This work is distributed as a Discussion Paper by the

STANFORD INSTITUTE FOR ECONOMIC POLICY RESEARCH

SIEPR Discussion Paper No. 12-006

Optimal Multi-Phase Transition Paths toward A Stabilized Global

Climate: Integrated Dynamic Requirements Analysis for the ‘Tech Fix’ By

Paul A. David and Adriaan van Zon

Stanford Institute for Economic Policy ResearchStanford UniversityStanford, CA 94305

(650) 725-1874

The Stanford Institute for Economic Policy Research at Stanford University supports research bearing on economic and public policy issues. The SIEPR Discussion Paper

Series reports on research and policy analysis conducted by researchers affiliated with the Institute. Working papers in this series reflect the views of the authors and

not necessarily those of the Stanford Institute for Economic Policy Research or Stanford University

OptimalMulti‐PhaseTransitionPathstowardAStabilizedGlobalClimate:

IntegratedDynamicRequirementsAnalysisforthe‘TechFix’

By

PaulA.DavidH andAdriaanvanZonI

HStanfordUniversity,AllSoulsCollege,OxfordandUNU‐MERIT(NL)IUNU‐MERITandSBEUniversityofMaastricht(NL)

Version1.1:28November2011Version2:5January2012Version:3:12June2012

Version:4:15September2012

ABSTRACT

Thispaperanalyzes the requirements fora socialwelfare‐optimized transitionpath towardacarbon‐freeeconomy,focusingparticularlyontheroleofR&Dandothertechnologicalmeasuresto achieve timely supply‐side transformations in the global production regime thatwill avertcatastrophic climate instability.We construct a heuristic integratedmodel ofmacroeconomicgrowthconstrainedbygeophysicalsystemwithclimate feedbacks, includingextremeweatherdamages from global warming driven by greenhouse gas emissions, and ‘tipping point’ forcatastrophic runaway warming. A variety of options for technology development andimplementation, and the dynamic relationships among them will be examined: interactionsamong the, each having a distinctive primary functionality CO2 emissions control. Thespecificationsrecognize(i)theendogeneityandembodimentoftechnical innovations,and(ii)theirreversibilityandlonggestationperiodsofrequiredintangible(R&D)andtangiblecapitalformation. Efficient exercise of these options is shown to involve sequencing differentinvestment and production activities in separate temporal “phases” that together form atransitionpathtoacarbonfreeeconomy.Tostudytherequirementsofatimely(catastrophe‐averting)transition,weformulateasequenceofoptimalcontrolsub‐problemslinkedtogetherby transversality conditions, the solution of which determines the optimum allocation ofresources and sequencing of the several phases implied by the options under consideration.Ours is “planning‐model” approach, which departs from conventional IAM exercises byeschewing assumptions about the behaviors of economic and political actors in response tomarketincentivesandspecificpublicpolicymeasures.Solutionsforeachofseveralmulti‐phasemodels yields the optimal phase durations and rates of investment and production thatcharacterize the transition path. Sensitivity experiments with parameters of economic andgeophysical sub‐systems provide insights into the robustness of the requirements analysisundervariationsinthetechnicalandgeophysicalsystemparameters.

JELcodes:Q540,Q550,O310,O320,O330

Keywords:globalwarming,tippingpoints,catastrophicclimateinstability,technologyfixoptions,R&Dinvestments,capital‐embodiedinnovations,optimalsequencing,IAMandDIRAMpolicydesignapproaches,multi‐phaseoptimalcontrol,sustainableendogenousgrowth

Contactauthorsbyemail:pad@stanford.edu; adriaan.vanzon@maastrichtuniversity.nl

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OptimalMulti‐PhaseTransitionPathstowardaStabilizedGlobalClimate:

IntegratedDynamicRequirementsAnalysisforthe‘TechFix’

1.BackgroundandMotivation

ClimatechangeisnowconvincinglylinkedtoincreasingatmosphericconcentrationsofGHG(greenhousegases).1Amongthosewhohaveexaminedtherelevantscientificdatatherehasbeenagrowingconsensusthatthisposesaninter‐relatedhostofworrisomeproblems.Moreover,ratherthanconvenientlygoingaway,orbeinggraduallybeingaccommodatedbyadaptationsonthepartoftheworld’speoples,theseproblemsarelikelytogrowworse.Whileonmanyspecificpointsofclimatescienceuncertainties,doubtsanddisagreementpersist,theunderlyingphysicalandchemicalprocessesresponsibleforanthropogenic“greenhouseeffects”warmingtheEarth’surfacearefirmlygrounded,asistheaccumulatingmassofempiricalobservationsattestingtotheriseinmeanglobaltemperaturethathastakenplaceduringthepasttwocenturies.2Together,thesefindingsfirmlyundergirdthescientificallyinformedwarningsabouttheconsequencesofcontinuing“businessasusual”basedonburningfossilfuels.

TherisinglevelsofmoisturethatwillaccompanythewarmingoftheEarth’ssurfaceinthetemperatelatitudescanbeexpectedtodrivemorefrequentandmoresevereweathercycles,bringingheavierprecipitation,strongerwindsandweather‐relateddamagestophysicalpropertyandlossesofhumanlife.Moreover,itismorethanconceivablethatatsomepointinthenotverydistantfuturecontinuingthecurrentrateofGHGemissionsandunimpededglobalwarmingwillusherinanageofcatastrophicallyabruptclimatechanges,characterizedbychaoticinstabilitiesinEarth’sclimates.Thiswoulddrasticallycurtailthefractionoftheworld’scurrentpopulationthatwouldbeabletoachieveastateof“adaptivesurvival”preservingsubstantialresemblancestopresentstateofcivilization.

MitigatingenvironmentaldamagesfromCO2,methaneanddangerousparticulateemissionsbymoreextensivelydeployingknowntechnologies,andreplacingcarbon‐

1SeeIPCC(2007)andtheSternReview(Stern,2006)foranextensiveoverviewofthecausesandconsequencesofclimatechange.

2SeeSolomonetal.(2007)onthephysicalbasisandscientificevidenceforIPCCconclusionsregardingglobalwarminganditssourcesinhumanactivity.“Climateskepticism”and“globalwarmingdenial”shouldnotbeconfusedwiththegenerallyacknowledgedscientificuncertaintiesthatstillsurroundthevaluessomekeyparametersinquantitativemodelsoftheprocessesinvolvedinanthropogenicglobalwarming‐‐e.g.,themagnitudeofthe“climatesensitivity”(linkingchangesinmeanglobalsurfacetemperaturetorelativegaininatmosphericGHGconcentration),andthecriticalleveloftemperaturegainthatwouldtriggerirreversiblerunawaywarmingandcatastrophicclimateinstability.Seefurtherdiscussionbelowanddetailsinfootnotes6‐7.

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basedproductionsystemswiththosethatutilizenewandeconomicallyefficient“carbon‐free”technologies,aretwoobviouscoursesofactionthatmaybeabletoavertthesegrimfutureprospects.Togetherwithstillmoreradicaladaptationsthatpossiblymaybeachievedbygeo‐engineeringtocaptureandsequesterexistingatmosphericcarbonandmethane,theseformthecoreofthetechnologicaloptionswhosedevelopmentanddeploymentarethefocusofthisexploratoryexaminationofthedynamicsofafeasibleandtimelytransitiontoasustainablelow‐carbonglobaleconomy.

AprogramofpublicandprivateR&Dinvestmentsyieldingdirectedtechnicalchangesofthiskindshouldbeviewedasa“supply‐sidestrategy”inthecampaigntostabilizeglobalGHGconcentrationsataviablelevel.Sucha“technologyfix”programwouldbecomplementarytocarbontaxes,andthe“capandtrade”schemesthathaveenvisagedthedevelopmentofaglobalmarketforcarbon‐emissionspermitsandderivativeinstruments.Theselattermechanismswouldworkprimarilybyraisingthecurrentandexpectedfuturerelativepricesofcarbon‐intensivegoodsandservices.Weretheytohavethateffect,thevolumedemandforgoodsandservices(e.g.,electricitysupply,transportation)thatareespeciallycarbon‐intensiveintheirmethodsofproduction,wouldbereduced.Whereasthe“carbonemissions‐pricing”policyproposalstomeetthechallengesofclimatechangearedirectedtowardachievingsameultimategoalasthe“technologyfix”strategy,theygiveatbestsecondarynoticetotherolethatpubliclysupportedR&Dinvestmentsdirectedtowardsthegenerationofcarbon‐freetechnologiesmayplayinthedynamicsofatransitiontoanew,radicallylesscarbon‐intensivesystemofproductionfortheworldeconomy.The“supply‐side”approachthatthispaperexplores,however,shouldnotbeunderstoodmerelyasanoptionalsupplementto“demand‐management”throughemissions‐pricingpoliciesthattodatehavereceivedpreponderantanalyticalandpracticalattentionintheeconomicsliteratureonclimatechange.PreviouseconomicargumentsforgreaterinvestmentinR&Danddiffusionoftechnologicallyenhancedproductionmethodsarerelativelyfewandfarbetweenintheeconomicsliteratureconcernedwithclimatechange,butitisnotablethatacademicpapersinthatgenrehadbeguntoemergeevenbeforethedebacleoftheattemptatthe2009CopenhagenConferencestrengthenandexpandtoKyotoTreatyprotocols.3Neithertheseearly

3InthewakeoftheSternReport(2007)andtheCopenhagenConferenceeconomicargumentsforgreaterpublicandprivateinvestmentinR&D,andsubsidiesforthediffusionofadvancedtechnologiesdirectedtoloweringCO2emissionsbeganappearingwithgreatermorefrequently–althoughthesetypicallyremainedunspecific.Earlystatementsproposinga“technologyfix”approachincludeArrow,Cohen,Davidetal.(2008),NelsonandSarewitz(2008),Klemperer(2007/2009),David(2009),David,Huang,SoeteandvanZon(2009),andHendry(2010).None,toourknowledge,venturetoexaminetherequiredbehaviorofdirectedR&DandcapitalformationembodyinginnovationsthatwouldlowertheglobalrateofGHGemissionsperunitofrealoutput,whichisthefocusofthepresentwork.Suchtheoreticaland

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foraysintothisarea,norotherworkstoourknowledge,venturedtoexaminetherequireddynamicsequencingofdirectedR&Dandothertechnologypoliciesaffectingpublicandprivatesectorinvestmentsrequiredforatimelytransitiontoalow‐GHGemittingproductionregime,thepolicyissuethatisthefocusoftheresearchreportedhere.

IthasbeensuggestedthatCO2emissionsabatementachievedbytheintroductionofcarbon‐taxesor“cap‐andtrade”schemesfor“pricing”transferableemissionspermits,couldindirectlyachievethewhatpublicresearchexpendituresandtaxsubsidiesforprivateinvestmentin“green”R&Dwouldotherwisehavebedirectedtoaccomplish.4Theargumentisthattheeffectofraisingthecostofcarbon‐intensiveprocessesandproductstoproducersandconsumerswillbetostimulateproducers’demandsforoffsetting,carbon‐freetechnologiesandinduceincrementalprivatefundingofthesearchofthoseinnovativemethods.

Althoughthatisapossibility,sotooistheoppositeoutcome.Ifraisingcarbontaxesdoeseffectivelycurtaildemandforhighlycarbon‐intensiveproducts,theresultmaywellbeagenerallyadverseexpectationsaboutmarketgrowthandaconsequentweakingofincentivesforR&Dinvestmentinallproductiontechniquesforinthoselinesofbusiness.5Moreover,theimpositionofcarbontaxationthatislikelytobemaintainedandevenriseinthefuturemightwellhavetheperversefirst‐ordereffectofdepressingthepresentvalueoffossilfuelstocksintheground,bringingforwardintimetheexploitationofthoseresourcedepositsandtendingtodepresspricesinmarketsfor

computationalmodellingthathasbeenundertakentodatehaslargelyfollowingthepioneeringworkofGoulderandSchneider(1999),whichisnotedbelow.

4Amongthepioneeringeffortstoexaminethepropositionthatraisingthepriceofcarbonwouldinducebeneficialprivateinvestmentsintechnologicalinnovation,seetheworkingpapersbyNordhaus(1997),andGoulderandMathai(1998).GoulderandSchneider(1999)carriedthislineofinquiryfartherbyusingacomputablemulti‐sectorgeneralequilibriummodeltoquantitativelyassesstheextenttowhichrealGDPcostsofimposingCO2abatementtaxeswouldbealteredbytheeffectsofendogenoustechnologicalimprovementsresultingfromR&Dinvestmentinducedbytheeffectofthetaxontherelativepricesofcarbonvs.non‐carboninputs.Thismodellingexercisedidnot,however,undertakeanevaluationoftheefficacyofthehypothesizedinducedtechnicalchangesintermsofthemagnitudeoftheabsoluteabatementoftheeconomy’saggregateCO2emissionsrate.Morerecentexercisesinintegratedclimatepolicyassessmenthaveundertakentodoso,asisnoticedbelow(insection2).

5Thequestioncannotbesettledontheoreticalgrounds,butseeGans(2009)foranalysisoftheconditionsunderwhichcarbon‐taxeswouldhaveaperverseeffectofdiscouragingcarbonemissions‐reducingtechnologyresearchandinnovation.Acemoglu,Aghion,Bursztynx,andHemous(2010)exploretheconditionsforcarbontaxesinagrowthmodelwithenvironmental(carbonemissions)constraintsandendogenousdirectedtechnicalchangedrivenbyprivatesectorR&Dinvestments.Theyfindthatwithasufficientlyhighelasticityofsubstitutionbetweenhighlycarbon‐intensityandlowcarbon‐intensityinputsinproduction,carbontaxescanswitchdirectedR&Dto“clean”(lowcarbonintensity)technologicalresearch,andthatthereisacomplementaryrolefor“temporaryR&Dsubsidies”.Butthecomplicationsof“GreenParadox”effectsofcarbontaxes(seefn.6,below),andcarbon‐savingR&Dsubsidiesarenotaddressedbytheirtheoreticalanalysis.

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carbonfuel.6Beyondthispossibledrawbackinrelianceuponcarbontaxes,thereisstillgreaterroomfordoubtsregardingtheefficacyofdownstreamcap‐and‐tradeschemes,whichpricetheemissionsfromburningcarbon,butnotthecarbonmaterialthatistheenergysource.

