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/

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ModelingUncertaintyinClimateChange:

AMulti‐ModelComparison1

KennethGillingham,WilliamNordhaus,DavidAnthoff,GeoffreyBlanford,ValentinaBosetti,PeterChristensen,HaewonMcJeon,JohnReilly,PaulSztorc

September17,2015

Abstract

Theeconomicsofclimatechangeinvolvesavastarrayofuncertainties,complicatingboththeanalysisanddevelopmentofclimatepolicy.Thisstudypresentstheresultsofthefirstcomprehensivestudyofuncertaintyinclimatechangeusingmultipleintegratedassessmentmodels.Thestudylooksatmodelandparametricuncertaintiesforpopulation,totalfactorproductivity,andclimatesensitivity.Itestimatesthepdfsofkeyoutputvariables,includingCO2concentrations,temperature,damages,andthesocialcostofcarbon(SCC).Onekeyfindingisthatparametricuncertaintyismoreimportantthanuncertaintyinmodelstructure.Ourresultingpdfsalsoprovideinsightsontailevents.

                                                            1TheauthorsaregratefultotheDepartmentofEnergyandtheNationalScienceFoundationforprimarysupportoftheproject.ReillyandMcJeonacknowledgesupportbytheU.S.DepartmentofEnergy,OfficeofScience.ReillyalsoacknowledgestheothersponsorstheMITJointProgramontheScienceandPolicyofGlobalChangelistedathttp://globalchange.mit.edu/sponsors/all.TheStanfordEnergyModelingForumhasprovidedsupportthroughitsSnowmasssummerworkshops.KennethGillinghamcurrentlyworksasaSeniorEconomistfortheCouncilofEconomicAdvisers(CEA).TheCEAdisclaimsresponsibilityforanyoftheviewsexpressedherein,andtheseviewsdonotnecessarilyrepresenttheviewsoftheCEAortheUnitedStatesgovernment.Noneoftheauthorshasaconflictofinteresttodisclose.KennethGillinghamandWilliamNordhausarecorrespondingauthors(kenneth.gillingham@yale.eduandwilliam.nordhaus@yale.edu).

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I. Introduction

Acentralissueintheeconomicsofclimatechangeisunderstandinganddealingwiththevastarrayofuncertainties.Theserangefromthoseregardingeconomicandpopulationgrowth,emissionsintensitiesandnewtechnologies,tothecarboncycle,climateresponse,anddamages,andcascadetothecostsandbenefitsofdifferentpolicyobjectives.

Thispaperpresentsthefirstcomprehensivestudyofuncertaintyofmajoroutcomesforclimatechangeusingmultipleintegratedassessmentmodels(IAMs).ThesixmodelsusedinthestudyarerepresentativeofthemodelsusedintheIPCCFifthAssessmentReport(IPCC2014)andintheU.S.governmentInteragencyWorkingGroupReportontheSocialCostofCarbonorSCC(USInteragencyWorkingGroup2013).Wefocusoureffortsinthisstudyonthreekeyuncertainparameters:populationgrowth,totalfactorproductivitygrowth,andequilibriumclimatesensitivity.Fortheestimateduncertaintyinthesethreeparameters,wedevelopestimatesoftheuncertaintyto2100formajorvariables,suchasemissions,concentrations,temperature,percapitaconsumption,output,damages,andthesocialcostofcarbon.

Ourapproachisatwo‐trackmethodologythatpermitsreliablequantificationofuncertaintyformodelsofdifferentsizeandcomplexity.Thefirsttrackinvolvesperformingmodelrunsoverasetofgridpointsandfittingasurfaceresponsefunctiontothemodelresults;thisapproachprovidesaquickandaccuratewaytoemulaterunningthemodels.Thesecondtrackdevelopsprobabilitydensityfunctionsforthechoseninputparameters(i.e.,theparameterpdfs)usingthebestavailableevidence.WethencombinebothtracksbyperformingMonteCarlosimulationsusingtheparameterpdfsandthesurfaceresponsefunctions.

Thismethodologyprovidesatransparentapproachtoaddressinguncertaintyacrossmultipleparametersandmodelsandcaneasilybeappliedtoadditionalmodelsanduncertainparameters.Animportantaspectofthismethodology,unlikevirtuallyallothermodelcomparisonexercises,isitsreplicability.Theapproachiseasilyvalidatedbecausethedatafromthecalibrationexercisesarerelativelycompactandarecompiledinacompatibleformat,thesurfaceresponsescanbeestimatedindependently,andtheMonteCarlosimulationscanbeeasilyruninmultipleexistingsoftwarepackages.

Thispaperisstructuredasfollows.Thenextsectiondiscussesthestatisticalconsiderationsunderpinningourstudyofuncertaintyinclimatechange.SectionIIIpresentsourmethodologyforthetwo‐trackapproach,whilethenextsectiondiscussesselectionofcalibrationruns.SectionVgivesthederivationofthe

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probabilitydistributions.SectionVIgivestheresultsofthemodelcalculationsandthesurfaceresponsefunctions,andsectionVIIpresentstheresultsoftheMonteCarloestimatesofuncertainties.WeconcludewithasummaryofthemajorfindingsinsectionVIII.TheAppendicesprovidefurtherbackgroundinformation.

II. StatisticalConsiderations

A. BackgroundonUncertaintyinClimateChange

Climatechangescienceandpolicyhavefocusedlargelyonprojectingthecentraltendenciesofmajorvariablesandimpacts.Whilecentraltendenciesareclearlyimportantforafirst‐levelunderstanding,attentionisincreasinglyontheuncertaintiesintheprojections.Uncertaintiestakeongreatsignificancebecauseofthepossibilityofnon‐linearitiesinresponses,particularlythepotentialfortriggeringthresholdsinearthsystems,inecosystem,orineconomicoutcomes.Tobesure,uncertaintieshavebeenexploredinmajorreports,suchastheIPCCScientificAssessmentReportsfromthefirsttothefifth.However,thesehavemainlyexamineddifferencesamongmodelsasatoolforassessinguncertaintiesaboutfutureprojections.Asweindicatebelow,ourresultssuggestthatparametricuncertaintyisquantitativelymoreimportantthandifferencesacrossmodelsformostvariables.

Inrecentreviewsofclimatechange,thereisanincreasingfocusonimprovingourunderstandingoftheuncertainties.Forexample,in2010theInter‐AcademyReviewoftheIPCC,theprimaryrecommendationforimprovingtheusefulnessofthereportwasaboutuncertainty:

Theevolvingnatureofclimatescience,thelongtimescalesinvolved,

andthedifficultiesofpredictinghumanimpactsonandresponsestoclimatechangemeanthatmanyoftheresultspresentedinIPCCassessmentreportshaveinherentlyuncertaincomponents.Toinformpolicydecisionsproperly,itisimportantforuncertaintiestobecharacterizedandcommunicatedclearlyandcoherently.(InterAcademyCouncil2010)

Inarecentreport,theU.S.CongressionalBudgetOfficealsovoiceditsconcernsaboutuncertainty:

Inassessingthepotentialrisksfromclimatechangeandthecostsofavertingit,however,researchersandpolicymakersencounterpervasiveuncertainty.Thatuncertaintycontributestogreat

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differencesofopinionastotheappropriatepolicyresponse,withsomeexpertsseeinglittleornothreatandothersfindingcauseforimmediate,extensiveaction.Policymakersarethusconfrontedwithawiderangeofrecommendationsabouthowtoaddresstherisksposedbyachangingclimate—inparticular,whether,how,andhowmuchtolimitemissionsofgreenhousegases.(CBO2005)

Thefocusonuncertaintyhastakenonincreasedurgencybecauseofthegreatattentiongivenbyscientiststotippingelementsintheearthsystem.AninfluentialstudybyLentonetal.(2008)discussedimportanttippingelementssuchasthelargeicesheets,large‐scaleoceancirculation,andtropicalrainforests.Someclimatologistshavearguedthatglobalwarmingbeyond2°CwillleadtoanirreversiblemeltingoftheGreenlandicesheet(Robinsonetal.2012).Onceuncertaintiesarefullyincluded,policieswillneedtoaccountfortheprobabilitythatpathsmayleadacrosstippingpoints,withparticularconcernforonesthathaveirreversibleelements.

Afurthersetofquestionsinvolvesthepotentialforfattailsinthedistributionofparameters,ofoutcomes,andoftheriskofcatastrophicclimatechange.(Afat‐orthick‐taileddistributionisonewheretheprobabilityofextremeeventsdeclinesslowly,sothetailofthedistributionisthick.Animportantexampleisthepower‐laworParetodistribution,inwhichthevarianceoftheprocessisunboundedforcertainparametervalues.)

Theissuearisesbecauseofthecombinationofoutcomesthatarepotentiallycatastrophicinnatureandprobabilitydistributionswithfattails.Thecombinationofthesetwofactorsmayleadtosituationsinwhichfocusingoncentraltendenciesiscompletelymisleadingforpolicyanalysis.Inaseriesofpapers,MartinWeitzman(seeespeciallyWeitzman2009)hasproposedadramaticallydifferentconclusionfromstandardanalysisinwhathehascalledtheDismalTheorem.Intheextremecase,thecombinationoffattails,unlimitedexposure,andhighriskaversionimpliesthattheexpectedlossfromcertainriskssuchasclimatechangeisunboundedandwethereforecannotperformstandardoptimizationcalculationsorcost‐benefitanalyses.

Therearetodatemanystudiesoftheimplicationsofuncertaintyforclimatechangeandclimate‐changepolicyorofuncertaintyinoneormanyparametersusingasinglemodel.SomenotableexamplesincludeReillyetal.(1987),PeckandTeisberg(1993),NordhausandPopp(1997),Pizer(1999),Webster(2002),Baker(2005),Hope(2006),Nordhaus(2008),Websteretal.(2012),AnthoffandTol(2013),andLemoineandMcJeon(2013).

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Todate,however,theonlypublishedstudythataimstoquantifyuncertaintyinclimatechangeformultiplemodelsistheU.S.governmentInteragencyWorkingGroupreportonthesocialcostofcarbon,whichispublishedinGreenstoneetal.(2013)andmoreextensivelydescribedinIAWG(2010).Thisstudyusedthreemodels,twoofwhichareincludedinthisstudy,toestimatethesocialcostofcarbonforU.S.governmentpurposes.However,whileitdidexamineuncertainty,thecross‐modelcomparisonfocusedonasingleuncertainparameter(equilibriumclimatesensitivity)foritsformaluncertaintyanalysis;allotheruncertainparametersinthemodelswereleftuncertainwiththemodelers’pdfs.Evenwiththissingleuncertainparameter,theestimatedsocialcostofcarbonvariesgreatly.The2015socialcostofcarbonintheupdatedIAWG(2013)is$38pertonofCO2usingthemedianestimateversus$109pertonofCO2usingthe95percentile(bothin2007dollarsandusinga3%discountrate),whichwouldimplyverydifferentlevelsofpolicystringency.TheIAWGanalysisalsousedcombinationsofmodelinputsandoutputsthatwerenotalwaysinternallyconsistent.Comparisonoftheuncertaintiesinaconsistentmannerindifferentmodelsisclearlyanimportantmissingareaofstudy.

B. Centralapproachofthisstudy

Thisprojectaimstoquantifytheuncertaintiesofkeymodeloutcomesinducedbyuncertaintyinimportantparameters.Wehopetolearnthedegreetowhichthereisprecisioninthepointestimatesofmajorvariablesthatareusedinmajorintegratedassessmentmodels.Putdifferently,theresearchquestionweaimtoanswerfromthisstudyis:Howdomajorparameteruncertaintiesaffectthedistributionofpossibleoutcomesofmajoroutcomes;andwhatisthelevelofuncertaintyofmajoroutcomevariables?

Wecallthisquestiononeof“classicalstatisticalforecastuncertainty.”Thestudyofforecastinguncertaintyanderrorhasalonghistoryinstatisticsandeconometrics.SeeforexampleClementsandHendry(1998,1999)andEricsson(2001).Thestandardtoolsofforecastinguncertaintyhavevirtuallyneverbeenappliedtomodelsintheenergy‐climate‐economyareasbecauseofthecomplexityofthemodelsandthenon‐probabilisticnatureofbothinputsandstructuralrelationships.

Keyuncertaintiesthatwewillexamineincludebothprojectionsandpolicyoutcomes.Forexample,whataretheuncertaintiesofemissions,concentrations,temperatureincreases,anddamagesinabaselineprojection?Whatistheuncertaintyinthesocialcostofcarbon?Howdouncertaintiesacrossmodelscomparewiththeuncertaintieswithinmodelsgeneratedbyparameteruncertainty?

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Oneofthekeycontributionsofthisworkisthatithasthepotentialtohighlightareaswherereducinguncertaintywillhaveahighpayoff.

