Oligopoly `a la Cournot-Nash in Markets with a Continuum of Traders∗ Giulio Codognato†and Sayantan Ghosal‡ July 2000

Abstract We show the existence of a Cournot-Nash equilibrium in pure strategies with trade in a model of noncooperative exchange with some large traders and many small traders in which traders are allowed to buy and sell each commodity. Moreover, we show, by an example, that, even with identical large traders of equal measure, our model yields Cournot-Nash equilibrium allocations which are not Walrasian. Journal of Economic Literature Classification Numbers: C72, D51.

1

Introduction

In a seminal paper, Shitovitz (1973) studied oligopoly in a general equilibrium framework by modelling trade as a cooperative game of exchange between large traders, represented as atoms, and small traders, represented by an ∗

We are indebted to Nick Baigent, Francis Bloch, Dikran Dikranjan, Jean Gabszewicz, Marcellino Gaudenzi, Peter Hammond, Jean-Fran¸cois Mertens, Nando Prati and Myrna Wooders for their comments and suggestions. In particular, the first author would like to thank Pradeep Dubey for a conversation which inspired the present paper. † Dipartimento di Scienze Economiche, Universit` a degli Studi di Udine, Via Tomadini 30, 33100 Udine, Italy, and SET, Universit` a degli Studi di Milano Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy. Financial support from CNR Research Contribution 98.01351.CT10 is gratefully acknowledged. ‡ Department of Economics, University of Warwick, Coventry CV4 7AL, United Kingdom.

1

atomless sector. A feature of the model analysed by Shitovitz (1973) is the counterintuitive possibility that core allocations are competitive despite the presence of atoms. In order to overcome this problem, following Shitovitz (1973), Okuno et al. (1980) proposed to study the Cournot-Nash equilibria of a model of simultaneous, noncooperative exchange between large traders and small traders as the appropriate setting for the analysis of oligopoly in general equilibrium. The model of noncooperative exchange they used belongs to a line of research initiated by Shapley and Shubik (1977) (see, for instance, Dubey and Shubik (1978), Postlewaite and Schmeidler (1978), Mas-Colell (1982), Amir et al. (1990), Peck and Shell (1990), (1991), Peck et al. (1992), Dubey and Shapley (1994)). The approach proposed and adopted by Okuno et al. (1980) also contrasts with a line of research initiated by Gabszewicz and Vial (1972) (see, for instance, Roberts and Sonnenschein (1977), Roberts (1980), Mas-Colell (1982), Dierker and Grodal (1986), Codognato and Gabszewicz (1993), Gabszewicz and Michel (1997), d’Aspremont et al. (1997)) In this class of models, large traders (firms) and small traders interact sequentially: while small traders are assumed to be price takers and submit Walras demand correspondences, large traders (firms) influence prices by manipulating the Walras price correspondence. Okuno et al. (1980) observe that these models “have been deficient in that they have simply assumed a priori that certain agents behave as price takers while others act noncompetitively, with no formal explanation being given as to why a particular agent should behave one way or the other. Shitovitz’s approach represents an important contribution in pointing to an explicit formulation leading to such differences in behavior... we seek to explore the use of this type of model in studying issues of oligopoly in a general equilibrium framework. A specific focus of our work is in illuminating how either perfectly or imperfectly competitive behavior may emerge endogenously in this model, depending on the characteristics of the agent and his place in the economy.” This underlying conceptual problem has an important consequence for the existence of equilibria in the models belonging to the line of research initiated by Gabszewicz and Vial (1972): as the Walras price correspondence may fail to be continuous, equilibria may not exist even in mixed strategies. In this paper, we follow the noncooperative approach to oligopoly in general equilibrium proposed by Okuno et al. (1980). In their paper, no trader is allowed to buy and sell a commodity. Further, their paper contains no 2

general existence result. Here, in contrast, we are able to show that (i) under standard assumptions on traders’ endowments and preferences and allowing for a general model of trade, a Cournot-Nash equilibrium in pure strategies with trade exists; (ii) identical atoms of equal measure yield noncooperative equilibria whose allocations are not Walrasian thus showing that the results obtained by Okuno et al. (1980) are robust. The model of noncooperative exchange we use was originally proposed by Lloyd S. Shapley, and analysed by Sahi and Yao (1989) in exchange economies with a finite number of traders and by Codognato and Ghosal (2000) in exchange economies with an atomless continuum of traders. In this model, traders send out bids, i.e., quantity signals, which indicate how much of each commodity they are willing to put up for trade. Every bid of any commodity is tagged by the name of some other commodity for which it has to be exchanged. The rule of price formation requires that a single price system, which equates the value of the total amount of bids for any commodity to the value of the total amount available of that commodity, is used to clear the markets. We use this model to study exchange with a mixed measure space of traders consisting of an atomic part, which represents the large traders, and a nonatomic part, which represents the small traders. The model has the feature that, since no small trader is able to manipulate prices, the best reply of each small trader attains a commodity bundle which maximizes his utility subject to his budget constraint at the prevailing market clearing prices. Our construction of the mixed measure space of traders allows us to synthetize, in the proof of our main theorem, the existence theorem, the techniques used by Sahi and Yao (1989) in finite exchange economies and the techniques used to show the existence of noncooperative equilibria in nonatomic games (see, for instance, Schmeidler (1973) and Khan (1985)). Our proof, however, requires us to solve new technical problems in taking limits. We overcome these problems by using a version of Fatou’s lemma in several dimensions proved by Artstein (1979). The paper is organized as follows. In Section 2, we construct the mathematical model and we state our main theorem. Section 3 is devoted to a discussion of the model. In Section 4, we prove the main theorem. Section 5 concludes.

