Topic 1: The Solow Model of Economic Growth

1 EC4010 Notes,2005(KarlWhelan)1 Topic1: TheSolow Model of EconomicGrowthMacroeconomicsis nota one-size- ts-alltype of wouldbe a duantingtaskto evenattemptto constructa modelthatexplainedallinterestingmacroeco nomicphenomena,andany such modelwouldundoubtedlybe complicatedandunwieldy, makingit di culttolearn(andteach).For thisreason,macroeconomiststendto adopta moreeclecticapproach,withmodelsoftenbein gdevelopedwiththeintentionof helpingto explainoneparticularaspectof rstmodelthatwe willlook at in thisclass,a modelof economicgrowthoriginallydeveloped by MIT'sRobertSolow in the1950s,is a good exampleof 'spurposein developingthemodelwas to deliberatelyignoresomeimportant aspectsof macroeconomics,such as short-run uctuationsin employment andsavingsrates.

2 In orderto developa modelthatattemptedto describe thelong-runevolutionof theeconomy. Theresultingpaper (AContributionto theTheoryof EconomicGrowth,QJE, 1956)remainshighlyin uentialeven today and,despiteits relative simplicity, it conveysa number of veryusefulinsights aboutthedynamicsof modelis alsoworthteachingfroma methodologicalperspective becauseit providesa simpleexampleof thetype of dynamicmodelthatis commonlyusedintoday'smoreadvancedmacroec onomictheory:Forthosebrave soulsamongstyouwhomay be interestedin studyingmoremacroeconomicsafterthiscours e,notethatsomeofthetricks andterminologyintroducedhere(such as theuselogsto derive growthrates,andtheconceptsof steady-stategrowthandconvergent dynamics)arewidespreadin Asideon NotationWe areinterestedin modellingchangesover timein usefulmathe-maticalshorthandthatsaves us fromhavingto writedownderivatives withrespectto timeeverywhereis to write_Yt=dYtdt(1)

3 Whatwe arereallyinterestedin,though,isgrowthrat esof series:If I tellyouGDPwasupby 5 millioneuros,thatmay soundlike a lot,butunlesswe scaleit by theoveralllevelof GDP, it' ,whatwe areinterestedin calculatingis_YtYt, andthisis ourmathematicalexpressionforthegrowthrat eof a ,2005(KarlWhelan)2 TheSolow Model 'sIngredientsThemodelassumesthatGDP is producedaccordingto anaggregateproductionfunctiontechnology. It is worth aggingthatmostof thekeyresultsforSolow'smodelcanbeobtaine dusingany of thestandardproductionfunctionsthatyousee in microeconomicproductiontheory.

4 However,forconcreteness,I amgoingto be speci candlimitus to thecasein which theproductionfunctiontakes theCobb-Douglasform:Yt=AtK tL1 t0< <1(2)whereKtis capitalinputandLtis higheroutputwithouthavingto callincreasesinAt\technologicalprogress" andwe willlooselyreferto thisas the\technology"term,butultimatelyAtis simplya measureof productive e ciency. BecauseanincreaseinAtincreasestheproduct ivenessof theotherfactors,it is alsosometimesknownasTotalFactorProductiv ity(TFP),andthisis thetermmostcommonlyusedin empiricalpapersthatattemptto additionto theproductionfunction,themodelhasfouroth erequations.

5 Capitalaccumulatesaccordingto_Kt=Yt Ct Kt(3)In otherwords,theadditionto thecapitalstock each period dependspositivelyonsavings(thisis a closed-economy modelso savingsequalsinvestment) andnegativelyondepreciation,which is assumedto take placeat rate . Labourinputgrowsat raten:_LtLt=n(4) Technologicalprogressgrowsat rateg:_AtAt=g A fractionsof outputis savedeach Ct=sYt(5)EC4010 Notes,2005(KarlWhelan)3We have notputtimesubscriptsontherateof populationgrowth,therateof tech-nologicalprogress,therateof depreciationof capitalor thesavingsrate,becausewe willgenerallyconsiderthesetobe constant.

