Valuation Equilibrium with Clubs · consumer puts weight 1 on one and only one commodity. With lotteries, the restriction is that the total probability weights sum to 1. Commodity

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Journal of Economic Theory � 2247

journal of economic theory 74, 19�39 (1997)

Valuation Equilibrium with Clubs*

Harold L. Cole

Research Department, Federal Reserve Bank of Minneapolis,250 Marquette Avenue, Minneapolis, Minnesota 55401-2171

and

Edward C. Prescott

Department of Economics, University of Minnesota, Minneapolis, Minnesota 55455; andResearch Department, Federal Reserve Bank of Minneapolis,250 Marquette Avenue, Minneapolis, Minnesota 55401-2171

Received February 8, 1995; revised July 12, 1996

This paper considers model worlds in which there is a continuum of individualswho form finite-sized associations to undertake joint activities. We show how,through a suitable choice of commodity space, restrictions on the composition offeasible groups can be incorporated into the specification of the consumption andproduction sets of the economy. We also show that if there are a finite number oftypes, then the classical results from the competitive analysis of convex finite-agenteconomies can be reinterpreted to apply. Journal of Economic Literature Classifica-tion Numbers: D51, H41. � 1997 Academic Press

1. INTRODUCTION

Economists are interested in situations that involve people voluntarilyforming associations in order to undertake joint activities. Important exam-ples of such associations include households, firms, and communities.Buchanan [1] has labeled these associations clubs. In this paper we extendthe general competitive framework to a large class of environments whereclubs can form.1 What distinguishes our competitive analysis of theseenvironments is our choice of commodity space. This choice allows us to

article no. ET962247

190022-0531�97 �25.00

Copyright � 1997 by Academic PressAll rights of reproduction in any form reserved.

* This research was supported in part by the National Science Foundation. We thank RokoAliprantis, Beth Allen, Fernando Alvarez, V. V. Chari, Edward Green, Larry Jones, TimothyKehoe, Richard Rogerson, Jan Werner, and especially an anonymous referee. The viewsexpressed herein are those of the authors and not necessarily those of the Federal ReserveBank of Minneapolis or the Federal Reserve System.

1 We do not survey this large literature, but instead leave that to three recent surveys: Rothand Sotomayor (1990) survey the matching, or two-person��club, literature. Wooders (1994)and Scotchmer (1994) survey the club literature.


fit a large class of club and matching environments into the Theory ofValue (Debreu [4]) framework.

The features of the class of environments that we consider are the follow-ing: First, there are finitely many types of individuals and a large numberof individuals of each type. Second, clubs are small relative to the numberof individuals of any given type. Third, individuals maximize the expectedvalue of a continuous utility function. Fourth, individuals have access to arandomizing device upon which they can condition their trades.

For a competitive analysis of this class of environments, a commodityspace that meets certain requirements is needed. First, the space must belinear. Second, a firm (or consumer) can determine whether or not it canproduce (or consume) a particular point in the commodity space withoutany knowledge of what any other firm or person is doing; that is, the firm'stechnology sets and consumers' consumption sets are subsets of the com-modity space. Third, the feasibility of plans for all people and firms, alongwith the resource balance constraint, implies that the allocation is feasible.

The key requirement for our commodity space is the second one, sincein club environments there are typically membership restrictions on groupformation. To deal with this, we index the commodity vector by consumertypes as well as by characteristics of the potential clubs. With this indexa-tion, consistency of membership composition is determined by restrictionson feasible plans. For example, a household club might consist of one maleand one female and might be characterized by its consumption level of theclub good. In this case, the consumption sets restrict persons to consumingonly club memberships of their own sex, while production sets impose theconstraint that the number of male and female club memberships must beequal for each club consumption level. These restrictions permit thisenvironment to be represented as an economy in the Theory of Value sense.

Because there can be a continuum of commodities and people can con-sume a finite quantity of an individual good, we follow Mas-Colell [7] inusing a commodity space that consists of signed measures. But we followPrescott and Townsend [9, 10], rather than Mas-Colell, in interpreting thecommodity vector from the viewpoint of the household as being a prob-ability measure. This is done to ensure that all gains from trade areexhausted. Without randomization, preferences would not be convex, sinceclub membership is discrete. Hence there could exist mutually beneficialgambles. (See [11].)

Randomization can be introduced into the Arrow�Debreu event-contingent analysis by introducing sunspots. (See [14].) The reason weadopt the lottery approach rather than the sunspot approach is that withlotteries the analysis is simpler. One simplification arises because there isno need to keep track of people's names. Another simplification arisesbecause preferences become convex; this in turn permits attention to be

20 COLE AND PRESCOTT


restricted to type-identical allocations. An implication of this is that all theresults for finite-agent convex economies hold for our large economies.

This commodity space does require that a rich and artificial menu ofcommodities be traded. However, here, just as with the Arrow�Debreu equi-librium, there are equivalent decentralizations that correspond more closelyto the actual commodities that we see exchanged. To make this point, weestablish the equivalence between lottery equilibria and the equilibria of analternative two-stage decentralization. Under this decentralization, peoplegamble at the first stage and match or form clubs in a deterministic fashionat the second stage.

