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For an exchange economy, under assumptions which did not bring about the existence of equilibrium with dividends as yet, we prove the non-emptiness of the Edgeworth rejective core. Then, via Konovalov, 1998 , Konovalov, 2005 ’s decentralization result, we solve the equilibrium with dividends existence problem. Adding to the same assumptions a weak non-satiation condition which differs from the weak non-satiation assumption introduced by Allouch and Le Van (2009) , we show in the last section the existence of a Walrasian quasiequilibrium. This result, which fits with exchange economies whose consumers’ utility functions are not assumed to be upper semicontinuous, complements the one obtained by Martins-da-Rocha and Monteiro (2009) .

In this paper, we propose a definition of Edgeworth equilibrium for a private ownership production economy with (possibly infinitely) many private goods and a finite number of pure public goods. We show that Edgeworth equilibria exist whatever be the dimension of the private goods space, and can be decentralized, in the finite and infinite dimensional cases, as Lindahl-Foley equilibria. Existence theorems for Lindahl-Foley equilibria are a by-product of our results., Résumé. Nous proposons dans cet article une définition de l'équilibre d'Edgeworth pour une économie de production de propriété privée. L'espace de biens privés peut être de dimension infinie, mais le nombre des biens publics (purs) est fini. Nous montrons qu'il existe des équilibres d'Edgeworth, quelle que soit la dimension de l'espace des biens privés, et que ces équilibres peuvent être décentralisés comme équilibres de Lindahl-Foley. Des théorèmes d'existence d'équilibres de Lindahl-Foley sont un sous-produit de nos résultats.

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

This paper addresses existence of equilibria for an overlapping generations exchange economy with incomplete markets, one-period real assets and bounded short sales, defined on an infinite event-tree where there is a continuum of branches at each node of the tree. The demographic structure used in the paper is quite general and allows for infinite-lived as well as finite-lived agents.

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.

The general equilibrium model with incomplete markets is here extended to infinite horizon economies populated by a finite number of infinitely lived agents. the crucial issue that divides the infinite horizon setting from the finite horizon setting is in the nature of borrowing constraints, which added to spot constraints, define a plausible budget set for individual agents. the paper relates seven alternative definitions of equilibrium and states corresponding equilibrium existence theorems when assets are one-period and purely financial.

For an exchange economy, under assumptions which did not bring about the existence of equilibrium with dividends as yet, we prove the non-emptiness of the fuzzy rejective core. Then, via Konovalov (1998, 2005)'s equivalence result, we solve the equilibrium with dividends existence problem. Adding to the same assumptions a weak non-satiation condition which differs from the weak non-satiation assumption introduced by Allouch-Le Van (2009), we show in a last section the existence of a Walrasian quasiequilibrium. This result, which fits with exchange economies whose consumers' utility functions are not assumed to be upper semicontinuous, complements the one obtained by Martins-da-Rocha and Monteiro (2009).

This paper investigates the existence of competitive equilibria in dynamic exchange models with countably many periods and countably many agents. At each period the commodity space can be finite or infinite dimensional. The preferences of agents are not assumed to be transitive or complete. A first equilibrium existence theorem is established under the classical assumption that there exists a finite set of non-negligible agents. In the particular case of an overlapping generations model, a second existence theorem allows simultaneously for finite-lived assets and infinite-lived assets and limits the previous assumption to infinite-lived assets. This theorem covers obviously the standard case of an overlapping generations model where the agents have no endowment outside their lifetime.

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.

A very simple proof is given for the existence of stochastic equilibria in a T -period economy with an incomplete financial structure.

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.

