Satellite City Formation for a Spatial Economic Model

Abstract

The economic agglomeration of one large city surrounded by satellite cities is observed worldwide and is a topic of keen economic interest. We theoretically investigate where such satellite cities emerge in a two-dimensional economic space in which discrete locations are evenly distributed in a regular-hexagonal domain. To elucidate this emergence, we introduce two viewpoints: (1) the bifurcation mechanism of the full agglomeration at the geographical center in this domain (mono-center), which produces satellite cities around this center, and (2) the existence of invariant patterns, which are equilibria for any value of the transport cost parameter. Theoretically-predicted agglomeration patterns are ensured to exist as stable equilibria for a spatial economic model proposed by Forslid and Ottaviano (2003). We theoretically find one large central city surrounded by hexagonal satellite cities that is a two-dimensional counterpart of the core-periphery pattern (Krugman 1991). Moreover, we demonstrate that spatial patterns of twin cities, three cities, and racetrack cities are absorbed into the mono-center as the transport cost decreases. These transitions are ubiquitously observed in the two-dimensional spatial platform with the geographical center.

Appendices

A. Detail of the FO Model

The fundamental logic and the governing equation of a multi-places version of the model by Forslid and Ottaviano (2003) are presented (see Akamatsu et al. 2012). The budget constraint is given by

$$\begin{aligned} p^\textrm{A}_i C_i^\textrm{A}+\sum _{j \in P} \int _0^{n_j}p_{ji}(\ell )q_{ji}(\ell )d\ell =Y_i, \end{aligned}$$

(24)

where $p^\textrm{A}_i$ represents the price of the A-sector good in place i, $C_i^\textrm{A}$ is the consumption of A-sector goods in place i, $P= \left\{ 1,...,K\right\}$, $n_j$ is the number of varieties produced in place j, $p_{ji}(\ell )$ denotes the price of a variety $\ell$ in place i produced in place j, $q_{ji}(\ell )$ is the consumption of variety $\ell \in [0, n_j]$ in place i produced in place j, and $Y_i$ is the income of an individual in place i. The incomes (wages) of skilled workers and unskilled workers are represented respectively by $w_i$ and $w_i^\textrm{L}$.

An individual at place i maximizes the utility in (5) subject to the budget constraint in (24). This maximization yields the following demand functions

$$\begin{aligned} C_i^\textrm{A}=(1-\mu )\frac{Y_i}{p_i^\textrm{A}}, \qquad C_i^\textrm{M}=\mu \frac{Y_i}{\rho _i}, \qquad q_{ji}(\ell )= \mu \frac{\rho _i^{\sigma -1}Y_i}{p_{ji}(\ell )^{\sigma }}, \end{aligned}$$

where $\rho _i$ denotes the price index of the differentiated products in place i, and is given by

$$\begin{aligned} \rho _i=\left( \sum _{j \in P} \int _0^{n_j}p_{ji}(\ell )^{1-\sigma }d\ell \right) ^{1/(1-\sigma )}. \end{aligned}$$

(25)

Because the total income in place i is $w_i \lambda _i+w_i^\textrm{L}$, the total demand $Q_{ji}(\ell )$ in place i for a variety $\ell$ produced in place j is given as

$$\begin{aligned} Q_{ji}(\ell )=\mu \frac{\rho _i^{\sigma -1}}{p_{ji} (\ell )^{\sigma }}(w_i\lambda _i+w_i^\textrm{L}). \end{aligned}$$

(26)

The A-sector is perfectly competitive and produces homogeneous goods under constant-returns-to-scale, and requires one unit of unskilled labor per unit of output. The A-sector good is traded freely across locations and is chosen as the numéraire. In equilibrium, $p_i^\textrm{A}=w_i^\textrm{L}=1$ for each i.

