Equilibria in a large production economy with an infinite dimensional commodity space and price dependent preferences
aa r X i v : . [ ec on . T H ] F e b Equilibria in a large production economywith an infinite dimensional commodity spaceand price dependent preferences ∗ Hyo Seok Jang † Sangjik Lee ‡ February 6, 2020
Abstract
We prove the existence of a competitive equilibrium in a production economy with infinitelymany commodities and a measure space of agents whose preferences are price dependent.We employ a saturated measure space for the set of agents and apply recent results for aninfinite dimensional separable Banach space such as Lyapunov’s convexity theorem and anexact Fatou’s lemma to obtain the result.
JEL Classification Numbers:
C62, D51.
Keywords:
Separable Banach space, Saturated measure space, Price dependent preferences,Lyapunov’s convexity theorem, Fatou’s lemma
The purpose of this paper is to prove the existence of a competitive equilibrium in a productioneconomy with infinitely many commodities and a measure space of agents whose preferences areprice dependent. In a seminal paper, Aumann [3] demonstrated the existence of a competitiveequilibrium for an exchange economy with a finite dimensional commodity space and a contin-uum of agents modeled as an atomless finite measure space by utilizing Lyapunov’s convexitytheorem to dispense with convex preferences. Aumann’s model in [3] was generalized to allowincomplete preferences by Schmeidler [35] and to include production by Hildenbrand [15].As Shafer[36] and Balasko [5] pointed out, price dependent preferences have been tradition-ally explained by consumers taking relative prices as an indication of quality. In addition, we ∗ We thank two anonymous referees for their comments and suggestions. This research was supported byHankuk University of Foreign Studies Research Fund. † Department of Mathematical Sciences, Seoul National University, Seoul, Korea. Email:[email protected] ‡ Division of Economics, Hankuk University of Foreign Studies, Seoul, Korea. Email: [email protected]. This was drawn to our attention by an anonymous referee.
Let
X, Y be topological spaces. A set-valued function or a correspondence F from Y to thefamily of non-empty subsets of Y is called upper semicontinuous if the set { x : X : F ( x ) ⊂ V } is open in X and said to be lower semicontinuous if the set { x : X : F ( x ) ∩ V = ∅} is openin X for every V of Y . When Y is a Banach space, F is norm upper semicontinuous if the set { x : X : F ( x ) ⊂ V } is open in X for every norm open subset V of Y . And F is called weaklyupper semicontinuous if the set { x : X : F ( x ) ⊂ V } is open in X for every weakly open subset V of Y . We say that F is norm lower semicontinuous if the set { x : X : F ( x ) ∩ V = ∅} is openin X for every norm open subset V of Y and F is said to be weakly lower semicontinuous if theset { x : X : F ( x ) ∩ V = ∅} is open in X for every weakly open subset V of Y .Let ( T, T , µ ) be a finite measure space and E be a Banach space. A measurable function f : ( T, T , µ ) → E is said to be Bochner integrable if there exists a sequence of simple functions { f n } n ∈ N such that lim n →∞ Z T k f n ( t ) − f ( t ) k dµ = 0 (2.1)where N denotes the set of natural numbers. For each S ∈ T the integral is defined to be R S f ( t ) dµ = lim n →∞ R S f n ( t ) dµ . Denote by L ( µ, E ) the space of (the equivalence classes of) E -valued Bochner integrable functions f : T → E normed by k f k = R T k f ( t ) k dµ .3he weak upper limit of a sequence { S n } of subsets in E is defined by w -Ls S n = { x ∈ E : ∃{ x n k } such that x = w - lim x n k , x n k ∈ S n k , for all k ∈ N } (2.2)where { x n k } is a subsequence of a sequence { x n } and w - lim n x n k denotes the weak limit pointof { x n k } .A correspondence F : T → E is said to be measurable if for every open subset V of E , the set { t ∈ T : F ( t ) ∩ V = ∅} ∈ T . The correspondence F is said to have a measurable graph if its graph G F = { ( t, x ) ∈ T × E : x ∈ F ( t ) } belongs to the product σ -algebra T ⊗ B ( E, w ), where B ( E, w )denotes the Borel σ -algebra of E generated by the weak topology. If correspondences from T to E are closed valued, measurability and graph measurability are equivalent when ( T, T , µ ) iscomplete and E is separable. A measurable correspondence F : T → E is integrably bounded ifthere exists a real-valued integrable function h on ( T, T , µ ) such that sup {k x k : x ∈ F ( t ) } ≤ h ( t )for almost all t ∈ T .A measurable function f from ( T, T , µ ) to E is called a measurable selection of the corre-spondence F if f ( t ) ∈ F ( t ) for almost all t ∈ T . By Aumann’s measurable selection theorem in[4], if ( T, T , µ ) is a complete finite measure space, F has a measurable graph, and E is separable,then F has a measurable selection. We denote by S F the set of all E -valued Bochner integrableselections for the correspondence F , i.e., S F = { f ∈ L ( µ, E ) : f ( t ) ∈ F ( t ) a.e. t ∈ T } . When F is also integrably bounded, it admits a Bochner integrable selection so that S F is non-empty.The integral of the correspondence F is defined by Z T F ( t ) dµ = { Z T f ( t ) dµ : f ∈ S F } . (2.3)A sequence of correspondences { F n } from T to E is said to be well-dominated if thereexists an integrably bounded and weakly compact-valued correspondence φ : T → E such that F n ( t ) ⊂ φ ( t ) a.e. t ∈ T for each n .Let E be an ordered Banach space equipped with ordering ≥ such that the positive cone E + = { x ∈ E : x ≥ } of E is closed. For x, y ∈ E , x > y means x − y ∈ E + and x = y . Wedenote by E ∗ the dual space of E , i.e., the space of all continuous linear functionals from E into R . For x ∈ E, p ∈ E ∗ , we write p · x for the value of p at x . We denote by E ∗ + the dual cone of E + , i.e., E ∗ + = { p ∈ E ∗ : p · x ≥ x ∈ E + } . We denote by B ( E ∗ , w ∗ ) the Borel σ -algebraof E ∗ generated by the weak* topology. For any set A in E , cl A stands for the norm closure of A and co A for the convex hull of A .Let ( T, T , µ ) be a finite measure space. Denote by L ( µ ) the the space of ( µ -equivalenceclasses of) real valued integrable functions on T . Let T S = { A ∩ S | A ∈ T } be the sub- σ -algebraof T restricted to S ∈ T and µ S be a restriction of µ to T S . We write L S ( µ ) for the vector See Theorem 8.1.4 in [2]. L ( µ ) which consists of each function in L ( µ ) restricted to S . Definition 1.
