Why does Protection for Intellectual Property
Rights Strengthen an Incentive?∗
Yasuhiro Arai† and Fumitoshi Moriya‡
June 26, 2011

Abstract
This article aims at comparing two hypotheses, i.e., sequential innovation and legal action, and at theoretically obtaining the testable
implications to specify which hypothesis is crucial in empirical evidence. Our main results are that we distinguish the two hypotheses,
depending on i) whether the cross-term coeﬃcient of (the number ofpatent) and (the dummy of patent law) is positive or negative, and ii)
whether to decreases the variance of the patent distribution or not.
KEYWORDS: Intellectual Property Rights, Sequential Innovation,
Multitask
JEL Classiﬁcation: D42, K39, L86∗
We are grateful to Koji Adachi, Sadao Nagaoka, Youichi Sugita, Satoru Tanaka, Yasunobu Tomoda, and the attendees of CTW and CTWE. Of course, we assume responsibility for any remaining errors.
†
Hitotsubashi University‡
Kobe City University of Foreign Studies

1

Introduction

A principal objective of patent law is the promotion of innovation by granting
monopoly power to inventors (Menell and Scotchmer (2007)). The objective
comes from the fact that innovative knowledge is public good; if the patentlaw would not exist, i) it would be diﬃcult for inventors to exclude other fromusing the inventions (i.e., non rival and nonexclusable), ii) in competitive
economy inventors could not aﬀord sunk costs such as the research anddevelopment (R&D) expenditure, iii) the level of investment in R&D isexcessively small low in society. To prevent the underinvestment, patent law
provides patent holder with “exclusive right,” that is, “the right to exclude
others from making, using, oﬀering for sale, or selling the invention.” (35U.S.C. 154) and deﬁne duration and breadth of patent.
Although it looks reasonable that enhancement of patent protection levelstimulates R&D activities of ﬁrms, the existing empirical studies present
something of a paradox: the data do not statistically conﬁrm it. For example, Lerner (2009) and Lerner (2002) examine 177 events (modiﬁcationsof patent law) in 51 counties and ﬁnd that enhancement of patent protection decreases the number of ﬁlings. Sakakibara and Branstetter (2001)
ﬁnds that extension of patent duration does not statistically increase R&D
expenditure in Japanese pharmaceutical ﬁrms. Qian (2007) consider the establishment of pharmaceutical patents in 26 developing countries and ﬁnd
the positive relationship between the establishment of patent law and thecitation-weighted number of ﬁlings by controlling for economic development,
educational attainment.
The theoretical studies oﬀer two possible factors for explaining the observation. One is “sequential innovation” (Bessen and Maskin 2009, Chang
2000, Denicolo 2000, Green and Scotchmer 1995, and Scotchmer 1991).
They begin with the fact that the technical progress is sometimes cumulative, that is, some invention (called by “application technology”) is necessary
for another invention (called by “basic technology”). In this case, ﬁrms haveto bargain for licensing the basic technology in order to engage in innovative
activity for the application technology, but the proﬁt is not divided to maximize their incentive, because 1) the inventor for basic technology cannot
recoup the sunk cost of the basic technology (Green and Scotchmer 1995);or 2) asymmetrical information entails information rent (Bessen and Maskin2009). Therefore, the ineﬃcient outcome of the bargaining for licensing reduces innovators’ incentive. Another (but informal) possible explanation is“legal action”(Jaﬀe and Lerner 2004). In reality, ﬁrms have an option tospend their eﬀorts to the prevention of illegal copies of patented technology through litigation. Such legal action is unproductive activity. However,ﬁrms can increase the proﬁt from the patented products by taking the legal action. Patent law promotes not only innovation but also unproductive
activity such as legal action. If the beneﬁt of the latter outweighs of the1former, enhancement of patent protection level decreases ﬁrms’ incentive forinnovation.
Which explanation is consistent with the puzzling empirical evidence?
How do we distinguish the two hypotheses? This article aims at theoreticallyobtaining the testable implications for judging which hypothesis is crucial.
To accomplish the purpose, we construct a model where there are two ﬁrms
named by ﬁrm 1 and 2. Firm 1 already holds patent A, while ﬁrm 2 does not.
The two ﬁrms compete for another new patent B. In sequential innovation
model, we assume ﬁrm 2 does not execute a new patent B without the useof patent A, that is , ﬁrm 2 does not develop patent B without permissionof ﬁrm 1. Alternatively, we introduce the legal action which patent holder(Firm 1) can take to improve the proﬁt of patent A by giving up developing
patent B. Examples of legal action include searching for illegal use of patent
A or suing illegal users for damages.
By comparing the results of both models, we obtain three results as follows. First, in both models, the enhancement of patent law does not always
stimulate the incentives of R&D. Second, it makes opposite eﬀects on incentives of ﬁrms in each model. In sequential innovation model, the enhancement increases the incentive of the patent holder, while decreases incentive of
non-patent holder. Greater proﬁt of patent B, brought by the enhancement,
discourages the patent holder to license patent A with non-patent holder
and thus the enhancement unable non-patent holder to develop patent B. In
legal action model, the enhancement increases the incentive of non-patent
holder but decreases that of patent holder. Because the enhancement increases the proﬁt of patent A, the patent holder is reluctant to develop a
new technology. That is why the enhancement makes heterogeneous eﬀects.
The ﬁnal result is concerned with empirical implications for distinguishing two hypotheses. We suggest to introduce as a control variable the number of patent which ﬁrm previously holds, and to examine whether the crossterm coeﬃcient of (the number of patent) and (the dummy of patent law) is
positive or negative. If the coeﬃcient is statistically positive, we infer thatthe sequential hypothesis is crucial in the subject of investigation, and viseverse. The empirical implication comes from the heterogeneous eﬀects ofpatent law.
The remainder of the paper is organized as follows. Section 2 develops
our model. We examine the sequential innovation hypothesis in section 3 andlegal action hypothesis in section 4. In section 5, the empirical implications
are obtained based on the result of section 3 and section 4. Section 5 provides
some concluding remarks on our argument.