6 TheSolow modeldoes notattempttoexplain uctuationsin ,we dowishto charaterisethedynamicsof themodelwell enoughto be ableto gureoutwhathappensif ,forinstance,we willbe interestedin whathappenswhenthereis a once-o increasein Digressionon the ProductionFunctionTwo well-knownfeaturesof theCobb-Douglasproductionfunctionarewort hrecappinghere: Constant returnsto scale(a doublingof inputsleadsto a doublingof outputs):At( Kt) ( Lt)1 = 1 Yt= Yt(6) Decreasingmarginalreturnsto factoraccumulation(addingextracapitalwhi lehold-inglabourinput xedyieldsever-smallerincreasesin output):@Y@K= AtK 1tL1 t(7) @2Y@K!

7 = ( 1)AtK 2tL1 t<0(8)Thisturnsoutto be thekeyelement of it is sensible:If a rmacquiresanextraunitof capital ,it the rmkeepspilingonextracapitalwithoutraisin gthenumber of workersavailableto usethiscapital,theincreasesin outputwillprobablytaper o .IntheCobb-Douglascase,theparameter dictatesthepaceof thistaperingo .ModellingDynamicsusingTricks withLogsOneof thepurposesof thisclassis to teach you sometricks 'sthe rst:Theuseof logarithimscanbe thefollowingproperty:d(logYt)dt=d(logYt) dYtdYtdt=_YtYt(9)EC4010 Notes,2005(KarlWhelan)4 Thegrowthrateof a seriesis thesameas thederivative of itslogwithrespectto time(notetheuseof chain-ruleof di erentiationin theabove equation.)

8 Two otherusefulpropertiesof logarithmsthatwillalsohelpus characterisethedynamicsof theSolow modelarethefollowing:log(XY) =logX+ logY(10)log XY =YlogX(11)We canapplyeach of thesepropertiesto geta usefulrepresentationof thegrowthrateofoutput:log(Yt) =log(AtK tL1 t)(12)=log(At) + log(K t) + log(L1 t)(13)=log(At) + log(Kt) + (1 ) log(Lt)(14)Now takingthederivative withrespectto time,we gettherequiredformula:_YtYt=_AtAt+ _KtKt+ (1 )_LtLt(15)Thistakes us fromtheCobb-Douglasformulainvolvinglevel sto a outputper worker is simply_YtYt _LtLt=_AtAt+ _KtKt _LtLt!

9 (16)Thus,therearetwo sourcesof increasesin outputper worker: Technologicalprogress Capitaldeepening( capitalper worker)TheSolow modelprovidesa usefulframeworkforunderstandinghow technologicalprogressandcapitaldeepening interactto determinethegrowthrateof outputper rstthingwe aregoingto dowiththeSolow modelis gureoutwhatthiseconomylookslike alonga pathonwhich outputgrowthis constant. Macroeconomistsreferto suchconstant don'tnecessarilywant to studyEC4010 Notes,2005(KarlWhelan)5onlyconstant-grow thpaths,butwe willseebelow thattheSolow-modeleconomy tendstoconvergeover ,givenconstant growthratesfortechnologyandlabourinput,a llvaria-tionsin outputgrowtharedueto variationsin thegrowthrateof capitalinput.

10 _YtYt=g+ _KtKt+ (1 )n(17)Soforoutputgrowthto be constant, we mustalsohave canalsoshow thatthesegrowthratesforcapitalandoutputm ustbe thesame,sothatthecapital-outputratiois constant alonga constant seethis,re-writethecapitalaccumulationeq uationas_Kt=sYt Kt(18)anddivideacrossbyKtonbothsides_KtK t=sYtKt (19)Thegrowthrateof thecapitalstock dependsnegativelyonthecapital-outputrati oKtYt. So,forthecapitalstock to be growingat a constant rate,thenKtYtmustbe constant. ButKtYtcanonlybe constant if thegrowthrateofKtis thesameas mind,we seethatthesteady-stategrowthratemustsati sfy_YtYt=g+ _YtYt+ (1 )n(20)Subtracting _YtYtfrombothsides,we get(1 )_YtYt=g+ (1 )n(21)So,thesteady-stategrowthrateis_YtY t=g1 +n(22)Onlythegrowthrateof technology,g, andthefactorcontrollingtheextent of diminishingmarginalreturnsto capital , , cana ectthegrowthrateof outputper a key result.