Chapter1
NAME
TheMarket
Introduction.Theproblemsinthischapterexaminesomevariationson
theapartmentmarketdescribedinthetext.Inmostoftheproblemsweworkwiththetruedemandcurveconstructedfromthereservationpricesoftheconsumersratherthanthe“smoothed”demandcurvethatweusedinthetext.
Rememberthatthereservationpriceofaconsumeristhatpricewhereheisjustindi?erentbetweenrentingornotrentingtheapartment.Atanypricebelowthereservationpricetheconsumerwilldemandoneapartment,atanypriceabovethereservationpricetheconsumerwillde-mandzeroapartments,andexactlyatthereservationpricetheconsumerwillbeindi?erentbetweenhavingzerooroneapartment.
Youshouldalsoobservethatwhendemandcurveshavethe“stair-case”shapeusedhere,therewilltypicallybearangeofpriceswheresupplyequalsdemand.Thuswewillaskforthethehighestandlowestpriceintherange.
1.1(3)Supposethatwehave8peoplewhowanttorentanapartment.Theirreservationpricesaregivenbelow.(Tokeepthenumberssmall,thinkofthesenumbersasbeingdailyrentpayments.)
Person
=ABCDEFGHPrice
=40
25
3035
10
18
15
5
(a)Plotthemarketdemandcurveinthefollowinggraph.(Hint:Whenthemarketpriceisequaltosomeconsumeri’sreservationprice,therewillbetwodi?erentquantitiesofapartmentsdemanded,sinceconsumeriwillbeindi?erentbetweenhavingornothavinganapartment.)
NAME3PersonABCDEFGHHighprice1818181825252525Lowprice
15
15
15
15
18
15
18
18
(b)Supposethatthereweretwopeopleateachreservationpriceand10apartments.Whatisthehighestpriceatwhichdemandequalssupply?
18.
Supposethatoneoftheapartmentswasturnedintoacondo-
minium.Isthatpricestillanequilibriumprice?
Yes.
1.3(2)Supposenowthatamonopolistownsalltheapartmentsandthatheistryingtodeterminewhichpriceandquantitymaximizehisrevenues.(a)Fillintheboxwiththemaximumpriceandrevenuethatthemonop-olistcanmakeifherents1,2,...,8apartments.(Assumethathemustchargeonepriceforallapartments.)Number12345678Price403530251815105Revenue
40
70
90
100
90
90
70
40
(b)WhichofthepeopleA–Fwouldgetapartments?
A,B,C,D.(c)Ifthemonopolistwererequiredbylawtorentexactly5apartments,whatpricewouldhechargetomaximizehisrevenue?$18.
(d)Whowouldgetapartments?
A,B,C,D,F.
(e)Ifthislandlordcouldchargeeachindividualadi?erentprice,andheknewthereservationpricesofalltheindividuals,whatisthemaximumrevenuehecouldmakeifherentedall5apartments?
$148.
(f)If5apartmentswererented,whichindividualswouldgettheapart-ments?
A,B,C,D,F.1.4(2)Supposethatthereare5apartmentstoberentedandthatthecityrent-controlboardsetsamaximumrentof$9.FurthersupposethatpeopleA,B,C,D,andEmanagetogetanapartment,whileF,G,andHarefrozenout.
2THEMARKET(Ch.1)
Price605040302010
012345
Apartments
678(b)Supposethesupplyofapartmentsis?xedat5units.Inthiscasethereisawholerangeofpricesthatwillbeequilibriumprices.Whatisthehighestpricethatwouldmakethedemandforapartmentsequalto5
units?
$18.(c)Whatisthelowestpricethatwouldmakethemarketdemandequalto5units?
$15.
(d)Withasupplyof4apartments,whichofthepeopleA–Hendupgettingapartments?
A,B,C,D.
(e)Whatifthesupplyofapartmentsincreasesto6units.Whatistherangeofequilibriumprices?
$10to$15.
1.2(3)Supposethatthereareoriginally5unitsinthemarketandthat1ofthemisturnedintoacondominium.
(a)SupposethatpersonAdecidestobuythecondominium.Whatwillbethehighestpriceatwhichthedemandforapartmentswillequalthesupplyofapartments?Whatwillbethelowestprice?Enteryouran-swersincolumnA,inthetable.ThencalculatetheequilibriumpricesofapartmentsifB,C,...,decidetobuythecondominium.
4THEMARKET(Ch.1)
(a)Ifsublettingislegal—or,atleast,practiced—whowillsublettowhominequilibrium?(Assumethatpeoplewhosubletcanevadethecityrent-controlrestrictions.)
