Duopoly innovation under product externalities

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1 Economc Research-Ekonomska Istražvanja ISSN: X (Prnt) (Onlne) Journal homepage: Duopoly nnovaton under product externaltes You-hua Chen & Pu-yan Ne To cte ths artcle: You-hua Chen & Pu-yan Ne (04) Duopoly nnovaton under product externaltes, Economc Research-Ekonomska Istražvanja, 7:, 3-43, DOI: 0.080/33677X To lnk to ths artcle: 04 The Author(s). Publshed by Taylor & Francs Publshed onlne: 8 Sep 04. Submt your artcle to ths journal Artcle vews: 40 Vew related artcles Vew Crossmark data Ctng artcles: Vew ctng artcles Full Terms & Condtons of access and use can be found at Download by: [ ] Date: 4 December 06, At: 04:44

2 Economc Research-Ekonomska Istražvanja, 04 Vol. 7, No., 3 43, Duopoly nnovaton under product externaltes You-hua Chen a * and Pu-yan Ne b a College of Economcs and Management and Guangdong Center for Rural Economc Studes, South Chna Agrcultural Unversty, Guangzhou, 5063, P.R. Chna; b Insttute of Industral Economcs, Jnan Unversty, Guangzhou, 5063, P.R. Chna (Receved 7 May 0; accepted 0 December 0) Ths study argues that product substtutablty and complementary have major effects on the relatonshp between nnovaton and competton and some nterestng conclusons are derved. Frst, nnovatve nvestment s reduced wth market power. The total quantty of products and socal welfare are ncreased wth market power whle decreased wth ncreasng of substtutablty or deceasng of complementary. Second, the equlbrum products and nnovatve nvestment are lower than those under socal optmalty. Fnally, by comparson wth Cournot quantty competton, Bertrand prce competton s keener. But the man conclusons are the same under both knds of compettons. Keywords: nnovaton; duopoly; product complementary; product substtutablty JEL classfcaton: D43, L3. Introducton Snce Schumpeter (94) proposed nnovaton theory, many people fxed ther attenton on nnovaton behavour and the relatonshp between nnovaton and competton became an mportant topc n economcs and management. Schumpeter (94) ssued that monopoly stmulates nnovaton, but 0 years later Arrow (96) declared that competton motvates nnovaton. Snce them, the debate about the relatonshp between competton and nnovaton s contnuous. Some people supported Schumpeter (e.g. Demsetz, 969; Glbert & Newbery, 98; Y, 999), some people sustaned Arrow (e.g. Recently Holmes, Levne & Schmtz, 0; Vves, 008), whle some others held ther own opnon (e.g. Aghon et al., 005; Sacco & Schmutzer, 0). What s the relatonshp between nnovaton and competton? Dfferent people have dfferent conclusons, so we do not try to gve the fnal answer to that queston. The purpose of ths study s to make efforts to reveal that relatonshp under some specal condtons.ths study addresses the nnovaton of the only frm wth cost advantage, whle n some lterature, all frms launch nnovatve nvestment. For example, Sacco and Schmutzer (0) dscussed that all frms launch nnovaton. Takng spllover nto account, D aspremont and Jacquemn (998) explored both cooperatve and non-cooperatve nnovaton theory. Wang and Yang (00) further developed ths cooperatve nnovaton theory under a vertcally related market structure. The man contrbutons of ths study le as follows. Frst, consderng externaltes 3 (ncludng substtutablty and *Correspondng author. Emal: chenyhua463.com 04 The Author(s). Publshed by Taylor & Francs. Ths s an Open Access artcle dstrbuted under the terms of the Creatve Commons Attrbuton Lcense mons.org/lcenses/by/3.0/, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. The moral rghts of the named author(s) have been asserted.

