Book contents
- Frontmatter
- Introduction
- 1 The coefficient of resource utilization
- 2 A social equilibrium existence theorem
- 3 A classical tax-subsidy problem
- 4 Existence of an equilibrium for a competitive economy
- 5 Valuation equilibrium and Pareto optimum
- 6 Representation of a preference ordering by a numerical function
- 7 Market equilibrium
- 8 Economics under uncertainty
- 9 Topological methods in cardinal utility theory
- 10 New concepts and techniques for equilibrium analysis
- 11 A limit theorem on the core of an economy
- 12 Continuity properties of Paretian utility
- 13 Neighboring economic agents
- 14 Economies with a finite set of equilibria
- 15 Smooth preferences
- 16 Excess demand functions
- 17 The rate of convergence of the core of an economy
- 18 Four aspects of the mathematical theory of economic equilibrium
- 19 The application to economics of differential topology and global analysis
- 20 Least concave utility functions
18 - Four aspects of the mathematical theory of economic equilibrium
Published online by Cambridge University Press: 05 January 2013
- Frontmatter
- Introduction
- 1 The coefficient of resource utilization
- 2 A social equilibrium existence theorem
- 3 A classical tax-subsidy problem
- 4 Existence of an equilibrium for a competitive economy
- 5 Valuation equilibrium and Pareto optimum
- 6 Representation of a preference ordering by a numerical function
- 7 Market equilibrium
- 8 Economics under uncertainty
- 9 Topological methods in cardinal utility theory
- 10 New concepts and techniques for equilibrium analysis
- 11 A limit theorem on the core of an economy
- 12 Continuity properties of Paretian utility
- 13 Neighboring economic agents
- 14 Economies with a finite set of equilibria
- 15 Smooth preferences
- 16 Excess demand functions
- 17 The rate of convergence of the core of an economy
- 18 Four aspects of the mathematical theory of economic equilibrium
- 19 The application to economics of differential topology and global analysis
- 20 Least concave utility functions
Summary
The observed state of an economy can be viewed as an equilibrium resulting from the interaction of a large number of agents with partially conflicting interests. Taking this viewpoint, exactly one hundred years ago, Léon Walras presented in his Eléments d'Economie Politique Pure the first general mathematical analysis of this equilibrium problem. During the last four decades, Walrasian theory has given rise to several developments that required the use of basic concepts and results borrowed from diverse branches of mathematics. In this article, I propose to review four of them.
The existence of economic equilibria
As soon as an equilibrium state is defined for a model of an economy, the fundamental question of its existence is raised. The first solution of this problem was provided by A. Wald (1935-1936), and after a twenty-year interruption, research by a large number of authors has steadily extended the framework in which the existence of an equilibrium can be established. Although no work was done on the problem of existence of a Walrasian equilibrium from the early thirties to the early fifties, several contributions, which, later on, were to play a major role in the study of that problem, were made in related areas during that period. One of them was a lemma proved by J. von Neumann (1937) in connection with his model of economic growth.
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- Information
- Mathematical EconomicsTwenty Papers of Gerard Debreu, pp. 217 - 231Publisher: Cambridge University PressPrint publication year: 1983
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