Pi zi0 ⇒ zi Pi zi1+++ ⇒ zi Pi0 zi1+++ , ⇒ zi1 Ri0 zi ⇒ zi1+++ Pi0 zi .

− −− −−− 2+ 2++ , zN , zN , zN , zN ∈ X |N | such that for all By (A.8), there exist zN i ∈ N, zi > zi− > zi−− > zi−−− Pi0 zi2++ > zi2+ > zi2 , (A.12)

and for all i ∈ L,

zi− − − Pi0 zi1+++ .

(A.13)

To sum up, for all i ∈ L, zi Pi0 zi− > zi− − > zi− − − Pi0 zi1+++ > zi1++ > zi1+ > zi1 Pi0 zi0 and zi− − − Pi0 zi2++ > zi2+ > zi2 whereas for all i ∈ Lc , zi1+++ > zi1++ > zi1+ > zi1 Pi0 zi0 Ri0 zi > zi− > zi− − > zi− − − Pi0 zi2++ > zi2+ > zi2 . Step 2: Derivation of a contradiction. By Weak Pareto and (A.4), 1 0 P(E) zN . Hence, by (A.1) and transitivity, zN 1 P(E) zN . zN

(A.14)

0 , Ω). By (A.14) and Unchanged-Contour Independence, Let E 0 = (RN 1 P(E 0 ) zN . zN

(A.15)

0 , msΩ) ∈ E be a ms-replica of E 0 . We partition Nms into Let E 0ms = (RN ms four subsets: for every i ∈ L, N1 contains (an arbitrary selection of) q clones of i and N3 contains the remaining ms − q clones; for every i ∈ Lc , N2 contains q

A.2. CHAPTER 5: FAIR DISTRIBUTION OF DIVISIBLE GOODS

249

clones of i and N4 contain ms − q such clones. Next, we introduce three other sets of additional agents, N5 , N6 and N7 with sizes (the agents’ indices do not matter) X X |N5 | = qν, |N6 | = qμ ni , |N7 | = qμ nk . k∈Lc

i∈L

By (A.15) and Replication, N1 zi1

N2 zi1

N3 zi1

N4 zi1

R(E 0ms )

N1 zi

N4 zi1++ zi

P(E 0ms ) P(E 0ms )

N1 zi1 zi− −

N2 zi

N3 zi

N4 zi .

By Strong Pareto, N2 zi1+ zi

N1 zi1+ zi

N3 zi− zi

N2 zi1 zi− −

N3 zi1 zi−

N4 zi1 , zi− .

By Theorem 3.1, R satisfies Priority among Equals. Therefore, one also has N1 zi− −

N2 zi− −

N3 zi−

N4 zi−

R(E 0ms )

N1 zi− −

N2 zi2++

N3 zi−

N4 zi1++ ,

by Lemma A.3, because for all j ∈ N2 , k ∈ N4 who are clones of the same i ∈ Lc , one has: zi1++ Pi0 zi− Pi0 zi−− Pi0 zi2++ . | {z } | {z } (consum ed by k)

(cons. by j)

By transitivity: N1 zi1+

N2 zi1+

N3 zi−

N4 zi1++

P(E 0ms )

N1 zi− −

N2 zi2++

N3 zi−

N4 zi1++ .

0 , msΩ). In the last two allocations, notice that for all Let E 00 = (RN 1 ∪N2 − 0 1+ i ∈ L, zi Pi zi , and for all i ∈ Lc , zi1++ Pi0 zi2++ . One can then apply Well-Off Separation and obtain:

N1 zi1+

N2 zk1+

00

R(E )

N1 zi− −

N2 zk2++ .

By Strong Pareto, N1 zi−−

N2 zk2++

P(E 00 )

N1 zi− − −

N2 zk2+ .

so that, by transitivity, N1 zi1+

N2 zk1+

P(E 00 )

N1 zi− − −

N2 zk2+ .

250

APPENDIX A. PROOFS

0 Let E 000 = (RN , msΩ), with the corresponding profiles of pref1 ∪N2 ∪N5 ∪N6 ∪N7 0 0 erences: RN5 = (Rj0 ∗ , ..., Rj0 ∗ ), RN = (qμni Ri )i∈L (meaning that every Ri is 6 0 replicated qμni times), RN7 = (qμni Ri )i∈Lc . We first note that j ∗ ∈ Lc and that for every i ∈ Lc , zi1++ Pi0 zi2+ . For every i ∈ L, zi Pi0 zi1+ . So that, by Well-Off Separation:

N1 zi1+

N2 zi1+

N5 zj1++ ∗

N6 zi

N7 N1 zi1+++ R(E 000 ) zi− − −

N2 zi2+

N5 zj1++ ∗

N6 zi

N7 zi1+++ .

By Lemma A.3 one has N1 zi1++

N2 zi1+

N5 zj1++ ∗

N6 zi1+++

N7 N1 zi1+++ P(E 000 ) zi1+

N2 zi1+

N5 zj1++ ∗

N6 zi

N7 zi1+++

because for all j ∈ N1 , k ∈ N6 who are clones of the same i ∈ L, one has zi Pi0 zi1+++ Pi0 zi1++ Pi0 zi1+ . | {z } | {z } (cons. by k)

By Strong Pareto, N1 zi−−−

N2 zi2+

N5 zj1++ ∗

N6 zi

N7 zi1+++

(cons. by j)

P(E 000 )

N1 zi2

N2 zi2

N5 x∗

N6 zi2

N7 zi2 ,

so that by transitivity, N1 zi1++

N2 zi1+

N5 zj1++ ∗

N6 zi1+++

N7 zi1+++

P(E 000 )

N1 zi2

N2 zi2

N5 x∗

N6 zi2

N7 zi2 .

Now, by (A.10), N1 zi1++

N2 zi1+

N5 zj1++ ∗

N6 zi1+++

N7 zi1+++

∈ Z(E 000 )

while by (A.7) and (A.11), N1 zi2

N2 zi2

N5 x∗

N6 zi2

N7 zi2

∈ S EW (E 000 ).

As a consequence, by Theorem 5.3, N1 zi2

N2 zi2

N5 x∗

N6 zi2

N7 zi2

R(E 000 )

N1 zi1++

N2 zi1+

N5 zj1++ ∗

N6 zi1+++

N7 zi1+++ ,

a contradiction. No axiom is redundant: 1) Drop Strong Pareto: The SOF yielding total indifference between all allocations.

