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Continuum economies with finite coalitions: Core, equilibria, and widespread externalities
JQURNAL
OF ECONOMIC
THEORY
49, 113-134
(1989)
ontinuum Economies with Finite Co ore, Equilibria, and Widesprea PETERJ. HAMMOND Department
of Ecotiomics, Stanford Unhersity, Stanford, California 94305
MAMORU KANEKO Department
of Economics, Blacksburg,
Virginia Virginia
Polytechnic 24061
Institute,
AND MYRNA HOLTZ WOODERS Department of Economics, Unhersity of Toronto, Toronto, Ontario, Canada MSS !Ai
Received January 22, 1987; revised May 8, 1988
We develop a new model of a continuum economy with coalitions consisting of on!y finite numbers of agents. The core, called the f-core. is the set of allocations that are stable against improvement by finite coalitions and feasible by trade within finite coalitions. Even with widespread externalities-preferences depend on own consumptions and also on the entire allocation up to the null set-we obtain the result that the f-core coincides with the Walrasian allocations. Without widespread externalities, the f-core, the Aumann core, and the Walrasian allocations all coincide; however. with widespread externalities there is no obvious natural definition of the Aumann core. Journal of Economic Literature Classification Numbers: 021, 022, 026. 0 1989 Academic Press, Inc.
1. INTRODUCTION
1.1. Motivation
and Results
The purpose of this paper is to give a new picture of with recontracting. A perfectly competitive economy large number of participants, each of whom actively pursues his interests and each of whom has negligible influence on econ aggregates. Recontracting involves individuals meeting face ts face an 113 OO22-0531/89 $3.00 Copyright 0 1989 by Academic Press. Inc. All rights of reproduction in any form reserved.
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making mutually advantageous agreements. This suggests that in our picture, individuals should be effective in pursuit of their own interests through recontracting and simultaneously be ineffective in influencing any broad economic totals. One innovative formulation of perfect competition with recontracting was given by Aumann [2]. There, the set of agents is a continuum-a nonatomic measure space-and coalitions of recontracting agents are subsets of positive measure. In this formulation each individual agent is negligible relative to the total economy and thus one aspect of the notion of perfect competition in exchange economies is elegantly captured. However, each agent is also negligible relative to any recontracting group; i.e., coalitions are of positive measure. This implies that Aumann’s formulation does not capture the aspect of the individual as effective in recontracting. If we interpret Aumann’s model in any way which makes the individual nonnegligible or significant in recontracting, then the individual must also be interpreted as nonnegligible relative to the entire continuum.’ Thus Aumann’s formulation does capture the aspect of the individual as negligible relative to the total but not the aspect of the individual as effective in recontracting. In this paper we formulate our picture as a model with a nonatomic measure space of agents and with finite coalitions, i.e., ones consisting of finite numbers of agents. In this formulation each individual agent is negligible relative to the total, as in Aumann’s formulation, but is nonnegligible in pursuit of his own interests through recontracting. Thus our formulation captures the two features of perfect competition with recontracting. The subtle aspects of the treatment of finite coalitions in the continuum will be discussed in the next subsection. Before discussing the basic problems of modeling finite coalitions in the continuum, we describe the model and its major economic aspects and results. The solution concept adopted in this paper is the core. An allocation is in the core of the economy, called the S-core, if it satisfies two properties: First, it cannot be improved upon by any finite coalition and second, it is achievable by trades only in finite coalitions in some partition of the player set into finite coalitions. The second requirement is necessary to make the outcomes of cooperation for the entire economy consistent with the requirement that only finite coalitions can form. (We will also require that the partitions of the player set into finite coalitions be consis’ Aumann himself interprets the continuum not as a continuum of agents but as a large, finite set of agents. He writes “one should not think of the points t of T (the agent set) as individual players: rather, one should think of an individual player as an infinitesimal subset dt of T” (-a subset of small, but nevertheless, positive measure) (Aumann [4], p. 614). Since the total agent set has a finite measure, it follows that one should think of the total player set as large but finite, and thus of the individual agent as nonnegligible relative to the total agent set. See also Aumann 12, p. 411 and Aumann and Shapley [S, pp. 176-1781.
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115
tent with the proportions given by the measure.) An aggregate outcome of recontracting is an allocation in the f-core.* Our framework allows us to show the equivalence of the f-core and the Walrasian allocations in the context of an economy with widespread externalities where the Walrasian allocations may be Pareto-i~e~cie~t. widespread externalities we mean that the preference depends on his own consumption of goods and also o averages of the consumptions in the economy; more allocation, up to null sets. For the special case of nalities, the equivalence of thef-core, the Aumann core, allocations is obtained. When there are widespread externa~ities~ the with the Aumann core is in defining the concept of “can improv rior to any consideration of equivalence. This wil% be discusse A general form of externalities could be that preferences depend on entire allocation of goods in the economy. In this case, the e~ter~a~~ fall into two types-local externalities and global, or wide nalities. lin this paper, we exclude local externalities ces to depend on the entire allocation, hut only u course, on own consumptions). Here, by local externalities, we mean situamptions of tions where the preferences of individuals depe sets of other individuals of measure zero-the out by the term ‘“widespread.” With widespread externalities we can that preferences depend on the standards or averages of t in the economy, and not on the consumptions of individuals. For example, preferences could epend fashions, or how much clothing and housing is consum average. With widespread externalities, even though an indi in a. sense, by the total environment, he need not explicitly take it into account in his individual decision making. From the v~ew~o~~t of the individual9 the average consumptions are ~~cha~~eab~e. This is quite different from local externalities which are noticeable because they m differ in different localities and the individual may well benefit by taki them into account. By excluding local externalities we rule out situations where preferences depend on, e.g., the taste for loud music of one’s nextdoor neighbors. With widespread externalities, as in perfectly competitive economies, the individual takes the entire allocation as given. The idea of perfect at the individual is negligible relative to the total, and t * The further genera! version
concepts of measurement-consistent discussed in Kaneko and Wooders to accommodate the model herein of an assignment game of Shapley
partitions and thef-core were introduced and are [ 10, 111 in a game-theoretic framework, sufficien!Ry and other economic situations such as a continuum and Shubik [!7].
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the Walrasian allocations with widespread externalities. The problem with widespread externalities is not the nonequivalence of the f-core and the A-core but the definition of the A-core, more precisely, of “can improve upon” with coalitions of positive measure. The problem in the definition is maintaining feasibility for the complementary agent set when a coalition of positive measure changes its own part of an allocation. To see the difficulty in defining “can improve upon” with widespread externalities, suppose that a coalition S of positive measure changes that part of an allocation under its control. Then the feasibility of the allocation received by the complementary agent set is violated almost necessarily. Therefore we need to consider the response of the complementary agent set to the change brought about by S. We cannot simply take their part of the allocation as fixed. In the case of widespread externalities, the effects of the behavior of S are not limited to the change in its own part of the allocation. Through the external effects, the coalition S incurs further changes in utilities through the violation in feasibility and resultant change in behavior of the complementary agent set. With no widespread externalities, S is unaffected by any outcome for the complementary agent set. The definition of the f-core is free from these problems of effects of the complementary agent set. Essentially, since a finite coalition S is of measure zero a change in its behavior will not affect the feasibility of the allocation for the complementary agent set. For a more precise argument, recall that an f-core allocation f is the limit of a sequence of exactly feasible allocations, say if”}, with associated measurement-consistent partitions (p”}. In other words, our feasibility concept is approximate. To discuss precisely the question of feasibility, we need to return to the sequences {f”} and {p”>. In any exactly feasible allocationf” E F(p”), if the finite coalition S forms and changes its part of the allocation then feasibility is violated for the coalitions in p” intersecting S. However, the number of these coalitions is finite and each contains only a finite number of agents, so feasibility is violated for only a finite number of players in total. Since these players constitute a set of measure zero, by assumption (A.O), the violation in feasibility and the resultant responses can be ignored.14 To define the A-core from the above argument we must make some hypotheses about the behavior of the complementary agent set in response to a change in the behavior of a coalition of positive measure. So far the two approaches most frequently used to define a core are the minimax criterion (more precisely, the a-core, p-core due to Aumann [3]) and the strong equilibrium (also due to Aumann [ 1)). For the reasons discussed above concerning the feasibility of a new allocation made by a coalition I4 For another, to the continuum,
more detailed explanation see Kaneko and Wooders
in the context [12].
of convergence
of finite
economies
CONTINUUM ECONOMICS WITH EXTERNALITIES
417
tions. A measurement-consistent partition is defined in a fashion consistent with both absolute magnitudes for finite coalitions and proportional magnitudes for economic totals. Roughly speaking, the measureme~tconsistency requirement is that the “integration” of the cardinal numbers of members of finite coalitions in a partition “recovers” the measure on the rtitions enables us total agent set. This notion of measurement-consistent same model. to interpret finite sums and integrals meaningfully in En addition to logical consistency of the model itself, an important question to be asked of the continuum model is how it is to be inter a finite world. This is the subject of Hammond [6], where the related to extant models without externalities, and of Ka Wooders [12], where convergence in solution and structures is studied in the presence of widespread externalities. In these papers the restriction of coalitions in the continuum economy is approximated by the restrictive of (permissible) coalitions to relatively small sets of agents in the finite economies. In the presence of widespread externalities (Kaneko Wooders [ 12]), the permissible coalitions must be further restricted in the squares of the bounds on permissible coalition sizes are re become small relative to the total economy. In this manner, the ness of the individual agent in coalitions is the same in the con in the finite economies while, relative to the total economy, the individual agent becomes negligible. Thus, the prominent features of the co~ti~~~ model with finite coalitions are reflected in large finite economies wi relatively small coalitions, This paper is organized as follows. In Section 2, we formniate o mathematical model, and then give our two equivalence theorems. T meaning of our model and results is carefully discusse of the theorem. Proofs are in Section 3, 2. THE MODEL AND THE T~o~E~~ 2. I Agents and Measurement-Consistent Partitions of Agents Let (A, ~4, Jo) be a measure space, where A is a ore1 subset of a c plete separable metric space; ~4, the a-algebra of all ore1 subsets of A; y, a nonatomic measure with 0 < p(A) < -I- ~0.~ Each element in A is cal an agent. The measure p expresses proportional magnitudes of the measurable components of the economy. The o-algebra J$ is necessary for measurability arguments but does not play any important game-t role. 4 Under our assumptions, (A, d) is measure-theoretically isomorphic to ([O, I I3 C), where G is the a-algebra of all Bore1 subsets of the interval CO,11 (see Parthasarathy [15], pp. 12-14).
