CH3.docx
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CH3.docx
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CH3
3GrowthandAccumulation
focusofthechapter
•Inthischapterwestudyhowpotentialoutputtheoutputthatwouldbeproducedifallfactorswerefullyemployedgrowsovertime.
•Tobetteraccomplishthis,welearngrowthaccountingandthefundamentalsofneoclassicalgrowththeory.Together,theytellusthatoutputgrowthresultsbothfromimprovementsintechnologyandfromincreasesinoneormoreoftheinputstotheproductionprocesscapital,labor,andnaturalresources.Neoclassicalgrowththeoryalsotellsusthatinthelongrun,growthinpotentialoutputresultsentirelyfromtechnologicalimprovement.
*Note:
Theauthorsinthischapterandthenextusetheterm“longrun”inawaythatisinconsistentwiththerestofthetextbook.Theyshouldbesaying“verylongrun.”
sectionsummaries
1.GrowthAccounting
Outputgrowsbecauseofincreasesinfactorsofproductionlikecapitalandlabor,andbecauseofimprovementsintechnology.Theproductionfunctionprovidesalinkbetweentheleveloftechnology(A),theamountofcapital(K),labor(N),andotherinputsused,andtheamountofoutput(Y)created.Thegenericformulafortheproductionfunctionis:
Y=AF(K,N)
TheCobb-Douglasproductionfunction,amorespecificformula,isfrequentlyusedaswell,asitprovidesagoodapproximationofproductionintheactualeconomy.TheformulafortheCobb-Douglasproductionfunctionis:
Y=AKN1
pronounced“theta”,representscapital’sshareofincometotalpaymentstocapital,asafractionofoutput,or(iK)/Y.
(1)islabor’sshareofincome,givenby(wN)/Y.Toderivetheseresultsalgebraically,youneedonemorefact:
Whenthemarketsforcapitalandlaborareinequilibrium(i.e.,whenthesupplyofcapitalequalsthedemandforcapital,andthesupplyoflaborequalsthedemandforlabor),capitalandlaborareeachpaidtheirmarginalproduct.
FortheCobb-Douglasfunction,themarginalproductofcapital(MPK)isAKN.
Themarginalproductoflabor(MPL)is
(1)AKN1
Wecanexpressourproductionfunctionintermsofgrowthratesratherthanlevels:
Y/Y=[
(1)xN/N]+[xK/K]+A/A
Thesymbolpronounced“delta”means“changein”.ThetermY/Y,then,shouldbeinterpretedasthegrowthrateofoutput.ThetermsN/NandK/Kshouldbeinterpretedasthegrowthratesoflaborandcapital,respectively.Thelastterm,A/Aistherateofimprovementoftechnology,oftencalledthegrowthrateoftotalfactorproductivity(TFP).Itistheamountthatoutputincreasesasaresultoftechnologicalprogressalone(pluginN/N=K/K=0,andyou’llseewhy).
BecausegrowthinGDPpercapita(outputperperson)tellsusmoreaboutincreasesinthestandardofliving,itisusefultosubtracttherateofpopulationgrowth(N/N)frombothsides,andwritetheaboveequationinpercapitaterms:
y/y=(xk/k)+A/A
Thetermsyandkrepresentoutputandcapitalperperson:
y=Y/N,k=K/N.(Thereisanimplicitassumptionherethatthefractionofthepopulationinthelaborforceisconstant.Thisiswhywecangetawaywithusingtheterms“population”and“laborsupply”interchangeably.)Theterms(y/y)and(k/k)arethegrowthratesofoutputandcapitalperperson:
y/y=Y/YN/N,andk/k=K/KN/N.ThetermK/Nisoftencalledthecapital-laborratio.
2.EmpiricalEstimatesofGrowth
Since1929,U.S.economicgrowthhasaveragedabout2.9percentayear.Ofthis,estimatessuggestthatabout1.09percenthasbeenduetoincreasesinthelaborsupply,0.32percenthasbeenduetocapitalaccumulation,and1.49percenthasbeentheresultoftechnologicalprogress.
Physicalcapitalandlaborarenottheonlyinputstoproduction.Twootherimportantfactorsofproductionarenaturalresourcesandhumancapitaltheskillsandtalentsofworkers.Thesharesofincome(alsocalledfactorshares)ofphysicalcapital,humancapital,andrawlaborareestimatedtoberoughly1/3each.
3.GrowthTheory:
TheNeoclassicalModel
Neoclassicalgrowththeorystudiesthewaythatgrowthinthecapitalstockperworkeraffectsthelong-runlevelofper-capitapotentialoutput.Akeyresultisthat,whiletherateofsavinghasasignificantimpactonthelevelofpercapitapotentialoutputinthelongrun,therateofimprovementintechnologyentirelydeterminesitsgrowthrate.
Figure31
theper-capitaproductionfunction
Tobuildourmodel,webeginwithafewsimplifyingassumptions:
(1)theleveloftechnologyisfixed,sothatthereisnogrowthintotalfactorproductivity;
(2)theproductionfunctionhasconstantreturnstoscale(seeReviewofTechnique4),sothatincreasingtheamounteveryinputusedinproductionwillincreaseoutputbythesameamount.
