Capital Allocation and International Equilibrium with Pollution Permits *

Transcription

1 Modrn conomy Publshd Onln March ( Captal Allocaton Intrnatonal qulbrum wth Polluton Prmts * Prr-André Jouvt Glls Rotllon conomx Unvrsty of Pars Oust-Nantrr Nantrr Franc Clmat conomcs Char Pars Franc mal: rotllon@u-pars.fr Rcvd Octobr ; rvsd Dcmbr ; accptd Dcmbr 9 ABSTRACT Snc th yoto Agrmnt th da of sttng up polluton rghts as an nstrumnt of nvronmntal polcy for th rducton of grnhous gass has progrssd sgnfcantly. But th crucal problm of allocatng ths prmts n a mannr accptabl to all countrs s stll unsolvd. Thr s a gnral consnsus that ths should b don accordng to som proportonal allocaton rul but opnons vary gratly about what would b th approprat proportonalty paramtr. In ths papr w analyz th conomc consquncs of dffrnt allocaton ruls n a gnral qulbrum framwork. W frst show th xstnc uncty of an ntrnatonal qulbrum undr th assumpton of prfct moblty of captal w charactrz ths qulbrum accordng to th dotatons of prmts. Thn w compar th conomc consquncs of thr typs of allocaton ruls whn th prmt markt s dsgnd to rduc total polluton. W show that a rul whch appls som form of grfathrng smply rducs producton mssons proportonally ffcntly. In contrast an allocaton rul proportonal to populaton s bnfcal for dvlopng countrs. Fnally pr capta allocaton ruls nduc sz ffct can rvrs ths rsults. ywords: Polluton Prmts; Captal Allocaton; Intrnatonal qulbrum. Introducton On of th most ntrstng dvlopmnts n nvronmntal polcy n rcnt yars has bn th mrgnc of global nvronmnt as a North-South ssu. Th clos lnk btwn global nvronmnt dvlopmnt calls for nw nsghts. In a world of global xtrnalts natonal polcs hav mportant ntrnatonal rprcussons through trad factor moblty. To b sur that th full mpact of nvronmntal polcs can b analyzd through to ts ultmat ffcts on factor markts ncom polluton a gnral qulbrum approach s ndd. Ths s th way ponrd by Copl Taylor [] Chchnlsky [3] who study th lnks btwn trad nvronmnt n a North-South contxt. Copl Taylor [4] xamn lnkags btwn natonal ncom polluton ntrnatonal trad n a smpl modl of North-South trad. By solatng th scal composton tchncal ffcts of ntrnatonal trad on polluton thy show that fr trad ncrass world polluton. Morovr an ncras n th North s producton possblts ncrass polluton whl smlar growth n th South lowrs polluton. In thr paprs polluton has only a local natur n th sns that damags ar con- * Ths papr orgnats from a rsarch projct launchd jontly wth Phlpp Mchl. fnd to th mttng country thy analyz th sam qustons wth transboundary polluton n Copl Taylor [5] whr countrs dffrd only n thr ndowmnt of ffcnt labor whch s th on prmary factor. Chchnlsky [3] consdr two prmary factors physcal captal nvronmntal rsourc focuss manly on th consquncs of dffrncs n proprty rghts on th common-proprty problm gvng answrs to th prsumd comparatv advantag n drty ndustrs for dvlopng countrs or th compatblty of trad polcs basd on tradtonal comparatv advantags wth nvronmntal prsrvaton. In ths papr w adopt th Copl-Taylor framwork wth global polluton producd jontly wth consumpton good but w ntroduc ntrnatonal markts for physcal captal polluton prmts. Snc tradabl msson prmts hav bn ntroducd n conomc thory by J. H. Dals [6] as a nw nstrumnt for nvronmntal polcy thy hav bn th objct of many studs (Ttnbrg [7]). Many of ths studs dal wth th comparson btwn msson prmts msson fs thr s now a growng body of ltratur Copl Taylor [4] dvlop a prfctly compttv gnral qulbrum modl wth trad n goods n msson prmts but wthout captal markt. Copyrght ScRs.