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Let 9 be the set of all finite subsets of A. Each element S in 9 is called a finite coalition or simply a coalition. As mentioned in Section 1, only finite subsets of agents can form coalitions in our model. Singleton sets are closed in A, so every coalition is measurable. Finite coalitions are negligible from the viewpoint of the measure (i.e., contain zero proportion of the economy); this means that finite coalitions are ignored by the measure. Their behavior will, however, be constrained and described by absolute magnitudes. Since only finite subsets of agents are allowed to form coalitions, allocations arising from recontracting are required to be attained by trades within the finite coalitions of a partition of the agent set A into finite coalitions. Partitions of agents into finite coalitions involve measurements both in absolute magnitudes and in proportional magnitudes. Partitions which reconcile these measurements are called measurement-consistent. We first define measurement-consistent partitions and then motivate these conditions via an example. A partition p of A is measurement-consistent iff for any positive integer k A;=U sEp,,s, =k S is a measurable subset of A, and each A: (k= 1,2, ...) has a partition (Ag,}:=,, where each A$, is measurable, with the following property: there are measure-preserving isomorphisms’ @kpl,$kpZ,.... ll/& from A;, to A;,, .... Aik, respectively, such that ($a, (a), .... ~)$~(a)} Ep for all aEA$1.6 (2-l) Note that (2.1) implies that for any SEP with ISI = k, we have S= ($gl(a), ....$&(a)> for some aE Ail. Therefore, for each integer k, Ai consists of all the members of k-agent coalitions and Aft consists of the tth members of these coalitions. The requirements that all the sets A:, have equal measure and that the isomorphisms be measure-preserving then capture the idea that coalitions of size k should have as “many” (i.e. the same measure) first members as second members, as many second members as third members, etc. Let II denote the set of measurement-consistent partitions. To exemplify the need for measurement-consistent partitions, we consider an economy where the set of agents is [0, 3) with Lebesgue measure. The agents indexed by real numbers in [0, 1) each own a right-hand a function rl, from a set B in a to a set C in a is a measure-preserving B to C iff (i) I/I is a measure-theoretic isomorphism, i.e., 3/ is 1 to 1, onto, and measurable in both directions, and (ii) p(T) = p(+( T)) for all Tc A with TE d. 6 In Kaneko and Wooders [ 11, Lemma A.11 a simple example of the construction of a measurement-consistent partition is provided. The example considers the case of k = 2 but the extension of the general case is clear. 5 Recall
that
isomorphismfrom
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EXTERNALITIES
glove (RHG) and those in [l, 3), a left-hand glove (LPI ( (a, I+ 2~): a E [O, 1)) is a partition of CO, 3) which pairs every agent owning a LHG with an agent owning a RHG. However, this partition violates the definition of a measurement-consistent partition since it matches, in a one-to-one way, one-third of the agents to two-thirds of the agents. Were such partitions allowed we could not consider the idea of relative scarcities of commodities reflected by the measure in economic models. Hn this economy, partition p = ( (a, 1 + u> : a {(u]: UE [Z, 3)), where th e agents in [0, 1) are arranged in agents in [ 1,2) and the others in [2, 3) remain single, is measureme~tconsistent.’ 2.2 The Economy and Allocations Let Z, be the set of nonnegative integers and let D = .Z: x co~su~~t~o~ set were I is a finite index set for indivisible commodities an D is a finite index set for divisible commodities with ID], the eardinahty o greater than zero. Let M= Iu D. We use the symbols for the orderings of elements of Q: l,=(l,
r--lMl -l 1, .~.)I),
and define 8, and 1, analogously. Finally, we will denote the space of measurable functions from the agent set A to $2 by k(A, Sz).” We model widespread externalities by having preferences depend on the allocation space L(A, 52) in addition to the commodity space a. Thus a preference relation > is defined to be a subset of (92 x k(A, (Q x L(A, Q)), and we typically write
CX? fl> Lx’,S’I to denote the preference between two members of the extended ~o~s~rn~tion set !S x L(A, Q). We assume )
if f(a) = g(a) = h(u) a.e. in A, then
[x,fj
> [v, 81 Q
This means that a preference relation is not affected by a change in t allocation on a null set of agents. (No other assumptions are required on the widespread externalities.) ’ The reader may notice from this example that a continuum version of an assignment game of Shapley and Shubik [17] is well formulated in our framework. For more detailed discussions of this, see Kaneko and Wooders [lo, 111. * Note that this is not to be taken as the space of equivalence classes of functions which are equal almost everywhere.
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HAMMOND,
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For each fixed f E L(A, Q) and preference relation >, we define the conditiona2 preference relative tof, >(f ), of > as the relation on 52 satisfying
We impose mathematical structure on preferences via conditions on conditional preferences. The space of possible conditional preference relations is taken to be PC = { >‘: >” is an irreflexive, transitive, and open binary relation on D x Q>. In particular; continuity and transitivity of conditional preferences is assumed, but not of > itself. The space PC is then given the topology induced by closed convergence on the class {Sz x Q\>‘: >” E PC} of closed subsets of ~2 x L2 as in Hildenbrand [8, pp. 96-981. The space of preference relations P is defined by P = { > : > satisfies (A.0) and >(f)
E PC for all
f E L(A,
a) >.
An economy E is a mapping E(a) = (>,, e(a)) from A to P x Szsuch that for any f E L(A, Q) the mapping Ef (a): = (>, (f ), e(a)) is measurable-i.e., for any Bore1 set T of PC x 52, we have ET’(T) E&. The economy E assigns to each agent a in A a preference relation >, and an initial endowment e(a). For each f in L(A, Q), the mapping from agents to their conditional preferences and their initial endowments is measurable. We assume that JA e is finite and strictly positive. We will now define feasible allocations for the economy E. A functionf in L(A, &I) is called an allocation with respect to a (measurement-consistent) partition p (p E IT) of A into finite subsets iff 1 f(a)<
C e(a)
for all
Sep.
(2.2)
An allocation f with respect to p can be attained by trading commodities only within coalitions in p. Define the sets F(p) (p~17), F, and F* by F(p) = (f~ W, F=
u
Qsz):f is an allocation with respect top};
(2.3) (2.4)
F(P);
Pen
F* = (f~ L(A, 52): for some sequence (f”} (f”}
converges in measure to f }.
in F, (2.5)
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121
Note that I?(p) c PC F* for all p E II and F* is the closure of F wit respect to convergence in measure. Also, since the endowment e satisfies (2.2), F(p) is not empty. Recall that “(f’) converges in measure to f" means that for any E>O, ~((uE A: d(f”(a), f(a))>&)) 40 as n -+ co.’ In interpretation we consider an element of F* as approximately feasible in the sense that it is in the closure of F and not necessarily in F itself.” The end of this subsection includes an example il~~strati~~ that I?’ may not be closed. We say that a coalition S in can improve upon a function f in L(A, a) if? there is a vector (x~),~~ such that XaEQ
for all
a E S;
1 x0 < 1 e(a); RES UES for all xa >-.(f)f(a)
(2.6) (2.7)
a E S.
(2.8)
course, if the members of S change their part of the allocation f to 1aes then the resulting allocation agrees with j” only on ta\S. However, from the assumption (A.0) that preferences are unaffeeted by a change m the allocation on a null set, the change in f on the finite set S can be ignored in condition (2.8). The f-core of the economy E is defined to be Cj.= {f E F*: there is a fullr’ set A” c A such that no finite coalition in J can improve upon f >. An allocation f in the f-core Cf is stable, roughly speaking, in the sense that no finite coalition can improve uponf, It is also approximately feas in the sense that f E I;*; i.e., f can be approximated by a sequence of exa feasible allocations. The exact definition takes a full subset 2 so that no finite coalition in A” can improve uponf. This intermediate step of a full set A” is made for simplicity of the statements of our results. E we could adopt a definition of the f-core requiring that no finite coalition could improve upon an allocation in thef-core. However, this 6ornp~i~at~s the statements of our results without adding any substance to the meaning. The following example illustrates that F may not be closed with respect to convergence in measure so it may be the case that Ff F*. 9 If {fnj converges in measure to f; then {f”) has a subsequence which converges pointwise to f a.e. and, conversely, if {PI converges pointwise to S a.e., then {f”) itself converges in measure to J lo In Kaneko and Wooders [ 121 we discuss the appropriateness of F* as the idealized feasible set for sequences of finite economies. ” A full set of agents is one whose complement is null.
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HAMMOND,
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EXAMPLE 2.2.1: F is not closed. Rc , ,u is Lesbesgue measure, and
e(a) =
i
AND
WOODERS
Suppose that A = [0, l), that D =
(4 0)
(O
(0, 1)
(a
1)
for some irrational number a. Consider the allocation f(a) = (a, 1 -a) (all SEA) which satisfies jA (f-e) = 0. Each finite coalition consisting of m agents with endowment (1,O) and of n agents with endowment (0, 1) has an aggregate endowment equal to (m, n). To achieve the allocation (a, 1 -a) for each of the m + n agents, one needs (m + n)(a, 1 - IX)< (m, n). This implies that (m+n)a
This subsection presents conditions on E under which the f-core and the set of Walrasian allocations W coincide. We first define the Walrasian allocations and then state the required additional conditions. A function f in L(A, Q) is called a Walrasian allocation iff for some PER?,
P . f(a) G P. e(a)
a.e. in A, x X-, (f)f(a)
forall
(2.10)
a.e. in A; x~Qwithp.x
(2.11)
Let W denote the set of Walrasian allocations. The first step in showing equivalence of the f-core and the Walrasian allocations is to demonstrate that f E F* iff f satisfies the mean excess demand condition (2.12). This is proved12 in Kaneko and Wooders [ll, Lemma 3.11. Therefore we have (2.13)
F*=(~EL(A,~):SA~~~~~~. In particular, I2 Under goods may
this implies that WC F*.
slightly less restrictive be zero.
assumptions
on 51, in particular,
the number
of divisible
CONTINUUM ECONOMICS WITH EXTERNALITIES
123
Additional conditions are required on the conditional preferences >,(J’) in the economy E: For all a E A and for any f~ i&4, a), (A.l) >,(f) is strictly monotone on RG, (the divisible goods); (A.2) For all (x,, x,) E Q = 2: x RD,, there is a yD E RD, such that lo,, rD> kAf)(xI, xd; and (A.3) e(a) >,(f)(x(, 0,) for all x,fZ!+. Assumption (A.l) is standard. Assumption (AL?) is simply that for any commodity bundle (x1, xD) there is a commodity bundle (O,, yD), with divisible goods only, such that the second bundle is preferred-~‘e~oug~ of the divisible goods is better than the given bundle. Our third ass~rn~t~o~ (A.3) is that the initial endowment is preferred to any commodity bundle with only indivisibIe goods. 1. Under assumptions (AO), (Al), (.4.2), and ( Cl. coincides with the set of Walrasian allocations W. THEOREM
Next we will consider the case of no widespread exter~aiitjes; for any aEA,
(A.@)
>,(f)=>,(g)
for allS,gEL(A,Q)
In this case, we write simply 2,. The following standard de~~~t~o~ of t A- (Aumann) core now applies. The A-core of ihe economy is the set of allocations C,={~EL(A,Q):~
f
eandnosubsetofAof A
positive measure can improve uponf),
(2.9)
where a subset S of positive measure is said to be able to kn~~ove UJW.IJf iff for some g E L(S, Q), we have IS g < IS e and da) >, f(a) 2. w= CA.
