NAME41(b)Onthegraphyouhavedrawn,shadeintheregionwherebothofthefollowinginequalitiesaresatis?ed:x1+2x2≥12and2x1+x2≥12.Atthebundle(x1,x2)=(8,2),oneseesthat2x1+x2=18
and
x1+2x2=
12.
Thereforeu(8,2)=
12.
(c)Useblackinktosketchintheindi?erencecurvealongwhichHarry’sutilityis12.Useredinktosketchintheindi?erencecurvealongwhichHarry’sutilityis6.(Hint:IsthereanythingaboutHarryMazzolathatremindsyouofMaryGranola?)
French fries8
Bluelines6
Black line4
Redline2
BluePencil linelines0
2
4
68Corn chips
(d)AtthepointwhereHarryisconsuming5unitsofcornchipsand2unitsoffrenchfries,howmanyunitsofcornchipswouldhebewillingtotradeforoneunitoffrenchfries?
2.
4.8(1)VannaBoogielikestohavelargeparties.Shealsohasastrongpreferenceforhavingexactlyasmanymenaswomenatherparties.Infact,Vanna’spreferencesamongpartiescanberepresentedbytheutilityfunctionU(x,y)=min{2x?y,2y?x}wherexisthenumberofwomenandyisthenumberofmenattheparty.Onthegraphbelow,letustrytodrawtheindi?erencecurvealongwhichVanna’sutilityis10.(a)Usepenciltodrawthelocusofpointsatwhichx=y.WhatpointonthisgivesVannaautilityof10?(10,10).Useblueinktodrawthelinealongwhich2y?x=10.Whenmin{2x?y,2y?x}=2y?x,
NAME43calculususers:Adi?erentiablefunctionf(u)isanincreasingfunctionofuifitsderivativeispositive.)(a)f(u)=3.141592u.
Yes.(b)f(u)=5,000?23u.
No.(c)f(u)=u?100,000.
Yes.
(d)f(u)=log10u.Yes.
(e)f(u)=?e?u.
Yes.(f)f(u)=1/u.
No.(g)f(u)=?1/u.
Yes.
4.10(0)MarthaModesthaspreferencesrepresentedbytheutilityfunc-tionU(a,b)=ab/100,whereaisthenumberofouncesofanimalcrackersthatsheconsumesandbisthenumberofouncesofbeansthatshecon-sumes.
(a)Onthegraphbelow,sketchthelocusofpointsthatMartha?ndsindi?erenttohaving8ouncesofanimalcrackersand2ouncesofbeans.Alsosketchthelocusofpointsthatshe?ndsindi?erenttohaving6ouncesofanimalcrackersand4ouncesofbeans.
Beans8
6
4
(6,4)2
(8,2)024
Animal crackers
68
42UTILITY(Ch.4)
thereare(morementhanwomen,morewomenthanmen)?
More
women.DrawasquigglyredlineoverthepartofthebluelineforwhichU(x,y)=min{2x?y,2y?x}=2y?x.ThisshowsallthecombinationsthatVannathinksarejustasgoodas(10,10)butwherethereare(more
menthanwomen,morewomenthanmen)?Morewomen.Nowdrawabluelinealongwhich2x?y=10.Drawasquigglyredlineoverthepartofthisnewbluelineforwhichmin{2x?y,2y?x}=2x?y.UsepenciltoshadeintheareaonthegraphthatrepresentsallcombinationsthatVannalikesatleastaswellas(10,10).
(b)Supposethatthereare9menand10womenatVanna’sparty.WouldVannathinkitwasabetterpartyoraworsepartyif5moremencame
toherparty?
Worse.
(c)IfVannahas16womenatherpartyandmorementhanwomen,andifshethinksthepartyisexactlyasgoodashaving10menand10women,howmanymendoesshehaveattheparty?22.IfVannahas16womenatherpartyandmorewomenthanmen,andifshethinksthepartyisexactlyasgoodashaving10menand10women,howmanymendoesshehaveatherparty?