The paper is organized as follows. The first section shows how a simple,prototypical matching example can be represented as an economy in theTheory of Value sense. We then extend this example to a club model inwhich the club is characterized, in part, by the number of its members;hence congestion can arise. In the third section, the analysis is generalizedto permit multiclub membership, and the welfare theorems, core equiv-alence, and existence for the case in which there are a finite number of clubtypes are established. Problems of extending the results to a continuum ofclub types are discussed. The fourth section establishes an equivalencebetween this decentralization and an alternative decentralization thatentails gambling at the first stage and no randomization of allocations atthe second stage.

2. A SIMPLE MATCHING EXAMPLE

The purpose of this section is to show that the simple model of house-hold formation in Cole, Mailath, and Postlewaite [2] can be representedas an economy in the sense of Debreu.

A. The Environment

There are two types of people, which we refer to as female or male anddenote by s # S=[ f, m]. When referring to a person of type s, we use s$ todenote the person's opposite sex. There is a continuum of each type, andthe measure of people having type s is *s>0. An individual is matchedif paired with a member of the opposite sex. Matched individuals canconsume both a pair-specific public or club good and, individually, theprivate consumption good. Unmatched individuals can consume onlythe private consumption good. A person of type s is endowed with |s ofthe private consumption good. There exists a technology for convertingunits of the private consumption good into half as many units of the clubgood.

21VALUATION EQUILIBRIUM WITH CLUBS


Let A and B denote the Euclidean spaces of private and club consumptionlevels, respectively, and let C=A_B denote the space of consumptions.We assume that individuals of type s have preferences defined upon thecompact set C s / C. In this section, we take C s � [0, d]2, where d issufficiently large that no feasible allocation exists for which all of any typehave private or club consumption level d or greater.

Preferences are identical for all people and are given by

E[u(a, b)],

where u : Cs � R is a continuous function.

B. Representation as an Economy

We need a linear commodity space with the property that the feasibleactions of a firm or a consumer are a subset of this space. A feasible alloca-tion of the economy is a specification of actions for each agent that areindividually feasible and mutually consistent. Further, this consistencyrequirement takes the form of a resource balance condition, which dependsonly on the sum over the actions (commodity vectors) of producers andconsumers and on the aggregate endowment.

The challenge posed by matching models is that, besides the usual needto specify production and consumption activities, we also need to specifythe allocation of individuals to different pairs (or, more generally, togroups). We handle this by indexing the commodity vector by (a, b, s),which permits us to distinguish between male and female consumption.With this indexation we can impose restrictions on group formation by acombination of restrictions on the technology and consumption sets.

For our simple example, we impose the restrictions that the productionof male and female commodities with club level b must be equal for everyb>0. We restrict the consumers to choosing a consumption vector thatputs positive weight only on commodities with the appropriate sex designa-tion. Without lotteries, the commodity vector chosen by an individualconsumer puts weight 1 on one and only one commodity. With lotteries,the restriction is that the total probability weights sum to 1.

Commodity Space. An (a, b, s) commodity is a contract to deliver to anindividual of type s a membership in a type b club or match along with aunits of the private consumption. A point in the commodity space is a vectorwhich indicates the number of each type of commodity that is delivered. Ifb=0, it is simply a contract to deliver a units of the private consumptiongood to a person of type s. In these cases, the implicit club consists of asingleton person.

Formally, the commodity space is L=M(C_S), where M( } ) denotes thespace of signed measures on the Borel-sigma algebra of a set. The implicit



topology on C_S is the product topology generated by the Euclideanmetric on C and the discrete metric on S. We follow the standard conven-tion in the general equilibrium literature: a firm's outputs are positive andinputs negative, and an individual's consumptions are positive and suppliesare negative.

Technology Set. Firms produce club memberships indexed by (a, b, s).For a production plan y # L to be feasible, there must exist a set of matches$ # M+(B"[0]) such that the following two conditions are satisfied:

(1) For all type b clubs, the number of males and females in suchclubs must be equal. This condition is

| y(a, D, f ) da=| y(a, D, m) da=$([D]) for all D # B(B"[0]), (1)

where B( } ) denotes the Borel-sigma algebra of a set.

(2) These are sufficient resources to produce the club goods. Thiscondition is

| a dy+| 2b d$�0. (2)

If there are any matches with positive levels of b produced, then given thatA is a set of non-negative numbers, some of the components of y must benegative for (2) to hold.

The production set of a firm is given by

Y=[ y # L : _$ # M+(B"[0]) such that (1) and (2) hold].

Note that the technology set is a convex cone. This implies that theaggregate technology set is Y as well.

Consumption Sets and Utility Functions. We follow Prescott andTownsend [9, 10] in interpreting components of a person's plan, x # L, asbeing a probability measure. To rule out randomization would be tan-tamount to ruling out certain event-contingent commodities. Indeed, aswas noted in [11], indivisibilities with respect to individual decisions typi-cally result in there being gains to randomized consumptions. Component(a, b, s) of x is the probability or likelihood of having private consumptiona and being a type s member of a club with club consumption level b.Clearly, the probability of an individual of type s being a type s${s mustbe zero. Hence a person of sex s has the consumption set

Xs=[x # L+: x(Cs_[s])=x(C_S)�1],



and the utility function, Us: Xs � R, is

Us(x)=| u(c) dx+u(0, 0) \1&| dx+ .

