Coalition Formation in a Global Warming Game: How the Design of Protocols Affects the Success of Environmental TreatyMaking

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1 Coalton Formaton n a Global Warmng Game: How the Desgn of Protocols Affects the uccess of Envronmental TreatyMang Frst draft: March, 23 Ths verson: November, 23 Johan Eycmans K.U.Leuven, Centrum voor Economsche tudën and EHAL Europese Hogeschool Brussel Mchael Fnus Department of Economcs, Unversty of Hagen, Proflstr. 8, 5884 Hagen, Germany Abstract We combne the newest concepts of noncooperatve coalton theory wth a computable general equlbrum model close to the semnal RICEmodel of Nordhaus and Yang (1996) to determne stable coalton structures n a global warmng game. We consder three coalton games that allow for the formaton of multple coaltons. The coalton games represent dfferent desgns of clmate treaty protocols. Counterntutvely, t turns out that treates based on a unanmous decson rule and exclusve membershp lead to superor outcomes than treates wth open membershp. We also demonstrate that f coalton formaton s not restrcted to a sngle coalton, as ths has been done prevously n the lterature, coalton structures wth multple coaltons wll emerge n equlbrum. Most of the regonal agreements are superor to sngle agreements. Moreover, our fndngs confrm those derved from smpler theoretcal models that a cleverly desgned transfer scheme can foster cooperaton and that from the number of partcpants the success of a treaty cannot be nferred. They also support a conjecture of theory that n the case of greenhouse gases stable coalton structures (partal cooperaton) can close the gap between the global optmum (full cooperaton) and the Nash equlbrum (no cooperaton) by a substantal amount. JELClasssfcaton: C68, C72, H41, Q25 Keywords: desgn of clmate treaty protocol, coalton formaton, noncooperatve game theory We would le to than Therry Bréchet (UcLCORE) for valuable comments on an earler draft of ths paper. Johan Eycmans s vstng research fellow at UcLCORE and acnowledges fnacal support by the Fund for centfc Research Flanders (Belgum) FWOVlaanderen and by the CLIMNEG 2 project funded by the Belgan Federal cence Polcy Offce.
2 1 1. Introducton In the lterature on the economcs of global warmng two strands of the lterature can be dstngushed. The emprcal strand comprses costbeneft analyses (CBA) and costeffectveness analyses (CEA). Important research topcs nclude methodologcal and conceptual ssues on estmatng abatement costs (e.g., Weyant 1999 or IPCC 21) and damage costs (e.g., Tol 22a, b), the determnaton of optmal global abatement levels and effcent allocaton of abatement burdens (e.g., d Arge et al and Nordhaus 1991) as well as dstrbutonal mplcatons of varous abatement polces (e.g., Chchlnsy and Heal 1994 and Eycmans et al. 1993). In recent years computatonal general equlbrum models (CGEmodels) have been developed that tae many dynamc aspects of clmate change as well as the relaton and feedbac between the clmate and the economc system nto account (see, e.g., Nordhaus and Yang 1996 for the semnal RICE model). The mportance of ths lterature les n dentfyng globally optmal and costeffcent abatement polces, provdng nformaton on expected mpacts of varous clmate polces on the economy and the ecologcal system, as well as dentfyng the ganers and losers n a cooperatve "global warmng game". However, n terms of stablty of clmate treates ths lterature has lttle to say. Though t s certanly true that a necessary condton for a country to accede to a treaty s that t benefts from partcpaton, t s by no means a suffcent condton. Even f countres would gan from sgnng a treaty they may nevertheless decde to pursue a noncooperatve polcy. Due to the publc good nature of the global warmng problem, freerdng s attractve snce the abatement contrbuton of a sngle country has only a margnal mpact on reducng damage costs from greenhouse gas concentraton but has a substantal mpact on ts abatement costs. In contrast, the theoretcal strand of the lterature comprses game theoretcal analyses on the formaton and stablty of nternatonal envronmental agreements (IEAs). 1 The mportance of ths lterature les n pontng out the dffcultes n desgnng selfenforcng agreements, tang nto account that cooperaton requres a hgh amount of consensus because of voluntary partcpaton of countres and a successful agreement must neutralze varous freerder ncentves. Repeated game models have stressed the mportance of credble threats of punshment to enforce complance n IEAs (e.g., Barrett 1994 and 2, Fnus/Rundshagen 1998, Fnus/Tjotta 22 and tähler 1996). One man fndng s that due to the lac of credble punshments complance can only be ensured for very moderate abatement targets and there mght be a tradeoff between effcency, effectveness and stablty. A conceptual drawbac of these models s that the coalton formaton process receves lttle attenton and 1 An extensve overvew s provded n Fnus (21 and 23).
3 2 that stoc pollutants cannot be analyzed. In contrast, coalton models analyze partcpaton n IEAs applyng cooperatve and noncooperatve game theoretcal stablty concepts. Cooperatve game theory, whch s manly normatvely orented, focused on transfer schemes that ensure stablty of the (effcent) grand coalton by applyng the stablty concept of the core (e.g., Chander and Tulens 1995 and 1997 and German et al. 23). A central result s that there exsts a transfer scheme, whch allocates the gans from cooperaton n proporton to the margnal benefts of reduced envronmental damages, ensurng stablty of an effcent envronmental polcy (see secton 3). A conceptual drawbac of the cooperatve approach s that t only tests stablty of the grand coalton, analyss stablty n terms of the aggregate payoff to coaltons and rests on very strong assumptons about the mplct punshment after freerdng of a group of countres. However, the amount of emprcal studes s relatvely large and most rely on a sound emprcal module (e.g., Eycmans and Tulens 23, German et al and Katala et al. 1995). Noncooperatve game theory, whch s manly postvely orented, tred to ratonalze the problems of formng large and effectve coaltons by applyng the concept of nternal and external stablty (e.g., Barrett 1994, Carraro/nscalco 1993, Hoel 1992 and Hoel/chneder 1997). A ey result s that whenever cooperaton (socal optmum) would generate large global welfare gans compared to a noncooperatve stuaton (Nash equlbrum), stable coaltons acheve only lttle and vce versa. Moreover, t s ponted out that from partcpaton one cannot conclude success of cooperaton: small coaltons may acheve more than large coaltons n terms of global welfare and emsson reducton. A conceptual drawbac of nternal&external stablty s that t exogenously restrcts coalton formaton to one (nontrval) coalton. Moreover, most models rely on very specfc assumptons to derve ther results. 2 Fnally, there are only few emprcal studes but they compromse on the emprcal part for tractablty (e.g., Botteon/Carraro 1997 and 1998 and Tol 21). Recently, there has been a development of new approaches and concepts n noncooperatve game theory that has been termed "new coalton theory" (Fnus 22). 3 Compared to the cooperatve approach t has two advantages. Frst, t seems natural to assume that, even f n the case of transfers, ratonal agents wll base ther decson of partcpaton n an IEA on 2 3 Typcal assumptons nclude for nstance a statc payoff structure and dentcal welfare functons (symmetrc players). An excepton s Rubo/Ulph (21) who consder a dynamc game but symmetrc players and Barrett (1997) who consders heterogeneous players but uses smulatons to derve results. For an excellent overvew see Bloch (1997) and Y (1997).
