European Journal of Operational Research

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1 European Journal of Operatonal Research 0 (010) Contents lsts avalable at ScenceDrect European Journal of Operatonal Research journal homepage: www. elsever. com/ locate/ ejor Innovatve Applcatons of O.R. Can nternatonal envronmental cooperaton be bought? Crstna Fuentes-Albero a, Santago J. Rubo b, * a Unversty of ennsylvana, Department of Economcs, 3718 Locust Walk, hladelpha, A 19104, USA b Unversty of Valenca, Department of Economc Analyss, Edfco Departamental Orental, Avda. de los Naranjos s/n, 460 Valenca, Span a r t c l e n f o a b s t r a c t Artcle hstory: Receved 10 July 007 Accepted 6 May 009 Avalable onlne 15 May 009 Keywords: Self-enforcng nternatonal envronmental agreements Lnear envronmental damages Emssons Transfers In ths paper a two-stage game of nternatonal envronmental agreement formaton wth asymmetrc countres s analytcally solved. The equlbrum of the game makes t possble to determne the sze and composton of a stable agreement. Two cases are studed. In the frst case, countres dffer only n abatement costs, whle n the second case, they dffer n envronmental damages. In both cases, two dfferent nsttutonal settngs, one wthout transfers and another wth transfers, are consdered. The results establsh that the asymmetry assumpton has no mportant effects on the scope of cooperaton n comparson wth the symmetrc case f transfers are not used or abatement costs represent the only dfference among countres. However, when the only dfference s n envronmental damages, the level of cooperaton that can be bought through a self-fnanced transfer scheme ncreases wth the degree of asymmetry. Ó 009 Elsever B.V. All rghts reserved. 1. Introducton * Correspondng author. E-mal address: Santago.Rubo@uv.es (S.J. Rubo). Clmate change, the depleton of the ozone layer, and the loss of bologcal dversty are some of the most mportant envronmental problems facng contemporary socetes. One of the man characterstcs of ths knd of problems s ther nternatonal dmenson because of the common property of global envronmental resources and the transboundary effects of many pollutng actvtes. Hence, managng envronmental ssues requres transnatonal cooperaton. However, the lack of a supranatonal authorty wth enough coercve power over soveregn natons determnes that nternatonal envronmental cooperaton must be reached by voluntary agreements. Therefore, Internatonal Envronmental Agreements (IEAs) should be desgned n such a way that they wll be not only proftable, but also self-enforcng,.e. there must be ncentves for countres, whle actng n ther own self-nterest, to jon or to reman part of an agreement. One of the earlest defntons of a self-enforcng agreement used n the lterature on IEAs was the stablty concept proposed by D Aspremont et al. (1983) n ther analyss of cartel formaton. Accordng to ths defnton an IEA wll be stable f no sgnatory country has ncentves to leave the agreement and f no non-sgnatory has ncentves to jon the IEA, takng the membershp decsons of all other countres as gven. 1 Models based on ths concept nclude the semnal papers by Carraro and Snscalco (1991, 1993) and Barrett (1994) where t s assumed that countres are dentcal. Carraro and Snscalco (1991) have shown that f sgnatory countres act n Cournot fashon wth respect to non-sgnatores, then a stable IEA conssts of three countres when margnal envronmental damage s constant (.e., when countres best-reply functons are orthogonal), and of two countres when margnal damage ncreases wth emssons (.e., when the best-reply functons have a negatve slope), n both cases regardless of the number of countres affected. In the paper publshed by Barrettn (1994), t s shown that when the best-reply functons have negatve slope and sgnatory countres act n Stackelberg fashon, a stable IEA may acheve a hgh degree of cooperaton, but only when the gans of cooperaton are 1 There s also a stream of lterature that adopted cooperatve games as an analytcal framework to deal wth ths ssue. Among the dfferent contrbutons to ths lterature we would lke to stand out the papers wrtten by Chander and Tulkens (1995, 1997). These authors use the gamma-core concept to argue that the grand coalton can be stable usng approprately defned transfers. Other papers that belong to ths lterature are Helm (001), Eyckmans and Tulkens (003), German et al. (003), Forgó et al. (005) and Flam (006). More recently Chander (007) has elaborated a renterpretaton of the gamma-core n terms of an nfntely repeated game. These results hnge on a partcular specfcaton of the net beneft functon from emssons. See Carraro and Snscalco (1991, pp. 0 1). Other specfcatons of the net functon have been studed by Fnus (001). Hs results show that although the level of partcpaton s senstve to the specfcaton of the net beneft functon and also to the parameter values, the number of sgnatores for the three types of net beneft functons studed by the author s not greater than three. Consult chapter 13 n Fnus book /$ - see front matter Ó 009 Elsever B.V. All rghts reserved. do: /j.ejor

2 56 C. Fuentes-Albero, S.J. Rubo / European Journal of Operatonal Research 0 (010) small. Nevertheless, Carraro and Snscalco (1993) have also shown that f the countres belongng to a stable agreement commt to cooperate, the agreement can be enlarged usng ex-post transfers. Ths means that after the coalton has formed, transfers are used to brng new sgnatores nto the agreement. Botteon and Carraro (1997) have extended ths analyss for the asymmetrc case, fndng that ex-post transfers may succeed n expandng a stable coalton even wthout commtment. They obtan ths result usng an emprcal model wth fve countres (regons) and constant margnal damages. Later on they adapted the emprcal model to nvestgate the scope of ther result wth ncreasng margnal damages. See Botteon and Carraro (001). More recently, Carraro et al. (006) have used the ntegrated assessment smulaton model of clmate change elaborated by Eyckmans and Tulkens (003) to show how optmal transfers may nduce almost all countres nto sgnng a self-enforcng IEA. An optmal transfer rule, accordng to the defnton gven by Eyckmans and Fnus (004), gves every sgnatory at least hs free-rder payoff and allocates the remanng surplus n a proportonal way. They also fnd that optmal transfers exhaust all possbltes for expanson by (ex-post) nternal transfers. In ths paper, we explot the defnton of what s a potentally nternally stable coalton, gven by Eyckmans and Fnus (004), to show that the stablty analyss can be developed wthout the necessty of specfyng prevously any knd of transfer rule. To know whether an agreement can be (nternally) stablzed through transfers, t s enough to check whether the countres that gan wth the agreement have enough resources to buy the cooperaton of the countres that lose through accedng to the agreement. Ths wll occur when the aggregate payoff of