Absolute vs Relative Risk Aversion

Question

Are there results that says the monotonicity of one measure of risk aversion implies the monotonicity of the other measure?

For example,

• Does constant relative risk aversion imply decreasing absolute risk aversion?
• Does constant absolute risk aversion imply decreasing relative risk aversion?

and so on.

Answer 1

Given a utility function $u(c)$, $c>0$, with $u'(c)>0, u''(c) <0$. Regarding the sign of the second derivative, if it is zero, then both measures are zero, if it is positive, it would imply increasing marginal utility, and I don't remember having seen these attitude-towards-risk measures for such a utility function.

Denote Absolute Risk Aversion ($ARA$) and Relative Risk Aversion ($RRA$) correspondingly by

$$A(c) = -\frac {u''(c)}{u'(c)},\;\;\; R(c) = cA(c), \;\; A(c) = \frac 1c R(c)$$

1) Monotonicity of $A(c)$ in relation to $R(c)$

$$\frac {\partial A(c)}{\partial c} = \frac {\partial [(1/c)R(c)]}{\partial c}= -\frac 1{c^2} R(c) + \frac 1{c}\frac {\partial R(c)}{\partial c} $$

So if $$\frac {\partial R(c)}{\partial c} \leq 0 \Rightarrow \frac {\partial A(c)}{\partial c} < 0$$

$RRA$ weakly decreasing $\Rightarrow ARA$ is strictly decreasing in $c$.

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2) Monotonicity of $R(c)$ in relation to $A(c)$

$$\frac {\partial R(c)}{\partial c} = \frac {\partial [cA(c)]}{\partial c}=A(c) + c\frac {\partial A(c)}{\partial c} $$

So if $$\frac {\partial A(c)}{\partial c} \geq 0 \Rightarrow \frac {\partial R(c)}{\partial c} > 0$$

$ARA$ weakly increasing $\Rightarrow RRA$ is strictly increasing in $c$.

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