Unique global solution of an integral-differential equation of Footloose Entrepreneur model in new economic geography

TL;DR

This paper establishes the existence and uniqueness of solutions for a Footloose Entrepreneur model in economic geography, analyzes stability, and explores how parameters influence solution patterns through numerical simulations.

Contribution

It introduces a novel integral-differential equation formulation for the model and proves a unique global solution using Banach fixed point theorem.

Findings

Numerical solutions converge to spike-shaped stationary solutions.

Number of spikes decreases with lower transport costs.

Strengthening preference for variety affects solution patterns.

Abstract

This paper studies the Footloose Entrepreneur model in new economic geography in continuous space. In an appropriate function space, the model is formulated as an initial value problem for an infinite-dimensional ordinary differential equation. A unique global solution is constructed based on the Banach fixed point theorem. The stability of a homogeneous stationary solution is then investigated and numerical simulations of the asymptotic behavior of the solution are performed. Numerical solutions starting near the unstable homogeneous stationary solution converge to spike-shaped stationary solutions, and the number of spikes decreases with decreasing transport costs and strengthening preference for variety.