On the winner-take-all principle in innovation races

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1 On the winne-take-all pinciple in innovation aces VincenzoDenicolòandLuigiAlbetoFanzoni Univesity of Bologna, Italy Novembe 2007 Abstact What is the optimal allocation of pizes in an innovation ace? Should the winne take all, o is it pefeable that fist inventos shae the maket with late independent duplicatos? Seveal papes have agued that the latte, moe pemissive egime is socially pefeable. We e-examine that issue, finding that the winne-take-all system can in fact be socially optimal in a boad set of cicumstances, much boade than claimed by the ealie liteatue. In ou baseline model, two fims ace fo an innovation in continuous time. In the winne-take-all system,assoonasonefiminnovatestheothestopsinvestinginr&d; in the altenative, moe pemissive system, the laggad keeps investing to duplicate the innovation and when it also succeeds the maket becomes a duopoly. The winne-take-all system is pefeable in highly innovative industies, wheeas the pemissive system is moe likely to be optimal in matue industies, whee poduct maket competition is stong and innovation is a elatively ae occuence. We analyze seveal extensions of this baseline model to bette claify why we aive at diffeent esults than the ealie liteatue. Coespondence to: Depatmentof Economics, Piazza Scaavilli2, Bologna, Italy; We ae gateful to Maco Ricolfi and semina audiences at Haifa, Hambug, Amstedam (ACLE), and Copenhagen fo helpful comments. Financial suppot fom the Univesity of Bologna(Pogetto stategico) and Micosoft is gatefully acknowledged. 1

2 1 Intoduction What is the optimal allocation of pizes in an innovation ace? Should the winne take all, o is it pefeable that fist inventos shae the maket with independent duplicatos, if and when they mateialize? Following La Manna, MacLeod and De Meza (1989), a ecent liteatue (eviewed below)hasaddessedthatissue, 1 aivingattheconclusionthata pemissive egime, in which late independent inventos ae allowed to pactice the innovation and compete with the fist, is geneally pefeable to the winne-take-allsystem. 2 Thispapecautionsthatthisconclusionisbased on estictive assumptions, and points out that the winne-take-all system caninfactbesociallyoptimalinaboadsetofcicumstances. Ou analysis sheds light on a numbe of contovesial legal ules and policy issues. Fo example, a majo diffeence between patents and othe potection mechanisms, such as copyights and tade secets, is that independent invention is a defense against copyight infingement o tade secet misappopiation, but is not a defense against patent infingement (Mau- 1 See Maue and Scotchme (2002), Cugno and Ottoz (2004), Shapio (2006), and Heny(2007). Fo a diffeent pespective, see Kultti et al. (2006, 2007). 2 In the last decade, economists have also devoted a lot of attention to the issue of the allocation of pizes in contests: see e.g. Taylo(1995), Fulleton and McAfee(1999), Moldovanu and Sela (2001) and Che and Gale (2003). Innovation aces diffe fom eseach contests in two ways. Fist, eseach contests end on a specified date, wheeas an innovation ace ends wheneve the innovation is achieved. As a consequence, in a contest the timing is fixed but the amount of innovative knowledge poduced is vaiable, wheeas in an innovation ace the R&D output is fixed but the timing of innovation is vaiable (Taylo,1995,p. 875). Second,thepizeinacontestistypicallyasumofmoney,wheeas inaninnovationaceitamountstosomedegeeofmaketpowe. Whilethesocialcostof aising a fixed sum of money is independent of its division, the deadweight losses caused by innovatos maket powe geneally depend on the degee of exclusivity they enjoy. 2

