Licensing Non- and (Super-)Drastic Innovations: The Case of the Inside Patent Holder *

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1 Licensing Non- and (Super-)Drastic Innovations: The Case of the Inside Patent Holder * Cuihong Fan Shanghai University of Finance and Economics cuihongf@mail.shufe.edu.cn Byoung Heon Jun Korea University, Seoul bhjun@korea.ac.kr Elmar G. Wolfstetter Humboldt-University at Berlin and Korea University, Seoul wolfstetter@gmail.com March, 016 Abstract The present paper reconsiders licensing by an inside innovator under incomplete information. Contrary to what was claimed in the literature, the optimal mechanism for non-drastic innovations prescribes royalty rates lower than the cost reduction and generally exhibits fixed fees and exclusion; for drastic innovations fixed-fee contracts are optimal under complete information, whereas under incomplete information, it is optimal to combine state dependent royalty rates with state dependent fixed fees, essentially because linking the license fee to a variable that is correlated with the licensee s private information tends to lower information rents. KEYWORDS: Innovation, licensing, industrial organization, linkage principle. JEL CLASSIFICATIONS: D1, D4, D44, D45 1 Introduction The literature on patent licensing in oligopoly markets draws a sharp distinction between licensing by an outside innovator and an inside patent holder who is a competitor in the product market of potential licensees. Whereas outside innovators are advised to either use fixed fee contracts or auction patent licenses to a limited number of licensees, inside patent holders are advised to employ pure output based royalty contracts without a fixed fee. The outside innovators licensing problem was explored by Kamien and Tauman (1984, 1986), Kamien (199), who showed that fixed-fee licensing and license auctions are more profitable than royalty contracts. However, this result was qualified by Giebe and Wolfstetter (008) and Fan, Jun, and Wolfstetter (01, 014), who showed in a variety of frameworks, ranging from private to common values as well as complete and incomplete information, that it is optimal to supplement license auctions with royalty contracts to those who lose the auction. *Research support by Korea University, the Deutsche Forschungsgemeinschaft (DFG), SFB Transregio 15, Governance and Efficiency of Economic Systems, and the National Natural Science Foundation of China (Grant: ) is gratefully acknowledged. 1

2 The inside innovator s licensing problem was initiated, among others, by Wang (1998) and Sen and Tauman (007). They showed that pure royalty contracts with a royalty rate equal to the cost reduction and without fixed fees are optimal. This way, the innovator neutralizes the licensee s cost advantage and thus protects his turf. While this literature assumed a common value innovation and complete information, Heywood, Li, and Ye (014) confirmed the optimality of pure royalty contracts under incomplete information, assuming a binary private values model with linear demand. 1 However, they considered innovations with a particular profile of cost reductions, where the innovator is at least as efficient as the licensee. Poddar and Sinha (010) consider an example in which the licensee is more efficient, assuming a common value innovation under complete information and linear demand. The greater efficiency of the licensee occurs because prior to the innovation the licensee has lower cost. The authors claim that the optimal license contract can be a two-part tariff, regardless of whether the innovation is drastic or non-drastic. However, they assume that the innovator cannot grant an exclusive right to use the innovation, which ignores the possibility of selling the patent, and do not allow for incomplete information. The present paper reconsiders the inside patent holders licensing problem, assuming incomplete information and general probability distributions of unit costs. In the case of non-drastic innovations we find that pure royalty contracts are generally not optimal. The optimal mechanism typically prescribes fixed fees, exhibits royalty rates lower than the cost reduction, and even negative royalty rates, and exclusion. Its detailed characteristics are significantly affected by incomplete information and they depend critically on whether the innovator or the licensee is able to make better use of the innovation. We also consider drastic innovations, and distinguish between licenses that grant either an exclusive or a non-exclusive right to use the innovation, and between drastic and super-drastic innovations. While super-drastic innovations propel monopoly even if the license does not grant an exclusive right, the optimal licensing of an innovation that is drastic but not super-drastic requires the sale of the patent. We show that under complete information the optimal license contract is a fixed-fee contract that grants the exclusive right to use the innovation if the innovation is drastic but not super-drastic, whereas under incomplete information, using output based royalties is essential because linking the license fee to a variable that is correlated with the licensee s private information tends to lower information rents. The plan of the paper is as follows: In Section we state the model. In Section we start the analysis with a simple example of a non-drastic innovation under complete information, assuming the licensee is more efficient. We show that, depending upon parameter values, the optimal contract is either a fixed fee or a two-part tariff or a pure royalty contract. In Section 4 we state the optimal mechanism design problem for a non-drastic innovation under incomplete information and characterize the optimal mechanism if the innovator is more efficient and if the licensee is more efficient in some range. In Section 5 we extend the analysis to drastic innovations. There we distinguish between drastic and super-drastic innovations, and exclusive and non-exclusive licenses, and explain why employing output dependent royalties increases the innovator s expected revenue. We close with a brief discussion. 1 Heywood, Li, and Ye (014) also consider ad valorem royalties and show that for most parameter values output based royalties are superior and fixed fees are never employed. An extension of this analysis to three firms is in Wang, Liang, and Chou (01).

