UNIVERSITY OF NOTTINGHAM. Discussion Papers in Economics BERTRAND AND COURNOT COMPETITIONS IN A DYNAMIC GAME

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1 UNIVERSITY OF NOTTINGHAM Dsusson Paers n Eonoms Dsusson Paer No. 03/06 BERTRAND AND COURNOT COMPETITIONS IN A DYNAMIC GAME y Arjt Mukherjee Arl 003 DP 03/06 ISSN

2 UNIVERSITY OF NOTTINGHAM Dsusson Paers n Eonoms Dsusson Paer No. 03/06 BERTRAND AND COURNOT COMPETITIONS IN A DYNAMIC GAME y Arjt Mukherjee Arjt Mukherjee s Leturer Shool of Eonoms Unversty of Nottngham Arl 003

3 Bertrand and Cournot omettons n a dynam game Arjt Mukherjee Unversty of Nottngham U.K. January 003 Astrat: Ths aer omares Bertrand and Cournot equlra n a horzontally dfferentated duooly market wth non-tournament R&D ometton. We onsder that suess n R&D s unertan. We show that whether frms nvest more under Cournot ometton or Bertrand ometton s amguous and deends on the roalty of suess n R&D ut does not deend on the degree of rodut dfferentaton. Whle stat welfare s hgher under Bertrand ometton we fnd that dynam welfare may e hgher under Cournot ometton for moderate R&D rodutvtes. Further we show that whether the dfferene etween the eeted welfare under Cournot and Bertrand omettons nreases wth hgher rodut dfferentaton s amguous and deends on the R&D rodutvty. Key Words: Bertrand Cournot Unertan R&D Welfare JEL Classfaton: D34 L3 O33 Corresondene to: Arjt Mukherjee Shool of Eonoms Eonoms & Geograhy Buldng Unversty of Nottngham Nottngham NG7 RD U.K. E-mal: arjt.mukherjee@nottngham.a.uk Fa:

4 Bertrand and Cournot omettons n a dynam game Introduton There s an estng deate on the relatonsh etween R&D nvestment and the ntensty of ometton n the rodut market. In a semnal work Shumeter 943 argued that market onentraton s a stmulus to the nnovaton. So a soety mght refer to sarfe stat effeny for the dynam effeny. The work y Arrow 96 hallenged ths vew. In hs work Arrow 96 argued that the nentve for nnovaton would e stronger for omettve ndustry than a monoolst. Contrary to the earler lterature fousng on the omettve market and monooly the reent ontrutons eamne the nentve for R&D under dfferent tyes of ometton n olgoolst markets. More sefally researhers eamne whether frms nvest more n R&D under Bertrand ometton or Cournot ometton see e.g. Delono and Denolo 990 and Qu 997. The urose of the resent aer s to reeamne the relatonsh etween the tye of rodut market ometton n the olgoolst market and the nentve for R&D and ts mlaton on welfare. In artular we onsder a two-stage duooly model of R&D and roduton where the frms rodue horzontally dfferentated roduts and suess n R&D s unertan whh deends on the R&D nvestment. Emloyng a model of non-tournament R&D ometton we show that whether R&D nvestment and welfare are hgher n a market wth more omettve ehavor deends on the R&D rodutvty. We fnd that R&D nvestments are more less under Cournot ometton omared to Bertrand ometton f the equlrum roaltes are greater lower than a rtal level. Whle stat welfare s always hgher under Bertrand ometton dynam welfare may e more under Cournot ometton for R&D rodutvtes. The resent aer s losely related to Delono and Denolo 990 Bester and Petraks 993 Qu 997 and Bonanno and Haworth 998. In a model of symmetr homogeneous olgooly Delono and Denolo 990 argue that R&D nvestment s One may refer to Delono and Denolo 990 for the meanng of stat and dynam welfare. By stat welfare we mean welfare e-ost R&D and y dynam welfare we mean the eeted welfare eante R&D. One may refer to Sngh and Vves 984 Vves 985 and Cheng 985 for works on Bertrand and Cournot ometton n a stat framework.e. where the frms fae same demand and ost struture n oth tyes of rodut market ometton.

