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In this paper we investigate a generalisation of a Boltzmann mean field game(BMFG) for knowledge growth, originally introduced by the economists Lucas andMoll. In BMFG the evolution of the agent density with respect to theirknowledge level is described by a Boltzmann equation. Agents increase theirknowledge through binary interactions with others; their increase is modulatedby the interaction and learning rate: Agents with similar knowledge learn morein encounters, while agents with very different levels benefit less fromlearning interactions. The optimal fraction of time spent on learning iscalculated by a Bellman equation, resulting in a highly nonlinearforward-backward in time PDE system.  The structure of solutions to the Boltzmann and Bellman equation dependsstrongly on the learning rate in the Boltzmann collision kernel as well as theutility function in the Bellman equation. In this paper we investigate themonotonicity behavior of solutions for different learning and utilityfunctions, show existence of solutions and investigate how they impact theexistence of so-called balanced growth path solutions, that relate toexponential growth of the overall economy. Furthermore we corroborate andillustrate our analytical results with computational experiments.

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