2.1 General Set-up

In this study, we consider an economic model composed of two symmetric regions, i.e., they share the same technology and consumer preferences and they have the same labor endowment (see below), denoted as Region 1 and Region 2. The model includes two key sectors: Agriculture ( $A$ ) and Manufacturing ( $M$ ), each of which plays a distinct role in the economy. The production process in these sectors relies on two primary factors of production: labor ( $L$ ) and capital ( $K$ ). The output of this economy consists of two corresponding goods: an agricultural good ( $A$ ) produced in the agricultural sector and a manufacturing good ( $M$ ) produced in the industrial sector.

Each firm in the manufacturing sector produces a single variety of the manufacturing good. In the absence of policy intervention, the production process requires each firm to employ $F$ units of $K$ and $\varsigma$ units of labor for each additional unit of output produced. With an environmental policy that we assume to be implemented only in Region 1, additional $\theta F$ units of the fixed input are required per firm, thereby increasing production costs. These $\theta F$ units contribute to the abatement of pollution emissions, and $\theta$ is a parameter fixed by the local government in Region 1 aiming to achieve some “green” standard. The wage rate of workers is denoted by $w$ , while the remuneration for one unit of $K$ is represented by $r$ .

The economy is populated by two types of households: workers ( $L$ ) and capital owners ( $E$ ). Capital owners are allowed to move between regions. The number of capital owners residing in Region 1 is $E_1$ , while those in Region 2 are $E_2$ . The per capita endowment of $K$ for each capital owner is given by $k = \frac {K}{E}$ . Capital is distributed between the two regions, with $\eta$ representing the share of total capital allocated to Region 1. Consequently, $1-\eta$ denotes the share allocated to Region 2. The spatial distribution of capital owners between the two regions is captured by the parameter $\lambda$ , which denotes the share of capital owners living in Region 1. Thus, $1-\lambda$ represents the share of capital owners living in Region 2, leading to $E_1 = \lambda E$ and $E_2 = (1-\lambda )E$ . Capital is sector-specific, while workers supply labor to both sectors $ A$ and $ M$ . The workforce is equally distributed between the two areas, with each region containing $\frac {L}{2}$ workers. These workers are assumed to be perfectly mobile between sectors but geographically immobile.Footnote ²

2.2 Consumption

The representative consumer’s utility function is given by:

(1) \begin{align} U(c_M, c_A, \varepsilon ) = u(c_M, c_A) - D(\varepsilon ), \end{align}

where $ u(c_M, c_A)$ represents the utility derived from consuming the manufacturing good and the agricultural good. Specifically, $ c_M$ is the consumption of a composite of manufacturing good varieties, and $ c_A$ is the consumption of the agricultural good. Moreover, $D(\varepsilon )$ captures the externality imposed by pollution on agents, who lack any direct means to control or mitigate their own exposure. Following Rauscher (Reference Rauscher2009) and Commendatore et al. (Reference Commendatore, Kubin, Sodini and Sushko2024), we assume a quasi-linear utility function:

(2) \begin{align} u(c_M, c_A) = c_M + c_A. \end{align}

The manufacturing good component $ c_M$ corresponds to:

(3) \begin{align} c_M = \Phi \left ( \sum _{i=1}^{N} c_i^{\frac {\sigma -1}{\sigma }} \right ), \end{align}

where $\sigma \gt 1$ represents the elasticity of substitution,Footnote ³ $c_i$ the consumption of variety $i$ , and $\Phi \gt 0$ is a scaling parameter. The specification in (3) is the simplest way to capture love-of-variety à la Dixit and Stiglitz (Reference Dixit and Stiglitz1977). This assumption simplifies the analysis and allows for more analytical results. Its main implication is that individual demand is independent of both individual income and the price index, thereby neglecting income effects and interactions among differentiated goods.

