<p>In EZ preferences, risk aversion comes from two things. First, from <span class="math-container">$\rho$</span>, like in the standard model where <span class="math-container">$\rho=\gamma$</span>.</p>
<p><span class="math-container">$\rho$</span> represents relative risk aversion (RRA) with respect to dated risks, i.e., RRA with respect to consumption at a single point in time. If you compute <span class="math-container">$c_t\frac{\partial^2U_t}{\partial c_t^2}$</span> you get back to <span class="math-container">$\rho$</span>.</p>
<p>This is not all of the story of risk aversion. Then, <span class="math-container">$\gamma$</span> is an additional term that makes EZ preferences with <span class="math-container">$(\gamma,\rho)$</span> more risk averse than preferences with <span class="math-container">$(\rho,\rho)$</span> (which is nothing but the standard additive discounted utility model) if and only if <span class="math-container">$\gamma &gt;\rho$</span>. If this is so, then EZ preferences now contain &quot;temporal risk aversion&quot;, which means: Aversion to risk on future utility, i.e. on <span class="math-container">$U_{t+1}$</span> (which itself depends on all future consumption...).</p>
<p>So total RRA depends on <span class="math-container">$\gamma$</span> AND <span class="math-container">$\rho$</span>, on <span class="math-container">$c_t$</span> but also on <span class="math-container">$c_{t+1},c_{t+2},...$</span>.</p>
<p>Now, what is meant by &quot;risk aversion&quot; in the case of EZ preferences? What is meant is something like that: Assume that you face the problem of</p>
<p><span class="math-container">$$V(W)=\max_{(\tilde{c}_0,\tilde{c}_1,...,\tilde{c}_T)}V_0(\tilde{c}_0,\tilde{c}_1,...,\tilde{c}_T)$$</span>
such that
<span class="math-container">$$\mathbb{E}[\tilde{c}_0+\delta \tilde{c}_1+...+\delta^T \tilde{c}_T]\leq W$$</span></p>
<p>where the tilde denotes random variables.</p>
<p>Now, RRA in EZ is simply RRA of the indirect utility function <span class="math-container">$V(W)$</span>. That is, RRA with respect to wealth, not consumption. This is the quantity you are looking for, that is discussed in applied papers regarding risk aversion. It will depend on <span class="math-container">$\gamma$</span> and <span class="math-container">$\rho$</span>.</p>

CC BY-SA 4.0
2024-09-03T16:44:11.023Z
2024-09-03T16:50:00.663Z
2024-09-03T16:50:00.663Z
Seneleh
<p><span class="math-container">$$
\newcommand{\E}{\mathbb{E}}
$$</span></p>
<h1>Preface</h1>
<p>This question is related to <a href="https://economics.stackexchange.com/questions/9296/calculate-the-elasticity-of-substitution-of-epstein-zin-preferences/9297#9297">this one about the elasticity of intertemporal substitution</a> and <a href="https://economics.stackexchange.com/questions/1816/definition-of-absolute-risk-aversion">this one about the definition of absolute risk aversion</a>. (It's related to the second one insofar as the definition of relative risk aversion can be motivated by the quantity that solves
<span class="math-container">$$
U(C(1-RRA/2)) = \E[U(C(1-\epsilon))\mid C].
$$</span></p>
<h1>Question</h1>
<p>In this question, I want to know how to compute the <em>relative risk aversion</em>
of Epstein-Zin preferences.</p>
<p>Let a consumption
sequence be given <span class="math-container">$C=(C_0, C_1,...)$</span> and let
<span class="math-container">$C_t^+ = (C_t, C_{t+1}, ...)$</span>.
Now, suppose I have Epstein-Zin preferences,
<span class="math-container">\begin{align*}
U_t(C_t^+) &amp;= f(C_t, q(U_{t+1}(C_{t+1}^+))) \\
U_t  &amp;= \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*}</span>
where <span class="math-container">$f$</span> is the time aggregator and <span class="math-container">$q$</span> is the conditional certainty
equivalent operator. That is,
<span class="math-container">$$
f(c,q) = ((1-\beta) c^{1-\rho} + \beta q^{1-\rho})^{\frac{1}{1-\rho}}
$$</span>
and
<span class="math-container">$$
q_t = q(U_{t+1}) = \left(\E_t[U_{t+1}^{1-\gamma}]\right)^{\frac{1}{1-\gamma}}.
$$</span>
How do I show that the coefficient of relative risk aversion is
<span class="math-container">$\gamma$</span>?</p>
<h1>Notes</h1>
<p>Applying the usual definition of relative risk aversion appears to require care. If we were to calculate <span class="math-container">$RRA = -c u''(c)/u'(c)$</span>, we would need to be careful
about the time subscripts on <span class="math-container">$c$</span>. Calculating these derivatives with respect to <span class="math-container">$C_t$</span> would not give us the correct answer. It should probably be
<span class="math-container">$$
RRA = - C_{t+1} \left . \frac{\partial^2 U_t}{\partial C_{t+1}^2} \middle / \frac{\partial U_t}{\partial C_{t+1}} \right. .
$$</span></p>

