On the Licensing of Innovations under Strategic Delegation

Transcription

1 On the Liensing of Innovations under Strategi Delegation Judy Hsu Institute of Finanial Management Nanhua University Taiwan and X. Henry Wang Department of Eonomis University of Missouri USA Abstrat This paper uses a three-stage liensing-delegation-quantity game to study the liensing of a ost-reduing innovation by a patent-holding firm to its ompetitor. It is shown that liensing is less likely to our under strategi delegation ompared to no delegation. JEL Classifiation: D45; L0; L0 Key Words: liensing; strategi delegation

2 . Introdution A patented innovation provides the innovator opportunities to reap a reward on his or her investment in researh and development. For an outside independent researh lab, this reward may be realized through liensing its innovation to the produing firms. For an inside firm, it may keep its innovation for its own use and gains an advantage in ompeting with its ompetitors. It may also liense the innovation to its ompetitors. Several important reasons have been advaned in the literature as to why a firm may want to liense an innovation to its ompetitors, overing both the profit motive and the strategi inentive. For example, Gallini (984) points out the inentive for an inumbent to liense to a potential entrant so as to redue the likelihood of the latter developing a better tehnology; Katz and Shapiro (985) regard the inentive to liense as an integral part of a firm s R&D deision in evaluating the profitability of a R&D projet; Eswaran (994) explores the possibility that liensees an serve as a barrier to entry; Lin (996) shows that liensing in the form of a fixed fee may serve as a failitating devie for ollusion among ompetitors. Despite the presene of these and other possible inentives for a firm to liense an innovation to its ompetitors, it remains well reognized by eonomists that liensing of innovations between ompeting firms does not happen very often. The most pronouned explanation for this is the existene of asymmetri information. Potential liensees lak detailed information about an innovation to assess its value prior to liensing so that they are not willing to pay the desired amount of ompensation demanded by the patent-holder. Other known reasons inlude the innovating firm s unwillingness to share pertinent information with ompetitors that might have bearing on other related and ongoing R&D, and the fat that it may be ostly or impossible for the liensor to monitor how the potential liensee uses the liensed innovation, inluding the monitoring of its output produed using the liensed innovation and the

3 possibility that it might re-liense to other firms. The goal of this paper is to point to another potentially important reason for the lak of liensing of innovations between ompeting firms. It has to do with the widely reognized fat of separation of ownership and ontrol in the modern orporation and the delegation of some deision making from owners to managers. We onsider liensing as part of a delegation-liensing-quantity game. Our game involves three stages and two ompeting duopoly firms. The first stage is the delegation stage. In this stage, owners of the two firms deide simultaneously inentive ontrats for their managers. The seond stage of the game is the liensing stage in whih the patent-holding firm hooses a liensing ontrat for its innovation and the other firm deides whether to aept the ontrat offer. The third stage is the quantity ompetition stage in whih the firms managers engage in an output ompetition. The main result of this paper is that liensing is less likely to our under strategi delegation than under no delegation. The delegation-liensing-quantity game studied here ombines two strands of literature. One is the strategi delegation literature. The most influential early work in this literature inludes Fershtman and Judd (987) and Sklivas (987). The other is the liensing literature. A seminal paper on tehnology liensing is Arrow (96). Kamien (99) ontains an exellent review of this literature. A paper losely related to the present paper is Saraho (00), who studies liensing by an independent researh lab to an oligopoly under strategi delegation. Her model has the same three stages as in our model. Her fous, however, is on the omparison of fixed-fee liensing and royalty liensing and her main result extends the finding in Wang (998) and Kamien and Tauman (00) that royalty an be superior to fixed fee for the patentee. See Shapiro (985) for an aount of these reasons.

