Research ArticleA Cooperative Stochastic Differential Game of TransboundaryIndustrial Pollution between Two Asymmetric Nations

Yongxi Yi Rongwei Xu and Sheng Zhang

School of Economics amp Management University of South China Hengyang China

Correspondence should be addressed to Yongxi Yi yyx19999126com

Received 23 December 2016 Accepted 13 March 2017 Published 18 April 2017

Academic Editor Huanqing Wang

Copyright copy 2017 Yongxi Yi et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Considering the fact that transboundary pollution control calls for the cooperation between interested parties this paper studies acooperative stochastic differential game of transboundary industrial pollution between two asymmetric nations in infinite-horizonlevel In this paper we model two ways of transboundary pollution one is an accumulative global pollutant with an uncertainevolutionary dynamic and the other is a regional nonaccumulative pollutant In our model firms and governments are separatedentities and they play a Stackelberg game while the governments of the two nations can cooperate in pollution reduction Wediscuss the feedback Nash equilibrium strategies of governments and industrial firms and it is found that the governments beingcooperative in transboundary pollution control will set a higher pollution tax rate and make more pollution abatement effort thanwhen they are noncooperative Additionally a payment distribution mechanism that supports the subgame consistent solution isproposed

1 Introduction

Global environment is a whole ecosystem that is intercon-nected and indivisible Any environmental pollution misuseof resources and ecological damage risk pressures are likelyto be cross-border Through pathways such as water orair pollutants can spread across an incredible distance tocause transboundary pollution Atmospheric pollution andacid rain ocean and river basin water pollution hazardouswaste transponder movement and other issues have beengradually a threat to human survival and development whichbecome a core issue with international concern With thedevelopment of the industrialization the transboundary pol-lution in the world has become increasingly serious To settlethe problem unilateral response on the part of one nationor region is often ineffective it calls for cooperation betweeninterested parties [1 2] In this paper we present a coop-erative stochastic differential game model of transboundaryindustrial pollution between two asymmetric nations Ourobjective is to find the optimal pollution abatement strategywhich maximizes the social welfare for the two nationsgaming for transboundary industrial pollution control

Differential games have been used as an effective tool tostudy transboundary pollution control For example Dock-ner and Van Long [3] model a simple dynamic game of twoneighboring countries which game for transboundary pollu-tion control it is found that when the governments of the twocountries are restricted to use of linear strategies noncooper-ative behavior may result in overall losses for both countriesIn 1998 Zagonari extends the model of Dockner and VanLong to analyze a pollution control game between envi-ronmentally concerned countries and consumption-orientedcountries In 1993 a taxsubsidy scheme is presented byMartin et al to combat the transboundary problem of globalclimate change Petrosjan and Zaccour [4] apply the Shapleyvalue to determine a fair distribution of the total cooperativecost incurred by countries in a cooperative game of pollutionreduction Bayramoglu [5] uses a dynamic and strategicframework to analyze the transboundary pollution betweenRomania and Ukraine it is found that among the threedifferent institutional systems such as noncooperative gameof countries uniform emission policy and constant emissionpolicy the noncooperative game can provide the highest levelof totalwelfare Joslashrgensen andZaccour [6] take emissions and

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 9492582 10 pageshttpsdoiorg10115520179492582

2 Mathematical Problems in Engineering

investments in abatement technology as control variables toanalyze the problem of cooperative transboundary industrialpollution control in their study a coordinated solution thatmaximizes joint welfare is derived together with a paymentdistribution mechanism that supports the subgame consis-tent solution Bertinelli et al [7] study the strategic behaviorof two countries facing transboundary CO2 pollution undera differential game setting the author considers that feedbackstrategy may lead to less social waste than when the countriestake open-loop strategy Taking emission permits trading intoaccount Li [8] studies a differential game of transboundaryindustrial pollution between two neighboring countries andthe focus of the authorrsquos analysis is to find the two countriesrsquononcooperative and cooperative optimal emission pathsBenchekroun and Martın-Herran [9] study the impact offoresight in a transboundary pollution game their studyshows that when all countries are myopic that is choose theldquolaisser-fairerdquo policy their payoffs are smaller than when allcountries are farsighted

Most cooperative environmental games do not distin-guish the government from the industrial sector in anynation Apart from these studies Yeung and Petrosyan [2]take industries and governments as separated entities anddevelop a cooperative differential gamemodel of transbound-ary industrial pollution and the time-consistent solutionsare derived Extending the work of Yeung and Petrosyan[2] Huang et al [10] develop a model in which there isa Stackelberg game between the industrial firms and theirlocal government while the governments can cooperate intransboundary pollution control and the feedback Nashequilibrium strategies of governments and industrial firmsare given In this paper we raise a stochastic differentialgame model of transboundary industrial pollution betweentwo asymmetric countries Our work can be viewed as acontinuing work of Huang et al [10] in the context oftransboundary industrial pollution control Compared withthe work of Huang et al [10] there are three significantfeatures in our paper (i) We present a stochastic differentialgame model of transboundary industrial pollution in whichuncertain pollution stock dynamics are taken into account Itis because that uncertain pollution stock dynamics is frequent[2 11ndash13] actually many factors like unexpected changes inabatement technologies the unexpected changes of decay rateof the pollutants and so forth may incur uncertain pollutionstock dynamics (ii) We extend the model set by Huang et al[10] from finite-horizon level to infinite-horizon level this isbecause in many situations the terminal time of the game119879 is either very far in the future or unknown to the playersA way to resolve the problem as suggested by Dockner andNishimura [14] is to set 119879 = infin so it makes sense to presenta model in infinite-horizon level to analyze transboundaryindustrial pollution control In our model we find that thedifference of the horizontal level does play an important effecton the behavior of game players From the governmentrsquos pointof view the plan is based on long-term interests so the plan-ning period for the infinite-horizon level is more reasonableFor example we find that in the infinite-horizon level theoptimal tax rate and the pollution abatement effort are rela-tively stable While in finite-horizon level Huang et al [10]

consider the optimal tax rate and the pollution abatementeffort decrease in the late game it means that a hyperopiagovernment can protect the environment better (iii) In ourmodel the asymmetry of participants in game is consideredwe assume that the industrial firms in one country have moregreen and energy-efficient technologies than the others andwe show that the one with technology advantage may seta higher pollution tax and make more pollution abatementeffort no matter what condition it would be in cooperationor noncooperation

The paper is organized as follows In Section 2 the gameformulation is provided In Section 3 we characterize thenoncooperative outcomes Cooperative arrangements andindividual rationality are analyzed in Section 4 We illustratethe results of a numerical example in Section 5 Section 6 isthe results summarizing the paper

2 Game Formulation

21 Government and Domestic Industrial Firm A StackelbergGame Consider a multinational economy which is com-prised of two nations There are 119899 and 119898 industrial firmsin nation 1 and nation 2 respectively In order to controlproduction pollution the government (the leader) of eachnation imposes a pollution tax on domestic industrial firmsand then the industrial firms (the followers) choose theiroptimal output (emissions) according to the pollution taxrate This leads to a Stackelberg equilibrium

Letsum119899119894=1 119906119894(119904) andsum119898119895=1 V119895(119904) denote the quantity of goodsproduced by 119899 firms in nation 1 and 119898 firms in nation 2 attime 119904 respectively Firm 119894rsquos (or 119895rsquos) revenue function120601119894 (or120601119895)is assumed to be concave and increasing with the followingsimple functional form

120601119894 = 1205731198941 (119906119894 (119904))12 120601119895 = 1205731198952 (V119895 (119904))12 (1)

where 1205731198941 1205731198952 gt 0 are constants the instantaneous profits ofindustrial firm in the two nations can be expressed as

120601119894 (119906119894 (119904)) minus 1198881 (119904) 119906119894 (119904) 120601119895 (V119895 (119904)) minus 1198882 (119904) V119895 (119904) (2)

where 119888119896(119904) is the pollution tax rate imposed on industrialfirms by government 119896 (119896 isin 1 2) at time 119904

Through the first-order condition of (2) the optimalyields for the firms in nation 1 and nation 2 could be obtainedas follows

119906lowast119894 (119904) = ( 120573119894121198881 (119904))2

Vlowast119895 (119904) = ( 120573119895221198882 (119904))2

(3)

22 Local and Global Environmental Impacts Transbound-ary pollution may damage the environment through two

Mathematical Problems in Engineering 3

ways that is an accumulative global pollutant and a regionalnonaccumulative pollutant For example some transbound-ary pollutants are caused by firms such as passing-by waste inwaterways wind-driven suspended particles in air unpleas-ant odour noise dust and heat which are all nonaccu-mulative pollutants We assume that the nonaccumulativepollutant for an output of 119906119894(119904) produced by the firm innation 1 would cause a short-term impact (cost) of 12057611119906119894(119904)and 12057621119906119894(119904) on nation 1 and nation 2 respectively Similarly anoutput of V119895(119904)produced by the firm innation 2would cause ashort-term (cost) of 12057612V119895(119904) and 12057622V119895(119904) on nation 1 and nation2 respectively In addition to the nonaccumulative pollutantthere is an accumulative pollutant such as green-house-gas CFC and atmospheric particulates these pollutantscan be maintained in environment for a long time andbuilt up existing pollution stocks to create long-term globalenvironmental impacts [2 5 10 15 16] Let 119909(119904) sub 119877+ denotethe level of pollution at time 119904 and the dynamics of pollutionstock are governed by the stochastic differential equation

119889119909 (119904) = 119899sum119894=1

1205721198941119906119894 (119904) + 119898sum119895=1

1205721198952V119895 (119904)minus 2sum119896=1

120574119896 (119868119896 (119904) (119909 (119904))12 minus 120575119909 (119904)) 119889119904+ 120590119909 (119904) 119889119911 (119904) 119896 isin 1 2

(4)

where 120590 denotes a noise parameter and 119911(119904) is a Wienerprocess 1205721198941 ge 0 and 1205721198952 ge 0 are the amount added tothe pollution stock by a unit of firms 119894rsquos and 119895rsquos output innation 1 and nation 2 respectively 119868119896(119904) denotes the pollutionabatement effort of nation 119896 120574119896 is the efficiency of theabatement 120574119896119868119896(119904)(119909(119904))12 denotes the amount of pollutionremoved by 119868119896(119904) unit of abatement effort of nation 119896 and 120590denotes the natural rate of decay of the pollutants

23 The Governmentsrsquo Objectives The instantaneous objec-tive of government 119896 (119896 isin 1 2) at time 119904 can be expressed as

119899sum119894=1

1205731198941 (119906lowast119894 (119904))12 minus 121198861 (1198681 (119904))2 minus 12057611 119899sum119894=1

119906lowast119894 (119904)minus 12057612 119898sum119895=1

Vlowast119895 (119904) minus ℎ1119909 (119904) (5a)

119898sum119895=1

1205731198951 (Vlowast119895 (119904))12 minus 121198862 (1198682 (119904))2 minus 12057621 119899sum119894=1

119906lowast119894 (119904)minus 12057622 119898sum119895=1

Vlowast119895 (119904) minus ℎ2119909 (119904) (5b)

where 119886119896 ℎ119896 gt 0 (119896 isin 1 2) are constants (12)119886119896(119868119896(119904))2denotes the cost of employing 119868119896(119904) amount of pollutionabatement effort and ℎ119896119909(119904) denotes the value of damage tonation 119896 from 119909(119904) amount of pollution

The governmentrsquos planning horizon is [1199050infin) The dis-count rate is 119903 Each government seeks to maximize theintegral of its instantaneous objectives (5a) and (5b) over theplanning horizon [1199050infin) subjected to pollution dynamics(4) with controls on the level of abatement effort andpollution tax

Substituting 119906119894(119904) and V119895(119904) from (3) into (4) and (5a)and (5b) one obtains a stochastic differential game in whichgovernment 119896 isin 1 2 seeks

max1198881(119904)1198681(119904)

1198641199050 intinfin1199050 119890minus119903119905(sum119899119894=1 (1205731198941)221198881 (119904) minus 121198861 (1198681 (119904))2

minus 12057611 sum119899119894=1 (1205731198941)2411988821 (119904) minus 12057612 sum119898119895=1 (1205731198952)2411988822 (119904)minus ℎ1119909 (119904))119889119904

(6a)

max1198882(119904)1198682(119904)

1198641199050 intinfin1199050 119890minus119903119905(sum119898119895=1 (1205731198952)221198882 (119904) minus 121198862 (1198682 (119904))2

minus 12057621 sum119899119894=1 (1205731198941)2411988821 (119904) minus 12057622 sum119898119895=1 (1205731198952)2411988822 (119904) minus ℎ2119909 (119904))119889119904(6b)

subject to

119889119909 (119904) = 119899sum119894=1

1205721198941 ( 120573119894121198881 (119904))2 + 119898sum119895=1

1205721198952 ( 120573119895221198882 (119904))2

minus 2sum119896=1

120574119896 (119868119896 (119904) (119909 (119904))12 minus 120575119909 (119904)) 119889119904+ 120590119909 (119904) 119889119911 (119904)

(7)

Since the payoffs of nations are measured in monetaryterms 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 noncooperativeframework

A feedback Nash equilibrium solution can be character-ized as follows

Definition 1 A set of feedback strategies [119888lowast119896 (119905) 119868lowast119896 (119905)] =[120601119888119896(119905 119909) 120601119868119896(119905 119909)] for 119896 isin 1 2 provides a Nash equilibriumsolution to the games (6a) (6b) and (7) if there exist suitably

4 Mathematical Problems in Engineering

smooth functions 119881119896(119905 119909) [1199050infin) times 119877 rarr 119877 119896 isin 1 2satisfying the following partial differential equations

1199031198811 (119905 119909) minus 12120590211990921198811119909119909 (119905 119909) = max1198881(119905)1198681(119905)

sum119899119894=1 (1205731198941)221198881 (119905)

minus 121198861 (1198681 (119905))2 minus 12057611 sum119899119894=1 (1205731198941)2411988821 (119905) minus 12057612 sum119898119895=1 (1205731198952)2411988822 (119905)minus ℎ1119909 (119905) + 1198811119909 (119905 119909)(sum119899119894=1 (1205731198941)2411988821 (119905) + sum119898119895=1 (1205731198952)2411988822 (119905)minus 2sum119896=1

120574119896 (119868119896 (119905) (119909 (119905))12 minus 120575119909 (119905)))

(8a)

1199031198812 (119905 119909) minus 12120590211990921198812119909119909 (119905 119909) = max1198882(119905)1198682(119905)

sum119898119895=1 (1205731198952)221198882 (119905)

minus 121198862 (1198682 (119905))2 minus 12057621 sum119899119894=1 (1205731198941)2411988821 (119905) minus 12057622 sum119898119895=1 (1205731198952)2411988822 (119905)minus ℎ2119909 (119905) + 1198812119909 (119905 119909)(sum119899119894=1 (1205731198941)2411988821 (119905) + sum119898119895=1 (1205731198952)2411988822 (119905)minus 2sum119896=1

120574119896 (119868119896 (119905) (119909 (119905))12 minus 120575119909 (119905)))

(8b)

Performing the indicated maximization in (8a) and (8b)yields

1206011198881 (119905 119909) = 12057611 minus 1198811119909 (119905 119909) 1206011198882 (119905 119909) = 12057622 minus 1198812119909 (119905 119909) (9a)

1206011198681 (119905 119909) = minus12057411198811119909 (119905 119909) 11990912 (119905)1198861 1206011198682 (119905 119909) = minus12057421198812119909 (119905 119909) 11990912 (119905)1198862

(9b)

Substituting the results of (9a) and (9b) into (8a) and (8b)we obtain the following

Proposition 2 The government 119896rsquos payoff during [1199050infin)119881119896(119905 119909) can be obtained as

119881119896 (119905 119909) = [119860119896119909 (119905) + 119861119896] for 119896 isin 1 2 (10)

where 119860119896 119861119896 satisfy the following set of constant coefficientquadratic ordinary differential equations

120574211198602121198861 + 12057422119860111986021198862 minus 1205741198601 minus 1205751198601 minus ℎ1 = 0 (11a)

120574221198602221198862 + 12057421119860111986021198861 minus 1205741198602 minus 1205751198602 minus ℎ2 = 0 (11b)

1198611 = 1119903 ( sum119899119894=1 (1205731198941)24 (12057611 minus 1198601) + (1198601 minus 12057612)sum119898119895=1 (1205731198952)24 (12057622 minus 1198602)2 ) (11c)

1198612 = 1119903 (sum119898119895=1 (1205731198951)24 (12057622 minus 1198602) + (1198602 minus 12057621)sum119899119894=1 (1205731198941)24 (12057611 minus 1198601)2 ) (11d)

The corresponding feedback Nash equilibrium of the games(8a) and (8b) can be obtained as

1206011198881 (119905 119909) = 12057611 minus 11986011206011198681 (119905 119909) = minus1205741119860111990912 (119905)1198861 1206011198882 (119905 119909) = 12057622 minus 11986021206011198682 (119905 119909) = minus1205742119860211990912 (119905)1198862

(12)

Equation (12) indicates that under noncooperation opti-mal pollution tax rate (1206011198881 or 1206011198882) may depend on the regionalpollutant (12057611 or 12057622) which is caused by the firms in homebut the regional pollutant (12057621 or 12057612) which is caused by thefirms in neighbormay have no effect on optimal pollution taxrate In addition the optimal abatement effort of the regiondoes not depend on the damage from the regional pollutantat home (12057611 or 12057622) this is because the abatement effort ofthe nation can only affect pollution stock and cannot affectcurrent pollution

4 Cooperative Arrangements

Now consider the case when both nations cooperate inpollution control To uphold the cooperative scheme bothgroup rationality and individual rationality are required to besatisfied at any time

41 Group Optimality and Cooperative State Trajectory Tosecure group optimality the participating two nations would

Mathematical Problems in Engineering 5

seek to maximize their joint expected payoff by solving thefollowing stochastic control problem

119882(119904 119909) = 1198641199050 intinfin0 sum119899119894=1 (1205731198941)221198881 (119904) + sum119898119895=1 (1205731198952)221198882 (119904)

minus 122sum119896=1

119886119896 (119868119896 (119904))2 minus (12057611 + 12057621)sum119899119894=1 (1205731198941)2411988821 (119904)minus (12057612 + 12057622)sum119898119895=1 (1205731198952)2411988822 (119904) minus (ℎ1 + ℎ2) 119909 (119904)119890minus119903119905119889119905

(13)

subject to (7)Invoking Bellmanrsquos technique a set of controls 119888lowast119896 119868lowast119896 =120601119888119896(119909) 120601119868119896(119909) constitutes an optimal solution to the stochas-

tic control problem (13) and (7) if there exists continuouslydifferentiable function 119882(119905 119909) [119905infin) times 119877 rarr 119877 119896 isin 1 2satisfying the following partial differential equations

119903119882 (119905 119909) minus 1212059021199092119882119909119909 (119905 119909) = max1198881 11986811198882 1198682

sum119899119894=1 (1205731198941)221198881 (119905)

+ sum119898119895=1 (1205731198952)221198882 (119905) minus 122sum119896=1

119886119896 (119868119896 (119905))2

minus (12057611 + 12057621)sum119899119894=1 (1205731198941)2411988821 (119905) minus (12057612 + 12057622)sum119898119895=1 (1205731198952)2411988822 (119905)minus (ℎ1 + ℎ2) 119909 (119905) + 119882119883 (119905 119909)(sum119899119894=1 1205721198941 (1205731198941)2411988821 (119905)+ sum119898119895=1 1205721198952 (1205731198952)2411988822 (119905)minus 2sum119896=1

120574119896 (119868119896 (119905) (119909 (119905))12 minus 120575119909 (119905)))

(14)

Performing the indicated maximization in (14) yields theoptimal controls under cooperation

1206011198881 (119905 119909) = 12057611 + 12057621 minus 119882119909 (119905 119909) 1206011198882 (119905 119909) = 12057612 + 12057622 minus 119882119909 (119905 119909) (15a)

1206011198681 (119909) = minus1205741119882119909 (119909) 119909121198861 1206011198682 (119909) = minus1205742119882119909 (119909) 119909121198862

(15b)

Substituting the results in (15a) and (15b) into (14) weobtain the following

Proposition 3 System (13) admits a solution

119882(119905 119909) = [119860119909 (119905) + 119861] (16)

where 119860 119861 satisfy the following set of constant coefficientquadratic ordinary differential equations

( 1205742121198861 + 1205742221198862)1198602 minus (119903 + 120575)119860 minus (ℎ1 + ℎ2) = 0 (17a)

119861 = 1119903 (sum119899119894=1 (1205731198941)2 + sum119898119895=1 (1205731198952)24 (12057611 + 12057622 minus 119860)2 ) (17b)

The corresponding feedback Nash equilibriums undercooperation can be obtained as

1206011198881 (119905 119909) = 12057611 + 12057622 minus 1198601206011198681 (119905 119909) = minus120574111986011990912 (119905)21198861 1206011198882 (119905 119909) = 12057611 + 12057622 minus 1198601206011198682 (119905 119909) = minus120574211986011990912 (119905)21198862

(18)

Substituting the optimal control strategy from (18) into(7) yields the dynamics of pollution accumulation undercooperation

119889119909 (119904) = (sum119899119894=1 1205721198941 (1205731198941)2 + sum119898119895=1 1205721198952 (1205731198952)24 (12057611 + 12057622 minus 119860)2+ 2sum119896=1

(120574119896)2 119860119909 (119904)2119886119896 minus 120575119909 (119904))119889119904 + 120590119909 (119904) 119889119911 (119904) 119909 (1199050) = 1199090

(19)

Solving the stochastic cooperative pollution dynamicsyields the cooperative state trajectory

119909lowast (119905) = 119890[int1199051199050 [(12057421 21198861+12057422 21198862)119860minus120575minus1205902]119889119904+int1199051199050 120590119889119911(119904)] times [[1199091199050+ int1199051199050(sum119899119894=1 1205721198941 (1205731198941)2 + sum119898119895=1 1205721198952 (1205731198952)24 (12057611 + 12057622 minus 119860)2 )

times 119890[int1199051199050 [12059022+120575minus(12057421 21198861+12057422 21198862)119860]119889120591+int1199051199050 120590119889119911(119904)]119889119904]] for 119905 isin [1199050infin)

(20)

42 Individually Rational and Time-Consistent Imputationand Payment Distribution Mechanism Let 119883lowast119905 denote the

6 Mathematical Problems in Engineering

set of realizable values of 119909lowast(119905) at time 119905 generated by (20)The term 119909lowast119905 is used to denote an element of the set 119883lowast119905 An agreed upon optimality principle would be sought toallocate the cooperative payoff In a dynamic frameworkindividual rationality has to be maintained at every instantof time within the cooperative duration [1199050infin) along thecooperative trajectory (20) For 120591 isin [1199050infin) let vector120585119896(119909lowast119905 ) = 1205851(119909lowast119905 ) 1205852(119909lowast119905 ) denote the solution imputation(payoff under cooperation) over the period [120591infin) to player119896 isin 1 2 given that the state is 119909lowast120591 isin 119883lowast120591 Individualrationality along the cooperative trajectory requires

120585(120591)119896 (120591 119909lowast119905 ) ge 119881(120591)119896 (120591 119909lowast119905 ) for 119896 isin 1 2 (21)

where 119881(120591)119896(120591 119909lowast119905 ) denotes the payoff to nation 119896 undernoncooperation over the period Let 119866(119904) = [1198661(119904) 1198662(119904)]denote the instantaneous payoff of the cooperative game attime 119904 isin [1199050infin) for the cooperative Γ119888(119909lowast1199050)Theorem 4 An instantaneous payment at time 120591 isin [1199050 119879]equaling

