An Optimal Rule for Patent Damages Under Sequential Innovation

Transcription

1 An Optimal Rule for Patent Damages Under Sequential Innovation by Yongmin Chen* and David E. M. Sappington** May 2017 Abstract: Weanalyzetheoptimaldesignofdamagesforpatentinfringementinsettings where the patent of an initial innovator may be infringed by a follow-on innovator. We consider relatively simple damage rules that are linear combinations of the popular ìlost proötî (LP) and ìunjust enrichmentî (UE) rules, coupled with a lump-sum transfer between the innovators. We identify conditions under which a linear rule can induce the socially optimal levels of sequential innovation and the optimal allocation of industry output. We also show that, despite its simplicity, the optimal linear rule achieves the highest welfare among all rules that ensure a balanced budget for the industry, and often secures substantially more welfare than either the LP rule or the UE rule. Keywords: Patents, sequential innovation, infringement damages, linear rules for patent damages. *DepartmentofEconomics,UniversityofColorado;Yongmin.Chen@colorado.edu ** Department of Economics, University of Florida; sapping@uá.edu We gratefully acknowledge helpful comments from Ted Bergstrom, Rich Gilbert, Michael Meurer, Michael Riordan, Marius Schwartz, Yair Tauman, John Turner, Dennis Yao, and seminar participants at Columbia University, Fudan University, Queens University, the State University of New York at Stony Brook, the University of California at Santa Barbara, the University of Georgia, the University of Groningen, the 2016 Workshop on IO and Competition Policy at the Shanghai University of Finance and Economics, and the 2016 Workshop on Industrial Organization and Management Strategy at the Hong Kong University of Science and Technology. 1

2 1 Introduction Innovation is a key driver of economic growth and prosperity. To encourage innovation, successful innovators often are awarded patents for their inventions. A patent grants an innovator exclusive rights to her invention for a speciöed period of time. An extensive literature analyzes optimal patent protection, focusing on issues such as the optimal strength and breadth of patents. 1 An important, but less developed, literature studies Önancial penalties (ìdamagesî) for patent infringement. To date, this literature has primarily analyzed the performance of individual damage rules that are employed in practice, including the lost proöt (LP) rule and the unjust enrichment (UE) rule. 2 In contrast, the purpose of this paper is to analyze the optimal design of patent damage rules under sequential innovation, where an initial innovatorís patent may be infringed by a follow-on innovator whose di erentiated product, possibly of higher quality, expands market demand. Sequential innovation is important to consider because it drives progress in many important industries. Scotchmer (1991) identiöes several products ñ including antibiotic drugs, incandescent lights, lasers, and computer operating systems ñ whose development was fueled by sequential innovation. 3 Todayís smartphones are estimated to embody innovations protected by as many as 250,000 patents that have been developed sequentially (Sparapani, 2015). 4 The design of damages for patent infringement is particularly subtle in the presence 1 Early studies of optimal patent protection include Nordhaus (1969), Gilbert and Shapiro (1990), Klemperer (1990), and Scotchmer (1991) 2 Anton and Yao (2007) examine the performance of the LP rule. Shankerman and Scotchmer (2001), Choi (2009), and Henry and Turner (2010) examine the performance of both the LP and the UE rules. (These rules are described below.) Choi and Henry and Turner also analyze the performance of the reasonable royalty damage rule, which is discussed in section 6. Schankerman and Scotchmer (2005) report that although the UE rule was commonly employed in the U.S. prior to the implementation of patent reforms in 1946, the LP rule has been employed relatively frequently since then. See Blair and Cotter (2005) for additional discussion of the use of patent damage rules in the U.S. and Reitzig et al. (2003) for corresponding discussion of international experience. 3 In addition, gene sequencing discoveries are valuable inputs in follow-on scientiöc research and in commercial applications (Sampat and Williams, 2015). Murray et al. (2016) examine the e ects of a ording academic researchers expanded access to newly discovered information about genetically engineered (transgenic) mice. The authors report that expanded access increases follow-on discoveries by new researchers and promotes diverse follow-on research methodologies without reducing the rate of innovation. 4 It has been observed that modern computing technologies like smart phones ìtightly and e ciently integrate the engineering of other companies and other earlier inventors, and are enhanced with new functions and 1

3 of sequential innovation because, although stringent damage rules can encourage early innovation, they may discourage subsequent innovation, especially when uncertainty prevails about whether follow-on innovations infringe earlier patents. 5 We consider a model in which innovation is not certain because of stochastic variation in innovation costs. Patent protection also is uncertain in our model, as it is in practice. 6 Hundreds of thousands of patents are granted annually, and patent descriptions can be vague and incomplete. 7 Therefore, in practice, it is often di cult to discern whether an innovation infringes an existing patent. 8 The parameter! 2 (0; 1] in our model denotes the probability that the patent of an initial innovator, Örm 1, is infringed by the di erentiated product of afollow-oninnovator,örm2.thevalueof! can be viewed as a measure of the strength of patent protection (e.g., Choi, 1998; and Farrell and Shapiro, 2008). We consider damage rules that are linear combinations of the LP rule and the UE rule, coupled with a lump-sum transfer between the innovators. The LP rule requires the infringer to compensate the patent holder for the reduction in proöt the latter su ers due to the infringement. 9 The UE rule requires the infringer to deliver its realized proöt to the patent features to attract consumer interest and drive demandî (Sparapani, 2015). Scotchmer (2004, p. 134) notes that although no product development Öts the basic model of sequential innovation perfectly, ìif one takes a more metaphorical view, [the model] Öts almost every important technology, including the laser and desktop software.î 5 Green and Scotchmer (1995) examine the patent length and division of surplus required to induce e cient sequential innovation. We extend their work in part by explicitly modeling competition between innovators in the presence of uncertainty about the applicability of existing patents. 6 Anton and Yao (2007), Choi (2009), and Henry and Turner (2010) also analyze probabilistic patent enforcement. 7 See, for example, Choi (1998), Lemley and Shapiro (2005), Bessen and Meurer (2008), and Farrell and Shapiro (2008). 8 The recent protracted patent infringement ligation between Apple and Samsung is a case in point (e.g., Vascellaro, 2012). In addition, Lemley and Shapiro (2005) report that the U.S. Patent and Trademark O ce issues nearly 200,000 patents annually, so the time that a patent o cer can devote to assessing the merits of any individual patent application is limited. Consequently, even after a patent is issued, its validity may be successfully contested in court. Uncertainty about prevailing patent protection also can complicate the licensing of innovations. See Kamien and Tauman (1986), Katz and Shapiro (1986), and Scotchmer (1991), for example, for analyses of the licensing of innovations. 9 U.S. patent law stipulates that the damage penalty for patent infringement must be ìadequate to compensate for the infringement, but in no event less than a reasonable royalty for the use made of the invention by the infringer, together with interest and costs as Öxed by the court.î Courts have interpreted this stipulation to require ìan award of lost proöts, or other compensatory damages, where the patentee can prove, and 2