3

2

The mathematical model and the main theorem

Traders are represented by the elements of the set T = T0 ∪ T1 , where T0 = [0, 1] and T1 = {2, . . . , m + 1}. If a, b ∈ T0 satisfy a ≤ b, let us write [a, b) = ∅ if a = b and [a, b) = {x ∈ T0 : a ≤ x < b} if a < b. Then, it is well known that the collection S0 = {[a, b) : a, b ∈ T0 and a ≤ b} is a semiring. Let S1 = P(T1 ) be the collection of all subsets of T1 . It is also well known that this collection is an algebra. We shall denote by µ0 and µ1 , respectively, the Lebesgue measure on S0 and the counting measure on S1 . The following proposition provides us with a first characterization of the set of traders as a measure space. Proposition 1. The triplet (T, S, µ), where T is the set of traders, S = {E ∈ T : E = A ∪ B, A ∈ S0 , B ∈ S1 }, µ : S → [0, ∞] is a set function such that, for each E ∈ S, µ(E) = µ0 (E ∩ T0 ) + µ1 (E ∩ T1 ), is a measure space. Proof. Since S0 is a semiring and S1 is an algebra, it easily follows that S is a semiring. Now, observe that µ(∅) = µ0 (∅) + µ1 (∅) = 0. Moreover, let {En } be a disjoint sequence of S with ∪∞ n=1 En ∈ S. Then, we have ∞ ∞ µ(∪n=1 En ) = µ0 ((∪∞ n=1 En ) ∩ T0 ) + µ1 ((∪n=1 En ) ∩ T1 ) ∞ = µ0 (∪∞ n=1 (En ∩ T0 )) + µ1 (∪n=1 (En ∩ T1 ))

= =

∞ X

n=1 ∞ X

µ0 (En ∩ T0 ) +

∞ X

n=1

µ1 (En ∩ T1 )

(µ0 (En ∩ T0 ) + µ1 (En ∩ T1 )) =

n=1

∞ X

µ(En ).

n=1

Hence, µ is a measure on S. Let µ∗ denote the Caratheodory extension of µ. Since µ∗ (T ) = 1 + m < ∞, the measure space (T, S, µ) is finite and this implies that a subset E of T is µmeasurable if and only if µ∗ (E)+µ∗ (E c ) = µ∗ (T ). Denote by T the collection of all µ-measurable subsets of T . The following proposition provides us with a characterization of the set of traders as a complete measure space. 4

Proposition 2. The triplet (T, T , µ∗ ) is a complete measure space and µ∗ is the unique extension of µ on a measure on T .

Proof. It follows from the Caratheodory Extension Procedure Theorem (see Aliprantis and Border (1994) p. 289), since the measure space (T, S, µ), being finite, is σ-finite.

Now, consider the triplet (T0 , S0 , µ0 ) which, as is well known, is a measure space. Denote by µ∗0 the Caratheodory extension of µ0 and by T0 the collection of all µ0 -measurable subsets of T0 . Then, it is also well known that the triplet (T0 , T0 , µ0∗ ) is a complete measure space and that µ∗0 is the unique extension of µ0 to a measure on T0 . Let TT0 = {E ∩ T0 : E ∈ T } be the restriction of T to T0 . The following proposition characterizes the restriction of the measure space (T, T , µ∗ ) to T0 . Proposition 3. The triplet (T0 , TT0 , µ∗ ) is a measure space such that TT0 = T0 and µ∗ = µ∗0 , when these measures are restricted to TT0 .

Proof. First, we shall show that, for every subset E of T0 , µ∗ (E) = µ∗0 (E). This can be proved as follows µ∗ (E) = inf{

∞ X

n=1

= inf{

∞ X

n=1 ∞ X

= inf{ =

n=1 ∗ µ0 (E).

µ(En ) : {En } ⊂ S, E ⊆ ∪∞ n=1 En }

µ0 (En ∩ T0 ) +

∞ X

n=1

µ1 (En ∩ T1 ) : {En } ⊂ S, E ⊆ ∪∞ n=1 En }

µ0 (En ∩ T0 ) : {En } ⊂ S, {En ∩ T1 } = ∅, E ⊂ ∪∞ n=1 (En ∩ T0 )}

Now, observe that, since T0 is µ-measurable, TT0 is a collection of µ-measurable subsets of T0 . Let E be a subset of T0 . By the σ-subadditivity of µ∗ , we have µ∗ (T ) ≤ µ∗ (E) + µ∗ (E c ) = µ∗ (E) + µ∗ ((E c ∩ T0 ) ∪ T1 ) ≤ µ∗ (E) + µ∗ (E c ∩ T0 ) + µ∗ (T1 ). On the other hand, since E is µ0 -measurable, we have µ∗ (E) + µ∗ (E c ∩ T0 ) + µ∗ (T1 ) = µ0∗ (E) + µ∗0 (E c ∩ T0 ) + µ∗ (T1 ) = µ∗0 (T0 ) + µ∗ (T1 ) = µ∗ (T0 ) + µ∗ (T1 ) = µ∗ (T ). 5

