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ANSWERS
1TheMarket
1.Supposethattherewere25peoplewhohadareservationpriceof$500,andthe26thpersonhadareservationpriceof$200.Whatwouldthedemandcurvelooklike
1.1.Itwouldbeconstantat$500for25apartmentsandthendropto$200.
2.Intheaboveexample,whatwouldtheequilibriumpricebeiftherewere24apartmentstorentWhatiftherewere26apartmentstorentWhatiftherewere25apartmentstorent
1.2.Inthe?rstcase,$500,andinthesecondcase,$200.Inthethirdcase,theequilibriumpricewouldbeanypricebetween$200and$500.
3.Ifpeoplehavedi?erentreservationprices,whydoesthemarketdemandcurveslopedown
1.3.Becauseifwewanttorentonemoreapartment,wehavetoo?eralowerprice.Thenumberofpeoplewhohavereservationpricesgreaterthanpmustalwaysincreaseaspdecreases.
4.Inthetextweassumedthatthecondominiumpurchaserscamefromtheinner-ringpeople—peoplewhowerealreadyrentingapartments.Whatwouldhappentothepriceofinner-ringapartmentsifallofthecondominiumpurchaserswereouter-ringpeople—thepeoplewhowerenotcurrentlyrentingapartmentsintheinnerring
1.4.Thepriceofapartmentsintheinnerringwouldgoupsincedemandforapartmentswouldnotchangebutsupplywoulddecrease.
5.Supposenowthatthecondominiumpurchaserswereallinner-ringpeople,butthateachcondominiumwasconstructedfromtwoapartments.Whatwouldhappentothepriceofapartments
1.5.Thepriceofapartmentsintheinnerringwouldrise.
6.Whatdoyousupposethee?ectofataxwouldbeonthenumberofapartmentsthatwouldbebuiltinthelongrun
1.6.Ataxwouldundoubtedlyreducethenumberofapartmentssuppliedinthelongrun.
7.SupposethedemandcurveisD(p)=100?2p.Whatpricewouldthemonopolistsetifhehad60apartmentsHowmanywouldherentWhatpricewouldhesetifhehad40apartmentsHowmanywouldherent
1.7.Hewouldsetapriceof25andrent50apartments.Inthesecondcasehewouldrentall40apartmentsatthemaximumpricethemarketwouldbear.ThiswouldbegivenbythesolutiontoD(p)=100?2p=40,whichisp?=30.
8.Ifourmodelofrentcontrolallowedforunrestrictedsubletting,whowouldendupgettingapartmentsintheinnercircleWouldtheoutcomebeParetoe?cient
1.8.Everyonewhohadareservationpricehigherthantheequilibriumpriceinthecompetitivemarket,sothatthe?naloutcomewouldbeParetoe?cient.(Ofcourseinthelongruntherewouldprobablybefewernewapartmentsbuilt,whichwouldleadtoanotherkindofine?ciency.)
2BudgetConstraint
1.Originallytheconsumerfacesthebudgetlinep1x1+p2x2=m.Thenthepriceofgood1doubles,thepriceofgood2becomes8timeslarger,andincomebecomes4timeslarger.Writedownanequationforthenewbudgetlineintermsoftheoriginalpricesandincome.
2.1.Thenewbudgetlineisgivenby2p1x1+8p2x2=4m.
2.Whathappenstothebudgetlineifthepriceofgood2increases,butthepriceofgood1andincomeremainconstant
2.2.Theverticalintercept(axis)decreasesandthehorizontalintercept(axis)staysthesame.Thusthebudgetlinebecomes?atter.
3.Ifthepriceofgood1doublesandthepriceofgood2triples,doesthebudgetlinebecome?atterorsteeper
2.3.Flatter.Theslopeis?2/3.
4.Whatisthede?nitionofanumerairegood
2.4.Agoodwhosepricehasbeensetto1;allothergoods’pricesaremeasuredrelativetothenumerairegood’sprice.
5.Supposethatthegovernmentputsataxof15centsagallonongasolineandthenlaterdecidestoputasubsidyongasolineatarateof7centsagallon.Whatnettaxisthiscombinationequivalentto
2.5.Ataxof8centsagallon.
6.Supposethatabudgetequationisgivenby
+=m.Thegovernmentdecidestoimposealump-sumtaxofu,aquantitytaxongood1oft,andaquantitysubsidyongood2ofs.Whatistheformulaforthenewbudgetline
2.6.(+t)+(?s)=m?u.