A.2. CHAPTER 5: FAIR DISTRIBUTION OF DIVISIBLE GOODS

251

2) Drop Transfer among Equals: The serial dictatorship of agents (e.g., 1 is the dictator, when she is indifferent then 2 takes over, and so on). 3) Drop Unchanged-Contour Independence: Consider R defined as follows. Let E CD be the subset of economies with Cobb-Douglas preferences. For E ∈ E CD let p(E) ∈ ΠΩ denote the unique price vector of the Walrasian allocation(s) ¡ ¢ 0 0 with equal budgets. Then for all zN , zN ∈ X N , zN R(E) zN iff up(E) (zi , Ri ) i∈N ≥lex ¢ ¡ 0 0 iff either zN PEW (E) zN or up(E) (zi0 , Ri ) i∈N . For E ∈ E \ E CD , zN R(E) zN EW 0 Ωlex 0 (E) zN and zN R (E) zN . zN I 4) Drop Selection Monotonicity: RΩlex . 5) Drop Well-Off Separation: Consider R defined as follows. Let p0 ∈ ΠΩ ∩ 0 0 R++ . For E = (RN , Ω) ∈ E, zN , zN ∈ X N , let zN R(E) zN iff (up (zi , Ri ))i∈N ≥lex 0 ∗ (up0 (zi , Ri ))i∈N , with p ∈ arg maxp∈ΠΩ mini∈N up (zi , Ri ) if there exists zN ∈ EW ∗ 0 (E) such that for all i ∈ N, zi Ri zi , and p = p otherwise; and similarly S for p0 . 6) Drop Replication: Consider R defined as follows. Let E = (RN , Ω) ∈ E, 0 zN , zN ∈ X N , and λ, λ0 ∈ R+ be such that λ = maxp∈ΠΩ mini∈N up (zi , Ri ) and 0 λ = maxp∈ΠΩ mini∈N up (zi0 , Ri ). If for all k ∈ Z++ , ¤ 1 £ ∈ / min{λ, λ0 }, max{λ, λ0 } , k

0 0 0 0 ⇔ zN RΩlex zN ; otherwise zN R(E) zN iff either zN PEW (E) zN then zN R(E) zN EW 0 Ωlex 0 or zN I (E) zN and zN R (E) zN . 0 Proof of Theorem 5.6. Let zN , zN ∈ X N be two allocations such that EW 0 0 zN P (E) zN . Suppose that, contrary to the result, one has zN R(E) zN . The latter fact, by Weak Pareto, implies the existence of some i0 ∈ N such that zi00 Ri0 zi0 . Let pz ∈ arg max min up (zi , Ri ), p∈ΠΩ i∈N

and a, b, c, d, e, f, g ∈ R be such that maxp∈ΠΩ mini∈N up (zi , Ri ) > a > b > c > d > e > maxp∈ΠΩ mini∈N up (zi0 , Ri ), g > f > max {maxi uΩ (zi , Ri ), maxi uΩ (zi0 , Ri )} . 0 0 Let E 0 = (RN , Ω) ∈ E, with RN being defined as follows. First, Ri0 0 is such that

U (zi0 , Ri0 0 ) = U (zi0 , Ri0 ), U (zi00 , Ri0 0 ) = C(zi00 , Ri0 ), for all 0 ≤ λ ≤ a, and

© ª U (λΩ, Ri0 0 ) = q ∈ R+ | pz q ≥ λ , © ª U (gΩ, Ri0 0 ) = q ∈ R+ | q ≥ gΩ .

For all i 6= i0 , let Ri0 be such that

U (zi , Ri0 ) = U (zi , Ri ), U (zi0 , Ri0 ) = U (zi0 , Ri ),

252

APPENDIX A. PROOFS

and

© ª U (gΩ, Ri0 ) = q ∈ R+ | q ≥ gΩ .

0 By Hansson Independence, zN R(E 0 ) zN . 1 Let zN be such that for all i 6= i0 , zi Pi zi1 and © ª U (zi1 , Ri ) ⊂ q ∈ R+ | pz q > a ,

1 0 1 whereas zi10 = aΩ. By Weak Pareto, zN P(E 0 ) zN , so that by transitivity, zN P(E 0 ) zN . 2 2 2 1 2 Let zN be such that for all i 6= i0 , zi = gΩ, while zi0 = bΩ. One has zi0 Pi0 zi0 and for all i 6= i0 , zi2 Pi zi1 . In addition, for all i 6= i0 ,

L(zi10 , Ri0 0 ) ∩ U (zi1 , Ri0 ) = ∅ (their indifference curves do not cross). This implies that by repeated application of Lemma A.1 in combination with Unchanged-Contour Independence and Weak Pareto (the detailed argument has the same structure as the proof of 3 1 1 2 “zN P(E 00 ) zN ” in the proof of Theorem 5.1), one shows that zN P(E 0 ) zN , and 0 0 2 therefore, by transitivity, zN P(E ) zN . Since max min up (zi0 , Ri ) < e, p∈ΠΩ i∈N

there exists an allocation z for all i ∈ N, and

1∗

and positive integers a1 , ..., an such that zi1∗ Pi0 zi0

P

1

i∈N ai

P

X

i∈N

ai yi0 ¿ eΩ.

0 0 Let k = i ai , and consider a k-replica E 0k = (RN k , kΩ) of E . Consider the 2∗ 2∗ 1∗ 2∗ allocation zN k such that zi1 = zi for all i ∈ N, and zj = gΩ for all “clones” j ∈ 2∗ 0k 0 γ −1 (i)\{i} (see Section 2.3 for these notions). By Weak Pareto, zN ) zN k P(E k, 2∗ 0k 2 and therefore, by Replication and transitivity, zN k P(E ) zN k . 3 3 3 −1 (i0 ) \ Let the allocation zN k be defined by zi0 = dΩ, zj = cΩ for all j ∈ γ 3 −1 2 0k 3 {i0 }, and zi = f Ω for all i ∈ / γ (i0 ). By Weak Pareto, zN k P(E ) zN k . Let the ∗ ∗ ∗ allocation zN k be defined by zi0 = eΩ and zi = gΩ for all i 6= i0 . By Lemma A.1, Unchanged-Contour Independence and Weak Pareto (same argument as 3 0k ∗ 2∗ 0k ∗ above), zN ) zN ) zN k P(E k , so that, by transitivity, zN k P(E k. 00k 00 00 Now, let E = (RN k , kΩ) with a new profile RN k defined by: Ri00 = Ri0 for all i ∈ N, and among the (k − 1)n remaining agents (the “clones”, who all have 2∗ ∗ gΩ in both zN k and zN k ), ai n − 1 of them have a preference relation equal to 0 Ri , for every i ∈ N . Since for all i ∈ N k ,

U (Ri00 , zi2∗ ) = U (Ri0 , zi2∗ ), U (Ri00 , zi∗ ) = U (Ri0 , zi∗ ), 2∗ 00k ∗ ) zN by Unchanged-Contour Independence, one has zN k P(E k. 1∗ 3∗ 0∗ Close to zN , in E there exist two allocations zN and zN such that for all i ∈ N, gΩ Pi0 zi0∗ À zi3∗ À zi1∗ ,

A.3. CHAPTER 6: SPECIFIC DOMAINS and

1 P

i ai

X i

253

ai zi0∗ ¿ eΩ.

0∗∗ 0∗∗ = zi3∗ for all i ∈ N, and In E 00k , let zN k be the allocation defined by zi zj0∗∗ = zi0∗ for all j among the ai n − 1 clones who have Rj00 = Ri0 . By Lemma A.1, Unchanged-Contour Independence and Weak Pareto (same argument as above), 0∗∗ 00k 2∗ 0∗∗ 00k ∗ zN ) zN ) zN k P(E k , so that zN k P(E k. ∗ k But zN is such that for all i ∈ N , k

zi∗ ≥ eΩ À P

1

j∈N

aj

X

j∈N

aj zj0∗ ≥

1 X 0∗∗ zj . nk k j∈N

¡ 00 P ¢ ∗ ∗ Moreover, zN k ∈ P RN k , i∈N k zi . This entails a contradiction with Proportional Efficient Dominance. Finally, we have to show that no axiom is redundant. 1) Drop Weak Pareto. Take the SOF yielding total indifference. 2) Drop Transfer among Equals. Take RRU . 3) Drop Proportional Efficient Dominance. Take RΩmin . 4) Drop Unchanged-Contour Independence. Take the SOF which coincides with REW if there are agents with identical preferences, and with RRU otherwise. 4) Dropping Replication. Take the SOF R defined by: 0 0 zN R(E) zN ⇔ V (zN ) ≥ V (zN ),

with V (zN ) = min

A.3

½

λ ∈ R+ | ∃q PN ∈

Q

|N |

U (zi , Ri ), aP N ∈ Z+ , a = |N | , λΩ ≥ i∈N i i∈N ai qi i∈N

¾

.