This implies that µ∗ (E) + µ∗ (E c ) = µ∗ (T ) and this, in turn, implies that E is µ-measurable, thereby showing that T0 ⊆ TT0 . Now, let E be a subset of TT0 . Since E is µ-measurable, we have µ∗ (E) + µ∗ (E c ) = µ∗ (T ). By the σ-additivity of µ∗ on T , it follows that µ∗ (E) + µ∗ (E c ∩ T0 ) + µ∗ (T1 ) = µ∗ (T0 ) + µ∗ (T1 ). This implies that µ∗0 (E) + µ∗0 (E c ∩ T0 ) = µ∗0 (T0 ) and this, in turn, implies that E is µ0 -measurable, thereby showing that TT0 ⊆ T0 . Hence, TT0 = T0 .

Now, consider the triplet (T1 , S1 , µ1 ). It is easy to show that µ∗1 = µ1 , where µ∗1 denotes the Caratheodory extension of µ1 , and that S1 is the collection of all µ1 -measurable subsets of T1 . This implies that the triplet (T1 , S1 , µ1 ) is a complete measure space. Let TT1 = {E ∩ T1 : E ∈ T } be the restriction of T to T1 . The following proposition characterizes the restriction of the measure space (T, T , µ∗ ) to T1 . Proposition 4. The triplet (T1 , TT1 , µ∗ ) is a measure space such that TT1 = S1 and µ∗ = µ1 , where these measures are restricted to TT1 .

Proof. First, by the same argument used in the proof of Proposition 3, it is possible to show that, for every subset E of T1 , µ∗ (E) = µ∗1 (E) = µ1 (E). Now, observe that, since S ⊆ T , S1 = S ∩ T1 ⊆ T ∩ T1 = TT1 . On the other hand, TT1 ⊆ P(T1 ) = S1 . Hence, S1 = TT1 .

The following definition introduces the concept of an atom of a measure space (see Aliprantis and Border (1994) p. 303). Definition 1. Let (X, Σ, µ) be a measure space. A measurable set A is called an atom if µ∗ (A) > 0 and for every subset B of A, either µ∗ (B) = 0 or µ∗ (A \ B) = 0. If (X, Σ, µ) has no atoms, then it is called an atomless measure space. If there exists a countable set A such that, for each a ∈ A, the singleton set {a} is measurable with µ∗ ({a}) > 0 and µ∗ (X \ A) = 0, then the measure space (X, Σ, µ) is called purely atomic. From now on, for simplicity, µ∗ , µ0∗ , µ∗1 will be denoted by µ, µ0 , µ1 , respectively. The space of traders will be denoted by the complete measure space (T, T , µ). By Propositions 3 and 4, it is straightforward to show that the 6

measure space (T0 , TT0 , µ) is atomless and that the measure space (T1 , TT1 , µ) is purely atomic. Moreover, it should be clear that, for each t ∈ T1 , the singleton set {t} is an atom of the measure space (T, T , µ). The words “integral”, “integrable” and “integrability” are to be understood in the sense of Lebesgue. Given a function f : T → R, we shall denote by 0 f and 1 f the restriction of f to T0 and T1 , respectively. The following proposition reminds us that the integrability of f is equivalent to the integrability of 0 f and 1 f . Proposition 5. A function f : T → R is integrable if and only if 0 f and 1 f are integrable over T0 and T1 , respectively. Proof. See Theorem 4 and Problem 6 in Kolmogorov and Fomin (1975) p. 298 and p. 302, respectively. R

R

R

It is also well known that R T f dµ = T0 0 fR dµ + T1 1 fR dµ, where, by ProposiR 0 P 1 f (t). tions 3 and 4, T0 f dµ = T0 0 f dµ0 and T1 1 f dµ = T1 1 f dµ1 = m+1 t=2 l A commodity bundle is a point in R+ . An assignment (of commodity bundles l to traders) is a function x: T → R+ , each coordinate of which is integrable. There is a fixed initial assignment w, satisfying the following assumption. Assumption 1. w(t) > 0, for all t ∈ T .

R

R

An allocation is an assignment x for which T x(t) dµ = T w(t) dµ. The preferences of each trader t ∈ T are described by an utility function ut : l → R, satisfying the following asuumptions. R+

l Assumption 2. ut : R+ → R is continuous, strictly monotonic and concave, for all t ∈ T . l Assumption 3. u : T × R+ → R given by u(t, x) = ut (x) is measurable.

We also need the following assumption (see Sahi and Yao (1989)).

