曼昆第10版讲义mankiw10e-irm-ch08

|CHAPTER8 EconomicGrowthI:CapitalAccumulationandPopulationGrowth

LectureNotes|

CHAPTER8

EconomicGrowthI:CapitalAccumulationandPopulationGrowth

NotestotheInstructor

ChapterSummary

ThischapterandthenextonepresenttheSolowgrowthmodel.Althoughtheyaretwoofthemoredifficultchaptersinthebook,theycovermaterialthatstudentsusuallyfindinteresting.Chapter8proceedsbyfirstholdingpopulationandtechnologyconstantandshowinghowtherateofsavingdeterminesthesteady-statecapital–laborratio.ThechapterthendiscussesthepositiveandnormativeimplicationsoftheGoldenRulelevelofaccumulation.Followingthis,themodelisexpandedtoconsiderpopulationgrowth.Thepurposeofthechapteristoteachstudentsaboutsomeofthedeterminantsofeconomicwell-beingandtooffersomeexplanationsofinternationaldifferencesinlivingstandards.Thethreesectionsofthechapterteachthefollowinglessons:

Therateofsavingdeterminesthesizeofacountry’scapitalstock.Increasesinthesavingrateleadtotemporarilyhighergrowthandapermanentlyhigherlevelofcapitalandoutput.

BecausetheU.S.economyhaslesscapitalthantheGoldenRulelevel,raisingconsumptionoffuturegenerationsentailsasacrificeofconsumptionbycurrentgenerations.

Highpopulationgrowthreducessteady-stateincomeperworkerbecauseitishardtomaintainhighcapitalperworkerwhenthenumberofworkersisgrowingquickly.

Comments

TeachingallthematerialinChapters8and9willprobablyrequirethreetofourlectures.Anabridgedversionofthesechapters(presentingthebasicmodelofthedeterminationofthecapitalstockanddiscussingpopulationgrowthandtechnologicalprogressinformally)couldbepresentedinonetotwolectures.Supplements8-9and9-7presentnotestoaidsuchapresentation.

ThelecturenotesemphasizetheconnectionsbetweentheSolowgrowthmodelandtheclassicalmodelofChapter3whereverpossible,remindingstudentsthattheeconomicforcesdiscussedintheclassicalmodelarestilloperative(forexample,therealinterestrateisstilltheequilibratingmechanismintheloansmarket).Thelecturenotesdistinguishcarefullybetweenthesteady-statecapitalstockandtheGoldenRule,emphasizingthatthelatterconcernsonlythetechnologicalpossibilitiesoftheeconomy,whereastheformerdependsonpeople’sbehavior(theirsavingdecisions).Sinceeconomicgrowthisintrinsicallyoneofthemoredifficulttopicscoveredinthetextbook,itcanbeachallengeeventoveryablestudents.Thefollowingaresomecommondifficultiesstudentsencounterwiththematerialinthischapterandwaystoovercomethem.

1. InterpretingaDynamicModel

TheSolowgrowthmodelislikelytobethefirstdynamicmodelthatmanyeconomicsstudentsencounter.Itisworthexplainingtheideaofadynamicmodelandmakingsurethatstudentsarecomfortablewithtime-seriesgraphs.Studentsneedtograspthedifferencebetweenlevelsandgrowthrates;theyshouldalsounderstandthemeaningofasteadystateandthebasicconceptofadifferentialequation.

2. ConfusionBetweenthePopulationandtheLaborForce

StudentsaresometimestroubledbythefactthattheSolowmodelusesthetermspopulationandlaborforcemoreorlessinterchangeably.ItmaybeusefultorefertoChapter2ofthetextandtomaketheexplicitassumptionthatthelabor-forceparticipationrateisconstant.Analternative—ifsomewhatoutdated—approachistodiscussthemodelintermsofhouseholdswithoneworkerperhousehold.

3. TheTransitionfromF(K,L)tof(k)

StudentssometimeshavedifficultyunderstandingtheanalysisoftheSolowmodelintermsoftheintensiveformproductionfunction.Iassurestudentsthatthisisjustausefulsimplificationthatmakesgraphicalanalysiseasierandthatthereisnotrickeryinvolved!Supplement8-9presentsnotesframedsolelyintermsofthefamiliarproductionfunctionandsocouldbeusedtoavoidtheproblementirely.

