Core and competitive equilibria of a coalitional exchange economy with infinite time horizon

Journal of Mathematical Economics 49 (2013) 234–244

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Core and competitive equilibria of a coalitional exchange economy with infinite time horizon Takashi Suzuki Department of Economics, Meiji-Gakuin University, 1-2-37 Shiroganedai, Minato-ku, Tokyo 108, Japan

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Article history: Received 15 October 2012 Received in revised form 22 February 2013 Accepted 28 February 2013 Available online 21 March 2013

abstract The core and competitive equilibria of a large exchange economy on the commodity space ℓ∞ will be discussed. We define the economy as a measure on the space of consumers’ characteristics following Hart and Kohlberg (1974), and prove the existence of competitive equilibria and their equivalence with the core without assuming the convexity of preferences. © 2013 Elsevier B.V. All rights reserved.

Keywords: Large exchange economy Coalitional form Competitive equilibrium Infinite dimensional commodity space

1. Introduction An exchange economy with an atomless measure space

(A, A, ν) of consumers was first introduced by Aumann (1964) on a finite dimensional commodity space. He described the economy by a map %: A → 2X ×X which assigns each consumer a ∈ A its preference and a map ω : A → X which assigns a its initial endowment, where X is the non-negative orthant of the commodity space which is identified as the consumption set, and showed that the set of core allocations coincides with that of allocations which are supported as competitive equilibria (core equivalence theorem). Hildenbrand (1974) established that the economy is defined by a measurable map E : A → P × Ω , where P is the set of preferences and Ω is the set of endowment vectors. Every element a ∈ A is interpreted as a ‘‘name’’ of a consumer, and each value of the map E (a) = (%a , ωa ) is the characteristics of the consumer a (the individual form of the economy). Aumann (1966) made the remarkable observation that for demonstrating the existence of the competitive equilibrium of such an economy, one does not have to assume the convexity on the preferences. This is a mathematical consequence of the Liapunoff theorem which asserts that the range of a finite dimensional vector measure is convex. (See Diestel and Uhl, 1977, Chapter IX for details.) The economy which has the commodity space ℓ∞ = {ξ = t (ξ ) | supt ≥1 |ξ t | < +∞}, the space of the sequences with a bounded supremum norm was introduced by Bewley (1970). As we

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will see in the next the space of all summable sequences, section, t ℓ1 = {p = (pt ) | ∞ t =1 |p | < +∞}, is a natural candidate of the price space. The value of a commodity ξ = (ξ t ) ∈ ℓ∞ evaluated by a price vector p =  (pt ) ∈ ℓ1 is then given by the natural ∞ t t ‘‘inner product’’ pξ = t =1 p ξ . Bewley (1970) established the existence of competitive equilibria for an economy with finite number of consumers on this commodity space. Thereafter, this commodity space has been applied to theories of intertemporal resource allocations and capital accumulation by Bewley (1982), Yano (1984), Suzuki (1996) among others; see also Suzuki (2009). Several authors have tried to unify the above results of Aumann and Bewley. For example, Bewley (1991), Noguchi (1997b) and Suzuki (2012) proved the equilibrium existence theorems for the economies with the measure space of consumers on the commodity space ℓ∞ . Khan and Yannelis (1991) and Noguchi (1997a) proved the existence of a competitive equilibrium for the economies with the measure space of agents in which the commodity space is a separable Banach space whose positive orthant has a norm interior point.1 Bewley, Khan–Yannelis and Suzuki worked with exchange economies, and Noguchi (1997a,b) proved his theorems for an economy with continua of consumers and producers. On the other hand, Bewley (1973) and Mertens (1991) proved the Aumann-type core equivalence theorems for exchange

1 Since the space ℓ∞ is not separable, these results are not considered as generalizations of Bewley (1970). However, Khan–Yannelis pointed out that their result includes the space L∞ (Σ ), the space of essentially bounded measurable function on a finite measure space Σ .

T. Suzuki / Journal of Mathematical Economics 49 (2013) 234–244

economies with the measure space of consumers and the commodity space ℓ∞ , Rustichini and Yannelis (1991a) and Podczeck (2004) proved such results for more general Banach spaces. These studies on the core equivalence, however, have been accumulated quite independently with those of the existence of equilibria. For instance, they did not assume the consumption sets to be bounded, nor the convexity of preferences. Hence one may suspect that their nice looking results could be vacuous. This means that the present state of our knowledge is sharply in contrast with that of Aumann in which the existence and the core equivalence were demonstrated in a single framework. In this situation, the work of Mas-Colell (1975) is an exception. He proved the both theorems together for an exchange economy whose commodity space is ca(K ) which is the space of countably additive set functions on a compact metric space K (the model of differentiated commodities). The purpose of this paper is to propose for an economy with the commodity space ℓ∞ a reasonable economic environment in which the proofs of the existence and the core equivalence theorems are simultaneously carried out, as Mas-Colell did for the economy on the space ca(K ). We will focus on the exchange economy, hence there exist no producers in it. The mathematical difficulty for demonstrating the existence of equilibrium for the individual form has been well known. Usually the proof is carried out by approximating the large, infinite dimensional economy by large, finite dimensional sub-economies, a technique which we will also utilize in this paper. In the course of the approximation one is expected to apply the Fatou’s lemma of several dimension. On the finite dimensional spaces, the limit set of the integrals of allocations arising from finite dimensional subeconomies is contained  in the integral of  the limit set. (One denotes this situation like Ls( A fn (a)dν) ⊂ A Ls(fn (a))dν .) The infinite dimensional version of Fatou’s lemma, however, only ensures that the former is contained in the integral of the  closed convex hull of the limit set of the sequences. (That is, Ls( A fn (a)dν) ⊂  coLs(fn (a))dν . For the case of Bochner integral, see Yannelis A (1991). For the Fatou’s lemma for the Gel’fand integral, see Suzuki, 2012, Fact 11.) Since the Liapunoff convexity theorem fails in the infinite dimensional spaces, this means that the convex valuedness of the demand correspondences themselves is needed. Indeed, Bewley, Khan–Yannelis and Suzuki assumed that the preferences are convex. Noguchi assumed that each commodity vector does not belong to the convex hull of its preferred set. These assumptions obviously weaken the impact of the Aumann’s classical result which revealed the ‘‘convexfying effect’’ of large numbers of the economic agents. In this paper, we define our economy as a probability measure µ on the set of agents’ characteristics P × Ω (the coalitional form or the distribution form). Then the competitive equilibrium of this economy is also defined as a probability measure ν (and a price vector p, see Definition 2) on X × P × Ω , where the set X ⊂ ℓ∞ is a consumption set which is assumed to be identical among all consumers. These definitions of the economy and the competitive equilibrium on it were first proposed by Hart and Kohlberg (1974), and applied to the model on the space ca(K ) by Mas-Colell (1975) mentioned above.2 We can interpret (ξ , % , ω) ∈ support (ν) in such a way that ξ ∈ X is the allocation assigned to νP ×Ω percent3 of consumers with the characteristics (%, ω) ∈ P × Ω . A mathematical advantage of the coalitional approach is that the finite dimensional approximation works without any use of Fatou’s lemma or Liapunoff’s theorem. Consequently, it is possible

2 Bewley (1991) also used this approach. 3 ν P ×Ω is the marginal distribution of ν on P × Ω .

235

to prove the existence of equilibria without any convexity-like assumptions on preferences.4 A conceptual advantage of the coalitional form is that the economies are described by smaller information than the individual form. From the economic point of view, what we are really interested in is the performance of the market itself rather than the behavior of each individual. For this it is enough to know the distribution of consumers’ characteristics, and we do not have to know who has which character. In other words, even if the economy is defined by the distribution µ rather than the map E , almost nothing is lost from the point of view of economic theorists and/or policy makers. Indeed when at least one of the numbers of agents or commodities is finite, the individual form is epistemologically powerful in the sense that we know everything of each individual in the economy at equilibria. However, we are now concerned with a ‘‘huge’’ market in which the both of the agents and the commodities are infinite. When the market scale is very large, it will be generally hard to get all information of the market, hence it is usually advisable to see the market from macro-economic point of view. In this case, the coalitional form seems to be more natural and appropriate. The philosophy which emphasizes the distribution more than each individual was already addressed by Hildenbrand (1974). In order to prove the existence of equilibria, we will follow Bewley (1991) and Noguchi (1997b) for the consumption set. Hence in this paper the consumption set X which is identical for all consumers will be a convex and bounded subset of the positive orthant of ℓ∞ , X = {ξ = (ξ t ) ∈ ℓ∞ | 0 ≤ ξ t ≤ β for all t ≥ 1}, for some β > 0. The bounded consumption set is not desirable for the models with a continuum of consumers, but this assumption will make the problem tractable. For the assumption for the initial endowments, however, we will depart from most of those authors. For example, Bewley (1991), Khan and Yannelis (1991), Noguchi (1997a,b), Rustichini and Yannelis (1991b) assumed that almost all consumers have his/her initial endowment in the (norm) interior of the consumption set. Obviously such an assumption is very strong in the economies with a continuum of traders.5 We will adopt the irreducible condition initiated by McKenzie (1959) and applied to the continuum of consumers model by Yamazaki (1981). It was also used in the model with a continuum of consumers and the infinite time horizon by Suzuki (2012). On this setting, we will prove the existence of a competitive equilibrium (Theorem 1). We will also show that the core allocations of this economy will be supported as a competitive equilibrium allocations (Theorem 2). The idea of the proof is the standard application of Hahn–Banach theorem (see Appendix) which is a natural extension of finite dimensional proofs using the hyper plane separation theorem (e.g., Hildenbrand, 1974, p. 133). For details of the proof, we are much indebted to Bewley (1973). A crucial assumption for his proof is

