A Robust Numerical Method for Solving Risk-Sharing. Problems with Recursive Preferences

A Robust Numerical Method for Solving Risk-Sharing Problems with Recursive Preferences

Pierre Collin-Dufresne, Michael Johannes, and Lars A. Lochstoer EPFL Columbia Business School

July 10, 2015

Abstract

We provide a robust and accurate numerical solution methodology for discrete-time, com- plete markets, general equilibrium risk-sharing problems when agents have recursive prefer- ences. The agents can di¤er both in terms of preference parameters and beliefs. The methodol- ogy can easily handle multiple aggregate shocks and state variables. To illustrate the method- ology, we consider an economy where aggregate consumption dynamics follow those in Bansal and Yaron (2004) and there are two Epstein-Zin agents with di¤erent preference parameters.

Any errors or omissions are our own. Contact info: Lars A. Lochstoer, 405B Uris Hall, Columbia Busi- ness School, Columbia University, 3022 Broadway, New York, NY 10027. E-mail: [email protected].

First draft: November 2013.

1 Introduction

Recursive preferences are now commonplace in asset pricing and macroeconomic studies (e.g., Bansal and Yaron (2004), Hansen and Sargent (2008), Backus, Routledge, and Zin (2009), Barro (2010), Routledge and Zin (2013), Gourio (2011)). The separation between risk aversion and the elasticity of intertemporal substitution and the ensuing added risk factor these preferences imply relative to time-additive power utility preferences (shocks to the value function) have shown promise in accounting for the joint stylized facts of aggregate quantities and asset prices.

However, the typical analysis features a representative agent. Heterogeneity in preference para- meters and/or beliefs are economically plausible and can lead to interesting endogenous dynamics in terms of risk pricing and savings behavior, as well as having implications for the volume of trade.

One might therefore expect complete markets heterogenous agent models with recursive utility to also be commonplace. This is not the case. In fact, there are only a few studies that feature heterogeneity in preference parameters or beliefs when agents have recursive utility. Garleanu and Panageas (2012) allow for di¤erent preference parameters across two sets of agents, while Borovicka (2013) considers the long-run wealth distribution in an economy with pessimists and optimists. In both cases, aggregate consumption growth is i.i.d., there is only one aggregate shock, markets are complete, and time is continuous.

These modeling choices are not made by accident. Multiple aggregate state variables and shocks makes the solution much more di¢ cult. Multiple state variables lead to PDEs (as opposed to ODEs) and multiple shocks means two assets no longer dynamically complete the market (in a continuous- time setting). In discrete time, it is hard to numerically solve even such relatively stylized models, due to the fact that the evolution of the endogenous state variable (the relative wealth of agents) is not known.

In this paper, we propose a robust numerical method for solving complete markets risk-sharing problems when agents have recursive utility. Agents can di¤er in (all) preference parameters and/or

beliefs. The solution method is accurate and it does not rely on approximations to the true prob- lem (e.g., expanding a polynomial around a steady-state solution). By ’accurate’we mean that as one decreases the coarseness of the grids for the state variables in the model at hand, the solution converges to the true solution. Further, the solution method does not increase in conceptual com- plexity as more state-variables and/or shocks are added, although of course adding more shocks or state-variables increases computation time. By ’robust’we refer to the fact that ’good’(lucky) guesses of initial functional forms are not needed to ensure convergence, neither for the value func- tion nor the endogenous evolution equations. This is a strong positive, as ’good’ in the context of existing methods for solving such models typically refers to guesses that are very close to the equilibrium dynamics you are trying to …nd and therefore do not presently know. In other words, existing methods are not readily implementable for general cases. On the negative side, the solution methodology we propose is subject to the curse of dimensionality.

The general methodology we propose here applies to a number of settings where agents have

’exotic’preferences, including, in addition to Epstein-Zin preferences, risk-sensitive utility (Hansen and Sargent (2008)), (generalized) disappointment aversion (Gul (1991), Routledge and Zin (2013)), and smooth ambiguity aversion (Klibano¤, Marinacci, and Mukerji (2005, 2009)).

The next section lays out the general problem and why it is hard to solve these risk-sharing problems when agents have recursive utility. We also note that perturbation solutions that rely on expanding around a steady-state often are not suitable to deal with models with preference and/or belief heterogeneity as these models often feature non-stationary dynamics. Thus, the steady-state simply does not exist. Further, as a general point, even if a model is stationary one would want to expand around the unknown stochastic steady state, which in models that match asset price data, and therefore feature a large amount of risk, can be far away from the known non-stochastic steady state solution. Taylor approximations are not in general good approximations globally, just locally. The Taylor approximation to ln (1 + x) around x0 = 0 is a classic example where the

Taylor approximation (of any order) is good locally, but not for values of x further away from x0

(e.g., x > 1). This is of course not to say approximate solution methods are inherently bad— in particular, polynomial expansions around a known solution are popular because they provide very fast solutions given the analytical expressions and thus allow for estimation of models. However, their widespread adoption has arguably moved ahead of justi…cations for their use. We provide a readily implementable method for assessing the accuracy of typical expansion solutions in interesting economic models, such as models with constraints, models that feature a signi…cant amount of risk, models with highly nonlinear features due to, e.g., di¤erences in beliefs or, say, risk aversion.

