Isoelastic utility

There is substantial debate in the economics and finance literature with respect to the empirical value of ${\displaystyle \eta }$. While relatively high values of ${\displaystyle \eta }$ (as high as 50 in some models)[2] are necessary to explain the behavior of asset prices, some controlled experiments have documented behavior that is more consistent with values of ${\displaystyle \eta }$ as low as one. For example, Groom and Maddison (2019) estimated the value of ${\displaystyle \eta }$ to be 1.5 in the United Kingdom,[3] while Evans (2005) estimated its value to be around 1.4 in 20 OECD countries.[4]

This utility function has the feature of constant relative risk aversion. Mathematically this means that ${\displaystyle -c\cdot u''(c)/u'(c)}$ is a constant, specifically ${\displaystyle \eta }$. 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.[5][6]

• ${\displaystyle \eta =0}$: this corresponds to risk neutrality, because utility is linear in c.
• ${\displaystyle \eta =1}$: by virtue of l'Hôpital's rule, the limit of ${\displaystyle u(c)}$ is ${\displaystyle \ln c}$ as ${\displaystyle \eta }$ goes to 1:

${\displaystyle \lim _{\eta \rightarrow 1}{\frac {c^{1-\eta }-1}{1-\eta }}=\ln(c)}$

which justifies the convention of using the limiting value u(c) = ln c when ${\displaystyle \eta =1}$.

• ${\displaystyle \eta }$ → ${\displaystyle \infty }$: this is the case of infinite risk aversion.

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

• Wakker, P. P. (2008), Explaining the characteristics of the power (CRRA) utility family. Health Economics, 17: 1329–1344.
• Closed form solution of a consumption savings problem with iso-elastic utility