Patent infringement, litigation, and settlement

Economic Modelling 51 (2015) 99–111

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Patent infringement, litigation, and settlement☆ Haejun Jeon ⁎ Ohnishi Laboratory, Graduate School of Economics, Osaka University, 1-7 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan

a r t i c l e

i n f o

Article history: Accepted 27 July 2015 Available online 24 August 2015 Keywords: Patent R&D Infringement Litigation Settlement Real options

a b s t r a c t We propose a model that integrates a series of events regarding patent rights based on real option framework. After the incumbent has acquired a patent, it can be infringed by the challenger, and the conflict between them can be resolved via litigation or settlement with endogenously determined triggers and royalties. The model explains why litigation is so unusual in the real world and why most of the lawsuits over patent rights cease before the court's judgment is made. It also clarifies why roughly a half of litigated patents are found to be invalid in court and in what circumstances the introduction of new technology or the infringement of patent is delayed. From the perspective of implications on patent system, the model shows that neither tightening the patent examination nor widening the patent scope guarantees the acceleration of R&D investment, and rather delays it in some cases. © 2015 Published by Elsevier B.V.

1. Introduction Intellectural property is drawing attention more than ever. The number of patent application is growing steadily,1 and we can easily hear the news of patent wars in global business, such as “Apple vs. Samsung” and “Microsoft vs. Motorola.” Even a firm called a “patent troll” which collects patents and makes profits from the litigation against the infringement has appeared in the market. The patent holders, however, do not always win the trial; roughly a half of all litigated patents are found to be invalid.2 Furthermore, not every conflict over patent rights involves a lawsuit. In fact, a majority of the disputes are settled; some of them are resolved peacefully before they go to court, and others are settled in the middle of the trial. Given these various scenarios, it is natural to attempt to integrate a series of events regarding patent rights and to clarify the mechanism behind them. In this paper, we propose a model that illustrates strategic behaviors of the incumbent and the challenger regarding the patent acquisition, ☆ This version: July 25th, 2015. This work was supported by the Ishii Memorial Securities Research Promotion Foundation. We would like to thank Prof. Kyoko Yagi from Akita Prefectural University for the thoughtful advices at the 23rd conference of Nippon Finance Association. We are also grateful to Prof. Takashi Shibata, Prof. Takahiro Watanabe, and Assoc. Prof. Tarishi Matsuoka from Tokyo Metropolitan University for their insightful comments at the Economics Seminar of Tokyo Metropolitan University. ⁎ Tel./fax: +81 6 6850 5277. E-mail address: [email protected]. 1 Lemley and Shapiro (2005) noted that the U.S. Patent and Trademark Office granted nearly 200,000 new patents in 2003, a number which has roughly doubled over the past 15 years. 2 Allison and Lemley (1998) reported that 46% of patents have been found to be invalid in their empirical research, and Moore (2000) also supported this finding. AIPLA (2011) reported that patent holders' average success rate for the cases in the top ten industries from 1995 to 2010 is approximately 31%.

http://dx.doi.org/10.1016/j.econmod.2015.07.019 0264-9993/© 2015 Published by Elsevier B.V.

the infringement of it, and the way to resolve the dispute, such as litigation and settlement. After the incumbent has acquired a patent, the challenger can infringe it and make profits that are supposed to belong to the patent holder.3 The incumbent has an option to litigate against this, but it is a heavy burden in terms of the costs and the uncertainty of duration, which can lead to the withdrawal of it. Furthermore, if the patent is found to be invalid in the court's ruling, the incumbent gets nothing in spite of the cost it will have paid. For these reasons, the incumbent can choose to mediate with the challenger, rather than to litigate, offering a contract that claims a portion of the challenger's profit as settlement money or royalty. The challenger, of course, can reject the offer if the required royalty is so high that it is better for the challenger to be litigated, expecting the patent to be found invalid. Even if the dispute results in a lawsuit, the incumbent still has a chance to make a settlement as an alternative to withdrawal of the lawsuit, and whether it results in a successful settlement depends on the reservation royalty. Not only the choice between litigation and settlement but the timing of the R&D investment, patent infringement, and litigation or settlement is determined endogenously; the events can get earlier or be delayed, and some of them can occur simultaneously depending on the circumstances. The present model gives us insight with which to comprehend the patent war observed in the real world from various angles. First of all, it explains why litigation is so unusual considering the huge number of registered patents and the prevalent infringement of them and why most of the lawsuits over patent infringement cease before the court's judgment is made. As a matter of fact, only 1.5% of all patents are ever litigated, and only 0.1% are litigated to trial (e.g., Lemley and Shapiro, 2005). The present model effectively integrates the choice between 3 Pakes (1986) noted that only 10% of patents are in force for the whole patent life, and similar findings can be found in Lanjouw (1998).

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litigation and settlement with endogenously determined royalties, and shows that the incumbent chooses to mediate with the challenger, not go to court, provided the royalty is high enough; even if they fail to reach a mutually attractive “ex ante” settlement, the ongoing lawsuit can be ended via an “ex post” settlement, which is an alternative of the withdrawal of it. The fact that litigation hardly occurs does not imply that its impact is negligible; it is the incumbent's threat to litigate that makes the settlement possible; thus, we would rather say that it plays a crucial role in resolving the conflict over patent rights, no matter what means are chosen to resolve the dispute. In addition, the present model can also illustrate why roughly a half of the litigated patents are found to be invalid (e.g., Allison and Lemley, 1998; Moore, 2000). At a glance, this seems to be an unnatural phenomenon, considering that the patents had already passed the authorities' examination. Yet, numerical analysis of the present model shows that conflict over a patent in which novelty is not clear, that is, one that has a high probability of being found invalid in the court's ruling, tends to involve legal procedures to resolve the dispute, while that with higher novelty, which hardly loses its validity through the trial, is resolved via a settlement between the stakeholders. The rationale behind this result is as follows: when the patent's novelty is so clear that its validity is upheld with high probability, the incumbent is willing to file a lawsuit aggressively, and the challenger, who recognizes this substantive threat, pays a high royalty to the incumbent, which makes reaching a settlement more likely. In other words, the fact that the conflict over the infringement has not been resolved via a settlement reveals that the patent's novelty is not clear enough, which explains why such a lot of patents are found to be invalid at the court's ruling. Meanwhile, a couple of events can occur simultaneously in the present model, and it helps us to comprehend in what circumstances the introduction of new technology or the infringement of patents is delayed. For instance, a patent with clear novelty not only makes the conflict resolvable via a settlement but delays the infringement significantly by the incumbent's strong threat to litigate. Also, the incumbent defers investing in R&D projects considerably when the degree of infringement is substantial or it costs too much to acquire patent rights; this corresponds to “submarine patents,” which refers to the intentional delay in the issuance of patents to take a mature industry by surprise (e.g., Graham and Mowrey, 2004; Moore, 2004). Furthermore, it is shown that the ex ante settlement tends to be triggered instantaneously by the challenger's delayed infringement, whereas the litigation usually occurs sequentially. This explains why we hear the news of infringement and litigation for such a long time; not only the duration of the process but the time it takes to be triggered is much longer in litigation than in settlement. The model also provides us normative implications on patent system. Given all the social losses involved with patent litigation, one might think that the authorities need to tighten patent examination so that only patents with clear novelty are issued and the conflict over them can be resolved promptly via settlement. Our model, however, shows that it can result in the distortion of the firm's incentive to introduce new technology. Tightening the patent examination usually involves the increase of costs needed for the acquisition of patent rights because not only the examination fees but also the R&D costs increase to develop a technology that is novel enough to pass a strict screening, and the comparative statics of the costs show that its increase can result in “submarine patents,” the last thing the authorities want, while the timing of R&D investment is independent of the patent's novelty. It is also easy to think that the authorities should widen the scope of patents to accelerate R&D investment, but the comparative statics show that the trigger is mostly independent of patent scope, and it can rather delays the introduction of new technology in some cases. It is the increase of the expected growth rate of R&D project that guarantees the acceleration of the investment, and this implies that what really matter is the fundamental features of the R&D project.

