\`Istherenoway,'saidI,`ofescapingCharybdis,andatthesametimekeepingScyl - PDF document

\`Istherenoway,'saidI,`ofescapingCharybdis,andatthesametimekeepingScyllaowhensheistryingtoharmmymen?'\`Youdare-devil,'repliedthegoddess,`youarealwayswantingtoghtsomebodyorsomething;youwillnotletyourselfbebeatenevenbytheimmortals.'"Homer,Odyssey,BookXII,trans.SamuelButler.Likeanygoodsailor,Odysseussoughttoavoideverypotentialcatastrophethatmightharmhimandhiscrew.But,asthegoddessCircemadeclear,althoughhecouldavoidthesix-headedseamonsterScyllaorthe\suckingwhirlpool"ofCharybdis,hecouldnotavoidboth.CirceexplainedthatthegreatestexpectedlosswouldcomefromanencounterwithCharybdis,whichshouldthereforebeavoided,evenatthecostofanencounterwithScylla.Wemodernmortalslikewisefacemyriadpotentialcatastrophes,somemoredauntingthanthosefacedbyOdysseus.Nuclearorbio-terrorism,anuncontrolledviralepidemiconthescaleofthe1918Spanish u,oraclimatechangecatastropheareexamples.Naturally,wewouldliketoavoidallsuchcatastrophes.Butevenifitwerefeasible,isthatgoaladvisable?Shouldweinsteadavoidsomecatastrophesandaccepttheinevitabilityofothers?Ifso,whichonesshouldweavoid?UnlikeOdysseus,wecannotturntothegodsforadvice.Wemustturninsteadtoeconomics,thetrulydismalscience.Thosereadershopingthateconomicswillprovidesimpleadvice,suchas\avertacatas-tropheifthebenetsofdoingsoexceedthecost,"willbedisappointed.Wewillseethatdecidingwhichcatastrophestoavertisamuchmoredicultproblemthanitmightrstappear,andasimplecost-benetruledoesn'twork.Suppose,forexample,thatsocietyfacesvemajorpotentialcatastrophes.Ifthebenetofavertingeachoneexceedsthecost,straightforwardcost-benetanalysiswouldsayweshouldavertallve.1Weshow,however,thatitmaybeoptimaltoavertonly(say)threeoftheve,andnotnecessarilythethreewiththehighestbenet/costratios.Thisresultmightatrstseem\strange"(hencethetitleofthepaper),butwewillseethatitfollowsfrombasiceconomicprinciples.Ourresultshighlightafundamental awinthewayeconomistsusuallyapproachpotentialcatastrophes.Considerthepossibilityofaclimatechangecatastrophe|aclimateoutcomesosevereintermsofhighertemperaturesandrisingsealevelsthatitwouldsharplyreduceeconomicoutputandconsumption(broadlyunderstood).Anumberofstudieshavetriedtoevaluategreenhousegas(GHG)abatementpoliciesbycombiningGHGabatementcostestimateswithestimatesoftheexpectedbenetstosociety(intermsofreducedfuture 1Althoughwewilloftentalkof`averting'or`eliminating'catastrophes,ourframeworkallowsforthepossibilityofonlypartiallyalleviatingoneormorecatastrophes,asweshowinSection4.1.1 thatwealsoneedtoknowthecost.Therearevariouswaytocharacterizesuchacost:axeddollaramount,atime-varyingstreamofexpenditures,etc.InordertomakecomparisonswiththeWTPmeasureofbenets,weexpresscostasapermanenttaxonconsumptionatrate,therevenuesfromwhichwouldjustsucetopayforwhateverisrequiredtoavertthecatastrophe.Nowsupposeweknow,foreachmajortypeofcatastrophe,thecorrespondingcostsandbenets.Moreprecisely,imaginewearegivenalist(1;w1),(2;w2),...,(N;wN)ofcosts(i)andWTPs(wi)associatedwithprojectstoeliminateNdierentpotentialcatastrophes.Thatbringsustooursecondquestion:WhichoftheNprojectsshouldweimplement?Ifwi�iforalli,shouldweeliminateallNpotentialcatastrophes?Notnecessarily.Weshowhowtodecidewhichprojectstochoosetomaximizesocialwelfare.Whentheprojectsareverysmallrelativetotheeconomy,andiftherearenottoomanyofthem,theconventionalcost-benetintuitionprevails:iftheprojectsarenotmutuallyexclusive,weshouldimplementanyprojectwhosebenetwiexceedsitscosti.Thisintu-itionmightapply,forexample,fortheconstructionofadamtoavert oodinginsomearea.Thingsaremoreinterestingwhenprojectsarelargerelativetotheeconomy,asmightbethecasefortheglobalcatastrophesmentionedabove,oriftheyaresmallbutlargeinnumber(sotheiraggregatein uenceislarge).Largeprojectschangetotalconsumptionandmarginalutility,causingtheusualintuitiontobreakdown:Thereisanessentialinterdependenceamongtheprojectsthatmustbetakenintoaccountwhenformulatingpolicy.Wearenotthersttonotetheinterdependenceoflargeprojects;earlyexpositionsofthispointincludeDasgupta,SenandMarglin(1972)andLittleandMirrlees(1974).(Morerecently,DietzandHepburn(2013)illustratethispointinthecontextofclimatechangepolicy.)Norarewethersttonotetheeectsofbackgroundrisk;see,e.g.,Gollier(2001)andGollierandPratt(1996).Buttoourknowledgethispaperisthersttoaddressthequestionofselectingamongasetoflargeprojects.Weshowhowthiscanbedone,andweuseseveralexamplestoillustratesomeofthecounterintuitiveresultsthatcanarise.Forinstance,oneapparentlysensibleresponsetothenon-marginalnatureoflargecatas-trophesistodecidewhichisthemostseriouscatastrophe,avertthat,andthendecidewhethertoavertothercatastrophes.Thisapproachisintuitiveandplausible|andwrong.Weillustratethisinanexamplewiththreepotentialcatastrophes.Thersthasabenetw1muchgreaterthanthecost1,andtheothertwohavebenetsgreaterthanthecosts,butnotthatmuchgreater.Naivereasoningsuggestsweshouldproceedsequentially:eliminatetherstcatastropheandthendecidewhethertoeliminatetheothertwo,butweshowthatsuchreasoningis awed.Ifonlyoneofthethreeweretobeeliminated,weshouldindeed3 