Industrial configuration in an economy with low transportation costs

Abstract

We examine how the spatial economy with multiple industries is shaped when interregional trade costs and intraregional commuting costs are low. All industries are characterized by increasing returns to scale and monopolistic competition, and they are differentiated by their trade costs and the degree of intra-industry competition measured by their firm numbers. We find some distinct rules in industrial location. First, at most, one industry disperses, while others agglomerate in a region according to their ratios of relative trade costs to firm numbers. Second, industries with stronger competition constitute a smaller region, while those with higher trade costs compose a larger region. The results are consistent with the classical Weberian location theory and suggest that the degree of intra-industry competition also becomes an essential factor to determine industrial location when transportation costs are small. Finally, the population differential between the regions monotonically decreases in the relative commuting cost.

Appendices

Appendix A: Proof of Proposition 1

We show that any distribution pattern in which $k(\ge \!\!\!2)$ types of industries disperse is unstable for a sufficiently small $\tau$. On the contrary, assume a stable equilibrium in which $k$ types of industries disperse. Number the dispersing industries by $1, \ldots ,k$, and consider matrix $\Delta \equiv (\delta _{ij})_{k\times k}$, where $\delta _{ij}$ is defined by (8). Because of the stability of this equilibrium with respect to dynamic system (9), all the real parts of the eigenvalues of $(-\Delta )$ must be negative, and, hence, $\left| \Delta \right| > 0$.

Let

$$\begin{aligned} \Upsilon \equiv \biggr (\delta _{ij}-\frac{\rho }{2}L_{j}\tau \biggr ) _{k\times k},\quad \Theta _{i}\equiv \frac{\rho }{2}L_{i}\tau \left( \begin{array}{c} 1 \\ \vdots \\ 1 \end{array} \right) _{k\times 1}, \end{aligned}$$

and let $(\Theta _{i},\Upsilon ^{-i})$ be the matrix $\Upsilon$ with its $i$ th column replaced by $\Theta _{i}$. It holds that

$$\begin{aligned} \left|\Delta \right|=\left|\Upsilon \right|+\sum _{i=1}^{k}\left|(\Theta _{i},\Upsilon ^{-i})\right|\text{.} \end{aligned}$$

(18)

In (18), $|\Upsilon |$ is a $\tau$-polynomial of degree $2k$, while $|(\Theta _{i},\Upsilon ^{-i})|$ is a $\tau$-polynomial of degree $2k-1$. According to Lemma 4, all terms with orders below $(2k\!-\!2)$ in (18) disappear. In other words, the terms of order $(2k\!-\!2)$ in the polynomials are significant if $\tau$ is small enough. Furthermore, according to Lemma 5, $|\Delta |\!<\!0$ holds when $\tau$ is small. The contradiction implies that plural industries could not disperse simultaneously.$\square$

Lemma 4

All terms with orders below $(2k-2)$ in (18) are zero.

Proof

First, we consider $\left|\Upsilon \right|$. The elements of $\Upsilon$ are composed of $\tau ^{2}$-terms ($\mu _{1}$, $\xi _{1}$, $\psi _{i}$) and $\tau$-terms ($\mu _{2}$, $\xi _{2}$). If $\left|\Upsilon \right|$ is decomposed into a sum of determinants, in which each column is composed of $\tau ^{2}$-terms only or $\tau$-terms only, the terms of order below $(2k-2)$ in $\left|\Upsilon \right|$ could then be represented as a sum of determinants with at least three columns that are composed of only $\tau$-terms. One of the determinants, for example, is

$$\begin{aligned}&\left|\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} -\omega _{1}\mu _{2}-\omega _{1}\frac{L_{1}}{L_{1}}\xi _{2}&-\omega _{2}\mu _{2}-\omega _{1}\frac{L_{2}}{L_{1}}\xi _{2}&-\omega _{3}\mu _{2}-\omega _{1}\frac{L_{3}}{L_{1}}\xi _{2}&\delta _{14}-\frac{\rho }{2} L_{4}\tau&\cdots&\delta _{1k}-\frac{\rho }{2}L_{k}\tau \\ -\omega _{1}\mu _{2}-\omega _{2}\frac{L_{1}}{L_{2}}\xi _{2}&-\omega _{2}\mu _{2}-\omega _{2}\frac{L_{2}}{L_{2}}\xi _{2}&-\omega _{3}\mu _{2}-\omega _{2}\frac{L_{3}}{L_{2}}\xi _{2}&\delta _{24}-\frac{\rho }{2} L_{4}\tau&\cdots&\delta _{2k}-\frac{\rho }{2}L_{k}\tau \\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots \\ -\omega _{1}\mu _{2}-\omega _{k}\frac{L_{1}}{L_{k}}\xi _{2}&-\omega _{2}\mu _{2}-\omega _{k}\frac{L_{2}}{L_{k}}\xi _{2}&-\omega _{3}\mu _{2}-\omega _{k}\frac{L_{3}}{L_{k}}\xi _{2}&\delta _{k4}-\frac{\rho }{2} L_{4}\tau&\cdots&\delta _{kk}-\frac{\rho }{2}L_{k}\tau \end{array} \right|\\&\qquad =\left|\begin{array}{cccc} -\omega _{1}\mu _{2}-\omega _{1}\frac{L_{1}}{L_{1}}\xi _{2}&-\omega _{2}\mu _{2}-\omega _{1}\frac{L_{2}}{L_{1}}\xi _{2}&-\omega _{3}\mu _{2}-\omega _{1}\frac{L_{3}}{L_{1}}\xi _{2}&\cdots \\ L_{1}(\frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}})\xi _{2}&L_{2}( \frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}})\xi _{2}&L_{3}(\frac{ \omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}})\xi _{2}&\cdots \\ \vdots&\vdots&\vdots&\vdots \\ L_{1}(\frac{\omega _{1}}{L_{1}}-\frac{\omega _{k}}{L_{k}})\xi _{2}&L_{2}( \frac{\omega _{1}}{L_{1}}-\frac{\omega _{k}}{L_{k}})\xi _{2}&L_{3}(\frac{ \omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}})\xi _{2}&\cdots \end{array} \right|\\&\qquad =\left|\begin{array}{cccc} -\omega _{1}\mu _{2}-\omega _{1}\frac{L_{1}}{L_{1}}\xi _{2}&(\frac{L_{2}}{ L_{1}}\omega _{1}-\omega _{2})\mu _{2}&(\frac{L_{3}}{L_{1}}\omega _{1}-\omega _{3})\mu _{2}&\cdots \\ L_{1}(\frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}})\xi _{2}&0&0&\cdots \\ \vdots&\vdots&\vdots&\vdots \\ L_{1}(\frac{\omega _{1}}{L_{1}}-\frac{\omega _{k}}{L_{k}})\xi _{2}&0&0&\cdots \end{array} \right|=0\text{.} \end{aligned}$$

Of course, this result holds even if the columns with $\tau$-terms are in different places.