The M-sector output is produced under increasing-returns-to-scale and Dixit–Stiglitz monopolistic competition. A firm incurs a fixed input requirement of $\alpha$ units of skilled labor and a marginal input requirement of $\beta$ units of unskilled labor. An M-sector firm located in place i chooses $( p_{ij}(\ell ) \mid j \in P)$ that maximizes its profit

$$\begin{aligned}&\Pi _i(\ell )=\sum _{j \in P} p_{ij}(\ell )Q_{ij}(\ell )-\left( \alpha w_i+\beta x_i(\ell )\right) , \end{aligned}$$

(27)

where $x_i(\ell )$ denotes the total supply of variety $\ell$ produced in place i and $\alpha w_i+\beta x_i(\ell )$ signifies the cost function introduced by Flam and Helpman (1987).

With the use of the iceberg form of the transport cost, we have

$$\begin{aligned} x_i(\ell )=\sum _{j \in P} \tau _{ij}Q_{ij}(\ell ). \end{aligned}$$

(28)

Then the profit function of the M-sector firm in place i, given in (27) above, can be rewritten as

$$\begin{aligned}&\Pi _i(\ell )=\sum _{j \in P} p_{ij}(\ell )Q_{ij}(\ell )- \left( \alpha w_i+\beta \sum _{j \in P} \tau _{ij}Q_{ij}(\ell ) \right) , \end{aligned}$$

which is maximized by the firm. The first-order condition for this profit maximization yields the following optimal price

$$\begin{aligned} p_{ij}(\ell )=\frac{\sigma \beta }{\sigma -1} \tau _{ij}. \end{aligned}$$

(29)

This result implies that $p_{ij}(\ell )$, $Q_{ij}(\ell )$, and $x_i(\ell )$ are independent of $\ell$. Therefore, the argument $\ell$ is suppressed subsequently.

In the short run, skilled workers are immobile between places, i.e., their spatial distribution $\varvec{\lambda }=(\lambda _i \mid i \in P)$ is assumed to be given. The market equilibrium conditions consist of three conditions: the M-sector goods market clearing condition, the zero-profit condition attributable to the free entry and exit of firms, and the skilled labor market clearing condition. The first condition is written as (28) above. The second one requires that the operating profit of a firm, given in (27), be absorbed entirely by the wage bill of its skilled workers. This gives

$$\begin{aligned} w_i =\frac{1}{\alpha }\left\{ \sum _{j \in P} p_{ij}Q_{ij} -\beta x_i \right\} . \end{aligned}$$

(30)

The third condition is expressed as $\alpha n_i=\lambda _i$ and the price index $\rho _i$ in (25) can be rewritten using (29) as

$$\begin{aligned} \rho _i=\frac{\sigma \beta }{\sigma -1} \left( \frac{1}{\alpha }\sum _{j \in P} \lambda _j d_{ji}\right) ^{1/(1-\sigma )}. \end{aligned}$$

(31)

The market equilibrium wage $w_i$ in (30) can be represented as

$$\begin{aligned} w_i =\frac{\mu }{\sigma } \sum _{j \in P} \frac{d_{ij}}{\Delta _j}(w_j \lambda _j+1) \end{aligned}$$

(32)

using $d_{ji}= \tau _{ji}^{1-\sigma }= \phi \, ^{m(i,j)}$, (26), (28), (29), and (31). Here, $\Delta _{j} = \sum _{k \in P} d_{kj}\lambda _k$. Equation (32) can be rewritten, using $\varvec{w}=(w_1, \ldots , w_K)$, as $\varvec{w}= \frac{\mu }{\sigma } \ D\Delta ^{-1} (\Lambda \varvec{w}+\varvec{1})$, which is solved for $\varvec{w}$ as

$$\begin{aligned} \varvec{w}= \frac{\mu }{\sigma } \left( I- \frac{\mu }{\sigma } \ D\Delta ^{-1}\Lambda \right) ^{-1} D\Delta ^{-1} \varvec{1} \end{aligned}$$

