Cournot And Bertrand Prices In A Model Of Differentiated Duopoly With R&D

Transcription

1 ournot And ertr Pres In A Model Of fferentated uooly Wth R& y Tarun Kara Indan Statstal Insttute, Kolkata, Inda And Soa Roy u u Motheel ollege, Kolkata, Inda July 00 Astrat We onstrut a odel of dfferentated duooly wth roess R& when goods are susttutes. In the frst stage frs dede ther tehnologes.e., the average osts of roduton n the seond stage they oete n uanttes or res. We have shown that not only the ournot frs nvest a larger aount on R& than the ertr frs, ut, ontrary to the result n the lterature, ournot re an e less than ertr re. Ths ours when the R& tehnology s relatvely neffent. Key words: Proess R&, ertr re, ournot re. JEL lassfaton: L3, 3. orresondene to: Tarun Kara, Eono Researh Unt, Indan Statstal Insttute, 03.T. Road, Kolkata Eal: tarunkara@hotal.o; Fax:

2 . Introduton In a stat olgooly odel frs have generally two alternatve strateges, uantty re. These gve ournot odel ertr odel, resetvely. Sne the ertr odel for hoogeneous good aears to have less redtve ower, eonosts reasonaly work wth dfferentated roduts, sefally wth erfetly susttute goods. Then a logal extenson of ths researh s to rovde an analyss ased on oarng the results of these two odels. A lass ontruton n ths ontext s the work y Sngh Vves 98. heng 985 rovdes a geoetral reresentaton of the results. Vves 985 has extended the analyss to the ase of general de ost strutures. The ortant results of Sngh Vves are the followng. Frst, eulru res are lower oututs are hgher under ertr oetton oared to ournot oetton. Seond, ertr eulru s ore effent than ournot eulru n the sense that oth onsuer surlus total surlus are greater under re oetton. Fnally, ournot rofts are larger than ertr rofts gven that the goods are susttutes. Sne then these results have een fang stee hallenges. In artular, eonosts have exaned the ueston of effeny n a dyna setu y ntrodung R& nteraton of frs ror to the roduton stage. The dea s that the oetng frs an, through a rodut or roess R&, alter the de ost strutures, therefore the outoes of the roduton stage s also affeted. Sne ournot rofts are hgher than ertr rofts, ournot frs wll have a stronger ost redung R& nentve. Hene there an e stuatons when ournot eulru s ore effent than ertr eulru n the sense that soal welfare under uantty oetton wll e hgher oared to re oetton. Let us refly outlne the lterature elow. elono enolo 990 have a erfetly susttute good duooly odel, wth R& n the for of a atent rae. The odel shows that the R& nentve are larger under re oetton than that under uantty oetton, ut the general welfare oarson s aguous. Motta 993 has a vertally dfferentated duooly wth ualty rovng R&. It shows that to relax fere arket oetton frs dfferentate ore under ertr than under ournot, that re oetton leads to hgher onsuer as well as soal surlus. On the other h, Qu 997 rovdes an extended odel of Sngh Vves 98 wth ntrodung a roess R& oetton. It also allows for R& sllovers. It s shown that ournot frs nvest ore on R& than the ertr frs that ournot eulru eoes ore effent f R& rodutvty s hgh, sllovers are large In fat, ths result holds also for the ase of oleentary goods. Arrow 96 has a oneerng work to the ueston of R& nentves. He shows that a oettve fr has a larger R& nentve than a onooly fr. ester Petraks 993 have studed ost redung nentve n a dfferentated duooly have shown that ournot oetton rovdes a stronger nentve to nnovate than ertr oetton f the degree of susttuton s low.

