中級(jí)微觀經(jīng)濟(jì)學(xué)習(xí)題及答案

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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