Next, we consider $\sum _{i=1}^{k}\left|(\Theta _{i},\Upsilon ^{-i})\right|$. If $\left|(\Theta _{i},\Upsilon ^{-i})\right|$ is decomposed into a sum of determinants, as in the above case, the term of order below $(2k-2)$ in $|(\Theta _{i},\Upsilon ^{-i})|$ could then be represented as a sum of determinants with at least two columns that are composed of $\tau$-terms only except for the $i$th column. One of the determinants, for example, is

$$\begin{aligned}&\left|\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }cc} \frac{\rho }{2}\tau L_{1}&-\omega _{2}\mu _{2}-\omega _{1}\frac{L_{2}}{ L_{1}}\xi _{2}&-\omega _{3}\mu _{2}-\omega _{1}\frac{L_{3}}{L_{1}}\xi _{2}&\delta _{14}-\frac{\rho }{2}L_{4}\tau&\cdots&\delta _{1k}-\frac{\rho }{ 2}L_{k}\tau \\ \frac{\rho }{2}\tau L_{1}&-\omega _{2}\mu _{2}-\omega _{2}\frac{L_{2}}{ L_{2}}\xi _{2}&-\omega _{3}\mu _{2}-\omega _{2}\frac{L_{3}}{L_{2}}\xi _{2}&\delta _{24}-\frac{\rho }{2}L_{4}\tau&\cdots&\delta _{2k}-\frac{\rho }{ 2}L_{k}\tau \\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots \\ \frac{\rho }{2}\tau L_{1}&-\omega _{2}\mu _{2}-\omega _{k}\frac{L_{2}}{ L_{k}}\xi _{2}&-\omega _{3}\mu _{2}-\omega _{k}\frac{L_{3}}{L_{k}}\xi _{2}&\delta _{k4}-\frac{\rho }{2}L_{4}\tau&\cdots&\delta _{kk}-\frac{\rho }{ 2}L_{k}\tau \end{array} \right|\\&\quad =\left|\begin{array}{cccc} \frac{\rho }{2}\tau L_{1}&-\omega _{2}\mu _{2}-\omega _{1}\frac{L_{2}}{ L_{1}}\xi _{2}&-\omega _{3}\mu _{2}-\omega _{1}\frac{L_{3}}{L_{1}}\xi _{2}&\cdots \\ 0&L_{2}(\frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}})\xi _{2}&L_{3}(\frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}})\xi _{2}&\cdots \\ \vdots&\vdots&\vdots&\vdots \\ 0&L_{2}(\frac{\omega _{1}}{L_{1}}-\frac{\omega _{k}}{L_{k}})\xi _{2}&L_{3}(\frac{\omega _{1}}{L_{1}}-\frac{\omega _{k}}{L_{k}})\xi _{2}&\cdots \end{array} \right|\\&\quad =\frac{\rho \tau L_{1}}{2}\left|\begin{array}{ccc} L_{2}(\frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}})\xi _{2}&L_{3}( \frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}})\xi _{2}&\cdots \\ \vdots&\vdots&\vdots \\ L_{2}(\frac{\omega _{1}}{L_{1}}-\frac{\omega _{k}}{L_{k}})\xi _{2}&L_{3}( \frac{\omega _{1}}{L_{1}}-\frac{\omega _{k}}{L_{k}})\xi _{2}&\cdots \end{array} \right|\\&\quad =0. \end{aligned}$$

Of course, this result holds even if the columns with $\tau$-terms, and $\Theta _{i}$ are in different places.

Therefore, we have shown that all terms of orders below $(2k\!-\!2)$ in$|\Delta |$ are zero. $\square$

Lemma 5

The term of order $2k-2$ in $|\Upsilon |$ is negative, while the term of order $(2k-2)$ in $\sum _{i=1}^{k}|(\Theta _{i},\Upsilon ^{-i})|$ is zero.

Proof

First, for $m\ne n\ne i\in \{1,\ldots ,k\}$, let

$$\begin{aligned} \mathcal{A }&\equiv \sum \limits _{i=3}^{k}\biggr (\frac{\omega _{3}L_{i}}{\omega _{i}L_{3}}\biggr )^{2}\biggr (\frac{\omega _{1}}{L_{1}}-\frac{\omega _{i}}{ L_{i}}\biggr ), \\ \mathcal{B }&\equiv \frac{\omega _{3}^{2}}{L_{3}}\psi _{3}+\xi _{1}\sum _{i=3}^{k}\biggr (\frac{\omega _{3}L_{i}}{\omega _{i}L_{3}}\biggr )^{2} \biggr (\frac{\omega _{i}^{2}}{L_{i}}-\frac{\omega _{1}^{2}}{L_{1}}\biggr ). \end{aligned}$$

Then

$$\begin{aligned}&\left|\begin{array}{ll} \frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}}&(\frac{\omega _{2}^{2} }{L_{2}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1} \\ \mathcal{A }&\mathcal{B } \end{array} \right|\nonumber \\&\quad = \frac{\omega _{3}^{2}\psi _{3}}{L_{3}}\biggr (\frac{\omega _{1}}{L_{1}}- \frac{\omega _{2}}{L_{2}}\biggr )+\sum _{i=3}^{k}\biggr (\frac{\omega _{3}L_{i} }{\omega _{i}L_{3}}\biggr )^{2}\left|\begin{array}{ll} \frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}}&(\frac{\omega _{2}^{2} }{L_{2}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1} \\ \frac{\omega _{1}}{L_{1}}-\frac{\omega _{i}}{L_{i}}&(\frac{\omega _{i}^{2} }{L_{i}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1} \end{array} \right|\nonumber \\&\quad = \frac{\omega _{3}^{2}\psi _{3}}{L_{3}}\biggr \{\frac{\omega _{1}}{L_{1}}- \frac{\omega _{2}}{L_{2}}+\frac{\xi _{1}}{L_{1}L_{2}}\sum _{i=3}^{k}\frac{1}{\omega _{i}^{2}\psi _{i} }[\omega _{1}\omega _{2}(\omega _{1}-\omega _{2})L_{i}\nonumber \\&\;\qquad +\ \omega _{2}\omega _{i}(\omega _{2}-\omega _{i})L_{1}+\omega _{i}\omega _{1}(\omega _{i}-\omega _{1})L_{2}]\biggr \}. \end{aligned}$$

(19)

For convenience, we introduce notation

$$\begin{aligned} \varpi (m,n,i)\equiv&\biggr (\frac{\omega _{m}}{L_{m}}-\frac{\omega _{n}}{ L_{n}}\biggr )\nonumber \\&\times [\omega _{m}\omega _{n}(\omega _{m}-\omega _{n})L_{i}+\omega _{n}\omega _{i}(\omega _{n}-\omega _{i})L_{m}+\omega _{i}\omega _{m}(\omega _{i}-\omega _{m})L_{n}]\nonumber \\ \end{aligned}$$

(20)

for $m,n,i\in \{1,2,\ldots ,k\}$. Then, it holds that

$$\begin{aligned} \varpi (m,n,i)+\varpi (n,i,m)+\varpi (i,m,n)=0. \end{aligned}$$

