The Trade-off between Intra- and Intergenerational Equity in Climate Policy: Can Carbon Leakage be Justified?

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1 CREE Woking Pape 9/2012 The Tade-off between Inta- and Integeneational Equity in Climate Policy: Can Cabon Leakage be Jutified? Snoe Kvendokk, Eic Nævdal and Linda Nøtbakken The CREE Cente acknowledge financial uppot fom The Reeach Council of Noway, Univeity of Olo and ue patne. ISBN: ISSN:

2 Augut 2012 The Tade-off between Inta- and Integeneational Equity in Climate Policy: Can Cabon Leakage be Jutified? 1 by Snoe Kvendokk 2, Eic Nævdal 3 and Linda Nøtbakken 4 Abtact Thi pape focue on two equity apect of climate policy, inta- and integeneational equity, and analyze the implication of equity pefeence on climate policy, and on the poduction and conumption patten in ich and poo countie. We develop a dynamic two-egion model, in which each egion uffe fom local pollution and global waming, but alo ha an inequality aveion ove cuent conumption allocation. Inequality aveion ove conumption lift the conumption path of the poo egion, while it lowe the conumption path of the ich egion, which mut take a geate hae of the climate buden. Theefoe, a high abatement in the ich egion i met by moe pollution in the poo egion, thu jutifying cabon leakage. Moeove, the poo egion may even be allowed to inceae emiion elative to buine a uual unde the optimal climate policy. Thee effect ae einfoced when intoducing tanfe between the egion. Howeve, loan to poo countie to educe inequality may eult in a debt cii, and debt emittance may be pat of the optimal climate policy. JEL code: C63, D31, D63, Q54. Keywod: Intageneational equity; integeneational equity; inequality aveion; climate policy; economic development; cabon leakage; intenational tanfe; debt cii 1 Thi pape i funded by the MILJØ2015 pogam at the Reeach Council of Noway. We have benefitted fom dicuion with Gei Aheim, Reye Gelagh, Johan Eyckman, Samuel Fankhaue, Haifang Huang, Itzia Lazkano and Emilon C.D. Silva, in addition to comment fom paticipant at the SURED confeence in June The autho ae aociated with CREE - the Olo Cente fo Reeach on Envionmentally Fiendly Enegy - which i uppoted by the Reeach Council of Noway. 2 Coeponding autho. The Ragna Fich Cente fo Economic Reeach, Gautadallèen 21, 0349 Olo, Noway. noe.kvendokk@fich.uio.no. 3 The Ragna Fich Cente fo Economic Reeach, Gautadallèen 21, 0349 Olo, Noway. eic.navdal@fich.uio.no. 4 Albeta School of Buine, Univeity of Albeta, 2-32D Buine, Edmonton, Albeta, Canada T6G 2R6. E- mail: linda.notbakken@ualbeta.ca. 1

3 1. Intoduction While climate change ha been ecognized a a theat to the futue by mot cientit and politician fo many yea, thee i till an ongoing debate a to what to do about it. Reeache may not agee on the optimal emiion eduction, even if they agee on the natual cience backgound, the impact and the cot of abating geenhoue ga emiion. To a lage extent, the eaon fo thi i that optimal emiion eduction depend on equity iue, and how we dicount the futue climate impact i paticulaly impotant. 5 Howeve, ethical iue have not been fully exploed in economic analye, a geenhoue ga abatement not only affect the welfae ditibution between peent and futue geneation, but alo the ditibution within a geneation uch a that between ich and poo countie. Thee two equity apect ae impotant when tudying optimal emiion eduction, and a we explain below, they may wok in diffeent diection. The pupoe of the cuent tudy i to analyze the tade-off between the two dimenion of equity in climate policy. The two dimenion of equity in climate change police can be efeed to a inta- and integeneational. The fit i pimaily about how we hould ditibute the buden within a geneation, eithe within the geneation living today o in the futue, ee Kvendokk and Roe (2008). Two example of thi can be: who would uffe fom climate change (inaction), and how hould the buden of mitigation (action) be ditibuted? In the yea to come, the wold may face lage climatic change uch a inceaed tempeatue, ea level ie, changed wind and pecipitation patten, moe exteme weathe, etc. (IPCC, 2007a). Nevethele, the damage aociated with climate change will not be evenly ditibuted among countie o within a given county. Studie uch a thoe by Tol et al. (2000), Tol (2002a,b) and Yohe et al. (2007) how that ome ecto will loe fom climate change while othe will benefit, with the pooe countie likely to face elatively tonge negative impact than the iche countie. Seveal economic tudie alo eveal that the cot of action will vay among countie and ecto, and that it i geneally moe expenive to abate the moe enegy efficient the economy i (IPCC, 2007b). Policy intument implemented to educe geenhoue ga emiion will impoe diffeent buden on people, and economic intument uch a cabon taxe will often be egeive, i.e. the buden will be the highet fo the pooet (ee, e.g. Bye et al., 2002). 5 See Dagupta (2008) fo an oveview and Anthoff and Tol (2009) fo illutation. 2

