2THEAMERICANECONOMICREVIEWMONTHYEARHelpman(1991),ourmodelcharacterizes - PDF document

2THEAMERICANECONOMICREVIEWMONTHYEARHelpman(1991),ourmodelcharacterizestechnologicalprogressasanexpansioninthenumberofinputvarieties.Thenumberofvarietiesevolvesendogenouslyinresponsetoproducers'incentivestoaddtotherangeofinputstheyuse,andincreasesinthenumberofvarietiesraisetheaveragelevelofproductivity.Ourcontributionistomakethemodelstochastic,sothatitcanbeusedtostudyitsimplicationsforoutputvolatility.Inparticular,weassumethateachvari-etycanbeaectedbyaproductivityshock;thustheexpansioninthenumberofvarietiescanprovidediversicationbenets,andhencereducethelevelofvolatilityoftheeconomy.1Inotherwords,thereductioninvolatilityarisesasalikelyby-productofrms'incentivestoincreaseproductivity.Assuch,ourmodelhighlightsahithertooverlookedimplicationofexpanding-varietygrowthmodels,whichmakesthemsuitabletoexplainthedeclineinvolatilitythataccompaniesthedevelopmentprocess.Wesay\suitabletoexplain"because,interestingly,oncetechnologicaldiver-sicationisembeddedinanendogenousgrowthmodelwithmultiplerms,itispossibletogenerateexampleswherevolatilityanddevelopmentdonotnecessarilymoveinoppositedirections.Thishappens,forexample,ifasignicantnumberofrmsadoptsaninputthatisalreadywidelyusedbyotherrms;theeconomyasawholemaythenbecomehighlytechnologicallyconcentratedandhenceexposedtoshockstothatparticularinput,leadingtoepisodicsurgesinvolatility|higherproductivityinthiscasecancomeatthecostofhighervolatility.Inpractice,however,developmentandvolatilitymoveinoppositedirectionsmostofthetime,andthisisindeedthecaseinvirtuallyallournumericalexperiments.Thisoc-cursbecausetheintroductionofanewvarietyintheeconomyalwaysincreasesthelevelofdevelopment,andraisesthedegreeoftechnologicaldiversicationbyreducingthecontributiontooutputofpreviouslyexistingvarieties(thusloweringvolatility).Acalibratedversionofthemodelcanyieldadeclineinvolatilitywithdevelopmentquantitativelycomparabletothatinthedata.Asimpleexampleofthemechanismoftechnologicaldiversicationisoeredbyacomparisonofaneconomyusingonlylabourandaneconomyusinglabourandcapital.Understandardassumptionsontechnology,thelatterwilltendtobemoreproductiveonaper-capitabasis.Ourpointisthatitwillalsobelessvolatile.Inparticular,anyshockthatreducesthesupplyoflabour(suchasanepidemic,ageneralstrike,etc.)willhaveabiggernegativeimpactontheeconomythatdoesnothavescopetosubstitutelabourwithcapital.Or,tothinkofacurrentlymorerealisticexample,considerleading-edgesteelproducersthathavethecapacitytoprocessironoreofarangeofqualitiesascomparedtomorebasicproducerswhocanonlyaccepthigh-qualityoresasinput.Clearlytheformeraremoreproductive,and,inaddition,theyshouldbelesssusceptibletoshockstothe(globalorlocal)supplyofhigh-qualityironore.21Inputvarietiesarebroadlyconstruedtoencompassbothtangibleandintangibleinputsortechnolo-gies.Shocksarevariety-specicandtotheextentthatthevarietiesareusedbyapositivemeasureofrms,theyleadtoaggregatevolatility.2Throughoutthepaper,wefocustheanalysisonthecaseinwhichdierentvarietiesaregross 4THEAMERICANECONOMICREVIEWMONTHYEARFurthermore,inourmodeltheprocessoftechnologicaldiversicationtakesplacewithintherm,notacrossrms:Finally,ourresultsdonothingeonnancialdevelopment.Thesetheoreticaldierencesleadtoimportantdierencesinempiricalimpli-cations.First,nancialdiversicationmodelspredictanincreaseinrm-levelvolatilitywiththelevelofdevelopment,andanegativecomovementbetweenag-gregateandrm-levelvolatility.Instead,ourmodelpredictsadeclineinrm-levelvolatilitywithdevelopmentandapositivecomovementbetweenaggregateandrm-levelvolatility.Onthesetwopredictions,ourmodelndssupportinrecentworkbyDavis,Haltiwanger,JarminandMiranda(2006),whodocumentthatintheUnitedStates,overtime,privatelyheldrmshaveexperiencedasubstan-tialdeclineinvolatility;theauthorsfurthershowthatthedeclineinaggregatevolatilityintheUnitedStateshasbeenoverwhelminglydrivenbythedeclineinrm-levelvolatility(andnotbytheaggregationofhighlyvolatilermsexposedtoincreasinglylesscorrelatedidiosyncraticshocks).Inthenextsectionwedis-cussnewevidencefor17othercountriesconrmingthetendencyforapositivecomovementbetweenaggregateandrm-levelvolatility.Asecondtestablepredictionofmodelsofnancialdiversicationisthatthedeclineinaggregatevolatilitywithdevelopmentisbroughtaboutbynancialdevelopment.Inourmodel,thedeclineinvolatilitytakesplaceindependentlyofthelevelofnancialdevelopment.Asweargueinthenextsection,thisimpli-cationiscorroboratedbytheevidence.Thestrongnegativecorrelationbetweenvolatilityanddevelopmenttakesplaceatalllevelsofnancialdevelopment.Putdierently,evencontrollingforthelevelofnancialdevelopment,thereremainsastrongnegativecorrelationbetweenvolatilityanddevelopmentthatneedsexpla-nation.Whileweviewbothmarginsofdiversicationfortherm,nancialandtechnological,ascomplementaryandempiricallyplausible,ourmodelwillfocusexclusivelyonthesecondone.6Asmentioned,ourmodelpositsnotradeobetweenproductivityandvolatil-ityatthemicroeconomiclevel.Theabsenceofatradeoismotivatedbythendingthatcountriesatearlystagesofdevelopmenttendtospecializeinlow-productivity,high-riskactivities,whereastheoppositepatternisobservedatlaterstagesofdevelopment.Moreover,evenwithinnarrowlydenedsectors,develop-ingcountriestendtofeaturebothlowerproductivityandhighervolatilitythandevelopedcountries.