119866119870 (120591) = 119903120585119896 (119909lowast120591 ) minus 121205902 (119909lowast120591 )2 120585119896119909lowast120591119909lowast120591 (119909lowast120591 ) minus 120585119896119909lowast120591

(119909lowast120591 )sdot ( 119899sum119894=1

1205721198941 ( 120573119894121198881 (120591 119909lowast120591 ))2 + 119898sum119895=1

1205721198952 ( 120573119895221198882 (120591 119909lowast120591 ))2

minus 2sum119896=1

120574119896119868119896 (120591 119909lowast120591 ) (119909lowast120591 )12 minus 120575119909lowast120591) for 120591 isin [1199050infin) 119909lowast120591 isin 119883lowast120591 119896 isin 1 2

(22)

yields a subgame consistent solution to the cooperative gameΓ119888(119909lowast1199050)Proof Along the cooperative trajectory 119909lowast(119905)119905ge1199050 we haveΥ119896 (120591 120591 119909lowast120591 ) = 120585119896 (119909lowast120591 )

= 119864120591 intinfin120591

119866119896 (119904) 119890minus119903(119904minus120591)119889119904 | 119909 (120591) = 119909lowast120591 Υ119896 (120591 119905 119909lowast119905 ) = 120585119896 (119909lowast119905 )

= 119864119905 intinfin119905

119866119896 (119904) 119890minus119903(119904minus119905)119889119904 | 119909 (119905) = 119909lowast119905 for 119896 isin 1 2 119909lowast119905 isin 119883lowast119905 119905 ge 120591

(23)

Note that

Υ119896 (120591 119905 119909lowast119905 ) = 119890minus119903(119904minus120591)120585 (119909lowast119905 ) = 119890minus119903(119904minus120591)Υ119896 (119905 119905 119909lowast119905 ) 119896 isin 1 2 119909lowast120591 isin 119883lowast120591 (24)

Since Υ119896(119905 119905 119909lowast119905 ) is continuously differentiable in 119905 and119909lowast119905 one can obtain

Υ119896 (120591 120591 119909lowast120591 ) = 119864120591 int120591+Δ119905120591

119866119896 (119904) 119890minus119903(119904minus120591)119889119904+ 119890minus119903(Δ119905)Υ119896 (120591 + Δ119905 120591 + Δ119905 119909lowast120591 + Δ119909lowast120591 ) | 119909 (120591)= 119909lowast120591 for 120591 isin [1199050infin) 119909lowast120591 isin 119883lowast120591 119896 isin 1 2

(25)

where

Δ119909lowast120591= 119899sum119894=1

1205721198941 ( 120573119894121198881 (120591 119909lowast120591 ))2 + 119898sum119895=1

1205721198952 ( 120573119895221198882 (120591 119909lowast120591 ))2

minus 2sum119896=1

120574119896 (119868119870 (120591 119909lowast120591 ) (119909lowast120591 )12 minus 120575119909lowast120591)Δ119905+ 120590 [119909lowast120591 ] Δ119911120591 + 119900 (Δ119905)

Δ119911120591 = 119911 (120591 + Δ119905) minus 119911 (120591) 119864120591 [119900 (Δ119905)]Δ119905 997888rarr 0

as Δ119905 997888rarr 0

(26)

From (24) and (25) one can obtain

Υ119896 (120591 120591 119909lowast120591 ) = 119864120591 int120591+Δ119905120591

119866119896 (119904) 119890minus119903(119904minus120591)119889119904+ Υ119896 (120591 120591 + Δ119905 119909lowast120591 + Δ119909lowast120591 ) | 119909 (120591) = 119909lowast120591

for 120591 isin [1199050infin) 119909lowast120591 isin 119883lowast120591 119896 isin 1 2 (27)

Therefore with Δ119905 rarr 0 (27) can be expressed as

119866119896 (120591) Δ119905 = minus [Υ119896119905 (120591 119905 119909lowast120591 )1003816100381610038161003816119905=120591] Δ119905 minus 121205902 (119909lowast120591 )2sdot [Υ119896119909lowast

120591119909lowast120591

(120591 119905 119909lowast120591 )1003816100381610038161003816119905=120591] Δ119905 minus [Υ119896119909lowast120591

(120591 119905 119909lowast120591 )1003816100381610038161003816119905=120591]times ( 119899sum119894=1

1205721198941 ( 120573119894121198881 (120591 119909lowast120591 ))2 + 119898sum119895=1

1205721198952 ( 120573119895221198882 (120591 119909lowast120591 ))2

minus 2sum119896=1

120574119896 (119868119870 (120591 119909lowast120591 ) (119909lowast120591 )12 minus 120575119909lowast120591)Δ119905 minus 119900 (Δ119905))

(28)

Dividing (28) throughout by Δ119905 with Δ119905 rarr 0 and takingexpectation one can obtain (22) HenceTheorem 4 follows

When the two nations adopt the cooperative strategiesthe rate of instantaneous payment that nation 119896 (119896 isin 1 2)

Mathematical Problems in Engineering 7

Table 1 The parameters of model used in the numerical solutions

1199050 12057311 12057321 12057312 12057322 119909(1199050) 12057211 12057221 12057212 12057222 1205741 12057420 60 90 40 70 20 04 03 05 04 04 02120575 120590 1198861 1198862 12057611 12057621 12057622 12057612 119903 ℎ1 ℎ2001 005 08 1 02 015 015 016 005 4 5

will realize at time 120591 with the state being 119909lowast120591 can be expressedas

1198781 (120591 119909) = 1199032 [1198601119909lowast120591 + 1198611] + [119860119909lowast120591 + 119861]minus [1198602119909lowast120591 + 1198612] minus 12 [1198601 + 119860 minus 1198602]sdot

sum119899119894=1 (1205731198941)2 + sum119898119895=1 (1205731198952)24 (12057611 + 12057622 minus 119860)2+ 2sum119896=1

1205742119896119860119909lowast1205912119886119896 minus 120575119909lowast120591

(29a)

1198782 (120591 119909) = 1199032 [1198602119909lowast120591 + 1198612] + [119860119909lowast120591 + 119861]minus [1198601119909lowast120591 + 1198611] minus 12 [1198602 + 119860 minus 1198601]sdot

sum119899119894=1 (1205731198941)2 + sum119898119895=1 (1205731198952)24 (12057611 + 12057622 minus 119860)2+ 2sum119896=1

1205742119896119860119909lowast1205912119886119896 minus 120575119909lowast120591

(29b)

5 Numerical Example

Consider a multinational economy consisting of two nationsand each with two industrial firms Industrial firmrsquos revenuefunction in nation 1 and nation 2 at time 119904 is 120601119894 = 1205731198941(119906119894(119904))12and 120601119895 = 1205731198952(V119895(119904))12 (119894 119895 = 1 2) respectively where12057311 = 60 12057321 = 90 12057312 = 40 and 12057322 = 70 The valueof 1205731198941 (119894 isin 1 2) is higher than 1205731198952 (119895 isin 1 2) it meansthat compared with industrial firms in nation 2 industrialfirms in nation 1 have more energy-efficient technology Inaddition industrial firms in nation 1 have technical advantageresult in 12057211 lt 12057212 12057221 lt 12057222 1205741 gt 1205742 As mentionedabove the optimal pollution emissions of the firms in nations1 and 2 are 119906lowast119894 (119904) = (120573119894121198881(119904))2 and Vlowast119895 (119904) = (120573119895221198882(119904))2respectively The dynamics of pollution stock are governedby the stochastic differential equation (4) where 119909(1199050) = 20120575 = 001 and 120590 = 005 We defined 119886119896 as the amountadded to the pollution stock by a unit of output in nation119896 (119896 = 1 2) They are a positive number less than 1 that isto say the stock of pollution will increase with the outputbut not higher than the amount of output According to

Table 2 The optimal pollution tax rate

1198881(119905) 1198882(119905) 1198881(119905) 1198882(119905)820 234 1090 1087

experience we assign 12057211 = 04 12057221 = 03 12057212 = 0512057222 = 04 120574119896 (119896 = 1 2) can also be considered as pollutionabatement effort efficiency which is linked to pollutioncontrol policies and environmental governance efficiency indifferent nations Generally speaking pollution abatementeffort efficiency is higher in developed countries We assign1205741 = 04 1205742 = 02 Abatement costs are 04(1198681(119904))12 and05(1198682(119904))12 for nations 1 and 2 respectively Local pollutionimpacts are closely linked to the level of production activitiesThe accumulation of pollution stock like greenhouse gas oftenconcerns the interactions between the natural environmentand the pollutants emitted So stochastic elements wouldappear According to what we defined before 1205761119896 1205762119896 (119896 = 1 2)is a pollution damage cost coefficients It is said that the biggerthey are the higher the cost of environmental damage forthis nation is We make the following assignments 12057611 = 0212057621 = 015 12057622 = 015 and 12057612 = 016 ℎ119896 (119896 = 1 2) arethe value of pollution damage to nation 119896 (119896 = 1 2) Thelarger they are the greater the impact of the environment iswhich is determined by the different regions And we makeassignments ℎ1 = 4 ℎ2 = 5 Other parameters are as follows119903 = 005 1199050 = 0 The parameters of the model which are usedin the numerical solution are presented in Table 1

Then we obtain a stochastic differential game in whichgovernment 1 and government 2 under noncooperation seek

max1198881(119904)1198681(119904)

intinfin0

119890minus119903119905 (58501198881 (119904) minus 0411986821 (119904) minus 58511988821 (119904) minus 162211988822 (119904)minus 4119909 (119904)) 119889119909

(30a)

max1198882(119904)1198682(119904)

intinfin0

119890minus119903119905 (32501198882 (119904) minus 0511986822 (119904) minus 4387511988821 (119904)minus 2437511988822 (119904) minus 5119909 (119904)) 119889119909

(30b)

Through the analysis mentioned above we have obtainedthe pollution tax rate abatement effort of each nationand the dynamics of pollution stock under noncooperativegame and cooperative game Now we will analyze theirdifference between the governments under noncooperationand cooperation through Table 2 and Figures 1ndash5

8 Mathematical Problems in Engineering

0 05 1 15 2 25 3 35 4 45 515

2

25

3

35

4

45

Pollu

tion

abat

emen

t effo

rt u

nder

non

coop

erat

ion

I2

I1

Time t

Figure 1 The pollution abatement effort under noncooperationwhere 1198681 and 1198682 denote the pollution abatement effort of govern-ments under noncooperation in nations 1 and 2 respectively

0 05 1 15 2 25 3 35 4 45 50

2

4

6

8

10

12

Pollu

tion

abat

emen

t effo

rt u

nder

coop

erat

ion

I4

I3

Time t

Figure 2 The pollution abatement effort under cooperation where1198683 and 1198684 are the pollution abatement effort of governments undercooperation in nations 1 and 2 respectively

From Table 2 we can see that for each governmentthe noncooperative optimal tax rates 1198881(119905) and 1198882(119905) are lessthan the cooperative tax rates 1198881(119905) and 1198882(119905) This meansthat cooperation would push to raise the tax rate to reduceproduction and cut pollution And for both governmentsthe optimal pollution tax rate is not changing with timethis is different from Huang et al [10] and Yeung andPetrosyan [2] because in their research they model afinite-horizon level game regardless of cooperative game ornoncooperative game and in the later stage of game bothgovernments would pay more attention to the economic

0 05 1 15 2 25 3 35 4 45 52

4

6

8

10

12

14

16

18

20

22

Pollu

tion

stock

x2

x1

Time t

Figure 3 The dynamics of pollution stock under noncooperationor cooperation where 1199091 and 1199092 denote the pollution stock undernoncooperation and cooperation respectively

0 05 1 15 2 25 3 35 4 45 5Time t

times106

Insta

ntan

eous

pay

men

ts of

natio

n 1

unde

r coo

pera

tion

61466861467

614672614674614676614678

61468614682614684614686

Figure 4 The instantaneous payments of nation 1 under coopera-tion

benefits than pollution abatementThen they would decreasethe pollution tax rate to encourage production and stimulateeconomic development [10] But in our model the gameis in infinite-horizon and there is no later stage of thegame so the optimal pollution tax rate will not decrease Inaddition under noncooperation the governments take theexternalities of production into account including their ownnation only However under a cooperation the externalitiesare considered by the governments which include their ownnation and the neighbors so the optimal pollution tax rate ishigher than under noncooperation

Figures 1 and 2 show the pollution abatement effort ofboth governments From them we find that under noncoop-eration the pollution abatement effort of both governmentsin the two nations could remain stable all the time butunder cooperation both governments would make a highpollution abatement effort at first and then the effort levelmay reduce to a certain degree after that it would tendto remain stable For each government the noncooperativepollution abatement effort is lower than the cooperativepollution abatement effort This is determined by externality

Mathematical Problems in Engineering 9In

stan

tane

ous p

aym

ents

of

natio

n 2

unde

r coo

pera

tion times10

6

0 05 1 15 2 25 3 35 4 45 5Time t

29685929686

296861296862296863296864296865296866296867296868

Figure 5 The instantaneous payments of nation 2 under coopera-tion

of pollution emissions Under cooperative game the govern-ments pay attention to the damage for the entire region somarginal damage of pollution emission would be greater andthe governments would be more concerned about pollutantreduction

Figure 3 shows the dynamics of pollution stock undernoncooperation or cooperation We can see that the pollu-tion stock under noncooperative game is higher than thatunder cooperative game It is due to the higher pollutiontax rate and higher pollution abatement effort of both gov-ernments under cooperative game Then in both economicand environmental terms cooperation is always better thannoncooperation

Now consider the case when the two nations want tocooperate To secure group optimality the two nations seek tomaximize their joint expected payoff by solving the followingstochastic control problem

max1198881(119904)1198882(119904)1198681(119904)1198682(119904)

1198640 intinfin0

(58501198881 (119904) + 32501198882 (119904) minus 04 (1198681 (119904))2

minus 05 (1198682 (119904))2 minus 102375(1198881 (119904))2 minus50375(1198882 (119904))2 minus 10119909 (119904))

sdot 119890minus005119889119905(31)

subject to

119889119909 (119904) = 349 minus 128119909 (119904) + 005119909 (119904) 119889119911 (119904) (32)

According toTheorem 4 we get the appropriate dynamicdistribution scheme that both group optimality and individ-ual rationality are satisfied at any time during cooperativegame The instantaneous payments of two nations 1198781(119905 119909lowast119905 )and 1198782(119905 119909lowast119905 ) at different time 119905 with given 119909lowast119905 are shown in(33a) and (33b) and Figures 3 and 4

1198781 (119905 119909lowast119905 ) = 1007100178948148626983247163840000000000000minus 12670021998361119890minus321199052578125000000 (33a)

1198782 (119905 119909lowast119905 ) = 77822033722351462743911532621440000000000000minus 1518945509601119890minus321199052519531250000 (33b)

It is shown in Figures 4 and 5 that the instantaneous pay-ment of both nations which satisfy the property of subgameconsistency shows an increasing trend at the start and thenremains stable In addition the instantaneous payment fornation 1 is always higher than that of nation 2This is becausethe industrial firms in nation 1 have more green and energy-efficient technology so they would get more instantaneouspayment

6 Conclusions

In this paper we have shown a stochastic differential game oftransboundary industrial pollution between two asymmetricnations in infinite-horizon level In our model the firmsand governments are separate entities As participants inthe games they take game behavior to optimize their ownpayment and the game equilibrium has been solved underthe two levels of firms and governments simultaneouslyThe uncertain evolution dynamic of pollution stock thathas a long-term global impact has been taken into accountin our models Otherwise another negative environmentalexternality a regional nonaccumulative pollutant that causesa short-term local impact has also been considered Wecharacterize the parameters of spaces where two neighboringgovernments can cooperate and give the feedback Nashequilibrium strategies of governments and industrial firmsThrough analysis we have shown that the optimal pollutiontax rate hinges on the two effects of regional damagethe source region and the global pollution stock undernoncooperative game while under cooperative game inaddition to the above two kinds of effects it also hingeson the additional effect of regional damage coming fromthe neighborhood so the latter is greater than the formerMoreover we have shown that the pollution abatement effortunder cooperation is more than under noncooperation Byanalyzing this stochastic differential game model we havealso provided a payment distribution mechanism whichsupports the subgame consistent solution and present anumerical example to illustrate the implementation of thismechanism

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge the financial supportfrom the ChinaNational Funding for Social Science ResearchProject (Project no 15CJY037) and the Philosophy and SocialScience Foundation of Hunan China (Project no 16YBA315)for this paper

10 Mathematical Problems in Engineering

References

[1] D W Yeung ldquoDynamically consistent cooperative solutionin a differential game of transboundary industrial pollutionrdquoJournal of Optimization Theory and Applications vol 134 no1 pp 143ndash160 2007

[2] D W Yeung and L A Petrosyan ldquoA cooperative stochasticdifferential game of transboundary industrial pollutionrdquo Auto-matica vol 44 no 6 pp 1532ndash1544 2008

[3] E J Dockner andNVanLong ldquoInternational pollution controlcooperative versus noncooperative strategiesrdquo Journal of Envi-ronmental Economics and Management vol 25 no 1 pp 13ndash291993

[4] L Petrosjan and G Zaccour ldquoTime-consistent Shapley valueallocation of pollution cost reductionrdquo Journal of EconomicDynamics amp Control vol 27 no 3 pp 381ndash398 2003

[5] B Bayramoglu ldquoTransboundary pollution in the black seacomparison of institutional arrangementsrdquo Environmental andResource Economics vol 35 no 4 pp 289ndash325 2006

[6] S Joslashrgensen and G Zaccour ldquoIncentive equilibrium strategiesand welfare allocation in a dynamic game of pollution controlrdquoAutomatica vol 37 no 1 pp 29ndash36 2001

[7] L Bertinelli C Camacho and B Zou ldquoCarbon capture andstorage and transboundary pollution a differential gameapproachrdquo European Journal of Operational Research vol 237no 2 pp 721ndash728 2014

[8] S Li ldquoA differential game of transboundary industrial pollutionwith emission permits tradingrdquo Journal of OptimizationTheoryand Applications vol 163 no 2 pp 642ndash659 2014

[9] H Benchekroun and G Martın-Herran ldquoThe impact of fore-sight in a transboundary pollution gamerdquo European Journal ofOperational Research vol 251 no 1 pp 300ndash309 2016

[10] XHuang PHe andW Zhang ldquoA cooperative differential gameof transboundary industrial pollution between two regionsrdquoJournal of Cleaner Production vol 120 pp 43ndash52 2016

[11] A Haurie J B Krawczyk and M Roche ldquoMonitoring coop-erative equilibria in a stochastic differential gamerdquo Journal ofOptimization Theory and Applications vol 81 no 1 pp 73ndash951994

[12] E Saltari and G Travaglini ldquoThe effects of environmental poli-cies on the abatement investment decisions of a green firmrdquoResource and Energy Economics vol 33 no 3 pp 666ndash685 2011

[13] S Athanassoglou and A Xepapadeas ldquoPollution control withuncertain stock dynamics when and how to be precautiousrdquoJournal of Environmental Economics and Management vol 63no 3 pp 304ndash320 2012

[14] E J Dockner and K Nishimura ldquoTransboundary pollution ina dynamic game modelrdquo Japanese Economic Review vol 50 no4 pp 443ndash456 1999

[15] F Bosello B Buchner and C Carraro ldquoEquity developmentand climate change controlrdquo Journal of the European EconomicAssociation vol 1 no 2-3 pp 601ndash611 2003

[16] A Bernard A Haurie M Vielle and L Viguier ldquoA two-leveldynamic game of carbon emission trading between RussiaChina and annex B countriesrdquo Journal of Economic Dynamicsand Control vol 32 no 6 pp 1830ndash1856 2008

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

ways that is an accumulative global pollutant and a regionalnonaccumulative pollutant For example some transbound-ary pollutants are caused by firms such as passing-by waste inwaterways wind-driven suspended particles in air unpleas-ant odour noise dust and heat which are all nonaccu-mulative pollutants We assume that the nonaccumulativepollutant for an output of 119906119894(119904) produced by the firm innation 1 would cause a short-term impact (cost) of 12057611119906119894(119904)and 12057621119906119894(119904) on nation 1 and nation 2 respectively Similarly anoutput of V119895(119904)produced by the firm innation 2would cause ashort-term (cost) of 12057612V119895(119904) and 12057622V119895(119904) on nation 1 and nation2 respectively In addition to the nonaccumulative pollutantthere is an accumulative pollutant such as green-house-gas CFC and atmospheric particulates these pollutantscan be maintained in environment for a long time andbuilt up existing pollution stocks to create long-term globalenvironmental impacts [2 5 10 15 16] Let 119909(119904) sub 119877+ denotethe level of pollution at time 119904 and the dynamics of pollutionstock are governed by the stochastic differential equation

119889119909 (119904) = 119899sum119894=1

1205721198941119906119894 (119904) + 119898sum119895=1

1205721198952V119895 (119904)minus 2sum119896=1

120574119896 (119868119896 (119904) (119909 (119904))12 minus 120575119909 (119904)) 119889119904+ 120590119909 (119904) 119889119911 (119904) 119896 isin 1 2

(4)

where 120590 denotes a noise parameter and 119911(119904) is a Wienerprocess 1205721198941 ge 0 and 1205721198952 ge 0 are the amount added tothe pollution stock by a unit of firms 119894rsquos and 119895rsquos output innation 1 and nation 2 respectively 119868119896(119904) denotes the pollutionabatement effort of nation 119896 120574119896 is the efficiency of theabatement 120574119896119868119896(119904)(119909(119904))12 denotes the amount of pollutionremoved by 119868119896(119904) unit of abatement effort of nation 119896 and 120590denotes the natural rate of decay of the pollutants

23 The Governmentsrsquo Objectives The instantaneous objec-tive of government 119896 (119896 isin 1 2) at time 119904 can be expressed as

119899sum119894=1

1205731198941 (119906lowast119894 (119904))12 minus 121198861 (1198681 (119904))2 minus 12057611 119899sum119894=1

119906lowast119894 (119904)minus 12057612 119898sum119895=1

Vlowast119895 (119904) minus ℎ1119909 (119904) (5a)

119898sum119895=1

1205731198951 (Vlowast119895 (119904))12 minus 121198862 (1198682 (119904))2 minus 12057621 119899sum119894=1

119906lowast119894 (119904)minus 12057622 119898sum119895=1

Vlowast119895 (119904) minus ℎ2119909 (119904) (5b)

where 119886119896 ℎ119896 gt 0 (119896 isin 1 2) are constants (12)119886119896(119868119896(119904))2denotes the cost of employing 119868119896(119904) amount of pollutionabatement effort and ℎ119896119909(119904) denotes the value of damage tonation 119896 from 119909(119904) amount of pollution

The governmentrsquos planning horizon is [1199050infin) The dis-count rate is 119903 Each government seeks to maximize theintegral of its instantaneous objectives (5a) and (5b) over theplanning horizon [1199050infin) subjected to pollution dynamics(4) with controls on the level of abatement effort andpollution tax

Substituting 119906119894(119904) and V119895(119904) from (3) into (4) and (5a)and (5b) one obtains a stochastic differential game in whichgovernment 119896 isin 1 2 seeks

max1198881(119904)1198681(119904)

1198641199050 intinfin1199050 119890minus119903119905(sum119899119894=1 (1205731198941)221198881 (119904) minus 121198861 (1198681 (119904))2

minus 12057611 sum119899119894=1 (1205731198941)2411988821 (119904) minus 12057612 sum119898119895=1 (1205731198952)2411988822 (119904)minus ℎ1119909 (119904))119889119904

(6a)

max1198882(119904)1198682(119904)