A huge number of papers have adopted the framework of real options for examining R&D investment, especially in the presence of preemptive incentives (e.g., Hsu and Lambrecht, 2007; Lambrecht and Perraudin, 2003; Leung and Kwok, 2011; Miltersen and Schwartz, 2004; Pawlina and Kort, 2006; Weeds, 2002). These studies, however, have paid little attention to the risk of litigation, which impacts firms' strategies significantly4; some of them did not identify R&D investment with the acquisition of patents, and others have neglected it for simplicity. Marco (2005) articulated the option value of patent litigation, but the model involved strong exogeneity, interpreting the decrease of a diffusion process by exogenous shocks as the infringement of a patent, which triggers litigation. In contrast, the present model clarifies a series of events that occur regarding patent rights in reality, such as patent acquisition, patent infringement, and resolution of the conflict via litigation or settlement, and the triggers of each event are determined endogenously. Much has been written about the stakeholders' choice between litigation and settlement. Earlier works such as Landes (1971), Gould (1973), and Posner (1973) described the economic incentives underlying the litigation process, and subsequent studies (e.g., Bebchuk, 1984; P'ng, 1983; Reinganum and Wilde, 1986; Salant, 1984; Schweizer, 1989; Shavell, 1982) introduced asymmetric information with respect to the probability of winning the trial, the amount of damages, or even whether the violation is genuine, and elucidated the parties' strategic behavior based upon them. In most cases, however, the set of firms' action is quite restricted; for instance, neither withdrawal of a lawsuit nor ex post settlement is integrated, and the settlement is dealt with an exogenous royalty. In contrast, the present model provides a fuller discussion in terms of firms' actions; not only the set of actions is wider but also some of them can occur simultaneously. A vast literature is dedicated to the optimal design of patent policy. The seminal works include Nordhaus (1969), Scherer (1972), and Nordhaus (1972), and their debates on the optimal “length” of a patent have led subsequent studies to investigate the impact of patent “length” and “breadth” upon social welfare. For instance, Gilbert and Shapiro (1990) claimed that the optimal length of a patent can easily be infinite because of the deadweight loss from increasing its breadth, and Klemperer (1990) clarified a couple of conditions under which infinitely-lived, narrow patents and short-lived, broad ones are optimal, respectively. Denicolò (1996) further extended these works to the case in which winner-take-all may not hold. Yet, there also have been a great deal of studies which cast doubt on whether a patent system accelerates R&D investment. Scotchmer and Green (1990), Takalo and Kanniainen (2000), and Hunt (2006) presented a theoretical framework which shows that the patent system, or even the reform of it, can result in the delay of innovation, and there is a growing body of empirical evidence that strongly supports these claims, such as Sakakibara and Branstetter (2001), Jaffe and Lerner (2004), Qian (2007), and Lerner (2009). They consistently reported the lack of positive impact of changes in patent policy on innovation, which reveals why we need to pay scrupulous attention to the reform of patent system. The remainder of the present paper is organized as follows. The setup of the model is explained in Section 2.1, and the model which does not include the settlement is demonstrated as a benchmark model in Section 2.2. The main model which encompasses both litigation and settlement is illustrated in Section 2.3. Based on the model and its solution given in Section 2, comparative statics and the discussion regarding the results are presented in Section 3. The parameters adopted for numerical calculation are introduced in Section 3.1, and the comparative statics with respect to the features of the R&D project and those regarding the litigation process are presented in Section 3.2 and Section 3.3, respectively. Section 3.4 presents a few implications of the patent system, and conclusion is given in Section 4. 4 Schankerman (2001) argued that the costs of engaging in litigation can lower firms' incentive to invest in R&D projects significantly.

H. Jeon / Economic Modelling 51 (2015) 99–111

2. The model and solutions 2.1. Setup Suppose that there are two risk-neutral firms in the market, the incumbent and the challenger,5 and that the incumbent has an irreversible investment opportunity to develop and commercialize a product of which demand shock is given by one-dimensional geometric Brownian motion as follows: dX t ¼ μX t dt þ σ X t dW t ;

