Negative Royalty in Duopoly and Definition of License Fee: General Demand and Cost Functions

Transcription

1 Intrnational Journal of usinss Economics 018 Vol 17 No Ngativ Royalty in Duopoly Dfinition of Licns F: Gnral Dm Cost Functions Masahiko Hattori Faculty of Economics Doshisha Univrsity Japan Yasuhito Tanaka Faculty of Economics Doshisha Univrsity Japan bstract W xtnd th analysis about ngativ royalty in a duopoly with an outsid innovator undr linar dm cost functions by Liao Sn (005) to a situation with gnral dm cost functions Morovr w considr a cas whr an innovator has an option to ntr th markt show that th optimal royalty rat for th innovator whn it dos not hav an option to ntr th markt is smallr than that whn it can ntr th markt Th sign of th optimal royalty rat dpnds on whthr firms goods ar stratgic substituts or stratgic complmnts W provid a concis xampl of wlfar analysis which suggsts that th prohibition of ntry of th innovator into th markt may b an appropriat policy 1 Introduction Liao Sn (005) analyz th problm of licnsing by a combination of a royalty pr output a fixd f undr an oligopoly with an outsid or an incumbnt innovator showing that whn thr ar on licns on non-licns th innovator imposs a ngativ royalty (thy call it a subsidy) with a positiv fixd f on th licns but thy assum linar dm cost functions W xtnd th analysis to a situation with gnral dm cost functions This is similar to Kamin Tauman (1986) who dfin th total licns f in th outsid innovator cas by th diffrnc btwn th profit of th licns whn it buys th licns its profit whn it dos not buy th licns that th othr incumbnt firm (non-licns) buys th licns Howvr if th outsid innovator has an option to sll a licns to th othr incumbnt firm at th sam tim ntr th markt whn th potntial licns rfuss to buy th licns thn w shall us a diffrnt dfinition of licns f Liao Sn (005) considr two licns schms (1) FR: Upfront f plus royalty Corrspondnc to: Faculty of Economics Doshisha Univrsity Kamigyo-ku Kyoto Japan yasuhito@maildoshishaacjp FX: W ar gratful to th rfrs for thir dtaild rviw of our papr insightful commnts constructiv suggstions

2 164 Intrnational Journal of usinss Economics () R: uction plus royalty In th FR policy Liao Sn (005) stat that th willingnss to pay for th licns is th diffrnc btwn its profit as a licns th profit whn no firm buys th licns This is smallr than th willingnss to pay in th R policy 1 Thus for th innovator th R policy yilds a highr payoff than th FR policy whn on firm buys th licns considring R policy is sufficint In this papr w considr two scnarios about th licns f Scnario 1: If th potntial licns rfuss th paymnt of a licns f th othr incumbnt firm buys th licns th innovator dos not ntr th markt thn th willingnss to pay for th licns is th diffrnc btwn its profit as a licns th profit of a non-licns This is th dfinition in Liao Sn (005) Scnario : If th potntial licns rfuss th paymnt of a licns f th othr incumbnt firm buys th licns at th sam tim th innovator ntrs th markt thn it may involv mor svr punishmnt than slling th licns to th othr firm without ntring th markt Th willingnss to pay for th licns is th diffrnc btwn its profit as a licns its profit whn th innovator ntrs th markt with a licns to th othr incumbnt firm W considr a modl of an oligopoly in which firms produc substitutabl goods undr gnral dm cost functions that th innovator has a nw cost-rducing tchnology that can b commonly usd by all firms W prsnt th following rsults 1 Th optimal royalty rat in Scnario 1 is smallr than that in Scnario If th goods of th firms ar stratgic substituts thn th optimal royalty rats in Scnario 1 Scnario ar ngativ 3 If th goods of th firms ar stratgic complmnts thn th optimal royalty rat in Scnario 1 may b positiv or ngativ but th on in Scnario is positiv 4 Whn th non-licns ops out of th markt at th optimal royalty rat Scnario 1 Scnario ar quivalnt In th rlatd litratur no study has vr considrd a ngativ royalty Katz Shapiro (1985) rprsnt on of th arlist paprs in th fild us gnral dm cost functions but only considr a fixd f Subsquntly Kamin t al (199) compar th pur auction fixd f royalty with gnral dm function linar cost functions Thir rsults show that a royalty is infrior to th othr two an auction is suprior to th othr two whn th magnitud of innovation is not too small Morovr Wang (1998) not that a pur royalty may b suprior to a fixd f in a modl of on incumbnt innovator on licns Similarly Sn (005) stat that a pur royalty may b suprior to th othr two licnsing schms for an outsid innovator Wang (1998) Sn (005) us linar dm cost functions whil Sn Tauman (007) discuss th optimal combinations of an auction royalty Howvr thir papr mploys linar dm cost functions but do not considr th thrat of ntry by th licnsor Fosfuri Roca (004) compar th royalty th fixd f Using a modl with linar dm cost functions thy show that although a royalty is optimal for