1198641199050 intinfin1199050 119890minus119903119905(sum119898119895=1 (1205731198952)221198882 (119904) minus 121198862 (1198682 (119904))2

minus 12057621 sum119899119894=1 (1205731198941)2411988821 (119904) minus 12057622 sum119898119895=1 (1205731198952)2411988822 (119904) minus ℎ2119909 (119904))119889119904(6b)

subject to

119889119909 (119904) = 119899sum119894=1

1205721198941 ( 120573119894121198881 (119904))2 + 119898sum119895=1

1205721198952 ( 120573119895221198882 (119904))2

minus 2sum119896=1

120574119896 (119868119896 (119904) (119909 (119904))12 minus 120575119909 (119904)) 119889119904+ 120590119909 (119904) 119889119911 (119904)

(7)

Since the payoffs of nations are measured in monetaryterms 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 noncooperativeframework

A feedback Nash equilibrium solution can be character-ized as follows

Definition 1 A set of feedback strategies [119888lowast119896 (119905) 119868lowast119896 (119905)] =[120601119888119896(119905 119909) 120601119868119896(119905 119909)] for 119896 isin 1 2 provides a Nash equilibriumsolution to the games (6a) (6b) and (7) if there exist suitably

4 Mathematical Problems in Engineering

smooth functions 119881119896(119905 119909) [1199050infin) times 119877 rarr 119877 119896 isin 1 2satisfying the following partial differential equations

1199031198811 (119905 119909) minus 12120590211990921198811119909119909 (119905 119909) = max1198881(119905)1198681(119905)

sum119899119894=1 (1205731198941)221198881 (119905)

minus 121198861 (1198681 (119905))2 minus 12057611 sum119899119894=1 (1205731198941)2411988821 (119905) minus 12057612 sum119898119895=1 (1205731198952)2411988822 (119905)minus ℎ1119909 (119905) + 1198811119909 (119905 119909)(sum119899119894=1 (1205731198941)2411988821 (119905) + sum119898119895=1 (1205731198952)2411988822 (119905)minus 2sum119896=1

120574119896 (119868119896 (119905) (119909 (119905))12 minus 120575119909 (119905)))

(8a)

1199031198812 (119905 119909) minus 12120590211990921198812119909119909 (119905 119909) = max1198882(119905)1198682(119905)

sum119898119895=1 (1205731198952)221198882 (119905)

minus 121198862 (1198682 (119905))2 minus 12057621 sum119899119894=1 (1205731198941)2411988821 (119905) minus 12057622 sum119898119895=1 (1205731198952)2411988822 (119905)minus ℎ2119909 (119905) + 1198812119909 (119905 119909)(sum119899119894=1 (1205731198941)2411988821 (119905) + sum119898119895=1 (1205731198952)2411988822 (119905)minus 2sum119896=1

120574119896 (119868119896 (119905) (119909 (119905))12 minus 120575119909 (119905)))

(8b)

Performing the indicated maximization in (8a) and (8b)yields

1206011198881 (119905 119909) = 12057611 minus 1198811119909 (119905 119909) 1206011198882 (119905 119909) = 12057622 minus 1198812119909 (119905 119909) (9a)

1206011198681 (119905 119909) = minus12057411198811119909 (119905 119909) 11990912 (119905)1198861 1206011198682 (119905 119909) = minus12057421198812119909 (119905 119909) 11990912 (119905)1198862

(9b)

Substituting the results of (9a) and (9b) into (8a) and (8b)we obtain the following

Proposition 2 The government 119896rsquos payoff during [1199050infin)119881119896(119905 119909) can be obtained as

119881119896 (119905 119909) = [119860119896119909 (119905) + 119861119896] for 119896 isin 1 2 (10)

where 119860119896 119861119896 satisfy the following set of constant coefficientquadratic ordinary differential equations

120574211198602121198861 + 12057422119860111986021198862 minus 1205741198601 minus 1205751198601 minus ℎ1 = 0 (11a)

120574221198602221198862 + 12057421119860111986021198861 minus 1205741198602 minus 1205751198602 minus ℎ2 = 0 (11b)

1198611 = 1119903 ( sum119899119894=1 (1205731198941)24 (12057611 minus 1198601) + (1198601 minus 12057612)sum119898119895=1 (1205731198952)24 (12057622 minus 1198602)2 ) (11c)

1198612 = 1119903 (sum119898119895=1 (1205731198951)24 (12057622 minus 1198602) + (1198602 minus 12057621)sum119899119894=1 (1205731198941)24 (12057611 minus 1198601)2 ) (11d)

The corresponding feedback Nash equilibrium of the games(8a) and (8b) can be obtained as

1206011198881 (119905 119909) = 12057611 minus 11986011206011198681 (119905 119909) = minus1205741119860111990912 (119905)1198861 1206011198882 (119905 119909) = 12057622 minus 11986021206011198682 (119905 119909) = minus1205742119860211990912 (119905)1198862

(12)

Equation (12) indicates that under noncooperation opti-mal pollution tax rate (1206011198881 or 1206011198882) may depend on the regionalpollutant (12057611 or 12057622) which is caused by the firms in homebut the regional pollutant (12057621 or 12057612) which is caused by thefirms in neighbormay have no effect on optimal pollution taxrate In addition the optimal abatement effort of the regiondoes not depend on the damage from the regional pollutantat home (12057611 or 12057622) this is because the abatement effort ofthe nation can only affect pollution stock and cannot affectcurrent pollution

4 Cooperative Arrangements

Now consider the case when both nations cooperate inpollution control To uphold the cooperative scheme bothgroup rationality and individual rationality are required to besatisfied at any time

41 Group Optimality and Cooperative State Trajectory Tosecure group optimality the participating two nations would

Mathematical Problems in Engineering 5

seek to maximize their joint expected payoff by solving thefollowing stochastic control problem

119882(119904 119909) = 1198641199050 intinfin0 sum119899119894=1 (1205731198941)221198881 (119904) + sum119898119895=1 (1205731198952)221198882 (119904)

minus 122sum119896=1

119886119896 (119868119896 (119904))2 minus (12057611 + 12057621)sum119899119894=1 (1205731198941)2411988821 (119904)minus (12057612 + 12057622)sum119898119895=1 (1205731198952)2411988822 (119904) minus (ℎ1 + ℎ2) 119909 (119904)119890minus119903119905119889119905

(13)

subject to (7)Invoking Bellmanrsquos technique a set of controls 119888lowast119896 119868lowast119896 =120601119888119896(119909) 120601119868119896(119909) constitutes an optimal solution to the stochas-

tic control problem (13) and (7) if there exists continuouslydifferentiable function 119882(119905 119909) [119905infin) times 119877 rarr 119877 119896 isin 1 2satisfying the following partial differential equations

119903119882 (119905 119909) minus 1212059021199092119882119909119909 (119905 119909) = max1198881 11986811198882 1198682

sum119899119894=1 (1205731198941)221198881 (119905)

+ sum119898119895=1 (1205731198952)221198882 (119905) minus 122sum119896=1

119886119896 (119868119896 (119905))2

minus (12057611 + 12057621)sum119899119894=1 (1205731198941)2411988821 (119905) minus (12057612 + 12057622)sum119898119895=1 (1205731198952)2411988822 (119905)minus (ℎ1 + ℎ2) 119909 (119905) + 119882119883 (119905 119909)(sum119899119894=1 1205721198941 (1205731198941)2411988821 (119905)+ sum119898119895=1 1205721198952 (1205731198952)2411988822 (119905)minus 2sum119896=1

120574119896 (119868119896 (119905) (119909 (119905))12 minus 120575119909 (119905)))

(14)

Performing the indicated maximization in (14) yields theoptimal controls under cooperation

1206011198881 (119905 119909) = 12057611 + 12057621 minus 119882119909 (119905 119909) 1206011198882 (119905 119909) = 12057612 + 12057622 minus 119882119909 (119905 119909) (15a)

1206011198681 (119909) = minus1205741119882119909 (119909) 119909121198861 1206011198682 (119909) = minus1205742119882119909 (119909) 119909121198862

(15b)

Substituting the results in (15a) and (15b) into (14) weobtain the following

Proposition 3 System (13) admits a solution

119882(119905 119909) = [119860119909 (119905) + 119861] (16)

where 119860 119861 satisfy the following set of constant coefficientquadratic ordinary differential equations

( 1205742121198861 + 1205742221198862)1198602 minus (119903 + 120575)119860 minus (ℎ1 + ℎ2) = 0 (17a)

119861 = 1119903 (sum119899119894=1 (1205731198941)2 + sum119898119895=1 (1205731198952)24 (12057611 + 12057622 minus 119860)2 ) (17b)

The corresponding feedback Nash equilibriums undercooperation can be obtained as

1206011198881 (119905 119909) = 12057611 + 12057622 minus 1198601206011198681 (119905 119909) = minus120574111986011990912 (119905)21198861 1206011198882 (119905 119909) = 12057611 + 12057622 minus 1198601206011198682 (119905 119909) = minus120574211986011990912 (119905)21198862

(18)

Substituting the optimal control strategy from (18) into(7) yields the dynamics of pollution accumulation undercooperation

119889119909 (119904) = (sum119899119894=1 1205721198941 (1205731198941)2 + sum119898119895=1 1205721198952 (1205731198952)24 (12057611 + 12057622 minus 119860)2+ 2sum119896=1

(120574119896)2 119860119909 (119904)2119886119896 minus 120575119909 (119904))119889119904 + 120590119909 (119904) 119889119911 (119904) 119909 (1199050) = 1199090

(19)

Solving the stochastic cooperative pollution dynamicsyields the cooperative state trajectory

119909lowast (119905) = 119890[int1199051199050 [(12057421 21198861+12057422 21198862)119860minus120575minus1205902]119889119904+int1199051199050 120590119889119911(119904)] times [[1199091199050+ int1199051199050(sum119899119894=1 1205721198941 (1205731198941)2 + sum119898119895=1 1205721198952 (1205731198952)24 (12057611 + 12057622 minus 119860)2 )

times 119890[int1199051199050 [12059022+120575minus(12057421 21198861+12057422 21198862)119860]119889120591+int1199051199050 120590119889119911(119904)]119889119904]] for 119905 isin [1199050infin)

(20)

42 Individually Rational and Time-Consistent Imputationand Payment Distribution Mechanism Let 119883lowast119905 denote the

6 Mathematical Problems in Engineering

set of realizable values of 119909lowast(119905) at time 119905 generated by (20)The term 119909lowast119905 is used to denote an element of the set 119883lowast119905 An agreed upon optimality principle would be sought toallocate the cooperative payoff In a dynamic frameworkindividual rationality has to be maintained at every instantof time within the cooperative duration [1199050infin) along thecooperative trajectory (20) For 120591 isin [1199050infin) let vector120585119896(119909lowast119905 ) = 1205851(119909lowast119905 ) 1205852(119909lowast119905 ) denote the solution imputation(payoff under cooperation) over the period [120591infin) to player119896 isin 1 2 given that the state is 119909lowast120591 isin 119883lowast120591 Individualrationality along the cooperative trajectory requires

120585(120591)119896 (120591 119909lowast119905 ) ge 119881(120591)119896 (120591 119909lowast119905 ) for 119896 isin 1 2 (21)

where 119881(120591)119896(120591 119909lowast119905 ) denotes the payoff to nation 119896 undernoncooperation over the period Let 119866(119904) = [1198661(119904) 1198662(119904)]denote the instantaneous payoff of the cooperative game attime 119904 isin [1199050infin) for the cooperative Γ119888(119909lowast1199050)Theorem 4 An instantaneous payment at time 120591 isin [1199050 119879]equaling

119866119870 (120591) = 119903120585119896 (119909lowast120591 ) minus 121205902 (119909lowast120591 )2 120585119896119909lowast120591119909lowast120591 (119909lowast120591 ) minus 120585119896119909lowast120591

(119909lowast120591 )sdot ( 119899sum119894=1

1205721198941 ( 120573119894121198881 (120591 119909lowast120591 ))2 + 119898sum119895=1

1205721198952 ( 120573119895221198882 (120591 119909lowast120591 ))2

minus 2sum119896=1

120574119896119868119896 (120591 119909lowast120591 ) (119909lowast120591 )12 minus 120575119909lowast120591) for 120591 isin [1199050infin) 119909lowast120591 isin 119883lowast120591 119896 isin 1 2

(22)

yields a subgame consistent solution to the cooperative gameΓ119888(119909lowast1199050)Proof Along the cooperative trajectory 119909lowast(119905)119905ge1199050 we haveΥ119896 (120591 120591 119909lowast120591 ) = 120585119896 (119909lowast120591 )

= 119864120591 intinfin120591

119866119896 (119904) 119890minus119903(119904minus120591)119889119904 | 119909 (120591) = 119909lowast120591 Υ119896 (120591 119905 119909lowast119905 ) = 120585119896 (119909lowast119905 )

= 119864119905 intinfin119905

119866119896 (119904) 119890minus119903(119904minus119905)119889119904 | 119909 (119905) = 119909lowast119905 for 119896 isin 1 2 119909lowast119905 isin 119883lowast119905 119905 ge 120591

(23)

Note that

Υ119896 (120591 119905 119909lowast119905 ) = 119890minus119903(119904minus120591)120585 (119909lowast119905 ) = 119890minus119903(119904minus120591)Υ119896 (119905 119905 119909lowast119905 ) 119896 isin 1 2 119909lowast120591 isin 119883lowast120591 (24)

Since Υ119896(119905 119905 119909lowast119905 ) is continuously differentiable in 119905 and119909lowast119905 one can obtain

Υ119896 (120591 120591 119909lowast120591 ) = 119864120591 int120591+Δ119905120591

119866119896 (119904) 119890minus119903(119904minus120591)119889119904+ 119890minus119903(Δ119905)Υ119896 (120591 + Δ119905 120591 + Δ119905 119909lowast120591 + Δ119909lowast120591 ) | 119909 (120591)= 119909lowast120591 for 120591 isin [1199050infin) 119909lowast120591 isin 119883lowast120591 119896 isin 1 2

(25)

where

Δ119909lowast120591= 119899sum119894=1

1205721198941 ( 120573119894121198881 (120591 119909lowast120591 ))2 + 119898sum119895=1

1205721198952 ( 120573119895221198882 (120591 119909lowast120591 ))2

minus 2sum119896=1

120574119896 (119868119870 (120591 119909lowast120591 ) (119909lowast120591 )12 minus 120575119909lowast120591)Δ119905+ 120590 [119909lowast120591 ] Δ119911120591 + 119900 (Δ119905)

Δ119911120591 = 119911 (120591 + Δ119905) minus 119911 (120591) 119864120591 [119900 (Δ119905)]Δ119905 997888rarr 0

as Δ119905 997888rarr 0

(26)

From (24) and (25) one can obtain

Υ119896 (120591 120591 119909lowast120591 ) = 119864120591 int120591+Δ119905120591

119866119896 (119904) 119890minus119903(119904minus120591)119889119904+ Υ119896 (120591 120591 + Δ119905 119909lowast120591 + Δ119909lowast120591 ) | 119909 (120591) = 119909lowast120591

for 120591 isin [1199050infin) 119909lowast120591 isin 119883lowast120591 119896 isin 1 2 (27)

Therefore with Δ119905 rarr 0 (27) can be expressed as

119866119896 (120591) Δ119905 = minus [Υ119896119905 (120591 119905 119909lowast120591 )1003816100381610038161003816119905=120591] Δ119905 minus 121205902 (119909lowast120591 )2sdot [Υ119896119909lowast

120591119909lowast120591

(120591 119905 119909lowast120591 )1003816100381610038161003816119905=120591] Δ119905 minus [Υ119896119909lowast120591

(120591 119905 119909lowast120591 )1003816100381610038161003816119905=120591]times ( 119899sum119894=1

1205721198941 ( 120573119894121198881 (120591 119909lowast120591 ))2 + 119898sum119895=1

1205721198952 ( 120573119895221198882 (120591 119909lowast120591 ))2

minus 2sum119896=1

120574119896 (119868119870 (120591 119909lowast120591 ) (119909lowast120591 )12 minus 120575119909lowast120591)Δ119905 minus 119900 (Δ119905))

(28)

Dividing (28) throughout by Δ119905 with Δ119905 rarr 0 and takingexpectation one can obtain (22) HenceTheorem 4 follows

When the two nations adopt the cooperative strategiesthe rate of instantaneous payment that nation 119896 (119896 isin 1 2)

Mathematical Problems in Engineering 7

Table 1 The parameters of model used in the numerical solutions

1199050 12057311 12057321 12057312 12057322 119909(1199050) 12057211 12057221 12057212 12057222 1205741 12057420 60 90 40 70 20 04 03 05 04 04 02120575 120590 1198861 1198862 12057611 12057621 12057622 12057612 119903 ℎ1 ℎ2001 005 08 1 02 015 015 016 005 4 5

will realize at time 120591 with the state being 119909lowast120591 can be expressedas

1198781 (120591 119909) = 1199032 [1198601119909lowast120591 + 1198611] + [119860119909lowast120591 + 119861]minus [1198602119909lowast120591 + 1198612] minus 12 [1198601 + 119860 minus 1198602]sdot

sum119899119894=1 (1205731198941)2 + sum119898119895=1 (1205731198952)24 (12057611 + 12057622 minus 119860)2+ 2sum119896=1

1205742119896119860119909lowast1205912119886119896 minus 120575119909lowast120591

(29a)

1198782 (120591 119909) = 1199032 [1198602119909lowast120591 + 1198612] + [119860119909lowast120591 + 119861]minus [1198601119909lowast120591 + 1198611] minus 12 [1198602 + 119860 minus 1198601]sdot

sum119899119894=1 (1205731198941)2 + sum119898119895=1 (1205731198952)24 (12057611 + 12057622 minus 119860)2+ 2sum119896=1

1205742119896119860119909lowast1205912119886119896 minus 120575119909lowast120591

(29b)

5 Numerical Example

Consider a multinational economy consisting of two nationsand each with two industrial firms Industrial firmrsquos revenuefunction in nation 1 and nation 2 at time 119904 is 120601119894 = 1205731198941(119906119894(119904))12and 120601119895 = 1205731198952(V119895(119904))12 (119894 119895 = 1 2) respectively where12057311 = 60 12057321 = 90 12057312 = 40 and 12057322 = 70 The valueof 1205731198941 (119894 isin 1 2) is higher than 1205731198952 (119895 isin 1 2) it meansthat compared with industrial firms in nation 2 industrialfirms in nation 1 have more energy-efficient technology Inaddition industrial firms in nation 1 have technical advantageresult in 12057211 lt 12057212 12057221 lt 12057222 1205741 gt 1205742 As mentionedabove the optimal pollution emissions of the firms in nations1 and 2 are 119906lowast119894 (119904) = (120573119894121198881(119904))2 and Vlowast119895 (119904) = (120573119895221198882(119904))2respectively The dynamics of pollution stock are governedby the stochastic differential equation (4) where 119909(1199050) = 20120575 = 001 and 120590 = 005 We defined 119886119896 as the amountadded to the pollution stock by a unit of output in nation119896 (119896 = 1 2) They are a positive number less than 1 that isto say the stock of pollution will increase with the outputbut not higher than the amount of output According to

Table 2 The optimal pollution tax rate

1198881(119905) 1198882(119905) 1198881(119905) 1198882(119905)820 234 1090 1087

experience we assign 12057211 = 04 12057221 = 03 12057212 = 0512057222 = 04 120574119896 (119896 = 1 2) can also be considered as pollutionabatement effort efficiency which is linked to pollutioncontrol policies and environmental governance efficiency indifferent nations Generally speaking pollution abatementeffort efficiency is higher in developed countries We assign1205741 = 04 1205742 = 02 Abatement costs are 04(1198681(119904))12 and05(1198682(119904))12 for nations 1 and 2 respectively Local pollutionimpacts are closely linked to the level of production activitiesThe accumulation of pollution stock like greenhouse gas oftenconcerns the interactions between the natural environmentand the pollutants emitted So stochastic elements wouldappear According to what we defined before 1205761119896 1205762119896 (119896 = 1 2)is a pollution damage cost coefficients It is said that the biggerthey are the higher the cost of environmental damage forthis nation is We make the following assignments 12057611 = 0212057621 = 015 12057622 = 015 and 12057612 = 016 ℎ119896 (119896 = 1 2) arethe value of pollution damage to nation 119896 (119896 = 1 2) Thelarger they are the greater the impact of the environment iswhich is determined by the different regions And we makeassignments ℎ1 = 4 ℎ2 = 5 Other parameters are as follows119903 = 005 1199050 = 0 The parameters of the model which are usedin the numerical solution are presented in Table 1

Then we obtain a stochastic differential game in whichgovernment 1 and government 2 under noncooperation seek

max1198881(119904)1198681(119904)

intinfin0

119890minus119903119905 (58501198881 (119904) minus 0411986821 (119904) minus 58511988821 (119904) minus 162211988822 (119904)minus 4119909 (119904)) 119889119909

(30a)

max1198882(119904)1198682(119904)

intinfin0

119890minus119903119905 (32501198882 (119904) minus 0511986822 (119904) minus 4387511988821 (119904)minus 2437511988822 (119904) minus 5119909 (119904)) 119889119909

(30b)

Through the analysis mentioned above we have obtainedthe pollution tax rate abatement effort of each nationand the dynamics of pollution stock under noncooperativegame and cooperative game Now we will analyze theirdifference between the governments under noncooperationand cooperation through Table 2 and Figures 1ndash5

8 Mathematical Problems in Engineering

0 05 1 15 2 25 3 35 4 45 515

2

25

3

35

4

45

Pollu

tion

abat

emen

t effo

rt u

nder

non

coop

erat

ion

I2

I1

Time t

Figure 1 The pollution abatement effort under noncooperationwhere 1198681 and 1198682 denote the pollution abatement effort of govern-ments under noncooperation in nations 1 and 2 respectively

0 05 1 15 2 25 3 35 4 45 50

2

4

6

8

10

12

Pollu

tion

abat

emen

t effo

rt u

nder

coop

erat

ion

I4

I3

Time t

Figure 2 The pollution abatement effort under cooperation where1198683 and 1198684 are the pollution abatement effort of governments undercooperation in nations 1 and 2 respectively

From Table 2 we can see that for each governmentthe noncooperative optimal tax rates 1198881(119905) and 1198882(119905) are lessthan the cooperative tax rates 1198881(119905) and 1198882(119905) This meansthat cooperation would push to raise the tax rate to reduceproduction and cut pollution And for both governmentsthe optimal pollution tax rate is not changing with timethis is different from Huang et al [10] and Yeung andPetrosyan [2] because in their research they model afinite-horizon level game regardless of cooperative game ornoncooperative game and in the later stage of game bothgovernments would pay more attention to the economic

0 05 1 15 2 25 3 35 4 45 52

4

6

8

10

12

14

16

18

20

22

Pollu

tion

stock

x2

x1

Time t

Figure 3 The dynamics of pollution stock under noncooperationor cooperation where 1199091 and 1199092 denote the pollution stock undernoncooperation and cooperation respectively

0 05 1 15 2 25 3 35 4 45 5Time t

times106

Insta

ntan

eous

pay

men

ts of

natio

n 1

unde

r coo

pera

tion

61466861467

614672614674614676614678

61468614682614684614686

Figure 4 The instantaneous payments of nation 1 under coopera-tion

benefits than pollution abatementThen they would decreasethe pollution tax rate to encourage production and stimulateeconomic development [10] But in our model the gameis in infinite-horizon and there is no later stage of thegame so the optimal pollution tax rate will not decrease Inaddition under noncooperation the governments take theexternalities of production into account including their ownnation only However under a cooperation the externalitiesare considered by the governments which include their ownnation and the neighbors so the optimal pollution tax rate ishigher than under noncooperation

