2.1 Diffusion

A central concept in any attempt to model the movement of variables associated with environmental and natural resources in continuous space is the concept of diffusion. A diffusion process describes a situation in which movements of individual objects such as pollutants or animals result in a regular macroscopic flow. Diffusion models can be derived from dispersal models of random walks, from Fick’s law, or from stochastic differential equations. Fick’s law states that in one-dimensional space (that is, a line), diffusion moves objects from locations of high concentration to locations of low concentration. If the concentration of a material at time $t\ge 0$ and spatial point $x\in O\subseteq \mathbb{R}$ , where $O$ is the spatial domain, is denoted by $y\left(t,x\right)$ , then Fick’s law states that the flux $J$ of the material is proportional to the gradient (that is, the derivative with respect to space $\frac{\mathrm{\partial }y\left(t,x\right)}{\mathrm{\partial }x})$ of the material, or $J\propto -\frac{\mathrm{\partial }y\left(t,x\right)}{\mathrm{\partial }x}$ , $J=-D\frac{\mathrm{\partial }y\left(t,x\right)}{\mathrm{\partial }x}$ . If the material’s net growth in a spatial location is determined by $f\left(y\left(t,x\right),u\left(t,x\right)\right)$ , then concentration dynamics under diffusion are given by the PDEFootnote ⁸

$\frac{\mathrm{\partial }y\left(t,x\right)}{\mathrm{\partial }t}=f\left(y\left(t,x\right),u\left(t,x\right)\right)+D\frac{\mathrm{\partial }{y}^2\left(t,x\right)}{\mathrm{\partial }{x}^2},y\left(0,x\right)=y_0\left(x\right).$(1)

In (1), $D$ is called diffusivity and is a constant indicating that the diffusion is linear. Diffusivity, as we will see later on in this section, could depend on the location or the concentration itself in the context of nonlinear diffusion. This is important in the context of one- and two-dimensional models of heat diffusion toward the poles (e.g., North, Reference North1975a, Reference North1975b; Ghil, Reference Ghil1976; North et al., Reference North, Cahalan and Coakley1981; North and Kim, Reference North and Kim2017). The function $u\left(t,x\right)$ represents a control, such as harvesting, emissions, or abatement. Equation (1), apart from the temporal initial condition, should be supplemented with spatial boundary conditions which provide information about what the concentration is expected to be at the boundary of the spatial domain $O$ at all times.Footnote ⁹

If we consider a vector $\mathbf{y}=\left(y_1,\dots ,y_n\right)$ of concentrations at time $t$ and location $x$ , which diffuse with diffusivities $\mathbf{D}=\left(D_1,\dots ,D_n\right)$ and interact among themselves, and a vector of controls $\mathbf{u}=\left(u_1,\dots ,u_n\right)$ , then (1) will represent a reaction-diffusion system. The Fickian diffusion framework can be further extended to include an advection term that represents a drift in the process caused by external forcing such as wind or currents (Murray, Reference Murray2003; Wilen, Reference Wilen2007). In this case, the advection term $-V\left(\mathrm{\partial }y\left(t,x\right)/\mathrm{\partial }x\right)$ is added to the right-hand side of (1).

Figure 1 presents the spatiotemporal evolution of resource biomass under linear diffusion.Footnote ¹⁰

Figure 1

(a) Biomass evolution in time and space $\left(t,x\right)$ of a population with logistic growth under linear diffusion. (b) Contours of the evolution surface. (c) Biomass evolution in time and space $\left(t,x\right)$ of two interacting populations with logistic growth under linear diffusion (reaction-diffusion system). (d) Case (a) with advection.

Figure 1 indicates that the role of diffusion is to generate a spatially heterogeneous pattern of concentration. In our example, this spatial pattern seems to persist over time. Persistent spatial heterogeneity raises policy questions regarding the need to design spatially heterogeneous policies if the emerging spatial pattern is not the desired one.

When diffusivity $D$ depends on the concentration or the spatial location, diffusion is nonlinear. Nonlinear diffusion in spatial models of climate change is modeled by the term $D\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left[(1-x^2)\frac{\mathrm{\partial }T(t,x)}{\mathrm{\partial }x}\right]$ , where $x\in \left[-1,1\right]$ is the sine of latitude and $T(t,x)$ is surface temperature at the latitude with sine $x$ . More details will be presented in Section 6.2.4.

2.2 Long-Range Transport

Spatial diffusion captures local or short-range spatial interactions (Murray, Reference Murray2003). In economics as well as in environmental and resource management, spatial interactions and spatial effects could be long range. This means that the rate of change of the concentration at a specific location $x$ is affected by the concentrations of all other locations $x^{\mathrm{\prime }}\in O$ . These long-range interactions can be expressed by the model

$\begin{array}{rl}\frac{\mathrm{\partial }y\left(t,x\right)}{\mathrm{\partial }t}& =f\left(y\left(t,x\right),u\left(t,x\right),Y\left(t,x\right)\right),y\left(0,x\right)=y_0\left(x\right)\end{array}$(2)

$\begin{array}{rl}Y\left(t,x\right)& =\mathbf{K}y\left(t,x\right):={\int }_{O}w\left(x-x^{\mathrm{\prime }}\right)y\left(t,x^{\mathrm{\prime }}\right)d{x}^{\mathrm{\prime }},\end{array}$(3)

and appropriate boundary conditions. In (3), $\mathbf{K}={\int }_{O}w\left(x-x^{\mathrm{\prime }}\right)d{x}^{\mathrm{\prime }}$ is a linear integral operator acting on a function $y\left(t,x\right)$ and $w\left(x-x^{\mathrm{\prime }}\right)$ is a kernel function that models the effect that location $x^{\mathrm{\prime }}$ has on location $x$ .Footnote ¹¹ Since one of the basic premises of spatial economics is that what happens near us matters more than what happens far from us, it is reasonable to assume that the kernel is declining with the distance $\left|x-x^{\mathrm{\prime }}\right|$ , and that the influence tends to zero when this distance becomes sufficiently large. Another usual assumption is that the effects are spatially symmetric. Spatial kernels could reflect positive effects such as knowledge or productivity spillovers, or negative effects such as congestion effects. Kernels are usually modeled by exponential functions. Figure 2(a) depicts the spatiotemporal evolution of biomass under long-range effects.Footnote ¹²

Figure 2

(a) Exponential kernel with positive and negative spatial effects. (b) Biomass evolution in time and space $\left(t,x\right)$ of a population with logistic growth and nonlinear predation ${}^{13}$ effects under long-range transport.

Figures 1 and 2 showFootnote ¹³ an interesting qualitative characteristic of the transport mechanisms in continuous time and space. In Figure 1 the spatial heterogeneity of the concentration is preserved with the passage of time, while in Figure 2(b) the initial spatial heterogeneity vanishes and the concentration becomes spatially homogeneous, or returns to a “flat earth” situation. The emergence of spatial heterogeneity – or spatial pattern formation – from an initial state of flat earth, or the convergence to flat from an initial state of spatial heterogeneity, is an important issue when the system is controlled optimally and is related to the design of efficient spatial policies. These issues are examined in Sections 4 and 4.1.

