nicholson微观经济理论（答案+ppt+习题库）-60-03-033593-0

DEMANDRELATIONSHIPSAMONGGOODSINCHAPTER5WEEXAMINEDHOWCHANGESINTHEPRICEOFAPARTICULARGOODSAY,GOODXAFFECTTHEQUANTITYOFTHATGOODCHOSENTHROUGHOUTTHEDISCUSSIONWEHELDTHEPRICESOFALLOTHERGOODSCONSTANTITSHOULDBECLEAR,HOWEVER,THATACHANGEINONEOFTHESEOTHERPRICESCOULDALSOAFFECTTHEQUANTITYOFXCHOSENFOREXAMPLE,IFXWERETAKENTOREPRESENTTHEQUANTITYOFAUTOMOBILEMILESTHATANINDIVIDUALDRIVES,THISQUANTITYMIGHTBEEXPECTEDTODECLINEWHENTHEPRICEOFGASOLINERISESORTOINCREASEWHENAIRANDBUSFARESRISEINTHISCHAPTERWEWILLUSETHEUTILITYMAXIMIZATIONMODELTOSTUDYSUCHRELATIONSHIPS6CHAPTERTHETWOGOODCASEFIGURE61PRESENTSTWOEXAMPLESOFHOWTHEQUANTITYOFXCHOSENMIGHTBEAFFECTEDBYACHANGEINTHEPRICEOFGOODYINBOTHPANELSOFTHEGURE,PYHASFALLEN,THEREBYSHIFTINGTHEBUDGETCONSTRAINTOUTWARDFROMI0TOI1INBOTHCASESALSOTHEQUANTITYOFGOODYCHOSENHASINCREASEDFROMY0TOY1ASARESULTOFTHEDECLINEINPY,ASWOULDBEEXPECTEDIFYISANORMALGOODFORGOODX,HOWEVER,THERESULTSSHOWNINTHETWOPANELSDIFFERINATHEINDIFFERENCECURVESARENEARLYLSHAPED,IMPLYINGAFAIRLYSMALLSUBSTITUTIONEFFECTADECLINEINPYDOESNOTINDUCEAVERYLARGEMOVEALONGU0ASYISSUBSTITUTEDFORXTHATIS,XDROPSRELATIVELYLITTLEASARESULTOFTHESUBSTITUTIONTHEINCOMEEFFECT,HOWEVER,REECTSTHEGREATERPURCHASINGPOWERNOWAVAILABLE,ANDTHISCAUSESTHETOTALQUANTITYOFXCHOSENTOINCREASEHENCE,H11128X/H11128PYISNEGATIVEXANDPYMOVEINOPPOSITEDIRECTIONSINFIGURE61BTHISSITUATIONISREVERSEDTHEREH11128X/H11128PYISPOSITIVETHERELATIVELYATINDIFFERENCECURVESINFIGURE61BRESULTINALARGESUBSTITUTIONEFFECTFROMTHEFALLINPYTHEQUANTITYOFXDECLINESSHARPLYASYISSUBSTITUTEDFORXALONGU0ASINFIGURE61A,THEINCREASEDPURCHASINGPOWERFROMTHEDECLINEINPYCAUSESMOREXTOBEBOUGHT,BUTNOWTHESUBSTITUTIONEFFECTDOMINATESANDTHEQUANTITYOFXDECLINESTOX1INTHISCASE,THEN,XANDPYMOVEINTHESAMEDIRECTIONAMATHEMATICALTREATMENTTHEAMBIGUITYINTHEEFFECTOFCHANGESINPYCANBEFURTHERILLUSTRATEDBYASLUTSKYTYPEEQUATIONBYUSINGPROCEDURESSIMILARTOTHOSEINCHAPTER5,ITISFAIRLYSIMPLETOSHOWTHATH11005H11005H20841UH11005CONSTANTH11002Y,61WHERETHERSTTERMONTHERIGHT,ASBEFORE,REPRESENTSTHESUBSTITUTIONEFFECTANDTHESECONDTERMISTHEINCOMEEFFECTNOTICETHATINTHEINCOMEEFFECT,THEDERIVATIVEH11128X/H11128IISNOWMULTIPLIEDBYTHEAMOUNTOFYPURCHASEDBECAUSETHATQUANTITYREECTSTHEEXTENTTOWHICHCHANGESINPYAFFECTPURCHASINGPOWERFORTHETWOGOODCASE,THETERMSONTHERIGHTSIDEOFEQUATION61HAVEDIFFERENTSIGNSASSUMINGTHATINDIFFERENCECURVESARECONVEX,THESUBSTITUTIONEFFECTH11128X/H11128PYH20919UH11005CONSTANTISPOSITIVEIFWECONNEOURSELVESTOMOVESALONGONEINDIFFERENCECURVE,INCREASESINPYINCREASEXANDDECREASESINPYDECREASETHEQUANTITYOFXCHOSENBUT,ASSUMINGXISANORMALGOOD,THEINCOMEEFFECTH11002YH11128X/H11128IISCLEARLYNEGATIVEHENCE,THECOMBINEDEFFECTINEQUATION61ISAMBIGUOUSH11128X/H11128PYCOULDBEEITHERPOSITIVEORNEGATIVEEVENINTHETWOGOODCASE,THEDEMANDRELATIONSHIPBETWEENXANDPYISRATHERCOMPLEXEXAMPLE61ANOTHERSLUTSKYDECOMPOSITIONFORCROSSPRICEEFFECTSINEXAMPLE54WEEXAMINEDTHESLUTSKYDECOMPOSITIONFORTHEEFFECTOFACHANGEINTHEPRICEOFXNOWLETSLOOKATTHECROSSPRICEEFFECTOFACHANGEINYPRICESONXPURCHASESREMEMBERTHATTHEUNCOMPENSATEDANDCOMPENSATEDDEMANDFUNCTIONSFORXAREGIVENBYDXPX,PY,IH1100562IH50072PXH11128XH5007H11128IH11128XH5007H11128PYH11128DXH5007H11128PYH11128XH5007H11128PY152PARTIICHOICEANDDEMANDANDHXPX,PY,VH11005,63RESPECTIVELYASWEHAVENOTEDINSEVERALPLACES,THEFACTTHATUNDERTHEUNCOMPENSATEDFUNCTIONH11005064SHOWSTHATCHANGESINYPRICESHAVENOEFFECTONXPURCHASESTOSEEWHYTHISISSO,WECANCOMPUTETHESUBSTITUTIONEFFECTFROMTHECOMPENSATEDDEMANDFUNCTIONASH20841UH11005CONSTANTH11005H1100565BUTPREVIOUSLYWEFOUNDTHATINDIRECTUTILITYISGIVENBYVH11005,66SOTHESUBSTITUTIONEFFECTFORTHISCASEISH20841UH11005CONSTANTH110056725IH5007PXPYH11128XH5007H11128PYIH50072P5XP5Y5VH5007P5XP5YH11128HXH5007H11128PYH11128XH5007H11128PYH11128DXH5007H11128PYVP5YH5007P5XCHAPTER6DEMANDRELATIONSHIPSAMONGGOODS153DIFFERINGDIRECTIONSOFCROSSPRICEEFFECTSINBOTHPANELSTHEPRICEOFYHASFALLENINASUBSTITUTIONEFFECTSARESMALLSOTHEQUANTITYOFXCONSUMEDINCREASESALONGWITHYBECAUSEH11128X/H11128PYH110210,XANDYAREGROSSCOMPLEMENTSINBSUBSTITUTIONEFFECTSARELARGESOTHEQUANTITYOFXCHOSENFALLSBECAUSEH11128X/H11128PYH110220,XANDYWOULDBETERMEDGROSSSUBSTITUTESQUANTITYOFXQUANTITYOFXQUANTITYOFYQUANTITYOFYAGROSSCOMPLEMENTSBGROSSSUBSTITUTESX0Y0Y1X1Y0Y1X1X0I0I0I1I1U0U1U0U1FIGURE61USINGTHEMARSHALLIANDEMANDFUNCTIONFORYYH11005I/2PY,WECANCALCULATETHEINCOMEEFFECTINTHISPROBLEMSEEEQUATION61ASH11002YH11005H11002H20898H20899H20898H20899H11005H1100268COMPARINGEQUATIONS67AND68,ITISCLEARTHATTHEINCOMEANDSUBSTITUTIONEFFECTSEXACTLYCANCELEACHOTHEROUTHERECHANGESINYPRICESHAVENOEFFECTONXPURCHASESBECAUSEOFTHISCANCELINGOFEFFECTS,NOTBECAUSETHETWOGOODSAREUNRELATEDASANUMERICALILLUSTRATION,SUPPOSEYPRICESRISEDRAMATICALLYFROM1TO4THECOMPENSATEDDEMANDFUNCTIONSHOWSTHATXPURCHASESSHOULDRISEFROM4TO8H110052H110802/5IFWEARETOHOLDINDIRECTUTILITYCONSTANTATVH110052THATIS,THEINDIVIDUALSHOULDSUBSTITUTEMANYUNITSOFXFORTHENOWMOREEXPENSIVEYBUTWITHACONSTANTNOMINALINCOMEOF2,INDIRECTUTILITYCANNOTREMAINATVH110052UTILITYMUSTFALLBECAUSEOFTHERISEINYPRICESTOVH110052/2H110805H110802H110051WITHTHISREDUCEDLEVELOFUTILITY,THECOMPENSATEDDEMANDFUNCTIONPREDICTSTHATXPURCHASESWILLREMAINAT4H110051H110802/5RATHERTHANRISINGTOREECTONLYTHESUBSTITUTIONEFFECTTHETWOEFFECTSARETHEREFOREPRECISELYOFFSETTINGWITHOTHERFUNCTIONALFORMSEITHEROFTHEEFFECTSCOULDDOMINATE,ANDH11128DX/H11128PYCOULDHAVEEITHERSIGNQUERYWHYWOULDITBEINCORRECTTOARGUETHATIFH11128DX/H11128PYH110050,XANDYHAVENOSUBSTITUTIONPOSSIBILITIESTHATIS,THEYMUSTBECONSUMEDINXEDPROPORTIONSISTHEREANYCASEINWHICHSUCHACONCLUSIONCOULDBEDRAWNSUBSTITUTESANDCOMPLEMENTSFORTHECASEOFMANYGOODS,ITISASIMPLEMATTERTOGENERALIZETHESLUTSKYANALYSISASREPRESENTEDBYEQUATIONS527AND61ASH11005H11005H20841UH11005CONSTANTH11002XJ69FORANYIANDJINCLUDINGIH11005JTHISSAYSTHATTHECHANGEINTHEPRICEOFANYGOODHEREGOODJINDUCESINCOMEANDSUBSTITUTIONEFFECTSTHATMAYCHANGETHEQUANTITYOFEVERYGOODDEMANDEDEQUATION69CANBEUSEDTODISCUSSTHEIDEAOFSUBSTITUTESANDCOMPLEMENTSINTUITIVELY,THESEIDEASARERATHERSIMPLETWOGOODSARESUBSTITUTESIFONEGOODMAY,ASARESULTOFCHANGEDCONDITIONS,REPLACETHEOTHERINUSESOMEEXAMPLESARETEAANDCOFFEE,HAMBURGERSANDHOTDOGS,ANDBUTTERANDMARGARINECOMPLEMENTS,ONTHEOTHERHAND,AREGOODSTHAT“GOTOGETHER,”SUCHASCOFFEEANDCREAM,SHANDCHIPS,ORBRANDYANDCIGARSINSOMESENSE“SUBSTITUTES”SUBSTITUTEFORONEANOTHERINTHEUTILITYFUNCTION,WHEREAS“COMPLEMENTS”COMPLEMENTEACHOTHERTHEREARETWODIFFERENTWAYSTOMAKETHESEINTUITIVEIDEASPRECISEONEOFTHESEFOCUSESONTHE“GROSS”EFFECTSOFPRICECHANGESBYINCLUDINGBOTHINCOMEANDSUBSTITUTIONEFFECTS,WHILETHEOTHERLOOKSATSUBSTITUTIONEFFECTSALONEBECAUSEBOTHDENITIONSAREUSED,WEWILLEXAMINEEACHINDETAILH11128XIH5007H11128IH11128XIH5007H11128PJH11128DIH5007H11128PJH11128XIH5007H11128PJ25IH5007PXPY1H50072PXIH50072PYH11128XH5007H11128I154PARTIICHOICEANDDEMANDGROSSSUBSTITUTESANDCOMPLEMENTSSUBSTITUTEANDCOMPLEMENTARYRELATIONSHIPSCANBEDENEDBYREFERRINGTOOBSERVEDPRICEREACTIONSASGROSSSUBSTITUTESANDCOMPLEMENTSTWOGOODS,XIANDXJ,ARESAIDTOBEGROSSSUBSTITUTESIFH110220610ANDGROSSCOMPLEMENTSIFH110210611THATIS,TWOGOODSAREGROSSSUBSTITUTESIFARISEINTHEPRICEOFONEGOODCAUSESMOREOFTHEOTHERGOODTOBEBOUGHTTHEYAREGROSSCOMPLEMENTSIFARISEINTHEPRICEOFONEGOODCAUSESLESSOFTHEOTHERGOODTOBEPURCHASEDFOREXAMPLE,IFTHEPRICEOFCOFFEERISES,THEDEMANDFORTEAMIGHTBEEXPECTEDTOINCREASETHEYARESUBSTITUTES,WHEREASTHEDEMANDFORCREAMMIGHTDECREASECOFFEEANDCREAMARECOMPLEMENTSEQUATION69MAKESITCLEARTHATTHISDENITIONISA“GROSS”DENITIONINTHATITINCLUDESBOTHINCOMEANDSUBSTITUTIONEFFECTSTHATARISEFROMPRICECHANGESBECAUSETHESEEFFECTSAREINFACTCOMBINEDINANYREALWORLDOBSERVATIONWECANMAKE,ITMIGHTBEREASONABLEALWAYSTOSPEAKONLYOF“GROSS”SUBSTITUTESAND“GROSS”COMPLEMENTSASYMMETRYOFTHEGROSSDENITIONSTHEREARE,HOWEVER,SEVERALTHINGSTHATAREUNDESIRABLEABOUTTHEGROSSDENITIONSOFSUBSTITUTESANDCOMPLEMENTSTHEMOSTIMPORTANTOFTHESEISTHATTHEDENITIONSARENOTSYMMETRICITISPOSSIBLE,BYTHEDENITIONS,FORX1TOBEASUBSTITUTEFORX2ANDATTHESAMETIMEFORX2TOBEACOMPLEMENTOFX1THEPRESENCEOFINCOMEEFFECTSCANPRODUCEPARADOXICALRESULTSLETSLOOKATASPECICEXAMPLEEXAMPLE62ASYMMETRYINCROSSPRICEEFFECTSSUPPOSETHEUTILITYFUNCTIONFORTWOGOODSXANDYISGIVENBYUX,YH11005LNXH11001Y612SETTINGUPTHELAGRANGIANEXPRESSIONH5112H11005LNXH11001YH11001H9261IH11002PXXH11002PYY613H11128XIH5007H11128PJH11128XIH5007H11128PJDEFINITIONCHAPTER6DEMANDRELATIONSHIPSAMONGGOODS155YIELDSTHEFOLLOWINGRSTORDERCONDITIONSH11005H5007X1H5007H11002H9261PXH110050614H110051H11002H9261PYH110050H11005IH11002PXXH11002PYYH110050MOVINGTHETERMSINH9261TOTHERIGHTANDDIVIDINGTHERSTEQUATIONBYTHESECONDYIELDSH11005615PXXH11005PY616SUBSTITUTIONINTOTHEBUDGETCONSTRAINTNOWPERMITSUSTOSOLVEFORTHEMARSHALLIANDEMANDFUNCTIONFORYIH11005PXXH11001PYYH11005PYH11001PYY617HENCE,PYYH11005IH11002PY618THISEQUATIONSHOWSTHATANINCREASEINPYMUSTDECREASESPENDINGONGOODYTHATIS,PYYTHEREFORE,SINCEPXANDIAREUNCHANGED,SPENDINGONXMUSTRISESOH110220,619ANDWEWOULDTERMXANDYGROSSSUBSTITUTESONTHEOTHERHAND,EQUATION618SHOWSTHATSPENDINGONYISINDEPENDENTOFPXCONSEQUENTLY,H110050620AND,LOOKEDATINTHISWAY,XANDYWOULDBESAIDTOBEINDEPENDENTOFEACHOTHERTHEYARENEITHERGROSSSUBSTITUTESNORGROSSCOMPLEMENTSRELYINGONGROSSMARKETBASEDRESPONSESTODENETHERELATIONSHIPBETWEENXANDYWOULDTHEREFORERUNINTOAMBIGUITYQUERYINEXAMPLE34WESHOWEDTHATAUTILITYFUNCTIONOFTHEFORMGIVENBYEQUATION612ISNONHOMOTHETICTHEMRSDOESNOTDEPENDONLYONTHERATIOOFXTOYCANASYMMETRYARISEINTHEHOMOTHETICCASENETSUBSTITUTESANDCOMPLEMENTSBECAUSEOFTHEPOSSIBLEASYMMETRIESINVOLVEDINTHEDENITIONOFGROSSSUBSTITUTESANDCOMPLEMENTS,ANALTERNATIVEDENITIONTHATFOCUSESONLYONSUBSTITUTIONEFFECTSISSOMETIMESUSEDH11128YH5007H11128PXH11128XH5007H11128PYPXH5007PY1H5007XH11128H5112H5007H11128H9261H11128H5112H5007H11128YH11128H5112H5007H11128X156PARTIICHOICEANDDEMANDNETSUBSTITUTESANDCOMPLEMENTS1XIANDXJARESAIDTOBENETSUBSTITUTESIFH20841UH11005CONSTANTH110220621ANDNETCOMPLEMENTSIFH20841UH11005CONSTANTH110210622THESEDENITIONS,THEN,LOOKONLYATTHESUBSTITUTIONTERMSTODETERMINEWHETHERTWOGOODSARESUBSTITUTESORCOMPLEMENTSTHISDENITIONISBOTHINTUITIVELYAPPEALINGBECAUSEITLOOKSONLYATTHESHAPEOFANINDIFFERENCECURVEANDTHEORETICALLYDESIRABLEBECAUSEITISUNAMBIGUOUSONCEXIANDXJHAVEBEENDISCOVEREDTOBESUBSTITUTES,THEYSTAYSUBSTITUTES,NOMATTERINWHICHDIRECTIONTHEDENITIONISAPPLIEDASAMATTEROFFACTTHEDENITIONSAREPERFECTLYSYMMETRICH20841UH11005CONSTANTH11005H20841UH11005CONSTANT623THESUBSTITUTIONEFFECTOFACHANGEINPIONGOODXJISIDENTICALTOTHESUBSTITUTIONEFFECTOFACHANGEINPJONTHEQUANTITYOFXICHOSENTHISSYMMETRYISIMPORTANTINBOTHTHEORETICALANDEMPIRICALWORK2THEDIFFERENCESBETWEENTHETWODENITIONSOFSUBSTITUTESANDCOMPLEMENTSAREEASILYDEMONSTRATEDINFIGURE61AINTHISGUREXANDYAREGROSSCOMPLEMENTS,BUTTHEYARENETSUBSTITUTESTHEDERIVATIVEH11128X/H11128PYTURNSOUTTOBENEGATIVEXANDYAREGROSSCOMPLEMENTSBECAUSETHEPOSITIVESUBSTITUTIONEFFECTISOUTWEIGHEDBYTHENEGATIVEINCOMEEFFECTAFALLINTHEPRICEOFYCAUSESREALINCOMETOINCREASEGREATLYAND,CONSEQUENTLY,ACTUALPURCHASESOFXINCREASEHOWEVER,ASTHEGUREMAKESCLEAR,IFTHEREAREONLYTWOGOODSFROMWHICHTOCHOOSE,THEYMUSTBENETSUBSTITUTES,ALTHOUGHTHEYMAYBEEITHERGROSSSUBSTITUTESORGROSSCOMPLEMENTSBECAUSEWEHAVEASSUMEDADIMINISHINGMRS,THEOWNPRICESUBSTITUTIONEFFECTMUSTBENEGATIVEAND,CONSEQUENTLY,THECROSSPRICESUBSTITUTIONEFFECTMUSTBEPOSITIVEINDEED,ITCANBESHOWNTHATTHERECANBEONLYA“FEW”H11128XJH5007H11128PIH11128XIH5007H11128PJH11128XIH5007H11128PJH11128XIH5007H11128PJDEFINITIONCHAPTER6DEMANDRELATIONSHIPSAMONGGOODS1571SOMETIMESTHESEARECALLED“HICKSIAN”SUBSTITUTESANDCOMPLEMENTS,NAMEDAFTERTHEBRITISHECONOMISTJOHNHICKS,WHOORIGINALLYDEVELOPEDTHEDENITIONS2THISSYMMETRYISEASILYSHOWNBYUSINGSHEPHARDSLEMMASEEFOOTNOTE5,CHAPTER5BECAUSEXIH11005H11128E/H11128PI,H20841UH11005CONSTANTH11005,BUTH11005H11005H20841UH11005CONSTANTBECAUSE,BYYOUNGSTHEOREMSEECHAPTER2,THEORDEROFPARTIALDIFFERENTIATIONISIRRELEVANTH11128XJH5007H11128PIH11128E2H5007H11128PIH11128PJH11128E2H5007H11128PJH11128PIH11128E2H5007H11128PJH11128PIH11128XIH5007H11128PJCOMPLEMENTARYRELATIONSHIPSINTHENETSENSE3NETSUBSTITUTIONISTHEPREVALENTRELATIONSHIPAMONGGOODSTHESYMMETRYOFNETCROSSSUBSTITUTIONEFFECTSEQUATION623ANDTHENEGATIVITYOFTHEOWNSUBSTITUTIONEFFECTDISCUSSEDINCHAPTER5ARETHEMAJORRESULTSOFTHETHEORYOFINDIVIDUALCHOICETHESERESULTSCAN,INPRINCIPLE,BETESTEDUSINGREALWORLDDATAHOWEVER,BECAUSEMOSTACTUALMARKETDATAINVOLVETHEBEHAVIOROFMANYDEMANDERSANDBECAUSEADDINGDEMANDACROSSINDIVIDUALSINVOLVESSOMEDIFCULTMETHODOLOGICALPROBLEMSSEECHAPTER7,THEREAREFEWCONVINCINGTESTSOFTHEPROPOSITIONSINSTEAD,THERESULTSAREUSEDMAINLYINTHEORETICALWORKANDINTHESPECICATIONOFHYPOTHETICALDEMANDFUNCTIONSTHATOBEYTHESEPROPERTIESCOMPOSITECOMMODITIESOURDISCUSSIONINTHEPREVIOUSSECTIONSHOWEDTHATTHEDEMANDRELATIONSHIPSAMONGGOODSAREQUITECOMPLICATEDINTHEMOSTGENERALCASE,ANINDIVIDUALWHOCONSUMESNGOODSWILLHAVEDEMANDFUNCTIONSTHATREECTNNH110011/2DIFFERENTSUBSTITUTIONEFFECTS4WHENNISVERYLARGEASITSURELYISFORALLTHESPECICGOODSTHATINDIVIDUALSACTUALLYCONSUME,THISGENERALCASECANBEUNMANAGEABLEITISOFTENFARMORECONVENIENTTOGROUPGOODSINTOLARGERAGGREGATESSUCHASFOOD,CLOTHING,SHELTER,ANDSOFORTHATTHEMOSTEXTREMELEVELOFAGGREGATES,WEMIGHTWISHTOEXAMINEONESPECICGOODSAY,GASOLINE,WHICHWEMIGHTCALLXANDITSRELATIONSHIPTO“ALLOTHERGOODS,”WHICHWEMIGHTCALLYTHISISTHEPROCEDUREWEHAVEBEENUSINGINMANYOFOURTWODIMENSIONALGRAPHS,ANDWEWILLCONTINUETODOSOATMANYOTHERPLACESINTHISBOOKINTHISSECTIONWESHOWTHECONDITIONSUNDERWHICHTHISPROCEDURECANBEDEFENDEDINTHEEXTENSIONTOTHISCHAPTER,WEEXPLOREMOREGENERALISSUESINVOLVEDINAGGREGATINGGOODSINTOLARGERGROUPINGSCOMPOSITECOMMODITYTHEOREMSUPPOSECONSUMERSCHOOSEAMONGNGOODS,BUTTHATWEAREONLYINTERESTEDSPECIFICALLYINONEOFTHEM,SAY,X1INGENERAL,THEDEMANDFORX1WILLDEPENDONTHEINDIVIDUALPRICESOFTHEOTHERNH110021COMMODITIESBUTIFALLTHESEPRICESMOVETOGETHER,ITMAYMAKESENSETOLUMPTHEMINTOASINGLE“COMPOSITECOMMODITY,”YFORMALLY,ISWELETP02P0NREPRESENTTHEINITIALPRICESOFTHESEGOODS,THENWEASSUMETHATTHESEPRICESCANONLYVARYTOGETHERTHEYMIGHTALLDOUBLE,ORALLDECLINEBY50PERCENT,BUTTHERELATIVEPRICESOFX2XNWOULDNOTCHANGENOWWEDENETHECOMPOSITECOMMODITYYTOBETOTALEXPENDITURESONX2XN,USINGTHEINITIALPRICESP02P0NYH11005P02X2H11001P03X3H11001H11001P0NXN624THISPERSONSINITIALBUDGETCONSTRAINTISGIVENBYIH11005P1X1H11001P02X2H11001H11001P0NXNH11005P1X1H11001Y625158PARTIICHOICEANDDEMAND3SEEJRHICKS,VALUEANDCAPITALOXFORDOXFORDUNIVERSITYPRESS,1939,P312ANDPROBLEM684TOSEETHIS,NOTICETHATALLSUBSTITUTIONEFFECTS,SIJ,COULDBERECORDEDINANNH11003NMATRIXHOWEVER,SYMMETRYOFTHEEFFECTSSIJH11005SJIIMPLIESTHATONLYTHOSETERMSONANDBELOWTHEPRINCIPALDIAGONALOFTHISMATRIXMAYBEDISTINCTLYDIFFERENTFROMEACHOTHERTHISAMOUNTSTOHALFTHETERMSINTHEMATRIXH500712H5007N2PLUSTHEREMAININGHALFOFTHETERMSONTHEMAINDIAGONALOFTHEMATRIXH500712H5007NBYASSUMPTION,ALLOFTHEPRICESP2PNCHANGEINUNISONASSUMEALLOFTHESEPRICESCHANGEBYAFACTOROFTTH110220NOWTHEBUDGETCONSTRAINTISIH11005P1X1H11001TP02Y2H11001H11001TP0NXNH11005P1X1H11001TY626CONSEQUENTLY,THEFACTOROFPROPORTIONALITY,T,PLAYSTHESAMEROLEINTHISPERSONSBUDGETCONSTRAINTASDIDTHEPRICEOFYPYINOUREARLIERTWOGOODANALYSISCHANGESINP1ORINTINDUCETHESAMEKINDSOFSUBSTITUTIONEFFECTSWEHAVEBEENANALYZINGSOLONGASP2PNMOVETOGETHER,WECANTHEREFORECONNEOUREXAMINATIONOFDEMANDTOCHOICESBETWEENBUYINGX1ORBUYING“EVERYTHINGELSE”5SIMPLIEDGRAPHSTHATSHOWTHESETWOGOODSONTHEIRAXESCANTHEREFOREBEDEFENDEDRIGOROUSLYSOLONGASTHECONDITIONSOFTHIS“COMPOSITECOMMODITYTHEOREM”THATALLOTHERPRICESMOVETOGETHERARESATISEDNOTICE,HOWEVER,THATTHETHEOREMMAKESNOPREDICTIONSABOUTHOWCHOICESOFX2XNBEHAVETHEYNEEDNOTMOVEINUNISONTHETHEOREMFOCUSESONLYONTOTALSPENDINGONX2XN,NOTONHOWTHATSPENDINGISALLOCATEDAMONGSPECICITEMSALTHOUGHTHISALLOCATIONISASSUMEDTOBEDONEINAUTILITYMAXIMIZINGWAYGENERALIZATIONSANDLIMITATIONSTHECOMPOSITECOMMODITYTHEOREMAPPLIESTOANYGROUPOFCOMMODITIESWHOSERELATIVEPRICESALLMOVETOGETHERITISPOSSIBLETOHAVEMORETHANONESUCHCOMMODITYIFTHEREARESEVERALGROUPINGSTHATOBEYTHETHEOREMIE,EXPENDITURESON“FOOD,”“CLOTHING,”ANDSOFORTHHENCE,WEHAVEDEVELOPEDTHEFOLLOWINGCOMPOSITECOMMODITYACOMPOSITECOMMODITYISAGROUPOFGOODSFORWHICHALLPRICESMOVETOGETHERTHESEGOODSCANBETREATEDASASINGLE“COMMODITY”INTHATTHEINDIVIDUALBEHAVESASIFHEORSHEWERECHOOSINGBETWEENOTHERGOODSANDTOTALSPENDINGONTHEENTIRECOMPOSITEGROUPTHISDENITIONANDTHERELATEDTHEOREMAREVERYPOWERFULRESULTSTHEYHELPSIMPLIFYMANYPROBLEMSTHATWOULDOTHERWISEBEINTRACTABLESTILL,ONEMUSTBERATHERCAREFULINAPPLYINGTHETHEOREMTOTHEREALWORLDBECAUSEITSCONDITIONSARESTRINGENTFINDINGASETOFCOMMODITIESWHOSEPRICESMOVETOGETHERMAYBERARESLIGHTDEPARTURESFROMSTRICTPROPORTIONALITYMAYNEGATETHECOMPOSITECOMMODITYTHEOREMIFCROSSSUBSTITUTIONEFFECTSARELARGEEXAMPLE63HOUSINGCOSTSASACOMPOSITECOMMODITYSUPPOSETHATANINDIVIDUALRECEIVESUTILITYFROMTHREEGOODSFOODX,HOUSINGSERVICESYMEASUREDINHUNDREDSOFSQUAREFEET,ANDHOUSEHOLDOPERATIONSASMEASUREDBYELECTRICITYUSEZDEFINITIONCHAPTER6DEMANDRELATIONSHIPSAMONGGOODS1595THEIDEAOFA“COMPOSITECOMMODITY”WASINTRODUCEDBYJRHICKSINVALUEANDCAPITAL,2NDEDOXFORDOXFORDUNIVERSITYPRESS,1946,PP312313PROOFOFTHETHEOREMRELIESONTHENOTIONTHATTOACHIEVEMAXIMUMUTILITY,THERATIOOFTHEMARGINALUTILITIESFORX2XNMUSTREMAINUNCHANGEDWHENP2PNALLMOVETOGETHERHENCE,THENGOODPROBLEMCANBEREDUCEDTOTHETWODIMENSIONALPROBLEMOFEQUATINGTHERATIOOFTHEMARGINALUTILITYFROMXTOTHATFROMYTOTHE“PRICERATIO”P1/TIFTHEINDIVIDUALSUTILITYISGIVENBYTHETHREEGOODCESFUNCTIONUTILITYH11005UX,Y,ZH11005H11002H11002H11002,627THELAGRANGIANTECHNIQUECANBEUSEDTOCALCULATEDEMANDFUNCTIONSFORTHESEGOODSASXH11005YH11005628ZH11005IFINITIALLYIH11005100,PXH110051,PYH110054,ANDPZH110051,THENTHEDEMANDFUNCTIONSPREDICTXH1100525629YH11005125ZH1100525HENCE,25ISSPENTONFOODANDATOTALOF75ISSPENTONHOUSINGRELATEDNEEDSIFWEASSUMETHATHOUSINGSERVICEPRICESPYANDHOUSEHOLDOPERATIONPRICESPZALWAYSMOVETOGETHER,WECANUSETHEIRINITIALPRICESTODENETHE“COMPOSITECOMMODITY”HOUSINGHHH110054YH110011Z630HEREWEALSOARBITRARILYDENETHEINITIALPRICEOFHOUSINGPHTOBE1THEINITIALQUANTITYOFHOUSINGISSIMPLYTOTALDOLLARSSPENTONHHH110054125H11001125H1100575631FURTHERMORE,BECAUSEPYANDPZALWAYSMOVETOGETHER,PHWILLALWAYSBERELATEDTOTHESEPRICESBYPHH11005PZH1100525PY632USINGTHISINFORMATION,WECANRECALCULATETHEDEMANDFUNCTIONFORXASAFUNCTIONOFI,PX,ANDPHXH11005H11005633ASBEFORE,INITIALLYIH11005100,PXH110051,ANDPHH110051,SOXH1100525SPENDINGONHOUSINGCANBEMOSTEASILYCALCULATEDFROMTHEBUDGETCONSTRAINTASHH1100575,BECAUSEHERESPENDINGONHOUSINGREPRESENTS“EVERYTHING”OTHERTHANFOODANINCREASEINHOUSINGCOSTSIFTHEPRICESOFYANDZWERETORISEPROPORTIONALLYTOPYH1100516,PZH110054WITHPXREMAININGAT1,PHWOULDALSORISETOPHH110054EQUATION633NOWPREDICTSTHATTHEDEMANDFORXWOULDFALLTOXH11005H11005634100H50077100H5007H50071H1