Climate treaties and approaching catastrophes

16 Pages·2013·1 MB·English

by  Scott Barrett

JournalofEnvironmentalEconomicsandManagement](]]]])]]]–]]]
ContentslistsavailableatSciVerseScienceDirect
Journal of
Environmental Economics and Management
journal homepage: www.elsevier.com/locate/jeem
$
Climate treaties and approaching catastrophes
Scott Barretta,b,n
aSchoolofInternationalandPublicAffairs&EarthInstitute,ColumbiaUniversity,NewYork,NY10027,UnitedStates
bPrincetonInstituteforInternationalandRegionalStudies,PrincetonUniversity,Princeton,NJ08544,UnitedStates
a r t i c l e i n f o a b s t r a c t
Articlehistory: If the threshold that triggers climate catastrophe is known with certainty, and the
Received7November2011 benefits of avoiding catastrophe are high relative to the costs, treaties can easily
coordinatecountries’ behavior so asto avoid thethreshold.Where thenetbenefits of
Keywords: avoidingcatastrophearelower,treatiestypicallyfailtohelpcountriescooperatetoavoid
Climatechange catastrophe,sustainingonlymodestcutsinemissions.Theseresultsareunaffectedby
Internationalenvironmentalagreements uncertainty about the impact of catastrophe. By contrast, uncertainty about the
Catastrophe catastrophicthresholdnormallycausescoordinationtocollapse.Whethertheprobability
Cooperation densityfunctionhas‘‘thin’’or‘‘fat’’tailsmakeslittledifference.
Coordination &2012ElsevierInc.Allrightsreserved.
Uncertainty
Enforcement
1. Introduction
Thereisuniversalagreement,codifiedintheFrameworkConventiononClimateChange,thatatmosphericconcentra-
tions of greenhouse gases should be stabilized ‘‘at a level that would prevent dangerous [my emphasis] anthropogenic
interference with the climate system.’’ The Kyoto Protocol tried to get countries to reduce their emissions without
identifyingathresholdfordangerousinterference,butKyotofailed,mainlyforlackofanenforcementmechanism.1The
recentlynegotiatedCopenhagenandCancunagreementsalsolackenforcement,andarelegallynon-bindingbesides,but
unlikeKyototheseagreementsidentifyadangerousthreshold—a21Celsiusincreaseinaverageglobaltemperature.Inthis
paperIask,Canthefearofcrossingacatastrophicthresholdovercometheenforcementchallenge?Canitmakeclimate
treatiesmoreeffective?
The theory of international environmental agreements has generally offered a gloomy prognosis for cooperation,
particularly on this issue.2 To this point, however, the literature has considered only continuous abatement benefit
functions.InthispaperItakethebenefitfunctiontobediscontinuousatathreshold.Thissimplechangegivesrisetoanew
result.Ifthethresholdisknownwithcertainty,andthelossfromcatastrophevastlyexceedsthecostsofavoidingit,then
the collective action problem changes fundamentally. Rather than cooperate to limit emissions, countries need only
$TheideaforthispapercametomeasIwasthinkingofhowtorespondtoaquestionposedbyWilliamNordhausataseminarIwasgivingatYalea
fewyearsago.IamgratefultoApurvaSanghi,ArthurCampbell,AstridDannenberg,ClaudeHenry,ErinMansur,GeirAsheim,KennethArrow,MarkCane,
MatthewKotchen,MartinWeitzman,RobertKeohane,RobertMendelsohn,RobertSocolowandThomasSchellingforcommentsonapreviousdraft.
Iamalsogratefultotheeditor,twoanonymousreferees,andparticipantsatseminarsgivenatColumbia,Harvard,Princeton,Stanford,theWorldBank,
andYale(alaterseminar)fortheircomments.AnearlierversionofthispaperwasgivenattheinvitationofMichaelFinusasakeynotelectureatthe
EnvironmentalProtectionandSustainabilityForum,UniversityofExeter,April2011.
nCorrespondenceaddress:SchoolofInternationalandPublicAffairs&EarthInstitute,ColumbiaUniversity,NewYork,NY10027,UnitedStates.
Fax:þ12128544782.
E-mailaddress:[email protected]
1SeeBarrett[7].
2Forsurveysoftheliterature,seeMichaelFinus[13],UlrichJ.Wagner[26],andBarrett[6,7].
0095-0696/$-seefrontmatter&2012ElsevierInc.Allrightsreserved.
http://dx.doi.org/10.1016/j.jeem.2012.12.004
Pleasecitethisarticleas:S.Barrett,Climatetreatiesandapproachingcatastrophes,JournalofEnvironmentalEconomics
andManagement(2013),http://dx.doi.org/10.1016/j.jeem.2012.12.004i
2 S.Barrett/JournalofEnvironmentalEconomicsandManagement](]]]])]]]–]]]
coordinatetoavoidcatastrophe.Underthesecircumstances,climatetreatiescansustaintheefficientoutcome.Essentially,
natureherselfenforcesanagreementtoavoidcatastrophe.
Here is why: when the net benefit of steering clear of the threshold is very high, avoiding catastrophe is a Nash
equilibrium.(Imainlyrestrictattentioninthispapertosymmetricequilibriainpurestrategies.)Sinceeachcountryistoo
smalltoavertcatastropheonitsown,however,therealsoexistsaParetoinferiorNashequilibriuminwhichthethreshold
is crossed. Under these circumstances, the challenge is for countries to coordinate on the mutually preferred
equilibrium—a task for which treaties are exquisitely suited. When these circumstances do not apply, there exists a
unique Nash equilibrium that sits on the wrong side of the threshold. Since there also exist other feasible, mutually
preferredoutcomes,countrieswillwanttocooperatesoastoimproveonthisinefficientequilibrium(asinaprisoners’
dilemmagame).Atreatycanhelphere,too,butcredibleenforcementmechanismsforsupplyingthiskindofglobalpublic
goodaregenerallyweak.3RelativetotheinefficientNashequilibrium,whichiscommontobothsituations,coordination,
whenpossible,almostalwayssucceedsbetterthancooperation.
Weitzman [27] has drawn our attention to ‘‘uncertain catastrophes with tiny but highly unknown probabilities,’’
showingthat,iftheprobabilityofeverlargercatastrophesdoesnotfallfasterthanwelfarelossesincreaseasweventure
deeperintothetailsoftheprobabilitydensityfunction,thentheexpectedgainfromapolicytoreduceemissionswillbe
infinite.Underthesecircumstances,policyshoulddoeverythingpossibletoreduceemissionsasquicklyaspossible,even
thoughdoingsocannotguaranteethatcatastrophewillbeavoided.4Hecallsthisresult,appropriatelyenough,‘‘thedismal
theorem.’’