Itisnotatallobviousthatputtingapriceontheemissionsfromtheburningofhydrocarbonsnecessarilywouldsufficetoraisethecombinedpriceofthematerialanditscombustion,sothepossibilityofaperverserelativepricechangecannotbedefinitivelyruledout.7Bycontrast,theproposalofan“up‐stream”implementationofthecapandtrademechanism(seeRepetto2011)isadministrativelyattractivebecauseeffectiverestrictionofthevolumeoffossilfuelsavailablefrom“firstsellers”itwouldnecessarilyraisethecostofcarbon‐basedenergysources.Furthermonitoringandenforcementofthe“caps”wouldbegreatlysimplifiedbythelimitednumberof“firstsellers”andtheeaseofidentifyingthem–especiallyincomparisonwiththemyriaddownstreamusersthatwouldrequireCO2emissionpermits.Theoreticalanalystsoftheproblematic“GreenPardox”effectofcarbontaxesactuallyincreasingtheextractionanduseofcarbonfuelsareagreedthatthisperverseoutcomewouldnotmaterializeifahighenoughadvaloremtaxratecouldbeimposedsoonerratherthanpromisedforthefuture.But,asefficientasadministrativelyfeasibleasitmightbeforcentralgovernmentauthoritiesinthelargesteconomicstoquicklyimposecoordinatedtightregulatory

6Discussionofthelatterproblemwasbroachedfirstbyinn(2008,2009:pp.10‐13)whoprovidedalabel‐‐“TheGreenParadox”–underwhichatheoreticalliteraturesoondeveloped(see,e.g.,vanderPloegandWithagen(2010),Hoel(2010).l,thattheprospectofinnovationsthatwouldprovidemoreattractivesubstitutesforfossilfuelasanenergysourcewoulddepressthevalueofthosenaturalresourcedeposits,leadingtheownerstoraisethecurrentrateofextractionanduseofcarbonasanenergysource–therebyoffsettingtheCO2‐mitigatingintentionofpoliciesofferingpublicsubsidiesfor“greeninnovation.”

7Thissourceofambiguityregardingtheeffectsofusingthe“capandtrade”mechanismtosetapositivepriceoncarbonemissions,itshouldbeemphasized,isdistinctfromtheconcernsaboutthepossibledisincentiveeffectsofcarbontaxes’negativeimpactsondemandsforcarbon‐intensivegoodsandservices–e.g.,thoserelyingheavilyonproductionprocessesusingenergysourcessuchascoal,oilandnaturalgas.Thetheoryofexhaustiblenaturalresourcepricing,followingtheclassicworkofHotelling(1931)tellsusthattheresourcevaluationmustriseattherateofinterest,andwithmarginalextractioncostsconstant,thepriceoftheflowofmaterialstradedinthemarketmustriseparipassuswiththatoftheresourcedeposit.Subsequenttheoreticalcontributionsonexhaustibleresourceeconomics,e.g.Heal(1976),Dasgupta,HealandMajumdar(1978),DasguptaandHeal(1980)‐‐havepointedoutthatdownwardpressureextertedbyanticipationsofadvancesin“backstop”technologiespermittingsubstitutionofcarbon‐freeenergysources(and,moregenerally,moreenergy‐efficientproductionprocessinnovations)wouldlowerthelevelsfromwhichthemarketpricesofcoal,oilandnaturalgaswouldbetrendingupwards.Thiseffectwouldworktoslow,ifnottermporarilyhaltthediffusionofthoseinnovations.Whetherthemagnitudeofthefirst‐orderoffsetonpricesincarbonmarketswouldbelargeenoughtoperverselyneutralizetheintendedimpactofaglobalcap‐and‐tradescheme,orsolargeastotemporarilylowertheafter‐taxunitcostofburningcarbonfuels,isadistinctempiricalquestion(ofobviouspolicy‐relevance)thatdeservesmoreattentionthanithasreceived.Onfirstconsideration,itseemsthattheanswermustturnontheextentoftheanticipateddemanddisplacementinrelationtothesizeoftheresourcereservesthatcouldbeaccessedfromthemarketsinquestion.WearegratefultoLawrenceGoulderobservationthatthelaterconsiderationmakethemagnitudeoftheoffset‐effectsgreaterinthecaseofhighqualityoilreservesthanitwouldbefortheworld’sexistingcoaldeposits.

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capsonfirstvendors,suchactionswouldbetantamounttodirectlyrestrictingextractionandimportationoffossilfuels.Thepoliticaleconomyofdomesticlegislationandinternationaltreatynegotiations,however,presentlyandforsometimetocomewouldmilitatestronglyagainstpracticalimplementationofthisapproachtofixingthegreenhousegasexternality,doesnotseemtogiveitanyadvantagevis‐à‐visthemorefamiliar“downstream”cap‐and‐trademechanism.8

Inthesecircumstancesitshouldberemarkedthatthepresentandlikelyfuturesocialandpoliticalresistancestostringentandenforceablecarbon‐pricingpoliciesmightbeweakenedweretherecrediblegroundsforanticipatingthattheeconomicwelfarecostsentailedinswitchingtoalternativeenergysourceswouldbesignificantreducedbyaconcertedmulti‐nationalprogramofdirectedtechnologicalinnovation.Whilethedynamicsofthistechno‐politiconexuswillneedtobethoughtthroughcarefully,theargumentforsystematiceconomicanalysisofthepotentialroletechnologypoliciesineffectingatimely,climatestabilizingtransitiontoaviable“lowcarbon”economytransitionthereforecan,atleastforthetimebeing,restinpartonthepossibletemporalasymmetryofcomplementary(orsuper‐modular)relationshipsbetweenthetwomajorpolicyapproaches.

Whenviewedinthatexpandedandexplicitlydynamicperspective,aprogramthatstartswith“techfix”maymakethatpolicyinitiativecomplementaryto(andnotasubstitutefor)fiscalandregulatoryinstrumentsdesignedtogetmarketpricesofcarbonfueltoreflecttheirmarginalsocialcosts.Thisconsiderationprovidesanothercompellingreasonforaccordinggreaterattentiontotherequirementsofa“technologypush”strategythanithasreceivedinpreviousandcontemporaryeconomicresearchonclimatechangepolicy.

2.Preliminariesformodellinga“TechFix”strategy’sdynamicrequirements

Allofthepreviouslymentionedsupply‐sideoptions,however,requirepayingexplicitattentiontomajorchangesinproductionsystemsand/orchangesinlife‐styleandconsumptionpatterns.Furthermore,toexplorethedynamicsoftransitionstowardsalesscarbon‐intensiveproductionregimeandtherolethatresearchpolicywouldhavetoplay,itisnecessarytoconsiderthetemporaldurationofR&Dactivitiesdirectedtoalteringtheperformancecharacteristicsofvariousclassesoftechnology,aswellasthedurationoftangiblecapitalformationprocessesrequiredbytechnologiesthatmustbe“embodied”innewphysicalplantandequipment.Suchquestionscanbemostusefully

8Indeed,itseemmorethanlikelythatgovernmentregulationsbanningunlicensedfirstsalesoffossilcarbonsourcesofenergyistantamountstateexpropriationandcontrolofthoseformsofnaturalresourcewealth,andwouldmeetwithstrongpoliticaloppositionfrompowerfuleconomicinterestsas“defactosocialization”oftheirprivatelyownedmineralwealth.

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exploredinthispaperwithintheframeworkofaheuristicmodelofendogenousglobaleconomicgrowth,whichistheresearchapproachpursuedhere..

Itisnolessessentialtobeginbyexplicitlytakingintoaccountthedynamicbehaviorofthegeophysicalsystemwithinwhichtheglobaleconomyisset.Thatlargerphysicalcontextquiteobviouslyimposesbothnaturalresourceandenvironmentalconstraintsupontheproductionregime,aswellasimpingingdirectlyuponhumanwelfare–inthisinstancethroughthecontinuinganthropogenicemissionsofgreenhousegasesandtheirdestabilizingimpactsuponglobalclimateandtheaccompanyingalterationofweatherpatterns.

2.1Geophysicalsystemconstraints

TherearenowfirmscientificfoundationsforthegrowingconcernsthattoallowGHGemissionstocontinueatanythingclosetotheirpresentratemaysoonsetinmotionirreversiblerunawayglobalwarming,withpotentiallycatastrophicconsequencesforglobalecosystemsandforhumanlifeasweknowit.Thisconclusionhassurvivedthepersistingdoubtsvoicedbyadwindlingfringeofacademic“climatechangesceptics”andoutrightdeniersoftheseriousnessoftheproblemsthatcontinuedglobalwarmingwillpose.9

ThemassofevidenceprovidedbychemicalanalysisofthegasesinthedeepicecoresthathavebeenextractedfromArcticandmountainglaciersduringthepastdecade‐and‐a‐halfhasallowedclimatescientiststodocumentahistoryofabruptalterationsoftheEarth’sclimate(s)duringpre‐Holoceneepoch.TherecordpointstothepossibilitythatevenmodestglobalwarmingdrivenbycontinuedanthropogenicemissionsofGHGcouldtriggertheonsetofirreversibleglobalwarming,butaformofclimateinstabilitythatduringthemostrecenttransitionbetweenglacialandstadialepochsfeatureda

9DespitetheevidencepresentedbySolomonetal.(2007),doubtshavebeenraisedaboutconclusionthatthewarmingtrendobservableduringthesecondhalfofthe20thcenturyresultedfromradiativeforcingduetohumanactivities.ThesedoubtsarerootedincriticismofIPCCmodelsofgeneralclimatecirculationmodels(GCMs)onaccountoftheirpoorperformanceinpredictingthehighfrequencyandhighrelativeamplitudevariationsinglobalclimateduringthepast50‐60years.See,e.g.,Scarfetta(2011)andreferencesthereinforstudiesshowingthatthelattervariationscanbemoreaccuratelypredictedbystatisticallyfittingharmonic(Fourierseries)regressionmodelsthatcombineindicatorsofmultipleastronomicalcycleswithdifferingfrequencies.Suchresearch,however,shedsscantlightonthedynamicsofthephysicalforcingprocessesthatwouldaccountcasuallyfortheobservedstatisticalcorrelation,nordoestheexistenceofunexplainedfluctuationsaroundatrendvitiatethepositivestatisticalsignificanceofthetrenditself.Recenttheoreticalefforts,suchasthatbyStockwell(2011),offergeneralphysicalbasisfortheskeptics’contentionthatperiodicradiativeforcingfromastronomicalsources,notaccountedforintheIPCC’sGCMs,couldberesponsibleforthelatter’sfailuresinpredictingrecenthighfrequencyvariationsobservedintheEarth’stemperature.Thetroublewiththislineofspeculativecriticismseemsobviousenough:variationsinsolarradiationarenotthoughttobeaconcurrentlyemergingfeatureinthehistoryofourplanet’sastronomicalenvironment.Consequentlyitishardtounderstandhowthatphenomenoncouldaccountforthe“hockey‐stick”up‐turnobservedinthelasthalf‐millennium’sworthofdataindicatingthelong‐termtrendintheEarth’stemperature.

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prolongederaof“climateflickering.”10Thislattertermreferstoaclimateregimecharacterizedbyhighfrequencyswitching(withperiodicitiesasshortas3to5years)betweenmarkedlywarmerandcooleraveragetemperatures‐‐thelatterdifferingbyasmuchashalfthe12oC.rangebetweenthelowestandhighestextremesrecordedduringEarth’sglacialandstadialepisodes.ThepossibilityofcatastrophicstatechangeofthatsortunderminesthesanguinesuppositionsthatthecontinuedemissionsofCO2wouldsimplyresultinasmoothtransitiontoawarmerglobalclimatetowhichhumansocietieswouldbeabletosuccessfullyadapt.

Thepositivefeedbacksthatwoulddriverunawayglobalwarmingdynamics‐‐anditsmanifoldentaileddamagestohumanwelfare–arethosegeneratedbytheamplificationofthelinkagebetweenelevatedGHGconcentrationlevelsofanthropogenicoriginsandstrongerradiativeforcingthatproducesfasterwarmingofEarth’ssurface.Thewarmingatvariouspointstriggersself‐reinforcingalterationsinthebehaviourofthegeophysicalsystem.ThesefeedbackprocesseswouldoperatetoincreasethestrengthoftheradiativeforcingresultingfromagivenatmosphericGHGconcentrationlevel;boostthesurfaceabsorptionofheat,andhencetherateofwarmingproducedbyagivenlevelofradiativeforcing;supplementdirectanthropogenicemissionsofCO2bytriggeringreleasesofmethane(amuchmorepowerfulGHG)–which,atlowertemperatures,wouldhaveremainednaturallysequesteredinthepermafrostbeneathglacialiceandinthemethanehydrateslyingontheshallowseabedatthenorthernedgeofthearcticocean.

Someoftheinterveningstepsofsuchsequences(depictedschematicallyinFigure1)haveacascade‐likesub‐structurethatservetoextendandfurtheracceleratethepositivefeedbackprocess.11Forexample,glacialretreat,byloweringalbedo(thefractionofsunlightreflectedbytheearth’ssurface)intheaffectedregionsleadsimmediatelytogreaterlocalheatabsorption.Thisinturnpromotesthawingoftheexposedpermafrostandtheformationof“thawlakes”fromwhich(likeotherwetlands

10HallandBuhl(2006)reviewthefindingsfromrecentadvancesinclimatescience,andpointoutthatitsimplicationsthoroughlyunderminethesuppositionlongmaintainedbymainstreamcontributorstotheliteratureonenergyandenvironmentaleconomics,namely,that“climatechange”drivenbyrisingGHGconcentrationlevelscouldbesatisfactorilymodelledasasmoothtransitiontoahigherequilibriumleveloftheglobalmeansurfacetemperature–ashasbeenassumedbytheintegratedassessmentmodels(IAMs)thathavefiguredprominentlyandremainsalientintheenergyandenvironmentaleconomicsfield.SeeHallandBehl(2006:pp.461‐462)foradetaileddiscussionofthisandrelatedfeaturesofthegeophysicalsub‐systeminNordhausandBoyer’s(2000)updatingoftheoriginal(Nordhaus1994)DICEmodel,TheassumptionthatradiativeforcingduetotheaccumulationofatmosphericCO2woulddriveasmoothtransitiontoahigherequilibriumtemperatureoftheEarth’ssurfaceisretainedbyNordhaus(2010)inthelatestupdateofDICE(2010),aswellasintheannualizedversionofthemodelthatCai,JuddandLontzek(2012)createasabasisforDSICE,astochasticversionofthemodel.

11SeeStern(2007),pp.11‐14,forashortoverviewofthepositivefeedbacks,includingreductionofalbedothroughreducedice‐coverageofthearcticregions,thawingofpermafrostandinducedreleaseofmethane,aswellasthereleaseofmethanefromoceanichydratestores.