There are several things to note here. First, with these consumption sets, anindividual of type s must put weight one or less on his or her own sex-designated consumptions and must put weight zero on the other sex-designated consumptions. Note that x(Cs_[s])=x(C_S) implies thatx(C_[s$])=0. When x(Cs_[s])<1, the residual weight is put on thezero-consumption commodity bundle (0, 0). This will turn out to be con-venient later when we discuss the existence of an equilibrium. Second, sinceXs is convex and Us is linear, preferences are convex. Third, with the weak-star topology, Xs is compact (since C_[s] is a compact metric space) andUs is continuous (since u(c) is continuous).

Endowments. We treat unmatched individuals as members of asingleton club. Consistent with this, we interpret endowments to besingleton clubs; that is, we assume that each individual is endowed with aclub that has zero units of b and |s units of a. Formally, the endowmentvector of a type s consumer is the commodity vector es # L, where es(D)=1if (|s, 0, s) # D and zero otherwise.

C. Valuation Equilibrium

A type-identical valuation equilibrium with a price system that has a dotproduct representation is a measurable function p : C_S � R and a type-identical allocation [x f, xm, y] that satisfy the following:

(i) xs maximizes U s(x) subject to [x # Xs : � p dx�� p des] for all s.

(ii) y maximizes � p dy subject to y # Y.

(iii) �s *s(xs&es)= y.

In the section on core equivalence, a more general definition of a com-petitive equilibrium is presented. However, as the following propositionmakes clear, our current restriction to a type-identical competitive equi-librium is innocuous.

Proposition 1. Given any equilibrium, there is an equivalent type-identi-cal equilibrium, where by equivalent we mean that the y and p are the sameand that the type-average commodity vectors are the same.

The proposition follows from the convexity of preferences. (See [5] fora formal proof.) The outline of a proof is as follows. Since only the type-average consumption vector enters the resource balance constraint, and



since consumption sets are convex, if we give each individual his or hertype average while keeping y the same, this allocation is also feasible. Withp the same, the convexity of individuals' consumption sets and the linearityof their utility functions us imply that a type's average consumption vectorsolves the maximization problem for that type. Finally, since y and p arethe same, profit maximization holds.

Remark 1. We do not need a law of large numbers result for a con-tinuum of identical and independently distributed random variables in ourspecification of condition (iii), the resource balance condition, because weare not assuming that the lotteries are independent. We simply assume thatthe lotteries are identically distributed for individuals of the same type.Implicitly, there can be thought to be one grand lottery that determineswhich individuals of a given type will receive which consumption.

3. CLUBS WITH MULTIPLE MEMBERS AND CONGESTION

One of the classical issues in the club literature is congestion effects. Herewe show how our decentralization can be extended to the case where thereis a variety of possible membership compositions and where individualscare about the composition of the membership of their club.

A. The Environment

We assume that clubs come in different combinations of males andfemales. Let N be a bounded subset of I_I, where I denotes the non-negative integers. We assume that singleton clubs are possible; that is,[(1, 0), (0, 1)]/N. The membership composition of a club will be denotedby n=(n f , nm) # N, where ns is the number of members of sex s. Theproduction of a single club involves a specification as to the input of theprivate good, a, used in the production of the club good, the level of theclub good, b, and the composition of the membership, n. The set of clubsthat can be produced is given by the compact set T/A_B_N.

Remark 2. In our prior simple matching example, N=[(1, 0), (0, 1),(1, 1)], and the set of feasible clubs is

T=[(a, b, n) # A_(B"[0])_[(1, 1)] : a=2b]

_ [A_[0]_[(0, 1), (1, 0)]].

Consumers have preferences not only over private and club consump-tion, but also over the membership composition; hence the space ofpossible consumptions is now Cs�C=A_B_N, and preferences aregiven by us : C s � R. Note that with this specification of preferences, the



club consumption being offered to an individual is not b but (b, n). Forexample, in consuming swimming, it is not just the size of the pool thatmatters, but the size of the pool along with the number of people in it. Inthe lexicon of the club literature, this is a model of differentiated crowding,since we do not restrict individuals to caring only about the total numberof people in the pool.

B. Representation as an Economy

Since the commodities that can be produced are determined by the setof feasible clubs, it will be convenient to distinguish between the measureof clubs, which we denote by $, and the signed measure of commodities,which we will continue to denote by y. We will determine the feasible com-modity measures that can be produced by requiring that they be consistentwith some feasible measure of clubs. The measure of commodities is anelement of the commodity space L=M(C_S).

The aggregate production set in terms of commodities offered andreceived is

Y=[y # L : _$ # M+(A_B_N) for which

(i) $(T )=$(A_B_N),

(ii) � a dy+� a d$�0, and

(iii) y(A_D_[s])=�A_D ns d$ for all D�B_N].

