Efficient Incentive Compatible Economies Are Perfectly Competitive*

Journal of Economic Theory 85, 169225 (1999) Article ID jeth.1998.2494, available online at http:ÂÂwww.idealibrary.com on

Louis Makowski

Department of Economics, University of California, Davis, California 95616 lmakowskiÄucdavis.edu

Joseph M. Ostroy

Department of Economics, University of California, Los Angeles, California 90095 ostroyÄecon.ucla.edu

and

Uzi Segal

Department of Economics, University of Western Ontario, London, Ontario, Canada N6A 5C2 segalÄsscl.uwo.ca

Received June 4, 1996; revised October 27, 1998

Efficient, anonymous, and continuous mechanisms for exchange environments with a finite number of individuals are dominant strategy incentive compatible if and only if they are perfectly competitive, i.e., each individual is unable to influence prices or anyone's wealth. Equivalently, in such a mechanism each individual creates no externalities for others by her announcement of a type. The characterization applies whether preferences are ordinal or quasilinear, and it also applies to continuum economies. Perfectly competitive mechanisms are non-generic (although non-vacuous) in finite economies and are generic (but non-universal) in continuum economies. We use these results to provide bridges to related work. Journal of Economic Literature Classification Numbers: C72, D51, D62. 1999 Academic Press

1. INTRODUCTION

Consider a pure exchange economy with a finite number of individuals, each with private information about his preferences. When can one design

* We thank the associate editor for numerous helpful comments and the referee for pointing out serious shortcomings in earlier versions. 169 0022-0531Â99 30.00 Copyright 1999 by Academic Press All rights of reproduction in any form reserved. 170 MAKOWSKI, OSTROY, AND SEGAL a continuous allocation mechanism that is Pareto efficient, anonymous, and dominant strategy incentive compatible (i.e., strategy-proof)? When preferences are convex, efficient allocations admit efficiency prices; hence any efficient mechanism will have to be Walrasian, possibly with lump-sum transfers. Call an efficient mechanism perfectly competitive if it satisfies two important extra conditions: v there are no-lump sum transfers; v no agent can change the Walrasian equilibrium price vector by changing his announced preferencesso no individual possesses any monopolyÂprice-making powers.

Our main result is that for any continuous, efficient and anonymous mechanism, the following are equivalent: (1) the mechanism is incentive compatible (IC), (2) the mechanism is perfectly competitive (PC), and (3) the mechanism permits no externalities in the sense that no one person can influence others' welfare by his action (announcement). Schematically,

Since in a PC mechanism no one can influence others' wealth or the prices others face, PC evidently implies no externalities. Further, it is not difficult to see that in an efficient mechanism, no externalities implies IC: When maximizing over all feasible allocations subject to the constraint that one's choice does not change the utility of others, there is no incentive to misrepresent one's preferences. Thus the heart of the characterization is in showing that IC O PC. Our results establish an important connection between general equilibrium and mechanism design. Internalization of externalities has been recognized in the VickreyClarkeGroves results as the key to efficiency and IC in partial equilibrium (see the remarks on the quasilinear model, below). This paper extends the conclusion to a fully ordinal model: we

File: 642J 249402 . By:SD . Date:12:03:99 . Time:09:37 LOP8M. V8.B. Page 01:01 Codes: 2348 Signs: 1813 . Length: 45 pic 0 pts, 190 mm EFFICIENT IC ECONOMIES 171 show that the general equilibrium concept of perfect competition is not simply one way that efficiency may be achieved with no externalities, it is the only way. Resolving the Conflict between Characterization and Existence Our characterization holds on any sufficiently rich domain, but it is non- vacuous only on carefully chosen, sufficiently rich domains. We explain why. In finite economies, there are impossibility theorems for efficient IC mechanisms on the universal domain consisting of all economies with a given finite number of individuals. Consequently, any non-vacuous characterization must involve ``domain tailoring.'' Nevertheless, an interesting characterization cannot involve too much tailoring. For example, if the domain chosen is a single population, then the existence of efficient incentive compatible mechanisms is trivial, but no unique characterization emerges. Therefore, to make our characterization both non-vacuous and interesting, we have had to specify a domain that is sufficiently small to admit the existence of an efficient incentive compatible mechanism, but sufficiently large to make the conditions for existence unique. The need for domain tailoring suggests that many characterizations of efficient incentive compatibility mechanisms may exist, each applying to some carefully chosen domain. For example, Hurwicz and Walker [14] give a characterization that is only remotely connected with perfect competition (see Section 7). Can one say that one characterization is better than another? In our view, the answer is ``yes.'' To support this position we show that our result is unique on the class of what we will call first-order mechanisms. Moreover, the characterization continues to hold in continuum economies, and in such economies it holds generically, not just exceptionally as in finite economies. We take it as a criterion for a ``good finite characterization'' that it should continue to hold generically in the continuum. Since our characterization satisfies this criterion, any other ``good finite characterization'' would have to agree with ours on a generic set of large economies. Even if it is granted that ours is the ``right'' characterization, nevertheless the question may be asked: Why should one be interested in efficient, IC mechanisms for finite economies since any such characterization will necessarily apply only on a meager domain? Our rationale is based on a desire to establish common threads. When held up for close examination, the conditions for efficiency and IC are remarkably similar in finite and continuum economies and in ordinal and quasilinear models, in terms of both their underlying geometry and their overall interpretation. In finite models, we identify conditions for ordinal economies that are analogous to the conditions for IC in the quasilinear model. The associated geometry 172 MAKOWSKI, OSTROY, AND SEGAL helps to explain why existence is so much easier to achieve in the continuum than in finite models.1

Bridging the Literature The existing literature is rather dichotomous, with special results for the quasilinear model, and with impossibility results predominating for finite economies and possibility results predominating in the continuum. We divide the literature into three categories.

1. Finite Numbers and Quasilinear Preferences. For models with quasilinear preferences, there exists a general characterization of all incentive compatible mechanisms satisfying a qualified notion of efficiency in which money transfers need not balance: they must be in the VickreyClarke Groves family (Vickrey [24], Clarke [4], Groves and Loeb [9], Green and Laffont [7], Holmstro m [12]). Impossibility results involve showing that money transfers typically do not balance, e.g., Hurwicz and Walker [14]. Our characterization shows that budget-balance failures arise from failures of perfect competition. To amplify, observe that VickreyClarke Groves schemes operate on the principle that individuals must internalize the external effects of their choices. Internalization of externalities for each individual, however, does not necessarily imply their elimination for the economy as a whole, as is evidenced by the budget-balance problem. For externalities to be eliminated, the sum of money payments must be zero. Our characterization implies that, provided the domain is sufficiently large, elimination will occur only under PC.2

2. Finite Numbers and Ordinal Preferences. Hurwicz [13] demonstrated that for 2-person, pure-exchange economies the Walrasian mechanism is manipulable. He shows the same holds for any 2-person, efficient, individually rational allocation mechanism. A related 2-person impossibility result, without the individual rationality assumption, is proved by Dasgupta, Hammond, and Maskin [5] and by Zhou [26]. Other impossibility results are demonstrated by Satterthwaite and Sonnenschein [23] and BarberaÁ and Jackson [1]. While we characterize possibility we show, as a corollary, that if the domain is sufficiently rich to permit the exercise

1 We show the characterization IC PC no externalities holds for finite economies, whether preferences are ordinal or quasilinear. But in the continuum we restrict our attention to ordinal economies and only show IC PC. See Makowski and Ostroy [17] for the connections between IC and no externalities in the continuum quasilinear model. 2 The equivalence between balanced Groves schemes and PC is shown in Makowski and Ostroy [16] assuming individual rationality. Here we show the result holds even without an individual rationality constraint, at least for anonymous mechanisms. EFFICIENT IC ECONOMIES 173 of monopoly poweri.e., most domains for finite-agent economiesthen impossibility follows. 3. Continuum Economies. Taking an asymptotic point of view, Roberts and Postlewaite [22] showed that the Walrasian mechanism becomes asymptotically incentive compatible; but they did not show the uniqueness of the Walrasian mechanism. For continuum economies, Hammond [10], Kleinberg [15], Champsaur and Laroque [2], McLennan [20], Mas- Colell [19], Nehring [21], and others have shown that the only efficient no-envyÂincentive-compatible allocations are Walrasian without transfers. Note, no-envy and incentive-compatible allocations are intimately related; see Section 6. There we show that the IC PC result for finite economies extends to the continuum, where existence holds on a generic domain. Our characterization unifies the literature by showing why impossibility results predominate for finite economiesperfect competition is rare (although not vacuous) in such economiesand why there is a turnaround for continuum economiesperfect competition is generic (although not universal) there. Two related contributions establish what appear to be rather different conclusions. Hurwicz and Walker [14] give a characterization of efficient incentive compatible mechanisms for exchange economies with finite numbers and quasilinear preferences emphasizing the absence of conflict. BarberaÁ and Jackson [1] characterize all incentive compatible mechanisms for exchange economies with finite numbers and ordinal preferences; they find that none of them comes close to being efficient, even as the number of individuals increases. We shall explain the apparent discrepancies after the presentation of our results. (See Section 7.) In addition to the benefit of the characterization for unifying the finite and continuum literatures, two other points are worth making.

(i) The Magnification Principle. Section 5 illustrates the sort of economies required for PC, and hence for IC, in the finite case: There need to be some appropriate flat segements in some individuals' indifference curves. As the number of individuals increases, the size of these necessary flats goes to zero; so in the continuum, smooth indifference curves suffice for PC and IC. The intuition is that in the continuum individuals are of infinitesimal size, but in finite economies they are discrete. Navigating along a flat segment for discrete individuals is equivalent to navigating locally along a smoothhence locally linearsurface for infinitesimal individuals. Alternatively expressed, the flats that are required for IC in the finite case magnify how the economy appears to a tiny perfect competitor in the continuum. Thus the picture of flats helps our intuition why PC, and hence IC, becomes the rule rather than the exception in the continuum: flat 174 MAKOWSKI, OSTROY, AND SEGAL surfaces in finite economies are exceptional while smooth surfaces in continuum economies are not. (ii) The Importance of Perfect Competition. Although PC is exceptional for finite economies, it is important to recognize that it is possible there. Similarly, although PC is typical in the continuum, it is important to recognize that its existence does not hold simply because of the continuum. Recall that a Walrasian mechanism is PC if it has two extra properties, no lump-sum transfers, and no individual can influence prices. In continuum economies, it is automatic that no individual can influence prices, provided the mechanism is price-continuous. When the characterization of IC in the continuum is stated for price-continuous mechanisms, it reads as follows: an IC mechanism must be Walrasian without lump-sum transfers. Because it is implicit, the importance of the second proviso, that no one can influence prices, can easily be lost sight of. The characterization in the finite model emphasizes this second necessary condition for IC.

The Private Goods Environment A private goods exchange economy can be formulated in two different ways: each individual may have a private initial endowment or, more simply, the economy may have an aggregate endowment. We employ the latter description. Hence we will identify ``no lump sum transfers'' with everyone having equal wealthtaking as our benchmark from which to measure transfers the endowment point in which everyone is given an equal share of the aggregate endowment. Thus, in our setup, ``IC is equivalent to PC'' means ``IC is equivalent to an equal-wealth Walrasian mechanism in which no one can change prices by her action.'' For environments with prespecified private endowments, the analogous result would be that the mechanism must be Walrasian with the added properties that no one can influence prices or others wealths calculated from their private endowments. [This conjecture for the finite case is based on known results for the continuum: for continuum economies, the equivalence between efficient, no-envy allocations and Walrasian allocations without transfers has been shown under either description.] The remainder of the paper is organized as follows. In Section 2, we describe a finite-numbers exchange environment consistent with either ordinal or quasilinear preferences. We treat the ordinal and quasilinear cases separately, but with an emphasis on their similarity. Section 3 presents the main results (Theorems 1 and 2), the finite ordinal characterization. Section 4 presents the analogous result for the quasilinear case (Theorem 3). The conditions for Theorem 3 are somewhat weaker; its proof combines results from Theorem 1 along with known results exploiting quasilinearity. In Section 5, the main result is illustrated by EFFICIENT IC ECONOMIES 175 example; using a term introduced earlier, the ``magnification principle'' is illustrated. Extensions to the continuum are given in Section 6. In the continuum, we address only the ordinal model. Interestingly, we show that by using a ``money-metric'' representation of preferences, intuitions and results from finite quasilinear economies can be transported to ordinal continuum economies. In the continuum we show that at all price-continuous points, a mechanism is efficient and incentive compatible if and only if it is equal- wealth Walrasian (characterization, Theorem 4), and that there exists an equal-wealth Walrasian mechanism that is price-continuous on a residual set (generic existence, Theorem 5). Based on Theorem 5, we also show an asymptotic result connecting the finite to the continuum (Propositions 2 and 3): most sequences of finite economies converging to a large limit economy will be asymptotically incentive compatible. The proof of Theorem 4 is of some independent interest. It shows IC O PC in large ordinal economies using the main lemma in Holmstrom's [12] demonstration that, for finite quasilinear economies, IC O a Groves mechanism. The proof provides further confirmation that our characterization bridges known results on efficient mechanism design in various set- tings. The proof of Theorem 5 is also of independent interest. It does notand as far as we know cannotrely on the concept of regular economies typically used to demonstrate genericity with respect to the continuity of an equilibrium selection (see Mas-Collel [18, ch. 5]); rather it builds on Fort's Theorem regarding the genericity of the continuity points of an upper- hemicontinuous correspondence (see Section 6.3). Section 7 provides a summary of the common threads in the various results and a discussion of the connections between our findings and some apparently contradictory results proved by others. The treatment below allows the mechanism to assign zero allocations to one or more individuals, and this complicates the analysis. For example, it has required us to demonstrate that the set of continuous quasiconcave functions is path connected (see Appendix A). Section 8 is devoted to a proof of Theorems 1 and its Corollaries using a simpler version of the key Lemma 3 when zero allocations are ruled out. Appendixes contain lemmas and proofs not included in the previous sections.

