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A Model of Growth Through Creative DestructionAuthor(s): Philippe Aghion and Peter HowittSource: Econometrica, Vol. 60, No. 2 (Mar., 1992), pp. 323-351Published by: The Econometric SocietyStable URL: http://www.jstor.org/stable/2951599 .

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Econometrica, Vol. 60, No. 2 (March, 1992), 323-351

A MODEL OF GROWTH THROUGHCREATIVE DESTRUCTIONBYPHILIPPE AGHION AND PETER HowirrT

A model of endogenous growth is developed in which vertical innovations, generatedby a competitive research sector, constitute the underlying source of growth. Equilibriumis determinedby a forward-lookingifference quation,accordingo which he amountofresearch in any period depends upon the expected amount of research next period. Onesource of this intertemporal relationship is creative destruction. That is, the prospect ofmore future research discourages current research by threatening to destroy the rentscreatedbycurrentresearch.The paper analyzes he positiveand normative ropertiesofstationaryequilibria, n which researchemployment s constant and GNP follows arandomwalkwithdrift,althoughunder some circumstancesyclicalequilibriaalso exist.Both the averagegrowthrateandthe varianceof thegrowthrateare increasing unctionsof the size of innovations, he size of the skilled labor force, and the productivity fresearch as measured by a parameter indicating the effect of research on the Poissonarrival rate of innovations; and decreasing functions of the rate of time preference of therepresentative individual. Under laissez faire the economy's growth rate may be more orless than optimal because, in addition to the appropriability and intertemporal spillovereffects of other endogenous growth models, which tend to make growth slower thanoptimal, the model also has effects that work in the opposite direction. In particular, thefact that private research firms do not internalize the destruction of rents generated bytheir innovations introduces a business-stealing effect similar to that found in thepartial-equilibrium patent race literature. When we endogenize the size of innovations wefind that business stealing also makes innovations too small.

KEYWORDS:Endogenous growth, nnovations, reativedestruction.

1. INTRODUCTIONTHE MAIN CONTRIBUTION of the literatureon endogenousgrowthpioneeredbyRomer(1986) and Lucas (1988)has been to endogenizethe underlying ourceof sustainedgrowthin per-capita ncome, namelythe accumulationof knowl-edge. There are manychannelsthroughwhichsocieties accumulateknowledge,including ormal education,on-the-jobtraining,basic scientificresearch, earn-ingby doing,processinnovations,andproduct nnovations.Thispaperexaminesa channel thathas received ittle attentionin the endogenousgrowth iterature,namelythat of industrial nnovationswhich improve he qualityof products.Thischannelintroduces nto endogenousgrowth heorythe factorof obsoles-cence;betterproductsrenderpreviousones obsolete. Obsolescenceexemplifiesan importantgeneralcharacteristic f the growthprocess,namelythat progresscreates losses as well as gains. It also embodiesSchumpeter'sdea of creative

'The authorswish to acknowledge he helpfulcomments and criticismsof Roland Benabou,Olivier Blanchard,PatrickBolton, Louis Corriveau,Mathias Dewatripont,Dick Eckaus, ZviGriliches,ElhananHelpman,RebeccaHenderson,LouisPhaneuf,WilliamScarth,Nancy Stokey,Patrick Rey, and the Co-Editor and referees of this journal.323


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324 PHILIPPE AGHION AND PETER HOWITTdestruction 1942, p. 83, his emphasis):

The fundamentalmpulsethat sets and keeps the capitalistenginein motioncomesfromthe newconsumers' oods, the new methodsof production r transportation,henew markets,.... [This process] incessantlyrevolutionizes he economic structurefrom within, ncessantlydestroying he old one, incessantly reatinga new one. Thisprocessof CreativeDestruction s the essentialfact aboutcapitalism.

The present paper constructs a simple model of growth through creativedestruction,bymodelling he innovationprocessas in the patent-race iteraturesurveyedby Tirole(1988, Ch. 10) and Reinganum 1989).The expected growthrate of the economydepends upon the economy-wideamountof research.Thepaper shows that equilibrium n such an economy is determinedby a forward-looking differenceequation, according o which the amount of research n anyperiod depends upon the expected amount of researchnext period, similartothe differenceequationthat definesequilibrium n the two-periodoverlapping-generationsmodel of money (Azariadis 1981),Grandmont1985)).More specifically, he model assumes, followingSchumpeter, hat individualinnovationsare sufficiently mportant o affect the entire economy.A period isthe time between two successive innovations.The length of each period israndom, because of the stochastic nature of the innovationprocess, but therelationshipbetween the amount of research n two successiveperiods can bemodelled as deterministic.The amount of research this period depends nega-tively upon the expected amountnext period, through wo effects.The first effect is that of creative destruction.The payoff from research thisperiodis the prospectof monopolyrents next period. Those rents will last onlyuntil the next innovationoccurs, at which time the knowledgeunderlying herentswill be renderedobsolete. Thus the expected present value of the rentsdepends negativelyupon the Poisson arrivalrate of the next innovation.Theexpectationof more researchnext period will increase that arrivalrate, andhencewill discourageresearchthis period.The second effect is a general equilibrium ffectworking hrough he wageofskilled abor,which can be used either in researchor in manufacturing.n orderto be consistentwith the conditionsfor labor-market quilibrium,he expecta-tion of moreresearchnext periodmustcorrespond o an expectationof higherdemandfor skilled labor in researchnextperiod,whichimpliesthe expectationof a higherrealwage of skilled labor.Higherwagesnextperiodwill reducethemonopolyrents that can be gained by exclusiveknowledgeof how to producethe best products.Thus the expectationof more research next period willdiscourage esearch hisperiod by reducing he flowof rentsexpectedto accrueto a successful nnovator.

This functionalrelationshipbetween research n two successiveperiodshas aunique fixed point, which defines a stationary equilibrium.The stationaryequilibrium xhibitsbalancedgrowth, n the sense that the allocationof skilledlabor between researchand manufacturingemainsunchangedwith each inno-


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MODEL OF GROWTH 325vation; the log of GNP follows a randomwalk with drift. This is not always,however,the only equilibriumn the model. As in the overlapping-generationsliterature he functionalrelationship an also be satisfiedby cyclical rajectories.One noteworthy mplicationof the negative dependencyof current researchupon future research is the possible existence of what we call a "no-growthtrap,"a cyclicalequilibriumn which the level of researchoscillatesdeterminis-tically between two levels each period, and in which the lower of these twolevels is zero. An economyin such an equilibriumwould stop growing n finitetime, because with no research there would be no innovation,and hence theperiodwith no researchwould nevercometo an end.The (rational)expectationthat the next innovationwould be followed by a very high level of researchwoulddiscourageanyonefrom undertaking hat innovation.Another implication s that the average growthrate of the economyis notnecessarily ncreasedby an increase in the productivity f research.In particu-lar, a parameterchangethat makes researchmoreproductiven some statesofthe world can discourageresearchin other states, by increasing he threat ofobsolescence faced by the productof research n those other states,to such anextent that the averagegrowthrate is reduced.From a normativepoint of view, the average growth rate in stationaryequilibriummaybe more or less than sociallyoptimalbecauseof the presenceof conflictingdistortionaryffects. Specifically, lthough he model includestheappropriabilitynd intertemporal pillovereffects which generate a less thanoptimalgrowthrate in Romer's 1990) model,it also has effectsthat work n theoppositedirection.In particular,here is a "business-stealing"ffectof the sortfamiliarfrom the patent-raceliterature(Tirole (1988, p. 399)). That is, re-searchersdo not internalizethe destructionof existingrents created by theirinnovations.When the size of innovations s taken as given,the businessstealingeffect can lead to too much growth.In addition,we find that when the size ofinnovations s endogenized,the businessstealingeffect tends to make innova-tions too small.Otherpapers n the endogenousgrowth iterature hat model verticalproductinnovations nclude Segerstrom,Anant, and Dinopoulos (1990), who assumethat the timebetween successive nnovations s deterministic.Theyhavea richerintersectoralstructurethan the present paper, and addressa differentset ofquestions. Stokey (1988)models verticalproduct nnovationsand obsolescencein a perfectly competitive model where innovationsare the unintentionalby-productof learning by doing. The cost-reducing nnovations of Shleifer(1986)can also be interpretedas verticalproduct nnovations.His model doesnot endogenizegrowth,however,exceptin a limited dichotomous ense;that is,the long-run average growthrate is fixedby the exogenouslyspecifiedrate ofinvention,exceptin the singularcasewhere no inventionsareeverimplementedandthe economystopsgrowing.Corriveau1988)has a discrete-timeanalysisof endogenousgrowthbasedoncost-reducingnnovationsas in Shleifer, n whichthe possibilityof simultaneousdiscoveriescreates a different kind of "business-stealing"ffect. In his model


