ChapterSixDemand需求函數的靜態比較分析ChapterSixDemand1WhatDoWeDoinThisChapter?Weconductcomparativestaticsanalysisofordinarydemandfunctions--thestudyofhowordinarydemandsx1*(p1,p2,y)andx2*(p1,p2,y)changeaspricesp1,p2andincomeychange.Theoretically,nothingnew.WhatDoWeDoinThisChapter?2Own-PriceChangesHowdoesx1*(p1,p2,y)changeasp1changes,holdingp2andyconstant?Supposeonlyp1increases,fromp1’top1’’andthentop1’’’.Own-PriceChangesHowdoesx1*(3x1*(p1’’’)x1*(p1’)x1*(p1’’)p1x1*(p1’)x1*(p1’’’)x1*(p1’’)p1’p1’’p1’’’x1*Own-PriceChangesOrdinary

demandcurve

forcommodity1Fixedp2andy.x1*(p1’’’)x1*(p1’)x1*(p1’’)p1x4x1*(p1’’’)x1*(p1’)x1*(p1’’)p1x1*(p1’)x1*(p1’’’)x1*(p1’’)p1’p1’’p1’’’x1*Own-PriceChangesOrdinary

demandcurve

forcommodity1p1price

offer

curveFixedp2andy.x1*(p1’’’)x1*(p1’)x1*(p1’’)p1x5Own-PriceChangesThecurvecontainingalltheutility-maximizingbundlestracedoutasp1changes,withp2andyconstant,isthep1-priceoffercurve.Theplotofthex1-coordinateofthep1-priceoffercurveagainstp1istheordinarydemandcurveforcommodity1.Own-PriceChangesThecurvecon6TheCaseofCobb-DouglasUtilityFunctionTake

Thentheordinarydemandfunctionsforcommodities1and2areTheCaseofCobb-DouglasUtili7Own-PriceChangesandNoticethatx2*doesnotvarywithp1sothe

p1priceoffercurveisflatandtheordinary

demandcurveforcommodity1isa

rectangularhyperbola.Own-PriceChangesandNoticetha8x1*(p1’’’)x1*(p1’)x1*(p1’’)Own-PriceChangesFixedp2andy.x1*(p1’’’)x1*(p1’)x1*(p1’’)Own9x1*(p1’’’)x1*(p1’)x1*(p1’’)p1x1*Own-PriceChangesOrdinary

demandcurve

forcommodity1

isFixedp2andy.x1*(p1’’’)x1*(p1’)x1*(p1’’)p1x10TheCaseofPerfect-ComplementsUtilityFunctionWhatdoesap1price-offercurvelooklikeforaperfect-complementsutilityfunction?Thentheordinarydemandfunctions

forcommodities1and2areTheCaseofPerfect-Complement11TheCaseofPerfect-ComplementsUtilityFunctionTheCaseofPerfect-Complement12Own-PriceChangesWithp2andyfixed,higherp1causes

smallerx1*andx2*.AsAsOwn-PriceChangesWithp2andy13p1x1*Ordinary

demandcurve

forcommodity1

isFixedp2andy.Own-PriceChangesx1x2p1’p1’’p1’’’y/p2p1x1*Ordinary

demandcurve

for14TheCaseofPerfect-SubstitutesUtilityFunctionThentheordinarydemandfunctions

forcommodities1and2areTheCaseofPerfect-Substitute15andand16Fixedp2andy.Own-PriceChangesx2x1p1x1*Fixedp2andy.p1’p2=p1’’p1’’’p1price

offercurveOrdinary

demandcurve

forcommodity1Fixedp2andy.Own-PriceChang17Own-PriceChangesUsuallyweask“Giventhepriceforcommodity1whatisthequantitydemandedofcommodity1?”Butwecouldalsoasktheinversequestion“Atwhatpriceforcommodity1wouldagivenquantityofcommodity1bedemanded?”Own-PriceChangesUsuallyweas18Own-PriceChangesTakingquantitydemandedasgivenandthenaskingwhatmustbepricedescribestheinversedemandfunctionofacommodity.Own-PriceChangesTakingquanti19InverseDemandFunctionACobb-Douglasexample:istheordinarydemandfunctionandistheinversedemandfunction.InverseDemandFunctionACobb-20IncomeChangesHowdoesthevalueofx1*(p1,p2,y)changeasychanges,holdingbothp1andp2constant?IncomeChangesHowdoestheval21IncomeChangesFixedp1andp2.y’<y’’<y’’’x1’’’x1’’x1’x2’’’x2’’x2’Income

offercurveIncomeChangesFixedp1andp2.22TheEngelCurveAplotofquantitydemandedagainstincomeiscalledanEngelcurve.TheEngelCurveAplotofquant23IncomeChangesFixedp1andp2.y’<y’’<y’’’x1’’’x1’’x1’x2’’’x2’’x2’Income

