ISSN (Print) 0473-453X

Discussion Paper No. 1205

ISSN (Online) 2435-0982

LEADING PATENT BREADTH,ENDOGENOUS QUALITY CHOICE,AND ECONOMIC GROWTH

Keishun Suzuki
Shin Kishimoto

March 2023

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Leading Patent Breadth, Endogenous Quality Choice, andEconomic Growth*
Keishun Suzuki†, Shin Kishimoto‡
March 13, 2023

Abstract
O’donoghue and Zweimüller (2004, J. of Econ. Growth), a seminal work, showed thatbroadening leading breadth in patent protection can stimulate innovation. However, theempirical literature has consistently found skeptical results on the positive eﬀect. To ﬁll thegap, we build another framework where the quality improvement size is derived as an interior solution. In our model, broadening leading breadth can negatively aﬀect innovation
because each innovator is incentivized to free-ride the other innovators’ quality improvements. As a further analysis, we quantitatively investigate the growth eﬀect of intervention
in patent licensing negotiation using two diﬀerent proﬁt division rules derived from a cooperative game. We ﬁnd that intervention in patent licensing negotiation increases the growthrate and stabilizes the economy.

Keywords: Patent protection, Leading breadth, Schumpeterian growth, Endogenous quality
increments, Cooperative game.
JEL-Classiﬁcation: C71, D45, O30.

* We thank many valuable comments from Ryousuke Shimizu, Tatsuro Iwaisako, Takeo Hori, Yuichi Furukawa,Akihiko Yanase, Real Arai, Kohei Hasui, and participants of the 2022 spring meeting of the Japanese Economic
Association and the seminars of Aichi University and Osaka Metropolitan University. Speciﬁcally, we are gratefulto Ryo Horii for his suggestions at the Joint Usage/Research Center at the Institute of Social and Economic Research(ISER), Osaka University. This research is ﬁnancially supported by the Japan Society for the Promotion of Science(JSPS) KAKENHI Grant Numbers JP20K13449 (Suzuki) and JP20K01542 (Kishimoto). All remaining errors are ourown.
† Graduate School of Social Sciences, Chiba University. 1-33, Yayoi-cho, Inage-ku, Chiba, Japan. Email:ksuzuki@chiba-u.jp.
‡ Graduate School of Social Sciences, Chiba University. 1-33, Yayoi-cho, Inage-ku, Chiba, Japan. Email:skishimoto@chiba-u.jp.

1

1

Introduction

Research and development (R&D) is a cumulative process. Innovation builds on previous
innovations, similar to economists extending previous studies when writing their papers. The
cumulativeness creates an exciting but complex problem in the presence of the leading breadth, ascope of patent protection against competition from higher-quality products.1 For a technology
to be patentable, it must have a suﬃcient novelty compared to past technologies. However,
the required novelty may be insuﬃcient to avoid infringing existing patents. Generally, theminimum improvement size for patentability does not equal the size of improvement needed
to avoid infringements of existing patents.2
Traditionally, strengthening patent protection had been recognized as an eﬀective policy topromote technological progress because such policy reforms increase the rewards of innovation.
Under a broader leading breadth, past innovators can claim patent infringement against new
innovators. Therefore, like other dimensions of patent protection (e.g., the length of patentterm, the lagging breadth, and the enforcement), the leading breadth enables patent holders
to appropriate the returns from inventions. The traditional idea that more rewards induce
more innovation is called the Schumpeterian eﬀect. Many existing studies show that patent
protection has been consistently strengthened worldwide (e.g., Ginarte and Park, 1997; Park,
2008; Papageorgiadis et al., 2014).
However, the empirical studies do not support the straightforward idea that strong patent
protection enhances innovation. Lerner (2002), an inclusive survey paper, lists many empiricalstudies and states that “these papers generally cast doubt on claims that enhancing patent policychanges spurs innovative behavior.” Boldrin and Levine (2009) also collect numerous empiricalstudies and summarize the ﬁndings as “they ﬁnd weak or no empirical evidence that strengthening patent regimes increases innovation.” This counterintuitive result is a longstanding
question in the literature called the patent puzzle.
The leading breadth is no exception to the patent puzzle, although the literature on endogenous growth theory has paid little attention to it. O’donoghue and Zweimüller (2004) (fromhere on, O&Z) is a pioneer work incorporating leading breadth into a canonical quality-laddermodel, like pension in the overlapping generations model. Speciﬁcally, a new innovator must
pay the license fees to the past innovators but can eventually obtain the license fees from thefuture innovators after the subsequent generational innovations occur. This feature yields theso-called backloading eﬀect, negatively aﬀecting on the present discounted value of R&D by delaying the timing of the future rewards. O&Z shows that the growth eﬀect of broadening leading
breadth is always positive when the discount rate is suﬃciently low such that the backloading
1This deﬁnition is from O’Donoghue et al. (1998). The concept of patent breadth includes both the lagging andleading breadths. While lagging breadth is a scope that protects an invention from imitation, leading breadth is a
scope that protects an invention from future innovation.
2For the details, see Scotchmer (2004). She introduces a history of Maser and Laser to explain this point.

2

eﬀect is smaller than the Schumpeterian eﬀect. However, there is no signiﬁcant evidence ofthe positive eﬀect in empirical studies that focus on a speciﬁc policy reform that broadened theleading breadth in the U.S. and Japan (e.g., Hall and Ziedonis, 2001; Sakakibara and Branstetter,2001).3
The Schumpeterian eﬀect in the O&Z model arises from (i) the exogenous size of the quality
improvement and (ii) the Bertrand competition where multiple ﬁrms on the quality laddercan form a price cartel. First, for simplicity, they consider a special case in which the quality
improvement size is always the same as the minimum requirement for patenting (i.e., thelegal lower bound of the quality improvement for the grant), which is exogenously given as apolicy parameter. In other words, due to the corner solution, their model neglects the eﬀectof broadening leading breadth on the quality improvement size. Second, in their model, a
new innovator colludes with the licensors (i.e., the licensors do not produce their goods) tomaximize the proﬁt of their cartel in the Bertrand competition. Because a broader leading
breadth increases the quality gap between the new innovator and the nearest rival outside thecartel, the new innovator can set a higher markup. This results in the broadening of leading
breadth yielding a strong Schumpeterian eﬀect in their model.
Our paper aims to explore the negative aspect of broadening leading breadth. We builda modiﬁed O&Z-type model in which each R&D ﬁrm chooses the quality improvement size
as an interior solution. We ﬁnd that the growth eﬀect of broadening leading breadth can be
negative by shrinking the quality improvement size. Our result is opposite to O&Z because thegrowth eﬀect is still negative even when the discount rate is close to zero. It is well-known thathigh-income countries tend to have a lower discount rate (e.g., Wang et al., 2016; Falk et al.,2018). We contribute to the literature by solving the patent puzzle in developed countries, suchas the U.S. and Japan.
The fundamental mechanism of the negative growth eﬀect is the free-ride on the past innovators’ quality improvement. In the O&Z-type models, some successive ﬁrms in a quality ladderform the cartel to enhance their monopolistic power. Our model assumes that the R&D successprobability decreases in the quality improvement size as a realistic assumption. Then, when
the past innovators in the cartel had suﬃciently widened the quality gap between the nearest
rival ﬁrm, R&D ﬁrms will shrink the quality improvement size to join the cartel quickly.
Our model predicts that the broadening leading breadth reduces each patent’s contribution
to economic growth. This seems consistent with the aggregate data in the U.S. because the
3Hall and Ziedonis (2001) focus on the doctrine of equivalents by the Court of Appeals for the Federal Circuit(CAFC) established in 1982 in the U.S. The doctrine of equivalents is a judicial treatment that even a new invention
that does not coincide with the claims of past patents could be considered infringing on them if the new invention is
intrinsically equivalent to them. As the doctrine of equivalents has expanded the scope of the leading breadth to theoutside of the claims, an invention has the propensity to infringe on similar past patents. Sakakibara and Branstetter(2001) focus on the Japanese patent reform in 1988. This policy reform enabled patent applicants to include multiple
claims in the same patent application. As claims deﬁne the technological scope of the patent, subsequent inventions
are likely to infringe on past patents of similar inventions.

3

(a) Number of patent applications & grants
in the United States(b) TFP growth rate in the United States%

800,000

Total Patent Applications

700,000

Total Patent Grants

600,000

4
3
2

500,000

1

400,000

-3

2019

2015

2011

2007

2003

1999

1995

1991

1987

1983

1979

1975

1971

1967

1963

2019

2015

2011

2007

2003

1999

1995

1991

1987

1983

1979

-2

0

1975

-1

100,000

1971

200,000

1967

0
1963

300,000-4

Figure 1: (a) Number of patent applications and grants in the United States and (b) TFP growthrate in the United States.
Source: The patent data is from the U.S. patent statistics chart (1963-2020) from the U.S. patent andtrademark oﬃce. The TFP data is from the Federal Reserve Bank of San Francisco.

total factor productivity (TFP) growth rate in the U.S. has not increased over time, whereas thenumber of patent applications and grants has dramatically increased since the 1980s, as shownin Figure 1. Hall and Ziedonis (2001) focus on the U.S. policy reforms in the 1980s and ﬁnd
no signiﬁcant evidence of accelerating the pace of innovation in the semiconductor industry
where innovation is very cumulative. They attribute the change in patenting strategy of ﬁrms
to the patent explosion since the 1980s. In high-tech industries where technologies are closely
related to each other, ﬁrms must be licensed with many patents separately held by independent
owners to commercialize their own technology. In private negotiations of patent licensing,
the relative bargaining power of each patent holder matters (e.g., Sakakibara, 2010). Hall andZiedonis (2001) emphasizes that ﬁrms have an incentive to “harvest” more patents from their
R&D to improve their negotiating position. Their view suggests that the patent policy reforms
have lowered the average quality of patents because ﬁrms began to apply for patents on trivial
technologies.