4 3. Model Setup The impat of strategi delegation on liensing is most transparent in the ontext of a homogeneous good Cournot duopoly with a linear demand and onstant unit ost of prodution. Assume the (inverse) market demand funtion is given by p = a Q, where p denotes prie and Q represents industry output. With the old tehnology, both firms produe at onstant unit prodution ost (0 < < a). The ost-reduing innovation by firm reates a new tehnology that lowers its unit ost and any liensee s unit ost by the amount of ε. For simpliity, our fous is on non-drasti innovations (i.e., ε < a ). Our game takes plae in three stages: delegation, liensing, and quantity ompetition, respetively. In the first stage, the firms owners deide simultaneously their inentive ontrat for their managers. In the seond stage, firm (the patent-holder) hooses a liensing ontrat and firm deides whether to aept firm s offer. In the third stage, the firms managers simultaneously hoose their output levels. The output hoie stage is essentially the standard Cournot game exept that the managers are not profit maximizers but rather that they maximize a weighted sum of profit and revenue. The delegation stage determines the firms inentive parameters for their managers, these parameter values determine the weights assigned to profit and revenue in the manager s optimization problem. The values of these parameters will depend on the firms unit osts of prodution. That is, the inentive parameters are An innovation is drasti if the post-innovation monopoly prie is equal to or below the pre-innovation ompetitive prie. That is, ε a. See, for example, Kamien (99) for definition. As it is well established that a drasti innovation will not be liensed to ompetitors by the patent-holding firm in the ase of no delegation (e.g., Katz and Shapiro, 985), in our model with strategi delegation suh an innovation will ertainly not be liensed.

5 4 funtions of the firms ost levels. The atual levels of the firms unit osts are determined by the liensing stage. To solve the delegation game, one need only find the solution to the last (output) stage of the game. 3 The payoffs to all partiipants (both owners and managers) in the three stage game are realized in the last stage of the game. 3. The Delegation-Liensing-Quantity Game We start by solving the output stage of the three stage game. For onveniene, we then move on to solve the delegation problem and finally the liensing problem. Quantity Competition To study the managers output hoies in the last stage of the game, we assume that firm has an unspeified onstant unit prodution ost of and firm has an unspeified onstant unit prodution ost of. These marginal ost levels will be determined by the first two stages of the game. Throughout the paper subsripts and denote for firms and, respetively. Firm s profit funtion is represented by Π = ( a q q ) q and its revenue funtion is R = ( a q q ) q. The manager for firm hooses q to maximize a weighted sum of its profit and revenue, namely, O = Π + ( α ) R α, () where α is the inentive (weight) parameter hosen by firm s owner in the first two stages of the game. Similarly, the manager for firm hooses q to maximize 3 It is implied that one need not solve the seond (liensing) stage game first in order to solve the first (delegation) stage game. Hene, the order of the first two stages is atually irrelevant to the final solution of the game.

6 5 O = Π + ( α ) R α, () where α is the inentive parameter hosen by firm s owner in the first two stages of the game, Π = ( a q q ) q and R = ( a q q ) q represent respetively firm s profit and revenue. As in Fershtman and Judd (987), we allow α and α to take any value. Maximizing O in () with respet to q gives firm s quantity reation funtion: q (a α q ) / = 0 if q if q a α > a α (3) Similarly, maximizing O in () with respet to q yields firm s quantity reation funtion: q (a α q) / = 0 if q if q a α > a α (4) These reation funtions have the usual interpretation of first-order onditions. That is, a firm s optimal output response is one in whih its marginal benefit is equal to marginal ost; and when marginal ost is always higher than marginal benefit the firm s optimal hoie of output is zero. The intersetion of the reation funtions (3) and (4) gives the firms equilibrium quantities in the third stage of the game as a funtion of hoies made in the first two stages of the game. As in the standard Cournot model with unequal marginal osts, the intersetion point of the reation funtions may be either an interior point with both quantities positive or a boundary point with one firm produing zero. In our model, firm as the innovating firm will always be at least as effiient as firm (i.e., ) and the only possible orner solution is where firm produes zero. The third stage output hoies as funtions of hoies in the first two stages of the game are summarized in Lemma. (Proofs of all lemmas and propositions are provided in the appendix.)

7 6 Lemma : Assuming that α α, the equilibrium output levels in the quantity game in whih firm i s manager maximizes O i (i =,) are q q = q i (a α (a α + α )/ ) / 3 if a + α α if a + α α > 0 0 (5) q q = q i (a α 0 + α )/ 3 if a + α α if a + α α > 0 0 (6) By Lemma, if + α α 0 then the third stage quantity game gives an interior a > i i solution, given by ( q, q ); and if a + α α 0 then the third stage game gives a orner solution, given by ( q, q ). The assumption α α will be verified later on to be satisfied in equilibrium. The Delegation Problem In the delegation stage of the game the two firms owners simultaneously hoose their inentive parameters α and α, knowing that the solution for the output stage of the game will be given aording to Lemma. Let π α, ) and π α, ) denote respetively firm s and firm s ( α ( α redued-form profit funtions derived based on the solution to the last stage of the game. Maximizing π with respet to α and π with respet to α yield the two firms inentive parameter reation funtions, as given in the next lemma. Lemma : Firm s inentive parameter reation funtion in the delegation stage of the game is given by α (6 α a)/(4 ) = (α a) / if α (a + ) /(3 if (a + ) /(3 ) α if α (a + ) /( ) ) (a + ) /( ) (7)

8 7 Firm s inentive parameter reation funtion (more preisely orrespondene) in the delegation stage of the game is given by α (6 α a) /(4 ) = [(a + α) /( ), ) if α if α > ( ( a) / a) / (8) Firm s inentive parameter reation urve has three segments. The first line of (7) orresponds to a dereasing segment; the middle line of (7) orresponds to a rising segment (on whih a + α = 0); and the last line of (7) orresponds to a vertial segment. Regarding firm s α inentive parameter reation map, it is important to observe that firm s best response on or above the line a + α α = 0 is not unique. This is beause when a + α α 0 firm s profit is zero due to a zero output level. Note that above the line a + α α = 0, a + α α < 0. Below the line a + α α = 0, firm s best response is unique and dereasing in α. The intersetion of these reation funtions gives the firms equilibrium values for the inentive parameters as funtions of their unit osts of prodution, as given by the following lemma. Lemma 3: (a) If a + 3, there is a unique equilibrium in the delegation stage of the game, given by 8 a α =, 5 8 a α =. (9) 5 (b) If a + < 3, there is a ontinuum of equilibria in the delegation stage of the game, given by the set:

9 8 4 a a + a E {( α, α) : α min{,}; α ; a + α α = 0.} 3 3 It may be noted here that the hoies of α and α as given by Lemma 3 satisfy the ondition in Lemma that α α. First, onsider the solution given by (9). From (9), α α = ( ), whih is less than or equal to zero sine with or without liensing. Next, onsider the set E. On this set, α α = ( a α ), whih is less than zero sine a > by assumption and α in equilibrium. The preeding results lead to the firms equilibrium quantity hoies as funtions of levels of marginal osts, as summarized in the following proposition. Proposition : In the quantity ompetition stage of the three-stage liensing-delegation-quantity game, (a) if a + > 3 then the equilibrium quantities are an interior solution, given by = (a + 3 ), q = (a + 3) ; (0) 5 5 q (b) if a + 3 then the following multiple orner solutions in quantity are obtained: a max{ a, } q (a ), q 0 3 =. () The interior solution given by (0) has the usual property that eah firm s output dereases in its own marginal ost but inreases in its ompetitor s unit ost. Comparing (0) with the well-known solution without delegation, given by ( a + ) / 3, (a + ) / 3 ), one obtains that industry (

10 9 output is higher under delegation. This onfirms the established result that firms under delegation are more aggressive in their output hoies than firms under no delegation. Based on Proposition, the firms redued-form profit funtions are π (, ) = (a 3 + ), π (, ) = (a 3 + ) () 5 5 in the ase of interior quantity solution, and π (, ) = ( a q ) q, π (, ) = 0 (3) in the ase of a orner quantity solution, where q is given by (). The Liensing Problem In the liensing stage of our three stage game, the patent-holding firm (firm ) first hooses a liensing ontrat, then the potential liensee (firm ) deides whether to aept the offer from the patent holder. The patent-holding firm s objetive is to maximize its total inome whih is the sum of the profit from its own prodution and the liensing revenue. In the following analysis, we onsider three forms of liensing ontrat: fixed fee only, royalty only, and fixed fee plus royalty. 4 We use F to denote a fixed fee that is independent of the liensee s output level and r to denote a royalty rate per unit of output. Firm s unit ost of prodution is = ε, firm s unit ost of prodution is = ε + when liensing ours and is = when liensing does not our. Obviously, the royalty rate r annot r exeed the magnitude of innovation (ε ). Firm hooses a liensing ontrat to maximize its total inome subjet to the onstraints that firm is willing to aept the liensing ontrat and that firm s output is non-negative. That is, firm solves the following problem:

11 0 Max {r, F} π ( ε, ε + r) + r q ( ε, ε + r) F (4) + s. t. π ( ε, ε + r) F π ( ε, ) q ( ε, ε + r) 0 In (4), the profit funtions and firm s output funtion are given by (0)-(3). Under fixed-fee liensing, firm hooses F while restriting r to be zero; under royalty liensing, firm hooses r while restriting F to be zero; under fee plus royalty liensing, firm hooses both F and r. The next proposition onerns the ourrene of liensing in the equilibrium of the three-stage game under eah of the three forms of liensing. Proposition : The equilibrium outome of the three-stage delegation-liensing-quantity game is given by the following: (a) under fixed-fee liensing, liensing ours if and only if ε < ( a )/; (b) under royalty liensing, liensing ours if and only if ε < ( a )/; () under fee plus royalty liensing, liensing ours if and only if ε < ( a )/ and will be in the form of royalty only when ourring. It has been shown in the literature that, for the linear Cournot model without delegation, fixed- 4 These three forms of liensing aount for almost all liensing in pratie. Rostoker (984) reported that fixed fee alone was used thirteen perent of the time, royalty alone thirty-nine perent, and royalty plus fixed fee forty-six

12 fee liensing ours for ε < ( a )/3 and royalty liensing and fee plus royalty liensing our for any non-drasti innovation (i.e., ε < a ). 5 Comparing this onlusion with the results in Proposition, it follows that in the linear model liensing ours at most half as likely under strategi delegation ompared to no delegation. Moreover, liensing ours only for small innovations under strategi delegation Conluding Remarks This paper has examined liensing of a ost-reduing innovation by a patent-holding firm to its ompetitor from the profit motive. Under strategi delegation, firms (managers) behave more aggressively than under standard quantity ompetition, reduing the inentive for the patent-holding firm to liense its innovation to the other firm. This is the result of two fores. On the one hand, the ostreduing innovation (if kept for own use) affords the patent-holding firm a bigger advantage over its ompetitor under strategi delegation than under no delegation. On the other hand, the potential liensing revenue is smaller due to a smaller potential for profit gain from liensing by the ompetitor under strategi delegation than under no delegation. Both fores work to redue the likelihood of liensing under strategi delegation relative to no delegation. The disussion above also indiates that the main onlusion of this paper that liensing is less likely to our under strategi delegation than under no delegation should survive extension of the simple homogenous good duopoly model with linear demand to more general settings. perent, among the firms surveyed. 5 See, for example, Katz and Shapiro (985) and Wang (998). 6 The onlusion in Proposition () extends the result in Wang (998) and Kamien and Tauman (00) that royalty is superior to fixed fee for the liensor to the situation with strategi delegation.

13 Referenes Arrow, K. (96) Eonomi welfare and the alloation of resoures for invention in The Rate and Diretion of Inventive Ativity, ed. R. R. Nelson, Prineton University Press. Eswaran, M. (994) Liensees as entry barriers Canadian Journal of Eonomis 7, Fershtman, C., and K. Judd (987) Equilibrium inentives in oligopoly Amerian Eonomi Review 77, Gallini, N. (984) Deterrene by market sharing: a strategi inentive for liensing Amerian Eonomi Review 74, Katz, M., and C. Shapiro (985) On the liensing of innovations Rand Journal of Eonomis 6, Kamien, M. (99) Patent liensing Chapter in Handbook of Game Theory, Eds. R. J. Aumann and S. Hart. Kamien, M., and Y. Tauman (00) Optimal liensing: the inside story The Manhester Shool 70, 7-5. Lin, P. (996) Fixed-fee liensing of innovations and ollusion Journal of Industrial Eonomis 44, Rostoker, M. (984) A survey of orporate liensing IDEA 4, Saraho, A. (00) Patent liensing under strategi delegation Journal of Eonomis & Management Strategy, 5-5. Shapiro, C. (985) Patent liensing and R&D rivalry Amerian Eonomi Review 75, Sklivas, S. (987) The strategi hoie of managerial inentives Rand Journal of Eonomis 8,

