中级宏观经济学曼昆英文版第七版第三章习题答案

1、CHAPTERNationalIncome:WhereItComesFromandWhereItGoesQuestionsforReviewThefactorsofproductionandtheproductiontechnologydeterminetheamountofoutputaneconomycanproduce.Thefactorsofproductionaretheinputsusedtoproducegoodsandservices:themostimportantfactorsarecapitalandlabor.Theproductiontechnologydetermi

2、neshowmuchoutputcanbeproducedfromanygivenamountsoftheseinputs.Anincreaseinoneofthefactorsofproductionoranimprovementintechnologyleadstoanincreaseintheeconomysoutput.Whenafirmdecideshowmuchofafactorofproductiontohireordemand,itconsidershowthisdecisionaffectsprofits.Forexample,hiringanextraunitoflabor

3、increasesoutputandthereforeincreasesrevenue;thefirmcomparesthisadditionalrevenuetotheadditionalcostfromthehigherwagebill.Theadditionalrevenuethefirmreceivesdependsonthemarginalproductoflabor(MPL)andthepriceofthegoodproduced(P).AnadditionalunitoflaborproducesMPLunitsofadditionaloutput,whichsellsforPd

4、ollarsperunit.Therefore,theadditionalrevenuetothefirmisPxMPL.ThecostofhiringtheadditionalunitoflaboristhewageW.Thus,thishiringdecisionhasthefollowingeffectonprofits:AProfit=ARevenue一ACost=(PxMPL)W.Iftheadditionalrevenue,PxMPL,exceedsthecost(W)ofhiringtheadditionalunitoflabor,thenprofitincreases.Thef

5、irmwillhirelaboruntilitisnolongerprofitabletodoso一thatis,untiltheMPLfallstothepointwherethechangeinprofitiszero.Intheequationabove,thefirmhireslaboruntiAprofit=0,whichiswhenPxMPL)=W.Thisconditioncanberewrittenas:MPL=W/P.Therefore,acompetitiveprofit-maximizingfirmhireslaboruntilthemarginalproductofla

6、borequalstherealwage.Thesamelogicappliestothefirmsdecisionregardinghowmuchcapitaltohire:thefirmwillhirecapitaluntilthemarginalproductofcapitalequalstherealrentalprice.Aproductionfunctionhasconstantreturnstoscaleifanequalpercentageincreaseinallfactorsofproductioncausesanincreaseinoutputofthesameperce

7、ntage.Forexample,ifafirmincreasesitsuseofcapitalandlaborby50percent,andoutputincreasesby50percent,thentheproductionfunctionhasconstantreturnstoscale.Iftheproductionfunctionhasconstantreturnstoscale,thentotalincome(orequivalently,totaloutput)inaneconomyofcompetitiveprofit-maximizingfirmsisdividedbetw

8、eenthereturntolabor,MPLxL,andthereturntocapital,MPKxK.Thatis,underconstantreturnstoscale,economicprofitiszero.ACobb-DouglasproductionfunctionfunctionhastheformF(K,L)=AKaLi_a.Thetextshowedthattheparameteragivescapitalsshareofincome.(Sinceincomeequalsoutputfortheoveralleconomy,itisalsocapitalsshareofo

9、utput.)Soifcapitalearnsone-fourthoftotalincome,thena=0.25.Hence,F(K,L)=AK0.25L0.75.Consumptiondependspositivelyondisposableincome一theamountofincomeafteralltaxeshavebeenpaid.Thehigherdisposableincomeis,thegreaterconsumptionis.Thequantityofinvestmentgoodsdemandeddependsnegativelyontherealinterestrate.

