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A fuzzy core equivalence theorem
Journal of Mathematical Economics 34 Ž2000. 143–158 www.elsevier.comrlocaterjmateco
A fuzzy core equivalence theorem Mitsunori Noguchi
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Department of Economics, Faculty of Commerce, Meijo UniÕersity, Tengoya 1-501 Shiogmaguchi, Nagoya, Japan Received 7 December 1998; received in revised form 2 July 1999; accepted 4 August 1999
Abstract We attempt to establish the equivalence of the fuzzy core, the Edgeworth core, and the set of Walrasian equilibria in economies with a measure space of agents, which is not necessarily non-atomic. The commodity spaces which we consider here include ordered separable Banach spaces whose positive cone admit an interior point, separable Banach lattices whose positive cone may lack an interior point, and L` endowed with the Mackey topology. Preference relations are assumed to be convex except for the non-atomic case. q 2000 Elsevier Science S.A. All rights reserved. JEL classification: D51 Keywords: Core; Fuzzy core; Edgeworth conjecture; Equivalence theorem; Infinite dimensional commodity space; Measure space of agents
1. Introduction The main objective of the present paper is to establish the equivalence of the fuzzy core ŽAubin, 1979. and the set of Walrasian equilibria in economies with a measure space of agents, which is not necessarily non-atomic. The commodity spaces which we consider here include ordered separable Banach spaces whose positive cone admits an interior point, separable Banach lattices whose positive cone may lack an interior point, and L` endowed with the Mackey topology. For
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Fax: q81-52-833-4767 Žoffice., q81-5613-8-8514 Žhome.; e-mail: noguchi@ meijo-u.ac.jp
0304-4068r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 9 9 . 0 0 0 3 6 - 1
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those economies with a commodity space whose positive cone admits an interior point, we obtain the equivalence of the set of Edgeworth equilibria Žsee Aliprantis et al., 1987. and the set of fuzzy core. We first consider economies with a separable commodity space X whose positive cone Xq admits an interior point. We label the consumption possibility set of consumer t by X Ž t . and consumption plan chosen by consumer t by x Ž t .. Throughout this paper we assume that for each t g T, X Ž t . s Xq. Thus we obtain a constant consumption correspondence t ™ Xq and a selection t ™ x Ž t .. We require x to be Bochner integrable and allow x Ž t . f Xq for those consumers forming a set of measure zero. Let % t ; Xq= Xq be the preference relation of consumer t. We allow satiation, i.e., h g Xq:h % t j 4 may be B for some t g T and j g X . We assume that for each t g T, the set h g Xq:h % t j 4 is norm open in Xq for every j g Xq. We also assume that for each t g T, % t satisfies the usual local non-satiation condition at every non-satiation point. We generally assume that for each t g T, the set h g Xq:h % t j 4 is a convex subset of Xq for every j g Xq, but for an atomless measure space of agents, we can dispense the convexity assumption. For an atomless measure space of agents, we obtain the classical core–Walras equivalence theorem with slightly more generality. Our argument relies on a new core concept called ‘open core’ which appeared in Sun Ž1997. in an attempt to extend the results of Florenzano Ž1990. about non-emptiness of core to the case in which preferences are lower semicontinuous. Sun’s basic constructions allow generalization to our measure theoretic framework. The results in this section of the present article resemble those appearing in the first part of Rustichini and Yannelis Ž1991.. Our assumptions are partly stronger than theirs due to our weaker hypotheses on consumption possibility sets and preferences. We next consider economies with a commodity space which is a separable Banach lattice whose positive cone may lack an interior point. Our assumptions in this section are nearly identical to those which appeared in Rustichini and Yannelis Ž1991., and under these assumptions we obtain, without the non-atomicity assumption, the equivalence of the fuzzy core and the set of Walrasian equilibria. The lack of interior points of the positive cone forces us to assume the existence of an extremely desirable commodity introduced by Yannelis and Zame Ž1986.. Also due to the lack of interior points, we are presently unable to prove the equivalence of the Edgeworth core and the fuzzy core. We remark that the crucial separation argument in Rustichini and Yannelis Ž1991. can be modified so that the same claim holds without the non-atomicity assumption. We finally consider economies with commodity space L` with the Mackey topology t Ž L` , L1 .. Our arguments in this section are based on the observation made by Bewley Ž1973. that the Mackey topology induces a Polish topology on bounded subsets and hence the Mackey topology admits a stronger Polish topol-
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ogy, where both topologies give rise to the same Borel sets. At the point where we show that an element in the fuzzy core gives rise to a quasi-equilibrium, we appeal to the measurable selection theorem in order to extract a measurable selection which restricts to a Gel’fand integrable selection over some set of positive measure. For simplicity we assume that our measure space is the unit interval with Lebesgue measure as in Bewley Ž1973.. We follow Mertens Ž1991. for showing that the countably additive part of the linear functional obtained by the separation theorem possesses the right properties for being an equilibrium price system.
2. Separable commodity space whose positive cone admits an interior point 2.1. Definitions For a linear space X and a subset A ; X , co A denotes the convex hull of A. For a topological space X and a subset A ; X , cl A denotes the closure of A and X™Y a int A denotes the interior of A. Let X , Y be any sets and f :X correspondence. Gf denotes the graph of f , i.e., Gf s Ž x, y . g X = Y : y g f Ž x .4 . Let X and Y be paired linear spaces. For a Banach space X , X U denotes the dual of X , and 5 P 5 x the norm of X . Let ŽT,I . be a measurable space. A correspondence f :T ™ Y is said to admit a measurable graph if Gf g I m BŽ Y ., where BŽ Y . denotes the Borel sigma algebra on Y . For sigma algebras t 1 and t 2 , t 1 m t 2 denotes the product sigma algebra. For measurable spaces ŽT,I ., Ž S, G ., and a function f :T ™ S, f is said to be Ž I, G .-measurable whenever it is measurable with respect to the sigma algebras I and G . Let ŽT,I, m . be a finite X . denotes the space of equivalence measure space and X a Banach space. L1Ž m ,X classes of X-valued Bochner integrable functions f :T ™ X normed by 5 f 5 s HT 5 f Ž t .5 x wfor details see Diestel and Uhl Ž1977.x. For a Banach space X and its dual X U , ² P ,P : denotes the natural paring. For a Banach space X and a X .: f Ž t . g f Ž t . m-almost correspondence f :T ™ X , define L1Ž m , f . s f g L1Ž m ,X everywhere Ža.e..4 . Let R denote the set of real numbers and Q the set of rational numbers. A correspondence f :T ™ X is said to be integrally bounded if there exists a function h g L1Ž m ,R. such that sup5 j 5 x : j g f Ž t .4 F hŽ t . m-a.e. 2.2. Theorems Our commodity space is an ordered separable Banach space X whose positive cone Xq is proper and admits an interior point. An economy E is a tuple wŽT,I, m ., X, P,e x, where 1. ŽT,I, m . is a measure space of consumers; 2. X:T ™ X is a consumption correspondence; 3. % t ; X Ž t . = X Ž t . is the preference relation of consumer t; 4. e:T ™ X is the initial endowment, where eŽ t . g X Ž t . for all t g T.
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U Let _ denote the set theoretic subtraction. Let Xq be the positive cone U U induced by Xq on X . A nonzero element p g Xq _ Ž0. is said to be a price system for E . Denote the budget set of consumer t at price system p by B Ž t, p . s j g X Ž t .:² p, j : F ² p, e Ž t .:4 . We also define B˚Ž t, p . s j g X Ž t .:² p, j : - ² p,eŽ t .:4. U Ž x, p . g L1Ž m , X . = Xq _ Ž0. is said to be a quasi-equilibrium if 1. for almost all t g T, x Ž t . g B Ž t, p ., h g X Ž t .:h % t x Ž t .4 l B˚Ž t, p . s B, and Xq. 2. HT x Ž t . y HT eŽ t . g yX An equilibrium is a quasi-equilibrium Ž x, p . such that for almost all t g T, h g X Ž t .:h % t x Ž t .4 l B Ž t, p . s B. Define A s simple I-measurable functions t :T ™ w0, 1x such that m Žt ) 04. ) 04 . Note that A can be viewed as a family of ‘generalized’ coalitions of consumers, in the following sense: t Ž t . represents the rate of participation of t to the coalition t ) 04 g I wsee Aubin Ž1979.x: Following Aubin Ž1979., we call t g A a fuzzy coalition. For each positive integer r, define A r s t g A:rt Ž t . g 0, PPP ,r 4 , ; t g T 4 . A r is the family of all coalitions of the r-fold replica E r of economy E wsee Florenzano Ž1990.x, where the same type of consumers are considered identical. We also define A Q s t g A:t Ž t . g w0,1x l Q, ; t g T 4 . ˆ s xg In what follows we adopt the following notations: X s L1Ž m , X .; X ˆ t s x t g X t :HT t Ž x t y e . g Xq 4 ; for t g A, X t s xt ) 04 X, X X:HT Ž x y e . g yX ˆ topen s x t g X t :HT t Ž x t y e . g yint Xq 4; for t g A, x g X, P t Ž x . s Xq 4 , X yX t t x g X : x t Ž t . % t x Ž t . m-a.e. in t ) 044 .
Remark 1. In the above definition of quasi-equilibrium, we have required Ž3. Xq instead of the standard market clearing condition Ž3.X HT x Ž t . y HT eŽ t . g yX HT x Ž t . s HT eŽ t .. Our proof works as well with Ž3.X and an alternative definition ˆ s x g X:HT Ž x y e . s 04 instead of the one given in the preceding paragraph. X The proof only needs a very slight modification.
