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In this paper, we prove a new version of the Second Welfare Theorem for economies with a finite number of agents and an infinite number of commodities, when the preference correspondences are not convex-valued and/or when the total production set is not convex. For this kind of nonconvex economies, a recent result, obtained by one of the authors, introduces conditions which, when applied to the convex case, give for Banach commodity spaces the well-known result of decentralization by continuous prices of Pareto-optimal allocations under an interiority condition. In this paper, in order to prove a different version of the Second Welfare Theorem, we reinforce the conditions on the commodity space, assumed here to be a Banach lattice, and introduce a nonconvex version of the properness assumptions on preferences and the total production set. Applied to the convex case, our result becomes the usual Second Welfare Theorem when properness assumptions replace the interiority condition. The proof uses a Hahn-Banach Theorem generalization by Borwein and Jofre (in Joper Res Appl Math 48:169–180, 1997) which allows to separate nonconvex sets in general Banach spaces

The general purpose of this paper is to prove quasiequilibrium existence theorems for production economies with general consumption sets in an infinite dimensional commodity space, without assuming any monotonicity of preferences or free-disposal in production. The commodity space is a vector lattice commodity space whose topological dual is a sublattice of its order dual. We formulate two kinds of properness concepts for agents' preferences and production sets, which reduce to more classical ones when the commodity space is locally convex and the consumption sets coincide with the positive cone. Assuming properness allows for extension theorems of quasiequilibrium prices obtained for the economy restricted to some order ideal of the commodity space. As an application, the existence of quasiequilibrium in the whole economy is proved without any assumption of monotonicity of preferences or free-disposal in production.

We present a simple and direct proof for a Leray-Schauder type alternative for a large class of condensing or compact set-valued maps containing convex as well as nonconvex maps. The aim of this note is to extend the Leray-Schauder type nonlinear alternative presented in [BI] to a condensing upper semicontinuous approximable set-valued map F: X -> E when X is a closed subset with nonempty interior of a locally convex topological vector space E. The proof presented here is even shorter and simpler than the one given in [BI]. In what follows E stands for a Hausdorff locally convex topological vector space with a fundamental basis JK. of convex, symmetric neighborhoods of the origin; if X, Y are nonempty subsets of E, then F: X -Y is a set-valued map with nonempty values (simply called map). The boundary, the interior, the closure, and the convex hull of a subset A in E are denoted by AA, int A, A, and co A respectively. Definition 1. F is said to be upper semicontinuous (u.s.c.) on X if and only if for any open subset V of Y, the set {x E X: F(x) C V} is open in X. Definitions 2 ([BD], [BI], see also [GGK] for metric spaces). (1) Given U, V E X, a function s: X -+ Y is said to be a (U, V)-approximative selection of F if for any xE X, S(x) E (F[(x+ U) nX] +V) n Y. (2) F: X -* Y is said to be approachable if it has a continuous (U, V)approximative selection for any (U, V) E KV x JK. A(X, Y) denotes the class of such maps. We write .A(X) for A(X, X). (3) F is said to be approximable if its restriction FIK to any compact subset K of X is approachable. Note that an approachable map is approximable (cf. [B]). Examples. It is well-known that if F is u.s.c. with nonempty convex values, then F is approachable provided X is paracompact and Y is convex (cf. [DG]). Obviously, F is approximable without conditions on X (see [C]). Received by the editors August 12, 1996 and, in revised form, January 16, 1997. 1991 Mathematics Subject Classification. Primary 47H04, 47H10, 54C60.

A definition and an existence result are given for quasiequilibria in abstract economies with a (possibly) countably infinite set of agents and infinite dimensional choice sets. Applied to the overlapping generations exchange model, this result allows us to prove the existence of equilibria when the commodity space in each period is some Euclidean space, L ∞ or some L p ( p ≥ 1). The preferences of the agents may be interdependent and are not required to be transitive or complete.

In this paper, we establish the existence of Berge's strong equilibrium for games with n persons in infinite dimensional strategy spaces in the case where the payoff function of each player is quasi-concave. Moreover, we study the continuity of Berge's strong equilibrium correspondence and prove that most of Berge's strong games are essential., Pour un jeu non-coopératif en forme normale, un vecteur de stratégies est un point d'équilibre fort de Berge si la coalition complémentaire de tout joueur n'a aucun intérêt à dévier. Dans ce papier, supposant les espaces de stratégie de dimension possiblement infinie et les fonctions de paiement quasi-concaves, nous établissons l'existence de l'équilibre fort de Berge pour les jeux à n personnes ainsi que pour les jeux généralisés ou économies abstraites où la définition du jeu est complétée par la donnée de correspondances de contrainte pour chacun des agents. Nous étudions la continuité de la correspondance d'équilibre fort de Berge et montrons que la plupart des jeux forts de Berge sont essentiels.

