中级微观经济学习题

1、ANSWERS1TheMarket1.Supposethattherewere25peoplewhohadareservationpriceof$500,andthe26thpersonhadareservationpriceof$200.Whatwouldthedemandcurvelooklike.Itwouldbeconstantat$500for25apartmentsandthendropto$200.Intheaboveexample,whatwouldtheequilibriumpricebeiftherewere24apartmentstorentWhatiftherewere

2、26apartmentstorentWhatiftherewere25apartmentstorent.Intherstcase,$500,andinthesecondcase,$200.Inthethirdcase,theequilibriumpricewouldbeanypricebetween$200and$500.Ifpeoplehavedierentreservationprices,whydoesthemarketdemandcurveslopedown.Becauseifwewanttorentonemoreapartment,wehavetooeralowerprice.The

3、numberofpeoplewhohavereservationpricesgreaterthanpmustalwaysincreaseaspdecreases.Inthetextweassumedthatthecondominiumpurchaserscamefromtheinner-ringpeoplepeoplewhowerealreadyrentingapartments.Whatwouldhappentothepriceofinner-ringapartmentsifallofthecondominiumpurchaserswereouter-ringpeoplethepeoplew

4、howerenotcurrentlyrentingapartmentsintheinnerring.Thepriceofapartmentsintheinnerringwouldgoupsincedemandforapartmentswouldnotchangebutsupplywoulddecrease.Supposenowthatthecondominiumpurchaserswereallinner-ringpeople,butthateachcondominiumwasconstructedfromtwoapartments.Whatwouldhappentothepriceofapa

5、rtments.Thepriceofapartmentsintheinnerringwouldrise.Whatdoyousupposetheeectofataxwouldbeonthenumberofapartmentsthatwouldbebuiltinthelongrun.Ataxwouldundoubtedlyreducethenumberofapartmentssuppliedinthelongrun.SupposethedemandcurveisD(p)=1002p.Whatpricewouldthemonopolistsetifhehad60apartmentsHowmanywo

6、uldherentWhatpricewouldhesetifhehad40apartmentsHowmanywouldherent.Hewouldsetapriceof25andrent50apartments.Inthesecondcasehewouldrentall40apartmentsatthemaximumpricethemarketwouldbear.ThiswouldbegivenbythesolutiontoD(p)=1002p=40,whichisp=30.Ifourmodelofrentcontrolallowedforunrestrictedsubletting,whow

7、ouldendupgettingapartmentsintheinnercircleWouldtheoutcomebeParetoecient.Everyonewhohadareservationpricehigherthantheequilibriumpriceinthecompetitivemarket,sothatthenaloutcomewouldbeParetoecient.(Ofcourseinthelongruntherewouldprobablybefewernewapartmentsbuilt,whichwouldleadtoanotherkindofineciency.)O

8、riginallytheconsumerfacesthebudgetlinep1x1+p2x2=m.Thenthepriceofgood1doubles,thepriceofgood2becomes8timeslarger,andincomebecomes4timeslarger.Writedownanequationforthenewbudgetlineintermsoftheoriginalpricesandincome.Thenewbudgetlineisgivenby2p1x1+8p2x2=4m.Whathappenstothebudgetlineifthepriceofgood2in

9、creases,butthepriceofgood1andincomeremainconstant.Theverticalintercept(axis)decreasesandthehorizontalintercept(axis)staysthesame.Thusthebudgetlinebecomesatter.Ifthepriceofgood1doublesandthepriceofgood2triples,doesthebudgetlinebecomeatterorsteeper.Flatter.Theslopeis2/3.4.Whatisthedenitionofanumeraire

10、good.Agoodwhosepricehasbeensetto1;allothergoodspricesaremeasuredrelativetothenumerairegoodsprice.Supposethatthegovernmentputsataxof15centsagallonongasolineandthenlaterdecidestoputasubsidyongasolineatarateofcentsagallon.Whatnettaxisthiscombinationequivalentto.Ataxof8centsagallon.6.Supposethatabudgete