4 While intageneational equity i impotant, mot of the equity debate on climate change iue in the economic liteatue ha been on integeneational equity iue, i.e. ditibution aco geneation, focuing on how lage emiion eduction we hould be aiming fo, o how lage the atmopheic geenhoue ga concentation o global mean tempeatue ceiling hould be. Thi affect the ditibution of buden between the cuent geneation and futue geneation, a the buden of mitigation - the cot - have to be taken by the peent geneation, while the benefit of mitigation will be felt by futue geneation. Thee ae eveal eaon fo extenive mitigation today uch a attitude towad ik a well a concen about catatophic event (Weitzman, 2007a), although mot of the dicuion have been about the appopiate dicount ate to ue fo climate change policy deciion, a the optimal level of abatement i vey enitive to the choice of dicount ate. Dicount ate ae weight put on the futue benefit of climate change policie in ode to compae them to the peent cot. If we meaue the cot and benefit in conumption unit, the main eaon fo dicounting ae that we may teat diffeent geneation diffeently (the pue ate of time pefeence), and that the benefit of a conumption unit diffe depending on the conumption level. The econd agument i that a high level of conumption give a low maginal utility of an additional unit, epeented by the elaticity of the maginal utility of conumption. Thu, a high conumption level fo futue geneation may be an agument fo paying le attention to thee geneation. The conumption dicount ate ued in economic analye combine thee two agument, which both epeent ethical choice. The choice of the appopiate dicount ate ha been a contoveial iue fo many yea. Fo intance, the Sten Review (Sten, 2007) ued a quite low conumption dicount ate, and theefoe found a high level of optimal abatement compaed to othe tudie uch a thoe of Nodhau (1993) and Nodhau and Boye (2000). Thi ceated a lively debate in which, e.g. Nodhau (2007) and Weitzman (2007b) agued againt the choice of the low dicount ate baed on the obeved value of the long-un etun to capital. Howeve, Heal (2009) how that equality between the conumption dicount ate and the etun to capital only hold unde idealized condition uch a a one-good model and pefect maket aumption including no extenal effect, which obviouly do not hold in the peence of climate change. The one-good model ha alo been citicized by Hoel and Stene (2007) and Stene and Peon (2008), who pointed out that if thee i conumption of an envionmental good in addition to a poduced conumption good, the envionmental good will become moe cace ove time 3

5 elative to the poduced good a we expeience moe climate change, which could be an agument fo uing a low and even negative conumption dicount ate fo envionmental evice. Latly, Nævdal and Vilie (2008) ague that the dicount ate ha a mino impact on the tabilization taget to avoid climate catatophe, and theefoe, a lage impact on futue geneation. Mot tudie focu on eithe an inta- o integeneational ditibutive jutice poblem, theeby implicitly auming that thee poblem ae autonomou, i.e. that they can be teated epaately. Howeve, choice that affect integeneational ditibution alo impact on the intageneational ditibution between ich and poo countie. A Heal (2009) point out, thee ae two way that pefeence fo equality affect the choice of climate action. Fit, a high elaticity of the maginal utility of conumption will lead to le aggeive action if we believe that conumption i gowing ove time. The eaon fo i that thi make futue geneation iche, and if we cae about inequality between the peent and futue geneation, we place a lowe value on the iche futue geneation (integeneational equity), but thee i alo anothe effect. The ich countie ae pimaily eponible fo the aggegate level of geenhoue gae in the atmophee, while a mentioned above, the poo countie ae likely to uffe the mot fom climate change. Hence, if we put a low weight on futue outcome, climate change i moe likely to occu, and poo countie may uffe moe (intageneational equity). Conequently, the gap between the welfae level of the ich and the poo may be highe, and baed on the latte effect, tonge pefeence fo equality hould go in the diection of moe action to help pevent climate change. Thee two effect of inequality aveion go in diffeent diection, and the impact of tonge pefeence fo equity on the level of geenhoue ga abatement ae ambiguou. The poblem i that the global model ued to detemine the optimal level of geenhoue ga emiion only take the fit effect into account (integeneational), meaning that tonge pefeence fo equality in thee model actually induce low abatement a found in, e.g. Nodhau and Boye (2000). A peviouly mentioned, economic analye have mainly concentated on epaately tudying the two afoementioned equity apect. One exception to thi i Baumgätne et al. (2012), who povide a geneal dicuion about economic analyi and the tade-off between inte- and intageneational jutice, while Glotzbach and Baumgätne (2012) give an analyi 4

6 of the elationhip between thee two jutice apect when it come to ecoytem management. Nonethele, we ae not awae of any tudie that take both inequality aveion into account when finding the optimal climate policy and thi pape cloe that gap. We ak the following quetion: How hould climate policie be deigned when policy make have pefeence fo both inta- and integeneational equity? A imple model of two egion, one ich and one poo, i et up to explicitly take into account pefeence fo the two apect of equality befoe finding the optimal climate policy. The integeneational apect i epeented by the tade-off between welfae in the peent and futue geneation due to the impact of global waming, while the intageneational equity concen in thi etup i puely a developmental iue, i.e. the conumption level of the poo compaed to the ich. Fo thi eaon, we do not tudy the vulneability to climate damage between the ich and poo countie. Ou main finding i that the pefeence fo intageneational equality will go in the diection of educed conumption in the ich egion and inceaed conumption in the poo egion. The optimal climate policy ubequently give the highet abatement buden to the ich wold, which again ha impact fo pollution a moe dity poduction i moved to the poo wold, theefoe jutifying cabon leakage. 6 Thi pape i oganized in the following way. In the next ection, we tudy the optimal climate policy contact in the peence of pefeence fo both inta- and integeneational equity, while Section 3 compae thi outcome with a ituation when a ocial contact i not poible. Inteaction between the egion uch a tanfe and intenational tade ae intoduced in Section 4 to ee how thi affect the ocial contact, and ou eult ae illutated with ome numeical imulation in Section 5. The final ection conclude. 2. Deciding on a Social Contact: A Model of Inequality Aveion We tat by conideing the optimal global climate contact. One way to think about thi i to follow Rawl (1971) and to aume a veil of ignoance ituation, in which epeentative individual fom popective countie aound the wold meet to deign the futue oganization of the ociety, the oiginal poition. All individual know that ome countie ae going to be ich and ome ae going to be poo, and that economic activity will eult in 6 The tem cabon leakage i often ued when thee i a limited paticipation in climate teatie (ee e.g. Felde and Ruthefod, 1993). In thi pape, we ue the tem when emiion eduction in one egion ae met by emiion inceae in anothe egion alo within a climate teaty. Altenatively, the tem pollution haven can be ued in thi cae. 5