4 A reason that Mas-Colell (1975) used the coalitional form was that he included the indivisible commodities in his model, hence actually he could not assume the convexity. Jones (1983) showed that the indivisibility and boundedness assumptions on the consumption set are not necessary for the equilibrium existence theorem for the economies of the coalitional form with the commodity space ca(K ). Ostroy and Zame (1994) further developed the works of Mas-Colell and Jones, and proved the existence and the core equivalence theorems for an economy of the individual form with the commodity space ca(K ) in which the consumption set is the positive orthant of ca(K ). However, they also had to assume the convexity on the preferences. For existence theorems without the convexity of preferences for the individual form, see Rustichini and Yannelis (1991b) and Podczeck (1997). 5 Precisely, Khan–Yannelis, Noguchi, Rustichini–Yannelis assumed that there exists ζ ∈ X such that ω − ζ belongs to the (norm) interior of X .

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T. Suzuki / Journal of Mathematical Economics 49 (2013) 234–244

the measurability of the agents’ characteristics, in particular that of the endowments which says that the map ω can be approximated by simple functions ωn , n = 1, 2, . . . in the norm topology. See Assumption (SM) and its remark. A similar assumption was also used in Mertens (1991). We define the resource feasibility by Gel’fand Integral as Suzuki (2012, 2013) did. However, we will utilize both the Gel’fand integral and the Bochner integral (see Appendix) in the proof. Since the Bochner integral commutes with the price vector in the space ba, an adequate total endowment ensures that some of the consumers have positive income. See the proof of Section 3. The model and the results will be presented in Section 2. Section 3 will be devoted to the proofs. Basically, the paper is written in a fully self-contained manner. Section 4 will give some remarks on the relationship between the individual and the coalitional forms. All mathematical concepts and results needed in the text will be found in Appendix. 2. The model and the results 2.1. Competitive equilibria of the economy As stated in the Introduction, the commodity space of the economy in this paper is set to be

  ℓ∞ = ξ = (ξ t ) | sup |ξ t | < +∞ , t ≥1

the space of the bounded sequences. It is well known that the space ℓ∞ is a Banach space with respect to the norm ∥ξ ∥ = supt ≥1 |ξ t | (Royden, 1988). For ξ = (ξ t ) ∈ Rℓ or ℓ∞ , ξ ≥ 0 means that ξ t ≥ 0 for all t and ξ > 0 means that ξ ≥ 0 and ξ ̸= 0. ξ ≫ 0 means that ξ t > 0 for all t. Finally for ξ = (ξ t ) ∈ ℓ∞ , we denote by ξ ≫ 0 if and only if there exists an ϵ > 0 such that ξ t ≥ ϵ for all t. It is also well known that the dual space of ℓ∞ is the space of bounded and finitely additive set functions on N which is denoted by ba, ba =

 π : 2N → R| sup |π (E )| < +∞, E ⊂N

 π(E ∪ F ) = π (E ) + π (F ) whenever E ∩ F = ∅ . Then we can show that the space ba is a Banach space with the norm

∥π ∥ = sup

 n 

 |π (Ei )| | Ei ∩ Ej = ∅ for i ̸= j, n ∈ N .

i=1

Since the commodity vectors are represented by sequences, it is more natural to consider the price vectors also as sequences rather than the set functions. Therefore the subspace ca of ba,

 π ∈ ba|π (∪∞ n =1 E n ) =

ca =

∞ 

π (En )

n=1

 whenever Ei ∩ Ej = ∅ (i ̸= j) , which is the space of the bounded and countably additive set functions on N is more appropriate as the price space. Indeed it is easy to see that the space ca is isometrically isomorphic to the space ℓ1 , the space of all summable sequences,

 ℓ = p = (p ) | 1

t

∞ 

 |p | < +∞ , t

t =1

∞

t which is a separable Banach space with the norm ∥p∥ = t =1 | p | . t ∞ Then the value of a commodity ξ = (ξ ) ∈ ℓ evaluated by a t 1 price vector ∞ pt =t (p ) ∈ ℓ is given by the natural ‘‘inner product’’ pξ = t =1 p ξ .

Let β > 0 be a given positive number. We will assume that the consumption set X of each consumer is the set of non-negative vectors whose coordinates are bounded by β , X = {ξ = (ξ t ) ∈ ℓ∞ | 0 ≤ ξ ≤ β 1},

where 1 = (1, 1, . . .).

Of course the β > 0 is intended to be a very large number. From Fact 11, we have τD = σ (ℓ∞ , ℓ1 ) = τ (ℓ∞ , ℓ1 ) on the set X . Since X is compact in τD (hence σ (ℓ∞ , ℓ1 ) and τ (ℓ∞ , ℓ1 )) topology, it is a complete and separable metric space. As usual, a preference % is a complete, transitive and reflexive binary relation on X . We denote (ξ , ζ ) ∈% by ξ % ζ . ξ ≺ ζ means that (ξ , ζ ) ̸∈%. Since X is a locally compact separable metric space, F (X × X ) is a compact metric space with respect to the topology of closed convergence τc by Fact 1, so that it is complete and separable. We denote the metric for the topology τc by δ (the Hausdorff distance). Let P ⊂ F (X × X ) be the collection of allowed preference relations which will be assumed to be compact in the topology τc and satisfy the following assumptions, Assumption (PR). (i) %∈ P is complete, transitive and reflexive, (ii) (monotonicity). For each ξ ∈ X and ζ ∈ X such that ξ < ζ , one has ξ ≺ ζ . Note that preferences are τD (hence σ (ℓ∞ , ℓ1 ) and τ (ℓ∞ , ℓ1 )) continuous, since P ⊂ F (X × X ). Hence nearby commodities are considered to be uniformly (since P is compact) good substitutes. An endowment vector is an element of ℓ∞ . We denote the set of all endowment vectors by Ω and assume that it is of the form

Ω = {ω = (ωt ) ∈ ℓ∞ | 0 ≤ ξ ≤ γ 1}, for some γ > 0. We assume that γ < β . The set Ω is also a compact metric space by the same reason as the space X . Definition 1. An economy is a probability measure on the measurable space (P × Ω , B (P × Ω )). We will denote the economy under consideration by µ. The marginals of µ will be denoted by subscripts, for instance, the marginal on P is µP and so on. We will use the similar notations for a distribution on X × P × Ω , see the next definition. A probability measure ν on X × P × Ω is called an allocation distribution if νP ×Ω = µ. An allocation distribution is called feasible  if X i(ξ )dνX ≤ Ω i(ω)dµΩ , where i is the identity map. A feasi ble allocation distribution is called exactly feasible if X i(ξ )dνX =  i(ξ ) = ξ for all ξ , hereafter we will denote Ω i(ω)dµΩ . Since  i(ξ )dνX = X ξ dνX , and so on. Note that the Gel’fand integrals X   ξ dνX and Ω ωdµΩ exist by virtue of Fact 14. X Definition 2. A pair (p, ν) of a price vector p ∈ ℓ1 with p > 0 and a probability measure ν on X × P × Ω is called a competitive equilibrium of the economy µ if the following conditions hold, (E-1) ν({(ξ , %, ω) ∈ X × P × Ω | pξ ≤ pω and ξ % ζ whenever pζ ≤ pω}) = 1, (E-2) X ξ dνX = Ω ωdµΩ , (E-3) νP ×Ω = µ. The condition (E-1) says that almost all consumers maximize their utilities under their budget constraints. The conditions (E-2) and (E-3) say that the distribution µ is an exactly feasible allocation distribution. The following assumption which means that every commodity is available in the market is standard. Assumption (PE) (Positive Total Endowment).



Ω

ωdµΩ ≫ 0.

Let I = [0, 1] be the unit interval on R and λ be the Lebesgue measure on I. A measurable map E : I → P × Ω such that µ = λ ◦ E −1 is called a representation of the economy µ. Since P × Ω

T. Suzuki / Journal of Mathematical Economics 49 (2013) 234–244

is a compact metric space, the representation of µ exists by Skorokhod’s theorem (Fact 7). Similarly, for every allocation distribution ν , there exists a representation (ξ , E ) : I → X × P × Ω . The map ξ : I → X can be called an allocation. A value of the representation E : I → P × Ω is written as E (a) = (%a , ω(a)). The map %: I → P , a → %a is sometimes called a preference assignment, and the map ω : I → Ω , a → ω(a) an endowment assignment (or an endowment map). An economy µ is called irreducible if it satisfies the next condition. Assumption (IR) (Irreducibility). Let ν be an arbitrary feasible allocation distribution of the economy µ. For every representation (ξ , E ) of ν and for every measurable partition {I1 , I2 } of I, there exist a measurable map φ and  a measurable set I3 ⊂ I1 with positive measure such that I (ω(a) − φ(a))dλ + I ξ (a)dλ ∈

 I3

2

{ζ ∈ X | ξ (a) ≺a ζ }dλ whenever

3

 I1

{ζ ∈ X | ξ (a) ≺a ζ }dλ ̸= ∅.