4 3 ndvdual payoffs and not on the aggregated payoff to ther coalton. econd, t explctly taes nto account spllovers between coaltons through a valuaton functon (see secton 2). nce spllovers are an mportant source of freerdng, neffcent agreements can be better ratonalzed. Compared to the noncooperatve concept of nternal & external stablty these new developments have the advantage that they do not restrct coalton formaton to a sngle nontrval coalton exante but allow for the coexstence of multple coaltons. Fnally, compared to the "classcal" cooperatve and noncooperatve approaches, new coalton theory draws a conceptual lne between the rules of coalton formaton and stablty. nce the rules of coalton formaton can be nterpreted as the nsttutonal settng n whch treaty formaton taes place (Carraro/Marchor/Oreffce 21 and Eccha/Marott 1998), polcy conclusons about the optmal desgn of protocols are possble. Moreover, the reacton of countres after a devaton do not follow from ad hoc assumptons but follow from the rules of coalton formaton and can therefore be related to the ratonal behavor of agents. The phlosophy of ths paper s to combne a fully fledged emprcal model wth aspects of new coalton theory as ths has been done recently by Eycmans (21) n order to analyze stable clmate change coaltons. The purpose s to relate the desgn of protocols to the success coalton formaton. We proceed as follows. In secton 2 we lay out the game theoretcal settng and n secton 3 we descrbe the emprcal module of our model. Equlbrum coalton structures are reported and evaluated n secton 4. ecton 5 summarzes our man fndngs and concludes wth some fnal remars. 2. The Game Theoretcal Model 2.1 Introducton Coalton formaton s modeled as a twostage game. In the frst stage players (.e., countres or regons) decde on ther membershp n a coalton, n the second stage coalton members choose ther economc strateges. The decson n the frst stage depends on the rules of coalton formaton that follow from the defnton of a coalton game. The defnton comprses three elements: 1) the set of players I={1,..., N} wth a partcular player denoted by ndex or j, 2) the set of coalton (or, alternatvely, membershp or announcement) strateges Σ=Σ 1 Σ 2 ΣN wth a partcular strategy of player I denoted by σ Σ, and 3) a coalton functon ψ that maps membershp strateges σ= ( σ1,..., σ N) nto coalton structures, ψ : Σ C: σ ψ( σ). A coalton structure where a partcular coalton s denoted by c, {1,..., M), c C where C s the set of all possble coalton structures. 1 M c = (c,..., c ) s a partton of players c l c = l, c = I and
5 4 The decson n the second stage depends on the rules how players choose ther economc strateges that follow from the defnton of the valuaton functon. The valuaton functon v(c) maps coalton structures nto a vector of ndvdual payoffs (.e., welfare) va an nstructon how players choose ther economc strateges s, = 1 2 N, for a gven coalton structure c. Hence, the valuaton functon s a composton of two functons v= w ε where ε: C : c ε(c) s a functon mappng coalton structures nto economc N strateges and w: R :s w(s) s a functon mappng economc strateges nto welfare levels. For the frst stage we consder three dfferent coalton games, representng three dfferent nsttutonal rules how coaltons form. For the second stage we consder only one rule that assgns a unque vector of ndvdual payoffs for each possble coalton structure. Ths mples that players choosng a membershp strategy n the frst stage now for each coalton game the mplcatons n the second stage. Hence, games can be solved by bacwards nducton. Consequently, we start n the followng by descrbng frst stage 2 and subsequently we move on to explan stage 1 of the coalton formaton game. 2.2 econd tage of the Coalton Formaton Game Let the welfare functon of player n the global warmng game be gven by w(s) where s s a vector of economc strateges. That s, welfare of player depends on hs own strategy s and those of other players, s. The exact relatonshp between economc strateges (e.g. emsson abatement and captal nvestment) and welfare as well as the elements n players welfare functon wll be outlned n secton 3. For the valuaton functon  mappng coalton structures nto payoffs  we mae the followng standard assumptons (Bloch 1997): Defnton 1: Valuaton Functon 1 M Fx a coalton structure c = ( c,...,c ), let v( c ) = { v(c)} I ={ w( ε (c))} I ={ w(s )} I and assume for nstructon ε that all players belongng to a coalton c c jontly maxmze the aggregate payoff to ther coalton. Let s denote the economc strategy vector of coalton c and s the economc strategy vector of all other coaltons c l c, l, and assume that equlbrum economc strategy vector * s = ε (c) for coalton structure c satsfes: c C, s : c * w(s ) c * w ( s, s ) where * s s a unque nteror equlbrum. Defnton 1 mples that the valuaton of player, v (c), s dentfed by the entre coalton structure c and not only by the coalton to whch he belongs. Players behave cooperatvely
6 5 wthn ther coalton but noncooperatvely aganst players belongng to other coaltons when choosng ther economc strateges. Put dfferently, members of a coalton act as one sngle player maxmzng the aggregate welfare to ther coalton and coaltons play a Nash equlbrum strategy n terms of economc strateges. Hence, economc strateges wthn a coalton are effcently chosen. Consequently, the sngleton coalton structure (grand coalton) mples an equlbrum economc strategy vector correspondng to the "classcal" Nash equlbrum (socal optmum). 