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Equilibrium existence theorems with closed preferences
Nonlinear Analysis 63 (2005) e1833 – e1840 www.elsevier.com/locate/na
Equilibrium existence theorems with closed preferences Kok-Keong Tan∗,1 , Zhou Wu, X.Z. Yuan Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5
Abstract We first prove an equilibrium existence theorem for an abstract economy in which the preference correspondences are upper semicontinuous with closed values. As applications, we provide two examples to illustrate how the general result established in this paper can be used to give the existence of equilibria of economic models such as a pure exchange model and the Arrow–Debreu model. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: N-person game; Equilibrium; Lower (upper) semicontinuous; Fixed point
1. Introduction Following Debreu [4], = (Xi , Fi , Ui )N i=1 is called a generalized N-person game (or an abstract economy) if for each person (agent) i = 1, . . . , N, Xi is a choice set, Fi : X = i∈I Xi → 2Xi is a constraint correspondence and Ui : X → R is a utility (payoff) function. The objective of the ith agent is for each xˆ ∈ X to choose an action xi that maximizes Ui (xˆi , . . . , xˆi−1 , ., xˆi+1 , . . . , xˆN ) subject to xi ∈ Fi (x). ˆ The vector xˆ = (xˆ1 , . . . , xˆn ) of actions is an equilibrium for if xˆi maximizes Ui (xˆ1 , . . . , xˆi−1 , ., xˆi+1 , xˆN ) subject to xi ∈ Fi (x) ˆ for each i = 1, . . . , N. This notion of equilibria is a natural extension
∗ Corresponding author. Tel.: +1 902 494 2166; fax: +1 902 494 5130.
E-mail address: [email protected] (K. Tan). 1 The author is partially supported by Natural Sciences and Engineering Research Council of Canada under
grant A-8096. 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.02.064
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of the concept of an equilibrium introduced by Nash [10] for non-cooperative N-person games. If each Xi is a non-empty compact convex subset of R l , each Fi is continuous with non-empty convex values and each Ui is continuous on X and quasi-convex in xi , then Debreu [4] showed that an equilibrium exists. Arrow and Debreu [2] used this result to prove the existence of a competitive equilibrium. Following Shafer [12], E = (Yi , wi , Vi )N i=1 is said to be an economy if N is the number of individuals, and for each i = 1, . . . , N, Yi ⊂ R l is a consumption set of the ith individual, l is the ith individual’s initial endowment vector, and V : Y → R is a utility wi ∈ R+ i i function that represents the preferences of the ith individual. Let Y = N i=1 Yi and = l : l p = 1} denote the set of normalized prices. A competitive equilibrium {p ∈ R+ i i=1 for E is a point (y, ˆ p) ˆ ∈ Y × such that the following three conditions are satisfied: (1) N w ; (2) pˆ yˆ = pw N y ˆ ˆ i for each i = 1, . . . , N; (3) for i = 1, . . . , N, yˆi is the i i i=1 i=1 ˆ i and yi ∈ Yi . solution to maximize Vi subject to py ˆ i pw We associate E with an (N + 1)-person generalized game = (Yi , Fi , Vi )N i=1 in the following manner. For each i = 0, 1, . . . , N, the ith person has a utility function Vi , a choice set Yi and a constraint correspondence Fi defined by Fi (y, p) = {yi ∈ Yi : py i pwi } for each (y, p) ∈ Y × . The 0th person, called a market player, has a utility (payoff) function V0 that is defined by V0 (y, p) = p(yi − wi ) for each (y, p) ∈ Y × , a choice set and a constraint correspondence F0 defined by F0 (y, p) = for each (y, p) ∈ Y × . Then, if each Yi is compact and convex, wi ∈ int Yi and each Vi is continuous and quasi-concave, this (N + 1)-person generalized game will satisfy the sufficient conditions mentioned above for the existence of an equilibrium. It is easy to see that (y ∗ , p∗ ) is an equilibrium for E if for each i = 0, . . . , N, yi∗ is a maximum of Vi . Note that the above argument remains valid if each agent’s utility function Vi is assumed to depend not only on his own consumption yi , but also on the consumptions of the other agents and on the prices p. Thus, Arrow and Debreu also showed how to prove the existence of competitive equilibrium with consumption externalities and price dependent preferences. We now follow Shafer’s idea to extend the above result to the existence of equilibrium in abstract economy. Given an N-person generalized game = (Xi , Fi , Ui )N i=1 , for each i = 1, . . . , N, consider the correspondence Pi : X → 2Xi defined by Pi (x) = {zi ∈ Xi : Ui (x1 , . . . , xi−1 , zi , xi+1 , . . . , xn ) > Ui (x)} ˆ ∩ Fi (x) ˆ = ∅ and for each x ∈ X. Note that xˆ is an equilibrium of if and only if Pi (x) ˆ for each i = 1, . . . , N. We shall consider abstract economies in which the xˆi ∈ Fi (x) individual preferences are given by preference correspondence Pi rather than by utility function. In this formulation, preferences, which depend on the actions of others, are not required to be transitive nor asymmetric. In this paper, we first prove an equilibrium existence theorem for an abstract economy in which the preference correspondences are upper semicontinuous with closed values. Then, as applications, we provide two examples to illustrate how the general result established in this paper can be used to give the existence of equilibria of economic models such as pure exchange model and the Arrow–Debreu model.
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2. An equilibrium theorem for an abstract economy We recall that an abstract economy = (Xi , Fi , Pi )N i=1 is defined by N ordered triples (Xi , Fi , Pi ), where Fi , Pi : X → 2Xi are correspondences. An equilibrium for is an ˆ ∩ Fi (x) ˆ = ∅. ˆ and Pi (x) xˆ ∈ X satisfying for each i = 1, ..., N , xˆi ∈ Fi (x) Theorem 1. Let =(Xi , Fi , Pi )N i=1 be an abstract economy and I0 ∪I1 ={1, . . . , N} where I0 and I1 are disjoint such that: (i) For each i = 1, . . . , N, Xi is a non-empty, compact and convex subset of R li . (ii) For each i = 1, . . . , N, Fi is a continuous correspondence with non-empty compact convex values. (iii) (a) For i ∈ I0 , Pi has an open graph and convex values, (b) for i ∈ I1 , Pi is upper semicontinuous with closed convex values. (iv) For each i = 1, . . . , N, xi ∈ / Pi (x) for all x ∈ X. Then has an equilibrium. Proof. Let X = N i=1 Xi . (1) Fix an i ∈ I0 . Let Ui = {x ∈ X : Fi (x) ∩ Pi (x) = ∅}. Since Fi is lower semicontinuous and Pi has open graph, Fi ∩ Pi is lower semicontinuous by Lemma 4.2 in [18]. Hence Ui is open. By Michael selection theorem [9], there exists a continuous function fi such that fi (x) ∈ Fi (x) ∩ Pi (x) for each x ∈ Ui . Define Gi : X → Xi by Gi (x) =
Fi (x) if x ∈ / Ui , {fi (x)} if x ∈ Ui .
