Journal of Economic Theory 85, 169225 (1999)
Article ID jeth.1998.2494, available online at http:www.idealibrary.com on
Efficient Incentive Compatible Economies
Are Perfectly Competitive*
Louis Makowski
Department of Economics, University of California, Davis, California 95616
lmakowskiucdavis.edu
Joseph M. Ostroy
Department of Economics, University of California, Los Angeles, California 90095
ostroyecon.ucla.edu
and
Uzi Segal
Department of Economics, University of Western Ontario, London, Ontario, Canada N6A 5C2
segalsscl.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
0022-053199 30.00
Copyright 1999 by Academic Press
All rights of reproduction in any form reserved.
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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
monopolyprice-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
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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 nonvacuous 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
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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], Holmstrom [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 Barbera 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], MasColell [19], Nehring [21], and others have shown that the only efficient
no-envyincentive-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.
Barbera 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
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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 equalwealth 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 settings.
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 upperhemicontinuous 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/R l. U denotes the set of admissible
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MAKOWSKI, OSTROY, AND SEGAL
utility functions u for each individual, where u: 0 Ä R. A population is a
vector of utility functions u=(u 1 , ..., u i , ..., u n ). The economy's aggregate
endowment is | # R l+ and there is no production; so the attainable allocations for any population u is given by
{
=
X= x=(x 1 , ..., x i , ...x n ) # 0 n : : x i =| .
i
Let DU n, with typical elements u, v, and w. A mechanism 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, u i (x)u i ( y) iff v i (x)v i ( 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
u i (x i )u i ( f i (u)) for each i and u i (x i )>u i ( f i (u)) for some i.
Anonymity.
For all u # D, u i =u j implies u i ( f i (u))=u j ( f j (u)).
For any population u, let u &i denote the population with the utility function of individual i deleted; and let (u &i , v i ) denote the same population
as u except the utility function of individual i is replaced by v i . The
mechanism ( f, D) is incentive compatible (IC) at u # D if, for each i,
u i ( f i (u))u i ( f i (u &i , v i ))
for all (u &i , v i ) # D.
The mechanism ( f, D) is incentive 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
individual with preferences u. A price vector is a p # R l++ that is normalized
so that p|=1 in the ordinal model, or is normalized so that p l =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) at x 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) at x. 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, x i # ( p, u i , p| ). A mechanism ( f, D) is an equal-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
assume that | # 0=R l++ , and U consists of all u on 0 satisfying both the
maintined assumptions and the boundary condition:
for all x # R l++ , [ y: u( y)u(x)] is closed in R l.
(See Mas-Colell [18, p. 68].) Denote this set Ub(R l++ ).
In the finite ordinal model, we assume | # R l++ and 0=R l++ _ [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)
support {1 u(x) for each x>
>0. Denote this set Ub, us(R l++ _ [0]). [If x=0,
we adopt the convention that 1 u(x) includes all normalized prices.]
The quasilinear model is distinguished by the fact that 0=R l&1
+ _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
(see Section 4). We denote this set by Uql (R l&1
+ _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 u i ( f i (u)) be the set of all normalized price vectors that
support f (u) at u. 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.
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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, f i (u) # ( p, u i , pf i (u)). In the finite ordinal
model, since supports are unique, there is only one normalized price
p( f, u).
3. FINITE ORDINAL ECONOMIES
Throughout this section we assume U=Ub, us(R l++ _ [0]). We also will
assume that the mechanism is continuous, at least in utilities. More
precisely, let u ki Ä u i mean the sequence of utility functions [u ki ] k=1, 2, ...
converges to u i uniformly on compacta. It is straightforward to verify that
any incentive compatible mechanism f will be continuous in the following
limited sense: Suppose u, u k # D, where u k =(u &i , u ki ). Then u ki Ä u i implies
u ki ( f i (u k )) Ä u i ( f i (u)). 4 We will make a stronger assumption.
Utility Continuity. Suppose u, u k # D, where u k =(u &i , u ki ). Then
u Ä u i implies u ki ( f i (u k )) Ä u i ( f i (u)) and u j ( f j (u k )) Ä u j ( f j (u)) for all
individuals j{i.
k
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=U n, 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 U n no IC mechanism exists (see also Barbera 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
Here is a sketch of the proof. Let x k = f i (u k ) and x= f i (u). Suppose u ki (x k ) Ä
% u i (x). Then
on a subsequence u ki(x k ) Ä a, where a{u i (x). Suppose a<u i (x). Then for some sufficiently large k, u ki(x)>u ki(x k ). This contradicts incentive compatibility, because consumer i
will be better off if he claims he is of type u i rather than u ki . Similarly, a>u i (x) leads to a
contradiction.
EFFICIENT IC ECONOMIES
179
The richness condition has two parts. For the first, call the utility function v i # U first order similar to u i at x i # R l++ if
{1 v i (x i )={1 u i (x i ).
That is, v i and u i have the same MRS's at x i . [If x i =0, call all v i # U
first order similar to u i at x i .] More generally, the population
v=(v 1 , ..., v i , ..., v n ) # U n is first order similar to the population u at x if, for
every i, v i is first order similar to u i at x i .
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 # U n that are first
order similar to the population u at f (u). The mechanism ( f, D) is first
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=U n is evidently first order; but fortunately for
the possibility of the non-vacuous characterization below, ( f, D) can be
first order even when D{U n. 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) to all 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 f i (u) is in i's Walrasian
demand correspondence at prices p( f, u) not only if i has utility function u i ,
but also if i has any utility function v i that is first order similar to u i at f i (u)
(recall i's MRS's are the same at f i (u) under both u i and v i ).
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 (IC O Walrasian).
for all u # D,
If ( f, D) is first order, then IC implies that
( 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
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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 , v i ): v i # 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, then D=U n; 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) is perfectly competitive around u if there is a p such
that for every individual i and every v i # U,
( f (u &i , v i ), p) # WE(u &i , v i ).
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 |&|. We call the economy u quasiregular if,
for each i, u &i is pseudoregular. 5
Theorem 1 (IC O 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
R i ( f, u &i )=[ f i (u &i , v i ): (u &i , v i ) # D].
Thus R i ( 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 , v i ) if and only if it assigns i his v i -best point in the set
R i ( f, u &i ); that is, it will be incentive compatible if and only if
f i (u &i , v i ) # arg max v i (x)
s.t. x # R i ( f, u &i ).
The idea is simple: If there exists y i = f i (u &i , w i ) such that v i ( f i (u &i , v i ))<
v i ( y i ), then it would not be incentive compatible for i to truthfully
announce v i .
Let H p denote the hyperplane through the origin with normal p, i.e.,
H p :=[ 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 |.
