  In this paper we study balanced growth path solutions of a Boltzmann mean
field game model proposed by Lucas et al [13] to model knowledge growth in an
economy. Agents can either increase their knowledge level by exchanging ideas
in learning events or by producing goods with the knowledge they already have.
The existence of balanced growth path solutions implies exponential growth of
the overall production in time. We proof existence of balanced growth path
solutions if the initial distribution of individuals with respect to their
knowledge level satisfies a Pareto-tail condition. Furthermore we give first
insights into the existence of such solutions if in addition to production and
knowledge exchange the knowledge level evolves by geometric Brownian motion.
