Perfect Competition in the Continuous Assignment Model

Journal of Economic Theory 88, 60�118 (1999)

Perfect Competition in the ContinuousAssignment Model*

Neil E. Gretsky-

Department of Mathematics, University of California, Riverside 92521

neg�math.ucr.edu

Joseph M. Ostroy�

Department of Economics, University of California, Los Angeles 90024

ostroy�econ.ucla.edu

and

William R. Zame�

Department of Economics, University of California, Los Angeles 90024

zame�econ.ucla.edu

Received March 25, 1998; revised April 9, 1999

This paper provides a rigorous formalization, in the context of the transferableutility assignment model, of perfect competition as the inability of individuals to(favorably) influence prices. The central tool for the analysis is the social gainsfunction; the central issue is differentiability of the gains function with respect to thepopulation. Seven conditions are shown to be equivalent to perfect competition.Perfect competition and imperfect competition are possible for both finite andcontinuum assignment economies, but most finite economies are imperfectly com-petitive, while most continuum economies are perfectly competitive. However, mostlarge finite assignment economies are approximately perfectly competitive. Journalof Economic Literature Classification Numbers: C61, C62, C78. � 1999 Academic

Press

Article ID jeth.1999.2540, available online at http:��www.idealibrary.com on

600022-0531�99 �30.00Copyright � 1999 by Academic PressAll rights of reproduction in any form reserved.

* We are grateful to the editor and several referees for very useful comments andsuggestions.

- Hospitality of the Department of Mathematics of the University of California, LosAngeles, during the preparation of an earlier version of this paper is gratefully acknowledged.

� Support from the National Science Foundation and the UCLA Academic Senate Commit-tee on Research is gratefully acknowledged.

1. INTRODUCTION

This paper provides a rigorous formulation, in the widely studied frame-work of the transferable utility assignment model, of perfect competition asthe inability of individuals to ( favorably) influence prices and of approxi-mate perfect competition as the inability of individuals to ( favorably)influence prices more than a little.1 Within this framework, we identifyconditions which are equivalent to perfect competition and identifya natural environment in which most continuum economies are perfect-ly competitive and most large finite economies are approximately per-fectly competitive. The assignment model is particularly suited to ourpurpose because it allows us to highlight in an especially clear and con-crete way the ability of individuals to extract their marginal products,a notion that is central to perfect competition, and because analysisof the assignment model points the way to a similar analysis for moregeneral settings.2

We analyze assignment economies through the social gains function.Given a population distribution + (a positive measure on agent charac-teristics), g(+) is the total of gains available from trade. (The assumptionof transferable utility means that these gains can be summarized as a singlenumber.) We show that the gains function is concave, superadditive andhomogeneous of degree 1, and that the subdifferential of the gains functionat a population distribution + can be identified with the core of theeconomy defined by +. Most importantly, we characterize perfect competi-tion as differentiability of the gains function with respect to the populationdistribution.

The gains function allows us to analyze finite and continuum economiesin a unified way. We show that perfect competition is possible (but rare)in finite economies, and (with the assumptions made here that houseslie in a compact set and that buyer valuation functions lie in anequicontinuous family) imperfect competition is possible (but rare) incontinuum economies. That is, imperfect competition is generic in finiteassignment economies and perfect competition is generic in (equicontinuous)

61COMPETITION IN THE ASSIGNMENT MODEL

1 Because the buyer of a commodity can always pay more than the market price, and theseller of a commodity can always accept less than the market price, it will always be the casethat some individuals can unfavorably influence prices.

2 For the classical version of the assignment model, with a finite number of objects and afinite number of individuals, see for instance Shapley [30], Gale [9], and Shapley and Shubik[31]; Roth and Sotomayor [29] provides a convenient general reference and historical over-view. For the version of the assignment model with a continuum of objects and a continuumof individuals, see the work of Kantorovich [17] on the transportation problem��a closerelative of the assignment problem��and Gretsky, Ostroy, and Zame [11].

continuum assignment economies.3 Because perfect competition is express-ible in terms of the gains function, we can connect perfect competition incontinuum economies with approximate perfect competition in large finiteeconomies. In particular, from the fact that most continuum assignmenteconomies are perfectly competitive, we are able to conclude that mostlarge finite assignment economies are approximately perfectly competitive.4

That there is a connection between perfect competition and differen-tiability of the social gains function can be seen clearly in finite economies.An individual's ability to manipulate is limited by his marginal product (hiscontribution to the entire social gain): an individual who extracts his or hermarginal product cannot gain by manipulation, because the remainder ofsociety could achieve the same utility without his or her cooperation. If themarginal products of all individuals add up to the social gain, then allagents can simultaneously extract their marginal product, so no individualcan manipulate. Because individual marginal products are ``discrete direc-tional derivatives'' of the social gains function, adding up of individualmarginal products is a kind of discrete differentiability. Thus discretedifferentiability implies perfect competition��the inability of individuals to(favorably) manipulate.

In general economies, an individual who is not extracting his marginalproduct might still be unable to (favorably) manipulate; thus, perfect com-petition need not imply discrete differentiability. The assignment model isspecial, however, in that it is always possible for an individual tomanipulate prices in such a way as to achieve the Walrasian price he mostprefers, and this most preferred Walrasian price guarantees the individualhis marginal product.5 These properties, together with a third, that discretedifferentiability coincides with differentiability of the gains function in theusual (infinitesimal) sense, lead (Theorem 1) to the following equivalentcharacterizations of perfect competition in the finite assignment model:

(i) individuals fully appropriate their marginal contributions;

(ii) marginal products of individuals add up to the social gain;

(iii) individuals face perfectly elastic demands and supplies;

62 GRETSKY, OSTROY, AND ZAME

3 As we show by example, without the assumption that buyer valuation prices lie in anequicontinuous family, imperfect competition may be a robust phenomenon in continuumassignment economies. See Ostroy and Zame [24] for another framework in which imperfectcompetition may be a robust possibility in continuum economies.

4 For general assignment economies, there may be a disconnection between continuumeconomies and large finite economies; see Gretsky, Ostroy, and Zame [12]. For a similardisconnection in general exchange economies, see Anderson and Zame [2].

5 The reader might suppose that the special characteristics of the assignment model makeit an unreliable guide to more general economies. As we shall argue in Section 8, however,exactly the opposite is the case.

(iv) the core is a singleton;(v) Walrasian prices are unique;

(vi) the social gains function is differentiable.6

Note that a familiar test of perfect competition��the coincidence of thecore with the set of Walrasian allocations��is missing from this list. Thereason is that, as a consequence of the special nature of the assignmentmodel, the core always coincides with the set of Walrasian allocations, evenwith a single buyer and a single seller��an environment that is trans-parently non-competitive. Thus, it is something more than the coincidenceof the core and Walrasian allocations per se that underlies thecompetitivity conclusions in continuum economies.

Although we can identify perfect competition in finite assignmenteconomies, it is exceedingly rare. An assignment model with k sellers andl buyers is specified by k+lk real parameters (the reservation value eachof the k sellers places on his or her own house and the value each of thel buyers places on each house), and the set of parameters for which theseequivalent conditions are satisfied is of lower dimension��and in particularis a set of Lebesgue measure zero.

Of course, we do not expect perfect competition in finite economies; if weseek (robust) perfect competition we must look to continuum economies.The continuum assignment model we use is developed in Gretsky, Ostroy,and Zame [11], hereafter referred to simply as GOZ1. In the finite assign-ment model the number of commodities (houses) is typically the same asthe number of sellers��each seller is a monopolist. In the continuum model,therefore, we allow for an infinite number of potential commodities andagain for each seller to be a monopolist. We limit the monopoly power ofsellers, however, by insisting that the space of houses be compact and thatbuyers find nearby houses to be nearly perfect substitutes; that is, werequire each buyer's schedule of house valuations to be a continuous func-tion. And we limit the monopsony power of buyers by insisting that theyhave similar valuations, i.e., that buyer valuation functions lie in a compact(hence equicontinuous) family.

Our plan for analysis of perfect competition in the continuum assign-ment model is the same as for the finite model, but our execution must ofnecessity be somewhat different. The most obvious reason for thisdifference is that in the continuum model there is no precise analog of asingle individual or of the influence of a single individual. A literalindividual is a set of measure zero in the population and hence��almost bydefinition��has no influence. To measure the influence of a singleindividual, we use small groups, measure influence in per capita terms, and


6 There are two possible notions of differentiability, but they coincide in this setting.

take limits as the group size tends to zero. In other words, we identify themarginal product of an individual with a derivative, rather than with a dis-crete difference quotient. (This is as it should be: in the continuum, anindividual is infinitesimal, so individual marginal products should beinfinitesimal difference quotients��that is, derivatives.)

Our analysis is complicated because the domain of the gains function isthe space of population distributions (non-negative measures) on buyerand seller characteristics. Because the space of population distributions hasempty interior in the underlying vector space of all measures on buyer andseller characteristics, we cannot rely on familiar results from finite dimen-sional concave analysis or on well-known infinite dimensional extensions.7

Instead we adapt recent work of Verona [32, 33] to our setting and exploitthe special nature of the gains function. An essential ingredient is thatGateaux and Fre� chet differentiability of the gains function coincide.8 Ouranalysis leads (Theorem 3) to the equivalent conditions below, parallel tothose identified for the finite assignment model, that characterize perfectcompetition in the continuum:

(1) small groups fully appropriate their marginal contributions;

(2) marginal products of small groups add up to the social gain;

(3) small groups face perfectly elastic demands and supplies;

(4) the core is a singleton;

(5) Walrasian prices are unique;

(6) the social gains function is Gateaux differentiable;

(7) the social gains function is Fre� chet differentiable.

In the finite assignment model, perfect competition is rare; in thecontinuum exactly the opposite is true (Theorem 4): the conditions (1)�(7)identified above are generically valid.

To understand the implications of these results about continuumeconomies for large finite economies, we consider sequences of finiteeconomies whose (normalized) population distributions converge to alimiting distribution. We show

v If + is perfectly competitive, in the sense that it satisfies theequivalent conditions (1)�(7), and (En) is a sequence of finite economies forwhich the (normalized) population distributions converge to +, then for


7 See Rockafellar [28] for concave analysis in the finite dimensional setting and Giles [10]and Phelps [26] for concave analysis in the infinite dimensional setting.

8 Gateaux and Fre� chet differentiability coincide for homogeneous, concave functionsdefined on a finite dimensional space but do not generally coincide for homogeneous, concavefunctions defined on an infinite dimensional space.

every n sufficiently large, the economy En is approximately perfectly com-petitive in the sense that no individuals in En can influence prices verymuch.

v If + is imperfectly competitive, in the sense that it fails theequivalent conditions (1)�(7), then there is some sequence (Fn) of finiteeconomies for which the (normalized) population measures converge to +and for which each economy fails to be approximately perfectly com-petitive, in the sense that, for every n, many individuals can influence pricesa lot.

Informally: most large finite assignment economies are approximatelyperfectly competitive.

Ostroy [22, 23], Makowski and Ostroy [19]; and Makowski, Ostroy,and Segal [20] establish connections among perfect competition, non-manipulability, and differentiability. The contributions of this paper are toplace the problem of perfect competition in what we believe to be its mostnatural setting (the infinite dimensional space of population measures), tobuild tools (infinite dimensional concave analysis) for studying perfectcompetition in that setting, and to use those tools to obtain a complete androunded picture of perfect competition in the continuous assignmentmodel:

v perfect competition=differentiability;

v perfect competition is typical for continuum economies;

v perfect competition for continuum economies entails almost perfectcompetition for large finite economies;

v almost perfect competition is typical for large finite economies.

The remainder of the paper is organized in the following way. Section 2discusses perfect competition in the finite assignment model, as backgroundand motivation for what is to come. Section 3 describes the basic con-tinuum framework and interprets the results of GOZ1 in the presentframework. Section 4 presents the equivalent conditions for perfect com-petition for continuum economies. Section 5 demonstrates the genericvalidity of perfect competition. Section 6 discusses the implications for largefinite models. Section 7 presents the (new) mathematical tools of concaveanalysis that are used throughout. Section 8 speculates about extensions ofthe present results for the assignment model to other environments. Indica-tions of proofs and methods are in the text, but most of the details arecollected in Section 9. Illustrative examples are provided at appropriateplaces in the text.


2. PERFECT COMPETITION IN THE FINITE MODEL

In this section we recall the Shapley�Shubik formulation of the finitetransferable utility assignment model (with notation slightly adapted forour use), formulate perfect competition as the inability of individuals toinfluence prices, and give a number of equivalent conditions.

The data of a finite assignment model are a set H of (indivisible) houses,a finite set S of sellers and a finite set B of buyers. Each seller s # S isendowed with a house ?(s) # H, for which her reservation value is _(s). (Weuse masculine pronouns for buyers and feminine pronouns for sellers.)Sellers have no use for any house other than their own. Buyers have noendowment of houses but desire to acquire one house; write ;(b, h) for thevalue buyer b places on house h # H. (Implicit in this formulation, and inthe sequel on continuum models, is the presence of a money good and theassumption that utility functions are quasi-linear in money. We suppressexplicit reference to the money good.) We assume that all reservationvalues lie in the interval [0, 1].

The assignment model can be analyzed as a linear programmingproblem, as a cooperative game, and as an exchange economy. Thefundamental result of Shapley�Shubik is that these formulations areequivalent: solutions to the linear programming problem are Walrasianallocations, and solutions to the dual linear programming problem are coreutilities and correspond to Walrasian prices.

If buyer b and seller s are matched (i.e., if b and s trade), the gain fromthe match is

V(b, s)=;(b, ?(s))&_(s).

The assignment problem is to match buyers and sellers (perhaps leavingsome buyers and sellers unmatched) so as to maximize the social gain. Theassignment problem can be written as an integer programming problem:

maximize :b

:s

V(b, s) xbs

subject to :b

xbs�1 for all s

and :s

xbs�1 for all b

and xbs is a non-negative integer, for all b, s.

