中级微观经济学习题及答案.docx
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中级微观经济学习题及答案
ANSWERS
1TheMarket
1.Supposethattherewere25peoplewhohadareservationpriceof$500,andthe26thpersonhadareservationpriceof$200.Whatwouldthedemandcurvelooklike?
1.1.Itwouldbeconstantat$500for25apartmentsandthendropto$200.
2.Intheaboveexample,whatwouldtheequilibriumpricebeiftherewere24apartmentstorent?
Whatiftherewere26apartmentstorent?
Whatiftherewere25apartmentstorent?
1.2.Inthefirstcase,$500,andinthesecondcase,$200.Inthethirdcase,theequilibriumpricewouldbeanypricebetween$200and$500.
3.Ifpeoplehavedifferentreservationprices,whydoesthemarketdemandcurveslopedown?
1.3.Becauseifwewanttorentonemoreapartment,wehavetoofferalowerprice.Thenumberofpeoplewhohavereservationpricesgreaterthanpmustalwaysincreaseaspdecreases.
4.Inthetextweassumedthatthecondominiumpurchaserscamefromtheinner-ringpeople—peoplewhowerealreadyrentingapartments.Whatwouldhappentothepriceofinner-ringapartmentsifallofthecondominiumpurchaserswereouter-ringpeople—thepeoplewhowerenotcurrentlyrentingapartmentsintheinnerring?
1.4.Thepriceofapartmentsintheinnerringwouldgoupsincedemandforapartmentswouldnotchangebutsupplywoulddecrease.
5.Supposenowthatthecondominiumpurchaserswereallinner-ringpeople,butthateachcondominiumwasconstructedfromtwoapartments.Whatwouldhappentothepriceofapartments?
1.5.Thepriceofapartmentsintheinnerringwouldrise.
6.Whatdoyousupposetheeffectofataxwouldbeonthenumberofapartmentsthatwouldbebuiltinthelongrun?
1.6.Ataxwouldundoubtedlyreducethenumberofapartmentssuppliedinthelongrun.
7.SupposethedemandcurveisD(p)=100−2p.Whatpricewouldthemonopolistsetifhehad60apartments?
Howmanywouldherent?
Whatpricewouldhesetifhehad40apartments?
Howmanywouldherent?
1.7.Hewouldsetapriceof25andrent50apartments.Inthesecondcasehewouldrentall40apartmentsatthemaximumpricethemarketwouldbear.ThiswouldbegivenbythesolutiontoD(p)=100−2p=40,whichisp∗=30.
8.Ifourmodelofrentcontrolallowedforunrestrictedsubletting,whowouldendupgettingapartmentsintheinnercircle?
WouldtheoutcomebeParetoefficient?
1.8.Everyonewhohadareservationpricehigherthantheequilibriumpriceinthecompetitivemarket,sothatthefinaloutcomewouldbeParetoefficient.(Ofcourseinthelongruntherewouldprobablybefewernewapartmentsbuilt,whichwouldleadtoanotherkindofinefficiency.)
2BudgetConstraint
1.Originallytheconsumerfacesthebudgetlinep1x1+p2x2=m.Thenthepriceofgood1doubles,thepriceofgood2becomes8timeslarger,andincomebecomes4timeslarger.Writedownanequationforthenewbudgetlineintermsoftheoriginalpricesandincome.
2.1.Thenewbudgetlineisgivenby2p1x1+8p2x2=4m.
2.Whathappenstothebudgetlineifthepriceofgood2increases,butthepriceofgood1andincomeremainconstant?
2.2.Theverticalintercept(
axis)decreasesandthehorizontalintercept(
axis)staysthesame.Thusthebudgetlinebecomesflatter.
3.Ifthepriceofgood1doublesandthepriceofgood2triples,doesthebudgetlinebecomeflatterorsteeper?
2.3.Flatter.Theslopeis−2
/3
.
4.Whatisthedefinitionofanumerairegood?
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-off?
2.7.Yes,sinceallofthebundlestheconsumercouldaffordbeforeareaffordableatthenewpricesandincome.
3Preferences
1.Ifweobserveaconsumerchoosing(
)when(
)isavailableonetime,arewejustifiedinconcludingthat(
)>(
)?