3 The model Consider a dynamic licensing game played between an inside innovator of a non-drastic process innovation (firm 1) and a potential licensee (firm ). In the first stage the innovator offers a license contract which the licensee either accepts or rejects. In the second stage firms play a Cournot game. Prior to the innovation firms unit costs are equal to c and prior to licensing the innovation, firms unit costs are (c 1,c ) = (d,c),c > d. Licensing reduces the unit cost of firm to x, which is that firm s private information. From the perspective of the innovator x is a random variable drawn from the c.d.f. F with support [x, x], 0 x < x c. F is assumed to be log-concave. The payoff functions of the duopoly games are: π 1 (q 1,q ;c 1 ) = (P(Q) c 1 )q 1, π (q 1,q ;c ) = (P(Q) c )q, Q := q 1 + q. For simplicity and in order to obtain closed form solutions, we assume linear demand with P(Q) := 1 Q. The innovator employs a direct mechanism, M := (t(x,q ),γ(x)), to license his innovation which consists of an allocation rule γ(x) and a transfer rule t(x,q ), as a function of the reported unit cost, x, and the output q that is observed after the oligopoly game has been played. Without loss of generality, we consider incentive compatible mechanisms with a deterministic allocation rule, γ(x) {0,1}, where γ(x) = 1 means that the license is awarded. The transfer rule prescribes firm to pay a royalty rate r(x) per output unit plus a fixed fee, f L (x), if the license is awarded and f N if no license is awarded: { f L (x) + r(x)q if γ(x) = 1 t(x,q ) = (1) f N if γ(x) = 0. The potential licensee reports his unit cost, and the innovator adopts the allocation and transfers prescribed by the mechanism for the reported cost. Royalty rates cannot exceed the licensee s cost reduction: r(x) c x (antitrust constraints). () Without this restriction the innovator could induce the licensee to reduce output by setting a high royalty rate and split the gains by offering a negative fixed fee. In the extreme this could implement the monopoly outcome even if the innovation is non-drastic. Quite appropriately, contracts of this form would likely be held to be illegal by antitrust authorities (Katz and Shapiro, 1985, p. 51). 4 This is why we refer to this constraint as the antitrust constraint. Example with complete information As a benchmark consider a simple example with complete information in which the licensee (firm ) is able to make better use of a non-drastic innovation than the inside innovator (firm 1). Specifically, assume the innovation reduces the unit cost of firm from c (0, 1/) to zero, while the unit cost of firm 1 is equal to d (0,c). 5 Many standard distribution functions are log-concave (see Bagnoli and Bergstrom, 005). 4 Licensing agreements for intellectual property are typically illegal if they involve naked price-fixing, output restrictions, or market division (U.S. Department of Justice and Federal Trade Commission, 1995). 5 This example is similar to the model by Poddar and Sinha (010) who assume a common value framework in which cost differences are due to different pre-innovation costs.