(33)

with I being the identity matrix, $\varvec{1}=(1,\ldots ,1)^\top$, and

$$\begin{aligned} D=(d_{ij}), \quad \Delta =\textrm{diag}(\Delta _1,\ldots ,\Delta _{K}), \quad \Lambda =\textrm{diag}(\lambda _1,\ldots ,\lambda _{K}). \end{aligned}$$

(34)

B. Classifications of Stationary Points

The mono-center is classified into a corner solution, for which some places have zero population. We can appropriately permute the components of $\varvec{\lambda }$, without loss of generality, to arrive at

$$\begin{aligned} \hat{\varvec{\lambda }}= (\varvec{\lambda }_+, \varvec{0}_{K-m}), \end{aligned}$$

where all components of $\varvec{\lambda }_+ = (\lambda _1, \ldots , \lambda _m)$ are positive and $\textbf{0}_{K-m}$ is the $(K-m)$-dimensional zero vector. $\varvec{0}_{K-m}$ is present for corner solutions ($K > m$). For corner solutions, the governing equation (4) and associated Jacobian matrix can be rearranged, respectively, as (Ikeda et al. 2018)

$$\begin{aligned} \hat{\varvec{F}}= \begin{pmatrix} {\varvec{F}}_+ (\hat{\varvec{\lambda }},\phi ) \\ {\varvec{F}}_0 (\hat{\varvec{\lambda }},\phi ) \end{pmatrix}, \qquad \hat{J} = \frac{\partial \hat{\varvec{F}}}{\partial \hat{\varvec{\lambda }}}= \begin{pmatrix} J_+ &{} J_{+0} \\ O &{} J_{0} \end{pmatrix}, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} J_+&= \textrm{diag}(\lambda _1, \lambda _2, \ldots , \lambda _{m}) \times \left( {\displaystyle \frac{\partial (v_i - \bar{v})}{\partial \lambda _j} } \mid i,j = 1, \ldots , m \right) , \\ J_{+0}&= \textrm{diag}(\lambda _1, \lambda _2, \ldots , \lambda _{m}) \times \left( \frac{\partial (v_i - \bar{v})}{\partial \lambda _j} \mid i = 1, \ldots , m; \ j = m + 1, \ldots , K \right) , \\ J_0&= \textrm{diag}(v_{m+1} - \bar{v}, \ldots , v_{K} - \bar{v}), \end{aligned} \end{aligned}$$

(35)

and diag$(\cdots )$ denotes a diagonal matrix with the entries in parentheses.

The stability condition of a stable corner solution is decomposed into two conditions:

$$\begin{aligned} \left\{ \begin{array}{ll} \text {Stability condition for } \varvec{\lambda}_+\!\!: &{} \text {all eigenvalues of }J_+\text { are negative.} \\ \text {Sustainability condition:}&{} \text {all diagonal entries of } J_0 \text { are negative.} \end{array}\right. \end{aligned}$$

Both of these two conditions are satisfied if and only if all eigenvalues of $\hat{J}$ are negative.¹³

Critical points are those which have one or more zero eigenvalue(s) of the Jacobian matrix $\hat{J}$. Critical points are classified into a bifurcation point with singular $J_+$ or $J_0$, and a limit point of $\phi$ with singular $J_+$. We classify bifurcation points into a break bifurcation point with singular $J_+$ and a corner bifurcation point with singular $J_0$. The corner bifurcation from the mono-center is theoretically investigated in this paper.

C. Theoretical Details

1.1 C.1 Proof of Lemma 1

Since $T(g)\varvec{\lambda }= \varvec{\lambda }$ for a subgroup $G'$ of G, we have $T(g)\varvec{v}(\varvec{\lambda }, \phi ) = \varvec{v}(\varvec{\lambda }, \phi )$ for $g \in G'$ by (10). This means that $v_i$ in the same orbit is permutable. This suffices for the proof.

