Nonpaternalistic Sympathy and the Inefficiency of Consistent Intertemporal Plans David G. Pearce∗ Princeton University, 1983

Abstract Perfect equilibria of intertemporal “cake-eating” models in which generations care about one another’s utilities, are shown to be inefficient. Systematic consumption bias is generated without the temporally asymmetric assumptions used in earlier papers where the same phenomenon arises (4, 16). The definition of subgame perfect equilibrium (13, 14) is substantially generalized to apply to the well-defined strategic situation analyzed, which is formally not a game.

1

Introduction

Strotz’ path-breaking paper (15) concerns the dilemma of an individual who realizes that he cannot trust himself to carry out (over time) the course of action that he currently views as optimal. The literature spawned by that work discusses a set of interrelated problems including taste formation, the desire for precommitment, and the existence and efficiency of consistent intertemporal plans (see, for example, Pollak (10), Phelps and Pollak (9), Blackorby, Primont, Nissen and Russell (1), Peleg and Yaari (8), Hammond (6), Yaari (16), and Goldman (4, 5)). As these authors recognize, identical issues arise when economic decisions are made by successive generations having interdependent preferences. In this context Goldman (4) gives conditions ∗

I wish to thank Dilip Abreu, Bob Anderson and Hugo Sonnenschein for their searching questions and invaluable technical advice. I enjoyed helpful conversation with Ed Green, Greg Mankiw, Barry Nalebuff, Evan Porteus, Richard Quandt and Dan Usher.

1

guaranteeing the inefficiency of an interior subgame perfect equilibrium (Selten (13, 14)) of an intergenerational “cake-eating” game and the existence of a “slower” consumption programme that Pareto dominates the equilibrium outcome. The latter result suggests a “bias” in the decision process, favouring overly rapid consumption. The model presented here is almost identical to Goldman’s, except that instead of being interested in one another’s consumption levels, generations care about one another’s utilities. Such preferences seem rather natural: if generation i is sympathetic to j, and j’s utility rises, this should please i, regardless of whether the increase in j’s utility results from increased consumption, or from an improvement in the welfare of some other generation k toward whom j is favourably disposed. This kind of interdependence might be called liberal, or nonpaternalistic, sympathy. In Section 2 I give conditions under which a system of utility functions displaying nonpaternalistic sympathy generates a unique vector of utilities, and behaves well in a sense to be made precise. Section 3 then describes the strategies of each of the T generations, and defines a subgame perfect outcome of the model. This is of special interest to students of strategic behaviour, because the strategic situation involved is actually not a game, and the definition of subgame perfect Nash equilibrium must be extended substantially to cover it. Section 4 establishes that under the conditions of Section 2, nonpaternalistic sympathy leads to the same results (inefficiency and “bias”) obtained in (4). Analyzing what drives his results, Goldman notes that the “ ‘overconsumption’ arises as a consequence principally of the future bias (A3) and monotonicity (A4).” ((4), page 624). Our respective axioms are not nested. However, my assumptions have the advantage of being symmetric in all players, thereby making it clear that the results emerge naturally from the strategic situation, without assistance from asymmetric restrictions on the utility functions. The theoretical structure developed in Section 2 has many interesting applications; Section 5 concentrates on one of these. I show that even when all members of a society exhibit nonpaternalistic sympathy, a “socially optimal” interior allocation (evaluated by any strictly increasing social welfare function) can never be achieved by appointing a dictator to distribute resources. The Conclusion, Section 6, includes a brief discussion of some extensions of the results of Section 4.

2

2

Nonpaternalistic Sympathy and Coherence

A number of difficulties arise when utilities are generated by a system of interdependent utility functions.1 These difficulties and their resolutions are discussed here without reference to the intergenerational strategic situation of ultimate interest; this should make it clear that the issues involved, such as the existence and uniqueness of mutually consistent utility numbers, are entirely distinct from questions of strategic equilibrium. Consider T individuals (or generations) having utilities defined by ut = Ut (ct , u1 , . . . , ut−1 , ut+1 , . . . , uT ),

t = 1, . . . , T

(∗)

where ct is the quantity of some commodity consumed by t. Thus player t’s utility is given by his utility function Ut (mapping R+ × RT −1 into R) and depends upon the level of his own consumption, and the utilities of other individuals. The most basic question that presents itself is the following: what assumptions guarantee that for any particular vector (c1 , . . . , cT ) there will exist utilities u1 , . . . , uT satisfying the system (∗)? That this question has some substance is illustrated by the following two person example. u1 = U1 (c1 , u2 ) = u2 + 1 u2 = U2 (c2 , u1 ) = u1 + 1

