# A Cooperative Stochastic Differential Game of Transboundary Industrial Pollution between Two Asymmetric Nations

Authors: Yongxi Yi; Rongwei Xu; Sheng Zhang
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

Considering the fact that transboundary pollution control calls for the cooperation between interested parties, this paper studies a cooperative stochastic differential game of transboundary industrial pollution between two asymmetric nations in infinite-horizon level. In this paper, we model two ways of transboundary pollution: one is an accumulative global pollutant with an uncertain evolutionary dynamic and the other is a regional nonaccumulative pollutant. In our model, firms and governments are separated entities and they play a Stackelberg game, while the governments of the two nations can cooperate in pollution reduction. We discuss the feedback Nash equilibrium strategies of governments and industrial firms, and it is found that the governments being cooperative in transboundary pollution control will set a higher pollution tax rate and make more pollution abatement effort than when they are noncooperative. Additionally, a payment distribution mechanism that supports the subgame consistent solution is proposed.


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## Body

## 1. Introduction

Global environment is a whole ecosystem that is interconnected and indivisible. Any environmental pollution, misuse of resources, and ecological damage risk pressures are likely to be cross-border. Through pathways such as water or air, pollutants can spread across an incredible distance to cause transboundary pollution. Atmospheric pollution and acid rain, ocean and river basin water pollution, hazardous waste transponder movement, and other issues, have been gradually a threat to human survival and development, which become a core issue with international concern. With the development of the industrialization, the transboundary pollution in the world has become increasingly serious. To settle the problem, unilateral response on the part of one nation or region is often ineffective; it calls for cooperation between interested parties [1, 2]. In this paper, we present a cooperative stochastic differential game model of transboundary industrial pollution between two asymmetric nations. Our objective is to find the optimal pollution abatement strategy which maximizes the social welfare for the two nations gaming for transboundary industrial pollution control.Differential games have been used as an effective tool to study transboundary pollution control. For example, Dockner and Van Long [3] model a simple dynamic game of two neighboring countries which game for transboundary pollution control; it is found that when the governments of the two countries are restricted to use of linear strategies, noncooperative behavior may result in overall losses for both countries. In 1998, Zagonari extends the model of Dockner and Van Long to analyze a pollution control game between environmentally concerned countries and consumption-oriented countries. In 1993, a tax/subsidy scheme is presented by Martin et al. to combat the transboundary problem of global climate change. Petrosjan and Zaccour [4] apply the Shapley value to determine a fair distribution of the total cooperative cost incurred by countries in a cooperative game of pollution reduction. Bayramoglu [5] uses a dynamic and strategic framework to analyze the transboundary pollution between Romania and Ukraine, it is found that, among the three different institutional systems such as noncooperative game of countries, uniform emission policy, and constant emission policy, the noncooperative game can provide the highest level of total welfare. Jørgensen and Zaccour [6] take emissions and investments in abatement technology as control variables to analyze the problem of cooperative transboundary industrial pollution control; in their study, a coordinated solution that maximizes joint welfare is derived together with a payment distribution mechanism that supports the subgame consistent solution. Bertinelli et al. [7] study the strategic behavior of two countries facing transboundary CO2 pollution under a differential game setting; the author considers that feedback strategy may lead to less social waste than when the countries take open-loop strategy. Taking emission permits trading into account; Li [8] studies a differential game of transboundary industrial pollution between two neighboring countries, and the focus of the author’s analysis is to find the two countries’ noncooperative and cooperative optimal emission paths. Benchekroun and Martín-Herrán [9] study the impact of foresight in a transboundary pollution game; their study shows that when all countries are myopic, that is, choose the “laisser-faire” policy, their payoffs are smaller than when all countries are farsighted.Most cooperative environmental games do not distinguish the government from the industrial sector in any nation. Apart from these studies, Yeung and Petrosyan [2] take industries and governments as separated entities and develop a cooperative differential game model of transboundary industrial pollution and the time-consistent solutions are derived. Extending the work of Yeung and Petrosyan [2], Huang et al. [10] develop a model in which there is a Stackelberg game between the industrial firms and their local government while the governments can cooperate in transboundary pollution control, and the feedback Nash equilibrium strategies of governments and industrial firms are given. In this paper, we raise a stochastic differential game model of transboundary industrial pollution between two asymmetric countries. Our work can be viewed as a continuing work of Huang et al. [10] in the context of transboundary industrial pollution control. Compared with the work of Huang et al. [10], there are three significant features in our paper. (i) We present a stochastic differential game model of transboundary industrial pollution in which uncertain pollution stock dynamics are taken into account. It is because that uncertain pollution stock dynamics is frequent [2, 11–13]; actually many factors like unexpected changes in abatement technologies, the unexpected changes of decay rate of the pollutants, and so forth may incur uncertain pollution stock dynamics. (ii) We extend the model set by Huang et al. [10] from finite-horizon level to infinite-horizon level; this is because, in many situations, the terminal time of the game, T, is either very far in the future or unknown to the players. A way to resolve the problem, as suggested by Dockner and Nishimura   [14], is to set T=∞, so it makes sense to present a model in infinite-horizon level to analyze transboundary industrial pollution control. In our model, we find that the difference of the horizontal level does play an important effect on the behavior of game players. From the government’s point of view, the plan is based on long-term interests, so the planning period for the infinite-horizon level is more reasonable. For example, we find that, in the infinite-horizon level, the optimal tax rate and the pollution abatement effort are relatively stable. While, in finite-horizon level, Huang et al. [10] consider the optimal tax rate and the pollution abatement effort decrease in the late game; it means that a hyperopia government can protect the environment better. (iii) In our model, the asymmetry of participants in game is considered; we assume that the industrial firms in one country have more green and energy-efficient technologies than the others, and we show that the one with technology advantage may set a higher pollution tax and make more pollution abatement effort, no matter what condition it would be in, cooperation or noncooperation.The paper is organized as follows. In Section2, the game formulation is provided. In Section 3, we characterize the noncooperative outcomes. Cooperative arrangements and individual rationality are analyzed in Section 4. We illustrate the results of a numerical example in Section 5. Section 6 is the results summarizing the paper.


