数理解析研究所講究録
第 1983 巻 2016 年 41-57

41

A Note on Patent Litigation and Cross Licensing1
海溶 (Haejun Jeon)
大阪大学経済学研究科
Graduate School of Economics
Osaka University$\leq\geqq$

1

Introduction

In this note, we investigate the interaction between the leader and the follower in developing

new technologies in the presence of litigation against patent infringement and cross licensing
as its alternative. After the leader has acquired a patent and manufactures a product basedon it, the follower can carry out the follow-on research to manufacture an advanced product,
which can partially substitute the leader’s product. Upon the losses of revenue, the leader canaccuse the follower of infringement to recover monopoly profit. The perpetrator of the alleged
infringement, however, is not always found guilty. When making a judgment, the court takes
into account how much the advanced technology overlaps with the existing one and how much
improvement has been made by the follow-on research, and occasionally the court considers thepatent to be invalid, which reveals the nature of “probabilistic patents.”2 For these reasons, theleader can opt for cross licensing over the costly lawsuit. Provided that both parties agree on it,
cross licensing allows them to utilize each other’s works without fear of infringement. That is,
the leader can take advantage of the follower’s advanced technology, of which novelty has been
made based on his preceding works. The choice between a lawsuit and cross licensing is made by
the firms, taking into account the nature of“cumulative innovation”’ and (probabilistic patents
This is examined in the framework of real options based on Huisman and Kort (2015), in which
not only the timing of each investment but also its capacity is endogenously determined. This
enables us to incorporate how much the follower will imitate the precursor’s work and how muchresources will be devoted to yield the improvement.
The present model helps us comprehend the patent war between competitors we can observein the real world, in which most innovation is sequential and cumulative. First of all, the modelshows that the competition in the burgeoning market usually entails a costly lawsuit to resolve
the conflict, while the competitors easily agree on cross licensing to take advantage of eachother’s works in the slow-growing market. This is a natural result because when the market isvery lucrative, the leader desires to monopolize the market in spite of the legal expenses and therisk of losing the patent’s validity in the court. If it does not seem profitable enough, however,the leader instead chooses to reconcile with the follower via cross licensing, raising extra revenue
from the advanced technology developed by the follower because the exercise of exclusive rights
paying the costs becomes less attractive. The fact that a more lucrative market is more likely
to entail a legal dispute is consistent with the argument from Cooter and Rubinfeld (1989) and1This manuscript is the abbreviated version of Jeon (2015).
2A great number of studies have acknowledged probabilistic validity of patents (e.g., Allison and Lemley (1998),
Aoki and Hu (1999), Lemley and Shapiro (2005), Llobet (2003), Farrell and Shapiro (2008), Choi (2010)).

42
Lemley and Shapiro (2005), and this finding is supported by empirical studies such as Lanjouw
and Schankerman (2001) and Somaya (2003).
Not only the way they resolve the dispute but also the timing of the initial innovation andthe follow-on research has significant implications, especially in terms of diﬀerentiated goods.
When the market demand is rapidly growing and the follower’s challenge is expected shortly
after the initial innovation, the leader delays his investment until the market grows enough tocompensate the losses of revenue from the competition so that the follow-on research is triggeredat the same time, and he files alawsuit over infringement after it becomes a more severe problem.
This implies that there is a distinct diﬀerence between the products of the leader and the follower
in the fast-growing market unless the follower is found guilty at the court later and driven outof the market. If the demand is expected to grow slowly and the competitors reconcile via
cross licensing, the leader delays the initial innovation substantially so that both the followon research and the agreement of cross licensing are triggered at the same time. This implies
that both parties takes advantage of each other’s works from the very beginning, leaving little
diﬀerence between their products in the market.
The present model also eﬀectively integrates the authorities’ patent policy by allowing theprobability that the patent is found to be valid to depend on the extent to which the patentscope is applied. The result shows that when the court interprets a patent in a narrow sense,
the leader and the follower tend to resolve the conflict via cross licensing. This is a natural
result because the leader is less likely to win the case over infringement, and thus opts for crosslicensing to raise extra revenue, even with a small market share. This relationship between thecredibility of threat of a lawsuit and the way the firms resolve the dispute is in line with Aoki
and Hu (1999). Being relieved of the fear of infringement, the follower focuses on making a
profit at a lower cost by imitating the existing technology, rather than enhancing the level oftechnology. As the patent scope becomes wider, the leader holds a more dominant position in
the negotiation of cross licensing, and thus takes a larger share in the market of the advanced
technology developed by the follower. If it becomes wider enough, the leader chooses to accuse
the follower encouraged by the judge’s hard line against infringement, and the follower devotes
most of his resources to develop his own technology not to be found guilty by the court.
Furthermore, the model clarifies that the policy on patent scope cannot induce the first-best
result in terms of the speed of innovation. If the patent scope is interpreted in a narrow sense
and the firms decide to resolve the conflict via cross licensing, the introduction of the secondgeneration technology is so delayed that the agreement of cross licensing is triggered at the sametime. This is because the follower is concerned about the expected losses of his revenue from thecompetition with the leader regarding the novel technology and chooses to wait until the market
grows enough to compensate for such losses. If the breadth of patent is wide enough to induce
the leader to file a lawsuit over infringement, now it is the initial innovation that is deferredsignificantly, and the follow-on research is triggered simultaneously. This result comes from theleader’s concern about losing monopoly profits by the follower’s challenge. Given these results,
we can argue that neither policy can induce the first-best outcome in which both technologies are
introduced without substantial delay. This argument is in line with Scotchmer (1991) and Green43
and Scotchmer (1995), which have shown that it is impossible for both the initial innovator andthe follow-on innovator to have adequate incentives under a patent system.
Still, we can evaluate the eﬀectiveness of patent policy by carrying out welfare analysis ina more comprehensive way. The total expected surplus of the consumer and that of producer,
and thus social welfare, depend not only on the timing of innovations but also on the prices
and the quantities of the products in the market, the way the competitors resolve the dispute,and the probability that the patent is found to be valid by the court provided that it ends
up with a lawsuit. The welfare analysis taking these aspects into account reveals that social
welfare is higher when the conflict is resolved via cross licensing because it does not involve
legal expenses and consumers can enjoy more products with the advanced technology at a lower
price. This result accords with Shapiro (2001), who noted that any cross license is superior
to a world in which the patentee fails to cooperate, and Denicol\‘o (2002), who argued thatcollusion between patentees via cross licensing can be socially beneficial even if the patents arecompeting. Given these arguments, one might conclude that the narrower the patent scope is,the more social welfare we can yield. Yet, this is only half the story. Even though the disputeis resolved out of court, the authorities’ stance on infringement still comes into play because
the leader’s bargaining power in the negotation of cross licensing depends on it. Namely, if thepatent scope is interpreted in a very narrow sense, the leader has little bargaining power in crosslicensing and cannot take advantage of the second-generation technology as much as he wants.
Thus, there will be fewer products with the advanced technology even at a higher price, whichleads to the decrease of social welfare.

