ⓘ Isoelastic utility
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 decisionmaker 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.
It is
u c = { c 1 − η − 1 − η ≥ 0, η ≠ 1 ln c η = 1 {\displaystyle uc={\begin{cases}{\frac {c^{1\eta }1}{1\eta }}&\eta \geq 0,\eta \neq 1\\\lnc&\eta =1\end{cases}}}where c {\displaystyle c} is consumption, u c {\displaystyle uc} the associated utility, and η {\displaystyle \eta } 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 ln c {\displaystyle \lnc} as below).
When the context involves risk, the utility function is viewed as a von NeumannMorgenstern utility function, and the parameter η {\displaystyle \eta } is the degree of relative 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.
1. Empirical parametrization
There is substantial debate in the economics and finance literature with respect to the empirical value of η {\displaystyle \eta }. While relatively high values of η {\displaystyle \eta } as high as 50 in some modelsTemplate:Mehra & Prescott; 1985; Equity Premium: a Puzzle are necessary to explain the behavior of asset prices, some controlled experiments have documented behavior that is more consistent with values of η {\displaystyle \eta } as low as one.
2. Risk aversion features
This and only this utility function has the feature of constant relative risk aversion. Mathematically this means that − c ⋅ u ″ c / u ′ c {\displaystyle c\cdot uc/uc} is a constant, specifically η {\displaystyle \eta }. In theoretical models this often has the implication that decisionmaking is unaffected by scale. For instance, in the standard model of one riskfree 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.
3. Special cases
 η = 0 {\displaystyle \eta =0}: this corresponds to risk neutrality, because utility is linear in c.
 η = 1 {\displaystyle \eta =1}: by virtue of lHopitals rule, the limit of u c {\displaystyle uc} is ln c {\displaystyle \ln c} as η {\displaystyle \eta } goes to 1
 η {\displaystyle \eta } → ∞ {\displaystyle \infty }: this is the case of infinite risk aversion.
 This feature explains why the exponential utility function is considered unrealistic. Though isoelastic utility exhibiting constant relative risk aversion
 utility function, the exponential utility function, and the isoelastic utility function. A utility function is said to exhibit hyperbolic absolute risk aversion
 alternative utility functions such as: CES constant elasticity of substitution, or isoelastic utility Isoelastic utility Exponential utility Quasilinear
 competition. Note the difference between CES utility and isoelastic utility the CES utility function is an ordinal utility function that represents preferences
 von Neumann Morgenstern expected utility index. Importantly, unlike VNM utility functions e.g. isoelastic utility Epstein  Zin preferences allow the
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