Cournot Meets the Chicago School

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1 Cournot Meets the Chicago School Mikko Packalen University of Waterloo 14 February 2006 Abstract This paper re-examines whether cooperation between complementary monopolists increases consumer surplus. We show that Cournot s celebrated double monopoly result can be overturned when each incumbent monopolist s incentive to induce entry into complementary markets is taken into account. We present a formal analysis of patent pooling in which renegotiable price commitment provides each incumbent the means for inducement of entry. As increased entry into a complementary market indirectly also increases the expected reward for entry into the monopolist s own market, an important step in the analysis is to show that inducement of entry can nevertheless be pro table if entry is uncertain. JEL Classi cation Codes: K11, K21, L41, M2. This is Chapter 1 of my Stanford University Dissertation. I m grateful to Roger Noll for guidance and encouragement, to Jay Bhattacharya and Jon Levin for advice and support, and to Tim Bresnahan, Liran Einav, Jeremy Bulow, Susan Athey, Steven Tadelis, Bruce Owen, Greg Rosston, Rob McMillan, Peter Coles, Paul Riskind, and Kelley Porter for discussions. I thank the ASLA-Fulbright Foundation, the John M. Olin Foundation, the Emil Aaltonen Foundation, the Yrjö Jahnsson Foundation, and the Finnish Cultural Foundation for nancial support. Only the author is culpable for any errors or omissions. packalen@uwaterloo.ca.

2 1 Introduction This paper re-examines whether cooperation between complementary monopolists necessarily increases consumer surplus. The premise of our approach is that when each incumbent monopolist faces potential entry, e ciency comparisons that follow the double-monopoly analysis of Cournot (1838) can be at odds with the implications of an established corollary of the Chicago School s single-monopoly-rent theorem. Cournot s double-monopoly result states that prices are lower and pro ts are higher when complementary goods are priced by a single monopolist compared to when the complementary goods are priced non-cooperatively by separate monopolists. The single-monopoly-rent theorem in contrast states that a monopolist cannot increase its pro ts by monopolizing a competitive complementary market. 1 According to a corollary of the single-monopoly-rent theorem, each incumbent monopolist bene ts from increased competition in a complementary market because the demand for one incumbent s product rises when increased competition lowers the quality-adjusted price of the complementary good. 2 This corollary in turn implies that each incumbent monopolist has an incentive to induce entry into a complementary market. Lower quality-adjusted prices from increased entry generally bene t nal consumers as well. Consequently, in this paper we show that because integration or cooperation by the incumbent monopolists can eliminate the incentive to induce entry, cooperation of complementary monopolists does not necessarily increase expected consumer surplus. 3 Analysis that relies on the inducement of entry obviously requires that the incumbent monopolists have the means to induce entry into a complementary market. The history of direct involvement in the provision of complementary goods is rich. Examples include the elimination of complementary river toll posts by force in the middle ages, subsidy of roads and maps by the automobile industry in the early twentieth century, and Intel s nancing of the development of computationally demanding applications (see Hecksher (1931), Brandenburger and Nalebu (1996), and Chesbrough (2003), respectively). In many contexts such direct involvement may be prohibitively expensive. In our analysis we focus on indirect involvement in the provision of complementary goods. An incumbent 1 The single-monopoly-rent theorem was presented rst in Director and Levi (1956, p. 290) and Bowman (1957, p. 21), and later in Posner (1976, p. 173) and Bork (1978, p. 373). 2 See Judge Richard Posner s opinion in Olympia Equipment Leasing Co. v. Western Union Telegraph Co., 797 F.2d 370, 374 (7th Cir. 1986), Whinston (1990, p. 840), and Farrell and Weiser (2003, section IV). 3 A novel conceptual contribution of our analysis is to show that two goods can be ex-ante substitutes and ex-post complements. According to Nalebu (2003) such a case has not come up in the antitrust literature, whereas the case in which two goods are ex-ante complements and ex-post substitutes arises fairly often (see also Ma (1997)). 1

3 monopolist can induce entry into complementary markets indirectly by decreasing the expected quality-adjusted price of its own good, by allowing other rms to develop complementary goods, or by disclosing technical information that makes the development of the complementary goods easier. In this paper we focus on price commitment as a means to induce entry into complementary markets. An incumbent s price commitment to induce entry into a complementary market also indirectly increases the expected reward for displacing the incumbent itself. Therefore, an important step in the analysis is to show that inducing entry can be pro table for each incumbent in equilibrium if entry is uncertain. We analyze the e ciency of cooperation formally in a model of patent licensing. While complementary monopolies arise in many contexts, fragmentation of patent rights has made the problem of complementary monopolists particularly prevalent in the context of patent licensing. Many commentators have suggested that this fragmentation of patent rights may lead to ine cient outcomes and forestall technical progress, especially in the information technology and biotechnology industries (see e.g. Heller and Eisenberg (1998), Merges (2001), Shapiro (2001), Hall and Ziedonis (2001), and Ziedonis (2004)). 4 Apart from abandoning or weakening patent rights, the suggested solution to the fragmentation of patent rights has been to encourage pooling of di erent patent owners complementary patents (see e.g. Shapiro (2001) and Gilbert (2004)). The analysis in this paper is in part intended to re-examine the desirability of pooling complementary patents. The premise of this analysis is that pricing of patent licenses a ects R&D investments and that, conversely, both the expectation and the outcome of R&D investments a ect prices of licenses for existing patents. The e ciency of pricing of patent licenses, therefore, cannot be evaluated in isolation from the decision to invest in R&D. We formulate a model with four essential ingredients that depict salient institutional characteristics of many technology markets. These ingredients are: 1) two incumbent monopolists own patents that are complementary; 2) the incumbent patent owners license their patents to a competitive downstream industry; 3) the incumbent patent owners are restricted to royalty licensing and can commit to lower than ex-post optimal royalty rates for licenses to their patents; and 4) each incumbent patent owner faces potential entrants who can invest in risky R&D and attempt to displace the incumbent patent owner. 4 The emergence of this potential danger bears similarities with the detrimental e ects that the fragmentation of transportation rights may historically have had on trade and economic growth (see e.g. Hecksher (1931) and North and Thomas (1973)). As has been noted by Buchanan and Yoon (2000), Adam Smith (1776) observed in passing in the context of river tolls that complementary rights are more valuable to a single rights holder than to multiple separate rights holders. Fragmentation of complementary transportation rights has since been analyzed in more detail by e.g. Ellet (1839), Karni and Chakrabarti (1997), and Feinberg and Kamien (2001). 2

