<p>Please bear with me since I was also looking for this transformation and with my limited understanding this is what I found.
Feel free to correct me and develop more on this question.</p>
<p>Besides the utility function, we also need to look for the coefficient of risk aversion, defined as
<span class="math-container">\begin{equation}
A(x)=-\frac{u''(x)}{u'(x)}
\end{equation}</span>
and relative risk aversion
<span class="math-container">\begin{equation}
R(x)=xA(x)=\frac{-xu''(x)}{u'(x)}
\end{equation}</span></p>
<p>Where <span class="math-container">$u’(x)$</span> and <span class="math-container">$u’’(x)$</span> are the first and second derivatives with respect of <span class="math-container">$x$</span> of <span class="math-container">$u(x)$</span>.</p>
<p>For example:
<span class="math-container">$u(x)=\alpha+\beta \textbf{ln}(x)$</span>, then <span class="math-container">$u’(x) = \beta/x$</span> and <span class="math-container">$u’’(x) =\beta/x^2$</span>. So that, <span class="math-container">$A(x) =1/x$</span>. (Note that <span class="math-container">$A(x)$</span> does not depend on <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span>.)</p>
<p>For HARA, risk aversion is an hyperbolic function (the H in HARA), such as<br />
<span class="math-container">\begin{equation}
A(x)=-\frac{u''(x)}{u'(x)} = \frac{1}{ac+b}
\end{equation}</span></p>
<p>When <span class="math-container">$b = 0$</span>, this is CRRA, where relative risk aversion is <span class="math-container">$xA(x) = 1/a = const$</span>.
<span class="math-container">\begin{equation}
\hat{\gamma}=-\frac{u''(x)}{u'(x)}x = \gamma \frac{x}{x+x_0} = \gamma(1+\frac{x_0}{x})
\end{equation}</span></p>
<p>When <span class="math-container">$a=0$</span>, HARA collapses to CARA, where absolute risk aversion is <span class="math-container">$A(x) =1/b = const$</span>.</p>
<p>Hope this helps to get a more detailed answer.</p>

CC BY-SA 4.0
2022-12-27T05:41:24.960Z
2022-12-27T05:41:24.960Z
pinpss
<p>I've been looking for an explanation why the HARA function as shown below:
<a href="https://i.sstatic.net/XmpkH.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/XmpkH.png" alt="enter image description here" /></a></p>
<p>can be this CRRA function below?</p>
<p><a href="https://i.sstatic.net/7ygug.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/7ygug.png" alt="enter image description here" /></a></p>
<p>I need to know what kind of special case that CRRA has compared to HARA. Hope my sentences make sense, english is not my first language.
thank you very much.</p>

<utility>
crra as special case of hara
Oei Evelyn Clarieta Kusuma
# crra as special case of hara

(Asked by: username_0 on 2022-12-22)

I've been looking for an explanation why the HARA function as shown below:
[![enter image description here](https://i.sstatic.net/XmpkH.png)](https://i.sstatic.net/XmpkH.png)

can be this CRRA function below?

[![enter image description here](https://i.sstatic.net/7ygug.png)](https://i.sstatic.net/7ygug.png)

I need to know what kind of special case that CRRA has compared to HARA. Hope my sentences make sense, english is not my first language.
thank you very much.

**Tags:** utility


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## 1 answers

### Answer by username_1 on 2022-12-27. Score: 1

Please bear with me since I was also looking for this transformation and with my limited understanding this is what I found.
Feel free to correct me and develop more on this question.

Besides the utility function, we also need to look for the coefficient of risk aversion, defined as
\begin{equation}
A(x)=-\frac{u''(x)}{u'(x)}
\end{equation}
and relative risk aversion
\begin{equation}
R(x)=xA(x)=\frac{-xu''(x)}{u'(x)}
\end{equation}

Where $u’(x)$ and $u’’(x)$ are the first and second derivatives with respect of $x$ of $u(x)$.

For example:
$u(x)=\alpha+\beta \textbf{ln}(x)$, then $u’(x) = \beta/x$ and $u’’(x) =\beta/x^2$. So that, $A(x) =1/x$. (Note that $A(x)$ does not depend on $\alpha$ and $\beta$.)

For HARA, risk aversion is an hyperbolic function (the H in HARA), such as  
\begin{equation}
A(x)=-\frac{u''(x)}{u'(x)} = \frac{1}{ac+b}
\end{equation}

When $b = 0$, this is CRRA, where relative risk aversion is $xA(x) = 1/a = const$.
\begin{equation}
\hat{\gamma}=-\frac{u''(x)}{u'(x)}x = \gamma \frac{x}{x+x\_0} = \gamma(1+\frac{x\_0}{x})
\end{equation}

When $a=0$, HARA collapses to CARA, where absolute risk aversion is $A(x) =1/b = const$.

Hope this helps to get a more detailed answer.


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