Thecontributionofthispaperislargelytheoretical:weprovideaframeworkforanalyzingdierenttypesofcatastrophesanddecidingwhichonesshouldbeincludedasatargetofgovernmentpolicy.Determiningtheactuallikelihoodofnuclearterrorismoramega-virus,aswellasthecostofreducingthelikelihood,isnoeasymatter.Nonetheless,wewanttoshowhowourframeworkmightbeappliedtoreal-worldgovernmentpolicyformulation.Tothatend,wesurveythe(verylimited)literatureforsevenpotentialcatastrophes,discusshowonecouldcomeupwiththerelevantnumbers,andthenuseourframeworktodeterminewhichofthesecatastrophesshouldorshouldnotbeaverted.1TwoSimpleExamplesWhyisitthat\large,"i.e.,non-marginalprojectsareinherentlyinterdependentandcannotbeevaluatedinisolation?Thefollowingsimpleexamplesshouldhelpconveysomeofthebasicintuition,andalsoclarifytheconnectionbetweenourworkandthepriorliterature.Therstexampleaddressesa(static)decisiontoundertakeasetofprojects,andshowshowthedecisionrulechangesiftheprojectsarelarge.Thesecondexampleaskswhetherresourcesshouldbesacricedtodaytoavertoneortwocatastrophesthatwillotherwiseoccurinthefuture.Itillustratestheeectofbackgroundrisk,theinterdependenceofWTPs,andtheconnectiontocost-benetanalysis.StaticExample.Supposewearedecidingwhethertoundertaketwoindependentprojects.5Tomakethebasicpointinthesimplestpossiblecase,weassumethattheseareyes/noprojects,sothattheresourcesexpendedonprojecti,ei,equalseither0orxi.Wecanapproximatenetwelfare,W,usingasecond-orderTaylorexpansion:W(e1;e2)W(0;0)+2Xi=1ei@W @eie1=e2=0+1 22Xi=12Xj=1eiej@2W @ei@eje1=e2=0:(1)Ifbothprojectsare\marginal,"i.e.thexiareverysmall,thenwecanignorethesecond-ordertermin(1),andtheoptimaldecisionistosetei=xiif@W=@eije1=e2=0�0andei=0otherwise.Inotherwords,thestandardcost-benetruleapplies:undertakeaprojectifdoingsoyieldsanincreaseinnetwelfare.Butiftheprojectsarenotmarginal,thenwecannotignorethesecond-ordertermin(1).Nowthestandardcost-benetrulefails.Why?Becauseofthesecondderivativeterms,thevalueofproject1dependsonwhether 5Aversionofthisexamplewassuggestedbyananonymousreferee,whomwethank.5 2.WTPsdon'tadd.Specically,w1;2w1+w2.Forexample,if=:5,w1;2=:60w1+w2=:80.Sacricing40%ofconsumptionsharplyincreasesthemarginalutilitylossfromanyfurthersacriceofconsumption.3.Naivecost-benetanalysiscanbemisleading.Morespecically,wemightnotavertacatastropheevenifthebenetofavertingit|consideredinisolation|exceedsthecost.Forexample,suppose=:5asbefore,sothatw1=w2=:4.If1=2=:35,thebenetofavertingeachcatastropheexceedsthecost.Butweshouldnotavertboth.Forifweavertneithercatastrophe,netwelfareisV0=�5;ifweavertone,netwelfareisW1=�4:62;andifweavertboth,netwelfareisW1;2=�4:73.Avertingbothisbetterthanavertingneither,butwedobestbyavertingexactlyone.Tounderstandthis,notethatifweavertonecatastrophe,whatmattersiswhethertheadditionalbenetfromavertingthesecondexceedsthecost,i.e.,whether(w1;2�w1)=(1�w1)&#x]TJ ;&#x-429;&#x.630; -1;.93;' T; [00;2.Weshouldnotavert#2because(w1;2�w1)=(1�w1)=:332=:35.Theseexampleshelpconnectourworktotheearlierliteratureandillustratewhylargeprojectsareinterdependent.Weturnnexttoafullydynamicmodelthatincludesuncertaintyoverthearrivalandimpactofmultiplepotentialcatastrophes,andthatletsusderiveakeyresultregardingthesetofcatastrophesthatshouldbeaverted.2TheModelwithOneTypeofCatastropheWerstconsiderasingletypeofcatastrophe.Itmightbeaclimatechangecatastrophe,amega-virus,orsomethingelse.Whatmattersisthatweassumefornowthatthisparticulartypeofcatastropheistheonlythingsocietyisconcernedabout.Wewanttodeterminesociety'sWTPtoavoidthistypeofcatastrophe,i.e.,themaximumfractionofconsumption,nowandthroughoutthefuture,thatsocietywouldsacrice.OfcourseitmightbethecasethattherevenuestreamcorrespondingtothisWTPisinsucienttoeliminatetheriskofthecatastropheoccurring,inwhichcaseeliminatingtheriskiseconomicallyinfeasible.Or,thecostofeliminatingtheriskmightbelowerthanthecorrespondingrevenuestream,inwhichcasetheprojectwouldhaveapositivenetsocialsurplus.TheWTPappliesonlytothedemandsideofgovernmentpolicy.Later,whenweexaminemultipletypesofcatastrophes,wewillalsoconsiderthesupply(i.e.,cost)side.TocalculateaWTP,wemustconsiderwhetherthetypeofcatastropheatissuecanoccuronceandonlyonce(ifitoccursatall),orcanoccurrepeatedly.Foraclimatecatastrophe,itmightbereasonabletoassumethatitwouldoccuronlyonce|theglobalmeantemperature,7 periodCGFlinearlyint.WeshowintheappendixthattheCGFisthen9()=g+�Ee�1�1
:(6)Giventhisconsumptionprocess,welfareisEZ101 1�e�tC1�tdt=1 1�Z10e�te(1�)tdt=1 1�1 �(1�);(7)where(1�)istheCGFofequation(6)with=1�.Notethatequation(7)isquitegeneralandappliestoanydistributionfortheimpact.Butnotealsothatwelfareisniteonlyiftheintegralsconverge,andforthisweneed�(1�)�0(Martin(2013)).Eliminatingthecatastropheisequivalenttosetting=0inequation(6).WedenotetheCGFinthiscaseby(1)().(Thisnotationwillproveconvenientlaterwhenweallowforseveraltypesofcatastrophes.)Soifwesacriceafractionwofconsumptiontoavoidthecatastrophe,welfareis(1�w)1� 1�1 �(1)(1�):(8)TheWTPtoeliminatetheevent(i.e.,set=0)isthevalueofwthatequates(7)and(8):1 1�1 �(1�)=(1�w)1� 1�1 �(1)(1�):Shouldsocietyavoidthiscatastrophe?Thisiseasytoanswerbecausewithonlyonetypeofcatastrophetoworryabout,wecanapplystandardcost-benetanalysis.Thebenetisw,andthecostisthepermanenttaxonconsumption,,neededtogeneratetherevenuetoeliminatetherisk.Weshouldavoidthecatastropheaslongasw�.Aswewillseeshortly,whentherearemultiplepotentialcatastrophesthebenetsfromeliminatingeachareinterdependent,causingthissimplelogictobreakdown.10 9Wecouldallowforct=gt�PN(t)n=1n,wheregtisanyLevyprocess,subjecttotheconditionthatensuresnitenessofexpectedutility.(Forthespecialcasein(5),gt=gtforaconstantg.)ThisonlyrequiresthatthetermgintheCGFsisreplacedbyg(),whereg()istheCGFofg1,soifthereareBrownianshockswithvolatility,andjumpswitharrivalrate!andstochasticimpactJ,theng()=+1 222+!�EeJ�1