ˆ t defined above may seem ‘too large’ compared to the Remark 2. The set X corresponding object in the usual definition of core since it allows excess supply of commodities in the coalition t ) 04 . If we go along with the widely accepted ˆ t should be x t g X t :HT t Ž x t y e . s 04 version of such set, the definition of X instead of the one given above. However, under the standard assumptions of core–Walras equivalence results such as in Aumann Ž1964., Rustichini and Yannelis Ž1991., these two definitions coincide; for example, if X Ž t . s Xq for all t g T, and the individual preference relation for t is transitive and monotone for all t g T, it can be easily shown that the above two definitions coincide. ˆ is called an attainable allocation. A fuzzy coalition t g A An element x g X ˆ if P t Ž x . l Xˆ t / B, and is said to open block x g Xˆ if is said to block x g X tŽ . t ˆ w P x l X open / B cf. Sun Ž1997.x. Following Aubin Ž1979., we define the fuzzy core of E , C f Ž E ., as the set of all attainable allocations which cannot be blocked f Ž E ., by any fuzzy coalition. Likewise, we define the open fuzzy core of E , Copen
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as the set of all attainable allocations which cannot be open blocked by any fuzzy coalition. Note that the core is the set C Ž E . of all attainable allocations which cannot be blocked by any fuzzy coalition in A 1. Let C r Ž E . be the set of all attainable allocations which cannot be blocked by any fuzzy coalition in A r. Following Aliprantis et al. Ž1987., x g Xˆ is said to be an Edgeworth equilibrium if x g l r G 1C t Ž E .. Let C e Ž E . be the set of all attainable allocations which cannot be blocked by any fuzzy coalition in A Q . Observe that C e Ž E . s r e Ž . Ž E ., Copen l r G 1C r Ž E .. We can also define Copen E in the obvious manner. We e Ž . fŽ . eŽ . f e Ž . eŽ . Ž E . ; Copen clearly have C E ; Copen E , C E ; C E , Copen E . Let W Ž E . Ž Wquasi Ž E .. be the set of all attainable allocations which can be an equilibrium U Žquasi-equilibrium. with a suitable choice of p g Xq _ Ž0.. In the sequel, we show f f that W Ž E . ; C Ž E . ŽTheorem 1., CopenŽ E . ; Wquasi Ž E . ŽTheorem 2., Wquasi Ž E . e Ž . f Ž E . ŽTheorem 10.. Consequently, we s W Ž E . ŽTheorem 9., and Copen E ; Copen f e e f Ž E . ; Wquasi Ž E . s W Ž E .. In obtain W Ž E . ; C Ž E . ; C Ž E . ; CopenŽ E . s Copen particular, we obtain the equivalence theorem W Ž E . s C e Ž E .. We now state the following set of assumptions on economy E . ŽA.1. ŽT,I, m . is a complete finite measure space. ŽA.2. X Ž t . s Xq for all t g T. ŽA.3. Ža. For each t g T, the set h g Xq:h % t j 4 is norm open in Xq for every j g Xq. Žb. For each t g T, the set h g Xq:h % t j 4 is a convex subset of Xq for every j g Xq. Žc. For each t g T, if j g Xq is not a satiation point for % t , then j g cl h g Xq:h % t j 4 . Žd. For each t g T, either h g Xq:h % t j 4 / B or j y eŽ t . g Xq holds for every j g Xq. Že. The set Ž t, h , j . g T = Xq= Xq:h % t j 4 belongs to I m BŽ Xq . m BŽ Xq .. ŽA.4. e is Bochner integrable. ŽA.5. For each t g T, eŽ t . is strictly positive Ži.e., ² p,eŽ t .: ) 0 for all U p g Xq _ Ž0... Under ŽA.1. – ŽA.5., we prove the following theorems: Theorem 1. If Ž x, p . is an equilibrium for E , then x g C f Ž E .. Proof of Theorem 1. Assume the contrary, that x f C f Ž E .. By definition, there ˆ t / B. Let x t g P t Ž x . l Xˆ t . Then exists t g A such that P t Ž x . l X x t Ž t . %x Ž t .
Ž 1.1 .
t
m-a.e. in t ) 04 , and
HTt Ž t . x
t
Ž t . FH t Ž t . e Ž t . .
Ž 1.2 .
T
Eq. Ž1.1. implies that x t Ž t . f B Ž t, p . m-a.e. in t ) 04 , and hence
HTt Ž t . ² p, x
t
Ž t . : ) H t Ž t . ² p,e Ž t . : . T
Ž 1.3 .
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By Theorem 6 in Diestel and Uhl Ž1977, p. 47., Eq. Ž1.2. gives
HTt Ž t . ² p, x
t
Ž t . : F H t Ž t . ² p,e Ž t . : .
Ž 1.4 .
T
Combining Eqs. Ž1.3. and Ž1.4., we reach a contradiction.
I
U f Ž E ., then there exists p g Xq Theorem 2. If x g Copen _ Ž0. such that Ž x, p . is a quasi-equilibrium of E . f Ž E .. Define a correspondence P:T ™ Xq by Proof of Theorem 2. Let x g Copen P Ž t . s h g Xq:h % t x Ž t .4 and let Tx s t g T : P Ž t . s B4 . Note that since ŽT,I, m . is complete, X Suslin, and GP g I m BŽ X . wŽA.3. Že.x, by Theorem III.23 in Castaing and Valadier Ž1977, p. 75., Tx s w Pg T Ž GP .x c g I, where Pg T :T = X ™ T is a projection map. We first assume that m ŽT _ Tx . s 0. We have
HTx FHTe.
Ž 2.1 .
U Let p g Xq _ Ž0.. We show that Ž x, p . is an equilibrium. Since P Ž t . s B m-a.e., we trivially have P Ž t . l B Ž t, p . s B m-a.e. By ŽA.3. Žd.,
² p, x Ž t . : G ² p, e Ž t . :
Ž 2.2 .
m-a.e. By Theorem 6 in Diestel and Uhl Ž1977, p. 47. applied to Eq. Ž2.1., we obtain
HT² p, x Ž t . : FHT² p, e Ž t . : .
Ž 2.3 .
Eqs. Ž2.2. and Ž2.3. imply that ² p, x Ž t . : s ² p, e Ž t . :
Ž 2.4 .
m-a.e. In particular, x Ž t . g B Ž t, p . m-a.e. Thus Ž x, p . is an equilibrium. We next assume that m ŽT _ Tx . ) 0. For the sake of simplicity we write T X s T _ Tx , IX s I < T X , and mX s m < I X . Define Ax s simple IX-measurable functions t :T X ™ w0,1x such that mX Žt ) 04. ) 04 and define AxU s t g Ax :t Ž t . g Ž0,1x, ; t g T X 4 / B. For each t g Ax , define Gt1 s HT X t Ž t . x Ž t . g X : x g L1Ž mX , P < T X .4 . Now we define Gt s Gt1 y HT X t Ž t . eŽ t . q int Xq. Lemma 3. Gt1 / B. See Appendix A for the proof. Hence, Gt / B for all t g AxU . Define G s j t g A Ux Gt . Lemma 4. G is conÕex. See Appendix A for the proof.
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Lemma 5. 0 f G. See Appendix A for the proof. Since int G / B, the separating hyperplane theorem implies that there exists p g X U _ Ž0. such that p P G G 0. Define H s j t g A xGt . Lemma 6. p P H G 0. See Appendix A for the proof. U Lemma 7. p g Xq _ Ž0.
See Appendix A for the proof. The above line of argument based on the construction of the sets G, H is attributed to Sun Ž1997.. Now we claim that Ž x, p . is a quasi-equilibrium. Let B g IX and let t s x B g Ax . Repeating the argument in the proof of Lemma 3 with U the ball of radius 1rn in X , we obtain x n g L1Ž mX , P < T X . such that 5 x Ž t . y x nŽ t .5 F Ž1rn. for all t g T X . By Lemma 6, we have HB ² p, x nŽ t . y eŽ t . G 0 for all n, and consequently, HB ² p, x Ž t . y eŽ t .: G 0. Since B g IX is arbitrary, considering ŽA.3. Žd., we obtain ² p, x Ž t . y eŽ t .: G 0 m-a.e. in T. Combining this with HT x F Ht e, we obtain x Ž t . g B Ž t, p . m-a.e. Let c 1Ž t . s P Ž t . l B˚Ž t, p .. Lemma 8. Gc 1 g I m BŽ X . and dom c 1 g I. See Appendix A for the proof. By Lemma 8 and Theorem lll.22 in Castaing and Valadier Ž1977, p. 74., we see that c 1 admits an measurable selection x:dom c 1 ™ X . If m Ž dom c 1 . ) 0, we can find a IX-measurable set B ; dom c 1 , m Ž B . ) 0, such that x is integrable over B. Let t s x B so that t g Ax . Now we have by Lemma 6 that p
HT t Ž x y e . X
G 0,
Ž 2.5 .
where x is now extended as an integrable selection to the entire T X . Eq. Ž2.5. can be written as
HB² p, x Ž t . : GHB² p, e Ž t . : .
Ž 2.6 .
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On the other hand, since x Ž t . g B˚Ž t, p . for all t g dom c 1 , we have
HB² p, x Ž t . : -HB² p, e Ž t . : , which contradicts with Eq. Ž2.6.. Thus, m Ž dom c 1 . s 0, i.e., P Ž t . l B˚Ž t, p . s B m-a.e. Theorem 9. EÕery quasi-equilibrium Ž x, p . of E is an equilibrium of E . Proof of Theorem 9. Let EB Ž t, p . s B Ž t, p . _ B˚Ž t, p .. Define c 3 Ž t . s P Ž t . l EB Ž t, p .. By ŽA.5., we have B˚Ž t, p . / B for all t g T. Since P Ž t . is norm open, we have dom c 3 ; dom c 1. By assumption, we have m Ž dom c 1 . s 0, and hence m Ž dom c 3 . s 0, i.e., P Ž t . l EB Ž t, p . s B m-a.e. I The following theorem for the case of E with finitely many consumers is attributed to Sun ŽLemma 2, 1997., and our generalization to E with a measure space of consumers is almost verbatim. e Ž . f Ž E .. Theorem 10. Copen E ; Copen e Ž . f Ž E .. By definition, Proof of Theorem 10. Let x g Copen E . Suppose x f Copen tŽ . t t ˆ ˆ topen . Then there exists t g A such that P x l X open / B. Let x g P t Ž x . g X t Ž . 5 5 4 HT t x y e g yint Xq. Let Be s j g X : j X - e such that
Be q t Ž x t y e . ; yint Xq .
HT
Ž 10.1 .
Choose k g R such that k e ) HT xt ) 04 5 x t y e 5 X . Then
HT Ž 1 q w kt x y kt . x
t ) 04
Ž x t y e . g kBe ,
Ž 10.2 .
where w kt xŽ t . denotes the largest integer less than kt Ž t .. Multiplying Eq. Ž10.1. by k, we obtain kBe q kt Ž x t y e . ; yint Xq .
HT
Ž 10.3 .
Combining Eqs. Ž10.2. and Ž10.3., we obtain
HT Ž 1 q w kt x . x
t ) 04
Ž x t y e . g yint Xq .
