Raouf Boucekkine

Giorgio Fabbri

Salvatore Federico

AND Fausto Gozzi

A DYNAMIC THEORY OF SPATIAL EXTERNALITIES

Keywords: Spatial externalities, environmental federalism, transboundary pollution, dierential games in continuous time and space, innite dimensional optimal control problems JEL classication: Q53, R12, O13, C72, C61, O44

In this paper, we revisit the theory of spatial externalities. In particular, we depart in several respects from the important literature studying the fundamental pollution free riding problem uncovered in the associated empirical works. First, instead of assuming ad hoc pollution diusion schemes across space, we consider a realistic spatiotemporal law of motion for air and water pollution (diusion and advection). Second, we tackle spatiotemporal non-cooperative (and cooperative) dierential games. Precisely, we consider a circle partitioned into several states where a local authority decides autonomously about its investment, production and depollution strategies over time knowing that investment/production generates pollution, and pollution is transboundary. The time horizon is innite. Third, we allow for a rich set of geographic heterogeneities across states while the literature assumes identical states. We solve analytically the induced non-cooperative dierential game under decentralization and fully characterize the resulting long-term spatial distributions. We further provide with full exploration of the free riding problem, reected in the so-called border eects. In particular, net pollution ows diuse at an increasing rate as we approach the borders, with strong asymmetries under advection, and structural breaks show up at the borders. We also build a formal case in which a larger number of states goes with the exacerbation of pollution externalities. Finally, we explore how geographic discrepancies aect the shape of the border eects.

Introduction

The theory of spatial externalities is an important and quite broad area in economic theory, public economics, and more recently, in the new economic geography. A key aspect in this literature turns out to be the potential free riding problem that might typically arise when actions taken within a jurisdiction have (negative) spillover eects on neighbors, which leads to ineciency of the decentralized equilibrium (assuming it exists). These are typical considerations for example in the literature of scal federalism à la [START_REF] Oates | Fiscal Federalism[END_REF].

They are also increasingly crucial in another rising literature, labeled environmental federalism by [START_REF] Konisky | Exporting Air Pollution? Regulatory Enforcement and Environmental Free Riding in theUnited States[END_REF]. In the context of transboundary pollution, the free riding problem has a clear geographic feature: it is associated with a neat border eect. Indeed, if pollution control is decentralized, local governments may be strongly tempted to locate the most polluting facilities near the jurisdictional borders [START_REF] Monogan | Gone with the Wind: Federalism and the Strategic Location of Air Polluters[END_REF] and/or to enforce less frequently at these borders pro-environmental policies as those promoted by federal or international acts and protocols (see [START_REF] Konisky | Exporting Air Pollution? Regulatory Enforcement and Environmental Free Riding in theUnited States[END_REF], on the local enforcement of the US federal Clean Air Act).

Earlier empirical assessments suggest indeed that as a consequence of the latter actions, pollution levels are systematically elevated near state borders relative to interior regions (Helland and Whitford, 2003[START_REF] Sigman | International Spillovers and Water Quality in Rivers: Do Countries Free Ride?[END_REF][START_REF] Sigman | Transboundary Spillovers and Decen tralization of Environmental Policies[END_REF]. The Helland and Whitford's 2003 paper is one of the most inuential in the area: using toxics release inventory (TRI) US data from 1987 to 1996, they nd that facilities' emissions into the air and water are systematically higher in counties that border other states. Focusing on air pollution, [START_REF] Monogan | Gone with the Wind: Federalism and the Strategic Location of Air Polluters[END_REF] have analyzed polluters location in the US using a spatial point pattern model. Again they nd that the main air polluters are by far more likely to be located near a state's downwind border than a control group of other industrial facilities.

More intriguing: when studying the enforcement of the US federal Clean Air Act from 1990 through 2000, Konisky and Wood (2010) nd that while there is a signicant negative correlation between enforcement levels and proximity to international borders (Canada, Mexico), proximity to state borders is not associated with fewer inspections or punitive actions. This need not contradict though the general nding outlined above that pollution levels are systematically higher near state borders as this nding is more likely to be caused by the location of the most polluting facilities near the jurisdictional borders (due to scal incentives mainly) than by enforcement policy. This said, the nding of [START_REF] Monogan | Gone with the Wind: Federalism and the Strategic Location of Air Polluters[END_REF] makes very clear that one of the main rationales behind environmental free riding by local governments is maximization of (local and regional) political support. A key subsequent problem to solve at the federal/national level is precisely to eliminate or at least to limit free riding, in order to ultimately reach a socially more ecient environmental federalism. This key issue is treated in several theoretical frames (see below for a brief view). The recent empirical literature reports several relevant natural experiments. In particular, [START_REF] Kahn | Water Pollution Progress at Borders: The Role of Changes in China's Political Promotion Incentives[END_REF] have for example studied the impact of a natural experiment set in China in 2005 when the central government changed the local political promotion criteria in order to reduce border pollution. They do nd evidence of water pollution reduction at province boundaries. 1

On the theoretical ground, the free riding problem described above has suggested a quite substantial literature. We refer here for simplicity to the extremely useful works of [START_REF] Hutchinson | State enforcement of federal standards: Implications for interstate pollution[END_REF], and [START_REF] Silva | Transboundary pollution control in federal systems[END_REF].

2 All these papers use statics frameworks, mostly game-theoretic. To give a quite interesting example which uses the same type of geographic space as we will do, [START_REF] Hutchinson | State enforcement of federal standards: Implications for interstate pollution[END_REF] consider a federation of identical states distributed around a latitudinal circle, each of them occupying an arc of length one. A continuum of identical polluting rms is distributed uniformly along the length of each state, and the mass of rms in each state is normalized to one. Pollution is transboundary as wind is blowing (here from west to east). The authors complete their story of pollution diusion by assuming an ad hoc downwind transfer coecient for emissions per location. In a rst stage, the authors show that, under decentralization, states tend to enforce less stringent environmental standards on rms located close to downwind borders, leading to excessive interstate pollution in equilibrium, which is the standard free riding result. Second, they examine how the interplay between the federal policy on standards and the state policies on enforcement may restore eciency.

In this paper, we depart from the latter abundant environmental federalism literature to dig much deeper in the theoretical foundations and characterization of the fundamental free riding problem highlighted in the associated empirical literature. In particular, instead of assuming ad hoc pollution diusion schemes across space, we start with a realistic spatiotemporal law of motion for pollution, that is a diusion equation (parabolic partial dierential equation) with and without advection. Advection allows to introduce nonhomogeneous diusion across space to account for currents or winds for example.

For simplicity of exposition, on the same line of [START_REF] Hutchinson | State enforcement of federal standards: Implications for interstate pollution[END_REF], we model the space as a circle.

3 The circle is partitioned into several states, which need not be identical, contrarily to [START_REF] Hutchinson | State enforcement of federal standards: Implications for interstate pollution[END_REF]. Each state is run by a 1 In the recent past, water pollution levels have been elevated at political boundaries in China. Using the Hebei province as a case, [START_REF] Duvivier | Transboundary Pollution in China: A Study of Polluting Firms' Location Choices in Hebei Province[END_REF] nd that dirty rms are more likely to locate in counties borders than in the interior. 2 Hutchinson and Kennedy's paper includes an excellent survey of related theory papers. 3 We must clarify here that the ndings of the paper still hold if we model the space as a sphere or a portion of it. This would allow to calibrate the model with spatial pollution data and to perform realistic simulations. Here we keep the one dimensional model since this allows to see and explain more clearly the arising stylized phenomena like the border eect.

a local authority which decides about its investment, production and depollution strategies over time knowing that investment/production generates pollution, and pollution is transboundary.

4 The time horizon is innite. We solve analytically the induced non- cooperative dierential game under decentralization, which is itself a far non-trivial task (see the relation to the technical literature below).

With the closed-form solution paths in hands, we are able to illustrate a series of implications of the model. First of all, we are able to generate the border eect, that's a specic equilibrium behavior near the borders of the states. In particular, we show that net pollution emission ow is increasing as we approach the borders, with strong asymmetries under advection, and that structural breaks show up at the borders. Beside being consistent with the basic theory of spatial externalities, analogous phenomena have been recently disclosed by [START_REF] Lipscomb | Decentralization and Pollution Spillovers: Evidence from the Redrawing of County Borders in Brazil[END_REF] in their empirical study on water pollution in Brazil. Second, we uncover the predictions of our theory regarding the evolution of the size of spatial externalities (or in other terms, the extent of ineciency)

when the number of states (or jurisdictions) rises. In particular, we pose a formal case in which, as the spatial externalities theory suggests, a larger number of jurisdictions goes with the exacerbation of pollution externalities. Third, we extend the analysis of Hutchinson and Kennedy by departing from the identical states assumption: instead, our analysis allows for a large set of discrepancies across states (starting with the size), which may matter in the shape of the border eects. Last but not least, denitely much easier than the decentralized equilibrium dierential game setting, we also characterize the outcomes of the cooperative equilibrium run by, say, a federal government. This is done to permanently outline the distance to eciency of the equilibrium counterpart.

Our setting applies not only to the large set of interesting questions raised by environmental federalism but also to the currently hot debate around supranational coordination of environmental policies. Our theoretical setting is general enough to accommodate the two global levels (federal and supranational). Even more important, as our approach allows for deep geographic discrepancies, it is perfectly suitable to study some of the fundamental questions in the international agenda, in particular those related to the North/South environmental divide and the associated debate on the compensations to be given to the South to reach a global deal. This is out of the scope of the current paper.

Relation to the existing technical literature. Here come a few technical references for the readers to grasp the scope of this paper from this point of view. Indeed, from the methodological point of view, the dierential game problem corresponding to the decentralized equilibrium is rewritten as a (non-zero sum) dierential game in an innite dimensional state space. Only very few papers in the mathematical literature (see, e.g., [START_REF] Nisio | On innite-dimensional stochastic dierential game[END_REF][START_REF] Nisio | On value function of stochastic dierential games in innite dimensions and its application to sensitive control[END_REF][START_REF] Baltas | Robust control of parabolic stochastic partial dierential equations under model uncertainty[END_REF][START_REF] Kocan | On Dierential Games for Innite-Dimensional Systems with Nonlinear, Unbounded Operators[END_REF], deal with such problems which, however, arise in a natural way.

