Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Bücher
Zeitschriften
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
MTZ-Fachkongress

Antriebe und Energiesysteme von morgen


Austausch von Automobil- und Energiewirtschaft

Am 14./15. Mai geht es in Chemnitz um die Mobilitätswende durch Elektro- und Wasserstofffahrzeuge.
Erweiterte Suche
Anmelden
nach oben

1991 | Buch

Integration of Banach-Valued Correspondence Kapitel lesen Erstes Kapitel lesen

Equilibrium Theory in Infinite Dimensional Spaces

herausgegeben von: Prof. Dr. M. Ali Khan, Prof. Dr. Nicholas C. Yannelis

Verlag: Springer Berlin Heidelberg

Buchreihe : Studies in Economic Theory

Enthalten in: Professional Book Archive

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Über dieses Buch

Apart from the underlying theme that all the contributions to this volume pertain to models set in an infinite dimensional space, they differ on many counts. Some were written in the early seventies while others are reports of ongoing research done especially with this volume in mind. Some are surveys of material that can, at least at this point in time, be deemed to have attained a satisfactory solution of the problem, while oth­ ers represent initial forays into an original and novel formulation. Some furnish alternative proofs of known, and by now, classical results, while others can be seen as groping towards and exploring formulations that have not yet reached a definitive form. The subject matter also has a wide leeway, ranging from solution concepts for economies to those for games and also including representation of preferences and discussion of purely mathematical problems, all within the rubric of choice variables belonging to an infinite dimensional space, interpreted as a commodity space or as a strategy space. Thus, this is a collective enterprise in a fairly wide sense of the term and one with the diversity of which we have interfered as little as possible. Our motivation for bringing all of this work under one set of covers was severalfold.

Inhaltsverzeichnis

Frontmatter

Mathematical Background

Frontmatter
Integration of Banach-Valued Correspondence
Abstract
We study the basic properties of the integral of a Banach-valued correspondence. In particular, we examine the convergence, compactness and convexity properties of the Bochner and Gel’fand integrals of a set-valued function. The above properties are applied to prove the existence of an equilibrium for an abstract economy with a continuum of agents.
Nicholas C. Yannelis
Set-Valued Functions of Two Variables in Economic Theory
Abstract
Several properties of set-valued functions of two variables are studied. Specifically, we study the existence of (i) Carathéodory-type selections, (ii) random fixed points and (iii) random maximal elements. An application to the problem of the existence of a random price equilibrium is also given.
Nicholas C. Yannelis

Equilibria, Core, and Pareto Optimality

Frontmatter
A Theorem on the Existence of Competitive Equilibria in a Market with a Finite Number of Agents and Whose Commodity Space is L∞
Abstract
The current effort to develop the theory of markets with an infinite dimensional commodity space has not yet yielded a very general theorem on the existence of competitive equilibria.1 The absence of such a result is perhaps explained by the great difficulty, if not impossibility, of applying the standard fixed point theorem arguments. The difficulty arises because the key steps in the argument deal with subsets of the commodity space and its dual. But when the commodity space becomes infinite dimensional, these subsets lose many of the nice properties they have when they lie in ℝn.
Truman F. Bewley
The Core of an Economy Without Ordered Preferences
Abstract
Core existence results are proved for exchange economies with an infinite dimensional commodity space. In particular, the commodity space may be any ordered Hausdorff linear topological space, and agents’ preferences need not be transitive, complete, monotone or convex; preferences may even be interdependent. Under these assumptions a quasi equilibrium may not exist.
Nicholas C. Yannelis
Fundamental Theorems of Welfare Economics in Infinite Dimensional Commodity Spaces
Abstract
The fundamental theorems of classical welfare economics state conditions for a competitive equilibrium allocation to be a Pareto optimal allocation and conversely. The first modern treatments of this equivalence between competitive and optimal allocations may be found in the seminal papers of Arrow [1951] and Debreu [1951]. They presented an axiomatic framework for the characteristics and properties of economies with a finite number of agents and commodities. The separating hyperplane theorem from convex analysis was the key to their demonstration of the second classical welfare theorem.
Robert A. Becker

Core Equivalence

Frontmatter
A Limit Theorem on the Core of an Economy with a Continuum of Commodities
Abstract
The Edgeworth-Scarf-Debreu [see Debreu-Scarf (1963)] theorem on the core of a pure exchange economy is extended to allow for a continuum of commodities.
Jean Jaskold Gabszewicz
An Equivalence Theorem for the Core of an Economy with Commodity Space L∞ — τ (L∞ ,L1)
Abstract
The equivalence of the core and the set of competitive equilibrium is proved, for an economy with an atomless measure space of agents and with a commodity space which is L. endowed with the Mackey-topology τ (L∞ ,L1).
Jean-François Mertens
The Principle of Equivalence
Abstract
A general principle of obtaining equivalence of core and quasiWalrasian allocations in nonatomic markets with an infinite number of commodities is formulated through four ingredients: the set of arbitrage, the coalitional representation, the space of allocations and the (weak) Lyapunov Convexity Theorem.
Harrison H. C. Cheng

The Existence of an Equilibrium in Economies with a Continuum of Agents

Frontmatter
A Very Weak Theorem on the Existence of Equilibria in Atomless Economies with Infinitely Many Commodities
Abstract
The equilibrium existence theorem we obtain resembles Robert Aumann’s (1966) Auxiliary Theorem, in which he assumes that preferences are commodity-wise saturated. Our result may therefore be looked upon as a first step towards a satisfactory existence theorem for ι ∞ (if such a theorem exists).
Truman F. Bewley
Equilibria in Markets with a Continuum of Agents and Commodities
Abstract
We prove the existence of an equilibrium for an exchange economy with a measure space of agents and with an infinite dimensional commodity space.
M.Ali Khan, Nicholas C. Yannelis
What is Perfect Competition?
Abstract
We provide a mathematical formulation of the idea of perfect competition for an economy with infinitely many agents and commodities. We conclude that in the presence of infinitely many commodities the Aumann (1964, 1966) measure space of agents, i.e., the interval [0,1] endowed with Lebesgue measure, is not appropriate to model the idea of perfect competition and we provide a characterization of the “appropriate” measure space of agents in an infinite dimensional commodity space setting. The latter is achieved by modeling precisely the idea of an economy with “many more” agents than commodities. For such an economy the existence of a competitive equilibrium is proved. The convexity assumption on preferences is not needed in the existence proof. We wish to thank Tom Armstrong for useful comments. As always we are responsible for any remaining errors.
Aldo Rustichini, Nicholas C. Yannelis

Correlated Equilibria

Frontmatter
On the Existence of Correlated Equilibria
Abstract
We provide sufficient conditions which guarantee the existence of correlated equilibria in noncooperative games with finitely many players.
Nicholas C. Yannelis, Aldo Rustichini
Existence of Correlated Weak Equilibria in Discontinuous Games
Abstract
This paper treats of non-zero-sum discontinuous games with compact Hausdorff strategy spaces. It is assumed that the payoff function of each player in the game is bounded, Borel measurable and is upper semicontinuous on his strategy space, for all fixed actions of the remaining players. It is shown that for each ε > 0, such games possess correlated weak ε-equilibria intro-duced by Moulin and Vial as extension of correlated equilibria in the sense of Aumann. The existence of Nash or correlated ε-equilibria is an open problem.
Andrzej S. Nowak
Communication Equilibria with Large State Spaces
Abstract
A definition of communication equilibrium of games for which players may have arbitrary [rather than finite] type spaces is examined. The revelation principle is proven, and the set of equilibria is compared with the sets of strategy and action correlated equilibria. The equilibrium correspondence is shown to be discontinuous with respect to the information structure of the game, in contrast with previous continuity results for strategy and action correlated equilibrium.
Kevin D. Cotter

Games with a Continuum of Players

Frontmatter
An Axiomatic Approach to the Efficiency of Non-Cooperative Equilibrium in Economies with a Continuum of Traders
Abstract
This is a reprint of Technical Report 274, IMMSSS, Stanford, 1978. Substantial parts of it have appeared in P. Dubey, A. Mas-Colell, and M. Shubik: “Efficiency Properties of Strategic Market Games: An Axiomatic Approach,” J. Econ. Theory 22, 1980, and in A. Mas-Colell: “On a Theorem of Schmeidler, ” J. Math. Econ., 1984. Needless to say, the author is gratified and thankful that the editors have found enough valuable left to request its publication in this volume.
Andreu Mas-Colell
On Symmetric Cournot-Nash Equilibrium Distributions in a Finite-Action, Atomless Game
Abstract
We show that in a finite action, atomless game, every Cournot-Nash equilibrium distribution can “besymmetrized.” This yields an elementary proof of a result of Mas-Colell.
M. Ali Khan, Ye Neng Sun
Equilibria in Random and Bayesian Games with a Continuum of Players
Abstract
We prove random Nash equilibrium existence theorems as well as Bayesian Nash equilibrium existence results for games with a measure space of players.
Erik J. Balder, Nicholas C. Yannelis

Sequential Equilibria

Frontmatter
Recursive Utility Under Uncertainty
Abstract
This paper provides an axiomatization of recursive utility functions in an infinite horizon stochastic setting. In addition, some recently developed atemporal non-expected utility theories are integrated axiomatically into an intertemporal framework. The key axioms deal with intertemporal consistency and attitudes towards the temporal resolution of uncertainty.
Soo H. Chew, Larry G. Epstein
Consistency and Continuity of Choice in a Sequence of Spot and Futures Markets
Abstract
This paper is the first of a competitive analysis of an exchange economy where markets are open at each of an infinite sequence of dates for spot trading and unconditional futures contracting. In the absence of institutional arrangements for handling bankruptcy, the consistency (determinateness) and continuity of agent choice becomes an issue. If an agent’s probabilistic opinions (expectations) regarding prices are consistent in an appropriate sense, then choice is consistent and demand is upper hemi-continuous for important price-action histories. In the second part of this analysis [Nachman, 1980], commonness and compatibility assumptions regarding agents’ opinions imply a specific support structure of these opinions. This structure entails that for important histories at a given date individual and aggregate demand for futures contracts are bounded below by resources at the subsequent date. Existence of a sequence of temporary equilibria then follows in a routine fashion.
David C. Nachman, Robert P. Kertz
Backmatter
Metadaten
Titel
Equilibrium Theory in Infinite Dimensional Spaces
herausgegeben von
Prof. Dr. M. Ali Khan
Prof. Dr. Nicholas C. Yannelis
Copyright-Jahr
1991
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-662-07071-0
Print ISBN
978-3-642-08114-9
DOI
https://doi.org/10.1007/978-3-662-07071-0