3 Masahiko Hattori Yasuhito Tanaka 165 th incumbnt innovator whn it slls licnss to all othr incumbnt firms a fixd f is optimal whn only som of th othr incumbnt firms buy licnss This prsnt rsarch considrs only th outsid innovator cas In th incumbnt innovator cas thr is no problm of a dfinition of licns f bcaus in that cas th innovator dos not hav an option whthr its ntrs a markt or not W also analyz only a problm of a possibility of a ngativ royalty with on licns on non-licns For an outsid innovator with two potntial licnss whthr it slls a licns to on firm or slls licnss to two firms is an important problm Howvr analyss of th innovator s optimal choic as wll as analyss of social wlfar public policy about licnsing may b complicatd undr gnral dm cost functions W will study such problms in futur rsarch Th rst of th papr runs as follows Sctoin dscribs th papr s modl Sction 3 prsnts th main rsults summarizd in thr propositions Sction 4 considrs a simpl xampl with linar dm cost functions offrs a brif analysis about social wlfar W will show that in such an xampl th social wlfar in Scnario 1 is largr than th social wlfar in Scnario Thus prohibition of ntry of th innovator into th markt may b an appropriat policy Sction 5 provids concluding rmarks Th Modl Thr ar thr firms: on outsid innovator two incumbnt firms Th innovator has a suprior cost rducing tchnology that can b commonly usd by all firms licnss its tchnology to th licns nothr incumbnt firm is th non-licns Th licns is calld Firm th non-licns is calld Firm th innovator is calld Firm I Firm I is an outsid innovator at prsnt but it may ntr th markt if Firm rfuss to buy th licns Thrfor w considr th possibility of ntry by Firm I so as to dtrmin th licns f imposd on Firm Firm I dos not rally ntr th markt Dnot th outputs of Firms by x x Th output of Firm I whn it ntrs th markt is x I Th firms produc substitutabl goods Th prics of th goods of Firms I ar p p p I th invrs dm functions ar writtn as p x x p x x or p x x x I I p x x x I p x x x I W assum that all partial drivativs of th invrs dm functions ar ngativ ar twic diffrntiabl Th cost function of Firm with a licns of th nw tchnology is c x Th cost function of Firm is c x th cost function of Firm I whn it ntrs th markt is c x I Thy ar incrasing twic diffrntiabl W assum c x c x c' x c' x for x x Firm I imposs a royalty pr output a fixd f on Firm W dnot th royalty rat by r W calculat th profit of Firm nt of th royalty th profit of Firm as: p x c x rx

4 166 Intrnational Journal of usinss Economics p x c x To dtrmin th total licns f w considr th two following scnarios Scnario 1: Th innovator dos not ntr th markt ccording to th auction policy by th innovator in Liao Sn (005) if Firm rfuss th paymnt of a licns f thn Firm buys th licns th willingnss to pay for Firm is th diffrnc btwn its profit as a licns th profit of a non-licns - that is Lt L b th fixd licns f thus w hav: L Th payoff of th innovator is th sum of th royalty th fixd licns f dnotd by: L rx p x c x p x c x 1 Scnario : Th innovator has an option to ntr th markt whn Firm rfuss to buy a licns Firm thn buys th licns at th sam tim Firm I ntrs th markt Firm must pay th diffrnc btwn its profit as a licns whn Firm I dos not ntr its profit whn Firm I ntrs th markt with a licns to Firm Th fixd licns f L is: L Hr dnots th profit of Firm whn Firm I ntrs th markt Firm buys th licns Firm is thn a non-licns Th ppndix includs a concis dscription of th quilibrium in that cas W dnot th payoff of th innovator in this cas by: L rx p x c x Not that is a constant - that is it dos not dpnd on th royalty rat whn Firm I dos not ntr bcaus th optimal royalty rat whn Firm I ntrs th markt is diffrnt from ( indpndnt of) that whn it dos not ntr On th othr h in 1 is not constant It dpnds on th optimal royalty rat without ntry of Firm I Svrity Crdibility of Punishmnt If thn th ntry of Firm I with a licns to Firm is mor svr punishmnt for Firm than a licns to Firm without ntry of Firm I whn it dos

