Cross-Licensing and Competition

Transcription

1 Cross-Licensing and Competition Doh-Shin Jeon and Yassine Lefouili y June 7, 2013 Very preliminary and incomplete - Please do not circulate Abstract We study bilateral cross-licensing agreements among N(> 2) rms that engage in competition after the licensing phase. It is shown that the most collusive cross-licensing royalty, i.e. the one that allows the industry to achieve the monopoly pro t, is sustainable as the outcome of bilaterally e cient agreements. When the terms of cross-licensing agreements are not observable to third parties, the monopoly royalty is the unique symmetric bilaterally e cient royalty. However, when the terms of the agreements are public, the most competitive royalty (i.e. zero) can also be bilaterally e cient. Policy implications regading the antitrust treatment of cross-licensing agreements are derived from these results. Keywords: Cross-Licensing, Collusion, Antitrust and Intellectual Property. Toulouse School of Economics. doh-shin.jeon@tse-fr.eu y Toulouse School of Economics. yassine.lefouili@tse-fr.eu 1

2 1 Introduction In several industries, the intellectual property rights necessary to market a product are held by a large number of parties. Such a situation, known as a patent thicket, arises for multiple reasons: rms desire to strengthen and broaden their patent rights (Hall and Ziedonis, 2001, and Galasso and Schankerman, 2010), the cumulative nature of technology (von Graevenitz et al., 2012), the lack of resources and misaligned incentives at patent o ces (Ja e and Lerner, 2004, and Bessen and Meurer, 2008), etc. Patent thickets are particularly prevalent in areas related to information technology such as semiconductors, audio-visual technology, telecommunications and optics (von Graevenitz et al., 2011). For instance, according to an estimate of Intel, by 2002 over 90,000 US patents related to central processing unit technologies had been awarded to more than 10,000 rms, universities, government labs and independent inventors (Detkin, 2002). Patent thickets raise many concerns and are considered as one of the most crucial intellectual property issues of the day (Shapiro, 2007, and Régibeau and Rockett, 2011). One well-known solution to the patent thicket problem is cross-licensing, an agreement between two rms that grants each the right to practice the other s patents (Shapiro, 2001, and Régibeau and Rockett, 2011). For instance, cross-licensing is widespread in the semiconductor industry (Grindley and Teece, 1997, Hall and Ziedonis, 2001 and Galasso, 2012). However, like other solutions to the patent thicket problem such as patent pools and cooperative standard setting, cross-licensing may negatively a ect competition. More precisely, it can reduce competition through high royalties. In particular, rms in a duopoly can sign a cross-licensing agreement with royalties high enough to replicate the monopoly pro t (Fershtman and Kamien, 1992). For this reason, competition authorities often prohibit the use of royalties that are disproportionate with respect to the market value of the license. 1 However, as long as bilateral cross-licensing agreements are signed in a industry with more than two competing rms, it is unclear whether two rms would agree on high royalties since this might weaken their competitive positions with respect to their rivals. In this paper, we investigate the formation of bilateral cross-licensing agreements among N(> 2) competing rms and assess their e ects on competition. Each rm is initially endowed with a patent portfolio and can get access to its rivals patents through cross-licensing agreements involving the payment of xed fees and royalties. After the cross-licensing phase, rms engage in quantity competition. As a rst step, this paper considers symmetric rms with symmetric patent portfolios and focuses on symmetric equilibria where all cross-licensing agreements specify the same royalty. We investigate the strategic e ects that drive the choice of royalties in a bilateral cross- 1 For instance, according to the Guidelines on the application of Article 81 of the EC Treaty to technology transfer agreements (2004),... Article 81(1) may be applicable where competitors cross license and impose running royalties that are clearly disproportionate compared to the market value of the licence and where such royalties have a signi cant impact on market prices. 2

3 licensing agreement. In the benchmark case of a multilateral licensing agreement signed by all N rms, the rms would agree on the royalty that allows them to achieve the monopoly outcome. When we consider a bilateral cross-licensing, the two rms in an agreement will also internalize the externalities exerted on each other. Is this the only e ect that matters when there are N(> 2) competing rms? If not, under which conditions is this e ect dominated by other (potentially pro-competitive) strategic e ects? The answers to these questions can generate interesting policy implications regarding the antitrust treatment of cross-licensing. In our model, we assume that the larger the set of patents to which a rm has access, the lower its marginal cost. Alternatively, we can assume that the larger the set of patents to which a rm has access, the higher the value of its product. We actually show that our model of cost-reducing patents can be equivalently interpreted as a model of value-increasing patents. We distinguish publicly observable cross-licensing agreements from secret agreements and focus on bilaterally e cient agreements. A set of cross-licensing agreements is said to be bilaterally e cient if each agreement maximizes the joint pro t of the pair of rms who signed the agreement given all other cross-licensing agreements in the industry. In the case of public agreements, we nd that when two rms decide bilateral royalties to maximize their joint pro t, they take into account two opposite e ects: the coordination e ect and the Stackelberg e ect. The coordination e ect refers to the fact that the coalition chooses its royalties to internalize the externalities to its members. The Stackelberg e ect captures the fact that rms can in uence the output levels chosen by their rivals through their choice of royalties. This e ect disappears when agreements are secret. Our main results are as follows. If agreements are secret, the unique bilaterally e cient set of symmetric cross-licensing agreements is the one involving the monopoly royalty, i.e. the one that induces the rms to achieve the monopoly outcome. If agreements are public, the outcome depends on which of the Stackelberg e ect and the coordination e ect dominates the other one: if the coordination e ect dominates, the unique symmetric bilaterally e cient set of agreements is the one involving the monopoly royalty; if the Stackelberg e ect dominates, there are two symmetric bilaterally e cient set of agreements: the one involving the monopoly royalty and the one involving zero royalty. In both cases, the Pareto-dominant bilaterally e cient royalty is the monopoly royalty. This implies that the most collusive outcome can be sustained even if cross-licensing agreements are decided bilaterally. We show that this result still holds in the case of cross-licensing agreements decided by coalitions of any size comprising more than two rms. Finally, we provide policy remedies to restore the most competitive outcome, i.e. the outcome with zero royalty. The remedies consist of three elements. First, the antitrust authorities should impose an upper bound on royalties, which is lower than the monopoly royalty. Second, the authorities should make disclosure of royalty terms compulsory. Third, the authorities should impose an appropriate upper bound on the size of coalitions, i.e. the number of rms who 3