The utility maximization problem is solved subject to the budget constraint:

(4) \begin{align} \sum _{i=1}^{n}p_i c_i + p_Ac_{A} = Y + p_A\overline {c_{A}}, \end{align}

where $p_i$ corresponds to the price of variety $i$ (with $i = 1, \ldots , N$ ), $p_A$ is the price of the agricultural good, $Y$ is the income of an individual consumer (worker or capital owner), that may include local taxation, and $\overline {c_{A}}$ is a given endowment of the agricultural good. Moreover, as it is conventional in NEG models, we choose the agricultural good as the numéraire, giving $p_A = 1$ . We obtain the demand for a single variety $ i$ :

\begin{equation*} c_i = \frac {\alpha }{p^\sigma _i}, \end{equation*}

where $\alpha = (\Phi \frac {\sigma -1}{\sigma })^\sigma$ . Note that since we are not yet considering space, in this expression there is not distinction between the price of a locally produced variety and the price of an imported one.

2.3 Production

In the absence of any environmental policy, the total number of firms in the economy is given by $N = \frac {K}{F}$ , where $K$ denotes the aggregate capital stock and $F$ the fixed capital requirement per firm. Firms are distributed across the two regions according to the capital share parameter $\eta$ , so that Region 1 hosts $N_1 = \frac {\eta K}{F}$ firms, and Region 2 hosts $N_2 = \frac {(1-\eta )K}{F}$ firms.

To introduce environmental regulation into the model, we consider a CAC policy implemented in Region 1 that mandates the adoption of cleaner production technologies to achieve a targeted reduction in emissions. This policy increases the amount of capital required per firm to operate, effectively increasing the fixed input requirement. We denote the stringency of the policy by the parameter $\theta \geq 0$ , which captures the increase in capital intensity due to compliance with environmental standards. A higher $\theta$ reflects more stringent requirements.

Consequently, the number of firms that can be supported in Region 1 becomes $N_1 = \frac {\eta K}{(1+\theta )F}$ while the number of firms in Region 2 remains unchanged at $N_2 = \frac {(1-\eta )K}{F}$ . The total number of firms in the economy is therefore:

\begin{equation*} N = \frac {K}{F} \left ( 1 - \eta \frac {\theta }{1+\theta } \right ). \end{equation*}

As expected, the overall number of firms declines with the stringency of environmental policy, since a higher $\theta$ increases the fixed input requirement in the regulated region.Footnote ⁴

The production structure adheres to the footloose capital (FC) framework as outlined by Martin and Rogers (Reference Martin and Rogers1995). In this framework, the mobile factor, capital, contributes to production only as a fixed component. The profit of a representative firm, disregarding the location, is given by

\begin{equation*} \pi = pq - (\varsigma qw + Fr), \end{equation*}

where $ q$ represents the output, $ \varsigma$ is the variable input requirement, $ F$ denotes the fixed amount of capital required for production, $ w$ is the nominal wage of workers, and $ r$ is the remuneration for capital. Moreover, following from perfect competition in the agricultural market and from mobility of workers between the sectors: $w = p_A = 1$ . With the introduction of environmental policy, regional profits $\pi _1$ for Region 1 and $\pi _2$ for Region 2 at the individual firm level are defined as:

(5) \begin{align}\pi _{1} & =(p_{11}-\varsigma )q_{11}+(p_{12}-\varsigma )\tau q_{12} -r_{1}(1+\theta )F+s,\end{align}

(6) \begin{align}\pi _{2} & =(p_{21}-\varsigma )\tau q_{21}+(p_{22}-\varsigma )q_{22}-r_{2}F,\end{align}

where $\tau \gt 1$ denotes the trade cost of the $M$ -good between regions, following the standard iceberg formulation,Footnote ⁵ $ p_{ij}$ denotes the mill price charged in region $ j$ by a firm located in region $ i$ , $ q_{ij}$ represents the quantity sold in region $ j$ by a firm located in region $ i$ , $r_j$ indicates the remuneration for capital in region $ j$ and $s$ the subsidy paid to a firm in region 1, with $ i, j = 1, 2$ .