22

Framework

Basic Framework We consider a market in which ﬁrm 1 and ﬁrm 2 whohave the competence for R&D. In this market, they potentially develop two
kinds of technology, A and B. The competition for the patents is represented
in the two stage game as Figure 1.
The patent concerning technology A is exogenously allocated between
the two ﬁrms in stage 1. Without loss of generality, we assume that ﬁrm 1
holds patent A. The patent concerning technology B is endogenously allocated between the two ﬁrms in stage 2. Firm i succeeds (respectively, fails)
in developing technology B with the probability of Di ∈ [0, 1] (respectively,
1 − Di ), by spending R&D cost ci Di /2. c > 0 represents a parameter concerning the marginal cost. The ﬁrm who succeeds in developing technology
B does not always obtains patent since the competitor may also succeed.
With consideration of that possibility, ﬁrm i acquires the new patent B with1
Di (1 − Dj ) + Di Dj .
2
We assume that ﬁrm i obtains patent B with probability 1/2 when both ﬁrm
succeed to obtain the new technology, which is represented in the secondterm.
To keep our analysis simple, we adopt the following two assumptions.
First, it is assumed that both patents generates the same monopoly proﬁt
M and the enhancement of patent protection increases the monopoly proﬁt.
The second assumption is that ﬁrms chooses the inner solution Di ∈ (0, 1),that is, 0 < M ≤ c.
In the subsequent section, we introduce the two factor, i.e., sequential
innovation and legal action. We call by sequential innovation the technological restriction that the ﬁrm who holds patent A cannot develop technology
B. In the other words, ﬁrm 2 chooses D2 = 0 if ﬁrm 2 fails to bargain forpatent A of ﬁrm 1. We deﬁne as legal action the unproductive action that
improves monopoly proﬁt of patent A. The example of legal action is thatpatent holders search for the patent infringement of their competitors andsue the illegal user for damages.
Benchmark result Before examining the eﬀects of sequential innovationand legal action, we establish the standard argument that the enhancementof patent protection stimulates the innovation. In the second stage, ﬁrm ifaces the following optimal problem,{}1cmax Di (1 − Dj ) + Di Dj M − Di2 .
(1)
Di
2
2