2

The model and solutions

2.1

Setup

Suppose that there are two risk-neutral firms in the market, the leader and the follower.3 The
leader has an option to develop technology which can be patented and to manufacture a productbased on it. The demand shock is given by a one-dimensional geometric Brownian motion as
follows:$dX(t)=\mu X(t)dt+\sigma X(t)dW(t)$

(2.1)

and
where
are constant coeﬃcients and $(W(t))_{t\geq 0}$ is a standard Brownian motion on a
. A risk-free rate is assumed to be a constantfiltered probability space$r>\mu$ for the sake of finiteness of value function. Given the demand shock, the price of the$\mu$$\sigma$$(\Omega, \mathcal{F}, \mathbb{F} :=(\mathcal{F}_{t})_{t\geq 0}, \mathbb{P})$product with the first-generation technology at time$t$is determined as follows:$P_{1}(t)=X(t)(1-\eta_{1}Q_{1}(t))$

(2.2)

where
and a constant $\eta_{1}>0$ denote total market output of the product with the firstgeneration technology at time and its price elasticity, respectively. If the demand shock exceeds$Q_{1}(t)$$t$3For simplicity, we assume that their roles are predetermined. The role can be endogenized to investigate
preemption incentive. See Chapter 9 of Dixit and Pindyck (1994) for detailed illustration regarding this issue.

44
a threshold, the leader invests in R&D and manufactures a product with capacity, whichincurs investment costs ofincluding the R&D costs, and he makes a profit at the rateof
. Note that the investment in R&D and the capacity is integrated in one stage forsimplicity and that the patent is assumed to be acquired immediately with no expiry date.
$Q_{1}^{L}$$c_{1}Q_{1}^{L}$$P_{1}(t)Q_{1}^{L}$

The basic technology developed by the leader lays the foundation for the follow-on research.
Namely, the follower has an option to develop advanced technology building on the existing
one and to manufacture a product with it. Further improvement is expected to be made, but
it can partially overlap the predecessor’s works. For instance, after a leading firm in the IT
industry has invented a smartphone, a rival firm can manufacture a smartphone with fingerprint
identification or a curved display. The latter is embedded with newer technologies and canreplace the former to a certain extent. The degree of overlapping and the improvement can be
represented by the scale of the follower’s investment in R&D and the capacities regarding thefirst second-generation technologies, denoted by, respectively. Note that the size
andof the capacities is assumed to be proportional to the scale of R&D investment because it isreasonable to suppose that the higher the quality of R&D investment is, the larger capacities
the firm would like to have. In the example of a smartphone industry, the follower invests not
only in the facilities to produce the components of an existing smartphone but also in thoseto develop and manufacture a fingerprint reader or a curved display, which correspond to, respectively. The former can be invested at a lower cost,andper unit with a constant$\gamma\in[0$ , 1 , because the patent has been granted to theleader in exchange for detailed public
disclosure of the invention, which enables the follower to imitate easily. Yet, the improvement intechnology can only be made at relatively higher costs, and we suppose that a unit ofcosts$Q_{1}^{F}$$Q_{2}^{F}$$Q_{1}^{F}$$Q_{2}^{F}$$\gamma c_{1}$$]$$Q_{2}^{F}$$c_{2}.$

Given the investments in the technologies with diﬀerent level of novelty, the follower’s revenue
consists of two channels; $P_{1}(t)Q_{1}^{F}+P_{2}(t)Q_{2}^{F}$ with
$P_{2}(t)=X(t)(1-\eta_{2}Q_{2}(t))$(2.3)

where $Q_{2}(t)$ and a constant $\eta_{2}>0$ are defined in the same manner as (2.2). The investment
regarding the existing technology makes duopoly profit
since he competes with theleader in terms of the total output $(i.e., Q_{1}(t)=Q_{1}^{L}+Q_{1}^{F})$ . Yet, the investment regarding thesecond-generation technology of which novelty can only be found in the follower’s product yields
monopoly revenue
Having his product partially substituted by that of the follower, the leader can accuse thefollower of infringement, which costs both parties , to recover monopoly profit. Yet, the patentee does not always win the trial. The patent occasionally turns out to be invalid at the court,
even though it has been granted by the authorities after appropriate examination. Innovation
is inherently cumulative, and thus the court takes into account how much the follower’s technology overlaps with the leader’s and how much improvement has been made in the subsequentinnovation when judging whether the follower has infringed on the precursor’s patent rights.4$P_{1}(t)Q_{1}^{F}$$P_{2}(t)Q_{2}^{F}.$$C_{L}$

4Chang (1995) addressed that courts sometimes refuse to find infringement if the allegedly infringing device
features major improvements. Green and Scotchmer (1995) and Llobet (2003) supposed that the amount of45
Yet, there is a room for discretion by the court in the sense that there can be a huge diﬀerence
regarding the interpretation of patent breadth depending on patent policy. With this in mind,

we suppose that the patent is found to be valid at the court with probability$p$given by
(2.4)

