The Multiregional Core-periphery Model: The Role of the Spatial Topology

Abstract

We use the multiregional core-periphery model of the new economic geography to analyze and compare the agglomeration and dispersion forces shaping the location of economic activity for a continuum of network topologies — spatial or geographic configuration — characterized by their degree of centrality, and comprised between two extremes represented by the homogenous (ring) and the heterogeneous (star) configurations. Resorting to graph theory, we systematically extend the analytical tools and graphical representations of the core-periphery model for alternative spatial configuration, and study the sustain and break points. We study new phenomena such as the infeasibility of the dispersed equilibrium in the heterogeneous space, resulting in the introduction of the concept pseudo flat-earth as a long-run equilibrium corresponding to an uneven distribution of economic activity between regions.

Appendices

Appendix A: Real Wages in a Multiregional Economy when One Region is Agglomerating

When only one region — say, region 1 — is agglomerating we set λ ₁ = 1 and λ _i = 0 ∀i ≠ 1 in Eq. 2, thereby obtaining:

$$y_{1} = \frac{1 + (N-1)\mu}{N}; \quad y_{i} = \frac{1-\mu}{N},\quad i=2,\dots,N. $$

Since by Eq. 9 the nominal wage of region 1 is equal to 1, we can substitute and get price indices (3):

$$g_{1} = 1; \quad g_{i} = \tau_{1i},\quad i=2,\dots,N. $$

Inserting the price indices and income we obtain nominal wages (4):

$$w_{1} = 1 $$

$$w_{i} = \left( \frac{1-\mu}{N} \tau_{1i}^{\sigma-1} + \frac{1+(N-1)\mu}{N} \tau_{i1}^{1-\sigma} + \sum\limits_{j=2\ne i}^{N} \frac{1-\mu}{N} \tau_{1j}^{\sigma-1} \tau_{ij}^{1-\sigma}\right)^{1/\sigma}. $$

as well as the real wage (5):

$$\omega_{1}^{\sigma} = 1 $$

$$\omega_{i}^{\sigma} = \frac{1-\mu}{N} \tau_{1i}^{\sigma-1-\mu\sigma} + \frac{1+(N-1)\mu}{N} \tau_{i1}^{1-\sigma} \tau_{1i}^{-\mu\sigma} + \frac{1-\mu}{N} \tau_{1i}^{-\mu\sigma} \sum\limits_{j=2\ne i}^{N} \tau_{1j}^{\sigma-1} \tau_{ij}^{1-\sigma} $$

Appendix B: Real Wages in a Multiregional Economy with N = 4

Following the same procedure as in Appendix A and setting N = 4, we obtain the following expressions of real wages:

$$\begin{array}{ll} \omega_{1}^{\sigma} &= 1 \\ \omega_{2}^{\sigma} &= \frac{1-\mu}{4} \tau_{12}^{\sigma-1-\mu\sigma} + \frac{1+3\mu}{4} \tau_{21}^{1-\sigma} \tau_{12}^{-\mu\sigma} + \frac{1-\mu}{4} \tau_{12}^{-\mu\sigma} \left( \tau_{13}^{\sigma-1} \tau_{23}^{1-\sigma} + \tau_{14}^{\sigma-1} \tau_{24}^{1-\sigma} \right) \\ \omega_{3}^{\sigma} &= \frac{1-\mu}{4} \tau_{13}^{\sigma-1-\mu\sigma} + \frac{1+3\mu}{4} \tau_{31}^{1-\sigma} \tau_{13}^{-\mu\sigma} + \frac{1-\mu}{4} \tau_{13}^{-\mu\sigma} \left( \tau_{12}^{\sigma-1} \tau_{32}^{1-\sigma} + \tau_{14}^{\sigma-1} \tau_{34}^{1-\sigma} \right) \\ \omega_{4}^{\sigma} &= \frac{1-\mu}{4} \tau_{14}^{\sigma-1-\mu\sigma} + \frac{1+3\mu}{4} \tau_{41}^{1-\sigma} \tau_{14}^{-\mu\sigma} + \frac{1-\mu}{4} \tau_{14}^{-\mu\sigma} \left( \tau_{12}^{\sigma-1} \tau_{42}^{1-\sigma} + \tau_{13}^{\sigma-1} \tau_{43}^{1-\sigma} \right) \end{array} $$

Appendix C: Price Index Derivative

Raising the price index Eq. 3 to 1−σ yields:

$$g_{i}^{1-\sigma} = \lambda_{i} w_{i}^{1-\sigma} + \sum\limits_{j=1\ne i}^{N} \lambda_{j} (w_{j} \tau_{ji})^{1-\sigma}, $$

