NAME21(d)Whatistheslopeofhisindi?erencecurveatthepoint(4,12)?
?1.
Berries40
30
Pencil Shading20
Red Curve10
Red ShadingBlue Curve0102030
Nuts
40
(e)Whatistheslopeofhisindi?erencecurveatthepoint(9,12)??2/3
atthepoint(4,16)?
?1.
(f)Dotheindi?erencecurvesyouhavedrawnforAmbroseexhibitdimin-ishingmarginalrateofsubstitution?
Yes.
(g)DoesAmbrosehaveconvexpreferences?
Yes.
3.3(0)ShirleySixpackisinthehabitofdrinkingbeereacheveningwhilewatching“TheBestofBowlerama”onTV.Shehasastrongthumbandabigrefrigerator,soshedoesn’tcareaboutthesizeofthecansthatbeercomesin,sheonlycaresabouthowmuchbeershehas.
(a)Onthegraphbelow,drawsomeofShirley’sindi?erencecurvesbe-tween16-ouncecansand8-ouncecansofbeer.Useblueinktodrawtheseindi?erencecurves.
NAME23Dimes8
6
RedshadingBlueshading4
2
Blacklines0246
Quarters
8
(a)IfElmohas2quartersandadimeinhispockets,hecanbuy1softdrink.Howmanysoftdrinkscanhebuyifhehas4quartersand2dimes?
2.
(b)Useredinktoshadeintheareaonthegraphconsistingofallcom-binationsofquartersanddimesthatElmothinksarejustindi?erenttohaving2quartersand1dime.(ImaginethatitispossibleforElmotohavefractionsofquartersorofdimes,but,ofcourse,theywouldbeuse-lessinthemachine.)NowuseblueinktoshadeintheareaconsistingofallcombinationsthatElmothinksarejustindi?erenttohaving4quartersand2dimes.NoticethatElmohasindi?erence“bands,”notindi?erencecurves.
(c)DoesElmohaveconvexpreferencesbetweendimesandquarters?
Yes.
(d)DoesElmoalwaysprefermoreofbothkindsofmoneytoless?No.
(e)DoesElmohaveablisspoint?
No.
(f)IfElmohadarrivedattheCokemachineonaSaturday,thedrugstoreacrossthestreetwouldhavebeenopen.Thisdrugstorehasasodafoun-tainthatwillsellyouasmuchCokeasyouwantatapriceof4centsanounce.Thesalespersonwilltakeanycombinationofdimesandquartersinpayment.SupposethatElmoplanstospendallofthemoneyinhispocketonCokeatthedrugstoreonSaturday.Onthegraphabove,usepencilorblackinktodrawoneortwoofElmo’sindi?erencecurvesbe-tweenquartersanddimesinhispocket.(Forsimplicity,drawyourgraph
22PREFERENCES(Ch.3)
8-ounce
8
6
Blue Lines4
2
Red Lines0246
16-ounce
8(b)LorraineQuichelikestohaveabeerwhileshewatches“MasterpieceTheatre.”Sheonlyallowsherselfan8-ounceglassofbeeratanyonetime.Sincehercatdoesn’tlikebeerandshehatesstalebeer,ifthereismorethan8ouncesinthecanshepourstheexcessintothesink.(Shehasnomoralscruplesaboutwastingbeer.)Onthegraphabove,useredinktodrawsomeofLorraine’sindi?erencecurves.
3.4(0)Elmo?ndshimselfataCokemachineonahotanddustySunday.TheCokemachinerequiresexactchange—twoquartersandadime.Noothercombinationofcoinswillmakeanythingcomeoutofthemachine.Nostoresareopen;nooneisinsight.Elmoissothirstythattheonlythinghecaresaboutishowmanysoftdrinkshewillbeabletobuywiththechangeinhispocket;themorehecanbuy,thebetter.WhileElmosearcheshispockets,yourtaskistodrawsomeindi?erencecurvesthatdescribeElmo’spreferencesaboutwhathe?nds.
24PREFERENCES(Ch.3)
asifElmo’sfractionalquartersandfractionaldimesareacceptedatthe
correspondingfractionoftheirvalue.)Describethesenewindi?erencecurvesinwords.
Linesegmentswithslope?2.5.
3.5(0)RandyRatpackhatesstudyingbotheconomicsandhistory.The
moretimehespendsstudyingeithersubject,thelesshappyheis.ButRandyhasstrictlyconvexpreferences.