4 holder. 10 Under the linear rules that we analyze, if Örm 2 is found to have infringed Örm 1ís patent, Örm 2 must deliver a damage payment (D) toörm1thathasthreecomponents: a lump sum monetary payment (m), a fraction (d 1 )oftheamountbywhichörm2ísoperation reduces Örm 1ís proöt, and a fraction (d 2 )oförm2ísproöt. Thus,linearrulesgeneralize the LP rule and the UE rule, including the former (with d 1 =1and m = d 2 =0)andthe latter (with d 2 =1and m = d 1 =0)asspecialcases. 11 These linear rules are relatively simple in nature. Nevertheless, we Önd that an optimallydesigned linear rule secures the highest welfare among all balanced damage rules (i.e., rules in which all payments are internal to the industry). The optimality of the linear rule reáects in part its links to both the (lost) proöt of the initial innovator and the (realized) proöt of the follow-on innovator. These links enable an optimally-designed linear rule to secure desired levels of industry proöt (which a ect aggregate investment incentives), distribute the proöt to foster desired levels of innovation by both Örms, and ensure desired allocations of industry output. We also show that the optimal linear rule typically di ers from both the LP rule and the UE rule, and often secures a substantial increase in welfare relative to both of these rules. Furthermore, when the maximum feasible level of industry proöt is su ciently large relative to innovation costs, the optimal linear rule can ensure the Örst-best outcome, under which each Örm innovates if and only if its innovation enhances welfare and industry output is allocated between the producers so as to maximize welfare. When the optimal linear rule achieves the Örst-best outcome, it resembles the LP rule more closely than the UE rule (so d 1 >d 2 )whenconsumersvaluetheproductoftheinitialinnovatorrelativelyhighly,while elects to prove, such damagesî (Frank and DeFranco, , p. 281). 10 The UE rule is sometimes employed when a design patent is infringed. The Supreme Court of the United States recently agreed to review a Federal Appeals Court decision that Apple is entitled to all of the proöt that Samsung earned from its sales of smartphones because Samsung ìinfringed three Apple design patents relating to a portion of the iphoneís outer shell and a single graphical-user-interface screenî (Amici Curiae Brief, 2016). 11 Linear rules also encompass proöt sharing rules, which require a patent infringer to pay the patent holder a portion of the proöt the infringer secures in the marketplace. 3

5 it resembles the UE rule more closely (so d 2 >d 1 )whenconsumersvalueörm2ísproduct relatively highly. 12 We also show how the optimal linear rule can limit the distortions in innovation incentives and output allocation when the Örst-best outcome is not attainable. The welfare-maximizing linear rule is optimal among balanced rules regardless of the strength of protection for Örm 1ís patent (i.e., for any given!>0). 13 This Önding suggests that although the appropriate strength and scope of patents can help to foster e cient levels of sequential innovation, the design of damages for patent infringement may play a particularly important role in this regard. The ensuing analysis proceeds as follows. Section 2 presents the model. Section 3 records equilibrium market outcomes. Section 4 characterizes and explains the key features of the optimal linear rule. Section 5 demonstrates that the welfare-maximizing linear rule is optimal among balanced damage rules. Section 5 also illustrates the (often substantial) welfare gains that the optimal linear rule can secure relative to popular damage rules. Section 6 summarizes our Öndings, discusses alternative modeling assumptions, and suggests directions for further research. Appendix A presents the proofs of all formal conclusions. Appendix B further illustrates the optimal linear rule and the welfare gains it can secure. 2 The Model We consider the interaction between two Örms. Firm 1 has the potential to innovate Örst and secure a patent on its product by incurring innovation cost k 1.Firm2hasthepotentialto innovate subsequently by incurring k 2,butonlyifÖrm1hasinnovatedsuccessfully. 14 These 12 As we demonstrate below, the optimal linear rule serves to shift equilibrium industry output toward the product that consumers value most highly. The linear rule a ects the equilibrium allocation of industry output by inducing the Örms to partially internalize each otherís proöt, which ináuences their pricing strategies. 13 It follows that the optimal linear rule also dominates ex ante e cient patent licensing, as we demonstrate in section Thus, we follow Chang (1995) in abstracting from stochastic innovation and from the possibility of simultaneous research and development activity by multiple Örms. We extend Changís work in part by considering product di erentiation and uncertainty about the applicability of existing patents. Bessen and Maskin (2009) allow for stochastic, simultaneous research and development by multiple Örms. They show that patent protection can reduce welfare in settings where the research activities of distinct Örms are complementary. 4

6 innovation costs are the realizations of independent random variables, with continuous and strictly increasing distribution functions F (k 1 ) and G(k 2 ); respectively, on support! 0; k ' " i for i =1; 2. WeassumethatinnovationcostsaretheonlycoststheÖrmsincur. 15 AmassofQ potential consumers are distributed uniformly on a line segment of length Q. If Örm 1 innovates, it locates at point 0 on the line segment, and Q =1if Örm 1 is the only innovator. If Örm 2 innovates, it locates at point Q = L " 1 on the line segment, where L is an exogenous parameter that captures the extent to which Örm 2ís innovation expands the market. This potential expansion adds to the standard Hotelling model another dimension on which the operation of a second innovator can enhance welfare. In practice, market expansion might arise, for instance, when the follow-on innovation attracts the attention of a new consumer demographic (e.g., young consumers or consumers in a new foreign market). 16 The product supplied by Örm i delivers value v i to consumers for i =1; 2. Thenetutility aconsumerderivesfromtravelingdistances to purchase a unit of the product from Örm i at price p i is v i # p i # s. Each consumer purchases at most one unit of the product, and purchases from the supplier that o ers the highest nonnegative net utility. This formulation admits both monopoly and duopoly industry structures. The duopoly structure extends the standard Hotelling model by allowing the follow-on innovation of Örm 2 to beneöt consumers not only through (standard) horizontal product di erentiation, 17 but also through quality improvement (when v 2 >v 1 ) and market expansion (when L>1). 18 To capture relevant uncertainty about whether Örm 2ís product infringes Örm 1ís patent, we let! 2 (0; 1] denote the probability that, after serving consumers, Örm 2 is ultimately 15 Thus, we adopt Green and Scotchmer (1995)ís ìideaî approach to modeling innovation, assuming that innovators naturally acquire ideas about new products and an innovator must incur a speciöed cost to implement the innovation. 16 Compared to consumers in the market before Örm 2ís innovation, new consumers that arrive after the innovation have a stronger innate preference for Örm 2ís product relative to Örm 1ís product. 17 Henry and Turner (2010) consider price competition in a standard Hotelling model. Anton and Yao (2007) and Choi (2009) analyze (Cournot) quantity competition with a homogeneous product. 18 As we explain in section 6, our primary Öndings persist in more general models of competition with product di erentiation. We focus on this Hotelling model with potential market expansion because the closed-form solutions it admits facilitate the demonstration of our Öndings. 5