E,whoiswillingtopayonly
$10foranapartmentwouldsublettoF,
whoiswillingtopay$18.
(b)Whatwillbethemaximumamountthatcanbechargedforthesubletpayment?
$18.
(c)Ifyouhaverentcontrolwithunlimitedsublettingallowed,whichoftheconsumersdescribedabovewillendupinthe5apartments?
A,
B,C,D,F.
(d)Howdoesthiscomparetothemarketoutcome?
It’sthe
same.
1.5(2)Inthetextwearguedthatataxonlandlordswouldnotgetpassedalongtotherenters.Whatwouldhappenifinsteadthetaxwasimposedonrenters?
(a)Toanswerthisquestion,considerthegroupofpeopleinProblem1.1.Whatisthemaximumthattheywouldbewillingtopaytothelandlordiftheyeachhadtopaya$5taxonapartmentstothecity?Fillintheboxbelowwiththesereservationprices.PersonABCDEFGHReservationPrice
35202530513100
(b)Usingthisinformationdeterminethemaximumequilibriumpriceifthereare5apartmentstoberented.
$13.
(c)Ofcourse,thetotalpricearenterpaysconsistsofhisorherrentplusthetax.Thisamountis
$18.
(d)Howdoesthiscomparetowhathappensifthetaxisleviedonthelandlords?
It’sthesame.
Chapter2
NAME
BudgetConstraint
Introduction.Theseworkoutsaredesignedtobuildyourskillsinde-scribingeconomicsituationswithgraphsandalgebra.Budgetsetsareagoodplacetostart,becauseboththealgebraandthegraphingareveryeasy.Wheretherearejusttwogoods,aconsumerwhoconsumesx1unitsofgood1andx2unitsofgood2issaidtoconsumetheconsumptionbun-dle,(x1,x2).Anyconsumptionbundlecanberepresentedbyapointonatwo-dimensionalgraphwithquantitiesofgood1onthehorizontalaxisandquantitiesofgood2ontheverticalaxis.Ifthepricesarep1forgood1andp2forgood2,andiftheconsumerhasincomem,thenshecana?ordanyconsumptionbundle,(x1,x2),suchthatp1x1+p2x2≤m.Onagraph,thebudgetlineisjustthelinesegmentwithequationp1x1+p2x2=mandwithx1andx2bothnonnegative.Thebudgetlineistheboundaryofthebudgetset.Allofthepointsthattheconsumercana?ordlieononesideofthelineandallofthepointsthattheconsumercannota?ordlieontheother.
Ifyouknowpricesandincome,youcanconstructaconsumer’sbud-getlineby?ndingtwocommoditybundlesthatshecan“justa?ord”anddrawingthestraightlinethatrunsthroughbothpoints.
Example:Myrtlehas50dollarstospend.Sheconsumesonlyapplesandbananas.Applescost2dollarseachandbananascost1dollareach.Youaretographherbudgetline,whereapplesaremeasuredonthehorizontalaxisandbananasontheverticalaxis.Noticethatifshespendsallofherincomeonapples,shecana?ord25applesandnobananas.Thereforeherbudgetlinegoesthroughthepoint(25,0)onthehorizontalaxis.Ifshespendsallofherincomeonbananas,shecana?ord50bananasandnoapples.Therforeherbudgetlinealsopassesthroughtthepoint(0,50)ontheverticalaxis.Markthesetwopointsonyourgraph.Thendrawastraightlinebetweenthem.ThisisMyrtle’sbudgetline.
Whatifyouarenottoldpricesorincome,butyouknowtwocom-moditybundlesthattheconsumercanjusta?ord?Then,iftherearejusttwocommodities,youknowthatauniquelinecanbedrawnthroughtwopoints,soyouhaveenoughinformationtodrawthebudgetline.
Example:Laurelconsumesonlyaleandbread.Ifshespendsallofherincome,shecanjusta?ord20bottlesofaleand5loavesofbread.Anothercommoditybundlethatshecana?ordifshespendsherentireincomeis10bottlesofaleand10loavesofbread.Ifthepriceofaleis1dollarperbottle,howmuchmoneydoesshehavetospend?Youcouldsolvethisproblemgraphically.Measurealeonthehorizontalaxisandbreadontheverticalaxis.Plotthetwopoints,(20,5)and(10,10),thatyouknowtobeonthebudgetline.Drawthestraightlinebetweenthesepointsandextendthelinetothehorizontalaxis.Thispointdenotestheamountof
NAME72.1(0)Youhaveanincomeof$40tospendontwocommodities.Com-modity1costs$10perunit,andcommodity2costs$5perunit.(a)Writedownyourbudgetequation.