3 Economc Research-Ekonomska Istražvanja 33 complementary), ths artcle examnes Arrow s (96) nnovaton theory. Second, takng externaltes nto account, ths study expands Vves s (008) research. Consderng the effects of externaltes, ths study fnds that nnovatve nvestment competton s fercer than that under product quantty competton n Vves s (008) study. Fnally, ths study expands the theory about substtutablty. Substtutablty not only effects product competton, but also effects frm s nnovaton nvestment. Takng product externaltes, ncludng substtutablty and complementary nto account, both under Cournot quantty competton and Bertrand prce competton, ths study further addresses the relatonshp between nnovaton and competton. Ths study argues that complementary and substtutablty have major effects on the relatonshp between nnovaton and competton. Market power has negatve effects on nnovatve nvestment. Both total quantty of products and socal welfare are reduced wth substtutablty. Hgh market power mproves the total quantty of products and socal welfare. Moreover, hgher substtutablty reduces profts of frms wth hgher costs of producton. Furthermore, we fnd that socal welfare under equlbrum s lower than under optmalty. Besdes, by comparng Bertrand wth Cournot, ths study argues that Bertrand competton s more ntense than that of Cournot competton. 4. Lterature revew Many people focus ther attenton on the relatonshp between nnovaton and competton. So before statng our own research, we wll ntroduce some major pror studes. Followng Arrow s dea, there exst many mportant papers. Recently, Holmes, Levne and Schmtz (008) developed a theory of swtchover dsruptons under the monopoly market structure, and ther conclusons support Arrow s dea. Vves (008) developed an nnovaton theory under competton and hs study took market structure as a major fact. Chen and Sappngton (00) further dscussed nnovaton under vertcally related market structures. Vves (008) and Chen and Sappngton (00) drew the same conclusons as Arrow (96). On the other hand, Arrow s proposal has been ntensely debated for many years. Perhaps the most famous crtque came from Demsetz (969) and Y (999). Demsetz (969) and Y (999) dscussed that Arrow s dea manfests that ncreased competton yelds less nnovaton. Another mportant crtque s proposed by Glbert and Newbery (98), n whch Arrow s assumptons were changed and t showed that a monopolst has a greater ncentve to adopt new technologes. In other words, those earler studes hold the smlar vewpont as Schumpeter (94). More sgnfcantly, some other people took the strategc effects of nnovaton nto account. Maybe the most mportant research came form Brander and Spencer (983) and Spence (984). Strategc effects means cost reducton nnovaton reduces output of ts compettors. Brander and Spencer (983) declared that strategc effects ncreases total amount of R&D and total output. Spence s (984) research consdered spllover of R&D. Then Bester and Petraks (993) nvestgated how product substtutablty nctes cost reducton nnovaton both under Betrand competton and Cournot competton. In ther model, two frms acted under a socal planner. As a result, two frms should produce heterogenety products, or the socal planner would only operate the more effcent frm. And they ssued that one cannot draw a general concluson about the relatonshp between nnovaton and substtutablty. That two frms should act under a socal planner could also be seen as a drawback of Bester and Petraks s (993) study, because all

