The role of centrality and market size in a four-region asymmetric new economic geography model

Abstract

In this paper, we put forward a four-region new economic geography footloose entrepreneur model in which regions are differentiated by their size and their geographical position along a line. There are two distinct trade blocs, each of them consisting of a pair of regions. Direct and indirect trade between all regions is allowed, whereas factor mobility can occur only between regions of the same bloc. Given this more general geographical structure, as compared to previous studies, we are able to disentangle two manifestations of the market access effect: firms can take advantage of locating both in a more central region (centrality effect) and/or in a bigger region (local market size effect). The model is able to generate a plethora of long-term outcomes, including four equilibria with full agglomeration in each trade bloc that can be ranked by factor owners. Equilibria where industry is dispersed or agglomerated in a bloc and dispersed in the other one, are also possible as well as more complex attractors. Finally, by allowing direct and indirect trade between regions, we are able to look at the effect of trade integration on transit traffic by evaluating in a preliminary analysis the consequences of policies aiming at limiting transport volumes in a model with shifting industry.

Appendices

Appendix A

In Eq. 15 profits π _i for $i=\overline {1,4},$ are defined as follows:

$$\pi_{1}=\frac{A_{1}(1-C_{2})+A_{2}C_{1}}{(1-B_{1})(1-C_{2})-B_{2}C_{1}},\ \ \pi_{2}=\frac{A_{2}(1-B_{1})+A_{1}B_{2}}{(1-B_{1})(1-C_{2})-B_{2}C_{1}}, $$

$$\pi_{3}=\frac{A_{3}(1-B_{4})+A_{4}B_{3}}{(1-C_{3})(1-B_{4})-B_{3}C_{4}},\ \ \pi_{4}=\frac{A_{4}(1-C_{3})+A_{3}C_{4}}{(1-C_{3})(1-B_{4})-B_{3}C_{4}}, $$

where

$$A_{1}=\left( {\Theta}_{1}+\phi_{C}\phi_{D}h_{b}\beta_{3}Q_{3}{\Theta}_{3}\right) Q_{1},\ A_{2}=\left( {\Theta}_{2}+\phi_{D}h_{b}\beta_{3}Q_{3}{\Theta}_{3}\right) Q_{2}, $$

$$A_{3}=\left( {\Omega}_{3}+\phi_{D}\widetilde{h}_{b}\beta_{2}M_{2}{\Omega}_{2}\right) M_{3},\ \ A_{4}=\left( {\Omega}_{4}+\phi_{C}\phi_{D}\widetilde{h }_{b}\beta_{2}M_{2}{\Omega}_{2}\right) M_{4}, $$

$$B_{1}=\left( {\phi_{C}^{2}}{\phi_{D}^{2}}{h_{b}^{2}}\beta_{1}\beta_{3}Q_{3}\right) Q_{1},\ \ B_{2}=\phi_{C}\beta_{1}(h_{a}+\phi_{D}^{2}{h_{b}^{2}}\beta_{3}Q_{3})Q_{2}, $$

$$B_{3}=\phi_{C}\beta_{4}(\widetilde{h}_{a}+{\phi_{D}^{2}}\widetilde{h} _{b}^{2}\beta_{2}M_{2})M_{3},\ \ B_{4}=\left( {\phi_{C}^{2}}{\phi_{D}^{2}} \widetilde{h}_{b}^{2}\beta_{2}\beta_{4}M_{2}\right) M_{4}, $$

$$C_{1}=\phi_{C}\beta_{2}(h_{a}+{\phi_{D}^{2}}{h_{b}^{2}}\beta_{3}Q_{3})Q_{1},\ \ C_{2}={\phi_{D}^{2}}{h_{b}^{2}}\beta_{2}\beta_{3}Q_{2}Q_{3}, $$

$$C_{3}={\phi_{D}^{2}}\widetilde{h}_{b}^{2}\beta_{2}\beta_{3}M_{2}M_{3},\ \ C_{4}=\phi_{C}\beta_{3}(\widetilde{h}_{a}+{\phi_{D}^{2}}\widetilde{h} _{b}^{2}\beta_{2}M_{3})M_{4}, $$

$$\begin{array}{@{}rcl@{}} {\Theta}_{1} &=&\alpha_{1}z_{1}+\phi_{C}\alpha_{2}z_{2}+\phi_{C}\phi_{D}\alpha_{3}z_{3}+{\phi_{C}^{2}}\phi_{D}\alpha_{4}z_{4}, \\ {\Theta}_{2} &=&(\phi_{C}\alpha_{1}+\alpha_{2})z_{2}+\phi_{D}\alpha_{3}z_{3}+\phi_{C}\phi_{D}\alpha_{4}z_{4}, \\ {\Theta}_{3} &=&(\phi_{C}\phi_{D}\alpha_{1}+\phi_{D}\alpha_{2}+\alpha_{3})z_{3}+\phi_{C}\alpha_{4}z_{4}, \end{array} $$

