Lecture-8choice-under-uncertainty(高级微观-武大高研-叶建亮).ppt

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1、Lecture8:choiceunderuncertaintyContentLotteriesandexpectedutilityRiskaversionMetricSubjectiveprobabilitytheoryLotteriesandexpectedutilityAlottery:Acompoundlottery:and.AsimplifiedlotteryofisorSeethefigLotteriesandexpectedutilityThepreferenceoflotteries:Continuous:thesetisclosureIndependenceaxiom

2、:Expectedutilityv.N-Mexpectedutilityfunction:Proposition1:autilityfunctionisanexpectedutilityfunctionifandonlyifit’sliner,thatiswehave:ExpectedutilityProposition2:isthev.N-Mexp.utilityfunctionofpreferenceonifandonlyif,isanotherv.N-MexpectedutilityfunctionExpectedutilityProposition3:iftheprefere

3、nceoncanberepresentedbyanexpectedutilityfunction,thensatisfiedindependentaxiom.Proposition4:(expectionutilitytheorem)ifthepolicymakertakeacontinuousandindependentpreferenceon,thenwecanfindav.N-Mexpectedutilityfunctiontorepresentit.Seethefig.RiskaversionAlotterywithmonetarypayoffs:continuousquan

4、tityofmoneyisarandomvariableAccumulateddistributionfunction:v.N-MexpectedutilityfunctionwhereisBernoulliutilityfunction.isincreasing,continuousandbounded.RiskaversionAriskaversionman:isasbetteratleastasalotterywithF(x).Jenson’sinequality:u(.)isconcaveorstrictlyconcaveifthemanis(strictly)riskave

5、rsion.Seethefig.RiskaversionCertaintyequivalence:ariskpremiumc(F,u)makeitindifferentwithalotterywithF(.).Probabilitypremium:anextraprobabilityovertheimpartialprobability,RiskaversionProposition5:apolicymakerisriskaversionifandonlyif:Proposition6:apolicymakerisriskaversionifandonlyif:Riskaversio

6、nExample1:insuranceExample2:demandforriskassets.MetricAgamblingacceptableset:IfAacceptx,Bnecessaryacceptitforthesamew,meansBismoreriskpreferencethanA,thatis:Seethefig.MetricThecurvatureofAismorethanBat(0,0).Differentialover:Wecanget:andDefinition:Arrow-Pratt’sabsoluteriskaversionindexMetricPrat

7、t’stheorem:i)ii)concaveandincreasing:iii)iv)v)Theyareequivalence.MetricExample3:PricingforriskassetsMetricRelativeriskaversionindex:ThewealthchangesinproportionDefinition:SubjectiveprobabilitytheoryAliasparadox:Thefirstlotteriesch