Price vs Quantity in a Duopoly with Technological Spillovers: A Welfare Re-Appraisal

Transcription

1 Prce vs Quantty n a Duopoly wth Technologcal Spllovers: A Welfare Re-Apprasal Luca Lambertn # and Andrea Mantovan #;x #Department of Economcs Unversty of Bologna Strada Maggore 45 I Bologna, Italy fax: e-mal: lambert@spbo.unbo.t e-mal: mantovan@spbo.unbo.t CORE, Unversté Catholque de Louvan 34, voe du Roman Pays B-1348 Louvan-la-Neuve, Belgum November 3, 2000

2 Abstract We analyse the problem of the choce of the market varable n a model where rms actvate R&D nvestments for process nnovaton. We establsh that () rms always choose the Cournot behavour; and () there exsts a set of the relevant parameters where a benevolent socal planner prefers quantty settng to prce settng. Ths happens when the margnal cost of R&D actvtessrelatvely low whle technologcal externaltesare relatvely hgh. In ths stuaton, the con ct between socal and prvate preferences overthe type of market behavour dsappears. J.E.L. class caton: L13, O31 Keywords: prce, quantty, R&D, spllovers

3 1 Introducton The nterplay betweentechnologcalchocesandmarket behavournolgopoly models has been studed along two man routes. The rst has emphaszed the lnk between the knd of competton prevalng on the market and rms ncentves to nvest ether n process or n product nnovaton. The second concerns the n uence of capacty constrantsonmarket equlbrum. Most lterature on R&D races n olgopoly deals wth the evaluaton of ncentves to undertake cost reducng nvestments as the number of rms changes. ThsSchumpeteranapproachholdsthat a maorfactordetermnng the pace of technologcal progress s market structure (amongst the countless contrbutons n ths ven, see Arrow, 1962; Loury; 1979; Lee and Wlde, 1980; Dasgupta and Stgltz, 1980; Delbono and Dencolò, 1991; for an overvew see Renganum, 1989). An establshedresult oncost reducng nvestment n olgopolstc markets under perfect certanty states that there s excess expendture n R&D under Cournot competton, and conversely under Bertrand competton, due to the opposte slopes of reacton functons at the market stage (Brander and Spencer, 1983; Dxon, 1985). Wth d erentated products, Bester and Petraks (1993) mantan that the ncentve to nvest n cost reducng nnovaton depends upon the degree of product substtutablty. Under both Cournot and Bertrandcompetton, undernvestment, ascomparedto the socal optmum, obtans when products are farly mperfect substtutes, whle the opposte may occur when products are su cently smlar. Cournot competton provdes a lower (respectvely, hgher) ncentve to nnovate than Bertrand competton f substtutablty s hgh (respectvely, low). Socal welfare may thenbe hgherundercournot thanunderbertrandcompetton (Delbono and Dencolò, 1990; Qu, 1997). Sngh and Vves (1984) nvestgate the choce of the market varable n a duopoly where rmsoperate costlessly. They ndthat, ndependently of the degree of product substtutablty, rms choose to be quantty setters at the subgame perfect equlbrum, whle socal welfare would be hgher under prce settng behavour. The opposte holds f products are demand complements. In ths paper, we extend Qu s analyss to account for the asymmetrc case where one rm s a quantty setter whle the other s a prce setter, and we derve the subgame perfect equlbrum of a two-stage game where rms operate R&D actvtes amed at reducng margnal producton costs and then compete at the marketng stage. Then, we recast Sngh and Vves s analyss n a three-stage game where rms choose whether to be prce or quantty setters at the rst stage, then nvest n cost-reducng R&D, and nally compete on the market. We establsh that, at the subgame perfect 1

