Financing of Innovation, Learning Across Peer Firms and
Innovation Waves
Abstract
This paper studies a model where innovation by a pioneering …rm spurs subsequent innovations by
peer …rms, leading to the emergence of innovation waves. In our model, innovation of a …rm is accessed
by peer …rms competing in similar markets. Peer …rms can either expropriate the innovation, or learn
from it to generate a subsequent innovation. As an innovation reaches a greater number of peer …rms
with a greater probability, each peer …rm’s expropriation incentives decline while its incentives to learn
from the innovation and to generate a subsequent innovation increase. These results are due to the
intuition that an increase in the number of …rms expropriating the same innovation makes expropriation
less desirable, inducing some …rms to deviate to innovation.
The resulting increase in the number of
…rms investing in innovation, in turn, makes expropriation even less attractive by reducing the life cycle
of an existing innovation, and hence, the payo¤ from its expropriation. As the extent of cross learning
across peer …rms becomes stronger, the ability of the pioneering …rm to raise capital for its innovation
improves. Similarly, the incentives of capital providers to invest in innovative technologies increase as
the degree of information dissemination across peer …rms increases. These results are consistent with
the empirical evidence presented in Matray (2014) that innovation by listed …rms spur innovation by
neighboring private …rms, and a more e¢ cient circulation of innovation among peer …rms have a positive
impact on the availability of capital such as venture capital …nancing for innovative …rms. Our paper also
suggests that the initial public o¤ering (IPO) of a …rm with a pioneering innovation would create an
opportunity for other …rms to learn from the innovation of the IPO …rm, and will allow the pioneering
IPO …rm to raise …nancing at more attractive terms.
1
Introduction
Firms considering to invest in innovation face an expropriation risk in terms of their innovation
output being accessed and expropriated by peer …rms, due to the absence of well-de…ned and
strongly protected property rights. The expropriation risk reduces the expected returns from
investing in innovation, and hence, may a¤ect negatively ex ante innovation incentives, resulting
in underinvestment in innovation (Arrow 1962). In addition, the possibility of expropriation by
peer …rms might have implications for how a …rm …nances its innovative project as well. It is
often argued, for instance, that raising capital from public capital markets may lead to greater
leakage of proprietary information and innovative knowledge of a …rm to its peers, reducing
the desirability of public capital markets over private capital markets for innovative …rms. The
expropriation risk of a …rm’s innovative knowledge being accessed and expropriated by peer …rms
is based on the assumption that peer …rms accessing a rival …rm’s innovation always …nd it
desirable to expropriate it. However, expropriation may neither be the only available strategy
nor the most desirable one. A breakthrough innovation by a …rm in an industry may provide new
information on which peer …rms can build and generate subsequent innovations. Consistent with
this view, Matray (2014) provides evidence that innovation by listed …rms spurs innovation by
neighboring private …rms where private …rms learn from the innovation output of their publicly
listed peer …rms. In addition, the extent of learning across peer …rms becomes stronger as the
intensity of information ‡ow becomes greater, and as the distance across peer …rms becomes
shorter. In this paper, we analyze a …rm’s choice between expropriating an existing innovation of
a peer …rm, and learning from the innovation to generate a subsequent innovation. We show that
as an innovation is accessed by a greater number of peer …rms, and with a greater probability,
each peer …rm’s incentive to expropriate the innovation becomes weaker while its incentive to
learn from it and advance it into a subsequent innovation gets stronger. This, in turn, leads to
the emergence of innovation waves where a …rm’s innovation spurs future innovations by its peer
…rms as well as it gets displaced by future innovations generated by its peer …rms.
In our model, an innovation generated by a …rm is accessed with a probability by peer …rms
1
competing in similar product markets. Peer …rms accessing the innovation have two mutually
exclusive strategies to choose from: to expropriate the innovation, or to innovate by learning from
the current innovation. Expropriation is less costly than innovation. When the current innovation
reaches a greater number of peer …rms, each …rm’s incentive to expropriate becomes weaker while
its incentive to innovate gets stronger. To see the intuition behind this result, suppose that we
have a pioneering …rm P with a current innovation, and two follower …rms, F1 and F2. Consider
a scenario where P’s current innovation is accessed only by F1, and F1 chooses to expropriate
the innovation since the expropriation payo¤ is greater than the innovation payo¤ net of the cost
of investing in innovation. Anticipating that its innovation will be expropriated by F1, P’s ex
ante incentives to innovate would be weaker relative to the scenario with no expropriation risk.
Consider now a second scenario where P’s innovation is accessed by both F1 and F2. Suppose
initially that both …rms choose expropriation. This leads to a smaller expropriation payo¤ for
F1, compared to the …rst scenario, given that now there are three …rms sharing the payo¤ from
the current innovation. When F1 evaluates the choice between the expropriation strategy and
the innovation strategy, it would have greater incentives to switch to the innovation strategy,
compared to the …rst scenario, for the simple reason that the expropriation payo¤ is now smaller.
Suppose that the payo¤ from the innovation strategy now exceeds that from the expropriation
strategy and F1 switches to innovation. Although F2 now is the only candidate to expropriate
P’s innovation, interestingly, its incentives to do so will be weaker, relative to F1’s expropriation
incentives in the …rst scenario. This is because the fact that F1 invests in innovation introduces
the possibility that a future innovation will make the current innovation obsolete, and eliminate
the payo¤ from expropriating it. The reduction in the expropriation payo¤, in turn, may induce
F2 to switch to innovation as well. Hence, it is possible that P’s innovation is always expropriated
when it is accessed only by F1, when it is accessed by both F1 and F2, it is not expropriated,
but is advanced into future innovations. Although future innovations by F1 and F2 displace P’s
innovation, P can be still better o¤ in the second scenario where its innovation is accessed by F1
and F2 together, than in the …rst scenario where its innovation is accessed only by F1. This is
2
because investing in innovation takes a longer time than expropriation, and F1 and F2 choosing
expropriation over innovation allows P to enjoy the full payo¤ from its innovation until it gets
displaced. This, in turn, provides P with greater incentives ex ante to invest in innovation.
