2010-2011學年上學期第二次S1授課教師LuMarginalRateof

2010-2011學年上學期第二次TheslopeofanindifferencecurveisitsmarginalMRSatx’MRSatx’0=dx2/dx1atMarginalRate

2010-2011學年上學期第二次

dx2=MRSdx1so,atx’,MRSistherateatwhichtheconsumerisonlyjustwillingtoexchangecommodity2x’forasmallamount 3Tradingatanexchange

2010-2011學年上學期第二次42010-2011學年上學期第二次MRS&Ind.CurveGoodTwogoodsanegativelyindifferenceMRS<Good MRS&Ind.CurveGood

2010-2011學年上學期第二次Onegoodandone positivelyslopedindifferencecurveMRS>Bad 2010-2011學年上學期第二次MRS&Ind.CurveGoodMRS=-MRSalwaysincreaseswith eslessnegative)ifandonlyifpreferencesarestrictlyGoodMRS=-Good7MRS&Ind.Curve

2010-2011學年上學期第二次 MRS=- MRS esmoreasx1increasesnonconvexpreferencesMRS=- MRS&Ind.Curve

2010-2011學年上學期第二次

MRSisnotalwaysincreasingasx1increasesnonconvexMRS=-=- MRS=- 2010-2011學年上學期第二次2010-2011學年上學期第二次Preferences-yRispreferredstrictlytoxy:xandyareequallyx:xispreferredatleastasmuchasis2010-2011學年上學期第二次UtilityApreferencerelationthatiscomplete,reflexive,transitiveandcontinuouscanberepresentedbyacontinuousutilityfunction.Continuitymeansthatsmallchangestoaconsumptionbundlecauseonlysmallchangestothepreferencelevel.Utility

2010-2011學年上學期第二次AutilityfunctionU(x)representsapreferencerelation~fandonlyif: px’

U(x’)>U(x’)<U(x’)=2010-2011學年上學期第二次Utilityisanordinal(i.e.ordering)E.g.ifU(x)=6andU(y)=2thenbundlexisstrictlypreferredtobundley.Butxisnotpreferredthreetimesasmuchasisy.Considerthebundles(4,1),(2,3)andSuppose (4,1)Assigntothesebundlesanynumbersthatthepreferencee.g.U(2,3)=6>U(4,1)=U(2,2)=Callthesenumbersutility 2010-2011學年上學期第二次UtilityFunctions&Indiff.AnindifferencecurvecontainsequallypreferredEqualpreferencesameutilityTherefore,allbundlesinanindifferencehavethesameutilitySothebundles(4,1)and(2,2)areinthecurvewithutilitylevelUButthebundle(2,3)isintheindiff.curvewithutilitylevelU6.Onanindifferencecurvediagram,preferenceinformationlooksas 2010-2011學年上學期第二次UtilityFunctions&Indiff. 2010-2011學年上學期第二次MonotonicAwaytopreserveIfU1>U2,thenExamples:f(u)=3u,f(u)=u+C,ApositiverateofMonotonic

2010-2011學年上學期第二次

2010-2011學年上學期第二次2010-2011學年上學期第二次ConstructingautilityfunctionfromindifferencecurvesMeasuresfrom3210 UtilityFunctions&

2010-2011學年上學期第二次nrwaytovisualizethissameistoplottheutilitylevelonaverticalU(2,3)=U(2,2)=U(4,1)=3Dplotofconsumption&utilitylevelsfor3 2010-2011學年上學期第二次UtilityFunctions&Indiff.This3Dvisualizationofpreferencescanbemademoreinformativebyaddingintoitthetwoindifferencecurves.2010-2011學年上學期第二次UtilityFunctions&Indiff.Comparingmorebundleswillcreatealargercollectionofallindifferencecurvesandabetterdescriptionoftheconsumer’spreferences.2010-2011學年上學期第二次UtilityFunctions&Indiff.Asbefore,thiscanbevisualizedin3Dbyplottingeachindifferencecurveattheheightofitsutilityindex.2010-2011學年上學期第二次UtilityFunctions&Indiff.Comparingallpossibleconsumptionbundlesgivesthecompletecollectionoftheconsumer’sindifferencecurves,eachwithitsassignedutilitylevel.Thiscompletecollectionofindifferencecurves yrepresentstheconsumer’spreferences.2010-2011學年上學期第二次UtilityFunctions&Indiff.Thecollectionofallindifferencecurvesforagivenpreferencerelationisanindifferencemap.Anindifferencemapisequivalenttoafunction;eachistheThereisnouniqueutilityfunctionrepresentationofapreferencerelation.2010-2011學年上學期第二次UtilityFunctions:someSupposeU(x1,x2)=x1x2representsapreferenceAgainconsiderthebundles(4,1),(2,3)andU(x1,x2)=x1x2,U(2,3)=6>U(4,1)=U(2,2)=thatis, (4,1)Forsomeconstantk,k=2010-2011學年上學期第二次UtilityFunctions:someU(x1,x2)= (4,1)2010-2011學年上學期第二次e= ThenV(x1,x2)=x2x V(2,3)=36>V(4,1)=V(2,2)=so (4,1)VpreservesthesameorderasUandsorepresentssame2010-2011學年上學期第二次2U+ThenW(x1,x2)=2x1x2+10W(2,3)=22>W(4,1)=W(2,2)=18. (4,1)WpreservesthesameorderasUandVandrepresentsthesameGoods,Bads

