The Trade-off between Intra- and Intergenerational Equity in Climate Policy

Transcription

1 The Trade-off beween Inra- and Inergeneraional Equiy in Climae Poliy Kverndokk S. E. Nævdal and L. Nøsbakken Posprin version This is a pos-peer-review pre-opyedi version of an arile published in: European Eonomi Review This manusrip version is made available under he CC-BY-NC-ND 4.0 liense see hp://reaiveommons.org/lienses/by-n-nd/4.0/ The definiive publisher-auheniaed and formaed version: is available a: Kverndokk S. E. Nævdal and L. Nøsbakken 2014 The Trade-off beween Inra- and Inergeneraional Equiy in Climae Poliy European Eonomi Review vol DOI: /j.euroeorev hps://doi.org/ /j.euroeorev Frish Cenre Gausadalléen Oslo Norway. hp:// Frish Cenre

2 January 2014 The Trade-off beween Inra- and Inergeneraional Equiy in Climae Poliy * by Snorre Kverndokk Eri Nævdal and Linda Nøsbakken Absra This paper fouses on wo equiy dimensions of limae poliy inra- and inergeneraional and analyzes he impliaions of equiy preferenes on limae poliy and on he produion and onsumpion paerns in rih and poor ounries. We develop a dynami wo-region model in whih eah region suffers from global warming bu also has an inequaliy aversion over urren onsumpion alloaions. Inequaliy aversion generally lifs he onsumpion pah of he poor region while he rih region mus ake a greaer share of he limae burden. Furhermore wih inequaliy aversion he opimal limae poliy generally leads o higher invesmen in lean apial in he Norh and in diry apial in he Souh hereby allowing he Souh o pollue more and develop faser. The opimal poliy may even require he poor region o inrease emissions relaive o he unoordinaed business-as-usual ase. Inroduing loal polluion and ransfers onfirm he main resuls. JEL odes: C63 D31 D63 Q54. Keywords: Inrageneraional equiy; inergeneraional equiy; inequaliy aversion; limae poliy; eonomi developmen; inernaional ransfers; loal polluion. * This paper is funded by he MILJØ2015 program a he Researh Counil of Norway. We have benefied from disussions wih Geir Asheim Johan Eykmans Samuel Fankhauser Reyer Gerlagh Bård Harsad Haifang Huang Iziar Lazkano Ale Seiersad and Emilson C.D. Silva in addiion o ommens from pariipans a he SURED onferene 2012 he CREE workshop in Oslo in Sepember 2012 he Annual Meeing of he Norwegian Assoiaion of Eonomiss in 2013 he AERE summer onferene 2013 EAERE 2013 EEA-ESEM 2013 as well as wo referees an assoiae edior and he ediors of he journal. The auhors are assoiaed wih CREE - he Oslo Cenre for Researh on Environmenally Friendly Energy - whih is suppored by he Researh Counil of Norway. Corresponding auhor. The Ragnar Frish Cenre for Eonomi Researh Gausadallèen Oslo Norway. snorre.kverndokk@frish.uio.no. The Ragnar Frish Cenre for Eonomi Researh Gausadallèen Oslo Norway. eri.navdal@frish.uio.no. Norwegian Shool of Eonomis Deparmen of Eonomis Helleveien Bergen Norway. linda.nosbakken@nhh.no. 1

3 1 Inroduion While mos sieniss and poliiians have reognized limae hange as a hrea o he fuure for many years here is sill an ongoing debae as o wha o do abou i. Researhers may no agree on he opimal emissions reduions even if hey agree on he naural siene bakground he impas and he oss of abaing greenhouse gas emissions. One imporan reason is ha opimal emission reduions depend on equiy issues and our disouning of fuure limae impas is pariularly imporan. However ehial issues have no been fully explored in eonomi analyses as greenhouse gas abaemen no only affes he welfare disribuion beween presen and fuure generaions bu also he disribuion wihin a generaion suh as beween rih and poor ounries. These wo equiy dimensions are imporan when sudying opimal emissions reduions and as we explain below hey may work in differen direions. The purpose of our sudy is o invesigae he rade-off beween he wo dimensions of equiy in limae poliy. We ask he following quesion: How should we design limae poliies when people have preferenes for boh equiy dimensions and wha are he impliaions for emissions and energy invesmens? These dimensions of equiy an be referred o as inra- and inergeneraional. The firs is primarily abou how we should disribue he burdens wihin a generaion eiher wihin he generaion living oday or wihin fuure generaions see Kverndokk and Rose (2008). Two examples of his an be: who would suffer from limae hange (inaion) and how should he burdens of miigaion (aion) be disribued? In he years o ome he world may fae large limai hanges suh as inreased emperaures sea level rise hanged wind and preipiaion paerns and more exreme weaher (IPCC 2013). However he assoiaed damages will no be evenly disribued among ounries or wihin a given ounry. Sudies by Tol e al. (2000) Tol (2002ab) and Yohe e al. (2007) show ha some seors will lose from limae hange while ohers will benefi. Poorer ounries are likely faing relaively sronger negaive impas han riher ounries. In addiion several sudies sugges ha he oss of aion will vary aross ounries and seors and ha abaemen is generally more expensive in more energy effiien eonomies (IPCC 2007). Poliy insrumens implemened o redue greenhouse gas emissions will impose differen burdens on people and eonomi insrumens 2