Assumption 4. There are at least two traders in T1 for whom w(t)  0; l l l ; {x ∈ R+ : ut (x) = u(w(t))} ⊂ R++ ut is continuously differentiable in R++ .

l A price vector is a vector p ∈ R+ . A Walras equilibrium is a pair (p∗ , x∗ ), consisting of a price vector p∗ and an allocation x∗ , such that, for all t ∈ T , l : px ≤ pw(t)}. x∗ (t) is maximal with respect to ut in t’s budget set {x ∈ R+ l For each p ∈ R+ , we introduce the following functions (see Aumann (1966)): l Bp : T0 → P(Rl ) such that, for each t ∈ T0 , Bp (t) = {x ∈ R+ : px ≤ pw(t)}; l l : for all y ∈ Cp : T0 → P(R ) such that, for each t ∈ T0 , Cp (t) = {x ∈ R+

7

Bp (t), ut (x) ≥ ut (y)}; Xp : T0 → P(Rl ) such that, for each t ∈ T0 , Xp (t) = Bp (t) ∩ Cp (t). The following proposition, which will be used in the proof of the main theorem, shows that there is a measurable selector from the function Xp . l such that p  0, there exists an integrable Proposition 6. For each p ∈ R+ l function xp : T0 → R+ such that, for each t ∈ T0 , xp (t) ∈ Xp (t).

l Proof. First, observe that, for each p ∈ R+ such that p  0, Assumption 2 6 ∅. Moreover, by Aumann (1966), we implies that, for each t ∈ T0 , Xp (t) = know that the function Xp is a Borel measurable function because the functions Bp and Cp are Borel measurable functions and {(t, x) : x ∈ Xp (t)} = {(t, x) : x ∈ Bp (t)} ∩ {(t, x) : x ∈ Cp (t)}. Finally, Xp is integrably bounded

Pl

pj w(t)

because xi ≤ j=1pi , i = 1, . . . , l, for all t ∈ T0 and for all x such that x ∈ Xp (t). But then, by Theorem 2 in Aumann (1965), there exists an integrable function xp such that, for each t ∈ T0 , xp (t) ∈ Xp (t). 2

We proceed to the definition of the Cournot-Nash equilibrium. Let b ∈ Rl be a vector such that b = (b11 , b12 , . . . , bl−1l , bll ). A strategy function is a function 2 2 B : T → P(Rl ) such that, for each t ∈ T , B(t) = {b ∈ Rl : bij ≥ 0, i, j = P 1, . . . , l; lj=1 bij ≤ wi (t), i = 1, . . . , l}. Notice that, for each t ∈ T , the set B(t) is nonempty, convex and compact. A strategy selection is a function 2 b : T → Rl , each coordinate of which is integrable, such that, for almost all t ∈ T , b(t) ∈ B(t). For each t ∈ T , bij (t), i, j = 1, . . . , l, is the amount of commodity i that trader t offers in exchange for commodity j. By neglecting, as usual, the distinction between integrable functions and equivalence classes 2 of such functions, we denote by L1 (µ, Rl ) the set of integrable functions 2 taking values in Rl and by L1 (µ, B(·)) the set of strategy selections (see Schmeidler (1973) and Khan (1985)). RGiven a strategy selection b, we define ¯ = ( T bij (t) dµ). Moreover, we denote by ¯ to be B the aggregate matrix B b \ b(t) a strategy selection obtained by replacing b(t) in b by b(t) ∈ B(t). Now, we are able to give the following definitions (see Sahi and Yao (1989)). Definition 2. Given a strategy selection b, we say that a price vector p exists if l l X X j i¯ ¯ ji ), j = 1, . . . , l. p  0, (1) p bij = p ( b i=1

i=1

By Lemma 1 in Sahi and Yao (1989), there is a unique, up to a scalar multiple, ¯ is irreducible. Denote by p(b) price vector p satisfying (1) if and only if B 8

¯ is the function which associates to each strategy selection b such that B irreducible the unique, up to a scalar multiple, price vector p satisfying (1). Given a strategy selection b such that p exists and is unique, up to a scalar multiple, consider the assignment determined as follows xj (t, b(t), p(b)) = wj (t) −

l X

bji (t) +

i=1

l X i=1

bij (t)

pi (b) , pj (b)

(2)

for all t ∈ T , j = 1, . . . , l. It is easy to verify that the assignemt x(t, b(t), p(b)) is an allocation. Given a strategy selection b, the final holdings of the traders are ( xj (t, b(t), p(b)) if p exists and is unique, j x (t) = (3) wj (t) otherwise, for all t ∈ T , j = 1, . . . , l. We would like to notice that this is a respecification of the model analysed by Sahi and Yao (1989) which allows to define a Cournot-Nash equilibrium for exchange economies with a continuum of traders (see Codognato and Ghosal (2000)). Now, we are able to define the equilibrium concept. ˆ ˆ such that B ¯ is irreducible is a CournotDefinition 3. A strategy selection b Nash equilibrium if ˆ ≥ ut (x(t, b(t), p(b ˆ \ b(t)))), ˆ p(b))) ut (x(t, b(t), for all almost all t ∈ T and for all b(t) ∈ B(t).

Now, we are ready to state our main theorem. Theorem. Under Assumptions 1, 2, 3 and 4, there is a Cournot-Nash equiˆ librium b.

