Cooperation on Climate-Change Mitigation†

Cooperation on Climate-Change Mitigation† Charles F Mason Department of Economics and Finance, University of Wyoming Stephen Polasky Department of Applied Economics, University of Minnesota Nori Tarui Department of Economics, University of Hawaii This draft: 17 November, 2008 Abstract We model greenhouse gas (GHG) emissions as a dynamic game Countries’ emissions increase atmospheric concentrations of GHG, which negatively affects all countries' welfare We analyze self-enforcing climate-change treaties that are supportable as subgame perfect equilibria A simulation model illustrates conditions where a subgame perfect equilibrium supports the firstbest outcome In one of our simulations, which is based on current conditions, we explore the structure of a self-enforcing agreement that achieves optimal climate change policy, what such a solution might look like, and which countries have the most to gain from such a agreement † The authors thank Stephen Salant, Akihiko Yanase, and the seminar participants at the University of Hawaii, the ASSA Meetings in 2007, Doshisha, Japan Economic Association Meeting, Hitotsubashi, Tokyo Tech, Tokyo, Tsukuba, Kobe, Keio University, and the Occasional Workshop on Environmental and Resource Economics The authors are responsible for any remaining errors Cooperation on Climate-Change Mitigation Abstract We model greenhouse gas (GHG) emissions as a dynamic game Countries’ emissions increase atmospheric concentrations of GHG, which negatively affects all countries' welfare We analyze self-enforcing climate-change treaties that are supportable as subgame perfect equilibria A simulation model illustrates conditions where a subgame perfect equilibrium supports the firstbest outcome In one of our simulations, which is based on current conditions, we explore the structure of a self-enforcing agreement that achieves optimal climate change policy, what such a solution might look like, and which countries have the most to gain from such a agreement Introduction Global environmental problems such as climate change, depletion of the ozone layer and loss of biodiversity have risen to the top of the world’s environmental agenda For each of these problems there is a large scientific literature warning of the dangers of failing to successfully address the issue and continuing business-as-usual For climate change, the Intergovernmental Panel on Climate Change (IPCC) concluded that continued emissions of greenhouse gases (GHG) would likely lead to significant warming over the coming centuries with the potential for large consequences on the global environment (IPCC 2007) Climate change and other global environmental problems, however, are particularly difficult to address because actions that address these problems impose private costs and generate a global public good Successfully addressing global public goods require concerted action by numerous sovereign countries Sovereignty of nations implies that any international environmental agreement must be selfenforcing for every country But designing self enforcement agreements is problematic given the nature of global public goods because the self interest of each country is best served by having other countries bear the cost of addressing the problem while they free-ride on these efforts A large number of prior studies have analyzed the benefits and costs of reducing greenhouse gas emissions (e.g., Cline 1992, Fankhouser 1995, Manne and Richels 1992, Mendelsohn et al 2000, Nordhaus 1991, 1994, Nordhaus and Yang 1996, Nordhaus and Boyer 2000, Tol 1995; see Tol 2005 and 2007 for recent summaries) The release of the Stern Review (Stern et al 2006), which argued for much swifter and deeper cuts in GHG emissions than the dominant view in the published economics literature, ignited a new round of discussion about optimal climate change policy (e.g., Dasgupta 2007, Mendelsohn 2007, Nordhaus 2007, Tol and Yohe 2006, Weitzman 2007, Yohe et al 2007) For the most part these studies not analyze equilibrium in which countries can choose emissions strategically Analyzing countries' strategic interactions is crucial for understanding climate-change policy As the negotiations over the Kyoto Protocol illustrate, countries can choose to participate or stay on the sidelines (e.g., US and China) Even if a country chooses to participate, there are limited An exception is Nordhaus and Yang (1996) which solved for an open-loop Nash equilibrium emissions allocation by countries as well as a