Solving Master Equations of Continuous-Time, Heterogeneous Agent Uninsurable Income Risk Models Using Deep Learning

Abstract

As rising wealth inequality is becoming a major component of many present-day economies, the study of dynamic general equilibrium economic models with heterogeneous agents is becoming increasingly important. However, the need for large computational resources due to the “curse of dimensionality” has often made it difficult to solve them using traditional numerical methods such as finite differences. Leveraging recent advances in deep learning, particularly in the physics-informed neural networks (PINNs) literature, we present a novel solution method for solving such models through their master equation formulation. This is done by first approximating the master equation with a large but finite number of agents, transforming it from an infinite-dimensional PDE into a high-dimensional one. We then introduce a PINN algorithm to solve the resulting high-dimensional PDE, whose architecture exploits the symmetry and permutation-invariance properties of the PDE. We show that our algorithm is able to solve with exceptional accuracy the uninsurable income risk model in [2] —a model for which a finite difference solution scheme also exists and serves as a benchmark for comparison. We also provide a preliminary demonstration of our algorithm’s capability to solve models outside the reach of the finite difference method by solving our own proposed uninsurable income risk model where risk aversion is wealth-dependent.