3

Patent

ProfitPatent A
Patent B
R&D

Stage 1PatentProfitStage 2Time

Figure 1: Timing of benchmark model

By First order condition, we obtain the best-response function of ﬁrm i, i.e.,
Di (Dj ) =M (2 − Dj ).
2c

Two ﬁrms faces strategic substitution, that is, Di′ (Dj ) < 0. Since an increasein eﬀort of ﬁrm j decreases the probability that ﬁrm i acquires the patent
B (i.e., 1 − Dj /2), ﬁrm i decreases its eﬀort. By solving the best-response
functions of both ﬁrms, we obtain the optimal eﬀort level and the proﬁt of
ﬁrm i as follows.
2M
,
(2)
Di∗ =
2c + M
2cM 2
πi∗ =
.
(3)
(2c + M )2
By diﬀerentiating (2) by M , we observe the positive relationship between
monopoly proﬁt and the incentive of ﬁrms, i.e.,
∂Di∗
4c
=
>0
∂M
(2c + M )2
where the inequality comes from the fact that 0 < M ≤ c. Therefore,high protection for intellectual property rights strengthens an incentive forinnovation.
Proposition 1. Suppose the model without sequential innovation and legal action. Firm i’s eﬀort level Di∗ is an increasing function of monopoly
proﬁt M .
In the subsequent sections, we extend the basic model to examine theeﬀects of sequential innovation and legal action on the incentive for innovation.

3

Sequential Innovation Hypothesis

In this section, we discuss R&D incentive under the sequential innovation
case. We assume that the successive innovation (patent B) builds on the4Patent

ProfitPatent A
Patent B
Licensing R&D

PatentStage 1

ProfitStage 2Time

Figure 2: Timing of sequential innovation

preceding one (patent A). To examine the eﬀect of sequential innovation,
we consider the game as Figure 2.
When we consider the sequential innovation case, we have to pay attention to the patent breadth of the preceding patent and the patentability ofthe second innovation. For example, Green and Scotchmer (1995) assumes
that the second innovation is patentable. They discuss how the division of
proﬁt between the ﬁrst innovator and the second innovator is aﬀected by theﬁrst patent’s breadth. Scotchmer (1996), on the other side, assumes thatthe ﬁrst patent’s breadth is large enough. She considers how the division of
proﬁt depends on the patentability of the second innovation. To focus on
the innovator’s incentive, we assume that the second products is patentable
and the breadth of the ﬁrst patent is large. The similar assumptions to ourmodels is made in Bessen and Maskin (2009).
Under these assumptions, ﬁrm 2 has an incentive to bargain for patent
A to develop technology B. The patent competition is modiﬁed as follows:
1. Firm 1 oﬀers a ﬁxed price contract T ≥ 0,
2. Firm 2 decides whether to accept it or not,3. If ﬁrm 2 accepts it, ﬁrm 2 can choose D2 ∈ [0, 1]; otherwise, ﬁrm 2
cannot develop technology B (D2 = 0).
We assume that ﬁrm 1 has to pay the negotiation cost αT (0 < α < 1)
to make a licensing contract.1 The deﬁnition of the negotiation cost meansthat the cost becomes large when the amount of contract T is large.

3.1

Without licensing contract

First, we consider the subgame in which ﬁrm 1 does not oﬀer the licensing
contract. In this case, the non-patent holder does not engages in R&D(D2 = 0), since patent A holder (ﬁrm 1) does not allow non-patent holder(ﬁrm 2) to use patent A’s technology. Then, this subgame is equivalent tothe benchmark at D2 = 0.
1

Laﬀont and Tirole (1993) adopts this type of costs.

