Solving the incomplete markets model with aggregate uncertainty using the Krusell–Smith algorithm
Introduction
This paper studies the properties of the solution to the incomplete markets model with aggregate uncertainty in Den Haan et al. (2009). Our solution method consists of two interconnected steps: the first is to solve the individual problem for a given aggregate behavior of the economy and the second is to compute the aggregate law of motion for the given individual policy rules. We iterate on these two steps until we find a fixed point at which the individual and aggregate policy rules are mutually consistent.
Step one is straightforward: the individual problem is the typical capital-accumulation problem with an occasionally binding borrowing constraint, and it can be solved by the standard numerical methods. We solve the individual problem by using a grid-based Euler-equation algorithm similar to that in Maliar and Maliar, 2005, Maliar and Maliar, 2006. We extend Maliar and Maliar, 2005, Maliar and Maliar, 2006 algorithm by incorporating a simple polynomial rule for constructing the grid, which allows us to vary the concentration of capital grid points on different parts of the domain, thus increasing the accuracy of approximation on non-linear parts of the policy rules. Our algorithm is also similar to the grid-based Euler-equation method used by Baxter et al. (1990) for solving the standard one-sector growth model. Furthermore, our algorithm is related to the parameterized expectations algorithm used in Den Haan and Marcet (1990), Den Haan (1997), Christiano and Fisher (2000), Maliar and Maliar (2003b), and Algan et al. (2008). However, the above papers parameterize an expectation term in the Euler equation and use a polynomial approximation, whereas we parameterize a capital function and compute a solution on a grid of pre-specified points.1
Step two is non-trivial. Decisions of each heterogeneous agent depend on the interest rate and wage rate, which in turn depend on the aggregate capital stock. Since the aggregate capital stock is determined by capital holdings of all heterogeneous agents, the whole capital distribution becomes a state variable.2 With a continuum of agents, this distribution is a function, and therefore, it cannot be used as an argument of the individual policy rules. To deal with this problem, Krusell and Smith (1998) propose to summarize the capital distribution by a discrete and finite set of moments.3 They solve the individual problem by using value iteration, and they compute the aggregate law of motion by simulating a panel for a large finite number of agents and by running regressions on the simulated data. In this paper, we follow the stochastic-simulation approach of Krusell and Smith (1998). Consequently, our solution procedure is a variant of the Krusell–Smith algorithm, specifically one in which the individual problem is solved by an Euler-equation method instead of Krusell and Smith's (1998) value function iteration. Our computer programs are written in MATLAB in an instructive manner and are provided on the JEDC web site (see the web pages of the authors for updated versions of the program).
An important advantage of the stochastic-simulation Krusell–Smith algorithm is that it is simple, intuitive and easy to program. As Algan et al. (2008) show, however, stochastic-simulation methods have two potential shortcomings. First, the introduction of stochastic simulations produces sampling noise, which makes the policy rules to depend on a specific random draw. Second, the simulated endogenous data are clustered around the mean, which implies that the accuracy of the approximation on the tails is low. They argue that replacing a stochastic simulation with a non-stochastic one can enhance the accuracy and speed of the algorithm. Therefore, it is of interest to assess the accuracy of the stochastic-simulation version of the Krusell–Smith algorithm and to compare it with a non-stochastic-simulation version.
We find that, despite the above shortcomings, the stochastic-simulation Krusell–Smith method produces sufficiently accurate solutions.4 This is true even under our relatively small panel of 10,000 agents and relatively short simulation length of 1,100 periods. For example, in an accuracy test where the model was simulated on a random realization of shocks of 10,000 periods, the average and maximum errors in our aggregate capital series were 0.050% and 0.156%, respectively. Furthermore, we consider a non-stochastic-simulation Krusell–Smith algorithm where simulations are performed on a grid of pre-specified points, as is described in the appendix in Den Haan (2009).5 We find that the benchmark stochastic-simulation version of the Krusell–Smith algorithm with a panel of 10,000 agents has approximately the same cost as the non-stochastic-simulation version with a grid of 1,000 points and produces solutions of comparable (or even higher) accuracy. Thus, in our case, the introduction of non-stochastic simulation does not lead to substantial improvements.
Section snippets
The individual problem
In this section, we describe an Euler-equation algorithm for finding a solution to the individual problem described in Den Haan et al. (2009). This is the standard capital-accumulation problem with an occasionally binding borrowing constraint. The Euler equation, the budget constraint, the borrowing constraint and the Kuhn–Tucker conditions, respectively, arewhere variables without and with primes refer to the current and
The stochastic-simulation algorithm
In this section, we discuss a version of the stochastic-simulation Krusell–Smith algorithm for solving the model with aggregate uncertainty. We parameterize the aggregate law of motion (ALM) for a set of moments of the capital distribution, m, by the following flexible functional form:where b is a vector of the ALM coefficients. Subsequently, we compute b by using the following iterative procedure. Algorithm 1 stochastic simulation Step I. Fix an initial vector of coefficients b. Generate and fix time series of
Stochastic versus non-stochastic simulation
In this section, we compare the stochastic- and non-stochastic-simulation versions of the Krusell–Smith method. To this purpose, we replace the procedure for simulating a panel of agents in our benchmark Krusell–Smith algorithm with a procedure for simulating the evolution of capital distribution on a grid of pre-specified points, as described in the appendix of Den Haan (2009). We outline the non-stochastic-simulation method below. Algorithm 2 non-stochastic simulation Step I. Fix an initial vector of coefficients b. Generate and
Acknowledgments
We thank editors Kenneth Judd, Michel Juillard and particularly, Wouter Den Haan for many useful comments and suggestions. This research was supported by the Center for Financial Studies in Frankfurt, the Stanford Institute for Theoretical Economics, the Paris School of Economics, the Hoover Institution at Stanford University, the Instituto Valenciano de Investigaciones Económicas and the Ministerio de Educación y Ciencia de España under the Grant SEJ 2007-62656.
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