Deep Equilibrium Nets

Abstract

We introduce deep equilibrium nets---a deep learning-based method to compute approximate functional rational expectations equilibria of economic models featuring a substantial amount of heterogeneity, significant uncertainty, and occasionally binding constraints. Deep equilibrium nets are neural networks that directly approximate all equilibrium functions and that are trained in an unsupervised fashion to satisfy all equilibrium conditions along simulated paths of the economy. Since the neural network approximates the equilibrium functions directly, simulating the economy is computationally cheap, and training data can be generated at virtually zero cost. We demonstrate that deep equilibrium nets can solve rich and economically relevant models accurately by applying them to solve three different models, all featuring a very high-dimensional state space. Specifically, we solve two overlapping generations models with aggregate and idiosyncratic uncertainty, illiquid capital, a one-period bond, and occasionally binding constraints. Additionally, we solve a Bewley-style model with a continuum of agents, aggregate and idiosyncratic risk, borrowing constraints, and recursive preferences.