Nordhaus[18]:21argues,tothecontrary,thatanyconceivablecatastrophicoutcomecanbeavoidedbypolicy,butthat
doingsorequires‘‘solvingtheglobalpublicgoodsproblembygatheringmostnationstogethertotakecollectiveaction.’’5
Weitzman does not address the collective action aspects of this challenge, but his theorem should carry through in a
decentralizedsetting.Ifitpaystheworldasawholealwaystodevotemoreresourcestoreducingathreatthatcannotbe
eliminated,nomatterthecost,thenitshouldpayindividualcountries(havingthesamepreferences,andfacingthesame
uncertainty as assumed by Weitzman) always to devote more resources to reducing the same existential threat, even
when the benefits of doing so are widely dispersed. That is, under Weitzman’s assumptions, coordination should be
unnecessary. Every country should want to put its economy on a ‘‘climate war’’ footing, irrespective of whether other
countriesjointheminthiseffort.
Underpinningthedismaltheoremisthestrongassumptionthatutilityisunbounded.6Ifthisassumptionisrelaxed,so
that preferences are not very risk averse as consumption approaches zero, a different outcome emerges, one that is
consistentwithWeitzman’spolicyrecommendationof‘‘arelativelymorecautiousapproach[emphasisadded]’’toreducing
greenhousegasemissions([27]:13).Inthispaper,forsimplicity,Itakeutilitytobelinear.Whilethisassumptionrulesout
Weitzman’s dismal theorem, I shall show that it nonetheless implies that countries may want to adopt a much more
cautiousapproachascomparedtooneinwhichthethreatofcatastrophecanbeignored.Unfortunately,underthesesame
circumstances, I also find that the uncertain prospect of approaching catastrophe has little if any effect on the non-
cooperative outcome, or the ability of an international agreement to sustain collective action. The reason is that
uncertainty(aboutthethresholdforcatastrophe)makestheexpecteddamagefunctioncontinuous,renderingcoordina-
tion ineffective, and restoring the main results reported previously in the literature—that cooperation is needed, but
difficulttosustain.Inthismodel,whethertheprobabilitydensityfunction(pdf)hasthinorfattailsmakeslittledifference.
Uncertaintyaboutthemagnitudeofcatastrophicdamagesisalsorelativelyunimportant.Itisuncertaintyinthethreshold
thatmatters.
Acertainthresholdcanbeinterpretedasadiscreteuniformdistributionhavingasinglevalue.Aninterestingextension
ofthismodel,whichtiesmyearlieranalysisofcertaintytogetherwithmylateranalysisofuncertainty,isthecontinuous
uniformdistribution.Thisdistributionexhibitsacriticaldiscontinuity(inthedistribution’srightsupport),beingfat-tailed
throughoutitsrangeandzero-tailedelsewhere.Ishowthat,forthisspecialpdf,theresultIobtainedpreviouslyforthe
modelofcertaintyextendstouncertainty—countriesmaybeabletocoordinateontheright-sidediscontinuitysoasto
guaranteeavoidingcatastrophe.However,ascomparedwiththecertaintycase,Ifindthatopportunitiesforcoordination
areextremelylimited.Evenunderthefavorableconditionsofthisspecialpdf,cooperationwillalmostcertainlybeneeded
and,asusual,difficulttosustain.
Myoverallconclusionisthatfearofapproachingcatastropheswithuncertainthresholdsstrengthenstheimperativeto
cooperatewithoutimprovingtheprospectsforcollectiveaction.
3Thisistrueeveninarepeatedgamesolongastreatiesmustberenegotiation-proof;seeBarrett[5,6,7].Theone-shotgamedevelopedinthispaper
obeysananalogousproperty.
4Iftheconditionsthatgiverisetothisresultarerelaxedevenslightly,thenthethreatofcatastropheraisesnumerouspolicyissues;seeKouskyetal.[15].
5ToNordhaus[18],threeconditionsmustholdforcatastrophetobecauseforconcern:policyfailure,asjustnoted,hightemperaturesensitivityto
concentrations,andextremelyconvexdamages.Inthispaper,becausetemperatureissubsumedintheanalysis(damagesarerelatedtoaggregate
emissions),thelattertwoconditionswillbesatisfiedwhentheabatementbenefitfunctionisdiscontinuous(or,intheuncertaintycase,whenthe
varianceisverysmall)andthelossduetocrossingthethreshold(parameterXinthispaper)isverylarge.
6Arrow[2]notesthat,forindividuals,theassumptionimpliesthat‘‘thevalueofstatisticallifeisinfinite,aconclusionclearlycontrarytoallempirical
evidenceandtoeverydayobservation.’’
Pleasecitethisarticleas:S.Barrett,Climatetreatiesandapproachingcatastrophes,JournalofEnvironmentalEconomics
andManagement(2013),http://dx.doi.org/10.1016/j.jeem.2012.12.004i
S.Barrett/JournalofEnvironmentalEconomicsandManagement](]]]])]]]–]]] 3
Althoughthefocusofthispaperisonclimatechange,themodelcanbeapplied,withsuitablemodifications,toother
situations.Considertheproblemofspacedebris.Ifthevolumeofdebrisinlowearthorbitexceedsacriticallevel,onemore
collision dramatically increases the probability of more collisions, creating a cascading effect that could render orbital
spaceunusable(KesslerandCour-Palais[14]).Here,catastrophecanbeavoidedbylimitingadditionstothestockofdebris,
orbydesigningsatellitesandrocketboosterssothat theycanbemaneuveredto a‘‘graveyard’’ordecayingorbitwhen
they become obsolete. Another example is antibiotic resistance. Resistant strains of a pathogen sometimes have high
evolutionaryfitnessinthepresenceofadrugtreatment,butsufferafitnesscostrelativetodrug-sensitivestrainswhenthe
drugisremoved(Smith[23]).Ifuseofthedrugislow,thesensitivestrainswillwinout.Ifuseishigh,theresistantstrains
willthrive.Inbetweenthereexistsacriticalthresholdfordruguse,withthepriorityforpublichealthbeingtostayonthe
goodsideofthethreshold.Consider,finally,exploitationofabiologicalresourcesuchasaspeciesoffishforwhichthere
existsastrictlypositiveminimumviablepopulationlevel.Economicoverexploitationoftheresourcemaybeaproblem
even at higher stock levels, but exploitation below the critical level would be disastrous; crossing it would cause the
speciestogoextinct.Thismaybeaparticularlyimportantissueforfishpopulationssuchastunathataggregate,ifschool
sizeisindependentofstockabundance,andharvesteconomicsdependonschooling(ClarkandMangel[11]).
Notwithstandingtheimportanceoftheseandotherapplications,mymainconcerninthispaperiswithclimatechange.
I shall introduce my model in Section 3, but it will help to begin by putting this model in the context of a previously
publishedexperiment,whichwasalsoconcernedwithclimatechange.
2. Anexperimentincatastropheavoidance
TheexperimentbyMilinskietal.[17]simulatesthe‘‘collective-risksocialdilemma’’inpreventing‘‘dangerousclimate
change.’’Intheirexperimenttherearesixplayers.Eachisgivenh40.Thegameisplayedovertenperiods.Ineachperiod,
everyplayermustchoosetocontributeh0,h2,orh4withoutcommunicating.If,attheendofthegame,atleasth120has
beencontributed,dangerousclimatechangeisavertedwithcertainty,andeachplayergetsapayoffequaltotheamountof
money he has left (there are no refunds in this game). If less than h120 has been contributed, each player loses all the
money she has left with probability 0.9. In their experiment, Milinski et al. [17] played the game with ten groups of
students,onlyhalfofwhichsucceededinavoidingthethreshold.