Clearly no utilities exist that can solve both equations; for any c = (c1 , c2 ) the map U c (u1 , u2 ) ≡ (U1 (c1 , u2 ), U2 (c1 , u2 )) = (u2 + 1, u1 + 1) has no fixed point. If mutually compatible utility numbers do exist, when will they be unique? A system such as U1 (c1 , u2 ) = u2 U2 (c2 , u1 ) = u1 1

Ed Green first alerted me to the problems discussed in this section.

3

certainly cannot predict definitively the utilities that result from a particular consumption vector, and the derivative of u1 with respect to c1 , for example, is consequently not well-defined. Finally, the system may or may not be “stable” in a certain sense. Suppose that U1 (c1 , u2 ) = −10 + c1 + 2u2 U2 (c2 , u1 ) = −10 + c2 + 2u1 Solving these equations for u1 and u2 yields c1 2c2 − 3 3 c2 2c1 u2 = 10 − − 3 3 u1 = 10 −

Notice that although the functions U1 and U2 seem to indicate that the individuals enjoy consumption and are mutually sympathetic, the reduced form indicates that both u1 and u2 are increased when c1 or c2 is decreased! This does not correspond to any plausible dynamic adjustment story, such as the following. If c1 is reduced by 1 unit, this should “initially” lower u1 by 1, which would then cause u2 to fall 2 units (via the function U2 ), diminishing u1 by a further 4 units. This downward spiral does not converge; the only finite solution of the equations entails a perverse increase in utilities for both individuals. Such a counter-intuitive state of affairs might be called instability, just as we consider unstable a bare-bones Keynesian model with a marginal propensity to consume (M.P.C.) of 2, giving an apparent (but 1 = −1. misleading) government expenditure multipler of 1−M.P.C. I shall establish that assumptions (A1)–(A3) below guarantee the existence of a unique set of utilities satisfying (∗): the resulting “reduced form” utility functions are continuously differentiable and respond positively to increases in consumption levels. Hence I refer to (A1)–(A3) as coherence conditions for (∗). Let Utt denote the partial derivative of Ut with respect to ct (holding all other utilities constant) and Uij denote the partial derivative of Ui with respect to Uj , i 6= j, holding all other utilities and ci constant. The functional arguments are suppressed below: (A1)–(A3) hold at all points in the domain of each function. (A1) Ut : R+ × RT −1 → R is continuously differentiable, t = 1, . . . , T . 4

(A2) Uij > 0,

i = 1, . . . , T ; j = 1, . . . , T .

(A3) There exist positive real numbers eji such that Uij ≤ eji , and   0 e21 . . . eT1  e1 0 eT2    2 E ≡  .. ..  . . 1 2 eT eT . . . 0 satisfies the Hawkins-Simon condition (6), that is, all the principal minors of I − E are positive (or equivalently, since E is a nonnegative square matrix, the largest characteristic root r of E is less than 1 (see Debreu and Herstein (2), Theorem IV)). Proposition 1. If equations (∗) satisfy (A1)–(A3), for any given vector (c1 , . . . , cT ), there exists a unique set of utilities satisfying (∗). Proof. Existence. For all c = (c1 , . . . , cT ) ∈ RT+ , and u = (u1 , . . . , uT ) ∈ RT , define   U1 (c1 , u2 , . . . , uT )   .. U c (u) =   . UT (cT , u1 , . . . , uT −1 ) Uic is the ith component of the vector U c . Let   |U1 (c1 , 0, . . . , 0)|   .. bc =   . |UT (cT , 0, . . . , 0)|

and Gc (u) = bc + Eu, where E is the matrix of uniform upper bounds on derivatives, in (A3). Notice that for u > 0, Utc (u) exceeds Utc (0) by P defined i at most i6=t et (ui − 0). Hence, for all u ≥ 0, U c (u) ≤ U (c, 0) + Eu ∴ U c (u) ≤ bc + Eu = Gc (u)

(1)

Since by (A3), the determinant of E is positive, I − E is invertible. Let ∴

B c = (I − E)−1 bc B c = bc + EB c = Gc (B c ), by the definition of Gc . 5

(2)

Thus B c is a fixed point of Gc . Letting subscripts indicate components of the vector B c , let Stc be the closed interval [−Btc , Btc ], t = 1, . . . , T , and Q S c = Tt=1 Stc . Recall that by (A2), U is strictly monotonic. For all u ∈ S c , |(c1 , u2 , . . . , uT )| . c c . U (u) ≤ U . |(cT , u1 , . . . , uT −1 )| ≤ U c (B c ) ≤ Gc (B c ) (by (1)) = B c (by (2)) A symmetric argument establishes that for all u ∈ S c , U c (u) ≥ −B c . Combining these results, U c (u) ∈ S c

for all u ∈ S c .