7 geenhoue ga emiion that will have a negative impact on them in tem of climate change. It i alo known that mitigation today may affect developing countie moe negatively than indutialized countie a it may ham economic gowth and thei poibility to develop. The epeentative individual ae concened about conumption inequality, but they alo have pefeence ove the welfae of futue geneation compaed to that of the peent geneation. Howeve, they do not know if they ae going to live in a ich o poo county, o if they ae going to live in a time with a high o low global envionmental quality poduced by the concentation of geenhoue gae in the atmophee. Behind the veil of ignoance, the epeentative individual agee on the ocial planne deciion poblem. Since climate change i a global poblem, the epeentative individual o countie would not have an incentive to puue the global ocial optimum olution without a contact. Thu, they commit to follow the policy that come out of the ocial planne optimization poblem, a thi will wok a inuance fo all. We take a conequentialit tandpoint and conide the welfae of the individual a the final aim of it action. Baed on thi, a good ociety i one that maximize a ocial welfae function. 2.1 The Baic of the Model To fomalize thi, conide two egion n and, whee n denote the developed egion (Noth) and the developing egion (South). The welfae of a epeentative conume/county in egion n, at time t i: t U u( c, D, S ) max c c,0 max c c,0,, in,, i. (1) t, t, t, t it, t, t, i, c i conumption, D i local pollution and S i the tate of the global envionment, wheea i epeent the othe epeentative conume/egion. ucds (,, ) i a tandad utility function, which i inceaing in c and S, but falling in D, and with: ucds (,, ) c fo c 0, 2 ucds (,, ) 0 c D and 2 ucds (,, ) 0. The latte condition implie that local pollution 2 D 6

8 become inceaingly damaging a the pollution level ie. Note that in thi imple model, we do not conide diffeent degee of vulneability to climate change within the two egion. Pefeence fo equality ae modeled a inequality aveion following Feh and Schmidt (1999), and baed on thi, conume have an inequality aveion in uch a way that they dilike having highe conumption than othe, although they dilike even moe having a lowe conumption than othe conume that they compae themelve to. Thi teamline the economic development pepective a the intageneational apect, while, in contat, the climate change pepective i the integeneational apect in thi model. The Feh and Schmidt famewok ha pimaily been ued in decibing pefeence fo income equality among individual, but may alo be ueful a a deciption of the ocial pefeence of policy make in diffeent egion, o long a the tanfe between egion ae not aumed to be due to tategic eaon only. 7 Following thi, let α be a paamete epeenting the negative feeling of being woe off than othe, while β i the paamete epeenting the negative feeling of being bette off. We then have that α β. We ignoe tategic inteaction by auming that each egion, both Noth and South, conit of many identical countie that do not have any maket powe and cannot individually affect the oveall global envionmental quality. 8 Without a lo of geneality, let u aume that the population ize of the two egion ae equal and nomalized to unity. Theefoe, c i alo pe capita conumption in egion. Futhemoe, we model each epeentative county' poduction of an aggegate good, Y,t, with two type of input, clean and dity, both of which ae pefect ubtitute: 9 Y Y Y (2) t, ct,, dt,, 7 Othe altenative ocial pefeence could be ued, but thi i not cucial to ou concluion. One example i Chane-Rabin pefeence applied by Koltad (2011) to tudy coalition in public good poviion. 8 One could alo ague that the ich wold not only ha pefeence fo the equality of conumption, but alo fo the total welfae of the poo wold uch a how climate change affect poo ocietie. One example of thi can be that the ich wold feel eponible fo a lage pat of hitoical geenhoue ga emiion, which alo ha a negative effect on the welfae of the poo. If the utility function wa egion pecific, thi could be thought of a being embodied in the utility function. Howeve, a mentioned above, we have choen to focu on the economic development pepective of intageneational equity. 9 One example can be electicity. 7

9 Y c A K (3) ct,, ct, ct,, Y d A K (4) dt,, dt, dt,, whee ubcipt c and d denote clean and dity. Hence, the clean input i poduced uing clean (geen) capital, while the dity input i poduced uing dity (bown) capital. We aume that 0,1 fo j cd,, which implie diminihing maginal etun. The poductivity of j dity capital i highe than fo clean capital, i.e. A d,t > A c,t fo all t. Fo implicity, we aume thee total facto poductivitie to be contant. Howeve, within thi famewok, we can eaily intoduce technological development a a poitive tend in the total facto poductivitie. We aume that the capital tock of dity capital i initially highe in the Noth than in the South, i.e. Knd,,1 Kd,,1. Even o, a county can invet in clean and dity capital, with capital dynamic given by: K,, 1 1 j t j K, j, t I, j, t, n,, j c, d, (5) whee j i the capital depeciation ate. The eouce containt ae: 10 Y,, t, ct, Ict,, Idt,, n (6) By ubtituting fo invetment, I ct,, and I,, a county in egion can be ewitten a: dt, uing equation (5), the eouce containt fo 1 1 Yt, Kct,, 1 c Kct,, Kdt,, 1 d Kdt,, ct,. (7) 10 Note that we have not explicitly modeled maket hee, which can be jutified by a equence of pot maket that ae enewed aco geneation. 8

10 Similaly to Silva and Zhu (2009), we aume that thee i a co-poduction of local and global pollution fom the ue of dity capital. 11 A a eult, thee pollutant ae coelated. Aume that l i a coefficient that eflect the local pollution pe unit of dity capital ued, while g eflect the global pollution (geenhoue ga emiion) pe unit of dity capital. Fo local pollution, we then have: D,,. t, lkdt,, n (8) The global envionment i modeled a a tock vaiable that deteioate with global pollution following fom the aggegate ue of the dity capital ove the two egion, and egeneate natually at a ate : t 1 t g, d, t S S 1 S K (9) The global envionmental quality i contained a follow: St 0, S, whee S i the quality in abence of pollution. Note that in thi cae, S will convege aymptotically to S. We theefoe teat climate change a a eveible poce in the vey long un. Latly, we aume that conumption i initially lage in the Noth than in the South, i.e. c c n,1,1, ince the Noth i endowed with moe initial poductive capital than the South. Becaue of thi, we can wite the welfae function of the two egion at time t = 1 a:,, 1 1,1 U u c D S c c (10) n,1 n,1 n,1 n n,,, 1 1,1 U u c D S c c (11),1,1,1 n, 11 Silva and Zhu (2009) aume that pollution follow fom the poduction of the dity good. In ou model, it follow fom the ue of dity capital. In the long un (teady tate), howeve, thee i a contant elationhip between poduction and the capital tock. 9