## 2. Game Formulation

### 2.1. Government and Domestic Industrial Firm: A Stackelberg Game

Consider a multinational economy, which is comprised of two nations. There aren and m industrial firms in nation 1 and nation 2, respectively. In order to control production pollution, the government (the leader) of each nation imposes a pollution tax on domestic industrial firms, and then the industrial firms (the followers) choose their optimal output (emissions) according to the pollution tax rate. This leads to a Stackelberg equilibrium.Let∑i=1nui(s) and ∑j=1mvj(s) denote the quantity of goods produced by n firms in nation 1 and m firms in nation 2 at time s, respectively. Firm i’s (or j’s) revenue function ϕi (or ϕj) is assumed to be concave and increasing, with the following simple functional form:(1)ϕi=βi1uis1/2,ϕj=βj2vjs1/2,where βi1,βj2>0 are constants; the instantaneous profits of industrial firm in the two nations can be expressed as:(2)ϕiuis-c1suis,ϕjvjs-c2svjs,where ck(s) is the pollution tax rate imposed on industrial firms by government k (k∈1,2) at time s.Through the first-order condition of (2), the optimal yields for the firms in nation 1 and nation 2 could be obtained as follows:(3)ui∗s=βi12c1s2,vj∗s=βj22c2s2.


### 2.2. Local and Global Environmental Impacts

Transboundary pollution may damage the environment through two ways, that is, an accumulative global pollutant and a regional nonaccumulative pollutant. For example, some transboundary pollutants are caused by firms such as passing-by waste in waterways, wind-driven suspended particles in air, unpleasant odour, noise, dust, and heat, which are all nonaccumulative pollutants. We assume that the nonaccumulative pollutant, for an output ofui(s) produced by the firm in nation 1, would cause a short-term impact (cost) of ε11ui(s) and ε12ui(s) on nation 1 and nation 2, respectively. Similarly, an output of vj(s) produced by the firm in nation 2, would cause a short-term (cost) of ε21vj(s) and ε22vj(s) on nation 1 and nation 2, respectively. In addition to the nonaccumulative pollutant, there is an accumulative pollutant, such as green-house-gas, CFC, and atmospheric particulates; these pollutants can be maintained in environment for a long time and built up existing pollution stocks to create long-term global environmental impacts [2, 5, 10, 15, 16]. Let x(s)⊂R+ denote the level of pollution at time s, and the dynamics of pollution stock are governed by the stochastic differential equation(4)dxs=∑i=1nαi1uis+∑j=1mαj2vjs-∑k=12γkIksxs1/2-δxsds+σxsdzs,k∈1,2,where σ denotes a noise parameter and z(s) is a Wiener process, αi1≥0 and αj2≥0 are the amount added to the pollution stock by a unit of firms i’s and j’s output in nation 1 and nation 2, respectively. Ik(s) denotes the pollution abatement effort of nation k, γk is the efficiency of the abatement, γkIk(s)(x(s))1/2 denotes the amount of pollution removed by Ik(s) unit of abatement effort of nation k, and σ denotes the natural rate of decay of the pollutants.


### 2.3. The Governments’ Objectives

The instantaneous objective of governmentk (k∈1,2) at time s can be expressed as(5a)∑i=1nβi1ui∗s1/2-12a1I1s2-ε11∑i=1nui∗s-ε21∑j=1mvj∗s-h1xs,(5b)∑j=1mβj1vj∗s1/2-12a2I2s2-ε12∑i=1nui∗s-ε22∑j=1mvj∗s-h2xs,where ak, hk>0 (k∈1,2) are constants, 1/2ak(Ik(s))2 denotes the cost of employing Ik(s) amount of pollution abatement effort, and hkx(s) denotes the value of damage to nation k from x(s) amount of pollution.The government’s planning horizon ist0,∞. The discount rate is r. Each government seeks to maximize the integral of its instantaneous objectives (5a) and (5b) over the planning horizon t0,∞ subjected to pollution dynamics (4), with controls on the level of abatement effort and pollution tax.Substitutingui(s) and vj(s) from (3) into (4) and (5a) and (5b), one obtains a stochastic differential game in which government k∈1,2 seeks (6a)maxc1s,I1sEt0∫t0∞e-rt∑i=1nβi122c1s-12a1I1s2-ε11∑i=1nβi124c12s-ε21∑j=1mβj224c22s-h1xsds,(6b)maxc2s,I2sEt0∫t0∞e-rt∑j=1mβj222c2s-12a2I2s2-ε12∑i=1nβi124c12s-ε22∑j=1mβj224c22s-h2xsdssubject to(7)dxs=∑i=1nαi1βi12c1s2+∑j=1mαj2βj22c2s2-∑k=12γkIksxs1/2-δxsds+σxsdzs.Since the payoffs of nations are measured in monetary terms, the games (6a) and (6b) are a transferable payoff game.