Therearetwosymmetricpurestrategyequilibria.Inone,everyplayercontributesh0everyperiod,givingeachplayer
anexpectedpayoffofh4.Intheother,everyplayercontributesh2everyperiod,givingeachacertainpayoffofh20.(Of
course,therealsoexistmanyasymmetricpurestrategyequilibriainwhichdifferentplayerscontributedifferentamounts,
possiblyindifferentperiods).Thelatterequilibriumisefficient;theformerisnot.
In contrast to the conventional representation of the climate change game, the game with catastrophic damages is
pleadingforcoordination.Why,then,didhalfthegroupsintheMilinskietal.[17]experimentfailtocoordinate?Themain
reasonisalmostsurelythat,byconstruction,theplayerswerenotallowedtocommunicate,letaloneformulateatreatyto
coordinatetheircontributions.7
Theusualwayofmodelinganinternationalenvironmentalagreementisinthreestages.8Instageone,countrieschoose
independentlywhethertobeapartyornon-partytotheagreement.Instagetwo,partieschoosetheiractions(inthiscase,
contributions) so as to maximize their collective payoff. Finally, in stage three, non-parties choose their actions with
the aim of maximizing their individual payoffs. A treaty is self-enforcing if, given the treaty and participation level,
non-participantsdonotwanttochangetheirbehavior;if,giventheparticipationlevel,partiestothetreatydonotwantto
changetheobligationsexpressedinthetreaty;andif,giventheparticipationdecisionsofothercountries,eachcountry
doesnotwanttochangeitsdecisionofwhethertobeapartyornon-partytothetreaty.
Applyingthisnotionofaself-enforcingtreatytotheMilinskietal.[17]game,itwillhelptoassumethatcontributions
aremadeinasingleperiodandthateachplayercancontributeanyamountuptohisorherendowment.Ifparticipation
werefull,thetreatywouldthentelleachcountrytocontributeh20,nettingeachcountryapayoffofh20.Wereacountry
todropoutofthisagreement,theremainingparties,choosingcollectively,wouldchangetheircontributions.Theywould
reason that, if they contributed an amount Y in total, then, taking this contribution as given, the non-party would
contributeanamountZ¼h120–Yforh120ZYZh84andZ¼h0forYoh84.Knowingthis,thefiveremainingsignatories
could do no better than to contribute Y¼h84 collectively (h16.80 each). This would net each of the five parties h23.20,
whereasthesolenon-signatorywouldgetjusth4.Recallthat,werethiscountrynottowithdraw,itwouldgetapayoffof
h20.Obviously,withthetreatywritteninthisway,nocountryhasanincentivetowithdraw,startingfromasituationin
7AsimilarexperimentbyTavonietal.[25]confirmstheimportanceoftheassumptionaboutcommunication.
8SeeBarrett[7].Thisthree-stageformulationimbuespartiestotheagreementwiththespecialabilitytocommittoplayingacollectiveabatement
level—theso-called‘‘Stackelbergleadership’’assumption.Ifnon-signatorieshavedominantstrategies,thisassumptionisindistinguishablefromthe
Nashassumption.IntheMilinskietal.model,however,thepayofftoavoidingcatastrophejumpsatthethreshold,implyingthatcountriesmaynothave
dominantstrategies.Themainjustificationfortheleadershipassumptionisthecustomarylawprinciple,pactasuntservanda:treatiesarebinding.Parties
toanagreementareexpectedtofulfilltheirobligations,withcustombeingenforcedoutsidethetreaty(thatis,outsidethemodel).Notethatthisisa
minimalcommitment,sinceinternationallawalsosaysthatcountriesarefreetoparticipateintreatiesornotastheyplease—anassumptioncapturedin
Stage1oftheabovemodel.Inthispaper,theleadershipassumptionisalsoimportantforequilibriumselection—itensuresthat,inacoordinationgame,
themoreefficientequilibriumwillbesupportedbyaself-enforcingagreement.
Pleasecitethisarticleas:S.Barrett,Climatetreatiesandapproachingcatastrophes,JournalofEnvironmentalEconomics
andManagement(2013),http://dx.doi.org/10.1016/j.jeem.2012.12.004i
4 S.Barrett/JournalofEnvironmentalEconomicsandManagement](]]]])]]]–]]]
which participation is full. Hence, a treaty comprising six signatories, each of which contributes h20, is self-enforcing.
Moreover,whilethereexisttwoNashequilibriainpurestrategiesintheunderlyinggame,theself-enforcingagreement
justnotedisuniqueandsupportstheParetoefficientequilibrium.
Tosumup,whileMilinskietal.[17]claimthatcountriesmayfailtoavertcatastrophewhendoingsoisfeasibleand
efficient,simpletheorysuggestthat,solongascountriesarepermittedtonegotiateatreaty,catastropheshouldberather
easy to avoid. In the remainder of this paper I inquire into whether this result could be expected to hold in richer
(analytical)environments.
3. Aclimatechangecatastrophegame
Assumethatcountrieshavesymmetricpayofffunctions.Denotingcountryi’sabatementbyq andaggregateabatement
i
byQ,whereQ¼PN q andN isthenumberofcountries,countryi’spayoffisassumedtobegivenby
i¼1 i
p ¼8<bQ(cid:2)cq22i ifQZQ : ð1Þ
i :bQ(cid:2)X(cid:2)cq2i ifQoQ
2
In (1), parameter b stands for each country’s marginal benefit of abating ‘‘gradual’’ climate change, cq represents
i
countryi’smarginalabatementcosts,andXdenotesthedamageseachcountrysuffersfrom‘‘catastrophic’’climatechange.
ThesedamagesareexperiencedifandonlyifglobalabatementfallsshortofQ,thethreshold.9IfweletX¼0,(1)collapses
tothestandardmodelintheliteratureoninternationalenvironmentalagreements.10
It is worth noting the important differences between (1) and Milinski et al. [17] model. First, in (1) contributions
(abatementlevels)arebeneficialevenifthethresholdisexceeded;abatementreduces‘‘gradual’’aswellas‘‘catastrophic’’
climatechange.Second,Milinskietal.varyX(morespecifically,theexpectedvalueofX),butnotthethreshold,and(1)
allowsustovaryboth.11Third,in(1),marginalabatementcostsareincreasing,whereasMilinksietal.implicitlyassume
that they are constant. Fourth, Milinski and coauthors take abatement (contributions) to be discrete, and (1) allows
abatementtobevariedcontinuously.Finally,while(1)assumesthatXiscertain,andMilinksietal.letXbeuncertain,later
inthispaperIshallletbothXandQ beuncertain.
IncommonwithMilinskietal.[17]andWeitzman[27],model(1)abstractsawayfromthedynamicsofclimatechange.
