Though isoelastic utility exhibiting constant relative risk aversion is considered more plausible (as are other utility functions exhibiting decreasing absolute risk aversion) exponential utility is particularly convenient for many calculations. Consumption example

For example suppose that consumption c is a function of labor supply x and a random term : c c(x) + . Then under exponential utility expected utility is given by: E(u(c)) E1 e a(c(x) + )

where E is the expectation operator. With normally distributed noise i.e. E(u(c)) can be calculated easily using the fact that Thus Multi-asset portfolio example Consider the portfolio allocation problem of maximizing expected exponential utility E e aW of final wealth W subject to where W0 is initial wealth x is a column vector of quantities placed in the n risky assets r is a random vector of stochastic returns on the n assets k is a vector of ones (so W0 x'k is the quantity placed in the risk-free asset) and rf is the known scalar return on the risk-free asset. Suppose further that the stochastic vector r is jointly normally distributed. Then expected utility can be written as where is the mean of the vector r and 2 is the variance of final wealth. Maximizing this is equivalent to minimizing which in turn is equivalent to maximizing Denoting the covariance matrix of r as V the variance 2 of final wealth can be written as x'Vx. Thus we wish to maximize the following with respect to the choice vector x of quantities to be placed in the risky assets: This is an easy problem in matrix calculus and its solution is From this it can be seen that (1) the holdings x* of the risky assets are unaffected by initial wealth W0 an unrealistic property and (2) the holding of each risky asset is smaller the larger is the risk aversion parameter a (as would be intuitively expected). This portfolio example shows the two key features of exponential utility: tractability under joint normality and lack of realism due to its feature of constant absolute risk aversion. See also Entropic risk measure Isoelastic (power) utility function References Arrow K.J.1965 "The theory of risk aversion" in Aspects of the Theory of Risk Bearing by Yrjo Jahnssonin Saatio Helsinki. Reprinted in: Essays in the Theory of Risk Bearing Markham Publ. Co. Chicago 1971 90-109. Pratt J. W. "Risk aversion in the small and in the large" Econometrica 32 JanuaryApril 1964 122-136.