Remark. Note that the condition (IR) is concerned with the maps %a and ω(a) both together. It was introduced by McKenzie (1959), and applied in a continuum of consumers model by Yamazaki (1981). This condition was also used for models with infinite time horizon by Boyd and McKenzie (1993). See also McKenzie (2002). Since we imposed an upper bound on X , I {ζ ∈ X | ξ (a) ≺a ζ } 1 dλ = ∅ when ξ (a) = β 1 on I1 . Actually in the proof, the set is nonempty by the budget conditions. See the proof of Lemma 5. By Assumption (PE), one obtains that p I ω(a)dλ > 0, where p is an equilibrium price vector. Hence λ({a ∈ I | pω(a) > 0}) > 0. The role of the irreducibility condition (IR) is that it makes λ({a ∈ I | pω(a) > 0}) = 1, in other words, ‘‘if some one has positive income, then every one does’’. In order to demonstrate the existence of competitive equilibria for the exchange economies of the coalitional form, Bewley (1991) and Suzuki (2013) assumed Assumption (IP) (Individually Positive Endowments). µΩ {ω ∈ Ω | ω ≫ 0} = 1, which is very strong. A purpose of this paper is to generalize (IP) to (IR). Indeed, under the monotonicity (PR) (ii), obviously the Assumption (IP) implies the Assumption (IR). The next simple example could give some illustration for the economic environment consisting of the Assumptions (PR), (PE) and (IR). Example. Suppose that β = 2 and γ = 1. Let eo = (1, 0, 1, 0, . . .) and ee = (0, 1, 0, 1, . . .). The endowment distribution of the economy µ∗ is specified by µ ∗Ω = (1/2)δeo + (1/2)δee , where δeo and δee are the Dirac measures. Let E (a) = (%a , ω(a)) be a representation of µ∗, and let Io = ω−1 (eo ) and Ie = ω−1 (ee ). Obviously {Io , Ie } be a measurable partition of I with λ(Io ) = λ(Ie ) = 1/2. The consumers in Io has one unit of the consumption good in odd periods t = 1, 3, . . . , and the consumers in Ie has it in even periods. Then any consumer vector  does not have  its endowment  in the interior of X . Since I ω(a)dλ = I ω(a)dλ + I ω(a)dλ = o e (1/2, 0, 1/2, . . .) + (0, 1/2, 0, . . .) = (1/2, 1/2, . . .), Assumption (PE) holds. We now check whether condition (IR) holds or not. It is obvious that if I ω(a)dλ ≫ 0 in the condition (IR), the condition will be 1 met under the monotonicity (PR) (ii). So set I1 = Io (hence I2 = Ie . The case I1 = Ie is similar) and let ξ (a) be a feasible allocation. Define I3 = {a ∈ I2 | ξ t (a) < 2 for some odd t }. Then λ(I3 ) > 0. For if not, one has I ξ t (a)dλ = 1 for every odd t, hence I ξ t (a)dλ ≥ 2  ξ t (a)dλ = 1 > 1/2 = I ωt (a)dλ, violating the feasibility. Under the monotonicity (PR) (ii), it is easy   to obtain a map φ : I1 → X satisfying I1 (ω(a) − φ(a))dλ + I3 ξ (a)dλ ∈ I3 {ζ ∈ X | ξ (a) ≺a ζ }dλ, hence Assumption (IR) also holds.



I2

237

The first main result of this paper reads Theorem 1. Let µ be an economy which satisfies the Assumptions (PR), (PE) and (IR). Then there exist a price vector p ∈ ℓ1 with p > 0 and a feasible allocation distribution ν such that (p, ν) is a competitive equilibrium for µ. 2.2. Core of the economy We now define the core. Let ν be a feasible allocation distribution and let (ξ , E ) : I → X × P × Ω be its representation. Definition 3. A measurable set C ⊂ I is said to block the allocation distribution ν if there exists a measurable map ζ : I → X such that (C-1) C ζ (a)dλ ≤ C ω(a)dλ, (C-2) ξ (a) ≺a ζ (a) on C .





As usual, a feasible allocation distribution ν is said to belong to a core of an economy µ if and only if there exist no measurable sets C ⊂ I with λ(C ) > 0 which blocks ν . Notice that Definition 3 depends on the representation used. Fortunately, Theorem 2 below will ensure that the core is determined independently of the ˆ be a norm compact subset of Ω , and Bˆ = representations. Let Ω ˆ ) be the σ -field generated by the norm topology. Bnorm (Ω For the core equivalence theorem we need a stronger measurability condition for µ. Assumption (SM) (Strong Measurability). µΩ is a Borel (probabilˆ , Bˆ ). ity) measure on (Ω Remark. If the economy µ satisfies Assumption (SM), then the endowment map ω is weakly measurable, since σ (ℓ∞ , ba)-topology is weaker than the norm topology, hence it is Pettis integrable. Furthermore, since compact normed (metric) spaces are separable, ω is strongly measurable by Fact 16. Hence it is Bochner integrable. Actually, the Assumption (SM) is too strong for proving Theorem 2. What we shall really use are that µΩ is weakly measurable on a σ (ℓ∞ , ba)-compact, norm separable subset of Ω on which σ (ℓ∞ , ba)-topology is metrizable. Recall that the economy µ∗ given by the example in Section 2.1 has support (µ ∗Ω ) = ˆ = {eo , ee }, µ∗ satisfies the Assump{eo , ee }, hence setting Ω tion (SM). The second main result of this paper now reads Theorem 2. Let µ be an economy which satisfies the Assumptions (PR), (PE), (IR) and (SM). A feasible allocation distribution ν belongs to the core of the economy µ if and only if there exists a price vector p ∈ ℓ1 with p > 0 such that (p, ν) is a competitive equilibrium for µ. As a Corollary, we obtain that the core is well defined. Corollary. Let µ be an economy which satisfies the Assumptions (PR), (PE), (IR) and (SM). Then the set of core allocations is nonempty and it is determined independently of the representations. 3. Proofs of theorems 3.1. Proof of Theorem 1 Let E : I → P × Ω be a representation of the economy µ. For each n ∈ N, let K n be the canonical projection of ℓ∞ to Rn , K n = {ξ = (ξ t ) ∈ ℓ∞ |ξ = (ξ 1 , ξ 2 , . . . , ξ n , 0, 0, . . .)}. Naturally we can identify K n with Rn , or K n ≈ Rn . We define X n = X ∩ K n, n

X n ×X n

P =P ∩2

%n =% ∩(X n × X n ),

,

and Ω n = Ω ∩ K n .

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T. Suzuki / Journal of Mathematical Economics 49 (2013) 234–244

For every ω = (ω1 , ω2 , . . . , ωn , ωn+1 , . . .) ∈ Ω , we denote ωn = (ω1 , ω2 , . . . , ωn , 0, 0, . . .) ∈ Ω n , the canonical projection of ω. They induce finite dimensional economies E n : A → P n × Ω n defined by E n (a) = (%na , ωn (a)), n = 1, 2, . . . . We have Lemma 1. E n (a) → E (a) a.e. n

Lemma 2. For each n, there exists a quasi-competitive equilibrium for the economy E n , or a price-allocation pair (πn , ξn (a)) which satisfies

(E-2n)

πn ξn (a) ≤ πn ωn (a) and ξn (a) %a ζ whenever πn ζ ≤ πn ωn (a) and πn ω  n (a) > 0 a.e., ξ (a)dλ ≤ I ωn (a)dλ. I n

Proof. See Theorem A1 in Suzuki (2013) or Khan and Yamazaki (1981, Proposition 2).  We define probability measures ν n on X × P × Ω by ν n = λ ◦ (ξn , E n )−1 . It is evident that support (ν n ) ⊂ X n × P n × Ω n for all n, and it follows from Lemma 2 that for all n, ν n ({(ξn , %n , ωn )|πn ξn ≤ πn ωn and ξn %n ζ whenever πn ζ ≤ πn ωn and πn ωn > 0}) = 1. n Without loss of generality, we can assume that πn 1 = t =1 t t pn = 1 for all n, where πn = (pn ) and 1 = (1, 1, . . .). Here we have identified πn ∈ Rn+ with a vector in ℓ1+ which is also denoted by πn as πn = (πn , 0, 0, . . .). The set ∆ = {π ∈ ba+ | ∥π ∥ = π 1 = 1} is weak∗ compact by the Alaoglu’s theorem (Fact 12). Similarly, since ν n are probability measures on a compact metric space X × P × Ω , we can assume by Fact 12 that there exist a converging subnet (πn(α) , ν n(α) ) → (π , ν) ∈ ba+ × M(X × P × Ω ) in the weak∗ topologies with π 1 = 1 and ν(X × P × Ω ) = 1. Since E n (a) → E (a) a.e. by Lemma 1, it follows from Facts 6 and n(α) 8 that νP ×Ω → νP ×Ω = µ in the weak∗ topology. Then by Facts 9 and 15 and the condition (E-2n), one obtains that