1.1

Related literature

The recent literature on Schumpeterian growth theory has emphasized the role of patent licensing on economic growth (e.g., Chu, 2009; Chu et al., 2012; Chu and Pan, 2013; Niwa, 2016, 2018;
Yang, 2018; Suzuki, 2020; Kishimoto and Suzuki, 2021; Klein, 2022).
Our paper closely relates to Klein (2022) that quantitatively ﬁnds that the broadening leading
breadth reduces the growth rate. Although this result overlaps with ours, the speciﬁcation isquite diﬀerent because we deﬁne the leading breadth as the extent of intra-industry patent4

infringement as O&Z. In contrast to O&Z, Klein (2022) deﬁnes the leading breadth as the extent
of inter-industry patent infringement.4 Although we do not consider such inter-industry patent
infringement, we allow a case in which a new innovator infringes the latest and older patents in
the industry. Therefore, our model can complement Klein (2022) by showing that the negative
growth eﬀect of the broadening leading breadth can emerge in a diﬀerent speciﬁcation.
Furthermore, our paper relates to Chu (2009) that quantitatively evaluates how much reducing the backloading eﬀect (e.g., lowering the license fees) increases the R&D in the O&Z-typemodel. The author ﬁnds that reducing the backloading eﬀect signiﬁcantly increases the R&D.
However, it remains unclear how much such policies are possible because the license fees are
determined in private negotiations between the relevant ﬁrms and are not directly controllable
variables for outsiders. In reality, competition authorities or standard-setting organizations canintervene in the negotiations to lower the license fees (Section 5 details this point). We quantitatively analyze how such interventions aﬀect the growth rate by calibrating the model to the U.S.
economy. We ﬁnd that, while the growth eﬀect of the intervention is quantitatively small, it canstabilize the macroeconomy by eliminating the multiplicity of equilibria. Thus, we contributeto the literature by considering more speciﬁc policies and showing the macroeconomic eﬀects.

1.2

Roadmap

The structure of our paper is as follows. Section 2 builds a modiﬁed O&Z-type model in which
each R&D ﬁrm chooses the size of quality improvement as an interior solution. Section 3 solves
the long-run equilibrium and shows that the R&D ﬁrms may choose a small improvement size
to enjoy a free-ride on the quality improvements by the other innovators. Section 4 analyticallyshows that the broadening leading breadth may decrease the growth rate and discusses thesocially optimal innovation size. Section 5 calibrates the U.S. economy and quantitatively showsthat the broadening leading breadth decreases the growth rate. We also ﬁnd that intervention
in patent licensing negotiation has a growth-enhancing eﬀect. Section 6 concludes.

2

A baseline model

We consider a canonical model in the literature of leading breadth (e.g., O’donoghue andZweimüller, 2004; Chu, 2009). Following these studies, we incorporate the leading breadth ofpatent and proﬁt division between several innovators into the quality-ladder model of Grossman
and Helpman (1991, Ch.4).

4Klein (2022) assumes that an innovation infringes the patents of the leading ﬁrms in a fraction of ϕ ∈ (0, 1) of aunit mass of industries. The author quantitatively ﬁnds that an increasing ϕ (i.e., the broadening leading breadth)is growth-reducing.

5

2.1

Households

We consider the following closed economy in continuous time. The economy consists of acontinuum of identical inﬁnitely living households of measure L > 0. There is no population
growth. A representative household has the following intertemporal utility function:
Ut =

Z ∞
0

exp(−ρt) ln ct dt,(1)

where ρ is the subjective discount rate and ct is a consumption index of ﬁnal goods at time t.
The economy has a unit-continuum of industries indexed by h ∈ [0, 1], and the household
consumes ﬁnal goods across all industries. In each industry, there are some countable ﬁrms
that produce ﬁnal goods that are diﬀerentiated in terms of quality. Let N (h) be the set of theﬁrms producing goods in industry h. Firm k ∈ N (h) produces a ﬁnal good whose quality is
q k ( h ).
The instantaneous period utility is given by!

Z 1

ln ct =

∑ qk (h)xkt (h)

ln

0

dh,

(2)

k ∈N (h)where xkt (h) is the consumption of ﬁrm k’s good in industry h at time t. According to theadditive speciﬁcation in the abovementioned period utility, all goods in each industry are
perfect substitutes for households.
Let et be the household’s expenditure in time t. Then, it is given by
Z 1

et =

0

!

∑

pkt (h) xkt (h)dh,k ∈N (h)where pkt (h) is the price of ﬁrm k’s good in industry h at time t. Every household supplies
a unit of labor inelastically and earns a wage in each period. The budget constraint for eachperiod is given byȧt = rt at + wt − et ,where at is the value of assets (equities), rt is the interest rate, and wt is the wage rate.
We solve the utility maximization problem in two steps: static problem and dynamic problem. First, given instantaneous expenditure level et , the household maximizes the period
utility function ln ct . Under the logarithmic utility function, the household spends the budget
equally across h ∈ [0, 1]. Moreover, for each industry, the household chooses the good withthe lowest quality-adjusted price. Assume that ﬁrm k̃ (h) produces such a good in industry h.
Then, the individual demand in industry h at time t is xk̃t (h) = et /pk̃t (h) and xkt (h) = 0 fork ∈ N (h)\{k̃ (h)}.
6Second, we solve the dynamic maximization problem. Every household decides the expenditure in each period to maximize the intertemporal utility function, Ut , subject to the intertemporal budget constraint. Their indirect period utility function is given by ln ct = ln et − ln Pt ,where Pt is the aggregate price index associated with the consumption index ct , which is deﬁned
asln Pt ≡



Z 1
0

ln

pk̃t (h)qk̃ (h)dh.

Each household spends to maximize the intertemporal utility. From the intertemporal utility
maximization, the household’s optimal time path for spending is ėt /et = rt − ρ. The aggregate
expenditure is Et ≡ et L. As L is constant, Ėt /Et = rt − ρ also holds. We normalize the aggregate
price index at each time so that Pt = 1. Then, et = ct holds for all time. Therefore, the familiar
Euler equation ċt /ct = rt − ρ also holds.
The total demand for the good that has the quality k̃ in industry h is Xk̃t (h) ≡ xk̃t (h) L =Et /pk̃t (h). Hereafter, the notations omit t and h in cases where there is no risk of misunderstanding.

2.2

Industries

Consider an industry in which there are many generations of innovators.5 More recent innovators can produce higher quality goods than older innovators. Let “ﬁrm 0” be the latest innovatorat the time. Similarly, let “ﬁrm i” be the (i + 1)-th latest innovator. To distinguish between pastinnovators and future innovators from the perspective of ﬁrm 0, let ﬁrm −i be the i-th innovator
after ﬁrm 0 has emerged in the industry. For simplicity, ﬁrm i or ﬁrm −i is sometimes denotedby “i” or “−i.” Panel (a) of Figure 2 illustrates the industry at a certain time.
We assume that each ﬁrm has the same linear production technology. They can produce one
unit of their own good by devoting one unit of labor. However, their goods are diﬀerentiatedin terms of quality. The quality of ﬁrm i’s good is λi -times higher than that of ﬁrm i + 1’s good(i.e., qi = λi qi+1 ), where λi > 1 is the quality improvement size between ﬁrm i and ﬁrm i + 1.
As discussed later, the size λi is endogenously determined by solving the ﬁrm i’s maximization
problem.

Patent breadth
Each ﬁrm has a patent for its own good. We assume that the lagging breadth is perfect in thesense that ﬁrm i can claim patent infringement against any other ﬁrm that produces a good withquality in (qi+1 , qi ] (i.e., the range of the quality improved by ﬁrm i). For the leading breadth,
we assume that ﬁrm i can claim patent infringement against any other ﬁrm that produces a5More formally, we suppose that the cardinality |N (h)| is suﬃciently large for all h ∈ [0, 1].

7

(b) Patent license between 𝜂 + 1 firms(a) Quality ladderFirm 0Firm 1

Firm 𝜂
Firm 𝜂 + 1

Firm 1𝜆"Firm 2

Patentlicense≈Licensefee

Firm 2⋯

Firm 0infringes
theirpatents

Firm 0𝜆!

Firm 𝜂

𝜆#Figure 2: (a) Quality ladder and (b) Patent license
Notes: Firm 0 is the latest innovator in the industry. Although ﬁrm 0 has the patent for the highest quality
good in the industry, because of the leading breadth, ﬁrm 0 infringes the patents of ﬁrm 1, ﬁrm 2, ..., andﬁrm η. Therefore, to produce its own good, ﬁrm 0 must be licensed by all of them through the patent
license negotiation (Panel (b)). λi is the ﬁrm i’s quality improvement size (Panel (a)).

good with quality in (qi , qi−η ]. This means that ﬁrm i’s patent is certainly protected from thesubsequent η innovations that occurred by ﬁrm i − 1, ﬁrm i − 2, ..., and ﬁrm i − η. We interpretthe positive integer η as the degree of the leading breadth.
Our speciﬁcation of the leading breadth is mathematically tractable because the optimal
size of quality improvement (λi ) can be analytically solved as an interior solution. Conversely,other speciﬁcations make it extremely diﬃcult. For example, O&Z assumes that ﬁrm i’s patent is
protected from subsequent innovations because no other ﬁrm can produce goods with quality in(qi , Kqi ], where K > 1 is the parameter of the leading breadth. Unfortunately, their speciﬁcation
makes it almost impossible to analytically ﬁnd the optimal size of quality improvement foreach innovator as an interior solution (see the Online Appendix for the details). To avoid thecomplication, O&Z considers a corner solution in which all innovators make only the minimum
quality improvements required to obtain a patent (i.e., λi = P for all i), where P > 1 is given
exogenously as a policy parameter. By setting K = Pα where α ∈ N, they eventually use α as
the degree of the leading breadth instead of K, as η in our model.
By solving for the optimal λi as an interior solution, we can analyze the growth eﬀect of
the broadening leading breadth, including changes in the innovation size. O&Z neglects thischannel because the innovation size is always constant in their model. As shown later, thischannel is the main driver of the negative growth eﬀect in our model. The Online Appendix
demonstrates that our main result can continue to hold even when the scope of the leading
breadth of ﬁrm i is given by (qi , Kqi ] and no minimum quality improvement is required toobtain a patent.
8Bertrand competition
As illustrated in Panel (b) of Figure 2, due to the patent infringements, ﬁrm 0 cannot produce
the good without being licensed by ﬁrm 1, ﬁrm 2, ..., and ﬁrm η. For notational ease, we deﬁne

L0 ≡ {1, 2, . . . , η } as the set of the licensors of ﬁrm 0. If all ﬁrms in L0 license their own patents
to ﬁrm 0, we call their consortium “pool” and denote the set by P0 ≡ {0} ∪ L0 .
All ﬁrms in the industry engage in Bertrand competition. Recall that their goods are perfect
substitutes for all households. Therefore, ﬁrm 0 takes the limit-pricing strategy that charges aprice low enough to exclude the nearest rival. Following O&Z, we assume that the competition
authority allows a price cartel by P0 . Namely, all ﬁrms in L0 do not produce their goods toconsolidate the market power of ﬁrm 0. Then, the nearest competitor for ﬁrm 0 is ﬁrm η + 1. Byη
the accumulation of the quality improvements, q0 = ∏i=0 λi · qη +1 holds (∏ is the symbol ofthe product). Recall that the lagging breadth prevents ﬁrm η + 1 from producing a good withhigher quality than qη +1 . By the standard argument of the limit-pricing strategy, to exclude
ﬁrm η + 1, ﬁrm 0 sets the following price:

p0 =



q0

w
!

q η +1
η

∏ λi

=

w.

