The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Convexity

  • Lawrence E. Blume
Reference work entry
DOI: https://doi.org/10.1057/978-1-349-95189-5_565

Abstract

Convexity is the modern expression of the classical law of diminishing returns, which was prominent in political economy from Malthus and Ricardo through the neoclassical revolution. Its importance today rests less on any utilitarian or behavioural psychological rationale or physical principle than on its utility as a tool of mathematical analysis. In general equilibrium and game theory, proofs of the existence of equilibrium, competitive and Nash, respectively, rely on the application of a fixed-point theorem to a set-valued, convex-valued map from a convex set to itself. Welfare economics provides another example: The second theorem of welfare economics, which asserts that optimal allocations can be supported by competitive prices, relies on an application of the supporting hyperplane theorem to an appropriate convex set.

Keywords

Convexity Duality Existence of equilibrium Hyperplanes Large economies Lyapunov’s theorem Optimization Quasi-concavity Quasi-convexity Shapley–Folkman theorem 
This is a preview of subscription content, log in to check access

Bibliography

  1. Edgeworth, F.Y. 1881. Mathematical psychics. London: C. Kegan Paul & Co.Google Scholar
  2. Hildenbrand, W. 1974. Core and equilibria of a large economy. Princeton: Princeton University Press.Google Scholar
  3. Simon, C., and L. Blume. 1994. Mathematics for economists. New York: W.W. Norton.Google Scholar

Copyright information

© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Lawrence E. Blume
    • 1
  1. 1.