In this talk I will discuss a mean-field game model proposed by R.E. Lucas and B. Moll 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. In a joint work with Luca Rossi, we proved the existence of critical traveling waves which yield balanced growth paths for the described economy, supposed to be the expected stable growth in the long run. I will describe one possible strategy to construct those (and other) traveling waves, mentioning existence and nonexistence results, and I will address several interesting questions which remain open.