Leveraged Frms, Patent Lcensng, and Lmted Lablty Kuang-Cheng Andy Wang Socal Scence Dvson Center for General Educaton Chang Gung Unversty and Y-Je Wang Department of Economcs Natonal Dong Hwa Unversty and Wen-Jung Lang Department of Economcs Natonal Dong Hwa Unversty and Mng-Che Tsa Department of Industral Economcs Tamkang Unversty and Chao-Cheng Ma Department of Industral Economcs Tamkang Unversty and Research Center for Humantes and Socal Scences Academa Snca Current Verson: August 7, 03 Correspondng Author: Wen-Jung Lang, Department of Economcs, Natonal Dong Hwa Unversty, Shoufeng, Hualen County, Tawan 9740, ROC. Tel: 886-3-863-5538, Fax: 886-3-863-5530, E-mal: wjlang@mal.ndhu.edu.tw
JEL Classfcaton: L4, G33 Key words: Leveraged Frms; Lmted Lablty; Outsder Patentee; Cournot Competton; Lcensng Means Abstract In a semnal paper on patent lcensng, Kamen and Tauman (986) show that fxed-fee lcensng s always superor to royalty lcensng for the outsder patentee. However, emprcal studes demonstrate that royalty lcensng s much more popular than fxed-fee lcensng n practce. Ths paper attempts to reconcle wth ths controversy wthn Kamen and Tauman s (986) theoretcal framework by takng nto account the fnancal structure. In partcular, ths paper focuses on an mportant feature of the fnancal structure of the frm n the modern corporaton that has receved no attenton n the patent lcensng lterature. The man contrbuton of the paper s as follows. Provded that the leveraged frms produce a homogeneous product and engage n Cournot competton, ths paper shows that the optmal lcensng contract for the outsder patentee s royalty lcensng when the mean-preservng varance of demand s large n the presence of debt fnancng, whle t s non-exclusve fxed-fee lcensng otherwse.
Leveraged Frms, Patent Lcensng, and Lmted Lablty. Introducton In a semnal paper on patent lcensng, Kamen and Tauman (986) show that fxed-fee lcensng s always superor to royalty lcensng for the outsder patentee ownng a cost-reducng nnovaton, when frms produce a homogeneous good and engage n Cournot competton n the commodty market. Subsequently, Kamen et al. (99) confrm the same result by usng a generaled demand functon. Nevertheless, the emprcal lterature, such as Rostoker (984), shows that royaltes alone account for 39%, fxed fees alone for 3%, and royaltes plus fxed fees for 46% of lcensng contracts. In addton, Macho-Stadler et al. (996) and Jensen and Thursby (00) also consstently pont to the prevalence of royalty lcensng. Thus, the fndngs of emprcal studes demonstrate that royalty lcensng s much more popular than fxed-fee lcensng. Ths creates sgnfcant nterest n explanng the ratonale for choosng royalty lcensng n lcensng contracts. The superorty of royalty over fxed-fee lcensng has been justfed n the lterature by appealng to the followng aspects. For the case where the patentee stands outsde the market, Beggs (99) emphases the role of asymmetrc nformaton; Muto (993) ponts out the mportance of product dfferentaton; Bousquet et al. (998) hghlght rsk sharng; Macho-Stadler et al. (996) and Jensen and Thursby
(00) focus on moral haard; Saracho (00) shows that a sales delegaton game s crucal; and Poddar and Snha (004) refer to the nfluence of spatal competton. For the case where the patentee s an nsder competng wth the lcensees n the ndustry, Kat and Shapro (985) and Wang (998) show that royalty lcensng s defntely superor to fxed-fee lcensng. Unfortunately, the above analyses completely gnore an mportant feature of the modern corporaton that has receved no attenton n the treatment of the frm n the patent lcensng lterature, namely, the fnancal structure of the frm. It s well recogned that frms usually ssue debt to fnance ther producton n a modern economy. In a poneerng paper, Brander and Lews (986) argue that the choce of fnancal structure can affect output markets through the lmted lablty effect of debt fnancng, n whch shareholders wll gnore reductons n returns n bankrupt states, snce debt holders become the resdual clamants. The second channel for fnances to affect output markets s the strategc bankruptcy effect, n whch frms mght make output market decsons that rase the chances of drvng ther rvals nto nsolvency. Brander and Lews (986) show that both the lmted lablty effect and the strategc bankruptcy effect may commt a leveraged frm to a more aggressve output stance. Other studes related to the ssue that the choce of fnancal structure can affect output markets nclude: Brander and Lews (988), Showalter (995), Hughes et al. (998), and Damana (999), etc.
