Patent policy, patent pools, and the accumulation of claims in sequential innovation

PATENT POLICY, PATENT POOLS, AND THE ACCUMULATION OF CLAIMS IN SEQUENTIAL INNOVATION GAST´ ON LLANES†AND STEFANO TRENTO‡ Abs ac . We p esen a dynamic model whe e he accumula ion o pa en s gene a es an inc easing numbe o claims on sequen ial inno a ion. We compa e inno a ion ac i i y unde h ee egimes –pa en s, no-pa en s, and pa en pools– and ind ha none o hem can each he i s bes . We ind ha he i s bes can be eached h ough a decen alized ax-subsidy mechanism, by which inno- a o s ecei e a subsidy when hey inno a e, and a e axed wi h subsequen inno a ions. This inding implies ha op imal ans- e s wo k in he exac opposi e way as adi ional pa en s. Finally, we conside pa en s o ini e du a ion and de e mine he op imal pa en leng h. Keywo ds: Sequen ial Inno a ion, Pa en Policy, Pa en Pools, An icommons, Double Ma ginaliza ion, Complemen a y Monopoly (JEL L13, O31, O34). Da e: No embe 10, 2010. †Escuela de Adminis aci´on, Pon i icia Uni e sidad Ca ´olica de Chile, [email p o ec ed]. ‡Depa men o Economics and Economic His o y, Uni e si a Au `onoma de Ba celona, s e ano. en[email p o ec ed]. We a e g a e ul o Michele Bold in o his guidance and ad ice. We hank An onio Cab ales, An onio Ciccone, Ma co Celen ani, And es E osa, Bel´en Je ez, Ge a d Llobe , Xa ie Vi es, and pa icipan s o semina s a Ha a d Business School, Uni e sidad Ca los III de Mad id, Uni e si a Au `onoma de Ba celona, and Cen e o Applied Economics o Uni e sidad de Chile o use ul commen s and sugges ions. All emaining e o s a e ou esponsibili y. We g a e ully acknowledge inancial suppo om he Minis y o Educa ion o Spain (Llanes, FPU g an AP2003-2204), he Minis y o Science and Technology o Spain (T en o, g an SEJ2006-00538), and he Comunidad Au ´onoma de Mad id (T en o). S e ano T en o is also a ilia ed o MOVE and Ba celona G adua e School o Economics. 1 2 LLANES AND TRENTO 1. In oduc ion Knowledge builds upon p e ious knowledge. This is ue o mos in- no a ions nowadays, especially in high- ech indus ies such as molecu- la biology, plan bio echnology, semiconduc o s, and so wa e. In some cases, he inno a ion consis s o an imp o emen o an olde e sion o he same good. In o he cases, he esea ch leading o he disco e y o he new good depends on he access o esea ch ools, echniques and inpu s ha we e p e ious inno a ions hemsel es. The sequen ial na u e o inno a ion in oduces he issue o how o di ide he e enues om he chain o in en ions among he di e en inno a o s. Suppose wo inno a ions may be in oduced sequen ially. I he i s inno a o ecei es a pa en , she may ob ain a claim o e pa o he second inno a o ’s e enues. Then, he policy make aces an impo an ade-o : i he pa en co e ing he i s inno a ion is s ong, he second inno a ion may become unp o i able, bu i ha pa en is weak, i may p o ide low incen i es o in oduce he i s inno a ion. The li e a u e on sequen ial inno a ion, pionee ed by Sco chme (1991), has s udied his p oblem in dep h. Usually, his li e a u e has analyzed he op imal di ision o p o i s be ween wo sequen ial inno- a o s. Bu wha happens when a con inuous sequence o inno a ions exis s, each building on all p e ious in en ions? Recen esea ch has sugges ed he possibili y ha he accumula ion o claims on sequen ial inno a ions may gene a e a agedy o he an i- commons (Helle , 1998; Helle and Eisenbe g, 1998). When oo many agen s ha e exclusion igh s o e he use o a common esou ce, his esou ce ends o be unde u ilized, in clea duali y wi h he agedy o he commons in which oo many agen s hold igh s o use and he esou ce ends o be o e used. In ou case, he an icommons could a ise i oo many pa en hold- e s ha e exclusi e claims on sepa a e componen s o he s a e-o - he- a echnology, c ea ing an obs acle o u u e esea ch. Howe e , he an icommons hypo hesis has no ye been s udied o mally in a dy- namic model wi h endogenous inno a ion. In pa icula , pa en s p o- duce claims on subsequen inno a ions ha may mo e han compensa e o he nega i e e ec o ha ing o pay licensing ees o p e ious in- no a o s. Se e al in e es ing ques ions a ise: wha is he ne e ec o pa en s on inno a ion incen i es? How should policy pa ame e s be se o maximize social wel a e? ACCUMULATION OF CLAIMS IN SEQUENTIAL INNOVATION 3 Two s eams o li e a u e p o ide pa ial answe s o hese ques ions.1 The li e a u e on sequen ial inno a ion is mainly conce ned wi h subs i- u e inno a ions, whe e la e inno a ions a e applica ions o imp o e- men s o ea lie inno a ions. This se ing allows li le oom o s udying he e ec s o he accumula ion o claims. On he o he hand, he li e - a u es o complemen a y monopoly, pa en hicke s, and pa en pools s udy he p oblem o he accumula ion o complemen a y pa en s, bu om a s a ic poin o iew. As Shapi o (2001) s a es: “The gene ic p oblem inhe en in he pa en hicke is well unde s ood as a ma e o economic heo y, a leas in i s s a ic e sion.” The main con ibu ion o ou pape is o de elop a dynamic model o s udy how he accu- mula ion o complemen a y claims a ec s inno a ion incen i es. We ela e he li e a u e on sequen ial inno