4 Thus, the hghest global welfare wll be obtaned n the grand coalton, the lowest n the sngle coalton structure and any welfare level n between n any other coalton structure. For the propertes of the welfare functon that we defne n secton 3 t turns out that s * s unque and les well wthn the boundares of the economc strategy space. Defnton 1 mplctly assumes no transfers. Ths s one opton that we consder n our smulatons below, whch we call the "no transfer case". Alternatvely, we consder a second opton to whch we refer to as the "transfer case". Of course, there are many possbltes how the gans of cooperaton could be dstrbuted among coalton members as for nstance the hapley value (Barrett 1997, Botteon and Carraro 1997) or the Chander/Tulens transfer scheme (Chander and Tulens 1997). We restrct ourselves to a modfed verson of the last transfer scheme of whch the detals wll be lad out n secton 3. At ths stage t suffces to pont out that ths transfer scheme assumes a partcular rule how the surplus from cooperaton wthn a coalton s dstrbuted. Hence, ˆv (c) = v(c) + t where t > mples that a player receves a transfer and t < that a player pays a transfer. 2.3 Frst tage of the Coalton Formaton Games In ths subsecton we defne and dscuss three coalton games that mply dfferent rules how coaltons can form. All games assume that countres smultaneously announce ther coalton strategy and allow for the coexstence of several coaltons. That s, coalton formaton s not exante restrcted to a sngle coalton as ths s assumed for the concepts of nternal&external stablty and the core that have been wdely appled n the lterature on IEAs. However, games dffer n the strategy set and most mportantly n the coalton functon that maps coalton strateges nto coalton structures. 4 The grand coalton s that coalton structure where all players are n one coalton and wll be denoted by c. The sngleton coalton structure mples that every player forms a (trval) coalton by hmself and s denoted by c N.
7 6 The frst game s called open membershp game (OMG) and s due to Y/hn (1995). In ths game players can freely form coaltons as long as no outsder s excluded from jonng a coalton. Players choose ther membershp by announcng an address,.e., a number between 1 and N. Players that have announced the same address form a coalton. For nstance, suppose N=6 and σ 1 =σ 2 =σ 3 = 1, σ 4 = 2, σ 5 =σ 6 = 3, then c={{1, 2, 3}, {4}, {5, 6}} forms. If player 3 were to announce σ 3 = 2 nstead, then c ={{1, 2}, {3, 4}, {5, 6}} would form. More formally, we have: 5 Defnton 2: Open Membershp Game (OMG) a) The set of coalton strateges of player I s gven by Σ = {1,..., N } where a partcular strategy σ s an announcement of an address. OMG b) Coalton functon ψ maps strategy vector σ nto coalton structure c as follows: c = {} {j σ = σ. j } Thus n the OMG a player can jon any coalton he wants. Ths strong assumpton, however, seems not entrely n lne wth the noton of voluntary partcpaton n all IEAs. For nstance, the U and developng countres cannot be forced nto the Kyoto Protocol. Hence, t seems natural to consder an extenson of the OMG where players have only unrestrcted open access to nontrval coaltons but requre the consent of a sngleton player f they ntend to jon hm. Ths extenson s called a restrcted open membershp game (ROMG) and has been proposed by Bloch (1997) and formalzed by Rundshagen (22). Conceptually, only a slght modfcaton of Defnton 2 s requred, addng to the strategy set an address and specfyng the coalton functon such that players announcng reman sngletons. Defnton 3: Restrcted Open Membershp Game (ROMG) a) The set of coalton strateges set of player s gven by Σ = {,1,...,N } where a partcular strategy σ s an announcement of an address. and c the coalton to whch m m fnally belongs, =1,.., N, then c = {} {j σ = σ j }. ROMG b) Coalton functon ψ maps strategy vector σ nto coalton structure c as follows: c = {} {j σ = σ j }. For nstance, recall our prevous example that assumed σ 1 =σ 2 =σ 3 = 1, σ 4 = 2, σ 5 =σ 6 = 3, so that c={{1, 2, 3}, {4}, {5, 6}} forms and where we argued that f player 3 5 The open membershp rule of ths game s smlar as mpled by nternal&external stablty, except that n the OMG multple coaltons may form. For detals see Fnus/Rundshagen (21a).
8 7 changes hs address to σ 3 = 2, then c ={{1, 2}, {3, 4}, {5, 6}} wll come about. In the ROMG player 4 can announce σ 4 = nstead of σ 4 = 2 so that no other player can force hm nto a coalton. However, also n the ROMG, any player not n coalton {1, 2, 3} can jon ths coalton and also any player not n coalton {5, 6} can accede to ths coalton. Ths s dfferent n the next game. In the exclusve membershp game (EMG), whch s due to Hart/Kurz (1983), players announce a lst of players wth whom they would le to form a coalton. 6 Those players wth the same lst form a coalton. For nstance, suppose N=6 and σ 1 =σ 2 = {1, 2, 3}, σ 3 = {3}, σ 4 = {4,5, 6} and σ 5 =σ 6 = {5, 6}, then c = {{1, 2},{3},{4}, {5, 6}} forms. Players 1 and 2 have the same lst and therefore form a coalton. The same s true for players 5 and 6. Players 3 and 4 reman sngletons. Player 3 s a sngleton by ts own wll and cannot be forced nto a coalton wth players 1 and 2 and player 4 s a sngleton aganst hs wll snce he s not on the lst of players 5 and 6. In both cases, players are sngletons snce membershp s exclusve. In other words, a coalton only forms by unanmous agreement. More formally, we have: Defnton 4: Exclusve Membershp Game (EMG) a) The set of coalton strateges of country I s gven by Σ = {c partcular strategy coalton. I c } where a σ s a lst of countres wth whch country would le to form a EM G b) Coalton functon ψ maps strategy vector σ nto coalton structure c as follows: c = {} {j σ = σ. j } In the same sprt as we defned an equlbrum n economc strateges n the second stage of the coalton formaton process, t remans to defne an equlbrum n coalton strateges n the frst stage of the game. Ths tas s straghtforward snce, agan, we use the defnton of a Nash equlbrum but now n the context of coalton strateges. Defnton 5: table Coalton tructures Consder a partcular coalton functon ψ and valuaton functon v. Coalton structure c* * s sad to be stable f there exsts a coalton strategy σ Σ generatng c * such that I * * and σ Σ : v( ψ( σ )) v( ψ( σ, σ )) Hence, a coalton structure s called stable f t can be supported by an announcement strategy vector that consttutes a Nash equlbrum for the frst stage of the coalton formaton game. 6 Our defnton corresponds to what s called game by Hart/Kurz (1983).
