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.”
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In the long run we are all dead.  Nevertheless, dynamic models require a variety of boundary conditions on long run forward-looking behavior such as transversality, no-bubble, and no-ponzi-scheme conditions.  While these are essential to ensure the problems are well-posed and that short-term behavior is disciplined by long-run expectations, they represent a computational challenge.  Calculating the long run behavior or ergodic sets is typically the most expensive and unstable part of solving a model, and the global nature of spectral solutions makes it the central speed limit on increase dimensionality.  Using a few classic models, we show numerically that with deep learning we can solve under-determined setups that automatically fulfill the appropriate long-run boundaries conditions aligned with our models and economic assumptions.  With this approach, we cannot efficiently calculate steady states and ergodic behavior. However, it allows us to accurately calculate short-term dynamics from a finite set of initial conditions which are still disciplined by long-run boundaries.  Not only are these solutions shockingly accurate with very little data and calculated in seconds, but they can even be used to solve transition dynamics with non-stationarity and steady-state multiplicity.  While the computer science theory has not yet caught up with the empiric to prove conditions under which these results would always hold, we provide an intuitive connection to the literature on “double-descent'” which shows an implicit bias towards min-norm solutions in over-parameterized deep-learning models.
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In models with irrecoverable investment and uncertainty in the output prices it is a well-established result that uncertainty increases the output price that a firm starts investment. This paper studies a model of irrecoverable investment (entry) where the variable cost and output price are characterized by two correlated geometric Brownian motions. The numerical results indicate that in the presence of high levels of correlation the impact of uncertainty in output price is ambiguous and depends on the level of variable cost. Specifically, increasing uncertainty in output prices increases the entry output price for low levels of variable cost and the reverse happens for high levels of variable cost. Therefore, in the presence of high levels of correlation the conventional result does not hold anymore. Moreover, this study establishes that increasing the correlation level decreases the entry output price.
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