As a matter of fact, the ssue that whether or not royalty lcensng s superor over fxed-fee lcensng s very mportant n theory and n practce. Ths paper attempts to reconcle wth these arguments wthn Kamen and Tauman s (986) theoretcal framework by takng nto account the fnancal structure, when the leveraged frms produce a homogeneous good and engage n Cournot competton n the commodty market. In partcular, we reason that the outsder patentee may decde to choose royalty lcensng n order to earn a larger lcensng proft caused by a more aggressve output strategy adopted by the leveraged frm, when the frm can ssue debt to the publc and fnancal nsttutons. To the best of our knowledge, a study on ths ssue has not yet been touched upon. Ths paper ams to fll ths gap. The man fndng of ths paper s as follows. By takng nto account the debt fnancng of the leveraged frms, the optmal lcensng contract for the outsder patentee s royalty lcensng n the case of a homogeneous product and Cournot competton when the mean-preservng varance of demand s large, whle t s non-exclusve fxed-fee lcensng otherwse. Thus, ths paper provdes a new explanaton to prevous lterature to justfy the superorty of the royalty over the fxed-fee lcensng contract. The remander of the paper s organed as follows. Secton sets up a basc model to analye the case where patent lcensng s absent. Secton 3 and Secton 4 3
examne the optmal number of lcenses under fxed-fee lcensng and the optmal royalty rate under royalty lcensng, respectvely. Secton 5 explores the outsder patentee s optmal lcensng contracts n terms of fxed-fee and royalty lcensng. The fnal secton concludes the paper.. The Basc Framework Consder two leveraged frms denoted as and n an ndustry that sell a homogeneous product wth a quadratc cost functon C = q /, =, where q s frm s output. The frms can ssue debt to the publc and to fnancal nsttutons, and engage n Cournot competton n the commodty market. In addton, there s a patent holder standng outsde the market and havng a process nnovaton, whch can reduce the producton cost of the lcensee by (εq ) where ε denotes the nnovaton se. The output decsons are made before the realaton of a random varable reflectng the varaton n demand. Once profts are determned, the leveraged frms are oblged to repay the debt they have ncurred by usng the operatng proft that they earn. When the operatng profts are nsuffcent to meet the debt oblgatons, the frms go bankrupt and wll then be taken over by the debt holders. In order to ensure an nteror soluton for the optmal debt level, the cost functon has to be quadratc n output. 4
Assume that the frms face demand uncertanty, n whch the nverse demand functon takes the lnear form as follows: p aq, () where a s a constant and a > ; p denotes prce; Q = q +q s market demand; and the random varable reflects the demand uncertanty. It s assumed that the random varable follows a unform dstrbuton over the nterval [, ], where and denotes the upper bound of the demand uncertanty, whose densty functon s f, for whch the mean s E 0 and the varance s 3 Var. It follows that a rse n the upper bound of the random varable can be denoted as a mean-preservng spread that leaves the mean fxed but ncreases the varance of demand. The game n queston conssts of three stages. In the frst stage, the outsder patentee chooses the optmal lcensng contract n terms of fxed-fee and royalty lcensng to maxme ts proft. Then, the frms decde whether or not to accept the lcense. By followng Brander and Lews (986), the owners choose optmal debt levels to maxme the total value of the frm n the second stage. Fnally, the leveraged frms determne ther outputs to maxme the equty value, engagng n Cournot competton n the commodty market n the thrd stage. The game can be solved by backward nducton, begnnng wth the fnal stage. 5