a ion o he s a ic li e a u es o complemen a y monopoly and pa en pools. Ex ending he analysis o complemen a y monopoly o a dynamic amewo k allows us o gain ele an insigh s in o he eme gence o pa en hicke s and he ne e - ec o di e en pa en policy egimes on inno a ion a di e en s ages o indus y ma u i y. We p esen a dynamic model o s udy he di ision o p o i s be ween sequen ial inno a o s when each inno a ion builds on se e al p io in- en ions. An in ini e sequence o inno a ions n= 1,2, . . . exis s, whe e inno a ion ncanno be in oduced un il inno a ion n−1 has been in oduced. Each inno a ion has a comme cial alue ( he p o i i gen- e a es as a inal good), which is andom and p i a e in o ma ion o he inno a o , and equi es a de e minis ic cos o R&D o be de eloped. Ou model p o ides a good desc ip ion o he inno a ion p ocess in se e al indus ies. Fo example, in he so wa e indus y, he i s p og ams we e w i en om sc a ch and he e o e buil on li le p io knowledge. As mo e and mo e p og ams we e de eloped, hey p o- g essi ely became mo e dependen on echnologies he i s p og ams had in oduced. Acco ding o Ga inkel e al (1991), mode n so wa e p og ams con ain housands o p e iously de eloped ma hema ical al- go i hms and echniques. Simila examples can be ound in o he high- ech indus ies. Fo mally, ou model is a mul i-s age game in disc e e ime wi h an unce ain end. In e es ingly, he p obabili y o eaching he nex pe iod is de e mined endogenously. The equilib ium concep we use is sub- game pe ec equilib ium wi h Ma ko ian s a egies (Ma ko pe ec equilib ium). 1Read sec ion 1.1 o mo e de ails. 4 LLANES AND TRENTO In he i s sec ions o he pape , we s udy equilib ium dynamics in h ee scena ios: pa en s, no pa en s, and pa en pools. Wi h pa en s, inno a ion becomes ha de and ha de wi h mo e complex inno a ions. The p obabili y o inno a ion goes o 0 as n→ ∞. The p obabili y o inno a ion is highe han in he s a ic case, bu no high enough o p e en he agedy o he an icommons. The e o e, we show ha complemen a y monopoly ine iciencies, o iginally explo ed by Cou no (1838), also ex end o a dynamic amewo k whe e we emo e he bound on he social alue complemen a y monopolis s sha e, as we explain in sec ion 1.1. Wi hou pa en s, he p obabili y o inno a ion is cons an and de- pends on he deg ee o app op iabili y o he inno a ion’s comme cial alue in he inal goods sec o . When pa en s p o ec ideas, he o ma ion o a pa en pool inc eases he p obabili y o inno a ion o all inno a ions. In e es ingly, he p ob- abili y o inno a ion wi h a pool is cons an and highe han i would be in he s a ic case. This esul s eng hens he indings o Shapi o (2001), Le ne and Ti ole (2004), and Llanes and T en o (2009) o s a ic models. We ind ha pools a e dynamically uns able: he emp a ion o e- main ou side he pool inc eases as he sequence o inno a ions ad ances, which means ea ly inno a o s ha e mo e incen i es han la e inno a- o s o en e he pool. The design o a mechanism o sol e he pool ins abili y p oblem, along he lines o B enne (2009), is beyond he scope o his pape . Howe e , we ind ha a scheme in which each in- no a o buys all pa en igh s om he p eceding inno a o , ins ead o paying only o he pe mission o use he idea, can eplica e he pa en pool ou come. The comple e sale o pa en igh s will he e o e gen- e a e highe inno a ion han licensing. An al e na i e scheme, leading o he same inno a ion ou come, is o allow subsequen compe i ion be ween he licensee and he o iginal licenso . This al e na i e scheme emo es he monopoly powe o all bu he las pa en , elimina ing he an icommons e ec . We s udy he op imal inno a ion policy ha maximizes he expec ed wel a e o he sequence o inno a ions and ind ha inno a ion is sub- op imal in he h ee policy egimes. In he no-pa en s egime, he e is a dynamic ex e nali y: inno a o s do no conside how hei decisions impac he echnological possibili ies o u u e inno a o s. In he wo o he policy egimes, he ine iciency s ems om asymme ic in o ma- ion and ma ke powe : pa en holde s do no know he exac alue o he inno a ion, bu hey know i s p obabili y dis ibu ion. The asym- me ic in o ma ion gene a es a downwa d-sloping expec ed demand o ACCUMULATION OF CLAIMS IN SEQUENTIAL INNOVATION 5 he use o ideas, and pa en holde s’ ma ke powe esul s in a p ice o old ideas abo e hei ma ginal cos . We also show ha he i s bes can be eached by decen alizing he inno a ion decision and implemen ing a ax-subsidy scheme. Su p is- ingly, his esul holds e en i he go e nmen and p e ious inno a o s do no know he alue o he inno a ion. Wi h espec o he im- ing o he op imal ans e s, we ind ha inno a o s should ecei e a subsidy o inno a e, and hen be axed wi h subsequen inno a ions. The e o e, op imal ans e s wo k in he exac opposi e way as pa en s, which equi e an inno a o o pay p e ious in en o s up on and hen be compensa ed by ollowing inno a o s. These indings ex end he esul s o E kal and Sco chme (2007), who show ha he sca ci y o ideas has a non- i ial impac on he op imal ewa d o he inno a o . In his sense, he wo pape s a e complemen a y. While E kal and Sco chme show ha he sca ci y o ideas a