2.3 Discrete Space: Metapopulation Models and Dispersion

A class of spatial models – metapopulation models – most often encountered in environmental and resource economics treats space as discrete, consisting of patches, and describes how populations or other objects move across patches.Footnote ¹⁴ In bioeconomics based on metapopulation models, harvesting takes place in a patchy spatial domain and populations on different patches are connected by the dispersal process. Dispersion is modeled by a system of ODEs which, for a spatial domain with $i=1,\dots ,n$ patches, can be written as

$\begin{array}{rl}\frac{d{y}_i\left(t\right)}{dt}& =f_i\left(y_i\left(t\right),u_i\left(t\right)\right)+\sum _{j=1,j\ne i}^n{d}_{ij}{y}_j\left(t\right)-D_i{y}_i\left(t\right),\\ y_i\left(0\right)& =y_{i0},\sum _{j=1,j\ne i}^n{d}_{ij}\le D_i,\end{array}$(4)

where $f_i\left(y_i\left(t\right),u_i\left(t\right)\right)$ represent population dynamics, and individuals disperse from patch $i$ at a rate $D_i\ge 0$ and arrive from patch $j$ at a rate $d_{ij}\ge 0$ (Cantrell and Cosner, Reference Cantrell and Cosner2003). Models like (4) can be extended to include density-dependent dispersal, multispecies interactions, bioinvasions, or pollution flows across patches. There is a considerable amount of literature that analyzes environmental and resource management issues using patchy environments and metapopulation models with dispersion across patches.Footnote ¹⁵

Dispersion models have been used to study the “acid rain game” (e.g., Mäler, Reference Mäler, Folmer and van Ierland1989; Kaitala et al., Reference Kaitala, Pohjola and Tahvonen1992; Mäler and de Zeeuw, Reference Mäler and de Zeeuw1998; Nagase and Silva, Reference Nagase and Silva2007) in which acid deposits damage regions due to acid rain generated by sulfur emissions in other regions. In Mäler and de Zeeuw (Reference Mäler and de Zeeuw1998), acid depositions in each of the $i=1,\dots ,n$ countries are given by $A\mathbf{e}$ , where $A=\left[a_{ij}\right],i,j=1,\dots n$ is a transportation matrix in which the element $a_{ij}$ denotes the fraction of country $j$ ’s emissions $e_j$ of sulfur or nitrogen oxides that is deposited in country $i$ , so $A\mathbf{e}$ is the vector of acid depositions in each country. Acid depositions deplete each country’s acid buffer stock if they exceed a critical load. The evolution of depletions is given by the system of ODEs

$\dot{\mathbf{d}}\left(t\right)=A\mathbf{e}\left(t\right)\mathbf{-}\mathbf{c}\mathbf{,}\mathbf{d}\mathbf{(}0{\mathbf{)}\mathbf{=}\mathbf{d}}_0,$

where $\mathbf{d}$ is the vector of depletion of each country’s acid buffer stock and $\mathbf{c}$ is the vector of critical loads. An increase in depletion means damage to the country’s soil. The objective is to choose emission paths to minimize the cost of reducing emissions plus damages from depletion.

Metapopulation models have also been used in the study of biological invasions. Albers et al. (Reference Albers, Fischer and Sanchirico2010) study the spread of invasive species over heterogeneous regions and compare optimal spatially heterogeneous policy to spatially uniform policy. Epanchin-Nieli and Wilen (Reference Epanchin-Nieli and Wilen2012) study optimal spatial control of biological invasions in spatiotemporal models in which the spatial domain is two-dimensional. Bioinvasion spreads from the invaded cell to adjacent cells in the absence of regulation. Epanchin-Nieli and Wilen develop optimal policy with a spatially explicit characterization.

Brock and Xepapadeas (Reference Brock and Xepapadeas2002) analyze a species competition for a limited resource in a patchy environment. They show that three different equilibrium species specialization patterns emerge – undisturbed nature with harvesting, and private optimal and social optimal with harvesting – and show that policy rules are spatially dependent.

Dispersion type of modeling has been used to study heat transport from the equator to the North Pole in the economics of climate change. This includes “two-box” models in which heat moves from the equatorial region to the North and causes Arctic amplification.Footnote ¹⁶

Spatial kernels can also be incorporated into spatial domains consisting of patches. In this case, (4) can be written as

$\frac{d{y}_i\left(t\right)}{dt}=f\left(y_i\left(t\right),u_i\left(t\right),\sum _{j=1,j\ne i}^n{w}_{ij}{y}_j\right),y_i\left(0\right)=y_{i0},$

where the $w_{ij}$ element of the kernel provides a measure of the influence of the state of the system at patch $j$ on the state of the system at patch $i$ .

Section 5 shows how the transport mechanisms presented in this section have been used to analyze specific issues in environmental and resource economics.

4.1 Optimal Pattern Formation and Policy Implications

In the case of optimized systems, the important differences between the “optimal spatial instability” – whether it is diffusion-driven or kernel-driven – and the celebrated Turing instability, which explains pattern formation in biological and chemical systems, are that: (a) contrary to the spirit of the Turing model, here the instability is driven by optimizing behavior, so it is the outcome of forward-looking optimizing behavior by economic agents and not the result of reaction-diffusion in chemical or biological agents; (b) the spatial patterns do not emerge between two state variables that in general reflect quantities, but between a state variable and its shadow price, thus the spatial pattern occurs in the price-quantity space; and (c) contrary again to the Turing approach, there is no need to have two or more diffusing/interacting state variables to generate patterns, but only one diffusing state. Optimization induces diffusion to the price system (the costate) and the interaction of the price-quantity system generates patterns.

The optimal Turing instability is quashed when the discount rate $\rho$ becomes zero (see Figure 3). This relation of the discount rate to Turing instability can be linked with general results from the turnpike literature and the role of the utility discount rate. In the classical turnpike theory of multisectoral capital theory (Cass and Shell, Reference Cass and Shell1976; McKenzie, Reference McKenzie1976), a discount rate close to zero is associated with unique steady-state equilibria and global asymptotic stability. When the discount rate is close to zero, the optimal Turing instability is expected to vanish as well. As indicated in Xepapadeas and Yannacopoulos (Reference Xepapadeas and Yannacopoulos2023) – in the context of an optimal growth model with spatial capital flows – the optimal Turing instability emerges from comparing benefits from moving towards a flat-earth optimal steady state after a perturbation. In systems without spatial diffusion, moving toward the optimal steady after a perturbation reduces the value loss. In systems with spatial diffusion, if the discount rate is sufficiently high, the benefits from moving towards a flat-earth steady state after a perturbation could be negative. From the social planner’s point of view, this can be interpreted as suggesting that it is preferable to let patterns emerge instead of controlling the system to the flat-earth optimal steady state. Thus the optimal Turing instability can be regarded as a new form of instability that may emerge if diffusion is present and the discount rate is sufficiently high. Low discount rates, on the other hand, induce positive benefits from moving towards the steady state and pattern elimination.Footnote ²² The role of the discount rate should also be important in a potential integration of optimal Turing theory with Pareto optima in general equilibrium theory, which could be an area of future research to extend the work of Bewley (Reference Bewley1982) in the integration of equilibrium theory and turnpike theory to spatial settings with diffusion.

