Endogenous Choice of Capacity and Product Innovation in a Di erentiated Duopoly

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1 Endogenous Choce of Capacty and Product Innovaton n a D erentated Duopoly Luca Lambertn - Ganpaolo Rossn Dpartmento d Scenze Economche Unverstà d Bologna Strada Maggore, 45 I Bologna fax lambert@spbo.unbo.t rossn@spbo.unbo.t November 16, 1998 Abstract We model a symmetrc duopoly where rms choose whether to be quantty setters or prce setters by decdng the optmal capacty; undertake R&D actvty to determne the degree of d erentaton; and nally compete n the market. Two games are proposed, where nvestment decsons follow d erent sequences. We assess prce and quantty decsons, ndng a set of equlbra where the choce of the market varable s a ected by both technologcal commtments. As a result, the acqured wsdom that quantty settng s a domnant strategy for rms, whle prce settng s a domnant strategy from a socal standpont, may not be con rmed. J.E.L. class caton: D43, L13, O31 Keywords: R&D, product nnovaton, capacty Acknowledgements. We thank semnar audence at Catholc Unversty n Mlan for useful comments and dscusson. The usual dsclamer apples. 1

2 1 Introducton The nterplay between technologcal choces and market behavour n olgopoly modelshas beenstuded along two alternatve routes. The rst has emphaszed the lnk between the shape of competton prevalng on the market and rms ncentves to nvest ether nprocess 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. Ths Schumpeteran approach holds that a major factor determnng the pace of technologcal progress s market structure (amongst the countless contrbutons n thsven, see Arrow, 1962; Loury; 1979; Lee and Wlde, 1980; Dasgupta and Stgltz, 1980; Delbono and Dencolò, 1991; for an overvew see Renganum, 1989). An establshed result on cost 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). An extenson of these resultsto the case of dfferentated products canbe foundnbesterandpetraks (1993). They mantan that the ncentve to nvest n cost reducng nnovaton depends upon the degree of product substtutablty. Under both Cournot and Bertrand competton, undernvestment, as compared to 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). As a result, socal welfare may be hgher under Cournot than under Bertrand competton (Delbono and Dencolò, 1990; Qu, 1997). Whle the R&D lterature nvestgates the n uence of market competton on the optmal nvestment, contrbutons on capacty constrants follow a reverse route. They analyse how plant sze determnes the ntensty of market competton (Levtan and Shubk, 1972; Kreps and Schenkman, 1983; Osborne and Ptchk, 1986; Davdson and Deneckere, 1986). When rms have enough capacty to serve the whole market a Bertrandequlbrumob- 2

3 tans. If capacty s bndng, the Cournot outcome emerges. Deneckere and Kovenock (1996) prove that, undercost asymmetry, there sanncentve for the more e cent rm to drve the opponent out of busness. Ths prevents the market from reachng a Cournot equlbrum. 1 The man purpose of ths paper s to brdge these two streams of lterature n order to nvestgate the e ect of technologcal choces on the shape of market competton,.e., on rms ncentves to set ether prces or quanttes. As far as R&D s concerned, a pror, there s no clearcut ntuton as to how the shape of market competton can a ect product nnovaton,.e., nvestment amed at reducng product substtutablty. Aprelmnary result can be found n Lambertn and Rossn (1998), who show that a Prsoner s Dlemma can be responsble for product homogenety n a bnary model of nvestment n product nnovaton, followed by ether Bertrand or Cournot competton. Here we extend the analyss to the case n whch both R&D and capacty are contnuous varables. The choce of capacty strategcally antcpates the equlbrum output set accordng to the market varable chosen. As a result, n our model, capacty never bnds, snce t s endogenously set. Therefore, the optmal choce of capacty plays the role of a commtment of rms as to the market varable. We propose two alternatve games. The rst descrbes a symmetrc duopoly where rms choose whether to be quantty setters or prce setters,.e., they x capacty, then determne the recprocal degree of d erentaton through R&D, and nally compete n the market. Inthe secondgame, the rst two stagesfollow a reverse order. We are then able to reassess the choce between prce and quantty. So far, the establshed wsdomn ths respect (Sngh and Vves, 1984) mantans that rms prefer to play a Cournot game because quantty settng s a domnant strategy for any gven degree of substtutablty. By ntroducng two addtonal stages, where rms undertake rreversble commtments, we nd a rcher set of equlbra, where the choce of the market varable s a ected by technology. In nether of the two games Sngh and Vves s result necessarly holds. When the R&D decson s taken at the second stage, goods are charactersed by d erent degrees of substtutablty n each subgame. There are parameter ranges wheren symmetrc Cournot behavour ether s not an equlbrum or s not unque. When, nstead, the R&D e ort takes 1 A subset of ths lterature concerns repeated games wth capacty constrants (Brock and Schenkman, 1985; Benot and Krshna, 1987; Lambson, 1987; Davdson and Deneckere, 1990). 3

