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

Transcription

1 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, 25 Abstract We provide a robust and accurate numerical solution methodology for discrete-time, complete markets, general equilibrium risk-sharing problems when agents have recursive preferences. The agents can di er both in terms of preference parameters and beliefs. The methodology can easily handle multiple aggregate shocks and state variables. To illustrate the methodology, we consider an economy where aggregate consumption dynamics follow those in Bansal and Yaron (24) and there are two Epstein-Zin agents with di erent preference parameters. Any errors or omissions are our own. Contact info: Lars A. Lochstoer, 45B Uris Hall, Columbia Business School, Columbia University, 322 Broadway, New York, NY LL269@columbia.edu. First draft: November 23.

2 Introduction Recursive preferences are now commonplace in asset pricing and macroeconomic studies (e.g., Bansal and Yaron (24), Hansen and Sargent (28), Backus, Routledge, and Zin (29), Barro (2), Routledge and Zin (23), Gourio (2)). 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 parameters 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 (22) allow for di erent preference parameters across two sets of agents, while Borovicka (23) 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 continuoustime 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 2

3 beliefs. The solution method is accurate and it does not rely on approximations to the true problem (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 complexity 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 function 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 (28)), (generalized) disappointment aversion (Gul (99), Routledge and Zin (23)), and smooth ambiguity aversion (Klibano, Marinacci, and Mukerji (25, 29)). 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 ( + x) around x = is a classic example where the 3

4 Taylor approximation (of any order) is good locally, but not for values of x further away from x (e.g., x > ). 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 erent preference parameters when aggregate consumption dynamics follows the speci cation in the Bansal and Yaron (24). 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 countercyclical 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. 4

5 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. While the method we will explain below conceptually extends to N agents, the wealth distribution in general will be characterized by N 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 C t = C (x t ), and let x t be a vector of exogenous observable Markov state variables with evolution x t = h (x t ; " t ) where " t is a vector of shocks. The aggregate resource constraint is: C t = C A;t + C B;t. () The preferences of agents A and B are given by: V i;t = C i i;t + i i i;t =i ; (2) where i ; i ; i <, i >, i;t E i t V i i;t+ =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 These 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. 5

6 beliefs are such that equilibrium exists and that if beliefs are updated, the updating equations are known. The cum-consumption wealth levels are W i;t = V i;t (C i;t =V i;t ) = i, respectively (see Epstein and Zin (989)). Since markets are assumed to be complete, the budget constraint can be written W i;t+ = (W i;t C i;t ) R Wi ;t+, where R Wi ;t+ 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 Z t we have that z t = Z t =C t. In this case, the value functions can be written: v i;t = c i i;t + i i i;t =i ; (3) where i;t E i t v i i;t+ (C t+=c t ) i = i and where the resource constraint is ca;t + c B;t =. The stochastic discount factor under agent i s probability measure can then be written (see Epstein and Zin (989)): E i t M i t+ R j t+ = for all t and j i Mt+ i ci;t+ = i c i;t i Ct+ C t vi;t+ i;t i i : (4) The state variables in this economy are x t 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 X t [x t (T t)] 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, c A;t. Thus, v i;t = f i (X t ; c A;t ). If the endogenous evolution equation of c A;t is known, we can now easily solve a standard value 6

7 function iteration problem on a grid for x t and c A;t 2 (; ): f A (X t ; c A;t ) = f B (X t ; c A;t ) = h c A A;t + A Et A [f A (X t+ ; c A;t+ ) A (C t+ =C t ) A ] A Ai = =A ; (5) h ( c A;t ) B + B Et B [f B (X t+ ; c A;t+ ) B (C t+ =C t ) B ] B Bi = =B : (6) Thus, the crux of the risk-sharing problem is nding the endogenous evolution equation for c A for all points in the state-space. 2. 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 c A 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 B A CA;t+j t (! t+j ) A = B CB;t+j t (! t+j ) C B A;t C B;t for all t and! t+j : (7) Applying the normalization and noting that current state variables are x t and c A;t Equation (7) yields: ~c A;t+ = B t (! t+j ) B A t (! t+ ) A Ct+ C t B A ~c A;t: (8) where ~c A;t c A A;t ( c A;t ) B. Since A ; B < and c A 2 (; ), ~c A is monotonically decreasing in c A. Thus, ~c A uniquely identi es c A. 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 c A given ~c A, but one can do this extremely fast and to arbitrary accuracy. Thus, it is in this case easy to implement the 7

