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MODELING UNCERTAINTY IN CLIMATE CHANGE: A MULTI‐MODEL COMPARISON By Kenneth Gillingham, William Nordhaus, David Anthoff, Geoffrey Blanford, Valentina Bosetti, Peter Christensen, Haewon McJeon, John Reilly, and Paul Sztorc September 2015 COWLES FOUNDATION DISCUSSION PAPER NO. 2022 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 208281 New Haven, Connecticut 06520-8281 http://cowles.yale.edu/ ModelingUncertaintyinClimateChange: AMulti‐ModelComparison1 KennethGillingham,WilliamNordhaus,DavidAnthoff,GeoffreyBlanford,Valentina Bosetti,PeterChristensen,HaewonMcJeon,JohnReilly,PaulSztorc September17,2015 Abstract Theeconomicsofclimatechangeinvolvesavastarrayofuncertainties, complicatingboththeanalysisanddevelopmentofclimatepolicy.This studypresentstheresultsofthefirstcomprehensivestudyof uncertaintyinclimatechangeusingmultipleintegratedassessment models.Thestudylooksatmodelandparametricuncertaintiesfor population,totalfactorproductivity,andclimatesensitivity.Itestimates thepdfsofkeyoutputvariables,includingCO2concentrations, temperature,damages,andthesocialcostofcarbon(SCC).Onekey findingisthatparametricuncertaintyismoreimportantthan uncertaintyinmodelstructure.Ourresultingpdfsalsoprovideinsights ontailevents. 1TheauthorsaregratefultotheDepartmentofEnergyandtheNationalScience Foundationforprimarysupportoftheproject.ReillyandMcJeonacknowledgesupportby theU.S.DepartmentofEnergy,OfficeofScience.Reillyalsoacknowledgestheother sponsorstheMITJointProgramontheScienceandPolicyofGlobalChangelistedat http://globalchange.mit.edu/sponsors/all.TheStanfordEnergyModelingForumhas providedsupportthroughitsSnowmasssummerworkshops.KennethGillinghamcurrently worksasaSeniorEconomistfortheCouncilofEconomicAdvisers(CEA).TheCEA disclaimsresponsibilityforanyoftheviewsexpressedherein,andtheseviewsdonot necessarilyrepresenttheviewsoftheCEAortheUnitedStatesgovernment.Noneofthe authorshasaconflictofinteresttodisclose.KennethGillinghamandWilliamNordhausare correspondingauthors([email protected]@yale.edu). 1 I. Introduction Acentralissueintheeconomicsofclimatechangeisunderstandingand dealingwiththevastarrayofuncertainties.Theserangefromthoseregarding economicandpopulationgrowth,emissionsintensitiesandnewtechnologies,tothe carboncycle,climateresponse,anddamages,andcascadetothecostsandbenefits ofdifferentpolicyobjectives. Thispaperpresentsthefirstcomprehensivestudyofuncertaintyofmajor outcomesforclimatechangeusingmultipleintegratedassessmentmodels(IAMs). ThesixmodelsusedinthestudyarerepresentativeofthemodelsusedintheIPCC FifthAssessmentReport(IPCC2014)andintheU.S.governmentInteragency WorkingGroupReportontheSocialCostofCarbonorSCC(USInteragencyWorking Group2013).Wefocusoureffortsinthisstudyonthreekeyuncertainparameters: populationgrowth,totalfactorproductivitygrowth,andequilibriumclimate sensitivity.Fortheestimateduncertaintyinthesethreeparameters,wedevelop estimatesoftheuncertaintyto2100formajorvariables,suchasemissions, concentrations,temperature,percapitaconsumption,output,damages,andthe socialcostofcarbon. Ourapproachisatwo‐trackmethodologythatpermitsreliablequantification ofuncertaintyformodelsofdifferentsizeandcomplexity.Thefirsttrackinvolves performingmodelrunsoverasetofgridpointsandfittingasurfaceresponse functiontothemodelresults;thisapproachprovidesaquickandaccuratewayto emulaterunningthemodels.Thesecondtrackdevelopsprobabilitydensity functionsforthechoseninputparameters(i.e.,theparameterpdfs)usingthebest availableevidence.WethencombinebothtracksbyperformingMonteCarlo simulationsusingtheparameterpdfsandthesurfaceresponsefunctions. Thismethodologyprovidesatransparentapproachtoaddressinguncertainty acrossmultipleparametersandmodelsandcaneasilybeappliedtoadditional modelsanduncertainparameters.Animportantaspectofthismethodology,unlike virtuallyallothermodelcomparisonexercises,isitsreplicability.Theapproachis easilyvalidatedbecausethedatafromthecalibrationexercisesarerelatively compactandarecompiledinacompatibleformat,thesurfaceresponsescanbe estimatedindependently,andtheMonteCarlosimulationscanbeeasilyrunin multipleexistingsoftwarepackages. Thispaperisstructuredasfollows.Thenextsectiondiscussesthestatistical considerationsunderpinningourstudyofuncertaintyinclimatechange.SectionIII presentsourmethodologyforthetwo‐trackapproach,whilethenextsection discussesselectionofcalibrationruns.SectionVgivesthederivationofthe 2 probabilitydistributions.SectionVIgivestheresultsofthemodelcalculationsand thesurfaceresponsefunctions,andsectionVIIpresentstheresultsoftheMonte Carloestimatesofuncertainties.Weconcludewithasummaryofthemajorfindings insectionVIII.TheAppendicesprovidefurtherbackgroundinformation. II. StatisticalConsiderations A. BackgroundonUncertaintyinClimateChange Climatechangescienceandpolicyhavefocusedlargelyonprojectingthe centraltendenciesofmajorvariablesandimpacts.Whilecentraltendenciesare clearlyimportantforafirst‐levelunderstanding,attentionisincreasinglyonthe uncertaintiesintheprojections.Uncertaintiestakeongreatsignificancebecauseof thepossibilityofnon‐linearitiesinresponses,particularlythepotentialfor triggeringthresholdsinearthsystems,inecosystem,orineconomicoutcomes.To besure,uncertaintieshavebeenexploredinmajorreports,suchastheIPCC ScientificAssessmentReportsfromthefirsttothefifth.However,thesehavemainly examineddifferencesamongmodelsasatoolforassessinguncertaintiesabout futureprojections.Asweindicatebelow,ourresultssuggestthatparametric uncertaintyisquantitativelymoreimportantthandifferencesacrossmodelsfor mostvariables. Inrecentreviewsofclimatechange,thereisanincreasingfocusonimproving ourunderstandingoftheuncertainties.Forexample,in2010theInter‐Academy ReviewoftheIPCC,theprimaryrecommendationforimprovingtheusefulnessof thereportwasaboutuncertainty: Theevolvingnatureofclimatescience,thelongtimescalesinvolved, andthedifficultiesofpredictinghumanimpactsonandresponsestoclimate changemeanthatmanyoftheresultspresentedinIPCCassessmentreports haveinherentlyuncertaincomponents.Toinformpolicydecisionsproperly,it isimportantforuncertaintiestobecharacterizedandcommunicatedclearly andcoherently.(InterAcademyCouncil2010) Inarecentreport,theU.S.CongressionalBudgetOfficealsovoicedits concernsaboutuncertainty: Inassessingthepotentialrisksfromclimatechangeandthecostsof avertingit,however,researchersandpolicymakersencounter pervasiveuncertainty.Thatuncertaintycontributestogreat 3 differencesofopinionastotheappropriatepolicyresponse,withsome expertsseeinglittleornothreatandothersfindingcausefor immediate,extensiveaction.Policymakersarethusconfrontedwitha widerangeofrecommendationsabouthowtoaddresstherisksposed byachangingclimate—inparticular,whether,how,andhowmuchto limitemissionsofgreenhousegases.(CBO2005) Thefocusonuncertaintyhastakenonincreasedurgencybecauseofthegreat attentiongivenbyscientiststotippingelementsintheearthsystem.Aninfluential studybyLentonetal.(2008)discussedimportanttippingelementssuchasthelarge icesheets,large‐scaleoceancirculation,andtropicalrainforests.Some climatologistshavearguedthatglobalwarmingbeyond2°Cwillleadtoan irreversiblemeltingoftheGreenlandicesheet(Robinsonetal.2012).Once uncertaintiesarefullyincluded,policieswillneedtoaccountfortheprobabilitythat pathsmayleadacrosstippingpoints,withparticularconcernforonesthathave irreversibleelements. Afurthersetofquestionsinvolvesthepotentialforfattailsinthedistribution ofparameters,ofoutcomes,andoftheriskofcatastrophicclimatechange.(Afat‐or thick‐taileddistributionisonewheretheprobabilityofextremeeventsdeclines slowly,sothetailofthedistributionisthick.Animportantexampleisthepower‐law orParetodistribution,inwhichthevarianceoftheprocessisunboundedforcertain parametervalues.) Theissuearisesbecauseofthecombinationofoutcomesthatarepotentially catastrophicinnatureandprobabilitydistributionswithfattails.Thecombination ofthesetwofactorsmayleadtosituationsinwhichfocusingoncentraltendenciesis completelymisleadingforpolicyanalysis.Inaseriesofpapers,MartinWeitzman (seeespeciallyWeitzman2009)hasproposedadramaticallydifferentconclusion fromstandardanalysisinwhathehascalledtheDismalTheorem.Intheextreme case,thecombinationoffattails,unlimitedexposure,andhighriskaversionimplies thattheexpectedlossfromcertainriskssuchasclimatechangeisunboundedand wethereforecannotperformstandardoptimizationcalculationsorcost‐benefit analyses. Therearetodatemanystudiesoftheimplicationsofuncertaintyforclimate changeandclimate‐changepolicyorofuncertaintyinoneormanyparameters usingasinglemodel.SomenotableexamplesincludeReillyetal.(1987),Peckand Teisberg(1993),NordhausandPopp(1997),Pizer(1999),Webster(2002),Baker (2005),Hope(2006),Nordhaus(2008),Websteretal.(2012),AnthoffandTol (2013),andLemoineandMcJeon(2013). 4 Todate,however,theonlypublishedstudythataimstoquantifyuncertainty inclimatechangeformultiplemodelsistheU.S.governmentInteragencyWorking Groupreportonthesocialcostofcarbon,whichispublishedinGreenstoneetal. (2013)andmoreextensivelydescribedinIAWG(2010).Thisstudyusedthree models,twoofwhichareincludedinthisstudy,toestimatethesocialcostofcarbon forU.S.governmentpurposes.However,whileitdidexamineuncertainty,thecross‐ modelcomparisonfocusedonasingleuncertainparameter(equilibriumclimate sensitivity)foritsformaluncertaintyanalysis;allotheruncertainparametersinthe modelswereleftuncertainwiththemodelers’pdfs.Evenwiththissingleuncertain parameter,theestimatedsocialcostofcarbonvariesgreatly.The2015socialcostof carbonintheupdatedIAWG(2013)is$38pertonofCO2usingthemedianestimate versus$109pertonofCO2usingthe95percentile(bothin2007dollarsandusinga 3%discountrate),whichwouldimplyverydifferentlevelsofpolicystringency.The IAWGanalysisalsousedcombinationsofmodelinputsandoutputsthatwerenot alwaysinternallyconsistent.Comparisonoftheuncertaintiesinaconsistent mannerindifferentmodelsisclearlyanimportantmissingareaofstudy. B. Centralapproachofthisstudy Thisprojectaimstoquantifytheuncertaintiesofkeymodeloutcomes inducedbyuncertaintyinimportantparameters.Wehopetolearnthedegreeto whichthereisprecisioninthepointestimatesofmajorvariablesthatareusedin majorintegratedassessmentmodels.Putdifferently,theresearchquestionweaim toanswerfromthisstudyis:Howdomajorparameteruncertaintiesaffectthe distributionofpossibleoutcomesofmajoroutcomes;andwhatisthelevelof uncertaintyofmajoroutcomevariables? Wecallthisquestiononeof“classicalstatisticalforecastuncertainty.”The studyofforecastinguncertaintyanderrorhasalonghistoryinstatisticsand econometrics.SeeforexampleClementsandHendry(1998,1999)andEricsson (2001).Thestandardtoolsofforecastinguncertaintyhavevirtuallyneverbeen appliedtomodelsintheenergy‐climate‐economyareasbecauseofthecomplexityof themodelsandthenon‐probabilisticnatureofbothinputsandstructural relationships. Keyuncertaintiesthatwewillexamineincludebothprojectionsandpolicy outcomes.Forexample,whataretheuncertaintiesofemissions,concentrations, temperatureincreases,anddamagesinabaselineprojection?Whatisthe uncertaintyinthesocialcostofcarbon?Howdouncertaintiesacrossmodels comparewiththeuncertaintieswithinmodelsgeneratedbyparameteruncertainty? 