#2013-009

Designing an optimal 'tech fix' path to global climate stability:  R&D in a multi‐phase climate policy framework 

Adriaan van Zon and Paul A. David                 Maastricht Economic and social Research institute on Innovation and Technology (UNU‐MERIT) email: info@merit.unu.edu | website: http://www.merit.unu.edu  Maastricht Graduate School of Governance (MGSoG) email: info‐governance@maastrichtuniversity.nl | website: http://mgsog.merit.unu.edu  Keizer Karelplein 19, 6211 TC Maastricht, The Netherlands Tel: (31) (43) 388 4400, Fax: (31) (43) 388 4499 

Working Paper Series 

UNU-MERIT Working Papers 

ISSN 1871-9872

Maastricht Economic and social Research Institute on Innovation and Technology, UNU-MERIT

Maastricht Graduate School of Governance

MGSoG

UNU-MERIT Working Papers intend to disseminate preliminary results of research

carried out at UNU-MERIT and MGSoG to stimulate discussion on the issues raised.                                                    

  

   

DesigninganOptimal'TechFix'PathtoGlobalClimateStability:R&DinaMulti‐PhaseClimatePolicyFramework

By

*AdriaanvanZon1&PaulA.David1,21SBEMaastrichtUniversity&UNU‐MERIT(Maastricht,NL)

2EconomicsDepartment&SIEPR,StanfordUniversity

Firstdraft:25August,2012Thisversion:31January2013

ABSTRACT

The research reported here gives priority to understanding the inter‐temporal resourceallocationrequirementsofaprogramoftechnologicalchangesthatcouldhaltglobalwarmingbycompleting the transition to a “green” (zero net CO2‐ emission) production regimewithin thepossiblybrief finite interval that remainsbeforeEarth’s climate isdrivenbeyonda catastrophictippingpoint. Thispaperformulatesamulti‐phase,just‐in‐timetransitionmodelincorporatingcarbon‐based and carbon‐free technical options requiring physical embodiment in durableproductionfacilities,andhavingperformanceattributes thatareamenabletoenhancementbydirectedR&Dexpenditures.Transitionpathsthatindicatethebestorderinganddurationsofthephases inwhich intangibleand tangiblecapital formation is takingplace,andcapital stocksofdifferent types are being utilized in production, or scrappedwhen replaced types embodyingsociallymoreefficienttechnologies,areobtainedfromoptimizingsolutionsforeachofatrioofrelated models that couple the global macro‐economy’s dynamics with the dynamics of theclimate system. They describe the flows of consumption, CO2 emissions and the changingatmospheric concentration of green‐house gas (which drives globalwarming), alongwith theinvestment dynamics required for the timely transformationof theproduction regime. Thesepaths are found as thewelfare‐optimizing solutions of three different “stackedHamiltonians”,eachcorrespondingtooneofourtrioofintegratedendogenousgrowthmodelsthathavebeencalibrated comparably to emulate the basic global setting for the “transition planning”framework of dynamic integrated requirements analysis modeling (DIRAM). As the paper’sintroductorysectionexplains,thisframeworkisproposedinpreferencetothe(IAM)approachthatenvironmentalandenergyeconomistshavemadefamiliarinintegratedassessmentmodelsof climate policies that would rely on fiscal and regulatory instruments ‐‐ but eschew anyanalysisoftheessentialtechnologicaltransformationsthatwouldberequiredforthosepoliciesto have the intended effect. Simulation exercises with our models explore the optimizedtransition paths’ sensitivity to parameter variations, including alternative exogenousspecificationsof the locationof apair of successive climate “tippingpoints”: the first of theseinitiates higher expected rates of damage to productive capacity by extreme weather eventsdrivenby the rising temperatureof theEarth’s surface;whereas the second, farmore serious“climate catastrophe” tipping point occurs at a still higher temperature (corresponding to ahigher atmospheric concentration of CO2). In effect, that sets the point before which thetransition to a carbon‐free global production regime must have been completed in order tosecurethepossibilityoffuturesustainabledevelopmentandcontinuedglobaleconomicgrowth.JELcodes:Q540,Q550,O310,O320,O330,O410,O440.

Keywords:globalwarming,tippingpoint,catastrophicclimateinstability,extremeweather‐relateddamages,R&Dbasedtechnicalchange,embodiedtechnicalchange,optimalsequencing,multi‐phaseoptimalcontrol,sustainableendogenousgrowth.

E‐mailcontacts:*Correspondingauthor‐‐Adriaan.vanZon@MaastrichtUniversity.NL;pad@stanford.edu

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1Introduction:ClimateInstabilityandEnvironmentalPolicyResearch

Economic developments thus far in human history have been linked closely withprogress inmethods for thebulk conversion intousefulworkof the energy stored in carbon‐basedfuels.1Burningwood,coal,oil,andnaturalgasgivesrisetoCO2‐emissionsthat,togetherwith releases of other greenhouse gasses (GHG’s) like methane, are now thought to beresponsible for the considerable warming of the earth’s atmosphere since the industrialrevolution of the late eighteenth century and the worrying prospect of that upward trendcontinuingforyearstocome.2

1.1Motivation:climatescienceandclimatepolicyThe implied future consequences are “bad news” on a number of related counts: sea

levelswill rise, tropical diseaseswill becomemorewide‐spread, stormswill bemore violent,patterns of rainfall will change (affecting agriculture), and fresh‐water supply shortages willbecomeaproblemduetoglobalglacierretreat,andsoon.3 Mostoftheseconsequentchangesrepresentsignificantcoststosocietyenroutetothepotentialemergenceofcatastrophicclimateinstability.4 But, more worrying still is the possibility that the environmental changes set inmotion by the warming of Earth’s land surface and oceans may impart a self‐reinforcingmomentumandthusacceleratetheriseofglobalmeantemperaturetoapace faster thanthatresultingsolelyfromanthropogenicreleasesofGHG.

Thepast20yearshavebroughtrevolutionaryadvances inthestudyofclimatehistorybasedonthedeep‐icecoresofthemillenniabetweenthelastglacialmaximumandtheopeningofthe(present)Holoceneera.5 Amongthemanyimportantfindingsarethosethathavegivengreater plausibility and disturbing palpability to the conjectured existence of global climate“tippingpoints”thatcantrigger“abrupt”changes intheclimatesystem.6 ToallowtheEarth’s

                                                            1Thatistosay,“work”inthesensethatthetermisusedinphysics.SeeCipolla(1972)foranelegantlyconciseandpenetratingpresentationofthisthemeineconomichistory.

2Weputasideherediscussionofremainingscientificuncertaintiesregardingthepaceofwarmingandtheextenttowhich it is attributable to anthropogenic sources, and observations on the role of active climate change denial(whetherfromignorance,innateskepticismorstrategicperceptionofmaterialself‐interest).SeethecommentsandreferencesinDavidandvanZon(2012:esp.sect.2).

3See,e.g.Dyer(2010)andLynas(2007)onthedisruptivesocialandpoliticalsequelae.Lynasprovidesajournalisticviewon the consequences for theworldofup to6degrees additionalwarming for eachdegreeof extrawarming,while Dyer provides scenarios regarding the struggle for resources necessary to fullfil basic needs, likewater andarable(andhabitable)land,asthesearelikelytoarisewithon‐goingglobalwarming.

4SeetheSternReview(Stern2007),andICPP(2007);alsoDavidandvanZon(2012:sect2.,1)forfurtherdiscussionof the current science consensusof the literatureonglobalwarming, ecologicaldisruptions, including andhumanwelfaredamages–uponwhichthefollowingparagraphdraws.5 Seee.g.,Alley(2002),Burroughs(2005),Cronin(2009).

6Theterm“tippingpoint”hasbeenusedindiscussionsofglobalchangetodescribeavarietyofphenomena,includingthe appearance of positive feedbacks, reversible phase transitions, phase transitions with hysteresis effects, andbifurcationswherethetransitionissmoothbutthefuturepathofthesystemdependsonhavingbeenperturbedatacriticalpoint. Onabruptclimatechanges and“earth tippingpoints”seeAlley ,Marotzke, Nordhaus, Overpecketal.(2003); NationalResearchCouncil (2003). Stern (2007: pp. 11‐14) provides an overviewof positive feedbackprocessesinvolvingreductionofalbedothroughreducedice‐coverageofthearcticregions,thawingofpermafrostandinducedreleaseofmethane.Thebroaderterm“tippingelements”isintroducedbyLenton,Held,Kriegleretal.(2008)todescribesubsystemsoftheEarth’sgeophysicalsystemthatareatleastsub‐continentalinscaleandundercertainconditionscanbeswitchedintoaqualitativelydifferentstatebysmallperturbations.Thetippingpointisastructuralfeatureofsuchasub‐system,correspondingtothecriticalpointintheforcingprocessatwhichthefuturestateoftheglobalclimatesystemwouldbequalitativelyaltered–butnotnecessarilyinvolvingaself‐reinforced,irreversiblestatechange. The Atlantic Meridional overturning in the thermohaline circulation, the West Antarctic ice sheet, theGreenlandicesheet,andtheAmazonrainforestandtheElNino/SouthernOscillationarethesub‐systemsidentified

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mean global temperature to rise above a critical threshold of that kind would launch a self‐reinforcing and irreversible warming process that could manifest itself eventually incatastrophicclimate“flickering”–characterizedbyrecurringabruptclimatechanges,switchingbackandforthathighfrequencybetweenspatiallyuncorrelatedboutsofpronouncedwarmingandcooling.7 More than simply adding another groundonwhich the “precautionaryprinciple” callsurgently for mitigation of CO2 emissions, that existential threat undermines sanguinepresuppositions that “adaptations" by contemporary societies to a continuing gradual rise inglobal mean temperature would eventually bring humanity to a significantly warmer butnonethelessviableequilibriumenvironment;thatthatcurrentclimatepolicyshouldbeaimedtoeffect a correspondingly gradual, economic welfare‐maximizing approach to that distant butattainablegoal.Thealternativeprospectnowmoreclearlyenvisagedbymanyclimatescientists–namely, that the rising trend ofmean global temperature could take Earth’s climate systembeyonda“catastrophetippingpoint”withouttheconsequencesmanifestingthemselvesplainlyforsometimetocome‐‐raisesgravedoubtsabouttheformulaiceconomicadvicethatcontinuetobewidelyespoused.Climatepolicy, it is said, shoulddevise instruments toguide theworldeconomyalonganoptimalpath thatbalances thepresentvalueof socialwelfaresacrificedbyactionstorestrictCO2emissions,againstthepresentvalueofthefuturenetsocialwelfaregainsresulting from slowing the accumulation of atmospheric CO2 – just enough to allow moderncivilization sufficient time to adapt itself to environments that will be on the whole ratherwarmer,butinwhichadaptationwillbringbenefitstosetagainstwhathasbeenlost.8 Thereismuchtobesaid,instead,fortimelyactionsaimedtosharplyslowandeventuallyhaltglobalwarmingbyreducingtheflowofGHGemissions,aswellastoincreasethecapacityofhumansocietiestocarryoninthe(hopefullyviable)circumstancesthatthepeoplesoftheworldwillfacewhentheysucceedinstabilizingitsmanyclimates.“Defensive”adaptationtocurtailtheextentoftheeconomicandsocialdamageswroughtbyincreasinglyfrequentandseverestorms,coastal flooding and drought must, undoubtedly, be made part of the response to globalwarming ‐‐ if only because much of the warming that will drive them, and accompanyingalterationsoffutureregionalweatherpatternsare“inthepipeline”already,asaconsequenceofpastanthropogenicGHGemissions.9WiththecurrentlevelofatmosphericconcentrationofCO2

                                                                                                                                                                                          most frequently experts as harboring “large‐scale discontinuities” – i.e., dangerous “tipping point” potentialitiesunderanthropogenicglobalwarming.

7See,HallandBehl(2006)onthe“ClathrateGun”hypothesisthathasbeenadvancedinexplanationofabruptclimatechangeandtheonsetofthephenomenonof“climateflickering”attheendofthelasticeage,andfurtherdiscussionwithotherreferencesinDavidandvanZon(2012:sect.2.1).

8 See Hall and Behl (2006) on the persisting failure of the economic literature on integrated assessment models(IAMs) to address the implications of climate scientists’ conclusions and explanatory conjectures based on thepaleoclimateevidenceof “climate flickering”; esp.,pp.461‐462, foradetailedcritiqueof therepresentationof theclimate sub‐system in Nordhaus and Boyer’s (2000) updating of the original (Nordhaus 1994) DICE model. TheassumptionthatradiativeforcingduetotheaccumulationofatmosphericCO2woulddriveasmoothtransitiontoahigherequilibriumtemperatureoftheEarth’ssurfaceisretainedinthelatestupdateofDICE(seeNordhaus2010),aswell as in the annualized version of themodel that Cai, Judd and Lontzek (2012) create en route to SDICE, theirstochasticcontrolversionofDICE–whichinotherrespectsconstitutesasignificantadvanceintheIAMsliterature.9  Aswillbeseen(insect.3.2),thethirdofthemodelspresentedhereallowsforwarming–drivendamagestoglobalproductivecapacity,butdoesnotalsoconsider theoptionofundertaking“defensive”capital formationthatwouldreducethosedamages.Suchinvestmentshouldbeviewedasaformof“adaptation”andalthoughitsimportancehasbecomemorewidelyappreciated,itremainsthecasethatmostoftheavailableintegratedpolicyassessmentmodelsdonot explicitly treat that option. Anoteworthy exception is “AD‐DICE”, an IAM that allowsexplicit quantitativeexaminationofthetrade‐offsbetweenadaptationandmitigationofCO2emissionsbymeansofcarbontaxes.SeedeBruin,DellinkandTol(2009),andnotesinsects.3.2and4.2,below,forfurtherdiscussion.

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and thehigh rateof continuingemissions, it is becomingmoreandmoredoubtful that globalwarmingcanbekeptfromaddingagainofmorethan2oKelvintothemeansurfacetemperaturethat prevailed throughout the recorded pre‐industrial millennium. Such a large temperaturegain is considered to be a threshold thatoughtnotbe crossed, for fearof triggering runawayglobalwarmingandtheattendantsocietalchaosandwidespreadhumanlosses.10 Unfortunately, the mere acknowledgement of the existence of climate catastrophetipping points is not enough to ensure that CO2 and otherGHG emissionswill be sufficientlyreduced. The BRIC group of industrially developing countries has maintained astonishinglyrapid rates of economic growth during the past two decades, relying heavily on fossil fuelsources of energy. Thepeoplesof thesenations think it entirely appropriate that they shouldcontinue not only to lift their remaining masses from poverty, but advance toward levels ofeconomicwelfarecomparabletothoseintheWest. ItgoeswithoutsayingthatthenationsoftheWestalsofeelsimilarlyentitled,notonlytomaintainbuttoalsopursuestillhigherwelfarelevels.Asaconsequence,theglobalprovisionofenergywillbeunderpressuretoaccommodatethesewidelysharedaspirations.This,unfortunately,isalreadyreflectedinthefactthatmanyoftherecentlybuiltelectricitygeneratingplantsinChinaandIndia,andhundredsmoreproposedforconstructionthereandelsewhere,aretobecoal‐fired.11Werethatprospectnotdiscouragingenough,the2011tsunamiandFukushimaDaiichinucleardisasterhasledauthoritiesinJapan,and the governments of other countries to begin taking steps to reduce their economies’dependenceonnuclearpower;theclosingofasignificantnumberofnuclearpowerstationshasbeen announced in Germany, raising concerns that the electric generation capacity thusabandonedwillbereplacedwithnewcoal‐firedplants.12Inshort,therearestrongpoliticalandeconomicpressures thatcontinue topromoteadherence toahighlycarbon‐intensivegrowth–pathfortheglobaleconomy‐‐withmountingrisksoftriggeringrunawayclimatechange.13

                                                            10See,e.g.,Hansen,Sato,Ruedyetal.(2006:p.1):“ComparisonofmeasuredseasurfacetemperaturesintheWesternPacific with paleoclimate data suggests that this critical ocean region, and probably the planet as a whole, isapproximatelyaswarmnowasattheHolocenemaximumandwithin≈1°Cofthemaximumtemperatureofthepastmillion years. We conclude that globalwarming ofmore than ≈1°C, relative to 2000,will constitute ‘‘dangerous’’climatechangeasjudgedfromlikelyeffectsonsealevelandexterminationofspecies.” 

11AccordingtoaWorldResourcesInstitutereport(seeYangandCui(2012))arecentWRIsurveyfound1,199coal‐firedplantscurrentlyproposedglobally in59countries,over two‐thirdsof theminChinaandIndiawithaggregatecapacityrepresentingaboutthree‐quartersoftheprojectedglobaladditionstoinstalledmega‐wattage.ThenumberofprojectsplannedinIndia(455)exceedsthoseinChina(363),butthelatter’splantsareonaveragelarger,sothattheir total installedmega‐wattagewouldbeabout10per cent greater than India’s (510KMW).Whether thesealltheseplanswillmaterializeisanothermatter,however:inChinastricternewregulationsinresponsetoairpollutionproblems,andprospectivewatershortages(forcooling)maymilitateagainstit,andtheIndianpower‐plantboomofthepastdecadeendedwithmanyprojectshavingbeendiscontinued.12 Intermsoftheimplicationsforglobalwarming,coalisamongtheworstofavailablefossilfueloptions:theratioofgramsofCO2perjouleofenergyobtainedfromcoal is1.69timeshigherthanthatfromnaturalgas (cf.Wikipedia:“Energydensity”).Thatcomparison,however,substantiallyunderstatesthecomparative“dirtiness”ofcoal‐firedvis‐à‐visnaturalgaselectricpowergenerationplantsovertheirrespectivefull life‐cycles(fromground‐breakingtofuelsources,wastemanagementandreturntogreenfieldsite):threemeta‐analysesofmanystudiesconcurinplacingthecorrespondingratioofCO2gperKWheforcoal‐firedplants(withscrubbers)tothatfornaturalgas‐firedplantsinthenarrowrange1.96‐2.17(cf.Wikipedia:“Life‐cyclegreenhouse‐gasemissionsofenergysources.” Inadditiontobeingclimatically“dirty,”andemittingsootandotherhealth‐injuryingpollutants,thetroublewithbuildingnewcoal‐firedpowerplants is that the low international price of coal is likely to be kept downby the expansionof still cheapersubstituteslikenaturalgas,andincreasingcoalextractionrates(inducedbyexpectationsoffuturecarbontaxes).Thiswould tendkeep thesedurablepower station’smarginal costs fromrisingas theyage, so that they could continueyieldingquasi‐rentsandremainincommercialoperationformanydecades,unlessforciblyshutdownbyregulatoryactions.

13Obviously,thatthreatmaybemitigatedbyfundamentalchangesinlifestyles,atleastforthewell‐to‐dopartoftheglobalpopulation thatpresently aremajorusersof energy from fossil fuels.But it is hard to imagine that suchanimpulsewould gather political force spontaneously from a transcendent ideologicalmovement, or that it could be

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With humanity today seemingly headed almost unavoidably towards a worrisomelyproblematic future climate, it is all themore pertinent to thoroughly explore what economictheoryinformedbyclimatesciencecantellus,firstlyaboutthebroadrequirementsforsatisfyingthe desire to sustain development and the future growth of economic welfare by allocatingresources so as to avert drifting irreversibly into a catastrophically transformed globalenvironment; secondly what the role of both existing and new technologies and theirembodimentinirreversibleinvestmentmaybeineffectingatransitionfrom“business‐as‐usualtoaviablylowcarbonproductionregimethatwouldstabilizetheclimate;andthirdlywhethersucha“techfix”programwouldbetechnicallyandeconomicallyfeasibletoexecutewithinthefinitetimelimitsthatmayremaintodesignandimplementthenecessarystepsinthatprogrambefore arriving at a tipping point into irreversible “runaway” climate change (TPIRCC,hereinafter). Only when we have a clearer vision of what would be required to travel thetechnological path to a stabilized global climate thatwouldbe least costly in termsof humanwelfare (i.e., “first‐best” welfare‐optimal), will we then be in a better position than now torealistically consider thepolicymeasures thatmightbe able to realize such aprogram;or, tostatingthenatureoflatterchallengemoreprecisely,tore‐focusnegotiationsaboutnationalandinternational agreements on continuously monitor‐able actions to implement a socially andpolitically feasible approximation to the idealized, first‐best welfare optimum path towardsustainabledevelopmentinstabilizedglobalclimate.14 Unfortunately,inourview,muchtoomuchtimealreadyhaspassedduringwhichseriouspeople approaching this dauntingly formidable policy‐design challenge have tried to avoidtacklingithead‐on,andhaveoptedinsteadforadifferentstrategy‐‐basedontryingtodealwiththechallengeofglobalwarmingbydirectlyaddressingitsrooteconomiccauses.Thesemaybefoundinhumans’historicalpropensitytoburncarbon,andtocontinuedoingsowithoutregardforthecost intermsofenvironmentalharmsthatsuchactivitiesimposeuponothers,andwilleventuallyimposefarmoreheavilyonstillunborngenerations.Suchcosts,fallinguponothers‐‐whether through immediate localized pollution of the nano‐particle laden air inhaled by thepopulationinurbanareas,ortherisingconcentrationofatmosphericGHGandallofitspresentand future global consequences–‐are labeled as “externalities”. They are not reflected fully in“freemarketprices”ofcarbonfuels,andhencedonotregisteramongtheprivatecostsofthoseengagedincombustingthosematerials. Starting from this diagnosis of the source of the climate problem, a substantialconsensus among environmental and energy economists has found the indicated course ofclimate policy response to be elegantly simple: directly address the “externality” of GHGemissionsbyusingfiscaloracombinationofregulatoryandfiscalinstrumentstosettaxessoastogeneratemarketpricesofcarbon(orforlicensestoburnfossilfuels)thatwillfullyreflectthemarginalsocialcostsofusingthosematerialsassourcesofenergy.Then,itissaid,theworkingsofthefreemarketwilltakecareoftherestoftheproblem;raisingthecarbonpricewillcreateprivate incentives to reduce CO2 emissions in ways that will be least‐costly to the actors;“gettingthepricesright”nowandinthefuturethereforewillbesufficienttofixthe“externality”thathasallowedeconomicallyself‐interestedhumanstosettheirplanetonitswarmingcourse. There have been real advances in analyzing the likely effects of using fiscal andregulatoryinstrumentsinthemostwelfareefficientwaystofixtheCO2emissions“externality,”

                                                                                                                                                                                          effective without the facilitation and reinforcement of extensive technological changes in the global productionregime.

14SeeSchelling(2009)onthedistinctionbetweensettingoutcometargetsandagreeingonscheduledactions.

