中級(jí)微觀經(jīng)濟(jì)學(xué)習(xí)題及答案

ANSWERSTheMarketSupposethattherewere25peoplewhohadareservationpriceof$500,andthe26thpersonhadareservationpriceof$200.Whatwouldthedemandcurvelooklike.Itwouldbeconstantat$500for25apartmentsandthendropto$200.Intheaboveexample,whatwouldtheequilibriumpricebeiftherewere24apartmentstorentWhatiftherewere26apartmentstorentWhatiftherewere25apartmentstorent.Intherstcase,$500,andinthesecondcase,$200.Inthethirdcase,theequilibriumpricewouldbeanypricebetween$200and$500.Ifpeoplehavedierentreservationprices,whydoesthemarketdemandcurveslopedown.Becauseifwewanttorentonemoreapartment,wehavetooeralowerprice.Thenumberofpeoplewhohavereservationpricesgreaterthanpmustalwaysincreaseaspdecreases.Inthetextweassumedthatthecondominiumpurchaserscamefromtheinner-ringpeople—peoplewhowerealreadyrentingapartments.Whatwouldhappentothepriceofinner-ringapartmentsifallofthecondominiumpurchaserswereouter-ringpeople—thepeoplewhowerenotcurrentlyrentingapartmentsintheinnerring.Thepriceofapartmentsintheinnerringwouldgoupsincedemandforapartmentswouldnotchangebutsupplywoulddecrease.Supposenowthatthecondominiumpurchaserswereallinner-ringpeople,butthateachcondominiumwasconstructedfromtwoapartments.Whatwouldhappentothepriceofapartments.Thepriceofapartmentsintheinnerringwouldrise.Whatdoyousupposetheeectofataxwouldbeonthenumberofapartmentsthatwouldbebuiltinthelongrun.Ataxwouldundoubtedlyreducethenumberofapartmentssuppliedinthelongrun.SupposethedemandcurveisD(p)=1002p.Whatpricewouldthemonopolistsetifhehad60apartmentsHowmanywouldherentWhatpricewouldhesetifhehad40apartmentsHowmanywouldherent.Hewouldsetapriceof25andrent50apartments.Inthesecondcasehewouldrentall40apartmentsatthemaximumpricethemarketwouldbear.ThiswouldbegivenbythesolutiontoD(p)=1002p=40,whichisp=30.

Ifourmodelofrentcontrolallowedforunrestrictedsubletting,whowouldendupgettingapartmentsintheinnercircleWouldtheoutcomebeParetoecient.Everyonewhohadareservationpricehigherthantheequilibriumpriceinthecompetitivemarket,sothatthenaloutcomewouldbeParetoecient.(Ofcourseinthelongruntherewouldprobablybefewernewapartmentsbuilt,whichwouldleadtoanotherkindofineciency.)BudgetConstraintOriginallytheconsumerfacesthebudgetlinep1x1+p2x2=m.Thenthepriceofgood1doubles,thepriceofgood2becomes8timeslarger,andincomebecomes4timeslarger.Writedownanequationforthenewbudgetlineintermsoftheoriginalpricesandincome..Thenewbudgetlineisgivenby2p1x1+8p2x2=4m.Whathappenstothebudgetlineifthepriceofgood2increases,butthepriceofgood1andincomeremainconstant.Theverticalintercept(,.Theverticalintercept(,axis)decreasesandthehorizontalinterceptCaxis)staysthesame.Thusthebudgetlinebecomesatter.Ifthepriceofgood1doublesandthepriceofgood2triples,doesthebudgetlinebecomeatterorsteeper.Flatter.Theslopeis2/3|_Whatisthedenitionofanumerairegood.Agoodwhosepricehasbeensetto1;allothergoods'pricesaremeasuredrelativetothenumerairegood'sprice.Supposethatthegovernmentputsataxof15centsagallonongasolineandthenlaterdecidestoputasubsidyongasolineatarateof7centsagallon.Whatnettaxisthiscombinationequivalentto.Ataxof8centsagallon.Supposethatabudgetequationisgivenbyp、i+_1=m.Thegovernmentdecidestoimposealump-sumtaxofu,aquantitytaxongood1oft,andaquantitysubsidyongood2ofs.Whatistheformulaforthenewbudgetline.