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 NotEnoughR&D?OrMaybeTooMuch? IntensityofKnowledgeSpilloversandOptimalR&DPolicyin SchumpeterianGrowthTheory ElieGray ∗ Abstract Thispaperpresentsanendogenousgrowthmodel `aAghion&Howitt(1992)inwhichweexplicitly formalizeknowledgespilloversintheinnovationprocess. WerevisittheissueoftheParetonon-optimality oftheSchumpeterian equilibriumbyrevealingthepart playedbytheintensityofknowledgespillovers. Basically,wehighlightthatthemarketincompletenesscharacterizingthistypeofdecentralizedeconomy (knowledgeisnotpriced)isallthemorelikelytoleadtoanunder-optimal(resp.over-optimal)R&Deffort astheintensityofknowledgespilloversishigh(resp.low).Thereasonbehindthisisthattheeffectsof thedistortionofR&Dincentivesresultingfrommarketincompletenessareamplifiedallthemoreasthis intensityisstrong.Complementarily,wederivetheoptimaltooldedicatedtocorrectthemarketfailure causedbymarketincompleteness,andwedemonstratethatitclearlydependsontheintensityofknowledge spillovers: thehigher(resp.lower)theintensityofknowledgespilloversis,themorelikelythispolicytool shouldconsistinasubsidy(resp.tax).Moreover,ifthisoptimaltoolhappenstobeasubsidy,thenthis subsidywillbeallthelargerastheintensityishigh. Keywords:Schumpeteriangrowththeory;Paretosub-optimality;Marketincompleteness;Knowl- edgespillovers;R&Dincentives;Optimalpolicytools Howtocitethispaper:Gray,E.(2022)NotEnoughRD?OrMaybeTooMuch?Theoretical EconomicsLetters,12,1539-1558.https://doi.org/10.4236/tel.2022.126084 Received:July27,2022 Accepted:November6,2022 Published:November9,2022 ∗ Correspondingauthor-TBSBusinessSchool,UniversitédeToulouse,Toulouse,France.20Bd.Lascrosses,BP701031068 ToulouseCedex7,France.Tel.:+33(0)5.61.29.49.25.Email:e.gray@tbs-education.fr. 1539 Theoretical Economics Letters, 2022, 12, 1539-1558 https://www.scirp.org/journal/tel ISSN Online: 2162-2086 ISSN Print: 2162-2078  1Introduction Thetheoryofendogenousgrowthbasedoninnovationunderlinesthepresenceofavoluntarymechanismat theoriginoftheaccumulationofknowledgefromwhichstemstechnicalprogress,engineoflong-rungrowth. Knowledgeisanon-rivalgood-astockofintellectualcapitaldistinctfromphysicalcapitalorhumancapital -thecreationofwhichdependsonaspecificandendogenousinvestment: knowledgeaccumulatesthroughthe activityofresearchanddevelopment(R&D).Theprocessofknowledgeaccumulationhasbeenformalizedby twoseminalparadigmswhichbasicallydifferinhownewknowledgeisinterpreted.WhereasinRomer(1990),it consistsofnewgoodsornewproductionprocesses(onegenerallyrefersto“horizontal”knowledgeaccumulation), inAghion&Howitt(1992),itconsistsofanimprovementofthequalityofanalreadyexistinggoodorprocess (onegenerallyrefersto“vertical”knowledgeaccumulation). Inordertodealwiththenon-rivalrypropertyofknowledge,thefundamentalpapersbyRomer(1990)and Aghion&Howitt(1992),focusedondecentralizedeconomieswithincompletemarketsandimperfectcompe- tition.Amarketandapricearespecifiedforgoodsthatincorporateknowledge,butnotforknowledgeitself. IncentivestoinvestinthecreationofknowledgegothroughthefactthatagentsinvestingR&Dexpecttoget somemarketpower. Indeed,R&Dactivityisindirectlyfundedbymonopolyprofitsresultingfromintellectual propertyrightsgrantedtoinnovators.Romer(1990)considersanequilibriuminwhicheachinnovatorobtains aninfinitely-livedpatentonthegoodembodyinghisinnovation.InAghion&Howitt(1992)ontheotherhand, theequilibriumisinspiredbySchumpeter’screativedestructionmechanisminsofarasthefirmwhichmanages toinnovateinasectorreplacesthepreviousinnovatorandmonopolizesthissectoruntilthenextinnovation occurs.Hence,suchequilibriaarecharacterizedbytwomarketfailures:firstly,marketpowerensuingfrom thepresenceofamonopolyoneachintermediategoodincorporatinganinnovation,andsecondly,apositive externalitywhichconsistsofthemarketincompletenessresultingfromthefactthatthereisnomarketfor knowledgesinceknowledgeisnotpriced. Becauseofthesetwomarketfailures,Paretonon-optimalityislikely toariseinthelaisserfaireequilibrium. EndogenousgrowththeoryhaslongemphasizedthisissueofParetonon-optimality.Inparticular,many R&D-basedendogenousgrowthmodelspredictthatintheabsenceofpublicpolicies,thedecentralizedeconomy canleadtoaneitherinsufficientorexcessivelevelofresourcesallocatedtoR&Dactivity,andthustoaeither sub-optimalorover-optimalgrowthrateoftheeconomy.Thiswell-knownresulthasbeenextensivelydiscussed inthegrowthliterature,bothinverticaldifferentiationclassofmodels(e.g.,Grossman&Helpman1991;Aghion &Howitt1992;Segerstrom1998;Li2003)andinexpandingvarietymodelsàlaRomer(e.g.,Benassy1998; Jones&Williams2000;Alvarez-Pelaez&Groth2005). Inthepresentpaper,wedevelopastandardendogenousgrowthmodel`aAghion&Howitt(1992)inwhich wemakeknowledgeaccumulationandknowledgediffusionexplicitintheinnovationprocess;thisfirstnovelty enablesustoclearlyformalizewhatwecoin“intensityofknowledgespillovers”.Intheendogenousgrowth literature, theexpression“knowledgespillovers’’referssimultaneouslytotwointertwinedissues whichhave beenstudiedinalargebodyofliterature(see,forinstance,Romer,1990;Aghion&Howitt1992,1998,2009; Segerstrom,1998;Li,2002;Peretto&Smulders,2002;Jones,2005;Sener,2008;orAcemoglu,2009). Thefirst issuerelatestothetechnologyusedinendogenousgrowthmodels:knowledgeproductionfunctionsgenerally assumethattheknowledgepreviouslycreatedinaparticularsectorspreads(“spillsover”)intotheeconomy, thusenhancingthecreationofnewknowledgeinothersectors.Thesecondissuesomehowensuesfromthe firstone:itrelatestotheconsidereddecentralizedeconomywhichischaracterizedbythepresenceofapositive externalityentailedbymarketincompletenesssinceknowledgeisnotpriced.Thisarticleobviouslyconsiders bothissues,nevertheless,weusetheexpression“knowledgespillovers”specificallytorefertothefactthatthe knowledgeinherentinanyinnovationdiffusesacrosssectors.Inparticular,weintroducetheconceptof“intensity ofknowledgespillovers”toformalizethefactthatknowledgemaydiffusemoreorlessbroadly. Oncethemodeldeveloped,wecomputetheassociatedfirst-bestsocialoptimum;then,westudytheoutcome ofaclassicSchumpeterianequilibrium.Thedevelopedframeworkisnoexceptiontotherule:inthelaisserfaire Schumpeterianequilibrium,theR&Deffort(representedbythequantityoflaborusedinR&Dinthepresent model)canbesub-optimalorover-optimalasmaybethecaseinstandardSchumpeteriangrowthframeworks descendingfromtheseminalpaperbyAghion&Howitt(1992).Severalcomplementaryapproachestryingto understandwhytheR&Deffortcaneitherbesub-optimalorover-optimalcanbefound.Aghion&Howitt(1992, 1998)focusonthevariousmarketfailuresinvolvesbytheequilibriumconsideredinordertounderstandwhy Paretonon-optimalitymayarise;theyexplainthattheproblemsofsurplusappropriabilityandofknowledge