10、Foraninvestmenttobeprofitable,itsreturnmustbegreaterthanitscost.Becausetherealinterestratemeasuresthecostoffunds,ahigherrealinterestratemakesitmorecostlytoinvest,sothedemandforinvestmentgoodsfalls.11 AnswerstoTextbookQuestionsandProblemsChapter3NationalIncome:WhereItComesFromandWhereItGoes Governmen

11、tpurchasesareameasureofthedollarvalueofgoodsandservicespurchaseddirectlybythegovernment.Forexample,thegovernmentbuysmissilesandtanks,buildsroads,andprovidesservicessuchasairtrafficcontrol.Alloftheseactivi-tiesarepartofGDP.Transferpaymentsaregovernmentpaymentstoindividualsthatarenotinexchangeforgoods

12、orservices.Theyaretheoppositeoftaxes:taxesreducehouseholddisposableincome,whereastransferpaymentsincreaseit.ExamplesoftransferpaymentsincludeSocialSecuritypaymentstotheelderly,unemploymentinsurance,andveteransbenefits.Consumption,investment,andgovernmentpurchasesdeterminedemandfortheeconomysoutput,w

13、hereasthefactorsofproductionandtheproductionfunctiondeterminethesupplyofoutput.Therealinterestrateadjuststoensurethatthedemandfortheeconomysgoodsequalsthesupply.Attheequilibriuminterestrate,thedemandforgoodsandservicesequalsthesupply.Whenthegovernmentincreasestaxes,disposableincomefalls,andtherefore

14、consumptionfallsaswell.Thedecreaseinconsumptionequalstheamountthattaxesincreasemulti-pliedbythemarginalpropensitytoconsume(MPC).ThehighertheMPCis,thegreateristhenegativeeffectofthetaxincreaseonconsumption.Becauseoutputisfixedbythefactorsofproductionandtheproductiontechnology,andgovernmentpurchasesha

15、venotchanged,thedecreaseinconsumptionmustbeofsetbyanincreaseininvestment.Forinvestmenttorise,therealinterestratemustfall.Therefore,ataxincreaseleadstoadecreaseinconsumption,anincreaseininvestment,andafallintherealinterestrate.ProblemsandApplicationsa.Accordingtotheneoclassicaltheoryofdistribution,th

16、erealwageequalsthemarginalproductoflabor.Becauseofdiminishingreturnstolabor,anincreaseinthelaborforcecausesthemarginalproductoflabortofall.Hence,therealwagefalls.Therealrentalpriceequalsthemarginalproductofcapital.Ifanearthquakedestroyssomeofthecapitalstock(yetmiraculouslydoesnotkillanyoneandlowerth

17、elaborforce),themarginalproductofcapitalrisesand,hence,therealrentalpricerises.Ifatechnologicaladvanceimprovestheproductionfunction,thisislikelytoincreasethemarginalproductsofbothcapitalandlabor.Hence,therealwageandtherealrentalpricebothincrease.Aproductionfunctionhasdecreasingreturnstoscaleifanequa

18、lpercentageincreaseinallfactorsofproductionleadstoasmallerpercentageincreaseinoutput.Forexample,ifwedoubletheamountsofcapitalandlabor,andoutputlessthandoubles,thentheproductionfunctionhasdecreasingreturnstoscale.Thismayhappenifthereisafixedfactorsuchaslandintheproductionfunction,andthisfixedfactorbe

19、comesscarceastheeconomygrowslarger.Aproductionfunctionhasincreasingreturnstoscaleifanequalpercentageincreaseinallfactorsofproductionleadstoalargerpercentageincreaseinoutput.Forexample,ifdoublinginputsofcapitalandlabormorethandoublesoutput,thentheproductionfunctionhasincreasingreturnstoscale.Thismayh

20、appenifspecializationoflaborbecomesgreaterasthepopulationgrows.Forexample,ifonlyoneworkerbuildsacar,thenittakeshimalongtimebecausehehastolearnmanydifferentskills,andhemustconstantlychangetasksandtools.Butifmanyworkersbuildacar,theneachonecanspecializeinaparticulartaskandbecomeveryfastatit.a.ACobb-Do

21、uglasproductionfunctionhastheformY=AKL1-aThetextshowedthatthemarginalproductsfortheCobb-Douglasproductionfunctionare:MPL=(1OYIL.MPK=aYIK.Competitiveprofit-maximizingfirmshirelaboruntilitsmarginalproductequalstherealwage,andhirecapitaluntilitsmarginalproductequalstherealrentalrate.Usingthesefactsandt