3 roduts are susttutes. The tradtonal effeny results follow when sllovers are sall, the degree of susttuton s low the R& ost s large. 3 Surrsngly enough, n all the works ontruted so far n the lterature, ournot res are unaguously hgher than ertr res. Thus, the frst result of Sngh Vves reans asolutely unhallenged. Even n Hakner 000 that shows the osslty of a hgher ertr re vsàvs the ournot re n a fraework of ore than two frs, ut that osslty arses only when the ualty dfferenes are large goods are oleentary; for susttutes the tradtonal result holds. Syeonds 00 has larger onsuer surlus, ut ths s due to ualty roveent; ournot re reans unaguously hgher than ertr re. Ths s our frst odest attet to show that ertr re an e hgher than ournot re. Ths osslty arses when the R& tehnology s relatvely neffent or less rodutve. Our odel s lose to the Qu 997 struture wth zero sllovers of R&. However, n one reset there s an ortant dfferene. In Qu 997, to start wth frs have dental tehnologes as gven y the sae argnal osts of roduton. Then n the frst stage frs do R& to redue ost of roduton. Thus the R& nvestent n Qu s deendent on the ntal argnal ost the hgher the ntal argnal ost, the lower s the nentve for R&. In ontrary, we assue that n the egnnng frs dede ther R& nvestent to oe u wth an arorate nnovaton, then they oete n the rodut arket ether n res or n uanttes. 5 Ths struture aears to us ore logal. efore we go to the next seton let us refly exlan the ntuton of the result. Frst, ournot frs have hgher nentves to nvest on roess R& eause of the roset of a hgher roft. Hene ournot frs wll oe u wth a lower argnal ost oared to ertr frs. Seondly, hgher s the rodutvty of the R& tehnology, hgher wll e the nnovaton nentve for a larger nnovaton. So, the tehnologal dfferene etween ournot ertr frs wll e less f the R& rodutvty s large. Then for an neffent R& tehnology argnal ost dfferenes wll e wder. In fat as R& neffeny grows, ertr frs R& nentves fall at a greater rate oared to ournot frs. Fnally, the ertr re s ore senstve to the hange of the argnal ost. Therefore, as argnal ost nreases, the ertr re nreases ore than the ournot re. All these fators together ly that the ournot re an e lower than the ertr re when the R& tehnology s less rodutve. The lan of the aer s the followng. We rovde the odel results of the aer n Seton. It ontans a nuer of susetons. The frst suseton resents Sngh Vves results. The seond suseton s the Sngh Vves odel wth a ror roess R& stage. Then two sususetons dsuss the tehnology hoes under ournot gae ertr gae resetvely. Then n the thrd suseton we oare argnal osts of the two gaes. Fnally, n the fourth suseton ournot ertr res are 3 Syeonds 00 n an otherwse Qu 997 odel allows R& oth n roess nnovaton ualty roveent. The dstntve feature of ths odel s that although res under ournot oetton are always hgher, onsuer surlus an e hgher too due to ualty roveent. Even wth zero sllovers, ournot re s hgher than ertr re n Qu For tehnology hoe n a onetural varaton duooly one ay look at Ma Hwang

4 oared. The last seton s a onluson. Algera detals are relegated n the Aendx.. Model. Sngh Vves 98 Results We onsder an otherwse Sngh Vves odel wth an ntroduton of R & nteraton y the frs n the reroduton stage. In the roduton stage frs oete ether n uanttes or n res, n the R & stage they dede ther otal roduton tehnologes reresented y the onstant unt osts of roduton. Let the nverse lnear arket de of the th fr e,,,,, where s the aount of the goods rodued y fr s the re of the th fr's rodut. We assue that the goods are susttutes. The rato atures the degree of susttuton 0 eans goods are ndeendent eans goods are erfet susttutes. The natural restrtons on the de araeters are 0 0 < <. enotng a, e wrtten as, the dret de funtons an a,,,,. Now, assung onstant returns to sale tehnology the syetr ost funton s gven y, 3 where s the onstant argnal average ost of roduton; <. Then gven, 3, we an edately solve for syetr ournot ertr res as gven y: orresondngly, rofts are resetvely,

5 5 Then Sngh Vves 98 results easly follow eause 6 that s, for the gven argnal ost ournot re ournot rofts are larger oared to ertr ase. We an further see that d d d d 7 Ths states that ournot frs have larger nentves to redue the argnal ost of roduton than ertr frs. The laton of 6 7 s the followng. Sne ournot frs has larger nentves to redue ost oared to ertr frs, ournot frs wll nvest ore on R & wll oe u wth a lower argnal ost than the ertr frs. When the ost dfferene wll e suffently large, ournot re an e lower than ertr re. The followng result learly shows the dfferental at of the hange of the argnal ost on the resetve ournot ertr re. Lea If argnal ost goes u falls, the ertr re wll nrease derease d d ore than ournot re.e., <. d d. Sngh Vves Model wth R & We onsder a twostage nonooeratve gae. In the frst stage frs ndeendently dede ther R & nvestent R that deternes roduton tehnology that s, argnal osts of roduton,. Then n the seond stage frs sultaneously nonooeratvely dede ther uanttes or res deendng on whether there s ournot oetton or ertr oetton. We solve the gae y the akward nduton ethod, that s, we frst solve the seond stage role then the frst stage role notng the ehavor of the seond stage. Qute naturally, n the ontext of the nd stage, R & ost s sunk, the argnal osts deterned n the frst stage are treated as onstant. Throughout the aer we assue syetr eulru. Let the R & tehnology of a fr e 5