Figures 1 and 2 show the pollution abatement effort ofboth governments From them we find that under noncoop-eration the pollution abatement effort of both governmentsin the two nations could remain stable all the time butunder cooperation both governments would make a highpollution abatement effort at first and then the effort levelmay reduce to a certain degree after that it would tendto remain stable For each government the noncooperativepollution abatement effort is lower than the cooperativepollution abatement effort This is determined by externality

Mathematical Problems in Engineering 9In

stan

tane

ous p

aym

ents

of

natio

n 2

unde

r coo

pera

tion times10

6

0 05 1 15 2 25 3 35 4 45 5Time t

29685929686

296861296862296863296864296865296866296867296868

Figure 5 The instantaneous payments of nation 2 under coopera-tion

of pollution emissions Under cooperative game the govern-ments pay attention to the damage for the entire region somarginal damage of pollution emission would be greater andthe governments would be more concerned about pollutantreduction

Figure 3 shows the dynamics of pollution stock undernoncooperation or cooperation We can see that the pollu-tion stock under noncooperative game is higher than thatunder cooperative game It is due to the higher pollutiontax rate and higher pollution abatement effort of both gov-ernments under cooperative game Then in both economicand environmental terms cooperation is always better thannoncooperation

Now consider the case when the two nations want tocooperate To secure group optimality the two nations seek tomaximize their joint expected payoff by solving the followingstochastic control problem

max1198881(119904)1198882(119904)1198681(119904)1198682(119904)

1198640 intinfin0

(58501198881 (119904) + 32501198882 (119904) minus 04 (1198681 (119904))2

minus 05 (1198682 (119904))2 minus 102375(1198881 (119904))2 minus50375(1198882 (119904))2 minus 10119909 (119904))

sdot 119890minus005119889119905(31)

subject to

119889119909 (119904) = 349 minus 128119909 (119904) + 005119909 (119904) 119889119911 (119904) (32)

According toTheorem 4 we get the appropriate dynamicdistribution scheme that both group optimality and individ-ual rationality are satisfied at any time during cooperativegame The instantaneous payments of two nations 1198781(119905 119909lowast119905 )and 1198782(119905 119909lowast119905 ) at different time 119905 with given 119909lowast119905 are shown in(33a) and (33b) and Figures 3 and 4

1198781 (119905 119909lowast119905 ) = 1007100178948148626983247163840000000000000minus 12670021998361119890minus321199052578125000000 (33a)

1198782 (119905 119909lowast119905 ) = 77822033722351462743911532621440000000000000minus 1518945509601119890minus321199052519531250000 (33b)

It is shown in Figures 4 and 5 that the instantaneous pay-ment of both nations which satisfy the property of subgameconsistency shows an increasing trend at the start and thenremains stable In addition the instantaneous payment fornation 1 is always higher than that of nation 2This is becausethe industrial firms in nation 1 have more green and energy-efficient technology so they would get more instantaneouspayment

6 Conclusions

In this paper we have shown a stochastic differential game oftransboundary industrial pollution between two asymmetricnations in infinite-horizon level In our model the firmsand governments are separate entities As participants inthe games they take game behavior to optimize their ownpayment and the game equilibrium has been solved underthe two levels of firms and governments simultaneouslyThe uncertain evolution dynamic of pollution stock thathas a long-term global impact has been taken into accountin our models Otherwise another negative environmentalexternality a regional nonaccumulative pollutant that causesa short-term local impact has also been considered Wecharacterize the parameters of spaces where two neighboringgovernments can cooperate and give the feedback Nashequilibrium strategies of governments and industrial firmsThrough analysis we have shown that the optimal pollutiontax rate hinges on the two effects of regional damagethe source region and the global pollution stock undernoncooperative game while under cooperative game inaddition to the above two kinds of effects it also hingeson the additional effect of regional damage coming fromthe neighborhood so the latter is greater than the formerMoreover we have shown that the pollution abatement effortunder cooperation is more than under noncooperation Byanalyzing this stochastic differential game model we havealso provided a payment distribution mechanism whichsupports the subgame consistent solution and present anumerical example to illustrate the implementation of thismechanism

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge the financial supportfrom the ChinaNational Funding for Social Science ResearchProject (Project no 15CJY037) and the Philosophy and SocialScience Foundation of Hunan China (Project no 16YBA315)for this paper

10 Mathematical Problems in Engineering

References

[1] D W Yeung ldquoDynamically consistent cooperative solutionin a differential game of transboundary industrial pollutionrdquoJournal of Optimization Theory and Applications vol 134 no1 pp 143ndash160 2007

[2] D W Yeung and L A Petrosyan ldquoA cooperative stochasticdifferential game of transboundary industrial pollutionrdquo Auto-matica vol 44 no 6 pp 1532ndash1544 2008

[3] E J Dockner andNVanLong ldquoInternational pollution controlcooperative versus noncooperative strategiesrdquo Journal of Envi-ronmental Economics and Management vol 25 no 1 pp 13ndash291993

[4] L Petrosjan and G Zaccour ldquoTime-consistent Shapley valueallocation of pollution cost reductionrdquo Journal of EconomicDynamics amp Control vol 27 no 3 pp 381ndash398 2003

[5] B Bayramoglu ldquoTransboundary pollution in the black seacomparison of institutional arrangementsrdquo Environmental andResource Economics vol 35 no 4 pp 289ndash325 2006

[6] S Joslashrgensen and G Zaccour ldquoIncentive equilibrium strategiesand welfare allocation in a dynamic game of pollution controlrdquoAutomatica vol 37 no 1 pp 29ndash36 2001

[7] L Bertinelli C Camacho and B Zou ldquoCarbon capture andstorage and transboundary pollution a differential gameapproachrdquo European Journal of Operational Research vol 237no 2 pp 721ndash728 2014

[8] S Li ldquoA differential game of transboundary industrial pollutionwith emission permits tradingrdquo Journal of OptimizationTheoryand Applications vol 163 no 2 pp 642ndash659 2014

[9] H Benchekroun and G Martın-Herran ldquoThe impact of fore-sight in a transboundary pollution gamerdquo European Journal ofOperational Research vol 251 no 1 pp 300ndash309 2016

[10] XHuang PHe andW Zhang ldquoA cooperative differential gameof transboundary industrial pollution between two regionsrdquoJournal of Cleaner Production vol 120 pp 43ndash52 2016

[11] A Haurie J B Krawczyk and M Roche ldquoMonitoring coop-erative equilibria in a stochastic differential gamerdquo Journal ofOptimization Theory and Applications vol 81 no 1 pp 73ndash951994

[12] E Saltari and G Travaglini ldquoThe effects of environmental poli-cies on the abatement investment decisions of a green firmrdquoResource and Energy Economics vol 33 no 3 pp 666ndash685 2011

[13] S Athanassoglou and A Xepapadeas ldquoPollution control withuncertain stock dynamics when and how to be precautiousrdquoJournal of Environmental Economics and Management vol 63no 3 pp 304ndash320 2012

[14] E J Dockner and K Nishimura ldquoTransboundary pollution ina dynamic game modelrdquo Japanese Economic Review vol 50 no4 pp 443ndash456 1999

[15] F Bosello B Buchner and C Carraro ldquoEquity developmentand climate change controlrdquo Journal of the European EconomicAssociation vol 1 no 2-3 pp 601ndash611 2003

[16] A Bernard A Haurie M Vielle and L Viguier ldquoA two-leveldynamic game of carbon emission trading between RussiaChina and annex B countriesrdquo Journal of Economic Dynamicsand Control vol 32 no 6 pp 1830ndash1856 2008

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

smooth functions 119881119896(119905 119909) [1199050infin) times 119877 rarr 119877 119896 isin 1 2satisfying the following partial differential equations

1199031198811 (119905 119909) minus 12120590211990921198811119909119909 (119905 119909) = max1198881(119905)1198681(119905)

sum119899119894=1 (1205731198941)221198881 (119905)

minus 121198861 (1198681 (119905))2 minus 12057611 sum119899119894=1 (1205731198941)2411988821 (119905) minus 12057612 sum119898119895=1 (1205731198952)2411988822 (119905)minus ℎ1119909 (119905) + 1198811119909 (119905 119909)(sum119899119894=1 (1205731198941)2411988821 (119905) + sum119898119895=1 (1205731198952)2411988822 (119905)minus 2sum119896=1

120574119896 (119868119896 (119905) (119909 (119905))12 minus 120575119909 (119905)))

(8a)

1199031198812 (119905 119909) minus 12120590211990921198812119909119909 (119905 119909) = max1198882(119905)1198682(119905)

sum119898119895=1 (1205731198952)221198882 (119905)

minus 121198862 (1198682 (119905))2 minus 12057621 sum119899119894=1 (1205731198941)2411988821 (119905) minus 12057622 sum119898119895=1 (1205731198952)2411988822 (119905)minus ℎ2119909 (119905) + 1198812119909 (119905 119909)(sum119899119894=1 (1205731198941)2411988821 (119905) + sum119898119895=1 (1205731198952)2411988822 (119905)minus 2sum119896=1

120574119896 (119868119896 (119905) (119909 (119905))12 minus 120575119909 (119905)))

(8b)

Performing the indicated maximization in (8a) and (8b)yields

1206011198881 (119905 119909) = 12057611 minus 1198811119909 (119905 119909) 1206011198882 (119905 119909) = 12057622 minus 1198812119909 (119905 119909) (9a)

1206011198681 (119905 119909) = minus12057411198811119909 (119905 119909) 11990912 (119905)1198861 1206011198682 (119905 119909) = minus12057421198812119909 (119905 119909) 11990912 (119905)1198862

(9b)

Substituting the results of (9a) and (9b) into (8a) and (8b)we obtain the following

Proposition 2 The government 119896rsquos payoff during [1199050infin)119881119896(119905 119909) can be obtained as

119881119896 (119905 119909) = [119860119896119909 (119905) + 119861119896] for 119896 isin 1 2 (10)

where 119860119896 119861119896 satisfy the following set of constant coefficientquadratic ordinary differential equations

120574211198602121198861 + 12057422119860111986021198862 minus 1205741198601 minus 1205751198601 minus ℎ1 = 0 (11a)

120574221198602221198862 + 12057421119860111986021198861 minus 1205741198602 minus 1205751198602 minus ℎ2 = 0 (11b)

1198611 = 1119903 ( sum119899119894=1 (1205731198941)24 (12057611 minus 1198601) + (1198601 minus 12057612)sum119898119895=1 (1205731198952)24 (12057622 minus 1198602)2 ) (11c)

1198612 = 1119903 (sum119898119895=1 (1205731198951)24 (12057622 minus 1198602) + (1198602 minus 12057621)sum119899119894=1 (1205731198941)24 (12057611 minus 1198601)2 ) (11d)

The corresponding feedback Nash equilibrium of the games(8a) and (8b) can be obtained as

1206011198881 (119905 119909) = 12057611 minus 11986011206011198681 (119905 119909) = minus1205741119860111990912 (119905)1198861 1206011198882 (119905 119909) = 12057622 minus 11986021206011198682 (119905 119909) = minus1205742119860211990912 (119905)1198862

(12)

Equation (12) indicates that under noncooperation opti-mal pollution tax rate (1206011198881 or 1206011198882) may depend on the regionalpollutant (12057611 or 12057622) which is caused by the firms in homebut the regional pollutant (12057621 or 12057612) which is caused by thefirms in neighbormay have no effect on optimal pollution taxrate In addition the optimal abatement effort of the regiondoes not depend on the damage from the regional pollutantat home (12057611 or 12057622) this is because the abatement effort ofthe nation can only affect pollution stock and cannot affectcurrent pollution

4 Cooperative Arrangements

Now consider the case when both nations cooperate inpollution control To uphold the cooperative scheme bothgroup rationality and individual rationality are required to besatisfied at any time

41 Group Optimality and Cooperative State Trajectory Tosecure group optimality the participating two nations would

Mathematical Problems in Engineering 5

seek to maximize their joint expected payoff by solving thefollowing stochastic control problem

119882(119904 119909) = 1198641199050 intinfin0 sum119899119894=1 (1205731198941)221198881 (119904) + sum119898119895=1 (1205731198952)221198882 (119904)

minus 122sum119896=1

119886119896 (119868119896 (119904))2 minus (12057611 + 12057621)sum119899119894=1 (1205731198941)2411988821 (119904)minus (12057612 + 12057622)sum119898119895=1 (1205731198952)2411988822 (119904) minus (ℎ1 + ℎ2) 119909 (119904)119890minus119903119905119889119905

(13)

subject to (7)Invoking Bellmanrsquos technique a set of controls 119888lowast119896 119868lowast119896 =120601119888119896(119909) 120601119868119896(119909) constitutes an optimal solution to the stochas-

tic control problem (13) and (7) if there exists continuouslydifferentiable function 119882(119905 119909) [119905infin) times 119877 rarr 119877 119896 isin 1 2satisfying the following partial differential equations

119903119882 (119905 119909) minus 1212059021199092119882119909119909 (119905 119909) = max1198881 11986811198882 1198682

sum119899119894=1 (1205731198941)221198881 (119905)

+ sum119898119895=1 (1205731198952)221198882 (119905) minus 122sum119896=1

119886119896 (119868119896 (119905))2

minus (12057611 + 12057621)sum119899119894=1 (1205731198941)2411988821 (119905) minus (12057612 + 12057622)sum119898119895=1 (1205731198952)2411988822 (119905)minus (ℎ1 + ℎ2) 119909 (119905) + 119882119883 (119905 119909)(sum119899119894=1 1205721198941 (1205731198941)2411988821 (119905)+ sum119898119895=1 1205721198952 (1205731198952)2411988822 (119905)minus 2sum119896=1

120574119896 (119868119896 (119905) (119909 (119905))12 minus 120575119909 (119905)))

(14)

Performing the indicated maximization in (14) yields theoptimal controls under cooperation

1206011198881 (119905 119909) = 12057611 + 12057621 minus 119882119909 (119905 119909) 1206011198882 (119905 119909) = 12057612 + 12057622 minus 119882119909 (119905 119909) (15a)

1206011198681 (119909) = minus1205741119882119909 (119909) 119909121198861 1206011198682 (119909) = minus1205742119882119909 (119909) 119909121198862

(15b)

Substituting the results in (15a) and (15b) into (14) weobtain the following

Proposition 3 System (13) admits a solution

119882(119905 119909) = [119860119909 (119905) + 119861] (16)

where 119860 119861 satisfy the following set of constant coefficientquadratic ordinary differential equations

( 1205742121198861 + 1205742221198862)1198602 minus (119903 + 120575)119860 minus (ℎ1 + ℎ2) = 0 (17a)

119861 = 1119903 (sum119899119894=1 (1205731198941)2 + sum119898119895=1 (1205731198952)24 (12057611 + 12057622 minus 119860)2 ) (17b)

The corresponding feedback Nash equilibriums undercooperation can be obtained as

1206011198881 (119905 119909) = 12057611 + 12057622 minus 1198601206011198681 (119905 119909) = minus120574111986011990912 (119905)21198861 1206011198882 (119905 119909) = 12057611 + 12057622 minus 1198601206011198682 (119905 119909) = minus120574211986011990912 (119905)21198862

(18)

Substituting the optimal control strategy from (18) into(7) yields the dynamics of pollution accumulation undercooperation

119889119909 (119904) = (sum119899119894=1 1205721198941 (1205731198941)2 + sum119898119895=1 1205721198952 (1205731198952)24 (12057611 + 12057622 minus 119860)2+ 2sum119896=1

(120574119896)2 119860119909 (119904)2119886119896 minus 120575119909 (119904))119889119904 + 120590119909 (119904) 119889119911 (119904) 119909 (1199050) = 1199090

(19)

Solving the stochastic cooperative pollution dynamicsyields the cooperative state trajectory

119909lowast (119905) = 119890[int1199051199050 [(12057421 21198861+12057422 21198862)119860minus120575minus1205902]119889119904+int1199051199050 120590119889119911(119904)] times [[1199091199050+ int1199051199050(sum119899119894=1 1205721198941 (1205731198941)2 + sum119898119895=1 1205721198952 (1205731198952)24 (12057611 + 12057622 minus 119860)2 )

times 119890[int1199051199050 [12059022+120575minus(12057421 21198861+12057422 21198862)119860]119889120591+int1199051199050 120590119889119911(119904)]119889119904]] for 119905 isin [1199050infin)

(20)

42 Individually Rational and Time-Consistent Imputationand Payment Distribution Mechanism Let 119883lowast119905 denote the

6 Mathematical Problems in Engineering

set of realizable values of 119909lowast(119905) at time 119905 generated by (20)The term 119909lowast119905 is used to denote an element of the set 119883lowast119905 An agreed upon optimality principle would be sought toallocate the cooperative payoff In a dynamic frameworkindividual rationality has to be maintained at every instantof time within the cooperative duration [1199050infin) along thecooperative trajectory (20) For 120591 isin [1199050infin) let vector120585119896(119909lowast119905 ) = 1205851(119909lowast119905 ) 1205852(119909lowast119905 ) denote the solution imputation(payoff under cooperation) over the period [120591infin) to player119896 isin 1 2 given that the state is 119909lowast120591 isin 119883lowast120591 Individualrationality along the cooperative trajectory requires

120585(120591)119896 (120591 119909lowast119905 ) ge 119881(120591)119896 (120591 119909lowast119905 ) for 119896 isin 1 2 (21)

where 119881(120591)119896(120591 119909lowast119905 ) denotes the payoff to nation 119896 undernoncooperation over the period Let 119866(119904) = [1198661(119904) 1198662(119904)]denote the instantaneous payoff of the cooperative game attime 119904 isin [1199050infin) for the cooperative Γ119888(119909lowast1199050)Theorem 4 An instantaneous payment at time 120591 isin [1199050 119879]equaling

119866119870 (120591) = 119903120585119896 (119909lowast120591 ) minus 121205902 (119909lowast120591 )2 120585119896119909lowast120591119909lowast120591 (119909lowast120591 ) minus 120585119896119909lowast120591

(119909lowast120591 )sdot ( 119899sum119894=1

1205721198941 ( 120573119894121198881 (120591 119909lowast120591 ))2 + 119898sum119895=1

1205721198952 ( 120573119895221198882 (120591 119909lowast120591 ))2

minus 2sum119896=1

120574119896119868119896 (120591 119909lowast120591 ) (119909lowast120591 )12 minus 120575119909lowast120591) for 120591 isin [1199050infin) 119909lowast120591 isin 119883lowast120591 119896 isin 1 2

(22)

yields a subgame consistent solution to the cooperative gameΓ119888(119909lowast1199050)Proof Along the cooperative trajectory 119909lowast(119905)119905ge1199050 we haveΥ119896 (120591 120591 119909lowast120591 ) = 120585119896 (119909lowast120591 )

= 119864120591 intinfin120591

119866119896 (119904) 119890minus119903(119904minus120591)119889119904 | 119909 (120591) = 119909lowast120591 Υ119896 (120591 119905 119909lowast119905 ) = 120585119896 (119909lowast119905 )

= 119864119905 intinfin119905

119866119896 (119904) 119890minus119903(119904minus119905)119889119904 | 119909 (119905) = 119909lowast119905 for 119896 isin 1 2 119909lowast119905 isin 119883lowast119905 119905 ge 120591

(23)

Note that

Υ119896 (120591 119905 119909lowast119905 ) = 119890minus119903(119904minus120591)120585 (119909lowast119905 ) = 119890minus119903(119904minus120591)Υ119896 (119905 119905 119909lowast119905 ) 119896 isin 1 2 119909lowast120591 isin 119883lowast120591 (24)

Since Υ119896(119905 119905 119909lowast119905 ) is continuously differentiable in 119905 and119909lowast119905 one can obtain

Υ119896 (120591 120591 119909lowast120591 ) = 119864120591 int120591+Δ119905120591

119866119896 (119904) 119890minus119903(119904minus120591)119889119904+ 119890minus119903(Δ119905)Υ119896 (120591 + Δ119905 120591 + Δ119905 119909lowast120591 + Δ119909lowast120591 ) | 119909 (120591)= 119909lowast120591 for 120591 isin [1199050infin) 119909lowast120591 isin 119883lowast120591 119896 isin 1 2

(25)

where

Δ119909lowast120591= 119899sum119894=1

1205721198941 ( 120573119894121198881 (120591 119909lowast120591 ))2 + 119898sum119895=1

1205721198952 ( 120573119895221198882 (120591 119909lowast120591 ))2

minus 2sum119896=1

120574119896 (119868119870 (120591 119909lowast120591 ) (119909lowast120591 )12 minus 120575119909lowast120591)Δ119905+ 120590 [119909lowast120591 ] Δ119911120591 + 119900 (Δ119905)

Δ119911120591 = 119911 (120591 + Δ119905) minus 119911 (120591) 119864120591 [119900 (Δ119905)]Δ119905 997888rarr 0

as Δ119905 997888rarr 0

(26)

From (24) and (25) one can obtain

Υ119896 (120591 120591 119909lowast120591 ) = 119864120591 int120591+Δ119905120591

119866119896 (119904) 119890minus119903(119904minus120591)119889119904+ Υ119896 (120591 120591 + Δ119905 119909lowast120591 + Δ119909lowast120591 ) | 119909 (120591) = 119909lowast120591

for 120591 isin [1199050infin) 119909lowast120591 isin 119883lowast120591 119896 isin 1 2 (27)

Therefore with Δ119905 rarr 0 (27) can be expressed as

119866119896 (120591) Δ119905 = minus [Υ119896119905 (120591 119905 119909lowast120591 )1003816100381610038161003816119905=120591] Δ119905 minus 121205902 (119909lowast120591 )2sdot [Υ119896119909lowast

120591119909lowast120591

(120591 119905 119909lowast120591 )1003816100381610038161003816119905=120591] Δ119905 minus [Υ119896119909lowast120591

(120591 119905 119909lowast120591 )1003816100381610038161003816119905=120591]times ( 119899sum119894=1

1205721198941 ( 120573119894121198881 (120591 119909lowast120591 ))2 + 119898sum119895=1

1205721198952 ( 120573119895221198882 (120591 119909lowast120591 ))2

minus 2sum119896=1

120574119896 (119868119870 (120591 119909lowast120591 ) (119909lowast120591 )12 minus 120575119909lowast120591)Δ119905 minus 119900 (Δ119905))

(28)

Dividing (28) throughout by Δ119905 with Δ119905 rarr 0 and takingexpectation one can obtain (22) HenceTheorem 4 follows

When the two nations adopt the cooperative strategiesthe rate of instantaneous payment that nation 119896 (119896 isin 1 2)

Mathematical Problems in Engineering 7

Table 1 The parameters of model used in the numerical solutions

1199050 12057311 12057321 12057312 12057322 119909(1199050) 12057211 12057221 12057212 12057222 1205741 12057420 60 90 40 70 20 04 03 05 04 04 02120575 120590 1198861 1198862 12057611 12057621 12057622 12057612 119903 ℎ1 ℎ2001 005 08 1 02 015 015 016 005 4 5

will realize at time 120591 with the state being 119909lowast120591 can be expressedas

1198781 (120591 119909) = 1199032 [1198601119909lowast120591 + 1198611] + [119860119909lowast120591 + 119861]minus [1198602119909lowast120591 + 1198612] minus 12 [1198601 + 119860 minus 1198602]sdot

sum119899119894=1 (1205731198941)2 + sum119898119895=1 (1205731198952)24 (12057611 + 12057622 minus 119860)2+ 2sum119896=1

1205742119896119860119909lowast1205912119886119896 minus 120575119909lowast120591

(29a)

1198782 (120591 119909) = 1199032 [1198602119909lowast120591 + 1198612] + [119860119909lowast120591 + 119861]minus [1198601119909lowast120591 + 1198611] minus 12 [1198602 + 119860 minus 1198601]sdot

sum119899119894=1 (1205731198941)2 + sum119898119895=1 (1205731198952)24 (12057611 + 12057622 minus 119860)2+ 2sum119896=1