5 Oneofthekeycontributionsofthisworkisthatithasthepotentialtohighlight areaswherereducinguncertaintywillhaveahighpayoff. C. Uncertaintyinabroadercontext Thereareseveraluncertaintiesinclimatechangethatfacebothnaturaland socialscientistsanddecisionmakers.Amongtheimportantonesare:(1)parametric uncertainty,suchasuncertaintyaboutclimatesensitivityoroutputgrowth;(2) modelorspecificationuncertainty,suchasthespecificationoftheaggregate productionfunction;(3)measurementerror,suchasthelevelandtrendofglobal temperatures;(4)algorithmicerrors,suchasonesthatfindtheincorrectsolutionto amodel;(5)randomerrorinstructuralequations,suchasthoseduetoweather shocks;(6)codingerrorsinwritingtheprogramforthemodel;and(7)scientific uncertaintyorerror,suchaswhenamodelcontainsanerroneoustheory. Thisstudyfocusesprimarilyonthefirstofthese,parametricuncertainty,and toalimitedextentonthesecond,modeluncertainty.Wefocusonthefirstbecause therearemajoruncertaintiesaboutseveralparameters,becausethishasbeenakey areaforstudyinearlierapproaches,andbecauseitisatypeofuncertaintythat lendsitselfmostreadilytomodelcomparisons.Inaddition,sinceweemploysix models,theresultsprovidesomeinformationabouttheroleofmodeluncertainty, althoughwedonotdevelopaformalapproachtomodeluncertainty.Werecognize thatparameterandmodeluncertaintiesarebuttwooftheimportantquestionsthat arise,butarigorousapproachtomeasuringthecontributionoftheseuncertainties willmakeamajorcontributiontounderstandingtheoveralluncertaintyofclimate change. Fromatheoreticalpointofview,themeasuresofuncertaintycanbeviewed asapplyingtheprinciplesofjudgmentalorsubjectiveprobability,or“degreeof belief,”tomeasuringfutureuncertainties.Thisapproach,whichhasitsrootsinthe worksofRamsey(1931),deFinetti(1937),andSavage(1954),recognizesthatitis notpossibletoobtainfrequentistoractuarialprobabilitydistributionsforthemajor parametersinintegratedassessmentmodelsorinthestructuresofthemodels.The theoryofsubjectiveprobabilityviewstheprobabilitiesasakintotheoddsthat informedscientistswouldtakewhenwageringontheoutcomeofanuncertain event.Forexample,supposetheeventwaspopulationgrowthfrom2000to2050. Thesubjectiveprobabilitymightbethattheinterquartilerange(25%,75%)was between0.5%and2.0%peryear.Inmakingtheassessment,thescientistwouldin effectsaythatitisamatterofindifferencewhethertobetthattheoutcomewhen knownwouldbeinsideoroutsidethatrange.Whileitisnotcontemplatedthatabet 6 wouldactuallyoccur(althoughthatisnotunprecedented),thewagerapproach helpsframetheprobabilitycalculation. III. Methodology A. Overviewofourtwotrackapproach Inundertakinganuncertaintyanalysis,theprojectcontemplatedtwo potentialapproaches.Inoneapproach,eachmodelwoulddoaMonteCarlo simulationinwhichitwoulddomanyrunswherethechosenuncertainparameters aredrawnfromajointpdf.Whilepotentiallyfeasibleforsomemodels,suchan approachisexcessivelyburdensomeandlikelyinfeasibleatthescalenecessaryto havereliableestimates. Wethereforedevelopedasecondapproachwhichwecallthe“two‐track MonteCarlo.”Thisapproachseparatesthemodelcalibrationrunsfromgeneration oftheparameterpdfsandtheMonteCarloestimates.Atthecoreoftheapproachare twoparalleltracks,whicharethencombinedtoproducethefinalresults.Thefirst trackusesmodelrunsfromsixparticipatingeconomicclimatechangeintegrated assessmentmodelstodevelopsurfaceresponsefunctions;theserunsprovidethe relationshipbetweenouruncertaininputparametersandkeyoutputvariables.The secondtrackdevelopsprobabilitydensityfunctionscharacterizingtheuncertainty foreachanalyzeduncertaininputparameter.Wecombinetheresultsofthetwo tracksusingaMonteCarlosimulationtocharacterizestatisticaluncertaintyinthe outputvariables. B. Theapproachinequations Itwillbehelpfultoshowthestructureoftheapproachanalytically.Wecan representamodelasamappingfromexogenousandpolicyvariablesand parameterstoendogenousoutcomes.Themodelscanbewrittensymbolicallyas follows: (1) Y m H m ( z, , u ) Inthisschema,Ymisavectorofmodeloutputsformodelm;zisavectorof exogenousandpolicyvariables; isavectorofmodelparameters;uisavectorof uncertainparameterstobeinvestigated;andHmrepresentsthemodelstructure.We emphasizethatmodelshavedifferentstructures,modelparameters,andchoiceof inputvariables.However,wecanrepresenttheargumentsofHwithoutreferenceto modelsbyassumingsomeareomitted. 7 Thefirststepintheprojectistoselecttheuncertainparametersforanalysis. Oncetheparametersareselected,eachmodelthendoesselectedcalibrationruns. Thecalibrationrunstakeasacentralsetofparametersthebaseorreferencecase foreachofthemodels.Itthenmakesseveralrunsthataddorsubtractspecified incrementsfromeachofthebasevaluesoftheuncertainparameters.Thisproduces asetofinputandoutputsforeachmodel. Moreprecisely,hereistheprocedureforthefirsttrackoftheapproach.Each modelhasabaselinerunwithbasevaluesforeachoftheuncertainparameters. Denotethebaseparametervaluesas (um,1 , um,2 , um,3 ). Thenextstepdeterminesagrid b b b ofdeviationvaluesoftheuncertainparametersthateachmodeladdsorsubtracts fromthebasevaluesoftheuncertainparameters.Denotethesedeviationvaluesas G (1,1,1 , 1,1,2 ,..., 5,5,5 ). The G vectorrepresents125=5x5x5deviationsfrom themodelers’baseparametervalues.So,forexample,thevector 1,1,1 would representoneofthe125gridvectorsthattakesthefirstvalueforeachuncertain parameter.Supposethat 1,1,1 (0.014, .02, 2). Thenthatcalibrationrunwould calculatetheoutcomesfor Y m H m ( z, , umb ,1 .014, umb ,2 .02, umb ,3 2) ,whereagain umb , k is thebasevalueforuncertainparameterkformodelm.Similarly, 3,3,3 (0, 0, 0). For thatdeviationvalue,thecalibrationrunwouldcalculatetheoutcomesfor Y m H m ( z, , umb ,1 , umb ,2 , umb ,3 ), whichisthemodelbaselinerun. Thethirdstepistoestimatesurfaceresponsefunctions(SRFs)foreachmodel andvariableoutcome.Symbolically,thesearethefollowingfunctions: (2) Y m R m (u1 umb ,1 , u2 umb ,2 , u3 umb ,3 ) R m (um,1 , um,2 , um,3 ) TheSRFsarefitovertheobservationsofthe u m , k fromthecalibrationexercises (125eachforthebaselineandforthecarbon‐taxcases).TheSRFsarelinear‐ quadratic‐interactionequationsasdescribedbelow. Thesecondtrackoftheprojectprovidesuswithprobabilitydensityfunctions foreachofouruncertainparameters, f k (uk ) .Thesearedevelopedonthebasisof externalinformationasdescribedbelow. Thefinalstepistoestimatethecumulativedistributionoftheoutput variables, G m (Y m ). Thesearethedistributionsoftheoutcomevariables Y m for 8 modelm,wherewenotethatthedistributionswilldifferbymodel.The distributionsarecalculatedbyMonteCarlomethods,forasamplesizeofN: (3) G m (Y m ) N 1 if H n 1 ( um,1 , num,2 , num,3 ) Y m , otherwise = 0 / N m n Thenotationhereisthat n u m , k isthenthdrawofrandomvariable uk inthe MonteCarloexperiment.Thisunintuitiveequationsimplystatesthatthecumulative distributionisequaltothefractionofoutcomesintheMonteCarlosimulation wheretheSRFyieldsavalueoftheoutcomevariablethatislessthan Y m . The distributionofoutcomesforeachvariableandmodelisconditionalonthemodel structureandontheharmonizeduncertaintyoftheuncertainparameters.Fora classicstudyofMonteCarlomethods,seeHammersleyandHandscomb(1964). C. IntegratedAssessmentModels Thechallengeofanalysisandpoliciesforglobalwarmingisparticularly difficultbecauseitspansmanydisciplinesandpartsofsociety.Thismany‐faceted naturealsoposesachallengetonaturalandsocialscientists,whomustincorporate awidevarietyofgeophysical,economic,andpoliticaldisciplinesintotheir diagnosesandprescriptions.Thetaskofintegratedassessmentmodels(IAMs)isto pulltogetherthedifferentaspectsofaproblemsothatprojections,analyses,and decisionscanconsidersimultaneouslyallimportantendogenousvariables.IAMs generallydonotpretendtohavethemostdetailedandcompleterepresentationof eachincludedsystem.Rather,theyaspiretohave,atafirstlevelofapproximation, modelsthatoperateallthemodulessimultaneouslyandwithreasonableaccuracy. Thestudydesignwaspresentedatameetingwheremanyoftheestablished modelerswhobuildandoperateIAMswerepresent.Allwereinvitedtoparticipate. Aftersomepreliminaryinvestigationsandtrialruns,sixmodelswereableto incorporatethemajoruncertainparametersintotheirmodelsandtoprovidemost oftheoutputsthatwerenecessaryformodelcomparisons.Thefollowingisabrief descriptionofeachofthesixmodels.TableA5intheappendixprovidesfurther detailsoneachmodel. TheDICE(DynamicIntegratedmodelofClimateandtheEconomy)wasfirst developedaround1990andhasgonethroughseveralextensionsandrevisions.The latestpublishedversionisNordhaus(2014)withadetaileddescriptioninNordhaus andSztorc(2014).TheDICEmodelisagloballyaggregatedmodelthatviewsthe economicsofclimatechangefromtheperspectiveofneoclassicaleconomicgrowth theory.Inthisapproach,economiesmakeinvestmentsincapitalandinemissions 9 reductions,reducingconsumptiontoday,inordertolowerclimatedamagesand increaseconsumptioninthefuture.Thespecialfeatureofthemodelistheinclusion ofallmajorelementsinahighlyaggregatedfashion.Themodelcontainsabout25 dynamicequationsandidentities,includingthoseforglobaloutput,CO2emissions andconcentrations,globalmeantemperature,anddamages.Theversionforthis projectrunsfor60five‐yearperiods.ItcanberunineitheranExcelversionorin thepreferredGAMSversion.TheversionusedforthisstudydatesfromDecember 2013andaddsloopstocalculatetheoutcomesfordifferentuncertainparameters. TherunswereimplementedbyWilliamNordhausandPaulSztorc. TheFUNDmodel(ClimateFrameworkforUncertainty,Negotiation,and Distribution)wasdevelopedprimarilytoassesstheimpactsofclimatepoliciesinan integratedframework.Itisarecursivemodelthattakesexogenousscenariosof majoreconomicvariablesasinputsandthenperturbsthesewithestimatesofthe costofclimatepolicyandtheimpactsofclimatechange.Themodelhas16regions andcontainsexplicitrepresentationoffivegreenhousegases.Climatechange impactsaremonetizedandincludeagriculture,forestry,sea‐levelrise,health impacts,energyconsumption,waterresources,unmanagedecosystems,andstorm impacts.Eachimpactsectorhasadifferentfunctionalformandiscalculated separatelyforeachofthe16regions.Themodelrunsfrom1950to3000intime stepsof1year.Thesourcecode,data,andatechnicaldescriptionofthemodelare public(www.fund‐model.org),andthemodelhasbeenusedbyothermodeling teams(e.g.,Reveszetal.