Epstein-Zin utility intuition

Question

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:

1. the value function with CRRA utility: $$ V_t = \frac{C_t^{1-\gamma}}{1-\gamma} + \beta E_t[V_{t+1}]$$

2. 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 ($w^2$).

1. The utility function under the CRRA case takes the shape (on the last period): $$V_2 = \frac{w_2^{1-\gamma}}{1-\gamma}$$

2. 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?

Answer 1

While the concavity of a felicity function $u$ affects choices a value function is not unique up to monotone transformations. It is an ordinal and not a cardinal concept. Consider your CRRA equation

$$V_t=\frac{C_t^{1-\gamma}}{1-\gamma}+\beta\mathbb{E}_t[V_{t+1}]$$

$$\Leftrightarrow V_t^{\frac{1}{1-\gamma}}=\left(\frac{C_t^{1-\gamma}}{1-\gamma}+\beta\mathbb{E}_t[V_{t+1}]\right)^{\frac{1}{1-\gamma}}$$

Setting $\tilde{V}_t=V_t^{\frac{1}{1-\gamma}}$

$$\tilde{V}_t=\left(\frac{C_t^{1-\gamma}}{1-\gamma}+\beta\mathbb{E}_t[\tilde{V}_{t+1}^{1-\gamma}]\right)^{\frac{1}{1-\gamma}}$$

Now $\tilde{V}_2$ is also linear but the equation defines the same ranking towards different policies than the original equation.

As you mentioned, EZ corresponds to CRRA in the special case $\gamma=\frac{1}{\psi}$. Plugging in

$$ V_t = \bigg \{(1-\beta) C_t^{1-\gamma}+\beta E_t[V_{t+1}^{1-\gamma}] \bigg \}^{\frac{1}{1-\gamma}} $$

Now at first sight this might look like different preferences than your standard CRRA specification. But let us rewrite the problem:

$$ V_t^{1-\gamma} = \bigg \{(1-\beta) C_t^{1-\gamma}+\beta E_t[V_{t+1}^{1-\gamma}] \bigg \}$$

or

$$ V_t^{1-\gamma}(1-\beta)^{-1}(1-\gamma)^{-1} = \bigg \{ \frac{C_t^{1-\gamma}}{1-\gamma}+\beta E_t[V_{t+1}^{1-\gamma}(1-\beta)^{-1}(1-\gamma)^{-1}] \bigg \}$$

Define $\hat{V}_t= V_t^{1-\gamma}(1-\beta)^{-1}(1-\gamma)^{-1}$. You get

$$ \hat{V}_t = \bigg \{ \frac{C_t^{1-\gamma}}{1-\gamma}+\beta E_t[\hat{V}_{t+1}] \bigg \}$$

Which has exactly the same mathematical form as your CRRA problem. From this you should realize that this is just your original CRRA problem but written in a slightly different way.