Economic Growth I Capital Accumulation and Population Growth 經(jīng)濟增長資本積累和群體生長

EconomicGrowthI:

CapitalAccumulationandPopulationGrowth7Inthischapter,youwilllearn…theclosedeconomySolowmodelhowacountry’sstandardoflivingdependsonitssavingandpopulationgrowthrateshowtousethe“GoldenRule”tofindtheoptimalsavingrateandcapitalstockCHAPTER7EconomicGrowthIWhygrowthmattersDataoninfantmortalityrates:20%inthepoorest1/5ofallcountries0.4%intherichest1/5InPakistan,85%ofpeopleliveonlessthan$2/day.One-fourthofthepoorestcountrieshavehadfaminesduringthepast3decades.Povertyisassociatedwithoppressionofwomenandminorities.

Economicgrowthraiseslivingstandardsandreducespoverty….CHAPTER7EconomicGrowthIIncomeandpovertyintheworld

selectedcountries,2000CHAPTER7EconomicGrowthIWhygrowthmattersAnythingthateffectsthelong-runrateofeconomicgrowth–evenbyatinyamount–willhavehugeeffectsonlivingstandardsinthelongrun.1,081.4%243.7%85.4%624.5%169.2%64.0%2.5%2.0%…100years…50years…25yearspercentageincreasein

standardoflivingafter…annualgrowthrateofincomepercapitaCHAPTER7EconomicGrowthIWhygrowthmattersIftheannualgrowthrateofU.S.realGDPpercapitahadbeenjustone-tenthofonepercenthigherduringthe1990s,theU.S.wouldhavegeneratedanadditional$496billionofincome

duringthatdecade.CHAPTER7EconomicGrowthIThelessonsofgrowththeory…canmakeapositivedifferenceinthelivesofhundredsofmillionsofpeople.Theselessonshelpusunderstandwhypoorcountriesarepoordesignpoliciesthat

canhelpthemgrowlearnhowourowngrowthrateisaffectedbyshocksandourgovernment’spoliciesCHAPTER7EconomicGrowthITheSolowmodelduetoRobertSolow,

wonNobelPrizeforcontributionsto

thestudyofeconomicgrowthamajorparadigm:widelyusedinpolicymakingbenchmarkagainstwhichmost

recentgrowththeoriesarecomparedlooksatthedeterminantsofeconomicgrowthandthestandardoflivinginthelongrunCHAPTER7EconomicGrowthIHowSolowmodelisdifferentfromChapter3’smodel1.

Kisnolongerfixed:

investmentcausesittogrow,

depreciationcausesittoshrink2.

Lisnolongerfixed:

populationgrowthcausesittogrow3. theconsumptionfunctionissimplerCHAPTER7EconomicGrowthIHowSolowmodelisdifferentfromChapter3’smodel4.

noGorT

(onlytosimplifypresentation;

wecanstilldofiscalpolicyexperiments)5.

cosmeticdifferencesCHAPTER7EconomicGrowthITheproductionfunctionInaggregateterms:Y=F(K,L)Define:y=Y/L=outputperworker k=K/L=capitalperworkerAssumeconstantreturnstoscale:

zY=F(zK,zL)foranyz>0Pickz=1/L.ThenY/L=F(K/L,1)y=F(k,1)y=f(k) wheref(k)=F(k,1)CHAPTER7EconomicGrowthITheproductionfunctionOutputper

worker,y

Capitalper

worker,k

f(k)Note:thisproductionfunctionexhibitsdiminishingMPK.1MPK=f(k+1)–f(k)CHAPTER7EconomicGrowthIThenationalincomeidentityY=C+I (remember,noG)In“perworker”terms:

y=c+i

wherec=C/Landi=I

/L

CHAPTER7EconomicGrowthITheconsumptionfunctions=thesavingrate,

thefractionofincomethatissaved(sisanexogenousparameter) Note:sistheonlylowercasevariable

thatisnotequalto

itsuppercaseversiondividedbyLConsumptionfunction:c=(1–s)y

(perworker)CHAPTER7EconomicGrowthISavingandinvestmentsaving(perworker) =y–c

=y–(1–s)y

=syNationalincomeidentityisy=c+i Rearrangetoget:i=y–c=sy

(investment=saving,likeinchap.3!)Usingtheresultsabove,

i=sy=sf(k)CHAPTER7EconomicGrowthIOutput,consumption,andinvestmentOutputper

worker,y

Capitalper

worker,k

f(k)sf(k)k1

y1

i1

c1

CHAPTER7EconomicGrowthIDepreciationDepreciationperworker,k

Capitalper

worker,k

k=therateofdepreciation=thefractionofthecapitalstockthatwearsouteachperiod1CHAPTER7EconomicGrowthICapitalaccumulationChangeincapitalstock =investment–depreciation k =i