A finite measure space ( T, T , µ ) is saturated if L S ( µ ) is non-separable for every S ∈ T with µ ( S ) > . A saturated measure space is also called “super-atomless” in Podczeck [33]. Other equivalentdefinitions for saturation are available in the literature; see [10], [11], [17], [20], and [33]. As men-tioned in Khan and Sagara [24], “a germinal notion of saturation already appeared in [19, 30],”and Kakutani [19] constructed a non-separable extension of the Lebesgue measure space whichcan be seen as a saturated extension of the Lebesgue interval. Examples of saturated measurespaces include the product spaces of the form [0 , κ and { , } κ , where κ is an uncountable car-dinal, [0 ,
1] is endowed with the Lebesgue measure and { , } the fair coin flipping measure. Thecardinalities of these two examples are greater than the continuum. Podczeck [33] constructed asaturated measure structure on the unit interval by “enriching” the Lebesgue σ -algebra. Thus,as is pointed out in [33], when we have a saturated measure space of agents, the cardinality ofthe set of agents is not necessarily larger than the continuum. The commodity space E is an ordered separable Banach Space with an interior point v in E + . For the space of agents, we employ a complete probability space ( T, T , µ ) which is saturated.Let X be a correspondence from T to E + . The consumption set of agent t ∈ T is given by X ( t ) ⊂ E + . The initial endowment of each agent is given by a Bochner integrable function e : T → E where e ( t ) ∈ X ( t ) for all t ∈ T . The aggregate initial endowment is R T e ( t ) dµ . Let Y be a correspondence from T to E . The production set of agent t is given by Y ( t ) ⊂ E . A price is p ∈ E ∗ + \{ } . Let ∆ = { p ∈ E ∗ + \{ } : p · v = 1 } be the price space. Then by Alaoglu’s theorem, ∆is weak* compact. Let E = [( T, T , µ ) , ( X ( t ) , Y ( t ) , U t , e ( t )) t ∈ T ] be a production economy where U t : X ( t ) × ∆ → R represents agent t ’s utility function. We also write U ( t, x, p ) = U t ( x, p ) for t ∈ T , x ∈ X ( t ) and p ∈ ∆. An allocation for E is a Bochner integrable function f : T → E + suchthat f ∈ S X and a production plan is a Bochner integrable function g : T → E such that g ∈ S Y .The budget set of agent t at a price p ∈ ∆ is B ( t, p ) = { x ∈ X ( t ) : p · x ≤ p · e ( t ) + max p · Y ( t ) } .A competitive equilibrium for E is a triple of a price p , an allocation f and a productionplan g such that1. p · f ( t ) ≤ p · e ( t ) + p · g ( t ) for almost all t ∈ T ,2. R T f ( t ) dµ ≤ R T e ( t ) dµ + R T g ( t ) dµ , We are grateful to an anonymous referee for drawing our attention to Kakutani [19]. The examples of this space include C ( K ), the set of bounded continuous functions on a Hausdorff compactmetric space K equipped with sup norm and a weakly compact subset of L ∞ ( µ ) where µ is a finite measure.
5. for any x ∈ X ( t ), U t ( x, p ) > U t ( f ( t ) , p ) implies that p · x > p · e ( t ) + p · g ( t ) for almost all t ∈ T ,4. p · g ( t ) = max p · Y ( t ) for almost all t ∈ T .We assume that the production economy E satisfies the following assumptions:A.1 X ( t ) is non-empty, closed, convex, integrably bounded and weakly compact for all t ∈ T .A.2 Y ( t ) is non-empty, closed, integrably bounded and weakly compact for all t ∈ T .A.3 There is an element η ( t ) ∈ X ( t ) such that e ( t ) − η ( t ) is in the norm interior of E + for all t ∈ T .A.4 (i) U t : X ( t ) × ∆ → R is a jointly continuous function on X ( t ) × ∆ for all t ∈ T where X ( t ) is equipped with the weak topology and ∆ with the weak* topology. (ii) If x ∈ X ( t )is a satiation point for U t ( · , p ), then x ≥ e ( t ) + y for any y ∈ Y ( t ); if x ∈ X ( t ) is nota satiation point for U t ( · , p ), then x belongs to the weak closure of the set { x ′ ∈ X ( t ) : U t ( x ′ , p ) > U t ( x, p ) } for every p ∈ ∆.A.5 U is jointly measurable with respect to T ⊗ B ( E, w ) ⊗ B ( E ∗ , w ∗ ).A.6 the correspondence X : T → E has a measurable graph, i.e., G X ∈ T ⊗ B ( E, w ).A.7 the correspondence Y : T → E has a measurable graph, i.e., G Y = { ( t, y ) ∈ T × E : y ∈ Y ( t ) } ∈ T ⊗ B ( E, w ).A.8 ∈ Y ( t ) for all t ∈ T where is the zero vector of E .In A.1 and A.2, we assume that both the consumption sets and the production sets areweakly compact. Although these assumptions seem strong, the weakly compact consumptionset assumption was employed in Khan and Yannelis [27], Podczeck [32] and Khan and Sagara[24]. With this assumption, Khan and Yannelis [27] made the set of feasible allocations weaklycompact, Podczeck [32] obtained a weakly compact-valued demand correspondence, and Khanand Sagara [24] were able to invoke the exact Fatou’s lemma for an infinite dimensional separableBanach space. We use the weak compactness assumption to apply the exact Fatou’s lemma forour results. A.4 (ii) is imposed in [24, 28, 32] and the second part plays a similar role to the“local nonsatiation” assumption. Since these three papers dealt with exchange economies, the production sets are irrelevant. Results
The following theorem is our main result:
Main Theorem.