It is, however, able to capture the key difference between ‘‘gradual’’ and ‘‘abrupt and catastrophic’’ climate change. It
assumesthatgradualclimatechangeissoslowthatmarginalbenefitsareconstant,andthatabruptclimatechangeisso
fastthattransitionisinstantaneous.12Themodelcanbeinterpretedascompressingperhapsacenturyofdecision-making
intoasingleperiod.Sincetheimpactsofabruptchangewillunfoldmoreslowlythanthis(someoftheimpactscouldtake
amillenniumormoretoplayout),Xshouldbeinterpretedascapturingthefullconsequences,includingintothedistant
future,ofdecisionstakenthiscenturytoavoidorexceedathreshold.Theparameterbshouldbeinterpretedsimilarlyfor
gradualclimatechange.
Inthefullcooperativeoutcome,theaggregatepayoffwillbe
8>><bQN(cid:2)Pc2q2i ifQZQ
PFC¼ i ð2Þ
>>:bQN(cid:2)XN(cid:2)Pcq22i ifQoQ:
i
IfX¼0,maximizationof(2)yieldsQFC¼bN2=c.AssumeQ4bN2=c(avoidingthethresholdrequiresabatingmorethan
isoptimalforaddressingonly‘‘gradual’’climatechange).13Thentherearetwopossibilities.Eitheritwillpayallcountries
collectivelytomeetthethresholdðQFC¼QÞ,just,oritwillnotpaytomeetthethresholdðQFC¼bN2=cÞ.Uponsubstituting
9Damagesarenormallyrelatedtotemperature,temperaturetoconcentrations,andconcentrationstoanemissionsprofile.However,thereisstrong
evidencethattemperaturecanberelateddirectlytocumulativeemissions(Allenetal.[1];Zickfeldetal.[28]).Inthispaper,Itakebusinessasusual
emissionsasgiven.Myfocusisonreductionsfromthislevel:thelevelofabatement.Thegreateristhelevelofabatement,thesmallerwillbecumulative
(total)emissions,andthelowerwillbetemperatureand,therefore,damages.Emissionswillneedtobereducedtoacriticallevel,denotedherebyQ,ifa
dangerousthreshold,suchasthe21Cchangeintemperature,istobeavoided.
10SeeespeciallyBarrett[7].
11Milinskietal.[17]varytheexpectedvalueofXbyvaryingtheprobabilitythatdamageswillbe‘‘catastrophic’’giventhatcontributionsfallshortof
thethreshold.
12Inamodelwithincreasingmarginalbenefits,internationalcooperationstillachievesverylittle[3],soitisnottheconstancyofmarginalbenefits
thatmatterssomuchascontinuityinthebenefitfunction.
13Thisassumptionimpliesthatafocusonlyon‘‘gradual’’climatechangecouldcausetheworldtocrossadangerousthreshold.InthewordsofStern
[24],whowarnsofpossibletippingpoints,‘‘Itisimportanttobeclearthatthe‘‘climatepolicyramp’’advocatedbysomeeconomistsinvolvesareal
possibilityofdevastatingclimaticchanges.’’Ofcourse,itisconceivablethatthefullcooperativeabatementlevelcouldbedeterminedsolelyby‘‘gradual’’
climate change ðQobN2=cÞ. Suppose, then, that a treaty addressing only ‘‘gradual’’ climate change sustained the abatement level, Qn. Then, if
QoQnrbN2=c,considerationof‘‘catastrophic’’climatechangewouldhavenoeffectontheusualanalysis;abatementtoavoid‘‘gradual’’climatechange
wouldavoidcatastropheinthebargain.If,tothecontrary,QnoQobN2=c,considerationgivento‘‘catastrophic’’climatechangemightimprovethe
prospectsforcollectiveaction,whilestillfallingshortofsustainingthefullcooperativeabatementlevel.Inthiscase,theanalysiswouldproceedinthe
samewayasinthispaper.
Pleasecitethisarticleas:S.Barrett,Climatetreatiesandapproachingcatastrophes,JournalofEnvironmentalEconomics
andManagement(2013),http://dx.doi.org/10.1016/j.jeem.2012.12.004i
S.Barrett/JournalofEnvironmentalEconomicsandManagement](]]]])]]]–]]] 5
thesevaluesinto(2),itiseasytoshowthatitwillpaytomeetthethresholdifandonlyif
b2N2 cQ2!
XZ (cid:2) bQ(cid:2) : ð3Þ
2c 2N2
Fig.1illustratestherelationship(whichisnon-linearduetomarginalcostsincreasing).Iftheimpactofcatastrophic
climatechange,X,is‘‘small,’’catastropheisworthavoidingonlyifQ,thethreshold,is‘‘small’’(sothatthecostsofavoiding
catastrophearelow).IfQ is‘‘large,’’catastropheisworthavoidingonlyifXis‘‘large.’’
In the non-cooperative outcome, each country i will maximize (1), taking as given the abatement choices of other
countries.ThereareatmosttwosymmetricNashequilibriainpurestrategies.Inone,everycountryiplaysq ¼b=c,and
i
the threshold is exceeded. In theother, every country plays q ¼Q=N, and thethreshold is avoided, just. Suppose every
i
countryjaiplaysq ¼Q=N.Theniwillplayeitherq ¼Q=Norq ¼b=c.Uponsubstitutingthesevaluesin(1),itiseasyto
j i i
show that country i will prefer to play the former abatement level (in which case, avoiding catastrophe is a Nash
equilibrium)ratherthanthelatterifandonlyif
b2 bQ cQ2!
XZ (cid:2) (cid:2) : ð4Þ
2c N 2N2
The top part of the Fig. 2 shows the space in which (3) and (4) both hold—that is, the space in which catastrophe
avoidanceisboth(i)collectivelyoptimaland(ii)asymmetricNashequilibrium.Thebottompart,unchangedfromFig.1,
Fig.1. Optimalcatastropheavoidance.
Fig.2. Whencoordinationsufficesandcooperationisneeded.
Pleasecitethisarticleas:S.Barrett,Climatetreatiesandapproachingcatastrophes,JournalofEnvironmentalEconomics
andManagement(2013),http://dx.doi.org/10.1016/j.jeem.2012.12.004i
6 S.Barrett/JournalofEnvironmentalEconomicsandManagement](]]]])]]]–]]]
showsthecombinationsofXandQ forwhichavoidingcatastropheisinefficient.Finally,sandwichedbetweenthesetwo
spacesisamiddleareainwhich(3)holdsbut(4)doesnothold—thatis,catastropheavoidanceiscollectivelyoptimalbut
cannotbesupportedasasymmetricNashequilibrium.(Notethatthismiddleareaonlyexistsbecauseabatementreduces
‘‘gradual’’aswellas‘‘catastrophic’’climatechange.)Thefigureshowsthatcatastropheavoidancecannotbesustainedby
coordinationifXisextremelysmall,evenifQ isclosetothefullcooperativeabatementlevelforgradualclimatechange.
OnlyifXis‘‘bigenough’’willitbepossibleforcountriestocoordinatetoavoida‘‘catastrophe.’’
In comparing (3) and (4), we see that coordination can always be relied upon to avoid catastrophe when doing so
is optimal provided either N¼1 (no externality) or b¼0 (no benefit to abatement apart from avoiding catastrophe).