For an exchange economy, under assumptions which did not bring about the existence of quasiequilibrium with dividends as yet, we prove the nonemptiness of the fuzzy rejective core. Then, via Konovalov (1998, 2005)'s equivalence result, we solve the equilibrium (with dividends) existence problem. In a last section, we show the existence of a Walrasian quasiequilibrium under a weak non-satiation condition which differs from the weak non-satiation assumption introduced by Allouch-Le Van (2009). This result, designed for exchange economies whose consumers' utility functions are not assumed to be upper semicontinuous, complements the one obtained by Martins-da-Rocha and Monteiro (2009)., L'équilibre avec dividendes est un équilibre dans lequel les agents, qui peuvent être saturés, ne dépensent pas nécessairement la totalité de leur revenu d'équilibre. Un concept de coeur, le coeur réjectif, défini par Konovalov (1998, 2005), correspond à ce concept d'équilibre, avec à la limite des allocations d'équilibre et des allocations du coeur flou rejectif. Dans ce papier, nous établissons, sous des hypothèses qui généralisent les résultats existants, la non-vacuité du coeur flou réjectif d'une économie d'échange, puis, via un théorème d'équivalence à la Konovalov, l'existence de l'équilibre avec dividendes. Nous montrons finalement l'existence d'un équilibre Walrasien sous une hypothèse de non-saturation des consommateurs plus faible que l'hypothèse standard de non-saturation en toute composante d'une allocation réalisable.

In this paper, we investigate the existence of a competitive equilibrium in an intertemporal pure exchange model with a countable number of periods and a countable number of agents. This model covers both the overlapping generations case and the case where there is an infinite number of infinite-lived agents or perhaps a mixture of both infinite-lived and finite-lived agents. At each period, the commodity space can be finite or infinite dimensional. A first equilibrium existence theorem is established for this general model under the classical assumption that there exists a finite set of non-negligible agents. Applied to the overlapping generations exchange model, this result implies for the agents the possibility of having endowments outside their life-time. The existence of an equilibrium for the overlapping generations model when the agents have no endowment outside their life-time is then proved. The last section extends this result to the case of commodity spaces without order-unit element.

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.

Focussing on their analysis of the optimal public goods provision problem, this paper follows the parallel development of equilibrium models and mechanism design after the accommodation of Samuelson's definition of collective goods to the general equilibrium framework. Both paradigms lead to the negative conclusion of the impossibility of a fully decentralized optimal public goods provision through market or market-like institutions., L'article suit sur plus de 50 ans le développement parallèle des modèles d'équilibre et de la théorie des mécanismes appliqués au problème de provision d'une quantité optimale de biens publics, depuis que deux papiers célèbres de Samuelson ont étendu le domaine de la souveraineté des consommateurs individuels à la détermination de la quantité de biens publics à mettre à la disposition de la société. Cette orientation initiale conduit à un même impasse théorique dans les deux paradigmes, c'est-à-dire à la conclusion de l'impossibilité d'une provision optimale complètement décentralisée.

Despite the large number of its references, this paper is less a survey than a systematic exposition, in an unifying framework and assuming convexity as well on the consumption side as on the production side, of the different equilibrium concepts elaborated for studying provision of public goods. As weak as possible conditions for their existence and their optimality properties are proposed. The general conclusion is that the drawbacks of the different equilibrium concepts lead to founding public economic policy either on direct Pareto improving government interventions or on state enforcement of decentralized mechanisms., Malgré le nombre de ses références, ce papier est moins un survey qu'un exposé systématique, dans un cadre unifié et sous des hypothèses classiques de convexité aussi bien des ensembles préférés que des ensembles de production, des différents concepts d'équilibre élaborés pour étudier la provision des biens publics. Des conditions aussi faibles que possible sont indiquées pour l'existence et les propriétés d'optimalité des équilibres. La conclusion générale est que les défauts de chacun des différents concepts d'équilibre conduit à fonder la politique publique en matière de provision des biens publics soit sur des interventions gouvernementales améliorantes, soit sur l'imposition de mécanismes décentralisés remplaçant le mécanisme du marché.