spilloversbothtendtoleadtowardsunder-investmentinR&D,andthattheeffectsofcreativedestructionand ofduplicationbothtendtoleadtowardsover-investmentinR&D.Acomplementaryapproachrelatesthefact 1540  thatthereistoolittleortoomuchR&Dtothe“sizeofinnovations”,thatistotheheightofthejumpson thequalityladder.AccordingtoGrossman&Helpman(1991),onlyintermediate-sizeinnovationsshouldbe subsidized;small-sizeandlarge-sizeinnovationsshouldbetaxed;whileaccordingtoSegerstrom(1998),itis optimaltosubsidizesmall-sizeinnovationsandtotaxlarge-sizeinnovations. Theseapproachesfocusonintra- sectoralknowledgespilloversbutdonotconsiderinter-sectoralknowledgespillovers. ModelsbyLi(2003)and bySener(2008)generalizetheanalysisofSegerstrom(1998)bytakingthemintoaccount.Sener(2008)confirms theresultsofSegerstrom(1998),whereasLi(2003)showsthatwheninter-sectoralknowledgespilloverseffects aresufficientlylarge,R&Dactivitiesshouldbesubsidized. Asamatteroffact,“itiseasytofindmorepapers inthisliteraturewithmajordifferencesinR&Dpolicyrecommendations”(Sener,2008). Theanalysisconductedinourpapershedsnewlightontheseissuesinsofarasthemodelwedevelopenables ustoidentifyaclearlinkbetweentheintensityofknowledgespillovers,theParetonon-optimalityoftheseminal SchumpeterianequilibriuminitiallyintroducedbyAghion&Howitt(1992),andtheoptimalR&Dpublicpolicy. Thekeypointmadeinthispaperliesinthatthemarketfailureinvolvedbymarketincompletenesscharacterizing theSchumpeterianequilibriumisallthemorelikelytoleadtoanunder-optimalR&Deffortastheintensity ofknowledgespilloversishigh.Conversely, itisallthemorelikelytoleadtoanover-optimalR&Deffort astheintensityislow.Infact,werevealthattheeffectsofthedistortionofR&Dincentivesresultingfrom marketincompletenessareamplifiedallthemoreastheintensityofknowledgespilloversishigh.Thisissue isalsoreflectedinthefactthattheoptimaltooldedicatedtocorrectthemarketfailurecausedbymarket incompletenessisclearlypositivelydependentontheintensityofknowledgespillovers.Indeed,weshowthat thehigher(resp. lower)theintensityofknowledgespilloversis,themorelikelythispolicytoolshouldconsist inasubsidy(resp.tax);andthatifitisasubsidy,thenthissubsidywillbeallthelargerastheintensityis high. Theremainderofthepaperisorganizedasfollows.InSection2,weexplicitlyformalizeknowledgespillovers inanendogenousgrowthmodelwithverticalinnovationsinlinewithAghion&Howitt(1992)andwecompute thefirst-bestsocialoptimum.InSection3,westudytheseminaldecentralizedeconomyintroducedbyAghion &Howitt(1992);specifically,wedefine,characterize,andcomputeastandardSchumpeterianequilibrium.In Section4,werevisittheissueofParetonon-optimalityoftheSchumpeterianequilibriumbyconsideringthe partplayedbytheintensityofknowledgespillovers;inparticular,wecomparethelaisserfaireequilibriumwith thefirst-bestsocialoptimum,weimplementthefirst-bestsocialoptimumintheSchumpeteriandecentralized economybycharacterizingtheoptimalpublictools,andwestudythepropertiesoftheoptimaltooldedicated tocorrectthemarketfailurecausedbymarketincompleteness.WeconcludeinSection5. 2ModelandFirst-bestSocialOptimum Thissectiondisplaysacanonicalcontinuous-timeendogenousSchumpeteriangrowthmodel,inwhichweexplic- itlyformalizeknowledgespillovers.InSection2.1,wepresentthetechnologiesandthepreferences;inparticular, wedetailthemechanismsunderlyingtheprocessofinnovation,namelythepartplayedbyknowledgeaccumu- lationanddiffusion.Then,inSection2.2,wecomputethefirst-bestsocialoptimum. 2.1TechnologiesandPreferences WeconsiderastandardendogenousgrowthmodelwithverticalinnovationsinlinewithGrossman&Helpman (1991)andAghion&Howitt(1992)inwhichweintroduceexplicitlyknowledgespillovers.Forthatpurpose,we deriveagenerallawofknowledgeaccumulation,inwhichknowledgespilloversmanifestthroughtwochannels. Firstly,inanytypeofsector,theR&Dactivityproducesinnovationsusingtheknowledgeinherentinpreviously createdinnovations. Theseinnovationsmayhavebeenproducedwithinthissector;orinothersectors,which canbemoreorlesstechnologicallydistantfromtheaforesaidsector.Secondlyandreciprocally,theknowledge producedinanygivensectorspillsoverintoR&Dactivityofothersectors;thisdiffusionbeingallthemore likelythatsectorsaretechnologicallyclose.Toformalizetheseknowledgespillovers,weusethecircularproduct differentiationmodelofSalop(1979):thereisacontinuumQ,ofmeasureN,ofintermediatesectorsuniformly distributedonaclockwiseorientedcircle.Ateachdatet, ineachsectori, i∈Q, anintermediategoodi isproducedinquantityx it ; besides, eachoftheseintermediategoodsisassociatedwithaspecificstockof 1541  knowledgeκ it .Thewholestockofknowledgeintheeconomyatdatetis 1 K t = Z Q κ it di.(1) EachsectorhasitsownR&Dactivitywhichusestwoinputs: arivalone(labor)andanonrivalone(astock ofknowledge). 2 Letuspresentthesetofbasicassumptionsunderlyingtheinnovationprocessinourmodel. First,asinmoststandardSchumpeteriangrowthmodels,theinnovationprocessisuncertain: Assumption1.Inanyintermediatesectori,i∈Q,innovationsoccurrandomlywithaPoissonarrivalrate λl it ,λ>0,wherel it istheamountoflabordevotedtoR&Datdatet. Second, aninnovationatdatetinany given sectoriconsists inan enhancementofthequalityof the intermediategoodproducedinthissector.Inotherwords,aninnovationcorrespondstoanincreaseinthe stockofknowledgeκ it andtotheincorporationofthisnewstockofknowledgeintheintermediategoodi.This appearsinAssumption2below,whichformalizesthattheinnovationprocessgoesalongwiththefactthat, ineachsectori,theR&Dactivityproducesinnovations(andthusnewknowledge)bymakinguseofapool comprisingpreviouslycreatedknowledge: Assumption2.Inanyintermediatesectori,i∈Q,ifaninnovationoccursatdatet,theincreaseinknowledge is∆κ it = θP it ,θ>0,whereP it isthepoolofknowledgefromwhichthissector’sR&Dactivitycandrawfrom inordertoinnovate. Third,foranygivensectori,thecompositionofthepoolofknowledgeP it atthedisposalofR&Dactivityi basicallydependsontheusabilityoftheknowledgecreatedbytheR&Dactivitiesofalltheothersectors.The importanceoftheinfluencethattheR&Dactivitiesofvarioussectorscanhaveoneachotherhasoftenbeen stressedbyempiricalstudies(e.g.,Griliches,1992and1995;Hall,Mairesse&Mohnen,2010;Hall,2004)and hasbeenatthecoreoftheseminalendogenousgrowththeory(e.g.,Romer,1990;Aghion&Howitt,1992and 1998;Howitt,1999;Jones,1999).Inparticular,ithasbeenemphasizedthattheR&Dactivityofonesectoris likelytoentailpositivespilloverseffectsinothersectors;moreover,“suchspilloversareallthemorelikelyand significantasthesenderandthereceiverarecloselyrelated”(Hall,Mairesse&Mohnen,2010). Basedontheseideas,weproposeasimpleformalizationofhowthesepoolsofknowledgeareformed.Inany sectori,theR&Dactivityisbothreceivingandsendingknowledge. Indeed,anygivenR&Dactivityimakes useoftheknowledgegeneratedbytheinnovationprocessoccurringinothersectors.LetQ R i denotethesubset ofsectorsofQproducingknowledgewhichentersthepoolP it .Besides,throughitsinnovationprocess,any givenR&DactivityiproducesknowledgethatspillsoverintoR&Dactivitiesofothersectors. LetQ S i denote thesubsetofsectorsofQthatcanusetheknowledgeproducedbyR&Dactivityi;wenamethemeasureofQ S i the“scopeofdiffusionofknowledgeκ it ”.Wemakethefollowingassumptionsonthewayknowledgediffuses acrosssectors: Assumption3.Inanyintermediatesectori,i∈Q,whenaninnovationoccurs,knowledgespillssymmetrically overthecircleQ. Assumption4.Thescopeofdiffusionofknowledgeisidenticalforallsectors;itisdenotedbyϑ,1 ≤ϑ≤N. Notonlythesetwoassumptionsenablesustomitigateintricacy,buttheyalsoensuefromtheassumptionof symmetryacrosssectorscommonlymadeinendogenousgrowthmodels. 