中級微觀經(jīng)濟學習題及答案

ANSWERS

1TheMarket

1.Supposethattherewere25peoplewhohadareservationpriceof

$500,andthe26thpersonhadareservationpriceof$200.Whatwould

thedemandcurvelooklike?

1.1.Itwouldbeconstantat$500for25apartmentsandthendropto

$200.

2.Intheaboveexample,whatwouldtheequilibriumpricebeifthere

were24apartmentstorent?Whatiftherewere26apartmentsto

rent?Whatiftherewere25apartmentstorent?

1.2.Inthefirstcase,$500,andinthesecondcase,$200.Inthethirdcase,

theequilibriumpricewouldbeanypricebetween$200and$500.

3.Ifpeoplehavedifferentreservationprices,whydoesthemarket

demandcurveslopedown?

1.3.Becauseifwewanttorentonemoreapartment,wehavetooffera

lowerprice.Thenumberofpeoplewhohavereservationpricesgreater

thanpmustalwaysincreaseaspdecreases.

4.Inthetextweassumedthatthecondominiumpurchaserscamefrom

theinner-ringpeople—peoplewhowerealreadyrentingapartments.

Whatwouldhappentothepriceofinner-ringapartmentsifallofthe

condominiumpurchaserswereouter-ringpeople—thepeoplewho

werenotcurrentlyrentingapartmentsintheinnerring?

1.4.Thepriceofapartmentsintheinnerringwouldgoupsincedemand

forapartmentswouldnotchangebutsupplywoulddecrease.

5.Supposenowthatthecondominiumpurchaserswereallinner-ring

people,butthateachcondominiumwasconstructedfromtwo

apartments.Whatwouldhappentothepriceofapartments?

1.5.Thepriceofapartmentsintheinnerringwouldrise.

6.Whatdoyousupposetheeffectofataxwouldbeonthenumberof

apartmentsthatwouldbebuiltinthelongrun?

1.6.Ataxwouldundoubtedlyreducethenumberofapartmentssupplied

inthelongrun.

7.SupposethedemandcurveisD(p)=100-2p.Whatpricewouldthe

monopolistsetifhehad60apartments?Howmanywouldherent?

Whatpricewouldhesetifhehad40apartments?Howmanywouldhe

rent?

1.7.Hewouldsetapriceof25andrent50apartments.Inthesecond

casehewouldrentall40apartmentsatthemaximumpricethemarket

wouldbear.ThiswouldbegivenbythesolutiontoD(p)=100-2p=40,

whichisp*=30.

8.Ifourmodelofrentcontrolallowedforunrestrictedsubletting,who

wouldendupgettingapartmentsintheinnercircle?Wouldthe

outcomebeParetoefficient?

1.8.Everyonewhohadareservationpricehigherthantheequilibrium

priceinthecompetitivemarket,sothatthefinaloutcomewouldbe

Paretoefficient.(Ofcourseinthelongruntherewouldprobablybe

fewernewapartmentsbuilt,whichwouldleadtoanotherkindof

inefficiency.)

2BudgetConstraint

1.Originallytheconsumerfacesthebudgetlineplxl+p2x2=m.Then

thepriceofgood1doubles,thepriceofgood2becomes8timeslarger,

andincomebecomes4timeslarger.Writedownanequationforthe

newbudgetlineintermsoftheoriginalpricesandincome.

2.1.Thenewbudgetlineisgivenby2plxl+8p2x2=4m.

2.Whathappenstothebudgetlineifthepriceofgood2increases,but

thepriceofgood1andincomeremainconstant?

2.2.Theverticalintercept(x2axis)decreasesandthehorizontal

intercept(x】axis)staysthesame.Thusthebudgetlinebecomesflatter.

3.Ifthepriceofgood1doublesandthepriceofgood2triples,does

thebudgetlinebecomeflatterorsteeper?

2.3.Flatter.Theslopeis-2p1/3p2.

4.Whatisthedefinitionofanumerairegood?

2.4.Agoodwhosepricehasbeensetto1;allothergoods/pricesare

measuredrelativetothenumerairegood/sprice.

5.Supposethatthegovernmentputsataxof15centsagallonon

gasolineandthenlaterdecidestoputasubsidyongasolineatarateof

7centsagallon.Whatnettaxisthiscombinationequivalentto?

2.5.Ataxof8centsagallon.