2. THE MODEL

There are n individuals, indexed by i or j, and l commodities. Each individual's consumption set is 0/Rl. U denotes the set of admissible 176 MAKOWSKI, OSTROY, AND SEGAL utility functions u for each individual, where u: 0 Ä R.Apopulation is a vector of utility functions u=(u1 , ..., ui , ..., un). The economy's aggregate l endowment is | # R+ and there is no production; so the attainable allocations for any population u is given by

n X= x=(x1 , ..., xi , ...xn)#0 : : xi=| . { i = Let D Un, with typical elements u, v, and w.Amechanism is a pair ( f, D) consisting of a domain D and an outcome function f: D Ä X.As already mentioned, to ensure that our results are non-vacuous, we will have to be careful in specifying the domain; hence the results below will always pertain to a pair ( f, D), rather than just to an f. Throughout we assume the outcome function f only depends on individuals' underlying preferences, that is, if u and v represent the same preference profile (i.e., if for each i, ui (x)ui ( y) iff vi (x)vi ( y)), then f(u)=f(v). We also assume the mechanism ( f, D) is efficient and anonymous: Pareto efficiency. For all u # D, there is no x # X such that ui (xi)ui ( fi (u)) for each i and ui (xi)>ui ( fi (u)) for some i.

Anonymity. For all u # D, ui=uj implies ui ( fi (u))=uj ( fj (u)).

For any population u, let u&i denote the population with the utility function of individual i deleted; and let (u&i , vi) denote the same population as u except the utility function of individual i is replaced by vi . The mechanism ( f, D)isincentive compatible (IC) at u # D if, for each i,

ui ( fi (u))ui ( fi (u&i , vi)) for all (u&i , vi)#D.

The mechanism ( f, D)isincentive compatible if it is incentive compatible at each u # D. Let A(x, u)=[y# 0 : u( y)u(x)] be the at-least-as-good-as-x set for an l individual with preferences u.Aprice vector is a p # R++ that is normalized so that p|=1 in the ordinal model, or is normalized so that pl=1 in the quasilinear model. That we restrict p to be strictly positive is without loss of generality since, for both the ordinal and quasilinear cases, we will impose a strict monotonicity assumption on preferences. The price vector p supports A(x, u)atx if

px=min pA(x, u):=min[py: y # A(x, u)].

Let 1 u(x) denote the set of normalized supports of A(x, u)atx. When 1 u(x) is a singleton, we denote the unique normalized support by {1 u(x). If EFFICIENT IC ECONOMIES 177 u happens to be differentiable at x, {1 u(x) just equals the normalized gradient of u at x.3 At prices p, the set of utility-maximizing choices for an individual with preferences u and wealth w is

(-) ( p, u, w)=arg max u(x) s.t. px=w. x # 0 The set of equal-wealth Walrasian equilibria for u, denoted WE(u), is the set of all pairs (x, p), where x # X and p is a normalized price vector such that, for each i, xi # ( p, ui , p|Ä ). A mechanism ( f, D)isanequal-wealth Walrasian mechanism if, for each u # D, there is some normalized price vector p(u) such that ( f(u), p(u)) # WE(u). The condition p #1 u(x) is readily seen to imply that x # ( p, u, px). The following conditions guarantee that the converse holds and that supporting prices always exist for efficient allocations in each of the three environments below. Throughout we maintain the assumptions that any u # U is continuous, quasiconcave, and strictly monotonic. In the continuum model, we will l assume that | # 0=R++ ,andU consists of all u on 0 satisfying both the maintined assumptions and the boundary condition:

l l for all x # R++ , [y: u( y)u(x)] is closed in R .

l (See Mas-Colell [18, p. 68].) Denote this set Ub(R++). l l In the finite ordinal model, we assume | # R++ and 0=R++ _ [0]. The zero outcome is included to demonstrate that the results are valid even when the mechanism can assign the zero allocation to an individual. For the finite model, U consists of all u on 0 satisfying both the maintained assumptions, the boundary condition, and having a unique (normalized) 1 l support {u(x) for each x>>0. Denote this set Ub, us(R++ _ [0]). [If x=0, we adopt the convention that 1 u(x) includes all normalized prices.] l&1 The quasilinear model is distinguished by the fact that 0=R+ _R,so individuals are not restricted in the quantity of commodity l they can supply. On this set, the class of utility functions consists of all u on 0 that satisfy the maintained assumptions and that are additive in commodity l l&1 (see Section 4). We denote this set by Uql (R + _R). The aggregate endowment | is taken to be strictly positive for all goods except the last, with |l=0. Let P( f, u)=i 1 ui ( fi (u)) be the set of all normalized price vectors that support f(u)atu. Denote an element of this set by p( f, u). The above

3 The notations 1 u and {1 u are the convex set analogs of the subdifferential and gradient mappings for concave functions. In the quasilinear model of Section 4 the relation is made explicit. 178 MAKOWSKI, OSTROY, AND SEGAL assumptions on utility functions and the aggregate endowment imply that if f(u) is an efficient allocation, there exists at least one normalized price p= p( f, u). Thus, for each i, fi (u)#( p, ui , pfi (u)). In the finite ordinal model, since supports are unique, there is only one normalized price p( f, u).

3. FINITE ORDINAL ECONOMIES

l Throughout this section we assume U=Ub, us(R++ _ [0]). We also will assume that the mechanism is continuous, at least in utilities. More k Ä k precisely, let ui ui mean the sequence of utility functions [ui ]k=1,2,... converges to ui uniformly on compacta. It is straightforward to verify that any incentive compatible mechanism f will be continuous in the following k k k k Ä limited sense: Suppose u, u # D, where u =(u&i , ui ). Then u i ui implies k k Ä 4 ui ( fi (u )) ui ( fi (u)). We will make a stronger assumption.

k k k Utility Continuity. Suppose u, u # D, where u =(u&i , u i ). Then k Ä k k Ä k Ä ui ui implies ui ( fi (u )) ui ( fi (u)) and uj ( fj (u )) uj ( fj (u)) for all individuals j{i.

3.1. Main Result Throughout the remainder of this section ( f, D) will denote an efficient, anonymous, and utility continuous mechanism. For a unique characterization to emerge from the added hypothesis that the mechanism is IC, D will have to be sufficiently rich. For example if D=[u], then any efficient mechanism will be IC. The strongest richness conditionthat D=Un, the universal domainis compatible with the richness condition adopted below; so one implication of our characterization will be that on the universal domain any IC mechanism must be PC. But in Section 3.3 we show that on Un no IC mechanism exists (see also BarberaÁ and Jackson [1]); so in this setting the characterization is not really meaningful (any more than it is meaningfulalthough not falseto say that all hares with horns lay eggs). Section 5 will show that our richness condition does not force the domain to be too large: the condition also is satisfied on smaller domains where a PC mechanism exists (so our characterization does not emerge from making inconsistent assumptions).

4 k k k k Ä Here is a sketch of the proof. Let x = fi (u ) and x= fi (u). Suppose u i (x ) % ui (x). Then k k Ä on a subsequence ui (x ) a, where a{ui (x). Suppose au i (x ). This contradicts incentive compatibility, because consumer i k will be better off if he claims he is of type ui rather than ui . Similarly, a>ui (x) leads to a contradiction. EFFICIENT IC ECONOMIES 179

The richness condition has two parts. For the first, call the utility func- l tion vi # U first order similar to ui at xi # R++ if 1 1 {vi (xi)={ui (xi).

That is, vi and ui have the same MRS's at xi .[Ifxi=0, call all vi # U first order similar to ui at xi .] More generally, the population n v=(v1 , ..., vi , ..., vn)#U is first order similar to the population u at x if, for every i, vi is first order similar to ui at xi . Now consider any u # D and x= f(u). The mechanism ( f, D) will be called first order at u if D also contains all populations v # Un that are first order similar to the population u at f(u). The mechanism ( f, D)isfirst order if it is first order at every population u # D. (We will somtimes say ``D is first order relative to f '' as a synonym for ``( f, D) is first order.'') Any mechanism with D=Un is evidently first order; but fortunately for the possibility of the non-vacuous characterization below, ( f, D) can be first order even when D{Un. To illustrate that a first order IC mechanism exists, let f(u) be any equal-wealth Walrasian allocation for u; and let D consist only of u and all populations v that are first order similar to u at f(u). Let f be the function on D that assigns f(u)toall v # D; hence, by construction, ( f, D) is not only first order, but also (trivially) continuous and IC. Such an ( f, D) is also anonymous and efficient because it is an equal-wealth Walrasian mechanism: observe that fi (u)isini's Walrasian demand correspondence at prices p( f, u) not only if i has utility function ui , but also if i has any utility function vi that is first order similar to ui at fi (u) (recall i's MRS's are the same at fi (u) under both ui and vi). Are there any first-order mechanisms that are IC but not equal-wealth Walrasian? Surprisingly, the answer is no. So the apparently weak richness condition that a mechanism be first order, when combined with IC, leads to a strong conclusion. This first major step toward our main result is proved as Lemma 3 in the appendix:

Lemma 3(ICO Walrasian). If ( f, D) is first order, then IC implies that for all u # D,

( f(u), p( f, u)) # WE(u).

Note that p( f, u) may vary with u. To put the result into perspective, we know that any anonymous efficient mechanism must be Walrasian, perhaps with lump sum transfers; the result tells us that any anonymous efficient mechanism that is first order and IC cannot give any lump-sum transfers. It is important to point out that Lemma 3 is only necessary, not sufficient, for IC. Although the first richness condition ensures no individual can influence his own or others' wealths, an individual still may be able to 180 MAKOWSKI, OSTROY, AND SEGAL influence the prices he and others face by altering his announced preferences in a non-first-order wayso he may not be a perfect competitor. By contrast, in the continuum, the analog of Lemma 3that no individual can influence his own or others' wealthswill imply that each individual is a perfect competitor because in a large economy no one individual can influences prices (this assumes price continuity, see Theorem 4). The test for an individual's ability to influence prices brings us to the second part of the richness condition.

Let [u&i , U]=[(u&i , vi): vi # U]. We will say that the domain D is comprehensive around u if it contains all one-person perturbations from u, that is, if

. [u&i , U]/D. i

We will call the mechanism ( f, D) comprehensive around u if D is comprehensive around u. The interpretation is that the mechanism designer is completely ignorant of the type of any one person drawn at random from the population u (although he may know something about the aggregate distribution of types). If the mechanism is comprehensive around every u # D,thenD=Un; this corresponds to the case of complete ignorance. While our characterization applies to this limiting case, it also applies to cases when the mechanism is only comprehensive around some u # D,a caveat which is important for making the characterization meaningful (i.e., non-vacuous). If ( f, D) is first order, Lemma 3 says that if it is IC, it must be an equal wealth Walrasian mechanism throughout its domain, i.e., for all u # D, ( f(u), p( f, u)) # WE(u). Suppose ( f, D) is comprehensive around u.We shall say that ( f, D)isperfectly competitive around u if there is a p such that for every individual i and every vi # U,

( f(u&i , vi), p)#WE(u&i , vi).

Hence, prices do not depend on the preferences of any single individual. The following result adds a significant restriction to Lemma 3 by showing that if a first order mechanism is IC, it must not only be equal wealth Walrasian, but also perfectly competitive around any population u where the domain is comprehensive. We will need one technical assumption. Given any population u, the economy excluding individual i, u&i , will be called pseudoregular if it has only a finite number of equal-wealth Walrasian equilibrium price vectors (but not necessarily a finite number of equilibrium allocations) when its EFFICIENT IC ECONOMIES 181 aggregate endowment equals |&|Ä . We call the economy u quasiregular if, 5 for each i, u&i is pseudoregular.

Theorem 1(ICO PC). Suppose ( f, D) is first order, D is comprehensive around u, and u is quasiregular. If ( f, D) is IC, then it is perfectly competitive around u. To give Theorem 1 a geometrical representation, it is useful to express the result slightly differently. For any given u # D, the range of the mechanism f for individual i is

Ri ( f, u&i)=[fi (u&i , vi): (u&i , vi)#D].

Thus Ri ( f, u&i) may be regarded as i's opportunity set at u&i . An immediate but important observation is that f will be incentive compatible for i at (u&i , vi) if and only if it assigns i his vi -best point in the set

Ri ( f, u&i); that is, it will be incentive compatible if and only if

fi (u&i , vi) # arg max vi (x) s.t. x # Ri ( f, u&i).

The idea is simple: If there exists yi= fi (u&i , wi) such that vi ( fi (u&i , vi))< vi ( yi), then it would not be incentive compatible for i to truthfully announce vi .

Let Hp denote the hyperplane through the origin with normal p, i.e., Hp := [y: py=0]. Our main result shows that the range of the mechanism for all individuals is contained in a common hyperplane through the economy's average endowment |Ä .

Alternative Statement of Theorem 1. Let ( f, D) satisfy the hypotheses of Theorem 1. If ( f, D) is IC, then there exists a p such that for every individual i,

Ri ( f, u&i)/Hp+[|Ä].

3.2. Theorem 1 Is a Characterization for First-Order Mechanisms There may be efficient, anonymous, and continuous mechanisms (g, D$) that are IC, but not PC or even equal-wealth Walrasianbecause (g, D$) is not first order. That is, there exists a population v # D$ and some w that is first order similar to v at g(v), but w ÂD$. Notice, however, that if the

5 The definition of regular economies (Debreu [6]) presumes that utility functions are strictly quasiconcave and C 2, whereas we assume only that they are quasiconcave and C 0 with unique supports. Regular economies have a finite number of equilibria; moreover, within the class of such utility functions, regular economies are typical (e.g., see Mas-Colell [18, Chapter 5]). 182 MAKOWSKI, OSTROY, AND SEGAL domain D$ were expanded until it is first-order relative to g (and, in the process, g were expanded to specify the outcomes on the new expanded domain), then the conclusions of Theorem 1 would apply. One either would have to try to modify g to make it equal-wealth Walrasianand PC around every u where the domain is comprehensiveor give up on achieving IC on the richer domain because the domain is inconsistent with perfect competition. Consider the following related question:6 Let ( f, D) satisfy the hypotheses of Theorem 1. Keeping the domain D fixed, could there exist another outcome function g on D that is continuous, efficient, anonymous, and IC? Could g be non-equal wealth Walrasian? In that case, Theorem 1 would be sufficient, but not necessary for IC; hence, it would not be a characterization. The next result shows that the implications of Theorem 1 hold for any such g.