In the economic literature, de Frutos and Martin-Herran (2019a, 2019b) and the draft of de Frutos et al. (2019) are to our knowledge the sole papers studying the strategic implications of transboundary pollution in a spatial model. While the dynamics of the pollution stock is given by a diusion equation like in our setting 5 , the underlying economic model is quite dierent and therefore the economic focus is far from the target of our paper.

In particular, the economic objectives of the research above is not rooted in the theorty of spatial externalities and the literature of environmental federalism.

Moreover, the mathematical approach is radically dierent. In contrast, the economic literature on spatiotemporal dynamics is rather substantial after the seminal contribution of [START_REF] Brito | The dynamics of growth and distribution in a spatially heterogeneous world[END_REF]. In particular, a number of geographic optimal growth models with capital spatiotemporal dynamics have been devised and studied (see [START_REF] Boucekkine | Spatial dynamics and convergence: The spatial AK model[END_REF][START_REF] Fabbri | Geographical structure and convergence: A note on geometry in spatial growth models[END_REF][START_REF] Boucekkine | Growth and agglomeration in the heterogeneous space: a generalized AK approach[END_REF]. Another contribution in the same vein but with constant (though space dependent) saving rates is due to [START_REF] Xepapadeas | Spatial Growth with Exogenous Saving Rates[END_REF]. In all these papers, capital ows across space following a parabolic partial dierential equations. Just like the transboundary problems and for the very same reason, the induced problem is innite-dimensional.

Despite such a complexity, we show here that, thanks to the special structure of the 6 In this very brief review, we abstract away from the huge literature on multi-country dynamic games with a common stock of pollution. See [START_REF] Dockner | International pollution control: cooperative versus noncooperative strategies[END_REF], and [START_REF] Boucekkine | Environmental quality versus economic performance: A dynamic game approach[END_REF] for an earlier contribution.

for the players, and, at the same time, to give a clear economic insight into the properties extracted along the way. The explicit form of the equilibrium also enables us to characterize it comprehensively and to illustrate some key economic ndings readily through complementary numerical exercises. This ultimately shows how eective the machinery of innite dimensional optimal control can be in studying such type of problems. See, for an account of the theory, the books of [START_REF] Li | Optimal Control Theory for Innite Dimensional Systems[END_REF] and [START_REF] Fabbri | Spatial eects and strategic behavior in a multiregional transboundary pollution dynamic game[END_REF].

This paper is organized as follows. Section 2 develops the decentralized equilibrium setting and solves analytically for the Nash equilibrium. Section 3 characterizes the cooperative equilibrium. Section 4 and 5 dig deeper in the concept of border eect alluded to above, combining conceptual and numerical analysis, and nally accounting for a rich variety of inter-state heterogeneities. While the previous sections only consider (pollution) diusion, Section 6 incorporates advection to clearly highlight the implications of non-homogeneous diusion across space. Section 7 concludes. All the proofs are reported in the Appendix together with a full explanation of the mathematical setting.

The non-cooperative game

We consider a dynamic general equilibrium model for a spatial economy subject to spatial spillovers driven by transboundary pollution dynamics.

Even if generalizations are possible 7 we limit our attention to the case of the circular spatial support S 1 :

S 1 := x ∈ R 2 : |x| R 2 = 1
that is the simplest spatial model being compact and without boundary and then having the signicant advantage, in terms of modeling characteristics, of preserving the global stock of pollutants during their diusion processes, without absorbing or reecting boundaries (as in the case of possible models on sub-domains with Dirichlet or Neumann boundary conditions). By doing so, we are also closer to the related economic literature, in particular to the geographical setting of [START_REF] Hutchinson | State enforcement of federal standards: Implications for interstate pollution[END_REF].

As usual, in the following we will often describe S 1 as the segment [0, 2π] with the identication of the two extreme points 0 and 2π. We denote by x the generic spatial point and with t ≥ 0 the continuous time coordinate. As to demographics, we assume the simplest conguration: one individual per location x at any time t, so that aggregate and per capita variables coincide at any location.

At any time t and location x the production of the nal good y(t, x) depends on the quantity of input used in the production i(t, x) according to a linear production function:

(2.1) y(t, x) = A(x) i(t, x), 7 One can for instance replicate most of the results to the case of the 2-dimensional sphere surface S 2 and indeed many of them can also be generalized to more abstract contexts.

where A(x) is the time-independent exogenous productivity at x. 8 The production activity is polluting and a policy of depollution could be advantageously implemented. At any location, output is produced and used for consumption, input and depollution locally. Denoting by c(x, t) and b(x, t) consumption and resources devoted to depollution respectively, we have the following resource constraint equation at any location x and time t:

(2.2) c(x, t) + i(t, x) + b(t, x) = y(t, x).

We further assume that using one unit of input produces one unit of emission ow, while a depollution eort b(t, x) can sequestrate a ow ηb(t, x) θ (with η ≥ 0 and θ ∈ (0, 1)) of pollutants. Consequently, net emissions are given by

n(t, x) = i(t, x) -η(b(t, x)) θ .
The spatio-temporal dynamics of the pollution stock is subject to two natural phenomena: a diusion process which tends to disperse the pollutants across the locations, and a location-specic decay δ(x). All in all the evolution of the pollution stock p(t, x) is driven by the following parabolic partial dierential equation:

(2.3)          ∂p ∂t (t, x) = σ ∂ 2 p ∂x 2 (t, x) -δ(x)p(t, x) + i(t, x) -ηb(t, x) θ , p(0, x) = p 0 (x), x ∈ S 1 ,
being σ > 0 the diusivity coecient measuring the speed of the spatial diusion of the pollutants and p 0 (x) the initial spatial distribution of the pollution. We shall introduce advection in Section 6. 9

Let us come back now to geography. The global territory S 1 is partitioned into a nite number of national states or states /regions within the same country. Each of them is governed by a local public authority (for instance a national, state or regional government)

which only takes into account the welfare of the people living in its own ground.

More formally we suppose that there are N intervals in the circle M j ⊂ S 1 with j = 1, ..., N such that

M j ∩ M h = ∅ for h = j, h, j = 1, ..., N.
8 The production input i(t, x) can be interpreted as a capital good but we do not allow for capital accumulation (or equivalently, we assume full depreciation of capital). This is instrumental to obtaining closed-form solutions while preserving the necessary breadth in the analysis of the environmental free riding problem under scrutiny. 9 We could have considered a space-dependent diustion parameter, σ(x). Our solution method is unaected by such a specication just like the many other space-dependent magnitudes incorporated.

These regions can cover the whole space S 1 but we we also admit the possibility that some residual part of the global territory M 0 := S 1 \ N j=1 M j is inhabited (for instance oceans).

Each local authority is in charge of production, consumption and depollution decisions in its own territory and so the authority j chooses i(t, x), b(t, x), c(t, x) for all t ∈ R + and x ∈ M j subject to production and resource constraints (2.1) and (2.2). In order to emphasize that these decisions only concern the region j we will denote them by i j (t, x), b j (t, x) and c j (t, x) so that their relation with the functions i(t, x) and b(t, x) appearing in (2.3) are indeed

(2.4) i(t, x), b(t, x) := (i j (t, x), b j (t, x)) , if x ∈ M j , 0, if x ∈ M 0 .
We will also denote by A j (x) the restriction of A(x) to M j . The utility of the local public authority j depends, as already mentioned, only on the characteristics of its own territory/population and it takes the following form

(2.5) ∞ 0 e -ρ j t M j c j (t, x) 1-γ j 1 -γ j -w j (x)p(t, x) dx dt,
where ρ j > 0 is the discount factor, γ j ∈ (0, 1) ∪ (1, ∞) the inverse of the elasticity of intertemporal substitution and w j (x) is a measure of the unitary location-specic disutility from pollution. The latter can be roughly interpreted as the measure of environmental awareness at location x in state j. It should be noted that we do not assume that all the inhabitants of territory j share the same environmental awareness, which is somehow more realistic. In contrast, the elasticity of substitution is assumed territory-dependent for simplicity.

10 Finally, each local authority may have a specic view of time discounting, thus the territory-dependent parameter ρ j .

Observe that, even if the expression above only concerns territory M j , it also depend on the choices of other authorities through the variable p(t) because, thanks to the diusion dynamics of pollution, its value at the points x ∈ M j depends on all the past production (and thus pollution) decisions of all other players (authorities). We will make explicit this fact in the notation through the index -j, which stands for all the index but j. Moreover, supposing that A is greater than 1 in all locations 11 , we use (2.1) and (2.2) to express c j (t, x) in terms of i j (t, x) and b j (t, x) as (A j (x) -1)i j (t, x) -b j (t, x) . Finally we can write (2.5) as (2.6)

J (i -j ,b -j ) j p 0 ; (i j , b j )) := ∞ 0 e -ρ j t M j (A j (x) -1)i j (t, x) -b j (t, x) 1-γ j 1 -γ j -w j (x)p(t, x)dx dt.
10 Allowing for parameter γ to depend on location x does not break down the analytical solution neither. 11 This assumption is required by the production function specication (2.1) for the ratio investment to production to be lower than 1 everywhere and at any time.

We now get to formalize how decentralization works in our setting. We suppose that the authorities/players engage in a non-cooperative Nash game. By construction, the latter is a dierential game where each state authority maximizes the spatiotemporal payo (2.6) under the state equation ( 2.3) subject to positivity constraints on i j , b j and c j . See Appendix A for a more detailed formal description of the Nash problem and the corresponding Denitions A.3 and A.4 for open and of Markovian Nash equilibria in the described context.

As repeatedly mentioned above, we are able to solve analytically for the Nash equilibrium involved. This is displayed in the main theorem of our paper here below.

Theorem 2.1. The unique open-loop Nash Equilibrium strategies for the described game is given, for j = 1, ..., N , by

(2.7) b * j (t, x) = [(A j (x) -1)ηθ] 1 1-θ , i * j (t, x) = α j (x) -1 γ j (A j (x) -1) 1-γ j γ j + (ηθ) 1 1-θ (A j (x) -1) θ 1-θ , (2.8)
where α j is the solution to the following ODE12

(2.9)

ρ j α j (x) -σα j (x) + δ(x)α j (x) = w j (x), x ∈ S 1 .
The welfare of player j at the equilibrium is ane in p 0 :

v j (p 0 ) = S 1 α j (x)p 0 (x)dx + q j ,
for a suitable constant q j (see Theorem C.9 in the Appendix for its explicit expression).