In what follows, we frst analye the equlbra for the case where patent lcensng s absent, and then the equlbra for the case where the nnovaton s lcensed n terms of fxed-fee and royalty lcensng, respectvely. Note that the game n the former case wll be reduced to a two-stage game, n whch the owners choose optmal debt levels n stage, and then the managers determne outputs n stage. In stage, as argued by Brander and Lews (986), gven the debt levels (D, D ), the managers are assumed to choose output levels wth the objectve of maxmng the equty value of the frm to the shareholders. Ths s what an owner-manager would choose to do, and s certanly what wealth-maxmng shareholders would want the managers to do. Assume that the shareholders of the frm are rsk neutral wth respect to the frm s returns and therefore have ther nterests served by the maxmaton of the equty value. The value to the shareholders s referred to as the equty value and s represented by V as: V N ˆ f ˆ D d a q q j q C D d,,, j, () where the subscrpt N denotes varables assocated wth the case where patent lcensng s absent; D s frm s debt level; s frm s operatng proft; and ẑ represents the break-even state of the demand uncertanty where frm can just meet 6
ts debt oblgatons wth nothng left over. Eq. () represents the expected profts net of debt oblgatons n good states ˆ ). In bad states ( ˆ ), the frm earns ero proft as all of ts earnngs are pad ( to debt holders. Defne the break-even state of the demand uncertanty, ẑ, n whch the leveraged frm s net proft equals nl as follows: ˆ D a q q ˆ q C D 0,,, j. (3) j By dfferentatng () wth respect to the output q, respectvely, and then lettng them equal ero, we can solve for the equty value maxmng outputs as follows: q N 4a 3ˆ ˆ,,, j. j (4) 6 Substtutng (4) nto (3), we can rewrte (3) as follows: a0 5ˆ 3ˆj4 a 3 ˆ ˆj D,,, j. (3.) 5 Eqs. (4) and (3.) show that, n stage, gven the fnancal composton (D, D ) determned n stage, a rse n frm s debt level ncreases (decreases) frm s (j s) output by ncreasng (decreasng) ˆ ( ˆ ). The ntuton s the same as that derved n j Brander and Lews (986) as follows. As debt rses, low margnal value (bad) states become rrelevant, for n those states the frm s turned over to the debt holders, and the equty holders receve ero n any case. Snce the frm restrcts ts attenton to 7
hgher margnal proft (good) states, t adopts a more aggressve output strategy. Thus, we have Lemma as follows: Lemma. Gven the fnancal composton (D, D ) determned n stage, a rse n frm s debt level ncreases (decreases) frm s (j s) output. Next, frm s debt value W can be expressed as: 3 W N ˆ ˆ D f D f d ˆ f d (5) ˆ d [ a q q j q C ] f d,,, j, where denotes frm s ero operatng proft state of the demand uncertanty. Followng Brander and Lews (986), the operatng proft earned by the leveraged frms s suffcent to remburse the debt n good states denoted by the frst term on the rght-hand sde of (5), whle t s smaller than the debt oblgaton n that the whole of the operatng proft s gven to the debt holders n bad states as represented by the second term. 4 The ero-operatng proft state can be derved as follows: a q q j q q 0,,, j. (6) Substtutng (4) nto (6), we can solve for as follows: 3 When the demand stuatons are n bad states where [, ] ero so that there s nothng left for the debt holders. 4 The bankruptcy cost s assumed to be ero n the paper. 8, the operatng proft s less than
a 0 7ˆ 3ˆ,,, j. j (7) 3 In stage, the owners determne the optmal debt levels to maxme the total value of the frm Y, whch s the sum of the equty value V and the debt value W. Thus, the total value of frm can be derved from () and (5) as follows: 5 Y N VN WN f ( ) d [( a q q j ) q C ] d,,, j. (8) As n Hughes et al. (998), we observe that ẑ s related to D through the break-even condton (3.). For convenence, we solve for equlbrum debt levels by dfferentatng (8) wth respect to ẑ nstead of D. Thus, by substtutng (4) and (7) nto (8), and then dfferentatng (8) wth respect to ẑ by settng them equal to ero, we can solve for the optmal ẑ as follows: ˆ N 0a 9,,, (9.) 9 By substtutng (9.) nto (4), (3.), (7), and (8), we can derve the equlbra under the no lcensng case as follows: 6 q N a,,, (9.) 9 4 D N (9.3) a 9,,, N 4a 5,,, (9.4) 9 94 Y 3 N a,,, (9.5) 3 9 It should be noted that (9.)-(9.5) apply to the case where,,. N 5 It s noteworthy that the total value of the frm equals the frm s expected operatng proft. 9