(a)Sketchanindi?erencecurveforRandywherethetwocommoditiesarehoursperweekspentstudyingeconomicsandhoursperweekspentstudyinghistory.Willtheslopeofanindi?erencecurvebepositiveornegative?
Negative.
(b)DoRandy’sindi?erencecurvesgetsteeperor?atterasyoumovefromlefttorightalongoneofthem?
Steeper.
Hours studying history8
6
Preferencedirection4
2
0
2
468Hours studying economics
3.6(0)FlossyToothsomelikestospendsometimestudyingandsometimedating.Infactherindi?erencecurvesbetweenhoursperweekspentstudyingandhoursperweekspentdatingareconcentriccirclesaroundherfavoritecombination,whichis20hoursofstudyingand15hoursofdatingperweek.Theclosersheistoherfavoritecombination,thehappiersheis.
NAME25(a)SupposethatFlossyiscurrentlystudying25hoursaweekanddating3hoursaweek.Wouldsheprefertobestudying30hoursaweekanddating8hoursaweek?Yes.(Hint:Remembertheformulaforthedistancebetweentwopointsintheplane?)
(b)Ontheaxesbelow,drawafewofFlossy’sindi?erencecurvesanduseyourdiagramtoillustratewhichofthetwotimeallocationsdiscussedaboveFlossywouldprefer.
Hours dating40
30
Preferencedirection20
(20,15)10
(30,8)(25,3)0
10
20
3040Hours studying
3.7(0)Joanlikeschocolatecakeandicecream,butafter10slicesofcake,shegetstiredofcake,andeatingmorecakemakesherlesshappy.Joanalwaysprefersmoreicecreamtoless.Joan’sparentsrequirehertoeateverythingputonherplate.Intheaxesbelow,useblueinktodrawasetofindi?erencecurvesthatdepictherpreferencesbetweenplateswithdi?erentamountsofcakeandicecream.Besuretolabeltheaxes.(a)SupposethatJoan’spreferencesareasbefore,butthatherparentsallowhertoleaveanythingonherplatethatshedoesn’twant.Onthegraphbelow,useredinktodrawsomeindi?erencecurvesdepictingherpreferencesbetweenplateswithdi?erentamountsofcakeandicecream.
Ice creamBlue curvesRed curvesPreferencedirection10
Chocolate cake
NAME273.9(0)MaryGranolalovestoconsumetwogoods,grapefruitsandavocados.
(a)Onthegraphbelow,theslopeofanindi?erencecurvethroughanypointwhereshehasmoregrapefruitsthanavocadosis?2.Thismeansthatwhenshehasmoregrapefruitsthanavocados,sheiswillingtogiveup
2
grapefruit(s)togetoneavocado.
(b)Onthesamegraph,theslopeofanindi?erencecurveatpointswhereshehasfewergrapefruitsthanavocadosis?1/2.Thismeansthatwhenshehasfewergrapefruitsthanavocados,sheisjustwillingtogiveup
1/2
grapefruit(s)togetoneavocado.
(c)Onthisgraph,drawanindi?erencecurveforMarythroughbundle(10A,10G).Drawanotherindi?erencecurvethrough(20A,20G).
Grapefruits40
30
Slope -220
10Slope -1/2450
10
20
3040Avocados
(d)DoesMaryhaveconvexpreferences?
Yes.
3.10(2)RalphRigidlikestoeatlunchat12noon.However,healsolikestosavemoneysohecanbuyotherconsumptiongoodsbyattendingthe“earlybirdspecials”and“latelunchers”promotedbyhislocaldiner.Ralphhas15dollarsadaytospendonlunchandotherstu?.Lunchatnooncosts$5.Ifhedelayshislunchuntilthoursafternoon,heisabletobuyhislunchforapriceof$5?t.Similarlyifheeatshislunchthoursbeforenoon,hecanbuyitforapriceof$5?t.(Thisistrueforfractionsofhoursaswellasintegernumbersofhours.)
(a)IfRalpheatslunchatnoon,howmuchmoneydoeshehaveperdaytospendonotherstu??
$10.
26PREFERENCES(Ch.3)
3.8(0)ProfessorGoodheartalwaysgivestwomidtermsinhiscommu-nicationsclass.Heonlyusesthehigherofthetwoscoresthatastudentgetsonthemidtermswhenhecalculatesthecoursegrade.