9 8 Denotng stable coalton structures n the coalton games by C(OMG), C (ROMG) and C(EM G), respectvely, then from theory t s nown that C(OMG) C(ROMG) C (EMG) holds (Fnus and Rundhagen 23). The ntuton of ths relaton s smple: n all games the amount of possble devatons s the same, except that n the ROMG players can not jon sngletons wthout ther consent, whch s possble n the OMG. By the same toen, n the EMG players cannot jon a coalton wthout ther consent, whch s possble n the ROMG (and n the OMG). Thus, f a coalton structure s not stable n the EMG, then t cannot be stable n the ROMG and f a coalton structure s not stable n the ROMG, then t cannot be stable n the OMG. However, what s not nown from theory (except for very restrctve assumptons on payoff functons) 7 s whether those coalton structures whch are stable n the ROMG (EMG) but not n the OMG (ROMG) wll be superor n welfare or ecologcal terms. Hence, t wll be of partcular nterest n secton 4 to evaluate and compare stable coalton structures n the varous games from ths perspectve. nce t wll turn out n secton 4 that n the EMG the set of stable coaltons s relatvely large n the "transfer case", we ntroduce an equlbrum refnement. Ths refnement s smple and requres from a stable coalton structure that t s Paretoundomnated by another stable coalton structure. Defnton 6: Paretoundomnated table Coalton tructures Let c C be stable coalton structure n some coalton game, then c s called a Paretoundomnated stable coalton structure f there s no other coalton structure c C where at least one player s better off and no player s worse off,.e., there s no c C such that I: v(c ) v(c ) and j I: v(c ) > v(c ). 3. The Emprcal Model 3.1. The CLIMNEG World mulaton Model j j The CLIMNEG World mulaton Model (n the sequel referred to as CWM) s an ntegrated assessment, economyclmate model that resembles closely the semnal RICE model by Nordhaus and Yang (1996). The man dfference compared to RICE s that utlty s lnear n consumpton whch s necessary to perform game theoretc analyses n a transferable utlty (TU) framewor. We gve here only a short overvew of CWM, a more detaled descrpton and motvaton can be found n Eycmans and Tulens (23). The ey functons and parameters are provded n the Appendx. An mportant feature of ntegrated assessment 7 ee for nstance Fnus/Rundshagen (21a). Restrctve assumptons are for nstance a statc payoff structure, dentcal payoff functons of countres (symmetrc players) and orthogonal reacton functons that do not hold n our emprcal global warmng game. ee secton 3.
10 9 models s the endogenous feedbac of clmate change damages on producton and consumpton possbltes. The economc part of CWM conssts of a longterm dynamc, perfect foresght Ramsey type of optmal growth model wth endogenous nvestment and carbon emsson reducton decsons. The carbon cycle and temperature change module are the same as n RICE. In the CWM, the World s dvded nto sx regons: UA, JPN (Japan), EU (European Unon), CHN (Chna), FU (Former ovet Unon) and ROW (Rest of the World). In each of these regons and n every tme perod t the followng regonal budget equaton descrbes how gross producton, Y,t, can be allocated to consumpton, Z,t, nvestment, I,t, emsson abatement costs, Y,tC ( µ,t), and clmate change damages, Y,tD( T) t : ( ) ( ) Y = Z + I + Y C µ + Y D T (1),t,t,t,t,t,t t Gross producton can be nterpreted as potental GDP, that s, what could be produced n the absence of the clmate change problem. Abatement costs are an ncreasng and convex functon of emsson abatement effort µ,t [,1]. Abatement effort measures the relatve emsson reducton compared to the Busnessasusual scenaro (BAU) wthout any abatment polcy. Clmate change damages are an ncreasng and convex functon of temperature change T t. Abatement costs, C( µ,t ), and clmate change damages, D( T), are treated as proportons of potental producton. Hence, total costs and damages are the product of costs and damages wth potental producton Y,t, respectvely. Rewrtng (1) allows nterpretng the lefthand sde of the budget equaton as green GDP, that s, producton net of clmate change damages and abatement costs: ( ) ( ) Y,t 1 C,t D T µ t = Z,t + I,t (2) Every regon s characterzed by a producton functon F that maps combnatons of captal stoc K,t and labour nput L,t nto output. The producton technology s assumed to satsfy constant returns to scale of the CobbDouglas type: ( ) ( ) Y = a F K,L F K (3),t,t,t,t,t,t where a,t s a technology shft parameter that s assumed to ncreases exogenously over tme. Labour supply s assumed to be nelastc. Incorporatng the technology parameter and labour nput nto the producton functon, gves a "new" producton functon F,t (K,t ). Captal accumulaton s descrbed n the standard way: [ ] K = 1 δ K + I (4),t+ 1 K,t,t