$$\begin{array}{@{}rcl@{}} {\Omega}_{2} &=&\phi_{C}\alpha_{1}\widetilde{z}_{1}+(\phi_{C}\phi_{D}\alpha_{4}+\phi_{D}\alpha_{3}+\alpha_{2})\widetilde{z}_{2}, \\ {\Omega}_{3} &=&\phi_{C}\phi_{D}\alpha_{1}\widetilde{z}_{1}+\phi_{D}\alpha_{2}\widetilde{z}_{2}+(\alpha_{3}+\phi_{C}\alpha_{4}) \widetilde{z}_{3}, \\ {\Omega}_{4} &=&{\phi_{C}^{2}}\phi_{D}\alpha_{1}\widetilde{z}_{1}+\phi_{C}\phi_{D}\alpha_{2}\widetilde{z}_{2}+\phi_{C}\alpha_{3}\widetilde{z} _{3}+\alpha_{4}\widetilde{z}_{4}, \end{array} $$

$$Q_{1}=\frac{1}{\frac{1}{\kappa }-\beta_{1}\left( 1+{\phi_{C}^{4}}\phi_{D}^{2}z_{4}\kappa \beta_{4}\right) },\ \ Q_{2}=\frac{1}{\frac{1}{\kappa } -\beta_{2}h_{a}},\ \ Q_{3}=\frac{1}{\frac{1}{\kappa }-\beta_{3}h_{b}}, $$

$$M_{2}=\frac{1}{\frac{1}{\kappa }-\beta_{2}\widetilde{h}_{b}},\ \ M_{3}=\frac{1}{\frac{1}{\kappa }-\beta_{3}\widetilde{h}_{a}},\ \ M_{4}=\frac{1}{\frac{1}{\kappa }-\beta_{4}\left( 1+{\phi_{C}^{4}}{\phi_{D}^{2}}\widetilde{z}_{1}\kappa \beta_{1}\right) }, $$

$$\alpha_{1}=\frac{L_{s}}{{\Delta}_{1}},\ \ \ \alpha_{2}=\frac{L_{b}}{{\Delta}_{2}},\ \alpha_{3}=\frac{L_{s}}{{\Delta}_{3}},\ \ \ \alpha_{4}=\frac{L_{b}}{{\Delta}_{4}}, $$

$$\beta_{1}=\frac{x}{{\Delta}_{1}}\frac{E}{2},\ \beta_{2}=\frac{1-x}{{\Delta}_{2}}\frac{E}{2},\ \beta_{3}=\frac{y}{{\Delta}_{3}}\frac{E}{2},\ \beta_{4}=\frac{1-y}{{\Delta}_{4}}\frac{E}{2}, $$

$$\kappa =\frac{2\mu }{\sigma E}, $$

$$\begin{array}{@{}rcl@{}} z_{1} &=&z_{4}\left( 1-\kappa \beta_{4}(1-{\phi_{C}^{4}}\phi_{D}^{2})\right) ,\ \ z_{2}=z_{4}\left( 1-\kappa \beta_{4}(1-\phi _{C}^{2}{\phi_{D}^{2}})\right) , \\ z_{3} &=&z_{4}\left( 1-\kappa \beta_{4}(1-{\phi_{C}^{2}})\right) ,\ \ z_{4}= \frac{1}{1-\kappa \beta_{4}}, \end{array} $$

$$\begin{array}{@{}rcl@{}} \widetilde{z}_{1} &=&\frac{1}{1-\kappa \beta_{1}},\ \widetilde{z}_{2}= \widetilde{z}_{1}\left( 1-\kappa \beta_{1}(1-{\phi_{C}^{2}})\right) , \\ \widetilde{z}_{3} &=&\widetilde{z}_{1}\left( 1-\kappa \beta_{1}(1-\phi_{C}^{2}{\phi_{D}^{2}})\right) ,\ \ \widetilde{z}_{4}=\widetilde{z}_{1}\left( 1-\kappa \beta_{1}(1-{\phi_{C}^{4}}{\phi_{D}^{2}})\right) , \end{array} $$

$$h_{a}= 1+{\phi_{C}^{2}}{\phi_{D}^{2}}z_{4}\kappa \beta_{4},\ \ h_{b}= 1+\phi_{C}^{2}z_{4}\kappa \beta_{4}, $$

$$\widetilde{h}_{a}= 1+{\phi_{C}^{2}}{\phi_{D}^{2}}\widetilde{z}_{1}\kappa \beta_{1},\ \ \widetilde{h}_{b}= 1+{\phi_{C}^{2}}\widetilde{z}_{1}\kappa \beta_{1}. $$

