Disaggregation of excess demand functions in incomplete markets

We are interested in general equilibrium incomplete markets, where the number of consumers is N, the number of goods is L, and the dimension of the space of admissible trades is K (the case of complete markets being then K = (L - 1)). We prove that, if N greater than or equal to K, any non-vanishing analytic function satisfying the natural extension of the Walras law is, locally at least, the excess demand function of such a market. To be precise, consider a map theta --> Phi(theta) associating with a T-dimensional parameter theta a K-dimensional linear subspace Phi(theta) of R-L, representing the set of market transactions allowed by theta. Given parameter values <(theta)over bar>(1), ..., <(theta)over bar>(T), and a non-vanishing analytic function Z defined on some neighbourhood of <(theta)over bar> with values in R-L, with X(theta) is an element of Phi(theta)For All theta, then there exist concave utility functions U-n, 1 less than or equal to n less than or equal to N and individual endowments omega(1), ..., omega(N), such that the corresponding aggregate excess demand function coincides with Z on a (possibly smaller) neighbourhood of <(theta)over bar>. If Z vanishes at <(theta)over bar>, the disaggregation is still possible, but requires (K + 1) agents. (C) 1999 Elsevier Science S.A. All rights reserved.