4. TheMeaningoftheLongRun

MystudentshavetoldmethatthematerialintheInstructor’sResourcesonthesubjectofthelongrun(Supplements3-1and8-1)ishelpful;Isuggesthandingtheseouttostudents.Inaddition,discussingthedecompositionofoutputintotrendandcyclehelpsillustratetheseideas;theMacroBytesmaterialonthewebsiteisusefulforthispurpose.

5. TheConnectionBetweentheSolowModelandtheClassicalModel

Relatedtopoint3,studentsmayhavedifficultyseeingtheconnectionsbetweenthemodelsinChapter3andthoseinChapters8and9.IpayattentiontothefactthatweusetheconclusionsfromChapter3whensettingupthemodelinChapter8.Itisalsoworthreiteratingthatsavingincludesbothprivateandgovernmentsaving;Supplements3-5and8-4maybehelpful.

RememberalsothatthechaptersontheSolowmodelaresomewhatself-containedandcancertainlybetaughtlaterinthecourse.

6. TheAlgebraofGrowthRates

StudentsmayfindtheSolowmodelconfusingiftheydonotunderstandthesimplealgebraofgrowthrates.IsuggestreviewingthismaterialandreferringstudentstotheFYIsectioninChapter2ofthetextbookonproductsandpercentagechanges.Supplement8-5mayhelpaswell.

7. WhyPopulationGrowthReducestheCapital–LaborRatiobynk

Thepresenceofthenktermintheequationgoverningtheevolutionofthecapital–laborratiocantroublestudents.Thesimplestformalderivationofthistermisprobablyasfollows:

Sincek=K/L,wecanusestandardresultsongrowthratestowrite

∆k/k=∆K/K–∆L/L.

Since∆L/L=n,wewrite

∆k/k=∆K/K–n;

multiplyingbykyields

∆k=(∆K/K)k–nk.

(Notethatwehavenowobtainedthetermthattendsmosttoconfusestudents.)

Usingk=K/L,wehave

∆k=∆K/L–nk,

andweobtaintheequationinthetextbynotingthat

∆K=sF(K,L)–δK

⇒∆K/L=sf(k)–δk.

8. ConfusionBetweentheSteadyStateandtheGoldenRule

Thisisoneofthebiggestproblemsthatstudentsencounter.Isuggestthefollowingremedies:

(a) DownplaythediscussionoftheGoldenRule,perhapspresentingitlessformallyaftertheanalysisofpopulationgrowthandtechnologicalprogress,asinSupplement9-7.

(b) StressthetechnologicalnatureoftheGoldenRule—itcanbedefinedindependentlyofanybehavioralassumptions.

(c) Makesurethatstudentsunderstandthedifferencebetweenf(k)andsf(k)inthegraphicalanalysis.

UseoftheWebsite

ThechapterexercisesontheSolowgrowthmodelcanseemalittleintimidatingtothestudentsatfirst,soitmaybeagoodideatofocussomeattentionjustonsteadystates.Forexample,studentscouldgraphcombinationsofpopulationgrowthratesandsavingratesthatkeepoutputperworkerconstant.

SincetheSolowgrowthmodelcanseemabstracttostudents,anotherideaistocalibratethemodelroughlytotheU.S.economyandthenaskquestionsaboutthelong-runeffectsofchangesinsavingratessuchasthoseobservedinthe1980s.

Theweb-basedsoftwarecanbeusedtolookattheGoldenRuleandtoshowthattheGoldenRuleisachievedasasteadystatewhenthesavingrateequalscapital’sshare.

Foraverydifficultquestion,onecanusethefactthatkistheonlysteady-statevariableinthemodel,soitispossibletoconsiderachangeinexogenousparametersinthemiddleofatransitionpathbyfindingthevalueofkandthenresettingtheinitialparameterstogivethatvalue.Forexample,onecouldaskthefollowingquestion:consideracountryinsteadystatewithn=0.01ands=0.1.Policymakerswanttodoubleoutputperworkeroverthenext20yearsandthenkeepitconstantatthenewlevel.Whatchangesinthesavingratenowandin20years’timeareneeded?