(3)

i =0

This is the result of the Bertrand competition.6
We assume that the competition authority prohibits the pool from including another ﬁrm(e.g., ﬁrm η + 1) in P0 because it does not entail any patent license to ﬁrm 0 and aims to increase
the markup. In other words, we assume that the competition authority does not allow theηmarkup strictly higher than ∏i=0 λi .
Let Π̃0 be the proﬁt of ﬁrm 0. By X0 = E/p0 and (3), the proﬁt becomesΠ̃0 = p0 X0 − wX0

=

2.3

1−

η

1

∏ i =0 λ i

!
E.

(4)

Proﬁt division in the pool

After the production, ﬁrm 0 shares Π̃0 with the ﬁrms in L0 as the payments of the license fees.
Let πi ≥ 0 be the net proﬁt of ﬁrm i ∈ P0 . We deﬁne s = (s0 , s1 , ..., sη ) as the proﬁt-sharing
6To understand why this result holds, let p0 and pη +1 be the price of ﬁrm 0’s good and the price of ﬁrm η + 1’s
good, respectively. Each household purchases a good with the lowest quality-adjusted price in the industry, as
all goods are perfect substitutes. Therefore, they purchase the ﬁrm 0’s good when q0 /p0 ≤ qη +1 /pη +1 . As the
production technology is linear, the lowest price that ﬁrm η + 1 can charge is the unit cost: pη +1 = w. Then, ﬁrm 0
can exclude ﬁrm η + 1 by charging (3).

9

η

η

vector where si ≡ πi /Π̃0 for all i ∈ P0 . We assume that ∑i=0 πi = Π̃0 holds. Then, ∑i=0 si = 1
also holds. Let S be the set of the feasible proﬁt-sharing vectors. More formally,S≡s ∈ [0, 1]

1+ η

η



∑ si = 1 .

i =0One inevitable problem when we discuss the proﬁt division is how we consider the determination process of the proﬁt-sharing vector. O&Z and other papers in the literature on the
Schumpeterian growth theory have not formally considered how ﬁrms negotiate, and basically,
they treat the license fees as exogenous for simplicity.7
We also treat the proﬁt-sharing vector as exogenous in the baseline model for simplicity.
However, in Appendix B, we formally consider how the ﬁrms in the pool share Π̃0 by employing
a game-theoretic framework. In the literature on game theory, several studies formulate license
negotiations as cooperative games (e.g., Tauman and Watanabe, 2007; Watanabe and Muto,
2008; Kishimoto and Watanabe, 2017) and solve the license fees by applying the solution of thegame. Appendix B will connect two literature streams that have been independently developed.

2.4

R&D

The potential ﬁrms can enter freely into the R&D.8 The R&D function depends on the size ofthe quality improvement and the number of employing researchers. We assume that if an R&D
ﬁrm chooses λ > 1 as the improvement size and employs zt researchers in the time interval dt,
the successful invention of the next-generation good happens with the probability (θzt /λ β )dt,
where β > 1 represents the diﬃculty of a large-size innovation. Note that two or more ﬁrms
never simultaneously succeed in their R&D because the probability of such an event is zero in
the continuous-time models.9 The R&D cost is (wt zt )dt, ﬁnanced by issuing stocks.
Consider an industry where the leading ﬁrm on the ladder is ﬁrm 1. Then, if an R&D ﬁrm
succeeds in inventing a superior good, it emerges in the industry as ﬁrm 0. Recall that λi is thequality improvement size of ﬁrm i. To avoid the complexity, we assume that the R&D ﬁrms
take the sequence of the future quality improvements, {λ−i }i∞=1 , as given.10
We deﬁne Vi (λ0 ) as the present discounted value of ﬁrm 0 when the subsequent innova7Kishimoto and Suzuki (2021) is an exception. They consider cooperative games that determine the license fee
to describe how license fees are endogenously determined in the negotiation. A new innovator is the licensor andcan earn the licensing revenue. In contrast, the present paper considers a diﬀerent situation where a new innovatoris the licensee and pays the license fees to other ﬁrms.
8All ﬁrms in the pool do not conduct R&D due to the Arrow’s replacement eﬀect. Their ﬁrm value is strictly
positive. Therefore, their incentive to innovate is strictly weaker than the potential innovators with zero value.
9When we pick up two real numbers from a subinterval in R at random, the probability that their value is thesame is zero.
10Without this assumption, we must consider a strategic interaction between the current innovators and the future
innovators because λ0 may aﬀect the expected survival time of ﬁrm 0 via the changes of {λ−i }i∞=1 . The analyticalderivation of the optimal improvement size (λ0 ) will be impossible.

10tions have occurred i times after ﬁrm 0 entered the industry. The ﬁrm value is a function of
λ0 because the proﬁt in (4) depends on it. Vi (λ0 ) changes to Vi+1 (λ0 ) when the (i + 1)-thsubsequent innovation occurs after ﬁrm 0 entered the industry. It happens with the probabilityof θz/(λ−i−1 ) β .
The optimization problem for each R&D ﬁrm is represented asθzmaxz≥0,λ0 >1 λ β0V0 (λ0 ) − wz,(5)where V0 (λ0 ) is the present discounted value of ﬁrm 0 when it succeeds in R&D. The objective function shows that, for λ0 , a trade-oﬀ exists between the value of R&D and the successprobability. A larger λ0 increases the ﬁrm value but decreases the success probability.
The linearity of R&D technology implies that the demand for researchers is unboundedβ

above (i.e., z → ∞) whenever (θ/λ0 )V0 (λ0 ) > w holds. However, it cannot happen in the labormarket equilibrium. Therefore, the free-entry condition for R&D is given byθβ

λ0

V0 (λ0 ) ≤ w.

(6)

If strict inequality holds in (6), then z = 0 holds because the R&D is not proﬁtable. A positive
and ﬁnite demand for researchers (i.e., z ∈ (0, ∞)) arises only if equality holds in (6). We
focus on the latter case. Then, (6) works as the zero-proﬁt condition for R&D. Although z isindeterminate in the optimization, it will be determined by the clearing condition of the labormarket.

3Long-run equilibria

In the decentralized equilibrium, all individuals solve their maximization problem, and all
markets are clear at each period. We focus on a situation in which equality in the free-entry
condition holds. We consider the following symmetric stationary situation:A) the proﬁt division s = (s0 , s1 , ..., sη ) is constant over time;B) any successful potential ﬁrm chooses the same size of the quality improvement (i.e.,
λi = λ∗ for all i);C) all variables are symmetric across industries.

11

3.1

Labor market equilibrium

The ﬁxed labor supply, L > 0, is devoted to production and R&D. Then, by the symmetry ofindustries, the labor market equilibrium condition at each time isE
λ 1+ η w

3.2+ z = L.

(7)

Net proﬁt stream

Let us consider the proﬁt stream of ﬁrm 0. Suppose the subsequent innovations have occurredi times after ﬁrm 0 entered the industry. Then, ﬁrm −i is the latest ﬁrm in the industry, andthe set of the ﬁrms in the pool is P−i = {−i, −i + 1, ..., 0, 1, ..., η − i }. By the argument of thelimit-pricing strategy, ﬁrm −i charges the price as follows:η −i

p −i =

!

∏ λℓ

−1

∏ λk

· λ0 ·

!
w.

k =−i

ℓ=1

Then, the proﬁt of ﬁrm −i isΠ̃−i =

1−

!

1
η −i

1
∏ℓ=1 λℓ · λ0 · ∏−k=−i λkE.
η

By the stationary of s, the stream of the net proﬁt of ﬁrm 0 is {πi }i=0 where
πi = si Π̃−i .

(8)Note that πi = 0 for all i > η because ﬁrm 0 is no longer a licensor of the latest ﬁrm at that time
when the subsequent innovations have occurred η + 1 times after ﬁrm 0 entered the industry.

3.3

No-arbitrage condition

Consider a household with a stock of ﬁrm 0 whose value is Vi (λ0 ). In time interval dt, thehousehold obtains the dividend πi dt. Vi (λ0 ) changes to Vi+1 (λ0 ) with the probability of(θz/(λ−i−1 ) β )dt. We assume that there is a perfectly risk-free asset market, and the interest rate on the safe assets is r. Let V̇i (λ0 )dt be the capital gain (or loss) of the stock. Then,standard arguments imply that the ﬁrm 0’s value satisﬁes the following no-arbitrage condition:rVi (λ0 ) = πi + V̇i (λ0 ) −θz( λ − i −1 ) β

for i = 0, ..., η. Note that Vi (λ0 ) = 0 for all i > η.

12

(Vi (λ0 ) − Vi+1 (λ0 )),

(9)

3.4

The balanced growth path

In the balanced growth path (BGP), the variables { At , wt , Et , Vi,t } grow at the same constant
speed for all i = 0, ..., η. Let the growth rate be g. Then, by the Euler equation, g = r − ρ holds
in the BGP. By using V̇i /Vi = g, the no-arbitrage condition (9) is rewritten as
ρVi (λ0 ) = πi −θz

( λ − i −1 ) β(Vi (λ0 ) − Vi+1 (λ0 )),(10)

for all t and i = 0, ..., η. By recursively solving the no-arbitrage conditions in (10), we can derive
V0 (λ0 ) as follows:V0 (λ0 ) =1
ρ + θz/(λ−1 ) β"ηπ0 + ∑i =1−1

πi · ∏ ψk

!#
,

(11)

k =−i

where ψk is a discount factor deﬁned as
β

ψk ≡

θz/λk

β

ρ + θz/λk−1.