sgnatores s greater than the aggregate payoff they would receve f they qut the agreement and act as free-rders. Moreover, ths condton can be checked wthout the necessty of specfyng a transfer rule, just checkng whether the surplus of the agreement over the free-rder payoffs of ts members s postve. Therefore, an agreement can be stable wth transfers f the prevous condton s satsfed, but there are not enough resources to buy the cooperaton of any non-sgnatory whch wll make the agreement potentally externally stable as well. Obvously, f these condtons are satsfed the transfer schemes that make possble to stablze the agreement belong to the class of optmal or almost deal transfer schemes defned by Eyckmans and Fnus (004). It s clear that n order to mplement the agreement t wll be necessary to specfy an optmal transfer rule, but as we show n ths paper, ths s not necessary to know whch can be the scope of cooperaton when heterogenous countres use transfers. Besdes, to analyze ths ssue, we also nvestgate the relatonshp between the degree of asymmetry, partcpaton and the gans to an IEA. For a model wth two types of countres and an arbtrary number of countres of each type, our fndngs establsh that the asymmetry among countres has no relevant effects on the scope of envronmental cooperaton n comparson wth the symmetrc case f transfers are not allowed. Wth transfers the effects depend on the knd of asymmetry consdered. If the countres dffer n abatement costs alone, the result s that only lmted cooperaton can be bought. However, when countres dffer n terms of envronmental damages, the level of cooperaton ncreases wth the heterogenety n margnal envronmental damages, whch mples that an agreement wth a hgh degree of partcpaton may be self-enforcng f the degree of asymmetry among the countres s suffcently large. What we fnd n our analyss s that although an ncrease n asymmetry rses the ncentves to devate for the sgnatores wth lower damages also rses the ncentves to stay for the sgnatores wth greater damages yeldng, once the degree of asymmetry s above a certan threshold, a postve surplus of the agreement over the ncentves to devate. We also fnd that ths threshold ncreases wth the number of sgnatores wth lower damages whch explans the postve relatonshp between the degree of asymmetry and the level of partcpaton. The same knd of result s obtaned wth three types of countres. Fnally, we would lke to hghlght that the level of cooperaton n our model only depends on the degree of asymmetry. Beng more precse, we fnd that the level of cooperaton only depends on the relatve value of the margnal damages so that f a change n the absolute values s consdered that does not alter ther relatve value, the change has no effect on partcpaton. Thus, we fnd that t s not possble to establsh any systematc relatonshp between the gans to full cooperaton and the partcpaton n the agreement snce an ncrease n the relatve value of the margnal damages can be accompaned wth dfferent varatons of the gans to full cooperaton. In fact, an ncrease n the degree of asymmetry can ncrease, decrease or keep constant the gans to cooperaton dependng on whch are the changes n the absolute values of the margnal damages assocated wth the ncrease n ther relatve value. Although a lot of papers have been publshed on the stablty of IEAs, only a few have addressed ths ssue wthn a theoretcal framework. These papers are now brefly revewed. A frst paper to quote s Hoel (199). In ths paper a numercal smulaton s used to show that wthout transfers the number of sgnatores n a self-enforcng IEA s low. The result s obtaned for a model where the countres only dffer n margnal envronmental damages that, on the other hand, are assumed constant. Usng a generalzaton of ths model, etraks and Xepapadeas (1996) extend the results of Carraro and Snscalco (1993) to the case n whch countres are completely asymmetrc. However, they mantan the stable coalton commtment assumpton n ther analyss. Hoel and Schneder (1997), also n a model wth lnear envronmental damages where all the countres are dentcal except for some non-envronmental costs that depend on the number of sgnatores and represent the effects of socal norms, pont out that the prospect of recevng a transfer tends to reduce the ncentve a country mght have to jon the agreement. On comparng ther results wth those obtaned n ths paper, t seems clear that ths depends crtcally on ther specfcaton of non-envronmental costs. Another contrbuton worth mentonng s Barrett (1997). In ths paper, a seven-naton model s solved numercally where four natons are low beneft, low cost (type 1) and the remanng three are hgh beneft, hgh cost (type ). Wth ex-ante transfers determned by Shapley values and sgnatores actng as a leader, the author fnds that a self-enforcng IEA always exsts although t never conssts of more than three sgnatores. More recently, Barrett (001) has nvestgated the possbltes of buyng cooperaton n a lnear model wth ex-ante transfers and an arbtrary number of countres of two types whch can only choose between two actons: to pollute or to abate. The countres have dentcal abatement costs, but dffer n the beneft stemmng from total abatement. In ths framework, Barrett concludes that nternatonal cooperaton can be bought and that transfers become the vehcle for ncreasng partcpaton. In ths paper, we extend ths result for a non-lnear model wth two types of countres that dffer not only n envronmental damages but also n abatement costs. Although we assume that envronmental damages are lnear, the non-lnearty of abatement costs s enough to change the nature of the game and the knd of equlbrum that characterzes the soluton n comparson wth Barrett s (001) model. Lnearty of the net beneft functon nduces a bang-bang soluton: f any sgnatory were to qut from the equlbrum agreement, all the remanng sgnatores would play pollute, lke the non-sgnatores. However, n the non-lnear model used n ths paper ths s not the case. Although the sgnatores ncrease ther emssons when one sgnatory quts the agreement, they do not choose the same level of emssons as non-sgnatores. As the two models are dfferent, both yeld dfferent predctons for the symmetrc case and