Chapter 6: Specific domains

Proof of Theorem 6.1. We prove the result using Opponent Separation instead of Separation. Theorem 3.2 still holds on E R (even replacing Separability by Opponent Separation–moreover, Opponent Separation makes the result hold for the whole domain E R and not just for economies with at least three agents), so that any R satisfying Weak Pareto, Nested-Contour Transfer and Opponent Separation also satisfies a variety of Nested-Contour Priority which 0 says the following: For all E = (RN , Ω) ∈ E R , and zN , zN ∈ X N , if there exist j, k ∈ N such that zj À zj0 À zk0 À zk ,

U (zj0 , Rj ) ∩ L(zk0 , Rk ) = ∅

254

APPENDIX A. PROOFS

0 and for all i 6= j, k, zi0 Pi zi , then zN P(E) zN . R 0 Let E = (RN , Ω) ∈ E and zN , zN ∈ X N be such that mini∈N c(zi , Ri ) > 0 0 mini∈N c(zi , Ri ). Assume that, contrary to the desired result, zN R(E) zN . 0 1 2 3 Let i0 ∈ N be chosen so that c(zi0 , Ri0 ) < mini∈N c(zi , Ri ). Let zN , zN , zN , 4 5 6 N 3 1 0 zN , zN , zN ∈ Xc be such that for all i ∈ N, zi Pi zi , zi Pi zi ; for all i ∈ N \{i0 } , zi1 Pi zi and

zi10 ¿ zi20 ¿ zi6 = zi60 ¿ zi5 ¿ zi4 ¿ zi3 = zi30 ¿ zi2 ¿ zi1 ; X X zi2 + zi3 ¿ (2 |N | − 1) zi60 . i∈N

(A.16)

i6=i0

1 0 3 1 3 P(E) zN and zN P(E) zN , so that zN P(E) zN . For any By Weak Pareto, zN i 6= i0 , let ˆı ∈ / N be an agent such that Rˆı = Ri . Let M = {ˆı | i ∈ N \ {i0 }} . t For t = 3, 4, 5, 6, let zM be defined by zˆıt = zit for all ˆı ∈ M. By Opponent Separation, ¡ 1 4 ¢ ¡ 3 5 ¢ zN , zM P((RN , RM ) , Ω) zN , zM .

By the above variant of Nested-Contour Priority, ³ ´ ¡ 1 4 ¢ 2 3 zi20 , zN \{i0 } , zM P((RN , RM ) , Ω) zN , zM

so that by transitivity, ³ ´ ¡ 3 5 ¢ 2 2 zi20 , zN , z \{i0 } M P((RN , RM ) , Ω) zN , zM .

¡ 3 5 ¢ ¡ 6 6 ¢ By Weak Pareto, zN , zM P((RN , RM ) , Ω) zN , zM . But Certainty Transfer, Weak Pareto and (A.16) imply that ³ ´ ¡ 6 6 ¢ 2 2 zN , zM P((RN , RM ) , Ω) zi20 , zN \{i0 } , zM ,

a contradiction. We check that no condition is redundant. Drop Weak Pareto. Take the SOF yielding total indifference.P Drop Nested-Contour Transfer. Take R1 sum , which relies on i∈N c(zi , Ri ). Drop Certainty Transfer. Take RΩm in . Drop Opponent Separation. Take the SOF which coincides with R1 sum when preferences are identical and linear, and with R1 min otherwise.

Proof of Theorem 6.3. Let us say that the orderings Ri and Rj are strictly different if for all x, x0 ∈ X N , I(x, Ri ) ∩ I(x0 , Ri0 ) contains no manifold of dimension −1. Let E RN be the subdomain of economies with profile RN , and consider a profile RN such that for some i, j ∈ N, Ri and Rj are strictly different. We first show that R = RΩNash on E RN . By Independence of Feasible Set, there is an ordering R such that R(E) = R for all E ∈ E RN . By Continuity and Pareto Indifference, it is represented by a 0 continuous function W such that for all zN , zN ∈ XN , 0 zN R zN ⇔ W ((ui (zi ))i∈N ) ≥ W ((ui (zi0 ))i∈N ),

A.3. CHAPTER 6: SPECIFIC DOMAINS

255

where ui can be taken to be an arbitrary homogeneous representation of Ri for every i ∈ N. By Restricted Separation and Continuity, and invoking the Debreu-Gorman theorem (Debreu 1959, Gorman 1968), there exists a list of continuous functions (ϕi )i∈N such that: W ((ui (zi ))i∈N ) =

X

ϕi (ui (zi )) .

i∈N

The fact that max|R(E) Z(E) = S EW (E) for all E entails that every ϕi is increasing. Indeed, suppose that for some i ∈ N, ϕi (a) ≤ ϕi (b) for a > b. There is E ∈ E RN such that ui (zi ) = a for zN ∈ S EW (E). Take zi0 = (b/a)zi . One then has (zN \{i} , zi0 ) R(E) zN , implying (zN \{i} , zi0 ) ∈ max|R(E) Z(E), which is / S EW (E). impossible as (zN\{i} , zi0 ) ∈ As every ϕi is increasing, it is almost everywhere differentiable. Let zN ∈ max|R(E) Z(E). One has zN ∈ S EW (E) and by Theorem 1 in Eisenberg (1961), for every E ∈ E H , the utility levels at S EW (E) are unique. And they are positive. When differentiability P of each ϕi holds at (ui (zi ))i∈N , the first-order condition of maximization of i∈N ϕi (ui (zi )) implies that for all k = 1, . . . , , ϕ0i (ui (zi ))

∂ui (zi ) = λk (Ω) , ∂zik

where λ (Ω) is the vector of Lagrange multipliers for the resource constraint P EW (E), one has λ (Ω) zi = λ (Ω) Ω/ |N | for all i∈N zi ≤ Ω. Since zN ∈ S i ∈ N. Besides, ui being homogeneous implies λ (Ω) zi =

X

k=1

ϕ0i (ui (zi ))

∂ui (zi ) zik = ϕ0i (ui (zi )) ui (zi ) . ∂zik

Summarizing, one has, for almost all Ω ∈ R++ , all zN ∈ S EW (RN , Ω), all i ∈ N, ϕ0i (ui (zi )) ui (zi ) = λ (Ω) Ω/ |N | . 0 Now, let Ω, Ω0 ∈ R++ be such that for zN ∈ S EW (RN , Ω), zN ∈ S EW (RN , Ω0 ), 0 there is i ∈ N such that ui (zi ) =¡ui (z . This λ (Ω) Ω = λ (Ω0 ) Ω0 and ¡ i0)¢¢ ¡ 0implies ¢ 0 0 therefore ϕj (uj (zj )) uj (zj ) = ϕj uj zj uj zj for all j ∈ N. As utility levels at S EW (E) are unique, let Ui (E) = ui (zi ) for zN ∈ S EW (E), i ∈ N. Take i, j ∈ N such that Ri and Rj are strictly different. There exists A ⊂ R++ such that