Optimal spatial patterns or agglomeration as persistent outcomes implies that the value of the spatially heterogeneous system exceeds the value of the flat-earth system. In terms of policy design, an interesting distinction could be made between privately optimal solutions that ignore the diffusive or the long-range externality, and the socially optimal solution that internalizes the spatial externality. If spatial transport phenomena produce different outcomes between the social optimum and the private optimum – for example, if the equilibrium outcome at the private optimum implies spatial heterogeneity while the social optimum implies different spatial patterns or even flat earth – then policy instruments that promote socially optimal patterns or suppress spatial heterogeneity should be designed. In the remainder of this Element, we use the analytical tools presented earlier to study the issue of characterizing and designing policies under spatial transport for typical environmental and resource problems.

4.2 Cooperative and Noncooperative Solution Concepts in Continuous Space

Consider – at time $t$ and location $x\in \left[0,L\right]=X$  – a benefit function $B\left(u(t,x)\right)$ , where $B(u)$ denotes gross benefits generated by a polluting activity $u$ , with $B^{\prime }\ge 0,B^{\prime \prime }<0,\underset{u\to 0}{lim}{B}^{\prime }(u)=\mathrm{\infty }$ , and $C(S)$ an increasing convex cost function of the accumulated pollution stock. Net benefits accrue to an economic agent located at $x$ . Each agent is characterized by “spatial myopia,” since the agent takes into account pollution costs associated with the stock of pollution accumulated only at its location and ignores the impact of its actions on the rest of the spatial domain. In contrast to each individual agent, a social planner that seeks to internalize the pollution externality would take into account the aggregate pollution costs associated with the stock of pollution accumulated in each location. Furthermore, pollution costs could exhibit spatial dependence emerging from the fact that damages at a location $x$ do not depend on stock accumulation at $x$ only but on accumulation in nearby locations $x^{\prime }$ as well, with the impact declining with distance. Using a kernel to capture proximity effects, the social damage function can be defined as

$\begin{array}{r}SD(t,x)={\int }_{x\in X}w(x-x^{\prime })C(S(t,x^{\prime })d{x}^{\prime }.\end{array}$

A general spatiotemporal evolution of the pollution stock can be written as

$\begin{array}{r}{\mathrm{\partial }}_{t}S(t,x)=u(x,t)+{\int }_{X\setminus x}\varphi (x-x^{\prime })u(t,x^{\prime })d{x}^{\prime }-bS(t,x)+D{\mathrm{\partial }}_{xx}S(t,x).\end{array}$(16)

The integral term in (16) reflects long-range effects of emissions generated at $x^{\prime }$ by agents other than the agent located at $x$ , in addition to the local diffusion effects of the stock on location $x$ .

Interpreting $S$ as the local abundance of a renewable resource, introducing a local growth function $F(S(t,x)$ , and interpreting $u(t,x)$ as local harvesting, the dynamics of (16) describe the spatiotemporal evolution of a renewable resource. In this case, the integral term represents harvesting in $x$ of harvesters located in $x^{\prime }$ , if this is possible under the property rights structure (e.g., open access).

A cooperative solution is the solution to the following social planner’s problem:

$\begin{array}{r}\underset{\{u(t,x)\}}{\text{max}}J=\underset{\{u(t,x)\}}{\text{max}}{\int }_0^{\mathrm{\infty }}{e}^{-\rho t}{\int }_X\left[B\left(u(t,x)\right)-SD(t,x)\right]dxdt,\end{array}$

subject to (16) with appropriate initial, boundary, and transversality conditions. In the renewable resource version of the same problem, $B(u)$ could be interpreted as net harvesting benefits and $SD$ as additional benefits (existence values) or costs (stock effects) from the resource stock.

Noncooperative solutions can be associated with extensions of open-loop Nash equilibrium (OLNE) and feedback Nash equilibrium (FBNE) solution concepts. In OLNE, each agent follows a spatially myopic strategy, takes the actions of all other agents located at $x^{\prime }\in X\setminus x$ as exogenous (i.e., $u(t,x)$ ), and commits to an emission path that optimizes its own objective. That is,

$\underset{\{u(t,x)\}}{\text{max}}{J}^{\text{O}}=\underset{\{u(t,x)\}}{\text{max}}{\int }_0^{\mathrm{\infty }}{e}^{-\rho t}\left[B\left(u(t,x)\right)-SD(t,x)\right]dxdt,$

subject to

${\mathrm{\partial }}_{t}S(t,x)=u(x,t)+{\int }_{X\setminus x}\varphi (x-x^{\prime })\bar{u}(t,x^{\prime })d{x}^{\prime }-bS(t,x)+D{\mathrm{\partial }}_{xx}S(t,x).$(17)

In FBNE, each agent does not recall the previous history of the system, as described by past values of the state $S(t,x)$ , and assumes that the other agents’ emissions are conditioned on the current pollution accumulation in the spatial domain. There are different ways to interpret the relationship between agents’ emissions and the accumulated pollutant, which describes the equilibrium feedback strategy. One way is to assume that $u(t,x)=\theta \left({\int }_{X}S(t,x)dx,x\right)$ so all agents condition their emissions on the aggregate current stock. Another way is to assume that $u(t,x)=\theta \left({\int }_X\varphi (x-x^{\prime })S(t,x^{\prime })\right)d{x}^{\prime }$ , with $\varphi (0)=1$ . This means that each agent conditions emissions on the pollution stock accumulated at its location. The optimization problem, for the case of FBNE, can be written as:

$\underset{\{u(t,x)\}}{\text{max}}{J}^{\text{F}}=\underset{\{u(t,x)\}}{\text{max}}{\int }_0^{\mathrm{\infty }}{e}^{-\rho t}\left[B\left(u(t,x)\right)-SD(t,x)\right]dxdt,$

subject to

${\mathrm{\partial }}_{t}S(t,x)=u(x,t)+{\int }_{X\setminus x}\theta \left({\int }_{X}S(t,x)dx,x\right)d{x}^{\prime }-bS(t,x)+D{\mathrm{\partial }}_{xx}S(t,x),$(18)

with the $\theta$ function representing the equilibrium feedback strategy.