(SeeKorenandTenreyro(2007)).7;8oursettingrisk-neutralrmsstillwantto\diversify"(i.e.,expandthenumberofinputsinproduction).6Technologicaldiversicationisalsocomplementarytoothernance-relatedmechanismsemphasizedintheliterature.Inparticular,shockscanbeampliedbyintroducingnancialfrictions,ataskwedonotundertakeintheinterestofclarityandsimplicity.Formodelswithnancialfrictions,see,amongothers,BernankeandGertler(1990),KiyotakiandMoore(1997),Aghion,Angeletos,BanerjeeandManova(2010).7Thesectoralcompositionoftheeconomyalonecannotaccountforthedierencesinvolatilitybetweendevelopedanddevelopingcountries;the\within"sectordeclineinvolatilityisatleastasimportantinexplainingvolatilitydierencesbetweendevelopedanddevelopingeconomies(KorenandTenreyro2007).8Indepartingfromanecessarytradeobetweenproductivityandvolatility,ourpaperisclosertoKraayandVentura(2007),thoughthemechanismsaredierent:intheirmodel,thekeyideaisthat 6THEAMERICANECONOMICREVIEWMONTHYEARcontributionofthepaper,themodelismorestylizedinotherdimensionsthathavebeenemphasizedintherealbusinesscycle(RBC)ortheNewKeynesianliterature.Thepaperdoes,ontheotherhand,speaktootherregularitiesthatarenotaddressedintheRBCorNewKeynesianliterature,whichwediscussinthenextsection.Thepaperisorganizedasfollows.SectionIdocumentsasetofempiricalobser-vationsthatmotivateourmodelanddierentiateitfromalternativeexplanations.SectionIIpresentsthemodeloftechnologicaldiversicationandderivesitsim-plicationsforaggregatedynamics.SectionIIIpresentsaquantitativeanalysisofthemodel.SectionIVoersconcludingremarks.TheWebAppendixprovidesadditionalevidencesupportingtheregularitiesinSectionI.Itthenpresentstheproofs,generalizesthemodel,anddiscussesitsrobustnessunderdierentassumptions.Inparticular,itstudiestheconditionsunderwhichtechnologicaldiversicationtakesplacewhenvarietiesaregrosscomplements,asisthecaseintheO-ringtheoryformulatedbyKremer(1993),anditworksouttheimpli-cationsofthemodelunderdierentassumptionsregardingtechnologyandriskpreferences.I.EmpiricalMotivationThissectionpresentsthemainempiricalobservationsthatmotivatethethe-oreticalmodel,andalongwhichweshalllaterevaluateit.Italsodiscussesasetofauxiliaryempiricalresultsthatjustifythesearchfornewmodels.Intheinterestofspace,mostofthesupportingtablesandguresarereportedintheWebAppendix.A.EmpiricalObservationsEmpiricalObservation1.GDPvolatilitydeclineswithdevelopment,bothinthecrosssectionandforagivencountryovertime.Thenegativeassociationbetweenaggregatevolatilityandthelevelofdevelop-ment,notedinLucas(1988)'sseminalpaper,isoneofthestylizedfactsinthemacro-developmentliteratureandthestartingmotivationofthispaper.There-lationissummarizedintherstcolumnofTable1;whichreportstheresultsfromaregressionofthe(log)levelofvolatility,measuredasthestandarddeviationoftheannualgrowthrateofrealGDPpercapitaovernonoverlappingdecadesfrom1960through2007,ontheaverage(log)levelofrealGDPpercapitaofthecor-respondingdecade.ThedatacomefromthePennWorldTables(PWT,version6.3)andareadjustedforpurchasing-powerparity(PPP).Thesecondcolumndisplaystheregressionresultsaftercontrollingforcountry-specicxedeects;itindicatesthatforagivencountryovertime,growthandchangesinvolatilityopmentareconsideredunrelatedphenomenaor,putdierently,thelevelofdevelopmentplaysnoroleindetermining uctuations. 8THEAMERICANECONOMICREVIEWMONTHYEARTable2|Firm-LevelVolatilityandSize Notes:Allvariablesareinlogs.Theequationsusethe5yearstandarddeviationofannual(real)salesgrowthratesfrom1975to2007.Thetwosizemeasures(numberofemployeesandvolumeofsales)arecomputedattheirmeanvaluesoverthelustrum.Yearxedeectsincludedinallregressions.Clustered(byrm)standarderrorsinbrackets.*Signicantat10%;**signicantat5%;***signicantat1%.fornonoverlapping5-yearperiodsfrom1975through2005.Thenegativecorre-lationremainsstrongevenifweincluderm-xedeectstoconsiderwithin-rmvariationonly.InTableA1oftheWebAppendixwereportnewevidenceonthecross-sectionalrelationbetweenrm-levelvolatilityandsizeforabroadgroupofcountriesatdierentstagesofdevelopment.Thesize-volatilityrelationshipisconsistentlynegativeinallcountries.Thereisalsoevidencethattheshareofsmallrmsintheeconomy(measuredintermsofoutputoremployment)correlatesnegativelywithincomepercapitabothacrosscountries(LeidholmandMead(1987)andBanerji(1978))andwithincountriesovertime(Little,MazumdarandPage(1987)andSteel(1993)).Thiswillbethecaseinourmodel:economieswithlowerincomepercapitahaveahighershareofsmallandhighlyvolatilerms(i.e.,rmsusingarelativelysmallnumberofvarieties).Aspreviouslystated,inourmodeltechnologicaldiversicationstemsfromthediversicationof(broadlyconstrued)inputs,notoutputs.Itishencepertinenttonotethatthedeclineinrmvolatilitywithsizeisnotdrivensimplybylargermsoperatinginabiggernumberofbusinesssegments;inotherwords,diversicationinoutputalonedoesnotaccountforthenegativecorrelation.WeinvestigatethisissueintheWebAppendix,whereweconrmthattheresultsinTable2arerobusttocontrollingforthenumberofbusinesssegmentsinwhichrmsoperateandthattheseresultsalsoholdforasampleofrmsoperatinginasinglebusinesssegment|thatis,rmswithnodiversicationalongtheoutputdimension.EmpiricalObservation3.Firm-levelandaggregatevolatilitytendtodisplayapositivecomovement. VOL.VOLNO.ISSUETECHNOLOGICALDIVERSIFICATION9Aggregatevolatilityandthevolatilityofprivatelyownedrmstendtocomovepositively.AsshownbyDavisetal.(2006),thedeclineinaggregatevolatilityintheU.S.economythattookplacefromthemid1980suntilthemid2000shasbeenoverwhelminglydrivenbythedeclineinvolatilityofnonlistedrmsandnotbytheaggregationofincreasinglymorevolatilermsdisplayingprogressivelylowercorrelationintheirperformance.14;15Asimilarlypositivecomovementbe-tweenrm-levelandaggregatevolatilityisdocumentedforFrancebyThesmarandThoenig(2011)andforGermanybyStrotmann,DopkeandBuch(2006).IntheWebAppendixwestudyarelativelylongtimeseriesofrm-leveldataforHungary,conrmingapositivecomovementbetweenrmandaggregatevolatility.TheresultsarereportedinFigureA1.Inaddition,theWebAppendixreportsfurtherevidencefor14othercountries,forwhichwehaveshortertimeseriesofrm-leveldata.Theresults,whileonlysuggestivegiventhedatalimitations,in-dicatethatrm-levelandaggregatevolatilitytendtomoveinthesamedirection.B.AlternativeExplanationsandAdditionalEvidenceFinancialdevelopment.Thepositivecomovementbetweenrm-levelandag-gregatevolatilityisonedistinguishingfeatureofourmechanismvis-a-vismodelsofnancialdiversication.Inaddition,innancialdiversicationmodels,thedeclineinaggregatevolatilitywithdevelopmentisbroughtaboutbynancialde-velopment.Inthedata,however,thevolatility-developmentrelationshipholdsatdierentlevelsofnancialdevelopment,measuredasprivatecreditoverGDP.16ThisrelationshipisillustratedinFigureA2oftheWebAppendix,wherewesplitthelevelofnancialdevelopmentintodierentquartiles.Theplotsshowthatthedeclineinvolatilitywithdevelopmentisnotsensitivetothecountry'slevelofnancialdevelopment,thekeymediatingmechanisminnancialdiversicationmodels.Putdierently,evencontrollingforthelevelofnancialdevelopment,thereremainsastrongnegativecorrelationbetweenvolatilityanddevelopmentthatneedsexplanation.Equallyimportant,whiletheunivariateregressionsbe-tweenvolatilityandnancialdevelopmentyieldanegativecoecient,thecorre-lationvanishesonceothercontrolsareaddedtothespecication.Instead,therelationbetweenvolatilityanddevelopmentappearsrobusttothesamecontrolsandisnotalteredbytheinclusionofaproxyfornancialdevelopment.ThisisillustratedinTablesA4andA5oftheWebAppendix,whichreporttheresultsfromregressionsofvolatilityonanumberofcovariates.1714CominandPhilippon(2005)hadpreviouslydocumentedthatpubliclytradedU.S.rmsexperiencedanincreaseinvolatilityduringthesameperiod.However,publiclytradedrmsareonlyasmallfractionofallrms.Sinceamajorityofrmsinmostcountriesareprivatelyheld,theevidencefromDavisetal.