(a)NancyLernerwantstomaximizehergradeinthiscourse.Letx1beherscoreonthe?rstmidtermandx2beherscoreonthesecondmidterm.WhichcombinationofscoreswouldNancyprefer,x1=20andx2=70orx1=60andx2=60?
(20,70).
(b)Onthegraphbelow,useredinktodrawanindi?erencecurveshowingallofthecombinationsofscoresthatNancylikesexactlyasmuchasx1=20andx2=70.Alsouseredinktodrawanindi?erencecurveshowingthecombinationsthatNancylikesexactlyasmuchasx1=60andx2=60.
(c)DoesNancyhaveconvexpreferencesoverthesecombinations?
No.
Grade on second midterm
80
60
Red40
curvesBlue curves20
Preferencedirection0
20
406080Grade on first midterm
(d)NancyisalsotakingacourseineconomicsfromProfessorStern.ProfessorSterngivestwomidterms.Insteadofdiscardingthelowergrade,ProfessorSterndiscardsthehigherone.Letx1beherscoreonthe?rstmidtermandx2beherscoreonthesecondmidterm.WhichcombinationofscoreswouldNancyprefer,x1=20andx2=70orx1=60andx2=50?
(60,50).
(e)Onthegraphabove,useblueinktodrawanindi?erencecurveshowingallofthecombinationsofscoresonhereconexamsthatNancylikesexactlyaswellasx1=20andx2=70.Alsouseblueinktodrawanindi?erencecurveshowingthecombinationsthatNancylikesexactlyaswellasx1=60andx2=50.DoesNancyhaveconvexpreferencesoverthesecombinations?
Yes.
28PREFERENCES(Ch.3)
(b)Howmuchmoneyperdaywouldhehaveleftforotherstu?ifheateat2P.M.?
$12.
(c)Onthegraphbelow,useblueinktodrawthebrokenlinethatshowscombinationsofmealtimeandmoneyforotherstu?thatRalphcanjusta?ord.Onthissamegraph,drawsomeindi?erencecurvesthatwouldbeconsistentwithRalphchoosingtoeathislunchat11A.M.
Money20
15
10
5
0
10
11
12
1
2Time
3.11(0)HenryHanoveriscurrentlyconsuming20cheeseburgersand20CherryCokesaweek.Atypicalindi?erencecurveforHenryisdepictedbelow.
Cherry Coke40302010010 20 30 40Cheeseburgers
NAME29(a)Ifsomeoneo?eredtotradeHenryoneextracheeseburgerforeveryCokehegaveup,wouldHenrywanttodothis?
No.
(b)WhatifitweretheotherwayaroundforeverycheeseburgerHenrygaveup,hewouldgetanextraCoke.Wouldheacceptthiso?er?
Yes.
(c)AtwhatrateofexchangewouldHenrybewillingtostayputathiscurrentconsumptionlevel?
2cheeseburgersfor1
Coke.
3.12(1)TommyTwitishappiestwhenhehas8cookiesand4glassesofmilkperday.Wheneverhehasmorethanhisfavoriteamountofeitherfood,givinghimstillmoremakeshimworseo?.Wheneverhehaslessthanhisfavoriteamountofeitherfood,givinghimmoremakeshimbettero?.Hismothermakeshimdrink7glassesofmilkandonlyallowshim2cookiesperday.Onedaywhenhismotherwasgone,Tommy’ssadisticsistermadehimeat13cookiesandonlygavehim1glassofmilk,despitethefactthatTommycomplainedbitterlyaboutthelast5cookiesthatshemadehimeatandbeggedformoremilk.AlthoughTommycomplainedlatertohismother,hehadtoadmitthathelikedthedietthathissisterforcedonhimbetterthanwhathismotherdemanded.
(a)Useblackinktodrawsomeindi?erencecurvesforTommythatareconsistentwiththisstory.
Milk1211109876(2,7)543(8,4)21
(13,1)0
1
2
3
4
5
6
7
8
9
10
11
1213
14
15
16
Cookies
NAME31(h)AreCoachSteroid’snewpreferencesre?exive?
Yes.
