Universal approximation theorem: Difference between revisions

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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 (economics)|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|HARA]] 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.
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It is
 
:<math>
u(c) = \begin{cases}
\frac{c^{1-\eta}-1}{1-\eta} & \eta>0\text{, }\eta \neq 1 \\
\log(c) & \eta = 1
\end{cases}
</math>
 
where <math>c</math> is consumption, <math>u(c)</math> the associated utility, and <math>\eta</math> is a non-negative constant.<ref>{{cite book |last=Ljungqvist |first=Lars |last2=Sargent |first2=Thomas J. |authorlink2=Thomas J. Sargent |title=Recursive Macroeconomic Theory |location=London |publisher=MIT Press |year=2000 |page=451 |isbn=0262194511 }}</ref> 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 log(''c'') as below).
 
When the context involves risk, the utility function is viewed as a [[von Neumann-Morgenstern utility function]], and the parameter <math>\eta</math> is a measure of 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>\eta</math>. While relatively high values of <math>\eta</math> (as high as 50 in some models){{Citation needed|date=December 2010}} are necessary to explain the behavior of asset prices, some controlled experiments{{Citation needed|date=December 2010}} have documented behavior that is more consistent with values of <math>\eta</math> as low as one.  
 
== Risk aversion features ==
 
This and only this utility function has the feature of constant relative risk aversion. Mathematically this means that <math>-c \cdot u''(c)/u'(c)</math> is a constant, specifically <math>\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.<ref>{{cite book |last=Arrow |first=K. J. |authorlink=Kenneth Arrow |year=1965 |chapter=The theory of risk aversion |title=Aspects of the Theory of Risk Bearing |publisher=Yrjo Jahnssonin Saatio |location=Helsinki |isbn= }} Reprinted in: {{cite book |title=Essays in the Theory of Risk Bearing |publisher=Markham |location=Chicago |year=1971 |pages=90–109 |isbn=0841020019 }}</ref><ref>{{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 }}</ref>
 
== Special cases ==
* <math>\eta=0</math>: this corresponds to [[risk neutrality]], because utility is linear in ''c''.
* <math>\eta=1</math>: by virtue of [[l'Hôpital's rule]], the limit of <math>u(c)</math> is <math>\log c</math> as <math>\eta</math> goes to one:
::<math>\lim_{\eta\rightarrow1}\frac{c^{1-\eta}-1}{1-\eta}=\log(c)</math>
:which justifies the convention of using the limiting value ''u''(''c'') = log ''c'' when <math>\eta=1</math>.
* <math>\eta</math> → <math>\infty</math>: this is the case of infinite risk aversion.
 
== See also ==
* [[Isoelastic function]]
* [[Constant elasticity of substitution]]
* [[Exponential utility]]
* [[Risk aversion]]
 
== References ==
{{Reflist}}
 
{{DEFAULTSORT:Isoelastic Utility}}
[[Category:Macroeconomics]]
[[Category:Microeconomics]]
[[Category:Finance]]
[[Category:Utility]]

Latest revision as of 22:06, 1 June 2014

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