MPRA Munch Personal RePEc Archve Welfare Analyss of Cournot and Bertrand Competton Wth(out) Investment n R & D Jean-Baptste Tondj Unversty of Ottawa 25 March 2016 Onlne at https://mpra.ub.un-muenchen.de/75806/ MPRA Paper No. 75806, posted 25 December 2016 01:32 UTC
Welfare Analyss of Cournot and Bertrand Competton Wth(out) Investment n R & D Jean-Baptste Tondj December 24, 2016 Abstract I consder the model of a dfferentated duopoly wth process R&D when goods are ether substtute, complements or ndependent. I propose a non-cooperatve two-stage game wth two frms producng dfferentated goods. In the frst stage, frms decde ther technologes and n the second stage, they compete n quanttes or prces. I evaluate the socal welfare wthn a framework of Cournot and Bertrand competton models wth or wthout nvestment n research and development. I prove that the Cournot prce can be lower than Bertrand prce when the R&D technology s relatvely neffcent; thus, Cournot market structure can generate larger consumer s surplus and welfare. JEL classfcaton codes: L13, D60, O32. Keywords: R&D, Cournot duopoly, Bertrand model, Welfare. Department of Economcs, Unversty of Ottawa, jtond063@uottawa.ca
2 1 Introducton The comparson of Cournot and Bertrand results n a statc olgopoly settng have extensvely been studed n the lterature. Ths paper focuses on welfare under Cournot and Bertrand competton n dfferentated olgopoles. From the lterature, t s known that Bertrand competton yelds lower prces and profts and hgher consumer surplus and welfare than Cournot competton (e.g; [2],[8],[9],[10] and [14]). Accordngly, explotng cost asymmetres ncludng some other factors lke research and development (R & D) or endogenous prvatzaton; some researchers lke [1], [4], [5] and [6] have constructed models where at least one of the latter conclusons fals to hold. One mportant scope of the regulator n the economy s to get a better model that maxmzes socal welfare (sum of total surplus net of total cost and other negatves externaltes). Welfare comparson wthn Cournot and Bertrand models can solve a socal planner s problem who wsh to choose a better model n market competton to mprove the well-beng of ndvduals n a socety. In ths paper, I consder a non-cooperatve, two-stage game wth two frms producng dfferentated goods. In the frst stage, frms ndependently decde ther R&D (no spllovers effects) nvestment that determnes producton technology and, n the second stage, they compete n quanttes or prces. The lnear demands are derved from the utlty maxmzng problem of the representatve consumer. Ths paper generalzes the work of [6] n the sense that R&D technology functon n ther paper s a specfc case of ths model. The man works of ths paper can be summarzed as follows: 1. Frstly, I compare the welfare under both models when frms are not nvestng n research and development and my results are consstent wth the lterature (e.g, [10]). As my frst contrbuton, I formally show that Cournot and Bertrand competton models equlbra concde f and only f the products are ndependent. Then, I compare the welfare when products are ether ndependent, substtutes or complements. 2. Secondly, takng nto account the nvestment n research and development makes the welfare comparson under both models more nterestng. After presentng the equlbrum outcomes of each model, I dscuss the well-beng of consumers under the two possbltes offered to the frms, nvestng or not n R&D. At the end, a dscusson s gven to evaluate the effects of R&D sze on dfferent equlbrum outcomes. These comparatve statcs lead to a condton on whch socal welfare s better under Cournot model relatve to Bertrand model. All the results found n ths framework can be classfed among stuatons where the poneerng fndngs of [10] do not necessarly hold. As my second contrbuton, I prove that Cournot prce can be lower than
3 Bertrand prce when R & D technology s relatvely neffcent; thus Cournot market structure can generate larger consumer s surplus and welfare. The remander of the paper proceeds as follows. Secton 2 provdes a bref lterature revew of comparng welfare through Cournot and Bertrand models usng R&D process. Secton 3.1 and 3.2 descrbe the equlbrums along wth ts mplcatons on the welfare wthn a framework of Cournot and Bertrand models. Secton 4 examnes the comparatve statcs. Secton 5 concludes. The man proofs are collected n the appendx. 2 Lterature revew In a compettve market, frms often compete aganst each other by nvestng n research and development n order to mprove product qualty (n the case of product R&D) and/or to reduce producton cost (n case of process R&D). The mportant dfference between process and product R&D as explaned by [13] s that the product R&D ectly affects gross consumer surplus. Ths s because product R&D rases product qualty, and qualty enters ectly nto each consumer s utlty functon. Process R&D affects gross consumer surplus only nectly through a reducton n margnal cost and a consequent ncrease n output. The tradtonal results (e.g; [2],[8],[9],[10], and [14]) mentoned above about the effcent equlbrum outcomes under Bertrand model relatve to Cournot model are generally obtaned under the assumpton that frms face the same demand and cost structures. In a dynamc envronment, f the R&D nvestments dffer across the Bertrand and Cournot models, the post-nnovaton demand and cost structures wll also dffer, even though they were dentcal before the nvestment. The queston whch follows s whether the tradtonal results are affected n any way. Gven the above dfference between process and product R&D, t s unclear whether the results from models wth process R&D carry over to the case of product R&D. Qu([9]) focused on cost-reducng R&D wth spllover effects and reevaluated the relatve effcency of Cournot and Bertrand equlbra. He consdered a non-cooperatve, two stage game wth two frms producng dfferentated goods. In the frst stage, whch s called R&D stage, each frm ndependently sets out cost-reducng R&D. In the second stage called market stage, both frms produce and compete n prces or quanttes. He found that Cournot competton nduces more R&D effort than Bertrand competton. The prce s low and output s larger n Bertrand than n Cournot model. The tradtonal effcency result holds f ether R&D productvty s low, or spllovers are weak, or