2010-2011學年上學期第二次acommodityunitwhichincreasesutility(givesamorepreferredbundle).Abadisacommodityunitwhichdecreases(givesalesspreferredAneutralisacommodityunitwhichdoesnotchangeutility(givesanequallypreferredbundle).Goods,BadsUnits

Units

2010-2011學年上學期第二次waterare

waterare Aroundx’units,alittleextrawaterisa 2010-2011學年上學期第二次SomeOtherUtilityFunctionsTheirIndifferenceInsteadofU(x1,x2)=x1x2V(x1,x2)=x1+Whatdotheindifferencecurvesforthissubstitution”utilityfunctionlook2010-2011學年上學期第二次PerfectSubstitutionIndifferencex1+x2=x1+x2=9x1+x2=5V(x1,x2)=x1+ 2010-2011學年上學期第二次PerfectSubstitution

x1+x2=x1+x2=9x1+x2=5V(x1,x2)=x1+ Allarelinearand 2010-2011學年上學期第二次SomeOtherUtilityFunctionsTheirIndifferenceGenerally,weaandbareconstantsthatmeasurestheofgoods1and2totheTheslopeofatypicalindifferencecurveisgivenby–a/b2010-2011學年上學期第二次SomeOtherUtilityFunctionsTheirIndifferenceInsteadofU(x1,x2)=x1x2orV(x1,x2)=x1+x2,nowweconsiderW(x1,x2)=Whatdotheindifferencecurvesforthis“perfectcomplementarity”utilityfunctionlooklike?RecallourexampleofleftshoeandrightOnlythenumberofpairs2010-2011學年上學期第二次W(x1,x2)= min{x1,x2}= min{x1,x2}= min{x1,x2}= Allareright-angledwithverticesona fromthe2010-2011學年上學期第二次SomeOtherUtilityFunctionsTheirIndifferenceIngeneral,weNow,considerautilityfunctionoftheU(x1,x2)=f(x1)+islinearinjustx2andiscalledquasi- U(x1,x2)=2x11/2+Quasi-linear

2010-2011學年上學期第二次

Eachcurveisaverticallyshiftedcopyofthe2010-2011學年上學期第二次SomeOtherUtilityFunctionsTheirIndifference12Anyutilityfunctionofthe12U(x1,x2)=

axwitha>0andb>0iscalledaCobb-utility U(x1,x2)=x11/2V(x1,x2)=x1

(a=b=(a=1,b=2010-2011學年上學期第二次Cobb-DouglasIndifference2Allcurvesarehyperbolic,asymptotingto,butnevertouchinganyaxis. 2010-2011學年上學期第二次MarginalMarginalmeansThemarginalutilityofcommodityiistherate-of-changeoftotalutilityasthe tyofcommodityiconsumedchanges;i.e.MUi

MarginalE.g.ifU(x1,x2)=

2010-2011學年上學期第二次 112x1x222MU22

11

2x1/22010-2011學年上學期第二次MarginalUtilitiesandMarginalThegeneralequationforanindifferencecurveisU(x1,x2)k,aconstant.TotallydifferentiatingthisidentityUdx1Udx2 2010-2011學年上學期第二次MarginalUtilitiesandMarginal rearranged Udx2 U rearrangeddxdThisisthe

U/x122U/2010-2011學年上學期第二次Marg.Utilities&Marg.Rates-of-Substitution;AnexamplexU(x1,x2)=x2

MRS MRS(1,8)=-8/1=-MRS(6,6)=-6/6=-6U=U= 2010-2011學年上學期第二次Marg.Rates-of-SubstitutionlinearUtilityAquasi-linearutilityfunctionisoftheformU(x1,x2)=f(x1)+x2.Uf(

U MRSdx2U/x1f(x1).d U/2010-2011學年上學期第二次 UASI-LINEARrerillyMRS=-

copyoftheMRSisaMRS=-

alonganylineforwhichx1is 2010-2011學年上學期第二次MonotonicTransformations&MarginalRates-of-Applyingamonotonictransformationtoautilityfunctionrepresentingapreferencerelationsimplycreatesanotherutilityfunctionrepresentingthesamepreferencerelation.Whathappenstomarginalrates-of-substitutionamonotonictransformationis2010-2011學年上學期第二次MonotonicTransformations&MarginalRates-of-ForU(x1,x2)=x1x2theMRS=-CreateV=U2;i.e.V(x1,x2)=x12x22.WhatisMRSforMRS