3 e and Scotchme, 2002). The patent system, that is to say, lagely woks on the winne-take-all pinciple, wheeas secets and copyights ae moe pemissive as potection mechanisms. Thus, by detemining whethe new technological fields ae eligible to patent potection o else innovatos must elyonsececyocopyights,policycanaffectthedivisionofthepize. 3 Even though the patent system is the pototypical winne-take-all egime, ecently the intoduction of an independent-invention defense into the patent lawhasitselfbecomeatopicalpolicyissue. 4 Moeove,theextenttowhich the patent system effectively exhibits the winne-take-all popety depends on the beadth of patent potection, which is detemined by vaious legal povisionsandchoicesmadebypatentofficesandthecouts. 5 3 Pominentexamplesaesoftwaepatents,businessmethodspatents,andgenepatents. All of these today exist in the US, wheeas in Euope business methods and softwae patents ae still contovesial, and gene patents ae much hade to uphold than in the US. Databases, by contast, ae potected by stong exclusive intellectual popety ights ineuope,butnotintheus. 4 Building on the economics liteatue cited in footnote 1, seveal law scholas have agued in favo of a geneal eduction of patent holdes peogatives with espect to infinging followes. To be sue, thee ae many difficulties in the pactical implementation of an independent-invention defense, as agued at length by Blai and Cotte (2002). Howeve, Vemont (2006) and Lemley (2007) counte that such difficulties may not be insumountable. A elated issue petains to patent holdes ight to exclude pio inventos who have not patented thei innovations. Bills intoducing a fist-invento defense have beenepeatedlyputonthefloointhecongessovethelastdecade,includinghr1908 which passed the House of Repesentatives on Septembe 7, It must be said, howeve, that an independent-invention defense would potect second inventos who duplicated patented innovations independently, wheeas a fist-invento defense(o pio use ight) would potect fist inventos who concealed thei innovations (pehaps because they did not believe them to be patentable) against the claims of second-invento patentees: see Denicolò and Fanzoni(2004). 5 If patents ae naow in scope, thee may be plenty of oom fo duplicating the innovation lawfully; if instead patents have a boad coveage, the monopoly they ceate is moe pesistent: see Gilbet and Shapio (1990), Klempee (1990), Gallini (1992) and Denicolò(1996). Let us conside, fo instance, the pevalence of so-called me-too dugs in phamaceuticals. Aguably, among the causes of this phenomenon is the naowness of phamaceutical patents, which usually potect innovative molecules athe than the 3

4 It is impotant to note that technological effects, such as leaning by doing o netwok extenalities, can also affect the ability of an innovating fim to captue the whole maket. Let us conside, fo instance, an industy chaacteized by netwok extenalities. Hee, a shot lead time may suffice to ceate an effective baie to enty, even in the absence of stong, exclusive intellectual popety ights: the stategy fo the fist invento is to build an installed base lage enough that late independent duplicatos can ente the maket only by designing compatible poducts. If inte-opeability infomationiskeptsecet,suchlateentymaybeveydifficult,ifnotimpossible, and the winne of the ace effectively obtains the whole pize. But eveninthiscasepolicycanplayaole: ifthefistinventoweecompelled to disclose inte-opeability infomation, followes could achieve compatibilitymuchmoeeasily,andthewinnewouldhavetoshaethemaketwith laggads. 6 Which egime is socially pefeable? Geneally, a switch fom one egime toanothewillimpactthesizeofthepizeaswellasitsdivision,andsowill change the oveall incentives to innovate. To assess the final effect on social welfae one needs some estimate of the elasticity of the supply of inventions (Denicolò, 2007). Following the ealie liteatue, hee we abstact fom this issueandfocusonthestuctueoftheewad. Tothisend,weassumethata theapeutic pocess itself. Boade patents (e.g., patents coveing theapeutic pocesses) might discouage duplicative eseach in the phamaceutical secto and, convesely, might ceate an incentive to devote moe esouces to eally innovative pojects. 6 Recent litigation between the Euopean Commission and Micosoft shows that such mandatoy disclosue of inte-opeability infomation is a vey concete isk fo fims holding dominant positions in Euope. 4