In the following section, we describe the proposed solution methodology. We then consider as an example the asset pricing and risk-sharing implications for the case of two agents with di¤er- ent preference parameters when aggregate consumption dynamics follows the speci…cation in the Bansal and Yaron (2004). Of note, the consumption-sharing rule in this case depends not only on aggregate consumption (as it would with power utility preferences), but also the conditional mean and volatility of consumption growth (which are the other state variables that determine aggregate consumption dynamics), in addition to the endogenous state variable governing the relative wealth of the agents. Further, even if agents have the same level of risk aversion, heterogeneity in either the elasticity of intertemporal substitution or the time-discounting parameter can lead to large counter- cyclical time-variation in the conditional price of risk and the risk premium in the economy due to endogenous relative wealth ‡uctuations. In the …nal section, we show how our solution method can be extended for the case of a production economy where aggregate consumption is endogenous. We focus here on belief heterogeneity to exhibit this feature of our solution methodology as well.

2 Recursive preferences in a two-agent, complete markets economy

In this Section, we describe the model where there are two (sets of) agents (A and B) with Epstein- Zin preferences with potentially di¤erent preferences as well as di¤erent beliefs about the exogenous aggregate consumption dynamics.¹ While the method we will explain below conceptually extends to N agents, the wealth distribution in general will be characterized by N 1 endogenous state variables. Thus, due to the curse of dimensionality, the two agent problem is in practice the most relevant case. We will consider the case of endogenous aggregate consumption in a production economy setting in the …nal section.

Denote aggregate consumption Ct = C (x_t), and let xt be a vector of exogenous observable Markov state variables with evolution xt= h (x_{t 1}; "_t)where "tis a vector of shocks. The aggregate resource constraint is:

C_t = C_A;t+ C_B;t. (1)

The preferences of agents A and B are given by:

V_i;t = C_i;tⁱ+ _{i i;t}^{i 1= i}; (2)

where i; _i; _i < 1, _i > 0, _i;t E_tⁱ V_i;t+1^{i 1= i} and i = fA; Bg : The superscript i and subscript t on the expectation operator denote an expectation taken using agent i’s beliefs at time t. The subscript t on the value function means it is a function of the time t values of the state variables in the economy. We will discuss these below. First. note that the relevant beliefs consist of a complete probability speci…cation for the dynamics of aggregate consumption, that is over future x’s and "’s. We do not require beliefs to be updated in a Bayesian fashion, but we do require that

1These agents act competitively, consistent with the notion that each agent is a representative agent for an in…nite mass of agents with the same beliefs and preference parameters, but potentially di¤erent wealth levels.

beliefs are such that equilibrium exists and that if beliefs are updated, the updating equations are known. The cum-consumption wealth levels are Wi;t = V_i;t(C_i;t=V_i;t)^{1= i}, respectively (see Epstein and Zin (1989)). Since markets are assumed to be complete, the budget constraint can be written W_i;t+1= (W_i;t C_i;t) R_Wi;t+1, where RWi;t+1 is the return on total wealth for agent i.

It is convenient to solve a normalized version of this model, where all variables are divided by aggregate consumption. Let lower case of variables denote the normalized counterpart. Thus, for an arbitrary variable Zt we have that zt= Z_t=C_t. In this case, the value functions can be written:

v_i;t = c_i;tⁱ + _{i i;t}^{i 1= i}; (3)

where _i;t E_tⁱ v_i;t+1ⁱ (Ct+1=Ct) ^{i 1= i} and where the resource constraint is cA;t + cB;t = 1. The stochastic discount factor under agent i’s probability measure can then be written (see Epstein and Zin (1989)):

E_tⁱ M_t+1ⁱ R^j_t+1 = 1 for all t and j M_t+1ⁱ = _i c_i;t+1

c_i;t

i 1

C_t+1 C_t

i 1

v_i;t+1

i;t

i i

: (4)

The state variables in this economy are xt and the relative wealth of the two agents. In what follows, time will also be a state variable due to a backwards recursion from a particular point T . Therefore, let Xt [x⁰_t (T t)]⁰ be the time-augmented vector of exogenous state variables. Since the relative consumption of agents is monotone in the relative wealth of agents, we use the relative consumption of agent A as the endogenous state variable, cA;t. Thus, vi;t = f_i(X_t; c_A;t).

If the endogenous evolution equation of cA;t is known, we can now easily solve a standard value

function iteration problem on a grid for xt and cA;t2 (0; 1):

f_A(X_t; c_A;t) = h

c_A;t^A + _AE_t^A[f_A(X_t+1; c_A;t+1) ^A(C_t+1=C_t) ^A] ^{A= A}i1= _A

; (5)

f_B(X_t; c_A;t) = h

(1 c_A;t) ^B + _BE_t^B[f_B(X_t+1; c_A;t+1) ^B(C_t+1=C_t) ^B] ^{B= B}i1= _B

: (6)

Thus, the crux of the risk-sharing problem is …nding the endogenous evolution equation for cA for all points in the state-space.