4 We compare e ciency when the incumbent patent owners set the terms of licenses to their patents non-cooperatively (non-cooperative pricing), when the incumbent patent owners set the terms of licences to their patents cooperatively and do not allow either incumbent to change the terms of licenses to their patents non-cooperatively (exclusive cooperative pricing), and when the incumbent patent owners cooperatively set the terms of licenses that cover the bundle of both incumbent patents and non-cooperatively set the terms of licenses to that cover only each incumbent s own patent (non-exclusive cooperative pricing). 5 The three pricing arrangements yield di erent outcomes because of the presence of two pricing externalities: the double-marginalization externality and the entry-inducement externality. The double-marginalization pricing externality captures the pricing externality examined by Cournot (1838). This pricing externality arises because the potential consumers of the nal good are heterogenous in their valuations for goods that make use of the technology covered by the two incumbent patents. The entry-inducement pricing externality in turn captures the aforementioned corollary of the Chicago School s single-monopoly-rent theorem. By committing to a lower than expost optimal price for a license to its patent, an incumbent can increase the reward to successful entry against the complementary incumbent because the lower price increases the number of active consumers of goods that make use of complementary patents. The higher reward to entry in turn increases the probability of entry. Each incumbent has an incentive to induce entry against the complementary monopolist because entry against the complementary incumbent enables the remaining incumbent to capture all of the monopoly pro ts from the combined complementary patents. In the non-cooperative pricing model each incumbent has an additional incentive to induce entry into a complementary market because the displacement of one incumbent monopolist eliminates double marginalization. Double marginalization is avoided in the two cooperative pricing models. The entryinducement pricing externality is internalized only in the exclusive cooperative pricing model. 5 Patent pools have existed for over a century (see Gilbert (2004) and Lerner et. al. (2003)). The recent increase in interest in patent pools is closely related to the recent increase in the importance of technology markets, which are addressed in detail in Arora et. al. (2001). Recent examples of pooling arrangements between complementary patent owners that correspond to the exclusive cooperative pricing model include: 1) The joint marketing of patents that cover technologies related to Single-Chain Antibodies between Enzon and Micromet; 2) Micromet s acquisition of patents from Curis; 3) the MP3 patent pool between Fraunhofer and Thomson; and 4) Acquisition of Wi-Max patents by Wi-Lan Inc. from Ensemble Communications. Examples of pooling arrangements between complementary patent owners that correspond to the non-exclusive cooperative pricing model include: 1) Two DVD patent pools (members include Hitachi, Sony, Time Warner, etc.); 2) Several MPEG patent pools (members include Apple, Canon, Toshiba, etc.); 3) The AMR patent pool (members are VoiceAge, Nokia and Ericson); and 4) WCDMA patent pools (members include ETRI, Fujitsu, NTT DoCoMo, etc.). 3

5 If neither incumbent is displaced by an entrant, cooperative pricing, by avoiding double marginalization, decreases prices and increases consumer surplus. Internalization of the entryinducement pricing externality, in contrast, increases prices and decreases the probability of entry, and therefore decreases expected consumer surplus. The incumbents expected pro ts are thus higher in the exclusive cooperative pricing model than in either of the other two pricing models. Each incumbent s incentive to commit to a lower than ex-post optimal price is in part determined by whether, in the absence of entry, the double-marginalization pricing externality is internalized or not. Hence, while a non-exclusive cooperative pricing arrangement cannot yield lower expected surplus than an exclusive cooperative arrangement, non-exclusive cooperative pricing does not necessarily yield higher expected consumer surplus than noncooperative pricing. The analysis in this paper shows that welfare comparisons among the three arrangements can be made using straightforward economic concepts, namely the shape of the demand curve for licenses to the incumbent patents absent entry, the probability of entry, and the elasticity of the probability of entry with respect to the expected revenue that a successful entrant receives. We rst compare expected consumer surplus in the three pricing models taking the incumbent innovations as given. We show that non-cooperative pricing can yield a more or less e cient outcome than exclusive cooperative pricing. Generally, though not always, nonexclusive cooperative pricing yields a more e cient outcome than either non-cooperative pricing or exclusive cooperative pricing. To capture the initial incentive of incumbents to innovate, we also examine the ratio of the di erence in expected consumer surplus and the di erence in the incumbents expected revenue. This comparison yields an approximation of the relative e ciency of two pricing models in terms of the reward elasticity of the probability that the incumbent innovations are invented in the rst place. These results again suggest that the non-exclusive cooperative pricing model yields a higher expected consumer surplus than either of the other two pricing models. Our results have the following policy implication concerning patent pooling. If an exclusive cooperative pricing arrangement does not have pro-competitive e ects other than elimination of double marginalization, the antitrust authorities should require that pooling of patents or copyrights be accomplished through a non-exclusive cooperative pricing arrangement, rather than an exclusive cooperative pricing arrangement such as a merger, the acquisition of complementary patents, or an exclusive patent pool. This policy conclusion illustrates the importance of taking the entry-inducement incentive into account in the analysis of cooperation of complementary incumbents, as the result 4