ec s he size o he op imal ewa d, we show ha i also a ec s he iming o his ewa d. Finally, we ind ha symme ic in o ma ion o e he alue o he inno a ion leads o he i s bes . We hen u n o he analysis o he op imal pa en leng h. Wi h espec o he op imal pa en leng h, we ind ha sho pa en s maximize he p obabili y o inno a ion, because in ou model, absen any o m o p ice collusion o ag eemen be ween pa en holde s, e- ducing pa en leng h is he only way o educe he complemen a y monopoly p oblem. This inding complemen s he indings o p e ious pape s ha ocus on he inal goods sec o and subs i u e inno a ions. Ano he impo an inding is ha he main esul s o he li e a- u e o complemen a y monopoly ex end o a dynamic amewo k wi h endogenous inno a ion. Also, pa en pools composed o complemen- a y pa en s a e wel a e imp o ing wi h endogenous inno a ion. This inding pa ially answe s a ques ion Le ne and Ti ole (2004) posed ega ding he desi abili y o pa en pools when hei ex-an e e ec on inno a ion ac i i y is aken in o accoun . 1.1. Rela ed Li e a u e. We ex end he li e a u e on sequen ial in- no a ion by analyzing he case in which pa en s gene a e cumula i e claims on subsequen inno a ions. This ex ension is impo an be- cause i allows us o s udy he eme gence o pa en hicke s. Many pape s o sequen ial inno a ions, such as G een and Sco chme (1995), Chang (1995), and Sco chme (1996), analyze he op imal dis ibu ion o p o i s be ween wo sequen ial inno a o s. I he i s inno a ion has low comme cial alue (basic esea ch o ins ance), g an ing he i s inno a o a s ong pa en is op imal. This is no necessa ily ue in 6 LLANES AND TRENTO ou model, whe e he accumula ion o pa en s gene a es a p oblem o an icommons. Abs ac ing om ansac ion cos s, o he possibili y ha one o mo e pa en holde s e use o license hei ideas he eby blocking inno- a ion, he agedy o he an icommons is simila o a complemen a y monopoly p oblem, i s analyzed by Cou no (1838). Cou no mod- eled a compe i i e p oduce o b ass who has o use coppe and zinc as inpu s in p oduc ion, and showed ha , when wo di e en monopolis s sell he inpu s, he o al cos o p oducing b ass is highe han when he same monopolis sells bo h inpu s. Sonnenschein (1968) showed ha complemen a y monopoly is he dual o a duopoly model wi h quan i y compe i ion and homogeneous goods, and Be gs om (1978) gene alized his esul o a gene al numbe no inpu s and any deg ee o complemen a i y among hem. Cha i and Jones (2000) showed ha he ma ke ou come in a complemen a y monopoly se ing is inc eas- ingly ine icien as he numbe o agen s inc eases. Recen ly Shapi o (2001) and Le ne and Ti ole (2004) applied he in- s umen s o complemen a y monopoly o he analysis o pa en pools. Thei esul s ein o ce he esul s on complemen a y monopoly: pa en pools (o , equi alen ly, a single monopolis owning all he pa en ed in- pu s) educe he cos o inno a ion when pa en s a e complemen s and inc ease he cos when pa en s a e subs i u es. Bold in and Le ine (2005) and Llanes and T en o (2009) also made use o complemen a y monopoly o show ha , as he numbe o complemen a y pa en s in- c eases, he p obabili y ha a u u e inno a ion will be p o i able goes o ze o. All o hese pape s, al hough hey make impo an con ibu ions, p esen s a ic models. In o he wo ds, he i s inno a ion has been in en ed al eady, so pa en s and pa en pools only a ec he p o i abil- i y o in oducing a second inno a ion. This s uc u e in oduces an impo an asymme y be ween p e ious and u u e inno a ions ha ou dynamic model elimina es. We belie e ha adding a dynamic di- mension is an impo an s ep owa ds a be e unde s anding o he mechanism o an icommons in sequen ial inno a ion. In pa icula , one would expec he complemen a y monopoly p ob- lem o be weake in a dynamic con ex o wo easons: i s , in he s a ic model he e is a limi on he e enues inpu p oduce s sha e; we elimina e his limi by ex ending he analysis o a dynamic ame- wo k wi h po en ially in ini e inno a ions. Second, se ing high license ees inc eases he p obabili y ha he inno a ion chain, o a pa ic- ula esea ch line, will come o a hal : a pa en holde would hen ha e an incen i e o mode a e he license ee o be able o eap pa ACCUMULATION OF CLAIMS IN SEQUENTIAL INNOVATION 7 o he e enues o u he inno a ions. We ind ha , in spi e o hese wo e ec s, he complemen a y monopoly p oblem is so s ong ha inno a ion e en ually becomes unp o i able. Ou pape is ela ed o O’Donoghue e al (1998) and Hopenhayn e al (2006), who also p esen models o cumula i e inno a ion. How- e e , Hopenhayn e al (2006) ha e no accumula ion o claims since only one pa en is alid a any gi en ime. O’Donoghue e al (1998), on he o he hand, ha e accumula ion o claims bu ule ou comple- men a y monopoly, as ba gaining among pa en holde s is e icien by assump ion. Also, in bo h pape s, inno a ions a e subs i u es: he in- oduc ion o a new p oduc au oma ically implies he disappea ance o old e sions om he ma ke . The subs i u abili y be ween inno- a ion in oduces a na u al limi on app op iabili y and p oduces an impo an ade-o , because g an ing a pa en o he i s inno a o limi s wha can be o e ed o he second