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coveringdecayingorganicmaterial)methanewillbubbletothesurface.Similarly,thewarmingoftheupperocean,particularlyintheshallowerArcticwaterscandestabilizethemethaneclathratecompoundsthathadformedthere–and,moregenerallyinthedeeperocean–duringanearlierepoch.Similarly,warmingofthelargepeatdepositsinthearcticSiberianshelfwouldaugmentthereleaseofmethanefromthatnaturalsource.Methaneintheatmosphereisover20timesmorepotentthanCO2initsgreenhouseeffects,andsocloudsofthegas,bubblinguptothesurfaceandflowingquicklytotheatmospherebeingwillcausetheCO2‐equivalentconcentrationlevelandthestrengthofradiantforcingtospikeupwards.But,methane(CH4)isanunstablemoleculethatissubjecttoimmediatedegradationbyexposuretothewatermoleculecalled“OHradical,”whichsetsinmotionachemicalchainreactionthatwithinseveraldayswillhaveconvertedthenewlyaddedmethanemoleculestowaterandCO2.12Consequently,thestrongeffectuponradiantforcingofasuddenjumpinatmosphericmethaneconcentration(unlikethatofgainsinCO2ppmv)isonlytransientanditspersistingdirectresultupontherateofglobalwarmingwillbeequivalenttothatoftheadditional(unabated)CO2producedbythechemicalreactionsthatdegradeCH4.Asurgeinwarmingresultingfromaprolongedelevationoftherateofmethaneflux,nevertheless,maybesufficientlyextendedtoinducefurtherpositivefeedbackeffectsthatareself‐reinforcingandcontributetoraisethepersistingatmosphericconcentrationofCO2.13

CloudsofmethanegascouldbeabruptlyreleasedbythewarmingofshalloweroceanwatersalongtheArticOcean’sedges,sinceitisexpectedthatariseofonly2∘C.inwatertemperaturewouldbeenoughtodestabilizethemethaneclathratecompoundsthatareholdingCH4trappedinitscage‐likemolecularstructure.This,thehypothesized“ClathrateGun”effect–whichwouldoperateanalogouslytoproduceabruptmethanereleaseswhenthepermafrostretreated,nowthoughttobeaprimarymechanismofthepre‐historicepisodesofpronouncedhighfrequencytemperatefluctuationsthathaveleftarecordinthedeeparticice‐cores.14

12SeeArcher(2011:Ch5)foralucidexpositionofthemethanecycle,drawinglargelyonJacob(1999).TheOHradical(denotedOH●)isproducednaturallybytheeffectofsunlightonwater,whichcausesthelatertoloseahydrogenatom,andtoimmediatelystealonefromanavailablemethanemolecule‐‐yieldingH2Oandthemethylradical(CH3●),whichachainofsimplechemicalreactionsendsupproducing2stablemoleculesofwaterandoneofcarbondioxide.

13Whileitappearsthatthereislittlelikelihoodofthishappeningspontaneouslythroughnaturalevents(suchasdisruptionsofthedeepoceanseabed)thatwouldtakeplaceonlargeenoughscaletohavecatastrophicclimateconsequences,thesizeandinstabilityoftheEarth’smethanehydratereservesandtheuncertaintiessurroundthepresentstateofscientificknowledgeleadasobreclimatescientistlikeArcher(2011)tocharacterizethosepotentialitiesas“frightening”.

14Evenquitemoderatewarmingoftheoceanwatersalongthecontinentalshelvesisthoughttobesufficienttodestabilizethemethaneclathratecompoundsthathaveformedontheseabedinnorthern

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Yetanotherperversefeedbackloopinvolvesforestdie‐back.ThisworriesscientistsstudyingtheborealforeststretchingacrossthenorthernlatitudesofthenorthAmericancontinentandRussia‐‐azonewhosetreesrepresentbetween25and30percentoftheworld’sforestcover,andwhichisreportedtohaveexperiencedthemostpronouncedtemperatureincreasesobservedanywhereontheplanetduringthelastquarteroftwentiethcentury.InthenorthernlatitudeofSiberiathepredominantneedle‐sheddinglarchforestsarebeingreplacedbyevergreenconifersthatgrowmorerapidlyinthesummerwarmth,buttheevergreentreesabsorbmoresunlightandtheirexpansionisthuscontributingtotheglobalwarmingtrend.15

Thelossofpredictabilityoflocalenvironmentalchangesaccompanyingprofoundecologicaldamageandlossesofhumanlifeandwelfare,andthesocio‐economicrepercussionsthatwouldexacerbatetheprocesstherebysetinmotion,canbelefttotheimaginationatthispoint.Sufficeittosaythattheprospect,howeverremote,impartsconsiderableurgencytothetaskofdevelopingaworkableapproachtoidentifyingtherequirementsofaneffective“technologyfix”toavertingitsrealization.Ourmodellingapproachinthispaperbuildsuponthepioneeringworkofeconomistsonclimatepolicyanalysisrepresentedinso‐calledintegratedassessmentmodels(IAMs)thatprovideasimplifiedcharacterizationoftheessentialfeaturesofthegeo‐physicalsystem.16ThesequantitythelinkagebetweentheflowofGHGemissionsfromeconomicactivityandtheaugmentedradiativeforcingthatdriveshighermeanglobaltemperaturesattheEarth’ssurface(the“green‐house”effect).ModellingthecarboncycletakesaccountofthenaturalsequestrationofCO2intheforestsandoceans,andtheconsequentlaggedeffectintheadjustmentoftheaccumulatedatmosphericstockofGHG(equivalentCO2ppmv);the“climatesensitivity”parameterdescribestheequilibriummeansurfacetemperature’sresponsetoadoublingoftheCO2‐e

latitudes.Onthe“ClathrateGun”hypothesis,andit’srelevanceasanexplanationofabruptclimatechangeandthephenomenonof“climateflickering”attheendofthelasticeage,seeKennettetal(2003),Maslinetal.(2004),HallandBehl(2006),ReaganandMoridis(2007).Thehighfrequencyofthetemperaturefluctuationsrecordedintheice‐coresareheldtomilitateagainsttheearliertheorythatthisabruptandcatastrophicalterationoftheclimatesystemcouldhavebeenproducedbya“thermohalinecollapse”,whichwouldresultclimatecyclesofmuchlowerfrequencies.

15WhetherthisiscompensatedbytheirgreatercapacityforabsorbingCO2isnotclear.Ifsuchisthecase,itisbeingoffsetinthesoutherndrierborealzone,wherewater‐stressedtreesinthesummergrowlessrapidlyandtheforestsarebeingreplacedbygrasslandsandpasture.http://en.wikipedia.org/wiki/Boreal_forest#Climate_change

16ForsurveysandreviewsoftheevolvingfieldofresearchonIAMs,afterNordhaus(1993),seeDowlatabadi(1995),KellyandKolstad(1999),Parker,Letcher,Jakemanetal(2003),Ackerman,DeCanio,HowarthandSheeran(2009).

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concentrationintheatmosphere.17Withinthatcharacterizationofthephysicalframework,ourmodeladdsspecificationsoftheproductionsectorofanendogenousgrowthmodelinwhichdirectedR&Dexpenditurescanraisetheproductivityofnewlyformedcapitalgoodsembodyingnovel“carbon‐free”technologies,andhenceenhancetheeconomicefficiencyofthoseadditionstoaggregate(sustainable)productioncapacity.18Thisisthenexusthroughwhichpublic“researchpolicy”interventionsthatboostinvestmentsinscienceandR&Dcanpositivelyaffectsocialwelfarebyloweringthecostsofswitchingtoproductionsystemscharacterizedbylow,orinthelimit“GHGemissions‐free”technologies.19

Butinthismatterthetimingaswellasthemagnitudeofremedialactionisacriticalconcern.TheIntergovernmentalPanelonClimateChange(IPCC)hasconcurredintheconclusionthatthereisacriticalthresholdat2degreesC.ofglobalwarmingfromthe

17Forpurposesofourmodeldescribedbelow,atmosphericGHG(ppmv)concentrationlevelsaregivenbyaninitialbaselinelevelplusascalarfunctionoftheintegralof(CO2‐e)emissionsfromthebaselinedate,thelatterbeingproportionaltothemeanrateofCO2emittedperunitofoutputproduced(proportionaltoutilizedcapacity)andnotbeingabsorbedbytheforestsandtheupperandlowerocean.ThissimplificationignoresthelageffectsofemissionsonchangesinatmosphericCO2thatresultfromthecarbon‐cyclediffusionofthegasbetweentheatmosphereandtheupperocean,betweentheupperoceanandthelowerocean,andtheupperoceanandtheatmosphere.Inadditionitlinearizestherelationshipbetweentheatmosphericconcentrationofcarbonrelativetoitslevelatabasedate(Et/E0,inournotation)andtheabsolutegaininradiativeforcingfromtheatmosphere,Ft.–F0.WidelyusedIAMs‐‐e.g.,versionsofDICEduetoNordhaus(2002,2007)–typicallyrepresentthechangeinatmosphericradiativeforcingastakingtheformΔFt=η[log(Et/E0)–log(2)]withadjustmentsforthedirectandindirecteffectsofradiativeforcingfromtheupperandloweroceans.Ignoringthelatter,andthelagsintherelationshipbetweenchangesintheradiativeforcingandthoseinEarth’ssurfacetemperature,theparameterηistheapproximate“climatesensitivity”:theexpectedlong‐rungaininTrelativetoitsbaselevelresultingfromdoublingtheatmosphericC02concentration.Oursimplifieddynamicsofthegeophysicalsubsystemismoreacceptableinmodelingthetransitionpathstoclimatestabilizationthanitwouldbeinsimulatingthecourseofcarbonemissionsandtemperaturechangesoverthe600yearlongtimespanofoptimally”moderated”CO2emissionsenvisagedbyDICE(2007)anditsprecursorIAMs.

18Presently,R&Doutcomesarecompletelydeterministicinourmodel,butweenvisagetheintroductionofstochasticityinseveralrelationships,includingthisone.

19InproceedinginthiswaywehavenotignoredthecontroversialsuggestionadvancedbyWeitzman(2009b)thatthescientificuncertaintysurroundingthebehaviorofthegeo‐physicalsystem,suchasthedistributionofthe“climatesensitivity”parameter,leavesopenthepossibilityofcatastrophicoutcomesthatmayoccurwithvanishinglysmall“fat‐tailed”probability.Thedismalimplicationforeconomicpolicydesignwouldseemtobetheextremeformofthe“precautionaryprinciple”inwhichallotherconsiderationsareoutweighedbythisthreatandthewillingnesstopaytoavertitbecomeinfinite.Toacceptthatviewwouldmilitateutterlyagainstanyefforttopursuesocialwelfareoptimizingpolicyanalysiswithintheframeworkofanequivalentdeterministicsystem.Butwhiletheexistenceofa“fat‐tailed”distributionfollowsfromBayesianstatistics,asshownbyWeitzman’smathematicalresult,theimplicationsofhis“dismaltheory”concerningtheeconomicpolicyimplicationshavebeenfoundtocomefromtheassumedtheconstantrelativeriskaversion(CRRA)propertyofthetraditionsocialutilityfunctionthathasbeenpositedbyWeitzman,alongwithothereconomistsworkingthewelfareanalysisofclimatepolicy.Millner(2011),inalucidandinsightfulreviewofthecontroversyignitedbyWeitzman(2009b),showsthatthe“harmonicabsoluteriskaversion”(HARA)utilityfunction‐‐ageneralizationthetraditional(CRRA)utilityfunctionthatiswidelyusedinthefinanceliterature‐‐notonlyyieldsfinitewelfaremeasureswhenrisksarefat‐tailed;underplausibleparameterconditionsitalsomakespolicyevaluationsrelativelyinsensitivetothetailsoftheconsumptiondistribution.

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earth’spreindustrialmeantemperaturebeyondwhichtheprobabilityofbeingabletoavertcatastrophicrunawayglobalwarmingbeginstodropbelow50‐50.DuetothecumulativevolumeofpersistingGHGemissionsupuntilthispointintime,however,theworldalreadyiscommittedtoafutureriseinmeanglobaltemperature(relativetothepresent)ontheorderof0.5to1degrees–evenifcarbonemissionswerecompletelystopped.Evenifthe”tipping‐point”weresetata3degreegain,thisleavesrelativelylittleroomforfurtherwarmingwithoutcatastrophicconsequences,astheSternReportpointedout(Stern,2006:p.15).Theimplicationisthatundera“businessasusual”rateofGHGemissionstheremaynotbeverymuchtimeleftbeforethecriticalthresholdwillbecrossedandtheproblemfacingtheworld’spopulationwillbetransformedtooneoflearningtoadaptasbestwecantothealmostunimaginablemountingdamageandsocietaldisruptionsentailedincopingwithadestabilizedclimatesystem.

Againstthisworrisomebackground,itisaparticularlyharshfactofpresenteconomiclifethattomakemajorchangesinproductionsystemsalsotakesconsiderabletime(andresources).Thisisbecausethetransitionfromcarbon‐basedproductiontocarbon‐freeproductioninvolvesfirstthe(further)developmentofcarbon‐freealternativesandsecondlythesubsequentimplementationofthesealternativesthroughinvestmentintangiblecapitalformationprojects,includinginfrastructuremodificationsthatwillhavelonggestationperiods.Furthermore,thedevelopmentofthenewtechnologiesrequiredwillentailthecommitmentofresourcestoR&Dprojectsofuncertainandpossibleextendeddurations.TheclaimsoftheseprogramsoftangibleandintangibleinvestmentthereforewillpressuponconsumptionlevelsforsomeperiodwithoutyieldingresourcesavingsorsignificantlyloweringtherateofGHGemissions.

Thus,therealityoftransitioningtowardsacarbon‐freeeconomyintimetoavoidcrossingthecriticaltemperaturethresholdintoclimateinstabilitywillbe,atbest,anuncomfortable,unremittingandperilousjourneyforhumanity.TheparticularanalyticalchallengethatistakenupinthispaperishowbesttoscheduleR&Dexpendituresandrelatedtangiblecapitalformationatthemacro‐levelsoastomaximisethesocietalwelfare(ormorerealisticallyminimizethedamagetosocialwelfare)associatedwiththeentiretransitionpathtowardsacarbon‐freeproductionregimeinaGHG‐stabilizedenvironment,takingintoaccountthediversionofresourcesawayfromconsumptionthatthoseinvestmentsentail.Towardsthisendwehaveconstructedamulti‐phase,multi‐technologyendogenousgrowthmodelthatallowsfortheexpenditureofR&Dresourcesonthecreationandimprovementof“carbon‐freetechnology”andacknowledgesthatthisformofinvestmentdivertsrealresourcesfromconsumptionandhenceadverselyaffectssocietalwelfareintheshortrun(doing‘nothing’wouldhavenegativewelfareeffectsinthelongrun,however).

Takentogether,theeconomicandgeo‐physicalsub‐systemsposeatimingproblem,sinceoldcarbon‐basedtechnologiesgenerateCO2emissions,andpostponingtheimplementationofnew,carbon‐freetechnologiesreducesthetimeavailableforbuilding

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upnewcapacitywhilecumulativeemissionsofCO2(andotherGHG)remainbelowtheassociatedcriticalthresholdconcentration(andassociatedmeanglobalsurfacetemperature)thatcantriggertheonsetofclimateinstability.StartingsoonenoughtoundertaketheR&D(whichtakesbothtimeandresources)canmakeavailableabetterperformingfamilyoflow‐carbonandcarbon‐freetechnologiesintimetoembodythatknowledgeinnewproductionfacilitiesandavoidcrossingthat“tippingpoint”,buthowsoonissoonenoughwilldependupontherateofR&Dinvestmentsandtheirimpactontheproductivityofcarbon‐freeproductionfacilities,aswellastheeconomy’scapacitytoreplace“dirty”carbon‐usingcapacitywithacarbon‐freecapitalstock.