THEOREM
c,=
for all a in S.
Under assumptions (A.O’), (A.l),
A.2), and (
Theorem 2 demonstrates the equivalence of tbe f-core, the W allocations, and the A-core without any widespread externahties.13 other hand, Theorem 1 demonstrates the equivalence only of the f-core an I3 For an exchange economy with indivisible goods but without externalities,. Mas-Colell [14] gave a counterexample to the existence of equilibrium but with a nonempty Aumann core, implying the nonequivalence. Our assumption (A.31 excludes this example (see also Remark 2.4.11, and, together with (A.l) and (A.2), ensures the equivalence.
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the Walrasian allocations with widespread externalities. The problem with widespread externalities is not the nonequivalence of the f-core and the A-core but the definition of the A-core, more precisely, of “can improve upon” with coalitions of positive measure. The problem in the definition is maintaining feasibility for the complementary agent set when a coalition of positive measure changes its own part of an allocation. To see the difficulty in defining “can improve upon” with widespread externalities, suppose that a coalition S of positive measure changes that part of an allocation under its control. Then the feasibility of the allocation received by the complementary agent set is violated almost necessarily. Therefore we need to consider the response of the complementary agent set to the change brought about by S. We cannot simply take their part of the allocation as fixed. In the case of widespread externalities, the effects of the behavior of S are not limited to the change in its own part of the allocation. Through the external effects, the coalition S incurs further changes in utilities through the violation in feasibility and resultant change in behavior of the complementary agent set. With no widespread externalities, S is unaffected by any outcome for the complementary agent set. The definition of the f-core is free from these problems of effects of the complementary agent set. Essentially, since a finite coalition S is of measure zero a change in its behavior will not affect the feasibility of the allocation for the complementary agent set. For a more precise argument, recall that an f-core allocation f is the limit of a sequence of exactly feasible allocations, say if”}, with associated measurement-consistent partitions (p”}. In other words, our feasibility concept is approximate. To discuss precisely the question of feasibility, we need to return to the sequences {f”} and {p”>. In any exactly feasible allocationf” E F(p”), if the finite coalition S forms and changes its part of the allocation then feasibility is violated for the coalitions in p” intersecting S. However, the number of these coalitions is finite and each contains only a finite number of agents, so feasibility is violated for only a finite number of players in total. Since these players constitute a set of measure zero, by assumption (A.O), the violation in feasibility and the resultant responses can be ignored.14 To define the A-core from the above argument we must make some hypotheses about the behavior of the complementary agent set in response to a change in the behavior of a coalition of positive measure. So far the two approaches most frequently used to define a core are the minimax criterion (more precisely, the a-core, p-core due to Aumann [3]) and the strong equilibrium (also due to Aumann [ 1)). For the reasons discussed above concerning the feasibility of a new allocation made by a coalition I4 For another, to the continuum,
more detailed explanation see Kaneko and Wooders
in the context [12].
of convergence
of finite
economies
CONTINUUM
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825
EXTERNALITIES
and the complementary coalition, the strong equilibrium concept carrot be naturally applied. As is well known, the minimax criterion is not particularly persuasive except for two-person zero sum games. For completeness, the minimax criterion is defined below for the case of preferences monotonically increasing on the externalities, i.e., for external economies. The definition according to the minimax criterion of “can improve upon” would be to assume that the complementary coalition disposes of all its private goods. An allocationfE F* is in the A-core of I$ denoted by C,, if there is no subset S of A with positive measure and no allocation
g E L(A,
l2) where ss
gd
da)
ss
= 0
e a.e. in
A\S
an
Ma), gl>, U(a), Sl
for all
a E S.
We observe that an allocation in the A-core is Pareto-ef~cie~t whereas those in thef-core (and Walrasian allocations) are not necessarily so. Thus, the A-core and the Walrasian allocations do not coincide except in very special situations including, of course, the no-widespread-externalities case. A counterexample to equivalence is given in Kaneko and Wooders [I Even under our simple monotonicity assumptions on the externalities, minimax criterion does not lead to a natural definition of the A-core in the sense that we cannot justify the resulting assumption that a complementary coalition disposes of all its private goods. With no widespread externalities, our resuhs are suggestive of t due to Schmeidler [ 161 and Mas-Cole11 [ 141. Schmeidler considers a tinuum economy where only coalitions of small positive measure are permitted, while Mas-Cole11 uses the framework of a finite ecQ~orny where only coalitions bounded in size (membership) are pe itted. Schmeidler obtains Aumann coreequilibrium equivalence, while as-Cole11 obtams approximate equivalence. These papers are similar to ours in that coreequilibrium equivalence is obtained when only “‘smah” coalitions are allowed. However, neither author restricts allocations arising from recontracting to be achievable by cooperation only within small coalitions. Also neither author considers an actual “limit model.” Although does not impose the feasibility for small coalitions constraint, quite close to a large finite analogue of ours.” I5 For a more detailed discussion of the finite analogue and the convergence of finite economies
with
small
coalitions
to continuum
economies
with
finite
coalitions
in an exchange
economy with widespread externalities, see Kaneko and Wooders 1121, where convergence of solutions
and game
structure
is shown.
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AND
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2.4 Remarks Remark 2.4.1. Our assumptions are similar to those of Khan and Yamazaki [9] and Yamazaki [lS]. (In fact, if the cardinality IDI of D, the set of divisible goods, is equal to one, our assumptions imply those of Khan and Yamazaki. If the cardinality of D is possibly greater than one, Yamazaki [lS] uses a stronger assumption than our (A.2) and also uses other assumptions for the existence of Walrasian equilibrium.) Khan and Yamazaki show the equivalence of the A-core to the set of “weakly competitive allocations,” and the non-emptiness of the A-core; they employ an example due to Mas-Cole11 [13] to illustrate that the set of Walrasian allocations may be empty. Therefore, with their assumptions the A-core and the set of Walrasian equilibrium allocations might not coincide. Our assumption (A.3) rules out Mas-Colell’s example. Without widespread externalities, in fact, we can prove, under assumptions (A.O), (A.l), (A-2), and (A.3), that
c,=
W=C,#@
by modifying our equivalence in Theorem 2 and using Khan and Yamazaki’s [9] existence of weakly competitive allocations. The modifications will be indicated in Section 3, where the proofs of Theorems 1 and 2 are provided. Remark 2.4.2. With widespread externalities, the existence of Walrasian equilibrium and the f-core can be obtained by imposing only a slight restriction on the externalities. Specifically, we assume that there is a given finite family of subsets of positive measure of A, say T1, .... T,, such that the externalities depend on an allocation only via the value of its integral over each of the subsets.i6 That is, we assume
@.O*) C~,fl~,~y~~l~C~~~~,f,...,~,~fl~,C~~jr,~~~~~~~~,~l for all Cx,fl, [Iv, 81EQ x W& a). Assume also that preferences are open in (a x Ry’“) x (Q x Ry’“). Then essentially the same proof (as the one herein) of the existence of equilibrium can be applied by using Liapunov’s theorem to convexify over the externalities. Remark 2.4.3. Although we have shown that without widespread nalities the Aumann core coincides with the JLcore, this does not that these two concepts are equivalent nor that Aumann’s theorem equivalent in interpretation to ours. The Aumann core permits only tions containing positive proportions of the total agent set to improve I6 This
enables
us to work
in a finite
dimensional
“commodity”
space.
extermean [2] is coaliupon
CONTINUUM
ECONOMICS
WITH
EXTERNALITIES
127
allocations; thus individual agents and finite coalitions are unable to have any affect on the outcome of cooperation. Bolh the surname core an competitive equilibrium build in completely negligible agents. Hn contrast, in our approach individual agents are nonnegiigible relative to themselves and to any finite group. We obtain, as a conclusion, that individual agents ices as given Even and finite coalitions can do no better than taking though finite coalitions are negligible relative to t e~Qnomy~ we have alrasian equilibrium equivalence. emark 2.4.4. A different type of consideration rather than those discussed in the introduction was the original source of one author’s interest in finite coahtions. This concerns the power of finite ~oa~~tio~s to ipulate resource allocation mechanisms in ~o~~in~~rn economies. en one considers manipulation the size of permissible ~~a~~tio~s-~~~~e~ or of positive measure-is critical. See Hammon mond [6], where another core concept, which ombines” finite coalitions and coalitions of positive measure, is introduce
In this section, we prove C,c W in the case of no widespread externalities. The proof that C, c W is a simple rnodi~~at~o~ of the proof of Cfc W. The proofs that C;=, W and that C, 3 W are straightforward and standard, so we omit them. Thus we will have C/= W= C, in the case of no widespread externalities. In the case of widespread externalities, we can obtain C,= W by applyin the eq~iva~e~~e result C,= W in the case of no widespread externalities as follows. Let g be a function in L(A, .G) and let E be an economy with widespread externalities. Then we have an economy ER: a -+ b>,(g), e(a)), without widespread externalities, induced from E by fixing g in each eference relation. Then the above equivalence result without wid ternahties states that C; = Wg, where Cf” and Wg are respecti S-core and the set of Walrasian allocations of the economy ER. The conditions for g to belong to C/ and W of the economy E can be written as .!sq
if and only if
g E Cr;
gcwg
if and only if
g E kl/.