4 can sign a cross-licensing agreement together. These remedies allow to induce cross-licensing agreements with zero royalty as the unique symmetric equilibrium outcome. 2 Related literature Our paper contributes to the literature on the competitive e ects of cross-licensing agreements and patent pools. In a pioneering paper, Priest (1977) shows how these two practices can be used as a disguise for cartel arrangements. Fershtman and Kamien (1992) develop a model in which two rms engage in a patent race for two complementary patents and use it to shed light on the social trade-o underlying cross-licensing agreements: One the one hand, cross-licensing improves the e ciency of the R&D investments by eliminating the duplication of e orts. But on the other hand, it favors price collusion between the rms. Eswaran (1994) shows that crosslicensing can enhance the degree of collusion achieved in a repeated game by credibly introducing the threat of increased rivalry in the market for each rm s product. Shapiro (2001) argues that patent pools tend to raise (lower) welfare when patents are perfect complements (substitutes), an idea which is generalized to intermediate levels of substitutability/complementarity by Lerner and Tirole (2004) and further extended to the case of uncertain patents by Choi (2010). To the best of our knowledge, the present paper is the rst formalized study of the competitive e ects of bilateral cross-licensing agreements in an industry comprised of more than two rms. Our paper is also related to the literature on strategic formation of networks surveyed in Goyal (2007) and Jackson (2008). In particular, the concept of bilateral e ciency we use is similar to the widespread re nement of pairwise stability in the network literature (Jackson and Wolinsky, 1996). 2 Goyal and Moraga-Gonzales (2001) and Goyal, Moraga-Gonzales, and Konovalov (2008) also study two-stage games where a network formation stage is followed by a competition stage. However, they examine ex ante R&D cooperation, while we study ex post licensing agreements, and, contrary to the present paper, they do not allow for transfers among rms. Bloch and Jackson (2007) develop a general framework to examine the issue of network formation with transfers among players. A crucial di erence between their framework and our setting is that we allow the agreements among rms to involve the payment of per-unit royalties on top of xed fees while they only consider the case of lump-sum transfers. More generally, our paper is related to the literature on negotiations and cooperative arrangements in industrial organization (e.g. Gans and Inderst, 2001, 2003, and, for a detailed survey, Gans and Inderst, 2007). Of particular interest to us is the paper by De Fontenay and Gans (2012) who provide an analysis of a non-cooperative pairwise bargaining game between agents in a network and show that the equilibria of their bargaining game are those that result in bilaterally e cient agreements. This nding supports our focus on bilaterally e cient agreements. 2 Our concept of bilateral e ciency is also similar to the bilateral contracting principle used in the vertical relations literature (Hart and Tirole, 1990; Whinston, 2006). 4

5 3 The Model 3.1 The model of cost-reducing patents Consider an industry consisting of N 3 symmetric rms producing a homogeneous good. Each rm owns one patent 3 covering a cost-reducing technology and can get access to its rivals patents through cross-licensing agreements. We assume that the patents are symmetric in the sense that the marginal cost of a rm only depends on the number of patents it has access to. Let c(n) be a rm s marginal cost when it has access to a number n 2 f1; :::; Ng of patents with c(n)( c) c(n 1) ::: c(1)( c) and let c (c 1 ; :::; c N ) be the vector representing the marginal costs of all rms in the industry. Denote c c c. Let us assume that, prior to engaging in Cournot competition, each pair of rms can sign a cross-licensing agreement whereby each party gets access to the patented technology of the other one. More precisely, the rms play the following two-stage game: Stage 1: Cross-licensing. Each pair of rms (i; j) decides whether to sign a cross-licensing agreement and determine the terms of the agreement if any. We assume that a bilateral cross-licensing agreement between rm i and rm j speci es a pair of royalties (r i!j ; r j!i ) 2 R+ 2 and a lump-sum transfer F i!j 2 R; where r i!j (resp. r j!i ) is the per-unit royalty paid by rm i (resp. rm j) to rm j (resp. rm i) and F i!j is a transfer made by rm i to rm j (which is negative if rm i receives a transfer from rm j). Assume that all bilateral negotiations occur simultaneously. For the purpose of this paper we do not need to specify the underlying pairwise bargaining game: we will focus on the negotiation outcomes (and whether these satisfy a given property) and set aside the way such outcomes are achieved. Stage 2: Cournot competition. Firms compete à la Cournot with the cost structure inherited from Stage 1. Depending on the observability of royalties signed by i and j to other rms, we distinguish between public cross-licensing and secret cross-licensing. In the case of public cross-licensing, all rms observe all the cross-licensing agreements signed at stage 1 before they engage in Coutnot competition. In contrast, in the case of secret cross-licensing, the terms of crosslicensing agreement signed between i and j remain known only to i and j and therefore each rm k should form a conjecture about the licensing terms signed between i and j. We assume that the rms face an inverse demand function P () satisfying the following standard conditions (Novshek, 1985): 3 Given that we consider that rms have symmetric patent portfolios, we can assume, without loss of generality, that each rm has one patent. 5

