Epstein-Zin Preferences and Timing of Uncertainty Resolution
Summary
The document sets out a simplified Epstein-Zin recursive utility specification and rewrites it using a transformed value variable. It then sets the discount factor to one and states that a parameter condition should imply a preference for learning about a future consumption outcome earlier rather than later.
To examine the claim, the author considers consumption that is zero except for a random payoff in a later period. The post derives expressions for the value under early and late resolution and concludes that they appear equal, despite the expected preference. It asks where the reasoning fails but supplies no answer. Thus, it provides a useful statement of the timing question and algebraic setup, while leaving the proposed preference condition and its demonstration unresolved.
Key ideas
- Epstein-Zin utility separates recursive valuation across time from current consumption.
- The author transforms utility into a value variable to simplify the recursion.
- The example places a random consumption payoff in a later period.
- The post compares cases where uncertainty is learned earlier or later.
- The derivation finds equal values in both cases and leaves the discrepancy unresolved.
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# How to use Epstein-Zin model to show the preference for early/late resolution?
# How to use Epstein-Zin model to show the preference for early/late resolution?
Consider the Epstein-Zin utility function: $$ U_t=\left\{\left(1-\beta\right)c_t^\rho+\beta \left[E_t \left(U_{t+1}^\alpha\right)\right]^\frac{\rho}{\alpha}\right\}^{\frac{1}{\rho}}. $$ To simplify the algebra, I modified it to $$ U_t=\left\{c_t^\rho+\beta \left[E_t \left(U_{t+1}^\alpha\right)\right]^\frac{\rho}{\alpha}\right\}^{\frac{1}{\rho}}. $$ Let $V_t=U_t^\rho$, we have $$ V_t=c_t^\rho+\beta\left[E_t\left(V_{t+1}^{\frac{1}{\gamma}}\right)\right]^\gamma $$ where $\gamma=\frac{\rho}{\alpha}$. To further simplify, I let $\beta=1$, so $$ V_t=c_t^\rho+\left[E_t\left(V_{t+1}^{\frac{1}{\gamma}}\right)\right]^\gamma. $$ If $\gamma>1$, then the decision maker should prefer early resolution.
The formal proof in Kreps and Porteus (1978) and Epstein and Zin (1989) are too technical for me. I tried to get a sense of it through a simple example, but failed. Consider a specific consumption lottery: $c_0=c_1=c_3=\cdots=0$ and $c_2$ is a random variable. Consider the following two cases.
The first is ``early resolution,'' that is, in period 1, the decision maker knows the value of $c_2$. So $$ E_1\left(V_2^{\frac{1}{\gamma}}\right)=V_2^{\frac{1}{\gamma}} $$ and thus $$ V_1=V_2. $$ Then $$ V_0=\left[E_0\left(V_1^{\frac{1}{\gamma}}\right)\right]^\gamma=\left[E\left(V_2^{\frac{1}{\gamma}}\right)\right]^\gamma. $$
The second is ``late resolution,'' that is, the decision maker does not know the value of $c_2$ until period 2. So $$ V_1=\left[E_1\left(V_2^{\frac{1}{\gamma}}\right)\right]^\gamma $$ and $$ V_0=\left[E_0\left(V_1^{\frac{1}{\gamma}}\right)\right]^\gamma=\left[\left(V_1^{\frac{1}{\gamma}}\right)\right]^\gamma=V_1=\left[E_1\left(V_2^{\frac{1}{\gamma}}\right)\right]^\gamma. $$
In sum, $V_0$ in both cases are the same, which should be wrong. Could you please help me to figure out my mistake? Thank you very much!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.