宏觀經(jīng)濟學課件 Chapter 7

CHAPTER7EconomicGrowthI:CapitalAccumulationandPopulationGrowth?MANKIW'SMACROECONOMICSMODULESAPowerPointTutorialToAccompany

MACROECONOMICS,7th.EditionN.GregoryMankiwTutorialwrittenby:MannigJ.SimidianB.A.inEconomicswithDistinction,DukeUniversityM.P.A.,HarvardUniversityKennedySchoolofGovernmentM.B.A.,MassachusettsInstituteofTechnology(MIT)SloanSchoolofManagement1TheSolowGrowthModelTheSolowGrowthModelisdesignedtoshowhowgrowthinthecapitalstock,growthinthelaborforce,andadvancesintechnologyinteractinaneconomy,andhowtheyaffectanation’stotaloutputofgoodsandservices.Let’snowexaminehowthemodeltreatstheaccumulationofcapital.2TheAccumulationofCapital3Theproductionfunctionrepresentsthetransformationofinputs(labor(L),capital(K),productiontechnology)intooutputs

(finalgoodsandservicesforacertaintimeperiod).Thealgebraicrepresentationis:Y

=

F(

K,L)TheProductionFunctionIncomeissomefunctionof

ourgiveninputsLet’sanalyzethesupplyanddemandforgoods,andseehowmuchoutputisproducedatanygiventimeandhowthisoutputisallocatedamongalternativeuses.

KeyAssumption:TheProductionFunctionhasconstantreturnstoscale.zzz4ConstantReturnstoScaleThisassumptionletsusanalyzeallquantitiesrelativetothesizeofthelaborforce.Setz=1/L.Y/

L

=

F(

K/L,

1)OutputPerworkerissomefunctionof

theamountofcapitalperworkerConstantreturnstoscaleimplythatthesizeoftheeconomyasmeasuredbythenumberofworkersdoesnotaffecttherelationshipbetweenoutputperworkerandcapitalperworker.So,fromnowon,let’sdenoteallquantitiesinperworkertermsinlowercaseletters.Hereisourproductionfunction: ,wheref(k)=F(k,1).y

=

f(k)

Thisisaconstantthatcanbeignored.5MarginalProductofCapital(MPK):MPK=f(k+1)–f(k)ykf(k)Theproductionfunctionshowshowtheamountofcapitalperworkerkdeterminestheamountofoutputperworkery=

f(k).Theslopeoftheproductionfunctionisthemarginalproductofcapital:ifkincreasesby1unit,yincreasesbyMPKunits.1MPKDiminishingMarginal6TheDemandforGoodsandtheConsumptionFunctionconsumptionperworkerdependsonsavingsrate(between0and1)Outputperworkerconsumptionperworkerinvestmentperworkery=c+i1)c=(1-s)y

2)y=(1-s)y+i3)4)i=syInvestment=savings.Therateofsavingsisthefractionofoutputdevotedtoinvestment.7GrowthintheCapitalStockandtheSteadyStateHerearetwoforcesthatinfluencethecapitalstock:

Investment:expenditureonplantandequipment.Depreciation:wearingoutofoldcapital;causescapitalstocktofall.Recallinvestmentperworkeri=sy.Let’ssubstitutetheproductionfunctionfory,wecanexpressinvestmentperworkerasafunctionofthecapitalstockperworker:

i=s

f(k)Thisequationrelatestheexistingstockofcapital

k

totheaccumulationofnewcapitali.8Investment,sf(k)Output,f(k)c(perworker)i(perworker)y(perworker)Output,Consumption,andInvestmentThesavingratesdeterminestheallocationofoutputbetweenconsumptionandinvestment.Foranylevelofk,outputisf(k),investmentissf(k),andconsumptionisf(k)–sf(k).yk9DepreciationImpactofinvestmentanddepreciationonthecapitalstock:Dk=i–dkChangeincapitalstockInvestmentDepreciationRememberinvestmentequalssavingsso,itcanbewritten:Dk=s

f(k)–dkdkkdkDepreciationisthereforeproportionaltothecapitalstock.10Thelong-runequilibriumoftheeconomyInvestmentanddepreciationCapitalperworker,ki*=dk*k*k1k2Atk*,investmentequalsdepreciationandcapitalwillnotchangeovertime.Belowk*,investmentexceedsdepreciation,sothecapitalstockgrows.Investment,sf(k)Depreciation,dkTheSteadyState,k*Abovek*,depreciationexceedsinvestment,sothecapitalstockshrinks.