4 suh as arbon axes will ofen be regressive so ha he poores will fae he highes burden (see e.g. Bye e al. 2002). While inrageneraional equiy is imporan mos of he equiy debae relaed o limae hange in he eonomi lieraure has been on inergeneraional issues. This debae has foused on he size of emissions reduions o aim for and on wha should be he upper limi on he amospheri greenhouse gas onenraion or he global mean emperaure. These quesions also affe he disribuion of burdens beween he urren generaion and fuure generaions as he oss of miigaion are borne by he urren generaion while fuure generaions benefi from i. Aording o he lieraure here are several reasons for exensive miigaion oday suh as aiudes oward risk and onerns abou aasrophi evens (Weizman 2007a). However mos of he disussions have been abou he appropriae disoun rae for limae poliy deisions as he opimal abaemen level is very sensiive o his parameer (Nordhaus 2007; Weizman 2007b; Dasgupa 2008) whih again represens ehial hoies. 1 Inergeneraional aspes of limae hange have also been sudied by John and Pehenino wih oauhors (John and Pehenino 1994; John e al. 1995) who fous on he radeoff beween eonomi growh and environmenal qualiy. Mos sudies rea inra- or inergeneraional equiy separaely. However hoies ha affe inergeneraional disribuion also affe he inrageneraional disribuion beween rih and poor ounries. As Heal (2009) poins ou here are a leas wo ways in whih preferenes for equaliy affe he hoie of limae aion. Firs if we believe ha onsumpion inreases over ime a high elasiiy of marginal uiliy of onsumpion leads o less aggressive aion. The reason is ha his makes fuure generaions riher and if we are abou inequaliy beween he presen and fuure generaions we plae a lower value on he fuure rih (inergeneraional equiy). There is however an addiional effe. The poor ounries are likely o suffer he mos from limae hange. Hene if we pu a low weigh on fuure ouomes limae hange is more likely o our and hi poor ounries hard 1 The onsumpion disoun rae used in eonomi analyses depends on he pure rae of ime preferene (uiliy disouning) and he elasiiy of he marginal uiliy of onsumpion whih boh represen equiy hoies. In addiion if a sok variable suh as he environmen eners he uiliy funion we ge anoher erm in he onsumpion disoun rae ha depends on he elasiiy of he marginal uiliy of onsumpion wih respe o he level of he sok see e.g. Heal (2007). Again his variable represens an equiy hoie. 3

5 (inrageneraional equiy). Consequenly he gap beween he welfare levels of he rih and he poor may be wider and based on he laer effe sronger preferenes for equaliy should go in he direion of more aion o preven limae hange. These wo effes of inequaliy aversion work in differen direions and he impas of sronger preferenes for equiy on he level of greenhouse gas abaemen are ambiguous. However global models used o deermine he opimal level of greenhouse gas emissions fous on he firs effe (inergeneraional) implying ha sronger preferenes for equaliy indue low abaemen (see e.g. Nordhaus and Boyer 2000). 2 Shelling (1992) suggesed one soluion o his by arguing ha he bes way o redue he impas of global warming is developmen of he poor region. The developed world is no as vulnerable o limae hange due o heir high level of eonomi developmen. We herefore an redue he vulnerabiliy of poorer ounries by leing hem develop. The resul may hen be ha he world is no hi as hard by limae hange while eonomi differenes beween regions are redued. Apar from Shelling few eonomiss have disussed he linkages beween he wo equiy dimensions. However reenly Baumgärner e al. (2012) provided a general disussion abou he rade-offs beween iner- and inrageneraional equiy in eonomi analysis while Glozbah and Baumgärner (2012) analyze he relaionship beween hese wo aspes in eosysem managemen. We are no aware of any sudies of opimal limae poliy ha ake boh ypes of inequaliy aversion ino aoun when invesigaing he impas on emissions and invesmens in lean and diry apial. Our paper aims o lose his gap. We se up a simple model wih wo regions a rih and a poor o expliily aoun for equiy preferenes along he wo dimensions. The inergeneraional aspe is he rade-off beween welfare for presen and fuure generaions due o he impas of global warming while he inrageneraional aspe is purely a developmenal issue as we ompare he onsumpion levels of he poor and he rih. We use he Fehr and Shmid (1999) framework o express he laer onern. A reen experimen wih pariipans who have been involved in inernaional limae poliy suppors his (Dannenberg e al. 2010). 2 These models ofen apply Negishi weighs ha freeze he urren disribuion of inome. Hene hey do no onsider inrageneraional disribuion (Sanon 2011). 4

6 We do no sudy differenes in vulnerabiliy o limae damage aross ounries as disussed by Shelling (1992) bu fous on he impliaions of eonomi developmen in he poor region for emissions and apial invesmens. Our main finding is ha preferenes for inrageneraional equaliy shif he limae burden oward he rih region; he poor region should generally use he more produive diry apial o speed up is developmen while he rih region should arry mos of he abaemen burden. Sine lean apial is less produive he onsumpion in he rih region falls and inreases in he poor region. Hene his resul suppors he laims made by developing ounries in global limae negoiaions ha emissions reduions will ause sebaks on he road o developmen. The paper is organized in he following way. In he nex seion we sudy he opimal limae onra when people have preferenes for boh inra- and inergeneraional equiy while Seion 3 ompares his ouome o he Business-as-Usual ase (no soial onra). In Seion 4 we inrodue some exensions and analyze he impliaions of dire ransfers and loal polluion under he soial onra. We illusrae our resuls wih numerial simulaions of he opimal limae poliy in Seion 5. The final seion onludes. 2 Deiding on he Soial Conra: A Model of Inequaliy Aversion As a saring poin we sudy he opimal global limae onra. To do his we ake a onsequenialis sandpoin and onsider he aggregae welfare of individuals as he soial objeive. Hene he soial onra maximizes a soial welfare funion. 2.1 The Basis of he Model Consider wo regions n and s where n denoes he developed region (Norh) and s he developing region (Souh). The welfare of a represenaive onsumer/ounry in region r n s a ime is: U u S max 0 max 0 r k n s r k (1) r r k r r k where r is onsumpion and S is he sae of he global environmen while k denoes he oher region. r u S is a sandard uiliy funion ha is inreasing and onave in r and 5

7 u r S S and has he propery: lim r 0 r. Furhermore we assume ha onsumpion and environmenal qualiy are omplemens: U ( r S ) 0. As menioned above we do no S r onsider differen degrees of vulnerabiliy o limae hange beween he wo regions. We model preferenes for equaliy as inequaliy aversion following Fehr and Shmid (1999). This implies ha people dislike having higher onsumpion han ohers bu hey dislike even more o onsume less han ohers. 3 This sreamlines he eonomi developmen perspeive as he inrageneraional aspe. In onras he limae hange perspeive is he inergeneraional aspe in our model. The Fehr and Shmid framework has primarily been used o desribe preferenes for inome equaliy among individuals bu may also be useful as a desripion of he soial preferenes of poliy makers in differen regions as long as he ransfers beween regions are no due o sraegi reasons only. 4 Following his le α be a parameer represening he negaive feeling of being worse off han ohers are while β is he parameer represening he negaive feeling of being beer off. We hen have ha 0. We ignore sraegi ineraions by assuming ha eah region Norh and Souh onsiss of many idenial ounries ha do no have any marke power and anno individually affe he overall level of global environmenal qualiy. Noe ha sine he uiliy funion is inreasing and onave in onsumpion a soial planner seeking o maximize he sum of welfare over he wo regions will redue inequaliy in onsumpion as his inreases aggregae welfare. Hene even wihou he Fehr-Shmid inequaliy aversion in he uiliy funion (1) here are gains from eliminaing inrageneraional inequaliy. However his is no driven by aversion oward inequaliy per se 3 This assumpion is in onras o he resul from one experimen wih pariipans who have been involved in inernaional limae poliy. Dannenberg e al. (2010) find ha pariipans dislike o a onsiderable exen being beer off han ohers are while heir aversion o being worse off han ohers is moderae. However his does no have any impliaions for he analyses below. 4 We ould use oher alernaive soial preferenes bu his is no ruial o our onlusions as long as hey express preferenes for equaliy in payoffs suh as onsumpion. One example is Charness-Rabin preferenes (Charness and Rabin 2002) applied by Kolsad (2011) o sudy oaliions in publi goods provision. 6