11 2.2 The Social Contact Within thi modeling famewok, and with a ocial planne who detemine poduction, conumption and capital invetment in evey county (egion), conumption in the South will neve exceed conumption in the Noth (pe capita). Hence, we ue the implified welfae function (10) and (11) alo fo t > 1. The ocial planne eek to maximize the um of dicounted welfae aco egion, whee welfae in peiod t i defined a: W u( c, D, S ) u( c, D, S ) c c (12) t nt, nt, t t, t, t n nt, t, The maximization poblem can then be expeed a: max (,, ) { cnt, ct, Knct1, Kct1, Kndt1, Kdt1, St1} t0 t uct, Dt, St cnt, ct,, (13) ubject to (2)-(4) and (7)-(9). ρ i the time pefeence ate and contant., which i a n The Lagangian of the maximization poblem can now be expeed a follow: L t1 o uc (, t, D, t, St ) cn, t c, t t1 Y ( K, K ) K 1 K K 1 K c t, t, ct,, dt,, ct,, 1 c ct,, dt,, 1 d dt,, t, t S t 1 S 1 S t gk, d, t (14) whee λ i the hadow pice on capital, while μ i the hadow pice on the pollution tock. Fit ode condition include: 10

12 [ c ]: nt, [ c ]:, t uc ( nt,, Dnt,, St) c nt, uc ( t,, Dt,, St) c t, nt,, t (15) (16), 1, 1 1, 1 1 [ Kdt,, 1] : uc ( t, Dt, St ) Yt l, t 1 1 d t 1 g, t, n, D t, 1 K dt,, 1 (17) [ K ]: ct,, 1 [ ] t 1 Y t, 1 1 t, 1 1 c t,, n, (18) Kct,, 1 uc (, D, S ) t (19) t t t S : 1, 1, t1 St 1 Clealy, an impotant quetion when analyzing the optimal olution i whethe conumption in the two egion will convege to the ame level in the long un. In Appendix 1, we how that thi will be the cae, which give u the following Lemma: Lemma 1: In the long un, the ocially optimal conumption level of Noth and South will convege. Note that thi eult i independent of inequality aveion in conumption, but that inequality aveion acceleate the convegence poce, ee dicuion below. 2.3 Optimal Policy Let u now conide how inequality aveion affect the optimal conumption and capital path of the two egion. We tat by looking at optimal conumption. Fom the fit ode condition (equation (15) and (16)), we know that the maginal utility fom good conumption and inequality ( ) mut equal the hadow value of capital. Fo the Noth, thi implie that an additional unit of conumption today yield highe utility fom conumption, but diutility fom moe inequality. In the South, we have the oppoite effect of inequality aveion: An additional unit of conumption today inceae utility, both fom conumption and fom educed inequality. Conequently, intoducing inequality aveion hift the conumption path of the Noth downwad, while the conumption path of the South hift upwad. We ummaize thi eult in Popoition 1: 11

13 Popoition 1: Compaed to the cae without inequality aveion, inequality aveion within one geneation will lead to a eduction in conumption in the Noth and an inceae in conumption in the South unde the ocial contact. Fo the Noth to educe conumption and the South to inceae conumption elative to the cae without inequality aveion, the Noth invet elatively moe in clean capital while the South invet elatively moe in dity capital. It follow that the Noth i taking a bigge hit to impove the envionment, which can be thought of a an indiect tanfe of welfae fom Noth to South. Thi alo implie that a tonge concen fo equality caue the two egion' conumption level to convege fate. We can then wite Popoition 2: Popoition 2: Inequality aveion within one geneation will inceae invetment in clean capital and educe invetment in dity capital in the Noth in the ocial optimum. Fo the South, we find the oppoite eult. Thi popoition how that inequality aveion within one geneation may jutify cabon leakage in which pollution i tanfeed fom the Noth to the South. While cabon leakage i uually conideed a a bad (ee e.g. Felde and Ruthefod, 1993; Böhinge et al., 2010), thi Popoition open the poibility that it may be the olution to balancing the tade-off between inta- and integeneational equity. To ee how the tanition wok, conide the optimal capital path. To implify the notation, we define the following: MP, and MU x u c, D, S c Y j i the maginal poductivity of capital j in egion, K, j, with x c, D, S, i the maginal utility of conumption, local pollution, and global envionmental evice, epectively, in egion. By uing thi, eliminating common tem fom the two egion' optimality condition fo clean capital (18) and eaanging, we obtain the following condition: MP MP 1 t, nt, ct,, nct,, t, 1 nt, 1. (20) 12

14 In the hot un, befoe the two egion convege, we know that the Noth i iche and theefoe ha a highe utility than the South, although the diffeence between the two egion i deceaing ove time. Regadle of whethe the optimal teady-tate policy involve a eduction o an inceae in the Noth utility elative to the initial tate, it mut be the cae MU MU MU MU until the South ha caught up with the Noth. In c c c c that nt, nt, 1 t, t, 1 c c addition, we know fom (15) and (16) that nt, MUnt, and t, MU t,. With thi infomation, we can how that the following inequality mut hold until the egion convege:. (21), t, t1 nt, nt, 1 Inequality (21), which how that the diffeence in the hadow value of capital between the South and the Noth deceae ove time, alo implie that the ight-hand ide of (20) i poitive. Hence, while the South catche up with the Noth, the maginal poduct of clean capital i highe in the South than in the Noth, which implie a highe tock of clean capital in the Noth than in the South, Knct,, Kct,,. Latly, let u conide the optimal invetment in clean and dity capital in the long un. To do thi, we ewite the fit ode condition (15)-(19) fo the teady-tate level of the vaiable uing the new notation: MU (22) c n c n MU (23) D 1 lmu MP, d 1 d g, n, (24) 1 MP, c 1 c, n, (25) S 1 MU 1 (26) By ubtituting fo the teady-tate hadow pice of capital ((22)-(23)) and the envionment (26) in the optimality condition fo clean and dity capital ((24)-(25)), we obtain the following elationhip that the maginal poductivitie mut atify in the optimum: 13