The unique open-loop equilibrium described is also a Markovian Nash equilibrium of the game.

Proof. See Theorem C.9 in the Appendix and its proof.

In Section 5 we will discuss in more detail the dependence of the strategies chosen by the players on the parameters of the model, looking at the eect of dierent sorts of spatial heterogeneity. Here we comment briey on the shape of the players' welfare. One can easily see that the welfare of player j is an increasing function of pollution decay rate (or Nature self-cleaning capacity), δ, and a decreasing function of the pollution disutility parameter, w j . These properties are rather intuitive when one looks at the objective functionals (for a given set of strategies, it is clearly true that J has this kind of behavior) but they are a priori not obvious in the context of a Nash equilibrium (see also Remark C.10 for a related comment). Indeed the variation of δ and/or w in one region also impacts the welfare in the others. More precisely, decreasing w j pushes player j to produce and pollute more (see Proposition C.6: observe that i is a decreasing function of α) so that the welfare of players dierent from j decreases when w j decreases (see again Remark C.10).

The eect of δ on the welfare of other players is more complex because it pushes player j

to pollute more but it also pushes for a quicker decay of global pollution.

Since the equilibrium values of i j and b j (and consequently of c j , n j and y j ) are timeindependent, we will avoid from now to write the time variable in their expressions. As a consequence of our main theorem, the equilibrium net pollution ow, the production and the consumption in the region j are given respectively by

(2.10)

n * j (x) = i * j (x) -η(b * j (x)) θ , (2.11) y * j (x) = A(x) i * j (x),

and

(2.12)

c * j (t) = y * j (x) -i * j (x) -b * j (x).
Once we have the set of strategies chosen by the players we can use them in the state equation ( 2.3) to get the corresponding dynamics of the pollution space distribution and getting the following asymptotic result.

Proposition 2.2. In the context described by Theorem 2.1 the spatial pollution density p(t, •) converges to the long-run pollution prole p ∞ which is given by the unique solution of the following elliptic equation:

σp ∞ (x) = δ(x)p ∞ (x) -n * (x).
Proof. See Corollary C.12 in Appendix C.

We shall use the expressions given above in our numerical exercises from Section 4 in which we will focus on the spatial distributions of the relevant variable (long-term distribution in the case of pollution). Before, we characterize the cooperative equilibirum, which will serve as a reference for these exercises in order, as we wrote in the introduction, to measure the distance to eciency of the Nash equilibrium outcomes.

The cooperative game

To design the cooperative solution, we suppose that all the players cooperate to maximize a social welfare function dened as the sum of the utility of all the states/territories of S 1 :

(3.1) N j=1 ∞ 0 M j e -ρ j t (A j (x) -1)i j (t, x) -b j (t, x) 1-γ j 1 -γ j -w j (x)p(t, x) dx dt.
We limit our attention to the case where the preference parameters are the same for all players: ρ j = ρ and γ j = γ for every j = 1, ..., N . In this case the functional (3.1) can be rewritten as

(3.2) ∞ 0 e -ρt S 1 (A(x) -1)i(t, x) -b(t, x) 1-γ 1 -γ -w(x)p(t, x) dx dt
where w is dened similarly to (2.4). Indeed, it is the standard Benthamite social functional since we suppose that at each location there is exactly one inhabitant.

The optimal control problem of maximizing (3.2) subject to (2.3) and the positivity constraints on i, b and c can be explicitly solved as described in the following theorem.

Theorem 3.1. The unique equilibrium for the described cooperation game is given by

(3.3) b * (t, x) = (A(x) -1)ηθ 1 1-θ , i * (t, x) = α(x) -1 γ (A(x) -1) θ 1-θ (3.4)
where α is the solution to

(3.5) ρα(x) -σα (x) + δ(x)α(x) = w(x), x ∈ S 1 .
The corresponding welfare is v(p 0 ) := S 1 α(x)p 0 (x)dx + q, for a suitable constant q (see Theorem E.1 in the Appendix).

Proof. See Theorem E.1 and Corollary E.2 in the Appendix and its proof.

As for the Nash equilibrium case, since at the equilibrium values i * , b * , n * , c * and y * are time-independent we will avoid to write the time variable in their expressions. A counterpart of the asymptotic result given in Proposition 2.2 is given, in the cooperative context, by the following proposition. Proposition 3.2. In the context described by Theorem 3.1 the spatial pollution density p(t, •) converges to the long-run pollution prole p ∞ which is given by the unique solution of the following elliptic equation:

σp (x) = δ(x)p(x) -n * (x),
where n * (x) = i * (x) -η(b * (x)) θ (and b * and i * are dened in Theorem 3.1).

Proof. See Appendix D.

Not surprisingly, the Nash equilibrium described in Theorem 2.1 is markedly dierent from the social optimum described above and it is suboptimal in terms of the chosen social target. The suboptimality is of course driven by the spatial pollution externality: the local authority does not completely internalize the damages of the emissions produced in its territory, part of it moves away from its territory and then it does not aect its utility but the utility of other territories, especially the closest. Interestingly enough, our comprehensive modelling of pollution diusion allows for an accurate appraisal of this externality. A crucial parameter in this respect is parameter σ: when σ drops, the diusion is slower. When σ is equal to 0, the spatial dynamics vanish. In this case the model is spatially degenerate in the sense that there is no interaction among the economies of the various locations and the dynamics of the model (both in the non-cooperative and in the cooperative case) reduces to the pointwise maximization of the functional

∞ 0 e -ρt (A(x) -1)i(t, x) -b(t, x) 1-γ 1 -γ -w(x)p(t, x) dt for any xed x ∈ S 1 subject to ∂p ∂t (t, x) = -δ(x)p(t, x) + i(t, x) -ηb(t, x) θ .
The solution of this maximization problem is given by the same value of b given in (2.7)

and i(x) = w(x) ρ + δ(x) -1 γ (A(x) -1) 1-γ γ + (ηθ) 1 1-θ (A(x) -1) θ 1-θ .
Since the pollution remains where it is produced, no spatially externality arises in the case σ = 0, hence, in such case both the cooperative equilibrium and the non-cooperative one coincide. Moreover, the nal result of Theorem 3.1 tells us more, i.e. that the no-diusion non-cooperative case is conceptually close to the cooperative case with any diusion (Theorem 3.1). Indeed, it can be shown that the two coincide in the particular case where δ and w are constant, in terms of chosen i, b and c. This intriguing property is disclosed in the proposition below.

Proposition 3.3. Suppose that δ and w are constant in space. Then the strategies b * j,0 (t, x) and i * j,0 (t, x) of player j in the Nash equilibrium described in Theorem 2.1 when σ = 0 are the same as the equilibrium strategies b * j (t, x) and i * j (t, x) of the cooperative case described in Theorem 3.1, or every σ ≥ 0.

Proof. See Corollary E.2 in the Appendix.

Border Effects

We now perform a series of numerical exercises to uncover the main features of the long-term equilibrium spatial distributions. This will allow us to visualize at glance the shape of the border eects associated with the free riding problem under decentralization.

Essentially, we will make clear that the shapes generated are quite consistent with the predictions of the basic (static) theory of spatial externalities, particularly in what concerns the specic equilibrium behaviour near the borders of the states: net pollution ows will be shown to be larger as the borders are approached, and structural breaks emerge at the borders. In this section, we consider that all the states are geographically identical in terms of various parameters of the model, except possibly for the size while the next section will dig deeper in the implications of other spatial heterogeneities.

We shall report at the same time the induced distributions for the cooperative game, which will make the border eects even more striking. We start by showing the situation of two territories while, in a second step, we theoretically address another implication of the basic theory of spatial externalities, that is the evolution of the size of spatial externalities (or in other terms, the extent of ineciency) when the number of states (or jurisdictions) rises.

4.1. The shape of border eects. We concentrate here our attention on cases where all the parameters are constant in space. Accordingly, the unique geographic discrepancies result from the partition in decentralized territories (in the Nash case), that is in the existence of borders and possible dierences in territory size.

We will compare the Nash equilibrium described in Theorem 2.1 with the benchmark described in Section 3. In particular, since, as announced above, δ and w are constant in space, we are always in the context of Proposition 3.3 so that the cooperative benchmark with diusion of Theorem 3.1 and the non-diusion benchmark are always equivalent. To lighten the notation we will omit the use of the indexes j, this is reasonable because we will mainly look at the aggregate eect of the choices of the agents. 4.1.1. Symmetric two-region cases. We start with the case where territories have equal size. In the common economic language, this amounts to studying the symmetric tworegion case. The case of two players each controlling one half of the circle. All the parameters are space independent: the productivity factor A is 1.6, the discount ρ is 0.03, the diusivity σ is 0.5, the natural decay of the pollution δ is 0.2, the weight of pollution in the utility parameter w is 1, the intertemporal substitution parameter γ is 0.5, the eciency of depollution expenditure factor φ is 0.2 while the scale returns of depollution activity θ is 0.4. The continuous (and blue) lines represent the Nash equilibrium while the dashed (and gray) lines reproduce the cooperative benchmark or, equivalently, the non-diusion benchmark.

In Figure 4.1 each local authority controls one half of the circle and then it is interested only in utility of its region (parameters' values are given in the caption of the gure). The continuous (and blu) lines represent the values of the variables in various locations at the Nash equilibrium while the dashed (and gray) lines reproduce the cooperative benchmark or equivalently the non-diusion benchmark. We represent four variables: the investment i, the net pollution n, the production y and the long-run pollution prole p ∞ . The latter is obtained in particular thanks to the representation of the solution in series given in Appendix D. We do not represent in the gure the depollution eort at the equilibrium since, given the particularly simple situation (all the parameters are constant in space), it is constant over space. For this reason the qualitative behavior of net emissions is the same as that of investment. It is also the same as production since the productivity A is space independent.

