曼昆宏观经济经济学第九版英文原版答案9.docx
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曼昆宏观经济经济学第九版英文原版答案9
曼昆-宏观经济经济学第九版-英文原版答案9
AnswerstoTextbookQuestionsandProblems
CHAPTER9 EconomicGrowthII:
Technology,Empirics,andPolicy
QuestionsforReview
1.IntheSolowmodel,wefindthatonlytechnologicalprogresscanaffectthesteady-staterateofgrowthinincomeperworker.Growthinthecapitalstock(throughhighsaving)hasnoeffectonthesteady-stategrowthrateofincomeperworker;neitherdoespopulationgrowth.Buttechnologicalprogresscanleadtosustainedgrowth.
2.Inthesteadystate,outputperpersonintheSolowmodelgrowsattherateoftechnologicalprogressg.Capitalperpersonalsogrowsatrateg.Notethatthisimpliesthatoutputandcapitalpereffectiveworkerareconstantinsteadystate.IntheU.S.data,outputandcapitalperworkerhavebothgrownatabout2percentperyearforthepasthalf-century.
3.TodecidewhetheraneconomyhasmoreorlesscapitalthantheGoldenRule,weneedtocomparethemarginalproductofcapitalnetofdepreciation(MPK–δ)withthegrowthrateoftotaloutput(n+g).ThegrowthrateofGDPisreadilyavailable.Estimatingthenetmarginalproductofcapitalrequiresalittlemoreworkbut,asshowninthetext,canbebackedoutofavailabledataonthecapitalstockrelativetoGDP,thetotalamountofdepreciationrelativetoGDP,andcapital’sshareinGDP.
4.Economicpolicycaninfluencethesavingratebyeitherincreasingpublicsavingorprovidingincentivestostimulateprivatesaving.Publicsavingisthedifferencebetweengovernmentrevenueandgovernmentspending.Ifspendingexceedsrevenue,thegovernmentrunsabudgetdeficit,whichisnegativesaving.Policiesthatdecreasethedeficit(suchasreductionsingovernmentpurchasesorincreasesintaxes)increasepublicsaving,whereaspoliciesthatincreasethedeficitdecreasesaving.Avarietyofgovernmentpoliciesaffectprivatesaving.Thedecisionbyahouseholdtosavemaydependontherateofreturn;thegreaterthereturntosaving,themoreattractivesavingbecomes.Taxincentivessuchastax-exemptretirementaccountsforindividualsandinvestmenttaxcreditsforcorporationsincreasetherateofreturnandencourageprivatesaving.
5.Thelegalsystemisanexampleofaninstitutionaldifferencebetweencountriesthatmightexplaindifferencesinincomeperperson.CountriesthathaveadoptedtheEnglishstylecommonlawsystemtendtohavebetterdevelopedcapitalmarkets,andthisleadstomorerapidgrowthbecauseitiseasierforbusinessestoobtainfinancing.Thequalityofgovernmentisalsoimportant.Countrieswithmoregovernmentcorruptiontendtohavelowerlevelsofincomeperperson.
6.Endogenousgrowththeoriesattempttoexplaintherateoftechnologicalprogressbyexplainingthedecisionsthatdeterminethecreationofknowledgethroughresearchanddevelopment.Bycontrast,theSolowmodelsimplytookthisrateasexogenous.IntheSolowmodel,thesavingrateaffectsgrowthtemporarily,butdiminishingreturnstocapitaleventuallyforcetheeconomytoapproachasteadystateinwhichgrowthdependsonlyonexogenoustechnologicalprogress.Bycontrast,manyendogenousgrowthmodelsinessenceassumethatthereareconstant(ratherthandiminishing)returnstocapital,interpretedtoincludeknowledge.Hence,changesinthesavingratecanleadtopersistentgrowth.
ProblemsandApplications
1.a.IntheSolowmodelwithtechnologicalprogress,yisdefinedasoutputpereffectiveworker,andkisdefinedascapitalpereffectiveworker.ThenumberofeffectiveworkersisdefinedasL⨯E(orLE),whereListhenumberofworkers,andEmeasurestheefficiencyofeachworker.Tofindoutputpereffectiveworkery,dividetotaloutputbythenumberofeffectiveworkers:
b.Tosolveforthesteady-statevalueofyasafunctionofs,n,g,andδ,webeginwiththeequationforthechangeinthecapitalstockinthesteadystate:
Δk=sf(k)–(δ+n+g)k=0.
Theproductionfunction
canalsoberewrittenasy2=k.Pluggingthisproductionfunctionintotheequationforthechangeinthecapitalstock,wefindthatinthesteadystate:
sy–(δ+n+g)y2=0.
Solvingthis,wefindthesteady-statevalueofy:
y*=s/(δ+n+g).
c.Thequestionprovidesuswiththefollowinginformationabouteachcountry:
Atlantis:
s=0.28Xanadu:
s=0.10
n=0.01n=0.04
g=0.02g=0.02
δ=0.04δ=0.04
Usingtheequationfory*thatwederivedinpart(a),wecancalculatethesteady-statevaluesofyforeachcountry.
Developedcountry:
y*=0.28/(0.04+0.01+0.02)=4
Less-developedcountry:
y*=0.10/(0.04+0.04+0.02)=1
2.a.Inthesteadystate,capitalpereffectiveworkerisconstant,andthisleadstoaconstantlevelof
WecansolvefortheGoldenRulecapital–outputratiousingthisequation.Ifwepluginthevalue0.07fortheGoldenRulesteady-statemarginalproductofcapital,andthevalue0.3forα,wefind
K/Y=0.3/0.07=4.29.