10x1+5x2=40.
(b)Ifyouspentallyourincomeoncommodity1,howmuchcouldyoubuy?
4.
(c)Ifyouspentallofyourincomeoncommodity2,howmuchcouldyoubuy?8.
Useblueinktodrawyourbudgetlineinthegraph
below.
x28
Blue LineRed Line6
Black Shading4
Black Line2
BlueShading024
6
8x1
(d)Supposethatthepriceofcommodity1fallsto$5whileeverythingelsestaysthesame.Writedownyournewbudgetequation.
5x1+5x2=
40.
Onthegraphabove,useredinktodrawyournewbudgetline.
(e)Supposethattheamountyouareallowedtospendfallsto$30,whilethepricesofbothcommoditiesremainat$5.Writedownyourbudgetequation.5x1+5x2=30.
Useblackinktodrawthisbudgetline.
(f)Onyourdiagram,useblueinktoshadeinthearearepresentingcom-moditybundlesthatyoucana?ordwiththebudgetinPart(e)butcouldnota?ordtobuywiththebudgetinPart(a).Useblackinkorpenciltoshadeinthearearepresentingcommoditybundlesthatyoucoulda?ordwiththebudgetinPart(a)butcannota?ordwiththebudgetinPart(e).
2.2(0)Onthegraphbelow,drawabudgetlineforeachcase.
6BUDGETCONSTRAINT(Ch.2)
aleLaurelcana?ordifshespendsallofhermoneyonale.Sincealecosts1dollarabottle,herincomeindollarsisequaltothelargestnumberofbottlesshecana?ord.Alternatively,youcanreasonasfollows.Sincethebundles(20,5)and(10,10)costthesame,itmustbethatgivingup10bottlesofalemakesherabletoa?ordanextra5loavesofbread.Sobreadcoststwiceasmuchasale.Thepriceofaleis1dollar,sothepriceofbreadis2dollars.Thebundle(20,5)costsasmuchasherincome.Thereforeherincomemustbe20×1+5×2=30.
Whenyouhavecompletedthisworkout,wehopethatyouwillbeabletodothefollowing:
?Writeanequationforthebudgetlineanddrawthebudgetsetonagraphwhenyouaregivenpricesandincomeorwhenyouaregiventwopointsonthebudgetline.
?Graphthee?ectsofchangesinpricesandincomeonbudgetsets.?Understandtheconceptofnumeraireandknowwhathappenstothebudgetsetwhenincomeandallpricesaremultipliedbythesamepositiveamount.
?Knowwhatthebudgetsetlookslikeifoneormoreofthepricesisnegative.?Seethattheideaofa“budgetset”canbeappliedtoconstrainedchoiceswherethereareotherconstraintsonwhatyoucanhave,inadditiontoaconstraintonmoneyexpenditure.
8BUDGETCONSTRAINT(Ch.2)
(a)p1=1,p2=1,m=15.(Useblueink.)(b)p1=1,p2=2,m=20.(Useredink.)(c)p1=0,p2=1,m=10.(Useblackink.)
(d)p1=p2,m=15p1.(Usepencilorblackink.Hint:Howmuchofgood1couldyoua?ordifyouspendyourentirebudgetongood1?)
x220
15
Blue LineBlack Line10
Red Line5
051015
20x1
2.3(0)Yourbudgetissuchthatifyouspendyourentireincome,you
cana?ordeither4unitsofgoodxand6unitsofgoodyor12unitsofxand2unitsofy.
(a)Markthesetwoconsumptionbundlesanddrawthebudgetlineinthegraphbelow.
y16
12
8
4
04812
16x
NAME9(b)Whatistheratioofthepriceofxtothepriceofy?
1/2.
(c)Ifyouspentallofyourincomeonx,howmuchxcouldyoubuy?
16.
(d)Ifyouspentallofyourincomeony,howmuchycouldyoubuy?
8.
(e)Writeabudgetequationthatgivesyouthisbudgetline,wherethepriceofxis1.
x+2y=16.
(f)Writeanotherbudgetequationthatgivesyouthesamebudgetline,butwherethepriceofxis3.
3x+6y=48.
2.4(1)Murphywasconsuming100unitsofXand50unitsofY.The
priceofXrosefrom2to3.ThepriceofYremainedat4.