Pareto optimal GHG emissions allocation -1- sanctions available to punish countries that not meet their climate change treaty obligations Addressing questions of whether a country will choose to participate in a climate change agreement or will choose to comply with an agreement requires an analysis of the strategic interests of each country involved in climate change negotiations A number of studies apply static or repeated games to consider countries' strategic choice of GHG emissions (Barrett 2003, Finus 2001) Bosello et al (2003), de Zeeuw (2008) and Eyckmans and Tulkens (2002) incorporate the dynamics of GHG stock to analyze an international agreement on climate change However, these studies focus on the stability of an environmental treaty by a subset of countries where the treaty members are assumed to cooperate even when cheating may improve a treaty member's welfare Nordhaus and Yang (1996) investigate a dynamic game but assume that countries adopt open-loop strategies (where countries commit to future emissions at the outset of the game, and so are not necessarily subgame perfect equilibria) Yang (2003) studies a dynamic game allowing for closed-loop strategies, but without including the role of punishment for potential defectors A desirable model of international environmental agreements as applied to problems such as climate change would include stock effects (so that the interaction is properly modeled as a dynamic, rather than a repeated, game), and would allow for countries to credibly punish defector states Such a model would by necessity focus on closed-loop or feedback strategies To our knowledge, only a few papers include these ingredients Dockner et al (1996) and Dutta and Radner (2000, 2005) find conditions under which cooperative equilibrium can be supported as a subgame perfect equilibrium through use of a trigger strategy In these models, once some country cheats on the agreement by over-emitting, punishment begins and continues forever In the climate change application, however, such a trigger-strategy profile would involve mutually assured over-accumulation of GHGs if punishment were ever called for A legitimate criticism of such strategies is that they are not robust against renegotiation upon a country's deviation because the countries restart cooperation once a temporary sanction is completed In addition, most international sanctions are temporary in nature, calling into question the empirical relevance of strategies involving perennial punishment.2 Based on 103 case studies of economic sanctions between World War I and 1984, Hufbauer et al (1985) find that the average length of successful and unsuccessful sanctions were 2.9 and 6.9 years Success of a sanction is defined in terms of the extent to which the corresponding foreign policy goal is achieved (p.79) -2- In this paper, we reconsider the problem of designing a self-enforcing international environmental agreement for climate change Our model presents a dynamic game in which each country in each period chooses its level of economic activity Economic activity generates benefits for the country but also generates emissions that increase atmospheric concentrations of GHG, which negatively affect the welfare of all countries Atmospheric concentrations evolve over time through an increase of concentrations from emissions of GHG and the slow decay of existing concentrations We analyze a strategy profile in which each country initially chooses emissions that generate a Pareto optimal outcome (first best or cooperative strategy) and continues to play cooperatively as long as all other countries so However, our strategies entail temporary punishment: If a country deviates from the cooperative strategy, all countries then invoke a two-part punishment strategy In the first phase, countries inflict harsh punishment on the deviating country by requiring it to curtail emissions In the second phase, all countries return to playing the cooperative strategy The design of the two-phase punishment scheme guarantees that the punishment is sufficiently severe to deter cheating, and that all countries will have an incentive to carry out the punishment if called upon to so We identify conditions under which this strategy can support the first-best outcome as a subgame perfect equilibrium, i.e., when a self-enforcing international environmental agreement can generate an efficient outcome We provide a simulation model to illustrate conditions when it is possible for a self-enforcing agreement to support an efficient