ξ dνX = lim X



ξ dνXn(α) = lim



ξn(α) (a)dλ  ≤ lim ωn(α) (a)dλ = lim ωdνΩn(α) α α I Ω   = ωdνΩ = ω d µΩ . α

Ω

Lemma 3. π

α

X

I

Ω



Ω

ω d µΩ =



Ω

Lemma 4. ν({(ξ , %, ω) ∈ X × P × Ω | π ζ < π ω implies that ξ % ζ }) = 1. Proof. Define

n

Proof. We show that X × X → X × X in the topology of closed convergence τc . It is clear that Li(X n × X n ) ⊂ Ls(X n × X n ) ⊂ X × X . Therefore it suffices to show that X × X ⊂ Li(X n × X n ). Let (ξ , ζ ) = ((ξ t ), (ζ t )) ∈ X × X , and set ξn = (ξ 1 , . . . , ξ n , 0, 0, . . .) and similarly ζn for ζ . Then (ξn , ζn ) ∈ X n × X n for all n and (ξn , ζn ) → (ξ , ζ ). Hence (ξ , ζ ) ∈ Li(X n × X n ). Then it follows that %n =% ∩(X n × X n ) →%. Obviously one obtains ωn → ω in the σ (ℓ∞ , ℓ1 )-topology. Consequently we have E n (a) → E (a) a.e. on I. 

(Q-1n)

By Lemma 3, Assumption (PE) and π 1 = 1, we have µΩ ({ω ∈

Ω | π ω > 0}) > 0. We now prove

π ωdµΩ . n(α)

Proof. Note that πn(α) = (p1n(α) , . . . , pn(α) , 0, 0, . . .) ∈ ℓ1 and πn(α)  ω  → π ω a.e. on Ω , which implies that Ω πn(α) ωdµΩ → πωdµΩ . Similarly we have Ω  Ω πn(α) ωdµΩ = πn(α) Ω ωdµΩ → π Ω ωdµΩ . Therefore π Ω ωdµΩ = Ω π ωdµΩ .  Remark. Although the proof of Lemma 3 is easy, the lemma itself is not obvious, since the elements of the space ba generally do not commute with the Gel’fand integral. With the Bochner integral or the Pettis integral (see Appendix), they do.

F = {(ξ , %, ω) ∈ X × P × Ω | π ζ < π ω implies that ξ % ζ } and Fn = {(ξn , %n , ωn ) ∈ X n × P n × Ω n | πn ζn < πn ωn implies that ξn %n ζn }. We will show that ν(F ) = 1. It suffices to show that Ls(Fn ) ⊂ F . Indeed, since F (X × P × Ω ) is a compact metric space, we can extract a converging sub-sequence (Fni ) of (Fn ) with Fni → F˜ in the topology of closed convergence. Then by Fact 5 and the condition (Q-1n), 1 = ν(F˜ ) = ν(Ls(Fni )) ≤ ν(Ls(Fn )) ≤ 1, since ν is a probability measure. Let (ξ , %, ω) ∈ Ls(Fn ) and take a sequence (ξn , %n , ωn ) ∈ Fn with (ξn , %n , ωn ) → (ξ , %, ω). We need to show that (ξ , %, ω) ∈ F . If π ω = 0, we do not have to prove anything. So assume that π ω > 0 and (ξ , %, ω) ̸∈ F . Then there exists a vector ζ = (ζ t ) ∈ X such that π ζ < π ω and ξ ≺ ζ . Let ζn = (ζ 1 , ζ 2 , . . . , ζ n , 0, 0, . . .) be the projection of ζ to X n . Since ζn → ζ in the weak∗ topology, π ≥ 0 and ζN ≤ ζ , we have π ζN ≤ π ζ < π ω and ξ ≺ ζN for an N large enough. Since πn(α) → π and ξn → ξ in the weak∗ topologies, it follows for some α0 with n(α0 ) ≡ n0 ≥ N that 0 ≤ πn0 ζN < πn0 ω = πn0 ωn0 and ξn0 ≺ ζN , or ξn0 ≺n0 ζN , here observe that since πn = (p1n , . . . , pnn , 0, 0, . . .) and ωn = (ω1 , ω2 , . . . , ωn , 0, 0, . . .), we have that πn ω = πn ωn for all n. This contradicts the assumption that (ξn , %n , ωn ) ∈ Fn .  Let (ξ , E ′ ) : I → X × P × Ω be a representation of ν . Note that the representation E ′ of νP ×Ω = µ would be possibly different from the representation E which was introduced at the beginning of the proof. We will, however, denote E ′ = (%a , ω(a)) for notational simplicity without any danger of confusion. Let P = {a ∈ I | π ω(a) > 0}. Let π = πc + πp be the Yosida– Hewitt decomposition and denote πc = p. By Lemma 3, we have λ(P ) > 0. Suppose that π ω(a) > 0 and ξ (a) ≺a ζ . Let ζn = (ζ 1 , ζ 2 , . . . , ζ n , 0, 0, . . .) be the projection of ζ to X n . Then we can assume that ξ (a) ≺a ζn for n sufficiently large, hence it follows from Lemma 4 that π ζn ≥ π ω(a) for n sufficiently large. Since πp is purely finitely additive, πp ({1, . . . , n}) = 0 for each n. If π ζn = (πc + πp )ζn = pζn = π ω(a), then since π ω(a) > 0 and % is weak∗ -continuous, we can take ζ ′ ∈ X n ⊂ X sufficiently close to ζn such that ξ (a) ≺a ζ ′ and π ζ ′ = pζ ′ < π ω(a), contradicting Lemma 4. Hence one obtains that π ζn > π ω(a) for n sufficiently large. It follows from ζn ≤ ζ and πc ≥ 0 that π ζn = pζn ≤ pζ . On the other hand, πp ≥ 0 and ω(a) ≥ 0 imply that π ω(a) = (πc + πp )ω(a) ≥ πc ω(a) = pω(a), and consequently we have pζ > pω(a). Summing up, we have verified that

ξ (a) ≺ ζ implies that pω(a) < pζ a.e. on P .

(1)

Since the preferences are locally non-satiated, there exists ζ ∈ X arbitrarily close to ξ (a) such that ξ (a) ≺ ζ , therefore we have pω(a) ≤ pξ (a) for almost all a ∈ P . On the other hand, for a ∈ A with π ω(a) = 0, one obtains that 0 ≤ pω(a) ≤ π ω(a) = 0 ≤ pξ (a), since p ≥ 0 and ω(a), ξ (a) ≥ 0.   It follows from X ξ dνX ≤ Ω ωdνΩ and Fact 9 that



pξ (a)dλ = I



pξ dνX ≤ X

 Ω

pωdνΩ =

 I

pω(a)dλ.

T. Suzuki / Journal of Mathematical Economics 49 (2013) 234–244

Therefore pξ (a) = pω(a) a.e. on I, or νX ×Ω ({(ξ , ω) ∈ X × Ω | pξ = pω}) = 1. The proof of theorem is almost completed by Lemma 5. νΩ ({ω ∈ Ω | π ω = 0}) = 0. Proof. Let Q = {a ∈ I | π ω(a) = 0}. We  will show that  λ(Q ) = 0. Suppose that λ(Q ) > 0. Hence 0 ≤ p Q ω(a)dλ = Q pω(a)dλ ≤

πω(a)dλ = 0. Since the economy µ is irreducible (the Assumption (IR)), there exists an allocation φ with    (ω(a) − φ(a))dλ + ξ (a)dλ ∈ {ζ ∈ X | ξ (a) ≺a ζ }dλ. 

Q

P

P

Q

Note that pξ (a) = pω(a) ≤ γ < β a.e. on P, hence P {ζ ∈ X | ξ (a) ≺a ζ }dλ ̸= ∅. Therefore there exists a map ζ on P to X such that ξ (a) ≺a ζ (a) a.e. on P with





ζ (a)dλ =



(ω(a) − φ(a))dλ +

ξ (a)dλ. P

Q

P



It follows from (1) that pω(a) < pζ (a) a.e. on P, hence



ξ (a)dλ = p

p P



ω(a)dλ < p



By Lemma 5 and νX ×Ω ({(ξ , ω) ∈ X × Ω | pξ = pω}) = 1, we have obtained that

νX ×P ×Ω ({(ξ , %, ω) ∈ X × P × Ω | pξ = pω and ξ % ζ whenever pζ ≤ pω}) = 1.  t  t Finally, since I ξ (a)dλ ≤ I ω (a)dλ ≤ γ < β for each t, there exists a positive amount of consumers with ξ t (a) < β . Then by t (PR) (ii),one obtains the monotonicity   that p > 0 for all t, hence ξ ( a ) d λ = ω( a ) d λ , or ξ d λ = ω d µ X Ω . This completes the I I X Ω 

3.2. Proof of Theorem 2 The proof that a competitive equilibrium distribution is a core distribution is standard. We skip it. Let ν be a core allocation, and (ξ , E ) : I → X × P × Ω its representation.  Without loss of generality, we can assume that ξ (a)dλ = I ω(a)dλ. Recall that β > 0 is the upper bound I for the allocations and γ > 0 is that for the endowments. Let 1 = (1, 1, . . .) ∈ ℓ∞ . Then we have Lemma 6. ξ (a) < β 1 a.e. in I. Proof. Suppose not. Then there exists a measurable set B ⊂ I with λ(B) > 0 and ξ (a) = β 1 fora ∈ B. We claim that λ(I \ B) > 0, since otherwise one obtains that I ξ (a)dλ = β 1 ≫ γ 1 ≥ I ω(a)dλ, a contradiction. Then we have



ξ (a)dλ =



I \B

ξ (a)dλ − βλ(B)1 I



ω(a)dλ +

= I \B





ω(a)dλ − βλ(B)1 B

ω(a)dλ + (γ λ(B)1 − βλ(B)1)

≤ I \B



ω(a)dλ.

