Optimal multi phase transition paths toward a global green economy Adriaan van Zon and Paul A. David

Transcription

1 Working Paper Series # Opimal muli phase ransiion pahs oward a global green economy Adriaan van Zon and Paul A. David Maasrich Economic and social Research insiue on Innovaion and echnology (UNU MERI) info@meri.unu.edu websie: hp:// Maasrich Graduae School of Governance (MGSoG) info governance@maasrichuniversiy.nl websie: hp://mgsog.meri.unu.edu Keizer Karelplein 19, 6211 C Maasrich, he Neherlands el: (31) (43) , Fax: (31) (43)

2 UNU-MERI Working Papers ISSN Maasrich Economic and social Research Insiue on Innovaion and echnology, UNU-MERI Maasrich Graduae School of Governance MGSoG UNU-MERI Working Papers inend o disseminae preliminary resuls of research carried ou a UNU-MERI and MGSoG o simulae discussion on he issues raised.

3 Opimal Muli Phase ransiion Pahs oward a Global Green Economy by Adriaan van Zon 1,*) and Paul A. David 2) Firs draf, Augus, UNU MERI, SBE Maasrich Universiy 2. UNU MERI, Sanford Universiy JEL codes: Q540, Q550, O310, O320, O330,O410,O440 Keywords: global warming, ipping poin, caasrophic climae insabiliy, exreme weaher relaed damages, R&D based echnical change, embodied echnical change, opimal sequencing, muli sage opimal conrol, susainable endogenous growh. * Corresponding auhor. address: Adriaan.vanZon@MaasrichUniversiy.NL 1

4 1 Inroducion Economic growh hus far has been closely linked o he bulk conversion of energy sored in carbon based fuels (wood, coal, oil, naural gas) ino useful work. 1 Burning such fuels gives rise o CO2 emissions. hese emissions, ogeher wih oher greenhouse gasses (GHG s) like mehane, are now hough o be responsible for a considerable warming of he earh s amosphere in he presen and for years o come. ha is bad news on a number of accouns: sea levels will rise, ropical diseases will become more wide spread, sorms will be more violen, paerns of rainfall will change (affecing agriculure), fresh waer supply shorages will become a problem due o global glacier rerea, and so on. Mos of hese changes represen significan coss o sociey. 2 here is much o be said hen for avoiding hese consequences of global warming, by reducing emissions or by adaping o he new siuaion. Adapaion will have o be par of he response o global warming, if only because much of he warming induced by man made CO2 emissions is sill in he pipeline. Wih he curren level of (cumulaive) emissions, i is doubful wheher global warming can be kep below an addiional 2 degrees Celsius compared o pre indusrial imes. he laer is considered o be a hreshold ha should no be crossed for fear of runaway global warming and corresponding runaway damages and he end of life as we know i. Unforunaely, he mere acknowledgemen of he exisence of a hreshold is no enough o ensure ha CO2 and oher GHG emissions will be sufficienly reduced. he BRICS counries have shown asonishing growh figures during he las wo decades, and hey feel eniled o welfare levels comparable o hose in he Wes. I goes wihou saying ha he Wes also feels eniled o such welfare levels. As a consequence, he provision of energy will have o keep pace wih he coninuing desire o grow. Unforunaely, many of he new power producion plans currenly buil in China are coal fired power plans, which is he wors possible opion from a CO2 emissions poin of view. 3 A he same ime, he sunami disaser in Fukushima has forced governmens o reconsider nuclear energy as a relaively carbon exensive power source. Only recenly, he German governmen has announced o close down a number of nuclear power plans, and i is feared ha his will enail a swich owards coal fired power producion again. However, he problem wih coal fired power plans is ha once hey are here, hey will be acive for a leas hree decades or more, because hese ypes of invesmens are irreversible in pracice. In shor, here are srong poliical pressures ha end o lead o a larger consumpion of carbon based energy which causes a persisen hrea of runaway climae change. 4 Bu even hough we seem o be almos unavoidably heading owards a problemaic climae fuure, i would be ineresing o know firs wha economic heory has o say abou combining he desire o grow wih he need o avoid disasrous climae change and wha o do o end up in a susainable growh siuaion insead, and secondly wha he role of boh exising and new echnologies 1 I.e. work in he physics sense of he word. 2 See, for example, he Sern Review (Sern, 2006) for an exensive overview of he causes and consequences of climae change. See also (IPCC, 2007). 3 he energy conen of coal is 32.5 MJ/Kg while ha of naural gas is 53.6 MJ/Kg, implying ha he amoun of energy per uni of CO2 is abou 65 per cen larger for naural gas han for coal. (cf. Wikipedia, Energy Densiy). 4 Obviously, such a hrea may be miigaed by fundamenal changes in lifesyles, a leas for he well o do par of he global populaion. Bu i is hard o imagine ha his will be poliically accepable, and so we will have o rely on echnology as a savior of las resor. 2