By the equality in (6), the optimal z is indeterminate in (5). Then, the problem in (5) can be
simply rewritten as follows:
max
λ0 >1

V0 (λ0 )
β

λ0

.

(12)

By solving (12) and using the symmetry (λi = λ∗ for all i), we obtain the following Lemma.
Lemma 1. In the symmetric equilibrium, the optimal quality improvement size is given by
∗

λ =



1
1+
β

1/(1+η )(13)

for any s ∈ S .
Proof. See Appendix A.1.
Lemma 1 shows that the broadening patent breadth (η ↑) shrinks the optimal size of thequality improvement. Intuitively, the R&D ﬁrms have an incentive of free-ride on the quality
improvements by the other innovators because the ﬁrm value depends on them, not only on λ0 .
To conﬁrm this, let us derive the Bertrand equilibrium price. By (3) and (13), in the symmetricequilibrium, the price becomes1+ η

p = λ∗ w


1
1+
w.
=
β
13

This shows that the markup does not depend on η.
By (4) and (13), the proﬁt of the pool is

Π̃i =

1
1+β


E ≡ Π̃ for all i.

(14)

Using (8), (13), and (14), we can rewrite (11) as follows:
V0
=
E



1
1+β



Ψ(z)
ρ + θz/λ∗ β


,

(15)

where function Ψ(z) is deﬁned as


i



 θz/λ∗ β  .
Ψ ( z ) ≡ s0 + ∑ s i 
∗β 
ρ
+
θz/λ

i =1
|
{z
}
η

(16)

≡ψ∗ (z)<1

Using (16), we characterize each proﬁt-sharing vector in S as follows:
Deﬁnition 1. Consider any two diﬀerent proﬁt-sharing vectors s′ , s′′ ∈ S . Holding ψ∗ (z) constant,
if the change from s′ to s′′ increases (decreases) Ψ(z), then s′′ is more frontloaded (backloaded) than s′under ψ∗ (z).
Intuitively, in a more frontloaded vector (e.g., s0 is higher), ﬁrm 0 can receive the reward of
the innovation in early timing. In contrast, in a more backloaded vector, ﬁrm 0 must wait for thearrival of the subsequent innovations for a long time to receive the reward of the innovation.
The most frontloaded vector is sF ≡ (1, 0, ..., 0) because it maximizes Ψ(z) for any given ψ∗ (z).
Similarly, the most backloaded vector is sB ≡ (0, 0, ..., 1) because it minimizes Ψ(z) for any givenψ ∗ ( z ).
When the proﬁt-sharing vector is suﬃciently frontloaded, the graph of (15) is a downwardsloping curve as shown “FV” in Panel (a) of Figure 3. However, when the proﬁt-sharing vector
is suﬃciently backloaded, the FV curve becomes an inverted-U shape as shown in Panel (b) of
Figure 3.11
By using (6), (7), and (13), we obtain
V0
λ∗ β
=
.
E
θ (1 + 1/β)( L − z)(17)

The graph of (17) is shown as the upward-sloping “LME” curve in Figure 3. Each intersectionof two curves is a long-run equilibrium in the model. In this general equilibrium model, only
the ratio of V0 and E is determined.
11Appendix 1 in O&Z also discusses a similar point.

14

𝑉!
𝐸

(a) Unique equilibrium(b) Multiple equilibria

𝑉!
𝐸

𝐿𝑀𝐸

𝐸

𝐿𝑀𝐸

𝐸#

𝐸$
𝐹𝑉

𝑧∗

𝐹𝑉
𝑧

𝑧$∗

𝐿

𝑧#∗

𝑧
𝐿

Figure 3: The long-run equilibria.
Notes: An intersection of the LME and FV curves is a long-run equilibrium. Panel (a) is a case of uniqueequilibrium, and Panel (b) is a case of multiple equilibria. When the proﬁt-sharing vector is suﬃciently
backloaded, the economy may have multiple equilibria as Panel (b).

The long-run equilibrium is unique if the FV curve is higher than the LME curve at z = 0.
By (15) and (17), we obtain a suﬃcient condition for the uniqueness of the long-run equilibrium
as follows:
s0 >

βρ
θL


1+

1
β

 β/(1+η ).

In other words, the long-run equilibrium is unique when the proﬁt-sharing vector is suﬃciently
frontloaded. Then, the phase diagram in Panel (a) of Figure 3 is almost the same as the one
in Grossman and Helpman (1991, Ch.4). The economy must lie on the LME curve. As the FV
curve is derived by the no-arbitrage and BGP conditions, V0 and E grow at g on the FV curve.
V̇0 /V0 < g = Ė/E holds below the FV curve and V̇0 /V0 > g = Ė/E holds above the FV curve.
If the economy is not on the FV curve, V0 /E changes over time. Therefore, point E in Panel(a) of Figure 3 is the unstable equilibrium. The unique equilibrium path is that the economy
immediately jumps to point E at t = 0 and remains there forever.12 Therefore, the economy has
no transitional dynamics when the proﬁt-sharing vector is suﬃciently frontloaded.
The economy may have multiple equilibria when the proﬁt-sharing vector is suﬃciently
backloaded. In this case, the expectation determines which equilibrium is realized. To understand the role of the expectation in the leading breadth model, consider a backloading
proﬁt-sharing vector. Then, to earn large net proﬁts, ﬁrm 0 must wait for the arrival of themany subsequent innovations. Suppose the individuals have optimistic expectations such thatthey predict the speed of the technological advances is fast (i.e., z is high). The optimistic
12As shown in Grossman and Helpman (1991, Ch.4), other paths do not satisfy the transversality condition or
rational expectations.

15

expectations increase the present value of innovation, V0 , because the waiting intervals of ﬁrm
0 become shorter. As a result, z rises; therefore, the initial optimistic expectations are selffulﬁlled. In contrast, when individuals have pessimistic expectations about z, the present valueof innovation is small, as the discounted sum of future proﬁts is small. Then, the incentive toinnovate weakens, and z truly falls. The initial pessimistic expectations are self-fulﬁlled in thiscase as well.
Under the multiple equilibria, the equilibrium path is not unique because all variables arejumpable in the model. Any equilibrium path can be selected depending on the expectation. To
avoid the complexity, we focus on only the equilibrium paths that the economy initially jumps
to either one of EL and EH and remains at the same point forever. O&Z conducts an equilibrium
selection such that it neglects EL . However, because the expectation is an essential macroeconomic feature generated by blocking patents, we do not neglect the pessimistic equilibrium
paths. This enables us to investigate the eﬀect on macroeconomic stability in Section 5.

4

Analytical discussion

First, this section investigates how the broadening leading breadth (η ↑) aﬀects the growth rate.
Second, we derive the socially optimal innovation size and resource allocation. Finally, weinvestigate the welfare eﬀect of the broadening leading breadth.

4.1

Growth eﬀect of the broadening leading breadth

In the BGP, the consumption index ct also grows at g. The standard argument implies that, by(2), the growth rate is calculated as
g=

ċ
=
c

θz∗
λ∗ β
|{z}
R&D
success rate×∗ln
| {zλ}.

(18)

Quality
improvementsize

We can decompose the growth eﬀect of the broadening leading breadth into the following threeeﬀects. First, the broadening leading breadth shrinks λ∗ by (13). This decreases the growth rate
via ln λ∗ in (18). Second, the shrinking of λ∗ raises the R&D success rate because the diﬃcultyof quality improvement is lowered. This increases the growth rate via λ∗ in the denominator of(18). Third, the broadening leading breadth changes z∗ which is determined by (15) and (17).
However, this channel is very complex because it is unclear how the FV curve moves.
Before the analyses, we impose the following assumption on the parameters:

16

Assumption 1. The discount rate is suﬃciently small such that


θL
β



β
1+β

 β/2

> ρ.

(19)

This parameter condition ensures that the growth rate in the decentralized economy is always
positive.13 Intuitively, if the discount rate is very large, the R&D investment is not proﬁtablebecause the present value V0 is very low. To build an innovation-driven growth model, we
assume that the discount rate is suﬃciently small.
We begin to analytically investigate the growth eﬀect by establishing some Lemmas. First,
we consider a special case in which the proﬁt-sharing vector is the most frontloaded (s = sF ). Of
course, this proﬁt-sharing vector is unrealistic because the licensors would not be incentivized
to license their patents without license fees. However, this extreme case helps us investigate thegrowth eﬀect in more general cases.
Lemma 2. Suppose that the proﬁt-sharing vector is the most frontloaded (s = sF ). Then,(i) the growth eﬀect of the broadening leading breadth is not always positive. It is either one of thefollowing two eﬀects depending on the discount rate;• a non-monotonic growth eﬀect (i.e., g and η are represented as an inverted U-shaped curve asshown in Figure 4) when the discount rate is moderate;• a negative growth eﬀect for all η ≥ 1 when the discount rate is suﬃciently small.
(ii) the growth rate converges to zero as η → ∞ regardless of the discount rate.
Proof. See Appendix A.2.
To investigate the growth eﬀect in more general cases, we establish the following Lemma.
Lemma 3. Let g : S → R be the function that assigns s ∈ S to the growth rate in the BGP. Then,g(sF ) > g(s) holds for any s ̸= sF and η ≥ 1 because the most frontloaded proﬁt-sharing vector (sF )maximizes z∗ .
Proof. By (16), the maximum value of Ψ(z) is 1, as ψ∗ < 1. Note that Ψ(z) = 1 holds only whens = sF and Ψ(z) < 1 holds when s ̸= sF . Therefore, by (15), the most frontloaded proﬁt-sharing
vector pushes the graph of the FV curve in Figure 3 up to the highest position. Then, s = sF
maximizes z∗ for all η ≥ 1, as the LME curve does not depend on s ∈ S . By Lemma 1, λ∗ isindependent of s ∈ S . As a result, we obtain the statement.
13Strictly speaking, this condition ensures that the growth rate is positive in the case of s = sF . As proved in
Lemma 3, any other s ∈ S yields a lower growth rate compared to sF . Therefore, if the growth rate is non-positive
when s = sF , then any s ∈ S cannot yield a positive growth rate. Assumption 1 avoids such an uninteresting
situation.