C. Uncertaintyinabroadercontext

Thereareseveraluncertaintiesinclimatechangethatfacebothnaturalandsocialscientistsanddecisionmakers.Amongtheimportantonesare:(1)parametricuncertainty,suchasuncertaintyaboutclimatesensitivityoroutputgrowth;(2)modelorspecificationuncertainty,suchasthespecificationoftheaggregateproductionfunction;(3)measurementerror,suchasthelevelandtrendofglobaltemperatures;(4)algorithmicerrors,suchasonesthatfindtheincorrectsolutiontoamodel;(5)randomerrorinstructuralequations,suchasthoseduetoweathershocks;(6)codingerrorsinwritingtheprogramforthemodel;and(7)scientificuncertaintyorerror,suchaswhenamodelcontainsanerroneoustheory. Thisstudyfocusesprimarilyonthefirstofthese,parametricuncertainty,andtoalimitedextentonthesecond,modeluncertainty.Wefocusonthefirstbecausetherearemajoruncertaintiesaboutseveralparameters,becausethishasbeenakeyareaforstudyinearlierapproaches,andbecauseitisatypeofuncertaintythatlendsitselfmostreadilytomodelcomparisons.Inaddition,sinceweemploysixmodels,theresultsprovidesomeinformationabouttheroleofmodeluncertainty,althoughwedonotdevelopaformalapproachtomodeluncertainty.Werecognizethatparameterandmodeluncertaintiesarebuttwooftheimportantquestionsthatarise,butarigorousapproachtomeasuringthecontributionoftheseuncertaintieswillmakeamajorcontributiontounderstandingtheoveralluncertaintyofclimatechange. Fromatheoreticalpointofview,themeasuresofuncertaintycanbeviewedasapplyingtheprinciplesofjudgmentalorsubjectiveprobability,or“degreeofbelief,”tomeasuringfutureuncertainties.Thisapproach,whichhasitsrootsintheworksofRamsey(1931),deFinetti(1937),andSavage(1954),recognizesthatitisnotpossibletoobtainfrequentistoractuarialprobabilitydistributionsforthemajorparametersinintegratedassessmentmodelsorinthestructuresofthemodels.Thetheoryofsubjectiveprobabilityviewstheprobabilitiesasakintotheoddsthatinformedscientistswouldtakewhenwageringontheoutcomeofanuncertainevent.Forexample,supposetheeventwaspopulationgrowthfrom2000to2050.Thesubjectiveprobabilitymightbethattheinterquartilerange(25%,75%)wasbetween0.5%and2.0%peryear.Inmakingtheassessment,thescientistwouldineffectsaythatitisamatterofindifferencewhethertobetthattheoutcomewhenknownwouldbeinsideoroutsidethatrange.Whileitisnotcontemplatedthatabet

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wouldactuallyoccur(althoughthatisnotunprecedented),thewagerapproachhelpsframetheprobabilitycalculation.

III. Methodology

A. Overviewofourtwotrackapproach

Inundertakinganuncertaintyanalysis,theprojectcontemplatedtwopotentialapproaches.Inoneapproach,eachmodelwoulddoaMonteCarlosimulationinwhichitwoulddomanyrunswherethechosenuncertainparametersaredrawnfromajointpdf.Whilepotentiallyfeasibleforsomemodels,suchanapproachisexcessivelyburdensomeandlikelyinfeasibleatthescalenecessarytohavereliableestimates.

Wethereforedevelopedasecondapproachwhichwecallthe“two‐trackMonteCarlo.”ThisapproachseparatesthemodelcalibrationrunsfromgenerationoftheparameterpdfsandtheMonteCarloestimates.Atthecoreoftheapproacharetwoparalleltracks,whicharethencombinedtoproducethefinalresults.Thefirsttrackusesmodelrunsfromsixparticipatingeconomicclimatechangeintegratedassessmentmodelstodevelopsurfaceresponsefunctions;theserunsprovidetherelationshipbetweenouruncertaininputparametersandkeyoutputvariables.Thesecondtrackdevelopsprobabilitydensityfunctionscharacterizingtheuncertaintyforeachanalyzeduncertaininputparameter.WecombinetheresultsofthetwotracksusingaMonteCarlosimulationtocharacterizestatisticaluncertaintyintheoutputvariables.

B. Theapproachinequations

Itwillbehelpfultoshowthestructureoftheapproachanalytically.Wecanrepresentamodelasamappingfromexogenousandpolicyvariablesandparameterstoendogenousoutcomes.Themodelscanbewrittensymbolicallyasfollows:

(1) ( , , )m mY H z u

Inthisschema,Ymisavectorofmodeloutputsformodelm;zisavectorofexogenousandpolicyvariables; isavectorofmodelparameters;uisavectorofuncertainparameterstobeinvestigated;andHmrepresentsthemodelstructure.Weemphasizethatmodelshavedifferentstructures,modelparameters,andchoiceofinputvariables.However,wecanrepresenttheargumentsofHwithoutreferencetomodelsbyassumingsomeareomitted.

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Thefirststepintheprojectistoselecttheuncertainparametersforanalysis.Oncetheparametersareselected,eachmodelthendoesselectedcalibrationruns.Thecalibrationrunstakeasacentralsetofparametersthebaseorreferencecaseforeachofthemodels.Itthenmakesseveralrunsthataddorsubtractspecifiedincrementsfromeachofthebasevaluesoftheuncertainparameters.Thisproducesasetofinputandoutputsforeachmodel.

Moreprecisely,hereistheprocedureforthefirsttrackoftheapproach.Eachmodelhasabaselinerunwithbasevaluesforeachoftheuncertainparameters.

Denotethebaseparametervaluesas ,1 ,2 ,3( , , ).b b bm m mu u u Thenextstepdeterminesagrid

ofdeviationvaluesoftheuncertainparametersthateachmodeladdsorsubtractsfromthebasevaluesoftheuncertainparameters.Denotethesedeviationvaluesas

1,1,1 1,1,2 5,5,5( , ,..., ).G The G vectorrepresents125=5x5x5deviationsfrom

themodelers’baseparametervalues.So,forexample,thevector 1,1,1 would

representoneofthe125gridvectorsthattakesthefirstvalueforeachuncertainparameter.Supposethat 1,1,1 ( 0.014, .02, 2). Thenthatcalibrationrunwould

calculatetheoutcomesfor ,1 ,2 ,3( , , .014, .02, 2)m m b b bm m mY H z u u u ,whereagain ,

bm ku is

thebasevalueforuncertainparameterkformodelm.Similarly, 3,3,3 (0,0,0). For

thatdeviationvalue,thecalibrationrunwouldcalculatetheoutcomesfor

,1 ,2 ,3( , , , , ),m m b b bm m mY H z u u u whichisthemodelbaselinerun.

Thethirdstepistoestimatesurfaceresponsefunctions(SRFs)foreachmodelandvariableoutcome.Symbolically,thesearethefollowingfunctions:

(2) 1 ,1 2 ,2 3 ,3 ,1 ,2 ,3( , , ) ( , , )m m b b b mm m m m m mY R u u u u u u R u u u

TheSRFsarefitovertheobservationsofthe ,m ku fromthecalibrationexercises

(125eachforthebaselineandforthecarbon‐taxcases).TheSRFsarelinear‐quadratic‐interactionequationsasdescribedbelow.

Thesecondtrackoftheprojectprovidesuswithprobabilitydensityfunctions

foreachofouruncertainparameters, ( )kkf u .Thesearedevelopedonthebasisof

externalinformationasdescribedbelow.

Thefinalstepistoestimatethecumulativedistributionoftheoutputvariables, ( ).m mG Y Thesearethedistributionsoftheoutcomevariables mY for

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modelm,wherewenotethatthedistributionswilldifferbymodel.ThedistributionsarecalculatedbyMonteCarlomethods,forasamplesizeofN:

(3) ,1 ,2 ,31

( ) 1 if ( , , ) , otherwise = 0 /N

m m m n n n mm m m

n

G Y H u u u Y N

Thenotationhereisthat ,n

m ku isthenthdrawofrandomvariable ku inthe

MonteCarloexperiment.ThisunintuitiveequationsimplystatesthatthecumulativedistributionisequaltothefractionofoutcomesintheMonteCarlosimulationwheretheSRFyieldsavalueoftheoutcomevariablethatislessthan .mY Thedistributionofoutcomesforeachvariableandmodelisconditionalonthemodelstructureandontheharmonizeduncertaintyoftheuncertainparameters.ForaclassicstudyofMonteCarlomethods,seeHammersleyandHandscomb(1964).

C. IntegratedAssessmentModels

Thechallengeofanalysisandpoliciesforglobalwarmingisparticularlydifficultbecauseitspansmanydisciplinesandpartsofsociety.Thismany‐facetednaturealsoposesachallengetonaturalandsocialscientists,whomustincorporateawidevarietyofgeophysical,economic,andpoliticaldisciplinesintotheirdiagnosesandprescriptions.Thetaskofintegratedassessmentmodels(IAMs)istopulltogetherthedifferentaspectsofaproblemsothatprojections,analyses,anddecisionscanconsidersimultaneouslyallimportantendogenousvariables.IAMsgenerallydonotpretendtohavethemostdetailedandcompleterepresentationofeachincludedsystem.Rather,theyaspiretohave,atafirstlevelofapproximation,modelsthatoperateallthemodulessimultaneouslyandwithreasonableaccuracy.

ThestudydesignwaspresentedatameetingwheremanyoftheestablishedmodelerswhobuildandoperateIAMswerepresent.Allwereinvitedtoparticipate.Aftersomepreliminaryinvestigationsandtrialruns,sixmodelswereabletoincorporatethemajoruncertainparametersintotheirmodelsandtoprovidemostoftheoutputsthatwerenecessaryformodelcomparisons.Thefollowingisabriefdescriptionofeachofthesixmodels.TableA5intheappendixprovidesfurtherdetailsoneachmodel.

TheDICE(DynamicIntegratedmodelofClimateandtheEconomy)wasfirstdevelopedaround1990andhasgonethroughseveralextensionsandrevisions.ThelatestpublishedversionisNordhaus(2014)withadetaileddescriptioninNordhausandSztorc(2014).TheDICEmodelisagloballyaggregatedmodelthatviewstheeconomicsofclimatechangefromtheperspectiveofneoclassicaleconomicgrowththeory.Inthisapproach,economiesmakeinvestmentsincapitalandinemissions

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reductions,reducingconsumptiontoday,inordertolowerclimatedamagesandincreaseconsumptioninthefuture.Thespecialfeatureofthemodelistheinclusionofallmajorelementsinahighlyaggregatedfashion.Themodelcontainsabout25dynamicequationsandidentities,includingthoseforglobaloutput,CO2emissionsandconcentrations,globalmeantemperature,anddamages.Theversionforthisprojectrunsfor60five‐yearperiods.ItcanberunineitheranExcelversionorinthepreferredGAMSversion.TheversionusedforthisstudydatesfromDecember2013andaddsloopstocalculatetheoutcomesfordifferentuncertainparameters.TherunswereimplementedbyWilliamNordhausandPaulSztorc.

TheFUNDmodel(ClimateFrameworkforUncertainty,Negotiation,andDistribution)wasdevelopedprimarilytoassesstheimpactsofclimatepoliciesinanintegratedframework.Itisarecursivemodelthattakesexogenousscenariosofmajoreconomicvariablesasinputsandthenperturbsthesewithestimatesofthecostofclimatepolicyandtheimpactsofclimatechange.Themodelhas16regionsandcontainsexplicitrepresentationoffivegreenhousegases.Climatechangeimpactsaremonetizedandincludeagriculture,forestry,sea‐levelrise,healthimpacts,energyconsumption,waterresources,unmanagedecosystems,andstormimpacts.Eachimpactsectorhasadifferentfunctionalformandiscalculatedseparatelyforeachofthe16regions.Themodelrunsfrom1950to3000intimestepsof1year.Thesourcecode,data,andatechnicaldescriptionofthemodelarepublic(www.fund‐model.org),andthemodelhasbeenusedbyothermodelingteams(e.g.,Reveszetal.(2014)).FUNDwasoriginallycreatedbyRichardTol(Tol,1997)andisnowjointlydevelopedbyDavidAnthoffandRichardTol.TherunswereimplementedbyDavidAnthoff.

TheGCAM(GlobalChangeAssessmentModel)isaglobalintegratedassessmentmodelofenergy,economy,land‐use,andclimate.GCAMisalong‐termglobalmodelbasedontheEdmondsandReillymodel(EdmondsandReilly1983a,b,c).GCAMintegratesrepresentationsoftheglobaleconomy,energysystems,agricultureandlanduse,withrepresentationsofterrestrialandoceancarboncycles,andasuiteofcoupledgas‐cycleandclimatemodels.TheclimateandphysicalatmosphereinGCAMisbasedontheModelfortheAssessmentofGreenhouse‐GasInducedClimateChange(MAGICC)(Meinshausenetal.2011).TheglobaleconomyinGCAMisrepresentedin14geopoliticalregions,explicitlylinkedthroughinternationaltradeinenergycommodities,agriculturalandforestproducts,andothergoodssuchasemissionspermits.Thescaleofeconomicactivityineachregionisdrivenbypopulationsize,age,andgenderaswellaslaborproductivity.Themodelisdynamic‐recursivelysolvedforasetofmarket‐clearingequilibriumpricesinallenergyandagriculturalgoodmarketsevery5yearsover2005‐2095.Thefull

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documentationofthemodelisavailableataGCAMwiki(Calvinandetal.2011).GCAMisopen‐source,butisprimarilydevelopedandmaintainedbytheJointGlobalChangeResearchInstitute.ThemodelrunswereperformedbyHaewonMcJeon.