4 equlbrum, rms always choose to set quanttes. However, we also nd that there exsts a parameter regon where quantty settng behavour s socally preferable, f margnal R&D costs are su cently low and spllover are su cently hgh. In such a stuaton, we have a second best equlbrum where the usual con ct over the choce of the market varable dsappears. The remander of the paper s organsed as follows. The setup s lad out n secton 2. Secton 3 descrbes R&D and market behavour. Prvate and socal preferences concernng the choce of the market varable are then nvestgated n secton 4. Secton 5 concludes. 2 The setup The demand sde s a smpl ed verson of Bowley (1924), subsequently adopted by Spence (1976), Dxt (1979) and Sngh and Vves (1984), nter ala. Assume the representatve consumer s charactersed by the followng utlty functon: U(q q )=q +q 1 2 ³ q 2 +q 2 +2 q q (1) whereq andq are the quanttes of goods and; respectvely. The resultng (symmetrc) demandfunctonsundercournot and Bertrand compettonare, respectvely p =1 q q (2) q = 1 1+ p 1 2+ p 1 2 (3) In the asymmetrc case, where rm s a quantty setter, whle rm s a prce setter, demandfunctons are: p =1 q + (p + q 1) (4) q =1 p q (5) Parameter 2(0;1] represents product substtutablty as percevedby consumers, dependng upon the products rms supply. If one supposes that margnal costs are constant andequal to c across rms, thenndvdual profts are ¼ IJ =(p c)q ; whereij 2 fpp;qq;pq;qpg ndcates that, at the market stage, rmsets varablei whle rm sets varablej. In ths case, the choce between prce and quantty behavour s summarsed by the reducedformrepresentednmatrx 1. 2

5 P P ¼ PP ;¼ PP Q ¼ QP ;¼ PQ Q ¼ PQ ;¼ QP ¼ QQ ;¼ QQ Matrx 1 Onthe bassof Matrx 1, Sngh and Vves (1984) conclude that quantty settng (resp., prce settng) s the domnant strategy (at least weakly) f goodsare demandsubsttutes (resp., complements). Asa consequence, rms play a Cournot (resp., Bertrand) equlbrum for all 2 (0;1] (resp., 2 [ 1;0)). In the remander, gven the symmetry of the model w.r.t. parameter ; we wll focus on the case of substtutes. Snghand Vves s result entals that there exsts a con ct between rms pro t ncentves and the socal ncentve towards welfare maxmzaton, whch requres prce settng behavour when goodsare substtutes. If we abandonthe assumptonof homogeneouscosts, the choce between prce and quantty s drven by the sgn of the followng expressons: ¼ QP (c ;c ) ¼ PP (c ;c ) (6) ¼ QQ (c ;c ) ¼ PQ (c ;c ) (7) Now observe that, relabellng1 c ; the d erence n productve e cency across rms s formally equvalent to a d erence n reservaton prces for goods and n the representatve consumer s preferences (see Häckner, 2000), whch would now wrte as follows: U(q q )= q + q 1 2 ³ q 2 +q2 +2 q q Hence, as proved by Sngh and Vves (1984), fq;qg s the unque equlbrum outcome. Gven that rms are ex ante symmetrc, we need an explanaton to ustfy any asymmetry n margnal costs. The reason forsuch anasymmetry canbe found n the d erent ncentves towards R&D nvestment that rms have n the four subgames fpp;qq;pq;qpg: Suppose rms play a non-cooperatve two-stage game, where the rst stage nvolves choosng the ndvdually optmal amount of R&D for process 3

6 nnovaton, whle the second s for marketng. De ne asx the R&D e ort produced by rm: The R&D technology s the same across rms. The cost of R&D actvty sk =f(x ); wthf 0 (x )=@x >0 andf 00 (x 2 f(x )=@x 2 >0: That s, we suppose that R&D actvty s charactersed by decreasng returns to scale. The resultng margnal cost sc c (x ;µx ); where µ 2[0;1] denotes the spllover receved from rm s R&D nvestment. The net pro ts accrung to rm; whenthe market subgame sij; are ª IJ =¼ IJ h ³ c x IJ ;µx JI ³ ;c x JI ;µx IJ ³ f x IJ In ths stuaton, the reduced form of the game s as n Matrx 2. P P ª PP ; ª PP Q ª QP ; ª PQ Q ª PQ ; ª QP ª QQ ; ª QQ (8) Matrx 2 Hence, the choce between P and Q s made accordng to the sgn of: ¼ QP [c ( ; );c ( ; )] ¼ PP [c ( ; );c ( ; )] f ³ x QP ³ +f x PP (9) ¼ QQ [c ( ; );c ( ; )] ¼ PQ [c ( ; );c ( ; )] f ³ ³ x QQ +f x PQ (10) The addtonal task conssts n reassessng socal preferences over the choce of market varables n ths new settng. A pror, one cannot presume that the con ct betweensocalandprvate ncentvesthat charactersesthe prevous settng extends to ths case. Indeed, we know from Qu (1997) that there are stuatons where socal welfare s hgher at the Cournot equlbrumthan at the Bertrand equlbrum. In order to carry out the analyss of ths problem, we model the R&D stage as n Qu (1997), by assumng that c =c x µx ; c 2(0;1) (11) K = ºx2 (12) 2 Frms play a non-cooperatve three-stage game. At the rst stage, they choose between prce and quantty. The followng two stages descrbe () the choce of R&D e orts and () marketng. As usual, we proceed by backward nducton, usng subgame perfectonas the solutonconcept. 4

