微观经济学-范里安 EUT

EUT:BuildtheEmpireontheSandLing,ChenSchoolofEconomics,SWUFE1ExpectedUtilityTheoremThepreferencesaredefinedoverprospects(lotteries),whichsatisfyfollowingaxioms:CompletenessTransitivityContinuityIndependence2FailureoftheEUT?EasytoconstructtestsofEUTthatpeople“fail”Whatdoes“failure”mean?Aretheysystematic?CanweidentifythedomainofEUT?3ViolationoftheIndependenceAxiomAllais(1953):Commonconsequenceeffects1:($1M,1) r1:($5M,0.1;$1M,0.89;0,0.01)-------------------------------------------------------------------------s2:($1M,0.11;0,0.89)r2:($5M,0.1;0;0.9) Ifs1ispreferredoverr1,thenEUTimpliesthats2ispreferredoverr2

However,mostsubjectsprefers1overr1,andr2overs2:ViolationoftheIndependence(why?)4CommonRatioEffectAllais(1953)s1:($3000,1)r1:($4000,0.8;0,0.2)-------------------------------------------------------------s2:($3000,0.25;0,0.75)r2:($4000,0.2;0,0.8)Again,Ifs1ispreferredoverr1,thenEUTimpliesthats2ispreferredoverr2However,mostsubjectsprefers1overr1,andr2overs2:ViolationoftheIndependence5Dowehaveawell-definedpreference?Preferencereversal:$-betvs.P-betP-bet:.99$4,.01$-1(EV=$3.95VAR=0.0895)$-bet:.33$16,.67$-2(EV=$3.94VAR=50.518)Question1:Whichonedoyouprefertoplay?MostgoforP-betQuestion2:Ifyouaresellingthem,whichonehasthehigherprice?Mostsubjectspricing$-bethigher6IntransitivityinterpretationAsymmetricreversal:Pbetspreferredto$bets$betssellingpricesarehigherthansellingpricesforPbetsSetWTAitoavaluebetweenthosestatedfor$andPbets$-bet>WTAi(basedonpricingtask)WTAi>P-bet(basedonpricingtask)ButP-bet>$-bet7FramingEffectTverskyandKahneman(1981,Science)ImaginethattheU.S.ispreparingfortheoutbreakofanunusualAsiandisease,whichisexpectedtokill600people.Twoalternativeprogramstocombatthediseasehavebeenproposed.Assumethattheexactscientificestimateoftheconsequencesoftheprogramsareasfollows:8FramingEffectIfprogramAisadopted,200peoplewillbesaved.IfprogramBisadopted,thereis1/3probabilitythat600peoplewillbesaved,and2/3probabilitythatnopeoplewillbesaved72%subjectschoseprogramA9FramingEffectIfprogramCisadopted,400peoplewilldie.IfprogramDisadopted,thereis1/3probabilitythatnobodywilldie,and2/3probabilitythat600peoplewilldie.78%samesubjectschoseprogramDButA=CandB=D.Thewayyoupresentthematerialswillchangethepreference(atleasttherevealedpreference)!10Rabin’scalibrationSupposeJohnisarisk-averseexpectedutilitymaximizer,andthathewillalwaysturndownthe50-50gambleoflosing$10orgaining$11.WhatelsecanwesayaboutJohn?11ContsSpecifically,whatisthebiggestYsuchthatweknowJohnwillturndowna50-50lose$100/win$Ybet?$110 $221$2000 $20,242$1.1M $2.5M AlwaysrejectnomatterwhatYisNeedmoreinformationabouthisutilityfunction12CalibrationResults(Rabin,2000)13SubjectiveProbabilitySubjectiveexpectedutilitytheorem(Savage1954)Ellsberg’sParadox(Ellsberg1961)Considertwourns,bothcontain100ballseitherinredoryellow.Urn1has50redballs,whilewedon’tknowthenumberofredballsinurn2.Gamble1:win$100inyoudrawredballoutofanurn/Whichurndoyouprefer?Gamble1:win$100inyoudrawyellowballoutofanurn/Whichurndoyouprefer?14Non-EUTModel:AGeneralIdeaExpectedPaymentEUT:EUTsimplytransformthepaymenttoutility,butkeeptheprobability.Whynotalsotransformprobability?Non-EUT:15AlternativestoEUTRank-dependentEUQuigginJEBO1982Allownon-linearpreferencesOriginalprospecttheoryKahneman&TverskyEconometrica1979EditingandthenevaluationCumulativeprospecttheoryStarmer&SugdenAnnalsofOR1989Tversky&KahnemanJRU199216ProspectTheoryKahnemanandTversky(1979):2002NobelPrizeAssumeu(x)=xα ifx≥0Assumeu(x)=-λ(-x)β ifx<0Assumew(p)=pγ/[pγ+(1-p)γ]1/γPU=∑k[w(pk)xUk]NewcomponentsDecisionweightsw(p)Utilitydefinedovergains&lossesLossaversionλ17Probabilityweights–thenKahneman&Tversky197918Probabilityweights–nowTversky&Kahneman199219Lossaversion20Rank-DependentEUTJohnQuiggin(1982,hisundergraduatethesis)Theprobabilityweightfunctionwherex1isthebestoutcomeandxnistheworst.RDEU:21WhyNon-EUTColinF.Camerer(2003):ProspectTheoryintheWild:EvidencefromtheFieldFinance:TheequitypremiumAveragereturntostocksishigher(8%peryear)thanbondsWhy?Risk-averseplayersneedpremium…MehraandPrescott(1985)askhowlargeadegreeofrisk-aversionistoexplainthat8%?Indifferencebetweenacoinflippayingeither$50,000or$100,000andasureamount$51,20922ContsBenartziandThaler(1997):PTbasedexplantionInvestorareaversetoloss(thechancethatreturnarenegative).23EUTvs.Non-EUTWhichoneisthebetterunderlyingmodeltoexplainthebehavior?EUTisaspecialcaseofprospecttheoryProspecttheoryhasmoreparametersthanEUT,henceitisnotsurprisethatithasmorepowertoexplainthedataEUTissimple,elegant,andaxiombased.24ExpectedUtilityTheoryand