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,
R i ( f, u &i )/H p +[| ].
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  D$. Notice, however, that if the
5
The definition of regular economies (Debreu [6]) presumes that utility functions are
strictly quasiconcave and C 2, whereas we assume only that they are quasiconcave and C 0 with
unique supports. Regular economies have a finite number of equilibria; moreover, within
the class of such utility functions, regular economies are typical (e.g., see Mas-Colell [18,
Chapter 5]).
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MAKOWSKI, OSTROY, AND SEGAL
domain D$ 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( f j (u), u j ) denote the aggregate at-least-as-good-as-f (u)
set for individuals other than i. If i 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 u j ( f j (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,
f i (u) from i's perspective equals |& f i (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 (IC O no externalities). Let ( f, D) and u satisfy the
hypotheses of Theorem 1. Then the mechanism is IC only if for each i and
v i # U,
f i (u &i , v i ) # arg max v i (x)
x#0
s.t. x # [|]&A &i ( f, u),
and for each j{i,
u j ( f j (u &i , v i ))=u j ( f j (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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MAKOWSKI, OSTROY, AND SEGAL
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 accommodated 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 U n follows readily from
Theorem 1 since, in most populations u # U n, 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 , v i ). 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 u i in u is strictly quasiconcave. Then there exists
v=(u &i , v i ) 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 willing to accommodate him at the original prices. This will be contrasted
7
The impossibility implications of Barbera 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
|&|nr|. 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
Throughout this section U=Uql (R l&1
+ _R), the set of all continuous,
strictly monotonic, quasiconcave functions with the standard quasilinear
form
u(x 1 , ..., x l )=u~(x 1 , ..., x l&1 )+x l .
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., budgetbalancing) 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 # U n having p( f, u) #
1 v i ( f i (u)) for all i.
For any population u, the gains from trade in u is given by
G(u)=max : u i (x i )
i
s.t. x # X.
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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 convex combinations of the utility functions u and v. 8 Let D(u &i , [u i , v i ])/
[u &i , U] be the set of v in which i's preferences are some convex combination of u i and v i . For mechanisms achieving the maximum gains from
trade, Holmstrom's [12] version of the VCG characterization may be
expressed:
Fact 1. Consider the domain D$=D(u &i , [u i , v i ]), and suppose the
mechanism ( f, D$) is efficient and anonymous on this restricted domain. Then
( f, D$) is IC only if, for all w i # [u i , v i ],
w i ( f i (u &i , w i ))=G(u &i , w i )&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 w i . 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 , w i )=
w i ( f i (u &i , w i ))+ j{i u j ( f j (u &i , w i )). Substituting, this implies that for all
w i # [u i , v i ],
: u j ( f j (u &i , w i ))=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 w i # [u i , v i ]:
(F.1A)
: u j ( f j (u &i , w i ))= : u j ( f j (u))
j=
% i
(F.1B)
j=
% i
w i ( f i (u &i , w i ))=G(u &i , w i )& : u j ( f j (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
Because u and v are quasilinear, each consists of a concave function u~ and v~ on R l&1
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 Holmstrom's domain D(u &i , [u i , v i ]) 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 (IC O 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,
R i ( f, u &i )/H p +[| ].
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 u i 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 # R l: u(x)& pxu( y)& py for all y # 0],
which implies p l =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 u j ( f j (u)). That is, for all w i # [u i , v i ]:
(F.1C)
f i (u &i , w i ) # arg max w i (x)
x#0
s.t. (|&x) # A &i ( f, u).
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 u i 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 R i ( 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.
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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 u j ( f j (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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EFFICIENT IC ECONOMIES
189
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$ in U n and let (x*, p) be any equalwealth Walrasian equilibrium for u$. In an epsilon neighborhood of x i*,
flatten i's indifference curve through x *,
keeping its slope equal to p. Let
i
u=(u 1 , ..., u n ) denote such a perturbed population. So, by construction,
there exists an =>0 such that for each i
y # B(x i*, =) & (H p +[| ]) O u i ( y)=u i (x *).
i
Notice that (x*, p) remains an equilibrium for this local flattening of the
original economy.
Let $ be the diameter of a ball centered at | that includes R l++ &
(H p +[| ]), that is,
$
\ 2+ .
R l++ & (H p +[| ])/B |,
See Figure 3. To ensure that no one will be able to influence prices, assume
n is sufficiently large that
=>
2$
.
n&2
(C)
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)#x i # arg max v i (x)
s.t. pxp|.
x#0
And for each j{i, let
f j (v)#x j =x j*&
z
,
n&1
where z=x i &x i*. Since z # B(0, $2), &z&<$. So the inequality (C) implies
z
$
=
"n&1"<n&1<2 .
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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.
That is, for all individuals {1 v i (x i )={1 u j (x j )={1 u j (x j*)= p, so (x, p) #
WE(v).
Let Ds be the set of all populations w  Dc 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 firstorder. 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
U n 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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EFFICIENT IC ECONOMIES
191
economies with a continuum of agents and smooth preferenceswithout
the need for any further ``flattening'' of preferences: generically, in the continuum each infinitesimal individual does face a linear opportunity set,
hence each infinitesimal individual can fully appropriate his social contribution as illustrated in Figure 1. We show this in the next section.
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 section U=Ub(R l++ ), the set of continuous, strictly monotonic, quasiconcave
utility functions on 0=R l++ 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
is important to observe that in this topology + k Ä + need not imply that
supp + k converges in the Hausdorff metric to supp +, although there will
exist sets Q k /supp + k such that Q k Ä supp + and + k(Q k ) Ä +(U). The
elements of supp + k not in Q k 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
feasibility constraint f (+, u) d+(u)=|, where | # R l++ 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 U n; (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
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MAKOWSKI, OSTROY, AND SEGAL
f(+, } ). More precisely, let S=[ p # R l++ : 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
+ k Ä + and any v # supp + k , there exist efficiency prices p k # P( f, + k ) such
that
pk Ä p
and
p k f (+ k , v) Ä 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  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 pricecontinuity? It easily can be checked that it is more restrictive than utilitycontinuity (i.e., for all + k Ä + and all u # U, u( f (+, u))=lim u( f (+ k , u))),
but it is less restrictive than allocation-continuity (i.e., for all + k Ä + 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 pricecontinuous 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 (IC O 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., p f (+, 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 Holmstrom'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'' Holmstrom'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
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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 MasColell [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
so that + n Ä + 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 M n as the subset of probability measures + n = : m $ um with finite
support, where each : m is an integer multiple of n &1 and $ u is the measure
with unit mass centered at u. M n is interpreted as the set of possible distributions 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 distinguish 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
+ n +2u :=+ n &n &1($ u &$ v ),
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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 , v i )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* # M n a perfectly
competitive population relative to the price selection ? if 2?(+ n*)=0; and let
M n*(?)/M n 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),
\(+ n , + n +2u)=1n. 10
Definition. Given any price selection ? and any sequence of finite
economies + n Ä +, 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
2?(+ n ) Ä 0
as n Ä .