15 MP MU MU MU D S S g n d, dv l C C MU v MU (27) MP c, v (28) c whee,, n captue the effect of inequality aveion fo both Noth and South, and v 1 1 i the dicount ate. The optimal level of clean capital equie it maginal poductivity to equal the um of the depeciation and dicount ate. Thee tem alo ente the condition fo the maginal poductivity of dity capital, but when conideing the optimal level of dity capital we mut alo account fo the welfae effect of local and global pollution. Conequently, the maginal poductivity of dity capital mut equal the um of depeciation and dicount ate, the value of educed local pollution and the value of impoved global envionmental quality. The two latte effect ae meaued in tem of inceaed conumption, with inequality conideation accounted fo ( ) a conumption level convege. Once convegence occu, the maginal poductivity of dity capital mut be the ame in all countie. Note that evey tem in the optimality condition fo dity capital (27) i poitive. Hence, even if the capital depeciation ate fo clean and dity capital ae equal, we would equie a highe maginal poductivity fom dity- than clean capital to invet. A dicued above, thi i becaue an invetment in dity capital mut alo compenate fo the welfae effect of local and global envionmental damage. Note that the inequality aveion paamete ente into the denominato of equation (27), thu patially deceaing the maginal poductivity of dity capital in the Noth while inceaing it in the South. Howeve, a conumption convege in the long un, and a the effect of global pollution ente equally in the utility function, we find that the level of dity capital in the two countie convege. How doe inequality aveion aco a geneation affect geenhoue ga emiion? The teady-tate condition fo global envionmental quality can be eaanged and expeed in tem of the hadow pice of the envionment: 14

16 S MUn MU 1 S 1 (29) It follow that the teady-tate value of the hadow pice of the global envionment ( ) inceae in the maginal utility of envionmental quality given by the numeato in equation (29), inceae in the eplenihment ate of the envionment and deceae in the dicount facto. Equation (29) alo eveal that the teady-tate level of global envionmental quality doe not depend on the egion pefeence fo equality ( ). Nevethele, the path of envionmental quality towad it teady-tate level doe depend on the egion inequality pefeence. The teady-tate level of global envionmental quality i given by: S g S Kd, (30) The envionmental quality path towad the teady-tate depend on the aggegate level of dity capital in the two egion. A een above, inequality aveion educe emiion in the Noth, while inceaing emiion in the South befoe the teady-tate i eached. If the emiion eduction in the Noth exceed the inceae in the South, the envionmental quality path would hift upwad (bette envionment). Still, without pecifying functional fom, it i not clea which of the two effect would dominate. Hence, the impact of inequality aveion on envionmental quality i ambiguou, and we will etun to thi in the numeical analyi in Section But what if a Contact i Not Poible? The Buine a Uual Cae The next quetion i what the action of the two egion of the wold would be if the ocial contact cannot be eached? A long a thee i no enfocement mechanim in place, the egion will be bette off by following thei own inteet and maximizing the welfae of a epeentative conume. Thi may be called the Buine a Uual poblem (BAU), i.e. the optimization poblem of the policy make when thee i no coodinated action o global envionmental ageement. 15

17 A aumed above, conumption in the Noth i lage than in the South at the tating point (t = 1), i.e., cn,1 c,1. Countie ae identical in all epect othe than conumption and capital tock level. If the initial conumption level follow fom optimization in the two egion, the South will neve have a highe conumption than the Noth. Theefoe, we can et up the BAU optimization poblem fo the oiginal model a conume maximizing: t t t1 t1 Un, t ucn, t, Dn, t, Stncn, t c, t (31) t1 t1 t t t1 t1 U, t uc, t, Dt,, Stcnt, ct, (32) t1 t1 epectively, given the technology (2)-(4), the eouce containt (7) and the local envionmental containt (8). The time pefeence ate, ρ, i et equal fo both egion in ode to not have the effect of inequality aveion confounded by the effect of dicounting. To find the non-coopeative Nah equilibia fo the two egion, we fit define the Lagangian. Fo the Noth, the Lagangian i: L t1 BAU uc ( nt,, Dnt,, St) ncnt, ct, t1 Y ( K, K ) K 1 K K 1 K c nt, nt, nct,, ndt,, nct,, 1 c nct,, ndt,, 1 d ndt,, nt, (33) whee λ n,t > 0 i the hadow pice on capital. Recall that we made the aumption of each epeentative county being o mall that it impact on the dynamic of global envionmental quality i appoximately zeo. A a eult, (33) i maximized ove conumption and dity and clean capital tock, in which the local envionment i taken into account, though global envionmental quality i taken a given. The fit ode condition can be witten a: c [ c ]: MUn n n, t (34) nt, 16