(21)

Now, we consider the $\tau ^{(2k-2)}$-term in $\left|\Upsilon \right|$, which can be represented as a sum of determinants that has only two columns with $\tau$-terms. One of the determinant, for example, is calculated as

$$\begin{aligned}&\left|\begin{array}{c@{\quad }cccc} -\omega _{1}\mu _{2}-\omega _{1}\frac{L_{1}}{L_{1}}\xi _{2}&-\omega _{2}\mu _{2}-\omega _{1}\frac{L_{2}}{L_{1}}\xi _{2}&\omega _{3}^{2}\mu _{1}+\omega _{1}^{2}\frac{L_{3}}{L_{1}}\xi _{1}&\cdots&\omega _{k}^{2}\mu _{1}+\omega _{1}^{2}\frac{L_{k}}{L_{1}}\xi _{1} \\ -\omega _{1}\mu _{2}-\omega _{2}\frac{L_{1}}{L_{2}}\xi _{2}&-\omega _{2}\mu _{2}-\omega _{2}\frac{L_{2}}{L_{2}}\xi _{2}&\omega _{3}^{2}\mu _{1}+\omega _{2}^{2}\frac{L_{3}}{L_{2}}\xi _{1}&\cdots&\omega _{k}^{2}\mu _{1}+\omega _{2}^{2}\frac{L_{k}}{L_{2}}\xi _{1} \\ -\omega _{1}\mu _{2}-\omega _{3}\frac{L_{1}}{L_{3}}\xi _{2}&-\omega _{2}\mu _{2}-\omega _{3}\frac{L_{2}}{L_{3}}\xi _{2}&\omega _{3}^{2}(\mu _{1}+\xi _{1}+\psi _{3})&\cdots&\omega _{k}^{2}\mu _{1}+\omega _{3}^{2} \frac{L_{k}}{L_{3}}\xi _{1} \\ \vdots&\vdots&\vdots&\ddots&\vdots \\ -\omega _{1}\mu _{2}-\omega _{k}\frac{L_{1}}{L_{k}}\xi _{2}&-\omega _{2}\mu _{2}-\omega _{k}\frac{L_{2}}{L_{k}}\xi _{2}&\omega _{3}^{2}\mu _{1}+\omega _{k}^{2}\frac{L_{3}}{L_{k}}\xi _{1}&\cdots&\omega _{k}^{2}(\mu _{1}+\xi _{1}+\psi _{k}) \end{array} \right|\nonumber \\&\quad = \prod _{i=1}^{k}L_{i}\left|\begin{array}{ccccc} -\frac{\omega _{1}}{L_{1}}\mu _{2}-\frac{\omega _{1}}{L_{1}}\xi _{2}&( \frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}})\mu _{2}&\frac{\omega _{3}^{2}}{L_{3}}\mu _{1}+\frac{\omega _{1}^{2}}{L_{1}}\xi _{1}&\cdots&\frac{\omega _{k}^{2}}{L_{k}}\mu _{1}+\frac{\omega _{1}^{2}}{L_{1}}\xi _{1} \\ (\frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}})\xi _{2}&0&(\frac{ \omega _{2}^{2}}{L_{2}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1}&\cdots&( \frac{\omega _{2}^{2}}{L_{2}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1} \\ (\frac{\omega _{1}}{L_{1}}-\frac{\omega _{3}}{L_{3}})\xi _{2}&0&(\frac{ \omega _{3}^{2}}{L_{3}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1}+\frac{\omega _{3}^{2}}{L_{3}}\psi _{3}&\cdots&(\frac{\omega _{3}^{2}}{L_{3}}-\frac{ \omega _{1}^{2}}{L_{1}})\xi _{1} \\ \vdots&\vdots&\vdots&\ddots&\vdots \\ (\frac{\omega _{1}}{L_{1}}-\frac{\omega _{k}}{L_{k}})\xi _{2}&0&(\frac{ \omega _{k}^{2}}{L_{k}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1}&\cdots&( \frac{\omega _{k}^{2}}{L_{k}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1}+\frac{ \omega _{k}^{2}}{L_{k}}\psi _{k} \end{array} \right|\nonumber \\&\quad =-\biggr (\frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}}\biggr )\mu _{2}\xi _{2}\prod _{i=1}^{k}L_{i} \nonumber \\&\qquad \,\,\times \left|\begin{array}{ccccc} \frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}}&(\frac{\omega _{2}^{2} }{L_{2}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1}&(\frac{\omega _{2}^{2}}{ L_{2}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1}&\cdots&(\frac{\omega _{2}^{2}}{L_{2}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1} \\ \frac{\omega _{1}}{L_{1}}-\frac{\omega _{3}}{L_{3}}&(\frac{\omega _{3}^{2} }{L_{3}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1}+\frac{\omega _{3}^{2}}{L_{3}} \psi _{3}&(\frac{\omega _{3}^{2}}{L_{3}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1}&\cdots&(\frac{\omega _{3}^{2}}{L_{3}}-\frac{\omega _{1}^{2}}{L_{1}} )\xi _{1} \\ \frac{\omega _{1}}{L_{1}}-\frac{\omega _{4}}{L_{4}}&(\frac{\omega _{4}^{2} }{L_{4}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1}&(\frac{\omega _{4}^{2}}{ L_{4}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1}+\frac{\omega _{4}^{2}}{L_{4}} \psi _{4}&\cdots&(\frac{\omega _{4}^{2}}{L_{4}}-\frac{\omega _{1}^{2}}{ L_{1}})\xi _{1} \\ \vdots&\vdots&\vdots&\ddots&\vdots \\ \frac{\omega _{1}}{L_{1}}-\frac{\omega _{k}}{L_{k}}&(\frac{\omega _{k}^{2} }{L_{k}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1}&(\frac{\omega _{k}^{2}}{ L_{k}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1}&\cdots&(\frac{\omega _{k}^{2}}{L_{k}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1}+\frac{\omega _{k}^{2} }{L_{k}}\psi _{k} \end{array} \right|\nonumber \\&\quad = -\biggr (\frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}}\biggr )\mu _{2}\xi _{2}\prod _{i=1}^{k}L_{i}\left|\begin{array}{ccccc} \frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}}&(\frac{\omega _{2}^{2} }{L_{2}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1}&0&\cdots&0 \\ \frac{\omega _{1}}{L_{1}}-\frac{\omega _{3}}{L_{3}}&(\frac{\omega _{3}^{2} }{L_{3}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1}+\frac{\omega _{3}^{2}}{L_{3}} \psi _{3}&-\frac{\omega _{3}^{2}\psi _{3}}{L_{3}}&\cdots&-\frac{\omega _{3}^{2}\psi _{3}}{L_{3}} \\ \frac{\omega _{1}}{L_{1}}-\frac{\omega _{4}}{L_{4}}&(\frac{\omega _{4}^{2} }{L_{4}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1}&\frac{\omega _{4}^{2}\psi _{4}}{L_{4}}&\cdots&0 \\ \vdots&\vdots&\vdots&\ddots&\vdots \\ \frac{\omega _{1}}{L_{1}}-\frac{\omega _{k}}{L_{k}}&(\frac{\omega _{k}^{2} }{L_{k}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1}&0&\cdots&\frac{\omega _{k}^{2}\psi _{k}}{L_{k}} \end{array} \right|\nonumber \\&\quad = -\biggr (\frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}}\biggr )\mu _{2}\xi _{2}\prod _{i=1}^{k}L_{i}\left|\begin{array}{ccccc} \frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}}&(\frac{\omega _{2}^{2} }{L_{2}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1}&0&\cdots&0 \\ \mathcal{A }&\mathcal{B }&0&\cdots&0 \\ \frac{\omega _{1}}{L_{1}}-\frac{\omega _{4}}{L_{4}}&(\frac{\omega _{4}^{2} }{L_{4}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1}&\frac{\omega _{4}^{2}\psi _{4}}{L_{4}}&\cdots&0 \\ \vdots&\vdots&\vdots&\ddots&\vdots \\ \frac{\omega _{1}}{L_{1}}-\frac{\omega _{k}}{L_{k}}&(\frac{\omega _{k}^{2} }{L_{k}}-\frac{\omega _{1}^{2}}{L_{1}})\xi _{1}&0&\cdots&\frac{\omega _{k}^{2}\psi _{k}}{L_{k}} \end{array} \right|\nonumber \\&\quad = -\biggr (\frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}}\biggr )\mu _{2}\xi _{2}L_{1}L_{2}\prod _{i=3}^{k}\omega _{i}^{2}\psi _{i}\biggr \{\biggr ( \frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}}\biggr ) \nonumber \\&\qquad +\frac{\xi _{1}}{L_{1}L_{2}}\sum _{i=3}^{k}\frac{1}{\omega _{i}^{2}\psi _{i} }[\omega _{1}\omega _{2}(\omega _{1}-\omega _{2})L_{i}+\omega _{2}\omega _{i}(\omega _{2}-\omega _{i})L_{1}+\omega _{i}\omega _{1}(\omega _{i}-\omega _{1})L_{2}]\biggr \} \end{aligned}$$