(At an optimal solution, xbs=1 if b, s are matched; xbs=0 otherwise.)Associated with this integer programming problem is a linear program-

ming problem, which differs only in that the variables xbs are constrained tobe non-negative real numbers, rather than to be non-negative integers. The


solutions to the linear programming problem constitute a convex set; theextreme points of this set are solutions to the integer programmingproblem.

Taking the point of view of linear programming suggests that we thinkof the right-hand sides of the constraint equations as variables; writing+b , +s for these variables leads to the following primal problem:

maximize :b

:s

V(b, s) xbs

subject to :b

xbs�+s for all s

and :s

xbs�+b for all b

and xbs�0 for all b, s.

The dual problem is as follows:

minimize :b

+bqb+:s

+sqs

subject to qb+qs�V(b, s) for all b, s

and qb�0, qs�0 for all b, s.

Informally, we think of the vector + # RB _ S+ as representing the ``number''

of buyers and sellers of each type; the ``true'' model corresponds to+=1=(1, ..., 1). The fundamental theorem of linear programming impliesthat the primal and dual problems have solutions and that the optimalvalues coincide; call this optimal value g(+). In parallel with the assignmentproblem itself, we refer to g as the social gains function. It is readily verifiedthat g is super-additive and positively homogeneous of degree 1 and istherefore concave. By definition, the subdifferential �g(+) consists of thoseq # RB _ S

+ such that q } += g(+) and q } +$�g(+$) for all +$ # RB _ S+ .

The assignment problem is easily viewed as a transferable utility marketgame. The characteristic function of the game is defined by

v(A)=max :i

V(bi , si),

where the summation is taken over an arrangement of distinct playersbi # A & B and si # A & S and the maximum is taken over all sucharrangements. The core C(1) of this game is the set of utility vectors q # RB _ S

+

such that �i # B _ S q(i)= g(1) and �i # A q(i)�v(A) for each A�B _ S.It is easily verified that the core coincides with the subdifferential of the

social gains function:


C(1)=�g(1).

To define a Walrasian equilibrium for the market economy, we introducesome notation. Let ;b be the vector in RH

+ whose h th coordinate is ;(b, h).Let eh be the vector in RH

+ whose h th coordinate is 1 and whose othercoordinates are all 0.

It is convenient to adjoin to the set of houses an additional fictitious house0� which no one owns or values; a buyer acquiring 0� will be interpreted asnot trading. By convention the price of house 0� is 0. In what follows wesuppress the 0� component of prices and the 0� component of consumptions.

A Walrasian equilibrium is a pair ( y, p) where y : I � RH+ is a housing

allocation and p # RH+ is a price system satisfying

v yb # �h # H� eh , and ys # [e?(s) , 0]

v :b # B

yb+ :s # S

ys= :s # S

e?(s)

v ;b } yb& p } yb=suph # H [;(b, h)& p } eh] 6 0 for all b # Bv p } ys&_(s)=[ p } e?(s)&_(s)] 6 0 for all s # S

For the assignment model, the core and the set of Walrasian equilibriaare equivalent. The equivalence can be seen most easily by recognizing thecore utilities as a way of splitting the gains from trade. Given core utilitiesq, the Walrasian price for a house h # H owned by seller s is p(h)=q(s)+_(s); the corresponding Walrasian allocation y gives houses to buyersaccording to the optimal solution to the assignment problem (leavinghouses in the hands of sellers who are not matched, and leaving unmatchedbuyers with no house). If there are houses which are not part of any seller'sendowment, we assign them prices sufficiently high so that no buyer desiresto acquire them.

Two special properties of Walrasian equilibria of assignment economiesare important. The first is interchangeability : if ( p, y), ( p$, y$) areWalrasian equilibria, so is ( p$, y). (This is most easily seen by the corre-spondence to the linear programming formulation.) The second is that thecollection of Walrasian prices is a lattice with a largest element p� and asmallest element p

�. (This follows from the observation that an allocation

that is utility maximizing at two different Walrasian price systems is alsoutility maximizing at the least upper bound and the greatest lower boundof the two price systems.)

If we order core utilities from the point of view of sellers (who preferhigh prices), it follows from core�equilibrium equivalence that C(1) is alattice; write q� , q

�for the largest and smallest elements, which correspond

respectively to the highest and lowest Walrasian prices.Because the core of an assignment economy is always equivalent to the

set of Walrasian equilibria, this equivalence provides no information about


perfect competition. However, a direct test in terms of manipulation, in thespirit of Nash equilibrium, is available.

We consider the possibility of an agent manipulating the Walrasianmechanism by misrepresenting his�her preferences; however, we do notallow a seller to misrepresent the type of house she has, or a buyer (respec-tively, seller) to misrepresent himself as a seller (respectively, buyer). Con-sider the situation for a buyer b relative to a Walrasian equilibrium ( p, y)(for the true economy). Buyer b's preferences are represented by theschedule ;(b, } ) of reservation values for houses; misrepresentation meansannouncing a (possibly) false schedule ;$(b, } ). Given the announcement ;$,and assuming truthful announcements by other agents, the Walrasianmechanism will arrive at an equilibrium ( p$, y$). Buyer b prefers this alter-nate equilibrium exactly when

;(b, y$b)& p$ } y$b>;(b, yb)& p } yb .

(Note that we evaluate both equilibria according to b's true preferences.)Misrepresentation for sellers is defined similarly. The economy 1 # RB _ S

+ isnon-manipulable if there is a Walrasian equilibrium ( p, y) at which noagent can gain by manipulating the outcome via misrepresentation. Moreprecisely, there does not exist an agent, a misrepresentation by that agent,and an equilibrium of the economy which the misrepresenting agentprefers. As we shall see, this is equivalent to the assertion that for everyWalrasian equilibrium, no agent can gain by misrepresentation.

How much manipulation is possible for an individual? The answer ismost easily expressed in terms of marginal products. For i # B _ S, write 1i

for the vector which has a 1 in the i th slot and 0 elsewhere. The marginalproducts of an individual are then defined as

MP+i (+)=g(++1i)& g(+)

MP&i (+)=g(+)& g(+&1i).

Since the mechanism always yields Walrasian equilibria for the announ-cements, and Walrasian equilibria are in the core, in the manipulatedeconomy, the total of gains to all the individuals except for themanipulator will be at least what they could get on their own: g(1&1i).Moreover, because we evaluate the equilibrium of the economy with mis-representation according to true preferences, the total gains to allindividuals including the manipulator cannot be greater than the gainswhen there is no misrepresentation: g(1). Hence, an absolute upper boundon the utility an individual i can obtain, no matter what he�she announces is

g(1)& g(1&1i)=MP&i (1).


In particular, individual i will certainly be unable to manipulate if agent iis already receiving his�her marginal product.

The following proposition, which asserts that sellers (respectively,buyers) obtain their marginal products at the biggest (respectively,smallest) core utility, is key to the understanding of perfect competition inthe finite assignment model.

Proposition 1. For all b # B, s # S

MP&b (1)=q

�(b)=max[q(b) : q # C(1)]

�min[q(b) : q # C(1)]=q� (b)=MP+b (1)

and

MP&s (1)=q� (s)=max[q(s) : q # C(1)]

�min[q(s) : q # C(1)]=q�(s)=MP+

s (1)

Proof. Consider the right-hand directional derivative

Dg(+; #)= limt � 0+

g(++t#)& g(+)t

.

Because g is concave, it follows from Rockafellar [28, Theorems 23.1 and23.4] that

&[ g(+&1i)& g(+)]�&Dg(+; &1i)�Dg(+; 1i)�[ g(++1i)& g(+)].

(1)

Since 1 is in the interior of the domain of g,

&Dg(1; &1i)=max[q } 1i : q # �g(1)]

and

Dg(1; 1i)=min[q } 1i : q # �g(1)].

A distinctive property of the assignment model (see Roth and Sotomayor[29]) is that these discrete marginal products coincide with theinfinitesimal marginal products:

MP&i (1)=&Dg(1; &1i)

(2)MP+

i (1)=+Dg(1; +1i).


Combining all these equalities with the inequalities in Eq. (1) yields thedesired conclusions. K

In a perfectly competitive economy, all agents face perfectly elasticdemands and supplies. In the present context, this would mean that priceswould not change if a single buyer or seller were to enter or leave themarket. Thus, to say that agents face perfectly elastic demands and suppliesat a given Walrasian equilibrium ( p, y) for the economy 1 is to say that premains a Walrasian price for the economies represented by 1+1i , 1&1i

in which an agent of type i has entered or left the market.The final ingredient is the implication that concavity of the gains func-

tion g has for differentiability: unicity of the subdifferential �g(+) at +,Gateaux differentiability of g at + and Fre� chet differentiability of g at + allcoincide.

These observations lead to the following result, which characterizesperfect competition in the finite assignment model; we defer the proof toSection 8.

Theorem 1. For a finite assignment economy 1 # RB _ S, each of thefollowing conditions is equivalent to non-manipulability:

(i) Full appropriation: there exists q such that �i # B _ S q(i)= g(1)and q(i)=MP&

i (1) for all i # B _ S.

(ii) Adding up: �i # B _ S MP+i (1)= g(1).

(iii) Perfectly elastic demands and supplies: there is a Walrasianequilibrium ( p, y) such that p # P(1+1i) and p # P(1&1i).

(iv) Core unicity: C(1) is a singleton.

(v) Uniqueness of Walrasian prices: Walrasian prices are unique.

(vi) Gateaux differentiability: g is Gateaux differentiable at 1.

(vii) Fre� chet differentiability: g is Fre� chet differentiable at 1.

As noted in the Introduction, perfect competition (non-manipulability) ispossible in finite assignment economies, but it is not very likely. To beprecise, consider economies with k buyers and l sellers. A complete descrip-tion of such an economy is given by k+lk numbers in the interval [0, 1]:the value each seller places on her own house, and the value each buyerplaces on the house of each of the sellers. Perfect competition obtainstrivially if each seller values her house as much as any buyer, so thatV(b, s) is identically 0 and there are no gains from trade. Considertherefore the set V/[0, 1](k+lk) of economies for which V(b, s) is notidentically 0; note that V is a relatively open set. Let VPC be the subset ofV consisting of perfectly competitive economies.


Proposition 2. The collection VPC /V of perfectly competitive finiteassignment economies with k sellers, l buyers and positive gains to trade is afinite union of closed, lower-dimensional submanifolds of V. In particular,VPC is closed, has no interior, and has Lebesgue measure 0.

3. THE CONTINUUM MODEL

3.1. Notation

First, we collect some standard notation. For a metric space X, writeC(X ) for the Banach space of bounded, real-valued continuous functionson X, equipped with the supremum norm. Write B(X ) for the _-algebra ofBorel sets and M(X ) for the Banach space of (regular, countably-additive,real-valued) Borel measures, equipped with the variation norm. In additionto the topology induced by the variation norm, we will also be using theweak* topology on M(X ), which is the weakest topology for which themaps

+ [ ( f, +)#|X

f (x) d+(x)

are continuous for each f # C(X ). Note that C(X ) and M(X ) are Banachlattices in their natural ordering; write C+(X ) and M+(X) for the respec-tive positive cones. Given a measure ! defined on a Cartesian productX_Y, the marginals !1 , !2 are defined by

!1(E)=!(E_Y) for E # B(X )

!2(F)=!(X_F ) for F # B(Y ).

3.2. Assignment Economies

We follow a familiar route, due to Hart, Hildenbrand, and Kohlberg[14], and describe an economy as a distribution on agent characteristics.The basic data of an assignment economy consists of a set of (indivisible)houses, a space of seller characteristics, a space of buyer characteristics,and a population measure on the space of buyers and sellers. Implicit inour transferable utility framework is the presence of a money good,available in positive and negative amounts, and the assumption that allagents have utility functions that are quasi-linear in money. As usual in thetransferable utility framework, our notation is chosen so as to suppress themoney good. We write H for the set of houses, which we assume to be acompact metric space; our point of view is typically to regard H as givenonce and for all. (In the examples given below, H=[0, 1].)


A seller is endowed with one house and desires only her own house,so the seller is completely characterized by the house owned and thereservation value placed on that house (the least amount of money forwhich the seller would be willing to sell her house). We assume that reser-vation values lie in [0, 1], so that the space of seller characteristics is

S=H_[0, 1].

Write ?, _ for the coordinate projections (which are continuous); thus aseller with characteristics s=(h, r) has the reservation value _(s)=r for herown house ?(s)=h. Note that S is a compact��hence complete��metricspace.

A buyer is endowed with no house and desires to acquire at most one;the characteristics of a buyer are therefore a complete schedule of reserva-tion prices for houses. We assume that reservation prices lie in [0, 1] andare continuous, so that the space of buyer characteristics is

B=[b # C(H ) : 0�b�1].

Equipped with the topology of uniform convergence, B is a completemetric space. The sets B and S are descriptions of the universe of possiblebuyers and sellers. When studying a particular environment, we will restrictour attention to compact sets B/B, S/S of buyers and sellers, respec-tively. We regard B, S as describing the ``universe of discourse'' and B, S asparameters of the particular economy under study. Because the universe Sof sellers is compact, the requirement that S be compact is equivalent tothe requirement that it be a closed subset of S. The universe B of buyers,however, is not compact; the requirement that B be compact is equivalentto the requirement that it be a closed and equicontinuous subset of B.

For s # S, b # B, b(?(s)) is the value b places on the house owned by s,and the valuation function

V(b, s)=b(?(s))&_(s)

describes the social value obtained by matching b and s. Note that V isjointly continuous in its two arguments. Because B is equicontinuous, thefamily [V(b, } ) : b # B] is also equicontinuous.