7 3 R&D and market subgames We borrow from Qu (1997) the charactersaton of subgames fpp;qqg;.e., Bertrand and Cournot. 3.1 Symmetrc subgames Bertrand equlbrum prces are: p PP = (2+ )(1 +c) (2+µ )x (2µ+ )x 4 2 (13) The assocatedpro tsare: ª PP = [(1 c)( 2 + 2) (2 µ 2 )x +( 2µ+µ 2 )x ] 2 (4 3 2 ) 2 ºx2 6 2 (14) Solvng the R&D stage, one obtans: x PP = 2(1 c)(2 µ 2 ) (15) PP PP = º(1+ )(2 ) ³ 4 2 2(1+µ) ³ 2 µ 2 (16) The resultng per- rmequlbrumquantty s Consumer surplus s q PP = º(1 c)(4 2 ) PP (17) so that socal welfare s CS PP =(1+ ) " º(1 c)(4 2 # 2 ) (18) PP SW PP = X ª PP +CS PP = (19) h º(1 c)2 º( )(4 2 ) 2 4(2 µ 2 ) 2 = ( PP ) 2 The followng holds: Lemma 1 (Qu, 1997, p. 217) The condtonº>1=c s 5

8 ] su cent but not necessary for PP >0; for allµand (stablty) ] necessary and su cent for post-nnovaton costs to be postve,.e.,c PP = c (1+µ)x PP >0 ] su cent to ensure@ 2 ª PP =@x 2 <0 µ=1 If nsteadµ 2 ª 2 <0,º> 2(2 µ 2 ) 2 (1 2 )(4 2 ) 2 Examne now the Cournot case. The equlbrum output s The assocatedpro tsare: q QQ = (1 c)(2 )+(2 µ )x +(2µ )x 4 2 (20) ª QQ = [(1 c)(2 )+(2 µ )x +(2µ )x ] 2 (4 2 ) 2 ºx2 2 (21) Proceedng backward to solve the R&D stage, one obtans: x QQ 2(1 c)(2 µ ) = (22) QQ QQ = º(2+ ) ³ 4 2 2(1+µ)(2 µ ) (23) The resultng per- rmequlbrumquantty s The resultng consumer surplus s so that socal welfare s q QQ = º(1 c)(4 2 ) QQ (24) CS PP =(1+ ) " º(1 c)(4 2 # 2 ) (25) QQ SW QQ = X ª QQ +CS QQ = (26) = The followng holds: h º(1 c)2 º(3+ )(4 2 ) 2 4(2 µ ) 2 ( QQ ) 2 6