4 products are very dfferent. However, f R&D productvty s hgh, spllovers are strong, and goods are close substtutes, then Bertrand equlbrum s less effcent than the Cournot equlbrum. The latter challenged the tradtonal effcency result. Symeonds [13] studed a varant of the standard lnear demand model. He focused on product R&D and used the qualty-augmented lnear structure proposed by [11], [12] and ntroduced a cost structure that allows for R&D spllovers. He found almost the same results as [9]: Indeed, R&D expendture, prces and frm s net profts are always hgher under Cournot competton than under Bertrand competton. Output, consumer surplus and total welfare are hgher under prce competton than under quantty competton f ether R&D spllovers are weak or products are suffcently dfferentated. Furthermore, f R&D spllovers are strong and products are not too dfferentated, then output, consumer surplus and total welfare are lower n the Bertrand case than n the Cournot case. Motta [7] analyzed two versons of a vertcal product dfferentaton model, one wth fxed and the other wth varable costs of qualty. The case of fxed cost of qualty mprovement may be thought of as a stuaton where frms should engage n R&D and advertsng actvtes to mprove qualty. He found that economy s better off when frms compete on prces (wth fxed costs of qualty, not only consumer but also producer surplus s hgher under prce competton). Compare to the model of Symeonds ([13]), the study of Motta ([7]) dd not allow for R&D spllovers. Delbono and Dencolo ([3]) compared the equlbrum R&D nvestment under prce and quantty competton n a symmetrc and homogenous olgopoly. They found that the welfare comparson s generally ambguous n the context of a homogenous product duopoly wth process R&D n the form of patent race. They have shown that: although the R&D nvestment s greater under prce competton (n fact, t s even hgher than the socal optmal level), socal welfare net of R&D cost, may be greater under Cournot competton. Ths paper focuses on dfferentated duopoly wth process R&D wth no spllovers when goods are ether substtutes, complements or ndependent. The R&D functon consdered n ths settng s the generalzed form of the one used by [6] n a non-cooperatve two-stage game wth two frms producng dfferentated goods. In the frst stage, frms decde ther technologes and n the second stage, they compete n quanttes or prces. I am comparng the socal welfare (sum of net consumer surplus and total producer surplus) wthn a framework of Cournot and Bertrand competton models wth or wthout nvestment n R&D. The results are consstent wth those cted above (tradtonal results) and wth those found by [6] for the specfc case of R&D. At the equlbrum,
5 Cournot frms nvest a larger amount on R&D than the Bertrand frms. Contrary to the tradtonal results, Bertrand prce can be hgher than Cournot prce. It s proven that ths phenomenon happens n the case where the R&D technology s relatvely neffcent; thus Cournot market structure can generate larger consumer s surplus and welfare. 3 The model I consder a non-cooperatve, two-stage game wth two frms producng dfferentated goods. In the frst-stage, frms ndependently decde ther R&D nvestment that determnes producton technology. In the second stage, frms smultaneously and non-cooperatvely decde ther quantty or prces dependng on whether there s Cournot competton or Bertrand competton. I am usng backward nducton to solve the game, that s, frst solve the second stage problem and then the frst stage, takng the behavor of frms at the second stage. Assumng that the R&D technology s gven by : R(c ) r c e e Where e 1, r s a postve parameter, c the frms s margnal cost of producton. It shows that the hgher the value of r, the hgher s the R&D cost, the lower the effcency. Followng [10], I assume that the representatve consumer s preferences are descrbed by the utlty functon: U(q 1, q 2 ) a(q 1 + q 2 ) b(q 2 1 + 2θq 1 q 2 + q 2 2)/2 + m, (2) wth q s the quantty of frm s producton, a > 0, b [0, 1], m s the numerare good (composte good) and θ [0, 1[ s a postve parameter. To derve the demand functon, we need to solve the followng utlty maxmzaton problem: (1) { max U(q1, q 2 ) a(q 1 + q 2 ) b(q q 1,q 1 2 + 2θq 1q 2 + q2 2 )/2 + m} 2 s.t I p 1 q 1 + p 2 q 2 + m where I s the consumer s ncome. Rearrangng the constrant of maxmzng problem (3) by wrtng m I p 1 q 1 p 2 q 2 and substtutng nto utlty functon gves the followng equvalent maxmzaton problem: max q 1,q 2 { U(q1, q 2 ) a(q 1 + q 2 ) b(q 2 1 + 2θq 1q 2 + q 2 2 )/2 + I p 1q 1 p 2 q 2 }. (4) The frst order condton for maxmzaton problem s gven by: U(q, q j ) q a b(q + θq j ) p 0;, j 1, 2 j. (5) (3)