Let kU s max t Ž t .) 0 Ž1 q w kt Ž t .x.. Define t g A Q by t Ž t . s kUy1 Ž1 q w kt Ž t .x. e Ž . ˆ topen , and x f Copen for t Ž t . ) 0, and t Ž t . s 0 otherwise. Then x t g P t Ž x . l X E . But this is a contradiction. I
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2.3. Atomless measure space of agents In this section we assume the assumptions ŽA.1. – ŽA.5. except for ŽA.3. Žb.. Theorem 11. The core–Walras equiÕalence theorem W Ž E . s C Ž E . holds. Proof of Theorem 11. Let x g C Ž E .. Define Tx as in the proof of Theorem 2. If m ŽT _ Tx . s 0, then x g W Ž E . as in the proof of Theorem 2. For m ŽT _ Tx . ) 0, define Ax s x B : B g IX with mX Ž B . ) 04 . Note that AUx s x T X 4 , and G s Gx T X s Gx1T X y HT X e q int Xq. Now by Theorem 1 in Khan Ž1985., cl Gx1T X is convex. Since cl Gx1T X y HT X e ; cl Ž Gx1T X y HT X e ., we have cl Gx1T X y HT X e s cl Ž Gx1T X y HT X e .. Note that cl Ž Gx1T X y HT X e . is convex, and 0 g G iff 0 g cl Ž Gx1T X y HT X e . q int Xq. If 0 g G, ˆ topen / B, and hence then by letting t s x T X in the proof of Lemma 5, P t Ž x . l X tŽ . t ˆ P x l X / B, and this is a contradiction. Thus we can find p g X U _ Ž0. such that p P G G 0. We define the set H as before, and can show that p P H G 0 and as U in Lemma 6, p g Xq _ Ž0.. Observe that the argument after Lemma 8 is still valid since t s x d o m c 1 g A 1. Hence x g Wquasi Ž E .. By Theorem 9, x g W Ž E .. I Remark 3. We remark that production can be incorporated into our framework with minor modifications. Let Ž S, G , p . be a measure space of producers and Y:S ™ X the production correspondence. Let u :T = S ™ R denote the share. Define Y s HS y Ž s . and Yt s HT t Ž t .HS u Ž t, s . y Ž t, s . g X : y g L1Ž m = p ,Y ., where Y is viewed as a correspondence defined on T = S4 . Now we modify the ˆ ˆ t as Xˆ s x g X:HT Ž x y e . g Y y Xq 4,Xˆ t s x t g X t :HT Ž x t y definitions of X,X t e . g Y y Xq 4 , respectively. The definition of Yt above can be justified as follows: Let E be a finite economy with T consumers and S producers. Then it is natural to define Yt s ÝTis1t i ÝSjs1 u i j yj g X :Yj g Yj , j s 1, PPP ,S4 , where t g w0, 1xT _ Ž0., u i j the share of consumer i in the profit of producer j, and Yj the production set of producer j. We assume that each Yj is convex. For each i, let yi j g Yj , j s 1, PPP ,S. Then y ' ÝTis1t i ÝSjs1 u i j yi j s ÝSjs1 ÝTis1t i u i j , yi j s Ý jX ÝTis1t i u i jX , yi jX , where jX runs through the indices such that ÝTis1t i u i j ) 0. Note that we can write y s Ý jX ŽÝ TiXs 1t iX u i jX . Ý Tis 1 u˜i jX y i jX , where u˜i jX s ŽÝTiXs1t iX u iX jX .y1t i u i jX G 0. Observe that for each jX , ÝTis1 u˜i jX s 1. Then, ÝTis1 u˜i jX yi jX s y˜ jX g co YjX s YjX . Hence, y s ÝTis1t i ÝSjs1 u i j Yj , where y j s Y˜j if ÝTis1t i u i j ) 0, and yj s any element of Yj if ÝTis1t i u i j s 0. Thus, we obtain an equivalent definition Yt s ÝTis1t i ÝSjs1 u i j yi j g X : for each i yi j g Yj , j s 1, PPP ,S4 . Under the standard assumptions on production wfor example, Ž S, G , p . is a complete finite measure space; Y:S ™ X is a convex valued correspondence such that G Y g G m BŽ X .; u is a I m G-measurable function such that for every s g S, HT u Ž t, s . s 1 and for every Ž t, s . g T = S, 0 F u Ž t, s . F 1; and 0 g Y Ž s . for almost all s g S x, it is merely a routine matter to check that our previous argument goes through. For the production economy version of Theorem 11, one needs an additional compactness assumption on L1Ž mX = p ,Y . for every mX s m < I X , where IX s I l T X for some T X g I. The details can be found in Noguchi Ž1998..
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2.4. Separable commodity space whose positiÕe cone may haÕe an empty interior In this section we consider economies E whose commodity space is a separable Banach lattice X . We allow Xq to have an empty norm interior and ŽT, I, m . to be atomic. With the non-atomicity assumption, Rustichini and Yannelis Ž1991. obtained substantial results. For the rest of the argument, we assume that % t is represented by a utility function u:G X ™ R. Our goal in this section is to show the equivalence W Ž E . s C f Ž E . by modifying the proof provided by Rustichini and Yannelis Ž1991. for core–Walras equivalence results for non-atomic economies. Definition. Õ g Xq_Ž0. is said to be an extremely desirable commodity if there exists an open neighborhood U at the origin such that for each j g Xq and each t g T, it is true that Ž j q a Õ y h . is strictly preferred to j whenever a ) 0, j q a Õ y h g Xq and h g a U wthe notion of an extremely desirable commodity was introduced by Yannelis and Zame Ž1986.x. In our situation we can clearly assume that U is an ideal. Define an open cone C by C s j a ) 0 Ž Õ q U . as in Rustichini and Yannelis Ž1991.. Then every element h g Ž C q j . l Xq is strictly preferred to j . We now state our assumptions. ŽA.1. ŽT, I, m . is a complete finite measure space. ŽA.2. X Ž t . s Xq for all t g T. ŽA.3. u:T = Xq™ R is a function such that: Ža. for each t g T, uŽ t,P .:X Xq™ R is a norm-continuous, quasi-concave, and monotonic Ži.e., for every j g Xq and Õ g Xq_Ž0., we have uŽ t, j q Õ . ) uŽ t, j ..; Žb. for each j g Xq, uŽP, j .:T ™ R is I-measurable. ŽA.4. e is Bochner integrable. ŽA.5. eŽ t . is strictly positive for all t g T. ŽA.6. X is a separable Banach lattice. ŽA.7. ŽExtremely desirable commodity. There exists an extremely desirable commodity Õ g Xq_Ž0.. ŽA.8. ŽAdditivity condition. wthis was found in Rustichini and Yannelis Ž1991.x Let d i , i s 1, PPP ,n be positive numbers with Ý nis1 d i s 1. In ŽA.7. we can arrange U so that it is an ideal of X Žas a Riesz space., and if j i g Xq, j i f d i U, i s 1, PPP ,n, then Ý nis1 j i f U. Theorem 12. Under ŽA.1. – ŽA.8. we haÕe the equiÕalence W Ž E . s C f Ž E .. Proof of Theorem 12. Since Theorem 1 and Theorem 9 in the previous section are still valid, we only need to show that the inclusion C f Ž E . ; W Ž E . holds. To this
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end, we proceed as in the proof of Theorem 2. Let x g C f Ž E . and note that Tx s B. Define Ax and AxU as before noting that T X s T now. We use C instead of int Xq in the definition of Gt . Observe that Lemma 3 trivially follows and no changes are necessary in the proof of Lemma 4. Claim. G l y C s B. Assume the above claim holds Žthe justification will be given in the end.. Then by the separation argument as before, we obtain Lemma 6. For Lemma 7, let j g Xq and let Õ g C. Choose t s x T g Ax . Since x q jx T g L1Ž m , P ., we have HT ² p, x q j y e : q ² p,Õ : G 0. Since C is a cone, we obtain ² p,j : G 0, and U hence p g Xq _ Ž0.. Lemma 8 is valid with no change, and the rest of the argument follows thereafter. Justification of Claim. We must show that Gt l y C s B for all t g AxU . Following closely the line of argument on pp. 321–323 in Rustichini and Yannelis Ž1991., for a y g L1Ž m , P ., we construct a sequence of functions y k g L1Ž m , P . such that y k ™ y in the norm topology, and HT t y k y HT t e f yC. Note that in our case, T s S holds in the second paragraph in p. 321. They choose a sequence of k y kx k X . to y. They simple functions y k s Ý mis1 which converges in L1Ž m ,X i Ti k k arrange T1 , PPP ,Tm k so that Tik are mutually disjoint, cover T, and possess the same positiÕe measure j ) 0. This last requirement may not be satisfied in our case since our ŽT,I,u. may have atoms. Thus we can only expect to have m ŽTik . s j i ) 0 instead and it turns out that this is all we need to complete their proof. In other words, the apparent dependency on the non-atomicity assumption in their proof of Claim 6.1 can be removed. Our assertion, which corresponds to their Claim 6.1 writes HT t y k y HT t e f yC. Observe that since t is a simple function, we can assume that t is constant on each Tik by taking a finer partition if necessary. Thus their expression on top of p. 322 may be replaced by Ý mis1t i yi j i y Ý mis1t i e i j i g ya Ž Õ q U ., where t i ) 0 Žrecall that t g AxU . is the Ž HT k e .. Note that Ý mis1t i e i j i s Ý mis1 HT t i x T k e s value of t on Tik and e i s jy1 i i i HT t e. Instead of their Ž6.8. on p. 322, we have Ý mis1 t i yi j i q w y u s Ý mis1t i e i j i , where w s a Õ, u g a U. Basically, the only notational difference here is that we have t i yi j i , t i e i j i in place of their yi , e i , respectively. Their expression Ž6.10. now writes Ý mis1 u i s u, u i F t i yi j i q u i w for all i, and the definition of y˜i Ž u . y1 y1 y1 writes y˜i Ž u . s yi q ty1 ˜i Ž u U . having the i j i u i w y t i j i u i . Then we obtain y U property that y˜ i Ž u . is strictly preferred to x Ž t . m-a.e. in Ti Žsuperscript k is omitted. for all i. As in p. 323, define y˜ s Ý mis1 y˜i Ž u U . x T i . Note that we have HT t y˜ s Ý mis1 y˜i Ž u U . HT tx T i s Ý mis1 y˜i Ž u U .t i j i s HT t e. This shows that y˜ g P t Ž x . ˆ t in our notation, and hence we reach a contradiction as in their proof on p. lX 323. Being helped by the fact that t i ) 0, j i ) 0 for all i, the rest of the argument on p. 323 goes through with almost no alteration. We only remark that in the part showing Ý mjs1 d j s 0, their arj becomes a in our case. I