4 We will show that, depending upon the parameters c,d, the optimal license contract is either a pure royalty contract, with a royalty rate equal to the licensee s cost reduction, or a pure fixed fee contract, or a two-part tariff. Licensing occurs for all parameter values (no exclusion). The general format of the license contract is a two-part tariff, with a fixed fee f and a royalty rate per output unit, r. The royalty rate cannot be greater than the licensee s cost reduction (equivalently, we require f 0), and, for simplicity, we also assume in the present example that the royalty rate must be non-negative. Denote the equilibrium outputs and profits as a function of the profile of (effective) unit costs, (c 1,c ), by q e i (c 1,c ), π e i (c 1,c ). 6 If the innovation is not licensed, the profile of unit costs is (c 1,c ) = (d,c), whereas if a license is issued, one has (c 1,c ) = (d,r). The optimal license contract maximizes the innovator s profit subject to the participation constraint, the antitrust constraint, and the non-negativity constraint: max {r, f } πe 1(d,r) + rq e (d,r) + f, s.t. π e (d,r) f π e (d,c), 0 r c. Straightforward computations yield the solution that is summarized in Figure 1, for all permitted values of the parameters c and d. In order to interpret these results, fix a parameter value c (say c = 0.), and gradually reduce d from d = c to d = 0. For high values of d, the licensee is far more efficient than the innovator. In that case it is optimal to shift output to the licensee by reducing the distortion, i.e., by setting r = 0, and then extract the licensee s profit by means of the fixed fee (fixed fee region). 7 As d is reduced (below d = 0.), it becomes profitable to distort the output decision of the licensee, which increases the own output of the innovator, and adopt a combination of a positive royalty rate and positive fixed fee (two-part tariff region). As d is further reduced, the efficiency gap shrinks further away and it becomes optimal to adopt a pure royalty contract without fixed fee and with a royalty rate equal to the cost reduction, r = c, just like in the case when the innovator is at least as efficient as the licensee (pure royalty region). This suggests that the optimality of pure royalty asserted in the literature is not driven by the fact that the innovator is an inside patent holder. Instead, it is driven by the assumption that the licensee is not more efficient than the innovator. Pure royalty contracts are not always optimal, but if they are optimal, the optimal royalty rate is equal to the licensee s cost reduction. Licensing occurs for all permitted parameter values. Under complete information the optimal contract extracts the full surplus of the licensee. Thereby, the royalty rate serves the purpose to maximize the sum of the innovator s profit and the licensee s surplus. It achieves this by inducing the output profile that maximizes the sum of profits. The only purpose of the fixed fee is to extract that part of the surplus that is not already extracted by the royalty income. However, the properties of the optimal contract change when we employ a model with incomplete information, with unrestricted patterns of cost reductions, and when we allow for drastic innovations. 6 Straightforward computation yields: q e i (c i,c j ) = (1 c i +c j )/,π e i (c i,c j ) = q e i (c i,c j ),i, j {1,},i j. 7 Generally, if one does not require non-negativity of the royalty rate, it is optimal to shift output even more aggressively by prescribing negative royalty rates (combined with higher fixed fees), for all d > 0.. 4

5 c Two Part Tariff f 0, 0 r c Fixed Fee f 0, r Pure Royalty f 0, r c d Figure 1: Example of optimal license contract (r: royalty rate, f : fixed fee) 4 Optimal licensing of non-drastic innovations under incomplete information Introducing incomplete information changes the role of the royalty rate and the fixed fee. It is no longer possible to extract the full surplus in all states, and optimal surplus extraction requires the use of incentive compatible contracts. Of course, incentive compatibility can be assured by a simple state independent contract. However such a contract cannot make use of the information that it induces to be revealed. In general, the innovator will employ a state dependent contract that uses exclusion (only those whose cost is below a certain threshold level are awarded a license). The royalty rate, the fixed fee, and the exclusion level are used to induce the licensee to reveal information and to make use of that information to extract surplus. The innovator maximizes his expected payoff (which is the sum of his own expected profit and expected license revenue), subject to incentive compatibility, participation, and antitrust constraints. In order to solve the set of incentive compatible mechanisms, we proceed as follows: first, we compute the payoffs of the licensee for all combinations of his true cost x and reported cost z. This requires that we solve all oligopoly subgames that may occur on and off the equilibrium path. We are then able to characterize incentive compatible mechanisms. Oligopoly subgames on and off the equilibrium path Suppose the licensee with cost x reports the cost z (possibly different from x) for which a license is awarded. In that case firm 1 believes to play a duopoly game with the cost profile (c 1,c ) = (d,z + r(z)). Denote the equilibrium of that game by (q 1 (z), q (z)), defined as q 1 (z) := argmaxπ 1 (q, q (z);d), q q (z) := argmaxπ (q 1 (z),q;z + r(z)). () q However, firm privately knows that the cost profile is (c 1,c ) = (d,x + r(z)) and therefore plays its best reply to q 1 (z) = (1 d+z+r(z))/, which gives: q (x,z) = argmaxπ (q 1 (z),q;x + r(z)) = 1 ( + d x z 4r(z)). (4) q 6 The associated equilibrium profit of firm is π (q 1 (z),q (x,z);x + r(z)) = q (x,z). (5) 5