22、heabovemarginalproductsfortheCobb-Douglasproductionfunction,wefind:W/P=MPL=(1a)Y/L.R/P=MPK=aY/K.Rewritingthis:(W/P)L=MPLxL=(1a)Y.(R/P)K=MPKxK=aY.b.Notethattheterms(W/P)Land(R/P)Karethewagebillandtotalreturntocapi-tal,respectively.Giventhatthevalueofa=0.3,thentheaboveformulasindicatethatlaborreceives

23、70percentoftotaloutput(orincome),whichis(1一0.3),andcapitalreceives30percentoftotaloutput(orincome).Todeterminewhathappenstototaloutputwhenthelaborforceincreasesby10percent,considertheformulafortheCobbDouglasproductionfunction:Y=AKaL1-a.LetYequaltheinitialvalueofoutputandYequalfinaloutput.Weknowthata

24、=0.3.WealsoknowthatlaborLincreasesby10percent:=AKO%.7.1=AKo.3(1.1L)o.7.2NotethatwemultipliedLby1.1toreflectthe10-percentincreaseinthelaborforce.Tocalculatethepercentagechangeinoutput,divideYbyY:AKo.3(1.1L)0.7AK0.3L0.7=(1.1)0.7=1.069.Thatis,outputincreasesby6.9percent.Todeterminehowtheincreaseinthela

25、borforceaffectstherentalpriceofcapital,considertheformulafortherealrentalpriceofcapitalR/P:R/P=MPK=aAKa-L】-a.Weknowthata=0.3.Wealsoknowthatlabor(L)increasesby10percent.Let(R/P)equaltheinitialvalueoftherentalpriceofcapital,and(R/P)equalthefinalrentalpriceofcapitalafterthelaborforceincreasesby10percen

26、t.Tofind(R/P)2,multiplyLby1.1toreflectthe10-percentincreaseinthelaborforce:(R/P)=0.3AK-o.7Lo.7.1(R/P)=0.3AK-o.7(1.1L)o.7.2Therentalpriceincreasesbytheratio(R/P=0.3AK-o.7(1.1L)o.7(R/P)10.3AK-o.7Lo.7=(1.1)0.7=1.069.Sotherentalpriceincreasesby6.9percent.Todeterminehowtheincreaseinthelaborforceaffectsth

27、erealwage,considertheformulafortherealwagW/P:W/P=MPL=(1a)AKL-.Weknowthata=0.3.Wealsoknowthatlabor(L)increasesby10percent.Let(W/P)equaltheinitialvalueoftherealwageand(W/P)equalthefinalvalueoftherealwage.Tofind(W/P),multiplyLby1.1toreflectthe10-percentincreaseinthelaborforce:(W/P)=(10.3)AK0.3L-0.3.1(W

28、/P)=(10.3)AKo.3(1.1L)-0.3.2Tocalculatethepercentagechangeintherealwage,divide(W/P)by(10.3)AKo.3(1.1L)-o.3(1一0.3)AK0.3L-0.3(W/P)i(W/P)1：(w/p)2=(1.1)-0.3=0.972.Thatis,therealwagefallsby2.8percent.Wecanusethesamelogicasinpart(b)toset=AK0.3L0.7.1=A(1.1K)0.3L0.7.2Therefore,wehave:=A(1.1K)o.3L0.7AK0.3L0.7

29、1=(1.1)0.3=1.029.Thisequationshowsthatoutputincreasesbyabout3percent.Noticethata0.5meansthatproportionalincreasestocapitalwillincreaseoutputbylessthanthesameproportionalincreasetolabor.Againusingthesamelogicasinpart(b)forthechangeintherealrentalpriceofcapital:(R/P=0.3A(1.1K)-0.7L0.7(R/P)0.3AK-0.7L0.