Thus, letting U˜ c be the restriction of U c to S c , we have U˜ c : sc → sc . Since U˜ c is continuous (by (A1)) and sc is nonempty, compact, and convex, Brouwer’s Fixed Point Theorem guarantees the existence of a vector v c ∈ S c , such that U˜ c (v c ) = v c and hence c c Ut (ct , v1c , . . . , vt−1 , vt+1 , . . . , vTc ) = vtc ,

t = 1, . . . , T.

Thus (cc1 , . . . , vTc ) satisfies (∗). Uniqueness. Suppose that distinct vectors (v1 , . . . , vT ) and (w1 , . . . , wT ) satisfy (∗). Without loss of generality, suppose that for some t, wt > vt . Let a, b, . . . , f be the components in which w strictly exceeds v, and F denote

6

the matrix obtained by deleting all rows and columns of E except those corresponding to a, . . . , f . F inherits (A3) from E. Now     wa − va wa − va     .. ..  ≤F  . . w f − vf wf − vf 

 wa − va   .. But this is impossible, because   > 0 and F satisfies the Hawkins. wf − vf Simon condition (A3) (see Debreu and Herstein (2), Lemma*, with s = 1). Consequently, it is impossible for two distinct utility vectors to satisfy (∗). Definition. For the system (∗) satisfying (A1)–(A3), define for each c = (c1 , . . . , cT ) Vt (c1 , . . . , cT ) = Ut (ct , v1 , . . . , vt−1 , vt+1 , . . . , vT ),

t = 1, . . . , T,

where v1 , . . . , vT are the unique utilities satisfying (∗). Proposition 2. Vt : RT+ → R is continuously differentiable, and Vij , the partial derivative of Vi with respect to cj is positive, for all i and j. Proof. By (A1), Ut is continuously differentiable, and (A3) guarantees that (I − A) is nonsingular, where   0 U12 . . . U1T U 1 0 . . . U T  2   2 A ≡  .. .. ..   . . .  1 2 UT UT . . . 0 Then by the Implicit Function Theorem, Vt is continuously differentiable, t = 1, . . . , T . Totally differentiating (∗), we have   U11 dc1   (I − A)dU =  ...  UTT dcT

7

Let det(I − A) be the determinant of I − A, and Ci,j denote the cofactor of the i − j th element of I − A. Then Cramer’s rule gives PT U i Ci,t dci dvt = i=1 i det(I − A) and in particular, Vij

Ujj Cj,i . = det(I − A)

Now Ujj > 0 for all j, by (A2), and det(I − A) > 0 by (A3). Furthermore, Cj,i > 0 because it can be shown that if not, A has a characteristic root p ≥ 1; since A and E are nonnegative, A ≤ E implies that E has a characteristic root r ≥ p ≥ 1 ((2), Theorem 1*), contradicting (A3). Therefore it cannot be the case that Ci,j ≤ 0 for some i and j. Thus Vij > 0, i = 1, . . . , T ; j = 1, . . . , T . If the utilities of the first t − 1 individuals are fixed artificially at u¯1 , . . . , u¯t−1 , the system of equations ui = Ui (ci , u¯i , . . . , u¯t−1 , ut , . . . , ui−1 , ui+1 , . . . , uT ),

i = t, . . . , T

(**)

generates unique utilities ut , . . . , uT , because the system (**) inherits properties (A1)–(A3) from (∗). This means that the following “partially reduced form” equations are well-defined, strictly monotonic, and continuously differentiable, as a corollary of Propositions 1 and 2. Definition. Let the functions U1 , . . . , UT be those of (∗), satisfying (A1)– (A3). For any (¯ u1 , . . . , u¯t−1 ) ∈ Rt−1 , and (c1 , . . . , cT ) ∈ RT+ , define vi,t (¯ u1 , . . . , u¯t−1 ; c1 , . . . , cT ) = Ui (ci , u¯1 , . . . , u¯t−1 , wt , . . . , wi−1 , wi+1 , . . . , wT ),

i = 1, . . . , T

where (w1 , . . . , wT ) is the unique utility vector satisfying the equations (**). Vi,tj denotes the partial derivative of Vi,t with respect to cj , j > t. Notice that Vi,t has the redundant arguments c1 , . . . , ct−1 ; this is simply a matter of what is notationally convenient in the definitions employing these functions.

