NAME101(a)HowmuchXdoesZogconsume?
30.
(b)IfthepriceofXfallsto$2.50,whileincomeandthepriceofYstayconstant,howmuchXwillZogconsume?
35.
(c)HowmuchincomemustbetakenawayfromZogtoisolatetheHicksianincomeandsubstitutione?ects(i.e.,tomakehimjustabletoa?ordtoreachhisoldindi?erencecurveatthenewprices)?
$75.
(d)Thetotale?ectofthepricechangeistochangeconsumptionfromthepoint
E
tothepoint
C.
(e)Theincomee?ectcorrespondstothemovementfromthepoint
F
tothepoint
C
whilethesubstitutione?ectcorre-
spondstothemovementfromthepoint
E
tothepoint
F.
(f)IsXanormalgoodoraninferiorgood?
Aninferior
good.
(g)Ontheaxesbelow,sketchanEngelcurveandademandcurveforGoodXthatwouldbereasonablegiventheinformationinthegraphabove.Besuretolabeltheaxesonbothyourgraphs.
Income300
225
3043x
NAME103Delphiniums40
30
aBlack line20
Redcurves10
Blue lineb0
10
20
3040Hollyhocks
(c)Nowletthepriceofhollyhocksfallto$3aunit,whilethepriceofdelphiniumsdoesnotchange.Drawhernewbudgetlineinblackink.Drawthehighestindi?erencecurvethatshecannowreachwithredink.LabelthepointshechoosesnowasB.
(d)HowmuchwouldMaude’sincomehavetobeafterthepriceofholly-hocksfell,sothatshecouldjustexactlya?ordheroldcommoditybundleA?
$120.
(e)Whenthepriceofhollyhocksfellto$3,whatpartofthechangeinMaude’sdemandwasduetotheincomee?ectandwhatpartwasduetothesubstitutione?ect?
Allsubstitutioneffect.
8.5(1)Supposethattwogoodsareperfectcomplements.Ifthepriceofonegoodchanges,whatpartofthechangeindemandisduetothesubstitutione?ect,andwhatpartisduetotheincomee?ect?
All
incomeeffect.
8.6(0)DouglasCorn?eld’sdemandfunctionforgoodxisx(px,py,m)=2m/5px.Hisincomeis$1,000,thepriceofxis$5,andthepriceofyis$20.Ifthepriceofxfallsto$4,thenhisdemandforxwillchangefrom
80
to
100.
(a)Ifhisincomeweretochangeatthesametimesothathecouldexactly
a?ordhisoldcommoditybundleatpx=4andpy=20,whatwouldhisnewincomebe?
920.
Whatwouldbehisdemandforxatthisnew
levelofincome,atpricespx=4andpy=20?
92.
102SLUTSKYEQUATION(Ch.8)
Price
432.521
30
35
x
8.4(0)Maudespendsallofherincomeondelphiniumsandhollyhocks.Shethinksthatdelphiniumsandhollyhocksareperfectsubstitutes;onedelphiniumisjustasgoodasonehollyhock.Delphiniumscost$4aunitandhollyhockscost$5aunit.
(a)Ifthepriceofdelphiniumsdecreasesto$3aunit,willMaudebuymoreofthem?
Yes.
Whatpartofthechangeinconsumptionisdue
totheincomee?ectandwhatpartisduetothesubstitutione?ect?
Allduetoincomeeffect.
(b)Ifthepricesofdelphiniumsandhollyhocksarerespectivelypd=$4andph=$5andifMaudehas$120tospend,drawherbudgetlineinblueink.Drawthehighestindi?erencecurvethatshecanattaininredink,andlabelthepointthatshechoosesasA.
104SLUTSKYEQUATION(Ch.8)
(b)Thesubstitutione?ectisachangeindemandfrom
80
to
92.
Theincomee?ectofthepricechangeisachangeindemandfrom
92to
100.