13.
(d)Vanna’sindi?erencecurvesareshapedlikewhatletterofthealpha-bet?
V.
y20
15
10
PencillineSquigglyred5
linesBluelines0
5
10
15
20x
4.9(0)Supposethattheutilityfunctionsu(x,y)andv(x,y)arerelatedbyv(x,y)=f(u(x,y)).Ineachcasebelow,write“Yes”ifthefunctionfisapositivemonotonictransformationand“No”ifitisnot.(Hintfor
44UTILITY(Ch.4)
(b)BerthaBrassyhaspreferencesrepresentedbytheutilityfunctionV(a,b)=1,000a2b2,whereaisthenumberofouncesofanimalcrack-ersthatsheconsumesandbisthenumberofouncesofbeansthatsheconsumes.Onthegraphbelow,sketchthelocusofpointsthatBertha?ndsindi?erenttohaving8ouncesofanimalcrackersand2ouncesofbeans.Alsosketchthelocusofpointsthatshe?ndsindi?erenttohaving6ouncesofanimalcrackersand4ouncesofbeans.
Beans
8
6
4
(6,4)2
(8,2)024
Animal crackers
68(c)AreMartha’spreferencesconvex?
Yes.
AreBertha’s?
Yes.
(d)Whatcanyousayaboutthedi?erencebetweentheindi?erencecurvesyoudrewforBerthaandthoseyoudrewforMartha?
Thereisno
difference.
(e)Howcouldyoutellthiswasgoingtohappenwithouthavingtodrawthecurves?
Theirutilityfunctionsonlydiffer
byamonotonictransformation.
4.11(0)WillyWheeler’spreferencesoverbundlesthatcontainnon-negativeamountsofx1andx2arerepresentedbytheutilityfunction
U(x1,x2)=x21+x2
2.(a)Drawafewofhisindi?erencecurves.Whatkindofgeometric?g-
urearethey?
Quartercirclescenteredatthe
origin.DoesWillyhaveconvexpreferences?
No.
NAME45x28
6
4
2
0
2
4
6
8x1
Calculus
4.12(0)JoeBobhasautilityfunctiongivenbyu(x1,x2)=x21+2x1x2+x22.
(a)ComputeJoeBob’smarginalrateofsubstitution:MRS(x1,x2)=
?1.
(b)JoeBob’sstraightcousin,Al,hasautilityfunctionv(x1,x2)=x2+x1.ComputeAl’smarginalrateofsubstitution.MRS(x1,x2)=
?1.
(c)Dou(x1,x2)andv(x1,x2)representthesamepreferences?Yes.CanyoushowthatJoeBob’sutilityfunctionisamonotonictransforma-tionofAl’s?(Hint:SomehavesaidthatJoeBobissquare.)
Notice
thatu(x1,x2)=[v(x1,x2)]2
.
4.13(0)Theideaofassigningnumericalvaluestodetermineapreferenceorderingoverasetofobjectsisnotlimitedinapplicationtocommoditybundles.TheBillJamesBaseballAbstractarguesthatabaseballplayer’sbattingaverageisnotanadequatemeasureofhiso?ensiveproductivity.Battingaveragestreatsinglesjustthesameasextrabasehits.Further-moretheydonotgivecreditfor“walks,”althoughawalkisalmostasgoodasasingle.Jamesarguesthatadoubleintwoat-batsisbetterthanasingle,butnotasgoodastwosingles.Tore?ecttheseconsiderations,Jamesproposesthefollowingindex,whichhecalls“runscreated.”LetAbethenumberofhitsplusthenumberofwalksthatabattergetsinasea-son.LetBbethenumberoftotalbasesthatthebattergetsintheseason.(Thus,ifabatterhasSsingles,Wwalks,Ddoubles,Ttriples,andH
NAME474.14(0)Thisproblemconcernstheruns-createdindexdiscussedintheprecedingproblem.Considerabatterwhobats100timesandalwayseithermakesanout,hitsforasingle,orhitsahomerun.