5 Masahiko Hattori Yasuhito Tanaka 167 not buy th licns Hr is th profit of a non-licns in a duopoly On th othr h is th profit of a non-licns in an oligopoly with thr firms in which two firms Firm I us th nw tchnology Thus w can s that is usually smallr than Whn w hav 1 Scnario is mor profitabl than Scnario 1 for Firm I Howvr its ntry is not ncssarily a crdibl thrat Th payoff of Firm I whn it ntrs th markt with a licns to Firm is th sum of its profit as a firm in an oligopoly th licns f imposd on Firm (not Firm ) If it is largr than 1 thn ntry into th markt is a crdibl thrat W assum that th output of Firm x is positiv whn th royalty rat is zro Howvr if th innovator imposs a ngativ royalty on Firm thn x may b zro W considr two cass about dm functions On cas is whn th goods ar stratgic substituts th othr cas is whn th goods ar stratgic complmnts W furthr considr two cass about th innovation Th first is whn th non-licns continus to oprat aftr a ngativ royalty is imposd on th licns th scond is whn th non-licns ops out of th markt with th optimal royalty rat In th formr cas th innovation is non-astic in th lattr cas it is astic W shall show that in th cas whr th non-licns ops out of th markt Scnario 1 Scnario ar quivalnt 3 Th Rsults 31 Firm havior Th first-ordr conditions for profit maximization of Firm Firm ar: p p x c' x r 0 (1) x p p x c' x 0 () x Thir scond-ordr conditions ar: x p p x c '' x 0 x x p p x c '' x 0 x

6 168 Intrnational Journal of usinss Economics Diffrntiating (1) () with rspct to r yilds: p p p p x '' 1 c x x x x x xx p p p p x x '' 0 c x x xx x x From thm w obtain: p p '' x c x x x Δ p p x x xx Δ whr p p p p Δ x '' '' c x x c x x x x x p p p p x x x xx x xx W assum that: x c x x x p p p p '' x x x x (4) x c x x x p p p p '' x x x x (5) Thy imply that:

7 Masahiko Hattori Yasuhito Tanaka 169 Δ 0 W driv ths assumptions from th stability conditions for a duopoly (s Sad (1980) Dixit (1986)) Equations (4) (5) man that th absolut valus of th slops of th firms raction curvs ar smallr than on Thrfor w gt: 0 Th goods of th firms ar stratgic substituts whn p x +( p xx ) x 0 stratgic complmnts whn p x + ( p x x ) x 0 Thus w stat th following 1 Whn th goods of th firms ar stratgic substituts 0 Whn th goods of th firms ar stratgic complmnts 0 3 Comparison of Two Scnarios Suppos that Firm dos not op out of th markt Scnario 1: Th innovator dos not ntr th markt Th condition for maximization of 1 with rspct to r is: d1 p p p x c' x x x x p p p x c' x x x x p p r x x 0 x x (6) W gt th optimal royalty rat for th innovator as follows 1 p p r1 x x x x (7) Scnario : Th innovator ntrs th markt whn Firm rfuss to buy a licns