Market clearing for manufacturing good varieties requires:

\begin{align*} q_{1,1} &= c_{1,1} \left ( \frac {L}{2} + \lambda E \right ), \qquad q_{1,2} = c_{1,2} \left ( \frac {L}{2} + (1 - \lambda ) E \right ), \\[5pt] q_{2,1} &= c_{2,1} \left ( \frac {L}{2} + \lambda E \right ), \qquad q_{2,2} = c_{2,2} \left ( \frac {L}{2} + (1 - \lambda ) E \right ). \end{align*}

Inserting these conditions in the above expressions for the regional profits and, finally, taking into account that prices at destination include trade costs:

(7) \begin{align}c_{1,1} &= \alpha p_{1,1}^{-\sigma }, \hspace {0.8 cm} c_{1,2} = \alpha p_{1,2}^{-\sigma }\tau ^{-\sigma },\end{align}

(8) \begin{align}c_{2,1} &= \alpha p_{2,1}^{-\sigma }\tau ^{-\sigma }, c_{2,2} = \alpha p_{2,2}^{-\sigma },\end{align}

we obtain the profit maximizing (mill) prices: $p_{i,j}=\frac {\sigma }{\sigma -1}\varsigma$ , where $\frac {\sigma }{\sigma -1}$ is the mark-up. It gives $p_{i,j}=1$ after setting for simplicity $\varsigma =\frac {\sigma -1}{\sigma }$ , with $i,j = 1, 2$ .

2.4 Subsidies and Taxation

The government in Region 1 may decide to pay a subsidy to firms to implement the green technology, breaking the symmetry between the two regions. These subsidies correspond to a fraction $0\leq x\leq 1$ of the cost increase suffered by a firm located in Region 1:

(9) \begin{equation} s=x\theta Fr_1. \end{equation}

Notice that the government knows the exact value of $s$ which is calculated at the short-run equilibrium. The subsidy is financed by an income tax ( $tax$ ) imposed on resident households. Thus the income of a representative household living in Region $i=1,2$ corresponds to

\begin{align*} & Y_{1}+\overline {c_{A}}-tax\\[5pt] & Y_{2}+\overline {c_{A}}, \end{align*}

where $tax$

(10) \begin{align} tax =\frac {N_{1}}{\frac {L}{2}+E_{1}}s=\frac {\eta \frac {K}{(1+\theta )F}}{\frac {L}{2}+E_{1}}s=\frac {\eta K}{\frac {L}{2}+\lambda E} \frac {\theta }{1+\theta }xr_1. \end{align}

In essence, the total taxation in Region 1 is determined by the number of firms, workers, and capital owners located in the region, along with the subsidy set by the government to finance the environmental policy. We take $x$ as policy parameter.

2.5 Pollution

Emissions are proportional to production. To incorporate the (negative) effect of the green technology on emissions per unit of output we set:

(11) \begin{align}\varepsilon _{1} & =\frac {1}{1+\theta }N_{1}q_{1},\end{align}

(12) \begin{align}\varepsilon _{2} & =N_{2}q_{2},\end{align}

where $q_{1}=q_{11}+\tau q_{12}$ and $q_{2}=\tau q_{21}+q_{22}$ . The damage function $ D(\varepsilon )$ , that enters in the utility function additively, captures the negative impact on utility from industrial pollution emissions, $ \varepsilon$ :

(13) \begin{align} D(\varepsilon ) = \frac {\delta }{2} \varepsilon ^2, \end{align}

where $ \delta \geq 0$ represents the intensity of the pollution effect on utility.Footnote ⁶

3.1 Regional Capital Remunerations

Starting from the zero-profit conditions (obtained by setting $\pi _1 = 0$ and $\pi _2 = 0$ in equations (5) and (6) and substituting the profit-maximizing prices) we derive the short-run equilibrium values for regional capital returns:

(14) \begin{align}r_1 &= \frac {1}{\sigma F[1 + (1 - x)\theta ]} q_1 \notag \\[5pt] &= \frac {\alpha }{\sigma F[1 + (1 - x)\theta ]} \left ( \frac {L}{2} + \lambda E + \phi \left ( \frac {L}{2} + (1 - \lambda ) E \right )\right ),\end{align}

(15) \begin{align}r_2 &= \frac {1}{\sigma F} q_2 = \frac {\alpha }{\sigma F} \left ( \phi \left ( \frac {L}{2} + \lambda E \right ) + \frac {L}{2} + (1 - \lambda ) E \right ).\end{align}

where we used the standard definition of trade freeness $\phi = \tau ^{1 - \sigma }$ , with $0 \leq \phi \lt 1$ . Notice that $r_1$ and $r_2$ depend on the market size (determined by $\lambda$ ) and production costs (affected by the policy $\theta$ ), and they do not depend on local competition (determined by $\eta$ ).

3.2 Capital Owner Regional Indirect Utilities

From (1) and (2), the utility of a representative capital owner living in Region 1 and in Region 2, respectively, is given by:

\begin{align*} u_1(c_{M,1}, c_A, \varepsilon _{1}) &= c_{M,1} + c_A - D(\varepsilon _{1})\\[5pt] u_2(c_{M,2}, c_A, \varepsilon _{2}) &= c_{M,2} + c_A - D(\varepsilon _{2}), \end{align*}

where from (3),

\begin{align*} c_{M,1} &= \Phi \left ( N_{1} c_{1,1}^{\frac {\sigma -1}{\sigma }} + N_{2} c_{2,1}^{\frac {\sigma -1}{\sigma }} \right )\\[5pt] c_{M,2} &= \Phi \left ( N_{1} c_{1,2}^{\frac {\sigma -1}{\sigma }} + N_{2} c_{2,2}^{\frac {\sigma -1}{\sigma }} \right ), \end{align*}

and from (13),

\begin{align*} D(\varepsilon _1) = \frac {\delta }{2} \varepsilon _1^2,\\[5pt] D(\varepsilon _2) = \frac {\delta }{2} \varepsilon _2^2 .\end{align*}

Replacing for utility maximizing consumption, considering (7) and (8) and that $p_{ij} = 1$ with $(i,j =1,2)$ , $\alpha = (\Phi \frac {\sigma -1}{\sigma })^\sigma$ , $N_1 = \frac {\eta }{1+\theta } \frac {K}{F}$ and $N_{2} = (1 - \eta ) \frac {K}{F}$ , the indirect utilities correspond to:

(16) \begin{align}u_1 &= \frac {\alpha \sigma }{\sigma - 1} \frac {K}{F} \left ( \frac {\eta }{1 + \theta } + \phi (1 - \eta ) \right ) + c_A - \frac {\delta }{2} \varepsilon _1^2,\end{align}

(17) \begin{align}u_2 &= \frac {\alpha \sigma }{\sigma - 1} \frac {K}{F} \left ( \phi \frac {\eta }{1 + \theta } + 1 - \eta \right ) + c_A - \frac {\delta }{2} \varepsilon _2^2.\end{align}

Observe that the first component of the indirect utilities, representing the consumption of the manufacturing good, is influenced by the allocation of firms across regions. When a larger share of product varieties is locally available, avoiding transport costs and thus reducing prices, the utility level increases. This reflects the impact of regional product availability on the cost of living and overall utility, which we refer to as the cost of living effect. Using the budget constraint (4) solved for $c_{A}$ and expressions (11), (12), (14) and (15), we obtain:

(18) \begin{align}u_{1} & =\overline {r}k-tax+\overline {c_{A}}+\left [\frac {\alpha }{\sigma -1}\frac {K}{F}\left ( \frac {\eta }{1+\theta }+\phi \left ( 1-\eta \right ) \right )\right ] -\frac {\delta }{2}\varepsilon _{1}^{2},\end{align}

(19) \begin{align}u_{2} & =\overline {r}k+\overline {c_{A}}+\left [\frac {\alpha }{\sigma -1}\frac {K}{F}\left ( \phi \left ( \frac {\eta }{1+\theta }\right ) +1-\eta \right )\right ] -\frac {\delta }{2}\varepsilon _{2}^{2},\end{align}

where $\overline {r}=\eta r_{1}+(1-\eta )r_{2}$ represents the average remuneration of a unit of capital.

4.1 The Dynamic Map

We assume that the mobility decisions of households and firms (the latter through capital allocation choices) are governed by an adaptive mechanism resembling replicator dynamics, a framework widely used in evolutionary game theory (Weibull, Reference Weibull1995) and commonly applied in NEG models (Fujita et al. Reference Fujita, Krugman and Venables1999; Baldwin et al. Reference Baldwin, Forslid, Martin, Ottaviano and Robert-Nicoud2003). The adoption of evolutionary dynamics is justified as it captures empirically observed patterns of relocation behavior. Firms tend to gradually relocate toward regions offering higher short-term returns, consistent with empirical evidence on firm responses to environmental and cost differentials (Eskeland and Harrison, Reference Eskeland and Harrison2003). Similarly, households migrate in response to current utility differentials, in line with documented evidence of pollution-induced mobility (Chen et al. Reference Chen, Oliva and Zhang2022). Furthermore, evolutionary dynamics account for cognitive and informational constraints, in accordance with the notion of bounded rationality. Firms often lack the capacity to fully anticipate the long-term effects of environmental policy interventions (Dosi and Nelson, Reference Dosi and Nelson1994), while households tend to react myopically to pollution exposure—relocating ex post rather than engaging in anticipatory migration—consistent with adaptive residential mobility models (Mulder, Reference Mulder1996).

Formally, the share of households residing in Region 1 (and, conversely, in Region 2) evolves according to the short-term utility differential $\Omega (\lambda ,\eta )=\frac {u_{1}-\overline {u}}{\overline {u}}$ as follows:

\begin{equation*} f(\lambda ,\eta )=\lambda \left ( 1+\gamma _{\lambda }\Omega (\lambda ,\eta )\right ) \text{,} \end{equation*}

where $\overline {u}=\lambda u_{1}+(1-\lambda )u_{2}$ and $\gamma _{\lambda }$ is the adjustment speed of household migration; and capital allocation depends on the short-term remuneration differential $\Psi (\lambda ,\eta )=\frac {r_{1}-\overline {r}}{\overline {r}}$ as follows:

\begin{equation*} g(\lambda ,\eta )=\eta \left ( 1+\gamma _{\eta }\Psi (\lambda ,\eta )\right ) \text{,} \end{equation*}

where $\gamma _{\eta }$ is the adjustment speed of firms mobility. After introducing the constraints on shares that typically characterize NEG models, the 2-D map governing the discrete-time evolution of the two main variables, $\lambda$ and $\eta$ , corresponds to

\begin{equation*} T:(\lambda ,\eta )\rightarrow (F(\lambda ,\eta ),G(\lambda ,\eta )), \end{equation*}

where

\begin{eqnarray*} F(\lambda ,\eta ) &=&\left \{ \begin{array}{ccc} 0 &\quad if &\quad f(\lambda ,\eta )\leq 0 \\ f(\lambda ,\eta ) &\quad if &\quad 0\lt f(\lambda ,\eta )\lt 1 \\ 1 &\quad if &\quad f(\lambda ,\eta )\geq 1\end{array}\right . \\[5pt] G(\lambda ,\eta ) &=&\left \{ \begin{array}{ccc} 0 &\quad if &\quad g(\lambda ,\eta )\leq 0 \\ g(\lambda ,\eta ) &\quad if &\quad 0\lt g(\lambda ,\eta )\lt 1 \\ 1 &\quad if &\quad g(\lambda ,\eta )\geq 1.\end{array}\right . \end{eqnarray*}

The $2$ -D map $T$ has a piecewise smooth definition with flat branches. It fully describes the dynamics of the two main state variables describing the evolution of the economic system: the regional shares of capital owners ( $\lambda$ ) and capital ( $\eta$ ). Its domain corresponds to the unit square $U=[0,1]\times \lbrack 0,1]$ .