3.14(0)TheBearfamilyistryingtodecidewhattohavefordin-ner.BabyBearsaysthathisrankingofthepossibilitiesis(honey,grubs,
Goldilocks).MamaBearranksthechoices(grubs,Goldilocks,honey),whilePapaBear’srankingis(Goldilocks,honey,grubs).Theydecidetotakeeachpairofalternativesandletamajorityvotedeterminethefamilyrankings.
(a)Papasuggeststhatthey?rstconsiderhoneyvs.grubs,andthenthewinnerofthatcontestvs.Goldilocks.Whichalternativewillbechosen?
Goldilocks.
(b)Mamasuggestsinsteadthattheyconsiderhoneyvs.Goldilocksandthenthewinnervs.grubs.Whichgetschosen?
Grubs.
(c)WhatordershouldBabyBearsuggestifhewantstogethisfavoritefoodfordinner?
GrubsversusGoldilocks,then
oneyversusthewinner.
(d)AretheBearfamily’s“collectivepreferences,”asdeterminedbyvot-ing,transitive?
No.
3.15(0)Olsonlikesstrongco?ee,thestrongerthebetter.Buthecan’tdistinguishsmalldi?erences.Overtheyears,Mrs.Olsonhasdiscoveredthatifshechangestheamountofco?eebymorethanoneteaspooninhersix-cuppot,Olsoncantellthatshedidit.Buthecannotdistinguishdi?erencessmallerthanoneteaspoonperpot.WhereAandBaretwodi?erentcupsofco?ee,letuswriteA??BifOlsonpreferscupAtocupB.LetuswriteA??BifOlsoneitherprefersAtoB,orcan’ttellthedi?erencebetweenthem.LetuswriteA～BifOlsoncan’ttellthedi?erencebetweencupsAandB.SupposethatOlsoniso?eredcupsA,B,andCallbrewedintheOlsons’six-cuppot.CupAwasbrewedusing14teaspoonsofco?eeinthepot.CupBwasbrewedusing14.75teaspoonsofco?eeinthepotandcupCwasbrewedusing15.5teaspoonsofco?eeinthepot.Foreachofthefollowingexpressionsdeterminewhetheritistrueoffalse.(a)A～B.
True.(b)B～A.
True.
30PREFERENCES(Ch.3)
(b)Tommy’smotherbelievesthattheoptimalamountforhimtoconsumeis7glassesofmilkand2cookies.Shemeasuresdeviationsbyabsolutevalues.IfTommyconsumessomeotherbundle,say,(c,m),shemeasureshisdeparturefromtheoptimalbundlebyD=|7?m|+|2?c|.ThelargerDis,theworseo?shethinksTommyis.UseblueinkinthegraphabovetosketchafewofMrs.Twit’sindi?erencecurvesforTommy’sconsumption.(Hint:BeforeyoutrytodrawMrs.Twit’sindi?erencecurves,wesuggestthatyoutakeapieceofscrappaperanddrawagraphofthelocusofpoints(x1,x2)suchthat|x1|+|x2|=1.)
3.13(0)CoachSteroidlikeshisplayerstobebig,fast,andobedient.IfplayerAisbetterthanplayerBintwoofthesethreecharacteristics,thenCoachSteroidprefersAtoB,butifBisbetterthanAintwoofthesethreecharacteristics,thenSteroidprefersBtoA.Otherwise,Steroidisindi?erentbetweenthem.WilburWestinghouseweighs340pounds,runsveryslowly,andisfairlyobedient.HaroldHotpointweighs240pounds,runsveryfast,andisverydisobedient.JerryJacuzziweighs150pounds,runsataveragespeed,andisextremelyobedient.
(a)DoesSteroidpreferWestinghousetoHotpointorviceversa?
He
prefersWestinghousetoHotpoint.
(b)DoesSteroidpreferHotpointtoJacuzziorviceversa?
He
prefersHotpointtoJacuzzi.
(c)DoesSteroidpreferWestinghousetoJacuzziorviceversa?
He
prefersJacuzzitoWestinghouse.
(d)DoesCoachSteroidhavetransitivepreferences?
No.
(e)Afterseverallosingseasons,CoachSteroiddecidestochangehiswayofjudgingplayers.Accordingtohisnewpreferences,SteroidprefersplayerAtoplayerBifplayerAisbetterinallthreeofthecharacteristicsthatSteroidvalues,andheprefersBtoAifplayerBisbetteratallthreethings.Heisindi?erentbetweenAandBiftheyweighthesame,areequallyfast,andareequallyobedient.Inallothercases,CoachSteroidsimplysays“AandBarenotcomparable.”(f)AreCoachSteroid’snewpreferencescomplete?No.(g)AreCoachSteroid’snewpreferencestransitive?