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2.5. Economies with commodity-price paring (L` , L1 ) Let Ž M, M , Õ . be a totally s-finite countably generated measure space ŽBewley, 1973.. We abbreviate L` Ž M, M , Õ ., L1 Ž M, M , Õ . as L` , L1 , respectively. In this section we treat economies with commodity space L` and price systems lying in L1 , where L` is given the Mackey topology t Ž L` , L1 .. In the sequel we simply write Lt` for L` with t Ž L` , L1 ., and Lw`U for L` with weakU topology. For a function f :T ™ L` such that ² a, f ŽP.: g L1Ž m ,R. for all a g L1 , HT f Ž t . denotes the Gel’fand integral of f wsee Diestel and Uhl Ž1977, p. 53.x. For Gel’fand integrable functions f, g, we introduce an equivalence relation ; by f ; g if and only if ² a, f ŽP.: s ² a, g ŽP.: m-a.e., for each a g L1. Define LG1 Ž m , L` . to be the set of equivalence classes under ; of Gel’fand integrable functions on T. Since L1 is a separable Banach space, if f ; g, then f s g m-a.e. Therefore, for a correspondence f :T ™ L` and f g LG1 Ž m , L` ., it makes sense to talk about f Ž t . g f Ž t . m-a.e. For a correspondence f :T ™ L` , define LG1 Ž m , f . s f g LG1 Ž m , L` .: f Ž t . g f Ž t . m-a.e.4 . We write L`,q for the standard positive cone of L` . A price system p is required to lie in L1,q_Ž0., where L1,q is the standard positive cone of L1. We state the following two sets of assumptions on economy E . ŽA.1. ŽT, I, m . is the unit interval with Lebesgue measure. ŽA.2. X Ž t . s L`,q for all t g T. ŽA.3. Ža. For each t g T, the set h g L`,q:h % t j 4 is a t Ž L` , L1 .-open subset of L`,q for every j g L`,q. Žb. For each t g T, the set h g L`,q:h % t j 4 is a convex subset of L`,q for every j g L`,q. Žc. For each t g T, if j g L`,q is not a satiation point for % t , then j q Õ % t j for every Õ g L`,q_Ž0.. Žd. For each t g T, either h g L`,q:h % t j 4 / B or j y eŽ t . g L`,q holds for every j g L`,q. Že. The set Ž t,h , j . g T = L`,q= L`,q:h % t j 4 belongs to I m BŽ Lt`,q . m BŽ Lt`,q .. ŽA.4. e is Gel’fand integrable. ŽA.5. For each t g T, eŽ t . g int L`,q, where int is taken with respect to the norm topology. There exists a positive number l ) 0 such that essinf m g M eŽ t .Ž m. G l for all t g T. Remark 4. We assume ŽA.1. above since for a measurable function with respect to Lt` , we may not be able to shrink the domain in order to assure Gel’fand integrability. This condition is directly taken from Bewley Ž1973, p. 385. so that one can appeal to Lusin’s theorem for the integrability. Theorem 13. Under either set of assumptions ŽA.1. – ŽA.5. we haÕe the equiÕalence W Ž E . s C e Ž E . s C f Ž E .. Proof of Theorem 13. We follow the steps in the proof provided in the earliest section for the separable commodity space whose positive cone admits an interior
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point. Theorem 1 holds in the present situation. We simply replace Bochner integral by Gel’fand integral and apply the definition of Gel’fand integral in place of Theorem 6 in Diestel and Uhl Ž1977, p. 47.. Since Lt` is Suslin wthe author is indebted to the paper by Mertens Ž1991. for this observationx, Theorem III.23 in Castaing and Valadier Ž1977, p. 75. is still valid and the previous argument for Theorem 2 holds up to Eq. Ž2.4.. Lemma 3 holds as a simple consequence of ŽA.3. Žc.. Lemma 4, Lemma 5 and Lemma 6 need no modification. Note that p lies in the dual space of L` , but not in L1. In the proof of Lemma 7, e Ž² p,HT X x y e :. still makes sense and obtain p g baq_Ž0., where baq denotes the standard positive cone of the dual space of L` . We must show that p g L1. To this end we follow the steps in the proof provided in Mertens Ž1991.. Let p s pc q pf be the decomposition in Yosida and Hewitt Ž1956., where pc is the countably additive part and pf is the purely finitely additive part. We choose an increasing Bi g M such that D `is1 Bi s M and pf Ž Bi . s 0. For a Gel’fand integrable selection x g LG1 Ž mX , P < T X ., define x i s x x B i so that for almost all t g T, x i Ž t . g P Ž t . for i G iŽ t . for some iŽ t .. Let E ; T X be an arbitrary measurable subset with m Ž E . ) 0. Following Mertens Ž1991., define En s t g E:;i G n, x i Ž t . g P Ž t .4 . ) Note that x i is measurable with respect to BŽ Lw` . s BŽ Lt` . wThis follows from the well-known fact that for any Hausdorff topology H which admits a stronger Polish topology P, we have BŽ H . s BŽ P ., and t Ž L` , L1 . under our hypothesis admits such P as a topological sum of the Polish topologies induced on disks by t Ž L` , L1 ., which is Polish ŽBewley, 1973.x. Since Lt` is Suslin and Ž E,I < E , m < E . is complete, by the projection theorem, we see that En s l i G n PrE Ž Gx i l Gp . g IX . We obtain an increasing sequence Ei which covers E and proceed as in Mertens Ž1991.. Let t s x E i g Ax so that pw HT X t Ž x y e .x G 0, which can be rewritten as pw x B i HE iŽ x y e .x G pwŽ1 y x B i .HE i e x.. Since p G pf and Ž1 y x B i .HE i e g L`,q, the right hand side of the previous inequality equals pf Ž HE i e . G l pf Ž M . m Ž Ei . G 0. Recall that we may assume m Ž Ei . / 0. As in Mertens Ž1992. we obtain by the monotone convergence theorem that pc w HE Ž x y e .x G l pf Ž M . m Ž E . G 0. In particular, pc g L1 _ Ž0.. In our case, pf may not be zero. Let Õ g L`,q_Ž0. and define x S s x q sÕ for s g Ž0,1x. x S is clearly Gel’fand integrable and by ŽA.3. Žc., x S is a measurable selection of P < T X , and hence pc w HE Ž x S y e .x G 0. By letting s ™ 0, we obtain ² pc , x Ž t . y eŽ t .: G 0 m-a.e. in T X , and combining this with ŽA.3. Žd., we obtain ² pc , x Ž t . y eŽ t .: s 0 m-a.e. in all of T. In particular, x Ž t . g B Ž t, pc . m-a.e. in T. For Lemma 8, note that g Ž t, j . s ² pc , j y eŽ t .: is I-measurable in t, and BŽ L`P .-measurable in j , where L`P is L` endowed with a Polish topology P which is stronger than t Ž L` , L1 .. Then g is jointly measurable in I and BŽ Lt` .. Some care must be taken for the argument after Lemma 8 since unlike in the case of Bochner integral, we may not be able to shrink the domain of a measurable function in order to assure integrability. We appeal to Lusin’s theorem ŽBewley, 1973. so that the measurable function Žwith respect to Lt` . will be bounded on a compact subset of positive measure. We obtain pw HB Ž x y e .x G 0 as before and
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deduce that pc w HB Ž x y e .x G 0 using the same trick as above. Hence, we obtain the expression Ž2.6. with p replaced by pc . We have shown that Ž x, pc . is a quasi-equilibrium. The proof of Theorem 9 is valid with pc instead of p. The proof of Theorem 10 needs some modification. To be more specific, the way to choose k for Eq. Ž10.2. must be changed as follows: N since t is a simple function, we can write t s Ý is1 t i x T i , t i ) 0, and Ti l Tj s N t BŽ i / j .. Choose k g R such that Ý is1 5 HT x T iŽ x y e .5 L` - k e . Then we have N 5 HT Ž1 q w kt x y kt . xt ) 0 Ž x t y e .5 L` s 5Ý is1 HT iŽ1 q w kt i x y kt i .Ž x t y e .5 L` F N <Ž t N Ý is1 1 q w kt i x y kt i .< 5Ž HT i x y e .5 L` F Ý is1 5Ž HT i x t y e .5 L` , and obtain Eq. Ž10.2.. This completes the proof. I
Acknowledgements The author is indebted to an anonymous referee of JME for valuable suggestions.
Appendix A Proof of Lemma 3. We must show that L1Ž mX , P < T X . / B. Recall from ŽA.3. Žc. that x Ž t . g cl P Ž t . for t g T X . Let U be the unit ball in X and define UŽ t . s x Ž t . q U. Define V Ž t . s UŽ t . l P Ž t . on T X . Observe that by Lemma III.14 in Castaing and Valadier Ž1977, p. 70., we have GU s Ž t, j . g T X = X : 5 x Ž t . y j 5 X F 14 g IX m BŽ X .. By ŽA.3. Že., we have GP g I m BŽ X .. Note that by Theorem E in Holmos Ž1950, p. 25., we obtain GP < T X s GP l ŽT X = X . g IX m BŽ X .. Consequently, G V g IX m BŽ X .. Since ŽT X , IX , mX . is complete, by Theorem III.22 in Castaing and Valadier Ž1977, p. 25., we obtain a measurable selection x of V. Since 5 x Ž t . y x Ž t .5 X F 1 for all t g T X , x clearly lies in L1Ž mX , P < T X .. I Proof of Lemma 4. Let g 1 g Gt11, g 2 g Gt12 , where t 1 , t 2 g AxU . For s g w0,1x, let t s st 1 q Ž1 y s .t 2 g AxU . Since g 1 s HT X t 1 j 1 , g 2 s HT X t 2 j 2 for some j 1 , j 2 g L1Ž mX , P < T X ., we have sg 1 q Ž1 y s . g 2 s HT X tj , where j s ty1 st 1 j 1 q ty1 Ž1 y s .t 2 j 2 . Since P Ž t . is convex, sg 1 q Ž1 y s . g 2 g Gt1. I Proof of Lemma 5. Assume the contrary that 0 g G. Then we have X
HT t Ž x y e . q Õ s 0 X
for some t X g AxU , x g L1Ž mX , P < T X ., and Õ g int Xq. Let t be the extension of t X by zero to the entire T so that t g A. Let x t be the extension of x by zero to the
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entire T. Note that for any x˜ g X, which agrees with x on T X , we have x t s xt ) 04 x˜ g X t . Thus we have HT t Ž x t y e . g int Xq, and consequently, ˆ topen / B. This is a contradiction. xt g P t Ž x . l X I Proof of Lemma 6. Let h g H and g g G. Then sg q Ž1 y s . h g G for all s g Ž0,1x. Thus s² p, g : q Ž1 y s .² p,h: G 0 for all s g Ž0,1x. Consequently, ² p,h: G 0. I Proof of Lemma 7. Let e ) 0. Let Õ g int Xq. Let t s ex T X g Ax . For any x g L1Ž mX , P < T X ., we have e Ž HT X ² p, x y e :. q ² p,Õ : G 0. Consequently, ² p,Õ G 0. I Proof of Lemma 8. Define a function g:T = X ™ R by g Ž t, j . s ² p, j : y ² p,e,Ž t .:. Since X is separable, Lemma III.14 in Castaing and Valadier Ž1977, p. 70. implies that g is I m BŽ X .-measurable. Note that GC1 s GP l g - 04 . Thus by ŽA.3. Že., we obtain GC1 g I m BŽ X .. Now by Theorem III.23 in Castaing and Valadier Ž1977, p. 75., dom C 1 s Pr T Ž GC 1 . g T T where PrT is the projection map onto T. I
References Aliprantis, C.D., Brown, D.J., Burkinshaw, O., 1987. Edgeworth equilibria in production economies. J. Econ. Theory 43, 252–291. Aubin, J.P., 1979. Mathematical Methods of Games and Economic Theory. North-Holland, Amsterdam. Aumann, R.J., 1964. Markets with a continuum of agents. Econometrica 22, 265–290. Bewley, T.F., 1973. The equality of the core and the set of equilibria in economies with infinitely many commodities and a continuum of agents. Int. Econ. Rev. 14, 383–394. Castaing, C., Valadier, M., 1977. Complex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, Vol. 480, Springer-Verlag, New York. Diestel, J., Uhl, J., 1977. Vector Measures, Mathematical Surveys, Vol. 15. American Mathematical Society, Providence, RI. Florenzano, M., 1990. Edgeworth equilibria, fuzzy core, and equilibria of a production economy without ordered preferences. J. Math. Anal. Appl. 153, 18–36. Holmos, P.R., 1950. Measure Theory. Van Nostrand, NJ. Khan, M.A., 1985. On the integration of set-valued mappings in a non-reflexive Banach space II. Simon Stevin 59, 257–268. Mertens, J., 1991. An equivalence theorem for the core of an economy with commodity space l` yt Ž l` , l1 .. In: Khan, M.A., Yannelis, N.C. ŽEds.., Equilibrium Theory in Infinite Dimensional Spaces. Springer-Verlag, Tokyo, pp. 189–196. Noguchi, M., 1998. Edgeworth’s Conjecture in Large-Square Economies. Mimeo, Meijo University, Japan. Rustichini, A., Yannelis, N.C., 1991. Edgeworth’s conjecture in economies with a continuum of agents and commodities. J. Math. Econ. 20, 307–326.