5 and heir embodimen in irreversible invesmen may be in such a ransiion process owards susainable growh, and hird wheher such a process would be echnically and/or poliically feasible a all, wihin he limis se by he exisence of an irreversible climae change hreshold (furher ICC for shor). If no, we may have o prepare ourselves for an ineresing endgame. 5 If so, we should a leas hink again and ry o ake beer informed decisions. Invesmen (eiher in knowledge or in hardware) is he condiio sine qua non for a successful ransiion owards a susainable fuure. he ransiion owards susainabiliy will herefore be deermined by finding he righ balance beween wo imporan aspecs of invesmen: on he one hand we have o face he fac ha he irreversibiliy of invesmen implies a cerain degree of ineria o change, while on he oher hand invesmen is lierally he carrier of echnological progress and so enables (produciviy ) change. his double role of invesmen underlines he imporance of he iming of invesmen decisions: i is unwise o inves oo early because one runs he risk of missing ou on poenial produciviy improvemens sill o come, and neiher should one inves oo lae because of he rising opporuniy cos of coninuing o use old echnologies insead of new, superior, ones. his seing naurally gives rise o such quesions as how long o coninue using and invesing in presen carbon based echnologies, how much and how long o spend effors on improving new carbon free echnology, and when o sop improving such carbon free echnologies and sar implemening and using hem, his all in he face of having o sop cumulaive emissions jus in ime and jus below he ICC. o find answers o hese quesions, we formulae an opimum conrol model ha borrows heavily from he AK model from he endogenous growh lieraure (cf. Rebelo (1991) in paricular), bu ha also expands upon his AK seing in a number of ways. Firs, we allow for differen echnologies ha can be used eiher nex o each oher or sequenially. Secondly, a echnology is characerized no only by is capial produciviy, bu also by CO2 emissions per uni of capaciy used. Obviously, a carbon free echnology will no only be characerized by a zero emissions coefficien, bu also by a lower capial produciviy. 6 hird, we allow for he deacivaion of exising echnologies, i.e. he scrapping of exising echnologies, as in he vinage lieraure. We show how he ime of deacivaion of exising capaciy depends on echnological parameers bu also on emission characerisics, in combinaion wih he shadow price of emissions. he laer suggess ha he posiion of he ICC direcly influences such replacemen decisions hrough is impac on he shadow price of emissions. A ighening of he ICC (for example if one would find evidence for runaway global warming occurring a less han 2 degrees warming) would end o drive up he shadow price of emissions, and would lead o an earlier deacivaion of carbon based echnologies. Fourh, we allow for endogenous R&D based echnical change. his requires a specificaion of he R&D funcion ha is differen from he ones found in Romer (1990), or Aghion and Howi (1991), for example, because echnical change in our model seing is no mean o compensae for he loss in capial produciviy resuling from capial accumulaion under neo 5 See, for example, Dyer (2010) and Lynas (2007) on his subjec. Lynas provides a journalisic view on he consequences for he world of up o 6 degrees addiional warming for each degree of exra warming, while Dyer provides scenarios regarding he sruggle for resources necessary o fulfil basic needs, like waer and arable (and habiable) land, as hese are likely o arise wih on going global warming. 6 he laer is a necessary assumpion o make he model consisen wih he observaion ha he presen sae of he economy is characerized by he inensive use of carbon based energy raher han carbon free energy. If he capial produciviy of he laer echnology would exceed ha of he former, hen he carbon free echnology would be superior o he carbon based echnology on all accouns, which is no really he case. 3

6 classical condiions, as in Lucas (1988) and Romer (1990), for insance. Raher, in an AK seing, capial produciviy is consan by assumpion, and so echnical change specified in he usual way would produce a coninuously acceleraing growh rae raher han jus a higher, bu sill consan, rae of growh. We will come back o his in more deail laer on. Fifh, we explicily focus on he iming of he swiches beween invesmen in he one echnology and in he oher, and on he iming of he deacivaion of old echnologies, since he deacivaion and acivaion of echnologies ha differ wih regard o heir emission coefficiens have a direc impac on he macro emission rae and herefore on he ime lef unil he ICC will be hi. he model ha we are going o presen exends he relaively small lieraure ha is concerned wih he opimal iming of swiches beween producion echnologies. ahvonen and Salo (2001), Valene (2009) and Schumacher (2011) are hree examples. ahvonen and Salo (2001) focus on he iming of he swich beween resource exracion echnologies, based on differences in exracion coss, while Valene (2009) focuses on he swich beween wo macro producion echnologies in a seing wihou any irreversible invesmen in producion capaciy. Moreover, Valene (2009) deermines jus one opimal swiching momen in a sandard dynamic opimizaion seing involving a CIES uiliy funcion, whereas he very requiremen of he accumulaion of physical producion capaciy BEFORE producion using ha capaciy can acually ake place, implies ha one should consider a leas wo opimal swiching momens, even if one has o swich beween jus wo echnologies. 7 Schumacher (2011) focuses on he iming of a swich owards a renewable resource regime induced by he increasing probabiliy of climae disasers under a non renewable producion regime. However, he employs a producion srucure in which renewables and non renewables form a complex of perfecly subsiuable inpus ha ogeher wih generic capial produce oupu. Schumacher herefore ignores he embodimen of echnologies in echnology specific capial goods and so downplays he need for a sufficien amoun of ime o build up he carbon free capaciy required o saisfy fuure consumpion and invesmen needs. A recen paper is ha by Boucekkine e al. (2012) who provide he heoreical backgrounds for formulaing an opimal conrol resource exracion problem wih irreversible ecological regimes. he presen paper summarizes par of he work conduced by he auhors on he consrucion (and furher exension) of a muli phase ransiion model ha incorporaes he noion of embodimen of echnical change on he one hand, he irreversibiliy of invesmen decisions, and he fac ha he smooh ransiion oward a carbon free fuure will need o be prepared by means of he accumulaion and subsequen run down of carbon based producion capaciy, simply because capial, wheher carbonbased or carbon free, is a produced means of producion. he focus of he presen paper is herefore on he selecive build up and deacivaion of differen ypes of capial socks, he ime his akes, and he implicaions his has for he developmen over ime of welfare specified in erms of he flow of consumpion. he seup of he paper is as follows. In secion 2, we presen a descripion of he opimum conrol model. We provide four differen versions. he simples version, called he Basic Model (BAM for shor), conains jus wo echnologies, a carbon based one and a carbon free echnology, and we look a he 7 here mus be a leas hree phases: a pure carbon phase, a mixed carbon and carbon free phase and a pure carbon free phase. he simple reason is ha capial is a produced means of producion, and he very firs unis of carbon free capaciy mus be produced using carbon based capaciy if we sar ou wih a pure carbon phase firs. 4