–k Since

i=sf(k),thisbecomes:k=s

f(k)

–k

Thebasicidea:Investmentincreasesthecapitalstock,depreciationreducesit.CHAPTER7EconomicGrowthITheequationofmotionforkTheSolowmodel’scentralequationDeterminesbehaviorofcapitalovertime……which,inturn,determinesbehaviorof

alloftheotherendogenousvariables

becausetheyalldependonk.E.g., incomeperperson: y=f(k) consumptionperperson: c=(1–s)

f(k)

k=s

f(k)

–k

CHAPTER7EconomicGrowthIThesteadystateIfinvestmentisjustenoughtocoverdepreciation

[sf(k)

=

k],thencapitalperworkerwillremainconstant:

k=0.Thisoccursatonevalueofk,denotedk*,

calledthesteadystatecapitalstock.k=s

f(k)

–k

CHAPTER7EconomicGrowthIThesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)kk*

CHAPTER7EconomicGrowthIMovingtowardthesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)kk*

k=sf(k)kdepreciationkk1investmentCHAPTER7EconomicGrowthIMovingtowardthesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)kk*

k1k=sf(k)kkCHAPTER7EconomicGrowthIMovingtowardthesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)kk*

k1k=sf(k)kkk2CHAPTER7EconomicGrowthIMovingtowardthesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)kk*

k=sf(k)kk2investmentdepreciationkCHAPTER7EconomicGrowthIMovingtowardthesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)kk*

k=sf(k)kkk2CHAPTER7EconomicGrowthIMovingtowardthesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)kk*

k=sf(k)kk2kk3CHAPTER7EconomicGrowthIMovingtowardthesteadystateInvestmentanddepreciationCapitalper

worker,k

sf(k)kk*

k=sf(k)kk3Summary:

Aslongask<k*,investmentwillexceeddepreciation,

andkwillcontinuetogrowtowardk*.CHAPTER7EconomicGrowthINowyoutry:DrawtheSolowmodeldiagram,

labelingthesteadystatek*.Onthehorizontalaxis,pickavaluegreaterthank*fortheeconomy’sinitialcapitalstock.Labelitk1.Showwhathappenstokovertime.

Doeskmovetowardthesteadystateor

awayfromit?CHAPTER7EconomicGrowthIAnumericalexampleProductionfunction(aggregate):Toderivetheper-workerproductionfunction,dividethroughbyL:Thensubstitutey=Y/Landk=K/LtogetCHAPTER7EconomicGrowthIAnumericalexample,cont.Assume:s=0.3=0.1initialvalueofk=4.0CHAPTER7EconomicGrowthIApproachingthesteadystate:

AnumericalexampleYear

k y c i k k1 4.000 2.000 1.400 0.600 0.400 0.2002 4.200 2.049 1.435 0.615 0.420 0.1953 4.395 2.096 1.467 0.629 0.440 0.1894 4.584 2.141 1.499 0.642 0.458 0.184…10 5.602 2.367 1.657 0.710 0.560 0.150…25 7.351 2.706 1.894 0.812 0.732 0.080…100 8.962 2.994 2.096 0.898 0.896 0.002…

9.000 3.000 2.100 0.900 0.900 0.000CHAPTER7EconomicGrowthIExercise:SolveforthesteadystateContinuetoassume

s=0.3,=0.1,andy=k1/2Usetheequationofmotion

k=sf(k)k

tosolveforthesteady-statevaluesofk,y,andc.CHAPTER7EconomicGrowthISolutiontoexercise:CHAPTER7EconomicGrowthIAnincreaseinthesavingrateInvestment

and

depreciationkks1f(k)Anincreaseinthesavingrateraisesinvestment……causingktogrowtowardanewsteadystate:s2f(k)CHAPTER7EconomicGrowthIPrediction:Highers

higherk*.Andsincey=f(k),

higherk*highery*.Thus,theSolowmodelpredictsthatcountrieswithhigherratesofsavingandinvestment

willhavehigherlevelsofcapitalandincomeperworkerinthelongrun.CHAPTER7EconomicGrowthIInternationalevidenceoninvestmentratesandincomeperperson1001,00010,000100,00005101520253035Investmentaspercentageofoutput(average1960-2000)Incomeperpersonin2000(logscale)CHAPTER7EconomicGrowthITheGoldenRule:IntroductionDifferentvaluesofsleadtodifferentsteadystates.