outcome We also parameterize the simulation model to mimic current conditions to show whether a self-enforcing agreement that achieves optimal climate change policy is possible, the structure of what such a solution might look like, and which countries have the most to gain from such a agreement (or to lose from failure to agree) The two-part punishment scheme that we use here to find subgame perfect equilibrium that supports an efficient solution is similar to that in previous studies that analyze cooperation in a dynamic game of harvesting a common property resource (Polasky et al 2006, Tarui et al 2008) However, unlike a harvesting game in which a player can always guarantee non-negative payoffs -3- (by simply not harvesting), the GHG emissions game can have arbitrarily large negative payoffs Damages increase with the stock of GHGs and the stock of GHGs is outside the control of any single country In addition, and again in contrast to Dutta and Radner’s model, we assume nonlinear damage effects of GHG stock on each country.3 Though there is a large degree of uncertainty about the economic effects of future climate change, scientists predict that the effects may be nonlinear in the atmospheric greenhouse gas concentration Studies predict nonlinear effects of climate change on agriculture (Schlenker et al 2006, Schlenker and Roberts 2006) Nonlinearity may also arise due to catastrophic events such as the collapse of the thermohaline circulation (THC) in the North Atlantic Ocean: climate change may alter the circulation, which would result in significant temperature decrease in Western Europe Our numerical example with quadratic functions captures this nonlinearity In what follows, section describes the assumption of the game and a two-part strategy profile with a simple penal code to support the cooperative outcome Using an example with quadratic functions, section discusses the condition under which the two-part strategy profile is a subgame perfect equilibrium In Section 4, we choose the parameter values of the quadratic functions based on previous climate-change models in order to illustrate the implication to climate-change mitigation Section concludes the paper Basic model 2.1 Assumptions There are N 2 players in a dynamic game, each representing a country In each period t = 0,1, each country i = 1,…,N chooses a GHG emission level xit 0 For each country i, we assume there is a maximum feasible emission level xi  The transition of the GHG stock in the atmosphere is given by Dockner et al (1996) assumes nonlinear damage functions and linear emission reduction costs Our model assumes both nonlinear damages and nonlinear emission reduction costs -4- S t 1 = g ( S t , xt ) = S   ( S t  S )   X t , where X t ixit ,   represents the natural rate of decay of GHG per period (  < ), S is the GHG stock level prior to the industrial revolution, and  is the retention rate of current emissions (   1 ).4 Let x i be a vector of emissions by all countries other than i and let X  i  j = ix j , the total emissions by all countries other than i We denote the period-wise return of country i with emission xi when the GHG stock is S by  i ( xi , S ) Each country’s emission level is linked to its output, and hence generates net benefits from consumption On the other hand, each country suffers flow damages associated with its emissions The function i summarizes the combination of these effects Because emissions are linked to net benefits, reducing emissions is costly for country i Lastly, each country suffers damages from GHG concentration; these damages are increasing in the GHG stock We assume that each country i's period-wise return equals the economic benefit from emissions Bi , which is a function of its own emissions, minus the climate damage Di , a function of the current GHG stock:  i ( xit , St ) = Bi ( xit )  Di ( St ) We assume that Bi is a strictly concave function with Bi (0) = , that it has a unique maximum xib  , and that Bi ' ( x) > for all x  (0, xib ) We call xib the “myopic business-as-usual” (myopic BAU) emission level of country i This level of emissions maximizes period-wise returns, without taking into account any future implications associated with contributions to the stock of GHGs The damage function is increasing and convex ( Di ' > 0, Di ' ' > ), which captures the nonlinear effects of climate change Countries have the same one-period discount factor   (0,1) We assume that the economic benefit and the damage for all countries grow at the same rate The discount