6 contrasts with both the existing literature and the current antitrust enforcement policy in the U.S. regarding pooling of intellectual property. The current policy regarding pooling of intellectual property is outlined in the U.S. F.T.C./D.O.J. Antitrust Guidelines for Licensing of Intellectual Property (1995) and in the U.S. D.O.J. Business Review Letters (1997, 1998, 1999). These documents indicate that the current policy is to allow both exclusive and nonexclusive cooperative pricing arrangements between owners of complementary patents, and that antitrust authorities believe that requiring the agreement to be non-exclusive mainly serves to protect against pools containing patents that are substitutes. The policy to allow cooperative pricing of complementary patents is largely based on the analysis of the double-monopoly problem by Cournot (1838). Recent articles on cooperative pricing of complements (Economides and Salop (1992)) and on patent pools (Shapiro (2001), Gilbert (2004) and Lerner and Tirole (2004)) have reiterated and reinforced this view. 6 Lerner and Tirole (2004) also show that requiring patent pools to allow independent licensing (non-exclusive patent pools) can act as a safeguard against the formation of welfare-decreasing patent pools by owners of patents that are substitutes without a ecting the revenue to members of a welfare-enhancing patent pool formed by owners of patents that are complementary. Our analysis is obviously closely related to the Chicago School s single-monopoly-rent theorem as applied to monopoly leveraging. One focus of research on monopoly leveraging is on tying of complementary goods by a rm with market power in at least one of two or more complementary markets (see e.g. Whinston (1990), Nalebu (2000), Choi and Stefanadis (2001), Gilbert and Katz (2003), and Carlton and Waldman (2002)). Whinston (1990) recognizes that an incumbent bene ts from increased competition in a complementary market but does not examine the incentives for cooperation between rms with market power in complementary markets or the case with entry threats in both markets. Carlton and Waldman (2002) recognize that entry in a complementary market can a ect entry in the essential good market, 7 but in their analysis the e ect comes through intertemporal cost e ciencies whereas in our analysis the e ect of entry in one market on entry in a complementary market comes purely from the e ect that entry in one market has on the 6 Pooling of intellectual property has also been addressed in many law review articles (see e.g. Gollier (1968), Tom and Newberg (1997), Merges (1996, 2001), and Carlson (1999)). These contributions echo the existing economics literature by making a separation between pooling of competing patents and pooling of complementary patents. In contrast, our analysis shows that patents cannot be categorized as being either competing or complementary because the same two patents can be ex-post complements but ex-ante substitutes. 7 An essential good is purchased by all active buyers whereas goods that are complementary to the essential good are purchased by only some active buyers. 5

7 rewards for entry in a complementary market. 8 Choi and Stefanadis (2001) examine technical tying of two goods by a rm that is a monopolist in both markets and faces potential entry but they do not examine cooperation of monopolists or the case of heterogenous consumers. In the analyses of bundling by a monopolist in Nalebu (1999) and Gilbert and Katz (2003), price commitment provides the means for entry deterrence. In Gilbert and Katz (2003) price commitments also facilitate complementary investments by buyers. In our analysis price commitment facilitates inducement of entry. Our analysis departs from much of the literature on tying and vertical foreclosure in that in our analysis tying is not physical but contractual as in Gilbert and Katz (2003). Our analysis is also di erent from the analysis of forward contracting and quantity competition by Allaz and Vila (1993) as we instead consider option contracting and price competition between complementary rms. In our analysis we assume that incumbent patent owners cannot invest in R&D, which amounts to assuming that the "Arrow e ect" is operative. 9 Similar to a price commitment, a patent owner could commit to a lower expected quality-adjusted price for its technology by investing in R&D to improve its technology. The e ect of R&D investments by incumbents on the R&D investments of potential entrants is ambiguous because the R&D investment incentives of the complementary incumbent patent owners would be a ected as well. The analysis of cooperation by incumbents when incumbents, too, can invest in R&D and face potential entry is left for future research. 10 This paper is organized as follows. In the next section we present the model. In the 8 Bresnahan (2004) notes that with network externalities the sudden development of a complementary market can increase the demand for the essential good so dramatically that the incumbent essential good monopolist s installed base advantage is threatened. This e ect of increased provision of complementary goods was rst understood by the Microsoft Corporation (see Bresnahan (2004)). 9 The "Arrow e ect" refers to circumstances in which the incentive to engage in R&D is greater for entrants than for incumbents, see Arrow (1962). For an explication of circumstances in which the opposite is the case, see Gilbert and Newbery (1982). The model explored here is a special case of a circumstance in which the equilibrium R&D e ort by entrants exceeds the equilibrium R&D e ort by an incumbent monopolist. Whether this is an innocuous assumption is ultimately an empirical question concerning whether innovation is more likely to come from incumbents or entrants. The theoretical literature on whether incumbents or entrants innovate more is extensive and yields ambiguous results (see e.g. Reinganum (1989) and Gallini and Scotchmer (2002)). Empirical research on innovation by entrants and incumbents is also extensive (see e.g. Henderson (1993), Lerner (1997), Anderson and Tushman (1986), and Christensen (1997)). However, these empirical analyses confound the incentive to undertake R&D with the decision to adopt and commercialize a technology in the nal goods market. Therefore, as commercialization in the form of licensing is unlikely to involve the same complexities as commercialization in a nal goods market, this empirical literature is not necessarily informative about incumbents and entrants relative propensities to innovate and commercialize when commercialization of an innovation by the innovating rm means licensing the innovation. 10 Previously the e ect of integration of two complementary rms that do not face potential entry on the level of their R&D investments has been addressed by Economides (1999). The e ect of an essential good monopolist s integration into complementary markets on R&D incentives of the monopolist and rms that are only active in the complementary market has been analyzed in Farrell and Katz (2001) and Heeb (2003). 6