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326 PHILIPPE AGHION AND PETER HOWITTthe payoff to currentresearch s independentof future researchbecause rentsto innovationsare assumedto accrue only in the same period as the researchfrom which they resulted.Grossmanand Helpman(1991) constructa model ofvertical product nnovation hat explicitly ntegrates he analysisof Segerstrom,Anant,and Dinopoulos(1990)with that of the present paper.Judd (1985) and Romer (1990) model growthbased on horizontal productinnovations, using the Dixit-Stiglitz (1977) model of product variety. Thesemodels involveno obsolescence;new productsare no better than existingones.They also involveno uncertainty.King and Rebelo (1988) introduceuncertaintyinto an endogenous growthmodel by assuminga randomrate of returnto theaccumulation f human capital under conditionsof perfect competition.Within the patent-race iterature the paper closest to the present is that ofReinganum(1985), which also emphasizesthe affinity o creativedestruction.The present paper addsto Reinganum'smodel the general equilibrium ffectsof future researchon the rents created by currentresearch,and of the level ofmanufacturing mploymenton the cost of research.The paper also generalizesthe Reinganum model by allowing the stream of innovationsto continueforever,and by explicitlyanalyzing he effect of the future level of researchonthe prospectivereward o currentresearch.2Section 2 below presents the basic model of the paper. This basic modelassumes for simplicity hat each innovationcreatesan economy-widemonopolyin the production of intermediate goods. Section 3 derives the functionalrelationshipbetween research in two successiveperiods that defines equilib-rium. It then analyzesthe determinantsof the average growthrate and thevariabilityof the growthrate in stationaryequilibrium.One of those determi-nants is the degree of market power possessed by an intermediate-goodmonopolist, which is parameterized n the model. Section 4 characterizes hewelfarepropertiesof stationary quilibrian the basicmodel,underthe assump-tion of a fixed size of innovations. Section 5 introduces the possibilityofnondrastic nnovations.Section 6 generalizesthe model by allowingresearchfirmsto choose the size of innovationsas well as their arrivalrate. Section 7dealswith a strategicmonopsonyeffectthat hasbeen ignoreduntilthispoint inthe argument,bywhich an intermediate irmcan extendthe expected ifetime ofits monopoly by hiringmore than the short-runprofit-maximizingmountofskilledlabor,at the cost of a higherrealwage.Section8 relaxesthe assumptionof a single economy-widemonopolyin the productionof intermediategoods.Section9 containsbriefconcludingremarks.

2That is, Reinganum's comparative-statics analysis follows the common practice of the patent-raceliterature in taking the reward to a successful innovation as given, whereas the following analysisshows that the effect of a parameter change on the time path of research involves feedback fromfuture research to current research working through the two above-mentioned channels: creativedestruction and the general equilibrium wage effect on profits, both of which flow through thereward to a successful innovation.


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MODEL OF GROWTH 3272. THE BASIC MODEL2.A. Assumptions

There are three classesof tradeableobjects: abor,a consumptiongood, andan intermediategood. There is a continuumof infinitely-livedndividuals,withidentical intertemporally dditivepreferencesdefined over lifetime consump-tion, and the constantrate of time preference r > 0. The marginalutilityofconsumptions assumedconstant; hus r is also the rate of interest.There is no disutilityfrom supplying abor. There are three categoriesoflabor:unskilled labor,which can be used only in producing he consumptiongood;skilledlabor,whichcan be used either in researchor in the intermediatesector;and specialized abor, whichcan be used onlyin research.Each individ-ual is endowed with a one-unit flow of labor. Let M, N, and R denoterespectively he massof unskilled,skilled,and specialized ndividuals.The consumptiongood is producedusing the fixed quantityM of unskilledlabor, and the intermediategood, subjectto constantreturns.Since M is fixed,the production unctioncan be writtenas(2.1) y =AF(x),where F' > 0, F"< 0, y is the flow outputof consumptiongood, x the flow ofintermediateinput, and A a parameter indicatingthe productivityof theintermediatenput.The intermediategood is producedusingskilled aboralone, according o thelinear technology(2.2) x=L,where L is the flow of skilled labor used in the intermediate ector.Researchproducesa random sequence of innovations.The Poisson arrivalrate of innovations n the economyat any instantis A4(n, R), where n is theflow of skilled labor used in research, A a constant parameter,and b aconstant-returns,oncaveproduction unction.Both A and b are givenby thetechnologyof research.Thereis no memory n this technology, incethe arrivalrate depends only upon the currentflow of input to research,not upon pastresearch.Assumethat skilled abor s an essentialfactor n research:4(0, R) = 0.Then an economy that allocates no skilled labor to researchwill not grow,becauseit willexperienceno innovations.The "linear"casewhere0(n, R) = n,that is, where R = 0, will be used frequently.)Time is continuous,and indexed by r> 0. The subscriptt = 0,1... denotesthe intervalstartingwith the tth innovation or with r= O n the case of t = 0)and endingjust before the t + 1st. The length of each interval s random.Allpricesandquantitiesare assumedto remainconstantwithin each interval.If ntis appliedto research n intervalt, the lengthof the intervalwill be exponen-tiallydistributedwith parameterA4(nt, R).Each innovationconsistsof the inventionof a new intermediategood,whoseuse as input allows more efficient methods to be used in producing the


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328 PHILIPPE AGHION AND PETER HOWITTconsumptiongood. Real-worldexamplesinclude such "input" nnovations3asthe steamengine,the airplane,and the computer,whoseuse madepossible newmethods of production n mining,transportation, nd banking,with economy-wide effects. An innovationneed not, however, be as revolutionaryas theseexamples,but mightconsist instead of a new generationof intermediategood,similar o the old one.Specifically,use of the new intermediategood increases the productivityparameterA in (2.1) by the factor y > 1. There are no lags in the diffusionoftechnology.4The most modernintermediategood is alwaysproduced,so that:(2.3) At =AOyt (t = 0,whereAo is the initialvaluegiven by history. Of course, it is alwayspossible toproduce the consumptiongood using an old technology,with a correspondinglyold intermediategood.)A successful nnovatorobtains a patentwhichit can use to monopolizetheintermediatesector.(Section 8 relaxes this assumptionby allowingfor a finitenumber of monopolisticcompetitors.)The patent is assumed to last forever.However,the monopoly asts only until the next innovation,at whichtime theintermediategood is replacedby the next vintage.All marketsare perfectlycompetitiveexceptthat for intermediategoods.

2.B. TheIntermediateMonopolist's Decision ProblemFor ease of presentation he analysisstartsby assuming hat innovationsarealwaysdrastic; hat the intermediatemonopolistis unconstrainedby potentialcompetition from the previous patent. This assumptionwill be relaxed inSection 5 below. The intermediatemonopolist'sobjectiveis to maximize theexpectedpresent value of profitsover the current interval.When the interval

ends so do the profits.The onlyuncertainty oncernsthe lengthof the interval.Except in Section 7 below, the monopolistis assumedto take as given theamountof researchat each time,and hence also takes as giventhe lengthof theinterval.