UseoftheEWebsite

UsetheEwebsitetodownloadannualdatafortheU.S.population16yearsandoldersince1950.Also,downloadannualdataforrealdisposablepersonalincomeoverthesametimeperiod.Computethegrowthrateofper-capitaincomeonaverageovereachdecade(1950s,1960s,etc.).Discusschangesinthegrowthrateofper-capitaincomeoverthesedecades.

ChapterSupplements

Thischaptercontainsthefollowingsupplements:

8-1 HowLongIstheLongRun?PartII

8-2 GrowthFacts

8-3 DoestheSolowModelReallyExplainJapaneseGrowth?(CaseStudy)

8-4 TheDeclineintheU.S.SavingRate

8-5 GrowthRates,Logarithms,andElasticities

8-6 Labor-ForceParticipation

8-7 BridgeJobsandtheTransitiontoRetirement

8-8 HowMuchVariationinper-CapitaOutputIsExplainedbysandn?(CaseStudies)

8-9 TheSolowGrowthModel:AnIntuitiveApproach—PartI

8-10 AdditionalReadings

LectureNotes

Introduction

Supplement8-1,“HowLongIstheLongRun?PartII”

Supplement8-2,“GrowthFacts”

Havinganalyzedtheoverallproduction,distribution,andallocationofnationalincome,wenowconsiderthedeterminantsoflong-rungrowth.Onestylizedfactofmacroeconomicsisthat,indevelopedeconomies,outputgrowsovertime.Thisgrowthisirregularandissometimesinterruptedbyperiodsoffallingoutput(recessions),buttheoveralltrendisindisputablyupward.RealGDPintheUnitedStateshastripledoverthepast50years,andper-capitarealGDPhasmorethandoubled.Lookingattheeconomicperformanceofdifferentcountries,itisalsoevidentthatdifferentcountriesenjoyverydifferentstandardsofliving.

Table8-1

Traditionally,macroeconomicsanalyzesthebehaviorofoutputintermsofbothitsoverallupwardtrendandthefluctuationsaroundthattrend.Thefluctuationsaroundthetrendareknownasthebusinesscycle.Muchofmacroeconomicsisdevotedtounderstandingtheseshort-runchanges;weturntothemlateron.Butmacroeconomistsarealsokeentounderstandthegrowthofthenaturalrateofoutputinthelongrun.Ourmodelofeconomicgrowth,knownastheSolowgrowthmodel,isanexplicitlydynamicanalysisthatshowshowthegrowthofoutputisaffectedbysaving,populationgrowth,andtechnologicalprogress.Thischapteranalyzestheroleofsavingandpopulationgrowth,whileChapter9examinestheroleoftechnologicalprogress.

8-1 TheAccumulationofCapital

Ourstartingpointistheproductionfunction:

Y=F(K,L).

Fromthisequation,weseethreepossiblesourcesoflong-runoutputgrowth.First,thecapitalstockmayincreaseovertime.Second,laboremployedmaychangeovertime,perhapsaspopulationchanges.Third,asdiscussedintheChapter9,theproductionfunctionitselfmaychangeovertime(technologicalprogress).Ouranalysisofeconomicgrowthconsidersallofthesefactorsbutfocusesprimarilyonthedeterminationofthecapitalstock.WhileourpreviousanalysisofnationalincomefixedKatK,wenowexaminethelong-rundeterminationofK.Forthepresent,wesupposenotechnologicalprogressandnopopulationgrowth.

TheSupplyandDemandforGoods

Supposethattheproductionfunctionhasconstantreturnstoscale.[Recallthatthismeansthat,foranypositivez,zF(K,L)=F(zK,zL).]Ifz=(1/L),

Y/L=F(K/L,1).

Thatis,wecanwritetheproductionfunctioninper-capitatermsandobtainafunctionofonlyonevariable—thecapital–laborratio—ratherthantwo.Lety=Y/Landk=K/L,andwritethisas

Figure8-1

y=f(k).

Asanexample,theCobb–DouglasfunctionY=(KL)1/2becomes

y=k1/2,

andthemoregeneralCobb–DouglasfunctionY=KαL1–αbecomes

y=kα.