3 C. Fuentes-Albero, S.J. Rubo / European Journal of Operatonal Research 0 (010) also for the asymmetrc case wthout transfers. For nstance, n Barrett s (001) paper, cooperaton wthout transfers depends on the degree of asymmetry, see hs examples n Secton 4. However, n our model the maxmum level of cooperaton wthout transfers conssts of three countres of the same type regardless of the degree of asymmetry. Moreover, we hghlght n ths paper that the stablty of an agreement can be checked wthout the prevous specfcaton of a transfer scheme as occurs n Barrett s (001) paper. Fnally, we would lke to menton the nterestng papers wrtten by Wekard (009) and McGnty (007), where specal cases n the class of optmal transfers are used to analyze the stablty of IEAs among asymmetrc countres. 3 Wekard (009) suggests, also for a model wth quadratc abatement costs and completely asymmetrc lnear benefts, that large agreements may well be stable and shows that the grand coalton can be stable f the degree of asymmetry s large enough. Note that n order to ascertan whether or not the grand coalton s stable, t s enough to check whether t s nternally stable. In ths paper, we complete Wekard s (009) analyss by showng that between the grand coalton and the fully non-cooperatve equlbrum there can be partal coaltons that can be also self-enforcng, dependng on the dfferences n envronmental damages between the countres. Moreover, we also show that the degree of asymmetry n envronmental damages, not n abatement costs, s determnant when t comes to explanng the level of cooperaton. We also nvestgate the potental scope of cooperaton under asymmetry wthout transfers. In McGnty s (007) paper a model wth twenty countres s numercally solved. Hs numercal smulatons show that symmetrc models may vastly understate the degree of abatement achevable by a stable agreement wth transfers, but on the other hand they confrm the result obtaned by Barrett (1994) for the symmetrc model whch establshes that there s a trade-off between the gans to an IEA and the number of sgnatores. However, we do not fnd any systematc relatonshp between the gans to full cooperaton and the number of sgnatores of a stable IEA. As we have just ponted out, our fndngs establsh that an ncrease n the gans to full cooperaton can ncrease (decrease) partcpaton provded that t comes wth an ncrease (decrease) n the degree of heterogenety. Nevertheless, on comparng our results wth those obtaned by McGnty (007), one should take nto account that besdes assumng a quadratc beneft functon, McGnty (007) also assumes that the agreement s the Stackelberg leader n the abatement game. Our conjecture s that t s ths last assumpton that could explan the dfferences n results, as t occurs for the symmetrc model. 4 A frst numercal example we have developed for the model wth quadratc envronmental damages gves support to ths conjecture. The numercal example shows that an ncrease n the gans to full cooperaton, accompaned wth an ncrease n the degree of asymmetry, ncreases the level of cooperaton. 5 Next, we would lke to devote a few lnes to the mplcatons of our results for nternatonal envronmental polcy. The frst thng that seems pretty obvous s that our results are more optmstc than those obtaned for the symmetrc case, as we fnd that asymmetry can nduce more cooperaton f accompaned by transfers 3 Besdes the defnton of the optmal transfer scheme and the potentally nternally stable coaltons, Eyckmans and Fnus (004) show that any transfer scheme from the class of optmal transfers can make the coalton wth the hghest global welfare stable. However, they do not analyze the potental scope of the cooperaton under ths class of transfer schemes. 4 Remember that the dfferences n the results on partcpaton obtaned by Carraro and Snscalco (1991) and Barrett (1994) are only justfed by the fact that Barrett assumes that sgnatores enjoy a frst-mover advantage n the abatement game, whereas Carraro and Snscalco (1991) compute a Nash equlbrum. 5 The soluton to the model wth quadratc envronmental damages and the numercal example are avalable as supplementary materal at the EJOR webste. provded that the degree of asymmetry s suffcently large. In other words, cooperaton can be bought. However, only one class of transfer schemes can elmnate the ncentves of some countres to act as free-rders. Ths ntroduces a new element nto the debate about the propertes that a transfer scheme should have. It should be an optmal transfer scheme. A frst contrbuton n ths lne s the paper by Carraro et al. (006) mentoned above. In ths paper, they compare smple transfer schemes such as the Shapley Value, the Nash Barganng soluton or the Chander and Tulkens transfer scheme wth optmal transfer schemes. The result s that the partcpaton n a self-enforcng IEA wth smple transfer schemes s lower than that whch could be obtaned usng an optmal transfer scheme. Ths result suggests the possblty of a conflct between equty and stablty, an ssue that would deserve more attenton n the future. In partcular, the ssue s of great sgnfcance when transfers are mplemented through a system of tradable polluton permts, as the allocaton of permts that maxmzes cooperaton may not satsfy other desrable propertes. 6 To conclude, we would lke to clarfy one ssue. Followng the approach adopted by Carraro and Snscalco (1993), Barrett (1994), Chander and Tulkens (1997) and by many scholars afterwards, we focus on the case where only one IEA s formed and the questons remanng are the sze and composton of the self-enforcng agreement. We are aware that ths approach elmnates the possblty of dfferent countres formng dfferent agreements,.e., the possblty of equlbrum wth more than one agreement. Nevertheless, we beleve that ths approach may be reasonable for global envronmental problems, such as clmate change, for whch two global envronmental agreements have been launched by the Unted Natons (the UNFCCC and the Kyoto rotocol). Formally, t could be nterpreted as an nsttutonal constrant. The structure of the paper s as follows. In Secton, the model s set up and the defnton of a stable IEA s presented. In Secton 3, the Nash equlbrum of the emsson game s solved for a gven number of sgnatores. In Sectons 4 and 5, the stablty of IEAs wth and wthout transfers s analyzed consderng, n Secton 4, that the only dfference among countres appears n abatement costs and, n Secton 5, n envronmental damages. Secton 6 summarzes our conclusons.. Self-enforcng nternatonal envronmental agreements.1. The model Consder N countres that pollute a common envronment and negotate the emsson control of a specfc pollutant. We defne x 0 as the level of emssons generated by a country, and X ¼ N x as total emssons generated by all N countres. Each country derves a gross beneft from ts emssons so that the reducton of the emssons for controllng polluton mples some abatement costs, denoted by ðc =Þðd x Þ. The value of parameter d > 0, that stands for the busness-as-usual emsson level, depends on natonal technology, economc structure and the level of development. arameter c > 0 represents the margnal cost of natonal abatement and depends on natonal technology and measures the ntensty of the use of the pollutant for the producton of goods and servces. Each country also suffers envronmental damages whch depend on total emssons accordng to the followng expresson m X, where m > 0 s the margnal envronmental damage whch depends on the country s envronmental (natural) 6 In McGnty s (007) paper, polluton permts are allocated n order to mplement an optmal transfer but the focus of the paper s only on the degree of cooperaton that can be reached through ths partcular transfer scheme not on the comparson wth other transfers schemes.

4 58 C. Fuentes-Albero, S.J. Rubo / European Journal of Operatonal Research 0 (010) endowment. Then, for each country, the cost functon s: 7 C ¼ 1 c ðd x Þ þ m X where ¼ 1;... ; N and x 6 d... A self-enforcng IEA We model the formaton of an IEA as a two-stage game. We wll descrbe each game brefly, n reverse order, as we compute the subgame-perfect equlbra of ths two-stage game by backward nducton...1. The emsson game Suppose that, as the outcome of the frst-stage game, an agreement K of sze n has formed. If n61; ths s called the sngleton coalton structure. If n ¼ N t s called the grand coalton. Then there are n sgnatores and N n non-sgnatores. Non-sgnatory countres choose ther emssons actng non-cooperatvely and takng the emssons of all other countres as gven n order to mnmze ts own costs of controllng polluton. On the other hand, sgnatory countres choose emssons also actng non-cooperatvely aganst non-sgnatores n order to mnmze the aggregate costs of the agreement. Sgnatores also take the strategy of nonsgnatores as gven. Thus, emssons are provded by the partal agreement Nash equlbrum wth respect to a coalton defned by Chander and Tulkens (1995). Fnally, sgnatores can also mplement transfers, T s ðkþ, among them. Then, the costs of sgnatores and non-sgnatores at the equlbrum are gven by C f ðkþ and C s ðkþ þ Ts ðkþ wth K Ts ðkþ ¼ 0 where f denotes that the country s a non-sgnatory and s that t s a sgnatory.... The membershp game We assume that n the frst-stage, countres play a smultaneous open membershp game wth a sngle agreement and full complance. In a sngle agreement formaton game, the strateges for each country are to sgn or not to sgn and the agreement s formed by all players who smultaneously have chosen to sgn. Under open membershp, any country s free to jon the agreement f nterested. Fnally, we assume that the sgnng of the agreement s bndng on sgnatores. They therefore acqure a commtment to stay and mplement the agreement durng the second-stage of the game so that full complance s acheved. The game fnshes when the emsson subgame s over and f t s the case, when transfers are mplemented so that countres do not have the possblty of enlargng the agreement wth a second round of (ex-post) transfers. Summarzng, the tmng of the game when transfers are not used s as follows: frst, countres decde smultaneously and non-cooperatvely whether to sgn or not to sgn an nternatonal envronmental agreement. If a country decdes to sgn the agreement t commts to choose emssons n the second-stage of the game n order to mnmze the aggregate costs of the agreement. Second, once the agreement s sgned, sgnatores, actng as a unque player, and non-sgnatores choose smultaneously and noncooperatvely the emssons. When the countres accept to use transfers, they also commt n the frst-stage to share the gans to cooperaton. In ths case, once emssons has been chosen and the cost realzed, transfers are mplemented and the game s over. 8 Next, we present the defnton of coaltonal stablty due to D Aspremont et al. (1983). Defnton 1. An agreement K wthout transfers s self-enforcng f C s ðkþ6cf ðk n fgþ8 K and Cf ðkþ6cs ðk [ fgþ 8 R K: The frst nequalty, whch s also known as the nternal stablty condton, smply means that any sgnatory country s at least as well-off stayng n the IEA as wthdrawng from t, assumng that all other countres do not change ther membershp decsons. The second nequalty, whch s also known as the external stablty condton, smlarly requres any non-sgnatory to be at least as well-off remanng a non-sgnatory as jonng the IEA, assumng once agan, that all other countres do not change ther membershp decsons. Next, we rewrte ths defnton to take nto account the possblty of transfers among sgnatores. Defnton. An agreement K s potentally self-enforcng f there s at least one self-fnanced transfer vector such that C s ðkþ þ Ts ðkþ < Cf ðk n fgþ 8 K and there s no self-fnanced transfer vector such that C f ðkþ Cs ðk [ fgþ þ Ts ðk [ fgþ8 R K. In other words, an agreement consstng of n countres s stable f the countres that gan from the agreement can buy the cooperaton of the countres that are nterested n sgnng the agreement only f they are adequately compensated, and when t s not possble to do the same for an agreement consstng of n þ 1 countres. We assume a strct nequalty n the frst part of the defnton, because t seems reasonable that, when transfers are ncluded n nternatonal negotatons, the countres that beneft from the agreement wll not be nterested n buyng cooperaton f ths means that all ther gans must be transferred to the other countres and that the countres that lose from the agreement wll not be nterested n sellng cooperaton f they do not obtan, at least, a margnal gan. In the second part of the defnton, we use a weak nequalty because f an agreement consstng of n þ 1 countres s not stable, the agreement does not form and transfers do not apply. As a result, t does not seem necessary to be as strct as n the other case. 9 Thus, the frst part of Defnton would operate as the nternal stablty condton wth transfers and the second part as the external stablty condton. Gven ths defnton t s straghtforward to show the followng: Lemma 1. An agreement K s potentally self-enforcng through transfers f and only f K ðcs ðkþ Cf ðk n fgþþ < 0 and K[fjg ðcs ðk [ fjgþ Cf ðkþþ 0 8j R K. roof. Suppose that for an agreement consstng of n countres K Cs ðkþ Cf ðk n fgþ < 0 but that the nternal stablty condtons of Defnton 1 are not satsfed for all sgnatores. In ths case there must be a set of sgnatores W K of sze ~n wth ~n ½1; nþ for whch C s ðkþ Cf ðk n fgþ < 0 and a set of sgnatores L K of sze n ~n for whch C s ðkþ Cf ðk n fgþ 0 such that W Cs ðkþ C f ðk n fgþþj > L C s ðkþ Cf ðk n fgþ > 0 holds. Then there must be at least one vector of self-fnanced transfers through whch the ~n sgnatores can buy the cooperaton of the rest of sgnatores n ~n. For these transfers C s ðkþ þ Ts ðkþ < Cf ðk n fgþ8 K and the agreement wth transfers s nternally stable. Now we check that f an agreement consstng of n sgnatores s nternally stable then condton K Cs ðkþ Cf ðk n fgþ < 0 must be satsfed. The nternal stablty condton of Defnton can be wrtten as T s ðkþ < Cf ðk n fgþ Cs ðkþ for all sgnatores so that addng terms we get K Ts ðkþ ¼ 0 < K Cf ðk n fgþ Cs ðkþ whch mples that K Cs ðkþ Cf ðk n fgþ < 0. Next, suppose that for K[fjg any agreement consstng of n þ 1 countres C s ðk [ fjgþ Cf ðkþ 0. If ths s the case although there could 7 Ths s the cost functon used by Botteon and Carraro (1997). 8 As we ponted out n the ntroducton we do not need to be more precse about how the countres decde the transfers at ths stage of the game because we want to hghlght n ths paper, that the condtons for stablty can be checked wthout the necessty of specfyng n the frst-stage of the game any knd of transfer rule. 9 Nevertheless, we would lke to pont out that the same results are obtaned f weak nequaltes are assumed, although n ths case a rule should be establshed to decde what the stable coalton would be f the two condtons were satsfed as an equalty.