3Scherer 1984)combinesprocess-and input-orientedR and D into a measureof "used"R andD, which he distinguishes rom pure product R and D. He estimates that during the period1973-1978 n U.S. industry he social rate of return o "used"R andD lay between71%and104%,whereas he return o pure productR and D was insignificant.4 Gradualdiffusion could be introducedby allowing the productivityparameter after eachinnovation o follow apredetermined ut gradualpathasymptoticallypproachinghe limitAt, andthen to jumpto At upon the next innovationand followa gradualpath approachingAt+1. Thiswouldproduce a cycle in researchwithin each interval,as the gradualrise in productivitywouldinducemanufacturingirms o hire more and moreworkers ut of researchuntil the next innovationoccurs.


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MODEL OF GROWTH 329Let xt be the flow of the intermediategood produced by the monopolistduring interval t. By (2.2), xt also equals employmentof skilled labor inmanufacturing.The inverse demand curve facing a monopolistchargingtheprice pt (relativeto the numeraireconsumptiongood) is the marginalproduct(2.4) Pt =AtF'(xt).

Thus the monopolist chooses xt to maximize [AtF'(xt) - wt]xt, taking as givenAt and the wage wt of skilled labor.Define the "productivity-adjustedage" as wt- wt/At, and the "marginal-revenuefunction"as G(X) F'(x) + xF"(x). Assumethat marginalrevenueisdownward-slopingnd satisfiesInada-typeconditions.ASSUMPTION 1: G)(x) < 0 for all x > 0, limxOG (X)= oo,limx, 0(x) = 0.Then for anypositivewt the monopolist'schoice of output xt is given by thefirst-order ondition

(2.5) >9t o( xt) 1or(2.6) xt =(wt),where x is the function 6 -'. The flow of monopoly profits is(2.7) -rt=A t(v),where ir(w) - (x(co))2F"(I(wG)).Note that x and vF are each strictly positive-valued and strictlydecreasing or all positivewt.An examplesatisfyingAssumption1 is the Cobb-Douglas xample,in whichthe consumption-good technology is F(x) = xa, 0 < a < 1, which yields(2.8) Pt = wt/a, Tt= ( a ,xt = (wt/a2)

2.C. ResearchThere are no contemporaneous pillovers n research; hat is, a firmemploy-ing the amountsz, s of the two factorsin researchwill experience nnovationswith a Poisson arrivalrate of A4(z, s), independentlyof the inputs of otherfirms.The objectiveof a firmin choosing z and s at each date is to maximize

the flow of expected profitsfrom research:(2.9) A4(z, s)Vt+1-wtz-w's,where Vt+ is the value of the t + 1st innovation,and w' is the wage rate ofspecialized abor.


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330 PHILIPPE AGHION AND PETER HOWITTBecause p has constantreturns,and because the total flow of specializedlabormustequal R in equilibrium, t follows fromthe Kuhn-Tuckeronditionsfor maximizing2.9) that

(2.10) wt> 'pf(nt)AVt+j, nt> 0, with at leastone equality,where p(nt) -- (nt, R), and nt is the economy-wide low of skilledlabor usedin researchduring nterval t. Note that(2.11) p(O)= 0, and (p'(n) > 0, (p"(n) < 0 for all n > 0.

As we shall see, all researchis conductedby outside researchfirms ratherthan by the incumbentmonopolist.Because of constantreturnsto scale thenumber of research firms is indeterminate.The value Vtl to an outsideresearch irm s the expectedpresentvalue of the flow of monopolyprofits ?t+1generatedby the t + 1st innovationover an intervalwhose length is exponen-tiallydistributedwith parameterAp(nt 1):(2.12) t+1= r+A(nt+1 )

The reasonwhy the monopolistchooses to do no research s that the value tothe monopolistof making the next innovationwould be Vt+I Vt, which isstrictly ess than the value Vt 1 to an outside firm. This is an exampleof theArroweffect, or replacementeffect (see Tirole (1988, p. 392)). The efficiencyeffect, or rent-dissipationeffect, accordingto which an outside firm mightreceivea smallerpayofffrom an innovation han wouldthe presentincumbent,becauseof having o compete with the presentincumbent, s absent in the caseof drastic nnovationsbecause the flowof profitrt + in (2.12)is independentofwhetherthe firmearningthose profitshas access to the previouspatent.SThere is an important ntertemporal pillover n this model. An innovationraisesproductivityorever. It allows each subsequent nnovation o raise At bythe same multipley, andwith the same probabilityAp(nt), but froma startingvalue that is higher by the multiple y than it would otherwise have been. Theproducerof an innovationcaptures(some of) the rents from that productivitygain, but only duringone interval.After that the rents are capturedby otherinnovators,building upon the basis of the present innovation,but withoutcompensating he present innovator.6This intertemporal pilloverplaysa rolein the welfareanalysisof Section4 below.

5If, instead of a constant-returns research technology, each firm had an identical researchtechnology with rising marginal cost, then the monopolist might do some research, but the Arroweffect would imply that the monopolist would do less research than each outside research firm, asshown by Reinganum (1985, Proposition 2) in a similar context.6 This is the spillover identified by Romer (1990, pp. S83-S85).


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MODEL OF GROWTH 331The model also embodiesSchumpeter'sdea of "creativedestruction."Eachinnovation s an act of creationaimedat capturingmonopolyrents.But it alsodestroys the monopoly rents that motivatedthe previous creation. Creativedestructionaccounts for the term Ap(nt+d)in the denominatorof Vt+1 in(2.12). More researchreduces the expected tenure of the currentmonopolist,and hence reducesthe expectedpresentvalue of its flowof rents.

2.D. CapitalMarketsThe structureof capitalmarketscanbe specified n manydifferentways.Oneis to supposethat there is a frictionlessWalrasian reditmarket n whichfutureexpectedconsumptioncan be discountedat the constantrate r. Another is tosupposethat there is no creditmarket.Accordingto the latter specificationallnonresearchworkersconsumetheir wage incomeat each instant,the ownersofthe monopolyintermediate irmconsumetheir flow of profitsat each instant,and researchworkersreceiveno payunless their firmsinnovate, at whichtimethey are paid in shares of the next intermediatefirm. Accordingto eitherspecification,all research irmscouldbe assumedto be ownedby theirworkers,and (2.9) would representthe expected flow of surplusto be divided amongthem. The crucialassumption hat utilityis linear in consumptionmakes thesedifferentspecificationsall equivalent,by removingany motive to use capitalmarkets or risk-sharing.

3. PERFECT FORESIGHT DYNAMICS AND BALANCED GROWTH3.A. Equilibrium

At anypoint in time there is only one decisionfor societyto make;namely,how to allocate the fixed flow N of skilled labor between manufacturing ndresearch.Combining(2.5), (2.7), (2.10), (2.12) and the equilibriumconditionN = nt +xt yields(3X1) ?1p(n r + A.(n ) , nt >0, with at least one equality.

Condition(3.1) determinesresearchemploymentat t as a functionof researchemploymentat t + 1:(3.2) nt = qi(nt+ )Iwhere ,l: [0,N)-1 R+ is a strictlydecreasingfunctionwhereverit is positive-valued.The functionalrelationship frbetween researchemployment n two succes-sive periods is illustrated in Figure 1, where c(nt) is the "marginal cost of


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332 PHILIPPE AGHION AND PETER HOWITTmarginal cost, benefit

(&(N)) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _r

c(nt) X,.

b(nt ,

!~~ ~ ~~~~~~~~~~~lim i(m)f(N)__ ____ ___Xr+x,I , , I ,

I

. . I , I I. , InoA jn researchO n n n nfn n N employment

FIGURE 1.-The effectof future researchon currentresearch:no= q,(n1)and n1= 4(n2). Thepair (0, ng) constitutesa no-growth rap.

research" and b(nt+l) the "marginal benefit of research," defined by6(N -nt)

A(p'(nt)yir(6(N - nt+ 1))b(nt+,) r+Ap(n)+1)

By (2.11) and Assumption 1, c is strictly increasing, b is strictly decreasing, andc(nt) -* ooas n - N. It follows that in the case illustrated n Figure 1, wherec(O) 0. In Figure 1 the sequence {no, n1, n2,... }constructed from the

7The criticalvalue niis definedby c(O) b(nl),unless limn Nb(n)> c(O), n which case n = N.