8

3

The Intergenerational Model

The model studied here is a T generation, one commodity storage economy. Generation 1 is endowed with 1 unit of a perfectly divisible resource (called “cake”) that is valued by all generations. 1 decides on an amount of cake to consume, and passes on the remainder to the second generation, which eats some portion of what has been left to it, leaving the rest for posterity. Each generation eats, bequeathes, and dies; no two generations are alive contemporaneously. No stipulations can be attached to any bequest. Thus, 1 cannot control the way in which the resource left to 2 will be distributed among generations 2, . . . , T . The pattern of consumption (c1 , . . . , ct−1 ) is known to t as a matter of historical record; the model is one of perfect information. The motivation for a generation’s considering passing on some cake is that all generations care about one another’s utilities. Specifically, the welfare of the T generations is determined by the system (∗) of the previous section, and (A1)–(A3) are assumed to hold. The fact that generations 1, . . . , t care about Ut+1 , . . . , UT , but do not live to see ct+1 , . . . , cT (on which the T -person utility system functionally depends) introduces a new complication: generation t’s welfare is determined by the resource allocation it expects will arise, not the distribution that actually occurs. Moreover, since Ut+1 has an argument ut , person t + 1 needs to deduce what generation t’s welfare was. Attempting to model this as an extensive form game, one finds that it is not a game: utilities depend upon more than the endpoint of the extensive form that is reached, as explained above. Can this complexity be finessed by using the reduced form functions V1 , . . . , VT , thereby disposing of the need to worry about “i’s belief about j’s utility”? Surprisingly, the answer is no. Simply evaluating the endpoints of the intergenerational model by the reduced form functions and solving for the subgame perfect outcome(s) will produce an incorrect solution, as the following example illustrates. Consider 3 generations with utilities given by U1 (c1 , u2 , u3 ) = c1 + 10u3 U2 (c2 , u1 , u3 ) = c2 + u1 U3 (c3 , u1 , u2 ) = c3 .

9

The associated reduced form is V1 (c1 , c2 , c3 ) = c1 + 10c3 V2 (c1 , c2 , c3 ) = c1 + c2 + 10c3 V3 (c1 , c2 , c3 ) = c3 . The functions U1 , U2 , U3 do not satisfy (A2), but could easily be altered to do so, with some loss of simplicity, but without changing the point of the example. Solving the “artificial” game (whose outcomes are evaluated by the functions V1 , V2 , V3 ) by backward induction, one notes that since generation 3 eats anything it is given, 2’s optimal strategy is to pass on to 3 everything 2 has (because, in the notation of Section 2, V23 = 10 > 1 = V22 ). Knowing that anything passed on to 2 will ultimately be consumed by 3, 1 eats nothing, bequeathing the entire cake to his successors (because V13 = 10 > 1 = V11 ). The unique subgame perfect outcome of this artificial game, then, is (c1 , c2 , c3 ) = (0, 0, 1), yielding utilities (10, 10, 1). But this solution bears no resemblance to the result of individual utility maximizing behaviour in the intergenerational model. When generation 2 makes its consumption decision, 1 is dead; utility u¯1 is fixed; a thing of the past. 2 cares about u¯1 , but cannot affect it. Thus 2’s objective function is U2 (c2 , u¯1 , u3 ) = V2,2 (¯ u1 ; c2 , c3 ) = u¯1 + c2 which generation 2 maximizes (treating u¯1 as a parameter) by eating everything it is bequeathed. Understanding this, and deriving no satisfaction from 2’s consumption, 1 eats the entire cake. This “consistent plan” (c∗1 , c∗2 , c∗3 ) = (1, 0, 0) yields utility (1, 1, 0), and is not remotely similar to the solution of the artificial game analyzed above. What is needed is a formalization of the intuition that lets us solve the preceding example. The required generalization of subgame perfect equilibrium must also confront the problem (not encountered in the example) of a generation’s optimal decision depending upon its perception of earlier generations’ utility levels. Before attacking these problems, I define strategies for the generations, and allocation functions that determine consumption vectors as a function of strategy profiles. The notation used for these two definitions follows closely that of Goldman (4) who cannot, however, be blamed for the labyrinthine complexity of the subsequent definitions. 10

