Finance: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 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
- [math]\displaystyle{ u(c) = \begin{cases} \frac{c^{1-\eta} - 1}{1-\eta} & \eta \ge 0, \eta \neq 1 \\ \ln(c) & \eta = 1 \end{cases} }[/math]
where [math]\displaystyle{ c }[/math] is consumption, [math]\displaystyle{ u(c) }[/math] the associated utility, and [math]\displaystyle{ \eta }[/math] is a constant that is positive for risk averse agents.[1] 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 [math]\displaystyle{ \ln(c) }[/math]; see Special cases below). Since the family contains both power functions and the logarithmic function, it is sometimes called power-log utility.[2]
When the context involves risk, the utility function is viewed as a von Neumann–Morgenstern utility function, and the parameter [math]\displaystyle{ \eta }[/math] 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 parametrization
There is substantial debate in the economics and finance literature with respect to the empirical value of [math]\displaystyle{ \eta }[/math]. While relatively high values of [math]\displaystyle{ \eta }[/math] (as high as 50 in some models)[3] are necessary to explain the behavior of asset prices, some controlled experiments[citation needed] have documented behavior that is more consistent with values of [math]\displaystyle{ \eta }[/math] as low as one. For example, Groom and Maddison (2019) estimated the value of [math]\displaystyle{ \eta }[/math] to be 1.5 in the United Kingdom ,[4] while Evans (2005) estimated its value to be around 1.4 in 20 OECD countries.[5]
Risk aversion features
This utility function has the feature of constant relative risk aversion. Mathematically this means that [math]\displaystyle{ -c \cdot u''(c)/u'(c) }[/math] is a constant, specifically [math]\displaystyle{ \eta }[/math]. 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.[6][7]
Special cases
- [math]\displaystyle{ \eta=0 }[/math]: this corresponds to risk neutrality, because utility is linear in c.
- [math]\displaystyle{ \eta=1 }[/math]: by virtue of l'Hôpital's rule, the limit of [math]\displaystyle{ u(c) }[/math] is [math]\displaystyle{ \ln c }[/math] as [math]\displaystyle{ \eta }[/math] goes to 1:
- [math]\displaystyle{ \lim_{\eta\rightarrow1}\frac{c^{1-\eta}-1}{1-\eta}=\ln(c) }[/math]
- which justifies the convention of using the limiting value u(c) = ln c when [math]\displaystyle{ \eta=1 }[/math].
- [math]\displaystyle{ \eta }[/math] → [math]\displaystyle{ \infty }[/math]: this is the case of infinite risk aversion.
See also
References
- ↑ Ljungqvist, Lars; Sargent, Thomas J. (2000). Recursive Macroeconomic Theory. London: MIT Press. p. 451. ISBN 978-0262194518.
- ↑ Kale, Jivendra K. (2009). "Growth maximisation and downside protection using power-log utility functions for optimising portfolios with derivatives" (in en). International Journal of Computer Applications in Technology 34 (4): 309. doi:10.1504/IJCAT.2009.024085. ISSN 0952-8091. http://www.inderscience.com/link.php?id=24085.
- ↑ Mehra, Rajnish; Prescott, Edward (1985). "The Equity Premium Puzzle". Journal of Monetary Economics 15: 145-161.
- ↑ Groom, Ben; Maddison, David (2019). "New Estimates of the Elasticity of Marginal Utility for the UK". Environmental and Resource Economics 72 (4): 1155–1182. doi:10.1007/s10640-018-0242-z. http://pure-oai.bham.ac.uk/ws/files/104961904/Groom_MaddisonPr2019_Article_NewEstimatesOfTheElasticityOfM.pdf.
- ↑ Evans, David (2005). "The Elasticity of Marginal Utility of Consumption: Estimates for 20 OECD Countries". Fiscal Studies 26 (2): 197–224. doi:10.1111/j.1475-5890.2005.00010.x. https://www.jstor.org/stable/24440019. Retrieved 2021-01-01.
- ↑ 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.
External links
- 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
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