Aconsequenceofthissecondassumptionisthatallfactorsofproductionmusthavediminishingmarginalproductsasmoreofoneinputisadded,andtheothersareheldconstant,eachunitcontributeslesstooutputthandidthepreviousone.(Buyingmoretractorsforyourconstructioncompanywithouthiringanymoreworkerstodrivethemwillnothelpincreaseyouroutputmuch.)
Wealsoneedtowriteourvariablesinpercapitaform;asbefore,y=Y/Nandk=K/N.Wewritethepercapitaproductionfunction:
y=f(k)
Figure32
steady-stateintheneoclassicalmodel
Wethenconsidertheflowsintoandoutkthestockofcapitalperworker.
Investmentincreasesthetotalstockofcapital(K),whichincreasesk.Itcanbethoughtofasaflowintoeachworker’spoolofcapital.
Bothdepreciationandpopulationgrowthdecreasekdepreciationbecauseitdecreasesthestockoffunctionalcapital,andpopulationgrowthbecauseitincreasesthenumberofworkerssharingthiscapital.Bothdepreciationandpopulationgrowthcanbevisualizedasflowsoutofeachworker’spoolofcapital.Whentheflowintothispoolisgreaterthantheflowsout,kgrows.Whentheflowsoutofthispoolaregreaterthantheflowin,kshrinks.Andwhentheflowsinandoutexactlybalance,thelevelofcapitalperworkerwillremainfixed.
Wecallthislastcasethesteady-state,becauseitisthepointatwhichthelevelofcapitalineachworker’spoolremainssteady,orstable.Itisthepointofequilibriuminourmodel;wewillfindthatthecapitalstockperworkergrowsorshrinkstowardthispoint,andthatonceitgetsthereitstays.Atleastuntilsomeshockforcesittomove.
Itisnotdifficulttoexpressthedynamicsdescribedaboveasanequation.Weknowthatsavingmustequalinvestment.Ifweassumethatpeoplesaveaconstantfraction(s)oftheirincomes,wecanwritetheflowintokas(sxy),or,usingourpercapitaproductionfunction,as(sxf(k)).Thestandardassumptionaboutdepreciationisthataconstantfraction(d)ofthecapitalstockbecomesobsoleteeachperiod.Usingthisassumption,wecanexpresstheflowoutofeachworker’spoolofcapitalthatresultfromdepreciationas(dxk).Similarly,whenpopulationgrowsataconstantrate(n),wecanexpresstheflowoutofthispoolresultingfrompopulationgrowthas(nxk).Puttingallofthesetermstogether,wegetthefollowingequation:
k=sf(k)(n+d)k
Wefindanexpressionforthesteady-statebysimplypluggingintherequirementk=0:
sf(k*)=(n+d)k*
k*representsthesteady-statevalueofk.Thesteady-statevalueofyisy*=f(k*).
Thegrowthprocesscanbestudiedgraphicallyaswell.Figure32graphstheflowsintoandoutofkagainstthelevelofk.Theoutflowsaregraphedasastraightline,withslope(n+d).Thisisoftencalledtheinvestmentrequirementline,asitshowstheamountthatmustbeinvested,ifthecapitalstockistoremainconstant.Theslopeofthelinethatrepresentstheflowintokshrinksaskincreases,becausewehaveassumedittohavediminishingmarginalreturns.Wherethesetwolinesintersect,theflowsintoandoutofkbalance,andkisatitssteady-state.Whetherthesavingslineliesabovetheinvestmentrequirementline,sothatkisincreasing,ortheinvestmentrequirementlineliesabovethesavingsline,sothatkisdecreasing,kalwaysmovestowardthesteady-state.
Usingthisgraph,wecanexaminetheconsequencesofchangesins,n,ord.Anincreaseinthesavingsrate(s)willshiftthesavingsline,sf(k),upward,increasingthesteady-statecapital-laborratio,andhencethesteady-statelevelofpercapitapotentialoutput.Itwillalso,temporarily,increasethegrowthrateofbothyandk(remember,thegrowthrateofkiszeroatthesteady-state,andwithoutimprovementsintechnologypercapita,outputhasnootherreasontogrow).Anincreaseineithertherateofdepreciationortherateofpopulationgrowthwillincreasetheslopeoftheinvestmentrequirementline,decreasingthesteady-statelevelsofkandy,andcausingbothto“grow”,temporarily,atanegativerate.
steady-state,
lowers
steady-state,
highers
Figure33
adecreaseinthesavingsratereducesthesteady-statecapital-laborratio
Whentheleveloftechnologyispermittedtochange,weaddinanotherterm:
“g”,orthegrowthrateoftechnology.Technologicalimprovementisrepresentedonourgraphasanupwardshiftinthesavingsline,nowwrittensxAf(k).Noticethatitcausesthesteady-statelevelsofkandytorise.
Noticealso,though,thatwiththeadditionoftechnologicalgrowthourproductionfunctionnolongerhasconstantreturnstoscaledoublingcapitalandlaborwillmorethandoubleoutput.Inordertofixthisproblem,growththeoristsoftenassumethattechnologyhasaveryparticularcharacteristic:
Itisassumedtobelaboraugmenting,sothattechnologicalimprovementsincreasetheproductivityspecificallyoflabor.
Theproductionfunction,underthisassumption,iswrittenasfollows:
Y=F(
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