7.Iftheincomeoftheconsumerincreasesandoneofthepricesdecreasesatthesametime,willtheconsumernecessarilybeatleastaswell-o?
2.7.Yes,sinceallofthebundlestheconsumercoulda?ordbeforearea?ordableatthenewpricesandincome.
3Preferences
1.Ifweobserveaconsumerchoosing(,)when(,)isavailableonetime,arewejusti?edinconcludingthat(,)>(,)
3.1.No.Itmightbethattheconsumerwasindi?erentbetweenthetwobundles.Allwearejusti?edinconcludingisthat(,)>(,).
2.ConsideragroupofpeopleA,B,Candtherelation“atleastastallas,〞asin“AisatleastastallasB.〞IsthisrelationtransitiveIsitcomplete
3.2.Yestoboth.
3.Takethesamegroupofpeopleandconsidertherelation“strictlytallerthan.〞IsthisrelationtransitiveIsitre?exiveIsitcomplete
3.3.Itistransitive,butitisnotcomplete—twopeoplemightbethesameheight.Itisnotre?exivesinceitisfalsethatapersonisstrictlytallerthanhimself.
4.AcollegefootballcoachsaysthatgivenanytwolinemenAandB,healwayspreferstheonewhoisbiggerandfaster.IsthispreferencerelationtransitiveIsitcomplete
3.4.Itistransitive,butnotcomplete.WhatifAwerebiggerbutslowerthanBWhichonewouldheprefer
5.Cananindi?erencecurvecrossitselfForexample,couldFigure3.2depictasingleindi?erencecurve
3.5.Yes.Anindi?erencecurvecancrossitself,itjustcan’tcrossanotherdistinctindi?erencecurve.
6.CouldFigure3.2beasingleindi?erencecurveifpreferencesaremonotonic
3.6.No,becausetherearebundlesontheindi?erencecurvethathavestrictlymoreofbothgoodsthanotherbundlesonthe(alleged)indi?erencecurve.
7.Ifbothpepperoniandanchoviesarebads,willtheindi?erencecurvehaveapositiveoranegativeslope
3.7.Anegativeslope.Ifyougivetheconsumermoreanchovies,you’vemadehimworseo?,soyouhavetotakeawaysomepepperonitogethimbackonhisindi?erencecurve.Inthiscasethedirectionofincreasingutilityistowardtheorigin.
8.Explainwhyconvexpreferencesmeansthat“averagesarepreferredtoextremes.〞
3.8.Becausetheconsumerweaklypreferstheweightedaverageoftwobundlestoeitherbundle.
9.Whatisyourmarginalrateofsubstitutionof$1billsfor$5bills
3.9.Ifyougiveupone$5bill,howmany$1billsdoyouneedtocompensateyouFive$1billswilldonicely.Hencetheansweris?5or?1/5,dependingonwhichgoodyouputonthehorizontalaxis.
10.Ifgood1isa“neutral,〞whatisitsmarginalrateofsubstitutionforgood2
3.10.Zero—ifyoutakeawaysomeofgood1,theconsumerneedszerounitsofgood2tocompensatehimforhisloss.
ANSWERSA13
11.Thinkofsomeothergoodsforwhichyourpreferencesmightbeconcave.
3.11.Anchoviesandpeanutbutter,scotchandKoolAid,andothersimilarrepulsivecombinations.
4Utility
1.Thetextsaidthatraisinganumbertoanoddpowerwasamonotonictransformation.WhataboutraisinganumbertoanevenpowerIsthisamonotonictransformation(Hint:considerthecasef(u)=u^2.)
4.1.Thefunctionf(u)=u^2isamonotonictransformationforpositiveu,butnotfornegativeu.
2.Whichofthefollowingaremonotonictransformations
(1)u=2v?13;(2)u=?1/v^2;(3)u=1/v^2;(4)u=lnv;(5)u=?e^?v;(6)u=v^2;(7)u=v^2forv>0;(8)u=v^2forv<0.
4.2.(1)Yes.(2)No(worksforvpositive).(3)No(worksforvnegative).(4)Yes(onlyde?nedforvpositive).(5)Yes.(6)No.(7)Yes.(8)No.
