<p>Given a utility function $u(c)$, $c&gt;0$, with $u'(c)&gt;0, u''(c) &lt;0$. 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.  </p>

<p>Denote Absolute Risk Aversion ($ARA$) and Relative Risk Aversion ($RRA$) correspondingly by </p>

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

<p><strong>1) Monotonicity of $A(c)$ in relation to $R(c)$</strong></p>

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

<p>So if 
$$\frac {\partial R(c)}{\partial c} \leq 0 \Rightarrow \frac {\partial A(c)}{\partial c} &lt; 0$$ </p>

<p>$RRA$ weakly decreasing  $\Rightarrow ARA$ is strictly decreasing in $c$.  </p>

<hr>

<p><strong>2) Monotonicity of $R(c)$ in relation to $A(c)$</strong></p>

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

<p>So if $$\frac {\partial A(c)}{\partial c} \geq 0 \Rightarrow \frac {\partial R(c)}{\partial c} &gt; 0$$ </p>

<p>$ARA$ weakly increasing $\Rightarrow RRA$ is strictly increasing in $c$. </p>

<hr>

CC BY-SA 4.0
2024-08-29T11:51:22.923Z
@HerrK. from my understanding, $u''<0$ follows from the assumption of risk-averseness of the investor. Assuming non-satiation i.e. $u'>0$, it follows that for a risk averse investor $A(w),R(w)>0$ for any $w$. If otherwise, the signs change accordingly
zaira
CC BY-SA 3.0
2014-12-04T23:56:21.730Z
Is $u''<0$ required to define either measure? If not, what happens if $R(c)$ or $A(c)$ is negative?
Herr K.
CC BY-SA 3.0
2014-12-04T23:15:50.670Z
2014-12-05T00:08:22.840Z
2014-12-05T00:08:22.840Z
Alecos Papadopoulos
<p>Are there results that says the monotonicity of one measure of risk aversion implies the monotonicity of the other measure?</p>

<p>For example,</p>

<ul>
<li>Does constant relative risk aversion imply decreasing absolute risk aversion?</li>
<li>Does constant absolute risk aversion imply decreasing relative risk aversion?</li>
</ul>

<p>and so on.</p>

<decision-theory><risk>
Absolute vs Relative Risk Aversion
Herr K.
# Absolute vs Relative Risk Aversion

(Asked by: username_0 on 2014-12-04)

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.

**Tags:** decision-theory, risk


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## 1 answers

### Answer by username_1 on 2014-12-04. Score: 4 | Accepted answer

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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> **Comments:**
> **username_0**: Is $u''<0$ required to define either measure? If not, what happens if $R(c)$ or $A(c)$ is negative?  
> **username_2**: @HerrK. from my understanding, $u''<0$ follows from the assumption of risk-averseness of the investor. Assuming non-satiation i.e. $u'>0$, it follows that for a risk averse investor $A(w),R(w)>0$ for any $w$. If otherwise, the signs change accordingly  

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