7 iming of he various aciviies ha need o ge us from he business as usual siuaion which is carbon inensive o a carbon free susainable fuure. In he second version we add endogenous R&D driven capial augmening echnical change o BAM. his version is called BAM+R&D. I can be shown ha BAM+R&D and BAM have boh hree phases. However, he way in which decisions in each phase are linked ogeher will differ in boh models. he final version is called BAM+R&D+UCL and i exends he BAM+R&D model by allowing for exra weaher relaed damages. We implemen his by inroducing anoher cumulaive emissions hreshold, above which echnical decay occurs a a faser rae, due o increased weaher relaed damages. In secion 3, we presen he oucomes generaed for he hree versions of he muliphase ransiion model. In secion 4 we provide a summary and some concluding remarks. 2 he Model 2.1 Inroducion We use he simples possible endogenous growh seing in which we have wo broad (linear) echnologies. One of hem is an esablished echnology (called he A echnology) wih a relaively high produciviy of capial ha uses carbon based energy and ha produces CO2 emissions in he process. As saed, hese emissions add o he sock of GHG s and so affec he probabiliy of he world geing ino a siuaion of runaway global warming and caasrophe in he end. he alernaive echnology (furher called he B echnology) does no generae CO2 emissions, bu has a relaively low capial produciviy ha needs o be furher developed (hrough R&D) and scaled up (hrough invesmen in physical capial) o a level in which i can conribue significanly o he consumpion needs of he populaion a large. he simples possible growh seing ha allows a relaively sraighforward coverage of he feaures direcly above is ha of an AK+BK model, where AK reflecs he old carbon based echnology, and BK he new carbon free one. We se up a sequenial Hamilonian sysem in which we allow for differen phases in he ransiion from carbon based o carbon free producion. In he firs phase, called he business as usual phase (BAU for shor), only he AK echnology is acive, bu he basic innovaions ha ogeher define he new BK echnology have already been done. During he BAU phase he new echnology is no acive because i is ouperformed by he old echnology, in he sense ha using he new echnology would be relaively cosly in erms of insananeous consumpion possibiliies foregone. However, as cumulaive emissions grow, coninuing o use he A echnology becomes ever less profiable relaive o using he carbon free B echnology. A echnology being acive implies wo hings. Firs, he basic feaures of such a echnology mus be known, while secondly hese feaures are embodied in new capial goods. A echnology can be deacivaed by no using he capial goods embodying ha echnology anymore. Because capial goods are echnology specific, his implies ha a new echnology can ake over from an old one only by acively invesing in he capial goods ha implemen he new echnology and by swiching producion from he old o he new echnology. In such a seing characerized by linear producion echnologies and linear cos funcions, i can be shown ha i is no opimal o inves in boh echnologies a he same ime, if hese echnologies differ wih respec o heir ne produciviy (i.e. ne of depreciaion). he reason is ha a uni of invesmen for 5

8 boh echnologies would represen he same marginal cos in erms of consumpion forgone, and so he echnology wih he highes ne marginal produc, would generae he highes ne marginal welfare gain. Hence, if invesmen in he A echnology is aking place, hen invesmen in he B echnology will be zero and he oher way around. We will also allow for he possibiliy ha here is a phase in which no invesmen in A akes place, even hough he exising capial sock is used o produce oupu while invesmen in and producion using he B echnology is happening a he same ime. his second phase will herefore be called he join producion phase (JPR for shor). In he final phase, only invesmen in and producion wih he B echnology occurs, while he A echnology capial sock has been deacivaed a he beginning of ha phase. his is he carbon free phase (CFR for shor). BAU JPR CFR E E E Y A +Y B II I Y B Y A Y A Y B BAU=0 JPR CFR Figure 1. ransiion Phases he effecs of he producion and invesmen aciviies during he separae phases disinguished in he model can be summarized as in Figure 1. BAU, JPR and CFR mark he momens in ime a which he BAU, JPR and CFR phases begin. In he BAU phase, he cumulaive emissions (labelled E) are increasing exponenially as he sock of carbon based capial is growing. In he JPR phase invesmen in echnology A sops, and oupu using echnology A (i.e. Y A ) is a is maximum level bu sars o decrease over ime, because of echnical decay. he sock of echnology B capial is buil up from scrach saring wih he arrival of he JPR phase. Producion using echnology B (i.e. Y B ) is a full capaciy and exponenially increasing during he JPR phase. Cumulaive emissions are sill increasing, bu a a decreasing rae as he sock of carbon based capial is run down. During he JPR phase, oal oupu is sill growing, bu a a slower rae han he growh of Y B. Phase CFR sars when cumulaive emissions E hi he cumulaive emissions hreshold. During he final phase, only invesmen in and producion wih echnology B is possible. Because he hreshold has been reached, carbon based capial needs o be discarded. his leads o a drop in oupu, since producion using he A echnology ceases, even hough 6

9 here is sill a posiive amoun of carbon based producion capaciy available. Hence, oal oupu jumps from poin I o poin II and sars growing from he laer poin again during he CFR phase. I should be noed ha he various phases in he model are qualiaively differen. he firs phase, i.e. he BAU phase, uses a high growh echnology which unforunaely quickly raises he sock of cumulaive CO2 emissions ha is bounded from above by he ICC. Before cumulaive emissions will hi his hreshold, invesmen in carbon free echnology mus have aken place during phase JPR o bring carbon free capaciy up o a level where he swich from carbon based o carbon free echnology would no force changes in consumpion levels ha are oo disrupive. he laer is implied by our assumpion ha consumers dislike consumpion shocks (hey wan o smooh changes in consumpion paerns over ime o some exen, given he fac ha hey have a posiive coefficien of relaive risk aversion (i.e. )). From CFR onwards he world is green, and will have o grow a a relaively low bu CO2 emission free rae. In addiion o he Basic Model, we implemen wo oher versions. In he second version of he model, he produciviy of he new echnology can be improved hrough (endogenously deermined) R&D before invesmen in he new echnology akes place. he final version combines he endogenous R&D feaures wih he exisence of an emissions hreshold ha, once pas, pushes he world ino a high weaher relaed damages regime/phase. he laer regime shif is modelled by means of a one ime jump in he raes of echnical decay. his regime change has an impac on he shadow price of emissions, and so feeds back o invesmen decisions prior o he high damages swich poin. Using hese models, we ll ry o find ou how R&D in combinaion wih he embodimen of echnology in invesmen in he face of he exisence of (a) climae hreshold(s), will deermine he iming and he lengh of he various phases disinguished in our model. 2.2 he Basic Model 8 he endogenous growh framework of BAM borrows heavily from he AK model by Rebelo (1991). However, conrary o he original AK seing, we disinguish beween wo ypes of capial: carbon based, or carbon based, capial furher denoed by K A and carbon free, or green, capial furher denoed by K B. he capial socks in his model are subjeced o exponenial decay a raes for,. Because of he lineariy of he producion funcions in an AK seing, and since one uni of capial akes one uni of consumpion foregone for he wo echnologies disinguished, i follows ha here will always by invesmen in jus one ype of capial a he ime. Hence, gross invesmen in a paricular echnology is eiher equal o zero, or i is equal o oal savings. Welfare in his seup comes from consumpion only, and we use he CIES ineremporal welfare funcion o describe he oal flow of welfare over ime. he aciviies during he differen phases are summarized in able 1. 8 BAM for shor. 7