5

The maximization problem of patent holder (ﬁrm 1) is given bycmax D1 M − D12 .
D1
2Then, optimal eﬀort level is
D1SN =M, D2SN = 0c(4)

Firms’ proﬁts are respectively given by
π1SN =

3.2

M 2 SN
,π
=0
2c 2

(5)

With licensing contract

In this case, ﬁrm 1 permits ﬁrm 2 to develop the new technology based
on ﬁrm 1’s patent. After signing the licensing contract and paying theﬁxed payment T , the patent competition is the same as that in benchmark.
Therefore, the optimal eﬀort level is given by
DiSL =

2M.
2c + M

Anticipating the R&D behavior of both ﬁrms, ﬁrm 1 holding patent A
faces
}
{
c
1 SL SLSLSL
M − (D1SL )2 + (1 − α)T,max D1 (1 − D2 ) + D1 D2
T
2
2
}
{
c
1
subject to D2SL (1 − D1SL ) + D1SL D2SL M − (D2SL )2 − T ≥ 0,2
2
where the constraint means that ﬁrm 2 accepts the patent licensing contract.
The optimal patent price T ∗ is
T∗ =

2cM 2
(2c + M )2

Firm 1’s proﬁt with licensing contract is
π1SL =

3.3

2cM 2+ (1 − α)T ∗ , π2SL = 0(2c + M )2

(6)

Analysis

We are prepared to discuss the ﬁrm 1’s licensing strategy under sequential
innovation hypothesis. Next proposition shows ﬁrm 1’s optimal strategy.
6

Firm 1

Firm 2
D

D

NL

D

SL

D
SL

D

NL

D

0

M 0

M

Figure 3: Eﬀort level under the sequential innovation model√
Lemma 1.
If M ≤ M S = 2( 2 − α − 1)c, ﬁrm 1 makes a licensing
contract. Otherwise, no licensing is optimal.
The optimal strategy is determined by a trade-oﬀ between loss of thepatent competition and the beneﬁt of cost-sharing. If ﬁrm 1 does not makea licensing contract, he enjoys monopoly proﬁt by avoiding the patent competition, while he bears all the R&D cost cDi2 /2. If ﬁrm 1 makes a licensing
contract, he suﬀers loss from the patent competition and bargaining, but
enjoys beneﬁt from sharing the R&D cost with ﬁrm 2, since the marginal
cost is increasing function to Di and a licensing contract allows ﬁrm 1 toobtain the partial beneﬁt of the cost-sharing.
Lemma 1 also shows that the loss of the competition gradually dominates
the beneﬁt of cost-sharing as the proﬁt of patent (M ) becomes larger. Recall
that the best-response function depends on the proﬁt of patent M (i.e.,
Di (Dj ) = M (2 − Dj )/2c). The slope of the best-response function becomessteeper as M increases. This means that the larger proﬁt of the patentinduces the more ineﬃcient R&D level (D1 , D2 ) (in comparison with that
under monopoly) through the patent competition. In sum, the larger M
causes the severer competition and thus the loss increases.
Next proposition shows the optimal R&D level of the ﬁrms.
Proposition 2.
(1) Firm 1’s R&D level is an increasing function of M .
(2) Firm 2’s R&D level is an increasing function of M if M ≤ 2(1)c. Otherwise, ﬁrm 2’s eﬀort level becomes zero.

√2 − α−

Figure 3 represents the optimal R&D level of the ﬁrms. Since ﬁrm 1
have less incentive to licenses ﬁrm 2 as M increases, ﬁrm 2 cannot develop
the new technology because this model assumes sequential innovation. On
7the other hands, no license contract gives ﬁrm 1 the monopoly power andﬁrm 1’s R&D jumps upward. The R&D levels of both ﬁrms thus jump at
the threshold M = M S .
Therefore, the switching from licensing to no licensing makes the R&Dlevel of ﬁrm 2 downward, while making ﬁrm 1’s upward. Because of theseeﬀects, we obtain the following proposition.2
Proposition 3. If α > 0, the probability of innovation 1−(1−D1 )(1−D2 )is not monotone increasing in M . In particular, the probability of innovation
discontinuously go downward at M = M S .

4

Legal Action Hypothesis

In this section, we discuss another scenario that may reduce innovators’
incentive. Firms usually spend their eﬀort not only for the development but
also for the legal action that prevents illegal use of their invention. Firm canenjoy the proﬁt M from developed patent through the legal action. If hedoes not monitor and prevent illegal copy of his patent, his proﬁt becomesM̂ that is smaller than M . However, the eﬀort to the legal action does notcontribute to develop the new technology. If ﬁrm has an option to spend his
eﬀort to the legal action, the eﬀort level to the development may decrease. In
this section, we assume that the ﬁrm who launch the legal action does notedevelop the new technology. We try to discuss the legal action hypothesisby extending the benchmark model. To examine the eﬀect of legal action,
we consider the two stage game as Figure 4.

4.1

With Legal Action

If ﬁrm 1 chooses a legal action, he can enjoy the monopoly proﬁt M inperiod 1. However, he can not develop the new innovation in period 2 since
he spends his eﬀort to the legal action. In period 2, the eﬀort level are
M
c

(7)

M2
2c

(8)

D1LA = 0, D2LA =Firms’ proﬁts in second period are given by
π1LA = 0, π2LA =2
One might think that this result does not capture the decrease of the patents inindustries or countries in the empirical studies, because this result does not mean thatthe probability of innovation is decreasing in M . However, this theoretical result canbe related to the empirical observation, because the patent law reforms observed in theempirical literature causes not continuous change of M but the binary change.

8Patent

ProfitPatent A
Patent B
R&D Patent
Legal action

Stage 1

ProfitStage 2Time

Figure 4: Timing of legal action model

If ﬁrm 1 applies the legal action, only ﬁrm 2 spends the eﬀort to the development. Firm 1’s proﬁt with legal action is
π1LA = M + 0.

(9)

Firm 1 can earn the monopoly proﬁt perfectly in period 1. However, his
proﬁt in second period becomes zero since he can not make new products.
Then, ﬁrm 1’s proﬁt under the legal action is sum of M and zero.

4.2

Without Legal Action

If ﬁrm 1 does not apply the legal action, both ﬁrms spend their eﬀort todevelop the new technology. In addition to that, they can launch the legal
actions in the second stage. Then, eﬀort level is given by
D1LN = D2LN =2M
2c + M(10)

When ﬁrm 1 does not spend his eﬀort to the legal action, his proﬁt becomesM̂ that is smaller than M because of the illegal activity such as piracy.3 In
this model, we also assume that M̂ < 7c/9. If M̂ violates this limit, ﬁrm 1
does not have an incentive to apply legal action. Then, ﬁrm 1’s total proﬁt
without legal action is
π1LN = M̂ +

4.3

2cM 2
(2c + M )2

(11)

Analysis

We can compare the ﬁrm 1’s proﬁt with legal action and that without legal
action. Next lemma says the ﬁrm 1’s strategy under this scheme.
Lemma 2. Let M L be the threshold satisfying π1SN = π1SL . If M ≤ M L ,ﬁrm 1 does not launch the legal action. Otherwise, he makes the legal action.
3

In this model, we assume that M̂ is independent with M . Here, the proﬁt without
legal action M̂ (the proﬁt with legal action M ) is interpreted as the lower (upper) boundof the proﬁt of patent.

9Firm 1

Firm 2

D

D

LA

D

LN

D

LA

LN

D

M 0

0

D

M

Figure 5: Eﬀort level under the legal action model

The optimal strategy is determined by a trade-oﬀ between the improvement on the proﬁt of initial patent (patent A) and the proﬁt by acquiring
the patent in the future (patent B). If ﬁrm 1 makes a legal action, he can
improve the proﬁt of patent A in the ﬁrst period, but give up acquiring thepatent in the future. As M increases, ﬁrm 1 more prefers legal action to
R&D activity. Because the expected proﬁt of patent B involves the possibility that ﬁrm 2 acquire the patent B, the marginal eﬀect of patent A always
outweighs the marginal eﬀect of the expected proﬁt of patent B.
Proposition 4.
(1) Firm 1’s eﬀort is an increasing function of M when ﬁrm 1 does notlaunch the legal action. Otherwise, ﬁrm 1’s eﬀort level becomes zero.
(2) Firm 2’s eﬀort is an increasing function of M .
Intuition of this proposition is similar to proposition 2. If ﬁrm 1 has thelegal action, his R&D level becomes zero since he can not develop the newtechnology in the second period by the assumption. Since no R&D of ﬁrm
1 gives ﬁrm 2 the monopoly power, the incentive of ﬁrm 2 discontinuously
goes upward. Then the optimal R&D level is summarized in ﬁgure 5.
Note that this eﬀect under the legal action hypothesis is opposite tothat under the sequential innovation hypothesis. While the incentive of
the holder of initial patent (ﬁrm 1) jumps downward under the legal action
hypothesis, it jumps upward under the sequential innovation. This comesfrom the diﬀerent functions of the initial patent between both hypotheses.
Under legal action hypotheses, the initial patent prevents the holder to takethe future patent. Under the sequential innovation hypotheses, the holder
can eliminate the competition of the future patent by using the initial patent.
Finally we obtain the following proposition.

10

Proposition5.
Suppose the model with legal action. If M̂ is smaller√
than (4 2 − 5)c, the probability of innovation, i.e., 1 − (1 − D1 )(1 − D2 ), is
not monotone increasing in M .

5Theoretical Implications and The existing Empirical Studies

In this section, we discuss the relationships between our results and theexisting empirical studies. We consider a patent reform which rises theprotection level of patents. Let M be the monopoly proﬁt before the reformand M̄ be the monopoly proﬁt after the reform. We assume M̄ ≥ M .
Proposition 3 and 5 show that the probability of innovation in an industry (1 − (1 − D1 )(1 − D2 )) is not monotone increasing with respect to theprotection level M in both sequential innovation model (thereafter, SI) andlegal action model (thereafter, LA). In particular, the probability decreases
if M̄ and M are intermediate.
This result corresponds to the existing empirical studies based on countries level data, if we identify the probability as the number of patents in theindustries or countries. Lerner (2009) and Lerner (2002) examine 177 events(modiﬁcations of patent law) in 51 counties and ﬁnd that enhancement ofpatent protection decreases the number of ﬁlings. Qian (2007) consider 26
developing countries from 1978 to 1995 and ﬁnd no signiﬁcant relationshipbetween applications of patent law to pharmaceutical ﬁrm and innovationsuch as citation-weighted U.S. patents award and R&D expenditure. Moreinterestingly, it is found that patent law has positive eﬀects on innovation
conditional on levels of GDP. If the GDP inﬂuences the level of M , ourmodel can suggest one reason of this result.4
Proposition 2 and 4 show that R&D behaviors of ﬁrms are diﬀerent
depending on whether ﬁrms hold initial patent A or not. Furthermore, The
adopted hypothesis, SI or LA, determines which ﬁrm reduces the R&D eﬀort
responding to the enhancement of patent. In SI model, the R&D eﬀort ofinitial patent holder, i.e., D1 , increases as protection level becomes higher,
while the R&D eﬀort of no holder (D2 ) can be decreasing (Figure 3). In
LA-Model, the diﬀerent behaviors between patent holder and non-patent
holder are also observed but the eﬀect of higher protection level is opposite(Figure 5).
Proposition 2 and 4 also have both positive and negative implications
for the empirical studies. The negative side is that the existing studies usingﬁrm-level data has a problem: while the empirical studies (e.g., Sakakibara
and Branstetter 2001) assume that reforms of patent law have the sameeﬀect on ﬁrms regardless of the number of their patent, the assumption can4

Of course, this observation can be explained from macro economic perspective. See
Boldrin and Levine (2008).