6 A1 P () is twice continuously di erentiable and P 0 () < 0 whenever P () > 0: A2 P (0) > c > c > P (Q) for Q su ciently high. A3 + QP 00 (Q) < 0 for all Q 0 with P (Q) > 0. These mild assumptions ensure the existence and uniqueness of a Cournot equilibrium (q i ) i=1;:::;n satisfying the following (intuitive) comparative statics properties, where c i denote rm i s marginal i < 0 j > 0 for any j i < 0 for any i, where Q = X i is the total equilibrium output; i < 0 j > 0 for any j 6= i, where i is rm i s equilibrium pro t (see e.g. Amir et al., 2012). 3.2 Alternative equivalent formulation: value-increasing patents Instead of assuming that access to more patents reduces a rm s marginal cost, we can assume that access to more patents increases the value of the product produced by the rm. We below show that our model of cost-reducing patents can be equivalently interpreted as a model of value-increasing patents. We consider a constant symmetric marginal cost c for all rms. Each rm has one patent. Let v(n) represent the value of the product produced by a rm when the rm has access to n 2 f1; :::; N g number of distinct patents with v(n) v(n 1) ::: v(1)( v). Let v (v 1 ; :::; v N ) be the vector representing the value of each rm s product after the licensing stage. We de ne Cournot competition for given v (v 1 ; :::; v N ) as follows. Each rm i simultaneously chooses its quantity q i. Given v (v 1 ; :::; v N ), q (q 1 ; :::; q N ) and Q = q 1 + ::: + q N the quality-adjusted equilibrium prices are determined by the following two conditions: indi erence condition: v i p i = v j p j for all (i; j) 2 f1; :::; Ng 2 ; q i market-clearing condition: Q = D(p) where p i = p + v i v: In other words, p is the price for the product of a rm which has access to its own patent only. The market clearing condition means that this price is adjusted to make the total supply equal to the demand. The indi erence condition implies that the price each rm charges is adjusted such that all consumers who buy any product are indi erent among all products. A microfoundation of this setup can be provided as follows. There is a mass one of consumers. Each consumer has a unit demand and hence buys at most one unit among all products. A consumer s gross utility 6

7 from having a unit of product of rm i is given by u+v i : u is speci c to the consumer while v i is common to all consumers. Let F (u) represent the cumulative distribution function of u: Then, by construction of quality-adjusted prices, any consumer is indi erent among all products and the marginal consumer indi erent between buying any product and not buying is characterized by u + v p = 0, implying D(p) = 1 F (p v): In equilibrium, p is adjusted such that 1 F (p v) = Q. Let P (Q) be the inverse demand curve. In the equilibrium, a rm s pro t is given by 0 i (Q) + v i v c X j6=i r i!j After making a change of variables as follows c (v i v) = c i ; the pro t can be equivalently written as 0 X i (Q) c i j6=i r i!j 1 1 A q i + X j6=i A q i + X j6=i r j!i q j ; r j!i q j : which is the pro t expression in our original model of cost-reducing patents. In particular, the change of variables implies c (v(n) v) = c(n); which in turn implies v(n + 1) v(n) = c(n) c(n + 1): Therefore, our model of cost-reducing patents can be equivalently interpreted as a model of value-increasing patents. In what follows, for our formal analysis, we stick to the model of cost-reducing patents. 4 Benchmark: multilateral licensing agreement We here consider the case of a multilateral licensing agreement signed by all N rms. We focus on a symmetric outcome where all rms pay the same royalty r to each other. 7

8 Let P m (c) be the monopoly price when each rm s marginal cost is c. It is characterized by P m (c) P m (c) c = 1 "(P m (c)) : (1) where "(:) is the elasticity of demand. Given a symmetric r, each rm s marginal cost is c + (N 1)r. The rms will agree on a royalty to achieve the monopoly price. Given a symmetric r, at stage 2, rm i chooses q i to maximize [P (Q i + q i ) c (N 1)r] q i + rq i where Q i is the quantity chosen by all other rms. Note that this is the same as maximizing [P (Q i + q i ) c (N 1)r] q i when Q i is considered by rm i as given. Let r m be the royalty that allows to achieve the monopoly price P m (c). Then, from the rst-order condition associated to rm i s maximization program, we have From (1) and (2), r m is determined by which is equivalent to P m (c) c (N 1)r m P m (c) N [P m (c) c (N 1)r m ] = P m (c) c; P m (c) N = 1 "(P m (c))n : (2) c = r m : (3) Now we examine under which condition the multilateral licensing agreement will lead to a higher downstream price compared to the situation before cross-licensing. Let Q (N; c) represent the industry output when N rms with the marginal cost c compete in quantity. Therefore, the equilibrium price prior to licensing is P (Q (N; c)). At P (Q (N; c))., we have Therefore, P m (c) = P (Q (N; c)) if P (Q (N; c)) P (Q (N; c)) c = 1 "(P (Q (N; c)))n N [P (Q (N; c)) c] = P (Q (N; c)) c; which is equivalent to c = (N 1) [P (Q (N; c)) c] : (4) Note that the right hand side of (4) does not depend on c. As the cost reduction from licensing increases, c is smaller and the monopoly price is smaller. Therefore, we must have P m (c) T P (Q (N; c)) if and only if c S (N 1) [P (Q (N; c)): c]. Proposition 1 (Multilateral licensing). Assume that A1-A3 hold and that all rms in the 8

9 industry jointly agree on a symmetric royalty. Then they agree on r m = (P m (c) c) =N to achieve the monopoly price P m (c). Moreover: (i) If c < (N 1) (P (Q (N; c)) c), the post-licensing monopoly price is higher than the pre-licensing equilibrium price. (ii) If c = (N 1) (P (Q (N; c)) c), the post-licensing monopoly price is the same as the pre-licensing equilibrium price. (iii) If c > (N 1) (P (Q (N; c)) c), the post-licensing monopoly price is lower than the pre-licensing equilibrium price. Note that when c = 0, i.e. when patents are perfect substitutes, the multilateral licensing agreement described above serves as a purely collusive device. 5 Characterization of the bilaterally e cient public agreements We now characterize the bilaterally e cient cross-licensing outcomes, i.e. the outcomes such that that each pair of rms (i; j) does not nd it jointly optimal to modify the terms of the agreement they signed. In other words, each cross-licensing agreement is required to maximize the joint payo of the two parties involved in it, given all other cross-licensing agreements. In this section we focus on cross-licensing agreements that are observable to all rms in the industry. 5.1 Preliminaries Assume that all rms are always active (i.e. produce a positive output) whatever the crosslicensing agreements that are signed. Under this assumption, any given pair of rms nd it (joitly) optimal to license each other. To see why, assume that rm i does not license its patent to j. These two rms can (weakly) increase their joint pro ts if i licenses its patent to j by specifying the payment of a per-unit royalty r j!i equal to the reduction in marginal cost allowed by j s use of i s patent. Such licensing agreement would not a ect the level of joint output but will (weakly) decrease the cost of rm j. It will therefore (weakly) increase their joint pro ts. Consider now a symmetric situation where all rms cross-license each other and let r denote the (common) per-unit royalty paid by any rm i to have access to the patent of any rm j 6= i with i; j = 1; :::; N. Let S(r; N) denote such a set of cross-licensing agreements. We below study the joint incentives of a coalition of two rms, say 1 and 2, to deviate from the symmetric royalty r. The next lemma shows that it is su cient to focus on deviations specifying identical royalties between the two rms in the coalition. Indeed the joint payo of any asymmetric deviation can be replicated by a symmetric one as the joint payo depends on the royalties paid by each rm of the coalition to the other only through their sum. 9