6 As a special case one obtains the on the equilibrium path strategies and profits that apply when firm reports truthfully (i.e., z = x): q 1 d + x + r(x) 1(x) := q 1 (x) =, π1 (x) = q 1(x) (6) q 1 + d (x + r(x)) (x) := q (x,x) =, π (x) = q (x). (7) In turn, if firm is not awarded a license, the two firms play the equilibrium strategies q N 1,qN that solve the requirements q N 1 = argmax q π 1 (q,q N ;d) = 1 d + c, q N = argmax q π (q N 1,q;c) = 1 c + d. (8) The associated equilibrium profits are π 1 ( q N 1,q N ;d) = (q N 1 ), π ( q N 1,q N ;c) = (q N ). For convenience, we define: Π N := π ( q N 1,q N ;c ) f N, Π L (x,z) := π (q 1 (z),q (x,z);x + r(z)) f L (z). There, L and N are mnemonic for licensing and no-licensing (exclusion). Incentive compatibility Using the above equilibria of the duopoly subgames for all combinations of z and x, the potential licensee s expected payoff in the licensing game is: Π (x,z) := γ(z)π L (x,z) + (1 γ(z))π N. (9) The mechanism is incentive compatible if: x = argmax z Π (x,z), x. Lemma 1. The following conditions are necessary for incentive compatibility, for all x,z ˆx x: { 1 if x ˆx γ(x) = (10) 0 otherwise f L(x) = 1 9 (1 + d x r(x))(1 + 4r (x)) (11) r(x) r(z) x z 1 4. (1) Proof. To prove (10), suppose the mechanism prescribes γ(x ) = 1,γ(x) = 0 for some x,x with x > x. Then, the licensee with cost x has an incentive to report z = x, because in that case he obtains the license and earns an even higher profit than type x. Incentive compatibility requires that z Π L (x,z) z=x = 0, for all x, which implies (11). Moreover, incentive compatibility also requires that, for all x,z ˆx: 0 Π L (x,x) Π L (x,z) + Π L (z,z) Π L (z,x) = 1 (x z)(x z + 4r(x) 4r(z)). 6 Rearranging gives (1). 6

7 Participation and antitrust constraints The mechanism has to assure voluntary participation: Π (x,x) π (q N 1,q N ;c), x (participation constraints). (1) Note that Π (x,x) is non-increasing, because (by incentive compatibility and the fact that Π (x,x ) is decreasing in x): x > x Π (x,x) Π (x,x ) Π (x,x ). Therefore, if the participation constraint is satisfied for some x, it is also satisfied for all x < x. It follows that incentive compatibility implies that participation constraints can be replaced by: f N 0, Π L ( ˆx, ˆx) π (q N 1,q N ;c). (14) Note that (1) implies r(x) r( ˆx) + ˆx x 4, for all x ˆx. Together with the antitrust constraint, r( ˆx) c ˆx, this implies r(x) c ˆx + x. (15) 4 Evidently, for all x ˆx, (15) implies (). Therefore, we replace the incentive compatibility constraint (1) and the antitrust constraints () by (15). Optimal mechanism The optimal mechanism maximizes the innovator s expected payoff subject to incentive compatibility, participation, and antitrust constraints. 4.1 Optimal licensing if the innovator is more efficient We first consider the case when the innovator is more efficient than the potential licensee, for all x, which has been assumed by the literature. We show that in this case it is optimal to adopt pure royalty licensing. This result generalizes Heywood, Li, and Ye (014, Proposition 1) who showed the optimality of pure royalty contracts in a linear model with a binary random cost of the potential licensee. Proposition 1. Suppose the innovator is more efficient, i.e., x d. The optimal mechanism is a pure royalty contract with: r(x) = c ˆx + x, f L (x) = f N = 0, ˆx > x. (16) 4 It exhibits a strictly decreasing royalty rate, with the highest distortion at the top (x = x) and the lowest distortion at the bottom (x = ˆx). A license is awarded with positive probability, ˆx > x and, depending upon parameter values and the distribution function, exclusion can occur, ˆx < x. Proof. It is obviously optimal to adopt the highest possible fixed fee, which allows us to replace the participation constraints by f N = 0, f L ( ˆx) = π (q 1( ˆx),q ( ˆx); ˆx + r( ˆx)) π (q N 1,q N ;c). (17) Therefore, by Lemma 1, the optimal mechanism solves the following maximization problem: ˆx max Π 1 = (π1 (x) + r(x)q ( (x) + f L (x))df(x) + π 1 q N 1,q N ;d ) (1 F( ˆx)) r, f L, ˆx x s.t. (11), (15), (17), (P1) 7