4 place at the rst stage, t does not n uence the choce of the market varable, unquely determned by the tradeo between revenue and the cost of capacty. Some unconventonal results n terms of socal welfare are due to the capacty-ntensve character of Bertrand competton. Ths may releve the Cournot equlbrumfromtssocal ne cency. Moreover, we dentfy parameter sets, where the duopoly equlbrum may also be socally optmal. The remanderof the papersorganzedasfollows. The basc modeland the market stage are descrbed n secton 2. The two games and optmal nvestment behavour are dealt wth n secton 3. In secton 4, the welfare analyss s carred out. Concludng remarks are n secton 5. 2 The model and the market stage We descrbe a duopoly where rms noncooperatvely play a one-shot threestage game where they rst choose the market varable, then determne ther respectve e orts n product nnovaton, and, nally, noncooperatvely optmze w.r.t. the market varable chosen at the rst stage. Frms play smultaneously n every stage. They may compete symmetrcally ethernquanttes or n prces, or asymmetrcally, one beng a quantty setterwhle the others a prce setter. We also consder an alternatve orderng of stages, where the determnaton of the R&D e ort occurs at the rst stage and the choce of the market varable s made at the second. As n the prevous case, the thrd stage descrbes the market game. The demand sde s a smpl ed verson of Dxt (1979) and Sngh and Vves (1984). The symmetrc demandfunctonsunder Cournot andbertrand competton are, respectvely p =1 q q j (1) q = 1 1+ p 1 2+ p j 1 2 (2) In the asymmetrc case, where rm s a quantty setter, whle rm j s a prce setter, demand functons are: p =1 q + (p j + q 1) (3) q j =1 p j q (4) 4

5 Parameter 2(0;1] represents product substtutablty as percevedby consumers, dependng upon the products rms supply. Wthout loss of generalty, we assume that unt cost s constant and equal to zero, so that ndvdual pro t gross of nvestment concdes wth revenue R IJ = p q ; where IJ 2 fpp;qq;pq;qpg ndcates the knd of competton prevalng at the market stage. The bnary choce between a prce or a quantty strategy entals settng up a capacty at the costx IJ 2 x PP ;x QQ ;x QP ;x PQª ;.e., choosng a plant sze whch s optmal n any possble market subgame, whch we now solve by backwardnducton. Straghtforward calculatonsleadto: q QQ = ;1 2 ; q PP = q PQ 2 2 = 2 0; 1 ; q QP R QQ = R PP = 1 1 (2+ ) (2 ) 2 (1+ ) 2 0; ; (0;1]; (5) = 2 2[0:454;1]; (6) ;1 ; (7) 4 0; 1 ; (8) 4 R QP = ( 2)2 (1 2 ) 2 ; R PQ (3 2 4) 2 j = ( 1)2 ( +2) 2 2 0; 1 : (3 2 4) 2 4 (9) On the bass of the above quanttes, the followng chan of nequaltes can be establshed: q QP >q PP >q QQ >q PQ 8 2(0;1]: (10) Ths reveals that, gven the varable chosen by rmj, the output of rm as a quantty setter s larger than her output as a prce setter. Moreover, ndustry output under symmetrc Bertrand behavour s hgher than n the alternatve settngs. Accordngly, the same sequence of nequaltes must hold for capactes and ther correspondng cost: x QP >x PP >x QQ >x PQ 8 2(0;1]; 2x PP x QP +x PQ >2x QQ 8 2(0;1]: (11) As to revenues t s known from Sngh and Vves (1984) that R QP R PP andr QQ >R PQ 8 2(0;1]; mplyng that quantty settng s the domnant strategy f the choce between prce and quantty reles on revenue only. 5