3 Consequently,thesubsetofQcompris- ingthesectorsthatusetheknowledgeκ it producedbyR&DactivityiisQ S i = [i−ϑ/2;i+ϑ/2].Besides,R&D activityimakesuseoftheknowledgeproducedbythesectorsbelongingtothesubsetQ R i = [i−ϑ/2;i+ϑ/2]; inotherwords,theknowledgeproducedbytheR&Dactivitiesofanysectorj∈[i−ϑ/2;i+ϑ/2]contributes 1 Knowledgeisassumedtobeanhomogenousgood.Besides,itsinitialstock,K 0 ,isnormalizedtoone. 2 Inthispaper,therivalgoodusedinR&Dislabor;alternatively,onecouldconsidertheuseofphysicalcapital,orofthefinal good(see,forinstance,inBarro&Sala-i-Martin,2003).Asdetailedbelow,thecompositionofthestockofknowledgemayinclude onlytheknowledgeproducedwiththesector,alltheknowledgeavailableintheeconomy,oranycaseinbetweenthesetwopolar cases(seethecommentsafterLemma1). 3 Weprovidemoredetailsonthesymmetryassumptionbelow,attheendofthissection.Asusual,thisstandardassumption isused inthepresentpapertocomputethefirst-bestsocial optimum(seethe proofof Proposition1 inSection2.2) andthe Schumpeterianequilibrium(seeSection3.2). 1542  tothepoolofknowledgeusedbyR&Dactivityofsectori. 4 Hence,ateachdatet,inanyintermediatesector i,thepoolofknowledgeusedbytheR&Dactivityis P it = Z Q R i κ ht dh,∀i∈Q,withQ R i = [i−ϑ/2;i+ϑ/2].(2) ThelawofknowledgeaccumulationinanysectoriisderivedfromAssumptions1,2,3,and4;itischarac- terizedinLemma1below. Lemma1.Ateachdatet,inanyintermediatesectori,knowledgeisproducedalongwith ˙κ it = λθl it P it ,∀i∈Q,whereP it isgivenin(2).(3) Proof.Letk,k∈N,bethenumberofinnovationsoccurringinagivenintermediatesectori,i∈Q,during atimeinterval(t,t+∆t).Thestockofknowledgeaccumulatedinsectoriatthebeginningoftheperiodis κ it .UnderAssumptions1and2,thestockofknowledgeattheendoftheperiod,κ it+∆t ,isarandomvariable takingthevalues {κ it +kθP it } k∈N withassociatedprobabilities R t+∆t t λl iu du k k! e − R t+∆t t λl iu du k∈N , whereP it ensuesfromAssumptions3and4,andisgivenin(2).Theexpectedstockofknowledgeatdatet+∆t isthus E[κ it+∆t ] = ∞ X k=0 R t+∆t t λl iu du k k! e − R t+∆t t λl iu du [κ it +kθP it ] = κ it ∞ X k=0 R t+∆t t λl iu du k k! +θP it Z t+∆t t λl iu du ! ∞ X k=1 R t+∆t t λl iu du k−1 (k−1)! e − R t+∆t t λl iu du = " κ it e R t+∆t t λl iu du +θP it Z t+∆t t λl iu du ! e R t+∆t t λl iu du # e − R t+∆t t λl iu du = κ it +λθ Z t+∆t t l iu du ! P it . Hence,denotingbyΛ iu aprimitiveofl iu withrespecttothetimevariableu,onehas E[κ it+∆t ]−κ it ∆t = λθ Λ it+∆t −Λ it ∆t P it . Letting∆ttendtozerointheNewton’sdifferencequotientsofE[κ it ]andofΛ it ,onegets ˙κ it ≡ ∂E[κ it ] ∂t = λθl it P it . Thisshowsthattheexpectedknowledgeinanysectori,i∈Q,isadifferentiablefunctionoftime.Itsderiva- tivegivesthelawofknowledgeaccumulationinsectoriasexhibitedinLemma1(theexpectationoperatoris droppedtosimplifynotations). Theformalizationpresentedabovegeneralizesthestandardinnovation-basedendogenousgrowththeoryin- sofarasthelawofknowledgeaccumulationderivedinLemma1isquitegeneral.Indeed,manystandardlawsof knowledgeaccumulationconsideredintheendogenousgrowththeoryliteratureturnouttobeparticularcases 4 Becauseoftheassumptionsofsymmetry,onehasQ S i = Q R i .Consideringamoregeneralframeworkinwhicheachsectorihas aspecificϑ i ,onewouldhaveQ S i = n h∈Q/|i−h|≤ ϑ i 2 o andQ R i = n h∈Q/|i−h|≤ ϑ h 2 o . 1543  of(3).AsexplainedinAghion&Howitt(1998),Howitt(1999),Jones(1999),Laincz&Peretto(2006),or Dinopoulos&Sener(2007),mostgrowthmodelsdiffermainlyinthespecificationoftheknowledgeproduction technology.Infact,Lemma1underlinesthatthemaindistinctionistobefoundintheconstitutionofthepools ofknowledgeusedbyR&Dactivities,andintheknowledgespilloverstheystemfrom. Inparticular,Lemma1illustratesthefactthattheR&Dactivityofagivensectoralwaysusestheknowledge accumulatedsofarinthissectorandpotentiallycapturespartofthemassoftheideascreatedinallothers;this subsetofK t ismoreorlessimportant,asthescopeofknowledgediffusionϑismoreorlesswide. Eventually, dependingonthechoice ofthe parameterϑ, oneobtainsalargecollectionofpools, P it , andthus models consideringvarioustypesofknowledgespillovers.Letusdisplaytwopolarcases. Globalknowledgespillovers.Itcanbeassumedthatallknowledgesystematicallyspillsintothewhole economy.Formally,ifoneassumesthatthescopeofknowledgediffusionismaximal(ϑ=N),onehasQ S i = Q R i = Q,∀i∈Q.Eachsectorimakesuseofthewholestockofknowledgeavailableintheeconomy:rewriting (2)andusing(1),onehasP it = R Q κ ht dh= K t ,∀i∈Q.Then,theresultingknowledgeproductionfunctionin eachsectoriis ˙κ it = λθl it K t ,∀i∈Q.Thisfirstpolarcaseexhibitsglobalknowledgespillovers. Itisinterestingtonotethatthislawofknowledgeaccumulationrelatestotheoneoriginallyintroduced intheseminalpaperofRomer(1990).Indeed,thepresentexpressionofthelawofknowledgeaccumulation- whichishereendogenouslyderivedfromassumptionsmadeinastochasticqualityladdersmodel-leadstoalaw ofmotionofthewholedisposableknowledgeformallyidenticaltotheknowledgeproductionfunctioninitially introducedbyRomer(1990). 