4 34 Y.-h. Chen and P.-y. Ne conclusons n ther study were constraned n plannng economcs. More nterestngly, Ne and Chen (0) nvestgated doupoly competton wth nput constrants. In theoretcal studes, there also exsts plenty of mportant lterature about the relatonshp between nnovatve nvestment and compettve pressure. Arrow (96) ntally advanced that monopolstc ndustres would be less nnovatve than compettve ones. Aghon et al. (005) further examned an nverted-u relatonshp between competton and nnovaton. Schmutzler (007) dscussed the relatonshp between R&D and competton wth a two-stage model. Under a lnear demand functon, Sacco and Schmutzer (0) confrmed the U-shaped relatonshp between competton and nnovaton wth numercal smulatons. Narajabad and Watson (0) studed dynamc nnovaton under producton dfferentaton. Furthermore, Regardng emprcal conclusons, usng the Statstcs Canada 999 Survey of Innovaton, Tang (006) examned that ths relatonshp s affected by many factors, such as substtutablty, constant arrval of competng products and so on. Tang (006) confrmed that market power has negatve effects on the nnovatve nvestment under substtutablty. Patel and Ward (0) estmated competton n nnovatve market wth patent ctaton patterns. Summarsng from those pror studes, we obtan the conclusons that there are three knds of ssues about the relatonshp between nnovaton and competton. Frst, competton stmulates nnovaton. Second, competton nhbts nnovaton. And the relatonshp between them s U or an nverted-u shape. But we thnk that the relatonshp between nnovaton and competton are depended. On the one hand, the relatonshp between nnovaton and competton depends on the descrptons of nnovaton, such as cost-reducng or qualty-nprovng nnovaton. On the other hand, whch depends on the descrptons of competton, such as market concentraton or substtutablty. So we wll nvestgate the relatonshp between nnovaton and competton under the specal state that frms produce wth dfferent effcency and only the low cost frm nvests nnovaton, but not to jon n the debate. Dfferent from other studes, the value nterval of γ n our study s from to, whch means γ can been regarded as substtutablty as well as complementary. The rest of ths artcle s organsed as follows. The model s establshed n Secton 3. In ths model, complementary and substtutablty along wth nnovatve nvestment are ntroduced. Then model analyses are present n Secton 4, whch contans two parts. The model of Cournot quantty competton s analysed n part A. By analysng the model, some useful conclusons are drawn. The model of Bertrand prce competton s addressed n part B. The prce and socal welfare under Bertrand are compared wth those under Cournot. Some remarks are presented n the fnal secton. 3. Model The model of duopoly nnovaton s establshed. Dfferent from other studes (Aghon, et al., 005; Sacco, Schmutzler, 0; Schmutzler, 007), ths study apples a one-stage model. 5 In our model, product externalty s fully addressed. Assume that there are two producers, and denoted by N = {, }. q represents the product quantty of frm, =,. Gven the prces of the two frms, p and p, the representatve consumer s (net) utlty functon s outlned by the followng functon. uðp ; p ; q ; q Þ¼Aðq þ q Þ p q p q ðq þ q Þ cq q : ()

5 Economc Research-Ekonomska Istražvanja 35 In the above, A > 0represents the maxmum prce that consumers are wllng to pay for ether good, and γ [, ] denotes the externalty parameter. The nverse lnear demand functons are outlned as follows, whch are drectly obtaned from (). p ¼ A q cq ; p ¼ A q cq : () γ = 0 means that two goods are ndependent; γ = ndcates perfect substtutes; γ = sgnfes perfect complements of the frms products. Furthermore, γ [, 0) means that one frm has postve externalty on the other frm, or both frms beneft from each other s producton, whle γ [0, ] represents that the products of the two frms are substtutes. We note that () s employed n Lu and Wang (Ln and Henry, 03), and Sacco and Schmutzer (0) wth γ [0, ]. The parameter γ [, ) s also regarded as the externalty of one frm exerted on the other. Ths study extends the models of Lu and Wang (0) and Sacco and Schmutzer (0). Intally, the frms margnal costs of producton are c and c, respectvely. We assume that c = c + τ, where τ > 0 denotes the cost advantage or market power of the frst frm. The frst frm launches nnovatve nvestment I at the same tme when t make output decson, 6 whle the second frm makes no nnovatve nvestment. Ths s the benchmark model of nnovaton of Arrow (96) or Holmes, Levne and Schmtz (008). The cost assocated wth nnovatve nvestment I s I. Proft functons of the two frms are gven as follows. p ¼½p c ði ÞŠq I ; (3) p ¼ðp c Þq : (4) Wth nnovatve nvestment I of the frst frm, the margnal cost of the frst frm becomes c (I ), where c (0) = c and c (I ) s convex and contnuously decreasng n I. We further assume that c ði Þ [. Snce the reducton n margnal cost ncurred by nnovaton s not too much, ths hypothess s reasonable. To smplfy the problem, we further assume that c (I )=c g(i )=(c τ)g(i ), where g(i ) > 0 s contnuously decreasng for all I and convex. Obvously, g(0) =. 4. Model analyses To compare dfferent models, we analyse the duopoly model based on quantty (or Cournot) competton and prce (or Bertrand) competton. Most conclusons are smlar n the two models, whch mean the conclusons are robust. Furthermore, we fnd that Bertrand competton seems fercer than Cournot competton. A. Results under Cournot quantty competton. The model s addressed n ths secton. The equlbrum s outlned and charactersed under Cournot. Under Cournot competton, the two frms compete n quantty. From () (4), proft functons are restated as follows: p C ¼½A q cq c ði ÞŠq I ; (5) p C ¼ðA q cq c Þq : (6)

6 36 Y.-h. Chen and P.-y. Ne The two frms maxmse (5) and (6), respectvely by choosng q, I and q. The equlbrum s dscussed next. Snce c (I ) s convex and contnuously decreasng, both (5) and (6) are concave functons n the respectve choce varables. Therefore, there exsts a unque equlbrum to (5) (6). The equlbrum s determned by the followng frst-order optmal condtons. p C q ¼ f ¼½A cq c ði ÞŠ q ¼ 0: (7) p C c ði Þ ¼ f ¼ q I ¼ 0: (8) p C ¼ f 3 ¼ðA cq c Þ q ¼ 0: (9) Equaton (8) manfests that the margnal cost ncurred by nnovatve nvestment s exactly equal to the margnal beneft caused by nnovatve nvestment. Obvously, from (7) and (9) we have the relatonshp that q > q. By vrtue of (7) (9), we have the followng concluson Proposton. Gven c, a larger cost advantage yelds lower nnovatve nvestment and hgher total quantty n the ndustry. Proof: See Appendx. Remarks: We have concluded that market power has a negatve relatonshp wth nnovatve nvestment under substtutablty, successfully explanng the emprcal results of Tang (006). Ths concluson s consstent wth those n Vves (008) and Arrow s dea (96). Conclusons of Proposton are more extensve than those of Tang (006). Apparently, q ncreases wth ts cost advantage whle q decreases wth frst frm s cost advantage. Besdes, q þ [ 0, whch means that the total products n ths ndustry are ncreasng wth cost advantage of the frst frm. Larger market power of the frst frm ncreases the quantty of products of the frst frm and the total quantty of products n the ndustry. Moreover, cost advantage of the frst frm has more effects on ts own products than that of the second frm because of j q j [ j j. The equlbrum quantty and prce are further descrbed next. (7) ndcates that q = p c (I ). (9) mples that q = p c. From (7) (9), we have Proposton. Under the equlbrum state, the total quantty of outputs q = q +q, q and p all decrease wth γ. Both I and q decrease wth γ f q > γq but ncrease wth γ f q γq. Proof: See Appendx. Remarks: The above proposton llustrates that the total quantty of products, q and p decrease wth the externalty parameter. Obvously, accordng to Proposton, the frst frm s nnovatve nvestment and ts quantty are reduced wth postve externalty (product complementary or γ < 0). The total quantty of producton of the two frms s further dscussed. Accordng to the above analyss, we have the nterestng relatonshp q c þ c \0, whch means that as γ ncreases, total outputs of the ndustry decrease. Because of larger substtutablty parameter, competton n the ndustry becomes fercer and the total demand for