8 In the following we first solve (P1) as an optimal control problem for given ˆx and then characterize the optimal ˆx. To translate (P1) into an optimal control problem define the control variable, u(x) := r (x), the state variables r(x), f L (x), the constraint functions g 1,g,k, the co-state variables, λ 1 (x),λ (x), η(x) and the Hamiltonian, H(x,r, f,u,λ 1,λ,η): H(x,r, f L,u,λ 1,λ,η) := h(x,r, f L,u) + λ 1 (x)g 1 + λ (x)g + η(x)k ( (1 ) ) d + x + r(x) 1 x r(x) + d h := + r(x) + f L (x) F (x) g 1 := u(x), g := 1 ˆx + x (1 + d x r(x))(1 + 4u(x)), k := c r(x). 9 4 The endpoints f L ( ˆx) and r( ˆx) are not free (see (17)) and the endpoints f L (x), r(x) are free. The maximum principle requires that the following conditions are satisfied: 8 0 = u H = λ 1 (x) 4λ (x) (1 + d x r(x)) 9 (18) λ 1(x) = r H = η λ (x)(1 + 4u(x)) + (5 d 4x 10r(x))F (x) 9 (19) λ (x) = fl H = F (x) (0) r (x) = λ1 H = u(x) (1) f L(x) = λ H = 1 (1 + d x r(x))(1 + 4u(x)) () 9 ( ) 1 ( ˆx + r( ˆx)) + d ( ) 1 c + d f L ( ˆx) = () λ 1 (x) = λ (x) = 0 (4) η(x) 0, k := c ˆx + x r(x) 0, η(x)k = 0. (5) 4 (0) and (4) yield λ (x) = F(x). Inserting this into (18) yields λ 1 (x) = 4F(x) (1 + d x r(x)). (6) 9 Differentiating (6) and equating with (19), after inserting λ (x), yields: r(x) = 6F(x) 9η F (x) + 1 5d + 4x. (7) We now show that the constraint k 0 is binding everywhere. Suppose it is not binding at some x. Then, for those x, η(x) = 0 by (5), and hence k(x) = c ˆx + x 4 r(x) η=0 = (c 1) + 10d ˆx 9x 4 F(x) F (x). This is negative because c 1 < d (non-drastic innovation) and d < x < x < ˆx (innovator is more efficient) imply (c 1) + 10d ˆx 9x < 0. This is a contradiction; hence k = 0 everywhere, and we conclude that r(x) = c ( ˆx+x)/4, as asserted. 8 The present control problem is subject to inequality constraints concerning the state variables and some free and fixed end-points. The corresponding conditions for optimality can be found in Kamien and Schwartz (1991, p. 0 ff.). 8

9 Inserting r (x) = 1/4 into () yields f L (x) = 0 and inserting r(x) into () yields f L( ˆx) = 0 and hence f L (x) = 0 for all x, as asserted. Using the optimal r and f L it is straightforward to confirm that Π (x,x) Π (x,z), for all x,z. Therefore, incentive compatibility is satisfied. The Arrow Sufficiency Theorem (Seierstad and Sydstæter, 1977, Theorem ) applies, because, at the optimum, λ 1 g 1 + λ g = 0 and H := max u H is concave in r and f L. Denote the maximum expected payoff of the innovator by Π 1 ( ˆx). A contract is awarded with positive probability because: Π 1 ( ˆx) ˆx=x = 1 (1 c + d)(c x)f (x) > 0. (8) Depending upon the parameters, the optimal ˆx is either a corner solution, ˆx = x, or an interior solution (exclusion). Remark. Interestingly, the first-order conditions for incentive compatibility do not assure that truth-telling is a global maximum for all x,z. In fact, if we solve the optimal contract and use only the first-order conditions (11) (together with the antitrust constraints), we find that the optimal mechanism sets r(x) = c x, and uses negative state-dependent fixed fees. However, with that mechanism truth-telling, z = x, minimizes the licensee s payoff Π (x,z) for all x. 4. Optimal licensing if the licensee is more efficient with positive probability While pure royalty contracts are optimal if the innovator is more efficient, they are generally not optimal if the licensee is more efficient, at least for some range of his cost x. Instead, it may be optimal to apply two-part tariffs with increasing and even negative royalty rates, in some range. In the following we illustrate this with an example in which the licensee is more efficient in some range of x. Example. Assume F is the uniform distribution with support [0, /5], and d = 1/4,c = 1/ (the licensee is more efficient for all x [0,d)). The optimal mechanism prescribes three régimes: 1) a two-part tariff régime for x [0, ˇx), with the increasing r and decreasing f L functions: r(x) = x, f L(x) = 55 6 ˆx 7 ˆx 88x + 58x ; (9) 5 ) a pure royalty régime for x [ ˇx, ˆx], with a decreasing r function, exactly as in Proposition 1; ) an exclusion régime for x ( ˆx, x]. 4) The cutoff values at which régime changes occur are ˆx = (7 )/48 and ˇx := (5 6 ˆx)/4 = (5 )/48. Proof of Example. The optimization problem is exactly the same as that considered in Proposition 1. The only difference is that, due to the parameter specification, the constraint k 0 is not always binding, which affects the solution, as follows. For x ˇx the proof of Proposition 1 applies without modification. This proves ) and ). For x < ˇx, the proof of Proposition 1 applies without modification up to the line ending with (7). However, the constraint k 0 is no longer binding. For, if it were binding, one would have k = 0, which implies η = 6 ˆx 5+4x 5 6 ˆx 6 x, by (7). However, η can only be non-negative if x 4 = ˇx. Therefore, for x < ˇx the constraint is not binding and η = 0. Inserting η = 0 into (7) yields the r 9

