Convexity

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.