2 88 P.-A. JOUVT T AL. on thr practcal applcaton (Noll [8] Hahn [9] t [] Hahn Stavns []). As suggstd by Chchnlsky Hal [] tradabl msson prmts ar also a mans to scur th bosphr Chchnlsky t al. [3] hav analyzd thr us as a polcy nstrumnt aganst grnhous warmng. Thy show that th mannr n whch msson rghts ar ntally dstrbutd dtrmns th possblty of th markt attanng a Parto ffcnt outcom (Jouvt t al. [4] Jouvt t al. [5]). Snc th yoto agrmnt of 997 th da of sttng up polluton rghts as an nstrumnt of nvronmntal polcy for th rducton of grnhous gass has progrssd sgnfcantly. urop whch had bn hostl to th craton of such an ntrnatonal markt for a long tm sms to hav convrtd to ths approach. In spt of th advantags whch polluton prmts sm to possss n comparson to othr systms of nvronmntal rgulaton (Bohm Russl [6]) th nsttutonalzton of an ntrnatonal markt of polluton prmts ntals svral problms (Baumol Oats [7] Croppr Oats [8] Parc Turnr [9]). Among ths dffcults th frst on to b awar of s wthout doubt th dfnton of an nvronmntal norm ncssary for th ntal ssu of prmts. In fact smngly ntractabl problms mrg as soon as w try to stablsh what would b th approprat proportonalty paramtr n ordr to mplmnt th ntal allocaton of prmts. Opnons vary gratly n ths rspct th lst of approprat paramtrs whch hav bn actually bn put forward n submssons to th Intrgovrnmntal Panl on Clmat Chang s vry larg (Müllr []). W hav manly th followng: Pr capta msson. Pr capta GDP. Rlatv hstorcal rsponsblty. L ara. Sz of populaton. Th man quston that rmans to b solvd concrns th conomcal consquncs of thos dffrnt ruls. Ths quston s partcularly rlvant n th North-South trad contxt whr dvlopng countrs ar unlkly to partcpat n th yoto agrmnt xpctng that thr costs xcd thr bnfts. For ths rason Bohm Larsn [] do not consdr dvlopng countrs. Thy valuat th dstrbutonal mplcatons of th rducton costs brought about by varous prmt allocatons n a tradabl prmt rgm for carbon mssons rductons for a rgon consstng of urop th stats of th formr Sovt Unon (FSU). Thy show that ntal prmt allocatons by populaton /or GDP ar unlkly to nduc th partcpaton of most countrs of astrn urop FSU bcaus of th nt costs nvolvd. Thy dntfy a st of ntal allocatons that would at last compnsat ths countrs. But thr analyss only focuss on th dstrbuton of th conomc burdn of abatmnt msss th gnral qulbrum mplcatons of th allocaton ruls. In th sam way outstaal [] focuss on th dsgn mplmntaton consquncs of a systm of tradabl carbon prmts to rduc grnhous gas mssons wthn th contxt of th uropan Unon. In ths papr w study an ntrnatonal qulbrum n a two-country modl wth captal prmt markt. W analyz th ffcts of allocaton ruls of prmts on captal allocaton ( consquntly on ntrnatonal qulbrum) by consdrng prmt allocaton ruls proportonal to producton mssons physcal captal (n lvl or pr capta) to populaton n a gnral qulbrum framwork. W us th stard tchnology of producton wth thr factors (captal labor msson) n th form proposd by Stoky [3]. W frst analyz th ntrnatonal qulbrum. A prmt markt dos not modfy th compttv world qulbrum wthout prmts whn th total allocaton s larg nough. Whn t s not thr xsts a unqu qulbrum wth undr-us of th tchnology or wth full us of th tchnology n th two countrs. Whn allocaton of prmts s not proportonal to th mssons n th world wthout prmts thr s a rducton factor of mssons whch rsults from th qulbrum allocaton of captal. Th qulbrum lvl of us of tchnology s th sam n th two countrs. It dpnds both on th total world dotaton of prmts ts dstrbuton among countrs. Th scond man part of th papr s dvotd to studyng th conomc consquncs of dffrnt prmt allocatons ruls. Thr dffrnt typs of conclusons hold. A lvl allocaton rul (proportonal to outputs mssons or physcal captal) rducs producton mssons n both countrs proportonally wth a chang n th tchnology usd. In ths cas ach country uss xactly ts dotaton of prmts th qulbrum allocaton of captal s th sam as n th conomy wthout prmts. In fact such an allocaton s ffcnt.. t allows maxmum producton for a gvn total world dotaton of prmts. Th lvl allocaton ruls proportonally dmnsh output n th two countrs whatvr thr rlatv walth. A North-South dstncton (Copl Taylor []) assums hghr lvl of ffcnt labor pr capta n th North. Ths mpls that populaton allocaton rul lads to a North-South rato of prmts smallr than th lvl allocaton. Ths allocaton s bnfcal for th dvlopng country ncrasng captal producton. Morovr th South s nt sllr of prmts whch gvs hm an Copyrght ScRs.