4.2 Long-Run Stationary Equilibria (Map T Fixed Points)

The long-run stationary equilibria of the dynamic model correspond to the fixed points of map $T$ . Due to the constraints on the shares, which confine the dynamics of the map within the unit square $u$ , we identify three types of fixed points:

• • Interior equilibria, located in the interior of $U$ ;

• • Border equilibria, located at the interior points of the sides of $U$ ;

• • Core-Periphery equilibria located on the corners of $U$ .

Now we give more details starting from the interior fixed points. From $f(\lambda , \eta ) = \lambda$ and $g(\lambda , \eta ) = \eta$ , we deduce that an interior fixed point satisfies the following conditions:

(20) \begin{equation} r_1(\lambda ) = r_2(\lambda ) ,\end{equation}

(21) \begin{equation} u_1(\lambda , \eta ) = u_2(\lambda , \eta ) ,\end{equation}

which imply equal regional capital returns (Equation (20)) and equal mobile households utilities across space (Equation (21)), respectively. There are at most two interior equilibria, which we denote $I_1 = (\lambda , \eta _1)$ and $I_2 = (\lambda , \eta _2)$ , where $\lambda$ directly solves the first of the above equations. The expressions for $\lambda$ , $\eta _1$ and $\eta _2$ , given their length, are reported in Appendices A1 and A2. Moreover, in those Appendices, we show conditions under which $I_1$ and $I_2$ exist. An important feature of interior equilibria is that both firms and mobile households are spread across countries. This distribution is asymmetric, except when neither region implements an environmental policy, as in Commendatore et al. (Reference Commendatore, Kubin, Sodini and Sushko2024).

We now consider the border fixed points. First, no equilibria can exist in the interior of the vertical borders of $U$ , for which $\lambda = 0$ and $\lambda = 1$ , as these values cannot satisfy equation (20).Footnote ⁷ The economic intuition behind is as follows: according to equations (14) and (15), the regional returns depend on the distribution of demand, weighted by the transport costs. When demand is fully concentrated in one region, $\lambda = 0$ or $\lambda = 1$ , firms in each region face different demand conditions, translating into different remunerations. Equal demand conditions, and hence equal remuneration, can only occur when equation (20) holds. Looking at the interior of the horizontal borders of $u$ , for which $\eta = 0$ or $\eta = 1$ , these are obtained by solving for $\lambda$ equation (21) after setting $\eta = 0$ and $\eta = 1$ , respectively. Again the explicit solutions, given their length, are provided in Appendix A3. In that Appendix, it is also shown that only one feasible equilibrium, denoted as $B$ , can exist on the lower horizontal border ( $\eta =0$ ). In the Appendix A3, it is also suggested that at most two feasible equilibria can exist on the upper horizontal border ( $\eta =1$ ). For the parameter combination we chose below for our numerical simulations, only one of these border equilibria exists, denoted as $B'$ . The economic interest of these equilibria is that when $\eta = 0$ or $\eta = 1$ , firms are concentrated in one region while capital owners are spread across space.

Finally, the Core-Periphery equilibria at the corners of the unit square always exist. These are denoted as $CP_{00} = (0,0)$ , $CP_{01} = (0,1)$ , $CP_{10} = (1,0)$ , $CP_{11} = (1,1)$ . From an economic perspective, recall that at the equilibria $CP_{00}$ and $CP_{11}$ , we encounter an extreme case in which both firms and capital owners are entirely concentrated in the same region. The equilibria $CP_{01}$ and $CP_{10}$ describe situations where firms concentrate in one region while all capital owners reside in the other.

