Computing Certain Consumption Equivalents with Epstein–Zin Preferences

Summary

The document asks how to calculate a certain consumption equivalent for an agent with Epstein–Zin preferences. It first defines the idea under CRRA utility: find a fixed consumption level that gives the same lifetime value as a stochastic consumption stream. It presents an expression for that level in terms of the agent’s value, discount factor, risk aversion, and horizon.

It then gives a recursive Epstein–Zin utility specification, which separates risk aversion from the elasticity of intertemporal substitution, and poses the analogous indifference question. The document does not answer how to compute the equivalent under this recursive preference structure. In particular, it gives no solution method, numerical example, or treatment of how to handle the expectation over future value. It is therefore a useful statement of a consumption-equivalent problem and its CRRA baseline, but not a complete derivation for Epstein–Zin preferences.

Key ideas

• A certain consumption equivalent is a fixed consumption level that matches the value of a stochastic consumption stream.
• Under CRRA utility, the document expresses this equivalent using the lifetime value and discounted horizon.
• Epstein–Zin preferences separate risk aversion from the elasticity of intertemporal substitution.
• The document poses, but does not solve, the corresponding equivalent-consumption calculation for Epstein–Zin preferences.

Tags

# How to get a Certain Consumption Equivalent using Epstein-Zin preferences?


# How to get a Certain Consumption Equivalent using Epstein-Zin preferences?












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?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.