Since CT = W” for all g, these conditions are iden In the following analysis of an economy wi 642/49/l-9
128
HAMMOND,
nalities, individual (l-lx L(A, a))‘.
KANEKO,
AND
WOODERS
preferences are regarded as a subset of 52 x Q instead of
Proof that C*c W Let f be an allocation aeA by
in the f-core. Define the set $(a) for each agent
(which is equivalent to Hildenbrand’s definition of $(a) in [8, Theorem 1, p. 133)). Then the following lemma holds. 1. There is no w E 1 Ic/ such that w 4 0.
LEMMA
Since j @ is convex by Liapunov’s theorem, from this lemma it follows that the set s $ can be separated from the strictly negative orthant of RM by a hyperplane. That is, there is a price vector p in R”, p > 0, such that forall
0Gp.w
wE
s
*
(3.1)
Proof ,of Lemma 1. Suppose there exists a vector w ES + such that w + 0. Then there exists a measurable function t: A + Z’x R” such that t(a) E +(a) a.e. in A and JA t = w (< 0. Define S= (aEA: t(a)+e(a)>,f(a)). Then p(S)>0 and x=J,t@O and the following claim holds. Claim. There is a finite partition function i: S-+ Z’x RD such that p(S,)=p(S,)= i(a) = t‘(d) i(a) = 0 tD ta)
s s
%- tD ca)
t(a) < 0.
(S,, S1, .... S,) of S and a simple
... =p(S,)>O; if
a, a’ 62S,(j=
for all and
(3.2) 1, .... m);
(3.3)
afzS,; iI
(3.4) = t,(a)
forall
aE ij Si; j=l (3.5)
Proof of Claim. We begin by partitioning a set consisting of “most” agents into subsets so that if a and a’ are in the same subset, then t(a) is approximately equal to t(d). For any positive integer n, define
CONTINUUM
ECONOMICS
Z,={-n,
-n+l,..., r--- 14 1 z’,=z,x ... xz,;
WITH
-l,O,l,...,
12
EXTERNALITI[ES
n-s>
1 1; Kn=i --If,-n+; )...)-&o,$ ....n--2n-i9’-~ I r-=- IDI... xK,
Kf=K,x
Define a simple function t” = (t:, t;): s -+ .P x ~~ ~JJ [f(a)= k,+kl,
if
L
t>(a) =
if -i,
tl(a) o i I
(3.6)
-nl,
fD(U)E I-I
(3.7)
dfD
otherwise. Then, for each a E A, there is an fi such -El M f t(a) 6 fil M. Therefore t g for all n 2 ii and converges to t(a j. This converges to js t G 0. Therefore there is a integer n, such that
For +,, k,) E Zfi, x K$ denote the set (ac A: tl(a) =zI an lD(a) < k, + (1/2”O) ID) by T(z[, k,). Then it follows from (3.6), (3.7), and (3.8) that (3.9) From
each T(z,, k,)
((z,, kD)EZL,,x
KE),
we can choose
a subset
S(z,, k,) so that P( S(z,, k,)) is a rational number;
(3.10)
and (3.11)
130
HAMMOND,
KANEKO,
AND
WOODERS
Since each p(S(zl, kn)) is a rational number (possibly zero), we can find a (S, , .... S,) of the finite partition subpartition (S(z,, k,): (z,, kD)e ZL,, x Kg and ,u(S(Z,, k,)) > 0} such that ,u(S,) = n.. = ,u(S,) > 0. Define S, and i by s,=s-
(j sj; j=l
P”(a)
if
a E fi S, j=l
0
otherwise.
i(a) =
It immediately follows that (S,, S,, .... S,) and i satisfy conditions (3.3), and (3.4). From (3.11) we have
(3.2),
This completes the proof of the claim. Now select one agent a,E Sj for each j= 1, .... m, and write c = (q, ...) a,}.17 Since iD (aj) $ tD (aj) and i,(aj) = t,(aj) for all j= 1, .... m by (3.4), we have, by Assumption (A.1 ),
ita,) + e(aj>>aj t(aj) + e(aj) The feasibility of the allocation (3.2) and (3.5); indeed
.gc /
(t‘(aj)
+
e(aj))
=
for all j = 1, .... m.
>-,f(aj)
(i(aj) + e(aj)),,
--&J
j!l
=&jp+
tiaj)
c follows from conditions
P(sj)
c
e(Uj)
e(aj) cl,E c
+
C
<
CZjCC
C e(aj)* LljEC
Therefore the finite coalition C can improve upon J For any full subset 2 of A we can also find a finite coalition C in A” so that C consists of only one agent in Sj for each j= 1, .... m. Then, C can improve upon J: This is a contradiction to the supposition that fis in C,-. This completes the proof of Lemma 1. 1 I7 To modify
the proof
of this theorem
to obtain
C, c W, replace
C by U,“= 1 S,.
CONTINUUM
ECONOMICS
WITH
EXTERNALITIES
131
Recall that from Lemma 1 it follows that there is a price vector p E RM (p > 0) satisfying (3.1). We will now show that p is an equilibrium price vector. It follows from (3.1) and Hildenbrand [g, p. 63: Proposition 61 that
It also follows from (3.1) that 0 < JA infp I I/I* Since $(a) contains 0, we have inf p . $(a) < 0. Hence we have jA infp . $ = 6bl which implies that infp . $(a) = 0 a.e. in A. Then this implies that a.e. in A,
(3.12) or equivalently,
LEMMA
2.
p.e(a)=p.f(
a ) a.e. in A (budget constraints are satisfied).
Proof. For all a E A, by Assumption (A.1 ) (strict monotonicity of preferences for divisible goods) we can choose a sequence ( y”} such t&t { y”> converges tof(a) and y” >cl f(a) for all n. Then we have, from (3.82) p .e(a)
3.
~~$0,.
Proof. First we prove that pD > 0,. Suppose pD = 0,. Since IA e 90 and p > 0 there is a subset 5’ of A such that p ‘e(a) > 0 for all a E S and p(S) >O. Choose an agent a from S for whom (3.12) holds
132
HAMMOND,
KANEKO,
AND
WOODERS
and pa f(a) =p. e(a) > 0. By Assumption (A.2), there is a vector y=(O,,y,)~fi such that y>af(a). Since pD=OD, we have O=p.y< p .f(a) =p .e(a), which contradicts (3.12). This shows that p. > 0,. Now let us show that p. $0,. Since fAe>lS,fby (2.13) andp.J,e= p . JA f by Lemma 2, we have, for all k E I u D, the usual “rule of free goods,” (3.13)
Pk>O=>
Since p. > 0, and JA e, B O,, it follows from (3.13) that p. -JAfD = pD .jA eD > 0. Therefore there is a subset ,%of A with positive measure such that PO. eD (a) =pD .fD (a) > 0 and (3.12) holds for all a E S. Consider an arbitrary agent a in S. We shall prove that
x>,f(a)
and
xI=fAa)*p.x>p-f(a).
(3.14)
This implies p. 9 0, ; indeed, if pd= 0 for some d.s D, then x, given by xd=fd(a)+l and x,=fk(a) for all kfd, violates (3.14) because of Assumption (A.l). Therefore let us now prove (3.14). Suppose on the contrary that x .>, f(a), x,= f](a), and p . x Qp .f(a) for some x. Because >a is continuous, there exists 2 E (0,l) near 0 so that 2(x,, 0) + (1 - n)x >-,f(a) and p.[l(x,,O)+(l-I)x]=~~~.x,+(l-~)p.x~~p.f~(a)+ (1 - 1)~ .f(a) =p -f(a) - lpD .fD(a) cp .f(a) where the last inequality follows because a E S, as defined above. This is a contradiction to (3.12). So PD~OD.
I
LEMMA 4. A.e. in A, p . x d p . e(a) =s-x >a f(a) the budget sets).
(utility maximization
on
Proof. It follows from Assumption (A.3) that e,(a) > 0, for all a E A. This together with Lemma 3 implies that p . e(a) > 0 for all a E A. Consider an arbitrary agent a for whom (3.12) (p . x
OF ECONOMIC
THEORY
49, 113-134
(1989)
ontinuum Economies with Finite Co ore, Equilibria, and Widesprea PETERJ. HAMMOND Department
of Ecotiomics, Stanford Unhersity, Stanford, California 94305
MAMORU KANEKO Department
of Economics, Blacksburg,
Virginia Virginia
Polytechnic 24061
Institute,
AND MYRNA HOLTZ WOODERS Department of Economics, Unhersity of Toronto, Toronto, Ontario, Canada MSS !Ai
Received January 22, 1987; revised May 8, 1988
We develop a new model of a continuum economy with coalitions consisting of on!y finite numbers of agents. The core, called the f-core. is the set of allocations that are stable against improvement by finite coalitions and feasible by trade within finite coalitions. Even with widespread externalities-preferences depend on own consumptions and also on the entire allocation up to the null set-we obtain the result that the f-core coincides with the Walrasian allocations. Without widespread externalities, the f-core, the Aumann core, and the Walrasian allocations all coincide; however. with widespread externalities there is no obvious natural definition of the Aumann core. Journal of Economic Literature Classification Numbers: 021, 022, 026. 0 1989 Academic Press, Inc.
1. INTRODUCTION
1.1. Motivation
and Results
The purpose of this paper is to give a new picture of with recontracting. A perfectly competitive economy large number of participants, each of whom actively pursues his interests and each of whom has negligible influence on econ aggregates. Recontracting involves individuals meeting face ts face an 113 OO22-0531/89 $3.00 Copyright 0 1989 by Academic Press. Inc. All rights of reproduction in any form reserved.