The former result is to be expected. The latter result is surprising.14 Intuitively, the incentives to avoid ‘‘catastrophic’’
climate change should be helped when the actions needed to do this also reduce ‘‘gradual’’ climate change. Imagine,
however,thateverycountryiplaysq ¼Q=N.Then,takingtheabatementlevelsofothercountriesasgiven,acountrythat
i
deviatesunilaterallywillsufferasmallerlosswhentheabatementbytheothercountriesreducesgradualclimatechange.
Ironically,theeffectofabatementinreducinggradualclimatechangeincreasestheincentiveforacountrytodeviatefrom
anagreementseekingtoavertcatastrophicclimatechange.
Tosummarize:
Proposition1. Wheninequality(3)doesnothold,sothattheimpactof‘‘catastrophic’’climatechangeislowrelativetothecost
ofavoidingcatastrophe,thecollectiveactionproblemisaprisoners’dilemmagameforreducing‘‘gradual’’climatechange.When
(4)holds,sothattheimpactofcatastrophicclimatechangeishighrelativetothecostofpreventingcatastrophe,thecollective
actionproblemisacoordinationgameforavoidingcatastrophe.Inbetweenthesesituations,(3)holdsbut(4)doesnothold,and
thecollectiveactionproblemisaprisoners’dilemmagameforavoidingcatastrophe.15
4. Climatetreatiesforavoidingcatastrophes
Whatroleisthereforclimatetreatiesinavertinganapproachingcatastrophe?Startfromasituationinwhichevery
country participates in a treaty requiring that each play q ¼Q=N. What conditions must hold for such a treaty to be
i
self-enforcing?
Considerfirsttheincentivesforacountrytowithdrawfromthetreaty.Uponwithdrawing,thedeviatingcountrywill
takeasgiventhebehavioroftheremainingN(cid:2)1cooperatingcountriesasthisisspecifiedintheself-enforcingagreement.
Supposethat,onceiwithdraws,theagreementinstructsthesecountriestoplayQ .Then,uponwithdrawing,iwilleither
(cid:2)i
wanttoplayq ¼Q(cid:2)Q ,toensurethatthethresholdismet,oritwillletthethresholdslipandplayq ¼b=c.Giventhat
i (cid:2)i i
theothercountriesabateQ ,iwillprefertoplaytheformerlevelratherthanthelatterif
(cid:2)i
c(cid:2) (cid:3)2 (cid:4) b(cid:5) c(cid:4)b(cid:5)2
bQ(cid:2) Q(cid:2)Q Zb Q þ (cid:2)X(cid:2) ð5Þ
2 (cid:2)i (cid:2)i c 2 c
Solvingthequadratic,theminimumvalueofQ whichensuresthat(5)ismetisgivenbyQ^ ¼Q(cid:2)b=c(cid:2)pffi2ffiffiXffiffiffi=ffiffifficffiffi.Hence,
(cid:2)i (cid:2)i
offtheequilibriumoffullparticipation,iftheN(cid:2)1cooperatingcountriesplayQ^ ,theniwillplayq ¼b=cþpffi2ffiffiXffiffiffi=ffiffifficffiffiand
(cid:2)i i
catastrophewillbeavoided.If,tothecontrary,theyplayQ(cid:2)ioQ^(cid:2)i,theniwillplayqi¼b=c,andthecatastrophicthreshold
willbecrossed.
How, then, will the N(cid:2)1 remaining signatories want to play? If they play Q(cid:2)ioQ^(cid:2)i, then they cannot do better
collectively than to play Q ¼bðN(cid:2)1Þ2=c. Knowing how the deviant country will behave, they will prefer to abate
(cid:2)i
Q ¼Q^ ratherthanQ ¼bðN(cid:2)1Þ2=cifandonlyif
(cid:2)i (cid:2)i (cid:2)i
c Q(cid:2)b=c(cid:2)pffi2ffiffiXffiffiffi=ffiffifficffiffi!2 bðN(cid:2)1Þ2 b! c(cid:4)bðN(cid:2)1Þ(cid:5)2
bQ(cid:2) Zb þ (cid:2)X(cid:2) : ð6Þ
2 N(cid:2)1 c c 2 c
If(6)holds,andifishouldwithdrawfromtheagreementwithfullparticipation,thetreatywillinstructtheotherN(cid:2)1
countries to play (not Q ¼QðN(cid:2)1Þ=N, the Nash assumption, but) Q ¼Q^ . Under these conditions, country i will not
(cid:2)i (cid:2)i (cid:2)i
wanttowithdrawif
!2 rffiffiffiffiffiffi!2
c Q c b 2X
bQ(cid:2) ZbQ(cid:2) þ : ð7Þ
2 N 2 c c
Rewriting, we see that inequality (7) requires b=cþpffi2ffiffiXffiffiffi=ffiffifficffiffiZQ=N; a country that withdraws is punished by being
movedtoabatemoreasanon-signatorythanasasignatory(theoppositeoffreeriding!).
14ForMilinskietal.’s[17]model,thereisnomiddlewedge;efficientcatastropheavoidanceisalwaysaNashequilibrium.
15Obviously,theseresultsapplyspecificallytomodel(1),whichislinear-quadratic(linearbenefitsandquadraticcosts).However,itiseasytosee
thattheseresultsalsoapplytothemodeldiscussedinSection2,whichislinear–linearwithb¼0,exceptthat,inthismodel,thereisnocollectiveaction
problemforlimitinggradualclimatechange.ThepaperbyBarrettandDannenberg[10],writtenafterthispaperwassubmitted,showsthatProposition1
alsoholdsforamodelwithlinearbenefits(withb40)andstepwise-linearmarginalcosts(theirmodelwasdevelopedforthepurposeofconductingan
experimentaltestofthetheorypresentedinthispaper);compareFig.1intheirpaperwithFig.2inthisone.
Pleasecitethisarticleas:S.Barrett,Climatetreatiesandapproachingcatastrophes,JournalofEnvironmentalEconomics
andManagement(2013),http://dx.doi.org/10.1016/j.jeem.2012.12.004i
S.Barrett/JournalofEnvironmentalEconomicsandManagement](]]]])]]]–]]] 7
If(6)and(7)hold,nocountrycangainbywithdrawingunilaterallyfromtheagreementthatavoidscatastrophe.Itis
easytoshow(seetheAppendix)that,if(7)holds,thensowill(6)hold.Moreover,inequality(7)reducesto(4),whichisthe
conditionforwhencatastropheavoidancecanbesustainedbycoordination.Hence,wehave:
Proposition 2. When (4) holds, so that the impact of catastrophic climate change is high relative to the cost of preventing
catastrophe, a self-enforcing international environmental agreement can sustain the full cooperative outcome, ensuring that
catastropheisavoided.Undertheseconditions,thetreatyservesasacoordinatingdevice.
Thisanalysisassumesfullparticipation.Itiswellknownthat,inmostcases,treatiescanonlysupportalowerlevelof
participation.So,mightcollectiveactiontoavoidcatastrophebesustainedevenif(7)failstohold?