Yes.
32PREFERENCES(Ch.3)
(c)B～C.True.(d)A～C.False.(e)C～A.False.(f)A??B.True.(g)B??A.True.(h)B??C.True.(i)A??C.False.(j)C??A.True.(k)A??B.False.(l)B??A.False.(m)B??C.False.(n)A??C.False.(o)C??A.
True.
(p)IsOlson’s“at-least-as-good-as”relation,??,transitive?
No.(q)IsOlson’s“can’t-tell-the-di?erence”relation,～,transitive?No.
(r)isOlson’s“better-than”relation,??,transitive.
Yes.
Chapter4
NAME
Utility
Introduction.Inthepreviouschapter,youlearnedaboutpreferences
andindi?erencecurves.Herewestudyanotherwayofdescribingprefer-ences,theutilityfunction.Autilityfunctionthatrepresentsaperson’spreferencesisafunctionthatassignsautilitynumbertoeachcommoditybundle.Thenumbersareassignedinsuchawaythatcommoditybundle(x,y)getsahigherutilitynumberthanbundle(x??,y??)ifandonlyiftheconsumerprefers(x,y)to(x??,y??).IfaconsumerhastheutilityfunctionU(x1,x2),thenshewillbeindi?erentbetweentwobundlesiftheyareassignedthesameutility.
Ifyouknowaconsumer’sutilityfunction,thenyoucan?ndtheindi?erencecurvepassingthroughanycommoditybundle.Recallfromthepreviouschapterthatwhengood1isgraphedonthehorizontalaxisandgood2ontheverticalaxis,theslopeoftheindi?erencecurvepassingthroughapoint(x1,x2)isknownasthemarginalrateofsubstitution.Animportantandconvenientfactisthattheslopeofanindi?erencecurveisminustheratioofthemarginalutilityofgood1tothemarginalutilityofgood2.Forthoseofyouwhoknowevenatinybitofcalculus,calculatingmarginalutilitiesiseasy.To?ndthemarginalutilityofeithergood,youjusttakethederivativeofutilitywithrespecttotheamountofthatgood,treatingtheamountoftheothergoodasaconstant.(Ifyoudon’tknowanycalculusatall,youcancalculateanapproximationtomarginalutilitybythemethoddescribedinyourtextbook.Also,atthebeginningofthissectionoftheworkbook,welistthemarginalutilityfunctionsforcommonlyencounteredutilityfunctions.Evenifyoucan’tcomputetheseyourself,youcanrefertothislistwhenlaterproblemsrequireyoutousemarginalutilities.)
Example:Arthur’sutilityfunctionisU(x1,x2)=x1x2.Letus?ndthe
indi?erencecurveforArthurthatpassesthroughthepoint(3,4).First,calculateU(3,4)=3×4=12.Theindi?erencecurvethroughthispointconsistsofall(x1,x2)suchthatx1x2=12.Thislastequationisequivalenttox2=12/x1.ThereforetodrawArthur’sindi?erencecurvethrough(3,4),justdrawthecurvewithequationx2=12/x1.Atthepoint(x1,x2),themarginalutilityofgood1isx2andthemarginalutilityofgood2isx1.ThereforeArthur’smarginalrateofsubstitutionatthepoint(3,4)is?x2/x1=?4/3.
Example:Arthur’suncle,Basil,hastheutilityfunctionU?(x1,xthatU?2)=
3x1x2?10.Notice(x1,x2)=3U(x1,x2)?10,whereU(x1,x2)isArthur’sutilityfunction.SinceU?isapositivemultipleofUminusacon-stant,itmustbethatanychangeinconsumptionthatincreasesUwillalsoincreaseU?(andviceversa).ThereforewesaythatBasil’sutilityfunctionisamonotonicincreasingtransformationofArthur’sutilityfunction.Let
NAME35u(x1,x2)MU1(x1,x2)MU2(x1,x2)MRS(x1,x2)
2x1+3x223?2/34x1+6x246?2/3ax1+bx2a
b?a/b2√
x1+x2√1
x1
1?√1x1lnx1+x21/x11?1/x1v(x1)+x2
v??(x1)1?v??(x1)x1x2
x2
x1
?x2/x1xa1xb2
axa1?1xb2
bxa1xb2
?1?2
x?