There is an additional reason why P bene…ts from its innovation being accessed by a greater
number of …rms at a greater e¢ ciency when we incorporate the …nancing of P’s innovative
project. When P bargains with an outside …nancier to raise …nancing for developing its innovation,
anticipating that its innovation will spur future innovations which will require additional outside
…nancing from the outside …nancier, it is able to raise …nancing at more attractive terms, especially
as the rate of information transmission from P to peer …rms becomes greater. This results
in greater expected pro…ts for P from investing in innovation where as an industry leader, its
innovation will allow peer …rms to learn from it and generate future innovations. Interestingly,
the ability to raise cheaper …nancing and enjoy greater pro…ts from its innovation implies that
P might prefer its innovation to be accessed freely by peer …rms even though it may have the
ability to completely prevent its circulation across peer …rms.
Consistent with the empirical results in Matray (2014), in our model, F1 and F2’s willingness
to learn from P’s innovation and to advance it into future innovations becomes stronger as the
intensity of information ‡ow from P to F1 and F2 increases, resulting in a greater probability
that P’s innovation is accessed by both F1 and F2. This result has an interesting implication for
…nancing an innovative project. As information spreads more e¢ ciently among a greater number
of peer …rms, the expropriation risk associated with raising capital from public capital markets
goes down. Hence, this result suggests that more developed public capital markets which facilitate
a more e¢ cient transmission of information among peer …rms will enhance learning incentives
across peer …rms where peer …rms build on each other’s innovation rather than expropriating
each other’s innovation. Hence, the potential negative e¤ect of being a public …rm on innovation
incentives will be smaller, and the incentives to go public will be greater when information
dissemination among peer …rms occurs at a greater e¢ ciency.
Our model also has implications for the innovation output of a …rm and industry after a
3
merger wave. A wave of consolidating mergers in an industry will reduce the number of standalone …rms, and increase each …rm’s incentive to expropriate a given innovation, with a negative
e¤ect on incentives to invest in innovation in the …rst place. Consistent with this view, Haucap
and Stiebale (2016) provide evidence that mergers between pharmaceutical …rms have a negative
e¤ect on the innovation output of stand-alone …rms.
In our model, it is not possible to prevent the expropriation of an innovation through property
rights or contractual mechanisms. Hence, a …rm with an innovation faces the risk of expropriation
if the innovation is accessed by other …rms. In our model, a more intense information ‡ow among
peer …rms may help resolve the risk of expropriation. The intuition that existence of competition
may prevent expropriation is also in the model in Anton and Yao (1994) which analyzes a problem
faced by an inventor when selling a valuable idea which can be easily expropriated due to the
absence of property rights. This paper shows that the inventor can capture the full value of its
invention by revealing the invention to a competitor of the buyer. The threat of competition
prevents the buyer of the invention from expropriating or stealing the invention due to the
inventor’s ability to sell the invention to a competing buyer. In our set-up, di¤erent from Anton
and Yao (1994), peer …rms can not only expropriate an innovation, but also learn from it and
generate future innovations. As the innovation of a pioneering …rm reaches a greater number of
…rms, each competing …rm’s incentives to expropriate decline and their incentives to learn from
the innovation and to advance it become stronger. Hence, in our model, it is not the threat of
revealing an innovation to a competitor which prevents expropriation, but rather the actual ‡ow
of innovation to competitors that makes it less likely for the innovation to be expropriated and
more likely to be advanced. There is no scope for current innovations to spur future innovations
and to lead to innovation waves in Anton and Yao (1994) because competitors can only steal
an existing innovation, with no ability to learn from it. Biais and Perotti (2008) focuses on the
complementarity between di¤erent dimensions of an innovative idea to mitigate the risk of the
idea being stolen. In their model, an entrepreneur needs experts to evaluate her innovative idea
along di¤erent dimensions. Sharing the idea creates the risk of the idea being stolen by experts.
4
Forming a partnership with the experts can prevent the idea from being stolen since it gives each
experts access to the other’s expertise. Interestingly,very valuable ideas cannot be shared since it
is too attractive for experts to steal them. In our model, as an innovative idea reaches a greater
number of competing …rms, and is shared among a greater number of peer …rms, the idea is less
likely to be expropriated, and more likely to spur future valuable ideas. This leads to a novel
insight that incentives to create innovative ideas depend positively on whether innovative ideas
‡ow freely and are shared among a large number of peer …rms.
In our model, an innovation of a …rm travels to other …rms which can learn from it to
generate a subsequent innovation. Stein (2008) studies a similar setting where …rms exchange
ideas with their competitors since doing so increases the probability of generating a new idea due
to complementarity in the information structure. In our paper, incentives to learn from a current
innovation and to generate subsequent innovations increase as the innovation circulates among a
greater number of …rms at a greater e¢ ciency. Hence, complementarity among current and future
innovations arises in economies with mechanisms supporting free circulation of ideas among a large
number of …rms. This feature of our model allows our paper to generate implications on relation
between the intensity of information ‡ow among peer …rms and the emergence of innovation waves
through learning across peer …rms, consistent with the recently documented empirical evidence
on the innovation spillovers from listed …rms to private peer …rms in Matray (2014).
In another related work, Glode and Green (2011) provides a rational explanation for performance persistence in the hedge fund industry based on potential information spillovers associated
with an innovative trading strategy. The incumbent manager of a hedge fund is willing to let
informed investors bene…t from future pro…tability to prevent informed investors revealing the
strategy to a non-incumbent manager since doing so would allow non-incumbent managers to
expropriate or exploit the trading strategy.
This paper is organized as follows. Section 2 presents the model. Section 3 analyzes the
model. Section 4 concludes.
All proofs are in the Appendix.
5
2
Model
We consider an economy with risk-neutral …rms and no discounting. The game has three dates,
denoted by t 2 f0; 1; 2g: There is a pioneering …rm P with a technological innovation. P develops
its innovation at t=0. Once the innovation is developed, peer …rms can access the innovation.
There are two peer …rms, F1 and F2, each of which can access P’s innovation with probability q:
This implies that with probability q 2 , …rm P’s innovation is accessed by both F1 and F2. With
probability 2q(1
q); it is accessed by one of the peer …rms. With probability (1
q)2 ; neither
peer …rm accesses …rm P’s innovation.
P’s payo¤ from its innovation depends on whether the innovation is accessed by the peer
…rms and their choice of action. If the innovation is not accessed by any of the peer …rms, P
obtains payo¤ 1 from its innovation in each of the dates at t = 1 and t = 2, with total payo¤ 2:
If the innovation is accessed by one or two of the peer …rms, P’s payo¤ depends on the actions
chosen by each peer …rm. Peer …rms have two mutually exclusive strategies to choose from.
They either choose to expropriate the innovation, or to innovate by learning from P’s innovation.
Expropriation refers to commercialization of P’s innovation, is less costly than innovation where
its cost is normalized to zero for simplicity, and generates an earlier payo¤ than innovation.
Innovation involves a greater monetary cost, it is risky and generates a payo¤ only at a later
date if successful. Although there is no discounting in our model, expropriation and innovation
strategies have important implications for payo¤s obtained by P, as explained below.
First consider the case where P’s innovation is accessed by only one peer …rm, say F1. If
F1 chooses to expropriate P’s innovation, each …rm obtains expropriation payo¤ e2 with 0
e2 < 1, at t = 1 and t = 2: Note that the assumption that both the pioneering …rm and the
expropriating …rm obtain the same payo¤ is only for analytical simplicity, and our model is robust
to a speci…cation where the pioneering …rm captures a greater payo¤ than the expropriating …rm.
If F1 chooses the innovation strategy, it incurs cost k at t = 0, and works on generating
a subsequent innovation. At t = 1, F1 is successful in generating a subsequent innovation with
probability p, and obtains payo¤ v at t = 2: Hence, under the innovation strategy, with probability
6
p, F1 obtains payo¤ v, and with probability 1
p
v + (1
p)
0
i = pv
p, it obtains zero, yielding an expected payo¤
k: P’s payo¤s from its innovation under F1’s innovation strategy, on
the other hand, are as follows. If F1 chooses to innovate, since it is busy working on generating
a subsequent innovation, it cannot expropriate P’s innovation. In other words, F1 must choose
one of the two strategies since it has the resources and time to invest in only one of them. An
important implication of this assumption is that if F1 chooses innovation over expropriation, P
commercializes its own innovation and obtains the full payo¤ 1 from its innovation at t = 1. Put
di¤erently, P faces no competition in internalizing the …rst payo¤ from its own innovation at
t = 1, if F1 chooses innovation. However, its second payo¤ from its innovation at t = 2 depends
on whether F1 is successful in generating a subsequent innovation or not. With probability p, F1
is successful and its innovation displaces P’s innovation, and drives its payo¤ to zero, resulting
in zero payo¤ for P at t = 2. With probability 1
p, F1 fails in generating an innovation, and
P obtains the full payo¤ 1 from its innovation at t = 2. Hence, P’s expected payo¤ at t = 0,
conditional on F1 choosing innovation, is given by 1 + (1
p)
1=2
p:
The formulation described above is meant to capture the notion that innovation, compared to
expropriation, is a longer term strategy with greater cost and risk. Since there is no discounting
in our model for simplicity, F1 choosing innovation over expropriation implies that F1 chooses
to learn from P’s innovation to generate a subsequent innovation. This, in turn, allows P to
internalize the full …rst payo¤ from its innovation at t = 1. At the same time, however, it leads
to the possibility that P’s innovation is displaced by F1’s subsequent innovation, and hence, P
obtains zero payo¤ at t = 2 if F1 is successful. Hence, from P’s perspective, expropriation strategy
by F2 implies an immediate reduction in its payo¤ from its innovation, while the innovation
strategy by F1 allows P to obtain the full …rst period payo¤ from its own innovation, while facing
the risk of losing the entire second period payo¤ from it. This formulation appears to capture the
real-life trade-o¤s between expropriation and innovation strategies. While investing in innovation
is a costly, risky and time-consuming endeavor, it usually involves a higher potential payo¤ if
successful. Expropriation, on the other hand, has a lower cost, and lower risk but also lower
7
return, relative to innovation.
