Computing the Cross-Sectional Distribution to Approximate Stationary Markov Equilibria with Heterogeneous Agents and Incomplete Markets

Dynamic stochastic general equilibrium models with heterogeneous agents and incomplete markets usually have to be solved numerically. Existing algorithms assume bounded rationality of the agents meaning that they only keep track of a limited number of moments of the cross-sectional distribution. In this paper, we do not take this assumption and derive a law of motion of the whole state distribution. The proposed algorithm jointly solves for a fixed point of the equilibrium's Euler equation and its distribution's law of motion. Convergence to the equilibrium's optimal policy and the stationary state distribution is shown. We compare our numerical solution to the most prominent existing algorithm, the Krusell-Smith algorithm. Apart from improved error distributions and Pareto-improved policies for our algorithm, we show that the Krusell-Smith algorithm does not converge when using the limit of its simulation step to obtain the law of motion. This paper presents a computational method for solving DSGE models with heterogeneous agents and incomplete markets mainly based on projection methods. One minor part of the algorithm relies on perturbation. Importantly, I show that this algorithm is a modification of the proximal point algorithm for which convergence results are well established. In this paper, I develop a novel solution algorithm to solve a wide group of DSGE models with heterogeneous agents and incomplete markets. There are three major differences to the existing algorithms, most prominently the \cite{KS1998} algorithm. Firstly, this algorithm does not require bounded rationality of the agents and hence, it does not rely on an additional model assumption. Instead I solve for the original DGMM-recursive equilibrium where agents observe the full cross-sectional distribution. Economically, this leads to higher levels of consumption far any agent and is therefore Pareto-improving. However, even though this effect significantly changes the wealth distribution, it does not change the shape of the Lorenz curve towards a more realistic one. Because I do not abstract from the cross-sectional distribution, the whole state distribution is an integral part of my algorithm. I derive the theoretical law-of-motion operator for the stationary state distribution. The main difference to the existing algorithms' law of motions is that this operator looks at the joint distribution of aggregate and idiosyncratic exogenous shocks as well as endogenous variables whereas existing algorithms condition on simulated data of aggregate exogenous shocks or use perturbations around the aggregate shocks. The third difference is crucial for convergence. Our algorithm solves for the optimal policy and the stationary state distribution which together form a coupled system. This system is solved as a whole rather than component-wise as is the case in existing algorithms. The component-wise iterative approach, however, might not converge to a solution. I show in the last section that convergence is indeed a problem for the limiting Krusell-Smith algorithm. In contrast, convergence of the solution algorithm presented herein is proven mathematically. Therefore, when the approximation error is small, the numerical solution is indeed close to the exact equilibrium solution.