Exponential utility

In economics and finance, exponential utility is a specific form of the utility function, used in some contexts because of its convenience when risk (sometimes referred to as uncertainty) is present, in which case expected utility is maximized. Formally, exponential utility is given by: In economics and finance, exponential utility is a specific form of the utility function, used in some contexts because of its convenience when risk (sometimes referred to as uncertainty) is present, in which case expected utility is maximized. Formally, exponential utility is given by: c {displaystyle c} is a variable that the economic decision-maker prefers more of, such as consumption, and a {displaystyle a} is a constant that represents the degree of risk preference ( a > 0 {displaystyle a>0} for risk aversion, a = 0 {displaystyle a=0} for risk-neutrality, or a < 0 {displaystyle a<0} for risk-seeking). In situations where only risk aversion is allowed, the formula is often simplified to u ( c ) = 1 − e − a c {displaystyle u(c)=1-e^{-ac}} . Note that the additive term 1 in the above function is mathematically irrelevant and is (sometimes) included only for the aesthetic feature that it keeps the range of the function between zero and one over the domain of non-negative values for c. The reason for its irrelevance is that maximizing the expected value of utility u ( c ) = ( 1 − e − a c ) / a {displaystyle u(c)=(1-e^{-ac})/a} gives the same result for the choice variable as does maximizing the expected value of u ( c ) = − e − a c / a {displaystyle u(c)=-e^{-ac}/a} ; since expected values of utility (as opposed to the utility function itself) are interpreted ordinally instead of cardinally, the range and sign of the expected utility values are of no significance. The exponential utility function is a special case of the hyperbolic absolute risk aversion utility functions. Exponential utility implies constant absolute risk aversion (CARA), with coefficient of absolute risk aversion equal to a constant: In the standard model of one risky asset and one risk-free asset, for example, this feature implies that the optimal holding of the risky asset is independent of the level of initial wealth; thus on the margin any additional wealth would be allocated totally to additional holdings of the risk-free asset. This feature explains why the exponential utility function is considered unrealistic. Though isoelastic utility, exhibiting constant relative risk aversion (CRRA), is considered more plausible (as are other utility functions exhibiting decreasing absolute risk aversion), exponential utility is particularly convenient for many calculations. For example, suppose that consumption c is a function of labor supply x and a random term ϵ {displaystyle epsilon } : c = c(x) + ϵ {displaystyle epsilon } . Then under exponential utility, expected utility is given by: where E is the expectation operator. With normally distributed noise, i.e.,