ð1Þ

where μ and σ are positive constants and (Wt)t ≥ 0 is a standard Brownian motion on a filtered probability space ðΩ; F ; F :¼ ðF t Þt ≥ 0 ; ℙÞ satisfying the usual conditions. A risk-free rate is assumed to be a constant r N μ to ensure the finiteness of value function. When the demand shock exceeds a threshold, the incumbent will invest in R&D and acquire a patent which allows the firm to monopolize the product. This investment involves a positive constant costs cP, and the incumbent makes a profit at the rate of Π1Xt. For simplicity, we assume that the patent is acquired immediately6 and it does not have an expiry date.7 Even though the legal rights to monopolize the product are granted to the incumbent, the challenger can still infringe the patent rights and participate in the market, expecting that it can be litigated by the incumbent. It costs the challenger a positive constant cI(≤ cP) to imitate the incumbent's technology, and the challenger makes a profit at the rate of ΠC2Xt while the incumbent's profit decreases from Π1Xt to ΠI2Xt. For simplicity, we assume that the sum of two firms' profit equals the monopoly profit and denote the degree of the infringement by θI (i.e., Π1 = ΠI2 + ΠC2 where ΠC2 = θIΠ1). Having lost a portion of its profit, the incumbent can consider a couple of ways to resolve the problem. One that carries legal binding power and can possibly make a full recovery of the original profit is to litigate the challenger for infringing the patent rights, but the litigation process is uncertain in many ways. First of all, a patent holder does not always win the trial. In the real world, it is by no means rare that the patent turns out to be invalid at the court's judgment (e.g., AIPLA, 2011; Allison and Lemley, 1998; Moore, 2000). Furthermore, it is usually uncertain how long it will take for the court to reach a ruling, which implies that the litigation cost is also uncertain and can be significant. With this in mind, we postulate that it takes an exponential time τ to reach the court's judgment, and it costs both the plaintiff and the defendant cL continuously until the court's decision is made.8 The exponential time τ is assumed to be independent from (Wt)t ≥ 0, and its rate parameter is denoted by λ, which implies that the expected duration of the litigation process is 1/λ. By the court's ruling, the patent is found to be valid with probability p,9 and the incumbent recovers the monopoly profit Π1Xt while the challenger cannot make a profit thereafter. For tractability, we assume that there is no compensation for the infringement the challenger has incurred. With probability (1 − p), however, the patent turns out to be invalid, and the profits of the incumbent and the challenger keep the status quo in spite of the litigation costs the incumbent will have paid. By the memoryless property of 5 We assume that their role is predetermined to avoid discussing the preemption incentive. 6 There are a number of papers that separate the stages of R&D investment and patent acquisition, those of patent acquisition and the commercialization on thereof, or even all of them (e.g., Hunt, 2006; Leung and Kwok, 2011; Takalo and Kanniainen, 2000). 7 Reiss (1998) also adopted this assumption. Gilbert and Shapiro (1990) and Tandon (1982) showed that the optimal length of a patent can easily be infinite. 8 The case in which each party bears his own costs is called the American system, and that in which the losers bear all the costs is called the British system. For a fuller discussion of the allocation of legal costs, see Shavell (1982) and Reinganum and Wilde (1986). 9 Farrell and Shapiro (2008) also presumed “probabilistic patents.” Llobet (2003) and Jeon (2015) endogenize the probability that the patent is found to be valid at the court by assuming that the follower makes a further progress of technology based on the leader's one and that the court takes the improvement into account.

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exponential distribution, the expected time left to the judgment is always 1/λ no matter how long it has been since the initiation of the litigation, and thus, it is natural to expect that the incumbent can withdraw the litigation before the court's ruling if its profit decreases significantly and the litigation cost becomes a heavy burden. Once the litigation is withdrawn, the incumbent can never litigate the challenger again. Meanwhile, the incumbent can also consider participating in mediation with the challenger to recover its profit because the settlement is relatively costless in the sense that not only the cost itself, denoted by cS, is much lower than the expected cost of the litigation cLλ/(r + λ)2, but the procedure is much simpler, and thus ends instantaneously. There are two different types of settlement; “ex post” settlement which is chosen instead of the withdrawal of ongoing lawsuit, and “ex ante” settlement which is adopted in lieu of the litigation process itself. In either case, the incumbent offers a contract which forces the challenger to give a portion of its profit to the incumbent as a royalty henceforth, but one can easily expect that the required royalty in “ex post” settlement, denoted by θS, is different from that in “ex ante” settlement, denoted by ^θS . The challenger, of course, can reject this offer if θS or θS is so high that it is better for the challenger to expect the litigation to be withdrawn or for the patent to be found invalid during litigation. The sequence of the stakeholders' actions is summarized in Fig. 1. 2.2. The benchmark model In this subsection, we suppose that a lawsuit and its withdrawal constitute the only way to recover the incumbent's monopoly profit and to ease the burden of litigation costs, respectively, to outline the framework succinctly. As usual, the option values of the incumbent and the challenger will be determined backwards. For now, we presume that each event in Fig. 1 occurs sequentially, and the cases in which some of them are triggered simultaneously will be presented later.

Fig. 1. The sequence of actions of the incumbent and the challenger.

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First, suppose that the incumbent has litigated the challenger for the patent infringement. The expected profit of the incumbent waiting for the court's ruling is as follows: Z Et

tþτ t

e−rðs−t Þ ΠI2 xs dsþ

Z

Z

∞ tþτ

n o  e−rfs−ðtþτÞg pΠ1 xs þ ð1−pÞΠI2 xs ds

 Z tþτ o   ¼ Et pΠ1 þ ð1−pÞΠI2 xs ds− e−rðs−tÞ p Π1 −ΠI2 xs ds ð2Þ t t " #   1 Π I2 1 I − xt ; þ p Π1 −Π2 ¼ r−μ r þ λ−μ r−μ ∞n

where xt is the solution of Eq. (1). Meanwhile, the expected cost of the litigation is as follows: Z

∞ 0

tcL λe−ðrþλÞt dt ¼

cL λ ðr þ λÞ2

:

2

∂v 1 2 2 ∂ v þ σ x ∂x 2 ∂x2

ð4Þ

of which solution takes the form as follows: vðxÞ ¼ Axα þ Bxβ

ð5Þ

where α and β are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     1 μ 1 μ 2 2r 1 μ 1 μ 2 2r − 2 þ 2 N 1; β ¼ − 2 − − 2 þ 2 b 0: ð6Þ α¼ − 2þ 2 σ 2 σ 2 σ 2 σ σ σ

As mentioned earlier, the expected time left to the court's ruling does not change no matter how long it has been since the initiation of litigation because of the memoryless property of exponential distribution. And thus, the incumbent has an incentive to withdraw the litigation going on when its profit decreases and hits a lower threshold, denoted by xW. Once withdrawn, the profits of them go back to the level before the litigation, and the incumbent does not have an option to litigate again. Based on the arguments in Eqs. (2), (3), and (5), the incumbent's value function facing the decision of withdrawal can be described as follows: 8 I Π x > > > 2 ; < r−μ W ( ) V I ðxÞ ¼ I   1 > 1 cL λ βλ > Π 2 þ p Π −ΠI > x þ BW − ; 1 : 2 I x − r−μ r þ λ−μ r−μ ðr þ λÞ2

x ≤ xW ; x N xW ;

ð7Þ where β λ denotes β in Eq. (6) with (r + λ) instead of r. The withdrawal trigger xW and the coefficient of the option value BW I are determined by value-matching and smooth-pasting conditions at the trigger, and can be written as follows:

xW ¼

 −1 β cλ 1 1  λ L  − ; ðβλ −1Þp Π 1 −Π I2 ðr þ λÞ2 r−μ r þ λ−μ "

BW I

 #   1 1 −β I − xW xW λ : ¼ −p Π1 −Π2 r−μ r þ λ−μ ðr þ λÞ2 cL λ

8 > ΠC x > > x ≤ xW ; < 2 ; r−μ  

VW C ðxÞ ¼ 1 1 1 cL λ > βλ > > ΠC2 x þ BW ; x N xW : : r−μ −p r−μ − r þ λ−μ C x − ðr þ λÞ2

ð10Þ The value matching condition at xW determines the coefficients of the challenger's option value as follows: "  #  1 1 cL λ −β − ΠC2 xW þ BW xW λ : C ¼ p r−μ r þ λ−μ ðr þ λÞ2