An assignment economy consists of a compact set B of buyers, a compactset S of sellers, and a population measure + # M+(B _ S). (B _ S is the dis-joint union of B, S; we equip B _ S with the distance function which agreeswith those of B, S for pairs of points in B and pairs of points in S and findsevery point of B to be of distance 1 from every point of S.) Write +B , +S

for the restrictions of + to the space of buyers and the space of sellers. Forconvenience, we sometimes write I=B _ S for the space of individuals.


In our approach, we have taken as primitives the spaces of houses andof seller and buyer characteristics, and a population measure. An alter-native (equivalent) formulation would be to take as primitives abstract(compact metric) spaces B, S of buyers and sellers, a population measure+ on B _ S, and valuation functions ; : B_S � [0, 1] and _ : S � [0, 1].This latter formulation, which identifies sellers with the houses they own,is used in GOZ1 and is more convenient for the description of examples.

In the following subsections, we view the assignment model as a linearprogramming problem (for which a solution to the primal problem is amatching of buyers and sellers that optimizes social gains, and a solutionto the dual problem is an assignment of utilities that minimizes the total),as an exchange economy (for which the appropriate solution concept isWalrasian equilibrium), and as a market game (for which the core is thenatural solution concept). These views lead to equivalent solutions. Wegive summaries below; for details, see GOZ1 and Section 9.

3.3. Linear Programming and the Gains Function

As we noted in Section 2, the assignment problem was originally posedas an application of linear programming. In our distributional formulation,given B/B, S/S and a population measure + # M+(B _ S), the primallinear programming problem is to find an assignment x # M+(B_S) soas to

maximize |B_S

V dx

subject to x(E_S)�+(E ) for all Borel E/B

and x(B_F )�+(F ) for all Borel F/S.

The dual linear programming problem is to find a function q # C+(B _ S)so as to

minimize |B _ S

q d+

subject to q(b)+q(s)�V(b, s) for all b, s.

A solution to the primal programming problem is an optimal assignment.It was shown in GOZ1 that the primal and dual linear programmingproblems have solutions (not necessarily unique) and that the optimalvalue for the primal problem and dual problem coincide. We thereforemake the following definition:


Definition. The gains function g : M+(B _ S) � R is defined as

g(+)=max {| V dx : x is feasible for +==min {| q d+ : q is feasible for += .

The gains function plays a central role in our analysis; its basicproperties are summarized in the following result.

Proposition 3. The gains function g is homogeneous of degree 1, con-cave, Lipschitz continuous (with respect to the norm), and weak* continuous.

As we shall see, smoothness of the gains function is connected to perfectcompetition. In preparation for differentiability results we record thefollowing definition.

Definition. The subdifferential �g(+) of the gains function g at themeasure + is the collection of q # C+(B _ S) such that (q, &&+) �g(&)&g(+) for all & # M+(B _ S). Equivalently, because g is homogeneous,(q, +)= g(+) and (q, &)�g(&) for all & # M+(B _ S).

3.4. Walrasian EquilibriumRecall that H is the given set of houses. It is convenient to adjoin to H

an additional fictitious house 0� which no one owns or values; a buyeracquiring 0� will be interpreted as not trading. Write H� =H _ [0� ]. Givenan assignment economy B, S, +, we write +S for the restriction of + toS=H_[0, 1] and +H for the marginal of +S on H; +H is the distributionof houses actually present in the economy. A price system (or just price) isa non-negative continuous function on the space of houses; i.e., a functionp # C+(H ). When convenient, we extend prices so that p(0� )=0.

As usual, a Walrasian equilibrium consists of prices and consumptionsfor which individuals optimize and markets clear. To simplify the connec-tion between Walrasian equilibrium and the core, and to clarify our pricingconvention, we find it useful to exploit the transferable utility frameworkand express optimization in terms of indirect utility functions.9

The utility of agent i for house h is

b(h) if i=b and h # Hvi (h)={r if i=s=(h, r) and h # H

0 if h=0�


9 The indirect utility functions are just the conjugates of the utility functions, in the usualsense of concave analysis; see Fenchel [8], and Rockafellar [28].

Fix a price system p # C+(H ) for all houses. If a seller s # S were to sell herhouse for the price p(?(s)), she would gain p(?(s))&_(s). Of course, theseller might not wish to sell her house at the price p. Thus, the indirectutility function for seller s, expressed in terms of gains, is

vs*( p)=max[0, p(?(s))&_(s)].

Similarly, the indirect utility function for a buyer b # B is

vb*( p)=max[0, suph # H

[b(h)& p(h)]].

We define the indirect utility map

8 : C+(H) � C+(B _ S) (3)

by

8( p)( } )=v*( } )( p).

Definition. Fix a population measure +. A Walrasian equilibrium is aprice system p and a housing distribution y # M+(I_H� ) such that

(i) y1=+;

(ii) y2=+H ;

(iii) +(B)= y([(b, h) # B_H� : b(h)& p(h)=vb*( p)]);

(iv) +(S)= y([(s, h) # S_H� : h=0� and vs*( p)=0])+ y([(s, h) # S_H� : h=?(s) and p(?(s))&_(s)=vs*( p)]).

The first condition says that y is population consistent with +; the secondthat the market for houses clears, and the third and fourth that almost allbuyers and sellers are maximizing utility subject to a budget constraintdefined by prices p. (Hence, because prices and utility functions are con-tinuous, all buyers and sellers in supp + are maximizing utility subject tothe budget constraint.)

As in the finite model, the transferable utility nature of our frameworkimplies interchangeability: if ( p, y), ( p$, y$) are Walrasian equilibria then soare ( p$, y), ( p, y$).

3.5. The Market Game and the Core

Fix a population measure + # M+(I ). The restriction of the gains func-tion g to [& # M+(I ) : 0�&�+], the ideal coalitions, defines a non-atomicideal game in the sense of Aumann and Shapley [4], which we interpret as


the induced market game. To avoid ambiguities, we consider utilities onlyof agents in supp +, who are ``present'' in the economy.10

Definition. The core C(+) of the market game induced by the measure+ consists of those functions q # C(supp +) such that � q d+= g(+) and� q d&�g(&) for 0�&�+.

As in the finite model, order C(+) from the seller's viewpoint, so thatq�q$ exactly when q(s)�q$(s) for almost all s. We show below that C(+)is a compact lattice, so has a largest element q� and a smallest element q

�(the best and worst core utilities, from the viewpoint of sellers).

3.6. Indeterminacy of Solutions

By definition, a Walrasian price system assigns prices to all houses in H.However, some houses may not be traded at equilibrium (certainly housesin H"supp +H , which are not in the endowment of any seller, are nottraded at equilibrium), and there is an indeterminacy about the price ofsuch houses: if ( p, y) is a Walrasian equilibrium in which some housesH0 /H are not traded, then raising the price of houses in H0 while main-taining the price of all other houses will lead to a Walrasian equilibriumwith the same consumptions and utilities, but different prices.

Similarly, a solution to the dual linear programming problem assignsutilities to all buyer and seller types b # B, s # S. However, some buyer andseller types may not be represented in the population measure +, and thereis an indeterminacy about the utilities of such types: if q is a solution to thedual linear programming problem and some buyer and seller types are notrepresented in +, then raising the utility of these types while maintainingthe utility of all other types will lead to a dual solution which differs fromq only in utilities of agents ``not present'' in the economy.

Similarly, too, an element of the subdifferential �g(+) assigns values toall buyer and seller types b # B, s # S, but if q # �g(+) and q$ # C(B _ S) issuch that q$�q and q$=q on supp + then q$ also satisfies the subdifferen-tial inequality, and differs from q only in directions not represented in +.


10 An alternative would be to define the market game as a non-atomic game on Borel sub-sets of supp + (rather than ideal sets) by defining v(A)= g(+ |A) for each Borel set A/B _ S.By definition, the core of this game is the set of finitely additive set functions . such that.(A)�v(A) for every A and .(supp +)=v(supp +). In GOZ1 it is shown that, in the presentcontext, every core set function is necessarily countably additive and absolutely continuouswith respect to +, and that the Radon�Nikody� m derivative of such a core set function is acontinuous function on B _ S. As a consequence, the non-atomic game and the ideal game areequivalent and lead to the same core. For simplicity, we restrict attention here to the idealgame.

To deal with these indeterminacies, it is convenient to adopt conventionsthat assign unique prices to all houses, unique dual utilities to all agents,and unique subdifferential values at every point.

To this end, fix compact sets B/B, S/S and a population measure+ # M+(B _ S). We say that Walrasian prices p, p$ # C+(H ) are equivalentif the induced indirect utilities are the same for all agents in the population:

vi*( p)=vi*( p$)

for every i # supp +. Note that if house h trades at equilibrium thenp(h)= p$(h) whenever p, p$ are equivalent prices.11 For every Walrasianprice p # C+(H ) there is a unique smallest equivalent price p, defined by

p(h)= infb # supp +B

max[0, b(h)&vb*( p)].

It is easily verified that p is non-negative and continuous (because B is anequicontinuous family) so p # C+(H ) is a price system. Moreover, 0�p�1(because buyer valuation functions lie in the same range), p(h)= p(h) if htrades at equilibrium (because, at equilibrium, buyers optimize in theirbudget sets and vi*( p)=vi*( p) for each i # supp +). As we have alreadynoted, p(h)= p(h) for every house h that trades. Moreover, p(h) is thesupremum of all prices at which house h would trade. We adopt theconvention of using the smallest equivalent Walrasian price.

We adopt a parallel convention for dual solutions and elements of thesubdifferential. Say that dual solutions (respectively, elements of the sub-differential �g(+)) q, q$ are equivalent if q(i)=q$(i) for every i # supp +. Forevery dual solution (respectively, element of the subdifferential) q there isa unique smallest equivalent dual solution (respectively, element of thesubdifferential) q, defined by

q(b)= infs # supp +S

[V(b, s)&q(s)]

q(s)= infb # supp +B

[V(b, s)&q(b)].

We adopt the convention of using the smallest equivalent dual solution(respectively, element of the subdifferential).

With these conventions in force, we write P(+) for the set of Walrasianprices12 and D(+) for the set of dual solutions; we continue to write �g(+)for the subdifferential.


11 Keep in mind that interchangeability guarantees that all Walrasian prices support thesame Walrasian allocations.

12 Because interchangeability guarantees that all Walrasian prices support all Walrasianallocations, we suppress the allocation part of equilibrium.

An important consequence of our conventions and equicontinuity of theset B of buyer valuation functions is equicontinuity of the sets of allpossible Walrasian prices and all possible dual solutions:

Proposition 4. If B/B, S/S are compact sets then

P=[ p # C+(H ) : p # P(+) for some + # M+(B _ S)]

D=[q # C+(B _ S) : p # D(+) for some + # M+(B _ S)]

are equicontinuous subsets of C+(H ), C+(B _ S), respectively.

We now obtain results on the structure of Walrasian equilibriumparalleling those in the finite case.

Proposition 5. For every population measure +, the collection ofWalrasian prices P(+) forms a compact convex sublattice of C+(H ), with alargest element p� and a smallest element p

�.

3.7. Equivalence of Solutions

In this section we summarize the connections between Walrasian equi-librium, the linear programming problem, the core, and the subdifferentialof the gains function; for details, see Section 9.

To make the connections between these concepts we introduce somenotation. Recall that we have defined in Eq. (3) the map

8 : C+(H ) � C+(B _ S),

which assigns to any price system the indirect utility function of the agentsfor those prices as

(8p)( } )=v*( } )( p).

Given a population measure +, define

\+ : C+(B _ S) � C+(supp +)

to be the restriction map

\+(q)=q | supp +

Fixing our attention on Walrasian price systems, define the price-subdifferential map

(+ : P(+) � �g(+)


by

(+=8 |P(+)

and the price-core map

8+ : P(+) � C(+)

by

8+=\+ b 8 |P(+).

The following result expresses the essential equivalence of all foursolution notions.

Theorem 2. Let + # M+(B _ S) be a population measure. Then

(i) D(+)=�g(+).

(ii) 8+ : P(+) � C(+) is a homeomorphism and preserves the latticeoperations.

(iii) (+ : P(+) � �g(+) is a homeomorphism.

(iv) The restriction mapping \+ : �g(+) � C(+) is a homeomorphism.

3.8. Matching

It is worth noting the connection between the primal and dual linearprogramming problems and the theory of matching markets; see Roth andSotomayor [29]. Consider a solution x to the primal linear programmingproblem and a solution q to the dual problem. It follows from dualitytheory for linear programming that if (b, s) is in the support of x, thenq(b)+q(s)=V(b, s). Moreover, q(b$)+q(s$)�V(b$, s$) for all b$ # B, s$ # S.Hence there is no buyer-seller pair who are not matched at x and whocould match with each other and divide the gains from their match so thatthey would both be better off than they are at x, q. That is, the buyer�sellermatches and utility divisions represented by x, q, provide stable matchingsin the usual sense of matching markets.

4. PERFECT COMPETITION IN THE CONTINUUM

In Section 2, we identified seven conditions equivalent to perfect com-petition (non-manipulability) in the finite assignment model; in this Sectionwe obtain analogs of these conditions in the continuum assignment modeland show that they remain equivalent. We do not, however, provide anextension of the notion of non-manipulability to the continuum framework,


because the description of manipulation in our distributional framework iscomplicated and adds little to our discussion. (See Gretsky, Ostroy, andZame [12, 13].) Instead, we define perfect competition in the continuum interms of the equivalent conditions analogous to those identified in thefinite model; as we shall see in Section 6, this will be sufficient for thepurpose of connecting continuum and finite economies and establishingapproximate perfect competition in large finite economies.

As we have noted before, the chief issue is the use of a small group (ina continuum economy) as a proxy for an individual (in a finite economy).A useful point of view to keep in mind here is that of perturbations; com-petitivity is expressed as the idea that small perturbations have small effects(in per capita terms). Our approach is to begin with a given populationmeasure, compute the per capita effect of perturbation by a small group,and let the size of the group tend to zero.