Definition. A strategy for generation t is a function ht such that ( ) " # t−1 t−1 X X ht : (c1 , . . . , ct−1 ) : ci ≤ 1 → 0, 1 − ci . i=1

i=1

Ht denotes the set of all strategies of t. Definition. The allocation functions xt : H1 × · · · × Ht → Rt+ are given inductively by x1 (h1 ) = h1 xt (h1 , . . . , ht ) = (xt−1 (h1 , . . . , ht−1 ), ht (xt−1 (h1 , . . . , ht−1 ))),

t = 2, . . . , T.

Definition. An allocation (c1 , . . . , cT ) is interior if ct > 0 for all t. P Definition. An allocation (c1 , . . . , cT ) is feasible if Ti=1 ci = 1. In a subgame perfect equilibrium, expectations at the beginning of any subgame (whether on or off the equilibrium path) about other players’ strategies are determined by the strategy profile induced on that subgame by the equilibrium strategy profile of the original game. In the intergenerational model with nonpaternalistically sympathetic preferences, players need to form beliefs not only about others’ strategies, but also about the utility levels other have attained, or will attain. The belief formation functions I shall introduce, inductively generate expectations about the utilities of others on the same principle as strategic expectations are implicitly generated in subgame perfect equilibrium. Given an equilibrium profile (h∗1 , . . . , h∗T ), an arbitrary history generated by (possibly disequilibrium) strategies h1 , . . . , ht−1 , and utilities u¯1 , . . . , u¯t−1 determined by the first t − 1 belief formation functions, t should expect the outcome xT (h1 , . . . , ht , h∗t+1 , . . . , h∗T ) to occur if t employs the strategy ht . Then t’s utility from playing ht (given the history xt−1 (h1 , . . . , ht−1 )) should be Vt,t (¯ u1 , . . . , u¯t−1 ; xT (h1 , . . . , ht , h∗t+1 , . . . , h∗T )), so this is the number that the tth belief formation function associates with the history xt (h1 , . . . , ht ) under equilibrium (h∗1 , . . . , h∗T ). In the complete formal definition that follows, the profile (h∗1 , . . . , h∗T ) can be thought of as an equilibrium, while the second profile (h1 , . . . , hT ) simply serves to generate an arbitrary history for each t. 11

Definition. For all pairs of strategy profiles (h∗1 , . . . , h∗T ) and (h1 , . . . , hT ), the belief formation functions f 1 (h1 ; h∗ ), . . . , f T (h1 , . . . , hT ; h∗ ) are defined inductively by f 1 (h1 ; h∗ ) = V1 (xT (h1 , h∗2 , . . . , h∗T ) f t (h1 , . . . , ht−1 ; h∗ ) = (f t−1 (h1 , . . . , ht−1 ; h∗ ), Vt,t (f t−1 (h1 , . . . , ht−1 ; h∗ ); xT (h1 , . . . , ht , h∗t+1 , . . . , h∗T ))),

t = 2, . . . , T.

Definition. A strategy profile h∗ = (h∗1 , . . . , h∗T ) is a pure strategy subgame perfect equilibrium (S.P.E.) if for each profile (h1 , . . . , hT ) and each t, Vt,t (f t−1 (h1 , . . . , ht−1 ; h∗ ); xT (h1 , . . . , ht−1 , h∗t , . . . , h∗T )) ≥ Vt,t (f t−1 (h1 , . . . , ht−1 ; h∗ ); xT (h1 , . . . , ht , h∗t+1 , . . . , h∗T )). Definition. An allocation (c1 , . . . , cT ) is an S.P.E. outcome if xT (h∗1 , . . . , h∗T ) = (c1 , . . . , cT ) for some S.P.E. (h∗1 , . . . , h∗T ). The existence of such an equilibrium is a complex issue, and is not discussed here. I hope to study existence questions for “generalized games” (of which the present model is an example) in a subsequent paper.