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

Unlike partial equilibrium analysis which study the equilibrium of a particular market under the clause "ceteris paribus" that revenues and prices on the other markets stay approximately unaffected, the ambition of a general equilibrium model is to analyze the simultaneous equilibrium in all markets of a competitive economy. Definition of the abstract model, some of its basic results and insights are presented. The important issues of uniqueness and local uniqueness of equilibrium are sketched ; they are the condition for a predictive power of the theory and its ability to allow for statics comparisons. Finally, we review the main extensions of the general equilibrium model. Besides the natural extensions to infinitely many commodities and to a continuum of agents, some examples show how economic theory can accommodate the main ideas in order to study some contexts which were not thought of by the initial model.

Unlike partial equilibrium analysis which study the equilibrium of a particular market under the clause "ceteris paribus" that revenues and prices on the other markets stay approximately unaffected, the ambition of a general equilibrium model is to analyze the simultaneous equilibrium in all markets of a competitive economy. Definition of the abstract model, some of its basic results and insights are presented. The important issues of uniqueness and local uniqueness of equilibrium are sketched ; they are the condition for a predictive power of the theory and its ability to allow for statics comparisons. Finally, we review the main extensions of the general equilibrium model. Besides the natural extensions to infinitely many commodities and to a continuum of agents, some examples show how economic theory can accommodate the main ideas in order to study some contexts which were not thought of by the initial model., Contrairement aux analyses d'équilibre partiel qui étudient le fonctionnement d'un marché particulier sous l'hypothèse que les revenus des agents et les prix sur les autres marchés resteront approximativement constants, l'ambition d'un modèle d'équilibre général est de permettre d'étudier l'équilibre simultané sur tous les marchés d'une économie compétitive. Cet article présente le modèle abstrait, les résultats fondamentaux en matière d'existence, d'optimalité et d'unicité locale de l'équilibre, les extensions du modéle initial qui ont fait l'histoire de cette théorie. Au-delà de l'extension naturelle à une infinité d'agents et à une infinité de biens, quelques exemples montrent comment la théorie économique peut adapter la problématique de l'équilibre général à des contextes économiques qui n'étaient pas prévus par le modèle initial.

This paper studies a simple stochastic two-period general equilibrium exchange model with money, an incomplete market of nominal assets, and a competitive banking system, intermediate between consumers and a Central Bank. There is a finite number of agents, consumers and banks. Default is not permitted. The public policy instruments are, besides real taxes implicit in the model, public debt and creation of money both implemented at the first period. The equilibrium existence is established under a Gains to trade hypothesis and the assumption that banks have a non zero endowment of money at each date-event of the model., Nous étudions dans cet article l'équilibre général d'un modèle simple, à deux périodes, avec money, un marché incomplet d'actifs nominaux et un système bancaire compétitif intermédiaire entre la Banque Centrale et les consommateurs. Le nombre des agents, consommateurs et banques, est fini. Le défaut n'est pas permis. Les instruments de politique publique sont, au-delà de taxes réelles, implicites dans le modèle, sur les ressources initiales des consommateurs, les montants de dette publique et de création monétaire fixés au début de la première période. L'existence de l'équilibre est établie sous l'hypothèse que l'autarcie n'est Pareto optimale en aucun des états du modèle et la supposition qu'également en tout état l'encaisse monétaire initiale globale des banques est positive.

We show that if K is a closed cone in a finite dimensional vector space X, then there exists a one-to-one linear operator T : X C [0,1] such that K is the pull-back cone of the positive cone of C [0,1], i.e., K = T -1 (C+ [0,1]). This problem originated from questions regarding arbitrage free prices in economics., Nous montrons que si K est un cone fermé dans un espace vectoriel de dimension finie X, il existe un opérateur linéaire injectif T : X C [0,1] tel que K est l'image inverse du cone positif de C [0,1], i.e., K = T-1 (C+ [0,1]). Ce problème a son origine dans des questions concernant les prix ne permettant pas l'arbitrage en économie.