Otherwse, the varable n (5) and (8) has to be replaced by,,. N Thus, we can derve the crtcal state of the upper bound of the demand uncertanty, for whch N,,, by usng (9.4) as follows:, 7a, (9.6) In order to ensure that the ero-operatng proft state,, s greater than the lower bound of the demand uncertanty, we fnd from (9.6) that the upper bound of the demand uncertanty has to be greater than,.e., N,,, when. Snce a rse n the upper bound of the demand uncertanty can be denoted as a rse n the mean-preservng varance of demand, t follows from (9.3) that a rse n the mean-preservng varance of demand ncreases frm s optmal debt level. The ntuton s as follows. A rse n the upper bound of the demand uncertanty enlarges the mean-preservng varance of demand and the range of the good states. Snce the leveraged frm cares only about good states, t wll ncrease the optmal debt level. Thus, we have: Proposton. A rse n the mean-preservng varance of demand ncreases the optmal debt levels of the two frms. 3. Fxed-fee Lcensng 0
Suppose that the outsder patentee lcenses the patent to frm only under exclusve fxed-fee lcensng. It follows that the producton costs of frm and frm change to C q q and C, respectvely, when frm accepts the lcensng. q Note that the game n queston turns nto a three-stage game, n whch a lcensng stage s added pror to the debt fnancng stage. Ths game can be solved by backward nducton. In stage 3, by substtutng the producton costs of the frms nto the equty value, we can solve for the optmal outputs as follows: O qf 4a 6 3 ˆ ˆ, (0.) 6 O qf 4a 3 ˆ ˆ, (0.) 6 where the superscrpt O denotes varables assocated wth the case where the patent s lcensed exclusvely; and the subscrpt F represents varables n the case of fxed-fee lcensng. In stage, by substtutng these producton costs of the frms nto the total value of the frm, we can solve for the optmal ẑ as follows: ˆ F 70a30 33, 493 (.) ˆ F 70a60 33. 493 (.) Eq. () shows that a rse n the nnovaton se reduces ˆ F whle ncreasng ˆO F. Ths result occurs because the lcensee becomes more effcent n
producton by acceptng the lcensng. Thus, the lcensee can earn a hgher proft, leadng to a lower break-even state of the demand uncertanty. On the contrary, the unlcensed frm wll ncur a loss and ths wll result n a hgher break-even state of the demand uncertanty. Next, by substtutng () and the cost functons C q q and C nto (3), (7), (8), and (0), we can obtan the equlbra under exclusve q fxed-fee lcensng as follows: 4 (.) 7 a 3 7 493, D O F 4 (.) 7 a 6 7 493, D O F 6 7a 3 7, (.3) 493 q O F 6 7a 6 7, (.4) 493 q O F 94 3 7a 3 7, (.5) 3 493 Y O F 94 3 7a 6 7, (.6) 3 493 Y O F F 38a3 55, 493 (.7) F 38a84 55. 493 (.8) By defnton, the nnovaton s drastc f the lcensee can drve the unlcensed frm out of the market and meanwhle charge the monopoly prce under exclusve fxed-fee lcensng. However, we fnd from (.4) that frm s output s always greater than ero due to ( < a). Thus, we can obtan the followng lemma:
Lemma. The nnovaton can never be a drastc nnovaton, regardless of the nnovaton se n the presence of debt fnancng. The result derved n Lemma s sharply dfferent from the exstng lterature, n whch there always exsts a suffcently large nnovaton se makng the nnovaton drastc. Ths result occurs because lmted lablty commts a leveraged frm to a more aggressve output stance, leadng to the result that the output of the unlcensed frm s always greater than ero. Lkewse, (.)-(.8) apply to the case where O,,. F Otherwse, the varable n (5) and (8) has to be replaced by. By (.7) and (.8), we defne: 7a 6 374, (.9) 3 7a 87. (.0) where s defned as the crtcal state for whch F, whle 3 s the crtcal O state for whch O. Thus, F f, whle O f. 3 O F F We fnd from (.) and (.) that a rse n the nnovaton se ncreases the optmal debt level of the lcensee, whle t decreases that of the unlcensed frm. The ntuton can be stated as follows. We have shown that a rse n the nnovaton se 3
