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