30、71=(1.1)-0.7=0.935.Therealrentalpriceofcapitalfallsby6.5percentbecausetherearediminishingreturnstocapital;thatis,whencapitalincreases,itsmarginalproductfalls.Finally,thechangeintherealwageis:(W/P、=0.7A(1.1K)0.3L-0.3(W/P)0.7AK0.3L-0.3(1.1)0.31.029.Hence,realwagesincreaseby2.9percentbecausetheaddedcap

31、italincreasesthemarginalproductivityoftheexistingworkers.(Noticethatthewageandoutputhavebothincreasedbythesameamount,leavingthelaborshareunchanged一afeatureofCobb-Douglastechnologies.)Usingthesameformula,wefindthatthechangeinoutputis:Y2=(1.1A)K0.3Q0.7YAK0.3L0.71=1.1.Thisequationshowsthatoutputincreas

32、esby10percent.Similarly,therentalpriceofcapitalandtherealwagealsoincreaseby10percent:(R/P=O.3(1.1A)Ko.7Lo.7(R/P)0.3AK-0-7L0-71=1.1.(W/P)2_0.7(1.1A)K0.3L-0.3(W/P)10.7AK0.3L-0.3_1.1.LaborincomeisdefinedasWTWLTOC o 1-5 h zL=.PPLaborsshareofincomeisdefinedas(WLVWLIP丿PYIfthisratioisaboutconstantat,say,av

33、alueof0.7,thenitmustbethecasethatW/P_0.7*Y/L.Thismeansthattherealwageisroughlyproportionaltolaborproductivity.Hence,anytrendinlaborproductivitymustbematchedbyanequaltrendinrealwages一otherwise,laborssharewoulddeviatefrom0.7.Thus,thefirstfact(aconstantlaborshare)impliesthesecondfact(thetrendinrealwage

34、scloselytracksthetrendinlaborproductivity)a.Accordingtotheneoclassicaltheory,technicalprogressthatincreasesthemarginalproductoffarmerscausestheirrealwagetorise.Therealwageforfarmersismeasuredasunitsoffarmoutputperworker.TherealwageisW/P,andthisisequalto($/worker)/($/unitoffarmoutput).Ifthemarginalpr

35、oductivityofbarbersisunchanged,thentheirrealwageisunchanged.Therealwageforbarbersismeasuredashaircutsperworker.TherealwageisW/P,andthisisequalto($/worker)/($/haircut).Ifworkerscanmovefreelybetweenbeingfarmersandbeingbarbers,thentheymustbepaidthesamewageWineachsector.IfthenominalwageWisthesameinboths

36、ectors,buttherealwageintermsoffarmgoodsisgreaterthantherealwageintermsofhaircuts,thenthepriceofhaircutsmusthaverisenrelativetothepriceoffarmgoods.WeknowthatW/P_MPLsothatW_PxMPL.ThismeansthatPMPL_PMPL,giventhatthenominalwagesarethesame.Sincethemarginalproductoflaborforbarbershasnotchangedandthemargin

37、alproductoflaborforfarmershasrisen,thepriceofahaircutmusthaverisenrelativetothepriceofthefarmoutput.Ifweputitingrowth-rateterms,thenthegrowthofthefarmprice+thegrowthofthemarginalproductofthefarmlabor_thegrowthofthehaircutprice.Bothgroupsbenefitfromtechnologicalprogressinfarming.6.a.b.c.d.Themarginal

38、productoflaborMPLisfoundbydifferentiatingtheproductionfunctionwithrespecttolabor:MPL=-dYdL1=-3K1/3H1/3L-2/3.Anincreaseinhumancapitalwillincreasethemarginalproductoflaborbecausemorehumancapitalmakesalltheexistinglabormoreproductive.ThemarginalproductofhumancapitalMPHisfoundbydifferentiatingtheproduct

39、ionfunctionwithrespecttohumancapital:MPH=dYdH1=-3K1/3L1/3H-2/3.Anincreaseinhumancapitalwilldecreasethemarginalproductofhumancapitalbecausetherearediminishingreturns.Thelaborshareofoutputistheproportionofoutputthatgoestolabor.Thetotalamountofoutputthatgoestolaboristherealwage(which,underperfectcompet

40、ition,equalsthemarginalproductoflabor)timesthequantityoflabor.Thisquantityisdividedbythetotalamountofoutputtocomputethelaborshare:TR(1K13H13L-23)LLaborShare=3K13H13L13=1.3Wecanusethesamelogictofindthehumancapitalshare:QK1/3L1/3H-2/)HHumanCapitalShare=_3K1/3H1/3L1/31=3-solaborgetsone-thirdoftheoutput