lowers the lcensee s break-even state of the demand uncertanty, whle t ncreases that of the unlcensed frm. Hence, the good states faced by the lcensee become larger, whle those of the unlcensed frm become smaller. Snce the leveraged frms only care about the good states, t follows that the lcensee wll ncrease the optmal debt level, whle the unlcensed frm wll decrease the debt level. Next, a rse n the upper bound of the demand uncertanty,.e., a rse n the mean-preservng varance of demand, ncreases both frms optmal debt levels. The ntuton derved n the case of the absence of lcensng carres over to ths case. Thus, we can establsh: Proposton. Provded that the outsder patentee chooses exclusve fxed-fee lcensng, a rse n the nnovaton se wll ncrease the debt level of the lcensee, whle reducng that of the unlcensed frm. Moreover, a rse n the mean-preservng varance of demand ncreases the optmal debt levels of the two frms. In stage, when the outsder patentee chooses exclusve fxed-fee lcensng, the patentee s proft equals the maxmal fxed-fee that the lcensee would lke to pay for acceptng the lcense. Followng the defnton by Kamen and Tauman (986), the outsder patentee s proft can be defned as the dfference n the lcensee s proft between acceptng and not acceptng the lcense, whch s dervable by subtractng 4
(9.5) from (.5) as: 867a 73 a 59, 676 O F O F (3) 3 493 where F and F denote the outsder patentee s proft and the lcensee s fxed-fee under exclusve fxed-fee lcensng, respectvely. We move on to the case where the outsder patentee lcenses ts patent non-exclusvely under fxed-fee lcensng. The producton costs of both frms change to C q q,, when both frms accept the lcensng. By the same procedures, we can derve the equlbra under non-exclusve fxed-fee lcensng as follows: ˆ T F 0a 0 9,,, (4.) 9 4 D a,,, (4.) T F 9 6 q T F a,,, (4.3) 9 94 Y T 3 F a,,, (4.4) 3 9 T F 4a 4 5,,, 9 (4.5) 354 T 867 56 3 F a a, 3 493 (4.6) where the superscrpt T denotes varables assocated wth the case where the patent s lcensed non-exclusvely. Smlarly, (4.)-(4.6) apply to the case where T F,,. By (4.5), we defne: 5
a, 4 7 (4.7) where 4 s defned as the crtcal state for whch T F,,. Thus, T F f. 4 Eq. (4.) shows that both a rse n the nnovaton se and a rse n the mean-preservng varance of demand ncrease the optmal debt levels of the two frms. The same ntuton as that derved n the case of exclusve fxed-fee lcensng apples to ths result. Thus, we have: Proposton 3. Provded that the outsder patentee chooses non-exclusve fxed-fee lcensng, both a hgher nnovaton se and a larger mean-preservng varance of demand ncrease the optmal debt levels of the two frms. We are now n a poston to examne the optmal number of lcenses under fxed-fee lcensng. By subtractng (4.6) from (3), we obtan: T 676 F F 867 3 a 5a83 0, 493 7 3 005 a 66 f 5. 7 3 005 (5) By takng nto account the restrcton mposed n the case of exclusve fxed-fee lcensng,.e., 7a 6 374, we can fgure out that the nequalty 5 73 005a66 73 005 n (5) s 6
nvald. Thus, the outsder patentee wll lcense ts nnovaton non-exclusvely under fxed-fee lcensng, when the mean-preservng varance of demand s relatvely large, 5 7 3 005 66 7 3 005. say, a Intutvely, when the nnovaton se relatve to the mean-preservng varance of demand s large, the competton between frms can be mtgated dramatcally f the patent s lcensed exclusvely. Ths wll result n hgher extra proft beng earned by the lcensee va acceptng the lcensng. Thus, the outsder patentee can capture a larger proft by lcensng the patent exclusvely rather than non-exclusvely under fxed-fee lcensng. On the contrary, t wll lcense ts nnovaton non-exclusvely under fxed-fee lcensng, when the mean-preservng varance of demand s relatvely large. Based on the above analyss, we can establsh the followng proposton: Proposton 4. The outsder patentee wll lcense ts nnovaton non-exclusvely under fxed-fee lcensng, when the mean-preservng varance of demand s 5 7 3 005 66 7 3 005. relatvely large, say, a 4. Royalty Lcensng In ths secton, we explore the outsder patentee s optmal royalty rate under royalty 7