We assume that conditional on accessing P’s innovation, the e¢ cient strategy by F1 is innovation. In other words, the total payo¤ obtained by P and F1 under expropriation given by 4e2
is lower than that under innovation given by 1 + (1
p)
1
k+p
v where pv
k is F1’s
expected pro…t from investing in innovation while 1 + (1
p) is P’s expected payo¤ if F1 chooses
innovation. This assumption translates into k < 2
v)
p(1
4e2 :
If P’s innovation is accessed by both peer …rms at the same time, that is, by F1 and F2, the
two …rms choose simultaneously between the expropriation and the innovation strategy. If both
…rms choose expropriation, P’s innovation is commercialized by three …rms where each of the
three …rms obtains expropriation payo¤ e3 at t = 1 and t = 2,with e3 < e2 : If one …rm, say F1,
chooses innovation, and the other …rm, F2, chooses expropriation, F2 (expropriating …rm ) and
P obtain expropriation payo¤ e2 at t = 1 from commercializing P’s innovation. Their payo¤ at
t = 2 depends on whether F1 succeeds in generating a subsequent innovation. If F1 succeeds,
which happens with probability p, then its innovation displaces P’s original innovation, and drives
its payo¤ to 0, implying that P and F2 obtain zero payo¤ at t = 2:
Finally, if both F1 and F2 choose innovation, they obtain zero payo¤ at t = 1, given that they
choose not to expropriate P’s innovation, and work on generating a subsequent innovation. Their
payo¤ at t = 2 depends on whether they are successful in generating a subsequent innovation or
not. If both of them succeed, which happens with probability p2 ;we assume that they compete
in the product market where each …rm obtains payo¤ cv at t = 2, with 0 < c
1. If either F1
or F2 is successful in generating an innovation, the successful …rm obtains the full payo¤ v from
its innovation at t = 2, and the failed …rm obtains zero payo¤. Finally, if both F1 and F2 fail
in generating a subsequent innovation, which happens with probability (1
p)2 ;they obtain zero
payo¤ at t = 2. P’s payo¤s if both F1 and F2 choose innovation are as follows. P obtains the
full payo¤ 1 from its own innovation at t = 1. At t = 2, with probability p2 + 2p(1
p); at least
one of the two …rms, F1 and F2 generates an innovation and displaces P’s innovation. Hence, P
obtains zero payo¤. With probability (1
p)2 ; both F1 and F2 fail in generating a subsequent
8
innovation, and hence, P obtains the full payo¤ 1 from its innovation at t = 2.
3
Analysis
Suppose P develops its innovation. With probability q 2 , its innovation is accessed by both F1 and
F2. With probability 2q(1
(1
q); it is accessed by one of the two peer …rms, and with probability
q)2 ; it is accessed by neither peer …rm.
Consider …rst the state of the world where it is accessed by one of the peer …rms, say F1.
After accessing P’s innovation, F1 compares its payo¤ from expropriation to its expected payo¤
from innovation to determine its optimal strategy. If F1 chooses to expropriate the innovation,
it obtains total payo¤ 2e2 : If it chooses to innovate, it incurs cost k, and obtains a subsequent
innovation with probability p at t = 2, with payo¤ v; yielding an expected payo¤ of
k+
pv. Comparing the payo¤s from expropriation and innovation, it is immediate that F1 chooses
expropriation for k > pv
2e2 ; and innovation for k
pv
2e2 : Assume that k > pv
2e2 so that
F1 chooses expropriation, which implies that P obtains payo¤ 2e2 from developing its innovation
in the state of the world where its innovation is accessed by only one of the peer …rms.
Now consider the state where P’s innovation is accessed by both F1 and F2. As before, …rms
accessing P’s innovation choose between expropriation and innovation. Importantly, with two
…rms accessing the innovation, it is possible that in equilibrium, both F1 and F2 choose to learn
from P’s innovation rather than expropriating it. If F1 and F2 choose to learn from P’s innovation,
and if at least one of them is successful in advancing it by generating a subsequent innovation,
P’s innovation would get displaced. Although being displaced by a subsequent innovation drives
down its payo¤ to zero, P could be still better o¤, compared to the alternative where its innovation
is accessed by only one peer …rm.
To see the formal analysis of why expropriation might be less likely when P’s innovation is
accessed by both peer …rms, suppose that after accessing P’s innovation, both F1 and F2 choose
expropriation. Since now three …rms compete over developing the same innovation, each …rm’s
per period payo¤ is diluted to e3 where each of the three …rms obtains e3 at t = 1 and t = 2:
9
E;E
Hence, F1’s and F2’s expropriation payo¤s, denoted by vFE;E
1 and vF 2 respectively are given by
E;E
vFE;E
1 = vF 2 = 2e3 . For the expropriation strategy by both …rms to be an equilibrium outcome,
neither …rm should …nd deviation to the innovation strategy desirable, given that the other …rm
chooses expropriation. Suppose F1 deviates to innovation, given that F2 chooses expropriation.
It spends k, and is successful with generating a subsequent innovation with probability p and
obtains payo¤ v: Hence, F1’s innovation payo¤, given that F2 chooses expropriation, denoted
I;E
by vFI;E
1 is given by vF 1
k + pv: Comparing the expected payo¤ from innovation to the
expropriation payo¤, F1 …nds it optimal to deviate to innovation if vFI;E
1 =
or equivalently if k
pv
k + pv
vFE;E
1 = 2e3 ;
2e3 : Recall that in the state where P’s innovation is accessed only by
F1, F1 chooses innovation over expropriation for k
py 2e2 :Given that e3 < e2 , it is immediate
to see that F1’s expropriation incentives are weaker in the state where P’s innovation is accessed
by both peer …rms. In other words, for values of k such that py
2e2 < k
py
2e3 , F1
chooses expropriation if it is the only …rm accessing P’s innovation, but chooses innovation if P’s
innovation is accessed by both F1 and F2.
If F1 chooses innovation, and F2 chooses expropriation, F2’s expected payo¤ from expropriI;E
ation, denoted by vFI;E
2 is given by vF 2
e2 + (1
p)e2 = (2
p)e2 . To see the derivation
of this payo¤, note that during the …rst period, F2 expropriates P’s innovation and obtains e2 ,
given that F1 is working on a subsequent innovation, and hence, there are only two …rms, P
and F1, competing over P’s innovation. However, in the second period, whether F2 would be
still able to obtain the expropriation payo¤ depends on whether F1 is successful in generating
a subsequent innovation, which displaces …rm 1’s innovation. F1 fails in obtaining a subsequent
innovation with probability 1
p, and hence, F2 can expropriate P’s innovation and obtain the
second payo¤ e2 only with probability 1
p. With probability p, F1 succeeds in generating a
subsequent innovation which displaces P’s innovation and drives its second period payo¤ to 0.
Since (2
p)e2 < 2e2 , expropriation is not as desirable as in the state where P’s innovation is
accessed and expropriated by only one …rm. This implies that F2 may …nd innovation more
desirable than expropriation despite the fact that it is the only …rm to expropriate P’s innova-
10
tion. To see this, let vFI;I2 denote F2’s expected payo¤ from deviating to innovation, given that
F1 chooses innovation. F2 spends k to generate an innovation and is successful with probability
p. Given that F1 also invests in innovation, and is successful with probability p; with probability
p2 ; both F1 and F2 are successful. In this state, we assume that the two …rms will compete in
the product market, and each …rm would obtain payo¤ cv: With probability (1
p)p, only F2 is
successful, and obtains payo¤ v from its innovation. Hence, F2’s expected pro…t from switching
to innovation, given that F1 also chooses innovation is given by
vFI;I2 = p2
cv + p(1
p)
v
k = pv(1
p(1
c))
k:
(1)
Comparing vFI;I2 with vFI;E
2 , it is immediate to see that F2 …nds it optimal to deviate to innovation
if
vFI;I2 = pv(1
p(1
c))
vFI;E
2 = (2
k
p)e2
(2)
,or equivalently if
k
Hence, for k
2e3 ; pv(1
pv
p(1
c))
2c3 and k
(2
pv(1
pv(1
p(1
p(1
c))
c))
(2
(2
p)e2 :
p)e2 ; or equivalently for k
(3)
minfpv
p)e2 g; the unique equilibrium of the subgame played by F1 and F2
at t = 0 conditional on both …rms accessing P’s innovation is such that both F1 and F2 choose
the innovation strategy where they learn from P’s innovation to advance it into a subsequent
innovation. The following lemma summarizes this result.
Lemma 1 Let k
minfpv
2e3 ; pv(1
p(1
c))
(2
p)e2 g: Conditional on P’s innovation
being accessed by both F1 and F2, the unique equilibrium of the subgame played by F1 and F2 at
t = 0 is such that both F1 and F2 choose the innovation strategy over the expropriation strategy.
Hence, from Lemma 1, we see that, perhaps surprisingly, when P’s innovation reaches a
greater number of …rms with the potential to expropriate it, expropriation may be less likely.
The intuition for this result is due to two factors: the …rst is that as P’s innovation reaches two
peer …rms, expropriation results in a smaller payo¤ for each …rm, reducing its desirability, and
11
inducing one of the …rms (say F1) to switch to innovation. This creates a second factor which
reduces the desirability of expropriation further: as the number of …rms investing in innovation
increases (i.e., that is, as F1 deviates to innovation), it becomes less desirable for F2 to expropriate
P’s existing innovation due to the fact that a subsequent innovation by F1 displaces the current
innovation and eliminates the payo¤ from its expropriation. Hence, while in the state where P’s
innovation is accessed by only one peer …rm, there might be no way to prevent expropriation,
in the state it is accessed by two peer …rms, each peer …rm might unilaterally …nd it optimal
to innovate and advance it into a subsequent innovation. The following proposition presents the
conditions under which there is always expropriation if P’s innovation is accessed by one peer
…rm, while there is no expropriation if it is accessed by two peer …rms.
Proposition 1 Let c
1
2
and pv
2e2 < k
minfpv
2e3 ; pv(1
p(1
c))
(2
p)e2 g. If P’s
innovation is accessed by one peer …rm, it is always expropriated, while if it is accessed by two
peer …rms, it is not expropriated, but is advanced into subsequent innovations by the peer …rms.
Having established the optimal strategies of F1 and F2 in di¤erent states based on whether
both …rms or only one …rm accesses P’s innovation, now we turn our attention to P’s ex ante
expected pro…ts from investing in innovation.
Assume that the conditions identi…ed in Proposition 1 hold so that P’s innovation is expropriated if it is accessed by only one of the peer …rms, and is not expropriated if it is accessed
by both peer …rms. Let
I
P
denote P’s ex ante expected pro…t from developing its innovation.
P’s innovation is accessed by only one of the peer …rms with probability 2q(1
propriated, yielding payo¤ 2e2 for P. With probability (1
q); and its ex-
q)2 ; it is not accessed by any of the
peer …rms, yielding P payo¤ 2: With probability q 2 ; it is accessed by both peer …rms where each
…rm chooses optimally the innovation strategy. This implies that P obtains full payo¤ 1 from
its innovation at t = 1, given that F1 and F2 cannot expropriate P’s innovation while working
towards a subsequent innovation. The second period payo¤ P obtains depends on whether F1 and
F2 are successful in generating a subsequent innovation. With probability (1
p)2 ; both …rms
fail, and P obtains the second payo¤ 1 from its innovation. With probability p2 + 2p(1
12
p); at
least one of the …rms, F1 or F2, generates an innovation, which displaces P’s innovation, yielding
P payo¤ 0. Hence, we obtain1
I
P
q 2 (1 + (1
p)2 ) + 2q(1
q)
2e2 + (1
q)2
2:
(4)
It follows that P’s expected pro…ts from investing in innovation and, hence, it’s willingness to
innovate increase in q for su¢ ciently high values of q, as presented in the following proposition.
Proposition 2 Let e2 <
1+(1 p)2
:
2
P’s expected pro…ts given by
I
P
is increasing in q for q >
4(1 e2 )
:
6+2(1 p)2 8e2
Proposition 2 implies that P bene…ts from its innovation being accessed by peer …rms with a
greater probability for high values of q, since a higher overall probability of the innovation being
accessed by the peer …rms leads to a lower likelihood of expropriation.
Similarly, the total expected pro…ts of the three …rms, P, F1 and F2 increase in q for higher
values of q: In addition, for su¢ ciently lower values of q, the total expected pro…ts of the three
…rm exceed those of …rm P if it can prevent circulation of its innovation across peer …rms by
setting q = 0: The following proposition summarizes this result.
Proposition 3 Let e2 <
1+(1 p)2 +2(p(1 p(1 c))y k)
:
2
are increasing in q for q >
The total expected pro…ts of the three …rms
4(1 e2 )
:
6+2(1 p)2 +4p(1 p(1 c))v 8e2 4k
The total expected pro…ts of the three
…rms are greater than those of P that P would obtain if it can prevent circulation of its innovation.
3.1
Financing of Innovation
In this section, we assume P needs to raise I from an outside …nancier to develop its innovation.
In addition, we assume that if P’s innovation is accessed by F1 and F2, and if they choose the
1
Note that, under the conditions identi…ed in Proposition 1, P’s expected pro…t given by
I
P
does not depend on
e3 . The reason is that F1 and F2 uniletarally …nd it optimal to learn from P’s innovation rather than to expropriate
it. Hence, there is no situation where P’s innovation is expropriated where each …rm obtains per period payo¤ e3
from it, and as a result, e3 does not show up in P’s ex ante expected pro…t given in (4).
13
innovation strategy and are successful in generating an innovation, they will need to raise i from
the outside …nancier to develop their innovation.
The possibility that P’s innovation will be accessed by the two peer …rms which will learn
from the innovation and generate subsequent innovations implies that the …nancier who provides
…nancing to P will earn rents not only from P, but also from F1 and F2 due to their future
…nancing needs in developing their innovation. This, in turn, implies that when P and the
…nancier negotiate the term of the …nancing, …rm P will be able to capture a portion of future
expected pro…ts of the …nancier from …nancing the subsequent innovations since F1 and F2 would
be able to learn from P’s innovation only if P’s innovation is developed and accessed by the peer
…rms.
Suppose P raises I from an outside …nancier where the terms of the …nancing is determined
through bargaining over the total expected surplus that the …nancier and P will realize from the
development of P’s innovation. As in the previous section, P’s innovation is accessed by each
peer …rm with probability q: Hence, as before, with probability q 2 , its innovation is accessed by
both peer …rms, F1 and F2. With probability 2q(1
and with probability (1
q); it is accessed by one of the peer …rms,
q)2 ; it is accessed by neither peer …rm.
Consider …rst the state of the world where it is accessed by one of the peer …rms, say F1.
After accessing P’s innovation, F1 compares its payo¤ from expropriation to its expected payo¤
from innovation to determine its optimal strategy. If F1 chooses to expropriate the innovation,
it obtains payo¤ e2 at t = 1 and t = 2, resulting in total payo¤ 2e2 : If it chooses to innovate, it
incurs cost k, and obtains a subsequent innovation with probability p at t = 2. Di¤erent from
the previous section, it needs to raise i from the external …nancier to develop its innovation with
payo¤ v > i: We assume that F1 and the …nancier bargain over the net surplus v
party has equal bargaining power and obtains y
F1 from choosing innovation is given by
v i
2 :
This implies that the expected payo¤ for
k + py. Comparing the payo¤s from expropriation and
innovation, it is immediate that F1 chooses expropriation for k > py
k
py
2e2 : Assume that k > py
i where each
2e2 ; and innovation for
2e2 so that F1 chooses expropriation, which implies that
14
P obtains payo¤ 2e2 from commercialization of its innovation in the state of the world where its
innovation is accessed by only one of the peer …rms.
Now consider the state where P’s innovation is accessed by both F1 and F2. Suppose that
both F1 and F2 choose expropriation where each of the three …rms obtains e3 at t = 1 and
t = 2: Hence, as in the previous section, F1’s and F2’s expropriation payo¤, denoted by vFE;E
1
E;E
E;E
and vFE;E
2 respectively are given by vF 1 = vF 2 = 2e3 . For the expropriation strategy by both
…rms to be an equilibrium outcome, neither …rm should …nd deviation to the innovation strategy
desirable, given that the other …rm chooses expropriation. Suppose F1 deviates to innovation,
given that F2 chooses expropriation. It spends k, and is successful with generating a subsequent
innovation with probability p. It raises …nancing i from the external investor and obtains payo¤
y: Hence, F1’s innovation payo¤, given that F2 chooses expropriation, denoted by vFI;E
1 is given
by vFI;E
1
k + py: Comparing the expected payo¤ from innovation to the expropriation payo¤,
F1 …nds it optimal to deviate to innovation if vFI;E
1 =
k
py
2e3 : Assume that k
py
expropriation is given by e2 + (1
k + py
vFE;E
1 = 2e3 ; or equivalently if
2e3 and F1 deviates to innovation. F2’s expected payo¤ from
p)e2 = (2
p)e2 , as before. F2 will compare its expected payo¤
from expropriation to that from innovation to determine its optimal strategy. Suppose it deviates
to innovation as well. With probability p2 , both F1 and F2 generate a subsequent innovation,
and with probability p(1
p), only F2 is successful in generating an innovation. If both F1 and
F2 are successful, we assume that …nancing both innovations has a greater NPV that …nancing
only one of them, that is, 2(cv
i) > v
i, or equivalently, (2c
1)v > i: Hence, the …nancier
provides …nancing for each …rm’s innovation in return for a total payo¤ cv2 i + cv2 i = cv
F1 and F2 obtain payo¤
cv i
2 :
We let C =
cv i
v i,
i where
and denote each …rm’s payo¤ from developing
its innovation as Cy, to preserve a parallel payo¤ form as in the previous section. Hence, with
probability p2 ; F2 obtains payo¤ Cy and with probability p(1
p), it obtains payo¤ y from the
development of its innovation, resulting in
vFI;I2 = p2
Cy + p(1
p)
y
15
k = p(1
p(1
C))y
k:
(5)
Comparing vFI;I2 with vFI;E
2 , it is immediate to see that F2 …nds it optimal to deviate to innovation
if
vFI;I2 = p(1
p(1
C))y
vFI;E
2 = (2
k
p)e2
(6)
,or equivalently if
k
Hence, for k
2e3 ; p(1
py
p(1
2c3 and k
C))y
(2
p(1
p(1
p(1
p(1
C))y
C))y
(2
(2
p)e2 :
(7)
p)e2 ; or equivalently for k
minfpy
p)e2 g; the unique equilibrium of the subgame played by F1 and F2
at t = 0 conditional on both …rms accessing P’s innovation is such that both F1 and F2 choose
the innovation strategy where they learn from P’s innovation to advance it into a subsequent
innovation.
