2.1 A continuous-time OLG model with multiple regions

We consider a continuous-time overlapping generations model with $n \geq 2$ regions. Let $S=\{1, 2, ..., n\}$ be the set of regions. A mass one of the initial agents at time 0 are distributed across the regions according to the exogenously given distribution $x_0 \in \Delta$ , where $\Delta = \left\{ x \in \mathbb{R}_+^n: \sum_{i \in S} x_i =1 \right\}$ , each of whom is endowed with an amount $k_0 > 0$ of capital. Each agent is replaced by a newborn, also endowed with an initial capital amount $k_0$ , according to a Poisson process with parameter $\lambda > 0$ . We normalize the time unit in such a way that $\lambda = 1$ . These processes are assumed to be independent, so that during each short time interval $[t, t+\mathrm{d} t)$ , a mass $\mathrm{d} t$ of agents are replaced by the same mass of entrants, where the total mass of the population is fixed to one over time. We call the agents born at time $\tau$ generation $\tau$ .

Upon birth, an individual makes a once-and-for-all location decision, that is, he chooses which region to live in and settles in that region throughout his life. Let $\alpha_i(\tau)$ denote the population share of generation $\tau$ in region i. The time-t mass of generation $\tau$ locating in region i is $e^{-( t-\tau)} \alpha_i(\tau)$ , and therefore, the total population distribution $x (t) = (x_1(t), \ldots, x_n(t)) \in \Delta$ at time t is given by

(1) \begin{equation} x_i(t) = e^{-t} x_{0i} + \int_{0}^t e^{-(t- \tau)} \alpha_i(\tau) \mathrm{d} \tau\end{equation}

for each region i. We denote the time path of x(t), to be determined in equilibrium, by $x (\cdot) = \left( x ( t ) \right)_{t \ge 0}$ .

2.1.1 Production

In each region, firms produce the consumption good with capital as the only input under AK technology, where both the consumption and capital goods are assumed to be non-transferable to other regions. ^{Footnote 6} Production is subject to externalities in the spirit of Romer (Reference Romer1986); specifically, it benefits from spillover effects within the region as well as from other regions. Intraregional spillovers will work as agglomeration forces, while interregional spillovers tend to mitigate the former. To simplify the argument, we assume that the capital productivity depends directly on the population distribution $x\in \Delta$ and, in particular, it is increasing in the population of the region where the production takes place. The aggregate production function in region i takes the AK form,

(2) \begin{equation} Y_i = Z_i(x) K_i,\end{equation}

where $K_i$ is the aggregate capital input, and the productivity factor $Z_i(x)$ depends on the populations, and hence the levels of production, of the own region as well as the other regions through

(3) \begin{equation}Z_i(x)= \sum_{j \in S} z_{ij} x_j\end{equation}

with $z_{ii} > 0$ and $z_{ij} \geq 0$ for all $j \neq i$ . Intra- and interregional spillover effects are captured by $z_{ii}$ and $z_{ij}$ , respectively, and in fact define a network over the regions, where links are weighted according to the spillover matrix $Z = (z_{ij})$ .

The term $Z_i(x)$ captures the externalities arising from interactions of individuals with knowledge spillovers within and across regions, which are typically decreasing in the physical and/or social distance between regions. Our assumption of its dependence on the population is for tractability: in particular, it allows us to solve for the market equilibrium capital stocks within each region (given a path of population distribution $x(\cdot)$ fixed), as to be done below. ^{Footnote 7 Footnote 8}

2.1.2 Consumption

Each agent of generation $\tau$ located in region i decides on the path of consumption $(c_i(\tau, t))_{t \geq \tau}$ to maximize his expected lifetime utility, where we assume that the instantaneous utility is given by $\ln c_i(\tau, t)$ . Since the lifespan is exponentially distributed with mean 1, the expected lifetime utility is

(4) \begin{equation}\int_0^\infty \int_\tau^{\tau +s} e^{-\rho (t - \tau)} \ln c_i (\tau, t )\mathrm{d} t e^{-s} \mathrm{d} s = \int_\tau^\infty e^{-(1 + \rho)(t- \tau)} \ln c_i(\tau, t) \mathrm{d} t, \end{equation}

where $\rho > 0$ is the common rate of time preference, while $1 + \rho$ is interpreted as the effective discount rate. We will be interested mainly in the case where agents are farsighted, that is, $\rho$ is close to zero. Note that the model is well defined whenever $\rho > -1$ .

Agents in region i earn the returns to their capital by renting it to the firms within region i. For consumption, we assume that there are congestion externalities in the form of iceberg-type intraregional transport costs. Specifically, in order to consume one unit of the good, an individual in region i has to purchase $\phi_i (x_i(t) )$ units, where $\phi_i(x_i) \ge 1$ is continuously differentiable on [0, 1], ^{Footnote 9} and satisfies $\phi_i'(x_i) > 0$ for all $x_i \in [0, 1]$ . ^{Footnote 10} The intertemporal budget constraint of generation $\tau$ is then given by

(5) \begin{equation}\dot{k}_i(\tau, t)= r_i(t) k_i(\tau, t) -\phi_i (x_i(t)) c_i(\tau, t),\quad k_i(\tau, \tau) = k_0, \end{equation}

where $k(\tau, t)$ denotes the capital holding with $\dot{k}_i(\tau, t) = \partial k_i(\tau, t)/\partial t$ , $r_i(t)$ denotes the rental rate of capital (to be determined in equilibrium), and the depreciation rate is assumed to be zero.

Given $x(\cdot)$ , $r_i(\cdot)$ , and $k_0$ , an agent of generation $\tau$ in region i maximizes (4) subject to (5). With the (current-value) Hamiltonian

\begin{align*}H_i(k_i, c_i, \eta_i, t)= \ln c_i + \eta_i \bigl[r_i(t) k_i - \phi_i(x_i(t)) c_i\bigr],\end{align*}

the necessary conditions for optimality are

(6) \begin{align}&\frac{\partial H_i}{\partial c_i}(k_i(\tau, t), c_i(\tau, t), \eta_i(\tau, t), t) = \frac{1}{c_i(\tau, t)} - \eta_i(\tau, t) \phi_i (x_i(t)) = 0, \end{align}

(7) \begin{align}&\dot{\eta}_i(\tau, t) = (1 + \rho) \eta_i(\tau, t) - \frac{\partial H_i}{\partial k_i}(k_i(\tau, t), c_i(\tau, t), \eta_i(\tau, t), t) \notag \\&\phantom{\dot{\eta}_i(\tau, t) } {}={} -\bigl[r_i(t) - (1 + \rho)\bigr] \eta_i(\tau, t), \end{align}

(8) \begin{align}&\lim_{t \to \infty} e^{-(1 + \rho) (t - \tau)} \eta_i(\tau, t) k_i(\tau, t) = 0, \end{align}

where $\eta_i$ is the adjoint variable, and (8) is the transversality condition (see, e.g., Kamihigashi (Reference Kamihigashi2001), Section 4.2). By (7), we have

(9) \begin{equation} \eta_i(\tau, t) = \eta_i(\tau, \tau) e^{-R_i(\tau, t)}, \end{equation}

where

(10) \begin{equation}R_i(\tau, t) = \int_{\tau}^t \bigl[r_i(s) - (1 + \rho)\bigr] \mathrm{d} s.\end{equation}

We also have

\begin{align*}\frac{\partial}{\partial t} \bigl(\eta_i(\tau, t) k_i(\tau, t)\bigr)&= \dot{\eta}_i(\tau, t) k_i(\tau, t) + \eta_i(\tau, t) \dot{k}_i(\tau, t) \\&= -\bigl[r_i(t) - (1 + \rho)\bigr] \eta_i(\tau, t) k_i(\tau, t) + \eta_i(\tau, t) \bigl(r_i(t) k_i(\tau, t) - \eta_i(\tau, t)^{-1}\bigr) \\&= (1 + \rho) \left(\eta_i(\tau, t) k_i(\tau, t) - \frac{1}{1 + \rho}\right),\end{align*}

where the second equality follows from (5), (6), and (7), so that

(11) \begin{equation} \eta_i(\tau, t) k_i(\tau, t) - \frac{1}{1 + \rho}= e^{(1 + \rho) (t - \tau)} \left(\eta_i(\tau, \tau) k_0 - \frac{1}{1 + \rho}\right).\end{equation}

But the transversality condition (8) holds only if $\eta_i(\tau, \tau) k_0 - \frac{1}{1 + \rho} = 0$ , and therefore by (11) we have $\eta_i(\tau, t) k_i(\tau, t) = \frac{1}{1 + \rho}$ for all $t \geq \tau$ . It follows that the optimal consumption and capital holding are given as

(12) \begin{equation} c_i(\tau, t) = \frac{1 + \rho}{\phi_i (x_i(t))} k_i(\tau, t)\end{equation}

and

(13) \begin{equation} k_i(\tau, t) = k_0 e^{R_i(\tau, t)}\end{equation}

by (6) and (9), respectively.