41、-andhumancapitalgetsone-thirdoftheoutput.Sinceworkersowntheirhumancapital(wehope!)-itwillappearthatlaborgetstwo-thirdsofoutput.Theratiooftheskilledwagetotheunskilledwageis:W,“，=MPL+MPHunskillednMPL+1K1/3L-2/3H1/3十-1K1/3L1/3H-2/3=33-1K1/3L-2/3H1/33=1+攵.HNoticethattheratioisalwaysgreaterthan1becausesk

42、illedworkersgetpaidmorethanunskilledworkers.Also,whenHincreasesthisratiofallsbecausethediminishingreturnstohumancapitalloweritsreturn,whileatthesametimeincreasingthemarginalproductofunskilledworkers.IfmorecollegescholarshipsincreaseH,thenitdoesleadtoamoreegalitariansociety.Thepolicylowersthereturnst

43、oeducation,decreasingthegapbetweenthewagesofmoreandlesseducatedworkers.Moreimportantly,thepolicyevenraisestheabsolutewageofunskilledworkersbecausetheirmarginalproductriseswhenthenumberofskilledworkersrises.7.Theeffectofagovernmenttaxincreaseof$100billionon(a)publicsaving,(b)privatesaving,and(c)natio

44、nalsavingcanbeanalyzedbyusingthefollowingrelationships:NationalSaving=PrivateSaving+PublicSaving=YTC(YT)+TG=YC(YT)G.a.b.c.d.PublicSaving一Thetaxincreasecausesa1-for-1increaseinpublicsaving.Tincreasesby$100billionand,therefore,publicsavingincreasesby$100billion.PrivateSaving一Theincreaseintaxesdecrease

45、sdisposableincome,Y一T,by$100billion.Sincethemarginalpropensitytoconsume(MPC)is0.6,consumptionfallsby0.6x$100billion,or$60billion.Hence,APrivateSaving=一$1006一0.6(一$1006)=一$406.Privatesavingfalls$40billion.NationalSaving一Becausenationalsavingisthesumofprivateandpublicsaving,wecanconcludethatthe$100bil

46、liontaxincreaseleadstoa$60billionincreaseinnationalsaving.Anotherwaytoseethisisbyusingthethirdequationfornationalsavingexpressedabove,thatnationalsavingequalsY一C(Y一T)一G.The$100billiontaxincreasereducesdisposableincomeandcausesconsumptiontofallby$60billion.SinceneitherGnorYchanges,nationalsavingthusr

47、isesby$60billion.InvestmentTodeterminetheeffectofthetaxincreaseoninvestment,recallthenationalaccountsidentity:=C(YT)+I(r)+G.Rearranging,wefindC(YT)G=I(r).Theleft-handsideofthisequationisnationalsaving,sotheequationjustsaysthatnationalsavingequalsinvestment.Sincenationalsavingincreasesby$60billion,in

48、vestmentmustalsoincreaseby$60billion.Investment,SavingHowdoesthisincreaseininvestmenttakeplace?Weknowthatinvestmentdependsontherealinterestrate.Forinvestmenttorise,therealinterestratemustfall.Figure3-1illustratessavingandinvestmentasafunctionoftherealinterestrate.Thetaxincreasecausesnationalsavingto

49、rise,sothesupplycurveforloanablefundsshiftstotheright.Theequilibriumrealinterestratefalls,andinvest-mentrises.Ifconsumersincreasetheamountthattheyconsumetoday,thenprivatesavingand,therefore,nationalsavingwillfall.Weknowthisfromthedefinitionofnationalsaving:NationalSaving=PrivateSaving+PublicSaving=Y

50、-T-C(Y-T)+T-G.Anincreaseinconsumptiondecreasesprivatesaving,sonationalsavingfalls.Figure3-2illustratessavingandinvestmentasafunctionoftherealinterestrate.Ifnationalsavingdecreases,thesupplycurveforloanablefundsshiftstotheleft,therebyraisingtherealinterestrateandreducinginvestment.Figure3-29.a.Privat

