Diffusion of knowledge and the lottery society

TL;DR

This paper analyzes a mean-field game model of economic growth driven by knowledge diffusion, revealing a long-term regime where most agents focus solely on learning, leading to a 'lottery society' dynamic.

Contribution

It provides a long-time convergence analysis of the Lucas-Moll system in a regime where balanced growth paths do not exist, introducing the concept of a 'lottery society' regime.

Findings

Agents predominantly focus on learning at large times.

Agent density propagates at Fisher-KPP speed.

Balanced growth paths are absent in this regime.

Abstract

The Lucas-Moll system is a mean-field game type model describing the growth of an economy by means of diffusion of knowledge. The individual agents in the economy advance their knowledge by learning from each other and via internal innovation. Their cumulative distribution function satisfies a forward in time nonlinear non-local reaction-diffusion type equation. On the other hand, the learning strategy of the agents is based on the solution to a backward in time nonlocal Hamilton-Jacobi-Bellman equation that is coupled to the aforementioned equation for the agents density. Together, these equations form a system of the mean-field game type. When the learning rate is sufficiently large, existence of balanced growth path solutions to the Lucas-Moll system was proved in~\cite{PRV,Porretta-Rossi}. Here, we analyze a complementary regime where the balanced growth paths do not exist. The main result is a long time convergence theorem. Namely, the solution to the initial-terminal value problem behaves in such a way that at large times an overwhelming majority of the agents spend no time producing at all and are only learning. In particular, the agents density propagates at the Fisher-KPP speed. We name this type of solutions a lottery society.