TheMERGEmodel(ModelforEvaluatingRegionalandGlobalEffectsofgreenhousegasreductionpolicies)isanintegratedassessmentmodeldescribingglobalenergy‐economy‐climateinteractionswithregionaldetail.ItwasintroducedbyManneetal.(1999)andhasbeencontinuallydevelopedsince;arecentlypublisheddescriptionisinBlanfordetal.(2014).MERGEisformulatedasamulti‐regiondynamicgeneralequilibriummodelwithaprocessmodeloftheenergysystemandareduced‐formrepresentationoftheclimate.ItissolvedinGAMSviasequentialjointnon‐linearoptimizationwithNegishiweightstobalanceinter‐regionaltradeflows.Theeconomyisrepresentedasatop‐downRamseymodelinwhichelectricandnon‐electricenergyinputsaretradedoffagainstcapitalandlaborandproductionisallocatedbetweenconsumptionandinvestment.Theenergysystemincludesexplicittechnologiesforelectricitygenerationandnon‐electricenergysupply,witharesourceextractionmodelforfossilfuelsanduranium.Theclimatemodelincludesafive‐boxcarboncycleandtracksallmajornon‐CO2greenhousegasesandnon‐CO2forcingagentsexplicitly.Temperatureevolvesasatwo‐boxlagprocess,whereuncertaintyaboutclimatesensitivityisconsideredjointlywithuncertaintyabouttheresponsetimeandaerosolforcing.Theversionusedforstudyincludes10modelregionsandrunsthrough2100,withclimatevariablesprojectedforanadditionalcentury.TherunswereimplementedbyGeoffreyBlanford.

TheMITIGSM(IntegratedGlobalSystemsModel)wasdevelopedintheearly1990’sandhasbeencontinuallyupdated.Itincludesageneralcirculationmodeloftheatmosphereanditsinteractionswithoceans,atmosphericchemistry,terrestrialvegetation,andthelandsurface.Itseconomiccomponentrepresentstheeconomyandanthropogenicemissions.ThefullIGSMisdescribedinSokolovetal.(2009)andWebsteretal.(2012).TheversionoftheeconomiccomponentappliedhereisdescribedinChenetal.(2015).Theearthsystemcomponentisasimplifiedgeneralcirculationmodelresolvedin46latitudebandsand11verticallayersintheatmospherewithan11layeroceanmodel.Thelandsystemincludes17vegetationtypes.Theeconomiccomponentisamulti‐sector,multi‐regionappliedgeneralequilibriummodel,anempiricalimplementationconsistentwithneo‐classicaleconomictheory.Forthecurrentproject,themodeloperatesinarecursivefashioninwhichtheeconomydrivestheearthsystemmodelbutwithoutfeedbacksofclimateimpactsontheeconomicsystem.Theeconomiccomponentissolvedfor5yeartimestepsinGAMS‐MPSGEandforthisexercisewasrunthrough2100.The

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earthsystemcomponentsolveson10minutetimesteps(thevegetationmodelonmonthlytimesteps).ThesimulationsforthisexercisewereconductedbyY.‐H.HenryChen,AndreiSokolov,andJohnReilly.

TheWITCH(WorldInducedTechnicalChangeHybrid)modelwasdevelopedin2006(Bosettietal.2006)andhasbeendevelopedandextendedsincethen.ThelatestversionisfullydescribedinBosettietal.(2014).Themodeldividestheworldinto13majorregions.TheeconomyofeachregionisdescribedbyaRamsey‐typeneoclassicaloptimalgrowthmodel,whereforward‐lookingcentralplannersmaximizethepresentdiscountedvalueofutilityofeachregion.Theseoptimizationstakeaccountofotherregions'intertemporalstrategies.Theoptimalinvestmentstrategyincludesadetailedappraisalofenergysectorinvestmentsinpower‐generationtechnologiesandinnovation,andthedirectconsumptionoffuels,aswellasabatementofothergasesandland‐useemissions.Greenhouse‐gasemissionsandconcentrationsarethenusedasinputsinaclimatemodelofreducedcomplexity(Meinshausenetal.2011).Theversionusedforthisprojectrunsfor30five‐yearperiodsandcontains35statevariablesforeachofthe13regions,runningontheGAMSplatform.TherunswereimplementedbyValentinaBosettiandGiacomoMarangoni.

IV. Choiceofuncertainparametersandgriddesign

A. Choiceofuncertainparameters

Oneofthekeydecisionsinthisstudywastoselecttheuncertainparameters.Thecriteriaforselectionwere(atleastafterthefact)clear.First,eachparametermustbeimportantforinfluencinguncertainty.Second,parametersshouldbeonesthatcanbevariedineachofthemodelswithoutexcessiveburdenandwithoutviolatingthespiritofthemodelstructure.Third,theparametersshouldbeonesthatcanberepresentedbyaprobabilitydistribution,eitheronthebasisofpriorresearchorfeasiblewithinthescopeofthisproject. Ataninitialmeeting,anexperimentwasundertakeninwhicheachofthemodelswasgivensixuncertainparametersorshockstotestforfeasibility.Attheendofthisinitialtestexperiment,twoofthemodelingteamsdecidednottoparticipatebecausetheinitialparameterscouldnotbeeasilyincorporatedinthemodeldesignorbecauseoftimeconstraints.Threeoftheparametersfulfilledtheabove‐mentionedcriteria,andtheseweretheonesthatwereincorporatedinthefinalsetofexperiments. Thefinallistofuncertainparameterswerethefollowing:(1)Therateofgrowthofproductivity,orpercapitaoutput;(2)therateofgrowthofpopulation;

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and(3)theequilibriumclimatesensitivity(equilibriumchangeinglobalmeansurfacetemperaturefromadoublingofatmosphericCO2concentrations).

Additionally,itwasdecidedtodotwoalternativepolicyscenarios.Onewasa“Base”runinwhichnoclimatepolicieswereintroduced;andthesecond,labelled“CarbonTax”(andsometimes“Ampere”)introducedarapidlyrisingglobalcarbontax.2Arunbasedoncarbonpriceswasselected(insteadofquantitativelimits)becausemanymodelshadundertakensimilarrunsinothermodelcomparisonprojects,sotheywererelativelyeasytoimplement.

Severalotherparameterswerecarefullyconsideredbutrejected.Apulseofemissionswasrejectedbecauseithadessentiallynoimpact.Aglobalrecessionwasrejectedforthesamereason.Itwashopedtoadduncertaintiesfortechnology(suchasthoseconcerningtherateofdecarbonization,thecostofbackstoptechnologies,orthecostofadvancedcarbon‐freetechnologies),butitprovedimpossibletofindonethatwasbothsufficientlycomprehensiveandcouldbeincorporatedinallthemodels.Uncertaintyaboutclimatedamageswasexcludedbecausehalfthemodelsdidnotcontaindamages.Afinalpossibilitywastoanalyzepolicyrunsthathadquantitativelimitsratherthancarbonprices.Forexample,somemodelshadparticipatedinmodelcomparisonsinwhichradiativeforcingswerelimited.Thisapproachwasrejectedbecausethecarbontaxprovedeasiertodefineandimplement.Additionally,earlierexperimentsindicatedthatquantitativelimitswereoftenfoundinfeasible,andthiswouldcloudtheinterpretationoftheresults.3

                                                            2TheCarbonTaxrunwasselectedfromtheAMPEREmodelcomparisonstoreducetheburdenonmanyofthemodelersandsothattheresultsfromthisstudycanbecomparedtothosefromtheAMPEREinter‐modelcomparisonstudy(Kriegleretal.2015).ThespecificscenariochosenisknownintheAMPEREstudyas"CarbonTax$12.50‐increasing.”ThefullAMPEREscenariodatabasecanbefoundonlineathttps://secure.iiasa.ac.at/web‐apps/ene/AMPEREDB.3SeeparticularlytheresultsforEnergyModelingForum22reportedinaspecialissueinEnergyEconomics(e.g.,seeClarkeandWeyant(2009)).Manymodelsfoundthattightconstraintswereinfeasiblefortheirbaseruns.Aquantitativelimitwouldalmostsurelyhavefoundthatlargenumbersofthe125scenarioswereinfeasibleforanytightlimitontemperatureorradiativeforcings.

      14 

B. Descriptionofuncertainparameters

Wenextdescribethethreeuncertainparameterscontainedinthestudy.Itturnedoutthatharmonizingtheseacrossmodelswasmorecomplicatedthanwasoriginallyanticipated,asdescribedbelow.

(1) Therateofgrowthofpopulation.Uncertaintyabouttherateofgrowthofpopulationwasstraightforward.Forglobalmodels,therewasnoambiguityabouttheadjustment.Theuncertaintywasspecifiedasplusorminusauniformpercentagegrowthrateeachyearovertheperiod2010‐2100.Forregionalmodels,theadjustmentwaslefttothemodeler.Mostmodelsassumedauniformchangeinthegrowthrateineachregion.

(2)Therateofgrowthofproductivity,orpercapitaoutput.Theoriginaldesignhadbeentoincludeavariablethatrepresentedtheuncertaintyaboutoveralltechnologicalchangeintheglobaleconomy(oraveragedacrossregions).Theresultsoftheinitialexperimentindicatedthatthespecificationsoftechnologicalchangedifferedgreatlyacrossmodels,anditwasinfeasibletospecifyacomparabletechnologicalvariablethatcouldapplyforallmodels.Forexample,somemodelshadasingleproductionfunction,whileothershadmultiplesectors.

Ratherthanattempttofindacomparableparameter,itwasdecidedtoharmonizeontheuncertaintyofglobaloutputpercapitagrowthfrom2010to2100.Eachmodelerwasaskedtointroduceagridofchangesinitsmodel‐specifictechnologicalparameterthatwouldleadtoachangeinpercapitaoutputofplusorminusagivenamount(tobedescribedinthenextsection).ThemodelersweretheninstructedtoadjustthatchangesothattherangeofgrowthratesinpercapitaGDPfrom2010to2100inthecalibrationexercisewouldbeequaltothedesiredrange.

(3)Theclimatesensitivity.Modelinguncertaintyaboutclimatesensitivityprovedtobeoneofthemostdifficultissuesofharmonizationacrossthedifferentmodels.WhileallmodelshavemodulestotracethroughthetemperatureimplicationsofchangingconcentrationsofGHGs,theydifferindetailandspecification.Themajorproblemwasthatadjustingtheequilibriumclimatesensitivitygenerallyrequiredadjustingotherparametersinthemodelthatdeterminethespeedofadjustmenttotheequilibrium;theadjustmentspeedissometimesrepresentedbythetransientclimatesensitivity.Thisproblemwasidentifiedlateintheprocess,afterthesecond‐roundrunshadbeencompleted,andmodelerswereaskedtomaketheadjustmentsthattheythoughtappropriate.Somemodelsmadeadjustmentsinparameterstoreflectdifferencesinlargeclimate

      15 

models.Othersconstrainedtheparameterssothatthemodelwouldfitthehistoricaltemperaturerecord.Thedifferingapproachesledtodifferingstructuralresponsestotheclimatesensitivityuncertainty,aswillbeseenbelow.

C. Griddesign

Inthefirsttrack,themodelingteamsprovideasmallnumberofcalibrationrunsthatincludeafullsetofoutputsforathree‐dimensionalgridofvaluesoftheuncertainparameters.Foreachoftheuncertainparameters,weselectedfivevaluescenteredonthemodel’sbaselinevalues.Therefore,for3uncertainparameters,therewere125runseachfortheBaseandtheCarbonTaxpolicyscenarios.

Onthebasisofthesecalibrationruns,thenextstepinvolvedestimatingsurface‐responsefunctions(SRFs)inwhichthemodeloutcomesareestimatedasfunctionsoftheuncertainparameters.ThehopewasthatiftheSRFscouldapproximatethemodelsaccurately,thentheycouldbeusedtosimulatetheprobabilitydistributionsoftheoutcomevariablesaccurately.AninitialtestsuggestedthattheSRFswerewellapproximatedbyquadraticfunctions.Wethereforesettherangeofthegridsothatitwouldspanmostofthespacethatwouldbecoveredbythedistributionoftheuncertainparameters,yetnotgosofarastopushthemodelsintopartsoftheparameterspacewheretheresultswouldbeunreliable.

Asanexample,takethegridforpopulationgrowth.Thecentralcaseisthemodel’sbasecaseforpopulationgrowth.Eachmodelthenusesfouradditionalassumptionsforthegridforpopulationgrowth:thebasecaseplusandminus0.5%peryearandplusandminus1.0%peryear.Thesewouldcovertheperiod2010to2100.Forexample,assumethatthemodelhadabasecasewithaconstantpopulationgrowthrateof0.7%peryearfrom2010to2100.Thenthefivegridpointsforpopulationgrowthwouldbeconstantgrowthratesof‐0.3%,0.2%,0.7%,1.2%,and1.7%peryear.Populationafter2100wouldhavethesamegrowthrateasinthemodeler’sbasecase.Theseassumptionsmeanthatpopulationin2100wouldbe(0.99)90,(0.995)90,1,(1.005)90,and(1.01)90timesthebasecasepopulationfor2100.