≪ I \B



Lemma 7. Let %: I → P , a → %a be a preference assignment. Then there exists a sequence of simple maps %n : I → P which converges to % a.e. on I. Proof. Since F (X × X ) is a compact metric space, it is a separable metric space. Hence so is P ⊂ F (X × X ). Therefore we can assume without loss of generality that the set {%a | a ∈ I } is itself separable. For every n = 1, 2, . . . , one has a countable family of open balls B(i,n) , i = 1, 2, . . . with the radius ≤ 1/n which covers {%a | a ∈ I }. Let the center of B(i,n) be %ni ∈ {%a | a ∈ I }. Since the preference assignment is Borel measurable, the set I(i,n) = {a ∈ I | %a ∈ B(i,n) } is Borel measurable for all i, n. Evidently, I = ∪∞ i=1 I(i,n) . 1 n Set I(∗i,n) = I(i,n) \ ∪ij− I , and define a map % : I → P by =1 (j,n) for a ∈ I(∗i,n) .

∞



proof of Theorem 1.

allocation.

∗ n Since I = i=1 I(i,n) , we have δ(%a , %a ) ≤ 1/n for all a ∈ I, where δ is a metric on P for the topology τc . It is easy to see (n,m) that for each n, there exists a sequence of simple maps %a with (n,m) %a → %na as m → ∞ a.e. on I. Hence the lemma follows. 

P

a contradiction.

Define an allocation ζ (a) = (ζ t (a)) by ζ t (a) = min{ξ t (a) + (λ(B)/λ(I \ B))(β − γ ), β} (t = 1, 2, . . .) for a ∈ I \ B. By the monotonicity (PR)   (ii), we have that ξ (a) ≺a ζ (a) a.e. on I \ B and ζ ( a ) d λ ≤ ω(a)dλ, contradicting the fact that ξ (a) is a core I \B I \B

%na = %ni

ζ (a)dλ P P  = p (ω(a) − φ(a))dλ + p ξ (a)dλ Q P  ≤ p ξ (a)dλ,

239



Lemma 7 and Fact 2 imply that the preference map %: I → (P , δ) is Lusin measurable with respect to the Hausdorff distance δ . Assumption (SM) and Fact 16 imply that the endowment map ω : I → Ω is also Lusin measurable with respect to the norm topology. It follows from the Lusin’s theorem (Fact 3) and Fact 4 that there exists a sequence of compact subsets Kn of I satisfying (i) (ii) (iii) (iv) (v)

% is continuous (in the topology of closed convergence) on Kn , ω is norm continuous on Kn , ξ is weak∗ -continuous on Kn , Kn is a support for λ on itself, λ(Kn ) ≥ 1 − 1/n.

Define a correspondence φ(a) : I → X by φ(a) = {ζ ∈ X | ξ (a) ≺a ζ }. By Lemma 6, φ(a) ̸= ∅ a.e. on I. Let   Z = co ∪∞ n=1 ∪a∈Kn (φ(a) − ω(a)) . We show Lemma 8. Z ∩ (−int(ℓ∞ + )) = ∅. Proof. Suppose not. Then there exists η′ ∈ int(ℓ∞ + ), a finite set of agents ai ∈ Kni , i = 1, . . . , m, ζi ∈ φ(ai ), and 0 ≤ αi ≤ 1 with m  m ′ i=1 αi = 1 such that −η = i=1 αi (ζi − ω(ai )). We can take ′ η ≤ η such that

ζi + η ∈ int(X ),

i = 1 . . . m,

where int(X ) means the interior of X with respect to the norm topology. Indeed, since the preferences are continuous, we can assume (taking smaller vectors if necessary) ζi with ∥ζi ∥ < β that m and ζi ∈ φ(ai ), i = 1 . . . m, and η′ = − i=1 αi (ζi − ω(ai )) ∈ ′ int(ℓ∞ + ). Take 0 < α ≤ 1 if necessary such that ζi + αη ∈ ′ int(X ), i = 1 . . . m, and set η = αη . Let zi = ζi − ω(ai ). Since ω is norm continuous on Kni , the set Ei = {a ∈ I | zi + η + ω(a) ∈ int(X )} is open in Kni . Since ai ∈ Ei , it is nonempty. Let Fi = {a ∈ Ei | ξ (a) ≺a zi + η + ω(a)}. Since ξ (a), ω(a) and %a are continuous on Kni , the set Fi is open in Ei and hence in Kni . Since ai ∈ Kni and Kni is a support for λ on itself by Fact 4, we see that λ(Fi ) > 0. Since Lebesgue measure has convex

240

T. Suzuki / Journal of Mathematical Economics 49 (2013) 234–244

range, we can find Gi ⊂ Fi such that Gi ∩ Gj = ∅ for i ̸= j and λ(Gi ) = ραi for some ρ > 0, i = 1 . . . m. Define an allocation ψ : I → X by

ψ(a) =



zi + η + ω(a) 0

ψ(a)dλ = ρ ∪Gi

m 

αi (zi + η) +



≤ρ

m 

αi (zi + η′ ) +

i=1



ω(a)dλ ∪Gi

i=1



ω(a)dλ ∪Gi

ω(a)dλ,

=

ξ (a) ≺a ζ implies that pω(a) < pζ a.e. on P .

(3)

In what follows, we can proceed exactly in the same way after the claim (1) in the proof of Theorem 1, and obtain that pξ (a) = pω(a) a.e. on I. It follows from Lemma 5 which also holds here and (3), we obtain that

if a ∈ Gi , otherwise.

Then we have



Summing up, we have verified that

∪Gi

so that the coalition ∪m i=1 Gi blocks ξ via ψ . This contradiction proves Lemma 8.  ∞ Since int(ℓ∞ + ) is nonempty and open convex subset in ℓ , by the Hahn–Banach theorem (Fact 13), there exists a non-zero vector π ∈ ba+ such that

  π Z ≥ −π int(ℓ∞ +) , which implies that π Z ≥ 0 and π ≥ 0. Since λ(∪∞ k=1 Kn ) = 1, this implies that π φ(a) ≥ π ω(a) a.e. on I or

ξ (a) ≺a ζ implies that π ω(a) ≤ π ζ a.e. on I . (2)  Let P = {a ∈ I | π ω(a) > 0}. Since I ω(a)dλ can be seen as a Bochner from Assumption (PE) and π 1 = 1 that  integral, it follows  0 < π I ω(a)dλ = I π ω(a)dλ, hence λ(P ) > 0. Remark. Notice that Lemma 3 does not apply here, since the price vector π was directly obtained as an element of the space ba by the Hahn–Banach theorem, not as the limit of a sequence of prices in Rn . Instead, the Bochner (Pettis) integrability of the endowment map ω worked. We now prove Lemma 9. ξ (a) ≺a ζ implies that π ω(a) < π ζ a.e. on P. Proof. If the lemma was false, there exists ζ (a) = (ζ t (a)) ∈ X such that π ζ (a) ≤ π ω(a) and ξ (a) ≺a ζ (a) on a subset of P with λ-positive measure. Let ζn (a) = (ζ 1 (a), . . . , ζ n (a), 0, 0, . . .) be the projection of ζ (a) to Rn , which is naturally identified as a finite dimensional subspace of ℓ∞ . Since ζn (a) → ζ (a) in the σ (ℓ∞ , ℓ1 )-topology, we have for sufficiently large N that π ζN (a) ≤ πζ (a) ≤ π ω(a) and ξ (a) ≺a ζN (a), since π ≥ 0, ζN (a) ≤ ζ (a) and preferences are continuous. Let π = πc + πp be the Yosida–Hewitt decomposition and denote πc = p. Since πp is purely finitely additive, πp ({1 . . . n}) = 0 for each n. If π ζn = (πc + πp )ζn = pζn = π ω(a), then since π ω(a) > 0 and % is weak∗ -continuous, we can take ζ ′ ∈ Rn ⊂ ℓ∞ sufficiently close to ζn such that ξ (a) ≺a ζ ′ and π ζ ′ = pζ ′ < π ω(a), contradicting (2).  Suppose that π ω(a) > 0 and ξ (a) ≺a ζ . Then we can assume that ξ (a) ≺a ζn for n sufficiently large, where as usual ζn is the projection of ζ to Rn . Hence it follows from Lemma 9 that π ζn > πω(a) for n sufficiently large. Since πp is purely finitely additive, πp ({1 . . . n}) = 0 for each n. It follows from this and πc ≥ 0 that

πζn = (πc + πp )ζn = πc ζn ≤ πc ζ = pζ , since ζn ≤ ζ . On the other hand, πp ≥ 0 and ω(a) ≥ 0 imply that πω(a) = (πc + πp )ω(a) ≥ πc ω(a) = pω(a), and consequently we have pζ > pω(a).