(2014)).FUNDwasoriginallycreatedbyRichardTol(Tol, 1997)andisnowjointlydevelopedbyDavidAnthoffandRichardTol.Theruns wereimplementedbyDavidAnthoff. TheGCAM(GlobalChangeAssessmentModel)isaglobalintegrated assessmentmodelofenergy,economy,land‐use,andclimate.GCAMisalong‐term globalmodelbasedontheEdmondsandReillymodel(EdmondsandReilly1983a,b, c).GCAMintegratesrepresentationsoftheglobaleconomy,energysystems, agricultureandlanduse,withrepresentationsofterrestrialandoceancarbon cycles,andasuiteofcoupledgas‐cycleandclimatemodels.Theclimateandphysical atmosphereinGCAMisbasedontheModelfortheAssessmentofGreenhouse‐Gas InducedClimateChange(MAGICC)(Meinshausenetal.2011).Theglobaleconomy inGCAMisrepresentedin14geopoliticalregions,explicitlylinkedthrough internationaltradeinenergycommodities,agriculturalandforestproducts,and othergoodssuchasemissionspermits.Thescaleofeconomicactivityineachregion isdrivenbypopulationsize,age,andgenderaswellaslaborproductivity.The modelisdynamic‐recursivelysolvedforasetofmarket‐clearingequilibriumprices inallenergyandagriculturalgoodmarketsevery5yearsover2005‐2095.Thefull 10 documentationofthemodelisavailableataGCAMwiki(Calvinandetal.2011). GCAMisopen‐source,butisprimarilydevelopedandmaintainedbytheJointGlobal ChangeResearchInstitute.ThemodelrunswereperformedbyHaewonMcJeon. TheMERGEmodel(ModelforEvaluatingRegionalandGlobalEffectsof greenhousegasreductionpolicies)isanintegratedassessmentmodeldescribing globalenergy‐economy‐climateinteractionswithregionaldetail.Itwasintroduced byManneetal.(1999)andhasbeencontinuallydevelopedsince;arecently publisheddescriptionisinBlanfordetal.(2014).MERGEisformulatedasamulti‐ regiondynamicgeneralequilibriummodelwithaprocessmodeloftheenergy systemandareduced‐formrepresentationoftheclimate.ItissolvedinGAMSvia sequentialjointnon‐linearoptimizationwithNegishiweightstobalanceinter‐ regionaltradeflows.Theeconomyisrepresentedasatop‐downRamseymodelin whichelectricandnon‐electricenergyinputsaretradedoffagainstcapitalandlabor andproductionisallocatedbetweenconsumptionandinvestment.Theenergy systemincludesexplicittechnologiesforelectricitygenerationandnon‐electric energysupply,witharesourceextractionmodelforfossilfuelsanduranium.The climatemodelincludesafive‐boxcarboncycleandtracksallmajornon‐CO2 greenhousegasesandnon‐CO2forcingagentsexplicitly.Temperatureevolvesasa two‐boxlagprocess,whereuncertaintyaboutclimatesensitivityisconsidered jointlywithuncertaintyabouttheresponsetimeandaerosolforcing.Theversion usedforstudyincludes10modelregionsandrunsthrough2100,withclimate variablesprojectedforanadditionalcentury.Therunswereimplementedby GeoffreyBlanford. TheMITIGSM(IntegratedGlobalSystemsModel)wasdevelopedintheearly 1990’sandhasbeencontinuallyupdated.Itincludesageneralcirculationmodelof theatmosphereanditsinteractionswithoceans,atmosphericchemistry,terrestrial vegetation,andthelandsurface.Itseconomiccomponentrepresentstheeconomy andanthropogenicemissions.ThefullIGSMisdescribedinSokolovetal.(2009)and Websteretal.(2012).Theversionoftheeconomiccomponentappliedhereis describedinChenetal.(2015).Theearthsystemcomponentisasimplifiedgeneral circulationmodelresolvedin46latitudebandsand11verticallayersinthe atmospherewithan11layeroceanmodel.Thelandsystemincludes17vegetation types.Theeconomiccomponentisamulti‐sector,multi‐regionappliedgeneral equilibriummodel,anempiricalimplementationconsistentwithneo‐classical economictheory.Forthecurrentproject,themodeloperatesinarecursivefashion inwhichtheeconomydrivestheearthsystemmodelbutwithoutfeedbacksof climateimpactsontheeconomicsystem.Theeconomiccomponentissolvedfor5 yeartimestepsinGAMS‐MPSGEandforthisexercisewasrunthrough2100.The 11 earthsystemcomponentsolveson10minutetimesteps(thevegetationmodelon monthlytimesteps).ThesimulationsforthisexercisewereconductedbyY.‐H. HenryChen,AndreiSokolov,andJohnReilly. TheWITCH(WorldInducedTechnicalChangeHybrid)modelwasdeveloped in2006(Bosettietal.2006)andhasbeendevelopedandextendedsincethen.The latestversionisfullydescribedinBosettietal.(2014).Themodeldividestheworld into13majorregions.TheeconomyofeachregionisdescribedbyaRamsey‐type neoclassicaloptimalgrowthmodel,whereforward‐lookingcentralplanners maximizethepresentdiscountedvalueofutilityofeachregion.Theseoptimizations takeaccountofotherregions'intertemporalstrategies.Theoptimalinvestment strategyincludesadetailedappraisalofenergysectorinvestmentsinpower‐ generationtechnologiesandinnovation,andthedirectconsumptionoffuels,aswell asabatementofothergasesandland‐useemissions.Greenhouse‐gasemissionsand concentrationsarethenusedasinputsinaclimatemodelofreducedcomplexity (Meinshausenetal.2011).Theversionusedforthisprojectrunsfor30five‐year periodsandcontains35statevariablesforeachofthe13regions,runningonthe GAMSplatform.TherunswereimplementedbyValentinaBosettiandGiacomo Marangoni. IV. Choiceofuncertainparametersandgriddesign A. Choiceofuncertainparameters Oneofthekeydecisionsinthisstudywastoselecttheuncertainparameters. Thecriteriaforselectionwere(atleastafterthefact)clear.First,eachparameter mustbeimportantforinfluencinguncertainty.Second,parametersshouldbeones thatcanbevariedineachofthemodelswithoutexcessiveburdenandwithout violatingthespiritofthemodelstructure.Third,theparametersshouldbeonesthat canberepresentedbyaprobabilitydistribution,eitheronthebasisofprior researchorfeasiblewithinthescopeofthisproject. Ataninitialmeeting,anexperimentwasundertakeninwhicheachofthe modelswasgivensixuncertainparametersorshockstotestforfeasibility.Atthe endofthisinitialtestexperiment,twoofthemodelingteamsdecidednotto participatebecausetheinitialparameterscouldnotbeeasilyincorporatedinthe modeldesignorbecauseoftimeconstraints.Threeoftheparametersfulfilledthe above‐mentionedcriteria,andtheseweretheonesthatwereincorporatedinthe finalsetofexperiments. Thefinallistofuncertainparameterswerethefollowing:(1)Therateof growthofproductivity,orpercapitaoutput;(2)therateofgrowthofpopulation; 12 and(3)theequilibriumclimatesensitivity(equilibriumchangeinglobalmean surfacetemperaturefromadoublingofatmosphericCO2concentrations). Additionally,itwasdecidedtodotwoalternativepolicyscenarios.Onewasa “Base”runinwhichnoclimatepolicieswereintroduced;andthesecond,labelled “CarbonTax”(andsometimes“Ampere”)introducedarapidlyrisingglobalcarbon tax.2Arunbasedoncarbonpriceswasselected(insteadofquantitativelimits) becausemanymodelshadundertakensimilarrunsinothermodelcomparison projects,sotheywererelativelyeasytoimplement. Severalotherparameterswerecarefullyconsideredbutrejected.Apulseof emissionswasrejectedbecauseithadessentiallynoimpact.Aglobalrecessionwas rejectedforthesamereason.Itwashopedtoadduncertaintiesfortechnology(such asthoseconcerningtherateofdecarbonization,thecostofbackstoptechnologies, orthecostofadvancedcarbon‐freetechnologies),butitprovedimpossibletofind onethatwasbothsufficientlycomprehensiveandcouldbeincorporatedinallthe models.Uncertaintyaboutclimatedamageswasexcludedbecausehalfthemodels didnotcontaindamages.Afinalpossibilitywastoanalyzepolicyrunsthathad quantitativelimitsratherthancarbonprices.Forexample,somemodelshad participatedinmodelcomparisonsinwhichradiativeforcingswerelimited.This approachwasrejectedbecausethecarbontaxprovedeasiertodefineand implement.Additionally,earlierexperimentsindicatedthatquantitativelimitswere oftenfoundinfeasible,andthiswouldcloudtheinterpretationoftheresults.3 2TheCarbonTaxrunwasselectedfromtheAMPEREmodelcomparisonstoreducethe burdenonmanyofthemodelersandsothattheresultsfromthisstudycanbecomparedto thosefromtheAMPEREinter‐modelcomparisonstudy(Kriegleretal.2015).Thespecific scenariochosenisknownintheAMPEREstudyas"CarbonTax$12.50‐increasing.”Thefull AMPEREscenariodatabasecanbefoundonlineathttps://secure.iiasa.ac.at/web‐ apps/ene/AMPEREDB. 3SeeparticularlytheresultsforEnergyModelingForum22reportedinaspecialissuein EnergyEconomics(e.g.,seeClarkeandWeyant(2009)).Manymodelsfoundthattight constraintswereinfeasiblefortheirbaseruns.Aquantitativelimitwouldalmostsurely havefoundthatlargenumbersofthe125scenarioswereinfeasibleforanytightlimiton temperatureorradiativeforcings. 13 B. Descriptionofuncertainparameters Wenextdescribethethreeuncertainparameterscontainedinthestudy.It turnedoutthatharmonizingtheseacrossmodelswasmorecomplicatedthanwas originallyanticipated,asdescribedbelow. (1) Therateofgrowthofpopulation.Uncertaintyabouttherateofgrowth ofpopulationwasstraightforward.Forglobalmodels,therewasnoambiguityabout theadjustment.Theuncertaintywasspecifiedasplusorminusauniform percentagegrowthrateeachyearovertheperiod2010‐2100.Forregionalmodels, theadjustmentwaslefttothemodeler.Mostmodelsassumedauniformchangein thegrowthrateineachregion. (2)Therateofgrowthofproductivity,orpercapitaoutput.Theoriginal designhadbeentoincludeavariablethatrepresentedtheuncertaintyaboutoverall technologicalchangeintheglobaleconomy(oraveragedacrossregions).The resultsoftheinitialexperimentindicatedthatthespecificationsoftechnological changedifferedgreatlyacrossmodels,anditwasinfeasibletospecifyacomparable technologicalvariablethatcouldapplyforallmodels.Forexample,somemodels hadasingleproductionfunction,whileothershadmultiplesectors. Ratherthanattempttofindacomparableparameter,itwasdecidedto harmonizeontheuncertaintyofglobaloutputpercapitagrowthfrom2010to2100. Eachmodelerwasaskedtointroduceagridofchangesinitsmodel‐specific technologicalparameterthatwouldleadtoachangeinpercapitaoutputofplusor minusagivenamount(tobedescribedinthenextsection).Themodelerswerethen instructedtoadjustthatchangesothattherangeofgrowthratesinpercapitaGDP from2010to2100inthecalibrationexercisewouldbeequaltothedesiredrange. (3)Theclimatesensitivity.Modelinguncertaintyaboutclimatesensitivity provedtobeoneofthemostdifficultissuesofharmonizationacrossthedifferent models.Whileallmodelshavemodulestotracethroughthetemperature implicationsofchangingconcentrationsofGHGs,theydifferindetailand specification.Themajorproblemwasthatadjustingtheequilibriumclimate sensitivitygenerallyrequiredadjustingotherparametersinthemodelthat determinethespeedofadjustmenttotheequilibrium;theadjustmentspeedis sometimesrepresentedbythetransientclimatesensitivity.Thisproblemwas identifiedlateintheprocess,afterthesecond‐roundrunshadbeencompleted,and modelerswereaskedtomaketheadjustmentsthattheythoughtappropriate.Some modelsmadeadjustmentsinparameterstoreflectdifferencesinlargeclimate 14 models.Othersconstrainedtheparameterssothatthemodelwouldfitthehistorical temperaturerecord.Thedifferingapproachesledtodifferingstructuralresponses totheclimatesensitivityuncertainty,aswillbeseenbelow. C. Griddesign Inthefirsttrack,themodelingteamsprovideasmallnumberofcalibration runsthatincludeafullsetofoutputsforathree‐dimensionalgridofvaluesofthe uncertainparameters.Foreachoftheuncertainparameters,weselectedfivevalues centeredonthemodel’sbaselinevalues.Therefore,for3uncertainparameters, therewere125runseachfortheBaseandtheCarbonTaxpolicyscenarios. Onthebasisofthesecalibrationruns,thenextstepinvolvedestimating surface‐responsefunctions(SRFs)inwhichthemodeloutcomesareestimatedas functionsoftheuncertainparameters.ThehopewasthatiftheSRFscould approximatethemodelsaccurately,thentheycouldbeusedtosimulatethe probabilitydistributionsoftheoutcomevariablesaccurately.Aninitialtest suggestedthattheSRFswerewellapproximatedbyquadraticfunctions.We thereforesettherangeofthegridsothatitwouldspanmostofthespacethatwould becoveredbythedistributionoftheuncertainparameters,yetnotgosofarasto pushthemodelsintopartsoftheparameterspacewheretheresultswouldbe unreliable. Asanexample,takethegridforpopulationgrowth.Thecentralcaseisthe