Forproductivitygrowth,thegridwassimilarlyconstructed,butadjustedsothatthegrowthinpercapitaoutputfor2100added‐1%,‐0.5%,0%,0.5%,and1%tothegrowthrateineachyearfortheperiod2010‐2100.

Fortheclimatesensitivity,themodelersweretoaddtothebaselineequilibriumclimatesensitivity‐3°C,‐1.5°C,0°C,1.5°C,and3°C.Itturnedoutthatthelowerendofthisrangecauseddifficultiesforsomemodels,andforthesethe

      16 

modelersreportedresultsonlyforthefourhigherpointsinthegridorsubstitutedanotherlowvalue.

Inprinciple,then,fortrackIeachmodelreported5x5x5modelresultsforboththeBasecaseandtheCarbonTaxpolicyassumptions.

V. Approachfordevelopingprobabilitydensityfunctions

A. Generalconsiderations

Thethreeuncertainparametershavebeenthesubjectofuncertaintyanalysisinearlierstudies.Foreachparameter,wereviewedearlierstudiestodeterminewhethertherewasanexistingsetofmethodsordistributionsthatcouldbedrawnupon.Thedesirablefeaturesofthedistributionsisthattheyshouldreflectbestpractice,thattheyshouldbeacceptabletothemodelinggroups,andthattheybereplicable.Itturnedoutthatthethreeparametersusedthreedifferentapproaches,aswillbedescribedbelow.

B. Population

Populationgrowthhasbeenthesubjectofprojectionsformanyyears,andnumerousgroupshaveundertakenuncertaintyanalysesforbothcountriesandatthegloballevel.Ourreviewfoundonlyoneresearchgroupthathadmadelong‐termglobalprojectionsofuncertaintyforseveralyears,whichwasthepopulationgroupattheInternationalInstituteforAppliedSystemsAnalysis(IIASA)inAustria.(Foradiscussion,seeO'Neilletal.(2001)).TheIIASAdemographygroupisunderthedirectionofdemographerWolfgangLutz.

TheIIASAstochasticprojectionsweredevelopedoveraperiodofmorethanadecadeandarewidelyusedbydemographers.Themethodologyissummarizedasfollows:“IIASA’sprojections…arebasedexplicitlyontheresultsofdiscussionsofagroupofexpertsonfertility,mortality,andmigrationthatisconvenedforthepurposeofproducingscenariosforthesevitalrates”(Seehttp://www.demographic‐research.org/volumes/vol4/8/4‐8.pdf)Thelatestprojectionsfrom2013(Lutzetal.2014)areanupdatetothepreviousprojectionsfrom2007and2001(Lutzetal.2008),2001).Themethodologyisdescribedasfollows:

Theforecastsarecarriedoutfor13worldregions.Theforecastspresentedherearenotalternativescenariosorvariants,butthedistributionoftheresultsof2,000differentcohortcomponentprojections.Forthesestochasticsimulationsthefertility,mortalityandmigrationpathsunderlyingtheindividualprojection

      17 

runswerederivedrandomlyfromthedescribeduncertaintydistributionforfertility,mortalityandmigrationinthedifferentworldregions.(Lutz,Sanderson,andScherbov2008)

Thebackgroundmethodsaredescribedasfollowsonpage219ofO'Neilletal.(2001):

TheIIASAmethodologyisbasedonaskingagroupofinteractingexpertstogivealikelyrangeforfuturevitalrates,where"likely"isdefinedtobeaconfidenceintervalofroughly90%(Lutz1996,Lutzetal.1998).Combiningsubjectiveprobabilitydistributionsfromanumberofexpertsguardsagainstindividualbias,andIIASAdemographersarguethatastrengthofthemethodisthatitmaybepossibletocapturestructuralchangeandunexpectedeventsthatotherapproachesmightmiss.Inaddition,inareaswheredataonhistoricaltrendsaresparse,theremaybenobetteralternativetoproducingprobabilisticprojections.

Forthisstudy,weareaimingforaparsimoniousparameterizationofpopulationuncertainty.Thisisnecessarybecauseofthelargedifferencesinmodelstructure.Wethereforeselectedtheuncertaintyaboutglobalpopulationgrowthfortheperiod2010‐2100asthesingleparameterofinterest.Wefittedthegrowth‐ratequantilesfromtheIIASAprojectionstoseveraldistributions,withnormal,log‐normal,andgammabeingthemostsatisfactory.Thenormaldistributionperformedbetterthananyoftheothersonfiveofthesixquantitativetestsoffitfordistributions.Basedontheseresults,wethereforedecidedtorecommendthenormaldistributionforthepdfofpopulationgrowthovertheperiod.

Inaddition,wedidseveralalternativeteststodeterminewhethertheprojectionswereconsistentwithothermethodologies.Onesetoftestsexaminestheprojectionerrorsthatwouldhavebeengeneratedusinghistoricaldata.Asecondtestlooksatthestandarddeviationof100‐yeargrowthratesofpopulationforthelastmillennium.AthirdtestexaminesprojectionsfromareportoftheNationalResearchCouncilthatestimatedtheforecasterrorsforglobalpopulationovera50‐yearhorizon(seeNRC(2000),AppendixF,p.344).Whiletheseallgaveslightlydifferentuncertaintyranges,theyweresimilartotheuncertaintiesestimatedintheIIASAstudy.

Onthebasisofthisreview,wedecidedtouseanormaldistributionforthegrowthrateofpopulationbasedontheIIASAstudythathasastandarddeviationoftheaverageannualgrowthrateof0.22percentagepointsperyearovertheperiod

      18 

2010‐2100.Moredetailswithabackgroundmemorandumontheresultsareavailablefromtheauthors.

C. ClimateSensitivity

Animportantparameterinclimatescienceistheequilibriumorlong‐runresponseintheglobalmeansurfacetemperaturetoadoublingofatmosphericcarbondioxide.Intheclimatesciencecommunity,thisiscalledtheequilibriumclimatesensitivity.Withreferencetoclimatemodels,thisiscalculatedastheincreaseinaveragesurfacetemperaturewithadoubledCO2concentrationrelativetoapathwiththepre‐industrialCO2concentration.ThisparameteralsoplaysakeyroleinthegeophysicalcomponentsintheIAMsusedinthisstudy.Intheremainderofthispaper,wewillfollowtheconventioninthegeosciencesandcallittheequilibriumclimatesensitivity(ECS).

GiventheimportanceoftheECSinclimatescience,thereisanextensiveliteratureestimatingprobabilitydensityfunctions.Thesepdfsaregenerallybasedonclimatemodels,theinstrumentalrecordsoverthelastcenturyorso,paleoclimaticdatasuchasestimatedtemperatureandradiativeforcingsoverice‐ageintervals,andtheresultsofvolcaniceruptions.Muchoftheliteratureestimatesaprobabilitydensityfunctionusingasinglelineofevidence,butafewpaperssynthesizedifferentstudiesordifferentkindsofevidence.

Wefocusonthestudiesdrawinguponmultiplelinesofevidence.TheIPCCFifthAssessmentreport(AR5)reviewedtheliteraturequantifyinguncertaintyintheECSandhighlightedfiverecentpapersusingmultiplelinesofevidence(IPCC2014).EachpaperusedaBayesianapproachtoupdateapriordistributionbasedonpreviousevidence(thepriorevidenceusuallydrawnfrominstrumentalrecordsoraclimatemodel)tocalculatetheposteriorprobabilitydensityfunction.Sinceeachdistributionwasdevelopedusingmultiplelinesofevidence,andinsomecasesthesameevidence,itwouldbeinconsistenttoassumethattheywereindependentandsimplytocombinethem.Further,sincewecouldnotreliablyestimatethedegreeofdependenceofthedifferentstudies,wecouldnotsynthesizethembytakingintoaccountthedependence.WethereforechosetheprobabilitydensityfunctionfromasinglestudyandperformedrobustnesscheckstousingtheresultsfromalternativestudiescitedintheIPCCAR5.

ThechosenstudyforourprimaryestimatesisOlsenetal.(2012).ThisstudyisrepresentativeoftheliteratureinusingaBayesianapproach,withapriorbasedonpreviousstudiesandalikelihoodbasedonobservationalormodeleddata,suchasglobalaveragesurfacetemperaturesorglobaltotalheatcontent.ThepriorinOlsenetal.(2012)isprimarilybasedonKnuttiandHegerl(2008).Thatprioristhen

      19 

combinedwithoutputvariablesfromtheUniversityofVictoriaESCMclimatemodel(Weaveretal.2001)todeterminethefinalorposteriordistribution.

Olsenetal.(2012)waschosenforthefollowingreasons.First,itwasrecommendedtousinpersonalcommunicationswithseveralclimatescientists.Second,itwasrepresentativeoftheotherfourstudiesweexaminedandfallsintothemiddlerangeofthedifferentestimates.4Third,sensitivityanalysesoftheeffectonaggregateuncertaintyofchangingthestandarddeviationoftheOlsenetal.(2012)resultsfoundthatthesensitivitywassmall(seethesectionbelowonsensitivityanalyses).Appendix1providesmoredetailsonOlsenetal.(2012)andalsopresentsafigurecomparingthisstudytotheotherstudiesintheIPCCAR5.

NotethattheUSgovernmentusedaversionoftheRoeandBakerdistributioncalibratedtothreeconstraintsfromtheIPCCforitsuncertaintyestimates(IAWG2010).Specifically,theIAWGReportmodifiedtheoriginalRoeandBakerdistributiontoassumethatthemedianvalueis3.0°C,theprobabilityofbeingbetween2and4.5°Cistwo‐thirds,andthereisnomassbelowzeroorabove10°C.ThemodifiedRoeandBakerdistributionhasahighermeanECSthananyofthemodels(3.5°C)andamuchhigherdispersion(1.6°Cascomparedto0.84°CfromOlsenetal.2012).

TheestimatedpdfforOlsenetal.(2012)wasderivedasfollows.Wefirstobtainedthepdffromtheauthors.Thispdfwasprovidedasasetofequilibriumtemperaturevaluesandcorrespondingprobabilities.Wethenexploredfamiliesofdistributionsthatbestapproximatedthenumericalpdfprovided.Wefoundthatalog‐normalpdffitstheposteriordistributionsextremelywell.

Tofindtheparametersofthefittedlog‐normalpdf,weminimizethesquareddifferencebetweentheposteriordensityfunctionfromOlsenetal.andthelog‐normalpdfoverthesupportofthedistribution(theL2orEuclidiannorm).Inotherwords,weminimizethesumofthesquareoftheverticaldifferencesbetweentheposteriorpdfandalog‐normalpdfoverallgridpointsvaluesintheOlsenetal.(2012)distribution.5Figure1showstheOlsenetal.(2012)pdf,alongwiththefittedlog‐normaldensityfunction.Thefitisextremelyclose,withthelog‐normaldistributionalwayswithin0.14%oftheOlsenetal.(2012)pdfforanygridpointvalue.

                                                            4Intests,wefoundthattheOlsenetal.(2012)distributionissimilartoasimplemixturedistributionofallfivedistributions.Wecalculatethismixturedistributionbytakingtheaverageprobabilityoveralldistributionsateachtemperatureincrease.5MorepreciselyweminimizeovertherangeoftheOlsenetal.distribution,[1.509,7.4876]°C,withagridpointspacingof0.1508°C.

      20 

D. TotalFactorProductivity

Uncertaintyinthegrowthofproductivity(oroutputpercapita)isknowntobeacriticalparameterindeterminingallelementsofclimatechange,fromemissionstotemperaturechangetodamages(Nordhaus2008).ClimatemodelsgenerallydrawtheirestimatesofemissionstrajectoriesfrombackgroundmodelsofeconomicgrowthsuchasscenariospreparedfortheIPCCorstudiesoftheEnergyModelingForum.Nomajorstudies,however,relyonstatistically‐basedestimatesofemissionsandeconomicgrowth.

Forecastsoflong‐runproductivitygrowthinvolveactivedebatesonissuessuchastheroleofnewtechnologiesandinventions(BrynjolfssonandMcAfee2012,Gordon2012),potentialincreasesintheresearchintensityandeducationalattainmentinemergingeconomies(FernaldandJones2014,Freeman2010),andinstitutionalreformandpoliticalstability(Acemogluetal.2005).Whiletheempiricalliteratureoneconomicgrowthhasprovidedevidenceinsupportofvariousunderlyingmodels,noexistingstudycontainssufficientinformationtoderiveaprobabilitydistributionforlong‐rungrowthrates.

 

 

Figure1.TheOlsenetal.(2012)probabilitydensityfunctionalongwiththefittedlog‐normaldistributionusedinouranalysis. 