inno a o . In ou pape , inno- a ions a e complemen a y and do no compe e wi h each o he in he inal goods sec o . This se ing elimina es he app op iabili y p oblem. In such a model, one would expec a pa en sys em o pe o m well. Howe e , he opposi e happens: g an ing oo many pa en igh s on sequen ial inno a ions p oduces a complemen a y monopoly p oblem ha hampe s inno a ion. Finally, ou pape is also ela ed o Menezes and Pi ch o d (2004), who p esen a dynamic model o an icommons. Menezes and Pi ch- o d model he case o a buye who has o combine complemen a y asse s om wo selle s. Selle s may ha e an incen i e o a oid en e ing in o nego ia ions wi h he buye because hey may ge a highe sha e o o al su plus by nego ia ing a e he buye ag ees wi h he o he selle . Holdou occu s i a leas one selle is no p esen in he i s ound o nego ia ions. The au ho s show ha complemen a i y is a necessa y condi ion o holdou , and also ha a ise in complemen a - i y leads o an inc ease in he possibili y o holdou . In ou case, he accumula ion o claims may lead o inc easing delays in he ag eemen be ween cu en and pas inno a o s, u he ing he wel a e loss caused by complemen a y monopoly. 2. The model We s udy a model wi h an in ini e sequence o inno a ions n= 1,2, . . . Each inno a ion canno be in oduced un il all p e ious in- no a ions ha e been in oduced. This inno a ion p ocess e lec s he ac ha ea lie inno a ions do no ha e a solid backg ound upon which 8 LLANES AND TRENTO o build, while u he inno a ions become mo e and mo e indeb ed o p e ious ones as he ma ke ma u es. A each s age, an inno a o ge s an idea o how o de elop a pa icu- la inno a ion. I he inno a o decides o pe o m he inno a ion, he game con inues and, in he ollowing s age, ano he inno a o will ge an idea o he nex inno a ion. I he inno a o decides no o in o- duce he inno a ion, wo hings may happen: (i) wi h p obabili y φ, he game con inues and in he ollowing s age ano he inno a o ies o pe o m he ailed inno a ion, and (ii) wi h p obabili y 1 −φ, he game ends and no o he inno a ions a e possible. The pa ame e φ∈[0,1] ep esen s he deg ee o sca ci y o ideas. I ideas a e mo e sca ce (lowe φ), each idea is mo e di icul o subs i u e, and ano he inno a o is less likely o ha e a di e en app oach o implemen a ailed inno a ion. Le n, j ep esen he j h inno a o ying o in oduce inno a ion n (j−1 inno a o s ha e al eady ied o in oduce inno a ion nwi hou success). A he beginning o he s age, he inno a o ge s an idea wi h andom alue n,j, which she may de elop by incu ing in a de e min- is ic R&D cos o ε. n,j ep esen s he e enues ob ained by selling he new p oduc in he inal-goods ma ke . To concen a e on he e ec s o pa en s on inno a ion ac i i y, we will assume he inno a o is a pe ec p ice dis- c imina o in he inal-goods ma ke , which means he p i a e alue o he inno a ion is equal o he social su plus he new p oduc gene a es. The alue o he inno a ion is p i a e in o ma ion o he inno a o . Pa en holde s only know n,j is d awn om a uni o m dis ibu ion be ween 0 and 1, wi h cumula i e dis ibu ion unc ion F( n,j) = n,j. The inno a o ’s decision on whe he o pe o m he inno a ion will depend no only on n,j and ε, bu also on he licensing e enues and cos ha may a ise depending on he pa icula pa en egime unde analysis. Gi en ha a each s age he inno a o will pe o m he inno a ion wi h a ce ain p obabili y, he game is a mul i-s age game wi h unce - ain end, in which he p obabili y ha he game con inues is de e - mined endogenously. 3. Inno a ion wi h pa en s In his case, pa en s wi h in ini e leng h and b ead h p o ec ideas (we will elax hese assump ions in sec ion 10), which means each inno- a o has o pay license ees o all p e ious in en o s (pa en holde s), in case she wan s o in oduce he inno a ion. The cos o inno a ion ACCUMULATION OF CLAIMS IN SEQUENTIAL INNOVATION 9 is he sum o he cos o R&D and he licensing ees paid o p e ious inno a o s. A pa en o in ini e leng h will also p o ec he new idea, which means he inno a o can eques licensing ees om all subsequen inno a o s. The o al e enues o he inno a ion equal he comme cial alue o he inno a ion plus u u e licensing e enues. The iming o he game wi hin each s age is he ollowing: (i) The n−1 pa en holde s se licensing ees pi n,j, (ii) Na u e ex ac s a alue o n,j om dis ibu ion F( n,j), (iii) he inno a o decides whe he o inno a e (In,j = 1) o no (In,j = 0). I he e enues om he inno a ion a e highe han he cos , inno a- o n, j will in oduce he inno a ion, and in he nex s age, inno a o n+ 1,1 will y o in oduce inno a ion n+ 1. I e enues a e lowe han cos , inno a o n, j will no in oduce he inno a ion, and in he ollowing s age ( eached wi h p obabili y φ), inno a o n, j + 1 will y o in oduce inno a ion nbased on a di e en app oach. This inno a- o n, j + 1 will ace he same n−1 pa en holde s and will ha e a new d aw o he alue o inno a ion, n,j+1. Le Ji n,j be he expec ed u u e licensing e enues o pa en holde ia ial jo inno a ion n, gi en ha s age n, j has been eached. Exp essed in a ecu si e way, Ji n,j =P n,j (pi n,j +β Ji n+1,1) + (1 −P n,j)φ β Ji n,j+1, whe e P n,j is he p obabili y ha inno a ion nis in oduced a ial j, gi en ha n−1 p io inno a ions ha e been in oduced and ha j−1 ials o in oduce inno a ion nha e al eady ailed. Wi h p obabili y P n,j, he pa en holde ge s he p ice pi n,j plus he con inua ion alue o he i s ial o he nex inno a ion, Ji n+1,1, discoun ed by a ac o β∈[0,1]. Wi h p obabili y (1 −P n,j)φ, he inno a ion is no in o- duced bu he game con inues, in which case he pa en holde ge s he con inua ion alue co esponding o he nex ial o he cu en inno- a ion, Ji n,j+1, discoun ed by he ac o β.