$p(Q_{1}^{F}, Q_{2}^{F}, d)= \exp(-\frac{Q_{2}^{F}}{Q_{1}^{F}}d)$where a constant $d\in(0, \infty)$ denotes the extent to which the “Doctrine of Equivalents” isapplied.5 If the court rules that the follower has infringed on the leader’s patent, which occurs
with probability , the follower is forced to cease production thenceforth, while the leader recovers
monopoly profits. With probability $1-p$ , however, the patent turns out to be invalid, the followon research is acknowledged as legitimate innovation, and the follower maintains his position in
hold; the formerandthe market. Note that
implies that the more the follower’s investment overlaps the leader’s one, the more it is likely$p$$\partial p(Q_{1}^{F}, Q_{2}^{F}, d)/\partial Q_{1}^{F}>0$$\partial p(Q_{1}^{F}, Q_{2}^{F}, d)/\partial Q_{2}^{F}<0$

to be found as infringement, while the latter suggests that the more improvement the follow-on
research has made, the more it is likely to be admitted as legitimate innovation. Regarding theholds, which clarifies the interpretation of ; the lower
court’s discretion,is, the wider the patent breadth is. Yet,andensure that the discretion is excluded in the extreme cases.
The litigation process is costly for the patentee in many ways; not only is the legal costa burden but also he gains nothing with probability $1-p$ in spite of the litigation expenses.
Thus, the leader can consider cross licensing as an alternative of a lawsuit. Namely, the leader,still behind in technology, can make an oﬀer that allows both parties to utilize each other’s
work without fear of infringement. If agreed upon, the leader invests with his first-best capacity
perto embrace the new technology in manufacturing his product, which incurs cost ofunit, and this yields additional duopoly revenue of. The oﬀer, however, can be rejectedby the follower if it is better for him to be sued from the perspective of the expected profits.
This is because not only does his revenue decrease due to the competition regarding the secondgeneration technology $(i.e., Q_{2}(t)=Q_{2}^{L}+Q_{2}^{F})$ but also there is a chance that he is found not
guilty at the court, which makes him continue to compete with the leader regarding the firstgeneration technology while raising monopoly revenue from the second-generation technology.
which makes
If this is the case, the leader can make an oﬀer with the second-best capacity
the follower indiﬀerent between cross licensing and litigation, as oIlg as the leader is better oﬀ
than involved in a lawsuit in terms of expected profits.
$d$

$d$

$\partial p(Q_{1}^{F}, Q_{2}^{F}, d)/\partial d<0$$\lim_{Q_{1}^{F}arrow 0}p(Q_{1}^{F}, Q_{2}^{F}, d)=0$$\lim_{Q_{2}^{F}arrow 0}p(Q_{1}^{F}, Q_{2}^{F}, d)=1$$C_{L}$$Q_{2}^{L}$$\gamma c_{2}$$P_{2}(t)Q_{2}^{L}$$Q_{2}^{L}$

$1$

2.2

The benchmark model

In this subsection, we suppose that litigation is the only way for the leader to recover his profits
so as to facilitate the understanding of the framework. As usual, value functions of both parties
improvement determines the patent’s validity, and Koo and Wright (2010) assumed that an invention based on
infringing research can be patented.
5Aoki and Hu (1999) noted that exactly what constitutes “equivalent is left to thejury to decide, and presumed
that the probability that the patent is found to be valid by the court is solely determined by the extent of theDoctrine of Equivalents. A more sophisticated assumption can be found in Llobet (2003), which further takes into
account how much improvement is made from the subsequent innovation.

46
will be determined backwards. For now, we assume that each event occurs sequentially, leaving
the case in which some of them take place simulaneously to be illustrated later.
First, suppose that the follower has carried out the follow-on research based on the existing

technology to manufacture an advanced product that partially substitutes the leader’s product.
As the market demand grows, the leader’s losses of revenue from the competition with thefollower become severe, and we can easily guess that there is an upper threshold, denoted bywhich triggers the leader’s litigation against infringement with legal costs offor both parties.
Ifis hit, the court judges with probability that the follow-on research has infringed upon
the predecessor’s work and forces the follower to cease production. Yet, the patent is found to
be invalid with probability $1-p$ , and the follower keeps his position in the market. Thus, theleader’s expected profit at time , provided thatis hit, can be expressed as follows:$x_{L},$$c_{L}$$x_{L}$$p$$t$$x_{L}$$\mathbb{E}[\int_{t}^{\infty}e^{-r(s-t)}\{p(Q_{1}^{L}X(s)(1-\eta_{1}Q_{1}^{L}))+(1-p)(Q_{1}^{L}X(s)\{1-\eta_{1}(Q_{1}^{L}+Q_{1}^{F}$$=p \frac{Q_{1}^{L}x(1-\eta_{1}Q_{1}^{L})}{r-\mu}+(1-p)\frac{Q_{1}^{L}x\{1-\eta_{1}(Q_{1}^{L}+Q_{1}^{F})\}}{r-\mu}$$\}ds|X(t)=x]$

.

(2.5)

Meanwhile, by the standard argument, the option value of the firm, $v(x)$ , satisfies the fol-lowing ordinary diﬀerential equation:(2.6)$rv= \mu x\frac{\partial v}{\partial x}+\frac{1}{2}\sigma^{2}x^{2}\frac{\partial^{2}v}{\partial x^{2}}$of which the solution takes the form as follows:(2.7)$v(x)=Ax^{\alpha}+Bx^{\beta}$

where$\alpha=\frac{1}{2}-\frac{\mu}{\sigma^{2}}+\sqrt{(\frac{1}{2}-\frac{\mu}{\sigma^{2}})^{2}+\frac{2r}{\sigma^{2}}}>1,$$\beta=\frac{1}{2}-\frac{\mu}{\sigma^{2}}-\sqrt{(\frac{1}{2}-\frac{\mu}{\sigma^{2}})^{2}+\frac{2r}{\sigma^{2}}}<0$

.