1205742119896119860119909lowast1205912119886119896 minus 120575119909lowast120591

(29b)

5 Numerical Example

Consider a multinational economy consisting of two nationsand each with two industrial firms Industrial firmrsquos revenuefunction in nation 1 and nation 2 at time 119904 is 120601119894 = 1205731198941(119906119894(119904))12and 120601119895 = 1205731198952(V119895(119904))12 (119894 119895 = 1 2) respectively where12057311 = 60 12057321 = 90 12057312 = 40 and 12057322 = 70 The valueof 1205731198941 (119894 isin 1 2) is higher than 1205731198952 (119895 isin 1 2) it meansthat compared with industrial firms in nation 2 industrialfirms in nation 1 have more energy-efficient technology Inaddition industrial firms in nation 1 have technical advantageresult in 12057211 lt 12057212 12057221 lt 12057222 1205741 gt 1205742 As mentionedabove the optimal pollution emissions of the firms in nations1 and 2 are 119906lowast119894 (119904) = (120573119894121198881(119904))2 and Vlowast119895 (119904) = (120573119895221198882(119904))2respectively The dynamics of pollution stock are governedby the stochastic differential equation (4) where 119909(1199050) = 20120575 = 001 and 120590 = 005 We defined 119886119896 as the amountadded to the pollution stock by a unit of output in nation119896 (119896 = 1 2) They are a positive number less than 1 that isto say the stock of pollution will increase with the outputbut not higher than the amount of output According to

Table 2 The optimal pollution tax rate

1198881(119905) 1198882(119905) 1198881(119905) 1198882(119905)820 234 1090 1087

experience we assign 12057211 = 04 12057221 = 03 12057212 = 0512057222 = 04 120574119896 (119896 = 1 2) can also be considered as pollutionabatement effort efficiency which is linked to pollutioncontrol policies and environmental governance efficiency indifferent nations Generally speaking pollution abatementeffort efficiency is higher in developed countries We assign1205741 = 04 1205742 = 02 Abatement costs are 04(1198681(119904))12 and05(1198682(119904))12 for nations 1 and 2 respectively Local pollutionimpacts are closely linked to the level of production activitiesThe accumulation of pollution stock like greenhouse gas oftenconcerns the interactions between the natural environmentand the pollutants emitted So stochastic elements wouldappear According to what we defined before 1205761119896 1205762119896 (119896 = 1 2)is a pollution damage cost coefficients It is said that the biggerthey are the higher the cost of environmental damage forthis nation is We make the following assignments 12057611 = 0212057621 = 015 12057622 = 015 and 12057612 = 016 ℎ119896 (119896 = 1 2) arethe value of pollution damage to nation 119896 (119896 = 1 2) Thelarger they are the greater the impact of the environment iswhich is determined by the different regions And we makeassignments ℎ1 = 4 ℎ2 = 5 Other parameters are as follows119903 = 005 1199050 = 0 The parameters of the model which are usedin the numerical solution are presented in Table 1

Then we obtain a stochastic differential game in whichgovernment 1 and government 2 under noncooperation seek

max1198881(119904)1198681(119904)

intinfin0

119890minus119903119905 (58501198881 (119904) minus 0411986821 (119904) minus 58511988821 (119904) minus 162211988822 (119904)minus 4119909 (119904)) 119889119909

(30a)

max1198882(119904)1198682(119904)

intinfin0

119890minus119903119905 (32501198882 (119904) minus 0511986822 (119904) minus 4387511988821 (119904)minus 2437511988822 (119904) minus 5119909 (119904)) 119889119909

(30b)

Through the analysis mentioned above we have obtainedthe pollution tax rate abatement effort of each nationand the dynamics of pollution stock under noncooperativegame and cooperative game Now we will analyze theirdifference between the governments under noncooperationand cooperation through Table 2 and Figures 1ndash5

8 Mathematical Problems in Engineering

0 05 1 15 2 25 3 35 4 45 515

2

25

3

35

4

45

Pollu

tion

abat

emen

t effo

rt u

nder

non

coop

erat

ion

I2

I1

Time t

Figure 1 The pollution abatement effort under noncooperationwhere 1198681 and 1198682 denote the pollution abatement effort of govern-ments under noncooperation in nations 1 and 2 respectively

0 05 1 15 2 25 3 35 4 45 50

2

4

6

8

10

12

Pollu

tion

abat

emen

t effo

rt u

nder

coop

erat

ion

I4

I3

Time t

Figure 2 The pollution abatement effort under cooperation where1198683 and 1198684 are the pollution abatement effort of governments undercooperation in nations 1 and 2 respectively

From Table 2 we can see that for each governmentthe noncooperative optimal tax rates 1198881(119905) and 1198882(119905) are lessthan the cooperative tax rates 1198881(119905) and 1198882(119905) This meansthat cooperation would push to raise the tax rate to reduceproduction and cut pollution And for both governmentsthe optimal pollution tax rate is not changing with timethis is different from Huang et al [10] and Yeung andPetrosyan [2] because in their research they model afinite-horizon level game regardless of cooperative game ornoncooperative game and in the later stage of game bothgovernments would pay more attention to the economic

0 05 1 15 2 25 3 35 4 45 52

4

6

8

10

12

14

16

18

20

22

Pollu

tion

stock

x2

x1

Time t

Figure 3 The dynamics of pollution stock under noncooperationor cooperation where 1199091 and 1199092 denote the pollution stock undernoncooperation and cooperation respectively

0 05 1 15 2 25 3 35 4 45 5Time t

times106

Insta

ntan

eous

pay

men

ts of

natio

n 1

unde

r coo

pera

tion

61466861467

614672614674614676614678

61468614682614684614686

Figure 4 The instantaneous payments of nation 1 under coopera-tion

benefits than pollution abatementThen they would decreasethe pollution tax rate to encourage production and stimulateeconomic development [10] But in our model the gameis in infinite-horizon and there is no later stage of thegame so the optimal pollution tax rate will not decrease Inaddition under noncooperation the governments take theexternalities of production into account including their ownnation only However under a cooperation the externalitiesare considered by the governments which include their ownnation and the neighbors so the optimal pollution tax rate ishigher than under noncooperation

Figures 1 and 2 show the pollution abatement effort ofboth governments From them we find that under noncoop-eration the pollution abatement effort of both governmentsin the two nations could remain stable all the time butunder cooperation both governments would make a highpollution abatement effort at first and then the effort levelmay reduce to a certain degree after that it would tendto remain stable For each government the noncooperativepollution abatement effort is lower than the cooperativepollution abatement effort This is determined by externality

Mathematical Problems in Engineering 9In

stan

tane

ous p

aym

ents

of

natio

n 2

unde

r coo

pera

tion times10

6

0 05 1 15 2 25 3 35 4 45 5Time t

29685929686

296861296862296863296864296865296866296867296868

Figure 5 The instantaneous payments of nation 2 under coopera-tion

of pollution emissions Under cooperative game the govern-ments pay attention to the damage for the entire region somarginal damage of pollution emission would be greater andthe governments would be more concerned about pollutantreduction

Figure 3 shows the dynamics of pollution stock undernoncooperation or cooperation We can see that the pollu-tion stock under noncooperative game is higher than thatunder cooperative game It is due to the higher pollutiontax rate and higher pollution abatement effort of both gov-ernments under cooperative game Then in both economicand environmental terms cooperation is always better thannoncooperation

Now consider the case when the two nations want tocooperate To secure group optimality the two nations seek tomaximize their joint expected payoff by solving the followingstochastic control problem

max1198881(119904)1198882(119904)1198681(119904)1198682(119904)

1198640 intinfin0

(58501198881 (119904) + 32501198882 (119904) minus 04 (1198681 (119904))2

minus 05 (1198682 (119904))2 minus 102375(1198881 (119904))2 minus50375(1198882 (119904))2 minus 10119909 (119904))

sdot 119890minus005119889119905(31)

subject to

119889119909 (119904) = 349 minus 128119909 (119904) + 005119909 (119904) 119889119911 (119904) (32)

According toTheorem 4 we get the appropriate dynamicdistribution scheme that both group optimality and individ-ual rationality are satisfied at any time during cooperativegame The instantaneous payments of two nations 1198781(119905 119909lowast119905 )and 1198782(119905 119909lowast119905 ) at different time 119905 with given 119909lowast119905 are shown in(33a) and (33b) and Figures 3 and 4

1198781 (119905 119909lowast119905 ) = 1007100178948148626983247163840000000000000minus 12670021998361119890minus321199052578125000000 (33a)

1198782 (119905 119909lowast119905 ) = 77822033722351462743911532621440000000000000minus 1518945509601119890minus321199052519531250000 (33b)

It is shown in Figures 4 and 5 that the instantaneous pay-ment of both nations which satisfy the property of subgameconsistency shows an increasing trend at the start and thenremains stable In addition the instantaneous payment fornation 1 is always higher than that of nation 2This is becausethe industrial firms in nation 1 have more green and energy-efficient technology so they would get more instantaneouspayment

6 Conclusions

In this paper we have shown a stochastic differential game oftransboundary industrial pollution between two asymmetricnations in infinite-horizon level In our model the firmsand governments are separate entities As participants inthe games they take game behavior to optimize their ownpayment and the game equilibrium has been solved underthe two levels of firms and governments simultaneouslyThe uncertain evolution dynamic of pollution stock thathas a long-term global impact has been taken into accountin our models Otherwise another negative environmentalexternality a regional nonaccumulative pollutant that causesa short-term local impact has also been considered Wecharacterize the parameters of spaces where two neighboringgovernments can cooperate and give the feedback Nashequilibrium strategies of governments and industrial firmsThrough analysis we have shown that the optimal pollutiontax rate hinges on the two effects of regional damagethe source region and the global pollution stock undernoncooperative game while under cooperative game inaddition to the above two kinds of effects it also hingeson the additional effect of regional damage coming fromthe neighborhood so the latter is greater than the formerMoreover we have shown that the pollution abatement effortunder cooperation is more than under noncooperation Byanalyzing this stochastic differential game model we havealso provided a payment distribution mechanism whichsupports the subgame consistent solution and present anumerical example to illustrate the implementation of thismechanism

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge the financial supportfrom the ChinaNational Funding for Social Science ResearchProject (Project no 15CJY037) and the Philosophy and SocialScience Foundation of Hunan China (Project no 16YBA315)for this paper

10 Mathematical Problems in Engineering

References

[1] D W Yeung ldquoDynamically consistent cooperative solutionin a differential game of transboundary industrial pollutionrdquoJournal of Optimization Theory and Applications vol 134 no1 pp 143ndash160 2007

[2] D W Yeung and L A Petrosyan ldquoA cooperative stochasticdifferential game of transboundary industrial pollutionrdquo Auto-matica vol 44 no 6 pp 1532ndash1544 2008

[3] E J Dockner andNVanLong ldquoInternational pollution controlcooperative versus noncooperative strategiesrdquo Journal of Envi-ronmental Economics and Management vol 25 no 1 pp 13ndash291993

[4] L Petrosjan and G Zaccour ldquoTime-consistent Shapley valueallocation of pollution cost reductionrdquo Journal of EconomicDynamics amp Control vol 27 no 3 pp 381ndash398 2003

[5] B Bayramoglu ldquoTransboundary pollution in the black seacomparison of institutional arrangementsrdquo Environmental andResource Economics vol 35 no 4 pp 289ndash325 2006

[6] S Joslashrgensen and G Zaccour ldquoIncentive equilibrium strategiesand welfare allocation in a dynamic game of pollution controlrdquoAutomatica vol 37 no 1 pp 29ndash36 2001

[7] L Bertinelli C Camacho and B Zou ldquoCarbon capture andstorage and transboundary pollution a differential gameapproachrdquo European Journal of Operational Research vol 237no 2 pp 721ndash728 2014

[8] S Li ldquoA differential game of transboundary industrial pollutionwith emission permits tradingrdquo Journal of OptimizationTheoryand Applications vol 163 no 2 pp 642ndash659 2014

[9] H Benchekroun and G Martın-Herran ldquoThe impact of fore-sight in a transboundary pollution gamerdquo European Journal ofOperational Research vol 251 no 1 pp 300ndash309 2016

[10] XHuang PHe andW Zhang ldquoA cooperative differential gameof transboundary industrial pollution between two regionsrdquoJournal of Cleaner Production vol 120 pp 43ndash52 2016

[11] A Haurie J B Krawczyk and M Roche ldquoMonitoring coop-erative equilibria in a stochastic differential gamerdquo Journal ofOptimization Theory and Applications vol 81 no 1 pp 73ndash951994

[12] E Saltari and G Travaglini ldquoThe effects of environmental poli-cies on the abatement investment decisions of a green firmrdquoResource and Energy Economics vol 33 no 3 pp 666ndash685 2011

[13] S Athanassoglou and A Xepapadeas ldquoPollution control withuncertain stock dynamics when and how to be precautiousrdquoJournal of Environmental Economics and Management vol 63no 3 pp 304ndash320 2012

[14] E J Dockner and K Nishimura ldquoTransboundary pollution ina dynamic game modelrdquo Japanese Economic Review vol 50 no4 pp 443ndash456 1999

[15] F Bosello B Buchner and C Carraro ldquoEquity developmentand climate change controlrdquo Journal of the European EconomicAssociation vol 1 no 2-3 pp 601ndash611 2003

[16] A Bernard A Haurie M Vielle and L Viguier ldquoA two-leveldynamic game of carbon emission trading between RussiaChina and annex B countriesrdquo Journal of Economic Dynamicsand Control vol 32 no 6 pp 1830ndash1856 2008

Submit your manuscripts athttpswwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

seek to maximize their joint expected payoff by solving thefollowing stochastic control problem

119882(119904 119909) = 1198641199050 intinfin0 sum119899119894=1 (1205731198941)221198881 (119904) + sum119898119895=1 (1205731198952)221198882 (119904)

minus 122sum119896=1

119886119896 (119868119896 (119904))2 minus (12057611 + 12057621)sum119899119894=1 (1205731198941)2411988821 (119904)minus (12057612 + 12057622)sum119898119895=1 (1205731198952)2411988822 (119904) minus (ℎ1 + ℎ2) 119909 (119904)119890minus119903119905119889119905

(13)

subject to (7)Invoking Bellmanrsquos technique a set of controls 119888lowast119896 119868lowast119896 =120601119888119896(119909) 120601119868119896(119909) constitutes an optimal solution to the stochas-

tic control problem (13) and (7) if there exists continuouslydifferentiable function 119882(119905 119909) [119905infin) times 119877 rarr 119877 119896 isin 1 2satisfying the following partial differential equations

119903119882 (119905 119909) minus 1212059021199092119882119909119909 (119905 119909) = max1198881 11986811198882 1198682

sum119899119894=1 (1205731198941)221198881 (119905)

+ sum119898119895=1 (1205731198952)221198882 (119905) minus 122sum119896=1

119886119896 (119868119896 (119905))2

minus (12057611 + 12057621)sum119899119894=1 (1205731198941)2411988821 (119905) minus (12057612 + 12057622)sum119898119895=1 (1205731198952)2411988822 (119905)minus (ℎ1 + ℎ2) 119909 (119905) + 119882119883 (119905 119909)(sum119899119894=1 1205721198941 (1205731198941)2411988821 (119905)+ sum119898119895=1 1205721198952 (1205731198952)2411988822 (119905)minus 2sum119896=1

120574119896 (119868119896 (119905) (119909 (119905))12 minus 120575119909 (119905)))

(14)

Performing the indicated maximization in (14) yields theoptimal controls under cooperation

1206011198881 (119905 119909) = 12057611 + 12057621 minus 119882119909 (119905 119909) 1206011198882 (119905 119909) = 12057612 + 12057622 minus 119882119909 (119905 119909) (15a)

1206011198681 (119909) = minus1205741119882119909 (119909) 119909121198861 1206011198682 (119909) = minus1205742119882119909 (119909) 119909121198862

(15b)

Substituting the results in (15a) and (15b) into (14) weobtain the following

Proposition 3 System (13) admits a solution

119882(119905 119909) = [119860119909 (119905) + 119861] (16)

where 119860 119861 satisfy the following set of constant coefficientquadratic ordinary differential equations

( 1205742121198861 + 1205742221198862)1198602 minus (119903 + 120575)119860 minus (ℎ1 + ℎ2) = 0 (17a)

119861 = 1119903 (sum119899119894=1 (1205731198941)2 + sum119898119895=1 (1205731198952)24 (12057611 + 12057622 minus 119860)2 ) (17b)

The corresponding feedback Nash equilibriums undercooperation can be obtained as

1206011198881 (119905 119909) = 12057611 + 12057622 minus 1198601206011198681 (119905 119909) = minus120574111986011990912 (119905)21198861 1206011198882 (119905 119909) = 12057611 + 12057622 minus 1198601206011198682 (119905 119909) = minus120574211986011990912 (119905)21198862

(18)

Substituting the optimal control strategy from (18) into(7) yields the dynamics of pollution accumulation undercooperation

119889119909 (119904) = (sum119899119894=1 1205721198941 (1205731198941)2 + sum119898119895=1 1205721198952 (1205731198952)24 (12057611 + 12057622 minus 119860)2+ 2sum119896=1

(120574119896)2 119860119909 (119904)2119886119896 minus 120575119909 (119904))119889119904 + 120590119909 (119904) 119889119911 (119904) 119909 (1199050) = 1199090

(19)

Solving the stochastic cooperative pollution dynamicsyields the cooperative state trajectory

119909lowast (119905) = 119890[int1199051199050 [(12057421 21198861+12057422 21198862)119860minus120575minus1205902]119889119904+int1199051199050 120590119889119911(119904)] times [[1199091199050+ int1199051199050(sum119899119894=1 1205721198941 (1205731198941)2 + sum119898119895=1 1205721198952 (1205731198952)24 (12057611 + 12057622 minus 119860)2 )

times 119890[int1199051199050 [12059022+120575minus(12057421 21198861+12057422 21198862)119860]119889120591+int1199051199050 120590119889119911(119904)]119889119904]] for 119905 isin [1199050infin)

(20)

42 Individually Rational and Time-Consistent Imputationand Payment Distribution Mechanism Let 119883lowast119905 denote the

6 Mathematical Problems in Engineering

set of realizable values of 119909lowast(119905) at time 119905 generated by (20)The term 119909lowast119905 is used to denote an element of the set 119883lowast119905 An agreed upon optimality principle would be sought toallocate the cooperative payoff In a dynamic frameworkindividual rationality has to be maintained at every instantof time within the cooperative duration [1199050infin) along thecooperative trajectory (20) For 120591 isin [1199050infin) let vector120585119896(119909lowast119905 ) = 1205851(119909lowast119905 ) 1205852(119909lowast119905 ) denote the solution imputation(payoff under cooperation) over the period [120591infin) to player119896 isin 1 2 given that the state is 119909lowast120591 isin 119883lowast120591 Individualrationality along the cooperative trajectory requires

120585(120591)119896 (120591 119909lowast119905 ) ge 119881(120591)119896 (120591 119909lowast119905 ) for 119896 isin 1 2 (21)

where 119881(120591)119896(120591 119909lowast119905 ) denotes the payoff to nation 119896 undernoncooperation over the period Let 119866(119904) = [1198661(119904) 1198662(119904)]denote the instantaneous payoff of the cooperative game attime 119904 isin [1199050infin) for the cooperative Γ119888(119909lowast1199050)Theorem 4 An instantaneous payment at time 120591 isin [1199050 119879]equaling

119866119870 (120591) = 119903120585119896 (119909lowast120591 ) minus 121205902 (119909lowast120591 )2 120585119896119909lowast120591119909lowast120591 (119909lowast120591 ) minus 120585119896119909lowast120591

(119909lowast120591 )sdot ( 119899sum119894=1

1205721198941 ( 120573119894121198881 (120591 119909lowast120591 ))2 + 119898sum119895=1

1205721198952 ( 120573119895221198882 (120591 119909lowast120591 ))2

minus 2sum119896=1

120574119896119868119896 (120591 119909lowast120591 ) (119909lowast120591 )12 minus 120575119909lowast120591) for 120591 isin [1199050infin) 119909lowast120591 isin 119883lowast120591 119896 isin 1 2

(22)

yields a subgame consistent solution to the cooperative gameΓ119888(119909lowast1199050)Proof Along the cooperative trajectory 119909lowast(119905)119905ge1199050 we haveΥ119896 (120591 120591 119909lowast120591 ) = 120585119896 (119909lowast120591 )

= 119864120591 intinfin120591

119866119896 (119904) 119890minus119903(119904minus120591)119889119904 | 119909 (120591) = 119909lowast120591 Υ119896 (120591 119905 119909lowast119905 ) = 120585119896 (119909lowast119905 )

= 119864119905 intinfin119905

119866119896 (119904) 119890minus119903(119904minus119905)119889119904 | 119909 (119905) = 119909lowast119905 for 119896 isin 1 2 119909lowast119905 isin 119883lowast119905 119905 ge 120591

(23)

Note that

Υ119896 (120591 119905 119909lowast119905 ) = 119890minus119903(119904minus120591)120585 (119909lowast119905 ) = 119890minus119903(119904minus120591)Υ119896 (119905 119905 119909lowast119905 ) 119896 isin 1 2 119909lowast120591 isin 119883lowast120591 (24)

Since Υ119896(119905 119905 119909lowast119905 ) is continuously differentiable in 119905 and119909lowast119905 one can obtain

Υ119896 (120591 120591 119909lowast120591 ) = 119864120591 int120591+Δ119905120591

119866119896 (119904) 119890minus119903(119904minus120591)119889119904+ 119890minus119903(Δ119905)Υ119896 (120591 + Δ119905 120591 + Δ119905 119909lowast120591 + Δ119909lowast120591 ) | 119909 (120591)= 119909lowast120591 for 120591 isin [1199050infin) 119909lowast120591 isin 119883lowast120591 119896 isin 1 2

(25)

where

Δ119909lowast120591= 119899sum119894=1

1205721198941 ( 120573119894121198881 (120591 119909lowast120591 ))2 + 119898sum119895=1

1205721198952 ( 120573119895221198882 (120591 119909lowast120591 ))2

minus 2sum119896=1

120574119896 (119868119870 (120591 119909lowast120591 ) (119909lowast120591 )12 minus 120575119909lowast120591)Δ119905+ 120590 [119909lowast120591 ] Δ119911120591 + 119900 (Δ119905)

Δ119911120591 = 119911 (120591 + Δ119905) minus 119911 (120591) 119864120591 [119900 (Δ119905)]Δ119905 997888rarr 0

as Δ119905 997888rarr 0

(26)

From (24) and (25) one can obtain

Υ119896 (120591 120591 119909lowast120591 ) = 119864120591 int120591+Δ119905120591

119866119896 (119904) 119890minus119903(119904minus120591)119889119904+ Υ119896 (120591 120591 + Δ119905 119909lowast120591 + Δ119909lowast120591 ) | 119909 (120591) = 119909lowast120591

for 120591 isin [1199050infin) 119909lowast120591 isin 119883lowast120591 119896 isin 1 2 (27)

Therefore with Δ119905 rarr 0 (27) can be expressed as

119866119896 (120591) Δ119905 = minus [Υ119896119905 (120591 119905 119909lowast120591 )1003816100381610038161003816119905=120591] Δ119905 minus 121205902 (119909lowast120591 )2sdot [Υ119896119909lowast

120591119909lowast120591

(120591 119905 119909lowast120591 )1003816100381610038161003816119905=120591] Δ119905 minus [Υ119896119909lowast120591

(120591 119905 119909lowast120591 )1003816100381610038161003816119905=120591]times ( 119899sum119894=1