βcan be in e p e ed bo h as he discoun ac o o , o a ixed discoun ac o , as he ime be ween inno a ions: lowe alues o βimply ideas a i e less equen ly. The inno a o ’s payo is In,j( n,j +βJn n+1,1−cn,j −ε), whe e cn,j = Pn−1 i=1 pi n,j is he sum o licensing ees paid o p e ious inno a o s. We will ocus on Ma ko s a egies. A s a egy o playe ispeci- ies an ac ion condi ioned on he s a e, whe e ac ions a e p ices and he s a e is simply n, j. The equilib ium concep is Ma ko pe ec equilib ium, which implies u u e p ices will be de e mined by a Nash equilib ium in he subsequen games. Thus playe s unde s and ha 16 LLANES AND TRENTO which does no depend on n. I , on he o he hand, inno a o n−1 decides o join he pool wi h he n−2 p e ious inno a o s, he expec ed e enue will be Jn=1−φβ −√1−φβp1−β+β(1 −φ)ε2 (n−1)(1 −φβ)β2(1 −φ)2, which is dec easing in n. This esul is due o he ac ha he pa en pool maximizes join p o i s, hus keeping he o al cos o inno a ion cons an . This cons an amoun mus be di ided among an inc easing numbe o inside s; he e o e, he expec ed e enue o an inside is dec easing in nand con e ges o 0 as n→ ∞. The e o e some inno a o n0>2 wi h incen i es o de ia e by emaining ou side he pool always exis s. Figu e 2 illus a es his inding and shows he gains om de ia ing om he pool as a unc ion o n, o ε= 0.1, φ= 0.5, and β= 0.95. The gains become posi i e a e inno a o 3, which means he ou h inno a o would gain by emaining ou side he pool. Figu e 2. Gains om no joining he pa en pool. ε= 0.1, φ= 0.5, and β= 0.95 Pa en pools can imp o e inno a ion ac i i y, bu a e dynamically uns able. Ea ly inno a o s ha e mo e incen i es o en e he pool han subsequen inno a o s. B enne (2009) inds an elegan mechanism o sol e he ins abili y p oblem o socially desi able pa en pools in a s a ic model. We lea e he design o an equi alen mechanism in he con ex o a dynamic model o u u e esea ch. Wi hou such a mechanism, howe e , pa en pools a e likely o be uns able. This ins abili y migh explain why go e nmen s some imes ha e o en o ce he c ea ion o pa en pools, as he U.S. go e nmen did in he adio and ai c a indus y, o example. ACCUMULATION OF CLAIMS IN SEQUENTIAL INNOVATION 17 9. Socially op imal inno a ion The ele an measu e o wel a e is he expec ed social alue gene a ed by he sequence o inno a ions. The social alue o an inno a ion is equal o he inc ease in consume su plus minus he cos o he esou ces spen in R&D. The e o e, when conside ing ial jo inno a ion n, he social alue gene a ed is n,j −εi he inno a ion is pe o med, and 0 o he wise. Conside he decision o pe o ming inno a ion n, j. I he alue o he inno a ion is g ea e han he cos , ob iously he inno a ion should be pe o med. Howe e , he social planne could s ill decide o pe o m an inno a ion wi h nega i e social alue, because in he opposi e case, he sequence o inno a ions will s op wi h a p obabili y o 1 −φ. The decision will depend, he e o e, on a compa ison be ween he cu en cos o pe o ming an inno a ion wi h nega i e social alue and he expec ed u u e bene i s o con inuing wi h he chain o inno a ions. Le Wm,k be he expec ed social wel a e om s age m, k onwa ds. Once we know he ealiza ion o n,j, expec ed wel a e is n,j −ε+ β Wn+1,j i he inno a ion is pe o med, and β φ Wn,j+1 i he inno a ion is no pe o med. The e o e, he inno a ion should be pe o med i n,j −ε+β Wn+1,j ≥β φ Wn,j+1. P oposi ion 1 shows he socially op imal inno a ion policy. P oposi ion 1 (Socially op imal inno a ion).To maximize expec ed social wel a e, inno a ion n, j should be pe o med i and only i n,j ≥ ∗, whe e ∗=(0i ε≤β 2 1−φ 1−β φ , β−1+√1−β φ √1−2β(1−(1−φ)ε−φ/2) β(1−φ)i ε > β 2 1−φ 1−β φ . P oo . Gi en he assump ions o he model, inno a ions n,j and m,k a e equi alen o any n, j, m, k. I ollows ha Wn,j =Wm,k =W, and he op imal decision is ime-in a ian : a alue ∗∈[0, ε] exis s such ha inno a ion n, j should be pe o med i and only i n,j ≥ ∗. By de ini ion, ∗sol es ∗−ε+β W =β φ W. The e o e, we need o de e mine he alue o W. In pa icula , Wm,k is gi en by Wm,k =P ( m,k ≥ ∗) (E( m,k −ε/ n,j ≥ ∗) + β Wm+1,k) + (1 −P ( m,k ≥ ∗)) β φ Wm,k+1. Imposing Wm,k =Wm+1,k =Wm,k+1 =W, and sol ing o W, we ge W=1− ∗ 1−β(1 −(1 −φ) ∗)1 + ∗ 2−ε. 18 LLANES AND TRENTO Subs i u ing his esul in o ∗−ε+β W =β φ W, and sol ing o ∗, we ge he op imal policy s a ed in he p oposi ion. P oposi ion 1 implies ha inno a ion will be subop imal in he h ee cases s udied abo e. The e a e h ee easons why his is so: dynamic ex e nali ies, ma ke powe , and asymme ic in o ma ion. The dynamic ex e nali y is bes desc ibed by analyzing he no-pa en s case. Wi