6 Hence, there exsts a trade-o between the cost of nstallng the plant and the revenues assocated wth sze tself. The above dscussoncan be summarzed n the followng: Lemma 1 Quantty settng revenue-domnates prce settng, whle prce settng capacty-domnates quantty settng. We are now n a poston to tackle the ssue of the R&D technology. A pror, parameter can be consdered as an empty box to be lled by rms choces. 2 At the second stage, rms may ether costlessly produce perfect substtutes (wth = 1), or d erentated products f at least one of them undertakes R&D actvty. We adopt a general characterzaton of the R&D technology, = (k ;k j ); where k s the R&D nvestment of rm, wth k 2 [0;k max ); k max de nng the amount of nvestment gvng rse to two ndependent monopoles. For smplcty of notaton, n the remanderof the paper we use and to ndcate@ =@k and@ 2 =@k; 2 respectvely. The R&D functon s symmetrc between the two rms, and 0; 0: Pro ts are¼ IJ =R IJ k x IJ : We assume that¼ IJ s contnuous and twce d erentable w.r.t. k for all 2(0;1]: Under the assumpton that, ntally, =1; n order for nvestment to take place, t must be@¼ IJ =@k >0 for at least one rm. An nteror soluton of the Nash game at the nvestment stage exsts f there s at least a par(k ;k j ) such that 2(0;1);@¼ IJ =@k 2 ¼ IJ =@k 2 0for both rms. A su cent condton for asymptotc stablty s that(@ 2 ¼ IJ =@k 2)2 (@ 2 ¼ IJ j ) 2 : 3 Two alternatve games Here we descrbe two alternatve games where ² At the rst stage, by choosng the strategc market varable from the set V = fp;qg; rms set capactyx IJ. At the second stage they choose the optmal R&D e ortk : At the last stage they compete n the market. ² At the rst stage, rms undertake R&D actvty. At the second stage they choose capacty, whle the thrd stage remans the same. 2 The unt nterval assumed for descrbes a symmetrc horzontal d erentaton equvalent to that arsng n Hotellng s (1929) lnear cty, as long as, n the latter, rms play smultaneously. See the dscusson n Harrngton (1995). 6

7 3.1 The rst game The tree for the rst game s llustrated n gure 1. Fgure 1 : Game I P * 1 H 2 HHHH P * k max k max H HH Q Hj * 1 H * H 2 * smultaneous market HH HH H HH subgames Hj Hj Hj 0 0 Q H HHHj * H HH Hj The explct soluton of every subgame of the thrd stage s already known from the prevous secton. Here we deal wth the second stage of the game, where rms decde the optmal level of nvestmentk n product nnovaton Incentve to nnovate under Cournot competton When both rms are quantty setters, the relevant revenue functons are gven by (7). Frst and second order condtons w.r.t. k QQ µ = 1+ 2 =0; (2+ ) 2 ¼ QQ =3 2 (2+ ) 0: (13) We can mplctly de ne optmal nvestment behavour by solvng (12), obtanng = (2+ ) 3 =2; whch s negatve for all 2 (0;1]: Substtutng and rearrangng, (13) smpl es to 3(2+ ) 5 =4: If we consder@¼ QQ =@k at both the lower and the upper bound of the admssble nvestment range, de ned respectvely byk =0 (and =1) andk max ( =0); we characterze 7