Suppose that the production economy E satisfies A.1-A.8. Then there existsa competitive equilibrium for E . The proof of the Main Theorem is provided in Section 6. As is well known, for x ∈ E and p ∈ ∆ the bilinear map ( p, x ) p · x is not jointly continuous if E is equipped with the weaktopology and ∆ with the weak* topology. But when E is equipped with the norm topology, thebilinear map is continuous. To utilize this property, we modify A.1 and A.2:A.1 ′ X ( t ) is non-empty, closed, convex, integrably bounded and norm compact for all t ∈ T .A.2 ′ Y ( t ) is non-empty, closed, integrably bounded and norm compact for all t ∈ T .We now introduce the following auxiliary result: Auxiliary Theorem.
Suppose that the production economy E satisfies A.1 ′ , A.2 ′ and A.3-A.8.Then there exists a competitive equilibrium for E . We provide the proof of the Auxiliary Theorem in Section 5. We follow the idea of [12] forthe proof of the Auxiliary Theorem. Greenberg et al. [12] applied Debreu’s [8] social equilibriumresult to prove the existence of a competitive equilibrium.We introduce a 3-person game Γ which consists of three sets K , K , K , and three corre-spondence A : K × K → K , A : K × K → K , A : K × K → K , and three functions u i : K × K × K → R ( i = 1 , , I = { , , } and let K − i = Π j = i K j ( i, j ∈ I ). We write k i for an element in K i and k − i for K − i .An equilibrium for Γ is k ∗ ∈ K × K × K such that for all i ∈ Ik ∗ i ∈ argmax k i ∈ A i ( k ∗− i ) u i ( k i , k ∗− i ) . (4.1)The following lemma is Debreu’s [8] social equilibrium theorem for a Banach space. Lemma 1.
Let Γ be a 3-person game and suppose Γ satisfies, for i ∈ I ,(i) K i is a non-empty, convex, and compact subset of a Banach space;(ii) A i is continuous, non-empty, closed and convex valued;(iii) u i is continuous and quasi-concave on K i .Then Γ has an equilibrium. See Aliprantis and Border [1] pp. 241-242. roof. By applying a standard argument to our Banach space, we can have the result.Based on Lemma 1, we will prove the Auxiliary Theorem. Toward this end, we specify ourΓ. Without loss of generality, we assume the values of U t are contained in [0 ,
1] for all t ∈ T .Let K = ∆, K = R T X ( t ) dµ × [0 , K = R T Y ( t ) dµ . For p ∈ K , ( x, α ) ∈ K and y ∈ K ,let A (( x, α ) , y ) = K , A ( p, y ) = { ( x, α ) ∈ K : ∃ f ∈ S X such that x = R T f ( t ) dµ, f ( t ) ∈ B ( t, p ) a.e t ∈ T, α = R T U t ( f ( t ) , p ) dµ } , A ( p, ( x, α )) = K , and u ( p, ( x, α ) , y ) = p · ( x − Z T e ( t ) dµ − y ) , u ( p, ( x, α ) , y ) = α, u ( p, ( x, α ) , y ) = p · y. (4.2) Lemma 2.
Under A.1 ′ and A.2 ′ , R T X ( t ) dµ and R T Y ( t ) dµ are norm compact and convex.Proof. By appealing to Proposition 1 in Sun and Yannelis [37], we have the results.
Lemma 3. B ( t, p ) is a non-empty and continuous correspondence in p when X ( t ) and Y ( t ) arenorm compact and ∆ is weak* compact.Proof. By A.8, it is clear that max p · Y ( t ) ≥
0. Then η ( t ) ∈ B ( t, p ) for any p ∈ ∆. Therefore, B ( t, p ) is non-empty.Let ψ t : ∆ → R be a function defined by ψ t ( p ) = max y ∈ Y ( t ) p · y. By Berge’s theorem, ψ t ( p )is continuous in p . We define a function z t : ∆ → R by z t ( p ) = p · e ( t ) + max p · Y ( t ) = p · e ( t ) + ψ t ( p ) . (4.3)Clearly, z t ( p ) is continuous in p . The budget correspondence can be rewritten as B ( t, p ) = { x ∈ X ( t ) : p · x ≤ z t ( p ) } . By A.3 and A.8, z t ( p ) > p ∈ ∆. Then a standard argument canbe adopted to show that B ( t, p ) is continuous in p .The following is the exact Fatou’s lemma for Banach spaces proved by Khan and Sagara[22]. Lemma 4 (Theorem 3.5 in [22]) . Let ( T, T , µ ) be a complete saturated finite measure spaceand E be a Banach space. If { f n } is a well-dominated sequence in L ( µ, E ) , then there exists f ∈ L ( µ, E ) such that(i) f ( t ) ∈ w -Ls { f n ( t ) } a.e. t ∈ T ,(ii) R f dµ ∈ w -Ls { R f n dµ } . Lemma 5.