Thiswayofwritingtheproductionfunctionismathematicallysimpleandhastheadvantageoffocusingourattentiononoutputperperson,whichisabettermeasureoflivingstandardsthantotaloutput.Keepinmind,though,thatitisjustausefultrick;wecouldalsocarryoutallthefollowinganalysiswiththeoriginalproductionfunctionandgetthesameanswers.

JustasintheanalysisofChapter3,themarginalproductofcapitalisimportantinthismodel;inthiscase,ittellsushowmuchextraoutputperworkerwillbeproducedifcapitalperworkerisincreased:

MPK=f(k+1)–f(k).

AsinChapter3,weexpectthemarginalproductofcapitaltodecreaseasthecapital–laborratioincreases.

TheSolowmodelisalong-runversionofourpreviousanalysisofnationalincomeand,likethatmodel,isbasedonequilibriuminthemarketsforgoodsandfactorsofproduction.Forsimplicity,wesupposethatthereisnogovernment(G=T=0).Wewriteeverythinginper-capitaterms:c=C/Y;i=I/Y.Wehaveequilibriuminthemarketforgoods:

y=c+i.

Wealsohaveasimpleconsumptionfunction:

c=(1–s)y.

Thismodelwritestheconsumptionfunctionintermsofthesavingrates.Weimmediatelyhavethefamiliarresultthatinvestmentequalssaving(i.e.,thereisequilibriuminthemarketforloanablefunds):

i=sy=sf(k).

Outputisdividedbetweenconsumptionandinvestment.Althoughitisnotexplicithere,therealinterestrateisadjustingtoensurethatsavingequalsinvestment,andthewagerateandrentalrateareadjustingtobringaboutequilibriuminthemarketsforfactorsofproduction,justasintheclassicalmodelofChapter3.

GrowthintheCapitalStockandtheSteadyState

Figure8-2

Investmentmeansthattheeconomyisacquiringnewfactories,machines,houses,andsoforth,whichtendstoincreasetheeconomy’scapitalstockandallowformoreproduction.But,atthesametime,someofthesemachinesandfactorieswearoutandhavetobereplaced.Thisdepreciationdecreasesthecapitalstock.Iftherewerenoinvestmentatall,thecapitalstockwoulddeclineovertime.Wesupposethatδpercentofthecapitalstockwearsouteachperiod,soifthecapitalstockatthestartoftheperiodisk,thedepreciationduringtheperiodisδk.Therateofdepreciationcanbeinterpretedintermsofthelifetimeofthetypicalpieceofcapital.If,say,thetypicalmachinelastsforfiveyears,thenthedepreciationrateis20percent(sinceafactorywith100machineswouldhavetoreplaceanaverageof20everyyear).Ingeneral,theaveragelifetimeofapieceofcapitalequals1/δ.Forsometypesofcapital,suchasbuildings,thedepreciationratemightbeverylow(perhaps1percentto2percent),whilefor,say,personalcomputers,itwouldbemuchhigher(perhaps10percentto20percent).

Figure8-3

Welookatasituationwheretheeconomyisinasteadystate,whichentailsfindingabalancebetweentheinvestmentthatiscarriedoutineachperiodandthedepreciationofthecapitalstockthatoccursovertime.Theoverallchangeinthecapitalstockistheneteffectofnewinvestmentanddepreciation:

∆k=sf(k)–δk.

Theeconomywillbeinsteadystateifthecapitalstockisconstant:∆k=0.Inthiscase,theonlyinvestmentbeingundertakenisreplacementinvestment.Theequilibriumconditionisjust

sf(k)=δk.

Thisequationdefinesthesteady-statevalueofk,whichwecallk*.

WecanworkoutanexplicitexamplefortheCobb–Douglascase:

y=k1/2.

Thismeansthatinthesteadystate,

sk1/2=δk

⇒s=δk1/2

⇒s/δ=k1/2

⇒k*=(s/δ)2.

Forexample,ifpeoplesave30percentoftheirincomeandthedepreciationrateis10percent,thenthesteady-statecapital–laborratiois(0.3/0.1)2=9.Thisillustratestwoimportantandintuitiveresults:increasesinthesavingrateincreasethesteady-statecapital–laborratio,whileincreasesinthedepreciationratedecreaseit.