11、quationisgivenby+=m.Thegovernmentdecidestoimposealump-sumtaxofu,aquantitytaxongood1oft,andaquantitysubsidyongood2ofs.Whatistheformulaforthenewbudgetline.(+t)+(s)=mu.7.Iftheincomeoftheconsumerincreasesandoneofthepricesdecreasesatthesametime,willtheconsumernecessarilybeatleastaswell-o.Yes,sinceallofth

12、ebundlestheconsumercouldaordbeforeareaordableatthenewpricesandincome.3Preferences1.Ifweobserveaconsumerchoosing(,)when(,)isavailableonetime,arewejustiedinconcludingthat(,)(,).No.Itmightbethattheconsumerwasindierentbetweenthetwobundles.Allwearejustiedinconcludingisthat(,)(,).2.ConsideragroupofpeopleA

13、,B,Candtherelation“atleastastaas,”asin“AisatleastastallasB.”IsthisrelationtransitiveIsitcomplete.Yestoboth.Takethesamegroupofpeopleandconsidertherelation“strictlytallerthan.”IsthisrelationtransitiveIsitreexiveIsitcomplete.Itistransitive,butitisnotcompleteheight.Itisnotreexivesinceitisfalsehimself.tw

14、opeoplemightbethesamethatapersonisstrictlytallerthan4.AcollegefootballcoachsaysthatgivenanytwolinemenAandB,healwayspreferstheonewhoisbiggerandfaster.IsthispreferencerelationtransitiveIsitcomplete.Itistransitive,butnotcomplete.WhatifAwerebiggerbutslowerthanBWhichonewouldhepreferCananindierencecurvecr

15、ossitselfForexample,couldFiguredepictasingleindierencecurve.Yes.Anindierencecurvecancrossitself,itjustcancrosstanotherdistinctindierencecurve.CouldFigurebeasingleindierencecurveifpreferencesaremonotonic.No,becausetherearebundlesontheindierencecurvethathavestrictlymoreofbothgoodsthanotherbundlesonthe

16、(alleged)indierencecurve.Ifbothpepperoniandanchoviesarebads,willtheindierencecurvehaveapositiveoranegativeslope.Anegativeslope.Ifyougivetheconsumermoreanchovies,youvemadehimworseo,soyouhavetotakeawaysomepepperonitogethimbackonhisindierencecurve.Inthiscasethedirectionofincreasingutilityistowardtheori

17、gin.8.Explainwhyconvexpreferencesmeansthat“averagesarepreferredtoextremes.”.Becausetheconsumerweaklypreferstheweightedaverageoftwobundlestoeitherbundle.9.Whatisyourmarginalrateofsubstitutionof$1billsfor$5bills.Ifyougiveupone$5bill,howmany$1billsdoyouneedtocompensateyouFive$1billswilldonicely.Henceth

18、eansweris5or1/5,dependingonwhichgoodyouputonthehorizontalaxis.10.Ifgood1isa“neutral,”whatisitsmarginalrateofsubstitutionforgood2.Zeroifyoutakeawaysomeofgood1,theconsumerneedszerounitsofgood2tocompensatehimforhisloss.ANSWERSA13Thinkofsomeothergoodsforwhichyourpreferencesmightbeconcave.Anchoviesandpea

19、nutbutter,scotchandKoolAid,andothersimilarrepulsivecombinations.UtilityThetextsaidthatraisinganumbertoanoddpowerwasamonotonictransformation.WhataboutraisinganumbertoanevenpowerIsthisamonotonictransformation(Hint:considerthecasef(u)=u2.).Thefunctionf(u)=u2isamonotonictransformationforpositiveu,butnot

20、fornegativeu.2.Whichofthefollowingaremonotonictransformations(1)u=2v13;(2)u=1/v2;(3)u=1/v2;(4)u=lnv;(5)u=ev;(6)u=v2;(7)u=v2forv0;(8)u=v2forvyoryx,whichmeansthatoneofthebundleshasmoreofbothgoods.Butifpreferencesaremonotonic,thenoneofthebundleswouldhavetobepreferredtotheother.4.Whatkindofpreferencesar