114
HAMMOND,
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AND
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making mutually advantageous agreements. This suggests that in our picture, individuals should be effective in pursuit of their own interests through recontracting and simultaneously be ineffective in influencing any broad economic totals. One innovative formulation of perfect competition with recontracting was given by Aumann [2]. There, the set of agents is a continuum-a nonatomic measure space-and coalitions of recontracting agents are subsets of positive measure. In this formulation each individual agent is negligible relative to the total economy and thus one aspect of the notion of perfect competition in exchange economies is elegantly captured. However, each agent is also negligible relative to any recontracting group; i.e., coalitions are of positive measure. This implies that Aumann’s formulation does not capture the aspect of the individual as effective in recontracting. If we interpret Aumann’s model in any way which makes the individual nonnegligible or significant in recontracting, then the individual must also be interpreted as nonnegligible relative to the entire continuum.’ Thus Aumann’s formulation does capture the aspect of the individual as negligible relative to the total but not the aspect of the individual as effective in recontracting. In this paper we formulate our picture as a model with a nonatomic measure space of agents and with finite coalitions, i.e., ones consisting of finite numbers of agents. In this formulation each individual agent is negligible relative to the total, as in Aumann’s formulation, but is nonnegligible in pursuit of his own interests through recontracting. Thus our formulation captures the two features of perfect competition with recontracting. The subtle aspects of the treatment of finite coalitions in the continuum will be discussed in the next subsection. Before discussing the basic problems of modeling finite coalitions in the continuum, we describe the model and its major economic aspects and results. The solution concept adopted in this paper is the core. An allocation is in the core of the economy, called the S-core, if it satisfies two properties: First, it cannot be improved upon by any finite coalition and second, it is achievable by trades only in finite coalitions in some partition of the player set into finite coalitions. The second requirement is necessary to make the outcomes of cooperation for the entire economy consistent with the requirement that only finite coalitions can form. (We will also require that the partitions of the player set into finite coalitions be consis’ Aumann himself interprets the continuum not as a continuum of agents but as a large, finite set of agents. He writes “one should not think of the points t of T (the agent set) as individual players: rather, one should think of an individual player as an infinitesimal subset dt of T” (-a subset of small, but nevertheless, positive measure) (Aumann [4], p. 614). Since the total agent set has a finite measure, it follows that one should think of the total player set as large but finite, and thus of the individual agent as nonnegligible relative to the total agent set. See also Aumann 12, p. 411 and Aumann and Shapley [S, pp. 176-1781.
CONTINUUM
ECONOMICS
WITH
EXTE~NA~~T~~§
115
tent with the proportions given by the measure.) An aggregate outcome of recontracting is an allocation in the f-core.* Our framework allows us to show the equivalence of the f-core and the Walrasian allocations in the context of an economy with widespread externalities where the Walrasian allocations may be Pareto-i~e~cie~t. widespread externalities we mean that the preference depends on his own consumption of goods and also o averages of the consumptions in the economy; more allocation, up to null sets. For the special case of nalities, the equivalence of thef-core, the Aumann core, allocations is obtained. When there are widespread externa~ities~ the with the Aumann core is in defining the concept of “can improv rior to any consideration of equivalence. This wil% be discusse A general form of externalities could be that preferences depend on entire allocation of goods in the economy. In this case, the e~ter~a~~ fall into two types-local externalities and global, or wide nalities. lin this paper, we exclude local externalities ces to depend on the entire allocation, hut only u course, on own consumptions). Here, by local externalities, we mean situamptions of tions where the preferences of individuals depe sets of other individuals of measure zero-the out by the term ‘“widespread.” With widespread externalities we can that preferences depend on the standards or averages of t in the economy, and not on the consumptions of individuals. For example, preferences could epend fashions, or how much clothing and housing is consum average. With widespread externalities, even though an indi in a. sense, by the total environment, he need not explicitly take it into account in his individual decision making. From the v~ew~o~~t of the individual9 the average consumptions are ~~cha~~eab~e. This is quite different from local externalities which are noticeable because they m differ in different localities and the individual may well benefit by taki them into account. By excluding local externalities we rule out situations where preferences depend on, e.g., the taste for loud music of one’s nextdoor neighbors. With widespread externalities, as in perfectly competitive economies, the individual takes the entire allocation as given. The idea of perfect at the individual is negligible relative to the total, and t * The further genera! version
concepts of measurement-consistent discussed in Kaneko and Wooders to accommodate the model herein of an assignment game of Shapley
partitions and thef-core were introduced and are [ 10, 111 in a game-theoretic framework, sufficien!Ry and other economic situations such as a continuum and Shubik [!7].
124
HAMMOND,
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the Walrasian allocations with widespread externalities. The problem with widespread externalities is not the nonequivalence of the f-core and the A-core but the definition of the A-core, more precisely, of “can improve upon” with coalitions of positive measure. The problem in the definition is maintaining feasibility for the complementary agent set when a coalition of positive measure changes its own part of an allocation. To see the difficulty in defining “can improve upon” with widespread externalities, suppose that a coalition S of positive measure changes that part of an allocation under its control. Then the feasibility of the allocation received by the complementary agent set is violated almost necessarily. Therefore we need to consider the response of the complementary agent set to the change brought about by S. We cannot simply take their part of the allocation as fixed. In the case of widespread externalities, the effects of the behavior of S are not limited to the change in its own part of the allocation. Through the external effects, the coalition S incurs further changes in utilities through the violation in feasibility and resultant change in behavior of the complementary agent set. With no widespread externalities, S is unaffected by any outcome for the complementary agent set. The definition of the f-core is free from these problems of effects of the complementary agent set. Essentially, since a finite coalition S is of measure zero a change in its behavior will not affect the feasibility of the allocation for the complementary agent set. For a more precise argument, recall that an f-core allocation f is the limit of a sequence of exactly feasible allocations, say if”}, with associated measurement-consistent partitions (p”}. In other words, our feasibility concept is approximate. To discuss precisely the question of feasibility, we need to return to the sequences {f”} and {p”>. In any exactly feasible allocationf” E F(p”), if the finite coalition S forms and changes its part of the allocation then feasibility is violated for the coalitions in p” intersecting S. However, the number of these coalitions is finite and each contains only a finite number of agents, so feasibility is violated for only a finite number of players in total. Since these players constitute a set of measure zero, by assumption (A.O), the violation in feasibility and the resultant responses can be ignored.14 To define the A-core from the above argument we must make some hypotheses about the behavior of the complementary agent set in response to a change in the behavior of a coalition of positive measure. So far the two approaches most frequently used to define a core are the minimax criterion (more precisely, the a-core, p-core due to Aumann [3]) and the strong equilibrium (also due to Aumann [ 1)). For the reasons discussed above concerning the feasibility of a new allocation made by a coalition I4 For another, to the continuum,
more detailed explanation see Kaneko and Wooders
in the context [12].
of convergence
of finite
economies
CONTINUUM ECONOMICS WITH EXTERNALITIES
417
tions. A measurement-consistent partition is defined in a fashion consistent with both absolute magnitudes for finite coalitions and proportional magnitudes for economic totals. Roughly speaking, the measureme~tconsistency requirement is that the “integration” of the cardinal numbers of members of finite coalitions in a partition “recovers” the measure on the rtitions enables us total agent set. This notion of measurement-consistent same model. to interpret finite sums and integrals meaningfully in En addition to logical consistency of the model itself, an important question to be asked of the continuum model is how it is to be inter a finite world. This is the subject of Hammond [6], where the related to extant models without externalities, and of Ka Wooders [12], where convergence in solution and structures is studied in the presence of widespread externalities. In these papers the restriction of coalitions in the continuum economy is approximated by the restrictive of (permissible) coalitions to relatively small sets of agents in the finite economies. In the presence of widespread externalities (Kaneko Wooders [ 12]), the permissible coalitions must be further restricted in the squares of the bounds on permissible coalition sizes are re become small relative to the total economy. In this manner, the ness of the individual agent in coalitions is the same in the con in the finite economies while, relative to the total economy, the individual agent becomes negligible. Thus, the prominent features of the co~ti~~~ model with finite coalitions are reflected in large finite economies wi relatively small coalitions, This paper is organized as follows. In Section 2, we formniate o mathematical model, and then give our two equivalence theorems. T meaning of our model and results is carefully discusse of the theorem. Proofs are in Section 3, 2. THE MODEL AND THE T~o~E~~ 2. I Agents and Measurement-Consistent Partitions of Agents Let (A, ~4, Jo) be a measure space, where A is a ore1 subset of a c plete separable metric space; ~4, the a-algebra of all ore1 subsets of A; y, a nonatomic measure with 0 < p(A) < -I- ~0.~ Each element in A is cal an agent. The measure p expresses proportional magnitudes of the measurable components of the economy. The o-algebra J$ is necessary for measurability arguments but does not play any important game-t role. 4 Under our assumptions, (A, d) is measure-theoretically isomorphic to ([O, I I3 C), where G is the a-algebra of all Bore1 subsets of the interval CO,11 (see Parthasarathy [15], pp. 12-14).