6 R 8 where 0 s a araeter hgher the value of, hgher s the R & ost or lower effeny. The funton learly reveals dnshng returns tehnology. 6.. Tehnology hoe under ournot Gae Proft funton of the th fr under ournot oetton s defned as: where [ ],,,, 9 s the argnal ost of the th fr. Then the nd stage role s ax The frst order ondtons are gven y,,. 0,. e., 0,,,, 0 y solvng we get where. The seond order stalty ondtons are also satsfed, that s 7 e 6 If we take a ore general funton, R, e, our results wll rean unhanged. 7. e 6

7 7 0 <. 0. Now, gven the nd stage soluton, the frst stage role s: ax suet to.,, 0 The frst order ondtons, 0, an e redued to 3 We assue that seond order stalty ondtons are satsfed. Let e the syetr eulru of ths frst stage. Ths wll orresond syetr eulru outut re solutons for, n the nd stage. In Aendx A we have resented the stes to solve the ournot gae. The ondtons desrng eulru of ths gae are:.. Tehnology hoe under ertr Gae Under ertr oetton the roft funton of a fr s: a a a,;, 5 where a a, s the argnal ost of the th ertr fr. Therefore, the nd stage role s:

8 ax,,. The frst order ondtons are y solvng for 0,.e., a 0, 6 we have 3 a a 7 where a a. The frst stage role s ax suet to 0,,. Fro the frst order ondtons, 0, we shall get 8 Seond order stalty ondtons are assued to e satsfed. Assung syetr eulru we have. Ths leads to n 7. Aendx has rovded stes to solve the ertr gae. The eulru solutons of the ertr gae are gven y the followng euatons, 9 3 We an easly hek that seond order stalty ondtons are satsfed, that s, <

9 a oarson etween Under ournot gae the otal tehnology, s solved fro, the otal tehnology under ertr gae, s solved fro 9. oarng these tehnology levels we an wrte the followng rooston. Prooston In eulru ournot frs have a lower argnal ost than the ertr frs.e., <. The roof s gven n Aendx. Intuton of the result s sle, ournot frs have larger nentves to nvest n R &. Hene R & nvestent under ournot oetton wll e larger, lyng that t wll have a lower argnal ost of roduton.. oarson etween Fro, 0 fro 9 9, a Therefore, Qute ovously, f, we have, whh s the Sngh Vves result see 6. In our ase sne see Prooston, we have the osslty that < f the dfferene etween s large. 9

10 efore we desre the nature of the relatons, let us frst onsder the effet of the hange of on. The oaratve stat results are gven n the followng Lea. Lea : d 0 d, d d 0 d d d. d Proof s gven n Aendx. Lea says that as the R & tehnology eoes ore ore neffent, argnal ost under eah of ournot ertr gae nreases, ut t nreases ore under ertr oetton. Now we are to oare. Fro 0 we have, resetvely, d d d d. Therefore, d d d 0, 0 d d d d 3 d Ths result tells that as argnal ost hanges, ertr re s affeted ore oared to ournot re. Fnally, d[ d ] d d d d d d d. d d d d. d Usng 3 through we ust have d d 0, 0 d d d d < 5 d d Therefore, oth are uward slong wth havng a greater sloe than. Ths eans, < < 6 Hene we have the followng rooston. 0

11 Prooston If re wll e lower than ertr re., that s, f the R & tehnology s relatvely neffent, ournot The result s ortrayed n Fgure. The result an e exlaned as follows. Low eans that R & ost of nnovatng a g szed tehnology s low. Hene oth ournot ertr frs wll oe u wth ute low argnal osts eah, therefore, the dfferene etween ther tehnologes wll e sall. For exale, n the extree, thnk that 0. Then oth ournot ertr frs wll oe u wth the ost effent tehnology,.e., In ths ase Sngh Vves result works, that s, 0 0. ut as goes u, argnal ost under ertr oetton nreases at a greater rate than that under ournot oetton. Therefore, for larger, the dfferene etween wll e larger. Moreover, y 5 as argnal ost nreases, ertr re nreases ore than ournot re. Ths exlans why ertr re an e larger for a large. Ths s a new result n the lterature. In other works, although ournot frs have larger R & nvestent, ut ournot re never falls elow ertr re. One laton of ths result s that f R & s relatvely ostly, ournot arket struture wll generate a larger onsuers' welfare. 3. onluson There s an extensve lterature oarng ournot ertr results. In a oneerng work Sngh Vves show n ters of a dfferentated duooly that ertr re s lower than ournot re that the ertr eulru s ore effent. The lterature thereafter has thoroughly exaned these fndngs n a nuer of alternatve set us has oe to the onluson that f goods are susttutes there an e stuatons when ournot eulru generates ore soal welfare. However, ournot re ontnues to e hgher than ertr re n the lterature. In an otherwse Sngh Vves fraework wth ntrodung roess R& n the frst stage ror to the roduton stage we have shown n the resent aer that ournot re an e lower than ertr re when the R& tehnology s relatvely neffent. Low R& rodutvty leads to a wde dfferene n argnal osts n eulru under ournot ertr gae, wth ertr argnal ost eng hgher than ournot argnal ost. Ths gves the osslty of a hgher ertr re than the ournot re. Whle we have roved the result assung a sef nnovaton funton, our result wll go through for a wde lass of R& tehnology so long as there are dnshng returns to R& nvestent. Fnally, t ay e entoned that n ths aer we have not rovded searately any analyss on the overall welfare effet; nstead, we have foused only on the re effet. However, t s understood that the welfare result wll e slar to the exstng lterature.

