We develop a deep learning algorithm for approximating functional rational expectations equilibria of dynamic stochastic economies in the sequence space. We use deep neural networks to parameterize equilibrium objects of the economy as a function of truncated histories of exogenous shocks. We train the neural networks to fulfill all equilibrium conditions along simulated paths of the economy. To illustrate the performance of our method, we solve three economies of increasing complexity: the stochastic growth model, a high-dimensional overlapping generations economy with multiple sources of aggregate risk, and finally an economy where households and firms face uninsurable idiosyncratic risk, shocks to aggregate productivity, and shocks to idiosyncratic and aggregate volatility. Furthermore, we show how to design practical neural policy function architectures that guarantee monotonicity of the predicted policies, facilitating the use of the endogenous grid method to simplify parts of our algorithm.
Presented at: Advances in Computational Economics and Finance,* University of Zurich (2025); Macro Reading Group, UNC Chapel Hill (2025); Torino Conference on Machine Learning for Economics and Finance,* Turin (2025); Macro Lunch Seminar,* Princeton University (2025).
Intergenerational Consequences of Rare Disasters, with Jan Žemlička, 2023.
We analyze the intergenerational consequences of rare disasters in a calibrated overlapping generations model featuring realistic household portfolios and equilibrium asset prices. Households own houses and trade in bonds and equity. In a disaster, young households suffer from reduced labor income and tightened borrowing constraints. Older households lose a large portion of their savings invested in risky assets. The relative winners are households shortly before retirement, who have a more stable labor income, are not borrowing constrained, and young enough to benefit from large returns of assets purchased during the disaster at depressed prices. In order to solve the model, we advance contemporary deep learning based solution methods along two complementary dimensions. First, we introduce an economics-inspired neural network architecture that, by construction, ensures that market clearing conditions are always satisfied. Second, we illustrate how to solve models with multiple assets by introducing them step-wise into the economy. These two innovations enable us to reduce the number of equilibrium conditions, which are not fulfilled exactly, and to substantially improve the stability of the training algorithm.
Supplementary material: Recording of the presentation at the DSE 2023 conference, method part only, on youtube, recordings of overview presentations at the ACM EC-24 conference at Yale University and the SUERF, ECB, Bank of Finland and Bank of Italy Workshop on the Use of AI in Economic Modelling and Forecasting workshop.
Links to a previous version, including only the methodological part: arxiv , ssrn.
Presented at: Reading Group, * UPenn (2022); CERGE-EI,* Prague (2023); Advances in Computational Economics and Finance,* UZH (2023); CEF Conference,* Nice (2023); DSE Conference, Lausanne (2023); EEA-ESEM Conference,* Barcelona (2023); Math Seminar,* ETH Zurich; Brown Bag Seminar, John Hopkins University (2023); Macro Lunch Seminar, UPenn (2023); Seminar, Toronto Metropolitan University (2024); Seminar, University of Copenhagen (2024); Seminar, University of North Carolina at Chapel Hill, (2024); Seminar, ESSEC Paris (2024); Seminar, IE Madrid (2024); Seminar, University of Sussex (2024); Seminar, National University of Singapore, (2024); SUERF, ECB, Bank of Finland and Bank of Italy Workshop on the Use of AI in Economic Modelling and Forecasting, (recording available on youtube, 2024); SED conference,* Barcelona (2024); ACM EC conference, Yale University (2024); ESIF AI+ML Conference, Cornell University (2024); Seminar, Wake Forest University (2024); SITE conference, Stanford (2024); SEA Conference, DC (2024); Seminar, University of Waterloo (2025); Seminar, Arizona State University (2025); T2M conference,* Paris (2025); SED conference,* University of Copenhagen (2025).
Asset Pricing in a Low Rate Environment, with Harold Linh Cole and Felix Kubler, 2023.
We examine asset prices in environments where the risk-free rate lies considerably below the growth rate. To do so, we introduce a tractable model of a production economy featuring heterogeneous trading technologies, as well as idiosyncratic and aggregate risk. We show that allowing for the possibility of firms exiting is crucial for matching key macroeconomic moments and, simultaneously, the risk-free rate, the market price of risk, and price-earnings ratios. In particular, our model allows us to consider calibrations that match the high observed market price of risk and average interest rates as low as 2-3.5 percent below the average growth rate. High values for risk aversion or non-standard preferences are not necessary for this. We use the model to examine the wealth distribution and asset prices in economies with very low real rates. We also examine under which conditions realistic calibrations allow for an infinite rollover of government debt. For our benchmark calibration, rollover is impossible even if the average risk-free rate lies 3.5 percent below the average growth rate.
Presented at: Macro Lunch Seminar,* Wharton (2023); Bristol Macro Workshop, University of Bristol (2023); EEA-ESEM Conference, Barcelona (2023); Stockman Workshop,* University of Rochester (2023); Macroeconomics Seminar,* University of Pittsburgh (2023); Macro-CRC Seminar,* University of Mannheim (2024); Seminar,* Ohio State University (2024); 18th Annual Conference on General Equilibrium and its Applications, Cowles Foundation, Yale University (2024); Seminar,* Tinbergen Institute (2024); SED conference, Barcelona (2024).
*: Presentation by coauthor