(c)Ontheaxesbelow,useblueinktodrawDouglasCorn?eld’sbudgetlinebeforethepricechange.LocatethebundlehechoosesatthesepricesonyourgraphandlabelthispointA.UseblackinktodrawDouglasCorn?eld’sbudgetlineafterthepricechange.LabelhisconsumptionbundleafterthechangebyB.
y
80
60
40
bBlack lineacBlack line20
Blue line04080120160200240280320x
(d)Onthegraphabove,useblackinktodrawabudgetlinewiththenewpricesbutwithanincomethatjustallowsDouglastobuyhisoldbundle,A.FindthebundlethathewouldchoosewiththisbudgetlineandlabelthisbundleC.
8.7(1)Mr.Consumerallowshimselftospend$100permonthoncigarettesandicecream.Mr.C’spreferencesforcigarettesandicecreamareuna?ectedbytheseasonoftheyear.
(a)InJanuary,thepriceofcigaretteswas$1perpack,whileicecreamcost$2perpint.Facedwiththeseprices,Mr.Cbought30pintsoficecreamand40packsofcigarettes.DrawMr.C’sJanuarybudgetlinewithblueinkandlabelhisJanuaryconsumptionbundlewiththeletterJ.
NAME105Ice cream100
90
80
70
Pencil budget60
line50
Black budget line40
30
Red budget lineAJFBlue20
budgetline10
0102030405060708090100
Cigarettes
(b)InFebruary,Mr.Cagainhad$100tospendandicecreamstillcost
$2perpint,butthepriceofcigarettesroseto$1.25perpack.Mr.Cconsumed30pintsoficecreamand32packsofcigarettes.DrawMr.C’sFebruarybudgetlinewithredinkandmarkhisFebruarybundlewiththeletterF.Thesubstitutione?ectofthispricechangewouldmakehimbuy(less,more,thesameamountof)
less
cigarettesand(less,more,
thesameamountof)moreicecream.Sincethisistrueandthetotalchangeinhisicecreamconsumptionwaszero,itmustbethattheincomee?ectofthispricechangeonhisconsumptionoficecreammakeshimbuy(more,less,thesameamountof)
less
icecream.Theincome
e?ectofthispricechangeislikethee?ectofan(increase,decrease)
NAME107mostlikelyanormaloraninferiorgood.Explainyouranswer.
Ifpotatoeswereanormalgood,boththefallinpotatopriceandtheriseinincomewouldincreasethedemandforpotatoes.Butpotatoconsumptiondidnotincrease.So
potatoesmustbeaninferiorgood.
(b)Canonealsotellfromthesedatawhetheritislikelythatpota-toeswereaGi?engood?
Ifpotatoeswerea
Giffengood,thenthefallinthepriceofpotatoeswoulddecreasedemandandtheriseinincomewouldalsodecreasedemandforpotatoes.Butpotatodemandstayed
constant.Sopotatoeswereprobablynota
Giffengood.
8.10(1)AgathamusttravelontheOrientExpressfromIstanbultoParis.Thedistanceis1,500miles.Atravelercanchoosetomakeanyfractionofthejourneyina?rst-classcarriageandtraveltherestofthewayinasecond-classcarriage.Thepriceis10centsamileforasecond-classcarriageand20centsamilefora?rst-classcarriage.Agathamuchprefers?rst-classtosecond-classtravel,butbecauseofamisadventureinanIstanbulbazaar,shehasonly$200leftwithwhichtobuyhertickets.Luckily,shestillhashertoothbrushandasuitcasefullofcucumbersand-wichestoeatontheway.Agathaplanstospendherentire$200onherticketsforhertrip.Shewilltravel?rstclassasmuchasshecana?ordto,butshemustgetallthewaytoParis,and$200isnotenoughmoneytogetherallthewaytoParisin?rstclass.
(a)Onthegraphbelow,useredinktoshowthelocusofcombinationsof?rst-andsecond-classticketsthatAgathacanjusta?ordtopurchasewithher$200.Useblueinktoshowthelocusofcombinationsof?rst-andsecond-classticketsthataresu?cienttocarryhertheentiredistancefromIstanbultoParis.Locatethecombinationof?rst-andsecond-classmilesthatAgathawillchooseonyourgraphandlabelitA.
