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Equilibrium models with heterogeneous agents under rational expectations and its numerical solution
Communications in Nonlinear Science and Numerical Simulation ( IF 3.9 ) Pub Date : 2020-12-26 , DOI: 10.1016/j.cnsns.2020.105673
Jonatan Ráfales , Carlos Vázquez

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.



中文翻译:

理性期望下具有异构主体的均衡模型及其数值解

在这项工作中,我们假设理性的期望为可以进入或退出该行业的异类公司提出一般均衡模型。更确切地说,我们假设代理商生产率的动态变化,包括文献中的主要动态变化,都采用通用的Ito过程。汉密尔顿-雅各比-贝尔曼(HJB)公式模拟了公司保留或退出该行业的内在决定。所有退出的公司都会立即被一组新的公司取代,从而使公司的概率密度函数满足带有源项的适当Kolmogorov-Fokker-Plank(KFP)方程。平衡模型通过家庭问题公式化和可行性条件完成。在渐进式和常规平稳设置中,无法使用解析或半解析公式,因此需要适当的数值方法。我们为演化问题的时间离散化提出了一个Crank-Nicolson方案。此外,我们将增强拉格朗日活动集(ALAS)方法与HJB公式的有限差分离散化和KFP问题的适当有限差分离散化结合使用。对于全局平衡问题,我们提出了一种Steffensen算法。数值算例说明了所提出的数值方法的性能以及所计算的经济变量的预期行为。我们将增强拉格朗日活动集(ALAS)方法与HJB公式的有限差分离散化和KFP问题的适当有限差分离散化相结合。对于全局平衡问题,我们提出了一种Steffensen算法。数值算例说明了所提出的数值方法的性能以及所计算的经济变量的预期行为。我们将增强拉格朗日活动集(ALAS)方法与HJB公式的有限差分离散化和KFP问题的适当有限差分离散化相结合。对于全局平衡问题,我们提出了一种Steffensen算法。数值算例说明了所提出的数值方法的性能以及所计算的经济变量的预期行为。

更新日期:2020-12-26
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