(22)

$$\begin{aligned}&\quad = -\biggr (\frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}}\biggr )^{2}\mu _{2}\xi _{2}L_{1}L_{2}\prod _{i=3}^{k}\omega _{i}^{2}\psi _{i}-\mu _{2}\xi _{1}\xi _{2}\sum _{i=3}^{k}\varpi (1,2,i)\prod _{l\ne 1,2,i}\omega _{l}^{2}\psi _{l}, \end{aligned}$$

(23)

where (22) is from (19), and (23) is from (20).

Similarly, the determinant having $\tau$-terms in the $m$th and $n$th columns is obtained as

$$\begin{aligned} -\biggr (\frac{\omega _m}{L_m}-\frac{\omega _n}{L_n}\biggr )^2\mu _2\xi _2L_1L_2 \prod _{i\ne m,n}\omega _i^2\psi _i -\mu _2\xi _1\xi _2 \sum _{i\ne m,n}^k\varpi (m,n,i) \prod _{l\ne m,n,i}\omega _l^2\psi _l \end{aligned}$$

The $\tau ^{(2k-2)}$-term in $\left|\Upsilon \right|$ is the sum of the above determinants for all nonordered pairs $(m,n)$ with $m\ne n\in \{1,\ldots ,k\}$, which is calculated as

$$\begin{aligned}&-\sum _{m=1}^{k-1}\sum _{n=m+1}^{k}\biggr (\frac{\omega _{m}}{L_{m}}-\frac{ \omega _{n}}{L_{n}}\biggr )^{2}\mu _{2}\xi _{2}L_{1}L_{2}\prod _{i\ne m,n}\omega _{i}^{2}\psi _{i} \nonumber \\&-\mu _{2}\xi _{1}\xi _{2}\sum _{m=1}^{k-1}\sum _{n=m}^{k}\sum _{i\ne m,n}^{k}\varpi (m,n,i)\prod _{l\ne m,n,i}\omega _{l}^{2}\psi _{l} \nonumber \\&= -\sum _{m=1}^{k-1}\sum _{n=m+1}^{k}\biggr (\frac{\omega _{m}}{L_{m}}-\frac{ \omega _{n}}{L_{n}}\biggr )^{2}\mu _{2}\xi _{2}L_{1}L_{2}\prod _{i\ne m,n}\omega _{i}^{2}\psi _{i} \nonumber \\&-\frac{1}{2}\mu _{2}\xi _{1}\xi _{2}\sum _{(m,n,i)|m\ne n\ne i}[\varpi (m,n,i)+\varpi (n,i,m)+\varpi (i,m,n)]\prod _{l\ne m,n,i}\omega _{l}^{2}\psi _{l} \nonumber \\&= -\sum _{m=1}^{k-1}\sum _{n=m+1}^{k}\biggr (\frac{\omega _{m}}{L_{m}}-\frac{ \omega _{n}}{L_{n}}\biggr )^{2}\mu _{2}\xi _{2}L_{1}L_{2}\prod _{i\ne m,n}\omega _{i}^{2}\psi _{i} \end{aligned}$$

(24)

$$\begin{aligned}&< 0, \end{aligned}$$

(25)

where (24) is from (21), and inequality (25) is from (13).

Next, we consider the $\tau ^{(2k-2)}$-term in $\sum _{i=1}^{k}\left|(\Theta _{i},\Upsilon ^{-i})\right|$. This term is the sum of determinants having only one column with $\tau$-terms besides the $i$th column. They can be calculated as in the case of $\left|\Upsilon \right|$. For example, one determinant of $|(\Theta _{1},\Upsilon ^{-1})|$ is