51、esavingistheamountofdisposableincome,Y-T,thatisnotconsumed:Sprivate=YTC=5,000-1,000-(250+0.75(5,000-1,000)=750.Publicsavingistheamountoftaxesthegovernmenthasleftoverafteritmakesitspurchases:Spublic=TG=1,000-1,000=0.Totalsavingisthesumofprivatesavingandpublicsaving:S=Sprivate+Spublic=750+0=750.b.Thee

52、quilibriuminterestrateisthevalueofrthatclearsthemarketforloanablefunds.Wealreadyknowthatnationalsavingis750,sowejustneedtosetitequaltoinvestment:S=I750=1,000-50rSolvingthisequationforr,wefind:r=5%.CWhenthegovernmentincreasesitsspending,privatesavingremainsthesameasbefore(noticethatGdoesnotappearinth

53、eSprivateabove)whilegovernmentsavingdecreases.PuttingthenewGintotheequationsabove:Sprivate=750Spublic=TG=1,0001,250=-250.Thus,S=Sprivate+Spublic=750+(-250)=500.Onceagaintheequilibriuminterestrateclearsthemarketforloanablefunds:S=I500=1,00050rSolvingthisequationforr,wefind:r=10%.Todeterminetheeffecto

54、ninvestmentofanequalincreaseinbothtaxesandgovern-mentspending,considerthenationalincomeaccountsidentityfornationalsaving:NationalSaving=PrivateSaving+PublicSaving=Y-T-C(Y-T)+T-G.WeknowthatYisfixedbythefactorsofproduction.Wealsoknowthatthechangeinconsumptionequalsthemarginalpropensitytoconsume(MPC)ti

55、mesthechangeindisposableincome.ThistellsusthatANationalSaving=AT一(MPCx(AT)+AT一AG=AT+(MPCxAT)+0=(MPC1)AT.TheaboveexpressiontellsusthattheimpactonsavingofanequalincreaseinTandGdependsonthesizeofthemarginalpropensitytoconsume.TheclosertheMPCisto1,thesmalleristhefallinsaving.Forexample,iftheMPCequals1,t

56、henthefallinconsumptionequalstheriseingovernmentpurchases,sonationalsavingYC(YT)Gisunchanged.TheclosertheMPCisto0(andthereforethelargeristheamountsavedratherthanspentforaone-dollarchangeindisposableincome),thegreateristheimpactonsaving.BecauseweassumethattheMPCislessthan1,weexpectthatnationalsavingf

57、allsinresponsetoanequalincreaseintaxesandgovernmentspending.ThereductioninsavingmeansthatthesupplyofloanablefundscurveshiftstotheleftinFigure33.Therealinterestraterises,andinvestmentfalls.Figure33Investment,SavingI,Sa.Thedemandcurveforbusinessinvestmentshiftsouttotherightbecausethesubsidyincreasesth

58、enumberofprofitableinvestmentopportunitiesforanygiveninterestrate.Thedemandcurveforresidentialinvestmentremainsunchanged.Thetotaldemandcurveforinvestmentintheeconomyshiftsouttotherightsinceitrepresentsthesumofbusinessinvestment,whichshiftsouttotheright,andresidentialinvestment,whichisunchanged.Asare

59、sulttherealinterestraterisesThetotalquantityofinvestmentdoesnotchangebecauseitisconstrainedbytheinelasticsupplyofsavings.Theinvestmenttaxcreditleadstoariseinbusinessinvestment,butanoffsettingfallinresidentialinvestment.Thatis,thehigherinterestratemeansthatresidentialinvestmentfalls(amovementalongthe

60、curve),whereastherightwardshiftofthebusinessinvestmentcurveleadsbusinessinvestmenttorisebyanequalamount.Figure3-5showsthischange.NoteBusinessResidentialinvestmentinvestmentInthischapter,weconcludedthatanincreaseingovernmentexpendituresreducesnationalsavingandraisestheinterestrate;itthereforecrowdsou