11

be theoretically inadequate. This suggests that “the number of patents”
should be introduced as control variable.
On the other hands, the positive side is that the diﬀerence of R&D
behavior allows us to distinguish which hypothesis ﬁts with observations.
To address the positive side, we consider the following situations.
• Consider a patent reform which rises the protection level of patents(from M to M̄ ).
• Let Dki be ﬁrm i’s optimal R&D eﬀort before patent enhancement
and D̄ik be i’s optimal R&D eﬀort after patent enhancement, wherek = {SI, LA} represents the adopted hypothesis.
• We assume that 1 − (1 − Dk1 )(1 − Dk2 ) ≥ 1 − (1 − D̄1k )(1 − D̄2k ) for anyk.
By the assumption (ﬁnal one), we focus on the situation in which the patent
reform reduces the probability of innovation.5 This assumption is adequate
because we are interested in why the enhancement of patent law induces lessnumber of patent in industries.
By using the results in section 3 and 4, we observe that the ﬁrm behavior
depends on the patent law reform and the number of patent which it holds.
The probability that non-patent holder (ﬁrm 2) acquires new patent B before
the reform is represented by
1
D2k (1 − D2k ) + D2k D1k .
2This probability changes depending on the reform dummy (λ) and the patent
number (ηi ) as table 1.
Table 1: The additional eﬀects in R&D behavior of ﬁrm i
Non patent holder (ηi = 0)
Patent holder (ηi = 1)

Before reform (λ = 0)–
Dk1 − Dk2

after reform (λ = 1)D̄2k (1 − 21 D̄1k ) − Dk2 (1 − 12 Dk1 )(D̄1k − D̄2k ) − (Dk1 − Dk2 )Therefore, we obtain the probability as
1
1
1
Dk2 (1 − Dk1 ) + λ[D̄2k (1 − D̄1k ) − Dk2 (1 − Dk1 )] + ηi (Dk1 − Dk2 )
2
2
2
k
k
k
k
+ ηi λ[(D̄1 − D̄2 ) − (D1 − D2 )].
The last term in the above expression represents the increase of probability
if the number of patent increases and the patent law is enhanced. The sign5

We also assume the existence of M̄ and M .

12

of the term is diﬀerent, depending on which hypothesis we adopt. In thesequential innovation hypothesis, the last term is positive because D̄1SI −≥ 0 and D̄2SI − DSI≤ 0 (Figure 3). The legal action hypothesis
DSI
1
2
LA
implies that the second term is positive because D̄1LA − DLA
1 ≤ 0 and D̄2 −LA
D2 ≥ 0 (Figure 5). Therefore, the sign of coeﬃcient on (the number ofpatent) × (the dummy of patent law reform) determines which hypothesis
is statistically accepted.6
Another positive implication for empirical studies is related to the change
in the variance of patent distribution. In the legal action hypothesis, variance of patent distribution in industries decreases since the initial patent
prevents the incentive of the patent holder for innovation. The variance increases in the sequential innovation hypothesis, as the initial patent prevents
the invention of the non-patent holder.

6

Concluding Remarks

In this paper, we have investigated the diﬀerence between the sequential
innovation hypothesis and the legal action hypothesis. Our main result isthat patent law makes the diﬀerent eﬀects on incentives of ﬁrms and two
testable implications are obtained. While we believe the importance of theseresults, some remarks should be noted.
First, we specify the bargaining cost as αT . The assumption allows us
to focus on the important cases. If the bargaining cost is ﬁxed for M , there
are two thresholds value in which ﬁrm 1 is indiﬀerent between no license
contract and license contract (see (12) in appendix). This means that the
optimal choice as to whether to oﬀer patent license or not changes fromlicensing to no licensing and from no licensing to licensing, as monopoly
proﬁt increases. Since our interest attaches to the case in which the patentlaw reduces the incentive of ﬁrms, the change from licensing to no licensing
is abstracted.
Second, our important assumption is that ﬁrms bargain for the initialpatent ex ante. The existing literature investigates both ex ante bargaining
and ex post bargaining. For example, Scotchmer (1996) consider ex post
bargaining and argue that the sequential innovation hurts the incentive ofinitial patent holder. Our model captures not this eﬀects but the eﬀect
discussed in Bessen and Maskin (2009).
Finally, only two patents are considered in our model. If we consider
more than two technologies, the results can depend on ownership and technological relationships among the patents. In our paper, technology B is
assumed to be an application of technology A. If we introduce technology
C, there are four possibilities: given that C is another basic of technologyB, i) one ﬁrm holds both technology A and B, ii) each ﬁrm holds either
6

See Hall and Ziedonis (2001).