5 Nointer-sectoralknowledgespilloversbutonlyintra-sectoralknowledgespillovers.Themodels ofGrossman&Helpman(1991),Segerstrom(1998),Peretto(1999),Acemoglu(2009-Ch.14),orAghion& Howitt(2009-Ch. 4)implicitlyconsiderspilloversonlywithineachsector. Ineachsector,thepoolofknowl- edgeusedbytheR&Dactivityincludesexclusivelytheknowledgepreviouslyaccumulatedwithinthissector.In ourformalization,thissecondpolarcaseamountstoassumingthatthescopeofknowledgediffusionisminimal (ϑ= 1);thenonehasP it = κ it ,∀i∈Q.Theresultinglawofknowledgeaccumulationis ˙κ it = λθl it κ it ,∀i∈Q. Furthermore,thelawofknowledgeaccumulationderivedinLemma1alsoallowstoconsiderotherstandard lawsofknowledgeaccumulationusedintheendogenousgrowththeorybyspecifyingthepoolsofknowledgeP it onanadhocbasis. “Leading-edgetechnology”.InthemodelsofAghion&Howitt(1992),Young(1998), Howitt(1999), Segerstrom(2000),orGarner(2010),itisassumedthatknowledgespilloversdependontheknowledgelevel reachedbythefrontierfirms(i.e.reachedinthemostadvancedsector).Inourformalization,thisamountsto assumingthatP it = κ max t = max{κ it ,i∈Q},∀i∈Q;then,onehas ˙κ it = λθl it κ max t ,∀i∈Q. Noknowledgespillovers.InBarro&Sala-i-Martin(2003-Ch. 6)orinPeretto(2007),theknowledge productiontechnologyusesfinalgoodonly;itisthusconsideredthatthereisneitherinter-sectoralnoreven intra-sectoralknowledgespillovers.Ourformalizationencompassesasimilarframeworkinwhichknowledge isproducedonlywithprivateinputs(laborinthepresentmodel); forthispurpose,itcanbeassumedthat P it = 1,∀i∈Q;thusonehas ˙κ it = λθl it ,∀i∈Q. Beforepresentingtheremainderofthemodel,letusprovidesomecommentsonAssumptions1,2,3,4,and onLemma1.Eachofthetreeparametersλ,θ,andϑaccountsfortheproductivityofR&Dactivities.Asseen inAssumption1,λstandsfortheproductivityofthelabordevotedtothecreationofinnovations.Asseenin Assumption2,θstandsfortheproductivityofthepoolofknowledgeintheproductionofinnovations(i.e. of newknowledge): θaccountsfortheextenttowhichthepoolofknowledgeP it contributestotheincreasesin knowledge∆κ it resultingfromaninnovation(∆κ it = θP it ).Inotherwords,θpartlymeasurestheintensityof knowledgespillovers.AsseeninAssumptions3and4,andintheexpressiongivenin(2)ofthepoolofknowledge usedbyeachR&Dactivity,thescopeofknowledgediffusion,ϑ,standsfortheoverallinfluenceoftheknowledge inherentinanyinnovationontheconstitutionofthepoolsofknowledge.Accordingly,ϑalsopartlymeasures theintensityofknowledgespillovers.Wewillreturntothesepointsbelow(seeinthecommentsofProposition1). Therestofthemodelisfairlystandardandinlinewithseminalendogenousgrowthmodels.Weconsider 5 Indeed,differentiating(1)withrespecttotime,onegets ˙ K t = R Q ˙κ it di=λθ R Q l it di K t ⇔ ˙ K t =λθL R t K t ,whereL R t = R Q l it diisthetotalamountoflaborusedinR&D. 1544  aninfinitelylivedrepresentativehouseholdwhohasthefollowingintertemporalpreferences: U= Z ∞ 0 u(c t )e −ρt dt,(4) wherec t isthepercapitalconsumption,u(c t )istheassociatedinstantaneousutilityatdatet,andρisthe subjectivediscountrate.Ateachdatet,thereareLidenticalhouseholds,eachonebeingendowedwithone unitoflaborwhichissuppliedinelastically.Inordertolimitthenumberofparameters,wemakethefollowing simplifyingassumptions.Weconsideru(c t ) = ln(c t );besides,thereisnopopulationgrowth,andthepopulation sizeisnormalizedtoone. 6 LaborisusedintheR&Dactivityofeachsectori(l it ,i∈Q)andtoproducethe finalgood(L Y t );hence,theconstraintonthelabormarketis 1 = L R t +L Y t ,whereL R t = Z Q l it diisthetotalquantityoflaborusedinR&Dactivity.(5) Anhomogeneousfinalgoodisproducedwithlaborcombinedwithallavailableintermediategoodsandthe knowledgeincorporatedineachofthem: Y t = (L Y t ) 1−α Z Q κ it (x it ) α di,0 <α<1.(6) Ateachdatet,thefinalgoodiseitherconsumedbytherepresentativehousehold(c t )orusedtoproducethe intermediategoods. Itisoftenassumed,thattheproductionfunctionofanygivenintermediategoodi,i∈Q, ischaracterizedbyincreasingcomplexity(i.e.thelargerthestockofknowledgeinherentinintermediategood i,themorefinalgoodisneededtoproducethisintermediategood).Weassumelikewise: x it = y it κ it ,∀i∈Q,(7) wherey it isthequantityoffinalgoodusedtoproducex it unitsofintermediategoodi.Thus,theconstraint onthefinalgoodmarketwrites Y t = c t + Z Q y it di.(8) Asusualinendogenousgrowththeory,computingthefirst-bestsocialoptimumaswellastheSchumpeterian equilibrium requires atsome point to make the standard assumption ofsymmetry across sectors (see, for instance,inAghion&Howitt1992or1998-Ch.3,orinPeretto&Smulders2002). 7 Formally,whenthetime comes,itwillbeassumedbothforcomputingtheoptimum(Section2.2)andforcomputingtheSchumpeterian equilibrium(Section3.2),thatallsectorshavethesameinitiallevelofknowledge(κ i0 =κ 0 ,∀i∈Q),that l it =l t ,∀i∈Q,andthatκ it =κ t ,∀i∈Q. 8 Finally,inallthatfollows,therateofgrowth, ˙z t z t ,ofanyvariable z t ,isdenotedg z t . 2.2First-bestSocialOptimum Thefirst-bestsocialoptimumisderivedbymaximizingtherepresentativehousehold’sdiscountedutility(4) subjectto(1),(2),(3),(5),(6),(7),and(8).Thecontrolvariablesarex it ,i∈Q,L Y t ,l it ,i∈Q,andc t ;besides, 6 ThekeyresultsofthepaperaremaintainedifoneusesaC.E.S.instantaneousutilityfunctionofparameterε,u(c t ) = c 1−ε t /(1−ε) and/orifoneassumesconstantpopulationgrowth. Thesemoregeneralassumptionsintroduceadditionalparametersbutdonot addanyrelevantinsighttotheissuesaddressedinthepresentpaper.Furthermore,consideringconstantpopulationgrowthcan easilybedonewithoutthemodelexhibitingthenondesirablescaleeffectsproperty.Indeed,toconsiderascale-invariantfully endogenousgrowthmodel,itissufficienttoallowforexpansioninthenumberofsectors(see,forexample,Dinopoulos&Thompson 1998,Peretto1998,Young1998,Howitt1999,Peretto1999,orAghion&Howitt2009-Ch.4);then,removingscaleeffectsin thepresentframeworkcouldbedonebyassumingthat,ateachdatet,thenumberofsectorsN t andthesizeofthepopulation L t areproportional(e.g.,N t = γL t ,γ>0).Thiscommonlyusedassumptionhasbeenjustifiedboththeoreticallyandempirically (Aghion&Howitt1998-Ch.12,Jones1999,Segerstrom2000,Laincz&Peretto2006,orDinopoulos&Sener2007).SeeinGray &Grimaud(2016)foramoregeneralversionofthepresentmodel. 7 Formoredetailsonthisassumptionofsymmetryacrosssectors,seeforinstanceinPeretto(1998,1999),orinCozzi,Giordani &Zamparelli(2007). 8 Bywayofexample,letusrewrite(1),(2),and(3)underthesymmetryassumption.Thewholestockofknowledgeinthe economyisK t = R Q κ it di= Nκ t .ThepoolofknowledgeusedinsectoriisP it = P t = ϑκ t ,∀i∈Q.Theresultinglawofknowledge accumulationis ˙κ it =˙κ t = λθϑl t κ t ,∀i∈Q. 1545  thereisacontinuumofstatevariablesκ it ,i∈Q(infact,thereisacontinuumofconstraintsrelativetothelaw ofmotionofknowledge).TheHamiltonianofthedynamicoptimizationproblemcanbewrittenas H= ln(c t )e −ρt +µ t Y t −c t − Z Q κ it x it di +ν t 1−L Y t − Z Q l it di + Z Q ζ it [˙κ it ]di, whereY t = (L Y t ) 1−α R Q κ it (x it ) α di,˙κ it = λθl it P it ,∀i∈Q,P it = R Q R i κ ht dh;andwhereµ t ,ν t ,andζ it ,i∈Q,are theco-statevariablesassociatedwiththefinalgoodresourceconstraint,thelaborconstraint,andthecontinuum ofstatevariables,respectively. Thefirst-bestsocialoptimumischaracterizedinthefollowingProposition.Fromnowon,thesuperscript “ o ” isusedfor“‘first-bestsocialoptimum”.Furthermore,letusintroducethenotationΘ≡θϑ; asdetailed belowinthecommentsofProposition1,Θisdefinedasthemeasureofthe“intensityofknowledgespillovers”. Proposition1.Atthefirst-bestsocialoptimum,thepartitionoflabor,thequantityofeachintermediategood i,andthegrowthratesare L Yo t = L Yo = ρN λΘ ;l o it = l o = 1 N − ρ λΘ ,∀i∈Q;x o it = x o = α 1 1−α ρN λΘ ,∀i∈Q; andg o c t = g o Y t = g o K t = g o κ it = g o = λΘl o = λ N Θ−ρ,∀i∈Q,∀t,withΘ ≡θϑ. Proof.Thefirst-orderconditionsofthemaximizationprogramare 9 ∂H ∂x it = 0,∀i∈Q⇔µ t α(L Y t ) 1−α κ it (x it ) α−1 −κ it = 0,∀i∈Q,(9) ∂H ∂L Y t = 0 ⇔µ t (1−α) Y t L Y t = ν t ,(10) ∂H ∂l it = 0,∀i∈Q⇔ζ it λθ Z Q R i κ ht dh= ν t ⇔ζ it λθP it = ν t ,∀i∈Q,(11) ∂H ∂c t = 0 ⇔c −1 t e −ρt = µ t ,(12) ∂H ∂κ it = − ˙ ζ it ,∀i∈Q⇔µ t (L Y t ) 1−α (x it ) α −x it + Z Q R i ζ ht λθl ht dh= − ˙ ζ it ,∀i∈Q.