I working a lot with Epstein-Zin utility (standard in asset pricing models). But I am having some issues wrapping my head around some intuition for how this utility function works.
Let's think about a standard problem of maximization of utility of a consumer who lives two periods. The agent is endowed with wealth of $w_1$ in period 1, and receives an income of $y_2$ in period. His savings for period 2 are given by $w_2$. The risk-free rate is $r_f$.
$$max_{c_t} u(c_t)$$
subject to: $w_2 = (w_1 - c_1)(1+r_f) + y_1$.
Now assume two utility functions:
the value function with CRRA utility: $$ V_t = \frac{C_t^{1-\gamma}}{1-\gamma} + \beta E_t[V_{t+1}]$$
Epstein-zin utility: $$ V_t = \bigg \{(1-\beta) C_t^{1-1/\psi}+\beta E_t[V_{t+1}^{1-\gamma}]^{\frac{1-1/\psi}{1-\gamma}} \bigg \}^{\frac{1}{1-1/\psi}} $$
Now here's my confusion. Let say I start solving this model by backward induction as is standard in the literature. On the last period of life ($t=2$) agents just consume all wealth available to them ($w^2$).
The utility function under the CRRA case takes the shape (on the last period): $$V_2 = \frac{w_2^{1-\gamma}}{1-\gamma}$$
The utility function under the EZ case takes the shape (on the last period): $$V_2 = (1-\beta)^{\frac{1}{1-1/\psi}} w_2$$
So in some sense in the first case the utility function is concave in wealth whereas on the latter case it is linear in wealth. So they have dramatically different shapes. This is even more puzzling if one things that if we set $\psi$ (the elasticity of intertemporal substitution) to be $\frac{1}{\gamma}$, the EZ utility should collapse to the CRRA case.
What am I missing?