Theorem 2. Let ( f, D) be any continuous, equal-wealth Walrasian mechanism that is also first order. On the same domain, let (g, D) be any continuous, efficient, anonymous mechanism. If (g, D) is IC, then it is an equal-wealth Walrasian mechanism with p(g, u)=p( f, u) for all u # D (so g must be utility equivalent to f on D).

To apply the result, suppose first that ( f, D) is first order and IC; hence, by Theorem 1, it satisfies the hypotheses of Theorem 2. Then Theorem 2 says that g must agree with f: g must be equal wealth Walrasian everywhere on D and further, around any u where D is comprehensive, g must be PC. On the other hand, if ( f, D) is first order but not IC, then Theorem 2 implies no other outcome function g exists that is IC on this domain.

3.3. The No-Externalities Connection The conclusion of Theorem 1 can be restated in terms of no externalities:

Given any profile for others, u&i , in an IC mechanism individual i must impose no externalities by his presence, in the sense that no other individual is either benefited or hurt by i's announced type. Let

A&i ( f, u):=j{i A( fj (u), uj) denote the aggregate at-least-as-good-as-f(u) set for individuals other than i.Ifi imposes no externalities on others and the mechanism is efficient, then the boundary of this set must be i's opportunity set. That is, the mechanism must act as if i completely controls the allocation of the economy's entire resources |, subject only to the constraint that each individual j{i must achieve at least utility uj ( fj (u)) (see Figure 1). In labeling the figure we use ``full appropriation'' as a synonym

6 The referee called our attention to this issue. EFFICIENT IC ECONOMIES 183

FIG. 1. Individual i acts as a full appropriator. The size of the Edgeworth box is |. Hence, fi (u) from i's perspective equals |& fi (u) from I"[i]'s perspective. for creating ``no externalities''; the idea is that if one fully appropriates the consequences of one's action then one creates neither positive nor negative spillovers on others. A mechanism which always gives the individual his most-preferred consumption bundle in [|]&A&i ( f, u) will obviously be incentive compatible. Theorem 1 readily implies that the converse is also true:

Corollary 1(ICO no externalities). Let ( f, D) and u satisfy the hypotheses of Theorem 1. Then the mechanism is IC only if for each i and vi # U,

fi (u&i , vi) # arg max vi (x) s.t. x # [|]&A&i ( f, u), x # 0 and for each j{i,

uj ( fj (u&i , vi))=uj ( fj (u)).

Combined with Theorem 1, we have arrived at our finite characterization: for any ( f, D) satisfying the hypothesis of Theorem 1, IC PC no externalities.

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3.4. Impossibility on the Universal Domain In a finite economy, if there is a one person change from u to v #[u&i , U], then efficiency typically demands that the change be accom- modated by a change in the allocation, i.e., f(u){ f(v). But the change in the allocation will typically conflict with the demands of Theorem 1 that prices do not change that i should have no monopoly power. The two demands are compatible only if the change in the allocation does not lead to a change in the MRS's of u&i , i.e., only if the preferences in u&i exhibit linear segments. This will be illustrated in Section 5, where an ( f, D) satisfying the hypotheses of the Theorem will be constructed. The impossibility of IC on the universal domain Un follows readily from Theorem 1 since, in most populations u # Un, some individuals will have monopoly power. Indeed, impossibility can be shown even on relatively small domains. The following is a sufficient condition for IC to be unachievable.7

Corollary 2. Let ( f, D) and u satisfy the hypotheses of Theorem 1.

Suppose p( f, u){p( f, v) for some v=(u&i , vi). Then f will fail to be IC.

The next result shows that the conditions described in Corollary 2 are ubiquitous. (See Hurwicz and Walker [14] for an analogous result in the quasilinear model.)

Corollary 3. Let ( f, D) and u satisfy the hypotheses of Theorem 1, and suppose that each ui in u is strictly quasiconcave. Then there exists v=(u&i , vi) such that p( f, u){p( f, v); hence, by Corollary 2, f fails to be IC.

Intuition for Corollary 3 comes from Figure 3. It illustrates that IC implies each individual i must face a linear opportunity set. But if all individuals j{i have strictly quasiconcave preferences then the boundary of A&i ( f, u) will be strictly convex. So, acting as a full appropriator, i will not face a linear budget line. The underlying idea is that if everyone else's preferences are strictly quasiconcave, then when i changes his quantities demanded, market-clearing prices must change since others will not be will- ing to accommodate him at the original prices. This will be contrasted

7 The impossibility implications of BarberaÁ and Jackson [1] rely on the assumption that the mechanism is non-bossy [if an individual changes his announcement and nothing happens to him, then nothing can happen to anyone else], whereas we assume that the mechanism is utility continuous [small changes in an individual's preferences have small utility consequences for anyone else]. EFFICIENT IC ECONOMIES 185 below with the continuum where, if all individuals have strictly quasiconcave preferences, the boundary of the at-least-as-good-as set for the continuum analog of A&i ( f, u) will, from an individual's perspective, be linear even if the boundary merely has a unique support at the analog of |&|Ânr|. The difference is explained by the fact that the scale at which the individual trades in the continuum is infinitesimal will respect to the aggregate endowment |, whereas in a finite economy the scale of an individual's trading is of the same order as |.

4. TRANSFERABLE UTILITY ECONOMIES

l&1 Throughout this section U=Uql (R + _R), the set of all continuous, strictly monotonic, quasiconcave functions with the standard quasilinear form

u(x1 , ..., xl)=u~(x1 , ..., xl&1)+xl . It can be shown that any quasiconcave, quasilinear function must be concave, hence U consists of only concave functions with the quasilinear form. We will call the last commodity ``money.'' For the quasilinear model, there is a well known characterization of mechanisms that are both IC and satisfy a qualified notion of efficiency: they must be in the VickreyClarkGroves (VCG) family (see Fact 1, below). VCG mechanisms may be only qualified efficient, not fully efficient, because they may not be budget-balancing. Superficially the Vickrey ClarkGroves characterization seems quite different from our ordinal characterization. Our goal in this section is to show that this impression is only superficial: any VCG mechanism that is fully efficient (i.e., budget- balancing) must be perfectly competitive. 4.1. Main Result for Quasilinearity Our method of proof for the quasilinear model builds on the Vickrey ClarkeGroves characterization, which allows us to dispense with two assumptions required for the ordinal model: utility continuity and the uniqueness of supports. But we still will require the mechanism to be first order. Because 1 u(x) need not be a singleton, the definition of ( f, D)as first-order becomes: for each u # D there is a selection from P( f, u), denoted p( f, u), such that D contains every v # Un having p( f, u)#

1 vi ( fi (u)) for all i. For any population u, the gains from trade in u is given by

G(u)=max : ui (xi) s.t. x # X. i 186 MAKOWSKI, OSTROY, AND SEGAL

As is well known, for TU (transferable utility) economies, efficient allocations are equivalent to ones that maximize the gains from trade. Let [u, v]=[w: w(x)=au(x)+(1&a)v(x), a #[0,1]], the set of all con- 8 vex combinations of the utility functions u and v. Let D(u&i ,[ui , vi])/ [u&i , U] be the set of v in which i's preferences are some convex combination of ui and vi . For mechanisms achieving the maximum gains from trade, Holmstro m's [12] version of the VCG characterization may be expressed:

Fact 1. Consider the domain D$=D(u&i ,[ui , vi]), and suppose the mechanism ( f, D$) is efficient and anonymous on this restricted domain. Then

( f, D$) is IC only if, for all wi #[ui , vi],

wi ( fi (u&i , wi))=G(u&i , wi)&8(u&i).

The VCG characterization is instructive because it highlights the importance of internalizing external effects for IC: each individual i must appropriate the whole gains from trade minus a lump sum 8(u&i), an amount independent of wi . However, because the characterization is stated as a person-by-person ``partial equilibrium'' condition, without explicitly requiring that money transfers sum to zero, it highlights only a qualified notion of efficiency. For full efficiency, the money transfers must sum to zero, which is equivalent to the condition that G(u&i , wi)= wi ( fi (u&i , wi))+j{i uj ( fj (u&i , wi)). Substituting, this implies that for all wi #[ui , vi],

: uj ( fj (u&i , wi))=8(u&i). j{i

Hence, no matter what the characteristics of individual i, his choice will leave others with the same total utility. That is, for full efficiency, Fact 1 implies for all wi #[ui , vi]:

(F.1A) : uj ( fj (u&i , wi))= : uj ( fj (u)) j=% i j=% i

(F.1B) wi ( fi (u&i , wi))=G(u&i , wi)& : uj ( fj (u)). j{i

Facts (F.1A) and (F.1B) emphasize that for each individual i to fully internalize his effect on the total gains from trade while money payments sum to zero, the share of those gains going to others must remain constant.

8 l&1 Because u and v are quasilinear, each consists of a concave function u~and v~on R+ plus a quantity of money; hence a convex combination is of the same form, i.e., [u, v]/U. EFFICIENT IC ECONOMIES 187

When the VCG outcome function f is restricted to fully efficient (i.e., budget-balancing) allocations and Holmstro m's domain D(u&i ,[ui , vi]) is enlarged to include i [u&i , U] and all first order similar populations relative to f, we arrive at the same result as in the ordinal model.

Theorem 3(ICO PC). Let ( f, D) and u satisfy the hypotheses of Theorem 1. If mechanism is IC, then there exists a price vector p # P( f, u) such that, for every individual i,

Ri ( f, u&i)/Hp+[|Ä].

This theorem is proved in Appendix B. Although differentiability plays no explicit role in the quasilinear model, it is sufficient for quasiregularity. The following is also proved in Appendix B.

Fact 2. If ui is differentiable for each i, then u will be quasiregular. Remark 1. In the quasilinear model, any u # U is concave. Its subdifferential at any point x # 0 is defined by

u(x)=[p# Rl: u(x)&pxu( y)&py for all y # 0], which implies pl=1 for every p in the subdifferential. It follows, given our normalization of prices in the quasilinear model, that u(x)=1 u(x). Further, from the theory of concave functions, the subdifferential at x contains only one p iff u is differentiable at x, in which case {1 u(x) equals the gradient of u at x, {u(x).

4.2. The No-Externalities Connection To make the no-externality connection, it will be convenient to express Fact 1 in the set theoretic terms of Corollary 1. (F.1B) says that i receives his most-preferred consumption bundle subject to others receiving at least utility j{i uj ( fj (u)). That is, for all wi #[ui , vi]:

(F.1C) fi (u&i , wi) # arg max wi (x) s.t. (|&x)#A&i ( f, u). x # 0 An incentive compatible mechanism must assign ``utility rights'' to others, defined by A&i ( f, u), and then allow each individual i to fully appropriate all the added gains that ui contributes to the social total, as illustrated in Figure 2. Theorem 3 tells us that this requirement can be achieved by all individuals simultaneously only when Ri ( f, u&i), the incentive compatible opportunity set for each i, is linear. In other words, only in perfectly competitive environments can all individuals simultaneously fully appropriate. 188 MAKOWSKI, OSTROY, AND SEGAL

FIG. 2. Individual i fully appropriates the gains from trade minus a lump sum, leaving all others at utility level j{i uj ( fj (u)).

For example, in Figure 2 individual i is fully appropriating. But because his opportunity set is not linear, in the allocation f(u) in Figure 2 not everyone can fully appropriateunless the mechanism does not maintain a balanced budget. We assert without proof that the analogs of Corollaries 2 and 3 for Theorem 1 also hold for Theorem 3.

5. PERFECTLY COMPETITIVE DOMAINS FOR FINITE ECONOMIES: AN EXAMPLE

In this section we construct an ( f, D) and u satisfying the hypotheses of Theorem 1 (or Theorem 3). To meet the required stipulations, the example will involve an equal-wealth Walrasian mechanism on a family of perfectly competitive economies. Corollary 3 implies that if everyone's preferences are strictly quasiconcave, monopoly power is inevitable in finite economies and therefore IC mechanisms cannot be achieved. Nevertheless, we will show that constructing an example meeting the required conditions is easy: any finite economy can be subjected to a ``local flattening'' such that it and a class of nearby economies form a domain that satisfies the requirements of Theorems 1 or 3.

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Below attention is limited to ordinal economies, with n3. In the construction, B(x, =) will denote an =-ball centered at x, i.e., B(x, =):= [y: &y&x&<=]. To begin, pick any population u$inUn and let (x*, p) be any equal- wealth Walrasian equilibrium for u$. In an epsilon neighborhood of xi*, flatten i's indifference curve through x*,i keeping its slope equal to p.Let u=(u1 , ..., un) denote such a perturbed population. So, by construction, there exists an =>0 such that for each i O y # B(xi*, =) & (Hp+[|Ä]) ui ( y)=ui (x*).i Notice that (x*, p) remains an equilibrium for this local flattening of the original economy. l Let $ be the diameter of a ball centered at |Ä that includes R++ & (Hp+[|Ä]), that is, $ Rl & (H +[|Ä])/B |Ä , . ++ p \ 2+ See Figure 3. To ensure that no one will be able to influence prices, assume n is sufficiently large that

2$ => .(C) n&2

We construct an IC mechanism ( f, D) that is first order and comprehensive around u. The outcome function f will be such that for every population v # D, p( f, v)=p and ( f(v), p)#WE(v). So ( f, D) is a perfectly competitive mechanism.

Let Dc=i [u&i , U]. First we construct the outcome function f on Dc . 1. Let f(u)=x*.