21、erepresentedbyautilityfunctionoftheformu(x1,x2)=Whatabouttheutilityfunctionv(x1,x2)=13x1+13x2.Bothrepresentperfectsubstitutes.Whatkindofpreferencesarerepresentedbyautilityfunctionoftheformu(x1,x2)=x1+Istheutilityfunctionv(x1,x2)=x21+2x1+x2amonotonictransformationofu(x1,x2).Quasilinearpreferences.Yes

22、.6.Considertheutilityfunctionu(x1,x2)=erencesdoesitrepresentIsthefunctionv(,transformationofu(,)Isthefunction.Whatkindofpref-)=amonotonicw(,)=amonotonictransformationofu(,).TheutilityfunctionrepresentsCobb-Douglaspreferences.No.Yes.7.Canyouexplainwhytakingamonotonictransformationofautilityfunctiondo

23、esntchangethemarginalrateofsubstitution.BecausetheMRSismeasuredalonganindierencecurve,andutilityremainsconstantalonganindierencecurve.ChoiceIftwogoodsareperfectsubstitutes,whatisthedemandfunctionforgood2.=0when,=m/when,andanythingbetween0andm/p2when=.Supposethatindierencecurvesaredescribedbystraight

24、lineswithaslopeofb.Givenarbitrarypricesandmoneyincomep1,p2,andm,whatwilltheconsumersoptimalchoiceslooklike.Theoptimalchoiceswillbex1=m/p1andx2=0ifp1/p2b,andanyamountonthebudgetlineifp1/p2=b.Supposethataconsumeralwaysconsumes2spoonsofsugarwitheachcupofcoee.Ifthepriceofsugarisp1perspoonfulandthepriceo

25、fcoeeisp2percupandtheconsumerhasmdollarstospendoncoeeandsugar,howmuchwillheorshewanttopurchase.Letzbethenumberofcupsofcoeetheconsumerbuys.Thenweknowthat2zisthenumberofteaspoonsofsugarheorshebuys.Wemustsatisfythebudgetconstraintz+z=m.Solvingforzwehave=.Supposethatyouhavehighlynonconvexpreferencesfori

26、cecreamandolives,likethosegiveninthetext,andthatyoufacepricesp1,p2andhavemdollarstospend.Listthechoicesfortheoptimalconsumptionbundles.Weknowthatyoulleitherconsumeallicecreamorallolives.Thusthetwochoicesfortheoptimalconsumptionbundleswillbex1=m/,x2=0,orx1=0,x2=m/.Ifaconsumerhasautilityfunctionu(x1,x

27、2)=x1x42,whatfractionofherincomewillshespendongood2.ThisisaCobb-Douglasutilityfunction,soshewillspend4/(1+4)=4/5ofherincomeongood2.Forwhatkindofpreferenceswilltheconsumerbejustaswell-ofacingaquantitytaxasanincometax.Forkinkedpreferences,suchasperfectcomplements,wherethechangeinpricedoesntinduceanych

28、angedemandin.DemandIftheconsumerisconsumingexactlytwogoods,andsheisalwaysspendingallofhermoney,canbothofthembeinferiorgoods.No.Ifherincomeincreases,andshespendsitall,shemustbepurchasingmoreofatleastonegood.Showthatperfectsubstitutesareanexampleofhomotheticpreferences.Theutilityfunctionforperfectsubs

29、titutesisu(,)=+.Thusifu(,)u(,),wehave+.Itfollowsthatt+tt+t,sothatu(t,t)u(t,t).3.ShowthatCobb-Douglaspreferencesarehomotheticpreferences.TheCobb-Douglasutilityfunctionhasthepropertythatu(t,t)=2=t2=t*u(x1,).Thusifu(,)u(,),weknowthatu(t,t)u(t,),sothatCobb-Douglaspreferencesareindeedhomothetic.Theincome

30、oercurveistotheEngelcurveasthepriceoercurveisto.Thedemandcurve.Ifthepreferencesareconcavewilltheconsumereverconsumebothofthegoodstogether.No.Concavepreferencescanonlygiverisetooptimalconsumptionbundlesthatinvolvezeroconsumptionofoneofthegoods.Arehamburgersandbunscomplementsorsubstitutes.Normallythey