8 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, x t. To see this, consider the complete markets requirement that the IMRS is equalized across states: A A ca;t+ t (! t+j ) A c A;t A Ct+ C t E A t v A;t+ v A A;t+ (C t+=c t ) A = A! A A = B ::: B ca;t+ t (! t+j ) B c A;t B Ct+ C t E B t v B;t+ v B B;t+ (C t+=c t ) B = B! B B : (9) Rearranging, we have: ~c A;t+ v A A A;t+ v B B B;t+ = B t (! t+j ) B A t (! t+j ) A Ct+ C t B A k t ~c A;t; () where again ~c A;t c A A;t ( c A;t ) B and, new to the general case: k t EB t E A t v B B;t+ (C t+=c t ) B B = B v A A;t+ (C t+=c t ) : () A A = A Unfortunately, since the value functions are what we are trying to solve for, v A;t+, v B;t+ 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, c A;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, 8

9 as well as an initial guess of the evolution equation of c A. 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 () to nd the evolution equation of the endogenous state variable, c A. With this in hand, we can trivially solve for the value function at time T, 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. 2 Guvenen (29) 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. 9

10 At time T, when the economy ends, the value functions reduce to: v A;T = c A;T ; (2) v B;T = c A;T : (3) Equations (2) and (3) give the boundary conditions for the value functions as a function of the relative consumption of agent A, c A : As mentioned earlier, it is convenient to use c A;t as the endogenous state-variable (one could equivalently use relative wealth of, say, agent A), in addition to the exogenous state variables, x t : Thus, the value function for agent i can be written v i;t = f i (x t ; c A;t ) : At time T, the complete markets requirement that agents IMRS are equalized across states implies that: A T jt A ca;t A c A;T CT C T A A v A A;T = ::: A;T (v A;T C T =C T ) B B ca;t T jt B c A;T CT C T B B v B B;T : (4) B;T (v B;T C T =C T ) Here i T jt at time T denotes the probability agent i assigns to a given state at time T given agent i s beliefs. First, de ne: k T = B;T (v B;T C T =C T ) B B A;T (v A;T C T =C T ) A A = EB t (( c A;T ) B (C T =C T ) B ) B = B E A t c A A;T (C T =C T ) A A = A : (5) Next, imposing the boundary values as given in Equations (2) and (3), we have that: A T jt A ca;t A c A;T A CT c A C T A A;T = :::

11 B k T B ca;t T jt B c A;T CT C T B ( c A;T ) B B c A A;T ( c A;T ) B = k T B T jt B A T jt m c A A;T A ( c A;T ) B CT C T B A : (6) Note rst that Equation (6) implies that, for a given state of the world at time T and value of state variables at time T, c A;T 2 (; ) is decreasing in k T (since i < ). Thus, for a given k T, Equation (6) uniquely determines c A;T for each state of the world at time T. Of course, from Equation (5), we only know k T as a function of c A;T. This suggests solving for k T for each value of the state variables at time T by nding a joint solution to Equations (5) and (6). The question then is whether the solution one obtains is unique. Taking a derivative of the numerator of Equation (5) with respect to c A;T we have: B t (( c A;T ) B (C T =C T ) B ) B = A;T = ::: ( B = B ) Et B (( c A;T ) B (C T =C T ) B ) B = B 2 B ( ) Et B ( c A;T ) B (C T =C T ) B : (7) Since the expectations terms are all positive, we can determine the sign this derivative in terms of the preference parameters:! t (( c A;T ) B (C T =C T ) B ) B = A;T = sign (( B = B ) B ( )) = sign ( ( B B )) : (8) 3 Think of this as increasing c A;T by a particular tiny amount in each state.

12 Similarly, we can sign the derivative of the denominator of Equation (5) with respect to c A;T : t c A A;T (C T =C T ) A A = A;T = sign (( A = A ) A ) = sign ( A A ) : (9) A su cient condition for k T to be monotone in c A;T is that the derivatives of the numerator and denominator of Equation (5) with respect to c A;T have di erent signs. 4 In these cases Equations (5) and (6) provide unique solutions for k T, as well as for c A;T for each state of the world at time T. It is also immediate from these equations that a solution exists where c A;T 2 (; ) and k T >. While the xed point problem for nding k T implicit in Equations (5) and (6) 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 c A;T as a function of k T and the state variables at T. In sum, using c A;T as the endogenous state variable, we now have numerically the evolution equation for c A, from T to time T. Next, we can now solve numerically for the normalized value functions at time T on a grid for the relevant state variables at time T, using: v i;t = h c i i;t + iet i v i i;t+ (C t+=c t ) i i i = =i i ; (2) where the state variables are x t and c A;t. Note again that solving numerically for the certainty equivalent of next period s value function requires the use of the evolution equation for c A. The second backwards iteration is then at time t = T 2. Again, we start with the requirement 4 Note 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. 2