2. For any i and v #[u&i , U], let

fi (v)#xi # arg max vi (x) s.t. pxp|Ä . x # 0

And for each j{i, let

z f (v)#x =x*& , j j j n&1 C where z=xi&xi*. Since z # B(0, $Â2), &z&<$. So the inequality ( ) implies z $ = < < . "n&1" n&1 2 190 MAKOWSKI, OSTROY, AND SEGAL

FIG. 3. Each individual's indifference curve through x*i has been flattened to ensure no one will be able to influence prices.

1 1 1 That is, for all individuals {vi (xi)={uj (xj)={uj (xj*)=p,so(x, p)# WE(v).

Let Ds be the set of all populations w ÂDc such that, for at least one v # Dc , w is first order similar to v at f(v). We now extend f to Ds

3. For any w # Ds , let f(w)= f(v)#x for exactly one v # Dc such that w is first order similar to v at f(v). It is easily verified that (x, p)#WE(w).

To conclude the example, let D=Dc _ Ds . Notice Dc /D implies the mechanism is comprehensive around u, while Ds /D implies it is also first- order. In particular, the last step ensures that for any w # Ds , D includes all populations first order similar to w at f(w) since it includes all populations first order similar to v at f(v), where by construction f(v)= f(w).9 The example shows that small flats for all individuals, as illustrated in Figure 3, result in each individual facing a large flat segment as his opportunity set, as illustrated in Figure 1. Also notice from the inequality (C) that as n Ä , = can be chosen to go to zero; hence, the population u approaches the population u$ with smooth and perhaps strictly convex preferences. Alternatively expressed, an arbitrary population selected from Un approaches a perfectly competitive population as n Ä . The example therefore helps us understand why a Walrasian mechanism is IC in

9 We did not verify that u was quasiregular; but that is sufficient condition for the Theorem which is not needed to establish IC in this example.

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6. EFFICIENCY WITH INCENTIVE COMPATIBILITY IN THE CONTINUUM

We now extend the finite characterization to a setting with a continuum of agents. The bonus is that, in the continuum, possibility will turn out to be generic rather than exceptional. Having shown that the assignment of zero allocations to some individuals in the finite model is incompatible with IC, in this section zero allocations are precluded. Throughout this sec- l tion U=Ub(R++), the set of continuous, strictly monotonic, quasiconcave l utility functions on 0=R++ satisfying the boundary condition.

6.1. Preliminaries As earlier, U is given the topology of C 0 uniform convergence on compacta. Let M be the set of probability measures on U endowed with the topology of weak convergence. Denote by supp + the support of + # M.It Ä is important to observe that in this topology +k + need not imply that supp +k converges in the Hausdorff metric to supp +, although there will Ä Ä exist sets Qk /supp +k such that Qk supp + and +k(Qk) +(U). The elements of supp +k not in Qk may be regarded as the non-representative members of +k in the sense that they are not close to the types of individuals appearing in +. To describe misrepresentation of preferences, measures with such non-representative types must be allowed. In the continuum, a mechanism is a function f: M_U Ä 0 satisfying the l feasibility constraint f(+, u) d+(u)=|, where | # R++ is the aggregate endowment. Interpret f(+, u) as the allocation to any individual who announces type u when the distribution of announced types is +. We call attention to the following: (a) now a mechanism is just an f rather than a pair ( f, D) because the mechanism is defined on the universal domain: M=M(U), the continuum analog of Un; (b) anonymity is built into the distribution approach since f(+, u) depends only on the person's announcement; (c) because individuals are infinitesimal, no one individual can affect the distribution + by her announcement. We will assume throughout that, at each + # M, the mechanism assigns a Pareto efficient allocation. So there are efficiency prices that support 192 MAKOWSKI, OSTROY, AND SEGAL

l f(+, } ). More precisely, let S=[p# R++ : p } |=1] be the set of normalized prices. The set of efficiency prices supporting f(+, } ), denoted P( f, +), consists of all p # S such that

f(+, u)#( p, u, pf(+, u)) for all u #supp+.

(See equation (-) in Section 2 for the definition of .) Notice that feasibility and efficiency only restrict the assignment f(+,}) for u # supp +. But to address IC, the mechanism must assign a consumption bundle to any possible u # U, not just to u #supp+. The interpretation is that the mechanism designer does not know a priori what the support of the economy is; he must elicit this information from the individuals in the economy in an incentive compatible way. For example, as far as the designer knows, the true economy may be a neighbor of + in which all types in U are represented. To pin down the mechanism's assignment outside the support of + we will introduce a continuity assumption. Call f price continuous at + with respect to p if p # P( f, +) and, for any Ä +k + and any v #supp+k , there exist efficiency prices pk # P( f, +k) such that

Ä Ä pk p and pk f(+k , v) pf(+, v).

Notice v must be in supp +k for each k, but need not be in supp +. Hence price continuity ensures that f(+, v)#( p, v, pf(+, v)) even for v Â supp +;it pins down the mechanism's choices outside the support of +. At first glance, price continuity in the continuum may appear similar to utility continuity in the finite model of Section 3. Actually, the latter is implied by price continuity with respect to small changes in an individual's utility function, whereas price continuity in this section is with respect to any change in an individual's utility function. How restrictive is price- continuity? It easily can be checked that it is more restrictive than utility- Ä continuity (i.e., for all +k + and all u # U, u( f(+, u))=lim u( f(+k , u))), Ä but it is less restrictive than allocation-continuity (i.e., for all +k + and all u # U, f(+, u)=lim f(+k , u)). The prospects for price continuity in the continuum are examined in Section 6.3. For any p # P( f, +), the pair ( f(+,}),p) may be viewed as a Walrasian equilibrium for +, perhaps with lump-sum transfers. Analogous to the finite definition, the mechanism is perfectly competitive (PC) at + if f(+,}) involves no lump-sum transfers and no one can influence prices. But in the continuum only the first requirement is restrictive, at least if f is price- continuous at +: Recall price continuity implies each individual will face the same prices p in + regardless of his announcement v. Since + is a probability measure, |d+=|. That is, an equal-wealth Walrasian mechanism EFFICIENT IC ECONOMIES 193 gives each individual an endowment |; so no lump-sum transfersi.e., PCmeans that for some p # P( f, +),

pf(+, u)=p| for all u # U.

The mechanism f is incentive compatible at + if for every u #supp+ and all v # U:

u( f(+, u))u( f(+, v)).

The mechanism is incentive compatible if it is incentive compatible at all + # M. Observe that f being incentive compatible at + is the same as the definition of f(+, } ) being an envy-free allocationif one assumes supp +=U. Thus IC and envy-free allocations are intimately related. Indeed, if f is price-continuous at + then there is a nearby economy with full support and an associated nearby utility allocation; so at such + the two concepts are almost identical. (See Remark 3.)

6.2. Characterization Suppose f is price-continuous at +. Since no individual can influence prices, any Walrasian allocation without lump-sum transfers is evidently IC at +. The converse also is true:

Theorem 4(ICO PC). Suppose f is incentive compatible, and f is price continuous at + with respect to some p. Then f must be perfectly competitive at +, i.e., pf(+, u)=p| for all u # U. Observe that supp + may consist of only two types. Nevertheless, price continuity implies no individual in + can influence his wealth regardless of what type he announces in U, a set with an infinity of types. Theorem 4 is the continuum analog of Lemma 3 in the sense that both say that incentive compatibility precludes lump sum transfers. The crucial difference is that the finite analog of price continuityi.e., for any change in an individual's utility, the individual cannot change pricesis not a typically property of the finite model, whereas it can be typical in the continuum. There is also a connection between Theorem 4 and incentive compatibility in the finite quasilinear model. Our method of proving Theorem 4 builds on Holmstro m's [12] version of the VCG characterization (recall Section 4). Although our utility functions are entirely ordinal, rather surprisingly, for continuum economies one can ``lift'' Holmstro m's proof to the ordinal level by using a money-metric representation of preferences (introduced below). The proof of Theorem 4 thus provides some evidence for the view that, very roughly and intuitively speaking, the money metric 194 MAKOWSKI, OSTROY, AND SEGAL representation is as close as one can come to an ordinal version of quasi linearity. This link with the quasilinear model is of independent interest. The proof is also closely related to demonstrations that a Walrasian allocation without transfers is the unique efficient, no-envy allocation in large economies with smoothly connected supports, e.g., the proof in Mas- Colell [18, Section 7.5]. An interesting difference relative to Mas-Colell's result is that, because a significant part of our domain consists of a convex set of concave functionsrather than an arbitrary, smoothly connected domainwe will not need to assume the mechanism is bounded; this will come out as a conclusion (see Lemma 11 in Appendix C.1). While assuming the domain contains such a convex set may be restrictive for a no-envy result, it is entirely acceptable for an incentive-compatibility result. 6.3. Existence Although the characterization is unchanged from the finite case, instead of illustrating existence by exceptional examples, in the continuum we can show that it is generic. Since an equal-wealth Walrasian mechanism is IC at all + where it is price-continuous, to establish genericity it will suffice to show that there is a selection from the equal-wealth Walrasian price correspondence that is generically continuous. The aggregate demand correspondence for the economy + when individuals are each endowed with | is

9( p, +)=| ( p, u, p|) d+(u).

Therefore,

6(+)=[p: | # 9( p, +)] is the set of Walrasian equilibrium prices for the population + when each individual is endowed with |.Let? denote a selection from the correspondence 6: M Ä S. Recall that a set is residual if it is the complement of a countable collection of nowhere dense sets. In a complete metric space, a set is known to be residual if and only if it contains a dense G$ set.

Theorem 5 (Generic Existence). There exists a price selection ? from 6 that is continuous on a residual subset of the complete metric space M. The conclusion of Theorem 5 is related to known results from the literature on regular economies; nevertheless, those results do not seem to be applicable here. Under more restrictive assumptions, a stronger conclusion has been proved for regular economies, namely, the existence of a EFFICIENT IC ECONOMIES 195 continuous selection on an open and dense set (e.g., see Mas-Colell [18, Proposition 5.8.18]). The more restrictive assumptions include: (1) U contains only strictly quasiconcave C 2 utility functions, (2) economies + # M must have compact supports, and (3) the topology on M is strengthened Ä so that +n + means not just weak convergence but also convergence of supp +n to supp + in the Hausdorff distance. The proof of our finite characterization requires linear (hence not strictly quasiconcave) utility functions in U. More importantly, even in the continuum, incentive compatibility requires individuals to be able to claim any type u # U, not just types in supp +; so the stronger topology used to demonstrate continuity on an open and dense set is inappropriate for our purpose. Remark 2(A Quasilinear Extension). In the presence of quasilinearity, a stronger conclusion can be established: If utility functions are C 1, the price correspondence 6 is single-valued, and therefore continuous at every + for which it is non-empty-valued. Moreover, in the quasilinear assignment model where all individuals have utilities function that are never differentiable, the set of ``non-manipulable'' Walrasian equilibria (a combination of PC and IC) form a residual set. See Gretsky, Ostroy, and Zame [8].

6.4. Asymptotics Here we show that the characterization of IC as PC in the continuum agrees with our finite characterization in yet another way, namely, that most large finite economies are nearly PC and, hence, nearly IC. So, if there is even a small cost of misrepresenting, an efficient allocation can be implemented in most large but finite economies.

Define Mn as the subset of probability measures +n=:m $um with finite &1 support, where each :m is an integer multiple of n and $u is the measure with unit mass centered at u. Mn is interpreted as the set of possible dis- tributions of preferences for populations consisting of n individuals, where each individual has weight n&1. Since it is regarded as a finite economy with n individuals, we should denote +n as, for example, (+n , n) to dis- tinguish it from a continuum economy having finite support. However, unless the contrary is explicitly stated, +n will denote a finite economy with n individuals. Let

J(+n)=[2u := ( u, v) # supp +n _U];

2u # J(+n) represents a perturbation of +n to the ``adjacent'' population

&1 +n+2u := +n&n ($u&$v), 196 MAKOWSKI, OSTROY, AND SEGAL

accessible by a one-person perturbation from +n . In relation to the notation used in the finite model, ut+n and (u&i , vi)t+n+2u. Further, since +n is a distribution, individual characteristics are automatically in anonymous form. Given any price selection ?, let

2?(+n)=diameter [?(+n+2u): 2u # J(+n)], i.e., the diameter of the smallest ball containing the set of market-clearing prices for +n and all of its adjacent populations. Call +n*#Mn a perfectly competitive population relative to the price selection ? if 2?(+n*)=0; and let

Mn*(?)/Mn denote all such perfectly competitive populations. From Corollary 2 and the example in Section 5, if +n* is perfectly competitive then some individuals u # supp +n* must have indifference curves with flat segments through x # ( p, u, p|), where p=?(+*).n First we show that if ? is continuous at +, all large finite economies near + will be nearly perfectly competitive. For this purpose, it will be convenient to work with an explicit metric on M: we will use the Prohorov metric \ (see Hildenbrand [11, p. 49]. Under this metric, given any n-person economy +n and any one-person perturbation 2u{(u, u), 10 \(+n , +n+2u)=1Ân.

Definition. Given any price selection ? and any sequence of finite Ä economies +n +, where +n is an n-person economy and + is a continuum economy, we will say that the economies +n are becoming increasingly competitive as the population increases if

Ä Ä 2?(+n) 0asn .

The continuum economy + is a perfectly competitive limiting economy Ä relative to ? if, for every sequence +n +, the economies +n are becoming increasingly competitive.

Proposition 2. If ? is continuous at +, then + is a perfectly competitive limiting economy.

The proposition implies that under the price selection ? in Theorem 5, nearly perfectly competitive economies are dense in M. More formally, = using this selection, for any =>0 define M PC=n [+n : 2?(+n)<=]. = Observe that if = is arbitrarily small then any economy +n # M PC is nearly a perfectly competitive economy, i.e., nearly a +n # M n*(?). Proposition 2

10 &1 Recall if 2u=(u, v) then +n+2u=+n&n ($u&$v). EFFICIENT IC ECONOMIES 197

= and Theorem 5 together immediately imply that for any =>0, M PC is dense in M. It only remains to show that, for any nearly perfectly competitive finite economy, the (price-continuous) Walrasian mechanism is nearly incentive compatible. For this purpose, it will be useful to choose a canonical function to represent ordinal preferences. Re-represent the preferences underlying u # U according to the money-metric utility function

u(x | p)=min[py: u( y)u(x)], where p # S. It is readily verified that for x>>0, v u(x | p)px v u(x | p)=px if and only if p #1 u(x). Hence, under the money metric utility representation, any utility-maximizing choice by any individual of type u facing prices p and having wealth w must yield him utility w:

( p, u, w)=[x: u(x | p)=px=w].