Cooperative and OLNE problems can be solved using the extension of the Pontryagin’s principle presented in Section 3 (e.g., Xepapadeas, Reference Xepapadeas2022). For the solution of FBNE problems, the use of the maximum principle encounters difficulties since the the equilibrium feedback strategy $\theta$ is not known a priori and should be determined endogenously. In this case, the dynamic programming approach is used in a similar way to how it is used for the analysis of the temporal only FBNE problems. The full solution of these problems, including potentially nonlinear feedbacks in dynamics, nonlinearities in diffusion, and long-range effects, is a very interesting area for further research. A starting point could be the solution of the Hamilton-Jacobi-Bellman (HJB) equation in infinite dimensional Hilbert space following the approach of Boucekkine et al. (Reference Boucekkine, Camacho and Fabbri2013) for a problem without strategic interactions, in which the solution for an isoelastic utility function is obtained by looking for solutions for the HJB with an isoelastic value function. A feedback solution to a spatial differential game with linear diffusion and isoelastic objective is provided by de Frutos et al. (Reference de Frutos, López-Perez and Martín-Herrán2021) by considering affine functions as a solution for the infinite-dimensional HJB equation. Boucekkine et al. (Reference Boucekkine, Fabbri, Federico and Gozzi2022b) in a similar context show that there exists a Markov perfect equilibrium, unique among the class of the affine feedbacks.

5.1 Fishery Management in Patchy Environments

In fishery management, the explicit introduction of space is implemented in the context of metapopulation models with subpopulations in patches and population dispersal among them due to natural forces (e.g., winds or currents).

Smith et al. (Reference Smith, Sanchirico and Wilen2009) present spatial dynamic processes and their applications to renewable resource management. Sanchirico and Wilen (Reference Sanchirico and Wilen1999) use the modeling approach in (4) to describe the evolution of fish biomass in patch $i=1,\dots ,n$ under harvesting modeled by the catch function $h_i\left(E_i\left(t\right),y_i\left(t\right)\right)$ , with $E_i\left(t\right)$ being the level of fishing effort. In an open-access patchy system, fishing effort and biomass in each patch could evolve as

$\begin{array}{rl}\frac{d{E}_i\left(t\right)}{dt}& =s_i{R}_i\left(y_i\left(t\right),E_i\left(t\right)\right)\\ & +\sum _{j=1,j\ne i}^n{s}_{ij}\left[R_i\left(y_i\left(t\right),E_i\left(t\right)\right)-R_j\left(y_j\left(t\right),E_j\left(t\right)\right)\right]\\ \frac{d{y}_i\left(t\right)}{dt}& =f_i\left(y_i\left(t\right)\right)y_i\left(t\right)+N{D}_i\left(y_1\left(t\right),\dots ,y_n\left(t\right)\right)-h_i\left(E_i\left(t\right),y_i\left(t\right)\right),\end{array}$

where $f_i\left(y_i\right)$ is per capita growth function; $N{D}_i$ is the net dispersal function in patch $i;$ $R_i\left(y_i,E_i\right)$ denotes rents in patch $i;$ $s_i$ is entry exit rates; and $s_{ij}\left(R_i\left(y_i,E_i\right)-R_j\left(y_j,E_j\right)\right)$ is fleet dispersal because of revenue differentials across patches. The biomass-effort steady state of the system is determined as $\left(E_i^{*},y_i^{*}\right):\left(d{E}_i/dt=0,d{y}_i/dt=0\right)$ . Sanchirico and Wilen find the equilibrium patterns of biomass and effort across the system to be dependent upon bioeconomic conditions within each patch, and the nature of the biological dispersal mechanism between patches. In terms of policy, they conclude that optimal instruments should reflect the interplay between the spatial gradient of rents and the spatial gradient of biological dispersal.

Sanchirico and Wilen (Reference Sanchirico and Wilen2001) study the creation of marine reserves in a patchy environment and show that, under certain conditions, creating a reserve by closing a patch for harvesting could increase aggregate biomass and harvest.Footnote ²⁷

Rassweiler et al. (Reference Rassweiler, Costello and Siegel2012) studied optimal fishery management and marine protected areas in a patchy environment. In this model, annual yield $Y_{it}$ in patch $i$ and year $t$ is given by

$Y_{it}=\frac{B_{Lit}{E}_{it}\left(1-e^{-\left(E_{it}+M\right)}\right)}{E_{it}+M},$

where $B_{Lit}$ , $E_{it}$ , $M$ are legal-sized fish biomass, harvesting effort, and natural mortality rate, respectively. Profit in each patch per year is

${\pi }_{it}=Y_{it}-{\theta }_i{E}_{it},$

where ${\theta }_i$ is cost per unit effort in patch $i$ . The objective is to choose how to distribute total fishing effort among patches to maximize profits. It is shown that fully optimized spatial management could increase nearshore fishery profits relative to those obtained with nonspatial management, with the magnitude of these increases varying across species.Footnote ²⁸

In a continuous-space fishery model, Behringer and Upmann (Reference Behringer and Upmann2014) find that in atomistic equilibrium, each agent exploits one location only and tends to harvest the resource to extinction in this location. This result also points to spatially structured policy interventions. Reaction-diffusion processes have also been used to model fishery management in a continuous spatial domain. Broadbridge and Hutchinson (Reference Broadbridge and Hutchinson2022) develop such a model in a heterogeneous environment with spatially dependent diffusivity, while Cui et al. (Reference Cui, Li, Mei and Shi2017) use reaction-diffusion modeling to study harvesting quotas and protection zones in fishery management.

5.2 Groundwater Management

In groundwater management, the early literature such as Gisser and Sanchez (Reference Gisser and Sanchez1980), Negri (Reference Negri1989), and Provencher and Burt (Reference Provencher and Burt1993) considered the underground aquifer as a homogeneous single-cell “bathtub” in which abstraction by one user caused an instantaneous impact on others. More recent literature recognizes the fact that hydrological factors such as seepage or aquifer transmissivity introduce a spatial pumping externality. In this case, pumping by a farmer affects and is affected by the pumping behavior of neighboring farmers through the emergence of overlapping cones of depression in the aquifer. Thus optimization problems which seek to maximize benefits from the underground aquifer acquire an explicit spatial structure (e.g., Saak and Peterson, Reference Saak and Peterson2007; Brozović et al., Reference Brozović, Sunding and Zilberman2010).

Pfeiffer and Lin (Reference Pfeiffer and Lin2012) model a “patchy” groundwater aquifer with water flowing across patches according to hydrological rules. The dynamics of water stock $y_i\left(t\right)$ in each patch are given by

$\frac{d{y}_i\left(t\right)}{dt}=-u_i\left(t\right)+g_i\left(u_i\right)+\sum _{j=1,j\ne i}^n{\theta }_{ij}{y}_j\left(t\right),$(19)

where $u_i$ is water pumping, $g_i\left(u_i\right)$ is recharge to patch $i$ , and flow parameters ${\theta }_{ij}$ are determined by Darcy’s law or ${\theta }_{ij}=\left(y_i-y_j\right)/x_{ij}$ , where $x_{ij}$ is the distance between patches. Groundwater dynamics (19) act as a constraint to the problem of a social planner seeking to maximize discounted aquifer benefits, or

$\underset{u_i\left(t\right)}{\text{max}}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\rho t}\left[\sum _{i=1}^n\left[R_i\left(u_i\right)-C\left(y_i\right)u_i\right]\right]dt.$

Results suggest that the spatial externality results in overpumping relative to the social optimum (Pfeiffer and Lin, Reference Pfeiffer and Lin2012), and that the spatial externality is important for large, unconfined groundwater aquifers (e.g., Brozović et al., Reference Brozović, Sunding and Zilberman2010). Kuwayama and Brozović (Reference Kuwayama and Brozović2013) consider the adoption of a spatially differentiated groundwater permit system to efficiently regulate when groundwater pumping affects the flow of surface water.