7 found to have infringed Örm 1ís patent. 19 In this event, Örm 2 is obligated to pay Örm 1 the amount D $ m + d 1 R 1 + d 2 1 2,whereR 1 is the amount by which Örm 1ís proöt is reduced by Örm 2ís operation, 1 2 is Örm 2ís proöt, and m is a lump sum monetary payment. The policy instruments, m, d 1 " 0, andd 2 " 0, arechosentomaximizeexpectedwelfare (the sum of consumer and producer surplus). We will call the damage rule associated with damage payment D the linear rule. The timing in the model is as follows. First,! is determined exogenously. Then d 1, d 2, and m are set to maximize expected welfare. Next, Örm 1 privately learns the realization of k 1 and decides whether to innovate. If Örm 1 decides not to innovate, the game ends. If Örm 1 innovates by incurring cost k 1,thenÖrm2privatelylearnstherealizationofk 2 and decides whether to incur this cost in order to secure the follow-on innovation. If Örm 2 innovates, it subsequently competes against Örm 1 for the patronage of consumers on [0;L]. If Örm 2 chooses not to innovate, then Örm 1 acts as a monopolist and serves consumers on [0; 1]. When both Örms compete in the marketplace, they set their prices simultaneously and non-cooperatively. 20 After consumers have been served, it becomes known whether Örm 2hasinfringedÖrm1íspatent.IfÖrm2isfoundtohaveinfringed1íspatent,Örm2makes the requisite payment to Örm 1. We assume throughout the ensuing analysis that consumer valuations of the two products are not too disparate, and are large relative to transportation costs. This assumption helps to ensure that all consumers purchase a unit of the product and that, under duopoly competition, both Örms serve customers in equilibrium We take as given any actions Örm 2 might undertake to avoid patent infringement. Gallini (1992) demonstrates that long patent durations can reduce welfare by encouraging Örms to ìinvent aroundî the patents of early innovators. Zhang and Hylton (2015) analyze the optimal design of infringement penalties in a setting where the potential infringer can undertake unobserved actions to limit the loss the patent holder su ers in the event of infringement. 20 Industry suppliers often can change the prices of their products relatively rapidly. Consequently, we assume that Örm 1ís potentially longer tenure in the market does not endow it with any Stackelberg leadership advantage in its retail competition with Örm 2. (Such an advantage seems unlikely to alter our key qualitative conclusions.) 21 This is the equilibrium on which we focus throughout the ensuing analysis of the optimal damage rule for patent infringement. We explain below why this focus is without loss of generality, given Assumption 1. 6

8 Assumption 1. min fv 1 ;v 2 g > 2 L and j. j <L,where. $ v 2 # v 1. 3 Market Equilibrium Having identiöed the key elements of the model, we now characterize equilibrium outcomes. To begin, consider the setting where only Örm 1 innovates and so operates as the monopoly supplier of the product. Because v 1 > 2 by assumption, Örm 1 maximizes its proöt p 1 min fv 1 # p 1 ; 1g by selling one unit to all potential customers at price p 1 = v 1 # 1. Firm 1ís monopoly proöt in this case is: 1 M 1 = v 1 # 1. (1) Now suppose both Örms innovate. Let p i denote the price that Örm i 2f1; 2g charges for its product. Then the location of the consumer on (0;L) who is indi erent between purchasing from Örm 1 and from Örm 2 is l = 1 [ L + v 2 1 # v 2 + p 2 # p 1 ]. Therefore, the demand functions facing Örms 1 and 2 when they both serve customers and all consumers purchase one unit of a product are, respectively: q 1 (p 1 ;p 2 ) = 1 2 [ L #.+p 2 # p 1 ] and q 2 (p 2 ;p 1 ) = 1 2 [ L +.+p 1 # p 2 ]. (2) In the absence of patent infringement, the proöts (not counting innovation costs) of Örms 1and2are,respectively: 1 N 1 = p 1 q 1 (p 1 ;p 2 ) and 1 N 2 = p 2 q 2 (p 2 ;p 1 ). (3) Firm 2ís operation reduces Örm 1ís equilibrium proöt by R 1 = 1 M 1 # 1 N 1. Therefore, because Örm 2 is required to pay D = m + d 1 R 1 + d 2 1 N 2 to Örm 1 if Örm 2 is found to have infringed Örm 1ís patent, the proöts of Örm 1 and Örm 2 in the event of infringement (excluding m) are,respectively: 1 I 1 = 1 N 1 + d 1! 1 M 1 # 1 N 1 " + d2 1 N 2 and 1 I 2 = 1 N 2 # d 1! 1 M 1 # 1 N 1 " # d2 1 N 2. (4) The Örmsí ex ante expected proöts (not counting innovation costs and m) whenthey both innovate are, respectively: 1 e 1 = [1#! ] 1 N 1 +!1 I 1 and 1 e 2 = [1#! ] 1 N 2 +!1 I 2. (5) 7

9 In equilibrium, Örm i 2f1; 2g chooses p i to maximize 1 e i,takingitsrivalíspriceasgiven. Letting! ís denote equilibrium outcomes, standard calculations reveal that equilibrium prices and quantities are as speciöed in Lemma 1. Lemma 1. Suppose both Örms innovate, all consumers purchase a unit of the product, and each Örm supplies a strictly positive level of output. Then, given d 1 and d 2 with! [ d 1 + d 2 ] < 1, equilibriumpricesandquantitiesare,for i; j =1; 2 (j 6= i): p! i = [1#!d j ] L [3#! (3 d i # d j )]+[v i # v j ][1#! (d 1 + d 2 )] [1#! (d 1 + d 2 )][3#! (d 1 + d 2 )], and (6) q! i = L [3# 2!d j ]+v i # v j 2[3#! (d 1 + d 2 )] 2 (0;L). (7) Furthermore, p! 2 R p! 1 as. R L([ d 2 " d 1 ] 2 " ( [ d 1 + d 2 ],andallconsumerswillindeedpurchaseaunit of the product in equilibrium if d 1 " 0 and d 2 " 0 are su ciently small. Additionally, any prices p! 1 and p! 2 in (6) can be induced by some d 1 and d 2 : Lemma 1 speciöes how d 1 and d 2 a ect equilibrium prices and outputs. The lemma implies that if d 1 = d 2 ; then Örm i will charge a higher price than Örm j in equilibrium (p! i >p! j)ifconsumersvalueörmiís product more highly than Örm jís product (i.e., if v i >v j ). Proposition 1 examines the impact of changes in d 1 and d 2 on equilibrium outcomes, including 1! i,theequilibriumexpectedproötoförmi 2f1; 2g. 22 Proposition 1. Under duopoly competition: (i) p! 1 and p! 2 both increase in d 1 and in d 2 ; (ii) p! 2#p! 1 and q! 1 increase in d 1 ; while p! 1#p! 2 and q! 2 increase in d 2 ; (iii) 1! $ 1! 1+1! 2 increases in d 1 if. ) L([ d 2 " d 1 ] 2 " ( [ d 1 + d 2 ] (so p! 1 " p! 2), while 1! increases in d 2 if. " L([ d 2 " d 1 ] 2 " ( [ d 1 + d 2 ] (so p! 2 " p! 1). Proposition 1 reáects the following considerations. As d 1 increases, Örm 2 is penalized 22!! i is obtained by substituting p! i and q! i from equations (6) and (7) into equation (5). 8

