《微观经济学》清华大学课件Ch4Utility

ChapterFourUtility效用ChapterFourUtility1WhatDoWeDoinThisChapter?Wecreateamathematicalmeasureofpreferenceinordertoadvanceouranalysis.WhatDoWeDoinThisChapter?2UtilityFunctionsApreferencerelationthatiscomplete,reflexive,transitivecanberepresentedbyautilityfunction.UtilityFunctionsApreference3UtilityFunctionsAutilityfunctionU(x)representsapreferencerelationifandonlyif:

x’x”U(x’)>U(x”)

x’x”U(x’)<U(x”)

x’~x”U(x’)=U(x”).~fppUtilityFunctionsAutilityfun4UtilityFunctionsUtilityisanordinal(i.e.ordering)concept.E.g.ifU(x)=6andU(y)=2thenbundlexisstrictlypreferredtobundley.Butxisnotpreferredthreetimesasmuchasisy.UtilityFunctionsUtilityisan5UtilityFunctions&Indiff.CurvesAllbundlesinanindifferencecurvehavethesameutilitylevel.U(x1,x2)=Constantistheequationofanindifferencecurve.UtilityFunctions&Indiff.Cu6UtilityFunctions&Indiff.CurvesUº6Uº4(2,3)

(2,2)

~

(4,1)x1x2pUtilityFunctions&Indiff.Cu7UtilityFunctions&Indiff.CurvesThecollectionofallindifferencecurvesforagivenpreferencerelationisanindifferencemap.Anindifferencemapisequivalenttoautilityfunction.UtilityFunctions&Indiff.Cu8UtilityFunctionsIfUisautilityfunctionthatrepresentsapreferencerelationandfisastrictlyincreasingfunction,thenV=f(U)isalsoautilityfunction

representing.~f~fUtilityFunctionsIf~f~f9Goods,BadsandNeutralsAgoodisacommodityunitwhichincreasesutility(givesamorepreferredbundle).Abadisacommodityunitwhichdecreasesutility(givesalesspreferredbundle).Aneutralisacommodityunitwhichdoesnotchangeutility(givesanequallypreferredbundle).Goods,BadsandNeutralsAgood10Goods,BadsandNeutralsUtilityWaterx’Unitsof

waterare

goodsUnitsof

waterare

badsAroundx’units,alittleextrawaterisaneutral.Utility

functionGoods,BadsandNeutralsUtilit11SomeOtherUtilityFunctionsandTheirIndifferenceCurvesConsider

V(x1,x2)=x1+x2.

Whatdotheindifferencecurvesforthis“perfectsubstitution”utilityfunctionlooklike?SomeOtherUtilityFunctionsa12PerfectSubstitutionIndifferenceCurves55991313x1x2x1+x2=5x1+x2=9x1+x2=13V(x1,x2)=x1+x2.PerfectSubstitutionIndiffere13PerfectSubstitutionIndifferenceCurves55991313x1x2x1+x2=5x1+x2=9x1+x2=13Allarelinearandparallel.V(x1,x2)=x1+x2.PerfectSubstitutionIndiffere14SomeOtherUtilityFunctionsandTheirIndifferenceCurvesConsider

W(x1,x2)=min{x1,x2}.

Whatdotheindifferencecurvesforthis“perfectcomplementarity”utilityfunctionlooklike?SomeOtherUtilityFunctionsa15PerfectComplementarityIndifferenceCurvesx2x145omin{x1,x2}=8358358min{x1,x2}=5min{x1,x2}=3W(x1,x2)=min{x1,x2}PerfectComplementarityIndiff16PerfectComplementarityIndifferenceCurvesx2x145omin{x1,x2}=8358358min{x1,x2}=5min{x1,x2}=3Allareright-angledwithverticesonaray

fromtheorigin.W(x1,x2)=min{x1,x2}PerfectComplementarityIndiff17SomeOtherUtilityFunctionsandTheirIndifferenceCurvesAutilityfunctionoftheform

U(x1,x2)=f(x1)+x2

islinearinjustx2andiscalledquasi-linear.E.g.U(x1,x2)=2x11/2+x2.SomeOtherUtilityFunctionsa18Quasi-linearIndifferenceCurvesx2x1Eachcurveisaverticallyshiftedcopyoftheothers.Quasi-linearIndifferenceCurv19SomeOtherUtilityFunctionsandTheirIndifferenceCurvesAnyutilityfunctionoftheform