Therefore, by (4), (12), and (13), we obtain the expected lifetime utility of agents born at time $\tau$ from locating in region i as

(14) \begin{equation}\int_\tau^\infty e^{-(1 + \rho) (s -\tau ) } \left\{ R_i(\tau, s) + \ln \frac{(1 + \rho)k_0}{\phi_i ( x_i(s ) )} \right\} \mathrm{d} s. \end{equation}

2.1.3 Market equilibrium

The capital market clearing condition at each time t is

(15) \begin{equation}K_i(t) = e^{-t} x_{0i} k_i(0, t) + \int_{0}^t e^{-(t- \tau)} \alpha_i(\tau) k_i(\tau, t) \mathrm{d} \tau,\end{equation}

where the right hand side is the aggregate supply obtained by aggregating the capital holdings of the agents in region i.

The equilibrium rental rate $r_i(t)$ is generally not equal to the marginal productivity $Z_i(x)$ , depending on how the capital holdings of the exiting agents are handled. Here we assume that only a fraction $\mu \in [0, 1]$ of each individual’s holding of capital is transferable within a region upon exit, due to transaction costs which amount $1 - \mu$ per unit (Heijdra and Mierau (Reference Heijdra and Mierau2012)), and that there is a perfectly competitive insurance market, so that

(16) \begin{equation}r_i(t)= Z_i (x(t)) + \mu \end{equation}

holds under free entry. ^{Footnote 11} The value of $\mu$ depends on the extent of intraregional mobility of capital. In the limiting case where $\mu = 0$ , the capital is considered as completely sunk, while in the other polar case where $\mu = 1$ as in the standard continuous-time OLG models (Yaari (Reference Yaari1965)), capital may be interpreted as general assets which are tradable (within the region).

2.2 Spatial equilibrium dynamics

We have characterized the optimal paths of consumption and capital holding given a path of population distributions $x(\cdot)$ . Thus, the next task is to define the equilibrium condition regarding the location choice. When the agents choose their locations, they compare the expected lifetime payoff from each location. By (10), (14), and (16), the expected lifetime payoff from region i is given by

(17) \begin{align}V_i(x(\cdot), \tau)&= (1 + \rho) \int_\tau^\infty e^{-(1 + \rho) (s -\tau ) } \left\{ \int_{\tau}^s \left[Z_i(x(\nu)) + \mu - (1 + \rho)\right] \mathrm{d} \nu + \ln \frac{(1 + \rho)k_0}{\phi_i ( x_i(s ) )} \right\} \mathrm{d} s \notag \\&= (1 + \rho) \int_\tau^\infty e^{-(1 + \rho) (s -\tau ) } \left( \frac{Z_i (x(s ))}{1 + \rho} - \ln \phi_i (x_i(s )) \right) \mathrm{d} s + C, \end{align}

where the second equality obtains by integration by parts, and $C = \frac{\mu - (1 + \rho)}{1 + \rho} + \ln (1 + \rho) k_0$ . The value $V_i(x(\cdot), \tau)$ of locating in region i is determined by the path $x(\cdot)$ of population distributions through the spillover effects $Z_i (x(\cdot))$ , which depend on the position of region i in the spillover network defined by Z, as well as the congestion effects $\phi_i (x_i(\cdot))$ . Note that the value is normalized by the effective discount rate $1 + \rho$ .

Our equilibrium dynamics falls in the class of perfect foresight dynamics studied in the context of sectoral choice by Matsuyama (Reference Matsuyama1991) as well as in game theory by Matsui and Matsuyama (Reference Matsui and Matsuyama1995), Hofbauer and Sorger (Reference Hofbauer and Sorger1999), and Oyama et al. (Reference Oyama, Takahashi and Hofbauer2008), among others. A spatial equilibrium path is a path $x^\ast(\cdot)$ along which agents optimally choose locations under the expectation of $x^\ast(\cdot)$ itself.

Definition 2.1. $x^\ast\colon [0, \infty) \to \Delta$ is a spatial equilibrium path, or equilibrium path in short, from $x_0 \in \Delta$ if it is Lipschitz continuous and satisfies $x^\ast(0) = x_0$ , and for all $i \in S$ ,

(18) \begin{equation}\dot{x}_i^\ast (t) > -x_i^\ast (t) \Rightarrow i \in arg\,max_{j\in S} V_j (x^\ast(\cdot), t)\end{equation}

for almost all $t \geq 0$ .

Recall from (1) that we have $\dot{x}_i(t) = \alpha_i(t) - x_i(t)$ . Thus, $\dot{x}_i^\ast (t) > -x_i^\ast (t)$ (i.e., $\alpha_i(t) > 0$ ) implies that some positive fraction of new entrants choose to locate in region i during short time interval $[t, t + \mathrm{d} t)$ . The condition says that region i must maximize $V_j(x^\ast(\cdot), t)$ with respect to j given $x^\ast(\cdot)$ itself.

The continuity of the integrand in (17), $\frac{Z_i (x)}{1 + \rho} - \ln \phi_i (x_i)$ , in $x \in \Delta$ guarantees the existence of an equilibrium path for each initial state $x_0$ (see, e.g., Oyama et al. (Reference Oyama, Takahashi and Hofbauer2008), Subsection 2.3).

2.3 Stationary spatial equilibrium states

We say that a path $x(\cdot)$ is stationary at $\bar{x}\in \Delta$ if $x(t)=\bar{x}$ for all $t\ge 0$ (i.e., x(t) is constant at $\bar{x}$ over time). If $x(\cdot)$ is stationary at $\bar{x}$ , the payoff from locating in region i is given by

\begin{align*}V_i(x(\cdot), \tau) = \frac{Z_i (\bar{x})}{1 + \rho} - \ln \phi_i(\bar{x}_i)\end{align*}

for all $i \in S$ and all $\tau \geq 0$ . We define the function $Q\colon \Delta \to \mathbb{R}^n$ by the right-hand side:

(19) \begin{equation}Q_i(x)= \frac{Z_i (x)}{1 + \rho} - \ln \phi_i(x_i) \end{equation}

for each $i \in S$ . Thus, the function V in (17) is written as

\begin{align*}V_i(x(\cdot), \tau)= (1 + \rho) \int_\tau^\infty e^{-(1 + \rho) (s -\tau ) } Q_i(x(s)) \mathrm{d} s + C.\end{align*}

Definition 2.2. $\bar{x} \in \Delta$ is a stationary spatial equilibrium state, or equilibrium state in short, if for all $i \in S$ ,

(20) \begin{equation}\bar{x}_i >0 \Rightarrow i \in arg\,max_{j\in S} Q_j (\bar{x}).\end{equation}

By construction, $\bar{x}$ is an equilibrium state if and only if the stationary path $x(\cdot)$ at $\bar{x}$ is an equilibrium path from $\bar{x}$ . By the continuity of Q(x) in $x \in \Delta$ , the existence of an equilibrium state follows from a standard fixed point argument. We say that $\bar{x}$ is a strict equilibrium state if for all $i \in S$ , $\bar{x}_i >0 \Rightarrow \{i\} = arg\,max_{j\in S} Q_j (\bar{x})$ . Equivalently, a strict equilibrium state is a full agglomeration state $e_{i^\ast}$ , a state in which all the agents are located in one region $i^\ast$ , such that $Q_{i^\ast}(e_{i^\ast}) > Q_j(e_{i^\ast})$ for all $j\neq i^\ast$ .