1205721198941 ( 120573119894121198881 (120591 119909lowast120591 ))2 + 119898sum119895=1

1205721198952 ( 120573119895221198882 (120591 119909lowast120591 ))2

minus 2sum119896=1

120574119896 (119868119870 (120591 119909lowast120591 ) (119909lowast120591 )12 minus 120575119909lowast120591)Δ119905 minus 119900 (Δ119905))

(28)

Dividing (28) throughout by Δ119905 with Δ119905 rarr 0 and takingexpectation one can obtain (22) HenceTheorem 4 follows

When the two nations adopt the cooperative strategiesthe rate of instantaneous payment that nation 119896 (119896 isin 1 2)

Mathematical Problems in Engineering 7

Table 1 The parameters of model used in the numerical solutions

1199050 12057311 12057321 12057312 12057322 119909(1199050) 12057211 12057221 12057212 12057222 1205741 12057420 60 90 40 70 20 04 03 05 04 04 02120575 120590 1198861 1198862 12057611 12057621 12057622 12057612 119903 ℎ1 ℎ2001 005 08 1 02 015 015 016 005 4 5

will realize at time 120591 with the state being 119909lowast120591 can be expressedas

1198781 (120591 119909) = 1199032 [1198601119909lowast120591 + 1198611] + [119860119909lowast120591 + 119861]minus [1198602119909lowast120591 + 1198612] minus 12 [1198601 + 119860 minus 1198602]sdot

sum119899119894=1 (1205731198941)2 + sum119898119895=1 (1205731198952)24 (12057611 + 12057622 minus 119860)2+ 2sum119896=1

1205742119896119860119909lowast1205912119886119896 minus 120575119909lowast120591

(29a)

1198782 (120591 119909) = 1199032 [1198602119909lowast120591 + 1198612] + [119860119909lowast120591 + 119861]minus [1198601119909lowast120591 + 1198611] minus 12 [1198602 + 119860 minus 1198601]sdot

sum119899119894=1 (1205731198941)2 + sum119898119895=1 (1205731198952)24 (12057611 + 12057622 minus 119860)2+ 2sum119896=1

1205742119896119860119909lowast1205912119886119896 minus 120575119909lowast120591

(29b)

5 Numerical Example

Consider a multinational economy consisting of two nationsand each with two industrial firms Industrial firmrsquos revenuefunction in nation 1 and nation 2 at time 119904 is 120601119894 = 1205731198941(119906119894(119904))12and 120601119895 = 1205731198952(V119895(119904))12 (119894 119895 = 1 2) respectively where12057311 = 60 12057321 = 90 12057312 = 40 and 12057322 = 70 The valueof 1205731198941 (119894 isin 1 2) is higher than 1205731198952 (119895 isin 1 2) it meansthat compared with industrial firms in nation 2 industrialfirms in nation 1 have more energy-efficient technology Inaddition industrial firms in nation 1 have technical advantageresult in 12057211 lt 12057212 12057221 lt 12057222 1205741 gt 1205742 As mentionedabove the optimal pollution emissions of the firms in nations1 and 2 are 119906lowast119894 (119904) = (120573119894121198881(119904))2 and Vlowast119895 (119904) = (120573119895221198882(119904))2respectively The dynamics of pollution stock are governedby the stochastic differential equation (4) where 119909(1199050) = 20120575 = 001 and 120590 = 005 We defined 119886119896 as the amountadded to the pollution stock by a unit of output in nation119896 (119896 = 1 2) They are a positive number less than 1 that isto say the stock of pollution will increase with the outputbut not higher than the amount of output According to

Table 2 The optimal pollution tax rate

1198881(119905) 1198882(119905) 1198881(119905) 1198882(119905)820 234 1090 1087

experience we assign 12057211 = 04 12057221 = 03 12057212 = 0512057222 = 04 120574119896 (119896 = 1 2) can also be considered as pollutionabatement effort efficiency which is linked to pollutioncontrol policies and environmental governance efficiency indifferent nations Generally speaking pollution abatementeffort efficiency is higher in developed countries We assign1205741 = 04 1205742 = 02 Abatement costs are 04(1198681(119904))12 and05(1198682(119904))12 for nations 1 and 2 respectively Local pollutionimpacts are closely linked to the level of production activitiesThe accumulation of pollution stock like greenhouse gas oftenconcerns the interactions between the natural environmentand the pollutants emitted So stochastic elements wouldappear According to what we defined before 1205761119896 1205762119896 (119896 = 1 2)is a pollution damage cost coefficients It is said that the biggerthey are the higher the cost of environmental damage forthis nation is We make the following assignments 12057611 = 0212057621 = 015 12057622 = 015 and 12057612 = 016 ℎ119896 (119896 = 1 2) arethe value of pollution damage to nation 119896 (119896 = 1 2) Thelarger they are the greater the impact of the environment iswhich is determined by the different regions And we makeassignments ℎ1 = 4 ℎ2 = 5 Other parameters are as follows119903 = 005 1199050 = 0 The parameters of the model which are usedin the numerical solution are presented in Table 1

Then we obtain a stochastic differential game in whichgovernment 1 and government 2 under noncooperation seek

max1198881(119904)1198681(119904)

intinfin0

119890minus119903119905 (58501198881 (119904) minus 0411986821 (119904) minus 58511988821 (119904) minus 162211988822 (119904)minus 4119909 (119904)) 119889119909

(30a)

max1198882(119904)1198682(119904)

intinfin0

119890minus119903119905 (32501198882 (119904) minus 0511986822 (119904) minus 4387511988821 (119904)minus 2437511988822 (119904) minus 5119909 (119904)) 119889119909

(30b)

Through the analysis mentioned above we have obtainedthe pollution tax rate abatement effort of each nationand the dynamics of pollution stock under noncooperativegame and cooperative game Now we will analyze theirdifference between the governments under noncooperationand cooperation through Table 2 and Figures 1ndash5

8 Mathematical Problems in Engineering

0 05 1 15 2 25 3 35 4 45 515

2

25

3

35

4

45

Pollu

tion

abat

emen

t effo

rt u

nder

non

coop

erat

ion

I2

I1

Time t

Figure 1 The pollution abatement effort under noncooperationwhere 1198681 and 1198682 denote the pollution abatement effort of govern-ments under noncooperation in nations 1 and 2 respectively

0 05 1 15 2 25 3 35 4 45 50

2

4

6

8

10

12

Pollu

tion

abat

emen

t effo

rt u

nder

coop

erat

ion

I4

I3

Time t

Figure 2 The pollution abatement effort under cooperation where1198683 and 1198684 are the pollution abatement effort of governments undercooperation in nations 1 and 2 respectively

From Table 2 we can see that for each governmentthe noncooperative optimal tax rates 1198881(119905) and 1198882(119905) are lessthan the cooperative tax rates 1198881(119905) and 1198882(119905) This meansthat cooperation would push to raise the tax rate to reduceproduction and cut pollution And for both governmentsthe optimal pollution tax rate is not changing with timethis is different from Huang et al [10] and Yeung andPetrosyan [2] because in their research they model afinite-horizon level game regardless of cooperative game ornoncooperative game and in the later stage of game bothgovernments would pay more attention to the economic

0 05 1 15 2 25 3 35 4 45 52

4

6

8

10

12

14

16

18

20

22

Pollu

tion

stock

x2

x1

Time t

Figure 3 The dynamics of pollution stock under noncooperationor cooperation where 1199091 and 1199092 denote the pollution stock undernoncooperation and cooperation respectively

0 05 1 15 2 25 3 35 4 45 5Time t

times106

Insta

ntan

eous

pay

men

ts of

natio

n 1

unde

r coo

pera

tion

61466861467

614672614674614676614678

61468614682614684614686

Figure 4 The instantaneous payments of nation 1 under coopera-tion

benefits than pollution abatementThen they would decreasethe pollution tax rate to encourage production and stimulateeconomic development [10] But in our model the gameis in infinite-horizon and there is no later stage of thegame so the optimal pollution tax rate will not decrease Inaddition under noncooperation the governments take theexternalities of production into account including their ownnation only However under a cooperation the externalitiesare considered by the governments which include their ownnation and the neighbors so the optimal pollution tax rate ishigher than under noncooperation

Figures 1 and 2 show the pollution abatement effort ofboth governments From them we find that under noncoop-eration the pollution abatement effort of both governmentsin the two nations could remain stable all the time butunder cooperation both governments would make a highpollution abatement effort at first and then the effort levelmay reduce to a certain degree after that it would tendto remain stable For each government the noncooperativepollution abatement effort is lower than the cooperativepollution abatement effort This is determined by externality

Mathematical Problems in Engineering 9In

stan

tane

ous p

aym

ents

of

natio

n 2

unde

r coo

pera

tion times10

6

0 05 1 15 2 25 3 35 4 45 5Time t

29685929686

296861296862296863296864296865296866296867296868

Figure 5 The instantaneous payments of nation 2 under coopera-tion

of pollution emissions Under cooperative game the govern-ments pay attention to the damage for the entire region somarginal damage of pollution emission would be greater andthe governments would be more concerned about pollutantreduction

Figure 3 shows the dynamics of pollution stock undernoncooperation or cooperation We can see that the pollu-tion stock under noncooperative game is higher than thatunder cooperative game It is due to the higher pollutiontax rate and higher pollution abatement effort of both gov-ernments under cooperative game Then in both economicand environmental terms cooperation is always better thannoncooperation

Now consider the case when the two nations want tocooperate To secure group optimality the two nations seek tomaximize their joint expected payoff by solving the followingstochastic control problem

max1198881(119904)1198882(119904)1198681(119904)1198682(119904)

1198640 intinfin0

(58501198881 (119904) + 32501198882 (119904) minus 04 (1198681 (119904))2

minus 05 (1198682 (119904))2 minus 102375(1198881 (119904))2 minus50375(1198882 (119904))2 minus 10119909 (119904))

sdot 119890minus005119889119905(31)

subject to

119889119909 (119904) = 349 minus 128119909 (119904) + 005119909 (119904) 119889119911 (119904) (32)

According toTheorem 4 we get the appropriate dynamicdistribution scheme that both group optimality and individ-ual rationality are satisfied at any time during cooperativegame The instantaneous payments of two nations 1198781(119905 119909lowast119905 )and 1198782(119905 119909lowast119905 ) at different time 119905 with given 119909lowast119905 are shown in(33a) and (33b) and Figures 3 and 4

1198781 (119905 119909lowast119905 ) = 1007100178948148626983247163840000000000000minus 12670021998361119890minus321199052578125000000 (33a)

1198782 (119905 119909lowast119905 ) = 77822033722351462743911532621440000000000000minus 1518945509601119890minus321199052519531250000 (33b)

It is shown in Figures 4 and 5 that the instantaneous pay-ment of both nations which satisfy the property of subgameconsistency shows an increasing trend at the start and thenremains stable In addition the instantaneous payment fornation 1 is always higher than that of nation 2This is becausethe industrial firms in nation 1 have more green and energy-efficient technology so they would get more instantaneouspayment

6 Conclusions

In this paper we have shown a stochastic differential game oftransboundary industrial pollution between two asymmetricnations in infinite-horizon level In our model the firmsand governments are separate entities As participants inthe games they take game behavior to optimize their ownpayment and the game equilibrium has been solved underthe two levels of firms and governments simultaneouslyThe uncertain evolution dynamic of pollution stock thathas a long-term global impact has been taken into accountin our models Otherwise another negative environmentalexternality a regional nonaccumulative pollutant that causesa short-term local impact has also been considered Wecharacterize the parameters of spaces where two neighboringgovernments can cooperate and give the feedback Nashequilibrium strategies of governments and industrial firmsThrough analysis we have shown that the optimal pollutiontax rate hinges on the two effects of regional damagethe source region and the global pollution stock undernoncooperative game while under cooperative game inaddition to the above two kinds of effects it also hingeson the additional effect of regional damage coming fromthe neighborhood so the latter is greater than the formerMoreover we have shown that the pollution abatement effortunder cooperation is more than under noncooperation Byanalyzing this stochastic differential game model we havealso provided a payment distribution mechanism whichsupports the subgame consistent solution and present anumerical example to illustrate the implementation of thismechanism

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge the financial supportfrom the ChinaNational Funding for Social Science ResearchProject (Project no 15CJY037) and the Philosophy and SocialScience Foundation of Hunan China (Project no 16YBA315)for this paper

10 Mathematical Problems in Engineering

References

[1] D W Yeung ldquoDynamically consistent cooperative solutionin a differential game of transboundary industrial pollutionrdquoJournal of Optimization Theory and Applications vol 134 no1 pp 143ndash160 2007

[2] D W Yeung and L A Petrosyan ldquoA cooperative stochasticdifferential game of transboundary industrial pollutionrdquo Auto-matica vol 44 no 6 pp 1532ndash1544 2008

[3] E J Dockner andNVanLong ldquoInternational pollution controlcooperative versus noncooperative strategiesrdquo Journal of Envi-ronmental Economics and Management vol 25 no 1 pp 13ndash291993

[4] L Petrosjan and G Zaccour ldquoTime-consistent Shapley valueallocation of pollution cost reductionrdquo Journal of EconomicDynamics amp Control vol 27 no 3 pp 381ndash398 2003

[5] B Bayramoglu ldquoTransboundary pollution in the black seacomparison of institutional arrangementsrdquo Environmental andResource Economics vol 35 no 4 pp 289ndash325 2006

[6] S Joslashrgensen and G Zaccour ldquoIncentive equilibrium strategiesand welfare allocation in a dynamic game of pollution controlrdquoAutomatica vol 37 no 1 pp 29ndash36 2001

[7] L Bertinelli C Camacho and B Zou ldquoCarbon capture andstorage and transboundary pollution a differential gameapproachrdquo European Journal of Operational Research vol 237no 2 pp 721ndash728 2014

[8] S Li ldquoA differential game of transboundary industrial pollutionwith emission permits tradingrdquo Journal of OptimizationTheoryand Applications vol 163 no 2 pp 642ndash659 2014

[9] H Benchekroun and G Martın-Herran ldquoThe impact of fore-sight in a transboundary pollution gamerdquo European Journal ofOperational Research vol 251 no 1 pp 300ndash309 2016

[10] XHuang PHe andW Zhang ldquoA cooperative differential gameof transboundary industrial pollution between two regionsrdquoJournal of Cleaner Production vol 120 pp 43ndash52 2016

[11] A Haurie J B Krawczyk and M Roche ldquoMonitoring coop-erative equilibria in a stochastic differential gamerdquo Journal ofOptimization Theory and Applications vol 81 no 1 pp 73ndash951994

[12] E Saltari and G Travaglini ldquoThe effects of environmental poli-cies on the abatement investment decisions of a green firmrdquoResource and Energy Economics vol 33 no 3 pp 666ndash685 2011

[13] S Athanassoglou and A Xepapadeas ldquoPollution control withuncertain stock dynamics when and how to be precautiousrdquoJournal of Environmental Economics and Management vol 63no 3 pp 304ndash320 2012

[14] E J Dockner and K Nishimura ldquoTransboundary pollution ina dynamic game modelrdquo Japanese Economic Review vol 50 no4 pp 443ndash456 1999

[15] F Bosello B Buchner and C Carraro ldquoEquity developmentand climate change controlrdquo Journal of the European EconomicAssociation vol 1 no 2-3 pp 601ndash611 2003

[16] A Bernard A Haurie M Vielle and L Viguier ldquoA two-leveldynamic game of carbon emission trading between RussiaChina and annex B countriesrdquo Journal of Economic Dynamicsand Control vol 32 no 6 pp 1830ndash1856 2008

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

set of realizable values of 119909lowast(119905) at time 119905 generated by (20)The term 119909lowast119905 is used to denote an element of the set 119883lowast119905 An agreed upon optimality principle would be sought toallocate the cooperative payoff In a dynamic frameworkindividual rationality has to be maintained at every instantof time within the cooperative duration [1199050infin) along thecooperative trajectory (20) For 120591 isin [1199050infin) let vector120585119896(119909lowast119905 ) = 1205851(119909lowast119905 ) 1205852(119909lowast119905 ) denote the solution imputation(payoff under cooperation) over the period [120591infin) to player119896 isin 1 2 given that the state is 119909lowast120591 isin 119883lowast120591 Individualrationality along the cooperative trajectory requires

120585(120591)119896 (120591 119909lowast119905 ) ge 119881(120591)119896 (120591 119909lowast119905 ) for 119896 isin 1 2 (21)

where 119881(120591)119896(120591 119909lowast119905 ) denotes the payoff to nation 119896 undernoncooperation over the period Let 119866(119904) = [1198661(119904) 1198662(119904)]denote the instantaneous payoff of the cooperative game attime 119904 isin [1199050infin) for the cooperative Γ119888(119909lowast1199050)Theorem 4 An instantaneous payment at time 120591 isin [1199050 119879]equaling

119866119870 (120591) = 119903120585119896 (119909lowast120591 ) minus 121205902 (119909lowast120591 )2 120585119896119909lowast120591119909lowast120591 (119909lowast120591 ) minus 120585119896119909lowast120591

(119909lowast120591 )sdot ( 119899sum119894=1

1205721198941 ( 120573119894121198881 (120591 119909lowast120591 ))2 + 119898sum119895=1

1205721198952 ( 120573119895221198882 (120591 119909lowast120591 ))2

minus 2sum119896=1

120574119896119868119896 (120591 119909lowast120591 ) (119909lowast120591 )12 minus 120575119909lowast120591) for 120591 isin [1199050infin) 119909lowast120591 isin 119883lowast120591 119896 isin 1 2

(22)

yields a subgame consistent solution to the cooperative gameΓ119888(119909lowast1199050)Proof Along the cooperative trajectory 119909lowast(119905)119905ge1199050 we haveΥ119896 (120591 120591 119909lowast120591 ) = 120585119896 (119909lowast120591 )

= 119864120591 intinfin120591

119866119896 (119904) 119890minus119903(119904minus120591)119889119904 | 119909 (120591) = 119909lowast120591 Υ119896 (120591 119905 119909lowast119905 ) = 120585119896 (119909lowast119905 )

= 119864119905 intinfin119905

119866119896 (119904) 119890minus119903(119904minus119905)119889119904 | 119909 (119905) = 119909lowast119905 for 119896 isin 1 2 119909lowast119905 isin 119883lowast119905 119905 ge 120591

(23)

Note that

Υ119896 (120591 119905 119909lowast119905 ) = 119890minus119903(119904minus120591)120585 (119909lowast119905 ) = 119890minus119903(119904minus120591)Υ119896 (119905 119905 119909lowast119905 ) 119896 isin 1 2 119909lowast120591 isin 119883lowast120591 (24)

Since Υ119896(119905 119905 119909lowast119905 ) is continuously differentiable in 119905 and119909lowast119905 one can obtain

Υ119896 (120591 120591 119909lowast120591 ) = 119864120591 int120591+Δ119905120591

119866119896 (119904) 119890minus119903(119904minus120591)119889119904+ 119890minus119903(Δ119905)Υ119896 (120591 + Δ119905 120591 + Δ119905 119909lowast120591 + Δ119909lowast120591 ) | 119909 (120591)= 119909lowast120591 for 120591 isin [1199050infin) 119909lowast120591 isin 119883lowast120591 119896 isin 1 2

(25)

where

Δ119909lowast120591= 119899sum119894=1

1205721198941 ( 120573119894121198881 (120591 119909lowast120591 ))2 + 119898sum119895=1

1205721198952 ( 120573119895221198882 (120591 119909lowast120591 ))2

minus 2sum119896=1

120574119896 (119868119870 (120591 119909lowast120591 ) (119909lowast120591 )12 minus 120575119909lowast120591)Δ119905+ 120590 [119909lowast120591 ] Δ119911120591 + 119900 (Δ119905)

Δ119911120591 = 119911 (120591 + Δ119905) minus 119911 (120591) 119864120591 [119900 (Δ119905)]Δ119905 997888rarr 0

as Δ119905 997888rarr 0

(26)

From (24) and (25) one can obtain

Υ119896 (120591 120591 119909lowast120591 ) = 119864120591 int120591+Δ119905120591

119866119896 (119904) 119890minus119903(119904minus120591)119889119904+ Υ119896 (120591 120591 + Δ119905 119909lowast120591 + Δ119909lowast120591 ) | 119909 (120591) = 119909lowast120591

for 120591 isin [1199050infin) 119909lowast120591 isin 119883lowast120591 119896 isin 1 2 (27)

Therefore with Δ119905 rarr 0 (27) can be expressed as

119866119896 (120591) Δ119905 = minus [Υ119896119905 (120591 119905 119909lowast120591 )1003816100381610038161003816119905=120591] Δ119905 minus 121205902 (119909lowast120591 )2sdot [Υ119896119909lowast

120591119909lowast120591

(120591 119905 119909lowast120591 )1003816100381610038161003816119905=120591] Δ119905 minus [Υ119896119909lowast120591

(120591 119905 119909lowast120591 )1003816100381610038161003816119905=120591]times ( 119899sum119894=1

1205721198941 ( 120573119894121198881 (120591 119909lowast120591 ))2 + 119898sum119895=1

1205721198952 ( 120573119895221198882 (120591 119909lowast120591 ))2

minus 2sum119896=1

120574119896 (119868119870 (120591 119909lowast120591 ) (119909lowast120591 )12 minus 120575119909lowast120591)Δ119905 minus 119900 (Δ119905))

(28)

Dividing (28) throughout by Δ119905 with Δ119905 rarr 0 and takingexpectation one can obtain (22) HenceTheorem 4 follows

When the two nations adopt the cooperative strategiesthe rate of instantaneous payment that nation 119896 (119896 isin 1 2)

Mathematical Problems in Engineering 7

Table 1 The parameters of model used in the numerical solutions

1199050 12057311 12057321 12057312 12057322 119909(1199050) 12057211 12057221 12057212 12057222 1205741 12057420 60 90 40 70 20 04 03 05 04 04 02120575 120590 1198861 1198862 12057611 12057621 12057622 12057612 119903 ℎ1 ℎ2001 005 08 1 02 015 015 016 005 4 5

will realize at time 120591 with the state being 119909lowast120591 can be expressedas

1198781 (120591 119909) = 1199032 [1198601119909lowast120591 + 1198611] + [119860119909lowast120591 + 119861]minus [1198602119909lowast120591 + 1198612] minus 12 [1198601 + 119860 minus 1198602]sdot

sum119899119894=1 (1205731198941)2 + sum119898119895=1 (1205731198952)24 (12057611 + 12057622 minus 119860)2+ 2sum119896=1

1205742119896119860119909lowast1205912119886119896 minus 120575119909lowast120591

(29a)

1198782 (120591 119909) = 1199032 [1198602119909lowast120591 + 1198612] + [119860119909lowast120591 + 119861]minus [1198601119909lowast120591 + 1198611] minus 12 [1198602 + 119860 minus 1198601]sdot

sum119899119894=1 (1205731198941)2 + sum119898119895=1 (1205731198952)24 (12057611 + 12057622 minus 119860)2+ 2sum119896=1

1205742119896119860119909lowast1205912119886119896 minus 120575119909lowast120591

(29b)

5 Numerical Example

Consider a multinational economy consisting of two nationsand each with two industrial firms Industrial firmrsquos revenuefunction in nation 1 and nation 2 at time 119904 is 120601119894 = 1205731198941(119906119894(119904))12and 120601119895 = 1205731198952(V119895(119904))12 (119894 119895 = 1 2) respectively where12057311 = 60 12057321 = 90 12057312 = 40 and 12057322 = 70 The valueof 1205731198941 (119894 isin 1 2) is higher than 1205731198952 (119895 isin 1 2) it meansthat compared with industrial firms in nation 2 industrialfirms in nation 1 have more energy-efficient technology Inaddition industrial firms in nation 1 have technical advantageresult in 12057211 lt 12057212 12057221 lt 12057222 1205741 gt 1205742 As mentionedabove the optimal pollution emissions of the firms in nations1 and 2 are 119906lowast119894 (119904) = (120573119894121198881(119904))2 and Vlowast119895 (119904) = (120573119895221198882(119904))2respectively The dynamics of pollution stock are governedby the stochastic differential equation (4) where 119909(1199050) = 20120575 = 001 and 120590 = 005 We defined 119886119896 as the amountadded to the pollution stock by a unit of output in nation119896 (119896 = 1 2) They are a positive number less than 1 that isto say the stock of pollution will increase with the outputbut not higher than the amount of output According to

