Existence of Equilibria

Existence of competitive equilibrium under financial constraints and increasing returns

The index theorem, introduced to economics by Dierker [1972], allows to answer the question of uniqueness, which is a property of the excess demand considered globally, from the local structure of the excess demand around its potential equilibria. To get the idea of the index theorem consider the following graph of an excess demand for some commodity. Suppose there are just two commodities so that a zero excess demand of the commodity considered in the graph is by Walras Law already a competitive equilibrium. Now suppose furthermore that the graph is continuous and that it satisfies the boundary behaviour. Then, as has been proved in the previous section an equilibrium needs to exist. Note that in this case this conclusion can already be drawn from the intermediate value theorem. The astonishing feature of the Figure 4.1 is that we also get some idea on the number of equilibria. First of all, as Figure 4.1 shows, there may be a continuum of equilibria when the graph of the excess demand is flat at zero. This however is not a generic situation. On perturbing the characteristics of the economy one finds that the graph is transversal to the zero line. Hence generically equilibria are locally unique and since the set of equilibria is compact generically there is a finite number of those. Moreover suppose you do not know the graph globally but you only know how the graph cuts through the zero line. Then knowing that it will always cut from above shows that there is a unique equilibrium.KeywordsImplicit Function TheoremExcess DemandCompetitive EquilibriumIndex TheoremBoundary BehaviourThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

6 - Existence of competitive equilibria

Experts against adverse selection: A note on the existence of equilibrium with costly appropriable information

More on equilibria in competitive markets with externalities and a continuum of agents

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.

On the Existence of Competitive Equilibrium in Frictionless and Incomplete Stochastic Asset Markets

Existence of Competitive Equilibrium in Unbounded Exchange Economies with Satiation: A Note

Satiation and existence of competitive equilibrium

This paper studies economies with indivisible goods and budget-constrained agents with unit-demand who act as both sellers and buyers. In prior literature on the existence of competitive equilibrium, it is assumed the indispensability of money, which in turn implies that budgets constraints are irrelevant. We introduce a new condition, Money Scarcity (MS), that considers agents' budget constraints and ensures the existence of equilibrium. Moreover, an extended version of Gale's top trading cycles algorithm is presented, and it is shown that under MS this mechanism is strategy-proof. Finally, we prove that this mechanism is the unique mechanism that minimizes money transactions at equilibrium.

Journal Article HILDENBRAND (W.) and KIRMAN (A. P.). Introduction to Equilibrium Analysis Get access Introduction to Equilibrium Analysis. By W. Hildenbrand and A. P. Kirman. (Amsterdam: North-Holland Publishing Company, 1975. Pp. viii + 216. $18.50, Dfl. 45.00.) P. G. Hare P. G. Hare University of Stirling Search for other works by this author on: Oxford Academic Google Scholar The Economic Journal, Volume 87, Issue 345, 1 March 1977, Pages 152–153, https://doi.org/10.2307/2231845 Published: 01 March 1977

This paper derives necessary and sufficient conditions for the existence and occurrence likelihood of competitive equilibria in urban land and housing markets: (i) when there is imperfect information causing costly search trips and (ii) when land and housing are differentiated according to betweenzone and within-zone heterogeneity. Such competitive equilibria are contingent on the possibility that consumers spill over into the markets of less preferred zones. The main conclusions are that markets tend to be partitioned into homogeneous sub-groups and when they do not, heterogeneity may hamper the effectiveness of search and hence work against the existence of competitive equilibria.

LIONEL McKENZIE ON THE EXISTENCE OF COMPETITIVE EQUILIBRIUM

This paper presents the classical theorem on the existence of equilibrium as it was proved in the 1950's with the various improvements that have been made since then.In particular, the elimination of the survival assumption and of the requirement of transitive preferences are carried through with a proof that uses a mapping of social demand.This approach favors intuitive understanding and generalization of the results.Finally, the role of the firm and the introduction of external economies are critically viewed. MYPURPOSE IS TO DISCUSS the present status of the classical theorem on existence of competitive equilibrium that was proved in various guises in the 1950's by Arrow and Debreu [1], Debreu [5, 6], Gale [8], Kuhn [14], McKenzie [17, 18, 19], and Nikaido [22].The earliest papers were those of Arrow and Debreu, and McKenzie, both of which were presented to the Econometric Society at its Chicago meeting in December, 1952.They were written independently.The paper of Nikaido was also written independently of the other papers but delayed in publication.The major predecessors of the papers of the fifties were the papers of Abraham Wald [31, 32] and John von Neumann [30], all of which were delivered to Karl Menger's Colloquium in Vienna during the 1930's.The paper of von Neumann was not concerned with competitive equilibrium in the classical sense but with a program of maximal balanced growth in a closed production model.However, he first used a fixed point theorem for an existence argument in economics and provided the generalization of the Brouwer theorem that was the major mathematical tool in the classical proofs.Wald achieved the first success with the general problem of the existence of a meaningful solution to the Walrasian system of equations.The proofs which were published used an assumption that later became known as the Weak Axiom of Revealed Preference.This axiom virtually reduces the set of consumers to one person, since it is equivalent to consistent choices under budget constraints.In a one consumer economy the existence of the equilibrium becomes a simple maximum problem and advanced methods are not needed.When many consumers with independent preference orders are present, it has been shown (Uzawa [29]) that fixed point methods are necessary.Wald also wrote a third paper whose main theorem was announced in a summary article [33], but which never reached

Topology and Economics: The Contribution of S. Smale