5 greater under Bertrand ometton. But wth a horzontally dfferentated rodut market Qu 997 argues that frms nvest more n R&D under Cournot ometton. However unlke these aers our analyss shows the osslty of hgher R&D nvestment under Bertrand ometton as well as under Cournot ometton deendng on the equlrum roalty of suess. In ontrast to Delnono and Denolo 990 and Qu 997 our results also dffer sgnfantly for the analyss on welfare under Bertrand and Cournot omettons. For eamle whle Delono and Denolo 990 need suffently large numers of frms for hgher welfare under Cournot ometton we show that ths s ossle even n a duooly market struture. Further whle the estene of knowledge sllover s neessary for Qu 997 to rove the osslty of hgher welfare under Cournot ometton the resent aer shows the osslty of hgher welfare under Cournot ometton n asene of knowledge sllover. Though we onsder a two-stage game wth R&D and roduton lke Delono and Denolo 990 and Qu 997 our onlusons are sgnfantly dfferent from these two aers. Hene at ths ont t s worth mentonng the man dfferenes etween the resent aer and those aers. Whle Qu 997 onsdered a model of R&D where ost of roduton redues wth hgher amount of R&D nvestment and suess n R&D s ertan we onsder a dfferent tye of R&D ometton where suess n R&D s unertan and nreases wth R&D nvestments ut the ost reduton from R&D s gven. Contrary to the model of R&D rae n Delono and Denolo 990 we onsder a model of non-tournament R&D ometton. So n our framework t s ossle for oth frms to have newly nvented tehnology. Thus our model of R&D ometton s smlar to the one onsdered n Cho 993. Further unlke Delono and Denolo 990 we onsder a model of rodut dfferentaton where homogeneous rodut s a seal ase of our framework. Bester and Petraks 993 Bonanno and Haworth 998 and Symeonds 003 also deal wth smlar questons addressed n the resent aer. However ontrary to the resent aer Bonanno and Haworth 998 and Symeonds 003 onsder a model of vertal rodut dfferentaton and argue that the nentve for R&D s hgher under Cournot ometton. Though n ther aer Bester and Petraks 993 onsder a model wth horzontal rodut dfferentaton they onsder R&D nvestment y only one frm and the suess n R&D s ertan. They found that the nentve for R&D nvestment would deend on the degree of rodut dfferentaton. It wll e evdent from our analyss 3

6 that even f only one frm does R&D n our framework equlrum R&D nvestment s hgher under Bertrand ometton. In what follows n seton we onsder a general framework for a dynam game of R&D ometton and roduton. We assume that eah frm s suess n R&D s unertan whh deends on the R&D nvestment of that frm. In our analyss we onsder symmetr roalty funton for these frms and fous on drast nnovaton only.e. f only one frm s suessful n R&D then only the suessful frm rodues n the market. Seton 3 omares the R&D nvestments under Bertrand and Cournot omettons and shows that whether R&D nvestment s more under Cournot or Bertrand ometton deends on the roalty of suess. Suose the equlrum roalty of suess s suffently low. Then gven the low roalty of suess of the omettor t nreases a frm s hane of eomng a sngle nnovator. Sne the gan from eomng a sngle nnovator s lower under Cournot ometton ths osslty enourages a frm to redue ts otmal R&D nvestment under Cournot ometton omared to Bertrand ometton gven the equlrum R&D nvestment of the omettor frm. The symmetry of the frms mles that equlrum R&D nvestments of these frms are lower under Cournot ometton omared to Bertrand ometton. Now onsder the stuaton where the equlrum roaltes of suess are suffently hgh. Gven the hgh roalty of suess of the omettor t redues the hane of eomng the sngle nnovator and a duooly market struture s the more lkely outome. Sne duooly roft s hgher under Cournot ometton omared to Bertrand ometton here the frms nvest more under Cournot ometton omared to Bertrand ometton. It s well known that lower res under Bertrand ometton hels to reate hgher stat welfare under Bertrand ometton omared to Cournot ometton. However our analyss shows that R&D nvestments are more under Cournot ometton when the equlrum roalty of suess eeeds a rtal level. Therefore n ths stuaton the roalty of gettng the nnovaton s hgher under Cournot ometton omared to Bertrand ometton. As a result dynam welfare may e more under Cournot ometton f the equlrum roalty of suess s greater than a rtal value. We show that f the R&D rodutvty s not suffently hgh then dynam welfare may e more under Cournot ometton omared to Bertrand ometton. 4

