How do I compute the relative risk aversion of Epstein-Zin preferences?

$$\newcommand{\E}{\mathbb{E}}$$

Preface

This question is related to this one about the elasticity of intertemporal substitution and this one about the definition of absolute risk aversion. (It's related to the second one insofar as the definition of relative risk aversion can be motivated by the quantity that solves $$U(C(1-RRA/2)) = \E[U(C(1-\epsilon))\mid C].$$

Question

In this question, I want to know how to compute the relative risk aversion of Epstein-Zin preferences.

Let a consumption sequence be given $C=(C_0, C_1,...)$ and let $C_t^+ = (C_t, C_{t+1}, ...)$. Now, suppose I have Epstein-Sin preferences, \begin{align*} U_t(C_t^+) &= f(C_t, q(U_{t+1}(C_{t+1}^+))) \\ U_t &= \left \{(1-\beta) C_t^{1-\rho} + \beta \left(\E_t[U_{t+1}^{1-\gamma}]\right)^{\frac{1-\rho}{1-\gamma}} \right\}^{\frac{1}{1-\rho}}, \end{align*} where $f$ is the time aggregator and $q$ is the conditional certainty equivalent operator. That is, $$f(c,q) = ((1-\beta) c^{1-\rho} + \beta q^{1-\rho})^{\frac{1}{1-\rho}}$$ and $$q_t = q(U_{t+1}) = \left(\E_t[U_{t+1}^{1-\gamma}]\right)^{\frac{1}{1-\gamma}}.$$ How do I show that the coefficient of relative risk aversion is $\gamma$?

Notes

Applying the usual definition of relative risk aversion appears to require care. If we were to calculate $RRA = -c u''(c)/u'(c)$, we would need to be careful about the time subscripts on $c$. Calculating these derivatives with respect to $C_t$ would not give us the correct answer. It should probably be $$RRA = - C_{t+1} \left . \frac{\partial^2 U_t}{\partial C_{t+1}^2} \middle / \frac{\partial U_t}{\partial C_{t+1}} \right. .$$

• Note that $\gamma$ only "keeps track" of risk aversion, in the sense that $U^1$ is more risk averse than $U^2$ if and only if $\gamma_1>\gamma_2$. But $\gamma$ is not strictly speaking equal to risk aversion. The RRA coefficient is more complicated and depends on $\rho$. I don't have a proof right now, but maybe looking at the Epstein and Zin (1989) paper may help... though it is not a paper that I would qualify as "simple" ;) But if you find something I'd be interested too. – Louis. B Nov 20 '15 at 22:30
• Actually after quickly looking at the Epstein and Zin's paper, they do not seem to compute the Arrow-Pratt risk aversion coefficients, it may not even exist in closed-form... – Louis. B Nov 20 '15 at 22:42