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Continuous space core-periphery model with transport costs in differentiated agriculture

Kensuke Ohtake

TL;DR

This work extends the core-periphery framework with differentiated agriculture to a continuous one-dimensional racetrack space, coupling Dixit–Stiglitz demand with transport costs via an integral-differential formulation. It derives a homogeneous stationary solution and uses Fourier analysis to characterize stability, revealing a redispersion phenomenon: the homogeneous state is unstable at intermediate manufacturing transport costs but can be stabilized by sufficiently low manufacturing transport costs or stronger farming variety preferences. Numerically, time-evolution settles into spike-like agglomerations, with the average spike count first decreasing and then increasing as manufacturing transport costs fall; lower agricultural transport costs and higher variety preference further promote agglomeration. Together, the results illuminate how transport costs and variety preferences shape spatial patterns in differentiated agricultural goods, offering a rigorous continuous-space perspective on agglomeration dynamics and self-organization in economic geography, supported by a detailed numerical scheme on a racetrack geometry.

Abstract

The core-periphery model with transport costs in differentiated agriculture is extended to continuous space. A homogeneous stationary solution is unstable but exhibits redispersion that it is stabilized by sufficiently low manufacturing transport costs or sufficiently strong preference for manufacturing variety. It is numerically observed that a solution starting from around the unstable homogeneous solution eventually forms a spike-like agglomeration. Furthermore, the redispersion also appears in the sense that the number of the spikes goes from decreasing to increasing as the manufacturing transport costs decrease. It is also observed that lower agricultural transport costs and stronger preference for agricultural variety promote agglomeration.

Continuous space core-periphery model with transport costs in differentiated agriculture

TL;DR

This work extends the core-periphery framework with differentiated agriculture to a continuous one-dimensional racetrack space, coupling Dixit–Stiglitz demand with transport costs via an integral-differential formulation. It derives a homogeneous stationary solution and uses Fourier analysis to characterize stability, revealing a redispersion phenomenon: the homogeneous state is unstable at intermediate manufacturing transport costs but can be stabilized by sufficiently low manufacturing transport costs or stronger farming variety preferences. Numerically, time-evolution settles into spike-like agglomerations, with the average spike count first decreasing and then increasing as manufacturing transport costs fall; lower agricultural transport costs and higher variety preference further promote agglomeration. Together, the results illuminate how transport costs and variety preferences shape spatial patterns in differentiated agricultural goods, offering a rigorous continuous-space perspective on agglomeration dynamics and self-organization in economic geography, supported by a detailed numerical scheme on a racetrack geometry.

Abstract

The core-periphery model with transport costs in differentiated agriculture is extended to continuous space. A homogeneous stationary solution is unstable but exhibits redispersion that it is stabilized by sufficiently low manufacturing transport costs or sufficiently strong preference for manufacturing variety. It is numerically observed that a solution starting from around the unstable homogeneous solution eventually forms a spike-like agglomeration. Furthermore, the redispersion also appears in the sense that the number of the spikes goes from decreasing to increasing as the manufacturing transport costs decrease. It is also observed that lower agricultural transport costs and stronger preference for agricultural variety promote agglomeration.
Paper Structure (32 sections, 11 theorems, 117 equations, 11 figures)

This paper contains 32 sections, 11 theorems, 117 equations, 11 figures.

Table of Contents

  1. Introduction
  2. The model
  3. Modeling
  4. Dixit-Stiglitz framework
  5. Multi-regional extension
  6. Continuous space model
  7. Racetrack setting
  8. Stationary solution
  9. Homogeneous stationary solution
  10. Linearization
  11. Eigenvalues in some extreme situations
  12. Eigenvalue in detail
  13. Critical points
  14. Critical curves
  15. Numerical examples of asymptotic behavior
  16. ...and 17 more sections

Key Result

Theorem 1

Under homphi, the homogeneous states are given by where $E_\alpha$ and $E_\beta$ are defined by respectively.

Figures (11)

  • Figure 1: Eigenvalues as a function of $\tau_M$ for $n=1$
  • Figure 2: Eigenvalues as functions of $\tau_M$ for $n=1$ to $6$
  • Figure 3: Heatmaps of the eignevalue in $(\tau_M, \tau_A)$-space
  • Figure 4: Critical curve and two domains
  • Figure 5: Critical curves for $n=1$ to $6$
  • ...and 6 more figures

Theorems & Definitions (11)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • ...and 1 more