11 1 where δ K denotes the captal deprecaton rate. Producton gves rse to emssons of greenhouse gases. In the CWM, emssons are propotonal to potental output: E,t =α,t 1 µ,t Y,t (5) where the exogenous parameter α,t denotes the emssonouput rato and s assumed to declne over tme due to exogenous energy effcency mprovements. Emssons accumulate n the atmosphere accordng to a standard lnear stoc externalty accumulaton process: [ ] M = 1 δ M +β E (6) t+ 1 M t,t N where M denotes carbon concentraton, δ M the natural decay rate and β the arborn fracton of emssons, that s, the fracton of emssons added to atmospherc concentratons. Carbon concentraton s translated nto temperature change accordng to an ncreasng functon G: t ( ) T = G M (7) Welfare of each country s measured by ts aggregate lfetme dscounted consumpton: w(s) = Z t Ω,t (8) t t= [ 1+ ρ ] where ρ stands for the dscount rate of regon and Ω denotes the tme horzon. The strategy vector s conssts of a tme path (35 decades, startng n 199) for emsson abatement and nvestment for all sx regons, s = {I,t, µ,t} N;t =,, Ω, and hence s of length 2x35x6= Computng Valuatons Recall from Defnton 1 that the valuaton functon maps coalton structures 1 M * c = (c,..., c ) C nto payoffs, v(c) = w((c)) ε = w(s), va an nstructon how players choose ther economc strateges s. For a fxed coalton structure c, Defnton 1 assumes that coalton members jontly maxmze the aggregate welfare to ther coalton and that equlbrum economc strateges consttute a Nash equlbrum. Thus, coalton c chooses an nvestment and emsson abatement strategy vector s = {I,t, µ,t} c ;t =,, Ω n order to maxmze the aggregate dscounted welfare over all perods for gven strateges s of other countres (that s, gven emsson paths E,t ). Formally:
12 11 c,t max t { I } t,t, µ,t c = [ 1+ρ ] c;t, =, Ω ( ) ( ) ( ( )) s.t. F K 1 C D G M µ = Z + I,t,t,t t,t,t c [ ] ( ) M = 1 δ M +β α 1 µ F K +β E t+ 1 M t,t,t,t,t j,t c j c Ω Z (9) Ths gves rse to two frst order condtons of country beng a member of coalton c n coalton structure c at tme t: 1 t [ ] ( µ Ω ) ( ) τ τ τ ( τ ),t,,, τ t 1 =β G τ( Mτ)[ 1 δm ] (1) τ α,t c τ= t+ 1 [ ] C F K D T 1+ρ 1+ρ That s, country at tme t chooses ts abatement effort such that dscounted margnal abatement costs equal the sum of all coalton members dscounted margnal clmate change damages, tang nto account the senstvty of the clmate system and the natural decay of atmospherc carbon concentratons. Ths s a dynamc verson of the amuelson rule for the optmal provson of publc goods, though here restrcted to members of coalton c. Coalton members nternalze only ther benefts of emsson control but not of outsders. The second frstorder condton from whch the optmal captal accumulaton s derved s gven by: (,t) C µ ρ = ( ) ( µ ) ( ) µ δ F,t K,t 1 C,t D Tt 1,t t K [ 1+ρ] Ths condton s a modfcaton of the standard Ramsey rule for optmal captal accumulaton. Wthout clmate change problem, condton (11) would smplfy to ρ = F,t (K,t ) δ K. That s, captal s allocated over tme such that margnal productvty of captal (net of deprecaton) s equal to the dscount rate. In the presence of clmate change externaltes, ths rule s modfed, mplyng a downward adjustment of the margnal productvty snce the term n curly bracets on the rghthand sde of (11) s smaller than one. Ths mples a slower rate of captal accumulaton, and ceters parbus of output growth and carbon emssons. Gven the publc bad nature of emssons, the economc strateges of countres are nterdependent. Therefore, we have to solve smultaneously the two frst order condtons for each country at each tme t for every coalton structure c. nce we assume complete nformaton, the Nash equlbrum strategy vectors can be nterpreted as open loop Nash equlbra. In order to calculate numercally Nash equlbra, we use a standard numercal algorthm proceedng nteratvely. The teraton process contnues untl the Eucldean dstance between strategy vectors n two consecutve teratons s smaller than a gven threshold value. (11)
13 12 For our data set we never encountered covergence problems and found always a unque equlbrum for each of the 23 possble coalton structures. Note that our settng mples that f we move from some coalton structure c to any other coalton structure c, ths means a change of strateges and affects welfare of countres (valuatons). Thus f for nstance two coaltons n coalton structure c merge, leadng to coalton structure c, they usually ncrease ther abatement efforts but ths may be matched by an expanson of emssons by countres not nvolved n ths merger. The amount of these leaage effects depends on the partcular coalton structure, on the parameters of the margnal damage and cost functons and on the physcal parameters of the clmate system. As ponted out n secton 2, we consder a scenaro wth transfers n whch case ˆv (c) = v(c) + t. We adopt the transfer scheme used n Eycmans and Tulens (23) whch s a modfcaton of the scheme proposed by Chander and Tulens (1995 and 1997). The scheme assumes only transfers wthn coaltons where the surplus of a coalton c from cooperaton s allocated accordng to a proportonal sharng rule: where shares λ are gven by: ( ) ( ) N N t = v (c ) v c v c v (c ) +λ c (12) λ = Ω t= c t= Y D ( T ) Ω,t t t [ 1+ρ] Y D ( T ),t t t [ 1+ρ] (13) The frst term n bg bracets n (12) sets every country bac to ts welfare level n the sngleton coalton structure, the second term allocates the total surplus of coalton c compared to the sngleton coalton structure n proporton to ther margnal damages. Hence, the second term n (12) favors countres wth relatvely hgh potental producton and margnal damage estmates and low dscount rates snce they are enttled to a larger share of ther coalton s aggregate payoff. However, the frst term ensures that members whch would loose from cooperaton wthout transfers because they contrbute much to jont abatement but beneft only lttle because of low margnal abatement and damage costs brea at least even (provded there s a surplus from cooperaton). Notce that f a coalton s surplus from jont cooperaton s postve, then the transfer scheme ensures that every country receves a hgher payoff than n the sngleton coalton structure. Generally, there may be no surplus f leaage effects are strong enough, though for our data set t turns out that ths s always the case.