18 [ Kct,, 1] : nt, nt, 1 MPnct,, 11 c (35) [ ]: Kdt,, 1 D nt, MU nt, 1 lnt, 1 MPndt,, 1 1 d (36) The fit ode condition fo the South ae imila apat fom the detemination of conumption, which i found fom: c [ c ]: MU, t, t (37), t Equation (34) and (37) how that in the optimum, the peent maginal benefit fom inceaed conumption hould equal the oppotunity cot of inceaing the conumption, i.e. the hadow pice of capital. Compaed to tandad pefeence, i.e. without any inequality aveion (α = β = 0, = n,), the maginal benefit of conumption i lowe in the Noth, while being highe in the South, theeby indicating a lowe level of conumption in the Noth and a highe level of conumption in the South. Thi give u Popoition 3: Popoition 3: In buine-a-uual, inequality aveion within one geneation will lead to educed conumption in the Noth and highe conumption in the South compaed to tandad pefeence. Let u now compae the optimal conumption level unde BAU to the ocially optimal level by compaing equation (15) and (16) to equation (34) and (37). The diffeence in the optimality condition fo conumption i that each county unde the ocial contact take into account both countie diutility fom inequality, and not jut it own diutility, i.e. it take into account the conumption extenality that it impoe on the othe egion. Hence, the Noth educe- and the South inceae conumption moe unde the ocial contact than in the buine-a-uual cae, with all ele being equal. In addition to thi, the egion do not take into account thei impact on the global envionment, which goe in the diection of lowe conumption fo both egion in the ocial optimum compaed to BAU. Fo the Noth, both incentive mean a lowe conumption level unde the ocial contact compaed to BAU, but fo the South we have oppoing incentive. We ummaize thi eult in Popoition 4: 17

19 Popoition 4: Inequality aveion within one geneation will lead to a lage eduction in conumption in the Noth unde the ocial contact than in buine-a-uual, while fo conumption in the South, the eult i ambiguou. Equation (35) and (36) detemine the invetment in clean and dity capital, epectively. The intepetation of thee equation ae that the benefit of inceaed capital in peiod t hould equal the ocial cot of the capital invetment. Fo both type of capital, the cot of invetment i lowe conumption in the cuent peiod, while fo the dity capital the cot alo include lowe utility due to a woening of the local envionment. Uing equation (35) and (36), we can eliminate nt, 1 and,which yield the following elationhip between clean and dity capital in egion n, : MP MP MU. (38) 1 D dt,, ct,, d c t, l Equation (38) tate that the county hould adjut it tock of clean and dity capital o that the diffeence in the maginal poductivitie between them equal the diffeence in capital depeciation ate and the exta ocial cot of uing dity capital due to the deteioation of the local envionment. Note that the lat tem in equation (38) i poitive. Conequently, if cleanand dity capital depeciate at the ame ate, the county hould adjut it capital tock o that the maginal poductivity of dity capital i highe than that of clean capital. How would inequality aveion affect invetment in clean and dity capital in the two egion, epectively? Conide the Noth fit, a in thi egion inequality aveion lead to lowe conumption, which will futhe lead to lowe invetment in dity capital and highe invetment in clean capital. The intuition behind thi i that a lowe conumption would not equie an invetment in the mot effective capital (dity), while an invetment in clean capital will educe the emiion of local pollutant, theeby impoving the envionment. A bette local envionment will inceae utility and theefoe wok a a ubtitute to loweing conumption in the utility function in the Noth. In the South, the intuition wok the othe way aound and invetment in dity capital will inceae, while invetment in clean capital will be lowe. Thu, inequality aveion till ceate cabon leakage fom Noth to South. 18

20 Let u next conduct the ame compaion between the ocial contact and BAU fo egional capital tock a we did fo conumption level, which involve compaing optimality condition (17) and (18) to condition (35) and (36). The optimality condition fo clean capital i the ame in the two cae. Howeve, the optimality condition fo dity capital unde the ocial contact include an additional tem compaed to the BAU cae. Thi tem epeent the maginal effect of moe dity capital on global envionmental quality, with the incluion of thi tem in the ocial contact cae implying a lowe invetment in dity capital compaed to BAU ince the envionmental extenality i intenalized. An inteeting concluion fom inequality aveion i that while the tock of dity capital will be lowe fo the Noth unde the ocial contact than in BAU, we cannot tell if the tock hould be lowe fo the South unde the ocial contact. Fo thi eaon, the implication i that poo countie hould not neceaily have a pollution containt unde a global climate teaty. Unde cetain condition it may actually be optimal that poo countie inceae thei emiion unde a climate teaty. Thi give u Popoition 5: Popoition 5: Inequality aveion within one geneation will lead to a eduction in emiion in the Noth unde the ocial contact compaed to buine-a-uual, while fo the South the eult i ambiguou, and emiion may actually be highe unde the ocial contact. 4. Inteaction between the Region Let u now tun once again to the ocial contact. In the model above, the only inteaction between the two egion came via the impact of pollution on the global envionment. We now intoduce intenational tanfe (uch a enhancing development in the poo wold) and intenational tade to ee how thi change the concluion. 4.1 Intenational Tanfe Fit conide the implication of allowing fo unlimited intenational tanfe between the two egion. Tanfe between the two egion can eaily be intoduced into the model peented above by adding the tem t, which epeent tanfe fom Noth to South in peiod t to the egion eouce containt. The two egion modified eouce containt then become: 19

21 Y ( K, K ) c K 1 K K 1 K (39) nt, nct,, ndt,, nt, t nct,, 1 c nc,, t n, d, t1 d n, d, t Y ( K, K ) c K 1 K K 1 K (40) t, ct,, dt,, t, t ct,, 1 c c,, t dt,, 1 d dt,, In addition, we intoduce a containt on Noth-South tanfe uch that t M in evey time peiod. The ocial planne mut now alo detemine the ize of the tanfe t ; theefoe, we obtain an additional fit ode condition fo thi vaiable. The fit ode condition fo the modified optimization poblem include the optimality condition tated above fo the cae without tanfe ((15)-(19)), along with the following condition fo the Noth-South tanfe:, (41) nt, t t, whee t 0 i the hadow pice on the tanfe containt ( t M ). The optimality condition how that the optimal tanfe policy i a mot apid appoach path towad equality. In the cae of unlimited tanfe, which mean that t 0 fo all t in (41), thee hould be a tanfe fom Noth to South in the fit peiod that completely eliminate inequality. Fom the econd peiod onwad, all countie ae equal and thee i no diutility fom inequality. If the tanfe containt i binding, condition (41) how that the hadow pice in the Noth i highe than the hadow pice in the South. In thi limited tanfe cae, the optimal policy i to tanfe a much a poible fom Noth to South ( t M ) until the conumption path convege. The mot apid appoach path to equality i the optimal policy egadle of whethe we account fo inequality aveion. The eult that conumption level hould be equalized a oon a poible i alo evident if we combine condition (41) with the optimality condition fo conumption in both the Noth and South. Allowing fo tanfe fom Noth to South enable the ocial planne to eliminate inequality oone. Futhemoe, the ocial planne eek to eliminate inequality between the two egion egadle of whethe the countie have inequality pefeence, ince the bet ue of available 20