Ui (RN , A) = {Ui (E) | E = (RN , Ω), Ω ∈ A} is a singleton whereas Uj (RN , A) is a non-degenerate interval [α, β] ⊂ R++ . This implies that ϕ0j (u) u is almost everywhere defined and constant for u ∈ [α, β]. By homotheticity, with A0 = (β/α) A one obtains that Ui (RN , A0 ) is a singleton while Uj (RN , A0 ) = [β, β 2 /α]. Therefore ϕ0j (u) u is also constant over

256

APPENDIX A. PROOFS

[β, β 2 /α]. Repeating this argument, one eventually concludes that ϕ0j (u) u is constant over R++ . Since ϕ0j (uj (zj )) uj (zj ) = λ (Ω) Ω/ |N | for zN ∈ S EW (RN , Ω), this implies that λ (Ω) Ω/ |N | is constant in Ω (whenever it is defined), and therefore that, for all i ∈ N, ϕ0i (u) u is constant (and the same constant for all i) wherever it is defined over R++ . As a consequence, and by continuity of ϕi , for all u ∈ R+ , ϕi (u) = a ln u + bi for some a ∈ R++ , bi ∈ R. This implies that R coincides with RΩNash on E RN . Notice that ui was taken to be any arbitrary homogeneous representation of Ri . The particular function uΩ (zi , Ri ) is one such representation, and for is clear that the criterion Pthis particular choice of ui (for all i ∈ N ) it Q a i∈N ln ui (zi ) coincides with RΩNash (defined by i∈N uΩ (zi , Ri )). But the P fact that a i∈N ln ui (zi ) coincides with RΩNash holds true for any other homogeneous representations (ui )i∈N . This is simply because all homogeneous representations of a given homothetic Ri are proportional to each other. Now we extend the result to E. Let E = (RN , Ω) ∈ E. Take i ∈ / N such that for some j ∈ N, Ri and Rj are strictly different. Then, R(RN ∪{i} , Ω) = RΩNash (RN∪{i} , Ω) by the above argument. By Restricted Separation, R(E) = RΩNash (E) over allocations with positive utilities. By Continuity, this extends to all X N . We check that no condition is redundant. 0 Drop Pareto Indifference. Take the SOF R such that zN R(E) zN iff either 0 0 zN PΩNash (E) zN or zN IΩNash (E) zN and zN ∈ Pr (Ω) . Drop Independence of Feasible Set. For any E = (RN , Ω), choose some p 0 supporting price vector for S EW (E). Then let zN R(E) zN iff X X up (zi , Ri ) ≥ up (zi0 , Ri ). i∈N

i∈N

Drop Restricted Separation. Let o n EW = zN ∈ X |N| | ∃Ω ∈ R+ , zN ∈ EW (RN , Ω)

and choose an arbitrary p0 ∈ R++ . Then evaluate every allocation zN by X p0 zi0 . max T 0 ∈EW ∩ zN ( i∈N L(zi ,Ri )) i∈N

Drop max|R(E) Z(E) = S EW (E). Take RΩ0 sum for some fixed Ω0 ∈ R++ . 0 0 iff either zN PΩNash (E) zN , Drop Continuity. Take the SOF R such that zN R(E) zN ΩNash 0 Ω0 lex 0 or zN I (E) zN and zN R (E) zN .

Proof of Theorem 6.4. The proof is divided in four steps. In the first step, we prove the equivalence between Consistency and a Strong Consistency axiom. In the second step, we prove that Strong Pareto, Independence of Preferences over Infeasible Bundles, Consistency imply that only values of um (zi , Ri ) matter. In the third step, we prove that the um (zi , Ri ) need to be aggregated according

A.3. CHAPTER 6: SPECIFIC DOMAINS

257

to a maximin. In the fourth step, we prove that this maximin needs to be a leximin. Step 1) We begin by proving that Consistency is equivalent to the following Strong Consistency axiom. It requires that social evaluation be unaffected by the addition of agents receiving the same bundles. Formally it amounts to requiring that the implication of the Consistency statement be replaced with an equivalence. Axiom Strong Consistency 0 For all E = (RN , A) ∈ D with |N | ≥ 3, and zN , zN ∈ X N , if there is i ∈ N such 0 that zi = zi = (ai , mi ), then 0 0 zN R(E) zN ⇔ zN \{i} R(RN\{i} , A \ {ai }) zN \{i} .

To prove the claim, let us assume that R satisfies Consistency but not Strong 0 0 whereas zN \{i} P(RN \{i} , A \ {ai }) zN\{i} . As Consistency, that is zN I(E) zN 0 0 zN R(E) zN , by Consistency, zN \{i} R(RN\{i} , A \ {ai }) zN \{i} , the desired contradiction. The proof of the converse statement is similar. Step 2) Let us first state the property formally. Axiom Money Equivalence 0 0 0 For all E = (RN , A), E 0 = (RN , A0 ) ∈ D, and zN , zN , yN , yN ∈ X N , if for all i∈N um (zi , Ri ) = um (yi , Ri0 ) and um (zi0 , Ri ) = um (yi0 , Ri0 ), then 0 0 ⇔ yN R(E 0 ) yN . zN R(E) zN

We claim that if R satisfies Pareto Indifference, Independence of Preferences over Infeasible Bundles, and Consistency, then it satisfies Money Equivalence. By Consistency, R also satisfies Strong Consistency. Let E = (RN , A), E 0 = 0 0 0 , A0 ) ∈ E ind , zN , zN ∈ Z(E), and yN , yN ∈ Z(E 0 ) be such that (RN um (zi , Ri ) = um (yi , Ri0 ) and um (zi0 , Ri ) = um (yi0 , Ri0 ) .

(A.17)

Let us assume that 0 . zN R(E) zN

(A.18)

0 . Let n = #N . We begin by constructing We have to prove that yN R(E 0 ) yN two sets of n bundles which are infeasible for E. Let ε > 0. Let m, e m e 0 ∈ RN be defined by m e i = min {−ε, um (zi , Ri ) − ε}

m e 0i

= m e i if zi Ii zi0 , m e i − ε if zi Pi zi0 , m e i + ε if zi0 Pi zi .

258

APPENDIX A. PROOFS

e ⊂ A be such that A e ∩ (A ∪ A0 ) = ∅ and #A e = n, so that we can find a Let A e Let N e be such that N e ∩ N = ∅ and #N e = n, so that we bijection σ : N → A. ³ ´Nh e×R e . Let zeN , ze0 ∈ A be defined by: for all can find a bijection ρ : N → N N i ∈ N, zeρ(i) 0 zeρ(i)

= (σ(i), m e i) , = (σ(i), m e 0i ) .

Let R ∈ RindN be such that for all i ∈ N, a, a0 ∈ A ∪ {ν}, m, m0 ∈ R, (a, m) Ri (a0 , m0 ) ⇔ (a, m) Ri (a0 , m0 ) , and for all i ∈ N , 0 zeN

0 zi I i zeρ(i) and zi0 I i zeρ(i) .

