Mean-field games for economic applications

Date & Time: Wednesday, September 29, 2026 | 11:00 - 13:30
Place: tba
Chairs: Johannes Brumm, Lukas Frank

Abstract

Mean-field games describe situations in which infinitely many "players" optimize a target quantity that may depend on the actions of the other players (Lasry and Lions 2007; Huang et al. 2006). Such problems typically lead to a forward backward system of differential equations. Natural applications arise in models of macroeconomics and finance, where a large number of people interact on markets and optimize utility.  In this mini-symposium, we explore recent developments in that interdisciplinary field, where research from mathematics, economics, and engineering has fruitfully influenced each other since its very beginnings about twenty years ago.

Confirmed Speakers: 

Title: Propagation of carbon price shocks through the value chain: the mean-field game of defaults
Speaker: Thorsten Schmidt (University of Freiburg)
Abstract: We introduce a new mean-field game framework to analyze the impact of carbon pricing in a multi-sector economy with defaultable firms. Each sector produces a homogeneous good, with its price endogenously  determined through market clearing. Firms act as price takers and maximize profits by choosing an optimal allocation of inputs-including labor, emissions, and intermediate goods from other sectors-while interacting through the endogenous sectoral price. Firms also choose their default timing to maximize shareholder value. Formally, we model the economy as an optimal stopping mean-field  game within each sector. The resulting system of coupled mean-field games admits a linear programming formulation that characterizes Nash equilibria in terms of population measure flows. We prove the existence  of a linear programming Nash equilibrium and establish uniqueness of the associated price system.
Numerical illustrations are presented for firms with constant elasticity of substitution (CES) production functions. In a stylized  single-sector economy, carbon price shocks induce substitution between emissions and labor. In a three-sector economy, the manufacturing sector faces consumer demand and requires inputs from a brown sector, which can be increasingly replaced by green-sector goods as carbon prices  rise. These experiments reveal that carbon price shocks can generate substantial spillover effects along the value chain, underscoring the importance of sectoral interdependencies in shaping effective decarbonization pathways.

Title: Finite difference methods for a continuous-time heterogeneous agent model with recursive utility
Speaker: Yves Achdou (University Paris Cité)
Abstract: We propose, analyze and test computational methods for solving a continuous-time heterogenous agent model with Epstein-Zin utility. Such recursive utilities allow the model to disentangle between risk aversion and intertemporal substitution. Having discretized the Hamilton-Jacobi-Bellman (HJB) equation arising in the model, we propose a Howard-Newton algorithm for the late resolution preference case, and a  Howard-Tarski-Kantorovich algorithm for the early resolution preference case. We prove the convergence of the iterative algorithms. We obtain as a consequence the existence of solutions to the discretized HJB equations. In the late resolution case, we supply a priori estimates between the unique solutions of the continuous and discretized HJB equations.

Title: Mastering Stochastic Overlapping Generations Models in Continuous Time
Speaker: Lukas Frank (KIT)
Abstract: We propose a comprehensive framework for solving overlapping-generations (OLG) models in continuous time with both idiosyncratic and aggregate risk. Our general characterization of equilibrium through the master equation operates on the joint distribution over the continuous idiosyncratic states, age and wealth. Our computational strategy is to take a finite-dimensional representation of this distribution as an input of a neural net which in turn outputs a finite-difference representation of the (conditional) value function. This idea can be applied generally to heterogeneous agent models with aggregate risk, and we call it finite-difference neural operator. Our method combines advantages from modern neural nets and traditional finite-difference methods: It is grid-free in the high-dimensional distribution, and retains control on boundary conditions in low-dimensional state variables. Moreover, our method is able to enforce shape constraints. We showcase its flexibility by solving a continuous-time OLG model with aggregate risk alone where we characterize the distribution by its supporting function; and to an OLG model with both types of risk.

Title: Tensor-Train Approximation for High-Dimensional Economic Models
Speaker: Jakob Hußmann (KIT)
Abstract: We introduce a scalable and broadly applicable method for computing global solutions of dynamic stochastic models. Tensor train approximation (TTA) identifies latent low-rank structure in high-dimensional functions, thereby capturing complex non-linearities while mitigating the curse of dimensionality --- the number of parameters grows only linearly in dimension. We show that our TTA approach can accommodate irregular ergodic sets, is compatible with the endogenous grid method, enables quasi-analytical expectations, and can solve continuous-time models via least-squares projection, all in high dimensions. We demonstrate TTA’s scalability by solving multi-country models of growing dimension with only moderate increases in compute time and no loss of accuracy. We show TTA’s versatility in heterogeneous-agent models with large aggregate shocks. To approximate the wealth distribution efficiently, we introduce a simulation-based moment-selection procedure. In a model with stochastic wealth taxation, using the first nine selected moments, instead of mean wealth only, reduces approximation errors tenfold.