As in the previous section, we fix a compact subset B/B of buyers anda compact subset S/S of sellers. We describe the population of theeconomy by specifying a measure + # M+(B _ S), and we describe a smallgroup as a measure & # M+(B _ S). Of particular interest are subpopula-tions: measures & # M+(B _ S) for which &�+. Perturbation of + meansadding and�or subtracting a measure &��but note that (+&&) is non-negative only when & is a subpopulation.

The main result of this section is Theorem 3, below, which shows thatthe various conditions are equivalent. Before stating this result, we collectsome definitions and preliminary results. We make free use of the equiv-alence of core utilities, solutions to the dual linear programming problem,and elements of the subdifferential of the gains function, and of the canonicalidentification between the core and the set of Walrasian prices as given bythe price-core map 8+ .

4.1. Stability

We first investigate when the solution concepts for the assignment modelare stable with respect to changes in the data of the economy. Stabilitymanifests itself as continuity of the solution correspondences, which weinterpret as perfectly elastic demand and supply. Stability is related todifferentiability of the gains function, which in turn is related to the abilityof small groups to fully appropriate their gains from trade.

In the finite assignment model, perfect elasticity of demand and supplymeans that prices do not change if a single individual (buyer or seller)enters or leaves the market. In the continuum assignment model, anindividual is infinitesimal, so we interpret perfect elasticity of demand andsupply as continuity of the price correspondence with respect to thepopulation.


Definition. Let + be the population measure. We say that demand andsupply are perfectly elastic if the price correspondence P is continuous at +(when both M+(B _ S) and C+(H) are given their norm topologies).

For functions whose domains are open sets, Gateaux and Fre� chetdifferentiation are quite familiar notions, but the gains function g is definedonly on the positive cone M+(B _ S), which has empty interior inM(B _ S) (with respect to the norm topology). We therefore use thefollowing definitions of the Gateaux and Fre� chet derivatives, adapted fromVerona [32, 33]. See Section 7 for further discussion and for discussion ofthe connection between differentiability and the subdifferential.

Definition. Let + # M+(B _ S) be an assignment economy. The gainsfunction g is Gateaux differentiable at + if there is a q # C(B _ S) such that

limt � 0+ } g(++t&)& g(+)&(q, t&)

t }=0

for each & # M(supp +) having the property that ++t&�0 for some t>0.Note that the restriction q | supp + is uniquely determined but that thebehavior of q on (B _ S)"supp + is arbitrary. Because the Gateauxderivative is an element of the subdifferential, we follow our conventionand use the smallest equivalent element of the subdifferential. The gainsfunction g is Fre� chet differentiable at + if there is a q # C(B _ S) such that

lim&&& � 0 }

g(++&)& g(+)&(q, &)&&& }=0,

where the limit is taken over & # M(supp +) with &�&+. Again, therestriction q | supp + is uniquely determined but the behavior of q on(B _ S)"supp + is arbitrary. Again, we follow our convention and use thesmallest equivalent element of the subdifferential.

The idea of differentiation leads directly to the notion of marginalproduct and to full appropriation.

Definition. Fix a population measure +. For a subpopulation & definethe per capita marginal products:

mp&+ (&)=

g(+)& g(+&&)&&&

mp++ (&)=

g(++&)& g(+)&&&

.


For any core utility q # C(+), it is readily verified that the per capitapayoff to any subpopulation is trapped between the marginal products:

mp&+ (&)�

(q, &)&&&

�mp++ (&).

Note the similarity to the finite model.The idea that small groups fully appropriate their marginal products is

embodied in the following definition.

Definition. The population measure + exhibits full appropriation ifthere exists q # C(B _ S) such that (q, +)= g(+) and

lim&&& � 0 }mp&

+ (&)&(q, &)

&&& }=0.

We could also define full appropriation by reference to the marginalcontributions mp+

+ ; as we shall see, the two notions are equivalent.By analogy with the finite model, we define adding up by considering, for

each i # B _ S, the marginal product of type i, defined as

MP+(i)= limt � 0+

g(++t$ i)& g(+)t

.

Note that this is just the directional derivative of g at + in the direction $i .

Definition. The economy + exhibits adding up if

| MP+(i) d+= g(+).

4.2. Characterization

The main result of this Section asserts the equivalence of theseconditions.

Theorem 3. For a fixed population measure +, the following areequivalent:

(1) + satisfies full appropriation;

(2) + exhibits adding up;

(3) demand and supply are perfectly elastic;

(4) the core is a singleton;

(5) Walrasian prices are unique;


(6) the gains function g is Gateaux differentiable at +;

(7) the gains function g is Fre� chet differentiable at +.

We call an assignment economy perfectly competitive if it satisfies theseequivalent conditions.

The proof of Theorem 3 rests on

v the equivalence (Theorem 2) of the core, the set of Walrasianprices, the set of solutions to the dual linear programming, and the sub-differential of the gains function;

v the work of Verona [32, 33] characterizing Gateaux and Fre� chetdifferentiability of concave functions in terms of continuity properties of thesubdifferential correspondence;

v the equivalence of Gateaux (respectively, Fre� chet) differentiabilityof the gains function with Gateaux (respectively, Fre� chet) differentiabilityof the restriction of the gains function to the space of absolutely continuousmeasures;

v the characterization, in our setting, of Gateaux differentiability ofthe gains function as unicity of its subdifferential;

v exploitation of the weak* continuity of the gains function to establishthat Gateaux differentiability and Fre� chet differentiability are equivalent inour setting.

The following examples illustrate Theorem 2. We present detailed com-putations only for the first, making use of a method that is quite widelyapplicable and of interest in itself. In these examples we use * to denoteLebesgue measure on the Borel sets of [0, 1].

Example 1. Let B=S=[0, 1]; set ;(b, s)=bs and _(s)=0, so thatV(b, s)=bs. Take the population measures +B=+S=*. We assert that theunique optimal assignment x� matches seller s with buyer b exactly whenb=s (that is, x� is - 2�2 times Lebesgue measure on the diagonal) and thatthe unique point q� in the core is given by

q� (s)=s2�2, q� (b)=b2�2.

In particular, this economy is perfectly competitive.To see this, we proceed in several steps. Let x be an optimal assignment

and let q be any point in the core.

Step 1. supp x does not contain points (b1 , s1), (b2 , s2) such thatb2>b1 and s2<s1 . [If not, let (b1 , s1), (b2 , s2) # supp x be such points.


Note that

b2 s1+b1s2&b2s2&b1 s1=(b2&b1)(s1&s2)>0.

Because q is a dual solution matching, this inequality entails

q(b1)+q(s2)+q(b2)+q(s1)�V(b1 , s2)+V(b2 , s1)

>V(b1 , s1)+V(b2 , s2)

=q(b1)+q(s1)+q(b2)+q(s2),

which is absurd.]

Step 2. Every buyer is matched at x; that is, for each b # B there is atleast one s # S such that (b, s) # supp x. [If not, because +(B)=+(S)=1, itfollows that there are also sellers who are unmatched. Matching unmatchedbuyers and would increase the social gain, contradicting the assumptionthat x is an optimal assignment.]

Step 3. For each b # B there is a unique sb # S such that (b, sb) # supp x.[Otherwise, we could find b1 # B, s1 , s2 # S such that s1<s2 and(b1 , s1), (b1 , s2) # supp x. In view of Step 1, it follows that (b1 , s) # supp xfor each s with s1�s�s2 . Feasibility of x means that the marginal of x onS is less than or equal to +S , so that

x([(b1 , s) : s1�s�s2])�+S([s1 , s2])=s2&s1>0.

However, this means that the marginal of x on B has positive mass at thepoint b1 and hence is not less than or equal to +B (which is Lebesguemeasure, and hence has no atoms). This is a contradiction.]

Step 4. Interchanging the roles of B, S we see that for every s # S thereis a unique bs # B such that (bs , s) # supp x.

Step 5. sb=b for each b # B, so that x is - 2�2 times Lebesgue measureon the diagonal. [To see this, consider the mapping b [ sb , which is afunction (by Step 3). The graph of this function is supp x, which is com-pact, so this function is continuous. Steps 1 and 4 together imply that thisfunction is strictly increasing.13 Feasibility of x implies that this functionsatisfies the equality +B([0, b])=+S([0, sb]).]


13 Thus, the matching is assortative, in the sense of Becker [5]. The reader will note thatthe only property of V used to this point is strict supermodularity: V(b1 , s2)+V(b2 , s1)>V(b1 , s1)+V(b2 , s2) whenever b2>b1 and s2<s1 .

Hence sb=b for each b # B.]

Step 6. We derive a differential equation for the core utility q as a func-tion on S. To this end, fix sellers s0 , s and the corresponding buyers b0 , bwith whom they are matched. Because q is a dual solution, q(b0)+q(s0)=V(b0 , s0)=b0s0 , q(b)+q(s)=V(b, s)=bs and q(s0)+q(b)�V(b, s0)=bs0 .Hence

q(s0)�bs0&q(b)=bs0&[bs&q(s)].

Simplifying and rearranging yields

q(s)&q(s0)�bs&bs0 .

Interchanging the roles of s0 , s and multiplying by &1 yields

b0 s&b0 s0�q(s)&q(s0).

Combining these inequalities and factoring yields

b0(s&s0)�q(s)&q(s0)�b(s&s0).

If s&s0>0 this yields

b0�q(s)&q(s0)

s&s0

�b

and if s&s0<0 it yields

b0�q(s)&q(s0)

s&s0

�b.

Taking the limit as s � s0 and recalling that (at equilibrium) b0=s0 andb=s yields

lims � s0

q(s)&q(s0)s&s0

=s0 .

We conclude that the core utility q is a differentiable function of s andsatisfies the differential equation q$(s)=s, whence

q(s)=s2

2+C

for some constant C.


Step 7. In the assignment x, buyer 0 and seller 0 are matched andV(0, 0)=0, so we obtain the initial condition

q(0)=0.

We conclude that C=0, so q(s)=s2�2 and q(b)=b2�2, as asserted. K

Example 2. As in Example 1, we let B=S=[0, 1], ;(b, s)=bs and_(s)=0, so that V(b, s)=bs, but we increase the population of buyersuniformly: +S=*, +B=(1+=) * (with 0<=<1).

Write $==�(1+=). Arguing as in Example 1, we see that the uniqueoptimal assignment x matches buyer b and seller s exactly when

b=s+=1+=

=(1&$) s+$;

equivalently,

s=(1+=) b&==b&$1&$

for b�$ (buyers b<$ remain unmatched).Proceeding just as in Example 1, we derive the differential equation for q= :

q$=(s)=s+=1+=

so

q=(s)=1

2(1+=)(s+=)2+C.

If s=0 then q(s)=0, so C=&=2�2(1+=), whence

q=(s)=s2

2+

=1+=

s \1&s2+

q=(b)={b2

2&

=2 _

11+=

&(1&b)2&0

for b�$

for b<$.

For each =�0 the core is a singleton, so the economy is perfectly com-petitive. The core utility q= depends continuously on =; as = increases (sothat the proportion of buyers becomes larger), sellers appropriate anincreasing share of the gains from trade. K


The final example of this Section illustrates a point made in the Intro-duction: in an environment in which buyer valuations are not continuous,the various notions of perfect competition need not coincide.

Example 3. Set B=S=[0, 1], +S=* and +B=2*. Set _(s)=0 foreach s and

;(b, s)={10

if s=botherwise.

Note that each buyer likes exactly one house and that there are twice asmany buyers as sellers. It is easily checked that the unique optimal assign-ment is - 2�2 times arc length measure on the diagonal s=b (so that allthe sellers trade but only half the buyers trade) and that the sellers obtainall the gains from trade; that is, the unique point q in the core is given by

q(b)=0, q(s)=1.

On the other hand, buyers do not extract their marginal products.Indeed, if B0 /B then the marginal product of the subpopulation + |B0

is12 +(B0) and the per capita marginal product is 1�2, independent of the sizeof B0 (because buyers in B"B0 have no use for the houses purchased bybuyers in B0). K

5. GENERIC PERFECT COMPETITION

In the finite assignment model, perfect competition is the exception. Inthis Section we show that, in the continuum assignment model, perfectcompetition is the rule.

We recall a standard notion of genericity. A subset S of a completemetric space X is called a G$ set if S is the intersection of a countablefamily of open sets. A property is generic in the space X if the propertyholds for every element in a dense G$ set of X. In view of the BaireCategory theorem (which asserts that the intersection of a countable familyof dense open subsets of a complete metric space is itself dense), genericityis usually accepted as an embodiment of the idea that a property is``typical.''14, 15


14 Genericity in this sense is substantially weaker than the requirement that the propertyholds on a dense open set.

15 It is tempting to make an analogy between genericity in this sense and being the comple-ment of a null set (with respect to some appropriate measure); Oxtoby [25] is a compellingtreatise about the reasons to avoid this temptation.

We have specified an assignment model by a population measure + onB _ S, where B/B and S/S are compact sets which describe the ``currentworld'' of discourse. The space M+(B _ S) of all such population measuresadmits two relevant topologies, the norm topology and the weak* topol-ogy, in each of which it is a complete metric space. Thus there are twopossible senses of genericity; we show that perfect competition is generic ineach sense.

Theorem 4. For each environment B _ S where B/B and S/S arecompact, the set of perfectly competitive economies is dense in the normtopology of M+(B _ S) and is a G$ set in the weak* topology of M+(B _ S).In particular, perfect competition is generic in M+(B _ S) with respect to thenorm topology and with respect to the weak* topology.