9 Lemma 2 (Qu, 1997, p. 216) The condtonº>1=c s ] su cent butnot necessary for for allµ and QQ > 0 2 ª 2 < 0 (concavty) ] necessary and su cent for post-nnovaton costs to be postve,.e.,c QQ = c (1+µ)x QQ >0 3.2 The asymmetrc subgames CasesQP andpq are symmetrc up to a permutaton of rms. Therefore, we con ne our attenton to the stuaton where rm s a quantty-setter, whle rm s a prce-setter. The demand functons are (4) and (5). The Nash equlbrum at the market stage s gven by: q QP p PQ = (2 )(1 c)+(2 µ )x +(2µ )x (27) = 1 c µx x 4 The assocatedpro tsare: + (28) [(2 )(1 c)+(2 µ )x +(2µ )x ] ª QP = (1 2 )[(1 c)(2 )+(2 µ )x +(2µ )x ] 2 (4 3 2 ) 2 ºx2 2 (29) ª PQ = [(1 c)( 2 + 2) (µ 2 + 2)x +( 2 +µ 2)x ] 2 (4 3 2 ) 2 ºx2 2 (30) At the rst stage, rms non-cooperatvely maxmse ther respectve pro ts w.r.t. R&D e ort levels, = 2(1 2 )(2 µ )(1 c)(2 ) (4 3 2 ) 2 + (31) + 2(1 2 )(2 µ )[(2 µ )x +(2µ )x ] (4 3 2 ) 2 ºx =0 7

10 @ª = 2(µ 2 + 2)(1 c)( 2 + 2) (4 3 2 ) 2 + (32) 2(µ 2 + 2) h (µ 2 + 2)x +( 2 +µ 2)x 2 (4 3 2 ) 2 ºx =0 whose soluton yelds the equlbrum R&D nvestments: x QP x PQ = 2(1 c)(1 2 )(µ 2) xqp (33) = 2(1 c)(1 )(4 3 2 )( 2 +µ 2) xpq (34) where: =2(1 µ) ³ 2 +µ 2 +º(2 ) ³ =2(1 µ) ³ 2 µ 2 +2 ³ 2 3µ+µ 2 º(2+ ) ³ xqp = ³ µ 1 h 2 5 ³ 4 2 µ 3 +16(1 µ ) +4 h 2 ³ 3 2 (7 3 )µ º h 16 ³ ³ ³ µ 6(6 )µ ³ µ ³ º 27 ³ xpq =4 ³ 1 2 ( 2µ)(2 µ ) ³ 2 +µ 2 ³ 2µ+µ ³ 1 2 (4 µ ) 2 º ³ 2 2 h ³ ³ µ +( +µ) The equlbrum output levels are: 1 º ³ q QP = (1 c)(3 2 4)[2(1 µ)( 2 +µ 2)+º(2 )(4 3 2 )] xqp (35) q PQ = (1 c)(1 )(4 3 2 )º[2(1 µ)(2 µ 2 )+ xpq (36) +2 ³ 2 3µ+µ 2 º(2+ )(4 3 2 ) xpq 1 For the sake of brevty, we omt the expressons of equlbrum prces, whch are avalable upon request. 8

11 The correspondng equlbrum pro ts are: ª QP = (1 c)2 ( 2 1)º[2(1 µ)( 2 +µ 2)+º(2 )(4 3 2 )] 2 [2(1+ ( xqp ) 2 (37) 2 )(2 µ ) 2 º(4 3 2 ) ª PQ = (1 c)2 (1 )º ( xqp ) 2 h 2(1 µ)(2 µ 2 )+2 ³ 2 3µ+µ 2 º(2+ ( xpq ) 2 (38) + )(4 3 2 )] 2 h 8(1 2 µ )+2( +µ) 2 2 º(4 3 2 ) ( xpq ) 2 On the bass of the above expressons, we cannot derve analytcally the equvalent of Lemmata 1-2 for the asymmetrc case. Therefore, we must rely upon numercal calculatons to ensure that () concavty and stablty condtons are sats ed; and () post-nnovaton margnal costs are non-negatve. However, on the bass of (33-34), the followng holds: Lemma 3 Gven acceptable values of fc; ; µ;ºg; we have that x QP x PQ > : Moreover, gven acceptable values of fc;µ;ºg; there exsts b 2(0;1); such that n x PQ =0; c PQ =c o : That s, the quantty-setter nvests more than the prce setter, and the latter does not nvest at all to reduce her own margnal cost, f product substtutablty s largerthan a crtcal threshold. The general behavourof c QP andc PQ for 2[0;1] s descrbed n Fgure 1. 9