3.1 Cournot and Bertrand competton wthout nvestment n R&D 6 Solvng (5) for p gves : Solvng (6) for q yelds : q p a b(q + θq j ),, j 1, 2 j. (6) a b(1 + θ) 1 b(1 θ 2 ) p θ + b(1 θ 2 ) p j;, j 1, 2 j. (7) More generally, the demand functon for good derved from ths type of utlty functon can be wrtten n the form: q α βp + γp j, j 1, 2; j. (8) I assume that β > γ and α > (β γ)c. In ths specfc case, I have : α a b(1 + θ), β 1 b(1 θ 2 ), γ θ b(1 θ 2 ). (9) These notatons wll be useful later. In what follows, I calculate the optmal output and prces under Cournot competton model and Bertrand model wth or wthout nvestment n R&D. In each competton model, I derve the optmal socal welfare and make some comparsons. 3.1 Cournot and Bertrand competton wthout nvestment n R&D 3.1.1 Cournot model wthout nvestment n R & D In ths set up, R(c ) 0, so that there s a one short game where frms maxmze smultaneously ther profts, each frm takng the quantty of other frm as gven. From the fact that the demand functon s gven by equaton (6), the nverse demand functon for each frm s gven as follows: p For smplfcaton, I wll denote ω α β γ α β γ, φ γ β 2 γ 2 q j γ β 2 γ 2, λ β β 2 γ 2 q, j. (10) β β 2 γ 2. (11) The values of ω, λ, φ are postve and the nverse demand functon can be rewrtten as follows: p ω φq j λq, j. (12) I assume that frms have the same margnal cost.e c c j c and c > a, so the followng lemma provdes the Cournot equlbrum prce, quantty and proft for each frm. Lemma 3.1. The Cournot equlbrum (q c, p c, π c ) s gven by q c π c 1 b [ a c 2+θ ]2. a c b(2+θ), pc a+c(1+θ) 2+θ and
3.1 Cournot and Bertrand competton wthout nvestment n R&D 7 From Lemma 3.1, the equlbrum prce s hgher than the margnal cost (p c c a c 2+θ > 0) and each frm makes postve proft. The equlbrum consumer surplus CS U(q 1, q 2 ) p 1 q 1 p 2 q 2, total producer proft (Π π 1 + π 2 ), and welfare (W total surplus) n Cournot competton are gven as follows : CS c (1 + θ) ( a c b 2 + θ )2 + m ; Π c 2 b ( a c 2 + θ )2 ; W c CS c + Π c (3 + θ) ( a c b 2 + θ )2 + m (13) 3.1.2 Bertrand model wthout nvestment n R&D In ths framework, I stll have R(c) 0, and frms solve smultaneously ther usual proft-maxmzaton problem, n order to obtan ther optmal prces, each frm takng the prce of other frm as gven. The fact that I assumed the same constant margnal cost mples that, at equlbrum, frms produce the same quantty, sell at the same prce and earn the same postve proft. The followng lemma generalzes the Bertrand equlbrums quantty, prce and proft. Lemma 3.2. The Bertrand equlbrum (q B, p B, π B ) s gven by : p B a(1 θ)+c 2 θ, q B a c b(1+θ)(2 θ) and πb 1 θ b(1+θ) [ a c 2 θ ]2. Accordng to lemma 3.2, p B c (1 θ)(a c) 2 θ, a > c and θ < 1, t follows that each frm sells products at the prce above margnal cost and earns a postve economc proft. The equlbrum consumer surplus CS B, the total producer proft Π B, and welfare (W B total surplus) n Bertrand competton are gven by: CS B 1 b(1 + θ) ( a c 2 θ )2 +m ; Π B 2(1 θ) b(1 + θ) ( a c 2 θ )2 ; W B CS B +Π B (3 2θ) b(1 + θ) ( a c 2 θ )2 +m The results (Lemmas 3.1 and 3.2) found above clearly show that the equlbrum at the market s dfferent n both Cournot and Bertrand models. In fact, n Cournot competton model, frms fnd the optmal quanttes to maxmze ther profts rather than solvng for the optmal prces as n the Bertrand competton model. Hence, socal welfare s dfferent accordng to each market equlbrum. What happens to these dfferent levels of socal welfare f both equlbrum outcomes concde? We mght be nterested n lookng under whch model, socety s better off, Cournot or Bertrand competton? In what follows, I provde one condton for whch the equlbrum outcomes concde and I study the consequences of ths condton on socal welfare. Theorem 3.1. The Cournot equlbrum and the Bertrand equlbrum concde f and only f the goods are ndependent,.e the parameter θ n the lnear demand functon (6) s zero. (14)
3.1 Cournot and Bertrand competton wthout nvestment n R&D 8 The proof of ths theorem s provded n the appendx. The result of Theorem 3.1 s consstent wth the tradtonal results found by [10] that at the equlbrum market, prces are lower and outputs are hgher under Bertrand competton compared to Cournot competton for substtutes or complements goods. Moreover, profts are larger, equal or smaller n Cournot than n Bertrand competton, dependng on whether the goods are substtutes, ndependent, or complements. Here, wth goods that are substtutes, I fnd that the hghest proft s acheved at the Cournot equlbrum. Notce that f the parameter θ 0, I can rewrte the demand functon as follows: q 1 b (a p ) and q j 1 b (a p j). The demand functon n ths case are ndependent. We have two monopoles frms n the market whch produce the same quantty q M a c 2b of dfferent products, sell them at the same monopoly prce p M a c 2 and earn the same proft π M 1 (a c) 2 b 4. At ths equlbrum, the consumer surplus s SC M 1 b ( a c 2 )2 + m, the total producer surplus Π M 2 b ( a c 2 )2 and the socal welfare W M 3 b ( a c 2 )2 + m. Frms earn hgher proft under monopoly competton than Bertrand or Cournot competton. Less competton n the market leads to hgher prces, lower producton and eventually to lower consumer surplus. Socety s better off under Bertrand competton or Cournot competton compared to monopoly market. It can be proved that CS M < CS B, CS M < CS c, Π M > Π B, Π M > Π c, W M < W B and W M < W c. The followng proposton generalzes the welfare comparson between Cournot and Bertrand models wthout nvestment n research and development. Proposton 3.1. CS B CS c (a c)2 b θ 2 [2 + θ + (1 + θ)(2 θ)] (1 + θ)(4 θ 2 ) 2, (15) and Π c Π B 4(a c)2 b θ 3 (1 + θ)(4 θ 2 ) 2, (16) W B W c (a c) 2 b(1 + θ)(4 θ 2 ) 2 θ2 (θ 2 + 2θ 4). (17) Snce θ [0, 1[ and a > c, usng the equatons (15), (16) and (17), I have CS B CS c, Π c Π B and W B W c. Lower prces and hgher quanttes are always better n welfare terms. Consumer surplus s decreasng and convex as a functon of prces. Therefore, n term of consumer surplus, the Bertrand equlbrum domnates the Cournot equlbrum as proved n equaton (15). Gven that the goods are substtutes, low prces mean lower proft whch mples that Cournot s total producer surplus s hgher than Bertrand s total producer surplus as shown by equaton (16). The same results were found by [10], that consumer surplus and total surplus U(q 1, q 2 ) are larger n