In view of Theorem 3, generic perfect competition is equivalent togeneric Gateaux differentiability of the gains function g. Verona [32, 33]establishes generic differentiability of a concave, Lipschitz function definedon a cone in a separable Banach space �� but M+(B _ S) is not separable,and Verona's results cannot be applied directly in our context. (Indeed, aconcave Lipschitz function on M+(B _ S) may fail to be Gateaux differen-tiable at any point.) Instead, in Section 7 we exploit the weak* continuityof g to show that Gateaux differentiability of g at & is equivalent toGateaux differentiability of the restriction g |L1(+)+

at & whenever+, & # M+(B _ S) are mutually absolutely continuous. (We use the Radon�Nikodym theorem to identify L1(+)+ as the cone of positive measureswhich are absolutely continuous with respect to +.) Because L1(+) is aseparable Banach space, Verona's results imply that the restriction g |L1(+)+

is generically Gateaux differentiable; exploiting the equivalence of Gateauxdifferentiability of g at & and of the restriction g |L1(+)+

at & leads to theconclusion that g is Gateaux differentiable at every point of a norm densesubset of M+(B _ S). Finally, exploiting Proposition 7 below leads to theconclusion that the set of points in M+(B _ S) at which g is Gateauxdifferentiable is a weak* G$ set.

In some environments we can say that all economies of a certain type areperfectly competitive. The following Proposition, for instance, guaranteesperfect competition whenever there are ``no gaps'' in the valuations of thesellers or of the buyers; a related idea has been used by Kamecke [16] toobtain unicity of the core in a particular continuous assignment problem.

Proposition 6. If either (a) there are no gaps in the reservation valuesof sellers, in the sense that [r : s=(h, r) # supp +S]=[0, 1] for allh # supp +H , or (b) there are no gaps in the reservation values of buyers, inthe sense that [b(h) : b # supp +B]=[0, 1] for every h # supp +H , then + isperfectly competitive.


A variation on Examples 1 and 2 may help to convey the flavor of thegenericity result.

Example 4. Fix 0�=<1. Set B=S=[0, 1], +S=*, +B=(1+=) *,;(b, s)=bs+m (for m>0), and _(s)=0. Write $==�(1+=). Arguing as inExamples 1 and 2, we can show that the unique optimal assignmentmatches seller s with buyer b=(1&$) s+$ and the core utilities for sellerssatisfy the differential equation

q$(s)=(1&$) s+$2

.

Integration will determine q up to a constant of integration.If =>0 we can solve uniquely for the constant of integration. Note that,

because there are more buyers than sellers, the buyers with the lowest num-bers are unmatched, and hence obtain utility 0 at every point in the core.Continuity of core utilities entails that buyer b=$, who is matched withseller s=0, also obtains utility 0 so that seller s=0 obtains all the gainsfrom trade with buyer b=$. Hence q(0)=m, and q is uniquely determined:

q(s)=(1&$) s2+$s

2+m

q(b)=b2&$b2(1&$)

.

Written differently, for buyer b matched with seller s, the gain from tradeis V(b, s)=bs+m and the core utilities are q(b)= 1

2 bs and q(s)= 12 bs+m.

On the other hand, in the knife-edge case in which ==0 (so that thereare exactly as many buyers as sellers) the constant of integration is indeter-minate, so the core consists of those functions qc (0�c�m) defined by

qc(s)=s2

2+c

qc(b)=b2

2+(m&c)

and this economy is not perfectly competitive. K

To this point we have seen only examples where the core is a singletonor one-dimensional. Proposition 5 guarantees that the core is norm com-pact, but, as the following example shows, it need not be finite-dimensional.


Example 5. Set B=S=[0, 1] and +B=*=Lebesgue measure. Let Cbe the Cantor middle-thirds set in [0, 1] and let f : C � [0, 1] be theCantor function, so that f is continuous and non-decreasing, f is constanton every omitted interval, and f (C)=[0, 1].16 Set +S=� i :i $xi

, whereeach xi is a dyadic rational value of the Cantor function f correspondingto an interval in its domain which is one of the intervals removed duringthe construction of C and :i is the length of that interval. Because there arecountably many such intervals and their total length is 1, it follows that&+S&=1. Finally set

;(b, s)=1&| f (b)&s|1�2

and _(s)=0.The unique optimal matching x is arclength on the horizontal segments

of the graph of f. (Note that the total length of these segments is 1.) Theconditions for a function q�0 to be in the core of this assignment gameare

(i) qB is continuous on B and qS is continuous on S;

(ii) �B qB(b) d*(b)+� qS(s) d&(s)=1;

(iii) qB(b)+qS(s)=1 for s= f (b), i.e., for matched b and s;

(iv) qB(b)+qS(s)�1&|s& f (b)|1�2 for all b and s.

Observe that conditions (ii) and (iii) are equivalent; qB must be constanton each of the intervals removed in the construction of the Cantor set C;and the Walrasian price system corresponding to the core point q is givenby p(s)=qS(s).

Hence, from condition (iv) we have for every s, b with s> f (b) that

qB(b)+ p(s)�1&(s& f (b))1�2

and from condition (iii) for s= f (b) that

qB(b)+ p( f (b))=1

and thus that for all s� f (b)

p( f (b))& p(s)�(s& f (b))1�2.


16 Recall that C is constructed by removing from [0, 1] the open intervals (1�3, 2�3),(1�9, 2�9), (7�9, 8�9), ... The function f is defined to be 1�2 on (1�3, 2�3), 1�4 on (1�9, 2�9), 3�4on (7�9, 8�9), ..., and then is extended to all of [0, 1] by continuity. See Aliprantis andBurkinshaw [1] for details.

A similar inequality holds for s< f (b) and thus we have that

| p( f (b))& p(s)|�|s& f (b)|1�2.

We extend this result by continuity to the closure of the range of f so thatwe have for any Walrasian price system p and any s, t # S that

| p(s)& p(t)|�|s&t|1�2.

Conversely, every continuous p : S � [0, 1] which satisfies this condition isa Walrasian price and determines a corresponding element of the core.In particular, the core for this example is infinite-dimensional since thecollection of Walrasian prices is.17 K

6. LARGE FINITE ECONOMIES

As we have discussed in the Introduction, our study of perfect competi-tion in continuum assignment models is motivated in large part by thedesire to understand approximate perfect competition in large finite assign-ment models. It is to this subject that we now turn. The general approachthat we follow is familiar from the literature on exchange economies (seeHildenbrand [15] for instance): Consider a sequence of finite assignmentmodels, (En). For each n, let |En | be the number of individuals in theeconomy, and let +En

be its normalized population measure. Note that thefinite economy En and the continuum economy described by the populationmeasure +En

have the same core and the same Walrasian equilibrium prices.The ability of a particular individual i to manipulate a given finite

economy can be expressed in terms of i 's best and worst core utilities. Inview of the discussion in Section 2, no manipulation can ever yield i morethan his�her best core utility (because this best core utility is i 's marginalproduct). Moreover, as we shall see in the proof of Theorem 1, the rightmanipulation will guarantee i his�her best core utility (which is i 's marginalproduct). Given a particular point q # C(+), the difference between i 's bestcore utility and q(i) is a measure of i 's ability to manipulate from q. Hencethe difference between the best and worst core utilities is a measure oflargest manipulation possible for i; the best and worst core utilities coincideprecisely when i cannot manipulate at all. The overall manipulability of agiven economy can therefore be measured by the diameter of the core, orequivalently by the distance between the largest and smallest core utilities,q� , q

�.


17 As the reader might suspect, the non-differentiability of V is a crucial feature here. IfV(b, s)=1& |s& f (b)|: for :>1, computations similar to those above could be used todemonstrate that Walrasian prices must be constant, so the core would be one-dimensional.

We are therefore led to define the average diameter of the core of the(finite or continuum) assignment economy + as

dave(+)=1

&+& | |q� &q�| d+

and the maximum diameter as

dmax(+)= maxi # supp +

|q� (i)&q�(i)|.

The following result establishes upper semi-continuity of the averagediameter (with respect to weak* convergence of populations) and of themaximum diameter (with respect to weak* convergence of populations andconvergence of supports).

Proposition 7. If (+n)/M+(B _ S) is a sequence of populationmeasures converging in the weak* topology to + # M+(B _ S), then

lim supn � �

dave(+n)�dave(+).

If, in addition, supp +n � supp + (in the Hausdorff metric) then

lim supn � �

dmax(+n)�dmax(+).

Proposition 7 provides the necessary means to connect perfect competi-tion in continuum economies to approximate perfect competition in finiteeconomies.18

Theorem 5. Let B/B, S/S be compact sets of buyers and sellers(respectively). Let + # M+(B _ S) be a normalized population measuredefining a continuum assignment model.

(i) If + is perfectly competitive and (En) is a sequence of finiteeconomies such that supp +En

/B _ S and +En� +, then for n sufficiently

large, En is approximately perfectly competitive, in the sense that most agentscannot manipulate very much: for every =>0 there is an index N such that,if n�N then fewer than = |En | individuals in En can manipulate by as muchas =.

(ii) If + is perfectly competitive and (En) is a sequence of finiteeconomies such that supp +En

/B _ S and +En� + and supp +En

� supp +,


18 As noted earlier, we do not discuss manipulation directly in the continuum, but seeGretsky, Ostroy, and Zame [12, 13].

then for n sufficiently large, En is approximately perfectly competitive, in thestronger sense that no agents can manipulate very much: for every =>0 thereis an index N such that if n�N then no individual in En can manipulate byas much as =.

(iii) If + is not perfectly competitive then there exists some sequence(En) of finite economies with |Fn | � � and +Fn

� + and supp +Fn� supp +

and which are not approximately perfectly competitive, in the sense that, forall sufficiently large n, many agents in Fn can manipulate a lot: there is a$>0 and an index M such that, for every n�M there are at least $ |Fn |individuals in Fn who can manipulate by at least $.

The first two parts of Theorem 5 follow easily from Proposition 7 andour remarks about the connection between manipulation and the best andworst core utilities. The proof of the last part is constructive: given acontinuum economy + we construct (in Section 9) finite economies En con-verging to + with the property that every core utility q # C(+) induces anelement of the core of En .

As we noted in the Introduction, when reservation values are not con-tinuous, the connection between finite and continuum economies is muchmore tenuous. The following example illustrates what can go wrong.

Example 6. As in Example 3, set B=S=[0, 1], +B=*, +S=2*,_(s)=0 and

;(b, s)={10

if s=botherwise.

The optimal assignment is - 2�2 times arc length measure on the diagonals=b, and the unique point q in the core is given by

q(b)=1, q(s)=0

so that the buyers appropriate all the gains from trade.Consider three sequences of finite economies, each of whose distributions

converges to +:

En : In the economy En there are n buyers and 2n sellers; 1 buyer ateach of the points 1�n, 2�n, ..., n�n and 2 sellers at each of the points1�n, 2�n, ..., n�n.

Fn : In the economy Fn there are n buyers and 2n sellers; 1 buyer ateach of the points 1�n, 2�n, ..., n�n, 1 seller at the point 1�n, 1 seller at thepoint 1�(n+n2), and 2 sellers at each of the points 2�(n+n2), ..., n�(n+n2).

Gn : In the economy Gn there are n buyers and 2n sellers; 1 buyer ateach of the points 1�n, 2�n, ..., n�n and 1 seller at each of the points 1�2n,2�2n, ..., 2n�2n.


The economy En consists of n independent copies of a market with 1 buyerand 2 identical sellers; the buyers appropriate all the gains from trade, soEn is perfectly competitive. The economy Fn consists of a market with 1active buyer and 1 active seller and 3n&2 dummies; the division of gainsbetween the active buyer and seller is arbitrary, so the maximum diameterof the core is 1, but the average diameter of the core is 2�3n. The economyGn consists of n independent copies of a market with 1 active buyer and 1active seller, and n dummies; the maximum diameter of the core is 1 andthe average diameter of the core is 2�3. K

7. CONCAVE ANALYSIS

Concave analysis in finite dimensional spaces is a familiar and well-developed subject; Rockafellar [28] is the standard reference. Concaveanalysis on open subsets of infinite dimensional spaces is undoubtedly lessfamiliar, but it too is a well-developed subject; Giles [10] and Phelps [26]are convenient references. However, the concave function of present interestto us��the gains function��is defined, not on an open set, but on a convexcone whose interior is empty. The elements of concave analysis in such asetting have been developed in recent work by Verona [32, 33]. We gatherhere those parts of Verona's work which are relevant to our interests.

Let X be a Banach space and let X* be its dual. Let Y/X be a convexset that is not contained in a translate of any proper closed subspace of X.Say that y # Y is a support point of Y if there is a non-zero linear functional. # X* such that .( y)=0 and . |Y�0; otherwise, y is a non-support pointor quasi-interior point. Note that every interior point of Y is a quasi-interiorpoint, but not vice versa. In particular, Y may possess quasi-interior pointseven when it has no interior points.19

Let Y0 be the quasi-interior of Y (i.e., the set of quasi-interior points), letU/Y0 be a (relatively) open subset, let g : U � R be a concave, Lipschitzfunction on U and let y # U. Following Verona [32, 33], define the sub-differential �g( y) of g at y to be the set of all linear functionals q # X*such that (q, z& y) �g(z)& g( y) for all z # Y. Say that g is Gateauxdifferentiable at y if there is a linear functional q # X* such that

limt � 0+ }g( y+tz)& g( y)&(q, tz)

t }=0


19 For instance, if * is a finite measure and is not the sum of a finite number of atoms, thenL1(*) is infinite dimensional, and the positive cone L1(+)+ /L1(+) has empty interior, butthe quasi-interior consists of all functions which are strictly positive almost everywhere.

for each z # X having the property that y+tz # Y for some t>0. If such aq exists, it is unique, and is called the Gateaux derivative of g at y. Say thatg is Fre� chet differentiable at y if there is a q # X* such that

lim&z& � 0 }

g( y+z)& g( y)&(q, z)&z& }=0,

where the limit is taken over z # X with y+z # Y. Again, if such a qexists, it is unique, and is called the Fre� chet derivative of g at y. Fre� chetdifferentiability implies Gateaux differentiability.

