曼昆宏观经济经济学第九版英文原版答案9.docx

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

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.ThenumberofeffectiveworkersisdefinedasLE（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,andthisleadstoaconstantlevelofoutputpereffectiveworker.Giventhatthegrowthrateofoutputpereffectiveworkeriszero,thismeansthegrowthrateofoutputisequaltothegrowthrateofeffectiveworkers（LE）.Weknowlaborgrowsattherateofpopulationgrowthnandtheefficiencyoflabor（E）growsatrateg.Therefore,outputgrowsatraten+g.Givenoutputgrowsatraten+gandlaborgrowsatraten,outputperworkermustgrowatrateg.ThisfollowsfromtherulethatthegrowthrateofY/LisequaltothegrowthrateofYminusthegrowthrateofL.

b.FirstfindtheoutputpereffectiveworkerproductionfunctionbydividingbothsidesoftheproductionfunctionbythenumberofeffectiveworkersLE:

Tosolveforcapitalpereffectiveworker,westartwiththesteadystatecondition:

Δk=sf（k）–（δ+n+g）k=0.

Nowsubstituteinthegivenparametervaluesandsolveforcapitalpereffectiveworker（k）:

Substitutethevalueforkbackintothepereffectiveworkerproductionfunctiontofindoutputpereffectiveworkerisequalto2.Themarginalproductofcapitalisgivenby

Substitutethevalueforcapitalpereffectiveworkertofindthemarginalproductofcapitalisequalto1/12.

c.AccordingtotheGoldenRule,themarginalproductofcapitalisequalto（δ+n+g）or0.06.Inthecurrentsteadystate,themarginalproductofcapitalisequalto1/12or0.083.Therefore,wehavelesscapitalpereffectiveworkerincomparisontotheGoldenRule.Asthelevelofcapitalpereffectiveworkerrises,themarginalproductofcapitalwillfalluntilitisequalto0.06.Toincreasecapitalpereffectiveworker,theremustbeanincreaseinthesavingrate.

d.DuringthetransitiontotheGoldenRulesteadystate,thegrowthrateofoutputperworkerwillincrease.Inthesteadystate,outputperworkergrowsatrateg.Theincreaseinthesavingratewillincreaseoutputpereffectiveworker,andthiswillincreaseoutputpereffectiveworker.Inthenewsteadystate,outputpereffectiveworkerisconstantatanewhigherlevel,andoutputperworkerisgrowingatrateg.Duringthetransition,thegrowthrateofoutputperworkerjumpsup,andthentransitionsbackdowntorateg.

3.Tosolvethisproblem,itisusefultoestablishwhatweknowabouttheU.S.economy:

•ACobb–Douglasproductionfunctionhastheformy=kα,whereαiscapital’sshareofincome.Thequestiontellsusthatα=0.3,soweknowthattheproductionfunctionisy=k0.3.

•Inthesteadystate,weknowthatthegrowthrateofoutputequals3percent,soweknowthat（n+g）=0.03.

•Thedepreciationrateδ=0.04.

•Thecapital–outputratioK/Y=2.5.Becausek/y=[K/（LE）]/[Y/（LE）]=K/Y,wealsoknowthatk/y=2.5.（Thatis,thecapital–outputratioisthesameintermsofeffectiveworkersasitisinlevels.）

a.Beginwiththesteady-statecondition,sy=（δ+n+g）k.Rewritingthisequationleadstoaformulaforsavinginthesteadystate:

s=（δ+n+g）（k/y）.

Plugginginthevaluesestablishedabove:

s=（0.04+0.03）（2.5）=0.175.

Theinitialsavingrateis17.5percent.

b.WeknowfromChapter3thatwithaCobb–Douglasproductionfunction,capital’sshareofincomeα=MPK（K/Y）.Rewriting,wehave

MPK=α/（K/Y）.

Plugginginthevaluesestablishedabove,wefind

MPK=0.3/2.5=0.12.

c.WeknowthatattheGoldenRulesteadystate:

MPK=（n+g+δ）.

Plugginginthevaluesestablishedabove:

MPK=（0.03+0.04）=0.07.

AttheGoldenRulesteadystate,themarginalproductofcapitalis7percent,whereasitis12percentintheinitialsteadystate.Hence,fromtheinitialsteadystateweneedtoincreasektoachievetheGoldenRulesteadystate.

d.WeknowfromChapter3thatforaCobb–Douglasproductionfunction,MPK=α（Y/K）.Solvingthisforthecapital–outputratio,wefind

K/Y=α/MPK.

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,wecanconcludethatin