Brock and Xepapadeas (Reference Brock and Xepapadeas2010) studied a semiarid system with reaction-diffusion characteristics in which plant biomass and soil water interact and diffuse in a continuous space with linear (Fickian) diffusion. The plant–soil water dynamics are given by

$\begin{array}{rl}{\mathrm{\partial }}_{t}P\left(t,x\right)& =g\left(W\left(t,x\right),P\left(t,x\right)\right)-bP\left(t,x\right)-h\left(t,x\right)+D_p{\mathrm{\partial }}_{xx}P\left(t,x\right)\\ {\mathrm{\partial }}_{t}W\left(t,x\right)& =f\left(P\left(t,x\right),R\right)-\upsilon \left(W\left(t,x\right),P\left(t,x\right)\right)-r_{W}W\left(t,x\right)\\ & +D_w{\mathrm{\partial }}_{xx}W\left(t,x\right)P\left(0,x\right),W(0,x)given,\\ P(t,0)& =P(t,L),W(t,0)=W(t,L)\mathrm{\forall }t,\end{array}$

where $P(t,x)$ is plant density (biomass); $W(t,x)$ is soil water at time $t\in [0,\mathrm{\infty })$ and location $x\in [0,L]$ ; $R$ is fixed rainfall; $h$ is harvesting of plant biomass through grazing; $g(W,P)$ is plant growth, increasing in soil water and plant density; $bP$ is plant senescence; $f(P,R)$ is water infiltration; $\upsilon (W,P)$ is water uptake by plants; $r_W$ is specific rate of water loss due to evaporation and percolation; and $D_P$ and $D_W$ are diffusion coefficients for plant biomass (plant dispersal) and soil water, respectively.Footnote ²⁹ The authors consider the problem of a myopic agent that optimizes profits by ignoring spatiotemporal dynamics, and the problem of a social planner that internalizes the spatial externality. They find that at the myopic solution, spatial patterns in plant–soil water are generated through the Turing mechanism but the socially optimal solution is spatially homogeneous. Spatially dependent instruments are required in order to internalize the spatial externality.Footnote ³⁰

5.3 Pollution Control

Goetz and Zilberman (Reference Goetz and Zilberman2000) consider pollution accumulation in a lake associated with the runoff from mineral fertilizers and animal manure. They employ a two-stage optimization, optimizing first across the spatial and then across the temporal dimension. The social optimum can be implemented with site-specific taxes on mineral fertilizers, manure, and large animal units.

Sigman (Reference Sigman2005) considers transboundary river pollution and examines whether US states that, under decentralized policies, control their Clean Water Act programs free ride on downstream states. Sigman does not include an explicit pollution transport mechanism along the river and defines water quality $W{Q}_{it}$ in location $i$ and time $t$ as

$\begin{array}{r}W{Q}_{it}=\sum _{h=1}^H\frac{P_{ht}}{F_{ht}}{\delta }^{h-i},\end{array}$

where $P_{ht}$ is pollution at upstream locations $h$ , $F_{ht}$ is flow that dilutes pollution, and $\delta <1$ is a spatial discount term that diminishes pollution effects with distance. Using econometric estimation, Sigman concludes that there is free riding, whose costs need, however, to be compensated with benefits derived from the flexibility introduced by decentralization.

Xabadia et al. (Reference Xabadia, Goetz and Zilberman2006, Reference Xabadia, Goetz and Zilberman2008) study policies for controlling agricultural stock pollution in a framework in which the spatial differentiation is related to the heterogeneity in the land quality of producers located at different sites. Pollution generated by the heterogeneous producers is accumulated in the environment and the optimal emission policy is site specific. Although there is no explicit spatial transport mechanism, this research studies a spatially distributed parameter problem in the space of qualities which are distributed in space and presents another approach to spatial issues.

Explicit pollution diffusion across space was introduced by Brock and Xepapadeas (Reference Brock and Xepapadeas2008) into the so-called shallow-lake problem. In this case, pollution (phosphorous) accumulates in time and space according to the PDE

$\begin{array}{r}{\mathrm{\partial }}_{t}P\left(t,x\right)=E\left(t,x\right)-mP\left(t,x\right)+f\left(P\left(t,x\right)\right)+D{\mathrm{\partial }}_{xx}P\left(t,x\right),\end{array}$(20)

with appropriate initial and spatial boundary conditions. In this setup, $P\left(t,x\right)$ denotes the stock of the pollutant at $t$ and $x$ ; $E$ is emissions by location $x;$ $m$ is the pollution decay rate; and $f\left(P\left(t,x\right)\right)$ is in general a convex-concave function indicating nonlinear feedbacks that underlie the lake dynamics. Using (20) as a constraint in the planner’s problem for maximizing benefits over the whole spatial domain, and realistic parametrization for the lake, it is shown that a steady state can be destabilized in the context of optimal Turing instability, and spatial patterns in the accumulation of the pollutant start emerging due to pollution diffusion. Since the spatial patterns for the pollutant imply spatial patterns for its shadow cost, this result suggests spatially differentiated emissions taxes.Footnote ³¹

Camacho and Pérez-Barahona (Reference Camacho and Pérez-Barahona2015) use the model of Gaussian plume to describe the spatial dynamics of pollution. They consider a pollution accumulation equation at location $x$ of the form

$\begin{array}{r}{\mathrm{\partial }}_{t}P\left(t,x\right)=E\left(x,t\right)+{\mathrm{\partial }}_{xx}P\left(t,x\right)\end{array}$

in a model of optimal land use in which the interaction between land use and the environment generates a spatially heterogeneous solution and abatement technology is central in pollution stabilization. Cornet and Camacho (Reference Cornet and Camacho2021) study soil pollution diffusion in an agricultural economy model. In this model, output is produced by fertile and polluted land with given technology, and pollution diffuses in space with a constant diffusion coefficient as in (20). They provide conditions for pattern formation between fertile and polluted land.

De Frutos and Martín-Herrán (Reference de Frutos and Martín-Herrán2019) consider a spatiotemporal pollution dynamics problem of the form shown in (20) without the nonlinear feedback to study optimal regulation in a transboundary pollution problem. By making a linear quadratic approximation and discretizing the space, they derive regional cooperative and noncooperative emission paths.

5.4 Urban Economics and Spatial Effects

Environmental externalities are prominent in the area of urban economics. Pollution from transportation or industrial activities affects the location decisions of both individuals and firms and significantly affects the spatial structure of a city. In this context, environmental policy is an important factor in the development of residential and industrial clusters, since strict environmental measures can discourage firms from polluting urban areas, while reduced pollution levels can encourage people to locate closer to industrial areas, thus reducing commuting costs.