10 Aciviies BAU Phase JPR Phase CFR Phase Invesmen Producion Capial... Accumulaion. CO2 Emissions.. 0 able 1. BAM aciviies In able 1, I x refers o he amoun of invesmen in capial of ype x, where, refers o carbon based capial and carbon free capial, respecively. Similarly, Y x refers o he flow of oupu using K x, where K x is he sock of capial of ype x. We also assume ha Y x is proporional o Kx wih a consan produciviy of capial as a facor of proporion. ypically, we use A and B o denoe he capial produciviies of echnologies A and B, implying ha Y A =A.K A, eceera. Finally, he insananeous flow of CO2 emissions is proporional o he capial sock in use wih a consan facor of proporion for,. Obviously, 0. Noe ha when producion on some ype of carbon based capial ceases, his very fac iniiaes anoher phase, since his inroduces a difference beween he composiion of aciviies beween phases J and F. So Y A =0 implicily defines he arrival ime of phase F, and he scrapping of carbon based capial a =F. 2.3 BAM: he ineremporal opimizaion seing I should be noed ha he fac ha he final phase of BAM is a pure AK seing, allows us o obain he opimum consumpion pah for he CFR phase direcly, given he iniial value for. 9 As a consequence, we can immediaely derive he opimum ime pah for welfare during he CFR phase for a given iniial value. he welfare generaed during he CFR phase depends herefore on he erminal value of K B a he end of he JPR phase. I follows ha he disribuion of a sae variable over he enire pah is opimal when he marginal coss of having o deliver an exra uni of he sae variable in is role as a erminal value a some ime * is exacly mached by he marginal benefis ha his exra uni of he sae variable generaes as he iniial value for he opimum coninuaion from *. Since hese marginal benefis and marginal coss are capured by he co sae variables (see Leonard and Van Long (1992), ch 4), he laer need o be coninuous along an opimum pah: saes and co saes don jump. An opimum pah can be hough of as a combinaion of an opimum firs sep and an opimum coninuaion (as in dynamic programming problems), which allows us o inerpre our muliphase ransiion model as a finie horizon opimum conrol problem wih a free endpoin and a scrap value funcion, as described in Leonard and Van Long (1992, ch 7, furher called LVL7). his is he siuaion ha is of direc relevance in our case, since we do no know on beforehand when he nex phase will sar. However, on an opimum pah, posponing or exending a paricular phase by an infiniesimal amoun of ime, shouldn change he valuaion of he enire pah. LVL7 show ha he derivaive of he value 9 See, Barro and Sala i Marin (2004), chaper 4 in paricular. 8

11 funcion (in our case he presen value of oal welfare) wih respec o he erminal dae (of a phase) maches he value of he Hamilonian a ha dae. his makes sense, as he Hamilonian a some momen in ime measures he conribuion o he value funcion of he opimal use of all resources available a ha momen in ime. Bu in a sequence of phases, lenghening he one phase by a uni of ime implies shorening he nex phase by he same uni of ime. So we should keep on posponing he arrival of he nex phase as long as he Hamilonian of he earlier phase exceeds ha of he laer phase. he opimum swiching momen beween any wo phases is herefore implicily defined by he requiremen ha he Hamilonians of wo adjoining phases mus be he same when evaluaed a he momen of he phase change. Again, his makes perfec economic sense, as he value of he Hamilonian a he end of he curren phase can be seen as he benefis of expanding he curren phase by a marginal uni of ime, while he Hamilonian a he beginning of he nex phase can be seen as he opporuniy cos of ha expansion. In pracice, he equaliy of he Hamilonians evaluaed under he condiions relevan in eiher of he phases jus before and jus afer a phase change, will resul in a condiion ha needs o be me by a se of saes and co saes evaluaed a he momen of he phase change. he differences in he naure of he aciviies a he momen of a phase change can be used o implicily describe he condiions ha should be me a he momen of a phase change. For example, a =CFR i mus be he case ha carbon based capial is deacivaed. For >=CFR is mus herefore be he case ha he shadow price of carbon based capial is zero, since he cumulaive emission hreshold has been reached and carbon based capial can herefore no be used anymore and has become worhless herefore. 2.4 BAM: formal srucure he overall welfare funcion consiss of a summaion of inegral welfare derived from he flow of consumpion during he hree phases disinguished in he BAM: (1) In equaion (1) W 0 measures he presen value of oal welfare a ime =0, a which, by assumpion, phase U begins. In equaion (1), is he rae of discoun, while 1 is he (consan) ineremporal elasiciy of subsiuion. C is consumpion a ime, while J and F are he momens in ime a which phases J and F begin. Given he exposiion on ineremporal opimizaion above, he ime pahs ha would maximize (1) can be obained by solving he ime pahs for he Hamilonian problems ha can be defined for he individual phases, while linking hose ime pahs ogeher by means of he requiremens of opimum phase lenghs (implying he equaliy of he Hamilonians for phases U and J a =J and for phases J and F a =F). Effecively his comes down o maximizing (1) wih regard o he flows of consumpion during each phase and wih regard o he phase lenghs hemselves, consrained by he echnologies ha are relevan in each phase, by he socks inheried from previous phases, and by he hresholds ha are relevan during he various phases. We will now solve he Hamilonian problems for each individual phase. 9