(a)HowmuchwouldMurphy’sincomehavetorisesothathecanstillexactlya?ord100unitsofXand50unitsofY?
$100.
2.5(1)IfAmyspentherentireallowance,shecoulda?ord8candybarsand8comicbooksaweek.Shecouldalsojusta?ord10candybarsand4comicbooksaweek.Thepriceofacandybaris50cents.Drawherbudgetlineintheboxbelow.WhatisAmy’sweeklyallowance?
$6.
Comic books
32
24
16
8
08121624
32Candy bars
NAME11(b)Ifheaccepts15sacksofgarbage,howmanyvideocassettescanhebuy?
5.
(c)Writedownanequationforhisbudgetline.
6C?2G=0.
(d)DrawEdmund’sbudgetlineandshadeinhisbudgetset.
Garbage20
15
10
Budget LineBudget Set5
0510
1520Video cassettes
2.8(0)IfyouthinkEdmundisodd,considerhisbrotherEmmett.Emmettconsumesspeechesbypoliticiansanduniversityadministrators.Heispaid$1perhourforlisteningtopoliticiansand$2perhourforlisteningtouniversityadministrators.(Emmettisingreatdemandtohelp?llemptychairsatpubliclecturesbecauseofhisdistinguishedappearanceandhisabilitytorefrainfrommakingrudenoises.)Emmettconsumesonegoodforwhichhemustpay.Wehaveagreednottodisclosewhatthatgoodis,butwecantellyouthatitcosts$15perunitandweshallcallitGoodX.Inadditiontowhatheispaidforconsumingspeeches,Emmettreceivesapensionof$50perweek.
Administrator speeches
100
75
50
25
02550
75100Politician speeches
10BUDGETCONSTRAINT(Ch.2)
2.6(0)InasmallcountryneartheBalticSea,thereareonlythreecommodities:potatoes,meatballs,andjam.Priceshavebeenremark-ablystableforthelast50yearsorso.Potatoescost2crownspersack,meatballscost4crownspercrock,andjamcosts6crownsperjar.(a)WritedownabudgetequationforacitizennamedGunnarwhohasanincomeof360crownsperyear.LetPstandforthenumberofsacksofpotatoes,Mforthenumberofcrocksofmeatballs,andJforthenumberofjarsofjamconsumedbyGunnarinayear.
2P+4M+6J=
360.
(b)Thecitizensofthiscountryareingeneralverycleverpeople,buttheyarenotgoodatmultiplyingby2.Thismadeshoppingforpotatoesexcru-ciatinglydi?cultformanycitizens.Thereforeitwasdecidedtointroduceanewunitofcurrency,suchthatpotatoeswouldbethenumeraire.Asackofpotatoescostsoneunitofthenewcurrencywhilethesamerel-ativepricesapplyasinthepast.Intermsofthenewcurrency,whatisthepriceofmeatballs?
2crowns.
(c)Intermsofthenewcurrency,whatisthepriceofjam?
3
crowns.
(d)WhatwouldGunnar’sincomeinthenewcurrencyhavetobeforhimtobeexactlyabletoa?ordthesamecommoditybundlesthathecoulda?ordbeforethechange?
180crowns.
(e)WritedownGunnar’snewbudgetequation.
P+2M+3J=
180.
IsGunnar’sbudgetsetanydi?erentthanitwasbeforethechange?
No.
2.7(0)EdmundStenchconsumestwocommodities,namelygarbageandpunkrockvideocassettes.Hedoesn’tactuallyeattheformerbutkeepsitinhisbackyardwhereitiseatenbybillygoatsandassortedvermin.Thereasonthatheacceptsthegarbageisthatpeoplepayhim$2persackfortakingit.Edmundcanacceptasmuchgarbageashewishesatthatprice.Hehasnoothersourceofincome.Videocassettescosthim$6each.
(a)IfEdmundacceptszerosacksofgarbage,howmanyvideocassettes
canhebuy?
0.
12BUDGETCONSTRAINT(Ch.2)
(a)Writedownabudgetequationstatingthosecombinationsofthethreecommodities,GoodX,hoursofspeechesbypoliticians(P),andhoursofspeechesbyuniversityadministrators(A)thatEmmettcoulda?ordtoconsumeperweek.
15X?1P?2A=50.
(b)Onthegraphabove,drawatwo-dimensionaldiagramshowingthelocusofconsumptionsofthetwokindsofspeechesthatwouldbepossibleforEmmettifheconsumed10unitsofGoodXperweek.