Even if (7) were violated, inequality (6) could still hold. That is, the N(cid:2)1 signatories could have an incentive to play
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi
Q(cid:2)b=c(cid:2) 2X=c, knowing that the sole non-signatory would then abate b=cþ 2X=coQ=N (in which case the non-
signatory would be a true free rider). Moreover, this agreement comprising N(cid:2)1 countries would be self-enforcing,
provideditcouldbesustainedbycoordination.However,itiseasytoshowthattheconditionsthatenableN(cid:2)1countriesto
coordinate to avoid catastrophe also enable N countries to coordinate to sustain the full cooperative outcome (see the
Appendix).Moreover,byextendingthislogictootherparticipationlevels,itisclearthat,if(7)failstohold,itwillnotbe
possibletosustaincoordinationtoavoidcatastrophebyanynumberofcountries(rN).
Ofcourse,when(7)holds,itcouldbepossibleforfewerthanNcountriestocoordinatesoastoavoidcatastrophe(thisisjust
anotherwayofsayingthattheremayexistalargenumberofasymmetricNashequilibriaforavoidingcatastrophe).However,
coordinationbyfewerthanNcountries(withtheothersactingasfreeriders)offersnoadvantageovercoordinationbythe
grandcoalition—onlyadisadvantage,sinceitismorecostlyforN(cid:2)1countriestosustainthethresholdthanforNcountriesto
doso(withmarginalcostsincreasing,cost-effectiveabatementrequiresthateverycountryabate thesamequantity).Since
everycountry’sexpectedpayoff(withtheexpectationtakenbeforeanagreementisadopted,whennocountryknowsifitwill
beasignatoryoranon-signatory)isgreaterforanagreementcomprisingNcountriesthanonecomprisingN(cid:2)1 (orfewer)
countries,providedbothagreementsareself-enforcing,inthenegotiationstageallcountrieswillfavorauniversalagreement.
Moreover,oncesuchanagreementhasenteredintoforce,if(7)holds,nocountrywillhaveanincentivetowithdraw.
Together,theseresultsimply:
Proposition3. Ifthereexistsaself-enforcinginternationalenvironmentalagreementthatavoidscatastrophebycoordinating
countries’abatement,thenthisagreementwillbeuniqueandsustainfullparticipation.
Whathappensifavoidingcatastropheisefficientandyetcoordinationfails?Thiswouldputusinthemiddleterritoryof
Fig.2.Countriescouldstilltrytocooperatetoavoidcatastropheinthisregion,byusingstrategiestoovercometheprisoners’
dilemma,asintheusualmodelofaninternationalenvironmentalagreement.Inthiscase,signatorieswouldreducetheir
abatementinthebeliefthat,bydoingso,thethresholdwouldbeblown.However,thecooperationproblemisnowdifferent
from the usual one, which assumes only ‘‘gradual’’ climate change. For model (1), a deviation triggers a harsher
punishment—the‘‘catastrophic’’lossduetocrossingthethreshold.Itmay,therefore,sustaingreatercooperation.16
ForittobecollectivelyrationalfortheN(cid:2)1signatoriestoplayQ ¼bðN(cid:2)1Þ2=cafterihaswithdrawn,inequality(6)
(cid:2)i
mustnothold.Knowingthis,countryiwillnotwanttowithdrawif
b2½2ðN(cid:2)1Þ2þ1(cid:3) cQ2!
XZ (cid:2) bQ(cid:2) : ð8Þ
2c 2N2
Condition(8)isthesameas(3)exceptfortheintercept.Itisstraightforwardtoshowthattheintercepttermin(8)isno
greaterthantheintercepttermin(3)forNr3.Thatis,whencoordinationfails,atreatycanstillbereliedupontosustain
the full cooperative outcome—meaning, avoid catastrophe—if Nr3. It is well known that, with the functional forms
assumedin(1),aself-enforcinginternationalenvironmentalagreementcansustainthefullcooperativeoutcomeevenin
theabsenceofcatastropheforNr3.Considerationofcatastrophethusdoesnothingtochangethisresult.
ForN43,however,matterscanbedifferent.Cooperationmayfailentirelytoavoidcatastrophe,oritmaysucceedin
avoidingcatastrophebutonlywithinaportionofthemiddlewedgeshowninFig.2.17Inthisregion,inequality(8)holds
but(6)doesnothold.
Inequality (6) is awkward to work with, but an example can illustrate where and why cooperation can be effective
within this middlewedge. Suppose N¼20, b¼c¼1, and Q¼600.18 Then the vertical distance of thecooperation wedge
shown in Fig. 2, as defined by inequalities (3) and (4), requires 420:54XZ50 (coordination succeeds for XZ420:5;
16‘‘Punishment’’appearsintheone-shotmodelofaninternationalenvironmentalagreementwhenthetreatyspecifiesthatcooperatingcountries
willlowertheirper-countryabatementasthenumberofparticipatingcountriesfalls.Wenormallythinkof‘‘punishment’’inthecontextofarepeated
game.Butthestagegamemodelofaninternationalenvironmentalagreementisparticularlysuitedtoexploringtheenforcementofparticipation,
whereas the repeated game model is better suited to exploring the enforcement of compliance. I have shown elsewhere that there is a natural
correspondencebetweenthesetwoapproaches[5–7].
17Usingotherfunctionalforms,self-enforcingagreementscansustainfullcooperationforN43withoutconsideringthepossibilityofcatastrophe.
Theseincludethequadratic-quadratic[3]andlinear-linear[6]models.However,inbothofthesecases,aself-enforcingagreementcansustainthefull
cooperativeoutcomeonlywhenthisoutcomeisnearlyidenticaltothenon-cooperativeoutcome.
18YoucanthinkofthisassumptionassuggestingthatcollectiveactionbytheG-20groupofcountriescouldpreventcatastrophicclimatechange.
Pleasecitethisarticleas:S.Barrett,Climatetreatiesandapproachingcatastrophes,JournalofEnvironmentalEconomics
andManagement(2013),http://dx.doi.org/10.1016/j.jeem.2012.12.004i
8 S.Barrett/JournalofEnvironmentalEconomicsandManagement](]]]])]]]–]]]
catastropheavoidanceisinefficientforXo50).Condition(8)requiresXZ211:5;theimpactofcatastrophemustbelarge
enoughtodeterwithdrawal.Ofcourse,XmustalsobesmallenoughthattheN–1countriesremainingintheagreement
are willing to cross the threshold should i decline to cooperate. Upon substituting we find that (6) will not hold if
p
pffi7ffiffi2ffiffiffi2ffiffiffiðffiXffiffiffiffiþffiffiffiffi4ffiffiffi1ffiffiffi8ffiffiffi:ffi5ffiffiffiÞffio599(cid:2) ffi2ffiffiXffiffiffi,anditcanbeshownthatthereisnovalueforXwithintherelevantrangeð420:54XZ211:5Þ
for which reversal of inequality (6) is satisfied. For these parameter values, cooperation to avoid catastrophe fails
completely. Cooperation will have a better chance of avoiding catastrophe if Q is bigger. Raising the threshold helps
because it makes credible the threat by the N(cid:2)1 remaining signatories to lower their abatement dramatically should
country i withdraw. For example, if the threshold is increased to 4000, the vertical range of the cooperation wedge
becomes19,800:54XZ16,200(coordinationsucceedsforXZ19,800.5;avoidingcatastropheisinefficientforXo16,200),
and cooperation can succeed in avoiding catastrophe—but only over a small portion of this range ð16,371:5ZXZ
16,361:5Þ.