factor δ Many studies have used this specification of GHG stock transition (Nordhaus and Yang 1996, Newell and Pizer 2003, Karp and Zhang 2004, Dutta and Radner 2004) -5- incorporates the growth rate of benefits and damages (see section 3.2 for more discussions about the discount factor) All countries' return functions are measured in terms of a common metric There is no uncertainty (i.e., countries have complete information) In each period, each country observes the history of GHG stock evolution and all countries' previous emissions 2.2 First best solution The first best emissions path solves the following problem  max  t  i ( xit , St ) t =0 i s.t S t 1 = g ( S t , xt ) for t = 0,1,  given S *  * * N The solution to this problem generates a sequence of emissions {xt }t = where xt = {xit }i =1 The corresponding value function solves the following functional equation V ( S ) = max  i ( xi , S )  V ( S ) x i s.t S  = g ( S , x) We assume the solution is interior The optimal emission profile given S , x* ( S ) = {x1* ( S ), x2* ( S ),  , x *N ( S )} , satisfies the following  i ( xi* ( S ), S )  V ( g ( S , X * ( S )))  = xi for all i The first term represents the marginal benefit of emissions in country i while the second term is the discounted present value of the future stream of marginal damages in all countries from the next period Thus, under the first best allocation, the marginal abatement costs of all countries in the same period must be equalized, and they equal the shadow value of the stock The unique steady state S * satisfies the following equation:  i ( xi* ( S * ), S * )   xi    j ( x *j ( S ), S ) j S = * Given S < S , the stock increases monotonically to the steady state S * For the rest of the paper -6- * we assume S < S In what follows, we describe a strategy profile that supports x* as a subgame perfect equilibrium 2.3 A strategy profile to support cooperation Consider the following strategy profile * , which may support cooperative emissions reduction with a threat of punishment against over-emissions.5 Strategy profile *  * Phase I: Countries choose {xi } If a single country j chooses over-emission, with resulting stock S  , go to Phase II j (S ) Otherwise repeat Phase I in the next period  j j j j Phase II j (S ) : Countries play x = ( x1 ,  , xi , , x N ) for T periods If a country k deviates with resulting stock S , go to Phase II k (S ) Otherwise go back to Phase I The idea of the penal code x j is to induce country j (that cheated in the previous period) to choose low emissions for T periods while the others enjoy high emissions Each sanction is temporary, and the countries resume cooperation once the sanction is complete The punishment for country j in Phase II j works in two ways, one through its own low emissions (and hence low benefits during Phase II) and the other through increases in its future stream of damages due to an increase in the other countries' emissions during Phase II Under some parameter values and with appropriately specified penal codes {x j } , each country's present-value payoffs upon deviation will not exceed the present-value payoffs upon cooperation We now turn to a discussion of the condition under which * is a subgame perfect equilibrium Sufficient conditions for first best sustainability C D Let V j (S , I) and V j (S , I) be country j 's payoff upon cooperation and the maximum payoff The design of the penal code to support cooperation is similar to those discussed in Abreu (1988) -7- C D upon deviation in Phase I given current stock S Similarly, let V j ( S , IIi ), V j ( S , IIi ) be j 's payoff upon cooperation and an optimal deviation starting in Phase II i given stock S With these notations, the above strategy profile is a subgame perfect equilibrium if the following conditions are satisfied for country j , j = 1, , N : C D Condition (1)Country j has no incentive to deviate in phase I: V j ( S , I) V j ( S , I) ; C D Condition (2)Country j has no incentive to deviate in phase II j : V j ( S , II j ) V j ( S , II j ) ; and Condition (3)Country j has no incentive to deviate in phase II k for all k = j : V jC ( S , IIk ) V jD ( S , IIk ) ; for all possible stock levels given initial stock S Because each player's periodwise return is bounded from above and the discount rate is positive, the principle of optimality for discounted dynamic programming applies to this game Hence, in order to prove that * is subgame perfect, it is sufficient to show that any one-shot deviation cannot be payoff-improving for any player (Fudenberg and Tirole 1991) Because this is a dynamic game, we need to verify that no player has an incentive to deviate from the prescribed strategy in any phase and under any possible stock level Because combined emissions are nonnegative and bounded by the maximum feasible level X i xi , the feasible stock levels lie between and S  where S satisfies S = S   (S  S )  X We can exploit a few properties to simplify the above three conditions for first best sustainability D Let xi (S ) be the optimal deviation that maximizes country i’s payoff upon deviation in D either phase I or II Under a reasonable assumption on xi , the following propositions hold (Proofs are relegated to the appendix.) Notice the focus is on a single deviation The reader may wonder what happens if more than one player deviates; in keeping with the usual tradition in dynamic games, a simple way to avoid the complication of considering multiple defections is to assume the game remains in Phase I if more than one player deviates; see Fudenberg and Tirole (1991, pp 157-160) for details See Dutta (1995a) for a similar analysis in a dynamic game context -8- Suppose the value to country i is given by iV where i 0 and  i i = Define a transfer  * where  i* ( S ) i  j ( x*j ( S ), S )   i ( xi* ( S ), S ) j for all S and all i By choosing x* and  * , the countries realize the first best outcome with the shares induced by  Table Condition (2) with heterogeneous countries Note: In all cases we assume  = 99 and vi = V/N for all i = 10.2 (10.0), bi = 5.0001 (5), d i  = Conditions (1) and (3) hold for all S S NL: Condition (2) does not hold when S is low relative to S * , NH: Condition (2) does not hold when S does hot hold for any S S * = 00002 (.000025) for High * for all countries in all cases Y: Condition (2) holds for all (Low) cases, and is close to S S * , S * , N: Condition (2) Table describes whether condition (2) holds under different discount factor values for a game with heterogeneous countries We assume two values -high and low -for each of the benefit and damage function parameters {ai , bi , d i } and the eight countries have different combinations of these parameter values The table assumes vi = V/N for all i , i.e the payoffs upon cooperation are divided equally across countries With this example, conditions (1) and (3) under which * is a subgame perfect equilibrium holds given all discount factor values considered (2.5 – 10%) Condition (2) also holds (and hence * is a subgame perfect equilibrium) when the discount rate is 2.5% For a larger discount rate, condition (2) is violated – first for the countries - 24 - with larger BAU emissions (i.e larger ) and low marginal damages d i , then for those with 2bi low BAU emissions and high marginal damages Given equal sharing of V , a country with a higher BAU emission and lower marginal damages has less to lose by deviation than countries with lower BAU emissions and higher marginal damages This example illustrates different incentives for controlling emissions by countries with different benefits and damages Discussion Climate change mitigation is a global public good where reducing GHG reduction is costly for each country while the GHG stock accumulation in the atmosphere is likely to cause damages to many countries In order for sovereign countries to cooperate through an international agreement to control GHGs, the agreement must be self-enforcing for each country We applied a dynamic game to illustrate an international agreement with a simple rule of sanctions in order to support the first best, cooperative climate-change mitigation Instead of a trigger strategy where all countries choose over-emissions forever upon some country's cheating, we considered a two-part penal code where the sanction against an over-emitter is temporary and where countries resume cooperation upon completion the sanction With numerical examples and illustration using a simple climate-change model, we examined the conditions under which such a simple two-part sanction scheme -where the country being sanctioned chooses a low emission and the others choose over-emissions for one period -is a subgame perfect equilibrium In particular, we studied how heterogeneous countries' incentive for cooperation may change over time given GHG stock dynamics and nonlinear effects of GHG on each country's payoff Our numerical example confirmed that each country's incentive to cooperate may change as the stock level changes We might expect that it may become easier for countries to avoid