Henderson (Reference Henderson1977) studied a problem in which industrial pollution diffuses towards the residential/central business district boundary. The optimal policy consists of Pigouvian taxes along with regulations controlling the allocation of land between polluting firms and individuals, and potential redistribution of tax receipts between heavily taxed and lightly taxed communities. The combination of optimal taxes with zoning policies prevents polluting firms from locating in residential areas. Verhoef and Nijkamp (Reference Verhoef and Nijkamp2002, Reference Verhoef, Nijkamp, Capello and Nijkamp2005) use a monocentric city model and study first- and second-best policies in order to control the effect of industrial pollution on residential areas. They show that environmental goals can be promoted either at the expense of or in favor of agglomeration economies.

Arnott et al. (Reference Arnott, Hochman and Rausser2008) study a circular city in which firms generate pollution and households commute at a cost and receive disutility from pollution. Pollution disperses according to a function $D\left(e\left(x\right),x-x^{\mathrm{\prime }}\right)$ , where $e\left(x\right)$ denotes emissions at $x$ and $x-x^{\mathrm{\prime }}$ is the distance between location $x$ and $x^{\mathrm{\prime }}$ . The kernels introduced in Section 2.2 could be a reasonable specification of such a function. In this setup, the optimal allocation can be decentralized by imposing a tax per unit of area of industrial land at a particular location equal to the total damage caused by the pollution from that unit area, evaluated at the optimum. Location-specific Pigouvian taxes that do not fully internalize the total damage caused by this site are inefficient.

Rossi-Hansberg et al. (Reference Rossi-Hansberg, Sarte and Owens2010) and Rossi-Hansberg and Sarte (Reference Rossi-Hansberg and Sarte2012) study housing externalities, defined as the effects that a house’s characteristics have on neighbors. They find that these externalities decay fast with distance, with the impact on location $x$ from housing services $H\left(x^{\mathrm{\prime }}\right)$ in nearby location $x^{\mathrm{\prime }}\in \left[-L,L\right]$ defined as:

$\begin{array}{r}\delta {\int }_{-L}^L{\mathrm{e}}^{-\delta \left|x-x^{\mathrm{\prime }}\right|}H\left(x^{\mathrm{\prime }}\right)d{x}^{\mathrm{\prime }}.\end{array}$

They conclude that the number of residents in a neighborhood depends on the quality of nearby housing and propose, as policy measures, minimum maintenance requirements or zoning policies.

Kyriakopoulou and Xepapadeas (Reference Kyriakopoulou and Xepapadeas2013, Reference Kyriakopoulou and Xepapadeas2017) consider a linear city with: (i) productivity spillovers that decline with distance, or

$\begin{array}{r}z\left(x\right)=\delta {\int }_0^L{\mathrm{e}}^{-\delta {\left(x-x^{\mathrm{\prime }}\right)}^2}\lambda \left(x^{\mathrm{\prime }}\right)lnL\left(x^{\mathrm{\prime }}\right)d{x}^{\mathrm{\prime }},\end{array}$

where ${\mathrm{e}}^{-\delta {\left(x-x^{\mathrm{\prime }}\right)}^2}$ is a normal dispersal kernel, $\lambda \left(x^{\mathrm{\prime }}\right)$ is the proportion of land occupied by firms at the spatial point $x^{\mathrm{\prime }}$ , and $L\left(x^{\mathrm{\prime }}\right)$ is labor input; and (ii) pollution, $P$ , that diffuses across the city and concentrates in specific locations according to

$\begin{array}{r}lnP\left(x\right)={\int }_0^L{\mathrm{e}}^{-\zeta {\left(x-x^{\mathrm{\prime }}\right)}^2}\lambda \left(x^{\mathrm{\prime }}\right)lnE\left(x^{\mathrm{\prime }}\right)d{x}^{\mathrm{\prime }},\end{array}$

where $E\left(x^{\mathrm{\prime }}\right)$ denotes industrial pollution. In this setup, equilibrium and optimal solutions regarding the spatial structure of the city are compared and optimal policy is derived. In Kyriakopoulou and Xepapadeas (Reference Kyriakopoulou and Xepapadeas2013), where a first-nature advantage assumption is made, it is shown that the equilibrium outcome leads to either a monocentric city or a polycentric city with the first-nature advantage site attracting the majority of economic activity. On the contrary, the socially optimal solution leads to a duocentric city, where neither of the two centers is formed around the natural advantage site. The authors show that sites with inherent advantages can lose their comparative advantage when the social cost of pollution is taken into account.

Kyriakopoulou and Xepapadeas (Reference Kyriakopoulou and Xepapadeas2017) consider a general equilibrium setup where there is: competition for land between industries and households, polluting industrial activity, production externalities, and costly commuting. In equilibrium, the center of the city is mixed residential/industrial, while at the social optimum there are distinct residential and industrial clusters across the city. The presence of spatial productivity spillovers and spatial pollution spillovers requires that the optimal policy be site-specific and consist of two instruments: pollution taxes to internalize the negative pollution externality and labor subsidies to internalize the productivity externality. Uniform instruments are suboptimal.

Regnier and Legras (Reference Regnier and Legras2018) use a model à la Fujita and Ogawa (Reference Fujita and Ogawa1982) to study the urban patterns derived in the presence of industrial pollution, which decreases environmental quality $E$ at spatial point $x$ according to

$\begin{array}{r}E(x)=\bar{E}-{\int }_X[e-\eta \left|x-y\right|]b(y)dy,\end{array}$

where $\bar{E}$ denotes environmental quality without pollution, $e$ is the quantity of pollution emitted by one firm, $\eta$ shows how pollution disperses in space, $x-y$ is the distance between firm and household, and $b(y)$ is the density of firms at $y$ . The authors show that the internalization of pollution forms more specialized areas in the city, which results in lower greenhouse gases from commuting.

There is also a growing literature on the internal structure of cities when pollution comes from commuting. Verhoef and Nijkamp (Reference Verhoef and Nijkamp2003) point out the importance of space in the analysis of urban air pollution, which is affected by aggregate commuting and not by the number of commuters. Schindler et al. (Reference Schindler, Caruso and Picard2017) study how traffic-induced pollution affects residential choices and find that higher pollution levels reduce the size and the population of the city. Pollution ( $P$ ), in that framework, increases with the traffic volume passing by $r$ , as

$\begin{array}{r}P(r)=1+a+b{\int }_r^{r_f}n(r)dr,\end{array}$

where $a$ and $b$ measure the impacts of regional and traffic-induced pollution in the city and $n(r)$ is the number of people crossing location $r$ . Finally, Denant-Boemont et al. (Reference Denant-Boemont, Gaigné and Gaté2018) show that polycentric cities imply higher welfare and lower pollution levels.