106SLUTSKYEQUATION(Ch.8)
decrease
inhisincome.Thereforetheinformationwehavesuggests
thaticecreamisa(n)(normal,inferior,neutral)
normal
good.
(c)InMarch,Mr.Cagainhad$100tospend.Icecreamwasonsalefor$1perpint.Cigaretteprices,meanwhile,increasedto$1.50perpack.DrawhisMarchbudgetlinewithblackink.Ishebettero?thaninJanuary,worseo?,orcanyounotmakesuchacomparison?Betteroff.Howdoesyouranswertothelastquestionchangeifthepriceofcigaretteshadincreasedto$2perpack?
Nowyoucan’ttell.
8.8(1)ThisproblemcontinueswiththeadventuresofMr.Consumerfromthepreviousproblem.
(a)InApril,cigarettepricesroseto$2perpackandicecreamwasstillonsalefor$1perpint.Mr.Consumerbought34packsofcigarettesand32pintsoficecream.DrawhisAprilbudgetlinewithpencilandlabelhisAprilbundlewiththeletterA.Washebettero?orworseo?thaninJanuary?
Worseoff.
Washebettero?orworseo?thanin
February,orcan’tonetell?
Betteroff.
(b)InMay,cigarettesstayedat$2perpackandasthesaleonicecreamended,thepricereturnedto$2perpint.Onthewaytothestore,how-ever,Mr.Cfound$30lyinginthestreet.Hethenhad$130tospendoncigarettesandicecream.DrawhisMaybudgetwithadashedline.With-outknowingwhathepurchased,onecandeterminewhetherheisbettero?thanhewasinatleastonepreviousmonth.Whichmonthormonths?
eisbetteroffinMaythaninFebruary.
(c)Infact,Mr.Cbuys40packsofcigarettesand25pintsoficecreaminMay.DoeshesatisfyWARP?
No.
8.9(2)Inthelastchapter,westudiedaprobleminvolvingfoodpricesandconsumptioninSwedenin1850and1890.
(a)Potatoconsumptionwasthesameinbothyears.Realincomemusthavegoneupbetween1850and1890,sincetheamountoffoodstaplespurchased,asmeasuredbyeithertheLaspeyresorthePaaschequantityindex,rose.Thepriceofpotatoesroselessrapidlythanthepriceofeithermeatormilk,andataboutthesamerateasthepriceofgrain?our.Sorealincomewentupandthepriceofpotatoeswentdownrelativetoothergoods.Fromthisinformation,determinewhetherpotatoeswere
108SLUTSKYEQUATION(Ch.8)
First-class miles1600
1200
Pencil lineBlue line800
bRedlinea400Black linec0400
80012001600Second-class miles
(b)Letm1bethenumberofmilesshetravelsby?rst-classcoachandm2bethenumberofmilesshetravelsbysecond-classcoach.Writedowntwoequationsthatyoucansolveto?ndthenumberofmilesshechoosestotravelby?rst-classcoachandthenumberofmilesshechoosestotravelbysecond-classcoach.
.2m1+.1m2=200,m1+m2=
1,500.
(c)Thenumberofmilesthatshetravelsbysecond-classcoachis
1,000.
(d)Justbeforeshewasreadytobuyhertickets,thepriceofsecond-classticketsfellto$.05whilethepriceof?rst-classticketsremainedat$.20.Onthegraphthatyoudrewabove,usepenciltoshowthecombinationsof?rst-classandsecond-classticketsthatshecana?ordwithher$200attheseprices.Onyourgraph,locatethecombinationof?rst-classandsecond-classticketsthatshewouldnowchoose.(Remember,sheisgoingtotravelasmuch?rst-classasshecana?ordtoandstillmakethe1,500miletripon$200.)LabelthispointB.Howmanymilesdoesshetravelbysecondclassnow?666.66.(Hint:Foranexactsolutionyouwillhavetosolvetwolinearequationsintwounknowns.)Issecond-classtravelanormalgoodforAgatha?
No.
IsitaGi?engoodforher?
Yes.