5 move fom one egime to anothe is accompanied by suitable policy changes that guaantee that the oveall incentive to innovate is unalteed. Fom this pespective, the elevant question becomes, Which egime minimizes thesocialcostpeunitofincentivetoinnovateitpovides? 7 To addess this issue, the ealie liteatue has almost invaiably equated the incentive to innovate with industy pofits. Since in a winne-take-all system the innovato obtains all of its pofits as a monopolist, wheeas in a moe pemissive egime pat of its oveall ewad will consist of oligopoly pofits, the question of which egime is moe efficient then boils down to thequestionofwhetheitislessdistotingtoaiseaneuoofpofitsunde oligopoly o unde monopoly. It tuns out that oligopoly is elatively less distoting, except when the demand function is extemely convex. Hence the conclusion that the winne-take-all system is geneally inefficient. We contend that in an innovation ace the incentive to innovate depends notonlyonindustypofits,butalsoonthedivisionofthepofitsbetween ealy and late innovatos. Typically, the equilibium R&D expenditue in an innovation ace inceases with both the pize to the winne (the pofit incentive) and the diffeence between the pize to the winne and to the loses(the competitive theat)(beath et al., 1989). One vitue of the winnetake-all system is that it maximizes the competitive theat by making the consolation pize in the innovation ace vanish. 7 ThisappoachwaspioneeedbyKaplow(1984),whoseanalysisledtothedevelopment of so-called atio tests, which compae the atio of deadweight losses to pofits unde vaious scenaios. 5

6 Anothe vitue of the winne-take-all system is that it pevents wasteful duplication of effots. In a egime in which late inventos ae allowed to shae the maket with the fist invento, thee is an incentive to invest in R&D even afte the fist invento has succeeded. If the late invention epoduces the oiginal innovation identically, these duplication effots ae completely wasteful fom the social viewpoint. Even if duplication esults in diffeentiated poducts o devices, the incentive to engage in duplicative activity may be excessively high fom a social viewpoint because of a business stealing effect, and so peventing such activity may be socially valuable (Gallini, 1992). Ou analysis accounts fo these effects and combines them with those aleady identified in the liteatue. In ou baseline model, two fims ace fo an innovation. Fims choose thei R&D expenditues, which detemine the expected date of successful completion of thei R&D pojects accoding to a Poisson discovey pocess. In the winne-take-all system, as soon as one fim innovates the othe stops investing in R&D since it will be pecluded fom exploiting the innovation anyway. In the altenative, moe pemissive system, the laggad may keep investing to duplicate the innovation. When it also succeeds, the maket becomes a duopoly. In this famewok, we develop a new atio test fo the dominance of the winne-take-all system. As it tuns out, this atio test is passed unde boad cicumstances. In paticula, the winne-take-all system is moe likely to be optimal if poduct maket competition is weak, the innovation ace is intense, and duplication costs 6

7 ae lage. Moving beyond the baseline model, we analyze seveal extensions allowing fo licensing, the possibility that duplicative activity may not be completely wasteful, fee enty, and the case in which R&D expenditues ae an up-font payment athe than flow expenditues. In some extensions the pemissive egime faes bette than in the baseline model, but the esults of the ealie liteatue ae e-obtained only in exteme cases. The emainde of the pape is oganized as follows. Section 2 develops the baseline model. Section 3 deives ou atio test and agues that the winnetake-all system is pefeable in a boad set of cicumstances. Section 4 discusses the elated liteatue and analyzes seveal extensions, showing how the atio test changes as ou assumptions ae elaxed. Section 5 concludes the pape. All poofs ae elegated to an Appendix. 2 Model outline In this section we develop ou baseline model. We fist descibe the innovation ace, and then the downsteam poduct maket. 2.1 The innovation ace Two symmetic fims, A and B, ace in continuous time to obtain an innovation. The timing of the innovation, the natue of which is exogenous, is a pobabilistic function of the amount invested in R&D. Fo each fim i, the R&D effot detemines the expected time of successful completion of 7