10 alargerportionoförm1íslostproöt(1 M 1 # 1 N 1 )iförm2isultimatelyfoundtohave infringed Örm 1ís patent. Firm 2 recognizes that it can reduce this penalty by competing less aggressively, thereby allowing Örm 1ís duopoly proöt (1 N 1 )toincrease. Thereduced aggression leads to higher equilibrium prices for both Örms. The increased congruence of the Örmsí preferences regarding higher proöt for Örm 1 results in expanded output for Örm 1, secured by a reduction in Örm 1ís price relative to Örm 2ís price. As d 2 increases, Örm 2 forfeits a larger portion of its proöt (1 N 2 )ifitisultimatelyfound to have infringed Örm 1ís patent. Firm 1 recognizes that it can secure a larger expected penalty payment from Örm 2 by competing less aggressively, thereby allowing 1 N 2 to increase. The reduced aggression leads to higher equilibrium prices for both Örms. The increased congruence of the Örmsí preferences regarding higher proöt for Örm 2 results in expanded output for Örm 2, secured by a reduction in Örm 2ís price relative to Örm 1ís price. 23 The increase in Örm 1ís equilibrium output (q! 1)andthecorrespondingreductioninÖrm 2ís output (q! 2)inducedbyanincreaseind 1 increase equilibrium industry proöt (1! $ 1! 1 + 1! 2)whenÖrm1ísproÖtmarginexceedsÖrm2ísmargin(i.e.,whenp! 1 >p! 2). Similarly, the increase in q! 2 and the reduction in q! 1 induced by an increase in d 2 increase 1! when p! 2 exceeds p! 1. Before proceeding to characterize the optimal linear damage rule, it is helpful to consider the policy that maximizes welfare and the policy that maximizes industry proöt following innovation by both Örms. These policies are characterized in Lemmas 2 and 3, respectively. Lemma 2 refers to equilibrium welfare in the duopoly setting: fw 12 = v 1 q! 1 # Z q! 1 0 y dy + v 2 q! 2 # Z q! 2 0 y dy = v 1 q! 1 # 1 2 q!2 1 + v 2 q! 2 # 1 2 q!2 2. (8) Lemma 2 also refers to d w 1,whichisdeÖnedinexpression(41)inAppendixA. Lemma 2. Suppose: d 2 = d w 2 (d 1 ) $ d [1#!d 1 ]! [ L +.]. (9) 23 See Anton and Yao (2007), Choi (2009), and Henry and Turner (2010) for related observations. 9

11 Then when both Örms innovate, the maximum feasible level of welfare (excluding innovation costs), v 2 L # 1 4 [2L. #.2 + L 2 ],isinduced,andequilibriumpricesare p! 1 = p! 2 = [1" (d 1 ][L 2 " ( 2 ] L [1" 2 (d 1 ] " (.EquilibriumindustryproÖt,1! = p! 1 L, increasesasd 1 or d w 2 (d 1 ) increases, reaching its maximum 1 w = p w L at d 1 = d w 1 < 1 2 ( with p! 1 = p! 2 = p w $ v 1 # 1 2 [ L #.]. Lemma 3 refers to ' d 1 and ' d 2,whicharedeÖnedinexpression(48)inAppendixA. Lemma 3. When d 1 = ' d 1 and d 2 = ' d 2, the maximum feasible level of duopoly proöt, 1=1 w + (2,isinduced.Furthermore,equilibriumpricesare 8 p 1 $ v 1 # 1 % L #. & and p $ v 2 # 1 2 with corresponding outputs q 1 2 (0;L) and q 2 = L # q % L +. & ; (10) 2 Lemmas 2 and 3 report that the welfare-maximizing duopoly prices typically di er from the prices that maximize industry proöt. (They coincide only when v 1 = v 2.) Duopoly welfare is maximized when the two Örms charge identical prices. In this event, consumers purchase from the Örm that o ers the largest di erence between product value and transportation cost, which ensures that welfare is maximized. In contrast, industry proöt is maximized when the Örm with the most highly-valued product charges a higher price than its competitor. Relative to the identical welfare-maximizing prices, this price structure results in greater extraction of consumer surplus. It can also be veriöed that the values of d 1 and d 2 that induce welfare-maximizing or proöt-maximizing duopoly prices decline as! increases. Thus, less stringent damages are required to secure key industry outcomes when stronger patent protection prevails. 4 Characterizing the Optimal Linear Rule Lemma 2 identiöes the welfare-maximizing allocation of industry output following innovation by both Örms. To characterize the welfare-maximizing damage rule for patent infringement, one must also consider the Örmsí innovation incentives. To do so, observe Örst 24 These outputs are derived by substituting p 1 and p 2 into the expressions identiöed in equation (2). 10

12 that welfare when only Örm 1 innovates is: W 1 = W f 1 # k 1 where W f 1 = v 1 # Z 1 0 y dy = v 1 # 1 2. (11) Let b k i denote the realization of Örm iís innovation cost (k i )forwhichtheörmwillinnovate if and only if k i ) b k i.if b k 1 ) k ' 1 and b k 2 ) k ' 2 ; then b k1 = G( b k 2 )[1! 1 +!m]+[1# G( b k 2 )]1 M 1 and b k 2 = 1! 2 #!m. (12) Equations (8), (11), and (12) imply that ex ante expected welfare (i.e., the level of welfare that is anticipated before innovation costs are realized) is W = F ( b k 1 ) G( b k 2 ) f W 12 + F ( b k 1 )[1# G( b k 2 )] f W 1 Z b k2 Z b k1 # F ( b k 1 ) k 2 dg(k 2 ) # k 1 df (k 1 ). (13) To examine Örm 2ís innovation incentives, let k w denote the increase in welfare secured by Örm 2ís innovation (not counting innovation costs). From equations (8) and (11): k w 2 $ f W 12 # f W 1 = v 1 q! 1 # 1 2 q!2 1 + v 2 q! 2 # 1 2 q!2 2 # ( v 1 # 1 2 ). (14) Incremental welfare is maximized when Örm 2 innovates if and only if its innovation cost (k 2 ) does not exceed k w 2.Intheensuinganalysis,k w! 2 will denote the increase in welfare secured by Örm 2ís innovation when d 2 is set (given d 1 )toensurethewelfare-maximizingallocation of industry output. To assess whether Örm 1ís innovation increases expected welfare, one must account for Örm 2ís subsequent innovation activity. Let k w 1 ( b k 2 ) denote the increase in expected welfare secured by Örm 1ís innovation, given that Örm 2 innovates if and only if k 2 ) b k 2.Formally: k w 1 ( b k 2 ) $ G( b k 2 ) f W 12 +[1# G( b k 2 )] f W 1 # Z b k2 0 k 2 dg(k 2 ). (15) Given Örm 2ís innovation activity, incremental welfare is maximized when Örm 1 innovates if and only if its innovation cost (k 1 )doesnotexceed k w 1 ( b k 2 ). These observations imply that the optimal linear rule is the solution to problem [P], 11