Figure8-4

Ifkisatitssteady-statevalue,itwillnotchange.Whathappensifkisatsomeothervalue?Theansweristhatthesteady-stateequilibriumisstable,meaningthat,ifyoustarttheeconomyfromanotherpoint,itwilltendtoapproachthesteadystate.Supposethatk<k*.Then,fromFigure8-4,investmentexceedsdepreciation.Weareaddingnewcapitalfasterthantheoldcapitaliswearingout,sothecapitalstockisincreasingovertime.Exactlytheoppositeistrueifwestartwithacapitalstockinexcessofk*.

ApproachingtheSteadyState:ANumericalExample

Considerthepreviousexample:y=k1/2;s=0.3;δ=0.1.Supposethattheinitialcapitalstockperworkeris4.Then,outputperworkerequals2,sosavingperworkerequals0.6(0.3×2)andtotaldepreciationperworkerequals0.4(0.1×4).Sincenewinvestmentexceedsdepreciation,thecapitalstockperworkeratthestartofthenextperiodwillbehigher,k=4.2.Thecapitalstockwillcontinuetoincreaseuntilwereachthesteadystate,which(ascalculatedearlier)isk*=9.

Table8-2

CaseStudy:TheMiracleofJapaneseandGermanGrowth

Supplement8-3,“DoestheSolowMethodReallyExplainJapaneseGrowth?”

AttheendofWorldWarII,thedefeatedcountriesofJapanandGermanywereinpooreconomicshapebecausethewarhaddestroyedalargepartoftheircapitalstocks.TheSolowgrowthmodelpredictsthat,ifacountryisinsteadystateandthenlosesalotofitscapital,itwillsufferanimmediatelossinoutputbutwillalsoexperiencerelativelyrapideconomicgrowthasitbuildsitscapitalstockbackuptothesteady-statelevel.ThisfitstheexperienceofJapanandGermanyinthedecadesimmediatelyfollowingWorldWarII,whenbothcountriesexhibitedveryhighgrowthratesinoutputpercapita—8.0percentperyearinJapanand6.5percentperyearinGermanyovertheperiod1946–1972,comparedwithonly2.1percentperyearintheUnitedStates.Followingtheirpostwargrowthspurt,bothJapanandGermanytransitionedtomoremoderateratesofgrowth,closertothatoftheUnitedStates.Overtheperiod1972–2000,outputpercapitagrew2.4percentperyearinJapanand1.8percentperyearinGermany,closertothe1.8percentperyearratefortheUnitedStates.

HowSavingAffectsGrowth

TheSolowgrowthmodelindicatesthatthesavingrateisanimportantdeterminantofthesteady-statecapitalstockandthelevelofoutput.Acountrywithahighersavingrateinvestsmoreand

Figure8-5

Supplement8-4,“TheDeclineintheU.S.SavingRate”

socansustainahighercapitalstock,implyinghigheroutputinturn.

Inthesteadystateinthemodelaspresentedsofar,thecapitalstockisnotgrowing.Whileahighersavingrateisassociatedwithahighersteady-statecapitalstockandahigherlevelofoutput,ahighersavingratedoesnotimplyahighergrowthrateinthelongrun.Itisapopularmisconceptionthatwemustincreaseoursavingratetoachievehigherlong-rungrowth.Highersavingwillincreasegrowthintheshortrunbut,accordingtotheSolowmodel,willnotdosointhelongrun.

8-2 TheGoldenRuleLevelofCapital

Whileahighercapitalstockimplieshigheroutput,thisdoesnotmeanthatahighercapitalstockisnecessarilydesirable.Tosustainahighlevelofthecapitalstock,alotofoutputmayhavetobedevotedtoreplacementinvestment,soitwillnotbeavailableforconsumption.Wenowcomparesteadystatesintermsofconsumption.

ComparingSteadyStates

ThesteadystatewiththehighestpossiblelevelofconsumptionisknownastheGoldenRulelevelofcapitalaccumulation.TheGoldenRuledependspurelyonthetechnologicalpossibilitiesoftheeconomy.Itcanbeanalyzedwithoutreferencetopeople’sbehavior(thatis,withoutreferencetothesavingrate).