??axbx1??(x1+2)(x2+1)2+1x1+2x(x1+a)(x2+b)
x2+b
x1+a
???x2+1
1+2x??2+b
xa1+xa
2
axa1
?1
axa2
?1
???x1x??+aa?1x1
2
34UTILITY(Ch.4)
us?ndBasil’sindi?erencecurvethroughthepoint(3,4).Firstwe?ndthatU?(3,4)=3×3×4?10=26.Theindi?erencecurvepassingthroughthispointconsistsofall(x1,x2)suchthat3x1x2?10=26.Simplifythislastexpressionbyadding10tobothsidesoftheequationanddividingbothsidesby3.You?ndx1x2=12,orequivalently,x2=12/x1.ThisisexactlythesamecurveasArthur’sindi?erencecurvethrough(3,4).Wecouldhaveknowninadvancethatthiswouldhappen,becauseiftwoconsumers’utilityfunctionsaremonotonicincreasingtransformationsofeachother,thentheseconsumersmusthavethesamepreferencerelationbetweenanypairofcommoditybundles.
Whenyouhave?nishedthisworkout,wehopethatyouwillbeabletodothefollowing:
?Drawanindi?erencecurvethroughaspeci?edcommoditybundlewhenyouknowtheutilityfunction.?Calculatemarginalutilitiesandmarginalratesofsubstitutionwhenyouknowtheutilityfunction.?Determinewhetheroneutilityfunctionisjusta“monotonictransfor-mation”ofanotherandknowwhatthatimpliesaboutpreferences.?Findutilityfunctionsthatrepresentpreferenceswhengoodsareper-fectsubstitutesandwhengoodsareperfectcomplements.?Recognizeutilityfunctionsforcommonlystudiedpreferencessuchasperfectsubstitutes,perfectcomplements,andotherkinkedindi?er-encecurves,quasilinearutility,andCobb-Douglasutility.
4.0WarmUpExercise.Thisisthe?rstofseveral“warmupex-ercises”thatyouwill?ndinWorkouts.Theseareheretohelpyouseehowtodocalculationsthatareneededinlaterproblems.Theanswerstoallwarmupexercisesareinyouranswerpages.Ifyou?ndthewarmupexerciseseasyandboring,goahead—skipthemandgetontothemainproblems.Youcancomebackandlookatthemifyougetstucklater.Thisexerciseasksyoutocalculatemarginalutilitiesandmarginalratesofsubstitutionforsomecommonutilityfunctions.Theseutilityfunctionswillreappearinseveralchapters,soitisagoodideatogettoknowthemnow.Ifyouknowcalculus,youwill?ndthistobeabreeze.Evenifyourcalculusisshakyornonexistent,youcanhandlethe?rstthreeutilityfunctionsjustbyusingthede?nitionsinthetextbook.Thesethreeareeasybecausetheutilityfunctionsarelinear.Ifyoudonotknowanycalculus,?llintherestoftheanswersfromthebackoftheworkbookandkeepacopyofthisexerciseforreferencewhenyouencountertheseutilityfunctionsinlaterproblems.
36UTILITY(Ch.4)
4.1(0)RememberCharliefromChapter3?Charlieconsumesapplesandbananas.Wehadalookattwoofhisindi?erencecurves.Inthisproblemwegiveyouenoughinformationsoyoucan?ndallofCharlie’sindi?erencecurves.WedothisbytellingyouthatCharlie’sutilityfunctionhappenstobeU(xA,xB)=xAxB.
(a)Charliehas40applesand5bananas.Charlie’sutilityforthebun-dle(40,5)isU(40,5)=
200.
Theindi?erencecurvethrough(40,5)
includesallcommoditybundles(xA,xB)suchthatxAxB=
200.
So
theindi?erencecurvethrough(40,5)hastheequationxB=200
thegraphbelow,drawtheindi?erencecurveshowingallofthexA
.On
bundlesthatCharlielikesexactlyaswellasthebundle(40,5).
