Introduction

This paper deals with the properties of demand functions derived from utility maximization. Such properties may be either “finite” (involving finite sets of points) or “infinitesimal” (involving the derivatives of the demand functions). The “revealed preference” approach, pioneered by P. A. Samuelson and developed by H. S. Houthakker, is of the “finite” type. In what follows, we shall confine ourselves to the “infinitesimal” type of analysis, dealing with “substitution terms.” This analysis has been carried out in terms of the “direct” demand functions (quantities taken as functions of prices and incomes) by Slutsky, Hicks, Allen, Samuelson, and the “indirect” demand function (relative prices as functions of the quantities taken) by the “rediscovered” Antonelli, and also by Samuelson. The present paper deals only with the “direct” demand functions.

Perhaps the most familiar result, found in Hicks's Appendix in Value and Capital as well as in Samuelson's Foundations, is the fact that utility maximization (subject, of course, to the budget constraint) implies the symmetry and negative semidefiniteness of the Slutsky–Hicks substitution term matrix (derived from the “direct” demand functions). But much more remarkable is the converse proposition that, under certain regularity assumptions, if the demand function has a symmetric, negative semidefinite substitution term matrix, then it is generated by the maximization of a utility function.

In his Foundations of Economic Analysis, p. 116, Samuelson formulates the converse proposition and provides suggestions for a proof.