Then Gi (x) is upper semicontinuous with non-empty compact convex values. (2) Fix an i ∈ I1 . Let Mi = {x ∈ X : Fi (x) ∩ Pi (x) = ∅}. Then Mi is closed since Fi ∩ Pi is upper semicontinuous by Lemma 2.2 in [17]. Note that for each x ∈ Mi , Fi (x)∩Pi (x) ⊂ Fi (x), by Theorem in [11, p. 781], there exists an upper semicontinuous mapping Gi : X → 2Xi with non-empty compact convex values such that Gi |Mi =Fi ∩Pi and Gi (x) ⊂ Fi (x) for each x ∈ X. (3) Now define G : X → 2X by G(x) = N i=1 Gi (x) for each x ∈ X, then G is upper semicontinuous with non-empty compact convex values (by Lemma 3 in [6]). By Fan-Glickberg fixed point theorem [6,8], G has a fixed point x ∈ X. Clearly, x is an equilibrium for . Remark 1. Suppose Fi (x) = Xi for each i = 1, . . . , N and for all x ∈ X. (1) If I1 = ∅, then Theorem 2 reduces to the fixed point theorem in [7]. (2) If I0 = ∅, Theorem 2 can be regarded as the dual of [7]. Remark 2. (1) If I1 = ∅, Theorem 2 reduces to the abstract result in [14]. (2) If I0 = ∅, Theorem 2 can be regarded as the dual of [14]. Remark 3. By Lemma 2.3 of Tan and Yuan [16, p. 1851], it follows that the condition (iii)(a) in Theorem 2 is equivalent to the condition that “Pi is lower semicontinuous with convex and open values (which may be empty)”.
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3. Application to economic models Suppose there are n traders. For each trader i ∈ I = {1, . . . , n}, the trading set Xi is a subset of R l and there is a preference mapping Pi from Xi to 2Xi . There is also a subset Y of R l which is called the technology of the economy [7]. An allocation x is an n-tuple (x1 , . . . , xn ) where xi ∈ Xi for each i ∈ I . Thus x ∈ X = i∈I Xi . An allocation is said to be feasible if ni=1 xi ∈ Y . For a given allocation x = (x1 , . . . , xn ), x0 denotes the sum ni=1 xi . Let be the unit (l − 1)-simplex. Each element p of will be called a price vector. In formulating equilibrium models, we describe the way in which an allocation x is associated with the price vector p as follows: For each i ∈ I , let the trader i have his/her income i (p) at the price p. In the pure exchange model, it is assumed that the trader i has an initial endowment vector wi and his income is then given by i (p) = pwi . The Arrow–Debreu model (1954) involves a more complicated set of income functions. It is assumed that the technology set Y is the sum of some sub-technologies Y1 , . . . , Yr which are to be thought of as firms, and trader i is provided with portfolio vector i = (i1 , . . . , ir ), where ij represents trader i’s share of the firm Yj . The Arrow–Debreu income functions are then given by i (p) = pw i + rj =1 ij sup pY j . In the present treatment, by following [7], we wish to allow for more general income function. For any p ∈ , define the profit function (p) by (p) = sup pY . Since Y maybe unbounded, it follows that may be infinite. We define ⊂ by = {p ∈ : (p) < ∞}. The following lemma is implicitly contained in the proof of Lemma 2 in [7]; here we shall give a detailed proof: Lemma 2. Suppose Y ⊂ R l is closed convex, contains the negative orthant and has a bounded intersection other than {0} with the positive orthant. Let e be a point in Y and l and z ∈ Yˆ such that pz = max Yˆ = {y : y ∈ Y and y e}. Suppose for some p ∈ R+ y∈Yˆ py. Then there is q ∈ such that qz = maxy∈Y qy. Proof. Since pz=maxy∈Yˆ py, the set D := {y ∈ R l : y > z}, which is non-empty open convex and disjoint from Yˆ . D is also disjoint from Y. Otherwise, let y ∈ D ∩Y . Then y > z e so that y ∈ Yˆ which contradicts pz = maxy∈Yˆ py.Since Y is convex, by Hahn–Banach theorem, we can find q ∈ R l such that qy > qy for all y ∈ D and y ∈ Y . Note that since Y l . Take y = z and y in contains the negative orthant, it follows that q = (q1 , . . . , qn ) ∈ R+ n D converging to z. We have qz = maxy∈Y qy. Obviously, q 0 and q = 0. If li=1 qi = 1, replace it with q/ li=1 qi ∈ , which is required. Our hypothesis on Y will guarantee that = ∅. One also verifies that is convex. We also assume the existence of n real-valued functions i on (to be called income functions) satisfying the following formula: ni=1 i (p) = (p) for all p ∈ . A Walrasian (or a competitive) equilibrium [1] for the model described above consists of a price vector pˆ in and an allocation xˆ such that (1) pˆ xˆ i (p) ˆ for all i ∈ I (budget ˆ i > pˆ xˆi (preference condition); (3) xˆ inequality); (2) for each i ∈ I , if xi ∈ Pi (xˆi ), then px is feasible (balance of supply and demand).
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The following lemma is contained in the proof of Theorem 4 in [5, pp. 707–710]; here we shall give an explicit proof based on Debreu’s idea: Lemma 3. Suppose X is a non-empty compact convex subset of R l and (p) is a continuous function on a non-empty compact subset Y of R l . If (p) > minx∈X px for all p ∈ Y , then correspondence A : Y → 2X defined by A(p) = {x ∈ X : p · x (p)} is continuous. Proof. Since A is non-empty and closed-valued, Y is compact and A can be shown to have a closed-graph, A is upper semicontinuous. It remains to show that A is also lower semicontinuous. Let p 0 be any point inY. To show A is lower semicontinuous at p 0 , we consider a sequence 0 0 0 (pi )∞ i=1 in Y converging to p and a point z ∈ A(p ). ∞ i i i Let w =(p ) for i =1, . . ., then (w )i=1 converges to w 0 =(p0 ). For i =0, 1, . . ., since i w = (p i ) > minx∈X p i x, (p i , wi ) ∈ D := {(p, w) ∈ R l+1 : minx∈X px w}. By Lemma 3 in [5], : D → X defined by (p, w) = {x ∈ X : px w} is continuous at (p0 , w0 ). So i i i i for z0 ∈ A(p 0 ) = (p 0 , w0 ), there exists a sequence (zi )∞ i=1 , where z ∈ (p , w ) = A(p ) 0 for each i = 1, . . ., converging to z . Thus A is lower semicontinuous. Now our main application is the following theorem: Theorem 4. The following conditions are sufficient for the existence of an equilibrium: (1) the set Y is closed convex, contains the negative orthant, and has a bounded intersection with the positive orthant; (2) for each i = 1, . . . , n, the set Xi is non-empty closed convex and bounded below; (3) for each i = 1, . . . , n, the preference Pi is upper semicontinuous with closed convex / Pi (xi ) for all x ∈ X); values and is irreflexive (i.e., xi ∈ (4) for each i = 1, . . . , n, the function i (p) is continuous and satisfies i (p) > inf pX i for all p ∈ . Proof. As each Xi is bounded below, it follows there exists a vector e such that for any non-empty subset S of {1, . . . , n}, we have Xi . e< i∈S
Without loss of generality, we may assume that e < 0 so that e ∈ Y . Now define Yˆ = {y ∈ Y : y e}. Note that Yˆ contains all feasible x0 = ni=1 xi and by the condition of Y, Yˆ is also bounded above as well as below. Thus there exists a vector f such that f > Yˆ . By the feasibility, x0 = nj=1 xi < f − e, so that xi < f − j =i xj < f − e. Following [7], we define ⊂ as follows: = {p ∈ : py = (p) for some y ∈ Yˆ }. Take any p ∈ . Since Yˆ is compact, there is z ∈ Yˆ such that p · z = supy∈Yˆ p · y. By Lemma 2, there is q ∈ such that q · z = supy∈Y q · y = (q) so that q ∈ . Thus is non-empty. It is easy to see is closed. Now let ∗ be the convex hull of . Then ∗ ⊂ since is convex and ∗ is also closed as is closed.