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indeed,advancesuponwhichthispaperisabletobuild.Yet,atthesametimetherehasbeenadiscernible and unwarranted reluctance among economists to examine the technologicalrequirementsofwhat“themarket”wouldhavetoaccomplishbyalteringprivate incentivestocurtail theemissionsofCO2. It isas if thoseresearchershadsaid,perhapsnotaloud: “I’maneconomist not an engineer, and since I have good cause to expect that markets will do anefficientjobofresourceallocationoncewehaveremovedthetroublesomeexternality,whydoIhave to be concerned about the details of the technological solutions that rational economicagentswillarriveatwhenguidedbytherightprices?” Theevidentproblemwiththislineofresponseisthat“gettingtheglobalpriceofcarbonright,” and keeping it right, are not so simple tasks for “institutional engineering” andinternational political economy as it is convenient to imagine while formulating and solvingthose integratedassessmentmodels (IAMs).15 Secondly, if economists’ attentionwere to turnaway from the problem of the CO2 emissions externality for a moment or two, they wouldremember that there are other, well‐known externalities that are likely to thwart thespontaneous emergence of socially efficient decentralized responses to rightly‐priced carbon. Two such externalities are so familiar that they may be mentioned without furtherelaboration:existingtechnologicalknowledgemustbeimplemented,andimplementationunderactual field conditions may require investment in researching incremental adaptations andinteractions between providers of equipment embodying that knowledge and its users. Butthere are knowledge benefits from learning‐by‐doing, and investments in research that aredifficult ifnot impossible forprivateagents incompetitivemarkets toappropriatecompletely,giving rise to well‐known information transaction externalities and “spill‐overs” that lead tosocially inefficient resource allocation.16 Likewise, investment decisions relating to theimplementationofexistingtechnologies(andofrecentinnovations,afortiori)maybedistortedbytheexistenceofotherknowledgespill‐oversfromlearningby‐doing,andlearning‐by‐using.17

                                                            

15 Beyond the issues of negotiating and enforcing coordinated international commitments to a cap‐and‐trademechanismsthatalreadyhavebeenexposedbytheexperiencewiththeUNKyotoProtocolsandtheattemptstogobeyond them at the 2009 Copenhagen Conference, there is the another specific problem the dependence of suchmechanismsonmarketsfor“licenses”orpermits(eitherpermitstoemitCO2andotherGHGsinthecaseofso‐calleddownstreamcap‐andtrade),orlicensesallowingfirstvendorstotransactincombustibleformsofcarbon(inthecaseofupstreamcap‐and‐trade).Transferablepermitsandlicensesconstituteafinancialasset.Theyaredesirablebecauseitisenvisagedthattheywouldbetradedinmarketsinwhichpriceadjustmentswouldbeendogenousandautomatic(not requiring repeated government actions, assuming the capswere set correctly at the outset). Financial assets,however,willpermitsecuritization,andbondswillengendertheformationofmarketsforderivatives,andsoon.Itisnotatrivialproblemtoarrangeforthemonitoringandregulatingoftheissueofthesenewfinancialinstruments,andto integrate globalmarkets for the varied types of licenses, so that the current and expected future prices of theunderlyingassetswillbeabletobecomewildlywrong.Indeed,thedifficultiesofachievingsuchadesirableresultandthecostsof failuretodoso(again)havebeendemonstratedinthedestructivefinancialcrisisof2008‐2010anditseconomically and socially painful aftermath. Aside from these practical institutional details, and the politicalproblems of arranging for national legislative bodies constituted of political representatives with short careerhorizons to commit themselves and their successors to distant future schedules of taxes on fossil fuels, there areseriousquestionsabout thegeneralequilibriumeffectsofacommitment toraise the futurerelativepriceof thoseenergysources.These includepossibleadverseeffectsonprivate incentivesto invest in improvingtheefficiencyofcarbon‐basedtechnologies,andperverse“anti‐conservation”responsesfromownersoffossilfueldepositsthatcouldvitiatethedirecteffectofthecarbontaxormightevenforcedowntheneartermaftertaxpriceofusingcarbonenergysources.ForreferencesandmoredetaileddiscussionseeDavid2009,DavidandvanZon(2012:sect.1),andvanderPloegandWithagen(2010),specificallyonpotentialanti‐conservation(“GreenParadox”)effectsofcarbon‐pricing.

16 For recognition of the existence of “the second externality” in the context of environmental policy, see thediscussionoftheso‐called“appropriabilityproblem”inJaffe,NewellandStavins(2003:section3,esp.pp.471‐473).

17Thishaslongbeenrecognizedintheliteratureontheeconomicsofscienceandtechnologypolicy,andparticularlyincontributionsthatreflect“evolutionary”economicanalysisandmodeling(see, fordiscussionandreferences,e.g.,David,1992;Aghion,DavidandForay,2008).Thecorollaryisthatwithmorethanoneexternalitytherewillbeaneedformorethanonepublicpolicymechanismtocorrect likelymisallocationofresources.Therelevanceofthis inthe

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1.2TheResearchapproachandmethods:AnoverviewThereisamorefundamentalreasonforeconomiststodevotemoreseriousattentionto

examiningtechnologypolicyoptions,andinthispaperitproceedssimplyfromthefactthatinonewayorothertechnologicalimplementationsofpolicymeasuresdesignedtocurtailwarmingwill be required. Investment (either in generating future knowledge or in tangible productionfacilitiesembodyingmatureandnoveltechnologies)isthe‘conditiosinequanon’forasuccessfultransitiontowardsasustainable future.Thetransitiontowardssustainabilitywill thereforebedeterminedby finding therightbalancebetweentwoimportantaspectsof investment:on theonehandwehavetofacethefactthattheirreversibilityofinvestmentimpliesacertaindegreeofinertiatochange,whileontheotherhandinvestmentisliterallythecarrieroftechnologicalprogressandso‘enables’(productivity‐)change. This“double‐role”ofinvestmentunderlinestheimportanceofthetimingofinvestmentdecisions:it isunwisetoinvesttooearlybecauseonerunstheriskofmissingoutonpotentialproductivity improvementsstilltocome,andneithershouldoneinvesttoo latebecauseoftherisingopportunitycostofcontinuingtouseoldtechnologiesinsteadofnew,superior,ones.Thissetting naturally gives rise to such questions as how long to continue using and investing inpresentcarbon‐basedtechnologies,howmuchandhowlongtospendeffortsonimproving‘new’carbon‐free technology, andwhen to stop improving such carbon‐free technologies and startimplementingandusingthem,thisallinthefaceofhavingtostopcumulativeemissionsjustintimeandjustbelowtheTPIRCC.Inotherwords,thedesignofemission‐mitigatingpoliciescallsfor“thinking‐in‐time”aboutcomplicatedquestionsaboutwhichinvestmentoptionstoconsider,howmuchinvestmenthastobeundertaken,andinwhatorderitwillbebesttodoit.Itcallsforbeginningastagebeforeassessingtheeffectsofalternativepolicychoices, thestagethatdealswith questions that engineers and planners of systemswith improved performance propertyrefertoas“requirementsanalysis”. It also will be instantly recognized as “planning analysis” posited on the assumedexistenceofasociallybenevolentandomniscientplanningagent,andthereforesettingasideforthepurposesoftheanalysisallconsiderationsoftheproblemsofhowtoimplementtheactionsrequiredtoachievedthesystem’sdesiredperformanceifthesysteminquestionhappenstobeone in which resource allocation is decentralized and reflects the distributed decisions andbehaviorsofmanyhumanagents.Optimalplanningmodels,whetherofthedeterministicorthestochastic variety, are familiar tools inmoderneconomicsandour researchsimply (ornot sosimply) extends theiruse in the literatureofmacroeconomicgrowthmodelsbyapplying it toexamineaglobaleconomywhoseplannerhas toanswer to resourceallocationquestions thatare complicated by having to take into account the dynamic interconnections between hiseconomyandthegeophysicalsysteminwhichithappenstobeinextricablyembedded. To find answers to these questions, we formulate an optimum control model thatborrowsheavilyfromtheAK‐modelfromtheendogenousgrowthliterature(cf.Rebelo(1991)inparticular),butthatalsoexpandsuponthisAK‐settinginanumberofways.First,weallowfordifferent technologies that can be used either next to each other or sequentially. Secondly, atechnology ischaracterizednotonlyby itscapitalproductivity,butalsobyCO2‐emissionsperunitrealoutput.Intheinitialbusiness‐as‐usualphase,“carbon‐free”technologieswillbetakento allowproductionwith zeronet emissionsofCO2, althoughdoing so at higherunit costs of

                                                                                                                                                                                          context of climate policy seems only quite recently to have impressed itself upon leading contributors to thetheoreticalliteratureonendogenouseconomicgrowth(seeAcemoguluetal.,2012).

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capital(i.e.,beingcharacterizedbyaloweraverageandmarginalproductivityofcapital).18Theterm“netemissions”herereferstotheflowofproduction‐generatedCO2thatisinexcessofthenaturalabatementcapacitiesof theEarth’soceansand forests,and, rather thanbeing trappedandstoredthere,eventuallyaddstotheconcentrationofatmosphericGHG. Third, we allow for the deactivation of existing technologies, i.e. the ‘scrapping’ ofexisting technologies, as in the vintage literature. We show how the time of deactivation ofexistingcapacitydependson technologicalparametersbutalsoonemissioncharacteristics, incombinationwith the shadow price of emissions. The latter suggests that the position of theclimatecatastrophe“tippingpoint”directlyinfluencestheoptimaltimingofsuchmovestowardscompletingtheswitchawayfromcarbon‐basedproduction‐‐throughitsimpactontheshadowprice of emissions. An exogenous shock, shrinking the space left for further accumulation ofatmosphericCO2 (suchaswouldcome in the formof compelling evidence that theonsetof arunaway process of warming could be triggered by a rise in the CO2 –equivalent GHGconcentration level beyond 430‐450 ppmv, now that we are already close to the 400 ppmv)wouldsendtheshadowpriceofemissionssuddenlyupwards,forcingdrasticactionstocurtailtheoutput of consumptiongoods toward subsistence‐satisfying levels so that asmuchof theoperating capacity that remained could be used to rapidly build up the stock of carbon‐freecapital. Fourth, we allow for endogenous R&D based technical change. This requires aspecification of the R&D function that is different from the ones found in Romer (1990), orAghion andHowitt (1991), for example, because technical change in ourmodel setting is notmeant to compensate for the loss in capital productivity resulting from capital accumulationunderneo‐classicalconditions,asinLucas(1988)andRomer(1990),forinstance.Rather,inanAK‐setting,capitalproductivityisconstantbyassumption,andsotechnicalchangespecifiedintheusualwaywouldproduceacontinuouslyacceleratinggrowthrateratherthanjustahigher,but still constant, rate of growth.Wewill comeback to this inmore detail later on. Fifth,weexplicitlyfocusonthetimingoftheswitchesbetweeninvestmentintheonetechnologyandintheother,andonthetimingofthedeactivationofoldtechnologies,sincethedeactivationandactivationoftechnologiesthatdifferw.r.t.theiremission‐coefficientshaveadirectimpactonthemacro‐emissionrateandthereforeonthetimeleftuntiltheTPIRCCwillbehit. As the foregoing overviewof our approach implies, a central issue in themodel to bepresentedhereisthefactthatthetransitiontowardsacarbon‐freeproductionsystementailsaswitch in the deployment of production technologies that require the buildup of carbon‐freecapacityandthesimultaneousrundownofcarbon‐basedcapacity,simplybecausetheonetypeof capacity cannot be changed into the other type of capacity without new tangible capitalformation. Thereforewe have opted for anAK‐model setting (cf. Rebelo, 1991) inwhichweallowfortwotechnologiesthatbothcanproduceoutput,oneofwhichresultsinCO2emissionswhereastheotherdoesnot. The present paper summarizes part of the work conducted by the authors on theconstruction (and further extension) of a multi‐phase transition models incorporating theconceptsofchangesintechnologiesthatareembodiedintangibleanddurablecapitalgoods,andof the irreversibility of investment decisions. From these basic premises it follows that the“smooth”’ transition toward a carbon‐free future will need to be prepared by means of the

                                                            18Thelatterisanecessaryassumptiontomakethemodelconsistentwiththeobservationthatthepresentstateoftheeconomyischaracterizedbytheintensiveuseofcarbon‐basedenergyratherthancarbon‐freeenergy.Ifthecapitalproductivity of the latter technology would exceed that of the former, then the carbon‐free technology would besuperiortothecarbon‐basedtechnologyonallaccounts,whichisnotreallythecase.

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accumulation and subsequent run downof carbon‐based production capacity, simply becausecapital,whethercarbon‐basedorcarbon‐free, isaproducedmeansofproduction.Thefocusoftheresultingmodelsthereforeisontheselectivebuild‐upanddeactivationofdifferenttypesofcapitalstocks,thetimethistakes,andtheimplicationsofthisprocessforthedevelopmentovertimeofwelfarespecifiedintermsoftheflowofconsumption. The premise that embodiment in physical capital goods is necessary to implementchangesinthetechnologiesthatwouldlowertheglobalproductionregime’scarbon‐intensity,aswell as to increase the capacity of the carbon‐based capital stock, therefore is a distinctivefeatureofthemodellingapproachpursuedhere. Althoughenergytechnologistsandengineershave in effect long recognized that the “embodiment” of techniques in fixed reproduciblestructures and equipment cannot be ignored when considering the impact of technicalinnovations inenergysupplysystems,19ourexplicitrecognitionofthisintheverystructureofthemodel presented here (and its account of the dynamics of the optimal climate‐stabilizingtransitionpath)representsamajordeparturefromthewaysinwhichtheeffectsofendogenoustechnological change have been treated in previous economic contributions to the integratedassessmentofclimatepolicymeasures.20 Ourmulti‐phasemodelingapproachalsoofferssomenoveladditionstotheexistingbutstillrelativelysmallliteratureconcernedwiththeoptimaltimingofswitchesamongalternativeproduction technologies. Several points of comparison with the present analysis are worthremarkingupon inregard to threenotableprecedingcontributions in thisvein, byTahvonenandSalo(2001),Valente(2009)andSchumacher(2011),21TahvonenandSalo(2001)focusonthe timing of the switch between alternative resource extraction technologies that differed intheir variable costs, whereas Valente (2009) examines the switch between two macro‐production technologies in a setting without irreversibility of investments in productioncapacitynorendogenoustechnicalchangesresultingfrominvestmentsinR&D.ThelocationofasingleoptimalswitchingmomentinValente’s(2009)analysisisdeterminedastandarddynamicoptimization setting involving a CIES utility function. By contrast, our supposition thatformation of physical production capacity embodying each technology must have occurred

                                                            19Thedevelopmentanduseofenergytechnologiesisviewedasanintegratedsystemcomprisingresearchdiscoveriesand inventions, the creation of commercial products and processes, their initial deployment and adoption intocommercialoperations,andsubsequentwiderdiffusion–theviewembracedrecentlybytheReportofthePresident’sCouncil of Advisors on Science and Technology (PCAST, 2012). Accordingly, Ernest Moniz (2012: p. 82), formerUndersecretaryoftheU.S.DepartmentofEnergyandaPCASTmember,emphasizestheimportanceoftangiblefixedcapital formation in considering policies designed to stimulate “energy technology innovation”: “Adoption anddiffusionarethestagesatwhichmaterialityof[novel]productsandprocessesarerealized(ornot).Innovation,asIuseithere,referstotheend‐to‐endsystemincludingmarketdiffusion,notfront‐endR&Dalone.”

20 Onemay compare the implicit assumption – common to each of the following salient research contribution onendogenous technological changeandclimatepolicyanalysis ‐‐ that innovationsresulting fromR&Dor learningbydoing are of the disembodied kind, and therefore their effects are not intermediated by timing and volume ofinvestments intangiblecapital formation:GoulderandSchneider(1999),Nordhaus(2002),Bounanno,CarraroandGaleotti (2003), Edenhofer, Carrero and Galeotti (2004), Popp (2004), Lessman, Kemfert et al. (2006), Sue Wing(2006).Ratherstrikingly,thediscussionoftheseandotheritemsintheliteraturesurveybyGillingham,NewellandPizer(2008),remainssilentonthedistinctionbetweenembodiedanddisembodiedtechnologicalchanges,andomitsmentionofadoptionasadeterminantofgeneralorenergysector‐specificchangeinproductivityorGHG‐emissionsintensity. While commenting on several aspectsof an earlier literature review byAzar andDowlatabadi (1999),Gillinghametal.(2008)givesnonoticeofitsusefuldiscussionoftheevidencedocumentingthecomparativelyslowpaceoftechnologydiffusionintheenergysector,theresponsivenessofprivateadoptiondecisionstheretochangesinperformance standards and subsidies, and the role of infrastructure externalities and uncertainty in setting highhurdleratesofreturnrequiredforlumpyinvestments.

21Arecentadditiontothetheoretical literaturebyBoucekkineetal.(2012)providesthemathematicalfoundationsfor formulating and solving an optimal control resource extraction problem with multiple irreversible ecologicalregimes.

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before that capacity actually can be utilized – a fundamental aspect of the “embodiment”condition – carries the implication that at least two optimal switching moments must beconsidered,eventhoughonlytwotechnologiesareinvolvedintheswitch.22Schumacher(2011)focusseson the timingof a switch towardsproductionbasedupona renewable resource thatwould be induced by the increasing probability of climate disasters under a non‐renewableproduction regime. His analysis, however, posits aproduction structure inwhich renewablesand non‐renewables form a ‘complex’ of perfectly substitutable inputs that, together with‘generic’ capital, produce the economy’s aggregate output. Schumacher thus ignores theembodimentoftechnologiesinspecifictypesofcapitalgoods,andsuppressesconsiderationoftheneedforasufficientamountoftimeduringwhichtobuildupthecarbon‐freecapacitythatisrequiredtosatisfyfutureconsumptionandinvestmentneeds. 1.3OrganizationofthePaper The organization of the presentation that follows is straight‐forward. Section 2introduces the basic form and features of the multi‐phase optimum control model (in sub‐section2.1)anddescribesthetwowaysinwhichit isextendedinthispaper.TheBasicModel(BAM for short) captures the essential features of the transition problem of an economy thatmust complete the switch from initial dependence on production facilities that embody acarbon‐using technology to producing exclusively with capital that embodies a “carbon‐free”technology, i.e., one that enables it to utilize only non‐carbon sources of energy. Sub‐sections2.2‐2.3setouttheformalstructureandtheanalysisoftheoptimaldynamicsofthistransition‐‐whichmust be completedbefore theCO2 emitted in theprocesshaspushed the atmosphericconcentrationofGHGtothe(TPIRCC)levelthatwouldtriggeracatastrophicallyunstableclimateregime. The inter‐temporal optimization described in sub‐section 2.4 involves solving the“stackedHamiltonians” (in 2.5) to obtain the durations of the three phasesmentioned above(seefootnote22),andthemagnitudeofthesequencedtangibleinvestmentsrequiredtobuildupeach of the kinds production facilities, utilize them jointly and eventually shut down un‐depreciated fossil‐fueled capacity before entering the final phase of sustainable “green”economicgrowthinastabilizedclimatesystem. Section3setsouttheformalstructureandanalysisofthetwomodelsthatextendBAM.The introduction of endogenous R&D‐driven capital augmenting changes in the carbon‐freetechnologyprior to its embodiment anddeployment is shown (in 3.1) to result in amodifiedthree‐phase model, referred to as “BAM+R&D”. A third model, labeled “BAM+R&D+UCL” isobtained(in3.2)byaddingaclimatechangefeedbackeffectintheformofheightenedexpectedannual rates of damage to the extant capital stock ‐‐ driven by the rising atmosphericconcentrationofCO2,theconsequentwarmingoftheearth’ssurfaceandincreasingmoistureinthe atmosphere due to the faster evaporation from the oceans’ surface, which brings morefrequent severe storms, seaboard and riverine flooding and droughts in interior regions. Theonset of a higher expected (proportional) rate of “unscheduled capital losses” (UCL) as directand indirect consequences of damages to reproducible capital, and ecosystem services, ismodeledheresimplyasendogenousjumpinthecapitalstock’srateoftechnicaldecay,triggeredwhencumulativeCO2emissionsreacha(joint)weather‐systemsandecosystemstippingpoint.The latter is positioned at an atmospheric concentration level below that of the climate

                                                            22Inourcasetoo,theremustbeatleastthreephases:apurecarbonphase,amixedcarbonandcarbon‐freephaseandapurecarbon‐freephase.Thesimplereasonisthatcapitalisaproducedmeansofproduction,andtheveryfirstunitsofcarbon‐freecapacitymustbeproducedusingcarbon‐basedcapacityifwestartoutwithapurecarbonphasefirst.

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catastrophe tipping point (TPIRCC) which is specified uniformly for BAM and successiveextensionsofthatmodel. Section 4 describes the calibration of the threemodels (in 4.1), and follows thatwiththree sub‐sections (4.2 – 4.4) that in turn present the optimal solutions obtained for thetransition path of BAM and the two extensions of that model, along with the correspondingresults for each version of some parameter sensitivity experiments. The paper concludes insection5withasummaryofthesalientfindingandcommentsonthebroaderinsightsofferedbythis approach to climate policy design, and its limitations that point to priorities for furtherwork.2 TheBasicModelandItsExtendedVersions

2.1 IntroductionWeusethesimplestpossibleendogenousgrowthsetting inwhichtherearetwobroad

classes of (linear) technologies. One of them is an established technology (called the A‐technology)witharelativelyhighproductivityofcapitalthatusescarbon‐basedenergyandthatproducesCO2emissionsintheprocess.Asstated,theseemissionsaddtothestockofGHG’sandso affect the probability of theworld getting into a situation of runaway globalwarming andcatastrophe in the end. The alternative technology (further called theB‐technology) does notgenerateCO2emissions, buthas a relatively low capital productivity thatneeds tobe furtherdeveloped (throughR&D) and scaledup (through investment inphysical capital) to a level inwhichitcancontributesignificantlytotheconsumptionneedsofthepopulationatlarge.

Themostelementarysettingforgrowthmodelsthatallowformalcharacterizationoftheforegoingtechnologicaloptionsisprovidedbymodelsofthe‘AK+BK’form,whereAKrepresentstheoutputpotentialofcapitalembodyingthecarbon‐basedtechnologies,andBKrepresentsthecorresponding capacity of the capital stock embodying technologies permitting reliance onrenewable, non‐carbon energy sources. In this framework, the variables A and B denote therespectiveaverageproductivitiesofthetwokindsofcapitalgoods.Weproceedbyformulatingasequential Hamiltonian system that describes three distinct phases in the transition fromcarbon‐based to carbon‐freeproduction. In the first phase, called the business‐as‐usualphase(BAU for short), only the already existing AK production capacity is active, and in the BasicModel it is assumed that during the BAU phase the technical innovations needed to createproduction capacity of the BK kind are available but have not been implemented by capitalformationofthekindthatwouldberequired,becausethelatterwouldbelessproductivethancapital that embodies the carbon‐using technology. In other words, the known alternativetechnologies would be relatively costly in terms of instantaneous consumption possibilitiesforegone,andthatdisadvantageisnotperceivedtobeoffsetbybeingcarbon‐free.AscumulativeCO2emissionsgrowwith thecontinuinguseof theA technology, the latterbecomesever less(socially) advantageous vis‐à‐vis the option to build and substitute capital embodying the Btechnology.