## 2.1. Government and Domestic Industrial Firm: A Stackelberg Game

Consider a multinational economy, which is comprised of two nations. There aren and m industrial firms in nation 1 and nation 2, respectively. In order to control production pollution, the government (the leader) of each nation imposes a pollution tax on domestic industrial firms, and then the industrial firms (the followers) choose their optimal output (emissions) according to the pollution tax rate. This leads to a Stackelberg equilibrium.Let∑i=1nui(s) and ∑j=1mvj(s) denote the quantity of goods produced by n firms in nation 1 and m firms in nation 2 at time s, respectively. Firm i’s (or j’s) revenue function ϕi (or ϕj) is assumed to be concave and increasing, with the following simple functional form:(1)ϕi=βi1uis1/2,ϕj=βj2vjs1/2,where βi1,βj2>0 are constants; the instantaneous profits of industrial firm in the two nations can be expressed as:(2)ϕiuis-c1suis,ϕjvjs-c2svjs,where ck(s) is the pollution tax rate imposed on industrial firms by government k (k∈1,2) at time s.Through the first-order condition of (2), the optimal yields for the firms in nation 1 and nation 2 could be obtained as follows:(3)ui∗s=βi12c1s2,vj∗s=βj22c2s2.


## 2.2. Local and Global Environmental Impacts

Transboundary pollution may damage the environment through two ways, that is, an accumulative global pollutant and a regional nonaccumulative pollutant. For example, some transboundary pollutants are caused by firms such as passing-by waste in waterways, wind-driven suspended particles in air, unpleasant odour, noise, dust, and heat, which are all nonaccumulative pollutants. We assume that the nonaccumulative pollutant, for an output ofui(s) produced by the firm in nation 1, would cause a short-term impact (cost) of ε11ui(s) and ε12ui(s) on nation 1 and nation 2, respectively. Similarly, an output of vj(s) produced by the firm in nation 2, would cause a short-term (cost) of ε21vj(s) and ε22vj(s) on nation 1 and nation 2, respectively. In addition to the nonaccumulative pollutant, there is an accumulative pollutant, such as green-house-gas, CFC, and atmospheric particulates; these pollutants can be maintained in environment for a long time and built up existing pollution stocks to create long-term global environmental impacts [2, 5, 10, 15, 16]. Let x(s)⊂R+ denote the level of pollution at time s, and the dynamics of pollution stock are governed by the stochastic differential equation(4)dxs=∑i=1nαi1uis+∑j=1mαj2vjs-∑k=12γkIksxs1/2-δxsds+σxsdzs,k∈1,2,where σ denotes a noise parameter and z(s) is a Wiener process, αi1≥0 and αj2≥0 are the amount added to the pollution stock by a unit of firms i’s and j’s output in nation 1 and nation 2, respectively. Ik(s) denotes the pollution abatement effort of nation k, γk is the efficiency of the abatement, γkIk(s)(x(s))1/2 denotes the amount of pollution removed by Ik(s) unit of abatement effort of nation k, and σ denotes the natural rate of decay of the pollutants.


## 2.3. The Governments’ Objectives

The instantaneous objective of governmentk (k∈1,2) at time s can be expressed as(5a)∑i=1nβi1ui∗s1/2-12a1I1s2-ε11∑i=1nui∗s-ε21∑j=1mvj∗s-h1xs,(5b)∑j=1mβj1vj∗s1/2-12a2I2s2-ε12∑i=1nui∗s-ε22∑j=1mvj∗s-h2xs,where ak, hk>0 (k∈1,2) are constants, 1/2ak(Ik(s))2 denotes the cost of employing Ik(s) amount of pollution abatement effort, and hkx(s) denotes the value of damage to nation k from x(s) amount of pollution.The government’s planning horizon ist0,∞. The discount rate is r. Each government seeks to maximize the integral of its instantaneous objectives (5a) and (5b) over the planning horizon t0,∞ subjected to pollution dynamics (4), with controls on the level of abatement effort and pollution tax.Substitutingui(s) and vj(s) from (3) into (4) and (5a) and (5b), one obtains a stochastic differential game in which government k∈1,2 seeks (6a)maxc1s,I1sEt0∫t0∞e-rt∑i=1nβi122c1s-12a1I1s2-ε11∑i=1nβi124c12s-ε21∑j=1mβj224c22s-h1xsds,(6b)maxc2s,I2sEt0∫t0∞e-rt∑j=1mβj222c2s-12a2I2s2-ε12∑i=1nβi124c12s-ε22∑j=1mβj224c22s-h2xsdssubject to(7)dxs=∑i=1nαi1βi12c1s2+∑j=1mαj2βj22c2s2-∑k=12γkIksxs1/2-δxsds+σxsdzs.Since the payoffs of nations are measured in monetary terms, the games (6a) and (6b) are a transferable payoff game.