      21 

Thehistoricalrecordprovidesausefulbackgroundforestimatingfuturetrends.However,itisclearfromboththeoreticalandempiricalperspectivesthattheprocessesdrivingproductivitygrowtharenon‐stationary.Forexample,estimatesofthegrowthofglobaloutputpercapitaforthe18th,19th,and20thcenturyare0.6,1.9,and3.7percentperyear(DeLong2015inhttp://holtz.org/Library/Social%20Science/Economics/Estimating%20World%20GDP%20by%20DeLong/Estimating%20World%20GDP.htm).Totheextentthatexpertsoneconomicgrowthpossessvalidinsightsaboutthelikelihoodandpossibledeterminantsoflong‐rungrowthpatterns,theninformationdrawnfromexpertscanaddvaluetoforecastsbasedpurelyonhistoricalobservationsordrawnfromasinglemodel.Combiningexpertestimateshasbeenshowntoreduceerrorinshort‐runforecastsofeconomicgrowth(BatchelorandDua1995).However,therearefewexpertstudiesonlong‐rungrowth(seeAppendix2fordiscussion)and,toourknowledge,therehasbeennosystematicanddetailedpublishedstudyofuncertaintyinlong‐runfuturegrowthrates.

Todevelopestimatesofuncertainties,theprojectteam,ledbyPeterChristensen,undertookasurveyofexpertsoneconomicgrowthtodetermineboththecentraltendencyandtheuncertaintyaboutlong‐rungrowthtrends.Oursurveyutilizedinformationdrawnfromapanelofexpertstocharacterizeuncertaintyinestimatesofglobaloutputfortheperiods2010‐2050and2010‐2100.WedefinedgrowthastheaverageannualrateofrealpercapitaGDP,measuredinpurchasingpowerparity(PPP)terms.Weaskedexpertstoprovideestimatesoftheaverageannualgrowthratesat10th,25th,50th,75th,90thpercentiles.

Beginninginthesummerof2014,wesentoutsurveystoapanelof25economicgrowthexperts.AsofJune2015,wecollected11completeresultswithfulluncertaintyanalysisfortheperiod2010‐2100.AsummaryoftheprocedureisprovidedinAppendix2,andacompletereportwillbepreparedseparately.

Therearemanydifferentapproachestocombiningexpertforecasts(Armstrong2001)andaggregatingprobabilitydistributions(ClemenandWinkler1999).Weassumethatexpertshaveinformationaboutthelikelydistributionoflong‐rungrowthrates.Theirinformationsetsaredefinedbyestimatesfor5differentpercentiles.Webeginbyassumingthattheestimatesareindependentacrossexpertsandthenexaminedthedistributionsthatbestfitthepercentilesforeachexpertandforthecombinedestimates(averageofpercentiles)acrossexperts.

WefounditusefulforthisprojecttocharacterizetheexpertpdfswithcommonlyuseddistributionssothattheMonteCarloestimatescouldbeeasilyimplemented.Intestingthedistributionsforeachexpert,wefoundthatmostexperts’estimatescan

      22 

becloselyfittedbyanormaldistribution;similarly,thecombineddistributioniswellfittedbyanormaldistribution.DetailsareprovidedinAppendix2.

Theresultingcombinednormaldistributionhasameangrowthrateof2.29%peryearandastandarddeviationofthegrowthrateof1.15%peryearovertheperiod2010‐2100.(ThemeangrowthrateofpercapitaGDPinthebaserunsofthesixmodelsisslightlylowerat1.9%peryearoverthisperiod.)Wetestdifferentapproachesforcombiningtheexpertresponsesandfindlittlesensitivitytothechoiceofaggregationmethod.Figure2showsthefittedindividualandcombinednormalpdfs(explainedinAppendix2).IntheMonteCarloestimatesbelow,wechoseastandarddeviationofthegrowthrateofpercapitaoutputof1.12%peryear(basedonthefirst11responses).Thisvalueisusedinthisdraft,butwillbeupdatedwiththeadditionoffurtherresponses.

 

Figure2.Individualandcombinedpdfsforannualgrowthratesofoutputpercapita,2010–2100(averageannualpercentperyear)Forthemethods,seeAppendix2.

 

      23 

Itisusefultocomparethesurveyresultswithhistoricaldata.Ifwetakethelong‐termestimatesfromMaddison(2003),the100‐yearvariabilityofgrowthoverthetencenturiesfrom1000to2000was1.5%peryear,witharangeof‐0.1%to3.7%peryear.Thevariabilityinthesecentury‐stepdataishigherthantheexperts’estimateof1.15%peryear.

Globalgrowthratesbasedondetailednationaldataareavailablesince1900.Thestandarddeviationofannualgrowthratesoverthisperiodwas2.9%peryear,whilethestandarddeviationof25‐yeargrowthrateswas1.2or1.4%peryeardependinguponthesource.Thevariabilityofgrowthinrecentyearswaslowerthanfortheentireperiodsince1900.Thestandarddeviationintheannualgrowthrateduringtheperiod1975‐2000was1.1%peryear.Wecannoteasilytranslatehistoricalvariabilitiesintocentury‐longvariabilitieswithoutassumingaspecificstochasticstructureofgrowthrates.

VI. ResultsofModelingStudies

A. Modelresultsandlatticediagrams

Webeginbyprovidingresultsonthecalibrationrunsandthesurfaceresponsefunctions.Foreachmodel,thereisavoluminoussetofinputsandoutputvariablesfrom2010to2100.Thefullset(consistingof46,150x22elements)clearlycannotbefullypresented.Werestrictourfocusheretosomeofthemostimportantresults,andconsignfurtherresultstoAppendix3,withthefullresultsavailableonlineattimeofpublication.

Tohelpvisualizetheresults,wehavedevelopedlatticediagramstoshowhowtheresultsvaryacrossuncertainvariablesandmodels.Figure3isalatticediagramfortheincreaseinglobalmeansurfacetemperaturein2100.Withineachoftheninepanels,they‐axisistheglobalmeansurfacetemperatureincreasein2100relativeto1900.Thex‐axisisthevalueoftheequilibriumtemperaturesensitivity.Goingacrosspanelsonthehorizontalaxis,thefirstcolumnusesthegridvalueofthefirstofthefivepopulationscenarios(whichisthelowestgrowthrate);themiddlecolumnshowstheresultsforthemodeler’sbaselinepopulation;andthethirdcolumnshowstheresultsforthepopulationassociatedwiththehighestpopulationgrid(orhighestgrowthrate).

Goingdownpanelsontheverticalaxis,thefirstrowusesthehighestgrowthrateforTFP(orthefifthTFPgridpoint);themiddlerowshowsTFPgrowthforthemodelers’baselines;andthebottomrowshowstheresultsfortheslowestgridpointforthegrowthrateofTFP.Notethatinallcases,themodelers’baselinevalues

      24 

generallydiffer,butthedifferencesinparametervaluesacrossrowsorcolumnsareidentical.

Tounderstandthislatticegraph,begininthecenterpanel.Thispanelusesthemodeler’sbaselinepopulationandTFPgrowth.Itindicateshowtemperaturein2100acrossmodelsvarieswiththeECS,withthedifferencesbeing1.5°CbetweentheECSgridpoints.AfirstobservationisthatthemodelsallassumethattheECSiscloseto3°Cinthebaseline.Next,isthattheresultingbaselinetemperatureincreasesfor2100arecloselybunchedbetween3.75and4.25°C.Allcurvesareupwardsloping,indicatingagreater2100temperaturechangeisassociatedwithahigherECS.

AstheECSvariesfromthebaselinevalues,themodeldifferencesaredistinct.ThesecanbeseenintheslopesofthedifferentmodelcurvesinthemiddlepanelofFigure3.Wewillseebelowthattheimpactofa1°CchangeinECSon2100temperaturevariesbyafactorof2½acrossmodels.Forexample,DICE,MERGE,andGCAMhaverelativelyresponsiveclimatemodules,whileIGSMandFUNDclimatemodulesaremuchlessresponsivetoECSdifferences.Thedifferenceacrossmodelsbecomeslargeraswemovefromthebottom‐lefttotheupperright‐handpanel,correspondingtoincreasingpopulationandTFPgrowthfrombottomlefttotopright.Thisresulthighlightskeydifferencesinboththeeconomicandclimatecomponentsofthedifferentmodels.

      25 

Anotherimportantrelationshiptoexamineishowdifferentmodelsreacttothecarbonprices.Figure4showsthepercentagereductioninCO2emissionsintheCarbonTaxscenariov.theBaserun.Thehorizontalaxisshowsthemagnitudeofthecarbontax.Onekeyfeatureofallmodelsisthatattainingzeroemissionswouldrequireextremelyhighcarbonprices.

 

 

 

 

Figure3.Latticediagramfor2100temperatureincreaseThislatticediagramshowsthedifferencesinmodelresultsfor2100globalmeansurfacetemperatureacrosspopulation,totalfactorproductivityandtemperaturesensitivityparameters.Thecentralboxusesthemodelers’baselineparametersandtheBasepolicy. 

      26 

Therearemanyotherresultsofthemodelingexercise.Appendix3containsfurtherlatticediagrams,includingthoseforpercapitaconsumption,emissions,anddamages,aswellasadditionaltablesofresults.However,theprimarypurposeofthepresentstudyistodeterminetheimpactofuncertainties,soweleavethemodelcomparisonsofmajoroutputsasideatthispoint.

B. Resultsoftheestimatesofthesurfaceresponsefunctions

RecallthattrackIprovidesthemodeloutcomes(suchasoutput,emissions,andtemperature)foreachgrid‐pointofa5x5x5x2gridofthevaluesoftheuncertainparametersandpolicies.Thenextstepintheanalysisistofitsurfaceresponsefunctions(SRFs)toeachofthemodeloutputs.TheseSRFsthenwillbeused,whencombinedwiththeTrackIIprobabilitydistributionsjustdiscussed,toprovideprobabilitydistributionsoftheoutcomevariablesforeachmodel.

 

  

Figure4.CarbontaxandemissionsreductionsbymodelModelsshowdifferingresponsetohighercarbonprices.Notethatthecarbonpricesareallassociatedwithgivendatesandarecommonforallmodels.Thepointstothefarleftarefor2010,whiletheonesatthefarrightarefor2100.Theseestimatesareforthemodelers’baselineparameters. 

0%

20%

40%

60%

80%

100%

120%

0 100 200 300 400 500

Percen

tage red

uction (A

mpere v base)

Carbon price ($/tCO2)

DICE FUND

GCAM IGSM

MERGE WITCH

      27 

WeundertookextensiveanalysisofdifferentapproachestoestimatingtheSRFs.Theinitialandeventuallypreferredapproachwasalinear‐quadratic‐interactions(LQI)specification.Thistookthefollowingform:

3 3

01 1 1

j

i i ij i ji j i

Y u u u

Inthisspecification, and i ju u aretheuncertainparameters.TheYarethe

outcomevariablesfordifferentmodelsanddifferentyears(e.g.,temperaturefortheFUNDmodelfor2100intheBaserunfordifferentvaluesofthe3uncertainparameters).Theparameters 0 , , and i i j aretheestimatesfromtheSRF

regressionequations.Wesuppressthesubscriptforthemodel,year,policy,andvariable.

Table1showsacomparisonoftheresultsfortemperatureandlogofoutputforthelinear(L)andLQIspecificationsforthesixmodels.AllspecificationsshowmarkedimprovementoftheequationfitintheLQIrelativetotheLversion.Lookingatthelogoutputspecification(thelastcolumninthebottomsetofnumbers),theresidualvarianceintheLQIspecificationisessentiallyzeroforallmodels.ForthetemperatureSRF,morethan99.5%ofthevarianceisexplainedbytheLQIspecification.Thestandarderrorsofequationsfor2100temperaturerangefrom0.05to0.18°CfordifferentmodelsintheLQIversion.

      28 

Theequationsarefitasdeviationsfromthecentralcase,socoefficientsarelinearizedatthecentralpoint,whichisthemodelers’baselinesetofparameters.LookingattheLQIcoefficientsfortemperature,notethattheeffectoftheECSon2100temperaturevariessubstantiallyamongthemodels.Atthehighend,thereisclosetoaunitcoefficient,whileatthelowendthevariationisabout0.4°Cper°Cin

 

  

 

Table1.LinearparametersinofSRFfortemperatureandlogoutputforlinear(L)andliner‐quadratic‐interactions(LQI)specifications

ThelinearparametersarethecoefficientsonthelineartermintheSRFregressions.Becausethedataaredecentered(removethemedians),thelineartermsinthehigher‐orderpolynomialsarethederivativesorlineartermsatthemedianvaluesoftheuncertainparameters.

 

      29 

ECSchange.ForTFP,theimpactsarerelativelysimilarexceptfortheWITCHmodel,whichismuchlower.ThisislikelyduetoimplementationoftheTFPchangesasinput‐neutraltechnicalchange(ratherthanchangesinlaborproductivity,asinseveralothermodels).Forpopulation,theLQIcoefficientsvarybyafactorofthree.

Forlogofoutput,severalmodelshavenofeedbackfromECStooutputandthusshowa0.000value.TheimpactofTFPisalmostuniformbydesign.Similarly,theimpactofpopulationonoutputisverysimilar.

