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
19 - The application to economics of differential topology and global analysis
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 recent introduction of differential topology into economics was brought about by the study of several basic questions that arise in any mathematical theory of a social system centered on a concept of equilibrium. The purpose of this paper is to present a detailed discussion of two of those questions, and then to make a rapid survey of some related developments of the last five years.
Let e be a complete mathematical description of the economy to be studied (e.g., for an exchange economy, e might be a list of the demand functions and of the initial endowments of the consumers). Assumptions made a priori about e (e.g., assumptions of continuity on the demand functions) define the space ℰ of economies to which the study is restricted. By a state of an economy we mean a list of specific values of all the relevant endogenous variables (e.g., prices and quantities of all the commodities consumed by the various consumers). We denote by S the set of conceivable states. Now a given equilibrium theory associates with each economy e in ℰ, the set E(e) of equilibrium states of e, a subset of S (see Figure 1).
- Type
- Chapter
- Information
- Mathematical EconomicsTwenty Papers of Gerard Debreu, pp. 232 - 241Publisher: Cambridge University PressPrint publication year: 1983