宏观经济学曼昆英文课件TutorialCHAP07

CHAPTER7EconomicGrowthII®MANKIW'SMACROECONOMICSMODULESAPowerPointTutorialToAccompany

MACROECONOMICS,

6th.ed.N.GregoryMankiwByMannigJ.Simidian1CHAPTER7®MANKIW'SMACROECONOMTheSolowGrowthModelTheSolowGrowthModelisdesignedtoshowhowgrowthinthecapitalstock,growthinthelaborforce,andadvancesintechnologyinteractinaneconomy,andhowtheyaffectanation’stotaloutputofgoodsandservices.Let’snowexaminehowthemodeltreatstheaccumulationofcapital.2TheSolowGrowthModelTheSoloTheAccumulationofCapital3TheAccumulationofCapital3Theproductionfunctionrepresentsthetransformationofinputs(labor(L),capital(K),productiontechnology)intooutputs

(finalgoodsandservicesforacertaintimeperiod).Thealgebraicrepresentationis:Y

=

F(

K,L)TheProductionFunctionIncomeissomefunctionof

ourgiveninputsLet’sanalyzethesupplyanddemandforgoods,andseehowmuchoutputisproducedatanygiventimeandhowthisoutputisallocatedamongalternativeuses.

KeyAssumption:TheProductionFunctionhasconstantreturnstoscale.zzz4TheproductionfunctionrepresConstantReturnstoScaleThisassumptionletsusanalyzeallquantitiesrelativetothesizeofthelaborforce.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.5ConstantReturnstoScaleThisMarginalProductofCapital(MPK):MPK=f(k+1)–f(k)ykf(k)Theproductionfunctionshowshowtheamountofcapitalperworkerkdeterminestheamountofoutputperworkery=

f(k).Theslopeoftheproductionfunctionisthemarginalproductofcapital:ifkincreasesby1unit,yincreasesbyMPKunits.1MPKDiminishingMarginal6MarginalProductofCapital(MTheDemandforGoodsandtheConsumptionFunctionconsumptionperworkerdependsonsavingsrate(between0and1)Outputperworkerconsumptionperworkerinvestmentperworkery=c+i1)c=(1-s)y

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

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

i=s

f(k)Thisequationrelatestheexistingstockofcapital

k

totheaccumulationofnewcapitali.8GrowthintheCapitalStockanInvestment,sf(k)Output,f(k)c(perworker)i(perworker)y(perworker)Output,ConsumptionandInvestmentThesavingratesdeterminestheallocationofoutputbetweenconsumptionandinvestment.Foranylevelofk,outputisf(k),investmentissf(k),andconsumptionisf(k)–sf(k).yk9Investment,sf(k)Output,f(kDepreciationImpactofinvestmentanddepreciationonthecapitalstock:Dk=i–dkChangeincapitalstockInvestmentDepreciationRememberinvestmentequalssavingsso,itcanbewritten:Dk=sf(k)–dkdkkdkDepreciationisthereforeproportionaltothecapitalstock.10DepreciationImpactofinvestmeThelong-runequilibriumoftheeconomyInvestmentanddepreciationCapitalperworker,ki*=dk*k*k1k2Atk*,investmentequalsdepreciationandcapitalwillnotchangeovertime.Belowk*,investmentexceedsdepreciation,sothecapitalstockgrows.Investment,sf(k)Depreciation,dkTheSteadyState,k*Abovek*,depreciationexceedsinvestment,sothecapitalstockshrinks.

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

andrearrangeitas:

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

c*=f(k*)-dk*.y-c+i13TheGoldenRuleLevelofCapitc*=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*gold14c*=f(k*)-dk*.AccordingtoLet’snowderiveasimpleconditionthatcharacterizestheGoldenRulelevelofcapital.RecallthattheslopeoftheproductionfunctionisthemarginalproductofcapitalMPK.Theslopeofthedk*lineisd.Becausethesetwoslopesareequalatk*gold,theGoldenRulecanbedescribedbytheequation:MPK=d.AttheGoldenRulelevelofcapital,themarginalproductofcapitalequalsthedepreciationrate.KeepinmindthattheeconomydoesnotautomaticallygravitatetowardtheGoldenRulesteadystate.Ifwewantaparticularsteady-statecapitalstock,suchastheGoldenRule,weneedaparticularsavingratetosupportit.15Let’snowderiveasimplecondPopulationGrowthThebasicSolowmodelshowsthatcapitalaccumulation,alone,cannotexplainsustainedeconomicgrowth:highratesofsavingleadtohighgrowthtemporarily,buttheeconomyeventuallyapproachesasteadystateinwhichcapitalandoutputareconstant.Toexplainthesustainedeconomicgrowth,wemustexpandtheSolowmodeltoincorporatetheothertwosourcesofeconomicgrowth.So,let’saddpopulationgrowthtothemodel.We’llassumethatthepopulationandlaborforcegrowataconstantraten.16PopulationGrowthThebasicSolTheSteadyStatewithPopulationGrowthLikedepreciation,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.17TheSteadyStatewithPopulatiInvestment,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*2.18Investment,Capitalk*1InvestmePopulationGrowth(n)Thechangeinthecapitalstockperworkeris:Dk=i–(d+n)kNow,let’ssubstitutesf(k)fori:Dk=(sfk)–(d+n)kThisequationshowshownewinvestment,depreciation,andpopulationgrowthinfluencetheper-workercapitalstock.Newinvestmentincreasesk,whereasdepreciationandpopulationgrowthdecreasek.Whenwedidnotincludethe“n”variableinoursimpleversion—wewereassumingaspecialcaseinwhichthepopulationgrowthwas0.19PopulationGrowth(n)ThechangInthesteadystate,thepositiveeffectofinvestmentonthecapitalperworkerjustbalancesthenegativeeffectsofdepreciationandpopulationgrowth.Oncetheeconomyisinthesteadystate,investmenthastwopurposes:Someofit,(dk*),replacesthedepreciatedcapital,Therest,(nk*),providesnewworkerswiththesteadystateamountofcapital.Capitalperworker,kk*k*'TheSteadyStateInvestment,sf(k)Break-evenInvestment,(d+n)kBreak-eveninvestment,

(d+n')kAnincreaseintherateofgrowthofpopulationwilllowerthelevelofoutputperworker.sf(k)20Inthesteadystate,thepositFinalPointsonSavingInthelongrun,aneconomy’ssavingdeterminesthesizeofkandthusy.Thehighertherateofsaving,thehigherthestockofcapitalandthehigherthelevelofy.Anincreaseintherateofsavingcausesaperiodofrapidgrowth,buteventuallythatgrowthslowsasthenewsteadystateisreached.Conclusion:althoughahighsavingrateyieldsahighsteady-statelevelofoutput,savingbyitselfcannotgeneratepersistenteconomicgrowth.21FinalPointsonSavingIntheSolowgrowthmodelSteadystateGoldenRulelevelofcapitalKeyConceptsofCh.722SolowgrowthmodelCHAPTER7EconomicGrowthII®MANKIW'SMACROECONOMICSMODULESAPowerPointTutorialToAccompany

MACROECONOMICS,

6th.ed.N.GregoryMankiwByMannigJ.Simidian23CHAPTER7®MANKIW'SMACROECONOMTheSolowGrowthModelTheSolowGrowthModelisdesignedtoshowhowgrowthinthecapitalstock,growthinthelaborforce,andadvancesintechnologyinteractinaneconomy,andhowtheyaffectanation’stotaloutputofgoodsandservices.Let’snowexaminehowthemodeltreatstheaccumulationofcapital.24TheSolowGrowthModelTheSoloTheAccumulationofCapital25TheAccumulationofCapital3Theproductionfunctionrepresentsthetransformationofinputs(labor(L),capital(K),productiontechnology)intooutputs

(finalgoodsandservicesforacertaintimeperiod).Thealgebraicrepresentationis:Y

=

F(

K,L)TheProductionFunctionIncomeissomefunctionof

ourgiveninputsLet’sanalyzethesupplyanddemandforgoods,andseehowmuchoutputisproducedatanygiventimeandhowthisoutputisallocatedamongalternativeuses.

KeyAssumption:TheProductionFunctionhasconstantreturnstoscale.zzz26TheproductionfunctionrepresConstantReturnstoScaleThisassumptionletsusanalyzeallquantitiesrelativetothesizeofthelaborforce.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.27ConstantReturnstoScaleThisMarginalProductofCapital(MPK):MPK=f(k+1)–f(k)ykf(k)Theproductionfunctionshowshowtheamountofcapitalperworkerkdeterminestheamountofoutputperworkery=

f(k).Theslopeoftheproductionfunctionisthemarginalproductofcapital:ifkincreasesby1unit,yincreasesbyMPKunits.1MPKDiminishingMarginal28MarginalProductofCapital(MTheDemandforGoodsandtheConsumptionFunctionconsumptionperworkerdependsonsavingsrate(between0and1)Outputperworkerconsumptionperworkerinvestmentperworkery=c+i1)c=(1-s)y