5.5 Pollution Diffusion and Growth

Spatial considerations in growth models associated with diffusion of capital in a spatial domain have been explored in the context of spatial growth theory. Spatial diffusion of capital and pollution, which is jointly generated by output production, is still an area that is not very well researched. La Torre et al. (Reference La Torre, Liuzzi and Marsiglio2015) develop Solow- and Ramsey-type models with capital and pollution diffusion and concave or convex-concave production function. The process of capital and pollution accumulation is described for the augmented Solow model in a domain $\left[x_0,x_1\right]$ by the system

$\begin{array}{rl}{\mathrm{\partial }}_{t}k(t,x)& =\frac{s(x)f\left(k(t,x\right)\left[1-\tau (x)\right]}{\alpha +\beta P(t,x{)}^2}-{\delta }_{k}k(t,x)+D_k{\mathrm{\partial }}_{xx}k(t,x)\\ {\mathrm{\partial }}_{t}P(t,x)& =\theta {\int }_{x_0}^{x_1}\left[1-u(x^{\prime })\right]f\left(k(t,x^{\prime })\right)\varphi (x^{\prime },x)d{x}^{\prime }\\ & -{\delta }_{P}P(t,x)+D_P{\mathrm{\partial }}_{xx}P(t,x),\end{array}$

where $k(x,t),P(t,x)$ denote capital stock and pollution at time $t$ and location $x$ , respectively, which depreciate at rates ${\delta }_k,{\delta }_P$ ; $f(\cdot )$ is a production function; $s(x)$ , $\tau (x)$ denote saving rate and taxation at location $x$ ; $\alpha +\beta P(x,t{)}^2$ stands for output-reducing pollution-damage function; $u(x^{\prime })$ denotes pollution reduction due to abatement at location $x^{\prime }$ ; $\varphi (x^{\prime },x)$ is a kernel indicating the impact of pollution at $x^{\prime }$ on neighboring areas, and $D_k,D_P$ are diffusion coefficients for capital and pollution respectively. Results obtained from numerical simulation suggest the emergence of a spatially heterogeneous distribution for the capital stock and pollution at a terminal time. The convex-concave production function indicates, under certain conditions, the possibility of elimination of poverty traps.

La Torre et al. (Reference La Torre, Liuzzi and Marsiglio2021) study a transboundary pollution problem with pollution diffusion across spaceFootnote ³² and explore local solutions corresponding to decentralized outcomes in which local planners ignore the transboundary externality and global outcomes in which a social planner takes this externality into account. By comparing solutions, environmental taxes are characterized which are explicitly spatially heterogeneous when spatial heterogeneity exists in the initial pollution distributions.

5.6 Infectious Diseases and the Spatial Dimension

Traditional mathematical models describing the spread of infectious diseases (e.g., Hethcote, Reference Hethcote, Levin, Hallam and Gross1989, Reference Hethcote2000) do not include the spatial dimension. Typically these models link flows between the passively immune class $M$ , the susceptible class $S$ , the exposed class $E$ , the infective class $I$ , and the recovered class $R$ , with the passively immune class $M$ and the latent period class $E$ often being omitted because they are not crucial for the susceptible–infective interaction. If recovery does not provide immunity, then the model is called a susceptible–infective–susceptible (SIS) model, since individuals move from the susceptible class to the infective class and then back to the susceptible class upon recovery, while if individuals recover with permanent immunity, then the model is a susceptible–infective–recovered (SIR) model. Most of these models do not consider spatial diffusion of different classes in a given spatial domain, and analyze convergence of the classes to a spatially homogeneous steady state.

A simple model for the spatial spread of an epidemic (Murray, Reference Murray2002, chapter 10) can be developed by augmenting a standard SIS model with diffusion terms describing the spread of susceptives and infectives on a one-dimensional space with fixed population, or

$\begin{array}{rl}{\mathrm{\partial }}_{t}S(t,x)& =-\beta I(t,x)S(t,x)+D{\mathrm{\partial }}_{xx}S(t,x)\end{array}$(21)

$\begin{array}{rl}{\mathrm{\partial }}_{t}I(t,x)& =\beta I(t,x)S(t,x)-\gamma I(t,x)+D{\mathrm{\partial }}_{xx}I(t,x),\end{array}$(22)

where $I(t,x),S(t,x)$ are infective and susceptive, respectively, at time $t$ and location $x$ , infectives have a disease-induced mortality rate $\gamma I$ , $D$ is the diffusion coefficient, and $\beta$ is a fixed parameter. Murray shows that this model generates a traveling epidemic wave of constant shape.

The spatial dimension can also be introduced using the concept of kernels introduced in Section 2.2. Consider an SIR model with $S+I+R$ fixed and independent of time $t$ , evolving in a one-dimensional spatial domain. Following Ruan (Reference Ruan, Takeuchi, Iwasa and Sato2007) in the description of the Kendall model,Footnote ³³ the rate of infection is assumed to be

$\beta {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}I(t,y)K(x-y)dy,$

where $\beta >0$ is a fixed parameter and the kernel $K(x-y)$ describes the impact of infected individuals at location $y$ on the infection of susceptible individuals at location $x$ , under the assumption that ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}K(y)dy=1$ . Then the SIR model becomes

$\begin{array}{rl}{\mathrm{\partial }}_{t}S(t,x)& =-\beta S(t,x){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}I(t,y)K(x-y)dy\\ {\mathrm{\partial }}_{t}I(t,x)& =\beta S(t,x){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}I(t,y)K(x-y)dy-\gamma I(t,x)\\ {\mathrm{\partial }}_{t}R(t,x)& =\gamma I(t,x).\end{array}$

It can be shown that an initial infection does not propagate if $\gamma /\beta \sigma \ge 1$ , where $\sigma$ is the initial density of susceptibles. An SIR model that combines the kernel and diffusion characteristics with constant populationFootnote ³⁴ can be written as

$\begin{array}{rl}{\mathrm{\partial }}_{t}S(t,x)& =-\beta S(t,x){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}I(t,y)K(x-y)dy+D{\mathrm{\partial }}_{xx}S(t,x)\\ {\mathrm{\partial }}_{t}I(t,x)& =\beta S(t,x){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}I(t,y)K(x-y)dy-\gamma I(t,x)+D{\mathrm{\partial }}_{xx}I(t,x)\\ {\mathrm{\partial }}_{t}R(t,x)& =\gamma I(t,x)+D{\mathrm{\partial }}_{xx}R(t,x).\end{array}$

Belik et al. (Reference Belik, Geisel and Brockmann2011) analyze the spatial spread of an epidemic in the context of a metapopulation reaction-diffusion system. Using an SIS model, the system becomes

$\begin{array}{rl}{\mathrm{\partial }}_t{S}_n(t)& =-\beta \frac{I_n(t)S_n(t)}{N_n^s}+\sum _m\left(w_{nm}{S}_m-w_{mn}{S}_n\right)\end{array}$(23)

$\begin{array}{rl}{\mathrm{\partial }}_t{I}_n(t)& =\beta \frac{I_n(t)S_n(t)}{N_n^s}-\gamma I_n(t)+\sum _m\left(w_{nm}{I}_m-w_{mn}{I}_n\right),\end{array}$(24)

where $w_{nm}$ is dispersal from population at location $m$ to $n$ , and $N_n^s$ is the number of individuals in population $n$ in diffusive equilibrium.

