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We analyze a mean-field game model proposed by economists R.E. Lucas and B.Moll (2014) to describe economic systems where production is based on knowledgegrowth and diffusion. This model reduces to a PDE system where a backwardHamilton-Jacobi-Bellman equation is coupled with a forward KPP-type equationwith nonlocal reaction term. We study the existence of traveling waves for thismean-field game system, obtaining the existence of both critical andsupercritical waves. In particular we prove a conjecture raised by economistson the existence of a critical balanced growth path for the described economy,supposed to be the expected stable growth in the long run. We also providenonexistence results which clarify the role of parameters in the economicmodel.

Traveling waves for a nonlocal KPP equation and mean-field game models of knowledge diffusion