3

Discussion of the model

Okuno et al. (1980) consider crucial to show that their noncooperative version of the mixed model is immune from the counterintuitive issues of the cooperative model analysed by Shitovitz (1973). In particular, they devote an example and a proposition to show that the Cournot-Nash equilibrium

9

allocations of a mixed exchange economy in which atoms have the same endowments and the same preferences, but not necessarily the same size, are not Walrasian, whereas, in this case, by Theorem B in Shitovitz (1973), the core allocations and the competitive allocations coincide. Since Codognato and Ghosal (2000) are able to show an equivalence `a la Aumann (see Aumann (1964)) between the Cournot-Nash equilibrium allocations and the Walras equilibrium allocations, it seems opportune to check that the same does not occur under the assumptions of Thoerem B in Shitovitz (1973). The next results should clarify that the model we use is an appropriate model of oligopoly because the small traders always have a price-taking, Walrasian behavior whereas the large traders keep market power even in those circumstances where the core outcome is competitive. The following proposition shows that each nonatomic trader is unable to influence prices and that the best reply of each nonatomic trader attains a point in his Walras demand correspondence. ¯ is irreducible and Proposition 7. For each strategy selection b such that B for each t ∈ T0 , (i) p(b) = p(b \ b(t)), for all b(t) ∈ B(t); (ii) x(t, b(t), p(b \ b(t))) ∈ Xp(b) , for all b(t) ∈ argmax{ut (x(t, b(t), p(b \ b(t)))) : b(t) ∈ B(t)}.

Proof. (i) It is an immediate consequence of Definition 2. (ii) It can be proved by the same argument used in the proof of part (i) of Theorem 2 in Codognato and Ghosal (2000).

The following example generalizes the example and the proposition proposed by Okuno et al. (1980) since, being a particularization of our model to a two good economy, it allows traders to buy and sell each commodity, whereas, in the noncooperative model of a two goods exchange economy analysed by Okuno et al. (1980), traders are not allowed to sell both goods. Example. Consider the following specification of the model of Section 2: l = 2; T1 = {2, 3}, w(2) = w(3), u2 (x) = u3 (x); w(t) = (0, 1), ut (x) = ˆ is a Cournot-Nash (x1 )α (x2 )1−α , 0 < α < 1, for all t ∈ T0 . Then, if b ˆ and x ˆ ˆ) such that pˆ = p(b) ˆ(t) = x(t, b(t), pˆ), for equilibrium, the pair (ˆ p, x all t ∈ T , is not a Walras equilibrium. ˆ be a Cournot-Nash equilibrium and suppose that the pair Proof. Let b ˆ and x ˆ p, x ˆ(t) = x(t, b(t), pˆ), for all t ∈ T , is a Walras (ˆ ˆ) such that pˆ = p(b) ˆ 21 (t) = α, for all t ∈ T0 . Since, for each atom, at a equilibrium. Clearly, b Nash equilibrium, the marginal price (see Okuno et al. (1980)) must be equal 10

to the marginal rate of substitution which, in turn, at a Walras equilibrium, must be equal to the relative price of good 1 in terms of good 2, we must have ˆ 12 (t) b dx2 = pˆ, = −ˆ p2 ˆ 21 (t) + α dx1 b for each t ∈ T1 . But then, we must have ˆ 21 (2) + α ˆ 21 (3) + α ˆ 21 (2) + b ˆ 21 (3) + α b b b = = . ˆ 12 (2) ˆ 12 + b ˆ 12 (3) ˆ 12 (3) b b(2) b

(4)

ˆ 21 (3) + α) and ˆ 21 (2) = t(b The last equality of (4) holds if and only if b ˆ ˆ b12 (2) = tb12 (3), with t > 0. But then, the first and the last members of (4) cannot be equal because ˆ 21 (3) + α ˆ 21 (3) + α) + α b t(b = 6 . ˆ 12 (3) ˆ 12 (3) b tb ˆ ˆ and x This implies that the pair (ˆ ˆ) such that pˆ = p(b) pˆ), p, x ˆ(t) = x(t, b(t), for all t ∈ T , cannot be a Walras equilibrium.

We would like to notice that our example is stronger than the proposition proved by Okuno et al. (1980) because atoms have not only the same endowments and the same preferences but also the same measure.

4

Proof of the main theorem

As in Sahi and Yao (1989), we shall first show the existence of a slightly perturbed Cournot-Nash equilibrium. Given  > 0, we define the aggregate R   ¯ ¯  is ir¯ bid matrix B to be B = ( T bij (t) dµ + ). Clearly, the matrix B reducible. The interpretation is that an outside agency places fixed bids of  for each pair of commodities (i, j). Given  > 0, we denote by p (b) the function which associates to each strategy selection b the unique, up to a scalar multiple, price vector which satisfies l X i=1

l X

¯ ij + ) = pj ( pi (b

¯ ji + ), j = 1, . . . , l. (b

i=1

11

(5)

ˆ  is an -Cournot-Nash Definition 4. Given  > 0, a strategy selection b equilibrium if ˆ  (t), p (b ˆ  ))) ≥ ut (t, b(t), p (b ˆ  \ b(t)))), ut (x(t, b for almost all t ∈ T and for all b(t) ∈ B(t).

The following fixed point theorem, which is proved by Fan (1952) and Glicksberg (1952), is the basic tool to show our main theorem. Theorem (Fan-Glicksberg). Let K be a nonempty, convex and compact subset of a locally convex space X. If φ is an upper semicontinuos mapping from K into K and if, for all x ∈ X, the set φ(x) is nonempty and convex, then there exists a point xˆ ∈ K such that xˆ ∈ φ(ˆ x). 2