model’sbasecaseforpopulationgrowth.Eachmodelthenusesfouradditional assumptionsforthegridforpopulationgrowth:thebasecaseplusandminus0.5% peryearandplusandminus1.0%peryear.Thesewouldcovertheperiod2010to 2100.Forexample,assumethatthemodelhadabasecasewithaconstant populationgrowthrateof0.7%peryearfrom2010to2100.Thenthefivegrid pointsforpopulationgrowthwouldbeconstantgrowthratesof‐0.3%,0.2%,0.7%, 1.2%,and1.7%peryear.Populationafter2100wouldhavethesamegrowthrateas inthemodeler’sbasecase.Theseassumptionsmeanthatpopulationin2100would be(0.99)90,(0.995)90,1,(1.005)90,and(1.01)90timesthebasecasepopulationfor 2100. Forproductivitygrowth,thegridwassimilarlyconstructed,butadjustedso thatthegrowthinpercapitaoutputfor2100added‐1%,‐0.5%,0%,0.5%,and1% tothegrowthrateineachyearfortheperiod2010‐2100. Fortheclimatesensitivity,themodelersweretoaddtothebaseline equilibriumclimatesensitivity‐3°C,‐1.5°C,0°C,1.5°C,and3°C.Itturnedoutthat thelowerendofthisrangecauseddifficultiesforsomemodels,andforthesethe 15 modelersreportedresultsonlyforthefourhigherpointsinthegridorsubstituted anotherlowvalue. Inprinciple,then,fortrackIeachmodelreported5x5x5modelresultsfor boththeBasecaseandtheCarbonTaxpolicyassumptions. V. Approachfordevelopingprobabilitydensityfunctions A. Generalconsiderations Thethreeuncertainparametershavebeenthesubjectofuncertaintyanalysis inearlierstudies.Foreachparameter,wereviewedearlierstudiestodetermine whethertherewasanexistingsetofmethodsordistributionsthatcouldbedrawn upon.Thedesirablefeaturesofthedistributionsisthattheyshouldreflectbest practice,thattheyshouldbeacceptabletothemodelinggroups,andthattheybe replicable.Itturnedoutthatthethreeparametersusedthreedifferentapproaches, aswillbedescribedbelow. B. Population Populationgrowthhasbeenthesubjectofprojectionsformanyyears,and numerousgroupshaveundertakenuncertaintyanalysesforbothcountriesandat thegloballevel.Ourreviewfoundonlyoneresearchgroupthathadmadelong‐term globalprojectionsofuncertaintyforseveralyears,whichwasthepopulationgroup attheInternationalInstituteforAppliedSystemsAnalysis(IIASA)inAustria.(Fora discussion,seeO'Neilletal.(2001)).TheIIASAdemographygroupisunderthe directionofdemographerWolfgangLutz. TheIIASAstochasticprojectionsweredevelopedoveraperiodofmorethana decadeandarewidelyusedbydemographers.Themethodologyissummarizedas follows:“IIASA’sprojections…arebasedexplicitlyontheresultsofdiscussionsofa groupofexpertsonfertility,mortality,andmigrationthatisconvenedforthe purposeofproducingscenariosforthesevitalrates”(See http://www.demographic‐research.org/volumes/vol4/8/4‐8.pdf)Thelatest projectionsfrom2013(Lutzetal.2014)areanupdatetothepreviousprojections from2007and2001(Lutzetal.2008),2001).Themethodologyisdescribedas follows: Theforecastsarecarriedoutfor13worldregions.Theforecastspresentedhere arenotalternativescenariosorvariants,butthedistributionoftheresultsof 2,000differentcohortcomponentprojections.Forthesestochasticsimulations thefertility,mortalityandmigrationpathsunderlyingtheindividualprojection 16 runswerederivedrandomlyfromthedescribeduncertaintydistributionfor fertility,mortalityandmigrationinthedifferentworldregions.(Lutz,Sanderson, andScherbov2008) Thebackgroundmethodsaredescribedasfollowsonpage219ofO'Neilletal. (2001): TheIIASAmethodologyisbasedonaskingagroupofinteractingexpertsto givealikelyrangeforfuturevitalrates,where"likely"isdefinedtobea confidenceintervalofroughly90%(Lutz1996,Lutzetal.1998).Combining subjectiveprobabilitydistributionsfromanumberofexpertsguardsagainst individualbias,andIIASAdemographersarguethatastrengthofthemethodis thatitmaybepossibletocapturestructuralchangeandunexpectedeventsthat otherapproachesmightmiss.Inaddition,inareaswheredataonhistorical trendsaresparse,theremaybenobetteralternativetoproducingprobabilistic projections. Forthisstudy,weareaimingforaparsimoniousparameterizationofpopulation uncertainty.Thisisnecessarybecauseofthelargedifferencesinmodelstructure. Wethereforeselectedtheuncertaintyaboutglobalpopulationgrowthfortheperiod 2010‐2100asthesingleparameterofinterest.Wefittedthegrowth‐ratequantiles fromtheIIASAprojectionstoseveraldistributions,withnormal,log‐normal,and gammabeingthemostsatisfactory.Thenormaldistributionperformedbetterthan anyoftheothersonfiveofthesixquantitativetestsoffitfordistributions.Basedon theseresults,wethereforedecidedtorecommendthenormaldistributionforthe pdfofpopulationgrowthovertheperiod. Inaddition,wedidseveralalternativeteststodeterminewhetherthe projectionswereconsistentwithothermethodologies.Onesetoftestsexaminesthe projectionerrorsthatwouldhavebeengeneratedusinghistoricaldata.Asecond testlooksatthestandarddeviationof100‐yeargrowthratesofpopulationforthe lastmillennium.AthirdtestexaminesprojectionsfromareportoftheNational ResearchCouncilthatestimatedtheforecasterrorsforglobalpopulationovera50‐ yearhorizon(seeNRC(2000),AppendixF,p.344).Whiletheseallgaveslightly differentuncertaintyranges,theyweresimilartotheuncertaintiesestimatedinthe IIASAstudy. Onthebasisofthisreview,wedecidedtouseanormaldistributionforthe growthrateofpopulationbasedontheIIASAstudythathasastandarddeviationof theaverageannualgrowthrateof0.22percentagepointsperyearovertheperiod 17 2010‐2100.Moredetailswithabackgroundmemorandumontheresultsare availablefromtheauthors. C. ClimateSensitivity Animportantparameterinclimatescienceistheequilibriumorlong‐run responseintheglobalmeansurfacetemperaturetoadoublingofatmospheric carbondioxide.Intheclimatesciencecommunity,thisiscalledtheequilibrium climatesensitivity.Withreferencetoclimatemodels,thisiscalculatedasthe increaseinaveragesurfacetemperaturewithadoubledCO2concentrationrelative toapathwiththepre‐industrialCO2concentration.Thisparameteralsoplaysakey roleinthegeophysicalcomponentsintheIAMsusedinthisstudy.Intheremainder ofthispaper,wewillfollowtheconventioninthegeosciencesandcallitthe equilibriumclimatesensitivity(ECS). GiventheimportanceoftheECSinclimatescience,thereisanextensive literatureestimatingprobabilitydensityfunctions.Thesepdfsaregenerallybased onclimatemodels,theinstrumentalrecordsoverthelastcenturyorso, paleoclimaticdatasuchasestimatedtemperatureandradiativeforcingsoverice‐ ageintervals,andtheresultsofvolcaniceruptions.Muchoftheliteratureestimates aprobabilitydensityfunctionusingasinglelineofevidence,butafewpapers synthesizedifferentstudiesordifferentkindsofevidence. Wefocusonthestudiesdrawinguponmultiplelinesofevidence.TheIPCC FifthAssessmentreport(AR5)reviewedtheliteraturequantifyinguncertaintyin theECSandhighlightedfiverecentpapersusingmultiplelinesofevidence(IPCC 2014).EachpaperusedaBayesianapproachtoupdateapriordistributionbasedon previousevidence(thepriorevidenceusuallydrawnfrominstrumentalrecordsora climatemodel)tocalculatetheposteriorprobabilitydensityfunction.Sinceeach distributionwasdevelopedusingmultiplelinesofevidence,andinsomecasesthe sameevidence,itwouldbeinconsistenttoassumethattheywereindependentand simplytocombinethem.Further,sincewecouldnotreliablyestimatethedegreeof dependenceofthedifferentstudies,wecouldnotsynthesizethembytakinginto accountthedependence.Wethereforechosetheprobabilitydensityfunctionfroma singlestudyandperformedrobustnesscheckstousingtheresultsfromalternative studiescitedintheIPCCAR5. ThechosenstudyforourprimaryestimatesisOlsenetal.(2012).Thisstudy isrepresentativeoftheliteratureinusingaBayesianapproach,withapriorbased onpreviousstudiesandalikelihoodbasedonobservationalormodeleddata,such asglobalaveragesurfacetemperaturesorglobaltotalheatcontent.Thepriorin Olsenetal.(2012)isprimarilybasedonKnuttiandHegerl(2008).Thatprioristhen 18 combinedwithoutputvariablesfromtheUniversityofVictoriaESCMclimatemodel (Weaveretal.2001)todeterminethefinalorposteriordistribution. Olsenetal.(2012)waschosenforthefollowingreasons.First,itwas recommendedtousinpersonalcommunicationswithseveralclimatescientists. Second,itwasrepresentativeoftheotherfourstudiesweexaminedandfallsinto themiddlerangeofthedifferentestimates.4Third,sensitivityanalysesoftheeffect onaggregateuncertaintyofchangingthestandarddeviationoftheOlsenetal. (2012)resultsfoundthatthesensitivitywassmall(seethesectionbelowon sensitivityanalyses).Appendix1providesmoredetailsonOlsenetal.(2012)and alsopresentsafigurecomparingthisstudytotheotherstudiesintheIPCCAR5. NotethattheUSgovernmentusedaversionoftheRoeandBakerdistribution calibratedtothreeconstraintsfromtheIPCCforitsuncertaintyestimates(IAWG 2010).Specifically,theIAWGReportmodifiedtheoriginalRoeandBaker distributiontoassumethatthemedianvalueis3.0°C,theprobabilityofbeing between2and4.5°Cistwo‐thirds,andthereisnomassbelowzeroorabove10°C. ThemodifiedRoeandBakerdistributionhasahighermeanECSthananyofthe models(3.5°C)andamuchhigherdispersion(1.6°Cascomparedto0.84°Cfrom Olsenetal.2012). TheestimatedpdfforOlsenetal.(2012)wasderivedasfollows.Wefirst obtainedthepdffromtheauthors.Thispdfwasprovidedasasetofequilibrium temperaturevaluesandcorrespondingprobabilities.Wethenexploredfamiliesof distributionsthatbestapproximatedthenumericalpdfprovided.Wefoundthata log‐normalpdffitstheposteriordistributionsextremelywell. Tofindtheparametersofthefittedlog‐normalpdf,weminimizethesquared differencebetweentheposteriordensityfunctionfromOlsenetal.andthelog‐ normalpdfoverthesupportofthedistribution(theL2orEuclidiannorm).Inother words,weminimizethesumofthesquareoftheverticaldifferencesbetweenthe posteriorpdfandalog‐normalpdfoverallgridpointsvaluesintheOlsenetal. (2012)distribution.5Figure1showstheOlsenetal.(2012)pdf,alongwiththefitted log‐normaldensityfunction.Thefitisextremelyclose,withthelog‐normal distributionalwayswithin0.14%oftheOlsenetal.(2012)pdfforanygridpoint value. 4Intests,wefoundthattheOlsenetal.(2012)distributionissimilartoasimplemixture distributionofallfivedistributions.Wecalculatethismixturedistributionbytakingthe averageprobabilityoveralldistributionsateachtemperatureincrease. 5MorepreciselyweminimizeovertherangeoftheOlsenetal.distribution,[1.509,7.4876] °C,withagridpointspacingof0.1508°C. 19 Figure1.TheOlsenetal.(2012)probabilitydensityfunctionalongwiththefitted log‐normaldistributionusedinouranalysis. D. TotalFactorProductivity Uncertaintyinthegrowthofproductivity(oroutputpercapita)isknowntobea criticalparameterindeterminingallelementsofclimatechange,fromemissionsto temperaturechangetodamages(Nordhaus2008).Climatemodelsgenerallydraw theirestimatesofemissionstrajectoriesfrombackgroundmodelsofeconomic growthsuchasscenariospreparedfortheIPCCorstudiesoftheEnergyModeling Forum.Nomajorstudies,however,relyonstatistically‐basedestimatesofemissions andeconomicgrowth. Forecastsoflong‐runproductivitygrowthinvolveactivedebatesonissuessuch astheroleofnewtechnologiesandinventions(BrynjolfssonandMcAfee2012, Gordon2012),potentialincreasesintheresearchintensityandeducational attainmentinemergingeconomies(FernaldandJones2014,Freeman2010),and institutionalreformandpoliticalstability(Acemogluetal.2005).Whilethe empiricalliteratureoneconomicgrowthhasprovidedevidenceinsupportof variousunderlyingmodels,noexistingstudycontainssufficientinformationto deriveaprobabilitydistributionforlong‐rungrowthrates. 