14 Wang, X. (998) Fee versus royalty liensing in a Cournot duopoly model Eonomis Letters 60, 55-6.

15 4 Appendix Proof of Lemma : For the interior solution given by the first line of (5) and the first line of (6), it is obtained straightforwardly by solving for q and q using the first line of (3) and the first line of (4). This solution also gives us the ondition to have an interior solution, namely q is positive (whih implies q is positive). For the orner solution, it is important to reognize that under the ondition α α the only possible orner solution is when the intersetion point of the firms quantity reation urves is on the q axis so that q = 0. Using this fat, the orner solution for q is obtained by substituting q = 0 into the first line of (3) and is given by the seond line of (5). Proof of Lemma : Based on (5) and (6), the firms redued-form profit funtions are π ( α, α (a ) = (a q i q q )q i )q i if a + α α if a + α α > 0 0 (A) π (a ( α, α ) = 0 q i q i )q i if a + α α if a + α α > 0 0 (A) i i Consider (7) first. Substituting q and q given respetively by (5) and (6) into the first line of (A) and maximizing the resulting profit funtion for firm with respet to α yield α = ( 6 α a) /(4), this is the first line of (7). For this solution to represent firm s best reation, the ondition in the first line of (A) must hold. The intersetion of the lines α = ( 6 α a) /(4) and

16 5 a + α α = 0 is (,α α ) = ( 4 a) /(3 ),(a )/(3 ) ). Thus, the first line of (7) is proved. ( + Substituting q in (5) into the seond line of (A) and maximizing with respet to α implies that α =. For this to hold, it must be true that + α α 0. Replaing α in this inequality by one gives α a ( a ) /( ). This proves the last line of (7). To show the middle line of (7), it suffies to + observe that for the range of α in the interval [( a + ) /(3 ), (a + ) /()], firm maximizes its profit funtion given by the seond line of (A) subjet to the onstraint that + α α 0. For a α [( a + ) /(3 ), (a + ) /()], the above onstraint is binding, yielding the middle line of (7). Next onsider (8). The first line of (8) is parallel to the first line of (7) and an be shown similarly by maximizing π α, ) in (A). For the seond line of (8), it suffies to observe that when a ( α + α α 0 firm s profit is zero implying an arbitrary hoie of α subjet to the onstraint that the above inequality holds, whih implies that α a + )/( ). ( α Proof of Lemma 3: (a). Solving the system of equations omposed of the first line of (7) and the first line of (8) for α and α gives immediately (9). From (9), a + α α = 6(a + 3 ) / 5. Hene, if a > + 3 then a α α + > 0 and if a + = 3 then a + α α = 0. In either ase, firm s inentive parameter reation urve given by (7) and firm s inentive parameter reation map given by (8) have a unique intersetion point, that is given by (9). In the ase a + > 3, this intersetion point is below the line a + α α = 0, while in the ase a + = 3, it is right on the line a + α α = 0.