5 C. Fuentes-Albero, S.J. Rubo / European Journal of Operatonal Research 0 (010) be ~n countres wth ~n ½0; n þ 1Þ for whch C s ðk [ fjgþ Cf ðkþ < 0; W Cs ðk [ fjgþ Cf ðkþ L C s ðk [ fjgþ Cf ðkþ > 0 does not hold where now W ¼ ø or W K [ fjg and L # K [ fjg. Then there s no self-fnanced transfer vector such that for all sgnatores C s ðk [ fjgþ þ Ts ðk [ fjgþ6cf ðkþ and any agreement consstng of n þ 1 sgnatores s nternally unstable. Thus, by defnton, the agreement consstng of n sgnatores s externally stable. Fnally, we check f an agreement consstng of n sgnatores s externally stable. If ths s the case, condton K[fjg ðcs ðk [ fjgþ C f ðkþþ 0 must be satsfed 8j R K. The external stablty condton establshes that for all possble self-fnanced transfer vectors C f ðkþ6cs ðk [ fgþ þ Ts ðk [ fgþ holds 8 R K. Ths can be wrtten as C f ðkþ Cs ðk [ fgþ6ts ðk [ fgþ but accordng to Defnton, country stands for a non-sgnatory of the agreement K that we have represented by j n ths Lemma, so that for addng wth respect to the sgnatores of the agreement K [ fg we have adapted the notaton and wrtten K[fjg C f ðkþ Cs ðk [ fjgþ 6 K[fjg Ts ðk [ fjgþ ¼ 0. Then K[fjg C s ðk [ fjgþ Cf ðkþ 0 8j R K. Thus, we can conclude that f condtons K Cs ðkþ C f ðk n fgþþ < 0 and K[fjg C s ðk [ fjgþ Cf ðkþ 0 8j R K are satsfed, the nternal and external stablty condtons hold and the agreement s self-enforcng. h Each natonal dfference, C s ðkþ Cf ðk n fgþ, establshes the maxmum payment that one country s wllng to pay for cooperaton f the dfference s negatve, or the mnmum payment that has to be receved to sell cooperaton f the dfference s postve. If the agreement s self-enforcng, then the sum of the maxmum payments that some countres are wllng to pay must be larger than the sum of the mnmum payments that the other countres demand for sellng cooperaton. Hence n ths case, there must be at least one transfer scheme that makes the agreement stable. In fact, there wll be a class of transfer schemes that makes the agreement self-enforcng. Ths class was called the Almost Ideal or Optmal Transfer Schemes by Eyckmans and Fnus (004) and Carraro et al. (006). 3. The Nash equlbrum of the emsson game In ths secton, we develop a model wth two types of countres: type 1 and type so that N ¼ N 1 þ N. Suppose that, as the outcome of the frst-stage game, an agreement K ¼ fn 1 ; n g of sze n ¼ n 1 þ n s formed. Then a representatve non-sgnatory solves mn C f fx f g ¼ 1 c d x f þ m X; ¼ 1; ; where X ¼ n x s þ ðn n Þx f. The F.O.C. yelds x f ¼ d ðm = c Þ; ¼ 1;. On the other hand, sgnatores are assumed to coordnate n order to mnmze ther aggregate costs takng the emssons of nonsgnatores as gven. mn Cðn 1 ; n Þ ¼ X n C s fx s g ¼ X 1 n c d x s þ m X : The F.O.C. yeld x s ¼ d m n c ; ¼ 1; : ð1þ Notce that sgnatory emssons decrease wth the sze of the agreement. Aggregate emssons are 0 Xðn 1 ; n Þ ¼ X n c 1 m n C A þ X ðn n Þ d m ; c whch after developng the addtons yelds the followng expresson Xðn 1 ; n Þ ¼ X N d m X c n c ðm ðn 1Þ þ m j n j Þ; j ¼ 1; ; j: ðþ Ths expresson decreases wth respect to the number of sgnatores of both types: 1 ; n Þ ¼ 1 ðm ðn 1Þ þ m j n j Þ þ n j m < c c j ; j ¼ 1; ; j; n 1: Fnally, we obtan the cost functons of sgnatores and nonsgnatores. C s ðn 1; n Þ ¼ 1 c X m n! þ m Xðn 1 ; n Þ; ¼ 1; ð3þ C f ðn 1; n Þ ¼ m c þ m Xðn 1 ; n Þ; ¼ 1; : ð4þ These expressons show that both non-sgnatores and sgnatory cost functons depend on the sze and composton of the agreement. Moreover, t s easy to check from (4) that non-sgnatory costs decrease wth the sze of the agreement as aggregate emssons decrease wth the number of sgnatores. Ths means that n ths game there are postve spllovers as an ncrease n the number of sgnatores reduces non-sgnatory costs. Consequently, we can conclude that f the nternal stablty condton s satsfed for a country, the proftablty condton s also satsfed. 11 Moreover, t s easy to check that for a gven number of sgnatores, non-sgnatory costs are lower than for sgnatores of the same type C s ðn 1; n Þ C f ðn 1; n Þ ¼ 1 c n 1 m þ n n j m m j þ n j m j > 0; for n 1; ¼ j ¼ 1; ; j. Notce that ths dfference ncreases wth respect to the number of sgnatores of both types. Fnally, we obtan that the gans to cooperaton are gven by: X ðn n Þ C f ð0; 0Þ Cf ðn 1; n Þ þ X n ðn n 1Þðn 1Þm þ ððn n Þn j c þn j ðn 1ÞÞm m j þ ðn j n j Þn j m j ¼ X n C f ð0; 0Þ Cs ðn 1; n Þ wth j ¼ 1; and j. In ths expresson C f ð0; 0Þ stands for the cost of a country of type at the fully non-cooperatve equlbrum, C f ðn 1; n Þ for the cost of a non-sgnatory of type and C s ðn 1; n Þ 10 It s also easy to show that ths s the case when the frst sgnatory of type enters the agreement. 11 Remember that the nternal stablty condton requres that for K ¼ fn 1 ; n g; C s ðkþ6cf ðk n fgþ or Cs ðkþ þ Ts ðkþ6cf ðk n fgþ, then as Cf ðk n fgþ < Cf ðfgþ where C f ðfgþ stands for the costs correspondng to the sngleton coalton structure or the fully non-cooperatve equlbrum, we obtan that C s ðkþ < Cf ðfgþ or C s ðkþ þ Ts ðkþ < Cf ðfgþ so that the proftablty condton holds. Remember that the proftablty condton requres that sgnatores costs must be lower or at least equal to the fully non-cooperatve equlbrum costs. ð5þ