The continuum economy + is a perfectly competitive limiting economy
relative to ? if, for every sequence + n Ä +, 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
Recall if 2u=(u, v) then + n +2u=+ n &n &1($ 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[ p 1 , ..., p l ].
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):
u(x | p)&u( y | p)<
d
.
:&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
any sequence + n Ä +. If ? is continuous at + (a generic property), then
Proposition 2 implies the economies + n are becoming increasingly competitive, i.e., 2?(+ n ) Ä 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 Compatibility 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
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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
because for any + n Ä +, 2?(+ n ) Ä 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-envyIC characterization of
the Walrasian mechanism in continuum models. In this Section, we summarize 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 Section 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 exhibiting 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)<p| for some types u (a Walrasian mechanism with lump
sum transfers), then individual uif he truthfully announced his type
would be creating a positive externality for the rest of the economy, giving
others some of his wealth p|.
By contrast, an equal-wealth Walrasian mechanism in a finite model
does not typically exhibit the same no externalities implication because
individuals typically create externalities through their influence on prices as
well as on others' wealths. To eliminate external effects in the finite model,
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MAKOWSKI, OSTROY, AND SEGAL
it is necessary and sufficient to have the kind of Walrasian equilibrium normally associated with the continuum; namely, one is which individuals are
perfect competitors, which brings us back to Theorems 1 and 3.
Whereas most of the mechanism design literature is naturally enough
concerned with the mechanism itself, for us the question is not just whether
a certain kind of mechanism exists, but why certain kinds of environments
do or do not admit certain kinds of mechanisms. Given (i) the evident conclusion that in perfectly competitive economies a mechanism that is both
efficient and IC exists, and (ii) the demonstrations that in finite economies
a mechanism that is both efficient and IC typically does not exist, we set
out to demonstrate the equivalence between efficient IC mechanisms and
those that are perfectly competitive.
In Section 1 we discussed the possibility of arriving at alternative, quite
different characterizations for efficient IC mechanisms in finite economies
depending on one's choice of a domain, i.e., one's ``domain tailoring.'' To
illustrate, Hurwicz and Walker [14] establish that budget-balance fails to
hold generically for any VCG scheme in finite quasilinear economies having public or private goods. In the course of their demonstrations, they
make the following observation. The failure of IC arises from the existence
of conflicts among consumers. Indeed, once these conflicts disappear, IC
mechanisms are possible. They prove this intuition in the following way.
The set of commodities [1, ..., l] is partitioned into non-empty subsets
I 1 , ..., I n , and each consumer i's utility function is assumed to be sensitive
to changes only in the quantities of commodities in I i . So the set Ui of
utility functions available to consumer i is [u: R l Ä R : c  I i O ux c #0,
and u is not constant]. For a vector of utility functions (u 1 , ..., u n ) #
> ni=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 PCno-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
and incentive compatible on a Cartesian product > ni=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 Barbera 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 fixedproportion 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. Fixedproportions 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.
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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 R i (u &i ) for R i ( f, u &i ), p(u) for p( f, u), etc. The demonstration of
Theorem 1, that R i (u &i ) lies in a common hyperplane, consists of three
steps: (1) The set R i (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 R i (u &i ) is not locally strictly concave (Lemmas 4 and 5).
Step 1. In Fig. 4, point x lies on the concave part of R i (u &i ), but
R i (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 R i (u &i ) at y.
The mechanism must choose one of these two points. Suppose it chooses
x. Let u ki be a sequence of preferences converging to u i , as illustrated.
Incentive compatibility requires the mechanism to choose y for all u ki , but
it chooses x for u i . 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.
R i (u &i ) cannot have a convex-shaped segment as illustrated.
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Lemma 1 (Continuity). Let u k =(u &i , u ki ). Also assume u i is strictly
quasiconcave and f i (u)>
>0. Then u ki Ä u i implies f i (u k ) Ä f i (u).
Proof. Let x= f (u), x k = f (u k ), p= p(u), p k = p(u k ). Suppose x ki Ä
% xi .
Then there would be a convergent subsequence, say s(k), on which
p s(k) Ä p 0, x s(k) Ä x 0, and x 0i {x i . Since each p k supports x k at u k, p 0 supports x 0 at u, i.e.,
n
p 0|p 0 : A j (x 0j , u j ).
j=1
Hence, since j x 0j =|,
p 0x 0j p 0A j (x 0j , u j ) for each individual j.
Further, since continuity implies u j (x kj ) Ä u j (x j ) for each j,
u j (x 0j )=u j (x j ) for each j.
Hence, p 0x 0j p 0x j for each j. And since u i is strictly quasiconcave in
the interior of 0, where, by efficiency, both x i and x 0i reside, p 0x 0i <p 0x i .
Summing shows p 0 j x 0j <p 0 j x j . But feasibility implies j x 0j = j x j ,
a contradiction. K
We now show that if i may have any first-order similar preferences at u
then the convex hull of R i(u &i ), denoted con R i (u &i ), is supported by p(u)
at f i (u). Thus R i (u &i ) cannot have a convex segment as illustrated in
Figure 4.
Lemma 2. If ( f, D) is first order, then IC implies that for any u # D and
>0:
all i with f i (u)>
p(u) } R i (u &i )p(u) } f i (u).
Proof. Let x= f i (u), p= p(u). Let u be any strictly quasiconcave utility
function in U whose indifference curve through x is nested inside u i 's indifference curve through x; so, u i ( y)u i (x) implies u( y)<u(x) unless y=x.
Notice that incentive compatibility implies f i (u &i , u)=x.
Suppose there were a y$ # R i (u &i ) such that py$>px. Then there would
be a strictly quasiconcave utility funtion v # U such that v(x)<v( y$) for
some y$ # R i (u &i ), but the indifference curve of v through x coincides with
the indifference curve of u through x in a ball around x with radius =>0,
i.e., in B(x, =) :=[x$: &x$&x&<=]. Notice that incentive compatibility
implies f i (u &i , v) Â B(x, =).