$$\begin{aligned}&\left|\begin{array}{ccccc} \frac{\rho \tau L_{1}}{2}&-\omega _{2}\mu _{2}-\omega _{1}\frac{L_{2}}{ L_{1}}\xi _{2}&\omega _{3}^{2}\mu _{1}+\omega _{1}^{2}\frac{L_{3}}{L_{1}} \xi _{1}&\cdots&\omega _{k}^{2}\mu _{1}+\omega _{1}^{2}\frac{L_{k}}{L_{1} }\xi _{1} \\ \frac{\rho \tau L_{1}}{2}&-\omega _{2}\mu _{2}-\omega _{2}\frac{L_{2}}{ L_{2}}\xi _{2}&\omega _{3}^{2}\mu _{1}+\omega _{2}^{2}\frac{L_{3}}{L_{2}} \xi _{1}&\cdots&\omega _{k}^{2}\mu _{1}+\omega _{2}^{2}\frac{L_{k}}{L_{2} }\xi _{1} \\ \frac{\rho \tau L_{1}}{2}&-\omega _{2}\mu _{2}-\omega _{3}\frac{L_{2}}{ L_{3}}\xi _{2}&\omega _{3}^{2}(\mu _{1}+\xi _{1}+\psi _{3})&\cdots&\omega _{k}^{2}\mu _{1}+\omega _{3}^{2}\frac{L_{k}}{L_{3}}\xi _{1} \\ \vdots&\vdots&\vdots&\ddots&\vdots \\ \frac{\rho \tau L_{1}}{2}&-\omega _{2}\mu _{2}-\omega _{k}\frac{L_{2}}{ L_{k}}\xi _{2}&\omega _{3}^{2}\mu _{1}+\omega _{k}^{2}\frac{L_{3}}{L_{k}} \xi _{1}&\cdots&\omega _{k}^{2}(\mu _{1}+\xi _{1}+\psi _{k}) \end{array} \right|\\&= \frac{\rho \tau \xi _{2}}{2}\prod _{i=1}^{k}L_{i}\left|\begin{array}{cccc} \frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}}&\bigr (\frac{\omega _{2}^{2}}{L_{2}}-\frac{\omega _{1}^{2}}{L_{1}}\bigr )\xi _{1}&\cdots&\bigr (\frac{\omega _{2}^{2}}{L_{2}}-\frac{\omega _{1}^{2}}{L_{1}}\bigr )\xi _{1} \\ \frac{\omega _{1}}{L_{1}}-\frac{\omega _{3}}{L_{3}}&\bigr (\frac{\omega _{3}^{2}}{L_{3}}-\frac{\omega _{1}^{2}}{L_{1}}\bigr )\xi _{1}+\frac{\omega _{3}^{2}\psi _{3}}{L_{3}}&\cdots&\bigr (\frac{\omega _{3}^{2}}{L_{3}}- \frac{\omega _{1}^{2}}{L_{1}}\bigr )\xi _{1} \\ \vdots&\vdots&\ddots&\vdots \\ \frac{\omega _{1}}{L_{1}}-\frac{\omega _{k}}{L_{k}}&\bigr (\frac{\omega _{k}^{2}}{L_{k}}-\frac{\omega _{1}^{2}}{L_{1}}\bigr )\xi _{1}&\cdots&\bigr (\frac{\omega _{k}^{2}}{L_{k}}-\frac{\omega _{1}^{2}}{L_{1}}\bigr )\xi _{1}+\frac{\omega _{k}^{2}\psi _{k}}{L_{k}} \end{array} \right|\\&= \frac{\rho \tau \xi _{2}}{2}L_{1}L_{2}L_{3}\prod _{i=4}^{k}\omega _{i}^{2}\psi _{i}\left|\begin{array}{cc} \frac{\omega _{1}}{L_{1}}-\frac{\omega _{2}}{L_{2}}&\bigr (\frac{\omega _{2}^{2}}{L_{2}}-\frac{\omega _{1}^{2}}{L_{1}}\bigr )\xi _{1} \\ \sum \limits _{i=3}^{k}\bigr (\frac{\omega _{1}}{L_{1}}-\frac{\omega _{i}}{L_{i}}\bigr ) \bigr (\frac{\omega _{3}L_{i}}{\omega _{i}L_{3}}\bigr )^{2}&\frac{\omega _{3}^{2}\psi _{3}}{L_{3}}+\xi _{1}\sum \limits _{i=3}^{k}\bigr (\frac{\omega _{i}^{2}}{ L_{i}}-\frac{\omega _{1}^{2}}{L_{1}}\bigr )\bigr (\frac{\omega _{3}L_{i}}{ \omega _{i}L_{3}}\bigr )^{2} \end{array} \right|\\&= \frac{\rho }{2}\xi _{2}\tau (\omega _{1}L_{2}-\omega _{2}L_{1})\prod _{i\ne 1,2}\omega _{l}^{2}\psi _{i}+\frac{\rho }{2}\xi _{1}\xi _{2}\tau \sum \limits _{i=3}^{k}\prod _{l\ne 1,2,i}\omega _{l}^{2}\psi _{l}\varrho (1,2,i) \end{aligned}$$

where

$$\begin{aligned} \varrho (1,2,i)=\omega _{1}\omega _{2}(\omega _{1}-\omega _{2})L_{i}+\omega _{2}\omega _{i}(\omega _{2}-\omega _{i})L_{1}+\omega _{i}\omega _{1}(\omega _{i}-\omega _{1})L_{2}. \end{aligned}$$

Note that $|(\Theta _{2},\Gamma ^{-2})|$ has a similar determinant with $\omega _{1}L_{2}-\omega _{2}L_{1}$ replaced by $\omega _{2}L_{1}-\omega _{1}L_{2}$, and $\varrho (1,2,i)$ replaced by $\varrho (2,1,i)$. Furthermore, it is easy to check that $\varrho (1,2,i)+\varrho (2,1,i)=0$ holds. Therefore, the sum of these two determinants is zero. For this reason, the $\tau ^{(2k-2)}$-term in $\sum _{i=1}^{k}\left|(\Theta _{i},\Upsilon ^{-i})\right|$ is zero. In conclusion, the $\tau ^{(2k-2)}$-term in $\left|\Delta \right|$ is negative.$\square$

Appendix B: Proof of Lemma 1

• (i) follows directly from (13).

• (ii) Since $\sum _{j=1}^{l-1}L_j>\sum _{j=l}^KL_j$, we have

  $$\begin{aligned} \sum _{j=1}^{l-1}\omega _j-\sum _{j=l}^K\omega _j>\frac{\omega _l}{L_l}\biggr ( \sum _{j=1}^{l-1}L_j-\sum _{j=l}^KL_j\biggr )>0, \end{aligned}$$

  where the first inequality is from (13). The first inequality can be rewritten as

  $$\begin{aligned} \omega _l\biggr (\sum _{j=1}^{l-1}L_j-\sum _{j=l}^KL_j\biggr )< L_l\biggr ( \sum _{j=1}^{l-1}\omega _j-\sum _{j=l}^K\omega _j\biggr ), \end{aligned}$$

  and, hence,

  $$\begin{aligned}&\frac{\sum _{j=1}^{l-1}\omega _j-\sum _{j=l}^K\omega _j}{\sum _{j=1}^{l-1}L_j- \sum _{j=l}^KL_j} >\frac{\sum _{j=1}^{l-1}\omega _j-\sum _{j=l}^K\omega _j+2 \omega _l}{\sum _{j=1}^{l-1}L_j-\sum _{j=l}^KL_j+2L_l}\nonumber \\&\quad \,=\frac{ \sum _{j=1}^l\omega _j-\sum _{j=l+1}^K\omega _j}{\sum _{j=1}^lL_j- \sum _{j=l+1}^KL_j}, \end{aligned}$$

  which derives $\rho ^{\dagger }(l, l)<\rho ^{\dagger }(l, l-1)$.