13 which is deöned to be: M aximize W; where W is speciöed in equation (13), prices and d 1 ;d 2 ;m quantities are speciöed in Lemma 1, and b k i is speciöed in equation (12) for i =1; 2. In the Örst-best outcome, each Örm innovates if and only if the associated increase in expected welfare exceeds the realized innovation cost, and industry output is allocated among the active producers to maximize welfare. Formally, in the Örst-best outcome: (i) Örm 2innovatesifandonlyif k 2 ) k o 2 $ minfk w! 2 ; k 2 g;(ii)örm1innovatesifandonlyif k 1 ) k o 1 $ minfk w 1 (k o 2); k 1 g;and(fromlemma2)(iii) q! 1 = 1 2 [ L #.]and q! 2 = 1 2 [ L +.] when both Örms innovate. Observation 1 considers the setting where b k 1 < k 1,soÖrm1doesnotalwaysinnovate because its maximum innovation cost is relatively large. The Observation reports that the penalty for patent infringement must include a lump sum payment (m >0) iftheörst-best outcome is to be achieved in this setting. Observation 1. Suppose b k 1 < k 1 and m = 0. Then Örm 1ís innovation incentive is ine ciently low (i.e., b k 1 <k w 1 ( b k 2 )). When Örm 1 innovates, it creates consumer surplus that it does not fully capture, regardless of whether Örm 2 innovates. Therefore, W1 f >1 M 1 and W f 12 >1! 2 + 1! 1,soaggregate welfare from Örm 1ís innovation always exceeds aggregate industry proöt. From expression (12), when m =0; Örm 2 innovates if and only if k 2 ) 1! 2 = b k 2,andÖrm1innovatesif and only if k 1 ) b k 1 = G( b k 2 ) 1! 1 +[1# G( b k 2 )]1 M 1 : Therefore, Örm 1 has ine ciently limited incentive to innovate when the penalty for patent infringement does not include a lump sum payment between the Örms. Although Örm 1ís innovation incentive is always ine ciently low when m =0,Örm2ís incentive can be excessive. If Örm 2ís innovation does not expand the market (so L =1), then Örm 2ís innovation reduces Örm 1ís proöt in the absence of patent infringement penalties. Consequently, Örm 2ís private beneöt from innovation can exceed the corresponding social beneöt (i.e., b k 2 >k2 w ). In contrast, Örm 2ís innovation incentive can be ine ciently low (i.e., 12

14 b k2 <k w 2 )ifl is su ciently large. Observation 2. Suppose v 1 = v 2 $ v and d 1 = d 2 = m = 0. Then: (i) k2 w < b k 2 if L =1;and (ii) k2 w > b h i k 2 if L>1 and v> 1 3 L 2 " 2. 4 L " 1 Despite the potential conáict between the social and private incentives for innovation, the optimal linear rule can sometimes fully align these incentives while inducing the welfaremaximizing allocation of industry output. As Proposition 2 and Corollary 1 report, the optimal linear rule secures the Örst-best outcome if innovation costs are su ciently small relative to the maximum feasible level of industry proöt, so inequality (16) holds. 1 w " ko 1 # 1 M 1 G(k o 2) + 1 M 1 + k o 2, G(k o 2)[1 w # k o 2 ]+[1# G(k o 2)]1 M 1 " k o 1. (16) Proposition 2. Suppose inequality (16) holds. Then ( d 1 ;d 2 ;m) can be chosen to induce the Örst-best outcome, with d 2 = d w 2 (d 1 ) and m = 1 ( [ 1! 2 # k o 2 ]. Underthis(optimal)linear rule, d 2 R d 1 as. R 0. Corollary 1. If v 1, v 2,andL are su ciently large to ensure k o 2 = k 2,thentheoptimal linear rule secures the Örst-best outcome if k 1 + k 2 ) L 2 [ v 1 + v 2 # L ]. This inequality is more likely to hold if k 1 and k 2 are small and if v 1, v 2,andL are large, ceteris paribus. Proposition 2 and Corollary 1 consider settings where innovations are highly valued, Örm 2ís innovation expands the market considerably, and/or innovation costs are low. In such settings, the substantial industry proöt that is potentially available can be divided between the suppliers to induce them both to always innovate even when damages are structured to induce the Örms to set the same prices and thereby maximize duopoly welfare. Proposition 2 reports that if v 1 >v 2,thentheinitialinnovatoríslostproÖtreceives more weight than the second innovatorís proöt in the optimal linear rule (i.e., d 1 >d 2 ).In contrast, if v 2 >v 1,thenthesecondinnovatorísproÖtreceivesmoreweightthantheÖrst innovatorís lost proöt (i.e., d 2 >d 2 ). In this sense, the optimal linear rule resembles the lost 13

15 proöt (LP) rule more than the unjust enrichment (UE) rule when the (vertical dimension of) quality of the initial innovation is relatively pronounced. In contrast, the optimal linear rule resembles the UE rule more than the LP rule when the quality of the follow-on innovation is relatively pronounced. To understand the rationale for this penalty structure, recall from Proposition 1 that under duopoly competition, the relative price of Örm 1ís product declines and its output increases as d 1 increases, whereas the relative price of Örm 2ís product declines and its output increases as d 2 increases. Therefore, the identiöed penalty structure helps to reduce the relative price of, and thereby shift equilibrium consumption toward, the product that consumers value most highly. Doing so ensures the welfare-maximizing allocation of industry output between the two suppliers. Corollary 2. If inequality (16) does not hold, then the optimal linear rule does not secure the Örst-best outcome. Corollary 2 reports that when inequality (16) does not hold, so the maximum feasible industry proöt (given the welfare-maximizing output allocation) is not su ciently large relative to innovation costs, then the optimal linear rule cannot induce welfare-maximizing innovation decisions while ensuring welfare-maximizing output allocations. The values of d 1 and d 2 required to induce the welfare-maximizing allocation of duopoly output do not generate the level of industry proöt required to induce both Örms to innovate whenever the social beneöt of innovation exceeds the private cost of innovation. 25 In this case, the optimal linear rule will increase industry proöt by eliminating the surplus of the marginal consumer. Formally, from equation (2): v 1 # p 1 # 1 2 [ L + v 1 # v 2 + p 2 # p 1 ] = 0 ) p 2 = v 1 + v 2 # p 1 # L: (17) 25 When the optimal linear rule cannot secure the Örst-best outcome, the critical problem is to induce both Örms to innovate more often. m can be adjusted to avoid settings where one Örm innovates too seldom and one Örm innovates too often relative to the Örst-best outcome. Furthermore, the logic that underlies Observation 1 explains why the critical problem is not to induce both Örms to innovate less often. 14