The rst evident element in the distribution of investment (and then in those of production and net pollution) are the big dierences among the locations of a same region. In particular we can observe that investment and economic activity are particularly strong near the borders of each region. This border eect is due to the spatial structure of the externalities: the negative eects of the emissions on the utility are less and less internalized by the local authority as the location gets closer to the border. That's because a greater part of the pollutants in these locations will ow into another territory and has therefore to be managed by another local authority. In the symmetric two-region case with positive diusivity that we have here, the emission at the boundary points are immediately equally shared by the two territories while the pollutants coming from a far interior point remains in the short run mostly in their native region. The source of the externality ineciency can be well visualized looking at the long-run distribution of the pollution: indeed the concentration of pollution at the boundary is much less pronounced than the corresponding peak of input because a signicant part of the pollutants leaves the locations where they are originally produced. when we vary the diusivity coecient σ that is when we change the speed of the diusion of emissions. We have again the same symmetric two-countries situation as in Figure 4.1 and we represent with a colored continuous (respectively, gray dashed) line the variables at the Nash equilibrium for various value of the parameter σ (respectively, at the benchmark).

We choose four possible values for σ: 0.4 (the lightest line), 0.8, 1.6, and 3.2 (the darkest line), all other parameters are the same as in Figure 4.1 so the values of the variable at the benchmark is the same.

Not surprising, the lower σ, the more the system tends to mimic the behaviour of the 0-diusion benchmark. Conversely, the higher the value of σ and the faster pollutions disseminates across locations: for σ very big the situation in each location is similar to the situation we have at the boundary since after a short period the produced pollution has an almost equal probability of being in the territory of both authorities. In other words the higher the value of σ, the lesser future negative eects on the utility are internalized by the local authority and then the higher the chosen level of input, production and emission in internal points. This mechanism highlights the intertemporal role of the parameter σ that will be emphasized even more in the following.

The two limits of the equilibrium prole of i * j when σ → 0 + and σ → +∞ can be computed explicitly and they are (see Proposition C.5 in the Appendix), for the general case specied in Theorem 2.1,

(4.1) i * ,0 j (x) = w j (x) ρ j + δ(x) -1 γ j (A j (x) -1) 1-γ j γ j + (ηθ) 1 1-θ (A j (x) -1) θ 1-θ , and (4.2) i * ,∞ j (x) = S 1 w j (x)dx S 1 (ρ j + δ(x))dx -1 γ j (A j (x) -1) 1-γ j γ j + (ηθ) 1 1-θ (A j (x) -1) θ 1-θ .
Notice that the latter expression depends on the space location x only through A(x).

In the case of 2 symmetric agents and spatial constants parameters, one gets the two following spatial-independent expressions (4.3)

i 0 = w ρ + δ -1 γ (A -1) 1-γ γ + (ηθ) 1 1-θ (A -1) θ 1-θ , and (4.4) i ∞ = 1 2 w ρ + δ -1 γ (A -1) 1-γ γ + (ηθ) 1 1-θ (A -1) θ 1-θ .
One can readily check that consistently with the intepretations given above, i ∞ > i 0 .

4.1.2. Non-symmetric two-region cases. We now move to asymmetric two-region cases. In Figure 4.3 we represent a situation where parameters are not space dependent and then spatial heterogeneities of agents' behavior is only due to borders and dierent importance that various agents attribute to the utility of people living in dierent locations. All the parameters are the same as in Figure 4.1 but here the dimensions of the regions governed by the two authorities are dierent: the rst controls three fourth of the circle while the second only governs on a fourth of the space. The eect of the new division of the territory with respect to the situation of Figure 4.1 is neat: the authority with the larger territory internalizes more the eect of its emissions because it anticipates that it will have them back to its part of the circle in the future. For the very opposite reason, the authority which has a fourth of the circle is less aected by its own emission and then produces and pollutes more than in Figure 4.1 (and a fortiori than the player controlling three fourth of the circle). A major implication of our circular" and geographically homogenous world is the bigger the country, the cleaner it is. This is a quite interesting result. Of course, by construction, if geographic heterogeneity is added (in technology or ecology for example), the latter result might be reversed. But it is important to visualize the benchmark result with only dierences in size and no further geographic discrepancy. Remark 4.1. The strength and the structure of the border eects we mentioned also depend on the model of the spatial structure we used. A structure as S 1 allows us to consider the global eects of pollution in the sense that, as already pointed out, the agents (especially those that govern larger areas) are led to consider the fact that a part of the emissions that leave the territory, contribute to increase a spatial stock of pollution that in a certain amount will return to the country in the future. Given the compactness of the support it can also happen that the pollution crosses the entire territory of the other player before returning. In other types of models the situation is dierent. For example if one considers the spatial model of the segment with absorbing borders the pollution that leaves the area controlled by the player will no longer return to it, if one considers a model of innite space the eect of compactness disappears and therefore the dierence of behaviors between the player controlling a large territory and the one controlling a small territory is reduced.

4.2. Number of players and the size of externalities. Exactly the same kind of behavior we have seen in the last numerical exercises arises from increasing the number of players: when several players with a per-capita small territory interact, each of them internalizes only a modest portion of the damages of her emission and then she increases input, production and pollution with respect to the two player case. In this way the total amount of pollution tends to increase with the number of players. We devote to this nding a deeper theoretical foundation here.

We consider again the case where the coecients of the problem are homogeneous in space. Among the sets of territories congurations described in Section 2 (see also

Appendix A), we introduce the following partial order relation: given two congurations

Π 1 = {M 1 1 , ..., M 1 N }, Π 2 = {M 2 1 , ..., M 2 K } we say that Π 1 Π 2 if ∀j = 1, ..., N, ∃i = 1, ...K : M 1 j ⊆ M 2 i .
This means that the territories conguration Π 1 is a fragmentation of the territories conguration Π 2 .

Proposition 4.2. Let Π 1 , Π 2 be two territories' congurations such that Π 1 Π 2 and let p 1, * , p 2, * the associated optimal pollution paths. Then p 1, * (t, x) ≥ p 2, * (t, x) for all (t, x) ∈

R + × S 1 . Proof. See Appendix C.
The result displayed by the proposition is quite sharp: when decisions in terms of investment, production and emissions of a certain territory move from a central entity to smaller sub-entities, pollution levels systematically increase in line with the dierent degree of internalization of externalities that we expect from the two players.

Again, this broadens the counterpart property in the basic static theory of spatial externalities. It is also (somehow by construction) consistent with the intuitions one can gain from the numerical exploration above. Note that the result is obtained under the assumption of geographic homogeneity. The next section is designed to highlight some the implications of adding geographic discrepancies into the theory, a feature not considered generally in the standard theory (see for example, [START_REF] Hutchinson | State enforcement of federal standards: Implications for interstate pollution[END_REF]. Our analytical approach allows for these discrepancies as our closed-form solutions do encompass them.

Geography and Heterogeneity

We look now at what happens when we introduce natural, technological or preference dierences among the regions. We stick to the symmetric two-region case since the relevant mechanisms are already at work there. To be able to disentangle various eects we will look at the eect of one parameter each time. We start with technology. 5.1. Geographic discrepancy in technology. In Figure 5.1 we consider the situation when in the economy we have two regions with dierent technological levels, that's withierent productivities. As we can see from (2.8), the eect on investment of varying A depends on the value of other parameters (in particular on γ, see Remark C.11 in the Appendix for technical details). Indeed a variation of A produces a typical incomesubstitution trade-o: on the one hand increasing A makes investment more productive leading to a higher increase in investment relative to consumption and depollution eort.

On the other hand, a higher level of A can guarantee a higher level of production and thus of consumption together with lower investment and subsequently lower emissions. The predominance of one of the two channels depends on the values of various parameters. In Figure 5.1 the rst eect is stronger. Conversely, as one can see from the expression of b j at the equilibrium given in (2.7), the impact of increasing A on the depollution expenditures is always positive: increasing output always gives more room for pro-environmental actions.

The eects on net pollution depends on the relative strength of the mechanisms described above. For our selected parametrization, the outcomes are represented in Figure The intuition is rather straightforward: an higher value of δ reduces, for a given investment strategy, the future stock ofpollution and then it reduces the marginal disutility of polluting with respect to marginal utility of consumption. So it tends to increase input use and thus production and consumption. Conversely, as it transpires from the expression of the optimal level of depollution expenditures (2.7), the latter are not impacted by variations on δ so dierences in input use drive mechanically the dierences in net emissions ow and production.

We can also note that δ enters in the expression of equilibrium investment (2.8) only through the values of α dened as the solution of (2.9). In this equation δ sounds as a substitute to ρ. This is not so surprising because both parameters have intertemporal implications: they act to discount the future eects of present actions . Nevertheless here substitutability is particularly large (1 to 1 at any time) because, dierently from standard growth model, the decay δ directly acts on a variable (pollution) that linearly appears in the utility function. So all the previous remarks on the eects of δ on various endogenous variables, can be replicated exactly for ρ.

Dierently from what we had for instance in the examples of Section 4 we can observe that in Figure 5.2 the two benchmarks we use (the cooperative case with the same σ and the no-diusion situation) give distinct long-term proles. The same behavior will appear in Figure 5.3. Both benchmarks are associated with less production and emissions than the Nash equilibrium. This is not surprising because in both cases the negative eects of pollution are completely internalized. The reason why the choice of the planner is to pollute more (compared with the zero-diusion benchmark) in the low-delta zone is that she knows that part of the emission will move to high-delta part and it will decay quickly. This mechanism also explains why the dierence between the two is higher, the closer to the border. A symmetric argument is enough to gure out why the investment-choice of the planner tends to be lower relative to the zero-diusion benchmark in the high-delta zone.

5.3. Geographic discrepancy in preferences. In Figure 5.3 we represent again a symmetric two-country example with all the parameters space-independent except the unitary disutility of pollution w which is 0.9 in the region controlled by the rst player and 1.1 for the other. As one can infer from Proposition C.6, pollution decreases when w is bigger: a larger marginal disutility from pollution leads the local planner to reduce investment and consumption to be able to reduce emissions. This fact also involves a reduction in production and in the net ow of emissions.

Figure 5.3. The case of two players controlling one half of the circle with dierent unitary disutility of the pollution w. The value of w in the region controlled by the rst player (on the left) is w 1 (constant it her territory) equal to 0.9 while the value of w 2 is 1.1. All other parameters (are constant over the space and) are the same as in Figure 4.1. Continuous (and colored) lines represent the Nash equilibrium, the dotted (and gray) lines are the cooperative benchmark (with the same σ) while the dashed (gray) lines are the zero-diusion benchmark.

