复旦大学-高级微观经济学课件1

JunjiXiaoFudanUniversityFall20101Preference1.Consumptionset:thesetofallalternatives,alsochoiceset.PropertiesofconsumptionsetXNonempty,;6=XRL+XisclosedXisconvex,x;y2X=)82[0;1]x+(1)y2X:02X.2.Preferences:tellsushowconsumerwouldrankanytwobasketsofgoodsamong,assumingthebasketwereavailableatnocost.1.1Preferencerelation1.LetXdenoteasetofalternatives/objects/goods.Apreferencerelationofaneconomicagentisabinaryrelation,denotedby,overX.Therelationdescribeswhattheagentwants.Foranyx;x02X,xx0meansthattheeconomicagentconsiderxtobeatleastasgoodasx0.(Iwillcalltheweakpreferencerelation.)12.Weassumethat,overall,people'spreferencesareinternallyconsistentorrational.(Ineconomicsthetwoconceptsmoreorlessmeanthesamething.)3.AxiomsofRationalPreferences:(a)Completeness:Foranytwobundlesxandx0,itisthateitherxx0,orx0x,orboth.(b)Transitivity:Foranythreechoicesx;x0,andx00,ifxx0andx0x00;thenxx00:Ifanagentlikesxbetterthanx0,andx0betterthanx00,thenhemustlikexbetterthanx00.4.Ifbothxx0andx0x,thentheagentisindierentbetweenxandx0.Weusethenotationtorepresenttheindierencerelation.Ifxx0andxx0,thentheagentstrictlyprefersxtox0.Weusetorepresentthestrictpreferencerelation.5.Thetransitivityofimpliesthetransitivityofandthetransitivityof,andviceversa.So,wecanequivalentlystateouraxiomsas(a)Completeness:Foranytwobundlesxandx0,itisthateitherxx0,x0x,orxx0.(b)Transitivity:Foranythreechoicesx;x0,andx00,ifxx0andx0x00;thenxx00;andifxx0andx0x00;thenxx00:26.ContinuityForanyconsumptionbundlex,thesetofconsumptionbundlesthatisweaklypreferredtoxisclosed.(Asetisclosedifforthelimitofanysequencethatbelongstothesetalsobelongstotheset.)7.Notallrationalpreferencesarecontinuous:Example:Lexicographicpreferences.Supposetherearetwogoods:x1andx2.Aconsumerprefersxtox0ifeitherx1x01orx1=x01andx2x02andisindierentbetweenxandx0ifx=x0.8.Considerthesequencexn=((1+1=n);1),n=1;2;::::Foralln,xnisstrictlypreferableto(1;2).[Trytoplottheutilitiesonarealline.]Foralln,xnmustliebetweentheutilityxn1andtheutilityof(1;2),andtheutilityof(1;1)islessthanthatof(1;2).Butyoucanseethatasngoestoinnity,theutilityisgoingtoconvergetothatof(1;1).Somehowthelexicographicpreferencerelationrequiresthattheutilitytomakeajumpat(1;1).9.Monotonicity,denitions(a)Monotonicity:Thepreferenceismonotoneif8x;y2X;yx=)yx:(b)Strictmonotonicity:Thepreferenceisstrictlymonotoneif8x;y2X;yx=)yx3whileyx=)yx:(c)Strongmonotonicity:Thepreferenceisstronglymonotoneif8x;y2X;yxandy6=x=)yx:10.Localnon-satiationThepreferenceislocallynonsatiatedifforallx2Xandforall0,thereexistsy2Xsuchthatjjyxjjandyx.Localnon-satiationruleoutblisspoint,weakerthanmonotonicity.11.Convexity:Averageispreferredtoextremes,andindierencecurvesarebowedtowardtheorigin.1.2Utilityfunction1.AutilityfunctiondenedoverXassignsarealnumbertoeachmemberofX.Wesayautilityfunctionu:Rn!Rrepresentsapreferencerelationifforanyobjectsxandx0,xx0iu(x)u(x0):2.Utilityfunctionisaconvenientwaytodescribeapreferencerelation.Forexample,ifItellyouthatmypreferencesoverapplesandbananasisu(a;b)=a0:5+b,thenyouwouldknowhowIwouldchoosebetweenanycombinationsofapplesandbananas.43.Theorem.Apreferencerelationcanberepresentedbyautilityfunctiononlyifitsatisescompletenessandtransitivity.Proof.Supposecanberepresentedbyu.Sinceforanyxandx0,eitheru(x)u(x0);oru(x0)u(x).Hencemustbecomplete.Sincetheorderingonrealnumberistransitive,thepreferencerelationmustbetransitive.4.ToensurethatapreferencerelationoveraconvexsubsetofRLisrepresentablebyautilityfunction,weneedanextraaxiom:5.Itisstraightforwardtoseethatthelexicographicpreferencerelationiscompleteandtransitive.Butthisrelationcannotberepresentedbyautilityfunction.Supposeweletu(x)=x1.While,thisutilityfunctionyieldstherightrelationbetweenallxandx0whenx16=x01,itdoesnotwhenx1=x01.However,wehavealreadyusedupalltherealnumbers|everyrealnumberwillbetheutilityofsomeconsumptionbundles.Thereisnothinglefttocapturetheeectofx2.6.Theorem.Apreferencerelationthatiscomplete,transitive,continuous,anddenedoveraconvexsubsetofRL,canberepresentedbyacontinuousutilityfunction.7.Theorem(Jehle&RenyTheorem1.1):Apreferencerelationthatiscomplete,transitive,continuousandstrictlymonotonic,thenthereexistsacontinuousu:RL+!Rthatrepresents.8.Step1.Lete(1;1;:::;1)denoteaunitconsumptionbundle(whichcontainsoneunitofeachgood).Foranyconsumptionbundlex,deneu(x)bytheequationu(x)ex:(u(x)e(u(x);:::;u(x)),i.e.,u(x)eistheconsumptionbundlethatconsistsofu(x)unitsofeachgood.)Step2.Bythecontinuityofpreferences,thesetsAft2R+:texg5andBft2R+:xtegarebothclosed.Bymonotonicity,A=ft2[t;1)gandB=t2[0;t] :Bycompleteness,anytmustbelongtoeitherAorB;thusA[B=R+.Wethereforecanconcludethattt;orthattheintersectionofAandBisnon-empty.Uniquenessfollowsfromthemonotonicityaxiom.Ift1exandt2ex,t1et2ebytransitivity.Thisimpliest1=t2.Step3.Foranytwoconsumptionbundlesxandx0,xx0,u(x)eu(x0)e(transitivity),u(x)u(x0)(strictmonotonicity).Hence,xx0ifandonlyifu(x)u(x0).Step4.(Dicult)Thelaststepismoretechnical.Afunctionuiscontinuousatx0iforall0,thereexistssome