Table 2 The optimal pollution tax rate

1198881(119905) 1198882(119905) 1198881(119905) 1198882(119905)820 234 1090 1087

experience we assign 12057211 = 04 12057221 = 03 12057212 = 0512057222 = 04 120574119896 (119896 = 1 2) can also be considered as pollutionabatement effort efficiency which is linked to pollutioncontrol policies and environmental governance efficiency indifferent nations Generally speaking pollution abatementeffort efficiency is higher in developed countries We assign1205741 = 04 1205742 = 02 Abatement costs are 04(1198681(119904))12 and05(1198682(119904))12 for nations 1 and 2 respectively Local pollutionimpacts are closely linked to the level of production activitiesThe accumulation of pollution stock like greenhouse gas oftenconcerns the interactions between the natural environmentand the pollutants emitted So stochastic elements wouldappear According to what we defined before 1205761119896 1205762119896 (119896 = 1 2)is a pollution damage cost coefficients It is said that the biggerthey are the higher the cost of environmental damage forthis nation is We make the following assignments 12057611 = 0212057621 = 015 12057622 = 015 and 12057612 = 016 ℎ119896 (119896 = 1 2) arethe value of pollution damage to nation 119896 (119896 = 1 2) Thelarger they are the greater the impact of the environment iswhich is determined by the different regions And we makeassignments ℎ1 = 4 ℎ2 = 5 Other parameters are as follows119903 = 005 1199050 = 0 The parameters of the model which are usedin the numerical solution are presented in Table 1

Then we obtain a stochastic differential game in whichgovernment 1 and government 2 under noncooperation seek

max1198881(119904)1198681(119904)

intinfin0

119890minus119903119905 (58501198881 (119904) minus 0411986821 (119904) minus 58511988821 (119904) minus 162211988822 (119904)minus 4119909 (119904)) 119889119909

(30a)

max1198882(119904)1198682(119904)

intinfin0

119890minus119903119905 (32501198882 (119904) minus 0511986822 (119904) minus 4387511988821 (119904)minus 2437511988822 (119904) minus 5119909 (119904)) 119889119909

(30b)

Through the analysis mentioned above we have obtainedthe pollution tax rate abatement effort of each nationand the dynamics of pollution stock under noncooperativegame and cooperative game Now we will analyze theirdifference between the governments under noncooperationand cooperation through Table 2 and Figures 1ndash5

8 Mathematical Problems in Engineering

0 05 1 15 2 25 3 35 4 45 515

2

25

3

35

4

45

Pollu

tion

abat

emen

t effo

rt u

nder

non

coop

erat

ion

I2

I1

Time t

Figure 1 The pollution abatement effort under noncooperationwhere 1198681 and 1198682 denote the pollution abatement effort of govern-ments under noncooperation in nations 1 and 2 respectively

0 05 1 15 2 25 3 35 4 45 50

2

4

6

8

10

12

Pollu

tion

abat

emen

t effo

rt u

nder

coop

erat

ion

I4

I3

Time t

Figure 2 The pollution abatement effort under cooperation where1198683 and 1198684 are the pollution abatement effort of governments undercooperation in nations 1 and 2 respectively

From Table 2 we can see that for each governmentthe noncooperative optimal tax rates 1198881(119905) and 1198882(119905) are lessthan the cooperative tax rates 1198881(119905) and 1198882(119905) This meansthat cooperation would push to raise the tax rate to reduceproduction and cut pollution And for both governmentsthe optimal pollution tax rate is not changing with timethis is different from Huang et al [10] and Yeung andPetrosyan [2] because in their research they model afinite-horizon level game regardless of cooperative game ornoncooperative game and in the later stage of game bothgovernments would pay more attention to the economic

0 05 1 15 2 25 3 35 4 45 52

4

6

8

10

12

14

16

18

20

22

Pollu

tion

stock

x2

x1

Time t

Figure 3 The dynamics of pollution stock under noncooperationor cooperation where 1199091 and 1199092 denote the pollution stock undernoncooperation and cooperation respectively

0 05 1 15 2 25 3 35 4 45 5Time t

times106

Insta

ntan

eous

pay

men

ts of

natio

n 1

unde

r coo

pera

tion

61466861467

614672614674614676614678

61468614682614684614686

Figure 4 The instantaneous payments of nation 1 under coopera-tion

benefits than pollution abatementThen they would decreasethe pollution tax rate to encourage production and stimulateeconomic development [10] But in our model the gameis in infinite-horizon and there is no later stage of thegame so the optimal pollution tax rate will not decrease Inaddition under noncooperation the governments take theexternalities of production into account including their ownnation only However under a cooperation the externalitiesare considered by the governments which include their ownnation and the neighbors so the optimal pollution tax rate ishigher than under noncooperation

Figures 1 and 2 show the pollution abatement effort ofboth governments From them we find that under noncoop-eration the pollution abatement effort of both governmentsin the two nations could remain stable all the time butunder cooperation both governments would make a highpollution abatement effort at first and then the effort levelmay reduce to a certain degree after that it would tendto remain stable For each government the noncooperativepollution abatement effort is lower than the cooperativepollution abatement effort This is determined by externality

Mathematical Problems in Engineering 9In

stan

tane

ous p

aym

ents

of

natio

n 2

unde

r coo

pera

tion times10

6

0 05 1 15 2 25 3 35 4 45 5Time t

29685929686

296861296862296863296864296865296866296867296868

Figure 5 The instantaneous payments of nation 2 under coopera-tion

of pollution emissions Under cooperative game the govern-ments pay attention to the damage for the entire region somarginal damage of pollution emission would be greater andthe governments would be more concerned about pollutantreduction

Figure 3 shows the dynamics of pollution stock undernoncooperation or cooperation We can see that the pollu-tion stock under noncooperative game is higher than thatunder cooperative game It is due to the higher pollutiontax rate and higher pollution abatement effort of both gov-ernments under cooperative game Then in both economicand environmental terms cooperation is always better thannoncooperation

Now consider the case when the two nations want tocooperate To secure group optimality the two nations seek tomaximize their joint expected payoff by solving the followingstochastic control problem

max1198881(119904)1198882(119904)1198681(119904)1198682(119904)

1198640 intinfin0

(58501198881 (119904) + 32501198882 (119904) minus 04 (1198681 (119904))2

minus 05 (1198682 (119904))2 minus 102375(1198881 (119904))2 minus50375(1198882 (119904))2 minus 10119909 (119904))

sdot 119890minus005119889119905(31)

subject to

119889119909 (119904) = 349 minus 128119909 (119904) + 005119909 (119904) 119889119911 (119904) (32)

According toTheorem 4 we get the appropriate dynamicdistribution scheme that both group optimality and individ-ual rationality are satisfied at any time during cooperativegame The instantaneous payments of two nations 1198781(119905 119909lowast119905 )and 1198782(119905 119909lowast119905 ) at different time 119905 with given 119909lowast119905 are shown in(33a) and (33b) and Figures 3 and 4

1198781 (119905 119909lowast119905 ) = 1007100178948148626983247163840000000000000minus 12670021998361119890minus321199052578125000000 (33a)

1198782 (119905 119909lowast119905 ) = 77822033722351462743911532621440000000000000minus 1518945509601119890minus321199052519531250000 (33b)

It is shown in Figures 4 and 5 that the instantaneous pay-ment of both nations which satisfy the property of subgameconsistency shows an increasing trend at the start and thenremains stable In addition the instantaneous payment fornation 1 is always higher than that of nation 2This is becausethe industrial firms in nation 1 have more green and energy-efficient technology so they would get more instantaneouspayment

6 Conclusions

In this paper we have shown a stochastic differential game oftransboundary industrial pollution between two asymmetricnations in infinite-horizon level In our model the firmsand governments are separate entities As participants inthe games they take game behavior to optimize their ownpayment and the game equilibrium has been solved underthe two levels of firms and governments simultaneouslyThe uncertain evolution dynamic of pollution stock thathas a long-term global impact has been taken into accountin our models Otherwise another negative environmentalexternality a regional nonaccumulative pollutant that causesa short-term local impact has also been considered Wecharacterize the parameters of spaces where two neighboringgovernments can cooperate and give the feedback Nashequilibrium strategies of governments and industrial firmsThrough analysis we have shown that the optimal pollutiontax rate hinges on the two effects of regional damagethe source region and the global pollution stock undernoncooperative game while under cooperative game inaddition to the above two kinds of effects it also hingeson the additional effect of regional damage coming fromthe neighborhood so the latter is greater than the formerMoreover we have shown that the pollution abatement effortunder cooperation is more than under noncooperation Byanalyzing this stochastic differential game model we havealso provided a payment distribution mechanism whichsupports the subgame consistent solution and present anumerical example to illustrate the implementation of thismechanism

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge the financial supportfrom the ChinaNational Funding for Social Science ResearchProject (Project no 15CJY037) and the Philosophy and SocialScience Foundation of Hunan China (Project no 16YBA315)for this paper

10 Mathematical Problems in Engineering

References

[1] D W Yeung ldquoDynamically consistent cooperative solutionin a differential game of transboundary industrial pollutionrdquoJournal of Optimization Theory and Applications vol 134 no1 pp 143ndash160 2007

[2] D W Yeung and L A Petrosyan ldquoA cooperative stochasticdifferential game of transboundary industrial pollutionrdquo Auto-matica vol 44 no 6 pp 1532ndash1544 2008

[3] E J Dockner andNVanLong ldquoInternational pollution controlcooperative versus noncooperative strategiesrdquo Journal of Envi-ronmental Economics and Management vol 25 no 1 pp 13ndash291993

[4] L Petrosjan and G Zaccour ldquoTime-consistent Shapley valueallocation of pollution cost reductionrdquo Journal of EconomicDynamics amp Control vol 27 no 3 pp 381ndash398 2003

[5] B Bayramoglu ldquoTransboundary pollution in the black seacomparison of institutional arrangementsrdquo Environmental andResource Economics vol 35 no 4 pp 289ndash325 2006

[6] S Joslashrgensen and G Zaccour ldquoIncentive equilibrium strategiesand welfare allocation in a dynamic game of pollution controlrdquoAutomatica vol 37 no 1 pp 29ndash36 2001

[7] L Bertinelli C Camacho and B Zou ldquoCarbon capture andstorage and transboundary pollution a differential gameapproachrdquo European Journal of Operational Research vol 237no 2 pp 721ndash728 2014

[8] S Li ldquoA differential game of transboundary industrial pollutionwith emission permits tradingrdquo Journal of OptimizationTheoryand Applications vol 163 no 2 pp 642ndash659 2014

[9] H Benchekroun and G Martın-Herran ldquoThe impact of fore-sight in a transboundary pollution gamerdquo European Journal ofOperational Research vol 251 no 1 pp 300ndash309 2016

[10] XHuang PHe andW Zhang ldquoA cooperative differential gameof transboundary industrial pollution between two regionsrdquoJournal of Cleaner Production vol 120 pp 43ndash52 2016

[11] A Haurie J B Krawczyk and M Roche ldquoMonitoring coop-erative equilibria in a stochastic differential gamerdquo Journal ofOptimization Theory and Applications vol 81 no 1 pp 73ndash951994

[12] E Saltari and G Travaglini ldquoThe effects of environmental poli-cies on the abatement investment decisions of a green firmrdquoResource and Energy Economics vol 33 no 3 pp 666ndash685 2011

[13] S Athanassoglou and A Xepapadeas ldquoPollution control withuncertain stock dynamics when and how to be precautiousrdquoJournal of Environmental Economics and Management vol 63no 3 pp 304ndash320 2012

[14] E J Dockner and K Nishimura ldquoTransboundary pollution ina dynamic game modelrdquo Japanese Economic Review vol 50 no4 pp 443ndash456 1999

[15] F Bosello B Buchner and C Carraro ldquoEquity developmentand climate change controlrdquo Journal of the European EconomicAssociation vol 1 no 2-3 pp 601ndash611 2003

[16] A Bernard A Haurie M Vielle and L Viguier ldquoA two-leveldynamic game of carbon emission trading between RussiaChina and annex B countriesrdquo Journal of Economic Dynamicsand Control vol 32 no 6 pp 1830ndash1856 2008

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 1 The parameters of model used in the numerical solutions

1199050 12057311 12057321 12057312 12057322 119909(1199050) 12057211 12057221 12057212 12057222 1205741 12057420 60 90 40 70 20 04 03 05 04 04 02120575 120590 1198861 1198862 12057611 12057621 12057622 12057612 119903 ℎ1 ℎ2001 005 08 1 02 015 015 016 005 4 5

will realize at time 120591 with the state being 119909lowast120591 can be expressedas

1198781 (120591 119909) = 1199032 [1198601119909lowast120591 + 1198611] + [119860119909lowast120591 + 119861]minus [1198602119909lowast120591 + 1198612] minus 12 [1198601 + 119860 minus 1198602]sdot

sum119899119894=1 (1205731198941)2 + sum119898119895=1 (1205731198952)24 (12057611 + 12057622 minus 119860)2+ 2sum119896=1

1205742119896119860119909lowast1205912119886119896 minus 120575119909lowast120591

(29a)

1198782 (120591 119909) = 1199032 [1198602119909lowast120591 + 1198612] + [119860119909lowast120591 + 119861]minus [1198601119909lowast120591 + 1198611] minus 12 [1198602 + 119860 minus 1198601]sdot

sum119899119894=1 (1205731198941)2 + sum119898119895=1 (1205731198952)24 (12057611 + 12057622 minus 119860)2+ 2sum119896=1

1205742119896119860119909lowast1205912119886119896 minus 120575119909lowast120591

(29b)

5 Numerical Example

Consider a multinational economy consisting of two nationsand each with two industrial firms Industrial firmrsquos revenuefunction in nation 1 and nation 2 at time 119904 is 120601119894 = 1205731198941(119906119894(119904))12and 120601119895 = 1205731198952(V119895(119904))12 (119894 119895 = 1 2) respectively where12057311 = 60 12057321 = 90 12057312 = 40 and 12057322 = 70 The valueof 1205731198941 (119894 isin 1 2) is higher than 1205731198952 (119895 isin 1 2) it meansthat compared with industrial firms in nation 2 industrialfirms in nation 1 have more energy-efficient technology Inaddition industrial firms in nation 1 have technical advantageresult in 12057211 lt 12057212 12057221 lt 12057222 1205741 gt 1205742 As mentionedabove the optimal pollution emissions of the firms in nations1 and 2 are 119906lowast119894 (119904) = (120573119894121198881(119904))2 and Vlowast119895 (119904) = (120573119895221198882(119904))2respectively The dynamics of pollution stock are governedby the stochastic differential equation (4) where 119909(1199050) = 20120575 = 001 and 120590 = 005 We defined 119886119896 as the amountadded to the pollution stock by a unit of output in nation119896 (119896 = 1 2) They are a positive number less than 1 that isto say the stock of pollution will increase with the outputbut not higher than the amount of output According to

Table 2 The optimal pollution tax rate

1198881(119905) 1198882(119905) 1198881(119905) 1198882(119905)820 234 1090 1087

experience we assign 12057211 = 04 12057221 = 03 12057212 = 0512057222 = 04 120574119896 (119896 = 1 2) can also be considered as pollutionabatement effort efficiency which is linked to pollutioncontrol policies and environmental governance efficiency indifferent nations Generally speaking pollution abatementeffort efficiency is higher in developed countries We assign1205741 = 04 1205742 = 02 Abatement costs are 04(1198681(119904))12 and05(1198682(119904))12 for nations 1 and 2 respectively Local pollutionimpacts are closely linked to the level of production activitiesThe accumulation of pollution stock like greenhouse gas oftenconcerns the interactions between the natural environmentand the pollutants emitted So stochastic elements wouldappear According to what we defined before 1205761119896 1205762119896 (119896 = 1 2)is a pollution damage cost coefficients It is said that the biggerthey are the higher the cost of environmental damage forthis nation is We make the following assignments 12057611 = 0212057621 = 015 12057622 = 015 and 12057612 = 016 ℎ119896 (119896 = 1 2) arethe value of pollution damage to nation 119896 (119896 = 1 2) Thelarger they are the greater the impact of the environment iswhich is determined by the different regions And we makeassignments ℎ1 = 4 ℎ2 = 5 Other parameters are as follows119903 = 005 1199050 = 0 The parameters of the model which are usedin the numerical solution are presented in Table 1

Then we obtain a stochastic differential game in whichgovernment 1 and government 2 under noncooperation seek

max1198881(119904)1198681(119904)

intinfin0

119890minus119903119905 (58501198881 (119904) minus 0411986821 (119904) minus 58511988821 (119904) minus 162211988822 (119904)minus 4119909 (119904)) 119889119909

(30a)

max1198882(119904)1198682(119904)

intinfin0

119890minus119903119905 (32501198882 (119904) minus 0511986822 (119904) minus 4387511988821 (119904)minus 2437511988822 (119904) minus 5119909 (119904)) 119889119909

(30b)

Through the analysis mentioned above we have obtainedthe pollution tax rate abatement effort of each nationand the dynamics of pollution stock under noncooperativegame and cooperative game Now we will analyze theirdifference between the governments under noncooperationand cooperation through Table 2 and Figures 1ndash5

8 Mathematical Problems in Engineering

0 05 1 15 2 25 3 35 4 45 515

2

25

3

35

4

45

Pollu

tion

abat

emen

t effo

rt u

nder

non

coop

erat

ion

I2

I1

Time t

Figure 1 The pollution abatement effort under noncooperationwhere 1198681 and 1198682 denote the pollution abatement effort of govern-ments under noncooperation in nations 1 and 2 respectively

0 05 1 15 2 25 3 35 4 45 50

2

4

6

8

10

12

Pollu

tion

abat

emen

t effo

rt u

nder

coop

erat

ion

I4

I3

Time t

Figure 2 The pollution abatement effort under cooperation where1198683 and 1198684 are the pollution abatement effort of governments undercooperation in nations 1 and 2 respectively

From Table 2 we can see that for each governmentthe noncooperative optimal tax rates 1198881(119905) and 1198882(119905) are lessthan the cooperative tax rates 1198881(119905) and 1198882(119905) This meansthat cooperation would push to raise the tax rate to reduceproduction and cut pollution And for both governmentsthe optimal pollution tax rate is not changing with timethis is different from Huang et al [10] and Yeung andPetrosyan [2] because in their research they model afinite-horizon level game regardless of cooperative game ornoncooperative game and in the later stage of game bothgovernments would pay more attention to the economic

0 05 1 15 2 25 3 35 4 45 52

4

6

8

10

12

14

16

18

20

22

Pollu

tion

stock

x2

x1

Time t

Figure 3 The dynamics of pollution stock under noncooperationor cooperation where 1199091 and 1199092 denote the pollution stock undernoncooperation and cooperation respectively

0 05 1 15 2 25 3 35 4 45 5Time t

times106

Insta

ntan

eous

pay

men

ts of

natio

n 1

unde

r coo

pera

tion

61466861467

614672614674614676614678

61468614682614684614686

Figure 4 The instantaneous payments of nation 1 under coopera-tion

benefits than pollution abatementThen they would decreasethe pollution tax rate to encourage production and stimulateeconomic development [10] But in our model the gameis in infinite-horizon and there is no later stage of thegame so the optimal pollution tax rate will not decrease Inaddition under noncooperation the governments take theexternalities of production into account including their ownnation only However under a cooperation the externalitiesare considered by the governments which include their ownnation and the neighbors so the optimal pollution tax rate ishigher than under noncooperation

Figures 1 and 2 show the pollution abatement effort ofboth governments From them we find that under noncoop-eration the pollution abatement effort of both governmentsin the two nations could remain stable all the time butunder cooperation both governments would make a highpollution abatement effort at first and then the effort levelmay reduce to a certain degree after that it would tendto remain stable For each government the noncooperativepollution abatement effort is lower than the cooperativepollution abatement effort This is determined by externality

Mathematical Problems in Engineering 9In

stan

tane

ous p

aym

ents

of

natio

n 2

unde

r coo

pera

tion times10

6

0 05 1 15 2 25 3 35 4 45 5Time t

29685929686

296861296862296863296864296865296866296867296868

Figure 5 The instantaneous payments of nation 2 under coopera-tion

of pollution emissions Under cooperative game the govern-ments pay attention to the damage for the entire region somarginal damage of pollution emission would be greater andthe governments would be more concerned about pollutantreduction

Figure 3 shows the dynamics of pollution stock undernoncooperation or cooperation We can see that the pollu-tion stock under noncooperative game is higher than thatunder cooperative game It is due to the higher pollutiontax rate and higher pollution abatement effort of both gov-ernments under cooperative game Then in both economicand environmental terms cooperation is always better thannoncooperation

Now consider the case when the two nations want tocooperate To secure group optimality the two nations seek tomaximize their joint expected payoff by solving the followingstochastic control problem

max1198881(119904)1198882(119904)1198681(119904)1198682(119904)

1198640 intinfin0

(58501198881 (119904) + 32501198882 (119904) minus 04 (1198681 (119904))2

minus 05 (1198682 (119904))2 minus 102375(1198881 (119904))2 minus50375(1198882 (119904))2 minus 10119909 (119904))

sdot 119890minus005119889119905(31)

subject to

119889119909 (119904) = 349 minus 128119909 (119904) + 005119909 (119904) 119889119911 (119904) (32)

According toTheorem 4 we get the appropriate dynamicdistribution scheme that both group optimality and individ-ual rationality are satisfied at any time during cooperativegame The instantaneous payments of two nations 1198781(119905 119909lowast119905 )and 1198782(119905 119909lowast119905 ) at different time 119905 with given 119909lowast119905 are shown in(33a) and (33b) and Figures 3 and 4

1198781 (119905 119909lowast119905 ) = 1007100178948148626983247163840000000000000minus 12670021998361119890minus321199052578125000000 (33a)

1198782 (119905 119909lowast119905 ) = 77822033722351462743911532621440000000000000minus 1518945509601119890minus321199052519531250000 (33b)

It is shown in Figures 4 and 5 that the instantaneous pay-ment of both nations which satisfy the property of subgameconsistency shows an increasing trend at the start and thenremains stable In addition the instantaneous payment fornation 1 is always higher than that of nation 2This is becausethe industrial firms in nation 1 have more green and energy-efficient technology so they would get more instantaneouspayment

6 Conclusions

In this paper we have shown a stochastic differential game oftransboundary industrial pollution between two asymmetricnations in infinite-horizon level In our model the firmsand governments are separate entities As participants inthe games they take game behavior to optimize their ownpayment and the game equilibrium has been solved underthe two levels of firms and governments simultaneouslyThe uncertain evolution dynamic of pollution stock thathas a long-term global impact has been taken into accountin our models Otherwise another negative environmentalexternality a regional nonaccumulative pollutant that causesa short-term local impact has also been considered Wecharacterize the parameters of spaces where two neighboringgovernments can cooperate and give the feedback Nashequilibrium strategies of governments and industrial firmsThrough analysis we have shown that the optimal pollutiontax rate hinges on the two effects of regional damagethe source region and the global pollution stock undernoncooperative game while under cooperative game inaddition to the above two kinds of effects it also hingeson the additional effect of regional damage coming fromthe neighborhood so the latter is greater than the formerMoreover we have shown that the pollution abatement effortunder cooperation is more than under noncooperation Byanalyzing this stochastic differential game model we havealso provided a payment distribution mechanism whichsupports the subgame consistent solution and present anumerical example to illustrate the implementation of thismechanism