Formally, the stationary payoff function $Q = (Q_i)_{i \in S}$ can be interpreted as a static population game (Sandholm (Reference Sandholm2010)), a game in which a continuum of homogeneous players choose among actions in $S = \{1, \ldots, n\}$ and their payoff function Q(x) depends only on the action distribution $x \in \Delta$ (rather than action profile) as well as one’s own action. Our (strict) equilibrium states are precisely the (strict) Nash equilibrium action distributions of the population game.

An important departure of our model from the existing literature of perfect foresight dynamics in population games (such as Matsui and Matsuyama (Reference Matsui and Matsuyama1995) and Hofbauer and Sorger (Reference Hofbauer and Sorger1999) in random matching settings and Matsuyama (Reference Matsuyama1991) and Oyama (Reference Oyama2009) in economic contexts) is that the stationary payoff function Q and hence the equilibrium states depend on the discount rate $\rho$ . Our model involves a stock variable, that is, capital stock, the returns to which constitute the first term in the expression (19), where the population distribution x affects the payoffs through the rate of return as spillovers. The second term in (19) represents the congestion costs, which by assumption are directly affected by the local population $x_i$ . The relative importance between these benefits and costs is governed by the discount rate $\rho$ .

As a preliminary, we discuss some intuitive properties of the equilibrium states of our model. First, if intraregional spillover effects net of congestion effects dominate interregional spillover effects for region i, then the full agglomeration state in region i is a strict equilibrium state, provided that the discounting of the future benefits from spillovers is sufficiently small.

Observation 2.1 For $i \in S$ , suppose that

(21) \begin{equation} z_{ii} - \ln \phi_i(1) > \max_{j \neq i} z_{ji}.\end{equation}

Then there exists $\rho_0(i) > 0$ such that if $\rho < \rho_0(i)$ , then the full agglomeration in i is a strict equilibrium state.

This immediately follows from the observation that the full agglomeration state in region i is a strict equilibrium state if and only if $\frac{z_{ii}}{1 + \rho} - \ln \phi_i(1) > \frac{z_{ji}}{1 + \rho}$ for all $j \neq i$ (recall that $\phi_j(0) = 1$ ), that is, the effective intraregional net spillovers $\frac{z_{ii}}{1 + \rho} - \ln \phi_i(1)$ are greater than the effective inter-regional spillovers $\frac{z_{ji}}{1 + \rho}$ ; then let $\rho_0(i) = \min_{j \neq i} (z_{ii} - z_{ji} - \ln \phi_i(1)) / \ln \phi_i(1) > 0$ .

Second, if the time discounting is sufficiently large, so that the congestion effects become dominant relative to the effective spillover terms, then there is a unique equilibrium state, and it is a full dispersion state, a state in which every region hosts a positive mass of agents. Formally, for sufficiently large $\rho$ , first, the payoff function Q satisfies the strict contractivity condition (Sandholm (Reference Sandholm, Young and Zamir2015)), ^{Footnote 12}

(22) \begin{equation} (y - x)' (Q(y) - Q(x)) < 0 \text{ for all $x, y \in \Delta$ with $x \neq y$},\end{equation}

in which case Q necessarily has a unique equilibrium state, and second, the unique equilibrium state lies in the interior of $\Delta$ .

Observation 2.2 There exists $\rho_1 > 0$ such that if $\rho > \rho_1$ , then the strict contractivity condition (22) holds, and $\bar{x}_i > 0$ for all i in the unique equilibrium state $\bar{x}$ .

Recall that, by (10), (13), and (16), at the full agglomeration state in region i, the growth rate of individual capital holding in each region j is $\frac{\dot{k}_j(\tau, t)}{k_j(\tau, t)} = z_{ji} + \mu - (1 + \rho)$ , which, under the condition (21), is maximized at $j = i$ . Thus, agglomeration makes the capital grow faster, which tends to increase the future benefits in i. On the other hand, by (12), the initial consumption is $\frac{ (1 + \rho)k_0}{\phi_i(1)}$ in i and $(1 + \rho)k_0$ in $j \neq i$ (where $\phi_j(0) = 1$ ), and it is minimized at region i. Thus, agglomeration causes congestion, which tends to reduce the current consumption. When the discount rate $\rho$ is small, or people are farsighted, they attach greater importance to the agglomeration benefits on the capital growth than to the congestion costs on the current consumption. As a result, agglomeration in region i is likely to attain. In particular, there are multiple strict equilibrium states if the condition in (21) holds for multiple regions and the discount rate is small accordingly. ^{Footnote 13} If the discounting is sufficiently large, or people are sufficiently myopic, in contrast, they care about congestion more than economic growth, and therefore agglomeration is less likely to attain and in fact, in a unique equilibrium state, the population is fully dispersed across the regions.

2.4 Long-run capital and income at equilibrium states

Before closing this section, we turn our attention to the long-run levels of capital and income at equilibrium states. Consider the equilibrium path that is stationary at $x\in \Delta$ and any region i such that $x_i > 0$ . Then by (16), the rental rate $r_i(t)$ in region i is equal to

(23) \begin{equation}r_i(t) = Z_i(x) +\mu\end{equation}

for all $t \geq 0$ . Thus, by (10) and (13), we have $k_i(\tau, t) = k_0 e^{[r_i - (1 + \rho)] (t - \tau)}$ , and therefore, by (15) with $\alpha_i(\tau) = x_i$ ,

(24) \begin{equation}K_i(t) =\begin{cases}\dfrac{(r_i - \rho - 1) e^{( r_i -\rho - 2) t} - 1}{r_i - \rho - 2} K_i(0) & \text{if $r_i - \rho - 2 \neq 0$}, \\(t + 1) K_i(0) & \text{otherwise},\end{cases}\end{equation}

where $K_i(0) = x_i k_0$ . Let $k_i(t)=K_i(t)/x_i$ and $y_i(t) = Y_i(t)/x_i = Z_i(x) k_i(t)$ , which are the per-capita capital and per-capita income in region i, respectively. If $r_i - \rho - 2 < 0$ , or $Z_i(x) < -\mu + \rho + 2$ , then $k_i(t) \to \frac{k_0}{-Z_i(x) - \mu + \rho + 2}$ and $y_i(t) \to \frac{Z_i(x) k_0}{-Z_i(x) - \mu + \rho + 2}$ as $t \to \infty$ , so that the region i tends to a steady state. On the other hand, if $r_i -\rho - 2 \geq 0$ , or $Z_i(x) \geq -\mu + \rho + 2$ , the per-capita capital grows without bound, where $\frac{\dot{k}_i(t)}{k_i(t)}, \frac{\dot{y}_i(t)}{y_i(t)} \to Z_i(x) + \mu - \rho - 2$ as $t \to \infty$ , so that the region rides on a balanced growth path asymptotically. Hence, we obtain the following result.

Proposition 2.3. Suppose that the equilibrium path is stationary at $x\in \Delta$ . Then,

\begin{equation*}\begin{cases}(k_i(t), y_i(t) ) \to \left( \frac{k_0}{-Z_i(x) - \mu + \rho + 2}, \frac{Z_i(x) k_0}{-Z_i(x) - \mu + \rho + 2} \right) & \text{if $Z_i(x) < -\mu + \rho + 2$} \\\frac{\dot{k}_i(t)}{k_i(t)}, \frac{\dot{y}_i(t)}{y_i(t)} \to Z_i(x) + \mu - \rho - 2 & \text{if $Z_i(x) \geq -\mu + \rho + 2$}\end{cases} \quad \text{as $t\to \infty$}.\end{equation*}

Since the productivity at an equilibrium state differs across regions in general, the proposition above implies that the $\sigma$ -convergence (Barro and Sala-i-Martin (Reference Barro and Sala-i-Martin1992)), which is the diminishing spatial variation of per-capita incomes over time, will not always occur. In our model, the $\sigma$ -divergence can also happen. In particular, it is possible that, while some regions will succeed in riding on balanced growth paths, some other regions will be stuck with fixed income levels. ^{Footnote 14} In fact, we present a numerical example for such a case in Subsection 4.2.