2.1 The power utility case

It is useful to …rst revisit the solution to the much simpler power utility case, where = . In this economy, the evolution equation for cA is known analytically. To see this, …rst note that since markets are complete, agents intertemporal marginal rates of substitution (IMRS) must be equal in each state (!). Denote the conditional probability agent i assigns at time t to state !t+j realized at time t + j as ⁱ_t(!_t+j). Then we have that:

A

t (!_t+j) _A C_A;t+j C_A;t

A 1

= ^B_t (!_t+j) _B C_B;t+j C_B;t

B 1

for all t and !t+j: (7)

Applying the normalization and noting that current state variables are xt and cA;t Equation (7) yields:

~

c_A;t+1=

B

t (!_t+j) _B

A

t (!_t+1) _A

C_t+1 C_t

B A

~

c_A;t: (8)

where ~c_A;t c_A;t^{A 1}(1 c_A;t)^{1 B}. Since _A 1; _B 1 < 0 and cA 2 (0; 1), ~c^A is monotonically decreasing in cA. Thus, ~c_A uniquely identi…es cA. In words, next period’s relative consumption of agent A is given as a known function of the outcomes of the exogenous shocks and today’s state variables. Granted, for general _A and _B one must solve numerically for cA given ~cA, but one can do this extremely fast and to arbitrary accuracy. Thus, it is in this case easy to implement the

value function iteration in Equation (5).

2.2 The general case

The problem with solving the risk-sharing problem with agents with recursive preferences is that the consumption-sharing rule depends on the value functions, which are what we are trying to solve for.

Thus, unlike the case of power utility, the evolution equation for the endogenous state-variable (the relative wealth, or consumption, of the agents) is not a known function of the evolution equation of the exogenous state-variables, xt. To see this, consider the complete markets requirement that the IMRS is equalized across states:

A

t (!_t+j) _A c_A;t+1 c_A;t

A 1

C_t+1 C_t

A 1

v_A;t+1

E_t^A v_A;t+1^A (C_t+1=C_t) ^{A 1= A}

! A A

=

::: ^B_t (!_t+j) _B 1 c_A;t+1 1 c_A;t

B 1

C_t+1 C_t

B 1

v_B;t+1

E_t^B v_B;t+1^B (C_t+1=C_t) ^{B 1= B}

! B B

: (9)

Rearranging, we have:

~

c_A;t+1v_A;t+1^{A A}v_B;t+1^{B B} =

B

t (!_t+j) _B

A

t (!_t+j) _A

C_t+1 C_t

B A

k_t~c_A;t; (10)

where again ~c_A;t c_A;t^{A 1}(1 c_A;t)^{1 B} and, new to the general case:

k_t E_t^B v_B;t+1^B (C_t+1=C_t) ^{B B= B 1} E_t^A v_A;t+1^A (C_t+1=C_t) ^{A A= A 1}

: (11)

Unfortunately, since the value functions are what we are trying to solve for, vA;t+1, vB;t+1 and thus k_t are not known. Thus, unlike for the power utility case as given in Equation (8), we no longer have the evolution equation for the endogenous variable, cA;t as an analytical function of known variables. A common way to proceed would be to start with an initial guess of the value functions,

as well as an initial guess of the evolution equation of cA. One can then use value function iteration to update the value function and the …rst order conditions of the agents’to update the evolution equation iteratively. This is not easy and there is no guarantee that the problem converges. In fact, in our experience, the value function iteration problem using what would seem like a reasonable guesses of the evolution equation (e.g., using the power utility case as a starting point) typically does not converge.²

3 A Robust Numerical Method: Backwards Recursion

As a solution to this problem, we propose a numerical solution scheme that relies on a backwards recursion algorithm. Thus, we consider an economy that has a terminal date T . If the transversality condition holds for both agents, the solution to the in…nite horizon problem is found by choosing a su¢ ciently large T .

The basic idea is that by having a terminal date we can do a backwards recursion with this date as a starting point, where due to the recursive nature, next period’s value functions are known.

Thus, this is di¤erent from a typical value function iteration, where one has to start from a guess that does not correspond to an equilibrium outcome. Further, with a known next period’s value function, we can use the …rst order condition of the agents, implicit in Equation (10) to …nd the evolution equation of the endogenous state variable, cA. With this in hand, we can trivially solve for the value function at time T 1, and so on. In showing how this method is robust, we show that the numerical exercises at each time step are well-behaved due to monotonicity properties of the functions we are trying to …nd. The following goes through the numerical procedure in some detail and the next Section applies the method to a speci…c example.

2Guvenen (2009) solves a model where agents have di¤erent preference parameters and recursive preferences.

He notes that the numerical solution, obtained in this fashion, depends crucially on initial values for the evolution equation and the value functions that are close to the true values. Thus, while possible to implement, this is a very hard and time-consuming problem to solve.

At time T , when the economy ends, the value functions reduce to:

v_A;T = c_A;T; (12)

v_B;T = 1 c_A;T: (13)

Equations (12) and (13) give the boundary conditions for the value functions as a function of the relative consumption of agent A, cA: As mentioned earlier, it is convenient to use cA;t as the endogenous state-variable (one could equivalently use relative wealth of, say, agent A), in addition to the exogenous state variables, xt: Thus, the value function for agent i can be written vi;t = f_i(x_t; c_A;t) :

At time T 1, the complete markets requirement that agents IMRS are equalized across states implies that:

A T jT 1 A

c_A;T c_{A;T 1}

A 1

C_T C_{T 1}

A 1

v_A;T

A;T 1(v_A;TC_T=C_{T 1})

A A

= :::

B T jT 1 B

1 c_A;T 1 c_{A;T 1}

B 1

C_T C_{T 1}

B 1

v_B;T

B;T 1(v_B;TC_T=C_{T 1})

B B

: (14)