(The money-metric representation will also be used in the proofs of Proposition 1, Lemmas 3 and 6, and Theorem 4, below.)

Proposition 3. Let d=2?(+n), let p=?(+n), and let :=min[p1 , ..., pl].

Suppose :>d. Then for any 2u=(u, v)#J(+n), any y # ( p, u, p|), and any x # (q, v, q|), where q=?(+n+2u):

d u(x | p)&u( y | p)< . :&d

The proposition says that the gain from any misrepresentation by any individual in +n is bounded by an amount that goes to zero with 2?(+n). To apply the proposition, consider the price selection of Theorem 5 and Ä any sequence +n +.If? is continuous at + (a generic property), then Proposition 2 implies the economies +n are becoming increasingly com- Ä petitive, i.e., 2?(+n) 0. Hence, Proposition 3 implies the gains from misrepresentation are going to zero as n Ä . Remark 3(The Definitions of Perfect Competition and Incentive Com- patibility in the Continuum). The Walrasian definition of perfectly competitive equilibrium focuses entirely on the coordination of demands and supplies in a single economy. By contrast, the definition of incentive compatibility involves restrictions with respect to nearby economiesresulting from misrepresentation. Similarly, perfect competition as we use it here 198 MAKOWSKI, OSTROY, AND SEGAL involves restrictions related to nearby economiesreached by some individual experimenting by altering his demands or supplies. Thus, to establish our asymptotic results, we view a continuum economy + as perfectly competitive not because individuals are of negliglible size, but Ä Ä because for any +n +, 2?(+n) 0. Similarly, the Walrasian mechanism is incentive compatible at + only if for all nearby finite economies +n , the gains from misrepresentation are small. The literature on envy-free allocations in the continuum represents an interesting way around the problem of explicitly dealing with perturbations by individuals. ``Envy'' is a virtual condition: would A rather have what he is given or what B gets. There is no need to ask whether it is possible to give A what B is getting without disturbing B or anyone else. Nevertheless, the issue of perturbations is treated implicitly. A caveat is made that the analysis should be undertaken using a continuous mechanism. See Champsaur and Laroque [3]. Our characterization of efficiency and incentive compatibility in the finite model along with this study into the asymptotic behavior of the Walrasian mechanism builds a bridge to the envy-free characterization of Walrasian equilibrium in the continuum, one that calls explicit attention to the importance of perfect competition.

7. REMARKS ON THE INDISPENSIBILITY OF PERFECT COMPETITION

In Section 1 we pointed to three branches of the mechanism design literature that are linked to our results on the indispensability of perfect competition: the VickreyClarkeGroves characterization and associated budget-balancing problems in finite quasilinear models, the impossibility theorems for finite ordinal models, and the no-envyÂIC characterization of the Walrasian mechanism in continuum models. In this Section, we sum- marize our conclusions and compare our results to two contributions containing apparently different conclusions. Recall the triangle of equivalences for an efficient allocation scheme connecting (1) IC to (2) PC to (3) no externalities. To emphasize the nexus between general equilibrium theory and mechanism design, we want to highlight the equivalence between PC and no externalities (i.e., no matter what his announced characteristics, no one individual can influence the welfare of others). The VCG characterization for IC in the finite quasilinear model was an important stepping stone in this connection. Recall from Sec- tion 4 that any VCG scheme may be thought of as assigning ``utility rights'' to the rest of the population, and then permitting the individual to choose his most desirable allocation subject to the constraint that others' utility rights must be respected (Fig. 2). Hence, the individual's choice imposes no EFFICIENT IC ECONOMIES 199 externalities on others. When this partial-equilibrium principle is simultaneously applied to all individuals in the finite quasilinear model, there is typically an unavoidable inconsistency, which shows up as the absence of ``budget balance.'' If, however, the environment is perfectly competitive (i.e., no matter what his characteristics and hence utility maximizing choice, no one individual can change prices or the wealths of others), externalities can be eliminated while markets clear, thereby guaranteeing budget balance in the money commodity. Theorem 3 demonstrates that under certain (non-vacuous) richness conditions on the domain of economies, the only way to construct a budget-balancing, efficient, and anonymous VCG scheme (ensuring consistent utility rights), is if individuals are perfect competitors. Conversely, the reason this is generically impossible to do (the asserted quasilinear analog of Corollary 3) is that Walrasian equilibria are generically not perfectly competitive in finite quasilinear economies. There is no known analog in the finite ordinal model of the partial equilibrium, VCG characterization. But Theorems 1 and 2, along with Corollary 1, establish that there is an ordinal general equilibrium analog: under certain (non-vacuous) richness conditions on the domain, the only way to construct an efficient, anonymous, and continuous scheme exhibit- ing IC is to ensure that externalities are eliminated, and the only way to do that is if there is perfect competition. Our presentation in Section 3 reversed the order, showing the equivalence of IC with perfect competition first and then with no externalities; but the order in which the equivalents are proved is immaterial. In the continuum, a price-continuous Walrasian mechanism is IC. More significant is the converse, Theorem 4: an efficient, anonymous, price-continuous mechanism is IC only if it is Walrasian. To relate this conclusion to results for finite economies, it was important to recognize that there are built-in properties of a price-continuous Walrasian mechanism in the continuum which typically are not shared by the finite model. One way to see the difference is through the notion of externalities. At points of price continuity, an equal-wealth Walrasian mechanism in the continuum exhibits no externalities: no one individual can influence prices or the wealths of others by his announced type u # U. If, however, the mechanism were such that pf(+, u)

I1 , ..., In , and each consumer i's utility function is assumed to be sensitive to changes only in the quantities of commodities in Ii . So the set Ui of l Ä O utility functions available to consumer i is [u: R R : c Â Ii uÂxc #0, and u is not constant]. For a vector of utility functions (u1 , ..., un)# n >i=1 Ui , the mechanism assigns each consumer all the available endowments of all the goods to which his utility may be sensitive (and throws away all goods no one wants). This is obviously an efficient incentive compatible mechanism, but the authors also prove that this is the only case where such a mechanism exists. The elimination of conflicts implies the elimination of externalities, but not conversely. Indeed, Section 4 demonstrated and Section 5 illustrated that PCÂno-externality mechanisms can exist in economies with gains from trade, hence potential conflicts (i.e., overlapping preferences for some commodities). Nevertheless, there is no real contradiction between the two characterizations; they apply to different domains. The H6W domain is ``larger'' in the sense that they require that a mechanism must be efficient n and incentive compatible on a Cartesian product >i=1 Ui of sets of utility functions, whereas we only require that it achieve these properties on a restricted neighborhood of u # D, a neighborhood which need not be a Cartesian product. On the other hand, the H6W domain is ``smaller'' in that any individual's announced type cannot overlap with others, whereas we EFFICIENT IC ECONOMIES 201 allow any one individual to announce any possible type given others' types (i.e., D is comprehensive around u). The H6W characterization shows that eliminating conflict is one way to avoid its possible inefficient consequences, but perfect competition is another much less restrictive way in which conflict can be efficiently resolved: Competition eliminates monopoly power, forcing each agent to bear (pay for) the full social cost of any resources she takes from others. Indeed, the finite economies offered by H6Wwith non-overlapping preferencesdo not converge to typical continuum economies. So while their characterization may be an efficient means for proving impossibility results for the finite case, the no-conflicts characterization does not build a robust bridge from the finite to the continuum, explaining why generic impossibility turns into generic possibility. Another characterization of incentive compatible mechanisms for finite ordinal models is offered by BarberaÁ and Jackson [1]. B6J mechanisms are also defined on a Cartesian product, but their mechanisms need not be efficient. They characterize all incentive compatible mechanisms as fixed- proportion trading schemes. On its face, their mechanisms appear to mimic trade according to prices and therefore to exhibit an underlying similarity to the results of this paper, but the resemblence is only superficial. Fixed- proportions schemes permit consumers to move only in a finite number of directions rather than the infinite directions possible with hyperplanes, as is the case when they are constrained by prices. A second difference is that these proportions are determined independently of individuals' declared utility functions, in contrast to perfect competition where prices depend on consumers' utilities. As B6J note, these two factors imply that their mechanisms do not converge to the general incentive compatible mechanism for continuum economies; similarly, the lack of efficiency is not reduced when the number of individuals increases. The disparity in asymptotic conclusions follows from B6J's requirement that incentive compatibility holds exactly throughout the entire domain. Note that the conclusions of Theorems 4 and 5 are only that IC and PC hold generically, not universally. A comparison of the conclusions of B6J with the conclusions in this paper and the work of others illustrates that the consequences of imposing exact incentive compatibility everywhere can be quite different from the standard adopted here and elsewhere of asymptotically achieving IC generically.

8. PROOF OF THEOREM 1 AND ITS COROLLARIES

The proof of Theorem 1 involves a geometrically appealing and relatively elementary argument. 202 MAKOWSKI, OSTROY, AND SEGAL

Since f is fixed throughout this section and most of the appendices, we henceforth omit it as a variable unless confusion might result. Hence we write Ri (u&i) for Ri ( f, u&i), p(u) for p( f, u), etc. The demonstration of

Theorem 1, that Ri (u&i) lies in a common hyperplane, consists of three steps: (1) The set Ri (u&i) is not locally strictly convex (Lemmas 1 and 2); (2) The mechanism must always assign an equal-wealth Walrasian allocation (Lemma 3); (3) The set Ri (u&i) is not locally strictly concave (Lem- mas 4 and 5).

Step 1. In Fig. 4, point x lies on the concave part of Ri (u&i), but

Ri (u&i) also has a convex-shaped segment. Our first step (Lemmas 12) is to rule out the convex possibility. To illustrate the idea, observe that in the figure the indifference curve of u through x is also tangent to Ri (u&i)aty. The mechanism must choose one of these two points. Suppose it chooses k x.Letui be a sequence of preferences converging to ui , as illustrated. k Incentive compatibility requires the mechanism to choose y for all ui , but it chooses x for ui . This is compatible with our assumption that the mechanism is continuous in terms of utilities because, even though there is a discontinuous change in the allocation, the individual's utility varies continuously. However, the jump from y to x changes the individual's wealth, which must discontinuously change some other individuals' utilities; a violation of the continuity assumption (See the proof of Lemma 2 for details.) We first show that for individuals with strictly convex preferences, our assumption that the outcome function f is continuous in utilities implies f also will be continuous in allocations.

FIG. 4. Ri (u&i) cannot have a convex-shaped segment as illustrated.

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Lemma k k 1 (Continuity). Let u =(u&i , u i ). Also assume ui is strictly k Ä k Ä quasiconcave and fi (u)>>0. Then ui ui implies fi (u ) fi (u). k k k k k Ä Proof.Letx= f(u), x = f(u ), p= p(u), p = p(u ). Suppose xi % xi . Then there would be a convergent subsequence, say s(k), on which s(k) Ä 0 s(k) Ä 0 0 k k k 0 p p , x x ,andxi {xi . Since each p supports x at u , p supports x0 at u, i.e.,

n 0 0 0 p |p : Aj (xj , uj). j=1

0 Hence, since j xj =|,

0 0 0 0 p xj p Aj (xj , uj) for each individual j. k Ä Further, since continuity implies uj (xj ) uj (xj) for each j,

0 uj (xj )=uj (xj) for each j.

0 0 0 Hence, p xj p xj for each j. And since ui is strictly quasiconcave in 0 0 0 0 the interior of 0, where, by efficiency, both xi and xi reside, p xi

Lemma 2. If ( f, D) is first order, then IC implies that for any u # D and all i with fi (u)>>0:

p(u)}Ri (u&i)p(u)}fi (u).

Proof.Letx= fi (u), p= p(u). Let u be any strictly quasiconcave utility function in U whose indifference curve through x is nested inside ui 's indifference curve through x;so,ui ( y)ui (x) implies u( y)

Notice that incentive compatibility implies fi (u&i , u)=x. Suppose there were a y$#Ri (u&i) such that py$>px. Then there would be a strictly quasiconcave utility funtion v # U such that v(x)0, i.e., in B(x, =):=[x$: &x$&x&<=]. Notice that incentive compatibility implies fi (u&i , v) Â B(x, =). Now, for : # [0,1], let v:( } ) be a family of strictly quasiconcave functions connecting v and u, with v0=v, v1=u, and the indifference curves 204 MAKOWSKI, OSTROY, AND SEGAL through x of v: and of u coincide in B(x, =).11 Increase : starting from :=0 : : as long as there is a y$#Ri (u&i) such that v ( y$)>v (x). Thus one finds an :* :* a* such that v ( y$)=v (x) for some y$#R&i (u&i) (where y${x), but :* :* k v ( y$)~v (x) for any y$#Ri (u&i). Let : be any sequence approaching k :k :* with each : <:*. IC implies fi (u&i , v ) Â B(x, =) for all k; hence :* k Lemma 1 implies fi (u&i , v ){x. On the other hand, let ; be any k ;k sequence approaching :* with each ; >:*. IC implies fi (u&i , v )=x for :* all k; hence Lemma 1 implies fi (u&i , v )=x, a contradiction. Step 2. Our next step is to show that any incentive compatible mechanism must satisfy an equal-wealth condition.

Lemma 3. If ( f, D) is first order, then IC implies that for all u # D,

( f(u), p(u)) # WE(u).

The basic idea behind this key lemma is simple. Technical complications arise only because, for the sake of generality, we (1) permit f to assign zero allocations to some individuals and (2) permit individuals' income expansion paths to by asymptotic to one of the axes as their wealths approach zero. To give the reader the main idea behind Lemma 3, here we present a proof of the lemma under the extra assumption that, for each individual fi (u)x>>0, where x is interpreted as a basic subsistence bundle; this extra assumptionÄ eliminatesÄ both sources of complication. Appendix A contains a proof of Lemma 3 without this simplifying assumption.