(2006)ismoreinformativeforourpurposes.15Modelsofnancialdiversicationrequireadecreaseincross-mcorrelationovertimeinordertogenerateadeclineinaggregatevolatility.Inourmodel,thiscorrelationwillbeconstant.16ThedatacomefromtheWorldBank'sFinancialStructureDatabase(v.4)andcorrespondtotheseriesprivatecreditbydepositmoneybanksandothernancialinstitutionsoverGDP.17Anadditionaldierencewithnancialmodelsconcernstherelationbetweenproductivityandvolatil-ityatthemicroeconomiclevel.KorenandTenreyro(2007)ndanegativecorrelationbetweenproduc- VOL.VOLNO.ISSUETECHNOLOGICALDIVERSIFICATION11tivecorrelationbetweensalesgrowthandgrowthintechnologydiversicationatalllevelsofaggregation.GambardellaandTorrisi(1998)measuretechnologicaldiversicationofthelargestU.S.andEuropeanelectronicsrmsbycalculat-ingtheHerndahlindexofeachrm'snumberofpatentsin1984{1991.Theirmainndingsarethatbetterperformance(intermsofsalesandprotability)isassociatedwithincreasedtechnologicaldiversicationandlowerproductdiver-sication.Theyconcludethattechnologicaldiversicationisthekeycovariatepositivelyrelatedwithvariousmeasuresofperformance.Feenstra,MarkusenandZeile(1992)provideevidencethatinputdiversicationleadstogrowthandproductivitygains.UsingdataonSouthKoreanconglomer-ates,theyndthattheentryofnewinput-producingrmsintoaconglomerateincreasestheproductivityofthatconglomerate.Infarming,therearemultipleexamplesofinputsleadingtoproductivitygainsandfastergrowth.TheWorldBank(2011)reportsthatinlargerscalecropproduction,thetwoshorttermin-terventionswiththegreatestimpactinproductivityaretheuseofhighqualityseedandchemicalfertilizers.Thesamestudylistsanumberofinputsthatbothincreaseagriculturalproductivityandlowervolatility,includingfertilizers,mod-ernseeds,agronomicskills,irrigationsystems,andcellphones(usefultotransmitinformationonweathernews).IntheWebAppendixwediscussadditionalstudiesandpresentevidencefrominput-outputtablesindierentcountriesshowingthatpurchases(directorindi-rect)byagivensectorfromitselfrelativetototalpurchasesbythatsectorhavefallensignicantlyovertimeinOECDcountriesfrom1970to2007.Thetrendtowardshigherusageofinputsfromothersectorsisanothermanifestationofthetechnologicaldiversicationmechanism.II.AModelofTechnologicalDiversicationBeforespecifyingthemodelindetail,weoerabriefinformalpreviewofthemainfeatures.Monopolisticallycompetitivermsproducegoodsusingavarietyofinputs(or,morebroadly,technologies).Thereisfreeentrybyrmsandnewrmsstartupwithnovarieties.Firmscanaddnewvarietiestotherangeofinputstheyusebyengaginginsomeadoptioneort(e.g.tolearnhowtouseit).Inparticular,theycaninvestresourcesinanadoptionprocess,whichsucceedsthesoonerthemoreresourcestherminvests.Indecidinghowmuchtoinvestinadoptioneachrmseekstomaximizethepresentdiscountedvalueofprots.(Sincermsarerisk-neutral,protmaximizationistheonlygoalofthisprocess.)Hencetheadoptionpartofthemodelisverysimilartostandardexpanding-varietymodels,exceptthattheadoptiongoesonsimultaneouslyinmultiplermsand,duetotherandomelementsofthemodel,itimpliesthatdierentrmswillhavedierentnumbersofvarietiesatagivenpointintime.Themodel'sinnovativefeatureisthatvarietiesaresubjecttoproductivityshocks.Inparticular,onceanewvarietyhasbeenaddedtoarm'srangeofinputsortechnologies,itbecomesapermanentpartofitsproductiveprocessuntil VOL.VOLNO.ISSUETECHNOLOGICALDIVERSIFICATION13WebAppendixderivestheconditionsunderwhichtechnologicaldiversicationcanleadtolowervolatilitywhen"1,thatis,wheninputsarecomplements.Onecouldre-interpretthevarietiesinourmodelnotasinputsbutasdisem-bodiedtechnologiestoturnlaborintooutput.Expandingthenumberofvarietiesofsuchtechnologiesisalsolikelytobothincreaseproductivityandprovidetech-nologicaldiversication.Hence,intherestofthepaperwerefertothevarietiesinterchangeablyasinputsandastechnologies.20Noticethatweareimplicitlyassumingthatthermuseseachvarietyincon-stantquantities,herenormalizedto1.Whatvariesisthenumberofvarieties,thequantityoflaborassignedtoeachofthem|capacityutilization|(bothofwhichdependontherm'sdecisions),andtheproductivityofeachvariety(whichwillberandom).Inreality,thequantityofeachinputvarietywillalsovary,butabstractingfromthisdecisionallowsustofocusontechnologicaldiversication,whichcomesfromanexpansioninthenumberofvarieties,withoutoverlycompli-catingtheanalysis.Underthetechnologyinterpretation,theassumptionwouldbeneasis.Weassumethatvarietieshaveaconstantproductivityduringtheirrandomlifetime;whenashockhitsthevariety,itceasestocontributetoproduction.