3.Weclaimedinthetextthatifpreferencesweremonotonic,thenadiagonallinethroughtheoriginwouldintersecteachindi?erencecurveexactlyonce.Canyouprovethisrigorously(Hint:whatwouldhappenifitintersectedsomeindi?erencecurvetwice)
4.3.Supposethatthediagonalintersectedagivenindi?erencecurveattwopoints,say(x,x)and(y,y).Theneitherx>yory>x,whichmeansthatoneofthebundleshasmoreofbothgoods.Butifpreferencesaremonotonic,thenoneofthebundleswouldhavetobepreferredtotheother.
4.Whatkindofpreferencesarerepresentedbyautilityfunctionoftheformu(x1,x2)=Whatabouttheutilityfunctionv(x1,x2)=13x1+13x2
4.4.Bothrepresentperfectsubstitutes.
5.Whatkindofpreferencesarerepresentedbyautilityfunctionoftheformu(x1,x2)=x1+Istheutilityfunctionv(x1,x2)=x21+2x1+x2amonotonictransformationofu(x1,x2)
4.5.Quasilinearpreferences.Yes.
6.Considertheutilityfunctionu(x1,x2)=.Whatkindofpref-erencesdoesitrepresentIsthefunctionv(,)=amonotonictransformationofu(,)Isthefunctionw(,)=amonotonictransformationofu(,)
4.6.TheutilityfunctionrepresentsCobb-Douglaspreferences.No.Yes.
7.Canyouexplainwhytakingamonotonictransformationofautilityfunctiondoesn’tchangethemarginalrateofsubstitution
4.7.BecausetheMRSismeasuredalonganindi?erencecurve,andutilityremainsconstantalonganindi?erencecurve.
5Choice
1.Iftwogoodsareperfectsubstitutes,whatisthedemandfunctionforgood2
5.1.=0when>,=m/when<,andanythingbetween0andm/p2when=.
2.Supposethatindi?erencecurvesaredescribedbystraightlineswithaslopeof?b.Givenarbitrarypricesandmoneyincomep1,p2,andm,whatwilltheconsumer’soptimalchoiceslooklike
5.2.Theoptimalchoiceswillbex1=m/p1andx2=0ifp1/p2<b,x1=0andx2=m/p2ifp1/p2>b,andanyamountonthebudgetlineifp1/p2=b.
3.Supposethataconsumeralwaysconsumes2spoonsofsugarwitheachcupofco?ee.Ifthepriceofsugarisp1perspoonfulandthepriceofco?eeisp2percupandtheconsumerhasmdollarstospendonco?eeandsugar,howmuchwillheorshewanttopurchase
5.3.Letzbethenumberofcupsofco?eetheconsumerbuys.Thenweknowthat2zisthenumberofteaspoonsofsugarheorshebuys.Wemustsatisfythebudgetconstraint
2z+z=m.
Solvingforzwehave
z=
4.Supposethatyouhavehighlynonconvexpreferencesforicecreamandolives,likethosegiveninthetext,andthatyoufacepricesp1,p2andhavemdollarstospend.Listthechoicesfortheoptimalconsumptionbundles.
5.4.Weknowthatyou’lleitherconsumeallicecreamorallolives.Thusthetwochoicesfortheoptimalconsumptionbundleswillbex1=m/,x2=0,orx1=0,x2=m/.
5.Ifaconsumerhasautilityfunctionu(x1,x2)=x1x42,whatfractionofherincomewillshespendongood2
5.5.ThisisaCobb-Douglasutilityfunction,soshewillspend4/(1+4)=4/5ofherincomeongood2.
6.Forwhatkindofpreferenceswilltheconsumerbejustaswell-o?facingaquantitytaxasanincometax
5.6.Forkinkedpreferences,suchasperfectcomplements,wherethechangeinpricedoesn’tinduceanychangeindemand.
6Demand
1.Iftheconsumerisconsumingexactlytwogoods,andsheisalwaysspendingallofhermoney,canbothofthembeinferiorgoods
6.1.No.Ifherincomeincreases,andshespendsitall,shemustbepurchasingmoreofatleastonegood.
2.Showthatperfectsubstitutesareanexampleofhomotheticpreferences.
6.2.Theutilityfunctionforperfectsubstitutesisu(,)=+.Thusifu(,)>u(,),wehave+>+.Itfollowsthatt+t>t+t,sothatu(t,t)>u(t,t).
3.ShowthatCobb-Douglaspreferencesarehomotheticpreferences.