(13) From(9),onegetstheusualsymmetricuseofintermediategoods: x it = x t = α 1 1−α L Y t ,∀i∈Q.(14) Using(1)and(14),thefinalgoodproductionfunction(6)rewrites Y t = α α 1−α L Y t K t .(15) Then,from(8),(14),and(15),oneobtains Y t = c t +α 1 1−α L Y t K t ⇔Y t = c t +αY t ⇔ c t Y t = 1−α.(16) Log-differentiating(12),(15),and(16)gives g c t +ρ= −g µ t ,g Y t = g L Y t +g K t ,andg c t = g Y t ,respectively.(17) Besides,plugging(15)in(10)andin(13),onegets µ t (1−α)α α 1−α K t = ν t .(18) andµ t (1−α)α α 1−α L Y t +λθ Z Q R i ζ ht l ht dh= − ˙ ζ it ,∀i∈Q.(19) 9 Plustheusualtransversalityconditionwhichwillbeusedtocharacterizethesteady-stateoptimum. 1546  Letusnowusetheusualassumptionofsymmetryacrosssectors(l it =l t andκ it =κ t ,∀i∈Q). 10 The wholestockofknowledge(1)andthepoolofknowledgeinsectori(2)thenwrite K t = Z Q κ it di= Nκ t ,andP it = Z Q R i κ ht dh= P t = ϑκ t ,∀i∈Q,respectively.(20) Hence,weobtainthegrowthrateofthestockofknowledgeinanysectoriandofthewholestockofknowledge intheeconomy: g κ it = g κ t = g K t = λθϑl t ,∀i∈Q.(21) Besides,thelaborconstraint(5)rewrites Nl t = 1−L Y t .(22) Underthesymmetryassumption,(11)canbeexpressedas ζ it = ζ t = ν t λθϑκ t ,∀i∈Q.(23) Then,using(18),(20),and(23),oneobtains ν t = µ t (1−α)α α 1−α Nκ t = ζ t λθϑκ t ⇔ µ t ζ t = λθϑ (1−α)α α 1−α N .(24) Log-differentiating(24),oneobtains g µ t = g ζ t .(25) Furthermore,rewriting(19)underthesymmetryassumptiongives µ t ζ t (1−α)α α 1−α L Y t +λθϑl t = − ˙ ζ t ζ t ;then,using (24)onegetsg ζ t = ˙ ζ t ζ t = − λθϑ (1−α)α α 1−α N (1−α)α α 1−α L Y t +λθϑl t = − λθϑ N L Y t +Nl t .Finally,using(22)and(25) gives g µ t = g ζ t = − λθϑ N .(26) Then,theoptimalgrowthrateofper-capitaconsumption,g o c ,canbederivedusing(17)and(26): g o c = λ N θϑ−ρ.(27) Finally,thefirst-bestsocialoptimumisthesolutionofthefollowingsystemofequations,whichsummarize theconditionsobtainedin(14),(17),(21),(22),and(27): g o c = λθϑ N −ρ(a) g o Y = g o c (b) g o L Y t = g o Y −g o K t (c) g κ o it = g κ o t = g K o t = λθϑl o t ,∀i∈Q (d) Nl o t = 1−L Yo t (e) x o it = x o t = α 1 1−α L Yo t ,∀i∈Q(f) Fromequations(a),(b),(c),(d)and(e),oneobtainsthefollowingdifferentialequationinL Yo t : g o L Y t = λθϑ N −ρ−λθϑ 1−L Yo t N ⇔g o L Y t = λθϑ N L Yo t −ρ. LetX t =1/L Yo t ,onegetsafirst-orderlineardifferentialequationinX t , ˙ X t −ρX t =− λθϑ N ,thesolutionof whichis X t = X 0 − λθϑ ρN e ρt + λθϑ ρN ⇔L Yo t = 1 1 L Yo 0 − λθϑ ρN e ρt + λθϑ ρN . 10 Seethecommentabove(endofSection2.1)onthesymmetryassumptioncommonlymadeinendogenousgrowthmodels. 1547  Usingthetransversalitycondition,weshowthatL Yo t instantlyreachesitssteady-statelevelL Yo ss = ρN/λθϑ; andthus,onehasL Yo t = L Yo ss ,∀t.Hence,oneobtainstheoptimalpartitionoflabor,andoptimalquantityof intermediategoods: L Yo = ρN λθϑ ,l o i = l o = 1 N − ρ λθϑ ,∀i∈Q,andx o i = x o = α 1 1−α ρN λθϑ ,∀i∈Q.(28) Theoptimalgrowthrateofknowledgeaccumulationinanysectoriandinthewholeeconomyaregivenby g κ o i = g κ o = g K o = λθϑ N −ρ,∀i∈Q.(29) Finally,(27),(28),and(29)proveProposition1. AsexplainedaboveinthecommentstoAssumptions1,2,3,4,andtoLemma1,thetreeparametersλ,θ, andϑalldeterminetheproductivityofR&Dactivities.Hence,asshownbytheresultsobtainedinProposition 1,theseparametersobviouslydeterminetheoptimalgrowthrateoftheeconomy.Atthefirst-bestoptimum,an increaseinλ,θ,and/orϑimpliesareallocationoflaborfromthefinalgoodproductiontowardR&Dactivity; thusleadingtoahighergrowthrate. ThepartplayedbyλisratherevidentasitdeterminestheproductivityoflaborinR&D(Assumption1): themoreproductivetheR&Dis,themoreinnovationsoccur,andthusthehighergrowthrateis. Therolesplayedbytheparametersθandϑarecloselyrelated:θdeterminestheproductivityofthepoolof knowledgeintheproductionofnewknowledgeinagivensector(Assumption2),andϑdeterminestheoverall influenceoftheknowledgeproducedinaparticularsectorontheconstitutionofthepoolsofknowledgeusedby theR&Dactivitiesoftheothersectors(Assumptions3and4,andequation(2)). Whenaninnovationoccurs inagivensector,θmeasurestheimpactoftheknowledgeinherentinthisinnovationontheproductionofnew knowledgeinothersectors;whereasϑmeasuresthesubsetofsectorswhichwillbeusingtheknowledgestemming fromthisinnovation. Nevertheless,θandϑbothclearlymeasuretheintensityofknowledgespillovers.Inthe presentmodel,asillustratedinProposition1,thesetwoparametersoftenappearintheformoftheproductθϑ; therefore,wehaveintroducedthenotationΘ≡θϑ,tostandforthe“intensityofknowledgespillovers”,which isakeydeterminantintheoptimalgrowthrate.OnehasthefollowingCorollarytoProposition1. Corollary.ThestrongertheintensityofknowledgespilloversΘintheeconomy,thehighertheoptimalR&D effortl o = 1 N − ρ λΘ ,andthusthehighertheoptimalgrowthrateg o = λ N Θ−ρ. Toconcludethissection,letusnotethatintheendogenousgrowthliterature,thedenomination“knowledge spillovers” relatessimultaneouslytotwoissues.First,a“technology-related” issueasitreferstothemecha- nismbywhichknowledgepreviouslycreatedspillsoverintotheeconomy,thusenhancingthecreationofnew knowledge.Second,an“equilibirum-related” issueasitreferstomarketincompleteness:inthedecentralized economiesgenerallyconsidered(e.g.,theequilibriastudiedbyRomer,1990,orbyAghion&Howitt,1992), knowledgeisnotpriced. Asdetailedintheintroductionofthispaper,knowledgespilloversrelatedissueshave beenextensivelystudied(e.g.,Romer,1990;Aghion&Howitt1992,1998,2009;Segerstrom,1998;Li,2002; Peretto&Smulders,2002;Jones,2005;Sener,2008;Acemoglu,2009). Inthissection,wehaveintroducedex- plicitlyaformalizationofknowledgespilloversinthetechnologyofproductionofknowledge(seeAssumptions 2,3,and4,andLemma1).InSection3below,westudyaSchumpeterianequilibrium`aAghion&Howitt (1992),andweshedanewlightonthelinkbetweenknowledgespilloversandtheParetonon-optimalityofthe Schumpeterianequilibriumwhichresultsfrommarketincompleteness. 3DecentralizedEconomy:SchumpeterianEquilibrium Inthis section, we studya decentralized economy inwhichthe fundingof R&Dactivity (andthusofthe productionofknowledge)isbasedonassumptionsinspiredbySchumpeter’screativedestructionmechanism. Formally,wedefine,characterize,andcomputethestandardSchumpeterianequilibrium`aAghion&Howitt (1992). 