Now, for : # [0,1], let v :( } ) be a family of strictly quasiconcave functions connecting v and u, with v 0 =v, v 1 =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$ # R i (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${x), but
v :*( y$)~v :*(x) for any y$ # R i (u &i ). Let : k be any sequence approaching
:* with each : k <:*. IC implies f i (u &i , v :k ) Â B(x, =) for all k; hence
Lemma 1 implies f i (u &i , v :* ){x. On the other hand, let ; k be any
sequence approaching :* with each ; k >:*. IC implies f i (u &i , v ;k )=x for
all k; hence Lemma 1 implies f i (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
>0, where x is interpreted as a basic subsistence bundle; this
fi (u)x >
extra assumption
eliminates
both sources of complication. Appendix A
contains a proof of Lemma 3 without this simplifying assumption.
>0 for
Lemma 3A. Let ( f, D) be first order and assume that f i (u)x >
all individuals i and all profiles u # D. Then IC implies that for all u # D
( f(u), p(u)) # WE(u).
Proof. Let p= p( f, u). Since p supports f (u) at u, we need only verify
that pf i (u)= p| for all i. Assume the contrary; we will show a contradiction.
Let $= px. Individual i's income expansion path when his utility function is u i , he faces prices p, and he has wealth of at least $ is given by
E(u i )#[x # 0 : {1 u i (x)=p and px$].
The boundary condition implies that there exists a closed convex cone in
R l++ _ [0], call it C, that contains _ i E(u i ).
Let u* be any utility function in U that is linear in C with gradient p,
i.e., u*(x)= px for all x # C. (Notice the boundary condition implies that,
11
For a way to construct such a family, see Proposition 1 in Appendix A.
EFFICIENT IC ECONOMIES
205
outside of C, u*'s indifference curves will become bowed as they approach
the axes.)
Feasibility implies ni=1 f i (u)=n|; hence there must be an individual,
say i=1, with pf 1(u)>p|. Let u 1 =(u*, u 2 , ..., u n ). Lemma 2 implies
pf 1(u 1 )= pf 1(u)>p| and {1 u*( f 1(u 1 ))= p; so p= p(u 1 ).
Since f (u 1 ) is feasible and pf 1(u 1 )>p|, there must be some other
individual, say i=2, with pf 2(u*)<p|. Let u 2 =(u*, u*, u 3 , ..., u n ). As
above, Lemma 2 implies pf 2(u 2 )= pf 2(u 1 )<p| and {1 u*( f 2(u 2 ))= p; so
p= p(u 2 ). Further, since individuals 1 and 2 have the same preferences in
u 2, anonymity implies pf 1(u 2 )= pf 2(u 2 )<p|.
Similarly, since f (u 2 ) is feasible, there must be a third consumer, say
i=3, for whom pf 3(u 2 )>p|. Let u 3 =(u*, u*, u*, u 4 ..., u n ). Lemma 2
implies pf 3(u 3 )= pf 3(u 2 )>p| and {1 u*( f 3(u 3 ))= p; so p= p(u 3 ). Further,
since individuals 1 through 3 have the same preferences in u 3, anonymity
implies pf 1(u 3 )= pf 2(u 3 )= pf 3(u 3 )>p|.
Repeat the above procedure n&3 more times to form u 4, ..., u k, ...u n,
where u k =(u*, ..., u*, u k+1 , ..., u n ). One thus finds, depending on whether n
is odd or even, that pf i (u n )>p| for all i or pf i (u n )<p| for all i, contradicting the feasibility of f (u n ), where u n =(u*, u*, ..., u*).
Step 3. The final step is to show that R i (u &i ) must be contained in
a hyperplane through |. For this, we will show that
1. | # R i (u &i ) (Lemma 4);
2. for all x i # R i (u &i ) there exists a supporting hyperplane H to
R i (u &i ) at x i such that | # H; hence H supports R i (u &i ) at |;
3. if H and H$ support R i (u &i ) at |, then, under a regularity assumption, H=H$ (Lemma 5).
Let w p denote any strictly quasiconcave function in U satisfying
1{w p(| )= p.
Lemma 4. If ( f, D) is first order and D is comprehensive around u, then
IC implies that for all i,
| # R i (u &i ).
Proof. Let p= p(u), v i =w p, v=(u &i , v i ), q= p(v), and y= f (v).
Lemma 3 implies (y, q) # WE(v). Hence, since the indifference curve of v i
through | is strictly convex, v i ( y i )>v i (| ) unless y i =|. But since
p={1 v i (| ) by construction, v i ( y i )>v i (| ) O py i >p| O y i  R i (u &i ) since
pR i (u &i )pf i (u)= p| (using Lemmas 23), a contradiction. So, y i =|. K
Now consider any x i = f i (u), and let p= p(u). Lemma 2 implies
H p +[| ] supports R i (u &i ) at x i , and Lemma 3 implies | # H p +[| ].
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MAKOWSKI, OSTROY, AND SEGAL
FIG. 5. If R i (u &i ) has a kink then the economy without i would have a continuum of
Walrasian equilibria.
Hence, Lemma 4 implies H p +[| ] supports R i (u &i ) at |. But we have not
yet excluded the possibility of R i (u &i ) having a kink at |. 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 |: the individual would
receive the same allocation, |, whether he announces u, v, or any preferences ``between'' them. Let p(:)=:p+(1&:)q and let u : =w p(:), so
p(:)={1 u :(| ). Let (x 1(:), ..., x i&1(:), x i+1(:), ..., x n(:)) be the allocation to
all other individuals as i's preferences vary while others' remain fixed.
Feasibility implies
(a) j{i x j (:)=(n&1) |,
while Lemma 3 implies
(b) p(:) } x j (:)= p(:) } | 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, R i (u &i ) will not have a
kink at |.
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 v i # U,
p(u &i , v i )= p(u).
File: 642J 249438 . By:SD . Date:12:03:99 . Time:09:38 LOP8M. V8.B. Page 01:01
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EFFICIENT IC ECONOMIES
207
Proof. Let v=(u &i , v i ), q= p(v), p= p(u), and R=R i (u &i ). By
Lemma 3, pf i (u)= p| and qf i (v)=q|; and by Lemma 4, | # R. Hence,
Lemma 2 implies p } con R<p| and q } con R<q|, i.e., both p and q
support the convex set con R at |. But the set of supports is convex;
so any p(:) # [ p, q] also supports con R at |. Suppose p{q. Consider
the populations w : =(u &i , u :i ), where u :i =w p(:) and : # [0, 1]. Since
fi (w : )=|, p(w : )= p(:). Now Lemma 3 implies ( f (w : ), p(:)) # WE(w : )
with f i (w : )=|. Hence, for any : # [0, 1], the pair (( f 1(w : ), ..., f i&1(w : ),
fi+1(w : ), ..., f n(w : )), p(:)) is an equal-wealth Walrasian equilibrium for the
economy u &i when its aggregate endowment equals |&|, contradicting
the quasiregularity of u. K
Proof of Theorem 1. Lemma 3 implies that for any i and v=(u &i , v i ),
( f(v), p(v)) # WE(v).
