A Reexamination of Utility Functions Derived from Demand Functions

Abstract

In this paper the existence of a continuous utility function generating a given demand function will be reconsidered. We shall see that Hurwicz’ and Uzawa’s hypotheses [5], together wih certain additional conditions, imply the existence of a continuous utility function on the range of the demand function. Specifically, taking results of A. Mas-Codell [7] and M. Jackson [6] further, it will be shown that non-inferior demand functions can also be generated by continuous utility functions when, more generally, the range S of these demand functions is a convex subset of ℝ ⁿ₊ . Therefore, the range of the demand functions may also be a ray from the origin. In the theory of revealed preference A. Mas-Colell has been concerned with the continuity of a generating utility function in the case when the demand function is non-inferior and has the range ℝ ⁿ₊₊ . In integrability theory, M.Jackson [6] assumed strong non-inferiority instead of non-inferiority in order to show the existence of a continuous utility function generating the given demand function when the demand function has the range ℝ ⁿ₊ (Theorem 2 in [6]). However, as we will see, our weaker condition is sufficient. In contrast to Jackson and Mas-Colell in this paper another method of proof, which is by supporting hyperplanes, is applied in order to show the existence of a continuous utility function generating a given demand function. This method of proof will be applied not only in integrability theory but also in the theory of revealed preference. Finally, we will establish two further conditions under which continuity of the generating utility function can be shown.