Exploiting Symmetry in High-Dimensional Dynamic Programming

52 Pages Posted: 5 Jul 2021 Last revised: 8 Sep 2024

See all articles by Mahdi Ebrahimi Kahou

Mahdi Ebrahimi Kahou

University of British Columbia (UBC)

Jesús Fernández-Villaverde

University of Pennsylvania - Department of Economics; National Bureau of Economic Research (NBER)

Jesse Perla

University of British Columbia (UBC)

Arnav Sood

Carnegie Mellon University

Date Written: July 2021

Abstract

We propose a new method for solving high-dimensional dynamic programming problems and recursive competitive equilibria with a large (but finite) number of heterogeneous agents using deep learning. We avoid the curse of dimensionality thanks to three complementary techniques: (1) exploiting symmetry in the approximate law of motion and the value function; (2) constructing a concentration of measure to calculate high-dimensional expectations using a single Monte Carlo draw from the distribution of idiosyncratic shocks; and (3) designing and training deep learning architectures that exploit symmetry and concentration of measure. As an application, we find a global solution of a multi-firm version of the classic Lucas and Prescott (1971) model of investment under uncertainty. First, we compare the solution against a linear-quadratic Gaussian version for validation and benchmarking. Next, we solve the nonlinear version where no accurate or closed-form solution exists. Finally, we describe how our approach applies to a large class of models in economics.

Suggested Citation

Ebrahimi Kahou, Mahdi and Fernández-Villaverde, Jesús and Perla, Jesse and Sood, Arnav, Exploiting Symmetry in High-Dimensional Dynamic Programming (July 2021). NBER Working Paper No. w28981, Available at SSRN: https://ssrn.com/abstract=3880214

Mahdi Ebrahimi Kahou (Contact Author)

University of British Columbia (UBC) ( email )

2329 West Mall
Vancouver, British Columbia BC V6T 1Z4
Canada

Jesús Fernández-Villaverde

University of Pennsylvania - Department of Economics ( email )

3718 Locust Walk
160 McNeil Building
Philadelphia, PA 19104
United States
215-898-1504 (Phone)
215-573-2057 (Fax)

National Bureau of Economic Research (NBER)

1050 Massachusetts Avenue
Cambridge, MA 02138
United States

Jesse Perla

University of British Columbia (UBC) ( email )

2329 West Mall
Vancouver, British Columbia BC V6T 1Z4
Canada

Arnav Sood

Carnegie Mellon University ( email )

Pittsburgh, PA 15213-3890
United States

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