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Sun, N., 1997. Quasi-equilibria in Banach Spaces with lower semi-continuous preferences. Mimeo, University of Tsukuba, Japan. Yannelis, N.C., Zame, W.R., 1986. Equilibria in Banach lattices without ordered preferences. J. Math. Econ. 15, 85–110. Yosida, K., Hewitt, E., 1956. Finitely additive measures. Trans. Am. Math. Soc. 72, 46–66.
A fuzzy core equivalence theorem Mitsunori Noguchi
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Department of Economics, Faculty of Commerce, Meijo UniÕersity, Tengoya 1-501 Shiogmaguchi, Nagoya, Japan Received 7 December 1998; received in revised form 2 July 1999; accepted 4 August 1999
Abstract We attempt to establish the equivalence of the fuzzy core, the Edgeworth core, and the set of Walrasian equilibria in economies with a measure space of agents, which is not necessarily non-atomic. The commodity spaces which we consider here include ordered separable Banach spaces whose positive cone admit an interior point, separable Banach lattices whose positive cone may lack an interior point, and L` endowed with the Mackey topology. Preference relations are assumed to be convex except for the non-atomic case. q 2000 Elsevier Science S.A. All rights reserved. JEL classification: D51 Keywords: Core; Fuzzy core; Edgeworth conjecture; Equivalence theorem; Infinite dimensional commodity space; Measure space of agents
1. Introduction The main objective of the present paper is to establish the equivalence of the fuzzy core ŽAubin, 1979. and the set of Walrasian equilibria in economies with a measure space of agents, which is not necessarily non-atomic. The commodity spaces which we consider here include ordered separable Banach spaces whose positive cone admits an interior point, separable Banach lattices whose positive cone may lack an interior point, and L` endowed with the Mackey topology. For
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Fax: q81-52-833-4767 Žoffice., q81-5613-8-8514 Žhome.; e-mail: noguchi@ meijo-u.ac.jp
0304-4068r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 9 9 . 0 0 0 3 6 - 1
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those economies with a commodity space whose positive cone admits an interior point, we obtain the equivalence of the set of Edgeworth equilibria Žsee Aliprantis et al., 1987. and the set of fuzzy core. We first consider economies with a separable commodity space X whose positive cone Xq admits an interior point. We label the consumption possibility set of consumer t by X Ž t . and consumption plan chosen by consumer t by x Ž t .. Throughout this paper we assume that for each t g T, X Ž t . s Xq. Thus we obtain a constant consumption correspondence t ™ Xq and a selection t ™ x Ž t .. We require x to be Bochner integrable and allow x Ž t . f Xq for those consumers forming a set of measure zero. Let % t ; Xq= Xq be the preference relation of consumer t. We allow satiation, i.e., h g Xq:h % t j 4 may be B for some t g T and j g X . We assume that for each t g T, the set h g Xq:h % t j 4 is norm open in Xq for every j g Xq. We also assume that for each t g T, % t satisfies the usual local non-satiation condition at every non-satiation point. We generally assume that for each t g T, the set h g Xq:h % t j 4 is a convex subset of Xq for every j g Xq, but for an atomless measure space of agents, we can dispense the convexity assumption. For an atomless measure space of agents, we obtain the classical core–Walras equivalence theorem with slightly more generality. Our argument relies on a new core concept called ‘open core’ which appeared in Sun Ž1997. in an attempt to extend the results of Florenzano Ž1990. about non-emptiness of core to the case in which preferences are lower semicontinuous. Sun’s basic constructions allow generalization to our measure theoretic framework. The results in this section of the present article resemble those appearing in the first part of Rustichini and Yannelis Ž1991.. Our assumptions are partly stronger than theirs due to our weaker hypotheses on consumption possibility sets and preferences. We next consider economies with a commodity space which is a separable Banach lattice whose positive cone may lack an interior point. Our assumptions in this section are nearly identical to those which appeared in Rustichini and Yannelis Ž1991., and under these assumptions we obtain, without the non-atomicity assumption, the equivalence of the fuzzy core and the set of Walrasian equilibria. The lack of interior points of the positive cone forces us to assume the existence of an extremely desirable commodity introduced by Yannelis and Zame Ž1986.. Also due to the lack of interior points, we are presently unable to prove the equivalence of the Edgeworth core and the fuzzy core. We remark that the crucial separation argument in Rustichini and Yannelis Ž1991. can be modified so that the same claim holds without the non-atomicity assumption. We finally consider economies with commodity space L` with the Mackey topology t Ž L` , L1 .. Our arguments in this section are based on the observation made by Bewley Ž1973. that the Mackey topology induces a Polish topology on bounded subsets and hence the Mackey topology admits a stronger Polish topol-
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ogy, where both topologies give rise to the same Borel sets. At the point where we show that an element in the fuzzy core gives rise to a quasi-equilibrium, we appeal to the measurable selection theorem in order to extract a measurable selection which restricts to a Gel’fand integrable selection over some set of positive measure. For simplicity we assume that our measure space is the unit interval with Lebesgue measure as in Bewley Ž1973.. We follow Mertens Ž1991. for showing that the countably additive part of the linear functional obtained by the separation theorem possesses the right properties for being an equilibrium price system.
2. Separable commodity space whose positive cone admits an interior point 2.1. Definitions For a linear space X and a subset A ; X , co A denotes the convex hull of A. For a topological space X and a subset A ; X , cl A denotes the closure of A and X™Y a int A denotes the interior of A. Let X , Y be any sets and f :X correspondence. Gf denotes the graph of f , i.e., Gf s Ž x, y . g X = Y : y g f Ž x .4 . Let X and Y be paired linear spaces. For a Banach space X , X U denotes the dual of X , and 5 P 5 x the norm of X . Let ŽT,I . be a measurable space. A correspondence f :T ™ Y is said to admit a measurable graph if Gf g I m BŽ Y ., where BŽ Y . denotes the Borel sigma algebra on Y . For sigma algebras t 1 and t 2 , t 1 m t 2 denotes the product sigma algebra. For measurable spaces ŽT,I ., Ž S, G ., and a function f :T ™ S, f is said to be Ž I, G .-measurable whenever it is measurable with respect to the sigma algebras I and G . Let ŽT,I, m . be a finite X . denotes the space of equivalence measure space and X a Banach space. L1Ž m ,X classes of X-valued Bochner integrable functions f :T ™ X normed by 5 f 5 s HT 5 f Ž t .5 x wfor details see Diestel and Uhl Ž1977.x. For a Banach space X and its dual X U , ² P ,P : denotes the natural paring. For a Banach space X and a X .: f Ž t . g f Ž t . m-almost correspondence f :T ™ X , define L1Ž m , f . s f g L1Ž m ,X everywhere Ža.e..4 . Let R denote the set of real numbers and Q the set of rational numbers. A correspondence f :T ™ X is said to be integrally bounded if there exists a function h g L1Ž m ,R. such that sup5 j 5 x : j g f Ž t .4 F hŽ t . m-a.e. 2.2. Theorems Our commodity space is an ordered separable Banach space X whose positive cone Xq is proper and admits an interior point. An economy E is a tuple wŽT,I, m ., X, P,e x, where 1. ŽT,I, m . is a measure space of consumers; 2. X:T ™ X is a consumption correspondence; 3. % t ; X Ž t . = X Ž t . is the preference relation of consumer t; 4. e:T ™ X is the initial endowment, where eŽ t . g X Ž t . for all t g T.
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U Let _ denote the set theoretic subtraction. Let Xq be the positive cone U U induced by Xq on X . A nonzero element p g Xq _ Ž0. is said to be a price system for E . Denote the budget set of consumer t at price system p by B Ž t, p . s j g X Ž t .:² p, j : F ² p, e Ž t .:4 . We also define B˚Ž t, p . s j g X Ž t .:² p, j : - ² p,eŽ t .:4. U Ž x, p . g L1Ž m , X . = Xq _ Ž0. is said to be a quasi-equilibrium if 1. for almost all t g T, x Ž t . g B Ž t, p ., h g X Ž t .:h % t x Ž t .4 l B˚Ž t, p . s B, and Xq. 2. HT x Ž t . y HT eŽ t . g yX An equilibrium is a quasi-equilibrium Ž x, p . such that for almost all t g T, h g X Ž t .:h % t x Ž t .4 l B Ž t, p . s B. Define A s simple I-measurable functions t :T ™ w0, 1x such that m Žt ) 04. ) 04 . Note that A can be viewed as a family of ‘generalized’ coalitions of consumers, in the following sense: t Ž t . represents the rate of participation of t to the coalition t ) 04 g I wsee Aubin Ž1979.x: Following Aubin Ž1979., we call t g A a fuzzy coalition. For each positive integer r, define A r s t g A:rt Ž t . g 0, PPP ,r 4 , ; t g T 4 . A r is the family of all coalitions of the r-fold replica E r of economy E wsee Florenzano Ž1990.x, where the same type of consumers are considered identical. We also define A Q s t g A:t Ž t . g w0,1x l Q, ; t g T 4 . ˆ s xg In what follows we adopt the following notations: X s L1Ž m , X .; X ˆ t s x t g X t :HT t Ž x t y e . g Xq 4 ; for t g A, X t s xt ) 04 X, X X:HT Ž x y e . g yX ˆ topen s x t g X t :HT t Ž x t y e . g yint Xq 4; for t g A, x g X, P t Ž x . s Xq 4 , X yX t t x g X : x t Ž t . % t x Ž t . m-a.e. in t ) 044 .
Remark 1. In the above definition of quasi-equilibrium, we have required Ž3. Xq instead of the standard market clearing condition Ž3.X HT x Ž t . y HT eŽ t . g yX HT x Ž t . s HT eŽ t .. Our proof works as well with Ž3.X and an alternative definition ˆ s x g X:HT Ž x y e . s 04 instead of the one given in the preceding paragraph. X The proof only needs a very slight modification.
