A mean field game model with non-local spatial interactions and resources accumulation
Daria Ghilli, Fausto Gozzi, Giovanni Zanco
TL;DR
The paper develops a mean field game model for time-space economic evolution with a continuum of agents whose dynamics include non-local spillovers via a distribution-dependent term $F(x,\mu)$ and a resource-accumulation mechanism. It formulates a coupled forward-backward PDE system consisting of a Hamilton-Jacobi-Bellman equation for the value function $V$ and a Fokker-Planck equation for the population distribution $\mu$, with a non-standard interaction and positivity constraints on human capital. The authors establish regularity and well-posedness of the HJB given $\mu$, prove strong well-posedness of the associated McKean-Vlasov dynamics and FP equation on a short time horizon, and then prove existence of a MFG equilibrium via a Schauder fixed-point argument. The work provides a rigorous local-in-time existence foundation for time-space spillover models in economics and lays the groundwork for future numerical analysis and economic interpretation of spatial human capital dynamics.
Abstract
We study a family of mean field games arising in modeling the behavior of strategic economic agents which move across space maximizing their utility from consumption and have the possibility to accumulate resources for production (such as human capital). The resulting mean field game PDE system is not covered in the actual literature on the topic as it displays weaker assumptions on the regularity of the data (in particular global Lipschitz continuity and boundedness of the objective are lost), state constraints, and a non-standard interaction term. We obtain a first result on the existence of solution of the mean field game PDE system.