Lemma 3A. Let ( f, D) be first order and assume that fi (u)x>>0 for all individuals i and all profiles u # D. Then IC implies that for allÄ u # D

( f(u), p(u)) # WE(u).

Proof.Letp= p( f, u). Since p supports f(u)atu, we need only verify that pfi (u)=p|Ä for all i. Assume the contrary; we will show a contradiction. Let $= px. Individual i's income expansion path when his utility func- Ä tion is ui , he faces prices p, and he has wealth of at least $ is given by 1 E(ui)#[x # 0 : {ui (x)=p and px$].

The boundary condition implies that there exists a closed convex cone in l R++ _ [0], call it C, that contains _ i E(ui). Let u* be any utility function in U that is linear in C with gradient p, i.e., u*(x)=px for all x # C. (Notice the boundary condition implies that,

11 For a way to construct such a family, see Proposition 1 in Appendix A. EFFICIENT IC ECONOMIES 205 outside of C, u*'s indifference curves will become bowed as they approach the axes.) n Feasibility implies i=1 fi (u)=n|Ä ; hence there must be an individual, 1 say i=1, with pf1(u)>p|Ä .Letu =(u*, u2 , ..., un). Lemma 2 implies 1 1 1 1 pf1(u )=pf1(u)>p|Ä and {u*( f1(u ))=p;sop= p(u ). 1 1 Since f(u ) is feasible and pf1(u )>p|Ä , there must be some other 2 individual, say i=2, with pf2(u*)p|Ä .Letu =(u*, u*, u*, u4 ..., un). Lemma 2 3 2 1 3 3 implies pf3(u )=pf3(u )>p|Ä and {u*( f3(u ))=p;sop= p(u ). Further, since individuals 1 through 3 have the same preferences in u3, anonymity 3 3 3 implies pf1(u )=pf2(u )=pf3(u )>p|Ä . Repeat the above procedure n&3 more times to form u4, ..., uk, ...un, k where u =(u*, ..., u*, uk+1, ..., un). One thus finds, depending on whether n n n is odd or even, that pfi (u )>p|Ä for all i or pfi (u )

Step 3. The final step is to show that Ri (u&i) must be contained in a hyperplane through |Ä . For this, we will show that

1. |Ä # Ri (u&i) (Lemma 4);

2. for all xi # Ri (u&i) there exists a supporting hyperplane H to

Ri (u&i)atxi such that |Ä # H; hence H supports Ri (u&i)at|Ä ;

3. if H and H$ support Ri (u&i)at|Ä , then, under a regularity assumption, H=H$ (Lemma 5). Let w p denote any strictly quasiconcave function in U satisfying {1 w p(|Ä )=p.

Lemma 4. If ( f, D) is first order and D is comprehensive around u, then IC implies that for all i,

|Ä # Ri (u&i).

p Proof.Letp= p(u), vi=w , v=(u&i , vi), q= p(v), and y= f(v).

Lemma 3 implies (y, q)#WE(v). Hence, since the indifference curve of vi through |Ä is strictly convex, vi ( yi)>vi (|Ä ) unless yi=|Ä . But since 1 O O p={vi (|Ä ) by construction, vi ( yi)>vi (|Ä ) pyi>p|Ä yi Â Ri (u&i) since K pRi (u&i)pfi (u)=p|Ä (using Lemmas 23), a contradiction. So, yi=|Ä .

Now consider any xi= fi (u), and let p= p(u). Lemma 2 implies

Hp+[|Ä] supports Ri (u&i)atxi , and Lemma 3 implies |Ä # Hp+[|Ä]. 206 MAKOWSKI, OSTROY, AND SEGAL

FIG. 5. If Ri (u&i) has a kink then the economy without i would have a continuum of Walrasian equilibria.

Hence, Lemma 4 implies Hp+[|Ä]supports Ri (u&i)at|Ä . But we have not yet excluded the possibility of Ri (u&i) having a kink at |Ä . We now will show that such a kink may be ruled out, at least for most economies. Figure 5 illustrates the consequences of a kink at |Ä : the individual would receive the same allocation, |Ä , whether he announces u, v, or any preferences ``between'' them. Let p(:)=:p+(1&:)q and let u:=w p(:),so 1 : p(:)={u (|Ä ). Let (x1(:), ..., xi&1(:), xi+1(:), ..., xn(:)) be the allocation to all other individuals as i's preferences vary while others' remain fixed. Feasibility implies

(a) j{i xj (:)=(n&1) |Ä , while Lemma 3 implies

(b) p(:)}xj (:)=p(:)}|Ä for all j{i. Conditions (a) and (b) say that the economy consisting of all individuals except i has a continuum of equal-wealth Walrasian equilibria, one for each price vector p(:). So, for quasiregular economies, Ri (u&i) will not have a kink at |Ä .

Lemma 5. If ( f, D) is first order, D is comprehensive around u, and u is quasiregular, then IC implies that for all i and all vi # U,

p(u&i , vi)=p(u).

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Proof.Letv=(u&i , vi), q= p(v), p= p(u), and R=Ri (u&i). By

Lemma 3, pfi (u)=p|Ä and qfi (v)=q|Ä ; and by Lemma 4, |Ä # R. Hence, Lemma 2 implies p }conR

Proof of Theorem 1. Lemma 3 implies that for any i and v=(u&i , vi), ( f(v), p(v)) # WE(v).

Further, Lemma 5 implies that p(v)=p(u). Hence, letting p= p(u), K fi (v)#Hp+[|Ä].

Proof of Corollary 1. Theorem 1 implies that for any i and v=(u&i , vi), ( f(v), p)#WE(v), where p= p(u). Since p remains constant when i switches to vi , for each j{i

uj ( fj (v))=uj ( fj (u)). Hence, efficiency implies

vi ( fi (v))vi (x) K for those x # 0 that are also in [|]&A&i (u). Proof of Corollary 2. Follows immediately from Lemma 5. K Proof of Corollary 3. In view of Lemmas 3 and 5, it will suffice to 1 1 show that ( f(v), p) Â WE(v). Since {ui ( fi (u)){{vi ( fi (u)), fi (u) is not i's

Walrasian demand when he is of type vi and faces prices p. But the demand of each individual j{i is unique at prices p; so markets will not clear at p K when i is of type vi .

APPENDIX A: PROOF OF LEMMA 3 AND THEOREM 2

To prove Lemma 3 and Theorem 2, we need to know that the set U is path connected, at least under a normalization; that is, under a normalization, 208 MAKOWSKI, OSTROY, AND SEGAL for any u and v in U there is a continuous function F from [0, 1] to U such that F(0)=u and F(1)=v.Ifu and v are concave, a path connecting them can be found by simply taking the convex combination of the two functions, i.e., by letting F(a)=(1&a)u+av. Unfortunately a convex com- binaton of quasiconcave functions need not be quasiconcave, so we must proceed by a different route.12 It will be convenient to normalize utility functions in money metric form (see Section 6.4), so let Up denote the set of utility functions u # U that are money metric with respect to the price vector p, where p>>0. (That is, u # Up iff u(x)=min[py: u( y)u(x)].)

Proposition 1 (Path connectedness). For any u, v # Up , there is a con- Ä tinuous function F(:): [0, 1] Up such that F(0)=u and F(1)=v. In particular, letting F(:)=u(},:), u(x, :)=px iff there is a y, z # 0 such that x=(1&:)y+:z, u( y)=py= px, and v(z)=pz= px.

Proof. For any utility level s #[0, ), let A(s, u)=[x: u(x)s] and define A(s, v) similarly. Let

A(s, :)=(1&:) A(s, u)+:A(s, v), which is convex since the sum of convex sets is convex. For any x # 0, let u(x, :) solve

(1) max s s.t. x # A(s, :).

Hence, by construction, u(},0)=u and u(},1)=v. Let

1(x, :)=[s #[0, ):u( y)s, v(z)s, :y+(1&:) zx].

Notice that (1) is equivalent to

(2) max s s.t. s # 1(x, :).

Since 1(x, :) is nonempty, compact for any (x, :), the max in (2) (and hence in (1)) exists, i.e., u(},:) is well defined. Further, since on any com- l Ä pact subset of 0_[0, 1], the correspondence 1: R++ _[0, 1] R+ can be shown to be continuous (i.e., both lower and upper hemicontinuous), the Theorem of the Maximum implies that u(x, :) is (jointly) continuous l n n Ä n n Ä on R++ _[0, 1]. Observing that u(x , : ) u(0, :)=0 as (x , : ) (0, :), we extend our conclusion to all 0_[0, 1].

12 Thanks to Yakkar Kannai for helpful discussion on this issue. EFFICIENT IC ECONOMIES 209

Notice next that s solves (1) iff x # bdry A(s, :). Hence u(},:)is l quasiconcave, strictly increasing in R++ , and satisfies the boundary condition, i.e., each u(},:)#U.

To verify that u(},:)#Up , suppose u(x, :)=s. Hence, since x # bdry A(s, :), there must be a y #bdryA(s, u) and x # bdry A(s, v) such that x=(1&:) y+:z.Lety$ # arg min pA(s, u) (hence y$#A(s, u) is on the hyperplane with normal p tangent to A(s, u)); similarly, let z$# arg min pA(s, v). By construction, u( y)=py$andv(z)=pz$. Let x$= (1&:) y$+:z$. By a standard argument, x$ # arg min pA(s, :); so u(x$, :)=s. Combining, we see u(x, :)=u(x$, :)=s=(1&:) u( y$)+:u(z$)=px$; so u(}, :) is indeed money metric. If u(x, :)=px#s, then there exists y, z such that x=(1&:) y+:z, u( y)=py=s, and v(z)=pz=s. Further, since x # arg min pA(s, u(},:)), by a standard argument y # arg min pA(s, u) and z # arg min pA(s, v). Conver- sely, if y # arg min pA(s, u)andz # arg min pA(s, v), then by a standard argument x # arg min pA(s, u(}, :)). A.1. Proof of Lemma 3

Let p= p( f, u), let d be any positive integer such that d>maxc(p|Âpc |c) (where c is an index over commodities), and let $= p|Â(2nd). The income expansion path of an individual having utility function u, facing prices p, and having wealth of at least $ is

E(u):=[x # 0 : {1 u(x)=p and px$].

The boundary condition implies there is a closed convex cone in l R++ _ [0], say C$, which contains all these income expansion paths for u:

. E(ui)/C$. i

l Let C be a closed convex cone in R++ _ [0] containing C$ and also [|&x : x0 and pxp|Âd].Letu* be any function in U that is linear in C with gradient p [i.e., u*(x)=px for all x # C] and strictly quasiconcave outside of C.

Since p supports f(u)atu, we need only verify that pfi (u)=p|Ä for all i. Assume the contrary, we will show a contradiction. In particular, we establish the inductive hypothesis that, for any k # [1, 2, ..., n], there is a k k k k population u =(u1 , u2 , ..., un)inD satisfying v k i E(ui )/C v p= p( f, uk) v k ui =u*fori=1, 2, ..., k v k pfi (u ){p|Ä for i=1, ..., k. 210 MAKOWSKI, OSTROY, AND SEGAL

n n Notice that for k=n the hypothesis implies pi=1 fi (u ){np|Ä , which n n n contradicts the feasibility of f(u ), i.e., the fact that i=1 fi (u )=|=n|Ä . n The hypothesis holds for k=1: Since feasibility implies i=1 fi (u)=n|Ä , 1 there must be an individual, say i=1, with pf1(u)>p|Ä .Letu = 1 (u*, u2 , ..., un). Since f1(u)#C, Lemma 2 implies pf1(u )=pf1(u)>p|Ä and 1 1 1 1 {u*( f1(u ))=p;sop= p( f, u ). Further, i E(ui )/C since u* is strictly quasiconcave outside of C. Let the hypothesis hold for an arbitrary k; we show how to constructed uk+1. There are three possible cases:

k (a) pfi (u ) Â [0, $) _ [p|Ä]for some i>k k k (b) pfi (u )#[0,$) _ [p|Ä] for all i>k and pfi (u )=p|Ä for some i>k. k (c) pfi (u )#[0,$) for all i>k. If case (a) applies, say for individual i=k+1, let uk+1= k k (u1 *, ..., uk+1*, uk+2, ..., u n), where each ui *=u*. Since u* is strictly k+1 k quasiconcave outside of C, i E(ui )/C. Since fk+1(u )#C, Lemma 2 k+1 k 1 k+1 implies pfk+1(u )=pfk+1(u ){p|Ä and {u*( fk+1(u ))=p;sop= p( f, uk+1). Further, since individuals 1 through k+1 have the same k+1 k+1 preferences in u , anonymity implies pfi (u ){p|Ä for i=1, ..., k+1. k Suppose instead that case (b) applies. Notice that in this case pfi (u )<$ n k k k for some i>k because i=1 fi (u )=n|Ä (by feasibility) and pi=1 fi (u ){ k kp|Ä (by hypothesis). Say that for individual n, pfn(u )=p|Ä . Suppose, k : without loss of generality, that un # Up , and let v for : # [0, 1] be Proposi- k tion 1's family of money metric utility functions connecting un with u*(so 1 1 : k : : v =u*). By construction, {v ( fn(u ))=p and E(v )/C for all :.Letv be k k : the population u with u n replaced by v . Observe that Lemma 2 implies p= p( f, v:) for each : # [0, 1]. Further, by anonymity,

1 1 1 pf1(v )=}}}=pfk(v )=pfn(v )=p|Ä .

