DP16285 Exploiting Symmetry in High-Dimensional Dynamic Programming

Author(s): Mahdi Ebrahimi Kahou, Jesús Fernández-Villaverde, Jesse Perla, Arnav Sood
Publication Date: June 2021
Keyword(s): Dynamic programming, Machine Learning
Programme Areas: Monetary Economics and Fluctuations
Link to this Page: cepr.org/active/publications/discussion_papers/dp.php?dpno=16285

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. The ``curse of dimensionality'' is avoided due to four 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; (3) sampling methods to ensure the model fits along manifolds of interest; and (4) selecting the most generalizable over-parameterized deep learning approximation without calculating the stationary distribution or applying a transversality condition. As an application, we solve 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 nonlinear versions with aggregate shocks. Finally, we describe how our approach applies to a large class of models in economics.