(A.19)

eρ(i) = Ri . R

(A.21)

Given the way zeN and were constructed, such preferences exist. By Independence of Preferences over Infeasible Bundles, Eq (A.18) implies ¡ ¢ 0 zN R RN , A zN . (A.20) ³ ´ e= N e , A, e R e ∈ E ind be defined by: for all i ∈ N, Let E By Consistency and Strong Consistency, Eq (A.20) implies ´ ´ ¡ ¡ ¢ ³³ ¢ 0 eh ,A ∪A e zN zN , ze h R RN , R , ze h . N

N

N

By Pareto Indifference, and Eqs (A.19) and (A.21), ´ ´ ³ ´ ¢ ³³ ¡ 0 eh A∪A e , z z e zeNh , zN R RN , R h N . N N

By Consistency,

³ ´ e ze0 . zeNh R RN , A h N

(A.22)

Let R ∈ RindN be the list of preferences which coincide with R on the bundles e ∪ {ν} as first component, and with R0 on the bundles having an element of A having an element of A0 ∪ {ν} as first component, that is, for all i ∈ N, a, a0 ∈ e ∪ {ν}, b, b0 ∈ A0 ∪ {ν} , m, m0 ∈ R, A (a, m) Ri (a0 , m0 ) ⇔ (a, m) Ri (a0 , m0 ) , and 0

0

(b, m) Ri (b , m ) ⇔ (b, m)

Ri0

0

(A.23)

0

(b , m ) .

By Independence of Preferences over Infeasible Bundles, Eq (A.22) implies ³ ´ e ze0 . zeNh R RN , A (A.24) h N

A.3. CHAPTER 6: SPECIFIC DOMAINS ee h Let R ∈ RindN be defined by for all i ∈ N,

ee R ρ(i) = Ri .

259

(A.25)

By Consistency and Strong Consistency, Eq (A.24) implies ¶ ³ ¶ µµ ´ ¢ ¡ ee 0 0 e RN , RNh , A ∪ A, zeN , y . zeNh , yN R N h

By Pareto Indifference and Eqs (A.17), (A.19), (A.21) and (A.25), ¶ ³ ¶ ´ µµ ´ ³ ee 0 0 e R e e . RN , R ∪ A, yN , zeN , A yN , zeN h N

By Consistency,

³ ´ 0 yN R RN , A0 yN .

By Independence of Preferences over Infeasible Bundles and Eq (A.23), 0 yN R (E 0 ) yN ,

the desired outcome. Step 3) We now prove that if R satisfies Strong Pareto, Independence of Preferences over Infeasible Bundles, Consistency, and Transfer among Equals, 0 then it s a maximin in um (zi , Ri ). Let E = (RN , A) ∈ E ind , zN , zN ∈ Z(E), be such that min {um (zi , Ri )} > min {um (zi0 , Ri )} . i∈N

i∈N

0 Assume, contrary to what needs to be proven, that zN R(E) zN . Let E 0 = 0 ind 0 0 (RN , A) ∈ E be such that for all j, k ∈ N, Rj = Rk and Rj0 has the property that for all a ∈ A, m, m0 ∈ R, and for all i ∈ N ,

(a, m) Ii (ν, m0 ) ⇒ (a, m) Rj0 (ν, m0 ) , which means that the willingness to pay for any object in A is greater for Rj0 0 than for any Ri . Given the restriction on Rj0 , there exist yN , yN ∈ Z(E 0 ) such that um (zi , Ri ) = um (yi , Ri0 ) and um (zi0 , Ri ) = um (yi0 , Ri0 ) .

(A.26)

By Money Equivalence, Eq (A.26) implies 0 R(E 0 ) yN . (A.27) yN ¡ 0 0¢ Let j ∈ N be such that um yj , Rj = mini∈N um (yi0 , Ri0 ). Let N be partitionned into N1 , and N2 such that #N1 = n1 , #N2 = n2 and

∀ i ∈ N1 : um (yi , Ri0 ) ≥ um (yi0 , Ri0 ), ∀ i ∈ N2 : um (yi , Ri0 ) < um (yi0 , Ri0 ).

260

APPENDIX A. PROOFS

0 Note that j ∈ N1 . If N2 = ∅, then, by Strong Pareto, yN R(E 0 ) yN , a contradiction. So, let us assume that N2 6= ∅. The remainder of the proof consists in 00 00 showing that it is possible to build a new allocation yN , such that yN P(E 0 ) yN 0 0 and N can still be partitionned into two sets, N1 and N2 , such that #N10 = n1 +1 and #N20 = n2 − 1. Repeating the argument n2 times eventually yields the contradiction with Strong Pareto. Note that each repetition of the argument typically requires that new preferences be defined, which, by Money Equivalence, is always possible. Let k ∈ N2 . Let u00j , u00k be such that ¡ ¢ um yj0 , Rj0 < u00j ≤ u00k < um (yl , Rl0 ) , ∀ l = j, k.

Let a, b ∈ A. We may and do assume that a 6= b, since #A ≥ 2. Moreover, Rj0 could have been defined in such a way that there is some ∆ > 0, and mj , mk ∈ R, such that yj0 Ij0 (a, mj ) , ¢ ¡ ν, u00j Ij0 (a, mj + ∆) , (ν, u00k ) Ik0 (b, mk − ∆) , yk0 Ik0 (b, mk ) .

00 We do assume that Rj0 has this property. Let y N , yN ∈ Z(E 0 ) be such that for 00 0 0 all i 6= j, k, y i = yi Ii yi , y j = (a, mj ), y k = (b, mk ), yj00 = (a, mj + ∆), and yk00 = (b, mk − ∆). By Pareto Indifference, 0 yN I(E 0 ) y N .

(A.28)

By the Transfer Principle among Equals, 00 yN P(E 0 ) y N .

(A.29)

By Eqs (A.27), (A.28) and (A.29), 00 P(E 0 ) yN , yN

the desired outcome. Step 4) To complete the proof, we show that if R satisfies Strong Pareto, Independence of Preferences over Infeasible Bundles, Consistency, Transfer among Equals, and Anonymity among Equals , then it coincides with Rmlex . Let R satisfy the axioms. It also satisfies Strong Consistency and Money Equivalence. Let E = (RN , A) ∈ E ind . By Money Equivalence, we can assume, w.l.o.g., that for all i, j ∈ N, Ri = Rj . (If this condition were not satisfied, then we could change the actual profile of preferences into a new profile satisfying the condition, without affecting the social preferences over money equivalent 0 vectors, as we did above.) Let zN = ((ai , mi ))i∈N , zN = ((a0i , m0i ))i∈N ∈ Z(E). We distinguish two cases. 0 Case 1: zN Imlex (E) zN . Let n = #N . Let σ : N → {1, . . . , n} be a bijection satisfying the property that for all i, j ∈ N , σ(i) ≥ σ(j) ⇒ um (zi , Ri ) ≥

A.3. CHAPTER 6: SPECIFIC DOMAINS

261

u (zj , Rj ). Let zσN ∈ Z(E) denote the allocation obtained from zN by permuting its component according to σ. Let σ 0 denote the similar bijection associ0 ated to zN , and zσ0 0 N the resulting allocation. By Anonymity among Equals, 0 0 zN I(E) zσN and zN I(E) zσ0 0 N . By construction, and given that zN IL (E) zN , 0 0 zσi Ii zσ0 i so that by Pareto Indifference, zσN I(E) zσ0 N . Gathering all the above 0 social indifferences yield zN I(E) zN , the desired outcome. mlex 0 Case 2: zN P (E) zN . Assume, contrary to the statement we have to prove, that 0 zN R(E) zN . (A.30) Let n = #N . Let σ, σ 0 , zσN , zσ0 0 N be defined as above. By Anonymity among 0 Equals, zN I(E) zσN and zN I(E) zσ0 0 N .Therefore, by Eq (A.30), zσ0 0 N R(E) zσN .