14 Reference mulatons Carbon emssons and atmospherc carbon concentratons Fgure 1 shows annual world carbon emssons n three scenaros: busnessasusual (BAU), mplyng no emsson reducton ( µ,t =, t), Nash equlbrum (NAH), correspondng to the sngleton coalton structure, and socal optmum, O, correpondng to the grand coalton. 8 We only consder carbon emssons orgnatng from fossl fuel use where world carbon emssons n 199 amount to approxmately 6 ggatons of carbon. It s evdent that BAUemssons contnuously grow, reach nearly 4 GtC by the year 21, and more than 62 GtC n 22. BAUemssons contnue growng throughout the entre tme horzon although the pace of growth gradually slows down. NAHemssons grow at a slghtly slower rate to reach about 38 GtC by the year 21 and 59 GtC by 22. Also NAHemssons contnue growng though growth decelerates. In contrast, Oemssons are substantally lower: n 21 they amount to roughly 24 GtC, and only 21 GtC n 22. Ths s about half of NAHemssons n 21 and almost one thrd of NAHemssons n 22. In contrast to NAHemssons, the Oemsson path rses untl 215, levels off at about 26 GtC and decreases afterwards. FIGURE 1 and FIGURE 2 about here Fgure 2 shows the atmospherc carbon concentraton n the three scenaros. In 199 atmospherc carbon concentraton amounts to approxmately 75 GtC. BAUconcentraton rses steadly and reaches about 1718 GtC n 21 and 3443 GtC n 22. Doublng of the concentraton wth respect to 199 taes place between 28 and 29. The NAHconcentraton path follows closely the BAUpath and contnues to grow steadly over the entre tme horzon. In contrast, Oconcentraton grows at a much slower rate and reaches 1368 GtC n 21 and 217 GtC n 22. Doublng of atmospherc carbon concentraton s postponed untl some tme between 211 and 212. The carbon concentraton levels off at about 2 GtC by the year 22. At the level of ndvdual countres there are substantal dfferences across regons. Tang averages of abatement efforts over tme, we fnd n the Nash equlbrum that CHN abates about 7.7%, followed by EU wth 7.24% and UA wth 6.44%. The lowest abatement effort s undertaen by ROW wth only 1.45%. World average abatement amounts to 3.74%. For 8 All fgures report data for a tme horzon of 199 to 225. However, all calculatons were conduct for the entre tme perod of 35 years.
15 14 ROW low abatement s due to strong freerder ncentves wthn ths heterogeneous regon. 9 For CHN hgh abatement s due to low margnal abatement costs and for EU ths s due to ther hgh clmate change damage valuaton. In the socal optmum world average abatement s 37.14%. CHN and ROW are requred to reduce ther emssons substantally more than other regons (68.13% and 55.5%, respectvely) due to ther low margnal abatement costs. Macroeconomc Magntudes Fgures 3 and 4 show the tme profles of world consumpton, Z, nvestment, I, abatement cost, YC, and damage costs, YD, for NAH and Oscenaro where potental producton, Y, s the sum of these components (see (1)). It s evdent that producton and consumpton profles are qute smlar n both scenaros. The small dfferences n terms of producton and consumpton stem from the fact that abatement costs and damage costs constute a small porton of total producton and consumpton. Hence, strong dfferences n the emsson and concentraton path n both scenaros do not alter much the producton and consumpton patterns and thus also not YC+YD. However, the composton of YC and YD s dfferent n the two scenaros. In the Nash equlbrum abatement costs are very small (they do show up n Fgure 3) but clmate change damage are hgh. In contrast, n the socal optmum damage costs are relatvely small but ths gan requres devotng part of the producton to emsson abatement. FIGURE 3 and FIGURE 4 about here for each * Ω Table 1 dsplays total dscounted consumpton v(c) = w(s) = Z [ 1+ρ ] t N N t=,t regon n the Nash equlbrum, v = v (c ), and n the socal optmum, v = v (c ). The last row World reveals the overall magntudes at stae. Dscounted consumpton amounts to 338,6 mllon $ n the Nash equlbrum and 339,831 mllon $ n the socal optmum. Though n absolute terms, the gan at the world level from full cooperaton s not a small number, n relatve terms t amounts to only.52%. Ths s due three to reasons. Frst, as ponted out above, abatement costs and clmate change damages are small compared to producton or consumpton. econd, dfferences between both scenaros n terms of welfare occure manly n the far future but receve less weght due to dscountng. Thrd, abatement costs are relatvely hgh compared to the benefts from reduced emssons for greenhouse gases, so that also n the socal optmum only moderate acton s requred. Nevertheless, we 9 As n Nordhaus and Yang (1996) we have revsed downward the clmate change damage parameter of ROW n all parttons n whch ROW s a sngleton n order to account for the fact that ths regon comprse of many countres.
16 15 should stress that the small dfference n dscounted consumpton hdes large dfferences between both scenaros n ecologcal terms,.e. n emsson patterns and correspondng atmospherc carbon concentratons (see Fgure 1 and 2 above). Table 1: World Dscounted Consumpton* N v v (v v )/ v N t ˆv (vˆ v )/ vˆ N UA 78,352 78, , JPN 42,99 43, , EU 12,73 13, , CHN 9,14 8, , FU 23,794 24, , ROW 81,135 81, , WORLD 338,6 339, , * Fgures are the dscounted sum of consumpton n mllon U$ 199. All varables as explaned n the text. In terms of ndvdual wnners and loosers, we fnd that CHN and, to a lesser extent, ROW would lose from full cooperaton wthout transfers. 1 Wth transfer ndvdual ratonalty holds for all countres n the socal optmum and as mentoned above for any other coalton structure dfferent from the grand coaltons for our data set as well. Hence, wthout transfers the grand coalton cannot be an equlbrum n any of the coalton formaton games defned n secton 2. However, ndvdual ratonalty s only a necessary condton for stablty and as we wll see from secton 4 by no means suffcent to guarantee stablty. 4. tablty Analyss In ths secton we report on results of our stablty analyss n the three coalton formaton games. In subsecton 4.1 we explan the nformaton contaned n Table 2 and 3, summarzng our results. In subsecton 4.2 we provde some ratonale that helps to explan membershp n stable coaltons and n subsecton 4.3 we evaluate and compare stable coalton structures n the three coalton formaton games. 4.1 Informaton n Table 2 and 3 Table 2 and 3 lst stable coalton structures n the no transfer and transfer case, respectvely. The frst column lsts the number of a partcular coalton structure where to each coalton structure a number between 1 and 23 s attached. Coalton structure 1 represents the sngleton coalton structure, correspondng to the "conventonal" Nash equlbrum, whereas 1 A more detaled analyss of the ncentve to cooperate s provded n subsecton 4.2.