free riding and cooperate as GHG stock increases; however, we found that a larger stock level does not necessarily imply that the sanction scheme is more likely to be a subgame perfect equilibrium Because sanctions are more severe when the number of countries is larger, the sanction scheme that is not an equilibrium for a given total number of players can be an equilibrium given a larger number of players - 25 - Our linear-quadratic climate-change model with parameter values from existing studies illustrates different incentives to support cooperation held by countries with different benefits from GHG emissions and different potential damages from increases in the GHG stock Given heterogeneous benefits from emissions, countries such as US and Europe, which have larger baseline GHG emissions than the other countries, have larger incentive to deviate from the first best emissions reduction than the others Considering heterogeneity in potential damages from climate change, we found that lower-middle income countries and Western European countries will have the most to gain from cooperation due to relatively larger vulnerability to climate change This finding about heterogeneity is similar in spirit to Mason and Polasky (2003), who found that an oil-producing country's OPEC membership is significantly associated with the country's larger oil reserves (implying larger benefits from cooperation) and smaller domestic oil consumption (implying smaller benefits in consumer surplus from non-cooperation) Our climate-change model also suggested the weak link for cooperation -those countries that have the most to gain from deviation -depends on how the burden of GHG emission control is distributed across countries In international negotiation and under Kyoto Protocol, policymakers argue that richer countries including US or those countries with large GHG emissions in the past should bear larger cost burden Our simulation suggests that US is unlikely to cooperate with the simple sanction scheme if the cost burden is proportional to GDP (because of its relatively large benefits from GHG emissions and relatively moderate potential damages from US) In contrast, Western Europe may have incentive to cooperate under the same cost sharing rule despite its relatively large GDP because Western Europe is likely to be more vulnerable to climate change than US These findings imply that the cost sharing rule must be correctly specified in order for climate-change mitigation to be self-enforcing to all countries and that the self-enforcing cost sharing rule may not coincide with a rule perceived to be fair in an international context Further sensitivity analysis will be necessary for our linear-quadratic climate-change model Future research should also address a number of assumptions that we made to keep our analysis simple We assumed that each country's periodwise return function is time-invariant However, the benefit from GHG emissions and damages from climate change will change over time due to - 26 - changes in population, economic growth and technological progress In addition, the benefits and damages may change at different rates for different countries Our analysis on heterogeneity does not consider these possibilities A natural extension of our model would be to incorporate uncertainty regarding climate change Another useful extension would be to consider sanctions through means other than increased emissions such as trade (Barret 2003) A temporary trade sanctions may be less costly for each country than sanctions with increased emissions, the effect of which will last for a long time because of the nature of GHG as a stock pollutant Future research may study the extent to which the availability of trade sanctions increases the likelihood of a self-enforcing treaty Our findings, despite their tentativeness, imply that dynamic-game formulation provides a useful framework for analyzing a self-enforcing treaty for climate-change mitigation and a useful insight that may not be available from static-game or repeated-game analysis Appendix Conditions for first-best sustainability Suppose T = Consider countries' incentive to deviate in Phase I In period t , given current stock St , country j 's payoff upon cooperation is V j ( S t ) Country j 's payoff upon deviation, D with over-emission x j in period t , is given by    j ( x Dj , St )  max 0,  j  i ( xi* ( S ), S )   j ( x*j ( S ), S )   j ( x jj ( St 1 ' ), St 1 ' )   2V j ( St 2 ' ) i   j D * where St 1 ' = g ( St , x j , x j ) and St 2 ' = g ( St 1 , x ( S t 1 )) This is a discounted sum of a current gain by over-emitting in period t , a low return in period t  due to punishment, and continuation payoffs with a larger GHG stock due to its own over-emission Therefore, no country deviates from Phase I if   V j ( S )  j ( x Dj , S )  max  0,  j  i ( xi* ( S ), S )   j ( x*j ( S ), S ) i     j ( x jj ( g ( S , x Dj , x* j ( S ))), g ( S , x Dj , x* j ( S )))   2V j ( g ( g ( S , x Dj , x* j ( S )), x j ( g ( S , x Dj , x* j ( S ))))) - 27 - D for all x j 0 , S S and all j In Phase II j , country j does not deviate from cooperation if  j ( x jj ( S ), S )  V j ( g ( S , x j ( S )))  j ( x Dj , S )   j ( x jj ( g ( S , x Dj , xj j ( S ))), g ( S , x Dj , xj j ( S )))   2V j ( g ( g ( S , x Dj , xj j ( S ))), x j ( g ( S , x Dj , xj j ( S )))) D * D for all x j 0 and all possible stock levels given S (i.e for all S  g ( S , ( x j ( S ), x j )) ) Similarly, in Phase II k , country j has no incentive to deviate if  j ( x kj ( S ), S )  V j ( g ( S , x k ( S )))  j ( x Dj , S )   j ( x jj ( g ( S , xk j , x Dj )), g ( S , xk j , x Dj ))   2V j ( g ( g ( S , xk j , x Dj ), x j ( g ( S , xk j , x Dj )))) D * D for all x j 0 and all S g ( S , ( x k , xk )) To summarize, the strategy profile is a subgame perfect equilibrium if conditions (), (), and () (for all k = j ) hold for all possible deviations, all S S and all j Proof of Proposition Let xˆ be the optimal deviation by country i in phase II i , v the optimal value function of country i , and X * the optimal total emission of N countries (all of which are functions of stock S ) Let G ( z , S t ) be the gain from devaition for country i in Phase II i ( S t ) :    G ( z , S t ) = B( xˆ )  D( S t )   B( z )  D( S td1 )   v( S td2 )  B ( z )  D( S t )  v( S t*1 )    = B( xˆ )  B( z )   B( z )  D( S td1 )   v( S td2 )  v( S t*1 ), S t  X ( S t )  z  xˆ , S S  X ( S ), and St*1 S t  X * ( St ) G = B ( xˆ )  (1   ) B ( z ) Step To show z The partial derivative of G with respect to z is given by G xˆ xˆ  xˆ    = B ( xˆ )  B ( z )  B ( z )  D ( S td1 )       v ( S td2 )   X *' ( S td1 )     z z z  z    where S d t 1 * d t 2 d t 1 * d t 1  = xˆ B ( xˆ )  D ( S td1 )   v ( S td2 )   X *' ( S td1 ) z   - 28 -      B ( z )  B ( z )  B ( xˆ )  B ( xˆ )  D ( S td1 )   v ( S td2 )   X *' ( S td1 ) where the first term equals zero due to the envelope theorem Therefore, G =  B ( xˆ )  D ( S td1 )   v ( S td2 )   X *' ( S td1 )  B ( xˆ )  (1   ) B ( z ) z Applying the envelope theorem once again, we have G = B ( xˆ )  (1   ) B ( z ) z Let xˆ I ( S t ) be the optimal deviation by country i in phase I ( S t ) and GI ( z, St ) the gain from deviation for country i in Phase I ( S t ) : G I ( z , S t ) = B( xˆ I )  D( S t )   B( z )  D( S td,1I )   v( S td,2I )  B ( x* )  D( St )  v ( St*1 )          = B( xˆ I )  B( x * )   B( z )  D( S td,1I )   v( S td,2I )  v( S t*1 ), d ,I * * d ,I d * d * * where S t 1 S t  X ( S t )  x ( S t )  xˆ I , St 2 St 1  X ( S t 1 ), and St 1 S t  X ( St ) GI = B( z ) Step To show z The partial derivative of G with respect to z is given by G I xˆ xˆ xˆ = B ( xˆ I ) I  B ( z )  D ( S td1 ) I   v ( S td2 )   X *' ( S td1 ) I z z z z   xˆ I B ( xˆ I )  D ( S td1 )   v ( S td2 )  B ( z ) = B ( z ), z where the last equality follows from the envelope theorem Step To show G ( z , S ) GI ( z, S ) for all S Note that G ( x * ( S ), S )  GI ( x * ( S ), S ) = 0, i.e the gains from deviation are the same in Phases I and II when z equals x* ( S ) Steps and imply  [G ( z, S )  G I ( z , S )] = B ( xˆ ) < z for all z  [0, x * ( S )] given any stock level as long as xˆ > z It follows from the last two equations that G GI for all z ▄ =   The first best solution of a quadratic example with heterogeneous countries Suppose country i 's periodwise return in period t is given by xit  bi xit2  ( g i S t  d i S  f i ), given emissions xit and stock St where , bi , d i > for all i (The parameters {g i , f i } may take positive or negative values.) Let a = (a1 , a2 ,  , a N ) , b = (b1 , b2 ,  , bN ) , and d = (d1 , d , , d N ) be column vectors Let G = ig i and F = i f i Let B be an N by N diagonal matrix whose diagonal entries are b Given stock S and an emissions profile x , The total periodwise return of N countries is ax  xBx  (GS  DS  F ), - 29 - where D = id i The state transition is given by St 1 = S   x  h where   (0,1) ,  a column vector where   i represents the short-run decay rate, and h (1   ) S is a constant Let V be the total value function of all N countries: V ( S ) = max