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By the definition of , we shall show that there is a finite set Xi ⊂ Xi such that min pX i < i (p) for all p ∈ ∗ . Indeed, for each p ∈ ∗ , let xp ∈ Xi be such that px p < i (p). By the continuity of i , it follows that there is, for each p ∈ ∗ , an open neighborhood Vp of p in ∗ such that qx p < i (q) for each q ∈ Vp . The family of open sets {Vp : p ∈ ∗ } covers ∗ and since ∗ is compact, there is a finite subcover {Vp1 , . . . , Vpk }. Then take Xi = {xp1 , . . . , xpk }. Let r be sufficiently large and we define Xˆ i = {xi ∈ Xi : xi r}. Without loss of generality, we may assume that Xi ⊂ Xˆ i for all i = 1, . . . , n. ˆ For each i = 1, . . . , n, define Fi : ∗ × Xˆ → 2Xi by Fi (p, x) = {zi ∈ Xˆ i : pzi i (p)} ∗ n ˆ where Xˆ = Xˆ i . By the construction of Xˆ i above and the for each (p, x) ∈ × X, i=1 condition (4), it follows that Fi (p, x) is non-empty and convex for each (p, x) ∈ ∗ × Xˆ and, moreover, by Lemma 3 in [5], Fi is continuous with non-empty closed and convex values. ∗ ˆ Of course, F0 Now define F0 : ∗ × Xˆ → 2 by F0 (p, x)=∗ for each (p, x) ∈ ∗ × X. ∗ ˆ is continuous with non-empty closed convex values for all (p, x) ∈ × X. Finally, define ∗ P0 : ∗ × Xˆ → 2 by P0 (p, x) = {q ∈ ∗ : q · (ni=1 xi ) − (q) > p · (ni=1 xi ) − (p)} ˆ It follows that P0 has an open graph, each P0 (p, x) is convex for each (p, x) ∈ ∗ × X. ˆ (maybe empty), and p ∈ / P0 (p, x) for all (p, x) ∈ ∗ × X. ˆ Identify Pi with P¯i : ∗ × Xˆ → Xˆ i defined by P¯i (p, x)=Pi (xi ) for each (p, x) ∈ ∗ × X. By condition (3), it follows that the abstract economy (Xˆ i , Fi , Pi )ni=0 satisfies all hypotheses of Theorem 1, where Xˆ 0 = ∗ . By Theorem 1 with I0 = {0} and I1 = {1, . . . , n}, there exists (p, ˆ x) ˆ ∈ ∗ × Xˆ such that pˆ ∈ ∗ and n n ∗ ∗ F0 (p, ˆ ˆ x) ˆ ∩ P0 (p, ˆ x) ˆ = ∩ q ∈ :q xˆi − (q) > pˆ xˆi − (p) i=1
i=1
= ∅, which implies that for each q ∈ ∗ we have n n ˆ xˆi − (q) pˆ xˆi − (p). q i=1
(*)
i=1
ˆ x) ˆ for i = 1, 2, . . . , n and Fi (p, ˆ x) ˆ ∩ Moreover, for each i = 1, 2, . . . , n, xˆi ∈ Fi (p, Pi (p, ˆ x) ˆ = ∅. Now we prove that ˆ x) ˆ is a Walrasian equilibrium. First we claim that xˆ0 = ni=1 xˆi is n(p, / Y . We may choose such feasible, i.e., xˆ0 = i=1 xˆi ∈ Y . Suppose the contrary, i.e, xˆ0 ∈ that < , y = e + (1 − )xˆ0 in Y and y = e + (1 − )xˆ0 ∈ / Y for all < . Since xˆ0 > e, by a similar argument as in the proof of Lemma 2 there exists q in such that qy = (q) and q xˆ0 > (q). Note that y ∈ Yˆ , and it follows that q ∈ ⊂ ∗ . p( ˆ xˆ0 ) − (p). ˆ Thus p( ˆ xˆ0 ) > (p). ˆ However, we do Therefore, we have q(xˆ0 ) − (q) have that p( ˆ xˆ0 )= ni=1 p( ˆ xˆi ) ni=1 i (p)=( ˆ p), ˆ which is a contradiction. Thus we must have xˆ0 = ni=1 xˆi ∈ Y . ˆ i > pˆ xˆi . Since Pi (p, ˆ x) ˆ ∩ Finally, we wish to show that for each xi ∈ Pi (xˆi ), we have px Fi (p, ˆ x) ˆ = ∅, it follows that for each xi ∈ Pi (p, ˆ xˆi ), xi ∈ / Fi (p, ˆ xˆi ), i.e, px ˆ i > i (p). ˆ As
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i (p) ˆ pˆ xˆi , it follows that px ˆ i > pˆ xˆi for all xi ∈ Pi (xˆi ). Therefore (p, ˆ x) ˆ is a Walrasian equilibrium and the proof is complete. 4. Generalization of Shafer–Sonnenschien’s result in [15] We now consider the pure exchange economy E with n consumers and l commodities. For each i = 1, ..., n, the ith consumer is specified by his consumption set Xi which is a l , his initial holdings which is a point in R l , and a preference indicator subset of R+ i + Pi . A preference indicator may take different forms. In economies without externalities, it may be either a utility function Ui : Xi → R or an irreflexive relation Pi ⊂ Xi × Xi which was used [7] and is more general than the utility function formulation since Pi is not required to be asymmetric or transitive. If Ui (x) > Ui (y), or alternatively if (x, y) ∈ Pi , we then say that the ith consumer prefers x to y. In economies with externalities, we allow the preference of each individual to depend not only on his own consumption, but also on the l . An allocation consumption of each consumer and price. A price vector p is a point in R+ x = (x1 , . . . , xn ) ∈ X = i∈I Xi specifies a consumption for each consumer. A competitive equilibrium [7] (see also [12]) for the economy E is defined as an allocation l × X such that for each i = 1, 2, . . . , n, (1) pˆ · xˆ pˆ · for all i price pair (p, ˆ x) ˆ ∈ R+ i i (preference condition), and ), then p ˆ · x > p ˆ · (budget inequality), (2) if x ∈ P ( p, ˆ x i i i i i (3) ni=1 xˆi ni=1 i (demand cannot exceed supply). We shall study the existence of competitive equilibrium for the pure economy E in which l × X → 2Xi is such that P (p, ·) is an irreflexive relation each preference indicator Pi : R+ i l , i.e., x ∈ in X × Xi for each fixed price vector p ∈ R+ i / Pi (p, x). Also we assume that Pi is upper semicontinuous with closed values instead of being lower semicontinuous or having open graph (which is extensively used in the literature, e.g., see [5,7,14,15] and references therein). Theorem 5. Let E = (Xi , i , Pi )ni=1 be an economy satisfying for each i = 1, . . . , n, (1) the consumption set Xi is a non-empty compact convex subset of R l , (2) the ith initial l × X → 2Xi is upper endowment i ∈ intXi , and (3) the preference indicator Pi : R+ l ×X, semicontinuous with closed convex values such that xi ∈ / Pi (p, x) for each (p, x) ∈ R+ m where X = i=1 Xi . Then there exists a competitive equilibrium for the economy E. l : l Proof. Let = {p ∈ R+ i=1 pi = 1}. For each i = 1, . . . , n, we define Fi : × X → 2Xi by Fi (p, x) = {zi ∈ Xi : pzi p i } for each (p, x) ∈ × X. Then Fi has nonempty closed convex values. As i ∈ int Xi , Fi is both upper and lower semicontinuous by Lemma 5. Now we define the mapping F0 : × X → 2 by F0 (p, x) = for each (p, x) ∈ × X and define another mapping P0 : × X → 2 by P0 (p, x) = {q ∈ : q(ni=1 xi − ni=1 i ) > p(ni=1 xi − ni=1 i )} for each (p, x) ∈ × X. Of course, F0 is continuous and P0 has an open graph. Also, p ∈ / P0 (p, x) for each (p, x) ∈ × X. Thus the family (Xi , Fi , Pi )ni=0 , where X0 = , satisfies all hypotheses of Theorem 2, where X0 = X. By Theorem 2, there exists (p, ˆ x) ˆ ∈ × X such that for each i = 1, . . . , n, we ˆ x) ˆ and Fi (p, ˆ x) ˆ ∩ Pi (p, ˆ x) ˆ = ∅, and pˆ ∈ F0 (p, ˆ x) ˆ and F0 (p, ˆ x) ˆ ∩ have that xˆi ∈ Fi (p, P0 (p, ˆ x) ˆ = ∅. That is, we have the following: (a) pˆ ∈ and pˆ · xˆi pw ˆ i for i = 1, . . . , n, (b)
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pˆ · ( ni=1 xˆi − ni=1 wi ) q( ni=1 xˆi − ni=1 wi ) for all q ∈ , and (c) Pi (pˆ i , x) ˆ ∩ {xi ∈ ˆ i } = ∅ for i = 1, . . . , n. Xi : pˆ · xi pw By (a), we have pˆ
n
xˆi pˆ
i=1
n
i .
(**)
i=1
n claim that ni=1 xˆi ni=1 i . Suppose not, let z = (z1 , . . . , zl ) =: i=1 xˆ i − We n . Then there must be some k such that z > 0. Take q = (q , . . . , q ) ∈ such that k 1 l i=1 i n n qz = z > 0 so that p( ˆ x ˆ − ) qk = 1 and qi = 0 for i = k. Then we have k i=1 i i=1 i > 0 ˆ x) ˆ by (b). This contradicts (**). So we have ni=1 xi ni=1 i . Further, by (a) and (c), (p, is an equilibrium for E. References [1] C.D. Aliprantis, D.J. Brown, O. Burkinshaw, Existence and Optimality of Competitive Equilibria, Springer, New York, 1989. [2] K. Arrow, G. Debreu, Existence of equilibrium for a competitive economy, Econometric 22 (1954) 265–290. [4] G. Debreu, A social equilibrium existence theorem, Proc. Natl. Acad. Sci. USA 38 (1952) 886–893. [5] G. Debreu, Existence of competitive equilibrium, in: Handbook of Mathematical Economics, vol. 2, NorthHolland Publishing Company, Amsterdam, 1982, Chapter 7, pp. 697–743. [6] K. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Natl. Acad. Sci. USA 38 (1952) 121–126. [7] D. Gale, A. Mas-Colell, An equilibrium theorem for a general model without ordered preferences, J. Math. Econ. 2 (1975) 9–15; D. Gale, A. Mas-Colell, An equilibrium theorem for a general model without ordered preferences, J. Math. Econ. 6 (1979) 297–298 (Correction). [8] I. Glickberg, A further generalization of the Kakutani fixed-point theorem with applications to Nash equlibrium points, Proc. Am. Math. Soc. 3 (1952) 170–174. [9] E. Michael, Continuous selections I, Ann. Math. 63 (1956) 361–382. [10] J.F. Nash, Equilibrium points in N-person games, Proc. Natl. Acad. Sci. USA 36 (1950) 48–49. [11] T. Pruszko, Completely continuous extensions of convex-valued selectors, Nonlinear Anal. Theory Methods Appl. 27 (1996) 781–784. [12] W. Shafer, Equilibrium in abstract economies without ordered preferences, J. Math. Econ. 2 (1975) 345–348. [14] W. Shafer, H. Sonnenschein, Equilibrium in abstract economies without ordered preferences, J. Math. Econ. 2 (1975) 345–348. [15] W. Shafer, H. Sonnenschein, Some theorems on the existence of competitive equilibrium, J. Econ. Theory 11 (1975) 83–93. [16] K.K. Tan, X.Z. Yuan, Lower semicontinuity of multivalued mappings and equilibrium points, V. Lakshmikantham (Ed.), Proceedings of the First World Congress of Nonlinear Analysts, 19–26, August 1992, Tampa, Florida, Walter de Gruyter, Berlin, pp. 1849–1860. [17] K.K. Tan, X.Z. Yuan, Maximal elements and equilibria for U-majorised preferences, Bull. Aust. Math. Soc. 49 (1994) 47–54. [18] N.C. Yannelis, Equlibria in noncooperative models of competition, J. Econ. Theory 41 (1987) 96–111.
Further Reading [3] N. Bourbaki, Topological Vector Spaces, Springer, New York, 1987. [13] W. Shafer, Equilibrium in economies without ordered preferences or free disposal, J. Math. Econ. 3 (1976) 135–137.