13 that the IMRS is equalized for each state for the two agents: A A ca;t+ t+jt A c A;t A Ct+ C t A v A A;t+ = ::: A;t (v A;t+ C t+ =C t ) B ca;t+ B t+jt B c A;t B Ct+ C t B v B B;t+ : B;t (v B;t+ C t+ =C t ) Note that v i;t+ = f i (X t+ ; c A;t+ ) is known from the previous step in the backwards recursion. Now, rewrite the above equation as: c A A;t+ ( c A;t+ ) B v A A A;t+ v B B B;t+ B t+jt = k B c A A;t t A t+jt A ( c A;t ) B Ct+ C t B A ; (2) where similar to the T case, k t = B;t(v B;t+ C t+ =C t) B B A;t (v A;t+ C t+ =C t) A A. Note that v A;t (v B;t ) is increasing (decreasing) in c A;t. A su cient condition for the left hand side of Equation (2) to be monotone in k t is that both agents weakly prefer early resolution of uncertainty, i.e. i i. Since v A;t+ v B;t+ is known as a function of x t+ and c A;t+, it is easy to numerically nd the value of c A;t+ corresponding to a particular outcome (x t ; " t+ ), given a value for k t. Finally, we need to solve for k t. Note that the previous equation gives the evolution equation for c A;t+ as a function also of k t (c A;t+ = g (X t ; c A;t ; k t ; " t+ )). We again nd k t as a xed point of the equation: k t = EB t [(v B;t+ (x t+ ; c A;t+ (k t ))) B (C t+ =C t ) B ] B = B E A t [(v A;t+ (x t+ ; c A;t+ (k t ))) A (C t+ =C t ) A ] A = A : (22) Thus, a su cient condition for Equations (2) and (22) to provide a unique solution to the evolution equation for c A;t to c A;t+ as a function of the aggregate exogenous state variables at x t and x t+ is that both agents weakly prefer early resolution of uncertainty. The normalized value function at time t can then be found using Equation (2). Further backwards recursions follow the same algorithm as that given for the case t = T 2. In practice, we nd that having T > 2 is su cient for typical calibrations (clearly, the time- 3

14 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 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 preferences in an economy where aggregate consumption are speci ed as in Bansal and Yaron (24): c t+ = + x t + t " t+ ; x t+ = x x t + ' t t+ ; 2 t+ = max + 2 t +! t+ ; : (23) Here. all the shocks are i.i.d., uncorrelated standard normal and > 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 (24). 5 The market is de ned as a claim to aggregate dividends, which have log growth rates: d t+ = c t+ + div p vt+ ; (24) 5 One could alternatively choose a, say, mean-reverting process for log variance in order to ensure positive variance. However, we chose the truncated AR() process in order to keep the analysis as close as possible to that in Bansal and Yaron (24). 4

15 where t+ 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 v A;t = f A (x t ; 2 t ; c A;t ) and v B;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 (2) and (3). Following Equations (5) and (6), we then have at time T : k T = E t (( c A;T ) B (C T =C T ) B ) B = B E t c A A;T (C T =C T ) A A = A ; (25) and c A A;T ( c A;T ) B = k B c A A;T T A ( c A;T ) B CT C T B A ; (26) which jointly give the evolution equation for the endogenous state variable, c A;T = g T x T ; 2 T ; c A;T ; " T. Given this, we can solve (numerically) for the value functions at time T. At T 2, corresponding to Equations (2) and (22), we have: c A A;t+ ( c A;t+ ) B v A A A;t+ v B B B;t+ = k B c A A;t t A ( c A;t ) B Ct+ C t B A ; (27) and k t = E t [(v B (x t+ ; c A;t+ (k t ))) B (C t+ =C t ) B ] B = B E t [(v A (x t+ ; c A;t+ (k t ))) A (C t+ =C t ) A ] A = A : (28) which now give the evolution equation for the endogenous state variable from T 2 to T. Letting t = T 2, we have c A;t+ = g t x t ; 2 t ; c A;t ; " t+ ; t+ ; t+. Note that in this case the shocks t+ 5