17

𝑔𝐬 = 𝐬𝐅𝐬 ≠ 𝐬𝐅
𝜂
1

2

3

4

Figure 4: The growth rate (g) and the leading breadth (η)
Notes: Lemma 2 shows that g is not monotonically increasing in η and converges to zero as η → ∞when s = sF . Lemma 3 shows that g is maximized when s = sF . Therefore, for any s ∈ S , g is not
monotonically increasing in η and converges to zero as η → ∞.

Then, Lemmas 2 and 3 derive the following statement.
Proposition 1. Consider a proﬁt-sharing vector s ∈ S . Then, by Lemma 2, the growth eﬀect of thebroadening leading breadth is not globally positive. The broadening leading breadth has either (i) anon-monotonic growth eﬀect or (ii) a negative growth eﬀect. Also, the growth rate converges to zero asη → ∞.
Proof. By Lemma 2, when s = sF , the relationship between g and η is either a non-monotonic(an inverted U-shaped curve) or a negative (a downward-sloping curve). Moreover, the growthrate converges to zero as η → ∞ when s = sF . Recall that Lemma 3 shows that s ̸= sF yields alower growth rate than sF . Then, the relationship between g and η cannot be globally positive(an upward-sloping curve) when s ̸= sF . Otherwise, it contradicts Lemma 3. As a result, thestatement in Proposition 1 holds.
Proposition 1 shows that the broadening leading breadth does not always increase thegrowth rate. Although the relationship between g and η is generally ambiguous, we can show
that, in a special case of ρ → 0, we can establish the following statement.
Proposition 2. For any s ∈ S and η ≥ 1, the growth eﬀect is always negative when ρ → 0.
Proof. Assume that ρ → 0. By (15) and (17), we can analytically solve the long-run equilibrium
as z∗ = L/(1 + β). As this is constant, the broadening leading breadth aﬀects the growth eﬀect
only via λ∗ . Note that (ln λ∗ )/λ∗ β is increasing in λ∗ ∈ (1, λ̄] where λ̄ ≡ (1 + 1/β)1/2 is the
maximum size of the quality improvement derived by (13). Therefore, we obtain the result as
the broadening leading breadth decreases λ∗ .

18This result is opposite to O&Z because they show that the growth eﬀect is positive when
the discount rate is suﬃciently small. In their model, the quality improvement size is constant,and the broadening leading breadth can unlimitedly increase the proﬁt of the pool. This is thedriver of the growth-enhancing eﬀect in their model. Recall that the backloading eﬀect, which
is a negative growth eﬀect, is weaker as ρ is smaller. Then, when the discount rate is suﬃciently
small, the former positive eﬀect dominates the backloading eﬀect.
In contrast, our model with endogenous quality improvement does not have the growthenhancing eﬀect via increased proﬁt. Instead, we show that the free-ride behavior of the R&D
ﬁrms (i.e., the shrinking of the quality improvement size) yields a new negative growth eﬀect.
This negative growth eﬀect does not depend on the discount rate. Although there is a positive
growth eﬀect via the R&D success probability, the negative growth eﬀect dominates it.
The Online Appendix demonstrates that this negative growth eﬀect can be replicated inanother speciﬁcation of the leading breadth in which the scope does not depend on η. We
formulate an overlapping generations (OLG) game between the R&D ﬁrms and show that theirfree-ride behavior can be rationalized as a Nash equilibrium in a special case.

4.2

Socially optimal innovation size and labor allocation

In this subsection, we investigate (i) whether the quality improvement size in the decentralized
economy (13) is large or small from the perspective of welfare and (ii) whether the resource
allocation in the decentralized equilibrium is socially optimal or not.
To address this problem, we ﬁrst derive the ﬁrst-best allocation in the economy. We evaluate
the welfare of an economy that is in a steady state at t = 0 and stays there forever. By the utility
function (2) and the labor market equilibrium condition (7), the representative household’s
period utility is rewritten as follows:
ln ct = g · t + ln( L − z) − ln L.
Note that ( L − z) equals the aggregate output, as the production technology is one-to-one.
Then, ( L − z)/L is per capita consumption in the equilibrium. By using this and integrating
the lifetime utility function (1) with respect to time, we obtain the representative household’swelfare as follows:
U=




1
 1 θz ln λ + ln( L − z) − ln L .
β{z}ρ ρ | λ{z } |(20)

Consumption

Growth

A social planner maximizes (20) with respect to λ and z. Let (λ# , z# ) be the solution. We assume
that the interior solution exists (i.e., λ# > 1, z# > 0).14 Then, by diﬀerentiating (20) with respect
14We consider an innovation-driven growth economy. Therefore, we avoid a trivial case in which a no-growth19

to λ and z, we obtain the welfare-maximizing pair of the quality improvement size and thenumber of researchers as follows: 1
λ = exp
,
β
 
ρβ
#
z = L−
exp(1).
θ#(21)

Then, the following result holds.
Proposition 3. In the decentralized economy,(i) λ∗ < λ# holds: the quality improvement size is smaller than the socially optimal size;(ii) z∗ is either larger or smaller than the socially optimal level z# .

Proof. See Appendix A.3.
The ﬁrst statement comes from the free-ride behavior of potential ﬁrms. From the perspective of welfare maximization, each potential ﬁrm should invest in R&D that will signiﬁcantly
improve quality. However, they are not incentivized to do that because the success probabilityof such a large R&D investment is very low. Under the leading breadth, each potential ﬁrm canjoin the pool in the industry even if the quality improvement size is small.
The second statement comes from the standard externalities in the model. First, eachpotential ﬁrm does not consider the positive externality of innovation on the households.
Second, each potential ﬁrm does not care about the capital loss of the existing ﬁrms (i.e., a proﬁt
destruction eﬀect). However, in addition to these normal externalities, the innovation size inthe decentralized economy also yields the gap between z∗ and z# . As λ∗ < λ# , the R&D successprobability is relatively high in the decentralized economy. It naturally works to increase z∗ .

4.3

Welfare eﬀect of the broadening leading breadth

How does the broadening leading breadth aﬀect welfare? By (20), we can decompose thewelfare eﬀect into two channels: (i) via growth and (ii) via consumption. The analysis of thewelfare eﬀect is more complex than one of the growth eﬀect because a higher growth rate does
not mean that the welfare is also high.
Although we will quantitatively show that the welfare eﬀect can be negative under plausible
parameters in the next section, we can analytically show the negative welfare eﬀect in the special
cases as follows:
economy is socially optimal by assuming that z# > 0. The parameter assumption for z# > 0 is stricter than one inAssumption 1 because exp(2) > (1 + 1/β) β holds for all β > 0. Therefore, the assumption for z# > 0 derives theparameter condition in Assumption 1.

20

Proposition 4. For any s ∈ S and η ≥ 1, the welfare eﬀect of the broadening leading breadth is negative
when ρ → 0. In the case of s = sF , the welfare eﬀect is not globally positive. In particular, there exists η̃such that the welfare eﬀect is negative in η ≥ η̃ + 1.
Proof. Proposition 2 shows that, for any s ∈ S and η ≥ 1, the growth eﬀect is always negative,
but z∗ is independent of η. Therefore, by (20), the welfare eﬀect is always negative. In the caseof s = sF , we showed the existence of η̃ such that the growth eﬀect is negative in η ≥ η̃ + 1. Asz∗ is increasing in η in the case of s = sF , the welfare eﬀect is negative in η ≥ η̃ + 1.

5

Further analyses: a quantitative illustration

5.1

Two theoretically grounded proﬁt-sharing vectors

This section quantitatively investigates the growth eﬀect of the broadening leading breadthby calibrating the U.S. economy. So far, we have treated a proﬁt-sharing vector as exogenous.
Although this simpliﬁcation is commonly used in the O&Z-type models, in the quantitative
analysis, we need to specify the proﬁt-sharing vector. Because we cannot directly observe the
proﬁt-sharing vector in reality, it is diﬃcult to ﬁnd the plausible proﬁt-sharing vector.
As a benchmark, we use the most backloaded vector and the equal-sharing vector deﬁned
as follows:


s ≡
E

sB ≡ (0, 0, ..., 1) .
1
1
1,
, ...,
1+η 1+η
1+η.

We can provide the theoretical basis for adopting these two proﬁt-sharing vectors by cooperative game theory. In Appendix B, we formulate a game of pool formation and proﬁt division
between the ﬁrms in an industry and derive sB and sE as two diﬀerent solution concepts.
First, Appendix B shows that sB is uniquely given by applying the core to the game. In
cooperative game theory, the core is widely used as a standard solution concept. The core isdeﬁned based on the incentives for negotiating ﬁrms to gain more proﬁts. We regard a proﬁt
allocation in the core representing license fees determined through negotiation of the ﬁrms in
the industry without any intervention from the outside.
Although sB is highly unfair to ﬁrm 0, it seems a natural result that the patent licensing
negotiation yields an exploitative licensing fee when the bargaining power of the licensee isvery weak. Sakakibara (2010) empirically shows that the relative bargaining powers betweenﬁrms signiﬁcantly aﬀect the license fees determined in their private negotiation. Patent holders
have the right to refuse the request for the license because the patents are the private property of21

each of them. Thus, the relative bargaining power of the patent holder tends to be high, and thelicense fees that the new innovator pays to the past innovators would be expensive. For example,
the Federal Trade Commission in the United States has recently insisted that Qualcomm has
illegally charged an excessive license fee in the market for smartphone chips.
However, in reality, standard-setting organizations (SSOs) intervene in the license negotiation to prevent the patent holders from refusing the license and charging an unfairly high licensefee. SSOs often request ﬁrms holding standard-essential industry patents to license under thefair, reasonable, and non-discriminatory (FRAND) conditions. We use sB as the benchmark resultof the private negotiation without any intervention.
Second, Appendix B shows that we have sE by applying the Shapley value to the game. The
Shapley value is a normative solution concept introduced by Shapley (1953). In the literature,
Layne-Farrar et al. (2007) and Dewatripont and Legros (2013) use the Shapley value as abenchmark for FRAND conditions in SSOs. We interpret the equal sharing vector sE as theresult of a negotiation with the intervention that aims to correct unfair license fees.
Specifying two proﬁt-sharing vectors enables us to investigate the growth eﬀect of theintervention in the license negotiation.

5.2

The long-run equilibria with and without intervention

Intervention case (sB )Suppose that the license negotiation is not intervened. Then, the oldest ﬁrm in the pool obtains
the entire proﬁt as the license fee. This means ﬁrm 0 can obtain the license fee after thesubsequent innovations occur η times. Then, (16) becomesΨ(z) =

θz/λ∗ βρ + θz/λ∗ βη.