.ThisletsushandleBrownianshocksandunavoidablecatastropheswithoutmodifyingtheframework.Sincethegeneralizationhasnoeectonanyofourqualitativeresults,westicktothesimplerformulation.10Arefereesuggestedthatwecouldhavealternativelyexpressedbenetsintermsofthegrowthrateofconsumption,ratherthanasapercentageofitslevel.ThenWTPwouldbethemaximumreductioninthegrowthratesocietywouldbewillingtosacricetoavertacatastrophe.Expressingbenetsthiswayiscertainlyreasonable.Ifthecostsofavertingcatastrophesarelikewisemodeledasrequiredreductionsinthegrowthrate(whichwethinkismuchlessreasonable),ourdynamicmodelcouldbewritteninastaticform.Modelingbenetsandcostsintermsoflevelsistheconventionalapproach,whichwehavechosentomaintain.9 Result1.TheWTPtoavertasubset,S,ofthecatastrophesislinkedtotheWTPstoaverteachindividualcatastropheinthesubsetbytheexpression(1�wS)1��1=Xi2S(1�wi)1��1:(12)Proof.Theresultfollowsfromarelationshipbetween(S)()andtheindividual(i)().Notethat(i)()=()�i�Ee�i�1
and(S)()=()�Pi2Si�Ee�i�1
.(Thisiseectivelythedenitionofthenotation(i)and(S).)ThusXi2S(i)()=jSj()�Xi2Si�Ee�i�1
=(jSj�1)()+(S)();wherejSjdenotesthenumberofcatastrophesinthesubsetS,andhenceXi2S�(i)(1�) �(1�)=(jSj�1)(�(1�))+(�(S)(1�)) �(1�):Using(11),wehavetheresult. If,say,thereareN=2typesofcatastrophes,thenResult1impliesthat1+(1�w1;2)1�=(1�w1)1�+(1�w2)1�:(13)ThuswecanexpresstheWTPtoeliminatebothtypesofcatastrophes,w1;2,intermsofw1andw2.ButnotethattheseWTPsdonotadd:sincethefunction(1�x)1�isconvex,equation(13)impliesthatw1;2w1+w2,byJensen'sinequality.Bythesamereasoning,itcanbeshownthatw1;2;:::;NPNi=1wi.Likewise,ifwedividetheNcatastrophesintotwogroups,1throughMandM+1throughN,thenw1;2;:::;Nw1;2;:::;M+wM+1;:::;N.TheWTPtoeliminateallNcatastrophesislessthanthesumoftheWTPsforeachoftheindividualcatastrophes,andlessthanthesumoftheWTPstoeliminateanytwogroupsofcatastrophes.3.1WhichCatastrophestoAvert?TheWTP,wi,measuresthebenetofavertingCatastropheiasthemaximumfractionofconsumptionsocietywouldsacricetoachievethisresult.Wemeasurethecorrespondingcostastheactualfractionofconsumptionthatwouldhavetobesacriced,viaapermanentconsumptiontaxi,togeneratetherevenueneededtoavertthecatastrophe.Thuswe11 whereifnocatastrophesareeliminated(i.e.,ifSistheemptyset)thentheobjectivefunctionin(16)istakentoequalone.Proof.IfwechoosesomesubsetSthen,usingResult1torewritethedenominatorofex-pression(14)intermsoftheindividualWTPs,wi,expectedutilityequalsQi2S(1�i)1� (1�)(�(1�))�1+Pi2S[(1�wi)1��1]
or,rewritingintermsofBiandKi,Qi2S(1+Ki) (1�)(�(1�))�1+Pi2SBi
:Since(1�)(�(1�))0,theoptimalsetSthatmaximizestheaboveexpressionisthesameasthesetSthatsolvestheproblem(16). Itisproblem(16)thatgeneratesthestrangeeconomicsofthetitle.Tounderstandhowtheproblemdiersfromwhatonemightnaivelyexpect,noticethatthesetSsolvesmaxSlog 1+Xi2SBi!�Xi2Slog(1+Ki):OnemightthinkthatifcostsandbenetsKiandBiareallsmall,then|sincelog(1+x)xforsmallx|thisproblemcouldbecloselyapproximatedbythesimplerproblemmaxSXi2S(Bi�Ki):(17)Thislinearizedproblemisseparable,whichvastlysimpliesitssolution:acatastropheshouldbeavertedifandonlyifthebenetofdoingso,Bi,exceedsthecost,Ki.Butthelinearizedproblemisonlyatolerableapproximationtothetrueproblemifthetotalnumberofcatas-trophesislimited,andinparticular,ifPi2SBiissmall.ItisnotenoughfortheBistobeindividuallysmall.Thereasonisthatavertingalargenumberofsmallcatastropheshasthesameaggregateimpactonconsumption(andmarginalutility)asdoesavertingafewlargecatastrophes.Weillustratethiswiththefollowingexample.Example1:ManySmallCatastrophes.Supposewehavealargenumberofidentical(butindependent)smallpotentialcatastrophes,eachwithBi=BandKi=K.ThenaiveintuitionistoeliminateallifB�K,andnoneifBK.AsResult3belowshows,thenaiveintuitioniscorrectinthelattercase;butifB�Kweshouldnoteliminateallofthecatastrophes.Instead,thenumbertoeliminate,m,mustsolvetheproblemmaxm1+mB (1+K)m:(18)13 (a)=2 (b)=3Figure1:Therearetwopotentialcatastrophes,with1=20%and2=10%.Theguresshow,forallpossiblevaluesofw1andw2,whichcatastrophesshouldbeaverted(incurlybrackets).Weshouldavertbothcatastrophesonlyforcombinations(w1;w2)inthemiddleshadedregion.Thatregionshrinksconsiderablywhenriskaversion,,increases.Example2:TwoCatastrophes.Toillustratethisresult,suppose1=20%and2=10%.Figure1showswhichcatastrophesshouldbeavertedfordierentvaluesofw1andw2.Whenwiiforbothcatastrophes(thebottomleftrectangle),neithershouldbeaverted.Weshouldavertbothonlyforcombinations(w1;w2)inthemiddlelozenge-shapedregion.Thatregionshrinksconsiderablywhenweincrease.Inthecontextofequation(19),thelargeristhelargerisB1,andthusthelargeristhehurdlerateforavertingthesecondcatastrophe.Considerthepoint(w1;w2)=(60%;20%)inFigure1(b).Asshown,weshouldavertonlytherstcatastropheeventhoughw2&#x-403;2.HereB1=5:25;B2=0:56,andK2=0:23,soB2=K2=2:391+B1=6:25.Equivalently,w1;2=61:7%,so(w1;2�w1)=(1�w1)=4:3%2=10%.TheadditionalbenetfromavertingCatastrophe2islessthanthecost.HowistheWTPtoavertCatastrophe1aectedbytheexistenceofCatastrophe2?Catastrophe2isakindof\backgroundrisk"that(a)reducesexpectedfutureconsumption;and(b)therebyraisesfutureexpectedmarginalutility.Becauseeachcatastrophiceventreducesconsumptionbysomepercentage,thersteectreducestheWTP;thereisless(future)consumptionavailable,sotheeventcausesasmallerabsolutedropinconsumption.ThesecondeectraisestheWTPbecausethelossofutilityisgreaterwhentotalconsump-tionhasbeenreduced.If&#x-277;1sothatexpectedmarginalutilityrisessucientlywhenconsumptionfalls,thesecondeectdominates,andtheexistenceofCatastrophe2willon15 netincreasethebenetofavertingCatastrophe1,andraiseitsWTP.3.3MultipleCatastrophesofArbitrarySizeWithmultiplecatastrophesofarbitrarysize,thesolutionofproblem(16)ismuchmorecomplicated.HowdoesonendthesetSinpractice?Ingeneral,onecansearchovereverypossiblesubsetofthecatastrophestondthesubsetthatmaximizestheobjectivefunctionin(16).WithNcatastrophesthereare2Npossiblesubsetstoevaluate.Thereisastarkcontrastherewithconventionalcost-benetanalysis,inwhichanindividualprojectcanbeevaluatedinisolation.Thenextresultshowsthatwecaneliminatecertainprojectsfromconsideration,beforecheckingallsubsetsoftheremainingprojects.Result3(Donoharm).Aprojectwithwiishouldneverbeimplemented.Proof.Letibeaprojectwithwii;thenbydenition,BiKi.LetSbeanysetofprojectsthatdoesnotincludei.Since1+Bi+Ps2SBs (1+Ki)Qs2S(1+Ks)| {z }obj.fn.in(16)ifweavertSandi(1+Bi)(1+Ps2SBs) (1+Ki)Qs2S(1+Ks)1+Ps2SBs Qs2S(1+Ks)| {z }obj.fn.ifweavertS;andsinceSwasarbitrary,itisneveroptimaltoavertcatastrophei. Intheotherdirection|decidingwhichprojectsshouldbeimplemented|thingsaremuchlessstraightforward.However,wehavethefollowingresult,whoseproofisintheappendix.Result4.(i)Ifthereisacatastropheiwhosewiexceedsitsithenwewillwanttoeliminatesomecatastrophe,thoughnotnecessarilyiitself.(ii)Ifitisoptimaltoavertcatastrophei,andcatastrophejhashigherbenetsandlowercosts,wj�wiandji,thenitisalsooptimaltoavertj.(iii)Ifthereisaprojectwithwi&#x-277;ithathasbothhighestbenetwiandlowestcosti,thenitshouldbeaverted.(iv)Fixf(i;wi)gi=1;:::;Nandassumethatwi&#x-277;iforatleastonecatastrophe.Forsu-cientlyhighriskaversion,itisoptimaltoavertexactlyonecatastrophe:theonethatmaximizes(1�i)=(1�wi),orequivalently(1+Bi)=(1+Ki).Ifmorethanonedisastermaximizesthisquantity,thenanyoneofthemaximizersshouldbechosen.16 (a)=1:01;w3=7% (b)=2;w3=7% (c)=2;w3=20% (d)=3;w3=20%Figure2:Therearethreecatastrophetypeswith1=20%,2=10%,and3=5%.Dierentpanelsmakedierentassumptionsaboutw3and.Numbersinbracketsindicatewhichcatastrophesshouldbeavertedfordierentvaluesofw1andw2.18 lessthan90%),weshouldavertCatastrophes1and2,butnot3;theadditionalbenetofalsoavertingCatastrophe3,i.e.,(w1;2;3�w1;2)=(1�w1;2),islessthanthecost,3.Finally,whenweincreaseto3,inFigure2d,therangeofvaluesofw1andw2forwhichallthreecatastrophesshouldbeavertedismuchsmaller.Wenowshowthatthepresenceofmanysmallpotentialcatastrophesraisesthehurdleraterequiredtopreventalargeone.Example5:MultipleSmallCatastrophesCanCrowdOutaLargeCatastrophe.Supposethattherearemanysmall,independent,catastrophes,eachwithcostkandbenetb,andonelargecatastrophewithcostKandbenetB.Thenwemustcomparemaxm1+mb (1+k)mwithmaxm1+B+mb (1+K)(1+k)m:Ignoringtheintegerconstraint,andassumingthatitisoptimaltoeliminateatleastonesmallcatastrophe,theoptimizedvaluesoftheseproblemsareb(1+k)1=b elog(1+k)andb(1+k)(1+B)=b e(1+K)log(1+k);respectively.ItfollowsthatweshouldeliminatethelargecatastropheifandonlyifB log(1+K)�b log(1+k):(20)Thusthehurdlerateforeliminationofthelargecatastropheisincreasedbythepresenceofthesmallcatastrophes.Figure3showsthisgraphically.Here=4andthesmallcatastrophes,indicatedoneachgurebyasmallsolidcircle,havewi=1%andi=0:5%(ontheleft)orwi=1%andi=0:25%(ontheright).Ifthelargecatastropheliesintheshadedregiondeterminedby(20),itshouldnotbeaverted.Incontrast,absentthesmallcatastrophes,themajoronewouldbeavertedifitliesanywhereabovethedashed45line.Example6:ChoosingAmongEightCatastrophes.Figure4showssomenumericalexper-iments.Eachpanelplotsrandomlychosen(fromauniformdistributionon[0;50%])WTPsandcosts,wiandi,foreightcatastrophes.Fixingthesewisandis,wecalculateBiandKiforarangeofvaluesof.Wethenndthesetofcatastrophesthatshouldbeeliminatedtomaximize(16).Theseareindicatedbybluedotsineachpanel;catastrophesthatshouldnotbeeliminatedareindicatedbyredcrosses.The45lineisshownineachpanel;pointsbelowithavewiiandhenceshouldneverbeaverted.Pointsabovethelinehavewi&#x-277;i,sothebenetofavertingexceedsthecost.Evenso,itisoftennotoptimaltoavert.19 (a)2[1;1:1] (b)2[1:2;1:4] (c)2[1:5;2:8] (d)2[2:9;3:9] (e)2[4;4:6] (f)2[4:7;1)Figure4:Eightcatastrophes.Optimalprojectchoiceatdierentlevelsofriskaversion,.21 