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