WetestedsevendifferentspecificationsfortheSRF:Linear(L),Linearwithinteractions(LI),Linearquadratic(LQ),Linear,quadratic,linearinteractions(LQI)asshownabove,3rddegreepolynomialwithlinearinteractions(P3I),4thdegreepolynomialswithseconddegreeinteractions(P4I2),andfourthdegreepolynomialwithfourthdegreeinteractionsandpolynomialthree‐wayinteractions(P4I4S3).Forvirtuallyallmodelsandspecifications,theaccuracyincreasedsharplyasfarastheLQIspecification.However,asisshowninFigure5,verylittlefurtherimprovementwasfoundforthemoreexoticpolynomials.Inadditiontothepolynomialinterpolations,weinvestigatedseveralalternativetechniques,includingChebyshevpolynomialsandbasis‐splines.Wefoundnoimprovementfromtheseotherapproaches.

 

  

Figure5.Residualvarianceforallvariables,models,andspecificationsindicatesthatfornearlyallmodels,thereislittletobegainedaddingfurtherpolynomialtermsbeyondLQI. 

0.00

0.02

0.04

0.06

0.08

0.10L LQ LI LQI LQI++

1‐R2

All

Temp(2100)

Conc(2100)

Y(2100)

      30 

Insummary,wefoundthatthelinear‐quadratic‐interaction(LQI)specificationofthesurfaceresponsefunctionperformedextremelywellinfittingthedatainourtests.Thereasonisthatthemodels,whilehighlynon‐linearoverall,aregenerallyclosetoquadraticinthethreeuncertainparameters.WearethereforeconfidentthattheyareareliablebasisfortheMonteCarlosimulations.

C.ReliabilityoftheMUPprocedureswithextrapolation

OneissuethatarisesinestimatingthedistributionsofoutcomevariablesistheextenttowhichthecalibrationrunsintrackIadequatelycovertherangeofthepdfsfromtrackII.Forbothpopulationandtheequilibriumtemperaturesensitivity,thecalibrationrunscoveratleast99.9%oftherangeofthepdfs.However,whensettingthecalibrationrangeforTFPbasedonearlierinformalestimates,weunderestimatedthevariabilityofthefinalpdfs.Asaresult,thecalibrationrunsonlyextendasfarasthe83percentileattheupperend,requiringustoextrapolatebeyondtherangeofthecalibrationruns.

Sinceitwasnotpossibletorepeatthecalibrationrunswithanexpandedgrid,wetestedthereliabilityoftheextrapolationandthetwotrackapproachwithtwomodels.WefirstexaminedthereliabilityforTFPwiththebasecaseintheDICEmodel.ThiswasdonebymakingrunswithincrementsofTFPgrowthupto3estimatedstandarddeviations(i.e.,uptoaglobaloutputgrowthrateof6.1%peryearto2100).Theserunscover99.7%ofthedistribution.Wethenestimatedasurfaceresponsefunctionfor2100temperatureoverthesameintervalasforthecalibrationexercisesandextrapolatedoutsidetherange.TheresultsshowedhighreliabilityoftheestimatedSRFfortemperatureincreaseuptoabout2standarddeviationsabovethebaselineTFPgrowthrate.Beyondthat,theSRFtendedtooverestimatethe2100temperature.(SimilarresultswerefoundforCO2concentrationsandthedamage‐outputratiointheDICEmodel.)Thereasonfortheoverestimateisthatcarbonfuelsbecomeexhaustedathighgrowthrates,soraisingthegrowthratefurtherabovethealready‐highratehasarelativelysmalleffectsonemissions,concentrations,2100temperature,andthedamageratio.NotethatthisimpliesthatthefaruppertailofthetemperaturedistributionusingthecorrectedSRFwillshowathinnertailthantheonegeneratedbytheSRFestimatedoverthecalibrationruns.

WealsoperformedamorecomprehensivecomparisonoftheMUPprocedureswithafullMonteCarlousingtheFUNDmodel.Forthis,wetookthepdfsforthethreeuncertainvariablesandranaMonteCarloforthefullFUNDmodelwith1milliondraws.Wethencomparedthemeansandstandarddeviationsofdifferent

      31 

variablesforthetwoapproaches.WetestedfourdifferentspecificationsoftheSRFstodeterminewhetherthesewouldproducemarkedlydifferentoutcomes.TheresultsindicatedthattheMUPprocedureprovidedreliableestimatesofthemeansandstandarddeviationsofallvariablesthatwetestedexceptFUNDdamages.Exceptingdamages,forthepreferredLQIestimate,theabsoluteaverageerrorofthemeanfortheMUPprocedurerelativetotheFUNDMonteCarlowas0.3%,whiletheabsoluteaverageerrorforthestandarddeviationwas1.2%.Fordamages,theerrorswere7%and44%,respectively.Additionally,thepercentileestimatesfortheMUPprocedure(againexceptfordamages)wereaccurateuptothe90thpercentile.And,aswillbenotedbelow,theestimatesfortheparametersofthetailsofthedistributionswereaccurateforallvariablesexceptdamages.Anoteprovidingfurtherdetailsonthecomparisonsisavailablefromtheauthors.

VII. ResultsoftheMonteCarlosimulations

A. Distributionsformajorvariables

FortheMonteCarlosimulations,wetooktheSRFsforeachparameter/model/year/policyandmade1,000,000drawsfromeachpdfforthethreeuncertainparameters.Wethenexaminedtheresultingdistributions.Thissamplesizewaschosenbecausetheresultswerereliableatthatlevel.Thebootstrapstandarderrorsofthemeansandthestandarddeviationsweregenerallylessthan0.1%ofthemeanorstandarddeviation.Theexceptionwasfordamages,wherethebootstrapstandarderroroftheestimatedstandarddeviationswasabout0.2%ofthevaluefortheFUNDmodel.Wetreateachpdfindependently,butrecognizethattheremaybesomecorrelationbetweenrealizationsofpopulationandGDP.However,explorationsintothisrevealedthatitdidnotsubstantiallyinfluenceourfindings.

Table2showsstatisticsofthedistributionofthedrawsforeachofthemajoroutcomevariables,withaveragestakenacrossallsixmodels.WealsoshowtheestimatesforthelinearandLQIversionstoillustratethesensitivityoftheresultstotheSRFspecification.Thelastcolumnshowsthecoefficientofvariationforeachvariable.Notethattheseestimatesarewithin‐model(parametricuncertainty)resultsanddonotincludeacross‐modelvariability.Theresultshighlightthatemissions,economicoutput,anddamageshavethehighestcoefficientofvariation,underscoringthattheuncertaintyintheseoutputvariablesisgreaterthanforothervariables,suchasCO2concentrationsandtemperature.Thisistheresultofboththeunderlyingpdfsusedandthemodelsthemselves.

      32 

Table3showsthepercentiledistributionforallmajorvariablesforallmodelswithresultsforthebasecase.Thedetailedresultsbymodelsareprovidedintheappendix.Akeyresultisthedistributionoftemperatureincreasefor2100.Themedianincreaseacrossallmodelsis3.79°Cabove1900levels.The95thpercentileoftheincreaseis5.46°C.Giventhesizeoftheinterquartilerange,theseresultsdefinitelyindicatethattherearesubstantialuncertaintiesinallaspectsoffutureclimatechangeanditsimpactsinallthemodelsinvestigatedhere.

 

 

 

Table2.ResultsofMonteCarlosimulationsforaveragesofallmodelsThetableshowsthevaluesofallvariablesfor2100,exceptforthesocialcostofcarbon,whichisfor2020.DamagesandSCCareforthreemodels. 

 

 

Table3.Distributionofallmajorvariables,averageofsixmodelsThedateforallvariablesis2100exceptfortheSCC,whichis2020.DamagesandSCCareforthreemodels. 

      33 

Table4showsthedistributionforglobaltemperatureincreasein2100bymodel.Thetemperaturedistributionsofthesixmodelsareonthewholereasonablyclose.Themedianrangesfrom3.6to4.2°C,withIGSMbeingthelowestandMERGEbeingthehighest.Theinterquartilerangevariesfrom0.99°C(FUND)to1.39°C(DICE).The10‐90%rangesfrom1.91°C(WITCH)to2.65°C(DICE).Sincethevariabilityintherandomparametersisthesame,thedifferencesareduetomodelstructures. Oneinterestingfeatureisthetemperaturedistributioninthetails.The99thpercentilerangesfrom5.6(WITCH)to7.1°C(MERGE),whilethefartailofthe99.9thpercentilerangesfrom6.2(WITCH)to8.5°C(MERGE).

Table5showsthedistributionoftheSCCforthethreemodelsthatprovidetheseestimates.Thesearetheestimatesofthepresentvalueoftheflowoffuturemarginaldamagesofemissionsin2020.Twoofthemodels(WITCHandDICE)usesimilarquadraticdamagefunctionsandareroughlycomparableinthemiddleofthedistribution,buttherangeismuchsmallerinWITCH.6TheFUNDmodelhasmuchlowerdamages(duetoadifferentdamagefunction),andtheSCCdistributionisanorderofmagnitudelowerthantheothertwomodels.NotethatthecentralestimateoftheSCChereis$13.30pertonofCO2.ThisismuchlowerthanthepreferredestimateoftheUSgovernmentfor2020,whichis$46pertonin2011$witha3%annualdiscountrate.However,thebasecasediscountratesintheMUPrunsforthemodelsthatreportaverage4½%peryearto2050.TheIAWGestimateata5%discountrateis$13pertonandthereforeconsistentwiththeestimatespresentedhere.

                                                            6InWITCHmultipleregionsaremodeled,hencetheglobalSCCistheresultoftheaggregationofregionalSCC.

 

 

Table4.DistributionoftemperaturechangeintheBasecase,2100,°C 

      34 

Figure6showstheresultsforthetemperaturedistributionsforthemodelsonapercentilescale.Theshapesofthedistributionsaresimilar,althoughtheydifferbyasmuchas1°Cinscaleacrossmostofthedistribution.

Animportantquestionthatthisstudycanaddressiswhether,basedonthecurrentmodelstructuresandtheassumptionsaboutuncertainparameters,thedistributionsofoutcomesarethinorfattailed.Forthesetests,wedefineafattaileddistributionasonethathasaninfinite‐varianceParetoorpower‐lawdistributioninthetails(basedonthediscussioninSchuster1984).VariableswithaParetodistributionhaveinfinitevariancewhentheshapeparameterisbelow2,andtheyhaveaninfinitemeanwithaparameterequaltoorlessthanone.Asaninformaltest,wecanexaminetheratioofthevaluesoftheoutputvariablesatthe99thand

 

 

Table5.Distributionofsocialcostofcarbon,2020(2005$pertonCO2) 

 

  

Figure6.Percentilesofthechangeintemperaturein2100acrossthesixmodels. 

0

1

2

3

4

5

6

7

8

9

 ‐  20  40  60  80  100

Temperature increase, 2100 (deg C)

Percentile of results

DICE FUND

GCAM IGSM

MERGE WITCH

      35 

99.9thpercentile.Foranormaldistribution,theratiooftheseis1.33.ForParetodistributionswithslopevaluesof2.0,1.8,and1.5,theratiosare3.7,3.9,and5.2.IfweexaminetheMonteCarloestimates,themaximumratiois1.56,whichoccursfordamagesintheDICEandFUNDmodels.Whilethissuggestsatailthatisslightlyfatterthanthenormaldistribution,itfallsfarshortoftheslopeassociatedwithaninfinite‐varianceParetoprocess.

Beforepresentingtheresults,wereiteratetheconcernthatthecalibrationrunsdonotextendfarintothetailsforTFP.ThisimpliesthattheresultsontailsreportedhererelyonextrapolationsoftheSRFoutsidethesamplerange.WecommentbelowonourreplicationofthetailestimateswiththeFUNDmodel,whicharegenerallyaccurate.Wealsoemphasizethattheestimatesofthetailsarederivedfromtheinteractionofthemodelswiththeassumedpdfs.Totheextentthatthemodelsomitdiscontinuitiesorsharpnon‐linearities,orthatourassumedpdfsaretoothin‐tailed,thenwemayunderestimatethethicknessofthetails.

WecanalsouseaformaltestoftheParetoshapeparameter,althoughthisismorecomplicatedbecauseitrequiresassumptionsabouttheminimumoftheParetoregion(statisticaltechniquesarefromRytgaard1990).Examiningthetop10%ofthedamagedistributionfortheDICEmodel(themostskewedofthevariables),wefindthattheparameteroftheParetodistributionabovethe1%righttailisestimatedtobe4.7(+0.047),whichiswellbelowtheinfinite‐variancethresholdof2.TheParetoparameterestimateforthe0.1%tailis7.03(+0.22).Thesetestsrejectthehypothesisthatthedistributionsarefat‐tailedinthesenseofbelongingtoaninfinite‐varianceParetodistribution.Theresultsareduetoboththestructuresofthemodelsandthenatureoftheshocks.Nothinginthemodelspreventsthegenerationoffattailsinthissituation,buttheymaymisscriticalnon‐linearities,sothetestsarenotbyanymeansconclusive.

WeexaminedthevalidityoftheresultsforthetailsusingthefullMonteCarloestimateoftheFUNDmodeldiscussedabove.Forthese,wecomparedtheinformaltests(ratioofthevariablesatthe99.9%iletothe99%ile).TheMUPcalculationswereveryaccurateforallvariablesexceptdamages,whereasfordamagestheMUPcalculationsunderestimatedtheskewness(overestimatedtheParetotail).WealsoexaminedtheParetoparameterinthefullFUNDMonteCarloandfoundthattheestimatewassignificantlyabovethethresholdofaninfinitevarianceprocess.