10 Lemma 1 Consider a symmetric set of cross-licensing agreements S(r; N). The joint payo a coalition fi; jg gets from a deviation to a cross-licensing agreement in which rm i (resp. rm j) pays a royalty r i!j (resp. r j!i ) to rm j (resp. rm j) depends on (r i!j ; r j!i ) only through the sum r i!j + r j!i. Proof. See Appendix. This lemma shows that to determine whether the set S(r; N) of cross-licensing agreements is bilaterally e cient, it is su cient to investigate the incentives of the coalition to deviate by changing the royalty they charge each other from r to ^r 6= r. For given (r; ^r), let Q (r; ^r) denote the total industry output and Q 12 (r; ^r) denote the sum of the outputs of rms 1 and 2 in the (second-stage) equilibrium of Cournot competition. Let Q 12 (r; ^r) Q (r; ^r) Q 12 (r; ^r) denote the total equilibrium output of their rivals. Then, the considered set of symmetric agreements is bilaterally e cient if and only if: where: r 2 Arg max ( 1 + 2) (r; ^r) ^r0 ( 1 + 2) (r; ^r) = [P (Q (r; ^r)) (c + () r)] Q 12(r; ^r) + 2rQ 12(r; ^r) = [P (Q 12(r; ^r) + BR 1;2 (Q 12(r; ^r))) (c + () r)] Q 12(r; ^r) + 2rBR 12 (Q 12(r; ^r)) where BR 12 (:) is de ned as follows. If N = 3; then BR 12 (:) is the best-response function of rm 3. If N 4, then BR 12 (:) is the aggregate response of the coalition s rivals, de ned as follows: For any joint output Q 12 = q 1 + q 2 of rms 1 and 2, BR 12 (Q 12 ) is the (unique) real number such that BR 12(Q 12 ) is the best-response of any rm j 3 to each rm i = 1; 2 producing q i and each rm j 3 producing BR 12(Q 12 ) : After observing the coalition s deviation to ^r 6= r, the rivals of the coalition expect the coalition to produce Q 12 (r; ^r) and will best respond to this quantity by producing BR 12(Q 12 (r; ^r)) in aggregate. In other words, from a strategic point of view, the coalition s deviation to ^r 6= r is equivalent to its committment to produce Q 12 (r; ^r) as a Stackelberg leader. We therefore de ne the following two-stage Stackelberg game. De nition: For any r 0 and N 3, the Stackelberg game G(r; N) is de ned by the following elements: Players: Coalition {1,2} and rms j = 3; :::; N; each player has a marginal production cost c (excluding royalties) and pays a per-unit royalty r to each of the other players. Actions: The coalition {1,2} chooses its (total) output Q 12 within the interval I(r; N) [Q 12 (r; 1); Q 12 (r; 0)] and each rm j 3 can choose any quantity q j 0. Timing: There are two stages. 10

11 - Stage 1: The coalition {1,2} acts as a Stackelberg leader and chooses Q 12 within the interval [Q 12 (r; 1); Q 12 (r; 0)] : - Stage 2: If N = 3, then rm 3 chooses its output q 3 (given Q 12 ). If N 4, then rms j 3 compete in quantities (given Q 12 ). The following lemma shows that the set of cross-licensing agreements S(r; N) is bilaterally e cient if and only if choosing Q 12 (r; r) is optimal for the coalition in the Stackelberg game G(r; N): Lemma 2 The symmetric set of cross-licensing agreements S(r; N) is bilaterally e cient if and only if choosing to produce Q 12 (r; r) is a subgame-perfect equilibrium strategy of the coalition {1,2} in the Stackelberg game G(r; N). Proof. See Appendix. 5.2 Incentives to deviate: downstream and upstream pro ts In what follows we identify two opposite e ects on the incentives of the coalition f1; 2g to marginally expand or contract its output with respect to Q 12 (r; r). The coalition s pro t can be rewritten as 12 (Q 12 ; r) = [P (Q 12 + BR 12 (Q 12 )) (c + (N 1) r)] Q {z 12 } D 12 (Q 12;r) + r [Q BR 12 (Q 12 )] {z } : U 12 (Q 12;r) (5) The term D 12 (Q 12; r) represents the coalition s downstream market pro t from selling its products when its marginal cost is given by c + (N 1) r. The term U 12 (Q 12; r) represents its upstream market pro t from licensing its patents and includes the payment made by each rm within the coalition to the other one. We below study the e ect of a (local) variation of Q 12 on each of the two sources of pro t. E ect on the downstream market pro t Starting with the e ect on the coalition s downstream market pro t D 12 (Q 12; r), we D 12 (Q 12 ; r) = P 0 (Q 12 + BR 12 (Q 12 )) Q 12 BR 0 12(Q 12 ) + P 0 (Q 12 + BR 12 (Q 12 )) Q 12 +(6) [P (Q 12 + BR 12 (Q 12 )) (c + (N 1) r)] which, when evaluated at Q 12 (r; r), is given D 12 (Q 12(r; r); r) = P 0 (Q (r; r)) Q 12(r; r)br 0 12(Q 12(r; r)) + P 0 (Q (r; r)) Q 12(r; r) + (7) [P (Q (r; r)) (c + (N 1) r)] : 11