8 bu by he desire o maximize aggregae uiliy. Alloaing resoures for onsumpion where hey yield he highes reurn ahieves his. Also hese gains are only presen in he soial planner ase: he onaviy of he uiliy funion does no give individual ounries inenives o redue inrageneraional inequaliy as ounries only are abou heir own welfare no aggregae global welfare. Wihou loss of generaliy le us assume ha he populaion sizes of he wo regions are equal and normalized o uniy. Therefore r is per apia onsumpion in region r in period. Furhermore eah represenaive ounry produes an aggregae good Y using lean and diry inpus Y j d whih are perfe subsiues. 5 For ease of exposiion we assume j ha he produion funions are onsan aross ime and regions: Y( K K ) Y ( K ) Y ( K ) r n s (2) r r d r d r d where subsrips and d denoe lean and diry respeively. The lean inpu is produed wih lean apial K r while he diry inpu is produed wih diry apial K assume diminishing marginal reurns in boh produion proesses. r d. We Iniially ( = 0) he Norh has more of boh ypes of apial han he Souh: Kn j0 Ks j0 for j d. Indeed given ha preferenes and ehnology are he same in boh regions Norh's larger iniial sok of apial defines i as he riher region bu he regions are idenial in all oher aspes. Eah ounry an inves in lean and diry apial wih apial dynamis given by: K 1 1 r j j K r j Ir j r n s j d (3) where j is he apial depreiaion rae. The ounries resoure onsrains are: 6 Y( Kr Kr d ) r Ir Ir d r n s. (4) 5 One example is eleriiy produed from eiher lean or diry soures. 6 Noe ha we do no expliily model markes. This an be jusified by imagining a sequene of spo markes ha are renewed aross generaions. 7

9 By subsiuing for invesmen I r and I r d onsrain of a ounry in region r as: from equaion (3) we an wrie he resoure Y( Kr Kr d ) Kr 1 1 Kr Kr d 1 1 d Kr d r r n s. (5) We model he global environmen as a sok variable ha deerioraes wih global polluion (e.g. limae hange) whih follows from he aggregae use of he diry apial assuming a onsan emissions faor 0 and regeneraes naurally a a rae 1 0: 1 r r d S S 1 S K (6) The equaion implies ha he global environmenal qualiy saisfies he following onsrain: S 0 S where S is he level in absene of polluion. Noe ha wihou polluion S onverges asympoially o S. We herefore rea limae hange as a reversible proess in he very long run. 2.2 The Soial Conra The soial planner seeks o maximize he sum of disouned welfare aross regions where welfare in period is given by: n s max n 0 max s 0 W u S u S s n (7) I is imporan o noe ha W is no differeniable when n s beause of he maxoperaors. However W is righ differeniable whih is all ha is required for onsrained maximizaion o work. We an alulae he following derivaives when n s : d d r d d k max r k 0 r k r k max 0 0 r k (8) The soial planner seeks o find he onsumpion and invesmen pahs for eah region ha maximize he sum of disouned welfare given by equaion (7): 8

10 max u r S max n s 0 max s n 0 n s Kn 1 Ks 1 Kn d 1 Ks d 1 S1 (9) 0 r subje o he produion funion (2) resoure onsrain (5) and he dynamis of environmenal qualiy (6) where 1/ (1 v) is a disoun faor v 0 is he pure ime preferene rae and 0 is a onsan. These parameers represen he main preferenes for iner- and inrageneraional equiy in he model. Noe ha he only ineraion beween he regions is hrough he impa of polluion on he global environmen. Thus we do no onsider dire ransfers here bu analyze he impliaions of ransfers in Seion 4 below. 7 We an now express he Lagrangian of he maximizaion problem (9) as follows: 8 so u r S max n s s n 0 r 0 ( ) max 0 Y ( K K ) K 1 K K 1 K r r r r d r 1 r r d 1 d r d r S 1 S Kr d S 1 r (10) given ha S0 S0 S Kr j0 Kr j0 and Kn j0 Ks j0 for j d and r n s. Furhermore r 0 is he shadow prie of apial in region r while 0 is he shadow prie of environmenal qualiy in period. Below we show ha he opimal onsumpion pahs of he wo regions mus saisfy n s and for ease of exposiion we use his o simplify he firs order ondiions. 9 These ondiions inlude: n : u( n S ) n n (11) 7 The role of dire ransfers in limae poliies is sill a heme in negoiaions. Even if he Copenhagen Aord opened for subsanial limae finane from he rih o he poor world he funding of hese ransfers is a major opi. 8 See e.g. Conrad and Clark (1987). 9 The omplee ondiions would require wo ondiions in (11) in he same manner as in equaion (12). 9

11 s : u( S ) s n s s s u( S ) s n s s s K r d 1 : Y Kr 1 K r d 1 1 r 1 d 1 r Kr d 1 1 r n s K r 1 : Y Kr 1 K r d 1 1 r 1 r Kr 1 1 r n s r S : 1 1 u( S ) (12) (13) (14) 1 (15) r S 1 In addiion he following ransversaliy ondiions mus hold: lim K 0 for r n s j d (16) r r j lim S 0 (17) Clearly an imporan quesion when analyzing he opimal soluion is wheher onsumpion in he wo regions will onverge o he same level in he long run. In Appendix 1 we show ha his will be he ase whih gives us Lemma 1. Lemma 1. In he long run he soially opimal onsumpion levels and apial soks of he wo regions will onverge independenly of inequaliy aversion. Proof: See Appendix 1. Noe ha he resul is independen of inequaliy aversion in onsumpion. As disussed above diminishing marginal uiliy of onsumpion and delining marginal produiviy will ensure equaliy in he long run. However inequaliy aversion affes he onvergene proess as we disuss below and show wih numerial simulaions in Seion Opimal Poliy We firs haraerize he soial planner soluion. Laer we use his as our basis when we analyze how inequaliy aversion affes he opimal onsumpion and apial pahs of he wo 10

