Boltzmann mean-field game model for knowledge growth: limits to learning and general utilities

In this paper we investigate a generalisation of a Boltzmann mean field game (BMFG) for knowledge growth, originally introduced by the economists Lucas and Moll [R.E. Lucas Jr. and B. Moll, J. Pol. Econ., 122(1):1–51, 2014]. In BMFG the evolution of the agent density with respect to their knowledge level is described by a Boltzmann equation. Agents increase their knowledge through binary interactions with others; their increase is modulated by the interaction and learning rate: Agents with similar knowledge learn more in encounters, while agents with very different levels benefit less from learning interactions. The optimal fraction of time spent on learning is calculated by a Hamilton–Jacobi–Bellman equation, resulting in a highly nonlinear forward-backward in time PDE system.

The structure of solutions to the Boltzmann and Hamilton-Jacobi-Bellman equation depends strongly on the learning rate in the Boltzmann collision kernel as well as the utility function in the Hamilton-Jacobi-Bellman equation. In this paper we investigate the monotonicity behaviour of solutions for different learning and utility functions, show existence of solutions and investigate how they impact the existence of so-called balanced growth path solutions, that relate to exponential growth of the overall economy. Furthermore we corroborate and illustrate our analytical results with computational experiments.

Keywords

Boltzmann-type equation, Hamilton-Jacobi-Bellman equation, mean-field games

2010 Mathematics Subject Classification

35Q20, 35Q91, 49J20, 49N90, 70H20

Full Text (PDF format)

Received 10 September 2022

Received revised 26 September 2023

Accepted 27 September 2023

Published 12 July 2024