2.9(0)JonathanLivingstoneYuppieisaprosperouslawyer.Hehas,inhisownwords,“outgrownthosecon?ningtwo-commoditylim-its.”Jonathanconsumesthreegoods,unblendedScotchwhiskey,de-signertennisshoes,andmealsinFrenchgourmetrestaurants.ThepriceofJonathan’sbrandofwhiskeyis$20perbottle,thepriceofdesignertennisshoesis$80perpair,andthepriceofgourmetrestaurantmealsis$50permeal.Afterhehaspaidhistaxesandalimony,Jonathanhas$400aweektospend.
(a)WritedownabudgetequationforJonathan,whereWstandsforthenumberofbottlesofwhiskey,Tstandsforthenumberofpairsoftennisshoes,andMforthenumberofgourmetrestaurantmealsthatheconsumes.
20W+80T+50M=400.
(b)Drawathree-dimensionaldiagramtoshowhisbudgetset.Labeltheintersectionsofthebudgetsetwitheachaxis.
M
8
5
20
T
W(c)Supposethathedeterminesthathewillbuyonepairofdesignertennisshoesperweek.Whatequationmustbesatis?edbythecombinationsofrestaurantmealsandwhiskeythathecoulda?ord?
20W+50M=
320.
2.10(0)Marthaispreparingforexamsineconomicsandsociology.Shehastimetoread40pagesofeconomicsand30pagesofsociology.Inthesameamountoftimeshecouldalsoread30pagesofeconomicsand60pagesofsociology.
NAME13(a)Assumingthatthenumberofpagesperhourthatshecanreadofeithersubjectdoesnotdependonhowsheallocateshertime,howmanypagesofsociologycouldshereadifshedecidedtospendallofhertimeonsociologyandnoneoneconomics?150pages.(Hint:Youhavetwopointsonherbudgetline,soyoushouldbeabletodeterminetheentireline.)
(b)Howmanypagesofeconomicscouldshereadifshedecidedtospendallofhertimereadingeconomics?
50pages.
2.11(1)HarryHypehas$5,000tospendonadvertisinganewkindof
dehydratedsushi.MarketresearchshowsthatthepeoplemostlikelytobuythisnewproductarerecentrecipientsofM.B.A.degreesandlawyerswhoownhottubs.Harryisconsideringadvertisingintwopublications,aboringbusinessmagazineandatrendyconsumerpublicationforpeoplewhowishtheylivedinCalifornia.
Fact1:Adsintheboringbusinessmagazinecost$500eachandadsintheconsumermagazinecost$250each.
Fact2:Eachadinthebusinessmagazinewillbereadby1,000recentM.B.A.’sand300lawyerswithhottubs.
Fact3:Eachadintheconsumerpublicationwillbereadby300recentM.B.A.’sand250lawyerswhoownhottubs.
Fact4:Nobodyreadsmorethanonead,andnobodywhoreadsonemagazinereadstheother.
(a)IfHarryspendshisentireadvertisingbudgetonthebusinesspub-lication,hisadwillbereadby
10,000
recentM.B.A.’sandby
3,000
lawyerswithhottubs.
(b)Ifhespendshisentireadvertisingbudgetontheconsumerpublication,hisadwillbereadby6,000
recentM.B.A.’sandby
5,000
lawyerswithhottubs.
(c)Supposehespenthalfofhisadvertisingbudgetoneachpublication.Hisadwouldbereadby8,000
recentM.B.A.’sandby
4,000
lawyerswithhottubs.
(d)Drawa“budgetline”showingthecombinationsofnumberofreadingsbyrecentM.B.A.’sandbylawyerswithhottubsthathecanobtainifhespendshisentireadvertisingbudget.Doesthislineextendallthewaytotheaxes?No.Sketch,shadein,andlabelthebudgetset,whichincludesallthecombinationsofMBA’sandlawyershecanreachifhespendsnomorethanhisbudget.
NAME15its?twobudgetconstraints.Remember,Haroldhastohaveenoughbluemoneyandenoughredmoneytopayboththeblue-moneycostandthered-moneycostofabundleofgoods.
Gum2015
10
Blue Lines5
Red Line0510
1520Ambrosia
(b)AnotherMungoan,Gladys,facesthesamepricesthatHaroldfacesandhasthesameredincomeasHarold,butGladyshasablueincomeof20.ExplainhowitisthatGladyswillnotspenditsentireblueincomenomatterwhatitstastesmaybe.(HintDrawGladys’sbudgetlines.)