Here ⁱ_{T jT 1} denotes the probability agent i assigns to a given state at time T given agent i’s beliefs at time T 1. First, de…ne:

k_{T 1} = ^{B;T 1}(v_B;TC_T=C_{T 1}) ^{B B}

A;T 1(v_A;TC_T=C_{T 1}) ^{A A}

= E_t^B((1 c_A;T) ^B (C_T=C_{T 1}) ^B) ^{B= B 1} E_t^A c_A;T^A (C_T=C_{T 1}) ^{A A= A 1}

: (15)

Next, imposing the boundary values as given in Equations (12) and (13), we have that:

A T jT 1 A

c_A;T c_{A;T 1}

A 1

C_T C_{T 1}

A 1

c_A;T^{A A} = :::

k_{T 1} ^B_{T jT 1 B} 1 c_A;T 1 c_{A;T 1}

B 1

C_T C_{T 1}

B 1

(1 c_A;T) ^{B B} m

c_A;T^{A 1}

(1 c_A;T) ^{B 1} = k_{T 1}

B T jT 1 A T jT 1

B A

c_A;T^{A 1}₁ (1 c_{A;T 1}) ^{B 1}

C_T C_{T 1}

B A

: (16)

Note …rst that Equation (16) implies that, for a given state of the world at time T and value of state variables at time T 1, cA;T 2 (0; 1) is decreasing in k^{T 1} (since i 1 < 0). Thus, for a given k_{T 1}, Equation (16) uniquely determines cA;T for each state of the world at time T .

Of course, from Equation (15), we only know kT 1 as a function of cA;T. This suggests solving for kT 1 for each value of the state variables at time T 1 by …nding a joint solution to Equations (15) and (16). The question then is whether the solution one obtains is unique. Taking a derivative of the numerator of Equation (15) with respect to cA;T we have:³

@E_t^B((1 cA;T) ^B (CT=CT 1) ^B) ^{B= B 1}

@c_A;T = :::

( _B= _B 1) E_t^B((1 c_A;T) ^B (C_T=C_{T 1}) ^B) ^{B= B 2} _B( 1) E_t^B (1 c_A;T) ^{B 1}(C_T=C_{T 1}) ^B : (17) Since the expectations terms are all positive, we can determine the sign this derivative in terms of the preference parameters:

sign @E_t^B((1 c_A;T) ^B (C_T=C_{T 1}) ^B) ^{B= B 1}

@c_A;T

!

= sign (( _B= _B 1) _B( 1))

= sign ( ( _{B B})) : (18)

3Think of this as increasing c_A;T by a particular tiny amount in each state.

Similarly, we can sign the derivative of the denominator of Equation (15) with respect to cA;T:

sign @E_t^A c_A;T^A (C_T=C_{T 1}) ^{A A= A 1}

@c_A;T

!

= sign (( _A= _A 1) _A)

= sign ( _{A A}) : (19)

A su¢ cient condition for kT 1 to be monotone in cA;T is that the derivatives of the numerator and denominator of Equation (15) with respect to cA;T have di¤erent signs.⁴ In these cases Equations (15) and (16) provide unique solutions for kT 1, as well as for cA;T for each state of the world at time T . It is also immediate from these equations that a solution exists where cA;T 1 2 (0; 1) and k_{T 1} > 0.

While the …xed point problem for …nding kT 1implicit in Equations (15) and (16) must be solved numerically for each point on a grid for the state variables, this is very fast given the monotonicity (e.g., a routine like zbrent works very fast). For particular choices of A and B, one can even solve analytically for cA;T as a function of kT 1 and the state variables at T 1. In sum, using cA;T 1 as the endogenous state variable, we now have numerically the evolution equation for cA, from T 1 to time T .

Next, we can now solve numerically for the normalized value functions at time T 1 on a grid for the relevant state variables at time T 1, using:

v_i;t =h

c_i;tⁱ + _iE_tⁱ v_i;t+1ⁱ (C_t+1=C_t) ^{i i= i}i1= _i

; (20)

where the state variables are xt and cA;t. Note again that solving numerically for the certainty equivalent of next period’s value function requires the use of the evolution equation for cA.

The second backwards iteration is then at time t = T 2. Again, we start with the requirement

4Note that this is not a necessary condition for monotonicity, but that this su¢ cient condition (which is easy to check) alone is likely to cover the most relevant applications. Basically, the condition says that agents should both weakly prefer early (or late) resolution of uncertainty.

that the IMRS is equalized for each state for the two agents:

A t+1jt A

c_A;t+1 c_A;t

A 1

C_t+1 C_t

A 1

v_A;t+1

A;t(v_A;t+1C_t+1=C_t)

A A

= :::

B t+1jt B

1 c_A;t+1 1 c_A;t

B 1

C_t+1 C_t

B 1

v_B;t+1

B;t(v_B;t+1C_t+1=C_t)

B B

:

Note that vi;t+1= f_i(X_t+1; c_A;t+1)is known from the previous step in the backwards recursion.

Now, rewrite the above equation as:

c_A;t+1^{A 1} (1 c_A;t+1) ^{B 1}

v_A;t+1^{A A} v_B;t+1^{B B} = k_t

B t+1jt A t+1jt

B A

c_A;t^{A 1} (1 c_A;t) ^{B 1}

C_t+1 C_t

B A

; (21)

where similar to the T 1 case, kt = ^B;t(^{vB;t+1Ct+1=Ct}) ^{B B}

A;t(^{vA;t+1Ct+1=Ct}) ^{A A}. Note that vA;t (vB;t) is increasing (decreasing) in cA;t. A su¢ cient condition for the left hand side of Equation (21) to be monotone in kt is that both agents weakly prefer early resolution of uncertainty, i.e. i i 0. Since

vA;t+1

vB;t+1 is known as a function of xt+1 and cA;t+1, it is easy to numerically …nd the value of cA;t+1

corresponding to a particular outcome (xt; "_t+1), given a value for kt.