ð11Þ

ð3Þ

By the standard argument, the option value of the firm, v(x), satisfies the following ordinary differential equation: rv ¼ μx

provided it is litigated by the incumbent can be obtained by the similar argument in Eq. (2), and the value function can be written as follows:

Now we proceed to the analysis regarding the incumbent's decision to litigate provided the challenger has infringed the patent rights. We can easily guess that there is an upper threshold which triggers the litigation, denoted by xL, and following the argument in Eq. (7), the incumbent's value function at this stage can be obtained as follows: 8 I > < Π2 x þ ALI xα ; x b xL ; V LI ðxÞ ¼ r−μ > : V W ðxÞ; x ≥ xL : I

ð12Þ

By value-matching and smooth-pasting conditions, the litigation trigger xL is determined by the following nonlinear equation:    1 1 cL λ βλ ð1−α Þp Π1 −ΠI2 ¼ 0; − xL þ ðβλ −α ÞBW I xL þ α r−μ r þ λ−μ ðr þ λÞ2

ð13Þ and the coefficient ALI is obtained as follows: "

ALI

#    1 1 cL λ I W βλ − xL þ BI xL − ¼ p Π1 −Π2 x−α L : ð14Þ r−μ r þ λ−μ ðr þ λÞ2

Having the litigation trigger determined by the incumbent, the challenger's value function at this stage can be written as follows: 8 C > < Π2 x þ ALC xα ; V LC ðxÞ ¼ r−μ > : V W ðxÞ; C

x b xL ;

ð15Þ

x ≥ xL ;

where the coefficient ALC is determined by value-matching condition at xL as follows: "  ALC ¼ p

#  1 1 cL λ βλ − ΠC2 xL þ BW x − x−α L : C L r þ λ−μ r−μ ðr þ λÞ2

ð16Þ

ð8Þ

ð9Þ

Having the withdrawal trigger determined by the incumbent, now we can calculate the value function of the challenger expecting the withdrawal of the ongoing lawsuit. The expected profit of the challenger

Next, we analyze the challenger's decision to infringe the patent rights and participate in the market provided the incumbent has invested in R&D and has acquired the patent. It is reasonable to guess that there is an upper threshold which triggers the infringement, denoted by xI, and following the similar argument, the challenger's value function can be obtained as follows: ( V IC ðxÞ ¼

x b xI ; AIC xα ; V LC ðxÞ−cI ; x ≥ xI ;

ð17Þ

H. Jeon / Economic Modelling 51 (2015) 99–111

and the infringement trigger xI and the coefficient AIC can be calculated by value-matching and smooth-pasting conditions as follows: xI ¼

α ðr−μ ÞcI

; C

ðα−1ÞΠ 2

" AIC ¼

# Π C2 xI þ ALC xαI −cI x−α I : r−μ

ð18Þ

Having the infringement trigger determined by the challenger, the incumbent's value function at this stage is obtained as follows: 8 < Π 1 x þ AI x α ; I I V I ðxÞ ¼ r−μ : L V I ðxÞ;

x b xI ;

ð19Þ

x ≥ xI ;

where the coefficient AII is obtained by value-matching condition as follows: AII ¼ ALI −

Π 1 −Π I2 1−α xI : r−μ

ð20Þ

Lastly, we proceed to the incumbent's decision to invest in R&D and acquire the patent. With an upper trigger denoted by xP, its value function can be obtained as follows: ( V PI ðxÞ ¼

brevity. It is possible that both Eqs. (24) and (25) hold, which implies that the challenger has an option to choose the timing of infringement, and the challenger will choose the trigger xI that maximizes its option value. Meanwhile, the following inequality can also hold depending on parameters: xI ≤xP ≤xL :

ð28Þ

This implies that the incumbent delays the R&D investment so that the challenger's infringement of the patent rights is triggered right after the patent acquisition. If this is the case, the incumbent's value function in Eq. (21) is replaced with ( V PI ðxÞ ¼

x b xP ; API xα ; V LI ðxÞ−cP ; x ≥ xP ;

API xα ; x b xP ; V II ðxÞ−cP ; x ≥ xP ;

ð21Þ

α ðr−μ ÞcP xP ¼ ; ðα−1ÞΠ 1

API

  Π 1 xP þ AII xαP −cP x−α ¼ P : r−μ

ð22Þ

So far, we have implicitly assumed that the following inequalities hold regarding the triggers: xW ≤ xL

ð23Þ

and xP ≤ xI ≤ xL :

ð24Þ

This assumption has enabled us to consider the sequential processes of patent acquisition, infringement, litigation, and withdrawal. Even though Eq. (23) always holds as one can easily deduce by comparing Eq. (7) with Eq. (12), this is not the case for Eq. (24). Namely, xP ; xL ≤xI

ð25Þ

can also hold, which implies that the challenger delays the infringement, and the litigation initiates as soon as the challenger participates in the market. Then, the value function of the challenger and the incumbent at the infringement stage change from Eq. (17) and Eq. (19) to x b xI ; AIC xα ; ð x Þ−c ; x ≥ xI ; VW I C

V IC ðxÞ

¼

V II ðxÞ

8 < Π 1 x þ AI x α ; I ¼ r−μ : W V I ðxÞ;

x b xI ;

ð26Þ

ð27Þ

x ≥ xI :

The trigger xI and the coefficients of the option value AIC and AII are determined in the same manner, and they are omitted here for

ð29Þ

where the investment trigger xP and the coefficient of option value API are determined in the same manner. Furthermore, the incumbent can delay the investment even more so that the following inequality holds: xI ; xL ≤ xP :

where the trigger xP and the coefficient API can be calculated by valuematching and smooth-pasting conditions as follows:

(

103

ð30Þ

In this case, the challenger's infringement and the incumbent's litigation against it are followed immediately as soon as the incumbent acquires the patent, whereas the litigation is triggered sequentially in Eq. (28), and the incumbent's value function is described as follows: ( V PI ðxÞ

¼

API xα ; VW I ðxÞ−cP ;

xbxP ; x≥xP ;

ð31Þ

where the trigger xP and the coefficient API are determined similarly. Note that it is the incumbent who induces the immediate infringement in Eqs. (28) and (30), whereas it is the challenger in Eq. (25), and these scenarios occur because of the challenger's strong incentive to infringe or the incumbent's burden of costs. In short, the incumbent has three cases to choose from: the one chosen from Eqs. (24) and (25) by the challenger, Eq. (28), and Eq. (30). The incumbent, of course, chooses the one that maximizes its option value, provided that the candidates hold for the parameters. 2.3. The main model Having outlined the framework, now we proceed to the main model which encompasses the incumbent's option to make a settlement. As has been noted earlier, the litigation process is costly in many ways; the duration of the litigation is uncertain, and thus, so are the costs. Furthermore, with probability (1 − p) the incumbent gains nothing in spite of the costs it has paid. For these reasons, the incumbent has an incentive to mediate with the challenger; in the settlement process, the cost is much lower than that of litigation and there is no uncertainty in the procedure. There are two different types of the settlement; “ex post” settlement which is chosen instead of the withdrawal of litigation, and “ex ante” settlement which is adopted in lieu of the litigation process itself. First, we analyze the case with the former because it can be easily illustrated by modifying the benchmark model, and then we examine the case with the latter which can be easily drawn from the former case. The whole picture that integrates all the possible scenarios will be described at the end of this subsection. The argument follows similar arguments, and thus, we only enumerate the value function and the value of the triggers and the coefficients to avoid repetition of the same words.