1.2 C.2 Proof of Lemma 2

We can rearrange the components of $\varvec{\lambda }_{P_l}$ to arrive at $\hat{\varvec{\lambda }} = (\frac{1}{m}\varvec{1}_m, \varvec{0}_{K - m})$, where ${\varvec{1}}_m$ is the m-dimensional all-one vector, and ${\varvec{0}}_{K - m}$ is the $(K-m)$-dimensional zero vector. The $m~(=N_l)$ places belonging to ${\varvec{\lambda }}_+ = \frac{1}{m}\varvec{1}_m$ are permuted each other by the geometrical transformation by an element of a subgroup of G and ${\varvec{\lambda }}_+$ is invariant with respect to the permutation. By equivariance (10), we have $v_1=\cdots =v_m$, as well as $\lambda _1=\cdots =\lambda _m=1/m$. Thus we have $v_i = \bar{v} \ (i \in P_l)$.

1.3 C.3 Proof of Proposition 1

By Lemma 2, we have $v_{i}-\bar{v}=0$ $(i=1,\ldots ,m)$. Thus, ${\varvec{F}}_+(\frac{1}{m}{\varvec{1}}_m,\textbf{0}_{K-m},\phi )=\textbf{0}_{m}$ holds. For $K-m$ places with no population, we have $\lambda _{j}=0$. Hence, ${\varvec{F}}_0(\frac{1}{m}{\varvec{1}}_m,\textbf{0}_{K-m}, \phi )=\textbf{0}_{K-m}$ holds. This shows that $({\varvec{\lambda }}_+,{\varvec{\lambda }}_0,\phi ) =(\frac{1}{m}\varvec{1}_m,\textbf{0}_{K-m}, \phi )$ serves as a solution to (4) for any $\phi$.

1.4 C.4 Proof of Proposition 2

By choosing $G=E$, which leaves each place unchanged, each place forms an orbit. Hence, the full agglomeration at any place i is an invariant pattern by Proposition 1.

1.5 C.5 Proof of Proposition 3

The bifurcating conditions of the Jacobian matrix (13) are given by as follows:

$$\begin{aligned} v_1&= 0, \end{aligned}$$

(36)

$$\begin{aligned} v_{\alpha i} - v_c&= 0, \quad i = 1,\ldots ,n_1, \end{aligned}$$

(37)

$$\begin{aligned} v_{\beta i} - v_c&= 0, \quad i = 1,\ldots ,n_2. \end{aligned}$$

(38)

However, no bifurcation solution with the direction $(1, 0, \ldots , 0)$ in the space $\sum ^K_{i=1} \lambda _i = 1$ emerges. Therefore, only (37) and (38) are the bifurcating conditions from the mono-center.

1.6 C.6 Proof of Lemma 3

By the product form of the replicator dynamics in (3), $\tilde{F}_i(\varvec{x},\psi )$ takes the product form: $\tilde{F}_i(\varvec{x},\psi )=x_i G_i(\varvec{x},\psi )$ $(i\in P_l)$. Since the group $\textrm{D}_6$ is generated by the elements r and s, it suffices to consider the symmetry condition $T(g)\tilde{\varvec{F}}(\varvec{x})=\tilde{\varvec{F}}(T(g)\varvec{x})$ for $g=r,s$. This condition for $g=r$ gives the form (19) for some function $R(\varvec{x},\psi )$ and that for $g=s$ gives the symmetry condition (20).

1.7 C.7 Proof of Proposition 4

To begin with, we consider $\varvec{x}=w(1,1,1,1,1,1)$. Then (19) reduces to a single condition $wR(w,w,w,w,w,w,\psi )=0$. Since $w=0$ corresponds to the pre-bifurcation solution, we focus on a relation $R(w,w,w,w,w,w,\psi )=0$. Because this relation in general has a solution of the form $\psi = a w +\mathrm{(higher \ order \ terms)}$ for some real constant a in the neighborhood of the bifurcation point, the bifurcating solution of the form $\varvec{x}=w(1,1,1,1,1,1)$ exists. The other five cases can be treated similarly.

1.8 C.8 The Detail of the Bifurcation Equation for the Orbit of Type $\beta i$

Using the following Lemma, we can obtain the bifurcating solutions for Type $\beta i$ (i.e., Proposition 5).