3.2 Cournot and Bertrand competton wth nvestment n R&D 9 Bertrand equlbrum than n Cournot competton except when the goods are ndependent. In ths latter case, consumer surplus and total surplus are equal under both competton models. Sngh and Vves ([10]) have shown that the converse of ths asserton s true f goods are complements, because n order to ncrease profts, frms have to lower prces from the Cournot levels to gan the market share. From equaton (17), the Bertrand equlbrum s more effcent than Cournot equlbrum. 3.2 Cournot and Bertrand competton wth nvestment n R&D 3.2.1 Cournot model wth nvestment n R&D In what follows, I use backward nducton to solve the subgame perfect equlbrum of the two-stage game descrbed at the begnnng of secton 3. Let π c π c denote frm s market proft at stage 2, then: (p c )q r c e e, where c means Cournot wth nvestment n R&D. At ths stage, frms choose smultaneously ther optmal level of producton takng the margnal cost as gven. So, frm solves the followng maxmzaton problem: { } max π c q (q, q j ) (p c )q r c e e s.t p a bq bθq j (18) In the frst stage, frms choose smultaneously ther optmal level of R&D cost consderng the optmal level of producton at the second stage. The followng proposton descrbes the equlbrum condtons of the game. Proposton 3.2. The condtons descrbng the two-stage Cournot equlbrum are summarzed n the followng equatons: 4 4 θ 2 qc rĉ e 1 (19) q c a ĉ b(2 + θ) (20) p c a + ĉ(1 + θ) 2 + θ (21) π c 1 b [ a ĉ 2 + θ ]2 r ĉ e e, e 1 (22) The proof of ths proposton s provded n appendx. From ths proposton, the equlbrum n symmetrc Cournot competton wth nvestment n R&D are as follows: consumer surplus CS c (1+θ) b (3+θ) b ( a ĉ ( a ĉ 2+θ )2 + m, the total producer proft Π c 2 b ( a ĉ 2+θ )2 2r ĉ e e, and welfare W c 2+θ )2 + m 2r ĉ e e, where ĉ s the optmal level of R&D cost chosen by frms at the frst stage.
3.2 Cournot and Bertrand competton wth nvestment n R&D 10 Usng equaton (14) n the Cournot model n secton 2, the welfare comparson under symmetrc Cournot model wth and wthout nvestment n R&D are gven by: CS c CS c 1+θ (c ĉ)(2a c ĉ) b(2+θ) 2 Π c Π c 2 (c ĉ)(2a c ĉ) 2r ĉ e b(2+θ) 2 e W c W c 3+θ (c ĉ)(2a c ĉ) 2r ĉ e b(2+θ) 2 e (23) Remark 3.1. It follows that CS c CS c f and only f c ĉ; Π c < Π c and W c < W c f ĉ c. If ĉ < c, the sgn of Π c Π c and W c W c are ambguous. A clear comparson of welfare can be provde when full nformaton about the sze of R and D cost R(ĉ) can be estmated. I may expect to have hgher level of welfare n case where ĉ < c because by nvestng more n R&D, frms lower ther margnal cost, produce more, and ths can ultmately mproves socal welfare. 3.2.2 Bertrand competton wth nvestment n R&D In ths secton, the demand curve s gven by equaton (7) or (8). I stll use backward nducton to solve the R&D problem maxmzaton as n the prevous secton. At stage 2, frms choose smultaneously prces to maxmze profts taken the R&D margnal cost as gven. Usng equaton (6), the frm s proft maxmzaton problem at stage 2 s set up as follows: { max p π B } (p, p j, c, c j ) (p c )q (p c )(α βp + γp j ) r c e, (24) e where B means Bertrand competton wth nvestment n R&D. In the frst stage, frms choose smultaneously ther optmal level of R&D cost, consderng the optmal level of producton at the second stage. I have the followng result: Proposton 3.3. The condtons descrbng ths two-stage Bertrand equlbrum are characterzed n the followng equatons: θ 2 b(1 θ 2 )(4 θ 2 ) (pb c) + a pb b(1 + θ) r c e 1 (25) p B a(1 θ) + c 2 θ (26) q B a c b(1 + θ)(2 θ) (27) π B 1 θ b(1 + θ) [ a c 2 θ ]2 r c e e, e 1 (28)
11 The proof of ths proposton s provded n the appendx. The equlbrum n symmetrc Bertrand competton wth nvestment n R&D are as follows: consumer surplus CS B 1 b(1+θ) ( a c 2 θ )2 + m, total producer proft Π B 2(1 θ) b(1+θ) ( a c 2 θ )2 2r c e e, and welfare W B 3 2θ b(1+θ) ( a c 2 θ )2 + m 2r c e e, where c s the optmal level of R&D cost chosen by frms at the frst stage. It s mportant to menton that the welfare comparson that I dd n the prevous secton can be done here as well. The better off stuaton depends on the gap (c c) between the dfferent margnal cost and /or the sze of R and D nvestment. Assumng that frms compete under Cournot competton, and frm nvests enough n the R&D n the frst stage. It turns out that ths nvestment n R&D reduces ts margnal cost at the second stage and ncreases ts output. Gven that the quanttes are strategc substtutes, the quantty produced by frm j s reduced, whch ncreases frm s proft. Gven that prces are strategc complements, I have the reverse result under Bertrand competton model. Proposton 3.4. In equlbrum, Cournot frms have larger ncentve to nvest n R and D,.e ĉ < c. The proof of ths result s gven n the appendx. Ths proposton s consstent wth the ones shown by Kabraj and Roy ([6]) for a specfc case R&D technology (when e 1). In the next secton, I study the effect of the sze of R&D technology on margnal costs, prces and socal welfare at equlbrum. 