(2.8)

With these in mind, we can delineate value function of the leader, having an option to litigate
over infringement. Given the demand shock and both parties’ investment capacities
and, it can be written as follows:$Q_{1}^{L}$$x$$Q_{1}^{F}$$V_{L}^{L}(x, Q_{1}^{L}, Q_{1}^{F})=\{\begin{array}{ll}\frac{Q^{L}x\{1-\eta_{1}(Q^{L}+Q_{1}^{F})\}}{r-\mu}+A_{L}^{L}x^{\alpha}, x<x_{L},p\frac{Q_{1}^{L}x(1-\eta_{1}Q_{1}^{L})}{r-\mu}+(1-p)\frac{Q_{1}^{L}x\{1-\eta_{1}(Q_{1}^{L}+Q_{1}^{F})\}}{r-\mu}-c_{L}, x\geq x_{L}.\end{array}$

The litigation trigger
and the coeﬃcient of option value
and smooth-pasting conditions as follows:$x_{L}$$x_{L}(Q_{1}^{L}, Q_{1}^{F})= \frac{-\mu)c_{L}}{(\alpha p\eta_{1}Q_{1}^{L}Q_{1}^{F}}$,$A_{L}^{L}$(2.9)

are determined by value-matching$A_{L}^{L}(Q_{1}^{L}, Q_{1}^{F})=( \frac{p\eta_{1}Q_{1}^{L}Q_{1}^{F}x_{L}}{r-\mu}-c_{L})x_{L^{-\alpha}}$.

(2.10)

Note that
andhold. The former shows that themore the follow-on research overlaps the patented technology, the more it is likely to provoke
a lawsuit, while the latter implies that the more the leader has devoted his resources to theinnovation, the more he is likely to accuse the follower of infringement.
$\partial x_{L}(Q_{1}^{L}, Q_{1}^{F})/\partial Q_{1}^{F}<0$$\partial x_{L}(Q_{1}^{L}, Q_{1}^{F})/\partial Q_{1}^{L}<0$47
Having the litigation trigger determined by the leader, we can describe value function of thefollower confronted with an upcoming legal dispute. Given the demand shock and investment
capacities, it can be obtained in the same manner as follows:
and$x$$Q_{1}^{L},$$Q_{1}^{F}$$Q_{2}^{F}$(2.11)$V_{F}^{L}(x, Q_{1}^{L}, Q_{1}^{F}, Q_{2}^{F})=\{\begin{array}{ll}\frac{Q_{1}^{F}x\{1-\eta_{1}(Q_{1}^{L}+Q_{1}^{F})\}+Q_{2}^{F}x(1-\eta_{2}Q_{2}^{F})}{r-\mu}+A_{F}^{L}x^{\alpha}, x<x_{L},(1-p)\frac{Q_{1}^{F}x\{1-\eta_{1}(Q_{1}^{L}+Q^{F})\}+Q^{F}x(1-\eta_{2}Q^{F})}{r-\mu}-c_{L}, x\geq x_{L}.\end{array}$

The coeﬃcient of option value$A_{F}^{L}$is determined by value matching condition as follows:$A_{F}^{L}(Q_{1}^{L}, Q_{1}^{F}, Q_{2}^{F})=[-p \frac{Q_{1}^{F}x_{L}\{1-\eta_{1}(Q_{1}^{L}+Q_{1}^{F})\}+Q_{2}^{F}x_{L}(1-\eta_{2}Q_{2}^{F})}{r-\mu}-c_{L}]x_{L}^{-\alpha}$.

(2.12)

Now, we proceed to the follower’s decision to carry out the follow-on research and to man-ufacture advanced products based on it. We can easily guess that there is an upper threshold,denoted by , which triggers the follow-on research (or (alleged infringement
provided thatthe basic technology has been developed by the predecessor. Innovation is inherently cumulative,
and thus the outcome of the follower’s research and development can partially substitute theleader’s product. The investment capacity regarding the first-generation technology, denoted by, costs
per unit, and it yields duopoly revenue. Yet, the second-generation technology can
only be found in the follower’s product, and thus he can raise monopoly profit from the investment inwhich incurs per unit. Note that the follower takes into account the forthcoming
legal dispute that depends on the ratio of the capacities of his investment. Thus, the follower’s
value function at the moment of investment given the demand shock is as follows:$x_{I}$$Q_{1}^{F}$$\gamma c_{1}$$Q_{2}^{F}$$c_{2}$$x$$\frac{Q_{1}^{F}x\{1-\eta_{1}(Q_{1}^{L}+Q_{1}^{F})\}+Q_{2}^{F}x(1-\eta_{2}Q_{2}^{F})}{r-\mu}+A_{F}^{L}x^{\alpha}-\gamma c_{1}Q_{1}^{F}-c_{2}Q_{2}^{F}$(2.13)

whereis given by (2.12). Maximizing (2.13) with respect toyields the optimalandcapacities
and, respectively. Following the similar argument,
value function of the follower given the demand shock and the leader’s investment capacity
can be described as follows:$A_{F}^{L}$$Q_{1}^{F}$$Q_{1}^{F^{*}}(x, Q_{1}^{L}, Q_{2}^{F})$$Q_{2}^{F}$$Q_{2}^{F^{*}}(x, Q_{1}^{L}, Q_{1}^{F})$$x$$Q_{1}^{L}$$V_{F}^{I}(x, Q_{1}^{L})=\{\begin{array}{ll}A_{F}^{I}x^{\alpha}, x<x_{I},V_{F}^{L}(x, Q_{1}^{L}, Q_{1}^{F^{*}}, Q_{2}^{F^{*}})-\gamma c_{1}Q_{1}^{F^{*}}-c_{2}Q_{2}^{F^{*}}, x\geq x_{I}.\end{array}$

The trigger of “alleged infringement”
and the coeﬃcient of option value
by value-matching and smooth-pasting conditions at the trigger as follows:$x_{I}$$x_{I}(Q_{1}^{L})= \frac{\alpha(r-\mu)(\gamma c_{1}Q_{1}^{F^{*}}+c_{2}Q_{2}^{F^{*}})}{(\alpha-1)[Q_{1}^{F^{*}}\{1-\eta_{1}(Q_{1}^{L}+Q_{1}^{F^{*}})\}+Q_{2}^{F^{*}}(1-\eta_{2}Q_{2}^{F^{*}})]}$,$A_{F}^{I}$(2.14)are determined(2.15)$A_{F}^{I}(Q_{1}^{L})=[ \frac{Q_{1}^{F^{*}}x_{I}\{1-\eta_{1}(Q_{1}^{L}+Q_{1}^{F^{*}})\}+Q_{2}^{F^{*}}x_{I}(1-\eta_{2}Q_{2}^{F^{*}})}{r-\mu}+A_{F}^{L}(Q_{1}^{L},Q_{1}^{F^{*}})x_{I}^{\alpha}-\gamma c_{1}Q_{1}^{F^{*}}-c_{2}Q_{2}^{F^{*}}]x_{I}^{-\alpha}.$(2.16)