Further, Lemma 5 implies that p(v)= p(u). Hence, letting p= p(u),
fi (v) # H p +[| ]. K
Proof of Corollary 1. Theorem 1 implies that for any i and v=(u &i , v i ),
( f (v), p) # WE(v),
where p= p(u). Since p remains constant when i switches to v i , for each
j{i
u j ( f j (v))=u j ( f j (u)).
Hence, efficiency implies
v i ( f i (v))v i (x)
for those x # 0 that are also in [|]&A &i (u). K
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
show that ( f (v), p) Â WE(v). Since {1 u i ( f i (u)){{1 v i ( f i (u)), f i (u) is not i 's
Walrasian demand when he is of type v i and faces prices p. But the demand
of each individual j{i is unique at prices p; so markets will not clear at p
when i is of type v i . K
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. If u 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 combinaton 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 continuous function F(:): [0, 1] Ä 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 compact subset of 0_[0, 1], the correspondence 1: R l++ _[0, 1] Ä 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
on R l++ _[0, 1]. Observing that u(x n, : n ) Ä u(0, :)=0 as (x n, : n ) Ä (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
quasiconcave, strictly increasing in R l++ , 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 # bdry A(s, u) and x # bdry A(s, v) such that
x=(1&:) y+:z. Let y$ # 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$ and v(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). Conversely, if y # arg min pA(s, u) and z # 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>max c(p|p c | c )
(where c is an index over commodities), and let $= p|(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$].
The boundary condition implies there is a closed convex cone in
R l++ _ [0], say C$, which contains all these income expansion paths for u:
. E(u i )/C$.
i
Let C be a closed convex cone in R l++ _ [0] containing C$ and also
[|&x : x0 and pxp|d]. Let u* 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) at u, we need only verify that pf i (u)= p| 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
population u k =(u k1 , u k2 , ..., u kn ) in D satisfying
v i E(u ki )/C
v p= p( f, u k )
v u ki =u* for i=1, 2, ..., k
v pf i (u k ){ p| for i=1, ..., k.
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MAKOWSKI, OSTROY, AND SEGAL
Notice that for k=n the hypothesis implies p ni=1 f i (u n ){np|, which
contradicts the feasibility of f (u n ), i.e., the fact that ni=1 f i (u n )=|=n|.
The hypothesis holds for k=1: Since feasibility implies ni=1 f i (u)=n|,
there must be an individual, say i=1, with pf 1(u)>p|. Let u 1 =
(u*, u 2 , ..., u n ). Since f 1(u) # C, Lemma 2 implies pf 1(u 1 )= pf 1(u)>p| and
{1 u*( f1(u 1 ))= p; so p= p( f, u 1 ). Further, i E(u 1i )/C since u* is strictly
quasiconcave outside of C.
Let the hypothesis hold for an arbitrary k; we show how to constructed
u k+1. There are three possible cases:
(a) pf i (u k ) Â [0, $) _ [ p| ] for some i>k
(b) pf i (u k ) # [0, $) _ [ p| ] for all i>k and pf i (u k )= p| for some
i>k.
(c) pf i (u k ) # [0, $) for all i>k.
If case (a) applies, say for individual i=k+1, let u k+1 =
(u 1 *, ..., u k+1 *, u kk+2 , ..., u kn ), where each u i *=u*. Since u* is strictly
)/C. Since f k+1(u k ) # C, Lemma 2
quasiconcave outside of C, i E(u k+1
i
k+1
k
implies pf k+1(u
)= pf k+1(u ){p| and {1 u*( f k+1(u k+1 ))= p; so p=
k+1
). Further, since individuals 1 through k+1 have the same
p( f, u
preferences in u k+1, anonymity implies pf i (u k+1 ){p| for i=1, ..., k+1.
Suppose instead that case (b) applies. Notice that in this case pf i (u k )<$
for some i>k because ni=1 f i (u k )=n| (by feasibility) and p ki=1 f i (u k ){
kp| (by hypothesis). Say that for individual n, pf n(u k )= p|. Suppose,
without loss of generality, that u kn # Up , and let v : for : # [0, 1] be Proposition 1's family of money metric utility functions connecting u kn with u* (so
v 1 =u*). By construction, {1 v :( f n(u k ))= p and E(v : )/C for all :. Let v : be
the population u k with u kn replaced by v :. Observe that Lemma 2 implies
p= p( f, v : ) for each : # [0, 1]. Further, by anonymity,
pf 1(v 1 )= } } } = pf k(v 1 )= pf n(v 1 )= p|.
Hence feasibility implies there must be at least one individual (say
i=k+1) with pf i (v 1 )>p|, or pf i (v 1 )= p| for all i. In the first event, let
u k+1 =(u 1 *, ..., u k+1 *, u kk+2 , ..., u kn&1 , v 1n ). Since f k+1(v 1 ) # C, as above one
easily verifies that u k+1 satisfies the inductive hypothesis. In the second
event, there must be some individual i>k, say i=k+1, with pf i (v 0 )<$
while pf i (v 1 )= p|; so the utility continuity of the mechanism implies that
for some : # (0,1), say :=;, $pf k+1(v ; )<p|. In this event, let u k+1 =
(u 1 *, ..., u k+1 *, u kk+2 , ..., u kn&1 , v n ; ). Similar to the first event, since
fk+1(v ; ) # C, one easily verifies that u k+1 satisfies the inductive hypothesis.