By using this and (15), the FV curve becomes as follows:
V0∗
=
E



1
1+β



θz/λ∗ βηρ + θz/λ∗ β 1+ η .

(22)

Therefore, the FV curve is an inverted-U shape as shown in Panel (b) Figure 3. As a result,multiple equilibria can emerge under no intervention.

22

No-intervention case (sE )Next, we assume that the license negotiation is intervened. Then, each ﬁrm in the pool equallyobtains the same payoﬀ following the Shapley value. In this case, (16) becomesi #θz/λ∗ β1+∑
Ψ(z) =∗βi =1 ρ + θz/λ" 1+ η #1
θz/λ∗ β=1−.
1+η
ρ + θz/λ∗ β


1
1+η

"

η



By using this and (15), the FV curve becomes as follows:
V0∗
1
=
E
ρ



1
1+η



1
1+β

"


1−

θz/λ∗ βρ + θz/λ∗ β 1+ η #.

(23)

This is decreasing in z. Therefore, the FV curve is downward-sloping as shown in Panel (a)Figure 3. As a result, the long-run equilibrium is unique under intervention. The discussion
yields the following statement:
Proposition 5. Consider the intervention (sB → sE ) described in Deﬁnition 5 in Appendix B. Theintervention in the licensing negotiation stabilizes the economy.

5.3

A quantitative analyses

We quantitatively investigate whether patent protection against future innovation enhances
economic growth. As the benchmark scenario, we consider a situation in which the license
negotiation is intervened, and therefore, ﬁrm 0 shares its proﬁt with other ﬁrms according to
the Shapley value.
We must set the structural ﬁve parameters (η, ρ, L, θ, β). For η, we use the fragmentationindex used by Ziedonis (2004) and Entezarkheir (2017). This measure is deﬁned as
J

Fi = 1 − ∑

j =1



citeijcitei

2, i ̸= jwhere citeij is the number of citations made by ﬁrm i in its patent documents to the patents ofﬁrm j and citei is the count of all the citations made by ﬁrm i to other ﬁrms’ patents. Fi = 0
holds if every citation is to the patents of one ﬁrm. Let us assume that ﬁrm i equally cites
the other ﬁrms’ patents (i.e., citeij = c holds for all j). Then, Jc = citei holds. Entezarkheir(2017) reports that the mean of the index is 0.7. Then, we obtain J ≃ 3.33. This means that,on average, a patent cites three diﬀerent ﬁrms’ patents. The number of citations measures how
many other patents are technologically close to the patent. Therefore, we adopt η = 3. For the23

With intervention(sE )No intervention
(sB )

λ∗
g
ι
z
gH
ιH
zH
gL
ιL
zL
g

Baseline
η=3
1.057
2.00%
0.359
0.06321.93%
0.346
0.06090.025%
0.005
0.0008
0.98%

η=4
1.046
1.66%
0.372
0.06271.58%
0.354
0.05970.034%
0.008
0.0013
0.81%

η=5
1.038
1.41%
0.380
0.06211.33%
0.357
0.05840.042%
0.011
0.0018
0.69%

η=6
1.032
1.23%
0.385
0.06151.14%
0.357
0.05700.049%
0.015
0.0024
0.59%

η=7
1.028
1.08%
0.387
0.06100.99%
0.354
0.05570.055%
0.020
0.0031
0.52%

η=8
1.025
0.96%
0.388
0.06040.86%
0.349
0.0542
0.061%
0.025
0.00380.46%

Table 1: The growth eﬀect of broadening leading breadth (η ↑).
Notes: As shown in Lemma 1, λ∗ does not depend on s ∈ S . The long-run equilibrium is unique when s = sE
but two equilibria emerge when s = sB . The subscript H means the optimistic equilibrium, while L means thepessimistic equilibrium. ι m ≡ θzm /λ β is the R&D success probability in each equilibrium (m = H, L). g is theaverage growth rate deﬁned as g ≡ ( g H + g L )/2.

discount rate, we use a conventional value ρ = 0.01. We set L = 0.334, as the U.S. populationis around 0.344 billion. According to Norrbin (1993) and Basu (1996), the empirically plausible
range of markup of price over marginal cost is in [1.05, 1.4]. From (3), the markup in the modelis (λ∗ )1+η = 1 + 1/β. We set β = 4 so that the markup becomes a moderate value in the range(1 + 1/β = 1.25). Finally, we set θ = 7.1 so that g ≃ 0.02, which is the average growth rate of
the U.S. economy.
Table 1 is the result. The broadening leading breadth decreases the growth rate when thenegotiation is intervened (s = sE ). The negative growth eﬀect comes from a reduction in λ∗and z∗ . By (18), if we hold z∗ constant, there is an inverted-U relationship between the growthrate and λ∗ , and the growth rate is maximized at λ∗ = exp(1/β). In our numerical example,
exp(1/β) ≃ 1.284 is larger than λ∗ reported in Table 1. This means that the reduction of λ∗works to decrease the growth rate in the numerical example. In addition, by (18), the reductionin z∗ also decreases the growth rate. As η is larger, the share 1/(1 + η ) in sE decreases while
the lifetime of ﬁrm 0 increases. Because the future revenues are discounted, the broadening
leading breadth also decreases the present value of R&D in the intervention case. As a result,
the FV curve (23) shifts downward, and the innovation rate z decreases. Hence, the broadening
leading breadth discourages the incentive for R&D.
When the negotiation is not intervened (s = sB ), the growth eﬀect diﬀers between thetwo equilibria. The broadening leading breadth decreases the growth rate in the optimistic
equilibrium (g H ) but increases the growth rate in the pessimistic equilibrium (g L ). The average
growth rate, deﬁned as ( g H + g L )/2, decreases in η. Therefore, it seems reasonable to consider
that, even with no intervention, the broadening leading breadth would have a negative growth24

eﬀect. The intuition of the negative growth eﬀect is the same as in the intervention case. Whens = sB , as η is larger, ﬁrm 0 must wait a long time (i.e., η times of future innovations) to obtain thelicense fee Π̃. This decreases the present value of R&D and shifts the FV curve (22) downward.
As a result, the broadening leading breadth decreases the innovation rate z H . In other words,
the broadening leading breadth enhances the backloading eﬀect in both cases.

6

Conclusion

This paper investigated how broadening leading breadth aﬀects economic growth in a Schumpeterian growth model with endogenous quality improvement. In our model, the optimal sizeof the quality improvement is determined as an interior solution, unlike the previous studies. We found that the growth eﬀect of broadening leading breadth is not always positive. In
particular, an extremely broadening leading breadth is harmful to growth.
The fundamental mechanism of the negative growth eﬀect is the free-ride behaviors ofinnovators. In previous studies, the increase in the number of ﬁrms in the price cartel increases
the markup because the quality improvement is always constant. The Schumpeterian eﬀect
was the main driver of the positive growth eﬀect. However, this channel vanishes in our model
because the R&D ﬁrms shrink their sizes of quality improvement. In other words, the R&D
ﬁrms can receive stronger positive externalities from many future innovators. Therefore, when
λ is endogenous, and the potential ﬁrms would like to free-ride, the broadening leading breadth
may decrease λ because of the strong positive externalities.
We also found that intervention in patent licensing negotiation stimulates innovation. Without any intervention, the patent license fee tends to be high because the negotiation power ofthe licensor is strong. We formulated a cooperative game played by all ﬁrms on a quality ladderand showed that the outcome of the proﬁt division, derived as the core of the game, becomesa unique unfair distribution. Following the literature, we considered intervention in the negotiation such that the ﬁrms are forced to distribute the proﬁt based on the Shapley value. Our
numerical example also showed that the intervention stabilizes the economy by eliminating thepessimistic equilibrium.

25Appendix A. Technical details of the baseline model
A.1 Proof of Lemma 1
Substituting (8) and (11) into (12), the maximization problem can be rewritten as follows:
max

"

1

λ0 >1 λ β
0

s0

1−

!

1

η

λ0 · ∏ℓ=1 λℓ"

η

E + ∑ si

1−

i =1

!

1
η −i

1
∏ℓ=1 λℓ · λ0 · ∏−k =−i E ( λk )##−1

E · ∏ E (ψk )

.

k=−i

The ﬁrst-order condition is
"
##
"
!
!
η
−1
β
1
1
− β +1 s 0 1 −+ ∑ si 1 − η −i
· ∏ E (ψk )η
1
λ0 · ∏ℓ=1 λℓ
λ0
E
(
λ
)
∏ℓ=1 λℓ · λ0 · ∏−
i =1
k =−i
k
k =−i
"
"
!
##
η
−1
1
1
s0
+ β
+ ∑ si· ∏ E (ψk ) = 0
η
η −i
2
1
λ0 λ0 · ∏ℓ=1 λℓ i=1∏ℓ=1 λℓ · λ20 · ∏−k =−i
k =−i E ( λk )
!
"
!
##
"
η
−1
1
1
+ ∑ si 1 − η −i
· ∏ E (ψk )⇔ β s0 1 −η
1
λ0 · ∏ℓ=1 λℓ
E(λ)∏ℓ=1 λℓ · λ0 · ∏−
i =1
k =−i
k
k=−i
"
!
#
η
−1
s0
1
− ∑ si−·E (ψk ) = 0.
∏η
η −i
λ0 · ∏ℓ=1 λℓ i=1
λ ℓ · λ 0 · ∏ −1 E ( λ k )
∏
k =−i
k=−i

ℓ=1

Using the symmetry (λi = λ0 , ∀i), we obtain
"
β s0

1−

1

!

"

+ ∑ si

η +1

λ0

η

1−

i =1

"

⇔ s0 β − (1 + β )
"

1

!

1

· ψ∗ (z)

η +1

λ0
#

η +1

λ0

⇔ β − (1 + β )

##
i

−

"

+ β − (1 + β )
#

1
η +1
λ0

η

s0
η +1

λ0
1

#

i =1

η

η +1

λ0

!
ψ∗ (z)i=0∑ si ψ ∗ ( z ) = 0

η +1

λ0

i

i =1

!