Hence feasibility implies there must be at least one individual (say 1 1 i=k+1) with pfi (v )>p|Ä ,orpfi (v )=p|Ä for all i. In the first event, let k+1 k k 1 1 u =(u1 *, ..., uk+1*, uk+2, ..., u n&1, vn). Since fk+1(v )#C, as above one easily verifies that uk+1 satisfies the inductive hypothesis. In the second 0 event, there must be some individual i>k, say i=k+1, with pfi (v )<$ 1 while pfi (v )=p|Ä ; so the utility continuity of the mechanism implies that ; k+1 for some : # (0,1), say :=;, $pfk+1(v )p|Ä for all ik (by k feasibility), while pfn(u )<$. Without loss of generality, suppose each EFFICIENT IC ECONOMIES 211 individual's preferences in uk are in money metric form relative to p. Again let v: be Proposition 1's family of money metric utility functions connecting k : 1 un with u* (hence, by construction, E(v )/C for all : and v =u*), and let : : : k k : v #(v1 , ..., vn ) be the population u with un replaced by v . Lemma 2 : 1 implies pfn(v )<$ for all :; so annonymity implies pfi (v )<$ for all ik, 0 while pfi (v )>p|Ä for these individuals. We first show that individual n cannot change prices, at least for announcements : which are sufficiently small. Let x:= f(v:). Since preferences are in money metric form, : : : : : i vi (xi )i pxi , with equality only if p= p( f, v ). So, if p{ p( f, v ), efficiency implies there can exist no feasible allocation y: such that (1) : : : : yi # ( p, vi , pyi )for all i, hence i vi ( yi )=p| (maximum achievable : : : : utility under the money metric representation), (2) vi ( yi )=vi (xi ) for all : : : : : i>k,and(3)y1 =}}}=yk *, hence vi ( yi )>vi (xi ) for all ik (using : : (1) and (2)). It follows that, as long as vi (xi )2$ for all i>k, p( f, v:)=p. [As long as the inequality holds, one can easily construct a feasible allocation y: having the above three properties. The construction is similar to, although simpler than, the one we use to prove Lemma 6 0 1 below.] Since pfi (v )p|Ä while pfi (v )<$ for all ik, feasibility implies that as : increases some individual i>k must achieve wealth beyond $.In particular, the utility continuity of the mechanism implies that for some : ; ; # [0,1], say :=;, pfi (vi )<2$ for all i>k and $pfi (vi )<2$ for at k+1 k least one individual i>k, say i=k+1. Let u =(u1 *, ..., uk+1*, uk+2, ..., k ; ; k+1 un&1, vn ). Since p= p( f, v ), as above one easily verifies that u satisfies the inductive hypothesis. K

A.2. Proof of Theorem 2 In preparation for Theorem 2, we first demonstrate that for any population u having p as its efficiency price vector at some allocation x, there exists another population v that is first order similar to u at x and p is the efficiency price vector for all efficient allocations in v.

Lemma 6. Let x=(x1 , ..., xn)>>0 be an efficient allocation for the pop- n ulation u=(u1 , ..., un)#U , and let p be the efficiency prices. Then there is n a population v=(v1 , ..., vn)#U such that the following exist:

(1) v is first order similar to u. In particular, for each vi , vi ( y)=vi (xi) 1 1 and {vi ( y)=p iff ui ( y)=ui (xi) and {ui ( y)=p. If vi ( y)=vi (xi) but 1 13 {vi ( y){p, then ui ( y)

13 So vi 's indifference curve throuth xi coincides with ui 's indifference curve throuth xi only on the latter's flat segment around xi (if any); otherwise, the indifference curve of ui through xi is nested inside that of vi . 212 MAKOWSKI, OSTROY, AND SEGAL

(2) x is an efficient allocation for v. (3) Every efficient allocation for v has efficiency prices p.

l Proof.LetC be a closed convex cone in R++ _ [0] that is sufficiently large so that for each individual i, xi+*| # C for any *0 and, for some =>0,

(:) xi&=| # C for all i.

(Such a cone exists since x>>0.) Let d==Â2n.

The following regions are used to define vi .Let

Ti=[x: pxp(xi+d|)] & C,

Mi=[x: p(xi&d|)

Bi=[x: pxp(xi&d|)] & C.

On the set Ti _ Bi , let vi (x)=px. In the corridor Mi , let vi (x) be quasicon- 1 cave and smooth, with the constraints that {vi (x)=p for all x on the cord[xi&d|, xi+d|] and the curvature of vi around xi satisfies (1). On the rest of the domain (i.e., outside the cone C), extend vi so that it is in U (in particular, so that it satisfies the boundary condition). By construction, v # Un and satisfies condition (1). Condition (2) is also 1 satisfied since each {vi (xi)=p;sox is an efficient allocation for v. The remainder of the proof is to show that v satisfies (3). Without loss of generality, henceforth suppose each vi # Up . It follows that, for any feasible allocation y, i vi ( yi)i pyi=p|; with strict inequality if any individual's indifference curve through yi does not have p for its support at yi . Hence, to prove (3), it will suffice to show that the maximum conceivable aggregate utility, p|, is achieved at any point on the economy's utility n possibility frontier. That is, for any U # R+ such that i Ui=p|, there is a feasible allocation y such that vi ( yi)=Ui for all i.

Consider any nonnegative U such that i Ui=p|.Let*i be implicitly defined by

p(xi+*i |)=Ui .

Since pi xi=p|=i Ui ,

: *i=0. i EFFICIENT IC ECONOMIES 213

Let the set T consist of all individuals i such that xi+*i | # Ti (``T'' for top); similarly let the set M consist of all i such that xi+*i | # Mi , and let the set B consist of all other individuals. If xi+*i | # C, let * i=*i (e.g., for i in T or in M); otherwise let * i satisfy

xi+* i | # bdry C.

Finally let

*i$=&*i+* i , which, by construction, must be nonnegative.

If *$=0i for all i, then letting yi=xi+*i | for each i, we are done. B So assume the contrary, that *$>0i for some i # B. Define * =i # B *i , T M B M T similarly define * and * , and let *$=i *$. Since 0=* +* +* = (* B&*$)+*M+*T, it follows that

*T&*$=&* B&*M>=Â2

(where the inequality follows from the fact that &* B= and *M

So there exist weights * j such that

: (*j&* j)=*$ j # T and *j&* j>d=(=Â2n) for all j # T (so each j in T remains outside his corridor Mj even after losing some goods in the given proportion). That is, there is a feasible allocation y^ such that y^ i=xi+*i | for all i # M, y^ i= xi+* i | for all i # B, y^ i=xi+* i | for all i # T.

By construction, i vi ( y^ i)=p|,soy^ is efficient. But vi ( y^ i)>Ui for those i # B with xi+*i | Â C, while vj ( y^ j)

(1) For any u # D and any i for whom ui (gi (u)){ui ( fi (u)),

p(g, u)}Ri (g, u&i)p(g, u)}gi (u).

[The proof is entirely analogous to that of Lemma 2. Since ( f, D) is first order and ui (gi (u)){ui ( fi (u)), one can find utility functions u and v in D with the required properties. In particular, one can ensure that u and v have the necessary slopes and curvature around gi (u), while keeping their supports equal to p( f, u)atfi (u) (ensuring they are in D).] We also have (2) For any u # D such that p(g, u)=p( f, u)#p,

(g(u), p)#WE(u).

[The proof is entirely analogous to that of Lemma 3, with the inductive k hypothesis being modified to say ``pgi (u ){p|Ä for i=1, ..., k'' rather than k ``pfi (u ){p|Ä for i=1, ..., k.'' The only qualification is that one must strengthen the inductive hypothesis in the proof to also include the requirement that ``uk is first order similar to u at f(u)''; as it turns out, the inductive proof goes through without modification even with this strengthening. (The role of the strengthening is to ensure that the various perturbations involved in the proof never take one outside of D, e.g., it ensures that k (u , uk+1*) is in D.) In the modified proof, (1) above is invoked instead of Lemma 2; notice if pgi (u){p|Ä then ui (gi (u)){ui ( fi (u)), so (1) applies.] Let u # D, let x= f(u), and let p= p( f, u) (the efficiency prices at f(u)). In view of (2), it will suffice to show p(g, u)=p. Define v as in Lemma 6, and let

k v =(u1 , ..., uk , vk+1, ..., vn)

0 n 1 for k=0, 1, ..., n,sov =v and v =u. Since for every i, {vi (xi)=p,it follows from the hypothesis that ( f, D) is first-order that vk # D for every k. Consider the inductive hypothesis ``For all populations vk$ such that k$k, p(g, vk$)=p.'' By Lemma 6, the hypothesis holds for k=0. Suppose it holds for an arbitrary k; we show it also holds for k+1. Let x$=g(vk) and let y= g(uk+1). By hypothesis, p(g, vk)=p. Hence, k 1 using (2), we have (x$, p)#WE(v )and(x, p)#WE(u). So {vj (x$)=j p and 1 1 1 {uj (xj)=p. Further, the construction of vj ensures {vj (x$)=j {uj (xj$), hence 1 1 {uj (xj$)={uj (xj).

IC at vk+1 implies

uj ( yj)uj (xj$) EFFICIENT IC ECONOMIES 215

k (otherwise j would claim to be of type vj). Similarly, IC at v implies

vj (xj$)vj ( yj).

The construction of vj implies that if the first IC constraint were not binding then vj ( yj)>vj (xj$), violating the second IC constraint. It also implies that if the first IC constraint were binding but uj's MRS's were different at yj and xj$, then again vj ( yj)>vj (xj$) violating the second IC constraint. We conclude that the first IC constraint must be binding and uj's MRS's must 1 1 be the same at yj and at xj$, i.e., {uj ( yj)={uj (xj$). Above we found 1 1 {uj (xj$)={uj (xj). Combining shows 1 1 {uj ( yj)={uj (xj).

j 1 1 But efficiency implies that p(g, u )={uj ( yj) and p={uj (xj). Hence p(g, u j)=p. That is, the inductive hypothesis holds, and hence p(g, u)=p. K

APPENDIX B: PROOF OF THEOREM 3

Analogous to the proof of Theorem 1 for ordinal economies, the proof of Theorem 3 involves three steps. However, Fact 1 in Section 4, embodied in (F.1AC), will allow us to verify the conclusions of Lemmas 2 and 3 (i.e., Steps 1 and 2 for ordinal economies) without assuming the continuity of f or the smoothness of preferences. Hence the dispensability of these assumptions in the TU setting. We first give the TU analogs for Lemmas 2 and 3 (Lemmas 7 and 8 below). p Let ui be the (linear) utility function defined by

p ui (x)=px for all x # 0.

Lemma p 7. If ( f, D(u&i ,[ui , ui ])) is IC, where p= p(u), then

pRi (u&i)pfi (u).

Proof.Letx= f(u) and let B=[|]&j{i A(xj , uj). By definition,

p|p : A(xj , uj). j

Hence pBpA(xi , ui) and, in particular, pBpxi . Thus, (F.1C) implies p pfi (u&i , ui )=pxi .Soifpy>pxi for some y # Ri (u&i), then p K pfi (u&i , ui )>pxi , a contradiction. 216 MAKOWSKI, OSTROY, AND SEGAL

We now display the analog of Lemma 3. The proof is similar to that of Lemma 3A, but without the smoothness or boundary assumptions.

Lemma 8. If ( f, D) is first order, then IC implies that for all u # D,

( f(u), p(u)) # WE(u).

Proof.Letx= f(u), p= p(u), and B=[|]&j{i A(xj , uj). Since p supports x at u, pxipA(xi , ui) for all i. Hence, we need only verify that pxi= p|Ä for all i. Assume the contrary. Then there is an individual, say i=1, with 1 p 1 px1>p|Ä .Letu =(u1 , u2 , ..., un). Lemma 7 implies pf1(u )=pf1(u)>p|Ä . 1 1 1 p 1 Further, p supports x #f(u )atu . To verify observe that u1(x1)= p 1 u1(x1)=px1 ; and, by (F.1A), j{i A(xj , uj)=j{i A(xj , uj). Thus, since 1 j xj =|, p|pj A(xj , uj) implies

1 1 p : xj p : A(xj , uj). j{i j{i And trivially,

1 1 p px1pA(x1 , u ).

Summing shows

1 1 1 p : xj = p|p : A(xj , uj ). j j

1 Hence, there must be an individual, say i=2, with px2p|Ä . Repeat the above procedure n&2 more times, to form u3, ..., un. One thus finds, depending on whether n is odd or even, that n n n pxi >p|Ä for all i or pxi

p p Let v (x)#v~(x1 , ..., xl&1)+xl be any continuously differentiable function in U satisfying v~p is strictly quasiconcave and {v p(|Ä )=p.

Lemma 9. If ( f, D) is first order and D is comprehensive around u, then IC implies that for all i,

|Ä # Ri (u&i). EFFICIENT IC ECONOMIES 217

Proof. See the proof of Lemma 4 for ordinal economies. K

Lemma 10. If ( f, D) is first order, D is comprehensive around u, and u is quasiregular, then IC implies that for all i and all vi # U,

p(u&i , vi)=p(u). Proof. See the proof of Lemma 5 for ordinal economies. K Proof of Theorem 3. Proceed as in the proof of Theorem 1 for ordinal economies. K As noted in Section 4, while smoothness plays no explicit role in the proof of Lemma 10, it is sufficient (although not necessary) for the quasiregularity of u. Proof of Fact 2. Suppose both (x, p) and (y, q) are equal-wealth

Walrasian equilibria for u&i when its aggregate endowment equals |&|Ä . Let |*=|&|Ä . Then by assumption

p|*p : A(xj , uj) j{i

q|*q : A( yj , uj)=q : A(xj , uj), j{i j{i where the last equality follows for the quasilinearity of preferences (i.e., the vertical parallelism of indifference curves). Thus, since j{i xj=|*, by a familiar argument:

pxj pA(xj , uj) for each j{i (1)

qxj qA(xj , uj) for each j{i.(2)

Suppose p{q. Hence for some commodity h (h{l), ph{qh . But since |h*>0, there must be an individual j such that xjh>0. Eq. (1) implies that for this individual u~j (xj)Âxjh= ph , while Eq. (2) implies u~j (xj)Âxjh=qh , a contradiction. K

APPENDIX C: PROOFS OF LARGE-ECONOMY RESULTS IN SECTION 6

C.1. Proof of Theorem 4 We first show that efficiency plus incentive compatibility implies no transfers on a very restricted domain consisting of all convex combinations 218 MAKOWSKI, OSTROY, AND SEGAL of two reference utility functions, u # U2 and v # U2 , where U2 is the subset * of U consisting of strictly concave, C 2 functions and U2 is the subset of U2 * consisting of functions u with negative definite Hessians 2u(x) for all l x # R++ .Let

[u, v]=[va : va(})=au(})+(1&a) v(}):a #[0,1]], the set of convex combinations of u and v.