Having the trigger determined by the follower, now we can calculate value function of theleader of which product can be partially substituted in the same manner as follows:$V_{L}^{I}(x, Q_{1}^{L})=\{\begin{array}{ll}\frac{Q_{1}^{L}x(1Q_{1}^{L})}{r}+A_{L}^{I}x^{\alpha}, x<x_{I)}V_{L}^{L}(x, Q_{1}^{L}, Q_{1}^{F^{*}}) , x\geq x_{I}.\end{array}$(2.17)

48
The coeﬃcient of option value
follows:$A_{L}^{I}$

is determined by value-matching condition at the trigger as$A_{L}^{I}(Q_{1}^{L})=A_{L}^{L}(Q_{1}^{L}, Q_{1}^{F^{*}})- \frac{\eta_{1}Q_{1}^{L}Q_{1}^{F^{*}}x_{I}^{1-\alpha}}{r-\mu}$(2.18).

Lastly, we proceed to the leader’s value function at the initial stage. We can also easily, which triggers the leader’s innovation
guess that there is an upper threshold, denoted by
per unit, and the leader makes
cost
and patent acquisition. The investment capacity
monopoly profit from it until the follower challenges it by making a product that can partiallysubstitute his product. The leader takes into account the upcoming challenge by the follower,
and thus his value function at the moment of investment given the demand shock is as follows:$x_{P}$$Q_{1}^{L}$$c_{1}$$x$(2.19)$\frac{Q_{1}^{L}x(1-\eta_{1}Q_{1}^{L})}{r-\mu}+A_{L}^{I}x^{\alpha}-c_{1}Q_{1}^{L}$

where
is given by (2.18). The optimal investment capacity
can be obtained by. Following the same argument, we can describe the leader’smaximizing (2.19) with respect tovalue function at the initial stage as follows:$Q_{1}^{L^{*}}(x)$$A_{L}^{I}$$Q_{1}^{L}$(2.20)$V_{L}^{P}(x)=\{\begin{array}{ll}A_{P}x^{\alpha}, x<x_{P},V_{L}^{I}(x, Q_{1}^{L^{*}})-c_{1}Q_{1}^{L^{*}} x\geq x_{P}.\end{array}$

The trigger of patent acquisition
and the coeﬃcient of option value
value-matching and smooth-pasting conditions at the trigger as follows:$x_{P}$$x_{P}= \frac{\alpha(r-\mu)c_{1}Q_{1}^{L^{*}}}{(\alpha-1)Q_{1}^{L^{*}}(1-\eta_{1}Q_{1}^{L^{*}})},$$A_{P}$are determined by$A_{P}=[ \frac{Q_{1}^{L^{*}}x_{P}(1-\eta_{1}Q_{1}^{L^{*}})}{r-\mu}+A_{L}^{I}(Q_{1}^{L^{*}})x_{P}^{\alpha}-c_{1}Q_{1}^{L^{*}}]x_{P}^{-\alpha}.$$(2.21)$

So far, we have implicity assumed that the following inequality holds regarding the triggers:$x_{P}<x_{I}<x_{L}$.

(2.22)

This has enabled us to focus on the case in which the events occur sequentially. However, thefollower can delay his investment so that$x_{P}, x_{L}<x_{I}$.

(2.23)

holds. Furthermore, the leader can also delay the initial investment so that$x_{I}<x_{P}<x_{L}$.

(2.24)

or$x_{I}, x_{L}<x_{P}$.

holds. The explanation on these cases is omitted here, and can be found in Jeon (2015).

(2.25)

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2.3

The main model

Having outlined the framework, now we proceed to the main model in which the leader can makean oﬀer of cross licensing. If agreed upon, the leader makes an investment with capacity
totake advantage of the second-generation technology developed by the follower, and it incurs acost ofper unit. The oﬀer, however, can be rejected by the follower, and furthermore it
might not even be oﬀered from the very beginning for the sake of the leader’s interests. Namely,
the leader might choose a lawsuit over an agreement of cross licensing if he is better oﬀ in termsof expected profits, even though the follower is willing to accept the oﬀer. For now, we supposethat the leader makes an oﬀer and it is accepted by the follower and that the events occur
sequentially, leaving the whole picture to be described later.
First, suppose that the subsequent innovation has been made by the follower and the leader’s
product has been partially substituted by that of the follower. Then, the leader can make an oﬀer
of cross licensing in order to raise extra revenue from the second-generation technology developedby the follower. We can guess that there is an upper threshold, denoted by, which triggers
cross licensing, and having
hit, the leader’s value function at the moment of investment given
the demand shock is as follows:$Q_{2}^{L}$$\gamma c_{2}$$x_{C}$$x_{C}$$x$$\frac{Q_{1}^{L}x\{1-\eta_{1}(Q_{1}^{L}+Q_{1}^{F})\}+Q_{2}^{L}x\{1-\eta_{2}(Q_{2}^{L}+Q_{2}^{F})\}}{r-\mu}-\gamma c_{2}Q_{2}^{L}$.

(2.26)

Maximizing (2.26) with respect toyields the leader’s optimal capacity regarding the secondgeneration technology, and the leader’s value function at this stage can be written
as follows:$Q_{2}^{L}$$Q_{2}^{L^{*}}(x, Q_{2}^{F})$$V_{L}^{C}(x, Q_{1}^{L}, Q_{1}^{F}, Q_{2}^{F})=\{\begin{array}{ll}\frac{Q_{1}^{L}x\{1-\eta_{1}(Q_{1}^{L}+Q_{1}^{F})\}}{r-\mu}+A_{L}^{C}x^{\alpha}, x<xc,\frac{Q_{1}^{L}x\{1-\eta_{1}(Q_{1}^{L}+Q_{1}^{F})\}+Q_{2}^{L^{*}}x\{1-\eta_{2}(Q_{2}^{L^{*}}+Q_{2}^{F})\}}{r-\mu}-\gamma c_{2}Q_{2}^{L^{*}} x\geq x_{C},\end{array}$(2.27)