3.1 Stability concepts

As we discussed in the previous section, our dynamic model may generally have multiple equilibrium states, and it is indeed the case when spillover effects dominate congestion effects. Furthermore, from an equilibrium state $\bar{x}$ , there may exist an equilibrium path, other than the stationary path at $\bar{x}$ , that departs away from $\bar{x}$ and converges to another equilibrium state. We regard such a state $\bar{x}$ as unstable or fragile. We are interested in equilibrium states that are globally stable in the following sense (Matsui and Matsuyama (Reference Matsui and Matsuyama1995)). For $x \in \Delta$ and $\varepsilon > 0$ , we let $B_{\varepsilon}(x) = \{y \in \Delta \mid \lvert y - x\rvert < \varepsilon\}$ denote the $\varepsilon$ -neighborhood of x in $\Delta$ , where $\lvert z \rvert = \max_{i \in S} \lvert z_i\rvert$ is the max-norm of $z \in \mathbb{R}^n$ .

In what follows, we aim to characterize an equilibrium state $\bar{x}$ that is absorbing and globally accessible for sufficiently small discount rates $\rho > 0$ . Recall that the equilibrium path is not necessarily unique for an initial state. Nevertheless, the absorption of $\bar{x}$ requires that any equilibrium path from a neighborhood of $\bar{x}$ converge to $\bar{x}$ . On the other hand, for the global accessibility, we require that from any initial state, there exist at least one equilibrium path that converges to $\bar{x}$ . By definition, if a state is absorbing (globally accessible, resp.), then no other state can be globally accessible (absorbing, resp.), and thus an absorbing and globally accessible state is unique if it exists.

Our dynamic equilibrium model is highly nonlinear and infinite dimensional and thus is difficult to analyze in general, in particular when there are more than two regions. Accordingly, to maintain our n-region setup, the analysis in this paper will be conducted under a certain symmetry assumption on the spillover matrix Z, as introduced in the next subsection.

3.2 Potential

In our analysis, the concept of potential will play an important role. Let $\overline{\Delta}$ be an open neighborhood of $\Delta$ in $\mathbb{R}^n$ .

Definition 3.2. A function $W: \overline{\Delta}\to \mathbb{R}$ is a potential function of Q if it is differentiable and satisfies

(25) \begin{equation}\frac{\partial W}{\partial x_i}(x) - \frac{\partial W}{\partial x_j}(x) = Q_i(x) - Q_j (x)\text{ for all $x \in \Delta$ and all $i, j \in S$}.\end{equation}

The function W is defined on $\overline{\Delta}$ only for its derivatives to be well defined on $\Delta$ ; otherwise, it is innocuous. By definition, W is a potential function of Q if and only if $z' \nabla W(x) = z' Q(x)$ for all $x \in \Delta$ and all $z \in T\Delta$ , where $\nabla W(x) = \left(\frac{\partial W}{\partial x_1}(x), \ldots, \frac{\partial W}{\partial x_n}(x)\right)'$ denotes the gradient vector of W at x, and $T\Delta = \{z \in \mathbb{R}^n \mid \sum_{i \in S} z_i = 0\}$ the tangent space of $\Delta$ .

This concept is a natural extension of that by Monderer and Shapley (Reference Monderer and Shapley1996), defined for finite-player normal form games, to population games (Sandholm (Reference Sandholm2001, Reference Sandholm2009, Reference Sandholm2010), Oyama (Reference Oyama2009)). Observe that if Q admits a potential function W, the set of equilibrium states of Q coincides with the set of Karush–Kuhn–Tucker (KKT) points for the optimization problem $\max W(x)$ subject to $x \in \Delta$ , that is, those points $x \in \Delta$ for which there exist $\nu \in \mathbb{R}$ and $\eta \in \mathbb{R}^n_+$ such that for all $i \in S$ ,

(26a) \begin{align}&\frac{\partial W}{\partial x_i}(x) = \nu - \eta_i \end{align}

(26b) \begin{align}&x_i > 0 \Rightarrow \eta_i = 0. \end{align}

It is clear from the definition that a potential function always exists when there are only two regions. For three or more regions, the existence of a potential function requires a nontrivial restriction on the parameters. In our model, Q admits a potential function if and only if the spillover matrix Z satisfies the triangular integrability condition:

(27) \begin{equation} z_{ij} + z_{jk} + z_{ki} = z_{i k} + z_{kj} + z_{ji} \text{ for all $i, j, k\in S$}.\end{equation}

See, for example, Oyama (Reference Oyama2009), Appendix A or Sandholm (Reference Sandholm2010), Section 3.2 for more details. Assume that the triangular integrability condition holds. Let $\widetilde{Z} = (\tilde{z}_{ij})$ be the $n \times n$ matrix defined by

\begin{align*}\tilde{z}_{ij} = z_{ij} + z_{j1} - z_{1j},\end{align*}

which is symmetric (i.e., $\tilde{z}_{ij} = \tilde{z}_{ji}$ for all $i, j\in S$ ) by the triangular integrability. Then, a potential function of Q is given by

(28) \begin{equation}W(x) = \frac{1}{2(1 + \rho)} x' \widetilde{Z} x -\sum_{i \in S} \int_0^{x_i} \ln \phi_i (y ) \mathrm{d} y,\end{equation}

for which, for all $x \in \Delta$ , we have

\begin{align*}\frac{\partial W}{\partial x_i}(x)&= \frac{1}{1 + \rho} (\widetilde{Z} x)_i - \ln \phi_i(x_i) \\&= \frac{1}{1 + \rho} (Z x)_i - \ln \phi_i(x_i) + \sum_{k=1}^n (z_{k1} - z_{1k}) x_k\end{align*}

and hence $\frac{\partial W}{\partial x_i}(x) - \frac{\partial W}{\partial x_j}(x) = Q_i(x) - Q_j(x)$ for all $i, j \in S$ . In a special case where Z is symmetric, we have $\widetilde{Z} = Z$ and $\nabla W(x) = Q(x)$ for all $x \in \Delta$ . Note that a potential function is unique on $\Delta$ up to a constant.

In our model, the stationary payoff function Q depends on the discount rate $\rho$ . We will denote it as well as its potential function by $Q(\cdot, \rho)$ and $W(\cdot, \rho)$ , respectively, when we want to make the dependence on $\rho$ explicit. Note that $Q(\cdot, \rho)$ and $W(\cdot, \rho)$ as expressed in (19) and (28) are well defined for all $\rho \in (-1, \infty)$ .

3.3 Stability theorem

In this subsection, we present our global stability result, which holds under the following assumptions. For $x \in \Delta$ , we let $supp(x) = \{i \in S \mid x_i > 0\}$ denote the support of x.

The triangular integrability condition (27) holds, so that the function $W(\cdot, \rho)$ defined by (28) is a potential function of $Q(\cdot, \rho)$ for each $\rho \in (-1, \infty)$ .

$W(\cdot, 0)$ has a unique maximizer $\bar{x}^0$ on $\Delta$ .

$\bar{x}^0$ is a regular equilibrium state of $Q(\cdot, 0)$ in the following sense, where we denote $C = supp(\bar{x}^0)$ :

1. (1) $\bar{x}^0$ is a quasi-strict equilibrium state of $Q(\cdot, 0)$ , that is, it is an equilibrium state of $Q(\cdot, 0)$ such that $Q_i(\bar{x}^0, 0) > Q_j(\bar{x}^0, 0)$ for all $i \in C$ and all $j \notin C$ .