NAME1098.11(0)WecontinuewiththeadventuresofAgatha,fromthepreviousproblem.Justafterthepricechangefrom$.10permileto$.05permileforsecond-classtravel,andjustbeforeshehadboughtanytickets,Agathamisplacedherhandbag.Althoughshekeptmostofhermoneyinhersock,themoneyshelostwasjustenoughsothatatthenewprices,shecouldexactlya?ordthecombinationof?rst-andsecond-classticketsthatshewouldhavepurchasedattheoldprices.Howmuchmoneydidshelose?
$50.Onthegraphyoustartedinthepreviousproblem,useblackinktodrawthelocusofcombinationsof?rst-andsecond-classticketsthatshecanjusta?ordafterdiscoveringherloss.Labelthepointthatshe
chooseswithaC.Howmanymileswillshetravelbysecondclassnow?
1,000.
(a)Finally,poorAgatha?ndsherhandbagagain.Howmanymileswillshetravelbysecondclassnow(assumingshedidn’tbuyanyticketsbeforeshefoundherlosthandbag)?666.66.Whenthepriceofsecond-classticketsfellfrom$.10to$.05,howmuchofachangeinAgatha’sde-mandforsecond-classticketswasduetoasubstitutione?ect?None.
Howmuchofachangewasduetoanincomee?ect?
?333.33.
Chapter9
NAME
BuyingandSelling
Introduction.Inpreviouschapters,westudiedthebehaviorofcon-sumerswhostartoutwithoutowninganygoods,butwhohadsomemoneywithwhichtobuygoods.Inthischapter,theconsumerhasaninitialen-dowment,whichisthebundleofgoodstheconsumerownsbeforeanytradesaremade.Aconsumercantradeawayfromhisinitialendowmentbysellingonegoodandbuyingtheother.
Thetechniquesthatyouhavealreadylearnedwillserveyouwellhere.To?ndouthowmuchaconsumerdemandsatgivenprices,you?ndhisbudgetlineandthen?ndapointoftangencybetweenhisbudgetlineandanindi?erencecurve.Todetermineabudgetlineforaconsumerwhoistradingfromaninitialendowmentandwhohasnosourceofincomeotherthanhisinitialendowment,noticetwothings.First,theinitialendowmentmustlieontheconsumer’sbudgetline.Thisistruebecause,nomatterwhatthepricesare,theconsumercanalwaysa?ordhisinitialendowment.Second,ifthepricesarep1andp2,theslopeofthebudgetlinemustbe?p1/p2.Thisistrue,sinceforeveryunitofgood1theconsumergivesup,hecangetexactlyp1/p2unitsofgood2.Thereforeifyouknowthepricesandyouknowtheconsumer’sinitialendowment,thenyoucanalwayswriteanequationfortheconsumer’sbudgetline.Afterall,ifyouknowonepointonalineandyouknowitsslope,youcaneitherdrawthelineorwritedownitsequation.Onceyouhavethebudgetequation,youcan?ndthebundletheconsumerchooses,usingthesamemethodsyoulearnedinChapter5.
Example:Apeasantconsumesonlyriceand?sh.Hegrowssomericeandsome?sh,butnotnecessarilyinthesameproportioninwhichhewantstoconsumethem.Supposethatifhemakesnotrades,hewillhave20unitsofriceand5unitsof?sh.Thepriceofriceis1yuanperunit,andthepriceof?shis2yuanperunit.Thevalueofthepeasant’sendowmentis(1×20)+(2×5)=30.Thereforethepeasantcanconsumeanybundle(R,F)suchthat(1×R)+(2×F)=30.
Perhapsthemostinterestingapplicationoftradingfromaninitialendowmentisthetheoryoflaborsupply.Tostudylaborsupply,weconsiderthebehaviorofaconsumerwhoischoosingbetweenleisureandothergoods.Theonlythingthatisatallnewor“tricky”is?ndingtheappropriatebudgetconstraintfortheproblemathand.Tostudylaborsupply,wethinkoftheconsumerashavinganinitialendowmentofleisure,someofwhichhemaytradeawayforgoods.