U(x1,x2)=x1a

x2b

witha>0andb>0iscalledaCobb-Douglasutilityfunction.E.g.U(x1,x2)=x11/2x21/2(a=b=1/2)

V(x1,x2)=x1x23(a=1,b=3)SomeOtherUtilityFunctionsa20Cobb-DouglasIndifferenceCurvesx2x1Allcurvesarehyperbolic,

asymptotingto,butnever

touchinganyaxis.Cobb-DouglasIndifferenceCurv21MarginalUtilitiesMarginalmeans“incremental”.Themarginalutilityofcommodityiistherate-of-changeoftotalutilityasthequantityofcommodityiconsumedchanges;i.e.

MarginalUtilitiesMarginalmea22MarginalUtilitiesandMarginalRates-of-SubstitutionThegeneralequationforanindifferencecurveis

U(x1,x2)ºk,aconstant.

Totallydifferentiatingthisidentitygives

MarginalUtilitiesandMargina23MarginalUtilitiesandMarginalRates-of-SubstitutionrearrangedisMarginalUtilitiesandMargina24MarginalUtilitiesandMarginalRates-of-SubstitutionrearrangedisAndThisistheMRS.MarginalUtilitiesandMargina25Marg.Rates-of-SubstitutionforQuasi-linearUtilityFunctionsAquasi-linearutilityfunctionisoftheformU(x1,x2)=f(x1)+x2.

soMarg.Rates-of-Substitutionfo26Marg.Rates-of-SubstitutionforQuasi-linearUtilityFunctionsMRS=-f

(x1)doesnotdependuponx2sotheslopeofindifferencecurvesforaquasi-linearutilityfunctionisconstantalonganylineforwhichx1isconstant.Whatdoesthatmaketheindifferencemapforaquasi-linearutilityfunctionlooklike?¢Marg.Rates-of-Substitutionfo27Marg.Rates-of-SubstitutionforQuasi-linearUtilityFunctionsx2x1Eachcurveisaverticallyshiftedcopyoftheothers.MRSisaconstant

alonganylineforwhichx1is

constant.

MRS=

-f(x1’)MRS=-f(x1”)x1’x1”Marg.Rates-of-Substitutionfo28MonotonicTransformations&MarginalRates-of-SubstitutionMoregenerally,ifV=f(U)wherefisastrictlyincreasingfunction,thenSoMRSisunchangedbyapositive

monotonictransformation.MonotonicTransformations&Ma29TheKeytothisChapterTheindifferencecurveofaconsumerpreferencecanberepresentedbyautilityfunctionbasedequation:U(x1,x2)=k,aconstant.TheKeytothisChapterThein30ChapterFourUtility效用ChapterFourUtility31WhatDoWeDoinThisChapter?Wecreateamathematicalmeasureofpreferenceinordertoadvanceouranalysis.WhatDoWeDoinThisChapter?32UtilityFunctionsApreferencerelationthatiscomplete,reflexive,transitivecanberepresentedbyautilityfunction.UtilityFunctionsApreference33UtilityFunctionsAutilityfunctionU(x)representsapreferencerelationifandonlyif:

x’x”U(x’)>U(x”)

x’x”U(x’)<U(x”)

x’~x”U(x’)=U(x”).~fppUtilityFunctionsAutilityfun34UtilityFunctionsUtilityisanordinal(i.e.ordering)concept.E.g.ifU(x)=6andU(y)=2thenbundlexisstrictlypreferredtobundley.Butxisnotpreferredthreetimesasmuchasisy.UtilityFunctionsUtilityisan35UtilityFunctions&Indiff.CurvesAllbundlesinanindifferencecurvehavethesameutilitylevel.U(x1,x2)=Constantistheequationofanindifferencecurve.UtilityFunctions&Indiff.Cu36UtilityFunctions&Indiff.CurvesUº6Uº4(2,3)

(2,2)

