<p>Epstein-Zin (EZ) preferences are a generalisation of dynamic CRRA preferences.</p>
<p>In standard CRRA preferences, i.e.
<span class="math-container">$$
U(c_0,\dots) = \mathbb{E} \left[ \sum_t \beta^t \frac{c_t^{1 - \rho}}{1 - \rho} \right]
$$</span>
The Arrow-Pratt relative risk aversion is <span class="math-container">$\rho$</span> and elasticity of intertemporal substitution is <span class="math-container">$\frac{1}{\rho}$</span> and thus is unable to be disentangled.</p>
<p>EZ solves this. Actually, the representation you have above is only for deterministic streams. EZ preferences have the recursive representation:
<span class="math-container">$$
   V_t = \left( (1-\beta)c_t^{1-\rho} + \beta \left[ \mathbb{E} V_{t+1}^{1-\alpha} \right]^{\frac{1-\rho}{1-\alpha}} \right)^{\frac{1}{1-\rho}}
$$</span>
Where <span class="math-container">$\alpha$</span> captures risk attitudes and <span class="math-container">$\rho$</span> the elasticity of intertemporal substitution.</p>
<p>In fact, they agree on deterministic streams of consumption (i.e. when there is no uncertainty). Take a (non-stochastic) stream <span class="math-container">$(c_0,c_1, \dots)$</span> and the expectation operator drops to
<span class="math-container">$$
   V_t = \left( (1-\beta)c_t^{1-\rho} + \beta V_{t+1}^{1-\rho} \right)^{\frac{1}{1-\rho}}
$$</span>
Define the monotone transform <span class="math-container">$U_t = V_t^{1-\rho}$</span> to get
<span class="math-container">$$
     U_t = (1-\beta)c_t^{1-\rho} + \beta U_{t+1}
$$</span>
in other words
<span class="math-container">$$
     U_0(c_0,c_1,\dots) = (1-\beta) \sum_t \beta^t c_t^{1 - \rho}
$$</span>
which are just CRRA!</p>

CC BY-SA 4.0
2022-02-28T18:41:53.983Z
2022-02-28T18:41:53.983Z
Walrasian Auctioneer
<p>Epstein-Zin (EZ) utility is the solution to:</p>
<p><a href="https://i.sstatic.net/fTXBG.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/fTXBG.png" alt="enter image description here" /></a></p>
<p>DEU is relatively simple: <span class="math-container">$\sum_t \delta ^t\mathbb E[u(c_t)]$</span>.</p>
<p>Is DEU a special case of EZ? How are those two models compared?</p>
<p>Since EZ is a solution of a complicated dynamic equation, I am having a very hard time figuring out the connection between the two models.</p>

<microeconomics><utility><preferences><decision-theory><risk>
Is Epstein-Zin utility a generalization of dynamic expected utility (DEU)?
High GPA
# Is Epstein-Zin utility a generalization of dynamic expected utility (DEU)?

(Asked by: username_0 on 2022-02-28)

Epstein-Zin (EZ) utility is the solution to:

[![enter image description here](https://i.sstatic.net/fTXBG.png)](https://i.sstatic.net/fTXBG.png)

DEU is relatively simple: $\sum\_t \delta ^t\mathbb E[u(c\_t)]$.

Is DEU a special case of EZ? How are those two models compared?

Since EZ is a solution of a complicated dynamic equation, I am having a very hard time figuring out the connection between the two models.

**Tags:** microeconomics, utility, preferences, decision-theory, risk

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## 1 answers

### Answer by username_1 on 2022-02-28. Score: 4 | Accepted answer

Epstein-Zin (EZ) preferences are a generalisation of dynamic CRRA preferences.

In standard CRRA preferences, i.e.
$$
U(c\_0,\dots) = \mathbb{E} \left[ \sum\_t \beta^t \frac{c\_t^{1 - \rho}}{1 - \rho} \right]
$$
The Arrow-Pratt relative risk aversion is $\rho$ and elasticity of intertemporal substitution is $\frac{1}{\rho}$ and thus is unable to be disentangled.

EZ solves this. Actually, the representation you have above is only for deterministic streams. EZ preferences have the recursive representation:
$$
V\_t = \left( (1-\beta)c\_t^{1-\rho} + \beta \left[ \mathbb{E} V\_{t+1}^{1-\alpha} \right]^{\frac{1-\rho}{1-\alpha}} \right)^{\frac{1}{1-\rho}}
$$
Where $\alpha$ captures risk attitudes and $\rho$ the elasticity of intertemporal substitution.

In fact, they agree on deterministic streams of consumption (i.e. when there is no uncertainty). Take a (non-stochastic) stream $(c\_0,c\_1, \dots)$ and the expectation operator drops to
$$
V\_t = \left( (1-\beta)c\_t^{1-\rho} + \beta V\_{t+1}^{1-\rho} \right)^{\frac{1}{1-\rho}}
$$
Define the monotone transform $U\_t = V\_t^{1-\rho}$ to get
$$
U\_t = (1-\beta)c\_t^{1-\rho} + \beta U\_{t+1}
$$
in other words
$$
U\_0(c\_0,c\_1,\dots) = (1-\beta) \sum\_t \beta^t c\_t^{1 - \rho}
$$
which are just CRRA!


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