(A.31)

0 Given that zN Pmlex (E) zN , there is j ∈ N such that for all i ∈ N such that 0 σ (i) < σ (j) , zσi Ii zσ0 i and zσj Pj zσ0 0 j . Then, N can be partitionned into N1 , N2 and N3 such that #N1 = n1 , #N2 = n2 , #N3 = n3 and j = n1 + 1, and

∀ i ∈ N1 : σ(i) ≤ n1 and um (zσi Ri ) = um (zσ0 0 i Ri ), ∀ i ∈ N2 : um (zσi , Ri ) > um (zσ0 0 i , Ri ), ∀ i ∈ N3 : um (zσi , Ri ) ≤ um (zσ0 0 i , Ri ). If N1 = ∅, then, given that R is maximin in money-equivalent, zσN P(E) zσ0 0 N , a contradiction. So let us assume that N1 6= ∅. Our strategy consists in using Consistency to remove agents in N1 from the economy, so that agent j has the smallest money equivalent, in contradiction to R being a maximin, but removing those agents may yield an infeasible allocation (not enough money ©P ª P 0 e e would be left). Let M = min i∈N1 mσi , i∈N1 mσ 0 i . Let A ⊂ A. Let N be e ∩N = ∅, #N e = n, A∩A e e = n. Let ze h = ((e ai , m e i ))i∈Nh ∈ such that N = ∅ and #A ³ ´N P h N ∗ ind e e e e i < M. Let E = R h , A ∈ E be such that (A × R) , be such that hm i∈N

N

e, R ei = Rj . By Money Equivalence, we can assume, w.l.o.g., that for all i ∈ N e for all i ∈ N , ³ ´ ¡ ¢ ei > um zσ0 0 j , Rj . um zei , R

(If this condition were not satisfied, then we could change the actual profile of preferences into a new profile satisfying the condition, without affecting the social preferences over money equivalent vectors.) Let z N , z 0N ∈ Z(E) be defined by ∀ i ∈ N2 ∪ N3 , z i ∀ i ∈ N1 , z i = z 0i

= zσi and z 0i = zσ0 0 i , X X = zσi if mσi ≤ m0σ0 i , i∈N1

zσ0 0 i

if

X

i∈N1

i∈N1

mσi >

X

i∈N1

m0σ0 i .

262

APPENDIX A. PROOFS

By Pareto Indifference, zσN I(E) z N and zσ0 0 N I(E) z 0N , so that Eq (A.31) implies z 0N R(E) z N . By Consistency and Strong Consistency (remember that Consistency is equivalent to Strong Consistency), ¡ 0 ¢ ¡ ¢ e z N , ze h . z N , zeNh R(E, E) N

By Consistency, ´¡ ´ ³³ ´ ³ ¢ e h , (A \ ∪i∈N1 {ai }) ∪ A e z N \N , ze h , z 0N\N1 , zeNh R RN \N1 , R 1 N N

which contradicts the fact that R is a maximin in money³equivalent,´as agent j now has the smallest money equivalent in the allocation z 0N\N1 , zeNh . We check that no condition is redundant. Drop Strong Pareto. Take the SOF R that minimizes the larger money equivalent. Drop Independence of Preferences over Unfeasible Bundles. Take the SOF R that applies the leximin to the following well-being index. Let a∗ ∈ A. Let index v be defined by v (zi , Ri ) = m ⇔ (a∗ , m) Ii zi . Drop Consistency. Take the SOF R that applies the leximin to the following well-being index. For a ∈ A∗ , let ma (Ri , zi ) = m ⇔ (a, m) Ii zi . For all E = (RN , A) ∈ E ind , index w is defined by w (Ri , zi , A) = m ⇔ m =

X

ma (Ri , zi ) .

a∈A∪{ν}

Drop Transfer among Equals. Take the SOF R that ranks allocations so as to maximize the sum of agents’ money equivalent. Drop Anonymity among Equals. Let ≥ denote a complete ordering on the names of the agents. Take the SOF R that coincides with Rmlex in case of strict preference, and which prefers, in case of a tie, the allocation where the name of agent with smallest money equivalent is the smaller.

A.4

Chapter 7: Extensions

Proof of Theorem 7.1. 1-i) Suppose that for all i ∈ N, zi Ri zi0 . Then {λ

∈ R+ | ∃qN ∈ S(RN , λΩ) s.t. zi0 Ri qi , ∀ i ∈ N } ⊆ {λ ∈ R+ | ∃qN ∈ S(RN , λΩ) s.t. zi Ri qi , ∀ i ∈ N },

A.4. CHAPTER 7: EXTENSIONS

263

0 which implies VS∗ (zN , E) ≤ VS∗ (zN , E). Similarly, 0 0 ∈ RN , ∃qN ∈ S(RN , λΩ) {λ ∈ R+ | ∃RN 0 0 s.t. ∀ i ∈ N, zi Ri qi and I(zi0 , Ri ) = I(zi0 , Ri0 )}

0 0 ⊆ {λ ∈ R+ | ∃RN ∈ RN , ∃qN ∈ S(RN , λΩ) 0 s.t. ∀ i ∈ N, zi Ri qi and I(zi , Ri ) = I(zi , Ri0 )}

0 , E) ≤ VS∗∗ (zN , E). Pareto Indifference follows directly. implies VS∗∗ (zN 1-ii) By upper-hemicontinuity, sup is in fact max in the definition of V ∗ . Indeed, by definition of sup there is λk → VS∗ (zN , E) such that [ for all k, there k ∈ S(RN , λk Ω) such that zi Ri zik for all i ∈ N. As Z(RN , λk Ω) is is zN k

k ∗ compact, there is a subsequence of zN which converges to some zN . By upper∗ ∗ hemicontinuity, zN ∈ S(RN , VS (zN , E)Ω) and by continuity of preferences, zi Ri zi∗ for all i ∈ N. In other words,

VS∗ (zN , E) ∈ {λ ∈ R+ | ∃qN ∈ S(RN , λΩ) s.t. zi Ri qi , ∀ i ∈ N }, implying that VS∗ (zN , E) = max{λ ∈ R+ | ∃qN ∈ S(RN , λΩ) s.t. zi Ri qi , ∀ i ∈ N }. 0 , E) ≤ VS∗ (zN , E). Suppose that Suppose zi Pi zi0 for all i ∈ N. Then VS∗ (zN ∗ 0 ∗ VS (zN , E) = VS (zN , E). By the “sup is max” argument, there is qN ∈ S(RN , VS∗ (zN , E)Ω) such that zi0 Ri qi for all i ∈ N. One has zi Pi qi for all i ∈ N. Let λk → VS∗ (zN , E) k such that λk > VS∗ (zN , E) for all k. By lower-hemicontinuity, there is qN ∈ k S(RN , λk Ω) such that qN → qN . By continuity of preferences, there is a finite k such that zi Pi qik , ∀ i ∈ N. For this particular k,