Finally, suppose case (c) applies, hence pf i (u k )>p| for all ik (by
feasibility), while pf n(u k )<$. Without loss of generality, suppose each
EFFICIENT IC ECONOMIES
211
individual's preferences in u k are in money metric form relative to p. Again
let v : be Proposition 1's family of money metric utility functions connecting
u kn with u* (hence, by construction, E(v : )/C for all : and v 1 =u*), and let
v : #(v 1 :, ..., v n : ) be the population u k with u kn replaced by v :. Lemma 2
implies pf n(v : )<$ for all :; so annonymity implies pf i (v 1 )<$ for all ik,
while pf i (v 0 )>p| 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 v i :(x i : ) i px i :, 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)
y i : # ( p, v i , py i : )for all i, hence i v i :( y i : )=p| (maximum achievable
utility under the money metric representation), (2) v i :( y i : )=v i :(x i : ) for all
i>k, and (3) y 1 : = } } } =y k *, hence v i :( y i : )>v i :(x i : ) for all ik (using
(1) and (2)). It follows that, as long as v i :(x i : )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
below.] Since pf i (v 0 )p| while pf i (v 1 )<$ for all ik, feasibility implies
that as : increases some individual i>k must achieve wealth beyond $. In
particular, the utility continuity of the mechanism implies that for some :
# [0,1], say :=;, pf i (v i ; )<2$ for all i>k and $pf i (v i ; )<2$ for at
least one individual i>k, say i=k+1. Let u k+1 =(u 1 *, ..., u k+1 *, u kk+2 , ...,
u kn&1 , v n ; ). Since p= p( f, v ; ), as above one easily verifies that u k+1 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=(x 1 , ..., x n )>
>0 be an efficient allocation for the population u=(u 1 , ..., u n ) # U n, and let p be the efficiency prices. Then there is
a population v=(v 1 , ..., v n ) # U n such that the following exist:
(1) v is first order similar to u. In particular, for each v i , v i ( y)=v i (x i )
and {1 v i ( y)= p iff u i ( y)=u i (x i ) and {1 u i ( y)= p. If v i ( y)=v i (x i ) but
{1 v i ( y){ p, then u i ( y)<u i (x i ). 13
13
So v i 's indifference curve throuth x i coincides with u i 's indifference curve throuth x i only
on the latter's flat segment around x i (if any); otherwise, the indifference curve of u i through
xi is nested inside that of v i .
212
MAKOWSKI, OSTROY, AND SEGAL
(2) x is an efficient allocation for v.
(3) Every efficient allocation for v has efficiency prices p.
Proof. Let C be a closed convex cone in R l++ _ [0] that is sufficiently
large so that for each individual i, x i +*| # C for any *0 and, for some
=>0,
(:)
x i &=| # C
for all i.
(Such a cone exists since x>
>0.) Let d==2n.
The following regions are used to define v i . Let
T i =[x: pxp(x i +d|)] & C,
M i =[x: p(x i &d|)<px<p(x i +d|)] & C,
and
B i =[x: pxp(x i &d|)] & C.
On the set T i _ B i , let v i (x)=px. In the corridor M i , let v i (x) be quasiconcave and smooth, with the constraints that {1 v i (x)=p for all x on the
cord[x i &d|, x i +d|] and the curvature of v i around x i satisfies (1). On
the rest of the domain (i.e., outside the cone C), extend v i so that it is in
U (in particular, so that it satisfies the boundary condition).
By construction, v # U n and satisfies condition (1). Condition (2) is also
satisfied since each {1 v i (x i )= p; so x 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 v i # Up . It follows that, for any feasible
allocation y, i v i ( y i ) i py i =p|; with strict inequality if any individual's indifference curve through y i does not have p for its support at y i .
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
possibility frontier. That is, for any U # R n+ such that i U i =p|, there is
a feasible allocation y such that v i ( y i )=U i for all i.
Consider any nonnegative U such that i U i =p|. Let * i be implicitly
defined by
p(x i +* i |)=U i .
Since p i x i =p|= i U i ,
: * i =0.
i
EFFICIENT IC ECONOMIES
213
Let the set T consist of all individuals i such that x i +* i | # T i (``T'' for
top); similarly let the set M consist of all i such that x i +* i | # M i , and let
the set B consist of all other individuals. If x i +* i | # C, let * i =* i (e.g., for
i in T or in M); otherwise let * i satisfy
x i +* i | # bdry C.
Finally let
* i$=&* i +* i ,
which, by construction, must be nonnegative.
If * $=0
for all i, then letting y i =x i +* i | for each i, we are done.
i
for some i # B. Define * B = i # B * i ,
So assume the contrary, that * $>0
i
T
M
similarly define * and * , and let *$= i *$. Since 0=* B +* M +* T =
(* B &*$)+* M +* T, it follows that
* T &*$=&* B &* M >=2
(where the inequality follows from the fact that &* B = and * M <dn==2).
So there exist weights * j such that
: (* j &* j )=*$
j#T
and * j &* j >d=(=2n) for all j # T (so each j in T remains outside his
corridor M j even after losing some goods in the given proportion). That is,
there is a feasible allocation y^ such that y^ i =x i +* i | for all i # M, y^ i =
x i +* i | for all i # B, y^ i =x i +* i | for all i # T.
By construction, i v i ( y^ i )=p|, so y^ is efficient. But v i ( y^ i )>U i for those
i # B with x i +* i | Â C, while v j ( y^ j )<U j for some j # T. To move from y^ to
y, decrease y^ i for the first group in a continuous way starting from y^ i and
moving along the boundary of C until y i is reached, where v i ( y i )=U i (say
to y i =y^ i &z i ); and simultaneously increase any one j # T in the opposite
direction until he reaches y j = y^ j +z i assuming v j ( y^ j +z i )U j . Otherwise
only increase j's allocation to y j = y^ j ++z, for + such that v j ( y^ j ++z)=U j ;
then move to some other j # T and increase his allocation in the direction
z until he reaches his level of utility in U, etc. (Since z is a direction along
the boundary of C, each y j will be in C since C is a cone.) Proceeding in
this way, y is constructed: Recall j # T (* j &* j )=*$, so all i # B will reach
utility levels U i precisely when all j # T reach utility levels U j . K
Proof of Theorem 2. While D is not necessarily first order relative to g,
qualified versions of Lemmas 2 and 3 still hold. In particular,
214
MAKOWSKI, OSTROY, AND SEGAL
(1) For any u # D and any i for whom u i (g i (u)){u i ( f i (u)),
p(g, u) } R i (g, u &i )p( g, u) } g i (u).
[The proof is entirely analogous to that of Lemma 2. Since ( f, D) is first
order and u i (g i (u)){u i ( f i (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 g i (u), while keeping their
supports equal to p( f, u) at f i (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
hypothesis being modified to say ``pg i (u k ){ p| for i=1, ..., k'' rather than
``pf i (u k ){ p| for i=1, ..., k.'' The only qualification is that one must
strengthen the inductive hypothesis in the proof to also include the requirement that ``u k 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
(u k, u k+1 *) is in D.) In the modified proof, (1) above is invoked instead of
Lemma 2; notice if pg i (u){p| then u i (g i (u)){u i ( f i (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
v k =(u 1 , ..., u k , v k+1 , ..., v n )
for k=0, 1, ..., n, so v 0 =v and v n =u. Since for every i, {1 v i (x i )= p, it
follows from the hypothesis that ( f, D) is first-order that v k # D for every
k. Consider the inductive hypothesis ``For all populations v k$ such that
k$k, p( g, v k$ )= 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$= g(v k ) and let y= g(u k+1 ). By hypothesis, p( g, v k )= p. Hence,
and
using (2), we have (x$, p) # WE(v k ) and (x, p) # WE(u). So {1 v j (x $)=p
j
1
{1 u j (x j )=p. Further, the construction of v j ensures {1 v j (x $)={
u
(x
$),
hence
j
j
j
{1 u j (x j$)={1 u j (x j ).