where smooth-fit condition at the trigger determines the following:$x_{C}(Q_{2}^{F})= \frac{\alpha(r-\mu)\gamma c_{2}}{(\alpha-1)\{1-\eta_{2}(Q_{2}^{L^{*}}+Q_{2}^{F})\}},$$A_{L}^{C}(Q_{2}^{F})=[ \frac{Q_{2}^{L^{*}}x_{C}\{1-\eta_{2}(Q_{2}^{L^{*}}+Q_{2}^{F})\}}{r-\mu}-\gamma c_{2}Q_{2}^{L^{*}}]x_{C}^{-\alpha}.$(2.28)
Having the trigger of cross licensing determined by the leader, we can delineate value functionof the follower of which novel technology can be utilized by his competitor. If agreed upon, hecompetes with the leader with regard to not only the first-generation technology but also thesecond generation technology from which he has raised monopoly profits, and thus the follower’s
value function can be written as follows:$V_{F}^{C}(x, Q_{1}^{L}, Q_{1}^{F}, Q_{2}^{F})=\{\begin{array}{ll}\frac{Q_{1}^{F}x\{1-\eta_{1}(Q_{1}^{L}+Q_{1}^{F})\}+Q^{F}x(1-\eta_{2}Q_{2}^{F})}{r-\mu}+A_{F}^{C}x^{\alpha}, x<x_{C},\frac{Q_{1}^{F}x\{1-\eta_{1}(Q_{\rceil}^{L}+Q_{1}^{F})\}+Q_{2}^{F}x\{1-\eta_{2}(Q_{2}^{L^{*}}+Q_{2}^{F})\}}{r-\mu}, x\geq x_{C},\end{array}$(2.29)

where value-matching condition at the trigger yields the following:$A_{F}^{C}(Q_{2}^{F})=- \frac{\eta_{2}Q_{2}^{L^{*}}Q_{2}^{F}x_{C}^{1-\alpha}}{r-\mu}$.

(2.30)The rest of the steps follow the same argument as the benchmark model: deriving valuefunctions of the two firms backwards for the follow-on and the initial investments. We omit
them here for brevity and they can be found in the original paper of Jeon (2015).

50
2.4

The whole picture

In the previous subsections, we illustrated a few possible scenarios from the present model separately. In Section 2.2, we focused on the benchmark model in which litigation over infringement
is the only way for the leader to recover his profits, and in Section 2.3, we integrated the agreement of cross licensing, which allows the stakeholders to utilize each other’s technology without
fear of infringement. In particular, the latter has been discussed under the condition that thefollower accepts the oﬀer, although the investment capacity of the leader might be either the
first-best or the second-best, and this implicitly presumes that the leader makes an oﬀer to thefollower.
In reality, however, this might not be the case. Namely, the leader might not oﬀer the contractof cross licensing and rather chooses to accuse the follower of infringement in spite of the legal
costs and the risks of losing the validity of his patent. To be more precise, the leader compares
his option value of litigation and that of cross licensing with the first-best capacity, if possible,

or with the second-best capacity, and chooses the one that maximizes his expected profits.
Furthermore, not only the way the firms resolve the dispute but also the sequential/simultaneous
occurrence of each event is determined for the sake of the stakeholders’ interests, provided thatthey hold for given parameters.

3

Comparative statics and discussion

3.1

Parameters

We adopt the following parameters as the benchmark case:$r=0.05$ ;$c_{1}=2$

3.2;$\mu=0.02$ ;$c_{2}=2$;$\sigma=0.2$$c_{L}=1$;;$\eta_{1}=0.4$$\gamma=0.7$;$\eta_{2}=0.6$$d=0.5$ ;;$x=0.1$ .

(3.1)

Market demand and investment decisions

2
2$t$$\wedge^{d’}$

$t$

di

$|$

(a) The level of triggers for each event(b) The investment capacities of the leader andthe follower

Figure 1: The comparative statics regarding the expected growth rate of market demand

51
Panel (a) of Figure 1 shows the level of triggers with regard to the expected growth rate, which
naturally reveals the sequential/simultaneous occurrence of the events. First of all, we can seethat the leader and the follower agree on cross licensing when the expected growth rate ofmarket demand is very low. It is obvious that when the market is not profitable enough, thefirms have less incentive to monopolize their technologies bearing the legal costs. Moreover, theintroduction of basic technology is so delayed that the follow-on research and cross licensing
are triggered simultaneously. That is, both parties start their business when the market is very
mature, and there is no diﬀerence in the level of technology adopted in their products.
As the expected growth rate increases, however, they fail to reach an agreement on crosslicensing and resolve the dispute over infringement at the court. Intuitively, they desire to takea larger share in the fast-growing market, which makes the agreement less likely to be reached.
Yet, the advent of basic technology is still deferred so that the follow-on research initiates assoon as the leader carries out the initial innovation, while the lawsuit is brought later on, andthis corresponds to (2.24). Panel (b) of Figure 1 represents the optimal investment capacities
of both parties, and we can see that as the law steps in the resolution of conflict, the follower
curtails his investment capacities in imitating the old technology significantly and augments
those devoted to enhancing the technology. This is because the follower knows that unless hedoes so, he is more likely to lose the case and to be driven out of the market. The implicationof this result becomes even clearer as the expected growth rate increases further.
If the expected growth rate of market demand is very high and the investment is lucrative
enough to reward the anticipated losses of profits from the competition with the follower, theleader does not delay his investment in innovation and patent acquisition. That is, (2.22) holds
when is very high. Furthermore, Panel (b) shows that as the expected growth rate increases
further, the follower devotes more investment capacity to yield improvement in technology rather
than (invent around” the existing one, and there are two reasons for this result. First, for given, the leader is more willing to accuse the follower of infringement as the market becomes more
profitable, and thus the follower focuses on developing his own technology so as not to provoke
a lawsuit that can possibly lead to a ban on his production. Recall that is endogenously
determined by (2.4), and that it decreases as more capacity of the subsequent investment isdevoted to yield improvement in technology. Second, if the market is lucrative enough, thefollower has stronger incentive to develop the technology of which novelty can only be foundin his product in order to raise monopoly revenue from it, rather than sharing profits from theold-fashioned technology with precursor.
We can also comprehend these results from the perspective of diﬀerentiated goods. If the
expected growth rate of market demand is so high that the competition between the firms ends
with a legal dispute, there is a distinct diﬀerence between the products manufactured by theleader and the follower unless the follower is found guilty at the court and driven out of themarket. The product of the leader, who has established the foundation from which progress
can be 1mde, is behind the technology, and the follower’s product is advanced in terms of thenovelty of technology adopted in it unless the subsequent innovation loses its legitimacy by theruling. Meanwhile, if the market demand is expected to grow very slowly, the advent of the
$\mu$