Constraint (i) is that only feasible clubs are produced, that is, that the sup-port of $ is T. Constraint (ii) is that the total amount of the private con-sumption good given out as private consumption minus the amount takenin from consumers plus the amount used up in producing the club goods(which is � a d$) must be less than or equal to zero. Constraint (iii) is thatthe number of consumers of sex s memberships in clubs with characteristics(b, n) produced must be consistent with the number of clubs operated.

The definitions of consumption sets, preferences, and the price system areunchanged, given our new definitions of C and Cs. For females, the endow-ment point e f (D)=1 if (| f, 0, (1, 0), f ) # D and is zero otherwise, while formales, the endowment point em(D)=1 if (|m, 0, (0, 1), m) # D and is zerootherwise.

Remark 3. The club consumption good here is (b, n), but henceforth wewill take b to be a vector that contains all the aspects of club type that arerelevant to consumers.



4. A MORE GENERAL CLASS OF ECONOMIES WITH CLUBS

The method of decentralization that we have considered is applicable toa richer class of environments than we have thus far considered. Here weindicate how one can accommodate a much wider range of models thathave a greater variety of types and in which a more extensive set of jointactivities are undertaken. We can also allow for multiple club membership;for example, an individual can be a member of both a household club anda production club.

A. The Economy

Set of People. There are a finite number of types of people indexed byi # I. The measure of type i is *i>0. We say the distinction between twotypes of people is basic if some club technology does not treat them as per-fect substitutes. In the previous section, the sex distinction is basic, but theendowment of the private good is not. Type i 's basic type is si # S. Notethat the dimension of S is, at most, I.

Commodities. The space of possible consumptions of the exchangeablegoods is a finite-dimensional Euclidean space, which we denote by A. Thereis a finite set of categories of clubs indexed by j=1, ..., J. Each person mustbe a member of one and only one club in each category. Since we allow fora single-member club that does nothing in each category, this membershiprequirement does not restrict the set of feasible allocations. We denote thespace of possible club consumption vectors in category j by Bj , which, likeA, is a finite-dimensional Euclidean space. We define the space of consump-tion to be C=A_B1_ } } } _BJ . The commodity space is L=M(C_S),the space of signed measures on the Borel-sigma algebra of C_S.

Consumption Sets, Preferences, and Endowments. A person of type i haspreferences over probability measures on the Borel-sigma algebra of C i, acompact measurable subset of C that is ordered by the expected value ofthe continuous function ui (c) defined on Ci. We assume that 0 # Ci. Typei 's consumption set is

Xi=[x # L+: x(Ci_[si])=x(C_S)�1].

Type i 's utility function is Ui: Xi � R, where

Ui (x)=| u i (c) dx+\1&| dx+ ui (0).

An individual is endowed with |i # C. The endowment point of a type iperson is denoted by ei # L, with ei being the measure that assigns measure



1 to sets containing (|i, si) # C_S and that assigns measure 0 to all othersets.

Club Technologies. A club type of category j=1, ..., J is defined by itsvector of exchangeable commodities a, the vector of club goods bj , and itsmembership n. Its membership vector n specifies the integer number of eachbasic type s in the club. The set of possible memberships is the finite set N.Thus a club type in category j is denoted by tj # A_Bj _N. The set ofpossible category j club types is given by the compact set Tj �A_Bj_N,for j=1, ..., J.

Aggregate Production Set. A commodity vector y belongs to Y if, foreach category j=1, ..., J, there is a distribution $j on Tj consistent with y.Formally, the aggregate production set is

Y=[ y # L : _$j # M+(A_Bj_N) for j=1, 2, ..., J for which

(i) $j (Tj)=$j (A_Bj_N),

(ii) � a dy+�j � a d$j�0, and

(iii) �A_D_N ns d$j=y(A_B1 _..._D_..._BJ_[s])

for all D # B(Bj), j, s],

where B(Bj) denotes the Borel subsets of Bj .Constraint (i) is that only technologically feasible clubs are operated.

Constraint (ii) is that the net input of the exchangeable goods a plus theamount of a used up in the production of the distribution of clubs is lessthan or equal to zero. Constraint (iii) is that the measure of people of types who are receiving club consumption bj # D in the output measure mustbe consistent with the distribution of clubs in category j. The aggregateproduction technology is a convex cone.

Price System and Competitive Equilibrium. A price system for oureconomy is a linear functional p : L � R, which has representation

p } x=| p(c, s) dx,

where p : C_S � R is measurable.A type-identical competitive equilibrium for our economy is a price system

p and an allocation [[xi], y] satisfying the following:

(i) For all i, xi maximizes Ui (x) subject to x # Xi and p } x� p } ei.

(ii) y maximizes p } y subject to y # Y.

(iii) �i *i (xi&ei)= y.



Remark 4. The set of feasible allocations, that is, the set [[[xi], y] :xi # Xi all i, y # Y, and �i *i (xi&ei)= y], is compact, since the consump-tion sets are compact.

B. The Welfare Theorems and Existence

The first welfare theorem of Debreu [3] applies here, since preferencesare convex, provided no type is satiated in equilibrium.