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2.3.1. With ex post settlement While the litigation is going on, the incumbent's profit can decrease significantly so that the cost involved in the lawsuit becomes a heavy burden. The incumbent can withdraw the litigation to hold down the costs, but it is also possible to mediate with the challenger; namely, the incumbent offers a contract that claims a portion of the challenger's profit, denoted by θS , as settlement money or royalty henceforth. If the settlement agreement is reached, the incumbent and the challenger I

C

make a profit at the rate of Π2 x :¼ ðΠI2 þ θS ΠC2 Þx and Π2 x :¼ ð1−θS ÞΠC2 x, respectively, from then on. The challenger, of course, can reject the offer if the incumbent requires too much royalty, that is, if θS is too high, and we denote the highest royalty that the challenger is willing 

to pay in the ex post settlement by θS . If the challenger does not agree on the settlement, the only way for the incumbent to hold down the litigation cost is to withdraw the lawsuit. For now, we only examine the case in which the challenger accepts the settlement offer, that is,  for θS ≤θS, and we will see the whole picture at the end of this subsection.

In fact, the argument is exactly the same as the benchmark model except that the firm value at the stage of withdrawal needs to be modified. The settlement is proposed by the incumbent, and thus, the firm value and the trigger of the incumbent in Eqs. (7) to (9) are replaced with the following: 8 I > > > Π 2 x −c ; > x ≤ xS ; S < r−μ V SI ðxÞ ¼ ( I  )   > Π2 1 1 cL λ S > I > x þ BI xβλ − ; x N xS ; > : r−μ þ p Π1 −Π2 r−μ − r þ λ−μ ðr þ λÞ2

ð32Þ

2.3.2. With ex ante settlement Now we proceed to the case in which the costly legal procedure is not involved and the settlement is made instead; the incumbent offers a contract that forces the challenger to give a fraction of its profit as a royalty before the litigation process begins. With the ex ante settlement, the incumbent can only make a partial recovery of its profits, but it is relatively costless in terms of the duration of the procedure and the monetary cost. As before, we only examine the case in which the challenger accepts the settlement offer for now, leaving the whole picture to be shown at the end of this subsection. Following a similar argument, the value function of the firms facing the option to mediate with each other before the initiation of a lawsuit can be represented as follows: 8 I Π2 x ^S α > > > < r−μ þ AI x ; x b ^xS ; ^S V I ðxÞ ¼ ^Ix >Π > > : 2 −cS ; x ≥ ^xS ; r−μ

ð37Þ

8 C Π x ^S α > > > 2 þA < Cx ; r−μ ^S V C ðxÞ ¼ C ^ x > Π > > : 2 −cS ; r−μ

ð38Þ

α ðr−μ ÞcS ^xS ¼  I ; ^ −Π I ðα−1Þ Π

where βλ

xS ¼

−cS ðr þ λÞ2 #;    1 1 θS Π C2 I − − ðβλ −1Þ p Π 1 −Π 2 r−μ r þ λ−μ r−μ "

cL λ

(

ð34Þ Having the trigger determined by the incumbent, the firm value of the challenger in Eqs. (10) and (11) is changed as follows: 8 C > Π x > > < 2 −cS ; S r−μ  

V C ðxÞ ¼ > 1 1 1 cL λ S > > −p − ΠC2 x þ BC xβλ − ; : r−μ r−μ r þ λ−μ ðr þ λÞ2

:

ð33Þ

¼

r−μ

 8 I ^ −Π I ^xS < Π 2 2 :

r−μ

9 = −α −cS ^xS ; ;

9 = −α −cS ^xS : ;

ð39Þ

ð40Þ

With these results of the stage of ex ante settlement, now we proceed to the argument regarding the challenger's infringement. To distinguish the values from the benchmark model, we denote a hat on the values here. Following the same argument, the value functions of the challenger and the incumbent at this stage can be obtained as follows: ( V IC ðxÞ ¼

x ≤ xS ; x N xS ;

^S A I

2

 8 C ^ −Π C ^xS < Π 2 2

^S ¼ A C

) #    1 1 θS Π C2 −β I − ¼ −cS − p Π1 −Π2 xS xS λ : − r−μ r þ λ−μ r−μ ðr þ λÞ2 "

S BI

!

x ≥ ^xS ;

where

2

cL λ

x b ^xS ;

V II ðxÞ

^ C xα ; A I

x b ^xI ;

^

V SC ðxÞ−cI; x ≥ ^xI ;

8 < Π1 x þ A ^ I xα ; I ¼ r−μ : ^S V I ðxÞ;

x b ^xI ;

ð41Þ

ð42Þ

x ≥ ^xI ;

ð35Þ where where "

S BC

#   θS Π C2 1 1 cL λ −β C − Π2 xS þ ¼ − xS þ p −cS xS λ :ð36Þ r−μ r þ λ−μ r−μ ðr þ λÞ2

The rest of the steps follow exactly the same as those in the benchmark model. So far, we have implicitly assumed that the challenger accepts the incumbent's offer, but it needs a condition to be held; the challenger agree on the ex post settlement if the option value of ex post settlement S BC ≥BW C .

in Eq. (36) is larger than that of withdrawal in Eq. (11), that is, The highest royalty that makes this condition hold is the reservation 

royalty, and is denoted by θS , which can be obtained by numerical calculation.