6.3.TheCobb-Douglasutilityfunctionhasthepropertythatu(t,t)==2=t2=t*u(x1,).Thusifu(,)>u(,),weknowthatu(t,t)>u(t,),sothatCobb-Douglaspreferencesareindeedhomothetic.
4.Theincomeo?ercurveistotheEngelcurveasthepriceo?ercurveisto...
6.4.Thedemandcurve.
5.Ifthepreferencesareconcavewilltheconsumereverconsumebothofthegoodstogether
6.5.No.Concavepreferencescanonlygiverisetooptimalconsumptionbundlesthatinvolvezeroconsumptionofoneofthegoods.
6.Arehamburgersandbunscomplementsorsubstitutes
6.6.Normallytheywouldbecomplements,atleastfornon-vegetarians.
7.Whatistheformoftheinversedemandfunctionforgood1inthecaseofperfectcomplements
6.7.Weknowthatx1=m/(p1+p2).Solvingforp1asafunctionoftheothervariables,wehavep1=mx1?p2.
8.TrueorfalseIfthedemandfunctionisx1=?p1,thentheinversedemandfunctionisx=?1/p1.
6.8.False.
7RevealedPreference
1.Whenpricesare(p1,p2)=(1,2)aconsumerdemands(x1,x2)=(1,2),andwhenpricesare(q1,q2)=(2,1)theconsumerdemands(y1,y2)=(2,1).Isthisbehaviorconsistentwiththemodelofmaximizingbehavior
7.1.No.ThisconsumerviolatestheWeakAxiomofRevealedPreferencesincewhenhebought(x1,x2)hecouldhavebought(y1,y2)andviceversa.Insymbols:
p1x1+p2x2=1×1+2×2=5>4=1×2+2×1=p1y1+p2y2
and
q1y1+q2y2=2×2+1×1=5>4=2×1+1×2=q1x1+q2x2.
2.Whenpricesare(p1,p2)=(2,1)aconsumerdemands(x1,x2)=(1,2),andwhenpricesare(q1,q2)=(1,2)theconsumerdemands(y1,y2)=(2,1).Isthisbehaviorconsistentwiththemodelofmaximizingbehavior
7.2.Yes.NoviolationsofWARParepresent,sincethey-bundleisnota?ordablewhenthex-bundlewaspurchasedandviceversa.
3.Intheprecedingexercise,whichbundleispreferredbytheconsumer,thex-bundleorthey-bundle
7.3.Sincethey-bundlewasmoreexpensivethanthex-bundlewhenthex-bundlewaspurchasedandviceversa,thereisnowaytotellwhichbundleispreferred.
4.WesawthattheSocialSecurityadjustmentforchangingpriceswouldtypicallymakerecipientsatleastaswell-o?astheywereatthebaseyear.Whatkindofpricechangeswouldleavethemjustaswell-o?,nomatterwhatkindofpreferencestheyhad
7.4.Ifbothpriceschangedbythesameamount.Thenthebase-yearbundlewouldstillbeoptimal.
5.Inthesameframeworkastheabovequestion,whatkindofpreferenceswouldleavetheconsumerjustaswell-o?ashewasinthebaseyear,forallpricechanges
7.5.Perfectcomplements.
8SlutskyEquation
1.Supposeaconsumerhaspreferencesbetweentwogoodsthatareperfectsubstitutes.Canyouchangepricesinsuchawaythattheentiredemandresponseisduetotheincomee?ect
8.1.Yes.Toseethis,useourfavoriteexampleofredpencilsandbluepencils.Supposeredpencilscost10centsapiece,andbluepencilscost5centsapiece,andtheconsumerspends$1onpencils.Shewouldthenconsume20bluepencils.Ifthepriceofbluepencilsfallsto4centsapiece,shewouldconsume25bluepencils,achangewhichisentirelyduetotheincomee?ect.
2.Supposethatpreferencesareconcave.Isitstillthecasethatthesubstitutione?ectisnegative
8.2.Yes.
3.Inthecaseofthegasolinetax,whatwouldhappeniftherebatetotheconsumerswerebasedontheiroriginalconsumptionofgasoline,x,ratherthanontheir?nalconsumptionofgasoline,x’
8.3.Thentheincomee?ectwouldcancelout.Allthatwouldbeleftwouldbethepuresubstitutione?ect,whichwouldautomaticallybenegative.
4.Inthecasedescribedintheprecedingquestion,wouldthegovernmentbepayingoutmoreorlessthanitreceivedintaxrevenues
8.4.Theyarereceivingtx’inrevenuesandpayingouttx,sotheyarelosingmoney.
