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I've heard that there is a lot of work being done recently that applies Epstein-Zin preferences. The Wikipedia page doesn't seem to be very full.
I think CompEcon covered most of the points that I was going to mention. Just a few last thoughts:
1) Why are Epstein-Zin preferences important?
The preferences are important because they allow you to separate two of the dimensions along which people care about their allocations; namely, risk aversion and intertemporal substitution.
Additionally, one short coming of standard (i.e. CRRA) is their inability to achieve the Hansen-Jagannathan lower bound for the ratio of the standard devation of the stochastic discount factor to its expected value, $\frac{\sigma(m)}{E(m)}$. In a 2000 paper (I think his Job Market paper), Tallarini showed that recursive preferences are able to come into the Hansen-Jagannathan bounds at less ridiculous levels of risk aversion (although still not totally feasible risk aversion).
2) How does recursive utility differ from other preference models in general? What do they capture that can't be captured otherwise?
It allows you to capture the risk aversion - intertemporal substitution differences.
They are a more general set of preferences than CRRA. I'm pretty sure you can actually write CRRA utility using Epstein-Zin preferences with the right parameters.
Would be interested in hearing if other people know more about this. I know thy can be have model misspecification interpretations. Would love to hear more about that.
3) What are some good resources to learn more about them?
Like I said earlier in a comment, I have found that Ljungqvist and Sargent provide a pretty good explanation of the main things happening in recursive utility.
Additionally, the paper mentioned by CompEcon in another question earlier is a pretty good resource. I'm actually working through this right now when I have some spare time.
This is only a quick answer, unfortunately. The key intuitive insight for Epstein-Zin is that they separate two distinct properties of preferences: risk aversion ("I'd prefer less uncertainty to more uncertainty*") and intertemporal substitution ("I may want to shift consumption forward or backwards in time**").
In the very popular Constant Relative Risk Aversion class of preferences (CRRA), risk aversion and intertemporal elasticity of substitution are tied together as inverses of one another. Recursive preferences, and specifically Epstein-Zin, use certainty equivalence in a clever way to split out the parameter which controls intertemporal substitution from the parameter which controls risk aversion in a static gamble.
The static risk aversion parameter is embedded in the function which imposes the certainty equivalence, and the intertemporal elasticity of substitution parameter is imposed over today's certain consumption and the certainty equivalence value for utility of consumption tomorrow.
That's my attempt to describe it intuitively/verbally. It's much more precise in mathematical form -- one good exposition that I like a lot is Francios Gourio's Asset Pricing field's course notes (EC745 is the course number). You can currently find these notes, titled "Lecture Notes on Macroeconomics and Finance Ec 745," on his website here; see section 8 on page 36.
Work through the math a few times and hopefully it will suddenly "click." The analytical idea is really quite clever. Gourio goes on to discuss how to actually estimate these models, which is very helpful.
(*The proper definitions involves preferences over "lotteries," but I think discussion of that would cloud what we care about here.)
(** by a certain percent, as a function of the interest rate.)
