I am faced with the following function, called CES (constant elasticity of substitution), of the continuously-distributed random variable $\epsilon$:
$f(\epsilon) = (a^\sigma + (b+\epsilon)^\sigma)^{1/\sigma}$.
where $a,b$ are positive constants and $\sigma < 1$ so that $f$ is increasing and concave in $\epsilon$.
I am looking for a distribution of $\epsilon$ such that $\mathbb{E}[f(\epsilon)]$ can be expressed in closed-form. So far I have had no luck coming up with one.
Thank you