BOUNDED UTILITY FUNCTIONS DEFINED ON (a
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The structure and analytical representation of investors' utility-ofwealth functions has long been of interest in portfolio theory. In proposing convenient analytical utility functions most economists have used (i) con? stant elasticity (power) functions, (ii) the negative exponential funetion. Both (i) and (ii), of course, restrict the preference structure. Moreover, one may object to (i) because such functions are not uniformly bounded on [0,00). And, as has been shown by Arrow [1], this is undesirable in an axiomatic system. The negative exponential funetion has no such disadvantage, but 2 objections may be raised on empirical grounds. Thus, no simple convenient specification of bounded utility functions on [a,00) is available. In fact, even polynomials in wealth of arbitrary order are restrictive since they immediately impose the requirement that moments of wealth are finite. (If the polynomial is of order n, then the n moment must be finite.) The purpose of this note is to point out that a bounded utility funetion, U(W), on [a,00) can be represented as a weighted average of negative exponen? tial functions. No requirement of differentiability is imposed on U, but it 3 is assumed U is continuous on [a,??). Furthermore, if F(W) denotes the proba? bility distribution of W, then assume F(a) = 0 impiying / U(W) |dF(W) . Hence, sums of negative exponential functions provide a rich class of preference maps which can be put to use in analytical work (see [3]). Formally, the precise 4 statement and proof are as follows (without loss, put a = 0):
[1] James A. Ohlson. Quadratic Approximations of the Portfolio Selection Problem When the Means and Variances of Returns are Infinite , 1977 .
[2] K. Arrow,et al. Aspects of the theory of risk-bearing , 1966 .
[3] W. Feller. An Introduction to Probability Theory and Its Applications , 1959 .
[4] J. Pratt. RISK AVERSION IN THE SMALL AND IN THE LARGE11This research was supported by the National Science Foundation (grant NSF-G24035). Reproduction in whole or in part is permitted for any purpose of the United States Government. , 1964 .