16 If v 1 = v 2 = v; then the maximum level of industry proöt that can be secured is 1 w = 1. From expression (48), the optimal linear rule sets d 1 = d 2 = 2 v " 3 L to induce the welfare- 4 ( [ v " L ] maximizing prices p 1 = p 2 = p = v # L,and m is set to distribute 1 between the two 2 Örms to maximize W.Ifv 1 6= v 2 ; then industry proöt can be increased above 1 w by allowing p 1 and p 2 = v 1 + v 2 # p 1 # L to diverge in order to ensure 1! = 1. Now consider how the optimal linear rule is structured when it cannot secure the Örstbest outcome. Initially, the values of p 1 and b k 2 that maximize expected welfare when p 2 is set to leave the marginal consumer with zero surplus in the duopoly setting are identiöed (see equations (18) and (19) below), and the values of d 1 and d 2 that induce these prices are determined. m is then set to induce the identiöed value of b k 2. (From equation (12), m = 1 ( [ 1 2 # b k 2 ].) The properties of the optimal balanced rule then depend upon whether the identiöed rule generates more or less than the maximum feasible industry proöt, as Proposition 3 indicates. The proposition refers to 1 2,whichisÖrm2ís expected proöt (not counting innovation costs) at the equilibrium identiöed in Lemma 3, where duopoly industry proöt is maximized. Proposition 3. Suppose inequality (16) does not hold. Also suppose ep 2 = v 1 + v 2 # ep 1 # L, and ep 1 and e k 2 2! k 2 ; k 2 " are the values of p1 and b k 2 that solve [ k w 1 ( b k 2 ) # b k 1 k F ( b k 1 )[k o 2 # b k 2 k F ( b k 1 ) G( b k 2 f W 1 = 0,and (18) [ k w 1 ( b k 2 ) # b k 1 k 1 b k 2 + F ( b k 1 )[k o 2 # b k 2 k 2 b k 2 = 0. (19) (i) If e 1=ep1 [ v 1 # ep 1 ]+[v 1 + v 2 # ep 1 # L ][L + ep 1 # v 1 ] < 1; then the optimal linear rule is ( e d 1 ; e d 2 ; em), where e d 1 and e d 2 are the values of d 1 and d 2 that induce ep 1 and ep 2, 26 and em = 1 ( [ e1 2 # e k 2 ],where e1 2 = ~p 2 q 2 (~p 1 ; ~p 2 ) is Örm 2ís equilibrium proöt. (ii) If e 1 " 1 for the identiöed ep1 and ep 2,thentheoptimallinearruleis (d 1 ; d 2 ; m), where 26 These values are identiöed in the proof of Lemma 1 in Appendix A. 15

17 d 1 and d 2 are speciöed in expression (48), e k 2 solves equation (19), and m = 1 ( [ 1 2 # e k 2 ]. Proposition 3 reáects the fundamental trade-o that arises when the optimal linear rule cannot secure the Örst-best outcome. When v 1 6= v 2 ; setting d 1 and d 2 to induce distinct duopoly prices (p 1 6= p 2 )canraiseindustryproöt,whichcanbeemployedtoenhanceinnovation incentives. However, the distinct prices induce outputs that do not maximize welfare. Proposition 3 reports that, in e ect, p 1 is optimally set to balance these two considerations (where, given p 1, p 2 ensures that the marginal consumer receives zero surplus) as long as the resulting duopoly industry proöt ( 1)doesnotexceedthemaximumfeasibleindustryproÖt e (1). 27 When 1 e " 1, proötcannotbeincreasedfurtherbyalteringequilibriumprices,so the maximum industry proöt constraint binds. The optimal prices maximize industry proöt in this case (i.e., p 1 = p 1 and p 2 = p 2 ). 28 k w 1 # b k 1 > 0 can be viewed as a measure of the extent to which Örm 1 has insu cient incentive to innovate. F ( b k 1 )[k2 w # b k 2 ] can be viewed as a measure of the extent to which Örm 2 is expected to have insu cient incentive to innovate, where the expectation reáects the probability that Örm 2 will have an opportunity to innovate (because Örm 1 has innovated). Equation (19) implies that the patent infringement penalty is optimally increased to the point where the marginal reduction in Örm 1ís innovation deöciency is equal to the increase in Örm 2ís expected innovation deöciency, taking into account the rate at which b k 1 declines and b k 2 increases as the penalty increases. 5 The Optimality of the Linear Rule We now demonstrate that the optimal linear rule achieves the highest expected welfare among all balanced patent infringement damage rules. A balanced damage rule can specify 27 Notice that if v 1 = v 2 ; then ep 1 = ep 2 = p w and e "="=" W, so only equation (19) is relevant. 28 Assumption 1 ensures that all consumers purchase a unit of the product and each Örm serves some customers in both the welfare-maximizing and proöt-maximizing outcomes. Consequently, Proposition 3 implies that when characterizing the optimal linear rule, there is no loss of generality in focusing on the duopoly equilibrium in which all consumers purchase a unit of the product and both Örms serve some customers. 16

