Contributions to the mathematical analysis and numerical simulation of stochastic models of general equilibrium with heterogeneous agents and fixed costs

Abstract

In the framework of general equilibrium models for heterogeneous agents under rational expectations, we analyze different problems to establish their mathematical model and numerical solution. The productivity is the only stochastic underlying factor, the dynamics of which either follows an Ito process or a Levy one. We assume the possibility of exit and entry of new firms in the sectors and we consider the case of one sector or two sectors. In this setting, for the problems of incumbent firms, the mathematical models are mainly formulated in terms of Hamilton-Jacobi-Bellman (HJB) PDEs or PIDEs, with obstacle inequality constraints on the solution. For the probability distribution of firms, the mathematical models are based on Kolmogorov-Fokker-Plank (KFP) PDEs or PIDEs. The global equilibrium models are completed with the household problem and the feasibility conditions. For the numerical solution, the appropriate discretizations of the involved PDEs or PIDEs are combined with an augmented Lagrangian active set (ALAS) method to treat the free boundaries in the incumbent problems. A fixed point iteration that sequentially solves the different subproblems included in the global one is applied. For the time-dependent models, a Crank-Nicolson method for the time discretization is incorporated. The numerical examples illustrate the performance of the proposed models and numerical methods for different problems and show the convergence of the solutions of the evolutive problems to the ones of the corresponding steady state problems.