Suppose that k
minfpy
2e3 ; p(1
p(1
C))y
(2
p)e2 g holds so that conditional on P’s
innovation being accessed by both peer …rms, each …rm chooses innovation over expropriation.
Conditional on F1 and F2 accessing P’s innovation, the expected payo¤ of the …nancier, denoted
by
f in
from …nancing the subsequent innovations of F1 and F2 is given by
f in
= p2
2Cy + 2p(1
p)
y = 2p(1
p(1
C))y:
(8)
The …nancier’s expected payo¤ given by (8) will play a role in determining the fraction of the
surplus P needs to share with the …nancier when it raises …nancing for developing its innovation.
In other words, P and the …nancier would bargain over the total surplus to be generated from
…nancing P’s innovation where each party has equal bargaining power. The total expected surplus,
denoted by
T,
that the …nancier and P would obtain from developing P’s innovation is given by
P +;f in
= q 2 (1 + (1
p)2 + 2p(1
p(1
C))y) + 2q(1
q)2e2 + (1
q)2 2
I;
(9)
implying that the expected pro…ts for P from its innovation is given by
P
=
q 2 (1 + (1
p)2 + 2p(1
p(1
C))y) + 2q(1
2
16
q)2e2 + (1
q)2 2
I
:
(10)
As in the previous section, P’s expected pro…t from developing its innovation is increasing in q
for higher values of q, implying that P bene…ts from its innovation being circulated at greater
e¢ ciency across peer …rms. A high value of q not only leads to P’s innovation being expropriated
with a smaller probability, but also promotes subsequent innovations through peer …rm learning,
and allows P to internalize a portion of their expected payo¤ through its bargaining with the
outside …nancier. Interestingly, P’s ability to enjoy a part of the surplus from future innovations
may imply that if P has a choice between preventing the ‡ow of its innovation to its peer …rms
and encouraging the free ‡ow of its innovation across peer …rms, it can be better o¤ when its
innovation is accessed by peer …rms easily. In other words, if P has the ability to choose between
q = 0, and q = 1; it may be better of when its innovation is freely accessed by the peer …rms.
The following two Propositions summarize these results.
Proposition 4 Let e2 <
creasing in q for q >
1+(1 p)2 +2p(1 p(1 C))y
:The
2
expected pro…ts of P given by (10) is in-
2(1 e2 )
:
3+(1 p)2 +2p(1 p(1 C))y 4e2
Note that since the …nancier and P obtain the same expected pro…ts, proposition 3 implies
that the …nancier’s expected pro…ts increase in q for higher values of q as well. This result is
consistent with the evidence presented in Matray (2014) that private …rms’ability to learn from
listed …rms’ innovation output has a positive impact on the availability of VC …nancing in the
economy. In our model, as innovation circulation occurs at a greater rate among peer …rms, it
becomes more pro…table to provide …nancing to the pioneering …rm and the peer …rms generating
subsequent innovations by learning from the pioneering …rm’s innovation output.
Proposition 5 Let p >
2(1 y)
1 2y(1 C) .
P is better o¤ when its innovation is accessed by both peer
…rms than if it can prevent its innovation from being accessed by the peer …rms. In other words,
P’s expected pro…ts given by
P
are greater at q = 1 than at q = 0:
17
4
Conclusions
This paper studies the role of e¢ cient information transmission across peer …rms in preventing
the expropriation of an innovation and facilitating in the emergence of innovation waves where
an innovation by a …rm leads to subsequent innovations by peer …rms. As the innovation of a
…rm is accessed by a greater number of …rms with a greater probability, each …rm’s incentives to
expropriate the innovation go down as a larger number of …rms expropriating the same innovation
results in smaller expropriation payo¤ for each …rm. This induces some …rms to choose to learn
from the current innovation and generate future innovations. As the likelihood of future innovations increases, expropriation of the current innovation becomes even less attractive given that
future innovations displace the current innovation and eliminate the payo¤ from expropriating it.
The role of e¢ cient transmission of information in mitigating the expropriation risk associated
with innovation also has implications for the …nancing of innovation. When a pioneering …rm
raises capital for developing its innovation, the possibility that its innovation will be accessed
by peer …rms and will spur future innovations increases the …nancier’s payo¤ from …nancing the
future innovations. This, in turn, allows the pioneering …rm to raise …nancing at a cheaper cost.