4

Intertemporal Inefficiency

Proposition 3. Suppose that T ≥ 3, and that utilities in the T generation model are given by (∗), and (∗) satisfies (A1)–(A3) (see Section 2). If c = (c1 , . . . , cT ) is an interior S.P.E. outcome, there exists ε > 0 such that (c1 , . . . , cT −2 , cT −1 − ε, cT + ε) is feasible, and Pareto dominates c; that is, for all t, VT (c1 , . . . , cT ) < VT (c1 , . . . , cT −2 , cT −1 − ε, cT + ε). Proof. Let u¯ = (¯ u1 , . . . , u¯T ) ≡ (V1 (c), . . . , VT (c)). Since c is an interior S.P.E. outcome, generation T − 1 has divided its inheritance between cT −1 and cT to maximize VT −1,T −1 (¯ u1 , . . . , u¯T −2 ; xT −1 , xT −2 ) subject to xT −1 + xT = PT −2 1 − t=1 xt ; xT −1 , xT ∈ [0, 1]. A necessary condition for this interior solution is −1 VTT−1,T u1 , . . . , u¯T −2 ; cT −1 , cT ) = VTT−1,T −1 (¯ u1 , . . . , u¯T −2 ; cT −1 , cZ ) −1 (¯

12

(3)

By totally differentiating (∗) holding the first T −2 consumptions and utilities constant at c1 , . . . , cT −2 and u¯1 , . . . , u¯T −2 respectively, one gets (suppressing the functional arguments) −1 UTT−1 1 − UTT−1 UTT −1 UTT−1 UTT = 1 − UTT−1 UTT −1

−1 VTT−1,T −1 =

VTT−1,T −1

which are well-defined because the denominators are non-zero (by (A3)), as the denominators are a principal minor of I −A, and A satisfies the HawkinsSimon condition (see the proof of Proposition 2). Then (3) implies that at the S.P.E. outcome, −1 UTT−1 = UTT−1 UTT (4) For each t let

dVt ds

be the derivative of Vt (c1 , . . . , cT −2 , cT −1 − s, cT + s)

t >0 with respect to s, evaluated at s = 0. It will be sufficient to show dV ds for all t; because then for ε sufficiently small, (c1 , . . . , cT −2 , cT −1 − ε, cT + ε) is feasible and Pareto dominates (c1 , . . . , cT ). Differentiating the system (∗) with respect to s, holding c1 , . . . , cT −2 constant, one obtains   0   ..   .   (I − A)dV =  , 0   T −1 −UT −1 ds UTT ds

that is 

1 −U21 .. .

−U12 1

. . . −U1T −1 −U2T −1

−U1T −U2T .. .



dV1





0 .. .



             ..      .  =   0      T −1 1 2 T −UT −1 −UT −1  −UT −1 ds 1 −UT −1   dVT UTT ds −UT1 −UT2 . . . −UTT −1 1

13

and by Cramer’s rule, for t 6= T , 1 −U12 . . . −U21 1 . . dVt 1 . = ds det(I − A) −UT1 −1 −UT2 −1 −UT1 −UT2 . . .

0 0 .. . −1 −UTT−1 UTT |{z}

tth column

T −UT −1 ... 1 ...

−U1T −U2T .. .

Subtracting UTT times the last column, from the tth column (which leaves the determinant unaltered), and using (4), yields (for t 6= T ) 1 −U12 . . . U1T . . . −U1T −U21 1 U2T −U2T . .. .. .. T . . UT dVt T = U T −2 ds det(I − A) 0 −UTT−1 0 ... 1 −UT1 −UT2 . . . |{z} th t

column

Expanding by cofactors down the tth column gives T −2

dVt X Uit Utt Ci,t = , ds det(I − A) i=1

t 6= T

where ci,j is the i − j th cofactor of I − A. But since A satisfies the HawkinsSimon conditions, det(I − A) > 0, and ci,t > 0 for all i, t. Moreover by (A2), Uit > 0 for all i, t. Thus dV > 0 for all t 6= T . ds

(5)

Finally, differentiate UT (cT + s, u1 , . . . , uT −1 ) with respect to s (evaluating at s = 0). T −1

X dVt duT dVT = = UTT + UTt > 0 (by (5) and (A2)). ds ds ds t=1 Therefore

dVt ds

> 0 for all t. 14

The intuition behind this result is quite straightforward. When T − 1 makes its consumption decision, the utilities of 1 to T − 2 are treated parametrically. If all players were assured that the allocation (c1 , . . . , cT −2 , cT −1 − ε, cT + ε) would be enforced, the “initial” impact on T − 1’s utility (through UT −1 ) would be an almost negligible negative change, per unit increase in ε, because his derivative with respect to this transfer is 0 at the equilibrium. T is made better off because final period consumption rises, and there is no significant compensating decline in UT −1 (the only conceivable objection T might have had to the transfer). Since UT has risen substantially per unit transferred, and UT −1 has fallen inappreciably per unit, the utilities u1 , . . . , uT −2 are increased, since U1 , . . . , UT −2 have positive derivatives in UT −1 and UT . A further feedback occurs: UT −1 is favourably disposed toward 1, . . . , T − 2, and hence is pleased by the increases in u1 , . . . , uT −2 . The matrix algebra of the proof confirms that this pleasure more than compensates T − 1 for its decreased share of the consumption, for sufficiently small ε. Thus, the proof works because the utilities of T − 2 generations are immutable when the second last generation’s decision is taken. It would work equally well if only one generation’s utility were fixed at that time; in other words, Proposition 3 could easily be extended to cover overlapping generations models, as long as generation 1 is dead before generation T − 1 eats. (But defining the belief formation functions would be a torturous business.) Some readers may feel that it would be more natural for each generation to care only about future generations’ utilities (and its own consumption) rather than past and future utilities. In this case a slightly altered version of Proposition 3 applies, and is proved below. There is no need for (A3), because the utility system is recursive rather than fully simultaneous. Proposition 4. Suppose that in the intergenerational model with T ≥ 3, utilities are generated not by (∗), but by the equations ut = Ut (ct , ut+1 , . . . , uT ),

t = 1, . . . , T,

where for all t, Ut is differentiable and Uij > 0 for i ≤ j. If c¯ = (¯ c1 , . . . , c¯T ) is an interior S.P.E. outcome, there exist s, a > 0 such that (¯ c1 −as, c¯2 , . . . , c¯T −2 , c¯T −1 − (1 − a)s, c¯T + s) is feasible, and Vt (¯ c1 − as, c¯2 , . . . , c¯T −2 , c¯T −1 − (1 − a)s, cT + s) > V (¯ c) for all t. Proof. Since c¯ is an interior S.P.E. allocation, −1 UTT−1 = UTT−1 UTT

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(6)

(as in the proof of Proposition 3). For some a > 0, define c1 (s) = c¯1 − as ct (s) = c¯t , t = 2, . . . , T − 2 cT −1 (s) = c¯T −1 − (1 − a)s cT (s) = c¯T + s t It is sufficient to show that for some a, dU > 0 for all t. Successively ds differentiating UT , UT −1 , . . . , U1 with respect to s, and evaluating at s = 0 (i.e., at the S.P.E. outcome)

dUT (¯ cT + s) = UTT > 0 ds

(7)

dUT −1 dUT −1 (¯ cT −1 − (1 − a)s, UT ) = (a − 1)UTT−1 + UTT−1 ds ds −1 −1 = aUTT−1 − UTT−1 + UTT−1 UTT −1 = aUTT−1 >0

(from (7))

(from (6))

For t = 2, . . . , T − 2, T X dUi dUt (¯ ct , ut+1 , . . . , uT ) = Uti >0 ds ds i=t+1

by induction. T

X dUi dU1 (¯ c1 − as, u2 , . . . , uT ) = −aU11 + U1i . ds ds i=2 Setting a = 0 in the above derivatives, one gets dUt > 0, ds

t 6= T − 1

dUT −1 = 0. ds Thus by the continuous differentiability of V1 , . . . , VT , there exists an interval (0, a ¯) such that a ∈ (0, a ¯) implies dUt > 0, ds

t = 1, . . . , T. 16

5

Sympathetic Dictators

Gibbard (3) and Satterthwaite (12) have shown that only dictatorial social choice functions can be implemented via dominant strategies (as long as the social choice function must have at least three elements in its range), and work of Roberts (11) implies the same conclusion for Nash implementation. One might ask: “If people are sufficiently sympathetic to one another, could the appointment of a dictator lead to a socially attractive outcome?” I show that if a society of nonpaternalistically sympathetic persons satisfies the “coherence” conditions of Section 2, no interior allocation chosen by a dictator can maximize any strictly increasing differentiable social welfare function. In the proposition that follows, the functions Vt are the reduced form utility functions of Section 2. Proposition 5. For T ≥ 2, consider a T -person society (possessing 1 unit of a single commodity) in which everyone lives contemporaneously, and utilities are generated by (∗), (Section 2) satisfying (A1)–(A3). Suppose the interior allocation c¯ = (¯ c1 , . . . , c¯T ) maximizes t’s utility over all feasible consumption vectors. For every strictly increasing differentiable social welfare function W (V1 , . . . , VT ) there exists some s > 0 such that c(s) ≡ (c1 − s, c2 + s s , . . . , cT + T −1 ) is feasible and T −1 W (V1 (c(s)), . . . , VT (c(s))) > W (V (¯ c), . . . , VT (¯ c)). Proof. Without loss of generality let i = 1, and suppose c¯1 , . . . , c¯T is a feasible interior allocation maximizing 1’s utility. Define c1 (s) = c¯1 − s ct (s) = c¯t +

