Isoelastic utility

In economics, the isoelastic function for utility, also known as the isoelastic utility function, constant relative risk aversion utility function, or power utility function, is used to express utility in terms of consumption or some other economic variable that a decision-maker is concerned with.

It is

$u(c) = \begin{cases} \frac{c^{1-\eta}-1}{1-\eta} & \eta \neq 1 \\ \log(c) & \eta = 1 \end{cases}$

where c is consumption, u(c) the associated utility, and η is a non-negative constant.^[1] Since additive constant terms in objective functions do not affect optimal decisions, the term –1 in the numerator can be, and usually is, omitted (except when establishing the limiting case of log(c) as below).

When the context involves risk, the utility function is viewed as a von Neumann-Morgenstern utility function, and the parameter η is a measure of risk aversion.

The isoelastic utility function is a special case of the hyperbolic absolute risk aversion (HARA) utility functions, and is used in analyses that either include or do not include underlying risk.

Empirical parametrization

There is substantial debate in the economics and finance literature with respect to the empirical value of η. While relatively high values of η (as high as 50 in some models)^{[citation needed]} are necessary to explain the behavior of asset prices, controlled experiments^{[citation needed]} lead to the conclusion that individuals have η close to one.

Risk aversion features

This and only this utility function has the feature of constant relative risk aversion. Mathematically this means that $-c \cdot u''(c)/u'(c)$ is a constant, specifically η. In theoretical models this often has the implication that decision-making is unaffected by scale. For instance, in the standard model^[2][3] of one risk-free asset and one risky asset, under constant relative risk aversion the fraction of wealth optimally placed in the risky asset is independent of the level of initial wealth.

Special cases

• η = 0: this corresponds to risk neutrality, because utility is linear in c.
• η = 1: by virtue of l'Hôpital's rule, the limit of u(c) is log c as η goes to one:

$\lim_{\eta\rightarrow1}\frac{c^{1-\eta}-1}{1-\eta}=\log(c)$

which justifies the convention of using the limiting value u(c) = log c when η = 1.

• η → $\infty$: this is the case of infinite risk aversion.

See also

• Isoelastic function
• Constant elasticity of substitution
• Exponential utility
• Risk aversion

References

1. ^ Ljungqvist & Sargent: Recursive Macroeconomic Theory, page 451
2. ^ 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.
3. ^ Pratt, J. W., "Risk aversion in the small and in the large," Econometrica 32, January–April 1964, 122-136.