8 third section we present the equilibrium analysis of each of the three pricing models. For each pricing model we determine conditions under which a symmetric subgame-perfect purestrategy Nash equilibrium exists, and conditions under which the incumbents commit to lower than ex-post optimal maximum prices for licenses to their patents. In the fourth section we compare e ciency in the three pricing models both when e ciency is measured by the expected consumer surplus and the incumbent innovations are taken as given, and when the e ect of equilibrium pricing on the incumbent innovations is taken into account as well. The fth section concludes. 2 The Model We consider a setting with two upstream incumbent patent owners (incumbents). Each incumbent owns one patent and faces the threat that one of many potential entrants will innovate around its patent. The incumbent patents are perfectly complementary in that a license to exploit either is valueless in the absence of a license to exploit the other or, in the case of successful entry, its substitute. The incumbents license their patents using a constant, nondiscriminatory royalty to one or more competitive downstream industries that sells nal goods. We assume that in the absence of royalties the downstream rms would have zero xed and marginal costs. Potential consumers of nal goods are heterogenous in their valuations for the nal goods that make use of any two complementary patents. Timing in the model is depicted in gure 1. Stage 1: Ex Ante Royalty Commitment Stage 2: R&D Investment Stage 3: Ex Post Royalty Competition Stage 4: Buyers Decide Nature Reveals R&D Outcome Figure 1: Timing. In the rst stage the incumbents can commit to royalty rates that they can later decrease but not increase. 11 In the second stage the potential entrants R&D investment decisions 11 Price commitments are commonplace in patent licensing. Here we assume that price commitments are 7

9 are determined and nature subsequently reveals the success of the potential entrants R&D investments. In the third stage the incumbents o er licenses and engage in Bertrand competition on royalty rates with any successful entrant, which implies that the equilibrium royalty is zero if two patents are perfect substitutes. In the fourth stage rms in the competitive downstream industry decide which technologies to include in the nal goods, acquire the requisite licenses, produce and sell nal goods to nal consumers, and pay licensing fees accordingly. In the rest of this section we present in more detail the features that are common for all three pricing models. The details of how the incumbents price licenses to their patents in stages 1 and 3 in each of the three pricing models are discussed in connection with the equilibrium analysis of each pricing model in sections The Technology Market Firm 1 owns incumbent patent 1 and rm 2 owns incumbent patent If a downstream rm sells goods that uses technology that is covered by either incumbent patent, it must obtain a license that covers that patent and pay the associated licensing fees. Each incumbent (and each entrant) is restricted to o ering only licensing contracts that condition payments on the licensee s sales of nal goods that make use of the technology covered by the license (royalty licensing). Royalty licensing is commonplace in patent licensing, and even in cases where patent licenses are not pure royalty contracts, royalties are still often an important component of a patent licensing contract. 13 Exclusive licenses with a lump-sum payment have the advantage that they avoid the double-marginalization problem associated with royalty licenses. Nevertheless, patent owners have several other reasons to prefer royalty licenses instead of licenses with lump-sum payments. One reason arises from two-sided asymmetric information between patent owners and rms in the downstream industry. If the patent owners have more information about either the value of their technology or the validity of their patents, each renegotiable, which also is in line with actual patent licensing arrangements. 12 As has been recently emphasized by Shapiro (2003), patent licensing fees are likely to re ect both the di culty of innovating around the patents and the patents strengths in litigation. In this paper we ignore patent validity, but our analysis can be straightforwardly extended to the case in which the incumbent patents are valid only with some probability and potential buyers can attempt to displace each incumbent patent owner through investments in risky litigation. 13 Because the incumbents only o er royalty contracts, it is costless for downstream rms to accept licenses in stage 1. Therefore, if an incumbent patent is covered by both a license o ered in stage 1 and a license o ered in stage 3, and a downstream rm sells goods that use the technology covered by that incumbent patent, the downstream rm is free to choose which license to use when determining the royalty payments to the owner of that patent in stage 4. 8