ξ (a) ≺a ζ implies that pω(a) < pζ a.e. on I . This completes the proof of Theorem 2.



4. Concluding remarks 1. Recalling the proofs in the previous section, we shall give some remarks on the relationship between the coalitional form and the individual form of the economy. Let E : I → P × Ω be the representation of the economy µ given at the beginning of the proof of Theorem 1. We could start the economy of the individual form by this map E from the outset. By Lemma 2, we obtained a sequence of the finite dimensional (quasi-)equilibria (πn , ξn (a)). If our goal was to prove the existence of the (standard) equilibrium of the individual form, then we have to find an equilibrium allocation ξ : I → X and an equilibrium price p which satisfy (Id-1)

pξ = pω(a) and ξ %a ζ whenever pζ ≤ pω(a) a.e. in I.

The allocation ξ would be obtained as some sort of limit of ξn . It is this step where we have to invoke the Fatou’s lemma Ls( I ξn (a)dλ) ⊂ I coLs(ξn (a))dλ and the best we could get would be a map satisfying ξ (a) ∈ coLs(ξn (a)) a.e.6 In order to show that ξ satisfies the condition (Id-1), we will need the convexity of the

preferences.7 Actually, the statement of Theorem 1 is to obtain an equilibrium of the coalitional form which satisfies the condition (E-1) in Definition 2. For this purpose, it suffices to obtain the limit distribution ν = lim λ ◦ (ξn , E n )−1 rather than the map ξ (a) ∈ coLs(ξn (a)), a ∈ I. The existence of ν is an immediate consequence of the Banach–Alaoglu’s theorem, and we do not have to use Fatou’s lemma here. The essential ingredient of (E-1) is contained in Lemma 4 for which no convexity assumptions for preferences are needed. The distribution ν has a representation (ξ , E ′ ). Using the map ′ E , we transformed the statement of Lemma 4 to that of the individual form (for the consumer with positive incomes), namely the condition (1). If the representation were unique, or E ′ = E , the standard  argument combined with the feasibility condition ξ dλ ≤ ωdλ would yield the (standard) equilibrium of the individual form. Actually, the representations are not unique.8 Although we showed the condition (Id-1) for the allocation ξ (see the condition (1), Lemma 5 and the statement before Lemma 5), it is an equilibrium condition for the economy E ′ , not for E . Hence all which we can obtain by this proof are the equilibria of the coalitional form satisfying the distributional form of the utility maximization condition (E-1) which is the same for E and E ′ . In general, let Eˆ : I → P × Ω be an economy of the individual form. It will induce uniquely the economy µ = λ◦ Eˆ −1 of the coalitional form. Let (p, ν) be a (coalitional) competitive equilibrium of

6 On the finite dimensional spaces, we can obtain an allocation ξ with ξ (a) ∈ Ls(ξn (a)) a.e. 7 This procedure was indeed carried out in Suzuki (2012) in which the existence of the competitive equilibrium under the assumption of the convexity of preferences was proved. 8 They cannot be unique; it is easy to construct a measurable isomorphism φ : I → I such that for any representation (ξ , E ) of µ, (ξ , E ) ◦ φ = (ξ ◦ φ, E ◦ φ) is also a representation.

T. Suzuki / Journal of Mathematical Economics 49 (2013) 234–244

the economy µ. Then it is weaker than the (individual) competitive equilibrium (p, ξ ) of the economy Eˆ in the following sense. Let R be the set of representations of ν, R = {(ξ , E ) | ν = λ ◦ (ξ , E )−1 }. Each (ξ , E ) ∈ R will give a coalitional equilibrium (p, ν) of µ, since the coalitional equilibrium is independent of the representations. Only elements (ξ , E ) ∈ R with E = Eˆ , however, can be the individual equilibrium (p, ξ ) of Eˆ . Theorem 1 in this paper claimed that we could prove a coalitional equilibrium without convexity of preferences. However, in order to obtain a stronger equilibrium of the individual form, we would need the convexity (at least) at the present stage of our knowledge. 2. Tourky and Yannelis (2001) and Podczeck (2003) showed that on non-separable Banach spaces, the core equivalence theorem fails when the resource feasibility is defined by the Bochner integral.9 Then it turns out that the core equivalence theorem of Rustichini and Yannelis (1991a) which was proved for an economy in which the commodity space is a separable Banach space and the resource allocations are Bochner integrable is the best result in such a setting. Podczeck (2004) further explored this result for the case of Pettis integrable allocations, and Podczeck (2005) established necessary and sufficient conditions on the commodity spaces of Banach lattices for the Bochner integrable allocations. The separable commodity spaces can be interpreted to be ‘‘not too large’’ relative to the size (population) of the economies, in particular the agents’ preferences cannot be too ‘‘dispersed’’. Those authors called this situation as ‘‘many more agents than commodities’’, and brought new light to an Aumann’s ‘‘hidden’’ assumption. Mathematically, these results made clear that the separability is a necessary condition for the core equivalence on the setting of Banach spaces (or lattices) and Bochner (or Pettis) integrable allocations. The present paper gave sufficient conditions for the existence and the equivalence theorems of the core to hold on a specific space ℓ∞ . The crucial conditions we have observed are that (a) the consumption sets are norm bounded, (b) the resource feasibility is defined by the Gel’fand integral, and (c) the initial endowment map is strongly measurable. The conditions (a) and (b) were used for the non-emptiness of the core (actually competitive equilibria), and the condition (c) for the core equivalence. One recognizes that all three conditions are in favor of the core equivalence on account of the above results. Among these, seemingly the strongest one is the condition (a). The norm boundedness made the consumption set to be compact and metrizable in the weak∗ topology, hence separable. For the condition (b), in order for allocations to be Gel’fand integrable, they are enough to be weak∗ measurable, and the weak∗ measurability is weaker than the weak measurability or the strong measurability on ℓ∞ . (Of course they are equivalent on finite dimensional spaces.) Intuitively, the weaker the measurability is, there are the more integrable allocations, hence it will be easier for blocking and the core will be smaller. The strong measurable condition (c) was necessary for an alternative condition of the assumption of the ‘‘extremely desirable commodities (properness)’’ in the proof of the core equivalence when the positive orthant has the empty interior (see below). On the other hand, the above authors have obtained the positive results for the core equivalence. However, none of them covers our result (Theorem 2), for the nonempty interior of the positive orthant and the separability of the commodity spaces cannot be attained by any single topology on ℓ∞ , and our resource feasibility is in terms of the Gel’fand integral. It is well known that the positive orthant of ℓ∞ has nonempty norm interior, but the space is not

9 The author is indebted to an anonymous referee of this journal for paying his attention to this result.

241

separable in the norm topology. It is separable in the Mackey topology, but this topology makes the interior of the positive orthant empty. Theorem 4.1 of Rustichini and Yannelis (1991a) and Theorem 6.1 of Tourky and Yannelis (2001) assumed both the separability and the nonempty interiority.10 Theorem 6.1 of Rustichini and Yannelis (1991a), Theorem 6 of Podczeck (2004) and Theorem 3 of Podczeck (2005) are compatible with the space ℓ∞ with the Mackey topology. However, in this case they needed the properness condition on the preferences11 which we did not. Instead, we used the strong measurability (c) of endowments and applied the Hahn–Banach theorem to the norm topology. Moreover all of these results are obtained under the setting of Bochner or Pettis integrable allocations.12 Notice that the bounded consumption sets provided us a nice topological property, but it also yielded its own problem for the condition on the initial endowments if we want to discard the strongly positive individual endowments (IP). Rustichini and Yannelis proved their theorem without (IP); the strictly positive total endowment (PE) was enough. The argument for this is well known (e.g., Rustichini and Yannelis, 1991a, p. 316). Let ξ (a) be the core allocation to be proved being supported by the (equilibrium) price vector p > 0which satisfies the condition (3) in the proof of Theorem 2. Since ω(a)dλ ≫ 0 and p ̸= 0, we have λ(P ) > 0, where P = {a ∈ A | pω(a) > 0}. If pt = 0 for some t, then the monotonicity of preferences implies that we can find ζ = (ζ t ) with ξ ≺a ζ and pζ ≤ pω(a) by letting ζ t > ξ t (a) and ζ s = ξ s (a)(s ̸= t ) for a ∈ P, a contradiction. Hence one obtains p ≫ 0. This will be enough to prove that ξ (a) is maximal in the budget set for a ∈ P. But p ≫ 0 implies that ξ (a) = 0 for a ∈ A \ P, which is trivially a maximal element in the budget set {0}. However, this arguments will no more work if the consumption set is bounded by β 1. For it would be possible that ξ t (a) = β on P a.e., we cannot make ζ t > ξ t (a). Applying the irreducibility Assumption (IR) to prove Lemma 5 seems to be the best way to handle this problem. Acknowledgments Earlier versions of the paper were presented at seminars held at Kobe University and Keio University. I thank participants of the seminars, in particular, Tohru Maruyama and Nobusumi Sagara. I also want to thank Mitsunori Noguchi. His comments have been most helpful at each stage of the research. An anonymous referee of this journal helped me to clarify several ambiguous points in earlier versions of the paper. Of course, remaining errors are my own. Appendix A.1. Some measure theory on metric spaces Let X be a complete and separable metric space with a metric d. The Borel σ -field of X which is defined as the σ -field generated by