(pi+t)X]+(p：s)X2=mu.Iftheincomeoftheconsumerincreasesandoneofthepricesdecreasesatthesametime,willtheconsumernecessarilybeatleastaswell-o.Yes,sinceallofthebundlestheconsumercouldaordbeforeareaordableatthenewpricesandincome.PreferencesIfweobserveaconsumerchoosing(m,x?)when勺i,y：)isavailableonetime,arewejustiedinconcludingthatg,工2)>(i,”).No.Itmightbethattheconsumerwasindierentbetweenthetwobundles.Allwearejustiedinconcludingisthat(1,x2)>(i，丫2).ConsideragroupofpeopleA,B,Candtherelation“atleastastas,“asin"AsatleastastallasB."IsthisrelationtransitiveIsitcomplete.Yestoboth.Takethesamegroupofpeopleandconsidertherelation“strictlytallerthan."IsthisrelationtransitiveIsitreexiveIsitcomplete.Itistransitive,butitisnotcomplete—twopeoplemightbethesameheight.Itisnotreexivesinceitisfalsethatapersonisstrictlytallerthanhimself.AcollegefootballcoachsaysthatgivenanytwolinemenAandB,healwayspreferstheonewhoisbiggerandfaster.IsthispreferencerelationtransitiveIsitcomplete.Itistransitive,butnotcomplete.WhatifAwerebiggerbutslowerthanBWhichonewouldhepreferCananindierencecurvecrossitselfForexample,couldFiguredepictasingleindierencecurve.Yes.Anindierencecurvecancrossitself,itjustcan’ctrossanotherdistinctindierencecurve.CouldFigurebeasingleindierencecurveifpreferencesaremonotonic.No,becausetherearebundlesontheindierencecurvethathavestrictlymoreofbothgoodsthanotherbundlesonthe(alleged)indierencecurve.Ifbothpepperoniandanchoviesarebads,willtheindierencecurvehaveapositiveoranegativeslope.Anegativeslope.Ifyougivetheconsumermoreanchovies,you’vemadehimworseo,soyouhavetotakeawaysomepepperonitogethimbackonhisindierencecurve.Inthiscasethedirectionofincreasingutilityistowardtheorigin.Explainwhyconvexpreferencesmeansthat“averagesarepreferredtoextremes.”.Becausetheconsumerweaklypreferstheweightedaverageoftwobundlestoeitherbundle.Whatisyourmarginalrateofsubstitutionof$1billsfor$5bills.Ifyougiveupone$5bill,howmany$1billsdoyouneedtocompensateyouFive$1billswilldonicely.Hencetheansweris5or1/5,dependingonwhichgoodyouputonthehorizontalaxis.Ifgood1isa“neutral,”whatisitsmarginalrateofsubstitutionforgood2.Zero—ifyoutakeawaysomeofgood1,theconsumerneedszerounitsofgood2tocompensatehimforhisloss.ANSWERSA13Thinkofsomeothergoodsforwhichyourpreferencesmightbeconcave..Anchoviesandpeanutbutter,scotchandKoolAid,andothersimilarrepulsivecombinations.UtilityThetextsaidthatraisinganumbertoanoddpowerwasamonotonictransformation.WhataboutraisinganumbertoanevenpowerIsthisamonotonictransformation(Hint:considerthecasef(u)=uA2.).Thefunctionf(u)=uA2isamonotonictransformationforpositiveu,butnotfornegativeu.Whichofthefollowingaremonotonictransformationsu=2v13;(2)u='八2；(3)u=1/丫A2；(4)u=Inv;(5)u=eAv;(6)u=vA2;(7)u=v、2forv>0;(8)u=v、2forv<0..(1)Yes.(2)No(worksforvpositive).(3)No(worksforvnegative).(4)Yes(onlydenedforvpositive).(5)Yes.(6)No.(7)Yes.(8)No.Weclaimedinthetextthatifpreferencesweremonotonic,thenadiagonallinethroughtheoriginwouldintersecteachindierencecurveexactlyonce.Canyouprovethisrigorously(Hint:whatwouldhappenifitintersectedsomeindierencecurvetwice).Supposethatthediagonalintersectedagivenindierencecurveattwopoints,say(x,x)and(y,y).Theneitherx>yory>x,whichmeansthatoneofthebundleshasmoreofbothgoods.Butifpreferencesaremonotonic,thenoneofthebundleswouldhavetobepreferredtotheother.Whatkindofpreferencesarerepresentedbyautilityfunctionoftheformu(x1,x2)=、、l二Whatabouttheutilityfunctionv(x1,x2)=13x1+13x2.Bothrepresentperfectsubstitutes.Whatkindofpreferencesarerepresentedbyautilityfunctionoftheformu(x1,x2)=x1+；x：/Istheutilityfunctionv(x1,x2)=x21+2x17+x2amonotonictransformationofu(x1,x2).Quasilinearpreferences.Yes.Considertheutilityfunctionu(x1,x2)='"：.WhatkindofpreferencesdoesitrepresentIsthefunctionv(?,_)=__amonotonictransformationofu(*],x')Isthefunctionwg,￡i)=x「x；famonotonictransformationofu(Xi,x?).TheutilityfunctionrepresentsCobb-Douglaspreferences.No.Yes.Canyouexplainwhytakingamonotonictransformationofautilityfunctiondoesn'tchangethemarginalrateofsubstitution.BecausetheMRSismeasuredalonganindierencecurve,andutilityremainsconstantalonganindierencecurve.ChoiceIftwogoodsareperfectsubstitutes,whatisthedemandfunctionforgood2.X2=0whenPOi,工工=m/良whenS<P】，andanythingbetween0andm/p2whenPi=P2.Supposethatindierencecurvesaredescribedbystraightlineswithaslopeofb.Givenarbitrarypricesandmoneyincomep1,p2,andm,whatwilltheconsumer'soptimalchoiceslooklike.Theoptimalchoiceswillbex1=m/p1andx2=0ifp1/p2<b,x1=0andx2=m/p2ifp1/p2>b,andanyamountonthebudgetlineifp1/p2=b.Supposethataconsumeralwaysconsumes2spoonsofsugarwitheachcupofcoee.Ifthepriceofsugarisp1perspoonfulandthepriceofcoeeisp2percupandtheconsumerhasmdollarstospendoncoeeandsugar,howmuchwillheorshewanttopurchase.Letzbethenumberofcupsofcoeetheconsumerbuys.Thenweknowthat2zisthenumberofteaspoonsofsugarheorshebuys.Wemustsatisfythebudgetconstraint2iz+卜），z=m.Solvingforzwehavemz=m卜p/Supposethatyouhavehighlynonconvexpreferencesforicecreamandolives,likethosegiveninthetext,andthatyoufacepricesp1,p2andhavemdollarstospend.Listthechoicesfortheoptimalconsumptionbundles..Weknowthatyoulleitherconsumeallicecreamorallolives.Thusthetwochoicesfortheoptimalconsumptionbundleswillbex1=m/pj,x2=0,orx1=0,x2=m/』.Ifaconsumerhasautilityfunctionu(x1,x2)=x1x42,whatfractionofherincomewillshespendongood2.ThisisaCobb-Douglasutilityfunction,soshewillspend4/(1+4)=4/5ofherincomeongood2.Forwhatkindofpreferenceswilltheconsumerbejustaswell-ofacingaquantitytaxasanincometax.Forkinkedpreferences,suchasperfectcomplements,wherethechangeinpricedoesn'tinduceanychdegeiind.DemandIftheconsumerisconsumingexactlytwogoods,andsheisalwaysspendingallofhermoney,canbothofthembeinferiorgoods.No.Ifherincomeincreases,andshespendsitall,shemustbepurchasingmoreofatleastonegood.Showthatperfectsubstitutesareanexampleofhomotheticpreferences..Theutilityfunctionforperfectsubstitutesisu(ifu(x],x：)>u(yi,v2),wehavexi+x2>yi+Itfollowsthattxi+t\2>tyj+&工sothatu(txi,txt)>u(tyi,ty?).ShowthatCobb-Douglaspreferencesarehomotheticpreferences.TheCobb-Douglasutilityfunctionhasthepropertythatu(tX],tK2)=(tx垃=tatljME1x21-a2=tx^x-t=t*u(x1,xz).Thusifu(xi,x：)>u(yi,y?),weknowthatu(tM,tx])>u(t力，sothatCobb-Douglaspreferencesareindeedhomothetic.TheincomeoercurveistotheEngelcurveasthepriceoercurveisto....Thedemandcurve.Ifthepreferencesareconcavewilltheconsumereverconsumebothofthegoodstogether.No.Concavepreferencescanonlygiverisetooptimalconsumptionbundlesthatinvolvezeroconsumptionofoneofthegoods.Arehamburgersandbunscomplementsorsubstitutes.Normallytheywouldbecomplements,atleastfornon-vegetarians.Whatistheformoftheinversedemandfunctionforgood1inthecaseofperfectcomplements.Weknowthatx1=m/(p1+p2).Solvingforp1asafunctionoftheothervariables,wehavep1=mx1p2.TrueorfalseIfthedemandfunctionisx1=p1,thentheinversedemandfunctionisx=1/p1..False.RevealedPreferenceWhenpricesare