Equilibrium existence theorems with closed preferences Kok-Keong Tan∗,1 , Zhou Wu, X.Z. Yuan Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5
Abstract We first prove an equilibrium existence theorem for an abstract economy in which the preference correspondences are upper semicontinuous with closed values. As applications, we provide two examples to illustrate how the general result established in this paper can be used to give the existence of equilibria of economic models such as a pure exchange model and the Arrow–Debreu model. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: N-person game; Equilibrium; Lower (upper) semicontinuous; Fixed point
1. Introduction Following Debreu [4], = (Xi , Fi , Ui )N i=1 is called a generalized N-person game (or an abstract economy) if for each person (agent) i = 1, . . . , N, Xi is a choice set, Fi : X = i∈I Xi → 2Xi is a constraint correspondence and Ui : X → R is a utility (payoff) function. The objective of the ith agent is for each xˆ ∈ X to choose an action xi that maximizes Ui (xˆi , . . . , xˆi−1 , ., xˆi+1 , . . . , xˆN ) subject to xi ∈ Fi (x). ˆ The vector xˆ = (xˆ1 , . . . , xˆn ) of actions is an equilibrium for if xˆi maximizes Ui (xˆ1 , . . . , xˆi−1 , ., xˆi+1 , xˆN ) subject to xi ∈ Fi (x) ˆ for each i = 1, . . . , N. This notion of equilibria is a natural extension
∗ Corresponding author. Tel.: +1 902 494 2166; fax: +1 902 494 5130.
E-mail address: [email protected] (K. Tan). 1 The author is partially supported by Natural Sciences and Engineering Research Council of Canada under
grant A-8096. 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.02.064
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of the concept of an equilibrium introduced by Nash [10] for non-cooperative N-person games. If each Xi is a non-empty compact convex subset of R l , each Fi is continuous with non-empty convex values and each Ui is continuous on X and quasi-convex in xi , then Debreu [4] showed that an equilibrium exists. Arrow and Debreu [2] used this result to prove the existence of a competitive equilibrium. Following Shafer [12], E = (Yi , wi , Vi )N i=1 is said to be an economy if N is the number of individuals, and for each i = 1, . . . , N, Yi ⊂ R l is a consumption set of the ith individual, l is the ith individual’s initial endowment vector, and V : Y → R is a utility wi ∈ R+ i i function that represents the preferences of the ith individual. Let Y = N i=1 Yi and = l : l p = 1} denote the set of normalized prices. A competitive equilibrium {p ∈ R+ i i=1 for E is a point (y, ˆ p) ˆ ∈ Y × such that the following three conditions are satisfied: (1) N w ; (2) pˆ yˆ = pw N y ˆ ˆ i for each i = 1, . . . , N; (3) for i = 1, . . . , N, yˆi is the i i i=1 i=1 ˆ i and yi ∈ Yi . solution to maximize Vi subject to py ˆ i pw We associate E with an (N + 1)-person generalized game = (Yi , Fi , Vi )N i=1 in the following manner. For each i = 0, 1, . . . , N, the ith person has a utility function Vi , a choice set Yi and a constraint correspondence Fi defined by Fi (y, p) = {yi ∈ Yi : py i pwi } for each (y, p) ∈ Y × . The 0th person, called a market player, has a utility (payoff) function V0 that is defined by V0 (y, p) = p(yi − wi ) for each (y, p) ∈ Y × , a choice set and a constraint correspondence F0 defined by F0 (y, p) = for each (y, p) ∈ Y × . Then, if each Yi is compact and convex, wi ∈ int Yi and each Vi is continuous and quasi-concave, this (N + 1)-person generalized game will satisfy the sufficient conditions mentioned above for the existence of an equilibrium. It is easy to see that (y ∗ , p∗ ) is an equilibrium for E if for each i = 0, . . . , N, yi∗ is a maximum of Vi . Note that the above argument remains valid if each agent’s utility function Vi is assumed to depend not only on his own consumption yi , but also on the consumptions of the other agents and on the prices p. Thus, Arrow and Debreu also showed how to prove the existence of competitive equilibrium with consumption externalities and price dependent preferences. We now follow Shafer’s idea to extend the above result to the existence of equilibrium in abstract economy. Given an N-person generalized game = (Xi , Fi , Ui )N i=1 , for each i = 1, . . . , N, consider the correspondence Pi : X → 2Xi defined by Pi (x) = {zi ∈ Xi : Ui (x1 , . . . , xi−1 , zi , xi+1 , . . . , xn ) > Ui (x)} ˆ ∩ Fi (x) ˆ = ∅ and for each x ∈ X. Note that xˆ is an equilibrium of if and only if Pi (x) ˆ for each i = 1, . . . , N. We shall consider abstract economies in which the xˆi ∈ Fi (x) individual preferences are given by preference correspondence Pi rather than by utility function. In this formulation, preferences, which depend on the actions of others, are not required to be transitive nor asymmetric. In this paper, we first prove an equilibrium existence theorem for an abstract economy in which the preference correspondences are upper semicontinuous with closed values. Then, as applications, we provide two examples to illustrate how the general result established in this paper can be used to give the existence of equilibria of economic models such as pure exchange model and the Arrow–Debreu model.
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2. An equilibrium theorem for an abstract economy We recall that an abstract economy = (Xi , Fi , Pi )N i=1 is defined by N ordered triples (Xi , Fi , Pi ), where Fi , Pi : X → 2Xi are correspondences. An equilibrium for is an ˆ ∩ Fi (x) ˆ = ∅. ˆ and Pi (x) xˆ ∈ X satisfying for each i = 1, ..., N , xˆi ∈ Fi (x) Theorem 1. Let =(Xi , Fi , Pi )N i=1 be an abstract economy and I0 ∪I1 ={1, . . . , N} where I0 and I1 are disjoint such that: (i) For each i = 1, . . . , N, Xi is a non-empty, compact and convex subset of R li . (ii) For each i = 1, . . . , N, Fi is a continuous correspondence with non-empty compact convex values. (iii) (a) For i ∈ I0 , Pi has an open graph and convex values, (b) for i ∈ I1 , Pi is upper semicontinuous with closed convex values. (iv) For each i = 1, . . . , N, xi ∈ / Pi (x) for all x ∈ X. Then has an equilibrium. Proof. Let X = N i=1 Xi . (1) Fix an i ∈ I0 . Let Ui = {x ∈ X : Fi (x) ∩ Pi (x) = ∅}. Since Fi is lower semicontinuous and Pi has open graph, Fi ∩ Pi is lower semicontinuous by Lemma 4.2 in [18]. Hence Ui is open. By Michael selection theorem [9], there exists a continuous function fi such that fi (x) ∈ Fi (x) ∩ Pi (x) for each x ∈ Ui . Define Gi : X → Xi by Gi (x) =
Fi (x) if x ∈ / Ui , {fi (x)} if x ∈ Ui .