(Avarietycanpotentiallybere-adoptedifrmsincurnewadoptioncosts|seebelow.)Thearrivalofshocksforagivenvarietyiiscommontoallrmsusingthisvariety,anditfollowsaPoissonprocesswitharrivalrate .Shocksareindependentacrossvarieties.BecauseashockarriveswithaPoissonprocess,theinput'sproductivelife-timefollowsanexponentialdistributionwithparameter .Hence,conditionalonvarietyiworkingattime0,thedistributionofi(t)isgivenbyi(t)=(1withprob.e� t;0withprob.1�e� t:LetaJ(bt)beaPoissonprocesswitharrivalratebandjumpsofsizea.21Withthisnotation,thedynamicsofavariety'sproductivitycanbewrittenas(3)di(t)=�i(t)dJi( t);wherethesubindexiinJi( t)highlightsthatthePoissonprocessesarevariety-specicandindependentacrossvarieties.Productivityisconstant(at1)beforeajumpoccurs,anditjumpsdowntozerowiththerstarrivalofdJi�0.20Anotherinterpretationisthattheproductionfunctiontakestheformy=APixi1�1=""=("�1)wherexiistheintermediategoodproducedbythermbytransforminglabourthroughxi=ili.Nothingsubstantialchangeseitheriftheinputsareproducedbyspecializedproducersandsoldtothermatarms'length;inthiscase,shockstoimapintoinputpriceshocks.21See,forexample,CoxandIsham(1980),Section3.1. VOL.VOLNO.ISSUETECHNOLOGICALDIVERSIFICATION15sourceofoutput uctuationsinthelasthalfcentury.Aninputdoesnotneedtobeanexhaustiblenaturalresourcetobecome(tem-porarilyorpermanently)unavailable.Inthe2011droughtinEastAfricaalargefractionofthelivestockhasdied,causingdrasticdropsinoutput.In1993anexplosioninaSumitomoplantinJapanledtotheannihilationoftwo-thirdsoftheworldsupplyofthehigh-gradeepoxyresinusedtosealmostcomputerchips,causingshortagesandpricehikesinthesemiconductorindustryforseveralmonths.Moregenerally,disastersofdiversenaturecandestroytheoutputofintermediategoodsproducers.Similarly,governmentpoliciescanhinderthepro-ductionoruseofcertainintermediateproducts.Humancapitalisnotimmunefromsuchshockseither:Pol-PotandMaoZedongwipedoutthehumancapitalofanentiregenerationintheirrespectivecountries.Evenifnottakenliterally,theprocessdescribedinequation(3)canalsobeseenasashort-cuttomodellessradicaldisruptions;inthatspirit,shockstoicanresult,forexample,fromchangesintaxesorregulatorypolicies,increasesinthecostofproductionortheimportpriceofavariety(orfromthepriceofaninputneededtousethatvariety,suchasthepriceoffuels),tradedisruptions,weather-relatedshocksthatrenderavarietyuselessorseverelyhinderitstransportationtoitsdestination,andsoon.24B.AFirm'sStaticDecisionsSincermsengageinmonopolisticcompetition,eachrmfacesaniso-elasticdemandwithelasticity":(5)y(j;t)=Y(t)p(j;t)�";whereaggregateoutputY(t)istakenasthenumeraire,andp(j;t)isthepricechargedbyrmjattimet.Theproductionfunction(4)pinsdownthenumberofworkersnecessarytosatisfythisdemand,l(j;t)=y(j;t)n(j;t)1=(1�")=A(t)=Y(t)p(j;t)�"n(j;t)1=(1�")=A(t):Firmswithmorevarietiesofinputsaremoreproductive(astandardlove-of-varietyeect)andhencecanproduceagivenlevelofoutputwithfewerworkers.Thermhiresworkersincompetitivelabormarkets.Attimetitfacesawageratew(t),whichdependsontheaggregatestateoftheeconomy,andistakenasgivenbyindividualrms.Flowprotsarerevenueminuslaborcost,sotheoperatingprotofthermis(6)(j;t)=Y(t)p(j;t)1�"�n(j;t)1=(1�")w(t)Y(t)p(j;t)�"=A(t):24Atransportationortradedisruptionmightmakeatechnologyorvarietytemporarilyunavailable,butthevarietycanpotentiallycomebackintouseafterreinvestment. VOL.VOLNO.ISSUETECHNOLOGICALDIVERSIFICATION17Lunitsofthenalgoodperunitoftimeinordertoadopttheirrsttechnologicalvariety.TheadoptionoftherstvarietywillthenbesuccessfulwithaPoissonarrivalrate,thatis,theexpectedwaitingtimeofanewentranttobecomeaproductivermis1=.Theentrantmayalsoexitatanypointintime.27Risk-neutralrmsareindierentastowhichvarietytochoose,sinceallva-rietiesentersymmetricallyintheirprotfunctionandtheirsolegoalisprotmaximization.Sincethechoiceisindeterminate,asatie-breaker,weassumethatrmstrytoadopttechnologieswithlowerindexesrst.Armofsizenhasthusaccesstotechnologies1;2;:::;nandwould,uponsuccess,adopttechnologyn+1next.Thistie-breakingconditioncapturesthenotionthatsometechnologiesorinputsareeasiertoadoptandhencetendtobeadoptedrstbymostrms.28Let(n)=f[I(n)=L;n]=n;forn�0,denotetheadoptionintensityofasize-nrm.Becausefishomogeneousofdegreeone,the owcostofthisadoptionintensityis(7)I(n)=g[(n)]Ln;n�0whereg(:)istheinverseoff(:;1),anincreasing,convexfunction.Forprospectiveentrantswithn=0,the owcostofadoptingtherstvarietyissimplyI(0)=L:Statevariables.Becausenistheonlyrm-levelstatevariable,weintroduceachangeofvariablesandindexrmsbyn.Atype-nrmhasexactlynworkingvarietiesatitsdisposal.Becauseweonlyneedtokeeptrackoftheworkingvarieties,wheneveravarietyishitbyashock,theindexofallvarietieswithahigherindexisreadjustedsoastoleavenoholesintheordering.29Deneasmk(t)themeasureofrmshavingexactlyk=0;1;2;:::workingvarietiesattimet.LetM(t)=fm0(t);m1(t);m2(t);:::gdenotetherm-sizemassdistributionattimet.Hence,thetotalmassofrmsattimetisgivenbyM(t)=P1k=0mk(t).30Eventhoughentrantshavezeroproductivityandhencedonotcontributetooutputoremployment,theymaybecomesuccessfulinadoptingtheirrstvariety,soitisimportanttotrackthem.ThemassdistributionM(t)sucientlycharacterizesthestateoftheeconomy,bothintermsofaggregateallocationsandprices,andintermsofdynamics.NotethatM(t)israndom:therm-sizemassdistributionwilldependontherealizationofadversetechnologyshocks.LetSdenotethesetofallpossiblerm-sizemassdistributions.WeassumethatM(t)followsaMarkovprocesswith27Inequilibrium,freeentrypinsdownthevalueofaprospectiveentrantatzero.Hencethemarginalentrantwillbeindierentbetweencontinuingtospendonadoptioncostsorexiting.28Thiscouldalternativelybemodeledbyassumingafunctionalformforxedcostsofinvestment,wherebydierentvarietieshavedierentcostsofinvestmentsandhencelower-costvarietiesareadoptedrst.Thecoreresultswillbesimilartotheorderingassumptioninthetext.29Thatis,ifaneconomyhasvarietiesk=1;2;3;4andvariety3fails,then,variety4isreindexed3andthenewsetofvarietieshasindexesk0=1;2;3:30NotethatM(t)isnotaprobabilitydistributionasthetotalmassM(t)isingeneraldierentfrom1;theprobability(share)ofrmswithkvarieties,isgivenbymk(t) M(t). VOL.VOLNO.ISSUETECHNOLOGICALDIVERSIFICATION19maximizethepresentvalueofconsumption,discountedattherate:UZ1t=0e�tC(t)dt:TheEulerequationpinsdowntherisklessrateofreturnintheeconomyatr(t)=.Investorsmaximizetheexpectedpresentvalueofprots,discountedattherate.Toensurenon-negativegrowthandanitevaluefortherm,weimposethefollowingparameterrestrictionson ,andthecostofadoption:(11)g0( ) andlimx! +g(x)=1;Therstconditionensuresthatavarietyisprotableenoughsothatitisworthinvestinginadoptioncostswhenavarietysuersashock.Thesecondconditionensuresthatadoptioniscostlyenoughsothatthegrowthrateoftheeconomywillneverexceed,thesubjectivediscountrate.Bellmanequation.LetV(n;M)denotethevalueofasize-nrmwhenthestateoftheeconomyisM.Itistheexpectedpresentvalueofthestreamoffutureprots,comingfromnetoperatingrevenuesminusthecostsofadoption,V(n;M)=maxfp;IgEZ1t=0e�tf[n(t);M(t)]�I[n(t)]gdtwhereM(t)andn(t)evolvesubjecttothelawsofmotiondescribedinequations(9)and(10),respectively.Fromtheperspectiveofarm,thereisarm-levelstatevariable,n,andanaggregatestatevariable,M,thetwoofwhichcontainalltheinformationrelevantinitsdecision.Thermchoosesthepriceofitsproduct(takingaggregatedemand,theproductionfunctionandwagesasgiven),andtheintensitywithwhichitinvestsinadoptingnewvarieties.Thepolicyvariablesarethuspand.Giventhe