ItcanalsobeshownthatifNcountriescannotsustaincooperationtoavoidcatastrophe,thenneithercanfewerthanN
countries do so.19 Of course, in the event that cooperation to avoid catastrophe fails, a self-enforcing international
environmentalagreementcomprisingthreecountries(forNZ3)couldstillsucceedinreducing‘‘gradual’’climatechange.
However, such an agreement would reduce emissions just a little relative to the non-cooperative outcome; as is well
known,eachofthethreesignatorieswouldabateq ¼3b=candeachnon-signatoryjustb=c.Cooperationtolimit‘‘gradual’’
s
climatechangewouldhaveonlyamodestimpactonemissions.
Tosumup:
Proposition4. When(3)holdsbut(4)doesnothold,cooperationmaysucceedinavertingcatastrophe,butonlyunderspecial
circumstances. Moreover, even when these special circumstances apply, cooperation improves welfare relatively little.20
Coordination,bycontrast,succeedsspectacularlywhenthenetbenefitsofavoidingcatastropheareparticularlylarge.
Itshouldnotsurpriseusthatcoordinationmakesallthedifference.TheMontrealProtocolsucceededpartlybecausethe
strategicapplicationoftraderestrictionstransformedprotectionoftheozonelayer,ordinarilyaprisoners’dilemma,intoa
coordination game [4,6]. Network externalities can play the same role in a treaty that strategically targets technology
standardsratherthanemissionlimits[6,8].Inbothofthesecases,aself-enforcingtreatysustainscoordinationwithfull
participation.21Thenoveltyinthepresentpaperisthatheretheopportunityforcoordinationisnotstrategicbutrathera
giftfromMotherNature.
Having established the preeminent role of coordination in averting catastrophe under conditions of certainty, in the
remainderofthispaperIshalllimitmyattentiontotheconditionsunderwhichcoordinationcanreduce,ifnoteliminate,
theprobabilityofcrossingacatastrophicthresholdunderconditionsofuncertainty.
5. Impactandthresholduncertainty
[27]emphasizesfat-taileduncertaintyinclimatesensitivity,compoundedbyuncertaintiesintranslatingtemperature
changes into welfare changes. Here I distinguish between uncertainty in the impact of catastrophe (X) and uncertainty
(cid:2) (cid:3)
aboutthethresholdthattriggerscatastrophe Q .
Thetwouncertaintiesarebothverysubstantial,butalsoverydifferent.Onewaytounderstandthenatureofthreshold
uncertaintyistoconsidertherecentattention-gettingpaperbyRockstro¨metal.[19].Theyidentifya‘‘planetaryboundary’’
intermsofatmosphericCO concentrationsof350ppmbyvolume(ppmv)soas‘‘toensurethecontinuedexistenceofthe
2
largepolaricesheets,’’whilenotingthatthepaleoclimaticrecordimplies‘‘thatthereisacriticalthresholdbetween350
and 550ppmv.’’ Uncertainty about this threshold is purely scientific (and much of it is substantially irreducible).
Uncertaintiesabouttheimpactsassociatedwiththelossofpolarice,bycontrast,arebothscientificandeconomic.Lossof
theGreenlandIceSheet,forexample,wouldlikelycausesealeveltorisebetween2and7moveraperiodof300toover
1000years(Lenton[16]).ThevalueofXwillobviouslydependontheextentofsealevelriseanditsrateofchange,butit
willalsodependonthevaluesattachedtothesechanges,includingdiscounting.Muchhasbeenmadeintheeconomics
literatureoftheimportanceofthesevaluesandtheiruncertainties,butIshallnowshowthat,asregardstheprospectsfor
collectiveaction,uncertaintyabouttheimpactofcrossingathresholdisrelativelyunimportant.Itisscientificuncertainty
aboutthethresholdthatreallymatters.
Considerfirstuncertaintyabouttheimpactofcatastrophe,X.Itisstraightforwardtodemonstratethatalltheresults
shownthusfarcarrythroughifwesubstitutetheexpectedvalueofX,E(X),forX(this,again,isassumingthatcountriesare
19Forexample,itwouldonlypayN(cid:2)1countriestocooperatesoastoensurethatthethresholdwasavoidedwhentheothercountry,thefreerider,
reducedemissionsbyb=c,ifbQ(cid:2)ðc=2ÞððQ(cid:2)b=cÞ=ðN(cid:2)1ÞÞ2ZbðbðN(cid:2)1Þ2=cþb=cÞ(cid:2)X(cid:2)ðc=2ÞðbðN(cid:2)1Þ=cÞ2:Butifthisconditionapplies,then(6)surelyholds,
whichmeansweareinacoordinationsituation.
20RecallthatthewedgeforcooperationinFig.2borderstheregioninwhichcatastropheavoidanceisinefficient.
21InBarrett[4,6],traderestrictionssustainafirstbest.InBarrett[6,8],technologystandardsmaysustainonlyasecondbest.Inthepresentpaper,
theprospectofcatastrophemaysustainafirstbest;itwouldsustainasecondbestonlyundertheconditionsdescribedtowardstheendoffootnote13.
Withthresholduncertainty,asexplainedinthenextsection,acoordinatingtreatycanonlysustainasecondbest,exceptforaspecialcasediscussedin
Section6.
Pleasecitethisarticleas:S.Barrett,Climatetreatiesandapproachingcatastrophes,JournalofEnvironmentalEconomics
andManagement(2013),http://dx.doi.org/10.1016/j.jeem.2012.12.004i
S.Barrett/JournalofEnvironmentalEconomicsandManagement](]]]])]]]–]]] 9
Fig.3. (a)Prospectofcatastrophehaslittleeffectonthefullcooperativeoutcome.(b)Prospectofcatastrophehasprofoundeffectonthefullcooperative
outcome.(c)Prospectofcatastrophewith‘‘thin’’and‘‘fat’’tails.
riskneutral).22Uncertaintyaboutthemagnitudeofcatastrophicdamagesdoesnotfundamentallyalterthenatureofthe
cooperation challenge. So long as the catastrophic threshold is certain, and E(X) is large, countries will be able to
coordinatesoastoavertcatastrophe.23
Consider now uncertainty about the threshold, Q. In particular, let Q be a random variable with a continuous
(cid:2) (cid:3)
cumulativeprobabilitydistributionFðQÞ¼Pr QrQ .Werecountriestocooperatefully,theywouldnowmaximize
E(cid:7)Pc(cid:8)¼bQN(cid:2)Xcq2i (cid:2)XN½1(cid:2)FðQÞ(cid:3), ð9Þ
2
i
whichrequires
bN(cid:2)cqþXNfðQÞ¼0, ð10Þ
i
wherefðQÞistheprobabilitydensityfunction(pdf).Notethat,ingeneral,Eq.(10)willnotbesufficientforamaximum.
Theeffectofuncertaincatastrophedependsonthefunctionf.Ifthepdfhasinfinitesupports,fwillalwaysbepositive,
andtheprospectofcatastrophewillcommendgreaterabatement,comparedtoasituationinwhichcatastropheisignored.