## 3. Noncooperative Outcomes

In this section, the solution of the noncooperative games (6a) and (6b) and (7) will be discussed under a noncooperative framework.A feedback Nash equilibrium solution can be characterized as follows.Definition 1.
A set of feedback strategies[ck∗(t),Ik∗(t)]=[ϕkc(t,x),ϕkI(t,x)], for k∈1,2, provides a Nash equilibrium solution to the games (6a), (6b), and (7), if there exist suitably smooth functions Vk(t,x):t0,∞×R→R, k∈1,2, satisfying the following partial differential equations:(8a)rV1t,x-12σ2x2Vxx1t,x=maxc1t,I1t∑i=1nβi122c1t-12a1I1t2-ε11∑i=1nβi124c12t-ε21∑j=1mβj224c22t-h1xt+Vx1t,x∑i=1nβi124c12t+∑j=1mβj224c22t-∑k=12γkIktxt1/2-δxt,(8b)rV2t,x-12σ2x2Vxx2t,x=maxc2t,I2t∑j=1mβj222c2t-12a2I2t2-ε12∑i=1nβi124c12t-ε22∑j=1mβj224c22t-h2xt+Vx2t,x∑i=1nβi124c12t+∑j=1mβj224c22t-∑k=12γkIktxt1/2-δxt.Performing the indicated maximization in (8a) and (8b) yields(9a)ϕ1ct,x=ε11-Vx1t,x,ϕ2ct,x=ε22-Vx2t,x,(9b)ϕ1It,x=-γ1Vx1t,xx1/2ta1,ϕ2It,x=-γ2Vx2t,xx1/2ta2.Substituting the results of (9a) and (9b) into (8a) and (8b), we obtain the following.Proposition 2.
The governmentk’s payoff during t0,∞, Vk(t,x), can be obtained as(10)Vkt,x=Akxt+Bk,fork∈1,2,where Ak, Bk satisfy the following set of constant coefficient quadratic ordinary differential equations:(11a)γ12A122a1+γ22A1A2a2-γA1-δA1-h1=0,(11b)γ22A222a2+γ12A1A2a1-γA2-δA2-h2=0,(11c)B1=1r∑i=1nβi124ε11-A1+A1-ε21∑j=1mβj224ε22-A22,(11d)B2=1r∑j=1mβj124ε22-A2+A2-ε12∑i=1nβi124ε11-A12.
The corresponding feedback Nash equilibrium of the games (8a) and (8b) can be obtained as(12)ϕ1ct,x=ε11-A1,ϕ1It,x=-γ1A1x1/2ta1,ϕ2ct,x=ε22-A2,ϕ2It,x=-γ2A2x1/2ta2.Equation (12) indicates that, under noncooperation, optimal pollution tax rate (ϕ1c or ϕ2c) may depend on the regional pollutant (ε11 or ε22) which is caused by the firms in home, but the regional pollutant (ε12 or ε21) which is caused by the firms in neighbor may have no effect on optimal pollution tax rate. In addition, the optimal abatement effort of the region does not depend on the damage from the regional pollutant at home (ε11 or ε22); this is because the abatement effort of the nation can only affect pollution stock and cannot affect current pollution.


## 4. Cooperative Arrangements

Now consider the case when both nations cooperate in pollution control. To uphold the cooperative scheme, both group rationality and individual rationality are required to be satisfied at any time.


### 4.1. Group Optimality and Cooperative State Trajectory

To secure group optimality the participating two nations would seek to maximize their joint expected payoff by solving the following stochastic control problem:(13)Ws,x=Et0∫0∞∑i=1nβi122c1s+∑j=1mβj222c2s-12∑k=12akIks2-ε11+ε12∑i=1nβi124c12s-ε21+ε22∑j=1mβj224c22s-h1+h2xse-rtdtsubject to (7).Invoking Bellman’s technique, a set of controlsc^k∗,I^k∗=ϕ^kc(x),ϕ^kI(x) constitutes an optimal solution to the stochastic control problem (13) and (7), if there exists continuously differentiable function W(t,x):t,∞×R→R, k∈1,2, satisfying the following partial differential equations:(14)rWt,x-12σ2x2Wxxt,x=maxc1,I1,c2,I2∑i=1nβi122c1t+∑j=1mβj222c2t-12∑k=12akIkt2-ε11+ε12∑i=1nβi124c12t-ε21+ε22∑j=1mβj224c22t-h1+h2xt+WXt,x∑i=1nαi1βi124c12t+∑j=1mαj2βj224c22t-∑k=12γkIktxt1/2-δxt.Performing the indicated maximization in (14) yields the optimal controls under cooperation:(15a)ϕ^1ct,x=ε11+ε12-Wxt,x,ϕ^2ct,x=ε21+ε22-Wxt,x,(15b)ϕ^1Ix=-γ1Wxxx1/2a1,ϕ^2Ix=-γ2Wxxx1/2a2.Substituting the results in (15a) and (15b) into (14), we obtain the following.Proposition 3.
System (13) admits a solution(16)Wt,x=A^xt+B^,where A^, B^ satisfy the following set of constant coefficient quadratic ordinary differential equations:(17a)γ122a1+γ222a2A^2-r+δA^-h1+h2=0,(17b)B^=1r∑i=1nβi12+∑j=1mβj224ε11+ε22-A^2.The corresponding feedback Nash equilibriums under cooperation can be obtained as(18)ϕ^1ct,x=ε11+ε22-A^,ϕ^1It,x=-γ1A^x1/2t2a1,ϕ^2ct,x=ε11+ε22-A^,ϕ^2It,x=-γ2A^x1/2t2a2.Substituting the optimal control strategy from (18) into (7) yields the dynamics of pollution accumulation under cooperation:(19)dxs=∑i=1nαi1βi12+∑j=1mαj2βj224ε11+ε22-A^2+∑k=12γk2A^xs2ak-δxsds+σxsdzs,xt0=x0.Solving the stochastic cooperative pollution dynamics yields the cooperative state trajectory:(20)x∗t=e∫t0tγ12/2a1+γ22/2a2A^-δ-σ/2ds+∫t0tσdzs×xt0+∫t0t∑i=1nαi1βi12+∑j=1mαj2βj224ε11+ε22-A^2×e∫t0tσ2/2+δ-γ12/2a1+γ22/2a2A^dτ+∫t0tσdzsds,fort∈t0,∞.