International audience; An F-cone is a pointed and generating convex cone of a real vector space that is the union of a countable family of finite dimensional polyedral convex cones such that each of which is an extremel subset of the subsequent one. In this paper, we study securities markets with countably many securities and arbitrary finite portfolio holdings. Moreover, we assume that each investor is constrained to have a non-negative end-of-period wealth. If, under the portfolio dominance order, the positive cone of the portfolio space is an F-cone, then Edgeworth allocations and non-trivial quasi-equilibria exist. This result extend the case where, as in Aliprantis et al.[JME 30 (1998a) 347-366], the positive cone is a Yudin cone.

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.

The Walrasian model of general equilibrium of prices, production and exchange is today formulated in the concept of equilibrium of a so-called private ownership economy. In the classical model, only a finite number of commodities are exchanged, produced or consumed. Taking prices as given, finitely many consumers (who are endowed with initial holdings of the different commodities and who collect given shares of the profits from production), consume the commodities available on the market by optimizing their preferences among all possible consumption plans which satisfy their resulting budget constraint. Finitely many producers, who also take prices as given, maximize their profit on their individual set of possible production plans. They produce the commodities which satisfy consumers’ demand in competition with the initial resources not used by production. Market clearing determines equilibrium prices and the quantities actually consumed and produced at a state of equilibrium.

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.

Recall that in general equilibrium a commodity is defined not only by its physical properties but also by the date, the location and the state of nature that precise the conditions of its availability. The classical general equilibrium model hypothesis of a finite number of commodities, maintained in Chapters 2, 3 and 4, implies that the economic activity extends over only finitely many dates, locations and events. Such an hypothesis, contrary to the framework of many applied models of location, growth and finance, limits dramatically the application of the results to understanding the real economic life.

This chapter is devoted to the relations between the price system and the achievement of optimal economic welfare. Based on some easy consequences of the convexity assumptions made on the economy, this study was renewed in the fifties under the name of “new welfare economics”.

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.

This paper addresses existence of equilibria for a classical overlapping generations exchange economy with incomplete markets, one-period real assets and bounded short sales, defined on an infinite event-tree where there is a continuum of branches at each node of the tree.

Let A be a convex subset of ℝ n . A point x ∈ A is an extreme point of A if and only if there exists no open segment contained in A and containing x, that is, if and only if there is no way to express x as a convex combination x = λy + (1 - λ)z such that y ∈A, z ∈ A and 0 < λ < 1, except by taking x = y = z. Easy computations show that an equivalent characterization for an extreme point x of A is that the relations x = (1/2)y + (l/2)z together with y ∈ A, z ∈ A imply x = y = z. Obviously, if A = a, then a is an extreme point of A.

Let x and y be two points of ℝ n . The set $$ [x,y]: = \{ z \in \mathbb{R}^n |z = \alpha x + (1 - \alpha )y,\alpha \in \mathbb{R}and0 \leqslant \alpha \leqslant 1\} $$ is called closed line segment joining x and y. One also denotes $$ \begin{gathered} [(x,y): = [x,y]\backslash \{ y\} = \{ z \in \mathbb{R}^n |z = \alpha x + (1 - \alpha )y,\alpha \in \mathbb{R}and0 < \alpha \leqslant 1\} , \hfill \\ [(x,y: = [x,y]\backslash \{ x\} = \{ z \in \mathbb{R}^n |z = \alpha x + (1 - \alpha )y,\alpha \in \mathbb{R}and0 \leqslant \alpha < 1\} . \hfill \\ \end{gathered} $$ The set $$ \begin{gathered} \left[ x \right.,\left. y \right): = \left[ {x,y} \right]\backslash \left\{ y \right\} = \left\{ {z \in \mathbb{R}^n \left| {z = \alpha x + \left( {1 - \alpha } \right)y,\alpha \in \mathbb{R} and 0 < \alpha \leqslant 1} \right.} \right\}, \hfill \\ = \left\{ {z \in \mathbb{R}^n \left| {z = \alpha x + \left( {1 - \alpha } \right)y, \alpha \in \mathbb{R} and 0 < \alpha < 1} \right.} \right\}. \hfill \\ \end{gathered} $$ is called open line segment joining x and y. It may also be noted ]x,y[.