CC BY-SA 3.0
2015-11-20T22:30:52.580Z
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
CC BY-SA 3.0
2015-11-20T22:42:42.473Z
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
<utility><risk-aversion>
How do I compute the relative risk aversion of Epstein-Zin preferences?
jmbejara
# How do I compute the relative risk aversion of Epstein-Zin preferences?

(Asked by: username_0 on 2015-11-20)

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

# Preface

This question is related to [this one about the elasticity of intertemporal substitution](https://economics.stackexchange.com/questions/9296/calculate-the-elasticity-of-substitution-of-epstein-zin-preferences/9297#9297) and [this one about the definition of absolute risk aversion](https://economics.stackexchange.com/questions/1816/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-Zin 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
$$
RRA = - C\_{t+1} \left . \frac{\partial^2 U\_t}{\partial C\_{t+1}^2} \middle / \frac{\partial U\_t}{\partial C\_{t+1}} \right. .
$$

**Tags:** utility, risk-aversion

> **Comments:**
> **username_1**: 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.  
> **username_1**: 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...  

---

## 1 answers

### Answer by username_2 on 2024-09-03. Score: 0

In EZ preferences, risk aversion comes from two things. First, from $\rho$, like in the standard model where $\rho=\gamma$.

$\rho$ represents relative risk aversion (RRA) with respect to dated risks, i.e., RRA with respect to consumption at a single point in time. If you compute $c\_t\frac{\partial^2U\_t}{\partial c\_t^2}$ you get back to $\rho$.

This is not all of the story of risk aversion. Then, $\gamma$ is an additional term that makes EZ preferences with $(\gamma,\rho)$ more risk averse than preferences with $(\rho,\rho)$ (which is nothing but the standard additive discounted utility model) if and only if $\gamma >\rho$. If this is so, then EZ preferences now contain "temporal risk aversion", which means: Aversion to risk on future utility, i.e. on $U\_{t+1}$ (which itself depends on all future consumption...).

So total RRA depends on $\gamma$ AND $\rho$, on $c\_t$ but also on $c\_{t+1},c\_{t+2},...$.

Now, what is meant by "risk aversion" in the case of EZ preferences? What is meant is something like that: Assume that you face the problem of

$$V(W)=\max\_{(\tilde{c}\_0,\tilde{c}\_1,...,\tilde{c}\_T)}V\_0(\tilde{c}\_0,\tilde{c}\_1,...,\tilde{c}\_T)$$
such that
$$\mathbb{E}[\tilde{c}\_0+\delta \tilde{c}\_1+...+\delta^T \tilde{c}\_T]\leq W$$

where the tilde denotes random variables.

Now, RRA in EZ is simply RRA of the indirect utility function $V(W)$. That is, RRA with respect to wealth, not consumption. This is the quantity you are looking for, that is discussed in applied papers regarding risk aversion. It will depend on $\gamma$ and $\rho$.


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