6.Supposethatabudgetequationisgivenby

PiXi+p2x2=m.Thegovernmentdecidestoimposealump-sumtax

ofu,aquantitytaxongood1oft,andaquantitysubsidyongood2of

s.Whatistheformulaforthenewbudgetline?

2.6.(pi+t)xi+(p2-s)x2=m-u.

7.Iftheincomeoftheconsumerincreasesandoneoftheprices

decreasesatthesametime,willtheconsumernecessarilybeatleast

aswell-off?

2.7.Yes,sinceallofthebundlestheconsumercouldaffordbeforeare

affordableatthenewpricesandincome.

3Preferences

1.Ifweobserveaconsumerchoosing的甩)when(y"2)isavailable

onetime,arewejustifiedinconcludingthat(x1,x2)>(yi，yz)?

3.1.No.Itmightbethattheconsumerwasindifferentbetweenthetwo

bundles.Allwearejustifiedinconcludingisthat(x1,x2)>(y1/y2).

2.ConsideragroupofpeopleA,B,CandtherelationMatleastastallas,”

asin"AisatleastastallasB."Isthisrelationtransitive?Isitcomplete?

3.2.Yestoboth.

3.Takethesamegroupofpeopleandconsidertherelation“strictly

tallerthan/*Isthisrelationtransitive?Isitreflexive?Isitcomplete?

3.3.Itistransitive,butitisnotcomplete—twopeoplemightbethesame

height.Itisnotreflexivesinceitisfalsethatapersonisstrictlytallerthan

himself.

4.AcollegefootballcoachsaysthatgivenanytwolinemenAandB,he

alwayspreferstheonewhoisbiggerandfaster.Isthispreference

relationtransitive?Isitcomplete?

3.4.Itistransitive,butnotcomplete.WhatifAwerebiggerbutslower

thanB?Whichonewouldheprefer?

5.Cananindifferencecurvecrossitself?Forexample,couldFigure3.2

depictasingleindifferencecurve?

3.5.Yes.Anindifferencecurvecancrossitself,itjustcan'tcrossanother

distinctindifferencecurve.

6.CouldFigure3.2beasingleindifferencecurveifpreferencesare

monotonic?

3.6.No,becausetherearebundlesontheindifferencecurvethathave

strictlymoreofbothgoodsthanotherbundlesonthe(alleged)

indifferencecurve.

7.Ifbothpepperoniandanchoviesarebads,willtheindifferencecurve

haveapositiveoranegativeslope?

3.7.Anegativeslope.Ifyougivetheconsumermoreanchovies,you/ve

madehimworseoff,soyouhavetotakeawaysomepepperonitoget

himbackonhisindifferencecurve.Inthiscasethedirectionof

increasingutilityistowardtheorigin.

8.Explainwhyconvexpreferencesmeansthat“averagesarepreferred

toextremes?/

3.8.Becausetheconsumerweaklypreferstheweightedaverageoftwo

bundlestoeitherbundle.

9.Whatisyourmarginalrateofsubstitutionof$1billsfor$5bills?

3.9.Ifyougiveupone$5bill,howmany$1billsdoyouneedto

compensateyou?Five$1billswilldonicely.Hencetheansweris-5

or-1/5,dependingonwhichgoodyouputonthehorizontalaxis.

10.Ifgood1isa"neutral,“whatisitsmarginalrateofsubstitutionfor

good2?

3.10.Zero—ifyoutakeawaysomeofgood1,theconsumerneedszero

unitsofgood2tocompensatehimforhisloss.

ANSWERSA13

11.Thinkofsomeothergoodsforwhichyourpreferencesmightbe

concave.

3.11.Anchoviesandpeanutbutter,scotchandKoolAid,andother

similarrepulsivecombinations.

4Utility

1.Thetextsaidthatraisinganumbertoanoddpowerwasa

monotonictransformation.Whataboutraisinganumbertoaneven

power?Isthisamonotonictransformation?(Hint:considerthecase

f(u)=uA2.)

4.1.Thefunctionf(u)=uA2isamonotonictransformationforpositiveu,

butnotfornegativeu.

2.Whichofthefollowingaremonotonictransformations?

(1)u=2v-13;(2)u=-l/vA2;(3)u=l/vA2;(4)u=Inv;(5)u=-eA-v;(6)u

=vA2;(7)u=vA2forv>0;(8)u=vA2forv<0.