16 and t+ 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 consumptionsharing 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. Numerical details First, choose grids for the exogenous state variables, x t and 2 t. Examples of potential grids (notation: [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: 2 grid = x_grid = h ; + 4!= p i 2 " 2'r + 4!= p 2 6 grid points = ( 2 x); 2'r + 4!= p # 2 = ( 2 x) 6 grid points The terminal date T should be su ciently far in the future so as not to matter for solution, e.g., T = 4+ years (48+ 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 c A should have more grid points, especially close to and : c A _grid = [e 5 ; :99999] 96 grid points 6

17 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 k t we use the zbrent algorithm. 6 The monotonicity and the fact that the xed point was shown to be unique, means that nding k T 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 k t using guesses of k t starting close to the value of k t+ at the same grid points for the state variables x; 2 and c A. In practice, k t will be very close to k t+ 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: A = B 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 (5) and (6), we 6 See "Numerical Recipes in C." 7

18 now have at time T : k T = E t (( c A;T ) (C T =C T ) ) B = E t c A;T (C T =C T ) A = ; (29) and c A;T ( c A;T ) = k B c A A;T T A ( c A;T ) B ; (3) which jointly give the evolution equation for the endogenous state variable, c A;T = g T x T ; 2 T ; c A;T. That is, the consumption sharing rule is locally deterministic. Of course, k T depends on x T and 2 T, as well as c A;T, 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 k T. Upon nding k T we, as before, solve numerically for the value functions at time T. At T 2, corresponding to Equations (2) and (22), we have: c A A;t+ ( c A;t+ ) B v A A;t+ v B B;t+ = k B c A A;t t A ( c A;t ) B ; (3) and k t = E t [(v B (x t+ ; c A;t+ (k t ))) (C t+ =C t ) ] B = E t [(v A (x t+ ; c A;t+ (k t ))) (C t+ =C t ) ] A = : (32) which now give the evolution equation for the endogenous state variable from T 2 to T. Letting t = T 2, we have c A;t+ = g t x t ; 2 t ; c A;t ; t+ ; t+. Note that, di erent from the general case, the shocks to realized consumption growth, " t+, do not enter into the evolution equation for c A;t+. 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 k t ) when A = B, the normalized value functions, v A and v B, 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 k t speeds 8

19 up and simpli es the numerical solution. Given k t and the evolution equation for c A;t+, 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 (994)). 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 gives the parameters of the exogenous dynamics, which are taken from Bansal and Yaron (24; 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 ; as in BY). In Model EIS we set A = B = :998 (as in BY), while we let A = :5 and B = 4 (corresponding to EIS s of 2 and :2 respectively, compared to :5 in BY). In Model we set A = :999 and B = :995 (compared to = :998 in BY), A = B = =3 (which implies an EIS of :5 as in BY) Results for Model EIS We solve the model using the numerical procedure outlined earlier with T = 5, 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 c A is a function of the current state variables and the long-run risk shocks 9

20 Table - Parameter values Table : 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) B (risk aversion parameter) A (EIS parameter) :5 =3 B (EIS parameter) 4 =3 A (time discounting) :998 :999 B (time discounting) :998 :995 Parameters for consumption dynamics: Value (mean consumption growth rate) :5 (persistence of x t ) :979 ' (volatility scaling of shocks to x) :44 v (mean of consumption growth variance) 6:84e-5 (persistence of 2 t ) :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: c A;t+ = g x t ; 2 t ; t+ ; t+. 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)) c A;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 shows the consumption sharing rule (c A;t+ ) as a function of the shock to x t (left panel) and the shock to 2 t (right panel); t+ and t+, respectively. The current value of the state 2

21 Figure - Model EIS: The Consumption Sharing Rule.5 Consumption share of Agent A.499 Consumption share of Agent A c A,t c A,t normalized shock to x t normalized shock to σ 2 t Figure : 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 x t and 2 t are set to their unconditional means, while the current relative consumption of agent A, c A;t, is set to :4942. Since agent A has a higher EIS (2 versus :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, c A;t, and either the annualized value of 2x t (upper left panel) or consumption growth volatility, p 2 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 <, which implies 2

22 Figure 2 - Model EIS: Conditional Moments of the Dividend Claim Annualized risk premium on dividend claim Annualized risk premium on dividend claim annualized value of x t. c A,t.5..4 annualized value of.2 σ t c A,t.5 Annualized Sharpe ratio of dividend claim Annualized Sharpe ratio of dividend claim annualized value of x t. c A,t annualized value of.2 σ t c A,t.5 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 :5 to :5. In sum, heterogeneity in agents elasticity of intertemporal substitution has quantitatively signi cant implications for conditional asset pricing moments of interest. This occurs 22