Finally, we need to solve for kt. Note that the previous equation gives the evolution equation for cA;t+1 as a function also of kt (cA;t+1= g (X_t; c_A;t; k_t; "_t+1)). We again …nd kt as a …xed point of the equation:

k_t= E_t^B[(v_B;t+1(x_t+1; 1 c_A;t+1(k_t))) ^B(C_t+1=C_t) ^B] ^{B= B 1} E_t^A[(vA;t+1(xt+1; cA;t+1(kt))) ^A(Ct+1=Ct) ^A] ^{A= A 1}

: (22)

Thus, a su¢ cient condition for Equations (21) and (22) to provide a unique solution to the evolution equation for cA;t to cA;t+1 as a function of the aggregate exogenous state variables at xt and xt+1

is that both agents weakly prefer early resolution of uncertainty. The normalized value function at time t can then be found using Equation (20).

Further backwards recursions follow the same algorithm as that given for the case t = T 2.

In practice, we …nd that having T > 1200 is su¢ cient for typical calibrations (clearly, the time-

discount factors _A and _B are particularly important in this regard). Further, programmed in C (or Fortran or similar) and using multiprocessing (multithreading is now available on all normal desktops and laptops) we …nd that problems with 4 state variables with continuous support can be solved in a day with excellent accuracy. Of course, such statements are problem-speci…c. Next, we consider a problem with 3 state variables, 2 exogenous and 1 endogenous, that we solve with excellent accuracy in about 3 hours.

4 Example: Heterogeneous agents in a Bansal and Yaron economy

In this Section, we consider the asset pricing and risk-sharing implications of heterogenous prefer- ences in an economy where aggregate consumption are speci…ed as in Bansal and Yaron (2004):

c_t+1 = + x_t+ _t"_t+1; x_t+1 = _xx_t+ ' _{t t+1};

2

t+1 = max + ²_t + ! _t+1; : (23)

Here. all the shocks are i.i.d., uncorrelated standard normal and > 0 is a lower bound for the conditional variance. The latter is set very close to zero, so as to have minimal impact on the variance process, while ensuring that variance is positive. We let = e ⁸, while the other parameters are assumed to be the same as those in Bansal and Yaron (2004).⁵ The market is de…ned as a claim to aggregate dividends, which have log growth rates:

d_t+1 = c_t+1+ _divp

v _t+1; (24)

5One could alternatively choose a, say, mean-reverting process for log variance in order to ensure positive variance.

However, we chose the truncated AR(1) process in order to keep the analysis as close as possible to that in Bansal and Yaron (2004).

where t+1 is a standard Normal shock uncorrelated with all the other shocks and where is a

’leverage’parameter.

There are two competitive agents (A and B) in this economy who agree on the aggregate endowment process as given in (23), but that have di¤erent preference parameters ( _i; _i; _i). Thus, using similar notation as that introduced in the previous section, we have that vA;t= f_A(x_t; ²_t; c_A;t) and vB;t = f_B(x_t; ²_t; c_A;t). In this economy, the transversality condition holds, so when choosing a su¢ ciently high terminal date T for the backwards recursion we have the solution to the in…nite horizon problem (we think). The time T boundary conditions are the same as those given in Equations (12) and (13).

Following Equations (15) and (16), we then have at time T 1:

kT 1 = E_t((1 c_A;T) ^B(C_T=C_{T 1}) ^B) ^{B= B 1} E_t c_A;T^A (C_T=C_{T 1}) ^{A A= A 1}

; (25)

and

c_A;T^{A 1}

(1 c_A;T) ^{B 1} = k_{T 1} ^B

A

c_A;T^{A 1}₁ (1 c_{A;T 1}) ^{B 1}

CT

C_{T 1}

B A

; (26)

which jointly give the evolution equation for the endogenous state variable, cA;T = g_{T 1} x_{T 1}; ²_{T 1}; c_{A;T 1}; "_T . Given this, we can solve (numerically) for the value functions at time T 1.

At T 2, corresponding to Equations (21) and (22), we have:

c_A;t+1^{A 1} (1 c_A;t+1) ^{B 1}

v_A;t+1^{A A}

v_B;t+1^{B B} = k_t ^B

A

c_A;t^{A 1} (1 c_A;t) ^{B 1}

C_t+1 C_t

B A

; (27)

and

k_t= E_t[(v_B(x_t+1; 1 c_A;t+1(k_t))) ^B (C_t+1=C_t) ^B] ^{B= B 1}

E_t[(v_A(x_t+1; c_A;t+1(k_t))) ^A(C_t+1=C_t) ^A] ^{A= A 1} : (28) which now give the evolution equation for the endogenous state variable from T 2to T 1. Letting t = T 2, we have cA;t+1 = g_t x_t; ²_t; c_A;t; "_t+1; _t+1; _t+1 . Note that in this case the shocks _t+1

and t+1 also enter in the evolution equation for the endogenous state variable. This is because the value function, which is a¤ected by the long-run risk shocks, now appears in the consumption- sharing rule, di¤erent from the power utility case, as given by Equation (8). Given this, we can solve (numerically) for the value functions at time t. The rest of the backwards recursion follows the same process.