ˆ t defined above may seem ‘too large’ compared to the Remark 2. The set X corresponding object in the usual definition of core since it allows excess supply of commodities in the coalition t ) 04 . If we go along with the widely accepted ˆ t should be x t g X t :HT t Ž x t y e . s 04 version of such set, the definition of X instead of the one given above. However, under the standard assumptions of core–Walras equivalence results such as in Aumann Ž1964., Rustichini and Yannelis Ž1991., these two definitions coincide; for example, if X Ž t . s Xq for all t g T, and the individual preference relation for t is transitive and monotone for all t g T, it can be easily shown that the above two definitions coincide. ˆ is called an attainable allocation. A fuzzy coalition t g A An element x g X ˆ if P t Ž x . l Xˆ t / B, and is said to open block x g Xˆ if is said to block x g X tŽ . t ˆ w P x l X open / B cf. Sun Ž1997.x. Following Aubin Ž1979., we define the fuzzy core of E , C f Ž E ., as the set of all attainable allocations which cannot be blocked f Ž E ., by any fuzzy coalition. Likewise, we define the open fuzzy core of E , Copen
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as the set of all attainable allocations which cannot be open blocked by any fuzzy coalition. Note that the core is the set C Ž E . of all attainable allocations which cannot be blocked by any fuzzy coalition in A 1. Let C r Ž E . be the set of all attainable allocations which cannot be blocked by any fuzzy coalition in A r. Following Aliprantis et al. Ž1987., x g Xˆ is said to be an Edgeworth equilibrium if x g l r G 1C t Ž E .. Let C e Ž E . be the set of all attainable allocations which cannot be blocked by any fuzzy coalition in A Q . Observe that C e Ž E . s r e Ž . Ž E ., Copen l r G 1C r Ž E .. We can also define Copen E in the obvious manner. We e Ž . fŽ . eŽ . f e Ž . eŽ . Ž E . ; Copen clearly have C E ; Copen E , C E ; C E , Copen E . Let W Ž E . Ž Wquasi Ž E .. be the set of all attainable allocations which can be an equilibrium U Žquasi-equilibrium. with a suitable choice of p g Xq _ Ž0.. In the sequel, we show f f that W Ž E . ; C Ž E . ŽTheorem 1., CopenŽ E . ; Wquasi Ž E . ŽTheorem 2., Wquasi Ž E . e Ž . f Ž E . ŽTheorem 10.. Consequently, we s W Ž E . ŽTheorem 9., and Copen E ; Copen f e e f Ž E . ; Wquasi Ž E . s W Ž E .. In obtain W Ž E . ; C Ž E . ; C Ž E . ; CopenŽ E . s Copen particular, we obtain the equivalence theorem W Ž E . s C e Ž E .. We now state the following set of assumptions on economy E . ŽA.1. ŽT,I, m . is a complete finite measure space. ŽA.2. X Ž t . s Xq for all t g T. ŽA.3. Ža. For each t g T, the set h g Xq:h % t j 4 is norm open in Xq for every j g Xq. Žb. For each t g T, the set h g Xq:h % t j 4 is a convex subset of Xq for every j g Xq. Žc. For each t g T, if j g Xq is not a satiation point for % t , then j g cl h g Xq:h % t j 4 . Žd. For each t g T, either h g Xq:h % t j 4 / B or j y eŽ t . g Xq holds for every j g Xq. Že. The set Ž t, h , j . g T = Xq= Xq:h % t j 4 belongs to I m BŽ Xq . m BŽ Xq .. ŽA.4. e is Bochner integrable. ŽA.5. For each t g T, eŽ t . is strictly positive Ži.e., ² p,eŽ t .: ) 0 for all U p g Xq _ Ž0... Under ŽA.1. – ŽA.5., we prove the following theorems: Theorem 1. If Ž x, p . is an equilibrium for E , then x g C f Ž E .. Proof of Theorem 1. Assume the contrary, that x f C f Ž E .. By definition, there ˆ t / B. Let x t g P t Ž x . l Xˆ t . Then exists t g A such that P t Ž x . l X x t Ž t . %x Ž t .
Ž 1.1 .
t
m-a.e. in t ) 04 , and
HTt Ž t . x
t
Ž t . FH t Ž t . e Ž t . .
Ž 1.2 .
T
Eq. Ž1.1. implies that x t Ž t . f B Ž t, p . m-a.e. in t ) 04 , and hence
HTt Ž t . ² p, x
t
Ž t . : ) H t Ž t . ² p,e Ž t . : . T
Ž 1.3 .
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By Theorem 6 in Diestel and Uhl Ž1977, p. 47., Eq. Ž1.2. gives
HTt Ž t . ² p, x
t
Ž t . : F H t Ž t . ² p,e Ž t . : .
Ž 1.4 .
T
Combining Eqs. Ž1.3. and Ž1.4., we reach a contradiction.
I
U f Ž E ., then there exists p g Xq Theorem 2. If x g Copen _ Ž0. such that Ž x, p . is a quasi-equilibrium of E . f Ž E .. Define a correspondence P:T ™ Xq by Proof of Theorem 2. Let x g Copen P Ž t . s h g Xq:h % t x Ž t .4 and let Tx s t g T : P Ž t . s B4 . Note that since ŽT,I, m . is complete, X Suslin, and GP g I m BŽ X . wŽA.3. Že.x, by Theorem III.23 in Castaing and Valadier Ž1977, p. 75., Tx s w Pg T Ž GP .x c g I, where Pg T :T = X ™ T is a projection map. We first assume that m ŽT _ Tx . s 0. We have
HTx FHTe.
Ž 2.1 .
U Let p g Xq _ Ž0.. We show that Ž x, p . is an equilibrium. Since P Ž t . s B m-a.e., we trivially have P Ž t . l B Ž t, p . s B m-a.e. By ŽA.3. Žd.,
² p, x Ž t . : G ² p, e Ž t . :
Ž 2.2 .
m-a.e. By Theorem 6 in Diestel and Uhl Ž1977, p. 47. applied to Eq. Ž2.1., we obtain
HT² p, x Ž t . : FHT² p, e Ž t . : .
Ž 2.3 .
Eqs. Ž2.2. and Ž2.3. imply that ² p, x Ž t . : s ² p, e Ž t . :
Ž 2.4 .
m-a.e. In particular, x Ž t . g B Ž t, p . m-a.e. Thus Ž x, p . is an equilibrium. We next assume that m ŽT _ Tx . ) 0. For the sake of simplicity we write T X s T _ Tx , IX s I < T X , and mX s m < I X . Define Ax s simple IX-measurable functions t :T X ™ w0,1x such that mX Žt ) 04. ) 04 and define AxU s t g Ax :t Ž t . g Ž0,1x, ; t g T X 4 / B. For each t g Ax , define Gt1 s HT X t Ž t . x Ž t . g X : x g L1Ž mX , P < T X .4 . Now we define Gt s Gt1 y HT X t Ž t . eŽ t . q int Xq. Lemma 3. Gt1 / B. See Appendix A for the proof. Hence, Gt / B for all t g AxU . Define G s j t g A Ux Gt . Lemma 4. G is conÕex. See Appendix A for the proof.
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Lemma 5. 0 f G. See Appendix A for the proof. Since int G / B, the separating hyperplane theorem implies that there exists p g X U _ Ž0. such that p P G G 0. Define H s j t g A xGt . Lemma 6. p P H G 0. See Appendix A for the proof. U Lemma 7. p g Xq _ Ž0.
See Appendix A for the proof. The above line of argument based on the construction of the sets G, H is attributed to Sun Ž1997.. Now we claim that Ž x, p . is a quasi-equilibrium. Let B g IX and let t s x B g Ax . Repeating the argument in the proof of Lemma 3 with U the ball of radius 1rn in X , we obtain x n g L1Ž mX , P < T X . such that 5 x Ž t . y x nŽ t .5 F Ž1rn. for all t g T X . By Lemma 6, we have HB ² p, x nŽ t . y eŽ t . G 0 for all n, and consequently, HB ² p, x Ž t . y eŽ t .: G 0. Since B g IX is arbitrary, considering ŽA.3. Žd., we obtain ² p, x Ž t . y eŽ t .: G 0 m-a.e. in T. Combining this with HT x F Ht e, we obtain x Ž t . g B Ž t, p . m-a.e. Let c 1Ž t . s P Ž t . l B˚Ž t, p .. Lemma 8. Gc 1 g I m BŽ X . and dom c 1 g I. See Appendix A for the proof. By Lemma 8 and Theorem lll.22 in Castaing and Valadier Ž1977, p. 74., we see that c 1 admits an measurable selection x:dom c 1 ™ X . If m Ž dom c 1 . ) 0, we can find a IX-measurable set B ; dom c 1 , m Ž B . ) 0, such that x is integrable over B. Let t s x B so that t g Ax . Now we have by Lemma 6 that p
HT t Ž x y e . X
G 0,
Ž 2.5 .
where x is now extended as an integrable selection to the entire T X . Eq. Ž2.5. can be written as
HB² p, x Ž t . : GHB² p, e Ž t . : .
Ž 2.6 .
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On the other hand, since x Ž t . g B˚Ž t, p . for all t g dom c 1 , we have
HB² p, x Ž t . : -HB² p, e Ž t . : , which contradicts with Eq. Ž2.6.. Thus, m Ž dom c 1 . s 0, i.e., P Ž t . l B˚Ž t, p . s B m-a.e. Theorem 9. EÕery quasi-equilibrium Ž x, p . of E is an equilibrium of E . Proof of Theorem 9. Let EB Ž t, p . s B Ž t, p . _ B˚Ž t, p .. Define c 3 Ž t . s P Ž t . l EB Ž t, p .. By ŽA.5., we have B˚Ž t, p . / B for all t g T. Since P Ž t . is norm open, we have dom c 3 ; dom c 1. By assumption, we have m Ž dom c 1 . s 0, and hence m Ž dom c 3 . s 0, i.e., P Ž t . l EB Ž t, p . s B m-a.e. I The following theorem for the case of E with finitely many consumers is attributed to Sun ŽLemma 2, 1997., and our generalization to E with a measure space of consumers is almost verbatim. e Ž . f Ž E .. Theorem 10. Copen E ; Copen e Ž . f Ž E .. By definition, Proof of Theorem 10. Let x g Copen E . Suppose x f Copen tŽ . t t ˆ ˆ topen . Then there exists t g A such that P x l X open / B. Let x g P t Ž x . g X t Ž . 5 5 4 HT t x y e g yint Xq. Let Be s j g X : j X - e such that
Be q t Ž x t y e . ; yint Xq .
HT
Ž 10.1 .
Choose k g R such that k e ) HT xt ) 04 5 x t y e 5 X . Then
HT Ž 1 q w kt x y kt . x
t ) 04
Ž x t y e . g kBe ,
Ž 10.2 .
where w kt xŽ t . denotes the largest integer less than kt Ž t .. Multiplying Eq. Ž10.1. by k, we obtain kBe q kt Ž x t y e . ; yint Xq .
HT
Ž 10.3 .
Combining Eqs. Ž10.2. and Ž10.3., we obtain
HT Ž 1 q w kt x . x
t ) 04
Ž x t y e . g yint Xq .