Appendix B

The borders of the unit square I ² are denoted as follows:

$$I_{x0}=\left\{ (x,y):y = 0\right\} ,\ \ I_{x1}=\left\{ (x,y):y = 1\right\},$$

$$I_{0y}=\left\{ (x,y):x = 0\right\} ,\ \ I_{1y}=\left\{ (x,y):x = 1\right\} . $$

It is easy to see that these borders are invariant under map Z. Moreover, on each of these borders 2D map Z is reduced to the corresponding 1D map. Namely,

- on border I _x0 map Z is reduced to a 1D map, denoted z _x0, which is defined as

$$z_{x0}:x\mapsto z_{x0}(x)=\left\{ \begin{array}{lll} 0 & if & Z_{h0}(x)<0, \\ Z_{h0}(x) & if & 0\leq Z_{h0}(x)\leq 1, \\ 1 & if & Z_{h0}(x)>1, \end{array} \right. $$

where

$$ Z_{h0}(x)=x\left[ 1+\gamma (1-x)\frac{{\Omega}_{h}(x,0)-1}{1+x({\Omega}_{h}(x,0)-1)}\right] ; $$

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- on border I _x1 map Z is reduced to a 1D map, denoted z _x1, which is defined as

$$z_{x1}:x\mapsto z_{x1}(x)=\left\{ \begin{array}{lll} 0 & if & Z_{h1}(x)<0, \\ Z_{h1}(x) & if & 0\leq Z_{h1}(x)\leq 1, \\ 1 & if & Z_{h1}(x)>1, \end{array} \right. $$

where

$$ Z_{h1}(x)=x\left[ 1+\gamma (1-x)\frac{{\Omega}_{h}(x,1)-1}{1+x({\Omega}_{h}(x,1)-1)}\right] ; $$

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- on border I _0y map Z is reduced to a 1D map

$$z_{0y}:y\mapsto z_{0y}(y)=\left\{ \begin{array}{lll} 0 & if & Z_{f0}(y)<0, \\ Z_{f0}(y) & if & 0\leq Z_{f0}(y)\leq 1, \\ 1 & if & Z_{f0}(y)>1, \end{array} \right. $$

where

$$ Z_{f0}(y)=y\left[ 1+\gamma (1-y)\frac{{\Omega}_{f}(0,y)-1}{1+y({\Omega}_{f}(0,y)-1)}\right] ; $$

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- and, finally, on border I _1y the map Z is reduced to a 1D map

$$z_{1y}:y\mapsto z_{1y}(y)=\left\{ \begin{array}{lll} 0 & if & Z_{f1}(y)<0, \\ Z_{f1}(y) & if & 0\leq Z_{f1}(y)\leq 1, \\ 1 & if & Z_{f1}(y)>1, \end{array} \right. $$

where

$$ Z_{f1}(y)=y\left[ 1+\gamma (1-y)\frac{{\Omega}_{f}(1,y)-1}{1+y({\Omega}_{f}(1,y)-1)}\right] . $$

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Appendix C

Jacobian matrix of map Z (in its central part, without constraints) is given by

$$DZ=\left( \begin{array}{cc} Z_{h}{~}_{x}^{\prime }(x,y) & Z_{h}{~}_{y}^{\prime }(x,y) \\ Z_{f}{~}_{x}^{\prime }(x,y) & Z_{f}{~}_{y}^{\prime }(x,y) \end{array} \right) , $$

where

$$Z_{h}{~}_{x}^{\prime }(x,y)= 1+\gamma \left( (1-2x)\frac{{\Omega}_{h}(x,y)-1}{ 1+x({\Omega}_{h}(x,y)-1)}+x(1-x)\frac{{\Omega}_{h}{~}_{x}^{\prime }(x,y)-({\Omega}_{h}(x,y)-1)^{2}}{\left( 1+x({\Omega}_{h}(x,y)-1)\right)^{2}} \right) , $$

$$Z_{h}{~}_{y}^{\prime }(x,y)=\gamma x(1-x)\frac{{\Omega}_{h}{~}_{y}^{\prime }(x,y)}{\left( 1+x({\Omega}_{h}(x,y)-1)\right)^{2}}, $$