7 As the R&D rodutvty nreases t nreases the R&D nvestment whh also nreases the roalty of suess n R&D. If the R&D rodutvty s suffently large so that the equlrum roalty of suess s greater than a rtal value then we fnd that the roalty of gettng the nnovaton s hgher under Cournot ometton. It reates the osslty of hgher dynam welfare under Cournot ometton. But f the R&D rodutvty s very hgh so that the dfferene etween the R&D nvestments under Cournot and Bertrand omettons s sgnfantly large then ths hgher amount of R&D nvestment under Cournot ometton reates sgnfantly large negatve effet on welfare under Cournot ometton. Hene for very hgh R&D rodutvty welfare s hgher under Bertrand ometton. Thus for very low R&D rodutvty where R&D nvestments are hgher under Bertrand ometton and for very hgh R&D rodutvty dynam welfare s hgher under Bertrand ometton. But for ntermedate R&D rodutvtes dynam welfare may e more under Cournot ometton. Wth an eamle we show that whether the dfferene etween the eeted welfare under Cournot and Bertrand omettons nreases wth hgher rodut dfferentaton s amguous and deends on the R&D rodutvty. As the roduts are gettng dfferentated t affets the welfare y hangng the oututs of the frms and also y hangng the R&D nvestments of these frms. Hene the result deends on how these two effets are hangng the welfare under Bertrand and Cournot omettons. The eamle shows that hgher rodut dfferentaton nreases redues the dfferene etween the eeted welfare under Cournot and Bertrand omettons for suffently low hgh R&D rodutvty. Rest of the aer s organzed as follows. The net seton gves a general framework for our analyss. In seton 3 we omare the equlrum R&D nvestments of these frms under Bertrand and Cournot omettons. Seton 4 eamnes the welfare mlatons under these two tyes of market omettons. Seton 5 onludes the aer. A general framework Consder an eonomy wth two frms alled and rodung horzontally dfferentated roduts. Assume that the frms have smlar tehnologes at the egnnng and eah of them faes onstant average ost of roduton. Eah frm an nvest n R&D and an mrove ts tehnology. We assume that eah of them tres to nvent a tehnology 5

8 orresondng to onstant average ost of roduton. 3 However the R&D roess s unertan and frm = an sueed wth an unondtonal roalty where the roalty of suess deends on the th frm s R&D nvestment. We onsder that > = and = 0 for =. Sne our urose s to < fous on the effets of rodut market ometton we assume that oth frms fae same roalty funton so that the results are not nfluened y the asymmetry n roalty funtons. Hene we do our analyss under the assumton that = =. Therefore our model of R&D ometton s smlar to Cho 993. Further for smlty we assume that there are no osts of dong R&D eet R&D nvestment. q We onsder a two-stage game. In stage oth frms smultaneously nvest n R&D. In stage these frms omete n the rodut market and smultaneously take ther deson n the rodut market. In the followng analyss we wll onsder two tyes of rodut market ometton vz. Bertrand ometton and Cournot ometton. For smlty n our analyss we wll assume that the new tehnology s drast n nature.e. only the suessful frm wll serve the rodut market n ase of unlateral suess n R&D. It s mortant to note that we need relatvely hgher amount of ost reduton from R&D to reate drast R&D as the degree of rodut dfferentaton nreases. Further the amount of ost reduton that reates drast R&D also deends on the tye of rodut market ometton see e.g. Bester and Petraks 993. Hene a ertan amount of ost reduton ould mly drast R&D for a artular degree of rodut dfferentaton or for a artular tye of rodut market ometton ut mght not mly drast R&D for dfferent degree of rodut dfferentaton or for dfferent tye of rodut market ometton. However as a smlfaton we assume that the ost reduton from R&D s suffently large to reate drast R&D for all degree of rodut dfferentaton eet for the stuaton of solated roduts. 4 We onsder the followng demand struture for our analyss. We assume that the reresentatve onsumer s utlty s a funton of onsumton q = q q y U q + m wth j where and q are the oututs of frm and frm and the numerare good m. It s gven 3 The new tehnologes ould e dfferent ut reatng same ost of roduton. 6

9 U q = a q + q q + γ qq + q where the term γ shows the degree of rodut dfferentaton and an take any value etween 0 and. If γ = 0 ths mles that the roduts of these frms are solated ut for γ = the roduts are erfet susttutes. We wll do our analyss for γ 0]. 5 We assume that a. If a = then the otmal outut and roft of these frms are 0 even f nether of them s suessful n R&D. We wll do our analyss for a > f not sefed otherwse. The aove utlty funton gven n generates the followng nverse demand funton for the th frm: P = a q γq j where j = j. Let us defne the otmal roft level of the th frm = n the rodut market.e. revenue mnus total ost of roduton for a gven degree of rodut dfferentaton y ; γ and ; γ resetvely for the stuatons where only the th frm s suessful n R&D where oth frms are suessful n R&D and where nether frm s suessful n R&D. The arguments n the roft funtons are showng the onstant average ost of roduton of these frms. Sne the suessful frm eomes a monooly n ase of unlateral suess n R&D we do not wrte the susrt and the degree of rodut dfferentaton for ths stuaton. In what follows we wll do a general analyss wth these redued form roft funtons and then n the net seton we wll eamne how the values deend on the artular tye of the rodut market ometton. The net roft of the th frm s gven y 4 It s easy to understand that suffently hgher re-nnovaton margnal ost of roduton nreases the osslty of drast R&D. For eamle f these frms do not have any tehnology to rodue the rodut wthout nnovaton then t s trval that any ost reduton from R&D mles drast R&D. 5 The assumton of drast nnovaton s justfale rovded the roduts are not solated. 7