8 170 Intrnational Journal of usinss Economics Th condition for maximization of with rspct to r is: d p p p x c' x x x x p r x 0 x (8) W thn gt th optimal royalty rat for th innovator as follows p r x 1 x (9) r Suppos that 1 r (8) is satisfid Hr ( p x ) x ( ) in (6) thos in (8) ar qual Thus w hav: d p x 1 x 0 r r Thrfor 1 W hav shown th following proposition Proposition 1 Th optimal royalty rat in Scnario 1 is smallr than that in Scnario If both optimal royalty rats ar ngativ thn th absolut valu of th optimal royalty in Scnario 1 is largr than that in Scnario From (7) (9) w can show th following rsult Proposition Suppos that Firm dos not op out of th markt 1 If th goods of th firms ar stratgic substituts thn th optimal royalty rats in Scnario 1 Scnario ar ngativ If th goods of th firms ar stratgic complmnts thn th optimal royalty rat in Scnario 1 may b positiv or ngativ but that in Scnario is positiv Proof 1 If th goods of th firms ar stratgic substituts w hav 0 thn r1 0 bcaus 0 W also hav r 0

9 Masahiko Hattori Yasuhito Tanaka 171 If th goods of th firms ar stratgic complmnts thn 0 w p p hav r1 0 or r1 0 dpnding on x x 0 x x p p x x x x If 0 x is sufficintly smallr than p p it is likly that x x 0 x x On th othr h 0 mans r 0 x thn vn whn Firm dos not op out r1 0 (QED) or Cas whr Firm Drops Out Suppos that at som royalty rat Firm ops out of th markt In Scnario 1 w thn hav: d1 p r x x 0 x From (8) also in Scnario w gt th sam rlation Thus whn Firm ops out of th markt Scnario 1 Scnario ar quivalnt w obtain th following rsults Proposition 3 In th cas whr Firm ops out of th markt w obtain th following rsults 1 If th goods of th firms ar stratgic substituts thn th optimal royalty rat is ngativ If th goods of th firms ar stratgic complmnts thn th optimal royalty rat is positiv Proof 1 If d p r x 0 x x 0 thn x 0 at th optimal stat for th innovator w hav th cas in th d prvious proposition whr is 1 or On th othr h if 0 whn x 0 thn th licns is a monopolist th optimal royalty rat for th innovator is on such that x 0 This is ngativ bcaus x 0 with zro

10 17 Intrnational Journal of usinss Economics royalty 0 d at x 0 thn x 0 at th optimal stat for th innovator If 0 d at w hav th cas in th prvious proposition On th othr h if 0 x 0 thn th licns is a monopolist th optimal royalty rat for th innovator is on such that 0 royalty 0 (QED) x This is positiv bcaus x 0 with zro 4 n Exampl Policy Implications Now lt us considr an xampl ssum that th goods of th firms ar homognous Th invrs dm function is: p a x x Th cost functions of Firm Firm aftr adoption of th nw tchnology by Firm ar rspctivly c x cx Hr c ar positiv constants such that c w assum ac Nxt w hav: x From this w gt: a c r a c r x ar: Th optimal royalty rat for th innovator in Scnario 1 that in Scnario a c r1 0 ac r 0 4

11 Comparing thm w hav: Masahiko Hattori Yasuhito Tanaka 173 ac r1 r 0 (10) 4 With r r1 th quilibrium valus of th outputs ar: x a c a x c 3 6 r r With th quilibrium valus of th outputs ar: x a c a x c 4 r r Whn 1 th total output is: 5a5c X1 x x 6 r r Whn th total output is: 3a3c X x x 4 Comparing thm w find: ac X1 X 0 1 Thus th total output in Scnario 1 is largr than that in Scnario bcaus th optimal royalty rat in Scnario 1 is smallr than that in Scnario as shown in (10) Th wlfar which is th sum of th consumrs surplus th total profits of th firms including th innovator in Scnario 1 is: 65a 130ac 65c 14a 14c 68 W1 7 Th wlfar in Scnario is: 1a 4ac 1c 60a 60c 5 W 3 Comparing thm w find:

12 174 Intrnational Journal of usinss Economics a c 71a 71c 98 W1 W 0 88 This mans that in this xampl th wlfar in Scnario is smallr than th wlfar in Scnario 1 Thus for xampl th prohibition of ntry of th innovator into markt may b an appropriat policy 5 Concluding Rmarks This papr has xtndd an analysis of a ngativ royalty in a duopoly undr linar dm cost functions by Liao Sn (005) to a situation with gnral dm cost functions W furthr hav considrd two cass for th dtrmination of licns fs: Scnario 1) th innovator dos not hav an option to ntr th markt; Scnario ) th innovator has an option to ntr th markt whn th licns rfuss to buy a licns W hav shown that th optimal royalty rat in Scnario 1 is smallr than that in Scnario that th sign of th optimal royalty rat dpnds on whthr th goods ar stratgic substituts or complmnts Th stratgy of ntry thrat by th licnsor usd in Scnario ) rducs th licnsor's profit is socially undsirabl In futur rsarch w want to analyz th optimal stratgy of th innovator - for xampl whthr it slls licnss to on firm or two firms in an oligopoly with two incumbnt firms W also want to xamin social wlfar public policy about licnsing undr gnral dm cost functions Nots 1 Whn two firms buy th licnss th FR policy th R policy coincid Sn Stamatopoulos (016) prsnt an analysis of a royalty fixd f in a duopoly undr gnral dm cost functions ppndix: Main Points of th Cas whr th Innovator Entrs th Markt with a Licns to Firm Suppos that Firm rfuss to buy th licns Firm I ntrs th markt with a licns to Firm Firm thn uss th nw tchnology Firm uss th old tchnology Dnot th output th profit of th innovator by x I I Th firms profits ar: I I I I p x c x p x c x rx

13 p x c x Masahiko Hattori Yasuhito Tanaka 175 In this cas Firm pays th licns f Th first-ordr conditions for profit maximization of Firms I ar: pi pi xi c' xi 0 (10) x I p p x c' x r 0 (11) x p p x c' x 0 (1) x Th scond-ordr conditions ar: I xi pi p x I c '' x I 0 x I x p p x c '' x 0 x x p p x c '' x 0 x Diffrntiating (10) (11) (1) with rspct to r yilds: I I I I 0 I I 1 I I 0 From thm w obtain: I I I Γ Γ I I I

14 176 Intrnational Journal of usinss Economics whr I I I Γ Γ I I I I I I I I I I pi pi I '' x I c x I xi xi p p '' x c x x x p p '' x c x x x I I I I p x pi pi x I x xi x pi pi x I x xi x p p x xi xi x p x p x x x p p x xi xi x p x x Firm must pay th following total licns f rx x If Firm rfuss to buy th licns thn Firm in turn buys th licns Th fixd f is Th payoff of Firm I is: I rx pi xi c xi px c x px c x

15 Th condition for maximization of Masahiko Hattori Yasuhito Tanaka 177 with rspct to r is: d pi p p I pi xi c ' xi x x? xi xi xi p pi p p x c ' x xi x? x x x p p p p x c ' x xi x x x x I p p p p x x x x x x I r I xi x I I pi p xi x 0 x x W xprss th optimal royalty rat for th innovator as: 1 p p I pi p r x x xi x xi xi x x pi p xi x x x W dnot th profit of Firm in this cas by Rfrncs Dixit (1986) Comparativ Statics for Oligopoly Intrnational Economic Rviw 7(1) Fosfuri E Roca (004) Optimal Licnsing Stratgy: Royalty or Fixd F? Intrnatioal Journal of usinss Economics 3(1) Kamin M I S S Orn Y Tauman (199) Optimal Licnsing of Cost-rducing Innovation Journal of Mathmatical Economics 1(5) Kamin T Y Tauman (1986) Fs vrsus Royaltis th Privat Valu of a Patnt Quartrly Journal of Economics 101(3) Katz M L C Shapiro (1985) On th Licnsing of Innovations R Journal of Economics 16(4) Liao C D Sn (005) Subsidy in Licnsing: Optimality Wlfar Implications Th Manchstr School 73(3) Sad J K (1980) Th Stability of Cournot Rvisitd Journal of Economic

16 178 Intrnational Journal of usinss Economics Thory 3(1) 15-7 Sn D (005) F Vrsus Royalty Rconsidrd Gams Economic havior Sn D G Stamatopoulos (016) Licnsing undr Gnral Dm Cost Functions Europan Journal of Oprations Rsarch 53(3) Sn D Y Tauman (007) Gnral Licnsing Schms for a Cost-rducing Innovation Gams Economic havior 59(1) Wang X H (1998) F Vrsus Royalty Licnsing in a Cournot Duopoly Modl Economics Lttrs 60(1) 55-6