5.Inthiscasewouldtheconsumersbebettero?orworseo?ifthetaxwithrebatebasedonoriginalconsumptionwereine?ect
8.5.Sincetheiroldconsumptionisa?ordable,theconsumerswouldhavetobeatleastaswell-o?.Thishappensbecausethegovernmentisgivingthembackmoremoneythantheyarelosingduetothehigherpriceofgasoline.
9BuyingandSelling
1.Ifaconsumer’snetdemandsare(5,?3)andherendowmentis(4,4),whatarehergrossdemands
9.1.Hergrossdemandsare(9,1).
2.Thepricesare(p1,p2)=(2,3),andtheconsumeriscurrentlyconsuming(x1,x2)=(4,4).Thereisaperfectmarketforthetwogoodsinwhichtheycanbeboughtandsoldcostlessly.Willtheconsumernecessarilypreferconsumingthebundle(y1,y2)=(3,5)Willshenecessarilypreferhavingthebundle(y1,y2)
9.2.Thebundle(y1,y2)=(3,5)costsmorethanthebundle(4,4)atthecurrentprices.Theconsumerwillnotnecessarilypreferconsumingthisbundle,butwouldcertainlyprefertoownit,sinceshecouldsellitandpurchaseabundlethatshewouldprefer.
3.Thepricesare(p1,p2)=(2,3),andtheconsumeriscurrentlyconsuming(x1,x2)=(4,4).Nowthepriceschangeto(q1,q2)=(2,4).Couldtheconsumerbebettero?underthesenewprices
9.3.Sure.Itdependsonwhethershewasanetbuyeroranetsellerofthegoodthatbecamemoreexpensive.
4.TheU.S.currentlyimportsabouthalfofthepetroleumthatituses.Therestofitsneedsaremetbydomesticproduction.CouldthepriceofoilrisesomuchthattheU.S.wouldbemadebettero?
9.4.Yes,butonlyiftheU.S.switchedtobeinganetexporterofoil.
5.Supposethatbysomemiraclethenumberofhoursinthedayincreasedfrom24to30hours(withluckthiswouldhappenshortlybeforeexamweek).Howwouldthisa?ectthebudgetconstraint
9.5.Thenewbudgetlinewouldshiftoutwardandremainparalleltotheoldone,sincetheincreaseinthenumberofhoursinthedayisapureendowmente?ect.
6.Ifleisureisaninferiorgood,whatcanyousayabouttheslopeofthelaborsupplycurve
9.6.Theslopewillbepositive.
10IntertemporalChoice
1.Howmuchis$1milliontobedelivered20yearsinthefutureworthtodayiftheinterestrateis20percent
10.1.AccordingtoTable10.1,$120yearsfromnowisworth3centstodayata20percentinterestrate.Thus$1millionisworth.03×1,000,000=$30,000today.
2.Astheinterestraterises,doestheintertemporalbudgetconstraintbe-comesteeperor?atter
10.2.Theslopeoftheintertemporalbudgetconstraintisequalto?(1+r).Thusasrincreasestheslopebecomesmorenegative(steeper).
3.Wouldtheassumptionthatgoodsareperfectsubstitutesbevalidinastudyofintertemporalfoodpurchases
10.3.Ifgoodsareperfectsubstitutes,thenconsumerswillonlypurchasethecheapergood.Inthecaseofintertemporalfoodpurchases,thisimpliesthatconsumersonlybuyfoodinoneperiod,whichmaynotbeveryrealistic.
4.Aconsumer,whoisinitiallyalender,remainsalenderevenafteradeclineininterestrates.Isthisconsumerbettero?orworseo?afterthechangeininterestratesIftheconsumerbecomesaborrowerafterthechangeishebettero?orworseo?
10.4.Inordertoremainalenderafterthechangeininterestrates,theconsumermustbechoosingapointthathecouldhavechosenundertheoldinterestrates,butdecidednotto.Thustheconsumermustbeworseo?.Iftheconsumerbecomesaborrowerafterthechange,thenheischoosingapreviouslyunavailablepointthatcannotbecomparedtotheinitialpoint(sincetheinitialpointisnolongeravailableunderthenewbudgetconstraint),andthereforethechangeintheconsumer’swelfareisunknown.
