Rationalization of market demand on finite domains

Abstract

The hypothesis that individuals behave as utility maximizers is the basis of the microeconomic theory of demand. This hypothesis, however, is not observable and therefore its content and strength have to be evaluated through its implications for the properties of demand functions. The implications for the properties of individual demand functions have been well known for some time, but it is only recently that substantial work on the features of aggregate behavior has been undertaken. In 1972 Sonnenschein (6) conjectured that Walras Law, homogeneity of degree zero in prices, and boundedness from below characterize market excess demand functions, i.e., functions defined on a set of normalized price vectors. A succession of papers by Sonnenschein [7], Mantel [4], Debreu [ 11, and MC Fadden et al. [3] established such a result in a variety of precise formulations. After these conributions it could be said that the structure of excess demand functions was well understood. An analogous problem concerning the decomposition of market demand functions, i.e., functions defined on a set of price-income vectors, was also posed by Sonnenschein [8]. By means of a simple example he showed that when the income distribution is constant it is not possible, in general, to decompose functions that satisfy the balance and zero homogeneity conditions if we require postivity of individual demands. He also conjectured that abstracting from restrictions which follow from the positivity of individual demand, any function satisfying the two previously mentioned conditions can be decomposed into a finite number of individual demand functions derived from utility maximization and sharing market income in some fixed proportion. A proof of the conjecture for the case of two commodities was also provided.