η

|

−∑

si

s0 + ∑ s i ψ ∗ ( z )i
i =1

{z

= 0,
}

+
η +1

where ψ∗ (z) is deﬁned in (16). By solving β − (1 + β)/λ0= 0, we obtain the result. □A.2 Proof of Lemma 2
Suppose the proﬁt-sharing vector is the most frontloaded (s = sF ). Then, by (15) and (17), we
obtain the interior solution of z as follows:
∗

z =



1
1+β




L−

26

ρβ
θ


λ

∗β


.

(24)

Note that z∗ > 0 holds by Assumption 1. Then, the growth rate is calculated as follows:θL− ρβ ln λ∗
λ∗ β
" 
#
 β/(1+η )β1ln(1 + 1/β)
θL
− ρβ
.
1+β
1+β
1+η
{z
}
|
|
{z
}


g =

=

1
1+β



positive andincreasing in η

decreasingin η

Note that the square bracket in the above equation is increasing in η, as β/(1 + β) < 1.
Additionally, the growth rate converges to zero as η → ∞, as the square bracket converges to aﬁnite value. Thus, we obtain the second statement in Lemma 2.
By diﬀerentiating g with respect to η, we obtain
dg >
=0
dη <

 β/(1+η ) 


β
1
1
1
>
⇔ ρ = θL−ln 1 +≡ Ω ( η ).
<
1+β
β 1+η
β
{z
}|
{z
}
|
increasing in η

(25)

positive and increasing in η

Note that the square bracket in the RHS of (25) is positive. This can be easily shown by using
the fact that (1 + 1/β) β is increasing in β > 0 and is strictly lower than exp(1) ≃ 2.71828. Then,Ω(η ) is strictly increasing in η and limη →∞ Ω(η ) = θL/β. By this fact and Assumption 1, thefollowing two cases arise depending on the value of the discount rate. The ﬁrst case is that ρis suﬃciently small such that ρ < Ω(η ) ⇔ dg/dη < 0 holds for all η ≥ 1. In this case, thebroadening leading breadth always decreases the growth rate.
The second case is that ρ is a moderate value. Suppose that Ω(η̃ ) < ρ < Ω(η̃ + 1) whereη̃ ≥ 2 is a threshold of η. Then, by (25), we obtain dg/dη > 0 for η = 1, 2, ..., η̃. This means thatthe broadening leading breadth in η ∈ {1, 2, ..., η̃ − 1} increases the growth rate. However, asρ < Ω(η ) ⇔ dg/dη < 0 holds for all η ≥ η̃ + 1, the broadening leading breadth in η ≥ η̃ + 1
decreases the growth rate. Note that the growth eﬀect is ambiguous when η̃ → η̃ + 1. As aresult, we obtain the ﬁrst statement in Lemma 2. □A.3 Proof of Proposition 3
The ﬁrst statement immediately holds by comparing (13) with (21).
Let us consider the second statement. In the decentralized economy, z∗ or z∗L can be zerodepending on the parameters. To understand this, consider the vertical intercepts of the two
curves in Figure 3. By (15) and (17), the two curves intersect on the vertical line (i.e., z∗ = 0 or27z∗L = 0 holds) if the following parameter condition holds:
s0
λ∗
=
.
ρ (1 + β )θ (1 + 1/β) L
Note that the parameter condition does not contradict Assumption 1. As a result, z∗ < z# canhold, as z# > 0.
Next, we consider whether z∗ > z# can hold or not. Lemma 3 shows that sF maximizes z∗ .
Moreover, the value of (24) is maximized when η → ∞. Therefore, the highest value of z∗ thatis achievable in the decentralized economy is given by
∗
zmax
≡



1
1+β



ρβ
L−
θ


.

∗
Assumption 1 ensures that zmax
> z# . As a result, we obtain the second statement.

Appendix B. The game of pool formation and proﬁt divisionThis Appendix formulates a game of pool formation and proﬁt division. Then, we derive thecore and the Shapley value.

B.1 Basic setup
Consider an industry in which ﬁrm 0 is the latest ﬁrm as Figure 2. Let N ≡ {0, 1, ..., ω } be theset of the ﬁrms in the industry. This means that, in the industry, the oldest innovator is ﬁrm ω,
and innovations have occurred ω times. We assume that the number of ﬁrms in the industry is
suﬃciently large:
Assumption 2. |N | = 1 + ω > 2η.15
Each event in the game occurs in the following order:(i) Pool formation. The ﬁrms in N can form several coalitions. If all the ﬁrms in a coalition enter into certain contracts and the coalition satisﬁes certain conditions, we call thecoalition pool, which will be formally deﬁned later.
(ii) Competition. All pools engage in the Bertrand competition. Due to the property of the
Bertrand competition on the quality ladder, only one pool earns a positive proﬁt.
(iii) Proﬁt division. The pool that won the Bertrand competition shares the proﬁt with theﬁrms in the pool.
15 |N | is the cardinality of N (i.e., the number of ﬁrms).

28

A coalition, denoted by T, is a non-empty subset of N . Every ﬁrm can form a coalition
consisting only of itself. Note that some coalitions may be unable to produce goods due toa lack of patent licenses or violation of the prohibition by the competition authority. Such a
coalition becomes inactive and cannot participate in the Bertrand competition. Conversely, there
may be an active coalition that can produce goods. To clarify this concept, we formally deﬁne
the pool as follows.
Deﬁnition 2. Consider a coalition of ﬁrms in which the youngest ﬁrm is ﬁrm i, and let Pi be the coalition(i.e., Pi ⊆ N and min Pi = i). Then, Pi is pool if it satisﬁes (i) and (ii) and every ﬁrm in Pi agrees on(iii) to (vi):(i) Pi includes all the licensors of ﬁrm i: Li ⊂ Pi .
– Li ≡ {i + 1, i + 2, ..., min{i + η, ω }} is the set of ﬁrms that have patents infringed by ﬁrm i.
(ii) Regulation by the competition authority: Pi ∩ (N \Li ) = {i }.
– The competition authority prohibits Pi from including any ﬁrm, not in Li because such a ﬁrm
does not license the patent to ﬁrm i and it just aims to expand the cartel size.
– Pi = {i } ∪ Li holds by (i) and (ii).
(iii) Exclusive patent license.
– Every ﬁrm in Li grants the patent licenses to ﬁrm i to enable it to produce the good.
– Every ﬁrm in Pi refuses the patent license to any ﬁrm in N \Pi .
(iv) Price cartel.
– Every ﬁrm in Pi \{i } does not produce its own good to increase the market power of ﬁrm i as
much as possible.
(v) Proﬁt division.
– If Pi earns a positive proﬁt, ﬁrm i shares it with Li in accordance with the proﬁt-sharing vector
already determined when Pi formed.
– No ﬁrm transfers the proﬁt to the ﬁrms in N \Pi .
(vi) Relationship Length.
– All ﬁrms in Pi continue the relationship until a new innovator enters the industry.

For each i ∈ N , we call Pi the proﬁtable pool if Pi obtains a positive proﬁt in the Bertrand
competition regardless of what pools are formed by the ﬁrms in N \Pi . Then, by this deﬁnition,
we can show the following statement:
Lemma 4. Pi is the proﬁtable pool if and only if i ∈ {0, 1, ..., η }.
29

Proof. Consider a pool Pi where i ∈ {η + 1, η + 2, ..., ω }. Then, Pi loses to a superior pool Pℓwhere ℓ ∈ {0, 1, ..., i − η − 1} in the Bertrand competition. Therefore, Pi is not the proﬁtable
pool by deﬁnition. Next, consider a pool Pi where i ∈ {0, 1, ..., η }. Note that the ﬁrms in{0, 1, ..., η }\Pi cannot form the pool by Deﬁnition 2 (i). This means that Pi can produce thegood with the highest quality in the industry. Therefore, Pi always earns a positive proﬁt in
the Bertrand competition.
Following the baseline model, we consider a symmetric situation where all ﬁrms in theindustry have equally chosen λ∗ given by (13). Then, by conditions (i) to (iv) in Deﬁnition 2, any
proﬁtable pool always earns the proﬁt Π̃ deﬁned in (14) because any proﬁtable pool has η + 1
ﬁrms and the competition authority does not allow the markup strictly higher than (λ∗ )1+η .
All ﬁrms regard Π̃ as a constant because they take E, an aggregate variable determined in the
BGP, as given. On the other hand, as shown in the proof of Lemma 4, the non-proﬁtable pools
gain nothing when the superior pools form. Thus, following von Neumann and Morgenstern(1944), we deﬁne the proﬁt of the non-proﬁtable pool as zero from a pessimistic viewpoint.
The function v : 2N → R, called the characteristic function in cooperative game theory,
assigns a proﬁt that each coalition T ⊆ N deﬁnitely gains in the Bertrand competition. Notethat the coalitions, not the pools, gain no proﬁt because of a lack of patent licenses or violationof the prohibition by the competition authority. Then, the characteristic function is given as16v( T ) = Π̃ if T = Pi for i ∈ {0, 1, . . . , η }, andv( T ) = 0 otherwise.
Let i ∈ {0, 1, . . . , η }, and assume that the proﬁtable pool Pi forms. Then, Pi earns the proﬁtΠ̃, and each ﬁrm j ∈ N \ Pi gains nothing. Thus, by condition (v) in Deﬁnition 2, a vectorπ ≡ π j j∈N of the net proﬁt of the ﬁrms in the industry is deﬁned as∑ π j = Π̃, andj∈Pi

π j = 0 for all j ∈/ Pi .

B.2 The core of the gameWe deﬁne the concept of blocking in the game as follows:
Deﬁnition 3. A coalition T ⊆ N blocks π ∗ through π = (π j ) j∈N if the following properties hold.
• Feasibility: ∑ j∈T π j ≤ v( T ).
• Improvement: π j > π ∗j holds for all j ∈ T.
16By convention, we deﬁne v(∅) = 0.

30

Using Deﬁnition 3, we deﬁne the core of the game as follows:
Deﬁnition 4. The core is the set of π that is not blocked by any coalition.
By Deﬁnition 3, any net proﬁt vector π ∗ such that π ∗j < 0 for some ﬁrm j ∈ N is blocked by{ j} because v({ j}) = π j = 0 holds. Thus, without loss of generality, we assume that the core isa subset of the set of net proﬁt vectors π such that π j ≥ 0 for each j ∈ N . With this setup, we
can show the following statement:
Proposition 6. Let i ∈ {0, 1, . . . , η }, and assume that Pi forms. Then, the net proﬁt vector that belongs
to the core of the game is uniquely given by π C = (π Cj ) j∈N such thatπηC = Π̃ and π Cj = 0 for all j ∈ N \{η }.