8 the su cent condtons for an nterorsoluton to exst. When = 1; there s an ncentve to start nvestng n d erentaton f@¼ QQ =@k j =1 >0;.e., j j > 27=2: If ths does not hold, a corner soluton may obtan atk = 0; nvolvng =1: If R&D productvty s not su cently hgh, an externalty e ect prevents rms from nvestng. Asproducts tendto become completely ndependent,.e., tendsto zero, we QQ lm = +4)<0=) j j<4: (14) If the above condton s volated, a corner soluton may arse ask tends to k max and tends to zero. When@¼ QQ =@k j =1 >0and lm =@k QQ a su cent condton for an nteror soluton to exst s¼ QQ j =1 lm ¼ QQ ;!0 whch holds fk max 5=36: Incentve to nnovate under Bertrand competton Under symmetrc prce settng behavour, the relevant revenue functons are gven by (8). Frst and second order condtons w.r.t. k PP = 3 ( ( 2) 3 (1+ ) 1=0; (15) ¼ PP =2 2 ( )+ ( ) 0: (16) From (15) we obtan = ( 2)3 ( +1) 2 ( ) 3 ( 1) ( 2) 3 ( +1) 2 <0 8 2(0;1]: (17) Pluggng (17) nto (16), the second order condton (SOC) smpl es as follows: 3( 2)4 ( +1) 2 ( ) >0 8 2(0:3611;1]: (18) The above condton mples that the SOC s always met for 2(0;0:3611]: We now characterze the su cent condtons for an nteror soluton to exst. Consder@¼ PP =@k atk =0(and =1) andk max ( =0): At the 8

9 outset, under complete substtutablty, rm starts nvestng n d erentaton PP =@k j =1 > 0;.e., j j > 2: Otherwse, a corner soluton may obtan atk =0; nvolvng =1: Asproducts tendto become completely ndependent,.e., tendsto zero, we have that PP =@k < 0 f j j < 4: Ths obvously concdes wth!0 (14), snce the requrement for products to become completely ndependent s unrelated wth the strategc varable. Agan, f ths condton s not met, a corner soluton may arse ask tends tok max and tends to zero. PP =@k j =1 >0 andlm =@k >0; a su cent condton for an PP soluton to obtan s¼ PP j =1 lm!0 ¼ PP ; whch holds fk max 1=4: Incentve to nnovate n the mxed settng We now deal wth the mxed case where, say, rm s a quantty setter and rmj sa prce setter. The FOCsat the nvestment stage are, QP = 8(8+2 3 ) 2 ( ) (3 2 4) 3 PQ j = 12 ( 2 + 2) 2 j + 2(1+2 )( 2 + 2) j 1=0: j (4 3 2 ) 3 (4 3 2 ) 2 From (20) t can be ver ed that@¼ PQ j =@k j j =1 = 1; whch mples that the prce settng rm ntally does not nvest. As to the quantty settng QP =@k j =1 > 0 for all j j > 1=2: Ths entals that, at the outset, the quantty setter nds t easer to start nvestng f the rval s a prce setter, than n the symmetrc stuatonwhere both rmsare quantty setters. Ths can be gven the followng nterpretaton. On the one hand, the quantty setter has a hgher ncentve sncer QP >R PQ j for all 2(0;1): Intutvely, when =1; the quantty setter s unable to nternalze the strategc advantage mplct n settng an output level n the market stage, because product homogenety drves the prce down to margnal cost and consequently the output to the level correspondng to perfect competton and zero pro ts. On the other hand, the prce setter never starts nvestng n d erentaton because she explots the postve spllover exerted by the R&D e ort of the quantty setter. 9

10 The SOC forthe quantty 2 ¼ 2 =3( ) : (21) Smple calculatonssu ce to establsh 2 ¼ 2 j =1 _ 21( ) 2 0; (22) whch s always true. To complete the characterzaton of the nteror soluton for the quantty setter, observe that obvously QP =@k < 0 f j j < 4: Fnally, QP =@k j =1 >0 andlm@¼ QP =@k >0; a su cent condton for an nteror!0 soluton to obtan s¼ QP j =1 lm¼ QP ; whch holds fk max 1=4:! The reduced form We are now n the poston to solve the rst stage of the game, whereby each rm chooses to act ether as a quantty setter or as a prce setter. The reduced form of the game s descrbed by matrx 1. P P ¼ PP ;¼ PP j Q ¼ QP ;¼ PQ j j Q ¼ PQ ;¼ QP j ¼ QQ ;¼ QQ j Matrx 1 Suppose that at least one of the subgames represented n the above matrx yelds an nteror soluton where ether one or both rms nvest n product d erentaton,.e., exclude the stuatons where = 1 or = 0 n all subgames. The choce of market varable s drven by the ncentve to nvest n product d erentaton, whch n turn depends on the margnal productvty of captal, and j : As to the nvestment behavour of rm, the relevant parameter space can be dvded nto three regmes: 10