17 16 coalton structure 23 s the grand coalton, correspondng to the "conventonal" socal optmum. Apart from stable coalton structures, these coalton structures are lsted because they represent benchmars. Column 2 lsts coalton structures n partton form where nontrval coaltons are ndcated bold. Column 3, 4 and 5 contan nformaton whether a coalton s stable n a coalton game (n=not stable, y=stable). We lst only those coalton structures that are stable and Paretoundomnated by any other stable coalton structure n at least one game. For nstance, n the transfer case (Table 3), there are 34 stable coalton structures n the exclusve membershp game (EMG) of whch 19 are Paretodomnated by other stable coalton structures and none of them s stable n the open membershp games (OMG and ROMG). Hence, only 15 coalton structures are lsted n Table 3. nce some of these coalton structures are Paretodomnated n the EMG but not n the ROMG, Paretodomnaton s ndcated by a gray cell. TABLE 2 and 3 about here Column 6 gves total dscounted welfare over all regons and the entre tme perod ( ). We tae ths feature to sort coalton structures n descendng order. Column 9 gves concentraton at the end of the tme perod and column 12 ndcates total emssons over all regons and the entre tme perod. In order to evaluate coalton structures, we compute two ndces. The frst ndex s called the "degree of externalty ndex" (DEX). Ths ndex measures the relatve dfferences between the outcome n a coalton structure (superscrpt C) and the socal optmum (superscrpt ) and s defned as follows: welfare: 1 emsson: 1 N N v(c ) = 1 = 1 N v(c ) = 1 Ω N Ω N C E,t t= = 1 t= = 1 Ω N E,t t= = 1 v(c) C MΩ M, concentraton: 1 MΩ E,t where all varables are those of secton 3. By defnton, n the socal optmum the degree of externalty s. The second ndex s called the "closng the gap ndex" (CGX). Ths ndex measures the dfference between the outcome n a coalton structure and the socal optmum n relaton to the dfference between the Nash equlbrum and the socal optmum. It s defned as C N 1 (1 DEX / DEX ) where the superscrpt N stand for Nash equlbrum. Ω
18 17 Thus, by defnton, ths ndex s 1 n the socal optmum and n the Nash equlbrum. All numbers have been rounded to "sensble" dgts for nterpretaton. 4.2 Ratonalzng Membershp n table Coalton tructures From a frst glance at the tables one s let to conjecture that ntuton s not confrmed by the results: many coaltons of stable coalton structures comprse countres of whch one would expect that they should show lttle nterest n formng a coalton and/or not wth those countres lsted n Table 2 and 3. For nstance, t may be suspected that "rest of the world" (ROW) should show lttle nterest n partcpatng n an agreement and that UA and ROW wll hardly be n one coalton because of dfferent nterests. However, a closer loo at the underlyng fundamentals resolves ths puzzle though t s dffcult to trace all effects n a general equlbrum model. Frst, clmate change damage parameters n our model are modfed estmates of RICE that may not be n lne wth the presumed percepton of damages n a country. For nstance, t has been argued that developng countres wll pay lttle attenton to envronmental damages due to strong preferences for economc growth. Ths would suggest for nstance that the perceved damage costs of ROW should be very low. In our model however, we use objectve damage costs that are relatvely hgh for ROW, despte we assume hgher dscount rates for ROW and CHN than for the rest of the countres (see Appendx). econd, the conjecture that countres formng a coalton wth other countres are the "good guys" and countres remanng sngletons are the "bad guys" s premature 11. Ths conjecture presumes that coalton members wll substantally reduce ther emssons compared to the noncooperatve benchmar and compared to outsders. However, ths s not always the case for at least two reasons. One reason s that f for nstance two countres form a coalton where both countres face hgh margnal abatement costs and low margnal damages costs, then jont welfare maxmzaton mples only a margnal reducton compared to the noncooperatve benchmar. In contrast, a sngleton wth low margnal abatement costs and hgh margnal damage costs may reduce emssons substantally by tself. A second reason s that due to noncooperatve behavor and externaltes across coaltons, leaage effects may lead to downgraded abatement efforts n equlbrum (as derved from the valuaton functon). Taen together, there may well be stable "coaltons of the lazy guys". Thrd, the ncentve to form a stable coalton follows from a subtle ncentve structure. In order to shed some lght on the fundamentals, we construct two ndces that are based on two values. The frst value ams at measurng the nterest of a country n clmate change polcy n terms of benefts. For ths we compute dscounted margnal damages n the Nash equlbrum for each country and evaluate 11 Wth good guys we mean here countres that care about the envronment and are nclned to jon IEAs.