118
HAMMOND,
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AND
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Let 9 be the set of all finite subsets of A. Each element S in 9 is called a finite coalition or simply a coalition. As mentioned in Section 1, only finite subsets of agents can form coalitions in our model. Singleton sets are closed in A, so every coalition is measurable. Finite coalitions are negligible from the viewpoint of the measure (i.e., contain zero proportion of the economy); this means that finite coalitions are ignored by the measure. Their behavior will, however, be constrained and described by absolute magnitudes. Since only finite subsets of agents are allowed to form coalitions, allocations arising from recontracting are required to be attained by trades within the finite coalitions of a partition of the agent set A into finite coalitions. Partitions of agents into finite coalitions involve measurements both in absolute magnitudes and in proportional magnitudes. Partitions which reconcile these measurements are called measurement-consistent. We first define measurement-consistent partitions and then motivate these conditions via an example. A partition p of A is measurement-consistent iff for any positive integer k A;=U sEp,,s, =k S is a measurable subset of A, and each A: (k= 1,2, ...) has a partition (Ag,}:=,, where each A$, is measurable, with the following property: there are measure-preserving isomorphisms’ @kpl,$kpZ,.... ll/& from A;, to A;,, .... Aik, respectively, such that ($a, (a), .... ~)$~(a)} Ep for all aEA$1.6 (2-l) Note that (2.1) implies that for any SEP with ISI = k, we have S= ($gl(a), ....$&(a)> for some aE Ail. Therefore, for each integer k, Ai consists of all the members of k-agent coalitions and Aft consists of the tth members of these coalitions. The requirements that all the sets A:, have equal measure and that the isomorphisms be measure-preserving then capture the idea that coalitions of size k should have as “many” (i.e. the same measure) first members as second members, as many second members as third members, etc. Let II denote the set of measurement-consistent partitions. To exemplify the need for measurement-consistent partitions, we consider an economy where the set of agents is [0, 3) with Lebesgue measure. The agents indexed by real numbers in [0, 1) each own a right-hand a function rl, from a set B in a to a set C in a is a measure-preserving B to C iff (i) I/I is a measure-theoretic isomorphism, i.e., 3/ is 1 to 1, onto, and measurable in both directions, and (ii) p(T) = p(+( T)) for all Tc A with TE d. 6 In Kaneko and Wooders [ 11, Lemma A.11 a simple example of the construction of a measurement-consistent partition is provided. The example considers the case of k = 2 but the extension of the general case is clear. 5 Recall
that
isomorphismfrom
CONTINUUM
ECONOMICS
WITH
119
EXTERNALITIES
glove (RHG) and those in [l, 3), a left-hand glove (LPI ( (a, I+ 2~): a E [O, 1)) is a partition of CO, 3) which pairs every agent owning a LHG with an agent owning a RHG. However, this partition violates the definition of a measurement-consistent partition since it matches, in a one-to-one way, one-third of the agents to two-thirds of the agents. Were such partitions allowed we could not consider the idea of relative scarcities of commodities reflected by the measure in economic models. Hn this economy, partition p = ( (a, 1 + u> : a {(u]: UE [Z, 3)), where th e agents in [0, 1) are arranged in agents in [ 1,2) and the others in [2, 3) remain single, is measureme~tconsistent.’ 2.2 The Economy and Allocations Let Z, be the set of nonnegative integers and let D = .Z: x co~su~~t~o~ set were I is a finite index set for indivisible commodities an D is a finite index set for divisible commodities with ID], the eardinahty o greater than zero. Let M= Iu D. We use the symbols for the orderings of elements of Q: l,=(l,
r--lMl -l 1, .~.)I),
and define 8, and 1, analogously. Finally, we will denote the space of measurable functions from the agent set A to $2 by k(A, Sz).” We model widespread externalities by having preferences depend on the allocation space L(A, 52) in addition to the commodity space a. Thus a preference relation > is defined to be a subset of (92 x k(A, (Q x L(A, Q)), and we typically write
CX? fl> Lx’,S’I to denote the preference between two members of the extended ~o~s~rn~tion set !S x L(A, Q). We assume )
if f(a) = g(a) = h(u) a.e. in A, then
[x,fj
> [v, 81 Q
This means that a preference relation is not affected by a change in t allocation on a null set of agents. (No other assumptions are required on the widespread externalities.) ’ The reader may notice from this example that a continuum version of an assignment game of Shapley and Shubik [17] is well formulated in our framework. For more detailed discussions of this, see Kaneko and Wooders [lo, 111. * Note that this is not to be taken as the space of equivalence classes of functions which are equal almost everywhere.
120
HAMMOND,
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WOODERS
For each fixed f E L(A, Q) and preference relation >, we define the conditiona2 preference relative tof, >(f ), of > as the relation on 52 satisfying
We impose mathematical structure on preferences via conditions on conditional preferences. The space of possible conditional preference relations is taken to be PC = { >‘: >” is an irreflexive, transitive, and open binary relation on D x Q>. In particular; continuity and transitivity of conditional preferences is assumed, but not of > itself. The space PC is then given the topology induced by closed convergence on the class {Sz x Q\>‘: >” E PC} of closed subsets of ~2 x L2 as in Hildenbrand [8, pp. 96-981. The space of preference relations P is defined by P = { > : > satisfies (A.0) and >(f)
E PC for all
f E L(A,
a) >.
An economy E is a mapping E(a) = (>,, e(a)) from A to P x Szsuch that for any f E L(A, Q) the mapping Ef (a): = (>, (f ), e(a)) is measurable-i.e., for any Bore1 set T of PC x 52, we have ET’(T) E&. The economy E assigns to each agent a in A a preference relation >, and an initial endowment e(a). For each f in L(A, Q), the mapping from agents to their conditional preferences and their initial endowments is measurable. We assume that JA e is finite and strictly positive. We will now define feasible allocations for the economy E. A functionf in L(A, &I) is called an allocation with respect to a (measurement-consistent) partition p (p E IT) of A into finite subsets iff 1 f(a)<
C e(a)
for all
Sep.
(2.2)
An allocation f with respect to p can be attained by trading commodities only within coalitions in p. Define the sets F(p) (p~17), F, and F* by F(p) = (f~ W, F=
u
Qsz):f is an allocation with respect top};
(2.3) (2.4)
F(P);
Pen
F* = (f~ L(A, 52): for some sequence (f”} (f”}
converges in measure to f }.
in F, (2.5)
CONTINUUM
ECONOMICS
WITH
EXTERNALITIES
121
Note that I?(p) c PC F* for all p E II and F* is the closure of F wit respect to convergence in measure. Also, since the endowment e satisfies (2.2), F(p) is not empty. Recall that “(f’) converges in measure to f" means that for any E>O, ~((uE A: d(f”(a), f(a))>&)) 40 as n -+ co.’ In interpretation we consider an element of F* as approximately feasible in the sense that it is in the closure of F and not necessarily in F itself.” The end of this subsection includes an example il~~strati~~ that I?’ may not be closed. We say that a coalition S in can improve upon a function f in L(A, a) if? there is a vector (x~),~~ such that XaEQ
for all
a E S;
1 x0 < 1 e(a); RES UES for all xa >-.(f)f(a)
(2.6) (2.7)
a E S.
(2.8)
course, if the members of S change their part of the allocation f to 1aes then the resulting allocation agrees with j” only on ta\S. However, from the assumption (A.0) that preferences are unaffeeted by a change m the allocation on a null set, the change in f on the finite set S can be ignored in condition (2.8). The f-core of the economy E is defined to be Cj.= {f E F*: there is a fullr’ set A” c A such that no finite coalition in J can improve upon f >. An allocation f in the f-core Cf is stable, roughly speaking, in the sense that no finite coalition can improve uponf, It is also approximately feas in the sense that f E I;*; i.e., f can be approximated by a sequence of exa feasible allocations. The exact definition takes a full subset 2 so that no finite coalition in A” can improve uponf. This intermediate step of a full set A” is made for simplicity of the statements of our results. E we could adopt a definition of the f-core requiring that no finite coalition could improve upon an allocation in thef-core. However, this 6ornp~i~at~s the statements of our results without adding any substance to the meaning. The following example illustrates that F may not be closed with respect to convergence in measure so it may be the case that Ff F*. 9 If {fnj converges in measure to f; then {f”) has a subsequence which converges pointwise to f a.e. and, conversely, if {PI converges pointwise to S a.e., then {f”) itself converges in measure to J lo In Kaneko and Wooders [ 121 we discuss the appropriateness of F* as the idealized feasible set for sequences of finite economies. ” A full set of agents is one whose complement is null.
122
HAMMOND,
KANEKO,
EXAMPLE 2.2.1: F is not closed. Rc , ,u is Lesbesgue measure, and
e(a) =
i
AND
WOODERS
Suppose that A = [0, l), that D =
(4 0)
(O
(0, 1)
(a
1)
for some irrational number a. Consider the allocation f(a) = (a, 1 -a) (all SEA) which satisfies jA (f-e) = 0. Each finite coalition consisting of m agents with endowment (1,O) and of n agents with endowment (0, 1) has an aggregate endowment equal to (m, n). To achieve the allocation (a, 1 -a) for each of the m + n agents, one needs (m + n)(a, 1 - IX)< (m, n). This implies that (m+n)a
This subsection presents conditions on E under which the f-core and the set of Walrasian allocations W coincide. We first define the Walrasian allocations and then state the required additional conditions. A function f in L(A, Q) is called a Walrasian allocation iff for some PER?,
P . f(a) G P. e(a)
a.e. in A, x X-, (f)f(a)
forall
(2.10)
a.e. in A; x~Qwithp.x
(2.11)
Let W denote the set of Walrasian allocations. The first step in showing equivalence of the f-core and the Walrasian allocations is to demonstrate that f E F* iff f satisfies the mean excess demand condition (2.12). This is proved12 in Kaneko and Wooders [ll, Lemma 3.11. Therefore we have (2.13)
F*=(~EL(A,~):SA~~~~~~. In particular, I2 Under goods may
this implies that WC F*.
slightly less restrictive be zero.
assumptions
on 51, in particular,
the number
of divisible
CONTINUUM ECONOMICS WITH EXTERNALITIES
123
Additional conditions are required on the conditional preferences >,(J’) in the economy E: For all a E A and for any f~ i&4, a), (A.l) >,(f) is strictly monotone on RG, (the divisible goods); (A.2) For all (x,, x,) E Q = 2: x RD,, there is a yD E RD, such that lo,, rD> kAf)(xI, xd; and (A.3) e(a) >,(f)(x(, 0,) for all x,fZ!+. Assumption (A.l) is standard. Assumption (AL?) is simply that for any commodity bundle (x1, xD) there is a commodity bundle (O,, yD), with divisible goods only, such that the second bundle is preferred-~‘e~oug~ of the divisible goods is better than the given bundle. Our third ass~rn~t~o~ (A.3) is that the initial endowment is preferred to any commodity bundle with only indivisibIe goods. 1. Under assumptions (AO), (Al), (.4.2), and ( Cl. coincides with the set of Walrasian allocations W. THEOREM
Next we will consider the case of no widespread exter~aiitjes; for any aEA,
(A.@)
>,(f)=>,(g)
for allS,gEL(A,Q)
In this case, we write simply 2,. The following standard de~~~t~o~ of t A- (Aumann) core now applies. The A-core of ihe economy is the set of allocations C,={~EL(A,Q):~
f
eandnosubsetofAof A
positive measure can improve uponf),
(2.9)
where a subset S of positive measure is said to be able to kn~~ove UJW.IJf iff for some g E L(S, Q), we have IS g < IS e and da) >, f(a) 2. w= CA.
THEOREM
c,=
for all a in S.
Under assumptions (A.O’), (A.l),
A.2), and (
Theorem 2 demonstrates the equivalence of tbe f-core, the W allocations, and the A-core without any widespread externahties.13 other hand, Theorem 1 demonstrates the equivalence only of the f-core an I3 For an exchange economy with indivisible goods but without externalities,. Mas-Colell [14] gave a counterexample to the existence of equilibrium but with a nonempty Aumann core, implying the nonequivalence. Our assumption (A.31 excludes this example (see also Remark 2.4.11, and, together with (A.l) and (A.2), ensures the equivalence.