Inmostapplicationswesetthepriceof“othergoods”at1.Thewagerateisthepriceofleisure.Therolethatisplayedbyincomeintheordinaryconsumer-goodmodelisnowplayedby“fullincome.”Aworker’sfullincomeistheincomeshewouldhaveifshechosetotakenoleisure.
110SLUTSKYEQUATION(Ch.8)
112BUYINGANDSELLING(Ch.9)
Example:Sherwinhas18hoursadaywhichhedividesbetweenlaborand
leisure.Hecanworkasmanyhoursadayashewishesforawageof$5perhour.Healsoreceivesapensionthatgiveshim$10adaywhetherheworksornot.Thepriceofothergoodsis$1perunit.IfSherwinmakesnotradesatall,hewillhave18hoursofleisureand10unitsofothergoods.ThereforeSherwin’sinitialendowmentis18hoursofleisureadayand$10adayforothergoods.LetRbetheamountofleisurethathehasperday,andletCbethenumberofdollarshehastospendperdayonothergoods.Ifhiswageis$5anhour,hecana?ordtoconsumebundle(R,C)ifitcostsnomoreperdaythanthevalueofhisinitialendowment.Thevalueofhisinitialendowment(hisfullincome)is$10+($5×18)=$100perday.ThereforeSherwin’sbudgetequationis5R+C=100.
9.1(0)AbishagApplebyowns20quincesand5kumquats.Shehasnoincomefromanyothersource,butshecanbuyorselleitherquincesorkumquatsattheirmarketprices.Thepriceofkumquatsisfourtimesthepriceofquinces.Therearenoothercommoditiesofinterest.
(a)Howmanyquincescouldshehaveifshewaswillingtodowithoutkumquats?
40.
Howmanykumquatscouldshehaveifshewaswilling
todowithoutquinces?
10.
Kumquats40
30
20
Red line10
cBlue lineSquigglyeline0
102030
40Quinces
(b)DrawAbishag’sbudgetset,usingblueink,andlabeltheendowmentbundlewiththeletterE.Ifthepriceofquincesis1andthepriceofkumquatsis4,writeAbishag’sbudgetequation.Q+4K=40.Ifthepriceofquincesis2andthepriceofkumquatsis8,writeAbishag’sbudgetequation.
2Q+8K=80.Whate?ectdoesdoublingboth
NAME113priceshaveonthesetofcommoditybundlesthatAbishagcana?ord?
Noeffect.
(c)SupposethatAbishagdecidestosell10quinces.Labelher?nalconsumptionbundleinyourgraphwiththeletterC.
(d)Now,aftershehassold10quincesandownsthebundlelabeledC,supposethatthepriceofkumquatsfallssothatkumquatscostthesameasquinces.Onthediagramabove,drawAbishag’snewbudgetline,usingredink.
(e)IfAbishagobeystheweakaxiomofrevealedpreference,thentherearesomepointsonherredbudgetlinethatwecanbesureAbishagwillnotchoose.Onthegraph,makeasquigglylineovertheportionofAbishag’sredbudgetlinethatwecanbesureshewillnotchoose.
9.2(0)Mariohasasmallgardenwhereheraiseseggplantandtomatoes.Heconsumessomeofthesevegetables,andhesellssomeinthemarket.EggplantsandtomatoesareperfectcomplementsforMario,sincetheonlyrecipesheknowsusethemtogetherina1:1ratio.Oneweekhisgardenyielded30poundsofeggplantand10poundsoftomatoes.Atthattimethepriceofeachvegetablewas$5perpound.
(a)WhatisthemonetaryvalueofMario’sendowmentofvegetables?
$200.
(b)Onthegraphbelow,useblueinktodrawMario’sbudgetline.Mario
endsupconsuming
20poundsoftomatoesand20poundsofeggplant.Drawtheindi?erencecurvethroughtheconsumptionbundlethatMariochoosesandlabelthisbundleA.