hou pa en s, he inno a o will pe o m he inno a ion when n≥ε/θ. Gi en ha ∗≤ε, he inno a o may decide no o pe - o m a socially desi able inno a ion, e en i θ= 1, because she igno es he e ec o he decision on he echnological possibili ies o u u e inno a o s. This e ec is well known in he li e a u e o sequen ial in- no a ion (Sco chme , 1991; Hopenhayn e al, 2006) and is simila o he one ound in he li e a u e o mo al haza d in eams (e.g., Holm- s om, 1982), whe e each agen in e nalizes only his ewa d om he e o exe ed. The solu ion o he i s p oblem would equi e in e empo al ans- e s. In sec ion 9.1, we show ha he i s bes can be eached by decen alizing he inno a ion decision and implemen ing a ax-subsidy scheme. Su p isingly, his esul holds e en i in o ma ion is asymme - ic, ha is, i nei he he go e nmen no p e ious inno a o s know he alue o he inno a ion. In he pa en s and pa en -pool cases, he ine iciency a ises om a di e en sou ce: ma ke powe and asymme ic in o ma ion. Be- cause pa en holde s ca e abou he s eam o u u e licensing e - enues hey will lose i he sequence o inno a ions s ops, hey in e nal- ize he dynamic ex e nali y. Howe e , asymme ic in o ma ion implies a downwa d-sloping expec ed demand o old inno a ions, and ma ke powe implies ine icien p icing o pa en s, which leads o subop imal inno a ion. As he numbe o holde s o igh s on inno a ion inc eases, he ine iciency due o ma ke powe inc eases (because o he com- plemen a y monopoly), which is why he pa en -pools case is mo e e icien han he pa en s case. In o de o show he impo ance o he asymme ic in o ma ion as- sump ion, in Sec ion 9.2, we show ha unde symme ic in o ma ion he e exis s an equilib ium ha eaches he i s bes .2This means ha wi hou asymme ic in o ma ion, he dynamic ex e nali y could be pe ec ly in e nalized, eaching he i s bes . 2A con inuum o equilib ia exis s unde symme ic in o ma ion. Some o hese equilib ia do no each he i s bes , bu he impo an ac is ha some o hem do. ACCUMULATION OF CLAIMS IN SEQUENTIAL INNOVATION 19 9.1. Op imal ans e s. In his sec ion, we show ha he i s bes can be eached by decen alizing he inno a ion decision and imple- men ing a ax-subsidy scheme. This is a su p ising inding, because i does no equi e he go e nmen o know he alue o he inno a ion in o de o be implemen ed. In addi ion, we ind ha he op imal iming o he ax-subsidy scheme is as ollows: inno a o s should ecei e a sub- sidy i hey inno a e, and hen be axed when he ollowing inno a ion is pe o med. The s uc u e o ans e s is he ollowing: i inno a o n, j decides o inno a e, she will ha e o pay a ans e n o he inno a o who success ully pe o med inno a ion n−1, bu she will also ha e he igh o ecei e a ans e n+1 om he inno a o who pe o ms inno a ion n+ 1. The e o e, gi en n,j, he inno a o will inno a e i n,j −ε− n+β Jn+1,1≥0, whe e (1) Jn+1,1=P n+1,1 n+1 + (1 −P n+1,1)β φ Jn+1,2. P oposi ion 2 shows he op imal in e empo al ans e ha imple- men s he i s bes . P oposi ion 2. The op imal ans e is cons an and equal o ∗=( ∗−ε) (1 −φ β ∗) 1−β(1 − ∗+φ ∗). ∗≤0 o any alue o he pa ame e s, and ∗<0i and only i ε > 0, φ β < 1. P oo . The p oblems o inno a o s n, j and m, k a e equi alen o he social planne o any n, j, m, k, which means n= n+1 = , and Jn,j =Jm,k =J o any n, j, m, k. Gi en ans e s, he p obabili y o inno a ion is P = 1−ε− +β J. We wan o make his p obabili y equal o he op imal p obabili y, which is P ∗= 1 − ∗. The op imal ans e hen sol es ∗=ε+ −β J. On he o he hand, om equa ion (1), we ge J= (1− ∗) /(1−φ β ∗). Subs i u ing he la e exp ession in o he o me , and sol ing o , we ge he op imal ans e s a ed in he p oposi ion. Finally, ∗≤0 because ∗≤ε om P oposi ion 1. No e ha ∗has a kink when ε=β 2 1−φ 1−β φ , so ∗will also ha e a kink a ha poin . An in e es ing ea u e o he op imal ans e is ha i is nega i e. The e o e, he inno a o should ecei e a subsidy o inno a e, and be axed wi h he ollowing inno a ion. Mos impo an ly, op imal ans e s wo k in he exac opposi e way as pa en s, which equi e an inno a o o pay p e ious in en o s up on and hen be compensa ed by ollowing inno a o s. 20 LLANES AND TRENTO 9.2. Symme ic in o ma ion. To analyze he easons o ine iciency in he di e en cases, in his sec ion we s udy he e ec s o emo ing he asymme ic in o ma ion assump ion while keeping he ma ke powe assump ion (i.e., pas inno a o s a e p ice-se e s, whe eas he cu en inno a o is a p ice ake ). A each s age, a alue o n,j is d awn om F( n,j), and p e ious inno a o s se he le els o licensing ees he inno a o will pay, jus as in he basic model. The di e ence is ha now, p e ious inno a o s lea n he ealiza ion o n,j, and use his in o ma ion when se ing hei licensing ees. In equilib ium, p e ious inno a o s will se a le el o ees ha will lea e he inno a o indi e en be ween inno a ing o no . O he wise, one o he p e ious inno a o s could aise he ee wi hou a ec ing he inno a ion decision, he eby aising he p o i s. The e o e, any sequence o p ices such ha n−1 X i=1 pi n,j = n,j −ε+β Jn n+1,1 is an equilib ium. Conside an equilib ium in which each inno a o pays a ee only o he p e ious inno a o : pn−1 n,j = n,j −ε+β Jn n+1,1. Fo