19 18 those numbers n relatve terms (ROW=1). From Table 4 t s evdent that  neglectng abatement costs  EU and ROW have the hghest and CHN the lowest nterest n reducng greenhouse gases. The second value ams at measurng the nterest of a country n cooperaton n terms of abatement contrbutons. As follows from the frst order condton n (1), countres wth low margnal abatement cost wll have to contrbute more to jont abatement than those wth hgh margnal abatement costs. nce t turns out for our data set that margnal abatement cost curves never ntersect, we can compute emsson reductons for an arbtrary fxed tax rate. We choose a unform tax rate of 2 U$ per ton of carbon n each perod. Agan, we evaluate emsson reductons n relatve terms and compute average relatve contrbutons for the entre tme perod (CHN=1). From Table 4 t s evdent that CHN and ROW have to mae the hghest contrbutons due to ther low margnal abatement costs. In order to depct the entre ncentve structure, we compute a freerder ndex 1 n row 3 that s defned as abatement contrbuton (row 2) dvded by margnal damages (row 1), evaluated n relatve terms (CHN=1). Freerder ndex 2 (row 4) rans these numbers that maes subsequent nterpretaton easer. From Table 4 t s evdent that CHN has by far the hghest freerder ncentve. That s, CHN has to contrbute very much to a jont abatement polcy but benefts only lttle from reduced damage costs. The opposte ncentve structure s that of the EU: she contrbutes less on average to cooperaton but benefts more than proportonally from reduced damages. Table 4: Incentve Index of Coalton Formaton UA JPN EU CHN FU ROW 1 margnal damages abatement contrbuton freerder ndex freerder ndex For the nterpretaton of the freerder ndces three remars are n order. 1) The ndces can only be a crude measure snce t represents only an "average ncentve structure" that generally dffers from those n partcular coalton structures. 2) Though t s certanly true that a hgh freerder ndex ndcates that a country has no nterest n formng a coalton, not only the absolute value s mportant but the relatve dstance between those values. That s, countres wth a hgh but smlar ndex may well form a coalton because of smlar nterests as ths s the case for nstance for FU and ROW n coalton structure 153 n Table 2. 3) The ndces can only be used n the case of no transfers snce transfers may mply a substantal
20 19 redstrbuton of welfare so that also hgh freerder ndex countres can have an ncentve to form a coalton. Keepng these caveats n mnd, we can now try to ratonalze membershp of some stable coalton structures lsted n Table 2 and 3. Wthout transfers (Table 2) t s evdent that CHN s never a member of a stable coalton because of ts strong freerder ncentve. Also EU s only a coalton member n one stable coalton structure (No. 155) snce countres have usually no nterest formng a coalton wth EU because of her hgh margnal damages. Ths would requre substantal emssons reductons by coalton partners combned wth lttle contrbuton of EU to jont efforts. The only excepton s a coalton wth ROW n coalton structure No. 155 snce ROW faces even hgher margnal damages than the EU. Coalton structures whch demonstrate that countres wth smlar nterests form a coalton are for nstance No. 26, 156, 2, 6, 153 and In the case of transfers the pcture s dfferent. Now, CHN s a frequent member n coaltons wthn stable coalton structures; n partcular n coaltons wth the UA, EU and ROW. Those countres have margnal damages above average and therefore have an nterest n ambtous emsson reductons. Due to hgh margnal abatement costs, they beneft from cheap emsson reducton of CHN. CHN s nterest n partcpaton stems from the fact that t t s compensated for ts hgh contrbuton and receves a far amount of the suplus from cooperaton. Taen together, transfers balance dfferent nterests and thus allow to reap effcency gans from cooperaton between unequal partners. 4.3 Evaluatng and Comparng table Coalton tructures General comments The grand coalton s not stable n the sense of defnton 5. In the no transfer case ths follows mmedately from the fact that CHN and ROW are worse off than n the sngleton coalton structure (see Table 1). However, ndvdual ratonalty s only a necessary but not a suffcent condton for stablty. Hence, also n the case of transfers, the grand coalton s not stable. In fact, only coalton structures comprsng small coaltons wth no more than three members are stable. The reason s that the ncentve to leave a coalton ncreases wth the number of coalton members. Nevertheless, also agreements wth only few countres can close the gap between frst best and no cooperaton by a substantal amount. For nstance, n 12 Usng freerder ncentve ndex 2 we fnd: 26: {2, 5, 4}, 156:{2, 3, 4}, 2: {3, 5, 4}, 6: {2, 4}, 153: {2, 3}, {5, 4} and 5: {3, 4} where the values are ndex numbers of coalton members n these coalton structures.
21 2 the EMG the stable coalton structure assocate wth the hghest global welfare closes the gap between Nash equlbrum and socal optmum by 6.7 and 74.5 percent n the case of no transfers and transfers, respectvely, though ths percentages are lower for concentraton and emssons. However, also the degrees of externalty for concentraton (138 %) and emssons (16 %) are substantally hgher than for welfare n the Nash equlbrum so that a 5 percentage reducton of externaltes n a stable coalton structure may be judged as a success. Interestngly, the "degree of cooperaton" s not necessarly an ndcator for success as s evdent from comparng coalton structure No. 133 (three coaltons of two countres) and coalton structures No. 63 to 64 (only two coaltons of two countres) n the case of transfers. The emprcal fndngs are perfectly n lne wth the theoretcal fndngs of noncooperatve coalton theory, applyng the concept of nternal&external stablty (see Introducton and Fnus 23). Frst, the number of partcpants or the degree of cooperaton s not an ndcaton of the success of coalton formaton. econd, whenever the degree of externalty s low, only small coaltons are stable but they acheve much and f the degree s hgh the opposte holds. In our context ths degree s small n welfare terms (though admttedly t s larger n ecologcal varables) snce abatement costs are relatvely hgh compared to the benefts from reduced damages. As Barrett (1994) ponted out, ths relaton may be reversed for CFCs, predctng large coaltons but less success for solvng the problem of the depleton of the ozone layer effectvely. In ths lght, the current Kyoto Protocol may be seen less pessmstc despte ts rather low partcpaton. Open versus exclusve membershp In the OMG no coalton structure s stable. The assumpton of unrestrcted accesson to any coalton causes great nstablty. Ths s smlar n the ROMG, though sngletons cannot be joned wthout ther consent. Hence, the problem of nstablty s mtgated to some extent and hence fve coalton structures are stable n the case of transfers. The fact that the sngleton coalton structure s stable n the ROMG follows from the defnton of ths game. If each country announces address, then a sngle devaton cannot lead to a dfferent coalton structure and hence a smultaneous announcement of strategy σ = by all countres s trvally a Nash equlbrum coalton structure. 13 From an theoretcal pont of vew ths may be regarded as an advantage snce t guarantees exstence of an equlbrum. From an appled pont of vew ths may be perceved as dsturbng. Why should the sngleton coalton 13 A smlar argument establshes that the sngleton coalton structure s always a Nash equlbrum coalton structure n the EMG.
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