s , T −1

t = 2, . . . , T.

Notice that (c1 (s), . . . , cT (s)) is feasible for |s| sufficiently small. Evaluating at s = 0, du1 =0 (8) ds because at s = 0, 1’s utility is at a maximum. It is sufficient to show that dW (V1 (c(s)), . . . , VT (c(s))) > 0. Totally differentiating the last T − 1 ds

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equations of (∗) with respect to s yields T

dut Utt 1 X i dUi = + , U ds T − 1 T − 1 i6=t t ds

t = 2, . . . , T

T Utt 1 X i dUi = + U T − 1 T − 1 i6=1,t t ds

(from (8)).

Arranging this as a T − 1 person matrix system, one obtains   du   U 2   2 2 1 −U23 . . . −U2T  . T − 1     ..   ds  1 −U32 . ..  =   .  .    . .   T   ..   UT  −UT −1 duT T −UT2 −UT3 . . . 1 ds T −1 Let Ci,j (1) denote the cofactor of the left-hand side matrix, involving the deletion of the ith and j th persons’ row and column respectively (the i − 1th row and j−1th column of the (T −1)×(T −1) matrix itself). All such cofactors can be shown to be positive as a result of the Hawkins-Simon condition (A3), as is the determinant C1,1 (a cofactor of I − A). Cramer’s rule yields T

1 dut X i = Ui Ci,t (1) , 1 ds (T − 1)C 1 i=1

t = 1, . . . , T

>0 Thus, du1 =0 ds dut > 0, ds

t = 2, . . . , T.

Consequently T

X ∂W dui dW (V1 (c(s)), . . . , VT (c(s))) = ds ∂Vi ds i=1 > 0. 18

6

Conclusion

Proposition 3 established the inefficiency of interior outcomes of a multiperiod cake-eating model, with nonpaternalistic sympathy amongst generations. Although I explained in Section 4 that the proposition extends to easily cover overlapping generations models, and proved that inefficiency arises even if sympathies are restricted to caring about future utilities, the reader may wonder about the generality of the results. Are they restricted to one commodity storage economies? Notice that Propositions 3 and 4 did not use in any essential way the fact that the resource is transferred linearly from period to period; the results apply equally to any situation in which the “storage functions” are strictly increasing and differentiable. These functions could encompass possibilities such as production, inventory holding costs, spoilage and so on. Generalization of the results to a k-commodity world is so immediate that the proof can be sketched in a few sentences. Starting at an interior S.P.E. outcome, fix the allocation of all commodities except the first, and regard the reduced form utility functions (generated by a system satisfying (A3), and (A1), (A2) in each argument) as functions of the first commodity only. These functions satisfy the conditions of Proposition 3, and hence there exists some small shift of commodity 1 from T −1 to T , that improves everyone’s welfare. Proposition 4 generalizes in the same way. A trivial implication of the Pareto inefficiency of outcomes in these models, is that relative to any interior S.P.E., the first generation would always strictly increase its utility if it were able to dictate a “slower” consumption stream (in the sense of Proposition 3 or 4). Interpreting the generations as one person living for T periods, one finds that under (A1)–(A3), an individual always wishes to precommit his future decisions, and invariably feels now that he could benefit from being more patient or self-disciplined in the future (going on a diet tomorrow, stopping smoking next Monday, and so on). An incidental benefit of investigating nonpaternalistic sympathy in an intergenerational model has been encountering a well-specified strategic problem that is not a game. This presented the opportunity to study the complications involved in extending the subgame perfect equilibrium concept to cover a situation in which all players observe all earlier players’ moves before moving themselves (suggesting “perfect information”) and yet most players do not observe which endpoint of the extensive form is reached.

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