Activeutilizationofa technology implies two things.First, thebasic featuresof suchatechnologymustbeknown,whilesecondlythesefeaturesareembodiedinnewcapitalgoods.Atechnology can be de‐activated by not using the capital goods embodying that technologyanymore.Becausecapitalgoodsaretechnologyspecific,thisimpliesthatanewtechnologycantakeoverfromanoldoneonlybyactivelyinvestinginthecapitalgoodsthatimplementthenewtechnologyandbyswitchingproductionfromtheoldtothenewtechnology.

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In such a setting characterized by linear production technologies and linear costfunctions,itcanbeshownthatitisnotoptimaltoinvestinbothtechnologiesatthesametime,ifthese technologies differ with respect to their net productivity (i.e. net of depreciation). Thereasonisthataunitofinvestmentforbothtechnologieswouldrepresentthesamemarginalcostintermsofconsumptionforgone,andsothetechnologywiththehighestnetmarginalproduct,wouldgeneratethehighestnetmarginalwelfaregain.Hence,ifinvestmentintheAtechnologyistakingplace,theninvestmentintheB‐technologywillbezeroandtheotherwayaround.WewillalsoallowforthepossibilitythatthereisaphaseinwhichnoinvestmentinAtakesplace,eventhoughtheexistingcapitalstockisusedtoproduceoutputwhileinvestmentinandproductionusingtheB‐technologyishappeningatthesametime.Thissecondphasewillthereforebecalledthejointproductionphase(JPRforshort).Inthefinalphase,onlyinvestmentinandproductionwiththeB‐technologyoccurs,whiletheA‐technologycapitalstockhasbeendeactivatedatthebeginningofthatphase.Thisisthecarbon‐freephase(CFRforshort).

The effects of the production and investment activities during the separate phasesdistinguished in themodelcanbesummarizedas inFigure1.TBAU,TJPRandTCFRmarkthemomentsintimeatwhichtheBAU,JPRandCFRphasesbegin.IntheBAUphase(whichstartsatt=TU=0), the cumulative emissions (labeled E) are increasing exponentially as the stock ofcarbon‐basedcapitalisgrowing.IntheJPRphaseinvestmentintechnologyAstops,andoutputusingtechnologyA(i.e.YA)isatitsmaximumlevelbutstartstodecreaseovertime,becauseoftechnical decay. The stock of technology B capital is built up from scratch starting with thearrival of the JPRphase at t=TJ. Productionusing technologyB (i.e. YB) is at full capacity andexponentiallyincreasingduringtheJPRphase.Cumulativeemissionsarestillincreasing,butatadecreasing rate as the stock of carbon‐based capital is run down.During the JPR phase, totaloutput is still growing, but at a slower rate than the growth of YB. Phase CFR starts whencumulative emissionsE hit (at t=TF) the cumulative emissions threshold at , just below theclimate catastrophe tipping point. During the final phase, only investments capital goodsembodyingtechnologyBarepossibleandcarbon‐basedproductionmuststopcompletelyeventhoughthereisun‐depreciatedcapacityofthelattertype.Thatshut‐downcausesadropinthelevel of output, which is shown in Figure 1 to jump discontinuously from point I to point IIbeforeresumingitsgrowthfromthelatterlevelduringtheCFRphase.

Figure1.TransitionPhases

TJ TFTU=0 

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Clearly,thevariousphasesinthemodelareseentobequalitativelydifferent.Inthefirst(BAU) phase, the high productivity carbon‐based capital is use to rapidly buildmore of thatproductioncapacity,inordertoproducenewcapitalembodyingthealternativetechnology.But,unfortunately,thatquicklyraisesthestockofcumulativeCO2emissionstowardsitsadmissibletheupperboundary.Beforecumulativeemissionsreach ,however,phaseJPRmusthaveseencarbon‐freeproductivecapacitybroughttoalevelthatwillallowashut‐downofcarbon‐basedproductionthatdoesn’tforcetoopunishingadropinconsumption.Thelatterisimpliedbyourassumption that the social planning agent has internalized consumers’ relative aversion tonegative consumption shocks (i.e., their representative “felicity function” has a positivecoefficientofrelativeriskaversion,i.e., >0),andthereforetheoptimizationofsocialwelfareseeks tosmoothchanges in the levelofconsumptionover timeasmuchaspossible,giventhemacroeconomicandclimatesystemconstraints.FromTFonwardstheworldeconomyis“green”(having entered theCRFphase), andwill have to grow at a relatively slowerpace due to thehigherunitcostofcapitalembodyingthecarbon‐freetechnology..

InadditiontotheBasicModel,thefollowsectionformulatestwomodelsthatextenditintwo directions, sequentially incorporating, first, another technology policy option and, next,introducing an adverse feedback effect from the climate system. In the second version of thetransitionmodel thatwill be analyzed, the productivity of capital embodying the carbon‐freetechnology can be raised through (endogenously determined) R&D expenditures beforeundertaking the capital formation that is necessary to deploy it in production. The thirdtransitionmodelcombinestheendogenousR&DfeaturesofthesecondversionandaconstraintthatwillbeimposedonthegrowthofproductivecapacitywhencumulativeCO2emissionspassathresholdlevelthatcausestheglobalenvironmenttotipintoaphasecharacterizedbyhigherexpected rates of weather‐related physical damages that result in “unscheduled losses” ofservices from the global stock of capital. The latter regime‐shift is modeled simply as anendogenouslytimedone‐timejumpinthetotalannualrateofscheduledphysicaldepreciationplus“unscheduledcapital losses”(UCL).Beinganticipated,this impendingstatechangewillbereflected in the current shadow price of CO2 emissions, and so feeds back to affect the(optimized_allocationoftangibleandintangibleinvestmentspriortothesystems’arrivalatthehigh‐damagestippingpoint. 2.2PhasestructureoftheBasicModel:

The endogenous growth framework of BAM borrows heavily from the AK‐model byRebelo(1991).However,contrarytotheoriginalAK‐setting,wedistinguishbetweentwotypesofcapital:carbon‐based,orblack,capitalfurtherdenotedbyKAandcarbon‐free,orgreen,capitalfurther denoted byKB. The capital stocks in thismodel are subjected to exponential decay atrates ,for ∈ , .BecauseofthelinearityoftheproductionfunctionsinanAK‐setting,andsince one unit of capital takes one unit of consumption foregone for the two technologiesdistinguished, it follows that therewillalwaysby investment in justone typeof capitalat thetime.Hence,grossinvestmentinaparticulartechnologyiseitherequaltozero,oritisequaltototal savings.Welfare in this setup comes fromconsumptiononly, andweuse theCIES inter‐temporalwelfarefunctiontodescribethetotalflowofwelfareovertime.TheactivitiesduringthedifferentphasesaresummarizedinTable1.

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Activities BAUPhase JPRPhase CFRPhaseInvestment 0 0 0Production . .

. 0.

CapitalAccumulation

. . .

.

CO2Emissions . . 0

Table1.BAMactivities

Inthistable,Ixreferstotheamountofinvestmentincapitaloftypex,where ∈ , refers to ‘Carbon‐based’capitalandcarbon‐freecapital, respectively.Similarly,Yx refers totheflow of output usingKx, whereKx is the stock of capital of type x.We also assume thatYx isproportionaltoKxwithaconstantproductivityofcapitalasafactorofproportion.Typically,weuseAandBtodenotethecapitalproductivitiesoftechnologiesAandB,implyingthatYA=A.KA,etcetera.Finally,theinstantaneousflowofCO2emissionsisproportionaltothecapitalstockinusewith a constant factor of proportion for ∈ , . Obviously, 0. Note that whenproductiononsometypeofcarbon‐basedcapitalceases, thisvery fact initiatesanotherphase,sincethisintroducesadifferencebetweenthecompositionofactivitiesbetweentheJPRandCFRphases.SoYA=0implicitlydefinesthearrivaltimeoftheCFRphase,andthescrappingofcarbon‐basedcapitalatt=TF.

2.3BAM:Theinter‐temporaloptimizationsetting

ItshouldbenotedthatthefactthatthefinalphaseofBAMisapureAK‐setting,allowsusto obtain the optimum consumption path for the CFR phase directly, given an initial valuefor .23ThewelfaregeneratedduringtheCFRphasedependsthereforeontheterminalvalueofKBattheendoftheJPRphase.Itfollowsthatthedistributionofastatevariableovertheentirepathisoptimalwhenthemarginalcostsofhavingtodeliveranextraunitofthestatevariableinitsroleasaterminalvalueatsometimet*isexactlymatchedbythemarginalbenefitsthatthisextraunitofthestatevariablegeneratesastheinitialvaluefortheoptimumcontinuationfromt*.Sincethesemarginalbenefitsandmarginalcostsarecapturedbytheco‐statevariables(seeLeonardandVanLong(1992:Ch.4),the latterneedtobecontinuousalonganoptimumpath:statesandco‐statesdon’tjump.24

Anoptimumpathcanbe thoughtof asa combinationofanoptimum first stepandanoptimumcontinuation(asindynamicprogrammingproblems),whichallowsustointerpretourmultiphasetransitionmodelasafinitehorizonoptimumcontrolproblemwitha freeendpointandascrapvaluefunction,asdescribedinLeonardandVanLong(1992:Ch.7),hereafterL&VL:ch.7). This is the situation that is of direct relevance in our case, since we do not know onbeforehand when the next phase will start. However, on an optimum path, postponing orextendingaparticularphasebyaninfinitesimalamountoftimeshouldn’tchangethevaluationoftheentirepath.L&VLshowthatthederivativeofthevaluefunction(inourcasethepresentvalueof totalwelfare)withrespect to the terminaldate(ofaphase)matches thevalueof theHamiltonian at that date. This makes sense, as the Hamiltonian at some moment in timemeasuresthecontributiontothevaluefunctionoftheoptimaluseofallresourcesavailableat

                                                            23See,BarroandSala‐i‐Martin(2004),chapter4inparticular.

24Inthecaseofpurestate‐constraints,however,co‐statescanjump(cf.LeonardandVanLong(1992:Ch.10).

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thatmoment intime.But inasequenceofphases, lengtheningtheonephasebyaunitoftimeimpliesshorteningthenextphasebythesameunitoftime.SoweshouldkeeponpostponingthearrivalofthenextphaseaslongastheHamiltonianoftheearlierphaseexceedsthatofthelaterphase.Theoptimumswitchingmomentbetweenanytwophasesisthereforeimplicitlydefinedby the requirement that the Hamiltonians of two adjoining phases must be the same whenevaluatedatthemomentofthephase‐change.Again,thismakesperfecteconomicsense,asthevalueoftheHamiltonianattheendofthecurrentphasecanbeseenasthebenefitsofexpandingthecurrentphasebyamarginalunitoftime,whiletheHamiltonianatthebeginningofthenextphase can be seen as the opportunity cost of that expansion. In practice, the equality of theHamiltoniansevaluatedundertheconditionsrelevantineitherofthephasesjustbeforeandjustafteraphasechange,will result ina condition thatneeds tobemetbyasetof statesandco‐statesevaluatedatthemomentofthephase‐change.

Thedifferences in thenatureof theactivitiesat themomentofaphasechangecanbeusedtoimplicitlydescribetheconditionsthatshouldbemetatthemomentofaphasechange.Forexample,at t=TF itmustbe thecase thatcarbon‐basedcapital isdeactivated.For t>=TF itmust therefore be the case that the shadow price of carbon‐based capital is zero, since thecumulativeemissionthresholdhasbeenreachedandcarbon‐basedcapitalcanthereforenotbeusedanymoreandhasbecomeworthlessfromthenon. 2.4BAM:Formaldescriptionoftheoptimal3‐phasetransitionpath

Theoverallwelfare function consists of a summation of integralwelfare derived fromtheflowofconsumptionduringthethreephasesdistinguishedinBAM:

∙ ∙ ∙ ∙ ∙ ∙

. (1)

Inequation(1)W0measuresthepresentvalueoftotalwelfareattimet=0,atwhich,by

assumption,phaseUbegins.Inequation(1), istherateofdiscount,while1⁄ isthe(constant)intertemporal elasticity of substitution. Ct is consumption at time t, while TJ and TF are themomentsintimeatwhichtheJPRandCFRphasesbegin.Giventheexpositiononinter‐temporaloptimizationabove,thetimepathsthatwouldmaximize(1)canbeobtainedbysolvingthetimepathsfortheHamiltonianproblemsthatcanbedefinedfortheindividualphases,whilelinkingthosetimepathstogetherbymeansoftherequirementsofoptimumphaselengths(implyingtheequalityoftheHamiltoniansforphasesUandJatt=TJandfortheJPRandCFRphasesatt=TF).Effectivelythiscomesdowntomaximizing(1)w.r.t.theflowsofconsumptionduringeachphaseand w.r.t. the phase‐lengths themselves, constrained by the technologies that are relevant ineach phase, by the stocks inherited from previous phases, and by the thresholds that arerelevant during the various phases. We will now solve the Hamiltonian problems for eachindividualphase.

TheBAUphaseUsing thesuperscriptsU(forBAU), J (for JPR)andF(forCFR) todenote thephase to

whichaparticularvariablepertains,whiledroppingthetimesubscriptforeaseofnotation,thepresentvalueHamiltonianHUisgivenby:

∙ ∙

∙ ∙ ∙ ∙ . (2)

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Inequation(2),Cistheonlycontrolvariable,whileKAandEarethestatevariablesandand arethecorrespondingco‐states.Asfirstorderconditionswehave:

∙ ∙ 0 ⇒ ∙ ∙

/ (3)

∙ ∙ (4)

0 (5)

∙ ∙ ∙/

(6)

∙ (7)

Equation (6) is obtained by means of substitution of equation (3) into the macro‐

economic budget constraint which states that output is used for consumption and (gross)investment purposes. Equations (4)‐(7) constitute a simultaneous system of differentialequations thatcanbesolved forward in time,givenasetof initialvalues for thevariousstateand co‐state variables. Thiswill give rise to terminal values of those same state and co‐statevariablesattheterminaldateofphaseU,i.e.att=TJ,thevalueofwhichisunknownsofar.

TheJPRphaseThe JPR phase differs from the BAU phase since investment in the A technology has

stoppedandthatintheB‐technologybegins.However,thecarbon‐basedcapitalstockKAisstillusedforproductionpurposes.ThepresentvalueHamiltonianfortheJPRphase,i.e.HJ,isnowgivenby:

∙ ∙

∙ ∙ ∙ ∙ ∙ ∙ ∙ .(8)

AsintheBAUphase,wehavejustonecontrol, i.e.C J,butthreestatesKA,KBandEand

correspondingco‐states , and .Asfirstorderconditionswehave:

∙ ∙ 0 ⇒ ∙ ∙

/ (9)

∙ ∙ ∙ (10)

∙ (11)

0 (12)

∙ (13)

∙ ∙ ∙ ∙/ (14)

∙ (15)

Equation(14)isagainobtainedbymeansofsubstitutionofoptimumconsumptionlevels

(as given by equation (9)) into themacro‐economic budget constraint. As before, equations

‐ 16 ‐  

(10)‐(15)constituteasimultaneoussystemofdifferentialequationsthatcanbesolvedforwardin time, given initial values for the various state and co‐state variables. Note that the initialvaluesintheJPRphaseforthosestateandco‐statevariablesthatbothsystemshaveincommon,arethesameastheterminalvaluesforthosevariablesattheendoftheBAUphasebecauseofthecontinuityofstate‐andco‐statevariablesalonganoptimumpath.ForagivenvalueofTF,this systemof differential equations allows the forward calculation of terminal values for thestatesandco‐statesintheJPRphaseattimet=TF,whichwillthenfunctionastheinitialvaluesforthestatesandco‐statesduringthecarbon‐freephase.

TheCFRphasePhase F differs from phase J in that the carbon‐based capital stock is discarded, and

consequently the flowofCO2emissionsdrops to zero.From t=TFproduction is totally green.ThepresentvalueHamiltonianforphaseF,i.e.HF,isnowgivenby:

∙ ∙∙ ∙ ∙ 0 (16)

AsintheBAUandJPRphase,wehavejustonecontrol,i.e.CF,buttwostatesKBandEand

thecorrespondingco‐statevariables and .Asfirstorderconditionswehave:

∙ ∙ 0 ⇒ ∙ ∙

/ (17)

∙ (18)

0 (19)

∙ ∙ ∙/ (20)

0 (21)

As before, equation (20) is obtained by substituting equation (17) into the macro‐

economic budget constraint. Again, equations (18)‐(21) constitute a simultaneous system ofdifferentialequationsthatcanbesolvedforwardintime,giveninitialvaluesforthevariousstateandco‐statevariablesinheritedfromphaseJ.However,inthiscasetheterminalvaluesforthestatesandco‐statesareimplicitlydescribedbythestandardtransversalityconditionsinanAKsetting that require thepresentvalueof the carbon‐free capital stock to approachzeroat theterminaldate,i.e.inthiscaseattimeinfinity.Forcumulativeemissions,theterminalvaluehadalreadybeenreachedatt=TJ,whencumulativeemissionshitthethreshold.

TransversalityconditionsFor theCFRphase the standard transversality condition (further calledTVC for short)

appliesregardingthevalueofcarbon‐freecapitalattimeinfinity:lim →

,∙ , 0 (22)

wherewehavenowaddedtime‐subscripts.Notethat(18)canbe integrateddirectlytoobtainthetimepathfor , whichcanthenbesubstitutedinto(20)toobtainthetimepathfor , .

Thisyields:

‐ 17 ‐  

, ,∙ ∙ (23)

,∙

,

⁄

,∙

,

⁄

(24)

Equations(23)and(24)canbesubstitutedintoTVC(22),andwefindthatinorderfor

TVC(22)toholdthefollowingconditionsmustbesatisfied:

0 (25)

,∙

,

⁄

. (26)

Whensubstituting(26)into(24),wefindthat:

, , ∙ . (27)Itfollowsfrom(27)thatifthestructuralparametersaresuchthat(25)ismetandifwe

pick consumptionat timeTF (hence , (seeequation (17)) such that (26) ismet, then the

carbon‐free capital stockwill grow at the steady state growth rate δB / from timet=TF.

ApartfromtheTVCabove,werequirethat:

,0 , (28)

whichsetstheshadowpriceofcarbon‐basedcapitaltozeroattheendoftheJPRphase(i.e.,atthemoment it is discarded), because an extra unit of capital added thenwould not produceanythingand,beinguseless,wouldbeworthnothing.

Lastly, thereare twoTVCs thatpertain to theoptimum lengthof theBAUand the JPRphase,andthatrequiretheequalityoftheHamiltoniansofthevariousphasesatdifferentpointsin time. For the optimum length of the BAU phase (given by the value of TJ, since TU=0 by

assumption),wemust have 25,whereas the optimum start date for the F phase is

determinedbytherequirementthat .UsingthedefinitionsoftheHamiltoniansin(2),(8)and(16),aswellasimposingtheF.O.C.regardingconsumptionin(3),(9)and(17)togetherwiththecontinuityconstraintsonstatesandco‐statesthatfeatureinbothadjoiningphases,weobtainimplicitdescriptionsofthearrivaldatesoftheJPRandCFRphases.Thesearegivenby:

, ,

(29)

∙, ∙ , . (30)

                                                            

25 isshortforthevalueoftheHamiltonianpertainingtophasePattimet.Intheequality itfollowsthatthe time‐coordinateof cannotbeexactly the sameas thatof sinceP+1 is thephase comingdirectlyafterphase P. Consequently, must be read as lim ↓ , but for simplicitywe stick to our originalnotation.

‐ 18 ‐  

Equation(29)statesthatinvestmentinthecarbon‐basedtechnologyshouldstopatthemomentthattheshadowpriceoftheAtechnologyisequalto(andthereafterdropsbelow)theshadowpriceof theB technology. Since themarginal costofobtaining aunit of capital is thesameinbothcases(i.e.oneunitofconsumptionforegone),equation(29)isconsistentwiththemaximizationofthe(presentvalue)welfaresurplusassociatedwithinvestment.Equation(30)statesthatproductionusingcarbon‐basedcapitalshouldstopthemomentthatthebenefitsfromcontinuingtouseaunitofcapital(theLHSofequation(30),asoneunitofcapitalproducesA

unitsofoutput,andeachunitofoutputisworth,inpresentvaluewelfaretermsatt=TF)

matches thecostofdoingthat(asgivenbytheRHSofequation(30),sinceoneunitofcapital

produces units of CO2 emissions, each at a cost of , 26). Equation (30) is therefore

consistentwiththezeroquasi‐rentconditionasanimplicitdescriptionoftheoptimummomenttoscrapexistingcapacityknownforalongtimefromtheclay‐clayvintageliterature.27

2.5BAM:SequentialnumericalsolutionofthedifferentialequationsystemsThethreesystemsofdifferentialequations,furthercalledSU,SJandSF,caninprinciple

be solved, as the initial and terminal valueswehaveavailable for the statevariables, and theTVCsthatprovideeithersomefixedpointsforthetimepathsoftheco‐states(cf.equation(28)),or link theco‐states toastate‐variablewhich timepathhasbeen fixed throughagiven initialvalue(cf.equation(26)),orthatlinksdifferentco‐statesatsomepointintime(cf.equations(29)and (30)) provide exactly enough information to obtain a fixed point for all time pathsconcerned.Toseethis,itshouldbenotedthatweneedtoobtainfixedpointsforthetimepathsforthreedifferentstatevariables(KA,KBandE),andfortheircorrespondingco‐statevariables,aswellastheoptimumvaluesofthephaselengthsofphasesUandJ.Hence,forBAMweneed8

piecesofinformation.Wehaveinitialvaluesavailablefor , , , , 0,and .

In addition,we have a terminal value for cumulative emissions: . The transversalityconditionsgivenbyequations(26),(28),(29)and(30)thenprovidetheremaininginformationthatisneeded.

AnumericalsolutionofSUcaneasilybeobtained,conditionalonsomeapriorivaluesof

,, andTJ,giventheinitialvaluesof , and .ThesolutionofSUthenprovidesterminal

valuesfor,and , that,onaccountofthecontinuityofstatesandco‐statesgiveriseto

initial values for SJ, since we must have that, ,

, , , , , , ,

, 0, .Weonlyneedaninitialvaluefor,aswellasanapriorivalueforTFto

be able to calculate SJ forward in time. That initial value is provided by the TVCs listed inequation(29).Giventheapriorivaluesfor

,, ,TJandfinallyTF,weareabletocalculate

theterminalvaluesfor,, , , , and ,which,againusingtherequirementofthe

continuity of states and co‐states, imply that , , , ,, ,

and

, , .Thetimepathsthusobtainedforallstatesandco‐states,incombinationwithour

guessesofthevariousphaselengths,cannowbeusedtoevaluatethedifferencesbetweenthe

                                                            26Notethattheshadowpriceofemissionsitself isnegative,sinceanadditionalunitofcumulativeemissionswouldreducepotentialwelfareratherthanincreasingit.