### 4.2. Individually Rational and Time-Consistent Imputation and Payment Distribution Mechanism

LetXt∗ denote the set of realizable values of x∗(t) at time t generated by (20). The term xt∗ is used to denote an element of the set Xt∗. An agreed upon optimality principle would be sought to allocate the cooperative payoff. In a dynamic framework, individual rationality has to be maintained at every instant of time within the cooperative duration t0,∞ along the cooperative trajectory (20). For τ∈t0,∞, let vector ξk(xt∗)=ξ1(xt∗),ξ2(xt∗) denote the solution imputation (payoff under cooperation) over the period τ,∞ to player k∈1,2 given that the state is xτ∗∈Xτ∗. Individual rationality along the cooperative trajectory requires(21)ξτkτ,xt∗≥Vτkτ,xt∗,fork∈1,2,where V(τ)k(τ,xt∗) denotes the payoff to nation k under noncooperation over the period. Let G(s)=[G1(s),G2(s)] denote the instantaneous payoff of the cooperative game at time s∈t0,∞ for the cooperative Γc(xt0∗).Theorem 4.
An instantaneous payment at timeτ∈[t0,T] equaling(22)GKτ=rξkxτ∗-12σ2xτ∗2ξxτ∗xτ∗kxτ∗-ξxτ∗kxτ∗∑i=1nαi1βi12c1τ,xτ∗2+∑j=1mαj2βj22c2τ,xτ∗2-∑k=12γkIkτ,xτ∗xτ∗1/2-δxτ∗,forτ∈t0,∞,xτ∗∈Xτ∗,k∈1,2yields a subgame consistent solution to the cooperative game Γc(xt0∗).Proof.
Along the cooperative trajectoryx∗(t)t≥t0, we have(23)Υkτ,τ,xτ∗=ξkxτ∗=Eτ∫τ∞Gkse-rs-τds∣xτ=xτ∗,Υkτ,t,xt∗=ξkxt∗=Et∫t∞Gkse-rs-tds∣xt=xt∗,fork∈1,2,xt∗∈Xt∗,t≥τ.
Note that(24)Υkτ;t,xt∗=e-rs-τξxt∗=e-rs-τΥkt;t,xt∗,k∈1,2,xτ∗∈Xτ∗.
SinceΥk(t;t,xt∗) is continuously differentiable in t and xt∗, one can obtain(25)Υkτ,τ,xτ∗=Eτ∫ττ+ΔtGkse-rs-τds+e-rΔtΥkτ+Δt;τ+Δt,xτ∗+Δxτ∗∣xτ=xτ∗,forτ∈t0,∞,xτ∗∈Xτ∗,k∈1,2,where(26)Δxτ∗=∑i=1nαi1βi12c1τ,xτ∗2+∑j=1mαj2βj22c2τ,xτ∗2-∑k=12γkIKτ,xτ∗xτ∗1/2-δxτ∗Δt+σxτ∗Δzτ+oΔt,Δzτ=zτ+Δt-zτ,EτoΔtΔt⟶0,asΔt⟶0.
From (24) and (25), one can obtain(27)Υkτ,τ,xτ∗=Eτ∫ττ+ΔtGkse-rs-τds+Υkτ;τ+Δt,xτ∗+Δxτ∗∣xτ=xτ∗,forτ∈t0,∞,xτ∗∈Xτ∗,k∈1,2.
Therefore, withΔt→0, (27) can be expressed as(28)GkτΔt=-Υtkτ;t,xτ∗t=τΔt-12σ2xτ∗2Υxτ∗xτ∗kτ;t,xτ∗t=τΔt-Υxτ∗kτ;t,xτ∗t=τ×∑i=1nαi1βi12c1τ,xτ∗2+∑j=1mαj2βj22c2τ,xτ∗2-∑k=12γkIKτ,xτ∗xτ∗1/2-δxτ∗Δt-oΔt.
Dividing (28) throughout by Δt, with Δt→0, and taking expectation, one can obtain (22). Hence Theorem 4 follows.
When the two nations adopt the cooperative strategies, the rate of instantaneous payment that nationk (k∈1,2) will realize at time τ with the state being xτ∗ can be expressed as(29a)S1τ,x=r2A1xτ∗+B1+A^xτ∗+B^-A2xτ∗+B2-12A1+A^-A2∑i=1nβi12+∑j=1mβj224ε11+ε22-A^2+∑k=12γk2A^xτ∗2ak-δxτ∗,(29b)S2τ,x=r2A2xτ∗+B2+A^xτ∗+B^-A1xτ∗+B1-12A2+A^-A1∑i=1nβi12+∑j=1mβj224ε11+ε22-A^2+∑k=12γk2A^xτ∗2ak-δxτ∗.


