# Constant absolute risk aversion and certainty equivalent

I need to prove that Constant Absolute Risk Aversion (CARA) is equivalent to $$\begin{gather} \int u'(x)dF(x) = u'(c(F,u)) \end{gather}$$

where $$u(x)$$ is a Bernoulli utility function, $$F$$ is the distribution of the lottery and $$c(F,u)$$ is the certainty equivalent.

I started from the fact that CARA is defined as $$-\frac{u''(x)}{u'(x)}=a$$, where $$a$$ is a constant and that the certainty equivalent is defined as $$u(c(F,u))=\int u(x)dF(x)$$.

I tried to mix up the two definitions but I am a bit lost with the interaction of integration and differentiation.

Do you have any hint?

This is perhaps not the best way to prove the statement. But notice that a CARA utility is equivalent to having the general functional form: $$$$u(x)=-\alpha \mathrm e^{-ax-b}+\beta,\quad\text{where a,\alpha>0 and b,\beta\in\mathbb R}.$$$$ Knowing this, the rest should be straightforward.

• This actually worked! Thanks! And if I had to work out the inequality related to NIARA (non-increasing), should I proceed in a different way? – PhDing Feb 11 '19 at 21:16
• @PhDing: Probably. It's hard to pin down the general functional form of a non-CARA utility function. So my method will most likely be of little use. – Herr K. Feb 11 '19 at 22:02