3 P.-A. JOUVT T AL. 89 addtonal ncom. Howvr th pr capta ncom rmans lowr n th South country than n th North country. Fnally pr capta allocaton ruls (proportonal to pr capta output mssons or physcal captal) nduc a sz ffct. If th populaton n th dvlopng country s lowr than th populaton n th dvlopd country ths ruls hav th sam ffcts as th populaton rul. But f t s largr th dvlopd country bnfts from th pr capta allocaton ruls. Th rmr of th papr procds as follow. Scton sts up th modl. In Scton 3 w study th ntrnatonal qulbrum wthout prmts n Scton 4 w stat th condtons undr whch an ntrnatonal qulbrum wth prmts xsts s unqu. Scton 5 dals wth th conomc consquncs of dffrnt prmt allocatons ruls Scton 6 prsnts our conclusons.. Th Modl W study th ntrnatonal qulbrum for two countrs n a smpl modl wth on rprsntatv frm n ach country. Ths frms produc th sam good wth th sam tchnology. W assum prfct moblty of captal but fxd nlastc ffcnt labor supply H = n ach country gvn total captal stock. W also assum that mssons of polluton s a jont product w ntroduc an ntrnatonal markt of mssons prmts. Gvn th quotas = for ach country th rprsntatv frms can buy or sll t on a prmt markt dcdng on thr mssons as f thr was a global world quota. But whn th prc of prmts s postv thr s a rallocaton thn th frm s rvnus ar modfd. Assumng thr xsts compttv labor markt n ach country wag corrsponds to th margnal productvty of labor th frm s rvnu nt of wags ncluds th nt bnft of th prmt markt. As a consqunc th rat of rturn of captal s dffrnt from th margnal productvty of captal as soon as thr ar transactons on th prmt markt. Wth prfct moblty of captal across countrs only th avrag rturns to captal ar qualzd to th margnal productvts. Indd th prmt markt modfy th nt rvnu of th frms thus thr valu. As a consqunc th qulbrum wth prfct moblty of captal wll lad to qualzng th valus of captal that tak nto account th nt gans on th prmt markt... Th Tchnology Two countrs produc th sam good wth th sam Cobb-Douglas producton tchnology gvn by Y = z A L = () whr L ar rspctvly captal ffcnt labor z an ndx of th tchnology usd wth z. Wth z = Y = A L s th potntal output. Th rato msson on producton Y s an ncrasng functon of z = bz > () Y whn z = th us of all productv possblty lads to th largst mssons polluton. Rmark. Ths on-good modl (s Stoky [3]) can b ntrprtd as a rducd form of th framwork n Copl Taylor []. In fact t s quvalnt to th followng thr factor producton functon = = = = Y Y A b L f ba L Y Y A L f ba L Ths functon Y =mn Y Y s homognous of dgr on contnuous concav wth rspct to captal labor mssons. It s dffrntabl xcpt at th ponts at whch Y = Y... Frm s Bhavor In ach country = a rprsntatv frm maxmz profts wth rspct to th us of tchnology z ffcnt labor L captal stock. In addton frm n country hold a gvn stock of prmts. Ths ntal allocaton s dffrnt from th frm s dm whch dpnd of th markt prc q of th prmt on th ntrnatonal markt. Dnot by w th wag n country. Th rvnu ncludng th nt gans on th prmt markt s thus gvn by Y wl q Usng rlaton () th problm of frm n country s wth A L z qbz w L q max < z L> Th frst ordr condtons ar m z qbz = w = m A L qb z = f z < Ths last condton gvs z =mn z q qb (3). (4) (5) (6) Copyrght ScRs.

4 9 P.-A. JOUVT T AL. Thus n (4) m = z q qbz q = m q ffcnt labor s pad at ts margnal productvty accordng to (4). Dcson on th us of tchnology only dpnds on th prc of prmts. Hnc n th two countrs th ndx of th tchnology usd s th sam z = z. Thus profts satsfy = q q bz (7) As long as th prc of prmts s low nough.. whn q n th two countrs th producton b s qual to ts potntal output z = whch lads to maxmum polluton n th two countrs. But as soon as th prc of prmts xcds th ndx of b tchnology usd s lss than on whch mpls a rducton n producton thus n polluton. Not that polluton s rducd n two ways : mssons dcras both wth producton th ndx of tchnology usd (quaton ()). Followng Hahn Solow [4] (pags 7-7)...w tak t to b charactrstc of captalst frms that thr profts go to th supplrs of captal. W assum thrfor that savngs...ar usd to buy shars n th gross opratng surplus of frms. Thrfor th total rturn pr unt of captal s dfnd by π = π (8) Ths mpls that π = q q bz (9) Ths nt rvnu s smlar to th gross opratng surplus dfnd by Hahn Solow. Not that whn th prc of prmts s postv th prmt markt modfy th frm s ncom so th rturn of captal whch s not qual to ts margnal productvty. Accordng to th prc q of prmts two cass occur : q z = = q q () b b q > z = b = q q qb 3. qulbrum q < In th absnc of moblty of labor n ach country th qulbrum n th labor markt mpls th qualty of th labor dm L th supply H. () In th world wthout prmts th dfnton of th qulbrum s stard. It s ffcnt gvs th maxmum of th world producton. = maxa H A H Ths maxmum s obt whn th allocaton of total captal = s proportonal to ffcnt labor ths lads to th potntal world output = A H H () Th corrspondng total mssons s thn also maxmum: by Y = b. mssons ar proportonal to ffcnt labor Y H = = = Y H wth th allocaton of prmts n country = thr s an addtonal markt w dnot q th qulbrum prc on ths markt. In addton ths markt ntract wth th captal markt. Th assumpton of prfct moblty of captal lads to qualty of th two rats of rturn π = π = π whch mpls = Fnally th prmt markt clars whch mans wth qualtyf q > (3) (4) At qulbrum mssons ar = bz A H. Thus th rato = only dpnds on th qulbrum rato of captal stocks H H = = = H wth =. H (5) In a world wthout prmts th qulbrum allocaton of captal mssons ar proportonal to ffcnt H labor gvn by = H H bah = =. H H Mor gnrally whn th sum of th allocaton of prmts s at last qual to th maxmum of mssons th qulbrum prc of prmts s zro total producton s qual to potntal world output. Ths holds f = = b. Copyrght ScRs.