Then Gi (x) is upper semicontinuous with non-empty compact convex values. (2) Fix an i ∈ I1 . Let Mi = {x ∈ X : Fi (x) ∩ Pi (x) = ∅}. Then Mi is closed since Fi ∩ Pi is upper semicontinuous by Lemma 2.2 in [17]. Note that for each x ∈ Mi , Fi (x)∩Pi (x) ⊂ Fi (x), by Theorem in [11, p. 781], there exists an upper semicontinuous mapping Gi : X → 2Xi with non-empty compact convex values such that Gi |Mi =Fi ∩Pi and Gi (x) ⊂ Fi (x) for each x ∈ X. (3) Now define G : X → 2X by G(x) = N i=1 Gi (x) for each x ∈ X, then G is upper semicontinuous with non-empty compact convex values (by Lemma 3 in [6]). By Fan-Glickberg fixed point theorem [6,8], G has a fixed point x ∈ X. Clearly, x is an equilibrium for . Remark 1. Suppose Fi (x) = Xi for each i = 1, . . . , N and for all x ∈ X. (1) If I1 = ∅, then Theorem 2 reduces to the fixed point theorem in [7]. (2) If I0 = ∅, Theorem 2 can be regarded as the dual of [7]. Remark 2. (1) If I1 = ∅, Theorem 2 reduces to the abstract result in [14]. (2) If I0 = ∅, Theorem 2 can be regarded as the dual of [14]. Remark 3. By Lemma 2.3 of Tan and Yuan [16, p. 1851], it follows that the condition (iii)(a) in Theorem 2 is equivalent to the condition that “Pi is lower semicontinuous with convex and open values (which may be empty)”.
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3. Application to economic models Suppose there are n traders. For each trader i ∈ I = {1, . . . , n}, the trading set Xi is a subset of R l and there is a preference mapping Pi from Xi to 2Xi . There is also a subset Y of R l which is called the technology of the economy [7]. An allocation x is an n-tuple (x1 , . . . , xn ) where xi ∈ Xi for each i ∈ I . Thus x ∈ X = i∈I Xi . An allocation is said to be feasible if ni=1 xi ∈ Y . For a given allocation x = (x1 , . . . , xn ), x0 denotes the sum ni=1 xi . Let be the unit (l − 1)-simplex. Each element p of will be called a price vector. In formulating equilibrium models, we describe the way in which an allocation x is associated with the price vector p as follows: For each i ∈ I , let the trader i have his/her income i (p) at the price p. In the pure exchange model, it is assumed that the trader i has an initial endowment vector wi and his income is then given by i (p) = pwi . The Arrow–Debreu model (1954) involves a more complicated set of income functions. It is assumed that the technology set Y is the sum of some sub-technologies Y1 , . . . , Yr which are to be thought of as firms, and trader i is provided with portfolio vector i = (i1 , . . . , ir ), where ij represents trader i’s share of the firm Yj . The Arrow–Debreu income functions are then given by i (p) = pw i + rj =1 ij sup pY j . In the present treatment, by following [7], we wish to allow for more general income function. For any p ∈ , define the profit function (p) by (p) = sup pY . Since Y maybe unbounded, it follows that may be infinite. We define ⊂ by = {p ∈ : (p) < ∞}. The following lemma is implicitly contained in the proof of Lemma 2 in [7]; here we shall give a detailed proof: Lemma 2. Suppose Y ⊂ R l is closed convex, contains the negative orthant and has a bounded intersection other than {0} with the positive orthant. Let e be a point in Y and l and z ∈ Yˆ such that pz = max Yˆ = {y : y ∈ Y and y e}. Suppose for some p ∈ R+ y∈Yˆ py. Then there is q ∈ such that qz = maxy∈Y qy. Proof. Since pz=maxy∈Yˆ py, the set D := {y ∈ R l : y > z}, which is non-empty open convex and disjoint from Yˆ . D is also disjoint from Y. Otherwise, let y ∈ D ∩Y . Then y > z e so that y ∈ Yˆ which contradicts pz = maxy∈Yˆ py.Since Y is convex, by Hahn–Banach theorem, we can find q ∈ R l such that qy > qy for all y ∈ D and y ∈ Y . Note that since Y l . Take y = z and y in contains the negative orthant, it follows that q = (q1 , . . . , qn ) ∈ R+ n D converging to z. We have qz = maxy∈Y qy. Obviously, q 0 and q = 0. If li=1 qi = 1, replace it with q/ li=1 qi ∈ , which is required. Our hypothesis on Y will guarantee that = ∅. One also verifies that is convex. We also assume the existence of n real-valued functions i on (to be called income functions) satisfying the following formula: ni=1 i (p) = (p) for all p ∈ . A Walrasian (or a competitive) equilibrium [1] for the model described above consists of a price vector pˆ in and an allocation xˆ such that (1) pˆ xˆ i (p) ˆ for all i ∈ I (budget ˆ i > pˆ xˆi (preference condition); (3) xˆ inequality); (2) for each i ∈ I , if xi ∈ Pi (xˆi ), then px is feasible (balance of supply and demand).
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The following lemma is contained in the proof of Theorem 4 in [5, pp. 707–710]; here we shall give an explicit proof based on Debreu’s idea: Lemma 3. Suppose X is a non-empty compact convex subset of R l and (p) is a continuous function on a non-empty compact subset Y of R l . If (p) > minx∈X px for all p ∈ Y , then correspondence A : Y → 2X defined by A(p) = {x ∈ X : p · x (p)} is continuous. Proof. Since A is non-empty and closed-valued, Y is compact and A can be shown to have a closed-graph, A is upper semicontinuous. It remains to show that A is also lower semicontinuous. Let p 0 be any point inY. To show A is lower semicontinuous at p 0 , we consider a sequence 0 0 0 (pi )∞ i=1 in Y converging to p and a point z ∈ A(p ). ∞ i i i Let w =(p ) for i =1, . . ., then (w )i=1 converges to w 0 =(p0 ). For i =0, 1, . . ., since i w = (p i ) > minx∈X p i x, (p i , wi ) ∈ D := {(p, w) ∈ R l+1 : minx∈X px w}. By Lemma 3 in [5], : D → X defined by (p, w) = {x ∈ X : px w} is continuous at (p0 , w0 ). So i i i i for z0 ∈ A(p 0 ) = (p 0 , w0 ), there exists a sequence (zi )∞ i=1 , where z ∈ (p , w ) = A(p ) 0 for each i = 1, . . ., converging to z . Thus A is lower semicontinuous. Now our main application is the following theorem: Theorem 4. The following conditions are sufficient for the existence of an equilibrium: (1) the set Y is closed convex, contains the negative orthant, and has a bounded intersection with the positive orthant; (2) for each i = 1, . . . , n, the set Xi is non-empty closed convex and bounded below; (3) for each i = 1, . . . , n, the preference Pi is upper semicontinuous with closed convex / Pi (xi ) for all x ∈ X); values and is irreflexive (i.e., xi ∈ (4) for each i = 1, . . . , n, the function i (p) is continuous and satisfies i (p) > inf pX i for all p ∈ . Proof. As each Xi is bounded below, it follows there exists a vector e such that for any non-empty subset S of {1, . . . , n}, we have Xi . e< i∈S
Without loss of generality, we may assume that e < 0 so that e ∈ Y . Now define Yˆ = {y ∈ Y : y e}. Note that Yˆ contains all feasible x0 = ni=1 xi and by the condition of Y, Yˆ is also bounded above as well as below. Thus there exists a vector f such that f > Yˆ . By the feasibility, x0 = nj=1 xi < f − e, so that xi < f − j =i xj < f − e. Following [7], we define ⊂ as follows: = {p ∈ : py = (p) for some y ∈ Yˆ }. Take any p ∈ . Since Yˆ is compact, there is z ∈ Yˆ such that p · z = supy∈Yˆ p · y. By Lemma 2, there is q ∈ such that q · z = supy∈Y q · y = (q) so that q ∈ . Thus is non-empty. It is easy to see is closed. Now let ∗ be the convex hull of . Then ∗ ⊂ since is convex and ∗ is also closed as is closed.