Under A.1 ′ and A.2 ′ , A i is continuous, non-empty, closed and convex valued for i = 1 , , . roof. We adopt the idea of the proof from [12]. It is clear that K = ∆ is non-empty andconvex. By Alaoglu’s theorem, it is weak* compact and thus, weak* closed. It follows that A is non-empty, closed and convex valued. From A.8, ∈ R T Y ( t ) dµ and thus K = R T Y ( t ) dµ isnon-empty. By Lemma 2, R T Y ( t ) dµ is convex and norm compact and thus, norm closed. Hence, A is non-empty, closed and convex valued. Clearly, A and A are continuous.We now turn to A . Since the initial endowment map e ( t ) ∈ B ( t, p ), A is non-empty. Note R T e ( t ) dµ ∈ R T X ( t ) dµ for all t ∈ T and R T U t ( e ( t ) , p ) dµ ∈ [0 , K = R T X ( t ) dµ × [0 ,
1] isnon-empty. By Lemma 2, R T X ( t ) dµ is norm compact and convex. It follows that K is compactand convex.We show the value of A is closed. We need to show ( x, α ) ∈ A ( p, y ) when x n → x in normand α n → α such that ( x n , α n ) ∈ A ( p, y ) for all n . Then there exists a sequence { f n } ⊂ S X such that x n = R T f n ( t ) dµ and α n = R T U t ( f n ( t ) , p ) dµ with f n ( t ) ∈ B ( t, p ) for all n . By virtueof A.1 ′ , { f n } is well-dominated. We can appeal to Lemma 4 to have f ∈ L ( µ, E ) such that f ( t ) ∈ X ( t ) , f ( t ) ∈ w -Ls { f n ( t ) } for a.e. t ∈ T, and R T f ( t ) dµ ∈ w -Ls { R f n dµ } . Thus we canextract a subsequence from { f n } (which we do not relabel) such that f n ( t ) → f ( t ) weakly for a.e. t ∈ T and R T f n ( t ) dµ → R T f ( t ) dµ weakly. Since B ( t, p ) is norm compact and f n ( t ) ∈ B ( t, p )for all n , it follows f ( t ) ∈ B ( t, p ). The weak limit R T f ( t ) dµ of the subsequence of { x n } mustbe equal to the norm limit x of the whole sequence { x n } . Because U t ( · , p ) is weakly continuous, U t ( f n ( t ) , p ) → U t ( f ( t ) , p ) for a.e. t ∈ T . On the other hand, let g n ( t ) = U t ( f n ( t ) , p ). Then fromthe boundedness of U , the sequence of functions { g n } is well-dominated. Lemma 4 implies thatthere exists g ∈ L ( µ ) such that g n ( t ) → g ( t ) for a.e. t ∈ T and α n = R T g n ( t ) dµ → R T g ( t ) dµ up to subsequence. Hence g ( t ) = U t ( f ( t ) , p ) for a.e. t ∈ T and α = lim n α n = R T g ( t ) dµ = R T U t ( f ( t ) , p ) dµ. Next, we show the upper semicontinuity of A . Since K is compact, in order to prove A is upper semicontinuous, it is sufficient to show that the graph of A is closed. Let p n → p inthe weak* topology and y n → y in the norm topology. We want to show that ( x, α ) ∈ A ( p, y )when x n → x in norm and α n → α with ( x n , α n ) ∈ A ( p n , y n ) for all n . There exists { f n } suchthat x n = R T f n ( t ) dµ and α n = R T U t ( f n ( t ) , p n ) dµ with f n ( t ) ∈ B ( t, p n ) for a.e t ∈ T for all n .Clearly { f n } is well-dominated.Let g n ( t ) = U t ( f n ( t ) , p n ) and φ n ( t ) = ( f n ( t ) , g n ( t )). Then it is clear that { g n } and { φ n } are both well-dominated. Consequently, there exists an integrable function φ on T such that φ ( t ) ∈ w -Ls { φ n ( t ) } a.e. t ∈ T and R T φdµ ∈ w -Ls { R T φ n dµ } by Lemma 4, where φ ( t ) =( f ( t ) , g ( t )) for some f ∈ L ( µ, E ) and g ∈ L ( µ ) with f ( t ) ∈ X ( t ) and g ( t ) ∈ R . Thenwe can extract a convergent subsequence { φ n } (we do not relabel) such that φ n ( t ) → φ ( t )weakly for a.e. t ∈ T and R T φ n ( t ) dµ → R T φ ( t ) dµ weakly. So we have f n ( t ) → f n ( t ) weaklyfor a.e. t ∈ T , g n ( t ) → g ( t ) weakly for a.e. t ∈ T , R T f n ( t ) dµ → R T f ( t ) dµ weakly and α n = R T g n ( t ) dµ → R T g ( t ) dµ . 9ecause x n = R T f n ( t ) dµ converges to x in norm, R T f ( t ) dµ = x . By the joint continuity of U t , U t ( f n ( t ) , p n ) → U t ( f ( t ) , p ) for a.e. t ∈ T. Hence, we have g ( t ) = U t ( f ( t ) , p ) a.e. t ∈ T and R T U t ( f ( t ) , p ) dµ = R T g ( t ) dµ = lim n α n = α. Now it remains to show f ( t ) ∈ B ( t, p ). Because X ( t ) is norm compact, f n ( t ) converges up tosubsequence to some limit in norm, which must be equal to f ( t ). It follows that for a.e. t ∈ T , p n · f n ( t ) → p · f ( t ). Since p n · f n ( t ) ≤ p n · e ( t ) + max p n · Y ( t ), we have p · f ( t ) ≤ p · e ( t ) + max p · Y ( t ) . (4.4)Therefore, f ( t ) ∈ B ( t, p ) for almost all t ∈ T . In sum, we showed that A is norm uppersemicontinuous.We now prove the lower semicontinuity of A . Suppose ( x, α ) ∈ A ( p, y ). In order to show A is lower semicontinuous, it suffices to find a sequence ( x n , α n ) such that ( x n , α n ) ∈ A ( p n , y n )converging to ( x, α ) in norm. Since ( x, α ) ∈ A ( p, y ), there exists a function f such that x = R T f ( t ) dµ and α = R T U t ( f ( t ) , p ). Notice that since