4.1 Numerical details

First, choose grids for the exogenous state variables, xt and ²_t. Examples of potential grids (no- tation: [lower limit, upper limit]number of grid points) are given below to give a sense of the number of total grid points required for the problem using log-linear interpolation between grid points. Of course, depending on the interpolation method used between grid points, functional approximation schemes, the distribution of the grid points, etc., the number of grid points required for a particular implementation can deviate signi…cantly from the below examples:

2grid = h

; + 4!=p

1 ²i

16grid points

x_grid =

"

2' r

+ 4!=p

1 ² = (1 ²_x); 2' r

+ 4!=p

1 ² = (1 ²_x)

#

16grid points

The terminal date T should be su¢ ciently far in the future so as not to matter for solution, e.g., T = 400+ years (4800+ quarters, since we calibrate at the monthly frequency). Note, though, the there is no iteration here, just a backwards recursion, so T being large is not all that costly in computation time –the size of T is instead more comparable to the number of iterations required for convergence in a …xed-point value function iteration problem. The endogenous grid for cA should have more grid points, especially close to 0 and 1:

cA_grid = [e ⁵; 0:99999]96grid points

Finally, the shocks ", , and can be approximated with a Gaussian quadrature using, say, 6 quadrature points for each shock, to facilitate easy numerical integration when taking expectations.

When solving the …xed point problem of …nding ktwe use the zbrent algorithm.⁶ The monotonic- ity and the fact that the …xed point was shown to be unique, means that …nding kT 1 to high levels of accuracy (which is important) is very fast. From then on, going backwards in time, we look for the …xed point solution for kt using guesses of kt starting close to the value of kt+1 at the same grid points for the state variables x; ² and cA. In practice, kt will be very close to kt+1 when these state variables are the same values, so …nding the …xed point is extremely quick. Note also that the problem is easily amenable to parallel processing using multiple cores or GPU cards.

Using these numerical parameters, one can solve for the two value functions as well as equilibrium quantities of interest in a day on a regular desktop computer. This requires using a programming language that executes for-loops fast, such as C, C++, or Fortan, as well as programming the code to use all cores in parallel (typical desktops have 4 cores, which means they have 8 threads that can be run in parallel, speeding up the code by a factor of (almost) 8 relative to using a single thread, which is the benchmark). With more e¢ cient grids and/or the use of GPU cards or more cores, the computing time can be reduced to less than 3 hours in our experience.

In sum, the numerical method yields accurate solutions quickly, even though the system is highly nonlinear, has three state variables with an endogenous state variable whose evolution equation must be found numerically, and multiple aggregate shocks.

4.2 Special case:

=

If the curvature of the certainty equivalent function is the same for both agents ( A = _B = ), the consumption sharing problem is signi…cantly simpli…ed. Following Equations (15) and (16), we

6See "Numerical Recipes in C."

now have at time T 1:

k_{T 1} = E_t((1 c_A;T) (C_T=C_{T 1}) ) ^{B= 1} E_t c_A;T (C_T=C_{T 1}) ^{A= 1}

; (29)

and

c_A;T¹

(1 c_A;T) ¹ = k_{T 1} ^B

A

c_A;T^{A 1}₁

(1 c_{A;T 1}) ^{B 1}; (30)

which jointly give the evolution equation for the endogenous state variable, cA;T = g_{T 1} x_{T 1}; ²_{T 1}; c_{A;T 1} . That is, the consumption sharing rule is locally deterministic. Of course, kT 1 depends on xT 1 and

2

T 1, as well as cA;T 1, so the consumption sharing rule is not deterministic more than one period in advance. Thus, we have an analytical expression for next period’s consumption given current state variables. This greatly simpli…es the …xed point problem of …nding kT 1. Upon …nding kT 1

we, as before, solve numerically for the value functions at time T 1.

At T 2, corresponding to Equations (21) and (22), we have:

c_A;t+1^{A 1} (1 c_A;t+1) ^{B 1}

v_A;t+1^A

v_B;t+1^B = kt B A

c_A;t^{A 1}

(1 c_A;t) ^{B 1}; (31)

and

k_t = E_t[(v_B(x_t+1; 1 c_A;t+1(k_t))) (C_t+1=C_t) ] ^{B= 1}

E_t[(v_A(x_t+1; c_A;t+1(k_t))) (C_t+1=C_t) ] ^{A= 1} : (32) which now give the evolution equation for the endogenous state variable from T 2to T 1. Letting t = T 2, we have cA;t+1 = g_t x_t; ²_t; c_A;t; _t+1; _t+1 . Note that, di¤erent from the general case, the shocks to realized consumption growth, "t+1, do not enter into the evolution equation for cA;t+1. The reason is the same as before. Since realized aggregate consumption never appears in the recursion for the consumption sharing rule (other than integrated out in kt) when A= _B, the normalized value functions, vA and vB, are not a¤ected by the shocks to realized aggregate consumption.

The fact that we have one less shock to integrate over when …nding the …xed point for kt speeds

up and simpli…es the numerical solution. Given kt and the evolution equation for cA;t+1, we can again solve (numerically) for the value functions at time t. The rest of the backwards recursion follows the same process.