2)y=(1-s)y+i3)4)i=syInvestment=savings.Therateofsavingsisthefractionofoutputdevotedtoinvestment.29TheDemandforGoodsandtheCGrowthintheCapitalStockandtheSteadyStateHerearetwoforcesthatinfluencethecapitalstock:

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

i=s

f(k)Thisequationrelatestheexistingstockofcapital

k

totheaccumulationofnewcapitali.30GrowthintheCapitalStockanInvestment,sf(k)Output,f(k)c(perworker)i(perworker)y(perworker)Output,ConsumptionandInvestmentThesavingratesdeterminestheallocationofoutputbetweenconsumptionandinvestment.Foranylevelofk,outputisf(k),investmentissf(k),andconsumptionisf(k)–sf(k).yk31Investment,sf(k)Output,f(kDepreciationImpactofinvestmentanddepreciationonthecapitalstock:Dk=i–dkChangeincapitalstockInvestmentDepreciationRememberinvestmentequalssavingsso,itcanbewritten:Dk=sf(k)–dkdkkdkDepreciationisthereforeproportionaltothecapitalstock.32DepreciationImpactofinvestmeThelong-runequilibriumoftheeconomyInvestmentanddepreciationCapitalperworker,ki*=dk*k*k1k2Atk*,investmentequalsdepreciationandcapitalwillnotchangeovertime.Belowk*,investmentexceedsdepreciation,sothecapitalstockgrows.Investment,sf(k)Depreciation,dkTheSteadyState,k*Abovek*,depreciationexceedsinvestment,sothecapitalstockshrinks.

33Thelong-runequilibriumofthHowSavingAffectsGrowthInvestmentanddepreciationCapitalperworker,ki*=dk*k1*k2*Depreciation,dkInvestment,s1f(k)Investment,s2f(k)TheSolowModelshowsthatifthesavingrateishigh,theeconomywillhavealargecapitalstockandhighlevelofoutput.Ifthesaving rateislow,theeconomywillhaveasmallcapitalstockanda lowlevelofoutput.Anincreaseinthesavingratecausesthecapitalstocktogrowtoanewsteadystate.34HowSavingAffectsGrowthInvesTheGoldenRuleLevelofCapitalThesteady-statevalueofkthatmaximizesconsumptioniscalledtheGoldenRuleLevelofCapital.Tofindthesteady-stateconsumptionperworker,webeginwiththenationalincomeaccountsidentity:

andrearrangeitas:

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

c*=f(k*)-dk*.y-c+i35TheGoldenRuleLevelofCapitc*=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*gold36c*=f(k*)-dk*.AccordingtoLet’snowderiveasimpleconditionthatcharacterizestheGoldenRulelevelofcapital.RecallthattheslopeoftheproductionfunctionisthemarginalproductofcapitalMPK.Theslopeofthedk*lineisd.Becausethesetwoslopesareequalatk*gold,theGoldenRulecanbedescribedbytheequation:MPK=d.AttheGoldenRulelevelofcapital,themarginalproductofcapitalequalsthedepreciationrate.KeepinmindthattheeconomydoesnotautomaticallygravitatetowardtheGoldenRulesteadystate.Ifwewantaparticularsteady-statecapitalstock,suchastheGoldenRule,weneedaparticularsavingratetosupportit.37Let’snowderiveasimplecondPopulationGrowthThebasicSolowmodelshowsthatcapitalaccumulation,alone,cannotexplainsustainedeconomicgrowth:highratesofsavingleadtohighgrowthtemporarily,buttheeconomyeventuallyapproachesasteadystateinwhichcapitalandoutputareconstant.Toexplainthesustainedeconomicgrowth,wemustexpandtheSolowmodeltoincorporatetheothertwosourcesofeconomicgrowth.So,let’saddpopulationgrowthtothemodel.We’llassumethatthepopulationandlaborforcegrowataconstantraten.38PopulationGrowthThebasicSolTheSteadyStatewithPopulationGrowthLikedepreciation,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.39TheSteadyStatewithPopulatiInvestment,break-eveninvestmentCapitalperworker,kk*1Investment,sf(k)(d+n1)kAnincreaseintherateofpopulationgrowthshiftsthelinerepresentingpopulationgrowthanddepreciationupward.Thenew steadystatehasalowerlevelofcapitalperworkerthanthe initialsteadys