The Use of Unbounded Utility Functions in Expected-Utility Maximization: Comment

Abstract

The standard solution to the problem of rational decision making under uncertainty is to postulate the existence of a von Neumann-Morgenstern-type utility function, the expected value of which the individual seeks to maximize. The necessary properties of such functions have been widely discussed in the literature, and several writers, for example Arrow 1 and Chernoff and Moses,2 have pointed out the necessity for such functions to be bounded if they are to be relied upon to discriminate correctly between alternative actions. This restriction of boundedness may be regarded as a drawback to the empirical implementation of the expected-utility criterion, since many of the most convenient functional forms (e.g., the logarithmic, the polynomials) seem inadmissible because of their unboundedness. The purpose of this article is to examine the practical limitations on the use of unbounded utility functions. The conclusion is reached that only under highly improbable conditions concerning the probability density function of returns do unbounded utility functions fail to discriminate correctly between alternative actions. Consider first the important class of polynomial utility functions.