27See,e.g.,Johansen(1959)andSolowetal.(1966).Morerecently,Boucekkine(2011)alsousedthiscondition,whichstatesthatavintageonceinstalledshouldbediscardedassoonasit’squasi‐rentsbecomenegative,arulethatholdsinthis case since the quasi‐rents consist of the social welfare value of discounted output less total variable CO2emissionscosts.

‐ 19 ‐  

RHSandtheLHSoftheterminalconstraint andthoseoftheTVCsnotusedsofar,i.e.of equations (26), (28) and (30). Obviously, if these constraints were satisfied by the valueschosenapriorifor

,, ,TJ,andTF,wewouldhavefoundthesolutionsforallthewelfare‐

maximizing time paths. Being the result of a guess, these differences generallywill not equalzeroattheoutset.But,inthatcase,asearchalgorithmsuchasthesteepestdescentmethodthatwethusfarhavebeenusingwithsuccess–shouldreadilyfindthesetofinitialvaluesthatwillsimultaneously satisfy all of the TVCs.28 It should be noted that a similar approach can befollowedforversionsofthemodelwithmorethan3phases,aswewilldescribefurtherbelow.

3.ExtensionsoftheBAMFramework

This section describes the modifications of the basic 3‐phase model that extend thestructure of BAM in twoways. The first introduces the option of investment in research anddevelopment directed to lowering the unit capital costs of production facilities that embodytechnologiesthatdonotusefossilfuelasanenergysource,andthesecondtakesaccountofanadverse feed‐backeffecton theglobaleconomyresulting from thewarming that accompaniesthetransition.

3.1BAM+R&D:Themodifiedmodel’sstructure To introduce R&D investment‐driven endogenous technical changes affecting the

economic performance of the carbon‐free technology,we posit that thesewill be confined toraising the average (and marginal) productivity of the capital goods that embodies the newtechnology. The latter lowers the real unit cost of carbon‐free production capacity byirreversiblyraisingBaboveitsinitialvalueintheopening(BAU)phase,anddoingsopriortothestartofthecapital formationthat isnecessaryto implementthis formof technicalchangeandbeginthetransitionawayfromaneconomyreliantuponburningfossilfuels. SincewearebuildingupontheBAMframeworkthatusesanAKgrowthmodelsetting,thefamiliarmodelingofendogenoustechnicalchangederivingfromR&Dactivitiesisnotreadilyapplicable.Theclassicexemplarsofthelaterintheliterature–foundinLucas(1988),orRomer(1990),or(moreimplicitly)inAghionandHowitt(1992)‐‐specifythatthedependenceoftheproportional rate of change in total input productivity upon the flow of R&D expenditures isdescribedbyasimplefunctionalrelationshipofthefollowingform:

B R B (31)

                                                            28Infact,giventhesimplicityofoursystem,itispossibletoobtainanalyticalsolutionsforalltimepaths,eventhoughtheseareinpartquiteintricatenon‐linearexpressionsinvolvinghypergeometricfunctions.Theseintegralequationslink thevarious initial and terminal conditionsaswell as thevaluesofTJ andTF together throughasimultaneoussystem of non‐linear equations. Using that system, we were able to find the fixed points for all time paths bynumerically solving the non‐linear system for the fixed points. The time paths for each of the state and co‐statevariables could then be obtained by substituting the numerical values thus found for the fixed points into theanalyticalsolutionsofalltimepathsinvolvingthesefixedpoints.Thisprocedureprovedtowork,butisrathertediousandtimeconsumingandit’sfeasibilitydependedcruciallyonthesimplicityofthemodel.MinordeviationsfromtheAK‐set‐up proved to make using the analytical method infeasible. This provided a strong incentive to use thesequentialnumericalsolutionmethodoutlinedinthemaintext.Bothmethodsdogeneratethesameresults,astheyshould, but the sequential numerical method is much more efficient in terms of both obtaining the system ofdifferentialequationstobesolved,andfindingthesolution.Nonetheless,moreworkisneededtoimproveupontherather crude steepest descent search routine that we are employing at the moment. The latter converges ratherslowly, ifatall, for themore intricateversionsofthemodelextensionswehavebuiltuptodateandwhicharenotreported here. Nonetheless, the numericalmodel does allow us to expand themodel inways thatwouldmake itimpossibletosolveusingtheanalyticalsolutionapproach.

‐ 20 ‐  

where R represents R&D resources measured here as consumption foregone. But, with thelinear, single‐factor specification of the AK model’s production function, the average (andmarginal) productivity of capital remains constant as the economy accumulates capital. Nocontributions from increases in other factors of production (say, the state of technicalknowledge,orhumancapital)areneededtopreventthecapitalstock’sgrowthfromdrivingitsmarginal productivity downwards. Were a specification such as (31) to be introduced in asimple AK‐setting (disregarding the embodiment requirement for the sake of simplicity inmakingthispointclear)theresultinggrowthpathofthesystemwouldbecomeexplosive,ratherthan one that could attain a steady‐state equilibrium rate.29 Hence, we have opted for aspecification of theR&Dprocess that allows for a decreasingmarginal product of R&D and agrowth rate of B that asymptotically approaches zero for any constant positive allocation ofresourcetotheR&Dactivity.ThisassuresthattherealmarginalrateofreturnfrommaintaininginvestmentinR&Dwilldiminishovertime.Thesimpleformforthisfunctionthereforeis:

( )B R B B (32)

where B is the asymptotic value for capital productivity and where 0 and 0 1 are

constants.30 Starting out with a basic technical design that permits productivity to be at B0,

00 B B , themarginalproductofR&Dinvestmentwillbe fallingasRincreasesandB rises

towardsitsupperbound. Several features of this specification argue for its’ use in the present applicationscontext.Firstly,ithastheadvantageofintroducingdecreasingreturnstoR&Dinasettingthat,unlike the conventional endogenous growth models, excludes the possibility of a specifictechnology being rendered infinitely productive (and so resulting in infinitely rapid growth)merely by the application of more and more massive R&D expenditures at any particularmomentintime.31AllowingdecreasingmarginalreturnsinR&Drecognizesthatatagivenstagein theadvanceofknowledge the stateof fundamental scientificunderstandingof thephysicalprocessesinvolvedmaystillbeinadequatetopermittheeffectiveapplicationofmoreandmoreresourcestothesolutionofaparticularpracticalproblem‐‐suchasthefurtherimprovementofthe productivity of a particular class of technology‐embodying capital facilities.32 That more

                                                            29ThisfollowsfromthefactthatwithaconstantlevelofRinequation(31)therisinglevelofcapital’smarginal(andaverage) productivitywould increase the attractiveness of engaging in R&Dwhile providing the extra investmentresourcesforthatpurposebyacceleratingoutputgrowth.

30Tokeepkeepthemattersimpleforcomputationalpurposesinthisinvestigation,wehaveusedR,thecurrentflowofresourcesasthecontrolvariableintheR&Dprocess.Butitwillbeconceptuallymoresatisfactory,andshouldnotbecomputationallyproblematic infutureworkwiththismodeltoreplaceRinequation(31)byalatentstatevariablerepresenting thedepreciated stockofknowledge thatat time t is germane to thisdirectedR&Dactivity. FollowingconventionalpracticeintheempiricalliteratureonproductivityeffectsofR&D,thecurentstateofknowledgeusefulformodifyingthestatevariableB,canbeindexedbythecurrentvalueoftheintegralofdepreciatedflowsofR,addedtoaninitialmeasureoftheknowledgestockavailableatt=0,thecommencementdatefortheactivity.ThisleavesRasthecontrolvariable,butwilhavetheeffectofsmoothingchangesinthegrowthrateofB.

31Theequilibrium(steady‐state)growthrateinastandardAK‐modelriseslinearlywiththeproductivityofcapital(cf.BarroandSala‐i‐Martin,2004),andthereforewiththeflowrateofR&Dexpenditures. Theexistenceoftechnology‐specific intrinsicproductivityboundsthatareset in the limit (by thephysicalpropertiesof thematerials, chemicalandelectricalprocessesentailedinproduction)makesthissoimplausiblethatevenitsassertionasametaphorisofdubioususefulness.Bycontrast,afurther,technicaladvantageofthespecificationgivenbyequation(32)isthattherateofimprovementintheproductivityofcapital is jointlyconcaveinBandR,assuringfulfillmentofanecessaryconditionforthewelfaremaximizationproblemtohaveasolution.

32Conceptually, this formulationof theeffectsof investment inR&Dactivitiesmaybe thought to reflect aPlatonicworld inwhicha finitenumberof solutionpossibilities for technical transformations arepresent from the start of

‐ 21 ‐  

restrictiveviewofthetransformativepowerof investmentinR&Disappropriatealsobecausethe concern in this context is not with the undirected global expansion of the technologicalopportunity set typically envisaged in theoretical growth models. Rather, the aim of the“directedR&D”inthepresentmodelistoenhancetheeconomicpropertiesofparticularkindsofprocess inventions, with new product inventions only insofar as alterations in productcharacteristics are consequential for raising the efficiency of capital inputs into carbon‐freeproductionprocesses.33 Introducing (32) into theBasicModel requires a number ofmodifications: the first ofthese extends the Hamiltonian by adding the value of increases in B due to R&D. Next, themacro‐economicbudgetconstraintthatdescribestheaccumulationofcapitalmustbeadjustedto reflect the fact that a unit of resource inputs in R&D, R, requires allocation of a unit ofaggregateoutput(i.e.,foregoingitsuseinconsumptionortangibleinvestment).InsteadofjusttheonecontrolvariableC,wenowhaveanadditionalcontrolvariableR,andanadditionalstatevariable, i.e.B(thecarbon‐freeproductivityparameterofBAM).Bhasbecomeastatevariablesince it can change as long as R&D is taking place, butmust remain constant after R&D hasceased. Leonard andVanLong (1992:Ch.7)describehow the latter situation canbehandled.TheirapproachistoregardtheHamiltonianasafunctionofB,i.e.H(B),whilesubstituting 0B asthedynamicconstraintonBduringthephaseswheninvestmentsinR&Darenotmade(andBthereforewillnotbechanging).Inallphaseswehavethattheequationofmotionfortheco‐statevariable associatedwith B is given by /B H B . The reason for explicitly specifying the

dynamic constraints on the co‐state for B during the phases where B itself is not changing(because R&D has halted) is that the terminal value of B (when the R&D process stops)willinfluencethegenerationofwelfareinthefollowingphases.Thus,throughthecontinuityofco‐statesandstates,informationaboutthefuturewelfareeffectsofhavingahighvalueofBintheJPRandCFRphases,caninfluenceallocationdecisionsduringthephasewithpositiveR&Dthatwoulddirectly involve the trade‐offbetweenR&Dandotherusesofoutput, like consumptionandtangiblecapitalinvestment.

Notethatitcanbeshownthatonceaninitialvaluefor 0B isavailable,thenitpaysnotto

wait to improve theproductivity of capital embodying carbon‐free technology by engaging inR&D (see Appendix A). Hence,without loss of generality,wemay assume that the basic ideaunderlyingthecarbon‐freetechnologyisavailablefromt=TU=0.Inthatcase,R&Dshouldstartatt=TU,anditshouldstopatt=TJ,thatbeingthemomentatwhichinvestmentintheformationof carbon‐free production capacity commences. Consequently, the BAM+R&D model has thesamenumberofphasesasBAM.FortheBAUphase,assumingthatR&Disdonefromtheverybeginning, the three following equations have to be added34: /B H B ,

                                                                                                                                                                                          time,buttheseasarulewillnotrevealthemselvesspontaneously.Theycanbeuncovered,however,andformulatedfor practical application through costly research and development procedures based upon the existing state offundamental scientific knowledge, rather than being created de novo and without limit by the expenditure ofresourcesintheperformanceofR&Dactivities

33Followingthisinterpretation,addingendogenoustechnologicalchangetotheBasicModelallowsustocharacterizetheoptimalpathofglobalR&D that isdirected to increasing theproductivityofgreencapital. Correspondingly, theimpactofR&Dinvestmentoneconomicwelfareismodeledasbeingfeltindirectly,ratherthandirectlyintheformofpure product quality enhancements. In otherwords, the welfare gains come through reduction of the sacrifice ofconsumptionutilityrequiredinthetransition,andforthesubsequentsustainedgrowthof(percapita)consumptionunderstabilizedclimaticconditions.

34Note that for ease of notationwe have dropped the time subscript and the phase superscript exceptwhere thepresenceofthetimesubscriptisneeded.

‐ 22 ‐  

( )B R B B and / 0H R , and the capital accumulation constraint takes the

correspondinglymodifiedform: ( ).A A AK A K C R .

Furthermore, the revised Hamiltonian for the BAU phase is now given by: . 1 /(1 ) {( ) }t

A A A BH e C A K C R B .

Since there is now an additional state variable, there is also an additional TVC. As usual, werequire that at infinity the present (utility) value of B should approach zero, i.e.

,lim 0t B t tB . However,t TJB B t TJ , since R&D has ceased at t=TJ which has

turned B into a constant for t . Consequently, the additional TVC is reduced to the

requirementthat ,lim 0t B t .Using(25),itcanbeshownthatthelatterTVCimplies:

, , ∙,∙

∙. (33)

It is hard to give an interpretation for equation (33), and a fortiori, for the revised

version of the transversality condition that now determines the optimum length of the BAUphase. That TVC differs from the corresponding condition in BAM, since the R&D process isactive during the BAU phase and inactive during subsequent phases. Consequently, theHamiltoniansoftheBAUandJPRphaseevaluatedatt=TJ,willnowinvolvetermscomingfromthe R&D function as well as the corresponding co‐state evaluated at t=TJ, resulting in acomplicated expression linking the various states and co‐states together at t=TJ. Because itcannot readilybe interpreted, theexpression isnotgivenhere.Note that (33) incombinationwith the initial value forB, i.e.B0, provideenough information to solve this revisedsystemofdifferentialequations.

3.2Allowingforwarming‐driven“UnscheduledCapitalLosses”:BAM+R&D+UCLThefinalversionconcernsacombinationofBAMandR&Dthatincludestheintroduction

ofa“tippingpoint”beyondwhichfurtherwarming,drivenbytherisingconcentrationofCO2intheatmosphere,willbringahigherexpectedannualrateof“unscheduledlosses”ofproductivecapacityduetothedirectandindirecteffectsofdamagestothecarbon‐basedcapitalstock‐‐asa result of more frequent extreme‐weather events, and more extensive coastal and riverineflooding,andmoreprolongeddroughtininteriorregions.35

Forpresentpurposeswehavemodeledthisinthefollowing(simplest)way,supposingthatthereisathresholdthatwhencrossedwilltriggertheonsetofa“highdamage”regime,anddefiningthat“extreme‐weathertippingpoint”tobeacriticallevelofcumulativeCO2emissionsthat isreachablewithintheBAUphase.Astep‐functionrise intheproportionalannualrateofphysical losses of capital services, in effect a jump in the decay parameter from the normalphysical rateof depreciation, takesplacewhen that tippingpoint is reached, therebysplittingtheBAUphase intoan initial “lowdamage”sub‐phaseandthesubsequentsub‐phaseof “high

                                                            35Therestrictionofexpectedannual“unscheduledlosses”ofproductivecapacitytothecarbon‐usingcapitalstock,implicitly assumes, firstly, that the inherited energy, transport and communications infrastructures, and areas ofindustrialconcentrationintheBAUphasereflectedlocationalchoicesmadeinanerabeforethegreatervulnerabilityofthoseregionstoextreme‐weatherduetoglobalwarmingwasforeseen.Secondly,aformofadaptationisimpliedinthe(optimistic)assumptionthatthesubsequentlyformedcarbon‐freecapitalstockhasbeen“defensively”designedand/orlocatedinthisrespect–someportionofthedirectedR&Dactivities,andtheactualdeploymentcost,reflectingthe incremental costsof therebyprotecting its expectedmarginal social rateof return frombeingreducedbyhighphysicaldamanges and temporary “outages”. See alsoBrock et al.(2012) for recentmodelingof regional climaticchanges due the geographic variations in warming: these should be considered in modeling damage levels anddefensiveadaptationneeds.

‐ 23 ‐  

damages”.All thephases following the latter part of theBAUphase are also characterizedbycontinuationof the “highdamage” regime.36Onceagain, this isa situation that isdescribed inLeonardandVanLong(1992:Ch.10),wheretheregimeshiftfromlowdamagestohighdamagesinitiatedbyastatevariablehittingaparticularthreshold, impliesa jumpinthecorrespondingco‐statevariable.

Since the rate at which CO2 emissions are accumulated depends on productiondecisions,themomentintimeatwhichthedamagethresholdwillbehit,dependsontheseveryproductiondecisionstoo.Hence,thearrivaltimeofthehighweatherrelateddamagesubphasesissubjecttochoiceandthereforetooptimaldecision‐making.37Consequently,wecanoptimallychoose themoment at which the high damage sub‐phasewill arrive.We can implement thisagainbyrequiringthattheHamiltoniansevaluatedatthemomentofarrivalofthehighdamagesubphasewillbethesameimmediatelybeforeandafteritsarrival.Thisimpliesthatthismodelwillhave fourdifferentphases insteadof three. It follows thatweneedtodetermineanextraphaselengthinadditiontothesizeofthejumpintheco‐stateforcumulativeemissionsatthetimeofarrivalof the firsthighdamagesub‐phase. InadditiontothegiveninitialandterminalvaluesoftheBAM+R&Dmodelaswellasthecorrespondingtransversalityconditions,wehavean additional terminal value in the form of the location of the damage threshold itself, incombination with the requirement of the equality of the Hamiltonians at both sides of thedamagephasechange. It turnsout that the transversality condition for thearrivalof thehighdamagesubphaseduringtheBAUphaseisgivenby:38

, ∙ ∙ ,,

,, . (34)

Equation(34)shouldholdexactlyatthearrivaltimeofthehighdamagesub‐phase,i.e.at

t=TUH.39Inequation(34),thesuperscriptsHandLrefertolowandhighdamagesubphases,while the superscripts U refers to the business‐as‐usual phase. The RHS of equation (34)contains the jumpintheshadowpriceofcumulativeemissions. Itmeasures thechange in thewelfarecostsassociatedwithemissionsperunitofcapitalbeforeandafterthejump.TheLHSof

                                                            36Althoughfurtherupwardstepsinthedamageratearetobeexpectedbeyondtheclimatecatastrophetippingpoint(i.e.theTPIRCC),itisnotnecessarytospecifythem—sincethemodelisdeterministicandgoalof“socialplanning”istoavoidcrossingthatthreshold.Itisimportant,therefore,thatthecalibrationofthemodelassuresthatthelatterofthesetippingpointsliesbeyondthecloseoftheBAUphase.

37TheintricaciesoftheforegoingdynamicsareamongthereasonswhywehavenotfollowedtheapproachtakeninbydeBruin,DellinkandTol(2008)inAD‐DICE,inordertoexplicitlyconsidertheoptionofdefensiveexpendituresforcurtailinganticipateddamagestoproductivecapacityresultingfromglobalwarming.Becausetechnicalchangesareassumedtobedisembodied in theDICEmodel,mitigationofC02 inducedbyrisingcarbontaxes)doesnotrequirespecificcapitalformationtoachieveloworzero‐emissionsproductioncapacity.Consequently,inAD‐DICEmitigationdoesnotcompetedirectlywithconcurrentadaptationexpenditures forgross investmentallocations. Althoughthetwo policies are substitutes when their inter‐temporal effects are considered, because effective early mitigationwould check the pace of warming and reduce the future need for defensive adaptations, this relationship is notsymmetric:earlyreductionsindamagetocapacitywouldthentoincreaseoutputandaccompanyingCO2,butwhilethat would call for more vigorous mitigation efforts the incrementally protect production capacity would not beneededtoimplementdisembodiedloweremissionstechnology.DICE’sassumptionsregardingtechnicalchangethusrendertheassessmentoftheinteractionsbetweenthosetwooptionsbothsimplertomodelandtransparenttoassessthanwouldbethecaseinthepresent(butinourviewmoreempiricallyrelevant)frameworkofanalysis.38Obviously,thearrivaltimeofthefirsthighdamagesub‐phasecouldalsobewithinthejointproductionphase.Butatthisstagewearemainlyinterestedinreportingontheprincipleinvolved,whichwouldbethesameinwhicheverphasethedamagesthresholdwouldbesituated.

39NotethatTUHalsodenotestheendofthelowdamagesub‐phaseoftheBAUphase.Notethereforethatinthiscase,TUHcomesoneinstantbeforethevalueofTUHthatrepresentsthebeginningof thehighdamagesub‐phaseoftheBAUphase.

‐ 24 ‐  

equation (34) measures the welfare cost associated with the extra decay per unit of capital.Equation(34)impliestheequalitybetweenthewelfarecostofusingaunitofcapitalbeforeandafterthearrivalofthehighdamagesub‐phase.

Weconcludethatthejumpintheco‐statemustbesuchthatthewelfarecostofusingaunitof capital remainsunchanged. Since thedepreciation costs arehigherafter the jump, theemission costs must be lower, implying a drop (in absolute terms) in the shadow price ofemissions. The latter is consistent with the observation that a faster rate of decrease of thecarbon‐based capital stock will lead to a reduction in the rate at which CO2 emissions areaccumulated,otherthingsremainingthesame.

4.ModelCalibration,SolutionResultsandParameterSensitivityAnalysis

4.1CalibratingtheBasicModelanditsextendedversionsInordertoshowhowthevariousmodelswork,theparametersofthemodelneedtobe

calibratedorfixedapriori.TothisendwehavemadeuseoftheNordausRICE2010data,40aswellassomedirectassumptionsnecessitatedbythestructuraldifferencethelattermodelandour DIRAM framework. Nordhaus (2010) updates the calibration of the essential neoclassicalgrowthmodel setting of the integratedDICE and RICE policy assessmentmodels that have asingle,prolongedphase,whereasweemployanextensionofthemoreprimitiveAK‐setting.Withmultiplephases.ThisimpliesthatitisnotpossiblesimplytoimportthecalibrationdataforDICEandRICEintoourmodel

Therefore, inorder to reproduceglobalgrowthrates andsaving rates thathaveaboutthe right size in relation to the “bench‐mark” values provided by Nordhaus (2010), we havestartedwiththeassumptionthattheappropriatecapital‐outputratiofortheBAMmodelintheBAUphaseisequalto4,avaluethatiswellabovethecapital‐outratiointhemultifactorglobalproductionsystemspecifiedinDICE(andRICE).Implicitlyourpresentnotionof“capital”mustbeamorecomprehensiveone,inasmuchasintheabsenceofexplicitspecificationoflaborasafactorofproduction,itsmagnitudemustallowforthe(proportionality)betweenconventionallydefinedtangiblecapitalandallhumancapital inputs. Wehavealsomadetheassumptionthatdepreciationcostsasafractionofoutputis15%.Thisalsoisrelativelyhighvalue,butactuallynotsounrealistic,giventhemuchbroadercapitalconcept.