18 the prices the Örms set, a transfer payment from Örm 2 to Örm 1 following innovation by both Örms, the probability that Örm 1 innovates, and the probability that Örm 2 innovates, following innovation by Örm By the revelation principle (e.g., Myerson, 1979), it is without loss of generality to consider truthful direct mechanisms, where Örms are induced to truthfully report their privately-observed innovation costs. Let? 1 (k 0 1) 2 [0; 1] denote the probability that Örm 1 is required to innovate when it reports its innovation cost to be k 0 1.Alsolet? 2 (k 0 1;k 0 2) 2 [0; 1] denote the probability that Örm 2 is required to innovate following innovation by Örm 1 when Örm i reports its innovation cost to be k 0 i,fori =1; 2. p i (k 0 1;k 0 2) will denote the corresponding price that Örm i 2f1; 2g must set and T (k 0 2;k 0 2) will denote the corresponding payment that Örm 2 must make to Örm 1whenbothÖrmsinnovate. p M 1 (k 0 1;k 0 2) will denote the corresponding price that Örm 1 must set when it is the sole innovator. Denote this vector of policy instruments by: Z(k 0 1;k 0 2) $ +? 1 (k 0 1);? 2 (k 0 1;k 0 2); p 1 (k 0 1;k 0 2); p 2 (k 0 1;k 0 2); p M 1 (k 0 1;k 0 2); T(k 0 1;k 0 2),. In addition, let q z i (k 0 1;k 0 2) denote the demand for Örm iís product following innovation by both Örms, given that Örm i 2f1; 2g sets price p i (k 0 1;k 0 2). Furthermore,let1 z i (k 0 1;k 0 2)= p i (k 0 1;k 0 2) q z i (k 0 1;k 0 2) denote the corresponding proöt of Örm i (not counting innovation costs). q Mz 1 (k 0 1;k 0 2) will denote the demand for Örm 1ís product when it is the monopoly supplier and it sets price p M 1 (k 0 1;k 0 2). 1 Mz 1 (k 0 1;k 0 2)=p M 1 (k 0 1;k 0 2) q Mz 1 (k 0 1;k 0 2) will denote Örm 1ís corresponding monopoly proöt (not counting innovation costs). Total welfare in this setting when innovation costs (k 1 ;k 2 ) are reported truthfully is: W Z = Z k1 0? 1 (k 1 ) f Z k2 0 f? 2 (k 1 ;k 2 )[ f W z 12(k 1 ;k 2 ) # k 2 ] +[1#? 2 (k 1 ;k 2 )] f W z 1 (k 1 ;k 2 ) g dg(k 2 ) # k 1 g df (k 1 ) where W fz 1 (k 1 ;k 2 ) = v 1 q1 Mz (k 1 ;k 2 ) # 1! q Mz 1 (k 1 ;k 2 ) " 2 2 and 29 The balanced damage rules that we analyze permit the speciöcation of prices, not quantities. However, the speciöcation of a su ciently high price for Örm 2ís product can e ectively preclude the sale of the product, thereby functioning like an injunction (e.g., Shapiro, 2016). 17

19 fw z 12(k 1 ;k 2 ) = v 1 q z 1(k 1 ;k 2 ) # 1 2 [ qz 1(k 1 ;k 2 )] 2 + v 2 q z 2(k 1 ;k 2 ) # 1 2 [ qz 2(k 1 ;k 2 )] 2. Firm 1ís expected payo when its innovation cost is k 1,itreportsthiscosttobek 0 1,and it anticipates that Örm 2 will report its innovation cost truthfully is: B 1 (k 0 1 j k 1 ) $? 1 (k 0 1)[H(k 0 1) # k 1 ],where H(k 0 1) $ Z k2 0 f? 2 (k 0 1;k 2 )[1 z 1(k 0 1;k 2 )+T (k 0 1;k 2 )] + [1#? 2 (k 0 1;k 2 )]1 Mz 1 (k 0 1;k 2 ) g dg(k 2 ). (20) Following report k1 0 by Örm 1, Örm 2ís expected payo when its innovation cost is k 2 and it reports this cost to be k2 0 is: B 2 (k2 0 j k 2 ; k1) 0 $? 2 (k1;k 0 2)[1 0 z 2(k1;k 0 2) 0 # T (k1;k 0 2) 0 # k 2 ]. (21) The welfare-maximizing damage policy in this setting is the solution to the following problem, denoted [P-Z]: Maximize Z(k 1 ;k 2 ) W Z ; subject to, for all k i ; k 0 i 2 [0; ' k i ](i 2f1; 2g): B 1 (k 1 j k 1 ) " maximum f 0, B 1 (k 0 1 j k 1 ) g, and (22) B 2 (k 2 j k 2 ; k 1 ) " maximum f 0, B 2 (k 0 2 j k 2 ; k 1 ) g. (23) Inequality (22) ensures that Örm 1 truthfully reports its realized innovation cost and secures a nonnegative expected payo by doing so. Inequality (23) ensures the corresponding outcomes for Örm 2, for any cost report by Örm 1. To characterize the solution to [P-Z], it is helpful to consider problem [P-Z] 0,whichis [P-Z] except that constraints (22) and (23) are replaced by: B 1 (k 1 j k 1 ) " 0 and B 2 (k 2 j k 2 ; k 1 ) " 0. (24) Observe that if constraints (22) and (23) are satisöed at a solution to [P-Z] 0, then the identiöed solution to [P-Z] 0 is a solution to [P-Z]. Lemma 4. p M 1 (k 1 ;k 2 ) = p M 1 = v # 1 for all k 1 2 [0; k 1 ] and k 2 2 [0; k 2 ] for which? 1 (k 1 ) > 0 and? 2 (k 1 ;k 2 )=0at a solution to [P-Z] 0.Moreover,q z i (k 1 ;k 2 )=q i (p 1 (k 1 ;k 2 ); 18

20 p 2 (k 1 ;k 2 )) as speciöed in equation (2) for i =1; 2, forallk 1 2 [0; k 1 ] and k 2 2 [0; k 2 ] for which? 1 (k 1 ) > 0 and? 2 (k 1 ;k 2 ) > 0 at a solution to [P-Z] 0. Lemma 4, which reáects Assumption 1, indicates that full market coverage is induced in both the monopoly and duopoly settings at the solution to [P-Z] 0. Lemma 5. At a solution to [P-Z] 0,thereexist ~ k 1 2 [0; k 1 ] and ~ k 2 2 [0; k 2 ] such that: (i)? 1 (k 1 ) = 1 for all k 1 2 [0; ~ k 1 ] and? 1 (k 1 ) = 0 for all k 1 2 ( ~ k 1 ; k 1 ]; (ii) for each k 1 2 [0; ~ k 1 ]:? 2 (k 1 ;k 2 )=1 for all k 2 2 [0; ~ k 2 ] and? 2 (k 1 ;k 2 )=0for all k 2 2 ( ~ k 2 ; k 2 ]; and (iii) p i (k 0 1;k 0 2)=p i (k 00 1;k 00 2) and T (k 0 1;k 0 2)=T (k 0 1;k 00 2) for all k 0 2;k [0; ~ k 2 ] for each k [0; ~ k 1 ]. Lemma 5 indicates that if innovation is induced, it is induced for the smaller realizations of the Örmsí innovation costs. Furthermore, stochastic innovation (? i (,) 2 (0; 1)) serves no useful purpose in the present setting. 30 In addition, because the Örmsí production costs do not vary with their innovation costs, there is no gain from inducing duopoly prices and output levels (and thus transfer payments, T (,)) thatvarywithreported innovation costs. Lemmas 4 and 5 imply that each Örm e ectively faces the simple choice of innovating or not innovating at the identiöed solution to [P-Z] 0. The choices are structured so that innovation is proötable for a Örm if and only if its realized innovation cost is su ciently small, i.e., if k i ) k ~ i. Therefore, the Örms have no incentive to misrepresent their realized innovation costs, so constraints (22) and (23) are satisöed at the identiöed solution to [P-Z] 0. Consequently, a solution to [P-Z] 0 that satisöes the properties identiöed in Lemmas 4 and 5 is a solution to [P-Z]. It follows that expected welfare in the present setting can be written as: fw Z = F ( ~ k 1 ) G( ~ k 2 ) f W 12 + F ( ~ k 1 )[1# G( ~ k 2 )] f W 1 30 Hart and Reny (2015) analyze a setting in which randomization in product assignment can enhance a sellerís expected revenue when he sells multiple products, but not when he sells a single product. 19