12 regions. We sar ou by haraerizing he wo regions opimal apial pahs. We summarize he main findings in Lemmas 2-5 before we disuss he impliaions of hese resuls. Lemma 2. : The shadow prie of he resoure onsrain is higher in he Souh n s han in he Norh along he opimal pah. Proof: This follows from he onaviy of he value funion in he opimizaion problem. n and s are he shadow pries of he aggregae apial levels of he wo regions. Beause he value funion is onave and he shadow pries are he derivaives of he value funion wih r respe o he sae variables we know ha 0 where Kr Kr Kr d. Nex we prove K r ha K K. Assume ha for some 0 we have Kn K n s s see Lemma 1. For he opimal pahs of he wo regions inluding heir apial pahs will be he same as he wo regions are now idenial in every aspe. Thus as we sar ou wih Kn0 Ks0 we an never have Kn Ks sine regional apial soks say equal one hey onverge. Thus we have ha K K and i mus be he ase ha. n s n s To undersand why Lemma 2 mus hold noe ha here are no onsrains on invesmen or disinvesmen. This means ha eah ounry an adjus is share of lean (and hus diry) apial as i wishes in any period. Hene eah ounry is onsrained by is oal sok of apial and he lower his apial sok is all else equal he higher he shadow prie of he resoure onsrain (apial). Sine he Norh is riher han he Souh he Souh s shadow prie of apial mus exeed he Norh s. Lemma 3. K K : The Norh arries mos of he limae burden by holding more n s lean apial han he Souh. Proof: We rewrie he opimaliy ondiions for lean and diry apial as follows: MP r r 1 1 (18) r 1 11

13 MP 1 r r d 1 1d (19) r 1 r 1 where we have used he noaion MP r j Y K K K r r d r j. Noing ha he erm 1 in (19) is independen of region we isolae his erm subsiue in for r and ombine ondiions (19) for r n s. This gives us he following relaionship ha mus hold along he opimal pah oward seady sae: n 1 MPs d 1 MPs 1 d MP MP s 1 n d 1 n 1 d. (20) We know from Lemma 2 ha he shadow prie of he resoure onsrain is higher in he poorer region hene. This implies ha he denominaor of (20) is larger han he n s numeraor also for he erm on he righ-hand side. Using his and simplifying yield: MP MP MP MP. (21) n s n d s d Now assume ha he lemma does no hold and ha Kn Ks for some. This would imply ha MPn MPs. However sine n s he Norh mus sill be riher han he Souh and hene Kn d Ks d so ha MPn d MPs d. However he inequaliy (21) does no hold for MPn MPs and MPn d MPs d. I follows ha K K. n s Noe ha equaions (18) and (19) imply ha even if he apial depreiaion raes for lean and diry apial are equal we will require a higher marginal produiviy from diry han lean apial o inves. To see his learer we an seup he firs-order ondiions for lean and diry apial in seady sae: MP v and MPd v.while he opimal level of lean apial requires is marginal produiviy o equal he sum of he depreiaion and disoun raes he marginal produiviy of diry apial mus in addiion over he welfare effes of inreased polluion. This is apured by he erm whih is he reduion in environmenal qualiy from an addiional uni of diry apial measured in onsumpion. 10 Before he sysem reahes he seady-sae equilibrium equaions (18) and (19) shows ha he 10 As onsumpion and apial levels in he regions onverge over ime all erms beome he same for he wo regions in seady sae. 12

14 opimal invesmen deision aouns for he rade-off beween invesing more oday whih inreases fuure onsumpion possibiliies and higher onsumpion oday as apured by he erm r r 1. One he wo regions apial levels onverge he marginal produiviy of diry apial will be he same in all ounries (f. Lemma 1). Lemma 4. : The differene in he shadow prie of he resoure onsrain s n 0 beween he Norh and he Souh dereases over ime. Proof: From opimaliy ondiion (18) for r n s and he resul ha Kn Ks (Lemma 3) we know ha he following mus hold:. (22) n s n 1 s 1 Sine n s (Lemma 2) his implies ha he growh rae of he shadow prie of he resoure onsrain is higher in he Norh han in he Souh hereby reduing he erm over ime. s n Lemma 5. : The onsumpion level of he Souh never exeeds ha of he Norh n s independenly of inequaliy aversion. Proof: Assume ha he lemma does no hold and ha n s in some period. The opimaliy ondiions for onsumpion (11) and (12) now beomes: MUn n and MU where MU s s x r u r S x 2 his implies ha he following mus hold: MU wih x S. Using ha n s from Lemma MU. (23) s n However his an never hold for 0 sine MU n MU s when n s due o diminishing marginal uiliy of onsumpion. Lemmas 1-5 haraerize he opimal apial and onsumpion pahs for he wo regions. Before onvergene we know ha he Norh has more apial and herefore onsumes more 13

15 han he Souh. This inequaliy in apial soks is apured by he differene in shadow pries of he wo regions resoure onsrains r r n s. As shown in he proof of Lemma 3 he shadow prie of he Norh s resoure onsrain is iniially lower bu grows faser (or delines slower) han he Souh s shadow prie. In addiion we know from Lemma 3 ha while he Souh ahes up wih he Norh he marginal produ of lean apial is higher in he Souh han in he Norh whih implies a higher sok of lean apial in he Norh han in he Souh K K n s. Hene he Norh sars ou riher han he Souh bu over ime he regions onverge oward he same equilibrium levels of lean and diry apial and hene onsumpion. When he aggregaed apial sok of he Souh ( Ks Ks d ) ahes up wih ha of he Norh he shadow pries of he resoure onsrains will also onverge. Finally based on he analysis above we an show ha he onsumpion inequaliy generally dereases over ime. To see his we an look a how he firs order ondiions for regional onsumpion levels hange over ime. This gives us: MU r 1 MU r r 1 r for r n s. Noe ha he erm drops ou when we ake he differene and hene his equaion beomes he same for boh regions. Using his relaionship for boh regions and rearranging yield: MU s 1 MU n 1 MU s MU n s 1 n 1 s n. (24) From Lemma 2 and equaion (22) we know ha he erm on he righ-hand side mus be negaive. Hene he differene beween he wo regions marginal uiliies from onsumpion mus also derease over ime implying ha he differene in onsumpion levels is falling How Inequaliy Aversion Affes he Opimal Poliy Having haraerized he opimal onsumpion and apial pahs of he wo regions le us now urn o he impliaions of inequaliy aversion. Noe ha inequaliy aversion does no hange he available resoures or produion sruure in he eonomies; sronger inequaliy aversion only inreases he non-peuniary os of onsumpion inequaliy. Hene sronger inequaliy aversion will inrease he inenives o eliminae differenes in onsumpion levels and will 11 There is a speial ase in whih equaion (24) may no imply lower onsumpion inequaliy over ime. Reall ha he marginal uiliies also depend on environmenal qualiy. Hene a rapid hange in environmenal qualiy over his period ould make he lef hand side of (24) negaive even if onsumpion inequaliy inreases slighly. s This will depend on he sign and magniude of u r for r n s. 14