Proof. Let Pi be the proﬁtable pool where i ∈ {0, 1, ..., η }. Consider a net proﬁt vector π suchthat πη < Π̃ holds. Then, Pi \{η } has some ﬁrms that obtain a positive net proﬁt. We deﬁne
η −1

i+η

π ≡ ∑k=i πk and π ≡ ∑k=η +1 πk . Suppose that π > 0. Then, Pη can block π through π ′ suchthat
π ′j = 0 for all j ∈ {0, 1, ..., η − 1},
π ′j = π j +πfor all j ∈ {η, η + 1, η + 2, ..., 2η }, and1+η

π ′j = π j for all j ∈ {2η, 2η + 1, ..., ω }.
Next, suppose that π > 0. Then, P0 can block π through π ′′ such that
π ′′j = π j +πfor all j ∈ {0, 1, ..., η }, and1+η

π ′′j = 0 for all j ∈ {η + 1, η + 2, ..., ω }.
As a result, a net proﬁt vector π such that πη < Π̃ holds cannot belong to the core.
Finally, consider the net proﬁt vector π C given in the statement. Suppose that a coalition
T can block π C through π. If v( T ) = 0, then by the deﬁnition of π C and the feasibility in
Deﬁnition 3, ∑ j∈T π j ≤ v( T ) = 0 ≤ ∑ j∈T π Cj ; thus, π does not satisfy the improvement in
Deﬁnition 3. Therefore, T that can block π C must be the proﬁtable pool (i.e., v( T ) = Π̃ > 0).
This means that, by Lemma 4, T contains ﬁrm η. Then, by the improvement in Deﬁnition 3,∑ j∈T π j > ∑ j∈T π Cj = Π̃ = v( T ), which contradicts the feasibility in Deﬁnition 3. Thus, no
coalition can block π C .
Proposition 6 shows that ﬁrm η obtains all the proﬁt, and the rest does not obtain a positive
net proﬁt in the core of the game. Intuitively, ﬁrm η is the pivotal ﬁrm in N because any pool31

without ﬁrm η is not proﬁtable. Therefore, the other ﬁrms in the proﬁtable pool do not haveany negotiation power.17
By Lemma 4 and Proposition 6, it is unclear whether P0 is chosen as the proﬁtable pool in{P0 , P1 , ..., Pη }. Because they equally yield πη = Π̃, the pivot (i.e., ﬁrm η) is indiﬀerent between
them. To deal with the problem, we assume that λi = λi+1 + ε for all i ∈ N , where ε > 0 is
inﬁnitesimal. Although it does not change any result in the baseline model because ε > 0 is anegligibly small number, P0 earns the largest proﬁt in {P0 , P1 , ..., Pη }. Note that ﬁrm η will nolonger be pivotal when the next innovation occurs. Because this is the last opportunity for ﬁrmη to obtain a positive net proﬁt, ﬁrm η will choose P0 .

B.3 The Shapley value
Following the literature, we also consider the next intervention as an approximation of thelicense under FRAND conditions:Deﬁnition 5. The intervention in the licensing negotiation is that(i) All ﬁrms in L0 are prohibited from refusing ﬁrm 0’s oﬀer of the patent licenses;– The property implies that P0 = {0} ∪ L0 is the proﬁtable pool in the industry.
(ii) The license fees should be fair in the sense that all ﬁrms in P0 must share the proﬁt Π̃ following
the Shapley value.
Deﬁnition 6. Let P0 be the proﬁtable pool. The Shapley value π Sj for each j ∈ P0 is given as follows:
π Sj ≡t!(η − t)!
(v({ j} ∪ T ) − v( T )),
(
1
+
η
)
!
T ⊆P \{ j}∑0where t is the cardinality of a coalition T (i.e., t = | T |).
For each j ∈ P0 and for each T ⊆ (P0 \{ j}), v({ j} ∪ T ) − v( T ) is called the marginalcontribution of ﬁrm j to coalition T. In the formation process for P0 , when a coalition T
already forms and a ﬁrm j joins that coalition, the ﬁrm demands and is promised to gain its
marginal contribution to T. Suppose the order in which each ﬁrm joins is determined withequal probability. Then, the Shapley value for each ﬁrm j ∈ P0 is the average of j’s marginalcontribution to the formation of P0 . In this sense, the Shapley value is a fair proﬁt allocation
among P0 .

17Even if ﬁrm i (̸= η ) in the proﬁtable pool suggests leaving the pool to decrease the proﬁt of the proﬁtable pool,ﬁrm η can form another pool that does not contain ﬁrm i but yields the same proﬁt Π̃ in the Bertrand competition.

32

Proposition 7. Suppose that P0 forms. The net proﬁt vector applied to the Shapley value of the game isuniquely given as π S = (π Sj ) j∈N such that
π Sj =

Π̃for all j ∈ P0 and π Sj = 0 for all j ∈/ P0 .
1+η

Proof. Note that because P0 forms under the intervention, each ﬁrm j ∈/ P0 gains nothing; thatis, under the proﬁt allocation based on the Shapley value, π Sj = 0 for each j ∈/ P0 . Furthermore,as mentioned above, v( T ) = 0 holds for each T ⊊ P0 due to the patent infringements. Thus,
when and only when ﬁrm j ∈ P0 joins P0 \{ j}, the marginal contribution of ﬁrm j is positive(i.e., T = P0 \{ j} if and only if v({ j} ∪ T ) − v( T ) = v(P0 ) = Π̃ > 0). Therefore, we obtain the
result.

References[1] Basu, S. (1996). “Procyclical productivity: increasing returns or cyclical utilization?” The
Quarterly Journal of Economics, 111(3), 719-751.
[2] Chu, A. C. (2009). “Eﬀects of blocking patents on R&D: A quantitative DGE analysis.”
Journal of Economic Growth, 14(1), 55-78.
[3] Chu, A. C., Cozzi, G., & Galli, S. (2012). “Does intellectual monopoly stimulate or stiﬂe
innovation?” European Economic Review, 56(4), 727-746.
[4] Chu, A. C., & Pan, S. (2013). “The escape-infringement eﬀect of blocking patents on innovation and economic growth.” Macroeconomic Dynamics, 17(4), 955-969.
[5] Dewatripont, M., & Legros, P. (2013). “ Essential Patents, FRAND Royalties and Technological Standards.” The Journal of Industrial Economics, 61(4), 913-937.
[6] Entezarkheir, M. (2017). “Patent thickets, defensive patenting, and induced R&D: an empirical analysis of the costs and potential beneﬁts of fragmentation in patent ownership.”Empirical Economics, 52(2), 599-634.
[7] Falk, A., Becker, A., Dohmen, T., Enke, B., Huﬀman, D., & Sunde, U. (2018). “Global
evidence on economic preferences.” The Quarterly Journal of Economics, 133(4), 16451692.
[8] Grossman, G., & Helpman, E. (1991). “Innovation and growth in the global economy.” The
MIT Press, Cambridge, Massachusetts.
[9] Hall, B. H., & Ziedonis, R. H. (2001). “The patent paradox revisited: an empirical studyof patenting in the US semiconductor industry, 1979-1995.” Rand Journal of Economics,
32(1), 101-128.
[10] Kishimoto, S., & Suzuki, K. (2021). “Growth and welfare eﬀects of intervention into patent
license negotiation.” MPRA Paper.
[11] Kishimoto, S., & Watanabe, N. (2017). “The kernel of a patent licensing game: The optimal
number of licensees.” Mathematical Social Sciences, 86, 37-50.
[12] Klein, M. A. (2022). “The reward and contract theories of patents in a model of endogenous
growth.” European Economic Review, 147, 104178.
33

[13] Layne-Farrar, A., Padilla, A. J., & Schmalensee, R. (2007). “Pricing patents for licensing instandard-setting organizations: Making sense of FRAND commitments.” Antitrust Law
Journal, 74(3), 671-706.
[14] von Neumann, J., & Morgenstern, O. (1944). “Theory of games and economic behavior.”
1st Edition. Princeton University Press, Princeton, NJ, [2nd Edn. 1947], [3rd Edn. 1953].
[15] Niwa, S. (2016). “Patent claims and economic growth.” Economic Modelling, 54, 377-381.
[16] Niwa, S. (2018). “Eﬀects of a blocking patent on R&D with endogenous survival activities.”
Journal of Economics, 125(3), 263-277.
[17] Norrbin, S. C. (1993).“The relationship between price and marginal cost in U.S. industry:
A contradiction,” Journal of Political Economy, 101(6), 1149-1164.
[18] O’donoghue, T., & Zweimüller, J. (2004). “Patents in a model of endogenous growth.”
Journal of Economic Growth, 9(1), 81-123.
[19] Sakakibara, M. (2010). “An empirical analysis of pricing in patent licensing contracts.”
Industrial and Corporate Change, 19(3), 927-945.
[20] Sakakibara, M., & Branstetter, L. (2001). “Do Stronger Patents Induce More Innovation?
Evidence from the 1988 Japanese Patent Law Reforms.” RAND Journal of Economics, 32(1),77-100.
[21] Shapley, L. S. (1953). “A value for n-person games.” Contributions to the Theory of Games,2(28), 307-317.
[22] Suzuki, K. (2020). “Patent protection, optimal licensing, and innovation with endogenous
entry.” Macroeconomics Dynamics, 24(8), 2033-2059.
[23] Tauman, Y., & Watanabe, N. (2007). “The Shapley value of a patent licensing game: theasymptotic equivalence to non-cooperative results.” Economic Theory, 30(1), 135-149.
[24] Watanabe, N., & Muto, S. (2008). “Stable proﬁt-sharing in a patent licensing game: Generalbargaining outcomes.” International Journal of Game Theory, 37, 505-523.
[25] Wang, M., Rieger, M. O., & Hens, T. (2016). “How time preferences diﬀer: Evidence from53 countries.” Journal of Economic Psychology, 52, 115-135.
[26] Yang, Y. (2018). “On the optimality of IPR protection with blocking patents.” Review of
Economic Dynamics, 27, 205-230.
[27] Ziedonis, R. H. (2004). “Don’t fence me in: Fragmented markets for technology and thepatent acquisition strategies of ﬁrms.” Management Science, 50(6), 804-820.

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