Let kU s max t Ž t .) 0 Ž1 q w kt Ž t .x.. Define t g A Q by t Ž t . s kUy1 Ž1 q w kt Ž t .x. e Ž . ˆ topen , and x f Copen for t Ž t . ) 0, and t Ž t . s 0 otherwise. Then x t g P t Ž x . l X E . But this is a contradiction. I
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2.3. Atomless measure space of agents In this section we assume the assumptions ŽA.1. – ŽA.5. except for ŽA.3. Žb.. Theorem 11. The core–Walras equiÕalence theorem W Ž E . s C Ž E . holds. Proof of Theorem 11. Let x g C Ž E .. Define Tx as in the proof of Theorem 2. If m ŽT _ Tx . s 0, then x g W Ž E . as in the proof of Theorem 2. For m ŽT _ Tx . ) 0, define Ax s x B : B g IX with mX Ž B . ) 04 . Note that AUx s x T X 4 , and G s Gx T X s Gx1T X y HT X e q int Xq. Now by Theorem 1 in Khan Ž1985., cl Gx1T X is convex. Since cl Gx1T X y HT X e ; cl Ž Gx1T X y HT X e ., we have cl Gx1T X y HT X e s cl Ž Gx1T X y HT X e .. Note that cl Ž Gx1T X y HT X e . is convex, and 0 g G iff 0 g cl Ž Gx1T X y HT X e . q int Xq. If 0 g G, ˆ topen / B, and hence then by letting t s x T X in the proof of Lemma 5, P t Ž x . l X tŽ . t ˆ P x l X / B, and this is a contradiction. Thus we can find p g X U _ Ž0. such that p P G G 0. We define the set H as before, and can show that p P H G 0 and as U in Lemma 6, p g Xq _ Ž0.. Observe that the argument after Lemma 8 is still valid since t s x d o m c 1 g A 1. Hence x g Wquasi Ž E .. By Theorem 9, x g W Ž E .. I Remark 3. We remark that production can be incorporated into our framework with minor modifications. Let Ž S, G , p . be a measure space of producers and Y:S ™ X the production correspondence. Let u :T = S ™ R denote the share. Define Y s HS y Ž s . and Yt s HT t Ž t .HS u Ž t, s . y Ž t, s . g X : y g L1Ž m = p ,Y ., where Y is viewed as a correspondence defined on T = S4 . Now we modify the ˆ ˆ t as Xˆ s x g X:HT Ž x y e . g Y y Xq 4,Xˆ t s x t g X t :HT Ž x t y definitions of X,X t e . g Y y Xq 4 , respectively. The definition of Yt above can be justified as follows: Let E be a finite economy with T consumers and S producers. Then it is natural to define Yt s ÝTis1t i ÝSjs1 u i j yj g X :Yj g Yj , j s 1, PPP ,S4 , where t g w0, 1xT _ Ž0., u i j the share of consumer i in the profit of producer j, and Yj the production set of producer j. We assume that each Yj is convex. For each i, let yi j g Yj , j s 1, PPP ,S. Then y ' ÝTis1t i ÝSjs1 u i j yi j s ÝSjs1 ÝTis1t i u i j , yi j s Ý jX ÝTis1t i u i jX , yi jX , where jX runs through the indices such that ÝTis1t i u i j ) 0. Note that we can write y s Ý jX ŽÝ TiXs 1t iX u i jX . Ý Tis 1 u˜i jX y i jX , where u˜i jX s ŽÝTiXs1t iX u iX jX .y1t i u i jX G 0. Observe that for each jX , ÝTis1 u˜i jX s 1. Then, ÝTis1 u˜i jX yi jX s y˜ jX g co YjX s YjX . Hence, y s ÝTis1t i ÝSjs1 u i j Yj , where y j s Y˜j if ÝTis1t i u i j ) 0, and yj s any element of Yj if ÝTis1t i u i j s 0. Thus, we obtain an equivalent definition Yt s ÝTis1t i ÝSjs1 u i j yi j g X : for each i yi j g Yj , j s 1, PPP ,S4 . Under the standard assumptions on production wfor example, Ž S, G , p . is a complete finite measure space; Y:S ™ X is a convex valued correspondence such that G Y g G m BŽ X .; u is a I m G-measurable function such that for every s g S, HT u Ž t, s . s 1 and for every Ž t, s . g T = S, 0 F u Ž t, s . F 1; and 0 g Y Ž s . for almost all s g S x, it is merely a routine matter to check that our previous argument goes through. For the production economy version of Theorem 11, one needs an additional compactness assumption on L1Ž mX = p ,Y . for every mX s m < I X , where IX s I l T X for some T X g I. The details can be found in Noguchi Ž1998..
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2.4. Separable commodity space whose positiÕe cone may haÕe an empty interior In this section we consider economies E whose commodity space is a separable Banach lattice X . We allow Xq to have an empty norm interior and ŽT, I, m . to be atomic. With the non-atomicity assumption, Rustichini and Yannelis Ž1991. obtained substantial results. For the rest of the argument, we assume that % t is represented by a utility function u:G X ™ R. Our goal in this section is to show the equivalence W Ž E . s C f Ž E . by modifying the proof provided by Rustichini and Yannelis Ž1991. for core–Walras equivalence results for non-atomic economies. Definition. Õ g Xq_Ž0. is said to be an extremely desirable commodity if there exists an open neighborhood U at the origin such that for each j g Xq and each t g T, it is true that Ž j q a Õ y h . is strictly preferred to j whenever a ) 0, j q a Õ y h g Xq and h g a U wthe notion of an extremely desirable commodity was introduced by Yannelis and Zame Ž1986.x. In our situation we can clearly assume that U is an ideal. Define an open cone C by C s j a ) 0 Ž Õ q U . as in Rustichini and Yannelis Ž1991.. Then every element h g Ž C q j . l Xq is strictly preferred to j . We now state our assumptions. ŽA.1. ŽT, I, m . is a complete finite measure space. ŽA.2. X Ž t . s Xq for all t g T. ŽA.3. u:T = Xq™ R is a function such that: Ža. for each t g T, uŽ t,P .:X Xq™ R is a norm-continuous, quasi-concave, and monotonic Ži.e., for every j g Xq and Õ g Xq_Ž0., we have uŽ t, j q Õ . ) uŽ t, j ..; Žb. for each j g Xq, uŽP, j .:T ™ R is I-measurable. ŽA.4. e is Bochner integrable. ŽA.5. eŽ t . is strictly positive for all t g T. ŽA.6. X is a separable Banach lattice. ŽA.7. ŽExtremely desirable commodity. There exists an extremely desirable commodity Õ g Xq_Ž0.. ŽA.8. ŽAdditivity condition. wthis was found in Rustichini and Yannelis Ž1991.x Let d i , i s 1, PPP ,n be positive numbers with Ý nis1 d i s 1. In ŽA.7. we can arrange U so that it is an ideal of X Žas a Riesz space., and if j i g Xq, j i f d i U, i s 1, PPP ,n, then Ý nis1 j i f U. Theorem 12. Under ŽA.1. – ŽA.8. we haÕe the equiÕalence W Ž E . s C f Ž E .. Proof of Theorem 12. Since Theorem 1 and Theorem 9 in the previous section are still valid, we only need to show that the inclusion C f Ž E . ; W Ž E . holds. To this
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end, we proceed as in the proof of Theorem 2. Let x g C f Ž E . and note that Tx s B. Define Ax and AxU as before noting that T X s T now. We use C instead of int Xq in the definition of Gt . Observe that Lemma 3 trivially follows and no changes are necessary in the proof of Lemma 4. Claim. G l y C s B. Assume the above claim holds Žthe justification will be given in the end.. Then by the separation argument as before, we obtain Lemma 6. For Lemma 7, let j g Xq and let Õ g C. Choose t s x T g Ax . Since x q jx T g L1Ž m , P ., we have HT ² p, x q j y e : q ² p,Õ : G 0. Since C is a cone, we obtain ² p,j : G 0, and U hence p g Xq _ Ž0.. Lemma 8 is valid with no change, and the rest of the argument follows thereafter. Justification of Claim. We must show that Gt l y C s B for all t g AxU . Following closely the line of argument on pp. 321–323 in Rustichini and Yannelis Ž1991., for a y g L1Ž m , P ., we construct a sequence of functions y k g L1Ž m , P . such that y k ™ y in the norm topology, and HT t y k y HT t e f yC. Note that in our case, T s S holds in the second paragraph in p. 321. They choose a sequence of k y kx k X . to y. They simple functions y k s Ý mis1 which converges in L1Ž m ,X i Ti k k arrange T1 , PPP ,Tm k so that Tik are mutually disjoint, cover T, and possess the same positiÕe measure j ) 0. This last requirement may not be satisfied in our case since our ŽT,I,u. may have atoms. Thus we can only expect to have m ŽTik . s j i ) 0 instead and it turns out that this is all we need to complete their proof. In other words, the apparent dependency on the non-atomicity assumption in their proof of Claim 6.1 can be removed. Our assertion, which corresponds to their Claim 6.1 writes HT t y k y HT t e f yC. Observe that since t is a simple function, we can assume that t is constant on each Tik by taking a finer partition if necessary. Thus their expression on top of p. 322 may be replaced by Ý mis1t i yi j i y Ý mis1t i e i j i g ya Ž Õ q U ., where t i ) 0 Žrecall that t g AxU . is the Ž HT k e .. Note that Ý mis1t i e i j i s Ý mis1 HT t i x T k e s value of t on Tik and e i s jy1 i i i HT t e. Instead of their Ž6.8. on p. 322, we have Ý mis1 t i yi j i q w y u s Ý mis1t i e i j i , where w s a Õ, u g a U. Basically, the only notational difference here is that we have t i yi j i , t i e i j i in place of their yi , e i , respectively. Their expression Ž6.10. now writes Ý mis1 u i s u, u i F t i yi j i q u i w for all i, and the definition of y˜i Ž u . y1 y1 y1 writes y˜i Ž u . s yi q ty1 ˜i Ž u U . having the i j i u i w y t i j i u i . Then we obtain y U property that y˜ i Ž u . is strictly preferred to x Ž t . m-a.e. in Ti Žsuperscript k is omitted. for all i. As in p. 323, define y˜ s Ý mis1 y˜i Ž u U . x T i . Note that we have HT t y˜ s Ý mis1 y˜i Ž u U . HT tx T i s Ý mis1 y˜i Ž u U .t i j i s HT t e. This shows that y˜ g P t Ž x . ˆ t in our notation, and hence we reach a contradiction as in their proof on p. lX 323. Being helped by the fact that t i ) 0, j i ) 0 for all i, the rest of the argument on p. 323 goes through with almost no alteration. We only remark that in the part showing Ý mjs1 d j s 0, their arj becomes a in our case. I