IntheGoldenRulesteadystate,thecapital–outputratioequals4.29,comparedtothecurrentcapital–outputratioof2.5.
e.Weknowfrompart(a)thatinthesteadystate
s=(δ+n+g)(k/y),
wherek/yisthesteady-statecapital–outputratio.Intheintroductiontothisanswer,weshowedthatk/y=K/Y,andinpart(d)wefoundthattheGoldenRuleK/Y=4.29.Plugginginthisvalueandthoseestablishedabove:
s=(0.04+0.03)(4.29)=0.30.
ToreachtheGoldenRulesteadystate,thesavingratemustrisefrom17.5to30percent.Thisresultimpliesthatifwesetthesavingrateequaltothesharegoingtocapital(30percent),wewillachievetheGoldenRulesteadystate.
4.a.Inthesteadystate,weknowthatsy=(δ+n+g)k.Thisimpliesthat
k/y=s/(δ+n+g).
Sinces,δ,n,andgareconstant,thismeansthattheratiok/yisalsoconstant.Sincek/y=[K/(LE)]/[Y/(LE)]=K/Y,wecanconcludethatinthesteadystate,thecapital–outputratioisconstant.
b.Weknowthatcapital’sshareofincome=MPK⨯(K/Y).Inthesteadystate,weknowfrompart(a)thatthecapital–outputratioK/Yisconstant.WealsoknowfromthehintthattheMPKisafunctionofk,whichisconstantinthesteadystate;thereforetheMPKitselfmustbeconstant.Thus,capital’sshareofincomeisconstant.Labor’sshareofincomeis1–[Capital’sShare].Hence,ifcapital’sshareisconstant,weseethatlabor’sshareofincomeisalsoconstant.
c.Weknowthatinthesteadystate,totalincomegrowsatn+g,definedastherateofpopulationgrowthplustherateoftechnologicalchange.Inpart(b)weshowedthatlabor’sandcapital’sshareofincomeisconstant.Ifthesharesareconstant,andtotalincomegrowsattheraten+g,thenlaborincomeandcapitalincomemustalsogrowattheraten+g.
d.DefinetherealrentalpriceofcapitalRas
R=TotalCapitalIncome/CapitalStock
=(MPK⨯K)/K
=MPK.
Weknowthatinthesteadystate,theMPKisconstantbecausecapitalpereffectiveworkerkisconstant.Therefore,wecanconcludethattherealrentalpriceofcapitalisconstantinthesteadystate.
Toshowthattherealwagewgrowsattherateoftechnologicalprogressg,define
TLI=TotalLaborIncome
L=LaborForce
Usingthehintthattherealwageequalstotallaborincomedividedbythelaborforce:
w=TLI/L.
Equivalently,
wL=TLI.
Intermsofpercentagechanges,wecanwritethisas
Δw/w+ΔL/L=ΔTLI/TLI.
Thisequationsaysthatthegrowthrateoftherealwageplusthegrowthrateofthelaborforceequalsthegrowthrateoftotallaborincome.Weknowthatthelaborforcegrowsatraten,and,frompart(c),weknowthattotallaborincomegrowsatraten+g.We,therefore,concludethattherealwagegrowsatrateg.
5.a.Theperworkerproductionfunctionis
F(K,L)/L=AKαL1–α/L=A(K/L)α=Akα
b.Inthesteadystate,Δk=sf(k)–(δ+n+g)k=0.Hence,sAkα=(δ+n+g)k,or,afterrearranging:
.
Pluggingintotheper-workerproductionfunctionfrompart(a)gives
.
Thus,theratioofsteady-stateincomeperworkerinRichlandtoPoorlandis
c.Ifαequals1/3,thenRichlandshouldbe41/2,ortwotimes,richerthanPoorland.
d.If
=16,thenitmustbethecasethat
whichinturnrequiresthatαequals2/3.Hence,iftheCobb–Douglasproductionfunctionputs2/3oftheweightoncapitalandonly1/3onlabor,thenwecanexplaina16-folddifferenceinlevelsofincomeperworker.Onewaytojustifythismightbetothinkaboutcapitalmorebroadlytoincludehumancapital—whichmustalsobeaccumulatedthroughinvestment,muchinthewayoneaccumulatesphysicalcapital.
6.HowdodifferencesineducationacrosscountriesaffecttheSolowmodel?
Educationisonefactoraffectingtheefficiencyoflabor,whichwedenotedbyE.(Otherfactorsaffectingtheefficiencyoflaborincludelevelsofhealth,skill,andknowledge.)Sincecountry1hasamorehighlyeducatedlaborforcethancountry2,eachworkerincountry1ismoreefficient.Thatis,E1>E2.Wewillassumethatbothcountriesareinsteadystate.
a.IntheSolowgrowthmodel,therateofgrowthoftotalincomeisequalton+g,whichisindependentoftheworkforce’slevelofeducation.Thetwocountrieswill,thus,havethesamerateofgrowthoftotalincomebecausetheyhavethesamerateofpopulationgrowthandthesamerateoftechnologicalprogress.
b.Becausebothcountrieshavethesamesavingrate,thesamepopulationgrowthrate,andthesamerateoftechnologicalprogress,weknowthatthetwocountrieswillconvergetothesamesteady-statelevelofcapitalpereffectiveworkerk*.ThisisshowninFigure9-1.
Hence,outputpereffectiveworkerinthesteadystate,whichisy*=f(
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