# crra as special case of hara

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.

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 $$$$A(x)=-\frac{u''(x)}{u'(x)}$$$$ and relative risk aversion $$$$R(x)=xA(x)=\frac{-xu''(x)}{u'(x)}$$$$

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
$$$$A(x)=-\frac{u''(x)}{u'(x)} = \frac{1}{ac+b}$$$$

When $$b = 0$$, this is CRRA, where relative risk aversion is $$xA(x) = 1/a = const$$. $$$$\hat{\gamma}=-\frac{u''(x)}{u'(x)}x = \gamma \frac{x}{x+x_0} = \gamma(1+\frac{x_0}{x})$$$$

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.