Thebluebudgetlineliesstrictlyoutsidetheredbudgetline,sotosatisfybothbudgets,onemustbestrictlyinsidetheredbudgetline.
(c)AgroupofradicaleconomicreformersonMungobelievethatthecurrencyrulesareunfair.“Whyshouldeveryonehavetopaytwopricesforeverything?”theyask.Theyproposethefollowingscheme.Mungowillcontinuetohavetwocurrencies,everygoodwillhaveabluepriceandaredprice,andeveryMungoanwillhaveablueincomeandaredincome.Butnobodyhastopaybothprices.Instead,everyoneonMungomustdeclareitselftobeeitheraBlue-MoneyPurchaser(a“Blue”)oraRed-MoneyPurchaser(a“Red”)beforeitbuysanythingatall.Bluesmustmakealloftheirpurchasesinbluemoneyattheblueprices,spendingonlytheirblueincomes.Redsmustmakealloftheirpurchasesinredmoney,spendingonlytheirredincomes.
SupposethatHaroldhasthesameincomeafterthisreform,andthatpricesdonotchange.Beforedeclaringwhichkindofpurchaseritwillbe,
?
WerefertoallMungoansbythegender-neutralpronoun,“it.”Al-thoughMungohastwosexes,neitherofthemisremotelylikeeitherofours.
14BUDGETCONSTRAINT(Ch.2)
(e)LetMstandforthenumberofinstancesofanadbeingreadbyanM.B.A.andLstandforthenumberofinstancesofanadbeingreadbyalawyer.ThisbudgetlineisalinesegmentthatliesonthelinewithequationM+2L=16.Witha?xedadvertisingbudget,howmanyreadingsbyM.B.A.’smusthesacri?cetogetanadditionalreadingbyalawyerwithahottub?
2.
MBA's x 100016
1210Budget linea8c6b4
BudgetSet02
34
58
Lawyers x 1000
12162.12(0)OntheplanetMungo,theyhavetwokindsofmoney,bluemoneyandredmoney.Everycommodityhastwoprices—ared-moneypriceandablue-moneyprice.EveryMungoanhastwoincomes—aredincomeandablueincome.
Inordertobuyanobject,aMungoanhastopaythatobject’sred-moneypriceinredmoneyanditsblue-moneypriceinbluemoney.(Theshopssimplyhavetwocashregisters,andyouhavetopayatbothregisterstobuyanobject.)Itisforbiddentotradeonekindofmoneyfortheother,andthisprohibitionisstrictlyenforcedbyMungo’sruthlessande?cientmonetarypolice.
?TherearejusttwoconsumergoodsonMungo,ambrosiaandbubblegum.AllMungoansprefermoreofeachgoodtoless.
?Thebluepricesare1bcu(bcustandsforbluecurrencyunit)perunitofambrosiaand1bcuperunitofbubblegum.
?Theredpricesare2rcus(redcurrencyunits)perunitofambrosiaand6rcusperunitofbubblegum.(a)Onthegraphbelow,drawtheredbudget(withredink)andthebluebudget(withblueink)foraMungoannamedHaroldwhoseblueincomeis10andwhoseredincomeis30.Shadeinthe“budgetset”containingallofthecommoditybundlesthatHaroldcana?ord,given
16BUDGETCONSTRAINT(Ch.2)
Haroldcontemplatesthesetofcommoditybundlesthatitcoulda?ordbymakingonedeclarationortheother.Letuscallacommoditybundle“attainable”ifHaroldcana?orditbydeclaringitselftobea“Blue”andbuyingthebundlewithbluemoneyorifHaroldcana?ordthebundlebydeclaringitselftobea“Red”andbuyingitwithredmoney.Onthediagrambelow,shadeinalloftheattainablebundles.
Gum
20
15
10
;;;;;;;;;;;;;;;;;;;;;;;;;;Blue Line;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;5
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;Red Line;;;;;;;;;;;;;;;;;;;;;;0510
15Ambrosia
20
2.13(0)AreMungoanbudgetsreallysofanciful?Canyouthinkofsit-uationsonearthwherepeoplemustsimultaneouslysatisfymorethanone
budgetconstraint?Ismoneytheonlyscarceresourcethatpeopleuseupwhenconsuming?
Consumptionofmanycommodities
takestimeaswellasmoney.Peoplehavetosimultaneouslysatisfyatimebudgetandamoneybudget.Otherexamples--peoplemayhaveacaloriebudgetoracholesterolbudgetoranalcohol-intakebudget.