5 P.-A. JOUVT T AL World qulbrum wth Rducton of mssons Whn th total dotaton of prmts dos not allow for th maxmum of polluton.. = < b Th followng study shows th xstnc of a unqu qulbrum thr wth undr-us of th tchnology or wth full us of th tchnology n th two countrs. Ths scond possblty occurs whn th allocaton of prmts s not proportonal to th mssons n th world wthout prmts. Thr s thn a rducton factor of mssons whch rsults from th qulbrum allocaton of captal. Th qulbrum lvl of us of tchnology s th sam n th two countrs. It dpnds both on th total world dotaton of prmts ts dstrbuton among countrs. 4.. qulbrum wth Undr-Us of Potntal Outputs W bgn wth som usful concpts n ordr to study th xstnc of an qulbrum wth undr-us of potntal outputs. qulbrum rato: At th qulbrum wth undr-us z of potntal outputs < mssons = ar proportonal to (rlaton (5)) captal stocks ar proportonal to ncoms (rlaton (3)) ncoms ar proportonal to (rlaton ()). Ths lads to an qulbrum rato = as a functon of = dpndng on = H. Th qu- H lbrum rato of mssons ncrass wth ts valu s locatd btwn. (for dtals s Appndx A Lmma 6). Proportonal allocaton: W hav a proportonal allocaton whn th allocaton of prmts s proportonal to ffcnt labor thn thr ar no transactons on th prmt markt =. Th ndx of tchnology usd z s smply dfnd by th lvl of total.. prmts = = z b whch rsult from th proportonalty proprts. Non-proportonal allocaton: Whn an allocaton s not proportonal to ffcnt labor thr ar prmt s transactons whch draw th conomy n th drcton of th proportonal allocaton. Snc th allocaton of factors ar not proportonal thn th sum of potntal outputs Y Y s smallr than th world potntal output w hav Y Y = whr s a rducton factor smallr than. At th qulbrum ths rducton factor s a functon of th qulbrum rato: = th rducton factor at qulbrum whr =. Wth = = = Ths rducton factor s smallr than for. Mor prcsly th largr th gap btwn th smallr th rducton factor at qulbrum. qulbrum: Gvn th qulbrum ndx of tchnology usd z s dtrmnd by = z z b (6) q s dtrmnd by q =. b Thus z < s quvalnt to b > To summarz w hav shown th followng. Proposton. Gvn th dotatons of prmts th total captal stock thr xsts an qulbrum wth undr-us of tchnology f only f th total dotaton of prmts s smallr than th product of maxmum of mssons wth th rducton factor. Th qulbrum rato of mssons s an ncrasng functon of th rato of dotaton dtrmns th rducton factor. 4.. qulbrum wth Full Us of Potntal Outputs Wth full us of potntal outputs postv prc of prmts w hav z = q >. In th proportonal cas = at qulbrum thr s no transactons on th prmt s markt. In th partcular cas whr = b any valu of th prmt s prc q lads to th b sam allocaton as n th conomy wthout prmts. (Appndx A Lmma ) In th non proportonal cas thr s a rducton factor wth = w hav = Ths mpls < b lu of vrfs = z b (7) th corrspondng vawhch dtrmns th qulbrum valu of =. Assum. Whn s larg nough ( b th qulbrum allocaton s proportonal to ffcnt labor = z = =. Whn t s small nough < b thr s undr us of potn- tal outputs th qulbrum rato s = th rducton factor s = z <. Ths functon frst ncrass rachs a maxmum qual to n th proportonal cas = thn dcrass (s Appndx A Lmmas 7 8). Copyrght ScRs.