4.2.(1)Yes.(2)No(worksforvpositive).(3)No(worksforvnegative).(4)

Yes(onlydefinedforvpositive).(5)Yes.(6)No.(7)Yes.(8)No.

3.Weclaimedinthetextthatifpreferencesweremonotonic,thena

diagonallinethroughtheoriginwouldintersecteachindifferencecurve

exactlyonce.Canyouprovethisrigorously?(Hint:whatwouldhappen

ifitintersectedsomeindifferencecurvetwice?)

4.3.Supposethatthediagonalintersectedagivenindifferencecurveat

twopoints,say(x,x)and(y,y).Theneitherx>yory>x,whichmeansthat

oneofthebundleshasmoreofbothgoods.Butifpreferencesare

monotonic,thenoneofthebundleswouldhavetobepreferredtothe

other.

4.Whatkindofpreferencesarerepresentedbyautilityfunctionofthe

formu(xl,x2)=x4-x2?Whatabouttheutilityfunctionv(xlzx2)=

13x1+13x2?

4.4.Bothrepresentperfectsubstitutes.

5.Whatkindofpreferencesarerepresentedbyautilityfunctionofthe

formu(xlzx2)=xl+Vx2?Istheutilityfunctionv(xl,x2)=x21+2xlVx2^

+x2amonotonictransformationofu(xl,x2)?

4.5.Quasilinearpreferences.Yes.

6.Considertheutilityfunctionu(xl/x2)=A/x1x2.Whatkindofpref-

22

erencesdoesitrepresent?Isthefunctionv(x1/x2)=x1x2amonotonic

transformationofu(x17x2)?Isthefunctionw(xi,x?Tx/x22a

monotonictransformationofu(xlzx2)?

4.6.TheutilityfunctionrepresentsCobb-Douglaspreferences.No.Yes.

7.Canyouexplainwhytakingamonotonictransformationofautility

functiondoesn'tchangethemarginalrateofsubstitution?

4.7.BecausetheMRSismeasuredalonganindifferencecurve,and

utilityremainsconstantalonganindifferencecurve.

5Choice

1.Iftwogoodsareperfectsubstitutes,whatisthedemandfunctionfor

good2?

，，

5.1.X2=0whenp2>Pix2=m/p2whenp2<Piandanythingbetween

0andm/p2when=p2.

2.Supposethatindifferencecurvesaredescribedbystraightlineswith

aslopeof-b.Givenarbitrarypricesandmoneyincomepl,p2,andm,

whatwilltheconsumer'soptimalchoiceslooklike?

5.2.Theoptimalchoiceswillbexl=m/plandx2=0ifpl/p2<b,xl=0

andx2=m/p2ifpl/p2>b,andanyamountonthebudgetlineifpl/p2

=b.

3.Supposethataconsumeralwaysconsumes2spoonsofsugarwith

eachcupofcoffee.Ifthepriceofsugarisplperspoonfulandtheprice

ofcoffeeisp2percupandtheconsumerhasmdollarstospendon

coffeeandsugar;howmuchwillheorshewanttopurchase?

5.3.Letzbethenumberofcupsofcoffeetheconsumerbuys.Thenwe

knowthat2zisthenumberofteaspoonsofsugarheorshebuys.We

mustsatisfythebudgetconstraint

2Plz+p2z=m.

Solvingforzwehave

m

Z-

2P1+P2

4.Supposethatyouhavehighlynonconvexpreferencesforicecream

andolives,likethosegiveninthetext,andthatyoufacepricespl,p2

andhavemdollarstospend.Listthechoicesfortheoptimal

consumptionbundles.

5.4.Weknowthatyou/lleitherconsumeallicecreamorallolives.Thus

thetwochoicesfortheoptimalconsumptionbundleswillbexl=m/plz

x2=0,orxl=0,x2=m/p2.

5.Ifaconsumerhasautilityfunctionu(xl,x2)=xlx42,whatfractionof

herincomewillshespendongood2?

5.5.ThisisaCobb-Douglasutilityfunction,soshewillspend4/(1+4)=

4/5ofherincomeongood2.

6.Forwhatkindofpreferenceswilltheconsumerbejustaswell-off

facingaquantitytaxasanincometax?

5.6.Forkinkedpreferences,suchasperfectcomplements,wherethe

changeinpricedoesn'tinduceanychangeindemand.

6Demand

1.Iftheconsumerisconsumingexactlytwogoods,andsheisalways

spendingallofhermoney,canbothofthembeinferiorgoods?