31、wouldbecomplements,atleastfornon-vegetarians.7.Whatistheformoftheinversedemandfunctionforgood1inthecaseofperfectcomplements.Weknowthatx1=m/(p1+p2).Solvingforp1asafunctionoftheothervariables,wehavep1=mx1p2.TrueorfalseIfthedemandfunctionisx1=p1,thentheinversedemandfunctionisx=1/p1.False.7RevealedPrefe

32、renceWhenpricesare(p1,p2)=(1,2)aconsumerdemands(x1,x2)=(1,2),andwhenpricesare(q1,q2)=(2,1)theconsumerdemands(y1,y2)=(2,1).Isthisbehaviorconsistentwiththemodelofmaximizingbehavior.No.ThisconsumerviolatestheWeakAxiomofRevealedPreferencesincewhenhebought(x1,x2)hecouldhavebought(y1,y2)andviceversa.Insym

33、bols:p1x1+p2x2=11+22=54=12+21=p1y1+p2y2andq1y1+q2y2=22+11=54=21+12=q1x1+q2x2.Whenpricesare(p1,p2)=(2,1)aconsumerdemands(x1,x2)=(1,2),andwhenpricesare(q1,q2)=(1,2)theconsumerdemands(y1,y2)=(2,1).Isthisbehaviorconsistentwiththemodelofmaximizingbehavior.Yes.NoviolationsofWARParepresent,sincethey-bundle

34、isnotaordablewhenthex-bundlewaspurchasedandviceversa.Intheprecedingexercise,whichbundleispreferredbytheconsumer,thex-bundleorthey-bundle.Sincethey-bundlewasmoreexpensivethanthex-bundlewhenthex-bundlewaspurchasedandviceversa,thereisnowaytotellwhichbundleispreferred.WesawthattheSocialSecurityadjustmen

35、tforchangingpriceswouldtypicallymakerecipientsatleastaswell-oastheywereatthebaseyear.Whatkindofpricechangeswouldleavethemjustaswell-o,nomatterwhatkindofpreferencestheyhad.Ifbothpriceschangedbythesameamount.Thenthebase-yearbundlewouldstillbeoptimal.5.Inthesameframeworkastheabovequestion,whatkindofpre

36、ferenceswouldleavetheconsumerjustaswell-oashewasinthebaseyear,forallpricechanges.Perfectcomplements.8SlutskyEquationSupposeaconsumerhaspreferencesbetweentwogoodsthatareperfectsubstitutes.Canyouchangepricesinsuchawaythattheentiredemandresponseisduetotheincomeeect.Yes.Toseethis,useourfavoriteexampleof

37、redpencilsandbluepencils.Supposeredpencilscost10centsapiece,andbluepencilscost5centsapiece,andtheconsumerspends$1onpencils.Shewouldthenconsumebluepencils.Ifthepriceofbluepencilsfallsto4centsapiece,shewouldconsume25bluepencils,achangewhichisentirelyduetotheincomeeect.Supposethatpreferencesareconcave.

38、Isitstillthecasethatthesubstitutioneectisnegative.Yes.Inthecaseofthegasolinetax,whatwouldhappeniftherebatetotheconsumerswerebasedontheiroriginalconsumptionofgasoline,x,ratherthanontheirnalconsumptionofgasoline,x.Thentheincomeeectwouldcancelout.Allthatwouldbeleftwouldbethepuresubstitutioneect,whichwo

39、uldautomaticallybenegative.Inthecasedescribedintheprecedingquestion,wouldthegovernmentbepayingoutmoreorlessthanitreceivedintaxrevenues.Theyarereceivingtxinrevenuesandpayingouttx,sotheyarelosingmoney.Inthiscasewouldtheconsumersbebetteroorworseoifthetaxwithrebatebasedonoriginalconsumptionwereineect.Si

40、ncetheiroldconsumptionisaordable,theconsumerswouldhavetobeatleastaswell-o.Thishappensbecausethegovernmentisgivingthembackmoremoneythantheyarelosingduetothehigherpriceofgasoline.9BuyingandSellingIfaconsumersnetdemandsare(5,3)andherendowmentis(4,4),whatarehergrossdemands.Hergrossdemandsare(9,1).Thepri