Thefollowingsectionofthepaperdemonstrateshowthisandrelatedquestionscanbeansweredbyformulatingandsolvingasequenceofoptimalcontrolsub‐problemsforeachdistinctivephaseinthetransitionfroma“business‐as‐usual”carbonbasedeconomytoasustainablecarbonfreeeconomy,andthentyingoptimizedphasestogetherin“stackedHamiltonians”bytheuseoftransversalityconditionsthatguaranteetheoptimalityofthetransitionpathasawhole.Thesub‐sectionsof2applythisapproachindevelopingincreasinglymorecomplicatedmodels,allconstructedonthebasisofthemostelementarygrowthmodel.Section3commentsonanumberofinsightsthatthethreepartialmodelsprovideaboutquestionsoftiminginexercisingvarioustechnology‐fixoptions.Italsoreportspreliminaryresultsfromtheuseofsensitivityanalysistoassesstheeffectsonoptimalinvestmentpathsofvariationsintheassumedlocationofclimate“tippingpoints”.ThepaperconcludesinSection4withabriefreviewofwhathasbeenlearned,whatremainstobestudied,andtheproximatenextstepsinpursuingthislineofresearchintothedesignandimplementationrequirementforasuccessfultechnologyfixforglobalclimateinstability.

2.Modelingthephasesofatimelytransitiontoa”carbon‐free”productionsystem

2.0Anincrementalmodel‐buildingagenda

Ratherthanundertakingfromtheoutsettospecifythestructureofanequivalentdeterministicdynamicalsystemthatincorporatesnumerousfeaturesofeachofthevariouspossibletechnologicalstrategiescouldbedeployedtostabilizeglobalclimate,weproceedtowardsthatgoalinastep‐by‐stepmanner,takingthreediscretepartialmodel‐buildingsteps,andinvestigatingwhatcanbelearnedaboutthedynamicsofanoptimaltransitionpathfromeachofthem,consideredseparately.

Westartfromabasicmodelinwhichtherearetwoavailabletechnologicaloptions,amaturecarbon‐intensivetechnologythatisembodiedinexistingfixedproductionfacilitiesandacarbon‐freetechnologythathasyettobedeployedbyappropriatecapitalformation.ThetableauinFigure1(below)providesasummaryoverviewofactivitiesthatdistinguishthethreephasesofthetransitionthattransformsthe

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economyofourBasicModelfromacarbon‐burning“businessasusual”regimetooneinwhichtheswitchtoproducingexclusivelywithcarbon‐freecapitalfacilitieshasstabilizedtheglobalclimate.Thetableau’shorizontalcolumnsindicatethetwodifferentproductiontechnologiesthatcanbeusedbythegenericeconomy,whereasthedifferentkindoftangible(andintangibleinvestments)thatmaybeundertakenareshownintherows.Theshading(inred,fortheBasicModel)showsthecombinationsofconcurrentinvestmentandproductionactivitiesthatarecompatiblewithefficientresourceallocationineachphaseofthetransitionpath.

Next,weintroduceintotheBasicModelset‐upthepossibilityofundertakingR&Dexpendituresduringthe“businessasusualphase”.Theseinvestmentsin“directedtechnologicalchange”areaimedtoreducetheunitcapitalcostsofproductionfacilitiesembodyingthecarbon‐freetechnology,raisingtheaverageproductivityofKBenoughtomatchthatofcapacitybasedonthecarbon‐usingtechnology.Inthiselementarymodelofathree‐phase“endogenousR&D‐driventransition”,onlywhenthattechnologicalgoalisattaineddoesR&Dexpenditures(andfurthertangibleinvestmentincarbon‐intensiveproductioncapacity)cometoahalt,andtangiblecapitalformationbeginsdeployingthecarbon‐freetechnology.ThisisindicatedinFigure2bythetableau’sblueshadedareas.

Figure1:BasicModel(Red)

Investment Production (Capacity in Operation)

 Old Carbon High emission rate

Greened CarbonLower emission rate 

Carbon Free Zero emission rate 

Carbon‐using capital KA 

production business as usual 

   

Carbon‐free capital KB 

joint production    joint production 

   carbon free production 

R&D on carbon‐free technology 

     

R&D on clim–mate eng.  

     

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Figure2:BasicModelwithR&D(Blue)

Inthethirdofthepartialmodelstobeexamined,thestructureofthebasicmodelisextendedbyallowingfortheexploitationofathirdclassoftechnologicalopportunities.ThesedonotrequiresignificantR&Dinvestments,sincetheymakeuseofknown“core”engineeringtechniquestoprovidean“intermediate”,lesscarbon‐intensivesetofproductiontechnologiesthatineffectserveto“green”existingcarbon‐usingdirectproductionfacilitiesandinfrastructure.BecausetheyarecharacterizedbylowerratesofCO2emissionsperunitofcapitalthantheoldcarbon‐basedtechnologies(evenifahighercapitalcostperunitofoutput),theyprovideareducedemissions‐outputratio.Thustheycan“buytime”tomakethetangibleinvestmentsneededtoswitchtoacarbon‐freeproductionregime.

Figures3.1and3.2displayintableauformthepairofalternative5‐phasetrajectoriesthatariseinthecaseofthebuy‐timemodel.Eachversioncanbesolveforanoptimaltransitionpath,buttofindtheoptimumoptimorumitisnecessarytoevaluateandcomparetheimpliedpresentvalueofthesocialwelfareindexassociatedwitheachofthem.20

20Notethattheproductionconstraintsinthemodel,includingthosederivedfromthetechnologicalparametervalues,maydeterminewhich,betweenalternativetrajectoriescansatisfytheoptimalityrequirements,sothatanswertothequestionofwhichtochoosebecomeanempiricalmatter.Seesection2.3forfurtherdiscussion.

Investment  Production (Capacity in Operation)

 Old Carbon High emission rate

Greened Carbon Lower emission rate 

Carbon Free Zero emission rate 

Carbon‐using capital KA 

production  business as usual 

   

Carbon‐free capital KB 

joint production   joint production 

   carbon free production 

R&D on carbon‐free technology 

production  business as usual 

   

R&D on climate engineering 

     

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Figure3.1:BuyTimeModelTrajectory1(DarkGreen)

Investment  Production (Capacity in Operation) 

 Old Carbon High emission rate 

Greened Carbon Lower emission rate 

Carbon Free Zero emission rate 

Carbon‐using capital KA 

production business as usual 

   

Carbon‐economizing KD (retrofitted KA)  

joint old & green carbon production 

joint old & green carbon production 

 

 Greened carbon production 

 

Carbon‐free capital KB 

 Joint greened carbon and carbon‐free production

Joint greened carbon and carbon‐free production 

    Carbon‐free regime 

Figure3.2:BuytimemodelTrajectory2(LightGreen)

Investment  Production (Capacity in Operation) 

 Old Carbon High emission rate 

Greened Carbon Lower emission rate 

Carbon Free Zero emission rate 

Carbon‐using capital KA 

production business as usual 

   

Carbon‐economizing KD (retrofitted KA) 

Joint old and green carbon production 

Joint old and green carbon production 

 

 Carbon‐free capital KB 

Joint old and green carbon, carbon‐free production 

Joint old and green carbon, carbon‐free production 

Joint old and green carbon, carbon‐free production 

 Green carbon and carbon‐free production 

Green carbon and carbon‐free production 

   Carbon‐free production 

Intheoptimalcontrolsolutionseachofthethreeforegoing“partial”modelsofaclimatestabilizingtransition,theinternallyoptimizedphasesofthemodelaretiedtogetherbytransversalityconditions.Thatmakesitpossibletoobtainthe“requirements”oftheentireoptimaltransitionpath,intermsoftheoptimumdurationsofitsphases,andtheoptimizedwithin‐phaselevelsofactivityforproduction,consumption,andinvestmentrates(boththeintangibleR&Dandthealternativetangibleformsoftechnology‐embodyingcapitalformation)–aswellastheimpliedflowsofCO2emissionsuptothepointatwhichtheglobaleconomy’sproductionregimereachesthegoalofacarbon‐freestatethatavertstheonsetofirreversibleclimateinstability.Eveninthesimplestofpossiblegrowth‐modelsettingthathasbeenspecified,thesolutionsofeachofthese

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successivelymorecomplicatedmodelstofindtheirrespectiveoveralloptimaltransitionpathsmustbeobtainedbynumericalanalysisoftheresultingsystemsof“stackedHamiltonians”–thedetailsofwhichfollowingsectiondiscussesadseriatim.

Thepresenceofdifferentphasesoftransitioninwhichproductioncapacityembodyingnew(comparatively“clean”)technologiesisbuiltupandthatbasedonold(“dirtier’)techniquesisrundownandeventuallydiscarded,isthefeaturethatrendersthismodellingframeworkdifferentfromthestandardAKgrowthmodel.Acentraltaskforouranalysisofeachofthemodels,therefore,mustbetofindtheoptimalconfigurationofspecificinvestmentandproductionactivities,incombinationwiththeoptimalityoftheirschedulingduringthecourseofthetransitiontoaviableandstabilizedclimate.

Thequestionsaddressedbythismodellingexercisearenotpredictive.Ratherthanventuringtothrowlightonwhatwillhappen,theyaskwhathastobedone,andwhenmustitbedoneinordertogetfromacarbon‐basedproductionsystemtoacarbonfreeproductionsystemthatwouldavertrunawayglobalwarming,whilemaintainingthehighestpresentvalueofsocialwelfarethatisconsistentwithattainingthatgoal.

WehavebegunbysettingupaHamiltoniansystemwithastandardCIESinter‐temporalutilityfunctioninwhichconsumption(percapita)isitsargument,sincepopulationandlaborinputs)aresuppressedandimplicitlyassumedtobeconstant.21Thelatterisconsistentwithourmakinguseofthesimplestpossibleoveralldynamicsetting,namely,theAK‐modelofendogenouseconomicgrowthduetoRebelo(1991),andBarroandSala‐i‐Martin(2004,ch.4).Wehaveextendedthelatterstructurebyreplacingitsspecificationofasingle,linearcapital‐usingproductiontechnologybytheassumptionthattherearemultiplediscrete(linear)technologiesthatmaybeusedeithersimultaneouslyorsequentially,includingonethatcanbeenhancedbydirectedinvestmentsinR&Dactivities.

Thereducedunitcostofcapitalembodyingthecarbon‐freetechnologyasaresultofdirectedR&Dexpenditures,however,isonlyrealizedthroughthetechnology’ssubsequentdeploymentingrosstangiblecapitalformation.Toputasomewhatsharperpointonthis,itisnotenoughtothinkaboutR&Dpoliciesandprograms.Mechanismsofdiffusionintouse,asdistinctfromthedisseminationofinformationabouttechnologicalinnovationsalsomustbefiguredexplicitlyamongtherequirementsofa“technologyfix.”

21Readersinterestedinthedetailsofthestructuralequationsandsolutionsofthemodelsdiscussedheremayconsultthe“TechnicalAnnex:ModelingOptimalMultiphaseTransitionPathstoSustainableGrowth”(July2011),whichisavailableathttp://siepr.stanford.edu/system/files/shared/OptimalMulti.pdf,orcanbedownloadedasasupplementtoanearlierversionofthispaper(presentedon12/14attheSIEPR‐GEEGSocialScienceandTechnologySeminar)at:http://siepr.stanford.edu/programs/SST_Seminars/index.html.

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AlthoughtheAK‐settingisextremelysimple,becauseweallowfordifferentphasesinthetransitionfromcarbonbasedtocarbonfreeproduction,theresultingmodelisabletogenerateasetoftimepathsforthetransitiontoastabilizedclimate(andstationarylevelofatmosphericGHGconcentration)that,inpractice,canbedeterminedonlybytheuseofnumericalmethods.22Afuture,morecomplicatedextensionofthisset‐upisenvisaged,inwhichthecumulativeincreaseinGHGconcentration,orthemeanglobalsurfacetemperaturegainassociatedwithit,enterstheutilityfunctionnegatively.23

Thetransitionphasesmentionedearlierarelinkedtogetherthroughtransversalityconditions(cf.LeonardandVanLong(1992),ch.7inparticular)thatimplicitlydefinethephasechangesintermsofrathergeneraloptimalityconditionsthatoftengiverisetoconditionsontheco‐statevariablesthathaveaclear‐cuteconomicinterpretation.Forexample,itcanbeshownthatitisneveroptimaltosimultaneouslyinvestintwoexistingtechnologiesthataredifferentwithrespecttotheircapitalproductivityandtheiremissioncharacteristics.

Thus,inthecurrentmodelsetting,theexistenceofacumulativeemissiontipping‐pointwillgeneratenotionalemissioncostsassociatedwiththeuseofcarbon‐basedcapital.24Investmentinthelattertypeofcapitalwillstopandthatincarbonfreecapitalwillstartthemomenttheshadowpriceofcarbonbasedcapitalfallsbelowthatofcarbonfree

22Giventhefactthatwedistinguishexplicitlybetweendifferentsub‐phasesduringthetransition,weendupwitha‘stack’ofHamiltonianproblemsthatarelinkedtogetherthroughasetoftransversalityconditions(TVC’s),seefurtherbelowinthetext.Thefirstorderconditions,incombinationwiththeTVC’sandthegiveninitialvaluesofthestatevariables(andthegiventerminalvalueofcumulativeemissions),giverisetoasetofstronglynon‐linearconstraintsontheremaininginitialvaluesofthemodel’simpliedtimepathsforco‐stateandcontrolvariables.Usingtheseinitialvalues,thesimplicityoftheAK‐settingallowsustoobtainclosedformanalyticalexpressionsbymeansofdirectintegrationforthetimepathsofallvariables—exceptforthecumulativestockofGHGemissions.Theresultingpathsthemselvesareinparthighlynon‐linearandlargelyintractable(exceptfortheirunderlyingroots/structuralequations)necessitatingrecoursetonumericalexercisesinordertoinvestigatethepropertiesofthemodel,someofwhicharenotintuitivelytransparentapriori.Moreover,thepathforcumulativeemissionscannotbeobtainedbyanalyticalmeansandrequiresnumericalintegrationstartingfromsomeinitialguessofthelengthoftheBAUphase.TheoptimumdurationoftheBAUphasethenisimpliedbytherequirementthattotalcumulativeemissionsovertheentiretransition(allphases)shouldjustmatchapre‐specified“threshold”value.Evidentlyadifferentapproachwillberequiredbyastochasticformulationofthetransitionprocess–ofwhichthemodeldiscussedherecanbeviewedastheequivalentdeterministicsystem.