7 Economc Research-Ekonomska Istražvanja 37 products n ths ndustry s reduced. By the proof of Proposton, we also have the sgnfcant relatonshp that j q c j\j c j; whch ndcates that γ has more effects on the quantty of the second frm than that of the frst frm. Here, the profts of two producers are remarked. By the envelope theorem, we have the followng concluson. Proposton 3. Under equlbrum state, the profts of the second frm satsfy: pc c \0: Proof. See Appendx. Remarks: Wth a large γ, the two frms compete drastcally and the second frm undertakes a loss. In other words, γ reduces the proft of the second frm. It s not sure about the effect of parameterγon the frst frm s profts. Actually, from (), (7) (9), we mmedately have the followng equatons p C ¼ q I and pc ¼ q. Obvously, pc c \0. It s dffcult to determne the effects of γ on p C ¼ q I : Socal welfare (SW) s addressed next. Socal welfare (SW) s the sum of consumer surplus (CS) and producer surplus (PS). CS s gven by () and PS s gven by (3) and (4). SW ¼ CS þ PS ¼ Aðq þ q Þ ðq þ q Þ cq q c ði Þq I c q : (0) The socal optmal soluton s determned by (0). The frst order optmal condtons are gven as follows: SW q ¼ A q cq c ði Þ¼0 () SW ¼ q c ði Þ I ¼ 0; () SW ¼ A q cq c ¼ 0: (3) Comparng (7) (9) wth () (3), we draw the followng conclusons: Proposton 4. The quantty of products under duopoly s lower than that of socal optmalty. Under equlbrum based on (7) (9), socal welfare n the ndustry s ncreased wth market power whle reduced wth substtutablty. Proof: Denote the soluton of (7) (9) by ðq C; ; q C; ; I C; Þ and the correspondng socal welfare by SW C,*. Let the socal optmal soluton to () (3) be ðq ; q ; I Þ and the correspondng socal welfare s S W. Apparently, A q C; cq C; c ði C; Þ¼q C; and A q C; cq C; c ¼ q C;. Therefore, q [ q C; and q [ q C;. From (), we have I [ I C;. Gven c, we have SW SW q ðq ; q ; I Þ¼ q þ SW þ SW c ði Þ ðq q ; q ; I Þ ¼ ðq þ q Þ ðq ; þ gði Þq ; q ; I Þ [ 0 and

8 38 Y.-h. Chen and P.-y. Ne SW SW q c ðq ; q ; I Þ¼ q c þ SW c þ SW c q q ðq ; q ; I Þ ¼ ðq þ q Þ ðq q q c ; q ; I Þ\0: Therefore, the socal welfare n ths ndustry s ncreased wth market power whle reduced wth hgher substtutablty or lower complementary. Conclusons are therefore acheved and the proof s complete. Remarks: Proposton 4 mples that the equlbrum products of the two frms are lower than those of the socal optmum. There exsts undernvestment n nnovaton. Under equlbrum, socal welfare s ncreased wth market power because of lower costs of the frst frm. Socal welfare s reduced wth hgher substtutablty or lower complementary because larger parameter γ yelds more ferce competton. From Propostons 4, we learn that the frms beneft from nnovaton and socal welfare s also promoted by nnovaton, whch means government and frms should share the cost of nnovaton. That s the reason why the government spends a lot of ther revenue n frms nnovatons every year. Accordng to (7) (9) and () (3), governmental subsdes n nnovaton can effcently mprove quantty of products and socal welfare. B. Results Under Bertrand Prce Competton. Here the Bertrand prce competton s dscussed, n whch the two frms compete n prces. If γ, () s restated as follows: q ¼ Að cþ p þ cp c ; q ¼ Að cþ p þ cp c : (4) If γ =,p = p under Bertrand competton, whch s an exstng concluson n the lterature. So γ = s not consdered n the followng. Hence, the substtutablty s partal n ths secton: By (3), (4) and (4), the two frms am to solve the followng problems: max p B p ;I ¼½p c ði ÞŠ Að cþ p þ cp c I ; (5) max p B p ¼ðp c Þ Að cþ p þ cp c : (6) Snce both (5) and (6) are concave, there exsts a unque soluton whch s determned by the frst order condtons of (5) (6). The correspondng frst order optmal condtons are lsted as follows: p B ¼ g ¼ Að cþþcp þ c ði Þ p c p ¼ 0; (7) c p B ¼ g ¼ Að cþþcp p c c ði Þ I ¼ 0; (8)