It is somewhat more complicated to establish the second welfaretheorem. If L is finite-dimensional and�or Y has an interior point, thenTheorem 2 in [3] can be applied to conclude that every Pareto optimumcan be supported as a quasi-competitive equilibrium with transfers. If thisis not the case, then one may be able to proceed to establish the secondwelfare theorem for an economy with a larger production set that has aninterior, but which has the same set of Pareto optima and competitiveequilibrium as our economy.

The second welfare theorem guarantees the existence of a quasi-com-petitive equilibrium in which the price system is a continuous linear func-tional on the commodity space. This does not imply that this price systemhas a dot product representation when the commodity space is infinite-dimensional. To establish the existence of a competitive equilibrium withtransfers and with a price system that has a dot product representation, weverify that assumptions 15.8$ and 15.9$ in Stokey and Lucas [15, p. 468]hold, and we apply their Theorem 15.9. Assumption 15.8$ requires that ifx # Xi, then the modification of x obtained by setting the x equal to zeroon some measurable subset of C_S is also in Xi. Assumption 15.9$ isessentially that preferences are weak-star continuous, which is the casegiven expected utility maximization and the fact that the utility functions,ui, are continuous functions defined on a compact set for all i # I.

If C is finite, then standard existence arguments may apply, where thecommodity space L is just the Euclidean space with dimension equal to thenumber of commodities.

Remark 5. In his discussion of the classical theorem of the existence ofa competitive equilibrium, McKenzie [8] lists the following conditions asbeing sufficient:

(i) The consumption sets are closed and convex.

(ii) The preference ordering is continuous and quasi-concave.

(iii) The aggregate production set is a closed convex cone.

(iv) Xi+[&ei] & Y is not empty for all i.

(v) Y & L+=[0].



(vi) There is a common point in the relative interiors of Y andX=�i *i (Xi+[&ei]).

(vii) For any two nonempty, disjoint partitions of I, I1 , and I2 , andfor xI1

=y&xI2, where xI1

# �i # I1*i (Xi+[&ei]), xI2

# �i # I2*i (Xi+[&ei]),

and y # Y, there exists an | # �i # I2*i (X i+[&ei]) such that x$I1

+xI2

+| # Y and such that x$I1can be decomposed into an allocation for I1

that is weakly preferred by all members of I1 and is strictly preferred by atleast one member-type of I1 .

As has already been noted, conditions (i)�(iii) are satisfied. Conditions(iv)�(vii) are satisfied for our matching example if the set of commoditiesis suitably restricted. The commodities are (A_B_S)"((0, 0, f ) _(0, 0, m)). The reason these two points are excluded is that these com-modities can be produced costlessly. Condition (iv) is satisfied becauseeach individual can always consume his or her own endowment. Condition(v) is satisfied because no commodity can be produced without someinputs. The following argument establishes that condition (vi) is satisfied.For all s, if xs (c, s)==�*s and if xs (c, s$)=0 for all c, then for any positive= that is sufficiently small, xs # X s. Hence the point y=�s (xs&es) *s is inthe relative interior of �s (Xs &[es])*s. Since y(c, [ f ])= y(c, [m]), thecondition in equation (2) is satisfied. Finally, if R is the total amount ofresources needed to produce measure one of every type of commodity, thenfor any positive = that is sufficiently small, =R<�s *s|s, and the conditionin equation (1) is satisfied. Thus y is in the relative interior of Y. Condition(vii), which ensures that all agents have positive income, is triviallysatisfied, since more private consumption is preferred to less private con-sumption in any feasible allocation.

If C is not finite, the commodity space is infinite-dimensional, and itis more difficult to establish the existence of a competitive equilibrium.A standard method of proof is to show that the limit of a sequence of equi-libria for finite-dimensional economies exists and is an equilibrium for theinfinite-dimensional limit economy. An outline of this approach, as appliedto our more general class of economies, is as follows. For r # [1, 2, 3, ...],restrict the aggregate production set to elements with support that belongsa to finite set Cr �C, where all points in C lie in a 1�r-neighborhood of atleast one point in Cr . For the economy [[X i, ui, ei], Yr], the commodityspace Lr=M(Cr_S) is finite-dimensional. The competitive equilibrium[[xi

r], yr , pr] for the rth economy can be used to find an equilibrium forthe economy with the general commodity space and the restricted produc-tion set. Given this equilibrium, the problem is to find a convergent sub-sequence with the property that the limit is an equilibrium for our economywith the unrestricted production set Y.



The key part of the proof is to find a uniformly convergent subsequenceof [ pr]. This ensures that the limits of the sequences xi

r are maximal, giventhe price system p=limr � � pr , where r now indexes elements of a sub-sequence for which allocations converge weak-star and prices convergeuniformly. For our economy, if the ui (c) have uniformly bounded slopes,then a sequence of competitive equilibria exists with the desired property.First, we normalize prices so that the sup | pr (c, s)|=1. Next, we let qr (c, s)be the maximal demand reservation price over all i. Note that the qr

functions are defined on all of C=�i C i. Note also that [[x], yr , qr] is acompetitive equilibrium for the economy with the commodity spaceL=M(C_S) and the aggregate technology set Yr . The functions qr (c, s)will have uniformly bounded slopes, given that the underlying utility func-tion ui (c) has a uniformly bounded slope. This completes an outline of theextension to economies in which C is a compact metric space.