11HowSavingAffectsGrowthInvestmentanddepreciationCapitalperworker,ki*=dk*k1*k2*Depreciation,dkInvestment,s1f(k)Investment,s2f(k)TheSolowModelshowsthatifthesavingrateishigh,theeconomywillhavealargecapitalstockandhighlevelofoutput.Ifthesaving rateislow,theeconomywillhaveasmallcapitalstockanda lowlevelofoutput.Anincreaseinthesavingratecausesthecapitalstocktogrowtoanewsteadystate.12TheGoldenRuleLevelofCapitalThesteady-statevalueofkthatmaximizesconsumptioniscalledtheGoldenRuleLevelofCapital.Tofindthesteady-stateconsumptionperworker,webeginwiththenationalincomeaccountsidentity:

andrearrangeitas:

c=y-i.Thisequationholdsthatconsumptionisoutputminusinvestment.Becausewewanttofindsteady-stateconsumption,wesubstitutesteady-statevaluesforoutputandinvestment.Steady-stateoutputperworkerisf(k*)wherek*isthesteady-statecapitalstockperworker.Furthermore,becausethecapitalstockisnotchanginginthesteadystate,investmentisequaltodepreciationdk*.Substitutingf(k*)foryand

dk*fori,wecanwritesteady-stateconsumptionperworkeras:

c*=f(k*)-dk*.y-c+i13c*=f(k*)-dk*.Accordingtothisequation,steady-stateconsumptioniswhat’sleftofsteady-stateoutputafterpayingforsteady-statedepreciation.Itfurthershowsthatanincreaseinsteady-statecapitalhastwoopposingeffectsonsteady-stateconsumption.Ontheonehand,morecapitalmeansmoreoutput.Ontheotherhand,morecapitalalsomeansthatmoreoutputmustbeusedtoreplacecapitalthatiswearingout.Steady-stateConsumptionTheeconomy’soutputisusedforconsumptionorinvestment.Inthesteadystate,investmentequalsdepreciation.Therefore,steady-stateconsumptionisthedifferencebetweenoutputf(k*)anddepreciationdk*.Steady-stateconsumptionismaximizedattheGoldenRulesteadystate.TheGoldenRulecapitalstockisdenotedk*gold,andtheGoldenRuleconsumptionisc*gold.dkkdkOutput,f(k)c*goldk*gold14Let’snowderiveasimpleconditionthatcharacterizestheGoldenRulelevelofcapital.RecallthattheslopeoftheproductionfunctionisthemarginalproductofcapitalMPK.Theslopeofthedk*lineisd.Becausethesetwoslopesareequalatk*gold,theGoldenRulecanbedescribedbytheequation:MPK=d.AttheGoldenRulelevelofcapital,themarginalproductofcapitalequalsthedepreciationrate.KeepinmindthattheeconomydoesnotautomaticallygravitatetowardtheGoldenRulesteadystate.Ifwewantaparticularsteady-statecapitalstock,suchastheGoldenRule,weneedaparticularsavingratetosupportit.15PopulationGrowthThebasicSolowmodelshowsthatcapitalaccumulation,alone,cannotexplainsustainedeconomicgrowth.

Highratesofsavingleadtohighgrowthtemporarily,buttheeconomyeventuallyapproachesasteadystateinwhichcapitalandoutputareconstant.Toexplainthesustainedeconomicgrowth,wemustexpandtheSolowmodeltoincorporatetheothertwosourcesofeconomicgrowth.So,let’saddpopulationgrowthtothemodel.We’llassumethatthepopulationandlaborforcegrowataconstantraten.16TheSteadyStatewithPopulationGrowthLikedepreciation,populationgrowthisonereasonwhythecapitalstockperworkershrinks.Ifnistherateofpopulationgrowthandd

istherateofdepreciation,then(d+n)kisbreak-even investment,whichistheamountnecessary tokeepconstantthecapitalstock perworkerk.

Investment,break-eveninvestmentCapitalperworker,kk*Break-eveninvestment,(d+n)kInvestment,sf(k)Fortheeconomytobeinasteadystate,investmentsf(k)mustoffsettheeffectsofdepreciationandpopulationgrowth(d+n)k.Thisisshownbytheintersectionofthetwocurves.Anincreaseinthesavingratecausesthecapitalstocktogrowtoanewsteadystate.17Investment,break-eveninvestmentCapitalperworker,kk*1Investment,sf(k)(d+n1)kAnincreaseintherateofpopulationgrowthshiftsthelinerepresentingpopulationgrowthanddepreciationupward.Thenew steadystatehasalowerlevelofcapitalperworkerthanthe initialsteadystate.Thus,theSolowmodel predictsthateconomieswithhigherrates ofpopulationgrowthwillhavelower levelsofcapitalperworkerand thereforelowerincomes.k*2(d+n2)kTheImpactofPopulationGrowthAnincreaseintherateofpopulationgrowthfromn1ton2reducesthesteady-statecapitalstockfromk*1tok*2.18PopulationGrowth(n)Thechangeinthecapitalstockperworkeris:Dk=i–(d+n)kNow,let’ssubstitutesf(k)fori:Dk=(sfk)–(d+n)kThisequationshowshownewinvestment,depreciation,andpopulationgrowthinfluencetheper-workercapitalstock.Newinvestmentincreasesk,whereasdepreciationandpopulationgrowthdecreasek.Whenwedidnotincludethe“n”variableinoursimpleversion—wewereassumingaspecialcaseinwhichthepopulationgrowthwas0.19Inthesteadystate,thepositiveeffectofinvestmentonthecapitalperworkerjustbalancesthenegativeeffectsofdepreciationandpopulationgrow