The Existence and Uniqueness of Global Solutions to the Initial Value Problem for the System of Nonlinear Integropartial Differential Equations in Spatial Economics: The Dynamic Continuous Dixit-Stiglitz-Krugman Model in an Urban-Rural Setting

Minoru Tabata,Nobuoki Eshima

doi:10.1155/2015/760136

Abstract

Assume that economic activities are conducted in a bounded continuous domain where workers move toward regions that offer higher real wages and away from regions that offer below-average real wages. The density of real wages is calculated by solving the nominal wage equation of the continuous Dixit-Stiglitz-Krugman model in an urban-rural setting. The evolution of the density of workers is described by an unknown function of the replicator equation whose growth rate is equal to the difference between the density of real wages and the average real wage. Hence, the evolution of the densities of workers and real wages is described by the system of the nominal wage equation and the replicator equation. This system of equations is an essentially new kind of system of nonlinear integropartial differential equations in the theory of functional equations. The purpose of this paper is to obtain a sufficient condition for the initial value problem for this system to have a unique global solution.

Highlights

The new economic geography (NEG) is a new branch of spatial economics that was initiated by Krugman in the early 1990s
In this paper we prove a sufficient condition for the initial value problem for the dcDSK system to have a unique global solution and obtain estimates of the solution
Where w = w(t, x) is an unknown function that denotes the density of nominal wages at time t ≥ 0 and at point x ∈ D

Summary

Introduction

The new economic geography (NEG) is a new branch of spatial economics that was initiated by Krugman in the early 1990s. Abstract and Applied Analysis (see [15] and [16, Chapter 3]) His dynamic model is very important in spatial economics since it describes economies of agglomeration in the case where workers move from one point to another to seek higher real wages within a finite set of points at which economic activities are conducted

The System of Equations

Result and Discussion

Solutions of the Nominal Wage Equation

The Iteration Scheme