9 Economc Research-Ekonomska Istražvanja 39 p B ¼ g 3 ¼ Að cþþcp þ c p c p ¼ 0: (9) c Consderng (7) (9) and checkng Propostons 4, we mmedately have the followng conclusons. Proposton 5. Propostons 4 all hold under Bertrand prce competton. Proof: Smlar to the proofs of Propostons 4. Based on (7) (9), the correspondng conclusons are drawn. Remarks: The above proposton manfests that the relatonshps between market power and prce as well as quantty under Bertrand competton are smlar to those under Cournot competton. Denote the equlbrum under Bertrand competton by ðp B; ; pb; ; I B; ; q B; ; qb; Þ and the correspondng socal welfare by SW B,*. We then have the followng relatonshps. Proposton 6. Comparng the equlbrum under Bertrand wth that under Cournot, we have p C; > p B; ; pc; > p B;, qc; 6 q B; ; qc; 6 q B; ; I c; 6 I B; and SW C; 6 SW B; 6 S W : Proof: See Appendx. Remarks: The above conclusons llustrate that Bertrand competton seems to be much fercer than Cournot competton both n prces and n nnovatve nvestment. Therefore, prces under Bertrand are lower than those under Cournot, whle quantty of outputs, nnovatve nvestment and socal welfare under Bertrand are all hgher than those under Cournot. Here an explanaton s presented about the above conclusons. Under Bertrand competton, frms compete drectly and ths competton seems fercer than Cournot competton. Therefore, prces under Bertrand are lower and quanttes are hgher than those under Cournot. 5. Concludng remarks The conclusons n ths artcle are consstent wth some emnent pror research, ncludng Arrow (96), Qu (997), Tang (006), and Vves (008), although ths research uses dfferent models and dfferent competton structures. That means the conclusons are qute robust. Ths artcle fnds that nnovaton decreases wth market power, 7 the same as Arrow (96). It also fnds hgher substtutablty yeld lower total quantty of products n the ndustry. Usng the Bowley lnear demand system, Vves (008) and Qu (997) reach smlar conclusons. Ths research reaches the same concluson as Qu (997) that prce s lower and output s larger n Bertrand than n Cournot competton. And you can also fnd emprcal support of the conclusons n Tang (006). And dfferent from Bester and Petraks (993), we need no socal planner. Ths study addresses the relatonshp between nnovaton and competton under product externalty based on Arrow s (96) nnovaton theory. For the dfferent measures of market power, we measure t wth cost advantage whle Vves (008) wth Lerner ndex, ths artcle contrasts wth Vves s (008) concluson about the effect of market power on nnovaton. By a duopoly model both under quantty (Cournot) and prce (Bertrand) competton, the conclusons, n whch market power has negatve effects on nnovatve nvestment, supports the emprcal evdence n Tang (006). Hgher substtutablty yelds lower total quantty of products n the ndustry and lower