Recall that ( p, va , w) is the Walrasian demand of a person with utility va facing prices p and wealth w. Suppose that f is price continuous at + with respect to p, where p is a vector of efficiency prices supporting f(+,}).

We now show the key result that when f(+, u)#( p, va , pf(+, va) for all a #[0,1],then

pf(+, va)=mÄ for all a #[0,1].

The result is then easily extended to the much larger domain U.

To simplify notation, for any b, a # [0, 1] let x(b)= f(+, vb) and let

U(b, a)=va(x(b)), the utility ``a'' receives when she announces that she is ``b''; hence, U(a, a)=va(x(a)) is a's utility when she announces truthfully, where x(a)= f(+, va). With this notation,

Fact. If f is price continuous at + with respect to p, then

(I) x(a)#( p, va , px(a)) for all a #[0,1].

Further, if f is incentive compatible at +, then

(II) U(a, a)U(b, a) for all a, b #[0,1].

(I) was established in the text. To verify (II), consider any two types Ä va , vb #[u, v] (not necessarily in supp +). Pick any sequence +k + with

+k(va)>0 and +k(vb)>0. Since both functions are strictly concave, price- continuity implies f(+k , va) converges to f(+, va) and f(+k , vb) converges to f(+, vb); and incentive compatibility implies, for each k, va[ f(+k , va)] va[ f(+k , vb)]. Hence the continuity of va implies va[ f(+, va)]va[ f(+, vb)]. Let X=[x(a): a #[0,1]].

Lemma 11 (Boundedness). X is bounded and its closure is contained in 0. Proof. Since the efficiency price p is fixed in the following, let

.(a, w)=( p, va , w). If X were not bounded then there would be a EFFICIENT IC ECONOMIES 219 sequence in [0, 1], an Ä a0 such that &xn& Ä , where xn#x(an); hence, wn#pxn Ä . Notice that for all n sufficiently large,

n n n n n 0 n 0 U(a , a )=van[.(a , w )]>van[.(a , w +1)]>van[.(a , w )]. Hence, letting n Ä and using the fact that .[},w] is continuous in a,

n n 0 0 0 0 lim inf U(a , a )>va0[.(a , w +1)]>U(a , a ). That is, there is an =>0 such that

lim inf [anu(an)+(1&an) v(an)]a0u(a0)+(1&a0) v(a0)+=.

So there is a sufficiently large n such that

a0u(an)+(1&a0) v(an)>a0u(a0)+(1&a0) v(a0), contradicting incentive compatibility. On the other hand, if the closure of X is not contained in 0 then there exists a sequence an Ä a0 with xn Ä x0 Â 0. Notice the boundary condition implies u(xn) Ä inf u(0)andv(xn) Ä inf v(0). So, for a sufficiently large n,

U(a0, an)>anu(xn)+(1&an) v(xn)#U(an, an), again contradicting incentive compatibility. K At this point, it will be useful to re-represent preferences using the money-metric utility function introduced in Section 6.4. Let va(} | p) be the money metric representation of va . That is, va(x | p)=py(x, a), where y(x, a) solves

min py s.t. va( y)va(x). y0

2 It is straightforward to verify that each va #au+(1&a) v # U . Indeed, 2 2 2 since u(x) is negative semi-definite while v(x) is negative definite, va is negative definite on 0 for any a # (0, 1). Thus, a routine application of the implicit function theorem shows that y(}, }) is C 1 on 0_(0, 1) (e.g., follow the proof of Proposition 2.7.2 in Mas-Colell [18]).

Let u(b, a)=va(x(b)|p). Define the wealth function w(b)=px(b) and the loss function L(b, a)=w(b)&u(b, a). Notice that, because of the money metric representation, L(b, a)0 for all b, a # [0, 1]; while efficiency implies L(a, a)=0. Hence, the efficiency condition (I) of Fact 3 may be restated:

a # arg max &L(b, a), for all a #[0,1]. (3) b #[0,1] 220 MAKOWSKI, OSTROY, AND SEGAL

On the other hand, since u(b, a)=w(b)&L(b, a), the incentive compatibility condition (II) above may be restated,

a # arg max w(b)&L(b, a), for all a # [0, 1]. (4) b #[0,1]

We want to show that these two conditions are compatible if and only if w(a) is constant on [0, 1]. To see the intuition, suppose L and w were differentiable. Then the first-order conditions for efficiency and incentive compatibility would be, respectively:

&D1 L(b, a)=0 at b=a, for all a #[0,1],

w$(b)&D1 L(b, a)=0 at b=a, for all a #[0,1].

Combining would yield w$(a)=0 for all a # [0, 1]; hence w is constant on [0, 1]. The proof below involves establishing there is enough differentiability to essentially mimic this heuristic argument. (It is instructive to see how similar this argument is to Holmstro m's [12, p. 1139] heuristic argument for the uniqueness of the Groves mechanism.) Let I be any closed subset of (0, 1), and let X (I) denote the closure of [x(a): a # I].

Lemma 12. The wealth function w(a) is Lipschitz on I.

Proof. First we show that

L(b, a) K< for all a, b # I.(5) } a }

Recall L(b, a)#px(b)&u(b, a)=px(b)&py(b, a). Hence,

L(b, a) y(b, a) = p . a a

Since y(b, a)Âa#y(x(b), a)Âa, and since the latter is a continuous function on R++ _(0, 1) while X (I)_I is a compact subset of its domain, all the partial derivatives on the RHS are bounded on I_I. This establishes (5).14

14 Conditions (3)(5) immediately imply our desired conclusionthat w(a) is constant on I (Lemma 13, below)via Holmstro m's main Lemma. However, we prefer to give a self-contained proof rather than appeal to Holmstro m's Lemma at this point. EFFICIENT IC ECONOMIES 221

Now suppose w(b)w(a), where a, b # I. Then

0w(b)&w(a)=L(b, a)&L(b, a)+w(b)&w(a) =L(b, a)+u(b, a)&u(a, a) L(b, a)=L(b, a)&L(b, b) K |b&a|, where the third line follows from incentive compatibility (recall L(b, b)=0); and the fourth follows from Eq. (9). So w is Lipschitz on I. K

Lemma 13. The wealth function w is constant on I. Proof. Since w is Lipschitz on I, it is differentiable a.e. on I. Recall .(a, w) denotes type a's Walrasian demand when facing prices p and wealth w. It is known that . is continuously differentiable on (0, 1)_(0, ) (e.g., see Mas-Colell [11, Proposition 2.7.2]. Since x(a)=( p, va , w(a)), it follows that x(a) is differentiable at any point where w is differentiable, that is, a.e. on I. Further, since

L(b, a)=w(b)&py[x(b), a],

D1 L(b, a) exists at any point b where w is differentiable. Hence recalling Eq. (3), at any differentiable point a, efficiency implies the first-order condition

D1 L(a, a)=0; similarly, recalling Eq. (4), incentive compatibility at a implies

w$(a)&D1 L(a, a)=0; hence w$(a)=0. Since w$=0 a.e. on I and w is Lipschitz on I, w(a) is constant on I. K The following lemma extends the conclusion of Lemma 13 to include all a # [0, 1], not just a # (0, 1). It also is the key to extending the conclusion to the much larger domain U (see the proof of Theorem 4 below). Let un Ä u mean un converges to u uniformly on compacta.

Lemma 14. Suppose u, un # U, un Ä u, and px(un)=mÄ , a positive constant, for all n. Then px(u)=mÄ . Proof. Since px(un)=mÄ for all n, there is a subsequence, say uk,andan x such that f(+, uk) Ä x and px=mÄ . Since demand is continuous, x is in u's 222 MAKOWSKI, OSTROY, AND SEGAL demand correspondence when facing prices p and wealth mÄ . Hence, if w(u)mÄ then u( f(+, u))>u(x); hence, for a sufficiently large k, uk( f(+, u))>uk(xk), again contradicting incentive compatibility. K

Proof of Theorem 4. We first show that pf(+, u)=mÄ , a constant, on U2. Fix any v # U2 and pick any u # U2. By Lemmas 1314, pf(+, v)=pf(+, u). * Since this holds for any u # U2, we are done. Now we extend the conclusion to all of U. Pick any u # U. There exists a sequence un in U2 s.t. u^ n Ä u^ , where u^ n is some (perhaps different) utility function representing the same preferences as un, and similarly u^ is some utility function representing the same preferences as u. (See Mas-Colell [18, Section 2.8.] By Lemmas 1314, pf(+, un)=pf(+, v)=mÄ for all n; hence pf(+, u^ n)=pf(+, un)=mÄ for all n. Lemma 14 now implies pf(+, u^ )=mÄ ; hence pf(+, u)=pf(+, u^ )=mÄ , as required. To complete the proof we need only show mÄ = p|. But feasibility implies f(+, u) d+(u)=|, hence pf(+, u) d+(u)=mÄd+= p|, hence mÄ = p|,as required. K

C.2. Proofs of Theorem 5 and Propositions 2 and 3

Proof of Theorem 5. First, we verify that M is a complete metric space by appeal to various results in Mas-Colell [18]. The set of continuous l 0 l functions on R++ , denoted C (R++), is a complete metric space. (See K.1.2, p. 50.). Further the subset consisting of all quasiconcave functions, 0 l denoted QC (R++), is closed and therefore also complete. Recall U is the 0 l subset of QC (R++) consisting of all strictly increasing functions satisfying the boundary condition. U can be shown to be a G$ subset and, therefore, topologically equivalent to a complete metric space. (See A.3.4, p. 10, and Proposition 2.4.5, p. 72.) Finally, the set of probability measures M on the complete metric space U is a complete metric space. (See E.3.1, p. 24.) Next we show that the Walrasian price correspondence 6 from M to S is continuous on a dense G$ set. It is well-known that 6 is upper hemicontinuous. A Theorem of Fort (see Hildenbrand [11, p. 31]) says that for any =>0, the =-continuity points of an upper hemicontinuous correspondence from a metric space into a totally bounded space are open and dense. S is evidently totally bounded. Let M1Ân be the open and dense set of points with ``jumps'' in the correspondence of at most 1Ân. Because

n M1Ân are the continuity points of the correspondence, they form a G$ set. Further, because M is complete, by the Baire Category Theorem the intersection of the countable collection of open and dense sets M1Ân is itself dense. EFFICIENT IC ECONOMIES 223

To conclude, we show that there exists a selection ? from 6 that is continuous on a dense G$ set. Define a partial order on u.h.c. correspondences from M to S by

61 o62 iff 6(+)$61(+)$62(+) for all + # M, with 61(+)#62(+) for some +. Let (6i)i # I be any chain ordered by o, and let 6 be defined by

6 (+)= & i # I 6i (+). Observe that 6 is a lower bound to the chain: Since Ä for every i, 6i (+) is compact, 6 (+){<. Also, if +n + and pn # 6i (+n) Ä such that pn p, then p # 6(+). The reason is that for every i, pn # 6i (+n). By u.h.c of 6i , p # 6i (+); hence, p # 6 (+). So 6 is u.h.c. It now follows from Zorn's Lemma that there exists a minimal u.h.c correspondence, say 15 6*. Further, by the above 6* is continuous on a dense G$ set, say M*.

We now show that 6* is single-valued on M*. Let +0 # M* and suppose M*(+0) is not a singleton. Let p1 , p2 # 6*(+0). Let $>0 such that 1 16 d(+, +0)<$ implies d(6*(+), 6*(+0))<3 d( p1 , p2). (Recall that 6*is continuous at +0 .) Define 6$by6$(+)=6*(+)ifd(+, +0)$. Otherwise, 1 6$(+)=[p# 6*(+):d( p, p2)3 d( p1 , p2)]. 6$(+) is not empty because 1 there are points in 6*(+) that are within 3 d( p1 , p2) distance from p1 .It is easy to verify that 6$ is u.h.c. By construction 6*o6$, contradicting the fact that 6* is minimal. To conclude the proof, define the selection ? out of 6* (and hence out of 6) in the following way. For + # M* let ?(+)=6*(+). For + Â M*, pick any element of 6*(+). Since 6* is continuous at each + # M*, it follows Ä Ä that +n + where + # M* implies 6*(+n) 6*(+). In particular, Ä K ?(+n) ?(+).

Ä Proof of Proposition 2. Let +n +. We need to show that for any =>0 there is an N such that n>N implies 2?(+n)<=. Let B(+, $) denote a ball centered at + with diameter $. The continuity of ? implies there is a $>0 such that +$#B(+, $) implies &?(+$)&?(+)&<=.

Let N be sufficiently large that N>2Â$ and \(+n , +)<$Â2 for all n>N.

Then for any n>N and any perturbation 2u # J(+n), \(+n , +n+2u)= 1Ân<$Â2. Hence, for any n>N, [+n+2u : 2u # J(+n)]/B(+n , $Â2)/ B(+, $). K

Proof of Proposition 3. Since q # S, q|=1. Thus any x # (q, v, q|)is contained in the set

l &1 &1 &1 Bq=[x$#R+ : qx$=1]=convex hull [q1 , q2 , ..., ql ].

15 Appeal to Zorn's Lemma is used by Mas-Colell (5.8.18, p. 234) in establishing the existence of a continuous selection on the space of regular economies. 16 There are three different metrics in this statement. The first is on M, the second is the Hausdorff metric on subsets of S, and the third is on S. 224 MAKOWSKI, OSTROY, AND SEGAL

Therefore,

u(x| p):=min[py: u( y)u(x)]max [px$: x$#Bq] p =max c c qc p

The next to last inequality follows from fact that &q& p&

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