(a)Letxbethenumberofsinglesandybethenumberofhomerunsin100at-bats.SupposethattheutilityfunctionU(x,y)bywhichweevaluatealternativecombinationsofsinglesandhomerunsistheruns-createdindex.ThentheformulafortheutilityfunctionisU(x,y)=
(x+y)(x+4y)/100.
(b)Let’stryto?ndoutabouttheshapeofanindi?erencecurvebetweensinglesandhomeruns.Hitting10homerunsandnosingleswouldgivehimthesameruns-createdindexashitting20singlesandnohomeruns.Markthepoints(0,10)and(x,0),whereU(x,0)=U(0,10).(c)Wherexisthenumberofsinglesyousolvedforinthepreviouspart,markthepoint(x/2,5)onyourgraph.IsU(x/2,5)greaterthanorlessthanorequaltoU(0,10)?
Greaterthan.
Isthisconsistentwith
thebatterhavingconvexpreferencesbetweensinglesandhomeruns?
Yes.
Home runs20
15
10
(0,10)Preferencedirection5
(10,5)(20,0)0
5
10
15
20Singles
46UTILITY(Ch.4)
homeruns,thenA=S+D+T+H+WandB=S+W+2D+3T+4H.)LetNbethenumberoftimesthebatterbats.Thenhisindexofrunscreatedintheseasonisde?nedtobeAB/NandwillbecalledhisRC.(a)In1987,GeorgeBellbatted649times.Hehad39walks,105singles,32doubles,4triples,and47homeruns.In1987,WadeBoggsbatted656times.Hehad105walks,130singles,40doubles,6triples,and24homeruns.In1987,AlanTrammellbatted657times.Hehad60walks,140singles,34doubles,3triples,and28homeruns.In1987,TonyGwynnbatted671times.Hehad82walks,162singles,36doubles,13triples,and7homeruns.WecancalculateA,thenumberofhitspluswalks,Bthenumberoftotalbases,andRC,therunscreatedindexforeachoftheseplayers.ForBell,A=227,B=408,RC=143.ForBoggs,A=305,B=429,RC=199.ForTrammell,A=265,B=389,RC=157.For
Gwynn,A=
300,B=383,RC=171.
(b)Ifsomebodyhasapreferenceorderingamongtheseplayers,basedonly
ontheruns-createdindex,whichplayer(s)wouldsheprefertoTrammell?
BoggsandGwynn.
(c)Thedi?erencesinthenumberoftimesatbatfortheseplayersaresmall,andwewillignorethemforsimplicityofcalculation.Onthegraphbelow,plotthecombinationsofAandBachievedbyeachoftheplayers.Drawfour“indi?erencecurves,”onethrougheachofthefourpointsyouhaveplotted.Theseindi?erencecurvesshouldrepresentcombinationsofAandBthatleadtothesamenumberofruns-created.
Number of total bases480
Boggs400
BellGwynnTrammell320
240
160
80
060120180240300360
Number of hits plus walks
48UTILITY(Ch.4)
Chapter5
NAME
Choice
Introduction.Youhavestudiedbudgets,andyouhavestudiedprefer-ences.Nowisthetimetoputthesetwoideastogetheranddosomethingwiththem.Inthischapteryoustudythecommoditybundlechosenbyautility-maximizingconsumerfromagivenbudget.
Givenpricesandincome,youknowhowtographaconsumer’sbud-get.Ifyoualsoknowtheconsumer’spreferences,youcangraphsomeofhisindi?erencecurves.Theconsumerwillchoosethe“best”indi?erencecurvethathecanreachgivenhisbudget.Butwhenyoutrytodothis,youhavetoaskyourself,“HowdoI?ndthemostdesirableindi?erencecurvethattheconsumercanreach?”Theanswertothisquestionis“lookinthelikelyplaces.”Wherearethelikelyplaces?Asyourtextbooktellsyou,therearethreekindsoflikelyplaces.Theseare(i)atangencybetweenanindi?erencecurveandthebudgetline;(ii)akinkinanindi?erencecurve;(iii)a“corner”wheretheconsumerspecializesinconsumingjustonegood.
