# CRRA Ultility， simple question

for CRRA, does increasing gamma leads to increase in risk-aversion?

Looking at the curve, I think increasing gamma leads to less in risk-aversion (since the risk preimum is less). But in terms of absolute risk aversion, CRRA = $$\gamma /X$$. Looks like increasing $$\gamma$$ leads to high risk-aversion. Which is right?

If the utility function $$W \mapsto U(W)$$ (where $$W$$ is wealth) is concave, then the individual is risk-averse and unwilling to accept any actuariually fair gamble.

We can distinguish between absolute risk aversion ($$ARA$$) and relative risk aversion ($$RRA$$)

$$ARA(W) = - \frac{U''(W)}{U'(W)},\quad RRA(W) = - W\frac{U''(W)}{U'(W)}$$

Here, $$ARA(W)$$ determines the absolute amount the individual is willing to pay to avoid a gamble of a given absolute size. Similarly, $$RRA(W)$$ determines the relative amount, i.e., fraction of wealth, the individual is willing to pay to avoid a gamble of a given size relative to wealth. A derivation for $$ARA$$ is given here and is easily modified for $$RRA$$ by replacing $$\epsilon$$ and $$\delta$$ with $$\epsilon/W$$ and $$\delta/W$$, respectively.

As you would expect, a $$CRRA$$ utility function has constant relative risk aversion $$\gamma$$,

$$RRA(W) = -W \frac{U''(W)}{U'(W)} = - W \frac{d}{dW} \log U'(W) = \gamma$$

Without loss of generality in terms of constants, we can solve for $$U$$ as

$$U(W) = \frac{W^{1-\gamma}-1}{1-\gamma}$$

To ensure concavity (risk aversion) we must have $$\gamma > 0$$. The case where $$\gamma = 0$$ corresponds to a linear utility function (risk neutrality) and in the limit as $$\gamma \to 1$$ we have , by L'Hopital's rule,

$$\lim_{\gamma \to 1}U(W) = \log W$$

With $$\gamma$$ fixed the fraction of wealth the individual pays to avoid a gamble is, of course, independent of wealth since this is $$CRRA$$. Nevertheless the fraction of wealth paid would increase as $$\gamma$$ increases.

However, since $$ARA(W) = RRA(W)/W$$, the absolute amount paid decreases with increasing wealth.