Advection

So far we have considered, for the natural spatiotemporal dynamics of pollution, a completely homogeneous diusion/spreading process. Indeed, if we abstract away from the agents' decisions, the dynamics described by (2.3) reduces to

∂p ∂t (t, x) = σ ∂ 2 p ∂x 2 (t, x) -δ(x)p(t, x).
In the right side of this expression only the second derivative term describes the spatial dynamics of pollutants while the decay term is, essentially, purely local.

A more general formulation is possible by adding to the right hand side above a term representing exogenous location-specic ow, i.e.

v(x) ∂p ∂x (t, x).

This new term is a vector eld on S 1 specifying at each location a further movement term (which adds to the already described diusion term) having speed v(x) at any point x. It is called advection term and it allows to take into account the fact that basic dispersion of pollutants need not to be space-homogeneous, for instance due to winds, currents or geographic characteristics. With this new term the evolution equation 2.3 for the pollution stock becomes (6.1)

         ∂p ∂t (t, x) = σ ∂ 2 p ∂x 2 + v(x) ∂p ∂x (t, x) -δ(x)p(t, x) + i(t, x) -ηb(t, x) θ , (t, x) ∈ R + × S 1 , p(0, x) = p 0 (x), x ∈ S 1 .
All the results provided in the previous sections can be generalized to the system including a generic advection term. In particular the counterpart of the Nash equilibrium described in Theorem 2.1 reads now as

(6.2) b ad, * j (t, x) = [(A j (x) -1)ηθ] 1 1-θ , i ad, * j (t, x) = α j (x) -1 γ j (A j (x) -1) 1-γ j γ j + (ηθ) 1 1-θ (A j (x) -1) θ 1-θ , (6.3)
where α ad j is the unique solution to the ODE (6.4)

ρ j α j (x) -σα j (x) -v(x)α j (x) + δ(x)α j (x) = w j (x), x ∈ S 1 .
This result is proved in Theorem C.9 in the Appendix.

Figure 6.1. The case of two players controlling one half of the circle. All the setting and the parameters values are the same as in Figure 4.1 but now a spatially constant advection term v(x) = 0.08 is introduced. The continuous (and blue) lines represent the Nash equilibrium while the dashed (and gray) lines are the cooperative benchmark or, equivalently, the non-diusion benchmark.

To understand how the results can be qualitatively aected by the presence of the advection term we see, in Figure 6.1 how the spatial proles reported in Figure 4.1 are altered when a constant advection term with v(x) = v > 0 is introduced (all other specications and parameterization remaining the same). This, concretely, means introducing a persistent current which is clockwise when we look at the S 1 as a subset of R 2 , that is, roughly speaking a current directed to the right in the gure. As a consequence, pollutants tend to move quicker to the other region if they are produced in a location which is close to the right borders (say est ) and vice versa. For this reason the authority internalizes even less the disutility due to emissions coming from eastern locations and so she has un incentive to produce and pollute more there. A symmetric argument explains why the production and the emission in the western locations are lower than in the no-advection case of Figure 4.1.

Conclusion

We have proposed to revisit the foundations of the spatial externalities theory in the case of the free-riding pollution problem, so heavily referred to in the most recent literature on environmental federalism (see for example [START_REF] Hutchinson | State enforcement of federal standards: Implications for interstate pollution[END_REF] but also in several recent much more applied works on transboundary pollution (see Lipscomb and Mobarak, 2017, as a representative example). In particular, precisely to close the gap between the theory and the latter empirical works, we consider a spatiotemporal framework where, instead of assuming ad hoc pollution diusion schemes across space, we use a realistic spatiotemporal law of motion for air and water pollution (diusion and advection). This has led us to what we believe to be a strong methodological innovation as we have ultimately to tackle and to solve spatiotemporal non-cooperative (and cooperative) dierential games, which is far more complicated than the counterpart static games in the benchmark theory. We also incorporate into the analysis a large set of discrepancies across state and jurisdictions, which broadens even more the scope of our theory and its practical interest. Also it's worth pointing out that since we solve for both the decentralized noncooperative and the cooperative equilibrium, our setting can be also used for policy investigations. As outlined in the introduction, a sizeable bunch of interesting questions traditionally raised in the environmental federalism literature can be addressed. Even more interestingly, our setting is general enough to accommodate the two global levels (federal and supranational). Furthermore, since our analytical method allows for deep geographic discrepancies, this in principle enable us to address some of the hot questions in the international agenda, in particular those related to the North/South environmental divide. This is the next step in our project.

Last but not least, because our framework enables us to account for several key spatial heteorgeneities without endangering the availability of closed-form solutions, it can easily accommodate applied work on real multi-regional or multi-country data. This is especially granted because, as we have mentioned already in the introduction, our method works on more realistic spatial sets like spheres.

denote by ψ t and ψ x , respectively, the derivative with respect to the rst (time) and the second (space) variable.

We consider the following multiagent problem in S 1 . We x a positive integer number N ≥ 1 representing the number of players. Each player j = 1, ..., N is endowed with her/his own part of territory M j ⊂ S 1 . Hence, M j ∩ M h = ∅ for h = j, h, j = 1, ..., N. Notice that there may be parts of S 1 which do not belong to any player. We assume that M j is connected and relatively open (hence an open interval in the circle) for every j = 1, . . . , N . set

M 0 := S 1 \ N j=1 M j .
Note that M 0 is closed and contains at least the boundary points of M j , for j = 1, . . . , N . From now on, when we take an index j without mentioning explicitly where it lies, we mean that j ∈ {1, . . . , N }.

Player j decides the investment policy i j (t, x) and the depollution policy b j (t, x) at time t ∈ R + and location x ∈ M j . Let p 0 , δ, v : S 1 → R, be given measurable functions and let σ > 0, θ ∈ (0, 1), η ≥ 0 be given constants.

The evolution of the state variable p(t, x), representing pollution, is formally given by the following parabolic PDE(13 ) (A.1) The payo functional of player j is

     p t (t, x) = σp xx (t, x) + v(x)p x (t, x) -δ(x)p(t, x) + i(t, x) -ηb(t, x) θ , (t, x) ∈ R + × S 1 , p(0, x) = p 0 (x), x ∈ S 1 , where (A.2) i(t, x), b(t, x) := (i j (t, x), b j (t, x)) , if x ∈ M j , j = 1, . . . , N 0, if x ∈ M 0 = S 1 \ N J=1 M j . Given j =
J (i-j ,b-j ) j p 0 ; (i j , b j )) := ∞ 0 e -ρj t Mj (A j (x) -1)i j (t, x) -b j (t, x) 1-γj 1 -γ j -w j (x)p(t, x) dx dt, (A.3)
where ρ j > 0, γ j ∈ (0, 1) ∪ (1, ∞), and A j , w j : M j → R + , are given measurable functions. The following assumptions will be standing in the remainder of this Appendix.14 . Assumption A.1.

:= S 1 |f (x)| 2 dx, f ∈ H, |f | 2 Hj := Mj |f (x)| 2 dx, f ∈ H j .
Finally denote by H + , H + j the positive cones of H and H j , respectively. Now, for every f ∈ H we can write

(B.1) f (x) := N j=0 f (x)1 Mj (x), x ∈ S 1 .
Since the restriction of f 1 Mj to H j is an element of H j we write, again with a slight abuse of notation

H = N j=0 H j , H + = N j=0 H + j .
B.2. The operators. Denote by L(H) the space of bounded linear operators on H. Consider the dierential operator (L, D(L)) in H, where

D(L) = W 2,2 (S 1 ; R); [Lϕ](x) = σϕ (x) + v(x)ϕ (x) -δ(x)ϕ(x), ϕ ∈ D(L).
The latter is a closed, densely dened, unbounded linear operator on the space H (see, e.g. Lunardi, 1995, page 72, Section 3.1.1). A core for it is the space C ∞ (S 1 ; R) (see, e.g., Engel and Nagel, 1995, pages 69-70). Integration by parts shows that

(B.2) Lϕ, ψ H = ϕ, L * ψ H , ∀ϕ, ψ ∈ C ∞ (S 1 ; R)
where D(L * ) = D(L) = W 2,2 (S 1 ; R) and

(B.3) [L * ψ](x) = σψ (x) -v(x)ψ (x) -(v (x) + δ(x))ψ(x), ψ ∈ C ∞ (S 1 ; R).
Since C ∞ (S 1 ; R) is a core for L and L * , (B.3) extends to all couples of functions in D(L). This shows that L is self-adjoint and dissipative if v ≡ 0.

Integration by parts also shows (B.4)

S 1 v(x)ϕ(x)ϕ (x)dx = - S 1 v(x)ϕ(x)ϕ (x)dx - S 1 v (x)|ϕ(x)| 2 dx, hence, (B.5) S 1 v(x)ϕ(x)ϕ (x)dx = - 1 2 S 1 v (x)|ϕ(x)| 2 dx
Next, we compute, again using integration by parts (in particular, (B.4) and (B.5)),

Lϕ, ϕ H = S 1 ([Lϕ](x)) ϕ(x)dx = - S 1 σ(x)|ϕ (x)| 2 dx - S 1 v(x)ϕ(x)ϕ (x)dx - S 1 (v (x) + δ(x))|ϕ(x)| 2 dx = - S 1 σ(x)|ϕ (x)| 2 dx - S 1 1 2 v (x) + δ(x) |ϕ(x)| 2 dx ≤ 1 2 v + δ ∧ 0 ∞ |ϕ| 2 H .
Then, we can rewrite (B.8) as (B.9)

∞ 0 e -ρj t (A j -1)I j (t) -B j (t)

1-γj 1 -γ j , 1
w j (x) := w j (x), if x ∈ M j , 0, if x / ∈ M j ,
we have The reformulated problem for the agent j consists then in maximizing the functional J j in (B.12), over the set A j and under the state equation (B.6). Note that, in this reformulation, the rst term of the functional J j is the only one which depends on the initial datum. such that, for every j = 1, ..., N and for all p 0 ∈ L 2 (S 1 , R + ),

J (I * -j ,B * -j ) j p 0 ; (I * j , B * j ) ≥ J (I * -j ,B * -j ) j p 0 ; (I j , B j ) , ∀(I j , B j ) ∈ A j .
C.1. The functions α j and their properties. By Assumption B.1, we see that ρ j belongs to the resolvent set of L for every j = 1, ..., N (see [START_REF] Engel | One-parameter Semigroups for Linear Evolution Equations[END_REF]. Hence, the operator

ρ j -L : D(L) -→ H is invertible with bounded inverse (ρ j -L) -1 : H → H and (C.1) (ρ j -L) -1 h = ∞ 0 e -(ρj -L)t h dt ∀h ∈ H.
We dene (C.2)

α j := (ρ j -L) -1 w j ∈ D(L) = W 2,2 (S 1 ; R).
By denition α j is therefore the unique solution in

D(L) = W 2,2 (S 1 ; R) of the abstract ODE (C.3) (ρ j -L) α j = w j .
More explicitly, α j , as dened in (C.2), is the unique solution in the class W 2,2 (S 1 ; R) to

(C.4) ρ j α j (x) -σα j (x) -v(x)α j (x) + δ(x)α j (x) = w j (x), x ∈ S 1 ,
meaning that it veries (C.4) pointwise almost everywhere in S 1 (this will be, from now on, the meaning of solution to such equation). 15 By Sobolev embedding W 2,2 (S 1 ; R) ⊂ C 1 (S 1 ; R), so α j ∈ C 1 (S 1 ; R).