2. (2) The matrix

   \begin{align*} \begin{pmatrix} D_x Q(\bar{x}^0, 0)_C & \textbf{1}_C \\ \textbf{1}'_{C} & 0 \end{pmatrix} \in \mathbb{R}^{(|C|+1)\times (|C|+1)} \end{align*}
   is nonsingular, where $D_x Q(\cdot, 0)_C\in \mathbb{R}^{|C|\times |C|}$ is the submatrix of the Jacobian matrix $D_x Q(\cdot, 0)$ of $Q(\cdot, 0)$ restricted to C and $\textbf{1}_C\in \mathbb{R}^{|C|}$ is the $|C|$ -dimensional column vector of ones.

The regularity concept in A3 is analogous to the concepts of regular Nash equilibrium in normal form games (van Damme (Reference van Damme1983)) and regular evolutionarily stable strategy in population games (Taylor and Jonker (Reference Taylor and Jonker1978)).

The Jacobian $D_x Q(\cdot, \rho)$ is written as

\begin{align*}D_x Q(x, \rho) = \frac{1}{1 + \rho} Z - \Psi(x),\end{align*}

where $\Psi(x)$ is the matrix such that $\Psi_{ii}(x) = \phi_i'(x)/\phi_i(x)$ and $\Psi_{ij}(x) = 0$ for all i and $j \neq i$ . With the existence of a potential function $W(\cdot, \rho)$ , the inequality in 3.31 is written as $\frac{\partial W}{\partial x_i}(\bar{x}^0, 0) > \frac{\partial W}{\partial x_j}(\bar{x}^0, 0)$ , which implies that $\bar{x}^0$ satisfies the KKT conditions (26) with strict complementary slackness (i.e., $\eta_i = 0$ if and only if $x_i > 0$ in place of (26b)), while the nonsingularity of the matrix in A3(2) is equivalent to the nonsingularity of the bordered Hessian of W(x, 0) at $\bar{x}^0$ on $\{x\in \Delta \mid \sum_{i\in C}x_i =1\}$ ,

\begin{align*} \begin{pmatrix} D^2_x W(\bar{x}^0, 0)_C & \textbf{1}_C \\ \textbf{1}'_C & 0 \end{pmatrix}. \end{align*}

In particular, a strict equilibrium state is necessarily a regular equilibrium state. Assumption A2 is a genericity condition which excludes, for example, perfect symmetry among regions. From A3 it follows in particular that, for $\rho$ sufficiently close to 0, $W(\cdot, \rho)$ has a unique maximizer on $\Delta$ (as shown in Lemma A.6 in the Appendix).

Under these assumptions A1–A3, we have the main theorem on the stability of equilibrium states under our equilibrium dynamics.

Recall from Observation 2.1 that when the discount rate $\rho$ is small, that is, agents are sufficiently farsighted or patient, there tend to exist multiple agglomeration equilibrium states provided that intraregional net spillover effects dominate interregional spillover effects. As agents become more farsighted, expectations that the economy will move from one equilibrium state to another become more likely to be self-fulfilling, making some of the equilibrium states fragile. Our theorem shows that for sufficiently small $\rho$ , all the equilibrium states but one, the state $\bar{x}^{\rho}$ that maximizes the potential function, become fragile. Whatever initial condition $x_0$ the history picks, there is some form of self-fulfilling expectations that leads the economy to $\bar{x}^{\rho}$ , that is, $\bar{x}^{\rho}$ is globally accessible. On the other hand, history also matters, in that if history picks the initial condition in a neighborhood of $\bar{x}^{\rho}$ , any form of self-fulfilling expectations cannot prevent the economy from converging to $\bar{x}^{\rho}$ , that is, $\bar{x}^{\rho}$ is absorbing. Thus, our result offers a natural criterion to select among multiple equilibrium states, and in fact the selected equilibrium is characterized by the maximizer of the potential function of $Q(\cdot, \rho)$ on $\Delta$ . Hence, our task is, then, to inspect the shape of the potential function. In the next subsection (Subsection 3.4), we discuss the globally stable equilibrium state for some simple cases, while in the next section (Section 4), we study stability in relation to the spillover network structure.

In the remainder of this subsection, we briefly discuss the proof of our theorem; the full proof is provided in Appendix A.1.

Suppose that $W(\cdot, \rho)$ is a potential function of $Q(\cdot, \rho)$ with a unique maximizer $\bar{x}^{\rho}$ on $\Delta$ . We utilize two results from the previous literature (Hofbauer and Sorger (Reference Hofbauer and Sorger1999), Oyama (Reference Oyama2009)) which apply to our model for a fixed discount rate $\rho$ . (i) First, for a given initial condition $x_0 \in \Delta$ , consider the dynamic optimization problem,

(29a) \begin{equation} \text{maximize } \mathcal{W}(x(\cdot), \rho) = (1 + \rho) \int_0^\infty e^{-\rho t}W(x(t), \rho) \mathrm{d} t\end{equation}

(29b) \begin{equation}\text{subject to } \dot{x}(t) = \alpha(t) - x(t),\ \alpha(t) \in \Delta,\ x(0) = x_0.\end{equation}

Then, any solution to this problem is an equilibrium path from $x_0$ (Hofbauer and Sorger (Reference Hofbauer and Sorger1999), Theorem 2 or Oyama (Reference Oyama2009), Lemma C.2). This may be seen as a dynamic analogue of the property that a solution to the static optimization problem $\max W(x, \rho)$ subject to $x \in \Delta$ is an equilibrium state of $Q(\cdot, \rho)$ . (ii) To state the second result, call a state $x^{\mathrm{c}} \in \Delta$ a critical point of $W(\cdot, \rho)$ if $\frac{\partial W}{\partial x_i}(x^{\mathrm{c}}, \rho) = \frac{\partial W}{\partial x_j}(x^{\mathrm{c}}, \rho)$ for all $i, j \in supp(x^{\mathrm{c}})$ , and denote the set of critical points of $W(\cdot, \rho)$ by $\mathcal{C}(\rho)$ . Then, for any equilibrium path $x(\cdot)$ , if there exists $t \geq 0$ such that $W(x(t), \rho) > W(x^{\mathrm{c}}, \rho)$ for all $x^{\mathrm{c}} \in \mathcal{C}(\rho) \setminus \{\bar{x}^{\rho}\}$ , then $x(\cdot)$ necessarily converges to $\bar{x}^{\rho}$ (Hofbauer and Sorger (Reference Hofbauer and Sorger1999), Lemma 4 or Oyama (Reference Oyama2009), Lemma C.6). The task is then to guarantee the existence of an $\varepsilon > 0$ such that for any sufficiently small $\rho > 0$ , the $\varepsilon$ -neighborhood $B_{\varepsilon}(\bar{x}^{\rho})$ of $\bar{x}^{\rho}$ in $\Delta$ isolates $\bar{x}^{\rho}$ from other critical points (in that $W(x, \rho) > W(x^{\mathrm{c}}, \rho)$ for all $x \in B_{\varepsilon}(\bar{x}^{\rho})$ and all $x^{\mathrm{c}} \in \mathcal{C}(\rho) \setminus \{\bar{x}^{\rho}\}$ ) and for any $x_0 \in \Delta$ , any solution to the problem (29) visits $B_{\varepsilon}(\bar{x}^{\rho})$ at least once.

Here, we have to notice the difference between the previous models and ours. In the previous models, where a static game is repeatedly played over time, the payoff function Q and hence the potential function W and the unique potential maximizer $\bar{x}$ are independent of the discount rate $\rho$ , so that, under an assumption that $\bar{x}$ is isolated from other critical states, ^{Footnote 15} one can first fix an $\hat{\varepsilon} > 0$ , again independent of $\rho$ , such that $W(x) > W(x^{\mathrm{c}})$ for all $x \in B_{\hat{\varepsilon}}(\bar{x})$ and all $x^{\mathrm{c}} \in \mathcal{C}(\rho) \setminus \{\bar{x}\}$ . Given this $\hat{\varepsilon} > 0$ , a version of the so-called Visit Lemma (Oyama (Reference Oyama2009), Lemma C.3) from turnpike theory shows that there exists $\hat{\rho} > 0$ such that for any $\rho \in (0, \hat{\rho}]$ and any $x_0 \in \Delta$ , if $x(\cdot)$ is an equilibrium path for $\rho$ and $x_0$ , then there exists $t \geq 0$ such that $x(t) \in B_{\hat{\varepsilon}}(\bar{x})$ . Combined with the results (i)–(ii) stated above, these imply that $\bar{x}$ is absorbing and globally accessible for $\rho \in (0, \hat{\rho}]$ in the previous setting.