V/

2x x

x2V/

2x2x whichisthesameastheMRSfor2010-2011學年上學期第二次MonotonicTransformationsMarginalRates-of-Moregenerally,ifV=f(U)wherefisastrictlyincreasingfunction,thenMRSV/x1f(U)U/V/ f'(U)U/U/x1U/SoMRSisunchangedbya 2010-2011學年上學期第二次Economic

2010-2011學年上學期第二次Theprincipalbehavioralpostulateisthatadecisionmakerchoosesitsmostpreferredalternativefromthoseavailabletoit.TheavailablechoicesconstitutethechoiceHowisthemostpreferredbundleinthechoicesetlocated?RationalConstrained

2010-2011學年上學期第二次RationalConstrained

2010-2011學年上學期第二次MorepreferredMoreRMore

2010-2011學年上學期第二次 xx2

Constrained

2010-2011學年上學期第二次(x1*,x2(x1*,x2*)isthemostpreferredaffordablebundle,whichisthencalledthe:“optimal

RationalConstrained

2010-2011學年上學期第二次Whenx1*>0andx2*>0thedemandedisIfbuying(x1*,x2*)costs$mthenthebudgetis

2010-2011學年上學期第二次onstrained(x1*,x2*)isinterior.(a)(x1*,x2*)exhauststhebudget;p1x1*+p2x2*=m.(b)Theslopeoftheindiff.curveat(x1*,x2*)equalstheslopeofthebudget

2010-2011學年上學期第二次RationalConstrained(x1*,x2*)satisfiestwo(a)thebudgetisp1x1*+p2x2*= (b)theslopeofthebudgetconstraint,-p1/p2,andtheslopeoftheindifferencecurvecontaining(x1*,x2*)areequalat(x1*,x2*).Ageneral

2010-2011學年上學期第二次Lu(x,

)(p

m u(x*,x*)

p u(x*,x*) 22 22

p px x

m u(

*,x

)/

p p px1,x2st.p1x1,x2st.p1x1p2xmaxu(x1,x21

,x*)/ 2010-2011學年上學期第二次ComputingDemands-aCobb-DouglasExample. Supposethattheconsumerhas

U(x1,x2)MU1MU2

axa1xb bxa 2010-2011學年上學期第二次ComputingDemands-aCobb-DouglasExample.SotheMRSMRSMRS21a1xbbxaxb1 At(x1*,x2*),MRS=-p1/p2*2p1x*bp2111ComputingDemands-aCobb-DouglasExample.x*bp121p1x*x*bp121p1x*px*1 p1xp1x*pbp1x*121Thissimplifiesto

2010-2011學年上學期第二次2010-2011學年上學期第二次ComputingDemands-aCobb-DouglasExample.xx*1 (ab)p1Substitutingforx1*pp1x*px*1 thenxx*2 (ab)p22010-2011學年上學期第二次ComputingDemands-aCobb-DouglasExample.SowehavediscoveredthatthemostpreferredaffordablebundleforaconsumerwithCobb-Douglaspreferences U(x1,x2) (x(x,x)*1*2ambm(a (a,Rational

2010-2011學年上學期第二次

x1*>0andx2*> (x1*,x2*)exhauststhe indifferencecurveshave‘kinks’,thedemandsareobtained p1x1*+p2x2*= (b)theslopesofthebudgetconstraint,-p1/p2,andoftheindifferencecurvecontaining(x1*,x2*)areequalat(x1*,x2*).2010-2011學年上學期第二次RationalConstrainedButwhatifx1*=Orifx2*=Ifeitherx1*=0orx2*=0thenthedemand(x1*,x2*)isatacornersolutiontotheproblem izingutilitysubjecttoabudget2010-2011學年上學期第二次ExamplesofCornerSolutions-PerfectSubstitutesxx*2p2MRS=-Slope=-MRS=-Slope=-1x* 12010-2011學年上學期第二次ExamplesofCornerSolutions--PerfectSubstitutesMRS=-Slope=-MRS=-Slope=-2x*2x*x*1p12010-2011學年上學期第二次ExamplesofCornerSolutions--PerfectSubstitutesSowhenU(x1,x2)=x1+x2,themostpreferredaffordablebundleis(x1*,x2*)(x*,x*)(x*,x*)m12

(x,x) (x,x) **12mp22010-2011學年上學期第二次ExamplesofCornerSolutions--PerfectSubstitutesMRS=-Slope=MRS=-Slope=-p1/p2mmm2010-2011學年上學期第二次ExamplesofCornerSolutions--PerfectSubstitutesm Allthebundlesinthecon