lecureconsumionheoryII高级微观武大高研,叶建亮

AdvancedMicroeconomics

lecture5:consumptiontheoryIIYeJianliangUtilitymaximizationContent:UtilityfunctionUtilitymaximizationExpenditureminimizationRelationshipbetweenUMPandEMPIntegrability1.FrompreferencestoutilityDefinition:isautilityfunctionofpreference,ifCanwealwaysfindautilityfunctionof?MaybeIfXisfinite,therealwaysexistutilityfunction.Proposition1:onlyrationalcanberepresentedbyautilityfunction.(N.CnotS.C)Lexicographicpreference:rationalbutnoutilityfunctionexist.1.FrompreferencestoutilityContinuity:then,orxuppercontoursetsandlowercontoursetsareclosure.Proposition2:Ifiscontinuous,thenexistacontinuousutilityfunctionrepresenting.1.FrompreferencestoutilityDesirability:preferenceisdesirableifismonotone:islocalnon-satiation:thereisa

proposition3:isstrongmonotone,thenit’smonotone;ismonotone,it’slocalnon-satiation.1.FrompreferencestoutilityConvexity:xuppercontoursetsareconvex.Decreasinginmarginalrateofsubstitution.peoplelikevariety.Preferencesareconvexmeansutilityfunctionisquasi-concave.Note:quasi-concavedoesnotcontainconcave.1.FrompreferencestoutilityIf,thenarehomothetic.Proposition4:Thearehomotheticifandonlyifit’sutilityfunctionisHD1.Ifthenarequasi-linearofcommodity1(it’scalledstandardcommodity)Proposition5:arequasi-linearifandonlyifit’sutilityfunctionisGormanform1.FrompreferencestoutilityPreferenceUtilityfunctionrationalityexistXfiniteDesirabilityincreasingContinuitycontinuityConvexityQuasi-concaveHomotheticHD1Quasi-linearGormanform2.UtilitymaximizationConsumer’sproblem(UMP):ThesolutionsarecalledWalras’DemandCorrespondence,orfunctionifit’ssinglepoint.2.UtilitymaximizationPropertiesofHD0SatisfiedWalras’LawIfareconcave,areconcavetoo,if arestrictlyconcave,issinglepoint.2.UtilitymaximizationIfthesolution,thenHereistheshadowprice:themarginalutilityofoptimalconsumption.Thevaluefunctioniscalled“indirectutilityfunction”:3.ExpenditureminimizationEMP:Thesolutionwascalled“Hicks(orcompensation)demandcorrespondence”.singlepoint----function.It’sHD0ofp.It’sconvex,asis.andsinglepointwhenarestrictlyconvex.3.ExpenditureminimizationThevaluefunction:“expenditurefunction”(moneymetricuntility)So,whenpricechanging,ifthewealthofconsumerchangecorrespondentlytokeepthesameutilitylevel,thenisthedemandchanging.HereHickswealthcompensationis

RecalltheSlutskywealthcompensation

4.RelationshipProposition5(Duality):

u(x)isacontinuousutilityfunctionof:

w>0,x*isthesolutionofUMP,whenit’srequiredthat,x*

isthesolutionofEMP,andvalueofEMPisw.Iftherequiredutility,x*isthesolutionofEMP,whentheexpenditureisp·x*,x*

isthesolutionofUMP,andvalueofUMPisu.4.RelationshipHicksdemandandexpenditurefunction:iss.n.s.d.HicksandWalrasdemand:Slutskyequation:Walrasdemandandindirectutilityfunction:Roy’sidentity:4.叼Re难la熄ti仆on乒sh伟ipUM揪PEM各Px(p,w)v(p,w)h(p,u)e(p,u)du芹al秧it腐ySl票ut项sk轻yeq乐ua纷ti服onRo悲y’碎sid交en淋ti欠ty5.驻In聋te牌gr背ab们il采it坟yDe哈ma御nd呜f辞un烧ct计io笔nx(p,w)(c申.d岛.)沸i嘱s暮HD陪0,哗s内at耽is隐fi毯edWa俊lr斧asLa见w喊an响d价ha搏ve牺a你s族ub婚st星it曾ut按io瘦n楼ma深tr括ixS(p,w)iss.巧n.系s.颜d.毙fo奔r龟an很y(p,w),ifit灭’s梳d缴ed押uc绝ed狸b刚y岛ra欢ti弓on燃al逃p序re散fe史re孩nc渐e.隆A碎nd乞i帽f绘we滨o哄bs胖er灶ve丽d砌anx(p,w)sa柏ti膜sf羞ie顾d况su槽ch双c滨on尘di袜ti舞on厘s,口c砖an刊w茎e芒fi仍nd呼a处p枣re照fe端re源nc嫁e艇to搜r少at听io学na哀li微za劲ti酱onx(p,w)?巴T宗ha雾t渔th细ein幅te适gr名ab激il使it战ypr粮ob笑le谈m.5.答In回te敞gr傍ab诵il奇it差yex雄pe迎nd党it旗ur誓e闻fu厉nc蹈ti城onpr昆ef起er附en绳ce坑.Pr秘op木os奋it阅io社n6余:滴di堤ff酷er严en宰ti缴ab真lee(p,u)is芬t就he情e惕xp需en少di粪tu彼re选f倦un谱ct沉io江n唇of冠s钓et间s:We层n傻ee壁d拌to宣p的ro遗vee(p,u)is王t陷he冰s阳up星po介rt桃f酱un危ct静io大n扣ofV(u),t茎ha诞t逼isse象e旁th级e酷fi钟g.5.型In萌te教gr梢ab勉il括it丸yDe仇ma默nd督ex搭pe劳nd喝it释ur持e僻fu佣nc断ti偿on块.Pa混rt元ia非l拨di熔ff帖er龟en俭ti访al凡e础qu自at固io蓝n:Th弹e佛ex踏is熊te佩nc算e酸of籍s治ol捞ut魔io午n忙me痛an遵s驼su热bs侦ti谅tu瞧ti肃on各m困at故ri启x枣is代s抚ym法me翅tr微ic筹:As音si输gn矩me驶ntex昏.7键.4将,选ex纷.8泄.7杰,e酿x.挣8.喇14qu