Nicholas C. Yannelis

EXISTENCE OF MAXIMAL ELEMENTS AND EQUILIBRIA IN LINEAR TOPOLOGICAL SPACES

Abstract We present some mathematical theorems which are used to generalize previous results on the existence of maximal elements and of equilibrium. Our main theorem in this paper is a new existence proof for an equilibrium in an abstract economy, which is closely related to a previous result of Borglin–Keiding, and Shafer–Sonneschein, but allows for an infinite number of commodities and a countably infinite number of agents.

The core of an economy with differential information

We introduce a new core concept for an exchange economy with differential information which is contained in the coarse core concept of Wilson (1978). We prove the existence of: (1) a core allocation for an exchange economy with differential information and; (2) an [alpha]-core strategy for a game in normal form with differential information.

Incentive compatibility and information superiority of the core of an economy with differential information

We analyze the coarse, the fine, and the private core allocation of an exchange economy with differential information. The basic questions that we address are whether the above concepts are: (i) coalitionally incentive compatible, i.e., does truthful revelation of information in each coalition occur; and (ii) taking into account the information superiority or information advantage of an agent. Moreover, the above three concepts are examined in the presence of externalities and a comparison and interpretation of all of three core notions is provided.

EQUILIBRIA IN BANACH LATTICES WITHOUT ORDERED PREFERENCES

This paper establishes a very general result on the existence of competitive equilibria for exchange economies (with a finite number of agents) with an infinite-dimensional commodity space. The commodity spaces treated are Banach lattices, but no interiority assumptions on the positive cone are made; thus, the commodity spaces covered by this result include most of the spaces considered in economic analysis. Moreover, this result applies to preferences which may not be monotone, complete, or transitive; preferences may even be interdependent. Since preferences need not be monotone, the result allows for non-positive prices, and an exact equilibrium is obtained, rather than a free-disposal equilibrium.

Edgeworth's conjecture in economies with a continuum of agents and commodities*

Abstract This paper contains the following results for economies with infinite dimensional commodity spaces. (i) We establish a core-Walras equivalence theorem for economies with an atomless measure space of agents and with an ordered separable Banach commodity space whose positive cone has a non-empty norm interior. This result includes as a special case the Aumann (1964) and Schmeidler-Hildenbrand [Hildenbrand (1974, p. 33)] finite dimensional theorems. (ii) We provide a counterexample which shows that the above result fails in ordered Banach spaces whose positive cone has an empty interior even if preferences are strictly convex, monotone weakly continuous and initial endowments are strictly positive. (iii) Using the assumption of an ‘extremely desirable commodity’ (which is automatically satisfied whenever preferences are monotone and the positive cone of the commodity space has a non-empty interior), we establish core-Walras equivalence in any arbitrary separable Banach lattice whose positive cone may have an empty (norm) interior.

Equilibria in noncooperative models of competition

Abstract An equilibrium in a game theoretic setting a la Debreu (Proc. Nat. Acad. Sci. U.S.A. 38 (1954), 886–893) and Shafer-Sonnenschein (J. Math. Econ. 2 (1975), 345–348) with a broader structure is proved. In particular, our framework is general enough to encompass both the Aumann (Econometrica 34 (1966), 1–17) economy of perfect competition and the nonordered preferences setting of Mas-Colell (J. Math. Econ. 1 (1974), 237–246). Moreover, since the dimensionality of the strategy space may be infinite it contains Bewley-type (J. Econ. Theory 4 (1972), 514–540) results and may be useful in obtaining existence results for economies with a measure space of agents and infinitely many commodities.

Equilibrium Theory in Infinite Dimensional Spaces

1 Mathematical Background.- Integration of Banach-Valued Correspondences.- Set-Valued Functions of Two Variables in Economic Theory.- 2 Equilibria, Core, and Pareto Optimality.- A Theorem on the Existence of Competitive Equilibria in a Market with a Finite Number of Agents and Whose Commodity Space is L?.- The Core of an Economy Without Ordered Preferences.- Fundamental Theorems of Welfare Economics in Infinite Dimensional Commodity Spaces.- 3 Core Equivalence.- A Limit Theorem on the Core of an Economy with a Continuum of Commodities.- An Equivalence Theorem for the Core of an Economy with Commodity Space L? ? ?(L?, L1).- The Principle of Equivalence.- 4 The Existence of an Equilibrium in Economies with a Continuum of Agents.- A Very Weak Theorem on the Existence of Equilibria in Atomless Economies with Infinitely Many Commodities.- Equilibria in Markets with a Continuum of Agents and Commodities.- What is Perfect Competition?.- 5 Correlated Equilibria.- On the Existence of Correlated Equilibria.- Existence of Correlated Weak Equilibria in Discontinuous Games.- Communication Equilibria with Large State Spaces.- 6 Games with a Continuum of Players.- An Axiomatic Approach to the Efficiency of Non-Cooperative Equilibrium in Economies with a Continuum of Traders.- On Symmetric Cournot-Nash Equilibrium Distributions in a Finite-Action, Atomless Game.- Equilibria in Random and Bayesian Games with a Continuum of Players.- 7 Sequential Equilibria.- Recursive Utility Under Uncertainty.- Consistency and Continuity of Choice in a Sequence of Spot and Futures Markets.- 8 References.

A Theory of Value with Non-linear Prices: Equilibrium Analysis beyond Vector Lattices

This paper presents a new theory of value with a personalized pricing system that naturally induces a family of non-linear prices. This affords a coordinate free theory of value in which the analysis is without any lattice theoretic considerations. When commodity bundles are perfectly decomposable the generalized prices become linear and the analysis specializes to the Walrasian model. This happens, for instance, whenever the commodity space is a vector lattice and consumption sets coincide with the positive cone. Our approach affords theorems on the existence of equilibrium and provides a value-based characterization of Pareto optimality and Edgeworth equilibrium where the Walrasian linear price-based characterization fails. The analysis has applications in the finite as well as the infinite dimensional setting. Journal of Economic Literature Classification Numbers: C62, C71, D46, D51, D61.

Markets with Many More Agents than Commodities: Aumann's “Hidden” Assumption

Abstract We address a question posed by J.-F. Mertens and show that, indeed, R. J. Aumanns classical existence and equivalence theorems depend on there being “many more agents than commodities.” We show that for an arbitrary atomless measure space of agents there is a fixed non-separable infinite dimensional commodity space in which one can construct an economy that satisfies all the standard assumptions but which has no equilibrium, a core allocation that is not Walrasian, and a Pareto efficient allocation that is not a valuation equilibrium. We identify the source of the failure as the requirement that allocations be strongly measurable. Our main example is set in a commodity–measure space pair that displays an “acute scarcity” of strongly measurable allocations—where strong measurability necessitates that consumer choices be closely correlated no matter the prevailing prices. This makes the core large since there may not be any strongly measurable improvements even though there are many weakly measurable strict improvements. Moreover, at some prices the aggregate demand correspondence is empty since disaggregated demand has no strongly measurable selections, though it does have weakly measurable selections. We note that our example can be constructed in any vector space whose dimension is greater than the cardinality of the continuum—that is, whenever there are at least as many commodities as agents . We also prove a positive core equivalence result for economies in non-separable commodity spaces. Journal of Economic Literature Classification Numbers: C62, C71, D41, D50.

Efficiency and incentive compatibility in differential information economies

We introduce several efficiency notions depending on what kind of expected utility is used (ex ante, interim, ex post) and on how agents share their private information, i.e., whether they redistribute their initial endowments based on their own private information, or common knowledge information, or pooled information. Moreover, we introduce several Bayesian incentive compatibility notions and identify several efficiency concepts which maintain (coalitional) Bayesian incentive compatibility.