C. Equivalence of Core and Equilibrium Allocations

In this section, we show that the standard arguments establishing theequivalence of the set of core and equilibrium allocations hold with a lot-tery commodity space. (See [6] for an exposition of the standardarguments in exchange economies.)

To specify a core allocation, we must allow for allocations in whichpeople of a given type do not receive the same commodity point. Hence wealter our definition of an allocation for type i individuals in this section; itwill now be a measure defined on our commodity space L that specifiesthe distribution of consumption vectors which are consumed by theseindividuals.

Having specified an allocation more generally, we show that any coreallocation must satisfy equal welfare among people of the same type. Next,we show that any equilibrium allocation is in the core. Finally, we showthat, given a core allocation, there exists a price system that supports it asa quasi-competitive equilibrium, and we discuss the conditions under whicha quasi-competitive equilibrium is a competitive equilibrium.

A general definition of an allocation is a set of measures [zi], wherezi : B(L) � R+. An allocation is feasible if

(i) For all i, the support of zi is a subset of Xi, and

(ii) y(z)#�i � (x&ei) dzi # y.

A feasible allocation [zi] is blocked by the feasible allocation [z~ i] if

(i) For all i, z~ i (L)�zi (L), and



(ii) For all (i, u� ),

z~ ([x : ui (x)�u� ])�z([x : ui (x)�u� ]),

with strict inequality for some (i, u� ).

A core allocation is a feasible allocation that cannot be blocked.Implicitly, the members of the blocking allocation are those receiving the

lowest utility under [zi]. Everyone in the blocking allocation receiveslarger utility under [z~ i], with some positive measure receiving strictlylarger utility.

Proposition 2. If [zi] is a core allocation, then all people of the sametype receive the same welfare level.

Proof. If utilities were not equal for almost all people of some type,then some positive measure of that type must receive utility strictly lessthan the average. Pick =>0 sufficiently small so that

zi ([x : ui (x)�u� i])�=*i for all i,

where u� i denotes the type i average utility under z. Then, for the population;i==*i for all i, the allocation [z~ i]��which puts mass ;i on the averagecommodity point under z, � x dzi, and zero elsewhere��is feasible andblocks z. K

A competitive equilibrium is a triplet [[zi], y, p] satisfying the following:

(i) For each i, zi (L)=*i.

(ii) The support of zi�argmax[Ui (x) : x # Xi and p } x� p } ei] foreach i.

(iii) y # argmaxY p } y.

(iv) �I � (x&ei) dzi= y.

Given a type-identical competitive equilibrium [[xi], y, p], we can con-struct its associated general competitive equilibrium [[zi], y, p] in theobvious manner: by having zi put weight *i on sets containing xi and putweight zero on sets not containing xi.

For a quasi-competitive equilibrium, condition (ii) in the definition of acompetitive equilibrium is replaced by the following:

(ii$) For each i, if x$ is in the support of zi, then for any x # Xi,Ui (x)�Ui (x$) implies that p } x� p } x$.

Proposition 3. If [[zi], y, p] is a competitive equilibrium with no typesatiated, then its allocation [zi] is a core allocation.



Proof. We will show that the existence of a z~ which blocks z leads toa contradiction. All points in the support of zi must yield the same utilityfor a member of type i, since all members of type i have the same maxi-mization problem. Points that yield higher utility must be more expensive;that is, if u� i is the type i utility under [zi], then ui (x)>u� i implies thatp } x> p } ei. Further, the convexity of preferences, along with the existenceof a point in Xi that yields utility exceeding u� i, implies that if ui (x)�u� i,then p } x� p } ei. Therefore, if [z~ i] blocks [zi], then, for every i,

p } | (x&ei) dz~ i�0,

with strict inequality for at least one i. If we sum over i, this implies thatp } y(z~ )>0, while the feasibility of [z~ i] implies that y(z~ ) # Y. But giventhat y is a cone, equilibrium profits must be zero; hence we have a con-tradiction. K

To show the existence of a price system which supports a core allocationas a quasi-competitive equilibrium, we apply the Hahn�Banach theorem.However, here, just as with the second welfare theorem, the potentialemptiness of the interior of the production set when the commodity spaceis infinite-dimensional precludes the direct application of the Hahn�Banachtheorem. There are two obvious resolutions to this problem: (i) to assumethat C is finite and hence L is finite-dimensional or (ii) to construct analternative production set Y$, which includes Y and for which the set ofPareto optimal allocations is the same, regardless of the distribution of thepopulation (the *'s). The condition in approach (ii) is stronger than thecondition in the second welfare theorem because we need to ensure that,given any blocking allocation that is feasible for the production set Y$,there is at least as good an allocation that is feasible for Y; hence theexpansion in the set of possible blocking allocations under Y$ has notshrunk the core. We pursue approach (i) below. Approach (ii) can beapplied in our prototype economy by incorporating a sufficiently expensivemethod of producing singleton clubs with positive levels of the club good.

