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Concept
Isoelastic utility
Summary
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.
It is
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, a term –1 is sometimes added in the numerator to make it mathematically consistent with the limiting case of , see Special cases below.
When the context involves risk, the utility function is viewed as a von Neumann–Morgenstern utility function, and the parameter 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.
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. For example, Groom and Maddison (2019) estimated the value of to be 1.5 in the United Kingdom, while Evans (2005) estimated its value to be around 1.4 in 20 OECD countries.
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.
Official source
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