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MODEL OF GROWTH 333counterclockwise piralstartingat no constitutesa PFE. A stationaryequilib-riumcorresponds o a PFE with nt constant. It is defined as the solution ton =nThereexists a uniquestationary quilibrium.As Figure1 shows, f c(O)< b(O)then n is positive,and is definedby

,j( N n) y 7( (1(N n)Ap()(n r + Ai(n)

In this case growthis positive because innovationsarrive at the Poisson rateA(n)> 0. If c(0) > b(O)then n = 0 andthere is no growth,because the Poissonarrival ate of innovationss Ap(O) 0. Henceforth,assumethat c(O)< b(O)andn > 0.

Other equilibriamay also exist. A two-cycle is a pair (n0,n1) such thatno = qi(nl) and nl = qi(n0). It definesa PFEof periodtwo. If both no and nl arepositive, the PFE is a "real" two-cycle.If either no or nl is zero, it is a"no-growth rap."In a real two-cycle,the prospectof high researchin oddintervalsdiscouragesresearchin even intervals,and the prospect of low re-searchin even intervalsstimulatesresearch n odd intervals.8A no-growth rapis the extremecase in whichthe prospectof high research n odd intervals hutsdown research completely in even intervals.Although the no-growthtrapdefinesaninfinitesequence{nt}0, he oscillationwillcease after one innovation.Fromthenon no growthwill occurbecauseno innovationswilloccur.It is clearfrom Figure 1 that a no-growth rap exists if lim, -f i(w) = 0 and r is smallenough. A real two-cyclewill exist if a no-growthtrap exists and9 c'(n)+b'() > O.Consider the Cobb-Douglas example: F(x) = x', with a linear research tech-nology p(n) n. From(2.8), the equation (3.3) defininga positiven is

1-a(3.4) 1= r+ A

8 Shleifer 1986)also findsdeterministicycles n a modelof multipleequilibriawith innovations.The source of multiplicityn Shleifer'smodel is a contemporaneous trategiccomplementarity,whereby the incentiveto innovate this period is strongerthe more innovationsare occurringelsewherein the economythis period.No such strategiccomplementarityxists in the presentmodel,in whichmore research hisperiodraisesthe marginal ost of researchwithoutaffectinghemarginal enefit.BecauseShleiferassumes hat imitationdestroys he return rom nnovations fterone period,his model does not exhibit the dependencyof currentresearchupon future researchwhichunderliesthe cycles, as well as the other equilibria, n the present model. Deneckere andJudd 1986)alsogeneratecycles n a modelof innovations.Theircyclesarise from ocal instability fa unique equilibriumather han from multipleequilibria.LikeShleifer, heyalso do not allowforanyeffectof futureresearchupon currentresearch.9Consider he second-iterate unction+2(n). Geometrically, 2 is definedby reversinghe spiralillustratedn Figure 1; thus no= /i2(n2). Supposea no-growth rap exists. Then0 = +f2(n) < n forsmallbutpositiven (see Figure 1). Becausec'(ni)+ b'(n)> 0, therefore he counterclockwisepiralin Figure1 spiralsout in the neighbourhood f ni,so that +/,2(n) > n for n close to butless than nu.By the continuityof qf2 there must exist an no strictlybetween0 and n, such that /i2(n0) = no.Evidentlyno,O(n0) constitutea real two-cycle.


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334 PHILIPPE AGHION AND PETER HOWITTand the condition for n to be positive is(3.5) 1


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MODEL OF GROWTH 335arrivalparameterdecreasesboth the marginalcost and the marginalbenefit ofresearch, because on the one hand it results in more "effective"units ofresearch for any given level of employment,but on the other hand it alsoincreases the rate of creative destructionduringthe next interval.The formereffectdominates.The above discussion of result (d) suggests an interestingimplication ofcreativedestructionhatcould ariseif the arrivalparameterAwerepermitted ovary from one interval to the next. Suppose, for example, that with eachsuccessful nnovationa new value of Awasdrawn romthe finite set {A1,..., Am}according o a fixed transitionmatrixB. Transition nto a high-Astate couldrepresent a fundamentalbreakthrougheading to a Schumpeterianwave ofinnovations,whereas transition o a low-Astate could representthe exhaustionof a line of research.Then a stationary quilibriumwould involvenot one levelof researchemploymentbut one for each state. Now considerthe effects of aceterisparibus increase in A2. This parameterchange might raise researchemployment n state 2, but it would tend to reduce research employment nother states, by increasing he rationallyexpected rate of creativedestructionduring the next interval. Furthermore,even though the parameterchangerepresents an unambiguous mprovement n the productivityof the researchtechnology, it might reduce the average level of research employment instationaryequilibrium. ndeed, Appendix1 worksout a numericalexampleinwhich, n the limit,as A2becomesinfinite,averageresearchemployment alls tozero.The linearCobb-Douglas xampleof (3.4) and(3.5)aboveyieldsan additionalcomparative-staticsesult by parameterizinghe degree of marketpower en-joyed by an intermediatemonopolist.Specifically,1 - a is the Lerner(1934)measure of monopoly power (price minus marginal cost divided by price),(1 - a)-1 is the elasticityof demandfacedby an intermediatemonopolist,and1 - a is the fractionof equilibrium evenuein the intermediate ectoraccruingto the monopolist, wt/(wt + w xt). Thus, by all measures, the degree of marketpoweris measured nverselyby the parametera.According o (3.4),an increase n the degreeof marketpower (decrease n a)increases the stationary-equilibriummount of research n whenever n ispositive.According o (3.5), givenfixedvalues of the parametersy, A, r, and N,the stationary-equilibriummount of researchwill be positive if and only ifthere is at least some minimaldegreeof marketpower;that is, if andonly if ais less than the criticalvalue

AyNa*-- < 1 .AyN + 1If the degreeof marketpowerfalls shortof thisminimalvalue,then the flow ofmonopoly profits from the next innovation would not be enough to inducepositiveresearchaimedat capturing hose rents even if they could be retainedforever,with no creativedestruction n the next interval.


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336 PHILIPPE AGHION AND PETER HOWITT3.C. Balanced Growth

Real output (i.e. the flow of consumption goods) in the economy duringintervalt is(3.6) Yt=AtF(N - ni),whichimplies(3.7) Yt+1= yytThus the time path of the log of real output ln y(r) will be a randomstep-function tartingat ln yo= ln F(N - ni)+ ln AO,with the size of each stepequalto the constant ny > 0, and with the time between eachstep {A1,A2' ... }a sequenceof iid variablesexponentiallydistributedwithparameterA0(nI).Thisstatementand (3.3) fullyspecifythe stochasticprocess drivingoutput,in termsof the parametersof the model.Not surprisingly, his stochasticprocess is nonstationary.Suppose observa-tions were made at discretepointsin time 1 unit apart.Then from(3.7),(3.8) lny(-+ 1) =lny(r) +E(ir) (T=0,1,...where 80r) is lny times the numberof innovationsbetween r and r + 1. Fromthe aboveanalysis

(E(0) E(1)Iny' In y ...

is a sequence of iid variablesdistributedPoissonwith parameterAp(h). Thus(3.8)can be writtenas(3.9) ln y(r + 1) =ln y(r) + Ap(n)lny + e(Tr) ( = 0, 1where e(X) G--() - A p(nh)n y. Note that e(X) is iid.,with(3.10) E(e('i)) = 0, vare(Ti) = Ap(nh)(lny)2.From (3.9) and (3.10), the discrete sequence of observationson the log ofoutputfollowsa randomwalk with constantpositivedrift. It also followsthatthe economy'saverage growthrate (AGR) and the varianceof the economy'sgrowthrate (VGR) are givenby(3.11) AGR = Ap(n) ln y, VGR = AP(h)(ln y)2.