12 The term P 0 (Q (r; r)) Q 12 (r; r)br0 12 (Q 12 (r; r)) > 0 in (7) captures the (usual) Stackelberg e ect: the leader has an incentive to increase its output Q 12 above the Cournot level Q 12 (r; r) because such an increase will be met with a decrease in the aggregate output of the followers (one can easily check that BR 0 12 (Q 12 (r; r)) < 0). It will turn out to be useful to rewrite this term as 2 [P (Q (r; r)) (c + (N 1) r)] BR 0 12 (Q 12), which is obtained from the F.O.C. of rm 1 (or 2) in the Cournot game: P 0 (Q (r; r)) Q 12 (r; r) + P (Q (r; r)) (c + (N 1) r) = 0: (8) 2 The term P 0 (Q (r; r)) Q 12 (r; r) + [P (Q (r; r)) (c + (N 1) r)] in (7) represents the marginal downstream pro t of the coalition in a setting where it would play a simultaneous quantitysetting game. This term captures a coordination e ect: the coalition has an incentive to reduce output below the Cournot level Q 12 (r; r) since the joint output of the coalition when each member chooses its quantity in a non-cooperative way is too high with respect to what maximizes its joint downstream pro t (in a simultaneous quantity-setting game). Indeed, using again (8), we have: P 0 (Q (r; r)) Q 12(r; r) + [P (Q (r; r)) (c + (N 1) r)] = [P (Q (r; r)) (c + (N 1) r)] < 0: Therefore, the overall marginal e ect of a local increase of Q 12 (above the Cournot level Q 12 (r; r)) on the coalition s downstream pro t is given D 12 (Q 12(r; r); r) = [P (Q (r; r)) (c + (N 1) r)] [1 + 2BR 0 12(Q 12(r; r))]: (9) The rst term between brackets in (9) is positive while the sign of the second term between brackets in (9) is ambiguous. If the aggregate response of the coalition s rival is relatively strong, such that 1 + 2BR 0 12 (Q 12 (r; r)) < 0, then the Stackelberg e ect dominates the coordination e ect and the coalition has an incentive to increase its quantity above Q 12 (r; r), D 12 (Q 12 (r; r); r) > 0. In contrast, if the aggregate response of the coalition s rival is relatively weak, such that 1 + 2BR 0 12 (Q 12 (r; r)) < 0, then the coordination e ect dominates the Stackelberg e ect and the coalition has an incentive to reduce its quantity below Q 12 (r; r), D 12 (Q 12 (r; r); r) < 0. E ect on the upstream market pro t Let us now turn to the e ect of a local variation in Q 12 on the coalition s upstream market pro t U 12 (Q 12): We U 12 (Q 12(r; r); r) = r 1 + 2BR 0 12(Q 12(r; r)) : (10) 12

13 One the one hand, a marginal increase in Q 12 results in an increase of r=2 in the licensing revenues each rm of the coalition gets from the other member. On the other hand, a marginal increase in Q 12 induces the rivals of the coalition to reduce their output by BR 0 12 (Q 12 ) and hence results in a reduction of r BR 0 12 (Q 12 ) in the licensing revenues that each rm of the coalition gets from the rivals. If the Stackelberg e ect dominates the coordination e ect such that 1 + 2BR 0 12 (Q 12) < 0 holds, then the e ect on the "extra-coalition" licensing revenues dominates the e ect on the "intra-coalition" licensing revenues such that an increase in Q 12 reduces the upstream licensing revenue. The reverse holds if 1 + 2BR 0 12 (Q 12) > 0. The e ect of an increase in Q 12 (above the Cournot level Q 12 (r; r)) on each of the coalition s downstream and upstream market pro ts is ambiguous. What is very important to notice in (9) and (10) is that the e ect on the downstream pro t is always opposite to the e ect on the upstream pro t (whenever 1 + 2BR 0 12 (Q 12 (r; r)) 6= 0). For instance, when the Stackelberg e ect dominates the coordination e ect (i.e., 1 + 2BR 0 1;2 (Q 12 (r; r)) < 0), the marginal e ect of Q 12 on the downstream pro t gives output-expanding incentives to the coalition, D 12 (Q 12 (r; r); r) > 0, while the marginal e ect on the upstream pro t yields incentives, i.e. U 12 (Q 12 (r; r); r) < 0. This is because the contraction in the output of the rivals increases the coalition s downstream pro t while reducing its upstream pro t. The reverse holds when when the coordination e ect dominates the Stackelberg e ect. 5.3 Incentives to deviate: four cases From the analysis of the previous subsection, by summing up (9) and (10), the total e ect of a marginal increase in Q 12 on the coalition s pro t can be described in a simple way 12 (Q 12(r; r); r) = [c + Nr P (Q (r; r))] [1 + 2BR 0 12(Q 12(r; r))]: (11) We can distinguish four cases depending on whether or not the Stackelberg e ect dominates the coordination e ect and which is more important between the e ect of a local deviation on the downstream pro t and the one on the upstream pro t. Stackelberg e ect vs. coordination e ect Let us rst examine the term 1 + 2BR 0 12 (Q 12 (r; r)) which determines whether or not the Stackelberg e ect dominates the coordination e ect. The FOC for the maximization program of any rm j 3, when the coalition produces a given quantity Q 12, can be written as: P 0 (Q 12 + BR 12 (Q 12 ) BR 12(Q 12 ) + P (Q 12 + BR 12 (Q 12 ) (c + (N 1) r) = 0: 13