10 function stated in 1). The associated f L function follows immediately from () and the fact that f L ( ˇx) = 0. Finally, to confirm the asserted optimal cutoff value ˆx, compute the innovator s expected payoff as a function of ˆx. It is a cubic function that has its unique maximum at the value stated in 4). r x, f L x r x 0. f L x c x 0.1 Exclusion x x x 0.1 Figure : Optimal mechanism if the licensee is more efficient in some range 0.4 x x 0. Exclusion x 0. Pure Royalty x 0.1 Two Part Tariff x Figure : Optimal mechanism for all possible levels of x Figure illustrates the optimal mechanism described in the example. Clearly, for x < ˇx the optimal royalty rate starts at a negative value and is increasing, while the fixed fee starts at a high positive level and is decreasing. At x = ˇx a régime change occurs, and the optimal mechanism is a pure royalty contract with a decreasing royalty rate and zero fixed fees. At ˆx another régime change occurs, above which no license is awarded (exclusion). The example assumes a low level of x. If one increases x, the two-part tariff region shrinks and finally vanishes altogether, whereas the pure royalty region at first moves upward but does not change its size, but when the two-part tariff region vanishes, it begins to shrink, and at some point the exclusion region vanishes, as illustrated in Figure. Altogether, this example shows that using fixed fees can make it possible to reverse the slope of the r function and set lower royalty rates for licensees with lower cost. This allows the innovator to extract more surplus by shifting output to the more efficient licensee. 10

11 5 Extension to drastic innovations An innovation is drastic for firm i if that firm s exclusive use of the innovation makes it a monopolist, i.e., if the best reply output of the other firm to the monopoly output of firm i is equal to zero. An innovation is either drastic for one firm or for both firms or for no firm. An innovation is super-drastic for firm i if that firm s non-exclusive use of the innovation makes it a monopolist. Licensing generally allows both the licensee and the innovator to use the innovation. In principle, a license contract could also stipulate that the innovator promises not to use the innovation or even to shut down his plant. This may, however, raise a credibility issue or be in conflict with antitrust regulations. Yet, these problems can be bypassed by selling the patent. Under complete information the optimal licensing of a drastic innovation is very simple: the innovator either issues no license or sells the patent to the licensee in exchange for a fixed fee. No license is issued if and only if the innovator s cost is lower than that of the licensee (d < x). If the patent is sold, firm becomes a monopolist and the fixed fee is equal to the difference between the monopoly profit of firm and the equilibrium profit of firm if no license is issued. If the innovation is super-drastic and the licensee is more efficient, instead of selling the patent it is sufficient to grant a (non-exclusive) fixed-fee license contract because such a contract can implement a monopoly and extract the full surplus even though the innovator is left free to also use the innovation. Drastic innovations in models of complete information were already considered by Wang (1998) and Poddar and Sinha (010). Unlike the present paper, they consider a common value framework. In that framework, a drastic innovation will obviously not be licensed if the innovator is at least as efficient as the licensee in which case the innovator becomes a monopoly (see Wang, 1998). In turn, one would think that the optimal licensing makes the licensee a monopoly if the innovation is drastic and the licensee is more efficient than the innovator. Contrary to this conjecture, Poddar and Sinha (010) showed that the optimal licensing prescribes a two-part tariff and implements a duopoly rather than a monopoly. However, this is true only if, after licensing the innovation, the innovator is still able to use his innovation, which ignores the possibility to sell the patent (in which case the innovator s unit cost returns to the pre-innovation level, c). The optimal sale of the patent does not involve a two-part tariff because including royalty payments would distort the output decision of the licensee and reduces the surplus that the innovator can extract from him. Note that, in the common value case with asymmetric firms considered by Poddar and Sinha (010) it cannot occur that an innovation is super-drastic. Therefore, in this case, if license contracts are not permitted to include output restrictions, a monopoly cannot be implemented without selling the patent. Having shown that under complete information, the optimal licensing of a drastic innovation does not involve two-part tariffs, we now show that under incomplete information two-part tariffs are essential for the licensing of a (super-)drastic innovation. Of course, licensing occurs only if the licensee is more efficient. In that case, it is optimal to sell the patent to the licensee. However, if the innovation is super-drastic the innovator may also award a non-exclusive license and does not need to sell the patent. 11