10 j j j + + ; ; γ γ 3 where and = j j.mamzng 3 we get the otmal R&D nvestment for frm gven the R&D nvestment of frm j. So the roft mamzng R&D nvestment for frm s ; ; = + γ γ j j j 4 or ] ; ; [ ] ; [ = γ γ γ j. 4 Seond order ondton for mamzaton s satsfed. Due to the symmetry of the roalty funtons we have smlar reaton funtons for these frms. Solvng these two reaton funtons we an fnd the otmal R&D nvestments of these frms. We assume that the roalty funtons are suh that we get a unque equlrum for R&D nvestments. Defne otmal R&D nvestments of these frms y and for frms and resetvely. Further the symmetr roalty funton onsdered n ths seton wll lead to same level of R&D nvestments y these frms. It s lear from 4 that the otmal R&D nvestment of the th frm deends on the dfferenes ] ; γ [ and ] ;γ ; γ [. It s easy to understand that the term ] ; γ [ s always ostve for any degree of rodut dfferentaton eet for the solated roduts ut the sgn of ] ; ; [ γ γ s not lear and deends on the degree of rodut dfferentaton. Lemma : If the rodut dfferentaton s elow aove a rtal level then ] ; ; [ γ γ s ostve negatve. Ths rtal level of rodut dfferentaton s lkely to e dfferent under Bertrand ometton and Cournot ometton. 8

11 Proof: If the roduts are almost solated.e. γ = ε where ε > 0 ut very small then the term [ ; γ ; γ ] tends to e [ ] < 0 sne eah frm wll e almost monooly for ts rodut. Now onsder the other etreme where the roduts are erfet susttutes.e. γ =. Here the term [ ; γ ; γ ] redues to > 0 under Bertrand ometton. Also we fnd that [ ; γ ; γ ] s ostve for Cournot ometton f the roduts are erfet susttutes. Hene we fnd that the term [ ; γ ; γ ] s ostve under Cournot ometton when the roduts are erfet susttutes. The rofts are ontnuous for γ [ ε δ ] where δ > 0 and very small under Bertrand ometton and for γ [ε] under Cournot ometton. Therefore we an fnd a rtal value of γ for Bertrand ometton and a rtal value of γ for Cournot ometton suh that the term [ ; γ ; γ ] s ostve negatve when the atual value of γ s more less than ths rtal value.e. when the degree of rodut dfferentaton s elow aove that rtal level of rodut dfferentaton. Sne the term [ ; γ ; γ ] takes dfferent values under Cournot and Bertrand ometton the rtal values under Cournot and Bertrand ometton s lkely to e dfferent. Q.E.D. We have found that R&D nvestment of the th frm deends on the dfferenes [ ; γ ] and [ ; γ ;γ ]. The net result shows how these terms affet the equlrum R&D nvestments of these frms. Lemma : Let s the otmal R&D nvestment of the th frm for any gven R&D nvestment of the j th frm j = and j. We fnd > 0 where z = [ ; γ ]. z We fnd that < 0 where y = [ ; γ ; γ ]. dy 9

12 Proof: Totally dfferentatng 4 for and z we fnd that z = z y j > 0 where j. Ths s eause the denomnator z y j s negatve due to the seond order ondton of roft mamzaton wth reset to the R&D nvestment. j Totally dfferentatng 4 for and y we fnd that < 0. y = z y j Q.E.D. At ths ont t s worth onsderng the mlaton of the aove lemma. Frst onsder a stuaton where only the th frm does R&D. Ths mles that = 0. So the otmal R&D nvestment of frm wll satsfy the ondton j [ ; γ ] =. 5 Suose s the otmal R&D nvestment that satsfes ondton 5. So from 5 t s lear that f only the th frm does R&D then ts otmal R&D nvestment as the value of z wll redue redues. Hene the stand-alone nentve for nvestng n R&D.e. the nentve for nvestng n R&D y a frm when the omettor does not nvest n R&D redues as the gan from R&D redues. Therefore due to the stand-alone nentve for dong R&D one frm s R&D nvestment wll e more under Bertrand ometton whh s n ontradton to the Bester and Petraks 993. Sne frm j also does R&D we need to onsder the strateg effet on R&D nvestment as well. The seond art of the aove lemma shows that as the strateg gan from R&D redues a frm s nentve for R&D nvestment nreases.e. gven the R&D nvestment of the j th frm the R&D nvestment of the th frm nreases due to the reduton n y only. Hene due to the strateg nentve a frm s R&D nvestment wll e more under Cournot ometton. 0