Theresultscanalsobeseeninboxplots.Figure7showstheboxplotfortemperatureincreaseto2100.Figure8showstheboxplotfortheCO2concentrationsfor2100.Bothoftheseunderscorethatwhiletherearedifferencesbetweenthemodelsinthewaythattheyarerunforthisstudy,theyareperhaps

      36 

smallerthanonemighthaveexpected–andaremuchsmallerthanthewithin‐modelvariation.Weshowthisformallyinthenextsection.

 

0

1

2

3

4

5

6

7

DICE FUND GCAM IGSM MERGE WITCH

Temperature increase, 2100 (deg C)

  

Figure7.Boxplotfortheincreaseintemperatureacrossmodelsin2100.Noteonboxplots:Dotismean.Horizontallineismedian.Shadedareaaroundlineis95%confidenceintervalofmedian(usuallytoosmalltosee).Boxcontainsinterquartilerange(IQRor25%ileto75%ile).Theupperstaple(horizontalbar)issetatthemedianplus2timestheIQR,whilelowerstapleissetatthemedianminus2timestheIQR.Theupperstableisapproximatelythe95%ileformostvariables.Becauseofskewnessofthevariables,thelowerstaplerepresentsfaroutliers,andisgenerallyaroundthe0.1%ile. 

      37 

B. Modeluncertaintyv.parametricuncertainty

Inexaminingtheuncertaintiesofclimatechangeandotherissues,acommonapproachhasbeentolookatthedifferencesamongforecasts,models,orapproaches(“ensembles”)andtoassumethattheseareareasonableproxyfortheuncertaintiesabouttheendresultorendogenousvariables.Intheareaofclimatemodels,forexample,researchershaveoftenlookedattheequilibriumclimatesensitivitiesindifferentclimatemodelsandassumedthatthedispersionwouldbeanaccuratemeasureoftheactualuncertaintyoftheECS.

Itisconceptuallyclearthattheensembleapproachisaninappropriatemeasureofuncertaintyofoutcomes.Thedifferenceamongmodelsrepresentsameasureofstructuraluncertainty.Forexample,alternativeclimatemodelsmighthavedifferentwaysofincludingcloudfeedbacks.Takingallthedifferencesamongthemodelswouldindicatehowstate‐of‐the‐artmodelsdifferontheprocessesandvariablesthattheyinclude.Evenhere,however,existingmodelsarelikelytohaveanincompleteunderstandingandwillthereforeunderestimatestructuraluncertainty.However,fromaconceptualvantagepoint,theygenerallydonot

 

200

400

600

800

1,000

1,200

1,400

1,600

1,800

DICE FUND GCAM IGSM MERGE WITCH

CO2 Concentrations, 2100, ppm

 

Figure8.BoxplotforCO2concentrations,2100.Forexplanationofboxplots,seeFigure7. 

      38 

explicitlymodelandconsiderparametricuncertainty.InIAMs,tocomeclosertohome,differencesinmodelsreflectdifferencesinassumptionsaboutgrowthrates,productionfunctions,energysystems,andthelike.Butfewmodelsexplicitlyincludeparametricuncertaintyaboutthesevariables.Differencesinpopulationgrowth,forexample,areverysmallrelativetomeasuresofuncertaintybasedonstatisticaltechniquesbecausemanymodelsusethesameestimatesoflong‐runpopulationtrends.

WecanusetheresultsoftheMonteCarlosimulationstoestimatetherelativeimportanceofparametricuncertaintyandmodeluncertainty.WecanwritetheresultsoftheMonteCarlosimulationsschematicallyasfollows.Assumethatthe

modeloutcomeforvariableiandmodelmis miY andthattheuncertainparameters

are and i ju u :

3 3

,1 1 1

jm m m m

i i i i i j i ji j i

Y u u u

Foragivendistributionofeachoftheuncertainparameters,thevarianceof iY

includingmodelvariationis:

3 32 2 2 2 2 2 2

,1 1 1

( ) ( ) ( ) ( ) ( ) ( ) ( )j

m mi i i i i j i j

i j i

Y u u u

Thefirsttermontherighthandsideisthevarianceduetomodeldifferences(orstructuraluncertainty),whilethesecondandthirdtermsarethevarianceduetoparameteruncertainty.Forthispurpose,weincludetheinteractionofthemodel

coefficients ,( and )m mi i j andtheparameteruncertainties 2[ ( )]iu asparametric

uncertaintybecausetheywouldnotbeincludedinensembleuncertainty.Theothertermsvanishbecauseweassumethattheparametricuncertaintiesareindependent.Whiledependencewilladdfurthertermsontheright‐handsideoftheequationforthevariance,itwillnotaffectthefractionduetostructuraldifferencesduetothefirstterm.

Wecaneasilyestimatethetotaluncertaintyandthestructuraluncertaintyfordifferentvariables.TheresultsareshowninTable6.Formostvariables,virtuallyallthevarianceisexplainedbyparametricuncertainty.Forexample,94%ofthevarianceofthe2100temperatureincreaseinallthemodelsisexplainedbyparametricuncertainty,andonly6%isexplainedbydifferencesinmodelmeans.ThisfactiseasilyseenintheboxchartsinFigures7and8.Theonlyvariablefor

      39 

whichmodeluncertaintyisimportantisthesocialcostofcarbon,forwhichfour‐fifthsofthetotalvarianceisduetomodeldifferences.

Wecanputtheseresultsintermsofthevariabilitiesduetodifferentfactors.Ifwetakethecalculatedtemperatureincreaseto2100,theoverallstandarddeviationis0.84°Cincludingbothmodelandparametricuncertainty.Thestandarddeviationofthemodelmeansaloneis0.21°C.Sothevariabilitymeasuredintermsofstandarddeviationsofthetemperatureincreaseisunderestimatedbyafactoroffourusingtheensembletechnique.

Theneteffectoftheseresultsissobering.Theyindicatethatthetechniqueofrelyinguponensemblesasatechniquefordeterminingtheuncertaintyoffutureoutcomesis(atleastforthemajorclimatechangevariables)highlydeficient.Ensembleuncertaintytendstounderestimateoveralluncertaintybyasignificantamount.

C. Sensitivityoftheresultstoparametervariability

Animportantquestionistheextenttowhichtheresultsaresensitivetotheindividualpdfsfortheuncertainparameters.Totestforsensitivity,weperformedanexperimentwhereweincreasedthestandarddeviationofeachofthepdfsbyafactorof2,bothoneatatimeandtogether.Foradoublingofthestandarddeviationofallparameters,theincreaseinthestandarddeviationof2100temperaturewasa

 

  

Table6.Fractionofuncertainty(variance)explainedbymodeldifferences. 

      40 

factorof1.83forallmodelstogether.Webelievethatthisislessthantwobecausetheshort‐runtemperatureimpactisnotproportionaltotheECS.

Table7showstheresultschangingtheuncertaintybyafactoroftwooneparameteratatimefortheaverageofthe6modelsforallvariableswhichareproducedbythesixmodels.Thenumbershowstheratioofthestandarddeviationofthe2100valueofthevariableinthesensitivitycaserelativetothecasewithassumedpdfs.Doublingalluncertaintiesproducesclosetoadoublingoftheoutputuncertainty,withsomedeviationsbecauseofnon‐linearities.

Doublingpopulationuncertaintyhasasmalleffectonallvariablesexceptpopulation.Doublingequilibriumtemperatureuncertaintyraisestheuncertaintyof2100temperatureby40%buthasnosignificanteffectonotheruncertainties.ThemajorsensitivityisTFPuncertainty.Doublingthisuncertaintyleadstoclosetodoublingoftheuncertaintyofothermajoreconomicvariables,andtoanincreaseof62percentintheuncertaintyof2100temperature.ThisresultissimilartoaresultinvanVuurenetal.(2008),whichsuggeststhatuncertaintyinGDPgrowthdominatestheuncertaintyinemissions.

Thesummaryonsensitivityoftheresultstothepdfsshowsanimportantandsurprisingresult.Onthewhole,theresultsareinsensitivetochangesinthepopulationgrowthpdf;aremoderatelysensitivetotheuncertaintyabout

 

  

Table7.Sensitivityofoutcomesforchangesinstandarddeviationofeachuncertainparameterbyfactorof2Thefiguregivestheratioofthestandarddeviationofthevariableatthetopofthecolumntothestandarddeviationinthebaserun.Forexample,doublingthestandarddeviationofpopulationincreasedthestandarddeviationof2100temperatureby6%.

      41 

equilibriumtemperaturesensitivityontemperature(aswellastodamagesandthesocialcostofcarbon,notshown);andareextremelysensitivetotheuncertaintyabouttherateofgrowthofproductivity.Whilelong‐runproductivitygrowthhasthegreatestimpactonuncertainty,itisalsotheleastcarefullystudiedofanyoftheparameterswehaveexamined.Thisresultsuggeststhatmuchgreaterattentionshouldbegiventodevelopingreliableestimatesofthetrendanduncertaintiesaboutlong‐runproductivity.

VIII. Conclusions

Thisstudyisthefirstmulti‐modelanalysisofparametricuncertaintyineconomicclimate‐changemodeling.Theapproachisbasedonestimatingclassicstatisticalforecastuncertainty.Thecentralmethodologyconsistsoftwotracks.TrackIinvolvesdoingasetofmodelcalibrationrunsforthesixmodelsandthreeuncertainparametersandestimatingasurfaceresponsefunctionfortheresultsofthoseruns.TrackIIinvolvesdevelopingpdfsforkeyuncertainparameters.ThetwotracksarebroughttogetherthroughasetofMonteCarlosimulationstoestimatetheoutputdistributionsofmultipleoutputvariablesthatareimportantforclimatechangeandclimate‐changepolicy.Thisapproachisreplicableandtransparent,andovercomesseveralobstaclesforexamininguncertaintyinclimatechange.

Herearethekeyresults.First,thecentralprojectionsoftheintegratedassessmentmodels(IAMs)areremarkablysimilaratthemodeler’sbaselineparameters.Thisresultisprobablyduetothefactthatmodelshavebeenusedinmodelcomparisonsandmayhavebeenrevisedtoyieldsimilarbaselineresults.However,theprojectionsdivergesharplywhenalternativeassumptionsaboutthekeyuncertainparametersareused,especiallyathighlevelsofpopulationgrowth,productivitygrowth,andequilibriumclimatesensitivity.

Second,despitethesedifferencesacrossmodelsforalternativeparameters,thedistributionsofthekeyoutputvariablesareremarkablysimilaracrossmodelswithdifferentstructuresandlevelsofcomplexity.Totakeyear2100temperatureasanexample,thequantilesofthedistributionsofthemodelsdifferbylessthan½°Cfortheentiredistributionuptothe95thpercentile.

Third,wefindthattheclimate‐relatedvariablesarecharacterizedbylowuncertaintyrelativetothoserelatingtomosteconomicvariables.Forthiscomparison,welookatthecoefficientofvariation(CV)oftheMonteCarlosimulations.AsshowninTable2,CO2concentrations,radiativeforcings,andtemperature(allfor2100)haverelativelylowCV.OutputanddamageshaverelativelyhighCV.Asexamples,themodel‐averagecoefficientofvariationforcarbondioxideconcentrationsin2100is0.28,whilethecoefficientofvariationfor

      42 

climate‐changedamagesis1.29.ThesocialcostofcarbonhasanintermediateCVwithinmodels,butwhenmodelvariationisincludedtheCVisclosetothatofoutputanddamages.Theseresultshighlighttheimportanceoffurtherresearchoneconomicvariablesanddamagefunctionsforreducinguncertaintyandimprovingpolicymaking(e.g.,seePizeretal.2014andDrouetetal.2015).

Fourth,wefindmuchgreaterparametricuncertaintythanstructural(acrossmodel)uncertaintyforalloutputvariablesexceptthesocialcostofcarbon.Forexample,inexaminingtheuncertaintyin2100temperatureincrease,thedifferenceofmodelmeans(ortheensembleuncertainty)isapproximatelyone‐quarterofthetotaluncertainty,withtherestdrivenbyparametricuncertainty.Whilelookingacrosssixmodelsbynomeansspansthespaceofmethods,thesixmodelsexaminedherearerepresentativeofthedifferencesinsize,structure,andcomplexityofIAMs.Thisresultisimportantbecauseofthewidespreaduseofensembleuncertaintyasaproxyforoveralluncertaintyandhighlightstheneedforare‐orientationofresearchtowardsexaminingparametricuncertaintyacrossmodels.

Afifthinterestingfindingofthisanalysisisthelackofevidenceinsupportoffattailsinthedistributionsofemissions,globalmeansurfacetemperature,ordamages.Populationgrowth,totalfactorproductivitygrowth,andclimatesensitivityareverylikelytobethreeofthekeyuncertainparametersinclimatechange.Yet,basedonbothinformalandformaltests,themodelsascurrentlyconstructedfindthatthetailsarerelativelythin.Thedeclineinprobabilitiesassociatedwithachangeinanyofthevariablesismuchlargerthanwouldbeassociatedwithaninfinite‐varianceParetoprocess.Asdiscussedabove,weemphasizethatthesefindingsshouldbeinterpretedinthecontextofthecurrentgroupofmodelsandtheassumedpdfs.Theresultsdonotruleoutfattails,buttheydoprovideempiricalevidenceagainstfattailsinoutcomesinvestigatedinthisstudyforthecurrentsetofmodelsandthedistributionsofthethreeuncertainvariablesconsideredhere.Theseresultstendtosupporttheuseofexpectedbenefit‐costanalysisforclimatechangepolicy,incontrasttosuggestionsbysomeauthorsthatneglectoffattaileventsmayvitiatestandardanalyses(Weitzman2009).