^xI ¼

α ðr−μ ÞcI ðα−1Π ÞC2

^C ¼ A I

;

! Π C2 ^xI ^ C α −α ^ þ AS xI −cI ^xI r−μ

ð43Þ

I

^ S − Π 1 −Π 2 ^x1−α : ^I ¼ A A I I I r−μ

ð44Þ

Lastly, the value function of the incumbent having the option to invest in R&D and acquire patent rights can be written as follows: ( V PI ðxÞ ¼

^ P xα ; A I V II ðxÞ−cP ;

x b ^xP ; x ≥ ^xP ;

ð45Þ

H. Jeon / Economic Modelling 51 (2015) 99–111

incumbent's value function changes as follows:

where ^xP ¼

105

α ðr−μ ÞcP ; ðα−1ÞΠ 1

^P ¼ A I

  Π 1 ^xP ^ I α −α þ AI ^xP −cP ^xP : r−μ

ð46Þ

So far, we have implicitly assumed that the following inequality holds among the triggers: ^xP ≤ ^xI ≤ ^ xS :

ð47Þ

This has enabled us to focus on the case in which each event occurs sequentially, but this is not the only case that exists. Depending on the parameters, the following inequality can also hold: ^xP ; ^xS ≤ ^xI ;

ð48Þ

8 P ^ xα ; > x b ^xP ; : 2 −cS −cP ; x ≥ ^xP ; r−μ

ð54Þ

where the trigger ^xP and the coefficient ÂPI are determined similarly. In brief, the incumbent has three cases to choose from: the one chosen from Eqs. (47) and (48) by the challenger, Eq. (51), and Eq. (53). The incumbent, of course, chooses the one that maximizes its option value from the candidates, provided they hold. As the case with ex post settlement, a similar condition needs to be satisfied for the challenger to accept the offer; the option value of ex ante settlement in Eq. (40) needs to be larger than that of litigation in Eq. (16) (i.e., ÂSC ≥ ALC), and the highest royalty that makes this condition  hold is denoted by ^θ , which can be obtained by numerical calculation. S

which implies that the challenger delays its infringement, and the ex ante settlement is made instantaneously after the infringement occurs.  Note that the reservation royalty ^θ for this case is different from that in S

Eq. (47). If this is the case, the value functions of the challenger and the incumbent in Eqs. (41) and (42) are changed as follows:

V IC ðxÞ

8 C ^ xα ; > : 2 −cS −cI ; r−μ

8 Π1 x ^I α > > > < r−μ þ AI x ; I V I ðxÞ ¼ ^Ix >Π > > : 2 −cS ; r−μ

x b ^xI ; x ≥ ^xI ;

ð49Þ

2.3.3. The whole picture So far, we have considered a couple of scenarios separately: The one without options to mediate with each other was illustrated as a benchmark model in Section 2.2, and those with ex post and ex ante settlement were examined in Section 2.3.1 and Section 2.3.2, respectively. In particular, we have examined in Section 2.3 under the conditions that make the challenger to accept the incumbent's offer, which implicitly assumes that the incumbent makes an offer to the challenger. In reality, however, the incumbent might be better off withdrawing   the litigation or litigating the challenger if θ or ^θ is not high enough, S

S

respectively. To be more precise, the incumbent compares its option x b ^xI ;

S

ð50Þ x ≥ ^xI :

where the trigger ^xI and the coefficient ÂCI and ÂII are determined following similar arguments. If both Eqs. (47) and (48) hold, the challenger will choose the one that maximizes its option value. Meanwhile, it is also possible that the incumbent delays the investment, possibly because of the challenger's strong incentive to infringe or the cost burden, and the following holds



value of withdrawal BW I and that of ex post settlement BI with θS and decides whether to withdraw the litigation or to make an offer of the ex post settlement. Having the ex post decision determined, the incumbent now compares its option value of litigation ALI and that of ex ante  settlement ÂSI with ^θ to determine whether to go to court or to particS

ipate in mediation with the challenger. Each decision is made based on the inequality of the triggers determined by maximizing the stakeholders' option values. This completes the whole picture. 3. Comparative statics and discussion 3.1. Parameters

^xI ≤ ^xP ≤ ^xS ;

ð51Þ

which implies that the infringement is triggered instantaneously by the incumbent's investment. If this is the case, the value function of the incumbent at the initial stage in Eq. (45) is changed as follows: ( V PI ðxÞ ¼

^ P xα ; A I ^ V SI ðxÞ−cP ;

x b ^xP ; x ≥ ^xP ;

ð52Þ

where the trigger ^xP and the coefficient ÂPI are determined in the same manner.Furthermore, the incumbent can delay its investment even more so that following inequality holds: ^xS ; ^xI ≤ ^ xP :

ð53Þ

In this case, the challenger's infringement and the settlement between the parties initiate as soon as the incumbent acquires patent rights, whereas the mediation occurs sequentially in Eq. (51), and the

We adopt the following parameters as the benchmark case: r ¼ 0:05; θI ¼ 0:3;

μ ¼ 0:02; p ¼ 0:5;

σ ¼ 0:3; λ ¼ 1=2:5;

cP ¼ 3; cL ¼ 1;

cI ¼ 1:5; cS ¼ 0:5:

The risk-free rate and the coefficients of a diffusion process are in the range of the usual ones in real options literatures. The rate parameter of exponential time is adopted to reflect the fact that litigation process usually takes two to three years to reach the court's ruling (e.g., AIPLA, 2011). As has been noted earlier, roughly a half of litigated patents are found to be invalid (e.g., Allison and Lemley, 1998; Moore, 2000), and thus, we suppose that the patent has a 50% chance of having its validity upheld by the court. In terms of the degree of infringement, we assume that the incumbent loses 30% of its profit by the challenger's infringement, and the cost parameters are also chosen at our discretion, preserving the fact that the expected cost of litigation is higher than that of settlement.10 10 We regret to admit that empirical analysis is not given in this study due to the lack of quantitative data regarding patent litigation and settlement. In particular, a lawsuit over infringement is a very rare event considering the huge amount of patents issued every year, which makes it even harder to carry out the analysis based on data. It is to be hoped that numerical analysis given in this section will intrigue the practitioners with the access to necessary data for empirical studies.

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H. Jeon / Economic Modelling 51 (2015) 99–111

One of the most important parameters in our model is p, the probability that the patent is found to be valid at the court's ruling, and we can interpret this as the patent's novelty. Fig. 2 presents the comparative statics regarding p in terms of the triggers, option values, and reservation royalty. Panel (a) of Fig. 2 shows that when p is low, the incumbent files a lawsuit to recover its profit, rather than participates in mediation with the challenger, and the timing of litigation gets earlier as p increases, which reveals the incumbent's confidence to win the trial. After p exceeds a certain level, however, they make an ex ante settlement to resolve the conflict over the infringement. Not only the way they resolve the problem but the timing of the challenger's infringement changes; provided the ex ante settlement is adopted, the challenger defers the infringement substantially so that the settlement begins as soon as the

challenger participates in the market, which corresponds to Eq. (48), and it is delayed even more as p increases thereafter. We can interpret this result as follows: when the patent is not novel enough and it has a high probability of losing its validity in court, the incumbent is reluctant to file a lawsuit, and the challenger, who recognizes this, is not willing to pay large royalty, which makes the settlement less likely to be reached. However, the reservation royalty increases as the patent's novelty becomes clearer because the challenger is aware of the fact that the incumbent's threat of litigation is getting stronger that it is likely to lose the case, and this makes the ex ante settlement attractive enough for the incumbent to choose. And the challenger defers the infringement because the required royalty is so high that it cannot afford to pay it when its profit is low; we can see from panel (b) that the required royalty in the ex ante settlement increases in p. In other words, the fact that the conflict over patent infringement is not resolved via a settlement reveals that the patent's novelty is not

Fig. 2. The comparative statics w.r.t. the probability that the patent is found to be valid.