10 40 Y.-h. Chen and P.-y. Ne profts of the frm wth hgher costs of producton. The equlbrum products and nnovatve nvestment are all lower than socal optmum. Socal welfare s also reduced wth hgher substtutablty because a larger externalty parameter yelds fercer competton. These conclusons are robust under dfferent models. There are some further researchng topcs followng ths work. Ths study dscusses the nnovaton of frms wth cost advantage and t s nterestng to extend to general stuatons. When two frms smultaneously launch nnovatve nvestment, t s nterestng to capture. Besdes, f postve externalty (or product complementary) s strong enough, frms wll be co-nnovaton or free-rdng. These are our further research topcs. Acknowledgement Our sncere thanks to the edtor and the anonymous revewers for ther helpful suggeston to mprove the presentaton. Fundng Ths work s partally supported by Hgh-level Talents n Hgher Educaton of Guangdong, GDUPS [0], Natonal Natural Scence Foundaton of PRC [grant number 7700, ], The Guangdong Socal Scence Foundaton [grant number GD3YLJ0], Natonal Socal Scence Fund [grant number 4AJY00] and Key Program of Natonal Natural Scence Foundaton of Chna [grant number ]. Notes. Cost advantage can also be seen as market power. A frm wth cost advantage has the power to change prce frst. The prce s gven n ths study, but cost advantage can be use as competton threat to the compettor.. Only the cost advantage frm nnovates s reasonable and the reason for the cost dsadvantage one does not nnovate s that t does not have enough money to nnovate or the revenues are less than ts costs when t nnovates. 3. Notng that externaltes here mean the nteracton between products produced by dfferent frms, ncludng substtutablty and complementary, whch are dfferent from the externalty of nnovaton n of other studes such as Spence (984) and Bester and Petraks (993). 4. Though Bertrand competton s fercer than Cournot competton, the conclusons under both models are robust. 5. On the one hand, only one frm nvests nnovaton n our model. On the other hand, we assume that frm makes nnovaton and output decson at the same tme (or the frm makes output decson soon after nnovaton). So we set up a one-stage game model. 6. Whch means we assumes smultaneous decson n the nnovaton stage between R&D and producton (non-strategc R&D). 7. Ths study gets the same concluson as Arrow (96) but market power n ths artcle s measured n dfferent way from Arrow. References Armn, S. (007). The relaton between competton and nnovaton why s t such a mess. Mmeo, Nov: Unversty of Zurch Borghan, N., & Randal, W. (0). The dynamcs of nnovaton and horzontal dfferentaton. Journal of Economc Dynamcs & Control, 35, Claude, D., & Alexs, J. (998). Cooperatve and noncooperatve R&D n duopoly wth spllovers. Amercan Economc Revew, 78, Daro, S., & Armn, S. (0). Is there a U-shaped relaton between competton and nvestment? Internatonal Journal of Industral Organzaton, 9,

11 Economc Research-Ekonomska Istražvanja 4 Darshak, P., & Mchael, W. R. (0). Usng patent ctaton patterns to nfer nnovaton market competton. Research Polcy, 40, Harold, D. (969). Informaton and effcency: Another vewpont. Journal of Law and Economcs,,. Helmut, B., & Emmanuel, P. (993). The ncentves for cost reducton n a dfferentated ndustry. Internatonal Journal of Industral Organzaton,, Henry, W. X., & Bll, Y. Z. (00). Cooperatve and noncooperatve R&D n vertcally related markets. Seoul Journal of Economcs, 5, 4. Holmes, Thomas J., Levne, Davd K., & Schmtz, James A, Jr. (0). Monopoly and the ncentve to nnovate when adopton nvolves swtchover dsruptons. Amercan Economc Journal- Mcroeconomcs, 4, 33. James, B. A., & Barbara, S. J. (983). Strategc commtment wth R&D: The symmetrc Case. The Bell Journal of Economcs, 4, Jan-mn, T. (006). Competton and nnovaton behavor. Research Polcy., 35, Kenneth, A. (96). Economc welfare and the allocaton of resources for nventons. In R. Nelson (Ed.), The Rate and Drecton of Inventve Actvty (pp ). Prnceton, NJ: Prnceton Unversty Press. Larry, Qu D. (997). On the dynamc effcency of Bertrand and Cournot equlbra. Journal of economc theory, 75, 3 9. Ln, L., & Henry, W. X. (03). Entry bas and product substtutablty, Journal of Insttutonal and Theoretcal