16 generally inrease onsumpion in he Souh and redue onsumpion in he Norh ompared o he ase wih less or no inequaliy aversion. However i is no neessarily he ase ha he enire onsumpion pah of he Norh shifs down while he enire onsumpion pah of he Souh shifs up as here may be inenives o redue onsumpion inequaliy in he shor run a he os of inreased apial inequaliy and herefore higher onsumpion inequaliy laer on. The radeoff beween inequaliy now or laer will be affeed by he level of inequaliy aversion. Hene while sronger inequaliy aversion generally shifs he onsumpion pah of he Norh down and he Souh up here may be periods of ime for whih his may no hold. Indeed for some periods (or saes of he world) sronger inequaliy aversion may in fa inrease he onsumpion inequaliy beween he regions. To explain his we sar by disussing he differen opporuniies for he soial planner o redue he welfare loss from inequaliy aversion if his beomes more osly. Firs from equaion (18) and (19) we know ha he marginal produiviy of diry apial relaive o is depreiaion rae exeeds ha of lean apial. Hene by inreasing he share of lean apial in he Norh and he share of diry apial in he Souh he Souh beomes more produive and an hene onsume more all else equal. We an do his adjusmen wihou sarifiing he environmen if we keep K r r d onsan. However o le he Souh grow faser i may be worhwhile o sarifie he environmen in he shor run hereby leing he Souh have an even higher share of he more produive diry apial. This seond opion for reduing inequaliy highlighs he radeoff beween reduing inequaliy oday and susaining environmenal qualiy for omorrow. In he long run however he equilibrium level of environmenal qualiy is unaffeed by inequaliy aversion. We reurn o his below. The las opion for reduing he welfare loss from inequaliy aversion is by hanging onsumpion oday by inreasing or dereasing invesmen. We an ahieve equaliy in onsumpion in any period by inreasing invesmen in he Norh suffiienly for is onsumpion level o equal ha of he Souh. While his is a possibiliy i an only be opimal in he shor run if he welfare loss from inequaliy oday is high ompared o he presen value of he welfare loss from inequaliy omorrow. This is beause lower onsumpion inequaliy oday omes a he os of higher apial inequaliy whih leads o more onsumpion inequaliy in he fuure. A some poin he Norh mus onsume he aumulaed apial sine he wo regions should onverge o he same apial level in he long run. Consequenly 15

17 he less value we plae on fuure welfare (high disoun rae) he more araive i is o redue onsumpion inequaliy oday despie he os of inreased fuure inequaliy. Noe ha he shor-run reduion in onsumpion inequaliy may be opimal even wihou inequaliy aversion ( 0 ). Wih a large degree of inequaliy in apial soks and onsumpion levels beween he Norh and he Souh iniially he marginal uiliy from one more uni of onsumpion is lower in he Norh. Therefore i may be beer o insead inves more in lean apial hereby improving he environmenal qualiy S 1 s Souh s uiliy from onsumpion (sine u 0 ). r whih inreases The only way o ahieve equiy in he long run is by shifing invesmens oward more lean apial in he Norh and more diry apial in he Souh. In he shor run however he soial planner an redue inequaliy by inreasing invesmens in he Norh and/or reduing invesmens in he Souh. Boh opions ompromise equiy and possibly environmenal qualiy in he longer run as more (less) apial means higher (lower) produion ha mus affe onsumpion a some poin of ime. Wheher his shor-erm fix for he equiy problem is opimal and o wha exen depends on he rade-off beween lower inequaliy in onsumpion oday and higher inequaliy in apial soks and possibly lower environmenal qualiy omorrow. The more value we pu on he welfare of fuure generaions relaive o ourselves (low v ) he smaller he shor erm reduion in onsumpion inequaliy sine his inreases he presen value of inreased inequaliy in he fuure. Aouning for eah of he opions for reduing inequaliy emporarily or permanenly gives us Proposiion 2.1. Proposiion 2.1: For some we an have n s inrease onsumpion inequaliy in some periods. 0 : Sronger inequaliy aversion may Proof: We show ha his holds for speifi parameer values in he numerial simulaions presened in seion 5. In pariular Figure 1 shows ha he onsumpion inequaliy ( n s ) is larger for 1.5 han for 1 from = 26 o =

18 To see why inequaliy aversion may inrease onsumpion inequaliy in some periods (Proposiion 2.1) we sar by oal differeniaing he firs order ondiions for onsumpion wih respe o. This resuls in he following: n 1 n s S 1 u n u n (25) s 1 s s S 1 u s u s. (26) n s While i is suffiien ha for some noe ha Proposiion 2.1 holds if n 0 and s 0. Imposing his on equaions (25) and (26) and rearranging yield he following ondiion: 1 n S 1 s 1 1 s s. (27) un us We know ha 1 s u r 0 for r n s. Now onsider he siuaion desribed above in whih he Norh has aumulaed apial o redue is onsumpion level (and onsumpion inequaliy) emporarily. The more apial i has aumulaed he lower he shadow prie of apial and he more he Norh mus subsequenly inrease onsumpion o redue is n 0 apial sok. The larger he emporary reduion in onsumpion inequaliy hrough apial aumulaion he more he Norh mus onsume laer on. Hene a he sage when he Norh onsumes is aumulaed apial we have ha 0. Turning o he Souh we know ha he more welfare reduing inequaliy aversion is (high ) he higher he value he apial poor Souh pus on apial and hene 0. If hese hanges in he regions shadow s pries are suffiienly srong he expression on he righ-hand side of (27) will be larger han he expression on he lef-hand side. n To jusify ha he impa of inequaliy aversion on environmenal qualiy an lie beween hese wo erms in he desribed siuaion we ake he oal derivaive of he opimaliy ondiion for environmenal qualiy (15) and rearrange: S s n s s 1 ss ss u n u s un u s. (28) 17