ax  xBx  (GS  DS  F )  V ( S ) x (4) s.t S  = S   x  h Because V is quadratic, let V ( S ) = PS  QS  R where P, Q, R are scalars We have V ( S ) = (S   x  h)P(S   x  h)  Q(S   x  h)  R = (S  h)P (S  h)  (S  h)P x  xbP (S  h)  xP x  QS  Q x  Qh  R So  V ( S ) = P(S  h)  P(S  h)  P x  Q x = 2 P (S  h)  2 P x  Q The first order condition is a  Bx   [2P (S  h)  2 P x  Q ] = Hence [2 B  2P ]x = a  2P (S  h)  Q , i.e x = (1/2)[ B  P ] 1[a  2P(S  h)  Q ] Substitute this expression into the functional equation and we obtain the following expression: P =  D  2 P   2 P  ( P)  , where  ( P) [ B  P ] Solve this function for P , and we can solve for the remaining unknown Q and R  Pa( P)   G  2 P h  ( P)   2hP Q= ,   P ( P)    1 R= { (a Q )( P )(a  Q )  (a Q ) ( P )Ph 1    P   ( P )  h  Qh  F  h P} Value functions with no transfers Here we compute each country's value function under the first best outcome when there are no transfers among countries Let xi* ( S ) =  i S  i be country i 's optimal emission given stock S 0 and X * ( S ) = S   the optimal aggregate emission given stock S 0 , where  i i and  ii S1* = S  X * ( S )  h = (   ) S     h, S 2* = S1  X * ( S1 )  h = (  ) S1    h = (  ) S0  (  )(   h)    h, , - 30 - * t t t S = (   ) S   (   ) i  (    h ) i =1   h {1  (  ) t }  (    ) Plug this into country i 's payoff under the first best outcome (with no transfers) and obtain = (   ) t S   vi ( S )  t i ( xit , St* ) =  t [ai xi ( S t* )  bi ( xi ( St* ))  {g i St*  d i S t*2  f i }] t =0 t =  t [ai ( i S t*  i )  bi ( i2 St*2  2 ii St*  i2 )  {g i St*  d i S t*2  f i }] t =  t [ J i S t*2  K i S t*  Li ], t 2 where J i  bi i  d i , K i {ai i  2bi ii  g i , and Li aii  bii  f i Note that St*2 = (  ) 2t S 02  2(  ) t S (   h){1  (  ) t }  (    )    h  t 2t   {1  2(   )  (  ) }   (   )  Plug this into the expression of vi , and we have J i (  )t S (   h) J i (  )t S (   h)(  )t J i S  K i S  Li = J i (  ) S    (    )  (    ) *2 t * t 2t 2    h  t 2t  Ji   {1  2(   )  (  ) }   (    )    h  K i (   ) t S  K i {1  (   ) t }  Li  (   ) So  vi ( S ) = S 02  t J i (   ) 2t  t  J (  ) t (   h) J i (   ) t (   h)(  ) t   S0  t  i   K i (    ) t   (    )  (    ) t       h     h t 2t t   t  J i  {1  2(     )  (     ) }  K {1  (     ) }  L  i i  (   ) t    (  )   Hence, country i 's value function vi satisfies vi ( S ) = pi S  qi S  ri , where Ji pi  ,   (   ) - 31 - J (    h) J i (    h) Ki qi  i    (    )   (    )  (    )   (    )   (    ) 2    h     h  L   h   h ri J i    Ki  Ki  i ,    (    )    (  )   (  )     (  )      (  )    (  ) for all S Note that value function p i i q =P, = Q , and i i r = R i i hold so that v i i = V , the aggregate Optimal deviation j * Assume T = , x j ( S )(z j ( S )) = x j ( S ) where  < , and xij ( S ) = xi* ( S )  xi* ( S ) (1   ) x *j ( S ) Note that *  k = jxk (S ) x jj ( S )  xkj ( S ) = x *j ( S )  xi* ( S )  (1   ) x *j ( S ) = X * ( S ), k = j k = j for all S Hence the total emission in Phase II j (S ) equals the socially optimal aggregate emission given stock S The optimal deviation given S soves the following problem   2 * * 2 max x  bi x  (d i S  g i S  f )   aixi ( S1 )  bi {xi ( S1 )}  (d i S1  g i S1  f )   vi ( S ), x 0 where S1 = S   ( X i  x)  h = S  X i  h  x , S = S1  X * ( S1 )  h = S1   (S1  )  h = (  )(S   X i  h)  (  )  x    h The first-order condition is  x* ( S ) S1 x* ( S ) S1  S S   =  2bi x    ai i  2bi xi* ( S1 )  i   2d i S1  g i   S1 x S1 x  x x    S    pi {(   )(S  X i  h)  (   ) x    h}  qi  x =  2bi x   ai i   2bi ( i (Ci  x)  i )  i   (2d i (Ci   x)   g i  )      pi {(  )Ci  (  ) x    h}  qi  (  )  , where Ci S  X i  h Arrange the terms and we have =   ai i   2bi ( i Ci  i ) i   (2d i Ci   g i  )       pi {(   )Ci     h}  qi  (  )   2bi x    2bi ( i x) i   (2d i x )     pi {(   ) x} (  )  Solving the condition for x , we obtain country i 's optimal deviation x d :     ai i   2bi ( i Ci  i ) i   (2d i Ci   g i  )    pi {(  )Ci     h}  qi  (  )  x (S ) = 2bi   2bi ( i  ) i   (2d i  )    pi {(  )  } (  )  Note that Ci depends on the stock level as well as the phase in which country i deviates: d i   - 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Section 4, we choose the parameter values of the quadratic functions based on previous climate-change models in order to illustrate the implication to climate-change mitigation Section concludes.. .Cooperation on Climate-Change Mitigation Abstract We model greenhouse gas (GHG) emissions as a dynamic game Countries’ emissions increase atmospheric concentrations of GHG, which