Combining 3.11)withProposition1 allowsone to signthe impactof parame-ter changeson the averagegrowthrate. Increases n the arrivalparameter, hesize of innovations,the size of skilled labor endowment,and (in the Cobb-Douglas example)the degreeof marketpowerall raise AGR. Increases n therate of interest owerit. The parameter hangeshavethe samequalitative ffecton VGR as on AGR. The effects are intuitiveandstraightforward.he effectofmarketpower, combined with the findingthat a minimaldegree of marketpoweris needed before growth s even possible,underlinesthe importanceofimperfectcompetition or the growthprocess.


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MODEL OF GROWTH 337The exampleof Appendix1 shows, however,that in a more generalsettingwherethe arrivalparameterA can varyfrom state to state, it is not always ruethat an unambiguousmprovementn the productivity f the researchtechnol-ogywill increasethe economy'saveragegrowthrate.Instead,an increase n thearrival parameter in one state can discourageresearch in other states byincreasing he rationallyexpectedrate of creativedestruction o such an extentthat the economy'saveragegrowthrate falls.

4. WELFARE PROPERTIES OF THE STATIONARY EQUILIBRIUMThis section comparesthe laissez-faireaveragegrowthrate derived abovewith the AGR that wouldbe chosenby a socialplannerwhose objectivewas tomaximize he expectedpresentvalue of consumptiony(r). Since everyinnova-

tion raises y(r) bythe samefactor y, the optimalpolicyconsistsof a fixed levelof research.Expectedwelfareis000(4.1) U e-rr EH(t, T)AtF( - n) dT,0 t=O

where H(t, r) equalsthe probability hat there will be exactlyt innovationsupto time T. Given that the innovationprocessis Poissonwith parameterApo(n),we have(4.2) H(t, r) = (A9(n)T)te-A@(n)TtFrom(4.1) and (4.2),

AOF( N-n)

Equation(4.3) identifiesU as the initial flow of output AOF(x) discountedatthe rate r - A(p(n)(y - 1). This "social discount rate" is less than the rate ofinterest r because the stream of output will be growingover time. Morespecifically,the social discount rate is the rate at which each risk-neutralindividualn the economywouldcapitalizea streamthatwasperpetually ubjectto increasesby the factor (y - 1) with a Poisson arrivalrate of A*p(n),andconstantotherwise.The socially optimal level of research n* maximizes U. The first-orderconditionfor an interiormaximums

44) F'(N -n*) (y -1)F(N -n*)Aspt(n*) r-Ap(n*)(y -1)

(If no solutionexiststo (4.4) then n* = 0.) This level of researchwouldproduce.an averagegrowthrate of Ap(n*)lny. Accordingly aissez-faireproducesanaverage growthrate more (less) than optimalif n > (


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338 PHILIPPE AGHION AND PETER HOWITTing the stationary quilibrium evel of researchni:

(N -fi) yi=(4( N -n))3 Ap'(fi) r + AP(n)

There are four differencesbetween (4.4) and (3.3). The first is that the socialdiscount rate r - A


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MODEL OF GROWTH 339The intertemporal-spillovernd appropriability ffects tend to make thelaissez-faireaveragegrowthrate less than optimal,whereas the business-steal-ing and monopoly-distortion ffects tend to make it greater than optimal.Because these effects conflictwith each other, the laissez-faireaveragegrowthrate may be more or less than optimal. This can be seen in the linearCobb-Douglas xample,where n* and n satisfy

1A(y-1) - (N-n*)(4.5) 1= r-An*(

1 -a)Ay -- (N n)(3.4) 1= r+AnIn this example,the appropriability nd monopoly-distortionffects are com-bined in the presence of the factor (1 - a) in (3.4); together they tend to makethe laissez-faireAGR less than optimal.Thiscombinedeffect togetherwith theintertemporal-spilloverffectdominateswhenthe size of innovationsy is large,in whichcase13n < n*. However,when there is muchmonopolypower (a closeto zero) and innovationsare not too large, the business-stealing ffect domi-nates, in which case'4 n > n*.

5. NONDRASTIC INNOVATIONSUntil thispointthe analysishas assumed hat innovationsaredrastic; hat theintermediatemonopolist s not constrainedby potential competition romown-ersof previouspatents.The presentsection showsthat the analysisof stationaryequilibriacan be generalized o the case where innovationsare nondrastic.Innovations renondrasticf and onlyif the previous ncumbent ould makea

positive profitwhen the currentone was chargingthe price pt=A,F'(1(o,))whichyields an unconstrainedmaximum o the currentincumbent'sprofit.Ifinnovationsare nondrastic, he current ncumbent ets the maximumpricethatgivesthe previous ncumbentnonpositiveprofits,and satisfiesall the demandatthatprice,leavingnone to the previous ncumbent.The previous incumbent could make a positive profit if and only if acompetitiveproducerof consumptiongoodscouldproduceat an averagecostofless than unity by combiningunskilled labor with the previous incumbent'sgood, buying he latter at a priceequalto its averagecost of productionwt;that13From(4.5),as y risesto the upperlimit 1 + r/AN, n* approachesN while,from(3.4), n'isboundedbelow N.14If 1/a> 1 + r/AN, then n> for all , whereas if ay l+oar/AN, then n*=O. Theseinequalitiesare compatiblewiththe conditionderivedbelow for innovations o be drastic,namely

that y > aa, as can be verified with the example: a = 1/2, y = r2, r/AN = 2(W2- 1).


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340 PHILIPPE AGHION AND PETER HOWITTis, if and only if the condition(5.1) C(w,w, W) y-1 + min {y2/a, a-1}is sufficientfor the monopolistto do no research. Note that this conditionissatisfiedwhen y is close to the value a - at which innovationsbecome drastic.If there is positive researchduring ntervalt, then, as before,(5.6)= +A56 w(P ' ) r + ASD(t+ I

15In the Cobb-Douglas xample, he unit-cost unction sc wM, p)-(1 (X -a)-apa(WM)I-a

It followsfrom this and (5.2) that if the incumbentchargedthe unconstrained rofit-maximizingprice a-1lw,, the unskilledwagewouldbe wfm (1 - a)a 2a/(1-a)A t(W/A t)-a/(l-a). Puttingthisinto (5.1) yields the condition y< a-a. Treating(5.1) as an equalityand solvingit and (5.2)simultaneouslyor(wM,p,) yieldsPt= yl/aWt


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MODEL OF GROWTH 341In a stationary quilibriumwithpositive growth, 5.4) and (5.6) imply(5.7) 1 =Ay(yl/a- 1)(N-n)

(5*7) 1 = - r + An^Equation 5.7) defines the stationary quilibrium evel of researchn. It is thesame as the equation (3.4) that applies in the drastic case, except that themarkupyl/a in (5.7) replacesthe markupa1 in (3.4). The comparative-staticsresultsof Proposition1 applyto the solution of (5.7). In addition, he stationaryequilibrium evel of research defined by (5.7) is increasedby an increase inmarket power (a decrease in a) as in the drastic case. Thus in the linearCobb-Douglas xampleall the comparative taticsresultsderived or the case ofdrastic nnovationsare valid also when innovationsare nondrastic.Comparison f (5.7) and (4.5) shows that the same welfare effectsanalyzed nSection4 operate in the case whereinnovationsare nondrastic,againwith theresult that research and growthunder laissez-fairemay be more or less thanoptimal.16As is customaryn the patent-race iteraturethis analysishas ruled out thepossibility hat the currentandprevious ncumbentmightcontract o share thehighermonopolyprofitsthat couldbe earned if the previous ncumbentagreednever to compete. For example,the previous ncumbentmight sell its patent tothe currentone; in the extreme case where the previous ncumbentalwayshad

no bargainingpower in negotiationwith the current one, competitionfrompreviousvintagesof the intermediate oodwouldneverconstrain hemonopolist,and the above analysisof drastic nnovationswould apply no matterhow smallthe innovationswere.6. ENDOGENOUS SIZE OF INNOVATIONS

This section generalizes the analysis of stationary equilibria by allowingresearch irms o choose not only the frequencybut also the size of innovations.It shows that under aissez-faire, nnovationswillbe too smallif theyare drastic.In the nondrasticcase, the tendencyto make innovations oo small is at leastpartly mitigated by the incentive for innovatorsto move away from theircompetitive ringe,whichtheycan do by increasing he size of innovations.Assume that the arrivalrate of innovations o a firmemployingthe factorcombination z, s) and aimingfor innovationsof size y is A4(z, s)v(y), wherev'(y) < 0; the bigger the innovation,the harder it is to discover. Assumev"(y)< 0; the marginalcost (in terms of lowerarrivalrate)of aimingfor largerinnovationsncreaseswith the size of innovations.Then the productyv(y) is aconcave functionof y.