10 downstream rm has an incentive not to sign licensing contracts with up-front payments that are independent of whether the downstream rm is actually able to sell any nal goods that make use of the technology in question. If each downstream rm has diseconomies of scale or if the incumbent patent owners are uncertain about the costs faced by a downstream rm producing goods that make use of its technology, a patent owner would also be likely to nd royalty contracts more pro table than licensing its patent only to a single downstream rm for a xed fee Potential Consumers in the Final Goods Market There is a continuum of potential consumers. The potential consumers are heterogenous in their valuation,, for goods that make use of the technology covered by the two incumbent patents. Hence, absent entry the demand curve for nal goods that make use of the technologies covered by the two incumbent patents is downward-sloping in the price of the nal good. This demand curve is denoted by D (p) ; where p is the price of the nal good. Because the downstream industry is competitive and would have zero costs in the absence of any licensing fees, the price p is equal to the licensing fees that a downstream rm must pay to the incumbent patent owners. 2.3 Potential Entry to the Technology Market Each incumbent patent owner faces the threat of being displaced by one of many potential entrants that can attempt to develop an uninfringing, patentable and non-drastic improvement of the incumbent s patented technology. We initially assume that the incumbents face symmetric entry threats, and that the potential entrants seeking to innovate around patent 1 are distinct from the potential entrants seeking to innovate around patent 2. However, we also present results for the case in which the entry threat is asymmetric and an integrated potential entrant can innovate around both incumbents patents. We adopt a reduced-form approach to modeling entry. 15 Speci cally, we assume that the 14 If patent owners o er xed-fee contracts to multiple downstream rms, price competition between the downstream rms would obviously prevent the downstream rms from collecting enough revenue to cover the licensing fees. Hence, a xed-fee license must be exclusive in a particular application to induce a downstream rm to accept it. A patent owner may, however, be limited in its ability to commit to not to o er licenses, either to the same patent or to patents that are close substitutes, to other potential licensees. Hence, another rationale for royalty licensing is that potential licensees may be unwilling to accept any licensing contracts with signi cant up-front fees because of the licensor s inability to commit to not to ood the market with subsequent licenses to other potential licensees after receiving an up-front payment from one licensee. 15 The reduced-form entry threats depict the envelope of the cost functions both in the case when each incumbent patent owner faces a single potential entrant whose entry costs are quadratic in the probability of 9

11 probability of entry in technology i 2 f1; 2g is i = + R i ; > 0; (1) where R i is the expected reward for successful entry against the incumbent i. 16 We assume that the technology of a successful entrant does not infringe on an incumbent patent, so that a downstream rm that incorporates an entrant s technology in its nal good does not need to pay royalties to the corresponding incumbent. The potential entrants attempt to develop patentable technologies, and we assume that while many potential entrants can attempt entry against a given incumbent, at most one of the entrant is successful in entry against each incumbent. Therefore, a downstream rm that sells goods that make use of an entrant s technology must pay a royalty to the successful entrant. The adoption of a successful entrant s technology by a downstream rm increases the value of the nal good by the size of the improvement, which is denoted by. We assume that this increase in value is the same for all potential consumers. Hence, if the valuation of a potential consumer for a nal good that makes use of the incumbent technology i is, then the potential consumer s valuation for a nal good that makes use of one incumbent technology and one entrant technology is + and the potential consumer s valuation for a nal good that makes use of two entrant technologies is + 2 (see gure 2). If the qualityadjusted prices of an incumbent technology and the corresponding entrant technology are the same, each active buyer is assumed to favor the entrant s technology over the incumbent s technology. 3 Equilibrium Analysis In the rst subsection of this section we show how the probabilities of entry that are determined in the R&D investment stage 2 depend on price commitments in stage 1. Other subsections of this section examine the properties of the optimum exclusive cooperative pricing arrangement and determine the existence and characteristics of a symmetric subgame-perfect success, and in the case when each incumbent patent owner faces many potential entrants that have di erent xed costs to achieve some probability of success and di erent increasing marginal costs of increasing their probabilities of success. Importantly, the adoption of a reduced form entry threat implies that no assumptions need be made about variables such as xed costs of R&D and the number of potential entrants, which are likely to be unobservable. 16 That the probability of innovation is increasing in the expected reward that a successful innovator receives follows the demand-pull innovation hypothesis of Schmookler (1966) and the induced-innovation hypothesis of Hicks (1932). An example of empirical work in this vein is Acemoglu and Linn (2004), who estimate that the elasticity of entry of new drugs with respect to the market size is between 1 and 4. 10

12 Valuation { { for 2 entrant patents for 1 incumbent patent and 1 entrant patent for 2 incumbent patents Number of potential consumers with a higher valuation Figure 2: E ect of entry on potential consumers valuations for nal goods. pure-strategy Nash equilibrium (SPPS-NE) both in the non-cooperative pricing model and in the non-exclusive cooperative pricing model. We restrict the scope of the analysis by imposing the following two assumptions on the demand curve D(p). These assumptions are used in proving the existence of an equilibrium in the non-cooperative pricing model and in the non-exclusive cooperative pricing model. Both assumptions are satis ed for linear demand. Assumption (A1) d2 [p i D(p i +p j )] dp j p j 0: 0 for all (p i ; p j ) such that p i = arg max p pd(p + p j ) and Assumption A1 implies that if neither incumbent is displaced by an entrant, the incumbent patents are not strategic complements. 17 This implies that in the model without potential entry in equilibrium each monopolist sets a licensee fee that is lower than the monopoly price p M arg max pd (p). Any non-trivial price commitment in stage 1 in the model with potential entry is therefore lower than the monopoly price p M. Assumption (A2) d2 [p i D(p i +p j )] < 0 for all (p i; p j ) such that p i ; p j 0. Assumption (A2) implies that if neither incumbent is displaced by an entrant, the revenue of incumbent i is concave in p i for any given p j. 17 If the incumbent patents were strategic complements (see Bulow et. al. (1985)) absent entry, both incumbents would bene t from one incumbent s pre-commitment to lower than ex-post optimal prices even without potential entry. Unlike in the analysis of Cournot quantity competition and forward contracting by Allaz and Vila (1993), here in a model of complementary monopolists that compete on price and can pre-commit by option contracting, the total equilibrium price would still be higher than the monopoly price and increasing the number of pre-commitment stages beyond one would not lead to even lower prices than would result with just one pre-commitment stage. 11