11 1. j j > 27=2: In ths regon, at least one rm nvests n product dfferentaton, ndependently of downstream competton. The e cency of technology s hgh enough to trgger nvestment regardless of the level of splloverexertedonthe rval. 2. j j 2(2;27=2]: In ths regon, nvestment s observed n all cases but n the symmetrc Cournot settng. 3. j j 2 (1=2;2]: In ths regon, mxed cases gve rse to nvestment at least by the quantty setter. When =1; the prce setter does not nvest. However, we cannot exclude that the behavour of the quantty setter trggers nvestment by the prce setter for some 2(0;1): 4. j j 2(0;1=2]: In ths regon no rm nvests. As a result, market competton takesplace n homogeneous products, due to extremely low margnal productvty of the R&D actvty. Astraghtforward consequence of the above dscussons: Lemma 2 At the outset, under product homogenety, the ncentve to nvest of a quantty setter s hgher when the rval s a prce setter. The opposte holds for the prce setter. As to the choce of the market varable, one may wonder whether the well known result of Sngh and Vves (1984) holds true n the present settng. We prove the followng: Theorem 1 (A) Suppose j j 2(0;1=2]: Hence, =1; and the rst stage of the game () s a Prsoner sdlemma wtha unque Nashequlbrum(P;P) f x QQ x PQ > 1=9; () s a coordnaton game wth two Nash equlbra, (P;P) and(q;q), fx QQ x PQ <1=9: (B) Suppose () j j 2(1=2;2]; () j 2 [0;2:3326]; and () R QP k > x QP x PP : The reduced form of the game s a chcken game wth two Nash equlbra, (P;Q) and(q;p); f R PQ 1=9 > x PQ x QQ : A su cent condton for ths to obtan s 2 (0;2=3): Proof. (A) If j j 2 (0;1=2]; no rm nvests n product d erentaton ndependently of the shape of market competton, whch takes place wth homogeneous products. The reduced form s represented by matrx 2. 11

12 P P x PP ; x PP Q x QP ; x PQ j Q x PQ ; x QP 1=9 x QQ ; 1=9 x QQ Matrx 2 Frst, observe that1=9 x QQ > x PP and x PP > x QP : Ths excludes a chcken game, gvng rse to ether a Prsoner s Dlemma or a coordnaton game. The former obtans f x QQ x PQ > 1=9; yeldng prce settng as a strctly domnant strategy generatng(p;p) as the unque equlbrum. The latter obtans fx QQ x PQ <1=9: In ths case, no domnant strategy exsts and both(p;p) and(q;q) are equlbra, wth(q;q) obvously domnatng (P;P). (B). Suppose () j j 2(1=2;2] and () the reducton n resultng from the quantty setter s nvestment does not nduce the prce setter to undertake R&D actvty. To ensure ths, consder the FOC (20) for the prce setter at the R&D stage, and solve t w.r.t. j to obtan: j = (4 3 2 ) 3 2( 2 + 2)[6 ( 2 + 2)+(1+2 )(4 3 2 )] (23) whch takes ts mnmum at =0:91693; where j =2:3326: As a consequence, j 2[0;2:3326] s a su cent condton for the prce setter not to nvest. Ths case sshownnmatrx 3. j P Q P x PP ; x PP R PQ x PQ ; R QP j k j x QP Q R QP k x QP ; R PQ j x PQ 1=9 x QQ ; 1=9 x QQ Matrx 3 IfR QP k >x QP x PP >0 andr PQ 1=9>x PQ x QQ ; a chcken game obtans. Notce that, asx PQ x QQ <0; a su cent condton forr PQ 1=9> x PQ x QQ to hold sr PQ 1=9=( 1) 2 ( +2) 2 =(3 2 4) 2 1=9>0; whch s true for all 2(0;2=3): 12