8 the R&D poject accoding to a Poisson discovey pocess with a hazad ateequaltox i (i=a,b). Whileexetingeffotx i,fimisustainsaflow costc(x i ).Thepojectsofthetwofimsaeindependent; thus,theaggegate instantaneous pobability of success is simply the sum of the individual pobabilities. In the winne-take-all egime, innovative activity ends as soon as the fist invento succeeds. In the pemissive egime, by contast, the innovation can be duplicated. Like innovation, duplication occus accoding to a Poisson pocess whose hazad ate y depends on the laggad s duplication effot. Let s(y) be the duplication cost function. One could imagine thats(.)=c(.),i.e., afim sinnovativecapabilitiesaenotaffectedbyits competito s success. Howeve, to keep the model moe geneal we allow thesecostfunctionstodiffe, 8 makingonlythestandadegulaityassumption that both ae twice diffeentiable, inceasing, and convex. That is, s (y)>0,c (x i )>0,s (y)>0andc (x i )>0,foi=A,B. Let us conside the innovation ace in moe details. The R&D effots x i ae detemined as the Nash equilibium of a simultaneous moves game betweenfimaandfimb. 9 Fimi sexpectedpofitis 8 Inpaticula,itmightbeaguedthattheaivaloftheinnovationmaypovideuseful infomation that facilitates the followe s eseach, educing the cost that must be bone to duplicate the innovation. 9 In pinciple, fims could choose diffeent levels of R&D effots at diffeent points in time. Note,howeve,thatthegameisstationayinthesensethatateachpointintime, givennosuccesstodate,fimsfaceexactlythesamepayofffunctiosasattime0. Subgame pefection then ensues that equilibium R&D effots will be constant ove time until one fim innovates: see Reinganum(1989) fo details. 8

9 Π i (x i,x j )= x i P W +x j P L c(x i ), (1) x i +x j + wheeistheinteestate,x i isownr&deffot,x j isthecompetito sr&d effot,p W istheewadtowinneoftheaceandp L tothelose,bothto be detemined pesently. The best eply function of fim i is implicitly given by: P }{{} W + (P W P L ) } {{ } x j c (x i ) (x i +x j +)+c(x i )=0. (2) pofit incentive competitive theat Thefisttemin(2)ispopotionaltothepofitincentiveP W ;thiswould be the only deteminant of the incentive to innovate if fim i aced alone (x j =0). ThesecondtemispopotionaltothecompetitivetheatP W P L,which,bycontast,captuestheincentivetotakeovethecompetito and tun fom second to the fist pize. The lage is the instantaneous pobability that fim j succeeds, elegating fim i to the second position, the moe impotant is the competitive theat. Sincefimsaesymmetic,inequilibiumx A =x B =x (Nti,1999).In such a symmetic equilibium, P W +(P W P L )x c (x ) (2x +)+c(x )=0 (3) Assuming that the equilibium is stable, 10 we have (the poof of this and othe esults is in the Appendix): 10 Stabilityoftheequilibiumequiesthecondition dx i <1foi,j=A,B,i j dx j (see LeeandWilde,1980,Beathetal.,1989,andNti,1999). 9

10 Lemma 1 Iftheequilibiumisstable(i.e.,if dx i <1),equilibiumR&D dx j effotsinceasewiththepizetothewinnep W anddeceasewiththepize tothelosep L : x P W >0and x P L <0. The consolation pize P L impacts negatively the incentive to innovate becauseitlowesthecompetitivetheat The poduct maket Wenowtuntothepoductmaketwheetheinnovationisused. Tokeep the analysis as geneal as possible, initially we do not make any specific assumption on the natue of the innovation, demand, and the competition in the poduct maket. We only assume that when the fist invento innovates, itstatstoeanaflowmonopolypofitπ m. Inthewinne-take-allegime, at some finite date (e.g., when patent potection expies) the innovation falls into the public domain and competition dives pofits to zeo. In the pemissive egime, by contast, when the laggad also succeeds the industy becomesaduopolyandeachfimobtainsπ d pepeiod. Fosimplicity,we assumethatinthepemissiveegimeduopolypofitslastsindefinitely TheLemmatakesthepizesP W andp Lasgiven,butvaiablesthataffecttheconsolationpizemayalsoindiectlyimpactthepizetothewinnebyspeedinguposlowing downtheduplicationpocess,asweshallseeinthenextsection. 12 One can easily extend ou esults to the case whee duopoly pofits may also end, because the innovation may be supeseded by exogenous technical pogess, o because it mayleakinthepublicdomain. Allthatmattesisthatduopolypofitsinthepemissive egime last, on aveage, moe than monopoly pofits in the winne-take-all egime: othewise, it would be impossible to compae the two egimes holding the incentive to innovate constant. 10