12 Phase U Using he superscrips U, J and F o denoe he phase o which a paricular variable perains, while dropping he ime subscrip for ease of noaion, he presen value Hamilonian H U is given by: (2) In equaion (2), C is he only conrol variable, while K A and E are he sae variables and and are he corresponding co saes. As firs order condiions we have: 0 0 / (3) (4) / (6) (5) (7) Equaion (6) is obained by means of subsiuion of equaion (3) ino he macro economic budge consrain which saes ha oupu is used for consumpion and (gross) invesmen purposes. Equaions (4) (7) consiue a simulaneous sysem of differenial equaions ha can be solved forward in ime, given a se of iniial values for he various sae and co sae variables. his will give rise o erminal values of hose same sae and co sae variables a he erminal dae of phase U, i.e. a = J, he value of which is unknown so far. Phase J Phase J differs from phase U since invesmen in he A echnology has sopped and ha in he B echnology begins. However, he carbon based capial sock K A is sill used for producion purposes. he presen value Hamilonian for phase J, i.e. H J, is now given by: (8) As in he U phase, we have jus one conrol, i.e. C J, bu hree saes K A, K B and E and corresponding co saes, and. As firs order condiions we have: 0 / (9) (10) (11) 10

13 0 / (14) (12) (13) (15) Equaion (14) is again obained by means of subsiuion of opimum consumpion levels (as given by equaion (9)) ino he macro economic budge consrain. As before, equaions (10) (15) consiue a simulaneous sysem of differenial equaions ha can be solved forward in ime, given iniial values for he various sae and co sae variables. Noe ha he iniial values in phase J for hose sae and co sae variables ha boh sysems have in common, are he same as he erminal values for hose variables a he end of phase U because of he coninuiy of sae and co sae variables along an opimum pah. For a given value of F, his sysem of differenial equaions allows he forward calculaion of erminal values for he saes and co saes a ime =F, which will hen funcion as he iniial values for he saes and co saes during he carbon free phase. Phase F Phase F differs from phase J in ha he carbon based capial sock is discarded, and consequenly he flow of CO 2 emissions drops o zero. From =F producion is oally green. he presen value Hamilonian for phase F, i.e. H F, is now given by: 0 (16) As in he U and J phase, we have jus one conrol, i.e. C F, bu wo saes K B and E and corresponding co sae variables and. As firs order condiions we have: 0 0 / (17) (18) / (20) 0 (19) (21) As before, equaion (20) is obained by subsiuing equaion (17) ino he macro economic budge consrain. Again, equaions (18) (21) consiue a simulaneous sysem of differenial equaions ha can be solved forward in ime, given iniial values for he various sae and co sae variables inheried from phase J. However, in his case he erminal values for he saes and co saes are 11

14 implicily described by he sandard ransversaliy condiions in an AK seing ha require he presen value of he carbon free capial sock o approach zero a he erminal dae, i.e. in his case a ime infiniy. For cumulaive emissions, he erminal value had already been reached a =J, when cumulaive emissions hi he hreshold. ransversaliy condiions For phase F he sandard ransversaliy condiion (furher called VC for shor) applies regarding he value of carbon free capial a ime infiniy: lim,, 0 (22) where we have now added ime subscrips. Noe ha (18) can be inegraed direcly o obain he ime pah for, which can hen be subsiued ino (20) o obain he ime pah for,. We ge:,,, (23) B, B FB, F, B (24) Equaions (23) and (24) can be subsiued ino VC (22), and we find ha in order for VC (22) o hold we should have ha: 0 (25), F, B (26) When subsiuing (26) ino (24), we find ha:,, B (27) I follows from (27) ha if he srucural parameers are such ha (25) is me and if we pick consumpion a ime F (hence, (see equaion (17)) such ha (26) is me, hen he carbon free capial sock will grow a he seady sae growh rae δb / from ime =F. Apar from he VC above, we require ha:, 0 (28) Equaion (28) saes ha he shadow price of carbon based capial a he end of he J phase (so a he momen i is discarded) should be zero, since having an addiional uni of capial ha will no produce anyhing is useless and herefore worhless. 12

15 Finally, here are wo VCs ha perain o he opimum lengh of phase U and phase J, and ha require he equaliy of he Hamilonians of he various phases a differen poins in ime. For he opimum lengh of phase U (given by he value of J, since U=0 by assumpion), we mus have ha, while he opimum arrival daa of he F phase is deermined by he requiremen ha. Using he definiions of he Hamilonians in (2), (8) and (16), as well as he FOCS regarding consumpion in (3), (9) and (17) ogeher wih he coninuiy consrains on saes and co saes ha feaure in boh adjoining phases, we obain implici descripions of he arrival daes of phases J and F, given by:,, (29),, (30) Equaion (29) saes ha invesmen in he carbon based echnology should sop a he momen ha he shadow price of he A echnology is equal o (and hen drops below) he shadow price of he B echnology. Since he marginal cos of obaining a uni of capial is he same in boh cases (i.e. one uni of consumpion foregone), equaion (29) is consisen wih he maximizaion of he (presen value) welfare surplus associaed wih invesmen. Equaion (30) saes ha producion using carbon based capial should sop he momen ha he benefis from coninuing o use a uni of capial (he LHS of equaion (30), as one uni of capial produces A unis of oupu, and each uni of oupu is worh, in presen value welfare erms a =F) maches he cos of doing ha (as given by he RHS of equaion (30), since one uni of capial produces unis of CO2 emissions, each a a cos of 10, ). Equaion (30) is herefore consisen wih he zero quasi ren condiion as an implici descripion of he opimum momen o scrap exising capaciy known from he vinage lieraure (cf Johansen (1995), Solow e al. (1966) and Boucekkine (2011), for example) ha saes ha a vinage, once insalled, should be discarded as soon as i s quasi rens (consising in his case of he presen value welfare value of oupu less oal variable (emission ) coss, also in presen value welfare erms) become negaive. 2.5 BAM: Sequenial numerical soluion of he sysems of differenial equaions he hree sysems of differenial equaions, furher called SU, SJ and SF, can in principle be solved, as he iniial and erminal values we have available for he sae variables, and he VCs ha provide eiher some fixed poins for he ime pahs of he co saes (cf. equaion (28)), or link he cosaes o a sae variable which ime pah has been fixed hrough a given iniial value (cf. equaion (26)), or ha links differen co saes a some poin in ime (cf. equaions (29) and (30)) provide exacly enough informaion o obain a fixed poin for all ime pahs concerned. o see his, i should be noed ha we need o obain fixed poins for he ime pahs for hree differen sae variables (KA,KB and E), and for heir corresponding co sae variables, as well as he opimum values of he phase lenghs of phases U 10 Noe ha he shadow price of emissions iself is negaive, since an addiional uni of cumulaive emissions would reduce poenial welfare raher han increase i. 13

16 and J. Hence, for BAM we need 8 pieces of informaion. We have iniial values available for,,,, 0, and. In addiion, we have a erminal value for cumulaive emissions. he ransversaliy condiions given by equaions (26),(28),(29) and (30) hen provide he res of he necessary informaion. A numerical soluion of SU can easily be obained, condiional on some a priori values of,, and J, given he iniial values of, and. he soluion of SU hen provides erminal values for, and, ha, on accoun of he coninuiy of saes and co saes give rise o iniial values for SJ, since we mus have ha, only need an iniial value for,,,,,,,,,, 0,. We as well as an a priori value for F o be able o calculae SJ forward in ime. ha iniial value is provided by he VCs lised in equaion (29). Given he a priori values for,,, J and finally F, we are able o calculae he erminal values for,,,,, and, which, again using he requiremen of he coninuiy of saes and co saes, imply ha,,,,,, and,,. he ime pahs hus obained for all saes and co saes, in combinaion wih our guesses of he various phase lenghs, can now be used o evaluae he differences beween he RHS and he LHS of he erminal consrain and hose of he VCs no used so far, i.e. of equaions (26), (28) and (30). Obviously, if hese consrains would be me by our a priori values of,,, J and F, hen we have found he soluions for all he ime pahs ha would maximize welfare. Being a guess, hese differences will generally no be equal o zero, bu we can use a search mehod (so far we have used a seepes descen algorihm) o find he iniial values ha would mee all he VCs simulaneously. 11 I should be noed ha a similar approach can be followed for versions of he model wih more han 3 phases, as we will describe furher below. 2.6 BAM+R&D: changes relaive o BAM A firs modificaion of BAM involves he inroducion of R&D driven endogenous echnical change wih respec o he carbon free echnology. We have made he assumpion ha i akes he form 11 In fac, given he simpliciy of our sysem, i is possible o obain analyical soluions for all ime pahs, even hough hese are in par quie inricae non linear expressions involving hyper geomeric funcions. hese inegral equaions link he various iniial and erminal condiions as well as he values of J and F ogeher hrough a simulaneous sysem of non linear equaions. Using ha sysem, we were able o find he fixed poins for all ime pahs by numerically solving he non linear sysem for he fixed poins. he ime pahs for each of he sae and cosae variables could hen be obained by subsiuing he numerical values hus found for he fixed poins ino he analyical soluions of all ime pahs involving hese fixed poins. his procedure proved o work, bu is raher edious and ime consuming and i s feasibiliy depended crucially on he simpliciy of he model. Minor deviaions from he AK se up proved o make using he analyical mehod infeasible. his provided a srong incenive o use he sequenial numerical soluion mehod oulined in he main ex. Boh mehods do generae he same resuls, as hey should, bu he sequenial numerical mehod is much more efficien in erms of boh obaining he sysem of differenial equaions o be solved, and finding he soluion. Noneheless, more work is needed o improve upon he raher crude seepes descen search rouine ha we are employing a he momen. he laer converges raher slowly, if a all, for he more inricae versions of he model exensions we have buil up o dae and which are no repored here. Noneheless, he numerical model does allow us o expand he model in ways ha would make i impossible o solve using he analyical soluion approach. 14

17 of increasing he value of B before he momen he echnology is acually implemened, hus aking ino accoun he noion of he embodimen of echnical change and he irreversibiliy of invesmen decisions. Since we are using an AK seing, however, he sandard specificaion for endogenous echnical change as used in he endogenous R&D lieraure is no readily applicable. For example, a Lucas (1988) or Romer (1990) or Aghion and Howi (1992) ype of R&D funcion, would involve a specificaion such as: B R B (31) where R represens R&D resources measured as consumpion foregone. However, in an AK seing, capial produciviy is consan by consrucion, so ha a coninuous increase of he effecive availabiliy of anoher facor (say knowledge, or human capial) is no necessary o keep he marginal produc of capial from falling as he economy is growing and capial is accumulaing. If we would use (31) in a simple AK seing, hen, disregarding problems of embodimen for he sake of simpliciy, growh would be explosive raher han seady. his is because for a posiive value of R, capial produciviy would end o grow, hus raising boh he incenive o engage in R&D and he possibiliy o allocae he resources, since he growh rae of oupu would be acceleraing in he process. Hence, we have oped for a specificaion of he R&D process ha allows for a decreasing marginal produc of R&D and for a growh rae of B ha asympoically approaches zero for a consan posiive allocaion of R&D resources, so ha he incenive o engage in R&D will diminish over ime. We have: B R ( B B) (32) where B is he asympoic value for capial produciviy and where 0 and 0 1 are consans. Saring ou wih a basic idea ha provides a value 0 B 0 B, he marginal produc of R&D is falling as R increases and as B increases, raher han rising wih B. 12 he noion underlying he specificaion of (32) is ha he R&D process gives rise o discoveries of more efficien ways of doing hings from a finie se of unknown possibiliies, as in a Plaonic world, wih one of hese possibiliies simply being he bes (and providing he value of B ). Equaion (31) provides a principally differen view, in ha i allows for an R&D process ha acively creaes new and beer ways of doing hings in an unbounded fashion, insead of discovering he limied number of opions ha naure has been hiding so far. Inroducing (32) ino he Basic Model gives rise o a number of modificaions. Firs, he Hamilonian has o be exended by adding he value of increases in B due o R&D. Secondly, he macroeconomic budge consrain ha describes he accumulaion of capial has o be adjused for he fac ha one uni of R&D resources R requires one uni of oupu (i.e. consumpion or invesmen foregone). Insead of jus he one conrol variable C, we now have an addiional conrol variable R, and an addiional sae variable, i.e. B (he carbon free produciviy parameer of BAM). B has become a sae 12 Noe ha (32) is joinly concave in R and B (as opposed o (31)), which is a necessary condiion for he welfare maximisaion problem o have a soluion. 15

18 variable since i can change as long as R&D is aking place, bu mus remain consan afer R&D has ceased. Leonard and Van Long (1992, ch.7) describe how he laer siuaion can be handled. heir approach is o regard he Hamilonian as a funcion of B, i.e. H(B), while subsiuing B 0 as he dynamic consrain for B for he phases wihou R&D (in which B doesn change, herefore). In all phases we have ha he equaion of moion for he co sae variable associaed wih B is given by B H / B. he reason for explicily specifying he dynamic consrains on he co sae for B during he phases where B iself is no changing (because here is no R&D anymore) is ha he erminal value of B when he R&D process sops, will influence he generaion of welfare in he following phases wihou R&D. hus, hrough he coninuiy of co saes and saes, informaion abou he fuure welfare effecs of having a high value of B in he join producion phase and in he carbon free phase, can influence allocaion decisions during he phase wih posiive R&D ha would direcly involve he rade off beween R&D and oher uses of oupu, like consumpion and angible capial invesmen. Noe ha I can be shown ha once a B 0 is available, hen i pays no o wai improving he B echnology by engaging in R&D (see Appendix A). Hence, wihou loss of generaliy, we may assume ha he basic idea underlying he carbon free echnology is available from =BAU=0. In ha case, R&D should sar a =BAU, and i should sop a =JPR, i.e. he momen ha invesmen in he carbon free echnology commences. Consequenly, he BAM+R&D model has he same number of phases as BAM. For he BAU phase, assuming ha R&D is done from he very beginning, hree equaions need o be added, i.e. 13 B H / B, B R ( B B) and H / R 0, while he capial accumulaion consrain mus be modified, giving K A A A ( A ). K C R. In addiion o his, he revised Hamilonian for he BAU phase is now given by:. 1 H e C /(1 ) A {( A A) K A C R} B B. Since here is now an addiional sae variable, here is also an addiional VC. As usual, we require ha a infiniy he presen (uiliy) value of B should approach zero, i.e. lim, B 0. However, B B JPR, since R&D has B ceased a =JPR, urning B effecively ino a consan for, and so he addiional VC is reduced o he requiremen ha lim B, 0. Using (25), i can be shown ha he laer VC implies: JPR,,, (33) I is hard o inerpre (33). his goes a foriori for he revised version of he ransversaliy condiion ha now deermines he opimum lengh of he BAU phase. ha VC is now differen from he one we had before, since he R&D process is acive during he BAU phase and inacive during subsequen phases. Consequenly, he Hamilonians of he BAU and JPR phase evaluaed a =JPR, will now involve erms coming from he R&D funcion as well as he corresponding co sae evaluaed a =JPR, resuling in a complicaed expression linking he various saes and co saes ogeher a =JPR. Because i canno 13 Noe ha for ease of noaion we have dropped he ime subscrip and he phase superscrip excep where he presence of he ime subscrip is needed. 16

19 readily be inerpreed, he expression is no given here. Noe ha (33) in combinaion wih he iniial value for B, i.e. B 0, provide enough informaion o solve his revised sysem of differenial equaions. 2.6 BAM+ R&D+UCL he final version concerns a combinaion of BAM and R&D ha includes he inroducion of a weaher relaed damage hreshold. We do his as follows. Suppose ha he damage hreshold lies a some value of cumulaive CO2 emissions ha is reached wihin he BAU phase. When he hreshold is reached, he jump in he decay parameers occurs. A his momen in ime, he BAU phase is effecively spli ino a low damage sub phase and a high damage sub phase. All he phases coming afer he high damage BAU phase are also high damage phases from hen on. Again, his is a siuaion ha is described in Leonard and Van Long, where he regime shif from low damages o high damages iniiaed by a sae variable hiing a paricular hreshold, implies a jump in he corresponding co sae variable. (cf. LVL, ch10). he only difference beween he low damage sub phase of he BAU phase and he high damage sub phase is of course a change in he values of he echnical decay parameers. Since he rae a which CO2 emissions are accumulaed depends on producion decisions, he momen in ime a which he damage hreshold will be hi, depends on hese very producion decisions oo. Hence, he arrival ime of he high weaher relaed damage sub phases is subjec o choice and herefore o opimal decisionmaking. Consequenly, we can opimally choose he momen a which he high damage sub phase will arrive. We can do his again by requiring ha he Hamilonians evaluaed a he momen of arrival of he high damage sub phase will be he same a eiher side of is arrival ime. his implies ha his model will have four differen phases insead of hree. I follows ha we need o deermine an exra phase lengh in addiion o he size of he jump in he co sae for cumulaive emissions a he ime of arrival of he firs high damage sub phase. In addiion o he given iniial and erminal values of he BAM+R&D model as well as he corresponding ransversaliy condiions, we have an addiional erminal value in he form of he locaion of he damage hreshold iself, in combinaion wih he requiremen of he equaliy of he Hamilonians a boh sides of he damage phase change. I urns ou ha he ransversaliy condiion for he arrival of he high damage sub phase during he BAU phase is given by: 14,,,,, (34) Equaion (39) should hold exacly a he arrival ime of he high damage sub phase, i.e. a =UH. 15 In equaion (39), he superscrips H and L refer o low and high damage sub phases, while he superscrips you refers o he business as usual phase. he RHS of equaion (34) conains he jump in he shadow price of cumulaive emissions. I measures he change in he welfare coss associaed wih emissions per uni of capial before and afer he jump. he LHS of equaion (34) measures he welfare cos associaed wih he exra decay per uni of capial. Equaion (34) implies he equaliy beween he welfare cos of 14 Obviously, he arrival ime of he firs high damage sub phase could also be wihin he join producion phase. Bu ae his sage we are mainly ineresed in reporing on he principles involved, which would be he same in whichever phase he damages hreshold would be siuaed. 15 Noe ha UH also denoes he end of he low damage sub phase of he BAU phase. In ha case, UH comes one insan before he value of UH ha represens he beginning of he high damage sub phase of he BAU phase. 17

20 using a uni of capial before and afer he arrival of he high damage sub phase. We conclude ha he jump in he co sae mus be such ha he welfare cos of using a uni of capial remains unchanged. Since he depreciaion coss are higher afer he jump, he emission coss mus be lower, implying a drop (in absolue erms) in he shadow price of emissions. he laer is consisen wih he observaion ha a faser rae of decrease of he carbon based capial sock will lead o a reducion in he rae a which CO2 emissions are accumulaed, oher hings remaining he same. 3 Oucomes 3.1 Model calibraion In order o show how he various models work, he parameers of he model need o calibraed or fixed a priori. o his end we have made use of he Nordhaus RICE 2010 daa 16, as well as some direc assumpions, since he Nordhaus model and ours have a differen naure. Nordhaus essenially uses a neoclassical growh seing wih jus one phase, while we use an AK seing wih muliple phases. his implies ha simply copying he Nordhaus numbers ino our model is no possible. In order, noneheless, o reproduce growh raes and saving raes ha have abou he righ size, we have made he assumpion ha he capial oupu raio in he BAU phase is equal o 4, which is much higher han in he Nordhaus model. hen again, our (implici) noion of capial is much broader, since i may be hough o also include human capial. We have also made he assumpion ha depreciaion coss as a fracion of oupu is 15 per cen. his number is relaively high oo, bu no unrealisic given he much broader capial concep. Using Nordhaus daa on FP growh as well as oupu per capia and populaion growh, we arrive a an implied growh rae of oupu (and of capial in an AK seing) equal o Using equaion (27) o derive he seady sae growh rae in an AK seing, we find ha if we make he assumpion ha oupu and (carbon based) capial sock grow a seady sae rae equal o , we mus have ha: (35) In equaion (35), we have used he assumpion ha depreciaion as a fracion of oupu is 15 per cen, implying ha he laer implies ha his. value for he depreciaion rae is much lower han he one used by Nordhaus, who uses a 10 per cen depreciaion rae. Equaion (35) implies combinaions of and given by: (36) 16 See hp://nordhaus.econ.yale.edu/ricemodels.hm for furher deails. In order o obain observaions for 2010 from he ones lised for 2005 and 2015 in he RICE daa, we use geomerical inerpolaion. 18

21 Plugging in he value ha Nordhaus also uses, we find ha his implies a much lower ineremporal elasiciy of subsiuion han is used by Nordhaus, who uses a value of equal o 1.5. In boh cases, however, 1, implying ha feliciy is negaive, bu rising in consumpion levels. Marginal uiliy would sill be posiive and falling, however. Since welfare is he inegral over (he presen value) of feliciy, welfare would be negaive as well. We herefore added an addiive erm o feliciy a all momens in ime equal o he negaive of feliciy a ime zero. his is equivalen o an upward shif of he feliciy funcion o he posiive quadran. Since is relaively large, feliciy will be relaively small, and so will be he presen value of welfare. his implies ha he numerical values of he co saes will be small as well, since hey represen he change in welfare due o a 1 uni change in he corresponding saes. For example, since Y 0 =68.95 rillion dollars of , our assumpions imply ha K 0 =275.8, bu also ha C0=(1 s).y0= Hence, feliciy a =0 is equal o herefore we have added a muliplicaive scaling facor o he feliciy funcion as well equal o 10, resuling in values for saes and co saes ha are no many orders of magniude apar. Even hough our implied ineremporal elasiciy of subsiuion is raher low, in combinaion wih he oher parameer values, accepable values for boh he growh rae, and he saving rae can be generaed. he implied value of he saving rae (s) is given by: (37) Wih respec o cumulaive CO2 emissions and he locaion of he climae ipping poin in erms of he cumulaive emissions generaed by our model, we have used he following procedure. From he Nordhaus daa, we can infer ha he raio beween he concenraion of CO2 in he amosphere measured in ppm and he concenraion of carbon in he amosphere measured in GCs is given by: (38) where refers o he firs difference operaor. he presen concenraion of CO2 in he amosphere amouns o 390 ppm. 18 he preindusrial concenraion of CO2 is 280 ppm. he climae ipping poin is generally associaed wih a emperaure rise of 2 Kelvin above preindusrial levels. he equaion describing he relaion beween emperaure rises and CO2 concenraions in ppm relaive o some base level is given by: 19 /. (47) In equaion (39), represens he emperaure rise relaive o a baseline emperaure a a baseline concenraion of CO2 given by ppm 0. I follows ha he ipping poin a 2 Kelvin relaive o preindusrial levels is given by 280 / For a 3 Kelvin emperaure rise, he corresponding 17 =0 refers o he daa for 2010 are obained by means of geomerical inerpolaion beween he Nordhaus daa for 2005 and See he websie on CO2 measuremens on Mauna Loa, Hawaii hp:// 19 See for more deails hp://en.wikipedia.org/wiki/radiaive_forcing. 19