We state an a priori estimate for the solution of the equation (λ -L)g = f , λ > 0, when f has a suitable regularity.

Proposition C.2. Let M ⊂ S 1 be open, nonempty, and connected. Let f ∈ L 2 (S 1 ; R) be such that f | M and f | S 1 \M are continuous and f > 0 on M and f = 0 on S 1 \ M . Finally, let g ∈ W 2,2 (S 1 ; R) be such that (λ -L)g = f . Then there exists κ > 0 such that

κ ≤ g ≤ max M f.
Proof. First we notice that, since g ∈ W 2,2 (S 1 ; R) ⊂ C 1 (S 1 ; R) and S 1 is compact, the function g admits maximum and minimum over S 1 . Estimate from below. We identify M with an open interval (a, b), so {a, b} = ∂M . The fact that (λ -L)g = f and the assumption σ > 0 yield

g (x) = 1 σ [(λ + δ(x))g(x) -v(x)g (x) -f (x)] , for a.e. x ∈ S 1 . Since g ∈ C 1 (S 1 ; R), it follows that g ∈ C 2 (S 1 \ ∂M ; R) and (C.5) g (x) = 1 σ [(λ + δ(x))g(x) -v(x)g (x) -f (x)] , ∀x ∈ S 1 \ ∂M.
Then, From (C.5) and Assumption A.1, we see that there exist nite g (a + ) := lim x→a + g (x) and g (b -) := lim x→b -g (x) and their value is

(C.6) g (a + ) = 1 σ [(λ + δ(a))g(a) -v(a)g (a) -f (a)] , g (b -) = 1 σ [(λ + δ(b))g(b) -v(b)g (b) -f (b)] ,
15 The latter ODE can be also viewed as on ODE on the interval [0, 2π] with periodic boundary conditions:

     ρ j α j (x) -σα j (x) -v(x)α j (x) + δ(x)α j (x) = w j (x), x ∈ (0, 2π),
α j (0) = α j (2π), α j (0) = α j (2π), falling into the Sturm-Liouville theory with periodic boundary conditions (see Coddington and Levinson, 2013). Let x * ∈ S 1 be a minimum point of g over S 1 and set κ := g(x * ). Clearly, since g ∈ C 1 (S 1 , R) it must be g (x * ) = 0. We distinguish three cases.

Case 1: x * ∈ M . We have g (x * ) = 0 and g (x * ) ≥ 0. Plugging this into (C.5) we get

(λ + δ(x * ))κ = σg (x * ) + f (x * ) > 0,
hence, we conclude κ > 0.

Case 2: x * ∈ {a, b}. Assume, without loss of generality, that x * = a. One has g (a) = 0 and α j (a + ) ≥ 0. Plugging this into (C.6) we get

(C.7) 0 ≤ g (a + ) = 1 σ [(λ + δ(a))κ -f (a)] ,
Since f (a) > 0, we get κ > 0.

Case 3: x * ∈ S 1 \ M . In this case, as f (x * ) = 0, arguing as before we get

(λ + δ(x * ))κ = σg (x * ) ≥ 0,
hence κ ≥ 0. If κ > 0, we have concluded. If κ = 0, then the fact that g(x * ) = g j (x * ) = 0 and (C.5) yield g ≡ 0 on S 1 \ M . By continuity of g, g , we have also g(a) = g (a) = 0. Moreover, it must be g ≥ κ = 0 on M , as x * is a minimum point over S 1 . Hence, it must be g (a + ) ≥ 0. Hence we agein get (C.7), a contradiction if k = 0 as f (a) > 0 by assumption.

Estimate from above. This part follows by arguments similar to the ones used for the estimate from below.

Proposition C.2 immediatly yields the following two corollaries.

Corollary Corollary C.4. Let Assumption A.1 hold. Let α j ∈ W 2,2 (S 1 ; R) ⊂ C 1 (S 1 ; R) be the solution to (C.4). There exists κ j > 0 such that, for every j = 1, ..., N , we have

C.3. Let λ > 0. The operator (λ -L) -1 : H → D(L) ⊂ H is poitivity preserving, i.e. f ∈ H, f ≥ 0 a.e. =⇒ (λ -L) -1 f ≥ 0 a.e.. Proof
κ j ≤ α j ≤ K j ,
where K j := max Mj 1 ρj w j .

Proof. Due to Assumption A.1, this is a direct application of Proposition C.2.

In the next results, we investigate the dependence of α j on the data. We start with a convergence result on the diusion coecient σ.

Proposition C.5. Let Assumption A.1 hold. Denote by α j,σ the unique solution to (C.4) when v(•) ≡ 0. We have

lim σ→0 + α j,σ (x) = w j (x) ρ j + δ(x) , lim σ→+∞ α j,σ (x) = S 1 w j (x)dx S 1 (ρ j + δ(x))dx , ∀x ∈ S 1 .
Proof. Case σ → 0 + . First, notice that under our assumptions, (C.4) reads as

(C.8) ρ j α j,σ (x) -σα j,σ (x) + δ(x)α j,σ (x) = w j (x), x ∈ S 1 , By Proposition C.4 we have (α j ) * (x) := lim inf σ→0 + α j,σ (z) : σ ≤ σ, z ∈ S 1 , |z -x| ≤ 1/σ ≥ 0, (α j ) * (x) := lim sup σ→0 + α j,σ (z) : σ ≤ σ, z ∈ S 1 , |z -x| ≤ 1/σ ≤ K.
Clearly (α j ) * ≤ (α j ) * . By stability of viscosity solutions (see e.g. Crandal-Ishii-Lions, 1992), the latter functions are, respectively, (viscosity) super-and sub-solution to the limit equation

ρ j α j,0 (x) + δ(x)α j,0 (x) = w j (x)
whose unique solution is

α j,0 (x) = w j (x) ρ + δ(x)
.

By standard comparison of viscosity solutions one has (α j ) * ≥ α j,0 ≥ (α j ) * . It follows that

∃ lim σ→0 + α j,σ (x) = (α j ) * (x) = (α j ) * (x) = α j,0 (x) ∀x ∈ S 1 .
Case σ → +∞. First, we rewrite (C.8) as (C.9)

α j (x) = 1 σ [ρ j α j (x) + δ(x)α j (x) -w j (x)] , x ∈ S 1 ,
Notice now that α j,σ is equi-bounded and equi-uniformly continuous withe respect to σ ≥ 1. Hence, by Ascoli-Arzelà Theorem we have that, from each sequence σ n → +∞ we can extract a subsequence σ n k such that lim k→+∞ α j,σn k = α j,∞ uniformly on x ∈ S 1 , for some α j,∞ ∈ C(S 1 ; R). Again by stability viscosity solutions we see that α j,∞ must solve the limit equation α j,∞ (x) = 0, x ∈ S 1 , hence, it must be α j,∞ ≡ c 0 for some c 0 ≥ 0. to nd the value of c 0 we may integrate (C.8) over S 1 getting

S 1 (ρ j + δ(x))α j,σ (x)dx = S 1 w j (x)dx.
Letting σ → +∞ above we get

c 0 = S 1 w j (x)dx S 1 (ρ j + δ(x))
.

As this value does not depend on the sequence σ n chosen, the claim follows.

Proposition C.6. Let Assumption A.1 hold and let α ρj ,δ(•), wj (•) j be the unique solution to (C.4) for given ρ j , δ(•), w j (•). Then α j is nonincreasing with respect to space homogeneous increments of ρ j +δ(•) meaning that

h, k ∈ R, h + k ≥ 0 ⇐⇒ α ρj +h,δ(•)+k,wj (•) j (x) ≤ α ρj ,δ(•)wj (•) j (x) ∀x ∈ S 1 .
Moreover, with the same meaning, α j is nondecreasing with respect to space homogeneous increments of w j (•).

Proof. We start proving the rst claim. Considering (C.1)(C.2) we have

α ρj ,δ(•), wj (•) j (x) -α ρj +h,δ(•)+k, wj (•) j (x) = ∞ 0
1 -e -t(h+k) e -tL w j dt, and the claim follows since e -tL w j is a positive operator, i.e. it maps nonnegative functions into nonnegative functions (see, e.g., Section 2 in Chapter II, of [START_REF] Ma | Introduction to the Theory of (Non-Symmetric) Dirichlet Forms[END_REF]).

The second claim follows by Corollary C.3.

The following proposition establish the dependence on the territory M j .

Proposition C.7. Let M j ⊂ M j ⊂ S 1 , let w j , w j coecients associated to M j , M j , respectively, and let α j , α j be the associated solutions to (C.4). Assume that w| Mj = w j . Then Proof. We only have to rewrite the second and the third term of J j in (B.12).

Using (C.1) and (C.2), the second term of J j in (B.12) can be rewritten as follows (recall the denition of w j in (B.10)) (C.13)

The claim easily follows by rearranging the terms.

The above result is crucial since, from the expression of the functional J j , we immediately see that player j needs to optimize only the term where α j is the unique solution to the ODE (C.17) ρ j α j (x) -σα j (x) -v(x)α j (x) + δ(x)α j (x) = w j (x), x ∈ S 1 , with w j dened as (C.18) w j (x) :=

w j (x), if x ∈ M j , 0, if x / ∈ M j .
Moreover, setting Finally, the welfare of player j is ane in p 0 :

v j (p 0 ) := J (i * -j ,b * -j ) j p 0 ; (i * j , b * j ) = S 1
α j (x)p 0 (x)dx + q j , where q j := α, e (i) n H e (i) n .