Bananas40
30
20
10
0102030
40Apples
(b)Donnao?erstogiveCharlie15bananasifhewillgiveher25apples.WouldCharliehaveabundlethathelikesbetterthan(40,5)ifhemakesthistrade?Yes.WhatisthelargestnumberofapplesthatDonnacoulddemandfromCharlieinreturnfor15bananasifsheexpectshimtobewillingtotradeoratleastindi?erentabouttrading?30.(Hint:IfDonnagivesCharlie15bananas,hewillhaveatotalof20bananas.Ifhehas20bananas,howmanyapplesdoesheneedinordertobeaswell-o?ashewouldbewithouttrade?)
4.2(0)Ambrose,whomyoumetinthelastchapter,continuestothriveonnutsandberries.Yousawtwoofhisindi?erenceferencecurvehadtheequationx2=20?4√
curves.Oneindif-x1,andanotherindi?erence
curvehadtheequationx2=24?4√
x1,wherex1ishisconsumptionof
NAME37nutsandx2ishisconsumptionofberries.NowitcanbetoldthatAm-brosehasquasilinearutility.Infact,bytheutilityfunctionU(x1,x2)=4√
hispreferencescanberepresented
x1+x2.
(a)Ambroseoriginallyconsumed9unitsofnutsand10unitsofberries.Hisconsumptionofnutsisreducedto4units,butheisgivenenoughberriessothatheisjustaswell-o?ashewasbefore.Afterthechange,howmanyunitsofberriesdoesAmbroseconsume?
14.
(b)Onthegraphbelow,indicateAmbrose’soriginalconsumptionandsketchanindi?erencecurvepassingthroughthispoint.Asyoucanverify,Ambroseisindi?erentbetweenthebundle(9,10)andthebundle(25,2).Ifyoudoubledtheamountofeachgoodineachbundle,youwouldhavebundles(18,20)and(50,4).Arethesetwobundlesonthesameindi?er-encecurve?No.(Hint:Howdoyoucheckwhethertwobundlesareindi?erentwhenyouknowtheutilityfunction?)
Berries20
15
10(9,10)5
051015
Nuts
20(c)WhatisAmbrose’smarginalrateofsubstitution,MRS(x1,x2),whenheisconsumingthebundle(9,10)?(Giveanumericalanswer.)?2/3.WhatisAmbrose’smarginalrateofsubstitutionwhenheisconsumingthebundle(9,20)?
?2/3.(d)WecanwriteageneralexpressionforAmbrose’smarginalrateofsubstitutionwhenheis√
consumingcommoditybundle(x1,x2).ThisisMRS(x1,x2)=?2/x1.AlthoughwealwayswriteMRS(x1,x2)asafunctionofthetwovariables,x1andx2,weseethatAmbrose’sutilityfunctionhasthespecialpropertythathismarginalrateofsubstitutiondoesnotchangewhenthevariable
x2
changes.
NAME39“Ernie,youarerightthatmymarginalrateofsubstitutionis?2.ThatmeansthatIamwillingtomakesmalltradeswhereIgetmorethan2glassesofmilkforeverycookieIgiveyou,but9glassesofmilkfor3cookiesistoobigatrade.Myindi?erencecurvesarenotstraightlines,yousee.”WouldBurtbewillingtogiveup1cookiefor3glassesofmilk?
Yes,U(3,9)=75>U(4,6)=72.
WouldBurtobjectto
givingup2cookiesfor6glassesofmilk?
No,U(2,12)=72=
U(4,6).
(e)Onyourgraph,useredinktodrawalinewithslope?3throughthe
point(4,6).ThislineshowsallofthebundlesthatBurtcanachievebytradingcookiesformilk(ormilkforcookies)attherateof1cookieforevery3glassesofmilk.OnlyasegmentofthislinerepresentstradesthatmakeBurtbettero?thanhewaswithouttrade.LabelthislinesegmentonyourgraphAB.
4.4(0)PhilRupp’sutilityfunctionisU(x,y)=max{x,2y}.
(a)Onthegraphbelow,useblueinktodrawandlabelthelinewhoseequationisx=10.Alsouseblueinktodrawandlabelthelinewhoseequationis2y=10.
(b)Ifx=10and2y<10,thenU(x,y)=10.
Ifx<10and2y=10,
thenU(x,y)=
10.
(c)Nowuseredinktosketchintheindi?erencecurvealongwhichU(x,y)=10.DoesPhilhaveconvexpreferences?
