Constant absolute risk aversion and certainty equivalent

Question

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?

Answer 1

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: \begin{equation} u(x)=-\alpha \mathrm e^{-ax-b}+\beta,\quad\text{where $a,\alpha>0$ and $b,\beta\in\mathbb R$}. \end{equation} Knowing this, the rest should be straightforward.