UsingNordhaus’dataonTFPgrowthaswellasoutputpercapitaandpopulationgrowth,wearriveatanimpliedgrowthrateofoutput(andofcapitalinanAK‐setting)equalto0.03436.Fromequation(27)wecanderivethesteadystategrowthrateinanAK‐setting,andfindthatonthe premise that total output and (carbon‐based) capital stock are growing on a steady statepath in theBAUphase,atanannual rateequal to0.03436(sic!), the followingconditionmusthold:

0.03436. ∗ . ∗ .

. (35)

Inequation(35),wehaveusedtheassumptionthatdepreciationasafractionofoutput

is15%,implyingthat∙

.0.15.Thelatterimpliesthat 0.15 ∙ 0.15 ∙ 0.25 0.0375.

This value for the depreciation rate turns out to be much lower than the 0.10 rate used byNordhaus.Equation(35)impliescombinationsof and givenby:                                                            40See http://nordhaus.econ.yale.edu/RICEmodels.htm for further details. Inorder to obtain observations for 2010fromtheoneslistedfor2005and2015intheRICEdata,weusegeometricalinterpolation.

‐ 25 ‐  

6.185 29.104 ∗ . (36)

Importingthevalue 0.015usedbyNordhaus,wefindthat 5.748.Thisimpliesamuch lower intertemporalelasticityof substitution than theparametervalueof =1.5 thatNordhaushasused.Inbothcases,however, 1.Thehighervalue,however, impliesthatthefunction conventionally interpreted as expressing an index of utility or “felicity,” namely, f =

1 /(1 )C ,mustbenegative,althoughitisincreasinginthelevelofpercapitaconsumption.

Marginal felicitywouldstillbepositiveanddecreasinginconsumption. Sincesocialwelfareistheintegralover(thepresentvalue)offelicity,thecorrespondingwelfareindexalsoisnegative.Butthescalingoftheseindexesisentirelyarbitrary,anditisapermissibleandsimpleoperationto remove thisunaccustomedanddisturbingnegativityof “welfare”by renormalizationof theindexof felicity: adding to it, at eachmoment of time, apositive termequal to the (negative)valueof the indexof theoriginal indexat timezero.This forcesanupwardshiftof the felicityfunction(andalongwithit,thecorrespondingwelfareindex)intothepositivequadrant.

Since is relatively large, felicity will be relatively small, as is also the case for thepresent value ofwelfare. The numerical values of the co‐states consequentlywill be small aswell, since they represent the change inwelfare due to a 1 unit change in the correspondingstates. For example, since Y0=68.95 trillion dollars of 2005,41 our assumptions imply that

K0=275.8,butalsothatC0=(1‐s).Y0=49.16.Hence,felicityatt=0isequalto 1.958 ∗

10 .Thereforewehaveaddedamultiplicativescalingfactorequalto10 tothefelicityfunctionaswell,resultinginvaluesforstatesandco‐statesthatarenotmanyordersofmagnitudeapart.

Even though our implied inter‐temporal elasticity of substitution is rather low, incombinationwiththeotherparametervalues,plausiblevaluesforboththegrowthrate,andthesavingratecanbegenerated.Theimpliedvalueofthesavingrate(s)isgivenby:

∙

0.287. (37)

WithrespecttocumulativeCO2emissionsandthelocationoftheclimatetippingpointinterms of the cumulative emissions generated by our model, we have used the followingprocedure. From theNordhausdata,we can infer that the ratiobetween the concentration ofCO2 in the atmospheremeasured in ppm and the concentration of carbon in the atmospheremeasuredinGTCsisgivenby:

0.4695 ∗ ∆ 0.4695 ∗ ∆ , (38)

where ∆ refers to the first difference operator. The present concentration of CO2 in theatmosphereamountsto390ppmv.42ThepreindustrialconcentrationofCO2is280ppmv.The

                                                            41t=0refersto2010.Thedatafor2010areobtainedbymeansofgeometrical interpolationbetweentheNordhausdatafor2005and2015.

42 See the U.S. Department of Commerce National Oceanic & Atmospheric Administration (NOAA) data (athttp://www.esrl.noaa.gov/gmd/ccgg/trends/)fortrendsinmonthlymeanCO2concentrationslevelsmeasuredonMaunaLoa,Hawaii.ForthemonthofOctober,themeanstoodat388.92ppmvin2011andat391.01in2012,givingthe annualmidpoint of 390 ppm. It should be noted that the trend of thesemonthlymean observations from theScrippsInstitutionofOceanographyhasbeenexponentiallyupwardsthroughouttheperiodfromthelate1950stothepresent.Overthe32years1980‐2012theaverageannualrateofincreasewas0.00476,andprojectingthisfromthe390 ppm annual 2011‐12 level, the concentration level will have reached 398 ppm by 2016. Yet , a NOAA newsreleaseon31May2012announcedthatatBarrow,Alaska–itsonlyremotenorthernsitewithcontinualatmosphericCO2monitoring‐‐theconcentrationlevelinthespringtouchedthe400ppmmarkforthefirsttime,inconcertwith

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climate tipping point is generally associated with a temperature rise of 2° Kelvin abovepreindustriallevels.TheequationdescribingtherelationbetweentemperaturerisesandaCO2concentrationsinppmrelativetosomebaselevelisgivenby:43

∗ ∆ / . . (39)

Inequation(39),∆ representsthetemperatureriserelativetoabaselinetemperatureatabaselineconcentrationofCO2givenbyppm0.It followsthatthetippingpointat2°Kelvin

relative to preindustrial levels is given by 280 ∗ . 446.8 . For a 3° Kelvintemperaturerise,thecorrespondingconcentrationofCO2wouldbe564.4ppm.ThecurrentrateatwhichtheCO2concentrationisrisingequalsabout2ppmv/year.44Assumingthatthisrateofincreasewillbefollowingtherateofgrowthofoutput,wefindthattheroomtoemitprovidedbythe2°Kelvintippingpoint isequal to446.8‐390=56.8, implyingthat therecanbea further56.8/2=28.4 ‘2010 size batches’ of gross emissions until the 2° Kelvin tipping point will bereached. According to Nordhaus, the gross emission rate in 2010 is about 10.63 GTC/year.Therefore,28.4batchesofemissionsrepresent28.4x10.63=301.9GTCofcumulativeemissionsin total. A fraction(1‐q)of theseemissionswillbeabsorbedby theEarth’soceansand forestcover,whiletheremainder(q)willendupaddingtotheconcentrationlevelintheatmosphere.Usingequation(38),thenetemissionsassociatedwithariseof56.8ppmisgivenby:

∆ 0.4695 ∗ ∆ ∗.

. ∗ .0.4. (40)

It follows that if we rescale the initial level of cumulative emissions to zero, the sum of netemissionsuntilthe2°Kelvintippingpointwouldbereachedisgivenby0.4 ∗ 301.9 120GTC.Similarly, the cumulative net emissions for temperature gains of 2.5° and 3° Kelvinwould beabout240GTCand370GTC,respectively.

For themagnitude of real gross global product in 2010we obtain the value 68.95 intrillionsof constant2005dollars, from theNordhaus’ (2010)DICEcalibrationdata. Since thecorrespondinginitialcapitalstockvalueistakentobe4timesthat,theflowofnetemissionsperunitofcarbon‐usingcapital(inGTCpertrilliondollar)isfoundfrom:

0.4 ∗ .

∙ .0.0154 . (41)

Additionalaprioriparametervaluesandadjustments:Forthecarbon‐freetechnologywehavemadetheassumptionthatdepreciationisequal

to that of the carbon‐based technology. In addition, for the Basic Model the average (andmarginal)productivityofcarbon‐freecapitalissetat 0.12.Further,inspecifyingtheR&Dimpact function we have made the following parameter assumptions: 0.5, 0.1, and

0.2.

                                                                                                                                                                                          the same “milestone” readings reported by remote northernmonitoring sites in Canada, Finland, Norway and anisland in the north Pacific. This has supported the consensus view among NOAA’s climatologists and theirinternational colleagues that by 2016 mean global CO2 concentrations will have risen to 400 ppm (cf.http://researchmatters.noaa.gov/news/Pages/arcticCO2.aspx.).

43Forfurtherdetails,seehttp://en.wikipedia.org/wiki/Radiative_forcing.

44 See the NOAA (http://www.esrl.noaa.gov/gmd/ccgg/trends/) data for the annual increases in the CO2concentrations levels measured on Mauna Loa, Hawaii. Calculated for the years 2002‐2012 (from the time‐seriesavailableatftp://ftp.cmdl.noaa.gov/ccg/co2/trends/co2_gr_mlo.txt),theaverageannualgainwas2.06ppm.

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Lastly,wehaveadjusted theclimate tippingpoint inordertohaveabusiness‐as‐usualphaselongerthanjust5or10yearsforthe2°Kelvintemperaturerise.Infact,wehaveusedavalueof325GTCnetwhichisconsistentwithatemperatureriseofabout2.75°Kelvin.Sothattheexpectedhigherrateof lossesofcapitalservices(duetowarming‐drivenextremeweatherevents)willemergewithintheendogenously lengthenedbusiness‐as‐usualphase,wehavesetthatthresholdlevelat87GTC(net)cumulativeCO2.

4.2BAMsolutionresultsPreliminary parameter sensitivity analyses performed using the models show model

reactions that are familiar from growth theory. Changes in the rate of discount or in theintertemporal elasticity of substitution all have the expected impact. This holds for theproductivity parameters as well. When cumulative emission thresholds are tightened, theshadow price of CO2 emissions rises (in absolute terms). When productivity parametersincrease,sodothecorrespondingco‐statesoftheassociatedstatevariables.

Rather interestingly, the linking of various sequential phases results in anticipatoryadjustmentsthatintroducetransitionaldynamicswhicharemissinginanordinarysingle‐phaseAK endogenous growth setting. The results obtained for the Basic Model (BAM) with theparametersetandtheinitialvaluesdescribedattheendoftheprecedingsub‐section(4.1)aredisplayedinFigures2.1and2.2.

Figure2.1BAMbaselineresultsforKA,KBandE

ThefirstrowofplotsinFigure2.1displaystheoutcomeswithrespecttocarbon‐basedcapitalKA.Theverticaldottedlinesmarkthearrivaltimesofthejointproductionphaseandthecarbon‐free phase. They are situated at TJ=23.78 andTF=40.17, implying that theBAUphasetakesslightlylessthan24years,whiletheJPRphaseisslightly longerthan16years.Thefirstgraph in this row shows the shadow price of carbon‐based capital decreasing steadilythroughoutbothphases.Thenextgraphshowsthebuildupofcarbon‐basedcapacityduringthe

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opening,BAUphaseandthesubsequentrundownofthatcapacityduringtheJPRphase.Theseeventsaremirroredinthelastgraphintherow,whichplotstheinstantaneousrateofchangeinthecarbon‐basedcapitalstock.Weseethatnetinvestmentincarbon‐basedcapitalacceleratestowardstheendofthebusiness‐as‐usualphase,andturnsnegativeduringthejointproductionphase‐‐althoughbecominglessnegativeovertimeastheabsoluteamountofcapitallostduetotechnicaldepreciationofgivensizeofstockKAisbecomingsmaller. ThesecondrowofgraphsinFigure2.1showsthecorrespondingeventsforcarbon‐freecapital.Sincetheaccumulationofcarbon‐freecapitalbeginsatthestartofthejointproductionphase, there isnow justonedottedvertical thatmarks thearrivalof thecarbon‐freephase. Itshouldbenotedthatnetinvestmentincarbon‐freecapacityisrapidlyincreasingduringthejointproductionphase‐‐inanticipationofthedropincapacitythatwilloccuratthetimeofarrivalofthe carbon‐free phase, when carbon‐based capital will be discarded. During the carbon‐freephase, net investment in the carbon‐free capital stock is much lower than during the jointproductionphase. In the third row, the left‐most graph depicts the exponential rise of the atmosphericconcentration of CO2 (accumulating from the past flow of emissions) during the business‐as‐usual phase. These continue to rise but at slowing growth rates over the course of the jointproductionphase, eventually stabilizing at the threshold levelwhen theeconomyhas enteredthecarbon‐freephaseofproduction.Thecorrespondingshadowpriceofcumulativeemissionsis negative, since it reflects the social cost of adding an incremental unit to the stock ofatmosphericCO2.Theconstancyofthisnegativeshadowpricethroughoutthetransitiontothecarbon‐freeproductionregimeisastrikingdeparturefromthefamiliaroptimalsolutionresultsobtainedbyNordhaus(1994,2000,2007)withtheDICEmodelanditsclosevariants,aswellasbyothercontributionstotheIAMSclimatepolicyassessmentliteraturethatfindatime‐pathforthe “social cost of carbon” ‐‐i.e., that of the current flow of CO2 emissions ‐‐ that gradually“rampsup”paripassuswiththerisingcumulativestockofgreenhousegasintheatmosphere.45

The explanation for this apparent “anomaly” is simply that, unlike most of the IAMstudies,inourmodelstheeffectsofcumulativeemissionsarenotrepresentedas“damages”thatdirectly and continuously enter the social welfare function. In BAM, the current level ofcumulative emissions, taken in combination with a pre‐specified level of the climate‐tippingpoint (marking the onset of the climate catastrophe that is to be averted), set the remainingpermissible volume of cumulative CO2 emissions. So long as that volume is positive (actualcumulative emissions being below the critical threshold), the complementary slacknesstheoremtellsusthattheproductoftheLagrangemultiplier anditscorrespondinginequalityconstraintsetbythethresholdlevelshouldbezeroatalltimes.Hence,solongasthereis“roomformaneuver” shouldbezero.InoptimumcontroltermsthistranslatesintothefactthatthederivativeoftheHamiltonianwithrespecttocumulativeemissionsequals 0,makingthetime‐derivative of the corresponding co‐state equal to zero too, thereby implying that the co‐

                                                            

45 The reference here is to optimal control simulations using deterministicmodels. Cai, Judd and Lonztek (2012),however,havefoundthattheoptimalsolutionsofstochasticcontrolmodel(SDICE)basedonanannualizedversionofDICE(2007),theoptimaltimepathoftheincremental“socialcostofcarbon”startsoutsubstantiallyhigherthanthatfoundwith the corresponding deterministic version and remains essentially constant, rather than rising graduallywith the passage of time and the rising in cumulative emissions. The intuitive explanation is that faced withuncertaintiesarisinginthefeedbackeffectsoftheflowofemissionsandinthecontrolvariable,andincreasingriskaversion, welfare optimization calls for earlymitigation efforts as a form of “insurance” against the risks of lateradverseshockstotheflowofconsumption.Asispointedoutbelow(insect.5).

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state itself(theshadowprice)shouldbeconstantovertime.46 Aswillbeseen(below, insect.4.2), theforegoingresultsarefoundalso intheextendedversionsofBAMthatrecognizesthatrisingatmosphericCO2concentrationlevelswilldrive“damages”fromtheincreasingfrequencyofsevereweathereventsandclimaticshiftscausingcoastalfloodingandinteriordroughts,allofwhichtranslateintoeffectivelossesofproductivecapacity.47

Figure2.2presentsthecorrespondingBAMpaths found for totaloutput,consumption,alongwiththeirrespectivegrowthrates,andtheindexesof“felicity”(F)andsocialwelfare(W).Note that the the rescaling of the “felicity function” has yielded positive values for both thatindexandW,aswillbeseenfromthebottom‐mostrowofthisFigure.

Figure2.2BAMbaselineresultsforY,C,FandW

The growth rate of output shows some anticipatory reactions to the changes that thearrivalofanewphasewillbring.Forexample,during the jointproductionphase, theaverageproductivityof capitalmust fall, as theamountof relativelyproductive carbon‐based capacitydecreases,andtheamountofrelativelyunproductivecarbon‐freecapacityincreases.Thisholds

                                                            46 Since, by construction, the threshold is not binding during the BAU and the JPR phase, the emission constrainthasn’t been explicitly introduced in the Hamiltonians pertaining to both phases. If they would have been,complementaryslacknesswouldhave‘neutralised’themforthereasonsgivenhere.

47Thedamages,however,arenotmodeledaslossesof“environmentalamenities”whichwouldimpingedirectlyupon“fecility” and therefore be treated as directly damaging social welfare. Instead, they impinge upon welfare onlyindirectly through the optimizing resource allocation adjustments that have to be made to compensate for theanticipatedlossesofthecapitalstockbyloweringpresentandfuturelevelsofconsumption. 

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afortioriforthearrivalofthecarbon‐freephase:whentheremainingcarbon‐basedcapacityisdiscarded, aggregate capital productivity suddenly drops to the level associatedwith carbon‐freecapacity.Inordertomitigatetheeffectsontheconsumptionpathofthecorrespondingdropin output, the buildup of carbon‐free capacity during the joint production phase should bespeededuptowardstheendoftheproductionphase. A similar pattern can be observed (in Figure 2.1) for the buildup of carbon‐basedcapacity during the business‐as‐usual phase, as an increase of the carbon‐using capital stockalsoallows a relativelyhigh rateof investment in carbon‐freecapacityduring thenextphase.This nicely illustrates a general implication of technical innovations that must be physicallyembodied in production facilities, one that often is overlooked in discussions of policies tomitigate CO2 emissions: because capital is a producedmeans of production, a fast buildup ofcarbon‐free capacity will require a large pre‐existing stock of carbon‐using capital – at leastduringitsinitialstage. InFigure2.3wereporttheconsequencesfortheshadowpricesofcumulativeemissionsandofcarbon‐basedcapitalofalteringthelocationofthe“climatecatastrophetippingpoint,”intheBAMsetting.ThissensitivityexperimentvariesthecriticalCO2concentrationlevelovertherange125‐375ppmv,which isconsistentwith temperaturegainsof2‐3degreesKelvinabovepreindustriallevels.FromthelowerleftplotisseenthatwhenistheTPIRCCissetatthebottomend of its range, i.e., at Emax =125‐130 corresponding to an atmospheric CO2 concentrationlevel (405‐410 ppmv) not far so above the 390 ppmv mark reached by the average recordreported for 2011‐12 by the observatories on Mauna Loa (cf. footnote 39), the remaining“emissionsbudget”issosmallthattherearelessthan10yearslefttoaccomplishwhateverhastobedonebeforeendofthebusiness‐as‐usualphase.Thereafter,thedurationoftheBAUphaseincreases almost linearlyand less thanproportionately in relation to the further relaxationoftheconstraintsetbythe“permittedbudget”fortotalcumulativeemissions,buteach“relaxing

Figure2.3SensitivityresultsBAM

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step”willberaisingthemeanglobaltemperaturethatprevailsatthebeginningoftheswitchtocarbon‐free capital formation, and by the time another 9 years have been allowed for the(optimized) BAU duration, the temperature gain in this simple illustrativemodelwould havewillhaveexceedthe“dangerous”2oC. Since the carbon‐free end‐phasehas a constant steady state growth rate,we can limitourselves to looking only 75 years into the future, and thus showing just the first part of thecorresponding time paths following the successful completion of the climate stabilizingtransition. Figures3.1Aand3.1.Bshowinvestment forvaluesof theemissions threshold thatare varied uniformly over this range. Low threshold values are associated with the lowfrequencypart of the rainbowcolor spectrum, andhigher valuesof the threshold correspondwithcorrespondinglyhighercolorfrequencies,endingwiththevioletpartofthecolorspectrumfor thehighest threshold valuewithin the range.Hence, the red lines correspondwitha tightemissionconstraintandthevioletlinesareassociatedwiththeloosestemissionconstraint.

Figure3.1.A :Emax=125‐375 Figure3.1.B :Emax=125‐375

We can observe that as the threshold becomes tighter the arrival of the carbon‐freephaseisspeededup.Figure3.1Ashowstwoblocksoflines.Theleftmostblockisassociatedwiththebusiness‐as‐usualphase.Therightmostblockisassociatedwiththejointproductionphase.Notethattheendpointofaparticulartimepathintheleftmostblockcoincidesintime,butnotnecessarilyinvalue,withtheinitialpointofthatsametimepathintherightmostblock.InFigure3.1.Bthesepointsnotonlysharethesametimecoordinate,butalsothesamevalueofthecapitalstock.Thisisbecausestatesandco‐statescannotjump,whereastimederivativescan.

OneseesimmediatelyfromFigure3.1.Bthatwhentheemissionsconstraintisloosened,thebusiness‐as‐usualphase lengthensand, foragivendurationof the jointproductionphase,thatimpliesandequal‐lenghtpostponementofthearrivaltimeofthecarbon‐freephase.Buttheeffectofrelaxingthatconstraintisthatthephaseofjointproduction(andinvestmentincarbon‐free capacity) alsobecomes longer. There is a strikingdifference, however, between thewaythatvaryingEmaxaffectsthenetinvestmentpatternsforcarbon‐basedcapital,andforcarbon‐freecapital. Astheemissionconstraint is loosened,thewholeof thenet investmentcurve forcarbon‐freecapitalisshiftedupwardsthroughouttheentirejointproductionperiod.Inthecaseof carbon‐based capital, however, a loosening of the emission constraint implies both alengthening of the business‐as‐usual phase and a downward shift of the net investment timepath. At the end of the business‐as‐usual phase, however, the downward shift is more thancompensated by the rise in net investment taking place over a longer stretch of time. Thecounterpartof thissequenceofevents isshown in figure3.1.Gwhichshowsthetimepathsof

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consumption. There we see that the BAU phase allows higher levels of consumption in thebeginning,andcorrespondinglylowerlevelsofnetinvestmentincarbon‐freecapacity.

Figure3.1.C :Emax=125‐375 Figure3.1.D :Emax=125‐375 Theimplicationsforthelevelofthecarbon‐freecapitalstockareshowninFigure3.1.D.With a tighter emission constraint the jointproductionphase comes earlier,while the capitalstockreachesalowerlevelatt=75andhencewillbeloweralsoat ∞.Onceagain,thisisaconsequence of the fact that capital is a produced means of production: if the carbon‐basedcapital stock is limited in size by the existence of more binding cumulative CO2 emissionconstraints, then the implication is that the carbon‐free capital stockswill bemore limited insizeaswell.

Figure3.1.ECumulativeemissions,E:Emax=125‐375

Obviously, more limiting emissions show up directly in the time paths of cumulativeemissions,bothintermsofthevalueoftheendpoints,butalsointermsoftheassociatedtimecoordinate, as can be seen from Figure 3.1.E. Events for output are more interesting, sinceoutputinthemostconstrainedcasereacheshigherlevelsattheendoftheBAUphase,butthenloses out to the less constrained cases, as shown in Figure 3.1.F (below). That twist in therelativelevelsofoutput,however,isnotmimickedbythetimepathsofconsumptionwhichareseenfromFigure3.1.Gtogrowthinparallel.Throughouttheentiredurationofthetransitionitis the less tightly constrained solutions that yield the persisting higher levels of per capitaconsumption.