Recallthatwhencatastropheiscertain,abatementinthefullcooperativeoutcomemayormaynotbeaffected.
Fig. 3a illustrates Eq. (10) for a plausible pdf (fðQÞ).24 In the figure, the bell curve placed on top of the aggregate
marginal benefit of avoiding ‘‘gradual’’ climate change represents XNfðQÞ, the expected aggregate marginal benefit of
catastrophe avoidance. For the pdf shown in this figure, concern about catastrophic climate change increases the full
cooperativeabatementlevelveryslightlyabovebN2=c,theoptimalabatementlevelforgradualclimatechangeignoring
catastrophic climate change. If abatement is increased beyond QFC, net benefits fall. They later increase before falling
again,butsolongasareaAexceedsareaB,asitdoesinFig.3a,thenetbenefitsofincreasingabatementbeyondQFCwillbe
negative,anditwillbeoptimaltolimitaggregateabatementtoQFC.
Fig.3blookssimilarto3a,butinthiscaseareaBexceedsareaA,andtheoptimalaggregateabatementlevelincreases
dramatically,relativetobN2=c,duetotheuncertainprospectofcatastrophe.Obviously,whatmattersinthesefiguresis
thesizeofareaArelativetoB.Theserelativevaluesdependinturnonthevaluesofb,X,andcaswellasthepdfforthe
threshold.
Does the possibility of ‘‘fat tails’’ matter? As shown in Fig. 3c, a thickening of the tails of the pdf changes the full
cooperative outcome very little, with the full cooperative abatement level increasing only slightly, from QFC to QFC.25
1 2
Weitzman,ofcourse,getsaverydifferentresult,butasnotedintheintroduction,heassumesthatsocialpreferencesare
stronglyriskaverse.
Thisbriefdiscussionsummarizesjustafewofmanypossibleconstructions.Myaimhereisnottobecomprehensive,
but to illustrate how sensitive or insensitive policy recommendations can be to small changes in the constituent parts
ofEq.(10).
22IamalsoassumingthatE(X)exists.Forsomefat-taileddistributions,expectedvaluedoesnotexist.
23ThisresultshouldhavebeenanticipatedfrommypreviousdiscussionoftheMilinskietal.[17]experiment.Recallthat,inthisexperiment,the
valueofX(andnotQ)wasuncertain.AnexampleofasituationinwhichthethresholdwascertainbuttheimpactswereuncertainistheMillenniumBug
orY2Kproblem.Ofcourse,thisturnedoutnottobethecatastropheexpected,thoughweshallneverknowwhetherthisisbecausetheactiontakento
avertacrisissucceededorearlierfearsofcalamitywereoverblown.Thetheorypresentedhereisconsistentwithbothinterpretations.
24Allthepdfsdrawninthispaperaresymmetric,butmydiscussionandanalysisdoesnotdependonthisassumption.Therearereasonstobelieve
thepdfforclimatesensitivityexhibitspositiveskewness(RoeandBaker[20]).
25Thediagramcanbethoughtofascomparingthenormalandt-distributions,theformerbeingthin-andthelatterthick-tailed.
Pleasecitethisarticleas:S.Barrett,Climatetreatiesandapproachingcatastrophes,JournalofEnvironmentalEconomics
andManagement(2013),http://dx.doi.org/10.1016/j.jeem.2012.12.004i
10 S.Barrett/JournalofEnvironmentalEconomicsandManagement](]]]])]]]–]]]
Tosumup,wehave:
Proposition5. Uncertaintyabouttheimpactof‘‘catastrophic’’climatechangehasnoeffectonthefullcooperativeabatement
level.Bycontrast,uncertaintyaboutthethresholdmaycauseabatementinthefullcooperativeoutcometoincreasealotrelative
tothefullcooperativeoutcomeforlimiting‘‘gradual’’climatechange.
Ifcountriesfailtocooperate,eachcountryiwillchooseq tomaximize
i
cq2
EðpÞ¼bQ(cid:2) i (cid:2)X½1(cid:2)FðQÞ(cid:3): ð11Þ
i 2
Thesolutionrequires
b(cid:2)cqþXfðQÞ¼08i: ð12Þ
i
Whatdeterminesthegap betweenthefullcooperative and non-cooperativeoutcomes?Lookfirst atFig. 4a. The top
portion of the figure shows the full cooperative outcome. Though positioned somewhat differently, this portion of the
figure is analogous to the situations depicted in Fig. 3a–c. The bottom portion of the figure shows the non-cooperative
outcome.Asdrawn,Fig.4ashowsthattheprospectofuncertaincatastrophehasasubstantialeffectonthefullcooperative
outcome(raisingitfrombN2=c toQFC), butvirtuallynoeffectonthenon-cooperativeabatementlevel(which,ignoring
catastrophe,isbN=c,andwhichincreasesjustalittletoQNC,assumingthatthepdfhasaninfinitesupportontheleftside).
Uncertaintyaboutthethresholdrobsusofanychanceforcoordinationonthefullcooperativelevelofabatement.Indeed,
comparedtothecertaintycase(assumingthatthecertainthresholdisequaltoitsexpectedvalueinFig.4a),uncertainty
aboutthethresholdmakescountrieswanttoabatemore,andyetcausesthemtoabateless.
Fig. 4b illustrates a situation in which the prospect of uncertain catastrophe increases abatement in both the full
cooperativeandnon-cooperativeoutcomes(again,relativetoasituationinwhichtherewasonlygradualclimatechange),
while at the same time narrowing the gap between the non-cooperative and full cooperative outcomes (ignoring
catastrophe, this gap is bN2=c(cid:2)bN=c; taking catastrophe into account, it is QFC(cid:2)QNC). In this case, even though the
thresholdisuncertain,theprospectofcatastrophecreatesanopportunityforcoordinationonthemutuallypreferred(but
stillinefficient)Nashequilibrium.Itisobviousfromthefigure,however,thatthecircumstancesthatsupportthismore
cheerfulsituationarenotnecessarilytobeexpected.Theyincludenotonlyahighimpact,X,butalsoaverylowvariancein
thepdfandasmallN—circumstancesthatarenotfavoredbyclimatechange.
Again,myaimherehasnotbeentobecomprehensive,buttoindicatewhencollectiveactionislikelytosucceedorfail.
Themainresultsmaybestatedsimply:
Proposition6. Comparedtothecertaintycase,uncertaintyabouttheimpactofcatastrophehasnoeffectoncollectiveaction.
By contrast, uncertainty about the threshold is critical, being very likely to transform the collective action problem from a
coordinationgameunderthresholdcertaintytoaprisoners’dilemmaunderthresholduncertainty.
Fig.4. (a)Uncertainprospectofcatastrophewidensthegapbetweenthenon-cooperativeandfullcooperativeoutcomes.(b)Uncertainprospectof
catastrophenarrowsthegapbetweenthenon-cooperativeandfullcooperativeoutcomes.
Pleasecitethisarticleas:S.Barrett,Climatetreatiesandapproachingcatastrophes,JournalofEnvironmentalEconomics
andManagement(2013),http://dx.doi.org/10.1016/j.jeem.2012.12.004i