Crucially, the existence and stability properties of the fixed points depend on the parameter configuration.Footnote ⁸ To deepen our analysis on the role of environmental regulation—focusing in particular on the impact of the parameter $\theta$ —it is essential to complement the exploration of these properties numerically. To this end, we adopt the following parameter set:Footnote ⁹

\begin{align*} \sigma &= 3,\quad E = 1,\quad K = 1,\quad L = 2,\quad F = 0.05, \\[5pt] \alpha &= 0.1,\quad \gamma _\lambda = 2,\quad \gamma _\eta = 2,\quad \delta = 0.1, \\[5pt] \bar {c}_A &= 1,\quad \phi = 0.35,\quad x = 0.2,\quad \theta \in [0, 0.5].\\[-20pt]\nonumber \end{align*}

In Figure 1 we present a diagram—called bifurcation diagram—showing the behavior of the fixed points of map $T$ by increasing $\theta$ in the interval $[0, 0.5]$ within the feasible range of $\lambda$ and $\eta$ , i.e. $[0,1]$ . Solid lines correspond to a locally stable fixed point, while dashed lines correspond to an unstable fixed point. When curves cross at a specific point—a so-called bifurcation point—the properties of the fixed points involved change. In Figure 1, both interior equilibria $I_1$ and $I_2$ are visible. Considering the equilibrium $I_1$ , it is always stable within the feasible range. Increasing sufficiently $\theta$ , it eventually disappears leaving the feasible range of $\eta$ at the bifurcation point $T_1$ (where $\eta = 0$ ), where it crosses the border equilibrium $B$ . Interestingly, at $T_1$ , the equilibrium $B$ gains stability.Footnote ¹⁰ Concerning the interior equilibrium $I_2$ , it is always unstable. When $I_2$ enters the feasible range ( $\lambda = 1$ ) at the bifurcation point $T_2$ , it crosses the border equilibrium $B'$ . After $T_2$ , $B'$ becomes locally stable. This equilibrium loses stability at the bifurcation point $T_3$ , where the Core-Periphery equilibrium $CP_{11}$ becomes locally stable. Finally, when $\theta$ becomes sufficiently large.Footnote ¹¹ $I_2$ exits the feasible range and the Core-Periphery equilibrium $CP_{11}$ loses stability at the bifurcation point $T_4$ .Footnote ¹²

Figure 1.

Bifurcation diagram of $\lambda$ and $\eta$ with respect to $\theta$ , showing the long-run stationary equilibria and their properties for the chosen parameter constellation.

According to Figure 1, there are values of the parameter $\theta$ for which more than one equilibrium is locally stable---that is, multiple attractors coexist. This is a notable property of dynamical systems, as it implies that history matters: the long-run equilibrium outcome depends on the initial conditions. In the context of our model, the initial spatial distribution of capital owners and firms can influence where industrial activities and mobile households ultimately locate. To clarify this aspect, we illustrate in Figure 2 how the basins of attraction of the fixed points of the map $T$ evolve as the parameter $\theta$ increases, that is, at different levels of policy regulation. The basin of attraction of an equilibrium comprises all initial conditions from which the system’s state variables, $\lambda$ and $\eta$ , will eventually converge to that equilibrium.

Figure 2 is based on the same parameter combination introduced above and is plotted for six values of $\theta$ (0, 0.1, 0.2, 0.25, 0.4, and 0.5), representing different intensities of policy regulation. The fixed points appearing in the various panels of Figure 2 are the same as those plotted in Figure 1. The interior equilibria ( $I_1$ and $I_2$ ) are determined by the intersection of the line $\Psi$ , which plots equation (20) in the $(\lambda , \eta )$ plane, and the curve $\Omega$ , which plots equation (21) in the same plane. The border equilibrium $B'$ is defined by the intersection of the curve $\Omega$ with the upper boundary of the unit square $U$ (where $\eta = 1$ ), while the border equilibrium $B$ is defined by its intersection with the lower boundary (where $\eta = 0$ ). Locally stable fixed points are represented by black dots and unstable ones by white dots.