for any p ∈ ∆, B ( t, p ) is a norm closedsubset of X ( t ), it is norm compact. Clearly it is convex.Consider p n → p in the weak* topology and, y n → y in the norm topology. Note that B ( t, p n ) is convex and norm compact. Thus one can choose f n ( t ) from B ( t, p n ) such that f n ( t )is the closest to f ( t ), i.e., k f n ( t ) − f ( t ) k ≤ k z − f ( t ) k for all z ∈ B ( t, p n ) . (4.5)We will show that f n is measurable. Note that B ( · , p ) has a measurable graph. To see this,we adopt [27]. For p ∈ ∆, define ξ p : T × E → [ −∞ , ∞ ] by ξ p ( t, x ) = p · x − p · e ( t ) − max p · Y ( t ).By Proposition 3 in [16] (p.60), max p · Y ( t ) is measurable in t . Then ξ p is measurable in t andcontinuous in x . By Proposition 3.1 in [39], ξ p ( · , · ) is jointly measurable. Notice that G B ( · ,p ) = { ( t, x ) ∈ T × X ( t ) : p · x ≤ p · e ( t ) + max p · Y ( t ) } = ξ − p ([ −∞ , ∩ G X (4.6)and thus the budget correspondence B ( · , p ) is graph measurable given p .By Castaing’s Representation Theorem in [39], there exists { h nm ( t ) : m ∈ N } whose normclosure is B ( t, p n ). LetΨ nm ( t ) = { z ∈ B ( t, p n ) : k z − f ( t ) k ≤ k h nm ( t ) − f ( t ) k} (4.7)and Ψ n ( t ) ≡ ∩ ∞ m =1 Ψ nm ( t ) . (4.8)From the fact that B ( t, p ) is norm compact and the continuity of k·k , it follows that Ψ nm ( t ) is anon-empty measurable correspondence. Then the correspondence Ψ n : T → E has a measurable10raph. Since the set { h nm ( t ) : m ∈ N } is dense in B ( t, p n ), only the closest point f n ( t ) to f ( t )belongs to Ψ n ( t ). Therefore Ψ n is a measurable function which is equal to f n for µ − almost all t ∈ T . Hence, f n is measurable for all n . It is now clear that f n ∈ S X for all n .We will show that R T f n ( t ) dµ → R T f ( t ) dµ in norm. Let ε >
0. Pick b ∈ B ( t, p ) ∩ N ε ( f ( t ))where N ε ( f ( t )) is a neighborhood of f ( t ) with the radius ε . Suppose b / ∈ B ( t, p n ) for infinitelymany n . Then p n · b > p n · e ( t ) + max p n · Y ( t ) . (4.9)For some ε ∈ (0 , p n · εb > p n · e ( t ) + max p n · Y ( t ) . (4.10)As n → ∞ , it follows p · εb ≥ p · e ( t ) + max p · Y ( t ) (4.11)which contradicts b ∈ B ( t, p ).Thus, there is a ¯ n such that b ∈ B ( t, p n ) for all n ≥ ¯ n . Because of the minimizing prop-erty (4.5) of f n ( t ) in B ( t, p n ), we have k f n ( t ) − f ( t ) k < ε . So lim n →∞ R T U t ( f n ( t ) , p n ) dµ = R T U t ( f ( t ) , p ) dµ . And the Dominated Convergence Theorem in [9] sayslim n →∞ Z T k f n ( t ) − f ( t ) k dµ = 0 . (4.12)Let x n = R T f n ( t ) dµ and α n = R T U t ( f n ( t ) , p n ) dµ . Then ( x n , α n ) ∈ A ( p n , y n ) for all n ≥ ¯ n .Moreover, k x n − x k = (cid:13)(cid:13)(cid:13)(cid:13)Z T f n ( t ) dµ − Z T f ( t ) dµ (cid:13)(cid:13)(cid:13)(cid:13) ≤ Z T k f n ( t ) − f ( t ) k dµ → . (4.13)The last inequality comes from Theorem 4 in [9] (p.46). Hence, x n → x in norm and α n → α .It follows that A is norm lower semicontinuous.We will show that A is convex valued. Pick ( x, α ) ∈ A ( p, y ) and ( x ′ , α ′ ) ∈ A ( p, y ). Thenthere is a function f : T → E such that R T f ( t ) dµ = x and R T U t ( f ( t ) , p ) dµ = α with f ( t ) ∈ B ( t, p ) a.e. t and a function f ′ : T → E such that R T f ′ ( t ) dµ = x ′ and R T U t ( f ′ ( t ) , p ) dµ = α ′ with f ′ ( t ) ∈ B ( t, p ) a.e t . Let Z = E × R and we define a function h : T → Z by h ( t ) =( f ( t ) , U t ( f ( t ) , p )) and a function h ′ : T → Z by h ′ ( t ) = ( f ′ ( t ) , U t ( f ′ ( t ) , p )). It is clear that h, h ′ ∈ L ( µ, Z ). Let ν be a measure defined by ν ( S ) = ( Z S h ( t ) dµ, Z S h ′ ( t ) dµ ) (4.14)for S ∈ T . Notice that ν ( ∅ ) = (( , , ( , ν ( T ) = (( x, α ) , ( x ′ , α ′ )). It follows fromTheorem 4.1 in [21] (Lyapunov’s convexity theorem) that the range of ν is convex. Thus there See Theorem 3 in [9] p. 45. S ∈ T such that ν ( S ) = λν ( T ) = (( λx, λα ) , ( λx ′ , λα ′ )) for λ ∈ (0 , f λ = f χ S + f ′ χ T \ S . Then R T f λ ( t ) dµ = R S f ( t ) dµ + R T \ S f ′ ( t ) dµ = λx + (1 − λ ) x ′ and R S U t ( f ( t ) , p ) dµ + R T \ S U t ( f ′ ( t ) , p ) dµ = λα + (1 − λ ) α ′ . It is clear that f λ ( t ) ∈ B ( t, p ). Therefore, A is a convexvalued correspondence. Lemma 6. Γ has an equilibrium.Proof. As we proved in the proof of Lemma 5, K , K and K are non-empty, convex andcompact. Therefore, (i) of Lemma 1 is satisfied. Lemma 5 shows that A i ( i = 1 , ,
3) satisfies(ii) of Lemma 1. It is easy to see that u i ( i = 1 , ,
3) is continuous and quasi-concave on K i . Hence, (iii) of Lemma 1 holds. Now we can appeal to Lemma 1 to have an equilibrium( p ∗ , ( x ∗ , α ∗ ) , y ∗ ) for Γ. We are now ready to provide the proof of the Auxiliary Theorem.
Proof of the Auxiliary Theorem.