20 Thehistoricalrecordprovidesausefulbackgroundforestimatingfuturetrends. However,itisclearfromboththeoreticalandempiricalperspectivesthatthe processesdrivingproductivitygrowtharenon‐stationary.Forexample,estimatesof thegrowthofglobaloutputpercapitaforthe18th,19th,and20thcenturyare0.6,1.9, and3.7percentperyear(DeLong2015in http://holtz.org/Library/Social%20Science/Economics/Estimating%20World%20 GDP%20by%20DeLong/Estimating%20World%20GDP.htm).Totheextentthat expertsoneconomicgrowthpossessvalidinsightsaboutthelikelihoodandpossible determinantsoflong‐rungrowthpatterns,theninformationdrawnfromexperts canaddvaluetoforecastsbasedpurelyonhistoricalobservationsordrawnfroma singlemodel.Combiningexpertestimateshasbeenshowntoreduceerrorinshort‐ runforecastsofeconomicgrowth(BatchelorandDua1995).However,thereare fewexpertstudiesonlong‐rungrowth(seeAppendix2fordiscussion)and,toour knowledge,therehasbeennosystematicanddetailedpublishedstudyof uncertaintyinlong‐runfuturegrowthrates. Todevelopestimatesofuncertainties,theprojectteam,ledbyPeterChristensen, undertookasurveyofexpertsoneconomicgrowthtodetermineboththecentral tendencyandtheuncertaintyaboutlong‐rungrowthtrends.Oursurveyutilized informationdrawnfromapanelofexpertstocharacterizeuncertaintyinestimates ofglobaloutputfortheperiods2010‐2050and2010‐2100.Wedefinedgrowthas theaverageannualrateofrealpercapitaGDP,measuredinpurchasingpower parity(PPP)terms.Weaskedexpertstoprovideestimatesoftheaverageannual growthratesat10th,25th,50th,75th,90thpercentiles. Beginninginthesummerof2014,wesentoutsurveystoapanelof25economic growthexperts.AsofJune2015,wecollected11completeresultswithfull uncertaintyanalysisfortheperiod2010‐2100.Asummaryoftheprocedureis providedinAppendix2,andacompletereportwillbepreparedseparately. Therearemanydifferentapproachestocombiningexpertforecasts(Armstrong 2001)andaggregatingprobabilitydistributions(ClemenandWinkler1999).We assumethatexpertshaveinformationaboutthelikelydistributionoflong‐run growthrates.Theirinformationsetsaredefinedbyestimatesfor5different percentiles.Webeginbyassumingthattheestimatesareindependentacross expertsandthenexaminedthedistributionsthatbestfitthepercentilesforeach expertandforthecombinedestimates(averageofpercentiles)acrossexperts. Wefounditusefulforthisprojecttocharacterizetheexpertpdfswithcommonly useddistributionssothattheMonteCarloestimatescouldbeeasilyimplemented.In testingthedistributionsforeachexpert,wefoundthatmostexperts’estimatescan 21 becloselyfittedbyanormaldistribution;similarly,thecombineddistributionis wellfittedbyanormaldistribution.DetailsareprovidedinAppendix2. Theresultingcombinednormaldistributionhasameangrowthrateof2.29% peryearandastandarddeviationofthegrowthrateof1.15%peryearoverthe period2010‐2100.(ThemeangrowthrateofpercapitaGDPinthebaserunsofthe sixmodelsisslightlylowerat1.9%peryearoverthisperiod.)Wetestdifferent approachesforcombiningtheexpertresponsesandfindlittlesensitivitytothe choiceofaggregationmethod.Figure2showsthefittedindividualandcombined normalpdfs(explainedinAppendix2).IntheMonteCarloestimatesbelow,we choseastandarddeviationofthegrowthrateofpercapitaoutputof1.12%peryear (basedonthefirst11responses).Thisvalueisusedinthisdraft,butwillbeupdated withtheadditionoffurtherresponses. Figure2.Individualandcombinedpdfsforannualgrowthratesofoutputpercapita, 2010–2100(averageannualpercentperyear) Forthemethods,seeAppendix2. 22 Itisusefultocomparethesurveyresultswithhistoricaldata.Ifwetakethelong‐ termestimatesfromMaddison(2003),the100‐yearvariabilityofgrowthoverthe tencenturiesfrom1000to2000was1.5%peryear,witharangeof‐0.1%to3.7% peryear.Thevariabilityinthesecentury‐stepdataishigherthantheexperts’ estimateof1.15%peryear. Globalgrowthratesbasedondetailednationaldataareavailablesince1900.The standarddeviationofannualgrowthratesoverthisperiodwas2.9%peryear,while thestandarddeviationof25‐yeargrowthrateswas1.2or1.4%peryeardepending uponthesource.Thevariabilityofgrowthinrecentyearswaslowerthanforthe entireperiodsince1900.Thestandarddeviationintheannualgrowthrateduring theperiod1975‐2000was1.1%peryear.Wecannoteasilytranslatehistorical variabilitiesintocentury‐longvariabilitieswithoutassumingaspecificstochastic structureofgrowthrates. VI. ResultsofModelingStudies A. Modelresultsandlatticediagrams Webeginbyprovidingresultsonthecalibrationrunsandthesurfaceresponse functions.Foreachmodel,thereisavoluminoussetofinputsandoutputvariables from2010to2100.Thefullset(consistingof46,150x22elements)clearlycannot befullypresented.Werestrictourfocusheretosomeofthemostimportantresults, andconsignfurtherresultstoAppendix3,withthefullresultsavailableonlineat timeofpublication. Tohelpvisualizetheresults,wehavedevelopedlatticediagramstoshowhow theresultsvaryacrossuncertainvariablesandmodels.Figure3isalatticediagram fortheincreaseinglobalmeansurfacetemperaturein2100.Withineachofthenine panels,they‐axisistheglobalmeansurfacetemperatureincreasein2100relative to1900.Thex‐axisisthevalueoftheequilibriumtemperaturesensitivity.Going acrosspanelsonthehorizontalaxis,thefirstcolumnusesthegridvalueofthefirst ofthefivepopulationscenarios(whichisthelowestgrowthrate);themiddle columnshowstheresultsforthemodeler’sbaselinepopulation;andthethird columnshowstheresultsforthepopulationassociatedwiththehighestpopulation grid(orhighestgrowthrate). Goingdownpanelsontheverticalaxis,thefirstrowusesthehighestgrowthrate forTFP(orthefifthTFPgridpoint);themiddlerowshowsTFPgrowthforthe modelers’baselines;andthebottomrowshowstheresultsfortheslowestgrid pointforthegrowthrateofTFP.Notethatinallcases,themodelers’baselinevalues 23 generallydiffer,butthedifferencesinparametervaluesacrossrowsorcolumnsare identical. Tounderstandthislatticegraph,begininthecenterpanel.Thispanelusesthe modeler’sbaselinepopulationandTFPgrowth.Itindicateshowtemperaturein 2100acrossmodelsvarieswiththeECS,withthedifferencesbeing1.5°Cbetween theECSgridpoints.AfirstobservationisthatthemodelsallassumethattheECSis closeto3°Cinthebaseline.Next,isthattheresultingbaselinetemperature increasesfor2100arecloselybunchedbetween3.75and4.25°C.Allcurvesare upwardsloping,indicatingagreater2100temperaturechangeisassociatedwitha higherECS. AstheECSvariesfromthebaselinevalues,themodeldifferencesaredistinct. Thesecanbeseenintheslopesofthedifferentmodelcurvesinthemiddlepanelof Figure3.Wewillseebelowthattheimpactofa1°CchangeinECSon2100 temperaturevariesbyafactorof2½acrossmodels.Forexample,DICE,MERGE,and GCAMhaverelativelyresponsiveclimatemodules,whileIGSMandFUNDclimate modulesaremuchlessresponsivetoECSdifferences.Thedifferenceacrossmodels becomeslargeraswemovefromthebottom‐lefttotheupperright‐handpanel, correspondingtoincreasingpopulationandTFPgrowthfrombottomlefttotop right.Thisresulthighlightskeydifferencesinboththeeconomicandclimate componentsofthedifferentmodels. 24 Figure3.Latticediagramfor2100temperatureincrease Thislatticediagramshowsthedifferencesinmodelresultsfor2100globalmean surfacetemperatureacrosspopulation,totalfactorproductivityandtemperature sensitivityparameters.Thecentralboxusesthemodelers’baselineparametersand theBasepolicy. Anotherimportantrelationshiptoexamineishowdifferentmodelsreactto thecarbonprices.Figure4showsthepercentagereductioninCO2emissionsinthe CarbonTaxscenariov.theBaserun.Thehorizontalaxisshowsthemagnitudeofthe carbontax.Onekeyfeatureofallmodelsisthatattainingzeroemissionswould requireextremelyhighcarbonprices. 25 Percentage reduction (Ampere v base) 120% 100% 80% 60% 40% 20% DICE FUND GCAM IGSM MERGE WITCH 0% 0 100 200 300 400 Carbon price ($/tCO2) 500 Figure4.Carbontaxandemissionsreductionsbymodel Modelsshowdifferingresponsetohighercarbonprices.Notethatthecarbonprices areallassociatedwithgivendatesandarecommonforallmodels.Thepointstothe farleftarefor2010,whiletheonesatthefarrightarefor2100.Theseestimatesare forthemodelers’baselineparameters. Therearemanyotherresultsofthemodelingexercise.Appendix3contains furtherlatticediagrams,includingthoseforpercapitaconsumption,emissions,and damages,aswellasadditionaltablesofresults.However,theprimarypurposeof thepresentstudyistodeterminetheimpactofuncertainties,soweleavethemodel comparisonsofmajoroutputsasideatthispoint. B. Resultsoftheestimatesofthesurfaceresponsefunctions RecallthattrackIprovidesthemodeloutcomes(suchasoutput,emissions, andtemperature)foreachgrid‐pointofa5x5x5x2gridofthevaluesofthe uncertainparametersandpolicies.Thenextstepintheanalysisistofitsurface responsefunctions(SRFs)toeachofthemodeloutputs.TheseSRFsthenwillbe used,whencombinedwiththeTrackIIprobabilitydistributionsjustdiscussed,to provideprobabilitydistributionsoftheoutcomevariablesforeachmodel. 26 Weundertookextensiveanalysisofdifferentapproachestoestimatingthe SRFs.Theinitialandeventuallypreferredapproachwasalinear‐quadratic‐ interactions(LQI)specification.Thistookthefollowingform: 3 3 j Y 0 i ui ij ui u j i 1 j 1 i 1 Inthisspecification, ui and u j aretheuncertainparameters.TheYarethe outcomevariablesfordifferentmodelsanddifferentyears(e.g.,temperatureforthe FUNDmodelfor2100intheBaserunfordifferentvaluesofthe3uncertain parameters).Theparameters 0 , i , and i j aretheestimatesfromtheSRF regressionequations.Wesuppressthesubscriptforthemodel,year,policy,and variable. Table1showsacomparisonoftheresultsfortemperatureandlogofoutput forthelinear(L)andLQIspecificationsforthesixmodels.Allspecificationsshow markedimprovementoftheequationfitintheLQIrelativetotheLversion.Looking atthelogoutputspecification(thelastcolumninthebottomsetofnumbers),the residualvarianceintheLQIspecificationisessentiallyzeroforallmodels.Forthe temperatureSRF,morethan99.5%ofthevarianceisexplainedbytheLQI specification.Thestandarderrorsofequationsfor2100temperaturerangefrom 0.05to0.18°CfordifferentmodelsintheLQIversion. 