Economcs, 69, Phlppe, A., Nck, B., Rchard, B., Rachel, G., & Peter, H. (005). Competton and nnovaton: An nverted-u relatonshp. Quarterly Journal of Economcs, 0, Pu-yan, N., & You-hua, C. (0). Duopoly compettons wth capacty constraned nput. Economc Modellng, 9, Rchard, G. J., & Davd, N. M. G. (98). Preemptve patentng and the persstence of monopoly. Amercan Economc Revew, 7, Sang-Seung, Y. (999). Market structure and ncentves to nnovate: The case of Cournot olgopoly. Economc Letters, 65, Schumpeter Joseph Alos. (94). Captalsm. New York: Socalsm and Democracy. Spence, M. (984). Cost Reducton, Competton, and Industry Performance. Econometrca, 5, 0. Xaver, V. (008). Innovaton and compettve pressure. The Journal of Industral Economcs, 56, Yongmn, C., & Davd, S. E. M. (00). Innovaton n vertcally related markets. The Journal of Industral Economcs, 58, Appendx Proof of Propostons Denote the Jacoban matrx of (7) (9) to be: f f f 3 3 q q q 6 f D ¼ ¼ 4 Denote 6 D s ¼ 4 f f 3 f f f 3 f q f q f 3 q f f f f f 3 f ¼ 4 cðiþ cðiþ 3 c 7 5 q c ði Þ ði Þ 0 c 0 3 ðiþ c q c ði Þ c 0 cðiþ

12 4 Y.-h. Chen and P.-y. Ne h h Apparently, ¼ q c ði Þ \0. þ ð 4 þ c Þþ cðiþ ði Þ Ths nequalty comes from the hypotheses that 0 [ cðiþ [ and γ [, ]. By the mplct functon theorem, there exsts a unque soluton to (7) (9), whch s dfferentable. Meanwhle, where ¼ s \0; s ¼ 4q c ði Þ q ¼ ¼ ¼ 4q dgði Þ di ¼ dgðiþ di 6 det4 det6 4 þ q c ði Þ c þ cðiþ c ði Þ gði Þ dgði q Þ c cðiþ ½4q q c c gði ÞŠ\0: cðiþ cðiþ q c ði Þ 0 q c ði Þ ði Þ 0 c 0 cðiþ cðiþ cðiþ q c ði Þ ði Þ 0 q c ði Þ 0 Obvously, we have ðqþqþ [ 0: Conclusons are therefore acheved and the proof s complete [ 0; 3 c 7 5 \0: Proof of Propostons Accordng to (7) (9) and the analyss n Proposton, we have 3 q 0 q 6 det cðiþ q c ði Þ ði Þ 5 q c ¼ c 0 h q c ði Þ ðq ði ¼ Þ cq Þ \0 q [ cq > 0 q 6 cq : Therefore, q decreases wth γ f q > γq whle ncreases wth γ f q γq. cðiþ 3 c 6 det cðiþ q c ði Þ ði Þ c ¼ q 0 q h h q c ði Þ c ðq ði ¼ Þ cq Þþq ði Þ \0: The above nequalty follows from q > q, γ < and j cðiþ j\ (or 0 [ cðiþ [ ). Apparently, because γ <, we have q c þ q c \0: Combned wth q = p c, we therefore have that both q = q + q and prce p decrease wth γ.

13 3 cðiþ c det4 q 0 q 5 c ¼ c 0 ¼ c ði Þ ðq cq Þ \0 q [ cq 0 q cq I decreases wth γ f q > γq but ncreases wth γ f q γq. Conclusons are therefore acheved and the proof s complete. Proof of Propostons 3 We frst show the relatonshp q c \q based on the above analyss. h q q c ði Þ h ðq c ¼ ði Þ cq Þ q c ði Þ ðq ði h h \ Þ cq Þ h q c ði Þ ð4 c Þþ cðiþ ði Þ q c ði Þ ð c Þ \q : ði Þ We therefore have p c ¼ q q q cq c \0. Conclusons are therefore acheved and the proof s complete. Proof of Propostons 4 Economc Research-Ekonomska Istražvanja 43 Here we show Proposton 4. By vrtue of (7) (9), we mmedately have the followng system of equatons, whch are the frst order optmal condtons of (7) (9). p B p ¼ pc q q p þ pc p þ pc p ¼ 0; pb ¼ q c ði Þ I ¼ 0; p B ¼ pc q þ pc þ pc ¼ 0: p q p p p Usng the above equaton, (), (5) and (4) yeld the relatonshp p C ðq B; ¼ 0: By vr- ;qb; B; ;I Þ \0. Accordng to the con- ¼ cq ;qb; B; ; q ðq ;I Þ p B; tue of the equaton pb ðq p B; cavty of p C, p C ðq q B; q C; q B; ¼ c ;qb; B; ;I Þ c ; q ðq p B; ;qb; ;qb; B; ;I Þ ;qb; B; ;I Þ ¼ 0, we therefore have pc \0, and pc. Smlarly, we have qc; q B; ðq q C; ;q C; ;I C; Þ. ¼ B; ;I Þ c : and pc ðq B; ðq q B; ;qb; B; ;I Þ ¼ 0, we mmedately have the relatonshp From (8), q C; q B; ; qc; q B; : and the convexty of c (I ) yelds. I c; I B;. q C; q B; ; q C; q B; ; I c; I B;. and () mply p C; > p B; and p C; > p B; : p C; > p B; ; pc; > p B;, qc; q B; ; qc; q B; and I c; I B; : Induce SW C,* SW B,*. Combnng wth the concluson of Proposton 5, we have:sw B; S W. Conclusons are therefore acheved and the proof s complete.