The Sonnenschein-Mantel-Debreu Results after Thirty Years

Abstract

The properties of aggregate excess demand or market demand play a key role in the existence, uniqueness, stability, and comparative statics of competitive general equilibrium. They are also important for the theory of imperfectly competitive general equilibrium and the microfoundations of macroeconomics. Because observations of demand are likely to occur at the aggregate level, whereas economic theory has been largely concerned with individual behavior, the properties of aggregate excess or market demand are central to econometric identification. The importance of these projects for the development of general equilibrium theory can be seen in Kenneth Arrow and Frank Hahn’s General Competitive Analysis (1971). Soon after their book was published, though, Hugo Sonnenschein (1973) posed the following problem. Consider a function that maps prices into quantities. What conditions must this function satisfy if it is to be the aggregate excess demand function of a well-behaved exchange economy? The existence theory highlighted the importance of the function’s satisfying continuity, homogeneity of degree zero, and Walras’s law. Were these all of the restrictions on aggregate excess demand implied by the usual