## 4.1. Group Optimality and Cooperative State Trajectory

To secure group optimality the participating two nations would seek to maximize their joint expected payoff by solving the following stochastic control problem:(13)Ws,x=Et0∫0∞∑i=1nβi122c1s+∑j=1mβj222c2s-12∑k=12akIks2-ε11+ε12∑i=1nβi124c12s-ε21+ε22∑j=1mβj224c22s-h1+h2xse-rtdtsubject to (7).Invoking Bellman’s technique, a set of controlsc^k∗,I^k∗=ϕ^kc(x),ϕ^kI(x) constitutes an optimal solution to the stochastic control problem (13) and (7), if there exists continuously differentiable function W(t,x):t,∞×R→R, k∈1,2, satisfying the following partial differential equations:(14)rWt,x-12σ2x2Wxxt,x=maxc1,I1,c2,I2∑i=1nβi122c1t+∑j=1mβj222c2t-12∑k=12akIkt2-ε11+ε12∑i=1nβi124c12t-ε21+ε22∑j=1mβj224c22t-h1+h2xt+WXt,x∑i=1nαi1βi124c12t+∑j=1mαj2βj224c22t-∑k=12γkIktxt1/2-δxt.Performing the indicated maximization in (14) yields the optimal controls under cooperation:(15a)ϕ^1ct,x=ε11+ε12-Wxt,x,ϕ^2ct,x=ε21+ε22-Wxt,x,(15b)ϕ^1Ix=-γ1Wxxx1/2a1,ϕ^2Ix=-γ2Wxxx1/2a2.Substituting the results in (15a) and (15b) into (14), we obtain the following.Proposition 3.
System (13) admits a solution(16)Wt,x=A^xt+B^,where A^, B^ satisfy the following set of constant coefficient quadratic ordinary differential equations:(17a)γ122a1+γ222a2A^2-r+δA^-h1+h2=0,(17b)B^=1r∑i=1nβi12+∑j=1mβj224ε11+ε22-A^2.The corresponding feedback Nash equilibriums under cooperation can be obtained as(18)ϕ^1ct,x=ε11+ε22-A^,ϕ^1It,x=-γ1A^x1/2t2a1,ϕ^2ct,x=ε11+ε22-A^,ϕ^2It,x=-γ2A^x1/2t2a2.Substituting the optimal control strategy from (18) into (7) yields the dynamics of pollution accumulation under cooperation:(19)dxs=∑i=1nαi1βi12+∑j=1mαj2βj224ε11+ε22-A^2+∑k=12γk2A^xs2ak-δxsds+σxsdzs,xt0=x0.Solving the stochastic cooperative pollution dynamics yields the cooperative state trajectory:(20)x∗t=e∫t0tγ12/2a1+γ22/2a2A^-δ-σ/2ds+∫t0tσdzs×xt0+∫t0t∑i=1nαi1βi12+∑j=1mαj2βj224ε11+ε22-A^2×e∫t0tσ2/2+δ-γ12/2a1+γ22/2a2A^dτ+∫t0tσdzsds,fort∈t0,∞.


## 4.2. Individually Rational and Time-Consistent Imputation and Payment Distribution Mechanism

LetXt∗ denote the set of realizable values of x∗(t) at time t generated by (20). The term xt∗ is used to denote an element of the set Xt∗. An agreed upon optimality principle would be sought to allocate the cooperative payoff. In a dynamic framework, individual rationality has to be maintained at every instant of time within the cooperative duration t0,∞ along the cooperative trajectory (20). For τ∈t0,∞, let vector ξk(xt∗)=ξ1(xt∗),ξ2(xt∗) denote the solution imputation (payoff under cooperation) over the period τ,∞ to player k∈1,2 given that the state is xτ∗∈Xτ∗. Individual rationality along the cooperative trajectory requires(21)ξτkτ,xt∗≥Vτkτ,xt∗,fork∈1,2,where V(τ)k(τ,xt∗) denotes the payoff to nation k under noncooperation over the period. Let G(s)=[G1(s),G2(s)] denote the instantaneous payoff of the cooperative game at time s∈t0,∞ for the cooperative Γc(xt0∗).Theorem 4.
An instantaneous payment at timeτ∈[t0,T] equaling(22)GKτ=rξkxτ∗-12σ2xτ∗2ξxτ∗xτ∗kxτ∗-ξxτ∗kxτ∗∑i=1nαi1βi12c1τ,xτ∗2+∑j=1mαj2βj22c2τ,xτ∗2-∑k=12γkIkτ,xτ∗xτ∗1/2-δxτ∗,forτ∈t0,∞,xτ∗∈Xτ∗,k∈1,2yields a subgame consistent solution to the cooperative game Γc(xt0∗).Proof.
Along the cooperative trajectoryx∗(t)t≥t0, we have(23)Υkτ,τ,xτ∗=ξkxτ∗=Eτ∫τ∞Gkse-rs-τds∣xτ=xτ∗,Υkτ,t,xt∗=ξkxt∗=Et∫t∞Gkse-rs-tds∣xt=xt∗,fork∈1,2,xt∗∈Xt∗,t≥τ.
Note that(24)Υkτ;t,xt∗=e-rs-τξxt∗=e-rs-τΥkt;t,xt∗,k∈1,2,xτ∗∈Xτ∗.
SinceΥk(t;t,xt∗) is continuously differentiable in t and xt∗, one can obtain(25)Υkτ,τ,xτ∗=Eτ∫ττ+ΔtGkse-rs-τds+e-rΔtΥkτ+Δt;τ+Δt,xτ∗+Δxτ∗∣xτ=xτ∗,forτ∈t0,∞,xτ∗∈Xτ∗,k∈1,2,where(26)Δxτ∗=∑i=1nαi1βi12c1τ,xτ∗2+∑j=1mαj2βj22c2τ,xτ∗2-∑k=12γkIKτ,xτ∗xτ∗1/2-δxτ∗Δt+σxτ∗Δzτ+oΔt,Δzτ=zτ+Δt-zτ,EτoΔtΔt⟶0,asΔt⟶0.
From (24) and (25), one can obtain(27)Υkτ,τ,xτ∗=Eτ∫ττ+ΔtGkse-rs-τds+Υkτ;τ+Δt,xτ∗+Δxτ∗∣xτ=xτ∗,forτ∈t0,∞,xτ∗∈Xτ∗,k∈1,2.
Therefore, withΔt→0, (27) can be expressed as(28)GkτΔt=-Υtkτ;t,xτ∗t=τΔt-12σ2xτ∗2Υxτ∗xτ∗kτ;t,xτ∗t=τΔt-Υxτ∗kτ;t,xτ∗t=τ×∑i=1nαi1βi12c1τ,xτ∗2+∑j=1mαj2βj22c2τ,xτ∗2-∑k=12γkIKτ,xτ∗xτ∗1/2-δxτ∗Δt-oΔt.
Dividing (28) throughout by Δt, with Δt→0, and taking expectation, one can obtain (22). Hence Theorem 4 follows.
When the two nations adopt the cooperative strategies, the rate of instantaneous payment that nationk (k∈1,2) will realize at time τ with the state being xτ∗ can be expressed as(29a)S1τ,x=r2A1xτ∗+B1+A^xτ∗+B^-A2xτ∗+B2-12A1+A^-A2∑i=1nβi12+∑j=1mβj224ε11+ε22-A^2+∑k=12γk2A^xτ∗2ak-δxτ∗,(29b)S2τ,x=r2A2xτ∗+B2+A^xτ∗+B^-A1xτ∗+B1-12A2+A^-A1∑i=1nβi12+∑j=1mβj224ε11+ε22-A^2+∑k=12γk2A^xτ∗2ak-δxτ∗.


