nicholson微观经济理论（答案+ppt+习题库）-30-03-033593-0

PREFERENCESANDUTILITYINTHISCHAPTERWELOOKATTHEWAYINWHICHECONOMISTSCHARACTERIZEINDIVIDUALSPREFERENCESWEBEGINWITHAFAIRLYABSTRACTDISCUSSIONOFTHE“PREFERENCERELATION,”BUTQUICKLYTURNTOTHEECONOMISTSPRIMARYTOOLFORSTUDYINGINDIVIDUALCHOICESTHEUTILITYFUNCTIONWELOOKATSOMEGENERALCHARACTERISTICSOFSUCHAFUNCTIONANDATAFEWSIMPLEEXAMPLESOFSPECICUTILITYFUNCTIONSWEWILLENCOUNTERTHROUGHOUTTHISBOOK3CHAPTERAXIOMSOFRATIONALCHOICEONEWAYTOBEGINANANALYSISOFINDIVIDUALSCHOICESISTOSTATEABASICSETOFPOSTULATES,ORAXIOMS,THATCHARACTERIZE“RATIONAL”BEHAVIORALTHOUGHANUMBEROFSETSOFSUCHAXIOMSHAVEBEENPROPOSED,ALLHAVESIMILARITIESINTHATTHEYBEGINWITHTHECONCEPTOF“PREFERENCE”WHENANINDIVIDUALREPORTSTHAT“AISPREFERREDTOB,”ITISTAKENTOMEANTHATALLTHINGSCONSIDERED,HEORSHEFEELSBETTEROFFUNDERSITUATIONATHANUNDERSITUATIONBTHISPREFERENCERELATIONISASSUMEDTOHAVETHREEBASICPROPERTIESICOMPLETENESSIFAANDBAREANYTWOSITUATIONS,THEINDIVIDUALCANALWAYSSPECIFYEXACTLYONEOFTHEFOLLOWINGTHREEPOSSIBILITIES1“AISPREFERREDTOB,”2“BISPREFERREDTOA,”OR3“AANDBAREEQUALLYATTRACTIVE”INDIVIDUALSARECONSEQUENTLYASSUMEDNOTTOBEPARALYZEDBYINDECISIONTHEYCOMPLETELYUNDERSTANDANDCANALWAYSMAKEUPTHEIRMINDSABOUTTHEDESIRABILITYOFANYTWOALTERNATIVESTHEASSUMPTIONALSORULESOUTTHEPOSSIBILITYTHATTHEINDIVIDUALCANREPORTBOTHTHATAISPREFERREDTOBANDTHATBISPREFERREDTOAIITRANSITIVITYIFANINDIVIDUALREPORTSTHAT“AISPREFERREDTOB”ANDTHAT“BISPREFERREDTOC,”THENHEORSHEMUSTALSOREPORTTHAT“AISPREFERREDTOC”THISASSUMPTIONSTATESTHATTHEINDIVIDUALSCHOICESAREINTERNALLYCONSISTENTSUCHANASSUMPTIONCANBESUBJECTEDTOEMPIRICALSTUDYGENERALLY,SUCHSTUDIESCONCLUDETHATAPERSONSCHOICESAREINDEEDTRANSITIVE,BUTTHATCONCLUSIONMUSTBEMODIEDINCASESWHERETHEINDIVIDUALMAYNOTFULLYUNDERSTANDTHECONSEQUENCESOFTHECHOICESHEORSHEISMAKINGBECAUSE,FORTHEMOSTPART,WEWILLASSUMECHOICESAREFULLYINFORMEDBUTSEETHEDISCUSSIONOFUNCERTAINTYINPARTIIIANDELSEWHERE,THETRANSITIVITYPROPERTYSEEMSANAPPROPRIATEASSUMPTIONTOMAKEABOUTPREFERENCESIIICONTINUITYIFANINDIVIDUALREPORTS“AISPREFERREDTOB,”THENSITUATIONSSUITABLY“CLOSETO”AMUSTALSOBEPREFERREDTOBTHISRATHERTECHNICALASSUMPTIONISREQUIREDIFWEWISHTOANALYZEINDIVIDUALSRESPONSESTORELATIVELYSMALLCHANGESININCOMEANDPRICESTHEPURPOSEOFTHEASSUMPTIONISTORULEOUTCERTAINKINDSOFDISCONTINUOUS,KNIFEEDGEPREFERENCESTHATPOSEPROBLEMSFORAMATHEMATICALDEVELOPMENTOFTHETHEORYOFCHOICEASSUMINGCONTINUITYDOESNOTSEEMTORUNTHERISKOFMISSINGTYPESOFECONOMICBEHAVIORTHATAREESPECIALLYIMPORTANTINTHEREALWORLDUTILITYGIVENTHEASSUMPTIONSOFCOMPLETENESS,TRANSITIVITY,ANDCONTINUITY,ITISPOSSIBLETOSHOWFORMALLYTHATPEOPLEAREABLETORANKINORDERALLPOSSIBLESITUATIONSFROMTHELEASTDESIRABLETOTHEMOST1FOLLOWINGTHETERMINOLOGYINTRODUCEDBYTHE66PARTIICHOICEANDDEMAND1THESEPROPERTIESANDTHEIRCONNECTIONTOREPRESENTATIONOFPREFERENCESBYAUTILITYFUNCTIONAREDISCUSSEDINDETAILINANDREUMASCOLELL,MICHAELDWHINSTON,ANDJERRYRGREEN,MICROECONOMICTHEORYNEWYORKOXFORDUNIVERSITYPRESS,1995NINETEENTHCENTURYPOLITICALTHEORISTJEREMYBENTHAM,ECONOMISTSCALLTHISRANKINGUTILITY2WEALSOWILLFOLLOWBENTHAMBYSAYINGTHATMOREDESIRABLESITUATIONSOFFERMOREUTILITYTHANDOLESSDESIRABLEONESTHATIS,IFAPERSONPREFERSSITUATIONATOSITUATIONB,WEWOULDSAYTHATTHEUTILITYASSIGNEDTOOPTIONA,DENOTEDBYUA,EXCEEDSTHEUTILITYASSIGNEDTOB,UBNONUNIQUENESSOFUTILITYMEASURESWEMIGHTEVENATTACHNUMBERSTOTHESEUTILITYRANKINGSBUTTHESENUMBERSWILLNOTBEUNIQUEANYSETOFNUMBERSWEARBITRARILYASSIGNTHATACCURATELYREECTSTHEORIGINALPREFERENCEORDERINGWILLIMPLYTHESAMESETOFCHOICESITMAKESNODIFFERENCEWHETHERWESAYTHATUAH110055ANDUBH110054ORTHATUAH110051,000,000ANDUBH1100505INEITHERCASETHENUMBERSIMPLYTHATAISPREFERREDTOBINTECHNICALTERMS,OURNOTIONOFUTILITYISDENEDONLYUPTOANORDERPRESERVING“MONOTONIC”TRANSFORMATION3ANYSETOFNUMBERSTHATACCURATELYREECTSAPERSONSPREFERENCEORDERINGWILLDOCONSEQUENTLY,ITMAKESNOSENSETOASK“HOWMUCHMOREISAPREFERREDTHANB”SINCETHATQUESTIONHASNOUNIQUEANSWERSURVEYSTHATASKPEOPLETORANKTHEIR“HAPPINESS”ONASCALEOF1TO10COULDJUSTASWELLUSEASCALEOF7TO1,000,000ABOUTALLTHATCANBEHOPEDFORISTHATAPERSONWHOREPORTSHEORSHEISA“6”ONTHESCALEONEDAYANDA“7”ONTHENEXTDAYISINDEEDHAPPIERONTHESECONDDAYUTILITYRANKINGSARETHEREFORELIKETHEORDINALRANKINGSOFRESTAURANTSORMOVIESUSINGONE,TWO,THREE,ORFOURSTARSTHEYSIMPLYRECORDTHERELATIVEDESIRABILITYOFCOMMODITYBUNDLESTHISLACKOFUNIQUENESSINTHEASSIGNMENTOFUTILITYNUMBERSALSOSHOWSWHYITISNOTPOSSIBLETOCOMPAREUTILITIESBETWEENPEOPLEIFONEPERSONREPORTSTHATASTEAKDINNERPROVIDESAUTILITYOF“5”ANDANOTHERREPORTSTHATTHESAMEDINNEROFFERSAUTILITYOF“100,”WECANNOTSAYWHICHINDIVIDUALVALUESTHEDINNERMOREBECAUSETHEYCOULDBEUSINGVERYDIFFERENTSCALESSIMILARLY,WEHAVENOWAYOFMEASURINGWHETHERAMOVEFROMSITUATIONATOSITUATIONBPROVIDESMOREUTILITYTOONEPERSONORTOANOTHERNONETHELESS,ASWEWILLSEE,ECONOMISTSCANSAYQUITEABITABOUTUTILITYRANKINGSBYEXAMININGWHATPEOPLEVOLUNTARILYCHOOSETODOTHECETERISPARIBUSASSUMPTIONBECAUSEUTILITYREFERSTOOVERALLSATISFACTION,SUCHAMEASURECLEARLYISAFFECTEDBYAVARIETYOFFACTORSAPERSONSUTILITYISAFFECTEDNOTONLYBYHISORHERCONSUMPTIONOFPHYSICALCOMMODITIES,BUTALSOBYPSYCHOLOGICALATTITUDES,PEERGROUPPRESSURES,PERSONALEXPERIENCES,ANDTHEGENERALCULTURALENVIRONMENTALTHOUGHECONOMISTSDOHAVEAGENERALINTERESTINEXAMININGSUCHINUENCES,USUALLYANARROWINGOFFOCUSISNECESSARYCONSEQUENTLY,ACOMMONPRACTICEISTODEVOTEATTENTIONEXCLUSIVELYTOCHOICESAMONGQUANTIABLEOPTIONSFOREXAMPLE,THERELATIVEQUANTITIESOFFOODANDSHELTERBOUGHT,THENUMBEROFHOURSWORKEDPERWEEK,ORVOTESAMONGSPECICCHAPTER3PREFERENCESANDUTILITY672JBENTHAM,INTRODUCTIONTOTHEPRINCIPLESOFMORALSANDLEGISLATIONLONDONHAFNER,18483WECANDENOTETHISIDEAMATHEMATICALLYBYSAYINGTHATANYNUMERICALUTILITYRANKINGUCANBETRANSFORMEDINTOANOTHERSETOFNUMBERSBYTHEFUNCTIONFPROVIDINGTHATFUISORDERPRESERVINGTHISCANBEASSUREDIFFH11032UH110220FOREXAMPLE,THETRANSFORMATIONFUH11005U2ISORDERPRESERVINGASISTHETRANSFORMATIONFUH11005LNUATSOMEPLACESINTHETEXTANDPROBLEMS,WEMAYNDITCONVENIENTTOMAKESUCHTRANSFORMATIONSINORDERTOMAKEAPARTICULARUTILITYRANKINGEASIERTOANALYZETAXINGFORMULASWHILEHOLDINGCONSTANTTHEOTHERTHINGSTHATAFFECTBEHAVIORTHISCETERISPARIBUSOTHERTHINGSBEINGEQUALASSUMPTIONISINVOKEDINALLECONOMICANALYSISOFUTILITYMAXIMIZINGCHOICESSOASTOMAKETHEANALYSISOFCHOICESMANAGEABLEWITHINASIMPLIEDSETTINGUTILITYFROMCONSUMPTIONOFGOODSASANIMPORTANTEXAMPLEOFTHECETERISPARIBUSASSUMPTION,CONSIDERTHEINDIVIDUALSPROBLEMOFCHOOSING,ATASINGLEPOINTINTIME,AMONGNCONSUMPTIONGOODSX1,X2,XNWESHALLASSUMETHATTHEINDIVIDUALSRANKINGOFTHESEGOODSCANBEREPRESENTEDBYAUTILITYFUNCTIONOFTHEFORMUTILITYH11005UX1,X2,XNOTHERTHINGS,31WHERETHEXSREFERTOTHEQUANTITIESOFTHEGOODSTHATMIGHTBECHOSENANDTHE“OTHERTHINGS”NOTATIONISUSEDASAREMINDERTHATMANYASPECTSOFINDIVIDUALWELFAREAREBEINGHELDCONSTANTINTHEANALYSISQUITEOFTENITISEASIERTOWRITEEQUATION31ASUTILITYH11005UX1,X2,XN32OR,IFONLYTWOGOODSAREBEINGCONSIDERED,UTILITYH11005UX,Y,32H11032WHEREITISCLEARTHATEVERYTHINGISBEINGHELDCONSTANTTHATIS,OUTSIDETHEFRAMEOFANALYSISEXCEPTTHEGOODSACTUALLYREFERREDTOINTHEUTILITYFUNCTIONITWOULDBETEDIOUSTOREMINDYOUATEACHSTEPWHATISBEINGHELDCONSTANTINTHEANALYSIS,BUTITSHOULDBEREMEMBEREDTHATSOMEFORMOFTHECETERISPARIBUSASSUMPTIONWILLALWAYSBEINOPERATIONARGUMENTSOFUTILITYFUNCTIONSTHEUTILITYFUNCTIONNOTATIONISUSEDTOINDICATEHOWANINDIVIDUALRANKSTHEPARTICULARARGUMENTSOFTHEFUNCTIONBEINGCONSIDEREDINTHEMOSTCOMMONCASE,THEUTILITYFUNCTIONEQUATION32WILLBEUSEDTOREPRESENTHOWANINDIVIDUALRANKSCERTAINBUNDLESOFGOODSTHATAREAVAILABLEATONEPOINTINTIMEONOCCASIONWEWILLUSEOTHERARGUMENTSINTHEUTILITYFUNCTION,ANDITISBESTTOCLEARUPCERTAINCONVENTIONSATTHEOUTSETFOREXAMPLE,ITMAYBEUSEFULTOTALKABOUTTHEUTILITYANINDIVIDUALRECEIVESFROMREALWEALTHWTHEREFOREWESHALLUSETHENOTATIONUTILITYH11005UW33UNLESSTHEINDIVIDUALISARATHERPECULIARSCROOGETYPEOFPERSON,WEALTHINITSOWNRIGHTGIVESNODIRECTUTILITYRATHER,ITISONLYWHENWEALTHISSPENTONCONSUMPTIONGOODSTHATANYUTILITYRESULTSFORTHISREASONEQUATION33WILLBETAKENTOMEANTHATTHEUTILITYFROMWEALTHISINFACTDERIVEDBYSPENDINGTHATWEALTHINSUCHAWAYASTOYIELDASMUCHUTILITYASPOSSIBLETWOOTHERARGUMENTSOFUTILITYFUNCTIONSWILLBEUSEDINLATERCHAPTERSINCHAPTER22WESHALLBECONCERNEDWITHTHEINDIVIDUALSLABORLEISURECHOICEANDWILLTHEREFOREHAVETOCONSIDERTHEPRESENCEOFLEISUREINTHEUTILITYFUNCTIONAFUNCTIONOFTHEFORMUTILITYH11005UC,H3468PARTIICHOICEANDDEMANDWILLBEUSEDHERECREPRESENTSCONSUMPTIONANDHREPRESENTSHOURSOFNONWORKTIMETHATIS,LEISUREDURINGAPARTICULARTIMEPERIODINCHAPTER23WESHALLBEINTERESTEDINTHEINDIVIDUALSCONSUMPTIONDECISIONSINDIFFERENTTIMEPERIODSINTHATCHAPTERWESHALLUSEAUTILITYFUNCTIONOFTHEFORMUTILITYH11005UC1,C2,35WHEREC1ISCONSUMPTIONINTHISPERIODANDC2ISCONSUMPTIONINTHENEXTPERIODBYCHANGINGTHEARGUMENTSOFTHEUTILITYFUNCTION,THEREFORE,WEWILLBEABLETOFOCUSONSPECICASPECTSOFANINDIVIDUALSCHOICESINAVARIETYOFSIMPLIEDSETTINGSINSUMMARYTHEN,WESTARTOUREXAMINATIONOFINDIVIDUALBEHAVIORWITHTHEFOLLOWINGDENITIONUTILITYINDIVIDUALSPREFERENCESAREASSUMEDTOBEREPRESENTEDBYAUTILITYFUNCTIONOFTHEFORMUX1,X2,XN,36WHEREX1,X2,XNARETHEQUANTITIESOFEACHOFNGOODSTHATMIGHTBECONSUMEDINAPERIODTHISFUNCTIONISUNIQUEONLYUPTOANORDERPRESERVINGTRANSFORMATIONECONOMICGOODSINTHISREPRESENTATIONTHEXSARETAKENTOBE“GOODS”THATIS,WHATEVERECONOMICQUANTITIESTHEYREPRESENT,WEASSUMETHATMOREOFANYPARTICULARXIDURINGSOMEPERIODISPREFERREDTOLESSWEASSUMETHISISTRUEOFEVERYGOOD,BEITASIMPLECONSUMPTIONITEMSUCHASAHOTDOGORACOMPLEXAGGREGATESUCHASWEALTHORLEISUREWEHAVEPICTUREDTHISCONVENTIONFORATWOGOODUTILITYFUNCTIONINFIGURE31THERE,ALLCONSUMPTIONBUNDLESINTHESHADEDAREAAREPREFERREDTOTHEBUNDLEX,YBECAUSEANYBUNDLEINTHESHADEDAREAPROVIDESMOREOFATLEASTONEOFTHEGOODSBYOURDENITIONOF“GOODS,”THEN,BUNDLESOFGOODSINTHESHADEDAREAARERANKEDMOREHIGHLYTHANX,YSIMILARLY,BUNDLESINTHEAREAMARKED“WORSE”ARECLEARLYINFERIORTOX,YSINCETHEYCONTAINLESSOFATLEASTONEOFTHEGOODSANDNOMOREOFTHEOTHERBUNDLESINTHETWOAREASINDICATEDBYQUESTIONMARKSAREDIFCULTTOCOMPARETOX,YBECAUSETHEYCONTAINMOREOFONEOFTHEGOODSANDLESSOFTHEOTHERASWEWILLSEE,MOVEMENTSINTOTHESEAREASINVOLVETRADEOFFSBETWEENTHETWOGOODSTRADESANDSUBSTITUTIONMOSTECONOMICACTIVITYINVOLVESVOLUNTARYTRADINGBETWEENINDIVIDUALSWHENSOMEONEBUYS,SAY,ALOAFOFBREAD,HEORSHEISVOLUNTARILYGIVINGUPONETHINGMONEYFORSOMETHINGELSEBREADTHATISOFGREATERVALUETOEXAMINETHISKINDOFVOLUNTARYTRANSACTION,WENEEDTODEVELOPAFORMALAPPARATUSFORILLUSTRATINGTRADESINTHEUTILITYFUNCTIONCONTEXTDEFINITIONCHAPTER3PREFERENCESANDUTILITY69INDIFFERENCECURVESANDTHEMARGINALRATEOFSUBSTITUTIONTODISCUSSSUCHVOLUNTARYTRADES,ITISEASIESTRSTTODEVELOPTHEIDEAOFANINDIFFERENCECURVEINFIGURE32THECURVEU1REPRESENTSALLTHEALTERNATIVECOMBINATIONSOFXANDYFORWHICHANINDIVIDUALISEQUALLYWELLOFFREMEMBERAGAINTHATALLOTHERARGUMENTSOFTHEUTILITYFUNCTIONAREBEINGHELDCONSTANTTHEINDIVIDUALISEQUALLYHAPPYCONSUMING,FOREXAMPLE,EITHERTHECOMBINATIONOFGOODSX1,Y1ORTHECOMBINATIONX2,Y2THISCURVEREPRESENTINGALLTHECONSUMPTIONBUNDLESTHATTHEINDIVIDUALRANKSEQUALLYISCALLEDANINDIFFERENCECURVEINDIFFERENCECURVEANINDIFFERENCECURVEOR,INMANYDIMENSIONS,INDIFFERENCESURFACESHOWSASETOFCONSUMPTIONBUNDLESAMONGWHICHTHEINDIVIDUALISINDIFFERENTTHATIS,THEBUNDLESALLPROVIDETHESAMELEVELOFUTILITYTHESLOPEOFTHEINDIFFERENCECURVEINFIGURE32ISNEGATIVE,SHOWINGTHATIFTHEINDIVIDUALISFORCEDTOGIVEUPSOMEY,HEORSHEMUSTBECOMPENSATEDBYANADDITIONALAMOUNTOFXTOREMAININDIFFERENTBETWEENTHETWOBUNDLESOFGOODSTHECURVEISALSODRAWNSOTHATTHESLOPEINCREASESASXINCREASESTHATIS,THESLOPESTARTSATNEGATIVEINNITYANDINCREASESTOWARD0THISISAGRAPHICALREPRESENTATIONOFTHEASSUMPTIONTHATINDIVIDUALSBECOMEPROGRESSIVELYLESSWILLINGTOTRADEAWAYYTOGETDEFINITION70PARTIICHOICEANDDEMANDMOREOFAGOODISPREFERREDTOLESSTHESHADEDAREAREPRESENTSTHOSECOMBINATIONSOFXANDYTHATAREUNAMBIGUOUSLYPREFERREDTOTHECOMBINATIONX,YCETERISPARIBUS,INDIVIDUALSPREFERMOREOFANYGOODRATHERTHANLESSCOMBINATIONSIDENTIEDBY“”INVOLVEAMBIGUOUSCHANGESINWELFAREBECAUSETHEYCONTAINMOREOFONEGOODANDLESSOFTHEOTHERQUANTITYOFXQUANTITYOFYPREFERREDTOX,YWORSETHANX,YXYFIGURE31MOREXINMATHEMATICALTERMS,THISSLOPEDIMINISHESASXINCREASESHENCE,WEHAVETHEFOLLOWINGDENITIONMARGINALRATEOFSUBSTITUTIONTHENEGATIVEOFTHESLOPEOFANINDIFFERENCECURVEU1ATSOMEPOINTISTERMEDTHEMARGINALRATEOFSUBSTITUTIONMRSATTHATPOINTTHATIS,MRSH11005H11002H20841UH11005U137WHERETHENOTATIONINDICATESTHATTHESLOPEISTOBECALCULATEDALONGTHEU1INDIFFERENCECURVETHESLOPEOFU1ANDTHEMRSTHEREFORETELLUSSOMETHINGABOUTTHETRADESTHISPERSONWILLVOLUNTARILYMAKEATAPOINTSUCHASX1,Y1,THEPERSONHASQUITEALOTOFYANDISWILLINGTOTRADEAWAYASIGNICANTAMOUNTTOGETONEMOREXTHEINDIFFERENCECURVEATX1,Y1ISTHEREFORERATHERSTEEPTHISISASITUATIONWHERETHEPERSONHAS,SAY,DYH5007DXDEFINITIONCHAPTER3PREFERENCESANDUTILITY71ASINGLEINDIFFERENCECURVETHECURVEU1REPRESENTSTHOSECOMBINATIONSOFXANDYFROMWHICHTHEINDIVIDUALDERIVESTHESAMEUTILITYTHESLOPEOFTHISCURVEREPRESENTSTHERATEATWHICHTHEINDIVIDUALISWILLINGTOTRADEXFORYWHILEREMAININGEQUALLYWELLOFFTHISSLOPEOR,MOREPROPERLY,THENEGATIVEOFTHESLOPEISTERMEDTHEMARGINALRATEOFSUBSTITUTIONINTHEGURETHEINDIFFERENCECURVEISDRAWNONTHEASSUMPTIONOFADIMINISHINGMARGINALRATEOFSUBSTITUTIONU1U1X1X2Y2Y1QUANTITYOFXQUANTITYOFYFIGURE32MANYHAMBURGERSYANDLITTLETODRINKWITHTHEMXTHISPERSONWOULDGLADLYGIVEUPAFEWBURGERSSAY,5TOQUENCHHISORHERTHIRSTWITHONEMOREDRINKATX2,Y2,ONTHEOTHERHAND,THEINDIFFERENCECURVEISATTERHERETHISPERSONHASQUITEAFEWDRINKSANDISWILLINGTOGIVEUPRELATIVELYFEWBURGERSSAY,1TOGETANOTHERSOFTDRINKCONSEQUENTLY,THEMRSDIMINISHESBETWEENX1,Y1ANDX2,Y2THECHANGINGSLOPEOFU1SHOWSHOWTHEPARTICULARCONSUMPTIONBUNDLEAVAILABLEINUENCESTHETRADESTHISPERSONWILLFREELYMAKEINDIFFERENCECURVEMAPINFIGURE32ONLYONEINDIFFERENCECURVEWASDRAWNTHEX,YQUADRANT,HOWEVER,ISDENSELYPACKEDWITHSUCHCURVES,EACHCORRESPONDINGTOADIFFERENTLEVELOFUTILITYBECAUSEEVERYBUNDLEOFGOODSCANBERANKEDANDYIELDSSOMELEVELOFUTILITY,EACHPOINTINFIGURE32MUSTHAVEANINDIFFERENCECURVEPASSINGTHROUGHITINDIFFERENCECURVESARESIMILARTOCONTOURLINESONAMAPINTHATTHEYREPRESENTLINESOFEQUAL“ALTITUDE”OFUTILITYINFIGURE33SEVERALINDIFFERENCECURVESARESHOWNTOINDICATETHATTHEREAREINNITELYMANYINTHEPLANETHELEVELOFUTILITYREPRESENTEDBYTHESECURVESINCREASESASWEMOVEINANORTHEASTDIRECTIONTHEUTILITYOFCURVEU1ISLESSTHANTHATOFU2,WHICHISLESSTHANTHATOFU3THISISBECAUSEOFTHEASSUMPTIONMADEINFIGURE31MOREOFAGOODISPREFERREDTOLESSASWASDISCUSSEDEARLIER,THEREISNOUNIQUEWAYTOASSIGNNUMBERSTOTHESEUTILITYLEVELSALLTHECURVESSHOWISTHATTHECOMBINATIONSOFGOODSONU3AREPREFERREDTOTHOSEONU2,WHICHAREPREFERREDTOTHOSEONU172PARTIICHOICEANDDEMANDTHEREAREINNITELYMANYINDIFFERENCECURVESINTHEXYPLANETHEREISANINDIFFERENCECURVEPASSINGTHROUGHEACHPOINTINTHEXYPLANEEACHOFTHESECURVESRECORDSCOMBINATIONSOFXANDYFROMWHICHTHEINDIVIDUALRECEIVESACERTAINLEVELOFSATISFACTIONMOVEMENTSINANORTHEASTDIRECTIONREPRESENTMOVEMENTSTOHIGHERLEVELSOFSATISFACTIONU1U2U3U1U3U2QUANTITYOFXQUANTITYOFYINCREASINGUTILITYFIGURE33INDIFFERENCECURVESANDTRANSITIVITYASANEXERCISEINEXAMININGTHERELATIONSHIPBETWEENCONSISTENTPREFERENCESANDTHEREPRESENTATIONOFPREFERENCESBYUTILITYFUNCTIONS,CONSIDERTHEFOLLOWINGQUESTIONCANANYTWOOFANINDIVIDUALSINDIFFERENCECURVESINTERSECTTWOSUCHINTERSECTINGCURVESARESHOWNINFIGURE34WEWISHTOKNOWIFTHEYVIOLATEOURBASICAXIOMSOFRATIONALITYUSINGOURMAPANALOGY,THEREWOULDSEEMTOBESOMETHINGWRONGATPOINTETHERE“ALTITUDE”ISEQUALTOTWODIFFERENTNUMBERS,U1ANDU2BUTNOPOINTCANBEBOTH100AND200FEETABOVESEALEVELTOPROCEEDFORMALLY,LETUSANALYZETHEBUNDLESOFGOODSREPRESENTEDBYPOINTSA,B,C,ANDDBYTHEASSUMPTIONOFNONSATIATION,“AISPREFERREDTOB”AND“CISPREFERREDTOD”BUTTHEINDIVIDUALISEQUALLYSATISEDWITHEITHERBORCTHEYLIEONTHESAMEINDIFFERENCECURVE,SOTHEAXIOMOFTRANSITIVITYIMPLIESTHATAMUSTBEPREFERREDTODBUTTHATCANNOTBETRUE,BECAUSEAANDDAREONTHESAMEINDIFFERENCECURVEANDAREBYDENITIONREGARDEDASEQUALLYDESIRABLEHENCE,THEAXIOMOFTRANSITIVITYSHOWSTHATINDIFFERENCECURVESCANNOTINTERSECTWETHEREFORESHOULDALWAYSDRAWINDIFFERENCECURVEMAPSASTHEYAPPEARINFIGURE33CONVEXITYOFINDIFFERENCECURVESANALTERNATIVEWAYOFSTATINGTHEPRINCIPLEOFADIMINISHINGMARGINALRATEOFSUBSTITUTIONUSESTHEMATHEMATICALNOTIONOFACONVEXSETASETOFPOINTSISSAIDTOBECONVEXIFANYTWOPOINTSWITHINTHESETCANBEJOINEDBYASTRAIGHTLINETHATISCONTAINEDCOMPLETELYWITHINTHESETTHEASSUMPTIONOFADIMINISHINGMRSISEQUIVALENTTOTHECHAPTER3PREFERENCESANDUTILITY73INTERSECTINGINDIFFERENCECURVESIMPLYINCONSISTENTPREFERENCESCOMBINATIONSAANDDLIEONTHESAMEINDIFFERENCECURVEANDTHEREFOREAREEQUALLYDESIRABLEBUTTHEAXIOMOFTRANSITIVITYCANBEUSEDTOSHOWTHATAISPREFERREDTODHENCE,INTERSECTINGINDIFFERENCECURVESARENOTCONSISTENTWITHRATIONALPREFERENCESU1U2QUANTITYOFXQUANTITYOFYEABCDFIGURE34ASSUMPTIONTHATALLCOMBINATIONSOFXANDY,WHICHAREPREFERREDTOORINDIFFERENTTOAPARTICULARCOMBINATIONX,Y,FORMACONVEXSET4THISISILLUSTRATEDINFIGURE35A,WHEREALLCOMBINATIONSPREFERREDTOORINDIFFERENTTOX,YAREINTHESHADEDAREAANYTWOOFTHESECOMBINATIONSSAY,X1,Y1ANDX2,Y2CANBEJOINEDBYASTRAIGHTLINEALSOCONTAINEDINTHESHADEDAREAINFIGURE35BTHISISNOTTRUEALINEJOININGX1,Y1ANDX2,Y2PASSESOUTSIDETHESHADEDAREATHEREFORE,THEINDIFFERENCECURVETHROUGHX,YIN35BDOESNOTOBEYTHEASSUMPTIONOFADIMINISHINGMRS,BECAUSETHESETOFPOINTSPREFERREDORINDIFFERENTTOX,YISNOTCONVEXCONVEXITYANDBALANCEINCONSUMPTIONBYUSINGTHENOTIONOFCONVEXITY,WECANSHOWTHATINDIVIDUALSPREFERSOMEBALANCEINTHEIRCONSUMPTIONSUPPOSETHATANINDIVIDUALISINDIFFERENTBETWEENTHECOMBINATIONX1,Y1ANDX2,Y2IFTHEINDIFFERENCECURVEISSTRICTLYCONVEX,THENTHECOMBINATIONX1H11001X2/2,Y1H11001Y2/2WILLBEPREFERREDTOEITHEROFTHEINITIALCOMBI74PARTIICHOICEANDDEMANDTHENOTIONOFCONVEXITYASANALTERNATIVEDENITIONOFADIMINISHINGMRSINATHEINDIFFERENCECURVEISCONVEXANYLINEJOININGTWOPOINTSABOVEU1ISALSOABOVEU1INBTHISISNOTTHECASE,ANDTHECURVESHOWNHEREDOESNOTEVERYWHEREHAVEADIMINISHINGMRSQUANTITYOFXQUANTITYOFYY1YXY2X2X1U1U1QUANTITYOFXBAQUANTITYOFYY1YXY2X2X1U1U1FIGURE354THISDENITIONISEQUIVALENTTOASSUMINGTHATTHEUTILITYFUNCTIONISQUASICONCAVESUCHFUNCTIONSWEREDISCUSSEDINCHAPTER2,ANDWESHALLRETURNTOEXAMINETHEMINTHENEXTSECTIONSOMETIMESTHETERMSTRICTQUASICONCAVITYISUSEDTORULEOUTTHEPOSSIBILITYOFINDIFFERENCECURVESHAVINGLINEARSEGMENTSWEGENERALLYWILLASSUMESTRICTQUASICONCAVITY,BUTINAFEWPLACESWILLILLUSTRATETHECOMPLICATIONSPOSEDBYLINEARPORTIONSOFINDIFFERENCECURVESNATIONS5INTUITIVELY,“WELLBALANCED”BUNDLESOFCOMMODITIESAREPREFERREDTOBUNDLESTHATAREHEAVILYWEIGHTEDTOWARDONECOMMODITYTHISISILLUSTRATEDINFIGURE36BECAUSETHEINDIFFERENCECURVEISASSUMEDTOBECONVEX,ALLPOINTSONTHESTRAIGHTLINEJOININGX1,Y1ANDX2,Y2AREPREFERREDTOTHESEINITIALPOINTSTHISTHEREFOREWILLBETRUEOFTHEPOINTX1H11001X2/2,Y1H11001Y2/2,WHICHLIESATTHEMIDPOINTOFSUCHALINEINDEED,ANYPROPORTIONALCOMBINATIONOFTHETWOINDIFFERENTBUNDLESOFGOODSWILLBEPREFERREDTOTHEINITIALBUNDLES,BECAUSEITWILLREPRESENTAMOREBALANCEDCOMBINATIONTHUS,STRICTCONVEXITYISEQUIVALENTTOTHEASSUMPTIONOFADIMINISHINGMRSBOTHASSUMPTIONSRULEOUTTHEPOSSIBILITYOFANINDIFFERENC