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2.5. Economies with commodity-price paring (L` , L1 ) Let Ž M, M , Õ . be a totally s-finite countably generated measure space ŽBewley, 1973.. We abbreviate L` Ž M, M , Õ ., L1 Ž M, M , Õ . as L` , L1 , respectively. In this section we treat economies with commodity space L` and price systems lying in L1 , where L` is given the Mackey topology t Ž L` , L1 .. In the sequel we simply write Lt` for L` with t Ž L` , L1 ., and Lw`U for L` with weakU topology. For a function f :T ™ L` such that ² a, f ŽP.: g L1Ž m ,R. for all a g L1 , HT f Ž t . denotes the Gel’fand integral of f wsee Diestel and Uhl Ž1977, p. 53.x. For Gel’fand integrable functions f, g, we introduce an equivalence relation ; by f ; g if and only if ² a, f ŽP.: s ² a, g ŽP.: m-a.e., for each a g L1. Define LG1 Ž m , L` . to be the set of equivalence classes under ; of Gel’fand integrable functions on T. Since L1 is a separable Banach space, if f ; g, then f s g m-a.e. Therefore, for a correspondence f :T ™ L` and f g LG1 Ž m , L` ., it makes sense to talk about f Ž t . g f Ž t . m-a.e. For a correspondence f :T ™ L` , define LG1 Ž m , f . s f g LG1 Ž m , L` .: f Ž t . g f Ž t . m-a.e.4 . We write L`,q for the standard positive cone of L` . A price system p is required to lie in L1,q_Ž0., where L1,q is the standard positive cone of L1. We state the following two sets of assumptions on economy E . ŽA.1. ŽT, I, m . is the unit interval with Lebesgue measure. ŽA.2. X Ž t . s L`,q for all t g T. ŽA.3. Ža. For each t g T, the set h g L`,q:h % t j 4 is a t Ž L` , L1 .-open subset of L`,q for every j g L`,q. Žb. For each t g T, the set h g L`,q:h % t j 4 is a convex subset of L`,q for every j g L`,q. Žc. For each t g T, if j g L`,q is not a satiation point for % t , then j q Õ % t j for every Õ g L`,q_Ž0.. Žd. For each t g T, either h g L`,q:h % t j 4 / B or j y eŽ t . g L`,q holds for every j g L`,q. Že. The set Ž t,h , j . g T = L`,q= L`,q:h % t j 4 belongs to I m BŽ Lt`,q . m BŽ Lt`,q .. ŽA.4. e is Gel’fand integrable. ŽA.5. For each t g T, eŽ t . g int L`,q, where int is taken with respect to the norm topology. There exists a positive number l ) 0 such that essinf m g M eŽ t .Ž m. G l for all t g T. Remark 4. We assume ŽA.1. above since for a measurable function with respect to Lt` , we may not be able to shrink the domain in order to assure Gel’fand integrability. This condition is directly taken from Bewley Ž1973, p. 385. so that one can appeal to Lusin’s theorem for the integrability. Theorem 13. Under either set of assumptions ŽA.1. – ŽA.5. we haÕe the equiÕalence W Ž E . s C e Ž E . s C f Ž E .. Proof of Theorem 13. We follow the steps in the proof provided in the earliest section for the separable commodity space whose positive cone admits an interior
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point. Theorem 1 holds in the present situation. We simply replace Bochner integral by Gel’fand integral and apply the definition of Gel’fand integral in place of Theorem 6 in Diestel and Uhl Ž1977, p. 47.. Since Lt` is Suslin wthe author is indebted to the paper by Mertens Ž1991. for this observationx, Theorem III.23 in Castaing and Valadier Ž1977, p. 75. is still valid and the previous argument for Theorem 2 holds up to Eq. Ž2.4.. Lemma 3 holds as a simple consequence of ŽA.3. Žc.. Lemma 4, Lemma 5 and Lemma 6 need no modification. Note that p lies in the dual space of L` , but not in L1. In the proof of Lemma 7, e Ž² p,HT X x y e :. still makes sense and obtain p g baq_Ž0., where baq denotes the standard positive cone of the dual space of L` . We must show that p g L1. To this end we follow the steps in the proof provided in Mertens Ž1991.. Let p s pc q pf be the decomposition in Yosida and Hewitt Ž1956., where pc is the countably additive part and pf is the purely finitely additive part. We choose an increasing Bi g M such that D `is1 Bi s M and pf Ž Bi . s 0. For a Gel’fand integrable selection x g LG1 Ž mX , P < T X ., define x i s x x B i so that for almost all t g T, x i Ž t . g P Ž t . for i G iŽ t . for some iŽ t .. Let E ; T X be an arbitrary measurable subset with m Ž E . ) 0. Following Mertens Ž1991., define En s t g E:;i G n, x i Ž t . g P Ž t .4 . ) Note that x i is measurable with respect to BŽ Lw` . s BŽ Lt` . wThis follows from the well-known fact that for any Hausdorff topology H which admits a stronger Polish topology P, we have BŽ H . s BŽ P ., and t Ž L` , L1 . under our hypothesis admits such P as a topological sum of the Polish topologies induced on disks by t Ž L` , L1 ., which is Polish ŽBewley, 1973.x. Since Lt` is Suslin and Ž E,I < E , m < E . is complete, by the projection theorem, we see that En s l i G n PrE Ž Gx i l Gp . g IX . We obtain an increasing sequence Ei which covers E and proceed as in Mertens Ž1991.. Let t s x E i g Ax so that pw HT X t Ž x y e .x G 0, which can be rewritten as pw x B i HE iŽ x y e .x G pwŽ1 y x B i .HE i e x.. Since p G pf and Ž1 y x B i .HE i e g L`,q, the right hand side of the previous inequality equals pf Ž HE i e . G l pf Ž M . m Ž Ei . G 0. Recall that we may assume m Ž Ei . / 0. As in Mertens Ž1992. we obtain by the monotone convergence theorem that pc w HE Ž x y e .x G l pf Ž M . m Ž E . G 0. In particular, pc g L1 _ Ž0.. In our case, pf may not be zero. Let Õ g L`,q_Ž0. and define x S s x q sÕ for s g Ž0,1x. x S is clearly Gel’fand integrable and by ŽA.3. Žc., x S is a measurable selection of P < T X , and hence pc w HE Ž x S y e .x G 0. By letting s ™ 0, we obtain ² pc , x Ž t . y eŽ t .: G 0 m-a.e. in T X , and combining this with ŽA.3. Žd., we obtain ² pc , x Ž t . y eŽ t .: s 0 m-a.e. in all of T. In particular, x Ž t . g B Ž t, pc . m-a.e. in T. For Lemma 8, note that g Ž t, j . s ² pc , j y eŽ t .: is I-measurable in t, and BŽ L`P .-measurable in j , where L`P is L` endowed with a Polish topology P which is stronger than t Ž L` , L1 .. Then g is jointly measurable in I and BŽ Lt` .. Some care must be taken for the argument after Lemma 8 since unlike in the case of Bochner integral, we may not be able to shrink the domain of a measurable function in order to assure integrability. We appeal to Lusin’s theorem ŽBewley, 1973. so that the measurable function Žwith respect to Lt` . will be bounded on a compact subset of positive measure. We obtain pw HB Ž x y e .x G 0 as before and
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deduce that pc w HB Ž x y e .x G 0 using the same trick as above. Hence, we obtain the expression Ž2.6. with p replaced by pc . We have shown that Ž x, pc . is a quasi-equilibrium. The proof of Theorem 9 is valid with pc instead of p. The proof of Theorem 10 needs some modification. To be more specific, the way to choose k for Eq. Ž10.2. must be changed as follows: N since t is a simple function, we can write t s Ý is1 t i x T i , t i ) 0, and Ti l Tj s N t BŽ i / j .. Choose k g R such that Ý is1 5 HT x T iŽ x y e .5 L` - k e . Then we have N 5 HT Ž1 q w kt x y kt . xt ) 0 Ž x t y e .5 L` s 5Ý is1 HT iŽ1 q w kt i x y kt i .Ž x t y e .5 L` F N <Ž t N Ý is1 1 q w kt i x y kt i .< 5Ž HT i x y e .5 L` F Ý is1 5Ž HT i x t y e .5 L` , and obtain Eq. Ž10.2.. This completes the proof. I
Acknowledgements The author is indebted to an anonymous referee of JME for valuable suggestions.
Appendix A Proof of Lemma 3. We must show that L1Ž mX , P < T X . / B. Recall from ŽA.3. Žc. that x Ž t . g cl P Ž t . for t g T X . Let U be the unit ball in X and define UŽ t . s x Ž t . q U. Define V Ž t . s UŽ t . l P Ž t . on T X . Observe that by Lemma III.14 in Castaing and Valadier Ž1977, p. 70., we have GU s Ž t, j . g T X = X : 5 x Ž t . y j 5 X F 14 g IX m BŽ X .. By ŽA.3. Že., we have GP g I m BŽ X .. Note that by Theorem E in Holmos Ž1950, p. 25., we obtain GP < T X s GP l ŽT X = X . g IX m BŽ X .. Consequently, G V g IX m BŽ X .. Since ŽT X , IX , mX . is complete, by Theorem III.22 in Castaing and Valadier Ž1977, p. 25., we obtain a measurable selection x of V. Since 5 x Ž t . y x Ž t .5 X F 1 for all t g T X , x clearly lies in L1Ž mX , P < T X .. I Proof of Lemma 4. Let g 1 g Gt11, g 2 g Gt12 , where t 1 , t 2 g AxU . For s g w0,1x, let t s st 1 q Ž1 y s .t 2 g AxU . Since g 1 s HT X t 1 j 1 , g 2 s HT X t 2 j 2 for some j 1 , j 2 g L1Ž mX , P < T X ., we have sg 1 q Ž1 y s . g 2 s HT X tj , where j s ty1 st 1 j 1 q ty1 Ž1 y s .t 2 j 2 . Since P Ž t . is convex, sg 1 q Ž1 y s . g 2 g Gt1. I Proof of Lemma 5. Assume the contrary that 0 g G. Then we have X
HT t Ž x y e . q Õ s 0 X
for some t X g AxU , x g L1Ž mX , P < T X ., and Õ g int Xq. Let t be the extension of t X by zero to the entire T so that t g A. Let x t be the extension of x by zero to the
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entire T. Note that for any x˜ g X, which agrees with x on T X , we have x t s xt ) 04 x˜ g X t . Thus we have HT t Ž x t y e . g int Xq, and consequently, ˆ topen / B. This is a contradiction. xt g P t Ž x . l X I Proof of Lemma 6. Let h g H and g g G. Then sg q Ž1 y s . h g G for all s g Ž0,1x. Thus s² p, g : q Ž1 y s .² p,h: G 0 for all s g Ž0,1x. Consequently, ² p,h: G 0. I Proof of Lemma 7. Let e ) 0. Let Õ g int Xq. Let t s ex T X g Ax . For any x g L1Ž mX , P < T X ., we have e Ž HT X ² p, x y e :. q ² p,Õ : G 0. Consequently, ² p,Õ G 0. I Proof of Lemma 8. Define a function g:T = X ™ R by g Ž t, j . s ² p, j : y ² p,e,Ž t .:. Since X is separable, Lemma III.14 in Castaing and Valadier Ž1977, p. 70. implies that g is I m BŽ X .-measurable. Note that GC1 s GP l g - 04 . Thus by ŽA.3. Že., we obtain GC1 g I m BŽ X .. Now by Theorem III.23 in Castaing and Valadier Ž1977, p. 75., dom C 1 s Pr T Ž GC 1 . g T T where PrT is the projection map onto T. I
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