Numerical Simulation of Nonoptimal Dynamic Equilibrium Models

Abstract

In this article, we propose a recursive equilibrium algorithm for the numerical simulation of nonoptimal dynamic economies. This algorithm builds upon a convergent operator over an expanded set of state variables. The fixed point of this operator defines the set of all Markovian equilibria. We study approximation properties of the operator. We also apply our recursive equilibrium algorithm to various models with heterogeneous agents, incomplete financial markets, endogenous and exogenous borrowing constraints, taxes, and money.

Analysis of Numerical Errors

“…Follow-ing Duffie et al (1994), the existence of a Markov equilibrium in a generalized space of variables is proved in Kubler and Schmedders (2003) for an asset pricing model with collateral constraints. Feng et al (2012) extend these existence results to other economies, and define a Markov equilibrium as a solution over an expanded state of variables that includes the shadow values of investment. The addition of the shadow values of investment as state variables facilitates computation of the numerical solution.…”

“…This formulation was originally proposed by Kydland and Prescott (1980), and later used in Marcet and Marimon (1998) for recursive contracts, and in Phelan and Stacchetti (2001) for a competitive economy with a representative agent. The main insight of Feng et al (2012) is to develop a reliable and computable algorithm for the numerical simulation of competitive economies with heterogeneous agents and market frictions including endogenous borrowing constraints, and study its approximation properties. Before advancing to the study of the theoretical issues involved, we begin with a few examples to illustrate some of the pitfalls found in the computation of non-optimal economies.…”

“…Again, if the equilibrium is always unique then these approximate solutions would converge uniformly to the continuous Markovian equilibrium law of motion. The following result is proved in Feng et al (2012):…”

“…In these economies it seems pertinent to compute the set of all sequential competitive equilibria. It is certainly an easy task to compute this simple model by the algorithm of Section 4 of Feng et al (2012). We presently illustrate the workings of this reliable algorithm in a stochastic economy with two types of agents.…”

“…It simply requires iterating simultaneously on pairs of candidate shadow values of investment and values for participation (the lifetime utility of never defaulting). This operator is monotone (in the set inclusion sense) and thus the approximation results of Section 4 still hold [see Feng, et al (2012)]. …”