The reduction in cost of …nancing is greater at greater levels of information transmission.
Our paper provides an explanation for why clusters of interconnected …rms that both compete
and learn from each other is a necessary condition for innovation incentives to take-o¤. This
is consistent with observed free ‡ow of ideas across competing …rms in Silicon Valley where
companies learn from each other while they intensely compete in similar product, labor and
capital markets. Finally, our analysis has implications for the innovation output of a …rm and
industry after a merger wave. A wave of consolidating mergers in an industry will reduce the
number of independent peer …rms in the industry, and increase each …rm’s incentive to expropriate
a given innovation, with a negative e¤ect on incentives to invest in innovation in the …rst place.
18
References
Anton, J.J. and D.A. Yao, 1994, Expropriation and Inventions: Appropriable Rents in the Absence of Property Rights, American Economic Review 84, 190-209.
Arrow, K., 1962, Economic Welfare and the Allocation of Resources for Inventions, in R. Nelson(ed), The rate and direction of inventive activity: Economic and social factors, Princeton
University Press, Princeton.
Glode, V and R.Green, 2011, Information Spillovers and Performance Persistence for Hedge
Funds, Journal of Financial Economics 101, 1-17.
Haucap, J., and J. Stiebale, 2016, How mergers a¤ect innovation: Theory and evidence from the
pharmaceutical industry, DICE discussion paper.
Biais, B. and E. Perotti, 2008, Entrepreneurs and New Ideas, The RAND Journal of Economics,
39, 1105-1125.
Matray, A., 2014, Local Innovation Spillovers of Listed Firms, working paper.
Stein, J.C., 2008, Conversation Among Competitors, American Economic Review, 98, 2150-2162.
19
Appendix
Proof of Lemma 1: From the discussion in the text, it follows that F1 and F2 unilaterally
…nd it optimal to invest in innovation for k
pv
2e3 and k
pv(1
p(1
c))
(2
p)e2 :
p)e2
Letting v > maxf 2ep3 ; p(1(2 p(1
c)) g makes sure that there exist values of k such that 0 < k
minfpv
2e3 ; pv(1
p(1
c))
(2
p)e2 g:
Proof of Proposition 1: From the discussion in the text, it follows that in the state where
P’s innovation is accessed by only F1, the innovation is expropriated for k > pv
2e2 . We also
know from Lemma 1 that if P’s innovation is accessed by both F1 and F2, it is not expropriated
for k
minfpv
that pv
pv(1
2e3 ; pv(1
2e2 < k
p(1
c))
p(1
minfpv
(2
c))
(2
2e3 ; pv(1
p(1
c))
p(1
c))
(2
pv 2e2 < pv(1 p(1 c)) (2 p)e2 for v <
p)e2 < pv
e2
p(1 c) :
1
2,
8e2 )
4(1
e2 ):
4(1 e2 )
6+2(1 p)2 8e2
I
P
@
@q
> 0 for q >
pv
I
P
2e2 < k < pv(1
2e3 always
e2
p(1 c)
(2 p)e2
p(1 p(1 c))
p(1
c))
<v<
(2
p)e2
always holds.
with respect to q yields
4(1 e2 )
:
6+2(1 p)2 8e2
2e3 <
2e3 : It is straightforward to show that
(2 p)e2
p(1 p(1 c))
the inequality
2e2 < k
Hence, choosing v such that
Proof of Proposition 2: Taking the derivative of
that
p)e2 g, …rst suppose that pv
makes sure that values of k such that k > 0 and pv
always exist. Note that for c
p)2 )
(2
p)e2 . Since e3 < e2 ; values of k such that pv
exist. Now suppose that pv(1
e2
p(1 c)
p)e2 g: To show the existence of values of k such
The condition e2 <
@ IP
@q
= q(6 + 2(1
1+(1 p)2
2
makes sure
< 1:
Proof of Proposition 3: The total expected pro…ts of the three …rms, denoted by
T;
are given
by
T
= q 2 (1 + (1
p)2 + 2(p(1
Taking the partial derivative of
c))y
8e2
4k)q
4(1
1+(1 p)2 +2(p(1 p(1 c))y k)
2
of
T
with
P (q
T
p(1
k)) + 4q(1
with respect to q yields
e2 ): It follows that
makes sure that
= 0) reveals that
c))v
T
>
@ T
@q
> 0 for q >
= (3 + (1
p)2 + 2p(1
p(1
P (q
C))y
= (6 + 2(1
= 0) for q >
4e2 )q
20
2(1
P
q)2 :
p)2 + 4p(1
p(1
4(1 e2 )
:
6+2(1 p)2 +4p(1 p(1 c))v 8e2 4k
4(1 e2 )
6+2(1 p)2 +4p(1 p(1 c))y 8e2 4k
Proof of Proposition 4: Taking the derivative of
@ P
@q
@ T
@q
q)e2 + 2(1
e2 <
< 1: Direct comparison
4(1 e2 )
:
3+(1 p)2 +2p(1 p(1 c))v 4e2 2k
given in () with respect to q yields
e2 ): It follows that
@ P
@q
> 0 for q >
2(1 e2 )
.
3+(1 p)2 +2p(1 p(1 C))y 4e2
Note that
2(1 e2 )
3+(1 p)2 +2p(1 p(1 C))y 4e2
Proof of Proposition 5: Comparing
for p >
P
< 1 for e2 <
at q = 0 and q = 1 yields that
2(1 y)
1 2y(1 C) .
21
1+(1 p)2 +2p(1 p(1 C))y
:
2
P (q
= 1) >
P (q
= 0)