he emainde o his sec ion, le pn,j =pn−1 n,j and Jn,j =Jn−1 n,j . I n,j −ε+β Jn+1,1<0, inno a o n−1 will no allow inno a o n, j o inno a e (she can do his by se ing any p ice abo e n,j −ε+β Jn+1,1). In case n,j −ε+β Jn+1,1≥0, on he o he hand, inno a o n−1 will allow inno a ion n, j only i pn,j ≥β φ Jn,j+1. P oposi ion 3 shows ha in equilib ium, inno a ions will be pe o med only i hey a e socially desi able. P oposi ion 3 (Symme ic in o ma ion).Unde symme ic in o ma- ion, he equilib ium in which each inno a o only pays a licensing ee o he p e ious inno a o is socially op imal. P oo . The p oblem a ial jis he same as he p oblem a ial j+ 1, which means he lowes alue o n,j ha inno a o n−1 will ole a e is cons an . Le ˆ indica e his alue. ˆ sol es ˆ −ε+β Jn+1,1=β φ Jn,j+1, and Jn,j+1 sol es Jn,j+1 =P E(pn,j+1/ n,j+1 ≥ˆ ) + (1 −P )β φ Jn,j+2, whe e P =P ( n,j+1 ≥ˆ ) = 1 −ˆ , and E(pn,j+1/ n,j+1 ≥ˆ ) = 1+ˆ 2−ε+Jn+1,1. The p oblem o a di e en nand/o jis equi alen , ACCUMULATION OF CLAIMS IN SEQUENTIAL INNOVATION 21 so Jn,j =Jm,k =J o all n, j, m, k. Using his esul in he abo e equa ions, we ob ain ˆ =ε−β(1 −φ β)J J= (1 −ˆ )1 + ˆ 2+β φ J+ ˆ φ β J. Sol ing his sys em o equa ions o ˆ and J, we ge ha ˆ = ∗. 9.3. S a ic e sus dynamic incen i es. P e ious models o comple- men a y monopoly, sequen ial inno a ion, and pa en pools we e s a ic (Shapi o, 2001; Le ne and Ti ole, 2004; Bold in and Le ine, 2005; Llanes and T en o, 2009). Looking a wha changes when we add he dynamic dimension is in e es ing. To see wha happens in he s a ic case, assume only one inno a ion is unde conside a ion. The inno a ion uses n−1 old ideas, which ha e al eady been in en ed. I he inno a ion is pe o med, he inno a o ob ains a alue om a uni o m dis ibu ion be ween 0 and 1, and incu s in a cos εo R&D. The p obabili y o inno a ion is P = 1−ε−cn, wi h pa en s o pa en pool and P = 1 −ε/φ wi hou pa en s. Wi h pa en s, he pa en holde ’s p oblem is o maximize P pi. As a esul , he equilib ium p ice is 1−ε nand he p obabili y o inno a ion is 1−ε n. We ha e shown ha in he dynamic model, he p obabili y o inno a ion is 1−ε n+βJn+1 −n−1 nφβJn, wi h Jn, Jn+1 >0. These ex a e ms a ise because he inno a o ge s licensing e enues om u u e inno a o s. Dynamic incen i es imply a highe p obabili y o inno a ion, bu he inc ease is no enough o p e en he p obabili y o inno a ion om con e ging o 0 as n→ ∞. A pa en pool would conside c oss-p ice e ec s, which would lead o a p ice o 1−ε 2 (n−1) and a p obabili y o inno a ion o 1−ε 2. The p obabili y o he co esponding dynamic model is 1−ε 2+n 2βJn+1 −n−1 2φβJn, wi h Jn, Jn+1 >0. In his case, he ex a e ms a ise due o no only he u u e licensing e enues o he inno a o , bu also o he pool’s conce n wi h keeping he u u e licensing e enues o cu en pa en holde s. Wi h espec o he no-pa en s case, he p o i -maximizing decision is he same as in he dynamic case. Inno a o s will he e o e pe o m he inno a ion i φ n≥ε, which leads o a p obabili y o P = 1 −ε/φ. Howe e , in he dynamic case, inno a ion is subop imal e en when φ= 1, which con as s wi h he s a ic case, whe e inno a ion is socially op imal because no in e empo al link be ween inno a ions exis s and he e o e nei he does any ex e nali y. 22 LLANES AND TRENTO 10. Fini e pa en s We ha e seen ha wi h pa en s o in ini e leng h, inno a ion is s i led as ninc eases due o he complemen a y monopoly p oblem. In his sec ion, we ask whe he app op ia ely educing he leng h o pa en s can p e en his p oblem.3 Fo simplici y, we assume φ= 0 and β= 1, which is he mos a o able case o pa en s: ideas a e sca ce and he discoun ac o is small, so, in p inciple, socie y could la gely bene i i inno a o s wi h a low alue o hei in en ions could ge addi ional e enues om cha ging o he inno a o s. Each s age co esponds o one pe iod and only one inno a ion is a emp ed a each pe iod. I he inno a o decides o in oduce he inno a ion, she ob ains a pa en o Lpe iods. The inno a o , he e- o e, has o pay licensing ees o Lp e ious inno a ions, bu she also cha ges licenses o L u u e inno a o s. The main di icul y o he p esen analysis is ha , unlike in he p e ious sec ions, he iden i y o he pa en holde s ma e s. The p ice and u u e expec ed licensing e enues will be di e en o di e en pa en holde s, depending on how long he pa en las s. The inno a o will in oduce he inno a ion i he e enues om inno a ion a e la ge han he cos : n+ n+L X m=n+1 pn m m Y k=n+1 P k≥ n−1 X i=n−L pi n+ε, which means he p obabili y o inno a ion is P n= 1 + n+L X m=n+1 pn m m Y k=n+1 P k− n−1 X i=n−L pi n−ε. The Lcu en pa en holde s di e in hei objec i e unc ions. Le Ji nbe he u u e expec ed e enues o pa en holde ia s age n, gi en ha s age nhas been eached. Then Ji n=P n(pi n+Ji n+1). The pa en holde cha ging a license o he las ime is pa en holde n−L, so Jn−L n+1 = 0. The pa en o n−L+ 1, on he o he hand, will 3We ha e also analyzed he e ec s o educing he b ead h o pa en s. Fo example, suppose new in en ions may in inge on old pa en s wi h ce ain p obabili y. Wi hin ou amewo k, he e ec s o educing b ead h a e simila o he e ec s o educing pa en leng h: a lowe b ead h