4.3 Results: Heterogeneity in EIS and

We now turn to results from two calibrated models. The aggregate consumption dynamics are as given above (Equation (23) and there are two (sets of) agents that act competitively and that have di¤erent preference parameters. In both models, we let A = _B = . Thus, risk aversion over timeless gambles is identical in the two models. The dynamics that arise from di¤erent levels of risk aversion are well understood from previous work (e.g., Wang (1994)). Instead, we focus on heterogeneity in the elasticity of intertemporal substitution (EIS; governed by _A and _B), as well as in the time-discounting parameter ( _A and _B).

Table 1 gives the parameters of the exogenous dynamics, which are taken from Bansal and Yaron (2004; hereafter BY), as well as the preference parameters in the two models. The model is calibrated at the monthly frequency, as in BY. In both models we set = 9 (risk aversion equal to 10; as in BY). In ’Model EIS’we set _A = _B = 0:998 (as in BY), while we let _A = 0:5 and

B = 4 (corresponding to EIS’s of 2 and 0:2 respectively, compared to 1:5 in BY). In ’Model ’ we set _A = 0:999 and _B = 0:995 (compared to = 0:998 in BY), _A = _B = 1=3 (which implies an EIS of 1:5 as in BY).

4.3.1 Results for ’Model EIS’

We solve the model using the numerical procedure outlined earlier with T = 5000, which is su¢ cient to make the e¤ect of the terminal date vanish. As discussed earlier, the evolution equation of the endogenous state variable cAis a function of the current state variables and the long-run risk shocks

Table 1 - Parameter values

Table 1: The Table shows the preference parameters used in ’Model EIS’ and ’Model ,’ as well as the parameters for the process governing aggregate consumption and dividend growth. The numbers correspond to the monthly frequency of the model calibration.

Preference parameters: Model EIS Model

A (risk aversion parameter) 10 10

B (risk aversion parameter) 10 10

A (EIS parameter) 0:5 1=3

B (EIS parameter) 4 1=3

A (time discounting) 0:998 0:999

B (time discounting) 0:998 0:995

Parameters for consumption dynamics: Value (mean consumption growth rate) 0:0015

(persistence of xt) 0:979

' (volatility scaling of shocks to x) 0:044 v (mean of consumption growth variance) 6:084e-5

(persistence of ²_t) 0:987

! (volatility of variance shocks) 2:3e-6

(dividend leverage parameter) 3

div (dividend volatility scaling parameter) 4:5

to the mean and variance of consumption growth: cA;t+1 = g xt; ²_t; _t+1; t+1 . Since risk aversion is the same across the two agents, the shock to realized consumption is not relevant for risk-sharing.

In fact, with power utility (see Equation (8)) cA;t is constant over time in this case. However, since the continuation utility shows up in marginal utility when agents have Epstein-Zin utility, the agents will …nd it optimal to buy and sell insurance against these shocks. In particular, the agent who is less sensitive to long-run shocks should sell insurance to the agent that is more sensitive.

This sensitivity is governed by _i i, which clearly is higher for agent A who has the higher EIS.

Figure 1 shows the consumption sharing rule (cA;t+1) as a function of the shock to xt (left panel) and the shock to ²_t (right panel); _t+1 and t+1, respectively. The current value of the state

Figure 1 - Model EIS: The Consumption Sharing Rule

-3 -2 -1 0 1 2 3

0.475 0.48 0.485 0.49 0.495 0.5 0.505 0.51

cA,t+1

normalized shock to x t Consumption share of Agent A

-3 -2 -1 0 1 2 3

0.49 0.491 0.492 0.493 0.494 0.495 0.496 0.497 0.498 0.499

c A,t+1

normalized shock toσ² t Consumption share of Agent A

Figure 1: The plots show the response of the consumption share of agent A to a normalized shock to expected aggregate consumption growth or consumption growth volatility. The agents di¤er with respect to their elasticity of intertemporal substitution, with agent A having the higher EIS.

variables xt and ²_t are set to their unconditional means, while the current relative consumption of agent A, cA;t, is set to 0:4942. Since agent A has a higher EIS (2 versus 0:2 for agent B), this agent buys insurance against negative shocks to expected consumption growth and positive shocks to consumption growth variance. This can be seen as the consumption share of agent A increases in these cases. Thus, the relative wealth of the two agents ‡uctuates with agent A become relatively wealthier in ’bad’times. This has interesting implications for the conditional moments of the dividend claim, as we will see next.

Figure 2 shows the conditional, annualized risk premium and Sharpe ratio of the dividend claim versus the current relative consumption of agent A, cA;t, and either the annualized value of 12xt

(upper left panel) or consumption growth volatility, p

12 _t (upper right panel). The risk premium is strongly increasing in the consumption share of agent A— from negative when agent B dominates to high and positive when agent A dominates. Partly this is due to the fact that agent A is more sensitive to long-run risk shocks. Partly this is due to agent B having an EIS < 1, which implies

Figure 2 - Model EIS: Conditional Moments of the Dividend Claim

0

0.5 1 -0.1

0 0.1 -0.05

0 0.05

c_A,t Annualized risk premium on dividend claim

annualized value of x _t 0

0.5 1 0

0.02 0.04-0.1 -0.05 0 0.05 0.1

c_A,t Annualized risk premium on dividend claim

annualized value of σ_t

0

0.5 1 -0.1

0 0.1 -0.5 0 0.5

c_A,t Annualized Sharpe ratio of dividend claim

annualized value of x

t 0

0.5 1 0

0.02 0.04 -0.5 0 0.5

c_A,t Annualized Sharpe ratio of dividend claim

annualized value of σ_t

Figure 2: The plots show the conditional, annualied risk premium and Sharpe ratio of the dividend claim plotted against the relative consumption share of agent A and the annualied level of x_t or consumption growth volatility, _t. The agents di¤er with respect to their elasticity of intertemporal substitution, with agent A having the higher EIS.