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Figure3.1.FOutput:Emax=125‐375 Figure3.1.GConsumption:Emax=125‐375

The twist in the relative levelsofoutput, justmentioned, is reflectedby thepatternof

growthratesdepictedinFigure3.1.H.Thevaluesofthegrowthrateofoutputattheendofthebusiness‐as‐usual phase are the same for all threshold valueswithin the range. However, theperiodsoftimeduringwhichthesegrowthratesaffectthelevelofoutputareverydifferent,somuchsothatthepositiveeffectonthe levelofoutput(seeFigure3.1.F)ofanextensionoftheBAUphasewiththerelaxationoftheEmaxconstraintoutweighstheintialnegativeeffectofthatchangeontherateofgrowthofoutput.Thefactthatthemodelconvergestothesamesteady‐state,albeitatdifferentpointsintime,isreflectedbythehorizontallineintherightmostpartofFigure3.1.H

Figure3.1.HGrowthrate / :Emax=125‐375 Figure3.1.IWelfareW:Emax=125‐375

The effect on welfare of less stringent constraints on cumulative CO2 emissions is

positive,asonewouldexpect,ascanbeseeninFigure3.1.I.Inaddition,thetimepathsbecomeclosertogetherintheverticaldirectionwiththerelaxationofthatconstraint.Theimplicationofthis is that both the welfare effects associatedwith the trade‐offs between consumption andinvestmentduringtheearlierBAUphase,andthosewelfareeffectsinthejointproductionphase,becomestrongerwhentighterconstraintsonemissionsshortentheBAUphase.

4.3SolutionresultsforBAM+R&D Thebase lineresults forBAM+R&Darepresented inFigure4.1.The lengthof theBAUphaseandtheJPRphasearenow27.36and13.45years.Hence,thearrivaldateofthecarbon‐freephasehasbeenever so slightlypostponed relative toBAM (TF=40.81 forBAM+R&Dand

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TF=40.17forBAM),butthemaindifferenceisintherelativelengthsoftheBAUandJPRphases.WithendogenousR&D,theBAUphaseislengthenedfromavalueof23.8to27.4years,whereasthe length of the JPR phase is reduced from 16.4 in BAM to 13.4 in BAM+R&D. This is anillustrationofthefactthataccumulationofphysicalcarbon‐freecapitalwithlowproductivityforlonger periods of time (as in BAM) is a substitute for the accumulation of high productivitycarbon‐free capital for shorter periods of time (as in BAM+R&D) plus the accumulation ofproductivitychangethroughR&DpriortotheJPRandCFRphases. Thebigdifferencebetweentheresultsfound(below)inFigure4.1andthoseshownforBAMinFigure2.1isthepresenceofthethirdrowofplotsthatisnowassociatedwiththecapitalproductivity B of the carbon‐free technology: the shadow price of capital productivity risesduringthebusiness‐as‐usualphase,andthenfalls.

Figure4.1BAM+R&DbaselineresultsforKA,KBandE

Thereasonwhytheshadowpricerisesatfirstisthatduringthebusiness‐as‐usualphasethe capacity to produce carbon‐free capital that will embody the new value of the capitalproductivity is rising. This represents an increase of the value of doing R&D. But since theproductivityofdoingR&Dishighforlowvaluesoftheproductivityofcapital,thecorrespondingimpactonR&Dlevelsispositivebut limited,aslongasBisrelativelylow.AsBapproachesitsasymptoticvalue,however,thelevelofR&Dactivityincreasesexponentially,butthishappensata relatively late stage in the business‐as‐usual phase. At the close of that phaseR&D activity

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ceasesandBremainsataconstantlevel(belowitsasymptoticvalue)whiletheshadowpriceofBwillbefallingfromthenonwards.

Figure4.2BAM+R&DbaselineresultsforY,C,R&D,FandW

Figure4.2showstheabsoluteR&Dexpenditures(RD)aswellasthepercentageshareofR&Dexpenditures in totaloutput, fromwhich it canbe seen that theexponential rise inR&DexpendituresattheendoftheBAUphaseisreflectedintheslowdownofthegrowthrateofYattheendof theBAUphase.The retardation inoutputgrowth is traceable to theslowingofnetinvestmentinthecarbon‐basedcapitalstocktoaccommodatetheincreaseinR&Dexpenditures.During the joint production phase the rate of net investment in carbon‐free capacity issignificantlyhigherinthecaseofBAM+R&DthanitwasintheBasicmodel,becausetheeffectofR&Dactivitiesduringtheprecedingphasehasraisedthemarginalrateofreturnoninvestmentincarbon‐freecapacity,relativetothelevelatwhichitwasexogenouslyfixedinthecaseofBAM.

ForBAM+R&Dwehaverunthesamesensitivityexperimentregardingthelocationofthecumulativeemission thresholdaswascarriedout for theBAMmodel. Figure4.3presents theinitial values for the co‐states of cumulative emissions ( , of carbon‐based capital

andof theproductivity of carbon‐free capital , aswell as the length of theBAUphase(Δ andtheJPRphase Δ .

The plots for the initial values of the co‐states of cumulative CO2 emissions, and ofcarbon‐basedcapitalareverysimilartotheoneswehadobtained forBAM,whereas theplotsfor the initial value of the shadow price of carbon‐free capital productivity show that tighteremissionconstraintstendtoraisethevalueofdoingR&D,asonewouldexpect.WealsofindthattheBAUphasehasbeenlengthenedbyacoupleofyears,acrosstheboard.Oneofthereasonsforthisresult is that thestronglyconcaverelationshipbetweenR&Dexpenditures in increases intheproductivityofcapitalembodyingthecarbon‐freetechnologyintroducesatendencyforR&Dinvestmenttobespreadoutovertime.Consequently,thelongertheBAUphase,thehigherthebenefitsthatcanbeobtainedbyspreadingoutR&Dexpenditures.

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Figure4.3SensitivityresultsBAM+R&D

IndiscussingtheoptimalsolutionfoundfortheBAMtransitionpath,itwaspointedoutthat the duration of the JPR phase depended only on the technology parameters, whichremainedfixedintheBAMsetting.EndogenousR&Dactivity,however,causesBtochangeovertime.Asa consequence,theeffectsofvaryingtheTPIRCC’slocationintheBAMmodelarequite

Figure5.1TimepathsofcapitalproductivityB

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different from those found in theBAMcasenotonly in theBAUphase, but in thedurationoffollowing joint productionphase. Figure 5.1 displayswhat happens to thedevelopment overtime of carbon‐free capital productivity in this model when the constraint imposed on CO2emissionsby the climate catastrophe tippingpoint is relaxed.One sees, first, from the spreadbetweenthespreadbetweentheredandthevioletbounds,thatanincreasinglydistanttippingpoint allows thepace of R&D‐drivenproductivity enhancements (duringBAU) rampupmoreslowly,stretchingoutthatfirstphaseofthetransition,butraisingtheterminalvalueofcarbon‐freecapitalproductivity(B).Thelattertranslatesintoaloweraggregatecapital‐outputratioandafasterrateofsteady‐stategrowthduringtheeventualepochofcarbon‐freeproduction.

Toappreciatethecomplicatedcollateralchangesinthebehavioroftangibleinvestmentduring theBAUphase, it ishelpful tostartwithacomparisonofFigures4.1.A,BwithFigures2.1.A, B. From this it is seen that the duration of the (BAU) phase‐‐when net investment isbuildingupthecarbon‐usingcapital‐‐isnoticeablymoreprolongedinthecaseinwhichtheR&Doptionisbeingexercised(BAM+R&D)thanwhenitnot(BAM).Nevertheless,intheBAMmodelahigherandacceleratinggrowthrateofthecarbon‐usingcapitalstockresultsintheBAUphaseendingwiththestockofthattypeofcapitalbeingsubstantiallybiggerthanitscounterpartintheoptimized solution for the BAM+R&D model. With regard to the behavior of tangible netinvestment,therefore,thereisamuchlessabruptpassageinthelattermodelbetweentheBAUphasethefollowing(jointproduction)phaseofthetransitionpath. Lookingnowatthecontrastbetweenthesolutionsforthesetwomodelswithregardtothe effects of variations in the location of the tipping point (TPIRCC =Emax), Figures 3.2.A,BshowsthatintheBAMcasethemoredistantlypositionedistheconstraintonEmax,thelongeristhe “stretch‐out” of net investment in carbon‐using capital, thus deferring the largest annualchanges to the lateryearsof themoreprolongedBAUphase.Whentheoption to improve theproductivityofcarbon‐freecapital throughR&Disbeingexercised,however,capital formationto increase carbon‐based productive capacity continues for a much shorter time. Othercomparisons reveal that the level of net investment in carbon‐based capacity similarly isloweredmoreinthecaseofBAM+R&DthanitisinthemodelwheretheoptiontoinvestinR&Disnotpresent. On the other hand, the effects of varying Emax on the terminal values of the carbon‐basedcapitalstockattheendoftheBAUphasearefoundtobesimilarinmagnitudes,andthetimepathsofthestockofcarbon‐basedcapitalremainpackedtogethermoretightlyinthecaseofBAM+R&DthantheyarewithBAMalone.Forthecarbon‐freecapitalstock,however,therateof net investment under BAM+R&D rises more rapidly over time than under BAM, and theperiodsduringwhichthebuildupofcarbon‐freecapacityisrealized,aresomewhatshorter.

Figure5.2.A :Emax=125‐375 Figure5.2.B :Emax=125‐375

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In all cases, however, the carbon‐free capital stock at the endof the JPRphaseunderBAM+R&D exceeds that of the corresponding terminal value under BAM. But, varying theclimate catastrophe tipping point generates a spread in terminal values for the carbon‐freecapitalstockatt=75thatismuchsmallerintheBAM+R&DmodelthanthosefoundwhenthereisnoR&D.

Figure5.2.Ccapital :Emax=125‐375 Figure5.2.D :Emax=125‐375 Withrespect tothetimepathofemissions,Figure5.2.Eexhibits the“shockabsorbing”effectofexercisingtheR&Doption,dampingtheimpactsofthetighterconstraintsimposedbysettingthetippingpointatlowerCO2concentrationlevels.Fromacomparisonwiththeresultsfor BAM (in Figure 3.2.E) it is evident that the time‐paths of cumulative emissions virtuallycoincident throughout a large part of the somewhat longer BAU phase, and the subsequent“feathering”ofthepathsisnarrower:arangeof40ppmv,comparedwith70ppmvinBAM.

Figure5.2.ECumulativeemissions,E:Emax=125‐375

The more even development over time under BAM+R&D than under BAM is alsoreflectedintheplotsregardingoutput.ThekinkygrowthpatternsobservedunderBAMarefarlesspronouncedinthecaseofBAM+R&D(compare.Figures5.2.Fand3.1.Faswellas5.2.Hand3.1.H),andthisholdsalsoforconsumption(Figures5.2.Gand3.1.G).

OnedifferencebetweenBAM+R&DandBAMisverynoticeablefromthecomparisonofFigures5.2.Hwith3.1.H.Firstofall,duringtheBAUphaseunderBAM+R&DtheaveragegrowthrateofoutputislowerthantheaveragegrowthrateunderBAM.Secondly,underBAM+R&D,theaverage growth rate slows down at the end of the business‐as‐usual phase. During the joint

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productionphase,however, the rangeofvariationof thegrowthrateofoutput ismuch largerunder BAM+R&D than under BAM, while, moreover, the steady state growth rate underBAM+R&D is higher than under BAM and slightly rising as emission constraints become lesstight.Allofthisleadstoamuchmoreevendevelopmentovertimeofconsumption,welfareandfelicity,as canbeseenbycomparingFigures5.2.Gand3.1.GaswellasFigures5.2.I and3.1.I.TheseFigureshighlightthefactthathavingthepossibilitytochangeproductivitythroughR&Dhelpstofightthenegativeeffectsoftighteningemissionconstraints.

Figure5.2.FOutput:Emax=125‐375 Figure5.2.GConsumption:Emax=125‐375

Figure5.2.HGrowthrate / :Emax=125‐375 Figure5.2.IWelfareW:Emax=125‐375

4.4SolutionsresultsforBAM+R&D+UCL:SensitivityExperiments Theresults for thisexperimenthavebeenobtainedas follows.Wehave firstmade theassumption that the high damage depreciation parameters are exactly the same as the lowdamageparameters, inwhich case themodel is reduced to theBAM+R&Dmodel, providing abase‐line against which to gauge the effects of recognizing higher rates of warming‐drivendamages. Thenwe set damage thresholdwellwithin theBAUphase of theBAM+R&Dmodel.Withtheonsetthresholdsituatedat87GTCnetcumulativeemissions fromcurrent levels, thehigh‐damage threshold as reached after 15 years. To perform the sensitivity analysis that isreportedhere,thedepreciationparameterforcarbon‐basedcapacitywasvariedovertherange0.0375‐0.075,i.e.twicetheinitialvalue.Inotherwords,thelargestincreaseintheexpectedrate

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ofannualgrossweather‐relateddamagesconsideredhere is3.75percentofrealgrossoutput,withthe1.875percentbeingthemid‐pointoftheparametervariationrange.48 Because thismodel is so similar to BAM+R&D,we do not report the outcomes of theclimate change threshold here. Rather, we focus on the experiment in which we account forextradamagesforcarbon‐basedcapital.ThedamagethresholdeffectivelysplitstheBAUphaseinto a low damage first sub‐phase and then a high damage sub‐phase. The joint productionphaseandthecarbon‐freephasenowwillnowalsobehighdamagephases.The lengthof theseveralphasesWiththephysicaldepreciationvalueof 0.0375are15.36forthelowdamagesubphaseofBAU(i.e.Δ ),12.0forthehighdamagesub‐phaseoftheBAUphase(i.e.Δ ),and12.83fortheJPRphase(i.e.Δ ). It isclearfromFigure5.3thatarise inthedecayparameter inthehighdamagephasewill bring about a rise (in absolute terms) in the initial shadow price of emissions. This isreflectedalsoinFigure5.4,whichdisplaysthemagnitudesoftheeffectontheshadowpriceofcumulativeCO2emissionsforthefirst15yearsofthetransition.InFigure5.4italsoisevidentthattheabsoluteelevationoftheshadowpriceofCO2emissionsisananticipatoryeffectofthefutureriseintherateoflossesfromthenetstockofcapital,andisconfinedtothelowdamagesub‐phase.FromthehighdamageBAUsubphaseonwards,theshadowpriceremainsunaffectedbyrisesintherateofdecay,suggestinganabsoluteriseoftheshadowpriceascomparedtotheBAM+R&D baseline results. The reason why this absolute rise occurs is that the net rate ofreturn on carbon‐based capital in the low damage sub‐phase has risen relative to the highdamagesubphase.Consequently,thereismoredemandforcarbon‐basedcapitalduringthelowdamage sub‐phase, and therefore also more derived demand for the existing room left forcumulativeemissionsbeforetheclimatecatastrophetippingpointwillbereached. It isclearfromFigure5.3thatarise inthedecayparameter inthehighdamagephasewill bring about a rise (in absolute terms) in the initial shadow price of emissions. This isreflectedalsoinFigure5.4,whichdisplaysthemagnitudesoftheeffectontheshadowpriceofcumulativeCO2emissionsforthefirst15yearsofthetransition.InFigure5.4italsoisevidentthattheabsoluteelevationoftheshadowpriceofCO2emissionsisananticipatoryeffectofthefutureriseintherateoflossesfromthenetstockofcapital,andisconfinedtothelowdamagesub‐phase.FromthehighdamageBAUsubphaseonwards,theshadowpriceremainsunaffectedbyrisesintherateofdecay,suggestinganabsoluteriseoftheshadowpriceascomparedtotheBAM+R&D baseline results. The reason why this absolute rise occurs is that the net rate of

                                                            48Whilethestep‐functionspecificationisanarbitrarysimplification,thechoiceofthisstep‐sizerangeisnotentirelyadhoc.Althoughthestep‐functionspecificationisanarbitrarysimplification,thechoiceofthisstep‐sizerangeisnotentirely ad hoc. One may starting with a continuous specification that makes the proportional (weather‐driven)damage rate (Dw/Y) a positive power‐function of the gain in global mean temparature ()T):Dw Y⁄ ="1 )T)t+"2 )T)"3t.ThisisthespecificationforthegrossdamagerateusedintheAD‐DICEmodel,withtheparametervalues"1=.0012,"2=.0023,"3=2.32.(SeedeBruin,KellyandTol2009:esp.68‐69forthecalibrationproceedureusedtofindthevalueoftheexponentthatwasconsistentwithsimulationoutputfromAD‐DICEgivingthebest fit toDICE simulations – the lattermodel being one that implicitly nets out adaptive benefits against damagecurtailment costs, leaving net(residual) damages. Using the standard relationship for the equilibrium gain intemperature corresponding to an increase in atmospheric CO2, denoted here by (Ct ‐ C0 )= Et , the proportinateincreaseintheconcentrationlevelmaybeexpressedasCt/C0=[(Et/E0)+1],allowingustofind:ΔT* = S[(ln{Et/E0} + 1)/(ln2)], where the equilibrium climate sensitivity parameter S = 3, consistent with themagnitudeacceptedbyNordhaus(2007,2010)incalibratingDICE.Fromthiswefindthefollowingcorrespondencesbetween temperature gains and expected proportional rates of weather‐related losses of carbon‐using productivecapacity(grossoutput,inourlinearproductionsystem): .0012att=0,with)T=1°Kelvin;.0087atc.2010,with)T=1.61°;at.0228with)T=2.53°;at .00517with)T=3.68°. Thecorrespondingrangeofvariationsinthe“high‐damage”rateconsideredinthesensitivityanalysisthereforeisEt=[125,375],orinabsoluteconcentrations(ppmv)levels(Et+280)=[405,655];themid‐pointofthisrangecorrespondsto .00373.Recall,then,thattheconstantUCLratespecifiedbyourmodelwasset(coincidentally)at3.75percentperannum.

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return on carbon‐based capital in the low damage sub‐phase has risen relative to the highdamagesubphase.Consequently,thereismoredemandforcarbon‐basedcapitalduringthelowdamage sub‐phase, and therefore also more derived demand for the existing room left forcumulativeemissionsbeforetheclimatecatastrophetippingpointwillbereached.

Figure5.3SensitivityresultsBAM+R&D+UCL

It isclearfromFigure5.3thatarise inthedecayparameter inthehighdamagephasewill bring about a rise (in absolute terms) in the initial shadow price of emissions. This isreflectedalsoinFigure5.4,whichdisplaysthemagnitudesoftheeffectontheshadowpriceofcumulativeCO2emissionsforthefirst15yearsofthetransition.InFigure5.4italsoisevidentthattheabsoluteelevationoftheshadowpriceofCO2emissionsisananticipatoryeffectofthefutureriseintherateoflossesfromthenetstockofcapital,andisconfinedtothelowdamagesub‐phase.FromthehighdamageBAUsubphaseonwards,theshadowpriceremainsunaffectedbyrisesintherateofdecay,suggestinganabsoluteriseoftheshadowpriceascomparedtotheBAM+R&D baseline results. The reason why this absolute rise occurs is that the net rate of

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return on carbon‐based capital in the low damage sub‐phase has risen relative to the highdamagesubphase.Consequently,thereismoredemandforcarbon‐basedcapitalduringthelowdamage sub‐phase, and therefore also more derived demand for the existing room left forcumulativeemissionsbeforetheclimatecatastrophetippingpointwillbereached.

Figure5.4ShadowpriceofCO2emissions Figure5.5Carbon‐freecapitalproductivity,B Notefurther,thathigherdamagesalsomakesitattractivetoallocategreaterinvestmentresourcetoR&Dactivity,whereas,duetothehigherexpectedratesofdamagesinthefuture,theeffectiverateofreturnoninvestmentinbuildingfuturecarbon‐basedproductivecapacitywillbereduced, leadingtoaslowerrateofphysicalcarbon‐freecapitalaccumulation.Thenegativeeffectsofthisonlong‐termcarbon‐freecapacitycanbemitigatedtosomeextentbymakingthelowervolumeof carbon‐freecapacitymoreproductive– throughgreater investments inR&D.ThatisexactlywhatcanbeobservedfromFigure5.3:oneseesthatthathigherweatherrelateddamages tend to rotate the time path of carbon‐free capital productivity upwards, while thearrivalofthejointproductionphaseisbroughtforwardintime,albeitonlyslightly. Ahigherrateofdecayalsomakesthebusiness‐as‐usualphaseslightlyshorter,especiallybecause the low damage sub‐phase decreases in length. The latter results from some of theinvestmentincarbon‐basedcapacityduringthehighdamagesub‐phasebeingbroughtforwardintime,ascanbeseeninfigure5.6.A.InthatFigure,weseethattheviolettimepathsareontopof the collection of low damage sub phase time paths, whereas during the high damage subphaseoftheBAUphase,theyareatthebottomofthecollection.49Ahigherrateofcarbon‐basedcapital accumulation in the low damage sub‐phase ultimately implies a higher volume of thecapital stock at the end of the lowdamage sub‐phase. Thiswould lead to a higher volume ofcapitaltobediscardedatalaterdate,unlesstheaccumulationprocessitselfstopsearlierthanbefore. Hence, during the joint production phase, carbon‐based capital is seen to depreciateconsiderablyfasterthaninthepreviousexperiments.Theforwardshiftintimeofcarbon‐basednet investment shows up as a distinct kink in the high decay parameter time paths. For lowdecayparametersvalues, theBAM+R&D+UCLresultsareveryclose to theBAM+R&Dbaselineresultsthatareidenticaltothereddesttimepathinfigure5.6.B. Theresultsfornetinvestmentincarbon‐freecapacityexhibitverydifferentbehaviorforlow and for high damages.When damages are low, the rate of capital accumulation is risingexponentially over time. When carbon‐based capital damages are high, the rate of net

                                                            49Notethattheredtimepathdoesn'tshowabreakatall,becauseitreflectsthebaselineoftheBAM+R&DmodelinwhichtherewasnodifferencebetweenthelowdamageandthehighdamagesubphaseoftheBAUphase.

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investment incarbon‐freecapacity is initiallyhigherthan inthe lowdamagephase,butat theendof the jointproductionphase therateofnet investment isevenslightly lower thanat thebeginning.Thereasonisthatduetoincreaseddamagesofcarbon‐basedcapital,theopportunity

Figure5.6.A : 0.0375 0.075 Figure5.6.B : 0.0375 0.075

Figure5.6.C : 0.0375 0.075 Figure5.6.D : 0.0375 0.075

ThepatterninnetinvestmentobservedinFigure5.6.CisreflectedinFigure5.6.D,whichshowstheslightlyhighercarbon‐freecapitalstockinthebeginningofthejointproductionphaseforthehighdamagetimepaths,althoughafterawhilethosehighdamagetimepathsshowlevelsforthecarbon‐freestockofcapitalabovethoseforthelowdamagetimepaths.Underthesehighdamageconditions,thearrivalofthecarbon‐freephaseispostponedsomewhat.