λk ∈ {λ ∈ R+ | ∃qN ∈ S(RN , λΩ) s.t. zi Ri qi , ∀ i ∈ N }, which implies that VS∗ (zN , E) ≥ λk > VS∗ (zN , E), a contradiction. 2) When S is Pareto efficient, one has VS∗ (zN , E), VS∗∗ (zN , E) ≤ 1 for all 0 zN ∈ Z(E). Indeed, if VS∗ (zN , E) > 1, there is λ > 1 and zN ∈ S(RN , λΩ) 0 0 such that zi Ri zi for all i ∈ N. This is impossible if zN ∈ P (RN , λΩ) and 0 0 0 zN ∈ Z(E). If VS∗∗ (zN , E) > 1, there is λ > 1, RN ∈ RN and zN ∈ S(RN , λΩ) 0 0 0 such that zi Ri zi for all i ∈ N (which is equivalent to zi Ri zi because I(zi , Ri ) = 0 0 I(zi , Ri0 )). This is impossible if zN ∈ P (RN , λΩ) and zN ∈ Z(E). ∗ When zN ∈ S(E) or is Pareto indifferent to some zN ∈ S(E), necessarily ∗ ∗∗ ∗ VS (zN , E), VS (zN , E) ≥ 1 and therefore VS (zN , E) = VS∗∗ (zN , E) = 1. k Conversely, suppose VS∗ (zN , E) = 1. There is λk → 1 and zN ∈ S(RN , λk Ω) k k such that zi Ri zi for all i ∈ N. There is at least a subsequence of zN which tends ∗ ∗ to some zN . As S is upper-hemicontinuous with respect to Ω, zN ∈ S(RN , Ω) and by continuity of preferences, zi Ri zi∗ for all i ∈ N. As S is Pareto efficient, necessarily zi Ii zi∗ for all i ∈ N. Therefore VS∗ (zN , E) = 1 iff zN is Pareto ∗ indifferent to some zN ∈ S(E). If all allocations that are Pareto indifferent to ∗ some zN ∈ S(E) are also in S(E), then R∗S rationalizes S.

264

APPENDIX A. PROOFS

¯ coincides with RΩlex for ¯ E) Proof of Theorem 7.2. 1) Assume that R( 0 ¯ single-profile comparisons of allocations. Let E = (N, Ω) ∈ E and zN , zN ∈ XN , 0 N 0 0 RN , RN ∈ R be such that (uΩ (zi , Ri ))i∈N = (uΩ (zi , Ri ))i∈N , and not all the values of (uΩ (zi , Ri ))i∈N are equal. There is no loss of generality in assuming 0 0 that zN , zN ∈ Pr (Ω) (which implies zN = zN ) and that z1 ≤ ... ≤ zn . Let Ri∗ ∗ be such that I (z1 , Ri ) = I (z1 , Ri ) and U (zn , Ri∗ ) = {q ∈ X | q ≥ zn } , and Ri∗∗ be such that I (z1 , Ri∗∗ ) = I (z1 , Ri0 ) and U (zn , Ri∗∗ ) = {q ∈ X | q ≥ zn } . We first focus on i = 1. By Cross-Profile Independence, ¯ E) ¯ ((z1 , ..., zn ) , (R1∗ , ..., Rn )) . ((z1 , ..., zn ) , (R1 , ..., Rn )) I( By RΩlex , ¯ E) ¯ ((zn , z2 , ..., zn−1 , z1 ) , (R1∗ , ..., Rn )) . ((z1 , ..., zn ) , (R1∗ , ..., Rn )) I( By Cross-Profile Independence, ¯ E) ¯ ((zn , z2 , ..., zn−1 , z1 ) , (R1∗∗ , ..., Rn )) . ((zn , z2 , ..., zn−1 , z1 ) , (R1∗ , ..., Rn )) I( By RΩlex , ¯ E) ¯ ((z1 , ..., zn ) , (R1∗∗ , ..., Rn )) . ((zn , z2 , ..., zn−1 , z1 ) , (R1∗∗ , ..., Rn )) I( By Cross-Profile Independence, ¯ E) ¯ ((z1 , ..., zn ) , (R10 , ..., Rn )) . ((z1 , ..., zn ) , (R1∗∗ , ..., Rn )) I( By transitivity, ¯ E) ¯ ((z1 , ..., zn ) , (R10 , ..., Rn )) . ((z1 , ..., zn ) , (R1 , ..., Rn )) I( We now focus on i = 2. By RΩlex , ¯ E) ¯ ((z2 , z1 , ..., zn ) , (R10 , R2 , ..., Rn )) . ((z1 , z2 , ..., zn ) , (R10 , R2 ..., Rn )) I( By the above reasoning, ¯ E) ¯ ((z2 , z1 , ..., zn ) , (R10 , R20 , ..., Rn )) . ((z2 , z1 , ..., zn ) , (R10 , R2 , ..., Rn )) I( By RΩlex , ¯ E) ¯ ((z1 , z2 , ..., zn ) , (R10 , R20 , ..., Rn )) . ((z2 , z1 , ..., zn ) , (R10 , R20 , ..., Rn )) I( By transitivity, ¯ E) ¯ ((z1 , z2 , ..., zn ) , (R10 , R20 , ..., Rn )) . ((z1 , z2 , ..., zn ) , (R1 , R2 , ..., Rn )) I( If n > 2, repeating the last four steps for i = 3, ..., n, one obtains ¯ E) ¯ ((z1 , ..., zn ) , (R10 , ..., Rn0 )) . ((z1 , ..., zn ) , (R1 , ..., Rn )) I(

A.4. CHAPTER 7: EXTENSIONS

265

From this fact and by application of RΩlex it follows directly that when 0 0 ¯ (zN , RN ) . (uΩ (zi0 , Ri0 ))i∈N >lex (uΩ (zi , Ri ))i∈N one must have (zN , RN ) P¯ (E) EW ¯ coincides with R ¯ E) 2) Assume that R( for single-profile comparisons of ¯ = (N, Ω) ∈ E and zN , z 0 ∈ X N , RN , R0 ∈ RN be such that allocations. Let E N N max min up (zi , Ri ) = max min up (zi0 , Ri0 ) .

p∈ΠΩ i∈N

p∈ΠΩ i∈N

Let p∗ ∈ arg maxp∈ΠΩ mini∈N up (zi , Ri ) and λ∗ = mini∈N up∗ (zi , Ri ) . First, assume that for all i ∈ N, Ri0 is the linear ¡ordering¢based on p∗ and 1 1 ¯ E) ¯ (zN , RN ) . up∗ (zi0 , Ri0 ) = λ∗ . Let zN be such that z11 À z1 and zN , RN I( 1 Let RN be such that I(z11 , R11 ) = I(z11 , R1 ), I(λ∗ Ω, R11 ) = I(λ∗ Ω, R10 ),

¡ 1 ¢ ¡ ¢ 1 ¯ E) ¯ z 1 , RN . and Rj1 = Rj for all j ∈ N. By Cross-Profile Independence, zN , RN I( N By construction, ¢ ¡ ¢ ¡ 1 1 1 ¯ E) ¯ (λ∗ Ω, z2 , ..., zn ) , RN I( . zN , RN By Cross-Profile Independence, ¢ ¡ ∗ 1 ¯ E) ¯ ((λ∗ Ω, z2 , ..., zn ) , (R10 , R2 , ..., Rn )) . I( (λ Ω, z2 , ..., zn ) , RN By transitivity,