13 3.1 Outcome of the R&D Investment Stage 2 A successful entrant s reward for entry depends on the equilibrium price for licenses to the entrant s technology and the number of active nal consumers. The assumption that all potential consumers have the same marginal valuation for the entrant technologies implies that in equilibrium all active downstream rms sell nal goods that make use of the same technologies and acquire the same licenses. The assumption that the potential entrants can only attempt to develop improvements that are non-drastic in the economic sense implies by de nition that the price of an entrant technology is always constrained by how much more valuable the entrant technology is relative to the respective incumbent technology. Because Bertrand competition causes the equilibrium royalty on an incumbent s patent to be zero if an entrant is successful, a successful entrant always sets the royalty equal to. If only incumbent i is displaced by a successful entrant, the number of active consumers in stage 4 depends only on the royalty that downstream rms must pay to the complementary incumbent j. In stage 1 in all three pricing models an incumbent can commit to a royalty that is lower than the ex-post optimal fee by o ering ex-ante licenses to its patent but incumbents are unable to commit to higher than ex-post optimal prices. Hence, the royalty p j that a downstream rm must pay an incumbent j for each unit it sells when only the incumbent i faces successful entry is the minimum of the remaining incumbent s ex-post optimal license price p M arg max p pd (p) and the license fee speci ed in the license that incumbent i o ered in stage 1. Denote by p j the royalty speci ed in a license o ered in stage 1 that covers the incumbent patent j: The assumption (A1) implies that p j p M. Because a successful entrant s license price always equals every potential consumer s marginal valuation for the entrant technology, the number of active consumers in the event that only the incumbent j is displaced by an entrant is D(p j ): When both incumbent patents are displaced by successful entrants, price competition in stage 3 drives down the royalties for the incumbent patents to zero and for each entrant s patent to : Therefore, when both incumbent patents are displaced by a successful entrant the number of active consumers is D(0). The expected reward of a successful entrant that displaces incumbent i is therefore R i = 1 j D(pj ) + j D (0) ; j 6= i: (2) Together with the expression (1) for i, this implies that the outcome of the R&D investment 12

14 stage 2, denoted by ( 1; 2), is characterized by 18 1 = + [(1 2) D(p 2 ) + 2D (0)] ; (3) and 2 = + [(1 1) D(p 1 ) + 1D (0)] : (4) To rule out the possibility that entry is certain in both technologies and to guarantee that the probabilities of entry are strictly positive, regardless of whether either incumbent commits to a lower than ex-post optimal price for a license to its patent, we make the following assumption: Assumption (A3) + D(0) < 1 and + D(p M ) > 0; where p M arg max p pd(p): The Assumption (A3) implies that d i =d j < 1, which in turn implies that each entry threat i is decreasing in the e ective license price p j for the complementary patent j in the event that entry against the complementary incumbent is not successful: d 1 dp 2 < 0 and d 2 dp 1 < 0; (5) and that each entry threat i is decreasing in the e ective license price p i for the corresponding incumbent patent i in the event that entry against the incumbent i is not successful: d 1 dp 1 < 0 and d 2 dp 2 < 0. (6) The properties (5) and (6) of the model imply that by committing to a lower than expost optimal license price p i for its patent in stage 1, incumbent i will directly induce entry against the complementary incumbent j and indirectly induce entry against itself We call ( 1; 2) an outcome instead of an equilibrium because we assume a reduced-form entry threat. However, we could have assumed that each incumbent faces one potential entrant with R&D costs that are quadratic in the probability of success i, and in this case it would have been valid to call the outcome of the investment stage an equilibrium. 19 A smaller makes the ratio K d i = d j smaller and therefore makes it more attractive for the incumbents to o er ex-ante licenses in equilibrium. However, a smaller also implies that the size d j of the e ect of a decrease in price p i on the probability of entry j is smaller. Conversely, while a larger makes the ratio K larger and therefore makes it less attractive for the incumbents to o er ex-ante licenses in equilibrium, a larger also implies that the size d j of the e ect of a commitment to a license price pi on the probability of entry j is larger. Therefore, one might expect that the incumbents ability to commit to lower than ex-post optimal prices for their patents in stage 1 can have a signi cant e ect on equilibrium 13