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By the definition of , we shall show that there is a finite set Xi ⊂ Xi such that min pX i < i (p) for all p ∈ ∗ . Indeed, for each p ∈ ∗ , let xp ∈ Xi be such that px p < i (p). By the continuity of i , it follows that there is, for each p ∈ ∗ , an open neighborhood Vp of p in ∗ such that qx p < i (q) for each q ∈ Vp . The family of open sets {Vp : p ∈ ∗ } covers ∗ and since ∗ is compact, there is a finite subcover {Vp1 , . . . , Vpk }. Then take Xi = {xp1 , . . . , xpk }. Let r be sufficiently large and we define Xˆ i = {xi ∈ Xi : xi r}. Without loss of generality, we may assume that Xi ⊂ Xˆ i for all i = 1, . . . , n. ˆ For each i = 1, . . . , n, define Fi : ∗ × Xˆ → 2Xi by Fi (p, x) = {zi ∈ Xˆ i : pzi i (p)} ∗ n ˆ where Xˆ = Xˆ i . By the construction of Xˆ i above and the for each (p, x) ∈ × X, i=1 condition (4), it follows that Fi (p, x) is non-empty and convex for each (p, x) ∈ ∗ × Xˆ and, moreover, by Lemma 3 in [5], Fi is continuous with non-empty closed and convex values. ∗ ˆ Of course, F0 Now define F0 : ∗ × Xˆ → 2 by F0 (p, x)=∗ for each (p, x) ∈ ∗ × X. ∗ ˆ is continuous with non-empty closed convex values for all (p, x) ∈ × X. Finally, define ∗ P0 : ∗ × Xˆ → 2 by P0 (p, x) = {q ∈ ∗ : q · (ni=1 xi ) − (q) > p · (ni=1 xi ) − (p)} ˆ It follows that P0 has an open graph, each P0 (p, x) is convex for each (p, x) ∈ ∗ × X. ˆ (maybe empty), and p ∈ / P0 (p, x) for all (p, x) ∈ ∗ × X. ˆ Identify Pi with P¯i : ∗ × Xˆ → Xˆ i defined by P¯i (p, x)=Pi (xi ) for each (p, x) ∈ ∗ × X. By condition (3), it follows that the abstract economy (Xˆ i , Fi , Pi )ni=0 satisfies all hypotheses of Theorem 1, where Xˆ 0 = ∗ . By Theorem 1 with I0 = {0} and I1 = {1, . . . , n}, there exists (p, ˆ x) ˆ ∈ ∗ × Xˆ such that pˆ ∈ ∗ and n n ∗ ∗ F0 (p, ˆ ˆ x) ˆ ∩ P0 (p, ˆ x) ˆ = ∩ q ∈ :q xˆi − (q) > pˆ xˆi − (p) i=1
i=1
= ∅, which implies that for each q ∈ ∗ we have n n ˆ xˆi − (q) pˆ xˆi − (p). q i=1
(*)
i=1
ˆ x) ˆ for i = 1, 2, . . . , n and Fi (p, ˆ x) ˆ ∩ Moreover, for each i = 1, 2, . . . , n, xˆi ∈ Fi (p, Pi (p, ˆ x) ˆ = ∅. Now we prove that ˆ x) ˆ is a Walrasian equilibrium. First we claim that xˆ0 = ni=1 xˆi is n(p, / Y . We may choose such feasible, i.e., xˆ0 = i=1 xˆi ∈ Y . Suppose the contrary, i.e, xˆ0 ∈ that < , y = e + (1 − )xˆ0 in Y and y = e + (1 − )xˆ0 ∈ / Y for all < . Since xˆ0 > e, by a similar argument as in the proof of Lemma 2 there exists q in such that qy = (q) and q xˆ0 > (q). Note that y ∈ Yˆ , and it follows that q ∈ ⊂ ∗ . p( ˆ xˆ0 ) − (p). ˆ Thus p( ˆ xˆ0 ) > (p). ˆ However, we do Therefore, we have q(xˆ0 ) − (q) have that p( ˆ xˆ0 )= ni=1 p( ˆ xˆi ) ni=1 i (p)=( ˆ p), ˆ which is a contradiction. Thus we must have xˆ0 = ni=1 xˆi ∈ Y . ˆ i > pˆ xˆi . Since Pi (p, ˆ x) ˆ ∩ Finally, we wish to show that for each xi ∈ Pi (xˆi ), we have px Fi (p, ˆ x) ˆ = ∅, it follows that for each xi ∈ Pi (p, ˆ xˆi ), xi ∈ / Fi (p, ˆ xˆi ), i.e, px ˆ i > i (p). ˆ As
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i (p) ˆ pˆ xˆi , it follows that px ˆ i > pˆ xˆi for all xi ∈ Pi (xˆi ). Therefore (p, ˆ x) ˆ is a Walrasian equilibrium and the proof is complete. 4. Generalization of Shafer–Sonnenschien’s result in [15] We now consider the pure exchange economy E with n consumers and l commodities. For each i = 1, ..., n, the ith consumer is specified by his consumption set Xi which is a l , his initial holdings which is a point in R l , and a preference indicator subset of R+ i + Pi . A preference indicator may take different forms. In economies without externalities, it may be either a utility function Ui : Xi → R or an irreflexive relation Pi ⊂ Xi × Xi which was used [7] and is more general than the utility function formulation since Pi is not required to be asymmetric or transitive. If Ui (x) > Ui (y), or alternatively if (x, y) ∈ Pi , we then say that the ith consumer prefers x to y. In economies with externalities, we allow the preference of each individual to depend not only on his own consumption, but also on the l . An allocation consumption of each consumer and price. A price vector p is a point in R+ x = (x1 , . . . , xn ) ∈ X = i∈I Xi specifies a consumption for each consumer. A competitive equilibrium [7] (see also [12]) for the economy E is defined as an allocation l × X such that for each i = 1, 2, . . . , n, (1) pˆ · xˆ pˆ · for all i price pair (p, ˆ x) ˆ ∈ R+ i i (preference condition), and ), then p ˆ · x > p ˆ · (budget inequality), (2) if x ∈ P ( p, ˆ x i i i i i (3) ni=1 xˆi ni=1 i (demand cannot exceed supply). We shall study the existence of competitive equilibrium for the pure economy E in which l × X → 2Xi is such that P (p, ·) is an irreflexive relation each preference indicator Pi : R+ i l , i.e., x ∈ in X × Xi