Let us compute the coecients of the series. As for n = 0, we notice that Consider again the family (D.1). In this case e 0 is still an eigenfunction of L * , but e

(i)
n are not eigenfunction of L * anymore for n ∈ N \ {0}. However, still H n := Span{e

n H -v o n w, e

(2) n H

(ρ + σn 2 + δ o ) 2 + (v o n) 2 ,

Figure

  Figure 4.1.The case of two players each controlling one half of the circle. All the parameters are space independent: the productivity factor A is 1.6, the discount ρ is 0.03, the diusivity σ is 0.5, the natural decay of the pollution δ is 0.2, the weight of pollution in the utility parameter w is 1, the intertemporal substitution parameter γ is 0.5, the eciency of depollution expenditure factor φ is 0.2 while the scale returns of depollution activity θ is 0.4. The continuous (and blue) lines represent the Nash equilibrium while the dashed (and gray) lines reproduce the cooperative benchmark or, equivalently, the non-diusion benchmark.

Figure 4

 4 Figure 4.2. The case of two players controlling one half of the circle varying the diusivity coecient. The values of the parameters (all constant over the whole space) are the same as in Figure 4.1 except the values of diusivity σ which takes now four values: 0.4 (the lightest line), 0.8, 1.6, and 3.2 (the darkest line). Continuous (and colored) lines represent the Nash equilibrium while the dashed (and gray) lines are the cooperative benchmark or, equivalently, the non-diusion benchmark.

Figure 4

 4 Figure 4.3. The case of two players controlling respectively one fourth and three fourth of the circle. The values of the parameters (all constant over the whole space) are the same as in Figure 4.1. Continuous (and colored) lines represent the Nash equilibrium while the dashed (and gray) lines are the cooperative benchmark or, equivalently, the non-diusion benchmark.

Figure 5 . 1 .

 51 Figure 5.1. The case of two players controlling one half of the circle with dierent levels of technology A. The productivity of the rst player (on the left) is A 1 (constant it her territory) equal to 1.8 while the value of A 2 is 1.4. All other parameters (are constant over the space and) are the same as in Figure 4.1. Continuous (and colored) lines represent the proles of the variables at the Nash equilibrium while the dashed (and gray) lines are related to the cooperative benchmark or, equivalently, the non-diusion benchmark.

5. 1 .

 1 The long-run spatial distribution of pollution is not explicitly represented in the gure but, similarly to what we have in the gures of Section 4 it is a smoothed version of the distribution of net emission ows (see Remark C.13 for a more technical observation on that).

Figure 5

 5 Figure 5.2. The case of two players controlling one half of the circle with dierent levels natural decay of the pollution δ. The value of δ in the territory of the rst player (on the left) is δ 1 (constant it her territory) equal to 0.15 while the value of δ 2 is 0.25. All other parameters (are constant over the space and) are the same as in Figure 4.1. Continuous (and colored) lines represent the Nash equilibrium, the dotted (and gray) lines are the cooperative benchmark (with the same σ) while the dashed (gray) lines are the zero-diusion benchmark.

  1, ..., N , we denote (i -j , b -j ) = ((i 1 , b 1 ), ..., (i j-1 , b j-1 ), (i j+1 , b j+1 ), ..., (i N , b N )) .

( 1 )

 1 p 0 ∈ L 2 (S 1 ; R + ); which render them separable Hilbert spaces. Denote by | • | H , | • | Hj the associated norm, i.e. |f | 2 H

  x)p(t, x)dx dt, Setting e -(ρj -L)t := e -ρj t e tL , t ≥ 0, and dening w j : S 1 → R as (B.10)

∞ 0 e 0 e 0 e

 000 -ρj t Mj w j (x)p(t, x)dx dt = ∞ 0 e -ρj t w j , P (t) H dt = ∞ -ρj t w j , e tL p 0 + t 0 e (t-s)L I(s) -ηB(s) θ ds identications, the original functional J j of agent j rewrites as J (I-j ,B-j ) j p 0 ; (I j , B j )) := ∞ -ρj t (A j -1)I j (t) -B j (t)

  Appendix C. The Nash equilibrium Here we provide the explicit form of the unique open loop Nash equilibrium to the reformulated problem which give, as a corollary, the solution of the original problem. First of all, we rephrase Denition A.3 in this new framework. Denition C.1. An open loop Nash equilibrium for the game is a family of couples (I * j , B * j ) j=1,...,N ∈ A

  . The claim immediately follows by Proposition C.2 due to density of C(S 1 ; R) in H.

0 e

 0 α j ≤ α j . Proof. It follows from Corollary C.3. C.2. The solution. Proposition C.8. Let Assumption A.1 hold. We haveJ (I-j ,B-j ) j p 0 ; (I j , B j ) = ∞ -ρj t (A j -1)I j (t) -B j (t) s α j | Mj , (I j (t) -ηB j (t) θ ) Hj ds (C.10) -α j , p 0 H -s α j | M k , (I k (t) -ηB k (t) θ ) H k ds.

0 e

 0 ρj -L)t p 0 dt H = w j , (ρ j -L) -1 p 0 H = (ρ j -L) -1 w j , p 0 H = α j , p 0 H ,In the remainder of the proof, for simplicity of notation, we dene the net emissions(C.11) K(t) := I(t) -ηB(t) θ .Now, using again (C.1) and (C.2), the third term of J j in (B.12), can be rewritten by exchanging the integrals as follows: t w j , e (t-s)L K(s) s w j , e -(ρj -L)(t-s) K(s) s w j , (ρ j -L)-1 K(s) H ds = ∞ -ρj s (ρ j -L) -1 w j , K(s) H ds = ∞ 0 e -ρj s α j , K(s) s α j 1 Mj , (I j (t) -ηB j (t) θ )1 Mj Hj ds s α j 1 M k , (I k (t) -ηB k (t) θ )1 M k H k ds.

(-α j 1

 1 -1)I j (t) -B j (t) Mj , (I j (t) -ηB j (t) θ ) Hj   dt which does not depend on the choices of the other players. This follows from the separable additive nature of the functional J j .Theorem C.9. Let Assumptions A.1 hold. Then the unique open-loop Nash Equilibrium for our problem is given, for j = 1, ..., N and (t,x) ∈ R + × S 1 , by (C.15) b * j (t, x) = B * j (t)(x) = [(A j (x) -1)ηθ]

  stationary optimal net emission as:N * := I * -(ηB * ) θ , the equilibrium state p * (t, •) = P * (t) is (C.19)P * (t) = e tL p 0 + t 0 e (t-s)L N * ds.

  j (x)(i * k (t, x) -ηb * k (t, x) θ )dx dt.D.1. The case without advection. We know that α solves the equation (ρ -L)α = w, with w dened as ) , n ∈ N \ {0}is an orthonormal basis on H. They are also eigenfunctions of L with associated eigenvalues(D.2) µ n = -δ o -σn 2 , n ∈ N.We can expand in Fourier series (D.3) α = α, e 0 H e 0 + n∈N\{0}, i=1,2

  α, (ρ -L)e 0 H = (ρ -L)α, e 0 H = w, e 0 H , hence,(D.4) α, e 0 H = (ρ + δ o ) -1 w, e 0 H = (ρ + δ o ) (i) n H = ρ + δ o + σn 2 -1 w, e (i) n H , ∀i = 1, 2, ∀n ∈ N \ {0}. x) sin (nx) dx = w o √ π 0 sin (nx) dx = w o x) cos (nx) dx = w o √ π 0 cos (nx) dx = w o ) (1 -cos (n )) + sin(n ) cos (nx) n (ρ + δ o + σn 2 ) , D.2. The case with advection. In this subsection we remove the assumption v ≡ 0. Recalling the expression of L * provided in (B.3), we have in the present case [L * ψ](x) = σψ (x) -v o ψ (x) + δ o ψ(x), ψ ∈ D(L * ).

  invariant subspaces for L * for each n ∈ N \ {0}. IndeedL * e (1) n = -σn 2 e (1) n -v o ne (2) n + δ o e (1) n ; L * e (2) n = -σn 2 e (2) n + v o ne (1) n + δ o e (2)n ; Arguing in a similar way as in the previous subsection we get, for each n ∈ N\{0}, the couple of equations(ρ+σn 2 +δ o ) α, e (1) n H +v o n α, e (2) n H = w, e (1) n H , (ρ+σn 2 +δ o ) α, e (2) n H -v o n α, e (1) n H = w, e (2) n H . yielding, for each n ∈ N \ {0},α, e(1) n H = (ρ + σn 2 + δ o ) w, e

  In de Frutos and Martin-Herran (2019a and 2019b) the continuous space-time model is not studied: an analogous

discrete-space model is solved delivering a feedback Nash equilibrium, which is in turn used to capture the spatial interactions among agents through a truly comprehensive set of carefuly designed numerical exercises. In the draft of de Frutos et al. (

2019

) the continuous space-time model is studied in a heuristic way as a departure point to perform some interesting numerical simulations. It should be noted here that in our setting, we perform a complete mathematical study of the continuous time-space model, providing the explicit form of the (unique) open loop Nash equilibrium, which indeed is also a Markovian (feedback) Nash equilibrium. 6

Note that we allow for heterogeneity also in the discount and risk aversion parameters of the local authorities, heterogeneity across states may also show up in local governance.

In (2.9) w j (x) is meant to be extended to 0 outside its initial domain M j .

1-γ γ

In the following, by p t , p x , p xx we denote, rspectively, the rst partial derivatives of p with respect to t, x, and the second derivative with respect to x.

For the denition of the Lebesgue spaces L q we refer the reader e.g. to[START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Dierential Equations[END_REF], Chapter 4. We recall that these spaces are done by equivalence classes of functions according to the equivalence relation which identies functions which are equal almost everywhere.

To be precise, the case σ = 0 should be treated separately, as it is out of our assumptions. Nonetheless, this case can be easily treated pointwiseless on x and gives raise, in the case under consideration here, to the solutions we illustrate below.