6 60 C. Fuentes-Albero, S.J. Rubo / European Journal of Operatonal Research 0 (010) for the cost of a sgnatory of type n both cases when the agreement conssts of n 1 countres of type 1 and n countres of type. It s mmedately observed that for n 1 (5) ncreases wth respect to margnal envronmental damages. Moreover, calculatng the frst dervatves of ths expresson wth respect to n 1 and n t can be concluded that the gans from cooperaton also ncrease wth the sze of the agreement. However, an ncrease n margnal abatement costs has a negatve mpact on the gans from cooperaton. 4. The Nash equlbrum of the membershp game wth heterogenety n abatement costs In ths secton we assume that m 1 ¼ m ¼ m; d 1 ¼ d ¼ d; and c 1 > c. Bearng these assumptons n mnd, the costs functons for sgnatores and non-sgnatores accordng to ther type can be calculated by means of the expressons n Secton 3. Usng these cost functons, we can conclude that the nternal stablty condtons wll be satsfed when the followng expressons are negatve or zero. C s 1 ðn 1; n Þ C f 1 ðn 1 1; n Þ ¼ m n n n 1 þ 3 n ; c 1 c C s ðn 1; n Þ C f ðn 1; n 1Þ ¼ m n n n þ 3 n 1 : c c 1 Multplyng and dvdng by c 1, these expressons can be wrtten as follows C s 1 ðn 1; n Þ C f 1 ðn 1 1; n Þ ¼ m c 1 ðn n n 1 þ 3 n cþ; C s ðn 1; n Þ C f ðn 1; n 1Þ ¼ m c 1 ðcðn n n þ 3Þ n 1 Þ; where c s the rato c 1 =c. The same occurs for the external stablty condtons. Thus, as both n n n 1 þ 3 and n n n þ 3 are postve when the agreement conssts of countres of both types, we have that for a gven agreement consstng of countres of both types, the sgn of these expressons only depends on the rato c. Ths means that changes n the absolute values of parameters c 1 and c that do not change ther relatve value wll not have any effect on the stablty or unstablty of an agreement so that t can be establshed that Lemma. The stablty for a gven agreement wthout transfers consstng of countres of both types depends only on the value of rato c. Usng these condtons, stablty has been analyzed for two dfferent settngs: wth and wthout transfers Stable IEA wthout transfers As shown n Secton 3, f nternal stablty condtons hold, proftablty condtons also hold. Consequently, n ths secton we develop the stablty analyss usng only the nternal and external stablty condtons. The result of ths analyss s summarzed n the followng proposton: 1 roposton 1. The maxmum level of cooperaton that can be reached through a self-enforcng IEA wthout transfers conssts of three countres of the same type regardless of the heterogenety n abatement costs. The logc behnd ths result s that the nternal stablty condton for type 1 countres s satsfed when the dfferences n margnal abatement costs are large enough, whereas the contrary s requred n order to satsfy the nternal stablty condton for type countres. The concluson s that there are no values for c that 1 All the proofs correspondng to the dfferent propostons of the paper can be found as supplementary materals at the EJOR webste. ð6þ ð7þ satsfy the nternal stablty condton for an agreement wth both types of countres. Ths result concdes wth that obtaned by Carraro and Snscalco (1991) for the symmetrc model wth orthogonal reacton functons. 4.. Stable IEA wth transfers Next, we explore the scope of cooperaton when countres can use transfers. Therefore, the am of ths secton s to fnd the values for n 1 and n that satsfy the condtons of Lemma 1. Usng Lemma 1 the followng result s obtaned: roposton. Only an IEA consstng of two sgnatores of dfferent types s potentally self-enforcng wth transfers regardless of the heterogenety n abatement costs. If the countres n the agreement are of the same type, the free rder ncentve s the same for all sgnatores and cooperaton cannot be expanded usng transfers. In ths case, roposton 1 apples. If the countres n the agreement are dfferent, type 1 countres can buy the cooperaton of type countres accordng to Lemma 1, provded that the costs of the agreement are lower than the aggregate costs the sgnatores would receve f they qut the agreement and act as free-rders. Usng (6) and (7) ths dfference yelds the followng expresson: n 1 C s 1 ðn 1; n Þ C f 1 ðn 1 1; n Þ þ n C s ðn 1; n Þ C f ðn 1; n 1Þ ¼ m ðn 4n þ 3Þðn 1 þ cn Þ ¼ m c 1 ðn 4n þ 3Þ n 1 þ n ð8þ c 1 c Thus, we fnd that when countres only dffer n abatement costs, the possblty of buyng cooperaton depends only on the sze of the agreement: n. In other words, although c determnes, accordng to Lemma, the sgn of the stablty condtons wthout transfers, t does not affect the sgn of condton (8). Moreover, f the sgn s negatve, the greater c 1 or the greater c, the less the resources avalable to buy the cooperaton. Obvously, ths s related to the fact that the gans to an agreement decrease wth margnal abatement costs as establshed n Secton The Nash equlbrum of the membershp game wth heterogenety n envronmental damages In ths secton we assume that d 1 ¼ d ¼ d; c 1 ¼ c ¼ c; and m 1 > m. Takng these assumptons nto account, the cost functons for sgnatores and non-sgnatores accordng to ther type can be calculated by means of the expressons n Secton 3. Usng these cost functons, we can conclude that the nternal stablty condtons wll hold when the followng expressons are negatve or zero. C s 1 ðn 1; n Þ C f 1 ðn 1 1; n Þ ¼ 1 n 1 c 4n 1 n þ 3 m 1 þðn 1 1Þn m 1 m þ n m ; C s ðn 1; n Þ C f ðn 1; n 1Þ ¼ 1 n c 4n n 1 þ 3 m þðn 1Þn 1 m 1 m þ n 1 1 m : Dvdng and multplyng by m, these expressons can be wrtten as follows C s 1 ðn 1; n Þ C f 1 ðn 1 1; n Þ ¼ m n 1 c 4n 1 n þ 3 m þðn 1 1Þn m þ n ; ð9þ C s ðn 1; n Þ C f ðn 1; n 1Þ ¼ m c n 4n n 1 þ 3 þðn 1Þn 1 m þ n 1 m ; ð10þ

7 C. Fuentes-Albero, S.J. Rubo / European Journal of Operatonal Research 0 (010) where m s the rato m 1 =m. The same occurs for the external stablty condtons. Thus the same knd of result that that obtaned n Secton 4, see Lemma, apples also when the countres dffer n envronmental damages: gven an agreement, the sgn of the stablty condtons only depend on the relatve value of margnal envronmental damages. Next, usng these condtons, stablty has been analyzed n two dfferent settngs as n the prevous secton Stable IEA wthout transfers Applyng the stablty condtons of Defnton 1 we obtan the followng result: roposton 3. The maxmum level of cooperaton that can be reached through a self-enforcng IEA wthout transfers conssts of three countres of the same type regardless of the heterogenety n envronmental damages. An agreement consstng of one type 1 country and one type country can also be self-enforcng f the dfferences n margnal envronmental damages are not very large. If the agreement conssts of countres of the same type, there are once agan no dfferences wth