19 The impa of inequaliy aversion on environmenal qualiy in a period will depend on how inequaliy aversion affes he shadow prie of he environmen and onsumpion levels in Norh and Souh. Aording o equaion (28) S is posiive (negaive) if he brakeed erm is negaive (posiive). Firs in he siuaion we onsider Norh onsumes more and Souh onsumes less beause of higher inequaliy aversion. Hene he sum of he wo las erms in brakes an be posiive or negaive depending on whih of hese wo effes is sronger. Nex he shadow prie of he environmen an inrease or derease as we raise. The marginal value of he environmen inreases as onsumpion levels inrease due o he omplemenariy of onsumpion and he environmen in he uiliy funion. However wih opposie onsumpion effes in he Norh and he Souh he shadow prie of he environmen an inrease or derease wih sronger inequaliy aversion as an hen he wo firs erms in brakes in (28). Consequenly S siuaion and hene (27) an hold. an be posiive negaive or zero depending on he I seems reasonable ha wih a bigger weigh on inequaliy aversion i beomes opimal o redue inequaliy aversion more in he shor run even if his ompromises he environmen. The reason is ha when onsumpion inequaliy beomes more welfare reduing on he margin (higher ) he relaive marginal value of improved environmenal qualiy falls all else equal. This makes i more likely ha soiey should sarifie environmenal qualiy in he shor run by leing he Souh inves even more in diry apial hereby speeding up is developmen. Hene we expe S 0 in he shor run. To summarize we should someimes le he rih (poor) region inves more (less) oday o redue equaliy emporarily a he os of more inequaliy in he fuure. This is a resul of disouning. This represens ye anoher example of he onfli beween iner- and inrageneraional equiy. The less weigh we pu on fuure generaions relaive o hose living oday (high disoun rae) he sronger he inenive o immediaely eliminae inequaliy beween people living oday hrough invesmen. However his means sarifiing inrageneraional equiy for erain fuure generaions as he apial soks of he wo regions mus onverge oward he same level in he long run (Lemma 1). For he Norh his implies 18

20 ha he apial ha was aumulaed o redue shor-run onsumpion mus be onsumed leading o a emporary bump in onsumpion and possibly inreased inequaliy for a period. We reurn o his in our numerial analysis in Seion 5. This disussion emphasizes he lose relaionship beween limae aion and developmen/growh. In inernaional negoiaions aimed a reahing a global limae agreemen developing ounries have long expressed a onern ha limiing heir greenhouse gas emissions will hamper heir developmen opporuniies. On his basis hey argue ha he developed world mus bear he majoriy of he os of reduing global emissions. Our analysis may jusify his laim made by developing ounries and suggess ha if we all are abou equaliy; we may have o sarifie environmenal qualiy in he shor run o allow he poorer region o grow faser by polluing more. Consequenly he rih region should bear he majoriy of he oss of improved environmenal qualiy. Finally le us onsider how inequaliy aversion affes he seady-sae apial and onsumpion levels. We have saed he firs-order ondiions for he seady-sae equilibrium in Appendix 1. Noe ha as he wo regions onverge o he same apial and onsumpion levels he opimaliy ondiion for regional onsumpion beomes: MU for r n s. While here is no welfare loss from inequaliy in seady sae he inequaliy parameer is inluded beause if any of he wo regions marginally raise heir onsumpion level from he seady-sae level his yields marginal loss due o inequaliy of. However we an hink of inequaliy aversion as a non-peuniary os ha does no affe produion possibiliies or resoure availabiliy. Hene in seady sae he soial planner will ensure ha apial levels environmenal qualiy and onsumpion are se o maximize welfare whih means ha will no affe he seady-sae equilibrium sine here is equaliy. r r Hene inequaliy aversion aross a generaion will no affe greenhouse gas emissions in he long run. To see his we an rearrange and express he seady-sae ondiion for global environmenal qualiy in erms of he shadow prie of he environmen: S MU n MU v Equaion (29) onfirms ha he seady-sae level of global environmenal qualiy does no depend on he regions preferenes for equaliy ( ). We also see ha i is inreasing in he 19 S s (29)

21 marginal uiliy of environmenal qualiy whih is given by he numeraor in equaion (29) while i dereases wih he replenishmen rae of he environmen and he disoun rae. Equaion (50) in Appendix 1 gives he seady-sae level of global environmenal qualiy. The pah of environmenal qualiy oward seady sae depends on he aggregae level of diry apial in he wo regions. As seen above inequaliy aversion affes he diry apial pahs and hus emissions before he sysem reahes seady sae. We reurn o his in he numerial analysis in Seion 5. 3 Wha if a Conra is no Possible? The Business-as-Usual Case The nex quesion is wha he aions of he wo regions would be if he soial onra anno be reahed? Wihou an enforemen mehanism in plae he regions are beer off following heir own ineres and maximizing he welfare of a represenaive onsumer. We refer o his as he Business-as-Usual problem (BAU) i.e. he opimizaion problem of loal poliy makers when here is no oordinaed aion or global environmenal agreemen wihin or aross he regions. 3.1 The BAU Pahs To find he BAU pahs for he wo regions we firs define he Lagrangians. We se he disoun faor ρ < 1 equal for he wo regions o avoid having he effes of inequaliy aversion onfounded by he effes of disouning. 12 The Lagrangian for region r is: r BAU u( r S ) max r k max k r Y ( K K ) K 1 K K 1 K r r r d r 1 r r d 1 d r d r for r k n s r k where Kr j0 Kr j0 and Kn j0 Ks j0 for j d and 0 is he shadow prie of apial. We assume ha eah ounry pereives ha is impa on he dynamis of global environmenal qualiy is approximaely zero. As a resul we maximize (30) over onsumpion and diry and lean apial soks aking global environmenal r (30) 12 For onveniene his implies using he same disouning as in he soial planner ase bu his does no maer for onlusions. We will also use he same symbol for he shadow prie of apial; λ. 20