lcensng. The same result as that derved n Kamen and Tauman (986) and Kamen et al. (99), where the outsder patentee wll defntely lcense the patent non-exclusvely under royalty lcensng, can be obtaned n ths paper. 6 Thus, n what follows we shall study the case of non-exclusve royalty lcensng drectly. Snce both frms have the patent, ther producton cost functons can be expressed as C q q rq,,, where r s the royalty rate and r. By takng nto account the cost functons under royalty lcensng, we can derve the equlbra as follows: ˆ T R 0a 0 0r 9,,, (6.) 9 4 D a r,,, (6.) T R 9 6 q T R a r,,, (6.3) 9 where the subscrpt R denotes varables assocated wth the case of royalty lcensng. In stage, the outsder patentee s proft-maxmng problem s to choose an optmal royalty rate to maxme ts proft under the constrant r, whch can be expressed as: max r st.. r. r q q T T T R R Rj (7) 6 The soluton procedures are avalable from the authors upon request. 8
Substtutng (6.3) nto (7), and then dfferentatng (7) wth respect to r, we obtan: T R, r 9 a r (8) Recall that a >, r and 0. It follows that the sgn of (8) s defntely postve. Thus, the outsder patentee s proft s monotoncally ncreasng n the royalty rate, leadng to the result that the optmal royalty rate equals the nnovaton se by appealng to the constrant n (7) as follows: T r. (9) By substtutng (9) nto (6.), we fnd that the optmal debt levels of the frms reman unchanged under royalty lcensng when the lcensees accept the lcensng. Ths result emerges because the producton costs of the frms also reman unchanged. Next, the optmal debt levels of the frms are ncreasng n the mean-preservng varance of demand. The same ntuton carres over to ths case. Thus, we have the followng proposton: Proposton 5. Provded that the outsder patentee chooses royalty lcensng, the optmal debt levels of the frms reman unchanged, regardless of the nnovaton se, whle they are ncreasng n the mean-preservng varance of demand. 9
By substtutng (9) nto (7), we can fgure out the outsder patentee s proft under royalty lcensng as follows: T / 9. (0) R a Note that the lcensees producton costs by acceptng royalty lcensng are dentcal to those n the absence of lcensng. It follows that the restrcton, that (6.)-(6.3) apply to the case where T R,,, wll be the same as (9.6). 5. The Optmal Lcensng Contract We are now n a poston to explore the outsder patentee s optmal lcensng contract n terms of fxed-fee and royalty lcensng. Frst of all, we analye the case where the ero operatng proft state of the demand uncertanty s greater than the lower bound of the demand uncertanty n all lcensng regmes, j.e.,,,, j O, T, k N, F, R. k Recall that the outsder patentee wll lcense the patent non-exclusvely under fxed-fee lcensng. By subtractng (4.6) from (0), we can obtan the dfference n the outsder patentee s proft between royalty and non-exclusve fxed-fee lcensng as: where T R T F 9 a 493 0,f, 354 867 3 6 a 56 a 3 () 0
6 7535a 379 59078640a 807847078a35685065 83 3744 3744. Eq. () shows that the dfference n the outsder patentee s proft between royalty and non-exclusve fxed-fee lcensng s postve when the mean-preservng varance of demand s large, say,, 6 whle t s negatve otherwse. However, by takng nto account the restrcton n exclusve fxed-fee lcensng,.e., 7a 6 374, the nequalty 6 wll be ruled out. Thus, the optmal lcensng contract s royalty lcensng, when the mean-preservng varance of demand s large, say,. 6 The ntuton behnd the result can be stated as follows. When the mean-preservng varance of demand s large, the competton between frms n the commodty market s severe because the leveraged frms wll commt to an aggressve strategy by ncreasng the debt levels. Ths stuaton emerges because a larger mean-preservng varance of demand ncreases the debt levels of the frms n all lcensng regmes, and a hgher debt level ncreases the output. Moreover, we have proved that the optmal debt levels reman unchanged regardless of the nnovaton se under royalty lcensng so that the aggressve effect caused by the nnovaton se vanshes, whle they are ncreasng wth the nnovaton se under non-exclusve fxed-fee lcensng. It follows that non-exclusve fxed-fee lcensng wll cause the competton to be more ntense than under royalty lcensng. As a