Proposition 4. If C is finite, then for any core allocation z in which atleast one type is not satiated, there exist p and y such that [z, y, p] is aquasi-competitive equilibrium.

Proof. The proof is by contradiction. With y= y(z), the resourcebalance condition is satisfied. Now we need only find a supportingprice system. We show that such a price system exists as follows. First,



Proposition 2 states that, for a core allocation, there is a utility level foreach type, which we denote by u� i. Next we define

X�={x # L : _[0�%i�*i, xi # Xi] for which ui (xi)�u� i for all i,

with strict inequality for some i, and x=: %i (xi&ei)= .

The set X� is nonempty, since at least one type is not satiated. We knowthat X� & Y=,, since z is a core allocation. We now show that X� isconvex. Let x� , x̂ be any elements belonging to X�, and let [%� i, x� i] and[%� i, x̂i] be a set of associated weights and consumption vectors for x� andx̂ in the definition of X� . For : # [0, 1], let %i=:%� i+(1&:) %� i. Let xi=(:%� ix� i+(1&:) %� ix̂i)�%i if %i>0, and let xi=x� i otherwise. These [%i, xi]satisfy the conditions for :x� +(1&:)x̂ to belong to X�. Thus X� is convex.

Given that X� and Y are nonempty and convex, that X� & Y=,, andthat C, being finite, implies that L is finite-dimensional, the Hahn�Banachtheorem guarantees the existence of a nontrivial hyperplane that separatesX� and Y, which we can represent by p # L.

Next we show that if xi is in the support of zi, then *i (xi&ei) belongsto the closure of X� . Let xr=*i (1&1�r)(xi&ei)+(1�r)(xj&e j), where( j, x j) is such that xj # X j and u j (x j)>u� j. For sufficiently large r, xr # X�.Furthermore, limr � � xr=*(xi&ei). Thus the result follows.

The fact that set Y is a cone implies that points in the separating hyper-plane must have value 0. Thus p } x�0 for all x in the closure of X�, andp } y�0 for all y # Y. An implication of these results is that p } y(z)�0��i � p } (x&ei) dzi. But y(z)=�i � (x&ei) dzi. Thus p } y(z)=0, and y(z) isprofit-maximizing. Similarly, for xi in the support of zi, p } *i (xi&ei)=0,which implies that p } (xi&ei)=0. This in turn implies that xi minimizesexpenditures subject to ui (xi)�u� i. K

Here, just as in the second welfare theorem, the existence of a cheaperpoint than the support of zi in i 's budget set for each i, along with theconvexity of preferences, implies that a quasi-competitive equilibrium is acompetitive equilibrium.

5. EQUIVALENCE OF LOTTERY AND GAMBLING EQUILIBRIA

The decentralization (commodity space and price system) that we havethus far considered has proven convenient from the standpoint of com-petitive analysis. However, as was noted earlier, this decentralization doesrequire that a rich menu of artificial commodities be traded. In this section,



we consider an alternative decentralization in which a somewhat morestandard menu of commodities is traded, and we show that the equilibriumallocations are equivalent. The decentralization that we consider here is atwo-stage process with gambles over value transfers in the first stage anda no-lottery competitive equilibrium in the second stage. To distinguishbetween these two decentralizations, we label our original economy as thelottery economy and the economy with the two-stage decentralization(which we consider here) as the gambling economy.

A. A Simple Matching Example

To anticipate the formal notation for the gambling economy, it is usefulto set out the choice problem of a person of type s in our simple matchingexample.

In the first stage, a type s individual picks a probability distribution overwealth transfers, the expected value of which is equal to zero, and then,conditional on his or her realized wealth level, the individual picks asecond-stage consumption level of the private good and of his or her sex-specific club good. If we denote the feasible wealth transfers by W and theperson's marginal distribution over wealth transfers by gw , then in the firststage, that person is picking gw # M+(W) such that gw(W)=1 and� w dgw�0.

In the second stage, the commodity space, production set, and pricesystem, and of the gambling economy are the same as in the lotteryeconomy. Individuals now differ not only with respect to their type, butalso with respect to their first-stage wealth transfers. The problem of anindividual of type s with wealth transfer w is to

maxc

us(c) subject to p(c, s)� p } es+w.

Since individuals' second-stage consumption sets are no longer convex, wecannot, without loss of generality, assume that all individuals of the sametype and wealth make the same consumption choice. Therefore, wedescribe the choices of individuals of type s and wealth w by the distribu-tion giw # M(C_S).

B. Definition of an Equilibrium in the Gambling Economy

In the gambling economy, the second-stage commodity space, produc-tion set, and price system are unchanged from the lottery economy. The setof possible value transfers in the first stage is denoted by W=R. To avoidmeasurability issues, we will denote a type's gambling allocation by thesingle measure gi # M+(W_C_S), where the gw above is simply the



marginal distribution over wealth and where giw is type i 's choices overconsumptions conditional on w. Therefore, a type i individual's consump-tion set is

Gi#[g # M+(W_C_S) : g(W_Cs_[s])= g(W_C_S)=1].