22 qualiy S as given. However he dynamis of he environmen sill follows (6). The firs order ondiions hen beome: 13 [ n ]: [ s ]: MU n s n n MU (31) n s n n MU n s s s MU (32) n s s s Kr 1 r r 1 MPr 1 1 [ ]: Kr d 1 r r 1 MPr d 1 1 d [ ]: (33) (34) We sar ou by haraerizing he BAU soluion based on he firs-order ondiions. Lemmas 6-9 below summarize he main resuls for onsumpion and apial dynamis. Nex we analyze and disuss he impliaions of he Lemmas. Lemma 6. n s : Consumpion in he Norh is higher or equal o onsumpion in he Souh along he opimal BAU pah. Proof: Assume ha s n. Aording o (31) and (32) his gives MUn n and MU. Following he same argumens as in he proof of Lemma 2 we know ha s s. This gives MU n MU s whih obviously anno hold for s n. n s Hene we ge ha n s along he opimal pah. Lemma 7. K K K K : Boh apial soks in he Norh are higher or equal n s n d s d o he apial soks in he Souh along he opimal BAU pah. Proof: We know from (33) and (34) ha MP 1 r r 1 1 and r 1 MP 1 r r d 1 1d. Assume ha he Lemma does no hold whih implies ha r 1 13 In addiion he ransversaliy ondiion (16) in Seion 2 sill holds. 21

23 . This means ha MPn 1 MPs 1 and MPn d 1 MPs d 1 and n s n 1 s 1 herefore Kn Ks and Kn d Ks d. This anno hold sine i implies ha he aggregae apial sok is lower in he Norh han he Souh and n s. Thus we have ha (35) n s n 1 s 1 whih gives Kn Ks and Kn d Ks d. In he soial planner ase sudied in he previous seion we found ha only he lean apial sok mus be higher in he Norh han he Souh along he opimal pah. In omparison he Norh mus also have a higher diry apial sok along he BAU pah. The reason for his is ha no ounry believes i an affe he environmen in he BAU ase and hene ounries in boh regions inves more in diry apial han hey would if his exernaliy was inernalized as in he soial planner ase. Lemma 8. Under BAU onsumpion and apial soks in he wo regions onverge o he same levels. Proof: Aording o (35) he differene beween n and s diminishes over ime as long as he aggregae apial sok is higher in he Norh. Thus in he long run he eonomies ener a seady-sae equilibrium where. We see hen from (33) and (34) * r r 1 r BAU ha MP and * 1 r d 1 d MP. This means ha he apial soks are equal in * 1 r 1 * * * seady sae; Kn j Ks j K j BAU j d. As apial soks and invesmens are onsan and equal aross regions in seady sae onsumpion is also onsan and equal beween he wo regions:. * * * n s BAU Lemma 9. MPr MPr d or MPr MPr d : The diry ehnology does no have o be more produive han he lean ehnology along he opimal BAU pah. Proof: We see from (33) and (34) ha: MP MP (36) r 1 r d 1 d 22

24 Thus he depreiaion raes deermine he differene in marginal produiviies beween lean and diry apial. If d hen MPr 1 MPr d 1 while if d hen MPr 1 MPr d 1. As opposed o he soial onra under whih diry apial mus be more produive han lean apial along he opimal pah (ignoring depreiaion effes) Lemma 8 saes ha his is no longer he ase under BAU. The reason is ha produers do no ake ino aoun he exernal environmenal oss and hene he depreiaion raes deermine he differene in marginal produiviies. 3.2 The Effes of Inequaliy Aversion on BAU pahs Lemmas 6-9 haraerize he opimal apial and onsumpion pahs for he wo regions. Noe ha hey hold for 0. Le us now sudy he effe of inequaliy aversion i.e. 0. Reall ha he Norh had wo main ools for helping he Souh in he soial planner ase: by reduing onsumpion and by onribuing o beer environmenal qualiy. Under BAU he only possibiliy is o redue onsumpion as he Norh an no longer affe he marginal uiliy of he Souh as eah ounry akes he qualiy of he environmen as given. Similarly he Souh an only affe inequaliy by inreasing is onsumpion. However reduing onsumpion in he Norh and raising onsumpion in he Souh would lead o higher apial aumulaion in he Norh and lower apial aumulaion in he Souh hereby inreasing he apial inequaliy beween he regions. Furhermore a some sage he Norh mus onsume is aumulaed apial; hene we would ge a emporary inrease in onsumpion laer on. Disouning reinfores his sine a higher disoun rae means ha we shif weigh from fuure o urren generaions. Hene we anno onlude ha inequaliy aversion shifs he Norh s (Souh s) onsumpion pah down (up) for he whole ransiion period oward seady sae ompared he ase wihou equaliy preferenes ( 0 ). To see his onsider a permanen inrease in β. In a similar way as in subseion 2.4 we find from (31): n 1 1 n s S u n u n. (37) 23

25 The firs erm on he righ hand side is he dire effe of inequaliy aversion whih is negaive and gives an inenive o redue onsumpion. The wo nex erms represen he indire effes on he aggregae apial sok and he environmen. The apial sok is affeed as less onsumpion means more apial hene n 0 and he effe on onsumpion is posiive (noe u 0 ) as having more aggregaed apial means more fuure n onsumpion whih has a negaive effe on inequaliy. Finally even if he regions do no ake ino aoun he environmen in heir deisions environmenal qualiy affes he BAU pahs as hey ake he sae of he environmen a ime ino aoun when making invesmen and onsumpion deisions a ime. Thus he effe of inequaliy aversion on onsumpion also depends on how inequaliy aversion affes he environmen. In subseion 2.4 we argued ha mos likely inequaliy aversion will redue he environmenal qualiy in he shor S run (i.e. 0 ). This means ha he effe of he environmen is negaive as i redues he marginal uiliy of onsumpion. In he shor run we herefore would expe he effe on onsumpion in he Norh o be negaive. However as apial aumulaes he seond effe in (37) may be large and we may ge a emporary inrease in onsumpion. For he Souh we ge similar effes bu wih he opposie sign. The disussion also shows ha wih inequaliy aversion in BAU for some iniial period we ge higher aumulaion of lean and diry apial in he Norh and lower aumulaion of lean and diry apial in he Souh ompared o sandard preferenes ( 0 ). However as he Norh mus onsume some of is apial aumulaion a some poin we anno rule ou ha in some inermediae periods apial aumulaion is lower wih inequaliy aversion. While diry apial has o be more produive han lean apial in he soial planner ase (ignoring depreiaion effes); his is no longer he ase under BAU. An impliaion of his is ha i is no longer possible o redue inequaliy by leing he Norh inves relaively more in lean apial while he Souh invess more in diry. Wha maers under BAU are he hanges in aggregae apial soks as here are no differenes in marginal produiviies of lean and diry apial as equaion (36) shows. 24