6.1.No.Ifherincomeincreases,andshespendsitall,shemustbe

purchasingmoreofatleastonegood.

2.Showthatperfectsubstitutesareanexampleofhomothetic

preferences.

6.2.Theutilityfunctionforperfectsubstitutesisu(xlzx2)=+x2.

Thusifu(xlzx2)>u(yi，y2),wehave+x2>yi+y2.Itfollowsthat

txi+1x2>tyi+ty2,sothatu(txlztx2)>u(tyi，ty2).

3.ShowthatCobb-Douglaspreferencesarehomotheticpreferences.

6.3.TheCobb-Douglasutilityfunctionhasthepropertythat

a1-aa1-aa1-aa1-a

u(txlztx2)=(txj)(tx2)=ttx±x22=tXjx22=

t*u(xl,x2).Thusifu(X],x2)>u(y17y2),weknowthatu(txlztx2)>u

(tylzty2),sothatCobb-Douglaspreferencesareindeedhomothetic.

4.TheincomeoffercurveistotheEngelcurveasthepriceoffercurve

isto...?

6.4.Thedemandcurve.

5.Ifthepreferencesareconcavewilltheconsumereverconsumeboth

ofthegoodstogether?

6.5.No.Concavepreferencescanonlygiverisetooptimalconsumption

bundlesthatinvolvezeroconsumptionofoneofthegoods.

6.Arehamburgersandbunscomplementsorsubstitutes?

6.6.Normallytheywouldbecomplements,atleastfornon-vegetarians.

7.Whatistheformoftheinversedemandfunctionforgood1inthe

caseofperfectcomplements?

6.7.Weknowthatxl=m/(pl+p2).Solvingforplasafunctionofthe

othervariables,wehavepl=mxl-p2.

8.Trueorfalse?Ifthedemandfunctionisxl=-pl,thentheinverse

demandfunctionisx=-1/pl.

6.8.False.

7RevealedPreference

1.Whenpricesare(pl,p2)=(1,2)aconsumerdemands(xl,x2)=(1,2),

andwhenpricesare(ql,q2)=(2,1)theconsumerdemands(yl,y2)=

(2,1).Isthisbehaviorconsistentwiththemodelofmaximizing

behavior?

7.1.No.ThisconsumerviolatestheWeakAxiomofRevealed

Preferencesincewhenhebought(xl,x2)hecouldhavebought(yl,y2)

andviceversa.Insymbols:

plxl+p2x2=lxl+2x2=5>4=lx2+2xl=plyl+p2y2

and

qlyl+q2y2=2x2+lxl=5>4=2xl+lx2=qlxl+q2x2.

2.Whenpricesare(pl,p2)=(2,1)aconsumerdemands(xl,x2)=(1,2),

andwhenpricesare(ql,q2)=(1,2)theconsumerdemands(yl,y2)=

(2,1).Isthisbehaviorconsistentwiththemodelofmaximizing

behavior?

7.2.Yes.NoviolationsofWARParepresent,sincethey-bundleisnot

affordablewhenthex-bundlewaspurchasedandviceversa.

3.Intheprecedingexercise,whichbundleispreferredbytheconsumer,

thex-bundleorthey-bundle?

7.3.Sincethey-bundlewasmoreexpensivethanthex-bundlewhenthe

x-bundlewaspurchasedandviceversa,thereisnowaytotellwhich

bundleispreferred.

4.WesawthattheSocialSecurityadjustmentforchangingprices

wouldtypicallymakerecipientsatleastaswell-offastheywereatthe

baseyear.Whatkindofpricechangeswouldleavethemjustaswell-off,

nomatterwhatkindofpreferencestheyhad?

7.4.Ifbothpriceschangedbythesameamount.Thenthebase-year

bundlewouldstillbeoptimal.

5.Inthesameframeworkastheabovequestion,whatkindof

preferenceswouldleavetheconsumerjustaswell-offashewasinthe

baseyear,forallpricechanges?

7.5.Perfectcomplements.

8SlutskyEquation

1.Supposeaconsumerhaspreferencesbetweentwogoodsthatare

perfectsubstitutes.Canyouchangepricesinsuchawaythattheentire

demandresponseisduetotheincomeeffect?