Focusing mainly on equilibrium existence results, this paper emphasizes the role of fixed point theorems in the development of general equilibrium theory, as well for its standard definition as for some of its extensions., Centré principalement sur les résultats d'existence de l'équilibre, cet article met en évidence le rôle des théorèmes de point fixe dans le développement de la théorie d'équilibre général, aussi bien dans sa version standard que dans quelques unes de ses extensions.

This short paper published in Games and Economic Behavior (July 2009) "In Memoriam" of David Gale, emphasizes the seminal role played by two lemmas of David Gale in the development of the foundations of General Equilibrium Theory., Ce cours article, publié dans Games and Economic Behavior (Juillet 2009) "In Memoriam" et en l'honneur de David Gale, analyse l'importance de deux lemmes dûs à David Gale pour le développement de la théorie de l'équilibre général, à ses débuts en 1955 puis vingt ans après.

The existence of equilibria is proved in a stochastic market with a finite number of agents and countably many periods.

The coincidence under suitable conditions between the set of Walrasian allocations and the set of Edgeworth equilibria for an economy with ordered preferences defined in a finite dimensional commodity space is a result by Debreu and Scarf. The aim of this paper is to extend this result to production economies without ordered preferences defined in a Hausdorff linear topological space

International audience; We present in this paper a generalized version of the celebrated Knaster–Kuratowski–Mazurkiewicz–Fan's principle on the intersection of a family of closed sets subject to a classical geometric condition and a weakened compactness condition. The fixed point formulation of this generalized principle extends the Browder–Fan fixed point theorem to set-valued maps of non-compact convex subsets of topological vector spaces.

The second welfare theorem and the core-equivalence theorem have been proved to be fundamental tools for obtaining equilibrium existence theorems, especially in an infinite dimensional setting. For well-behaved exchange economies that we call proper economies, this paper gives (minimal) conditions for supporting with prices Pareto optimal allocations and decentralizing Edgeworth equilibrium allocations as non-trivial equilibria. As we assume neither transitivity nor monotonicity on the preferences of consumers, most of the existing equilibrium existence results are a consequence of our results. A natural application is in Finance, where our conditions lead to new equilibrium existence results, and also explain why some financial economies fail to have equilibrium., Résumé. Le second théorème de l'économie du bien-être et le théorème d'équivalence coeur-équilibre sont des outils fondamentaux pour obtenir des théorèmes d'existence de l'équilibre, spécialement quand l'espace des biens n'est pas de dimension finie. Pour des économies d'échange que nous appelons propres, cet articles donne des conditions (minimales) de décentralisation par des prix des allocations Pareto-optimales et des équilibres d'Edgeworth. Comme nous ne supposons ni transitivité, ni monotonicité des préférences des consommateurs, la plupart des théorèmes existants d'existance de l'équilibre sont une conséquence de nos résultats. Une appication naturelle de ces résultats est en Finance où nos conditions conduisent à des résultats nouveaux, en même temps qu'elles expliquent pourquoi certains modèles financiers n'ont pas d'équilibre.

In a finite or in an infinite dimensional setting, every general competitive equilibrium existence proof lies on a fixed point theorem. Our aim in this chapter is to give a systematic account of some topics in fixed point theory that lie on the border line of topology and equilibrium theory.

In the previous chapter, we study the minimization problem of a convex function under convex constraints. In this chapter, we study the minimization problem for two other classes of functions. The first ones are quasi-convex functions. We know (see Chapter 5) that for a convex function f, any level set of f is convex. But the converse is not true. The functions which have the property that every level set is convex are precisely called quasi-convex functions. In the first section we give necessary and sufficient conditions for a point to be a solution to a minimization problem of a quasi-convex function under convex constraints.

In this chapter we want to solve the problem minf(x) | x ∈ C, where f is a convex function on ℝ n , and C is a convex, nonempty subset of ℝ n . A point x* ∈ C is a global solution, or more simply a solution to this problem, or a minimizer of f on C, if f(x*) ≤ f(x), ∀x ∈ C. We say that x* is a local solution to this problem if there exists a relatively open set U of C such that f(x*) ≤ f (x), ∀x ∈ U.

We prove an infinite dimensional extension of the Gale–Nikaido–Debreu lemma which includes all necessary limiting processes and allows a proof of the existence of equilibria under standard assumptions in an economy with infinitely many commodities which exactly parallels the proof of Debreu (1959) for the finite dimensional case.

The purpose of this paper is to extend Himmelberg's fixed-point theorem, replacing the usual convexity in topological vector spaces with an abstract topological notion of convexity that generalizes classical convexity as well as several metric convexity structures found in the literature. We prove the existence, under weak hypotheses, of a fixed point for a compact approachable map, and we provide sufficient conditions under which this result applies to maps whose values are convex in the abstract sense mentioned above.

This paper gives a short proof of a non-trivial extension of the Gale-Nikaido-Debreu lemma.