Our model, in contrast, involves stock variables, and as a consequence, the potential function does depend on the discount rate $\rho$ , so that in general there is no guarantee that we can take an isolating $\varepsilon$ as above uniformly for all (sufficiently small) values of $\rho$ . It would not be possible, in particular, if the trajectory of critical points bifurcates as $\rho$ changes from $\rho = 0$ to $\rho > 0$ . This is where our regularity condition A3 comes in: By A3(1), for $\rho$ ’s sufficiently close to zero and in a neighborhood of $\bar{x}^0$ , the critical states have the same support as $\bar{x}^0$ and hence are precisely the solutions to the system of equations $\frac{\partial W}{\partial x_i}(x, \rho) = \frac{\partial W}{\partial x_j}(x, \rho)$ for all $i, j \in C = supp(\bar{x}^0)$ , $\sum_{i \in S} x_i = 1$ , and $x_i = 0$ for all $i \notin C$ , and A3(2) then allows us to apply the Implicit Function Theorem to this system to show that, under A2, there exist a continuous (in fact $C^1$ ) function $\phi(\rho)$ defined on a neighborhood J of $\rho = 0$ and $\bar{\varepsilon} > 0$ such that for all $\rho \in J$ , $\phi(\rho)$ is a unique maximizer of $W(\cdot, \rho)$ on $\Delta$ , and $W(x ,\rho) > W(x^{\mathrm{c}}, \rho)$ for all $x \in B_{\bar{\varepsilon}}(\phi(\rho))$ and all $x^{\mathrm{c}} \in \mathcal{C}(\rho) \setminus \{\phi(\rho)\}$ (Lemmas A.6–A.7 in the Appendix). Then, this implies, first, by result (ii) above that the potential maximizer $\phi(\rho)$ is absorbing for any $\rho > 0$ contained in J. Second, for $\bar{\varepsilon} > 0$ as obtained, our version of the Visit Lemma (Lemma A.9) shows that there exists $\bar{\rho} > 0$ in J such that for any $\rho \in (0, \bar{\rho}]$ and any initial condition $x_0 \in \Delta$ , any solution to the optimization problem (29), which is an equilibrium path from $x_0$ by result (i), must visit the $\bar{\varepsilon}$ -neighborhood of $\phi(\rho)$ , so that it converges to $\phi(\rho)$ by the choice of $\bar{\varepsilon}$ and result (ii). This means that $\phi(\rho)$ is globally accessible for $\rho \in (0, \bar{\rho}]$ .

3.4 Agglomeration and dispersion as stable equilibrium states

In this subsection, under the triangular integrability assumption (27), we study the shape of the potential function as given in (28) to characterize the stable equilibrium state of our model. Specifically, we consider sufficient conditions under which the potential function becomes convex or concave.

The potential function $W(\cdot, \rho)$ is strictly convex (concave, resp.) on $\Delta$ if and only if $(y - x)' (\nabla W(y, \rho) - \nabla W(x, \rho)) > 0$ ( $< 0$ , resp.) holds for all $x, y \in \Delta$ with $x \neq y$ , where by the definition of a potential function, we have $(y - x)' (\nabla W(y, \rho) - \nabla W(x, \rho)) = (y - x)' (Q(y, \rho) - Q(x, \rho))$ .

First, the following observation gives us a sufficient condition for the potential function to be strictly convex on $\Delta$ for $\rho = 0$ .

Observation 3.2 Assume A.1. If

(30) \begin{equation} z_{ii} - \max_{y \in [0, 1]} \frac{\phi_i'(y)}{\phi_i(y)} > \frac{1}{2} \sum_{j \neq i} (z_{ij} + z_{ji})\text{ for all $i \in S$},\end{equation}

then $W(\cdot, 0)$ is strictly convex on $\Delta$ .

The condition (30) says that $D_x Q(x, 0) + D_x Q(x, 0)'$ has a positive dominant diagonal for all $x \in \Delta$ . It implies that $D_x Q(x, 0)$ is positive definite for all $x \in \Delta$ , which in turn implies that $(y - x)' (\nabla W(y, 0) - \nabla W(x, 0)) = (y - x)' (Q(y, 0) - Q(x, 0)) > 0$ for all $x, y \in \Delta$ , $x \neq y$ , and hence, $W(\cdot, 0)$ is strictly convex on $\Delta$ .

Thus, the potential function is strictly convex when $\rho=0$ if, for each region, the agglomeration economy $z_{ii}$ net of the congestion force $\phi_i'(x_i)/\phi_i(x_i)$ is sufficiently large and/or the spillover effect from the other regions is sufficiently small.

The maximizer of a strictly convex potential function on $\Delta$ is a full agglomeration state (i.e., a vertex of $\Delta$ ) and is a strict equilibrium state, which automatically satisfies the regularity condition A.3. Thus, by Theorem 3.1, we obtain the following result for the case of strict convexity.

Being farsighted, people agglomerate in a single region if the positive effect of agglomeration on the economic growth is sufficiently large relative to its congestion effect on the current consumption (so that the potential function is strictly convex). In particular, if the spillover matrix Z is symmetric (i.e., $z_{ij} = z_{ji}$ for all i, j) and if $z_{11} = \cdots = z_{nn}$ , the global maximizer is attained when people agglomerate in region $i^\ast$ where $\int_0^1 \ln \phi_{i^\ast}(y) \mathrm{d} y <\int_0^1 \ln \phi_{j}(y) \mathrm{d} y$ for all $j\neq i^\ast$ . Thus, the stable population distribution is the full agglomeration in the region with the smallest congestion force. On the other hand, again under the symmetry of Z, if $\int_0^1 \ln \phi_{1}(y) \mathrm{d} y = \cdots = \int_0^1 \ln \phi_{n}(y) \mathrm{d} y$ , which trivially holds when $\phi_1=\cdots = \phi_n$ , the global maximizer is attained when people agglomerate in region $i^\ast$ where $z_{i^\ast i^\ast} >z_{jj}$ for all $j\neq i^\ast$ . Thus, the stable population distribution is the full agglomeration in the region with the largest agglomeration benefit. ^{Footnote 16}

Next, we consider the case where people are nearly myopic, or $\rho$ is very large. Suppose that $\rho$ is sufficiently large as in Observation 2.2. Then the strict contractivity condition (22) implies that the potential function $W(\cdot, \rho)$ is strictly concave on $\Delta$ . In this case, the unique maximizer $\bar{x}^{\rho}$ of W on $\Delta$ , which is a full dispersion state (i.e., $\bar{x}^{\rho}_i > 0$ for all i) by Observation 2.2, in fact attracts all the equilibrium paths, as shown in Proposition A.10 in Appendix A.1.4.

By definition, the unique equilibrium state is also absorbing and globally accessible in this case. We have already discussed the intuitions behind the fact that a fully dispersed population distribution is attained at a unique equilibrium state when $\rho$ is large, that is, myopic people care about congestion more than agglomeration economies. The proposition above confirms that this unique equilibrium state is globally stable.

