# Does CRRA-utility imply higher risk-aversion for lower wealth?

Consider the utility function $u(W)=\dfrac{1}{1-\gamma}W^{1-\gamma}$, where $\gamma=0.5$

Since this function will exhibit decreasing marginal utility of wealth, is it correct to say that for any given level of wealth $W_1$, risk aversion is higher than for any given level of wealth $W_2>W_1$? My intuition is as follows:

Given an initial amount of wealth $W_I$, since the marginal utility of wealth is decreasing, the utility from an amount of wealth gained $W*$, must be higher than the absolute value of the utility from loosing $W*$. (At least if not gained instantaneously)

Moreover, the decreasing marginal utility of wealth implies that loosing wealth over time becomes increasingly worse compared to gaining wealth over time (in terms of utility), the lower initial wealth is.

Hence, it is increasingly worse loosing an amount of $W*$ over time than gaining an amount of $W*$ over time when initial wealth is $W_1$, compared to when initial wealth is $W_2$. Therefore, risk aversion should be higher for an initial level of wealth $W_1$, than for an initial level of wealth $W_2$.

Also, I suspect I am missing a time notation in the utility function, so please help me in the direction of a more correct utility function if that is the case.

It depends on what you mean by risk aversion. The utility function you mention is called "CRRA - Constant Relative Risk Aversion Utility". As the name implies it has constant relative risk aversion (but not absolute).

How economists define risk aversion:

Arow–Pratt measure of absolute risk-aversion (ARA): $$A(W) = -\frac{u''(W)}{u'(W)} = \frac{\gamma}{W}$$

So as wealth increases absolute risk aversion decreases. Or mathematically:

$$\frac{\partial A(W)}{\partial W} < 0$$

Arow–Pratt measure of relative risk-aversion (ARA): $$R(W) = -W \frac{u''(W)}{u'(W)} = \gamma$$

However relative risk aversion is constant. Mathematically:

$$\frac{\partial R(W)}{\partial W} = 0$$

• Great answer, thank you! I take it $R(W)=\gamma$, follows from $R(W)$ and $u'(W)=W^{-\gamma}$ and $u''(W)=-\gamma W^{-\gamma-1}$? – user33475 Apr 17 '18 at 15:45
• Yes, that is correct. – phdstudent Apr 17 '18 at 15:46
• Also, might be slightly of topic, but is there any direct relation between $\gamma$ and the market price of risk (as in the sharpe-ratio)? Let me know if I should create a new post.. – user33475 Apr 17 '18 at 15:48
• You should definitely create a new post for that one. Comment box is not enough for that one :) – phdstudent Apr 17 '18 at 15:49
• Cool, will do:) – user33475 Apr 17 '18 at 15:52