Figure8-6

Atagivenvalueofthecapital–laborratio(k),therewillbedepreciationequaltoδk.Tomaintainthislevelofthecapital–laborratio,anamountofoutputequaltoδkmustbesetasideforreplacementinvestment.Theoutputthatthenremains[f(k)–δk]isthelevelofconsumptionthatcanbesustainedatthisvalueofk.

ToidentifytheGoldenRulevalueofthecapital–laborratio,considertheeffectsofchangesinkonsustainableconsumption.Aone-unitincreaseinkraisesoutputbythemarginalproductofcapital.Italsoimpliesthatanextraδunitsofoutputmustbesetasidetomaintainthecapital–laborratioatitsnewhigherlevel.IfMPK>δ,thenincreasesinthecapital–laborratiowillincreaseoutputbymorethantherequiredincreaseindepreciation,soconsumptionalsoincreases.IfMPK<δ,increasesinthecapital–laborratioactuallyreducesustainableconsumption.TheGoldenRulelevelofcapitalaccumulation(k*gold)iswherethemarginalproductofcapital,netofdepreciation,equalszero:

MPK–δ=0.

Graphically,itisthepointwherethef(k)lineisatthemaximumdistanceabovetheδkline.

Figure8-7

FindingtheGoldenRuleSteadyState:ANumericalExample

Policymakerswhowishtoinfluencethemarginalproductofcapitalcanenactpoliciesaimedatinfluencingthesavingrate.

Supposethat,bymeansofappropriatetaxpolicies,thegovernmentcaneffectivelychoosethesavingrate.Byanappropriatechoiceofs,itcouldplacetheeconomyattheGoldenRule.Thatis,itcouldchoosessuchthatthesteadystateoftheeconomy(k*)correspondedtotheGoldenRule(k*gold).Forexample,ify=k1/2,thentheGoldenRuleoccurswhens=1/2.Ifthedepreciationrateis10percent,then,atthisequilibrium,k*=k*gold=25;y=5;andc=2.5.

Table8-3

TheTransitiontotheGoldenRuleSteadyState

SupposethatpolicymakersdecidethattheywouldliketomovetheeconomytotheGoldenRule.Therearetwopossibilities:westartoffeitherwithmorecapitalthanattheGoldenRuleorwithlesscapitalthanattheGoldenRule.First,considerthelessrealisticcase,wherewehavemorecapitalthanattheGoldenRule,sothesavingrateistoohigh.Supposethatattimet0thesavingrateissuddenlyreduced.Tostart,wehavethesameamountofoutputandwearesavingless,soweareabletoconsumemoreimmediately.Gradually,depreciationwillstarttoeatintothecapitalstock,sinceitisnowwearingoutfasterthanwearereplacingit.Asthishappens,output,andthusconsumption,willfall.But,bythedefinitionoftheGoldenRule,consumptionwillbehigherattheGoldenRulethanitwasbeforethechangeinthesavingrate.Thus,consumptionishigherateverypointintime.

Figure8-8

IftheeconomyisbelowtheGoldenRuleandpolicymakerswanttomovetotheGoldenRule,theymustincreasethesavingrate.Initially,thismeansthatwehavelessoutputforconsumptionpurposes.Butinitially,wedonotgetanybenefitfromthehighersavingrate.Thebenefitcomesaboutonlygradually,asthecapital–laborratioincreases.Thus,intheshortrun,consumptionfalls.

Figure8-9

GiventhattheU.S.economyisbelowtheGoldenRule(asshownlater),shouldpolicymakersbetryingtoencouragesaving?Ifweincreasethesavingrate,theshort-runconsequenceisadeclineinlivingstandards,sinceconsumptionmustfallimmediately,butoutputwillgrowonlyslowlyovertime.Sowetradeofflowerconsumptioninthepresentforhigherconsumptioninthefuture.Itisnotself-evidentthatthisisdesirable.Overtime,peopledieandnewgenerationsareborn.Currentgenerationsmakethesacrifice,whilefuturegenerationsreapthebenefit.TheGoldenRuledoesnottellustheoptimallevelofcapitalaccumulationbutsimplypicksoutonepointofinterest.