4.1 Clustering versus reach

In this subsection, we compare two different kinds of networks: in one network, connections are strong locally but weak globally, while in the other, connections are weak locally but strong globally. The networks are given in Figure 1, where c is a constant to be specified below. That is, their net-spillover matrices $\hat{Z}^{\mathrm{local}}$ and $\hat{Z}^{\mathrm{global}}$ are given by $\hat{Z}^{\mathrm{local}}_{ii} = c$ if $i \in \{1, 2, 3\}$ , $\hat{Z}^{\mathrm{local}}_{ii} = c - 0.01$ if $i \in \{4, 5, 6\}$ , $\hat{Z}^{\mathrm{local}}_{ij} = 1.3$ if $i \neq j$ and [ $i, j \in \{1, 2, 3\}$ or $i, j \in \{4, 5, 6\}$ ], and $\hat{Z}^{\mathrm{local}}_{ij} = 0$ otherwise; and $\hat{Z}^{\mathrm{global}}_{ii} = c$ if $i \in \{1, 2, 3\}$ , $\hat{Z}^{\mathrm{global}}_{ii} = c - 0.01$ if $i \in \{4, 5, 6\}$ , $\hat{Z}^{\mathrm{global}}_{ij} = 1.2$ if $i \neq j$ and [ $i, j \in \{1, 2, 3\}$ or $i, j \in \{4, 5, 6\}$ ], $\hat{Z}^{\mathrm{global}}_{14} = \hat{Z}^{\mathrm{global}}_{41} = 0.6$ and $\hat{Z}^{\mathrm{global}}_{ij} = 0$ otherwise. In these networks, we think of regions 1, 2, 3 and regions 4, 5, 6 as clusters, respectively. In network $\hat{Z}^{\mathrm{local}}$ , the two clusters are completely isolated, but connections within each of the clusters are strong. In network $\hat{Z}^{\mathrm{global}}$ , on the other hand, connections within each of the clusters are weaker than in the left network, but the two clusters are connected through regions 1 and 4. Therefore, it is possible to reach farther regions in $\hat{Z}^{\mathrm{global}}$ than in $\hat{Z}^{\mathrm{local}}$ . We call network $\hat{Z}^{\mathrm{local}}$ a locally dense network while network $\hat{Z}^{\mathrm{global}}$ a globally connected network.

Figure 1. Clustering versus reach.

In this example, we illustrate how which of the two networks achieves the higher steady-state lifetime utility, as given in (19), is affected by the intraregional net spillover effects through the population distribution at the stable equilibrium state. To this end, we assume $\hat{z}_{ii}= c$ for $i=1, 2, 3$ while $\hat{z}_{ii}=c - 0.01$ for $i=4, 5, 6$ in both networks and compute the stable equilibrium states via the optimization of the potential function and compare the utility levels under the two networks for different values of c. We slightly differentiate intraregional net spillover effects for the two clusters to guarantee that the potential function has a unique maximizer. Note that the total strength of interregional spillover effects (i.e., the sum of the off-diagonal entries of the spillover matrix) is the same between the two networks so that there is no scale effect.

We consider two values for c: $c=-0.5$ and $c=-3$ . ^{Footnote 17} First, let $c=-0.5$ . ^{Footnote 18} The maximizers of the potential function are summarized in Table 1. ^{Footnote 19} The equilibrium utility $u^*$ is $0.7$ in $\hat{Z}^{\mathrm{local}}$ whereas $0.6333$ in $\hat{Z}^{\mathrm{global}}$ . In this case, as the intraregional spillover effects are strong and/or the congestion effects are weak (relative to the case of $c = -3$ that follows), only one of the two clusters is populated at the stable equilibrium states. Thus, the locally dense network gives a higher utility level, where the agents benefit from the larger magnitude of within-cluster spillover coefficients than in the globally connected network.

Table 1. Stable equilibrium states and utility levels for $c=-0.5$ and $c=-3$

Next, let $c=-3$ . The maximizers of the potential function, which is strictly concave for each network in this case, are summarized in Table 1, where, as opposed to the previous case, $\hat{Z}^{\mathrm{global}}$ achieves the higher equilibrium utility ( $u^* = -0.064$ ) than $\hat{Z}^{\mathrm{local}}$ ( $u^* = -0.0675$ ). With weak intraregional spillover effects and/or strong congestion effects, the population is dispersed across the regions in both networks. While the three regions in each cluster have an equal fraction of agents in the locally dense network, the “hub” regions 1 and 4 in the globally connected network attract larger fractions than the others. As a consequence, these hubs generate larger spillover benefits, compensating the smaller magnitude of within-cluster spillover coefficients, in the latter network than in the former. ^{Footnote 20}

4.2 $\sigma$ -Divergence

In Section 2.4, we show the possibility that some regions perpetually grow whereas the others eventually stop growing at an equilibrium state (Proposition 2.3). In this subsection, we provide an example where such a situation actually takes place in the stable equilibrium state. Let us consider the net-spillover network $(\hat{z}_{ij})$ in Figure 2, where we let $z_{11}= 3$ , $z_{22}=2$ , and $z_{33}=2$ , and $\kappa_1=6$ , $\kappa_2=3$ , and $\kappa_3=4$ . We also set $\mu = 1$ . The unique potential maximizer, hence the stable equilibrium state, is computed as $(\bar{x}_1, \bar{x}_2, \bar{x}_3)=(0.2392, 0.5064, 0.2545)$ , where all regions are populated, and the production coefficients are $Z_1(\bar{x})= 1.1232, Z_2(\bar{x})=1.2071$ , and $Z_3(\bar{x}) = 0.7059$ . Hence, $Z_1(\bar{x})$ and $Z_2(\bar{x})$ are greater than 1 ( $= -\mu + \rho + 2)$ , whereas $Z_3(\bar{x})$ is less than 1. Therefore, by Proposition 2.3, at the stable equilibrium state, regions 1 and 2 will ride on balanced growth paths with the growth rate $Z_i(\bar{x}) - 1$ , $i = 1, 2$ , in the limit as $t \to \infty$ , while region 3 will converge to a steady state in the long run with the per-capita capital stock $k_3 = \frac{k_0}{1 - Z_3(\bar{x})}$ . Moreover, notice that $\bar{x}_1 <\bar{x}_3$ . That is, a stagnating region has a larger population than a growing region. This implies that spatial agglomeration and economic growth is not always positively correlated.

Figure 2. $\sigma$ -Divergence.

4.3 Katz–Bonacich centrality

In network theory, various measures have been proposed to measure the centrality of each node. Among others, what we consider here is the Katz–Bonacich centrality (see, e.g., Zenou (Reference Zenou, BramoullÉ, Galeotti and Rogers2016)). Let us consider network M, and denote its spectral radius by r(M). Let $\delta >0$ be such that $\delta r(M) < 1$ , so that $\sum_{k=0}^\infty \delta^k M^k$ is well defined (and is equal to $(I - \delta M)^{-1}$ ). Then, the vector of Katz–Bonacich centralities in network M with decay factor $\delta$ is defined as

(34) \begin{equation} b (M, \delta) = \sum_{k=0}^\infty \delta^k M^k \textbf{1}\ \left(= (I - \delta M)^{-1} \textbf{1} \right)\!,\end{equation}

where 1 is the vector of ones with the same dimension as M. The ith element of $\delta^k M^k \textbf{1}$ is the weighted sum of walks of length k of network M that start from node i. ^{Footnote 21} Hence, the Katz–Bonacich centrality of node i, which is denoted by $b_i(M, \delta)$ , is the weighted sum of all walks that start from node i. ^{Footnote 22}

Ballester et al. (Reference Ballester, CalvÓ-Armengol and Zenou2006) consider a finite-player game with linear best responses and derive a sufficient condition under which its Nash equilibrium is unique and is represented as a (normalized) vector of Katz–Bonacich centralities of a network, whose adjacency matrix is obtained by appropriately transforming the matrix of the coefficients in the payoff function to express the interdependencies among the players as a nonnegative matrix. In this subsection, we conduct an analogous exercise for our population game, that is, we derive a sufficient condition under which an equilibrium state is represented as a (normalized) vector of Katz–Bonacich centralities through transforming the net-spillover matrix $\hat{Z}$ .

To simplify our argument, we impose the following assumptions.

A4. (0) $z_{ij} \geq 0$ for all $i \neq j$ .