27 Table1.LinearparametersinofSRFfortemperatureandlogoutputforlinear(L) andliner‐quadratic‐interactions(LQI)specifications ThelinearparametersarethecoefficientsonthelineartermintheSRFregressions. Becausethedataaredecentered(removethemedians),thelineartermsinthe higher‐orderpolynomialsarethederivativesorlineartermsatthemedianvaluesof theuncertainparameters. Theequationsarefitasdeviationsfromthecentralcase,socoefficientsare linearizedatthecentralpoint,whichisthemodelers’baselinesetofparameters. LookingattheLQIcoefficientsfortemperature,notethattheeffectoftheECSon 2100temperaturevariessubstantiallyamongthemodels.Atthehighend,thereis closetoaunitcoefficient,whileatthelowendthevariationisabout0.4°Cper°Cin 28 ECSchange.ForTFP,theimpactsarerelativelysimilarexceptfortheWITCHmodel, whichismuchlower.ThisislikelyduetoimplementationoftheTFPchangesas input‐neutraltechnicalchange(ratherthanchangesinlaborproductivity,asin severalothermodels).Forpopulation,theLQIcoefficientsvarybyafactorofthree. Forlogofoutput,severalmodelshavenofeedbackfromECStooutputand thusshowa0.000value.TheimpactofTFPisalmostuniformbydesign.Similarly, theimpactofpopulationonoutputisverysimilar. WetestedsevendifferentspecificationsfortheSRF:Linear(L),Linearwith interactions(LI),Linearquadratic(LQ),Linear,quadratic,linearinteractions(LQI) asshownabove,3rddegreepolynomialwithlinearinteractions(P3I),4thdegree polynomialswithseconddegreeinteractions(P4I2),andfourthdegreepolynomial withfourthdegreeinteractionsandpolynomialthree‐wayinteractions(P4I4S3). Forvirtuallyallmodelsandspecifications,theaccuracyincreasedsharplyasfaras theLQIspecification.However,asisshowninFigure5,verylittlefurther improvementwasfoundforthemoreexoticpolynomials.Inadditiontothe polynomialinterpolations,weinvestigatedseveralalternativetechniques,including Chebyshevpolynomialsandbasis‐splines.Wefoundnoimprovementfromthese otherapproaches. L LQ LI LQI LQI++ 0.10 All Temp(2100) 0.08 Conc(2100) Y(2100) 1‐R2 0.06 0.04 0.02 0.00 Figure5.Residualvarianceforallvariables,models,andspecificationsindicatesthat fornearlyallmodels,thereislittletobegainedaddingfurtherpolynomialtermsbeyond LQI. 29 Insummary,wefoundthatthelinear‐quadratic‐interaction(LQI) specificationofthesurfaceresponsefunctionperformedextremelywellinfitting thedatainourtests.Thereasonisthatthemodels,whilehighlynon‐linearoverall, aregenerallyclosetoquadraticinthethreeuncertainparameters.Wearetherefore confidentthattheyareareliablebasisfortheMonteCarlosimulations. C.ReliabilityoftheMUPprocedureswithextrapolation Oneissuethatarisesinestimatingthedistributionsofoutcomevariablesis theextenttowhichthecalibrationrunsintrackIadequatelycovertherangeofthe pdfsfromtrackII.Forbothpopulationandtheequilibriumtemperaturesensitivity, thecalibrationrunscoveratleast99.9%oftherangeofthepdfs.However,when settingthecalibrationrangeforTFPbasedonearlierinformalestimates,we underestimatedthevariabilityofthefinalpdfs.Asaresult,thecalibrationrunsonly extendasfarasthe83percentileattheupperend,requiringustoextrapolate beyondtherangeofthecalibrationruns. Sinceitwasnotpossibletorepeatthecalibrationrunswithanexpandedgrid, wetestedthereliabilityoftheextrapolationandthetwotrackapproachwithtwo models.WefirstexaminedthereliabilityforTFPwiththebasecaseintheDICE model.ThiswasdonebymakingrunswithincrementsofTFPgrowthupto3 estimatedstandarddeviations(i.e.,uptoaglobaloutputgrowthrateof6.1%per yearto2100).Theserunscover99.7%ofthedistribution.Wethenestimateda surfaceresponsefunctionfor2100temperatureoverthesameintervalasforthe calibrationexercisesandextrapolatedoutsidetherange.Theresultsshowedhigh reliabilityoftheestimatedSRFfortemperatureincreaseuptoabout2standard deviationsabovethebaselineTFPgrowthrate.Beyondthat,theSRFtendedto overestimatethe2100temperature.(SimilarresultswerefoundforCO2 concentrationsandthedamage‐outputratiointheDICEmodel.)Thereasonforthe overestimateisthatcarbonfuelsbecomeexhaustedathighgrowthrates,soraising thegrowthratefurtherabovethealready‐highratehasarelativelysmalleffectson emissions,concentrations,2100temperature,andthedamageratio.Notethatthis impliesthatthefaruppertailofthetemperaturedistributionusingthecorrected SRFwillshowathinnertailthantheonegeneratedbytheSRFestimatedoverthe calibrationruns. WealsoperformedamorecomprehensivecomparisonoftheMUP procedureswithafullMonteCarlousingtheFUNDmodel.Forthis,wetookthepdfs forthethreeuncertainvariablesandranaMonteCarloforthefullFUNDmodelwith 1milliondraws.Wethencomparedthemeansandstandarddeviationsofdifferent 30 variablesforthetwoapproaches.WetestedfourdifferentspecificationsoftheSRFs todeterminewhetherthesewouldproducemarkedlydifferentoutcomes.The resultsindicatedthattheMUPprocedureprovidedreliableestimatesofthemeans andstandarddeviationsofallvariablesthatwetestedexceptFUNDdamages. Exceptingdamages,forthepreferredLQIestimate,theabsoluteaverageerrorofthe meanfortheMUPprocedurerelativetotheFUNDMonteCarlowas0.3%,whilethe absoluteaverageerrorforthestandarddeviationwas1.2%.Fordamages,the errorswere7%and44%,respectively.Additionally,thepercentileestimatesforthe MUPprocedure(againexceptfordamages)wereaccurateuptothe90thpercentile. And,aswillbenotedbelow,theestimatesfortheparametersofthetailsofthe distributionswereaccurateforallvariablesexceptdamages.Anoteproviding furtherdetailsonthecomparisonsisavailablefromtheauthors. VII. ResultsoftheMonteCarlosimulations A. Distributionsformajorvariables FortheMonteCarlosimulations,wetooktheSRFsforeach parameter/model/year/policyandmade1,000,000drawsfromeachpdfforthe threeuncertainparameters.Wethenexaminedtheresultingdistributions.This samplesizewaschosenbecausetheresultswerereliableatthatlevel.Thebootstrap standarderrorsofthemeansandthestandarddeviationsweregenerallylessthan 0.1%ofthemeanorstandarddeviation.Theexceptionwasfordamages,wherethe bootstrapstandarderroroftheestimatedstandarddeviationswasabout0.2%of thevaluefortheFUNDmodel.Wetreateachpdfindependently,butrecognizethat theremaybesomecorrelationbetweenrealizationsofpopulationandGDP. However,explorationsintothisrevealedthatitdidnotsubstantiallyinfluenceour findings. Table2showsstatisticsofthedistributionofthedrawsforeachofthemajor outcomevariables,withaveragestakenacrossallsixmodels.Wealsoshowthe estimatesforthelinearandLQIversionstoillustratethesensitivityoftheresultsto theSRFspecification.Thelastcolumnshowsthecoefficientofvariationforeach variable.Notethattheseestimatesarewithin‐model(parametricuncertainty) resultsanddonotincludeacross‐modelvariability.Theresultshighlightthat emissions,economicoutput,anddamageshavethehighestcoefficientofvariation, underscoringthattheuncertaintyintheseoutputvariablesisgreaterthanforother variables,suchasCO2concentrationsandtemperature.Thisistheresultofboththe underlyingpdfsusedandthemodelsthemselves. 31 Table2.ResultsofMonteCarlosimulationsforaveragesofallmodels Thetableshowsthevaluesofallvariablesfor2100,exceptforthesocialcostof carbon,whichisfor2020.DamagesandSCCareforthreemodels. Table3showsthepercentiledistributionforallmajorvariablesforallmodels withresultsforthebasecase.Thedetailedresultsbymodelsareprovidedinthe appendix.Akeyresultisthedistributionoftemperatureincreasefor2100.The medianincreaseacrossallmodelsis3.79°Cabove1900levels.The95thpercentile oftheincreaseis5.46°C.Giventhesizeoftheinterquartilerange,theseresults definitelyindicatethattherearesubstantialuncertaintiesinallaspectsoffuture climatechangeanditsimpactsinallthemodelsinvestigatedhere. Table3.Distributionofallmajorvariables,averageofsixmodels Thedateforallvariablesis2100exceptfortheSCC,whichis2020.Damagesand SCCareforthreemodels. 32 Table4showsthedistributionforglobaltemperatureincreasein2100by model.Thetemperaturedistributionsofthesixmodelsareonthewholereasonably close.Themedianrangesfrom3.6to4.2°C,withIGSMbeingthelowestandMERGE beingthehighest.Theinterquartilerangevariesfrom0.99°C(FUND)to1.39°C (DICE).The10‐90%rangesfrom1.91°C(WITCH)to2.65°C(DICE).Sincethe variabilityintherandomparametersisthesame,thedifferencesareduetomodel structures. Oneinterestingfeatureisthetemperaturedistributioninthetails.The99th percentilerangesfrom5.6(WITCH)to7.1°C(MERGE),whilethefartailofthe 99.9thpercentilerangesfrom6.2(WITCH)to8.5°C(MERGE). Table4.DistributionoftemperaturechangeintheBasecase,2100,°C Table5showsthedistributionoftheSCCforthethreemodelsthatprovide theseestimates.Thesearetheestimatesofthepresentvalueoftheflowoffuture marginaldamagesofemissionsin2020.Twoofthemodels(WITCHandDICE)use similarquadraticdamagefunctionsandareroughlycomparableinthemiddleofthe distribution,buttherangeismuchsmallerinWITCH.6TheFUNDmodelhasmuch lowerdamages(duetoadifferentdamagefunction),andtheSCCdistributionisan orderofmagnitudelowerthantheothertwomodels.Notethatthecentralestimate oftheSCChereis$13.30pertonofCO2.Thisismuchlowerthanthepreferred estimateoftheUSgovernmentfor2020,whichis$46pertonin2011$witha3% annualdiscountrate.However,thebasecasediscountratesintheMUPrunsforthe modelsthatreportaverage4½%peryearto2050.TheIAWGestimateata5% discountrateis$13pertonandthereforeconsistentwiththeestimatespresented here. 6InWITCHmultipleregionsaremodeled,hencetheglobalSCCistheresultofthe aggregationofregionalSCC. 33 Table5.Distributionofsocialcostofcarbon,2020(2005$pertonCO2) Figure6showstheresultsforthetemperaturedistributionsforthemodels onapercentilescale.Theshapesofthedistributionsaresimilar,althoughtheydiffer byasmuchas1°Cinscaleacrossmostofthedistribution. 9 DICE FUND 7 GCAM IGSM 6 MERGE WITCH Temperature increase, 2100 (deg C) 8 5 4 3 2 1 0 ‐ 20 40 60 Percentile of results 80 100 Figure6.Percentilesofthechangeintemperaturein2100acrossthesixmodels. Animportantquestionthatthisstudycanaddressiswhether,basedonthe currentmodelstructuresandtheassumptionsaboutuncertainparameters,the distributionsofoutcomesarethinorfattailed.Forthesetests,wedefineafattailed distributionasonethathasaninfinite‐varianceParetoorpower‐lawdistributionin thetails(basedonthediscussioninSchuster1984).VariableswithaPareto distributionhaveinfinitevariancewhentheshapeparameterisbelow2,andthey haveaninfinitemeanwithaparameterequaltoorlessthanone.Asaninformal test,wecanexaminetheratioofthevaluesoftheoutputvariablesatthe99thand 34 99.9thpercentile.Foranormaldistribution,theratiooftheseis1.33.ForPareto distributionswithslopevaluesof2.0,1.8,and1.5,theratiosare3.7,3.9,and5.2.If weexaminetheMonteCarloestimates,themaximumratiois1.56,whichoccursfor damagesintheDICEandFUNDmodels.Whilethissuggestsatailthatisslightly fatterthanthenormaldistribution,itfallsfarshortoftheslopeassociatedwithan infinite‐varianceParetoprocess. Beforepresentingtheresults,wereiteratetheconcernthatthecalibration runsdonotextendfarintothetailsforTFP.Thisimpliesthattheresultsontails reportedhererelyonextrapolationsoftheSRFoutsidethesamplerange.We commentbelowonourreplicationofthetailestimateswiththeFUNDmodel,which aregenerallyaccurate.Wealsoemphasizethattheestimatesofthetailsarederived fromtheinteractionofthemodelswiththeassumedpdfs.Totheextentthatthe modelsomitdiscontinuitiesorsharpnon‐linearities,orthatourassumedpdfsare toothin‐tailed,thenwemayunderestimatethethicknessofthetails. WecanalsouseaformaltestoftheParetoshapeparameter,althoughthisis morecomplicatedbecauseitrequiresassumptionsabouttheminimumofthe Paretoregion(statisticaltechniquesarefromRytgaard1990).Examiningthetop 10%ofthedamagedistributionfortheDICEmodel(themostskewedofthe variables),wefindthattheparameteroftheParetodistributionabovethe1%right tailisestimatedtobe4.7(+0.047),whichiswellbelowtheinfinite‐variance thresholdof2.TheParetoparameterestimateforthe0.1%tailis7.03(+0.22). Thesetestsrejectthehypothesisthatthedistributionsarefat‐tailedinthesenseof belongingtoaninfinite‐varianceParetodistribution.Theresultsareduetoboththe structuresofthemodelsandthenatureoftheshocks.Nothinginthemodels preventsthegenerationoffattailsinthissituation,buttheymaymisscriticalnon‐ linearities,sothetestsarenotbyanymeansconclusive. WeexaminedthevalidityoftheresultsforthetailsusingthefullMonteCarlo estimateoftheFUNDmodeldiscussedabove.Forthese,wecomparedtheinformal tests(ratioofthevariablesatthe99.9%iletothe99%ile).TheMUPcalculations wereveryaccurateforallvariablesexceptdamages,whereasfordamagestheMUP calculationsunderestimatedtheskewness(overestimatedtheParetotail).Wealso examinedtheParetoparameterinthefullFUNDMonteCarloandfoundthatthe estimatewassignificantlyabovethethresholdofaninfinitevarianceprocess. Theresultscanalsobeseeninboxplots.Figure7showstheboxplotfor temperatureincreaseto2100.Figure8showstheboxplotfortheCO2 concentrationsfor2100.Bothoftheseunderscorethatwhiletherearedifferences betweenthemodelsinthewaythattheyarerunforthisstudy,theyareperhaps 35 smallerthanonemighthaveexpected–andaremuchsmallerthanthewithin‐ modelvariation.Weshowthisformallyinthenextsection. Temperature increase, 2100 (deg C) 7 6 5 4 3 2 1 0 DICE FUND GCAM IGSM MERGE WITCH Figure7.Boxplotfortheincreaseintemperatureacrossmodelsin2100. Noteonboxplots:Dotismean.Horizontallineismedian.Shadedareaaroundlineis 95%confidenceintervalofmedian(usuallytoosmalltosee).Boxcontains interquartilerange(IQRor25%ileto75%ile).Theupperstaple(horizontalbar)is setatthemedianplus2timestheIQR,whilelowerstapleissetatthemedianminus 2timestheIQR.Theupperstableisapproximatelythe95%ileformostvariables. Becauseofskewnessofthevariables,thelowerstaplerepresentsfaroutliers,andis generallyaroundthe0.1%ile. 36 CO2 Concentrations, 2100, ppm 1,800 1,600 1,400 1,200 1,000 800 600 400 200 DICE FUND GCAM IGSM MERGE WITCH Figure8.BoxplotforCO2concentrations,2100. Forexplanationofboxplots,seeFigure7. B. Modeluncertaintyv.parametricuncertainty Inexaminingtheuncertaintiesofclimatechangeandotherissues,acommon approachhasbeentolookatthedifferencesamongforecasts,models,or approaches(“ensembles”)andtoassumethattheseareareasonableproxyforthe uncertaintiesabouttheendresultorendogenousvariables.Intheareaofclimate models,forexample,researchershaveoftenlookedattheequilibriumclimate sensitivitiesindifferentclimatemodelsandassumedthatthedispersionwouldbe anaccuratemeasureoftheactualuncertaintyoftheECS. Itisconceptuallyclearthattheensembleapproachisaninappropriate measureofuncertaintyofoutcomes.Thedifferenceamongmodelsrepresentsa measureofstructuraluncertainty.Forexample,alternativeclimatemodelsmight havedifferentwaysofincludingcloudfeedbacks.Takingallthedifferencesamong themodelswouldindicatehowstate‐of‐the‐artmodelsdifferontheprocessesand variablesthattheyinclude.Evenhere,however,existingmodelsarelikelytohave anincompleteunderstandingandwillthereforeunderestimatestructural uncertainty.However,fromaconceptualvantagepoint,theygenerallydonot 37 explicitlymodelandconsiderparametricuncertainty.InIAMs,tocomecloserto home,differencesinmodelsreflectdifferencesinassumptionsaboutgrowthrates, productionfunctions,energysystems,andthelike.Butfewmodelsexplicitlyinclude parametricuncertaintyaboutthesevariables.Differencesinpopulationgrowth,for example,areverysmallrelativetomeasuresofuncertaintybasedonstatistical techniquesbecausemanymodelsusethesameestimatesoflong‐runpopulation trends. WecanusetheresultsoftheMonteCarlosimulationstoestimatetherelative importanceofparametricuncertaintyandmodeluncertainty.Wecanwritethe resultsoftheMonteCarlosimulationsschematicallyasfollows.Assumethatthe m modeloutcomeforvariableiandmodelmis Yi andthattheuncertainparameters are ui and u j : 3 3 j Yi m im im ui im, j ui u j i 1 j 1 i 1 Foragivendistributionofeachoftheuncertainparameters,thevarianceof Yi includingmodelvariationis: 3 3 j 2 (Yi ) 2 ( i ) ( im ) 2 2 (ui ) ( im, j ) 2 2 (ui ) 2 (u j ) i 1 j 1 i 1 Thefirsttermontherighthandsideisthevarianceduetomodeldifferences(or structuraluncertainty),whilethesecondandthirdtermsarethevariancedueto parameteruncertainty.Forthispurpose,weincludetheinteractionofthemodel coefficients ( m i and m i, j ) andtheparameteruncertainties [ 2 (ui )] asparametric uncertaintybecausetheywouldnotbeincludedinensembleuncertainty.Theother termsvanishbecauseweassumethattheparametricuncertaintiesareindependent. Whiledependencewilladdfurthertermsontheright‐handsideoftheequationfor thevariance,itwillnotaffectthefractionduetostructuraldifferencesduetothe firstterm. Wecaneasilyestimatethetotaluncertaintyandthestructuraluncertaintyfor differentvariables.TheresultsareshowninTable6.Formostvariables,virtuallyall thevarianceisexplainedbyparametricuncertainty.Forexample,94%ofthe varianceofthe2100temperatureincreaseinallthemodelsisexplainedby parametricuncertainty,andonly6%isexplainedbydifferencesinmodelmeans. ThisfactiseasilyseenintheboxchartsinFigures7and8.Theonlyvariablefor 38 whichmodeluncertaintyisimportantisthesocialcostofcarbon,forwhichfour‐ fifthsofthetotalvarianceisduetomodeldifferences. Wecanputtheseresultsintermsofthevariabilitiesduetodifferentfactors. Ifwetakethecalculatedtemperatureincreaseto2100,theoverallstandard deviationis0.84°Cincludingbothmodelandparametricuncertainty.Thestandard deviationofthemodelmeansaloneis0.21°C.Sothevariabilitymeasuredinterms ofstandarddeviationsofthetemperatureincreaseisunderestimatedbyafactorof fourusingtheensembletechnique. Theneteffectoftheseresultsissobering.Theyindicatethatthetechniqueof relyinguponensemblesasatechniquefordeterminingtheuncertaintyoffuture outcomesis(atleastforthemajorclimatechangevariables)highlydeficient. Ensembleuncertaintytendstounderestimateoveralluncertaintybyasignificant amount. Table6.Fractionofuncertainty(variance)explainedbymodeldifferences. C. Sensitivityoftheresultstoparametervariability Animportantquestionistheextenttowhichtheresultsaresensitivetothe individualpdfsfortheuncertainparameters.Totestforsensitivity,weperformed anexperimentwhereweincreasedthestandarddeviationofeachofthepdfsbya factorof2,bothoneatatimeandtogether.Foradoublingofthestandarddeviation ofallparameters,theincreaseinthestandarddeviationof2100temperaturewasa 39 factorof1.83forallmodelstogether.Webelievethatthisislessthantwobecause theshort‐runtemperatureimpactisnotproportionaltotheECS. Table7showstheresultschangingtheuncertaintybyafactoroftwoone parameteratatimefortheaverageofthe6modelsforallvariableswhichare producedbythesixmodels.Thenumbershowstheratioofthestandarddeviation ofthe2100valueofthevariableinthesensitivitycaserelativetothecasewith assumedpdfs.Doublingalluncertaintiesproducesclosetoadoublingoftheoutput uncertainty,withsomedeviationsbecauseofnon‐linearities. Doublingpopulationuncertaintyhasasmalleffectonallvariablesexcept population.Doublingequilibriumtemperatureuncertaintyraisestheuncertaintyof 2100temperatureby40%buthasnosignificanteffectonotheruncertainties.The majorsensitivityisTFPuncertainty.Doublingthisuncertaintyleadstocloseto doublingoftheuncertaintyofothermajoreconomicvariables,andtoanincreaseof 62percentintheuncertaintyof2100temperature.Thisresultissimilartoaresult invanVuurenetal.(2008),whichsuggeststhatuncertaintyinGDPgrowth dominatestheuncertaintyinemissions. Table7.Sensitivityofoutcomesforchangesinstandarddeviationofeachuncertain parameterbyfactorof2 Thefiguregivestheratioofthestandarddeviationofthevariableatthetopofthe columntothestandarddeviationinthebaserun.Forexample,doublingthe standarddeviationofpopulationincreasedthestandarddeviationof2100 temperatureby6%. Thesummaryonsensitivityoftheresultstothepdfsshowsanimportantand surprisingresult.Onthewhole,theresultsareinsensitivetochangesinthe populationgrowthpdf;aremoderatelysensitivetotheuncertaintyabout 40 equilibriumtemperaturesensitivityontemperature(aswellastodamagesandthe socialcostofcarbon,notshown);andareextremelysensitivetotheuncertainty abouttherateofgrowthofproductivity.Whilelong‐runproductivitygrowthhasthe greatestimpactonuncertainty,itisalsotheleastcarefullystudiedofanyofthe parameterswehaveexamined.Thisresultsuggeststhatmuchgreaterattention shouldbegiventodevelopingreliableestimatesofthetrendanduncertainties aboutlong‐runproductivity. VIII. Conclusions Thisstudyisthefirstmulti‐modelanalysisofparametricuncertaintyin economicclimate‐changemodeling.Theapproachisbasedonestimatingclassic statisticalforecastuncertainty.Thecentralmethodologyconsistsoftwotracks. TrackIinvolvesdoingasetofmodelcalibrationrunsforthesixmodelsandthree uncertainparametersandestimatingasurfaceresponsefunctionfortheresultsof thoseruns.TrackIIinvolvesdevelopingpdfsforkeyuncertainparameters.Thetwo tracksarebroughttogetherthroughasetofMonteCarlosimulationstoestimatethe outputdistributionsofmultipleoutputvariablesthatareimportantforclimate changeandclimate‐changepolicy.Thisapproachisreplicableandtransparent,and overcomesseveralobstaclesforexamininguncertaintyinclimatechange. Herearethekeyresults.First,thecentralprojectionsoftheintegrated assessmentmodels(IAMs)areremarkablysimilaratthemodeler’sbaseline parameters.Thisresultisprobablyduetothefactthatmodelshavebeenusedin modelcomparisonsandmayhavebeenrevisedtoyieldsimilarbaselineresults. However,theprojectionsdivergesharplywhenalternativeassumptionsaboutthe keyuncertainparametersareused,especiallyathighlevelsofpopulationgrowth, productivitygrowth,andequilibriumclimatesensitivity. Second,despitethesedifferencesacrossmodelsforalternativeparameters, thedistributionsofthekeyoutputvariablesareremarkablysimilaracrossmodels withdifferentstructuresandlevelsofcomplexity.Totakeyear2100temperatureas anexample,thequantilesofthedistributionsofthemodelsdifferbylessthan½°C fortheentiredistributionuptothe95thpercentile. Third,wefindthattheclimate‐relatedvariablesarecharacterizedbylow uncertaintyrelativetothoserelatingtomosteconomicvariables.Forthis comparison,welookatthecoefficientofvariation(CV)oftheMonteCarlo simulations.AsshowninTable2,CO2concentrations,radiativeforcings,and temperature(allfor2100)haverelativelylowCV.Outputanddamageshave relativelyhighCV.Asexamples,themodel‐averagecoefficientofvariationfor carbondioxideconcentrationsin2100is0.28,whilethecoefficientofvariationfor 41 climate‐changedamagesis1.29.ThesocialcostofcarbonhasanintermediateCV withinmodels,butwhenmodelvariationisincludedtheCVisclosetothatofoutput anddamages.Theseresultshighlighttheimportanceoffurtherresearchon economicvariablesanddamagefunctionsforreducinguncertaintyandimproving policymaking(e.g.,seePizeretal.2014andDrouetetal.2015). Fourth,wefindmuchgreaterparametricuncertaintythanstructural(across model)uncertaintyforalloutputvariablesexceptthesocialcostofcarbon.For example,inexaminingtheuncertaintyin2100temperatureincrease,thedifference ofmodelmeans(ortheensembleuncertainty)isapproximatelyone‐quarterofthe totaluncertainty,withtherestdrivenbyparametricuncertainty.Whilelooking acrosssixmodelsbynomeansspansthespaceofmethods,thesixmodelsexamined herearerepresentativeofthedifferencesinsize,structure,andcomplexityofIAMs. Thisresultisimportantbecauseofthewidespreaduseofensembleuncertaintyasa proxyforoveralluncertaintyandhighlightstheneedforare‐orientationofresearch