# Is Epstein-Zin utility a generalization of dynamic expected utility (DEU)?

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.

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!