• (iii) In a similar manner to (ii), we have

  $$\begin{aligned} 0<\omega _{l}\biggr (\sum _{j=1}^{l}L_{j}-\sum _{j=l+1}^{K}L_{j}\biggr )<L_{l} \biggr (\sum _{j=1}^{l}\omega _{j}-\sum _{j=l+1}^{K}\omega _{j}\biggr ) \end{aligned}$$

since $\sum _{j=1}^{l}L_{j}>\sum _{j=l+1}^{K}L_{j}$. Therefore, the last term of (14) is positive, and, thus, $\rho ^{\dagger }(i,l)>0$ holds for any $i$.$\square$

Appendix C: Proof of Lemma 2

(Necessity) Let $\varvec{\lambda }^{*}$ be an equilibrium of pattern (a), in which $l$ industries $i_{1},\ldots ,i_{l}$ agglomerate in region $H$ and $K-l$ industries $i_{l+1},\ldots ,i_{K}$ agglomerate in region $F$. Then, for each industry $i$, it holds that

$$\begin{aligned} V_{iH}(\varvec{\lambda }^{*})&-V_{iF}(\varvec{\lambda }^{*})=-\frac{ 1}{2}\sum \limits _{j=1}^{l}\delta _{ii_{j}}+\frac{1}{2}\sum \limits _{j=l+1}^{K}\delta _{ii_{j}} \\&= -\frac{1}{2}\biggr \{\frac{\rho }{2}\biggr (\sum _{j=1}^{l}L_{i_{j}}- \sum _{j=l+1}^{K}L_{i_{j}}\biggr )-\frac{2ab\overline{L}}{(2b-c)^{2}}\biggr [b \biggr (\sum _{j=1}^{l}\omega _{i_{j}}-\sum _{j=l+1}^{K}\omega _{i_{j}}\biggr )\\&+(2b-c)\biggr (\sum _{j=1}^{l}L_{i_{j}}-\sum _{j=l+1}^{K}L_{i_{j}}\biggr ) \frac{\omega _{i}}{L_{i}}\biggr ]\biggr \}\tau +O(\tau ^{2}). \end{aligned}$$

For a sufficiently small $\tau$, the sign of the above expression is determined by the coefficient of $\tau$. Because of (11), the populations of two regions are different, and, hence, $\sum _{j=1}^{l}L_{i_{j}}>\sum _{j=l+1}^{K}L_{i_{j}}$, because of our naming that $H$ is not smaller than $F$. Since industries $i_{1},\ldots ,i_{l}$ are in $H$, the stability conditions requires $V_{i_{j}H}(\varvec{\lambda } ^{*})-V_{i_{j}F}(\varvec{\lambda }^{*})\ge 0$ for $j=1,\ldots ,l$, which implies

$$\begin{aligned} \rho \le \frac{4ab\overline{L}}{2b-c}\biggr [\frac{\omega _{i_{k}}}{L_{i_{k}} }+\frac{b}{2b-c}\frac{\sum \nolimits _{j=1}^{l}\omega _{i_{j}}-\sum \nolimits _{j=l+1}^{K}\omega _{i_{j}}}{\sum \nolimits _{j=1}^{l}L_{i_{j}}-\sum \nolimits _{j=l+1}^{K}L_{i_{j}}}\biggr ] \text{ for} k=1,\ldots ,l, \end{aligned}$$

(26)

On the other hand, since industries $i_{l+1},\ldots ,i_{K}$ are in $F$, we have $V_{i_{j}H}(\varvec{\lambda }^{*})-V_{i_{j}F}(\varvec{\lambda } ^{*})\le 0$ for $j=l+1,\ldots ,K$, which implies

$$\begin{aligned} \rho \ge \frac{4ab\overline{L}}{2b-c}\biggr [\frac{\omega _{i_{k}}}{L_{i_{k}} }+\frac{b}{2b-c}\frac{\sum \nolimits _{j=1}^{l}\omega _{i_{j}}-\sum \nolimits _{j=l+1}^{K}\omega _{i_{j}}}{\sum \nolimits _{j=1}^{l}L_{i_{j}}-\sum \nolimits _{j=l+1}^{K}L_{i_{j}}}\biggr ] \text{ for} k=l+1,\ldots ,K.\qquad \end{aligned}$$

(27)

The following inequality holds directly from (26) and (27):

$$\begin{aligned} \min \biggr \{\frac{\omega _{i_{1}}}{L_{i_{1}}},\ldots ,\frac{\omega _{i_{l}} }{L_{_{l}}}\biggr \}\ge \max \biggr \{\frac{\omega _{i_{l+1}}}{L_{i_{l+1}}} ,\ldots ,\frac{\omega _{i_{K}}}{L_{i_{K}}}\biggr \}. \end{aligned}$$

(28)

From (13), (28) tells us that $i_{l}=l\ge l_{\sharp }$, the industries located in $H$ turn out to be $1,\ldots ,l$, and the industries located in $F$ turn out to be $l+1,\ldots ,K$. Therefore, (26) and ( 27) are summarized as $\rho \in [\rho ^{\dagger }(l+1,l),\rho ^{\dagger }(l,l)]$. Note that Lemma 1 (i) and (iii) imply that $0<\rho ^{\dagger }(l+1,l)<\rho ^{\dagger }(l,l)$.

(Sufficiency) Let $l$ be an industry satisfying $l\ge l_{\sharp }$ and $\rho \in (\rho ^{\dagger }(l+1,l),\rho ^{\dagger }(l,l))$, and let $\varvec{ \lambda }^{*}$ be the distribution in which industries $1,\ldots ,l$ agglomerate in region $H$ and industries $l+1,\ldots ,K$ agglomerate in region $F$. Then,

$$\begin{aligned} V_{iH}(\varvec{\lambda }^{*})&-V_{iF}(\varvec{\lambda }^{*})=-\frac{ 1}{2}\sum \limits _{j=1}^{l}\delta _{ij}+\frac{1}{2}\sum \limits _{j=l+1}^{K}\delta _{ij} \\&= -\frac{1}{2}\biggr \{\frac{\rho }{2}\biggr (\sum _{j=1}^{l}L_{j}- \sum _{j=l+1}^{K}L_{j}\biggr )-\frac{2ab\overline{L}}{(2b-c)^{2}}\biggr [b \biggr (\sum _{j=1}^{l}\omega _{j}-\sum _{j=l+1}^{K}\omega _{j}\biggr ) \\&+(2b-c)\biggr (\sum _{j=1}^{l}L_{j}-\sum _{j=l+1}^{K}L_{j}\biggr )\frac{\omega _{i}}{L_{i}}\biggr ]\biggr \}\tau +O(\tau ^{2}), \end{aligned}$$

whose sign is determined by the coefficient of $\tau$ for a sufficiently small $\tau$. From $\rho \in (\rho ^{\dagger }(l+1,l),\rho ^{\dagger }(l,l))$, we have

$$\begin{aligned} \rho&< \frac{4ab\overline{L}}{2b-c}\biggr [\frac{\omega _{i}}{L_{i}}+\frac{b }{2b-c}\frac{\sum \nolimits _{j=1}^{l}\omega _{j}-\sum \nolimits _{j=l+1}^{K}\omega _{j}}{\sum \nolimits _{j=1}^{l}L_{j}-\sum \nolimits _{j=l+1}^{K}L_{j}}\biggr ] \text{ for} i=1,\ldots ,l, \\ \rho&> \frac{4ab\overline{L}}{2b-c}\biggr [\frac{\omega _{i}}{L_{i}}+\frac{b }{2b-c}\frac{\sum \nolimits _{j=1}^{l}\omega _{j}-\sum \nolimits _{j=l+1}^{K}\omega _{j}}{\sum \nolimits _{j=1}^{l}L_{j}-\sum \nolimits _{j=l+1}^{K}L_{j}}\biggr ] \text{ for} i=l+1,\ldots ,K. \end{aligned}$$