17 6 (b). The proof for Part (a) indiates that if a + < 3 then firm s inentive parameter reation urve given by (7) and firm s inentive parameter reation map given by (8) do not have an intersetion point below the line a + α α = 0. In this ase they interset on a line segment of the line a + α α = 0. Straightforward derivations imply that this line segment orresponds to the set E. Proof of Proposition : (a). Results in (0) follow immediately by substituting (9) into the first line of (5) and the first line of (6). (b). By Lemma 3(b), if a + < 3 the delegation stage of the game results in hoies of α ) given by the set E. Using (6), it is obvious that on this set q = 0. The range of q is obtained (,α by substituting q = 0 and the range of values for α in the set E into (3). In partiular, orresponding to α = ( 4 a) /(3) we have q = (a ) / 3 and orresponding to α = ( a) / or we have q = a or q = ( a ) /. These values for q onfirm the equilibrium range for q in (). Proof of Proposition : (a). Based on (0) and (), both firms produe the quantity of ( a + ε )/5 and earn the profit of ( a + ε) /5 under fixed-fee liensing. If liensing does not our, based on Lemma 3, an interior solution is obtained if ε < ( a )/ and a orner solution is obtained if ε ( a )/. Consider first the ase of ε < ( a )/. If liensing does not our, by (), firm s profit is ( a + 3ε) /5 and firm s profit is ( a ε) /5. The maximum liensing fee firm an harge

18 7 firm is equal to the differene between firm s post-liensing profit and its profit if no liensing ours. Thus, F = ( a + ε) / 5 ( a ε) /5, and under fixed-fee liensing, firm s total inome is ( a + ε) /5+F. The differene between this and firm s profit under no liensing is ( a + ε) /5 + F ( a + 3ε) /5 = ε [ (a ) ε ]/5, whih is greater than zero only for ε < ( a )/. That is, under fixed-fee liensing and strategi delegation, liensing is profitable for firm for ε (0, ( a )/) but not profitable for ε (( a )/, ( a )/). Consider next the ase of ε ( a )/. In this ase firm will produe zero output and make zero profit if liensing does not our. Thus, the maximum fixed fee firm an harge is equal to ( a + ε) /5 and firm s total inome under liensing is 4( a + ε) /5. If firm does not liense to firm, by (3), its profit is equal to ( a + ε q ) q, where by () q varies from a to ( a + ε )/3. It is straightforward to verify that ( a + ε q ) q > 4 ( a + ε) /5 for all q [ a, ( a + ε )/3] when ε ( a )/. That is, under fixed-fee liensing and strategi delegation, liensing is not profitable for firm for ε [( a )/, a ). We have thus proved the assertion made in Proposition (a). (b). By using (0) and (), firm s total inome under royalty liensing is M = π ( ε, ε + r) + r q ( ε, ε + r) = (a + ε + r) + r (a + ε 3r). 5 5 Differentiating M with respet to r yields M = r (9(a 5 + ε) r). Non-negative equilibrium output for firm implies that 3 r a + ε. Hene, the feasible hoie of r is from zero to min{ε, ( a + ε )/3}. It is easy to see that for r [0, min{ε, ( a + ε )/3}], M / r >0.

19 8 It follows that firm s optimal hoie of r is equal to min{ε, ( a + ε )/3}. The optimal r is equal to ε if ε < ( a )/. In this ase, liensing ours and firm produes a positive output in equilibrium. The optimal r is equal to ( a + ε )/3 if ε ( a )/. In this ase, liensing does not our sine firm produes zero output in equilibrium. We have thus proved Proposition (b). (). Obviously, the liensor s optimal F is suh that the first onstraint in (4) holds in equality. Solving for F from this equality and substituting it into the objetive funtion in (4), M = π ( ε, ε + r) + r q ( ε, ε r) + π ( ε, ε + r) π (,). By using (0) and (), + ε M = (a + ε + r) + r (a + ε 3r) + (a + ε 3r) (a ε) Differentiating M with respet to r yields M = r (3(a 5 ) + 3ε 4r) > 0, where the inequality follows from the fat r ε < on the interval [0, ε ] of feasible hoies for r. a. Namely, M is a stritly inreasing funtion of r Two impliations follow immediately from the fat that the best hoie of royalty rate for the liensor is r = ε. One, in the optimal fee plus royalty ontrat, the optimal fee is zero. This is beause firm s marginal ost of prodution is unhanged by liensing and therefore its profit is unhanged. Seond, firm s equilibrium output ( ε,) = ( a ε )/5 is positive only for ε < ( a ) /. q These two onlusions together establish the two statements in Proposition ().