owprotfunctioninequation(6),thecostfunctionforadoptioninequation(7),andthelawofmotionforMinequation(9),theBellmanequationfortherm'sprotmaximizationproblemcanbewrittenas:V(n;M)=maxp;If(p;n;M)�I+(n)[V(n+1;M)�V(n;M)]+ nXi=1[V(n�1;M+Gi(M))�V(n;M)]++ 1Xi=n+1[V(n;M+Gi(M))�V(n;M)]+VMF(M)):(12)incentivetodiversifydoesnothingeonthenancialstructureoftheeconomyorthedegreeofriskaversion(thoughquantitativelytheymayaecttheseincentives). VOL.VOLNO.ISSUETECHNOLOGICALDIVERSIFICATION21DenotebyN(M)theaggregatenumberofvarietiesdenedas:(16)N(M)ZMndM=1Xi=1imi;whereRMdMstandsfortheLebesgueintegraloverrmswithdierentsizes,withrespecttotherm-sizemeasureM.34AggregateproductivityA(M)isgivenbyapositivefunctionA(N;m0)�0thatdependsonthetotalnumberofvarietiesNusedbyproductiverms,andonthemassofzero-sizermsm0,whichdonotcontributetoproduction.WeassumethatA(N;m0)satisesthefollowingproperties:N(M)0,m0(M)0and(17)1 "�1+N(M)1�m0(M);whereN(M)@lnA(N;m0) @lnNistheelasticityofA(N;m0)withrespecttoN(hold-ingm0xed),andm0(M)isthecorrespondingelasticitywithrespecttom0(holdingNxed).TheseconditionsarejointlysucientfortheexistenceanduniquenessofaBEGP.TheassumptionthatAisnondecreasinginN(N(M)0)embedstheideathattherecanbeknowledgespilloversacrossproductiverms.Notethattheinequalityisweak,soN(M)=0isapossibility.Theassump-tionthatAisdecreasinginm0(m0(M)0)impliesthatforagivennumberofvarietiesN,whenevertherearetoomanynewentrantsrelativetoequilibrium,protsperrmfall,reducingtheincentivestoenter.Intuitively,unproductivermswithnovarietiescontributenegativelytotheaveragestockofknowledgeoftheeconomy.ThenalinequalityconditionensuresthatthecontributionofnewrmstoGDPgrowthvanishesastheeconomygrowsandguaranteesapositivemeasureofnewrmsinequilibrium.Notethattheseconditionsaresucient,butnotnecessaryforaBEGP.Inparticular,ifN(M)=m0(M)=0,theeconomyfeaturesaBEGPwhen"=2.35Similarly,ifm0(M)=0,theeconomyfeaturesaBEGPwhenN(M)=1�1 "�1.Inthebaselinequantitativeexercise,weallowforverysmallexternaleects,consistentwiththeempiricalliterature.Theentrymarginm0adjustssoastopreventexplosivegrowthinthecaseoflowsubstitutability("2),ortoprevent34ThesumP1i=1imiwillbenitewithprobabilityoneatanypointintime,aslongaswestartfromaninitialrm-sizemassdistributionM0withniteN.Thisisbecauseatanypointintimet;N(t)hasaniteupperbound.Fromcondition(11),adoptionintensitybyincumbentscannotbegreaterthan +foranyrm.Hencevarietiesusedbyincumbentrmscanatmostgrowattherate +.Asitwillbecomeclear,thegrowthstemmingfromthecreationofnewrmsism0 N(M),whichisboundedfromaboveby" ,whereisnite.Hence,foranyt;N(t)N(0)e( ++" )t:Notethatafterapositiveamountoftimehaspassed,nwillhavefullsupportwithprobabilityone.ThisisbecausesuccessfuladoptionfollowsaPoissonprocess,whichmakesthenumberofnewvarietiesaPoisson-distributedrandomvariable.35IfN=m0=0and"=2;theaggregatedemandexternalityandthecompetitioneectcanceloutandprotspervarietyareconstant,whichissucientforaBEGP. VOL.VOLNO.ISSUETECHNOLOGICALDIVERSIFICATION23implicitlydenedby(21)g0()= :WagesandnaloutputarelinearinN.Wagesare(22)w(M)=("�1)N(M);naloutputis(23)Y(M)="N(M)L;where=(+ �) +g()ispercapitaprotspervariety,whichisconstant.Firmpricesare:(24)p(n;M)=N(M) n1=("�1):Themassofnewentrantsm0satises(25)=1 "N(M)"�2 "�1A(N;m0):ThelawofmotionforMisMarkovwithFiandGikdenedasFi(M)=((i�1)mi�1�imiifi�1;m0�m1ifi=1:(26)Gik(M)=8�&#x]TJ ;� -1; .63; Td;&#x [00;:mi+1�miifki;mi+1ifk=i+1;0ifk&#x]TJ ;� -1; .63; Td;&#x [00;i+1:(27)TheproofofthisandallotherpropositionsareintheWebAppendix.Equation(20)showsthermvalueasafunctionoftheadoptioncostofnewentrants.EachnewentrantspendsLforanexpected1=unitsoftimebeforebecomingaproductivermandachievingavalueofV(1;M)=v.Therestofthermvaluefunctionislinearinn.Equation(21)istherst-orderconditionforoptimaladoption.Thisconditionpinsdownaunique,constantthatisindependentofn.Equation(22)showshowwagesdependontheaggregatenumberofvarieties.Whentheeconomyusesmoreinputvarieties,aggregatelaborproductivityishigher,andwagesarehigherintermsofthenalgood(thenumeraire).WehavealreadysubstitutedouttheequilibriumvalueoftheexternalproductivityA(equation(25)).Equation(23)expressesnaloutputasafunction VOL.VOLNO.ISSUETECHNOLOGICALDIVERSIFICATION25nvarietieston+1varieties.Recallthatasize-nrmadoptsnewvarietieswitharrivalrate(n)=nforn1and(0)=.Ateverypointintime,ameasureZM(n)dMdt=N(M)dt+m0(M)dtofrmsbecomessuccessfulinadoptingthenextvariety.Thesecondtypeofshockisthefailureofaparticulartechnologyk.Thisshockdecreasesthenumberofvarietiesby1forallrmsthatusevarietyk.Becausethereisapositivemassoftheserms,thisshockinducesaninstantaneousjumpinN.Theaggregateimpactoftheshock(and,ultimately,aggregatevolatility)willdependonthemeasureofrmsusingtechnologyk.Notethatbecausetechnologyshocksarecommonacrossrms,theywillalsoinducecorrelationsacrossrms.Thisiswhythereisaggregateuncertaintyevenwithacontinuumofrms.38LetMidenotethemassofrmsusingvarietyi.Becausermsadoptlower-indexedvarietiesrst,thisisthesameasthemassofrmswithiormorevarieties,Mi=P1k=imk.Then,usingtheaggregationaboveandtheMarkovdynamicsofMasgivenbyequations(26)-(27),wecanwritethedynamicsofNasfollows:(30)dN=[N+m0]dt�1Xi=1MidJi( t):Thersttwotermsaretheeectofinnovation.Eachrm'sadoptionissubjecttoanindependentPoissonprocess,thesumofwhichisadeterministicprocessbythelawoflargenumbers.Thesecondtermcapturesadverseproductivityshocks,whicharecommonacrossrmsandarenotwashedoutbyaggregation.BecausevarietyiisusedbyameasureMiofrms,ashockdJireducesthetotalnumberofvarietiesbyMi.Proposition3.Inarecursiveequilibrium,theexpectedgrowthrateofthenum-berofvarietiesN(andhenceofoutputY)is(31)E(dN=N) dt=+m0 N� ;anditsinstantaneousvarianceis(32)Var(dN=N) dt= 