Fig. 3. The comparative statics w.r.t. the expected growth rate of R&D project.

3.2. The features of R&D projects

H. Jeon / Economic Modelling 51 (2015) 99–111

107

panel (a), we can see that the timing of litigation gets earlier as the expected growth rate of the project increases, which is in line with Cooter and Rubinfeld (1989),11 and the reason why the conflict leads to litigation is that the highest royalty the challenger is willing to pay is not attractive enough to the incumbent. As the expected growth rate increases, however, the reservation royalty increases because the incumbent's threat to litigate is getting stronger (panel (b)), and after μ exceeds a certain level, they reach a mutually attractive settlement. As observed in the former analysis, the challenger defers the infringement so that the settlement is initiated as soon as it participates in the market; the infringement trigger, however, decreases thereafter, whereas it increases in the case of the patent's novelty. The comparative statics regarding σ are given in Fig. 4, and we can see from panel (a) that the triggers increase in the volatility of the projects. This result is in line with other studies on real options such as Dixit and Pindyck (1994). As observed in the former analysis, the way they resolve the conflict switches from litigation to settlement after σ exceeds a certain level, and the timing of infringement is delayed further as σ increases thereafter. Unlike the other studies, however, we can see from panel (b) that the incumbent's option value decreases in σ, and this is because the probability that the patent is infringed by the challenger increases as the volatility increases. 3.3. The features of the litigation process

Fig. 4. The comparative statics w.r.t. the volatility of R&D project.

clear enough, which explains why roughly a half of the litigated patents are found to be invalid (e.g., Allison and Lemley, 1998; Moore, 2000). In the range of our numerical calculation, the ex post settlement is always chosen by the incumbent as the way to hold down the litigation costs, and this is consistent with what we observe in the real world; when litigation is initiated, most cases settle (e.g., Farrell and Shapiro, 2008; Lemley, 2001). Still, we left the trigger of withdrawal in the figures to clarify the fact that the cessation of litigation occurs faster via settlement. Also, we can observe in panel (c) that the highest royalty the challenger is willing to pay is much higher in the ex ante settlement than that in the ex post settlement. This is a natural result because in the former case, the incumbent is threatening the challenger to take legal procedures to recover its profit, but in the latter case, it is suggesting an alternative before calling the lawsuit off to hold down its cost burden. Fig. 3 presents the comparative statics with respect to μ, and we can comprehend the results in view of what Lemley and Shapiro (2005) noted; the patents involved in litigation are those that are important enough commercially to justify the costs of litigation and for which the parties were unable to reach a mutually attractive settlement. In

Uncertainty inherent in the litigation process plays a pivotal role in our framework. It induces the incumbent's incentive to withdraw the ongoing lawsuit and to make a settlement despite it does not yield full recovery of the original profits. Thus, it is essential to examine the impact of the uncertainty in the litigation process on various issues, which can be shown by the comparative statics regarding λ, the rate parameter of the exponential time to reach the court's judgment Fig. 5. We can see from panel (a) of Fig. 5 that when the expected duration of litigation is short, the conflict over the patent infringement tends to be resolved via ex ante settlement. This is because the sooner the lawsuit is expected to end, the stronger the incumbent's threat to litigate becomes, and this raises the challenger's reservation royalty, which leads to a mutually attractive settlement. Note that during this phase, the challenger delays the timing of infringement so that the settlement is made as soon as the infringement occurs, which corresponds to Eq. (48). As the expected duration of litigation lengthens, however, the incumbent's threat to litigate weakens, and the challenger, who is aware of this, lowers the reservation royalty, which makes the settlement less likely to be reached. After it goes beyond a certain level, they fail to settle and get involved in legal procedures to resolve the conflict, and the events are triggered sequentially. Having these results, one might think that shortening the duration of a lawsuit reduces the uncertainty of litigation process, and thus, eventually leads the parties to an ex ante settlement. This, however, implicitly assumes that the cost involved in the litigation is consistent in spite of all the efforts needed to shorten the legal procedures preserving the fairness and objectivity of them, which does not seem plausible in the real world. Thus, it is worth investigating the impact of increased litigation costs on the parties' behaviors. Panel (a) of Fig. 6 can be interpreted in a similar context as the previous analyses; the incumbent's threat to litigate the challenger is stronger when the cost of legal procedures is low, and thus, the challenger is willing to pay a higher royalty, which leads to a successful settlement. As the cost rises, however, the incumbent hesitates to litigate the challenger because of the cost burden, and the challenger, who is aware of this, lowers the reservation royalty, and finally they draw on the court's judgment to resolve the conflict. Note that not only the 11 They identified a few key determinants of litigation and noted that the probability of litigation rises with the size of the stakes.

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H. Jeon / Economic Modelling 51 (2015) 99–111

reservation royalty is so high that the challenger cannot afford to pay it when its profit is low, yet it is still better off paying the royalty than being litigated by the incumbent. In this sense, one can comprehend the ex ante settlement as (non-exclusive) licensing of patent rights, but we stick to using the term “ex ante settlement” to avoid confusing the reader. 3.4. The implications on patent systems One of the most interesting results from the previous subsections was that the conflict over the patent of which novelty is crystal clear tends to be resolved via a mutually attractive settlement, whereas that with lower novelty usually entails legal procedures to put an end to the dispute. Considering the role of patent system, one might think that the authorities should tighten the patent examination so that only the R&D project with higher novelty can acquire patent rights and the conflict regarding it can be resolved without costly legal procedures. In this context, Lemley and Shapiro (2005) expressed concern

Fig. 5. The comparative statics w.r.t. the expected duration of litigation process.

uncertainty of the duration but the significant monetary cost that it takes per unit time makes the litigation a heavy burden. Throughout the comparative statics, we can see that the patent holders have strong incentives to settle, and this explains why litigation is so unusual considering the huge number of registered patents and the prevalent infringement of them and why most of the lawsuits over patent infringement cease before the court's judgment is made (e.g., Farrell and Shapiro, 2008; Lemley, 2001; Lemley and Shapiro, 2005). Yet, the fact that litigation hardly occurs does not mean that its impact is negligible; it is the incumbent's threat to litigate that leads them to settle, and thus, we would rather say that it plays a crucial role in resolving the conflict over patent rights, no matter what means are chosen to dissolve the dispute. In addition, we can also observe that when they reach a mutually attractive settlement before they go to court, the challenger tends to delay the infringement so that the settlement initiates instantly, whereas the litigation occurs sequentially. This is because the

Fig. 6. The comparative statics w.r.t. the litigation costs.