6 9 P.-A. JOUVT T AL. In th ntrmdat cas b < b thr s full us of potntal outputs but t rmans a rducton factor whch s smallr than largr than. Th qulbrum rato of mssons s ntrmdat btwn. Indd s postv for < ngatv for > (s Appndx A Lmma ). To summarz w obtan : Proposton. Assum that allocaton of prmts s not proportonal to ffcnt labor total allocaton s blow th maxmum of polluton. Thn thr xsts a mnmum lvl of total allocaton for whch th world qulbrum uss potntal outputs th prc of prmts s postv. Agan th qulbrum rato of mssons s locatd btwn th rato of ffcnt labor th rato of dotatons. Mor prcsly t s locatd btwn th valu. As shown n th Appndx A w hav f > < < < f < < < Th uncty of qulbrum rsults from th thr prcdng propostons. Th thr prcdng propostons ar llustratd n Fgur blow. In th plan w hav drawn rgons corrspondng to th dffrnt qulbra. In rgon A total dotaton of prmts s at last qual to th maxmum of mssons q = (Proposton ) n rgon B total dotaton of prmts s smallr than th product of maxmum of mssons wth th rducton factor thr s b b C B A C maxmum of total mssons b b Fgur. Rgons corrspondng to dffrnt typs of qulbrum. undr-us of potntal output (Proposton ) n rgon C thr s full us of potntal output th prc of prmts s postv (Proposton 3). 5. Th conomc Consquncs of Allocaton Ruls of Prmts In ordr to study th consquncs of dffrnt allocaton ruls of prmts w compar th qulbrum wth prmts to th qulbrum wthout prmts. Wthout prmts th qulbrum valus of captal stocks producton Y mssons ar proportonal to ffcnt labor suppls H. Profts pr unt of captal ar qual n th two countrs (prfct moblty of captal) qual to th margnal productvty of captal. As shown n Scton 3 th qulbrum wth prmts concds wth th qulbrum wthout prmts whn th total dotaton of prmts allow for th potntal world output.. b polluton s maxmum n ths cas: = = b. Ths s our bnchmark cas dfnd by Y H = = = = (8) Y H W assum now that th total dotaton of prmts dos not allow for th maxmum of polluton.. = < b (9) w consdr thr typs of allocaton ruls. 5.. Lvl Allocaton Ruls Th proportonalty at th qulbrum wthout prmts of captal output mssons ffcnt labor (quaton (8)) mpls that any allocaton of prmts proportonal to on of ths lvls lads to th sam allocaton whch w call th lvl allocaton ruls. Ths ruls can b vwd as som form of grfathrng 3. All ths ruls ar quvalnt thy mply that th rato = s qual to = H. H Ths mpls that th qulbrum rducton factor = =. Undr (9) th qulbrum valu of th tchnology ndx s (Proposton wth = ) z = < b 3 In th smpl grfathrng allocaton all countrs rcv prmts n proporton to thr basln mssons. 4 For furthr comparson w dnot z Y th qulbrum valus wth =. 4 Copyrght ScRs.

7 P.-A. JOUVT T AL. 93 Th captal stocks rman unchangd = productons mssons ar rducd H zy Y = za = = bz Y = bz Y z = =. Th prc of prmts q = s postv but b z thr ar no transactons on th prmt markt. A lvl allocaton rul smply rducs proportonally producton mssons by applyng th tchnology ndx z. Ths s a consqunc of th assumpton that th tchnology of producton th corrspondng msson functon ar th sam n th two countrs. Bcaus of th ffct of th ndx of polluton mssons dmnsh mor than th producton: = bz < mpls Y Y Y <. Y W hav th followng rsult of ffcncy of ths allocaton rul: t lads to th maxmum of th world producton for gvn total captal stock total msson (s Prat [5]). Proposton 3. Gvn th total captal stock th maxmum of th world producton subjct to a total mssons constrant s rachd at th qulbrum obt by an allocaton rul whch s proportonal to ffcnt labor. Proof. Consdr frst any allocaton > > of =. Y = A H s th potntal producton n country. Th maxmum of zy zy subjct to = bz Y bz Y lads to z = z. Ths rsults from th concavty of th problm th maxmzaton on th Lagrangan = zy zy bz Y bz Y As a consqunc th maxmum of world producton can b formulatd as follow: Maxmz wth rspct to z Y = zy Y Y wth Y = A H = subjct to = bz Y Y =. = b Rplacng z xmz Y Y to th soluton Y Y ths lads to ma- = = =. W h av shown that for any allocaton of captal = th maxmum of th world producton Y Y subjct to s obt wth th sam ndx of tchnology usd z for th two countrs tha t th rducton factor s qual to on. 5.. Populaton Allocaton Rul A populaton allocaton rul lads to an allocaton of prmts proportonal to populaton. Indpndntly of th sz of populaton n th two countrs N = a rasonabl masur of stard of lvng pr capta s ffcnt labor pr capta. Thus as n Copl Taylor [] th North-South dstncton arss from an assumd hghr lvl of ffcnt labor n th North.. a largr ffcnt labor pr capta. W assum that country s a dvlopng country bcaus t has a lowr ffcnt labor pr capta than n country say a dvlopd country. H H h = < = h N N Thn an allocaton rul proportonal to populaton mpls N H = = > = N H W compar th ffcts of ths rul of allocaton to th prcdng rul proportonal to wth th sam dotaton of prmts = vrfyng (9). Whn th qulbrum rducton factor s smallr than thr ar two possblts for th qulbrum accordng to f s largr or smallr than b. If th qulbrum rducton factor s not too low b > th qulbrum holds wth nonus of potntal output z <. If not b th qulbrum holds wth us of potntal output z =. Mor prcsly as a functon of = s dcrasng wth rspct to for > admts a fnt lmt whn tnds to (Appndx A Lmma ). Thus If b thn for all > s smallr than b th ntrnatonal qulbrum holds wth z <. If b vrfs (9) thr xsts a thrshold # # soluton of b such that th ntrnatonal qulbrum holds wth z < # f < wth z # = f. Lt us dfn th thrshold ê such that at qulbrum z # < f only f < ˆ. Ths th rshold s ˆ = f > b f not ˆ =. Pro poston 4. Wth th p opulaton allocaton rul th world producton s rducd; th dvlopng country s Copyrght ScRs.