Chapter3
NAME
Preferences
Introduction.Intheprevioussectionyoulearnedhowtousegraphsto
showthesetofcommoditybundlesthataconsumercana?ord.Inthissection,youlearntoputinformationabouttheconsumer’spreferencesonthesamekindofgraph.Mostoftheproblemsaskyoutodrawindi?erencecurves.
Sometimeswegiveyouaformulafortheindi?erencecurve.Thenallyouhavetodoisgraphaknownequation.Butinsomeproblems,wegiveyouonly“qualitative”informationabouttheconsumer’spreferencesandaskyoutosketchindi?erencecurvesthatareconsistentwiththisinformation.Thisrequiresalittlemorethought.Don’tbesurprisedordisappointedifyoucannotimmediatelyseetheanswerwhenyoulookataproblem,anddon’texpectthatyouwill?ndtheanswershidingsomewhereinyourtextbook.Thebestwayweknowto?ndanswersisto“thinkanddoodle.”Drawsomeaxesonscratchpaperandlabelthem,thenmarkapointonyourgraphandaskyourself,“Whatotherpointsonthegraphwouldtheconsumer?ndindi?erenttothispoint?”Ifpossible,drawacurveconnectingsuchpoints,makingsurethattheshapeofthelineyoudrawre?ectsthefeaturesrequiredbytheproblem.Thisgivesyouoneindi?erencecurve.Nowpickanotherpointthatispreferredtothe?rstoneyoudrewanddrawanindi?erencecurvethroughit.
Example:Jocastalovestodanceandhateshousecleaning.Shehasstrictlyconvexpreferences.Sheprefersdancingtoanyotheractivityandnevergetstiredofdancing,butthemoretimeshespendscleaninghouse,thelesshappysheis.Letustrytodrawanindi?erencecurvethatisconsistentwithherpreferences.Thereisnotenoughinformationheretotellusexactlywhereherindi?erencecurvesgo,butthereisenoughinformationtodeterminesomethingsabouttheirshape.Takeapieceofscratchpaperanddrawapairofaxes.Labelthehorizontalaxis“Hoursperdayofhousecleaning.”Labeltheverticalaxis“Hoursperdayofdancing.”Markapointalittlewaysuptheverticalaxisandwritea4nexttoit.Atthispoint,shespends4hoursadaydancingandnotimehousecleaning.Otherpointsthatwouldbeindi?erenttothispointwouldhavetobepointswhereshedidmoredancingandmorehousecleaning.Thepainoftheextrahousekeepingshouldjustcompensateforthepleasureoftheextradancing.Soanindi?erencecurveforJocastamustbeupwardsloping.Becauseshelovesdancingandhateshousecleaning,itmustbethatsheprefersallthepointsabovethisindi?erencecurvetoallofthepointsonorbelowit.IfJocastahasstrictlyconvexpreferences,thenitmustbethatifyoudrawalinebetweenanytwopointsonthesameindi?erencecurve,allthepointsontheline(excepttheendpoints)arepreferredtotheendpoints.Forthistobethecase,itmustbethattheindi?erencecurveslopesupwardevermoresteeplyasyoumovetotherightalongit.Youshouldconvinceyourselfofthisbymakingsomedrawingsonscratch
NAME19Bananas40
30
Red CurvePencil Shading20
10
Blue CurveBlue Shading0102030
40
Apples
ForeachofthefollowingstatementsaboutCharlie’spreferences,write“true”or“false.”(c)(30,5)～(10,15).True.(d)(10,15)??(20,5).True.(e)(20,5)??(10,10).True.(f)(24,4)??(11,9.1).False.(g)(11,14)??(2,49).
True.
(h)Asetisconvexifforanytwopointsintheset,thelinesegmentbetweenthemisalsointheset.IsthesetofbundlesthatCharlieweaklyprefersto(20,5)aconvexset?
Yes.
(i)IsthesetofbundlesthatCharlieconsidersinferiorto(20,5)aconvexset?
No.
(j)TheslopeofCharlie’sindi?erencecurvethroughapoint,(xA,xB),isknownashismarginal
rate
of
substitution
atthatpoint.
18PREFERENCES(Ch.3)
paper.Drawanupward-slopingcurvepassingthroughthepoint(0,4)andgettingsteeperasonemovestotheright.