41、cesare(p1,p2)=(2,3),andtheconsumeriscurrentlyconsuming(x1,x2)=(4,4).Thereisaperfectmarketforthetwogoodsinwhichtheycanbeboughtandsoldcostlessly.Willtheconsumernecessarilypreferconsumingthebundle(y1,y2)=(3,5)Willshenecessarilypreferhavingthebundle(y1,y2).Thebundle(y1,y2)=(3,5)costsmorethanthebundle(4,

42、4)atthecurrentprices.Theconsumerwillnotnecessarilypreferconsumingthisbundle,butwouldcertainlyprefertoownit,sinceshecouldsellitandpurchaseabundlethatshewouldprefer.Thepricesare(p1,p2)=(2,3),andtheconsumeriscurrentlyconsuming(x1,x2)=(4,4).Nowthepriceschangeto(q1,q2)=(2,4).Couldtheconsumerbebetterounde

43、rthesenewprices.Sure.Itdependsonwhethershewasanetbuyeroranetsellerofthegoodthatbecamemoreexpensive.The.currentlyimportsabouthalfofthepetroleumthatituses.Therestofitsneedsaremetbydomesticproduction.Couldthepriceofoilrisesomuchthatthe.wouldbemadebettero.Yes,butonlyifthe.switchedtobeinganetexporterofoi

44、l.Supposethatbysomemiraclethenumberofhoursinthedayincreasedfrom24to30hours(withluckthiswouldhappenshortlybeforeexamweek).Howwouldthisaectthebudgetconstraint.Thenewbudgetlinewouldshiftoutwardandremainparalleltotheoldone,sincetheincreaseinthenumberofhoursinthedayisapureendowmenteect.Ifleisureisaninfer

45、iorgood,whatcanyousayabouttheslopeofthelaborsupplycurve.Theslopewillbepositive.10IntertemporalChoiceHowmuchis$1milliontobedelivered20yearsinthefutureworthtodayiftheinterestrateis20percent.AccordingtoTable,$120yearsfromnowisworth3centstodayata20percentinterestrate.Thus$1millionisworth.031,000,000=$30

46、,today.Astheinterestraterises,doestheintertemporalbudgetconstraintbe-comesteeperoratter.Theslopeoftheintertemporalbudgetconstraintisequalto(1+r).Thusasrincreasestheslopebecomesmorenegative(steeper).Wouldtheassumptionthatgoodsareperfectsubstitutesbevalidinastudyofintertemporalfoodpurchases.Ifgoodsare

47、perfectsubstitutes,thenconsumerswillonlypurchasethecheapergood.Inthecaseofintertemporalfoodpurchases,thisimpliesthatconsumersonlybuyfoodinoneperiod,whichmaynotbeveryrealistic.Aconsumer,whoisinitiallyalender,remainsalenderevenafteradeclineininterestrates.Isthisconsumerbetteroorworseoafterthechangeini

48、nterestratesIftheconsumerbecomesaborrowerafterthechangeishebetteroorworseo.Inordertoremainalenderafterthechangeininterestrates,theconsumermustbechoosingapointthathecouldhavechosenundertheoldinterestrates,butdecidednotto.Thustheconsumermustbeworseo.Iftheconsumerbecomesaborrowerafterthechange,thenheis

49、choosingapreviouslyunavailablepointthatcannotbecomparedtotheinitialpoint(sincetheinitialpointisnolongeravailableunderthenewbudgetconstraint),andthereforethechangeintheconsumerisunknown.Whatisthepresentvalueof$100oneyearfromnowiftheinterestrateis10%Whatisthepresentvalueiftheinterestrateis5%.Ataninter

50、estrateof10%,thepresentvalueof$100is$.Atarateof5%thepresentvalueis$.11AssetMarketsSupposeassetAcanbesoldfor$11nextperiod.IfassetssimilartoAarepayingarateofreturnof10%,whatmustbeassetA.AssetAmustbesellingfor11/(1+=$10.Ahouse,whichyoucouldrentfor$10,000ayearandsellfor$110,000ayearfromnow,canbepurchase