8 94 P.-A. JOUVT T AL. nt sllr of prmts rcvs mor captal producs mor thus mts mor polluton. Th dvlopd country s nt buyr of prmts rcvs lss captal producs lss mts lss polluton. Proof. Consdr frst th cas < ˆ thn z < th ntrnatonal qulbrum vrfs (Proposton Appndx A Lmmas 9 ) z = < < = > b snc <. World producton s rducd bcaus ts maxmum for gvn s rachd at th qulbrum wth allocaton =. Th captal rat o s larg r than bcaus w hav from rlaton (5) = > = > But th sum s th sam: = =. As a consqunc. Th ncras n z mpls an ncras n producton for country. Y = z A H > za H = Y Ths also mpls an ncras n mssons >. Snc th world producton dcrass Y dcrass (mor than Y ncrass) Y < Y Y Y < Y mss ons also dcras: < (th sum s constant) Morovr = < = pls that th dvl- m op ng country s a nt sllr th dvlopd country a nt buyr on th prmt s markt. Consdr now th cas ˆ. Thn z =. At ths qulbrum = s th soluton of = b t vrfs < < (Proposton ). Th prcdng argumnts thn appls wthout modfcaton. Clarly th allocaton rul proportonal to populaton s n favor of th dvlopng country ncrasng captal producton. An addtonal advantag s th ncom from sllng prmts. Th stuaton of th dvlopd country s th complt oppost: t looss n all rspct: captal producton z ar rducd t must buy mor prmts. W should also rmark that producton pr capta r- N mans largr n th dvlopd country whn = N snc Y = = < = N Y N Morovr w hav Proposton 5. Th pr capta ncom rmans lowr n th dvlopng country than n th dvlopd country H N Proof. Whn < = w hav H N < < Th rato of total ncom s Y q = = Yq Bcaus Y = q = w bz b z hav < < = N N whch mpls that pr capta ncom n th dvlopng country s smallr than n th dvlopd country Pr Capta Allocaton Ruls Pr capta allocaton ruls lad to an allocaton of prmts proportonal to pr capta outputs mssons or physcal captal. N W not th rato of populaton. Th thr pr capta allocaton ruls ar quvalnt lad to a rato of prmts =. Indd from quaton (8) w hav Y N N N = = = = Y N N N Pr capta allocaton ruls nduc a sz ffct rlatv to th lvl allocaton ruls xcpt whn =. In ths cas th two knd of allocaton ruls lad to = w hav th sam rsults as n Subscton 4.. Whn sz ffct xsts. If populaton n country (th dvlopng country) s lowr than populaton n country w hav < pr capta allocaton ruls mply >. Thus all th conclusons of th subscton 5. hold a dvlopng country wll prfr pr capta allocaton ruls to lvl allocaton ruls. On th contrary f populaton n country s largr than populaton n country w hav > pr capta allocaton ruls mply <. N Copyrght ScRs.

9 P.-A. JOUVT T AL. 95 Ths s quvalnt to >. as Rlabll ng c ountrs th dvlopng country as th dvlopd country th analyss of subscton 5. hold wthout othr modfcatons. Ths s to say that now th dvlopng country s a nt buyr of p rmts rcvs lss captal producs lss mts lss polluton. In ths cas pr capta allocaton ruls ar n unfavor of th dvlopng country. 6. Conclusons Th Scond Assssmnt Rport of th IPCC (Bruc t al. [6]) contans th rsults of a study apprasng th conomc ffcts of two allocaton ruls th grfathrng rul th populaton rul. Dvlopd countrs would b nt bnfcars f should quotas grfathrng b adoptd undr th populaton allocaton rul th nt bnfcars would b th dvlopng countrs. Our analyss allows us to b mor spcfc on th conomc consquncs of ths dffrnt allocaton ruls. Th lvl allocaton ruls (whch nclud th grfathrng rul) ar ffcnt lad to maxmum world output onc total mssons ar gvn. Thy mply proportonal rducton of polluton n all countrs hav no ffct on ntrnatonal captal allocaton undr th assumpton of th sam tchnology n all countrs. Th populaton allocaton rul confrms th bnfts for dvlopng countrs n vry rspcts: producton movmnt of captal ncom from th prmt markt. Nvrthlss pr capta ncom rmans lowr n th dvlopng country. Pr capta allocaton ruls hav dffrnt sz ffcts dpndng on th rato of populaton n th two countrs. Wth th sam lvl of populaton th pr capta ruls lad to th ffcncy