10 Tourky and Yannelis (2001) did not assume explicitly the separability, but the case in their minds is that of separable Banach spaces. 11 Rustichini and Yannelis called it the assumption of ‘‘extremely desirable commodities’’, Podczeck called the ‘‘bounded marginal rate of substitutions’’. 12 Although we stated the Gel’fand integral is intuitively more preferable for the core equivalence, this does not mean that theorems for Bochner or Pettis integrable allocations settings imply those for a Gel’fand integrable allocations setting. In fact, Podczeck (2004, pp. 432–3) stated that ‘‘. . . if the commodity space is actually a dual Banach space, then the notion of the Gel’fand integral may be more appropriate than that of the Pettis integral to define feasibility of allocations. This is so in particular for the space M (Ω ) (which is the space of regular bounded Borel measures on a compact Hausdorff space Ω . ℓ∞ is also a dual Banach space (Suzuki)) as a model of commodity differentiation. An investigation of the Core–Walras equivalence problem (also that of the existence (Suzuki)) in the Gel’fand integrable allocations setting will be the topic of future research . . . ’’.

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open subsets of X is denoted by B (X ). Let (I , I, λ) be the Lebesgue space on the unit interval I = [0, 1]. We denote the set of all closed subsets of a set X by F (X ). The topology τc on F (X ) of closed convergence is a topology which is generated by the base

[K ; G1 . . . Gn ] = {F ∈ F (X )|F ∩ K = ∅, F ∩ Gi ̸= ∅, i = 1 . . . n} as K ranges over the compact subsets of X and Gi are arbitrarily finitely many open subsets of X . The fundamental fact is Fact 1 (Hildenbrand, 1974, Theorem 2, p. 19). If X is locally compact separable metric space, then F (X ) is compact and metrizable. Moreover, a sequence Fn converges to F ∈ F (X ) if and only if Li(Fn ) = F = Ls(Fn ), where Li(Fn ) denotes the topological limes inferior of {Fn } which is defined by ξ ∈ Li(Fn ) if and only if there exists an integer N and a sequence ξn ∈ Fn for all n ≥ N and ξn → ξ (n → ∞), and Ls(Fn ) is the topological limes superior which is defined by ξ ∈ Ls(Fn ) if and only if there exists a sub-sequence Fnq with ξnq ∈ Fnq for all q and ξnq → ξ (q → ∞), see Hildenbrand (1974, pp. 15–19) for details. Let (X , d) be a metric space. A sequence of measurable maps fn : I → X , n = 1, 2, . . . is said to converge to a map f : I → X in measure if and only if given ϵ > 0, λ({a ∈ I | d(fn (a), f (a)) ≥ ϵ}) → 0. It is well known that Fact 2 (Hildenbrand, 1974, (21), p. 47). Every sequence of measurable maps fn : I → X which converges to f : I → X almost everywhere converges to f in measure. A map f : I → X is called simple if there exist a (finite) measurable partition {Ii }ni=1 of I with ∪ni=1 Ii = I and vectors ξi ∈ X , i = 1 . . . n such that f (a) = ξi for a ∈ Ii . A measurable map f : I → X is called Lusin measurable (with respect to d) if it is the limit in measure of a sequence of simple functions fn . Then we have the next theorem known as Lusin’s theorem. Fact 3 (Halmos, 1974, p. 242). If f : I → (X , d) is Lusin measurable, then for every ϵ > 0, there exists a compact set K ⊂ I such that λ(K ) > 1 − ϵ and f is continuous on K . Let K ⊂ I be compact and λ(K ) > 0. We say that K is a support for λ on itself if whenever U ⊂ I is open and U ∩ K ̸= ∅, then λ(K ∩ K ) > 0. It is well known that Fact 4. Every compact set K ⊂ I contains a compact set K ′ such that λ(K ′ ) = λ(K ) and K ′ is a support for λ on itself. Let K be a compact metric space, and B (K ) the Borel σ -field on K (the smallest σ -field generated by open sets of K ). The set ca(K , B (K )) of bounded countably additive set functions (signed measures) on K is defined analogously as the space ca given in Section 2.1. It is a Banach space by the norm defined in the same way as ca. Let C (K ) be the set of all continuous functions on K . C (K ) is also a Banach space with respect to the norm ∥f ∥ = sup{|f (t )| | t ∈ K }. Then Riesz representation theorem (Royden, 1988, p. 357) asserts that the dual space of C (K ) is ca(K , B (K )), or C ∗ (K ) = ca(K , B (K )). A net (να )in ca(K , B (K )) converges to ν in the weak∗ topology if and only if K f (t )dνα → K f (t )dν for every f ∈ C (K ). An element π of ca(K , B (K )) is called non-negative if π(B) ≥ 0 for all B ∈ B (K ). A non-negative element of ca(K , B (K )) is called a measure. Let M (K ) be the set of all measures on K . The smallest closed subset of K which gives the ν -full measure is called the support of the measure ν and is denoted by support (ν). A measure π is called a probability measure if π (K ) = 1. For t ∈ K , the Dirac measure δt is defined by δt (E ) = 1 if t ∈ E , δt (E ) = 0, otherwise. The next proposition is due to Mas-Colell (1975). We give a complete proof, since he did not give it.

Fact 5. Let K be a compact metric space. If Fn is a sequence of closed subsets of K such that Fn → F in the topology of closed convergence and µn is a sequence of probability measures on K such that µn (Fn ) = 1 for all n and µn → µ, then µ(F ) = 1. Proof. Suppose not. Then ν(K \ F ) > 0. Since K is a compact metric space, the family of Baire sets Ba (K ) coincides with B (K ), and the Borel measure ν is also a Baire measure on K (Royden, 1988, p. 334). Since every Baire measure is regular (Royden, 1988, p. 340), for every ϵ > 0, there exists a compact set C ⊂ K \ F with ν(K \ F ) − ν(C ) ≤ ϵ , or ν(C ) ≥ ν(K \ F ) − ϵ > 0 for ϵ small enough. Since K \ F is open in K , we can assume that ν(interior C ) > 0. Consider an open neighborhood of F in F (K ), [C ; K \ C ] = {H ∈ F (K ) | H ∩ C = ∅, H ∩ (K \ C ) ̸= ∅}. Since Fn → F in the closed convergence, Fn ∈ [C ; K \ C ] for n large enough, hence Fn ∩ C = ∅ for n large enough. Since νn → ν , we have lim infn νn (interior C ) ≥ ν(interior C ) > 0 (Hildenbrand, 1974, (26), p. 48), hence lim infn νn (Fn ) < 1. This contradicts that νn (Fn ) = 1 for all n.  Let (K , d) and (M , d′ ) be compact metric spaces and κ a measure on K × M. A marginal distribution of κ on K is the measure κK defined by κK (B) = κ(B × M ) for every B ∈ B (K ). A marginal distribution κM on M is defined similarly. Fact 6 (Hildenbrand, 1974, (27), pp. 48–9). Let (K , d) and (M , d′ ) be compact metric spaces and (κ n ) a weak∗ -converging sequence of measures on K × M with the limit κ . Then the marginal n converges in the weak∗ topology to κK and distributions κKn and κM κM , respectively. The next facts are also well known for the mathematical economics. Fact 7 (Skorokhod’s Theorem, Hildenbrand, 1974, (37), p. 50). Let

(K , d) be a compact metric space and (κ n ) a weak∗ -converging sequence of measures on K with the limit κ . Then there exist measurable mappings f and fn (n ∈ N) on the unit interval I = [0, 1] to K such that κ = λ ◦ f −1 , κ n = λ ◦ fn−1 , and fn → f a.e. in I, where λ is the Lebesgue measure on I. Fact 8 (Hildenbrand, 1974, (39), p. (52)). Let (A, A, µ) be a probability space and (K , d) be a separable metric space. If fn , f are (Borel) measurable functions from A to K and d(fn (a), f (a)) → 0 a.e. in µ, then µ ◦ fn−1 → µ ◦ f −1 in the weak∗ topology. Fact 9 (Change-of-Variable Formula, Hildenbrand, 1974, (36), p. 50). Let (A, A, µ) be a probability space and (K , d) be a metric space, φ a (Borel) measurable mapping from A to K and h a (Borel) measurable −1 real valued function on K . Then if and only  h is µ ◦−1φ -integrable  if h ◦ φ is µ-integrable and K hdµ ◦ φ = A h ◦ φ dµ. A.2. Topologies on locally convex topological vector spaces The set function π ∈ ba is called purely finitely additive if ρ = 0 whenever ρ ∈ ca and 0 ≤ ρ ≤ π . The relationship between the ba and ca is made clear by the next fundamental theorem, Fact 10 (Yosida and Hewitt, 1956). If π ∈ ba and π ≥ 0, then there exist set functions πc ≥ 0 and πp ≥ 0 in ba such that πc is countably additive and πp is purely finitely additive and satisfy π = πc + πp . This decomposition is unique. On the space ℓ∞ , we can consider the several topologies. One is of course the norm topology τnorm which was explained above. It is the strongest topology among the topologies which appear in this paper.