Whenyouhavecompletedthisworkout,wehopethatyouwillbeabletodothefollowing:
?Giventheformulaforanindi?erencecurve,drawthiscurve,and?nditsslopeatanypointonthecurve.?Determinewhetheraconsumerprefersonebundletoanotherorisindi?erentbetweenthem,givenspeci?cindi?erencecurves.?Drawindi?erencecurvesforthespecialcasesofperfectsubstitutesandperfectcomplements.?Drawindi?erencecurvesforsomeonewhodislikesoneorbothcom-modities.?Drawindi?erencecurvesforsomeonewholikesgoodsuptoapointbutwhocanget“toomuch”ofoneormoregoods.?Identifyweaklypreferredsetsanddeterminewhetherthesearecon-vexsetsandwhetherpreferencesareconvex.?Knowwhatthemarginalrateofsubstitutionisandbeabletodeter-minewhetheranindi?erencecurveexhibits“diminishingmarginalrateofsubstitution.”?Determinewhetherapreferencerelationoranyotherrelationbe-tweenpairsofthingsistransitive,whetheritisre?exive,andwhetheritiscomplete.3.1(0)Charlielikesbothapplesandbananas.Heconsumesnothingelse.TheconsumptionbundlewhereCharlieconsumesxAbushelsofapplesperyearandxBbushelsofbananasperyeariswrittenas(xA,xB).Lastyear,Charlieconsumed20bushelsofapplesand5bushelsofbananas.Ithappensthatthesetofconsumptionbundles(xA,xB)suchthatCharlieisindi?erentbetween(xA,xB)and(20,5)isthesetofallbundlessuchthatxB=100/xA.Thesetofbundles(xA,xB)suchthatCharlieisjustindi?erentbetween(xA,xB)andthebundle(10,15)isthesetofbundlessuchthatxB=150/xA.
(a)Onthegraphbelow,plotseveralpointsthatlieontheindi?erencecurvethatpassesthroughthepoint(20,5),andsketchthiscurve,usingblueink.Dothesame,usingredink,fortheindi?erencecurvepassingthroughthepoint(10,15).
(b)UsepenciltoshadeinthesetofcommoditybundlesthatCharlieweaklypreferstothebundle(10,15).UseblueinktoshadeinthesetofcommoditybundlessuchthatCharlieweaklyprefers(20,5)tothesebundles.
20PREFERENCES(Ch.3)
(k)RememberthatCharlie’sindi?erencecurvethroughthepoint(10,10)
hastheequationxB=100/xA.Thoseofyouwhoknowcalculuswillrememberthattheslopeofacurveisjustitsderivative,whichinthiscaseis?100/x2A.(Ifyoudon’tknowcalculus,youwillhavetotakeourwordforthis.)FindCharlie’smarginalrateofsubstitutionatthepoint,(10,10).
?1.
(l)Whatishismarginalrateofsubstitutionatthepoint(5,20)?
?4.
(m)Whatishismarginalrateofsubstitutionatthepoint(20,5)?
(?.25).
(n)Dotheindi?erencecurvesyouhavedrawnforCharlieexhibitdimin-ishingmarginalrateofsubstitution?
Yes.
3.2(0)Ambroseconsumesonlynutsandberries.Fortunately,helikesbothgoods.TheconsumptionbundlewhereAmbroseconsumesx1unitsofnutsperweekandx2unitsofberriesperweekiswrittenas(x1,x2).Thesetofconsumptionbundles(x1,x2)suchthatAmbroseisindi?erentbetween(x1,x2)and(1,16)isthesetofbundlessuchthatx1≥0,x2≥0,
andx2=20?4√
x1.Thesetofbundles(x1,x2)suchthat(x1,x2)～
(36,0)isthesetofbundlessuchthatx1≥0,x2≥0andx2=24?4√
x1.(a)Onthegraphbelow,plotseveralpointsthatlieontheindi?erencecurvethatpassesthroughthepoint(1,16),andsketchthiscurve,usingblueink.Dothesame,usingredink,fortheindi?erencecurvepassingthroughthepoint(36,0).
(b)UsepenciltoshadeinthesetofcommoditybundlesthatAmbroseweaklypreferstothebundle(1,16).Useredinktoshadeinthesetofallcommoditybundles(x1,x2)suchthatAmbroseweaklyprefers(36,0)tothesebundles.IsthesetofbundlesthatAmbroseprefersto(1,16)aconvexset?
Yes.
(c)WhatistheslopeofAmbrose’sindi?erencecurveatthepoint(9,8)?(HintRecallfromcalculusthewaytocalculatetheslopeofacurve.Ifyoudon’tknowcalculus,youwillhavetodrawyourdiagramcarefullyandestimatetheslope.)
?2/3.