8.1.Yes.Toseethis,useourfavoriteexampleofredpencilsandblue

pencils.Supposeredpencilscost10centsapiece,andbluepencilscost

5centsapiece,andtheconsumerspends$1onpencils.Shewouldthen

consume20bluepencils.Ifthepriceofbluepencilsfallsto4centsa

piece,shewouldconsume25bluepencils,achangewhichisentirely

duetotheincomeeffect.

2.Supposethatpreferencesareconcave.Isitstillthecasethatthe

substitutioneffectisnegative?

8.2.Yes.

3.Inthecaseofthegasolinetax,whatwouldhappeniftherebateto

theconsumerswerebasedontheiroriginalconsumptionofgasoline,x,

ratherthanontheirfinalconsumptionofgasoline,x'?

8.3.Thentheincomeeffectwouldcancelout.AllthatwouldbeIe代

wouldbethepuresubstitutioneffect,whichwouldautomaticallybe

negative.

4.Inthecasedescribedintheprecedingquestion,wouldthe

governmentbepayingoutmoreorlessthanitreceivedintaxrevenues?

8.4.Theyarereceivingtx'inrevenuesandpayingouttx,sotheyare

losingmoney.

5.Inthiscasewouldtheconsumersbebetterofforworseoffifthetax

withrebatebasedonoriginalconsumptionwereineffect?

8.5.Sincetheiroldconsumptionisaffordable,theconsumerswould

havetobeatleastaswell-off.Thishappensbecausethegovernmentis

givingthembackmoremoneythantheyarelosingduetothehigher

priceofgasoline.

9BuyingandSelling

1.Ifaconsumer'snetdemandsare(5,-3)andherendowmentis(4,4),

whatarehergrossdemands?

9.1.Hergrossdemandsare(9,1).

2.Thepricesare(pl,p2)=(2,3),andtheconsumeriscurrently

consuming(xl,x2)=(4,4).Thereisaperfectmarketforthetwogoods

inwhichtheycanbeboughtandsoldcostlessly.Willtheconsumer

necessarilypreferconsumingthebundle(yl,y2)=(3,5)?Willshe

necessarilypreferhavingthebundle(yl,y2)?

9.2.Thebundle(yl,y2)=(3,5)costsmorethanthebundle(4,4)atthe

currentprices.Theconsumerwillnotnecessarilypreferconsuming

thisbundle,butwouldcertainlyprefertoownit,sinceshecouldsellit

andpurchaseabundlethatshewouldprefer.

3.Thepricesare(pl,p2)=(2,3bandtheconsumeriscurrently

consuming(xl,x2)=(4,4).Nowthepriceschangeto(ql,q2)=(2,4).

Couldtheconsumerbebetteroffunderthesenewprices?

9.3.Sure.Itdependsonwhethershewasanetbuyeroranetsellerof

thegoodthatbecamemoreexpensive.

4.TheU.S.currentlyimportsabouthalfofthepetroleumthatituses.

Therestofitsneedsaremetbydomesticproduction.Couldtheprice

ofoilrisesomuchthattheU.S.wouldbemadebetteroff?

9.4.Yes,butonlyiftheU.S.switchedtobeinganetexporterofoil.

5.Supposethatbysomemiraclethenumberofhoursintheday

increasedfrom24to30hours(withluckthiswouldhappenshortly

beforeexamweek).Howwouldthisaffectthebudgetconstraint?

9.5.Thenewbudgetlinewouldshiftoutwardandremainparalleltothe

oldone,sincetheincreaseinthenumberofhoursinthedayisapure

endowmenteffect.

6.Ifleisureisaninferiorgood,whatcanyousayabouttheslopeofthe

laborsupplycurve?

9.6.Theslopewillbepositive.

10IntertemporalChoice

1.Howmuchis$1milliontobedelivered20yearsinthefutureworth

todayiftheinterestrateis20percent?

10.1.AccordingtoTable10.1,$120yearsfromnowisworth3cents

todayata20percentinterestrate.Thus$1millionis

worth.03x1,000,000=$30,000today.

2.Astheinterestraterises,doestheintertemporalbudgetconstraint

be-comesteeperorflatter?

10.2.Theslopeoftheintertemporalbudgetconstraintisequalto-(1+r).

Thusasrincreasestheslopebecomesmorenegative(steeper).

3.Wouldtheassumptionthatgoodsareperfectsubstitutesbevalidin

astudyofintertemporalfoodpurchases?