Appendix A. Formulation of the problem and main assumptions Let S 1 be the unitary circle in R 2 :

Hereafter, we often identify S 1 ∼ = 2πR/Z and, according to this identication, we identify functions ϕ : S 1 → R with 2π-periodic function ϕ : R → R. Moreover, given a function ψ : R + × S 1 → R, we

(2) δ ∈ C(S 1 ; R + );

(3) v ∈ C 1 (S 1 ; R);

(4) A j ∈ L ∞ (M j ; R + ) and there exists a constant l > 0 such that l ≤ A j (x) for all j = 1, . . . , N ;

(5) for each j = 1, ..., N , one has w j ∈ C(M j ; R + ) and w j be extended to a function w j ∈ C(M j ; R) such that w j (x) > 0 for each x ∈ M j .

The function p in the objective functional (A.3) above is the solution, in a sense to be specied, to (A.1) corresponding to the initial datum p 0 and to the strategies ((i j , b j )) j=1,...,N of the N players.

We consider two classes of admissible strategies: open loop and Markovian (or closed loop) (see Section 4.1 in [START_REF] Dockner | Dierential Games in Economics and Management Science[END_REF]). Let, for j = 1, . . . , N ,

and (A j (x) -1)i j (t, x) -b j (t, x) ≥ 0 for a.e. (t, x) ∈ R + × M j .

(A.4) If we are given, for every j = 1, . . . , N , an element (i j , b j ) ∈ A j , we obtain a couple (i, b) : R + × S 1 → R 2 dened as in (A.2) and

Observe that, choosing any (i, b) ∈ A, the state equation (A.1) has a unique solution in the mild sense for each p 0 ∈ L 2 (S 1 ; R + ) (see Remark A.5 and equation (B.7)).

Denition A.2. The class of open loop strategies is the set

An open loop Nash equilibrium for the game starting at p 0 is a family of couples

such that, for j = 1, . . . , N ,

We also consider Markovian strategies with perfect state information as follows. Let

, N, and let

A

). An admissible Markovian strategy starting at p 0 is an N -tuple of couples

where

admits (in the mild sense of equation (B.7), see Remark A.5) a unique solution p (φ,ψ) , and the functional

computed by substituting in (A.3)

is well dened for each j = 1, ..., N ; in particular

We denote the class of admissible Markovian strategies by A cl (p 0 ). Remark A.5. Notice that the equation in Denition A.4 is a PDE with nonlocal terms in the x variable, as φ(t, x, •) and ψ(t, x, •) depend, in general, on the structure of p(t, •) on S 1 , not only on its value at (t, x). The precise meaning of this equation and the concept of solution will be provided in the innite dimensional lifting of the problem that we will perform in the next subsection.

Given (φ, ψ) ∈ A cl and j = 1, ..., N, we denote (φ -j , ψ -j ) := ((φ 1 , ψ 1 ), ..., (φ j-1 , ψ j-1 ), (φ j+1 , ψ j+1 ), ...,

ad is said a Markovian Nash equilibrium starting at p 0 if, for each j = 1, .., N , it holds

for every

Remark A.7. The (unique) open loop Nash equilibrium characterized in the next sections is also a Markovian Nash equilibrium in the sense of Denition A.6 (see Section 4.1 in [START_REF] Dockner | Dierential Games in Economics and Management Science[END_REF]).

On the other hand, proving that this equilibrium is unique also in the class of Markovian Nash equilibria seems not trivial. Remark A.8. Here the space variable of the model lives in the one dimensional circle S 1 . Extensions to dierent space structures are possible but the one dimensional case allows to compute more easily the solution and to perform a more precise analysis of its behavior.

Appendix B. Reformulation of the problem in infinite dimensional spaces

To solve the problem it is useful to rewrite it in suitable innite dimensional spaces. We now provide the various ingredients of this reformulation: the spaces, the operators, and the reformulation itself.

B.1. The spaces. Consider the spaces

and, for j = 0, . . . , N ,

Hence, the operator L is pseudo-dissipative, and so is L * . Therefore, by [START_REF] Engel | One-parameter Semigroups for Linear Evolution Equations[END_REF] (see in particualr Chapter II), we see that L generates a strongly continuous semigroup (e tL ) t≥0 ⊂ L(H). We introduce therefore the following assumption, which will ensure that ρ j belongs to the resolvent set of the operator L.

Assumption B.1. For all j = 1, . . . , N we have

Reformulation of the state equation (A.1). Setting

we see that (I j , B j ) : R + → H + j × H + j and rewrite the set A 0 j , given in (A.4) (and then, consequently, A 0 ), as follows:

Note that the last inequality above means that, for a.e. t ≥ 0, the function in H + j given by x → (A j (x) -1)i j (t, x) -b j (t, x) is nonnegative for a.e. x ∈ M j . The fact the such function belongs to H + follows by Assumption A.1-(1).

Given (I j , B j ) j=1,...,N ∈ A, we set

Proof. Observe that (C.14) can be rewritten as

Hence, if for every (t, x) ∈ R + × M j , we maximize the integrand and the maximum point is unique, this will give the unique optimal strategy of player j, independently of the strategies of the other players. Fix (t, x) ∈ R + × M j . By strict concavity of the integrand function with respect to i j (t, x) and b j (t, x), the unique maximum point can be found just by rst order optimality conditions. The resulting system is

The claims then follows from straightforward computations and by (C.10).

Remark C.10. It is clear that, for every j = 1, . . . , N , the cost functional J j is decreasing with respect to w j ; this follows from the fact that a lower cost of pollution makes the welfare bigger, hence also the corresponding welfare function v j is decreasing in w j . Moreover, if, for i = j, w i increases, then J j , and so v j also increase. This can be seen looking at the decomposition of Proposition C.8 or simply observing that the increase in w i does not modies the strategy for the j-th agent, but makes the p to globally decrease since the agent i will pollute less.

Finally, since p is decreasing with respect to δ (this comes from the fact that a higher self-cleaning capacity makes the pollution lower), for every j = 1, . . . , N , the cost functional J j is decreasing with respect to δ, hence also the corresponding welfare function v j is decreasing in δ. Remark C.11. We now look at the dependence of the optimal net emissions, i.e. n(t, x) := i * j (t, x)η(b * j (t, x)) θ , on the data. By a simple computation we see that for every

From this computation, one can analyze the monotonicity of net emissions with respect to some parameters. For example: when γ j > 1, the value of n(t, x) is decreasing with respect to A j (x); when

the value of n(t, x) is increasing with resect to A j (x); by Proposition C.6, n(t, x) is increasing with respect to space homogeneous increments of ρ j +δ(•) in the sense specied in the same proposition. Proof of Proposition 4.2. Just for simplicity of notatation, we prove the claim in a very special case, i.e. when

The proof of the general claim is a straightforward generalization. Let n 1, * , n 2, * be the optimal net emissions associated to Π 1 , Π 2 , respactively. They dier only for the terms containing α. Now, with clear meaning of the symbols, we have

Then the claim follows from (2.10), (2.7), (2.8), (B.7), and since the operator positivity e tL is positive, i.e. maps nonnegative functions into nonnegative functions (see, e.g., Section 2 in Chapter II, of [START_REF] Ma | Introduction to the Theory of (Non-Symmetric) Dirichlet Forms[END_REF]).

Corollary C.12. Let Assumption A.1 hold. Furthermore, assume that there exists δ > 0 such that

Then

where P * ∞ is the unique solution in H to the ODE LP * ∞ + K * = 0, i.e.

(C.24)

Proof. Let us split

where

and

We can rewrite P * (t) = e -δt e t Lp 0 + t 0 e -δ(t-s) e (t-s) LK * ds, and take the limit above when t → ∞. By (C.23), L is dissipative, hence e s L is a contraction. Therefore, the rst term of the right hand side converges to 0 in H, whereas the second one converges to

e -δs e s LK * ds in H. Then, the limit state P * ∞ can be expressed using again [START_REF] Engel | One-parameter Semigroups for Linear Evolution Equations[END_REF], Proposition 3.14, page 82 and Chapter II, Theorem 1.10, as

Remark C.13. The dependence of P ∞ on A j (•) follows from what observed in Remark C.11. Indeed, since L does not depend on the A j (•), P ∞ depends on it only through the stationary optimal net emissions, K * .

Appendix D. Series expansion of the α j 's

In this section

We use the identication S 1 ∼ = 2πR/Z and assume, without loss of generality that M j = (0, j ). Finally, to save notation, we suppress the subscript j. We are going to study the Fourier series expansion of α = α j in the case without or with advection. We notice that the convergence of the series is uniform on S 1 due to the smoothness of α.

α, e (2) n H = (ρ + σn 2 + δ o ) w, e

(2)

Using (D.3)(D.5) and the expressions above, we have for x ∈ S 1 α(x) = 1 2π

Appendix E. The cooperative game case

In this case we consider the problem when the N players are cooperative and have the same discount factor ρ j = ρ and preference parameter γ j = γ for every j = 1, ..., N . We suppose they coordinate to maximize the sum of their utilities:

over ((i j , b j )) j=1,...,N ∈ A and under the state equation (A.1). As we see in Theorem E.1 below, this problem is equivalent to the problem of a unique player acting on N j=1 M j . We know the solution of the latter problem, as we only need to implement the results of Theorem C.9 in the case of a unique player. Dening

and letting α be the solution to

Theorem C.9 provides the solution for the unique player:

.,N ∈ A maximizing (E.1), i.e. the unique optimal choice of the cooperative game when the N players have the same discount factor ρ j = ρ and preference parameter γ j = γ for every j = 1, ..., N , is given by

where α is the solution to (E.2).The corresponding welfare is

with b * (t, x) and i * (t, x) given by (E.3) and (E.4). Sketch of proof. We observe that the sum of utilities given in (E.1) is exactly equal to the utility of the unique player. Since the constraints and the state equation are the same, the equivalence follows.

Corollary E.2. Let Assumptions A.1 hold and assume that δ(•) ≡ δ o > 0, w(•) ≡ w o > 0, and ρ j = ρ for all j = 1, ..., N . Then

which is (constant and) independent of σ; moreover, if σ = 0( 16 ), then Email address: giorgio.fabbri@univ-grenoble-alpes.fr S. Federico, Università degli Studi di Siena, Dipartimento di Economia Politica e Statistica. Piazza San Francesco 7-8, 53100 Siena , Italy.

Email address: salvatore.federico@unisi.it F. Gozzi, Dipartimento di Economia e Finanza, Libera Università degli Studi Sociali Guido Carli, Roma Email address: fgozzi@luiss.it