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Transcript
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
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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
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Notes:SWF=socialwelfarefunction,CES=constantelasticityofsubstitution.For
variablesincludedthekeyis:
1=GDP,population
2=CO2emissions,CO2concentrations
3=globaltemperature
4=multipleregions
5=mitigation
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7=non‐CO2GHGs
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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.
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twomodelsforwhichtheworstequationhasasignificantreductioninresidual
variationfromLQItoLQI++aretheIGSMandWITCHmodels.
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