Whetherornotpolicymakersshouldencouragesavingthusdependsonhowweweightherelativeinterestsofcurrentandfuturegenerations.Sinceaninfinitenumberoffuturegenerationswouldbemadebetteroff,itmightseemworththeshort-runsacrifice.But,conversely,historyteachesthattechnologicalprogressaddstoeconomicgrowth,socontinuingimprovementsintechnologicalprogressarelikelytomakefuturegenerationsbetteroffanyway.Thedebateabouteconomicgrowthisthusmoresubtlethancasualreadingofthenewspaperswouldindicate.Itisnotself-evidentthatwewanttoencourageeconomicgrowth;ourdecisionsondesirablelevelsofgrowthmustbemotivatedinpartbyconsiderationsofintergenerationaldistribution.

8-3 PopulationGrowth

TheSolowmodelteachesusthatwecannotexplainsustainedeconomicgrowthintermsofgrowthincapitalperworkersincetheeconomywilltendtowardasteadystatewherecapitalperworkerisconstant.Wenowconsiderpopulationchangeasapossibleexplanationofsustainedeconomicgrowth.

Wewillassumeapopulationgrowthrateequalton.Forexample,ifn=0.02,thenthepopulationincreasesby2percenteveryyear.Ifitis100milliononeyear,itwillbe102millionthenextyear.Ifitis250millionthisyear,itwillbe255millionnextyear.

TheSteadyStatewithPopulationGrowth

Thedifferencethismakestothemodelisthatthechangeinthecapitalstockbecomes

∆k=i–δk–nk,

sincepopulationgrowthdecreasestheamountofcapitalperworker,otherthingsbeingequal.Tokeepthecapital–laborratioconstant,wenotonlyneedinvestmenttoreplacedepreciatedcapital,wealsoneedinvestmentwithwhichtoequipnewworkers;weneedtosupplythennewworkerswithkunitsofcapitaleach.Sothesteady-statecapitalstockisnowdefinedby

i=sf(k)=(n+δ)k.

Figure8-10

Withthisonechange,ouranalysisproceedsmuchasbefore.Graphically,wesimplylookfortheintersectionofsf(k)with(n+δ)k.

TheEffectsofPopulationGrowth

Supplement8-5,“GrowthRates,Logarithms,andElasticities”

Wenowhaveanexplanationforsteady-stategrowthabsentfromthefirstversionoftheSolowmodel.Ifpopulationisgrowing,theninthesteadystatewewillobserveoutputandthecapitalstockalsogrowingattheraten.(Recallthattheproductionfunctionisconstantreturnstoscale,whichmeansthatifKisgrowingattheratenandLisgrowingattheraten,thenYmustalsobegrowingattheraten.)

PopulationGrowth=0

PopulationGrowth=n

Lisconstant

Lgrowsatraten

Kisconstant

Kgrowsatraten

kisconstant

kisconstant

yisconstant

yisconstant

Yisconstant

Ygrowsatraten

Figure8-11

Supplement8-6,“LaborForceParticipation”

Populationgrowthisanotherpossiblecauseofincomedifferencesacrosscountries.TheSolowgrowthmodelpredictsthat,otherthingsequal,countrieswithhigherratesofpopulationgrowthwillhavelowersteady-statecapital–laborratios.

TheintroductionofpopulationgrowthalsoalterstheGoldenRule.WherepreviouslyourconditionfortheGoldenRulewasMPK–δ=0,itnowbecomesMPK–δ=n.

Supplement8-7,“BridgeJobsandtheTransitiontoRetirement”

CaseStudy:InvestmentandPopulationGrowthAroundtheWorld

Thesteady-statefortheSolowmodelisdefinedby

sf(k)=(n+δ)k.

AssumingthattheproductionfunctionisCobb–Douglasandgivenbyy=f(k)=k,wecansubstituteforf(k)andkinthesteady-staterelationshipandexpressincomeperworkeras

y=[s/(n+δ)]/(1-).

Thisequationrelatesthesteady-statelevelofincome(y)positivelytotherateofsavingandinvest