12 Aendx Aendx A : An Outlne of ournot Gae Soluton Payoff funtons under ournot gae :,;,, ] [ For any gven, the seond stage soluton s gven y see seton.., that s, a A. Then for the frst stage role the frst order ondtons are, 0,.e. 0 A. Now, the frst order ondtons of the nd stage gae are see 0 where 0. Insertng these values n A. we shall get 0 A.3 We assue that seond order stalty ondtons for the frst stage gae are also satsfed. Then syetr eulru les. Ths wll lead to syetr uantty eulru. Hene usng A. A.3 we haraterze the eulru solutons of the ournot gae y the euatons through n seton...

13 Aendx : An Outlne of ertr Gae Soluton The roft funton of th fr s : a a, a a,,,; For any gven that s,, the seond stage soluton s gven y 7 see seton.., a a. Then the frst order ondtons of the frst stage role are 0,,.e., a a 0. The frst order ondtons of the nd stage gae see 6 are a 0 Fro ths, the oaratve stat results are: where 0 Usng the aove,. eoes 0.3 Agan we assue that seond order stalty ondtons of ths stage gae are satsfed. Then the assuton of syetr eulru leads to. Hene fro..3, the eulru solutons of the ertr gae are haraterzed y the euatons 9 through 9 n seton... 3

14 Aendx : Proof of Gven the syetr eulru for any artrary argnal ost, we have fro.3 9] [usng a. We shall now evaluate at. We ay reeer that s solved fro. Usng,. Therefore, 0 ] [ ] [ ] [ 3.3 Aendx : Proof of Lea Fro we have

15 where K. K fro 9 K. where K Therefore fro. d d K 3.3 fro. d d K 3. We an now hek that the seond order ondtons of the frst stage ournot or s ertr gae ly 3 where s,. onsder frst the ournot gae. We already know fro A.3 that The seond order ondtons reure 3 < 0 Sne n eulru we have satsfyng, that s 0 5

16 6 we have 3 3 K Hene. 3 0 <. Slarly, we shall get, K < Hene fro.3. 0 d 0 d d d.5 Agan fro.3 usng. we have K d d 3 Slarly, fro. usng. we have K d d 3 Fnally, therefore, d d d d.6 Thus,.5.6 together gve Lea.

17 Referenes Arrow, K., 96, Eono welfare the alloaton of resoures for nventon, n R. Nelson, ed., The Rate reton of Inventve Atvty, Prneton Unversty Press, Prneton. ester, H. E. Petraks, 993, The nentves for ost reduton n a dfferentated ndustry, Internatonal Journal of Industral Organzaton, heng, L., 985, oarng ertr ournot eulra: a geoetr aroah, R Journal of Eonos 6, 65. elono, F. V. enolo, 990, R& nvestent n a syetr hoogeneous olgooly, Internatonal Journal of Industral Organzaton 8, Hakner, J., 000, A note on re uantty oetton n dfferentated olgooles, Journal of Eono Theory 93, Ma, H. Hwang, 999, Tehnology hoe under duooly, Paf Eono Revew, 3. Motta, M., 993, Endogenous ualty hoe: Pre versus uantty oetton, Journal of Industral Eonos, 33. Qu, L.., 997, On the dyna effeny of ertr ournot eulra, Journal of Eono Theory 75, 39. Sngh, N. X. Vves, 98, Pre uantty oetton n a dfferentated duooly, R Journal of Eonos 5, Syeonds, G. 00, oarng ournot ertr eulra n a dfferentated duooly wth rodut R&, susson Paer # 539, eartent of Eonos, Unversty of Essex. Vves, X., 985, On the effeny of ertr ournot eulra wth rodut dfferentaton, Journal of Eono Theory 36,

18 , 0 Fgure : oarson of ournot ertr res 8