Approximation of excess demand on the boundary and euilibrium price set

Abstract

When preferences may not be homothetic but satisfy other regularity conditions such as monotonicity, the market excess demand function is characterized by continuity and Walras’ law on almost entire region of the price simplex. In particular, Mas-Colell (1977) shows that for a continuous function f defined on the interior of the price simplex satisfying Walras’ law and the boundary condition, there exists an exchange economy ℰ whose excess demand function is approximately equal to f and the equilibrium price set of ℰ is exactly equal to the one of f. This paper shows that if f may be finite on the boundary of the price simplex, ℰ can be chosen so that the equilibrium price set of ℰ is approximately equal to the one of f. Theorem 3 in Wong (1997), showing the equivalence between Brouwer’s fixed-point theorem and Arrow-Debreu’s equilibrium existence theorem, follows from this result.

This research is financially supported by Waseda University 21COE-GLOPE and Grant-in-Aid for Scientific Research #15530125 from JSPS.

The author thanks Professors Kazuya Kamiya and Akira Yamazaki for their useful comments on an earlier version. He is also benefited from the comments and suggestions of two anonymous referees and a co-editor of the journal. Any remaining errors are independent.