Equilibrium models with heterogeneous agents under rational expectations and its numerical solution

Abstract

[Abstract]: In this work we assume rational expectations to pose general equilibrium models with heterogeneous firms that can enter or exit the industry. More precisely, we assume a general Ito process for the dynamics of the agents productivity, including the main dynamics in the literature. A Hamilton-Jacobi-Bellman (HJB) formulation models the endogenous decision of firms to remain or exit the industry. All firms that exit are immediately replaced by a group of new ones, so that the probability density function of firms satisfies an appropriate Kolmogorov-Fokker-Plank (KFP) equation with source term. Equilibrium models are completed with the household problem formulation and the feasibility conditions. In the evolutive and general stationary settings, analytical or semi-analytical formulas are not available, so that appropriate numerical methods are required. We propose a Crank-Nicolson scheme for the time discretization of the evolutive problems. Moreover, we use an augmented Lagrangian active set (ALAS) method combined with a finite difference discretization for the HJB formulation and a suitable finite differences discretization for the KFP problem. For the global equilibrium problem we propose a Steffensen algorithm. Numerical examples illustrate the performance of the proposed numerical methodologies as well as the expected behaviours of the computed economic variables.