In many asset pricing models we use CRRA preferences and Epstein-Zin preferences.
Let's say I have an agent that lives $T$ periods with CRRA preferences:
$$ V_0 = \sum_{t=0}^{T} \beta^t \frac{C_t^{1-\gamma}}{1-\gamma}$$
For a given agent who is optimizing this utility function I can get its value at $t=0$.
So I can ask what is the certain consumption stream $\bar{C}$ that would make the agent indifferent between the stochastic consumption stream $C_t$ and the fixed one, i.e.:
$$ V_0 = \sum_{t=0}^{T} \beta^t \frac{\bar{C}^{1-\gamma}}{1-\gamma}$$
Which implies:
$$\bar{C} = \bigg [ \frac{(1-\gamma)V_0}{\sum_{t=0}^{T} \beta^t} \bigg ]^{\frac{1}{1-\gamma}}$$
Now suppose I have Epstein-Zin preferences. And I want to compute the certain equivalent. I.e. the preference of the agent are now given by:
$$V_t = \bigg \{ (1-\beta) C_t^{1-1/\psi} + \beta E_t( V_{t+1}^{1-\gamma})^{\frac{1-1/\psi}{1-\gamma}} \bigg \}^{\frac{1}{1-1/\psi}} $$
How would I go about computing this certain equivalent?