Hereishowyou?ndapointoftangencyifwearetoldtheconsumer’sutilityfunction,thepricesofbothgoods,andtheconsumer’sincome.Thebudgetlineandanindi?erencecurvearetangentatapoint(x1,x2)iftheyhavethesameslopeatthatpoint.Nowtheslopeofanindi?erencecurveat(x1,x2)istheratio?MU1(x1,x2)/MU2(x1,x2).(Thisslopeisalsoknownasthemarginalrateofsubstitution.)Theslopeofthebudgetlineis?p1/p2.Thereforeanindi?erencecurveistangenttothebudgetlineatthepoint(x1,x2)whenMU1(x1,x2)/MU2(x1,x2)=p1/p2.Thisgivesusoneequationinthetwounknowns,x1andx2.Ifwehopetosolveforthex’s,weneedanotherequation.Thatotherequationisthebudgetequationp1x1+p2x2=m.Withthesetwoequationsyoucansolvefor(x1,x2).?
Example:AconsumerhastheutilityfunctionU(x1,x2)=x2good2isp1x2.The
priceofgood1isp1=1,thepriceof2=3,andhisincomeis180.Then,MU1(x1,x2)=2x1x2andMU2(x1,x2)=x2forehismarginalrateofsubstitutionis?MU1.There-x1(x1,x2)/MU2(x1,x2)=?21x2/x21=?2x2/x1.Thisimpliesthathisindi?erencecurvewillbetangenttohisbudgetlinewhen?2x2/x1=?p1/p2=?1/3.Simplifyingthisexpression,wehave6x2=x1.Thisisoneofthetwoequationsweneedtosolveforthetwounknowns,x1andx2.Theotherequationisthebudgetequation.Inthiscasethebudgetequationisx1+3x2=180.Solvingthesetwoequationsintwounknowns,we?ndx1=120and
?Somepeoplehavetroublerememberingwhetherthemarginalrateofsubstitutionis?MU1/MU2or?MU2/MU1.Itisn’treallycrucialtorememberwhichwaythisgoesaslongasyourememberthatatangencyhappenswhenthemarginalutilitiesofanytwogoodsareinthesameproportionastheirprices.NAME51Bananas40
Redcurves30
Black curve20
Pencil line10
Blueebudgetlinea0102030
40Apples
(b)CanCharliea?ordanybundlesthatgivehimautilityof150?Yes.(c)CanCharliea?ordanybundlesthatgivehimautilityof300?
No.
(d)Onyourgraph,markapointthatCharliecana?ordandthatgiveshimahigherutilitythan150.LabelthatpointA.
(e)Neitheroftheindi?erencecurvesthatyoudrewistangenttoCharlie’sbudgetline.Let’stryto?ndonethatis.Atanypoint,(xA,xB),Charlie’smarginalrateofsubstitutionisafunctionofxAandxB.Infact,ifyoucalculatetheratioofmarginalutilitiesforCharlie’sutilityfunction,youwill?ndthatCharlie’smarginalrateofsubstitutionisMRS(xxA,xB)=?B/xA.Thisistheslopeofhisindi?erencecurveat(xA,xB).TheslopeofCharlie’sbudgetlineis
?1/2
(giveanumericalanswer).(f)Writeanequationthatimpliesthatthebudgetlineistangenttoanindi?erencecurveat(xA,xB).?xB/xA=?1/2.Therearemanysolutionstothisequation.Eachofthesesolutionscorrespondstoapointonadi?erentindi?erencecurve.Usepenciltodrawalinethatpassesthroughallofthesepoints.
50CHOICE(Ch.5)
x2=20.Thereforeweknowthattheconsumerchoosesthebundle(x1,x2)=(120,20).
