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.


1 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 $c$.

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 $c$.

  • $\begingroup$ Is $u''<0$ required to define either measure? If not, what happens if $R(c)$ or $A(c)$ is negative? $\endgroup$
    – Herr K.
    Dec 4, 2014 at 23:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.