12 Fgure 1 : Margnal costs and product substtutablty c ;c 6 c c PQ c QP 0 b 1 - Frst of all, notce that, as substtutablty ncreases, the margnal cost of the prce-setterbecomesncreasngly largerthanthe margnal cost borne by the quantty-setter, for all 2[0; b ). 2 Ths re ects the hgher ncentve towardsnvestment nprocessnnovatonforthe quantty-settercomparedto the prce-setter, nlne wth prevous ndngs by Brander and Spencer (1983), Dxon (1985), BesterandPetraks (1993). Moreover, at = b we have that c PQ = c and, therefore, the prce-setterstops nvestng n R&D. That s, for b ; we set x PQ = 0 and recalculate x QP from (31). Ths reveals that the quantty settng rm reduces her nvestment n R&D n response to the fact that the prce settng rval s not nvestng at all. As a consequence, c QP ncreases n the degree of product substtutablty, for all 2 [b ;1]: When = 1;.e., products are homogeneous, also the quantty-setter stops nvestng and both rms operate atc: In general, the soluton to rm s nvestment problem, when rm does 2 For example, when c = 3 4 ; º = 1:34 ; 2 [0 ; 1] ; µ = ; concavty and stablty condtons are met and we have b = 0:83622: 10

13 not nvest s gven by: x QP ³ x PQ =0 = 2(1 2 )( µ 2)(1 c)(2 ) 2(1 2 )(2 µ) 2 º(4 3 2 ) 2 (39) Fnally, socal welfare n the asymmetrc case s: SW PQ =SW QP = X ª IJ +CS IJ (40) wherecs IJ =(1+ ) ³ 2=4: q IJ +q JI 4 The rst stage: prvate vs socal preferences Equlbrum pro tsª PP,ª QQ,ª PQ andª QP can be plugged nto Matrx 2 to yeld the reduced form of the rst stage of the game, where rms noncooperatvely choose whether to be prce- or quantty-setters. We obtan the followng: Clam 1 For all admssble values of parameters fc; ;µ;ºg; we have that Q ºP: Hence, the Cournot equlbrums unque and results from (at least weakly) domnant strateges. Explct calculatonsoverthe relevant nequaltes,.e.: ª QQ ª PQ >0 (41) ª QP ª PP >0 (42) are omtted for the sake of brevty. Verfyng that (41-42) hold for all postve n x IJ o s a matter of smple albet tedous algebra. 3 Havng done that, the extenson to the case wherex QP >0 andx PQ =0 s mmedate, n that the prce-setter s pro ts ª PQ are decreasng over 2 [b ;1]; for two reasons. The rst s the ncrease n product substtutablty. As ncreases towards one, t becomes ncreasngly harder for the prce settng rm to keep her prce above margnal cost. The second reason s that the quantty-setter keeps nvestng n cost-reducng R&D for all 2 [b ;1). 3 As antcpated n secton 3.2, the only complcaton conssts n verfyng the concavty and stablty condtons, and the postvty of post nnovaton costs, for the asymmetrc case, together wth thecorrespondng condtonsfor the symmetrc casesasfromlemmata 1-2. Ths can only be done numercally. Calculatons are avalable upon request. 11

14 Clam 1 extends Sngh and Vves s ndngs to the case where rms nvest n R&D to reduce margnal costs. The nterpretaton of ths result s that the lower ncentve to nvest that characterse a prce-setter as compared to a quantty-setter, (see Brander and Spencer, 1983; Dxon, 1985; Bester and Petraks, 1993), s nsu cent to generate equlbra where at least one rm s a prce-setter, over the whole parameter space. Now we are n a poston to compare prvate and socal ncentves concernng the choce between P and Q: We know that the followng holds: Proposton 1 (Qu, 1997, p. 223) Supposeµ2(0;1);º>1=c; and Then, gven ; ether º> 2(2 µ 2 ) 2 (1 2 )(4 2 2 for all 2(0;1): ) ] SW PP >SW QQ for allº andµ; or ] there exsts a unque bº>1; such that ² for allº bº and allµ 2(0;1); we havesw PP >SW QQ ² for allº< bº; there exsts b µ 2(0;1) such that SW PP SW QQ >0 for allµ< b µ =0 forµ= b µ <0 for allµ> b µ III] For!0; [] holds; for!1; [] holds. Now, consder Clam1 and Proposton1 ontly. If Proposton 1[] holds, and, n partcular, º < bº andµ > b µ; then rms play fq;qg whch s also the socally preferred equlbrum. Therefore, we have our nal result: Proposton 2 Suppose ² µ 2(0;1);µ> b µ; ² º 2 Ã max ( 1 c ; 2(2 µ 2 ) 2 )! (1 2 )(4 2 ) 2 ; bº for all 2(0;1): If so, fq;qg s a second best equlbrum where socal and prvate preferences over the choce of the market varable concde. 12