IC at v k+1 implies
u j ( y j )u j (x $)
j
EFFICIENT IC ECONOMIES
215
(otherwise j would claim to be of type v j ). Similarly, IC at v k implies
v j (x j$)v j ( y j ).
The construction of v j implies that if the first IC constraint were not bindviolating the second IC constraint. It also implies
ing then v j ( y j )>v j (x $),
j
that if the first IC constraint were binding but u j's MRS's were different at
y j and x $,
j then again v j ( y j )>v j (x $)
j violating the second IC constraint. We
conclude that the first IC constraint must be binding and u j's MRS's must
i.e., {1 u j ( y j )={1 u j (x j$). Above we found
be the same at y j and at x $,
j
{1 u j (x j$)={1 u j (x j ). Combining shows
{1 u j ( y j )={1 u j (x j ).
But efficiency implies that p( g, u j )={1 u j ( y j ) and p={1 u j (x j ). 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).
Let u i p be the (linear) utility function defined by
u i p(x)=px
for all x # 0.
Lemma 7. If ( f, D(u &i , [u i , u ip ])) is IC, where p= p(u), then
pR i (u &i )pf i (u).
Proof.
Let x= f (u) and let B=[|]& j{i A(x j , u j ). By definition,
p|p : A(x j , u j ).
j
Hence pBpA(x i , u i ) and, in particular, pBpx i . Thus, (F.1C) implies
pf i (u &i , u ip )= px i . So if py>px i for some y # R i (u &i ), then
pf i (u &i , u ip )>px i , a contradiction. K
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. Let x= f (u), p= p(u), and B=[|]& j{i A(x j , u j ). Since p
supports x at u, px i pA(x i , u i ) for all i. Hence, we need only verify that
px i = p| for all i.
Assume the contrary. Then there is an individual, say i=1, with
px 1 >p|. Let u 1 =(u 1p , u 2 , ..., u n ). Lemma 7 implies pf 1(u 1 )= pf 1(u)>p|.
Further, p supports x 1 #f (u 1 ) at u 1. To verify observe that u 1p(x 11 )=
u 1p(x 1 )= px 1 ; and, by (F.1A), j{i A(x 1j , u j )= j{i A(x j , u j ). Thus, since
j x 1j =|, p|p j A(x j , u j ) implies
p : x 1j p : A(x 1j , u j ).
j{i
j{i
And trivially,
px 11 pA(x 11 , u p ).
Summing shows
p : x 1j = p|p : A(x 1j , u 1j ).
j
j
Hence, there must be an individual, say i=2, with px 12 <p|. Let
u =(u 1p , u 2p , u 3 , ..., u n ). Lemma 7 implies pf 2(u 2 )= pf 2(u 1 )<p|. Further,
p supports x 2 #f (u 2 ) at u 2 (verified as above).
Since individuals 1 and 2 have the same preferences in u 2, anonymity
implies px 21 = px 22 <p|. Thus there must be a third consumer, say i=3, for
whom px 23 >p|. Repeat the above procedure n&2 more times, to form
u 3, ..., u n. One thus finds, depending on whether n is odd or even, that
px ni >p| for all i or px ni <p| for all i, contradicting the feasibility of x n,
where x n = f (u n )= f (u 1p , ..., u np ). K
2
Let v p(x)#v~ p(x 1 , ..., x l&1 )+x l be any continuously differentiable function in U satisfying v~ p is strictly quasiconcave and {v p(| )= p.
Lemma 9. If ( f, D) is first order and D is comprehensive around u, then
IC implies that for all i,
| # R i (u &i ).
217
EFFICIENT IC ECONOMIES
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 v i # U,
p(u &i , v i )= 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 |&|.
Let |*=|&|. Then by assumption
p|*p : A(x j , u j )
j{i
q|*q : A( y j , u j )=q : A(x j , u j ),
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 x j =|*, by a
familiar argument:
px j pA(x j , u j )
for each j{i
(1)
qx j qA(x j , u j )
for each j{i.
(2)
Suppose p{q. Hence for some commodity h (h{l), p h {q h . But since
there must be an individual j such that x jh >0. Eq. (1) implies that
| *>0,
h
for this individual u~ j (x j )x jh = p h , while Eq. (2) implies u~ j (x j )x jh =q h ,
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 # U 2 and v # U 2 , where U 2 is the subset
*
of U consisting of strictly concave, C 2 functions and U 2 is the subset of U 2
*
consisting of functions u with negative definite Hessians 2u(x) for all
x # R l++ . Let
[u, v]=[v a : v a ( } )=au( } )+(1&a) v( } ): a # [0, 1]],
the set of convex combinations of u and v.
Recall that ( p, v a , w) is the Walrasian demand of a person with utility
v a 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, v a , pf (+, v a ) for all
a # [0, 1], then
pf (+, v a )=m
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 (+, v b ) and let
U(b, a)=v a (x(b)), the utility ``a'' receives when she announces that she is
``b''; hence, U(a, a)=v a (x(a)) is a's utility when she announces truthfully,
where x(a)= f (+, v a ). With this notation,
Fact. If f is price continuous at + with respect to p, then
(I)
x(a) # ( p, v a , 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
v a , v b # [u, v] (not necessarily in supp +). Pick any sequence + k Ä + with
+ k(v a )>0 and + k(v b )>0. Since both functions are strictly concave, pricecontinuity implies f (+ k , v a ) converges to f (+, v a ) and f (+ k , v b ) converges to
f(+, v b ); and incentive compatibility implies, for each k, v a[ f (+ k , v a )]
v a[ f (+ k , v b )]. Hence the continuity of v a implies v a[ f (+, v a )]v a[ f (+, v b )].
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, v a , w). If X were not bounded then there would be a
EFFICIENT IC ECONOMIES
219
sequence in [0, 1], a n Ä a 0 such that &x n& Ä , where x n #x(a n ); hence,
w n #px n Ä . Notice that for all n sufficiently large,
U(a n, a n )=v an[.(a n, w n )]>v an[.(a n, w 0 +1)]>v an[.(a n, w 0 )].
Hence, letting n Ä and using the fact that .[ } , w] is continuous in a,
lim inf U(a n, a n )>v a0[.(a 0, w 0 +1)]>U(a 0, a 0 ).
That is, there is an =>0 such that
lim inf [a nu(a n )+(1&a n ) v(a n )]a 0u(a 0 )+(1&a 0 ) v(a 0 )+=.