14 Di erentiating the latter with respect to Q 12 yields (dropping the argument (r; r)) BR 0 12(Q 12 ) = P 00 (Q)BR 12 (Q 12 ) + () P 00 (Q)BR 12 (Q 12 ) + (N 1) which, combined with < 0, proves that later), and, when evaluated at Q 12 = Q 12, also yields that 1 < BR 0 12 (Q 12) < 0 (a result that will be useful 1 + 2BR 0 12(Q 12) = = P 00 (Q )BR 12 (Q 12 ) + (N 3)P 0 (Q ) P 00 (Q )BR 12 (Q 12 ) + (N 1)P 0 (Q ) N P 00 (Q )Q + (N 3)P 0 (Q ) N P 00 (Q )Q + (N 1)P 0 (Q ) because BR 12 (Q 12 ) = N Q as the corresponding equilibrium is symmetric. Distinguishing between the two scenarios P 00 (Q ) 0 and P 00 (Q ) > 0 and using the fact that BR 12 (Q 12 ) Q, one can easily show that A3 implies that the denominator is always negative. Therefore, from P 0 (Q ) < 0, it follows that 1 + 2BR 0 12(Q 12) Q 0 () Q P 00 (Q ) P 0 (Q ) R N(N 3) (12) This shows that the coordination e ect dominates the Stackelberg e ect (1 + 2BR 0 12 (Q 12 ) > 0) if and only if the slope of the inverse demand is positive (P 00 (Q ) > 0) and su ciently elastic ( Q P 00 (Q ) N(N 3) P 0 (Q ) < ). The main intuition behind this result follows from the fact that a local increase in Q 12 a ects the marginal revenue q j +P (Q) of a rm j 3 outside the coalition through two channels: it has an e ect on both the inverse demand P (Q) and its slope. If the magnitude of the e ect on, which is proportional to P 00 (Q), is high relative to the magnitude of the e ect on P (Q), which is proportional to, then the magnitude of the adjustment that each rival of the coalition has to make to equalize its marginal revenue to its marginal cost will be relatively low. In particular, for N 4, under A3, the Stackelberg e ect always dominates the coordination e ect. Downstream pro t vs. upstream pro t Let us now examine the term f(r; N) c+nr P (Q (r; r)). The e ect of an increase in Q 12 on the downstream pro t is more important than the e ect on the upstream pro t if f(r; N) < 0. For instance, when r = 0, there is no upstream pro t and we have f(0; N) = c P (Q (0; 0)) < 0. Intuitively, we expect that the upstream pro t becomes more important as r increases, which turns out to be true as we below (r; N) P 0 (Q ) > 0: Adding the FOCs for all rms maximization program, when all of them produce at the e ective marginal cost 14

15 c + (N 1)r, yields: P 0 (Q ) Q + NP (Q ) N (c + (N 1)r) = 0: Di erentiating the latter with respect to r, P 0 (Q ) + P 00 (Q ) Q + N P 0 (N 1) = 0: From P 0 (Q ) + P 00 (Q ) Q < 0 (by < 0, it follows that P 0 (N 1) < 0, which (r; N) > 0 for any N 3. Since f(r; N) strictly increases with r, we expect that there exists a unique r > 0 at which c + Nr P (Q (r; r)) = 0. Surprisingly, it turns out that the unique r at which f(r; N) = 0 is r m in (3). At r = r m, we have that P (Q (r m ; r m )) = P m and, therefore, c + Nr m P (Q (r m ; r m )) = 0. Thus, for r < r m, the downstream pro t is more important than the upstream one and the reverse holds for r > r m. 5.4 Bilaterally e cient royalties From the previous analysis of local deviations, we know that there are four possible cases. First, when the Stackelberg e ect dominates the coordination e ect and the downstream pro t e ect is more important than the upstream pro t e ect (i.e. r belongs to [0; r m ), the coalition has an incentive to decrease its royalty in order to reduce the rivals outputs, which generates r = 0 as the unique potential bilateral e cient royalty among royalties r in [0; r m ). Second, when the Stackelberg e ect dominates the coordination e ect but the upstream pro t e ect is more important than the downstream pro t e ect (i.e. r > r m ), the coalition has an incentive to increase its royalty to boost the rivals production and thereby the licensing revenue from the rivals. At r = r m, the downstream pro t e ect is equal to the upstream pro t e ect and the coalition has no incentive to deviate at least locally. In summary, when the Stackelberg e ect dominates the coordination e ect, there are two potential bilaterally e cient royalties: r = 0 and r = r m. Third, when the coordination e ect dominates the Stackelberg e ect and the downstream pro t e ect is more important than the upstream pro t e ect (i.e. r belongs to (0; r m )), the coalition has an incentive to increase its royalty to make the retail price as close as possible to the monopoly price. In a symmetric way, when the coordination e ect dominates the Stackelberg e ect but the upstream pro t e ect is more important than the downstream pro t e ect (i.e. r > r m ), the coalition has an incentive to decrease its royalty to make the retail price as close as possible to the monopoly price. In summary, when the coordination e ect dominates the Stackelberg e ect, we have a unique potential bilaterally e cient royalty: r = r m. Summarizing, we have: 15

16 Lemma 3 Under A1-A3, from the local analysis of the Stackelberg game G(r; N), we obtain the following candidates for the bilaterally e cient symmetric royalties: (i) If the Stackelberg e ect dominates the coordination e ect QP 00 (Q) > N(N 3), then there are two candidate royalties, r = 0 and r = r m, for which S(r; N) can be bilaterally e cient. (ii) If the coordination e ect dominates the Stackelberg e ect QP 00 (Q), then < r = r m is the only possible value of r for which S(r; N) can be bilaterally e cient. N(N 3) In order to determine the values of r for which S(r; N) is indeed bilaterally e cient, we need to switch from a local analysis (which provides necessary conditions for S(r; N) to be bilaterally e cient) to a global analysis (which con rms or denies that the potential candidates are bilaterally e cient). The following proposition characterizes the bilaterally e cient symmetric agreements depending on whether the Stackelberg e ect dominates the coordination e ect. Proposition 2 (public cross-licensing) Under A1-A3, the following holds: (i) If the Stackelberg e ect dominates the coordination e ect QP 00 (Q) > S(r; N) is bilaterally e cient if and only if r 2 f0; r m g. (ii) If the coordination e ect dominates the Stackelberg e ect QP 00 (Q) < S(r; N) is bilaterally e cient if and only if r = r m. N(N 3), then N(N 3), then Proof. See Appendix. This proposition shows that the monopoly royalty is the Pareto-dominant bilaterally e - cient royalty regardless of whether the Stackelberg e ect dominates or is dominated by the coordination e ect. 6 Secret cross-licensing In the previous section, we considered public cross-licensing agreements. In this section, we consider the alternative scenario of secret cross-licensing: each bilateral cross-licensing agreement is only observable to the two rms involved in it. Hence, at the beginning of stage 2, rm i is aware of only the royalties of the licensing contracts that it itself signed. Regarding the royalties agreed on between rm j(6= i) and k (6= i), i should form an expectation. In a symmetric equilibrium in which every pair of rms agree on the same royalty r, every rm should believe that every pair of rms agreed on r on the equilibrium path. We here make an assumption of passive beliefs: regardless of the terms of cross-licensing signed between i and j in the licensing stage, i maintains the same belief about the terms signed by any pair of rivals and believes that every pair of rivals agreed on r. Then, we can show that Lemma 1 continues to hold in the case of secret cross-licensing and hence, without loss of generaltiy, we can restrict a coalition to deviate with a symmetric royalty. Then, a result similar to Lemma 2 holds in that the symmetric set of cross-licensing agreements 16