1. (1) $z_{in} = z_{ni} = 0$ for all $i \neq n$ .

2. (2) $\hat{z}_{nn} < 0$ .

Assumption A4(0) is our standing assumption that the spillover coefficients are nonnegative, which is included for reference. A4(1) makes region n, completely isolated from other regions, serve as an outside option. Under A4(1), A4(2) prevents the full agglomeration in region n from being an equilibrium state.

In the following, we focus on a fully dispersed equilibrium state (i.e., an equilibrium state whose support equals $S = \{1, \ldots, n\}$ ). Assuming the existence of such an equilibrium state amounts to imposing some structural assumption on the net-spillover matrix $\hat{Z}$ , as summarized in the proposition below. We denote by $\hat{Z}_{S\setminus\{n\}}$ the $(n-1) \times (n-1)$ submatrix of $\hat{Z}$ restricted to $S\setminus\{n\}$ , by $I_{n-1}$ the $(n-1) \times (n-1)$ identity matrix, and by $\textbf{1}_{n-1}$ the $(n-1)$ dimensional vector of ones.

1. (a) There exists an equilibrium state $\bar{x} \in \Delta$ such that $\bar{x}_n > 0$ .

2. (b) There exists an equilibrium state $\bar{x} \in \Delta$ such that $\bar{x}_i > 0$ for all $i \in S$ .

3. (c) There exists $\xi \in \mathbb{R}_{++}^{n-1}$ such that $(-\hat{Z}_{S \setminus \{n\}}) \xi = (-\hat{z}_{nn}) \textbf{1}_{n-1}$ .

4. (d) All the leading principal minors of $-\hat{Z}_{S \setminus \{n\}}$ are positive.

5. (e) $-\hat{Z}_{S \setminus \{n\}}$ is nonsingular, and $(-\hat{Z}_{S \setminus \{n\}})^{-1}$ exists and is nonnegative.

6. (f) $-\hat{Z}_{S \setminus \{n\}}$ is a nonsingular M-matrix, that is, it is a matrix written as $s I_{n-1} - B$ with $s > 0$ and $B \geq O$ for which $s > r(B)$ .

Proof. $(a) \Rightarrow (b)$ : Let $\bar{x}$ be an equilibrium state such that $\bar{x}_n > 0$ . We show that $\bar{x}_i > 0$ for all $i \in S$ . Fix any $i \neq n$ . By the equilibrium condition, we have $(\hat{Z} \bar{x})_i \leq (\hat{Z} \bar{x})_n$ , where $(\hat{Z} \bar{x})_n = \hat{z}_{nn} \bar{x}_n < 0$ by A4(1) and A4(2). On the other hand, we have $(\hat{Z} \bar{x})_i = \hat{z}_{ii} \bar{x}_i + \sum_{j \neq i} z_{ij} \bar{x}_j \geq \hat{z}_{ii} \bar{x}_i$ (by A4(0)). Hence, we must have $\bar{x}_i > 0$ (and $\hat{z}_{ii} < 0$ ).

$ (b) \Rightarrow (a)$ : Obvious.

$(c) \Rightarrow (b)$ : Given an equilibrium state $\bar{x} \gg 0$ , let $\xi = \frac{1}{\bar{x}_n} \bar{x}_{S \setminus \{n\}} \gg 0$ . Then we have $\hat{Z}_{S \setminus \{n\}} \xi= \frac{1}{\bar{x}_n} \hat{Z}_{S \setminus \{n\}} \bar{x}_{S \setminus \{n\}}= \frac{1}{\bar{x}_n} \hat{z}_{nn} \bar{x}_n \textbf{1}_{n-1} = \hat{z}_{nn} \textbf{1}_{n-1}$ by the equilibrium condition.

$ (c) \Rightarrow (b)$ : Given $\xi \gg 0$ such that $\hat{Z}_{S \setminus \{n\}} \xi = \hat{z}_{nn} \textbf{1}_{n-1}$ , let $\bar{x} = \frac{1}{1 + \textbf{1}_{n-1}' \xi} (\xi, 1) \gg 0$ . Then we have $\hat{Z}_{S \setminus \{n\}} \bar{x}_{S \setminus \{n\}}= \frac{1}{1 + \textbf{1}_{n-1}' \xi} \hat{z}_{nn} \textbf{1}_{n-1} = \hat{z}_{nn} \bar{x}_n \textbf{1}_{n-1}$ , and hence $\bar{x}$ is an equilibrium state.

Given that $-\hat{Z}_{S \setminus \{n\}}$ is a Z-matrix ^{Footnote 23} (A4(0)), the equivalence among conditions (c)–(f) is well known from the theory of M-matrices (see, e.g., Berman and Plemmons (Reference Berman and Plemmons1979), Theorem 6.2.3).

Suppose that there exists an equilibrium state $\bar{x}$ such that $\bar{x}_n > 0$ , or equivalently, assume any (hence all) of the conditions in Proposition 4.1. As is clear from the proof of the proposition, it is in fact a unique equilibrium state, uniquely determined by $\bar{x}_{S \setminus \{n\}} = \frac{1}{1 + \textbf{1}_{n-1}' \xi} \xi$ (and $\bar{x}_n = 1 - \textbf{1}_{n-1}' \bar{x}_{S \setminus \{n\}}$ ), where $\xi = (-\hat{Z}_{S \setminus \{n\}})^{-1} (-\hat{z}_{nn}) \textbf{1}_{n-1}$ . We also have $\hat{z}_{ii} < 0$ for all i.

Now let

(35) \begin{equation} \beta = \max_{i \in S \setminus \{n\}} (-\hat{z}_{ii}) > 0,\end{equation}

which can be interpreted as the maximum net congestion effect among the regions in $S \setminus \{n\}$ , and define the $(n-1) \times (n-1) $ nonnegative matrix, or network, G by

(36) \begin{equation} G = \hat{Z}_{S \setminus \{n\}} + \beta I_{n-1},\end{equation}

so that $-\hat{Z}_{S \setminus \{n\}} = \beta^{-1} (I_{n-1} - \beta^{-1} G)$ . By condition 4.1 in Proposition 4.1, $(I_{n-1} - \beta^{-1} G)^{-1}$ exists and is nonnegative, which implies that $r(\beta^{-1} G) < 1$ , or $r(G) < \beta$ , by the Perron–Frobenius theorem (or by condition (f)), where r(M) denotes the spectral radius of a matrix M. Thus, $\xi$ is written as

\begin{align*}\xi = \beta^{-1} (I_{n-1} - \beta^{-1} G)^{-1} (-\hat{z}_{nn}) \textbf{1}_{n-1}= \beta^{-1} (-\hat{z}_{nn}) b(G, \beta^{-1}),\end{align*}

through which the equilibrium state is represented as (a multiple of) the vector $b(G, \beta^{-1})$ of Katz–Bonacich centralities of the network G with decay factor $\beta^{-1}$ , as defined in (34). To summarize:

Proposition 4.2. Assume A4. Suppose that there exists an equilibrium state with region n in its support and denote it by $\bar{x}$ . Let $\beta$ be defined by (35) and G be the $(n-1) \times (n-1)$ nonnegative matrix defined by (36). Then we have

\begin{align*}\bar{x}_{S \setminus \{n\}}= \frac{-\hat{z}_{nn}}{\beta + \textbf{1}_{n-1}' b(G, \beta^{-1})} b(G, \beta^{-1}).\end{align*}

In particular, the share of the population of regions $i \in S \setminus \{n\}$ is equal to the share of the Katz–Bonacich centrality of that region.

Finally, since $\hat{Z}$ is symmetric as assumed, condition (d) in Proposition 4.1, under Assumption A4, implies that $\hat{Z}$ is negative definite, and hence the potential function $W(x) = \frac{1}{2}x'\hat{Z} x$ is strictly concave. Therefore, by Proposition A.10, the unique equilibrium state $\bar{x}$ has a strong global stability property, that from any state in $\Delta$ , any equilibrium path converges to $\bar{x}$ .