offercurvex1*x2*yyx1’’’x1’’x1’x2’’’x2’’x2’y’y’’y’’’y’y’’y’’’Engel

curve;good2Engel

curve;good1IncomeChangesFixedp1andp2.24IncomeChangesandCobb-DouglasPreferencesAnexampleofcomputingtheequationsofEngelcurves;theCobb-Douglascase.TheordinarydemandequationsareIncomeChangesandCobb-Dougla25IncomeChangesandCobb-DouglasPreferencesRearrangedtoisolatey,theseare:Engelcurveforgood1Engelcurveforgood2IncomeChangesandCobb-Dougla26IncomeChangesandCobb-DouglasPreferencesyyx1*x2*Engelcurve

forgood1Engelcurve

forgood2IncomeChangesandCobb-Dougla27IncomeChangesandPerfectly-ComplementaryPreferencesTheordinarydemandequationsareIncomeChangesandPerfectly-C28IncomeChangesandPerfectly-ComplementaryPreferencesRearrangedtoisolatey,theseare:Engelcurveforgood1Engelcurveforgood2IncomeChangesandPerfectly-C29IncomeChangesx1*x2*yyx2’’’x2’’x2’y’y’’y’’’y’y’’y’’’x1’’’x1’’x1’Engel

curve;good2Engel

curve;good1Fixedp1andp2.IncomeChangesx1*x2*yyx2’’’x2’30IncomeChangesandPerfectly-SubstitutablePreferencesAnotherexampleofcomputingtheequationsofEngelcurves;theperfectly-substitutioncase.TheordinarydemandequationsareIncomeChangesandPerfectly-S31IncomeChangesandPerfectly-SubstitutablePreferencesIncomeChangesandPerfectly-S32IncomeChangesandPerfectly-SubstitutablePreferencesSupposep1<p2.ThenandandIncomeChangesandPerfectly-S33IncomeChangesandPerfectly-SubstitutablePreferencesyyx1*x2*0Engelcurve

forgood1Engelcurve

forgood2IncomeChangesandPerfectly-S34IncomeChangesIneveryexamplesofartheEngelcurveshaveallbeenstraightlines?

Q:Isthistrueingeneral?A:No.Engelcurvesarestraightlinesiftheconsumer’spreferencesarehomothetic.IncomeChangesIneveryexample35HomotheticityAconsumer’spreferencesarehomotheticifandonlyif

foreveryk>0.Thatis,theconsumer’sMRSisthesameanywhereonastraightlinedrawnfromtheorigin.?(x1,x2)(y1,y2)(kx1,kx2)(ky1,ky2)

ppHomotheticityAconsumer’spref36IncomeEffects--ANonhomotheticExampleQuasilinearpreferencesarenothomothetic.

Forexample,IncomeEffects--ANonhomothe37IncomeChanges;QuasilinearUtilityx2x1x1~x1*x2*yyx1~Engel

curvefor

good2Engel

curvefor

good1IncomeChanges;QuasilinearUt38IncomeEffectsAgoodforwhichquantitydemandedriseswithincomeiscallednormal.Thereforeanormalgood’sEngelcurveispositivelysloped.IncomeEffectsAgoodforwhich39IncomeEffectsAgoodforwhichquantitydemandedfallsasincomeincreasesiscalledincomeinferior.Thereforeanincomeinferiorgood’sEngelcurveisnegativelysloped.IncomeEffectsAgoodforwhich40IncomeChanges;Goods

1&2Normalx1’’’x1’’x1’x2’’’x2’’x2’Income

offercurvex1*x2*yyx1’’’x1’’x1’x2’’’x2’’x2’y’y’’y’’’y’y’’y’’’Engel

curve;good2Engel

curve;good1IncomeChanges;Goods

1&2No41IncomeChanges;Good2IsNormal,Good1BecomesIncomeInferiorx2x1x1*x2*yyEngelcurve

forgood2Engelcurve

forgood1IncomeChanges;Good2IsNorm42OrdinaryGoodsAgoodiscalledordinaryifthequantitydemandedofitalwaysincreasesasitsownpricedecreases.OrdinaryGoodsAgoodiscalled43OrdinaryGoodsFixedp2andy.x1x2p1price

offer

curvex1*Downward-sloping

demandcurveGood1is

ordinary?p1OrdinaryGoodsFixedp2andy.x44GiffenGoodsIf,forsomevaluesofitsownprice,thequantitydemandedofagoodrisesasitsown-priceincreasesthenthegoodiscalledGiffen.GiffenGoodsIf,forsomevalue45OrdinaryGoodsFixedp2andy.x1x2OrdinaryGoodsFixedp2andy.x46OrdinaryGoodsFixedp2andy.x1x2p1priceoffercurvex1*Demandcurvehas

apositivelyslopedpartGood1is

Giffen?p1OrdinaryGoodsFixedp2andy.x47Cross-PriceEffectsIfanincreaseinp2increasesdemandforcommodity1thencommodity1isagrosssubstituteforcommodity2.

reducesdemandforcommodity1thencommodity1isagross