lecureconsumionheoryII高級微觀武大高研,葉建亮

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難la熄ti仆on乒sh偉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項sk輕yeq樂ua紛ti服onRo悲y’碎sid交en淋ti欠ty5.駐In聾te牌gr背ab們il采it墳yDe哈ma御nd嗚f辭un燒ct計io筆nx(p,w)(c申.d島.)沸i囑s暮HD陪0,嘩s內(nèi)at耽is隱fi毯edWa俊lr斧asLa見w喊an響d價ha搏ve犧a你s族ub婚st星it曾ut按io瘦n樓ma深tr括ixS(p,w)iss.巧n.系s.顏d.斃fo奔r龜an很y(p,w),ifit滅’s梳d繳ed押uc絕ed貍b剛y島ra歡ti弓on燃al逃p序re散fe史re孩nc漸e.隆A碎nd乞i帽f繪we濱o哄bs胖er灶ve麗d砌anx(p,w)sa柏ti膜sf羞ie顧d況su槽ch雙c濱on塵di襪ti舞on厘s,口c磚an刊w莖e芒fi仍nd呼a處p棗re照fe端re源nc嫁e艇to搜r少at聽io學(xué)na哀li微za勁ti醬onx(p,w)?巴T宗ha霧t漁th細ein幅te適gr名ab激il使it戰(zhàn)ypr糧ob笑le談m.5.答In回te敞gr傍ab誦il奇it差yex雄pe迎nd黨it旗ur誓e聞fu厲nc蹈ti城onpr昆ef起er附en繩ce坑.Pr秘op木os奮it閱io社n6余:滴di堤ff酷er嚴en宰ti繳ab真lee(p,u)is芬t就he情e惕xp需en少di糞tu彼re選f倦un譜ct沉io江n唇of冠s釣et間s:We層n傻ee壁d拌to宣p的ro遺vee(p,u)is王t陷he冰s陽up星po介rt桃f醬un危ct靜io大n扣ofV(u),t莖ha誕t逼isse象e旁th級e酷fi鐘g.5.型In萌te教gr梢ab勉il括it丸yDe仇ma默nd督ex搭pe勞nd喝it釋ur持e僻fu傭nc斷ti償on塊.Pa混rt元ia非l撥di熔ff帖er龜en儉ti訪al凡e礎(chǔ)qu自at固io藍n:Th彈e佛ex踏is熊te佩nc算e酸of籍s治ol撈ut魔io午n忙me痛an遵s駝su熱bs偵ti諒tu瞧ti肅on各m困at故ri啟x棗is代s撫ym法me翅t(yī)r微ic籌:As音si輸gn矩me駛ntex昏.7鍵.4將,選ex紛.8泄.7杰,e釀x.掙8.喇14qu