Verona shows that analysis of concave Lipschitz functions on the quasi-interior of Y parallels almost exactly the more familiar analysis of concavefunctions on open subsets of X.20 In what follows, we fix a Banach spaceX with dual X*, a convex subset Y/X that generates X, with quasi-inte-rior Y0 , a relatively open subset U/Y0 and a concave Lipschitz functiong : U � R. Verona establishes the following results concerning the sub-differential correspondence and its connections with Gateaux and Fre� chetdifferentiability.

V1 For each y0 # U, the subdifferential �g( y0) is a non-empty, convex,weak* compact subset of X*. Moreover, the subdifferential correspondence

y [ �g( y) : U � X*

is norm-to-weak* upper hemi-continuous.21

V2 For y0 # U, the following are equivalent:

(i) the subdifferential �g( y0) is a singleton;

(ii) the subdifferential correspondence y [ �g( y) is norm-to-weak*continuous at y0 ;

(iii) g is Gateaux differentiable at y0 .

V3 For y0 # U, the following are equivalent:

(i) the subdifferential correspondence y [ �g( y) is norm-to-normcontinuous at y0 ;

(ii) g is Fre� chet differentiable at y0 .

V4 If X is separable the set of points y0 # U at which g is Gateauxdifferentiable is a dense G$ .22


20 On open sets, concave functions are necessarily Lipschitz; on convex sets without interiorpoints, the Lipschitz property is an additional assumption.

21 By norm-to-weak* upper hemi-continuous we mean upper hemi-continuous with respectto the norm topology on U/X and the weak* topology on X*.

22 For a different notion of generic differentiability, see Anderson and Zame [3].

In our setting, we would like to take X=M(B _ S) and Y=M+(B _ S).Unfortunately, if B _ S is uncountable then the quasi-interior of the coneM+(B _ S) is empty.23 Hence the cited results of Verona are not directlyapplicable in our setting. A natural modification is available, however.Keeping in mind that M(B _ S) is the dual of C(B _ S), we view M(B _ S)as equipped with the weak* topology. In the weak* topology, M(B _ S) isseparable24 and the quasi-interior points (the positive measures of full sup-port) are dense in the positive cone M+(B _ S). For such points, thedefinitions above extend in a natural way. Because we are also concernedwith population measures + that do not have full support, we wish toextend the definitions to cover that case also. And here we find an addi-tional difficulty: If + # M+(B _ S) is a population measure and supp + is aproper subset of B _ S, then the notions of Gateaux and Fre� chet differen-tiability of g at + are well-formed only for ``directions'' & # M(supp +)(which we view as the subspace of M(B _ S) consisting of measures whosesupport is contained in supp +.) These considerations lead to thedefinitions of Gateaux and Fre� chet differentiability given in Section 4.

To build a bridge between Verona's results and our needs, fix compactsets B/B, S/S and a population measure + # M+(B _ S). If + # M+

(B _ S) we use the Radon�Nikodym theorem to view L1(+) as the sub-space of M(B _ S) consisting of measures absolutely continuous withrespect to +. (If & # M(B _ S) is absolutely continuous with respect to +then the Radon�Nikodym derivative d&�d+ belongs to L1(+). We abusenotation and identify an element of L1(+) interchangeably as a function oras a measure.) Note that & # L1(+)+ is a quasi-interior point of L1(+)+ ifand only if the Radon�Nikodym derivative d&�d+ is strictly positive almosteverywhere (with respect to +) if and only if + is absolutely continuous withrespect to &. Note that if +, & are mutually absolutely continuous then theyhave the same support (but not vice versa). Write g+= g |L1(+)+

for therestriction of the gains function g : M+(B _ S) � R to the positive cone ofL1(+). The following result, which follows from GOZ1, exploits the weak*continuity of the gains function. Recall that C(+) is the core of the idealgame for the population distribution +.

Proposition 8. Let B/B, S/S be compact sets, and let + # M+

(B _ S) be a population measure. If & # L1(+)+ is a quasi-interior point andq # �g+(&) then q is continuous on supp +=supp &, and q | supp + # C(&).


23 Given a measure + # M+(B _ S), choose any point i # B _ S for which +([i])=0. (Sucha point exists because B _ S is assumed uncountable.) Let . # M(B _ S)* be the linearfunctional given by integration against the characteristic function /[i] . Then .(+)=0 and. | M+(B _ S)�0. This does not contradict the cited results of Verona because M(B _ S) is notseparable.

24 Assuming, as we have done here, that B, S are compact metric spaces.

Proposition 8 enables us to show that Gateaux and Fre� chetdifferentiability coincide for the gains function g, and to reduce it todifferentiability of the restriction g+= g |L1(+)+

.25

Proposition 9. Let B/B, S/S be compact sets, let + # M+(B _ S)be a population measure, and let & # L1(+)+ be a non-support point. Thefollowing are equivalent:

(i) The restriction g+ is Gateaux differentiable at &.

(ii) The restriction g+ is Fre� chet differentiable at &.

(iii) g is Gateaux differentiable at &.

(iv) g is Fre� chet differentiable at &.

8. CONCLUDING REMARKS

In this paper, the traditional meaning of perfect competition��theinability of individuals to influence prices��has been given a rigorous for-mulation in the assignment model, and conditions equivalent to perfectcompetition have been identified: full appropriation, adding up, perfectlyelastic demand and supply, unicity of the core, uniqueness of Walrasianprices, and Gateaux and Fre� chet differentiability of the gains function.Much of the analysis is based on perturbations: In the finite assignmentmodel, the relevant perturbations are by individuals; in the continuum therelevant perturbations are by small groups. Perfect competition is rare infinite models, but generic in continuous continuum models. Perfect com-petition in such continuum models is the reflection of approximate perfectcompetition in large finite models: the inability of individuals to affectprices very much.

It remains to speculate about extensions of these results. In Gretsky,Ostroy, and Zame [12], we explore extensions to assignment models inwhich buyer reservation functions are upper semi-continuous but notnecessarily continuous. As in the continuous environment, unicity of thecore is associated with Gateaux differentiability of the gains function.26

Perfect competition, however, is associated with Fre� chet differentiabilityof the gains function and continuity of the core correspondence. In the


25 Our argument makes use of the fact that g is weak* continuous. Isaac Namioka andRobert Phelps have pointed out to us a general abstract argument that Gateaux and Fre� chetdifferentiability coincide for a weak* continuous concave function defined on an open set ofa dual Banach space��but g is only defined on a cone, not on an open set, and the generalargument does not seem to apply.

26 To be precise, unicity of the core is equivalent to Gateaux differentiability of the restric-tion of the gains function to the cone of absolutely continuous positive measures.

continuous environment, Fre� chet differentiability of the gains function andcontinuity of the core correspondence are consequences of unicity of thecore, but this is not the case in the upper semi-continuous environment.Consequently, unicity of the core is generic in upper semi-continuousenvironments but perfect competition is not. (Ostroy and Zame [24]provide another environment in which robust imperfect competition��eveninequivalence of the core with the set of Walrasian allocations��is possiblein continuum models.)

The assignment model is a particular instance of a transferable utilityexchange economy, but it is special in several ways, which we haveexploited here.

v The separation of the market into buyers and sellers, each wishingto trade at most one unit of an indivisible commodity, implies that the corealways coincides with the set of Walrasian allocations.

v Individual marginal products and directional derivatives coincide.

v For each individual there is a Walrasian equilibrium at which thatindividual extracts his marginal product.

These properties make the analysis of perfect competition in the assignmentmodel easier than in a general transferable utility exchange economy.Nevertheless, the analysis in the assignment model points to the possibilityof a similar analysis for general transferable utility economies. As theassignment model suggests, perhaps the most crucial issue is genericFre� chet differentiability of the gains function for continuum economies. Theproof we have given here exploits the structure of the assignment model,but we conjecture that this conclusion will obtain whenever the gains func-tion is weak* continuous, a property that will be satisfied in quite generaltransferable utility economies (assuming compactness of the space of com-modities and equicontinuity of preferences). As in the assignment model,Fre� chet differentiability of the gains function for a given continuumeconomy E will have strong implications for approximate perfect competitionof a sequence of large finite economies En converging to E. In particular,because Fre� chet differentiability entails continuity of the subdifferentialcorrespondence, we will be able to conclude that for large n, the subdif-ferential of the gains function in the economy En has small diameter.Approximate perfect competition for En will then follow if it is possible toestablish that, for large n, each individual in En approximately extracts hismarginal product at some Walrasian equilibrium.

Indeed, the analysis of perfect competition in the assignment model is insome ways more challenging than in other transferable utility models. Inthe assignment model, goods are indivisible and utility functions are neverdifferentiable; differentiability of the social gains function must arise


entirely through aggregation. In general transferable utility models,divisibility of goods and differentiability of utility functions should onlyease the task of establishing generic differentiability of the social gainsfunction (perfect competition).

Analysis of general exchange economies, in which utility is not transferable,will of necessity be more complicated. However, some of the groundworkis already in place. In particular, an analog of full appropriation��calledno-surplus��has been formulated and connections to differentiability andperfect competition have been pointed out in Ostroy [23]. Continuity ofthe Walrasian correspondence (which implies uniqueness of Walrasianequilibrium in the assignment model) is compatible with multiple equilibriain the general model, but has important connections with regulareconomies (see Debreu [7], Mas-Colell [21]), which in turn is connectedto non-manipulability in transferable utility environments (see Roberts andPostlewaite [27]). For the relation between perfect competition and non-manipulability in more general exchange economies where utility is nottransferable but the number of commodities is finite, see Makowski,Ostroy, and Segal [20].

9. PROOFS

Because we will derive much of Theorem 1 from Theorem 3, we defer theproof of Theorem 1.

Proof of Proposition 2. In view of Theorem 1, perfect competition forthe economy specified by the parameters ;, _ is equivalent to adding up:

g(1)= :i # B _ S

MPi (1).

By definition of the gains function and of marginal products:

MPi (1)=g(1)& g(1&1i)

=max :j

V(bj , s j)& maxbj , sj{i

:j

V(bj , sj).

Each summation is taken over an arrangement of distinct players bj , sj ; thefirst maximum is taken over all such arrangements, the second summationand maximum are taken over all such arrangements that exclude individuali. Adding up can thus be written as:

max :j

V(bj , sj)=:i _max :

j

V(bj , sj)& maxbj , sj{i

:j

V(bj , sj)& . (4)


By definition,

V(b, s)=;(b, s)&_(s).

Hence, the left side of Eq. (4) is the maximum of a finite number of func-tions that are algebraic in the parameters defining the economy; the rightside of Eq. (4) is the sum of differences of such functions. In particular,both sides of Eq. (4) are continuous and semi-algebraic (see Blume andZame [6]) so the set of economies that satisfy Adding Up is closed and isthe union of a finite number of closed submanifolds.

To complete the proof it remains only to show that the set of perfectlycompetitive economies in V does not contain any open set. To this end,let V # V be a perfectly competitive economy. According to Theorem 1,the core of this economy is a singleton. By assumption, there is at leastone buyer�seller pair for whom the gain from trade is strictly positive.Choose a pair b1 # B, s1 # S for whom the gain from trade V(b1 , s1) ismaximal among matched pairs. Let q be the unique core utility, so thatq(b1)+q(s1)=V(b1 , s1). For =>0 define ;= , _= by27

;=(b, s)={(1&=) ;(b1 , s1)+=(1&=) ;(b, s)

if (b, s)=(b1 , s1)otherwise

_=(s)={(1&=) _(s1)_(s)

if s=s1

otherwise.

It is easily checked that the core of this economy consists of all functionsq: (0�:�=), where

q:(b)={(1&=) q(b)+:(1&=) q(s)

if b=b1

otherwise

q:(s)={(1&=) q(s)+(=&:)(1&=) q(s)

if s=s1

otherwise.

In view of Theorem 1, this economy is not perfectly competitive, so theproof is complete. K

Before going further, it is convenient to establish equicontinuity ofWalrasian prices and dual solutions.

Proof of Proposition 4. Fix B, S and =>0. Equicontinuity of B meansthat we can choose $>0 such that if h1 , h2 # H and dist (h1 , h2)<$ then|b(h1)&b(h2)|<= for each b # B.


27 The leading factor (1&=) guarantees that the function ; takes values in [0, 1].

To establish equicontinuity of P, fix a population measure + # M+

(B _ S) and a Walrasian equilibrium price p # P(+). The definition of equi-librium and of indirect utilities and our conventional pricing constructionguarantee that

p(h1)= infb # supp +

[b(h1)&vb*( p)]

p(h2)= infb # supp +

[b(h2)&vb*( p)],

whence

| p(h1)& p(h2)|� supb # supp +

|b(h1)&b(h2)|�=

whenever dist(h1 , h2)<$. We conclude that P is an equicontinuous family.

To establish equicontinuity of D on B, fix b1 , b2 # B with dist(b1 , b2)<$.Fix a population measure + # M+(B _ S) and a dual solution q # D(+).Our convention for dual solutions guarantees that there is an s1 # supp +S

such that

q(b1)=V(b1 , s1)&q(s1).

Our convention and equicontinuity also guarantee that

q(b2)= infs # supp +S

[V(b2 , s)&q(s)]�V(b2 , s1)&q(s1)<V(b1 , s1)+=&q(s1).

Combining these yields

q(b2)<q(b1)+=.

Reversing the roles of b1 , b2 yields

|q(b1)&q(b2)|<=

whenever dist(b1 , b2)<$, so D is equicontinuous on B.To establish equicontinuity of D on S, fix s1 , s2 # S with dist(s1 , s2)<$.