12 Lemma. Suppose the licensee is awarded a monopoly license. The following conditions are necessary for incentive compatibility, for all x,z ˆx x: { 1 if x ˆx γ(x) = (0) 0 otherwise f L(x) = 1 (1 x r(x))r (x) (1) r(x) r(z) x z Proof. The proof of (0) is the same as in Lemma () The equilibrium profit of the licensee with cost x and reported cost z ˆx is: ( ) 1 x r(z) Π (x,z) = f L (z). Incentive compatibility requires z Π (x,z) z=x = 0 which implies (1). Moreover, incentive compatibility requires that r is non-decreasing, because Π (x,x) Π (x,z) + Π (z,z) Π (z,x) 0 1 (x z)(r(x) r(z)) 0. Proposition. Suppose the licensee is more efficient, x d, and the innovation is drastic, c > (1+d)/. The optimal mechanism has the following properties (where q M (x) denotes the monopoly output of firm ): r(x) = F(x) ˆx ( ) F(y) F (x), f L(x) = q M ( ˆx) + q M (y) x F dy, f N = 0. () (y) It exhibits an increasing royalty rate, a decreasing fixed fee, no distortion at the top (r(x) = 0), and full extraction of the surplus at the bottom ( f ( ˆx) = π M ( ˆx)). If F is the uniform distribution with support [0, x], one has r(x) = x, f L (x) = (1 ˆx(1 ˆx) x(1 x))/4, and ˆx = min{d/, x}, which implies exclusion ( ˆx < x) if x > d/. Proof. As one can easily confirm, (1) implies that Π (x,x) is monotone decreasing in x. Therefore, the participation constraints (1) can be replaced by the constraint: Π ( ˆx, ˆx) 0. (4) By () r is non-decreasing. Therefore, the antitrust constraints can be binding at most once, at x = ˆx, i.e., r( ˆx) c ˆx. (5) This allows us to replace the antitrust constraints () by the constraint (5). It is obviously optimal to adopt the highest possible fixed fee, which allows us to replace the participation constraints by ( ) 1 ˆx r( ˆx) f N = 0, f L ( ˆx) =. (6) 1

13 Using the incentive compatible allocation rule (0), the optimal mechanism must solve the following maximization problem (where q M denotes the monopoly output of firm and πm 1 the monopoly profit of firm 1): max Π 1 = r, f L, ˆx ˆx x ( r(x)q M (x) + f L (x) ) F (x)dx + (1 F( ˆx))π M 1, s.t. (1), (6), (5). (P) We first take ˆx as given and ignore the antitrust constraint (5), which does not affect the optimal r and f L functions. We solve the thus reduced program as an optimal control problem, and finally characterize the optimal ˆx, which can be affected by the constraint (5). The control variable is u(x) := r (x), the state variables are r(x), f L (x), the constraint functions are g 1,g, the co-state variables, λ 1 (x),λ (x), and the Hamiltonian is H(x,r, f L,u,λ 1,λ ): H(x,r, f L,u,λ 1,λ ) := ( r(x)q M (x) + f L (x) ) F (x) + λ 1 g 1 + λ g q M (x) := 1 (1 x r(x)), g 1 := u(x), g := 1 (1 x r(x))u(x). The endpoints f L ( ˆx), r( ˆx) are not free (by (6)), and the endpoints f L (0), r(0) are free. The maximum principle requires that the following conditions are satisfied: 0 = u H = λ 1 (x) λ (x) (1 x r(x)) (7) λ 1(x) = r H = 1 ( (x 1 + r(x))f (x) λ (x)u(x) ) (8) λ (x) = F (x) (9) r (x) = λ1 H = u(x) (40) f L(x) = λ H = 1 (1 x r(x))u(x) (41) ( ) 1 ˆx r( ˆx) f L ( ˆx) = (4) λ 1 (x) = λ (x) = 0. (4) (9) and (4) yield: λ (x) = F(x); inserting this into (7) yields λ 1 (x) = F(x) (1 x r(x)). (44) Differentiating (44) and equating with (8), after inserting λ (x), yields the asserted r function. To find f L (x) we use the fact that f L (x) f L ( ˆx) ˆx the asserted f L function. x f L (y)dy together with (41) and (4) and find Evidently, the assumed log-concavity of F assures that r is strictly increasing and f L strictly decreasing. Having derived the optimal mechanism we confirm its incentive compatibility and characterize the optimal ˆx. Because r (x) > 0 one has zx Π (x,z) = r (z)/ > 0. Therefore, Π (x,z) is pseudoconcave in z, which assures that the solution is incentive compatible. 9 9 The function Π (x,z) is pseudoconcave in z if, for all x, it is increasing to the left of its stationary point and decreasing to the right. Due to the first-order condition for incentive compatibility Π (x,z) has a stationary point at z = x, i.e., z Π (x,z) z=x = 0 for all x. Evidently, pseudoconcavity assures that the stationary points are global maxima. 1