result, n order to mtgate the competton between frms so that the outsder patentee can wrest a larger proft from the lcensees, the outsder patentee wll choose royalty lcensng. Next, as the above ntuton carres over to other cases where at least one of the ero operatng proft state of the demand uncertanty s no greater than the lower bound of the demand uncertanty,.e.,,, the detaled dervatons of the optmal lcensng contracts n these cases are contaned n Appendx A for savng on space. We llustrate the outsder patentee s optmal lcensng contracts by usng Fgure and Table. The optmal lcensng contracts derved n areas I-V of Fgure can be summared n Table. We can conclude from Table that the optmal lcensng s royalty lcensng when the mean-preservng varance of demand s large, whle t s non-exclusve fxed-fee lcensng otherwse. Thus, we can establsh: (Insert Fgure here) (Insert Table here) Proposton 6. Suppose that the leveraged frms manufacture a homogeneous product and engage n Cournot competton. The optmal lcensng contract for the outsder patentee n terms of fxed-fee and royalty lcensng s royalty lcensng when the mean-preservng varance of demand s large n the presence of debt fnancng,
whle t s non-exclusve fxed-fee lcensng otherwse. The result derved n Proposton 6 s sharply dfferent from that of Kamen and Tauman (986) as well as Kamen et al. (99), n whch the optmal lcensng contract of the outsder patentee s always non-exclusve fxed-fee lcensng when frms produce homogeneous products and engage n Cournot competton. The dfference emerges because the leveraged frms wll become more aggressve n terms of ther output strategy n the presence of debt fnancng so that the outsder patentee may choose royalty lcensng to reduce the competton between frms. 6. Concludng Remarks Ths paper has developed a lcensng model to reconcle wth the controversy of royalty lcensng beng superor over fxed-fee lcensng n theory and n practce wthn Kamen and Tauman s (986) theoretcal framework by takng nto account the fnancal structure of the frm. The focus of the paper s on the lmted lablty effect of debt fnancng and the strategc effect of the lcensng decson on output strateges. Several strkng results are derved as follows. Frst of all, provded that the leveraged frms produce a homogeneous product and engage n Cournot competton, ths paper shows that the optmal lcensng 3
contract for the outsder patentee n terms of fxed-fee and royalty lcensng s royalty lcensng when the mean-preservng varance of demand s large n the presence of debt fnancng, whle t s non-exclusve fxed-fee lcensng otherwse. Ths paper thus provdes a new explanaton to justfy the superorty of royalty lcensng over the fxed-fee lcensng contract. Secondly, the larger mean-preservng varance of demand always results n larger debt levels, regardless of the lcensng means. Thrdly, an ncrease n the nnovaton se ncreases the debt level of the lcensee under fxed-fee lcensng, whle t lowers that of the unlcensed frm. However, the debt levels reman unchanged under royalty lcensng. Fourthly, the outsder patentee wll lcense ts nnovaton non-exclusvely under fxed-fee lcensng, when the mean-preservng varance of demand s relatvely large. Lastly, the nnovaton can never be drastc, regardless of the nnovaton se n the presence of debt fnancng. 4