3.1DefinitionoftheSchumpeterianEquilibriumandAgents’Behaviors Consideranysectori,i∈Q.Onceaninnovationoccursinsectori,itsproducerisgivenaninfinitely-lived patent. Then,ineachsectori,thelatestinnovatorhasamonopolyontheintermediategoodiuntilreplaced 1548  bythenextinnovatorupgradingthequalityofthisintermediategood.Hence,theR&Dactivityofeachsector -whichbasicallyproduceknowledge-isindirectlyfinancedbyasuccessionofmonopolyonanintermediate goodthequalityofwhichsequentiallyincreasesastheknowledgeitincorporatesaccumulatesinthesector. Suchadecentralizedeconomyischaracterizedbytwomarketfailures.First,marketpowerimpliedbythe presenceofamonopolyoneachintermediategoodmarket.Second,marketincompleteness:knowledgecreation isindirectlyfundedbymonopolyprofitssincethere isnomarket forknowledge(knowledgeisnotpriced). Hence,theSchumpeterianequilibriumconsideredinthepresentpaperislikelytobeParetonon-optimal;in particular,atthelaisserfaireequilibrium,thequantityoflaborusedinR&D(i.e.theR&Deffort)caneither besub-optimalorover-optimal.Wethusconsidertwotoolsdedicatedtomitigatethesetwomarketfailures. LetΥ P denotethetooltocorrectmarketpower;itiswellknownthatmonopolypowercanbecorrectedby anadvaloremsubsidyoneachintermediategooddemand.LetΥ I denotethepublictooltoalleviatemarket incompleteness;asitwillbeprovenbelow,thistoolcanconsisteitherinasubsidyorinataxontheprofitsof eachR&Dactivity. LetusnowdefineformallythesetofSchumpeterianequilibriaasfunctionsofthepolicytoolsvector Υ P ,Υ I . Wedenotethewagebyw t ,thepriceofintermediategoodibyq it ,i∈Q,theinterestratebyr t ;andwenormalize thepriceofthefinalgoodtoone. Definition.Ateachvectorofpublicpolicytools Υ P ,Υ I isassociatedaparticularSchumpeterianequilibrium whichconsistsoftimepathsofsetofprices n w t Υ P ,Υ I , q it Υ P ,Υ I i∈Q ,r t Υ P ,Υ I o ∞ t=0 andofquantities n L Y t Υ P ,Υ I , l it Υ P ,Υ I i∈Q , κ it Υ P ,Υ I i∈Q , x it Υ P ,Υ I i∈Q ,c t Υ P ,Υ I ,Y t Υ P ,Υ I o ∞ t=0 suchthat:therepresentativehouseholdmaximizeshisutility;firmsmaximizetheirprofits; thelabormarket, thefinancialmarket,andthefinalgoodmarketareperfectlycompetitiveandclear;oneachintermediategood market,theinnovatorisgrantedaninfinitely-livedpatentandmonopolizestheproductionandsaleuntilreplaced bythenextinnovator;andthereisfreeentryoneachR&Dactivity(i.e.thezeroprofitconditionholdsforeach R&Dactivity). Inordertofullycharacterizethedecentralizedeconomy,wenowpresentindetailtheagents’behaviors.For thepurposeofsimplifyingnotationsthroughoutthefollowingcomputations,wemomentarilydropthe Υ P ,Υ I argumentsforallvariables. Representativehousehold. Therepresentativehouseholdmaximizeshisintertemporalutilitygivenby(4)subjecttohisbudgetconstraint, ˙ b t =w t +r t b t −c t −T t ,whereb t denotesthepercapitafinancialassetandT t isalump-sumtaxusedbythe governmenttofinancepublicpolicies.OnegetstheusualKeynes-Ramseycondition: r t = g c t +ρ.(30) Finalgoodproducer. Thefinalgoodmarketisassumedtobecompetitive;theprofitofthefinalgoodproducer(recall,thepriceof thefinalgoodisnormalizedtoone)writes π Y t = (L Y t ) 1−α Z Q κ it (x it ) α di−w t L Y t − Z Q (1−Υ P )q it x it di. Thefirst-orderconditionsoftheprofitmaximizationprogramare ∂π Y t ∂L Y t = 0 ⇔w t = (1−α) Y t L Y t ,and(31) ∂π Y t ∂x it = 0,∀i∈Q⇔ 1−Υ P q it = α(L Y t ) 1−α κ it (x it ) α−1 ,∀i∈Q,∀i∈Q.(32) 1549  Intermediategoodsproducers. Ineachsectori,i∈Q,thelatestinnovatorhasamonopolyontheproductionandsaleofintermediategood i.GiventhepublicinterventiononR&Dactivity(formalizedbythepublictoolΥ I introducedabove),the incumbentmonopolymaximizestheinstantaneousnetprofit π x i t = (1+Υ I )(q it x it −y it ),(33) wherethedemandforintermediategoodi,x it ,isgivenby(32).Using(7),themonopolymaximizationprogram canbewritten: Max x it π x i t = (1+Υ I )(q it x it −x it κ it )subjectto 1−Υ P q it = α(L Y t ) 1−α κ it (x it ) α−1 . Thefirst-orderconditionwithrespecttox it is ∂π x i t ∂x it = 0 ⇔ ∂ ∂x it (1+Υ I ) α(L Y t ) 1−α κ it (x it ) α 1−Υ P −x it κ it = 0 ⇔x it = α 2 1−Υ P 1 1−α L Y t ; replacingin(32),onegets q it = α 1−Υ P L Y t x it 1−α κ it = α 1−Υ P 1−Υ P α 2 1 1−α ! 1−α κ it = κ it α . Therefore,oneobtainstheusualsymmetricuseofintermediategoodsinthefinalgoodproductionandthe usualmark-uponthepriceofintermediategoods: x it = x t = α 2 1−Υ P 1 1−α L Y t andq it = κ it α ,∀i∈Q.(34) Then,using(34)and(1),wecanrewritethemonopolyprofitinanysectori(33),thefinalgoodproduction function(6),andtheexpressionofthewagegivenin(31)as: π x i t = (1+Υ I ) 1−α α α 2 1−Υ P 1 1−α L Y t κ it ,∀i∈Q,(35) Y t = α 2 1−Υ P α 1−α L Y t K t ,and(36) w t = (1−α) α 2 1−Υ P α 1−α K t .(37) R&Dactivities. Ineachsectori,i∈Q,theincumbentinnovatorhavinginnovatedatdatethasamonopolyonintermediate goodi,andreceives,atanydateτ>t,theinstantaneousnetprofitπ x i τ withprobabilitye − R τ t λl iu du (i.e.as longasthereisnootherinnovationinsectoribetweendatetanddateτ). 11 WedenotebyΠ x i t thevalueat datetofthelatestinnovationinsectori,itisthesumofthepresentvaluesoftheincumbent’sexpectednet profitsonthesaleofintermediategoodi: Π x i t = Z ∞ t π x i τ e − R τ t (r u +λl iu )du dτ,whereπ x i τ isgivenby(33).(38) Then,thearbitrageconditioninR&Dactivityiisobtainedbydifferentiating(38)withrespecttotime;one gets r t +λl it = ˙ Π x i t Π x i t + π x i t Π x i t ,∀i∈Q.(39) 11 AsdetailedinAssumption1,innovationsinsectorioccursaccordingtoaPoissonarrivalrateλl it . 1550  GivenAssumption1,innovationsarriveaccordingtoaPoissonprocessofrateλl it ; thusthetotalexpected revenueatdatetwhenoneunitoflaborisinvestedinR&DisλΠ x it .Besides,thecostofoneunitoflaboris w t .Consequently,thefreeentryconditioninanyR&Dactivityiis w t = λΠ x i t .(40) Then,using(37),onegetsthefollowingvalueofthelatestinnovationinsectoriatdatet: Π x i t = Π x t = 1−α λ α 2 1−Υ P α 1−α K t ,∀i∈Q.(41) 3.2CharacterizationoftheSchumpeterianEquilibrium Log-differentiating(36),onehas g Y t = g L Y t +g K t .(42) Using(1),(7),and(34),theconstraintonthefinalgoodmarket(8)rewrites Y t = c t + α 2 1−Υ P 1 1−α L Y t K t .(43) Then,from(36)and(43),oneobtains c t = 1− α 2 1−Υ P Y t .(44) Log-differentiatingthisexpressiongives g c t = g Y t .(45) Furthermore,log-differentiating(41)gives ˙ Π x i t Π x i t =g K t ;then,using(35)and(41),thearbitragecondition(39) canthenberewritten r t +λl it = g K t + 1+Υ I 1−Υ P λαL Y t κ it K t ,∀i∈Q.(46) Asexplainedabove,computingtheSchumpeterianequilibriumrequirestoconsiderthestandardassumption ofsymmetryacrosssector: l it =l t ,∀i∈Qandκ it =κ t ,∀i∈Q. 12 Accordingly,thewholestockofknowledge (1),thepoolofknowledgeinanysectori(2),thegrowthratesofthesestocksofknowledge,andthelabor constraint(5)canberewrittenasfollows: 13 K t = Z Q κ it di= Nκ t ;P it = Z Q R i κ ht dh= P t = ϑκ t ,∀i∈Q;(47) g κ it = g κ t = g K t = λθϑl t ,∀i∈Q;(48) Nl t = 1−L Y t .