23SeeArrow(2009)andWeitzman(2009a,b)onthesignificanceofspecifyinganadditivetemperatureeffectthatisnegative,interpretingthisasadirect“environmental”effectuponsocialwelfare.Analternativeformulationisavailable‐‐inwhichthenegativeterminthesocialutilityfunctionistheinverseofthedifferencethecritical“threshold,”or“tippingpoint”temperature(beyondwhichthewarmingprocessisexpectedtobecomeself‐reinforcing)andEarth’sprevailingglobalmeansurfacetemperature.Thisformulationwouldhaveessentiallythesameconsequencesforinter‐temporalresourceallocation,butitwouldadmittheadversepsychiceffectofapproachingtheexpectedpointatwhichhumanitywill,forallintentsandpurposeshavelostitsabilitytostabilizethewarmingtrendandavertthecatastrophiconsetofclimateinstability.

24Weusetheterm“notional”heretodenotetheexistenceofarealcost,thatis,however,generallynot,oratleastnotfully,paidforinpractice.

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capital.Asthecostofinvestmentisrepresentedbythewelfarelossassociatedwithconsumptionforegone,andasoneunitofinvestmentgeneratesoneunitofconsumptionforegoneinbothcases,thistransversalityconditionimpliestherequirementthatinvestmentwilltakeplaceinthetechnologythatgeneratesthehighestmarginalnetwelfareperunitofinvestment.Likewise,thediscardingofexistingcapacityistypicallyanactivitythatsignalstheendofaparticularphaseandthebeginningofthenextone.

Theoptimumtimingofsuchaphasechange,andhencethephases’durations,willbegovernedbyageneraltransversalitycondition,namelythattheshadowpriceofanincrementalunitofproductioncapacityembodyingtheparticulartypeoftechnologyshouldbezeroattheexactmomentthatitistakenoutofproduction;otherwise,ifithadapositiveproductiveuse,whydiscardit?Thistransversalityconditionturnsouttobeequivalenttotherequirementthataunitofexistingcarbon‐basedcapitalshouldbediscardedatthemomentthatthemarginalwelfarebenefitsofusingthatunitofcarbon‐basedcapacitydropbelowthecorrespondingmarginalemissioncosts.25

ThefeaturesoftheBasicModelof“technologyswitching”thathavebeenpresentedabovemaybeformallysummarizedintheequationssetoutbelow,whichdescribetheobjective(socialwelfarefunction,U,systemofintertemporalproductionconstraintsandthetransversalityconditionsthatdeterminetheoptimalsolutionforthemulti‐phasetransitionpath(seeTable0).This,thesimplestofthemodels,isdiscussedinfurtherdetail,andthenextendedinthesectionsthatfollow.

2.1TheBasicModel

Thefirstandfoundationalversionofoursuiteofmulti‐phasetransitionmodelsiscalled(appropriatelyenough)theBasicModel.Inthistherearetwoalreadyavailableproductiontechnologies,onethatiscarbon‐basedandalreadyinuse,andacarbon‐freetechnologythatattheoutsetisyettobedeployed.26Thisset‐upcreatesthepossibilityoftherebeingthreedistinctphases.

25Thisconditioncloselyresemblesthenegativequasi‐rentconditionthatgovernstheeconomiclifetimeofclay‐clayvintagesinaperfectcompetitionsetting(seeMalcomson(1975),forexample).

26Forpurposesofthisanalysisitissupposedthattheperiodunderconsiderationcommenceswhenacarbon‐free(B)technologybecomesavailable,althoughitaverageproductivityislowerthanthatofthecapital(KA)embodyingthecarbon‐usingtechnology;or,equivalently,thatthecostoftheproductioncapacity(KB)perunitofoutputexceedsthatofKA.27AstheBAUtechnologyhashighercapitalproductivitythantheCFRtechnologybyassumption,welfarewouldnotbemaximisediftheflowofCO2emissionswouldstopbeforethecumulativeemissionthresholdwouldbereached.

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Table0____________________________________________________________________________________________________

Phase‐SpecificEquationsoftheBasic3‐PhaseTransitionModel

(production,BAU,JPR)

(netinvestment,BAU)

(netinvestment,JPR)

(emissions,BAU,JPR)

(productionJPR,CFR)

. (netinvestmentCFR)

. (netinvestmentJPR)

+ (Cum.Emissions)

. / 1 (intertemp.socialwelfare,ALLphases)

⟺ (TVCdefiningTJPR(equality Hamiltonians))

⟺ (TVCdefiningTCFR:deactivationYA)

=0 (TVCvalueofdeactivatedcapital=0)

→ . 0 (TVC,standardAKTVCCFRphase)

Infirstphase,whichcanconvenientlybelabelledBAU(for‘BusinessasUsual’)thecarbon‐freetechnologyremainsun‐deployed,andtheonlypositivetangiblecapitalformationtakingplaceaddstothestockofcarbon‐usingproductionfacilities,whicharethecurrentsourceofthesystem’srealgrossoutput.Wehaveassumedthattheavailablecarbon‐freeproductiontechnologycannotmatchthecarbon‐basedalternativeintheaverageproductivityofthecapitalfacilitiesthatembodyit.Tostartdeployingandusingitimmediatelythereforewouldentailanimmediatesacrificeofconsumptionthatcouldbeenjoyedbycontinuingtoinvestandrelyuponcarbon‐basedcapacity.Better,therefore,toplantosacrificefutureconsumption,thepresentutilityvaluewillbediscounted–untilthefuturenegativeshadowpriceoftheassociatedCO2emissionsbeginstoapproximatetheincrementalutilityoftheextraconsumptionflow.ThisistheeconomiclogicofprolongingtheBAUphase,upuntiltheapointwhenitceasestobe

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optimaltogofurtheralongthatpathandgrosscapitalformationswitchesfromcarbon‐usingtocarbon‐freeproductionfacilities.

AsecondphaseopenswiththestartofpositiveinvestmentinfacilitiesembodyingtheCFRtechnology,andthecessationofBAUcapitalformation,thereforemarkstheopeningoftheBasicmodel’ssecondphase.Fromthispointforwardthereisnegativenetinvestmentincarbon‐usingfacilities,sincethephysicaldepreciationoftheexistingstockisnolongerbeingoffsetbypositivegrossinvestmentincapitalofthattype.ThisphaseoftheBasicmodelisreferredtoas‘thejointproduction’(JPR)phase,ascapitalgoodsembodyingbothclassesoftechnologybeingusedtogenerateoutput.Thetransitiontoacarbon‐freeproductionregimeandclimatestabilizationiscompletewiththeshut‐downoftheremainingcarbon‐usingfacilitiesand,hence,stoppageoftheflowofCO2emissionsjustasthecumulativestockofemissions(andthetemperature)approachtheirrespectivecriticalthresholdlevels.Thatmarksthebeginningofthethirdandfinalphaseofsystem’stransitiontosustainable,carbon‐freegrowth.27

Theintensityoftheflowsduringeachphase,aswellasthemomentsintimeatwhichthevariousphasechangesarescheduled,allfollowfromthefirst‐orderconditions(FOCS),thetransversalityconditionsandfromthegiveninitialandterminalvaluesforthestate‐variablesthatarepartoftheoptimumcontrolproblem.Eventsandphase‐

27AstheBAUtechnologyhashighercapitalproductivitythantheCFRtechnologybyassumption,welfarewouldnotbemaximisediftheflowofCO2emissionswouldstopbeforethecumulativeemissionthresholdwouldbereached.

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changesinthecontextoftheBasicModelaresummarisedinFigure4(below),which

showsthedevelopmentovertimeofthestockofcarbon‐basedcapitalKA,thestockofcarbon‐freecapitalKBandcumulativeemissions,GCO2

Thepointsmarkedonthetime‐axisoftheFigure(TU,TJ,TF)indicatetheoptimalmomentsatwhichtherespectivephases(BAU,JPR,CFR)begin.ItmaybenotedthatalthoughthecumulativestockofemissionscontinuestoriseduringtheJPRphase,isdoessoatadecreasingrateasthestockofcarbon‐basedcapitalKAisrundownandreplacedbyitscarbon‐freesubstitute.Thefinal,carbon‐freeproductionphasestartswhencumulativeemissionsatmosphericconcentration(GCO2)almostreachestheexpectedthrclimate“tipping‐point,andthecurvemarkedCO2becomesflattotherightofTF.

Evidently,thevariousphasesinthemodelarequalitativelydifferent.Thefirstphase,i.e.theBAUphase,usesahighgrowthtechnologywhichunfortunatelyquicklyraisesthestockofcumulativeCO2emissionsthatisboundedfromabovebythecumulativethreshold.Beforethatlevelisreached,investmentincarbonfreetechnologymusthavetakenplaceduringphaseJPRtobringcarbonfreeproductioncapacityuptoalevelwheretheswitchfromusingcarbon‐basedcapacitytousingcarbonfreecapacitywouldnotforcechangesinconsumptionlevelsthataretoodisruptive,sinceconsumersdislikeconsumptionshocks,asisimpliedbyouruseofasocialwelfarefunctionthatdepends

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uponthepresent(discounted)discountedlevelsofpercapitaconsumption,andisofthe(CIES)formallowingforrelativeriskaversion.

Fort>TFtheworldis“green”,and,giventhemoreexpensivecarbon‐freeproductionfacilitiesthatarerequiredtostabilizetheatmosphericCO2‐equivalentconcentrationlevel,theoutput(percapita)willhavetogrowforsomewhileattherelativelyslowpaceatwhichtheKBisaccumulatingwiththegrosscarbon‐freecapitalformationrateconstrainednottocuttooheavilyintoconsumption.Eventually,however,thebuild‐upofthecarbon‐freecapitalstockissufficienttosupportbothrisingconsumptionandahigherrateofinvestmentinKB,whichthereafteraccumulatesataquickenedpace.28

2.2ABasicModelwithEndogenousR&D

InadditiontotheBasicModel,wehavespecifiedaversioninwhichR&DcanbeundertakentoimprovetheproductivityoftheCFRtechnologybeforeitsactualimplementationthroughgrossinvestmentintheCFRtechnology.R&DrequiresresourcestobeinvestednowinreturnforafutureintangibleassetintheformofknowledgeofhowtoembodyCFRproductiontechniquesinproductionfacilitieswhosecapitalcostperunitofoutputwillbelowerthanisthecaseatpresent.Surprisingly,thisviewoftheroletobeplayedbyR&DincontrollingfutureGHGemissionsissomethingofanoveltyinthesmalleconomicliteraturethathasemployedintegratedassessmentmodellingofclimatepolicyoptions,becauseinsofarasthecontributionofinvestmentsinR&Dto“directed”technologychangeandinnovationhasbeenconsideredatall,the“direction”hasbeentakentobetheloweringofCO2emissionsperunittotaloutputintheeconomy.29Tosharpenthecontrastwiththelatterapproach,Section2.3(below)

28ThestructureoftheBasicModelhassomecommonalitieswiththeanalysisinValente(2003)ofatwo‐phaseendogenousgrowthmodelinwhichproductionbasedonessential(energy)inputsobtainedfromexhaustibleresourcesswitchestoa“backstoptechnology”thatprovidesaconstantsupplyofsustained(e.g.,solar)energy.R&Dinvestmentspermitgrowththroughproductivityimprovementsthatoccuratthesameratewitheithertechnologies(althoughthelevelsofproductivitycandiffer),andtheoptimaltimingoftechnologyswitchingisdeterminedbywelfaremaximizationinwhichutilitydependsupondiscountedpercapitaconsumption.Valentesimilarlyobtainsoptimalcontrolsolutionsforeachstageandtiesthesepathstogetherwithtransversalityconditions.But,unlikeinthepresentanalysis,thepointatwhichtherenewabletechnologycanbeefficientlyembodiedintangiblecapitalusedinproductionisnotendogenouslydeterminedbydirectedR&Dexpendituresandthediversionofoutputtobuildingthe“renewables‐base”capitalstock.NorisapositiveshadowpriceexplicitlyattachedtousingtheexhaustibleresourceinValente’sanalysis,becauseitabstractsfromthegeophysicalclimate‐systemconstraintsthataffecttheoptimaltransitionpath.Aswillbeseen,theresearchapproachhereexaminesmuchmorecomplicated,multi‐stagetransitions.

29Tothebestofourknowledge,theearliestpreviousinvestigationoftheeffectsofallowingforendogenoustechnologicalchangeinacomputationalclimatepolicyassessmentmodelappearsintherelatedpapersbyGoulderandMathai(1998),andGoulderandSchneider(1999),whichspecifytheeffectofR&Dexpendituresonchangesinaneconomy‐widetotalfactorproductivitycoefficient.InthesepioneeringinvestigationsoftheimpactofinducedtechnologicalchangeontheattractivenessofCO2abatementpolicies,absolutechangesinproductivity(orrealmarginalcost)levels,ratherthantheproportionaterateofchangeinproductivitywasspecifiedasapositiveincreasing(decreasing)functionofR&Dexpenditures.InGoulderandSchneider(1999:pp.216,223)thisfunctionalrelationshipnot

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examinestheoptionofachievinggreaterefficienciesintheuseofcarbon‐basedenergycanbeachievedwithoutR&Dexpenditures,butbyinvestmentsintheengineeringimplementationofexistingtechnologicalknowledgethatwouldresultinhigherunit(tangible)capitalcostsofcarbon‐basedproductionfacilities.30

Furthermore,inspecifyingtheimpactofsuchR&DinvestmentsontheunitcostofCFRcapitalgoods(KB)wedeliberatelydepartfromthe“R&Dproductionfunction”formulationthatisfamiliarinendogenousgrowthmodelsfollowingLucas(1988),Romer(1990)andAghionandHowitt(1991).Ratherthantakingtheinstantaneousrate

ofimprovementintheproductivityofCFB‐embodyingcapital, /B B ,tobeproportionaltothecurrentabsoluteflowofR&Dresourceinputs,R,asin

0 1B R B ,

analternativespecificationisproposedhere:

( ), 0 1B R B B ,

whereςisaconstantproductivityparameter,and representsthemaximumvaluethatBcanattain.31

Thereareseveralfeaturesofthisspecificationthatargueforits’useinthepresentapplicationscontext.Firstly,ithastheadvantageofintroducingdecreasingreturnsto

furtherrestrictedintheirmathematicalanalysisofa2‐periodmodel;whereasthelinearspecification,B R wasusedintheirdynamicmulti‐periodgeneralequilibriumsimulationstudies.Nordhaus(2002)takesadifferentapproach,ineffectassumingthatinducedresearcheffortfocusessolelyonreducingtheCO2emissionsintensityofproduction.Nordhauspositsaseparateproductionfunctionforcarbon‐basedenergyinputswhichareusedinfixedproportiontothenon‐energyinputsinaneconomy‐wideproductionfunction.Hethenspecifiesthattherateofchangeofthecarbon‐intensitycoefficientinenergyproductionisgivenby(dσ/σ)=‐[ψRβ–Ω],whereΨ>0isaconstantandΩ>0isa(constant)rateofobsolenceofthetechnicalknowledgegainedthroughR&Dandreflectedinσ.TheempiricalfindingsofGriliches(1973),Hall(1995)andothersonthelinkbetweencommercialR&DandindustrialproductivitygrowtharecitedbyNordhausinsupportofthisspecification,althoughingeneralthetime‐spanstowhichthedataonindustry‐levelandfirm‐leveldatarelatearefarshorter,andhardlyoftheglobalscopetypicallycontemplatedinIAMmodels.

