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.