23 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 x t and 2 t, and not from shocks to realized aggregate consumption growth 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 = =3, A = :999 and B = :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.498 Consumption share of Agent A.4975 Consumption share of Agent A c A,t c A,t normalized shock to x t normalized shock to σ 2 t 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 23

24 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 >. 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, Z t. There is a representative rm with production function: Y t = (Z t N t ) Kt ; where hours worked is set constant (N t = ), the capital share is, and capital is K t. Aggregate 24

25 Figure 4 - Model : Conditional Moments of the Dividend Claim Annualized risk premium on dividend claim Annualized risk premium on dividend claim annualized value of x t. c A,t annualized value of σ t c A,t Annualized Sharpe ratio of dividend claim Annualized Sharpe ratio of dividend claim annualized value of x.2.5 t annualized value of σ c t. A,t c A,t.5 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+ = ( ) K t + (I t =K t ) K t ; where capital adjustment costs are implicitly given by the concave function that governs the 25

26 transformation of investment to capital: (I t =K t ) = I t K t 2 a It ; (33) 2 K t where a >. Thus, there are standard quadratic adjustment costs to capital, and () is a concave function with () = a (I t =K t ). 7 The preferences of agents A and B are given by: h i V A;t = V A (C A;t ; V A;t+ ) = ( ) C A;t + EA t VA;t+ = = ; h i V B;t = V B (C B;t ; V B;t+ ) = ( ) C B;t + EB t VB;t+ = = : Let X t denote a vector of exogenous state variables governing beliefs and aggregate technology dynamics. There will be two endogenous state variables, aggregate capital stock, K t, and a state variable related to the relative wealth of the two agents. We let the latter be the relative consumption of agent A, c A;t C A;t =C t. 5. Numerical Method: Backwards Recursion Here, we instead assume that the economy ends at time T. For the below, just consider two agents (A and B) that have di erent beliefs about aggregate consumption dynamics. Both agents beliefs are summarized in a state vector X t whose evolution is analytically known and exogenous to the consumption sharing rule (i.e., agents update from observing exogenous aggregate consumption growth). Notation is otherwise as before (i.e., c i = C i =C where C denotes the exogenous aggregate consumption). In the nal period, all output is consumed and all capital will be sold. It is convenient to apply 7 One could let a be a function of investment (e.g., higher adjustment costs for disinvestments, I < ), or in general consider more exible speci cations of (), but we abstract from this for now. 26

27 the following normalizations: c i;t C i;t C t ; v i;t V i;t Z t, y t Y t Z t, i i I t Z t, k t K t Z t : (34) Again, we use a backwards recursion algorithm where the nal period is denoted T. We start with the requirement that agents IMRS are equalized across states: A T jt CA;T C A;T V A;T = ::: (35) A;T (V A;T ) A T jt = A T jt ca;t c A;T ca;t c A;T = B ca;t T jt c A;T ct c T ct c T ct c T ZT Z T ZT Z T ZT Z T v A;T Z T =Z T A;T (v A;T Z T =Z T ) v A;T A;T (v A;T Z T =Z T ) v B;T ; B;T (v B;T Z T =Z T ) which gives the evolution equation: ca;t c A;T va;t v B;T = B T jt A T jt ca;t c A;T A;T (v A;T Z T =Z T ) : (36) B;T (v B;T Z T =Z T ) Since at time T : v i;t = (( ) (c i;t c T ) ) = = ( ) = c i;t c T ; (37) 27

28 we have the evolution equation: ca;t ( ) = c A;T c T! = ::: (38) c A;T ( ) = ( c A;T ) c T ca;t c A;T m B T jt ca;t A T jt = B T jt m c A;T c A;T = A T jt B T jt A T jt c A;T ca;t c A;T E B T E A T ( ) = (c A;T c T Z T =Z T ) = C ( ) = (( c A;T ) c T Z T =Z T ) = A E A T ((c A;T c T Z T =Z T ) ) = E B T ((( c A;T ) c T Z T =Z T ) ) =! (39)! =( ) ca;t t ; (4) c A;T where: t = E A T [(c A;T c T Z T =Z T ) = ] ET B [(( c A;T ) c T Z T =Z T ) : (4) ] The stochastic discount factor under agent A s probability measure is: M A;t+ = ca;t+ c A;t ct+ c t Zt+ Z t v A;t+ (42) A;t (v A;t+ Z t+ =Z t ) The time T sdf is therefore: M A;T = = ca;t c A;T ca;t c A;T ct c T ct c T ZT Z T ZT Z A;T ( ) = c A;T c T A ( ) = c A;T c T Z T =Z T c A;T E A T [(c A;T (c T =c T ) Z T =Z T ) ] =! (43) 28