124
HAMMOND,
KANEKO,
AND
WOODERS
the Walrasian allocations with widespread externalities. The problem with widespread externalities is not the nonequivalence of the f-core and the A-core but the definition of the A-core, more precisely, of “can improve upon” with coalitions of positive measure. The problem in the definition is maintaining feasibility for the complementary agent set when a coalition of positive measure changes its own part of an allocation. To see the difficulty in defining “can improve upon” with widespread externalities, suppose that a coalition S of positive measure changes that part of an allocation under its control. Then the feasibility of the allocation received by the complementary agent set is violated almost necessarily. Therefore we need to consider the response of the complementary agent set to the change brought about by S. We cannot simply take their part of the allocation as fixed. In the case of widespread externalities, the effects of the behavior of S are not limited to the change in its own part of the allocation. Through the external effects, the coalition S incurs further changes in utilities through the violation in feasibility and resultant change in behavior of the complementary agent set. With no widespread externalities, S is unaffected by any outcome for the complementary agent set. The definition of the f-core is free from these problems of effects of the complementary agent set. Essentially, since a finite coalition S is of measure zero a change in its behavior will not affect the feasibility of the allocation for the complementary agent set. For a more precise argument, recall that an f-core allocation f is the limit of a sequence of exactly feasible allocations, say if”}, with associated measurement-consistent partitions (p”}. In other words, our feasibility concept is approximate. To discuss precisely the question of feasibility, we need to return to the sequences {f”} and {p”>. In any exactly feasible allocationf” E F(p”), if the finite coalition S forms and changes its part of the allocation then feasibility is violated for the coalitions in p” intersecting S. However, the number of these coalitions is finite and each contains only a finite number of agents, so feasibility is violated for only a finite number of players in total. Since these players constitute a set of measure zero, by assumption (A.O), the violation in feasibility and the resultant responses can be ignored.14 To define the A-core from the above argument we must make some hypotheses about the behavior of the complementary agent set in response to a change in the behavior of a coalition of positive measure. So far the two approaches most frequently used to define a core are the minimax criterion (more precisely, the a-core, p-core due to Aumann [3]) and the strong equilibrium (also due to Aumann [ 1)). For the reasons discussed above concerning the feasibility of a new allocation made by a coalition I4 For another, to the continuum,
more detailed explanation see Kaneko and Wooders
in the context [12].
of convergence
of finite
economies
CONTINUUM
ECONOMICS
WITH
825
EXTERNALITIES
and the complementary coalition, the strong equilibrium concept carrot be naturally applied. As is well known, the minimax criterion is not particularly persuasive except for two-person zero sum games. For completeness, the minimax criterion is defined below for the case of preferences monotonically increasing on the externalities, i.e., for external economies. The definition according to the minimax criterion of “can improve upon” would be to assume that the complementary coalition disposes of all its private goods. An allocationfE F* is in the A-core of I$ denoted by C,, if there is no subset S of A with positive measure and no allocation
g E L(A,
l2) where ss
gd
da)
ss
= 0
e a.e. in
A\S
an
Ma), gl>, U(a), Sl
for all
a E S.
We observe that an allocation in the A-core is Pareto-ef~cie~t whereas those in thef-core (and Walrasian allocations) are not necessarily so. Thus, the A-core and the Walrasian allocations do not coincide except in very special situations including, of course, the no-widespread-externalities case. A counterexample to equivalence is given in Kaneko and Wooders [I Even under our simple monotonicity assumptions on the externalities, minimax criterion does not lead to a natural definition of the A-core in the sense that we cannot justify the resulting assumption that a complementary coalition disposes of all its private goods. With no widespread externalities, our resuhs are suggestive of t due to Schmeidler [ 161 and Mas-Cole11 [ 141. Schmeidler considers a tinuum economy where only coalitions of small positive measure are permitted, while Mas-Cole11 uses the framework of a finite ecQ~orny where only coalitions bounded in size (membership) are pe itted. Schmeidler obtains Aumann coreequilibrium equivalence, while as-Cole11 obtams approximate equivalence. These papers are similar to ours in that coreequilibrium equivalence is obtained when only “‘smah” coalitions are allowed. However, neither author restricts allocations arising from recontracting to be achievable by cooperation only within small coalitions. Also neither author considers an actual “limit model.” Although does not impose the feasibility for small coalitions constraint, quite close to a large finite analogue of ours.” I5 For a more detailed discussion of the finite analogue and the convergence of finite economies
with
small
coalitions
to continuum
economies
with
finite
coalitions
in an exchange
economy with widespread externalities, see Kaneko and Wooders 1121, where convergence of solutions
and game
structure
is shown.
126
HAMMOND,
KANEKO,
AND
WOODERS
2.4 Remarks Remark 2.4.1. Our assumptions are similar to those of Khan and Yamazaki [9] and Yamazaki [lS]. (In fact, if the cardinality IDI of D, the set of divisible goods, is equal to one, our assumptions imply those of Khan and Yamazaki. If the cardinality of D is possibly greater than one, Yamazaki [lS] uses a stronger assumption than our (A.2) and also uses other assumptions for the existence of Walrasian equilibrium.) Khan and Yamazaki show the equivalence of the A-core to the set of “weakly competitive allocations,” and the non-emptiness of the A-core; they employ an example due to Mas-Cole11 [13] to illustrate that the set of Walrasian allocations may be empty. Therefore, with their assumptions the A-core and the set of Walrasian equilibrium allocations might not coincide. Our assumption (A.3) rules out Mas-Colell’s example. Without widespread externalities, in fact, we can prove, under assumptions (A.O), (A.l), (A-2), and (A.3), that
c,=
W=C,#@
by modifying our equivalence in Theorem 2 and using Khan and Yamazaki’s [9] existence of weakly competitive allocations. The modifications will be indicated in Section 3, where the proofs of Theorems 1 and 2 are provided. Remark 2.4.2. With widespread externalities, the existence of Walrasian equilibrium and the f-core can be obtained by imposing only a slight restriction on the externalities. Specifically, we assume that there is a given finite family of subsets of positive measure of A, say T1, .... T,, such that the externalities depend on an allocation only via the value of its integral over each of the subsets.i6 That is, we assume
@.O*) C~,fl~,~y~~l~C~~~~,f,...,~,~fl~,C~~jr,~~~~~~~~,~l for all Cx,fl, [Iv, 81EQ x W& a). Assume also that preferences are open in (a x Ry’“) x (Q x Ry’“). Then essentially the same proof (as the one herein) of the existence of equilibrium can be applied by using Liapunov’s theorem to convexify over the externalities. Remark 2.4.3. Although we have shown that without widespread nalities the Aumann core coincides with the JLcore, this does not that these two concepts are equivalent nor that Aumann’s theorem equivalent in interpretation to ours. The Aumann core permits only tions containing positive proportions of the total agent set to improve I6 This
enables
us to work
in a finite
dimensional
“commodity”
space.
extermean [2] is coaliupon
CONTINUUM
ECONOMICS
WITH
EXTERNALITIES
127
allocations; thus individual agents and finite coalitions are unable to have any affect on the outcome of cooperation. Bolh the surname core an competitive equilibrium build in completely negligible agents. Hn contrast, in our approach individual agents are nonnegiigible relative to themselves and to any finite group. We obtain, as a conclusion, that individual agents ices as given Even and finite coalitions can do no better than taking though finite coalitions are negligible relative to t e~Qnomy~ we have alrasian equilibrium equivalence. emark 2.4.4. A different type of consideration rather than those discussed in the introduction was the original source of one author’s interest in finite coahtions. This concerns the power of finite ~oa~~tio~s to ipulate resource allocation mechanisms in ~o~~in~~rn economies. en one considers manipulation the size of permissible ~~a~~tio~s-~~~~e~ or of positive measure-is critical. See Hammon mond [6], where another core concept, which ombines” finite coalitions and coalitions of positive measure, is introduce
In this section, we prove C,c W in the case of no widespread externalities. The proof that C, c W is a simple rnodi~~at~o~ of the proof of Cfc W. The proofs that C;=, W and that C, 3 W are straightforward and standard, so we omit them. Thus we will have C/= W= C, in the case of no widespread externalities. In the case of widespread externalities, we can obtain C,= W by applyin the eq~iva~e~~e result C,= W in the case of no widespread externalities as follows. Let g be a function in L(A, .G) and let E be an economy with widespread externalities. Then we have an economy ER: a -+ b>,(g), e(a)), without widespread externalities, induced from E by fixing g in each eference relation. Then the above equivalence result without wid ternahties states that C; = Wg, where Cf” and Wg are respecti S-core and the set of Walrasian allocations of the economy ER. The conditions for g to belong to C/ and W of the economy E can be written as .!sq
if and only if
g E Cr;
gcwg
if and only if
g E kl/.
Since CT = W” for all g, these conditions are iden In the following analysis of an economy wi 642/49/l-9
128
HAMMOND,
nalities, individual (l-lx L(A, a))‘.
KANEKO,
AND
WOODERS
preferences are regarded as a subset of 52 x Q instead of
Proof that C*c W Let f be an allocation aeA by
in the f-core. Define the set $(a) for each agent
(which is equivalent to Hildenbrand’s definition of $(a) in [8, Theorem 1, p. 133)). Then the following lemma holds. 1. There is no w E 1 Ic/ such that w 4 0.
LEMMA
Since j @ is convex by Liapunov’s theorem, from this lemma it follows that the set s $ can be separated from the strictly negative orthant of RM by a hyperplane. That is, there is a price vector p in R”, p > 0, such that forall
0Gp.w
wE
s
*
(3.1)
Proof ,of Lemma 1. Suppose there exists a vector w ES + such that w + 0. Then there exists a measurable function t: A + Z’x R” such that t(a) E +(a) a.e. in A and JA t = w (< 0. Define S= (aEA: t(a)+e(a)>,f(a)). Then p(S)>0 and x=J,t@O and the following claim holds. Claim. There is a finite partition function i: S-+ Z’x RD such that p(S,)=p(S,)= i(a) = t‘(d) i(a) = 0 tD ta)
s s
%- tD ca)
t(a) < 0.