allocaton thus prforms xactly lk th lvl allocatons ruls. Wth a dffrnt lvl of populaton th dvlopng country bnfts f only f t has a lowr lvl of populaton than n th dvlopd country whch bnfts n th oppost cas. Our rsults shd som lght on th rcurrnt dscusson btwn countrs about th ntal dstrbuton of prmts n a tradabl markt. Rgardng ffcncy th lvl allocaton ruls sms to b th bst. But t dos not allow for any voluton of th rlatv ncom btwn countrs. Ths shows that ths allocaton should b lnkd to rdstrbuton polcs. Furthr rsarch wll analyz th wlfar ffct of th abatmnt of polluton th allocaton rul. RFRNCS [] B. R. Copl M. C. Taylor North-South Trad th nvronmnt Quatrly Journal of conomcs Vol. 9 No pp do:.37/84 [] B. R. Copl M. C. Taylor Trad Transboundary Polluton Amrcan conomc Rvw Vol. 85 No pp [3] G. Chchnlsky North-South Trad th Global nvronmnt Amrcan conomc Rvw Vol. 84 No pp [4] B. R. Copl M. C. Taylor Intrnatonal Trad th nvronmnt: A Framwork for Analyss NBR Workng Papr 854. [5] B. R. Copl M. C. Taylor Fr Trad Global Warmng: A Trad Thory Vw of th yoto Protocol NBR Workng Papr [6] J. H. Dals Polluton Proprty Prcs Toronto Unvrsty Prss Toronto 968. [7] T. Ttnbrg Th Tradabl Prmts Approach to Protctng th Commons: What Hav W Larnd? In:. Ostrom t al. Th Drama of th Commons Natonal Rsarch Councl Commtt on th Human Dmnsons of Global Chang Natonal Acadmy Prss Washngton DC pp [8] R. Noll Implmntng Marktabl msson Prmts Amrcan conomc Rvw Paprs Procdngs Vol. 7 No. 98 pp. -4. [9] R. W. Hahn conomc Prscrptons for nvronmntal Problms: How th Patnt Followd th Doctor s Ordrs Journal of conomc Prspctvs Vol. 3 No. 989 pp [] N. t Th US Acd Ran Control Allowanc Tradng Systm Clmat Chang Dsgnng a Tradabl Prmt Systm Pars OCD 99. [] R. W. Hahn R. Stavns Tradng n Grnhous Prmts: A Crtcal xamnaton of Dsgn Implmntaton Issus Faculty Rsarch Workng Papr R93-5 Harvard Unvrsty Cambrdg 993. [] G. Chchnlsky G. Hal Scurtzng th Bosphr W.P. arth Insttut Columba Unvrsty Columba 997. [3] G. Chchnlsky G. Hal D. Starrtt Intrnatonal msson Prmts: quty ffcncy Annual Congrss of ASSA San Francsco 996. [4] P. A. Jouvt Ph. Mchl G. Rotllon Optmal Growth wth Polluton: How to Us Polluton Prmts? Journal of conomc Dynamcs Control Vol. 9 No. 9 5 pp do:.6/j.jdc [5] P. A. Jouvt Ph. Mchl G. Rotllon Compttv Markts for Polluton Prmts: Impact on Factor Incom Intrnatonal qulbrum nvronmntal Modlng Assssmnt Vol. 5 No. pp. -. do:.7/s [6] P. Bohm C. S. Russl Comparatv Analyss of Altrnatv Polcy Instrumnts n Hbook of Natural Rsourcs nrgy conomcs In: A. V. ns J. L. Swny ds. North-Holl Nw-York 985 pp [7] W. Baumol W.. Oats Th Thory of nvron- Copyrght ScRs.

10 96 P.-A. JOUVT T AL. mntal Polcy Cambrdg Unvrsty Prss Cambrdg 998. [8] M. L. Croppr W.. Oats nvronmntal conomcs: A Survy Journal of conomcs Ltratur Vol. 3 No. 99 pp [9] D. W. Parc R.. Turnr conomc of Natural Rsourcs th nvronmnt Harvstr Whatshaf 99. [] B. Müllr Justc n Global Warnng Ngotatons. How to Obtan a Procdurally Far Comproms Oxford Insttut for nrgy Studs V [] P. Bohm B. Larsn Farnss n a Tradabl-Prmt Traty for Carbon mssons Rductons n urop th formr Sovt Unon nvronmn tal Rsourcs conomcs Vol. 4 No pp do:.7/bf6935 [] P. outstaal conomc Polcy Clmat Chang: Tradabl Prmts for Rducng Carbon mssons d- ward lgar 997. [3] N. L. Stoky Ar Thr Lmts to Growth? Intrnatonal conomc Rvw Vol. 39 No. 989 pp. -3. do.org/.37/578 [4] F. Hahn R. Solow A Crtcal ssay on Macroco- nomc Thory Blackwll Publshrs Oxford 995. [5] A. Prat ffcncy Proprts of a Constant-Rato Mchansm for th Dstrbuton of Tradabl msson Prmts In: G. Chchlnsky G. Hal ds. nvronmntal Markts Columba Unvrsty Prss Columba. [6] J. P. Bruc H. L F. Hats ds. Clmat Chang 995: conomc Socal Dmnsons of Clmat Chang Contrbutons of Workng Group III to th nd Assssmnt Rport of th Intrgovrnmntal Panl on Clmat Chang Cambrdg Unvrsty Prss Cambrdg 996. Copyrght ScRs.