4 Comparatve statcs The followng comparatve statc result shows that, as the nvestment n research and development becomes more and more neffcent, margnal cost under each Cournot or Bertrand model ncreases, but t ncreases more under Bertrand competton. The followng lemma shows the varaton of optmal technology c and ĉ wth the sze r of R&D. We recall that these technologes depend on the parameter r. Hgher r mples that the R&D nvestment becomes more neffcent. The followng propostons also generalze the results found by [6]. Proposton 4.1 evaluates the effect of R&D technology sze on the equlbrum margnal costs of both models and Proposton 4.2 provdes the effect of nvestment n R&D technology on equlbrum prces. Proposton 4.1. dĉ d c dĉ > 0, > 0 and < d c. Proposton 4.2. dpc > 0, dpb > 0 and dpb dpc
12 The proofs of these propostons are provded n the appendx. The latter result (Proposton 4.2) descrbes the shape of curves p c (r) and p B (r). It shows that both curves are upward slopng wth p B (r) havng a greater slope than p c (r). From Proposton 4.2, I can derve the followng corollary about the comparson of prces under Cournot and Bertrand model wth nvestment n R&D. Corollary 4.1. There exsts r such that, p B > p c f and only f r > r. Ths corollary tells us that, f the R&D technology s neffcent (hgher r), then Cournot prce wll be lower than Bertrand prce and vce versa. Large r ncreases margnal cost (see proposton 4.1), but t ncreases more under Bertrand competton, leadng to hgher prces relatve to Cournot prces. Ths result s consstent wth the results found by Qu([9]) and Kabraj and Roy ([6]). They have proved that when R&D technology s more effcent (lower r), the Cournot prces are greater than Bertrand prces at equlbrum. When r goes up, frms nvest more n R&D under the Cournot model, then the margnal cost ncreases at a lower rate than under Bertrand model whch lead to lower prces. Ths latter result s stll consstent n ths general settng. One nterestng mplcaton of these results s that, neffcent R&D technology wll generate larger consumer s surplus and socal welfare under the Cournot competton model. But, t s also mportant to menton that consumers surpluses under both models decrease wth the sze of R&D technology. We can show that dcsc 2(1+θ) b(2+θ) 2 usefulness of these latter expressons are stll beng studed. dĉ (a ĉ) and dcsb 2 d c b(1+θ)(2 θ) 2 (a c). The 5 Concluson The am of ths paper was to compare Cournot and Bertrand models on effcency of results n term of socal welfare. The mportant result that challenged the tradtonal result on effcency of Bertrand equlbrum outcome s that at the equlbrum, not only the Cournot frms nvest a larger amount on R&D than the Bertrand frms, but Bertrand prce can be hgher than Cournot prce. I prove that ths occurs when the R&D technology s relatvely neffcent; thus a Cournot market structure can generate larger consumer s surplus and total welfare. In ths paper, all payoff functons and costs are parametrc and there are only two frms. One emnent project s to generalze ths research by allowng a competton between a large number of frms, and by usng non-parametrc functons. I expect that ths general case can provde somethng lke a necessary and/or suffcent condton that could provde more practcal gudance. Moreover, I dd not report the effects of R&D
REFERENCES 13 sze on producer surplus or total welfare because they are stll beng studed. Some future research wll be based on evaluatng these effects on one sde, and comparng Cournot and Bertrand models on other separate ssues from R&D. References [1] Arghya, G. and Manpushpak, M., 2008, Comparng Bertrand and Cournot Outcomes n The Presence of Publc Frms, Workng paper, 16. [2] Cheng, L., 1985, Comparng Bertrand and Cournot equlbra: a geometrc approach, Rand Journal of Economcs, 16, pp. 146-152. [3] Delbono, F. and Dencolo, V., 1990, R & D Investment In a Symmetrc And Homogeneous Olgopoly, Bertrand vs Cournot, Internatonal Journal of Industral Organzaton, 8, pp. 297-313. [4] Hackner, J., 2000, A Note on Prce and Quantty Competton n Dfferentated Olgopoles, Journal of Economc Theory, 93, pp. 233-239. [5] Hsu, J. and Wang, X. H., 2005, On Welfare under Cournot and Bertrand Competton n Dfferentated Olgopoles, Revew of Industral Organzaton, 27, pp. 185-191. [6] Kabraj, T. and Roy, S., 2002, Cournot and Bertrand Prces In A Model Of Dfferentated Duopoly Wth R and D, Workng paper. [7] Motta, M., 1993, Endogenous Qualty Choce Prce vs Quantty Competton, The Journal of Industral Economcs, 41, 2, pp. 113-131. [8] Okuguch, K.,1987, Equlbrum Prces n the Bertrand and Cournot Olgopoles, Journal of Economc Theory, 42, pp. 128-139. [9] Qu, L. D., 1997, On the dynamc effcency of Bertrand and Cournot equlbra, Journal of Economc Theory, 75, pp. 213-229. [10] Sngh, N. and X. Vves, 1994, Prce and quantty competton n a dfferentated duopoly, Rand Journal of Economcs, 15, pp. 546-554. [11] Sutton, J., 1997, One Smart Agent, RAND Journal of Economcs 28, pp. 605-628.
REFERENCES 14 [12] Sutton, J., 1998, Technology and Market Structure, M.I.T Press, MA. [13] Symeonds, G., 2003, Comparng Cournot and Bertrand equlbra n a dfferentated duopoly wth product R&D, Internatonal Journal of Industral Organzaton, 21, pp. 39-45. [14] Vves, X., 1985, On the effcency of Bertrand and Cournot equlbra wth product dfferentaton, Journal of Economc Theory, 36, pp. 166-175.