Appendx A. The Optmal Lcensng Contracts n Areas of Fgure In Fgure, area I denotes the stuaton where the ero operatng proft state s larger than the lower bound of the demand uncertanty n all lcensng regmes,.e.,,,, j O, T, k N, F, R, whch s restrcted by. j k The optmal lcensng contract for ths area s royalty lcensng, whch s determned by (). Area II s for whch s restrcted by 4. Area III s for F, T, and, and s restrcted by. 4 Area IV s for F F T F, F, and N, and s restrcted by 3, and area V s for T F, F, F, N, and ˆ T F, and s restrcted by 7 3. The crtcal value of the state 7 wll be defned later n ths secton. The remanng area s for and s restrcted by 0 7, where the frms can not survve n ths T ẑ F, stuaton, because they ncur a loss n every state of the demand uncertanty. In what follows, we examne the optmal lcensng contract n areas II-V, respectvely. For area II, snce and the ero operatng proft state can not be smaller F than, the endogenous varable F n (.7) s thus replaced by By F. substtutng nto the model and then redong the calculaton, we can obtan F the dfference n the outsder patentee s proft between royalty and non-exclusve fxed-fee lcensng as follows: 5
T R T F 94 3 5 3 8 3 a a a 9 4389 33 83. (A.) We fnd from (A.) that ths proft dfference s postve (negatve) f 8. 7 Thus, the optmal lcensng contract for area II s royalty lcensng f the mean-preservng varance of demand s large, say,, 8 whle t changes to non-exclusve fxed-fee lcensng otherwse. The ntuton s the same as that n (). When the mean-preservng varance of demand s large, and the competton between frms n the commodty market s severe, the outsder patentee wll choose royalty lcensng to mtgate the competton. On the contrary, when the mean-preservng varance of demand s small, the competton between frms s not so ntense, wth the result that the outsder patentee wll select non-exclusve fxed-fee lcensng. For area III, snce the endogenous varable T F, and F, F n (.7) and T F n (4.5) are replaced by By substtutng T F F. nto the model and then redong the calculaton, we can obtan: T F F T R T F a 7 3 a 5a 3 8. 3 (A.) 9 83 Eq. (A.) shows that ths proft dfference s postve (negatve) f 9. 8 7 The exposton of 8 s too tedous. We do not present t n the paper, but t s avalable from the authors upon request. 8 The exposton of 9 s too tedous. We do not present t n the paper, but t s avalable from the authors upon request. 6
Thus, the optmal lcensng contract for area III s royalty lcensng f the mean-preservng varance of demand s large, say,, 9 whle t changes to non-exclusve fxed-fee lcensng otherwse. T For area IV, snce,, and, the endogenous varables F F N n (.7), F T F n (4.5), and T N n (9.4) are replaced by F F N. By substtutng T nto the model and then redong the calculaton, F F N we can obtan: T R T F 6 a 7 3 a 5a 3 8. 3 (A.3) 83 Smlarly, (A.3) shows us that ths proft dfference s postve (negatve) f 0. 9 However, the nequalty 0 s ruled out by the restrcton N,.e., 7a. Thus, the optmal lcensng contract for area IV s non-exclusve fxed-fee lcensng. For area V, snce T,,, and, the endogenous F F N F varables n (.7), F n (4.5), n (9.4), and T F N F n (.8) are replaced T T by F F N F. By substtutng F F N F nto the T model and then redong the calculaton, we fnd that the restrcton ˆ, holds f a. Moreover, we derve: 7 9 The exposton of 0 s too tedous. We do not present t n the paper, but t s avalable from the authors upon request. 7
T T 30a 336 R F 0. (A.4) 305 Eq. (A.4) shows that the optmal lcensng contract for area V s non-exclusve fxed-fee lcensng. It should be noted that the crtcal value of the state 7 5. By referrng to ths relatonshp, we fnd from (5) and Fgure that the outsder patentee wll lcense ts nnovaton non-exclusvely under fxed-fee lcensng n areas I-V. Thus, t s not a problem for us to gnore the outsder patentee s proft n the case of exclusve fxed-fee lcensng n areas I-V whle comparng the dfference n proft between royalty and non-exclusve fxed-fee lcensng. 8
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a V V 7 3 4 9 8 0 Fgure. The Optmal Lcensng Contracts Table. Summary of Optmal Lcensng Contracts n Areas I-V of Fgure Area Restrcton on Restrcton on The optmal lcensng contract I none Royalty II 4 Royalty f 8 F Non-exclusve fxed-fee f 4 8 III T 4 Royalty f 9 4 F F Non-exclusve fxed-fee f 9 IV T 3 Non-exclusve fxed-fee F F N V T F F N F 7 3 Non-exclusve fxed-fee 3