¯ E) ¯ ((λ∗ Ω, z2 , ..., zn ) , (R10 , R2 , ..., Rn )) . (zN , RN ) I( Repeating this argument for i = 2, ..., n, one obtains 0 ¯ E) ¯ ((λ∗ Ω, ..., λ∗ Ω) , RN ). (zN , RN ) I( 0 0 ¯ E) ¯ ((λ∗ Ω, ..., λ∗ Ω) , R0 ) , one obtains (zN , RN ) I( ¯ E) ¯ (z 0 , R0 ) . , RN ) I( As (zN N N N 0 Second, assume that for all i ∈ N, Ri is such that

U (λ∗ Ω, Ri0 ) = {q ∈ X | q ≥ λ∗ Ω} ,

and zi0 = λ∗ Ω. For all i ∈ N, let Ri∗ be the linear ordering based on p∗ . By the previous step, we know that (zN , RN ) 0 0 (zN , RN )

∗ ¯ E) ¯ ((λ∗ Ω, ..., λ∗ Ω) , RN I( ), ∗ ∗ ∗ ¯ ¯ I(E) ((λ Ω, ..., λ Ω) , RN ) .

¯ E) ¯ (z 0 , R0 ) . By transitivity, (zN , RN ) I( N N ¯ E) ¯ (z 0 , R0 ) without making special asFinally, we prove that (zN , RN ) I( N N 0 0 ∗ ∗ sumptions about (zN , RN ) . Let (zN , RN ) be such that for all i ∈ N, zi∗ = λ∗ Ω and Ri∗ is such that U (λ∗ Ω, Ri∗ ) = {q ∈ X | q ≥ λ∗ Ω} .

¯ E) ¯ (z ∗ , R∗ ) and that (z 0 , R0 ) I( ¯ E) ¯ (z ∗ , R∗ ) . By the previous step, we know that (zN , RN ) I( N N N N N N 0 0 ¯ ¯ By transitivity, (zN , RN ) I(E) (zN , RN ) .

266

A.5

APPENDIX A. PROOFS

Chapter 10: Unequal Skills

Proof of Theorem 10.5. We omit the first part and focus on the sec0 ond. Let E = (sN , RN ), zN , zN ∈ X N be such that mini∈N usmin (zi , Ri ) > 0 mini∈N usmin (zi , Ri ). As is now usual, we can focus on the case in which there is i0 such that for all i 6= i0 , usmin (zi0 , Ri ) > usmin (zi , Ri ) > usmin (zi0 , Ri0 ) > usmin (zi00 , Ri0 ). Fix an arbitrary zi000 such that zi0 Pi0 zi000 Pi0 zi00 . Let R∗ be such that ( , c) R∗ ( 0 , c0 ) if and only c−(smin − ε) ≥ c0 −(smin − ε) 0 , where 0 < ε < mini∈N usmin (zi , Ri )− mini∈N usmin (zi0 , Ri ). As max|R∗ B(s, x) ≥ max|R B(s, x) for all R ∈ R, s ∈ S, x ∈ X, one has R∗ &M I R for all R ∈ R. Let ta , tb , tc , td be such that ta − tb = tc − td and min usmin (zi , Ri ) − smin ε > ta > tb > tc > td > usmin (zi0 , Ri0 ). i∈N

Let us focus on i0 and a particular j 6= i0 . Let Rj0 be such that I(zj , Rj0 ) = I(zj , Rj ), I(zj0 , Rj0 ) = I(zj0 , Rj ), I((1, ta + smin ), Rj0 ) = I((1, ta + smin ), R∗ ), 0 I((1, tb + smin ), Rji ) = I((1, tb + smin ), R∗ ). Let z c = max|Ri B(smin , (0, tc )), 0 z d = max|Ri B(smin , (0, td )). 0 As explained in Section 10.6, Strong Pareto, Equal-Preferences Transfer and Unchanged-Contour Independence imply Equal-Preferences Priority. Therefore ¡ ¡ ¢ ¢¢ ¡ 0 zj , (1, tb + smin ) . (zj , (1, ta + smin )) R (sj , smin ) , Rj0 , Rj0 By Unchanged-Contour Independence, ¡ ¡ ¢ ¢¢ ¡ 0 zj , (1, tb + smin ) . (zj , (1, ta + smin )) R (sj , smin ) , Rj , Rj0 Similarly,

¡ ¢ ¡ 00 d ¢ zi0 , z R ((si0 , smin ) , (Ri0 , Ri0 )) zi00 , z c .

By &M I -Equal-Skill Transfer, ¡ ¢ ¢ ¡ (1, tb + smin ), z c R ((smin , smin ) , (R∗ , Ri0 )) (1, ta + smin ), z d .

By Unchanged-Contour Independence, ¢ ¡ ¢¢ ¡ ¢ ¡ ¡ (1, ta + smin ), z d . (1, tb + smin ), z c R (smin , smin ) , Rj0 , Ri0

By Separation, one deduces from the previous equations that ¡ 0 ¡ ¢ ¡ ¢¢ ¡ 0 0 ¢ zi0 , zj , (1, ta + smin ), z c R (si0 , sj , smin , smin ) , Ri0 , Rj , Rj0 , Ri0 zi0 , zj , (1, tb + smin ), z c , ¡ ¢ ¡ ¢¢ ¡ 0 ¢ ¡ 00 zi0 , zj , (1, ta + smin ), z c , zi0 , zj , (1, ta + smin ), z d R (si0 , sj , smin , smin ) , Ri0 , Rj , Rj0 , Ri0 ¡ ¢ ¡ ¢¢ ¡ 00 ¢ ¡ 00 zi0 , zj , (1, ta + smin ), z d . zi0 , zj , (1, tb + smin ), z c R (si0 , sj , smin , smin ) , Ri0 , Rj , Rj0 , Ri0

A.5. CHAPTER 10: UNEQUAL SKILLS

267

By transitivity, ¡ 00 ¡ ¢ ¡ ¢¢ ¡ 0 0 ¢ zi0 , zj , (1, tb + smin ), z c R (si0 , sj , smin , smin ) , Ri0 , Rj , Rj0 , Ri0 zi0 , zj , (1, tb + smin ), z c . By Separation again, ¡ 00 ¡ ¢ ¢ zi0 , zj R ((si0 , sj ) , (Ri0 , Rj , )) zi00 , zj0 , ´ ¡¡ ´ ³ ¢ ¡ ¢¢ ³ 0 0 0 0 , s , s , R , R , z , z R s , R z zi000 , zj , zN\{i i0 j N \{i0 ,j} i0 j N\{i0 ,j} i0 j N\{i0 ,j} . 0 ,j}

Repeating this step for all i 6= i0 , one eventually obtains that for some zi∗0 such that zi0 Pi0 zi∗0 Pi0 zi00 , ¡ ∗ ¢ 0 zi0 , zN\{i0 } R (E) zN . ¡ ¢ 0 By Strong Pareto, zN P (E) zi∗0 , zN\{i0 } and therefore by transitivity zN P (E) zN .

268

APPENDIX A. PROOFS

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