The locally convex space we shall working in is L1 (µ, Rl ) endowed with its weak topology. The following lemma provides us with the required properties of the set L1 (µ, B(·)). Lemma 1. The set L1 (µ, B(·)) is nonempty, convex and weakly compact. P

l Proof. For each i = 1, . . . , l, let λij ≥ 0, j=1 λij = 1. Since w is an 2 l assignment, the function b : T → R+ such that, for each t ∈ T , bij (t) = 2 λij wi (t), i, j = 1, . . . , l belongs to L1 (µ, B(·)). The fact that L1 (µ, Rl ) is a vector space and the fact that, for each t ∈ T , B(t) is convex imply that may be L1 (µ, B(·)) is convex. Finally, the weak compactness of L1 (µ, B(·)) R proved following Khan (1985). First, notice that supb∈L1 (µ,B(·)) T |bij | dµ < each j = 1, . . . , l, there exists a δj > 0 ∞, i, j = 1, . . . , l. Let  > 0. For R (depending upon ) such that | E wj (t) dµ| ≤ , for all measurable sets E with µ(E) ≤ δj (see Problem 18.6 in Aliprantis and Burkinshaw (1990b) p. 127). This implies that, if µ(E) ≤ δ = min{δ1 , . . . , δl }, then, for all R b ∈ L1 (µ, B(·)), E |bij (t)| dµ ≤ , i, j = 1, . . . , l and this, in turn implies, by the Dunford-Pettis theorem (see Diestel (1984) p. 93), that L1 (µ, B(·)) has a weakly compact closure. Now, let {bn } be a Cauchy sequence of 2 L1 (µ, B(·)). Since L1 (µ, Rl ) is complete {bn } converges in the mean to an integrable function b. But then, there exists a subsequence {bkn } of {bn } such that bkn → b a.e. (see Theorem 21.5 in Aliprantis and Burkinshaw (1990a) p. 159). Since, for each t ∈ T , B(t) is compact, this implies that b ∈ L1 (µ, B(·)). Hence L1 (µ, B(·)) is norm closed and, since it is also convex, it is weakly closed (see Corollary 4 in Diestel (1984) p. 12).

12

Given  > 0, let α : L1 (µ, B(·)) → L1 (µ, B(·)) be a mapping such that α(b) = {b ∈ L1 (µ, B(·)) : b(t) ∈ αt (b), for almost all t ∈ T } where, for each t ∈ T , the mapping αt : L1 (µ, B(·)) → B(t) is such that αt (b) = argmax{ut (x(t, b(t), p (b \ b(t)))) : b(t) ∈ B(t)}. The following lemma provides us with the required properties of the mapping α. Lemma 2. Given  > 0, the mapping α : L1 (µ, B(·)) → L1 (µ, B(·)) is an upper semicontinuous mapping such that, for all b ∈ L1 (µ, B(·)), the set α(b) is nonempty and convex. Proof. Let  > 0 be given. Consider a trader t ∈ T1 . By Lemma 4 in Sahi and Yao (1989), we know that αt is an upper semicontinuos mapping such that, for all b ∈ L1 (µ, B(·)), αt (b) is nonempty, compact and convex. Now, consider a trader t ∈ T0 . Given b ∈ L1 (µ, B(·)), Proposition 7 implies that ut (x(t, b(t), p (b \ b(t)))) = ut (x(t, b(t), p (b))), for all b ∈ B(t). Therefore, for all b ∈ L1 (µ, B(·)), αt is nonempty and compact, by the continuity of the function ut (x(t, b(t), p (b))) over the compact set B(t), and convex, by Assumption 2. The upper semicontinuity of αt is a straightforward consequence of the Maximum Theorem (see Berge (1997), p. 116). Now, given a strategy selection b ∈ L1 (µ, B(·)), by Proposition 6, there exists an integrable l such that, for each t ∈ T0 , xp (b) (t) ∈ Xp (b) (t). function xp (b) : T0 → R+ By Lemma 5 in Codognato and Ghosal (2000), for each t ∈ T0 , there exist P λj (t) ≥ 0, lj=1 λj (t) = 1, such that xjp (b) (t)

j

= λ (t)

Pl

j=1

j

pj (b)0 w (t) j = 1, . . . , l. pj (b)

l Define a function λ : T0 → R+ , such that, λ(t) = λ(t), for each t ∈ 0 T0 . Since xp (b) and w are integrable functions with respect to µ0 and Pl j 0 j j=1 p (b) w (t)  0, for all t ∈ T0 , λ is a function which is integrable ∗ l2 with respect to µ0 . Now, define a function 0 b : T0 → R+ such that 0 b∗ (t) = 0 wi (t)λj (t), i, j = 1, . . . , l, for all t ∈ T . The function 0 b∗ is 0 ij integrable with respect to µ0 and hence, by Proposition 3, with respect to µ. Moreover, by Theorem 2 in Codognato and Ghosal (2000), b∗ (t) ∈ αt (b), ∗ ∗ l2 be a function such that 1 b (t) ∈ αt (b), for each t ∈ T0 . Let 1 b : T1 → R+ ∗ for each t ∈ T1 . The function 1 b is integrable with respect to µ1 and hence, by Proposition 4, with respect to µ. But then, by Proposition 5, α(b) is nonempty. The convexity of α(b) is a straightforward consequence of the

