Isoelastic utility

File:Isoelastic.svg

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 (constant relative risk aversion) utility function. In statistics, the same function is called the Box-Cox transformation.

It is

: $u(c) = \begin{cases}
\frac{c^{1-\eta} - 1}{1-\eta} & \eta \ge 0, \eta \neq 1 \\
\ln(c) & \eta = 1
\end{cases}$

where $c$ is consumption, $u(c)$ the associated utility, and $\eta$ is a constant that is positive for risk averse agents.{{cite book |last1=Ljungqvist |first1=Lars |last2=Sargent |first2=Thomas J. |author-link2=Thomas J. Sargent |title=Recursive Macroeconomic Theory |location=London |publisher=MIT Press |year=2000 |page=451 |isbn=978-0262194518 }} Since additive constant terms in objective functions do not affect optimal decisions, the –1 is sometimes omitted in the numerator (although it should be kept if one wishes to preserve mathematical consistency with the limiting case of $\ln(c)$ ; see Special cases below). Since the family contains both power functions and the logarithmic function, it is sometimes called power-log utility.{{Cite journal |last=Kale |first=Jivendra K. |date=2009 |title=Growth maximisation and downside protection using power-log utility functions for optimising portfolios with derivatives |url=http://www.inderscience.com/link.php?id=24085 |journal=International Journal of Computer Applications in Technology |language=en |volume=34 |issue=4 |pages=309 |doi=10.1504/IJCAT.2009.024085 |issn=0952-8091|url-access=subscription }}

When the context involves risk, the utility function is viewed as a von Neumann–Morgenstern utility function, and the parameter $\eta$ 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.

Empirical value

There is substantial debate in the economics and finance literature with respect to the true value of $\eta$ . While extremely high values of $\eta$ (of up to 50 in some models){{cite journal |last1=Mehra |first1=Rajnish|last2=Prescott |first2=Edward |title=The Equity Premium Puzzle |journal=Journal of Monetary Economics |date=1985 |volume=15 |pages=145–161}} are needed to explain the behavior of asset prices, most experiments document behavior that is more consistent with values of $\eta$ only slightly greater than 1. For example, Groom and Maddison (2019) estimated the value of $\eta$ to be 1.5 in the United Kingdom,{{cite journal|title=New Estimates of the Elasticity of Marginal Utility for the UK|last1=Groom|first1=Ben|last2=Maddison|first2=David|journal=Environmental and Resource Economics|year=2019 |volume=72 |issue=4|pages=1155–1182|doi=10.1007/s10640-018-0242-z |s2cid=254474366 |doi-access=free|url=http://pure-oai.bham.ac.uk/ws/files/104961904/Groom_MaddisonPr2019_Article_NewEstimatesOfTheElasticityOfM.pdf}} while Evans (2005) estimated its value to be around 1.4 in 20 OECD countries.{{cite journal|url=https://www.jstor.org/stable/24440019|last=Evans|first=David|title=The Elasticity of Marginal Utility of Consumption: Estimates for 20 OECD Countries|journal=Fiscal Studies|year=2005 |volume=26|issue=2|pages=197–224|doi=10.1111/j.1475-5890.2005.00010.x |jstor=24440019 |access-date=2021-01-01|url-access=subscription}} The utility of income can also be estimated using subjective well-being surveys. Using six national and international such surveys, Layard et al. (2008) found values between 1.19 an 1.34 with a combined estimate of 1.26.{{cite journal|url=https://www.sciencedirect.com/science/article/pii/S0047272708000248|last1=Layard|first1=Richard|last2=Mayraz|first2=Guy|last3=Nickell|first3=Steve|title=The Marginal Utility of Income|journal=Journal of Public Economics|year=2008|volume=92|pages=1846-1857|doi=10.1016/j.jpubeco.2008.01.007|access-date=2024-03-17|url-access=subscription}}

Risk aversion features

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 {{nobreak| $\eta$ .}} 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.{{cite book |last=Arrow |first=K. J. |author-link=Kenneth Arrow |year=1965 |chapter=The theory of risk aversion |title=Aspects of the Theory of Risk Bearing |publisher=Yrjo Jahnssonin Saatio |location=Helsinki }} Reprinted in: {{cite book |title=Essays in the Theory of Risk Bearing |publisher=Markham |location=Chicago |year=1971 |pages=90–109 |isbn=978-0841020016 }}{{cite journal |last=Pratt |first=J. W. |title=Risk aversion in the small and in the large |journal=Econometrica |volume=32 |issue=1–2 |year=1964 |pages=122–136 |jstor=1913738 |doi=10.2307/1913738 }}

Special cases

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

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

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

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

References

External links

[https://dx.doi.org/10.1002/hec.1331 Wakker, P. P. (2008), Explaining the characteristics of the power (CRRA) utility family. Health Economics, 17: 1329–1344].
[https://ssrn.com/abstract=2354226 Closed form solution of a consumption savings problem with iso-elastic utility]

Category:Financial risk modeling

Category:Utility function types