15 6 Appendx Proof of lemma 3.1: The margnal cost of each frm s c, then frm s proft - maxmzaton problem s set up as: max q {π (q, q j ) (p c)q } s.t p ω φq j λq (A1). The frst order condton for maxmzaton problem s gven by: and the second dervatve of proft s gven by: π q ω φq j 2λq c 0 (A2), 2 π q 2 2λ. In order to maxmze the proft, the second dervatve should be negatve. Snce, the sgn of λ s postve, I conclude that the second condton s satsfed. Solvng equaton (A2) for q gves : q ω φq j c, 2λ and usng the fact that by symmetry q c qc j qc, I obtan and solvng for q c gves : 2λq c + φq c ω c, q c ω c 2λ + φ (A3). Now, I obtan the equlbrum prce for each frm by substtutng the equlbrum quantty (A3) n the nverse demand functon n equaton (12). Furthermore, by symmetry p c pc j pc, then p c ω (φ + λ)q c. It follows that: p c (ω c) ω (φ + λ) 2λ + φ) (2λ + φ)ω (φ + λ)(ω c) 2λ + φ (2λ + φ)ω (φ + λ)ω + c(φ + λ) 2λ + φ ω(2λ + φ φ λ) + c(φ + λ) 2λ + φ λω + cφ + cλ 2λ + φ p c λω + c(φ + λ) (A4). 2λ + φ
16 Therefore, p c c λω + c(φ + λ) c 2λ + φ λω + (φ + λ 2λ φ)c 2λ + φ λ(ω c) 2λ + φ. As above, frms have the same proft π c (p c c)q c. Substtutng for the value of q c and p c n the proft functon gves : Snce I denoted ω α β γ, φ and π c n terms of α, β and γ. 1. I have q c ω c 2λ+φ ω c 2λ + φ π c from (18). γ β 2 γ 2, λ α β γ c α c(β γ) β γ ; 2β + γ β 2 γ 2 β 2 γ 2 λ(ω c) 2λ + φ ω c 2λ + φ λ( ω c 2λ + φ )2. 2β+γ β 2 γ 2 It follows that q c ω c 2λ+φ α c[β γ] (β γ) q c [α c(β γ)](β+γ) 2β+γ (A5). 2. I have p c λω+c(φ+λ) 2λ+φ from (A4). β+γ φ + λ (β+γ)(β γ) 1 β γ ; λω β β 2 γ 2 αβ αβ, so λω + c(φ + λ) (β γ)(β 2 γ 2 ) Then, p c λω+c(φ+λ) 2λ+φ αβ+c[β2 γ 2 ] obtan p c αβ+c[β2 γ 2 ] (β γ)(2β+γ) (A6). 3. I have π c λ( ω c 2λ+φ )2 from (19). ω c 2λ+φ α c(β γ) β γ λ( ω c 2λ+φ )2 (β γ)(β+γ) 2β+γ n (11), I want to smplfy the expressons of q c, p c 2β+γ (β γ)(β+γ) ; (β γ)(β 2 γ 2 ) (β2 γ 2 ) 2β+γ (β γ)(β+γ) 2β+γ, by smplfyng the factor (β γ), I get (β γ)(β 2 γ 2 ) + [α c(β γ)](β+γ) 2β+γ ; β α c(β γ) (β γ)(β+γ) (β + γ)(β + γ)[ 2β+γ ] 2. c β γ αβ+c(β2 γ 2 ) (β γ)(β 2 γ 2 ) ;, and by smplfyng the factor (β2 γ 2 ), I By smplfyng the factor (β + γ) n the latter expresson, I obtan π c β(β+γ) β γ [ α c(β γ) 2β+γ ] 2 (A7). From equaton (9), substtutng the expressons of parameters α, β and γ n terms of parameters a, b, θ n the expressons (A5), (A6) and (A7), I obtan β + γ 1 2+θ b(1 θ) ; 2β + γ b(1 θ 2 ) and α (β γ)c a c b(1+θ). Therefore, I deduce that the Cournot equlbrum quantty s qc Cournot equlbrum p c a+c(1+θ) 2+θ and the postve Cournot equlbrum proft π c 1 b [ a c 2+θ ]2. Proof of lemma 3.2: The frm s proft-maxmzaton problem s: a c b(2+θ), the
17 max {π (p, p j ) (p c)q (p c)(α βp + γp j )} (A8). p The frst order condton of problem (A8) s gven by: π p α βp + γp j β(p c) 0 (A9), and the second order condton s: 2 π p 2 2β < 0 snce β > 0. Solvng for equaton (A9) wth respect to p gves : 2βp α + γp j + βc (A10). By symmetry, the equlbrum prce for each frm s such that p B p B j p B. Then, usng these latter equaltes and substtutng n the equlbrum prce s equaton lead to 2βp B α + γp B + βc, rearrangng terms and solvng the latter equaton for prce p B gves p B α+βc 2β γ (A11). Now, I fnd the quantty of each frm at the equlbrum. By symmetry, q B q B j q B, then substtutng the equlbrum prce n demand functon (8) yelds: q B α (β γ)p B α (β γ) α + βc 2β γ (β γ)(α + βc) α 2β γ α(2β γ) (β γ)(α + βc) 2β γ α(2β γ) α(β γ) βc(β γ) 2β γ α(2β γ β + γ) βc(β γ) 2β γ αβ βc(β γ) 2β γ q B β[α (β γ)c] (A12). 2β γ Symmetry assumpton mples that π B π B j π B. Gven that π B (p B c)q B (A13), I use
18 the above expressons of p B and q B to obtan the complete form of π B. Usng equaton (A11), I get: p B c α + βc 2β γ c p B c α + βc 2βc + γc 2β γ α βc + γc 2β γ α (β γ)c (A14). 2β γ From equaton (A14), snce α > (β γ), I conclude that p c > c. Substtutng equatons (A12) and (A14) n equatons (A13) above gves π B β[ α (β γ)c 2β γ ] 2 (A15). Snce α a b(1+θ), β 1 b(1 θ 2 ), γ θ, t follows that : b(1 θ 2 ) [ α (β γ)c 2β γ ] [ β[α (β γ)c] 2β γ ], and then π B α + βc a(1 θ) + c b(1 θ 2 ), 2β γ 2 θ b(1 θ 2 ) and α (β γ)c a c b(1 + θ). Therefore, I conclude that p B a(1 θ)+c 2 θ, q B a c b(1+θ)(2 θ) and πb Proof of theorem 3.1: The Cournot equlbrum (q c, p c, π c ) s gven by : 1 θ b(1+θ) [ a c 2 θ ]2. p c a + c(1 + θ), q c a c 2 + θ b(2 + θ), πc (a c)2 b(2 + θ) 2. and the Bertrand equlbrum (q B, p B, π B ) s gven by : p B a(1 θ) + c, q B b(1 + θ) 2 θ (2 θ), πb 1 θ b(1 + θ) (a c) 2 (2 θ) 2. I also have: and p c p B a c 4 θ 2 θ2 q B q c π c π B a c b(1 + θ)(4 θ 2 ) θ2 2(a c) 2 b(1 + θ)(4 θ 2 ) 2 θ3 From these equatons, I show that p c p B, q c q B and π c π B f and only f the parameter θ s equal zero. Moreover, gven that a > c, f θ > 0, I have p c > p B, q B > q c and π c > π B. Let π c Proof of proposton 3.2: denote frm s market proft at stage 2, then: π c (p c )q r c e e. At ths stage, frms
19 choose smultaneously ther optmal level of producton takng the margnal cost as gven. So, frm solves the followng maxmzaton problem: { } max π c q (q, q j ) (p c )q r c e e s.t p a bq bθq j The frst order condton for maxmzaton problem s gven by: π c q a bθq j 2bq c 0; j;, j 1, 2 (A16); and the second dervatve of proft s gven by: 2 π c q 2 2b < 0. From equaton (A16), I have the followng system: Solvng ths system for q and q j gves : q c q c j a b(2 + θ) + 2c θc j b(θ 2 4) (A17) a b(2 + θ) + 2c j θc b(θ 2 4) (A18) 2bq + bθq j a c. bθq + 2bq j a c j The frm s proft at stage 1 s gven by: π c(qc, qc j, c, c j ) (p c c )q c r c e e rewrtten as π c(qc, qc j, c, c j ) (a bq c bθqj c c )q c r c e e. whch can be At ths stage, frms choose smultaneously ther optmal level of R&D cost consderng the optmal level of producton at the next stage. Then frm s proft maxmzaton problem s gven by: { } max π c c (qc, qc j, c, c j ) (a bq c bθqj c c )q c r c e e s.t (A17) and (A18) The frst order condton for maxmzaton problem s gven by: dπ c dc πc q c qc c + πc qj c qc j c and the second dervatve of proft s gven by: + πc 4 c θ 2 4 qc + rc e 1 0, d 2 π c c 2 8 b(θ 2 (e + 1)rc e 2 4) 2. I assume that the second dervatve s satsfed. Let also assume the symmetrc equlbrum at frst stage, then c c c c j ĉ < a, then qc q c qj c and p c p c p c j. Usng the same algebras as n
20 the secton 2, I obtan the requred condtons. Proof of proposton 3.3: Frm s proft maxmzaton problem at stage 2 s set up as: { max p π B (p, p j, c, c j ) (p c )q (p c )(α βp + γp j ) r c e e The frst order condton of problem (31) s gven by: and the second order condton s: π p α 2βp + γp j + βc 0, j,, j {1, 2} (A19), 2 π p 2 2β < 0 (snce β > 0). From equaton (A19), I retreve the followng system: Solvng ths system for p and p j gve s: } 2βp γp j α + βc. γp + 2βp j α + βc j p B α 2β γ + 2β2 c + βγc j 4β 2 γ 2 (A20) The frm s proft at stage 1 s gven by: p B j α 2β γ + 2β2 c j + βγc 4β 2 γ 2 π B (p B (A21), p B j, c, c j ) (p B c )q c r c e e, or π B (p B, p B j, c, c j ) (p B c )(α βp B + γp B j ) r c e e. At ths stage, frms choose smultaneously ther optmal level of R&D cost consderng the optmal level of prce at the next stage. Thus, frm solves the followng proft maxmzaton problem: { } max π c c (qc, qc j, c, c j ) (p B c )(α βp B + γp B j ) r c e e s.t (A20) and (A21) The frst order condton for maxmzaton problem s gven by: dπ B dc πb p B pb c + πb p B j pb j c + πc γ2 β c 4β 2 γ 2 (pb c ) (α βp B +γp B j )+rc e 1 0,