51、dfor$100,000.Whatistheswelscurrentprateofreturnonthishouse.Therateofreturnisequalto(10,000+10,000)/100,000=20%.Thepaymentsofcertaintypesofbonds.,municipalbonds)arenottaxable.Ifsimilartaxablebondsarepaying10%andeveryonefacesamarginaltaxrateof40%,whatrateofreturnmustthenontaxablebondspay.Weknowthatthe

52、rateofreturnonthenontaxablebonds,r,mustbesuchthat(1t)=r,therefore(1*=0.06=r.Supposethatascarceresource,facingaconstantdemand,willbeexhaustedin10years.Ifanalternativeresourcewillbeavailableatapriceof$40andiftheinterestrateis10%,whatmustthepriceofthescarceresourcebetoday.Thepricetodaymustbe40/(1+0.10)

53、10=$.UncertaintyHowcanonereachtheconsumptionpointstotheleftoftheendowmentinFigure.Weneedawaytoreduceconsumptioninthebadstateandincreaseconsumptioninthegoodstate.Todothisyouwouldhavetosellinsuranceagainstthelossratherthanbuyit.2.Whichofthefollowingutilityfunctionshavetheexpectedutilityproperty(a)u(c1

54、,c2,1,2)=a(+1c12c2),(b)u(c1,c2,1,2)=+1c12c22,(c)u(c1,c2,1,2)=1lnc1+2lnc2+17.Functions(a)and(c)havetheexpectedutilityproperty(theyareanetransformationsofthefunctionsdiscussedinthechapter),while(b)doesnot.Arisk-averseindividualisoeredachoicebetweenagamblethatpays$1000withaprobabilityof25%and$100withap

55、robabilityof75%,orapaymentof$325.Whichwouldhechoose.Sinceheisrisk-averse,hepreferstheexpectedvalueofthegamble,$325,tothegambleitself,andthereforehewouldtakethepayment.Whatifthepaymentwas$320.Ifthepaymentis$320thedecisionwilldependontheformoftheutilityfunction;wecantsayanythingingeneral.5.Drawautilit

56、yfunctionthatexhibitsrisk-lovingbehaviorforsmallgamblesandrisk-aversebehaviorforlargergambles.Yourpictureshouldshowafunctionthatisinitiallyconvex,butthenbecomesconcave.Whymightaneighborhoodgrouphaveahardertimeselfinsuringforooddamageversusredamage.Inordertoself-insure,therisksmustbeindependent.Howev

57、er,thisdoesnotholdinthecaseofooddamage.Ifonehouseintheneighborhoodisdamagedbyaooditislikelythatallofthehouseswillbedamaged.13RiskyAssetsIftherisk-freerateofreturnis6%,andifariskyassetisavailablewithareturnof9%andastandarddeviationof3%,whatisthemaximumrateofreturnyoucanachieveifyouarewillingtoaccepta

58、standarddeviationof2%Whatpercentageofyourwealthwouldhavetobeinvestedintheriskyasset.Toachieveastandarddeviationof2%youwillneedtoinvestx=2/3ofyourwealthintheriskyasset.Thiswillresultinarateofreturnequalto(2/3)+(12/3)=8%.2.Whatisthepriceofriskintheaboveexercise.Thepriceofriskisequalto(rmrf)/additional

59、percentofstandarddeviationm=(936)/=1.Thatis,foreveryyoucangain1%ofreturn.Ifastockhasaof,thereturnonthemarketis10%,andtherisk-freerateofreturnis5%,whatexpectedrateofreturnshouldthisstockoeraccordingtotheCapitalAssetPricingModelIftheexpectedvalueofthestockis$100,whatpriceshouldthestockbesellingfortoda

60、y.AccordingtotheCAPMpricingequation,thestockshouldoeranexpectedrateofreturnofrf+(rmrf)=+=0.125or%.Thestockshouldbesellingforitsexpectedpresentvalue,whichisequalto100/=$.14ConsumersplSursAgoodcanbeproducedinacompetitiveindustryatacostof$10perunit.Thereare100consumersareeachwillingtopay$12eachtoconsum