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge the financial supportfrom the ChinaNational Funding for Social Science ResearchProject (Project no 15CJY037) and the Philosophy and SocialScience Foundation of Hunan China (Project no 16YBA315)for this paper

10 Mathematical Problems in Engineering

References

[1] D W Yeung ldquoDynamically consistent cooperative solutionin a differential game of transboundary industrial pollutionrdquoJournal of Optimization Theory and Applications vol 134 no1 pp 143ndash160 2007

[2] D W Yeung and L A Petrosyan ldquoA cooperative stochasticdifferential game of transboundary industrial pollutionrdquo Auto-matica vol 44 no 6 pp 1532ndash1544 2008

[3] E J Dockner andNVanLong ldquoInternational pollution controlcooperative versus noncooperative strategiesrdquo Journal of Envi-ronmental Economics and Management vol 25 no 1 pp 13ndash291993

[4] L Petrosjan and G Zaccour ldquoTime-consistent Shapley valueallocation of pollution cost reductionrdquo Journal of EconomicDynamics amp Control vol 27 no 3 pp 381ndash398 2003

[5] B Bayramoglu ldquoTransboundary pollution in the black seacomparison of institutional arrangementsrdquo Environmental andResource Economics vol 35 no 4 pp 289ndash325 2006

[6] S Joslashrgensen and G Zaccour ldquoIncentive equilibrium strategiesand welfare allocation in a dynamic game of pollution controlrdquoAutomatica vol 37 no 1 pp 29ndash36 2001

[7] L Bertinelli C Camacho and B Zou ldquoCarbon capture andstorage and transboundary pollution a differential gameapproachrdquo European Journal of Operational Research vol 237no 2 pp 721ndash728 2014

[8] S Li ldquoA differential game of transboundary industrial pollutionwith emission permits tradingrdquo Journal of OptimizationTheoryand Applications vol 163 no 2 pp 642ndash659 2014

[9] H Benchekroun and G Martın-Herran ldquoThe impact of fore-sight in a transboundary pollution gamerdquo European Journal ofOperational Research vol 251 no 1 pp 300ndash309 2016

[10] XHuang PHe andW Zhang ldquoA cooperative differential gameof transboundary industrial pollution between two regionsrdquoJournal of Cleaner Production vol 120 pp 43ndash52 2016

[11] A Haurie J B Krawczyk and M Roche ldquoMonitoring coop-erative equilibria in a stochastic differential gamerdquo Journal ofOptimization Theory and Applications vol 81 no 1 pp 73ndash951994

[12] E Saltari and G Travaglini ldquoThe effects of environmental poli-cies on the abatement investment decisions of a green firmrdquoResource and Energy Economics vol 33 no 3 pp 666ndash685 2011

[13] S Athanassoglou and A Xepapadeas ldquoPollution control withuncertain stock dynamics when and how to be precautiousrdquoJournal of Environmental Economics and Management vol 63no 3 pp 304ndash320 2012

[14] E J Dockner and K Nishimura ldquoTransboundary pollution ina dynamic game modelrdquo Japanese Economic Review vol 50 no4 pp 443ndash456 1999

[15] F Bosello B Buchner and C Carraro ldquoEquity developmentand climate change controlrdquo Journal of the European EconomicAssociation vol 1 no 2-3 pp 601ndash611 2003

[16] A Bernard A Haurie M Vielle and L Viguier ldquoA two-leveldynamic game of carbon emission trading between RussiaChina and annex B countriesrdquo Journal of Economic Dynamicsand Control vol 32 no 6 pp 1830ndash1856 2008

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0 05 1 15 2 25 3 35 4 45 515

2

25

3

35

4

45

Pollu

tion

abat

emen

t effo

rt u

nder

non

coop

erat

ion

I2

I1

Time t

Figure 1 The pollution abatement effort under noncooperationwhere 1198681 and 1198682 denote the pollution abatement effort of govern-ments under noncooperation in nations 1 and 2 respectively

0 05 1 15 2 25 3 35 4 45 50

2

4

6

8

10

12

Pollu

tion

abat

emen

t effo

rt u

nder

coop

erat

ion

I4

I3

Time t

Figure 2 The pollution abatement effort under cooperation where1198683 and 1198684 are the pollution abatement effort of governments undercooperation in nations 1 and 2 respectively

From Table 2 we can see that for each governmentthe noncooperative optimal tax rates 1198881(119905) and 1198882(119905) are lessthan the cooperative tax rates 1198881(119905) and 1198882(119905) This meansthat cooperation would push to raise the tax rate to reduceproduction and cut pollution And for both governmentsthe optimal pollution tax rate is not changing with timethis is different from Huang et al [10] and Yeung andPetrosyan [2] because in their research they model afinite-horizon level game regardless of cooperative game ornoncooperative game and in the later stage of game bothgovernments would pay more attention to the economic

0 05 1 15 2 25 3 35 4 45 52

4

6

8

10

12

14

16

18

20

22

Pollu

tion

stock

x2

x1

Time t

Figure 3 The dynamics of pollution stock under noncooperationor cooperation where 1199091 and 1199092 denote the pollution stock undernoncooperation and cooperation respectively

0 05 1 15 2 25 3 35 4 45 5Time t

times106

Insta

ntan

eous

pay

men

ts of

natio

n 1

unde

r coo

pera

tion

61466861467

614672614674614676614678

61468614682614684614686

Figure 4 The instantaneous payments of nation 1 under coopera-tion

benefits than pollution abatementThen they would decreasethe pollution tax rate to encourage production and stimulateeconomic development [10] But in our model the gameis in infinite-horizon and there is no later stage of thegame so the optimal pollution tax rate will not decrease Inaddition under noncooperation the governments take theexternalities of production into account including their ownnation only However under a cooperation the externalitiesare considered by the governments which include their ownnation and the neighbors so the optimal pollution tax rate ishigher than under noncooperation

Figures 1 and 2 show the pollution abatement effort ofboth governments From them we find that under noncoop-eration the pollution abatement effort of both governmentsin the two nations could remain stable all the time butunder cooperation both governments would make a highpollution abatement effort at first and then the effort levelmay reduce to a certain degree after that it would tendto remain stable For each government the noncooperativepollution abatement effort is lower than the cooperativepollution abatement effort This is determined by externality

Mathematical Problems in Engineering 9In

stan

tane

ous p

aym

ents

of

natio

n 2

unde

r coo

pera

tion times10

6

0 05 1 15 2 25 3 35 4 45 5Time t

29685929686

296861296862296863296864296865296866296867296868

Figure 5 The instantaneous payments of nation 2 under coopera-tion

of pollution emissions Under cooperative game the govern-ments pay attention to the damage for the entire region somarginal damage of pollution emission would be greater andthe governments would be more concerned about pollutantreduction

Figure 3 shows the dynamics of pollution stock undernoncooperation or cooperation We can see that the pollu-tion stock under noncooperative game is higher than thatunder cooperative game It is due to the higher pollutiontax rate and higher pollution abatement effort of both gov-ernments under cooperative game Then in both economicand environmental terms cooperation is always better thannoncooperation

Now consider the case when the two nations want tocooperate To secure group optimality the two nations seek tomaximize their joint expected payoff by solving the followingstochastic control problem

max1198881(119904)1198882(119904)1198681(119904)1198682(119904)

1198640 intinfin0

(58501198881 (119904) + 32501198882 (119904) minus 04 (1198681 (119904))2

minus 05 (1198682 (119904))2 minus 102375(1198881 (119904))2 minus50375(1198882 (119904))2 minus 10119909 (119904))

sdot 119890minus005119889119905(31)

subject to

119889119909 (119904) = 349 minus 128119909 (119904) + 005119909 (119904) 119889119911 (119904) (32)

According toTheorem 4 we get the appropriate dynamicdistribution scheme that both group optimality and individ-ual rationality are satisfied at any time during cooperativegame The instantaneous payments of two nations 1198781(119905 119909lowast119905 )and 1198782(119905 119909lowast119905 ) at different time 119905 with given 119909lowast119905 are shown in(33a) and (33b) and Figures 3 and 4

1198781 (119905 119909lowast119905 ) = 1007100178948148626983247163840000000000000minus 12670021998361119890minus321199052578125000000 (33a)

1198782 (119905 119909lowast119905 ) = 77822033722351462743911532621440000000000000minus 1518945509601119890minus321199052519531250000 (33b)

It is shown in Figures 4 and 5 that the instantaneous pay-ment of both nations which satisfy the property of subgameconsistency shows an increasing trend at the start and thenremains stable In addition the instantaneous payment fornation 1 is always higher than that of nation 2This is becausethe industrial firms in nation 1 have more green and energy-efficient technology so they would get more instantaneouspayment

6 Conclusions

In this paper we have shown a stochastic differential game oftransboundary industrial pollution between two asymmetricnations in infinite-horizon level In our model the firmsand governments are separate entities As participants inthe games they take game behavior to optimize their ownpayment and the game equilibrium has been solved underthe two levels of firms and governments simultaneouslyThe uncertain evolution dynamic of pollution stock thathas a long-term global impact has been taken into accountin our models Otherwise another negative environmentalexternality a regional nonaccumulative pollutant that causesa short-term local impact has also been considered Wecharacterize the parameters of spaces where two neighboringgovernments can cooperate and give the feedback Nashequilibrium strategies of governments and industrial firmsThrough analysis we have shown that the optimal pollutiontax rate hinges on the two effects of regional damagethe source region and the global pollution stock undernoncooperative game while under cooperative game inaddition to the above two kinds of effects it also hingeson the additional effect of regional damage coming fromthe neighborhood so the latter is greater than the formerMoreover we have shown that the pollution abatement effortunder cooperation is more than under noncooperation Byanalyzing this stochastic differential game model we havealso provided a payment distribution mechanism whichsupports the subgame consistent solution and present anumerical example to illustrate the implementation of thismechanism

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge the financial supportfrom the ChinaNational Funding for Social Science ResearchProject (Project no 15CJY037) and the Philosophy and SocialScience Foundation of Hunan China (Project no 16YBA315)for this paper

10 Mathematical Problems in Engineering

References

[1] D W Yeung ldquoDynamically consistent cooperative solutionin a differential game of transboundary industrial pollutionrdquoJournal of Optimization Theory and Applications vol 134 no1 pp 143ndash160 2007

[2] D W Yeung and L A Petrosyan ldquoA cooperative stochasticdifferential game of transboundary industrial pollutionrdquo Auto-matica vol 44 no 6 pp 1532ndash1544 2008

[3] E J Dockner andNVanLong ldquoInternational pollution controlcooperative versus noncooperative strategiesrdquo Journal of Envi-ronmental Economics and Management vol 25 no 1 pp 13ndash291993

[4] L Petrosjan and G Zaccour ldquoTime-consistent Shapley valueallocation of pollution cost reductionrdquo Journal of EconomicDynamics amp Control vol 27 no 3 pp 381ndash398 2003

[5] B Bayramoglu ldquoTransboundary pollution in the black seacomparison of institutional arrangementsrdquo Environmental andResource Economics vol 35 no 4 pp 289ndash325 2006

[6] S Joslashrgensen and G Zaccour ldquoIncentive equilibrium strategiesand welfare allocation in a dynamic game of pollution controlrdquoAutomatica vol 37 no 1 pp 29ndash36 2001

[7] L Bertinelli C Camacho and B Zou ldquoCarbon capture andstorage and transboundary pollution a differential gameapproachrdquo European Journal of Operational Research vol 237no 2 pp 721ndash728 2014

[8] S Li ldquoA differential game of transboundary industrial pollutionwith emission permits tradingrdquo Journal of OptimizationTheoryand Applications vol 163 no 2 pp 642ndash659 2014

[9] H Benchekroun and G Martın-Herran ldquoThe impact of fore-sight in a transboundary pollution gamerdquo European Journal ofOperational Research vol 251 no 1 pp 300ndash309 2016

[10] XHuang PHe andW Zhang ldquoA cooperative differential gameof transboundary industrial pollution between two regionsrdquoJournal of Cleaner Production vol 120 pp 43ndash52 2016

[11] A Haurie J B Krawczyk and M Roche ldquoMonitoring coop-erative equilibria in a stochastic differential gamerdquo Journal ofOptimization Theory and Applications vol 81 no 1 pp 73ndash951994

[12] E Saltari and G Travaglini ldquoThe effects of environmental poli-cies on the abatement investment decisions of a green firmrdquoResource and Energy Economics vol 33 no 3 pp 666ndash685 2011

[13] S Athanassoglou and A Xepapadeas ldquoPollution control withuncertain stock dynamics when and how to be precautiousrdquoJournal of Environmental Economics and Management vol 63no 3 pp 304ndash320 2012

[14] E J Dockner and K Nishimura ldquoTransboundary pollution ina dynamic game modelrdquo Japanese Economic Review vol 50 no4 pp 443ndash456 1999

[15] F Bosello B Buchner and C Carraro ldquoEquity developmentand climate change controlrdquo Journal of the European EconomicAssociation vol 1 no 2-3 pp 601ndash611 2003

[16] A Bernard A Haurie M Vielle and L Viguier ldquoA two-leveldynamic game of carbon emission trading between RussiaChina and annex B countriesrdquo Journal of Economic Dynamicsand Control vol 32 no 6 pp 1830ndash1856 2008

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9In

stan

tane

ous p

aym

ents

of

natio

n 2

unde

r coo

pera

tion times10

6

0 05 1 15 2 25 3 35 4 45 5Time t

29685929686

296861296862296863296864296865296866296867296868

Figure 5 The instantaneous payments of nation 2 under coopera-tion

of pollution emissions Under cooperative game the govern-ments pay attention to the damage for the entire region somarginal damage of pollution emission would be greater andthe governments would be more concerned about pollutantreduction

Figure 3 shows the dynamics of pollution stock undernoncooperation or cooperation We can see that the pollu-tion stock under noncooperative game is higher than thatunder cooperative game It is due to the higher pollutiontax rate and higher pollution abatement effort of both gov-ernments under cooperative game Then in both economicand environmental terms cooperation is always better thannoncooperation

Now consider the case when the two nations want tocooperate To secure group optimality the two nations seek tomaximize their joint expected payoff by solving the followingstochastic control problem

max1198881(119904)1198882(119904)1198681(119904)1198682(119904)

1198640 intinfin0

(58501198881 (119904) + 32501198882 (119904) minus 04 (1198681 (119904))2

minus 05 (1198682 (119904))2 minus 102375(1198881 (119904))2 minus50375(1198882 (119904))2 minus 10119909 (119904))

sdot 119890minus005119889119905(31)

subject to

119889119909 (119904) = 349 minus 128119909 (119904) + 005119909 (119904) 119889119911 (119904) (32)

According toTheorem 4 we get the appropriate dynamicdistribution scheme that both group optimality and individ-ual rationality are satisfied at any time during cooperativegame The instantaneous payments of two nations 1198781(119905 119909lowast119905 )and 1198782(119905 119909lowast119905 ) at different time 119905 with given 119909lowast119905 are shown in(33a) and (33b) and Figures 3 and 4

1198781 (119905 119909lowast119905 ) = 1007100178948148626983247163840000000000000minus 12670021998361119890minus321199052578125000000 (33a)

1198782 (119905 119909lowast119905 ) = 77822033722351462743911532621440000000000000minus 1518945509601119890minus321199052519531250000 (33b)

It is shown in Figures 4 and 5 that the instantaneous pay-ment of both nations which satisfy the property of subgameconsistency shows an increasing trend at the start and thenremains stable In addition the instantaneous payment fornation 1 is always higher than that of nation 2This is becausethe industrial firms in nation 1 have more green and energy-efficient technology so they would get more instantaneouspayment

6 Conclusions

In this paper we have shown a stochastic differential game oftransboundary industrial pollution between two asymmetricnations in infinite-horizon level In our model the firmsand governments are separate entities As participants inthe games they take game behavior to optimize their ownpayment and the game equilibrium has been solved underthe two levels of firms and governments simultaneouslyThe uncertain evolution dynamic of pollution stock thathas a long-term global impact has been taken into accountin our models Otherwise another negative environmentalexternality a regional nonaccumulative pollutant that causesa short-term local impact has also been considered Wecharacterize the parameters of spaces where two neighboringgovernments can cooperate and give the feedback Nashequilibrium strategies of governments and industrial firmsThrough analysis we have shown that the optimal pollutiontax rate hinges on the two effects of regional damagethe source region and the global pollution stock undernoncooperative game while under cooperative game inaddition to the above two kinds of effects it also hingeson the additional effect of regional damage coming fromthe neighborhood so the latter is greater than the formerMoreover we have shown that the pollution abatement effortunder cooperation is more than under noncooperation Byanalyzing this stochastic differential game model we havealso provided a payment distribution mechanism whichsupports the subgame consistent solution and present anumerical example to illustrate the implementation of thismechanism

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors would like to acknowledge the financial supportfrom the ChinaNational Funding for Social Science ResearchProject (Project no 15CJY037) and the Philosophy and SocialScience Foundation of Hunan China (Project no 16YBA315)for this paper

10 Mathematical Problems in Engineering

References

[1] D W Yeung ldquoDynamically consistent cooperative solutionin a differential game of transboundary industrial pollutionrdquoJournal of Optimization Theory and Applications vol 134 no1 pp 143ndash160 2007

[2] D W Yeung and L A Petrosyan ldquoA cooperative stochasticdifferential game of transboundary industrial pollutionrdquo Auto-matica vol 44 no 6 pp 1532ndash1544 2008

[3] E J Dockner andNVanLong ldquoInternational pollution controlcooperative versus noncooperative strategiesrdquo Journal of Envi-ronmental Economics and Management vol 25 no 1 pp 13ndash291993

[4] L Petrosjan and G Zaccour ldquoTime-consistent Shapley valueallocation of pollution cost reductionrdquo Journal of EconomicDynamics amp Control vol 27 no 3 pp 381ndash398 2003

[5] B Bayramoglu ldquoTransboundary pollution in the black seacomparison of institutional arrangementsrdquo Environmental andResource Economics vol 35 no 4 pp 289ndash325 2006

[6] S Joslashrgensen and G Zaccour ldquoIncentive equilibrium strategiesand welfare allocation in a dynamic game of pollution controlrdquoAutomatica vol 37 no 1 pp 29ndash36 2001

[7] L Bertinelli C Camacho and B Zou ldquoCarbon capture andstorage and transboundary pollution a differential gameapproachrdquo European Journal of Operational Research vol 237no 2 pp 721ndash728 2014

[8] S Li ldquoA differential game of transboundary industrial pollutionwith emission permits tradingrdquo Journal of OptimizationTheoryand Applications vol 163 no 2 pp 642ndash659 2014

[9] H Benchekroun and G Martın-Herran ldquoThe impact of fore-sight in a transboundary pollution gamerdquo European Journal ofOperational Research vol 251 no 1 pp 300ndash309 2016

[10] XHuang PHe andW Zhang ldquoA cooperative differential gameof transboundary industrial pollution between two regionsrdquoJournal of Cleaner Production vol 120 pp 43ndash52 2016

[11] A Haurie J B Krawczyk and M Roche ldquoMonitoring coop-erative equilibria in a stochastic differential gamerdquo Journal ofOptimization Theory and Applications vol 81 no 1 pp 73ndash951994

[12] E Saltari and G Travaglini ldquoThe effects of environmental poli-cies on the abatement investment decisions of a green firmrdquoResource and Energy Economics vol 33 no 3 pp 666ndash685 2011

[13] S Athanassoglou and A Xepapadeas ldquoPollution control withuncertain stock dynamics when and how to be precautiousrdquoJournal of Environmental Economics and Management vol 63no 3 pp 304ndash320 2012

[14] E J Dockner and K Nishimura ldquoTransboundary pollution ina dynamic game modelrdquo Japanese Economic Review vol 50 no4 pp 443ndash456 1999

[15] F Bosello B Buchner and C Carraro ldquoEquity developmentand climate change controlrdquo Journal of the European EconomicAssociation vol 1 no 2-3 pp 601ndash611 2003

[16] A Bernard A Haurie M Vielle and L Viguier ldquoA two-leveldynamic game of carbon emission trading between RussiaChina and annex B countriesrdquo Journal of Economic Dynamicsand Control vol 32 no 6 pp 1830ndash1856 2008

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

References

[1] D W Yeung ldquoDynamically consistent cooperative solutionin a differential game of transboundary industrial pollutionrdquoJournal of Optimization Theory and Applications vol 134 no1 pp 143ndash160 2007

[2] D W Yeung and L A Petrosyan ldquoA cooperative stochasticdifferential game of transboundary industrial pollutionrdquo Auto-matica vol 44 no 6 pp 1532ndash1544 2008

[3] E J Dockner andNVanLong ldquoInternational pollution controlcooperative versus noncooperative strategiesrdquo Journal of Envi-ronmental Economics and Management vol 25 no 1 pp 13ndash291993

[4] L Petrosjan and G Zaccour ldquoTime-consistent Shapley valueallocation of pollution cost reductionrdquo Journal of EconomicDynamics amp Control vol 27 no 3 pp 381ndash398 2003

[5] B Bayramoglu ldquoTransboundary pollution in the black seacomparison of institutional arrangementsrdquo Environmental andResource Economics vol 35 no 4 pp 289ndash325 2006

[6] S Joslashrgensen and G Zaccour ldquoIncentive equilibrium strategiesand welfare allocation in a dynamic game of pollution controlrdquoAutomatica vol 37 no 1 pp 29ndash36 2001

[7] L Bertinelli C Camacho and B Zou ldquoCarbon capture andstorage and transboundary pollution a differential gameapproachrdquo European Journal of Operational Research vol 237no 2 pp 721ndash728 2014

[8] S Li ldquoA differential game of transboundary industrial pollutionwith emission permits tradingrdquo Journal of OptimizationTheoryand Applications vol 163 no 2 pp 642ndash659 2014

[9] H Benchekroun and G Martın-Herran ldquoThe impact of fore-sight in a transboundary pollution gamerdquo European Journal ofOperational Research vol 251 no 1 pp 300ndash309 2016

[10] XHuang PHe andW Zhang ldquoA cooperative differential gameof transboundary industrial pollution between two regionsrdquoJournal of Cleaner Production vol 120 pp 43ndash52 2016

[11] A Haurie J B Krawczyk and M Roche ldquoMonitoring coop-erative equilibria in a stochastic differential gamerdquo Journal ofOptimization Theory and Applications vol 81 no 1 pp 73ndash951994

[12] E Saltari and G Travaglini ldquoThe effects of environmental poli-cies on the abatement investment decisions of a green firmrdquoResource and Energy Economics vol 33 no 3 pp 666ndash685 2011

[13] S Athanassoglou and A Xepapadeas ldquoPollution control withuncertain stock dynamics when and how to be precautiousrdquoJournal of Environmental Economics and Management vol 63no 3 pp 304ndash320 2012

[14] E J Dockner and K Nishimura ldquoTransboundary pollution ina dynamic game modelrdquo Japanese Economic Review vol 50 no4 pp 443ndash456 1999

[15] F Bosello B Buchner and C Carraro ldquoEquity developmentand climate change controlrdquo Journal of the European EconomicAssociation vol 1 no 2-3 pp 601ndash611 2003

[16] A Bernard A Haurie M Vielle and L Viguier ldquoA two-leveldynamic game of carbon emission trading between RussiaChina and annex B countriesrdquo Journal of Economic Dynamicsand Control vol 32 no 6 pp 1830ndash1856 2008

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of