5.Whatisthepresentvalueof$100oneyearfromnowiftheinterestrateis10%Whatisthepresentvalueiftheinterestrateis5%
10.5.Ataninterestrateof10%,thepresentvalueof$100is$90.91.Atarateof5%thepresentvalueis$95.24.
11AssetMarkets
1.SupposeassetAcanbesoldfor$11nextperiod.IfassetssimilartoAarepayingarateofreturnof10%,whatmustbeassetA’scurrentprice
11.1.AssetAmustbesellingfor11/(1+0.10)=$10.
2.Ahouse,whichyoucouldrentfor$10,000ayearandsellfor$110,000ayearfromnow,canbepurchasedfor$100,000.Whatistherateofreturnonthishouse
11.2.Therateofreturnisequalto(10,000+10,000)/100,000=20%.
3.Thepaymentsofcertaintypesofbonds(e.g.,municipalbonds)arenottaxable.Ifsimilartaxablebondsarepaying10%andeveryonefacesamarginaltaxrateof40%,whatrateofreturnmustthenontaxablebondspay
11.3.Weknowthattherateofreturnonthenontaxablebonds,r,mustbesuchthat(1?t)=r,therefore(1?0.40)*0.10=0.06=r.
4.Supposethatascarceresource,facingaconstantdemand,willbeexhaustedin10years.Ifanalternativeresourcewillbeavailableatapriceof$40andiftheinterestrateis10%,whatmustthepriceofthescarceresourcebetoday
11.4.Thepricetodaymustbe40/(1+0.10)^10=$15.42.
12Uncertainty
1.HowcanonereachtheconsumptionpointstotheleftoftheendowmentinFigure12.1
12.1.Weneedawaytoreduceconsumptioninthebadstateandincreaseconsumptioninthegoodstate.Todothisyouwouldhavetosellinsuranceagainstthelossratherthanbuyit.
2.Whichofthefollowingutilityfunctionshavetheexpectedutilityproperty(a)u(c1,c2,π1,π2)=a(π1c1+π2c2),(b)u(c1,c2,π1,π2)=π1c1+π2c22,(c)u(c1,c2,π1,π2)=π1lnc1+π2lnc2+17.
12.2.Functions(a)and(c)havetheexpectedutilityproperty(theyarea?netransformationsofthefunctionsdiscussedinthechapter),while(b)doesnot.
3.Arisk-averseindividualiso?eredachoicebetweenagamblethatpays$1000withaprobabilityof25%and$100withaprobabilityof75%,orapaymentof$325.Whichwouldhechoose
12.3.Sinceheisrisk-averse,hepreferstheexpectedvalueofthegamble,$325,tothegambleitself,andthereforehewouldtakethepayment.
4.Whatifthepaymentwas$320
12.4.Ifthepaymentis$320thedecisionwilldependontheformoftheutilityfunction;wecan’tsayanythingingeneral.
5.Drawautilityfunctionthatexhibitsrisk-lovingbehaviorforsmallgamblesandrisk-aversebehaviorforlargergambles.
12.5.Yourpictureshouldshowafunctionthatisinitiallyconvex,butthenbecomesconcave.
6.Whymightaneighborhoodgrouphaveahardertimeselfinsuringfor?ooddamageversus?redamage
12.6.Inordertoself-insure,therisksmustbeindependent.However,thisdoesnotholdinthecaseof?ooddamage.Ifonehouseintheneighborhoodisdamagedbya?ooditislikelythatallofthehouseswillbedamaged.
13RiskyAssets
1.Iftherisk-freerateofreturnis6%,andifariskyassetisavailablewithareturnof9%andastandarddeviationof3%,whatisthemaximumrateofreturnyoucanachieveifyouarewillingtoacceptastandarddeviationof2%Whatpercentageofyourwealthwouldhavetobeinvestedintheriskyasset
13.1.Toachieveastandarddeviationof2%youwillneedtoinvestx=σx/σm=2/3ofyourwealthintheriskyasset.Thiswillresultinarateofreturnequalto(2/3)0.09+(1?2/3)0.06=8%.
2.Whatisthepriceofriskintheaboveexercise
13.2.Thepriceofriskisequalto(rm?rf)/σm=(9?6)/3=1.Thatis,foreveryadditionalpercentofstandarddeviationyoucangain1%ofreturn.