$$Z_{f}{~}_{x}^{\prime }(x,y)=\gamma y(1-y)\frac{{\Omega}_{f}{~}_{x}^{\prime }(x,y)}{\left( 1+y({\Omega}_{f}(x,y)-1)\right)^{2}}, $$

$$Z_{f}{~}_{y}^{\prime }(x,y)= 1+\gamma \left( (1-2y)\frac{{\Omega}_{f}(x,y)-1}{ 1+y({\Omega}_{f}(x,y)-1)}+y(1-y)\frac{{\Omega}_{f}{~}_{y}^{\prime }(x,y)-({\Omega}_{f}(x,y)-1)^{2}}{\left( 1+y({\Omega}_{f}(x,y)-1)\right)^{2}} \right)\!. $$

The eigenvalues are

$$\lambda_{1,2}(x,y)= 0.5\left( Z_{h}{~}_{x}^{\prime }(x,y)+Z_{f}{~}_{y}^{\prime }(x,y)\pm \sqrt{(Z_{h}{~}_{x}^{\prime }(x,y)-Z_{f}{~}_{y}^{\prime }(x,y))^{2}+ 4Z_{h}{~}_{y}^{\prime }(x,y)Z_{f}{~}_{x}^{\prime }(x,y)}\right) . $$

We use the following notations: λ ₁(x, y) = λ _h (x, y) and λ ₂(x, y) = λ _v (x, y). Eigenvalues of the fixed points of map Z are as follows:

$$C{}P_{00}:\ \ \ \lambda_{h}(0,0)= 1+\gamma ({\Omega}_{h}(0,0)-1),\ \ \lambda_{v}(0,0)= 1+\gamma ({\Omega}_{f}(0,0)-1), $$

$$C{}P_{11}:\ \ \ \lambda_{h}(1,1)= 1-\gamma \left( 1-\frac{1}{{\Omega}_{h}(0,0)} \right) ,\ \ \lambda_{v}(1,1)= 1-\gamma \left( 1-\frac{1}{{\Omega}_{f}(0,0)} \right) , $$

$$C{}P_{01}:\ \ \ \lambda_{h}(0,1)= 1+\gamma ({\Omega}_{h}(0,1)-1),\ \ \lambda_{v}(0,1)= 1-\gamma \left( 1-\frac{1}{{\Omega}_{f}(0,1)}\right) , $$

$$C{}P_{10}:\ \ \ \lambda_{h}(1,0)= 1-\gamma \left( 1-\frac{1}{{\Omega}_{h}(1,0)} \right) ,\ \ \lambda_{v}(1,0)= 1+\gamma ({\Omega}_{f}(1,0)-1), $$

$$B{}P_{0a}: \ \lambda_{h}(0,a)= 1+\gamma ({\Omega}_{h}(0,a)-1),\ \ \lambda_{v}(0,a)= 1+\gamma a(1-a){\Omega}_{f}{~}_{y}^{\prime }(0,a), $$

$$B{}P_{1a}:\ \lambda_{h}(1,a)= 1-\gamma \left( 1-\frac{1}{{\Omega}_{h}(1,a)} \right) ,\ \lambda_{v}(1,a)= 1+\gamma a(1-a){\Omega}_{f}{~}_{y}^{\prime }(1,a), $$

$$B{}P_{a0}:\ \lambda_{h}(a,0)= 1+\gamma a(1-a){\Omega}_{h}{~}_{x}^{\prime }(a,0),\ \lambda_{v}(a,0)= 1+\gamma ({\Omega}_{f}(a,0)-1), $$

$$B{}P_{a1}: \ \lambda_{h}(a,1)= 1+\gamma a(1-a){\Omega}_{h}{~}_{x}^{\prime }(a,1),\ \lambda_{v}(a,1)= 1-\gamma \left( 1-\frac{1}{{\Omega}_{f}(a,1)} \right) , $$

$$\begin{array}{@{}rcl@{}} I{}P_{ab}: \lambda_{h,v}(a,b)= 1 + 0.5\gamma (a(1-a){\Omega}_{h}{~}_{x}^{\prime }(a,b)+b(1-b){\Omega}_{f}{~}_{y}^{\prime }(a,b)\pm \\ \sqrt{(a(1-a){\Omega}_{h}{~}_{x}^{\prime }(a,b)-b(1-b){\Omega}_{f}{~}_{y}^{\prime }(a,b))^{2}+ 4a(1-a)b(1-b){\Omega}_{h}{~}_{y}^{\prime }(a,b){\Omega}_{f}{~}_{x}^{\prime }(a,b)}). \end{array} $$