Furthermore, since $\sum _{j=1}^{l}L_{j}>\sum _{j=l+1}^{K}L_{j}$, we soon obtain $V_{iH}(\varvec{\lambda }^{*})-V_{iF}(\varvec{\lambda }^{*})>0$ for $i=1,\ldots ,l$ and $V_{iH}(\varvec{\lambda }^{*})-V_{iF}( \varvec{\lambda }^{*})<0$ for $i=l+1,\ldots ,K$; therefore, $\varvec{ \lambda }^{*}$ is a stable equilibrium.$\square$

Appendix D: Proof of Lemma 3

(Necessity) Consider a stable equilibrium $\varvec{\lambda }^{*}$, in which industries $i_{1},\ldots ,i_{l_{1}}$ are located in region $H$ and industries $j_{1},\ldots ,j_{l_{2}}$ are located in region $F$, while industry $l$ disperses with population $\lambda _{l}^{*}L_{l}$ in region $H$. Then, $l_{1}+l_{2}+1=K$ and $V_{lH}(\varvec{\lambda }^{*})-V_{lF}( \varvec{\lambda }^{*})=0$, from which we have

$$\begin{aligned} \lambda _{l}^{*}&= \frac{1}{2}-\frac{1}{2\delta _{ll}}\biggr ( \sum \limits _{k=1}^{l_{1}}\delta _{li_{k}}-\sum \limits _{k=1}^{l_{2}}\delta _{lj_{k}}\biggr ) \nonumber \\&= -\frac{1}{2\delta _{ll}}\biggr [\mu _{1}\Omega _{1}-\mu _{2}\Omega _{2}+ \frac{\Lambda }{L_{l}}\big (\omega _{l}^{2}\xi _{1}-\omega _{l}\xi _{2}+\frac{ 1}{2}\rho \tau L_{l}\big )\biggr ]+\frac{1}{2} \end{aligned}$$

(29)

$$\begin{aligned}&= \frac{\mathcal{C }+\mathcal{D }\tau }{\mathcal{E }+\mathcal{F }\tau }, \end{aligned}$$

(30)

where

$$\begin{aligned} \Omega _{1}&\equiv \sum _{k=1}^{l_{1}}\omega _{i_{k}}^{2}-\sum _{k=1}^{l_{2}}\omega _{j_{k}}^{2},\ \ \Omega _{2}\equiv \sum _{k=1}^{l_{1}}\omega _{i_{k}}-\sum _{k=1}^{l_{2}}\omega _{j_{k}}, \Lambda \equiv \sum _{k=1}^{l_{1}}L_{i_{k}}-\sum _{k=1}^{l_{2}}L_{j_{k}}\qquad \quad \end{aligned}$$

(31)

$$\begin{aligned} \mathcal{C }&\equiv 4ab^{2}\overline{L}L_{l}(\Omega _{2}-\omega _{l})-(\Lambda -L_{l})(2b-c)[(2b-c)L_{l}\rho -4ab\overline{L}\omega _{l}] \nonumber \\ &= (2b-c)^{2}L_{l}(\Lambda -L_{l})[\rho ^{\dagger }(l,l-1)-\rho ] \end{aligned}$$

(32)

$$\begin{aligned} \mathcal{D }&\equiv \overline{L}\{2b^{2}(b-c)L_{l}(\omega _{l}^{2}-\Omega _{1})+\omega _{l}^{2}b(2b-c)[c\overline{L}+2(b-c)(L_{l}-\Lambda )]\} \nonumber \\ \mathcal{E }&= 2L_{l}[(2b-c)^{2}L_{l}\rho -4ab(3b-c)\overline{L}\omega _{l}] \nonumber \\ &= \mathcal{C }+(2b-c)^{2}L_{l}(\Lambda +L_{l})[\rho -\rho ^{\dagger }(l,l)] \\ \mathcal{F }&\equiv 2b\omega _{l}^{2}\overline{L}L_{l}\biggr [2(b-c)(3b-c)+ \frac{c(2b-c)\overline{L}}{L_{l}}\biggr ] \nonumber \end{aligned}$$

(33)

Noteworthily, $\mathcal{E }=0$ holds only if $\rho =4ab(3b-c)\overline{L} \omega _{l}/[(2b-c)^{2}L_{l}\rho ]$, which implies $\mathcal{C }=4ab^{2} \overline{L}(L_{l}\Omega _{2}-\Lambda \omega _{l})\ne 0$ according to (12), so that (30) is infinitely large for small $\tau$. Meanwhile, $\lambda _{l}^{*}$ of (30) is inside $[0,1]$. Therefore, we can exclude the case of $\mathcal{E }=0$ for the discussion on the case of a small $\tau$, which we focus on. Let

$$\begin{aligned} \bar{\lambda }_{l}\equiv \lim _{\tau \rightarrow 0}\lambda _{l}^{*}=\frac{\mathcal{C }}{\mathcal{E }}. \end{aligned}$$

(34)

For equilibrium $\varvec{\lambda }^{*}$ to be stable, one necessary condition is that

$$\begin{aligned} \frac{\partial [V_{lH}(\varvec{\lambda }^{*})-V_{lF}(\varvec{ \lambda }^{*})]}{\partial \lambda _{l}}=-\delta _{ll}=-\frac{\tau }{ 4(2b-c)^{2}L_{l}}(\mathcal{E }+\mathcal{F }\tau )\le 0. \end{aligned}$$

(35)

Note that the above inequality also implies $\mathcal{E }>0$.

By the naming of regions, $H$ is larger than $F$, which is expressed as follows:

$$\begin{aligned} \sum _{k=1}^{l_{1}}L_{i_{k}}+\lambda _{l}^*L_{l}\ge \sum _{k=1}^{l_{2}}L_{j_{k}}+(1-\lambda _{l}^*)L_{l}. \end{aligned}$$

(36)

For sufficiently small $\tau$, (36) is rewritten as

$$\begin{aligned} \frac{\Lambda }{L_l}\ge 1-2\bar{\lambda }_l=\lim _{\tau \rightarrow 0}\frac{1}{ \delta _{ll}} \biggr (\sum _{k=1}^{l_1}\delta _{li_k}-\sum _{k=1}^{l_2} \delta _{lj_k}\biggr ) =\lim _{\tau \rightarrow 0}-\frac{\displaystyle -\mu _2\Omega _2-\frac{\omega _l}{L_l}\xi _2\Lambda +\frac{\rho }{2}\tau \Lambda }{ \displaystyle -\omega _l(\mu _2+\xi _2)+\frac{\rho }{2}L_l\tau }, \end{aligned}$$

which turns out to be

$$\begin{aligned} \frac{\Omega _2}{\Lambda }>\frac{\omega _{l}}{L_l} \end{aligned}$$

(37)

according to (35) and the assumption of (12).