4.1PartialAlleviationofCatastrophesAsapracticalmatter,thecompleteeliminationofsomecatastrophesmaybeimpossibleorprohibitivelyexpensive.Amorefeasiblealternativemaybetoreducethelikelihoodthatthecatastrophewilloccur,i.e.,toreducethePoissonarrivalrate.Forexample,Allison(2004)suggeststhattheannualprobabilityofanuclearterroristattackis:07.Whilereducingtheprobabilitytozeromaynotbepossible,wemightbeabletoreducesubstantiallyatacostthatislessthanthebenet.Shouldwedothat,andhowwouldtheanswerchangeifwearealsoconsideringreducingthearrivalratesforotherpotentialcatastrophes?Ouranalysisofmultiplecatastrophesmakesthisproblemeasytodealwith.Weconsiderthepossibilityofreducingthearrivalrateofsomecatastrophefromto(1�p),whichwecall\alleviatingthecatastrophebyprobabilityp."Wewritew1(p)fortheWTPtodojustthatforthersttypeofcatastrophe.Thusw1,inourearliernotation,isequaltow1(1).Weconsidertwoformsofpartialalleviation.First,supposetherearespecicpoliciesthatalleviateagivencatastrophetypebysomeprobability;anexampleistherigorousinspectionofshippingcontainers.Thisimpliesadiscretesetofpoliciestoconsider,andthepreviousanalysisgoesthroughessentiallyunmodied.Second,weallowtheprobabilitybywhichthecatastropheisalleviatedtobechosenoptimally.Perhapssurprisingly,thediscrete avorofourearlierresultsstillholds,andthoseresultsarealmostunchanged.4.1.1DiscretePartialAlleviationTondtheWTPtoalleviatethersttypeofcatastrophebyprobabilityp,thatis,w1(p),wemakeuseofapropertyofPoissonprocesses.Wecansplitthe`type-1'catastropheintotwosubsidiarytypes:1a(arrivingatrate1a1p)and1b(arrivingatrate1b1(1�p)).13ThuswecanrewritetheCGF(10)intheequivalentform:()=g+1a�Ee�1�1
| {z }type1a,arrivingatrate1a+1b�Ee�1�1
| {z }type1b,arrivingatrate1b+NXi=2i�Ee�i�1
| {z }allothertypes;sothatalleviatingcatastrophe1byprobabilitypcorrespondstosetting1atozero,andalleviatingcatastrophe1byprobability1�pcorrespondstosetting1btozero.Thiststhepartialalleviationproblemintoourframework.Forexample,Result1impliesthat1+(1�w1(1))1�=(1�w1(p))1�+(1�w1(1�p))1�,andtheargumentbelowequation 13ThemathematicalfactinthebackgroundisthatifwestartwithasinglePoissonprocesswitharrivalrate,andindependentlycoloreacharrivalredwithprobabilitypandblueotherwise,theredandblueprocessesareeachPoissonprocesses,witharrivalratespand(1�p)respectively.22 multipliers jtotheconstraintspj�10andjtotheconstraints�pj0,wehavethefollowingnecessaryandsucientconditions:forallj,wehave j0andj0,andBj 1+PipiBi�k0j(pj)= j�jwhere j(pj�1)=0andjpj=0:Togofurther,weconsidertwoalternativeassumptionsaboutthecostfunctionski(p).Alternative1:Inada-typeconditionsonki(p).Supposethatk0j(0)=0andk0j(1)=1.Thenwecanruleoutcornersolutions,soallLagrangemultipliersarezeroandBj k0j(pj)=1+XipiBiforeachj:(21)Ifitisoptimaltoavertatleastonecatastrophe,then1+PipiBi�QNj=1(1+Kj(pj))andhence1+PipiBi�1+Kj(pj)forallj.Butthen,usingthefactthatk0j(pj)=K0j(pj)=[1+Kj(pj)],condition(21)impliesthatBj�K0j(pj)atanyinterioroptimum.14ComparethiswiththecorrespondingconditioninthenaiveproblemmaxpjPjBj(pj)�PjKj(pj),whichisthatBj=K0j(pj).Onceagain,thepresenceofmultiplecatastrophesraisesthehurdlerate,butnowforanincreaseinpj,i.e.,greateralleviation.Alternative2:Abenchmarkfunctionalformforki(p).Supposethat(1�i(p))(1�i(q))=1�i(p+q)forallp,q,andi;sothat`alleviatingbyp'andthen`alleviatingbyq'isascostlyas`alleviatingbyp+qinonego.'(Thismightholdif,e.g.,adeadlyviruscomesfromgoatsorchimps,andfundscanbedevotedtogoatresearch,chimpresearch,orboth.Itwouldnotholdif,e.g.,thereisanitecostofalleviatingby0.5butaninnitecostoffullyaverting.)Thisassumptionpinsdowntheformofthecostfunction:15writingi(1)=i,wemusthavei(p)=1�(1�i)por,equivalently,1+Ki(p)=(1+Ki)p.Thenthefunctionski(
)denedabovearelinear:ki(pi)=log(1+Ki(pi))=piki;(22)wherekilog(1+Ki).Thusk0i(pi)=ki,anexogenousconstantindependentofpi. 14RememberthatBj=Bj(1)isanumber,notafunction;sinceBj(pj)=pjBj,fromResult6,wecanalsointerpretBjasthemarginalbenetofanincreaseinpj,thatis,B0j(pj).15Toseethis,notethatthedeningassumptioncanberewrittenash(p)+h(q)=h(p+q),whereh(x)=log(1�i(x)).ThisisCauchy'sfunctionalequation,whosesolution(giventhati(x)andhenceh(x)ismonotonic)isthath(x)=cxforsomeconstantc.Theresultfollowsonimposingi(1)=i.24 (a)=2 (b)=3Figure5:ModifyingtheexampleillustratedinFigure1toallowforpartialalleviationwithcostfunctionsasin(22).Therearetwopotentialcatastrophes,with1=20%and2=10%.Unnumberedzonesareareaswhereoneofthecatastrophesshouldbepartiallyalleviated(anditshouldbeobviousfromthelocationwhichone).Forexample,wemaywanttobundlenuclearandbio-terrorismtogetherintoasinglecatastrophetypethatcanbeavertedatsomecost.Whenaterroristattackoccurs,itmaybeeitherabiologicalattackoranuclearattack.Thedistributionofdamagesassociatedwithbiologicalattacksmaydierfromthedistributionofdamagesassociatedwithnuclearattacks;theresultingdistributionfor,thelossassociatedwiththe`bundled'catastrophe,isthensimplyamixtureofthetwodistributions.Toillustratehowourframeworkcanaccommodatethispossibility,supposenuclearandbio-terrorismaretheonlytwotypesofcatastrophe,witharrivalrates1and2andstochasticimpacts1and2respectively.Ifthetwoareentirelyindependent,andpoliciestoavertthemareindependent(aswehavebeenimplicitlyassumingthusfar)thentheCGFis()=g+1�Ee�1�1
+2�Ee�2�1
:(23)Alternatively,ifwebelievethatthesamepolicyactionwillavertbothnuclearandbio-terrorism,wecanthinkoftherebeingasinglecatastrophe17thatarrivesatrate1+2,andsuchthatafractionp1=(1+2)ofarrivalscorrespondtonuclearattackswith 17Asinfootnote13,ifwehavea`red'Poissonprocesswitharrivalrate1anda`blue'Poissonprocesswitharrivalrate2,wecandenea`color-blind'stochasticprocessthatdoesnotdistinguishbetweenblueandredarrivals.ThisstochasticprocessisalsoaPoissonprocess,witharrivalrate1+2.26 stochasticimpact1,andafraction1�p=2=(1+2)correspondtobio-attackswithstochasticimpact2.Thisensuresthatthearrivalrateofnuclearattacksis1,asbefore,andsimilarlyforbio-attacks.ThenwecanthinkoftheCGFas()=g+�Ee��1
:(24)Equations(23)and(24)describethesameCGF,sinceEe�=pEe�1+(1�p)Ee�2.Ifpoliciestoavertnuclearandbio-terrorismarebestthoughtofseparately,thenitisnaturaltoworkwith(23);avertingnuclearterrorismcorrespondstosetting1=0.If,ontheotherhand,apolicytoavertnuclearterrorismwillalsoavertbio-terrorism,thenitismorenaturaltoworkwith(24);avertingbothcorrespondstosetting=0.Lastly,wecancombinetheresultsofthissectionandSection4.1.1toallowasinglepolicytoavertmultiplecatastrophespartially(andpotentiallybydierentprobabilities).Considerapolicythatalleviatescatastrophetype1withprobabilityp1andtype2withprobabilityp2.Thensplittypes1and2intofourseparatetypes:types1aand1bhavearrivalrates1p1and1(1�p1)respectively,andtypes2aand2bhavearrivalrates2p2and2(1�p2).Nowviewtypes1aand2aasanamalgamatedPoissonprocesswitharrivalratee1p1+2p2(withimpactdistributionequaltoamixtureofdistributions1and2withweights1p1=(1p1+2p2)and2p2=(1p1+2p2)).Thepolicyoptionthenistosetetozero,andthepreviousresultsgothroughunchanged.4.3BonanzasOurframeworkalsoappliestoprojectsthatmayleadtogoodoutcomes.Forsimplicity,supposethatlogconsumptionisct=gtintheabsenceofanyaction.Therearealsoprojectsj=1;:::;mthatcanbeimplemented.Ifprojectjisimplemented,logconsumptionisaugmentedbytheprocessPQj(t)i=1j;i;iftheyareallimplemented,logconsumptionfollowsct=gt+Q1(t)Xi=11;i+


+Qm(t)Xi=1m;i;wheretheprocessesQ1(t);:::;Qm(t)arePoissonprocessesasbefore.Forconsistencywithprevioussections,wedene()=gtobetheCGFoflogconsumptiongrowthifnopoliciesareimplemented,(j)()=g+j�Eej;1�1
tobetheCGFoflogconsumptiongrowthifprojectjisimplemented,and(S)()=g+Pj2Sj�Eej;1�1
tobetheCGFoflogconsumptiongrowthifprojectsj2Sareimplemented.Ifnoprojectsareimplemented,expectedutilityis1=[(1�)(�(1�))].Ifprojectsj2Sareimplemented,expectedutilityis1=(1�)(�(S)(1�)).TheWTPforthe27 aresummarizedinTable1alongwiththecalculatedvaluesofwi,Bi,andKi.ThelastrowofthetableshowstheWTPtoavertallsevencatastrophes(w1;:::;7Piwi)andthecorrespondingbenetandcostinutilityterms,B1;:::;7=PiBiand1+K1;:::;7=Qi(1+Ki).Notethatforbothvaluesof,B1;:::;7&#x]TJ/;ཀ ;.9;Ւ ;&#xTf 1;.48; 8.;除&#x Td[;K1;:::;7,butaswewillsee,itisnotoptimaltoavertallsevencatastrophes.Theestimatesofi,i,andiareexplainedinAppendixC.Forsomeofthecatastrophes( oods,storms,andearthquakes),theestimatesarebasedonarelativelylargeamountofdata.Forothers(e.g.,nuclearterrorism),theyarebasedonthesubjectiveestimatesofseveralauthors,andreadersmaydisagreewithsomeofthenumbers.Asaresult,theyshouldbeviewedasspeculativeandlargelyillustrative.Someofthecatastrophesweconsiderinvolvedeathasopposedtoadropinconsumption.InMartinandPindyck(2014),weshowthattheWTPtoavertthedeathofafractionofthepopulationismuchgreaterthantheWTPtoavertadropinconsumptionbythesamefraction.19Thisshouldnotbesurprising;mostpeoplewouldpayfarmoretoavoida5%chanceofdyingthantheywouldtoavoida5%dropinconsumption.ThedierenceinWTPsdependsonthevalueofalifelost,whichisoftenproxiedbythe\valueofastatisticallife"(VSL).EstimatesoftheVSLareintherangeof3to10timeslifetimeconsumption.WendthataVSLof6timeslifetimeconsumptionimpliesthattheWTPtoavoidaprobabilityofdeathofisequaltotheWTPtoavoidadropinconsumptionofatleast5.Weusethismultipletotranslateafordeathintoawelfare-equivalentforlostconsumption.Theestimatesofwi,Bi,andKiinTable1dependonand.Whatarethe\correct"valuesofthesetwoparameters?Wehavechosenvaluesconsistentwiththemacroeconomicsandnanceliteratures,butweviewandaspolicyparameters,i.e.,re ectingthechoicesofpolicymakers.Thustherearenosinglevaluesthatwecandeem\correct."20WhichofthesevenpotentialcatastrophessummarizedinTable1shouldbeaverted?WecananswerthisusingResult2.AlthoughB1;:::;7&#x]TJ/;ཀ ;.9;Ւ ;&#xTf 1;.48; 8.;除&#x Td[;K1;:::;7,itisnotoptimaltoavert 19Toourknowledge,theliteratureonclimatechange,andinparticulartheuseofIAMstoassessclimatechangepolicies,utilizeconsumption-baseddamages,i.e.,climatechangereducesGDPandconsumptiondirectly(asinNordhaus(2008)andStern(2007)),orreducesthegrowthrateofconsumption(asinPindyck(2012)).Millner(2013)discusseswelfareframeworksthatincorporatedeath.20Therateoftimepreferencemattersbecausecatastrophiceventsareexpectedtooccurinfrequently,solongtimehorizonsareinvolved.Themacroeconomicsandnanceliteraturessuggest2to5percent.Someeconomists,e.g.,Stern(2008),arguethatonethicalgrounds,shouldbezero.Likewise,re ectsaversiontoconsumptioninequalityacrossgenerations.Intheend,andare(implicitly)chosenbypolicymakers,whomightormightnotbelieve(orcare)thattheirdecisionsre