As above, our pricing convention guarantees that there is a b1 # supp +B

such that

q(s1)=V(b1 , s1)&q(b1)

and

q(s2)= infb # supp +B

[V(b, s2)&q(b)]�V(b1 , s2)&q(b1)<V(b1 , s1)+=&q(b1).


Combining these yields

q(s2)<q(s1)+=.

Reversing the roles of s1 , s2 yields

|q(s1)&q(s2)|<=

whenever dist(s1 , s2)<$, so D is equicontinuous on S. K

We now establish the basic properties of the gains function.

Proof of Proposition 3. Concavity and homogeneity of g are obvious.To see that g is Lipschitz, fix population measures :, ; # M+(B _ S); weassert that

| g(:)& g(;)|�&:&;&.

In order to estimate | g(:)& g(;)| we find it convenient to first estimate| g(:)& g(: 7 ;)|; this will be easier, because : 7 ;�:, and in particular,:7 ; is absolutely continuous with respect to :.

Write #=: 7 ;. Note that #=:&(:&;)+, so that 0�#�: and&:&#&�&:&;&. Let x # M+(B_S) be an optimal assignment for thepopulation measure :; by definition of the gains function

g(:)=|B_S

V(b, s) dx(b, s).

We construct an assignment y which is feasible with respect to the popula-tion measure # and for which &x& y&�&:&#&. From y we will obtain anupper bound for g(:)& g(#).

Because 0�#�:, # is absolutely continuous with respect to : and theRadon�Nikodym derivative f =d#�d: has the property that 0� f�1(almost everywhere with respect to :). Given Borel sets E/B, F/S, define

y(E_F )=|E_F

f (b) f (s) dx(b, s).

By the Carathe� odory extension theorem, y admits a unique extension to aBorel measure on B_S. Because 0� f�1, the measure y is a feasibleassignment for the population measure #.


Keeping in mind the feasibility condition for x and that 1& f=d(:&#)�d:, we obtain

&x& y&=x(B_S)& y(B_S)

=|B_S

[1& f (b) f (s)] dx(b, s)

=|B_S

[1& f (b)] dx(b, s)+|B_S

f (b)[1& f (s)] dx(b, s)

�|B_S

[1& f (b)] dx(b, s)+|B_S

[1& f (s)] dx(b, s)

�|B

[1& f (b)] d:B(b)+|S

[1& f (s)] d:S(s)

=|B _ S

[1& f ] d:

=&:&#&

�&:&;&.

Because 0�V(b, s)�1, it follows that

|B_S

V(b, s) dx(b, s)&|B_S

V(b, s) dy(b, s)�&x& y&�&:&;&.

Because #�;, it follows that g(#)�g(;). Moreover, because y is a feasibleassignment for #, the definition of the total gains function implies that

g(;)�g(#)�|B_S

V(b, s) dy(b, s).

Putting these together implies that

g(:)& g(;)�&:&;&.

Finally, reversing the roles of :, ; yields the desired Lipschitz inequality:

| g(:)& g(;)|�&:&;&.

To show that g is weak* continuous, note first that the positive coneM+(B _ S) is metrizable in the weak* topology, so it suffices to show thatif (+n)/M+(B _ S) is a sequence converging to + # M+(B _ S) in theweak* topology then g(+n) � g(+). To this end choose, for each n a dualsolution qn # D(+n). In view of Proposition 4, the set of all dual solutions


is an equicontinuous family, so, passing to a subsequence if necessary, wemay assume that there is a continuous function q # C+(B _ S) such thatqn � q uniformly. Because +n � + weak* and qn � q uniformly,

|B _ S

qn d+n � |B _ S

q d+. (5)

We assert that q is a dual solution for the population measure +. Toverify feasibility note that, because each qn is a dual solution, convergenceof the qn 's implies that

q(b)+q(s)�V(b, s)

for each b # B, s # S. To verify that q minimizes the integral, suppose thatq$ # C+(B _ S) is feasible and

|B _ S

q$ d+<|B _ S

q d+.

Using continuity of q$, weak* convergence of +n and (5), we see that

|B _ S

q$ d+n<|B _ S

qn d+n

for n sufficiently large, which contradicts the fact that qn is a dual solutionfor the population measure +n . We conclude that q is a dual solution for+ as asserted.

But now we have

g(+n)=|B _ S

qn d+n � |B _ S

q d+= g(+)

as desired. We conclude that g is weak* continuous, as asserted. K

Proof of Proposition 5. To see that P(+) is a lattice, fix Walrasianprices p1 , p2 ; we must show that the supremum p1 6 p2 and the infimump1 7 p2 are equilibrium prices. Because of the interchangeability property,there is a housing distribution y such that ( y, p1), ( y, p2) are bothWalrasian equilibria; we claim that ( y, p1 6 p2), ( y, p1 7 p2) are alsoWalrasian equilibria. Because y is a feasible allocation, we must only seethat consumption choices are in budget sets at prices p1 6p2 , p1 7 p2 andthat consumers optimize given prices p1 6 p2 , p1 7p2 . To this end, fix apair (i, h) # supp y, so that h # H� is the house that individual i consumes atthe allocation y. We need to check budget feasibility and optimization.


Budget feasibility: Because ( y, p1), ( y, p2) are both Walrasian equilibria,h is in i 's budget set at both prices p1 , p2 and hence��because i consumesonly one house and no other commodities��is in i 's budget set at thehigher of these two prices, and a fortiori at the lower of these two prices;that is, h is budget feasible for i at prices p1 6p2 and p1 7 p2 .

Optimization: Consider an alternative house h$; without loss, assumep1(h$)�p2(h$), so that ( p1 6p2)(h$)= p2(h$) and ( p1 7 p2)(h$)= p1(h$). Ifh$ is in i 's budget set at price p1 6 p2 then it is in i 's budget set at pricep2 ; by assumption, h is an optimal choice at price p2 so h is weaklypreferred to h$. If h$ is in i 's budget set at price p1 7 p2 then it is in i 'sbudget set at price p1 ; by assumption, h is an optimal choice at price p1

so h is weakly preferred to h$. Hence h is an optimal choice for i at pricesp1 6 p2 and p1 7 p2 .

We conclude that p1 6 p2 , p1 7p2 are equilibrium prices, so P(+) is alattice, as asserted.

In view of Proposition 14, P(+) is an equicontinuous family; it isevidently bounded and closed in C+(H ), so is a compact set.

Because P(+) is a lattice, the supremum and infimum of any finite set ofelements of P(+) are again in P(+); compactness implies that thesupremum and infimum of all elements of P(+) (which are limits of finitesuprema and infima) are again in P(+), so P(+) has a largest element p� anda smallest element p

�. K

We now establish equivalence of solutions for continuum economies.

Proof of Theorem 2. We show first that dual solutions and elements ofthe subdifferential coincide. To see this, first fix q # �g(+), so that (q, +) =g(+) and (q, &) �g(&) for each & # M+(B _ S). The latter inequality entailsin particular that q�0. For b # B, s # S, setting &=$b+$s yields

q(b)+q(s)=(q, $b+$s)

�g($b+$s)

=V(b, s).

Hence q solves the dual linear programming problem for +. Conversely, ifq solves the dual linear programming problem for + then (q, +) = g(+)and q(b)+q(s)�V(b, s) for all b, s. For & # M+(B _ S), the definition ofthe gains function is

g(&)=inf[(q$, &) : q$(b)+q$(s)�V(b, s) for all b, s].

The function q belongs to the set over which the infimum is taken, so(q, &)�g(&). Hence q # �g(+). We conclude that D(+)=�g(+), which is (i).


We show next that 8 : C+(H ) � C+(B _ S) is continuous. To this end,let pn � p in C+(H ). Given =>0, choose an index N so that n�N O&pn& p&<=. For b # B and n�N we have

|8( pn)(b)&8( p)(b)|=|vb*( pn)&vb*( p)|

=|max[0, suph # H

[b(h)& pn(h)]]

&max[0, suph # H

[b(h)& p(h)]]|

� | pn(h)& p(h)|

<=.

Hence &8( pn)(b)&8( p)(b)&�= whenever n�N. We conclude that 8 iscontinuous, as asserted.

Now fix a population measure +. Let ( p, y) be a Walrasian equilibrium.For i # B _ S, optimization implies that the indirect utility vi*( p) is theutility actually obtained by i at the equilibrium ( p, y). Of course,Walrasian utilities are in the core, so we conclude that 8( p) | supp +=8+( p) # C(+). Hence 8+ maps P(+) into C(+).

To see that 8+ is one-to-one and onto, we exhibit its inverse9+ : C(+) � P(+), which is defined by

9+(q)(h)= supb # supp +B

[b(h)&q(b)].

To see that 9+(q) # P(+) whenever q # C(+), write p=9+(q), fix an optimalassignment x and let y be the induced housing distribution; we claim that( p, y) is a Walrasian equilibrium. Because x is an assignment, y satisfiesmarket clearing. The core property guarantees that, for each buyerb # supp +B and seller s # supp +S ,

q(b)+q(s)�b(?(s))&_(s),

with equality when b, s are matched in x (equivalently, when b consumeshouse ?(s) in y). It follows that sellers in supp +S are optimizing at theprice p and that buyers in supp +B are optimizing at the price p whenrestricted to choose houses in supp +H (i.e., houses that are actuallyavailable). Our pricing convention then guarantees that all buyers insupp +B are optimizing even if they are not restricted to choose houses thatare actually available. Thus ( p, y) is a Walrasian equilibrium, as desired.

Hence 9+ is well-defined. The construction, together with the definitionof indirect utilities, implies that 8+ b 9+(q)=q for each q # C(+) and that9+ b 8+( p)= p for each p # P(+). That is, 8+ and 9+ are inverses; inparticular, 8+ is one-to-one and onto.


Because P(+) is compact, so is its image 8+[P(+)]=C(+). Because theinverse of a one-to-one continuous mapping between compact spaces iscontinuous, it follows that 9+ is also continuous.

To see that 8+ preserves the lattice operations, fix prices p, p$ # P(+).Interchangeability means that p, p$ support the same housing allocations;let y be any Walrasian housing allocation, so that ( p, y), ( p$y) are bothWalrasian equilibria. Now fix a seller s # supp +. If s consumes her housein y, then she does not wish to sell it at either price p, p$, whence

vs*( p)=0=vs*( p$).

On the other hand, if s does not consume her house in y then she does wishto sell it at either price p, p$ whence

vs*( p)= p(?(s))&_(s), vs*( p$)= p$(?(s))&_(s).

Thus, in either case we have

vs*( p 7p$)=vs*( p) 7 vs*( p$)

vs*( p 6p$)=vs*( p) 6 vs*( p$).

Our convention ensures that the same equalities also hold fors # S"supp +S . Because we order core utilities from the viewpoint of thesellers, we conclude that 8+ preserves the lattice operations. This completesthe proof of (ii).

Now fix a population measure +. Let \+ : C(B _ S) � C(supp +) be therestriction map. Our observations about the core, the subdifferential andsolutions to the dual linear programming problem, together with our con-vention about values of functions in �g(+) off supp +, imply that\+ : �g(+) � C(+) is one-to-one and onto. Because 8+=\+ b 8, it followsthat 8[P(+)]=�g(+) and hence that �g(+) is compact. As noted earlier,a continuous one-to-one onto mapping between compact spaces has a con-tinuous inverse; since 8 and \+ are continuous, this completes the proof of(iii) and (iv). K

Before we can establish equivalence of the various notions of perfectcompetition for continuum economies, we establish the results of Section 7,which connect Verona's results to our setting.

Proof of Proposition 8. This follows immediately from Theorems 5 and6 in GOZ1. K


Proof of Proposition 9. Because weak* compact sets are closed, weak*compact subsets of norm compact sets are norm compact. Hence a corre-spondence which is norm-to-weak* continuous and whose range iscontained in a norm compact set is norm-to-norm continuous. In view ofProposition 8, the restriction to supp + of the subdifferential of g+ at & isa subset of C(&)/C(supp &)=C(supp +). In view of Proposition 4 andTheorem 2, the range of the subdifferential correspondence is contained ina norm compact subset of C(supp +) and hence in a norm compact subsetof L�(+). It follows that, for the subdifferential correspondence, norm-to-norm continuity coincides with norm-to-weak* continuity. In view of V2and V3, this means that Gateaux differentiability of g+ at & coincides withFre� chet differentiability of g+ at &. That is, (i) � (ii).

It is evident that Fre� chet differentiability of g at & entails Gateauxdifferentiability of g at & and that Gateaux differentiability of g at & entailsGateaux differentiability of g+ at &, so (iv) O (iii) O (ii).

Finally, suppose that g+ is Fre� chet differentiable at &, and let q be theFre� chet derivative. We have shown that q is continuous. To establish thatg is Fre� chet differentiable at &, let =>0 be given and choose $>0 so that&$ # L1(+)+ , &&$&&&<$ implies that

| g(&)& g(&$)&(q, &&&$) |&&&&$&

<=.

Let # # M+(supp +) with &&&#&<$. Choose a sequence (#n)/L1(+)+

such that #n � # in the weak* topology and &&&#n& � &&&#&. Weak*continuity of the gains function g entails that g(#n) � g(#) and continuityof the function q entails that (q, #n) � (q, #) so

| g(&)& g(#n)&(q, &&#n) |&&&#n&

�| g(&)& g(#)&(q, &&#) |

&&&#&.

Hence

| g(&)& g(#)&(q, &&#) |&&&#&

�=.

We conclude that g is Fre� chet differentiable at &, so (ii) O (iv). K

With all the preliminaries out of the way, we turn to the proof ofTheorem 3 and then to Theorem 1.


Proof of Theorem 3.

(4) � (5): This follows from part (ii) of Theorem 2.

