nicholson微观经济理论（答案+ppt+习题库）-nicholson solutions manual

1CHAPTERCHAPTER22THETHEMATHEMATICSMATHEMATICSOFOFOPTIMIZATIONOPTIMIZATIONTHEPROBLEMSINTHISCHAPTERAREPRIMARILYMATHEMATICALTHEYAREINTENDEDTOGIVESTUDENTSSOMEPRACTICEWITHTAKINGDERIVATIVESANDUSINGTHELAGRANGIANTECHNIQUES,BUTTHEPROBLEMSINTHEMSELVESOFFERFEWECONOMICINSIGHTSCONSEQUENTLY,NOCOMMENTARYISPROVIDEDALLOFTHEPROBLEMSARERELATIVELYSIMPLEANDINSTRUCTORSMIGHTCHOOSEFROMAMONGTHEMONTHEBASISOFHOWTHEYWISHTOAPPROACHTHETEACHINGOFTHEOPTIMIZATIONMETHODSINCLASSSOLUTIONSSOLUTIONS2122,43UXYXYA86UUX,YXYB8,12C86UUDUDXDYXDXYDYXYDFOR0860DYDUXDXYDYDX8463DYXXDXYYE1,2413416XYUF412/332DYDXGU16CONTOURLINEISANELLIPSECENTEREDATTHEORIGINWITHEQUATION,SLOPEOFTHELINEATX,YIS224316XY43DYXDXY22APROFITSAREGIVENBY2240100RCQQ44010DQQDQ22104010100100BSOPROFITSAREMAXIMIZED224DDQC702DRMRQDQ230DCMCQDQ22SOLUTIONSSOLUTIONSMANUALMANUALSOQ10OBEYSMRMC5023SUBSTITUTION21SOYXFXYXX120FXX0505,025X,YFNOTETHISISALOCALANDGLOBALMAXIMUM20FLAGRANGIANMETHOD1XYXYY0XX0YSO,XYUSINGTHECONSTRAINTGIVES05,025XYXY24SETTINGUPTHELAGRANGIAN025XYXY11YXXYSO,XYUSINGTHECONSTRAINTGIVES2025,05XYXXY25A20540FTGTT40400,DFGTTDTGBSUBSTITUTINGFORT,205404040800FTGGGG2800FTGGCDEPENDSONGBECAUSETDEPENDSONG212FTGSO222408000505FTGGGD,AREDUCTIONOF08NOTICETHAT8003225,8003212492SOA01INCREASEINGCOULDBEPREDICTEDTOREDUCE228008003208GHEIGHTBY008FROMTHEENVELOPETHEOREMCHAPTERCHAPTER2/THE2/THEMATHEMATICSMATHEMATICSOFOFOPTIMIZATIONOPTIMIZATION3326ATHISISTHEVOLUMEOFARECTANGULARSOLIDMADEFROMAPIECEOFMETALWHICHISXBY3XWITHTHEDEFINEDCORNERSQUARESREMOVEDBAPPLYINGTHEQUADRATICFORMULATOTHISEXPRESSIONYIELDS22316120VXXTTTTODETERMINETRUE2216256144161060225,1112424XXXXXTXXMAXIMUMMUSTLOOKATSECONDDERIVATIVEWHICHISNEGATIVEONLY221624VXTTFORTHEFIRSTSOLUTIONCIFSOVINCREASESWITHOUTLIMIT33330225,0670405068TXVXXXXDTHISWOULDREQUIREASOLUTIONUSINGTHELAGRANGIANMETHODTHEOPTIMALSOLUTIONREQUIRESSOLVINGTHREENONLINEARSIMULTANEOUSEQUATIONSATASKNOTUNDERTAKENHEREBUTITSEEMSCLEARTHATTHESOLUTIONWOULDINVOLVEADIFFERENTRELATIONSHIPBETWEENTANDXTHANINPARTSAC27ASETUPLAGRANGIANYIELDSTHEFIRSTORDERCONDITIONS1212LNXXKXX122121000XXXKXXHENCE,WITHK10,OPTIMALSOLUTIONIS2215OR5XX125XXBWITHK4,SOLVINGTHEFIRSTORDERCONDITIONSYIELDS215,1XXCOPTIMALSOLUTIONISANYPOSITIVEVALUEFORX1REDUCESY120,4,5LN4XXYDIFK20,OPTIMALSOLUTIONISBECAUSEX2PROVIDESADIMINISHING1215,5XXMARGINALINCREMENTTOYWHEREASX1DOESNOT,ALLOPTIMALSOLUTIONSREQUIRETHAT,ONCEX2REACHES5,ANYEXTRAAMOUNTSBEDEVOTEDENTIRELYTOX128THEPROOFISMOSTEASILYACCOMPLISHEDTHROUGHTHEUSEOFTHEMATRIXALGEBRAOFQUADRATICFORMSSEE,FOREXAMPLE,MASCOLELLETAL,PP937939INTUITIVELY,BECAUSECONCAVEFUNCTIONSLIEBELOWANYTANGENTPLANE,THEIRLEVELCURVESMUSTALSOBECONVEXBUTTHECONVERSEISNOTTRUEQUASICONCAVEFUNCTIONSMAYEXHIBIT“INCREASINGRETURNSTOSCALE”EVENTHOUGHTHEIRLEVELCURVESARECONVEX,THEYMAYRISEABOVETHETANGENTPLANEWHENALLVARIABLESAREINCREASEDTOGETHER29A11210FXX11220FXX44SOLUTIONSSOLUTIONSMANUALMANUAL2111110FXX2122210FXX111212210FFXXCLEARLY,ALLTHETERMSINEQUATION2114ARENEGATIVEBIF12YCXXSINCE,0,X2ISACONVEXFUNCTIONOFX1/1/21XCXCUSINGEQUATION298,22222222222112211122211FFFXXXXWHICHISNEGATIVEFOR12222121XX210ASINCE,THEFUNCTIONISCONCAVE0,0YYBBECAUSE,AND,EQUATION298ISSATISFIEDANDTHE1122,0FF12210FFFUNCTIONISCONCAVECYISQUASICONCAVEASISBUTISNOTCONCAVEFOR1ALLOFTHESERESULTSYYCANBESHOWNBYAPPLYINGTHEVARIOUSDEFINITIONSTOTHEPARTIALDERIVATIVESOFY5CHAPTERCHAPTER33PREFERENCESPREFERENCESANDANDUTILITYUTILITYTHESEPROBLEMSPROVIDESOMEPRACTICEINEXAMININGUTILITYFUNCTIONSBYLOOKINGATINDIFFERENCECURVEMAPSTHEPRIMARYFOCUSISONILLUSTRATINGTHENOTIONOFADIMINISHINGMRSINVARIOUSCONTEXTSTHECONCEPTSOFTHEBUDGETCONSTRAINTANDUTILITYMAXIMIZATIONARENOTUSEDUNTILTHENEXTCHAPTERCOMMENTSCOMMENTSONONPROBLEMSPROBLEMS31THISPROBLEMREQUIRESSTUDENTSTOGRAPHINDIFFERENCECURVESFORAVARIETYOFFUNCTIONS,SOMEOFWHICHDONOTEXHIBITADIMINISHINGMRS32INTRODUCESTHEFORMALDEFINITIONOFQUASICONCAVITYFROMCHAPTER2TOBEAPPLIEDTOTHEFUNCTIONSINPROBLEM3133THISPROBLEMSHOWSTHATDIMINISHINGMARGINALUTILITYISNOTREQUIREDTOOBTAINADIMINISHINGMRSALLOFTHEFUNCTIONSAREMONOTONICTRANSFORMATIONSOFONEANOTHER,SOTHISPROBLEMILLUSTRATESTHATDIMINISHINGMRSISPRESERVEDBYMONOTONICTRANSFORMATIONS,BUTDIMINISHINGMARGINALUTILITYISNOT34THISPROBLEMFOCUSESONWHETHERSOMESIMPLEUTILITYFUNCTIONSEXHIBITCONVEXINDIFFERENCECURVES35THISPROBLEMISANEXPLORATIONOFTHEFIXEDPROPORTIONSUTILITYFUNCTIONTHEPROBLEMALSOSHOWSHOWSUCHPROBLEMSCANBETREATEDASACOMPOSITECOMMODITY36INTHISPROBLEMSTUDENTSAREASKEDTOPROVIDEAFORMAL,UTILITYBASEDEXPLANATIONFORAVARIETYOFADVERTISINGSLOGANSTHEPURPOSEISTOGETSTUDENTSTOTHINKMATHEMATICALLYABOUTEVERYDAYEXPRESSIONS37THISPROBLEMSHOWSHOWINITIALENDOWMENTSCANBEINCORPORATEDINTOUTILITYTHEORY38THISPROBLEMOFFERSAFURTHEREXPLORATIONOFTHECOBBDOUGLASFUNCTIONPARTCPROVIDESANINTRODUCTIONTOTHELINEAREXPENDITURESYSTEMTHISAPPLICATIONISTREATEDINMOREDETAILINTHEEXTENSIONSTOCHAPTER439THISPROBLEMSHOWSTHATINDEPENDENTMARGINALUTILITIESILLUSTRATEONESITUATIONINWHICHDIMINISHINGMARGINALUTILITYENSURESADIMINISHINGMRS310THISPROBLEMEXPLORESVARIOUSFEATURESOFTHECESFUNCTIONWITHWEIGHTINGONTHETWOGOODS66SOLUTIONSSOLUTIONSMANUALMANUALSOLUTIONSSOLUTIONS31HEREWECALCULATETHEMRSFOREACHOFTHESEFUNCTIONSAMRSISCONSTANT31XYMRSFFBMRSISDIMINISHING05050505XYYXMRSFFYXYXCMRSISDIMINISHING05051XYMRSFFXDMRSISINCREASING22052205052052XYMRSFFXYXXYYXYEMRSISDIMINISHING2222XYXYYXYXYXXYMRSFFYXXYXY32BECAUSEALLOFTHEFIRSTORDERPARTIALSAREPOSITIVE,WEMUSTONLYCHECKTHESECONDORDERPARTIALSANOTSTRICTLYQUASICONCAVE112220FFFBSTRICTLYQUASICONCAVE112212,0,0FFFCSTRICTLYQUASICONCAVE1122120,0,0FFFDEVENIFWEONLYCONSIDERCASESWHERE,BOTHOFTHEOWNSECONDORDERPARTIALSXYAREAMBIGUOUSANDTHEREFORETHEFUNCTIONISNOTNECESSARILYSTRICTLYQUASICONCAVEESTRICTLYQUASICONCAVE112212,00FFF33A,0,0,XXXYYYUYUUXUMRSYXB22222,2,2,2,XXXYYYUXYUYUXYUXMRSYXC221,1,1,1,XXXYYYUXUXUYUYMRSYXTHISSHOWSTHATMONOTONICTRANSFORMATIONSMAYAFFECTDIMINISHINGMARGINALUTILITY,BUTNOTTHEMRS34ATHECASEWHERETHESAMEGOODISLIMITINGISUNINTERESTINGBECAUSEIFTHE111222121212,2,22UXYXKUXYXUXXYYXXLIMITINGGOODSDIFFER,THENHENCE,1122YXKYXSOTHEINDIFFERENCECURVEISCONVEX1212/2AND/2XXKYYKCHAPTERCHAPTER3/PREFERENCE3/PREFERENCEANDANDUTILITYUTILITY77BAGAIN,THECASEWHERETHESAMEGOODISMAXIMUMISUNINTERESTINGIFTHEGOODSDIFFER,SOTHEINDIFFERENCECURVEIS11221212/2,/2YXKYXXXKYYKCONCAVE,NOTCONVEXCHERESOINDIFFERENCECURVEIS11221212/2,/2XYKXYXXYYNEITHERCONVEXORCONCAVEITISLINEAR35A,2,05UHBMRMINHBMRBAFULLYCONDIMENTEDHOTDOGC160D210ANINCREASEOF31PERCENTEPRICEWOULDINCREASEONLYTO1725ANINCREASEOF78PERCENTFRAISEPRICESSOTHATAFULLYCONDIMENTEDHOTDOGRISESINPRICETO260THISWOULDBEEQUIVALENTTOALUMPSUMREDUCTIONINPURCHASINGPOWER36A,UPBPBB20UXCOKECFORP1ANDALLX,1,UPXUXDFORKD,UKXUDXESEETHEEXTENSIONSTOCHAPTER388SOLUTIONSSOLUTIONSMANUALMANUAL37ABANYTRADINGOPPORTUNITIESTHATDIFFERFROMTHEMRSATWILLPROVIDETHE,XYOPPORTUNITYTORAISEUTILITYSEEFIGURECAPREFERENCEFORTHEINITIALENDOWMENTWILLREQUIRETHATTRADINGOPPORTUNITIESRAISEUTILITYSUBSTANTIALLYTHISWILLBEMORELIKELYIFTHETRADINGOPPORTUNITIESANDSIGNIFICANTLYDIFFERENTFROMTHEINITIALMRSSEEFIGURE38A11/UXXYMRSYXUYXYTHISRESULTDOESNOTDEPENDONTHESUMWHICH,CONTRARYTOPRODUCTIONTHEORY,HASNOSIGNIFICANCEINCONSUMERTHEORYBECAUSEUTILITYISUNIQUEONLYUPTOAMONOTONICTRANSFORMATIONBMATHEMATICSFOLLOWSDIRECTLYFROMPARTAIFTHEINDIVIDUALVALUESXRELATIVELYMOREHIGHLYHENCE,FORXY1DYDXCTHEFUNCTIONISHOMOTHETICINAND,BUTNOTINXANDY0XX0YY39FROMPROBLEM32,IMPLIESDIMINISHINGMRSPROVIDING120F1122,0FFCONVERSELY,THECOBBDOUGLASHAS,BUTALSOHASADIMINISHINGMRS1211220,0FFFSEEPROBLEM38A310ASOTHISFUNCTIONISHOMOTHETIC111/UXXMRSYXUYYBIF1,MRS/,ACONSTANTIF0,MRS/Y/X,WHICHAGREESWITHPROBLEM38CFOR0,SOMRSDIMINISHESDFOLLOWSFROMPARTA,IFXYMRS/CHAPTERCHAPTER3/PREFERENCE3/PREFERENCEANDANDUTILITYUTILITY99EWITH055,99949MRS051111105MRSWITH21,9981MRS21111121MRSHENCE,THEMRSCHANGESMOREDRAMATICALLYWHEN1THANWHEN5THELOWERIS,THEMORESHARPLYCURVEDARETHEINDIFFERENCECURVESWHEN,THEINDIFFERENCECURVESARELSHAPEDIMPLYINGFIXEDPROPORTIONS10CHAPTERCHAPTER44UTILITYUTILITYMAXIMIZATIONMAXIMIZATIONANDANDCHOICECHOICETHEPROBLEMSINTHISCHAPTERFOCUSMAINLYONTHEUTILITYMAXIMIZATIONASSUMPTIONRELATIVELYSIMPLECOMPUTATIONALPROBLEMSMAINLYBASEDONCOBBDOUGLASANDCESUTILITYFUNCTIONSAREINCLUDEDCOMPARATIVESTATICSEXERCISESAREINCLUDEDINAFEWPROBLEMS,BUTFORTHEMOSTPART,INTRODUCTIONOFTHISMATERIALISDELAYEDUNTILCHAPTERS5AND6COMMENTSCOMMENTSONONPROBLEMSPROBLEMS41THISISASIMPLECOBBDOUGLASEXAMPLEPARTBASKSSTUDENTSTOCOMPUTEINCOMECOMPENSATIONFORAPRICERISEANDMAYPROVEDIFFICULTFORTHEMASAHINTTHEYMIGHTBETOLDTOFINDTHECORRECTBUNDLEONTHEORIGINALINDIFFERENCECURVEFIRST,THENCOMPUTEITSCOST42THISUSESTHECOBBDOUGLASUTILITYFUNCTIONTOSOLVEFORQUANTITYDEMANDEDATTWODIFFERENTPRICESINSTRUCTORSMAYWISHTOINTRODUCETHEEXPENDITURESHARESINTERPRETATIONOFTHEFUNCTIONSEXPONENTSTHESEARECOVEREDEXTENSIVELYINTHEEXTENSIONSTOCHAPTER4ANDINAVARIETYOFNUMERICALEXAMPLESINCHAPTER543THISSTARTSASANUNCONSTRAINEDMAXIMIZATIONPROBLEMTHEREISNOINCOMECONSTRAINTINPARTAONTHEASSUMPTIONTHATTHISCONSTRAINTISNOTLIMITINGINPARTBTHEREISATOTALQUANTITYCONSTRAINTSTUDENTSSHOULDBEASKEDTOINTERPRETWHATLAGRANGIANMULTIPLIERMEANSINTHISCASE44THISPROBLEMSHOWSTHATWITHCONCAVEINDIFFERENCECURVESFIRSTORDERCONDITIONSDONOTENSUREALOCALMAXIMUM45THISISANEXAMPLEOFA“FIXEDPROPORTION”UTILITYFUNCTIONTHEPROBLEMMIGHTBEUSEDTOILLUSTRATETHENOTIONOFPERFECTCOMPLEMENTSANDTHEABSENCEOFRELATIVEPRICEEFFECTSFORTHEMSTUDENTSMAYNEEDSOMEHELPWITHTHEMINFUNCTIONALNOTATIONBYUSINGILLUSTRATIVENUMERICALVALUESFORVANDGANDSHOWINGWHATITMEANSTOHAVE“EXCESS”VORG46THISPROBLEMINTRODUCESATHIRDGOODFORWHICHOPTIMALCONSUMPTIONISZEROUNTILINCOMEREACHESACERTAINLEVEL47THISPROBLEMPROVIDESMOREPRACTICEWITHTHECOBBDOUGLASFUNCTIONBYASKINGSTUDENTSTOCOMPUTETHEINDIRECTUTILITYFUNCTIONANDEXPENDITUREFUNCTIONINTHISCASETHEMANIPULATIONSHEREAREOFTENQUITEDIFFICULTFORSTUDENTS,PRIMARILYBECAUSETHEYDONOTKEEPANEYEONWHATTHEFINALGOALIS48THISPROBLEMREPEATSTHELESSONSOFTHELUMPSUMPRINCIPLEFORTHECASEOFASUBSIDYNUMERICALEXAMPLESAREBASEDONTHECOBBDOUGLASEXPENDITUREFUNCTIONCHAPTERCHAPTER4/UTILITY4/UTILITYMAXIMIZATIONMAXIMIZATIONANDANDCHOICECHOICE111149THISPROBLEMLOOKSINDETAILATTHEFIRSTORDERCONDITIONSFORAUTILITYMAXIMUMWITHTHECESFUNCTIONPARTCOFTHEPROBLEMFOCUSESONHOWRELATIVEEXPENDITURESHARESAREDETERMINEDWITHTHECESFUNCTION410THISPROBLEMSHOWSUTILITYMAXIMIZATIONINTHELINEAREXPENDITURESYSTEMSEEALSOTHEEXTENSIONSTOCHAPTER4SOLUTIONSSOLUTIONS41ASETUPLAGRANGIAN1001025TSTS0505/10/25STTTSS10010250TSRATIOOFFIRSTTWOEQUATIONSIMPLIES2525TTSSHENCE10010T25S50SS2T5UTILITY10BNEWUTILITYORTS1010AND255408TS58STSUBSTITUTINGINTOINDIFFERENCECURVE25108SS216S4T25COSTOFTHISBUNDLEIS200,SOPAULNEEDSANOTHERDOLLAR1212SOLUTIONSSOLUTIONSMANUALMANUAL42USEASIMPLERNOTATIONFORTHISSOLUTION2/31/3,300UFCIFCA2/31/3300204FCFC1/32/32/3/201/3/4CFFFCCHENCE,52,25CCFFSUBSTITUTIONINTOBUDGETCONSTRAINTYIELDSF10,C25BWITHTHENEWCONSTRAINTF20,C25NOTETHISPERSONALWAYSSPENDS2/3OFINCOMEONFAND1/3ONCCONSUMPTIONOFCALIFORNIAWINEDOESNOTCHANGEWHENPRICEOFFRENCHWINECHANGESCINPARTA,INPARTB,23132313,1025135UFCFC2313,2025215UFCTOACHIEVETHEPARTBUTILITYWITHPARTAPRICES,THISPERSONWILLNEEDMOREINCOMEINDIRECTUTILITYISSOLVINGTHIS23132313232313215231323204FCIPPIEQUATIONFORTHEREQUIREDINCOMEGIVESI482WITHSUCHANINCOME,THISPERSONWOULDPURCHASEF161,C401,U2154322,20183UCBCCBBA|U202C0,C10CU186B0,B3BSO,U127BCONSTRAINTBC52201835CCBBCB202C0C186B0B5CB0C3B1SOB3B15,B1,C4,U79CHAPTERCHAPTER4/UTILITY4/UTILITYMAXIMIZATIONMAXIMIZATIONANDANDCHOICECHOICE1313442205,UXYXYMAXIMIZINGU2INWILLALSOMAXIMIZEUA225034XYXY232X30XX22Y40YY5003X4YFIRSTTWOEQUATIONSGIVESUBSTITUTINGINBUDGETCONSTRAINTGIVESX6,43YXY8,U10BTHISISNOTALOCALMAXIMUMBECAUSETHEINDIFFERENCECURVESDONOTHAVEADIMINISHINGMRSTHEYAREINFACTCONCENTRICCIRCLESHENCE,WEHAVENECESSARYBUTNOTSUFFICIENTCONDITIONSFORAMAXIMUMINFACTTHECALCULATEDALLOCATIONISAMINIMUMUTILITYIFMRBALLSPENDSALLINCOMEONX,SAY,U50/345,2,UMUGVMINGVANOMATTERWHATTHERELATIVEPRICEAREIE,THESLOPEOFTHEBUDGETCONSTRAINTTHEMAXIMUMUTILITYINTERSECTIONWILLALWAYSBEATTHEVERTEXOFANINDIFFERENCECURVEWHEREG2VBSUBSTITUTINGG2VINTOTHEBUDGETCONSTRAINTYIELDSOR2GVPVPVIGVIV2PPSIMILARLY,GV2IG2PPITISEASYTOSHOWTHATTHESETWODEMANDFUNCTIONSAREHOMOGENEOUSOFDEGREEZEROINPG,PV,ANDICSO,2UGVINDIRECTUTILITYIS,GVGVIVPPI2PPDTHEEXPENDITUREFUNCTIONISFOUNDBYINTERCHANGINGIEANDV,2GVGVEPPVPPV1414SOLUTIONSSOLUTIONSMANUALMANUAL46AIFX4Y1UZ02IFZ1U0SINCEXY0IFZ01SAYX9/2536,Y9U36595115189WHICHISLESSTHANUZ0BATX4Y1Z0XYXYM/M/1PPUUZZM/1/2PUSO,EVENATZ0,THEMARGINALUTILITYFROMZIS“NOTWORTH“THEGOODSPRICENOTICEHERETHATTHE“1”INTHEUTILITYFUNCTIONCAUSESTHISINDIVIDUALTOINCURSOMEDIMINISHINGMARGINALUTILITYFORZBEFOREANYISBOUGHTGOODZILLUSTRATESTHEPRINCIPLEOF“COMPLEMENTARYSLACKNESSDISCUSSEDINCHAPTER2CIFI10,OPTIMALCHOICESAREX16,Y4,Z1AHIGHERINCOMEMAKESITPOSSIBLETOCONSUMEZASPARTOFAUTILITYMAXIMUMTOFINDTHEMINIMALINCOMEATWHICHANYFRACTIONALZWOULDBEBOUGHT,USETHEFACTTHATWITHTHECOBBDOUGLASTHISPERSONWILLSPENDEQUALAMOUNTSONX,Y,AND1ZTHATIS1XYZPXPYPZSUBSTITUTINGTHISINTOTHEBUDGETCONSTRAINTYIELDS2132ZZZZPZPZIORPZIPHENCE,FORZ0ITMUSTBETHECASETHAT2OR4ZIPI471,UXYXYATHEDEMANDFUNCTIONSINTHISCASEARESUBSTITUTINGTHESE,1XYXIPYIPINTOTHEUTILITYFUNCTIONGIVES1,1XYXYXYVPPIIPIPBIPPWHERE11BBINTERCHANGINGIANDVYIELDS11,XYXYEPPVBPPVCTHEELASTICITYOFEXPENDITURESWITHRESPECTTOISGIVENBYTHEEXPONENTTHATIS,XPTHEMOREIMPORTANTXISINTHEUTILITYFUNCTIONTHEGREATERTHEPROPORTIONTHATEXPENDITURESMUSTBEINCREASEDTOCOMPENSATEFORAPROPORTIONALRISEINTHEPRICEOFXCHAPTERCHAPTER4/UTILITY4/UTILITYMAXIMIZATIONMAXIMIZATIONANDANDCHOICECHOICE151548ABWITHTORAISEUTILITYTO30505,2XYXYEPPUPPU1,4,2,8XYPPUEWOULDREQUIREE12THATIS,ANINCOMESUBSIDYOF4CNOWWEREQUIRESOTHATIS,EACH0505058243OR81223XXEPP49XPUNITMUSTBESUBSIDIZEDBY5/9ATTHESUBSIDIZEDPRICETHISPERSONCHOOSESTOBUYX9SOTOTALSUBSIDYIS5ONEDOLLARGREATERTHANINPARTCDWITHRAISINGUTO30307,184XYXYEPPUPPU1,4,2,971XYPPUEWOULDREQUIREEXTRAEXPENDITURESOF486SUBSIDIZINGGOODXALONEWOULDREQUIREAPRICEOFTHATIS,ASUBSIDYOF074PERUNITWITHTHISLOWPRICE,THISPERSON026XPWOULDCHOOSEX112,SOTOTALSUBSIDYWOULDBE82949AFORUTILITYMAXIMIZATION1XYU/XMRSXYP/PU/YHENCE,11WHERE11XYXYX/YPPPPBIF0,SOYXXYXYPPPXPYCPARTASHOWS1XYXYPXPYPPHENCE,FORTHERELATIVESHAREOFINCOMEDEVOTEDTOGOODXISPOSITIVELY1CORRELATEDWITHITSRELATIVEPRICETHISISASIGNOFLOWSUBSTITUTABILITYFORTHE1RELATIVESHAREOFINCOMEDEVOTEDTOGOODXISNEGATIVELYCORRELATEDWITHITSRELATIVEPRICEASIGNOFHIGHSUBSTITUTABILITYDTHEALGEBRAHEREISVERYMESSYFORASOLUTIONSEETHESYDSAETER,STROM,ANDBERCKREFERENCEATTHEENDOFCHAPTER5410AFORXPXYPPCDINCREASESINISHIFTSDEMANDFORXOUTWARDREDUCTIONSINPYDONOTAFFECTDEMANDFORXUNTILTHENDEMANDFORXFALLSTOZERO83XYP052AUTILITYMAXIMIZATIONREQUIRESPB2JANDTHEBUDGETCONSTRAINTIS05PB1J3SUBSTITUTIONGIVESPB30,J15BIFPJ15SUBSTITUTIONNOWYIELDSJ12,PB24CTOCONTINUEBUYINGJ15,PB30,DAVIDWOULDNEEDTOBUY3MOREOUNCESOFJELLYAND6MOREOUNCESOFPEANUTBUTTERTHISWOULDREQUIREANINCREASEININCOMEOF31560575CHAPTERCHAPTER5/INCOME5/INCOMEANDANDSUBSTITUTIONSUBSTITUTIONEFFECTSEFFECTS1919DESINCEDAVIDNUSESONLYPBJTOMAKESANDWICHESINFIXEDPROPORTIONS,ANDBECAUSEBREADISFREE,ITISJUSTASTHOUGHHEBUYSSANDWICHESWHEREPSANDWICH2PPBPJINPARTA,PS20,QS15INPARTB,PS25,QS12INGENERAL,SOTHEDEMANDCURVEFORSANDWICHESISAHYPERBOLASS3QPFTHEREISNOSUBSTITUTIONEFFECTDUETOTHEFIXEDPROPORTIONACHANGEINPRICERESULTSINONLYANINCOMEEFFECT53AASINCOMEINCREASES,THERATIOSTAYSCONSTANT,ANDTHEUTILITYMAXIMIZATIONXYPPCONDITIONSTHEREFOREREQUIRETHATMRSSTAYCONSTANTTHUS,IFMRSDEPENDSONTHERATIO,THISRATIOMUSTSTAYCONSTANTASINCOMEINCREASESTHEREFORE,SINCEYXINCOMEISSPENTONLYONTHESETWOGOODS,BOTHXANDYAREPROPORTIONALTOINCOMEBBECAUSEOFPARTA,SOGIFFENSPARADOXCANNOTARISE0XI54ASINCE,03,07XYXIPYIP37373737XYXYUIPPBIPPTHEEXPENDITUREFUNCTIONISTHEN137XYEBUPPBTHECOMPENSATEDDEMANDFUNCTIONIS771/3CXYXXEPPPBCITISEASIESTTOSHOWSLUTSKYEQUATIONINELASTICITIESBYJUSTREADINGEXPONENTSFROMTHEVARIOUSDEMANDFUNCTIONS,1,1,7,03CXXXPXIXXPEEESHENCE,OR107031CXXXPXXIXPEESE2020SOLUTIONSSOLUTIONSMANUALMANUAL56A2,21XXXXSIXIXXPXIIPXIPXIIEEIPXIIPXIF,FOREXAMPLE,15,05XXISIEEB,1XXXXXXXSPXPXXPXIPPXPXIEEPPXIIXIF,FOREXAMPLE,075,025XXXXPSPEECBECAUSEIMAYBECANCELLEDOUTOFTHEDERIVATIONINPARTB,ITISALSOTHECASETHAT,1XXXPXPXPEED,XYYYXYYYXSPXPYXXYPPXPPIPPXIXEEPPXIIPXPXEUSEPARTB1,2111XXKKKKYXYXKKSPXYXKKKKYXYXKPPKPPEPPPPPPPTOSIMPLIFYALGEBRA,LETKKYXDPPHENCENOWUSETHESLUTSKYEQUATION,11111XXXXPSPKDKDDEEDDREMEMBERINGTHAT,1XIE,1111111CXXXPXXXPKDDDKEESSDDD57ABECAUSEOFTHEFIXEDPROPORTIONSBETWEENHANDC,KNOWTHATTHEDEMANDFORHAMISHENCEHCHIPP,2HHHHCHHPHHCHCPPPPPHIEPHPPIPPSIMILARALGEBRASHOWSTHATSO,IF,CCHPHCPEPP,05HCHCHPHPPPEEBWITHFIXEDPROPORTIONSTHEREARENOSUBSTITUTIONEFFECTSHERETHECOMPENSATEDPRICEELASTICITIESAREZERO,SOTHESLUTSKYEQUATIONSHOWSTHAT,005XXPXESCWITH,212PARTASHOWSTHAT,33HCHCHPHPPPEEDIFTHISPERSONCONSUMESONLYHAMANDCHEESESANDWICHES,THEPRICEELASTICITYOFDEMANDFORTHOSEMUSTBE1PRICEELASTICITYFORTHECOMPONENTSREFLECTSTHEPROPORTIONALEFFECTOFACHANGEINTHEPRICEOFTHECOMPONENTONTHEPRICETHEWHOLECHAPTERCHAPTER5/INCOME5/INCOMEANDANDSUBSTITUTIONSUBSTITUTIONEFFECTSEFFECTS2121SANDWICHINPARTA,FOREXAMPLE,ATENPERCENTINCREASEINTHEPRICEOFHAMWILLINCREASETHEPRICEOFASANDWICHBY5PERCENTANDTHATWILLCAUSEQUANTITYDEMANDEDTOFALLBY5PERCENT58A,1HENCE1XYXYXPXXYPXYXPYPESSESSEETHESUMEQUALS2TRIVIALLYINTHECOBBDOUGLASCASEBRESULTFOLLOWSDIRECTLYFROMPARTAINTUITIVELY,PRICEELASTICITIESARELARGEWHENISLARGEANDSMALLWHENISSMALLCAGENERALIZATIONFROMTHEMULTIVARIABLECESFUNCTIONISPOSSIBLE,BUTTHECONSTRAINTSPLACEDONBEHAVIORBYTHISFUNCTIONAREPROBABLYNOTTENABLE59ABECAUSETHEDEMANDFORANYGOODISHOMOGENEOUSOFDEGREEZERO,EULERSTHEOREMSTATES10NIIJJJXXPIPIMULTIPLICATIONBYYIELDSTHEDESIREDRESULT1IXBPARTBANDCAREBASEDONTHEBUDGETCONSTRAINTIIIPXIDIFFERENTIATIONWITHRESPECTTOIYIELDS1IIIPXIMULTIPLICATIONOFEACHTERMBY,YIELDS1IIIIIIXIXISECDIFFERENTIATIONOFTHEBUDGETCONST