11 P.-A. JOUVT T AL. 97 Appndxs Charactrzaton of an Intror qulbrum (z* < ) = Dfn q = qb Dotaton of prmts th total captal ths con dton s quvalnt to =. stock ar gvn. s dcrasng wth rspct to Assum q > z = q < > < < Wth captal stock mssons prof- ts n country ar For fxd postv valus of = b q A H ncrass from to whn ncrass from to. Thus thr xsts a unqu soluton of = = s ncrasng wth rs- = q pct to. In addton w hav = th us Th qulbrum condton (3) on th captal markt mpls = s th unqu soluton of =. Assum > thn > = = = w hav (A) =( ) >= = Thus vrfs < < Th qulbrum condton (4) on th prmt markt Smlarly f < vrfs < < wth q > mpls Lmma 7. If th r xsts an qulbrum wth q > = (A) q < ths qulbrum s unqu t vrfs: = = Lmma 6. quatons (A) (A) mply that = z = q = = vrfs = whr = (A4) b q = = whr (A3) = (A5) Proof. Th qulbrum vrfs (A) (A). Th Th quaton = admts a unqu soluton > = s ncrasng wth rspct to valu of th rato rsults from (A). Th qulb-. If = thn = =. If > (rsp. < ) thn = vrfs < < rum condton = mpls = (rsp. < < ) Proof. Wth = = th qulbrum = = bz A H condton on th prmts markt (A) mpls: AH H = b = = = z Dfnng accordn g to (A5) w obtan th va- =. Thus (A) mpls lu of b q gvn by (A4). Copyrght ScRs.

12 98 P.-A. JOUVT T AL. Lmma 8. Th functon dfnd by (A5) s ncrasng for < dcrasng for > ; ts ma- xmum s qual to. Th functon = s also ncrasng f or < dcrasng for >. Th lmts of whn tnds to ( rsp. ) ar fnt corrspond to dotaton of all prmts to country (rsp. country ). Proof. Comput ng th drvatv of ln lads to Thus = has th sam sgn as = whch s postv for < ngatv for >. Snc s ncrasng wth rspct to = s ncras ng for < d- crasng for >. Th lm t of whn go s to (rsp. ) s th soluton of = = (rsp. = =) Ths lmts ar fnt th corrspondng lmts of = = ar postv smallr than. Th lmt valus of corrspond to dotatons of all prmts to on of th two countrs ( = f = = f = o ). Ths dotatons lad t an qulbrum wth = (rsp. = ) wth z < f only f < b (rsp. b. < Charactrzaton of an qulbrum wth z * = q * > Dotaton of prmts th total captal stock ar gvn. Assum q > z =. Wth captal stocks ˆ ˆ ˆ = mssons ar ˆ ˆ Ê = ba H = thr rato ˆ = ˆ wth vrfs (s quaton (5)) ˆ ˆ ˆ = ˆ = ˆ ˆ = = ˆ Thus wth ˆ ˆ ˆ = ˆ ˆ = H H H = w hav = A H H ˆ Aˆ H = (A6) At th qulbrum on th prmts markt = ˆ rfy ˆ ˆ = = = ˆ ˆ v ˆ ˆ ˆ H = = ba b (A7) whr = s th sam fun- cton as dfnd n Appndx (s quaton (A5)) Th qulbrum condton 3 on th captal ma rkt mpls (s quaton ()) ˆ ˆ ˆ ˆ ˆ = ˆ = = = ˆ ˆ x ˆ ˆ x (A8) whr xˆ = vrfs from quaton () ˆx bq Lmma 9 Thr xsts an qulbrum wth z = q > f only f thr xsts a soluton ê of (A7) a soluton ˆx of ˆ x = whr x = x x 5 Proof. Th xstnc of an qulbrum wt h z = q > mpls that = vrfs (A 7) that x = v rfs (A8) whch s quvalnt to bq x =. Convrsly consdr ˆ > vrfyng (A7) xˆ vrfyng x ˆ ˆ =. Dfn ˆ = wth (A7) = ˆ = q = >. Ths valus vrfy th qulbrum b xˆ condtons on both markts of prmts captal wth z =. Thus an qulbrum wth z = q > xsts. Lmma. Thr xsts an qulbrum wth z = q > = f only f = = b. Thn = = dfns an qulbrum wth = 5 Ths functon s smlar of th functon n Appndx A for x t concds wth : Thus Copyrght ScRs.

13 P.-A. JOUVT T AL. 99 = any q > Proof. x = q b dos not dp nd on x. Thus f = s an qulbrum wth z = q > x = mpls = (A7) mpl s = b snc =. Convrsly undr ths cond tons ˆ = any ˆx vrfy th xstnc condtons of Lmma 9. Lmma. If thr xsts an qulbrum wth = q > f only f b < b whr z vrfs: f > < < =. ths qulbrum s unqu < < f < Proof. Th drvatv s of vrfy : > = x. Assum thr xsts an qulbrum wth z = q > a nd >. x nc to whn x rass from ncrass from t o. Th xstnc of x x = s quvalnt to soluton of = And < mpls < < (Lmma 6) > snc < for >. (L mma 8) Wth (A7) for ˆ = w obtan th ncssary cond- tons of Lmma th uncty of > soluton of (A5) of x soluton of x =. xstnc rsults from Lmma 9. Assum thr xsts an qulbrum wth z = q > <. x dcrass from to whn x ncrass from t o. Th xstnc of x x = s thn quvalnt to soluton of Wth < t mpls < < > snc > for <. Thus th sam conclusons as n th cas > apply. Th proof s complt snc = s xcludd whn (Lmma ). Copyrght ScRs.