## 5. Numerical Example

Consider a multinational economy consisting of two nations, and each with two industrial firms. Industrial firm’s revenue function in nation 1 and nation 2 at times is ϕi=βi1(ui(s))1/2 and ϕj=βj2(vj(s))1/2 (i,j=1,2), respectively, where β11=60, β21=90, β12=40, and β22=70. The value of βi1 (i∈1,2) is higher than βj2 (j∈1,2); it means that, compared with industrial firms in nation 2, industrial firms in nation 1 have more energy-efficient technology. In addition, industrial firms in nation 1 have technical advantage result in α11<α12, α21<α22, γ1>γ2. As mentioned above, the optimal pollution emissions of the firms in nations 1 and 2 are ui∗(s)=(βi1/2c1(s))2 and vj∗(s)=(βj2/2c2(s))2, respectively. The dynamics of pollution stock are governed by the stochastic differential equation (4), where x(t0)=20, δ=0.01, and σ=0.05. We defined ak as the amount added to the pollution stock by a unit of output in nation k (k=1,2). They are a positive number less than 1; that is to say, the stock of pollution will increase with the output but not higher than the amount of output. According to experience, we assign α11=0.4, α21=0.3, α12=0.5, α22=0.4. γk (k=1,2) can also be considered as pollution abatement effort efficiency, which is linked to pollution control policies and environmental governance efficiency in different nations. Generally speaking, pollution abatement effort efficiency is higher in developed countries. We assign γ1=0.4, γ2=0.2. Abatement costs are 0.4(I1(s))1/2 and 0.5(I2(s))1/2 for nations 1 and 2, respectively. Local pollution impacts are closely linked to the level of production activities. The accumulation of pollution stock like greenhouse gas often concerns the interactions between the natural environment and the pollutants emitted. So stochastic elements would appear. According to what we defined before, εk1,εk2 (k=1,2) is a pollution damage cost coefficients. It is said that the bigger they are, the higher the cost of environmental damage for this nation is. We make the following assignments: ε11=0.2, ε12=0.15, ε22=0.15, and ε21=0.16. hk (k=1,2) are the value of pollution damage to nation k (k=1,2). The larger they are, the greater the impact of the environment is, which is determined by the different regions. And we make assignments: h1=4, h2=5. Other parameters are as follows: r=0.05, t0=0. The parameters of the model, which are used in the numerical solution, are presented in Table 1.Table 1
The parameters of model used in the numerical solutions.
t
0
β
11
β
21
β12
β22
x
(
t
0
)
α11
α21
α12
α22
γ1
γ2
0
60
90
40
70
20
0.4
0.3
0.5
0.4
0.4
0.2
δ
σ
a1
a2
ε11
ε12
ε22
ε21
r
h1
h2
0.01
0.05
0.8
1
0.2
0.15
0.15
0.16
0.05
4
5Then we obtain a stochastic differential game in which government 1 and government 2 under noncooperation seek(30a)maxc1s,I1s∫0∞e-rt5850c1s-0.4I12s-585c12s-162.2c22s-4xsdx,(30b)maxc2s,I2s∫0∞e-rt3250c2s-0.5I22s-438.75c12s-243.75c22s-5xsdx.Through the analysis mentioned above, we have obtained the pollution tax rate, abatement effort of each nation, and the dynamics of pollution stock under noncooperative game and cooperative game. Now we will analyze their difference between the governments under noncooperation and cooperation through Table2 and Figures 1–5.Table 2
The optimal pollution tax rate.
c
1
(
t
)
c
2
(
t
)
c
^
1
(
t
)
c
^
2
(
t
)
8.20
2.34
10.90
10.87Figure 1
The pollution abatement effort under noncooperation, whereI1 and I2 denote the pollution abatement effort of governments under noncooperation in nations 1 and 2, respectively.Figure 2
The pollution abatement effort under cooperation, whereI3 and I4 are the pollution abatement effort of governments under cooperation in nations 1 and 2, respectively.Figure 3
The dynamics of pollution stock under noncooperation or cooperation, wherex1 and x2 denote the pollution stock under noncooperation and cooperation, respectively.Figure 4
The instantaneous payments of nation 1 under cooperation.Figure 5
The instantaneous payments of nation 2 under cooperation.From Table2, we can see that, for each government, the noncooperative optimal tax rates c1(t) and c2(t) are less than the cooperative tax rates c^1(t) and c^2(t). This means that cooperation would push to raise the tax rate to reduce production and cut pollution. And, for both governments, the optimal pollution tax rate is not changing with time; this is different from Huang et al. [10] and Yeung and Petrosyan [2], because, in their research, they model a finite-horizon level game regardless of cooperative game or noncooperative game, and, in the later stage of game, both governments would pay more attention to the economic benefits than pollution abatement. Then they would decrease the pollution tax rate to