29 The value function is then: v i;t = ( ) c i;t c t + Et i v i;t+ (Z t+ =Z t ) = = ; (44) which in the next to nal period becomes: v i;t = ( ) c i;t ct + ( ) EiT [(c i;t c T ) (Z T =Z T ) ] = = (45) By the aggregate resource constraint, C T = Y T I T. In the nal period, it is optimal to transform all remaining capital to consumption goods. Thus, I T is given by the condition: K T = (I T =K T ) K T (46) = I T + a 2K T I 2 T (47) which implies that I T K T = p + 2a : (48) a To verify that this is reasonable, take limit as a goes to zero: lim a > + p + 2a a = lim a > + ( + 2a) =2 = ; (49) which is the correct solution for the investment rate I T =K T when there are no capital adjustment costs. 29

30 Further, C T = Y T I T = Z T (K T ( ) + (I T =K T ) K T ) ::: p + 2a (K T ( ) + (I T =K T ) K T ) (5) a. Thus, C T =C T = f (Z T ; I T ; ca T ; Z T ; K T ; c A;T ; X T ), where I T is the endogenous choice variable a decision made by the rm to maximize rm value. The rm s problem with one period left is is to maximize unlevered rm value (present value of free cash ows): max VT F I T = max I T Y T I T + E T [M T (Y T I T )] ; (5) where Y T = Z T (( ) K T + (I T =K T ) K T ) : (52) Substituting the above in, the objective function can be written: 2 6 B max I T + E T 4M Z T (( ) K T + (I T =K T ) K T ) ::: I T (( ) K T + (I T =K T ) K T ) p +2a a 3 C7 A5 : (53) The rst order condition of rm investment is: E T M T (I T =K T ) (K T =Z T ) p + 2a = : (54) a E i T M i T R I;T = ; (55) 3

31 where the return to investment is: p RT I = (I T =K T ) (K T =Z T ) + 2a (56) a p = ( a (I T =K T )) (K T =Z T ) + 2a : (57) a Given the quadratic adjustment costs, at su ciently high levels of investment a marginal dollar invested leads to less future capital. Thus, in equilibrium, = ai t =K t >, which implies that I t =K t < =a. Further, the lowest admissible investment rate in the periods before the nal period is determined by requiring positive capital stock next period (as will be enforced in equilibrium). Thus: < ( ) K t + (I t =K t ) K t =) I t K t > p + 2a ( ) : (58) a These bounds are useful when nding optimal investment numerically. The solution method is as follows: First, guess i on grid, which is well-de ned. Find kt (which is a monotone xed point problem given i), nd i using rm s FOC given the kt. Compare with actual i. Do this for all i on admissible investment grid. Numerical solution methodology at T : From earlier, the evolution equation for the relative consumption of agent A is now: ca;t c A;T va;t v B;T = B T jt A T jt ca;t c A;T A;T (v A;T Z T =Z T ) : B;T (v B;T Z T =Z T ) Further, the rm s FOC yields: E A T 2 M A T R I;T = ; (59) 3

32 where R I;t+ = (I t =K t ) (K t+ =Z t+ ) + + (I t+=k t+ ) (I t+ =K t+ ) I t+ K t+ : (6) Again, we make sure that the rm s rst order condition is satis ed in a manner similar to the above. The updating of the value functions and the continuing recursion backwards in time are now discussed in previous sections. For an application of this solution methodology to a production economy setting, see Collin-Dufresne, Johannes, and Lochstoer (23). 6 Conclusion We provide a robust and accurate numerical solution methodology for discrete-time, general equilibrium risk-sharing problems when agents have recursive preferences. The agents can di er both in terms of preference parameters and beliefs. The methodology relies on a backwards recursion algorithm that ensures next period s value function and the evolution equation for the endogenous state variable is known. We illustrate the potentially large e ects of heterogeneity in the elasticity of intertemporal substitution and the time-discounting parameter, as opposed to heterogeneity in the level of risk aversion, for conditional asset pricing moments, as well as the applicability of our solution methodology to a production economy setting with di erences in beliefs. In future work we aim to compare the results of our accurate solution techniques to common perturbation methods used in the literature. 32

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