10.3.Ifgoodsareperfectsubstitutes,thenconsumerswillonlypurchase

thecheapergood.Inthecaseofintertemporalfoodpurchases,this

impliesthatconsumersonlybuyfoodinoneperiod,whichmaynotbe

veryrealistic.

4.Aconsumer;whoisinitiallyalender,remainsalenderevenaftera

declineininterestrates.Isthisconsumerbetterofforworseoffafter

thechangeininterestrates?Iftheconsumerbecomesaborrowerafter

thechangeishebetterofforworseoff?

10.4.Inordertoremainalenderafterthechangeininterestrates,the

consumermustbechoosingapointthathecouldhavechosenunderthe

oldinterestrates,butdecidednotto.Thustheconsumermustbeworse

off.Iftheconsumerbecomesaborrowerafterthechange,thenheis

choosingapreviouslyunavailablepointthatcannotbecomparedtothe

initialpoint(sincetheinitialpointisnolongeravailableunderthenew

budgetconstraint),andthereforethechangeintheconsumer/swelfare

isunknown.

5.Whatisthepresentvalueof$100oneyearfromnowiftheinterest

rateis10%?Whatisthepresentvalueiftheinterestrateis5%?

10.5.Ataninterestrateof10%,thepresentvalueof$100is$90.91.Ata

rateof5%thepresentvalueis$95,24.

11AssetMarkets

1.SupposeassetAcanbesoldfor$11nextperiod.IfassetssimilartoA

arepayingarateofreturnof10%,whatmustbeassetA'scurrent

price?

11.1.AssetAmustbesellingfor11/(1+0.10)=$10.

2.Ahouse,whichyoucouldrentfor$10,000ayearandsellfor

$110,000ayearfromnow,canbepurchasedfor$100,000.Whatisthe

rateofreturnonthishouse?

11.2.Therateofreturnisequalto(10,000+10,000)/100,000=20%.

3.Thepaymentsofcertaintypesofbonds(e.g.,municipalbonds)are

nottaxable.Ifsimilartaxablebondsarepaying10%andeveryonefaces

amarginaltaxrateof40%,whatrateofreturnmustthenontaxable

bondspay?

11.3.Weknowthattherateofreturnonthenontaxablebonds,r,must

besuchthat(l-t)rt=r,therefore(1-0.40)*0.10=0.06=r.

4.Supposethatascarceresource,facingaconstantdemand,willbe

exhaustedin10years.Ifanalternativeresourcewillbeavailableata

priceof$40andiftheinterestrateis10%,whatmustthepriceofthe

scarceresourcebetoday?

11.4.Thepricetodaymustbe40/(1+0.10)A10=$15,42.

12Uncertainty

1.Howcanonereachtheconsumptionpointstotheleftofthe

endowmentinFigure12.1?

12.1.Weneedawaytoreduceconsumptioninthebadstateand

increaseconsumptioninthegoodstate.Todothisyouwouldhavetosell

insuranceagainstthelossratherthanbuyit.

2.Whichofthefollowingutilityfunctionshavetheexpectedutility

：

property?(a)u(cl/c2,nl/n2)=a(nlcl+712c2),(b)u(cl,c2,nL,Ti2)=Ttl,cl+

112c22,Incl+n2Inc2+17.

12.2.Functions(a)and(c)havetheexpectedutilityproperty(theyare

affinetransformationsofthefunctionsdiscussedinthechapter),while

(b)doesnot.

3.Arisk-averseindividualisofferedachoicebetweenagamblethat

pays$1000withaprobabilityof25%and$100withaprobabilityof

75%,orapaymentof$325.Whichwouldhechoose?

12.3.Sinceheisrisk-averse,hepreferstheexpectedvalueofthegamble,

$325,tothegambleitself,andthereforehewouldtakethepayment.

4.Whatifthepaymentwas$320?

12.4.Ifthepaymentis$320thedecisionwilldependontheformofthe

utilityfunction;wecan'tsayanythingingeneral.

5.Drawautilityfunctionthatexhibitsrisk-lovingbehaviorforsmall

gamblesandrisk-aversebehaviorforlargergambles.

12.5.Yourpictureshouldshowafunctionthatisinitiallyconvex,but

thenbecomesconcave.

6.Whymightaneighborhoodgrouphaveahardertimeselfinsuring

forflooddamageversusfiredamage?

12.6.Inordertoself-insure,therisksmustbeindependent.However,

thisdoesnotholdinthecaseofflooddamage.Ifonehouseinthe

neighborhoodisdamagedbyaflooditislikelythatallofthehouseswill

bedamaged.