The preferences of agent i over Gi are given by � ui (c) dg.A gambling equilibrium is defined as a gambling allocation [[gi], y] and

a price system p that satisfy the following:

(i) gi maximizes � ui (c) dg over Gi subject to

(1) � w dg�0 and

(2) c # argmax[ui (c) : p(c, s)� p } ei+w], if (w, c, s) is in the sup-port of gi.

(ii) y maximizes p } y subject to y # Y.

(iii) �i *i[� gi (dw_D)&ei (D)]= y(D) for all D # B(C_S).

We will refer to an equilibrium in our lottery economy as being equiv-alent to an equilibrium in the gambling economy if the price system,production vector, and ex ante distributions over consumption for eachtype are equal (that is, if xi ( } , } )=� gi (dw, } , } ) for all i). This implies thatthe ex ante welfare levels of people of a given type are also the same. Thefollowing proposition implies that the sets of lottery and gambling equi-libria are equivalent:

Proposition 5. Any lottery equilibrium is equivalent to a gambling equi-librium and vice versa.

Proof. The proof involves constructing two mappings, % and ,, whichare such that, if the gambling allocation [[gi], y] and the price system pare a gambling equilibrium, then the lottery allocation [[%(gi)], y] andthe price system p are an equivalent lottery equilibrium, and if [[xi], y]and the price system p are a lottery equilibrium, then the gambling alloca-tion [[,(xi, p, ei)], y] and p are an equivalent gambling equilibrium.

We define %(g) for g # � Gi to be

%(g)(D)=| g(dw_D) for all D # B(C_S).

The map % is simply the ex ante distribution over consumption impliedby g.



We define ,(x, p, e) for x # L, p(c, s) measurable, and for e # L by therequirement that

,(xi, p, ei)(D)=xi ([(c, s) : ( p(c, s)& p } ei, c, s) # D])

for all D # B(W_C_S).

The map , is a slightly more complicated object than %, since we implicitlymap into a higher dimensional space. The wealth transfer required to buycommodity (c, s) is w=p(c, s)& p } ei. Loosely speaking, we can select gi sothat gi ( p(c, s)& p } ei, c, s)=xi (c, s).

We now want to establish a key property of these mappings. If gi iswithin i 's consumption and budget sets in the gambling economy, then%(gi) is within i's consumption and budget sets in the lottery economy.Similarly, if xi is within i 's consumption and budget sets in the lotteryeconomy, then ,(xi, p, ei) is within i 's consumption and budget sets in thegambling economy. To see this, recall that type i 's budget constraint forthe lottery equilibrium is

p } xi& p } ei�0 (3)

while type i 's budget constraints for the gambling equilibrium are

| w dgi�0 (4)

and

p(c, s)�w+ p } ei for all (w, c) in the support of gis . (5)

To see that %(gi) satisfies individual i 's budget constraint in the lotteryeconomy, note that

0�| w dgi�| (p(c, s)&p } ei) dgi=| (p(c, s)&p } ei ) d%(gi ).

To see that condition (3) of the lottery economy implies that condition (4)and condition (5) of the gambling economy are satisfied by ,(xi, p, ei ),note that, by construction,

| w d,(xi, p, ei)=| (p(c, s)&p } ei ) dxi=p } xi&p } ei�0.

Thus the corresponding gambling elements satisfy their budget constraintsif the set of lottery elements satisfies its budget constraints with equality.



It is apparent that this method of mapping one type of equilibrium intothe other type preserves the feasibility and profit maximization conditionsof the respective definitions of equilibrium, since the ex ante distributionsof consumptions for each type and for the production vector are the same.Further, utility levels are the same, given that the distributions of consump-tions are the same for each type i. What remains is to show that if xi is asolution to type i 's problem in the lottery economy, then its associated ele-ment under the mapping , is a solution to i 's problem in the gamblingeconomy and vice versa for gi and %.

The proof that these mappings preserve optimality is a proof by con-tradiction: assume that gi is a solution to type i 's problem in the gamblingeconomy. Assume also that there exists an x̂i which is within type i 'sbudget set in the lottery economy and is preferred to %(gi). Then, since,(x̂i, p, ei) also satisfies type i 's gambling economy wealth and second-period budget constraints, individual i could have done better by choosing,(x̂i, p, ei). This contradicts our assumption that gi is optimal for i.Similarly, we can establish by contradiction that if xi is a solution to typei 's problem in the lottery economy, then ,(xi, p, ei) is a solution to type i 'sproblem in the gambling economy. K

Remark 6. This result makes it clear that the contribution of lotteries inthe lottery economy is simply to allow individuals to engage in implicitwealth gambles. Individuals have an incentive to gamble because theirindirect utility function over wealth is not concave everywhere, but ratherhas convex regions induced by the underlying indivisibility with regard toclub memberships.

Remark 7. If local nonsatiation holds for all equilibria, then the budgetconstraints hold with equality and the map between equilibria of the lot-tery and the gambling decentralizations is a bijection.

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Documents

Valuation Equilibrium with Clubs · consumer puts weight 1 on one and only one commodity. With lotteries, the restriction is that the total probability weights sum to 1. Commodity