15 Therefore, the ntroducton of anaddtonal stage descrbng cost-reducng R&D actvtes nto Sngh and Vves s framework produces a subgame perfect equlbrum where, provded margnal R&D costs are su cently low and spllover are su cently hgh, quantty settng behavour s preferred from both the socal and the prvate standpont. Obvously, there remans the ne cency assocated wth the pro t-maxmsng decsons of rms at the market stage, entalng a dstorton n output and prce levels as compared to the socal optmum. 5 Concludng remarks The foregong analyss recasts the problemof the choce of the market varable rst nvestgated by Sngh and Vves (1984) nto a pcture where an addtonal stage descrbes rms R&D nvestments n cost-reducng actvtes, as n Qu (1997). Ths allows us to establsh that () rms always choose the Cournot behavour; and () there exsts a set of the relevant parameters where a benevolent socal planner prefers quantty settng to prce settng. Ths happens when the margnal cost of R&D actvtes s relatvely low whle technologcal externaltes are relatvely hgh. In ths stuaton, the overnvestment nr&d assocated wthcournot behavour (Brander andspencer, 1983) s welcome n that t produces postve welfare e ects, to such an extent that the con ct between socal and prvate preferencesoverthe type of market behavourdsappears. 13

16 References [1] Arrow, K. (1962), Economc Welfare and the Allocaton of Resources for Inventons, n Nelson, R. (ed.), The Rate and Drecton of Inventve Actvty, Prnceton, NJ, Prnceton Unversty Press. [2] Bester, H. and E. Petraks (1993), The Incentves for Cost Reducton n a D erentated Industry, Internatonal Journal of Industral Organzaton, 11, [3] Bowley, A.L. (1924), The MathematcalGroundwork of Economcs, Oxford, OxfordUnversty Press. [4] Brander, J. andb. Spencer (1983), Strategc Commtment wthr&d: The Symmetrc Case, Bell Journal of Economcs, 14, [5] Dasgupta, P. and J. Stgltz (1980), Uncertanty, Industral Structure, and the Speedof R&D, BellJournalof Economcs, 11, [6] Delbono, F. and V. Dencolò (1990), R&D Investment n a Symmetrc and Homogeneous Olgopoly: Bertrand vs Cournot, Internatonal Journal of Industral Organzaton, 8, [7] Delbono, F. and V. Dencolò (1991), Incentves to Innovate n a Cournot Olgopoly, Quarterly Journal of Economcs, 106, [8] Dxt, A. (1979), A Model of Duopoly Suggestng a Theory of Entry Barrers, Bell Journal of Economcs, 10, [9] Dxon, H.D. (1985), Strategc Investment n a Compettve Industry, Journal of Industral Economcs, 33, [10] Häckner, J. (2000), A Note on Prce and Quantty Competton n D erentated Olgopoles, Journal of Economc Theory, 93, [11] Lee, T. and L. Wlde (1980), Market Structure and Innovaton: A Reformulaton, Quarterly Journal of Economcs, 94, [12] Loury, G. (1979), Market Structure andinnovaton, Quarterly Journal of Economcs, 93, [13] Qu, L. (1997), On the Dynamc E cency of Bertrand and Cournot Equlbra, Journal of Economc Theory, 75,

17 [14] Renganum, J. (1989), The Tmng of Innovaton: Research, Development and D uson, n Schmalensee, R. and R. Wllg (eds.), Handbook of Industral Organzaton, vol. 1, Amsterdam, North-Holland. [15] Sngh, N. and X. Vves (1984), Prce and Quantty Competton n a D erentated Duopoly, RAND Journal of Economcs, 15, [16] Spence, A.M. (1976), Product D erentatonand Welfare, Amercan Economc Revew, 66,