towardsexaminingparametricuncertaintyacrossmodels. Afifthinterestingfindingofthisanalysisisthelackofevidenceinsupportof fattailsinthedistributionsofemissions,globalmeansurfacetemperature,or damages.Populationgrowth,totalfactorproductivitygrowth,andclimate sensitivityareverylikelytobethreeofthekeyuncertainparametersinclimate change.Yet,basedonbothinformalandformaltests,themodelsascurrently constructedfindthatthetailsarerelativelythin.Thedeclineinprobabilities associatedwithachangeinanyofthevariablesismuchlargerthanwouldbe associatedwithaninfinite‐varianceParetoprocess.Asdiscussedabove,we emphasizethatthesefindingsshouldbeinterpretedinthecontextofthecurrent groupofmodelsandtheassumedpdfs.Theresultsdonotruleoutfattails,butthey doprovideempiricalevidenceagainstfattailsinoutcomesinvestigatedinthisstudy forthecurrentsetofmodelsandthedistributionsofthethreeuncertainvariables consideredhere.Theseresultstendtosupporttheuseofexpectedbenefit‐cost analysisforclimatechangepolicy,incontrasttosuggestionsbysomeauthorsthat neglectoffattaileventsmayvitiatestandardanalyses(Weitzman2009). Sixth,wefindthatwithinawiderangeofuncertainty,changesindispersion oftwooftheuncertainparameterstakensinglyhavearelativelysmalleffectonthe uncertaintyoftheoutputvariables,thesebeingpopulationgrowthandequilibrium temperaturesensitivity.However,uncertaintyaboutproductivitygrowthhasa majorimpactontheuncertaintyofallthemajoroutputvariables.Thereasonfor thisisthattheuncertaintyofproductivitygrowthfromtheexpertsurvey compoundsgreatlyoverthe21stcenturyandinducesanextremelylargeuncertainty 42 aboutoutput,emissions,concentrations,temperaturechange,anddamagesbythe endofthecentury. Asinanystudy,thisanalysisisintentionallysharplyfocused.Byanalyzing parametricuncertaintyinthreekeyparameters,wedonotclaimtobecapturingall uncertaintiesinclimatechange.Aswedescribeabove,therearemanyuncertainties thatcannotbecapturedusingthestatisticalframeworkdevelopedhere.Butby providingdetailedestimatesofuncertaintyacrossarangeofIAMsthatarecurrently beingusedinthepolicyprocess,webelievethatwehavesignificantlyimprovedthe understandingofuncertaintyinclimatechange.Moreover,ournewtwo‐track methodologyiswell‐suitedforexpansiontoadditionalparametersandmodels,and canbereadilyusedtoexploreadditionalconcerns,suchastheinteractionbetween carbonpoliciesanduncertainty. 43 Appendix1.FurtherDetailsontheChoiceofECSDistribution Thisappendixexplainstheprocedurefordevelopingthepdfforclimate sensitivity.Thestudybeganbyreviewingthefiveprobabilitydensityfunctionsfor equilibriumclimatesensitivity(ECS)usedintheIPCCAR5thatdrawuponmultiple linesofevidence.TheseareAldrinetal.(2012),LibardoniandForest(2013),Olsen etal.(2012),AnnanandHargreaves(2006),andHegerletal.(2006).FigureA1 illustratesthelog‐normalfitstoeachofthesedistributions(fitsbythepresent authors). FigureA1.Log‐normaldistributionsfittotheprobabilitydensityfunctionscitedin theIPCCAR5.ThedistributionshownhereisfromtheupdatedLibardoni&Forest (2013)figures. Ourchosenstudy,Olsenetal.(2012),isrepresentativeofthestudiesinboth itsmethodologyandresults.ItusesaBayesianapproach.Thepriordistributionwas constructedtofitthe“mostlikely”valuesand“likely”rangesinFigure3inKnutti andHegerl(2008)basedonthesummarystatisticsofthe“currentmeanclimate state”and“LastGlacialMaximummodels.”Olsenetal.assumeaninverseGaussian (Wald)distributionandobtainthispriorbyassumingindependencebetweenthe 44 currentmeanclimatestateandthelastglacialmaximummodels,andthen computingthemixturedistribution. TheposteriordistributionisthencalculatedbyusingaMarkovChainMonte Carlosimulationtoupdatethepriorwithalikelihoodfunction.Thelikelihoodis basedonseveraldifferenttracers,suchasglobalaverageatmospheric surface/oceansurfacetemperaturesandglobaltotalheatcontent.Thesetracers comefromtheUniversityofVictoriaESCMclimatemodel,whichconsistsofathree‐ dimensionaloceangeneralcirculationmodelcoupledwitha thermodynamic/dynamicsea‐icemodel.Theauthorsassumeindependence,sothat thelikelihoodofbothobservationsisequaltotheproductofthelikelihoods. Theparametersofthelog‐normaldistributionfittoOlsenetal.areμ= 1.10704andσ=0.264.Themajorsummarystatisticsofthereferencedistributionin thestudyarethefollowing:mean=3.13,median=3.03,standarddeviation=0.843, skewness=0.824,andkurtosis=4.23.InimplementingtheMonteCarloforeach model,weretainedthemeanECSforthatmodel.Wethenimposedalog‐normal distributionthatretainedthearithmeticstandarddeviationoftheECS(i.e.,a standarddeviationof0.843)basedontheOlsenetal.(2012)distribution. 45 Appendix2.ExpertSurveyonTotalFactorProductivity Akeycomponentoftheprojectwasdeterminingtheuncertaintyin productivity(or,asoperationallydefined,outputpercapita).Areviewofexisting studiesindicatedthattherewerenodetailedstudiesoffutureoutputuncertainties outto2100thatwecouldrelyon.Wethereforedecidedtoundertakeanexpert elicitation.Thedetailedresultsofthesurveywillbeshortlyavailableseparatelyasa workingpaper.Thisappendixsketchesthemethodsandsummarizesthe preliminaryresults.Notethatthecurrentresultsincludeonly11oftherespondents, andthecompletesurveyresultswillbeusedforthefinalpublication. 2.1 SurveyDesign Indeterminingtheprobabilitydistributionoffutureproductivitygrowth,a majordifficultyisthenon‐stationarityofthisvariable.Itisclearlynon‐stationaryif oneexamineshistoricaldata.Fromatheoreticalpointofview,wewouldexpect non‐stationaritybecausethemajordeterminantsoflong‐rungrowth–invention andtechnologicalchange–involvenewanddifferentprocessesratherthan replicationofsomeunderlyingprocess.Forthisreason,itisimportanttooverlay anyempiricalstudywithexpertviews. Expertopinionhasbeenusedsystematicallyinaverylimitednumberof studiesofeconomicgrowth.Forexample,Websteretal.(2002)analyzeuncertainty intheGDPgrowthrateoutto2100(asaproxyforchangesinlaborproductivity) usingestimatescollectedfromanelicitationof5expertsfromasingleinstitution. Thisseemedtoothinabaseforthepresentstudy. Inthisstudy,weconductedasurveyofexpertpredictionsaboutuncertainty inglobalannualgrowthratesfortheperiod2010‐2100.Expertsprovidedresponses usinganonlinesurvey(seeFigureA2fortheresponseformat).Thepanelofexperts wasselectedthroughaprocessofnominationbyleadingeconomists. Weaskedexpertsaboutgrowthratesinhigh‐,medium‐,andlow‐income countries,aswellasaboutglobalaggregaterates.Aspartofthesurvey,wealerted expertstoproblemsofoverconfidenceandincludeawarm‐upsectionthatwas designedtoincreaseawarenessoftheirpersonaloverconfidence.Inaddition,we askedexpertsaboutanyambiguitiesthattheyexperiencedinthesurveyand 46 providedthemwithhistoricaldataongrowthratesfortheperiod1900‐2000from Barro‐Ursua(2010)andMaddison(2003).7 FigureA2.ResponseFormatforExpertSurvey Thesurveywascomprisedof4setsofquestionsaboutgrowthrates:(1) centralestimates(50thpercentile)forgrowthratesfor2010‐2050and2010‐2100, (2)estimatesofuncertaintybasedonprovidingthe10th,25th,75th,and90th percentilesofthegrowthrates,(3)theprojectedmagnitudeofeffectsofpositive andnegativeshockstotheeconomy,and(4)near‐termpredictions(forthe followingyear).Weaskedeachexperttodescribetherationalefortheirresponseas wellasanexplanationofmajorpositiveandnegativeshocks.Thesurveyalsoasked expertstoidentifyoutsidesourcesofinformationthatwereusedtogenerate forecastsandtoranktheirownexpertiseoverallandforparticularregions. 2.2 CombiningExpertDistributions Weusetwomethodstoestimatethemeanandstandarddeviationforthe best‐fittingcombinednormaldistributionofgrowthratesfortheperiod2010‐2100. Thefirstmethodassumesthatexpertshaveestimatesofquantilesofthe distributionoflong‐rungrowthrates.Thecombinedpdfisthenthedistributionthat minimizesthesumofsquareddifferencesbetweenthecombinednormal 7Barro‐UrsuaMacroeconomicDataavailableat:rbarro.com/data‐sets/.Maddisonisfrom AngusMaddison(2003).Availableat:http://www.theworldeconomy.org/statistics.htm. 47 distributionateachquantileandtheaverageofthequantileestimatesofthe experts.Thesecondmethodbeginswithestimatesoftheparametersofthebest‐ fittingnormaldistributionforeachexpert;andthentakesthesamplemeansofthe parametersoftheexpertsforthecombinednormaldistribution. Wefindverylittledifferencebetweenthetwomethods.Forthepreliminary sample,themeangrowthratesofpercapitaoutputforthetwomethodsare2.29 and2.30,respectivelyformethods1and2.Thecombinedstandarddeviationsare 1.15and1.17,respectively. Thecombinedpdfsalongwith11preliminaryresponsesareshowninFigure 2inthemaintext.Thecurrentprocedureusesthesamplemeanofthestandard deviationfortheMonteCarloestimates,butweareconsideringusingarobust estimatorforthefinalreport. 48 Appendix3.AdditionalLatticeDiagrams Weincludeherefurtherlatticediagrams.Thestructureisasdescribedinthe text.Theonlydifferenceistheoutputvariable,whichisshownatthetopofthe graph. Notethatthefirstgroupofdiagramsisforthebaseruns,whilethesecond groupisfortherunswithcarbontaxes(CarbonTaxorAmpereruns). 49 50 51 52 53 54 Appendix4.AdditionalTablesandGraphs TableA1.Overviewofglobalintegratedassessmentmodelsincludedinthisstudy. Model DICE FUND GCAM IGSM MERGE Number Time Variables KeyCharacteristics of Horiz Included Economic on Regions 1 2010‐ 1,2,3,5,6 Optimalgrowthmodel, 2300 endogenousGDPand temperature,exogenous population,SWFisCES withrespectto consumption. 16 1950‐ 1,2,3,4,5,6, Multi‐region,multi‐gas, 3000 7 detaileddamagefunctions, exogenousscenarios perturbedbymodel 14 2005‐ 1,2,3,4,5,7 Integratedenergy‐land‐ 2095 climatemodelwith technologydetail; exogenouspopulationand GDP;endogenousenergy resources,agriculture,and temperature;economic costsarecalculatedfor producerandconsumer surpluschange 16 2100 1,2,3,4,5,7 Fullgeneralcirculation modellinkedtoamulti sector‐multiregion generalequilibriummodel oftheeconomywith explicitadvanced technologyoptions 10 2100 1,2,3,4,5,7 Ramseymodelcoupled Selected Reference s (Nordhaus andSztorc 2014) (Anthoff andTol 2010, 2013) (Calvinand etal.2011) (Chenetal. 2015, Sokolovet al.2009, Websteret al.2012) (Blanford 55 withenergyprocess etal.2014) model,multipleregions, endogenousGDPand temperature,exogenous population WITCH 13 2150 1,2,3,4,5,6 Optimalgrowthmodel, (Bosettiet ,7 endogenousGDPand al.2006) temperature,exogenous population,SWFisCES withrespectto consumption. Notes:SWF=socialwelfarefunction,CES=constantelasticityofsubstitution.For variablesincludedthekeyis: 1=GDP,population 2=CO2emissions,CO2concentrations 3=globaltemperature 4=multipleregions 5=mitigation 6=damages 7=non‐CO2GHGs 56 ResultsofMonteCarlosimulationsformodelsandmajorvariables [Allvariablesare2100exceptSCC,whichis2020] 57 CO2 emissions, 2100 (billions tons CO2) 500 400 300 200 100 0 -100 DICE FUND GCAM IGSM MERGE WITCH FigureforboxplotsforCO2emissions,2100.Fordiscussionofboxplots,seeFigure 7. 58 DICEFUNDWITCH Figureforboxplotsforsocialcostofcarbon,2020.Fordiscussionofboxplots,see Figure7. 59 Estimatesfromsurfaceresponsefunctionsbyvariableandmodel. 60 GoodnessoffitofworstfittingLQIvariablebymodel. 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