1Xk=1s2k;wheresk=Mk=NmeasuresthecontributionofvarietyktoGDP.38Becausepositiveshocks(technologyadoptions)areindependentacrossrms,whilenegativeshocks(technologyfailures)arecommon,aggregateshockswillhaveanegativelyskeweddistribution.ThisisconsistentwithevidencepresentedbyJovanovic(2006)fortheU.S.economy.SeeWebAppendixforfutherreferencesaswellasnewevidenceonskewnessinothercountries. VOL.VOLNO.ISSUETECHNOLOGICALDIVERSIFICATION27Proposition4.Forallinitialvarietydistributions,M(0)=M0,therecursiveequilibriumconvergestoabalanced-expected-growthpath.Theexpectedlong-rungrowthratexisimplicitlydenedby(33)g0( +x)= ;withx2[0;)and givesthepercapitarmvaluepervariety.Inthelong-run,asN!1,thecontributionofentrantsm0togrowthvanishesasm0 N!0:TheproofmakesuseofthesucientconditionsN(M)0,m0(M)0and(17).Asstressedearlier,theseconditionsarejointlysucientbutnotnecessaryforaBEGP.Note,inparticular,thattherearetwoalternativewaystoachieveaBEGPinoursetup.Therst,whichdoesnotrelyonexternaleects,istoimposeaparametricrestrictionontheelasticityofsubstitution.Specically,whenN(M)=m0(M)=0;aBEGPexistsif"=2:43Thesecondalternativethatdoesnotrelyonexternaleectsfromentry(i.e.,m0(M)=0)istoimposetherestrictionthatN(M)=1�1 "�1.44ThesetwooptionscauseoutputtobelinearinN,thekeyconditionforaBEGP.Inthebaselinequantitativeexerciseweallowforsmallexternaleects,consistentwiththeliterature,andintheWebAppendixwepresenttheresultswithoutexternaleects.G.GDPDynamicsalongtheBalanced-Expected-GrowthPathSinceatanytimet(instantaneous)GDPgrowthdY(t)=Y(t)isarandomvari-able,itnotonlyhasanexpectedvaluebutalsoavariance.Thisvarianceisnotconstant,evenontheBEGP.(Noticethatifitwere,themodelwouldhavenohopeofexplainingthecross-sectionalpatternsofvolatilityanddevelopmentthatmotivatethepaper.)Instead,itdependsonthesetoftechnologiesinuse,aswellastheirdistributionamongrms.Ingeneral,thesedependontheparticularhistoryofshocksthathavehittheeconomy,sothevariancemustbecomputedbynumericalsimulation.Beforeweturntothistask,weoersometheoreticalresultsthatbothhelptounderstandthesimulationsandprovidesomeintuitiononthemainmechanismatplay.ThevolatilityofN(andhenceofY)dependsonthewholedistributionofvarietiesusedbyrms.Ifsomevarietiesareusedbymorermsthanothers,thenshocksaectingthesevarietiesaregoingtohavealargerimpactonGDP.Throughtheintroductionofnewvarieties,technologicalprogressincreasesthede-greeoftechnologicaldiversication(andhencelowersvolatility)whileincreasingthelevelofdevelopment.Thisimpartsanaturaltendencyforanegativecorrela-tionbetweenvolatilityanddevelopmentthatwillbeprevalentinournumericalanalysis.Note,however,thatinprincipletherelationshipbetweenvolatilityanddevelopmentdoesnotalwaysneedtobestrictlynegative.Tounderstandthis43Inthiscase,protspervarietyareconstantandthereisnoentryofrms:m0=0:44Asbefore,inthiscase,thereisnoentryandm0=0: VOL.VOLNO.ISSUETECHNOLOGICALDIVERSIFICATION29pathwithrate� .Thedeclineinvolatilitythusresultsasaby-productofthedevelopmentprocess.Beforemovingtothequantitativeresults,acommentontheasymmetryinthesourcesofaggregatevolatilityisinorder.Inthemodel,positiveshocksatthemicrolevelaverageuptoasmoothaggregateprocess,whereasnegativeshocksatthemicrolevelgenerateaggregatevolatility.Whileofcoursethisisamodellingsimplication,theasymmetryleadstonegativeskewnessinthedistributionofgrowthrates,apredictionthatisconsistentwiththedata(seeWebAppendix).III.VolatilityandDevelopment:AQuantitativeAssessmentOuranalysissofarhasshownthatvolatilitydeclinesmonotonicallywiththedegreeoftechnologicaldiversicationandthat,ceterisparibus,theintroductionofanewvarietyintheeconomyincreasesthelevelofdevelopmentandthedegreeoftechnologicaldiversication,thusloweringvolatility.Wehavealsoarguedthatthegrowthprocess,throughtheexpansioninthenumberofvarieties,tendstoimpartanegativecorrelationbetweenvolatilityanddevelopment,thoughthistendencymaybeoverturnedundercertainhistoriesofshocks;specically,itisconceivablethatcountriesthatuseafewvarietiesveryintenselydisplaybotharelativelyhighlevelofdevelopmentandhighvolatilityduetotheirlackofdiversication.Toestablishwhethertheseoccurrencesarefrequentorrare,onehastosimulatethemodel.Ourstrategyistogeneratearticialdatabysimulatingthemodel1,000timesfor64dierenteconomies(countries)from1870through2007.Alleconomiesstartatthestageofdevelopmenttheywerein1870,accordingtoMaddison(2010).(Thereare64countriesatdierentstagesofdevelopmentwithdataonGDPpercapitain1870;seeWebAppendixforthelistofcountries.)Aninitial(single-parameter)logarithmicrm-sizedistributionforeacheconomyiscalibratedsoastomatchthelevelofdevelopmentofthecountryin1870(weshallelaborateonthislater).Allparameterscharacterizingtheevolutionoftheeconomiesareiden-tical.However,shocksarecountry-specicanddierentrealizationsofshocksleadtopotentiallydierentgrowthpaths.Weanalyzetherelationbetweenvolatilityandthelevelofdevelopmentforthesimulatedeconomiesandcomparepatternsofvolatilityanddevelopmentinthelast48yearsofoursimulationstothecorre-spondingpatternsinthecross-sectionaldatathatwealreadyexaminedinSectionI,coveringtheperiod1960-2007.NotethatbecausethevolatilityofaggregateGDPdependsonthedistributionoftechnologiesacrossrms,oursimulationsneedtokeeptrackoftheentiredistributionoftechnologyusageacrossrmsatallpointsintime.Weemphasizethatinrealitythereareseveraladditionalmechanismsdrivingacountry'seconomicdevelopmentanditspatternsofvolatility.Thegoalofthisnumericalexerciseistostudyhowthemodelbehaveswithreasonableparametervalues,nottorunahorseraceamongpotentialexplanations. VOL.VOLNO.ISSUETECHNOLOGICALDIVERSIFICATION31numberofvarietiesintheeconomyanddecreaseswiththenumberofentrants.WedonothavedirectestimatesofNandm0;theliteraturehaspointedtosmallbutnonzeroexternaleects.Forexample,Combes,DurantonandGobillon(2011a)and(2011b)estimatebothpositiveurbanexternalitiesandcongestionscoststobeoftheorderof0:03.HencewehaveselectedN=�m0=0:03asthebaseline.(NotethatthemagnitudesdonotneedtocoincidetosatisfythesucientconditionsforBEGP.)Forrobustness,weexperimentwithdierentvaluesforNand(�m0)between0and1satisfying(17).Wefoundthatthechoiceoftheseparameterswasnotimportantfortherelationbetweenvolatilityanddevelopment;therobustnessresultsareavailableintheWebAppendix.(NotethatinthelimitcaseinwhichN=m0=0,anecessaryandsucientconditionforbalancedgrowthisthat"=2:TheresultsforthiscaseareinTableA12oftheWebAppendix.)Initialconditions.Weinitializethemodelin1870andassumethatineachcountry,theinitialrm-sizedistributionislogarithmic.(Thisisthedistributionofrms'sizeinKletteandKortum(2004),discussedthoroughlyinthecontextofrms'sizesbyIjiriandSimon(1977).49)Wecalibratetheparameterofthecountry-specicdistributionsoastomatchthecountry'slevelofdevelopmentin1870,accordingtoMaddison(2010).Hence,allcountriesstartatthelevelofdevelopmenttheyhadin1870.Morespecically,thelogarithmicdistributionisgivenby:(34)pk=�1 ln(1�)k k;k1wherepkisthefractionofrmsusingkvarieties.Weassumeallcountriesstartwithaunitmassofproductiverms,M1,in1870,andletthedistributionofrmsvaryacrosscountrieswithacountry-specicparameterc.50Thus,pkmapsintomk,themassofrmswithk1varieties.Themeanofthesizedistributionis(35)1Xk=1k
