When analysing short-term dynamics in economic models, it is essential to consider long-run conditions that shape forward-looking behaviour. However, enforcing long-run boundary conditions complicates solving dynamic models, particularly high-dimensional ones. This column shows that machine learning, in particular deep learning, can satisfy these conditions without explicit enforcements, allowing for global solutions in many models of interest. It establishes this using two foundational models in economics and financial economics – linear asset pricing and the neoclassical growth model.
When analysing short-run dynamics in economic models – such as transition dynamics in growth models or the investment decisions of individuals and firms – it is essential to consider long-run conditions that shape forward-looking behaviour. These conditions, such as the transversality condition, play a paradoxical role: they hold only asymptotically, or ‘when we are all dead’, yet they also shape local and short-run dynamics.
Intuitively, these conditions prevent explosive trajectories in the model’s state (or co-state) variables. For example, the no-bubble condition rules out explosive price paths, while in the neoclassical growth model, the transversality condition prevents the marginal utility of consumption from diverging. In linear models, boundary conditions are typically framed in terms of stability, ruling out explosive solutions (Blanchard and Kahn 1980, Klein 2000).
However, enforcing long-run boundary conditions complicates solving dynamic models, particularly high-dimensional ones. Algorithms that guarantee global solutions while satisfying these conditions are difficult to design, as researchers often lack a clear understanding of their implications. A common workaround, perturbation methods, avoids this issue by seeking only locally valid solutions.
In our recent work (Ebrahimi Kahou et al. 2024), we show that machine learning (ML), particularly deep learning, can satisfy these conditions without explicit enforcement, allowing for global solutions in many models of interest. This ability stems from the inductive bias of machine learning – its built-in ‘Occam’s razor’ – which favours stable and non-explosive solutions. This bias provides a foundation for modelling forward-looking agents with self-consistent short-run expectations.
Deep learning relies on functional approximations, such as computing the value function of an agent in a dynamic model, using optimisation processes with millions – or even trillions, in the case of large language models – of parameters. Given this extreme over-parameterisation, multiple sets of parameters can solve the functional approximation problem.
The key question is which solution the model converges toward in practice. As Belkin (2021) illustrates, rather than overfitting or selecting an arbitrary interpolating function, machine learning algorithms tend to converge toward the ‘simplest’ interpolating solutions. This tendency, known as inductive bias, underpins machine learning’s ability to respect long-run constraints without explicit enforcement.
The classic interpretation of inductive bias is Occam’s razor – the principle that the simplest solution is the most likely. But what does ‘simple’ mean in this context? Here, simplicity can be understood as functions that are flat or have small derivatives with respect to their inputs. 1
Explosive solutions that violate long-run boundary conditions tend to have large derivatives. Thus, it is reasonable to conjecture that machine learning-based methods, particularly deep learning, naturally filter out these solutions due to their inductive bias. We confirm this hypothesis in our paper. By examining two foundational models in economics and financial economics – linear asset pricing and the neoclassical growth model – we establish that deep learning inherently respects long-run stability constraints.
The risk-neutral linear asset pricing model has traditionally served a pedagogical role in studying long-run boundary conditions for models with forward-looking agents (e.g. Ljungqvist and Sargent 2018). In this model, today’s price depends on forecasts of future prices. An agent can rationally anticipate a price bubble that grows indefinitely, leading to deviations from fundamentals (e.g. dividends). To prevent this, an economic long-run boundary condition – the no-bubble condition – ensures that the discounted price asymptotically approaches zero.
We solve this problem using deep neural networks without explicitly encoding the no-bubble condition. Figure 1 presents our results. In the left panel, the dashed curve represents the closed-form baseline for prices (i.e. prices based on fundamentals), the solid curve shows the median approximate solution for prices generated by the deep neural network, and the shaded region captures the 10th to 90th percentiles across 100 different random initialisations of the optimisation algorithm. In the right panel, the solid curve represents the median relative error between the approximate path and fundamental-based prices, while the dashed vertical line marks the boundary between extrapolation and interpolation regions.
Figure 1 Deep learning solution for the linear asset pricing model, using an ensemble of 100 random initial parameters for the neural network
We obtain highly accurate short-run approximations. Even in the extrapolation region – covering time periods beyond 30 – the approximations remain robust, despite this not being our primary objective. As shown in both panels, the difference between the fundamental-based price and the approximate price is imperceptible in the short run. While the optimisation procedure could select any interpolating solution, including those corresponding to different rational bubbles, the inductive bias inherently favours solutions without bubbles.
The neoclassical growth model highlights the crucial role of transversality conditions in eliminating suboptimal paths. We examine these conditions, focusing on their behavioural foundations and connection to inductive bias. The equations governing the agent’s intertemporal choices – balancing marginal costs and benefits over time – admit multiple solutions. Those that violate the transversality condition lead to a consumption path that asymptotically converges to zero, resulting in excessive capital accumulation. As a result, the optimal capital path is flatter than those violating the transversality condition. Consequently, the inductive bias of deep learning-based solutions naturally favours the optimal path.
Figure 2 presents the results obtained using deep neural networks without explicitly enforcing the transversality condition. The left panel compares the median of the approximate solutions for capital and consumption to a benchmark (dashed curves) derived from value function iteration. Despite lacking explicit knowledge of the transversality condition, the approximation consistently captures the correct dynamics. The right panel shows the median errors relative to the baseline, with the shaded region representing the 10th to 90th percentiles for both capital and consumption.
Figure 2 Deep learning solution for the neoclassical growth model, using an ensemble of 100 random initial parameters for the neural network
Given our emphasis on short- and medium-run dynamics, one concern might be cases where there is steady-state multiplicity, as is often found in models of economic growth, spatial, and urban economics. To this end, we also investigate the neoclassical growth model with a non-concave production function, similar to those used in the economic development literature (e.g. Azariadis and Drazen 1990). The non-concavity of the production function leads to steady-state multiplicity. One of the most surprising findings is that this approach remains effective even in the presence of steady-state multiplicity.
Figure 3 presents the results of this experiment for various initial conditions for capital in the economy. The left panel shows the capital paths, while the right panel displays the corresponding consumption paths. The inductive bias selects solutions that converge to the correct set of steady states, even in the presence of steady-state multiplicity.
Figure 3 Deep learning solution for non-concave production function
Although intentionally low-dimensional, these examples provide valuable insights into how economists can use deep learning to solve high-dimensional models while respecting long-run boundary conditions.
Our findings suggest that inductive bias can serve as a micro-foundation for modelling forward-looking agents. They also lay the theoretical groundwork for applying deep learning to equilibrium computation in dynamic models. This emerging literature includes works such as Ebrahimi Kahou et al. (2021), Azinovic et al. (2022), Maliar et al. (2021), Barnett et al. (2023), and Payne et al. (2024). Moreover, these results support the use of machine learning methods in solving high-dimensional models that require accurate short-run dynamics, such as spatial models for climate change analysis (e.g. Cruz and Rossi-Hansberg 2024).
Source : VOXeu
