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

TL;DR

This paper investigates traveling wave solutions in a nonlocal KPP equation coupled with a Hamilton-Jacobi-Bellman equation within a mean-field game framework, confirming the existence of critical growth paths and clarifying parameter effects.

Contribution

It establishes the existence of both critical and supercritical traveling waves in a mean-field game model of knowledge diffusion, confirming a conjecture about stable economic growth paths.

Findings

Existence of critical and supercritical traveling waves confirmed.

A conjecture on the stable growth path is proven.

Nonexistence results highlight parameter influences.

Abstract

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.