11 Letvbetheflowsocialvalueoftheinnovationonceitisinthepublic domain. If the innovation is used exclusively by one fim, society suffes a flowmonopolydeadweightloss m andsothesocialbenefitfomtheinnovationisonlyv m pepeiod. Whenbothfimspacticetheinnovation and the poduct maket is a duopoly, the deadweight loss is geneally lowe, d m. Apat fom any dynamics associated with duplication o patent expiation, the envionment is stationay, with a constant discount ate equal to. Fo the time being, we assume that the fist invento does not license the innovative knowledge to the othe fim; this assumption will be elaxed below. 3 Theatiotest We now deive the equilibium in the two policy egimes and develop the welfae compaison. We stat with the pemissive egime in which independent inventos ae allowed to pactice the innovation and compete with the fist invento. 3.1 The pemissive egime How ae the ewads P W and P L detemined? In the pemissive egime, the lose eans discounted duopoly pofits upon duplication, less duplication costs: [ y π d PL IID =max y s(y) y+ ], (4) 11

12 wheeiidstandsfoindependent-inventiondefense. Lety =agmax y [ y π d denote the optimal duplication effot, which, by implicit diffeentiation, inceaseswithπ d. Defineq= y y + <1asthe discountingadjusted pobability of duplication: with a Poisson duplication pocess, the innovation will eventually be duplicated with pobability one, but since thee is discounting, a delayed duplication counts less than instant duplication. Then, we can e-wite(4) as: PL IID =q π d ) (1 q)s(y. (5) Tuningtothewinne sewad,thisisgivenby s(y) y+ ] P IID W = π m+y π d y + = (1 q) π m +qπ d. (6) The fist line of (6) says that the winne eans monopolypofits until the loseduplicates,whichhappenswithinstantaneouspobabilityy. Afteduplication,bothfimsobtainduopolypofitsπ d. Thesecondlineshowsthat the innovato s ewad can be egaded as a weighted aveage of monopoly and duopoly discounted pofits, with weights eflecting the discounting adjusted pobability of duplication. Clealy, P IID W > PIID L ; moe pecisely, the competitive theat is: [ ] PW IID PL IID πm ) =(1 q) +s(y. (7) 12

13 3.2 The winne-take-all egime In the winne-take-all egime, by definition the pize fo the second innovato iszeo(p WTA L =0). Thepizefothefistinnovato,assumingthatitholds amonopolyfoatimepeiodoflengtht,is T PW WTA = 0 π m e t dt=τ π m, (8) wheeτ 1 e T isthenomalizedlengthofthefistinvento smonopoly. 3.3 Welfae We use the standad definition of social welfae in a patial equilibium famewok, namely the sum of consume and poduce suplus. In the winne-take-all egime, expected social welfae is [ W WTA = 2x 2x τ v m +(1 τ) v ] 2c(x ) + 2x +, (9) since monopoly ends when the innovation falls into the public domain. The tem inside squae backets is the discounted social value of the innovation, accounting fo the monopoly distotions that pevail fo a peiod of discounted length τ. The coefficient 2x 2x + is the discounting adjusted pobabilitythattheinnovationisachieved(thehazadateis2x sincetwo fims ae acing but who innovates is a matte of indiffeence fo society), and the last tem is the total discounted R&D expenditue. In the pemissive egime, things ae slightly moe complicated. Expected 13