encourage production and stimulate economic development [10]. But, in our model, the game is in infinite-horizon, and there is no later stage of the game, so the optimal pollution tax rate will not decrease. In addition, under noncooperation, the governments take the externalities of production into account including their own nation only. However, under a cooperation, the externalities are considered by the governments, which include their own nation and the neighbors, so the optimal pollution tax rate is higher than under noncooperation.Figures1 and 2 show the pollution abatement effort of both governments. From them we find that, under noncooperation, the pollution abatement effort of both governments in the two nations could remain stable all the time, but, under cooperation, both governments would make a high pollution abatement effort at first, and then, the effort level may reduce to a certain degree; after that, it would tend to remain stable. For each government, the noncooperative pollution abatement effort is lower than the cooperative pollution abatement effort. This is determined by externality of pollution emissions. Under cooperative game, the governments pay attention to the damage for the entire region, so marginal damage of pollution emission would be greater and the governments would be more concerned about pollutant reduction.Figure3 shows the dynamics of pollution stock under noncooperation or cooperation. We can see that, the pollution stock under noncooperative game is higher than that under cooperative game. It is due to the higher pollution tax rate and higher pollution abatement effort of both governments under cooperative game. Then, in both economic and environmental terms, cooperation is always better than noncooperation.Now consider the case when the two nations want to cooperate. To secure group optimality the two nations seek to maximize their joint expected payoff by solving the following stochastic control problem:(31)maxc1s,c2s,I1s,I2sE0∫0∞5850c1s+3250c2s-0.4I1s2-0.5I2s2-1023.75c1s2-503.75c2s2-10xse-0.05dtsubject to(32)dxs=3.49-1.28xs+0.05xsdzs.According to Theorem4, we get the appropriate dynamic distribution scheme that both group optimality and individual rationality are satisfied at any time during cooperative game. The instantaneous payments of two nations, S1(t,xt∗) and S2(t,xt∗), at different time t with given xt∗ are shown in (33a) and (33b) and Figures 3 and 4. (33a)S1t,xt∗=1007100178948148626983247163840000000000000-12670021998361e-32t/2578125000000,(33b)S2t,xt∗=77822033722351462743911532621440000000000000-1518945509601e-32t/2519531250000.It is shown in Figures4 and 5 that the instantaneous payment of both nations which satisfy the property of subgame consistency shows an increasing trend at the start and then remains stable. In addition, the instantaneous payment for nation 1 is always higher than that of nation 2. This is because the industrial firms in nation 1 have more green and energy-efficient technology, so they would get more instantaneous payment.


## 6. Conclusions

In this paper, we have shown a stochastic differential game of transboundary industrial pollution between two asymmetric nations in infinite-horizon level. In our model, the firms and governments are separate entities. As participants in the games, they take game behavior to optimize their own payment, and the game equilibrium has been solved under the two levels of firms and governments simultaneously. The uncertain evolution dynamic of pollution stock that has a long-term global impact has been taken into account in our models. Otherwise, another negative environmental externality, a regional nonaccumulative pollutant that causes a short-term local impact, has also been considered. We characterize the parameters of spaces where two neighboring governments can cooperate and give the feedback Nash equilibrium strategies of governments and industrial firms. Through analysis, we have shown that the optimal pollution tax rate hinges on the two effects of regional damage: the source region and the global pollution stock, under noncooperative game, while, under cooperative game, in addition to the above two kinds of effects, it also hinges on the additional effect of regional damage coming from the neighborhood, so the latter is greater than the former. Moreover, we have shown that the pollution abatement effort under cooperation is more than under noncooperation. By analyzing this stochastic differential game model, we have also provided a payment distribution mechanism, which supports the subgame consistent solution, and present a numerical example to illustrate the implementation of this mechanism.



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