From Figure 5.6.E one can see that the extra net investment in the beginning of thebusiness‐as‐usualphaseunderhighweather‐relateddamageconditionsrotatesthecumulativeemissiontimepathupwardsforthelargestpartofthephasesbeforethecarbon‐freephase.Therate atwhich the flowof emissions decreases on the highdamage time paths is, however, solargethattheupwardshiftinthebeginningofthecurveismorethancompensatedattheendofthecurve,leadingtoanintersectionofthehighdamagecumulativeemissionscurveandthelowdamagecumulativeemissionscurveafewyearsbeforethebeginningofthecarbon‐freephase.

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Figure5.6.ECumulativeemissionsE: 0.0375 0.075

ThetimepathsforoutputareprovidedinFigure5.6.F.Thereisamajordisruptionwhenthe carbon‐free phase begins, but this doesn't really affect the development over time of thecorresponding consumptionpaths, aswehave seenbefore, andnowalso inFigure5.6.G. Theonly slight hiccup in consumption occurs at the end of the high damage sub‐phase of thebusiness‐as‐usualphase.

Figure5.6.F.Output: 0.0375 0.075 Figure5.6.G.Consumption: .0375 .075

Figure5.6.HGrowthrate / Figure5.6.IWelfareW: 0.0375 0.075

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Figure 5.6.H shows the time paths of the growth rate of output. For the high decayparameterpathsthedropinthegrowthrateofoutputatthebeginningofthejointproductionphase is much higher than for the low decay parameter time paths. Nevertheless, welfare ishardlyaffected,ascanbeseenfromFigure5.6.I.

4SummaryandSomeConcludingComments

Thispaperhaspresentsamultiphasetransitionmodelthatcanbesolvedtoobtaintheoptimum timing and magnitudes of tangible and intangible investments that are required toeffecttheintegratedsystem’stransitionfromacarbon‐basedeconomytoonewhoseproductionactivitiesareessentially“carbon‐free”‐‐inthesenseofhavingceasedtogenerateCO2emissionsexceedingtheEarth’snaturalabatementcapacity‐‐andthereforewillhavestabilizedtheglobalclimate system. That transition is shown to be optimal when it is completed just‐in‐time,becauseourmodelingapproachpositstheexistenceofacatastrophictippingpointintheEarth’sclimatesystem.Tohave failed tostopshortof that critical threshold,which isdefined for thepurpose of our models in terms of an exogenous terminal bound on the atmospheric CO2concentration level, would have initiated the onset of irreversible runaway global warmingfeaturingabruptclimatechangesthatwouldcreateanenvironmentinhospitabletothesurvivalofcivilization,andthegreaterpartoftheEarth’shumanpopulation.

Onthelatterpremise,forwhichadvancesinclimatescienceandpaleoclimatologynowprovides a considerable measure of support, asserting the relevance of the precautionaryprinciplefordesigninganappropriatepolicyresponsesufficestorecasttheeconomicproblemofclimatestabilizationasthatofeffectingawelfareoptimizingtransitionfromacarbon‐basedeconomytoazeronetCO2emissionproductionregimeintimetoavertdrivingtheatmosphericconcentration to the catastrophe tipping point. A technical virtue of this reformulation is itsexplicit acknowledgment that the behavior of the coupled economic and geophysical systemsbeyond the tipping point would be marked by the high frequency recurrence of abruptdiscontinuities in temperature and economic and demographic system‐dynamics. Propertheoreticalmodelingwould include thosenon‐convexities, and thus vitiate the use of optimalcontrolanalysis.

In addition avoiding that particular cul‐de‐sac, this paper has taken an analyticalapproach that diverges in a secondway from the one that has come to characterize researchcontributionsinthefieldofintegratedassessmentmodeling(IAM)ofclimatepolicy.Itaddressesthe problem of finding the sequence of technological options whose development anddeployment would be required to stabilize the global climate in the finite but endogenouslydeterminedtimespanofanoptimaltransitiontoanessentiallycarbon‐freeregimeofproductionthatiscompatiblewithsustainableeconomicdevelopmentandcontinuingeconomicgrowth.

Acentralfeatureinourapproachtothatproblemisthesuppositionthatthetransitiontowardsthedesiredstateofcarbon‐freeproductionwillrequireaswitchinthedeploymentofalternative capital‐embodied technologies. This will entail building carbon‐using productionfacilities that are sufficient to allow the production of enough capital of the kind in whichcarbon‐free technologies are embodied so that the latter can completely displace thetroublesomeCO2‐emittingproductionfacilities.Accordingly,wehaveoptedforabasicgrowthmodel setting that features two distinct classes of technology, both of which can producemalleable(homogeneous)outputfromwhicheithercarbon‐basedorcarbon‐freecapitalgoodsmaybeformed.Utilizingtheformerclassofproductionassets,however,resultsinthereleaseofCO2 emissions in excess of the natural abatement capacity of the Earth’s forests and oceans,

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therebycontributingtoglobalwarming.Weassume,notunrealistically,thatattheoutsetofthetransitionprocessthe“dirty”(carbon‐using)modeofproductioncharacterizedbyahigherlevelof productivity than the “green” alternative: its technologies offer average unit capital costslowerthanthoseforproductionfacilitiesthatembodytheavailablecarbon‐freetechnologies.

Thesetwo(linear)technologyoptionsareintroducedintotheotherwisesimplesettingof the AK endogenous growth model by Rebelo (1991), but complicating the latter by theimplied requirement that a technical change in the state of the production system requires aspecifickindoftangiblecapitalformationtoimplementthe(carbon‐free)alternativetechnology.Thisisadistinctivefeatureofthemodellingapproachpursuedhere,andrestssquarelyonthetechnologicalpremisethatembodimentinphysicalcapitalgoodswillbenecessarytoimplementswitchesnon‐carbontechniques,theincreasingrelativediffusionofwhichwouldtendtolowerthe global production regime’s overall carbon‐intensity more rapidly the more closely theaverage productivity of carbon‐free facilities came to approach (and surpass) that of carbon‐basedcapitalstock.

The “embodiment” assumption and its implications are especially appropriate in ourview, and perhaps more importantly that expressed by engineers with experience‐basedexpertisewith innovationprocesses in energy supply systems. Yet, our explicit recognition ofthisconstitutesanimportantdepartureofthepresentanalysis(anditsaccountofthedynamicsof the optimal climate‐stabilizing transition path) from the way in which the effects ofendogenous technological change have been treated in previous economic contributions tointegrated modelling of climate policy measures – where assuming that technical change isdisembodiedhaslongremained.

Focusingonsequencingthedeploymentofdifferentcategoriesoftechnologypolicy,ourimposition of minimally restrictive efficiency conditions leads to a multi‐phase optimizationframeworkinwhichtheBasicModeloftechnologyswitchingrecognizesthreeseparatephasesof production, each of them characterized by different technologies or combinations oftechnologiesbeingactive. It isshownthatduetothe linearproductiontechnologiesemployedtherewillalwaysbeanactivetangiblecapitalformationprocesstoimplementoneortheotherof these technologies ‐‐ even though output can be produced using both technologiesconcurrently as long as there is still room to emit CO2 from the facilities that use energyobtainedbyburningfossilfuels.Thefirstofthethreephasesiscalledthe‘Business‐as‐usual’orBAUphase, inwhich carbon‐based capacity is still beingbuilt up, and theoutput produced iscompletelycarbon‐based.Atsomepointintimethenextphasearrives,inwhichinvestmentincarbon‐basedcapitalceasesandthebuildupandconcurrentuseofcarbon‐freecapacitybegins.During that phase, production using carbon‐based capital continues but the production levelfalls over timeas the capital stock isworndowndue to technical decay.The secondphase iscalledthejointproductionphase(JPRphaseforshort),asbothtechnologiesareusedtoproduceoutput.Atthebeginningofthethirdphase,calledthe‘carbon‐free’phase(CFRphaseforshort)the remaining carbon‐based capacity is scrapped and production is from then on completelycarbon‐free:thegreenfuturehasarrived(just‐in‐time).

We extend the basic three‐phase BAM setting in two ways. First we introduceendogenous R&D driven technical change that improves the productivity of the carbon‐freetechnology up to the moment of the installation of the first unit that embodies the newtechnology.Thereasontodothisisthattheembodimentoftechnologyinfixedcapitalimpliesthatthebuildupofsignificantstocksofcarbon‐freecapitalwilltaketimewhichmayturnouttobeinshortsupplyontheonehand,whileontheotherhanditwillalsodrawupontheexistingcarbon‐basedcapacityattheexpenseofconsumption,whichisthesolesourceofwelfareinthe

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standardAK‐modelandinourmodeltoo.Tokeepmatterssimple,wemaketheassumptionthatR&Ddriventechnicalchangestopsthemoment thenewtechnologystarts tobe implemented. Thebuild‐upofcarbon‐freecapacitytoalevelthatissufficienttomitigatetheaggregateproductivity drop associated with the scrapping of remaining carbon‐based capacity at thebeginning of the carbon‐free phase will take time. That fact is one of the main reasons forfocusing attention on the optimum timing of the changes between thedifferent phases of themodel. The specificationof thewayR&Dalters theproductivity of capital embodying carbon‐freetechnologies,whenincorporatedinanAK‐settingwithalimitingconstantmarginalproductof capital, generates productivity levels for the class of “alternative” technologies that arebounded from above (asymptotic technical change) ‐‐ reflecting the ‘fishing‐out’ effectrecognized inmuch of the endogenous growth literature since Jones (1995). The latter effectseemsespeciallyrelevantinregardtothefurtherdevelopmentofaspecificclassoftechnologies,as opposed to innovation and realized technical changes occurring at the macro‐level asconsequenceofexpansionoftechnicalconstraintsaffectingabroadanddiversearrayofgoodsandprocesses.

ThesecondextensionoftheBAMmodelpertainstotheintroductionofaweatherrelateddamage threshold. For this purpose we proceed simply, bymaking the assumption that highratesoflossofproductivecapacityduetoexpected,butunscheduledratesoflossinproductivecapacity will occur after the system reaches an “extreme weather tipping point” (defined intermsofmeanglobaltemperature,or,equivalently,aCO2concentrationlevel).Thelatterissetbelow the climate catastrophe tipping point, and triggers an upward jump in the expectedproportionalrateoftechnicaldecayofthecapitalstock.Theeffectistantamounttoraisingtheannualrateofphysicaldepreciationofthecapitalstocks.

Thetimingofthephasechangespresentinthevariousmodelversionsaregovernedbytransversality conditions following from the optimality condition that the values of theHamiltoniansmustbe identicalwhenevaluatedat themomentof thephasechangeunder theconditionspertainingtothephasesjustbeforeandjustafterthepassagebetweenphases.IntheBasic Model (without R&D and extra weather related damages) the condition defining theoptimum length of the BAUphase turns out to be the requirement that the shadowprices ofcarbon‐based and carbon‐free capacity should be the same at themoment investment in thelattertechnologytakesoverfrominvestmentintheformer.Thismakesperfecteconomicsense,astheopportunitycostofaunitofinvestmentintermsofconsumptionforegoneisthesameinboth cases. For the timing of the change from the JPRphase to theCFRphase, the optimalitycondition results ina transversality condition thatstates that themoment that theCFRphaseshouldbeginisdefinedbytherequirementthatthebenefitsofcontinuingtousecarbon‐basedcapacity (the utility value of its output) is outweighedby the cost of doing so (the (negative)utilityvalueofthecorrespondingemissions),whichcloselyresemblesthe‘negativequasi‐rent’scrapping condition known for a long time now from the vintage literature with fixed factorproportionsexpost(i.e.putty‐clayandclay‐claymodels).50Fortheotherversionsofthemodel,morecomplicatedtransversalityconditionsariseoutofthesamegeneraloptimalityconditionspertainingtotheequalityofHamiltoniansatthemomentofaphasechange.

BAMcontainstwofurthertransversalityconditions,i.e.thestandardoneassociatedwiththepureAK‐settingoftheCFRphase51andtheoneimpliedbythefactthatfromthestartofthe

                                                            50Cf.Johansen(1959)andSolowetal.(1966).

51IntheendogenousR&Dversionofthemodel,stillanothertransversalityconditionisthatastimegoestoinfinity,thepresentvalueofthewelfarevalueofcarbon‐freecapitalproductivityshouldapproachzero.

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CFRphasetheshadowpriceofcarbon‐basedcapitalshouldbezero,ascarbon‐basedcapacityisworthlessfromthenon.Usingtherequirementofthecontinuityofstateandco‐statevariablesat themomentofaphase change, the transversality conditions in combinationwith thegiveninitial and terminal values for the state variables in the model allow us to use a (steepestdescent)searchmethodthat,forapriori“guesses”ofthestillmissinginitialvaluesofasubsetofco‐states and of the phase lengths of the BAU and the JPR phases, solves the systems ofdifferentialequationsresultingfromthefirstorderconditionsoftheHamiltonianproblemsforeachoftheindividualphases.Giventhissolutionthatiscontingentontheaprioriinitialvalues,wecanevaluatetotheextenttowhichthedifferenttransversalityconditionsandtheboundaryconditionforcumulativeemissionsaremet.Ifoneormoreofthesetransversalityconditionsortheboundaryconditionarenotmet,theinitialvalue(s)needtobeadjusted;ifalltheconditionsaremetthentheoptimumpathhasbeenfound.

WehavecalibratedtheparametersandinitialvaluesoftheBAMmodel,basedtoalargeextentonNordhaus’2010(DICEand)RICEmodeldatasetandaddedsomeaprioriassumptionsregarding thecapitalproductivityparametersand theparametersof theR&D function. Usingthis setup, we have run a number of simulations to investigate the sensitivity of the optimalsolutionsofthevariousmodelversionstochangesinthestructuralparametersfeaturingintheutilityfunctionandintheproductionfunctions,whichallshowedtheexpectedoutcomesknownfromthestandardendogenousgrowthmodels.

We then turned to a sensitivity analysis that involved the location (on the scale ofatmospheric CO2 concentration levels) of the tipping point for irreversible runaway climatechange,andthelowertippingpointfortheonsetofgreaterexpectedratesofextreme‐weatherdamagestoproductivecapacity.Inthesimulationwherewesystematicallyreducedtheclimatechange threshold in order to see what a more stringent application of the precautionaryprinciplewouldmean,wefindthatinBAMitwillbeoptimaltoaccumulatecarbon‐basedcapitalatafasterpacethanwithalesstightcumulativeemissionsthreshold,sothatthearrivaloftheCFR phase is speeded up. To facilitate the latter, a quick buildup of the carbon‐based capitalstockisrequiredtobeabletoswitchrelativelyearlytoinvestmentincarbon‐freecapacityandtoenable considerable ratesofboth investmentandconsumptiononce investment in carbon‐basedproductionand lateronproductionusingcarbon‐basedcapacityhasceased.Tighteningthe cumulative emissions constraint raises (in absolute terms) the shadow price of CO2emissions, but also that of carbon‐free capital. But somewhat unexpectedly perhaps, it alsoraises the shadow price of carbon‐based capital, simply because the value attributed to thecarbon‐basedcapitalstockisforanimportantpartderivedfromthevalueofthecarbon‐freestockthatitisabletoproduce.

WhenweallowforendogenousR&Dreactionsinthissetting,weobservethatthelengthoftheBAUphaseincreasesrelativetotheBAMcase.ThelatterallowsR&Dactivitytobemoreevenlyspreadovertime,which,duetotheconcavityofcarbon‐freecapitalproductivityinR&Defforts raises the overall effectiveness of a given R&D budget. There is also more time toaccumulatecarbon‐basedcapital,sothatattheendoftheBAUphasetherecanbeaconsiderableterminalvalueofcarbon‐basedcapital: theextensionoftheBAUphaseallowstheeconomytoeatitscarbon‐basedcakeandstillhaveaconsiderableamountofit leftatthebeginningoftheJPRphaseintheformofcarbon‐basedcapital.IncontrasttotheBAMcase,theJPRphaseisnowshortenedasthecumulativeemissionsthresholdbecomestighter.TheoveralleffectisthattheCFRphasecomesslightlyearlierthaninBAM,whilethedispersioninthewelfareeffectsislessthanunderBAM: thepossibility tochangeproductivity throughR&Dhelps to fight thenegativewelfareeffectsoftighteningemissionconstraints.

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Twoquantitative aspects of theoptimal transitionpaths found for these threemodelsshould not passwithout notice here: both aspects are exhibited in the following comparativecompilationofthedurationsfoundforthephasesoftheclimatestabilizingtransition.Thefirstandperhapsthemoststrikingfeatureoftheseresultsisthecloseconsilienceofthelengthofthecompleted transitions that emerge from the welfare‐optimizing solutions of the differentmodels. As seen from the bottom‐most line of the tableau, the durations of the optimizedtransition remain tightlyclusteredaroundthe40‐yearmark, theunderlyingshifts inBAUandJPRphases’absoluteandrelativelengthsnotwithstanding

Thesenumbersgiveausefuldegreeofconcrete‐nesstothepointmadequalitativelyinSection 4’s discussion of the effect that introducing the R&D option has in extending thebusiness‐as‐usualphase,whilecompressingthetimetaken(duringthejointproductionphase)to implement of the actual switch from carbon‐using to carbon free production capacity.Comparing the results for BAM with those for BAM+R&D in order to gauge the effect ofexercising the R&D investment option, the right‐most column shows the largely offsettingmovements of the two component phases. Introducing R&D investments stretches out theoptimaldurationoftheBAUphase,vis‐à‐visthatfoundwiththeBasicModel.But,theireffectinraising the average productivity (and, by that token, lowering the unit capital costs) of thesubsequently installed carbon‐free capital stock, in combination with the somewhat largercarbon‐usingcapitalstockthatcanbebuiltupduringtheextra4.4year,permitsafasterswitch‐overduring the JPRphase.R&Dthereby lessening the transition’ tollonconsumptionand thelevelofsocialwelfare.52 It is important to emphasize the general insight that the welfare‐enhancing effect ofintroducingR&Dactivitiesinthisplanninganalysisframework,and,moregenerallyofoptimallydeployingstillotheroptions fromthe largeranddiverseportfolioofavailable techniques thatwillmitigateCO2emissions,arepropertythatisthedualoftheessentialconstancythatisseenin the endogenously determined transition durations. This is not to say that either thespecificationsof theseveralmodelsor theircalibrationswereselectedwithaviewtoyieldingthecommonpropertiesoftheiroptimalsolutionsthathavebeenhighlightedhere. Theclusteringofthelengthsofthetransitionsintheneighborhoodofthe40yearmark,forthisgroupofmodelshasacausethatmaybegraspedintuitively.Eachmodelstartsfromthesame implicit level of atmospheric CO2 concentration and has the identical “budget” for

                                                            52TheidenticaldurationsoftheBAUphaseinthemodelswithR&DaddedtotheBAMmodel,whetherornotallowanceismadeofweather‐related“unscheduledcapacityloses”(UCL)reflectstheassumptionthatR&DactivitiesdonotentailCO2emissions,andthusdonotdirectlycontributetoexhaustingthebudgetofallowableemissions..Further,whileAnticipatedcapacitylossesfrominthe“highdamage”sub‐phaseofBAUactuallywilllowerthetheannualflowsofoutputandemissions,additionalcarbon‐basedcapitalformationoverBAUphasecompensatesforthat.TotheextentthatR&Ddoesusefossilfuel‐basedhightemperatureovensandelectricitysupplies,therisingtrendoffinalgoodsandservicesavailableforconsumptionisoverstated,especiallyinthelatteryearsofBAU.

Comparison of Alternative Models’ Optimal Phase Durations (years) 

                Models:  Phases:                     v   

BAM+R&D+UCL  BAM+R&D  BAM  {BAM+R&D} – BAM 

Effect of R&D: 

Business‐as‐Usual: BAU  27.36  27.36  23.80  + 3.56 

Joint Production: JPR  12.38  13.45  16.39  ‐ 2.94 

Completed Transition:                              BAU+ JC 

39.74  40.81  40.17  +0.64 

  Source:Seetext,above:sub‐sects.4.3para.1and4.4para.2.

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permissible further cumulative emissions –because the social planner must in every casecompletethetransitionbeforereachingthesame“tippingpoint”(atc.447ppmv), inordertoavoid exceeding a globalmean temperature gain of 2o Kelvin above pre‐industrial levels. Theuniformity of the “hard budget constraint” in regard to net emissions generated during thetransitionisapropertyofthecommonclimatesystemthatonemustpositinordertorendertheoptimizationresultscomparable.Obviously,thisconditionholdsforany[start,stop]pairofCO2concentrationlevels.Therefore,thewaytoseetheessentialsofwhatisgoingonintheseDIRACtransition path‐optimization problems is this. The social planner will maximize the globalpresentvalueofeconomicwelfarebytakingwhateveractionsarerequiredtospendthatbudgetefficiently in exploitingwhatever technologicalmeans already exist or canbemade available.Butshealwayswillcontrivetohavecompletelyexhaustedthefixedemissionsbudgetjustbeforetheeconomyentersthephaseofzero(net)emissions.Giventheanticipatedconstraintsimposeduponoptimizationby thedynamicsof theclimate system,better technological tools thereforewill be translated into greater real income and increased social welfare. Of course, if theemissions budget already had been squandered before the planner arrived on the scene, thegamemightwellbeoverbeforeanymeliorativeactionscouldbetaken. Themainpurposeinbuildingandanalyzingmodelssuchasthosepresentedhereistoexpose and illuminate thedynamic interrelationships entailed in those actions, rather than topredicttherequireddurationofthecompletedtransition,orofitsconstituentsub‐phases.But,comingnow to the second striking aspectof themodels’ outcomes in that regard, one cannothelpbutbesurprised(andnota‘tickled’)tofindthatthe“required”transitionperiodonwhichthey concur lasts four decades from the t=0 point,which our calibrations set to correspondwith2010(see,above,footnote41).Thus,ourmodelsturnouttobealignedinthatrespectwiththeconclusionsofrecentcomprehensivestudiesofthefeasibilityandextentofthetechnologicalandphysicaltransformationinmodernenergysystems,whichhavefocused2010‐2050astheperiodduringwhicha concerted technologicaleffort couldmanage toend the contributionofanthropogenicGHGemissionstoglobalwarming.53 When we take account within this framework of the greater expected frequency ofwarming‐driven “extreme weather” damages to the flow of real output from vulnerableproductive capacity, such that the rate of unscheduled losses of KA is irreversibly shiftedupwardsatsomepointduringtheBAUphase,anextraphaseisaddedtotheBasicModel’sthreephases. The extended version of the Basicmodel then has a low damage BAU phase, a highdamageBAUphase, and (high damage) JPR and CFRphases.54 Theweather related damages

                                                            53 For example, the National Research Council (2010) report of its Committee on America’s Energy Future madedetailedtechnicalassessmentsoftheenergy‐supplyandend‐usetechnologiesthatwerejudgedmostlikelytohavemeaningfulimpactsontheU.S.energysystemduringthethreetimeintervals:2009–2020for(deploymentofexistingtechnologies,2020–2035fordevelopmentofnewtechnologies forcommercialdeployment),2035–2050for furtherdevelopment andwidespread deployment of advanced technologies (cf. Full Report, pp.134‐135. Amember of theAEFCommittee, RobertW.Fri(2013:p.6), remarkingmorerecentlyonthescaleandcomplexityoftheU.S.energysystem,hasstressedthepointthat“afewdecadesisnotaverylongtimetooverhaul it tothepointwhereitemitsessentiallynogreenhousegases”....Butitisdifficulttocomeupwithahigh‐probabilityscenariothatdoesnotexhausttheemissionsbudgetby2050.”Weviewthesecoincidingreferencesto2050asthetransition’sendpointtobelittlemorethan“coincidental.”