3.Ifastockhasaβof1.5,thereturnonthemarketis10%,andtherisk-freerateofreturnis5%,whatexpectedrateofreturnshouldthisstocko?eraccordingtotheCapitalAssetPricingModelIftheexpectedvalueofthestockis$100,whatpriceshouldthestockbesellingfortoday
13.3.AccordingtotheCAPMpricingequation,thestockshouldo?eranexpectedrateofreturnofrf+β(rm?rf)=0.05+1.5(0.10?0.05)=0.125or12.5%.Thestockshouldbesellingforitsexpectedpresentvalue,whichisequalto100/1.125=$88.89.
14Consumer’sSurplus
1.Agoodcanbeproducedinacompetitiveindustryatacostof$10perunit.Thereare100consumersareeachwillingtopay$12eachtoconsumeasingleunitofthegood(additionalunitshavenovaluetothem.)WhatistheequilibriumpriceandquantitysoldThegovernmentimposesataxof$1onthegood.Whatisthedeadweightlossofthistax
14.1.Theequilibriumpriceis$10andthequantitysoldis100units.Ifthetaxisimposed,thepricerisesto$11,but100unitsofthegoodwillstillbesold,sothereisnodeadweightloss.
2.SupposethatthedemandcurveisgivenbyD(p)=10?p.Whatisthegrossbene?tfromconsuming6unitsofthegood
14.2.Wewanttocomputetheareaunderthedemandcurvetotheleftofthequantity6.Breakthisupintotheareaofatrianglewithabaseof6andaheightof6andarectanglewithbase6andheight4.Applyingtheformulasfromhighschoolgeometry,thetrianglehasarea18andtherectanglehasarea24.Thusgrossbene?tis42.
3.Intheaboveexample,ifthepricechangesfrom4to6,whatisthechangeinconsumer’ssurplus
14.3.Whenthepriceis4,theconsumer’ssurplusisgivenbytheareaofatrianglewithabaseof6andaheightof6;i.e.,theconsumer’ssurplusis18.Whenthepriceis6,thetrianglehasabaseof4andaheightof4,givinganareaof8.Thusthepricechangehasreducedconsumer’ssurplusby$10.
4.Supposethataconsumerisconsuming10unitsofadiscretegoodandthepriceincreasesfrom$5perunitto$6.However,afterthepricechangetheconsumercontinuestoconsume10unitsofthediscretegood.Whatisthelossintheconsumer’ssurplusfromthispricechange
14.4.Tendollars.Sincethedemandforthediscretegoodhasn’tchanged,allthathashappenedisthattheconsumerhashadtoreducehisexpenditureonothergoodsbytendollars.
15MarketDemand
1.IfthemarketdemandcurveisD(p)=100?.5p,whatistheinversedemandcurve
15.1.TheinversedemandcurveisP(q)=200?2q.
2.Anaddict’sdemandfunctionforadrugmaybeveryinelastic,butthemarketdemandfunctionmightbequiteelastic.Howcanthisbe
15.2.Thedecisionaboutwhethertoconsumethedrugatallcouldwellbepricesensitive,sotheadjustmentofmarketdemandontheextensivemarginwouldcontributetotheelasticityofthemarketdemand.
3.IfD(p)=12?2p,whatpricewillmaximizerevenue
15.3.RevenueisR(p)=12p?2p2,whichismaximizedatp=3.
4.SupposethatthedemandcurveforagoodisgivenbyD(p)=100/p.Whatpricewillmaximizerevenue
15.4.RevenueispD(p)=100,regardlessoftheprice,soallpricesmaximizerevenue.
5.TrueorfalseInatwogoodmodelifonegoodisaninferiorgoodtheothergoodmustbealuxurygood.
15.5.True.Theweightedaverageoftheincomeelasticitiesmustbe1,soifonegoodhasanegativeincomeelasticity,theothergoodmusthaveanelasticitygreaterthan1togettheaveragetobe1.
16Equilibrium
1.Whatisthee?ectofasubsidyinamarketwithahorizontalsupplycurveWithaverticalsupplycurve
16.1.Theentiresubsidygetspassedalongtotheconsumersifthesupplycurveis?at,butthesubsidyistotallyreceivedbytheproducerswhenthesupplycurveisvertical.
2.Supposethatthedemandcurveisverticalwhilethesupplycurveslopesupward.Ifataxisimposedinthismarketwhoendsuppayingit
16.2.Theconsumer.
3.Supposethatallconsumersviewredpencilsandbluepencilsasperfectsubstitutes.Supposethatthesupplycurveforredpencilsisupwardsloping.Letthepriceofredpencilsandbluepenci
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