(49) Hence,inanysectori,i∈Q,thearbitrageconditioninR&D(46)rewrites r t +λl t = λθϑl t + 1+Υ I 1−Υ P λα N L Y t .(50) Eventually,thesetofSchumpeterianequilibriaischaracterizedby(30),(34),(36),(37),(42),(45),(48),(49), and(50);itisdisplayedinProposition2below. Proposition2.Ateachdatet,thesetofSchumpeterianequilibriaàlaAghion&Howittischaracterizedas follows. 12 See,attheendofSection2.1,thecommentonthesymmetryassumptioncommonlymadeinendogenousgrowthmodels. Furthermore,therelevancyofthesymmetricequilibriumisdiscussedinCozzi,Giordani&Zamparelli(2007). 13 Thesamereasoninghasbeenmadewhencomputingthefirst-bestsocialoptimum;see,(22),(20),and(21). 1551  •Thelaborpartitionandthequantitiesofintermediategoodsare L Y t Υ P ,Υ I = L Y Υ P ,Υ I = ρ+ λ N λ N 1+ 1+Υ I 1−Υ P α −1 ,∀t; l it Υ P ,Υ I = l Υ P ,Υ I = 1 N 1−L Y Υ P ,Υ I ,∀i∈Q,∀t; x it Υ P ,Υ I = x Υ P ,Υ I = α 2 1−Υ P 1 1−α L Y Υ P ,Υ I ,∀i∈Q,∀t. •Thegrowthratesofpercapitaconsumption,ofthefinalgoodoutput,ofthewholestockofknowledgeinthe economy,andofthestockofknowledgeineachsectorare g c t Υ P ,Υ I = g Y t Υ P ,Υ I = g K t Υ P ,Υ I = g Υ P ,Υ I ,∀t; g κ it Υ P ,Υ I = g Υ P ,Υ I ,∀i∈Q,∀t;whereg Υ P ,Υ I = λΘl Υ P ,Υ I . •Thestockofknowledgeintheeconomy,thequantityoffinalgood,andthelevelofpercapitaconsumptionare K t Υ P ,Υ I = e g ( Υ P ,Υ I ) t ,∀t, Y t Υ P ,Υ I = α 2 1−Υ P α 1−α L Y Υ P ,Υ I K t Υ P ,Υ I ,∀t, andc t Υ P ,Υ I = 1− α 2 1−Υ P Y t Υ P ,Υ I ,∀t. •Theprices(wage,priceofintermediategoods,andinterestrate)are w t Υ P ,Υ I = (1−α) α 2 1−Υ P α 1−α K t Υ P ,Υ I ,∀t, q it Υ P ,Υ I = q t Υ P ,Υ I = K t Υ P ,Υ I αN ,∀i∈Q,∀t, andr t Υ P ,Υ I = r Υ P ,Υ I = g Υ P ,Υ I +ρ,∀t. Proof.From(30),(42),(45),(48),and(50),oneobtains g L Y t +λθϑl t +ρ+λl t = λθϑl t + 1+Υ I 1−Υ P λα N L Y t . Using(49)andrearrangingthetermsresultsinthefollowingdifferentialequationinL Y t : g L Y t +ρ+ λ N (1−L Y t ) = 1+Υ I 1−Υ P λα N L Y t ⇔g L Y t − λ N 1+ 1+Υ I 1−Υ P α L Y t = − ρ+ λ N . UsingthevariablesubstitutionX t = 1/L Y t ,oneobtains ˙ X t − ρ+ λ N X t = − λ N 1+ 1+Υ I 1−Υ P α . Thesolutionofthisfirst-orderlineardifferentialequationis X t = e ( ρ+ λ N ) t X 0 − 1 ρ+ λ N λ N 1+ 1+Υ I 1−Υ P α ! + 1 ρ+ λ N λ N 1+ 1+Υ I 1−Υ P α . 1552  Consequently,onehas L Y t = 1 e ( ρ+ λ N ) t 1 L Y 0 − 1 ρ+ λ N λ N h 1+ 1+Υ I 1−Υ P α i + 1 ρ+ λ N λ N h 1+ 1+Υ I 1−Υ P α i . ThetransversalityconditionintheprogramoftherepresentativehouseholdimpliesthatL Y t immediatelyjumps toitssteady-statelevelL Y ss ;onehasL Y t = L Y ss ,∀t. Reintroducingthe Υ P ,Υ I arguments,theequilibrium quantityoflaborinthefinalgoodproductionis L Y t Υ P ,Υ I = L Y Υ P ,Υ I = ρ+ λ N λ N 1+ 1+Υ I 1−Υ P α −1 ,∀t.(51) Hence,onehasg L Y t = 0.Therefore,onecannowderivealltheequilibriumquantities,growthrates,andprices. Replacing(51)in(49)givesthequantityoflaborusedineachsectorR&Dactivity, l it Υ P ,Υ I = l Υ P ,Υ I = 1 N 1−L Y Υ P ,Υ I ,∀i∈Q,∀t.(52) Replacing(51)in(34)givesthequantityofeachintermediategoodusedinthefinalgoodproduction,x it Υ P ,Υ I = x Υ P ,Υ I ,∀i∈Q.From(45),(48),and(52),onegetsthegrowthrateofthestockofknowledgeineachsec- tor,g κ it Υ P ,Υ I =g Υ P ,Υ I ,∀i∈Q; thegrowthrateofthewholestockofknowledgeintheeconomy, g K t Υ P ,Υ I =g Υ P ,Υ I ;thegrowthrateoffinalgood,g Y t Υ P ,Υ I =g Υ P ,Υ I ;andthegrowthrateof percapitaconsumption,g c t Υ P ,Υ I = g Υ P ,Υ I .Then,thewholestockofknowledgeintheeconomyis K t Υ P ,Υ I = e g ( Υ P ,Υ I ) t ,∀t,whereg Υ P ,Υ I = λθϑl Υ P ,Υ I .(53) From(36),(44),(51),and(53),onegetstheequilibriumlevelsoffinalgoodandofpercapitaconsumption, Y t Υ P ,Υ I andc t Υ P ,Υ I .Replacing(53)in(37)andin(34),onegetstheequilibriumwage,w t Υ P ,Υ I , andtheequilibriumpriceofintermediategoodsq it Υ P ,Υ I =q t Υ P ,Υ I ,∀i∈Q.Finally,using(30)and (53),onegetstheequilibriuminterestrater t Υ P ,Υ I = r Υ P ,Υ I .ThisprovesProposition2. 4SchumpeterianEquilibriumandParetoNon-Optimality Asmentionedin3.1,intheabsenceofpublicpolicies,thedecentralizedeconomyconsideredinthispaperis likelytobeParetonon-optimal.Thepresentsectionaddressesthisissue.Notably,in4.1,werevisitthefact thatinthelaisserfaireequilibrium,theR&Deffort(i.e.thequantityoflaborusedinR&D)caneitherbe sub-optimalorover-optimal.Then,in4.2,weimplementthefirst-bestsocialoptimumintheSchumpeterian decentralizedeconomybycharacterizingtheoptimalpolicytools;thisenablesustoshowhowtheoptimaltool dedicatedtocorrectthemarketfailurecausedbymarketincompletenessdependsontheintensityofknowledge spillovers.Finally,in4.3,wehighlightthelinkbetweentheissueofParetonon-optimalityandtheintensityof knowledgespillovers. 4.1LaisserfaireSchumpeterianequilibrium FromProposition2, wecanderivestraightforwardlythelaisserfaire Schumpeterianequilibriumbysetting down Υ P ,Υ I = (0,0). Corollary.Ateachdatet,thelaisserfaireSchumpeterianequilibriumàlaAghion&Howittischaracterized asfollows. •Thelaborpartitionandthequantitiesofintermediategoodsare L Ylf = L Y (0,0) = ρ+ λ N λ N (1+α) = ρN λ +1 1+α ; l lf i = l lf = l(0,0) = 1 N − ρ+ λ N λ(1+α) ,∀i∈Q; x lf i = x lf = x(0,0) = α 2 1−α ρ+ λ N λ N (1+α) ,∀i∈Q. 1553  •Thegrowthratesofpercapitaconsumption,ofthefinalgoodoutput,andofthewholestockofknowledgein theeconomyare g lf c t = g lf Y t = g lf K t = g lf = g(0,0) = λΘl(0,0) = λΘ 1 N − ρ+ λ N λ(1+α) ! . Thecomparisonofthelaisserfaire SchumpeterianequilibriumobtainedinthisCorollarywiththefirst- bestsocialoptimum(derivedinProposition1)enablesustorevisittheissueofParetonon-optimalityofthe SchumpeterianequilibriumbyhighlightingthecentralroleplayedbytheintensityofknowledgespilloversΘ. TheSchumpeterianequilibriumlaisserfairegrowthrate,g lf =λΘl lf ,andtheoptimalgrowthrate,g o = λΘl o ,bothdependpositivelyonΘbutinadifferentwaybecausethepartitionoflabor,andthusthequantitiesof laborallocatedtoR&Dl lf andl o differfromeachother.ThequantityoflaborinthelaisserfaireSchumpeterian equilibrium,l lf ,doesnotdependontheintensityofknowledgespilloversΘ.Onthecontrary, theoptimal quantityoflaborinR&D,l o ,clearlydependspositivelyonΘ.Hence,thehigherΘis,thehighertheR&D effortshouldbeinordertomaintainoptimality.However,l lf isindependentofΘ. BecausetheSchumpeterianequilibriumexhibitsincompletemarkets,eachR&Dactivitydoesnotinternalize thepositiveimpactoftheknowledgeitcreatesonotherR&Dactivities;moreover,thisimpactisallthemore significantastheintensityofknowledgespilloversΘ isstrong.Therefore,ifΘ ishigh(resp.low),itislikelythat R&Deffortwillbeinsufficient(resp.excessive)withrespecttowhatwouldbeitsoptimallevel.Eventually, thisexplainswhythelaisserfairegrowthrateg lf canbelowerorhigherthantheoptimalgrowthrateg o . Furthermore,itispossibletodetermineathreshold ˜ Θsuchthatg lf islower(resp. greater)thang o ifand onlyifΘisabove(resp.below)thisthreshold: 14 g lf Sg o ⇔Θ T ˜ Θ,where ˜ Θ = (1+α)ρ ρ+ λ N .