pk=�1 ln(1�) 1�;whichisincreasingin.ThismeanmapsintoN,whichislinearlyrelatedtoGDPpercapita:Y(M)=L="N(M).HencefromdataonrealGDPpercapitain1870(GDPPCc;1870),foragiven",wecanobtainNc;1870,whereNc;1870is49Thelogarithmicdistributionisappealinginthiscontext,bothbecauseitcanmatchimportantfea-turesoftherm-sizedistribution(KletteandKortum(2004))andbecauseitreliesonasingleparameter,whichwecancalibrateusingaggregatedata.50Overtime,thesizeofM1(t)willadjustthroughtheentrymargin. VOL.VOLNO.ISSUETECHNOLOGICALDIVERSIFICATION33duringthe138yearsofsimulation.Atanypointintime,wecantakeasnapshotoftheeconomybycountingthenumberofrmsineachsizebin,m1t;m2t;:::RealGDPpercapitaattimetcanthenbecalculatedas"Pnmaxi=1imit.ToconstructstatisticsthathavethesameinterpretationasthoseintheempiricalanalysisofSectionI,wecomputedecadeaveragesof(thelogof)GDPpercapitaand(logsof)standarddeviationsofpercapitaGDPgrowth(ourmeasureofvolatility).RecallthatinSectionIwerunaregressionofcountry-levelvolatilityonincomeforthenearlyvedecadesbetween1960and2007.Torunasimilarregressiononoursimulatedpanelofcountriesweusedthelast48yearsofdatageneratedinoursimulations.Toreducesimulationerror,wereportthemeansfrom1,000simulations.B.ResultsThissectionpresentsanddiscussestheresultsfromthebaselinecalibration,alongwiththethreemainregularitiesmotivatingthemodel.1.GDPvolatilitydeclineswithdevelopment,bothinthecrosssectionandforagivencountryovertime.TherstsetofrowsinTable3showsforeachvalueof ,themodel-generatedslopecoecientsandthecorrespondingstandarderrorsfromOLSregressionsofdecade(log)volatilityontheaverage(log)GDPpercapitaofthedecade,poolingdatafromallsimulatedcountriesinthelastvedecades.ThelasttwocolumnsintherowshowthecorrespondingguresusingtwosamplesofPPP-adjusteddatafromthePWT.TherstsampleusesthesetofcountriesforwhichMaddison'sdataareavailablein1870|thecountriesforwhichwepindowntheinitialconditions.WerefertothissubsampleastheMaddisonsampleandtheresultsarereportedinthenext-to-lastcolumn.Thelastcolumnusesthewholesample,reproducingtheresultsintherstcolumnofTable1:Thesecondsetofrowsshows,correspondingly,thewithin-countryslopesandthestandarddeviationsresultingfromthemodel-generateddatafordierent s,aswellastheempiricalresultsbasedontheMaddisonsample,andthewholesample(thelattercorrespondingtothesecondcolumnofTable1):Asinthedata,thetime-seriesslopesgeneratedbythemodeltendtobelargerinmagnitudethanthecorrespondingcross-sectionalslopes.Theseresultsindicatethatfortheparametervaluesanalyzed,thecoecientsinboththepooledandwithin-countryregressionsarenegativeandsignicantatstandardcondencelevelsandquantitativelycomparabletothoseinthedata.Toassesswhetherandtowhatextentthemodelcanaccountforthedeclineinvolatilitywithdevelopmentseeninthedata,itisimportanttoknownotonlytheslopecoecientsbutalsothedegreeofdispersioninGDPgeneratedbythemodel.Inthe1960s,thestandarddeviationof(log)percapitaGDPacrosscountriesinthedatawas0.970(thecorrespondingvaluewas0.977inMaddison'ssample).5353Thisisthestandarddeviationacrosscountriesofthedecade-average(log)GDPpercapita. VOL.VOLNO.ISSUETECHNOLOGICALDIVERSIFICATION35Table3|VolatilityandDevelopment:QuantitativeResultsforDifferent Notes:Thetableshows,correspondingly,thecross-sectionalandwithin-countryslopecoecientsandstandarddeviations(inparentheses)fromregressionsof(log)volatilityofannualizedquarterlygrowthratescomputedovernon-overlappingdecadesontheaverage(log)levelofdevelopmentinthedecade;aconstant(notreported)isincludedineachregression.Thecrosssectionalregressionsarebasedonpooleddatafor5decades.ThethirdsetofrowsshowsthestandarddeviationofaverageloggedGDPpercapitaoverthewholedecade(andthestandarddeviationover1,000simulations).Thefourthlineshowsthepercentvariationinvolatilitygeneratedbya1standarddeviationincreaseintheloggedGDPpercapita.MaddisonsampleisasubsetofthewholesamplethatincludesthecountrieswithdataonGDPin1870(fromMaddison,2010).Theresultscorrespondtothebaselinecalibration;theparametervaluesare:=3;N=�m0=0:03;and=0:10.Seetextforexplanations.theeconomycorrelatesnegativelywithincomepercapita.Thisisalsothecaseinourmodel.Aregressionoftheshareofsmallrms56onlogGDPpercapitainthemodel,yieldsnegativeandsignicantcoecients,rangingfrom-0.049(s.e.=0.011)for =0:05to-0.025(s.e.=0.009)for =0:20.3.Firm-levelandaggregatevolatilitytendtodisplaypositivecomovement.Inthemodel-generateddata,rm-levelvolatility,measuredasthestandarddeviationofsalesgrowthforthemedianrm,andaggregatevolatilityareposi-tivelycorrelated.Themeancorrelations(andthestandarddeviationsover1,000simulations|inparentheses)are,correspondingly0.489(0.048)for =0:05;0.421(0.042)for =0:10;0.359(0.055)for =0:15;and0.274(0.059)for =0:20:Interestingly,theresultsalsosuggestthatwhenthevolatilityofshocksishigher(thatis, ishigher);thecorrelationbetweenmicroandmacrovolatilitybecomesweaker.Thismodelpredictioncanpotentiallybetestedinthefuture,aslongertimeseriesonrm-leveldatafordierentcountriesaregathered.Inall,thepositivecomovementgeneratedbythemodelisconsistentwiththeavailableevidence(seeSectionIandtheWebAppendix).Finally,asnotedearlier,inamajorityofcountries,thedistributionofgrowthratesisnegativelyskewed(skewnessismeasuredasthesamplethirdstandardizedmoment).Themodeliscapableofgeneratingthisnegativeskewness,asonlynegativeshocksatthemicrolevelcontributetoaggregatevolatility(positivemicroeconomicshocksadduptoadeterministicaggregateprocess).Theaverage56Smallrmsaredenedasthosehavingveorfewerinputvarieties: 