(S,, S1, .... S,) of S and a simple
... =p(S,)>O; if
a, a’ 62S,(j=
for all and
(3.2) 1, .... m);
(3.3)
afzS,; iI
(3.4) = t,(a)
forall
aE ij Si; j=l (3.5)
Proof of Claim. We begin by partitioning a set consisting of “most” agents into subsets so that if a and a’ are in the same subset, then t(a) is approximately equal to t(d). For any positive integer n, define
CONTINUUM
ECONOMICS
Z,={-n,
-n+l,..., r--- 14 1 z’,=z,x ... xz,;
WITH
-l,O,l,...,
12
EXTERNALITI[ES
n-s>
1 1; Kn=i --If,-n+; )...)-&o,$ ....n--2n-i9’-~ I r-=- IDI... xK,
Kf=K,x
Define a simple function t” = (t:, t;): s -+ .P x ~~ ~JJ [f(a)= k,+kl,
if
L
t>(a) =
if -i,
tl(a) o i I
(3.6)
-nl,
fD(U)E I-I
(3.7)
dfD
otherwise. Then, for each a E A, there is an fi such -El M f t(a) 6 fil M. Therefore t g for all n 2 ii and converges to t(a j. This converges to js t G 0. Therefore there is a integer n, such that
For +,, k,) E Zfi, x K$ denote the set (ac A: tl(a) =zI an lD(a) < k, + (1/2”O) ID) by T(z[, k,). Then it follows from (3.6), (3.7), and (3.8) that (3.9) From
each T(z,, k,)
((z,, kD)EZL,,x
KE),
we can choose
a subset
S(z,, k,) so that P( S(z,, k,)) is a rational number;
(3.10)
and (3.11)
130
HAMMOND,
KANEKO,
AND
WOODERS
Since each p(S(zl, kn)) is a rational number (possibly zero), we can find a (S, , .... S,) of the finite partition subpartition (S(z,, k,): (z,, kD)e ZL,, x Kg and ,u(S(Z,, k,)) > 0} such that ,u(S,) = n.. = ,u(S,) > 0. Define S, and i by s,=s-
(j sj; j=l
P”(a)
if
a E fi S, j=l
0
otherwise.
i(a) =
It immediately follows that (S,, S,, .... S,) and i satisfy conditions (3.3), and (3.4). From (3.11) we have
(3.2),
This completes the proof of the claim. Now select one agent a,E Sj for each j= 1, .... m, and write c = (q, ...) a,}.17 Since iD (aj) $ tD (aj) and i,(aj) = t,(aj) for all j= 1, .... m by (3.4), we have, by Assumption (A.1 ),
ita,) + e(aj>>aj t(aj) + e(aj) The feasibility of the allocation (3.2) and (3.5); indeed
.gc /
(t‘(aj)
+
e(aj))
=
for all j = 1, .... m.
>-,f(aj)
(i(aj) + e(aj)),,
--&J
j!l
=&jp+
tiaj)
c follows from conditions
P(sj)
c
e(Uj)
e(aj) cl,E c
+
C
<
CZjCC
C e(aj)* LljEC
Therefore the finite coalition C can improve upon J For any full subset 2 of A we can also find a finite coalition C in A” so that C consists of only one agent in Sj for each j= 1, .... m. Then, C can improve upon J: This is a contradiction to the supposition that fis in C,-. This completes the proof of Lemma 1. 1 I7 To modify
the proof
of this theorem
to obtain
C, c W, replace
C by U,“= 1 S,.
CONTINUUM
ECONOMICS
WITH
EXTERNALITIES
131
Recall that from Lemma 1 it follows that there is a price vector p E RM (p > 0) satisfying (3.1). We will now show that p is an equilibrium price vector. It follows from (3.1) and Hildenbrand [g, p. 63: Proposition 61 that
It also follows from (3.1) that 0 < JA infp I I/I* Since $(a) contains 0, we have inf p . $(a) < 0. Hence we have jA infp . $ = 6bl which implies that infp . $(a) = 0 a.e. in A. Then this implies that a.e. in A,
(3.12) or equivalently,
LEMMA
2.
p.e(a)=p.f(
a ) a.e. in A (budget constraints are satisfied).
Proof. For all a E A, by Assumption (A.1 ) (strict monotonicity of preferences for divisible goods) we can choose a sequence ( y”} such t&t { y”> converges tof(a) and y” >cl f(a) for all n. Then we have, from (3.82) p .e(a)
3.
~~$0,.
Proof. First we prove that pD > 0,. Suppose pD = 0,. Since IA e 90 and p > 0 there is a subset 5’ of A such that p ‘e(a) > 0 for all a E S and p(S) >O. Choose an agent a from S for whom (3.12) holds
132
HAMMOND,
KANEKO,
AND
WOODERS
and pa f(a) =p. e(a) > 0. By Assumption (A.2), there is a vector y=(O,,y,)~fi such that y>af(a). Since pD=OD, we have O=p.y< p .f(a) =p .e(a), which contradicts (3.12). This shows that p. > 0,. Now let us show that p. $0,. Since fAe>lS,fby (2.13) andp.J,e= p . JA f by Lemma 2, we have, for all k E I u D, the usual “rule of free goods,” (3.13)
Pk>O=>
Since p. > 0, and JA e, B O,, it follows from (3.13) that p. -JAfD = pD .jA eD > 0. Therefore there is a subset ,%of A with positive measure such that PO. eD (a) =pD .fD (a) > 0 and (3.12) holds for all a E S. Consider an arbitrary agent a in S. We shall prove that
x>,f(a)
and
xI=fAa)*p.x>p-f(a).
(3.14)
This implies p. 9 0, ; indeed, if pd= 0 for some d.s D, then x, given by xd=fd(a)+l and x,=fk(a) for all kfd, violates (3.14) because of Assumption (A.l). Therefore let us now prove (3.14). Suppose on the contrary that x .>, f(a), x,= f](a), and p . x Qp .f(a) for some x. Because >a is continuous, there exists 2 E (0,l) near 0 so that 2(x,, 0) + (1 - n)x >-,f(a) and p.[l(x,,O)+(l-I)x]=~~~.x,+(l-~)p.x~~p.f~(a)+ (1 - 1)~ .f(a) =p -f(a) - lpD .fD(a) cp .f(a) where the last inequality follows because a E S, as defined above. This is a contradiction to (3.12). So PD~OD.
I
LEMMA 4. A.e. in A, p . x d p . e(a) =s-x >a f(a) the budget sets).
(utility maximization
on
Proof. It follows from Assumption (A.3) that e,(a) > 0, for all a E A. This together with Lemma 3 implies that p . e(a) > 0 for all a E A. Consider an arbitrary agent a for whom (3.12) (p . x
0,, and x >,f(a) for some x. By continuity of >-, and Lemma 3 we can find an x’ such that p . x’
133
CONTINUUMECONOMICSWITHEXTE~NALHTIES
librium we do not use this argument.) >, we can find an x’ such that xb
xi= e,(a)
Since e,(a) > and
D, by continuity
of
x’ >, f(a).
3, we have p . x’
which contradicts
ACKNOWLEDGMENTS Research support from the National Science Foundation, the Office of Naval Research Contract NOOO14-77-0518, and the Social Sciences and Humanities Research Council and the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. We are also indebted to the Cowles Foundation, Yale University, and IMSSS, Stanford University, for their hospitality. We express our thanks to G. Debreu, B. Grodal, and M. Shubik, to an associate editor of this journal, and to the members of the audiences at severai seminars where this work was presented for helpful comments.
REFERENCES 1. R. 3. AUMANN, Acceptable points in general cooperative n-person games, in “‘Contributions to the Theory of Games Vol. IV,” (A. W. Tucker and R. D. Lute, Eds.), Annals of Mathematics Studies, Vol. 40, pp. 207-324. Princeton University Press, Princeton, NJ, 1959. 2. R. J. AUMAWN, Markets with a continuum of traders, Econometrica 32 (1964), 39-50. 3. R. J. AUMANN, Cooperative games without side payments, in “Essays in Mathematical Economics in Honor of Oskar Morgenstern” [M. Shubik, Ed.). Princeton University Press, Princeton, NJ 1967. 4. R. J. AUMANN, Values of markets with a continuum of traders, Econometrica 43 {1975), 61 l-646. 5. ,R. J. AUMANN AND L. S. SHAPLEY, “Values of Nonatomic Games.” Princeton University Press, Princeton, NJ, 1974. 6. P. J. HAMMOND,“The power of small coalitions in large economies.” Economics Technical Report No. 501, Stanford University Institute for Mathematical Studies in the Sodal Sciences, 1987. 1. P. j. HAMMOND, “Markets as constraints: Multilateral incentive compatibiiity in continuum economies,” Rev. Econ. Stud, 54 (1987), 399412. 8. W. MILDENBRAND, ‘Core and equilibria of a large economy.” Princeton University Press, Princeton, NJ, 1974. 9. M. A. KHAN AND A. YAMAZAKI, On the cores of economies with indivisible commodities and a continuum of traders, J. Econ. Theory 24 (1981), 218-225. 10. M. KANEKO AND M. H. WEEDERS, “Nonemptiness of the core of a game with a continuum of players and finite coalitions,” University of Tsukuba Institute of SocioEconomic Planning Working Paper, 1985. Il. M. KANEKO AND M. H. WOODERS,The core of a game with 2 continuum of piayers and finite coalitions: the model and some results, Math. Sot. Sci. PZ (1986). 105-137.
134
HAMMOND,
KANJZKO,
AND
WOODERS
12. M. KANEKO AND M. H. WOODERS,The core of a continuum economy with widespread externalities. From finite to continuum economies, J. Econ. Theory, on press. 13. A. MAS-COLELL, Indivisible commodities and general equilibrium theory, J. Econ. Theory 16 (1977),
443456.
14. A. MAS-COLELL, A refinement of the core equivalence theorem, Econ. Lett. 3 (1979), 307-310.
15. K. R. PARTHASARATHY,“Probability Measures on Metric Spaces.” Academic Press, New York, 1967. 16. D. SCHIVEIDLER,A remark on the core of an atomless economy, Econometrica 40 (1972), 579-580. 17. L. S, SHAPLEY 18.
AND M. SHUBIK, The assignment game I: The core, Int. J. Game Theory 1 (1972), 111-130. A. YAMAZAKI, Diversified consumption characteristics and conditionally dispersed endowment distribution: regularizing effect and existence of equilibria, Econometrica 49 (1981),
639-654.