Forequilibriumatkinksoratcorners,wedon’tneedtheslopeoftheindi?erencecurvestoequaltheslopeofthebudgetline.Sowedon’thavethetangencyequationtoworkwith.Butwestillhavethebudgetequation.Thesecondequationthatyoucanuseisanequationthattellsyouthatyouareatoneofthekinkypointsoratacorner.Youwillseeexactlyhowthisworkswhenyouworkafewexercises.
Example:AconsumerhastheutilityfunctionU(x1,x2)=min{x1,3x2}.
Thepriceofx1is2,thepriceofx2is1,andherincomeis140.Herindi?erencecurvesareL-shaped.ThecornersoftheL’sallliealongthelinex1=3x2.Shewillchooseacombinationatoneofthecorners,sothisgivesusoneofthetwoequationsweneedfor?ndingtheunknownsx1andx2.Thesecondequationisherbudgetequation,whichis2x1+x2=140.Solvethesetwoequationsto?ndthatx1=60andx2=20.Soweknowthattheconsumerchoosesthebundle(x1,x2)=(60,20).
Whenyouhave?nishedtheseexercises,wehopethatyouwillbeabletodothefollowing:
?Calculatethebestbundleaconsumercana?ordatgivenpricesandincomeinthecaseofsimpleutilityfunctionswherethebesta?ord-ablebundlehappensatapointoftangency.?Findthebesta?ordablebundle,givenpricesandincomeforacon-sumerwithkinkedindi?erencecurves.?Recognizestandardexampleswherethebestbundleaconsumercana?ordhappensatacornerofthebudgetset.?Drawadiagramillustratingeachoftheabovetypesofequilibrium.?Applythemethodsyouhavelearnedtochoicesmadewithsomekindsofnonlinearbudgetsthatariseinreal-worldsituations.
5.1(0)WebeginagainwithCharlieoftheapplesandbananas.RecallthatCharlie’sutilityfunctionisU(xA,xB)=xAxB.Supposethatthepriceofapplesis1,thepriceofbananasis2,andCharlie’sincomeis40.
(a)Onthegraphbelow,useblueinktodrawCharlie’sbudgetline.(Usearulerandtrytomakethislineaccurate.)Plotafewpointsontheindi?erencecurvethatgivesCharlieautilityof150andsketchthiscurvewithredink.Nowplotafewpointsontheindi?erencecurvethatgivesCharlieautilityof300andsketchthiscurvewithblackinkorpencil.
52CHOICE(Ch.5)
(g)ThebestbundlethatCharliecana?ordmustliesomewhereonthelineyoujustpenciledin.Itmustalsolieonhisbudgetline.Ifthepointisoutsideofhisbudgetline,hecan’ta?ordit.Ifthepointliesinsideofhisbudgetline,hecana?ordtodobetterbybuyingmoreofbothgoods.Onyourgraph,labelthisbesta?ordablebundlewithanE.ThishappenswherexA=20andxB=10.Verifyyouranswerbysolvingthetwosimultaneousequationsgivenbyhisbudgetequationandthetangencycondition.
(h)WhatisCharlie’sutilityifheconsumesthebundle(20,10)?
200.
(i)Onthegraphabove,useredinktodrawhisindi?erencecurvethrough(20,10).Doesthisindi?erencecurvecrossCharlie’sbudgetline,justtouchit,ornevertouchit?
Justtouchit.
5.2(0)Clara’sutilityfunctionisU(X,Y)=(X+2)(Y+1),whereXisherconsumptionofgoodXandYisherconsumptionofgoodY.(a)WriteanequationforClara’sindi?erencecurvethatgoesthroughthepoint(X,Y)=(2,8).Y=36
X+2?1.
Ontheaxesbelow,sketch
Clara’sindi?erencecurveforU=36.
Y
16
U=3612
11
8
4
048
1112
16X
(b)Supposethatthepriceofeachgoodis1andthatClarahasanincomeof11.Drawinherbudgetline.CanClaraachieveautilityof36withthisbudget?