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convexity of αt (b), for all t ∈ T . Finally, the upper semicontinuity of α may be proved following Khan (1985). Since L1 (µ, B(·)) is weakly compact, we can show the upper semicontinuity of α by showing that its graph is closed in L1 (µB(·)) × L1 (µB(·)) (see the Corollary in Berge (1997) p. 112). Let {bν , b∗ν } be a net converging to (b, b∗ ) where b∗ν ∈ α(bν ). The set {bν , b∗ν } ∪ (b, b∗ ), being a subset of L1 (µ, B(·) × L1 (µ, B(·)) (see Theorem 2.11 in Aliprantis and Border (1994) p. 30), is relatively weakly compact. By the Eberlein-Smulian Theorem (see Aliprantis and Border (1994) p. 200), the set {bν , b∗ν } ∪ (b, b∗ ) is also relatively weakly sequentially compact and this, in turn, implies that there exists a sequence {bn , b∗n }, extracted from the net {bν , b∗ν }, which converges weakly to (b, b∗ ) (see Problem 17L in Kelley and Namioka (1963) p. 165). Now, for each t ∈ T , denote by Ls {b∗n (t)} the set of limit points of the sequence {b∗n (t)} and by coLs {b∗n (t)} the set of convex combinations of these limit points. For each t ∈ T , the fact that αt is compact-valued and upper semicontinuous and the fact that B(t) is compact imply that Ls {b∗n (t)} ⊆ αt (b) and this, together with the fact that αt (b) is convex, in turn, implies that coLs {b∗n (t)} ⊆ αt (b). Since the sequence {b∗n } converges weakly to b∗ and it is uniformly integrable (see Hildenbrand (1974) p. 52), by Proposition C in Artstein (1979), b∗ (t) ∈ coLs {b∗n (t)}, for almost all t ∈ T , and we are done. Now, we can prove the existence of an -Cournot-Nash equilibrium.

ˆ. Lemma 3. For each  > 0, there is an -Cournot-Nash equilibrium b Proof. It is a straightforward consequence of Lemmas 1 and 2 and the Fan-Glicksberg Theorem. As in Sahi and Yao (1989), we introduce the concept of δ-positivity. 2

Definition 6. For δ > 0, the function Bδ : T → Rl is a δ-positive strategy P P 2 function if Bδ (t) = B(t) ∩ {b ∈ Rl : i6∈J j∈J (bij + bji ) ≥ δ, for each J ⊆ {1, . . . , l}}, for each t ∈ T1 with w(t)  0; Bδ (t) = B(t), for the remaining traders t ∈ T . ˆ  is called δ-positive if, for almost all t ∈ T , An -Cournot-Nash equilibrium b  δ ˆ (t) ∈ B (t). For each t ∈ T1 , let δ ∗ (t) = 1 min{w1 (t), . . . , wl (t)} and b m ∗ δ ∗ = min{δ ∗ (t) : δ ∗ (t) > 0, t ∈ T1 }. Given  > 0, let αδ : L1 (µ, B(·) → ∗ L1 (µ, B(·)) be a mapping such that αδ (b) = {b ∈ L1 (µ, B(·) : b(t) ∈ ∗ ∗ ∗ αtδ (b), for almost all t ∈ T } where, for each t ∈ T , αtδ (b) = αt (b) ∩ Bδ (t). The following lemma is a strengthening of Lemma 4. 14

Lemma 4. For each  > 0, there is a δ ∗ -positive -Cournot-Nash equilibrium ˆ . b Proof. Let  > 0 be given. By Lemma 6 in Sahi and Yao (1989), we ∗ know that, for each b ∈ L1 (µ, B(·)), αtδ (b) is nonempty, for each t ∈ T1 ∗ with w(t)  0. But then, by the same argument of Lemma 4, αδ (b) is ∗ nonempty. The convexity of αδ (b) is a straightforward consequence of the ∗ convexity of αt (b) and Bδ (t), for all t ∈ T . The upper semicontinuity ∗ of αδ can be proved using the same argument of Lemma 4 since, for all ∗ t ∈ T , αtδ is upper semicontinuos, by the upper semicontinuity of αt and the nonemptyness and compactness of B(t) (see Theorem 2’ in Berge (1997) p. 114). The proof is complete since all the assumptions of the Fan-Glicksberg Theorem are satisfied. Let n = n1 , n = 1, 2, . . .. By Lemma 4, for each n = 1, 2, . . ., there is ˆ n . The fact that the sequence aRδ ∗ -positive -Cournot-Nash equilibrium b  n ˆ (t) dµ0 } belongs to the compact set W = {bij ∈ Rl2 : 0 ≤ bij ≤ { T0 0 b R 1 ˆ n i T0 w (t) dµ0 , i, j = 1, . . . , l}, the sequence { b } belongs to the compact set Q δ∗ ˆ n ), for each n = 1, 2, . . ., ˆn , where pˆn = p(b t∈T1 B (t) and the sequence p belongs, by Lemma 9 in Sahi and Yao (1989), to a compact set P, imR ˆ n (t) dµ0 , 1 b ˆ n , pˆn } belongs to the compact plies that the sequence { T0 0 b Q ∗ set W × t∈T1 Bδ (t) × P and this, in turn, implies that it has a subsequence (which we denote in the same way to save in notation) which converges to Q ∗ an element of the set W × t∈T1 Bδ (t) × P (see Problem D in Kelley (1955) ˆ n } satisfies the assumptions of Theorem A p. 238). Since the sequence {0 b ˆ such that 0 b(t) ˆ is a limit point in Artstein (1979), there is a function 0 b R  n 0 ˆ ˆ n (t) dµ0 } of b (t) for almost all t ∈ T0 and such that the sequence { T0 0 b R ∗ ˆ dµ0 . Moreover, 0 b(t) ˆ converges to T0 0 b(t) ∈ Bδ (t), for almost all t ∈ T0 , ˆ is the limit of a subsequence of {0 b ˆ n (t)}, for almost all t ∈ T0 . because 0 b(t) ˆ ∈ Qt∈T Bδ∗ (t), the seˆ n } converges to a point 1 b Since the sequence {1 b 1 R ˆ dµ1 . But then, by Proposition ˆ n (t) dµ1 } converges to R 1 b(t) quence { T1 1 b T1 R ˆ n (t) dµ} must converge to R b(t) ˆ dµ. Since, the se5, the sequence { T b T n quence R{ˆ p } converges to a price vector pˆ ∈ P , by the continuity of (5), ˆ dµ must satisfy (1). Moreover, since, by Lemma 9 in Sahi pˆ and T b(t) and Yao (1989), pˆ  0, Lemma 1 in Sahi and Yao (1989) implies that ˆ ˆ ∈ L1 (µ, Bδ∗ (·)), by Remark ¯ is completely reducible. But then, since b B 15