implies ha he inno a o will ha e o pay ewe licensing ees, bu i also means ewe u u e in en ions will in inge on he pa en . ACCUMULATION OF CLAIMS IN SEQUENTIAL INNOVATION 23 las o one mo e pe iod, so Jn−L+1 n+1 =P n+1 pn−L+1 n+1 . In his way, we can cons uc he u u e expec ed e enues o he Lpa en holde s. The p o i maximiza ion p oblem is max pi n Ji n=P n(pi n+Ji n+1). The i s -o de condi ion is −pi n−Ji n+1 +P n= 0, so pi n+Ji n+1 =P n and Ji n=P 2 n o all i, which also implies pn−L n=P n. We a e in e es ed in s a iona y equilib ia, which means P n=P o all n. S a iona i y, oge he wi h he i s o de condi ion, implies pi n=P (1 −P ) o i≥n−L. Subs i u ing he equilib ium p ices in o he p obabili y o inno a ion, we ge P = 1 + n+L X m=n+1 pn m m Y k=n+1 P k− n−1 X i=n−L pi n−ε = 1 + L−1 X m=1 P (1 −P )P m+P P L−(L−1)P (1 −P )−P −ε. Sol ing o P , we ge : P =L+ 1 −p(L−1)2+ 4Lε 2L, which is he s a iona y equilib ium p obabili y o inno a ion. Figu e 3 shows he p obabili y o inno a ion as a unc ion o he pa en leng h o ε= 0.2. We can see he p obabili y o inno a ion de- c eases wi h L, which means pa en s hu mo e han bene i he inno- a o , because he inno a o has o pay licenses o he pa en holde s. Fu u e licensing e enues a e unce ain, howe e , as hey depend on u u e inno a ions being pe o med. Also no e ha P →0 when L→ ∞ and P →1−εwhen L→0, which co esponds o he p e iously analyzed pa en s and no-pa en s cases (wi h θ= 1). 10.1. Re enues depend on pa en leng h. We ha e assumed ha he e enues om selling he new p oduc in he inal goods ma ke a e independen o pa en leng h. In his subsec ion, we analyze wha happens when we elax his assump ion. Assume he e enues o he inno a o a e ψ(L) n, wi h ψ0(L)≥0, ψ00(L)≤0,limL→0ψ(L) = ψand limL→∞ ψ(L) = 1. He e, ψis he ac ion o social su plus he inno a o would app op ia e wi hou any pa en p o ec ion due o ade sec e s o i s -mo e ad an ages. 24 LLANES AND TRENTO Figu e 3. P obabili y o inno a ion and pa en leng h. In his case, he inno a o will inno a e i ψ(L) n+ n+L X m=n+1 pn m m Y k=n+1 P k≥ε+ n−1 X i=n−L pi n. Applying a p ocedu e simila o ha in he p e ious case, we ob ain he p obabili y o inno a ion in he s a iona y equilib ium: P =L+ 1 −p(L−1)2+ 4Lε/ψ(L) 2L. The e ec o pa en leng h on he p obabili y o inno a ion depends on he unc ional o m o ψ(L). Le ψ(L) = 1 −1−ψ (L+1)γ, whe e γmea- su es he speed a which e enues g ow when Linc eases. Figu e 4a shows ha when ψis mo e conca e (γ= 1), he p obabili y o inno a- ion i s inc eases and hen dec eases wi h pa en leng h. The op imal leng h is posi i e and ini e (in his case L= 1). Figu e 4b shows ha , o a lowe deg ee o conca i y o ψ(L), comple ely emo ing pa en s is op imal. The e o e, he esul s do no change signi ican ly when he e enues in he inal-goods sec o depend on pa en leng h. In his model, sho pa en s he e o e pe o m be e han long pa en s. O’Donoghue e al (1998) ind ha when pa en b ead h is in ini e, which is always he case in his model, long pa en s s imula e inno a- ion ac i i y. These appa en ly di e en esul s a ise because we a e looking o solu ions o di e en p oblems. We analyze he e ec o pa en policy on he complemen a y monopoly p oblem, dis ega ding i s e ec on he inal-goods ma ke . O’Donoghue e al (1998) do ex- ac ly he opposi e. The e o e, ou indings a e no opposed o hei s, bu a he a e complemen a y. ACCUMULATION OF CLAIMS IN SEQUENTIAL INNOVATION 25 (a) ψ= 0.2, ε = 0.1, γ = 1. (b) ψ= 0.2, ε = 0.1, γ = 0.1. Figu e 4. P obabili y o inno a ion as a unc ion o pa en leng h. 11. Conclusion In his pape , we build a dynamic model whe e he accumula ion o pa en s gene a es an inc easing numbe o claims on cumula i e inno- a ion. The model is in ended o ep oduce he cen al ea u e o inno- a ion ac i i y in high- ech indus ies: new p oduc s a e mo e complex han old p oduc s, because hey build on a la ge s ock o p e iously accumula ed knowledge. We s udy he policy ha maximizes expec ed social wel a e and com- pa e i wi h he ou come o h ee pa en -policy egimes: pa en s, pa en pools, and no pa en s. We ind ha , e en abs ac ing om he mo- nopolis ic ine iciencies o pa en s, none o hese policies a ains he op imum. Wi h pa en s, he inno a o has o pay an inc easing numbe o li- cense ees o p e ious inno a o . Asymme ic in o ma ion on he alue o he inno a ion and uncoo dina ed ma ke powe o licenso s c ea e an an icommons e ec ha educes he incen i es o inno a e as inno- a ion becomes mo e complex. The an icommons e ec is weake han in he s a ic case, bu i is s ill s ong enough o d i e he p obabil- i y o inno a ion o ze o as he numbe o licenses g ows. En o cing a pa en pool sol es he lack o coo dina ion bu no he asymme ic- in o ma ion p oblem. As a esul , he ou come o pa en pools is mo e desi able bu s ill does no achie e he i s bes . Elimina ing pa en p o ec ion sol es he wo p oblems bu in oduce a non-in e nalized ex e nali y: p e ious inno a ions se he ounda ions o u u e inno a- ions. The e o e he social cos o one inno a ion may be highe han i s ins an aneous social alue ( he social alue he inno a ion c ea es pe se), and ye he inno a ion may be socially desi able because i