Let H = x ∈ ℝ n | p · x = α be a hyperplane in ℝ n (p ∈ ℝ n \ {0}, α ∈ ℝ). The sets F = x ∈ ℝ n | p · x ≤ α and G = x ∈ ℝ n | p · x ≥ α are called closed half-spaces associated with (or determined by) H. Obviously, they are convex and topologically closed as the inverse images of the closed intervals of ℝ: ] - ∞, α] and [α,+ ∞[ respectively by the (continuous) linear functional: x → p · x. The complementary sets V = x ∈ℝ n | p · x > α and U = x ∈ℝ n | p · x < α are obviously convex and topologically open; they are called open half-spaces associated with (or determined by) H.

The purpose of this paper is to study how the equilibrium prices vary with respect to the initial endowments in a linear exchange economy with a continuum of agents. We first state the model and give conditions of an increasing strength for existence, uniqueness and continuity of equilibrium prices. Then, if we restrict ourselves to economies with essentially bounded initial endowments and if we assume that there is, from the point of view of preferences, only a finite number of types of agents, we show that, on an open dense subset of the space of initial endowments, the equilibrium price vector is an infinitely differentiable function of the initial endowments. The proof of this claim is based on a formula allowing to compute the equilibrium price vector around a so-called “regular” endowment where it is known.

The purpose of this paper is to explain the role of financial assets in allowing individual agents of an economy to make at time 0 some limited commitments into the future which, at some extent, redistribute their revenue among several time periods and different states of the world. It is done studying in different contexts the general equilibrium of a simple two-period exchange model, under weaker assumptions and in a more general setting than the ones usually described in the literature.Several equilibrium existence theorems are stated and proved. Even in this simple framework, they often require a rather sophisticated mathematical background and are of a deep economic significance. Moreover, they are a necessary step towards further developments (including infinite horizon, continuous time, continuum of states of the world, defaut and collateral securities, ...).

A financial equilibrium existence theorem is proved under very mild assumptions on a two-period exchange model with bounded portfolios. We also characterize the asset trading that supports allocations, prices and financial transfers at the limit, when bounds are relaxed to infinity.

This paper proves core-equivalence theorems for exchange economies without ordered preferences, defined on locally convex Riesz commodity spaces such that the price space is a lattice. Properness assumptions are borrowed from some recent equilibrium existence results.

In this paper, we first give a direct proof of the existence of Edgeworth equilibria for exchange economies with consumption sets which are (possibly) unbounded below. The key assumption is that the individually rational utility set is compact. It is worth noticing that the statement of this result and its proof do not depend on the dimension or the particular structure of the commodity space. In a second part of the paper, we give conditions under which Edgeworth allocations can be decentralized by continuous prices in a finite dimensional and in an infinite dimensional setting. We then show how these results apply to some finance models., Résumé. Nous donnons en premier lieu une preuve directe de l'existence d'allocations d'Edgeworth pour une économie d'échange pour laquelle les ensembles de consommation des agents ne sont pas nécessairement bornés inférieurement. Un intérêt de ce résultat est de ne dépendre ni de la dimension ni de la structure de l'espace des biens du modèle. Nous montrons ensuite que, sous des conditions adéquates,les allocations d'Edgeworth peuvent être décentralisées par des prix continus, en dimension finie comme en dimension infinie. Ces résultats sont appliqués à quelque modèles de finance.

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.

In this paper we discuss the asymptotic properties of equilibrium quantities of a Leontief economy when agents have additively separable utilities. We estimate the rate of growth of equilibrium quantities and prices that either tend to zero or infinity or remain bounded below and above by strictly positive vectors. In the case of homogeneous preferences we prove that the quantities of any equilibrium converge to those of an equilibrium that grows at a constant rate.

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