(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.Insymbols:plxl+p2x2=1X1+2X2=5>4=1X2+2Xl=p1y1+p2y2andqlyl+q2y2=2乂2+1乂1=5>4=2x1+1乂2=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-bundleisnotaordablewhenthex-bundlewaspurchasedandviceversa.Intheprecedingexercise,whichbundleispreferredbytheconsumer,thex-bundleorthey-bundle.Sincethey-bundlewasmoreexpensivethanthex-bundlewhenthex-bundlewaspurchasedandviceversa,thereisnowaytotellwhichbundleispreferred.WesawthattheSocialSecurityadjustmentforchangingpriceswouldtypicallymakerecipientsatleastaswell-oastheywereatthebaseyear.Whatkindofpricechangeswouldleavethemjustaswell-o,nomatterwhatkindofpreferencestheyhad.Ifbothpriceschangedbythesameamount.Thenthebase-yearbundlewouldstillbeoptimal.Inthesameframeworkastheabovequestion,whatkindofpreferenceswouldleavetheconsumerjustaswell-oashewasinthebaseyear,forallpricechanges.Perfectcomplements.SlutskyEquationSupposeaconsumerhaspreferencesbetweentwogoodsthatareperfectsubstitutes.Canyouchangepricesinsuchawaythattheentiredemandresponseisduetotheincomeeect.Yes.Toseethis,useourfavoriteexampleofredpencilsandbluepencils.Supposeredpencilscost10centsapiece,andbluepencilscost5centsapiece,andtheconsumerspends$1onpencils.Shewouldthenconsume20bluepencils.Ifthepriceofbluepencilsfallsto4centsapiece,shewouldconsume25bluepencils,achangewhichisentirelyduetotheincomeeect.Supposethatpreferencesareconcave.Isitstillthecasethatthesubstitutioneectisnegative.Yes.Inthecaseofthegasolinetax,whatwouldhappeniftherebatetotheconsumerswerebasedontheiroriginalconsumptionofgasoline,x,ratherthanontheirnalconsumptionofgasoline,x’.Thentheincomeeectwouldcancelout.Allthatwouldbeleftwouldbethepuresubstitutioneect,whichwouldautomaticallybenegative.Inthecasedescribedintheprecedingquestion,wouldthegovernmentbepayingoutmoreorlessthanitreceivedintaxrevenues.Theyarereceivingtx’inrevenuesandpayingouttx,sotheyarelosingmoney.Inthiscasewouldtheconsumersbebetteroorworseoifthetaxwithrebatebasedonoriginalconsumptionwereineect.Sincetheiroldconsumptionisaordable,theconsumerswouldhavetobeatleastaswell-o.Thishappensbecausethegovernmentisgivingthembackmoremoneythantheyarelosingduetothehigherpriceofgasoline.BuyingandSellingIfaconsumer’snetdemandsare(5,3)andherendowmentis(4,4),whatarehergrossdemands.Hergrossdemandsare(9,1).Thepricesare(p1,p2)=(2,3),andtheconsumeriscurrentlyconsuming(x1,x2)=(4,4).Thereisaperfectmarketforthetwogoodsinwhichtheycanbeboughtandsoldcostlessly.Willtheconsumernecessarilypreferconsumingthebundle(y1,y2)=(3,5)Willshenecessarilypreferhavingthebundle(y1,y2).Thebundle(y1,y2)=(3,5)costsmorethanthebundle(4,4)atthecurrentprices.Theconsumerwillnotnecessarilypreferconsumingthisbundle,butwouldcertainlyprefertoownit,sinceshecouldsellitandpurchaseabundlethatshewouldprefer.Thepricesare(p1,p2)=(2,3),andtheconsumeriscurrentlyconsuming(x1,x2)=(4,4).Nowthepriceschangeto(q1,q2)=(2,4).Couldtheconsumerbebetterounderthesenewprices.Sure.Itdependsonwhethershewasanetbuyeroranetsellerofthegoodthatbecamemoreexpensive.The.currentlyimportsabouthalfofthepetroleumthatituses.Therestofitsneedsaremetbydomesticproduction.Couldthepriceofoilrisesomuchthatthe.wouldbemadebettero.Yes,butonlyifthe.switchedtobeinganetexporterofoil.Supposethatbysomemiraclethenumberofhoursinthedayincreasedfrom24to30hours(withluckthiswouldhappenshortlybeforeexamweek).Howwouldthisaectthebudgetconstraint.Thenewbudgetlinewouldshiftoutwardandremainparalleltotheoldone,sincetheincreaseinthenumberofhoursinthedayisapureendowmenteect.Ifleisureisaninferiorgood,whatcanyousayabouttheslopeofthelaborsupplycurve.Theslopewillbepositive.IntertemporalChoiceHowmuchis$1milliontobedelivered20yearsinthefutureworthtodayiftheinterestrateis20percent.AccordingtoTable,$120yearsfromnowisworth3centstodayata20percentinterestrate.Thus$1millionisworth.03x1,000,000=$30,today.Astheinterestraterises,doestheintertemporalbudgetconstraintbe-comesteeperoratter.Theslopeoftheintertemporalbudgetconstraintisequalto(1+r).Thusasrincreasestheslopebecomesmorenegative(steeper).Wouldtheassumptionthatgoodsareperfectsubstitutesbevalidinastudyofintertemporalfoodpurchases.Ifgoodsareperfectsubstitutes,thenconsumerswillonlypurchasethecheapergood.Inthecaseofintertemporalfoodpurchases,thisimpliesthatconsumersonlybuyfoodinoneperiod,whichmaynotbeveryrealistic.Aconsumer,whoisinitiallyalender,remainsalenderevenafteradeclineininterestrates.Isthisconsumerbetteroorworseoafterthe

changeininterestratesIftheconsumerbecomesaborrowerafterthechangeishebetteroorworseo.Inordertoremainalenderafterthechangeininterestrates,theconsumermustbechoosingapointthathecouldhavechosenundertheoldinterestrates,butdecidednotto.Thustheconsumermustbeworseo.Iftheconsumerbecomesaborrowerafterthechange,thenheischoosingapreviouslyunavailablepointthatcannotbecomparedtotheinitialpoint(sincetheinitialpointisnolongeravailableunderthenewbudgetconstraint),andthereforethechangeintheconsumerisunknown.Whatisthepresentvalueof$100oneyearfromnowiftheinterestrateis10%Whatisthepresentvalueiftheinterestrateis5%.Ataninterestrateof10%,thepresentvalueof$100is$.Atarateof5%thepresentvalueis$.AssetMarketsSupposeassetAcanbesoldfor$11nextperiod.IfassetssimilartoAarepayingarateofreturnof10%,whatmustbeassetA.AssetAmustbesellingfor1/1(1+=$10.swelscurrentpAhouse,whichyoucouldrentfor$10,000ayearandsellfor$110,000ayearfromnow,canbepurchasedfor$100,000.Whatistheswelscurrentprateofreturnonthishouse.Therateofreturnisequalto(10,000+10,000)/100,000=20%.Thepaymentsofcertaintypesofbonds.,municipalbonds)arenottaxable.Ifsimilartaxablebondsarepaying10%andeveryonefacesamarginaltaxrateof40%,whatrateofreturnmustthenontaxablebondspay.Weknowthattherateofreturnonthenontaxablebonds,r,mustbesuchthat(1t)?=r,therefore(1*=0.06=r.Supposethatascarceresource,facingaconstantdemand,willbeexhaustedin10years.Ifanalternativeresourcewillbeavailableatapriceof$40andiftheinterestrateis10%,whatmustthepriceofthescarceresourcebetoday.Thepricetodaymustbe40/(1+0.10)八10=$.UncertaintyHowcanonereachtheconsumptionpointstotheleftoftheendowmentinFigure.Weneedawaytoreduceconsumptioninthebadstateandincreaseconsumptioninthegoodstate.Todothisyouwouldhavetosellinsuranceagainstthelossratherthanbuyit.Whichofthefollowingutilityfunctionshavetheexpectedutilityproperty(a)u(c1,c2,兀1,兀2)=a+加2c2)(b)u(c1,c2,兀1,兀2)=+兀1c1兀2c22,(c)u(c1,c2,兀1,兀2)=兀1lnc1+兀2lnc2+17..Functions(a)and(c)havetheexpectedutilityproperty(theyareanetransformationsofthefunctionsdiscussedinthechapter),while(b)doesnot.Arisk-averseindividualisoeredachoicebetweenagamblethatpays$1000withaprobabilityof25%and$100withaprobabilityof75%,orapaymentof$325.Whichwouldhechoose.Sinceheisrisk-averse,hepreferstheexpectedvalueofthegamble,$325,tothegambleitself,andthereforehewouldtakethepayment.Whatifthepaymentwas$320.Ifthepaymentis$320thedecisionwilldependontheformoftheutilityfunction;wecan’tsayanythingingeneral.Drawautilityfunctionthatexhibitsrisk-lovingbehaviorforsmallgamblesandrisk-aversebehaviorforlargergambles..Yourpictureshouldshowafunctionthatisinitiallyconvex,butthenbecomesconcave.Whymightaneighborhoodgrouphaveahardertimeselfinsuringforooddamageversusredamage.Inordertoself-insure,therisksmustbeindependent.However,thisdoesnotholdinthecaseofooddamage.Ifonehouseintheneighborhoodisdamagedbyaooditislikelythatallofthehouseswillbedamaged.RiskyAssetsIftherisk-freerateofreturnis6%,andifariskyassetisavailablewithareturnof9%andastandarddeviationof3%,whatisthemaximumrateofreturnyoucanachieveifyouarewillingtoacceptastandarddeviationof2%Whatpercentageofyourwealthwouldhavetobeinvestedintheriskyasset.Toachieveastandarddeviationof2%youwillneedtoinvestx==2/3ofyourwealthintheriskyasset.Thiswillresultinarateofreturnequalto(2/3)+(12/3)=8%.Whatisthepriceofriskintheaboveexercise.Thepriceofriskisequalto(rmrf)/crm=(936)/1.Thatis,foreveryadditionalpercentofstandarddeviationyoucangain1%ofreturn.Ifastockhasa[3of,thereturnonthemarketis10%,andther-skfreerateofreturnis5%,whatexpectedrateofreturnshouldthisstockoeraccordingtotheCapitalAssetPricingModelIftheexpectedvalueofthestockis$100,whatpriceshouldthestockbesellingfortoday.AccordingtotheCAPMpricingequation,thestockshouldoeranexpectedrateofreturnofrf+[3(rmrf)=+=0.125or%.Thestockshouldbesellingforitsexpectedpresentvalue,whichisequalto100/=$.Consumer’spSlursAgoodcanbeproducedinacompetitiveindustryatacostof$10perunit.Thereare100consumersareeachwillingtopay$12eachtoconsumeasingleunitofthegood(additionalunitshavenovaluetothem.)WhatistheequilibriumpriceandquantitysoldThegovernmentimposesataxof$1onthegood.Whatisthedeadweightlossofthistax.Theequilibriumpriceis$10andthequantitysoldis100units.Ifthetaxisimposed,thepricerisesto$11,but100unitsofthegoodwillstillbesold,sothereisnodeadweightloss.SupposethatthedemandcurveisgivenbyD(p)=10p.Whatisthegrossbenetfromconsuming6unitsofthegood.Wewanttocomputetheareaunderthedemandcurvetotheleftofthequantity6.Breakthisupintotheareaofatrianglewithabaseof6andaheightof6andarectanglewithbase6andheight4.Applyingtheformulasfromhighschoolgeometry,thetrianglehasarea18andtherectanglehasarea24.Thusgrossbenetis42.Intheaboveexample,ifthepricechangesfrom4to6,whatisthechangeinconsumer’ssurplus.Whenthepriceis4,theconsumer’ssurplusisgivenbytheareaofatrianglewithabaseof6andaheightof6;.,theconsumer’ssurWhenthepriceis6,thetrianglehasabaseof4andaheightof4,givinganareaof8.Thusthepricechangehasreducedconsumer’ssurplusb$10.Supposethataconsumerisconsuming10unitsofadiscretegoodandthepriceincreasesfrom$5perunitto$6.However,afterthepricechangetheconsumercontinuestoconsume10unitsofthediscretegood.Whatisthelossintheconsumer’ssurplusfromthispricechange.Tendollars.Sincethedemandforthediscretegoodhasn’tchangthathashappenedisthattheconsumerhashadtoreducehisexpenditureonothergoodsbytendollars.MarketDemandIfthemarketdemandcurveisD(p)=100.5p,whatistheinversedemandcurve.TheinversedemandcurveisP(q)=2002q.Anaddict’sdemandfunctionforadrugbmeavyeryinelastic,butthemarketdemandfunctionmightbequiteelastic.Howcanthisbe.Thedecisionaboutwhethertoconsumethedrugatallcouldwellbepricesensitive,sotheadjustmentofmarketdemandontheextensivemarginwouldcontributetotheelasticityofthemarketdemand.IfD(p)=122p,whatpricewillmaximizerevenue.RevenueisR(p)=12p2p2,whichismaximizedatp=3.SupposethatthedemandcurveforagoodisgivenbyD(p)=100/p.Whatpricewillmaximizerevenue.RevenueispD(p)=100,regardlessoftheprice,soallpricesmaximizerevenue.TrueorfalseInatwogoodmodelifonegoodisaninferiorgoodtheothergoodmustbealuxurygood..True.Theweightedaverageoftheincomeelasticitiesmustbe1,soifonegoodhasanegativeincomeelasticity,theothergoodmusthaveanelasticitygreaterthan1togettheaveragetobe1.EquilibriumWhatistheeectofasubsidyinamarketwithahorizontalsupplycurveWithaverticalsupplycurve.Theentiresubsidygetspassedalongtotheconsumersifthesupplycurveisat,butthesubsidyistotallyreceivedbytheproducerswhenthesupplycurveisvertical.Supposethatthedemandcurveisverticalwhilethesupplycurveslopesupward.Ifataxisimposedinthismarketwhoendsuppayingit.Theconsumer.Supposethatallconsumersviewredpencilsandbluepencilsasperfectsubstitutes.Supposethatthesupplycurveforredpencilsisupwardsloping.LetthepriceofredpencilsandbluepencilsbePandM-.Whatwouldhappenifthegovernmentputataxonlyonredpencils.Inthiscasethedemandcurveforredpencilsishorizontalatthepricei-i,sincethatisthemostthattheywouldbewillingtopayforaredpencil.Thus,ifataxisimposedonredpencils,consumerswillenduppaying:forthem,sotheentireamountofthetaxwillendupbeingbornebytheproducers(ifanyredpencilsaresoldataH-itcouldbethatthetaxwouldinducetheproducertogetoutoftheredpencilbusiness).TheUnitedStatesimportsabouthalfofitspetroleumneeds.SupposethattherestoftheoilproducersarewillingtosupplyasmuchoilastheUnitedStateswantsataconstantpriceof$25abarrel.Whatwouldhappentothepriceofdomesticoilifataxof$5abarrelwereplacedonforeignoil.Herethesupplycurveofforeignoilisatat$25.Thusthepricetotheconsumersmustrisebythe$5amountofthetax,sothatthenetpricetotheconsumersbecomes$30.Sinceforeignoilanddomesticoilareperfectsubstitutesasfarastheconsumersareconcerned,thedomesticproducerswillselltheiroilfor$30aswellandgetawindfallgainof$5perbarrel.Supposethatthesupplycurveisvertical.Whatisthedeadweightlossofataxinthismarket.Zero.Thedeadweightlossmeasuresthevalueoflostoutput.Sincethesameamountissuppliedbeforeandafterthetax,thereisnodeadweightloss.Putanotherway:thesuppliersarepayingtheentireamountofthetax,andeverythingtheypaygoestothegovernment.Theamountthatthesupplierswouldpaytoavoidthetaxissimplythetaxrevenuethegovernmentreceives,sothereisnoexcessburdenofthetax.Considerthetaxtreatmentofborrowingandlendingdescribedinthetext.Howmuchrevenuedoesthistaxsystemraiseifborrowersandlendersareinthesametaxbracket.Zerorevenue.Doessuchataxsystemraiseapositiveornegativeamountofrevenuewhenti<b.Itraisesnegativerevenue,sinceinthiscasewehaveanetsubsidyofborrowing.AuctionsConsideranauctionofantiquequiltstocollectors.Isthisaprivate-valueoracommon-valueauction.Sincethecollectorslikelyhavetheirownvaluesforthequilts,anddonparticularlycareabouttheotherbiddersvalues,itisaprivate-valueauction.Supposethatthereareonlytwobidderswithvaluesof$8and$10foranitemwithabidincrementof$1.Whatshouldthereservationpricebeinaprot-maximizingEnglishauction.Followingtheanalysisinthetext,therearefourequallylikelycongurationsofbidders:(8,8),(8,10),(10,8),and(10,10).Withzeroreservationprice,theoptimalbidswillbe(8,9,9,10),resultinginexpectedprotof$9.Theonlycandidateforareservationpriceis$10,whichyieldsexpectedprotof30/4=$7.50.Hencezeroisaprot-maximizingreservationpriceinthisauction.SupposethatwehavetwocopiesofIntermediateMicroeconomicstoselltothree(enthusiastic)students.Howcanweuseasealed-bidauctionthatwillguaranteethatthebidderswiththetwohighestvaluesgetthebooks.Haveeachpersonwritedownavalue,thenawardthetwobookstothestudentswiththetwohighestvalues,butjustchargethemthebidofthethirdhigheststudent.ConsidertheUcomexampleinthetext.WastheauctiondesignecientDiditmaximizeprots.Itwasecientinthesensethatitawardedthelicensetothermthatvalueditmosthighly.Butittookayearforthistohappen,whichisinecient.AVickreyauctionoranEnglishauctionwouldhaveachievedthesameresultmorequickly.AgametheoristllsajarwithpenniesandauctionsitoontherstdayofclassusinganEnglishauction.Isthisaprivate-valueoracommon-valueauctionDoyouthinkthewinningbidderusuallymakesaprot.Thisisacommon-valueauctionsincethevalueoftheprizeisthesametoallbidders.Normally,thewinningbidderoverestimatesthenumberofpenniesinthejar,illustratingthewinner'scurse.TechnologyConsidertheproductionfunctionf(xi,K2)=x：x9.Doesthisexhibitconstant,increasing,ordecreasingreturnstoscale.Increasingreturnstoscale.］門I1Considertheproductionfunctionf(x1,x2)=4xr\D.Doesthisexhibitconstant,increasing,ordecreasingreturnstoscale.Decreasingreturnstoscale.TheCobb-Douglasproductionfunctionisgivenbyf(x1,x2)=Aq.Itturnsoutthatthetypeofreturnstoscaleofthisfunctionwilldependonthemagnitudeofa+b.Whichvaluesofa+bwillbeassociatedwiththedierentkindsofreturnstoscale.Ifa+b=1,wehaveconstantreturnstoscale,a+b<1givesdecreasingreturnstoscale,anda+b>1givesincreasingreturnstoscale.Thetechnicalrateofsubstitutionbetweenfactorsx2andx1is4.Ifyoudesiretoproducethesameamountofoutputbutcutyouruseofx1by3units,howmanymoreunitsofx2willyouneed.4x3=12units.TrueorfalseIfthelawofdiminishingmarginalproductdidnothold,theworld’sfoodsupplycouldgrboewninaowerpot..True.Inaproductionprocessisitpossibletohavedecreasingmarginalproductinaninputandyetincreasingreturnstoscale.Yes.ProtMaximizationIntheshortrun,ifthepriceofthexedfactorisincreased,whatwillhappentoprots.Protswilldecrease.Ifarmhadeverywhereincreasingreturnstoscale,whatwouldhappentoitsprotsifpricesremainedxedandifitdoubleditsscaleofoperation.Protwouldincrease,sinceoutputwouldgoupmorethanthecostoftheinputs.Ifarmhaddecreasingreturnstoscaleatalllevelsofoutputanditdividedupintotwoequal-sizesmallerrms,whatwouldhappentoitsoverallprots.Ifthermreallyhaddecreasingreturnstoscale,dividingthescaleofallinputsby2wouldproducemorethanhalfasmuchoutput.Thusthesubdividedrmwouldmakemoreprotsthanthebigrm.Thisisoneargumentwhyhavingeverywheredecreasingreturnstoscaleisimplausible.Agardenerexclaims:“Foornly$1inseedsI’vgerownover$20inpro-duce!”Besidesthefactthatmostoftheproduceisintheformofzucchini,whatotherobservationswouldacynicaleconomistmakeaboutthissituation.Thegardenerhasignoredopportunitycosts.Inordertoaccuratelyaccountforthetruecosts,thegardenermustincludethecostofherowntimeusedintheproductionofthecrop,evenifnoexplicitwagewaspaid.Ismaximizingarm’psrotsalwaysidenticaltomaximizingtherm’sstockmarketvalue.Notingeneral.Forexample,considerthecaseofuncertainty.IfpMP1>w1,thenshouldthermincreaseordecreasetheamountoffactor1inordertoincreaseprots.Increase.Supposearmismaximizingprotsintheshortrunwithvariablefactorx1andxedfactorx2.Ifthepriceofx2goesdown,whathappenst