21 Z k ~ 2 Z k ~ 1 # F ( k ~ 1 ) k 2 dg(k 2 ) # k 1 df (k 1 ). (25) Furthermore, when ~ k i ) k i for i =1; 2: 0 0 [ 1! 1 + T ] G( ~ k 2 )+1 M 1 [1# G( ~ k 2 )] = ~ k 1 and 1! 2 # T = ~ k 2, (26) where T is a constant and 1! i = p i q i (p 1 ;p 2 ) does not vary with k 1 or k 2,fori 2f1; 2g. Because the corresponding industry proöt is 1! = 1! 1 + T +(1! 2 # T )= 1! 1 + T + ~ k 2, expression (26) can be written as: [1! # ~ k 2 ] G( ~ k 2 )+1 M 1 [1# G( ~ k 2 )] = ~ k 1 and 1! 2 # T = ~ k 2. (27) Therefore, problem [P-Z] 0 can be written as problem [P-Z] 00 : Maximize p 1 ;p 2 ;T fw Z,where f W Z is speciöed in equation (25), the Örmsí outputs are speciöed in expression (2), and ~ k i is speciöed in expression (27) when ~ k i ) k i,fori =1; 2. Consequently, problem [P-Z] 00 is identical to Problem [P], which ensures the following conclusion holds. Proposition 4. The optimal linear rule achieves the highest welfare among all balanced damage policies. Proposition 4 reáects the fact that linear rules can link damage payments to the proöts of both Örms and can specify lump-sum transfer payments between the Örms. Consequently, linear rules provide widespread latitude to induce desired allocations of industry output while implementing desired conögurations of industry proöt and corresponding innovation incentives. We now illustrate the welfare gains that the optimal linear rule can secure relative to the LP rule and the UE rule. We do so in the following baseline setting, whereconsumers value the two Örmsí products symmetrically, the innovation costs for the two Örms have the same uniform distribution, and Örm 2ís innovation increases the size of the market by eighty percent. Baseline Setting. v 1 = v 2 =7:5, L =1:8,! =0:50, andf (k 1 ) and G(k 2 ) are uniform 20

22 distributions with k i =0 and k i =5,for i =1; 2. Table 1 identiöes the optimal linear rule for patent infringement (d! 1;d! 2;m! ) in the baseline setting and as product valuations change. 31 The table also reports the level of expected welfare that arises under the optimal linear rule (W! ), under the UE rule (W UE ), under the LP rule (W LP ), and in the Örst-best outcome (W FB ). v 1 v 2 d! 1 d! 2 m! W! W UE W LP W FB 5 5 0:72 0:72 0:21 3:03 2:49 2:58 3: :25 1:5 5:56 7:94 6:96 5:35 7:94 7:5 7:5 0:81 0:81 0 7:69 5:98 6:22 7: :5 0:25 #5:56 7:94 6:37 6:16 7: :81 0: :19 10:03 9:45 12:19 Table 1. The E ects of Changing Product Valuations. Four elements of Table 1 warrant emphasis. First, there are many settings where the optimal linear rule ensures the Örst-best outcome. Second, the optimal linear rule often secures a substantial increase in welfare above the levels generated by the LP and UE rules. This is the case both when the optimal linear rule achieves the Örst-best outcome and when it does not do so. To illustrate, W! exceeds max fw UE ;W LP g by more than 21% when v 1 = v 2 =10,andbymorethan17% when v 1 = v 2 =5. Also observe that W! exceeds W LP by more than 48% when v 1 =7and v 2 =8. Third, the lump-sum payment from Örm 2 to Örm 1 under the optimal linear rule (m! ) can be positive, negative, or zero. For the settings in Table 1 where v 1 + v 2 = 15, m! is positive (negative) when consumers value Örm 2ís (Örm 1ís) product relatively highly and is zero when v 1 = v As the product valuations, v 1 and v 2, change in Table 1, the values of all other parameters remain at their values in the baseline setting. 32 Observe that when v 1 + v 2 = 15, W! is higher in Table 1 when v 1 6= v 2 than when v 1 = v 2. Welfare is higher in the presence of asymmetric product valuations here because consumers purchase more of the product they value most highly and less of the product they value less highly. 21

23 Fourth, the optimal linear rule does not always resemble the UE rule more than the LP rule (in the sense that d! 2 >d! 1)whentheUErulegeneratesahigherlevelofwelfarethan the LP rule. 33 This is the case because d! 1 and d! 2 a ect multiple determinants of welfare ñ both output allocations and innovation decisions ñ in nonlinear fashion. Consequently, even though welfare might be higher when, say, d 2 =1and d 1 = m =0than when d 1 =1and d 2 = m =0,itdoesnotfollowthatd 2 will exceed d 1 under the optimal linear rule. Appendix B illustrates how the optimal linear rule and its performance vary as other model parameters (k 1 ; k 2 ;L,and!) change.thenumericalsolutionsreportedinappendix Bindicate,forexample,thattherelativeperformanceoftheoptimallinearrule(W! =W LP and W! =W UE )tendstoincreaseasl (the extent to which Örm 2ís innovation expands the market) increases. 6 Concluding Remarks We have introduced and characterized the optimal linear rule for patent infringement damages under sequential innovation. We have demonstrated that this rule often secures substantially higher welfare than common rules like the LP rule and the UE rule. We have also identiöed conditions under which a linear rule can induce both e cient incentives for sequential innovation and the e cient allocation of industry output. Moreover, we have shown that the linear rule is optimal among all balanced patent infringement damage rules. We have emphasized the welfare gains that the optimal linear rule can secure relative to the LP rule and the UE rule. However, the optimal linear rule can also secure substantially higher levels of welfare than other popular damage rules, including the reasonable royalty (RR) rule. Under the RR rule, when Örm 2 is found to have infringed Örm 1ís patent, Örm 2 is required to deliver to Örm 1 the royalty payments that Örm 1 would have collected if the two Örms had negotiated a licensing/royalty agreement, knowing that Örm 2ís product infringes Örm 1ís patent (e.g., Henry and Turner, 2010). 34 Absent contracting frictions, the Örms 33 See the fourth row of data in Table 1 where v 1 =8and v 2 =7. 34 As noted above, U.S. patent law stipulates that the damage penalty for patent infringement must be ìadequate to compensate for the infringement, but in no event less than a reasonable royalty for the use 22