Yes.
NAME53(c)Atthecommoditybundle,(X,Y),Clara’smarginalrateofsubstitu-tionis
?Y+1X+2.(d)IfwesettheabsolutevalueoftheMRSequaltothepriceratio,wehavetheequation
Y+1
X+2=1.
(e)Thebudgetequationis
X+Y=11.
(f)Solvingthesetwoequationsforthetwounknowns,XandY,we?ndX=
5
andY=
6.
5.3(0)Ambrose,thenutandberryconsumer,hasautilityfunction
U(x1,x2)=4√
x1+x2,wherex1ishisconsumptionofnutsandx2ishisconsumptionofberries.
(a)Thecommoditybundle(25,0)givesAmbroseautilityof20.Otherpointsthatgivehimthesameutilityare(16,4),(9,
8
),(4,
12
),(1,16),and(0,20).Plotthesepointson
theaxesbelowanddrawaredindi?erencecurvethroughthem.(b)Supposethatthepriceofaunitofnutsis1,thepriceofaunitofberriesis2,andAmbrose’sincomeis24.DrawAmbrose’sbudgetlinewithblueink.Howmanyunitsofnutsdoeshechoosetobuy?
16
units.
(c)Howmanyunitsofberries?
4units.
(d)Findsomepointsontheindi?erencecurvethatgiveshimautilityof25andsketchthisindi?erencecurve(inred).
(e)Nowsupposethatthepricesareasbefore,butAmbrose’sincomeis34.Drawhisnewbudgetline(withpencil).Howmanyunitsofnutswillhechoose?
16units.
Howmanyunitsofberries?
9units.
NAME55(a)Onthegraphbelow,drawa“budgetline”showingthevariouscom-binationsofscoresonthetwoexamsthatshecanachievewithatotalof1,200minutesofstudying.Onthesamegraph,drawtwoorthree“indif-ferencecurves”forNancy.Onyourgraph,drawastraightlinethatgoesthroughthekinksinNancy’sindi?erencecurves.LabelthepointwherethislinehitsNancy’sbudgetwiththeletterA.DrawNancy’sindi?erencecurvethroughthispoint.
Score on test 280
\60
indifferencecurves40
a20
Budget line0
20
40
60
80
100
120
Score on test 1
(b)WriteanequationforthelinepassingthroughthekinksofNancy’sindi?erencecurves.
x1=x2.
(c)WriteanequationforNancy’sbudgetline.
10x1+20x2=
1,200.
(d)Solvethesetwoequationsintwounknownstodeterminetheintersec-tionoftheselines.Thishappensatthepoint(x1,x2)=
(40,40).
(e)Giventhatshespendsatotalof1,200minutesstudying,Nancywillmaximizeheroverallscorebyspending400
minutesstudyingforthe
?rstexaminationand800
minutesstudyingforthesecondexamina-
tion.
5.5(1)Inhercommunicationscourse,Nancyalsotakestwoexamina-tions.Heroverallgradeforthecoursewillbethemaximumofherscoresonthetwoexaminations.Nancydecidestospendatotalof400minutesstudyingforthesetwoexaminations.Ifshespendsm1minutesstudying
54CHOICE(Ch.5)
Berries20
15
10
Red curve5
Blue lineRedcurveBluePencil lineline0
5
10
15
202530Nuts
(f)Nowletusexploreacasewherethereisa“boundarysolution.”Sup-posethatthepriceofnutsisstill1andthepriceofberriesis2,butAmbrose’sincomeisonly9.Drawhisbudgetline(inblue).Sketchtheindi?erencecurvethatpassesthroughthepoint(9,0).Whatistheslopeofhisindi?erencecurveatthepoint(9,0)?
?2/3.
(g)Whatistheslopeofhisbudgetlineatthispoint?
?1/2.
(h)Whichissteeperatthispoint,thebudgetlineortheindi?erencecurve?
