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<p>In this article, we develop a general equilibrium macroeconomic model with heterogeneous agents under rational expectations. The model captures productivity dynamics in two sectors using jump-diffusion processes, allowing for the incorporation of sudden and significant economic shocks. Firms’ decisions to remain in or exit each sector are modelled endogenously as obstacle-type problems, formulated as Hamilton-Jacobi-Bellman (HJB) partial integro-differential equations (PIDEs). The evolution of the firm distribution in each sector is governed by a non-homogeneous Kolmogorov-Fokker-Planck (KFP) PIDE. The model is closed with household problems and feasibility conditions. Given the lack of (semi-)analytical solutions, we propose a suite of numerical techniques. Time discretization is handled with the Crank-Nicolson method, while integral terms are treated explicitly using the Adams-Bashforth scheme. Obstacle problems are solved using augmented Lagrangian active set (ALAS) methods, combined with finite-difference schemes for the HJB equations. The KFP equations are also discretized using finite-difference methods. To solve the global nonlinear equilibrium, we propose a Steffensen algorithm. Numerical examples prove the performance of the proposed numerical methodologies and illustrate the expected behaviour of computed economic variables.</p>

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Two-productive sector equilibrium problems with heterogeneous agents under jump-diffusion models

  • 期刊论文

  • 发表日期：

    2026年1月16日

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  • Jonatan Ráfales,
  • Carlos Vázquez

摘要

In this article, we develop a general equilibrium macroeconomic model with heterogeneous agents under rational expectations. The model captures productivity dynamics in two sectors using jump-diffusion processes, allowing for the incorporation of sudden and significant economic shocks. Firms’ decisions to remain in or exit each sector are modelled endogenously as obstacle-type problems, formulated as Hamilton-Jacobi-Bellman (HJB) partial integro-differential equations (PIDEs). The evolution of the firm distribution in each sector is governed by a non-homogeneous Kolmogorov-Fokker-Planck (KFP) PIDE. The model is closed with household problems and feasibility conditions. Given the lack of (semi-)analytical solutions, we propose a suite of numerical techniques. Time discretization is handled with the Crank-Nicolson method, while integral terms are treated explicitly using the Adams-Bashforth scheme. Obstacle problems are solved using augmented Lagrangian active set (ALAS) methods, combined with finite-difference schemes for the HJB equations. The KFP equations are also discretized using finite-difference methods. To solve the global nonlinear equilibrium, we propose a Steffensen algorithm. Numerical examples prove the performance of the proposed numerical methodologies and illustrate the expected behaviour of computed economic variables.