$p$

$p$

52
first-generation technology is so delayed that the follow-on research and the reconciliation via
cross licensing are triggered at the same time. This implies that the features of the productsin the market converge upon each other, leaving little diﬀerence between them from the very
beginning. This is consistent with what we can observe in the real world. There are a numberof diﬀerentiated goods in the burgeoning market, while the products become similar with eachother when the market loses its momentum in its growth.
To facilitate better understanding, let us take an example of the market of smartphones again.
A few years ago, the market started growing rapidly, and the giant IT firms such as Apple andSamsung manufactured their smartphones with distinctly diﬀerent features. Furthermore, theyused to accuse each other of infringement, even regarding the design of software, leading toa year-long patent war. As the growth of the market becomes slower, however, they chose tosettle the ongoing lawsuits and began taking advantage of the features of each other’s products,
flooding the market with similar products in terms of size, design, hardward, and even software
embedded in them. We can observe the similar tendency in the market of tablet PCs, where
demand has grown rapidly in the wake of smartphones. From these observations, we can easily
expect that the same sequence of events will occur in the market of smart wearables in which
the demand is about to explode in the very near future.
The present model eﬀectively integrates price-elasticity into the timing of investment, its
capacity, and the resolution of conflict. Thus, it is worth investigating the comparative statics
with regard to the price elasticity.

55$arrow\nearrow^{\nearrow}\prime’\nearrow\nearrow^{\nearrow\nearrow}$

45$R-\llcorner|.||m$$35$(a) The level of triggers for each event(b) The investment capacities of the leader andthe follower

Figure 2: The comparative statics regarding the price elasticity of the first-generation technology

Figure 2 presents the comparative statics with regard to , the price elasticity of the firstgeneration technology developed by the leader. Panel (a) shows that when the demand on thefirst-generation technology is very inelastic, they fail to reach an agreement of cross licensing.
Intuitively, when customers are willing to pay a high price regarding the first-generation technology, the leader has strong bargaining power in the negotiation, and thus claims a large sharein the market of the second-generation technology, which makes the agreement of cross licensing
less likely to be reached. As it becomes more price-elastic, the follower reduces his resources de$\eta_{1}$53
voted to imitate the first-generation technology from which less revenue is expected to be made,and this leads to the downward slope of(Panels (b) ). Note that the competition with thefollower in the market makes the leader defer his investment until the market becomes mature.
Afterexceeds a certain level, however, the firms succeed in resolving the problem without
a legal dispute because the leader requires less share in the market of the second-generation
technology. The follow-on research is delayed significantly so that the cross licensing is triggered
simultaneously, while the initial innovation is made individually. Furthermore, as we can see from
Panel (b), the follower curtails his investment devoted to enhancing the technology significantly
and rather chooses to invent around the old technology at a lower cost because he is no longer
afraid of a lawsuit over infringement.
If the price elasticity regarding the first-generation technology increases further, the follower
reduces the capacity of the first-generation technology and raises that of the second-generation
technology. This is a natural result because the investment in the first-generation technology
becomes less attractive, yielding less profits and raising the possibility of being driven out of
the market. In contrast, the leader reduces the investment capacities in both technologies. The
decrease of investment in the first-generation technology is obvious, and that in the secondgeneration technology results from the fact that the leader has less bargaining power in thenegotiation of cross licensing as the investment in the basic technology becomes less attractive.
In this subsection, we have examined the impact of market demand on the way the stakeholders resolve the conflict from various perspectives; its expected growth rate and the consumer’s
willingness to pay a high price. The bottom line was that the more lucrative the market is, themore the competitors are likely to go to court, and this finding is consistent with the argumentof a number of previous studies. For instance, Cooter and Rubinfeld (1989) delineated key determinants of legal disputes among the firms and argued that the probability of litigation risesin the size of the stakes. Lemley and Shapiro (2005) also noted that the patents involved ina lawsuit are those that are commercially important enough to endure the costs of litigation.
Empirical studies carried out by Lanjouw and Schankerman (2001) showed that more “valuable”
patents tend to involve alawsuit with much higher probability, and Somaya (2003) also provided
empirical evidence showing that the lawsuits are less likely to be settled if the stakeholders have$Q_{1}^{F}$$\eta_{1}$large stakes in the litigated patent.

3.3

Patent system and welfare analysis

One of the most significant features of the present model is that it endogenously determines how
much the subsequent innovation overlaps with the existing technology and how much improvement is made from it. This feature draws on the assumption that the probability of the existing
patent found to be valid by the court depends on the degree of overlapping and improvement,
and thus the follower takes this into account in deciding how his resources will be allocatedin the follow-on research in order to maximize his expected profits. Yet, there is still a roomfor discretion by the court represented by , except for the extreme cases in which there is no
imitation or no novelty $(i.e., Q_{1}^{F}=0 or Q_{2}^{F}=0)$ , and it can be read as the authorities’ patent
policy, that is, the extent to which the scope of patent is applied. It is of special interest how
$d$

54

$-\cup tk$$*_{*\sim_{\sim}}$(a) The level of triggers for each event(b) The investment capacities of the leader andthe follower(c) The probability that the patent is found to (d) Total expected surplus of each agent andbe valid at the courtsocial welfare

