Editing Risk aversion (section)

===Absolute risk aversion===
The higher the curvature of <math>u(c)</math>, the higher the risk aversion. However, since expected utility functions are not uniquely defined (are defined only up to [[affine transformations]]), a measure that stays constant with respect to these transformations is needed rather than just the second derivative of <math>u(c)</math>. One such measure is the '''Arrow–Pratt measure of absolute risk aversion''' ('''ARA'''), after the economists [[Kenneth Arrow]] and [[John W. Pratt]],<ref name="Arrow">{{cite book |last=Arrow |first=K. J. |year=1965 |title=The Theory of Risk Aversion|chapter-url=https://books.google.com/books?id=hnNEAAAAIAAJ|chapter=Aspects of the Theory of Risk Bearing |publisher=Yrjo Jahnssonin Saatio |location=Helsinki }} Reprinted in: [https://books.google.com/books?id=KkMoAQAAMAAJ ''Essays in the Theory of Risk Bearing''], Markham Publ. Co., Chicago, 1971, 90–109.</ref><ref name="Pratt">{{cite journal |last1=Pratt |first1=John W. |title=Risk Aversion in the Small and in the Large |journal=Econometrica |date=January 1964 |volume=32 |issue=1/2 |pages=122–136 |doi=10.2307/1913738 |jstor=1913738 }}</ref> also known as the '''coefficient of absolute risk aversion''', defined as

:<math>A(c)=-\frac{u''(c)}{u'(c)}</math>

where <math>u'(c)</math> and <math>u''(c)</math> denote the first and second derivatives with respect to <math>c</math> of <math>u(c)</math>. For example, if <math> u(c)=  \alpha +  \beta ln(c),</math> so <math>u'(c) = \beta/c</math> and <math>u''(c) =  -\beta/c^2,</math> then <math>A(c) = 1/c.</math> Note how <math>A(c)</math> does not depend on <math>\alpha</math> and <math>\beta,</math> so affine transformations of <math>u(c)</math> do not change it.

The following expressions relate to this term:
* [[Exponential utility]] of the form <math>u(c)=1-e^{-\alpha c}</math> is unique in exhibiting ''constant absolute risk aversion'' (CARA): <math>A(c)=\alpha</math> is constant with respect to ''c''.
* [[Hyperbolic absolute risk aversion]] (HARA) is the most general class of utility functions that are usually used in practice (specifically, CRRA (constant relative risk aversion, see below), CARA (constant absolute risk aversion), and quadratic utility all exhibit HARA and are often used because of their mathematical tractability). A utility function exhibits HARA if its absolute risk aversion is a [[hyperbola]], namely

:<math> A(c) = -\frac{u''(c)}{u'(c)}=\frac{1}{ac+b}</math>

The solution to this differential equation (omitting additive and multiplicative constant terms, which do not affect the behavior implied by the utility function) is:

:<math> u(c) = \frac{(c-c_s)^{1-R}}{1-R}</math>

where <math> R=1/a </math> and <math> c_s = -b/a </math>.
Note that when <math> a = 0 </math>, this is CARA, as <math> A(c) = 1/b = const </math>, and when <math> b=0 </math>, this is CRRA (see below), as <math> c A(c) = 1/a = const </math>.
See <ref>{{cite web |url=http://leeds-faculty.colorado.edu/zender/Fin7330/1-RiskAversion.doc |title=Zender's lecture notes }}</ref>
* ''Decreasing/increasing absolute risk aversion'' (DARA/IARA) is present if <math>A(c)</math> is decreasing/increasing. Using the above definition of ARA, the following inequality holds for DARA:

:<math>\frac{\partial A(c)}{\partial c} = -\frac{u'(c)u'''(c) - [u''(c)]^2}{[u'(c)]^2} < 0</math>

and this can hold only if <math>u'''(c)>0</math>. Therefore, DARA implies that the utility function is positively skewed; that is, <math>u'''(c)>0</math>.<ref>{{cite book |first=Haim |last=Levy |year=2006 |title=Stochastic Dominance: Investment Decision Making under Uncertainty |location=New York |publisher=Springer |edition=2nd |isbn=978-0-387-29302-8 }}</ref> Analogously, IARA can be derived with the opposite directions of inequalities, which permits but does not require a negatively skewed utility function (<math>u'''(c)<0</math>). An example of a DARA utility function is <math>u(c)=\log(c)</math>, with <math> A(c)=1/c</math>, while <math>u(c)=c-\alpha c^2,</math> <math>\alpha >0</math>, with <math>A(c)=2 \alpha/(1-2 \alpha c)</math> would represent a quadratic utility function exhibiting IARA.
*Experimental and empirical evidence is mostly consistent with decreasing absolute risk aversion.<ref>{{cite journal |last1=Friend |first1=Irwin |last2=Blume |first2=Marshall|author-link2=Marshall E. Blume |year=1975 |title=The Demand for Risky Assets |journal=[[American Economic Review]] |volume=65 |issue=5 |pages=900–922 |jstor=1806628 }}</ref>
* Contrary to what several empirical studies have assumed, wealth is not a good proxy for risk aversion when studying risk sharing in a principal-agent setting. Although <math>A(c)=-\frac{u''(c)}{u'(c)}</math> is monotonic in wealth under either DARA or IARA and constant in wealth under CARA, tests of contractual risk sharing relying on wealth as a proxy for absolute risk aversion are usually not identified.<ref>{{cite journal |last1=Bellemare |first1=Marc F. |last2=Brown |first2=Zachary S. |title=On the (Mis)Use of Wealth as a Proxy for Risk Aversion |journal=American Journal of Agricultural Economics |date=January 2010 |volume=92 |issue=1 |pages=273–282 |doi=10.1093/ajae/aap006 |hdl=10161/7006 |s2cid=59290774 |hdl-access=free }}</ref>