ectthevaluesofvoters.Forinterestingdiscussionsofsocialdiscounting,seeCaplinandLeahy(2004)andGollier(2013).Forawide-rangingandinsightfuldiscussionofeconomicpolicy-makingunderuncertainty,seeManski(2013).29 (a)=2 (b)=4Figure6:TheguresshowwhichofthesevencatastrophessummarizedinTable1shouldbeaverted.Catastrophesthatshouldbeavertedareindicatedbybluedotsineachpanel;catastrophesthatshouldnotbeavertedareindicatedbyredcrosses.isconsistentwithrecentassessments,butthoseassessmentsarewidelydispersed.Also,weassumedthearrivalrateforaclimatecatastropheisconstant,butitismorelikelytoincreaseovertime.Nonetheless,theresultsinTable1andFigure6illustrateourkeypoints:policiestoavertmajorcatastrophesshouldnotbeevaluatedinisolation,notallcatastrophesshouldnecessarilybeaverted,andthechoiceofwhichonestoavertiscomplex.Figure7makesthislastpointinadierentway,byshowinghowthesetofcatastrophesthatshouldbeaverteddependsonriskaversion,,andthetimepreferencerate,.Onlyifandarelowshouldallsevenbeaverted,andtheoptimalchoicevarieswidelyforlargervaluesofand.6ConclusionsHowshouldeconomistsevaluateprojectsorpoliciestoavertmajorcatastrophes?Wehaveshownthatifsocietyfacesmorethanjustonecatastrophe(whichitsurelydoes),conventionalcost-benetanalysisbreaksdown;ifappliedtoeachcatastropheinisolation,itcanleadtopoliciesthatarefarfromoptimal.Thereasonisthatthecostsandbenetsofavertingacatastrophearenotmarginal,inthattheyhavesignicantimpactsontotalconsumption.Thiscreatesaninterdependenceamongtheprojectsthatmustbetakenintoaccountwhenformulatingpolicy.Infact,aswedemonstratedinExample1,cost-benetanalysiscanevenfailwhenappliedtosmallcatastrophesiftheyhaveanon-marginalaggregateimpact.UsingWTPtomeasurebenetsandapermanenttaxonconsumptionasthemeasureofcost(bothapercentageofconsumption),wederivedadecisionrule(Result2)todeterminetheoptimalsetofcatastrophesthatshouldbeaverted.Andwehaveshownthatthisdecision31 Figure7:Thesetofcatastrophestobeaverteddependsinacomplexwayonand.V:virus;C:climate;N:nuclearterrorism;B:bioterrorism;F: oods;S:storms;Q:quakes.rulecanyield\strange"results.Forinstance,aswedemonstratedinExample3,althoughnaivereasoningwouldsuggestusingasequentialdecisionrule(e.g.,avertthecatastrophewiththelargestbenet/costratio,thendecideontheonewiththenext-largestratio,etc.),sucharuleisnotoptimal.Ingeneral,infact,thereisnosimpledecisionrule.Instead,determiningtheoptimalpolicyrequiresevaluatingtheobjectivefunction(16)ofResult2foreverypossiblecombinationofcatastrophes.Inastrongsense,then,thepolicyimplicationsofdierentcatastrophetypesareinextricablyintertwined.Giventhatthecompleteeliminationofsomecatastrophesmaybeimpossibleorpro-hibitivelyexpensive,amorerealisticalternativemaybetoreducethelikelihoodthatthecatastrophewilloccur,i.e.,reducethePoissonarrivalrate.Wehaveshownhowthatalternativecaneasilybehandledinourframework.Intheprevioussectionweexaminedthecostsandbenetsofcompletelyavertingsevencatastrophes,butwecouldhavejustaseasilyconsideredprojectstoreducethelikelihoodofeach,andgiventheamountsofreductionandcorrespondingcosts,determinedtheoptimalsetofprojectstobeundertaken.Thetheorywehavepresentedisquiteclear.(Wehopemostreaderswillagree.)Butthereremainimportantchallengeswhenapplyingitasatoolforgovernmentpolicy,asshouldbe32 evidentfromSection5.First,onemustidentifyalloftherelevantpotentialcatastrophes;weidentiedseven,buttheremightbeothers.Second,foreachpotentialcatastrophe,onemustestimatethemeanarrivalratei,andtheprobabilitydistributionfortheimpacti.Finally,onemustestimatethecostofavertingoralleviatingthecatastrophe,whichweexpressedasapermanenttaxonconsumptionattheratei.Asweexplained,forsomecatastrophes( oods,stormsandearthquakes),arelativelylargeamountofdataareavailable.Butforothers(nuclearandbio-terrorism,oramega-virus),estimatesofi,iandiarelikelytobesubjectiveandperhapsspeculative.Ontheotherhand,onecanuseourframeworktodetermineoptimalpoliciesforrangesofparametervalues,andtherebydeterminewhichparametersareparticularlycritical,andshouldbethefocusofresearch.33 ormaxSYi2Si Xj2S1�j!1 �1:Nowweusethefactthatforarbitrarypositivex1;:::;xN,wehavelim!1�x1+


+xN
1 =maxixi.Thismeansthatforsucientlylarge,theproblemisequivalenttomaxSmaxk2S1 kYi2Si:NoticethatforaxedsetS,maxk2S1 kYi2Simaxk2Sk k;becausei1foralli.SogivenacandidatesetS,wecanincreasetheobjectivefunctionbyavertingonlythecatastrophek2Sthatmaximizesk=k.ThisholdsforarbitraryS,sotheunconstrainedoptimumisachievedbyavertingonlyasinglecatastrophethatmaximizesk=k.Thisisequivalenttotheconditionsprovidedinthestatementoftheresult.ProofofResult5.Withlogutility,wecanusethepropertyoftheCGFthat0t(0)=ElogCttowriteexpectedutilityasEZ10e�tlogCtdt=Z10e�t0t(0)dt=0(0)Z10te�tdt=0(0) 2:Ifweeliminatecatastrophes1throughNcostlessly,expectedutilityis(1;:::;N)0(0)=2.SoWTPssatisfylog(1�w1;:::;N)=0(0)�(1;:::;N)0(0) andlog(1�wi)=0(0)�(i)0(0) :(27)ExploitingthesamerelationshipbetweenCGFsasbefore,wendthatNXi=10(0)�(i)0(0) =0(0)�(1;:::;N)0(0) ;andsoNXi=1log(1�wi)=log(1�w1;:::;N):(28)Now,supposeweeliminatecatastrophes1throughNatcosti(i.e.,asbefore,consump-tionismultipliedby(1�i)toeliminatecatastrophei),thenexpectedutilityisEZ10e�tloghC(1;:::;N)t(1�1)


(1�N)idt=1 2(1;:::;N)0(0)+1 [log(1�1)+