(c)SupposethatbeforeMariomakesanytrades,thepriceoftomatoesrisesto$15apound,whilethepriceofeggplantstaysat$5apound.WhatisthevalueofMario’sendowmentnow?$300.Drawhisnewbudgetline,usingredink.Hewillnowchooseaconsumptionbundleconsistingof
15
tomatoesand
15
eggplants.
(d)SupposethatMariohadsoldhisentirecropatthemarketforatotalof$200,intendingtobuybacksometomatoesandeggplantforhisownconsumption.Beforehehadachancetobuyanythingback,thepriceoftomatoesroseto$15,whilethepriceofeggplantstayedat$5.Drawhisbudgetline,usingpencilorblackink.Mariowillnowconsume10
poundsoftomatoesand
10
poundsofeggplant.
NAME115(d)SupposethatbeforeLucettahasmadeanytrades,thepriceofgoodBfallsto1,andthepriceofgoodAstaysat1.DrawLucetta’sbudgetlineatthesepricesonyourgraph,usingblueink.(e)DoesLucetta’sconsumptionofgoodBriseorfall?Itrises.
Byhowmuch?25units.
WhathappenstoLucetta’sconsumption
ofgoodA?
Itdecreasesby100units.
Good B600
500
400
300
Blue budget line200
Red budget linee100
075150225300Good A
(f)SupposethatbeforethepriceofgoodBfell,Lucettahadexchangedallofhergiftsformoney,planningtousethemoneytobuyherconsumptionbundlelater.HowmuchofgoodBwillshechoosetoconsume?
250
units.
HowmuchofgoodA?
250units.
(g)Explainwhyherconsumptionisdi?erentdependingonwhethershewasholdinggoodsormoneyatthetimeofthepricechange.
In
theformercase,thefallinpBmakesherpoorerbecausesheisanetsellerofgoodB.Inthelattercase,herincomedoesn’t
114BUYINGANDSELLING(Ch.9)
(e)Assumingthatthepriceoftomatoesroseto$15from$5beforeMariomadeanytransactions,thechangeinthedemandfortomatoesduetothesubstitutione?ectwas
0.
Thechangeinthedemandfor
tomatoesduetotheordinaryincomee?ectwas
?10.
Thechange
inthedemandfortomatoesduetotheendowmentincomee?ectwas
+5.
Thetotalchangeinthedemandfortomatoeswas
?5.
Eggplant40
Red line30
20
aBlue line10
Black line01020
3040Tomatoes
9.3(0)Lucettaconsumesonlytwogoods,AandB.Heronlysourceofincomeisgiftsofthesecommoditiesfromhermanyadmirers.Shedoesn’talwaysgetthesegoodsintheproportionsinwhichshewantstoconsumethem,butshecanalwaysbuyorsellAatthepricepA=1andBatthepricepB=2.Lucetta’sutilityfunctionisU(a,b)=ab,whereaistheamountofAsheconsumesandbistheamountofBsheconsumes.(a)SupposethatLucetta’sadmirersgiveher100unitsofAand200unitsofB.Inthegraphbelow,useredinktodrawherbudgetline.LabelherinitialendowmentE.
(b)WhatareLucetta’sgrossdemandsforA?250units.
Andfor
B?
125units.
(c)WhatareLucetta’snetdemands?
150ofAand?75of
B.
116BUYINGANDSELLING(Ch.9)
change.
9.4(0)Priscilla?ndsitoptimalnottoengageintradeatthegoingpricesandjustconsumesherendowment.Priscillahasnokinksinherindi?erencecurves,andsheisendowedwithpositiveamountsofbothgoods.Usepencilorblackinktodrawabudgetlineandanindi?erencecurveforPriscillathatwouldbeconsistentwiththesefacts.Supposethatthepriceofgood2staysthesame,butthepriceofgood1falls.Useblueinktoshowhernewbudgetline.Priscillasatis?estheweakaxiomofrevealedpreference.CouldithappenthatPriscillawillconsumelessofgood1thanbefore?Explain.
No.Ifp1falls,thenwiththenewbudget,shecanstill
affordheroldbundle.
Shecouldafford
thebundleswithlessofgood1thanherendowmentattheoldprices.ByWARP,shewon’tchoosethemnow.