ProspectTheory:

OneWeddingandADecentFuneralGlennW.Harrison&E.ElisabetRutström(ExperimentalEconomics2008)25GeneralmethodologicalpointHypothesistestsassumejustonedatageneratingprocessWhicheverDGPexplainsmoreofthedataisdeclared“the”DGP,andtheothersdiscardedConsiderlotterychoicebehaviorAssumeEUT&ProspectTheoryAssumecertainfunctionalformsforthemodelsgeneratingthedataAllowformultipleDGP,unitedusingmixturemodelsandagrandlikelihoodfunctionSolvethemodelIdentifywhichsubjectsarebetterdescribedbywhichDGP26GeneralmethodologicalpointHypothesistestsassumejustonedatageneratingprocessWhicheverDGPexplainsmoreofthedataisdeclared“the”DGP,andtheothersdiscardedConsiderlotterychoicebehavior:theChapelinVegasAssumeEUT&ProspectTheory:theBride&GroomAssumecertainfunctionalformsforthemodelsgeneratingthedata:thePrenuptialAgreementAllowformultipleDGP,unitedusingmixturemodelsandagrandlikelihoodfunction:theWeddingSolvethemodel:ConsummatingthemarriageIdentifywhichsubjectsarebetterdescribedbywhichDGP:aDecentFuneralfortheRepresentativeAgent27ScreenDisplayinGainFrame28ScreenDisplayinMixedFrame29TheirLab30TheRandomNumberGenerators31TheBride–EUTAssumeU(s,x)=(s+x)rAssumeprobabilitiesforlotteryasinducedEU=∑k[pkxUk]Definelatentindex∆EU=EUR-EULDefinecumulativeprobabilityofobservedchoiceusingthelogisticG(∆EU)Conditionallog-likelihoodofEUTthendefined:∑i[(lnG(∆EU)|yi=1)+(ln(1-G(∆EU))|yi=0)]Needtoestimater32TheCDFassumption33TheGroom–PTAssumeU(x)=xα ifx≥0AssumeU(x)=-λ(-x)β ifx<0Assumew(p)=pγ/[pγ+(1-p)γ]1/γPU=∑k[w(pk)xUk]Definelatentindex∆PU=PUR-PULDefinecumulativeprobabilityofobservedchoicebylogisticG(∆PU)Conditionallog-likelihoodofPTthendefined: ∑i[(lnG(∆PU)|yi=1)+(ln(1-G(∆PU))|yi=0)]Needtoestimateα,β,λandγ34TheNuptialGrand-likelihoodisjusttheprobabilityweightedconditionallikelihoodsProbabilityofEUT: ρEUTProbabilityofPT: