crra as special case of hara

Question

I've been looking for an explanation why the HARA function as shown below:

can be this CRRA function below?

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.

Answer 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.