26 Le us now ompare he opimal onsumpion levels under BAU o he soially opimal levels. Proposiion 3.1 summarizes his. Proposiion 3.1: When inequaliy aversion wihin a generaion redues onsumpion in he Norh and inreases onsumpion in he Souh we ge a larger onsumpion reduion in he Norh under he soial onra han under BAU while in he Souh we ge a larger onsumpion inrease under he soial onra han under BAU. Proof: From subseion 2.4 we know ha he effe of inequaliy aversion on he soially opimal r is independen of wheher we inrease α or β as wha maers is he sum:. However we see from (31) ha n is only indirely affeed by an inrease in and in a similar way i follows from (32) ha s is only indirely affeed by an inrease in. Thus he effe of inequaliy aversion on r will be higher under he soial onra han under BAU. Therefore he onsumpion inrease (derease) will be higher for he Souh (Norh) under he soial onra han under BAU. The inuiion behind he Proposiion is ha eah region under he soial onra akes ino aoun boh regions disuiliy from inequaliy and no jus is own disuiliy. Under BAU however he regions only are abou heir own disuiliy while hey ignore he disuiliy hey impose on he oher region. This represens an equaliy exernaliy. Consequenly sronger inequaliy aversion indues a larger onsumpion reduion in he Norh and a larger inrease in he Souh under he soial onra han in he BAU ase all else equal. Wihou inequaliy aversion onsumpion will in his ase be lower in boh he Norh and he Souh under he soial onra han under BAU. The reason is ha he marginal produiviy of diry apial has o be higher in he soial planner ase han under BAU due o he environmenal exernaliy. Thus in he soial planner ase boh regions mus inves less in diry apial ompared o wha was opimal under BAU and he available resoures will hus be lower. If we inrodue inequaliy aversion we know from Proposiion 3.1 ha onsumpion generally goes down in he Norh boh under BAU and he soial onra bu ha he reduion is sronger under he soial onra. Therefore he onsumpion level under he soial onra is sill lower han under BAU. However for he Souh his is no longer 25

27 lear as he inrease in onsumpion under he soial onra is higher han he inrease under BAU. This means ha i is possible ha he soial onra will lead o higher onsumpion levels in he Souh even if i akes ino aoun he effes on he environmen. Le us nex ompare regional apial soks under he soial onra and BAU. We sar wih diry apial and hene emissions as hey are proporional o diry apial. In subseion 2.4 we argued ha iniially he Norh redues diry apial invesmens while he Souh inreases hem when inroduing inequaliy aversion in he soial planner ase. Addiionally we argued above ha inequaliy aversion in BAU iniially inreases invesmen in diry apial in he Norh while reduing suh invesmen in he Souh. Thus he soial onra yields an iniial reduion in diry apial invesmen in he Norh while he effe in he Souh is ambiguous and emissions may aually be higher under he soial onra. The inuiion is as follows. The opimaliy ondiion for diry apial under he soial onra inludes an addiional erm ompared o he BAU ase see equaion (13). This erm represens he marginal effe of more diry apial on global environmenal qualiy and implies a lower invesmen in diry apial ompared o BAU in boh regions. However we also have he effe of inequaliy aversion whih yields lower invesmen in diry apial in he Norh and higher invesmen in he Souh. While boh effes redue diry apial aumulaion and emissions in he Norh we anno deermine he effe on he apial sok in he Souh. Hene poor ounries should no neessarily have a polluion onsrain under a global limae reay. Under erain ondiions i may aually be opimal o le poor ounries inrease heir emissions under suh reay. Based on his he resuls for lean apial are sraighforward as i follows from he disussion in subseion 2.4 and above ha inequaliy aversion generally redues lean apial aumulaion in he Souh under boh BAU and he soial onra while i inreases lean apial aumulaion in he Norh in boh ases. This means ha when we ake ino aoun preferenes for equaliy he Norh has o make a larger onribuion o omba global warming boh when i omes o lean apial invesmens and emissions reduions. Again our resuls show ha we in some ases should enourage poorer ounries o use more diry apial han hey oherwise would o speed up heir developmen. 26

28 4 Exending he Basi Model Le us now urn one again o he soial onra. Thus far he only ineraion beween he regions has ome hrough he impa of polluion on he global environmen. Now we open up for inernaional ransfers suh as developmen aid or limae finane from he rih o he poor region. In addiion we inrodue loal polluion as loal and global polluion are ofen inerrelaed so ha an inrease in diry ehnology may inur an exra os o he polluing ounry. 4.1 Ineraions beween he Regions When he soial planner faes resriion on ransfers beween he regions as above he opimal onra is a seond-bes poliy. Consequenly when we relax he onsrain by allowing for some or unlimied inernaional ransfers he resul mus be ha we ge loser o he firs-bes soluion and ahieve higher aggregae welfare. We inrodue ransfers by adding he erm whih represens ransfers from Norh o Souh in period o he regions resoure onsrains. The wo regions modified resoure onsrains hen beome: Y( K K ) K 1 K K 1 K (38) n n d n n 1 n n d 1 d n d Y( K K ) K 1 K K 1 K (39) s s d s s 1 s s d 1 d s d In addiion we inrodue a onsrain on Norh-Souh ransfers suh ha M in every period. There are several jusifiaions for suh a onsrain. Firs here are learly limis o how muh one ounry is willing o aep as ransfers o oher ounries in any given period. Seond even if rih ounries are willing o ransfer "whaever i akes" o poor ounries in order o eliminae inequaliy here are reasons o suspe ha very large ransfers exeed he reipien ounries' abiliy o absorb hese funds produively in a manner similar o e.g. he absorpion of resoure windfalls see van der Ploeg and Venables (2013). These onsideraions are however ouside he sope of our model. Appendix 2 presens he modified Lagrangian for he opimizaion problem. Now he soial planner mus also deermine he opimal size of he ransfers and hene we obain an addiional firs order ondiion for he Norh-Souh ransfer: 27