  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 knowledge
growth and diffusion. This model reduces to a PDE system where a backward
Hamilton-Jacobi-Bellman equation is coupled with a forward KPP-type equation
with nonlocal reaction term. We study the existence of traveling waves for this
mean-field game system, obtaining the existence of both critical and
supercritical waves. In particular we prove a conjecture raised by economists
on 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 provide
nonexistence results which clarify the role of parameters in the economic
model.
