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Global Solutions to Master Equations for Continuous Time Heterogeneous Agent Macroeconomic Models

Zhouzhou Gu, Mathieu Laurière, Sebastian Merkel, Jonathan Payne

TL;DR

This work approximate the agent distribution so that equilibrium in the economy can be characterized by a high, but finite, dimensional non-linear partial differential equation and represents the value function using a neural network and train it to solve the differential equation using deep learning tools.

Abstract

We propose and compare new global solution algorithms for continuous time heterogeneous agent economies with aggregate shocks. First, we approximate the agent distribution so that equilibrium in the economy can be characterized by a high, but finite, dimensional non-linear partial differential equation. We consider different approximations: discretizing the number of agents, discretizing the agent state variables, and projecting the distribution onto a finite set of basis functions. Second, we represent the value function using a neural network and train it to solve the differential equation using deep learning tools. We refer to the solution as an Economic Model Informed Neural Network (EMINN). The main advantage of this technique is that it allows us to find global solutions to high dimensional, non-linear problems. We demonstrate our algorithm by solving important models in the macroeconomics and spatial literatures (e.g. Krusell and Smith (1998), Khan and Thomas (2007), Bilal (2023)).

Global Solutions to Master Equations for Continuous Time Heterogeneous Agent Macroeconomic Models

TL;DR

This work approximate the agent distribution so that equilibrium in the economy can be characterized by a high, but finite, dimensional non-linear partial differential equation and represents the value function using a neural network and train it to solve the differential equation using deep learning tools.

Abstract

We propose and compare new global solution algorithms for continuous time heterogeneous agent economies with aggregate shocks. First, we approximate the agent distribution so that equilibrium in the economy can be characterized by a high, but finite, dimensional non-linear partial differential equation. We consider different approximations: discretizing the number of agents, discretizing the agent state variables, and projecting the distribution onto a finite set of basis functions. Second, we represent the value function using a neural network and train it to solve the differential equation using deep learning tools. We refer to the solution as an Economic Model Informed Neural Network (EMINN). The main advantage of this technique is that it allows us to find global solutions to high dimensional, non-linear problems. We demonstrate our algorithm by solving important models in the macroeconomics and spatial literatures (e.g. Krusell and Smith (1998), Khan and Thomas (2007), Bilal (2023)).
Paper Structure (51 sections, 3 theorems, 134 equations, 14 figures, 8 tables, 4 algorithms)

This paper contains 51 sections, 3 theorems, 134 equations, 14 figures, 8 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Economic Model
  3. Environment
  4. Recursive Representation of Equilibrium
  5. Model Generality
  6. Finite Dimensional Master Equation
  7. Finite Agent Approximation
  8. Discrete State Space Approximation
  9. Projection onto Basis
  10. Relationship to Continuous Time Perturbation Approaches
  11. Comparison Between Our Approximation Approaches
  12. Solution Method
  13. Neural Network Approximations
  14. Solution Algorithm
  15. Sampling
  16. ...and 36 more sections

Key Result

Proposition 1

If the KFE eq:generic-KFE is of the form $dg_t = \mathcal{L}^{KF}g_t dt$ with a (constant) linear operator $\mathcal{L}^{KF}$, $b_0=g^{ss}$, $b_1$, ..., $b_N$ are eigenfunctions of $\mathcal{L}^{KF}$ with eigenvalues $\lambda_0=0$, $\lambda_1$, ..., $\lambda_N$, and $g_0 - g^{ss} = \sum_{n=1}^{N}\va

Figures (14)

  • Figure 1: Simulations for the Krusell-Smith Model. The top left plot is the TFP shock path, the top right panel is the aggregate relative capital change. The second row left plot shows the relative change in the capital return and the second row right plot shows the relative change in the wage rate. The plots on rows three and four show the distribution at different times in the simulation. The labels "NN, FA", "NN, DS", and "NN, P" refer to solutions from the finite agent, discrete state, and projection neural networks respectively. "FV" refers to the solution from fernandez2023financial. Subscript sss refers to the stochastic steady state at $z=0$.
  • Figure 2: Forecasted aggregate capital dynamics starting from the stochastic steady state (sss) for the Krusell-Smith Model. The left plot is the fan chart for the TFP shock path, generated from the OU process with initial condition $z_0 = 0$. The right panel is the time series plot for relative change in aggregate capital at percentiles 10%, 30%, 50%, 70%, 90% (from the lowest to the highest). The labels "NN, FA", "NN, DS", and "NN, P" refer to solutions from the finite agent, discrete state, and projection neural networks respectively. "FV" refers to the solution from fernandez2023financial.
  • Figure 3: Comparison between the neural network and finite difference solutions for the Aiyagari model. The top left plot shows the consumption policy. The top right shows the derivative of the value function, the bottom left shows the pdf, and the bottom right shows the cdf. The labels "NN,FA", "NN,FS", and "FD" refers to solutions from the finite agent neural network, the discrete state space neural network, and finite difference respectively.
  • Figure 4: Illustration of model solution for the firm dynamics model. The left panel depicts the investment policy in the stochastic steady state and the depreciation line (dashed) as a function of firm's own capital level. The right panel depicts the (marginal) ergodic density for high and low productivity firms.
  • Figure 5: Illustration of model solution for the dynamic spatial model. The left panels depict the wage distribution in the stochastic steady state, $g=g^{sss}$, $z=0$, as a function of location $j$ (top) and location-specific productivity $\beta_{j}$ (bottom). The right panels depict the impact effects of a hypothetical "recession shock" that moves the aggregate state to $(z,g)=(-0.02,g^{sss})$. The top right panel shows the relative change in wages in a recession and the bottom right panel the resulting net migration flows as a proportion of the population ($\mu_{g,j,t}/g_t(j)$). In all panels but the bottom left, locations $j$ are sorted by their sensitivity to aggregate productivity ($\chi_j$) in ascending order. Blue bars/dots depict periphery locations and red bars/dots depict central locations.
  • ...and 9 more figures

Theorems & Definitions (10)

  • Proposition 1
  • proof
  • Proposition 2: Master Equation
  • proof
  • Proposition 3: Master Equation
  • proof
  • proof : Proof of the Kolmogorov Forward Equation
  • proof : Proof of Proposition \ref{['prop:linear_kfe_eigenfunction_basis']}
  • proof : Proof of Proposition \ref{['prop:master_eqn:firms']}
  • proof : Proof of Proposition \ref{['prop:spatial_model:master_equation']}