X2
Bluee budget lineBlackbudgetlineX1
9.5(0)PotatoesareaGi?engoodforPaddy,whohasasmallpotatofarm.Thepriceofpotatoesfell,butPaddyincreasedhispotatoconsump-tion.At?rstthisastonishedthevillageeconomist,whothoughtthatadecreaseinthepriceofaGi?engoodwassupposedtoreducedemand.ButthenherememberedthatPaddywasanetsupplierofpotatoes.Withthehelpofagraph,hewasabletoexplainPaddy’sbehavior.Intheaxesbelow,showhowthiscouldhavehappened.Put“potatoes”onthehor-izontalaxisand“allothergoods”ontheverticalaxis.LabeltheoldequilibriumAandthenewequilibriumB.DrawapointCsothatthe
NAME117Slutskysubstitutione?ectisthemovementfromAtoCandtheSlutskyincomee?ectisthemovementfromCtoB.Onthissamegraph,youarealsogoingtohavetoshowthatpotatoesareaGi?engood.Todothis,drawabudgetlineshowingthee?ectofafallinthepriceofpotatoesifPaddydidn’townanypotatoes,andonlyhadmoneyincome.LabelthenewconsumptionpointunderthesecircumstancesbyD.(Warning:Youprobablywillneedtomakeafewdryrunsonsomescratchpapertogetthewholestorystraight.)
All other goods
dacbePotatoes
9.6(0)RecallthetravailsofAgatha,fromthepreviouschapter.Shehadtotravel1,500milesfromIstanbultoParis.Shehadonly$200withwhichtobuy?rst-classandsecond-classticketsontheOrientExpresswhenthepriceof?rst-classticketswas$.20amileandthepriceofsecond-classticketswas$.10amile.SheboughtticketsthatenabledhertotravelallthewaytoParis,withasmanymilesof?rstclassasshecoulda?ord.Aftersheboardedthetrain,shediscoveredtoheramazementthatthepriceofsecond-classticketshadfallento$.05amilewhilethepriceof?rst-classticketsremainedat$.20amile.Shealsodiscoveredthatonthetrainitwaspossibletobuyorsell?rst-classticketsfor$.20amileandtobuyorsellsecond-classticketsfor$.05amile.Agathahadnomoneylefttobuyeitherkindofticket,butshedidhavetheticketsthatshehadalreadybought.
(a)Onthegraphbelow,usepenciltoshowthecombinationsofticketsthatshecoulda?ordattheoldprices.Useblueinktoshowthecombina-tionsofticketsthatwouldtakeherexactly1,500miles.MarkthepointthatshechooseswiththeletterA.
NAME119Consumption240
200
Red budget line160
120
Black budget line80
Blue budget line40
a0
4
8
12
16
20
24Leisure
(b)SupposethatMr.Cog’swageraterisesto$12anhour.Useredinktodrawhisnewbudgetline.(Hestillhas$16adayinnonlaborincome.)Ifhecontinuedtoworkexactlyasmanyhoursashedidbeforethewageincrease,howmuchmoremoneywouldhehaveeachdaytospendonconsumption?
$32.
Butwithhisnewbudgetline,hechoosestowork
813hours,andsohisconsumptionincreasesby
$36.
(c)SupposethatMr.Cogstillreceives$8anhourbutthathisnonlaborincomerisesto$48perday.Useblackinktodrawhisbudgetline.Howmanyhoursdoeshechoosetowork?
6.
(d)SupposethatMr.Coghasawageof$wperhour,anonlaborincomeof$m,andthathehas18hoursadaytodividebetweenlaborandleisure.Cog’sbudgetlinehastheequationC+wR=m+18w.Usingthesamemethodsyouusedinthechapterondemandfunctions,?ndtheamountofleisurethatCogwilldemandasafunctionofwagesandofnonlaborincome.(Hint:Noticethatthisisthesameas?ndingthedemandforRwhenthepriceofRisw,thepriceofCis1,andincomeism+18w.)Mr.Cog’sdemandfunctionforleisureisR(w,m)=
9+(m/2w).118BUYINGANDSELLING(Ch.9)
First-class miles1600
1200
Pencil lineBlue line800
a400
Red line0
400
800Second-class miles
12001600(b)Useredinktodrawalineshowingallofthecombinationsof?rst-classandsecond-classtravelthatshecana?ordwhensheisonthetrain,bytradingherendowmentofticketsatthenewpricesthatapplyonboardthetrain.