Lemma 4

Bifurcation equation for Type $\beta i$ orbit is derived as follow: some i, which comprises twelve points, the bifurcation equation becomes:

$$\begin{aligned} \tilde{\varvec{F}}_1(\varvec{x},\psi )= & {} x_1 R(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_{10},x_{11},x_{12},\psi )=0, \\ \tilde{\varvec{F}}_2(\varvec{x},\psi )= & {} x_2 R(x_2,x_1,x_{12},x_{11},x_{10},x_9,x_8,x_7,x_6,x_5,x_4,x_3,\psi )=0, \\ \tilde{\varvec{F}}_3(\varvec{x},\psi )= & {} x_3 R(x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_{10},x_{11},x_{12},x_1,x_2,\psi )=0, \\ \tilde{\varvec{F}}_4(\varvec{x},\psi )= & {} x_4 R(x_4,x_3,x_2,x_1,x_{12},x_{11},x_{10},x_9,x_8,x_7,x_6,x_5,\psi )=0, \\ \tilde{\varvec{F}}_5(\varvec{x},\psi )= & {} x_5 R(x_5,x_6,x_7,x_8,x_9,x_{10},x_{11},x_{12},x_1,x_2,x_3,x_4,\psi )=0, \\ \tilde{\varvec{F}}_6(\varvec{x},\psi )= & {} x_6 R(x_6,x_5,x_4,x_3,x_2,x_1,x_{12},x_{11},x_{10},x_9,x_8,x_7,\psi )=0, \\ \tilde{\varvec{F}}_7(\varvec{x},\psi )= & {} x_7 R(x_7,x_8,x_9,x_{10},x_{11},x_{12},x_1,x_2,x_3,x_4,x_5,x_6,\psi )=0, \\ \tilde{\varvec{F}}_8(\varvec{x},\psi )= & {} x_8 R(x_8,x_7,x_6,x_5,x_4,x_3,x_2,x_1,x_{12},x_{11},x_{10},x_9,\psi )=0, \\ \tilde{\varvec{F}}_9(\varvec{x},\psi )= & {} x_9 R(x_9,x_{10},x_{11},x_{12},x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,\psi )=0, \\ \tilde{\varvec{F}}_{10}(\varvec{x},\psi )= & {} x_{10} R(x_{10},x_9,x_8,x_7,x_6,x_5,x_4,x_3,x_2,x_1,x_{12},x_{11},\psi )=0, \\ \tilde{\varvec{F}}_{11}(\varvec{x},\psi )= & {} x_{11} R(x_{11},x_{12},x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_{10},\psi )=0, \\ \tilde{\varvec{F}}_{12}(\varvec{x},\psi )= & {} x_{12} R(x_{12},x_{11},x_{10},x_9,x_8,x_7,x_6,x_5,x_4,x_3,x_2,x_1,\psi )=0, \end{aligned}$$

where $\varvec{x}=(x_1,\ldots ,x_{12})=\{\lambda _i \mid i \in \beta i\}$ (Fig. 8(b)) and $\psi =\phi -\phi ^\textrm{c}_l$.

Proof

By the product form of the replicator dynamics in (3), $\tilde{F}_i(\varvec{x},\psi )$ takes the product form: $\tilde{F}_i(\varvec{x},\psi )=x_i R_i(\varvec{x},\psi )$ $(i\in P_{\beta i})$. Since the group $\textrm{D}_6$ is generated by the elements r and s, it suffice to consider the symmetry condition $T(g) \tilde{\varvec{F}}(\varvec{x})=\tilde{\varvec{F}}(T(g)\varvec{x})$ for $g=r,s$. This condition for $g=r$ gives the form for $\tilde{\varvec{F}}_{i}$ $(i=1,3,\ldots ,11)$ for some function $R(\varvec{x},\psi )$ and that for $g=s$ gives the form for $\tilde{\varvec{F}}_{i}$ $(i=2,4,\ldots ,12)$.