Sixth,wefindthatwithinawiderangeofuncertainty,changesindispersionoftwooftheuncertainparameterstakensinglyhavearelativelysmalleffectontheuncertaintyoftheoutputvariables,thesebeingpopulationgrowthandequilibriumtemperaturesensitivity.However,uncertaintyaboutproductivitygrowthhasamajorimpactontheuncertaintyofallthemajoroutputvariables.Thereasonforthisisthattheuncertaintyofproductivitygrowthfromtheexpertsurveycompoundsgreatlyoverthe21stcenturyandinducesanextremelylargeuncertainty

      43 

aboutoutput,emissions,concentrations,temperaturechange,anddamagesbytheendofthecentury. Asinanystudy,thisanalysisisintentionallysharplyfocused.Byanalyzingparametricuncertaintyinthreekeyparameters,wedonotclaimtobecapturingalluncertaintiesinclimatechange.Aswedescribeabove,therearemanyuncertaintiesthatcannotbecapturedusingthestatisticalframeworkdevelopedhere.ButbyprovidingdetailedestimatesofuncertaintyacrossarangeofIAMsthatarecurrentlybeingusedinthepolicyprocess,webelievethatwehavesignificantlyimprovedtheunderstandingofuncertaintyinclimatechange.Moreover,ournewtwo‐trackmethodologyiswell‐suitedforexpansiontoadditionalparametersandmodels,andcanbereadilyusedtoexploreadditionalconcerns,suchastheinteractionbetweencarbonpoliciesanduncertainty.

      44 

Appendix1.FurtherDetailsontheChoiceofECSDistribution

Thisappendixexplainstheprocedurefordevelopingthepdfforclimatesensitivity.Thestudybeganbyreviewingthefiveprobabilitydensityfunctionsforequilibriumclimatesensitivity(ECS)usedintheIPCCAR5thatdrawuponmultiplelinesofevidence.TheseareAldrinetal.(2012),LibardoniandForest(2013),Olsenetal.(2012),AnnanandHargreaves(2006),andHegerletal.(2006).FigureA1illustratesthelog‐normalfitstoeachofthesedistributions(fitsbythepresentauthors).

Ourchosenstudy,Olsenetal.(2012),isrepresentativeofthestudiesinbothitsmethodologyandresults.ItusesaBayesianapproach.Thepriordistributionwasconstructedtofitthe“mostlikely”valuesand“likely”rangesinFigure3inKnuttiandHegerl(2008)basedonthesummarystatisticsofthe“currentmeanclimatestate”and“LastGlacialMaximummodels.”Olsenetal.assumeaninverseGaussian(Wald)distributionandobtainthispriorbyassumingindependencebetweenthe

 

  

FigureA1.Log‐normaldistributionsfittotheprobabilitydensityfunctionscitedintheIPCCAR5.ThedistributionshownhereisfromtheupdatedLibardoni&Forest(2013)figures. 

      45 

currentmeanclimatestateandthelastglacialmaximummodels,andthencomputingthemixturedistribution.

TheposteriordistributionisthencalculatedbyusingaMarkovChainMonteCarlosimulationtoupdatethepriorwithalikelihoodfunction.Thelikelihoodisbasedonseveraldifferenttracers,suchasglobalaverageatmosphericsurface/oceansurfacetemperaturesandglobaltotalheatcontent.ThesetracerscomefromtheUniversityofVictoriaESCMclimatemodel,whichconsistsofathree‐dimensionaloceangeneralcirculationmodelcoupledwithathermodynamic/dynamicsea‐icemodel.Theauthorsassumeindependence,sothatthelikelihoodofbothobservationsisequaltotheproductofthelikelihoods.

Theparametersofthelog‐normaldistributionfittoOlsenetal.areμ=1.10704andσ=0.264.Themajorsummarystatisticsofthereferencedistributioninthestudyarethefollowing:mean=3.13,median=3.03,standarddeviation=0.843,skewness=0.824,andkurtosis=4.23.InimplementingtheMonteCarloforeachmodel,weretainedthemeanECSforthatmodel.Wethenimposedalog‐normaldistributionthatretainedthearithmeticstandarddeviationoftheECS(i.e.,astandarddeviationof0.843)basedontheOlsenetal.(2012)distribution.

      46 

Appendix2.ExpertSurveyonTotalFactorProductivity

Akeycomponentoftheprojectwasdeterminingtheuncertaintyinproductivity(or,asoperationallydefined,outputpercapita).Areviewofexistingstudiesindicatedthattherewerenodetailedstudiesoffutureoutputuncertaintiesoutto2100thatwecouldrelyon.Wethereforedecidedtoundertakeanexpertelicitation.Thedetailedresultsofthesurveywillbeshortlyavailableseparatelyasaworkingpaper.Thisappendixsketchesthemethodsandsummarizesthepreliminaryresults.Notethatthecurrentresultsincludeonly11oftherespondents,andthecompletesurveyresultswillbeusedforthefinalpublication.

2.1 SurveyDesign

Indeterminingtheprobabilitydistributionoffutureproductivitygrowth,amajordifficultyisthenon‐stationarityofthisvariable.Itisclearlynon‐stationaryifoneexamineshistoricaldata.Fromatheoreticalpointofview,wewouldexpectnon‐stationaritybecausethemajordeterminantsoflong‐rungrowth–inventionandtechnologicalchange–involvenewanddifferentprocessesratherthanreplicationofsomeunderlyingprocess.Forthisreason,itisimportanttooverlayanyempiricalstudywithexpertviews.

Expertopinionhasbeenusedsystematicallyinaverylimitednumberofstudiesofeconomicgrowth.Forexample,Websteretal.(2002)analyzeuncertaintyintheGDPgrowthrateoutto2100(asaproxyforchangesinlaborproductivity)usingestimatescollectedfromanelicitationof5expertsfromasingleinstitution.Thisseemedtoothinabaseforthepresentstudy.

Inthisstudy,weconductedasurveyofexpertpredictionsaboutuncertaintyinglobalannualgrowthratesfortheperiod2010‐2100.Expertsprovidedresponsesusinganonlinesurvey(seeFigureA2fortheresponseformat).Thepanelofexpertswasselectedthroughaprocessofnominationbyleadingeconomists.

Weaskedexpertsaboutgrowthratesinhigh‐,medium‐,andlow‐incomecountries,aswellasaboutglobalaggregaterates.Aspartofthesurvey,wealertedexpertstoproblemsofoverconfidenceandincludeawarm‐upsectionthatwasdesignedtoincreaseawarenessoftheirpersonaloverconfidence.Inaddition,weaskedexpertsaboutanyambiguitiesthattheyexperiencedinthesurveyand

      47 

providedthemwithhistoricaldataongrowthratesfortheperiod1900‐2000fromBarro‐Ursua(2010)andMaddison(2003).7

Thesurveywascomprisedof4setsofquestionsaboutgrowthrates:(1)centralestimates(50thpercentile)forgrowthratesfor2010‐2050and2010‐2100,(2)estimatesofuncertaintybasedonprovidingthe10th,25th,75th,and90thpercentilesofthegrowthrates,(3)theprojectedmagnitudeofeffectsofpositiveandnegativeshockstotheeconomy,and(4)near‐termpredictions(forthefollowingyear).Weaskedeachexperttodescribetherationalefortheirresponseaswellasanexplanationofmajorpositiveandnegativeshocks.Thesurveyalsoaskedexpertstoidentifyoutsidesourcesofinformationthatwereusedtogenerateforecastsandtoranktheirownexpertiseoverallandforparticularregions.

2.2 CombiningExpertDistributions

Weusetwomethodstoestimatethemeanandstandarddeviationforthe

best‐fittingcombinednormaldistributionofgrowthratesfortheperiod2010‐2100.

Thefirstmethodassumesthatexpertshaveestimatesofquantilesofthedistributionoflong‐rungrowthrates.Thecombinedpdfisthenthedistributionthatminimizesthesumofsquareddifferencesbetweenthecombinednormal

                                                            7Barro‐UrsuaMacroeconomicDataavailableat:rbarro.com/data‐sets/.MaddisonisfromAngusMaddison(2003).Availableat:http://www.theworldeconomy.org/statistics.htm.

 

  

FigureA2.ResponseFormatforExpertSurvey 

      48 

distributionateachquantileandtheaverageofthequantileestimatesoftheexperts.Thesecondmethodbeginswithestimatesoftheparametersofthebest‐fittingnormaldistributionforeachexpert;andthentakesthesamplemeansoftheparametersoftheexpertsforthecombinednormaldistribution.

Wefindverylittledifferencebetweenthetwomethods.Forthepreliminarysample,themeangrowthratesofpercapitaoutputforthetwomethodsare2.29and2.30,respectivelyformethods1and2.Thecombinedstandarddeviationsare1.15and1.17,respectively.

Thecombinedpdfsalongwith11preliminaryresponsesareshowninFigure2inthemaintext.ThecurrentprocedureusesthesamplemeanofthestandarddeviationfortheMonteCarloestimates,butweareconsideringusingarobustestimatorforthefinalreport.

      49 

Appendix3.AdditionalLatticeDiagrams

Weincludeherefurtherlatticediagrams.Thestructureisasdescribedinthetext.Theonlydifferenceistheoutputvariable,whichisshownatthetopofthegraph.

Notethatthefirstgroupofdiagramsisforthebaseruns,whilethesecondgroupisfortherunswithcarbontaxes(CarbonTaxorAmpereruns).

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      51 

      52 

      53 

      54 

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Appendix4.AdditionalTablesandGraphs

TableA1.Overviewofglobalintegratedassessmentmodelsincludedinthisstudy.

Model Numberof

EconomicRegions

TimeHorizon

VariablesIncluded

KeyCharacteristics SelectedReference

s

DICE 1 2010‐2300

1,2,3,5,6 Optimalgrowthmodel,endogenousGDPandtemperature,exogenouspopulation,SWFisCESwithrespecttoconsumption.

(NordhausandSztorc2014)

FUND 16 1950‐3000

1,2,3,4,5,6,7

Multi‐region,multi‐gas,detaileddamagefunctions,exogenousscenariosperturbedbymodel

(AnthoffandTol2010,2013)

GCAM 14 2005‐2095

1,2,3,4,5,7 Integratedenergy‐land‐climatemodelwithtechnologydetail;exogenouspopulationandGDP;endogenousenergyresources,agriculture,andtemperature;economiccostsarecalculatedforproducerandconsumersurpluschange

(Calvinandetal.2011)

IGSM 16

2100 1,2,3,4,5,7 Fullgeneralcirculationmodellinkedtoamultisector‐multiregiongeneralequilibriummodeloftheeconomywithexplicitadvancedtechnologyoptions

(Chenetal.2015,Sokolovetal.2009,Websteretal.2012)

MERGE 10 2100 1,2,3,4,5,7 Ramseymodelcoupled (Blanford

      56 

withenergyprocessmodel,multipleregions,endogenousGDPandtemperature,exogenouspopulation

etal.2014)

WITCH 13 2150 1,2,3,4,5,6,7

Optimalgrowthmodel,endogenousGDPandtemperature,exogenouspopulation,SWFisCESwithrespecttoconsumption.

(Bosettietal.2006)

Notes:SWF=socialwelfarefunction,CES=constantelasticityofsubstitution.Forvariablesincludedthekeyis:1=GDP,population2=CO2emissions,CO2concentrations3=globaltemperature4=multipleregions5=mitigation6=damages7=non‐CO2GHGs

      57 

ResultsofMonteCarlosimulationsformodelsandmajorvariables[Allvariablesare2100exceptSCC,whichis2020]

      58 

FigureforboxplotsforCO2emissions,2100.Fordiscussionofboxplots,seeFigure7.

 

-100

0

100

200

300

400

500

DICE FUND GCAM IGSM MERGE WITCH

CO2 emissions, 2100 (billions tons CO2)

      59 

DICEFUNDWITCH

Figureforboxplotsforsocialcostofcarbon,2020.Fordiscussionofboxplots,seeFigure7.

      60 

Estimatesfromsurfaceresponsefunctionsbyvariableandmodel.

      61 

GoodnessoffitofworstfittingLQIvariablebymodel.

Tableshowstheresidualvariance(1‐R2)fortheworstfittingoftheequations.Forexample,intheLQIspecification,theworstSRFfortheDICEmodelistheequationforpopulation,whichhasaresidualvarianceof0.00706.FortheMERGEmodel,theworstequationisforCO2emissions.NoteaswellthattheonlytwomodelsforwhichtheworstequationhasasignificantreductioninresidualvariationfromLQItoLQI++aretheIGSMandWITCHmodels.

      62 

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