Indifferencecurve.(i)CanAmbrosea?ordanybundlesthathelikesbetterthanthepoint(9,0)?
No.
5.4(1)NancyLerneristryingtodecidehowtoallocatehertimeinstudyingforhereconomicscourse.Therearetwoexaminationsinthiscourse.Heroverallscoreforthecoursewillbetheminimumofherscoresonthetwoexaminations.Shehasdecidedtodevoteatotalof1,200minutestostudyingforthesetwoexams,andshewantstogetashighanoverallscoreaspossible.Sheknowsthatonthe?rstexaminationifshedoesn’tstudyatall,shewillgetascoreofzeroonit.Forevery10minutesthatshespendsstudyingforthe?rstexamination,shewillincreaseherscorebyonepoint.Ifshedoesn’tstudyatallforthesecondexaminationshewillgetazeroonit.Forevery20minutesshespendsstudyingforthesecondexamination,shewillincreaseherscorebyonepoint.
56CHOICE(Ch.5)
forthe?rstexamination,herscoreonthisexamwillbex1=m1/5.Ifshespendsm2minutesstudyingforthesecondexamination,herscoreonthisexamwillbex2=m2/10.
(a)Onthegraphbelow,drawa“budgetline”showingthevariouscombi-nationsofscoresonthetwoexamsthatshecanachievewithatotalof400minutesofstudying.Onthesamegraph,drawtwoorthree“indi?erencecurves”forNancy.Onyourgraph,?ndthepointonNancy’sbudgetlinethatgivesherthebestoverallscoreinthecourse.
(b)Giventhatshespendsatotalof400minutesstudying,Nancywillmaximizeheroverallscorebyachievingascoreof80
onthe?rst
examinationand
0
onthesecondexamination.
(c)Heroverallscoreforthecoursewillthenbe
80.
Score on test 2
80
60
40
Max. 20
overallscorePreferencedirection0
20
40
6080Score on test 1
5.6(0)Elmer’sutilityfunctionisU(x,y)=min{x,y2}.(a)IfElmerconsumes4unitsofxand3unitsofy,hisutilityis4.(b)IfElmerconsumes4unitsofxand2unitsofy,hisutilityis4.(c)IfElmerconsumes5unitsofxand2unitsofy,hisutilityis
4.
(d)Onthegraphbelow,useblueinktodrawtheindi?erencecurveforElmerthatcontainsthebundlesthathelikesexactlyaswellasthebundle(4,2).
NAME57(e)Onthesamegraph,useblueinktodrawtheindi?erencecurveforElmerthatcontainsbundlesthathelikesexactlyaswellasthebundle(1,1)andtheindi?erencecurvethatpassesthroughthepoint(16,5).
(f)Onyourgraph,useblackinktoshowthelocusofpointsatwhichElmer’sindi?erencecurveshavekinks.Whatistheequationforthiscurve?
x=y2.
(g)Onthesamegraph,useblackinktodrawElmer’sbudgetlinewhenthepriceofxis1,thepriceofyis2,andhisincomeis8.WhatbundledoesElmerchooseinthissituation?
(4,2).
y16
Blue curve12
Blue curves8
Black4
lineChosenbundle(16,5)Black curve04812162024x
(h)Supposethatthepriceofxis10andthepriceofyis15andElmerbuys100unitsofx.WhatisElmer’sincome?1,150.(Hint:At?rstyoumightthinkthereistoolittleinformationtoanswerthisquestion.Butthinkabouthowmuchyhemustbedemandingifhechooses100unitsofx.)
5.7(0)LinushastheutilityfunctionU(x,y)=x+3y.
(a)Onthegraphbelow,useblueinktodrawtheindi?erencecurvepassingthroughthepoint(x,y)=(3,3).Useblackinktosketchtheindi?erencecurveconnectingbundlesthatgiveLinusautilityof6.
