中级微观经济学习题及答案.docx

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,howmuchwi