Starting from Figure 2a, in this scenario with no policy regulation, $\theta = 0$ , only the interior equilibrium $I_1$ exists representing the global attractor for the dynamics starting from the interior points of $U$ , with the corresponding basin of attraction colored in yellow. When $\theta = 0$ the interior equilibrium involves a symmetric distribution of capital and capital owners between the two countries.

When the stringency of the policy is low—as in Figure 2b where $\theta = 0.1$ — $I_1$ remains the only stable equilibrium, but symmetry is broken. The environmental policy affects both households and firms through two distinct channels. Specifically, in Region 1, where the policy is implemented, both the share of capital and the proportion of capital owners increase.Footnote ¹³ This scenario can be described as a moderate policy intervention. As long as the expansion of market size in Region 1 outweighs the cost burden imposed by the environmental regulation, this region continues to attract an increasing share of capital.

As $\theta$ is further increased, as shown in Figure 2c, where $\theta = 0.2$ , an interesting scenario emerges. The bifurcation at point $T_2$ (see Figure 1) has occurred: the interior equilibrium $I_2$ now exists, and the border equilibrium $B'$ has become stable. In this configuration, two attracting fixed points coexist: the interior equilibrium $I_1$ and the boundary equilibrium $B'$ (located on the side where $\eta = 1$ ). The basin of attraction for $I_1$ appears as a yellow tongue, while the brown region represents initial conditions that converge to $B'$ . In this context, as noted earlier, “history matters”: initial conditions dictate which long-run equilibrium the system will reach. If the initial state lies within the basin of attraction of $I_1$ , the system converges to an interior equilibrium where less than 40% of capital and approximately 70% of capital owners are located in Region 1. Conversely, if the initial state lies within the basin of attraction of $B'$ , the system converges to a near-corner solution where nearly all capital and almost the entire population of capital owners are concentrated in Region 1.

This coexistence of stable equilibria also opens the possibility of path dependence driven by shocks: a sufficiently large disturbance that shifts the initial values of $\lambda$ and $\eta$ from one basin of attraction to another can lead to a dramatically different long-run spatial distribution of industry and households.

Figure 2d, where $\theta = 0.25$ , displays a structure similar to that of Figure 2c, with two distinct stable equilibria. However, the bifurcation point $T_3$ (see Figure 1) has now been crossed: the border equilibrium $B'$ has disappeared, and the Core-Periphery equilibrium $CP_{11}$ has gained stability. The basin of attraction, represented by the grey region, includes all initial conditions that converge to this equilibrium. In $CP_{11}$ , not only is all industry concentrated in Region 1, but all capital owners are located there as well.

Figure 2.

Basins of attraction of the locally stable fixed points as the parameter $\theta$ varies. Black dots indicate locally stable fixed points, white dots represent unstable fixed points.

In Figure 2e, with a further increase in $\theta$ to a relatively high value ( $\theta = 0.4$ ), the system reaches a configuration beyond the bifurcation point $T_1$ (see Figure 1). The interior fixed point $I_1$ has collided with the lower boundary of the unit square $U$ (where $\eta = 0$ ) and crossed the border equilibrium $B$ . As a result, $I_1$ disappears and $B$ gains stability, with the basin of attraction of this equilibrium colored in magenta. In this scenario, the only stable fixed points lie on the boundaries of $U$ : the Core-Periphery equilibrium $CP_{11}$ , located at the corner of $U$ , and the border equilibrium $B$ , located on the line $\eta = 0$ . In the first equilibrium, all capital and all capital owners are concentrated in Region 1. In the second equilibrium, all capital is located in Region 2, while nearly all capital owners remain in Region 1.

Finally, in Figure 2f, we return to a configuration with a single stable equilibrium: the border equilibrium $B$ at $\eta = 0$ . At this high level of regulation ( $\theta = 0.5$ ), all other equilibria have vanished or lost stability (see Figure 1 beyond the bifurcation point T4). In this outcome, all capital is located in Region 2, while nearly all capital owners remain in Region 1. The entire set of initial conditions falls within the basin of attraction of $B$ , shown in magenta.