3.1.No.Itmightbethattheconsumerwasindifferentbetweenthetwobundles.Allwearejustifiedinconcludingisthat(
)>(
).
2.ConsideragroupofpeopleA,B,Candtherelation“atleastastallas,”asin“AisatleastastallasB.”Isthisrelationtransitive?
Isitcomplete?
3.2.Yestoboth.
3.Takethesamegroupofpeopleandconsidertherelation“strictlytallerthan.”Isthisrelationtransitive?
Isitreflexive?
Isitcomplete?
3.3.Itistransitive,butitisnotcomplete—twopeoplemightbethesameheight.Itisnotreflexivesinceitisfalsethatapersonisstrictlytallerthanhimself.
4.AcollegefootballcoachsaysthatgivenanytwolinemenAandB,healwayspreferstheonewhoisbiggerandfaster.Isthispreferencerelationtransitive?
Isitcomplete?
3.4.Itistransitive,butnotcomplete.WhatifAwerebiggerbutslowerthanB?
Whichonewouldheprefer?
5.Cananindifferencecurvecrossitself?
Forexample,couldFigure3.2depictasingleindifferencecurve?
3.5.Yes.Anindifferencecurvecancrossitself,itjustcan’tcrossanotherdistinctindifferencecurve.
6.CouldFigure3.2beasingleindifferencecurveifpreferencesaremonotonic?
3.6.No,becausetherearebundlesontheindifferencecurvethathavestrictlymoreofbothgoodsthanotherbundlesonthe(alleged)indifferencecurve.
7.Ifbothpepperoniandanchoviesarebads,willtheindifferencecurvehaveapositiveoranegativeslope?
3.7.Anegativeslope.Ifyougivetheconsumermoreanchovies,you’vemadehimworseoff,soyouhavetotakeawaysomepepperonitogethimbackonhisindifferencecurve.Inthiscasethedirectionofincreasingutilityistowardtheorigin.
8.Explainwhyconvexpreferencesmeansthat“averagesarepreferredtoextremes.”
3.8.Becausetheconsumerweaklypreferstheweightedaverageoftwobundlestoeitherbundle.
9.Whatisyourmarginalrateofsubstitutionof$1billsfor$5bills?
3.9.Ifyougiveupone$5bill,howmany$1billsdoyouneedtocompensateyou?
Five$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.Whataboutraisinganumbertoanevenpower?
Isthisamonotonictransformation?
(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(onlydefinedforvpositive).(5)Yes.(6)No.(7)Yes.(8)No.
3.Weclaimedinthetextthatifpreferencesweremonotonic,thenadiagonallinethroughtheoriginwouldintersecteachindifferencecurveexactlyonce.Canyouprovethisrigorously?
(Hint:
whatwouldhappenifitintersectedsomeindifferencecurvetwice?
)
4.3.Supposethatthediagonalintersectedagivenindifferencecurveattwopoints,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-erencesdoesitrepresent?
Isthefunctionv(
)=
amonotonictransformationofu(
)?
Isthefunctionw(
)=
amonotonictransformationofu(
)?
4.6.TheutilityfunctionrepresentsCobb-Douglaspreferences.No.Yes.
7.Canyouexplainwhytakingamonotonictransformationofautilityfunctiondoesn’tchangethemarginalrateofsubstitution?
4.7.BecausetheMRSismeasuredalonganindifferencecurve,andutilityremainsconstantalonganindifferencecurve.
5Choice
1.Iftwogoodsareperfectsubstitutes,whatisthedemandfunctionforgood2?
5.1.
=0when
>
=m/
when
<
andanythingbetween0andm/p2when
=
.
2.Supposethatindifferencecurvesaredescribedbystraightlineswithaslopeof−b.Givenarbitrarypricesandmoneyincomep1,p2,andm,whatwilltheconsumer’soptimalchoiceslooklike?
5.2.Theoptimalchoiceswillbex1=m/p1andx2=0ifp1/p2b,andanyamountonthebudgetlineifp1/p2=b.
3.Supposethataconsumeralwaysconsumes2spoonsofsugarwitheachcupofcoffee.Ifthepriceofsugarisp1perspoonfulandthepriceofcoffeeisp2percupandtheconsumerhasmdollarstospendoncoffeeandsugar,howm