13RiskyAssets

1.Iftherisk-freerateofreturnis6%,andifariskyassetisavailable

withareturnof9%andastandarddeviationof3%,whatisthe

maximumrateofreturnyoucanachieveifyouarewillingtoaccepta

standarddeviationof2%?Whatpercentageofyourwealthwouldhave

tobeinvestedintheriskyasset?

13.1.Toachieveastandarddeviationof2%youwillneedtoinvestx=

ox/om=2/3ofyourwealthintheriskyasset.Thiswillresultinarateof

returnequalto(2/3)0.09+(1-2/3)0.06=8%.

2.Whatisthepriceofriskintheaboveexercise?

13.2.Thepriceofriskisequalto(rm-rf)/om=(9-6)/3=1.Thatis,for

everyadditionalpercentofstandarddeviationyoucangain1%ofreturn.

3.IfastockhasaPof1.5,thereturnonthemarketis10%,andthe

risk-freerateofreturnis5%,whatexpectedrateofreturnshouldthis

stockofferaccordingtotheCapitalAssetPricingModel?Ifthe

expectedvalueofthestockis$100,whatpriceshouldthestockbe

sellingfortoday?

13.3.AccordingtotheCAPMpricingequation,thestockshouldofferan

expectedrateofreturnofrf+P(rm-rf)=0.05+1.5(0.10-0.05)=0.125or

12.5%,Thestockshouldbesellingforitsexpectedpresentvalue,which

isequalto100/1.125=$88.89.

14Consumer'sSurplus

1.Agoodcanbeproducedinacompetitiveindustryatacostof$10

perunit.Thereare100consumersareeachwillingtopay$12eachto

consumeasingleunitofthegood(additionalunitshavenovalueto

them.)Whatistheequilibriumpriceandquantitysold?The

governmentimposesataxof$1onthegood.Whatisthedeadweight

lossofthistax?

14.1.Theequilibriumpriceis$10andthequantitysoldis100units.If

thetaxisimposed,thepricerisesto$11,but100unitsofthegoodwill

stillbesold,sothereisnodeadweightloss.

2.SupposethatthedemandcurveisgivenbyD(p)=10-p.Whatisthe

grossbenefitfromconsuming6unitsofthegood?

14.2.Wewanttocomputetheareaunderthedemandcurvetotheleft

ofthequantity6.Breakthisupintotheareaofatrianglewithabaseof

6andaheightof6andarectanglewithbase6andheight4.Applying

theformulasfromhighschoolgeometry,thetrianglehasarea18andthe

rectanglehasarea24.Thusgrossbenefitis42.

3.Intheaboveexample,ifthepricechangesfrom4to6,whatisthe

changeinconsumer'ssurplus?

14.3.Whenthepriceis4,theconsumer'ssurplusisgivenbytheareaof

atrianglewithabaseof6andaheightof6;i.e.,theconsumer'ssurplus

is18.Whenthepriceis6,thetrianglehasabaseof4andaheightof4,

givinganareaof8.Thusthepricechangehasreducedconsumer's

surplusby$10.

4.Supposethataconsumerisconsuming10unitsofadiscretegood

andthepriceincreasesfrom$5perunitto$6.However,aftertheprice

changetheconsumercontinuestoconsume10unitsofthediscrete

good.Whatisthelossintheconsumer'ssurplusfromthisprice

change?

14.4.Tendollars.Sincethedemandforthediscretegoodhasn/tchanged,

allthathashappenedisthattheconsumerhashadtoreducehis

expenditureonothergoodsbytendollars.

15MarketDemand

1.IfthemarketdemandcurveisD(p)=100-.5p,whatistheinverse

demandcurve?

15.1.TheinversedemandcurveisP(q)=200-2q.

2.Anaddict'sdemandfunctionforadrugmaybeveryinelastic,butthe

marketdemandfunctionmightbequiteelastic.Howcanthisbe?

15.2.Thedecisionaboutwhethertoconsumethedrugatallcouldwell

bepricesensitive,sotheadjustmentofmarketdemandontheextensive

marginwouldcontributetotheelasticityofthemarketdemand.

3.IfD(p)=12-2p,whatpricewillmaximizerevenue?

15.3.RevenueisR(p)=12p-2P2,whichismaximizedatp=3.

4.Supposethatthedemandcurveforagoodisgivenby