In economics, the isoelastic function for utility, also known as the isoelastic 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. The isoelastic utility function is a special case of hyperbolic absolute risk aversion and at the same time is the only class of utility functions with constant relative risk aversion, which is why it is also called the CRRA utility function.
where is consumption, the associated utility, and is a constant that is positive for risk averse agents. 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 as below).
There is substantial debate in the economics and finance literature with respect to the empirical value of . While relatively high values of (as high as 50 in some models) are necessary to explain the behavior of asset prices, some controlled experiments have documented behavior that is more consistent with values of as low as one.
Risk aversion features
This and only this utility function has the feature of constant relative risk aversion. Mathematically this means that is a constant, specifically . In theoretical models this often has the implication that decision-making is unaffected by scale. For instance, in the standard model 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.
- : this corresponds to risk neutrality, because utility is linear in c.
- : by virtue of l'Hôpital's rule, the limit of is as goes to 1:
- which justifies the convention of using the limiting value u(c) = ln c when .
- → : this is the case of infinite risk aversion.
- Ljungqvist, Lars; Sargent, Thomas J. (2000). Recursive Macroeconomic Theory. London: MIT Press. p. 451. ISBN 978-0262194518.
- Mehra & Prescott; 1985; Equity Premium: a Puzzle[full citation needed]
- Arrow, K. J. (1965). "The theory of risk aversion". Aspects of the Theory of Risk Bearing. Helsinki: Yrjo Jahnssonin Saatio. Reprinted in: Essays in the Theory of Risk Bearing. Chicago: Markham. 1971. pp. 90–109. ISBN 978-0841020016.
- Pratt, J. W. (1964). "Risk aversion in the small and in the large". Econometrica. 32 (1–2): 122–136. doi:10.2307/1913738. JSTOR 1913738.