for each fixed price vector p ∈ R+ i / Pi (p, x). Also we assume that Pi is upper semicontinuous with closed values instead of being lower semicontinuous or having open graph (which is extensively used in the literature, e.g., see [5,7,14,15] and references therein). Theorem 5. Let E = (Xi , i , Pi )ni=1 be an economy satisfying for each i = 1, . . . , n, (1) the consumption set Xi is a non-empty compact convex subset of R l , (2) the ith initial l × X → 2Xi is upper endowment i ∈ intXi , and (3) the preference indicator Pi : R+ l ×X, semicontinuous with closed convex values such that xi ∈ / Pi (p, x) for each (p, x) ∈ R+ m where X = i=1 Xi . Then there exists a competitive equilibrium for the economy E. l : l Proof. Let = {p ∈ R+ i=1 pi = 1}. For each i = 1, . . . , n, we define Fi : × X → 2Xi by Fi (p, x) = {zi ∈ Xi : pzi p i } for each (p, x) ∈ × X. Then Fi has nonempty closed convex values. As i ∈ int Xi , Fi is both upper and lower semicontinuous by Lemma 5. Now we define the mapping F0 : × X → 2 by F0 (p, x) = for each (p, x) ∈ × X and define another mapping P0 : × X → 2 by P0 (p, x) = {q ∈ : q(ni=1 xi − ni=1 i ) > p(ni=1 xi − ni=1 i )} for each (p, x) ∈ × X. Of course, F0 is continuous and P0 has an open graph. Also, p ∈ / P0 (p, x) for each (p, x) ∈ × X. Thus the family (Xi , Fi , Pi )ni=0 , where X0 = , satisfies all hypotheses of Theorem 2, where X0 = X. By Theorem 2, there exists (p, ˆ x) ˆ ∈ × X such that for each i = 1, . . . , n, we ˆ x) ˆ and Fi (p, ˆ x) ˆ ∩ Pi (p, ˆ x) ˆ = ∅, and pˆ ∈ F0 (p, ˆ x) ˆ and F0 (p, ˆ x) ˆ ∩ have that xˆi ∈ Fi (p, P0 (p, ˆ x) ˆ = ∅. That is, we have the following: (a) pˆ ∈ and pˆ · xˆi pw ˆ i for i = 1, . . . , n, (b)
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pˆ · ( ni=1 xˆi − ni=1 wi ) q( ni=1 xˆi − ni=1 wi ) for all q ∈ , and (c) Pi (pˆ i , x) ˆ ∩ {xi ∈ ˆ i } = ∅ for i = 1, . . . , n. Xi : pˆ · xi pw By (a), we have pˆ
n
xˆi pˆ
i=1
n
i .
(**)
i=1
n claim that ni=1 xˆi ni=1 i . Suppose not, let z = (z1 , . . . , zl ) =: i=1 xˆ i − We n . Then there must be some k such that z > 0. Take q = (q , . . . , q ) ∈ such that k 1 l i=1 i n n qz = z > 0 so that p( ˆ x ˆ − ) qk = 1 and qi = 0 for i = k. Then we have k i=1 i i=1 i > 0 ˆ x) ˆ by (b). This contradicts (**). So we have ni=1 xi ni=1 i . Further, by (a) and (c), (p, is an equilibrium for E. References [1] C.D. Aliprantis, D.J. Brown, O. Burkinshaw, Existence and Optimality of Competitive Equilibria, Springer, New York, 1989. [2] K. Arrow, G. Debreu, Existence of equilibrium for a competitive economy, Econometric 22 (1954) 265–290. [4] G. Debreu, A social equilibrium existence theorem, Proc. Natl. Acad. Sci. USA 38 (1952) 886–893. [5] G. Debreu, Existence of competitive equilibrium, in: Handbook of Mathematical Economics, vol. 2, NorthHolland Publishing Company, Amsterdam, 1982, Chapter 7, pp. 697–743. [6] K. Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proc. Natl. Acad. Sci. USA 38 (1952) 121–126. [7] D. Gale, A. Mas-Colell, An equilibrium theorem for a general model without ordered preferences, J. Math. Econ. 2 (1975) 9–15; D. Gale, A. Mas-Colell, An equilibrium theorem for a general model without ordered preferences, J. Math. Econ. 6 (1979) 297–298 (Correction). [8] I. Glickberg, A further generalization of the Kakutani fixed-point theorem with applications to Nash equlibrium points, Proc. Am. Math. Soc. 3 (1952) 170–174. [9] E. Michael, Continuous selections I, Ann. Math. 63 (1956) 361–382. [10] J.F. Nash, Equilibrium points in N-person games, Proc. Natl. Acad. Sci. USA 36 (1950) 48–49. [11] T. Pruszko, Completely continuous extensions of convex-valued selectors, Nonlinear Anal. Theory Methods Appl. 27 (1996) 781–784. [12] W. Shafer, Equilibrium in abstract economies without ordered preferences, J. Math. Econ. 2 (1975) 345–348. [14] W. Shafer, H. Sonnenschein, Equilibrium in abstract economies without ordered preferences, J. Math. Econ. 2 (1975) 345–348. [15] W. Shafer, H. Sonnenschein, Some theorems on the existence of competitive equilibrium, J. Econ. Theory 11 (1975) 83–93. [16] K.K. Tan, X.Z. Yuan, Lower semicontinuity of multivalued mappings and equilibrium points, V. Lakshmikantham (Ed.), Proceedings of the First World Congress of Nonlinear Analysts, 19–26, August 1992, Tampa, Florida, Walter de Gruyter, Berlin, pp. 1849–1860. [17] K.K. Tan, X.Z. Yuan, Maximal elements and equilibria for U-majorised preferences, Bull. Aust. Math. Soc. 49 (1994) 47–54. [18] N.C. Yannelis, Equlibria in noncooperative models of competition, J. Econ. Theory 41 (1987) 96–111.
Further Reading [3] N. Bourbaki, Topological Vector Spaces, Springer, New York, 1987. [13] W. Shafer, Equilibrium in economies without ordered preferences or free disposal, J. Math. Econ. 3 (1976) 135–137.