Uzawa's Preference Axioms: A Comment - Fuchs-Seliger, S. (Review of Economic Studies, 1980)
Much attention in the theory of revealed preference has been devoted to the problem of demand functions generated from continuous utility functions. First Samuelson (1938), the originator of the theory of revealed preference, presented assumptions for P2+. Later Houthakker (1950) developed this model of consumer's behaviour for the n-dimensional case. A gap in Houthakker's proof has been recently closed by B. Stigum (1973). Uzawa (1960) presented a different version of Houthakker's theorem. His conditions AI-AIV and the Strong Axiom of Revealed Preference establish the existence of an upper semicontinuous utility function generating the given demand function. Uzawa's query whether these conditions guarantee the existence of a continuous utility function was answered in the negative by a counterexample of Hurwicz and Richter (1971). At approximately the same time E. Gordon (1971) published an article in the Review of Economic Studies where he tried to demonstrate that the axioms AI-AIV and the Strong Axiom do imply the existence of a continuous utility function. Unfortunately the proof of his Proposition 3 (p. 327) contains an error which led to this wrong conclusion. The purpose of this paper is to correct Gordon's theorem by adding conditions which are essentially due to Stigum. We will see that supporting hyperplanes play an important part in the method of the proof. The correction of Gordon's proof, based on results of Uzawa, turns out to be another method to prove Houthakker's theorem.