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RiskandUncertaintyIIBenjaminChiaoMarch31,2011(afterclassversion)LectureNotesforUndergraduateLevelInformationEconomics,ShanghaiUniversityofFinanceandEconomics1/19IDe…nitionsofRiskAttitudesIMeasurementofRiskAttitudesIComparisonofDistributions2/19IDe…nitionsofRiskAttitudes!IMeasurementofRiskAttitudesIComparisonofDistributions3/19IDe…nitionsofRiskAttitudes!4/19RiskAversev.RiskLovingISettingIMonetaryoutcomes.DenotemoneyasacontinuousrandomvariablexdistributedaccordingtothecumulativedistributionfunctionF(x).IThelotteryspaceisallpossiblec.d.fovernonnegativeamountofmoneyLISothecontinuousversionoftheEUtheoremgives:U(F)=u(x)dF(x)(1)ZITheBernoulliutilityfunctionu(x)isassumedtobeincreasingandcontinuous.IAndalsoboundedness(referstotheSt.Petersburg-Mengerparadox)5/19RiskAttitudeDe…nitionsIDe…nitionArisklovingpersonweaklyprefersalotteryF()overthelotterythatgivesasureoutcomehavingthevaluethesameastheexpectedvaluexdF(x).IIfthepersonisindi¤erent,he’scalledRriskneutralIIfthepersonpreferstheotherdirection,he’scalledriskaverseIIntheexpectedutilitycontext,itfollowsfromtheabovethatariskaversepersonsatis…estheJensen’sinequality(orconcavityofu):F:u(x)dF(x)u(xdF(x))(2)8ZZISeeFigure.6/19IDe…nitionICertaintyequivalentc(F,u)istheamountofmoneythatsatis…esu(x)dF(x)=u(c(F,u))(3)ZIThatis,youareindi¤erenttotakingcorthegambleF.ISeeFig.IExercise1:Provethatapersonisriskaversei¤c(F,u)xdF(x)R7/19IDe…nitionIProbabilitypremiumπ(x,e,u)istheprobabilitythatsatis…es,foralle>0,11u(x)=(+π(x,e,u))u(x+e)+(π(x,e,u))u(xe)22 IGivenasureoutcomex,theprobabilitypremiumπ(x,e,u)istheprobabilityyouwouldsubstractfrom1/2ofprobabilityofgettingaslightlyworseo¤outcome(xe)andaddtoa1/2probabilityofgettingaslightlybettero¤ outcome(x+e)sothatyouareindi¤erentbetweenthisandx.ISeeFig.IExercise2:Provethatapersonisriskaversei¤π(x,e,u)08/19IDe…nitionsofRiskAttitudesIMeasurementofRiskAttitudes!IComparisonofDistributions9/1