30Ofcourse,thisdistinctionisnotnecessarybecauseonecanmodelthesituationinwhichR&Dexpendituresareallocatedbetweenthetwo“directions”oftechnologicalchange,loweringraisingtheratioofgrossoutputperunitofCO2emissions,andraisingtheproductivityofcarbon‐freetangiblecapitalusedinproduction.Butbecausethepresentinvestigationisconfinedtomodelingthepathstosuccessfulclimatestabilization,andthetimescaleforthatisfarshorterthanthehundredsofyearscontemplatedintheexistingvariantsoftheNordhaus(1994,1999,2002,2007)DICEmodel,itisnotunreasonabletosupposethatthepresentlyexistingstockofimplementabletechniquesforenhancingtheproductivityofcarbon‐energyinputscouldsufficeforaconsiderablenumberofdecadeswithoutrequiring“refreshment”byfocusedinvestmentofR&Defforts.Seefurtherdiscussionofthe“buytime”optioninsection2.3,below.

31With , , asconstantpositiveparameters,BandRthereforebecomeadditionalstate‐andcontrolvariablesinourdynamicsystem.

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R&Dinasettingthat,unliketheconventionalendogenousgrowthmodels,excludesthepossibilityofaspecifictechnologybeingrenderedinfinitelyproductive(andsoresultingininfinitelyrapidgrowth)merelybytheapplicationofmoreandmoremassiveR&Dexpendituresatanyparticularmomentintime.32AllowingdecreasingmarginalreturnsinR&Drecognizesthatatagivenstageintheadvanceofknowledgethestateoffundamentalscientificunderstandingofthephysicalprocessesinvolvedmaystillbeinadequatetopermittheeffectiveapplicationofmoreandmoreresourcestothesolutionofaparticularpracticalproblem‐‐suchasthefurtherimprovementoftheproductivityofaparticularclassoftechnology‐embodyingcapitalfacilities.

Secondly,thisformulationoftheeffectsofinvestmentinR&DactivitiesmaybethoughttoreflectaPlatonicworldinwhichafinitenumberofsolutionpossibilitiesfortechnicaltransformationsarepresentfromthestartoftime,buttheseasarulewillnotrevealthemselvesspontaneously.Theycanbeuncovered,however,andformulatedforpracticalapplicationthroughcostlyresearchanddevelopmentproceduresbasedupontheexistingstateoffundamentalscientificknowledge,ratherthanbeingcreateddenovoandwithoutlimitbytheexpenditureofresourcesintheperformanceofR&Dactivities.33ThatmorerestrictiveviewofthetransformativepowerofinvestmentinR&Dandisappropriatenotonlyfortheforegoinggeneralreasons,butalsobecausetheconcerninthiscontextisnotwiththeundirectedglobalexpansionofthetechnologicalopportunitysettypicallyenvisagedintheoreticalgrowthmodels.Rather,theaimofthe“directedR&D”inthepresentmodelistoenhancetheeconomicpropertiesofparticularkindsofprocessinventions,withnewproductinventionsonlyinsofarasalterationsinproductcharacteristicsareconsequentialfortheraisingtheefficiencyofcapitalinputsintocarbon‐freeproductionprocesses.34

Withintheframeworkcreatedbyintroducingthis(oranyother)R&DproductionfunctionintotheBasicmodel,thereareagainthreedistinctphasesofthetransitiontoastabilizedclimate.Table1(below)comparesthephasesintheoriginalbasicmodelwiththeversionintroducingR&D.ItshouldbenotedthatwhiletheR&Dmodelresembles

32Theequilibrium(steady‐state)growthrateinastandardAK‐modelriseslinearlywiththerategrowthintheproductivityofcapital(cf.BarroandSala‐i‐Martin,2004),andthereforewiththeflowrateofR&Dexpenditures.Theexistenceoftechnology‐specificintrinsicproductivitylimitsset–inthelimit‐‐bythephysicalpropertiesofthematerials,chemicalandelectricalprocessesentailedinproductionmakesthissoimplausiblethatevenitsassertionasametaphorisofdubioususefulness.

33ItalsohastheadvantageofbeingjointlyconcaveinBandR,whichisanecessaryconditionforthewelfaremaximizationproblemtohaveasolution.

34Followingthisinterpretation,addingendogenoustechnologicalchangetotheBasicModelallowsustocharacterizetheoptimalpathofglobalR&DthatisdirectedtoincreasingtheproductivityofCFRcapital.CorrespondinglytheimpactofR&Dinvestmentoneconomicwelfareismodeledasbeingfeltindirectly,ratherthandirectlyintheformofpureproductqualityenhancements.Inotherwords,thewelfaregainscomethroughreductionofthesacrificeofconsumptionutilityrequiredinthetransition,andforthesubsequentsustainedgrowthof(percapita)consumptionunderstabilizedclimaticconditions.

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theBasicModel,becauseithasthreephasesandtwooptimumswitchingmoments,itdiffersinhavingthreestatevariablesandtwocontrolvariables.

Table1.ComparisonoftheBasicandEndogenousR&DModels

Basic model

Phase BAU1  Business as usual 

JPR1  Joint production 

CFR1 

Carbon‐free 

Investment in  Carbon‐based capital Carbon‐free capital Carbon‐free capital

Output using  Carbon‐based capital Both carbon‐based and 

reduced emission capital 

Carbon‐free capital 

only 

Endogenous R&D Model

Phase BAU2  Business as usual 

JPR2  Joint production 

CFR2 

Carbon‐free 

Investment in  Carbon‐based capital Carbon‐free capital Carbon‐free capital

R&D 

investment in Carbon‐free technology  Carbon‐free capital  Carbon‐free capital 

Output using  Carbon‐based capital Both carbon‐based and 

carbon‐free capital 

Carbon‐free capital 

only 

ThereasonisthatitturnsouttobeoptimaltostartdoingR&DtoimproveaparticulartechnologyfromtheveryfirstmomentthatinformationaboutaworkableCFR‐technologybecomesavailable.OntheassumptionthattheCFRtechnologyisdiscoveredattimezerobutislessproductivethanthematurecarbon‐usingtechnologywhenembodiedinequallycostlyproductionfacilities,R&Dactivitythenbeundertaken(only)duringthephasewhentangiblecapitalformationandproductionadheretothecarbon‐dependentfeaturesoftheBasicmodel’sBAUphase.Followingthat,aJPRphasewillcommencewithgrossinvestmentdirectedtodeployingtheimprovedCFRtechnologyinKB‐typeproductionfacilities.

2.3A“Buy‐TimeModel”–GreeningCarbon‐basedCapitalintheBasicModel

Wenowturntothethirdofthepartialmodels,inwhichthestructureofthebasicmodelisenrichedbyintroducingathirdcategoryoftechnologies,namelyaknown“intermediate”classofcarbon‐usingproductiontechniquesthatarecharacterizedbylowerratesofCO2emissionsperunitofcapitalbutcapitalproductivityperunitofoutputthatislowerthanofthematurecarbon‐basedtechnologies,butnotasmuchlowerthanthatoftheinitial(pre‐R&D)versionsofthecarbon‐freetechniquesof

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production.35Thisconceptualizationcorrespondsbroadlytothevarietyofwell‐groundedengineeringmethodsmaybeusedtoupgradecarbon‐usingproductionfacilities,whetherbyimprovingtheenergyefficiencyofresidential,businessandgovernmentofficebuildingsbybetterinsulation,heatingandcoolingsystems,andreducingthepassiveconsumptionelectricitybyelectricalandelectronicappliancesbyconfiguringthemtoswitchoffcompletelywhennotinuse,andsoforth.ButitalsosubsumesproposalsfortheexpandedexploitationofnaturalgasasareducedGHG‐emissionssourceofenergy,includingtheacknowledgementthatincrementalcostsperBTUwouldberequiredtocurtailtheenvironmentaldamagecurrentlyassociatedwith“fracking”methods.

Break‐throughresearchandnovelengineeringprinciples,however,arenotrequiredtoexploitthisclassofcarbon‐basedenergysourcesandproductiontechnologieshere,butspecificapplicationsofcoreengineeringknowledgeadaptationofexistingdesignstolocalcontextswillraisetheaverageunitcapitalcostscarbon‐usingplantequipmentthathasundergonethiskindof“green‐upgrading,”aswellasthatofa“lessdirty”energysourcesuchasnaturalgas,andsafernuclearpower‐plantswithprovisionsforlong‐termsequestrationoftheirtoxicwaste.Thegaintobehadbyseizingthis“low‐hangingfruit”comesinthereducedrateofCO2,whichmeansundertakingtheentailedincrementalcapitalexpendituresconstitutesawayto“buyingtime”earlyinthetransitioninordertobeabletosubsequentlyproceedmoreslowlyinreplacingthewholecarbon‐basedproductionregimewithonebasedonnew,carbon‐freeproductionfacilities.

35Onthelatter,seetherecentnoticedexampleoftheannualconsumptionofover$3billionworkofelectricpowerintheU.S.byHDTV“set‐top”boxessuppliedbycablecompanies,duetothechoiceoflowcostdesignsthatthatdonotactuallystopdrainingpowerwhentheyareswitchedoff(see,“AtopTVSets,aPowerDrainthatRunsNonstop,”TheNewYorkTimes,June26,2011:p.1.).TheMcKinseyGlobalInstitutedevotedconsiderableattentionintheyearsbeforethefinancialcrisistostudiesofcurrentandnear‐termoptionsforenergyefficiencyroutestoCO2emissions‐reductionsandprivatecostssavingthroughupgradingofproductionanddistributionsystems.See,e.g.,FarrelandRennes(2008);GroveandR.Burgelman(2008).IntheU.S.,numerousproposalsforthiskindof“retrofitting”havehaddifficultygainingpolicy‐tractionduetotheprevailingpolicybiastowardsresearchsubsidizestosupport“innovation”inrenewableenergytechnologies.”Morerecently,thecaseforexpandedexploitationofnaturalgasandgreaterinvestmentinotheropportunitiestoimprovetheproductivityofcarbon‐basedtechnologiesthat(similarly)offerhigheroutputpervolumeofGHGemittedhasbeencogentlyandvigorouslyadvancedbyBurtonRichter(2010).DuringthecrisisandrecessionyearstheattentionoftheEuropeanCommissionshiftedawayfromtheexpensiveandlonger‐termresearchenvisagedbyitsambitiousStrategicEnergyTechnologiesPlan[SET,COM(2007)723];itfocusedinsteadonavarietyofshorter‐termtacticsaimedatstimulatingaggregatedemandinwaysthatwouldimplementalreadyavailabletechnologiesfor“green”purposes,notably:retro‐fittingbuildingsforgreaterenergyefficiency,supportingtheautomotiveindustrytoincreaseproductionoflow‐CO2vehiclesusingelectricbatteriesandsecondgenerationbio‐fuels(seediscussioninDavid,op.cit.(2009).IntheU.S.,numerousproposalsforthiskindof“retrofitting”havehaddifficultygainingpolicy‐tractionduetotheprevailingpolicybiastowardsresearchsubsidizestosupport“innovation”inrenewableenergytechnologies.

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Notsurprisingly,therefore,theextenttowhichthe“buytimeoption”willbeattractivetoexploitinthecontextofourBasicmodel,bybuildingupproductioncapacity(KD)inthisintermediate“greened”formratherthancapitalthatembodiesthemature,morecarbon‐intensivetechnology,dependsnotonlyontheassociatedcapitalproductivitybutalsoonthereductiongainedinemissionsperunitofoutput.Itturnsoutthatintroducingthepossibilityofutilizingthisthirdclassoftechnologiesgivesrisetoamodelinwhichthereare5distinctphasesthatcanbearrangedineitheroftwoalternativetransitiontrajectories,asindicatedbyFigures3.1and3.2anddetailedinTable2.36

Moreover,whichtrajectorywillbetheonethatisoptimalforthissimpleeconomytofollowdependsupontheparameterconfigurationthatsetstheBTMtechnology’soutput/emissionratio.Giventhese‘technicaldata’,thechoicebetweenthealternativetrajectoriesthatwillbefollowedinexploitingthe“buytimeoption”isprescribedbycomparingthewelfarevaluationsofthetwosolutionsofthetwooptimalcontrolprograms.37Trajectory1,asdescribedbythefollowingTable,isfoundtobethewelfaredominantmemberofthepairwhenthethatratioishigh,whereaswhentheratioislowitisbettertofollowTrajectory2,whichcallsforashorterperiodofinvestmentinbuildingstocksofKDandanearlierswitch(inthethirdphaseratherthanthefourth)togrosscapitalformationembodyingthecarbon‐freetechnologies.

InTable2(below),theheaderlinescontainthelabelsforthesub‐phasesineachtrajectory.38Theentriesbeloweachphaselabelbelongingtoatrajectorycontainashorthanddescriptionoftheactivitiestakingplaceduringoratthebeginningofthesub‐phases.Noticethatthemaindifferencebetweenbothtrajectoriesisthetimeatwhichthecarbon‐usingA‐technologyisde‐activated.Asaconsequence,outputduringsub‐phaseBTM22oftrajectory2comesfromthreedifferentsources,sotheequationdescribingtheaccumulationofBTMcapitalismorecomplicatedthanthatfortrajectory1.39

36Notethattheactof‘buyingtime’involvesusingatechnologywitharelativelyhighoutput/emissionratio.Hence,logicallyspeaking,buyingtimeisassociatedwithproductionusingthattechnologyratherthanwithinvestmentinthattechnology.Therefore,inthiscase,theBTMphaseissubdividedintothreesub‐phases,inwhichwehavepositiveoutputfromtheBTMtechnology(hereindexed‘D’),andtheCFRphasestartswhentheBTM‐technologyisdeactivated.Typically,thej‐thBTMsub‐phaseoftrajectorykislabeledBTM.

37Thatthemodelitselfdirectsattentioninthiswaytotherelevanceofempiricalinformationaboutcertainparametersisworthnotice,becauseitmayhelpinprioritizingareaswarrantingmoreconcreteanddetailedempiricalresearchinengineeringandappliedeconomics.

38Typically,thej‐thBTMsub‐phaseoftrajectorykislabeledBTMj,k.

39Althoughthismakestheentiremodelconsiderablymoredifficulttosolve,itwassolvableforMathematica.

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Table2.Alternativetrajectoriesinvolvingthebuytime(BTM)technology

Trajectory 1

Phase 

BAU1  Business as 

usual 

BTM11  Buying time 

BTM21  Buying time 

BTM31  Buying time 

CFR1 

Carbon‐free 

Investment in Carbon‐

based capital 

Reduced 

emission 

capital 

Reduced 

emission 

capital 

Carbon‐free 

capital 

Carbon‐free 

capital 

Output using Carbon‐

based capital 

Both carbon‐

based and 

reduced 

emission 

capital 

Reduced 

emission 

capital only 

Carbon‐free 

capital and 

reduced 

emission 

capital 

Carbon‐free 

capital only 

Trajectory 2

Phase 

BAU2  Business as 

usual 

BTM12  Buying time 

BTM22  Buying time 

BTM32  Buying time 

CFR2 

Carbon‐free 

Investment in Carbon‐

based capital 

Low emissions 

capital 

Carbon‐free 

capital 

Carbon‐free 

capital 

Carbon‐free 

capital 

Output using Carbon‐

based capital 

Both carbon‐

based and 

reduced 

emission 

capital 

All three types 

of capital 

Carbon‐free 

capital and 

reduced 

emission 

capital only 

Carbon‐free 

capital only 

Bysolvingthestackedoptimumcontrolproblemsassociatedwiththemodelversionsoutlinedabove,wearriveatacompleteandintertemporallyconsistentdescriptionofthenatureandtimingofthevariousphases,aswellastheshapeofthetime‐pathsoftangibleandintangiblecapitalinvestment,levelsofproductionandconsumption,aswellascumulativeemissionsasafunctionofthestructuralparametersofthemodel.Theseparametersincludetheprobablelocationofthecumulativeemissionsthreshold,theproductivityoftheR&Dprocessaswellastheproductivitydifferencesbetweencarbon‐basedandcarbon‐freetechnologies,andthe‘standard’preferenceparametersfortheconsumptioncomponentofthesocialutilityfunction,i.e.therateoftimediscountandtheintertemporalelasticityofsubstitution.

3.Preliminaryresults,experimentswithphysicalsysteminteractions

Tothispointinthepresentresearchprogram,themodelsdescribedintheforegoingpageshavenotbeencalibratedonactualdatafortheglobaleconomy.NorhavewecompletedourworkontheintegrationoftheBTMmodelwiththeendogenousR&D

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model.ThelatterisamatterofparticularinterestbecausebeingabletoextendthelengthoftimeduringwhichR&Dexpendituresaremaintainedis(asfaraswepresentlycansee)likelytobethemostproductiveuseoftheaddedtime“bought”byCO2emission‐ratereductionseffectedbyexercisingtheBTMoption.Usingknowntechniquestoupgradingcarbon‐usinginfrastructuresanddirectlyproductivecapitalfacilitiescertainlywouldbeasociallymoreproductivepurposethanhasteningtheinevitabledescentintoclimateinstabilityjustforthesakeofprolongingcurrentenjoymentofthehigherconsumptionlevelsassociatedwith“businessasusual.”

Comparisonsoftheresultsobtainedwiththedifferentpartialmodels,however,yieldsanumberofusefulinsightsabouttheroleofdirectedR&Dinvestmentinthetransitiontoclimatestabilization.OnestrikingandintuitivelyunderstandablefindingthatemergesfromtheoptimizedsolutionoftheendogenousR&Dmodelishighlightedbyitsjuxtapositionwiththefeaturesofthetransitionpathfoundforthebasicmodel.Inthelattercase,business‐as‐usualinvestmentincarbon‐usingcapital(KA)comestoahaltwhencarbon‐freetechnologycanbeembodiedinproductionfacilities,regardlessoftheirgreaterunitcapitalcost;whereasintheformermodeltheBAUphaseendsonlywhenR&DsucceedsinrenderingtheunitcostsofproductionfacilitiesembodyingCFRtechniquescompetitivewiththatofKA.40

AllowingforthepossibilityofinvestmentinR&Ddirectedtowardsloweringtheunitcapitalcostsofcarbon‐freeproductionprocesseshastheeffectofraisingthetangibleinvestmentincarbon‐usingproductionfacilities,shorteningtheabsoluteandrelativedurationoftheBAUphase‐‐leavingthelengthofthesubsequentjointproductionphaseessentiallythesame.Inotherwords,beingabletoinvestintheresearchrequiredfora“technicalfix”hastheexpectedresultofspeedingthebeginningofCFRproduction,thecessationofinvestmentincarbon‐usingproductionfacilities,andcorrespondingly,theretirementoftheoldcarbon‐basedcapitalstock.

TheunderlyingeconomiclogichereisthattheanticipationofbeingabletobuildmoreproductiveCFRcapacityinthefuturegeneratesaheightenedderiveddemandforBAUcapacity,andhenceahigherrateoftangiblecapitalformationincarbon‐basedfacilities‐‐sincethedesiredstockofCFRcapitalisaproducedmeansofproduction.The

40Strictlyspeaking,thelastpartofthisstatementisnotquitetrueinthemodelonwhichthisdiscussionisbased,becausetheupperlimitonthecapitalproductivityoftheCFRtechnology,B‐barissetbelowthethatforthecarbon‐usingtechnology.R&DremovesasmuchaspossibleoftheproductivitydeficitofKBrelativetoKAbeforeimplementingtheCFRtechnologytangiblecapital.B.Butthisisinnowayessential,andtherearenoapriorireasonsfornotentertainingthepossibilitythatCRFtechnologiescan(eventually)havelowerunitcapitalcoststhancarbon‐basedtechnologies,sothatiftheexponentandconstantintheproductivitychange‐R&Dinputfunctionarequitesmall,themodelisthemodelisnotinconsistentwiththeobservationthattheworldweliveinis(still)adependentuponacarbon‐usingregimeofproduction.

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resultinghighervolumeofKAgeneratesalargeroutputflow,providingagreaterpoolofresourcesthatcanbespentonR&DinvestmentduringtheBAUphase,andsubsequentlyforCFR‐embodyingcapitalformationintheJPRphase.Itshouldbenoted,however,thatthiswouldbeadangerousplantohavepursuedwerethetechnicalimprovementsincarbon‐freeproductionsystemsthathadbeenexpectedtoresultfromR&Dexpendituretofallshortofexpectations.Insuchcircumstances–hardlyunrealisticinviewoftheuncertaintiessurroundingtheperformanceofR&D‐‐itwouldbenecessarytoeventuallymaketherequiredswitchtomuchmorecostlyCFRproductioncapacity,attendedbyconsequentlygreaterlossesofconsumptionandwelfare.

ThissuggeststhepracticalrelevanceofcombiningtheendogenousR&DmodelandtheBTMmodel,because,byaddingthe“buytimeoption”ofreducingtheCO2emissionsratesonthecarbon‐usingproductioncapacity,therewouldbemoretimelefttobuild‐upthenecessarycarbon‐freecapitalstock.Sincethatcapitalformationprocesswouldweighallthemoreheavilyofconsumptionlevels,exercisingthe“buytimeoption”canserveasapartialinsuranceagainsttheadversewelfareconsequencesofdisappointedexpectationsregardingtheeffectivenessofR&Dinvestmentinenhancingtheproductivityproductionfacilitiesembodyingcarbon‐freetechnologies.Althoughthetwooptionsoftenseemtobediscussedasalternative,substitute“climatepolicies”theyaremoreproperlyseentobecomplementsinanexpanded“technologyfix”portfolio.

Itisafairlystraightforwardmattertoconductsensitivityexperimentswiththesepartialmodels,inordertogetaqualitativeimpressionoftheimpactofparametervariationsthat,giventhedeterministicformthemodelcanbegintosuggesttherangeofdistributionsinthedynamicsthesystemexhibitwerethemodelcomponentreformulatedmorerealisticallyinstochasticterms.Moreover,suchsensitivityexperimentsalsoshedlightonthewayinwhichtherequirementsforasuccessfulclimate‐stabilizingtransitionwillbeaffectedby‐‐andthereforeneedtomakeexanteallowfor‐‐alterationsintheperceivedandactualdynamicfeedbacksarisingfrominteractionsbetweentheeconomicandthegeo‐physicalsubsystems.41

Sinceatthisstageofourworktheresultsofthesolutionofthemodelconsideringboththebuy‐timeoptionandR&Ddirectedtoinnovationsincarbonfreeproductiontechnologyarestillbeinganalysed,wereporttheresultsofusingsensitivityexperimentswiththe“external”physicalspecificationsofourpartialeconomicmodelsasapreliminarymeansofexploringtheimpactsoftheirinteractions.Amongthevarietyofcomputationalexperimentsthatcanbereadilyexecuted,thefollowingpairhasyieldedresultsthatareofprincipalinterest:(i)loweringthecumulativeGHGemissionstipping‐point,and(ii)increasingtheannualrateofphysicaldepreciationofcarbon‐basedproductioncapacity.

41Forexample,amorecautionarystancetowardsthedangersofsurpassingthecumulativeemissionsthresholdmayinvolvealoweringofthe‘model’‐thresholdbelowits‘realworld’expectedlevel.

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ThefirstoftheseoffersasimplewaytoassessthegrosseffectsofmakingprecautionaryallowancesfortheambiguitysurroundingtheexactleveloftheatmosphericGHGconcentrationlevelthatwillbea“tippingpoint”intothedomainofirreversibleclimateinstability.42Therobustresultwefindforallvariantsofloweringthecriticalthresholdfortemperaturegain(orequivalently,thecumulativeemissionstippingpoint)isthattherequiredlengthofthetransitiontoastabilizedclimatewouldbeshortenedbyabridgementoftheBAUphase,leavingthedurationofthejointproductionphaseessentiallyunaltered.Initially,outputisreducedbutconsumptionisraised,sothatinvestmentintangiblecapitalformationispostponed;althoughconsumptionsubsequentlywillbereduced,therealsoislessinvestmentinthenewCFRtechnology,and,indeedandlowerlevelsofactivityacrosstheboard–comparedtothosefoundforthe(basic)referencemodelwhenthetemperature‐gainthresholdissetatahigher,lessconstraininglevel.

Thegeneralfinding,then,isthatwithlessscopeforCO2emissionsbeforetheexpectedcriticalthresholdwillbereached,productionwithcarbon‐usingcapacitymustbecurtailedandthelengthofthetransitionabridged;withlesstimetobuildupcarbon‐freecapitalgoods,theeventualgrowthofoutputandconsumptionhastobedeferreduntilclimatestabilizationwithacarbon‐freeproductionregimehasbeenachieved.Afurther,consistentresultappliesinthespecificcasesoftheendogenousR&DandBMTmodels:becausealowercriticalthresholdleaveslesstimetodoR&Dortobuytime,ordertocounterthateffecttosomeextent,thedistributionofR&DactivityovertimemustbeshiftedtoinfavourofR&DnowandagainstR&Dlater.

Thus,theplanningmessagestobetakenfromthisisthatsettingalowerGHGppmtarget‐‐inkeepingwithgreateraversiontoriskambiguityandconsequentadherencetothemaximinstrategydictatedbythe“precautionaryprinciple,”callsforrapidlyrampinguptherateofR&Dexpenditures.Similarly,itwarrantsboostingattheoutsettherateofcapitalformationinproductionfacilitieswithupgradedoutput/emissionsperformance.Later,bothtypesofinvestmentwillhavetodecline,firstrelativetooutputandthenabsolutely,oncethebuild‐upofcarbonfreeproductioncapacitygetsunderway.

42Theimplicationsofallowingfortheambiguityinthelocationofthe“tippingpoint”beyondwhichwarmingbecomesaself‐reinforcingprocessevenwhereGHGemissionsofimmediateanthropogenicorginshaveceasedcompletelywillneedtobeassessedwithinthecontextofafullyintegratedstochasticversionofourmodel.LeomineandTraeger(2011)introduceprobabilisticfunctionsforcrossingaclimateregimetippingpointintoarecursiveformulationoftheDICEintegratedassessmentmodelforestablishingoptimalclimatepolicies.Thetippingpoint'sprobabilityandtimingthusbecomeendogenouslydeterminedbythechosenemissionpolicy.Recognizingthattheprobabilitydistributionforthe“temperatureThreshold”correspondingtothe“tippingpoint”isnotknownwithconfidence,policydecisionscanbemadethatreflect“ambiguityaversion”byselectingstrategiesthatreducethedownsidevariancearoundtheexpected“thresholdtemperature.LemoineandTraeger’ssimulationsshowthatunderreasonableparameterizationoftheDICEmodel,allowancefortippingpointsinthiswaycanraisethenear‐termsocialcostofcarbonemissionsbyasmuchas50%.

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Oursecondsetofexperimentsprovideasimplifiedwayofemulatingthehigherfrequencyofdamagesthatcanbeexpectedtoresultfromextremeweathereventsassociatedwithchangingclimate(s),andtoassesstheimpactontheoptimizedtransitionplanofanticipatingtheconsequencesofwarmingthatalreadyis“inthepipeline,”duetothepasthistoryofGHGemissions.Anincreasedrateofunscheduledcapitallosses(damagesfromincreasedclimateinstability)hastheeffectofpostponingthearrivaloftheCFRphase,primarilybecauseproductionusingthecarbonbasedtechnologyisnegativelyaffected,leadingtoalowerflowofCO2emissionsintheprocess.Thisis“thegoodnews”,but,unfortunatelyitisnotthewholestory.

Sinceitseemsreasonableforpurposesofthisanalysistoassumethatthesenegativeclimateeffectswillimpactoldcarbon‐basedfacilitiesmostseverely,ifnotexclusively,43therateofreturnoninvestmentincarbon‐usingfacilitieswillbelowerthanisthecaseintheabsenceofallowanceforthisfeedbackofclimatedestabilization(i.e.,inthebasemodel).The“badnews”isthatthisresultsinlowerinitialratesofcarbon‐usingcapitalformation(grossinvestmentinKAandKD)andhigherratesofconsumptioninanextendedBAUphase.ConsequentlytheconstraintplaceonthegrowthofproductivecapacityduringthisearlyphaseofthetransitioncurtailstherateofR&Dinvestment,andtheglobaleconomy’ssubsequentabilitytorapidlyaccumulatethenecessarystockofcapitalembodyingcarbon‐freetechnologies.

Oneimplicationoftheforegoingfindingswouldseemtobethatexpendituresaimedataverting“unscheduledlosses”inproductioncapacityduetoclimateinstability‐relateddamages,maybequitea“goodinvestment”.Itshouldbenotedthatundertheassumptionofincreaseddamages,thelevelofR&Dactivityalsoisaffectednegatively.But,consideringthatinafullyendogenousintegratedmodel,climate‐relatedcapital‐losseswillbedrivenbyrisingGHGconcentrationlevelsandhighermeansurfacetemperatures,theyarelikelytotakeaheaviertollonexistingproductioncapacityatdatesfartherinthefuture,ratherthanfromtheoutsetofthetransition.

Thatparticularformofnon‐stationaritycanbeemulated(albeitroughly)byafurthersimulationexperimentinwhichthephysicaldepreciationrateoncarbon‐usingcapitaljumpstothehigherlevelonlyafterthetangibleinvestmentinthattypeofcapitalhasstopped.Wemaysupposethattheanticipationofthosefuturecapitallosses,asbefore,willexertadepressingeffect(albeitlessheavily)upontheexpectednetrateofreturntoinvestmentinKAduringtheBAUphase.Consequently,althoughtheinducednear‐termreductionintangible(carbon‐technologyusing)capitalfacilities,andtheriseof

43Theconsiderationunderlyingthisassumptionisthealargeportionofthecarbon‐basedinfrastructureanddirectlyproductivitycapitalstockwillbelegaciesfromtheBAUregimeandthehistoricallyantecedentstate,itwillbelocatedintemperatelatitudesandbuiltinfashionsthatwillleaveitmorevulnerabletotheeffectsofcoastalandriverineflooding,andservewind‐damagefromhurricanesandmonsoons,thantherecentlyconstructedcarbon‐freefacilities.Thelatter,ideally,willhavebeendesignedandsitedwiththoserisksinmind.

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consumptionpercapitawilloccurduringtheBAUphase,thatwouldnothappenfromthephase’soutset.Instead,itwouldbeconcentratedintheperiodjustbeforethecarbon‐usingstockattainsitmaximum,when(undertheassumedtimingoftheincreasedrateofunscheduledcapitallosses)theweather‐createddamagesstarttooccurwithgreaterfrequency.44Theoveralleffectofthesealterationsinthetimingofintangible(R&D)investmentsuponontheterminalvalueoftheproductivityofcapitalembodyingCFRtechnologiesisbounded,however,becausetheBAUphaseduringwhichR&Distakingplacewillhavebeenstretchedout.ThatchangecompensatesforthecumulativeeffectofadecreaseintheflowrateofR&Dinputs,andthesoamelioratethelatter’snegativeimpactupontheextentofproductivityimprovementsinKB.

Amajormessagethatemergesfromtheforegoingmodellingexerciseandparameterizationexercisesisonethatmightwellhavebeenanticipatedatitsoutset.Whatwelearnfromheuristicmodelbuildingofthepresentkindishowtothinkabouttheproblem,andnotnecessarywhattoconcludeaboutthebestcourseofactioninmeetingthechallengeitposes.Eventhehighlysimplifieddynamicalsystemthatwearestudyinghassufficientinterconnectedandmutuallyinterdependent“movingparts”totaxone’sunassistedintuitionsastothewaysthatvariationsintheparametersofthegeophysicalandeconomicsubsystemswillaltertheoptimizedtransitionpathstoastabilizedglobalclimate.

Moreover,whilethesimplicityoftheheuristicgrowthmodelmakesmoretransparentthelogicofchangesinthedirectionsofinvestmentandproductionactivityintheirvariousforms,thesequencedphasestructureofthetransitionisnotsoimmediatelyobvious.Norcantheimpactsofparametricvariationsontheoptimalphases’relativedurationsbeascertainedwithoutundertakingexplicitlyquantitativeanalyses.Thelattercanservetoprioritizeareasforempiricalresearch,byidentifyingtechnicalparameters(andinmodelswithricherspecificationsofagents’behaviors,criticalbehaviouralparameters)‐‐themagnitudesofwhicharefoundtostronglyimpactthewelfarepropertiesandshapetheresourceallocationrequirementsintheearlystagesoftheoptimaltransitionpath.Buildingsuchmodels,andinvestigatingtheimplicationsofmoresophisticated,integratedsystemsofeconomic‐climateinteractions,shouldnotbeseenasapursuitintendedtoproduceasubstitutefortheexerciseofintuitivejudgementsaboutmattersofeconomicpolicydesign.Intheend,thelatterwillhavetoweighmanyimportantpracticalconsiderationsofhumanbehavior,cultureandpoliticsthanwillresistaccuratecaptureintractablequantitativemodels.Instead,weregard

44Thisdifferenceintiminghastheeffectofreleasingresourcesthatallowforafasterbuild‐upofcarbon‐freecapital(byassumption,designedsoastobenotsusceptibletotheelevatedseverityoftheweather).Inaddition,duetothespecificationadoptedfortheR&Dproductionfunctioninourmodel,thereleaseofthoseresourcesatalaterpointinthefuture,afterconsiderableR&Dexpenditureshavetakenplace,islesscostly(intermsofCFRproductivitygainsforegone)thanisthecasewhenthevolumeofR&Disloweredfromtheoutset.

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theexerciseofexperience‐basedintuitionsandquantitativeanalysestobecomplementaryingredientsinthepolicydesignprocess;andbelievetheywillworkmoreeffectivelyandreliablywheneachisallowedtoinform,sharpenandqualifytheconclusionstowhichoneisledbytheirjointemployment.

4.Whatistobelearnedfromthenextstageinthisresearchprogram?

Thepresentpaperfocusesonunderstandingthedeterministicsystemofthemodellingframeworkwehaveconstructed,andhasindicatedthedirectionsinwhichwecanproceedtocompletetheintegrationofthepartialmodelswithinacompleteendogenouseconomicandgeo‐physicalsystem.

Inthenextmajorstageoftheprogram,itwillbeimportanttomovetoastochasticintegratedmodel,byspecifyingtheprobabilitydistributionsgoverningthe“climatesensitivity”ofthephysicalsystem,whichistheprincipalnexusbetweenthealteredproductionsystemandthewelfareconsequencesoftheclimatechangesbroughtaboutbyglobalwarming.45Similarly,wealsomustrecognizethestochasticnatureoftheoutputofR&Dexpenditureinputsandthepayoffsoftechnologicaladvancespermittinghigherproductivityintheavailablestockofcapitalembodyingcarbon‐freetechnologies.Butforthepresenttherearemanyinterestingthingstolearn,andquestionstoanswerbyexploringtheequivalentdeterministicversionofourmodel.

Fordifferentparameterconstellationsweareabletoshowhowthetimingofthephaseswillchange,andhowtheevolutionovertimeofwelfarepercapitawillbeaffected,butalsohowtherequiredR&Dexpendituresandthevolumeofgrosscapitalformationwillchange,bothintermsoftheirdistributionandintensityovertime.Wealsoshowhowuncertaintyregardingthepositionoftheclimatetippingpointincombinationwithdifferentdegreesofprecautionarybehaviourwouldaffecttheoptimumtimingandintensitiesofthevariousactivitiesdistinguishedinthemodel.

Theflipsideofthelattercomputationalanalysisisourabilitytoexplorethewelfaredamageof“policyimplementationfailures”.StartingfromtheoptimalprogramforthetransitiontoaviableandstableGHGconcentrationlevel,itisfeasibletoshowtheeffectsofbudgetaryorpoliticalconstraintsthatresultinspecificallytimedpostponementsofpublicinvestmentinR&D,oruncompensatedprivatecutsintheupgradingofcarbon‐basedproductionfacilities,orintheroll‐outofthecarbon‐free

45Thenatureofthedistributionofthefeedbackparameterdescribingthephysicalsystem’sresponse(intermsofglobalmeansurfacetemperaturetoadoublingoftheatmosphericGHGconcentrationlevelissurroundedbyverysubstantialuncertaintiesarisingfromthecomplexitiesoftheinteractionsamongthephysicalandchemicalprocessesthatunderliethefeedbacksequencesindicatedintheintroduction.Inthepresentcontextthereisanimportantissueastowhetheritisreasonabletoassumethattheeffectsonthedistributionaroundthemean“climatesensitivity”ofchangesinthemeanGHGconcentrationlevelwillbevariance‐preserving.

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technologywhen,undernormalbusinessconditionsitwouldbeadvantageoustoundertakeinstallationoftherequiredcapitalinvestments.Howmuchthetimingofactionmattersinthissystem,andthepenaltiesfordeviationfromtheoptimalpathcanbeshownundertheassumptionthatnoeffortismadetore‐optimizefromthepositionthathasresultedintheaftermathofsuch“shocks”,andalsoundertheassumptionoftryingto“catchup”afteraparticularrangeofdelaysbymeansofre‐optimizingthetransitionpath.Thusitispossiblecomputationallywithinthisheuristicmodellingframeworktodescribetherelationshipbetweenthelengthsofimplementationdelaysduringspecificphasesofasuccessfultransitiontoclimatestability,andtheconsequentcostsintermsofnet‐welfareforegone.

5.Getting“TechFix”intotheclimatepolicymix:Abeginning,notaconclusion:Thisinitialstageinourprojectedexplorationsofthe“requirements”forasociallyoptimaltransitiontosustainableglobaleconomicgrowthinaviablestabilizedclimatehasbeenmotivatedprimarilybythebeliefthatitisvitallyimportanttogive“technologyfix”optionsacentralplaceinstructuredanalysisofpoliciesthatcanrespondeffectivelytothechallengeofclimatedestabilization.Environmentalandenergyeconomistshavebeenquicktoemphasizetheallocationalefficiencyofvariousfiscalandregulatorymeansofraisingthemarketpricesofcarbonfuels,andsuggestedthattheefficacyofthatpolicyapproachwouldbeenhancedbytheireffectsininducing“carbon‐saving”technicalinnovationfromprivatesector.

Ashasbeenpointedout,however,thelatterpropositionpresupposesthatraisingthecostoffossilfuelsourcesofenergywouldnotnegativeincomeeffectsthatdampenedprivateR&Dexpendituresbyenergy‐intensivesectorsoftheeconomy.Furthermore,theintegratedassessmentmodelsthateconomistshavedevelopedasquantitativetoolstoguidepoliciesaimingtoraisethepriceofcarbontypicallyfailtoshowtheconditionsunderwhichtheywouldelicitasufficientflowofinnovationsthatweredirectedspecificallytocurtailingandeventuallydisplacingfossilfuelsasadominantglobalsourceofenergy.Toaddressthis“policyassessmentgap”wouldentailgivinggreaterattentiontospecifyingtherelevantcharacteristicsofthearrayofenergy‐usingtechnologies,andthecorresponding“endogenoustechnologicalprogress”sub‐system,andtoconnectthemwiththeothercomponentsoftheaggregateeconomicgrowthmodelsthatwillgovernthepaceandextentofdeploymentofimprovedproductiontechniques.

Thedynamicintegratedrequirementsanalysismodeling(DIRAM)approachthathasbeenintroducedheremaybeviewedasapreliminarysteptowardsamoreilluminatingrepresentationwithinthecontextoffamiliarIAMsofthepotentialroleoftechnologicalmeasuresinthetransitionfromthepresentdominanceofcarbon‐basedenergytoalowcarbonglobalregimeofproduction.Itsurelywilloccurtosomereaderstoquestion

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accordingprioritytoelaboratingthetechnologicalspecificsofexistingIAMs,onthegroundthatmanyotherfeaturesofthesehighlystylizedmodelsalsowarrantfurtherelaboration,andgreaterpolicyrelevancecallsforaspatiallydisaggregatedapproachthatwouldtakeaccountofgeographicalandecologicalvariationsaffectingboththeglobaleconomyandtheclimatesystem.Indeed,therealreadyhavebeenmanyeffortsalongjustsuchlines,impelledbyapolicyinterestinassessingthedifferentialregionalandnationalincidenceofthecostsofcurtailingglobalGHGemissions.Othersmightarguefortheimportanceofrecognizingtheendogeneityofchangesinthesize,agestructureandspatialdistributionofglobalpopulation,andmoresophisticatedwelfareframeworktoaccountfordemographicaswellaseconomicchangesinthetransitiontoaviablestabilizedclimate.

evenwerethelatterqualificationtobewavedaway,thereshouldberoomtoconsiderthatarrivingatdomesticandinternationalagreementstoimposefutureincreasingpricesforfossilfuelswouldbepoliticallylessarduouswheretheregoodprospectsforsignificantmitigationofsuchcommitments’adverseimpactsonfuturerealprofitratesandconsumptionlevels.Justthatcontextwouldbecreatedbyapriorcommitmentonthepartofthealreadydevelopedandscientificallyandtechnicallyadvancecountriestomajorcoordinatedtechnologicalprograms‐‐suchasthoseaimedatsignificanlyincreasingtheefficiencyofenergydistributionanduse,andatloweringtherealunitcostsofnon‐fossilfuelsourcesofenergy.

Risingpricesforcarbon‐basedenergywouldencourageanyformcost‐savinginnovationactivitiesand,bythattoken,itwouldraisetheperceivedmarginalsocialpayofffromexpandingpublicsupportforscienceandtechnologyresearch–ifonlytocreateaknowledgeinfrastructurethatwouldlowerthecostsofdirectedinnovationintheaffectedsectorsandlinesofbusiness.Nevertheless,theaugmentedpublicR&Dfundingwouldhavetobeforthcoming,andadditional,differentialsubsidymeasureswouldneedtobeintroducedtodirectprivateR&D(andsubsequentdeploytheresults)towardloweringtheunitcostsofcarbon‐basedandalternativeenergysources.Alternatively,awellthoughtoutsupply‐sideclimatepolicythatstartedwiththegoalofexpandingavarietyofapplications‐orientedR&Dprogramswouldtendtocreatecredibleexpectationsofsubstantialfutureresourcesavingswithcarbon‐freeproductionfacilities,andaconcomitantlysmallersacrificeofmaterialwelfareentailedinrestrictingGHGemissions.ThatcouldcontributetoweakeningeconomicallymotivatedresistancetoascheduleofgraduallyrisingcarbontaxesandthereforeimpartwiderandstrongercommitmentstonationalandinternationalagreementsoncoordinatedandveriableactionsthatwouldcurtailGHGemissions.Bundlingproposedcommitmentsfromlowerincomedevelopingeconomieswithreciprocalloansubsidiesandcostconcessionsinthetransfertonew“greener”andcarbon‐freetechnologieswouldconstituteacrediblepackagefornegotiationsthatwouldgrowinitsattractivenessastheR&Dprogramsmatured.

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AcknowledgementsThepresentpaperisanextensiverevisionandexpansionofthework(Davidetal.,2011)preparedforpresentationatthe6‐8July2011conferenceoftheInternationalEnergyWorkshop,heldatStanfordUniversity.IthasbenefitedfromthediscussionbyparticipantsintheIEWconferencesessionon“R&DInvestment,ClimateandEnergyPolicy,”andfromcomments(onsubsequentversions)bythoseattheSIEPR‐CEEGSocialScienceandTechnologySeminaron14December2011,andfacultyanddoctoralstudentsattendingtheCEMIseminarsatÉcolePolytechniqueFédéraledeLausanneon17and30May2012.ThecontributionsofBronwynHallandLucSoetetoourpreparatorydraftsonthissubject(Davidetal.,2010)havebeenofenduringvalue,andthemanyperceptivecommentsandsuggestionsofferedbyLawrenceGoulderandThomasWeberdeservespecialacknowledgment.ThanksalsotoAshishArora,DominiqueForay,WardHanson,andWarrenSanderson.Davidgratefullyacknowledgesthesupportofa“seedgrant”fromtheStanfordInstituteforEconomicPolicyResearch.

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