We will prove that for an equilibrium for Γ, there is acompetitive equilibrium for the economy.Suppose that ( p ∗ , ( x ∗ , α ∗ ) , y ∗ ) is an equilibrium for Γ. Hence there exist f ∗ ∈ S X such thatthat x ∗ = R T f ∗ ( t ) dµ with f ∗ ( t ) ∈ B ( t, p ∗ ) and g ∗ ∈ S Y such that y ∗ = R T g ∗ ( t ) dµ . We will showthat ( p ∗ , f ∗ , g ∗ ) is a competitive equilibrium for the economy.(i) We show that g ∗ is a profit maximization production plan.By the definition of u , p ∗ · y ∗ = p ∗ · R T g ∗ ( t ) dµ ≥ p ∗ · y for any y ∈ R T Y ( t ) dµ . Therefore, p ∗ · R T g ∗ ( t ) dµ = max p ∗ · R T Y ( t ) dµ . By Proposition 6 in [16] (p.63), we have max p ∗ · R T Y ( t ) dµ = R T max p ∗ · Y ( t ) dµ . Thus p ∗ · g ∗ ( t ) = max p ∗ · Y ( t ) for almost all t ∈ T . Note that Proposition6 in [16] works in our commodity space E .(ii) Let us prove p ∗ · f ∗ ( t ) ≤ p ∗ · e ( t ) + p ∗ · g ∗ ( t ) a.e. t ∈ T .Note that f ∗ ( t ) ∈ B ( t, p ∗ ) = { x ∈ X ( t ) : p ∗ · x ≤ p ∗ · e ( t ) + max p ∗ · Y ( t ) } for almost all t ∈ T . From p ∗ · g ∗ ( t ) = max p ∗ · Y ( t ) for a.e. t ∈ T , we have the desired result.(iii) We show that U t ( x, p ∗ ) > U t ( f ∗ ( t ) , p ∗ ) implies p ∗ · x > p ∗ · e ( t ) + p ∗ · g ∗ ( t ) for almost all t ∈ T .By way of contradiction, suppose there exists a non-empty subset S ∈ T which is of positivemeasure and let F be a correspondence from S to X ( t ) defined by F ( t ) = { x ∈ X ( t ) : U t ( x, p ∗ ) >U t ( f ( t ) , p ∗ ) and p ∗ · x ≤ p ∗ · e ( t ) + p ∗ · g ∗ ( t ) } for all t ∈ S . Recall that U t ( · , p ∗ ) is measurableon the graph of X . Recall also that B ( · , p ∗ ) and X have measurable graphs. Therefore, F has a measurable graph. Moreover, since X is integrably bounded, so is F . Hence, there isa Bochner integrable selection f ′ of F . We now define f ′′ = f ′ χ S + f ∗ χ T \ S . It is clear that12 T U t ( f ′′ ( t ) , p ∗ ) dµ = R S U t ( f ′ ( t ) , p ∗ ) dµ + R T \ S U t ( f ∗ ( t ) , p ∗ ) dµ > R T U t ( f ∗ ( t ) , p ∗ ) dµ = α ∗ which isa contradiction.(iv) We prove that ( f ∗ , g ∗ ) is a feasible allocation and a production plan.We know that p ∗ · f ∗ ( t ) ≤ p ∗ · e ( t ) + p ∗ · g ∗ ( t ) a.e. t ∈ T . By aggregating over T , we have p ∗ · ( R T f ∗ ( t ) dµ − R T e ( t ) dµ − R T g ∗ ( t ) dµ ) ≤
0. From the definition of the equilibrium of Γ, itfollows that for any p ∈ ∆, p · ( Z T f ∗ ( t ) dµ − Z T e ( t ) dµ − Z T g ∗ ( t )) dµ ≤ p ∗ · ( Z T f ∗ ( t ) dµ − Z T e ( t ) dµ − Z T g ∗ ( t ) dµ ) ≤ . (5.1)Therefore, − ( R T f ∗ ( t ) dµ − R T e ( t ) dµ − R T g ∗ ( t ) dµ ) ∈ E + which leads to R T f ∗ ( t ) dµ ≤ R T e ( t ) dµ + R T g ∗ ( t ) dµ . We provide the proof of the Main Theorem. The proof follows Noguchi [31] by considering a netof truncated subeconomies, whose consumption and production sets are norm compact, whichis in line with Toussaint [38] and Khan and Yannleis [27]. From the Auxiliary Theorem, wehave a net of competitive equilibria for the subeconomies. We then construct a sequence ofcompetitive equilibria. Finally, by invoking the exact Fatou’s lemma for infinite dimensionalseparable Banach spaces, we obtain a competitive equilibrium for the original economy.
Proof of the Main Theorem.
