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

Abstract

We analyze a mean-field game model proposed by economists Lucas and Moll [J. Political Econ. 122 (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.

In order to prove these results, we build fixed point arguments on the sets of critical waves for the forced speed problem arising from the coupling in the KPP-type equation. To this purpose, we provide a full characterization of the whole family of traveling waves for a new class of KPP-type equations with nonlocal and nonhomogeneous reaction terms. This latter analysis has independent interest since it shows new phenomena induced by the nonlocal effects and a different picture of critical waves, compared to the classical literature on Fisher–KPP equations.