NAME59Money20
15
Red curves10
5
Blue budget set0
1011121
2
Time
(a)UseblueinktoshowRalph’sbudgetset.Onthisgraph,thehorizontalaxismeasuresthetimeofdaythatheeatslunch,andtheverticalaxismeasurestheamountofmoneythathewillhavetospendonthingsotherthanlunch.Assumethathehas$20totaltospendandthatlunchatnooncosts$10.(HintHowmuchmoneywouldhehaveleftifheateatnoon?at1P.M.?at11A.M.?)
(b)RecallthatRalph’spreferredlunchtimeis12noon,butthatheiswillingtoeatatanothertimeifthefoodissu?cientlycheap.Drawsomeredindi?erencecurvesforRalphthatwouldbeconsistentwithhischoosingtoeatat11A.M.
5.9(0)JoeGradhasjustarrivedatthebigU.Hehasafellowshipthatcovershistuitionandtherentonanapartment.Inordertogetby,Joehasbecomeagraderinintermediatepricetheory,earning$100amonth.Outofthis$100hemustpayforhisfoodandutilitiesinhisapartment.Hisutilitiesexpensesconsistofheatingcostswhenheheatshisapartmentandair-conditioningcostswhenhecoolsit.Toraisethetemperatureofhisapartmentbyonedegree,itcosts$2permonth(or$20permonthtoraiseittendegrees).Touseair-conditioningtocoolhisapartmentbyadegree,itcosts$3permonth.Whateverisleftoverafterpayingtheutilities,heusestobuyfoodat$1perunit.
58CHOICE(Ch.5)
Y
16
12
8
4
Red line(3,3)BlackcurveBluecurve048
12
16X
(b)Onthesamegraph,useredinktodrawLinus’sbudgetlineifthepriceofxis1andthepriceofyis2andhisincomeis8.WhatbundledoesLinuschooseinthissituation?
(0,4).
(c)WhatbundlewouldLinuschooseifthepriceofxis1,thepriceofyis4,andhisincomeis8?
(8,0).
5.8(2)RememberourfriendRalphRigidfromChapter3?Hisfavoritediner,FoodforThought,hasadoptedthefollowingpolicytoreducethecrowdsatlunchtime:ifyoushowupforlunchthoursbeforeorafter12noon,yougettodeducttdollarsfromyourbill.(Thisholdsforanyfractionofanhouraswell.)
60CHOICE(Ch.5)
Food120
100
December
SeptemberAugust
80
Blue budget constraintBlack budget constraintRed budget constraint60
40
20
0102030405060708090100
Temperature
(a)WhenJoe?rstarrivesinSeptember,thetemperatureofhisapartmentis60degrees.Ifhespendsnothingonheatingorcooling,thetemperatureinhisroomwillbe60degreesandhewillhave$100lefttospendonfood.Ifheheatedtheroomto70degrees,hewouldhave
$80
lefttospend
onfood.Ifhecooledtheroomto50degrees,hewouldhave$70lefttospendonfood.Onthegraphbelow,showJoe’sSeptemberbudgetconstraint(withblackink).(Hint:YouhavejustfoundthreepointsthatJoecana?ord.Apparently,hisbudgetsetisnotboundedbyasinglestraightline.)
(b)InDecember,theoutsidetemperatureis30degreesandinAugustpoorJoeistryingtounderstandmacroeconomicswhilethetemperatureoutsideis85degrees.Onthesamegraphyouusedabove,drawJoe’sbudgetconstraintsforthemonthsofDecember(inblueink)andAugust(inredink).
(c)Drawafewsmooth(unkinky)indi?erencecurvesforJoeinsuchawaythatthefollowingaretrue.(i)Hisfavoritetemperatureforhisapartmentwouldbe65degreesifitcosthimnothingtoheatitorcoolit.(ii)JoechoosestousethefurnaceinDecember,air-conditioninginAugust,andneitherinSeptember.(iii)Joeisbettero?inDecemberthaninAugust.(d)InwhatmonthsistheslopeofJoe’sbudgetconstraintequaltotheslopeofhisindi?erencecurve?
AugustandDecember.