17 S(r; N) is bilaterally e cient if and only if choosing to produce Q 12 (r; r) is optimal to the coalition {1,2}. The major di erence between public cross-licensing and secret cross-licensing is that in the latter case, there is no Stackelberg e ect: formally speaking, the analysis under secret cross-licensing can be derived from that of public cross-licensing by setting BR 0 12 (Q 12) to zero. This implies that S(r m ; N) is the unique bilaterally e cient set of symmetric agreements. Proposition 3 (secret cross-licensing) Under A1-A3, in the two-stage game of secret crosslicensing followed by Cournot competition, S(r m ; N) is the unique bilaterally e cient set of symmetric cross-licensing agreements. Therefore, secret cross-licensing strengthens the nding under public cross-licensing in Proposition 2 since the monopoly outcome is the unique symmetric outcome under secret crosslicensing. 7 Robustness to coalitions of any size In this section, we show that the main results previously obtained by considering a coalition of size 2 extend to a coalition of any given size k in f3; :::; Ng. We will say that a set of crosslicensing agreements is k-e cient if no coalition of size k nds it optimal to change the terms of the cross-licensing agreements between the members of the coalition. Since the case of k = N was studied in the benchmark of multilateral licensing by all rms in the industry, we study a coalition of size k in f3; :::; N the case of secret cross-licensing. 1g, rst in the case of public cross-licensing and then consider Consider the deviation of a coalition composed of f1; :::; kg in the licensing stage. Lemma 1 continues to hold in the case of coalition of size k and hence, without loss of generality, we can restrict attention to deviations involving a symmetric royalty ^r. For given (r; ^r), let Q (r; ^r) denote the total industry output and Q k (r; ^r) denote the sum of the outputs of the rms in the coalition in the (second-stage) equilibrium of Cournot competition. Let Q k (r; ^r) Q k (r; ^r) denote the total equilibrium output of the coalition s rivals. Denoting Q k the total quantity produced by the considered coalition and r the common Q (r; ^r) royalty paid to the rms outside the coalition, the coalition s pro t can be rewritten as k (Q k ; r) = [P (Q k + BR k (Q k )) (c + (N 1) r)] Q {z k } D k (Q k;r) + r [(k 1)Q k + kbr k (Q k )] {z } : U k (Q k;r) (13) Expression (13) generalizes (5). Suppose that the coalition marginally expands or contracts its output Q k with respect to Q k (r; r). Then, we D k (Q (r; r); r) = [P (Q (r; r)) (c + (N 1) r)] [k 1 + kbr 0 k (Q k (r; r))]: k 17

18 Summing up the two terms leads U k (Q k (r; r); r) = r k 1 + kbr0 k (Q k (r; r)) k (Q (r; r); r) = [c + Nr P (Q (r; r))] [k 1 + kbr 0 k (Q k (r; r))]: (14) k Expression (14) generalizes (11). In particular, the rst bracket term is the same in both equations does not depend on k while the second bracket term in (14) depends on k. Stackelberg e ect dominates the coordination e ect if and only if k 1 + kbr 0 k (Q k (r; r)) < 0. We have k 1 + kbr 0 k (Q k (r; r)) S 0 i QP 00 (Q) T N(k + 1) : N k The important point is that at r = r m, the rst bracket term is zero regardless of the coalition size: c + Nr m P (Q (r m ; r m )) = 0. Therefore, we have the following result Theorem 1 Assume that A1-A3 hold. In the two-stage game of public cross-licensing followed by Cournot competition; (i) If the Stackelberg e ect dominates the coordination e ect S(r m ; N) is k-e cient if and only if r 2 f0; r m g. (ii) If the coordination e ect dominates the Stackelberg e ect S(r m ; N) is k-e cient if and only if r = 0. QP 00 (Q) > QP 00 (Q) < The N(k+1) N k, then N(k+1) N k, then Theorem 1 generalizes Proposition 2 to any given size of coalition. In addition, this theorem implies that even if we allow for coalitions of di erent sizes (for instance, rm 1 signs a licensing agreement with rm 2 and at the same time signs an agreement with rms f3; 4g), the monopoly outcome survives. The theorem also implies that Proposition 3 of secret cross-licensing generalizes to coalitions of any size since secret cross-licensing is a particular case of public cross-licensing in which the Stackelberg e ect is absent: Theorem 2 Assume that A1-A3 hold. In the two-stage game of secret cross-licensing followed by Cournot competition, S(r m ; N) is the unique k-e cient set of symmetric agreements. 8 Policy implications The previous analysis reveals that market forces induce rms to sign cross-licensing agreements allowing them to achieve the monopoly outcome regardless of the size of coalitions and the information structure (i.e. whether the licensing terms are public or secret). In this section, we derive two di erent policy implications. First, we interpret our results in terms of desirability 18