14 The Arrow Sufficiency Theorem (Seierstad and Sydstæter, 1977, Theorem ) applies, because, at the optimum, λ 1 g 1 + λ g = 0, and H := max u H is concave in r and f L. The choice of ˆx must satisfy the antitrust constraint (5). Denote Π 1 ( ˆx) = max r, f L Π 1. The optimal ˆx maximizes Π 1 subject to the constraint ˆx c F( ˆx)/F ( ˆx). If F is the uniform distribution, one can easily confirm the asserted r and f L functions. Using ( the optimal r and f L functions, one finds that the innovator s expected profit is equal to: Π 1 ( ˆx) = x(1 d) + ˆx(6d d + ˆx( ˆx )) ) /(1 x). Maximizing Π 1 yields the optimal cutoff value ˆx = min{d/, x}. The constraint ˆx c F( ˆx)/F ( ˆx) is not binding in this case. Evidently, it is not optimal to sell the patent for a pure cash price. Instead, the cash price has to be supplemented by output based royalties. This can be interpreted as an implication of the linkage principle. According to that principle, linking the license fee to a variable that is correlated with the licensee s private information, such as output, tends to lower information rents (as shown in a different context by Milgrom, 1987). 6 Discussion The literature on patent licensing by an inside innovator claimed that the optimal license contract for a non-drastic innovation does not employ fixed fees and typically adopts a pure royalty contract. While it is generally true that the innovator may wish to charge a positive royalty rate in order to maintain a strategic cost advantage, it can however be more profitable to reduce that cost advantage by subsidizing the marginal cost of the licensee. The pivotal issue is whether the licensee or the innovator is able to make better use of the innovation. Our analysis indicates that the optimality of pure royalty contracts asserted in the literature is driven by the assumption that the inside innovator is more efficient than the licensee, rather than by the assumption that the innovator is an inside patent holder. In the case of a drastic innovation we showed that it is essential to distinguish between a license that grants the licensee the exclusive or the non-exclusive right to use the innovation. Moreover, it is important to distinguish between drastic and super-drastic innovations. Whereas an innovation that is drastic but not super-drastic propels a monopoly only if the license awards the exclusive right to the innovation (which makes the innovator s cost return to the pre-innovation level), a super-drastic innovation propels a monopoly even if the license is non-exclusive. We show that under complete information the optimal license contract of a (super-)drastic innovation is a fixed-fee contract rather than a two-part tariff. However, under incomplete information, output dependent royalties are used even if the license implements a monopoly, essentially because linking the license fee to a variable that is correlated with the licensee s private information contributes to lower information rents, which is reminiscent to the well known linkage principle in auction theory. Altogether our results in this paper, together with our work on the outside innovator s licensing problem (referred to in the introduction), indicate that there is no sharp distinction between licensing by an outside and an inside innovator. Generally, both outside and inside innovators tend to employ output based royalties and fixed fees. These results are broadly in line with observed licensing practice. In a study of US firms Rostoker (1984, p. 64) reported that: A down payment with running royalties method was used 46% of the time, while straight royalties and paid-up licenses accounted for 9% and 1%, respectively See also Vishwasrao (007) on foreign technology licensing to Indian firms. 14

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