So there is a sufficiently large n such that
a 0u(a n )+(1&a 0 ) v(a n )>a 0u(a 0 )+(1&a 0 ) v(a 0 ),
contradicting incentive compatibility.
On the other hand, if the closure of X is not contained in 0 then there
exists a sequence a n Ä a 0 with x n Ä x 0 Â 0. Notice the boundary condition
implies u(x n ) Ä inf u(0) and v(x n ) Ä inf v(0). So, for a sufficiently large n,
U(a 0, a n )>a nu(x n )+(1&a n ) v(x n )#U(a n, a n ),
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 v a ( } | p) be the
money metric representation of v a . That is, v a (x | p)= py(x, a), where
y(x, a) solves
min py
y0
s.t. v a ( y)v a (x).
It is straightforward to verify that each v a #au+(1&a) v # U 2. Indeed,
since 2u(x) is negative semi-definite while 2v(x) is negative definite, 2v a
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)=v a (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),
b # [0, 1]
for all a # [0, 1].
(3)
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:
&D 1 L(b, a)=0
at b=a,
for all a # [0, 1],
w$(b)&D 1 L(b, a)=0
at b=a,
for all a # [0, 1].
Combining would yield w$(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 Holmstrom'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<
a
}
for all a, b # I.
(5)
Recall L(b, a)#px(b)&u(b, a)= px(b)& py(b, a). Hence,
y(b, a)
L(b, a)
=p
.
a
a
Since y(b, a)a#y(x(b), a)a, 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 Holmstrom's main Lemma. However, we prefer to give a self-contained proof rather than appeal to Holmstrom'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, v a , 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],
D 1 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
D 1 L(a, a)=0;
similarly, recalling Eq. (4), incentive compatibility at a implies
w$(a)&D 1 L(a, a)=0;
hence w$(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
u n Ä u mean u n converges to u uniformly on compacta.
Lemma 14. Suppose u, u n # U, u n Ä u, and px(u n )=m, a positive constant, for all n. Then px(u)=m.
Proof. Since px(u n )=m for all n, there is a subsequence, say u k, and an
x such that f (+, u k ) Ä x and px=m. Since demand is continuous, x is in u's
222
MAKOWSKI, OSTROY, AND SEGAL
demand correspondence when facing prices p and wealth m. Hence, if
w(u)<m then u( f (+, u))<u(x), which implies for a sufficiently large k,
u( f (+, u))<u( f (+, u k )), contradicting incentive compatibility. On the other
hand if w(u)>m then u( f (+, u))>u(x); hence, for a sufficiently large k,
u k( f (+, u))>u k(x k ), again contradicting incentive compatibility. K
Proof of Theorem 4. We first show that pf (+, u)=m, a constant, on U 2.
Fix any v # U 2 and pick any u # U 2. By Lemmas 1314, pf (+, v)= pf (+, u).
*
Since this holds for any u # U 2, we are done.
Now we extend the conclusion to all of U. Pick any u # U. There exists
a sequence u n in U 2 s.t. u^ n Ä u^, where u^ n is some (perhaps different) utility
function representing the same preferences as u n, and similarly u^ is some
utility function representing the same preferences as u. (See Mas-Colell [18,
Section 2.8.] By Lemmas 1314, pf (+, u n )= pf (+, v)=m for all n; hence
pf (+, u^ n )= pf (+, u n )=m for all n. Lemma 14 now implies pf (+, u^ )=m;
hence pf (+, u)= pf (+, u^ )=m, as required.
To complete the proof we need only show m = p|. But feasibility implies
f (+, u) d+(u)=|, hence pf (+, u) d+(u)= m d+= p|, hence m = 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
functions on R l++ , denoted C 0(R l++ ), is a complete metric space. (See
K.1.2, p. 50.). Further the subset consisting of all quasiconcave functions,
denoted QC 0(R l++ ), is closed and therefore also complete. Recall U is the
subset of QC 0(R l++ ) 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 M 1n be the open and dense set
of points with ``jumps'' in the correspondence of at most 1n. Because
n M 1n 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 M 1n 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
6 1 o 6 2 iff 6(+)$6 1(+)$6 2(+) for all + # M, with 6 1(+)#6 2(+) for
some +. Let (6 i ) i # I be any chain ordered by o , and let 6 be defined by
6(+)= & i # I 6 i (+). Observe that 6 is a lower bound to the chain: Since
for every i, 6 i (+) is compact, 6(+){<. Also, if + n Ä + and p n # 6 i (+ n )
such that p n Ä p, then p # 6(+). The reason is that for every i, p n # 6 i (+ n ).
By u.h.c of 6 i , p # 6 i (+); 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
6*. 15 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 p 1 , p 2 # 6*(+ 0 ). Let $>0 such that
d(+, + 0 )<$ implies d(6*(+), 6*(+ 0 ))< 13 d( p 1 , p 2 ). 16 (Recall that 6* is
continuous at + 0 .) Define 6$ by 6$(+)=6*(+) if d(+, + 0 )$. Otherwise,
6$(+)=[ p # 6*(+) : d( p, p 2 ) 13 d( p 1 , p 2 )]. 6$(+) is not empty because
there are points in 6*(+) that are within 13 d( p 1 , p 2 ) distance from p 1 . It
is easy to verify that 6$ is u.h.c. By construction 6* o 6$, 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
that + n Ä + where + # M* implies 6*(+ n ) Ä 6*(+). In particular,
?(+ n ) Ä ?(+). K
Proof of Proposition 2. Let + n Ä +. 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>2$ and \(+ n , +)<$2 for all n>N.
Then for any n>N and any perturbation 2u # J(+ n ), \(+ n , + n +2u)=
1n<$2. Hence, for any n>N, [+ n +2u : 2u # J(+ n )]/B(+ n , $2)/
B(+, $). K
Proof of Proposition 3. Since q # S, q|=1. Thus any x # (q, v, q|) is
contained in the set
&1
&1
B q =[x$ # R l+ : qx$=1]=convex hull [q &1
1 , q 2 , ..., q l ].
15
Appeal to Zorn's Lemma is used by Mas-Colell (5.8.18, p. 234) in establishing the existence of a continuous selection on the space of regular economies.
16
There are three different metrics in this statement. The first is on M, the second is the
Hausdorff metric on subsets of S, and the third is on S.
224
MAKOWSKI, OSTROY, AND SEGAL
Therefore,
u(x| p) :=min [ py: u( y)u(x)]max [ px$: x$ # B q ]
=max
c
<max
c
pc
qc
pc
p c &d
:
.
:&d
The next to last inequality follows from fact that &q& p&<d<: and the
last from the fact that p c :. So, u(x| p)&1<:(:&d )&1=d(:&d ).
Observing that u( y| p)= p|=1 completes the proof. K
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