Figure 3: The comparative statics regarding the extent to which the patent scope is interpreted
for the base casethe policy on patent rights aﬀects the firms’ investment timing, their capacities, the way they
resolve the conflict, and thus the level of social welfare. Figure 3 represents the comparative
statics with regard to
We can see from Panel (a) that when the scope of patent is interpreted in a very narrow
sense so that its validity is acknowledged in a very limited case $(i.e.,$ when $d is$ very high) ,the stakeholders resolve the conflict via cross licensing. This is a natural result because theleader is less likely to win the case over infringement, and thus he rather opts for cross licensing
to raise extra revenues, even with a small market share as described in Panel (b). This canbe read in the context of Aoki and Hu (1999), which showed that a patentee may decide tolicense the technology to prevent imitation if the legal costs and probability of winning makethe patentee unable to credibly threaten the challenger with a lawsuit. Note that given theleader’s willingness to take advantage of the second-generation technology, the follower delays his
investment significantly until the market grows enough to reward the losses of revenue from thecompetition, and thus the follow-on research and the agreement of cross licensing are triggeredat the same time.
As the breadth of patent becomes wider $(i.e., as d$ decreases) , the leader holds a more
$d.$

55
dominant position in the negotiation of cross licensing, and this is represented by the increase ofand the decrease of
in Panel (b). When decreases further and falls below a certain level,
the leader chooses to accuse the follower, encouraged by the judge’s hard line against patent
infringement. Panel (c) shows that as decreases, the probability that the patentee wins thecase increases for a while, which is a natural result. When gets much lower, however, hasa tendency to decrease because the follower reduces
at a more rapid rate than. Thatis, when the patent protection is so strong that alledged infringement is found guilty even withlittle imitation of the patented technology, the follower devotes most of his resources to develop
his own technology. This leads to the increase of litigation trigger described in Panel (a).
Furthermore, Panel (a) shows that if the patent scope is wide enough $(i.e., d is low$ enough)to induce the leader to choose a lawsuit over cross licensing, the advent of the basic technology
is so delayed that it triggers the subsequent innovation at the same time because of the expected
losses of revenue from the follower’s challenge, while the legal dispute is triggered separately.
This is in sharp constrast with the case in which both parties agree on the cross licensing due
to the weak protection of patent rights $(i.e.,$ high value $of d)$ . Recall that, in that case, it is thefollow-on research that is delayed significantly so that both parties begin to take advantage ofthe advanced technology at the same time, while the initial innovation is carried out without
triggering any other events. This can be read in the view of the erosion of monopoly profits
and the firms’ response to it. In the former case, it is the leader of which monopoly revenue isthreatened by the competitor’s challenge, and thus the leader delays his investment until thedemand grows enough. In the latter case, however, now it is the follower of which monopoly
profit from his own technology is expected to be eroded by the competitor, and thus the follower
defers the subsequent innovation until the market becomes mature. This result has significant
implication from the perspective of public policy and social welfare. It implies that the policy
on patent scope cannot yield the first-best result in terms of the pace of technological progress.
This is in line with Scotchmer (1991) and Green and Scotchmer (1995), who have shown that,
under appropriate assumptions, it is impossible for both the initial innovator and the follow-on
innovator to have adequate incentives under a patent system.
The timing of the advent of technologies, however, is not the only determinant in the estimation of social welfare. That is, we need to take into account other aspects of the change inthe market as well in order to evaluate social welfare in a more accurate way. After a tediousalgebra, we can evaluate consumer surplus, producer surplus, and social welfare. We omit them
here, and they can be found in the original paper of Jeon (2015).
Panel (d) of Figure 3 depicts welfare analysis regarding the patent policy represented by. By comparing with Panel (a), we can easily see that consumer surplus and social welfare
are much higher when the dispute is resolved via cross licensing, and this is because it does
not involve legal costs and there are many more products with novel technology at much lower
prices. This finding is in line with Shapiro (2001), who argued that a royalty-free cross license
is the first-best result from the perspective of competition and that any cross license is superior to a world in which the stakeholders fail to cooperate. U.S. Department of Justice and theFederal Trade Commission (1995) also shed light on the procompetitive features of cross licens$Q_{2}^{L}$$d$$Q_{2}^{F}$

$d$

$d$

$Q_{1}^{F}$

$d$

$p$

$Q_{2}^{F}$

56
ing, noting that it helps the firms to integrate complementary technologies by clearing blocking
positions and to ease the burden of transaction costs and legal expenses. Denicol\‘o (2002) addressed that collusion between successive patentees through cross-licensing agreements might besocially beneficial under circumstances much less limited than in Chang (1995), which provided
only limited support for the permission on collusion, even if the patents are competing rather
than complementary or blocking.
Given these results, one might conclude that the narrower the patent scope is, the more
exceeds a certainsocial welfare we can yield. Yet, we can see from Panel (d) that after
level at which the firms agree on cross licensing, consumer surplus and social welfare gradually
decrease as increases further, that is, as the scope of patent becomes even narrower. At aglance, this might seem unnatural because they do not go to the court in which directly
matters. However, still comes into play if the agreement of cross licensing is made based on
the leader’s second-best capacity, which makes the follower indiﬀerent between litigation andcross licensing. Namely, when is relatively low yet still in the range that ensures the agreementof cross licensing, the patent is interpreted in a relatively wide sense and the leader has more
$d$

$d$

$d$

$d$

$d$

bargaining power in the negotiation, which enables him to raise investment capacity regarding
the second-generation technology. We can observe from Panel (b) that as increases further,, and this is because the follower has
decreases at a more rapid rate than the increase of
an incentive to manufacture fewer products to keep the price as high as possible. If is very
high and the patent can maintain its validity only in a very limited case, however, the follower,
who is aware of this, requires more rewards to accept the oﬀer of cross licensing, which leads
to less products with the advanced technology at a higher price. Thus, we can claim that the$d$$Q_{2}^{L}$$Q_{2}^{F}$$d$authorities’ stance against infringement still matters even if the dispute is resolved out of court
and that the narrow interpretation of patent rights to induce the agreement of cross licensing
does not always guarantee the improvement of social welfare.

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Graduate School of Economics
Osaka University, Toyonaka 560-0043, Japan-mail address: haejun.jeon@gmail.com$E$$\star\ovalbox{\tt\small REJECT}\ulcorner x\star_{\mp}^{A}.$$4^{\backslash }\prime\prime\backslash \not\geq_{\grave{(}}f^{\frac{\backslash \backslash \prime}{\neq-}}ffl_{y_{L}}^{*}7\}$

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