13 3 Comarson etween Bertrand and Cournot omettons The revous seton has develoed a model of R&D ometton n a general framework and eamned the effets of stand-alone nentve and strateg nentve. In ths seton we wll see how dfferent tyes of rodut market ometton vz. Bertrand and Cournot omettons affet the R&D nvestments of these frms. The net seton wll omare the welfare effets of these rodut market omettons. Before eamnng the effets of Bertrand and Cournot omettons we wll rove the followng lemma omarng the oututs and the rofts of these frms e-ost R&D whh wll e helful for our followng analyss. Lemma 3: Consder that ether oth frms sueed n R&D or nether frm sueeds n R&D and γ 0]. The otmal outut of eah frm s more under Bertrand ometton than Cournot ometton.e. q ; γ > q ; γ and q ; γ > q ; γ =. The rofts are greater under Cournot ometton omared to Bertrand ometton for eah of these frms.e. ; γ > ; γ and ; γ > ; γ =. Proof: Assume that γ 0. Gven the demand and ost sefatons the otmal outut and roft of the th frm = under Cournot and Bertrand omettons are resetvely a k q a k = and = 6 + γ + γ and a k q = and γ + γ a k γ = 7 γ γ where k = when oth frms sueed n R&D and k = when nether frm sueeds n q < q > R&D. It s easy to hek from 6 and 7 that and.

14 If γ = then the eressons n 6 show the otmal oututs and rofts under Cournot ometton. But n ths stuaton the otmal outut and roft of the th frm ak under Bertrand ometton are gven y and 0. Therefore also n ths stuaton the otmal outut s more under Bertrand ometton and the roft s more under Cournot ometton. Thus we rove the result of ths lemma. Q.E.D. Now we are n a oston to omare the R&D nvestments under Bertrand and Cournot omettons. From Lemma 3 t s lear that oth [ ; γ ] and [ ; γ ; γ ] are greater under Bertrand ometton omared to the Cournot ometton. Therefore from Lemma we fnd that oth frms wll nvest more n R&D under Bertrand ometton due to the stand-alone nentve. But gven the R&D nvestment of the omettor strateg effet enourages a frm to nvest less n R&D under Bertrand ometton. So total effet on the equlrum R&D nvestment s amguous and as the followng rooston shows t deends on the roalty of suess n R&D. Prooston : The equlrum roalty of suess n R&D under oth Bertrand ometton and Cournot ometton s ether greater than less than or equal to. If the equlrum roalty of suess under Bertrand and Cournot omettons s greater less than a rtal value.e. say the equlrum R&D nvestments are more less under Cournot ometton omared to Bertrand ometton. Proof: Suose under Bertrand ometton and are the equlrum R&D nvestments of these frms. So and satsfy the ondton 4 under Bertrand ometton. Straghtforward alulaton wll show that gven the R&D nvestment the left hand sde of 4 for frm under Cournot ometton wll e greater than rovded ; γ ; γ ] < [ ; γ ; γ + ; γ ; γ ]. 8 [

15 or [ ; γ ; γ ] > = [ ; γ ; γ + ; γ ; γ ]. 8 If tends to then the ondton 8 s satsfed. But f tends to 0 then the ondton 8 does not hold. Further the left hand sde of 8 s ontnuous and nreasng n over [ 0 ]. Therefore f s greater than a rtal value say then gven the R&D nvestment the otmal R&D nvestment of frm under Cournot ometton s more than. Sne these frms are symmetr we have a smlar ondton for frm also. Further symmetry of the frms mly that these frms wll nvest same amount n R&D. Therefore we an say that f = > the otmal R&D nvestments of these frms under Cournot ometton say and wll e more than the otmal R&D nvestments of these frms under Bertrand ometton. But f the ondton 8 does not hold.e. f = < the otmal R&D nvestments of these frms are lower under Cournot ometton omared to Bertrand ometton. We have seen that f = > the equlrum R&D nvestment s more under Cournot ometton and hene the roalty of suess s hgher under Cournot ometton. But for = < the equlrum R&D nvestment s lower under Cournot ometton. Hene n ths stuaton the equlrum roalty of suess s lower under Cournot ometton omared to Bertrand ometton. We have same equlrum R&D nvestment and roalty of suess under Bertrand and Cournot ometton when = =. Therefore the equlrum roalty of suess n R&D under oth Bertrand ometton and Cournot ometton s ether greater than less than or equal to. It follows from the aove argument that f = the roalty of > suess and the otmal R&D nvestments of these frms are more under Cournot ometton omared to Bertrand ometton. The ooste stuaton arses for =. Ths roves the result. Q.E.D. < 3

