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

TL;DR

This paper extends a Boltzmann mean-field game model for knowledge growth, analyzing how different learning rates and utility functions influence solution behavior, existence, and economic growth paths through analytical and computational methods.

Contribution

It introduces a generalized BMFG model for knowledge growth, exploring solution properties and their dependence on learning and utility functions, including existence and growth path analysis.

Findings

Solution monotonicity varies with learning and utility functions.

Existence of solutions depends on model parameters.

Balanced growth paths are influenced by the structure of the equations.

Abstract

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. 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 Bellman equation, resulting in a highly nonlinear forward-backward in time PDE system. The structure of solutions to the Boltzmann and Bellman equation depends strongly on the learning rate in the Boltzmann collision kernel as well as the utility function in the Bellman equation. In this paper we investigate the monotonicity behavior 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.