22 eouce equie that maginal poductivitie ae equal aco all countie at all time. Hence, it i deiable to have the ame level of clean and dity capital in both egion. Thi eult alo uppot the concluion of highe conumption and moe dity poduction in the South. Nonethele, note that in thi cae the cabon leakage fom Noth to South will follow independently of the inequality aveion. We ummaize the eult on tanfe in Popoition 6: Popoition 6: Tanfe between the egion will enhance the convegence poce unde the ocial contact. Tanfe fom Noth to South hould follow the mot apid appoach path to equality between the egion. A a eult, tanfe inceae pollution in the South and deceae pollution in the Noth. Figue 1 illutate the movement in the dity capital tock. The capital level will convege and the dity capital tock will fall in the Noth, while inceaing in the South. Note, howeve, that with technological gowth, i.e. a poitive tend in the total facto poductivitie, we cannot ule out the poibility of an inceaing dity capital tock in the Noth, but tanfe will unambiguouly inceae the dity capital tock of the South, again jutifying cabon leakage. Figue 1: The path of dity capital with limited tanfe compaed to no tanfe 21

23 4.2 Intenational Tade Thu fa, we have dicued pue tanfe fom the Noth to South. Let u now ee how the eult change if we allow fo intenational tade in the input facto. Define Edt, a expot fom Noth to South of the dity input facto, while Ect, i expot fom Noth to South of the clean input facto, meaning that the South expot i equal to E dt, and, E ct epectively. The available amount of input ae then: Y Y ( K ) E, j c, d (42) njt,, njt,, njt,, jt, Y Y ( K ) E, jc, d jt,, jt,, jt,, jt, (43) In thi imple etup, we impoe maket cleaing without conideing pice. Thi can be jutified in the ocial optimum, a we do not conide any tategic inteaction that may affect pice. The tempoal budget containt ae: 1 1 c Y ( K ) E Y ( K ) E K K K K nt, ndt,, ndt,, dt, nc,, t n, c, t c, t nct,, 1 c nct,, ndt,, 1 d nd,, t 1 1 c Y ( K ) E Y ( K ) E K K K K t, dt,, dt,, dt, c,, t, c, t c, t ct,, 1 c ct,, dt,, 1 d, d, t (44) (45) Latly, we have the intetempoal tade balance: t 1 t (1 v) E E 0 dt, ct, (46) whee v i the inteet ate. Note, howeve, that in thi model the inteet ate equal the time pefeence ate, ee equation (18). Thi mean that the intenational tade balance can be witten a: t1 1t E dt, ct, 22 E 0 (47)

24 In a ocial optimum, tade i detemined diectly by the ocial planne; a a eult, tade can be viewed a tanfe that have to be paid back with inteet. Intoducing thi into ou model yield the following fit ode condition whee φ > 0 i the hadow pice on the intetempoal tade balance: 2(1 t) t, nt, (48) Note that without the tade balance containt, we find that nt, t,, i.e. the eult with unlimited tanfe. The development of the conumption path ae a follow. At time t = 1, we have, thu the diffeence between the hadow pice of capital i at it minimum and t, nt, i equal to the hadow pice of the tade balance, which epeent the payback fom South to Noth. Howeve, note that with a poitive dicounting, the ight-hand ide of (48) inceae exponentially and convege to a t. Theefoe, we obtain a Geek dieae, in which it i bette to equalize conumption a fa a poible now and to take the cot of paying back in the futue, though thee cot will explode ove time. Yet once again, thi eult einfoce ou peviou eult with cabon leakage; an inceae in dity capital and pollution in the South that allow the South to inceae it conumption level and pay back it debt. Thi give u the final popoition: Popoition 7: Intenational tade between egion in unde the ocial contact will educe the diffeence in conumption between them in the nea tem, but with a poitive dicount ate, the gap in conumption will inceae ove time until it convege. To help ee thi, let u tat with the left-hand ide of (48), which mut alo convege to infinity a t. Thi mean that the diffeence in the hadow pice of capital between the South and the Noth mut appoach infinity. We know that in the long un, the South mut expot good to the Noth fo the tade balance containt to hold. A a conequence of thi, the South conumption level decline, while the Noth conumption inceae ove time. 23

25 Fom the fit ode condition fo conumption (equation (15) and (16)) and the aumption lim MU c0 c t,, it i clea that fo the tem, t n, t to appoach infinity a t, conumption in the South mut appoach zeo. Hence, in the long un, conumption appoache zeo in the South, while it inceae towad a finite uppe bound in the Noth, which i illutated in Figue 2: Figue 2: Geek dieae (poitive dicount ate) Figue 3: Geek dieae (no dicounting ) In the pecial cae without dicounting ( 1), the optimality condition fo tade implifie to. A with dicounting, thee hould be tade fom Noth to South in the fit t, nt, peiod o that the diffeence between the two hadow pice of capital equal the hadow pice of balanced tade. To eliminate the two egion tade balance uplue and deficit in the long un, the South mut expot good to the Noth fom peiod 2 and onwad. Becaue of thi, the two egion will neve convege completely. Intead, tade allow the two egion to immediately adjut to the long-un level with ome inequality, while that inequality i maintained ove time to achieve balanced tade, which i illutated in Figue 3. Note that the le tade that i initially needed to eliminate inequality, the le containing the tade balance containt will be, and the malle the diffeence between the countie in the long un (i.e. malle ). A i evident fom Figue 2 and 3, the dicount facto i a citical deteminant of the dynamic of the two egion. Regadle of the dicount facto, tade fom Noth to South 24