Howdoweknowwhichisthe“best”steadystate?The“best”steadystatehasthehighestpossible

consumptionperperson:c*=(1–s)f(k*).Anincreaseins

leadstohigherk*andy*,whichraisesc*

reducesconsumption’sshareofincome(1–s),

whichlowersc*.

So,howdowefindthesandk*thatmaximizec*?CHAPTER7EconomicGrowthITheGoldenRulecapitalstocktheGoldenRulelevelofcapital,

thesteadystatevalueofk

thatmaximizesconsumption.Tofindit,firstexpressc*intermsofk*: c* =y*

i* =f

(k*)

i*

=f

(k*)

k*

Inthesteadystate: i*

=

k*becausek=0.CHAPTER7EconomicGrowthIThen,graph

f(k*)andk*,

lookforthe

pointwhere

thegapbetweenthemisbiggest.TheGoldenRulecapitalstocksteadystateoutputanddepreciationsteady-statecapitalper

worker,k*

f(k*)

k*CHAPTER7EconomicGrowthITheGoldenRulecapitalstockc*

=f(k*)k*

isbiggestwheretheslopeoftheproductionfunction

equals

theslopeofthedepreciationline:steady-statecapitalper

worker,k*

f(k*)

k*MPK=

CHAPTER7EconomicGrowthIThetransitiontothe

GoldenRulesteadystateTheeconomydoesNOThaveatendencytomovetowardtheGoldenRulesteadystate.AchievingtheGoldenRulerequiresthatpolicymakersadjusts.Thisadjustmentleadstoanewsteadystatewithhigherconsumption.Butwhathappenstoconsumption

duringthetransitiontotheGoldenRule?CHAPTER7EconomicGrowthIStartingwithtoomuchcapitalthenincreasingc*requiresafallins.InthetransitiontotheGoldenRule,consumptionishigheratallpointsintime.timet0ciyCHAPTER7EconomicGrowthIStartingwithtoolittlecapitalthenincreasingc*requiresan

increaseins.Futuregenerations

enjoyhigherconsumption,

butthecurrent

oneexperiences

aninitialdrop

inconsumption.timet0ciyCHAPTER7EconomicGrowthIPopulationgrowthAssumethatthepopulation(andlaborforce)growatraten.(nisexogenous.)EX:SupposeL=1,000inyear1andthepopulationisgrowingat2%peryear(n=0.02).ThenL=n

L=0.02

1,000=20,

soL=1,020inyear2.CHAPTER7EconomicGrowthIBreak-eveninvestment(

+

n)k=break-eveninvestment,

theamountofinvestmentnecessary

tokeepkconstant.Break-eveninvestmentincludes:

k

toreplacecapitalasitwearsoutn

k

toequipnewworkerswithcapital (Otherwise,kwouldfallastheexistingcapitalstockwouldbespreadmorethinlyoveralargerpopulationofworkers.)CHAPTER7EconomicGrowthITheequationofmotionforkWithpopulationgrowth,

theequationofmotionforkisbreak-eveninvestmentactualinvestmentk=s

f(k)

(

+

n)

kCHAPTER7EconomicGrowthITheSolowmodeldiagramInvestment,break-eveninvestmentCapitalper

worker,k

sf(k)(

+

n

)

kk*

k

=

sf(k)

(+n)kCHAPTER7EconomicGrowthITheimpactofpopulationgrowthInvestment,break-eveninvestmentCapitalper

worker,k

sf(k)(

+n1)

kk1*

(

+n2)

kk2*

Anincreaseinncausesanincreaseinbreak-eveninvestment,leadingtoalowersteady-statelevelofk.CHAPTER7EconomicGrowthIPrediction:Highern

lowerk*.Andsincey=f(k),

lowerk*

lowery*.Thus,theSolowmodelpredictsthatcountrieswithhigherpopulationgrowthrateswillhavelowerlevelsofcapitalandincomeperworkerinthelongrun.CHAPTER7EconomicGrowthIInternationalevidenceonpopulationgrowthandincomeperperson1001,00010,000100,000012345PopulationGrowth(percentperyear;average1960-2000)IncomeperPersonin2000(logscale)CHAPTER7EconomicGrowthITheGoldenRulewithpopulationgrowthTofindtheGoldenRulecapitalstock,

expressc*intermsofk*: c* =y*

i* =f

(k*)

(

+n)

k*

c*ismaximizedwhen

MPK=

+n

orequivalently,

MPK

=nIntheGolden

Rulesteadystate,

themarginalproduct

ofcapitalnetofdepreciationequalsthepopulation

growthrate.CHAPTER7EconomicGrowthIAlternativeperspectivesonpopulationgrowthTheMalthusianModel(1798)PredictspopulationgrowthwilloutstriptheEarth’sabilitytoproducefood,l