36THEAMERICANECONOMICREVIEWMONTHYEARskewnessforcountriesinthemodel-generateddatarangesfrom�0:285when =0:05to�0:076when =0:20:When =0:10;skewnessis�0:141.Theaverageskewnesscoecientinthedataishigher:�0:390.Themodelcouldyieldhigherskewnessifnegativeshockswerenotindependentacrossvarieties.Inall,thequantitativeexerciseleadsustoconcludethatthetechnological-diversicationmodel,thoughstylized,canpotentiallyaccountforasubstantialpartofthedeclineinvolatilitywithdevelopmentobservedinthedata.Themodeloersanalternativechanneltoaccount(atleastpartially)forthevolatility-sizerelationshipobservedatthermlevel,andgeneratesapositivecorrelationbetweenrm-leveldataandaggregatevolatilitythatappearsinlinewithrecentempiricalndingsinthisarea.IV.ConcludingRemarksWearguethattechnologicaldiversicationoersapromising(yetsofarover-looked)explanationforthenegativerelationbetweenvolatilityanddevelopment.Wedosobyproposingamodelinwhichtheproductionprocessmakesuseofdif-ferentvarietiessubjecttoimperfectlycorrelatedshocks.AsinRomer(1990)andGrossmanandHelpman(1991),technologicalprogresstakesplaceasanexpan-sioninthenumberofinputvarieties,increasingproductivity.Thenewinsightinthemodelisthattheexpansionininputvarietiescanalsoleadtolowervolatilityinproduction.First,aseachindividualvarietymatterslessandlessinproduc-tion,thecontributionofvariety-specic uctuationstooverallvolatilitydeclines.Second,eachadditionalvarietyprovidesanewadjustmentmargininresponsetoexternalshocks,makingproductivitylessvolatile.Inthemodel,thenumberofvarietiesevolvesendogenouslyinresponsetoprotincentivesandthedecreaseinvolatilityresultsasaby-productofrms'incentivestoincreaseprots.Wesimulatethemodelforplausibleparametervaluesandndthatitcanquanti-tativelyaccountforasubstantialfractionofthestatisticalvariationinvolatilitywithrespecttodevelopmentobservedinthedata.Therearethreenaturaldirectionsforfurtherinvestigation.First,extendingthesetuptoamulti-sectormodelthatexplicitlydistinguishesbetweenwithin-andacross-sectordiversication.Second,allowingforinternationaltradetoanalyzethetradeobetweenhighersectoralspecialization(possiblybroughtaboutbyincreasedtradeopenness),andthescopeforinputortechnologydiversicationfacilitatedbytrade.57ThethirddirectionentailsextendingthemodeltomatchtheregularitiesemphasizedintheRBCliterature,withafocusonpoorcountries.Someofthefrictions(andshocks)neededtoaugmentourmodelwillbesimilartotheextensionsmadetotheRBC(orNewKeynesian)model.Thekeycontributionofourmodelwillbeontheendogenouslinkbetweenacountry'sdevelopmentanditssusceptibilitytoshocks,alinkthatisnotaddressedbytheRBCliterature.57SeeCaselli,Koren,LisickyandTenreyro(2010)foranexplorationofthesemechanisms. 38THEAMERICANECONOMICREVIEWMONTHYEARmakingthetheoriesfacethefacts."JournalofMonetaryEconomics,51(1):39{83.Comin,Diego,andThomasPhilippon.2005.TheRiseinFirm-levelVolatil-ity:CausesandConsequences.Cambridge,MA:NBERMacroeconomicsAnnual.GertlerandRogo(eds.).Cox,David,andVitIsham.1980.PointProcesses.London:ChapmanandHall.Davis,Steven,JohnHaltiwanger,C.J.Krizan,RonaldJarmin,JavierMiranda,AlfredNucci,andKristinSandusky.2009.\MeasuringtheDy-namicsofYoungandSmallBusinesses:IntegratingtheEmployerandNonEm-ployerBusinesses."InProducerDynamics:NewEvidencefromMicroData.,ed.Dunne,JensenandRoberts.NBER.Davis,Steven,JohnHaltiwanger,RonJarmin,andJavierMiranda.2006.VolatilityandDispersioninBusinessGrowthRates:PubliclyTradedver-susPrivatelyHeldFirms.Vol.21,Cambridge,MA:NBERMacroeconomicsAn-nual.Acemoglu,Rogo,andWoodford(eds.).Ethier,Wilfred.1982.\NationalandInternationalReturnstoScaleintheMod-ernTheoryofInternationalTrade."AmericanEconomicReview,73(3):389{405.Feenstra,Robert,JamesMarkusen,andWilliamZeile.1992.\Account-ingforGrowthwithNewInputs:TheoryandEvidence."AmericanEconomicReview,82(2):415{21.Gabaix,Xavier.2011.\Thegranularoriginsofaggregate uctuations."Econo-metrica,79(3):733{772.Gambardella,Alfonso,andSalvatoreTorrisi.1998.\Doestechnologicalconvergenceimplyconvergenceinmarkets?Evidencefromtheelectronicsin-dustry."ResearchPolicy,27:445{463.Granstrand,Ove.1998.\Towardsatheoryofthetechnology-basedrm."Re-searchPolicy,27(5):465{489.Granstrand,Ove,KeithPavitt,andPariPatel.1997.\Multi-technologycorporations:Whytheyhavedistributedratherthandistinctivecorecompeten-cies."CaliforniaManagementReview,39:8{25.Greenwood,Jeremy,andBoyanJovanovic.1990.\FinancialDevelop-ment,Growth,andtheDistributionofIncome."JournalofPoliticalEconomy,98(5):1076{1107.Griliches,Zvi.1957.\Hybridcorn:Anexplorationintheeconomicsoftechno-logicalchange."Econometrica,25(4):501{522.Grossman,Gene,andElhananHelpman.1991.InnovationandGrowthintheGlobalEconomy.Cambridge,MA:TheMITPress.Grossman,Gene,andEstebanRossi-Hansberg.2010.\ExternalEconomiesandInternationalTradeRedux."QuarterlyJournalofEconomics,125(2):829{858. 40THEAMERICANECONOMICREVIEWMONTHYEARRamey,Garey,andValerieRamey.1995.\Cross-countryevidenceonthelinkbetweenvolatilityandgrowth."AmericanEconomicReview,85(5):1138{51.Romer,Paul.1986.\IncreasingReturnsandLong-RunGrowth."JournalofPoliticalEconomy,94(5):1002{1037.Romer,Paul.1990.\EndogenousTechnologicalChange."JournalofPoliticalEconomy,98(5):71{102.Rossi-Hansberg,Esteban,andMarkWright.2007.\EstablishmentSizeDynamicsintheAggregateEconomy."AmericanEconomicReview,97(5):1639{1666.Saint-Paul,Gilles.1992.\TechnologicalChoice,FinancialMarketsandEco-nomicDevelopment."EuropeanEconomicReview,36:763{781.Sichel,Daniel.1993.\BusinessCycleAsymmetry:ADeeperLook."EconomicEnquiry,31(2):224{36.Steel,William.1993.\SmallEnterprisesinIndonesia:Role,Growth,andStrategicIssues."Workingpaper194.JakartaDevelopmentStudiesProjectII.Strotmann,Harald,JorgDopke,andClaudiaBuch.2006.\Doestradeopennessincreaserm-levelvolatility?"EconomicStudies2006-40,DeutscheBundesbank.Sutton,John.2002.\TheVarianceofFirmGrowthRates:TheScalingPuzzle."Physica,312(3-4):577{590.Thesmar,David,andMathiasThoenig.2011.\ContrastingTrendsinFirmVolatility."AmericanEconomicJournal:Macroeconomics,3(4):143{80.WorldBank.2011.\RisingFoodandEnergyPricesinEuropeandCentralAsia."WorldBank.WorkingPaper61097.