On the other hand, the utility differential is calculated as

$$\begin{aligned} V_{iH}(\varvec{\lambda }^*)-V_{iF}(\varvec{\lambda }^*)&= -\frac{1}{2} \sum \limits _{k=1}^{l_1}\delta _{ii_k}+\frac{1}{2}\sum \limits _{k=1}^{l_2} \delta _{ij_k}+ \frac{\delta _{il}}{2\delta _{ll}}\biggr ( \sum \limits _{k=1}^{l_1}\delta _{li_k}- \sum \limits _{k=1}^{l_2}\delta _{lj_k}\biggr ) \\&= -\frac{1}{2\delta _{ll}}\biggr ( \sum \limits _{k=1}^{l_1}\left| \begin{array}{cc} \delta _{ii_k}&\delta _{il} \\ \delta _{li_k}&\delta _{ll} \end{array} \right|- \sum \limits _{k=1}^{l_2}\left| \begin{array}{cc} \delta _{ij_k}&\delta _{il} \\ \delta _{lj_k}&\delta _{ll} \end{array} \right|\biggr ) \\&= \frac{1}{2\delta _{ll}}\biggr [\frac{\Lambda }{L_l} \biggr (\frac{\omega _i}{L_i }-\frac{\omega _l}{L_l}\biggr ) \biggr (\frac{\Omega _2}{\Lambda }-\frac{\omega _l }{L_l}\biggr ) \mu _2\xi _2+O(\tau ^3)\biggr ] \end{aligned}$$

for any $i\ne l$. Therefore, two other stability conditions of equilibrium $\varvec{\lambda }^{*}$

$$\begin{aligned} \begin{array}{ll} V_{iH}(\varvec{\lambda }^{*})-V_{iF}(\varvec{\lambda }^{*})\ge 0&\text{ for} i=i_1,\ldots , i_l \\ V_{jH}(\varvec{\lambda }^{*})-V_{jF}(\varvec{\lambda }^{*})\le 0&\text{ for} j=1,\ldots ,l_{2} \end{array} \end{aligned}$$

(38)

are simply rewritten as

$$\begin{aligned} \frac{\omega _{i}}{L_{i}}>\frac{\omega _{l}}{L_{l}}>\frac{\omega _{j}}{L_{j}} , \text{ for} \text{ any} i=i_{1},\ldots ,i_{l_{1}} \text{ and} j=j_{1},\ldots ,j_{l_{2}}. \end{aligned}$$

Therefore, the industries in region $H$ turn out to be $1,\ldots ,l-1$, and the industries in region $F$ turn out to be $l+1,\ldots ,K$. Furthermore, $l\ge l_{\sharp }$ because of (36).

Now, we derive (17). First, $\rho >\rho ^{\dagger }(l,l)$ holds from (33), (16) and the facts of $\bar{\lambda }_{l}\le 1$. If $l>l_{\sharp }$, then $l-1\ge l_{\sharp }$ so that $\Lambda >L_l$ holds from the first inequality of (15). Then, $\rho <\rho ^{\dagger }(l,l-1)$ holds from (32), (16) and the fact of $\bar{\lambda }_{l}\ge 0$.

(Sufficiency) Consider a distribution $\varvec{\lambda }^*$ such that industry $l\ge l_{\sharp }$ disperses with population $\lambda _{l}^{*}L_{l}$ in region $H$, industries $1,\ldots ,l-1$ agglomerate in $H$, industry $l+1,\ldots ,K$ agglomerate in $F$, and (17) holds. According to (13), we have $L_l\omega _i>L_i\omega _l$ and $L_l\omega _j<L_j\omega _l$ for $i=1,\ldots , l-1$ and $j=l+1, \ldots , K$, which implies

$$\begin{aligned} \sum _{i=1}^{l-1}L_l\omega _i-\sum _{j=l+1}^KL_l\omega _j> \sum _{i=1}^{l-1}L_i\omega _l-\sum _{j=l+1}^KL_j\omega _l, \end{aligned}$$

proving (37), in which $\Omega _2$ and $\Lambda$ are defined in (31) with industry names $i_k=k$ and $j_k=l+k$. We consider two cases separately.

Case (i): If $l>l_\sharp$, then $l-1\ge L_\sharp$ so that $\Lambda >L_l$ from the first inequality of (15). Then, $\mathcal{E }>\mathcal C >0$ hold from (32) and (33), implying $\bar{\lambda }_l\in (0,1)$ , defined by (34). On the other hand, $\delta _{ll}>0$ and ( 38) are immediate results. Therefore, distribution $\varvec{ \lambda }^*$ is a stable equilibrium.

Case (ii): If $l=l_{\sharp }$, then $\Lambda <L_{l}$ from the second inequality of (15). Therefore,

$$\begin{aligned} \frac{\Omega _{2}-\omega _{l}}{\Lambda -L_{l}}-\frac{\Omega _{2}+\omega _{l} }{\Lambda +L_{l}}=\frac{2(\Omega _{2}L_{l}-\omega _{l}\Lambda )}{\Lambda ^{2}-L_{l}^{2}}<0 \end{aligned}$$

holds from (37), implying $\rho ^{\dagger }(l,l-1)<\rho ^{\dagger }(l,l)$. Thus, $\rho >\rho ^{\dagger }(l,l-1)$ holds, and we obtain $\mathcal{C }>0$ and $\mathcal{E }>0$ again from (32) and (33 ). As in case (i), we have $\bar{\lambda }_{l}\in (0,1)$, $\delta _{ll}>0$ and (38).

Finally, from (8), (29), and (31), we have

$$\begin{aligned} \frac{\partial \lambda _{l}^{*}}{\partial \rho }=-\frac{\tau }{4}\frac{ \left( \sum _{k=1}^{l_{1}}L_{i_{k}}+\lambda _{l}^{*}L_{l}\right) -\left[ \sum _{k=1}^{l_{2}}L_{j_{k}}+(1-\lambda _{l}^{*})L_{l}\right] }{\delta _{ll}}\text{.} \end{aligned}$$

We know that the numerator is nonnegative from (36) and that $\delta _{ll}$ is also nonnegative from (35) if $\varvec{\lambda }^{*}$ is a stable equilibrium. Thus, we conclude that $\partial \lambda _{l}^{*}/\partial \rho \le 0$ when $\varvec{\lambda }^{*}$ is a stable equilibrium. Note that this inequality holds even if $\tau$ is large. $\square$