21 and the second dervatve of proft s gven by: d 2 π B dc 2 2β[ 2β2 γ 2 4β 2 γ 2 ]2 (e + 1)rc e 2. I assume that the second dervatve s satsfed. Let also assume the symmetrc equlbrum at frst stage, then c B c B j c < a, then p B p B p B j and q B q B qj B. After dong some algebras and substtutng α, β and γ by ther expressons n terms of a, b and θ, I obtan the equlbrum condtons. Proof of proposton 3.4: By symmetry of equlbrum for any c satsfyng (25) and after dong some algebras, I obtan: dπ B dc (2θ 2 4)(a c) b(1 + θ)(2 θ)(4 θ 2 ) + rc e 1 By frst order condton, I have dπb dc c c 0. Also, dπb dc cĉ 2θ(a ĉ)(θ2 +θ+2) b(4 θ 2 ) 2 (1+θ) By agan usng symmetry of equlbrum for any c satsfyng equaton (19), I have: dπ c dc 4(a c) b(4 θ 2 )(2 + θ) + rc e 1 By the frst order condton, dπc dc cĉ 0 and dπc dc c c 2θ3 (a c) b(4 θ 2 ) 2 (1+θ) whch s postve. whch s negatve. It follows that frm have more ncentve to nvest n R&D under Cournot equlbrum rather than Bertrand equlbrum, and then ĉ < c. Proof of proposton 4.1: Case 1: dĉ > 0. From equatons (19) and (20), I obtan the followng equaton: 4 b(2 + θ)(4 θ 2 ) (a ĉ)ĉe+1 r (A22). Dfferentatng equaton (A22) wth respect to r and solvng for dĉ gves : dĉ b(2 + θ)(4 θ2 ) ĉ e 4 [(e + 1)(a ĉ) ĉ] (A23). Let denote C(θ) b(2+θ)(4 θ2 ) 4. Gven that C(θ) s postve, I just need to show that (e + 1)(a ĉ) ĉ s also postve. To prove that, I use the second order condton from proposton 3.2 (stage 1 maxmzaton problem). From there, I have the second order condton: 2 π c c 2 1)r c e 1 c have qc c < 0. Gven that I assume symmetry equlbrum, c ĉ and q c qc ĉ 1 qc b(2+θ). Substtutng c 4 θ 2 4 q c c (e + q c ; from equaton (20), I and r from equaton (A22) n the second order condton,
22 and smplfy, I get 2 π c ĉ 2 4 (e + 1)(a ĉ) ĉ > 0 and dĉ > 0. Case 2: d c > 0. b(4 θ 2 )(2+θ) [ ĉ (e+1)(a ĉ) ĉ ]. Snce 2 π c ĉ 2 < 0, then ĉ < (e + 1)(a ĉ).e Usng equaton (26) from Bertrand equlbrum, I have pb r 1 d c 2 θ. Dfferentatng equaton (25) wth respect to r and solvng for d c yelds: d c θ 2 (1 θ) b(1 θ 2 )(4 θ 2 )(2 θ) c c e c e c b(1+θ)(2 θ) + (e + 1)r c e 1 B. Now, I show that B > 0. I use equatons (25), (26), the second order condton derved from proposton 3.3 (stage 1 maxmzaton problem) and symmetrc equlbrum. Usng (26), I get p B c (1 θ)(a c) 2 θ to the followng expresson: and a p B a c 2 θ. Substtutng these latter expressons nto equaton (25) lead r c e 1 θ 2 (1 θ)(a c) a c b(1 θ 2 )(4 θ 2 + ) 2 θ b(2 θ)(1 + θ) (A23). Substtutng r c e 1 from equaton (A23) nto expresson B and dong some algebras gves: B 2(2 θ 2 ) b(1 + θ)(4 θ 2 [(e + 1)(a c) c]. )(2 θ) The second order condton for proposton 3.3 can be rewrtten as follows: 2 π B c 2 θ 2 b(1 θ 2 )(4 θ 2 ) [ pb equlbrum, I have p B c 1] + 1 b(1 θ 2 ) p B c θ b(1 θ 2 ) p B j c r(e + 1) c e 2 < 0. By symmetrc p B, c c for {1, 2} and pb c 1 2 θ. Usng these latter nformaton, the second order condton can be rewrtten as: 2 π B c 2 θ 2 (1 θ) c b(1 θ 2 )(4 θ 2 )(2 θ) + ths latter second order condton lead to [ 2(2 θ 2 ) b(1+θ)(4 θ 2 )(2 θ) c b(1+θ)(2 θ) r(e + 1) c e 1 < 0. Substtutng equaton (A23) nto θ 2 (1 θ) b(1 θ 2 )(4 θ 2 )(2 θ) 1 b(1+θ)(2 θ) ](e + 1)(a c) c) < 0.e [(e + 1)(a c) c] < 0 or B < 0. It follows that B > 0, then d c > 0. Note that B postve means (e + 1)(a c) c s also postve, gven that the expresson d c can be rewrtten as follows: Let denote B(θ) b(1+θ)(4 θ2 )(2 θ) 2(2 θ 2 ). Case 3: d c > dĉ. d c b(1 + θ)(4 θ2 )(2 θ) c e 2(2 θ 2 ) [(e + 1)(a c) c]. From some expressons n cases 1 and 2, I wrte: d c / dĉ B(θ) C(θ) (ĉ c )e+1 (e + 1)[ a ĉ 1] (e + 1)[ ã c 1]. 2(2 θ 2 ) > 0. Fnally, b(1+θ)(4 θ 2 )(2 θ)
23 I have B(θ) C(θ) 2(1+θ)(2 θ) (2+θ)(2 θ 2 ) > 1 and (e+1)[ a ĉ 1] 1] > 1 snce c > ĉ. Assumng that the rato of margnal (e+1)[ ã c cost s close to one whch means that the term ( ĉ c )e+1 s not so low to reduce the product of terms on the rght sde of the rato of margnal effect of R&D sze on margnal cost far to one, I conclude that d c / dĉ d c > 1 or > dĉ d c. Wthout ths assumpton, I may also have < dĉ n case where Bertrand frms nvest too much n R&D for nstance, or n case where the rato of margnal cost s too low. Proof of proposton 4.2: From equaton (21), I have p c a+ĉ(1+θ) 2+θ), so dpc 1+θ dĉ dĉ dpc 2+θ. By proposton 4.1, > 0, then > 0. From equaton (26), p B a(1 θ)+ c 2 θ, so dpb 1 d c dpb 2 θ. Usng agan proposton 4.1, > 0 snce d c > 0. I have dpb dpc 1 d c 2 θ 1+θ dĉ 2+θ. By dong some algebras, 1 2 θ 1+θ 2+θ θ2 4 θ 2 non negatve. Snce, I have shown that dĉ > dĉ, hence dpb dpc 0. whch s