(4) � (6): If C(+) is not a singleton then �g(+) is not a singleton(Theorem 2) and g is not Gateaux differentiable at +. Conversely, if g is notGateaux differentiable at + then g+ is not Gateaux differentiable at +(Proposition 9) so �g+(+) is not a singleton (V2) whence C(+) is not asingleton (Theorem 2).

(6) � (7): This follows from Proposition 9.

(2) O (4): Let q # C(+). We have shown that q(i)�MP++ (i) for each

i # B _ S; combined with the definition of the core and adding up, thisyields

g(+)=|B _ S

q(i) d+(i)�|B _ S

MP++ (i) d+(i)= g(+).

Thus, q(i)=MP++ (i) for almost all i. Hence if q, q$ # C(+) then q(i)=

MP++ (i)=q$(i) for almost every i # supp +. Because q, q$ are continuous,

agreement almost everywhere entails agreement everywhere (on supp +).That is, the core is a singleton.

(7) O (2): Let q be the Fre� chet derivative of g at +, and let i # supp +;we assert that MP+

+ (i)=q(i). To see this, choose open sets Vn /B _ Swhich shrink to i. For each n, set

&n=+ |Vn

+(Vn)

so that &n # L1(+)+ , &&n&=1 and &n � $i (in the weak* topology). Now fix=>0 and use the definition of the Fre� chet derivative to choose $>0 so that

| g(++&)& g(+)&(q, &) |<= &&&

whenever & # M+(supp +) and &&&<$. Then for t<$, substituting &=t&n

in the above yields

t=>| g(++t&n)& g(+)&(q, &n) |

� | g(++t$i)& g(+)&(q, $i) |

=| g(++t$i)& g(+)&q(i)|.

Dividing by t and taking the limit as t � 0+ yields

|MP++ (i)&q(i)|�=.


Because =>0 was arbitrary, we conclude that MP++ (i)=q(i), as asserted.

By definition of the Fre� chet derivative,

|B _ S

MP++ (i) d+(i)=|

B _ Sq(i) d+(i)= g(+),

which is adding up.

(7) O (1): This is immediate from the definition of Fre� chet differen-tiability.

(1) O (4): Full appropriation means that there is a q # C(B _ S) suchthat (q, +)= g(+) and

lim&&& � 0 }mp&

+ (&)&(q, &)

&&& }=0.

At every core point q$ the per capita payoff to any subpopulation is nolarger than its marginal product:

(q$, &)�mp&+ (&).

Hence q$�q. But

|B _ S

q$ d+= g(+)=|B _ S

q d+.

In particular, the core is a singleton.

(3) O (7): If the price correspondence P is norm-to-norm continuousat + then the subdifferential of the restriction g+ is norm-to-norm con-tinuous at + (Theorem 2), whence g+ is Fre� chet differentiable at & (V3),whence g is Fre� chet differentiable at & (Proposition 9).

(5) O (3): Let (+n)/M+(B _ S) be a sequence of populationdistributions converging to +; for each n, let pn # P(+n) be a Walrasianprice, and let yn be the corresponding Walrasian allocation. We haveshown (Proposition 4) that the set P of all Walrasian prices is equicon-tinuous, so some subsequence of ( pn) converges in norm to a price p.Because the Walrasian allocations yn lie in a bounded subset ofM+((B _ S)_H� ), some subsequence converges weak* to a measure y. It iseasily checked that ( p, y) is a Walrasian equilibrium for the economy +; inparticular, p # P(+). By assumption, P(+) is a singleton, so it follows thatthe price correspondence P is norm-to-norm continuous at +. K


With Theorem 3 in hand, the proof of Theorem 1 is now easy.

Proof of Theorem 1. Note first that the gains function, the core, andthe set of Walrasian prices of a finite assignment economy coincide with thecorresponding objects for the continuum economy with the same popula-tion distribution. Moreover, for each individual i, the directionalderivatives of the gains function in the directions +i, &i coincide with thediscrete marginal products; see Eq. (2) and the discussion of discrete andinfinitesimal marginal products in the proof of Proposition 1. Hence theequivalence of the numbered statements follows from the equivalence ofthe numbered statements in Theorem 2. It remains only to show thatthe numbered statements are equivalent to the inability of individuals tomanipulate.

Suppose then that the core is a singleton. Full appropriation implies thatthe unique core utility gives to each individual his�her marginal contribu-tion. As we have observed, no manipulation can ever yield an individualobtain more than his�her marginal contribution. Hence, this implies thatno individual can manipulate.

Finally, suppose that Walrasian prices are not unique. Let ( p, y) be anyWalrasian equilibrium. Let p� , p

�be the highest and lowest Walrasian prices.

Interchangeability guarantees that ( p� , y) and (p�, y) are Walrasian equi-

libria. By assumption, p� {p�. If it were the case that p� (h)=p

�(h) for all

houses that are traded at y, then our pricing convention would guaranteethat p� (h)=p

�(h) for all houses h. Hence there is some house h which is

traded at y for which p� (h)>p�(h). Let s be any seller who sells house h in

the allocation y and let b be any buyer who consumes house h at theallocation y. At least one of the following two cases must obtain:

Case 1. p� (h)>p(h). Because seller s sells house h, it must be the casethat p(h)�_(s). In this case, seller s could successfully manipulate byannouncing the higher reservation value _$(s)= p� (h); such an announce-ment will result in an equilibrium in which house h trades at the price p� (h),which is certainly better for seller s.

Case 2. p(h)>p�(h). Because buyer b buys house h, it must be the case

that p(h)�b(h). In this case, buyer b could successfully manipulate byannouncing the schedule

b$(h$)={p�(h$)

b(h)if b=b$otherwise.

Such an announcement will result in an equilibrium in which househ trades at the price p

�(h), which is certainly better for buyer b. K


The last paragraph of the preceding proof justifies a remark made inthe definition of manipulability in finite economies: if some Walrasianequilibrium is manipulable then all Walrasian equilibria are manipulable.

It is convenient to establish Proposition 7 at this point, in order to makeuse of it in the proof of Theorem 3.

Proof of Proposition 7. We show first that the core correspondence isweak*-to-norm upper hemi-continuous. Let +n � + in the weak* topology.For each n, let qn be a core utility; in view of Theorem 2, we may find aWalrasian equilibrium ( pn , yn) giving rise to this core utility. Since thecollection P of all Walrasian prices is norm compact and ( yn) lies in abounded set, we may assume (passing to a subsequence if necessary) thatpn � p in norm and yn � y weak*. It follows easily that ( p, y) is aWalrasian equilibrium; let q be the corresponding core utility. In view ofTheorem 2, qn � q in norm. We conclude that the core correspondence isweak*-to-norm upper hemi-continuous.

We now turn to the average diameter. Let +n � + in the weak* topology.For each n, let q� n , q

�n be the best and worst core points (from the point of

view of sellers). Passing to a subsequence if necessary, we may assume thatq� n � q1, q

�n � q2, with q1, q2 # C(+). Let q� , q

�be the best and worst points in

C(+), so that

q� (s)�q1(s)�q2(s)�q�(s)

for all sellers s # supp +S and

q� (b)�q1(b)�q2(b)�q�(b)

for all buyers b # supp +B . Because population distributions are positivemeasures and +n � + in the weak* topology, &+n & � &+&. Combining thisfact with weak* convergence of the population distributions and uniformconvergence of the core utilities yields

dave(+n)=1

&+n& |B _ S

|q� n(i)&q�

n(i)| d+n(i)

�1

&+& |B _ S

|q1(i)&q2(i)| d+(i)

�1

&+& |B _ S

|q� (i)&q�(i)| d+(i)

=dave(+).

Hence dave is weak* upper semi-continuous.


If in addition supp +n � supp +, choose in # supp +n for which

dmax(+n)=|q� n(in)&q�

n(in)|.

Passing to a subsequence if necessary, we may assume in � i # supp +.Because q� n � q1 uniformly and q

�n � q2 uniformly, it follows that q� n(in) �

q1(i) and q�

n(in) � q2(i). Hence

dmax(+n)=|q� n(in)&q�

n(in)|

� |q1(i)&q2(i)|

�|q� (i)&q�(i)|

�dmax(+),

which is the desired conclusion. K

Proof of Theorem 4. Let PC/M+(B _ S) be the set of perfectlycompetitive economies. In view of Theorem 3, PC is the set of populationdistributions + for which C(+) is a singleton. Equivalently, PC is the set ofpopulation distributions + for which dave(+)=0. Proposition 7 tells us thatdave is an upper semi-continuous real-valued function, so for each positiveinteger n the set

Wn={& # M+(B _ S) : dave(&)<1n=

is weak* open in M+(B _ S). Because

PC= ,�

n=1

Wn

it follows that PC is a weak* G$ .To see that PC is norm dense, fix a measure + # M+(B _ S). Set

D(+)=[& # L1(+)+ : g+ is Gateaux differentiable at &].

In view of V4, D(+) is norm dense in L1(+)+ . In particular, there aremeasures & # L1(+)+ arbitrarily close to & at which g+ is Gateaux differen-tiable. In view of Proposition 9, g is Gateaux differentiable at each such &;in view of Theorem 3, each such & belongs to PC. Hence PC is normdense. K


Proof of Proposition 6. Suppose there is no gap in the reservationvalues of sellers. Let ( p, y) be a Walrasian equilibrium, and fix a househ0 # supp +H . We claim that

p(h0)=inf[r : (h0 , r) # supp +S , (h0 , r) � supp y]

where p(h0) is 1 if the set of such sellers is empty. To see this, note that ifp(h0) is greater than this supremum some seller who owns h0 and whosereservation value is less that p(h0) would wish to sell, while if p(h0) is lessthan this supremum then some seller is selling for a price less than hisreservation value. Either of these scenarios contradicts optimization. Hencethe equilibrium price for every house in supp +H is determined by theequilibrium allocation. Interchangeability means that all Walrasian pricessupport all Walrasian allocations, so it follows that Walrasian prices areunique. The argument in the case in which there is no gap in the reserva-tion values of buyers is similar, and left to the reader. K

Proof of Theorem 5. Suppose that + is perfectly competitive and (En) isa sequence of finite economies with supp +En

/B _ S and +En� +. Fix =>0;

for each n, write

En=[i # En : |q� n(i)&q�

n(i)|�=].

Proposition 7 guarantees that there is an index N so that dave(+En)<=2 for

n�N. Because the core of the finite economy En and the core of thedistributional economy +En

coincide, it follows that

1|En |

:i # En

|q� n(i)&q�

n(i)|<=2.

In particular, |En |<= |En | for n�N. Individuals in the complement of En

cannot manipulate by as much as =, so we have (i).If in addition supp +En

� supp +, Proposition 7 guarantees that there isan index N$ so that dmax(+En

)<= for n�N$, whence

|q� n(i)&q�

n(i)|<=

for every i # En . Hence no individual can manipulate by as much as =, whichis (ii).

To prove (iii), fix a population measure + of total mass 1 which is notperfectly competitive. We construct a sequence of finite economies Fn

whose population distributions +Fnconverge to + and which have the

property that every element of C(+) induces an element of the core of Fn .The latter property will guarantee that many agents can manipulate.


To this end, fix an optimal assignment x for +. Write &B , &S for themarginals of x on B, S. Feasibility of x means that &B�+B and &S�+S . Weuse x as a template for constructing the finite economies; the constructionis a little delicate because we must account for the possibility that &B {+B

or &S {+S (or both). Set &=&B+&S and `=+&&; feasibility of x impliesthat `�0.

Fix an index n. Choose a finite partition [Aj] of supp x into sets ofhaving positive x-measure and diameter less than 1�n; for each j, choose apoint aj=(bj , sj) # Aj . Choose strictly positive rational numbers :j<x(Aj)so that � |x(Aj)&:j |<1�n. Set

xn=: :j $aj

and

&n=: :j ($bj+$sj

).

Choose a finite partition [Zk] of supp ` into sets of having positive`-measure and diameter less than 1�n; for each k, choose a point zk # Zk .Choose strictly positive rational numbers ;k so that � |`(Zk)&;k |<2�nand � :j+� ;k=1. Set

`n=: ;k $zk.

Write +n=&n+`n. Because the coefficients :j , ;k are rational and sum to1, +n is the population distribution of a finite economy; let Fn be any suchfinite economy. Because we have chosen the sets Aj , Zk to have smalldiameter and � |x(Aj)&:j |<1�n and � |`(Zk)&;k |<2�n, it follows that+n � + in the weak* topology and supp +n � + in the Hausdorff metric.

We claim that many agents in the economy Fn can manipulate a lot.To see this, consider the best and worst core utilities q� , q

�# C(+). By

assumption, + is not perfectly competitive so the core is not a singleton; inparticular q� {q

�. Choose =>0 so that the open set

U=[i # supp + : q� (i)&q�(i)>=]

has measure +(U )>0 and the set

F=[i # supp + : q� (i)&q�(i)==]

has measure +(F )=0. Because q� , q�

both support the optimal assignment x,all agents in U must be matched in x, so that `(U )=0. Write Un for theset of individuals who belong to U and to the economy Fn ; because U con-sists of buyer�seller pairs who are matched in x, Un consists of buyer�seller


pairs who are matched in xn. Let (b, s) be any such matched pair. Writeq� n, q

�n for the largest and smallest elements of C(+Fn

)=core(Fn). Therestrictions q� (n), q

�(n) belong to core(Fn), so

q� n(s)&q�

n(s)�q� (n)(s)&q�

(n)(s)�=

and

q�

n(b)&q� n(b)�q�

(n)(b)&q� (n)(b)�=.

Hence, for any q # core(Fn), at least one of b, s can manipulate q by at least=�2. Because the boundary of U (which is contained in F ) has measure 0,+n(U ) � +(U). Hence, for large n, almost the fraction +(U )�2 of individualsin En can manipulate any given core allocation by at least =�2. Taking$=min[+(U)�3, =�2] yields (iii). K

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