An inverse problem in the economic theory of demand

Abstract

Given an exchange economy consisting of k consumers, there is an associated collective demand function, which is the sum of the individual demand functions. It maps the price system p to a goods bundle x(p). Conversely, given a map p→x(p), it is natural to ask whether it is the collective demand function of a market economy. We answer that question in the case when k is less than the number of goods n. The proof relies on finding convex solutions to a strongly nonlinear system of partial differential equations.