that when this agent dominates, the wealth e¤ect also dominates in the economy. A positive shock to expected consumption growth now decreases the price-dividend ratio. When agent B dominates su¢ ciently, this e¤ect is strong enough to overcome the direct e¤ect of the short-run correlation between dividend and consumption. Thus, the return on the dividend claim is negative when the shock to expected consumption growth is positive, which yields a negative risk premium. A similar e¤ect occurs for shocks to volatility.

The lower panels show similar patterns for the Sharpe ratio of the dividend claim, which ranges from 0:5 to 0:5. In sum, heterogeneity in agents’ elasticity of intertemporal substitution has quantitatively signi…cant implications for conditional asset pricing moments of interest. This occurs

even though risk aversion is the same across the two agents and the e¤ect is associated with shocks to the value function, here from shocks to xt and ²_t, and not from shocks to realized aggregate consumption growth.

4.3.2 Results for ’Model ’

Next, we consider a model where both the risk aversion parameter and the elasticity of intertemporal substitution is the same across both agents, but where the time-preference parameter is di¤erent.

In particular, A = _B = 9, _A = _B = 1=3, _A = 0:999 and _B = 0:995. Thus, in this case, both the risk aversion and the degree of preference for early resolution of uncertainty are the same across agents. Nevertheless, long-run risk shocks still has a di¤erential e¤ect as the discounting of the shock in the continuation utility is higher for agent B. So, again, agent A is more sensitive to long-run risk shocks.

Figure 3 - Model : The Consumption Sharing Rule

-4 -2 0 2 4

0.493 0.4935 0.494 0.4945 0.495 0.4955 0.496 0.4965 0.497 0.4975 0.498

c A,t+1

normalized shock to x t Consumption share of Agent A

-4 -2 0 2 4

0.4945 0.495 0.4955 0.496 0.4965 0.497 0.4975

c A,t+1

normalized shock to σ² t Consumption share of Agent A

Figure 3: The plots show the response of the consumption share of agent A to a normalized shock to expected aggregate consumption growth or consumption growth volatility. The agents di¤er with respect to their rate of time preference ( ), with agent A having a higher .

Figures 3 and 4 shows the consumption sharing rule as a function of the long-run risk shocks, as

well as the conditional moments of the dividend claim, as in the discussion of ’Model EIS.’Again we see that agent B insures agent A against adverse long-run risk shocks and again the shock to realized consumption growth does not matter for risk-sharing.

In the case of di¤erent rate of time preference parameters, the risk premium does not go negative when agent B dominates as also for this agent the substitution e¤ect dominates given her EIS > 1.

Still for what seems like a relatively small di¤erence in the rate of time preference, the variations in the risk premium and the Sharpe ratio are quantitatively large. Thus, with long-run risk shocks preference in both the time-discounting parameter and the elasticity of intertemporal substitution can lead to large ‡uctuations in conditional asset pricing moments. In all cases, the economy tends to display a higher conditional risk premium and Sharpe ratio in bad times due to the endogenous consumption sharing arrangement.

5 Risk-sharing in a production economy

In a production economy setting there is an additional complication due to the fact that aggregate consumption is endogenous. Here we show how the numerical solution method is applied to this case. Here, we focus on the case of di¤erences in beliefs as opposed to heterogeneity in preference parameters. The main di¤erence in this case is that the …rm’s …rst order conditional must also be brought into the solution.

Again there are two agents (A and B) with Epstein-Zin preferences and di¤erent beliefs about the exogenous dynamics of aggregate technology, Zt. There is a representative …rm with production function:

Y_t = (Z_tN_t)¹ K_t;

where hours worked is set constant (Nt = 1), the capital share is , and capital is Kt. Aggregate

Figure 4 - Model : Conditional Moments of the Dividend Claim

0

0.5 1

-0.1 0 0.1 0.025 0.03 0.035 0.04 0.045

cA,t Annualized risk premium on dividend claim

annualized value of x

t 0

0.5 1

0 0.02 0.04 -0.05 0 0.05 0.1 0.15

cA,t Annualized risk premium on dividend claim

annualized value of σ_t

0

0.5 1

-0.1 0 0.1 0.2 0.25 0.3

cA,t Annualized Sharpe ratio of dividend claim

annualized value of x

t 0

0.5 1

0 0.02 0.04 -0.2 0 0.2 0.4 0.6

cA,t Annualized Sharpe ratio of dividend claim

annualized value of σ_t

Figure 4: The plots show the conditional, annualied risk premium and Sharpe ratio of the dividend claim plotted against the relative consumption share of agent A and the annualied level of x_t or consumption growth volatility, t. The agents di¤er with respect to their rate of time preference ( ), with agent A having a higher .

consumption equals aggregate output minus aggregate gross investment:

C_t = C_A;t+ C_B;t = Y_t I_t.

The aggregate capital accumulation equation is:

K_t+1 = (1 ) K_t+ (I_t=K_t) K_t;

where capital adjustment costs are implicitly given by the concave function that governs the