16Supposey = C2 and a = 1/2. To get hn n* let r approach V's- 1)ANfromabove; hen n*approaches N whereas niis bounded below N. To get n > n* let r = 2(V2 - 1)AN; then n* = 0 andn > 0. In either case y = a-, but the example s robust o a smalldecrease n y that wouldsatisfythe necessaryand sufficientcondition 5.3) for innovations o be nondrasticwithoutviolatingthesufficient ondition 5.5)for the monopolist o do no research.


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342 PHILIPPE AGHION AND PETER HOWITrThe analysis focuses again on stationary equilibriawith positive growth.Considerfirst the case of drastic nnovations.By the same logic as before, thepayoffto the t + 1st innovator s

(6.1) At1+= ( A)V(A)where 5 is the stationary-equilibriumalue of y. If the t + 1st innovationhassize y, not necessarily equal to A,then A+ 1 - yAt and Vt 1= yVt. Thereforethe expected flowof profitsto the research irm in intervalt is(6.2) AA(z, s)v(y)y7Vt-wtz-ws.The firm takes Vt as given. Thus its profit-maximizinghoice of y also maxi-mizes the product v(y)y. Because this product is a concave function of y,therefore5 is definedby the condition17(6.3) v(9) + Av'(9) = 0.The first-order ondition for profitmaximizationwith respect to skilled labor,togetherwith(6.1)producesan expressionanalogous o (3.3):

Ji(N -fn) v(A)A( dN -A))(6.4) ApA'(n) r + Ap(n)v()The comparativetatics analysisof Section 3 carries hroughunchanged, ince 5is determinedby (6.3) independentlyof all parameters hat do not enter thefunction v, with the obvious exception that it is no longer permissibletoinvestigate he effects of a changein y.As in Section4, the expected presentvalueof consumption quals(6.5) U= ~~ AOF(N-n)(6.5)~ U= rAip(n) v(y) (y -1where the denominators the socialdiscountrate. Therefore, ndependentlyofthe choice of n, the social planner will choose y so as to maximize theexpressionv(y)(y - 1). The sociallyoptimalvalue y* is then definedby(6.6) v(y*) + y*v'(y*) - v'(y*) = 0.By concavityof yv(y),18 5 a-.18Since v' < 0, (6.6) implies that v(y)y is locally decreasing at y*, so that y* exceeds the point yat which v(y)y is maximized.


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MODEL OF GROWTH 343The sociallyoptimal level of researchemploymentn* satisfies the condition67) F'(N-n*) v(y*) (y* -1) F(N -n*)ACp'(n*) r -Ap(n*)v(y*)(y* 1)

Comparisonbetween (6.4) and (6.7) reveals the same welfare effects as in theanalysisof Section 4. In addition,the fact that y


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344 PHILIPPE AGHION AND PETER HOWITTTherefore a sufficient condition for < y* is that (-2 - 1)x(d,y)y =(y - 1)g(y) with g'(y) < 0. The latteris true,9 with g(y) (y + 1)y/4j2y4.

7. STRATEGIC MONOPSONY EFFECTIn this section the i'ntermediate irm is assumed to take into account itsinfluenceon the amountof currentresearch and therebyon the probability fits replacement. n particular,by increasing ts demandxt for skilled labor,themonopolistcan raisethe wage rate that must also be paid to skilled workers nresearch.The effect is to reduce the equilibriumamount of researchnt andconsequentlyo delaythe arrivalof the (t + 1)stinnovation.The monopolistwilltradethis gainoff againstthe higherwagesit must pay its own skilledlabor.The analysis focuses on stationaryequilibriawith positive growth. The

monopolistduring ntervalt chooses xt to maximize he expected presentvalueof profits:= (AtF'(x) - wt)x

r+Ap(N-x)subjectto(* ) wt AAp'(N-x)Vt+,where (*) follows from (2.10). The magnitudesxt, nt, Vt/At, and wt/At areconstant,at the equilibrium alues x, Ti,V, and -. Thereforex solves

(7.1) [F'(x) - Ay~p'(N -x)V]x(7.1) V= max r A1p(N-)V]xThe first-order ondition s(7.2) F'( x) +xF"(x) = AV[(y- 1)p'(N-x) -xyp"(N-x)J.From the constraint *),(7.3) - = AyqP'(N-x)V.From(7.2), (7.3), andthe definitionof 6,(7.4) [y -1- y(N -)qp"(1)/AP'(O) j6(N-i) - (n).It follows that the stationary-equilibriumevel of research Tiis given by theanalogueto (3.3):

6@(N-n)F1 1 ______(7 Ap'(i) j y -1- y(N - fi)qP"(ni)/P'(n) r + AP(n)19Because it compares y* with f, this analysis would apply even if the research firm ignored thelegative effect that its choice of y has on the value of an innovation by reducing the equilibriumevel of x,+1 and hence raising the rate of creative destruction next period.


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MODEL OF GROWTH 345wherethe function7F s definedas(7.6) rrT(ni)-F'(N -nT) - (T)] (N - n-)

Assume that the expression(N n)'p"(n) s nondecreasing n n. Then theleft-handside of (7.5) is increasingn n-.If the right-hand ide is stilldecreasing,then the solutionto (7.5) is unique. It is straightforwardo verifythat all thecomparativetaticsresults n Proposition1 apply o this solution,andthat in theCobb-Douglasexample the solution is an increasing unctionof the degree ofmarketpower.Welfareanalysisof stationary quilibria s somewhataffectedby the strategicmonopsonyeffect, but the overallresult remains,namely that the laissez-faireaveragegrowthratemaybe moreor less thanoptimal.Comparison f (7.5)with(4.4)revealsthe sameintertemporal-spillover,ppropriability,usiness-stealing,and monopoly-distortion ffects as before, although the monopoly-distortioneffect will be quantitatively ifferentbecause Fr(n) i+(J(N - n)). There is anadditionaleffect,however, rom the presenceof the term

y - 1 - y( N -n) " /'()Jon the left-hand side of (7.5). This additionaleffect is the "monopsony-distor-tion"effect.

In the linear case, where so" s zero, the constraint (*) indicates that theintermediate irm'swage rate is independentof the amount of skilled laborithires,so the monopsony-distortionffectinducesit to hire more skilledworkersin order to reduce the amountof currentresearch,and hence the amountofcreative destruction.The effectjust cancelsthe business-stealing ffect, as canbe seen by multiplyingboth sides of (7.5) by (y - 1)/y. Thus in the linearCobb-Douglasexample, research and growth are unambiguously ess thanoptimal.In the generalcase where Sp" 0 the monopsony-distortionffect is ambigu-ous, because hiringmore skilled labor increasesthe intermediatefirm'swagerate at the same time that it reduces creative destruction.Because of theseconflicting endencies it is straightforwardo constructexamplesin which theoverallmonopsony-distortionffectvanishes.Morespecifically,givenany speci-ficationof the model it is possibleto perturb he researchfunctionSo n such away20that n and n* remainunchangedand the solution nito (7.5) becomesequalto n/.Sincen/can be more or less than n* it followsthat nican be more orless than n*.The rest of the paper ignores the strategic monopsonyeffect, by assumingthat intermediate irmstake as giventhe wage of skilledlaborand the amount

20Justperturb p n such a waythat p(ni),q(n*), 9'(n'),and 9'(n*) remainunchanged,but e"(h)is made equal to -9'(ni)/y(N - n). This can be done withoutaltering he sign of '' and '" on[0, N). According o (3.3) and (4.4), ni and n* will be unchanged.This perturbationmakes thesecondfactoron the left-hand ideof (7.5)equalto unitywhen ni = n',and makes ,(ni) = 4Kr((Nni)).Sinceni solves(3.3) it will now also solve(7.5).