Taking logs:

$$(1-\sigma)\ln g_{i} = \ln \left( \lambda_{i} w_{i}^{1-\sigma} + \sum\limits_{j=1\ne i}^{N} \lambda_{j} (w_{j} \tau_{ji})^{1-\sigma} \right) $$

and taking the derivative:

$$(1-\sigma)\frac{dg_{i}}{g_{i}} = \frac{d \left( \lambda_{i} w_{i}^{1-\sigma} + {\sum}_{j=1\ne i}^{N} \lambda_{j} (w_{j} \tau_{ji})^{1-\sigma} \right)}{\lambda_{i} w_{i}^{1-\sigma} + {\sum}_{j=1\ne i}^{N} \lambda_{j} (w_{j} \tau_{ji})^{1-\sigma}} $$

The denominator of the right-hand side is $g_{i}^{1-\sigma }$, which can be brought to the left side:

$$(1-\sigma)\frac{dg_{i}}{g_{i}^{\sigma}} = d \left( \lambda_{i} w_{i}^{1-\sigma} + \sum\limits_{j=1\ne i}^{N} \lambda_{j} (w_{j} \tau_{ji})^{1-\sigma} \right). $$

Totally differentiating the right-hand side, we arrive at Eq. 12:

$$\begin{array}{ll} (1-\sigma)\frac{dg_{i}}{g_{i}^{\sigma}} = w_{i}^{1-\sigma} d\lambda_{i} + (1-\sigma)\lambda_{i}w_{i}^{-\sigma}dw_{i} +\\ \quad+\quad{\sum}_{j=1\ne i}^{N} \left( (w_{j} \tau_{ji})^{1-\sigma} d\lambda_{j} + (1-\sigma)\lambda_{j} \tau_{ji}^{1-\sigma} w_{j}^{-\sigma} dw_{j} \right), \end{array} $$

Appendix D: Wage Derivative

Raising wage Eq. 4 to σ yields:

$$w_{i}^{\sigma} = y_{i} g_{i}^{\sigma-1} + \sum\limits_{j=1\ne i}^{N} y_{j} g_{j}^{\sigma-1} \tau_{ij}^{1-\sigma}, $$

Taking logs:

$$\sigma \ln w_{i} = \ln \left( y_{i} g_{i}^{\sigma-1} + \sum\limits_{j=1\ne i}^{N} y_{j} g_{j}^{\sigma-1} \tau_{ij}^{1-\sigma} \right), $$

and taking derivatives:

$$\sigma \frac{dw_{i}}{w_{i}} = \frac{d \left( y_{i} g_{i}^{\sigma-1} + {\sum}_{j=1\ne i}^{N} y_{j} g_{j}^{\sigma-1} \tau_{ij}^{1-\sigma} \right)}{y_{i} g_{i}^{\sigma-1} + {\sum}_{j=1\ne i}^{N} y_{j} g_{j}^{\sigma-1} \tau_{ij}^{1-\sigma}}. $$

The denominator of the right-hand side is $w_{i}^{\sigma }$, so it can be brought to the left side:

$$\sigma \frac{dw_{i}}{w_{i}^{1-\sigma}} = d \left( y_{i} g_{i}^{\sigma-1} + \sum\limits_{j=1\ne i}^{N} y_{j} g_{j}^{\sigma-1} \tau_{ij}^{1-\sigma} \right) $$

Totally differentiating the right-hand side, we get Eq. 13:

$$\begin{array}{ll} \sigma \frac{dw_{i}}{w_{i}^{1-\sigma}} = g_{i}^{\sigma-1}dy_{i} + (\sigma-1)y_{i}\sigma_{i}^{\sigma-2}dg_{i} \\ \quad\quad\quad\quad+{\sum}_{j=1\ne i}^{N} \left( g_{j}^{\sigma-1}\tau_{ij}^{1-\sigma}dy_{j} + (\sigma-1)y_{j} \tau_{ij}^{1-\sigma} g_{j}^{\sigma-2} dg_{j} \right), \end{array} $$

Appendix E: Real wage Derivative

Totally differentiating Eq. 5 yields:

$$d\omega_{i} = g_{i}^{-\mu}dw_{i} - \mu w_{i} g_{i}^{-\mu-1}dg_{i}. $$

Multiplying both sides by $g_{i}^{\mu }$:

$$g_{i}^{\mu} d\omega_{i} = dw_{i} - \mu w_{i} g_{i}^{-1}dg_{i}, $$

results in Eq. 14

$$g_{i}^{\mu} d\omega_{i} = dw_{i} - \mu w_{i} \frac{dg_{i}}{g_{i}}. $$