respect to the symmetrc case. However, an agreement consstng of one country of each type can be stable f the dfference n envronmental damages n relatve terms s no greater than 40%. The reason for ths result s that f the dfferences n envronmental damages are not very large, cooperaton between two countres of dfferent types can be proftable for both, as the reducton n emssons that both have to bear wll not be very large ether. In ths case, the proftablty of the agreement mples ts nternal stablty, as the alternatve to beng part of a blateral agreement s bearng the costs correspondng to the fully non-cooperatve equlbrum whch n ths case wll be greater for both types of countres. However, wth a second type country n the agreement, the alternatve for the type countres to beng part of the agreement s to be a free rder of an agreement wth two sgnatores whch wth postve externaltes yelds lower costs makng the agreement nternally unstable. Thus, an agreement consstng of one country of each type s externally stable and gven that t s also nternally stable, we can conclude that t s self-enforcng. Hoel (199) also obtans that only two countres cooperate n equlbrum n a model where all the countres have dfferent margnal envronmental damages but dentcal beneft functons. However, n hs paper sgnatores agree upon the emsson level whch s the medan value of ther most preferred emsson levels. 5.. Stable IEA wth transfers We have just concluded that our canddates for beng stable agreements wthout transfers are coaltons of three or two countres. In ths secton, we analyze f larger coaltons can be stable when transfers between countres are taken nto account. The results of our analyss can be summarzed as follows: roposton 4. The level of cooperaton that can be bought wth transfers ncreases wth the heterogenety n envronmental damages provded that cooperaton s only bought by one or two countres of type 1. As n the case studed n the prevous secton, the possblty of type 1 countres buyng the cooperaton of type countres depends on the sgn of the followng dfference: C s 1 ðn 1; n Þ C f 1 ðn 1 1; n Þ þ n ðc s ðn 1; n Þ C f ðn 1; n 1ÞÞ n 1 c n 1ðn 1 ðn Þ n þ 3Þm þ n 1 ðn n 1 Þðn Þm þðn n 1 Þðn 4n þ 3 n 1 ðn ÞÞ : ¼ m ð11þ Table 1 Self-enforcng IEA for dfferent degrees of heterogenety. n Agreements m Agreements m 3 {1,} (1,4.4] {,1} (1,7.87] 4 {1,3} (4.4,6.74] {,} (7.87,18.46] 5 {1,4} (6.74,9.0] {,3} (18.16,3.43] 6 {1,5} (9.0,11.63] {,4} (3.43,50.69] 7 {1,6} (11.63,14.06] {,5} (50.69,7.94] 8 {1,7} (14.06,16.49] {,6} (7.94,99.0] 9 {1,8} (16.49,18.91] {,7} (99.0,19.45] 10 {1,9} (18.91,1.33] {,8} (19.45,163.70] Comparng ths expresson wth that obtaned n the prevous case, see expresson (8) n Secton 4, one realzes that now the sgn of ths dfference depends on the sze of the agreement as n the prevous case but also on the composton of the agreement and on the degree of asymmetry between both types of countres. Observe that the sze and composton of the agreement determne the coeffcents of a polynomal of second degree for m. Moreover, t s easy to check that the coeffcent of m s negatve and that the coeffcent of m s postve when n the agreement there s only one or two countres of type 1. In that case, there wll exst a lower bound for m above whch (11) s negatve and the cooperaton can be bought through transfers. Thus, for any agreement wth a maxmum of two sgnatores of type 1, t s possble to fnd an nterval of values for m for whch the agreement s stable. Fnally, ths expresson also defnes a postve relatonshp between the level of partcpaton n the agreement and the lower bound for m so that the hgher the heterogenety n envronmental damages, the hgher the level of cooperaton that can be bought usng transfers. 13 Table 1 presents the relatonshp between the sze and composton of the agreements and the values of m that allow to stablze the agreement through transfers. We used expresson (11) to obtan these values. The frst column represents the sze of the agreements and the second and fourth ther composton where the frst fgure stands for the number of type 1 sgnatores and the second for the number of type sgnatores. In the thrd column, the nterval of values for m that supports the correspondng level of cooperaton when the agreement conssts of only one type 1 sgnatory and n 1 type sgnatores s represented, and the ffth dsplays the nterval of values for m that supports the correspondng level of cooperaton when the agreement conssts of two type 1 sgnatores and n type sgnatores. For each nterval, the frst fgure s a lower bound that must be strctly satsfed n order for the type 1 country to have enough resources to buy the cooperaton of type countres. For nstance, one type 1 country wll only have enough resources to buy the cooperaton of two type countres f m s hgher than 1, and wll only be able to buy the cooperaton of three type countres f m s hgher than 4.4, and so on. 14 The same can be sad when there are two type 1 countres n the agreement. Ths means that, gven a sze and a composton of the agreement, ths wll be nternally and externally stable only f m belongs to the correspondng nterval of Table 1. For nstance, f m s equal to 5, one type 1 country wll have enough resources to buy the cooperaton of two type countres and also the cooperaton of three. Then, the nternal stablty condton wth transfers of Defnton wll be satsfed for n ¼ 3 and also for n ¼ 4, but ths wll not be the case for the external stablty condton. The agreement wth 13 Ths s a sketch of the proof of roposton 4. The complete proof can be found as supplementary materal at the EJOR webste. The supplementary materals also contans a proof of roposton 4 for the case of three types of countres and a numercal example. 14 Notce that for n 1 ¼ 1 and n ¼ 3, (11) yelds 3m ðm 4m 1Þ=c that s zero for m ¼ 4:4 and negatve for m > 4:4:

8 6 C. Fuentes-Albero, S.J. Rubo / European Journal of Operatonal Research 0 (010) three sgnatores s not externally stable because the type 1 country can buy the cooperaton of another type country. Thus, the second part of Defnton does not hold. Consequently, an agreement consstng of one type 1 country and two type countres s only externally and nternally stable when m s greater than 1, but equal or lower than 4.4. The same argument apples for the other levels of partcpaton and for an agreement that ncludes two type 1 countres. Table 1 shows that cooperaton ncreases n both cases wth the heterogenety n envronmental damages, as establshed n roposton 4. However, the grand coalton cannot be self-enforcng unless there are only one or two type 1 countres. Fnally, we would lke to pont out that the reason why only one or two type 1 countres can belong to a self-enforcng agreement s very smlar to the reason that explans the second part of roposton 3: an agreement consstng of two countres s always nternally stable f t s proftable. Next we study the relatonshp between the degree of asymmetry, partcpaton and the gans to cooperaton. Frst, usng (5) we calculate the gans to full cooperaton obtanng the followng expresson X N C f ð0; 0Þ Cs ðn 1; N Þ ¼ 1 c N 1m 1 þ N m þ ðn ÞðN 1m 1 þ N m Þ ; that multplyng and dvdng by m can be wrtten as TCð0; 0Þ TCðN 1 ; N Þ ¼ m c ðn 1m þ N þ ðn ÞðN 1 m þ N Þ Þ; ð1þ where TCð0; 0Þ ¼ N C f ð0; 0Þ and TCðN 1; N Þ ¼ N C f ðn 1; N Þ. Ths expresson says us that gven m the gans to cooperaton depend on the absolute value of parameter m. Thus, a change n n m, can ncrease, decrease or keep constant the gans to cooperaton dependng on the change n the absolute value of parameter m we consder. Now, usng also (5) we calculate the gans to a gven agreement ðn 1 ; n Þ that are gven by X ðn n Þ C f ð0; 0Þ Cf ðn 1; n Þ þ X ¼ 1 c n 1m 1 þ n m nðn 1m 1 þ n m Þ þðn 1ÞðN 1 m 1 þ N m Þðn 1 m 1 þ n m ÞÞ n C f ð0; 0Þ Cs ðn 1; n Þ that multplyng and dvdng by m can be wrtten as TCð0; 0Þ TCðn 1 ; n Þ ¼ m n 1 m þ n nðn 1 m þ n Þ c þðn 1ÞðN 1 m þ N Þðn 1 m þ n ÞÞ; ð13þ where TCðn 1 ; n Þ ¼ ðn n ÞC f ðn 1; n Þ þ n C s ðn 1; n Þ, so that the relatve gans to cooperaton for a gven agreement depends only on the heterogenety of countres: TCð0; 0Þ TCðn 1 ; n Þ TCð0; 0Þ TCðN 1 ; N Þ ¼ n 1m þ n nðn 1 m þ n Þ þ ðn 1ÞðN 1 m þ N Þðn 1 m þ n Þ N 1 m þ N þ ðn ÞðN 1 m þ N Þ : ð14þ Thus a change of m keepng constant m wll affect the gans to full cooperaton but, accordng to roposton 4, wll not have any effect on the level of cooperaton but besdes, accordng to (14), wll not have any effect on the relatve gans of cooperaton ether so that we can conclude that Corollary 1. Any varaton n the absolute gans to full cooperaton that t s not assocated wth a change n heterogenety nether alter the level of cooperaton nor the relatve gans acheved by the agreement. In fact, a reducton n the gans to full cooperaton could yeld an ncrease n the level of cooperaton provded that ths reducton s accompaned wth an ncrease n heterogenety. Summarzng, our analyss establshes that heterogenety s the determnant of the level of cooperaton ndependently of whch are the gans to full cooperaton assocated wth the degree of heterogenety of countres. Thus, an ncrease n heterogenety wll have the same effect on the level of cooperaton both f the varaton n heterogenety comes wth an ncrease n the gans to full cooperaton and f t has no effect on such gans. Fnally, we nvestgate whch are the effects of a change n the absolute value of parameter m that does not change the relatve value m. From (9) (11) s mmedate that when m rases for a constant m, the ncentves to jon or ext from the agreement are gong to ncrease as well as the resources avalable to buy the cooperaton. Thus, the greater the mean of the envronmental damages for a gven m, the greater the necessary mnmum transfers to stablze the agreement. Nevertheless, type 1 countres wll have more resources to buy the cooperaton as well. Moreover, accordng to (1), when m rases, the gans to full cooperaton wll also rase yeldng an ncrease n the absolute gans for each level of cooperaton. Notce that the relatve gans, accordng to Corollary 1, wll not change so that an ncrease n the gans to full cooperaton wll mply an ncrease n the absolute gans acheved by the agreement. Next we present two numercal examples that llustrate these results. In the frst example we have consdered changes n heterogenety that do not modfy the gans to full cooperaton. Ths example s shown n Table. In order to buld the table, we have selected some values for m belongng to the dfferent ntervals that appear n Table 1 and for each one of these values, that are n the frst column of Table, we have calculated usng (1) the correspondng value of m that yelds the gans to full cooperaton of the symmetrc case. For these calculatons we have gven the followng values to the parameters: N 1 ¼ 3; N ¼ 9 and c ¼ 0:5, and m 1 ¼ m ¼ 1 for the symmetrc case. Thus, for all the combnatons ðm; m Þ n the table, the gans to full cooperaton are 145, the gans of the symmetrc case. Then we have calculated the gans from cooperaton assumng that only one type 1 country buys the cooperaton of type countres. The absolute gans to the agreement (AG), calculated accordng to (13), appear n column four and the relatve gans (RG), calculated accordng to (14) n column fve. Moreover, n columns sx and seven, we show the fgures correspondng to (9) and (10). Fnally, the sum of the nternal stablty condtons for the sgnatores gven by (11) appears n the last column. The table shows that both the partcpaton and the gans from cooperaton ncrease wth the heterogenety of countres. Moreover, t also shows that the maxmum payment that a type 1 country s wllng to pay for cooperaton ncreases wth respect to m and that the surplus that can be dstrbuted among the dfferent sgnatores also ncreases although at a dfferent rate because the neces- Table ure changes n heterogenety. N 1 ¼ 3; N ¼ 9; c ¼ 0:5. m m Agreements AG RG ð%þ IEC 1 IEC Surplus {1,} {1,3} {1,4} {1,5} {1,6} {1,7} {1,8} {1,9}