~

(4,1)x1x2pUtilityFunctions&Indiff.Cu37UtilityFunctions&Indiff.CurvesThecollectionofallindifferencecurvesforagivenpreferencerelationisanindifferencemap.Anindifferencemapisequivalenttoautilityfunction.UtilityFunctions&Indiff.Cu38UtilityFunctionsIfUisautilityfunctionthatrepresentsapreferencerelationandfisastrictlyincreasingfunction,thenV=f(U)isalsoautilityfunction

representing.~f~fUtilityFunctionsIf~f~f39Goods,BadsandNeutralsAgoodisacommodityunitwhichincreasesutility(givesamorepreferredbundle).Abadisacommodityunitwhichdecreasesutility(givesalesspreferredbundle).Aneutralisacommodityunitwhichdoesnotchangeutility(givesanequallypreferredbundle).Goods,BadsandNeutralsAgood40Goods,BadsandNeutralsUtilityWaterx’Unitsof

waterare

goodsUnitsof

waterare

badsAroundx’units,alittleextrawaterisaneutral.Utility

functionGoods,BadsandNeutralsUtilit41SomeOtherUtilityFunctionsandTheirIndifferenceCurvesConsider

V(x1,x2)=x1+x2.

Whatdotheindifferencecurvesforthis“perfectsubstitution”utilityfunctionlooklike?SomeOtherUtilityFunctionsa42PerfectSubstitutionIndifferenceCurves55991313x1x2x1+x2=5x1+x2=9x1+x2=13V(x1,x2)=x1+x2.PerfectSubstitutionIndiffere43PerfectSubstitutionIndifferenceCurves55991313x1x2x1+x2=5x1+x2=9x1+x2=13Allarelinearandparallel.V(x1,x2)=x1+x2.PerfectSubstitutionIndiffere44SomeOtherUtilityFunctionsandTheirIndifferenceCurvesConsider

W(x1,x2)=min{x1,x2}.

Whatdotheindifferencecurvesforthis“perfectcomplementarity”utilityfunctionlooklike?SomeOtherUtilityFunctionsa45PerfectComplementarityIndifferenceCurvesx2x145omin{x1,x2}=8358358min{x1,x2}=5min{x1,x2}=3W(x1,x2)=min{x1,x2}PerfectComplementarityIndiff46PerfectComplementarityIndifferenceCurvesx2x145omin{x1,x2}=8358358min{x1,x2}=5min{x1,x2}=3Allareright-angledwithverticesonaray

fromtheorigin.W(x1,x2)=min{x1,x2}PerfectComplementarityIndiff47SomeOtherUtilityFunctionsandTheirIndifferenceCurvesAutilityfunctionoftheform

U(x1,x2)=f(x1)+x2

islinearinjustx2andiscalledquasi-linear.E.g.U(x1,x2)=2x11/2+x2.SomeOtherUtilityFunctionsa48Quasi-linearIndifferenceCurvesx2x1Eachcurveisaverticallyshiftedcopyoftheothers.Quasi-linearIndifferenceCurv49SomeOtherUtilityFunctionsandTheirIndifferenceCurvesAnyutilityfunctionoftheform

U(x1,x2)=x1a

x2b

witha>0andb>0iscalledaCobb-Douglasutilityfunction.E.g.U(x1,x2)=x11/2x21/2(a=b=1/2)

V(x1,x2)=x1x23(a=1,b=3)SomeOtherUtilityFunctionsa50Cobb-DouglasIndifferenceCurvesx2x1Allcurvesarehyperbolic,

asymptotingto,butnever

touchinganyaxis.Cobb-DouglasIndifferenceCurv51MarginalUtilitiesMarginalmeans“incremental”.Themarginalutilityofcommodityiistherate-of-changeoftotalutilityasthequantityofcommodityiconsumedchanges;i.e.

MarginalUtilitiesMarginalmea52MarginalUtilitiesandMarginalRates-of-SubstitutionThegeneralequationforanindifferencecurveis

U(x1,x2)ºk,aconstant.

Totallydifferentiatingthisidentitygives

MarginalUtilitiesandMargina53MarginalUtilitiesandMarginalRates-of-SubstitutionrearrangedisMarginalUtilitiesandMargina54MarginalUtilitiesandMarginalRates-of-SubstitutionrearrangedisAndThisistheMRS.MarginalU