54 This admittedly simplistic specification – which avoids creating still more phases (needed to accommodate anumberofdiscreteupwardstepsinthecapitallossrate)mayberationalizedalongthefollowinglines.ThelocationoffixedinfrastructurefacilitiesintheBAUwillbelegaciesofapreviouserainwhichsealevelsandmoisturelevelsintheatmospherewerelower,andexistingKAwillnothavebeendesignedtoreducetheirvulnerabilitytothedistributionofextremeweathereventsthatwillensuefromthewarmingthatalreadyhasoccurred.Moreover,furtheradditionsto

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introduce an additional incentive to engage in R&D. The reason is that the weather‐relateddamagespertaintophysicalcapital,whiletheproductivityofunaffectedunitsofcapacity(andmore importantly that of the carbon‐free capital stock to be built in the future)will remainuntouched.Consequently,intheseanticipatedcircumstances,investmentsdirectedtoraisetheproductivity of carbon‐free capital is a naturalmeans ofmaintaining a reasonable net capitalformation. In addition to the extraR&Dactivity,we also observe extra investment in carbon‐basedcapacityduring the lowdamageBAUphase.The reason is that theexpectationof extraweather‐relateddamagesintroducesawedgeintherateofreturnoncarbon‐basedinvestmentduringthelowandhighdamagesubphasesoftheBAUphase.Anincreaseintheextradamagesrate also leads to a more even distribution of carbon‐free investment over time, in order tomitigate thenegativeeffectsonconsumptionof increasedweather relateddamages.With lowdamagerates,investmentincarbon‐freecapacityduringtheJPRphaseisstronglyincreasinginanticipation of having to cushion the drop in output associated with the scrapping of theremainingcarbon‐basedcapitalstockatthebeginningoftheCFRphase.Withhigherdamages,thedropwillnecessarilybeless,ceterisparibus.

Concludingremarks:So,whatarethemainlessonstobedrawnfromtheforegoingwelterofspecificdetails

about theoptimal transitiondynamicsdisplayedby thesevery simplemodelsofaglobaleco‐climate system? First, one must stress the general point that the embodiment of technicalchangeinphysicalunitsofcapitalunderlinesthepracticalimportanceofthenotionofcapitalasaproducedmeansofproduction.Recognitionof thispoints totheneedtobuildormaintainacarbon‐based capital production system that is adequate to produce the right amount andquality of the carbon‐free production units on which future welfare will exclusively come todepend; and to do so within the remaining time‐span allowed by the accumulating stock ofatmospheric CO2. The results also draw attention to the implication that taking into accountembodied technical change in the design of a “tech fix” climate policy a worryingly shortdurationfortheBAUphase.

Secondly, R&D emerges from this analysis as an important means of cushioning thenegativewelfareeffectsoftighteremissionthresholdsandincreasingweatherrelateddamages.Thereason for this is thatphysicalcapital investment incarbon‐basedcapacity isat thesametimebothasubstitute forandacomplementofR&Dinvestment. It isasubstitute fromapureproductionpointofview,whileitisacomplementbecauseoftheembodiednatureoftechnicalchange that needs physical investment to turn potential productivity improvements into realones. Because the return to R&D and physical investment in carbon‐free capacity dependpositivelyoneachother,output itself,andthereforeconsumptionand investmentpossibilitiesarepositivelyaffectedbyhavingthepossibilityofengaginginR&D.Hence,R&Deffortsprovideanadditionalmeanstobothreducethedispersioninwelfareoutcomesandtomaintainorevenincrease future carbon‐free output levels and growth rates in the face of increasingly volatileweathereventsandcorrespondingdamagesandarisingprobabilityofrunawayglobalwarming.

                                                                                                                                                                                          thestockofcarbon‐basedinfrastructurearelikelytocontinuetobesitedessentiallyatthesametypesoflocations,toexploitexistingengineeringdesignslowercostsofservicingexistingdirectlyproductivecapacity. WhileKB ,beingnewlyformed,couldbesitedsoastobelessvulnerabletotheweathervariationsinthealteredclimateconditions,uncertainties combinedwith greater costs could combinewith increasing severity of storms and flooding (at stillfurther elevated mean global temperature) not only throughout the JPR and CFR phases. A later specification,however,will do awaywith thenotionofone (ormore) temperature triggerpoints for increasedexpected capitallosses,bymakingthelatteracontinuousfunctionoftheendogenouslevelofmeanglobaltemperature.

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Thirdly, the idealized “optimal planning” framework for endogenousmacroeconomicgrowth seen tobewell suited to incorporating representationsof the technical aspectsof thearrayofexistingandpotentialtechnologicaloptionsandtheirrespectiveresourcerequirements,aswell as those required tooperationalizing awelfare‐optimal transitionpath.Application ofmulti‐phaseoptimalcontrolanalysisprovidesDIRAMsolutionsthatdescribetheoptimalflowsoftangibleandintangiblecapitalformation,alongwiththeproductionflowsusingcarbon‐basedor alternative technologies in each of the phases, as well as the sequencing and respectivedurationsofthelatterwhichcompletelydescribethecompletedtransitionpath.

This very concreteway of setting outwhat has to be accomplished technologically ineachof thephases, in our view, canprovideauseful startingpoint for thinking abouthow todesignandcoordinatethemultiplicityofdiversetasksthatwillneedtobeundertakeninorderto adequately respond to the daunting existential challenges posed by global warming andclimate instability. More intricate and computationally demanding modeling and analyses oftemporallyextendedmulti‐phasetransitionpathssurelywillbenecessarytoshedgreaterlighton thecomplexitiesentailed inworkingout theproperdynamicsequencing for the integrateddevelopmentandexploitationofthevarietyofcomplementaryandcompetingtechnologypolicyoptions‐‐beyondtheveryfewthathavefiguredintheprecedingpages.Amongthemyriad“ofotheroptions”thatremaintobeconsidered,thefollowinghandfuldeserveprioritypositionsontheagendaforcontinuedfutureDIRAMresearch–byvirtueoftheirvariedfunctionalattributes:

(i) harvesting “low‐hanging fruit” ‐‐ by investments that implement core engineeringknow‐how to incrementally retrofit existing carbon‐burning infrastructure and directproductive capital goods, allowing these to switch to the “less dirty” fossil fuels as energysources (e.g.,naturalgas, inplaceofoil forheatingand(diesel) internal combustionengines),and/orupgrade theenergyefficiencyofexistingbuildingsandproductionequipment,and thesamemightwellbesaidforcurtailingreleasesofmethaneandotherpotentGHGs;

(ii) R&D expenditures on risky exploratory research programs that have longer‐timehorizonsthanthenormforappliedprojects(suchasthoseconsideredhere)becausetheyseeklow‐frequency“break‐through”discoveriesandinventionsthatwoulddrasticallylowertheunitcapitalcostsofreliableproductionfacilitiesthatusealternative,non‐carbonenergysources;

(iii)Long‐termresourcesupportforexperimentalgeo‐engineeringprojectsthatworkinparallel on the development and field‐testing of safely scalable “back‐stop” technologies foratmospheric carbon capture and sequestration (ACCS), and locally deployable solar radiationmanagement(SRM)techniques.55

(iv) Capital formation for reforestation with fast‐growing leafy trees in order toefficientlyraisethenaturalcapacityCO2abatementcapacityoftheEarth’spresentforestcoveras far as possible. Although it is highly unlikely that this measure could compensate for thedegraded abatement capacity of the oceans that would result from continuing warming,

                                                            55 Although under the suppostions of our deterministic optimal controlmodels of the climate‐stablizing transitionpath those techniques neverwould need to deployed in a “back‐stop” role, evenwere to some among them to beeconomicallyaswellas technically feasibleandenvironmentallysafe to implement.But in the lattercircumstancesthe investment in finding those method could nevertheless have a substantial social pay‐off, especially when thetransitiontoalowcarbonglobalproductionregimehadtobecompletedinreasonablyshortorder.Inthatcaseitismost likely that a considerable stock of operational carbon‐based capital would have to de‐activated before thesustainable economicgrowthphasebegan. But,bydeployingACS techniques thatwere (exhypothesis) technicallyeffectiveandeconomicallypractical,theconcentrationlevelofCO2couldbegraduallyloweredfarenoughtoallowingthe“moth‐balled”carbon‐basedproductionfacilitiestobebroughtbackintoproduction,therebyyieldingafiniteflowofsocialquasi‐rents.Afasttransitionalsowouldimplythatthelatter’spresentvaluewouldnothavebeensoseverelyreducedbydiscounting.Ofcourse,inastochasticcontrolsetting,“back‐stop”geo‐engineeringinvestmentwouldhavepositiveinsurancevalueevenweretheresearchresultstoturnoutnevertobeneeded,ordeemedtooriskyintheirpotentialenvironmentalside‐effectstobedeployedonlytocapturetheprivatequasi‐rents..

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pursuingthisoptionshouldbeseennotas“afix”butasa“buytime”strategy(similartoitem(i)above) in its effects on the net volume of CO2 that is added to the atmosphere by burningcarbon.Policymeasuresaimedatslowingoractuallyhaltingtheclear‐cuttingof forestswouldwork in the same direction, and might also involve compensatory capital formation to raiseagriculturalyieldsandthelivestockcarryingcapacityofalreadyclearedlands,therebytendingtoreducetheeconomicpressuresthataredrivingdeforestationduetohumanagency.

(v) “Defensive” expenditures for engineeringdesign researchand capital formation toimplementphysical reinforcementsandadditions toexisting infrastructure thatwouldreducethe latter’s own vulnerability to damage caused by extreme weather events and extensiveflooding.Thiswouldservetomitigatedirectandindirectlossesofproductivecapacityaswellasproviding ameasureof protection from lossof lives and livelihoods in some formsofnaturaldisastersthatarelikelytobecomemoredestructiveduetoregionalclimatechanges.

It will be no mean task for future research to develop informative heuristicrepresentations of the foregoing dynamic processes, and explore the way(s) to sequence theexercise of the enlarged set of policy options in an integrated multiphase model that wouldextendthepartial“BAM+(applied)R&D+UCL”structurethathasbeenanalyzed(insection4.4).56Moreover, it is a task thatwill be renderedmore feasibleby tackling itwithin the simplifyingframework of a “social planningmodel” that greatly reduces the number of assumptions thatneedtobemade(andempirically justified)‐‐aboutthemarketbehaviorsofprivateeconomicagents,andtheabilityofpoliticalauthoritiestocoordinateandimplementcoherentregulatorymeasuresandinstitutionalenforcementmechanismsonaglobalscale.

Fourthly,havingappreciatedthedifficultiesofcarryingthroughsuchanundertaking,itisallthemoreimportanttoappreciatethatitscompletionwouldstopwellshortofthefurthersteps thatwould lead towards articulating the available practical policy instruments, and thepossiblewaystheymightbeusedtoreachsuccessivemilestonesontheDIRAMmulti‐phasepaththatanomniscientandomnipotent socialplannerwould indicateas “required” toachieve theoptimal transition. That poses a research problem farmore daunting than anything that wehavecontemplatedtackling;indeed,onethatmaywellresistplausibleformulationsforsolutionusing optimal control methods (whether of the deterministic or stochastic variety). To thealready complex macroeconomic and geophysical system constraints considered in a DIRAMframework,itisnownecessarytoaddfurtherconstraintsimposedbypublicsectorpoliticalandadministrativeprocesses,and,ofcourse,thebehavioralpatternsofprivateeconomicagentsinresponsetomarketsignalsandregulatoryinducementsandrestraints.

Moreover, it is to be expected that the design of the optimal inter‐temporal policyimplementationprogram–wereonetoemergefromtheanalysis,woulddepartsubstantiallyinmanyrespects fromtherequirements indicatedbythesocialplanningoptimum. Becauseit isdealingwithaprocesstobeplayedoutinmajorpartwithinadecentralizedmarketsetting,theimplementationprogramwouldinvolverecoursetotheuseofpolicyinstrumentsinadditionto(inevitably interactive with) public‐private joint ventures and government led “technologypush” measures. The plural here is inescapable in the relevant practical context of a global

                                                            56 A start was made on this research agenda by David and van Zon(2012:section 3), in working out the phasestructuresofthetransitionpath(s)foramodelthatintegrated“BAM+R&D”witha“buytime”optionthatinvolvedincremental“retrofittinginvestments”(forappliedengineeringandcapitalequipment)toupgradetheenvironmentalperformance of carbon‐using capital. The specific functional purpose of the latter class of options is to lower CO2emissionsperunitofoutput in theselected facilitiesy,eventhoughtheupgradehadraised theirunit capital costs.Preliminarycomputationalresultsfortheresulting(uncalibrated)optimalsequencingwerepresentedforthatoption.Furthermore,thatpapersetsoutalist(insection4)ofdesireablemodificationsinthemodelingoftheclimatesystem,oftheproductionsystem,andotherspecificationchanges.Theprecedingtexthasofferedashorterandlessdetailedsketchesinthesamespirit.

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system with multiple sovereign political authorities‐‐however far one can hope to go byabstracting fromthemessinessof theprocessesbywhichnationalpoliciesemerge fromoftenchaoticdomesticconflictsamongconflictingpoliticallymediatedinterests.

Fifthly,withthissituationinmind,itmightbeausefulwayforwardforpracticalpolicydesignresearchtofocusonilluminatinghowandtowhatextentstheintroductionofostensibly“practical” means to implement the indicated allocation of resources (within each of thesuccessivephasesoftheoptimaltransitionpathfoundbytheDIRAMplanner)woulddistorttheallocation that materialized in the first phase, making it necessary to alter the indicatedrequirements for thephase thatwould follow, andsoon. Inprinciple, at least, thedistortionsand the welfare losses entailed by a well‐specified policy implementation program could becalculable. This step on the path toward stochastic control modeling that allows mid‐coursecorrections when expectations are not realized, however, generally would require taking aninterveningandnon‐trivialstep:fullyspecifyingamodelofthebehaviorofprivatesectoractorsin response to thearrayof relativepriceand cost changes, and subsidies, taxesandpenaltiesintroducedbyfiscalandregulatorymeasuresdesignedbyapublicactorinordertoimplementthe policy in question. Further allowance would need to be made for adjustment costs ofalterations in pre‐existing tax, subsidy and regulatory structures, including such newinstitutionaldevelopmentsthatwouldbenecessary.

Consequently, although it is interesting and informative to trace the complex inter‐temporal adjustments that are set inmotionbyadding to thevarietyof technologicaloptionsand corresponding control variables in the heuristic DIRAM framework presented in thesepages, the policy relevance of showing how an enlightened socialwelfare‐optimizing plannermightruna“techfix”programtostabilizetheglobalclimatesystemundervarioustechnologicalandclimaticsystemconstraintsisboundedbytherelevanceofthecircumstancesenvisagedbythosemodels.Theextent towhichtheplanner’soptimalprogramofferspracticalguidanceforthedesignofclimatepoliciesthatisofusetopolicy–makingbyindividualactorsandagenciesintheworldasitis,remainsanunresolvedquestion.Yet,suchguidancewillbeneeded,evenifitisnot presently wanted by decision‐makers and public actors and agencies that lack anythingapproximating themythical socialplanner’spowers to shape theallocationof resources; theywillhave tocopeasbest theycanbyapplying indirectandcomparativelyweakanduncertaininstrumentsinhighlydecentralizedpoliticalandeconomicsettings.

WeclosethesecommentsbyremarkingonaparticularaspectoftheDIRAMtransition

resultspresentedinsection4thatiscommontothesolutionsfoundforeachofthemodels,andwhich may serve to underscore the need for interpretive caution and restraint from overlycasual drawing concrete “policy insights” fromheuristic exercises such as those conducted inthispaper.Ineverycase,aswasnoticed,theoptimaltime‐pathfoundfortheco‐stateofE(t)‐‐whichistosaythe(negative)marginalvaluesofcurrentperiodflowofemissionsthatwilladdtotheatmosphericconcentrationofCO2–appearsasahorizontallinethroughoutthetransition,up until its completion and the beginning of the carbon‐free phase of economic growth.Applying the label “social cost of carbon” is an entirely valid way to interpret the economicwelfaresignificanceof that “shadowprice” (marginalvaluation)of thedifferencebetween thecurrently stockofatmosphericCO2and its pre‐industrial level ‐‐thatbeing theway the statevariable E(t) is defined. But it would be entirely inappropriate to casually take the furtherinterpretivestepandthinkofthesocialcostofcarbonfoundfromthesolutionstothestackedHamiltoniansinthepresentanalysisasindicativeofthesizeof“anoptimalcarbontax.”

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Thatinterpretationisaperfectlyvalidstepandonethatistakenasamatterofcoursein(IAM)climatepolicyassessmentexercisesofthekindthathavebeenmadefamiliarbytheDICEand RICE models applied developed by Nordhaus (e.g., 1994, 2000, 2007, 2010), and otherintegrated assessment studies that have drawn directly and indirectly on his pioneeringcontributionstothatapproach.Butitdoesn’tfollowinthepresentcontextforasimplereason:the DIRAM structure developed here differs radically from that class of IAM structures,mostobviously in not having considered the optimal design of climate policy instruments such asscheduled taxes on fossil fuelmaterials, ormarket‐determined prices for tradable licenses torelease volumes of CO2 generated by burning carbon. The shadow prices of the stock ofatmospheric CO2 on the optimal “tech fix” transition path do not describe the time‐path of acontrolvariableintheformofatax,andsotheirbehavioriswithoutanyintrinsic,independentsignificanceforthedesignofsuchpolicyinstruments.

Itshouldbeunderstood,instead,thattheconstancyofthemarginalsocialcostofcarbonemissionsalongthetransitionpathsleadingthephaseofcarbon‐freeproductioninourmodelsismerelytheformalreflectionofthefactthatthroughoutthatspantotimethecumulativestockof CO2 (by construction) has been kept from reaching the (pre‐specified) “tipping point” into“climate‐systemcatastrophe,”Themarginalsocialcoststhatareentailedateachmomentuntilthe economy is safely in a viable carbon‐free state are those measured in terms of theunchangingvalueofthe“felicity”foregonebyhavingtodivertenoughoutputfromconsumptionto other (investment) activities that will avert adding the marginal unit of CO2 the existingatmosphericconcentrationofgreenhousegases.

Futureanalysisoftheproblemofpolicydesignwithportfoliosofdistincttechnologicaloptions should certainlymove towards buildingmodels inwhich carbon‐pricingmechanismsare available for use as policy instruments, and apply such modes to the analysis of “mixedcontrolsystems”thatusethosetools,alongwithavarietyofothermeansbypublicpolicyandprivate non‐profit initiatives that would contribute to speeding costly transformations of theglobalregimeofproductionintheworld’s“mixedeconomies.”Thattaskwouldseemtoopenatemptinglylargeandpolicyrelevantfieldforsystematicexplorationsusingmultiphaseoptimalcontrol techniques.Yet, atpresent it appear tobeone thatmaybe tackledmoreproductivelyoncethefrontierofDIRAMresearchhasadvancedfarenoughtopermitfullcharacterizationsofthe optimal transition paths for technological system‐settings that afford a richer, andconsiderablymore complicatedportfolio of options than those considered in this preliminaryexploration.

‐ 56 ‐  

AppendixA.TheTimingofR&D

ConsidertheavailabilityofB0>0att=TU.Consideralsothepossibilityofwaitingawhile,saytillT*,beforestartingR&DactivitieswhilestillrelyingonlyonA‐typeproductionfacilities..ThenatT*theBAUphasewouldbeeffectivelysplitintotwosub‐phases,firstU0followedbyU1.Assume,moreover,thatthesplitatT*isoptimal.Inthatcase,theonlydifferencebetweenthesesub‐phases involves the existence in U1 of an R&Dprocess that needs resourcing from someportionof currentproduction to raise the average (andmarginal) productivity of carbon‐freecapital,B,abovelevel(B0)availablewithtechnologyofthatkindthatwasalreadyknownatthestartoftheBAUphase.So,theHamiltonianatT*forsub‐phaseU0wouldbegivenby:

}){()1/( ***1

*.0

* TA

TAA

TTtU

T CKACeH . (A.1)

Since the first order condition for consumption can be solved for C, giving rise to

* *( , *)AT TC f T ,theHamiltoniancanberewrittenas:

0

* * * * *( ( , *)) {( ) ( , *)}U A A A A AT T T T TH g f T A K f T . (A.2)

ForU1,wethereforehavetheHamiltonian:

1* * * * * * *( ( , *)) {( ) ( , *) } ( )U A A A A A R

T T T T T T TH g f T A K f T R R B B , (A.3)

andfindthatthefirst‐orderconditionforanoptimalallocationofR&Dresourcesimpliesthat

* * * * ( )A RT T T TR R B B . (A.4)

Hence,fortheretobeapositivelevelofR&DactivitytakingplacefromT*on,itmustbethecasethat:

1 0 0* * * * *(1 ) ( )U U R U

T T T T TH H R B B H . (A.5)

Itfollowsthat 0 1* * 0U U

T TH H for 0* TR ,whichimpliesthatT*wassettoolateinthe

BAU phase and should bemoved to an earlier date. 57. In this case, because that differencebetweenthetwoHamiltonianswouldnotvanish, thatadjustmentprocesswouldhavetopushthearbitrarystartpointoftheU1sub‐phaseallwaythebacktoT*=TU,i.e.,tothebeginningofsub‐phaseU0.ThatimpliesthattheR&Dprocessoptimallywouldstartatthemomentthatthebasic idea(s) became available to how to conduct research and development on a practicalcarbon‐free alternative to the A‐type technology, i.e., one that could be embodied in capitalgoodswithpositiveproductivity.NotethatwehaveassumedthatB0>0att=TU=0.

                                                            57ByreducingT*by1unitoftime,i.e.lettheU1phasebeginoneunitoftimeearlier,welosetheHamiltonianattheendofU0andgaintheHamiltonianatthebeginningofU1,whichinthiscasewouldleadtoanetgain.

‐ 57 ‐  

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