(54) TheseresultsareillustratedinFigure1. Figure1.Paretonon-optimalityoftheLaisserfaireSchumpeterianequilibriumandintensityofknowledge spillovers. 4.2Implementationofthefirst-bestsocialoptimal Asdetailedabove,theSchumpeterianequilibriumislikelytobeParetonon-optimalbecauseitinvolvestwo marketfailures;thenceforth,thefirst-bestsocialoptimalcanbeimplementedbytheuseoftwotools.The optimalsetoftools Υ Po ,Υ Io can,forinstance,bedeterminedbyidentifyingtheequilibriumquantitiesof intermediategoodsandoflaborinR&D,x Υ P ,Υ I andl Υ P ,Υ I ,withtheoptimalones,x o andl o .One getsProposition3below. Proposition 3. Thefirst-best socialoptimum canbe implemented inthe Schumpeterianequilibrium.The optimalsetofpublictools Υ Po ,Υ Io isgivenby Υ Po = 1−αandΥ Io = Θ 1+ λ ρN −2.(55) 14 Theproofisstraightforward:g lf Sg o ⇔λΘ 1 N − ρ+ λ N λ(1+α) S λ N Θ−ρ⇔Θ T (1+α)ρ ρ+ λ N ≡ ˜ Θ. 1554  Proof.InProposition2,wehavecharacterizedx Υ P ,Υ I andl Υ P ,Υ I ;inProposition1,wehavecomputed x o andl o .TheoptimaltoolsΥ Po andΥ Io mustsatisfy x Υ Po ,Υ Io = x o andl Υ Po ,Υ Io = l o . Fromx Υ Po ,Υ Io = x o ,onegets α 2 1−Υ Po 1 1−α L Y Υ Po ,Υ Io = α 1 1−α ρN λθϑ ,whereL Y Υ Po ,Υ Io = L Yo = ρN λθϑ ⇔ α 2 1−Υ Po 1 1−α ρN λθϑ = α 1 1−α ρN λθϑ ⇔ α 2 1−Υ Po = α⇔Υ Po = 1−α. From,l Υ Po ,Υ Io = l o onegets 1 N 1− ρ+ λ N λ N 1+ 1+Υ Io 1−Υ Po α −1 ! = 1 N − ρ λθϑ ⇔ρ+ λ N = ρ θϑ 1+ 1+Υ Io 1−Υ Po α ; then,usingΥ Po = 1−α,oneobtains ρ+ λ N = ρ θϑ 1+ 1+Υ Io α α ⇔Υ Io = Θ 1+ λ ρN −2. ThisprovesProposition3. TheresultsderivedinProposition3corroboratetheanalysisofthelaisserfaireconductedin4.1.Regarding theoptimaltooltocorrectthemarketfailureentailedbythepresenceofamonopoly,Υ Po ,werecoveraresult whichisstandardintheendogenousgrowthliterature.Withregardtotheoptimaltooltomendtheexternality resultingfrommarketincompleteness;wehaveshowninProposition3that-asexpected-itcanconsistina subsidy(Υ Io >0)orinatax(Υ Io <0),dependingontheparametersofthemodel. ThispropertyoftheoptimaltoolΥ Io echoestothefactthat,asexplainedabove,instandardSchumpeterian growthmodels,thedecentralizedR&Deffortcaneitherbesub-optimalorover-optimal.Thekeypointrevealed hereliesinthatΥ Io isanincreasingfunctionofΘ.Thishighlightsthekeyroleplayedbytheintensityof knowledgespillovers.Indeed, weobtainthecriticalresultaccordingtowhichthestrongertheintensityof knowledgespilloversΘis,themorelikelytheoptimaltoolΥ Io dedicatedtoR&Dshouldconsistinasubsidy; furthermore,thissubsidywillbeallthehigherasΘishigh.ThereasonforthisliesinthefactthatR&D incentivesareskewedbymarketincompleteness:ifΘishigh(resp.low),theR&Deffortwillmorelikelybe insufficient(resp.excessive),thisiswhyR&Dshouldprobablybesubsidized(resp.taxed). 4.3Optimaltooltocorrectmarketincompleteness,Paretooptimality,andinten- sityofknowledgespillovers LetusnowstudyinmoredetailsthepropertiesoftheoptimaltoolΥ Io .Forthatpurpose,letusfocusonthe SchumpeterianequilibriuminwhichthemonopolydistortionisoptimallycorrectedbysettingΥ P =Υ Po = 1−α,andinwhichthereislaisserfaireregardingR&D(Υ I = 0).Theassociatedgrowthratewrites g Υ Po ,0 = λΘl Υ Po ,0 = λΘ 1 N 1−L Y Υ Po ,0 = Θ λ N − ρ+ λ N 2 ! . Thecomparisonwiththeoptimalgrowthrateyieldsthefollowingresult: 15 g Υ Po ,0 Sg o ⇔Υ Io T0.(56) 15 Theproofisstraightforward: g Υ Po ,0 Sg o ⇔Θ λ N − ρ+ λ N 2 ! S λ N Θ−ρ⇔ρ− ρ+ λ N 2 Θ S0 ⇔Θ 1+ λ ρN −2 T0 ⇔Υ Io T0. 1555  In(56),weprovetheintuitiveresultthatoncethemarketfailureentailedbymarketpowerisoptimallycorrected, theoptimaltooltocorrectmarketincompleteness,Υ Io ,isasubsidy(resp.atax)ifandonlyiftheallocationof laborinR&Dactivity-andthusthegrowthrate-issub-optimal(resp.over-optimal). Again,theunderlying reasonliesinthatmarketincompletenessdistortsR&DincentivesleadingtoanonoptimalR&Deffort(too littleortoomuchlaborusedinR&D). Besides,similarlyasin4.1,herealso,onecandetermineathreshold ˜ ˜ Θsuchthat 16 g Υ Po ,0 Sg o ⇔Θ T ˜ ˜ Θ,where ˜ ˜ Θ = 2ρ ρ+ λ N .(57) Theresultsobtainedin(56)and(57)aresummarizedinProposition4andillustratedinFigure2. Figure2.Optimaltooltocorrectmarketincompletenessandintensityofknowledgespillovers. Proposition4.Thereexistsalevelofintensityofknowledgespillovers ˜ ˜ Θsuchthat Θ T ˜ ˜ Θ = 2ρ ρ+ λ N ⇔g Υ Po ,0 Sg o ⇔Υ Io T0.(58) Thesefinal resultshighlight thefactalready arguedabove thattheintensity ofknowledge spilloversis akey determinantintheissue ofParetonon-optimalityoftheSchumpeterian equilibriumandthusinthe characterizationoftheoptimaltooldedicatedtocompensateformarketincompleteness. 5Conclusion Inthispaper,wedevelopedastandardendogenousgrowthmodel`aAghion&Howitt(1992)inwhichweexplic- itlyintroduced(inafairlysimpleway)theconceptofintensityofknowledgespillovers.Thesubsequentanalysis enabledustoshednewlightontheissueofParetonon-optimalityoftheseminalSchumpeterianequilibrium initiallyintroducedbyAghion&Howitt(1992)byrevealinghowtheintensityofknowledgespilloversdeter- minestheimpactofthedistortionofR&Dincentivesduetomarketincompleteness(nomarketforknowledge isconsideredintraditionalSchumpeterianequilibria).Thekeyresultderivedinthispaperisthatthehigh- er(resp.lower)theintensityofknowledgespilloversis,themorelikelymarketincompletenesswillinducean under-optimal(resp.over-optimal)R&Deffort,andthusthemorelikelytheoptimalpolicyaimingatcorrecting thismarketfailureshouldbetosubsidy(resp.totax)theR&Dactivities.Furthermore,weshowedthatif theoptimaltooldoesconsistinasubsidy,thelevelofthissubsidyshouldbeallthehigherastheintensityof knowledgespilloversisstrong. Theformalizationdevelopedinthepaperremainssomehowsimpleinsofarasweconsiderhomogeneityin theintensityofknowledgespilloversacrosssectors(thatis,homogeneitybothintheincrementinknowledge 16 Theproofisstraightforward:g Υ Po ,0 Sg o ⇔Θ λ N − ρ+ λ N 2 S λ N Θ−ρ⇔Θ T 2ρ ρ+ λ N ≡ ˜ ˜ Θ. 1556  resultingfromaninnovationandinthescopeofdiffusionofknowledgeintheeconomy).Nevertheless,the resultsobtainedcanstillbeseenasafirststepindevelopingbasicargumentsinsupportofthefactthatvarious R&Dactivitiesshouldprobablybetargetedbydifferentpublicpolicies,dependingontheintensityofknowledge spilloversemanatingfromthem.Forexample,consideringanextensionofthismodelinwhicheachsectorwould becharacterizedbyaspecificlevelofintensityofknowledgespilloverscouldenablesustoobtainanalytically resultsinlinewithAkcigit,Hanley,&Serrano-Velarde(2016)whoshowquantitativelythatatype-dependent R&Dsubsidypolicyenablesthesocialplannertoachievehigherlevelsofwelfare. 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