ˆ ¯ must be irreducible. In order to conlcude 3 in Sahi and Yao (1989), B ∗ ˆ is a δ -positive -Cournot-Nash equilibrium, we have to show that that b ˆ ˆ \ b(t)))), for almost all t ∈ T and for all ut (x(t, b(t), pˆ)) ≥ ut (x(t, b(t), p(b ˆ \ b(t) denote the aggregate matrix corresponding to the ¯ b(t) ∈ B(t). Let B  ˆ ˆ \ b(t) and let B ¯ n \ b(t) denote the aggregate matrix corstrategy selection b ˆ n \ b(t), for each n = 1, 2, . . .. As in responding to the strategy selection b Sahi and Yao (1989), we proceed by considering the following possible cases. ˆ ¯ \ b(t) is completely reCase 1. t ∈ T1 and b(t) ∈ B(t) is such that B n ˆ ¯ \ b(t) is irreducible, for each n = 1, 2, . . ., and so is ducible. Clearly, B ˆ ˆ n \ ¯ \ b(t), by Remark 3 in Sahi and Yao (1989). Since the sequence {RT b B R ˆ \ b(t)(t) dµ b(t)(t) dµ} converges, by the same argument given above, to T b and since, by Lemma 2 in Sahi and Yao (1989), prices are cofactors, the seˆ n \ b(t))} converges to p(b ˆ \ b(t)). Consequently, the sequence quence {pn (b n ˆ n ˆ \ b(t))). The fact that {x(t, b(t), p (b \ b(t)))} converges to x(t, b(t), p(b   ˆ n (t), pˆ n )} converges to x(t, b(t), ˆ the sequence {x(t, b pˆ) and the fact that n n n ˆ ˆ ut (x(t, b (t), pˆ )) ≥ ut (x(t, b(t), p (b \ b(t)))), for each n = 1, 2, . . ., allow ˆ ˆ\ us to conclude, by Assumption 2, that ut (x(t, b(t), pˆ)) ≥ ut (x(t, b(t), p(b b(t)))). ˆ ¯ \ b(t) in not completely reCase 2. t ∈ T1 and b(t) ∈ B(t) is such that B n ˆ ˆ ducible. The fact that the sequence {x(t, b (t), pˆn )} converges to x(t, b(t), pˆ) n n ˆ and the fact that ut (x(t, b (t), pˆ ) ≥ ut (w(t)), for each n = 1, 2, . . ., imply, ˆ ˆ by Assumption 2, that ut (x(t, b(t), pˆ)) ≥ ut (w(t)) = ut (x(t, b(t), p(b\b(t)))). ˆ ¯ \ b(t) is irreducible Case 3. t ∈ T0 and b(t) ∈ B(t). Clearly, the matrix B n ˆ n n ˆ n and, by Proposition 7, p (b \ b(t)) = p (b ), for each n = 1, 2, . . ., and ˆ is a limit point of the sequence {b ˆ n (t)}, it is a ˆ \ b(t)) = p(b). ˆ Since b(t) p(b limit of a subsequence (which we denote in the same way to save in notation) of this sequence. But then, the fact that the sequence {x(t, b(t), pˆn )} conˆ n (t), pˆn )} converges verges to x(t, b(t), pˆ), the fact that the sequence {x(t, b   ˆ n (t), pˆ n ) ≥ ut (x(t, b(t), pˆn )), for ˆ to x(t, b(t), pˆ) and the fact that ut (x(t, b ˆ each n = 1, 2, . . ., imply that ut (x(t, b(t), pˆ)) ≥ ut (x(t, b(t), pˆ)). This completes the proof of the main theorem.

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5

Conclusion

Following Okuno et al. (1980), we provide a model of oligopoly in general equilibrium in which it is possible to prove the existence of a noncooperative equilibrium in pure strategies with trade. We do this by modelling trade as a game of simultaneous, noncooperative exchange between large traders, represented as atoms, and small traders, represented by an atomless sector. The model of noncooperative exchange we use allows traders to buy and sell each commodity and it has the feature that the best reply of each small trader picks a commodity bundle that maximizes his utility subject to his budget constraint at the prevailing market clearing prices.

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