16 Thus we fnd that whether the frms nvest more n R&D under a more or less omettve envronment deends on the equlrum roalty of suess n R&D. Ths ontradts the revous results of ths lterature where the authors have shown that equlrum R&D nvestments are always hgher under a artular mode of ometton see e.g. Deloo and Denolo 990 and Qu 997. In Prooston we have onsdered the effets of dfferent tyes of ometton on R&D nvestments for a gven degree of rodut dfferentaton. Now we wll eamne the nfluene of rodut dfferentaton on our result. Prooston : Whether R&D nvestment s hgher under Cournot ometton or Bertrand ometton does not deend on the degree of rodut dfferentaton. Proof: We have seen n Prooston that R&D nvestments are more under Cournot Bertrand ometton f the equlrum roalty of suess s greater less than the rtal value. From 6 7 and 8 we fnd that s ndeendent of the degree of rodut dfferentaton sne ; γ ; γ ; γ ; γ a = a. 9 Hene t mles that the omarson of R&D nvestments under Cournot and Bertrand omettons does not deendent of the degree of rodut dfferentaton. Q.E.D. Thus n ontrary to the revous result of ths lterature see Bester and Petraks 993 we show that the omarson etween the equlrum R&D nvestments under Bertrand and Cournot omettons does not deend on the degree of rodut dfferentaton. 3. An eamle In ths suseton we onsder a sef roalty funton faed y these frms and rovde an eamle for Prooston. We onsder that these frms fae same roalty 4

17 funton and s gven y = µ = roduts are homogenous.e. γ =.. Further n ths eamle we assume that the If the roduts are homogeneous then the rtal value of the roalty of suess n R&D s gven y ; γ = =. 0 ; + ; Usng the ondton 4 and due to symmetry we fnd that the th frm s = otmal R&D nvestment and the orresondng roalty of suess under Bertrand and Cournot omettons are resetvely µ γ = = and + µ [ ;] µ γ = = + µ [ ; ;] µ γ = = and + µ µ [ ;] γ = =. + µ [ ; ;] From 0 and we fnd that oth γ = and γ = are less greater than γ = rovded µ ; < > ;. Therefore otmal R&D nvestments are more less under Bertrand ometton omared to Cournot ometton rovded µ ; < > ;. The dret omarson of the eressons n also rovdes the same onluson. Thus we fnd that f the R&D rodutvty s suffently low hgh.e. µ s low hgh the otmal R&D nvestment s more less under Bertrand ometton omared to Cournot ometton. 4 Welfare omarson It s well known that stat welfare s hgher under Bertrand ometton omared to Cournot ometton sne the deadweght loss s lower under Bertrand ometton 5

18 omared to Cournot ometton. However the aove analyss shows that f the roalty of suess s suffently hgh then R&D nvestments are more under Cournot ometton omared to Bertrand ometton. Hene there mght e a onflt etween the stat and dynam welfare. We defne the welfare of the eonomy as the summaton of onsumer surlus and the ndustry roft net of R&D nvestments. If the roalty of suess s suffently low then the analyss of seton 3 shows that the hane of gettng the nnovaton under Bertrand ometton s hgher omared to Cournot ometton. However n ths stuaton a hgher R&D nvestment under Bertrand ometton has a negatve mat on welfare. But sne we are onsderng a stuaton where the R&D rodutvty s very low ths negatve mat s suffently low and the eeted welfare s most lkely to e hgher under Bertrand ometton omared to Cournot ometton. Hene f the roalty of suess s suffently low then oth stat and dynam welfare are hgher under Bertrand ometton omared to Cournot ometton. If the roalty of suess s suffently hgh then we have found that the R&D nvestments are hgher under Cournot ometton omared to Bertrand ometton. Hene the roalty of suess s hgher under Cournot ometton omared to Bertrand ometton. So n ths stuaton the stat welfare s hgher under Bertrand ometton ut the hane of gettng the nnovaton s hgher under Cournot ometton. As a result the dynam welfare may e more under Cournot ometton whle the stat welfare s more under Bertrand ometton. Generally seakng the omarson of welfare under Bertrand and Cournot omettons s umersome n our framework. In the followng analyss we wll onsder the sef roalty funton that has een onsdered n suseton 3. and wll rovde eamles to show the effets of the R&D rodutvty.e. µ and the degree of rodut dfferentaton.e. γ on welfare. The net rooston wll onsder an eamle of homogeneous rodut to show the osslty of hgher and lower welfare under Cournot ometton omared to Bertrand ometton. Prooston 3: Assume that a = = 0 [.5 ] γ = µ [03 ] and oth frms have same roalty funton = µ where =. 6