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346 PHILIPPE AGHION AND PETER HOWITTof research. This assumption is based on our belief that the effect is notimportant,because it derivesfrom the simplifying ssumption hat there is onlyone intermediate irm n the economy.If therewere manycompeting ntermedi-ate firms,as in the nextsection, each mightplausibly egard tself as too small toaffectthe skilled wage or the amount of research.

8. MANY INTERMEDIATEGOODSThis section relaxes the simplifyingassumptionof a single economy-widemonopolyin intermediategoods. Suppose instead that there are m differentintermediate sectors. Output of the consumption good is y, = ETl1Ai,F(xi,),where xi, denotes the flow of output of the ith intermediate good duringinterval t, and where F has all the propertiesassumed above. (This requiresthat each sectorhave its own specializedbrand of unskilled abor.)FollowingShleifer (1986), supposethat innovationsarrive n differentsectorsin a deterministicorder.21Specifically, he innovatingsector is alwaysthe onewith the lowest productivityparameterAit, Each innovator becomes a localmonopolist in that sector for a period of m successive innovations,and isreplaced by the last of those m innovations.Let At denote the productivityparameter n the leading sector,where an innovationhas arrivedmost recently.Assumethat Ait = y(l-k)/mA when i is the kth most advancedsector.Then yis againthe size of each innovationrelative to the previousvintageof good inthe same sector.Let xk denote the stationary-equilibriummploymentof skilledlaborin thekth most advancedsector.Then xk maximizes he flow of profits:

[Aty(l k)/mFf ( Xk) - WtI XkTherefore,

Xk =X( W),wherex is defined as aboveand w is againthe stationary-equilibriumroductiv-ity-adjustedwage, w,/At. The productivity-adjustedlow of profitsin the kth-leading sector is

vr k/At = y(1-k)/ m . (y (k l )/m,)

21The alternative of allowing innovations to be randomly distributed across sectors is analyzed inan Appendix to an earlier version of this paper, available from the authors upon request. ThisAppendix assumes a continuum of sectors and a continual flow of innovations. Whoever innovates atdate r is thereby allowed to enter a randomly chosen sector with the "leading technology" A(r),where A(ir) grows continuously at the exponential rate o-An. The possibility that the same sectormight receive two innovations in rapid succession, before the leading technology has advanced bymuch, implies that in stationary equilibrium some positive (and endogenous) fraction of innovationswill be nondrastic. The model yields all the comparative-statics results of Proposition 1 above,except possibly for result (b).


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MODEL OF GROWTH 347The productivity-adjustedage is the solutionto the equilibrium ondition

mN=n + Ex (Y(ki-)/m),k==1whichcan be writtenas

Ct= c(N-n),where the functiond^ atisfiesAssumption1 above.Let the technology of research be the same as before, with an arrivalparameterequalto mA. The lengthof each interval s distributed xponentiallywith parametermAqp(n),and each local monopoly asts for m intervals.Let rtdenote the time of the tth innovation.Then the value of the tth innovation s

m kVt= E: E( e - r(Tt+k I

- Td rt+kmA()1k=1 \r +mA~p(n)J

m mA~p(n) kl ()(-/)Ak=1tr+ mA(p(n) r+mAp(n).)'and the condition for a positivestationary-equilibriumevel of research is theanalogueto (3.3):

(8.) (N-) m ( mA(n) k-1 y1/m+[y(k-1)/m&(Nh ^)]mAp'(n) k=1 r+mAp()) r+mA(p(n)

In the linearCobb-Douglas xamplethis condition s the analogueto (3.4):m kl/m W-1AnEY(k-1)/m(a-1) 'Y/(mA )

/ 1-a! \ 1 ~~~(+ mAn^)k(8.2) 1 = mA (N -n) m.a E 7(k-1)/m(a-1)1

Since the right-handside of (8.1) is a decreasingfunction22of n and.theleft-handside an increasing unction,the solutionto (8.1), if it exists,is unique.If no solution exists, then the equilibrium evel of researchis zero. All thecomparative-staticsesultsof Proposition1 applyto the solutionof (8.1), withthe possible exceptionof (b), the effect of y, the size of innovations.In the22 Note that

a m (mA(p(n))kl1rTdn k-i (r+mAp(n))

(( - )mpf))k%-2 (m (f))k-1~-mAp'(n) E ( k k+ 1 )k= 1 (r + mAp(n)) (r + mAp(n))

m?1 (k - 1)(mAgo(n ))k2( 1k -Fk-1 )-k=2 (r + mAp(n))

where T> 2 > .>. m> m O+1-.


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348 PHILIPPE AGHION AND PETER HOWIlTlinearCobb-Douglasexample,however, t can be shown that (b) remains true,provided hat the social discountrate (the denominator f U below) s positive.23In stationaryequilibriumeach innovation raises the entire cross-sectionalprofileof productivity arametersby the factory'l/m. It thereforeraisesGNP byy /m and raises the log of GNP by the factor (1/m)ln y. Since the Poissonarrivalrate of innovation s mAn, therefore the economy'saveragegrowthrateis AnIny, exactly as before. The variance of the growth rate is An(lny)2/m;aggregationacross many sectors reduces variabilitythrough a law of largenumbers.By the same logic as in Section4, socialwelfare is measuredby

mE Y,(1-k)ImF( Xk )

U=A0 1r - mAp(n) (y/m - 1)A social plannerwould choose (x1,..., Xmn) to maximize U subjectto theconstraint,(8.3) N-n =xl + +xm,and n > 0. The first-order onditions or an interiormaximumare(8.4) F'(Xk) =pY(k-l)/m (k = 1,.. . m),(8.5) mApD'(n)yl/m - 1)U = Ao,where,u is a Lagrangemultiplier.Let ,i(N - n) denote the valueof ,usuchthatthe solutions(x1,..., xm)to (8.4) solve (8.3). Then (8.5) can be expressedas:(8.6) mAp(n>) k=1(Y )ArP (n*)(Y /m 1))In the linearCobb-Douglas xample,

(8.7) r-A7 -1m r((-1mAn -Y1/Comparisonof (8.6) with (8.1) reveals the same four effects as before. Themonopoly-distortion effect is still present because ,i(N - n) > d^(N n). The23It suffices o show thatthe right-hand ide of (8.2)is increasingn y. The ratioof sumsin thisexpression anbe regardedas the expectedvalueof the discreterandomvariable

Zk= [lA An,1 + rImAn]/under he truncated eometricdistributionwithparameter l/m(a -1) < 1. The effect on this ratioofa marginal ncreasein y is the sum of the effect on each Zk and the effect of changingtheparameter f the distribution. he former s positive.The latterwouldalso be positive,byfirst-orderstochasticdominance,f Zk were decreasingn k, whichit is if the social discountrate is positive.


