Models and numerical methods for equilibrium problems with heterogeneous agents involving two productive sectors

Abstract

[Abstract]: In this work, we assume rational expectations to propose new general equilibrium models with heterogeneous firms that can enter or exit two different productive sectors. More precisely, we consider a first sector of small establishments, where firms can remain in the sector, exit the industry, or move to the second sector of large establishments. In the second sector, firms can remain there or move to the first sector. We assume respective general Itô processes for the stochastic productivity dynamics corresponding to each of the two sectors. Obstacle-type problems associated with Hamilton–Jacobi–Bellman (HJB) partial differential equations (PDEs) model the endogenous decision of firms to remain in or leave each productive sector. Moreover, the probability density function of firms in each sector satisfies a Kolmogorov–Fokker–Planck (KFP) PDE with a source term. Equilibrium models for the steady-state and time-dependent regimes are completed with appropriate household problem formulations and feasibility conditions. For the numerical solution, we propose the Crank–Nicolson method for time discretization. Furthermore, we use augmented Lagrangian active set (ALAS) methods to solve unilateral and bilateral obstacle problems, jointly with finite-difference discretizations for HJB formulations. Additionally, appropriate finite difference discretizations for the KFP problems are considered. For the global non-linear equilibrium problem, we propose a Steffensen algorithm. Finally, numerical examples illustrate the performance of proposed numerical methodologies as well as the expected behaviour of economic variables.