No.
y20
15
Blue10
lines5
2y=10Redindifferencecurvex=100
5
10
15
20x
38UTILITY(Ch.4)
4.3(0)Burt’sutilityfunctionisU(x1,x2)=(x1+2)(x2+6),wherex1isthenumberofcookiesandx2isthenumberofglassesofmilkthatheconsumes.
(a)WhatistheslopeofBurt’sindi?erencecurveatthepointwhereheisconsumingthebundle(4,6)??2.Usepencilorblackinktodrawalinewiththisslopethroughthepoint(4,6).(Trytomakethisgraphfairlyneatandprecise,sincedetailswillmatter.)Thelineyoujustdrewisthetangentlinetotheconsumer’sindi?erencecurveatthepoint(4,6).(b)Theindi?erencecurvethroughthepoint(4,6)passesthroughthepoints(10,0),(7,2),and(2,12).Useblueinktosketchinthisindi?erencecurve.Incidentally,theequationforBurt’sindi?erencecurvethroughthepoint(4,6)isx2=
72/(x1+2)?6.
Glasses of milk16
12b8
Red Linea4
Blue curveBlack Line04812
16Cookies
(c)Burtcurrentlyhasthebundle(4,6).Ernieo?erstogiveBurt9glassesofmilkifBurtwillgiveErnie3cookies.IfBurtmakesthistrade,hewouldhavethebundle
(1,15).
Burtrefusestotrade.Wasthis
awisedecision?Yes,U(1,15)=63 Markthebundle(1,15)onyourgraph. (d)ErniesaystoBurt,“Burt,yourmarginalrateofsubstitutionis?2.Thatmeansthatanextracookieisworthonlytwiceasmuchtoyouasanextraglassofmilk.Io?eredtogiveyou3glassesofmilkforeverycookieyougiveme.IfIo?ertogiveyoumorethanyourmarginalrateofsubstitution,thenyoushouldwanttotradewithme.”Burtreplies, 40UTILITY(Ch.4) 4.5(0)Asyoumayrecall,NancyLerneristakingProfessorStern’seconomicscourse.Shewilltaketwoexaminationsinthecourse,andherscoreforthecourseistheminimumofthescoresthatshegetsonthetwoexams.Nancywantstogetthehighestpossiblescoreforthecourse.(a)WriteautilityfunctionthatrepresentsNancy’spreferencesoveral-ternativecombinationsoftestscoresx1andx2ontests1and2re-spectively.U(x1,x2)= min{x1,x2},oranymonotonic transformation. 4.6(0)RememberShirleySixpackandLorraineQuichefromthelastchapter?Shirleythinksa16-ouncecanofbeerisjustasgoodastwo8-ouncecans.Lorraineonlydrinks8ouncesatatimeandhatesstalebeer,soshethinksa16-ouncecanisnobetterorworsethanan8-ouncecan. (a)WriteautilityfunctionthatrepresentsShirley’spreferencesbetweencommoditybundlescomprisedof8-ouncecansand16-ouncecansofbeer.LetXstandforthenumberof8-ouncecansandYstandforthenumberof16-ouncecans. u(X,Y)=X+2Y. (b)NowwriteautilityfunctionthatrepresentsLorraine’spreferences. u(X,Y)=X+Y. (c)WouldthefunctionutilityU(X,Y)=100X+200YrepresentShirley’spreferences? Yes. WouldtheutilityfunctionU(x,y)=(5X+10Y)2 representherpreferences? Yes. WouldtheutilityfunctionU(x,y)= X+3Yrepresentherpreferences? No. (d)GiveanexampleoftwocommoditybundlessuchthatShirleylikesthe?rstbundlebetterthanthesecondbundle,whileLorrainelikesthesecondbundlebetterthanthe?rstbundle. Shirleyprefers (0,2)to(3,0).Lorrainedisagrees. 4.7(0)HarryMazzolahastheutilityfunctionu(x1,x2)=min{x1+2x2,2x1+x2},wherex1ishisconsumptionofcornchipsandx2ishisconsumptionoffrenchfries. (a)Onthegraphbelow,useapenciltodrawthelocusofpointsalongwhichx1+2x2=2x1+x2.Useblueinktoshowthelocusofpointsforwhichx1+2x2=12,andalsouseblueinktodrawthelocusofpointsforwhich2x1+x2=12.