26 hould be ued to educe inequality today. The gain fom educed inequality in the hot un i balanced againt the lo fom inequality in the long un. Without dicounting, the South conume le than the Noth becaue of the expot needed to achieve balanced tade in the long un. A we intoduce dicounting, the South conumption level will not jut be lowe than that of the Noth, it will appoach zeo in the long un. A highe dicount ate mean that we cae le about futue conumption and inequality, but it alo implie that the South mut educe conumption moe when it pay back it tade debt in the futue. Thu, it i ocially optimal to help the South today to eliminate inequality between the egion, even though the South will collape in the vey long un. It follow that even when people cae about equality and uch pefeence ae accounted fo by a welfae maximizing ocial planne, tade doe not enue equality in the long un. Clealy, the intoduction of an additional containt in the optimization poblem of the ocial planne can only woen the outcome. Unlimited tanfe without a (binding) tade balance containt would theefoe yield highe welfae than the tade olution dicued above, which ha eveal policy implication. Fit, thi uppot debt emittance to developing countie to help educe inequality. The developed wold can then tanfe wealth to the developing wold, both by emitting debt and by uing moe clean technology, theeby allowing pooe countie to ue moe poductive though ditie technology to develop fate. Eventually, howeve, a the inequality between diffeent egion diminihe, the poo countie mut alo ue moe clean technologie to educe aggegate pollution. Second, ou finding imply that wealth tanfe fom ich to poo countie, uch a development aid and foeign diect invetment, can be welfae impoving, paticulaly in the hot un to help peed up the poce of eliminating inequality between countie. 5. Numeical Simulation In ode to illutate ou eult, a numbe of imulation have been pefomed whee we examine how inequality aveion affect the ocial contact. In the imulation we have aumed that the Noth and the South have the ame utility function given by: a 1 Uc, D, S c S t b 1 D c t t t t (49) 25

27 The ocial contact ha been imulated fo thee value of aggegate inequality aveion 0,1,1.5 ove a time hoizon of 150 yea. A thee imulation ae fo illutation only, the paamete value choen ae baed on bet-gue, apat fom the total facto poductivitie in the poduction function (3) and (4) that ae taken fom Acemoglu et al. (2012). See Appendix 2 fo paamete value. 12 Conumption level in the Noth and the South have been plotted in Figue 4. A expected, the conumption patten change a inequality aveion inceae. Thee ae eveal thing to notice about thee change. The imulation confim that conumption level in the Noth and the South convege ove time, and the point of time when inequality i completely eliminated deceae in inequality aveion,. An inteeting apect of the conumption dynamic i that thee i a dip in Noth' conumption jut afte t 1. Thi dip become moe ponounced a inceae. Note that given ou paametic choice, both egion chooe to build down thei tock of dity capital. 13 When 0, the initial dip in conumption i caued by depeciation of dity capital befoe the tock of clean capital ha accumulated ufficiently to compenate. Thi can be een in Figue 5. A inceae, it become moe and moe impotant to educe the diffeence between Noth and South' conumption. Thi confim Popoition 1. When 1.5, thi implie apid convegence between conumption level. Howeve, in thi paticula cae, afte an initial peiod of convegence conumption patten divege befoe finally conveging at a late point in time. Thi divegence may appea puzzling, but i actually an end tem effect a the imulation ae un within a given time fame. A een fom Figue 5, the highe the inequality aveion, the moe the Noth invet in clean capital. Thi i not becaue the capital accumulation i diectly economically beneficial, but moe a a device to educe conumption to make it moe in line with conumption in the South. When conumption patten late divege, it i becaue the Noth ha accumulated exce capital and need to conume it at ome point in time. Thi conumption i potponed a dicounting make the diffeence in conumption level le cotly when potponed. Figue 5 alo confim the theoetical eult that inequality aveion lead to highe level of dity capital in the South. Although dity capital initially depeciate at the ame ate 12 The Matlab code ued to un the imulation ae available fom the autho upon equet. 13 In ou imulation, the exponent in the poduction function fo dity and clean input ae et equal (0.5), ee Appendix 2. An inceae in the elative poductivity of dity capital would eult in moe invetment in thi type of capital. 26

28 egadle of egion and inequality aveion, cabon leakage late become an iue and moe o the highe the inequality aveion. Finally, the figue confim ou theoetical eult of long un convegence in the clean and dity capital tock fo the egion. 9 = 0 9 = 1 9 = 1.5 C n C Conumption, C time Figue 4: Conumption in the Noth (black) and the South (blue) fo diffeent level of inequality aveion Clean capital tock, K c Dity capital tock, K d = 0 = 1 = time time Figue 5: Accumulation of clean and dity capital unde diffeent degee of inequality aveion: Noth (black) and South (blue) 27

29 The effect of inequality aveion on global envionmental evice can be een in Figue 6. Becaue of the cabon leakage to the South, a een in Figue 5, the quality of the global envionment i lowe when inequality aveion i high. Thi effect i howeve tanitoy. In the long un we ee that the envionmental quality i independent of inequality aveion, which alo confim ou peviou theoetical eult Envionmental quality, S = 0 = 1 = time Figue 6: Global envionmental quality 6. Concluion Thi pape tudie the tade-off between inta- and integeneational equity a epeented by pefeence to educe futue climate damage and to inceae economic development in the poo wold today. We find that inequality aveion within a geneation will educe conumption and geenhoue ga emiion in the ich egion of the wold, while conumption and emiion will inceae in the poo egion. Thi follow fom a tanfe fom dity to clean capital in the ich egion, while the poo egion pedominantly invet in moe poductive dity capital to catch up with the ich Noth. Thi eult jutifie cabon leakage fom Noth to South a a eult of pomoting economic development. Intoducing tanfe between the egion einfoce thi eult. Whethe total geenhoue ga emiion will 28

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