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MODEL OF GROWTH 349appropriabilityffect applies sector-by-sector, nd is amplifiedby the fact that

>-l/m > mAp(n)r + mAp(n) Jif the social discountrate is positive. As before, researchand growthunderlaissez-fairemay be more or less than optimal; in the linear Cobb-Douglasexample,nt< n* if y is large,24but n1> n*if a is small and y is not too large.25

9. CONCLUSIONThe paperhas presenteda modelof economicgrowthbasedon Schumpeter'sprocess of creative destruction.Growthresults exclusively rom technologicalprogress,which in turn results from competition among researchfirms thatgenerateinnovations.Each innovationconsistsof a new intermediategood thatcan be used to producefinaloutputmoreefficiently han before.Researchfirmsare motivatedby the prospect of monopolyrents that can be capturedwhen asuccessful nnovation s patented.But those rents in turnwill be destroyedbythe next innovation,whichwill renderobsolete the existing ntermediategood.It wouldbe useful to generalizeand extendthe analysis n several directions,such as assuming hat technology s ultimatelybounded,thereby requiring hesize of innovations ventually o fall.The model would gainrichnessand realismif capitalwere introduced,either physicalor humancapital embodyingechnicalchange,or R and D capitalthataffectsthe arrival ate of innovations.Allowingunemployment,by introducingsearch into the labor market,would facilitatestudy of the reciprocal interaction between-technological change and thebusinesscycle.All these extensionsseem feasible becauseof the simplicityandtransparency f the basic model.EuropeanBank for Reconstructionand Development,122 LeadenhallSt.,LondonEC3 V4QL,UK andDepartment f Economics,TheUniversity f WesternOntario,London,OntarioN6A 5C2, Canada

Manuscripteceivedune,1989; inalrevision eceiveduly,1991.

APPENDIX 1AN EXAMPLE WITH A RANDOM ARRIVAL PARAMETER

Let A follow a two-stateMarkovprocesson the space {A1,A2}with all transitionprobabilitiesequal to 1/2. A stationary quilibriums an equilibriumn which the productivity-adjustedageratedependsonlyon the stateof the world,not on time. Let AtVjbe the valueof making he tth24 From(8.7),as yl/m risesto the upper imit1 + r/mAN, n* approachesN while,from(8.2),niis boundedbelow N.25 If 1/a > 1+ rIAN then ni> 0 for all y,whereas f yllm < 1 + arlmAN, then n* = 0.


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350 PHILIPPE AGHION AND PETER HOWITTinnovation nd moving nto statej. Assumethe linearcase of qp(n)= n. In anystate i, the marginalexpectedreturn o research n interval is AiAt+?(Vl + V2)/2. Thiswill equal the wageif positiveresearchoccurs n state i. If researchoccurs n all states,the Vi'smustsatisfy he analogue o (2.12):

Vi r + A i[N-y(Aiy(Vl + V2)/2)] (i = 1, 2).Define ni= N - i(Aiy(Vl + V2)/2). Then the average level of research employment is

n = nlql + n2q2,whereqi is the asymptoticractionof timespentin state i. It is easilyverified hat

ql = 1 -q2 = A2n2/(Alnl + A2n2).To complete the example, take the Cobb-Douglas case (F(x) = xa), and suppose a = y = 1/2and r =N= Al= 1. Since k(co) = (co/la2)l/(a -) and r(wo) ((1 - a)/a)wA(w), the formula foreach Vican be rewrittenas:

V, = [16(Vl + V2)] - {1 - [4(Vi + 2)] 1

andV2 = [16A2(V1 + V2)]' - A2{1 - [4A2(V1 + V2)] 2.

When A2= 1, the solution to these equations is V, = V2= VT/8v'_, which implies n, = n2 == 1/3.As A2 * oc, the solution approaches V, = 1/4, V2 =0, which implies n, =0, n2 = 1, and q1 =n2/(n2 + Alnl/A2) = 1; hence ni = 0.The economy'saveragegrowthrate equals f ln y, where f is the asymptotic requencyofinnovations:f = Alnlql + A2n2q2- Al + A21]

Thus when A2= 1, f ln y = (1/3) ln 2 > 0; and as A2-+ oo, Iny approaches 0.

APPENDIX2DERIVATION OF CONDITION (5.5)

In the stationaryequilibriumdescribed n Section5, the monopolisthas no incentive to doresearch f(A.1) Wt> A VtM+- Vt)where

()//a- 1)wt(N -t r+ARis the value of the monopolist's urrentpatentand

[min(y 2/a,a 1) - 1]w A1(N-)Vt+l= ~~r+An

wouldbe the grossvalueof the next innovationo the currentmonopolist,orwhomthe innovationwould be drastic f y2 > a-', if next periodthe level of researchwas the stationary-equilibriumvalue nR.n fact more than this level of researchwouldbe conducted f the monopolistwere toinnovate,becausethe monopolist ould thenchargea markuphigher hanyl/a, so the value wouldactuallybe less than Vtm+Y.ubstitutinghese expressionsor Vtand Vtm+nto (A.1) andusingthe


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MODEL OF GROWTH 351fact that wt+1 = y wt produces

A[yi(y2/a,al) - y - (yl/a - _A(A.2) ( foll minm date from(A a 1)](N5.7)r+AnCondition (5.5)follows immediately from (A.2) and (5.7).

REFERENCESAGHION, P., AND P. HowiTr (1988): "Growth and Cycles through Creative Destruction," Unpub-lished, University of Western Ontario.AZARIADIS, C. (1981): "Self-Fulfilling Prophecies," Journal of Economic Theory, 25, 380-396.CORRIVEAU,L. (1988): "Entrepreneurs, Growth, and Cycles," Unpublished, University of WesternOntario.DENECKERE, R. J., AND K. L. JUDD (1986): "Cyclical and Chaotic Behavior in a DynamicEquilibrium Model, with Implications for Fiscal Policy," Unpublished, Northwestern University.DIXIT, A., AND J. STIGLITZ(1977): "Monopolistic Competition and Optimum Product Diversity,"American Economic Review, 67, 297-308.GRANDMONT, J.-M. (1985): "On Endogenous Competitive Business Cycles," Econometrica, 53,995-1045.GROSSMAN,G. M., AND E. HELPMAN(1991): "Quality Ladders in the Theory of Growth," Review fEconomic Studies, 58, 43-61.JUDD, K. L. (1985): "On the Performance of Patents," Econometrica, 53, 567-585.KING, R. G., AND S. T. REBELO(1988): "Business Cycles with Endogenous Growth," Unpublished,University of Rochester.LERNER, A. P. (1934): "The Concept of Monopoly and the Measurement of Monopoly Power,"Review of Economic Studies, 1, 157-175.LUCAS, R. E. JR. (1988): "On the Mechanics of Economic Development," Journal of MonetaryEconomics, 22, 3-42.MORTENSEN,D. T. (1982): "Property Rights and Efficiency in Mating, Racing and Related Games,"American Economic Review, 72, 968-979.REINGANUM,J. (1985): "Innovation and Industry Evolution," QuarterlyJournal of Economics, 100,81-99.(1989): "The Timing of Innovation: Research, Development and Diffusion," in Handbook ofIndustrialOrganization,Vol. I, ed. by R. Schmalensee and R. Willig. Amsterdam: North-Holland.ROMER, P. M. (1986): "Increasing Returns and Long-Run Growth," Journal of Political Economy,94,1002-1037.(1990): "Endogenous Technological Change," Journal of Political Economy, 98, S71-S102.SCHERER,F. M. (1984): Innovation and Growth: Schumpeterian Perspectives.Cambridge, MA: MITPress.SCHUMPETER,J. A. (1942): Capitalism, Socialism and Democracy. New York: Harper and Brothers.SEGERSTROM,P. S., T. C. A. ANANT, AND E. DINOPOULOS 1990): "A Schumpeterian Model of theProduct Life Cycle," American Economic Review, 80, 1077-1091.SHLEIFER,A. (1986): "Implementation Cycles," Journal of Political Economy, 94, 1163-1190.STOKEY,N. L. (1988): "Learning by Doing and the Introduction of New Goods," Journal of PoliticalEconomy, 96, 701-717.TIROLE,J. (1988): The Theory of Industrial Organization. Cambridge, MA: M.I.T. Press.

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