nicholson微观经济理论（答案+ppt+习题库）-40-03-033593-0

UTILITYMAXIMIZATIONANDCHOICEINTHISCHAPTERWEWILLEXAMINETHEBASICMODELOFCHOICETHATECONOMISTSUSETOEXPLAININDIVIDUALSBEHAVIORTHATMODELASSUMESTHATINDIVIDUALSWHOARECONSTRAINEDBYLIMITEDINCOMESWILLBEHAVEASIFTHEYWEREUSINGTHEIRPURCHASINGPOWERINSUCHAWAYASTOACHIEVETHEHIGHESTUTILITYPOSSIBLETHATIS,INDIVIDUALSAREASSUMEDTOBEHAVEASIFTHEYMAXIMIZEDUTILITYSUBJECTTOABUDGETCONSTRAINTALTHOUGHTHESPECICAPPLICATIONSOFTHISMODELAREQUITEVARIED,ASWEWILLSHOW,ALLOFTHEMAREBASEDONTHESAMEFUNDAMENTALMATHEMATICALMODEL,ANDALLARRIVEATTHESAMEGENERALCONCLUSIONTOMAXIMIZEUTILITY,INDIVIDUALSWILLCHOOSEBUNDLESOFCOMMODITIESFORWHICHTHERATEOFTRADEOFFBETWEENANYTWOGOODSTHEMRSISEQUALTOTHERATIOOFTHEGOODSMARKETPRICESMARKETPRICESCONVEYINFORMATIONABOUTOPPORTUNITYCOSTSTOINDIVIDUALS,ANDTHISINFORMATIONPLAYSANIMPORTANTROLEINAFFECTINGTHECHOICESACTUALLYMADE4CHAPTERUTILITYMAXIMIZATIONANDLIGHTNINGCALCULATIONSBEFORESTARTINGAFORMALSTUDYOFTHETHEORYOFCHOICE,ITMAYBEAPPROPRIATETODISPOSEOFTWOCOMPLAINTSNONECONOMISTSOFTENMAKEABOUTTHEAPPROACHWEWILLTAKEFIRSTISTHECHARGETHATNOREALPERSONCANMAKETHEKINDSOF“LIGHTNINGCALCULATIONS”REQUIREDFORUTILITYMAXIMIZATIONACCORDINGTOTHISCOMPLAINT,WHENMOVINGDOWNASUPERMARKETAISLE,PEOPLEJUSTGRABWHATISAVAILABLEWITHNOREALPATTERNORPURPOSETOTHEIRACTIONSECONOMISTSARENOTPERSUADEDBYTHISCOMPLAINTTHEYDOUBTTHATPEOPLEBEHAVERANDOMLYEVERYONE,AFTERALL,ISBOUNDBYSOMESORTOFBUDGETCONSTRAINT,ANDTHEYVIEWTHELIGHTNINGCALCULATIONCHARGEASMISPLACEDRECALL,AGAIN,FRIEDMANSPOOLPLAYERHEORSHEALSOCANNOTMAKETHELIGHTNINGCALCULATIONSREQUIREDTOPLANASHOTACCORDINGTOTHELAWSOFPHYSICS,BUTTHOSELAWSSTILLPREDICTTHEPLAYERSBEHAVIORSOTOO,ASWESHALLSEE,THEUTILITYMAXIMIZATIONMODELPREDICTSMANYASPECTSOFBEHAVIOREVENTHOUGHNOONECARRIESAROUNDACOMPUTERWITHHISORHERUTILITYFUNCTIONPROGRAMMEDINTOITTOBEPRECISE,ECONOMISTSASSUMETHATPEOPLEBEHAVEASIFTHEYMADESUCHCALCULATIONS,SOTHECOMPLAINTTHATTHECALCULATIONSCANNOTPOSSIBLYBEMADEISIRRELEVANTALTRUISMANDSELSHNESSASECONDCOMPLAINTAGAINSTOURMODELOFCHOICEISTHATITAPPEARSTOBEEXTREMELYSELSHNOONE,ACCORDINGTOTHISCOMPLAINT,HASSUCHSOLELYSELFCENTEREDGOALSALTHOUGHECONOMISTSAREPROBABLYMOREREADYTOACCEPTSELFINTERESTASAMOTIVATINGFORCETHANAREOTHER,MOREUTOPIANTHINKERSADAMSMITHOBSERVED,“WEARENOTREADYTOSUSPECTANYPERSONOFBEINGDECIENTINSELSHNESS”1,THISCHARGEISALSOMISPLACEDNOTHINGINTHEUTILITYMAXIMIZATIONMODELPREVENTSINDIVIDUALSFROMDERIVINGSATISFACTIONFROMPHILANTHROPYORGENERALLY“DOINGGOOD”THESEACTIVITIESALSOCANBEASSUMEDTOPROVIDEUTILITYINDEED,ECONOMISTSHAVEUSEDTHEUTILITYMAXIMIZATIONMODELEXTENSIVELYTOSTUDYSUCHISSUESASDONATINGTIMEANDMONEYTOCHARITY,LEAVINGBEQUESTSTOCHILDREN,OREVENGIVINGBLOODONENEEDNOTTAKEAPOSITIONONWHETHERSUCHACTIVITIESARE“SELSH”OR“SELESS”SINCEECONOMISTSDOUBTPEOPLEWOULDUNDERTAKETHEMIFTHEYWEREAGAINSTTHEIROWNBESTINTERESTS,BROADLYCONCEIVEDANINITIALSURVEYTHEGENERALRESULTSOFOUREXAMINATIONOFUTILITYMAXIMIZATIONCANBESTATEDSUCCINCTLYUTILITYMAXIMIZATIONTOMAXIMIZEUTILITY,GIVENAXEDAMOUNTOFINCOMETOSPEND,ANINDIVIDUALWILLBUYTHOSEQUANTITIESOFGOODSTHATEXHAUSTHISORHERTOTALINCOMEANDFORWHICHTHEPSYCHICRATEOFTRADEOFFBETWEENANYTWOGOODSTHEMRSISEQUALTOTHERATEATWHICHTHEGOODSCANBETRADEDONEFORTHEOTHERINTHEMARKETPLACEOPTIMIZATIONPRINCIPLE92PARTIICHOICEANDDEMAND1ADAMSMITH,THETHEORYOFMORALSENTIMENTS1759REPRINT,NEWROCHELLE,NYARLINGTONHOUSE,1969,P446THATSPENDINGALLONESINCOMEISREQUIREDFORUTILITYMAXIMIZATIONISOBVIOUSSINCEEXTRAGOODSPROVIDEEXTRAUTILITYTHEREISNOSATIATIONANDSINCETHEREISNOOTHERUSEFORINCOME,TOLEAVEANYUNSPENTWOULDBETOFAILTOMAXIMIZEUTILITYTHROWINGMONEYAWAYISNOTAUTILITYMAXIMIZINGACTIVITYTHECONDITIONSPECIFYINGEQUALITYOFTRADEOFFRATESREQUIRESABITMOREEXPLANATIONBECAUSETHERATEATWHICHONEGOODCANBETRADEDFORANOTHERINTHEMARKETISGIVENBYTHERATIOOFTHEIRPRICES,THISRESULTCANBERESTATEDTOSAYTHATTHEINDIVIDUALWILLEQUATETHEMRSOFXFORYTOTHERATIOOFTHEPRICEOFXTOTHEPRICEOFYPX/PYTHISEQUATINGOFAPERSONALTRADEOFFRATETOAMARKETDETERMINEDTRADEOFFRATEISARESULTCOMMONTOALLINDIVIDUALUTILITYMAXIMIZATIONPROBLEMSANDTOMANYOTHERTYPESOFMAXIMIZATIONPROBLEMSITWILLOCCURAGAINANDAGAINTHROUGHOUTTHISTEXTANUMERICALILLUSTRATIONTOSEETHEINTUITIVEREASONINGBEHINDTHISRESULT,ASSUMETHATITWERENOTTRUETHATANINDIVIDUALHADEQUATEDTHEMRSTOTHERATIOOFTHEPRICESOFGOODSSPECICALLY,ASSUMETHATTHEINDIVIDUALSMRSISEQUALTO1,THATHEORSHEISWILLINGTOTRADE1UNITOFXFOR1UNITOFYANDREMAINEQUALLYWELLOFFASSUMEALSOTHATTHEPRICEOFXIS2PERUNITANDOFYIS1PERUNITITISEASYTOSHOWINTHISCASETHATTHEINDIVIDUALCANBEMADEBETTEROFFGIVEUP1UNITOFXANDTRADEITINTHEMARKETFOR2UNITSOFYONLY1EXTRAUNITOFYWASNEEDEDTOKEEPTHEINDIVIDUALASHAPPYASBEFORETHETRADETHESECONDUNITOFYISANETADDITIONTOWELLBEINGTHEREFORE,THEINDIVIDUALSSPENDINGCOULDNOTHAVEBEENALLOCATEDOPTIMALLYINTHERSTPLACEASIMILARMETHODOFREASONINGCANBEUSEDWHENEVERTHEMRSANDTHEPRICERATIOPX/PYDIFFERTHECONDITIONFORMAXIMUMUTILITYMUSTBETHEEQUALITYOFTHESETWOMAGNITUDESTHETWOGOODCASEAGRAPHICALANALYSISTHISDISCUSSIONSEEMSEMINENTLYREASONABLE,BUTITCANHARDLYBECALLEDAPROOFRATHER,WEMUSTNOWSHOWTHERESULTINARIGOROUSMANNERAND,ATTHESAMETIME,ILLUSTRATESEVERALOTHERIMPORTANTATTRIBUTESOFTHEMAXIMIZATIONPROCESSFIRSTWETAKEAGRAPHICANALYSISTHENWETAKEAMOREMATHEMATICALAPPROACHBUDGETCONSTRAINTASSUMETHATTHEINDIVIDUALHASIDOLLARSTOALLOCATEBETWEENGOODXANDGOODYIFPXISTHEPRICEOFGOODXANDPYISTHEPRICEOFGOODY,THENTHEINDIVIDUALISCONSTRAINEDBYPXXH11001PYYH11349I41THATIS,NOMORETHANICANBESPENTONTHETWOGOODSINQUESTIONTHISBUDGETCONSTRAINTISSHOWNGRAPHICALLYINFIGURE41THEINDIVIDUALCANAFFORDTOCHOOSEONLYCOMBINATIONSOFXANDYINTHESHADEDTRIANGLEOFTHEGUREIFALLOFIISSPENTONGOODX,ITWILLBUYI/PXUNITSOFXSIMILARLY,IFALLISSPENTONY,ITWILLBUYI/PYUNITSOFYTHESLOPEOFTHECONSTRAINTISEASILYSEENTOBEH11002PX/PYCHAPTER4UTILITYMAXIMIZATIONANDCHOICE93FIRSTORDERCONDITIONSFORAMAXIMUMTHISBUDGETCONSTRAINTCANBEIMPOSEDONTHEINDIVIDUALSINDIFFERENCECURVEMAPTOSHOWTHEUTILITYMAXIMIZATIONPROCESSFIGURE42ILLUSTRATESTHISPROCEDURETHEINDIVIDUALWOULDBEIRRATIONALTOCHOOSEAPOINTSUCHASAHEORSHECANGETTOAHIGHERUTILITYLEVELJUSTBYSPENDINGSOMEOFTHEUNSPENTPORTIONOFINCOMETHEASSUMPTIONOFNONSATIATIONIMPLIESTHATAPERSONSHOULDSPENDALLOFHISORHERINCOMEINORDERTORECEIVEMAXIMUMUTILITYFROMITSIMILARLY,BYREALLOCATINGEXPENDITURES,THEINDIVIDUALCANDOBETTERTHANPOINTBPOINTDISOUTOFTHEQUESTIONBECAUSEINCOMEISNOTLARGEENOUGHTOPURCHASEDITISCLEARTHATTHEPOSITIONOFMAXIMUMUTILITYISATPOINTC,WHERETHECOMBINATIONX,YISCHOSENTHISISTHEONLYPOINTONINDIFFERENCECURVEU2THATCANBEBOUGHTWITHIDOLLARSNOHIGHERUTILITYLEVELCANBEBOUGHTCISAPOINTOFTANGENCYBETWEENTHEBUDGETCONSTRAINTANDTHEINDIFFERENCECURVETHEREFOREATC,SLOPEOFBUDGETCONSTRAINTH11005H5007H11002PPYXH5007H11005SLOPEOFINDIFFERENCECURVEH11005H5007DDXYH5007H20841UH11005CONSTANT42ORH11005H11002H20841UH11005CONSTANTH11005MRSOFXFORY43DYH5007DXPXH5007PY94PARTIICHOICEANDDEMANDTHEINDIVIDUALSBUDGETCONSTRAINTFORTWOGOODSTHOSECOMBINATIONSOFXANDYTHATTHEINDIVIDUALCANAFFORDARESHOWNINTHESHADEDTRIANGLEIF,ASWEUSUALLYASSUME,THEINDIVIDUALPREFERSMORERATHERTHANLESSOFEVERYGOOD,THEOUTERBOUNDARYOFTHISTRIANGLEISTHERELEVANTCONSTRAINTWHEREALLOFTHEAVAILABLEFUNDSARESPENTEITHERONXORONYTHESLOPEOFTHISSTRAIGHTLINEBOUNDARYISGIVENBYH11002PX/PYQUANTITYOFX0QUANTITYOFYIH11005PXXH11001PYYIPXIPYFIGURE41OURINTUITIVERESULTISPROVEDFORAUTILITYMAXIMUM,ALLINCOMESHOULDBESPENTANDTHEMRSSHOULDEQUALTHERATIOOFTHEPRICESOFTHEGOODSITISOBVIOUSFROMTHEDIAGRAMTHATIFTHISCONDITIONISNOTFULLLED,THEINDIVIDUALCOULDBEMADEBETTEROFFBYREALLOCATINGEXPENDITURESSECONDORDERCONDITIONSFORAMAXIMUMTHETANGENCYRULEISONLYANECESSARYCONDITIONFORAMAXIMUMTOSEETHATITISNOTASUFCIENTCONDITION,CONSIDERTHEINDIFFERENCECURVEMAPSHOWNINFIGURE43HEREAPOINTOFTANGENCYCISINFERIORTOAPOINTOFNONTANGENCYBINDEED,THETRUEMAXIMUMISATANOTHERPOINTOFTANGENCYATHEFAILUREOFTHETANGENCYCONDITIONTOPRODUCEANUNAMBIGUOUSMAXIMUMCANBEATTRIBUTEDTOTHESHAPEOFTHEINDIFFERENCECURVESINFIGURE43IFTHEINDIFFERENCECURVESARESHAPEDLIKETHOSEINFIGURE42,NOSUCHPROBLEMCANARISEBUTWEHAVEALREADYSHOWNTHAT“NORMALLY”SHAPEDINDIFFERENCECURVESRESULTFROMTHEASSUMPTIONOFADIMINISHINGMRSTHEREFORE,IFTHEMRSISASSUMEDTOBEDIMINISHING,THECONDITIONOFTANGENCYISBOTHANECESSARYANDSUFCIENTCONDITIONFORAMAXIMUM2WITHOUTTHISASSUMPTIONONEWOULDHAVETOBECAREFULINAPPLYINGTHETANGENCYRULECHAPTER4UTILITYMAXIMIZATIONANDCHOICE95AGRAPHICALDEMONSTRATIONOFUTILITYMAXIMIZATIONPOINTCREPRESENTSTHEHIGHESTUTILITYLEVELTHATCANBEREACHEDBYTHEINDIVIDUAL,GIVENTHEBUDGETCONSTRAINTTHECOMBINATIONX,YISTHEREFORETHERATIONALWAYFORTHEINDIVIDUALTOALLOCATEPURCHASINGPOWERONLYFORTHISCOMBINATIONOFGOODSWILLTWOCONDITIONSHOLDALLAVAILABLEFUNDSWILLBESPENTANDTHEINDIVIDUALSPSYCHICRATEOFTRADEOFFMRSWILLBEEQUALTOTHERATEATWHICHTHEGOODSCANBETRADEDINTHEMARKETPX/PYQUANTITYOFX0QUANTITYOFYIH11005PXXH11001PYYYXDCABU1U1U2U2U3U3FIGURE422INMATHEMATICALTERMS,BECAUSETHEASSUMPTIONOFADIMINISHINGMRSISEQUIVALENTTOASSUMINGQUASICONCAVITY,THENECESSARYCONDITIONSFORAMAXIMUMSUBJECTTOALINEARCONSTRAINTAREALSOSUFCIENT,ASWESHOWEDINCHAPTER2CORNERSOLUTIONSTHEUTILITYMAXIMIZATIONPROBLEMILLUSTRATEDINFIGURE42RESULTEDINAN“INTERIOR”MAXIMUM,INWHICHPOSITIVEAMOUNTSOFBOTHGOODSWERECONSUMEDINSOMESITUATIONSINDIVIDUALSPREFERENCESMAYBESUCHTHATTHEYCANOBTAINMAXIMUMUTILITYBYCHOOSINGTOCONSUMENOAMOUNTOFONEOFTHEGOODSIFSOMEONEDOESNOTLIKEHAMBURGERSVERYMUCH,THEREISNOREASONTOALLOCATEANYINCOMETOTHEIRPURCHASETHISPOSSIBILITYISREECTEDINFIGURE44THEREUTILITYISMAXIMIZEDATE,WHEREXH11005XANDYH110050ANYPOINTONTHEBUDGETCONSTRAINTWHEREPOSITIVEAMOUNTSOFYARECONSUMEDYIELDSALOWERUTILITYTHANDOESPOINTENOTICETHATATETHEBUDGETCONSTRAINTISNOTPRECISELYTANGENTTOTHEINDIFFERENCECURVEU2INSTEAD,ATTHEOPTIMALPOINTTHEBUDGETCONSTRAINTISATTERTHANU2,INDICATINGTHATTHERATEATWHICHXCANBETRADEDFORYINTHEMARKETISLOWERTHANTHEINDIVIDUALSPSYCHICTRADEOFFRATETHEMRSATPREVAILINGMARKETPRICESTHEINDIVIDUALISMORETHANWILLINGTOTRADEAWAYYTOGETEXTRAXBECAUSEITISIMPOSSIBLEINTHISPROBLEMTOCONSUMENEGATIVEAMOUNTSOFY,HOWEVER,THEPHYSICALLIMITFORTHISPROCESSISTHEXAXIS,ALONGWHICHPURCHASESOFYARE0HENCE,ASTHISDISCUSSIONMAKESCLEAR,ITISNECESSARYTOAMENDTHERSTORDERCONDITIONSFORAUTILITYMAXIMUMABITTOALLOWFORCORNER96PARTIICHOICEANDDEMANDEXAMPLEOFANINDIFFERENCECURVEMAPFORWHICHTHETANGENCYCONDITIONDOESNOTENSUREAMAXIMUMIFINDIFFERENCECURVESDONOTOBEYTHEASSUMPTIONOFADIMINISHINGMRS,NOTALLPOINTSOFTANGENCYPOINTSFORWHICHMRSH11005PX/PYMAYTRULYBEPOINTSOFMAXIMUMUTILITYINTHISEXAMPLETANGENCYPOINTCISINFERIORTOMANYOTHERPOINTSTHATCANALSOBEPURCHASEDWITHTHEAVAILABLEFUNDSINORDERTHATTHENECESSARYCONDITIONSFORAMAXIMUMTHATIS,THETANGENCYCONDITIONSALSOBESUFCIENT,ONEUSUALLYASSUMESTHATTHEMRSISDIMINISHINGTHATIS,THEUTILITYFUNCTIONISSTRICTLYQUASICONCAVEQUANTITYOFXQUANTITYOFYIH11005PXXH11001PYYACBU1U1U2U2U3U3FIGURE43SOLUTIONSOFTHETYPESHOWNINFIGURE44FOLLOWINGOURDISCUSSIONOFTHEGENERALNGOODCASE,WEWILLSHOWHOWTHISCANBEACCOMPLISHEDTHENGOODCASETHERESULTSDERIVEDGRAPHICALLYINTHECASEOFTWOGOODSCARRYOVERDIRECTLYTOTHECASEOFNGOODSAGAINITCANBESHOWNTHATFORANINTERIORUTILITYMAXIMUM,THEMRSBETWEENANYTWOGOODSMUSTEQUALTHERATIOOFTHEPRICESOFTHESEGOODSTOSTUDYTHISMOREGENERALCASE,HOWEVER,ITISBESTTOUSESOMEMATHEMATICSFIRSTORDERCONDITIONSWITHNGOODS,THEINDIVIDUALSOBJECTIVEISTOMAXIMIZEUTILITYFROMTHESENGOODSUTILITYH11005UX1,X2,XN,44SUBJECTTOTHEBUDGETCONSTRAINT3IH11005P1X1H11001P2X2H11001H11001PNXN45ORIH11002P1X1H11002P2X2H11002H11002PNXNH11005046CHAPTER4UTILITYMAXIMIZATIONANDCHOICE97CORNERSOLUTIONFORUTILITYMAXIMIZATIONWITHTHEPREFERENCESREPRESENTEDBYTHISSETOFINDIFFERENCECURVES,UTILITYMAXIMIZATIONOCCURSATE,WHERE0AMOUNTSOFGOODYARECONSUMEDTHERSTORDERCONDITIONSFORAMAXIMUMMUSTBEMODIEDSOMEWHATTOACCOMMODATETHISPOSSIBILITYQUANTITYOFXQUANTITYOFYU1U2U3EXFIGURE443AGAIN,THEBUDGETCONSTRAINTHASBEENWRITTENASANEQUALITYHEREBECAUSE,GIVENTHEASSUMPTIONOFNONSATIATION,ITISCLEARTHATTHEINDIVIDUALWILLSPENDALLAVAILABLEINCOMEFOLLOWINGTHETECHNIQUESDEVELOPEDINCHAPTER2FORMAXIMIZINGAFUNCTIONSUBJECTTOACONSTRAINT,WESETUPTHELAGRANGIANEXPRESSIONH5112H11005UX1,X2,XNH11001H9261IH11002P1X1H11002P2X2H11002H11002PNXN47SETTINGTHEPARTIALDERIVATIVESOFH5112WITHRESPECTTOX1,X2,XNANDH9261EQUALTO0YIELDSNH110011EQUATIONSREPRESENTINGTHENECESSARYCONDITIONSFORANINTERIORMAXIMUMH11005H11002H9261P1H11005048H11005H11002H9261P2H110050H11005H11002H9261PNH110050H11005IH11002P1X1H11002P2X2H11002H11002PNXNH110050THESENH110011EQUATIONSCANUSUALLYBESOLVEDFORTHEOPTIMALX1,X2,XNANDFORH9261SEEEXAMPLE41TOBECONVINCEDTHATSUCHASOLUTIONISPOSSIBLEEQUATIONS48ARENECESSARYBUTNOTSUFCIENTFORAMAXIMUMTHESECONDORDERCONDITIONSTHATENSUREAMAXIMUMARERELATIVELYCOMPLEXANDMUSTBESTATEDINMATRIXTERMSSEETHEAPPENDIXTOCHAPTER2HOWEVER,THEASSUMPTIONOFSTRICTQUASICONCAVITYADIMINISHINGMRSINTHETWOGOODCASEISSUFCIENTTOENSURETHATANYPOINTOBEYINGEQUATIONS48ISINFACTATRUEMAXIMUMIMPLICATIONSOFFIRSTORDERCONDITIONSTHERSTORDERCONDITIONSREPRESENTEDBYEQUATIONS48CANBEREWRITTENINAVARIETYOFINTERESTINGWAYSFOREXAMPLE,FORANYTWOGOODS,XIANDXJ,WEHAVEH1100549BUTINCHAPTER3WESHOWEDTHATTHERATIOOFTHEMARGINALUTILITIESOFTWOGOODSISEQUALTOTHEMARGINALRATEOFSUBSTITUTIONBETWEENTHEMTHEREFORE,THECONDITIONSFORANOPTIMALALLOCATIONOFINCOMEBECOMEMRSXIFORXJH11005410THISISEXACTLYTHERESULTDERIVEDEARLIERINTHISCHAPTERTOMAXIMIZEUTILITY,THEINDIVIDUALSHOULDEQUATETHEPSYCHICRATEOFTRADEOFFTOTHEMARKETTRADEOFFRATEINTERPRETINGTHELAGRANGIANMULTIPLIERANOTHERRESULTCANBEDERIVEDBYSOLVINGEQUATIONS48FORH9261H9261H11005H11005H11005H11005411H11128U/H11128XNH5007PNH11128U/H11128X2H5007P2H11128U/H11128X1H5007P1PIH5007PJPIH5007PJH11128U/H11128XIH5007H11128U/H11128XJH11128H5112H5007H11128H9261H11128UH5007H11128XNH11128H5112H5007H11128XNH11128UH5007H11128X2H11128H5112H5007H11128X2H11128UH5007H11128X1H11128H5112H5007H11128X198PARTIICHOICEANDDEMANDORH9261H11005H11005H11005H11005THISEQUATIONSAYSTHATATTHEUTILITYMAXIMIZINGPOINT,EACHGOODPURCHASEDSHOULDYIELDTHESAMEMARGINALUTILITYPERDOLLARSPENTONTHATGOODEACHGOODTHEREFORESHOULDHAVEANIDENTICALMARGINALBENETTOMARGINALCOSTRATIOIFTHISWERENOTTRUE,ONEGOODWOULDPROMISEMORE“MARGINALENJOYMENTPERDOLLAR”THANSOMEOTHERGOOD,ANDFUNDSWOULDNOTBEOPTIMALLYALLOCATEDALTHOUGHTHEREADERISAGAINWARNEDAGAINSTTALKINGVERYCONFIDENTLYABOUTMARGINALUTILITY,WHATEQUATION411SAYSISTHATANEXTRADOLLARSHOULDYIELDTHESAME“ADDITIONALUTILITY”NOMATTERWHICHGOODITISSPENTONTHECOMMONVALUEFORTHISEXTRAUTILITYISGIVENBYTHELAGRANGIANMULTIPLIERFORTHECONSUMERSBUDGETCONSTRAINTTHATIS,BYH9261CONSEQUENTLY,H9261CANBEREGARDEDASTHEMARGINALUTILITYOFANEXTRADOLLAROFCONSUMPTIONEXPENDITURETHEMARGINALUTILITYOF“INCOME”ONENALWAYTOREWRITETHENECESSARYCONDITIONSFORAMAXIMUMISPIH11005412FOREVERYGOODITHATISBOUGHTTHISEQUATIONSAYSTHATFOREVERYGOODTHATANINDIVIDUALBUYS,THEPRICEOFTHATGOODREPRESENTSHISORHEREVALUATIONOFTHEUTILITYOFTHELASTUNITCONSUMEDTHEPRICEOBVIOUSLYREPRESENTSHOWMUCHTHEINDIVIDUALISWILLINGTOPAYFORTHATLASTUNITINCHAPTER5ANDELSEWHEREWEWILLMAKECONSIDERABLEUSEOFTHISRESULTWHENDISCUSSINGTHEVALUEOFAGOODTOACONSUMERANDTHE“CONSUMERSURPLUS”RECEIVEDBYSOMEPURCHASERSWHENTHEYAREABLETOBUYAGOODFORLESSTHANTHEMAXIMUMAMOUNTTHEYWOULDBEWILLINGTOPAYCORNERSOLUTIONSTHERSTORDERCONDITIONSOFEQUATIONS48HOLDEXACTLYONLYFORINTERIORMAXIMAFORWHICHSOMEPOSITIVEAMOUNTOFEACHGOODISPURCHASEDWHENCORNERSOLUTIONSSUCHASTHOSEILLUSTRATEDINFIGURE44ARISE,THECONDITIONSHAVETOBEMODIEDSLIGHTLY4INTHISCASE,EQUATIONS48BECOMEH11005H11002H9261PIH113490IH110051N,413AND,IFH11005H11002H9261PIH110210,414THENXIH110050415H11128UH5007H11128XIH11128H5112H5007H11128XIH11128UH5007H11128XIH11128H5112H5007H11128XIMUXIH5007H9261MUXNH5007PNMUX2H5007P2MUX1H5007P1CHAPTER4UTILITYMAXIMIZATIONANDCHOICE994FORMALLY,THESECONDITIONSARECALLEDTHE“KUHNTUCKER”CONDITIONSFORNONLINEARPROGRAMMINGFORAMORECOMPLETEEXPLANATION,SEEAKDIXIT,OPTIMIZATIONINECONOMICTHEORY,2NDEDNEWYORKOXFORDUNIVERSITYPRESS,1990TOINTERPRETTHESECONDITIONS,WECANREWRITEEQUATION414ASPIH11022H11005416HENCE,THEOPTIMALCONDITIONSAREASBEFORE,EXCEPTTHATANYGOODWHOSEPRICEPIEXCEEDSITSMARGINALVALUETOTHECONSUMERMUXI/H9261WILLNOTBEPURCHASEDXIH110050THUS,THEMATHEMATICALRESULTSCONFORMTOTHECOMMONSENSEIDEATHATINDIVIDUALSWILLNOTPURCHASEGOODSTHATTHEYBELIEVEARENOTWORTHTHEMONEYALTHOUGHCORNERSOLUTIONSDONOTPROVIDEAMAJORFOCUSFOROURANALYSISINTHISBOOK,THEREADERSHOULDKEEPINMINDTHEPOSSIBILITIESFORSUCHSOLUTIONSARISINGANDTHEECONOMICINTERPRETATIONTHATCANBEATTACHEDTOTHEOPTIMALCONDITIONSINSUCHCASESEXAMPLE41COBBDOUGLASDEMANDFUNCTIONSASWESHOWEDINCHAPTER3,THECOBBDOUGLASUTILITYFUNCTIONISGIVENBYUX,YH11005XH9251YH9252,417WHERE,FORCONVENIENCE,5WEASSUMEH9251H11001H9252H110051WECANNOWSOLVEFORTHEUTILITYMAXIMIZINGVALUESOFXANDYFORANYPRICESPX,PYANDINCOMEISETTINGUPTHELAGRANGIANEXPRESSIONH5112H11005XH9251YH9252H11001H9261IH11002PXXH11002PYY418YIELDSTHERSTORDERCONDITIONSH11005H9251XH9251H110021YH9252H11002H9261PXH110050419H11005H9252XH9251YH9252H110021H11002H9261PYH110050H11005IH11002PXXH11002PYYH110050TAKINGTHERATIOOFTHERSTTWOTERMSSHOWSTHATH11005420ORPYYH11005PXXH11005PXX,421WHERETHENALEQUATIONFOLLOWSBECAUSEH9251H11001H9252H110051SUBSTITUTIONOFTHERSTORDERCONDITIONINEQUATION421INTOTHEBUDGETCONSTRAINTGIVESIH11005PXXH11001PYYH11005PXXH11001PXXH11005PXXH208981H11001H20899H11005PXX4221H5007H92511H11002H9251H5007H92511H11002H9251H5007H92511H11002H9251H5007H9251H9252H5007H9251PXH5007PYH9251YH5007H9252XH11128H5112H5007H11128H9261H11128H5112H5007H11128YH11128H5112H5007H11128XMUXIH5007H9261H5007H11128H11128XUIH5007H5007H9261100PARTIICHOICEANDDEMAND5NOTICETHATTHEEXPONENTSINTHECOBBDOUGLASUTILITYFUNCTIONCANALWAYSBENORMALIZEDTOSUMTOONEBECAUSEU1/H9251H11001H9252ISAMONOTONICTRANSFORMATIONSOLVINGFORXYIELDSXH11005423ANDASIMILARSETOFMANIPULATIONSWOULDGIVEYH11005424THESERESULTSSHOWTHATANINDIVIDUALWHOSEUTILITYFUNCTIONISGIVENBYEQUATION417WILLALWAYSCHOOSETOALLOCATEH9251PERCENTOFHISORHERINCOMETOBUYINGGOODXTHATIS,PXX/IH11005H9251ANDH9252PERCENTTOBUYINGGOODYPYY/IH11005H9252ALTHOUGHTHISFEATUREOFTHECOBBDOUGLASFUNCTIONOFTENMAKESITVERYEASYTOWORKOUTSIMPLEPROBLEMS,ITDOESSUGGESTTHATTHEFUNCTIONHASLIMITSINITSABILITYTOEXPLAINACTUALCONSUMPTIONBEHAVIORBECAUSETHESHAREOFINCOMEDEVOTEDTOPARTICULARGOODSOFTENCHANGESSIGNICANTLYINRESPONSETOCHANGINGECONOMICCONDITIONS,AMOREGENERALFUNCTIONALFORMMAYPROVIDEINSIGHTSNOTPROVIDEDBYTHECOBBDOUGLASFUNCTIONWEILLUSTRATEAFEWPOSSIBILITIESINEXAMPLE42NUMERICALEXAMPLEFIRST,HOWEVER,LETSLOOKATASPECICNUMERICALEXAMPLEFORTHECOBBDOUGLASCASESUPPOSETHATXSELLSFOR25ANDYSELLSFOR100ANDTHATTOTALINCOMEIS200SUCCINCTLYTHEN,ASSUMETHATPXH1100525,PYH110051,IH110052SUPPOSEALSOTHATH9251H11005H9252H1100505SOTHATTHISINDIVIDUALSPLITSHISORHERINCOMEEQUALLYBETWEENTHESETWOGOODSNOWTHEDEMANDEQUATIONS423AND424IMPLYXH11005H9251I/PXH110055I/PXH1100552/25H110054425YH11005H9252I/PYH110055I/PYH110