14 social welfae is W IID = = 2x 2x + 2x 2x + [ v m s(y )+y v d y + [ (1 q) v m s(y ) ] 2c(x ) 2x + ] + +q v d 2c(x ) 2x +. (10) The tem inside squae backets eflects the fact that now monopoly pevails only until the innovation is duplicated, which occus with an instantaneous pobabilityy. Afteduplication,themaketbecomesaduopolyandsociety suffesalowedeadweightloss, d,foeve. Aslongasduplicationhasnot occuedyet,howeve,thelaggadalsosustainstheduplicationcosts(y ). The second line of (10) expesses these effects in tems of the discounting adjusted pobability of duplication q. 3.4 Compaison Geneally speaking, the compaison between W WTA and W IID is complicated by the fact that the equilibium R&D effot x may diffe acoss egimes. Since we ae inteested in ascetaining which ule, winne-take-all o independent-invention defense, povides incentives to innovate moe efficiently, we develop the welfae compaison assuming that the incentives to innovate, and hence the R&D effots, ae the same in both egimes. Specifically, we assume that the level of potection in the winne-take-all egime, τ,isadjustedsoastoyieldthesamelevelofequilibiumr&dexpenditue as in the pemissive egime. 14

15 To poceed, note that the consolation pize can be ewitten as whee P IID L =q π d (1 Σ), (11) Σ= s(y ) y π d (12) can be intepeted as a elative index of the costliness of duplication. It epesents theshae of expected discounted duopoly pofits, q π d, which is absobedbyduplicationcosts. Σangesfom0(costlessduplication) 13 to1 (duplication is so costly as to absob all expected evenues). With this definition, we can state: Poposition 1 (The Ratio Test) The winne-take-all egime is pefeable tothepemissiveegimeintemsofsocialwelfaeif ( d +Σ 1 p ) m π d π m (1 p) m π m. (13) wheep= x x + isthestand-alone,discounting-adjustedpobabilityofsuccess in the innovation ace. When is the atio test moe likely to be passed? Is the optimality of the winne-take-all pinciple a mee theoetical cuiosum, as suggested by the ealie liteatue, o is it a ealistic possibility? Unfotunately, inequality (13) looks complicated. Moeove, some of the vaiables that appea in(13), 13 The case Σ=0aises, fo instance, when duplication is completely costless up to a cetain level y and is infinitely costly beyond thatlevel. Then, duplicative activity will alwaysoccuatatey andσ=0. 15

16 suchaspandσ,aeendogenous, 14 andothes,suchas d π d and m πm,depend on the shape of the demand cuve and the intensity of competition. To poceed, one has to make futhe assumptions. 3.5 Linea demand Letusconside,foinstance,thecaseofapoductinnovationwithalinea demandfunction. 15 Aftesuitablenomalization,thedemandfunctioncan bewittenasp =1 Q,wheeP ispiceandq=q A +q B istotaloutput. Theunitpoductioncostofthenewpoductisnomalizedto0,anddemand andcostsaeassumedtobestationay. 16 At the monopoly equilibium, we have π m = 1 4 and m πm = 1 2. As fo duopoly, the equilibium depends on the intensity of poduct maket competition. Using a conjectual vaiations educed-fom model whee the conjectual vaiations paamete ρ vaies fom 1 to 1, we can allow fo any degee of the intensity of competition between collusion(ρ = 1) and Betandcompetition(ρ=1). 17 Thecaseρ=0coespondstotheCounot equilibium. Standad calculations show that Q = 2 3 ρ, π d = 1 ρ (3 ρ) 2 and 14 Intuitively, the speed of innovation, x, and of duplication, y, depend positively on the magnitude of pospective pofits and negatively on the difficulty of achieving the innovation, as captued by shift paametes in the cost functions c(x) and s(y). Using quadaticspecificationssuchasc(x)=αx 2 ands(y)=βy 2,foinstance,onecouldeasily e-expess the atio test(13) in tems of exogenous vaiables only, but the outcome would hadly be moe tanspaent o easie to intepet. 15 One eason why this is a useful benchmak is that in this case the atio test aived atbytheealieliteatue,namely d 2π d m πm,isnevepassed: seesection4below. 16 Inthisexample,theflowsocialvalueoftheinnovationisv= InthelimitingcaseofBetandcompetition(ρ=1),withhomogeneouspoductsthe duplicato s pofits vanish and so thee is no investment in duplication anyway. The two egimes ae then indistinguishable. 16