19 a If the value of.e. the re-nnovaton ost s not very hgh then the eeted welfare s hgher under Bertrand ometton for all R&D rodutvtes. Suose the value of s suffently hgh. The eeted welfare s hgher under Bertrand ometton for very low and very hgh R&D rodutvtes ut the eeted welfare s hgher under Cournot ometton for moderate R&D rodutvtes. Proof: When γ = we fnd that the equlrum roalty of suess n R&D under Cournot ometton s equal to for µ = 3. Further gven a = the nnovaton s drast under Bertrand and Cournot ometton when = 0 and [.5 ]. Gven the assumtons of ths rooston and the demand funton mentoned n the eeted welfare under Cournot and Bertrand ometton are resetvely W 4 [6µ 9 4 7µ 9 4 = + 7µ µ µ ] 36[7 + µ 5 4 ] 3 and 4 [ µ + µ µ ] W =. 4 [8 + µ ] We lot the eresson W W n Fgure. 6 The nseton of the Fgure roves the result. For etter understandng n Fgure and Fgure 3 we onsder the value of as. 6 and. 9 resetvely wth all other assumtons of ths rooston. Q.E.D. The reasons for the aove results are followng. We know that stat welfare s always hgher under Bertrand ometton. If R&D rodutvty s very low so that R&D nvestments are hgher under Bertrand ometton then the roalty of suess s hgher under Bertrand ometton. But hgher R&D nvestment under Bertrand ometton has 6 We use The Mathemata 4 for the fgures related to ths rooston and Prooston 4.e. fgures and 6. 7

20 a negatve mat on the welfare. Sne the R&D rodutvty s very low ths negatve mat on welfare s very small and the eeted welfare s hgher under Bertrand ometton. On the other etreme f the R&D rodutvty s very hgh then the R&D nvestment and the roalty of suess n R&D are hgher under Cournot ometton. Sne now we are onsderng a stuaton wth suffently hgher R&D rodutvty the negatve mat of hgher R&D nvestment under Cournot ometton an e suffently large and the eeted welfare an e hgher under Bertrand ometton even f the R&D nvestment s hgher under Cournot ometton. However f R&D rodutvty s suffently large to reate hgher R&D nvestment under Cournot ometton ut f t s not very hgh to reate suffently large negatve mat on welfare we fnd that whether the eeted welfare s hgher under Cournot ometton deends on the re-nnovaton ost. Gven the roalty funton of Prooston 3 t s lear from that the R&D nvestment under Bertrand ometton does not deend on the re-nnovaton ost ut the R&D nvestment under Cournot ometton nreases wth the re-nnovaton ost. Therefore the eneft from hgher R&D nvestment under Cournot ometton nreases as the re-nnovaton ost nreases. Hene f the re-nnovaton ost s suffently low not suffently low then for these R&D rodutvtes the roalty of suess s not suffently hgher suffently hgher under Cournot ometton to outwegh the effet of hgher stat welfare under Bertrand ometton. Hene n ths stuaton the eeted welfare s hgher under Cournot Bertrand ometton for suffently hgher lower re-nnovaton osts. In the aove rooston we show that welfare under Cournot ometton ould e hgher omared to Bertrand ometton. In our model we dd not onsder the osslty of knowledge sllover and hene ths fndng ontradts the result of Qu 997 whh shows that welfare s always hgher under Bertrand ometton omared to Cournot ometton when there s no knowledge sllover under R&D. Further n ontrary to Delono and Denolo 990 whh shows the osslty of hgher welfare under Cournot ometton f the numer of frms n the ndustry s suffently large we show that welfare under Cournot ometton ould e hgher omared to Bertrand ometton even f we onsder a duooly market. Further unlke the revous ontrutons we fnd that welfare under Cournot ometton s hgher under moderate R&D rodutvtes. The aove rooston shows the mat of R&D rodutvty on welfare under Bertrand and Cournot ometton. In the net rooston we wll rovde an eamle to 8

21 show the mortane of the rodut dfferentaton. We wll show that whether the dfferene etween the eeted welfare under Cournot and Bertrand omettons nreases or dereases wth hgher rodut dfferentaton s amguous and an deend on the R&D rodutvty. Prooston 4: Assume that a = = = 0 µ 0 ] and onsder that γ 0 and oth frms have same roalty funton = µ where =. The dfferene etween the eeted welfare under Cournot and Bertrand omettons nreases redues wth hgher rodut dfferentaton f the R&D rodutvty s suffently low hgh. Proof: We fnd that f the roduts are solated.e. γ = 0 then the roalty of suess n R&D under Cournot ometton equals to when µ =. Therefore f µ 0 ] then for any degree of rodut dfferentaton roalty of suess n R&D s lower than under Bertrand and Cournot omettons. Further the nnovaton s drast under Bertrand and Cournot ometton when a = = and = 0. Gven the assumtons of ths rooston and the utlty funton sefed n we fnd that the eeted welfare under Cournot and Bertrand omettons are gven y 4 4 [4µ + γ 3 + γ + 3µ + γ 8 + γ 4µ 8µ + γ ] W = 5 4[8 + γ + µ γ 4 + γ ] and W 4 [4µ γ γ + γ 3 γ + µ γ γ 8 γ γ 4 4µ γ 8µ γ γ ] = 4[8 γ γ + µ γ γ 4 γ ]. 6 Sutratng 6 from 5 we an fnd the eresson for W W. In Fgure 4 we lot ths eresson W W for µ [0 ] and γ [0]. The nseton of Fgure 4 roves the result. 9

