On a Boltzmann mean field model for knowledge growth

Burger M., Lorz A., Wolfram M.

Research article (journal) | Peer reviewed

Abstract

In this paper we analyze a Boltzmann-type mean field game model for knowledge growth, which was proposed by Lucas et al. [J. Political Econ., 122 (2014), pp. 1-51]. We discuss the underlying mathematical model, which consists of a coupled system of a Boltzmann-type equation for the agent density and a Hamilton-Jacobi-Bellman equation for the optimal strategy. We study the analytic features of each equation separately and show local in time existence and uniqueness for the fully coupled system. Furthermore we focus on the construction and existence of special solutions, which relate to exponential growth in time - so-called balanced growth path solutions. Finally, we illustrate the behavior of solutions for the full system and the balanced growth path equations with numerical simulations.