T. Suzuki / Journal of Mathematical Economics 49 (2013) 234–244

The weakest topology in this paper is the product topology τD which is induced from the metric D(ξ , ζ ) =

∞  t =1

|ξ t − ζ t | for ξ = (ξ t ), ζ = (ζ t ) ∈ ℓ∞ . 2t (1 + |ξ t − ζ t |)

The product topology is nothing but the topology of coordinatewise convergence, or ξ = (ξ t ) → 0 if and only if ξ t → 0 for all t ∈ N. A net (ξα ) on ℓ∞ is said to converge to 0 in the weak∗ topology or σ (ℓ∞ , ℓ1 )-topology if and only if pξα → 0 for each p ∈ ℓ1 . The weak∗ topology is characterized by the weakest topology on ℓ∞ which makes (ℓ∞ )∗ = ℓ1 , where L∗ is the dual space (the set of all continuous linear functionals on L) of a normed linear space L. Then it is stronger than the product topology, since the latter is characterized by ξα → 0 if and only if et ξα → 0 for all for each et = (0, . . . , 0, 1, 0, . . .) ∈ ℓ1 , where 1 is in the t-th coordinate. A subbase of the neighborhood system of 0 ∈ ℓ∞ in the weak∗ topology is the family of the sets U of the form

  U = ξ ∈ ℓ∞ | |pξ | < ϵ ,

ϵ > 0, p ∈ ℓ1 .

The strongest topology on ℓ∞ which makes (ℓ∞ )∗ = ℓ1 is called the Mackey topology τ (ℓ∞ , ℓ1 ). It is characterized by saying that a net (ξα ) on ℓ∞ is said to converge to 0 in τ (ℓ∞ , ℓ1 )-topology if and only if sup{|pξα ||p ∈ C } → 0 on every σ (ℓ1 , ℓ∞ )-compact, convex and circled subset C of ℓ1 , where a set C is circled if and only if rC ⊂ C for −1 ≤ r ≤ 1, and the topology σ (ℓ1 , ℓ∞ ) is defined analogously as σ (ℓ∞ , ℓ1 ), namely that a net (pα ) on ℓ1 is said to converge to 0 in the σ (ℓ1 , ℓ∞ )-topology if and only if pα ξ → 0 for each ξ ∈ ℓ∞ . The topology τ (ℓ∞ , ℓ1 ) is weaker than the norm topology. Hence we have τD ⊂ σ (ℓ∞ , ℓ1 ) ⊂ τ (ℓ∞ , ℓ1 ) ⊂ τnorm . Similarly, a net (πα ) on ba is said to converge to 0 in the weak∗ topology or σ (ba, ℓ∞ )-topology if and only if πα ξ → 0 for each ξ ∈ ℓ∞ . We can use the next useful proposition on bounded subsets of the space ℓ∞ . Fact 11 (Bewley, 1991, p. 226). Let Z be a (norm) bounded subset of ℓ∞ . Then on the set Z , the Mackey topology τ (ℓ∞ , ℓ1 ) coincides with the product topology τD . Bounded subsets of ℓ∞ are σ (ℓ∞ , ℓ1 )-weakly compact, namely that the weak∗ closure of the sets are weak∗ -compact by Banach– Alaoglu’s theorem. Fact 12 (Rudin, 1991, pp. 68–70). If L is a Banach space, then the unit ball of L∗ , B = {π ∈ L∗ |∥π ∥ ≤ 1} is compact in the σ (L∗ , L)topology. In general, let L be a normed vector space and L∗ its dual space. A net (ξα ) in L converges to ξ ∈ L in the σ (L, L∗ )-topology or weak topology if and only if π ξα → π ξ for every π ∈ L∗ . A net (πα ) in L∗ converges to π ∈ L∗ in the σ (L∗ , L)-topology or weak∗ topology if and only if πα ξ → π ξ for every ξ ∈ L. (We already encountered the space ca(K , B (K )) with the weak∗ topology.) These are examples of the locally convex topological vector space. It is defined as a vector space endowed with the compatible topology (vector space operations are continuous with respect to this topology) whose every neighborhood of {0} includes a convex neighborhood of {0}. The dual space of a topological vector space L is also denoted by L∗ . Let A be a subset of a locally convex topological vector space. We denote by co(A) the convex hull of A. It is the set of finite convex combinations of the elements of A. co(A) ≡ co(A) is its closure. An element p ∈ L∗ and a real number α ∈ R determine the hyper plane H (α) = {x ∈ L | px = α}. The upper and the lower half space H+ (α) and H− (α) are defined by H+ (α) = {x ∈ L | px ≥ α} and H− (α) = {x ∈ L | px ≤ α}, respectively. The hyper plane H (α) separates two sets A, B ⊂ L

243

if either A ⊂ H+ (α) and B ⊂ H− (α) or if A ⊂ H− (α) and B ⊂ H+ (α). The hyper plane H (α) properly separates A and B if it separates them and A ∪ B is not included in H (α). The famous Minkowski’s separation hyper plane theorem is extended to locally convex topological vector spaces as the Hahn–Banach theorem(s). We will use a subsequent version of them. Fact 13 (Aliprantis and Border, 1994, p. 143). For disjoint nonempty convex subsets A and B of a (locally convex) topological vector space L can be properly separated by a non-zero continuous linear functional p ∈ L∗ , if (at least) one of them has an interior point. A.3. Integrations of vector valued maps Let (A, A, µ) be a finite measure space. A map f : A → ℓ∞ is said to be weak∗ -measurable if for each p ∈ ℓ1 , pf (a) is measurable. A map f : A → ℓ∞ is weakly measurable if for each π ∈ ba, π f (a) is measurable. A weak∗ -(weakly)measurable map f (a) is said to be Gel’fand (Pettis) integrable if there exists an element  ξ ∈ ℓ∞ such that for each p ∈ ℓ1 (π ∈ ba), pξ= pf (a)dµ (π ξ = π f (a)dµ). The vector ξ is denoted by f (a)dµ and called Gel’fand (Pettis) integral of f . Fact 14 (Diestel and Uhl, 1977, pp. 53–4). If f : A → ℓ∞ is weak∗ (weakly) measurable and pf (a)(π f (a)) is integrable function for all p ∈ ℓ1 (π ∈ ba), then f is Gel’fand (Pettis) integrable. Fact 15. Let K be a compact metric and φ : K → ℓ∞ be weak∗ continuous map, and (ν α ) a net of Borel probability measures on K  with ν α → ν . Then it follows that K φ(t )dν α → K φ(t )dν in the weak∗ topology. Proof. Let q ∈ ℓ1 . Then qφ(t ) is a continuous function on K . Since ν α → ν in the weak∗ topology of probability measures, we have



φ(t )dν α =

q K

hence



qφ(t )dν α → K

 K



qφ(t )dν = q K

φ(t )dν α →

 K



φ(t )dν, K

φ(t )dν in the weak∗ topology.



A map f : A → ℓ is called strongly measurable if there exists a sequence of simple functions {fn } such that limn→∞ ∥fn (a) − f (a)∥ = 0 a.e. on A. The next theorem is known as the Pettis’ measurability theorem. ∞

Fact 16 (Diestel and Uhl, 1977, p. 42). A map f : A → ℓ∞ is strongly measurable if and only if (i) f is weakly measurable and (ii) f is µalmost everywhere separable valued, that is, there exists E ⊂ A with µ(E ) = 0 such that f (A \ E ) is a norm separable subset of ℓ∞ . A strongly measurable function f : A → ℓ∞ is said to be Bochner integrable if there exists a sequence of simple functions  {fn } such that limn→∞ A ∥fn (a) − f (a)∥dµ = 0. In this case, we define for each measurable set E ∈ A the Bochner integral to be   f (a)dµ = limn→∞ E fn (a)dµ. It can be shown (Diestel and Uhl, E 1977, p. 45) that a strongly measurable function f : A → ℓ∞ is  Bochner integrable if and only if A ∥f (a)∥dµ < +∞. It is obvious that if f : A → ℓ∞ is Bochner integrable, then it is also Pettis integrable and Gel’fand integrable, and all three integrals coincide. References Aliprantis, C.D., Border, K., 1994. Infinite Dimensional Analysis. Springer-Verlag, Berlin, New York. Aumann, R.J., 1964. Markets with a continuum of traders. Econometrica 32, 39–50. Aumann, R.J., 1966. Existence of competitive equilibria in markets with a continuum of traders. Econometrica 34, 1–17. Bewley, T.F., 1970. Existence of equilibria with infinitely many commodities. Journal of Economic Theory 4, 514–540.

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