As in Noguchi [31], we construct the norm compact subsetsof X ( t ) and Y ( t ). Let F = { K : T → E | K = co( K X ∪ K Y ) where K X = co( ∪ mi =1 ϕ i ) , K Y =co( ∪ lj =1 ψ j ) such that ϕ i : T → E and ψ j : T → E are measurable with ϕ i ( t ) ∈ X ( t ) and ψ j ( t ) ∈ Y ( t ) for all t ∈ T ; e ( t ) , η ( t ) ∈ K X ( t ) and 0 ∈ K Y ( t ) for all t ∈ T } .Consider K = co( K X ∪ K Y ) such that K X ( t ) = co( e ( t ) ∪ η ( t )) for all t ∈ T and K Y ( t ) =0 for all t ∈ T . Then K ∈ F and thus F is non-empty. Let K , K ∈ F . Then it is clear thatco( K ∪ K ) ∈ F , which implies that F is directed under the inclusion. Notice that for every t ∈ T , K X ( t ) = co( ∪ mi =1 ϕ i ( t )) and K Y ( t ) = co( ∪ lj =1 ψ j ( t )) are norm compact and thus K ( t ) isalso norm compact (see Jameson [18] p.208). Now it follows that for K ∈ F , K X and K Y arenon-empty, convex and norm compact valued, respectively. By Theorem III. 30 in [7], K X and K Y are graph measurable.We define a truncated economy E K = [( T, T , µ ) , ( K X ( t ) , K Y ( t ) , U Kt , e ( t )) t ∈ T ] where U Kt isthe utility function U t whose first domain is restricted to K X ( t ). Since K X ( t ) is convex andnorm closed, by the separation theorem it is weakly closed. Thus it belongs to the Borel σ -algebra generated by the weak topology of E . It is clear that U K is measurable.It is easy to see that E K satisfies all the assumptions of the Auxiliary Theorem. Therefore,we appeal to the Auxiliary Theorem to obtain a competitive equilibrium ( p K , f K , g K ) for E K .13otice that { ( p K , f K , g K ) : K ∈ F } is a net directed by inclusion. For all K ∈ F , K X ( t ) ⊂ X ( t )and, by A.1, X is integrably bounded and weakly compact valued. Thus { f K } is well-dominated.We apply the same logic to K Y and Y to see { g K } is also well-dominated.Since X and Y are non-empty closed valued correspondences by A.1 and A.2, ( T, T , µ ) acomplete probability space, E a complete separable metric space, by Theorem III. 30 in [7] thereare two sequences of measurable functions ϕ i : T → E and ψ i : T → E such thatcl { ϕ i ( t ) } i ∈ N = X ( t ) and cl { ψ j ( t ) } j ∈ N = Y ( t ) for all t ∈ T. (6.1)We then construct K Xm ( t ) using { ϕ i ( t ) } mi =1 and K Yl ( t ) using { ψ j ( t ) } lj =1 . Let us define n =min { m, l } where m, l are the numbers of ϕ i and of ψ j in K , respectively. Then consider asequence of truncated subeconomies {E n } consisting of K Xn ( t ) and K Yn ( t ) for all t ∈ T . By theAuxiliary Theorem, we now have a sequence of competitive equilibria ( p n , f n , g n ) for E n .We appeal to Lemma 4 to have f ∈ L ( µ, E ) and g ∈ L ( µ, E ) such that f ( t ) ∈ X ( t ), f ( t ) ∈ w -Ls { f n ( t ) } a.e t ∈ T and R T f dµ ∈ w -Ls { R T f n dµ } as well as g ( t ) ∈ Y ( t ), g ( t ) ∈ w -Ls { g n ( t ) } a.e. t ∈ T and R T gdµ ∈ w -Ls { R T g n dµ } . Therefore, f is an allocation and g is a productionplan. Since p n belongs to ∆ which is weak* compact, it has a subsequence still denoted by p n weak* converging to p .We will now show that ( p, f, g ) is a competitive equilibrium for E .Step 1: Let us show that for x ∈ X ( t ), U t ( x, p ) > U t ( f ( t ) , p ) implies p · x > p · e ( t ) + max p · Y ( t ) for almost all t ∈ T. (6.2)We follow Khan and Sagara [24] for this proof. By method of contradiction, suppose thatthere exists S ∈ T of positive measure with the following property: for every t ∈ S thereexists ˆ x ∈ X ( t ) such that U t (ˆ x, p ) > U t ( f ( t ) , p ) and p · ˆ x ≤ p · e ( t ) + max p · Y ( t ). Since p · e ( t ) + max p · Y ( t ) > U t that U t ( ε ˆ x, p ) > U t ( f ( t ) , p ) and p · ε ˆ x < p · e ( t ) + max p · Y ( t ) for some ε ∈ (0 , t ∈ S there exists ˆ x ∈ X ( t ) such that U t (ˆ x, p ) > U t ( f ( t ) , p ) and p · ˆ x < p · e ( t ) + max p · Y ( t ). Let us define the correspondenceΛ : S → E byΛ( t ) = { x ∈ X ( t ) | U t ( x, p ) > U t ( f ( t ) , p ) , p · x < p · e ( t ) + max p · Y ( t ) } . Λ is an integrably bounded correspondence and ˆ x ∈ Λ( t ). We now show that Λ is graphmeasurable. Let Λ ( t ) := { x ∈ X ( t ) | U t ( x, p ) > U t ( f ( t ) , p ) } and Λ ( t ) := { x ∈ E | p · x
U t ( x ′ , p ) } ∩ (( G X ) × E ) ∩ proj T × E × E ( G ζ ) . Then in view of A.5 and A.6, H belongs to T ⊗B ( E, w ) ⊗B ( E, w ). Let proj T × E be the projectionof ( T × E ) × E onto T × E . Since G Λ = proj T × E ( H ), we again appeal to the projection theoremto argue that G Λ belongs to T ⊗ B ( E, w ).Therefore, Λ has a measurable selection by Aumann’s measurable selection theorem in [4].Let h : S → E be a measurable selection from Λ. By Theorem III. 30 in [7], we can choose asequence of measurable selections h n : S → E such that h n ( t ) ∈ X ( t ) converges to h ( t ) in normfor all t ∈ S . By Lemma 4, there exists a Bochner integrable function ˆ h : S → E such thatˆ h ( t ) ∈ w-Ls { h n ( t ) } and ˆ h ( t ) ∈ X ( t ) a.e. t ∈ S . Hence, there is a subsequence of { h n ( t ) } in E converging weakly to ˆ h ( t ) for a.e. t ∈ S . It is clear that ˆ h ( t ) = h ( t ) a.e. t ∈ S and we also have( f ( t ) , h ( t )) ∈ w-Ls { ( f n ( t ) , h n ( t )) } a.e. t ∈ S .Suppose now that the following set defined by [ n ∈ N { t ∈ S | U nt ( h n ( t ) , p n ) > U nt ( f n ( t ) , p n ) , p n · h n ( t ) < p n · e ( t ) + max p n · K Yn ( t ) } is of measure zero. Then for each n , U nt ( f n ( t ) , p n ) ≥ U nt ( h n ( t ) , p n ) or p n · h n ( t ) ≥ p n · e ( t ) +max p n · K Yn ( t ) a.e. t ∈ S . Notice for any y ∈ Y ( t ), there is a sequence of measurable functions y n : T → E such that y n ( t ) ∈ K Yn ( t ) converges to y in norm for all t ∈ T by Theorem III. 30 in [7].When p n converges to p in the weak* topology, we have p n · h n ( t ) → p · h ( t ) and p n · y n ( t ) → p · y .Passing to the limit yields U t ( f ( t ) , p ) ≥ U t ( h ( t ) , p ) or p · h ( t ) ≥ p · e ( t ) + max p · Y ( t ) a.e. t ∈ S . But this is a contradiction to the fact that h is a measurable selection from Λ. Hence, thereexists n such that { t ∈ S | U nt ( h n ( t ) , p n ) > U nt ( f n ( t ) , p ) , p n · h n ( t ) < p n · e ( t ) + max p n · K Yn ( t ) } is of positive measure. However, this contradicts the fact that p n and f n are a price and anallocation of a Walrasian equilibrium for E n . We proved (6.2).Indeed, we can further show that p · f ( t ) ≥ p · e ( t ) + max p · Y ( t ) (6.3)for almost all t ∈ T .By A.4 (ii), if f ( t ) is a satiation point, (6.3) follows. If f ( t ) is not a satiation point, f ( t )belongs to the weak closure of the upper contour set { x ′ ∈ X ( t ) : U t ( x ′ , p ) > U t ( f ( t ) , p ) } for any see Theorem III.23 in [7]. Note that for x ∈ K Xn ( t ), U nt ( x, p ) = U t ( x, p ) for any p ∈ ∆. ∈ ∆. Thus, (6.2) implies (6.3).Step 2: We show that f is a feasible allocation and g is a feasible production plan.Since ( p n , f n , g n ) is a competitive equilibrium for E n , it is clear that R T f n ( t ) dµ ≤ R T e ( t ) dµ + R T g n ( t ) dµ . Recall that for { f n } and { g n } Lemma 4 holds. Thus we can extract subsequences(which we do not relabel) from { f n } and { g n } such that R T f n ( t ) dµ → R T f ( t ) dµ weakly and R T g n ( t ) dµ → R T g ( t ) dµ weakly. Now from R T f n ( t ) dµ ≤ R T e ( t ) dµ + R T g n ( t ) dµ we obtain Z T f ( t ) dµ ≤ Z T e ( t ) dµ + Z T g ( t ) dµ. (6.4)Step 3: We prove that p · f ( t ) ≤ p · e ( t ) + p · g ( t ) for almost all t ∈ T .From (6.3), we have p · f ( t ) ≥ p · e ( t ) + p · g ( t ) (6.5)for almost all t ∈ T . By integrating (6.5) over T , Z T [ p · f ( t ) − p · e ( t ) − p · g ( t )] dµ = p · Z T [ f ( t ) − e ( t ) − g ( t )] dµ ≥ . (6.6)But from (6.4) it follows that p · Z T [ f ( t ) − e ( t ) − g ( t )] dµ = Z T [ p · f ( t ) − p · e ( t ) − p · g ( t )] ≤ . (6.7)Hence, we can conclude R T [ p · f ( t ) − p · e ( t ) − p · g ( t )] = 0 . Therefore, we have p · f ( t ) = p · e ( t ) + p · g ( t ) (6.8)for almost all t ∈ T .Step 4: Let us prove p · g ( t ) = max p · Y ( t ) a.e. t ∈ T .From (6.3) and (6.8), we have the following inequality:max p · Y ( t ) ≤ p · f ( t ) − p · e ( t ) = p · g ( t ) (6.9)for almost all t ∈ T . Obviously, we have max p · Y ( t ) ≥ p · g ( t ). Hence, the conclusion follows. Remark 1
We can appeal to Galerkin approximations, as suggested by Khan and Sagara [24],to construct a sequence of truncated subeconomies {E n } with finite dimensional commodity16paces. Each E n can have a competitive equilibrium ( p n , f n , g n ) due to Greenberg et al. [12].We then apply the exact Fatou’s lemma to the sequence of { f n , g n } to obtain f and g . Bythe weak* compactness of ∆, we can extract a subsequence from { p n } that weak* convergesto p ∈ ∆. Applying similar arguments as in Khan and Sagara [24], we can show that ( p, f, g )satisfies the properties of a competitive equilibrium. Remark 2
The Auxiliary theorem can be seen as a direct proof of Greenberg et al. [12] forinfinite dimensional commodity spaces without taking the approximation approach.
Remark 3
We can replace the weak compactness of production sets by the following condition:Let A Y be a set defined by A Y = { g ′ ∈ S Y : ∃ f ′ ∈ S X s.t. R T f ′ ( t ) dµ ≤ R T e ( t ) dµ + R T g ′ ( t ) dµ } .We assume A Y is weakly compact.Applying our approach in the proof of the main theorem, we construct a sequence of truncatedsubeconomies {E n } and obtain a sequence of competitive equilibria { p n , f n , g n } . Since X ( t ) isintegrably bounded and weakly compact, we apply the exact Fatou’s lemma to { f n } to have f and since A Y is weakly compact, we have g n → g in the weak topology. Also p n → p ∈ ∆ in theweak* topology. We are then able to prove that ( p, f, g ) is a competitive equilibrium.For the sequence { f n } , we need two results: (by passing to a subsequence) f n → f and f n ( t ) → f ( t ) for almost all t ∈ T . These results make f a feasible allocation and f ( t ) a maximalelement for the agent t . To our best knowledge, there are two ways to obtain these results:invoking the exact Fatou’s Lemma or appealing to Theorem 5.1 in Khan and Yannelis [27].Both approaches require weak compact subsets of the consumption sets. Therefore, even whenwe relax the weak compactness assumption of the consumption sets, we still need some weakcompact subsets of the consumption sets which contain the set of maximal elements. We are grateful to an anonymous referee for drawing our attention to Khan and Sagara [24]. Remark 3 in [12] provided an equilibrium existence result with non-compact consumption and productionsets for their economy. eferences [1] Aliprantis, C. and K. Border (2006), Infinite Dimensional Analysis , 3rd edition, New York,Springer-Verlag.[2] Aubin, J.-P. and H. Frankowska (1990),
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