22 Ths fndng s more romnent n Fgure 5 and Fgure 6 where we lot W W for µ = and µ =. 5 along wth other assumtons of ths rooston. Q.E.D. If the roduts are gettng dfferentated then t hanges the oututs and R&D nvestments of these frms under Bertrand and Cournot omettons. Gven the assumtons of ths rooston t s easy to hek from 4 that as the roduts are gettng dfferentated the R&D nvestments nrease. If the R&D rodutvty s suffently low so that the R&D nvestments are suffently low the effets through R&D nvestments tend to e small. Hene n ths stuaton the dfferene etween the eeted welfare under Cournot and Bertrand ometton nreases wth hgher rodut dfferentaton as the dfferene of oututs etween Bertrand and Cournot omettons redues wth hgher rodut dfferentaton. But f the R&D rodutvty s suffently hgh so that the R&D nvestments are suffently hgh the effets through R&D nvestments tend to e larger. Due to the hgh R&D rodutvty the effet on R&D nvestment s suffently large. Ths reates a negatve mat on welfare due to the hgh R&D nvestment and a ostve effet on welfare due to the hgher hane of gettng the nnovaton. Sne here the roalty of suess under Cournot ometton s already suffently hgh the latter effet s relatvely week under Cournot ometton. Further hgher rodut dfferentaton redues the dfferene of oututs etween Bertrand and Cournot omettons. On the alane the negatve mat of hgher R&D nvestment under Cournot ometton tends to domnate the result. Hene the dfferene etween the eeted welfare under Cournot and Bertrand ometton redues wth hgher rodut dfferentaton. 5 Conluson Whether R&D nvestment nreases under hgher or lower omettve envronment s an estng deate. Whle the ntal works have looked at monooly and omettve markets to fnd the answer reent ontrutons have foused on olgoolst markets. In ths aer we eamne ths ssue n a duooly market and show that sgnfantly dfferent onlusons an e otaned f we onsder non-tournament R&D ometton etween frms where eah frm affets ts roalty of suess n R&D through ts own R&D nvestment. 0

23 We show that whether equlrum R&D nvestment s more under Bertrand or Cournot ometton deends on the equlrum roalty of suess. Consderng symmetr roalty funton we show that f the equlrum roalty of suess s suffently low hgh then the equlrum R&D nvestment s more under Bertrand Cournot ometton. Hene ths fndng may have mortant mlatons for desgnng government oles to enourage R&D atvtes n an eonomy. Sne the equlrum R&D nvestment s more under Cournot ometton when the roalty of suess s suffently hgh here the hane of gettng the nnovaton s hgher under Cournot ometton omared to Bertrand ometton. Thus the eeted gan from the new nnovaton may reate hgher dynam welfare under Cournot ometton omared to Bertrand ometton. In ths reset we show the mortane of the R&D rodutvty and the degree of rodut dfferentaton.

24 Referenes Arrow K. 96 Eonom welfare and the alloaton of resoures for nventon n R. Nelson Ed. The Rate and dreton of nventve atvty Prneton Unversty Press Prneton. Bester H. and E. Petraks 993 The nentves for ost reduton n a dfferentated ndustry Internatonal Journal of Industral Organzaton : Bonanno G. and B. Haworth 998 Intensty of ometton and the hoe etween rodut and roess nnovaton Internatonal Journal of Industral Organzaton 6: Cheng L. 985 Comarng Bertrand and Cournot equlra: a geometr aroah RAND Journal of Eonoms 6: Cho J. P. 993 Cooeratve R&D wth rodut market ometton Internatonal Journal of Industral Organzaton : Delono F. and V. Denolo 990 R&D nvestment n a symmetr and homogeneous olgooly Internatonal Journal of Industral Organzaton 8: Qu L. D. 997 On the dynam effeny of Bertrand and Cournot equlra Journal of Eonom Theory 75: 3 9. Shumeter J. 943 Catalsm soalsm and demoray Allan and Unwn London. Sngh N. and X. Vves 984 Pre and quantty ometton n a dfferentated duooly RAND Journal of Eonoms 5: Symeonds G. 003 Comarng Cournot and Bertrand equlra n a dfferentated dyuooly wth rodut R&D Internatonal Journal of Industral Organzaton :

25 Vves X. 985 On the effeny of Bertrand and Cournot equlra wth rodut dfferentaton Journal of Eonom Theory 36:

26 µ Fgure : Sutratng 4 from µ Fgure : Sutratng 4 from 3 for =.6. 4

27 µ Fgure 3: Sutratng 4 from 3 for = γ µ Fgure 4: Sutratng 6 from 5 and for µ [0 ] and γ [0]. 5

28 γ Fgure 5: Sutratng 6 from 5 and for µ = and γ [0] γ Fgure 6: Sutratng 6 from 5 and for µ =. 5 and γ [0]. 6