19 (or undesirability) of cross-licensing of complementary patents (substituable patents). Second, we provide simple policy remedies to restore the most competitive outcome (i.e. cross-licensing with zero royalties). In our model, c c c has some close connection with whether patents are close substitutes or complements. In the case of close substitutes, c is expected to be very small: in the case of strong complements, c is expected to be large. When we combine this interpretation with the main result that under laissez-faire cross-licensing leads to the monopoly outcome, we obtain the following result from Proposition 1. Proposition 4 (i) When patents are relatively substitutes (i.e. c < (N 1) (P (Q (N; c)) c)), cross-licensing leads to a decrease in consumer surplus. (ii) When patents are relatively complements (i.e. c > (N 1) (P (Q (N; c)) c)), crosslicensing leads to an increase in consumer surplus. Therefore, absent any other policy remedies, a consumer-surplus-maximizing competition authority should allow cross-licensing only if patents are complementary enough. We now turn to simple policy remedies which take advantage of competitive forces existing in the market. Our policy remedies have three components. First, the antitrust authorities should prohibit any royalty above the monopoly royalty r m. More precisely, the authority should impose an upper bound on royalty, denoted by r satisfying r < r m. However, using our previous analysis of unconstrained agreements, we can show that this restrcition is not enough to restore the competitive outcome: under secret cross-licensing, the rms will end up choosing r = r. Therefore, antitrust authorities should make the disclosure of royalty terms compulsory. Then, as long as the Stackelberg e ect dominates the coordination e ect, competition among the rms induces all rms to end up sigining zero royalty. Third, from Theorem 1, it follows that the Stackelberg e ect becomes less likely to dominate the coordination e ect as the coalition size increases. Note also that, if only coalitions made of two rms are allowed, the Stackelberg e ect dominates the coordination e ect for any N 4 (due to assumption A3). Therefore, it is important to allow only coalitions made of small number of rms. Proposition 5 Assume that N 4: Imposing an upper bound on royalty r satisfying r < r m, making disclosure of royalty terms compulsory and allowing agreements only among a small number of rms (i.e. k satisfying QP 00 (Q) N(k+1) > N k ) induces the most competitive outcome: S(0; N) is then the unique k-e cient set of symmetric agreements. 9 Concluding remarks This paper is a rst step toward understanding bilateral cross-licensing agreements among N 3 competing rms. We plan to address several extensions in future research. 19

20 One extension is to introduce some asymmetries in our setting to study issues which cannot be investigated with a symmetric model. In particular, we can include in the set of players nonpracticing entities (NPEs) which do not compete in the downstream market. This would allow us to study the conditions under which NPEs weaken competition and (when these conditions are met) to isolate the anticompetitive e ects generated by NPEs from the e ects resulting from cross-licensing in the absence of NPEs. The issue of how NPEs a ect competition and innovation is of substantial current interest to policymakers (Morton, 2012). Another interesting asymmetry we can introduce is to consider, in addition to incumbent rms, entrants with no (or weak) patent portfolios. This would allow us to study whether crosslicensing can be used to raise barriers to entry (Hall et al., 2012). Note that NPEs and entrants involve completely opposite asymmetries. The former are present in the upstream market of patent licensing and are absent in the downstream (product) market while the second are absent (or have very weak presence) in the upstream market and are present in the downstream market. 10 Appendix Proof of Lemma 1 Assume, without loss of generality, that (i; j) = (1; 2):The joint payo rms 1 and 2 derive from a deviation to a licensing agreement involving the payment of (r 1!2 ; r 2!1 ) is: = [P (Q ) c r 1!2 () r] q 1 + r 2!1 q 2 + rq 12 + [P (Q ) c r 2!1 () r] q 2 + r 1!2 q 1 + rq 12 = [P (Q ) c r] (q 1 + q 2) + 2rQ 12 which can be rewritten as = [P (Q ) c () r] Q Q rQ 12 (15) Denoting c i the marginal cost of rm i (including the royalties paid to the other rms), the FOC for rm i 0 s maximization program is: P (Q ) c i + q i P 0 (Q ) = 0 Summing the FOCs for i = 1; 2; ::; N yields: NP (Q ) X c i + Q P 0 (Q ) = 0 i1 which shows that Q depends on (c 1 ; c 2 ; ::; c N ) only through X i1c i : Moreover, summing the FOCs 20

21 for i = 3; ::; N yields () P (Q ) X c i + Q 12P 0 (Q ) = 0 i3 which implies that Q 12 depends on (c 1; c 2 ; :::; c N ) only through X c i and X i. From (c 1 ; c 2 ; c 3 ; :::; c N ) = i1 i3c (c + r 1!2 + () r; c + r 2!1 + () r; c + (N 1) r; :::; c + (N 1) r) it then follows that both Q and Q 12 depend on (r 1!2; r 2!1 ) only through r 1!2 + r 2!1, which, combined with (15), implies that depends on (r 1!2; r 2!1 ) only through r 1!2 + r 2!1 : Proof of Lemma 2 Since Q 12 only if: Q 12(r; r) 2 (r; ^r) is strictly decreasing and continuous in ^r then r 2 Arg max ( ) (r; ^r) if and ^r2[0;] Arg max Q 12 2[Q 12 (r;r);q 12 (r;0)] 12 (Q 12 ) [P (Q 12 + BR 12 (Q 12 )) (c + () r)] Q 12 +2rBR 12 (Q 12 ) which means that Q 12 (r; r) is a subgame-perfect equilibrium strategy of the coalition f1; 2g in the game G(r; N): Proof of Proposition 2 Let us rst show some general preliminary results which will be useful for the subsequent speci c analysis of the four considered scenarios. We 12 (Q 12 ; r) = P 0 (Q 12 + BR 1;2 (Q 12 )) Q 12 (1 + BR 0 12(Q 12 )) + [P (Q 12 + BR 1;2 (Q 12 )) (c + (N 1) r)] + r 1 + 2BR 0 12(Q 12 ) = [c + Nr P (Q 12 + BR 1;2 (Q 12 ))] 1 + 2BR 0 12(Q 12 ) + 2 P 0 (Q 12 + BR 1;2 (Q 12 )) Q P (Q 12 + BR 1;2 (Q 12 )) (c + (N 1) r) 1 + BR 0 1;2 (Q 12 ) {z } J(Q 12 ;r) {z } D((Q 12 ;r) Let us show that J(Q 12 ; r) is decreasing in Q 12. We 12 ; r) = P 00 (Q) Q BR 0 1;2(Q 12 ) {z } >0 + 2 {z } <0 Since P 00 (Q) Q < max [P 00 (Q) Q + ; ] < 0 then J(Q 12 ; r) is decreasing in Q 12 : This, combined with the fact that the FOC for each rm i = 1; 2, satis ed at the symmetric 21