(c)Onyourgraph,showthepointthatshechoosesafter?ndingoutaboutthepricechange.Doesshechoosemore,less,orthesameamountofsecond-classtickets?
Thesame.
9.7(0)Mr.Cogworksinamachinefactory.Hecanworkasmanyhoursperdayashewishesatawagerateofw.LetCbethenumberofdollarshespendsonconsumergoodsandletRbethenumberofhoursofleisurethathechooses.
(a)Mr.Cogearns$8anhourandhas18hoursperdaytodevotetolabororleisure,andhehas$16ofnonlaborincomeperday.Writeanequationforhisbudgetbetweenconsumptionandleisure.C+8R=160.Useblueinktodrawhisbudgetlineinthegraphbelow.Hisinitialendowmentisthepointwherehedoesnoworkandenjoys18hoursofleisureperday.MarkthispointonthegraphbelowwiththeletterA.(RememberthatalthoughCogcanchoosetoworkandthereby“sell”someofhisendowmentofleisure,hecannot“buyleisure”bypayingsomebodyelsetoloafforhim.)IfMr.CoghastheutilityfunctionU(R,C)=CR,howmanyhoursofleisureperdaywillhechoose?10.
Howmany
hoursperdaywillhework?
8.
120BUYINGANDSELLING(Ch.9)
Mr.Cog’ssupplyfunctionforlaboristherefore18?R(w,m)=
9?
m/2w.
9.8(0)Fredhasjustarrivedatcollegeandistryingto?gureouthowtosupplementthemeagerchecksthathegetsfromhome.“Howcananyoneliveon$50aweekforspendingmoney?”heasks.Butheaskstonoavail.“Ifyouwantmoremoney,getajob,”sayhisparents.SoFredglumlyinvestigatesthepossibilities.Theamountofleisuretimethathehasleftafterallowingfornecessaryactivitieslikesleeping,brushingteeth,andstudyingforeconomicsclassesis50hoursaweek.HecanworkasmanyhoursperweekatanearbyTacoBellfor$5anhour.Fred’sutilityfunctionforleisureandmoneytospendonconsumptionisU(C,L)=CL.
(a)Fredhasanendowmentthatconsistsof$50ofmoneytospendon
consumptionand50hoursofleisure,someofwhichhemight“sell”formoney.ThemoneyvalueofFred’sendowmentbundle,includingbothhismoneyallowanceandthemarketvalueofhisleisuretimeistherefore
$300.
Fred’s“budgetline”forleisureandconsumptionislikeabudget
lineforsomeonewhocanbuythesetwogoodsatapriceof$1perunitofconsumptionandapriceof$5perunitofleisure.Theonlydi?erenceisthatthisbudgetlinedoesn’trunallthewaytothehorizontalaxis.
(b)Onthegraphbelow,useblackinktoshowFred’sbudgetline.(Hint:Findthecombinationofleisureandconsumptionexpendituresthathecouldhaveifhedidn’tworkatall.Findthecombinationhewouldhaveifhechosetohavenoleisureatall.Whatotherpointsareonyourgraph?)Onthesamegraph,useblueinktosketchtheindi?erencecurvesthatgiveFredutilitylevelsof3,000,4,500,and7,500.
(c)IfyoumaximizedFred’sutilitysubjecttotheabovebudget,howmuchconsumptionwouldhechoose?$150.(Hint:RememberhowtosolveforthedemandfunctionofsomeonewithaCobb-Douglasutilityfunction?)
(d)TheamountofleisurethatFredwillchoosetoconsumeis30
hours.Thismeansthathisoptimallaborsupplywillbe
20
hours.