17 d π d = 1 2 (1 ρ). Inseting the above fomulas into(13), the condition fo the winne-takeall system to be pefeable becomes ρ 2Σ+p(1 Σ). (14) Inspection of(14) eveals that the winne-take-all system is moe likely to be optimal: (i) the lowe the intensity of poduct maket competition, ρ; (ii) the geate the costliness of duplication, Σ; (iii) the geate the intensity of the innovation ace, p (and hence the lowe the inteestate,, and the lowe the expected waiting time to discovey, x ). 1 These compaative statics esults ae illustated in Figue 1. The winnetake-all system is pefeable below the upwad sloping paallel lines. All these lines have equation(14), taken as an equality. Two such lines, coespondingtoσ=0andσ= 1 3,aedepicted. WhenΣ 1 2,thewinne-takeall system dominates the pemissive system fo all values of the intensity of competition and of the intensity of the ace. The winne-take-all system is also pefeable when the intensity of competition is Counot o weake (ρ 0), iespective of the costliness of duplication and the speed of the innovation ace. Next,letusfocusonthecasewheeΣ< 1 2 andcompetitionistoughe thancounot(ρ>0). Figue1showsthatwhenpislage,thewinne-take- 17

18 Figue 1: The winne-take-all system is pefeable below the upwad sloping staight lines, which collapse to the noth-east vetex of the ectangle when Σ 1 2. all system can be optimal even if competition is substantially moe intense that Counot competition even with zeo duplication costs. To assess the likelihood that the winne-take-all system is still pefeable, it is theefoe impotant to get a sense of the magnitude of p. The vaiable p = x x + depends on the eal inteest ate and the instantaneous pobability of discoveyx. Taking10%asanuppeboundfotheealinteestateand 2%asaloweboundfotheinstantaneouspobabilityofsuccess, 18 onegets aloweboundfopof0.2. Inhighlyinnovativeindusties,howeve,pcan 18 1 Note that x is the expected time to discovey, and so a value of x = 2% means thatafiminnovatesevey50yeas. Althoughx isostensiblyanendogenousvaiable,it mustbesaidthatpivatefimsaelyengageineseachwhenthechancesofsuccessae so emote. 18

19 besignificantlylage. Withp=0.5, foinstance, theatiotestis passed when ρ 1 2, iespective of the magnitude of duplication costs, o when Σ 1 3, iespectiveoftheintensityofpoductmaketcompetition. Thus, the baseline model suggests that it is quite likely that the winne-take-all pinciple may be pefeable in highly innovative industies. Convesely, the pemissive egime is moe likely to be optimal in matue industies, whee poduct maket competition is stong and innovation is a elatively ae occuence(x low). 3.6 Ceative duplication So fa we have assumed that poducts ae homogeneous and so duplication is entiely wasteful. If the innovations tageted by the two fims ae not identical, howeve, duplication may incease poduct vaiety and hence may be socially valuable. To allow fo this possibility, we now suppose that the duplicato supplies a poduct that is diffeent fom that supplied by the innovato. Moe pecisely, we assume that the two poducts ae hoizontally diffeentiated. To addess the issue of ceative duplication in its puest fom, we assume that the fist invento cannot seek to discove the othe vaiety and is esticted to poduce only its own. This means that thee ae now two souces of monopoly deadweight losses: high pices and low vaiety Clealy,theassumptionthatthefistinventocanpoduceonlyitsownvaietybiases the compaison against the winne-take-all system: if this assumption wee elaxed, the appopiate atio test would be easie to pass. 19

