## Recursive Utility ##
The traditional approach to consumption-based asset pricing includes **time separable (additive) expected utility** function, 
$$U(C_t,C_{t+1})=u(C_t)+\beta \mathbb{E}_t[u(C_{t+1})],$$ where $\beta<1$ measures impatience (subjective discount factor). That's the first equation in Chapter 1.1. in Cochrane's stellar [asset pricing book](https://www.amazon.co.uk/Asset-Pricing-John-H-Cochrane/dp/0691121370). This obviously results in the standard SDF $$M_{t,t+1}=\beta\frac{u'(C_{t+1})}{u'(C_t)},$$ or in the case of plain power utility, $u(C_t)=C_t^{1-\gamma}$, $$M_{t,t+1}=\beta\left(\frac{C_{t+1}}{C_t}\right)^{-\gamma},$$ where $\gamma\geq0$ measures the investor's risk aversion (concavity of the utility function).

[Epstein and Zin's (1989, Ecta)](https://www.jstor.org/stable/1913778?seq=1#metadata_info_tab_contents) **recursive utility** instead defines today's utility to be $$U_t=\left((1-\beta)C_t^\alpha + \beta \mathbb{E}_t\left[U_{t+1}^{1-\gamma}\right]^\frac{\alpha}{1-\gamma}\right)^{\frac{1}{\alpha}},$$
where $\beta<1$ is the subjective discount factor, $\gamma\geq0$ the risk aversion coefficient and $\Psi=\frac{1}{1-\alpha}\geq0$ is the elasticity of intertemporal substitution (EIS). Note that time additive utility is a special case with $\alpha=1-\gamma$ and $\Psi=\frac{1}{\gamma}$. Recursive utility functions are more general then the above parametrisation but this one is arguably the most common one.

Agents are clearly risk-averse (dislike variation across different states) and clearly prefer an early resolution of uncertainty. However, EIS and risk aversion are inversely related via  $\Psi=\frac{1}{\gamma}$ for standard time additive expected utility functions which is counter-factional. That's one reason why these models struggle to generate a reasonable equity premium. Recursive utility functions, on the other hand, have no problem in separating risk aversion and EIS.

## Asset Pricing with Recursive Utility ## 

The SDF for the above utility function is
$$ M_{t,t+1} = \beta^\theta \left(\frac{C_{t+1}}{C_t}\right)^{-\frac{\theta}{\Psi}} \left(R_{t+1}^W\right)^{\theta-1},$$
where $\theta=\frac{1-\gamma}{1-\frac{1}{\Psi}}$ and $R_{t+1}^W$ is the gross return on the wealth portfolio (which pays aggregate consumption as dividends), which is of course different to the observable market return. A derivation is in chapter 6.4.4 of Munk's great [Financial Asset Pricing book](https://www.amazon.co.uk/Financial-Asset-Pricing-Theory-Claus/dp/0199585490) and, of course, in Epstein and Zin (1989).

The two SDFs for recursive utility and power utility don't look too different. Again, they are identical if $\theta=1$ (and thus $\Psi\gamma=1$). Essentially, all advanced consumption-based models write
$$M_{t,t+1} = \beta\left(\frac{C_{t+1}}{C_t}\right)^{-\gamma} Y_t,$$ where the variable $Y_t$ somehow makes the SDF more volatile by capturing different kinds of risk. This applies to long run risk models, habit formation, rare disasters, etc. They address the equity premium puzzle from [Mehra and Prescott (1985, JME)](https://www.sciencedirect.com/science/article/pii/0304393285900613) and the bound from [Hansen and Jagannathan (1991, JPE)](https://www.jstor.org/stable/2937680?seq=1#metadata_info_tab_contents). [Cochrane (2017, RF)](https://academic.oup.com/rof/article/21/3/945/3060346) provides a great summary of this literature.


Clearly, the same Euler equation applies to models with recursive utility and we can price assets by looking at $$\mathbb{E}_t\left[M_{t,t+1}R_{t+1}\right]=\mathbb{E}_t\left[\beta^\theta \left(\frac{C_{t+1}}{C_t}\right)^{-\frac{\theta}{\Psi}} \left(R_{t+1}^W\right)^{\theta-1}R_{i,t+1}\right]=1,$$
or more conveniently,
$$\beta^\theta\mathbb{E}_t\left[\exp\left(-\frac{\theta}{\Psi}\Delta c_{t+1} +(\theta-1) r_{t+1}^W +r_{i,t+1}\right)\right]=1,$$
where $\Delta c_{t+1}=\ln\left(\frac{C_{t+1}}{C_t}\right)$, $r^W_{t+1}=\ln(R^W_{t+1})$ and $r_{i,t+1}=\ln(R_{i,t+1})$.

## Long Run Risk and Log-Linearisation ##

As @fesman suggests in the comments, the standard model for recursive utility is [Bansal and Yaron's (2004, JF)](https://onlinelibrary.wiley.com/doi/full/10.1111/j.1540-6261.2004.00670.x) seminal long run risk model. Having said that, recursive utility is used in many models today. For example, [Chen (2016, JFE)](https://www.sciencedirect.com/science/article/pii/S0304405X16000088) focuses mostly on the production side and still includes recursive utility for the households in his model. [My choice here is completely random, Chen's paper is simply the top one on my desk. I only want to illustrate that recursive utility is common nowadays.]

I'm no t going to solve the entire 2004 model here but give you an overview what happens. Following [Campbell and Shiller (1988, RFS)](https://academic.oup.com/rfs/article/1/3/195/1580239), we conjecture a log-linear form $$\ln R_{t+1}^W \approx \kappa_0+\kappa_1z_{t+1}-z_t+\Delta c_{t+1},$$ where $z_t=\ln(P_t)-\ln(C_t)$ is the log price-consumption ratio, $\kappa_0,\kappa_1$ constants. That's equation 2 in BY 2004. The Euler equation turns to
$$\beta^\theta\mathbb{E}_t\left[\exp\left(-\frac{\theta}{\Psi}\Delta c_{t+1} +(\theta-1) \left(\kappa_0+\kappa_1z_{t+1}-z_t+\Delta c_{t+1}\right) +r_{i,t+1}\right)\right]=1.$$
That's great thus far because Bansal and Yaron's model tells us the dynamics of $\Delta c_{t+1}$. We next conjecture that $z_t$ is also linear, i.e. $$z_t\approx A_0+A_1x_t+A_2\sigma_t^2,$$ where $x_t$ and $\sigma_t^2$ are two further state variables with given dynamics in the model. In fact, $x_t$ is the long run risk component and $\sigma_t^2$ the conditional volatility of log consumption growth. See equation (8) for the model description. Because it all boils down to log-normal distributions, you can compute the expectation in the Euler equation: take the already known variables $x_t$ and $\sigma_t^2$ out of the conditional expectation and use $\mathbb{E}[e^{m+sZ}]=e^{m+0.5s^2}$ for $Z\sim N(0,1)$ for the rest. The appendix to Bansal and Yaron's paper contains all the details. Munk also presents the solution in his book. You can then get expressions for $A_0$, etc. in terms of the model parameters.


## A recent paper on Log-Linearisation ##

[Pohl, Schmedders and Wilms (2018, JF)](https://onlinelibrary.wiley.com/doi/full/10.1111/jofi.12615) show that long-linerisation, which is essentially a first-order Taylor polynomial, can result in very wrong solutions as higher order terms are neglected. Following other papers, they instead suggest more robust numerical methods using for instance two-dimensional Chebyshev polynomials in $x_t$ and $\sigma_t^2$.