13

technology A or B, and iii) C is another application of technology B. Whateﬀects the ownership and technology relationship make? What diﬀers between SI-model and LA-model? This direction is an interesting for our
future research.

7

Appendix

Proof of lemma 1
We compare the ﬁrm 1’s proﬁt with the licensing contract and that without
the licensing contract. From (5) and (6), we obtain the following equation.
M 2 {(M + 2c)2 − (8 − 4α)c2 }(12)2c(2c + M )2√
This equation is positive if and only if M > 2( 2 − α − 1)c, because i)(M + 2c)2 − (8 − 4α)c2 is increasing function for M ≥ 0, ii) (M + 2c)2 − (8 −224α)c2 < 0 when M = 0, and√ iii) (M + 2c) − (8 − 4α)c > 0 when M = c.
Note that the threshold 2( 2 − α − 1)c is smaller than c. Q.E.D.
π1SN − π1SL =Proof of Proposition 3
The diﬀerence of probability of innovation between with license contract andwithout license contract is given by
D1SN − [1 − (1 − D1SL )(1 − D2SL )] =M {(M + 2c)2 − 8c2 }c (2c + M )2

(13)

By applying the similar argument in the proof of lemma 1, we show thatthere
√ exists an unique threshold making the above expression zero. If M <2( 2 − 1)c, the probability of innovation with licensing contract is larger
than that without licensing contract, and vice versa.
By comparing the threshold in (12) and the threshold in (13), for any
α>0√√2( 2 − 1)c > 2( 2 − α − 1)c.
√Since this means that D1SN < [1 − (1 − D1SL )(1 − D2SL )] at M = 2( 2 − α −1)c, we obtain the proof. Q.E.D.
Proof of Lemma 2
We compare the ﬁrm 1’s proﬁt with the legal action and that without thelegal action. From (9) and (11), we obtain the following equation.
π1LN − π1LA = M̂ −M (M 2 + 2M c + 4c2 )(M + 2c)214(14)

Let X(M ) be the above expression. By diﬀerentiating X(M ), it yields∂X(M )
M 3 + 6cM 2 + 4c2 M + 8c3=−≤0∂M(M + 2c)3for any M ≥ 0. Since M > M̂ and 7c/9 > M̂ , we observe that i) X(0) ≥ 0
and ii) X(c) < 0. Therefore, there exists a unique threshold that π1LN −π1LA = 0. Q.E.D.
Proof of Proposition 5
Threshold value of innovation The diﬀerence between both schemes isgiven by
D2LA − [1 − (1 − D1LN )(1 − D2LN )] =M {(M + 2c)2 − 8c2 }c (2c + M )2

(15)

By applying the similar argument in proof of lemma 1, there exists an
uniquezero. Note that the threshold√√ threshold making the above expression(2 2 − 2)c is smaller than c. If M < (2 2 − 2)c, the probability of innovation with licensing contract is larger than that without licensing contract,and vice versa.
Comparison between threshold of payoﬀ and threshold of innovation Suppose that M̂ − 7c/9 < 0. Since the uniqueness of threshold of
payoﬀ and the monotone decreasing function of π1LN − π1LA are shown inLNLAthe proof of Lemma 2, the remaining√ part of proof is whether π1 − π1 ispositive or negative when M√= (2 2 − 2)c.
By substituting M = (2 2 − 2)c into the equation (14), the proﬁt difference between two schemes is given by the next equation.
√
π1LN − π1LA = M̂ − (4 2 − 5)c(16)√
If M̂ is smaller than (4 2−5)c, the ﬁrm 1’s proﬁt with legal action becomeslarger than that without legal action, and vice versa. Q.E.D.

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