15 Together with the observations that patent owners are often limited to royalty licensing and that patent owners commit to renegotiable license prices for their patents, this entry inducement mechanism forms the basis of the formal analysis in this paper. Notice also that with symmetric entry threats it is essential that the entry technology involves uncertainty as otherwise the inducement of entry that displaces the complementary incumbent patent owner would always also induce the displacement of both incumbents. 3.2 Equilibrium in the Non-Cooperative Pricing Model In stages 1 and 3 the incumbents non-cooperatively set the terms of licenses to their patents. In stage 3 the incumbents also compete on price with any successful entrants. We solve for the equilibrium pricing by backward induction Equilibrium in Stage 3 Consider rst the case when neither incumbent is displaced by an entrant. The optimal license price for the incumbent i is ^p i (p j ) arg max p i fp i D (p i + p j )g ; (7) when p j is the license price each downstream rm must pay to the complementary incumbent j 6= i for each unit that they sell. If neither incumbent has o ered any licenses in stage 1, the equilibrium license prices (p 1; p 2) in stage 3 satisfy conditions p 1 = ^p 1 (p 2) and p 2 = ^p 2 (p 1) : (8) Let p DM denote the symmetric equilibrium license price that satis es these two conditions. 20 Whether price commitments p 1 and p 2 in stage 1 a ect the equilibrium prices in stage 3 when neither incumbent is displaced by an entrant depends on whether either incumbent has committed to a price p i that is lower than its best-response to the complementary incumbent s best-response. If ^p 1 (^p 2 (p 1 )) p 1 and ^p 2 (^p 1 (p 2 )) p 2 then neither incumbent is constrained by its price commitment and consequently in equilibrium in stage 3 p 1 = p 2 = p DM : If outcomes only for intermediate values of the parameter : The comparisons of equilibrium outcomes in the three pricing models in section 4.1 con rm this intuition. 20 The analysis of the double-monopoly problem by Cournot (1838) showed that the sum p DM + p DM is greater than the monopoly price p M because neither monopolist internalizes the positive externality that decreasing its price has on the complementary monopolist s demand. Note also that the assumption (A1) implies that p DM p M. 14

16 ^p i (^p j (p i )) > p i ; incumbent i s license price p i in stage 3 will be constrained by its price commitment p i. If ^p 1 (^p 2 (p 1 )) p 1 and ^p 2 (^p 1 (p 2 )) p 2 then in equilibrium in stage 3 p 1 = p 1 and p 2 = p 2. If ^p i (^p j (p i )) p i and ^p j (^p i (p j )) p j then in equilibrium in stage 3 p i = ^p i (p j ) and p j = p j. 21 When only the incumbent patent i is displaced by an entrant, price competition between the entrant and incumbent i in stage 3 drives down the price of a license to the incumbent patent i to zero and the price of a license to the entrant s patent to. Absent any price commitments in stage 1 by the complementary incumbent j that wasn t displaced by an entrant, the incumbent j will set in stage 3 the royalty equal to the monopoly price p M : However, if the incumbent j has committed to a license price p j p M in stage 1, the incumbent j is restricted in stage 3 by this commitment. In stage 3 the incumbent j therefore o ers licenses at price p j = min fp j ; p M g. If both incumbents are displaced by successful entrants, price competition between entrants and incumbents drives down the prices of licenses to incumbent patents to zero, p 1 = p 2 = 0, and the prices of licenses to each entrant patent to : This outcome of the price competition in stage 3 is independent of whether the incumbents have o ered any licenses in stage Equilibrium in Stage 1 In stage 1 each incumbent can commit to a lower than ex-post optimal license price p i for its patent by o ering an ex-ante license. Because in all four possible states of the world in stage 3 the ex-post royalty does not exceed the monopoly price p M, we can restrict attention to ex-ante licenses with license prices equal to or lower than the monopoly price. The best-response correspondence of incumbent i in stage 1 is given by p NC i (p j ) arg max p i p M i (p i ; p j ) ; (9) where 8 >< i (p i ; p j ) = >: i;1 (p i ; p j ) if p j p DM and p i p DM i;2 (p i ; p j ) if p j ^p (p i ) and p i p DM i;3 (p i ; p j ) if p j ^p (p i ) and p i ^p (p j ) i;4 (p i ; p j ) if p j p DM and p i ^p (p j ) ; (10) 21 These results make use of Assumption (A2) which implies that p i D (p i + p j ) is decreasing in p i for all p i < ^p i (p j ). 15

17 where and ^p i (p j ) arg max p i D (p i + p j ) and ^p j (p i ) arg max p j D (p i + p j ) ; (11) p i p j i;1 (p i ; p j ) = (1 i ) 1 j pdm D (p DM + p DM ) + jp i D (p i ) (12) i;2 (p i ; p j ) = (1 i ) 1 j pi D (p i + ^p j (p i )) + jp i D (p i ) i;3 (p i ; p j ) = (1 i ) 1 j pi D (p i + p j ) + jp i D (p i ) i;4 (p i ; p j ) = (1 i ) 1 j ^pi (p j ) D (^p i (p j ) + p j ) + jp i D (p i ) : While it is crucial for the analysis in this paper that the incumbents take into account the dependence of the probabilities of entry 1 and 2 on the license prices p 1 and p 2 set in stage 1, this dependence is omitted in the above expressions for notational convenience. A symmetric equilibrium p NC ; p NC in stage 1 is characterized by the conditions p NC = p NC 1 p NC and p NC = p NC 2 p NC : (13) In appendix 6.1 we de ne Conditions (C1), (C2), and (C3) that ensure the existence of a SPPS-NE. The Conditions (C1), (C2) and (C3) are restrictions on the curvature of the demand curve D(p) and are satis ed for linear demand. The following proposition is proved in the appendix Proposition 1 When Assumptions (A1), (A2) and (A3), and Conditions (C1), (C2) and (C3) hold, there exists a symmetric SPPS-NE in the Non-Cooperative Pricing Model. The necessary condition for the incumbents to set p NC 1 = p NC 2 = p M in equilibrium in stage 1 is that d i;1(p i ;p j ) pi =p M ;p j =p M 0: A corollary of the Claim 1.2 in the proof of proposition 1 is that this is also a su cient condition for p NC 1 = p NC 2 = p M to be an equilibrium. Hence the following corollary holds. Corollary 1.1 When assumptions (A1), (A2) and (A3) and conditions (C1), (C2) and (C3) hold, the necessary and su cient condition for the incumbents to o er ex-ante licenses 22 The symmetry in the proposition refers to the royalties set in stage 1. The equilibrium license prices in stage 3 are asymmetric if only one incumbent is displaced by an entrant. The proof of proposition 1 is non-trivial because the incumbents best-response correspondences p NC 1 (p 2 ) and p NC 2 (p 1 ) are not necessarily convex. In the appendix we prove proposition 1 by showing that either the best-response correspondence in stage 1 is a continuous function for all p j 2 [p DM ; p 0 j ], where pnc pj 0 > pdm and either p NC pj 0 = pdm or p 0 j = p M ; or that the best-response correspondence in stage 1 is a continuous function for all p j 2 [p 0 j ; p DM ], where p NC (p DM ) < p DM and either p NC pj 0 = pdm or p 0 j = 0: 16