+log(1�N)]35 Pindyck(2014)),weshowthattheWTPtoavertaneventthatkillsarandompercentofthepopulationismuchlargerthantheWTPtoavertaneventthatreduceseveryone'sconsumptionbythesamefraction.ThelastmajorpandemictoaectdevelopedcountrieswastheSpanish uof1918{1919,whichinfectedroughly20percentoftheworld'spopulationandkilled3to5percent.Becausepopulationstodayhavegreatermobility,asimilarviruscouldspreadmoreeasily.Wetaketheaveragemortalityrateofthenextpandemictobe3.5percent,whichweestimateisequivalentinwelfaretermstoaroughly17.5percentdropinconsumption.21Thiscorrespondstoavalueof.825/.175=4.7for,whichweroundto5.Weassume=:02,i.e.,thereisroughlya20percentchanceofapandemicoccurringinthenext10years.Whatwouldberequiredtoavertsuchanevent?Thereisnothingthatcanbedonetopreventnewvirusesfromevolvingandinfectinghumans(mostlikelyfromananimalhost).Ifanewvirusisextremelyvirulentandcontagious,containmentinvolves(1)theimplementationofsystemstoidentifyandisolateinfectedindividuals(e.g.,beforetheyboardplanesortrains);and(2)therapidproductionofavaccine(whichwouldrequireyet-to-bedevelopedtechnologies,andgovernmentinvestmentinalarge-scaleproductionfacility).Bothoftheseelementsinvolvesubstantialongoingcosts,butwearenotawareofanyestimatesofhowlargethosecostswouldbe.Asabestguess,wewillassumethatthosecostscouldamountto2%ormoreofGDP,andset=:02.Climate:Aconsensusestimateoftheincreaseinglobalmeantemperaturethatwouldbecatastrophicisabout5to7C.Asummaryof22climatesciencestudiessurveyedbyIntergovernmentalPanelonClimateChange(2007)putstheprobabilityofthisoccurringbytheendofthecenturyataround5to10%.Preliminarydraftsofthe2014IPCCreportsuggestasomewhathigherprobability.Weitzman(2009,2011)arguedthattheprobabil-itydistributionisfat-tailed,makingtheactualprobability10%ormore.Wewillusethe\pessimistic"endoftherangeandassumethatthereisa20%chancethatacatastrophicclimateoutcomecouldoccurinthenext50or60years,whichimpliesthat=:004.Whatwouldbetheimpactofacatastrophicincreaseintemperature?Estimatesoftheeectivereductionin(world)GDPfromcatastrophicwarmingrangefrom10%to30%;wewilltakethemiddleofthisrange,whichputsat.20(sothat=4).22Whatwouldbethecostofavertingaclimatecatastrophe?SomehavearguedthatthiswouldrequirelimitingtheatmosphericGHGconcentrationto450ppm,andestimatesofthecostofachievingthistargetvarywidely.AstartingpointwouldbetheGHGemission 21Weuseamultipleof5,which,asdiscussedinMartinandPindyck(2014),wouldapplyifweusethe\valueofastatisticallife"(VSL)torepresentthevalueofalifelost,andtaketheVSLtobesixtimeslifetimeconsumption.AgreatmanystudieshavesoughttoestimatetheVSLusingdataonrisk-of-deathchoicesmadebyindividuals,andtypicallyndthattheVSLisontheorderof3to10timeslifetimeincomeorlifetimeconsumption.See,forexample,Viscusi(1993)andCropperandSussman(1990).22See,e.g.,Pindyck(2012)andStern(2007,2008,2013).37 (2013)ofnaturaldisastersandtheireconomicimpact.Theyutilizedadatasetcovering196countriesovertheperiod1970to2008,whichcombinedWorldBankdataonrealGDPpercapitawithdataonnaturaldisastersandtheirimpactsfromtheEM-DATdatabase.24Cavalloetal.(2013)estimatedtheeectsofdisastersoccurringin196countriesduring1970to1999onthecountries'GDPinthefollowingyears.Therewere2597disastersduring1970{2008,outofatotalof39196=7664country-yearobservations,whichimpliesanaverageannualrateof2597/7664=.34.Ofthesedisasters,abouthalfwere oods,about40%storms(includinghurricanes),andabout10%earthquakes.Thusweset=.17,.14,and.03for oods,storms,andearthquakesrespectively.Thesedisastersresultedindeaths,butthenumberwasalmostalwaysverysmallrelativetothecountry'spopulation.(Forexample,HurricaneKatrinacaused1833deaths,whichwaslessthan.001%oftheU.S.population.)WethereforeignorethedeathtollsfromtheseeventsandfocusontheimpactonGDP.Cavalloetal.(2013)foundthatonlythelargestdisasters(the99thpercentileintermsofdeathspermillionpeople)hadastatisticallysignicantimpactonGDPtenyearsfollowingtheevent(reducingGDPby28%relativetowhatitwouldhavebeenotherwise).Butalthoughnotstatisticallysignicant,smallerdisasters(atthe75thpercentile)reducedGDPby5to10%.Assumingthateventsbelowthe75thpercentilehadnoimpact,wetaketheaverageimpactforallthreetypesofdisasterstobea1%dropinconsumption,whichimplies=100.Thus oods,stormsandearthquakesarerelativelycommoncatastrophes,buthaverelativelysmallimpactsonaverage.Stormscannotbeprevented,buttheirimpactcanbereducedorcompletelyaverted.Thisinvolvesrelocatingcoastlinehomesandotherbuildings,retrottinghomes,puttingpowerlinesunderground,etc.Similarstepswouldhavetobetakentoaverttheimpactof oods.Weassumethecostofcompletelyavertingeachofthesedisastersisabout2%ofconsumption.Thecostofavertingtheimpactofearthquakesshouldbelower|weassume1%ofconsumption|becausemanybuildingsinvulnerableareashavealreadybeenbuilttowithstandearthquakes.Thusweset=:02forstormsand oods,and.01forearthquakes.OtherCatastrophicRisks:Muchlesslikely,butcertainlycatastrophic,eventsincludenuclearwar,gammaraybursts,anasteroidhittingtheEarth,andunforeseenconsequencesofnanotechnology.Foranoverview,seeBostromandCirkovic(2008),andseePosner(2004)forafurtherdiscussion,includingpolicyimplications.Weignoretheseotherrisks. 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