A common characteristic of these models is that they do not involve any optimization and the main objective is to characterize the solution of the dynamical system given initial and boundary conditions and parameter values. These models acquire economic characteristics if an objective to be optimized is introduced and the dynamical system acts as a constraint set to the optimization problem. La Torre (Reference La Torre, Liuzzi and Marsiglio2024) develop an SIS epidemic-economic model with spatial diffusion for the susceptibles and infectives as in (21)–(22) for COVID control.Footnote ³⁵ The objective is to choose optimal lockdown policies that limit social interactions to minimize aggregate costs consisting of lost output due to lockdown measures and disease prevalence.

Ferraccioli et al. (Reference Ferraccioli, Stilianakis and Veliov2024) develop a spatiotemporal model with heterogeneous mixed populations which are located at different locations. Using optimal control methods, they derive optimal policies in terms of social contact restrictions and partial lockdown that will maximize economic output net of epidemic costs.

Adda et al. (Reference Adda, Boucekkine and Thuilliez2024) develop a joint model of disease diffusion and mental health. By linking mental health to preferences for mobility, they provide better micro-foundations of epidemic dynamics and use the model to provide empirical estimates using high-frequency and geolocalized dataFootnote ³⁶ on mobility and psychotropic drugs.

Brock and Xepapadeas (Reference Brock and Xepapadeas2025) develop an integrated model with two geographical regions, North and South, which captures interactions between the economy and the natural world and links climate, land use, and infectious diseases represented by SIS or SIR models. The SIS version of the integrated two-region $i=1,2$ model with population normalized to one is written as

$\begin{array}{rl}& {\dot{S}}_i\left(t\right)=-{\beta }_i\left(t\right)I_i\left(t\right)S_i\left(t\right)+{\gamma }_{i}I\left(t\right),S_i(0)>0\end{array}$(25)

$\begin{array}{rl}& {\dot{I}}_i\left(t\right)={\beta }_i\left(t\right)I_i\left(t\right)S_i\left(t\right)-{\gamma }_i{I}_i\left(t\right),I_i(0)>0\end{array}$(26)

$\begin{array}{rl}& I_i\left(t\right)+S_i\left(t\right)=1,i=1,2.\end{array}$(27)

In this model, the quantity ${\sigma }_i\left(t\right)={\lambda }_i\left(t\right)/{\gamma }_i$ is the regional basic reproduction number, which is defined as the average number of secondary infections produced when one infected individual is introduced into a host population where everyone is susceptible. Spatial interactions are realized through the contact number. The susceptible steady state is defined as

$\begin{array}{rl}{S}_i\left(t\right)& =min\left\{1,1/{\sigma }_i(t)\right\}\end{array}$(28)

$\begin{array}{rl}\frac{1}{{\sigma }_i\left(t\right)}& ={\varphi }_{0i}\left(R_1(t),T_1(t)\right)+{\varphi }_{1i}\left[b_i{v}_i\left(t\right)-m_i^{as}{S}_i\left(t\right)-q_j\left(1-S_{jt}\right)\right]\\ i,j& =1,2,i\ne j.\end{array}$(29)

In (29), ${\varphi }_{0i}\left(R_1(t),T_1(t)\right)$ is the part of the inverse of the contact number that depends on variables evolving in the long run, namely land use and temperature in region 1, which is the infectious disease hot spot. The second term on the right-hand side of (29) indicates short-run effects from disease containment policies, where ${\varphi }_{1i}$ characterizes the overall containment effectiveness, $v_i\left(t\right)$ is containment control such as vaccination or social distancing, $b_i$ is the effectiveness of such control policies, and $m_i^{as}$ denotes the rate of potential spread of the disease by asymptomatic infecteds. Individuals from one region can make short visits to the other by regular means of transportation (e.g., airplanes, ships). Infected individuals from region $j$ traveling to region $i$ infect individuals in region $i$ proportionally to those infected in region $j$ and vice versa, with proportionalities $\left(q_j,q_i\right)$ respectively. The model is closed with an objective function which is defined in terms of a consumption composite and the natural environment. This objective is maximized subject to disease dynamics, climate dynamics, and land-use choices. The results suggest that the emergence of infectious diseases associated with land-use change and climate change points towards policies that will preserve the natural world (e.g., payments for ecosystem services); upwards adjustment of the social cost of carbon to capture the climate change–infectious disease link; and support of land augmenting innovations in agriculture that will slow down conversion and promote ecosystem conservation. Furthermore, the potential value of the ecosystem in mitigating future infectious diseases should be included in valuation studies based on stated preferences methods.

6.1 Regulating a Transboundary Externality

We consider a simple transboundaryFootnote ³⁷ (or cross-border) – but not global – externality with local damages in a two-region model, which is a special case of the general dispersion models described above with $d_{12}=D>0$ and $d_{21}=0$ . A very simple model of transboundary pollution is used in order to stress the fact that optimal policies should have a spatial structure even when regions are symmetric in their fundamentals. Our approach, despite its simplicity, makes clear the impact of spatial transport on optimal environmental policies, and the core model presented below can be extended along many different lines.

Let $u_i\left(c_i\right)=ln\left(y_i{E}_i^{\alpha }{\mathrm{e}}^{-{\upsilon }_i\left(P_i\right)}\right),i=1,2$ , denote utility in region $i$ from using emissions or energy $E_i$ in production net of pollution damages, where $y_i$ is an exogenous process incorporating the impact of other factors of production. Capital accumulation is not considered in order to simplify dynamics. The use of $E$ accumulates a pollutant $P_i$ in each region. Some of the pollutant accumulated in region 1 is transported to region 2 through natural forces (e.g., river flows, winds). The accumulated pollutant in each region generates damages according to a convex damage function ${\upsilon }_i\left(P_i\right),{\upsilon }_i^{\mathrm{\prime }}>0,{\upsilon }_i^{\mathrm{\prime }\mathrm{\prime }}\ge 0$ . Pollution dynamics, with the explicit dependence on $t$ omitted to ease notation, can be written as

$\begin{array}{rl}{\dot{P}}_1& =-B{P}_1-D{P}_1+E_1,P_1\left(0\right)=P_{10}\text{given}\end{array}$(30)

$\begin{array}{rl}{\dot{P}}_2& =-B{P}_2+D{P}_1+E_2,P_2\left(0\right)=P_{20}\text{given,}\end{array}$(31)

where $D>0$ is the pollution transportation coefficient, or diffusivity, and $B>0$ is a pollution depreciation rate. A regulator or a social planner will determine emission paths and emission taxes by maximizing the sum of discounted regional utilities subject to pollution dynamics.

The Hamiltonian representation of the regulator’s problem with a quadratic damage function, ${\upsilon }_i\left(P_i\right)=c_{1i}{P}_i+\left(c_{2i}/2\right)P_i^2$ , $i=1,2$ , can then be written as