18 with prices p NC 1 = p NC 2 < p M in the symmetric SPPS-NE of the Non-Cooperative Pricing Model in which the incumbents expected revenue is the highest is that d i;1 (p i ; p j ) The condition (14) can also be written as < 0 (14) pi =p M ;p j =p M K [(1 ) p DM D (p DM + p DM ) + p M D (p M )] (1 ) [p M D (p M ) p DM D (p DM + p DM )] < 0; (15) where 23 (p M ; p M ) and K d i = d j = [D(0) D(p M )] : (16) The rst term on the left-hand side of condition (15) represents the relative size of the e ect that a decrease in p i has on the expected revenue of incumbent i through the e ect that the price change has on the probability i that the incumbent i itself is displaced by an entrant. The second term on the left-hand side of condition (15) represents the relative size of the e ect that a decrease in p i has on the expected revenue of incumbent i through the e ect that the price decrease has on the probability j that the complementary incumbent j is displaced by an entrant. The e ect that a decrease in p i has on the expected revenue of incumbent i in the event that only incumbent j is displaced by an entrant does not enter the condition (15) because this e ect is only a second-order e ect around p i = p M. Likewise, the e ect that a decrease in p i potentially has on the expected revenue of incumbent i in the event that neither incumbent is displaced by an entrant does not enter the condition (15) because around p i = p M this e ect is non-existent if p DM < p M and only a second-order e ect if p DM = p M : However, these two additional e ects determine in part how low the price commitments are in equilibrium. 3.3 Equilibrium in the Exclusive Cooperative Pricing Model The incumbent patent owners now set royalties for their patents cooperatively in stages 1 and 3 to maximize their total expected revenue from the licensing of the incumbent patents. We assume that in both stages the incumbents can o er licenses that cover only one incumbent patent and licenses that cover the bundle of both incumbent patents. We solve for 23 The probability is the probability of entry when neither incumbent commits to any lower than ex-post optimal license prices. The constant K represents the ratio of the e ect that a change in p i has on i and the e ect that a change in p i has on j : Assumption A3 implies that 2 (0; 1) and K 2 (0; 1) : 17

19 equilibrium by backward induction Equilibrium in Stage 3 Consider rst the case when neither incumbent patent is displaced by an entrant. the incumbents have not committed to lower than ex-post optimal royalties in stage 1, in equilibrium in stage 3 the incumbents o er licenses that cover both incumbent patents at a royalty equal to the monopoly price, p M : 24 If the incumbents have committed in stage 1 either to prices p 1 and p 2 for licenses to individual incumbent patents or to a price p B for a license to the bundle of both incumbent patents, then in equilibrium in stage 3 the incumbents o er bundled licenses at price p B = min fp M; p B ; p 1 + p 2 g : Consider now the case when only the incumbent patent i is displaced by an entrant. If the incumbents have not committed to lower than ex-post optimal royalties in stage 1, price competition between the entrant and the incumbents results in the entrant setting its license price equal to and and the incumbents set the price of a license to the bundle of both incumbent patents to p M. If the incumbents have committed in stage 1 to lower than ex-post optimal prices p j or p B, in stage 3 the entrant sets the price of a license to its patent equal to and the incumbents o er bundled licenses at price p B = min fp M; p B ; p j g. 25 When both incumbents are displaced by successful entrants, price competition between the incumbents and the entrants drives the price of licenses to each incumbent patent to zero, p B = p 1 = p 2 = 0; and the prices of licenses to the entrant equal : If Equilibrium in Stage 1 In stage 1 the incumbents set p 1 ; p 2 and p B to maximize the incumbents expected total revenue. Formally, the incumbents solve where max (p 1 ; p 2 ; p B ) p 1 ;p 2 ;p B p M s.t. p 1 p B ; p 2 p B ; and p B p 1 + p 2 ; (p 1 ; p 2 ; p B ) = (1 1) (1 2) p B D (p B ) (17) + (1 1) 2p 1 D(p 1 ) + 1 (1 2) p 2 D(p 2 ): 24 The equilibrium outcome is the same if the incumbents instead o er separate licenses to each incumbent patent and set the total license price equal the monopoly price, p 1 + p 2 = p M. 25 Equivalently, incumbents can o er licenses to the incumbent patent j and set p j = min fp M ; p B ; p j g. 18