20 Following the classic fomulation of Singh and Vives(1984), let the invese demand functions be: P A =1 q A θq B, and P B =1 q B θq A. (15) wheeθ [0,1]isapaametethatcaptuesthedegeeofsubstitutability betweenpoducts. Thetwogoodsaeindependentfoθ=0,andaepefect substitutes fo θ = 1(the case consideed above). Again, poduction costs ae nomalized to 0. When both poducts ae supplied and pices ae set equal to maginal costs, we have q A = q B = 1 1+θ and so social suplus is v = 1 1+θ. Pices, howeve, will geneally exceed maginal costs, educing the benefit society obtains fom the innovations. Using again a conjectual vaiations educedfom solution, we obtain d = 1 ρθ π d 1+θ, (16) whee the conjectual vaiations paamete ρ now anges fom ρ = 1(collusion)toρ=θ(Betandcompetition). 20 As fo monopoly, accounting fo both souces of deadweight losses(high 20 1 Standadcalculationsshowthatq A=q B =. Duopolyequilibiumpofitsae 2+θ ρθ 1 ρd 1+θ π d =, while consume suplus can be easily calculated as CS (2+θ ρθ) 2 d =. (2+θ ρθ) 2 Subtacting the sum of poduce and consume suplus unde duopoly fom the fist best benchmak, we get (1 ρθ) 2 d = (1+θ)(2+θ ρθ) 2, whence equation(16) immediately follows. 20

21 picesandlowvaiety),nowwehave: 21 m = 5 3θ π m 2(1+θ). (17) Fosimplicity,letusfocusonthecaseΣ=0. Inseting(16),(17)and Σ=0into (13), it tuns out that the winne-take-allsystemis pefeable when ρ< 2 (5 3θ)(1 p). (18) 2θ As in the case of homogeneous poducts, less competition in the poduct maket and a faste ace favo the winne-take-all system. What is the effect of a change in the degee of poduct diffeentiation? Geate poduct diffeentiation affects the compaison between the two egimes in two opposing ways. On the one hand, it makes duplication moe valuable duplication enlages the vaiety of poducts available to consumes. On the othe hand, fo any given level of the conjectual vaiations paamete, geate poduct diffeentiation elaxes the competition between the fist and the second invento,theebyinceasingtheatio d π d. Whicheffectpevailsdependsonthe intensity of the innovation ace, as can be easily confimed by diffeentiating the ight-hand side of(18): Poposition 2 With ceative duplication, an incease in the degee of poduct diffeentiation militates in favo of the winne-take-all system when the ace is not vey intense, i.e., p < 3 5, othewise, it militates against the winne-take-all system. 21 Moe pecisely, monopoly pofit is still π m = 1, but now the monopoly deadweight 4 lossis m = d 8 21

22 Figue 2 depicts the egion whee the winne-take-all system is pefeable as a function of p and θ fo values of the intensity of competition ρ coesponding to collusion, Counot competition, and Betand competition, espectively. Remakably, the winne-take-all system can be desiable even if poducts ae completely independent(θ = 0), povided that the innovation ace is sufficiently intense: to be pecise, the condition is p> 3 5. The intuitive eason is that with θ = 0, in a pemissive system each fim would conduct the eseach as a monopolist. The winne-take-all system intoduces competition in eseach among the two fims, speeding up the innovation ace. Although ou assumptions make the social costs of such a policy especially lage (one poduct is neve developed), the oveall effect on social welfae can still be positive. 4 Discussion and extensions In this section we compae ou atio test (13) with that developed in the ealie liteatue. To bette claify why we aive at diffeent esults, we also analyze seveal extensions of the baseline model. 4.1 Liteatue eview Thee ae two main diffeences between ou baseline model and the analysis developed in the ealie liteatue. Fist, the ealie liteatue assumes that thee is no duplicative activity in equilibium. Second, it equates the incen- 22

23 Figue 2: The winne-take-all system is pefeable above the cuves coesponding to vaious degees of the intensity of competition. tive to innovate to industy pofits. With these assumptions, the question ofwhichegimeismoeefficientboilsdowntothequestionofwhetheitis less distoting to aise an euo of pofits unde oligopoly o unde monopoly. Accodingly, the winne-take-all system is pefeable only if: d 2π d m π m. (19) This atio test is vey difficult, if not impossible, to pass. With homogenous poducts, fo instance, inequality(19) is always violated with linea o constantelasticitydemand,andcanbemetonlywhenthedemandfunctionis extemely convex(shapio, 2006). Tocompaeouatiotestwiththetaditionalone,letusstatfomthe 23

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