H. Jeon / Economic Modelling 51 (2015) 99–111

that the Patent and Trademark Office issues far too many questionable patents that are unlikely to be found valid based on a thorough review, and further noted that when the patents are granted covering technologies that are already known or obvious, the resulting patents cause social costs without offsetting benefits. This, however, is only half the story; the stricter the authorities make the examination process, the more it would cost the incumbent to acquire patents. Not only would the patent examination fee increase but the R&D cost would increase in order to develop a new technology that is novel enough to pass a strict screening, and we should consider how the argument changes as these costs rise. As we can see from panel (a) of Fig. 2, the trigger of patent acquisition is independent of tp, and thus, we fix p here and see how the results change as cP increases Fig. 7. We can see from panel (a) of Fig. 7 that the incumbent delays R&D investment as the cost increases, while the timing of the challenger's infringement does not change; after it exceeds a certain level, the incumbent decides to defer the investment significantly, resulting in the instantaneous infringement, which corresponds to Eq. (28). Note that it is the incumbent who induces the instantaneous infringement, which makes the incumbent's option values match at the threshold (panel (b)), and this corresponds to “submarine patents.” This refers to the intentional delay of patent issuance to take a mature industry by surprise (e.g., Graham and Mowrey, 2004), and as Moore (2004) pointed out, there is no social benefit whatsoever to this. Panel (c) shows that the reservation royalty does not depend on cP, and thus, they fail to reach a mutually attractive settlement even though the cost rises because of the strict examination. It is obvious that these results are not what the authorities wish for. Reiss (1998) also noted that the firm's optimal strategies regarding patent acquisition depend upon R&D expenditures and that the increase of them makes the firm “wait and see.” With regard to the patent examination, Lemley (2001) argued that the authorities are “rationally ignorant” of the actual validity of a patent because extra resources devoted to determining the validity of a patent are largely wasted in the cases in which the patent is neither litigated nor licensed for a royalty, and that the costs of litigating a few commercially important patents may well be smaller than the costs of more thoroughly examining a great deal of patent applications. Also, Lemley and Shapiro (2005) pointed out that reducing litigation uncertainty is not a goal in designing the patent system, and further noted that litigation over patent validity could be eliminated entirely simply by making the authorities' validity determinations final, though they firmly concluded that few would advocate such a course. We can also fathom the present model in terms of patent design. While the concept of “length” of a patent leaves no room for disagreement, “width” of it is interpreted in various ways.12 Yet, what all these definitions have in common is that widening the scope of patents implies less damage from infringement, and this corresponds to lowering θI in our model. This effort, however, may not guarantee the improvement of social welfare, and thus, we should pay scrupulous attention to this issue. Fig. 8 presents the comparative statics regarding θI, the degree of patent infringement, and panel (a) shows that the timing of both the infringement and the litigation gets earlier as the degree of infringement increases, which is a natural result. After the infringement worsens and goes beyond a certain level, the incumbent decides to delay the R&D investment significantly, which corresponds to Eq. (28), and obviously this is a severe social loss. These “submarine patents” occur 12 Nordhaus (1972) and Takalo and Kanniainen (2000) measured it by the fraction of the cost reduction that does not spill out to other competitors, and Gilbert and Shapiro (1990) identified the breadth of a patent with the flow rate of profit available to the patentee. In Klemperer (1990), it refers to the area of differentiated product space covered by a patent and is measured by the degree of shifts of demand curve, and Gallini (1992) defines it in terms of the imitation costs. Denicolò (1996) encompasses these interpretations and measured it as the degree of dissemination of technological knowledge.

109

Fig. 7. The comparative statics w.r.t. the patent acquisition costs.

because the expected losses from the infringement are so huge that it is better for the incumbent to wait until the market's demand rises enough to compensate for all the expected losses from the infringement. As the expected losses from the infringement get even worse, the incumbent's threat to litigate gets stronger, and the challenger, who is aware of this, is willing to pay a higher royalty, making the ex ante settlement attractive for both parties. The timing of R&D investment returns to the original level, and now it is the challenger who induces instantaneous infringement, which corresponds to Eq. (48). From the perspective of patent scope, we can observe from panel (a) that the trigger of R&D investment is mostly independent of θI, and in some cases, adjusting the level of it can rather induces “submarine patent,” the worst case scenario. In the present model, it is the increase of μ that makes the investment get earlier (Fig. 3); that is, what really matter is the fundamental features of the R&D project, not the patent system. There are numerous studies that cast a doubt on the efficiency of the patent system or the reform of it. For instance, Scotchmer and Green (1990), Takalo and Kanniainen (2000), and Hunt (2006)

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endogenously, and some of the actions can be triggered simultaneously depending on the conditions in which the stakeholders are situated. This model helps us to comprehend the patent war observed in the real world. First of all, it explains why litigation is so unusual given the huge number of registered patents and the prevalent infringement, and why most of the lawsuits over patent rights cease before the court's ruling is reached. The model also clarifies why roughly half of litigated patents are found to be invalid in court, and in what circumstances the introduction of new technology or the infringement of a patent is delayed. Last but not least, the model gives us practical implications with respect to the patent system; neither tightening the patent examination nor widening the patent scope guarantees the acceleration of the introduction of new technology, and can rather delay it in some cases. This result should be of interests to the policy makers of which objective is to maximize social welfare by incentivizing the firms' R&D investment and innovation. Still, there remains a range of problems to be tackled, and it is to be hoped that this paper will serve as a platform for future works on R&D investment and patent system. For instance, more attention should be paid to the bargaining upon the settlement between stakeholders. We assumed that the leader has full bargaining power, yet it can be relaxed by incorporating Nash bargaining in the model. In particular, the bargaining upon the settlement under asymmetric information must be reserved for a more extensive study. It is reasonable to consider that there is information asymmetry regarding the profitability of the patented technology or the probability that the patent is found to be valid at the court. In general, information asymmetry distorts the allocation of sources in the market, and further studies on these issues are needed for the sake of more efficient patent system.

References

Fig. 8. The comparative statics w.r.t. the degree of infringement.

presented a theoretical framework which shows that the patent system might not work as intended, and a number of empirical researches such as Jaffe and Lerner (2004), Qian (2007), Sakakibara and Branstetter (2001), and Lerner (2009), support these claims. They found a negative correlation between tightening patent rights and innovation, and argued that patents may even be counterproductive, incurring additional application costs and promoting litigation.

4. Conclusion In this paper, we proposed a theoretical model that integrates a series of events regarding patent rights based on a real options framework. After the patent is infringed, the incumbent can litigate the challenger to recover its original profit, but the lawsuit can be withdrawn before the court's ruling is reached because of the cost burden, or a settlement can be made. They can even make a settlement before they go to court. The triggers and the royalty in a settlement are determined

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