13 The asymmetry observed at equlbrum n terms of capacty n case B s drven by the expectaton on the part of the prce setter of recevng a postve externalty from the quantty setter s decson to nvest n product d erentaton at the ensung R&D stage. 3 The above dscusson s su cent to show that there may exst a non trval parameter range where quantty settng snot a domnant strategy, so that the choce of the market varable may not follow the rule traced by Sngh and Vves (1984). Analogous consderatons hold n the remander of the range,.e., for j j>2: E.g., suppose that j j 2(2;27=2]; n ths range, nvestment s observed n all cases but the symmetrc Cournot one. For the latter to be the unque outcome of the game, as a result of an equlbrum n domnant strateges, the followng nequaltes must smultaneously hold: ¼ QP ¼ PP 0; 1 9 ¼PQ x QQ : (24) If bothnequaltes above are volated the unque equlbrums(p;p): If only one of the two nequaltes holds we end up wth ether a chcken game or a coordnatongame. Ourconclusonsmpnge uponthe endogensatonof the d erentaton parameter whch s n uenced by the nature of downstream competton. The d erent ncentves to nvest n product d erentaton determne a result whch s related to the relatve sze of revenues and/or capactes. 3.2 Thesecond game Consder the alternatve game, where the choce of the R&D e ort takes place at the rst stage of the game, whle the choce between P and Q;.e., the choce of capacty, s located at the second stage. Suppose, at the rst stage, rms set a spec c equlbrum level ofk andk j. In so dong, they x the numercal value of n the unt nterval. Then, the choce between P andqs descrbed by matrx 4. 3 Ths resembles the analyss carred out by Fudenberg and Trole (1984) and Bulow, Geanakoplos and Klemperer (1985) concernng entry deterrence. 13

14 P P R PP x PP ; R PP j x PP j Q R PQ x PQ ; R QP j x QP Q R QP x QP ; R PQ j x PQ R QQ x QQ ; R QQ j x QQ Matrx 4 Observe that the nvestment n product d erentaton s omtted from matrx 4, as t s common to all payo s accrung to rm; and consequently t s rrelevant as to the choce between P and Q: The nspecton of matrx 4 reveals that the equlbrum of ths game depends on the nterplay between the capacty-domnance of prce-settng and the revenue-domnance of quantty settng. Sngh and Vves s (1984) concluson, that symmetrc Cournot behavourshouldobtan n the unque equlbrumof thergame, draws upon the revenue-domnance of quantty settng, whch can be o set by the costly acquston of capacty. As to the lterature on Cournot equlbra under capacty constrants (LevtanandShubk, 1972; KrepsandSchenkman, 1983; Davdson and Deneckere, 1986; Osborne and Ptchk, 1986), the above analyss hghlghts the opportunty cost of buldng up capacty, and ts role n shapng endogenously market competton. As an llustraton, consder the followng nequaltes: x QP x PP >R QP R PP >0; x QQ x PQ >R QQ R PQ >0: (25) If the above nequaltes hold, the unque equlbrum s (P;P). Otherwse the game n matrx 4 can have(q;q) as the unque equlbrum or be ether a chcken game, or a coordnaton game. 4 Welfare analyss The foregong analyss of rms behavour opens the queston whether the well establshed result that Bertrand competton yelds the hghest welfare level should be expected to carry over to a settng where rms nvest n R&D and productve capacty. Indeed, some contrbutons pont out that ths may not happen when rms R&D e orts are drected towards the attanment of a process nnovaton (Delbono and Dencolò, 1990; Qu, 1997). We show below that smlar results emerge when capacty s endogenously decded. 14

15 De ne socal welfare as the sum of consumer surplus plus net pro ts,.e.,sw IJ =CS IJ + IJ ; where IJ =¼ IJ +¼ IJ j : The establshed wsdom states that prce settng behavour enhances consumer surplus, wthcs PP CS PQ =CS QP >CS QQ 8 2(0;1]. However, from (5-6), we know that mplyng 2q PP q QP +q PQ >2q QQ 8 2(0;1]; (26) 2x PP x QP +x PQ >2x QQ 8 2(0;1]: (27) As a consequence, we can state the followng Lemma 3 From a socal standpont, prce settng surplus-domnates quantty settng, whle quantty settng capacty-domnates prce settng. Although we cannot derve explct conclusons for both games and all the relevant parameter subsets, we can unambguously determne that there exsts at least one case where symmetrc Bertrand behavour s not socally e cent. Consder the rst game, where the choce of capacty takes place at the rst stage. The followng obtans: Proposton 1 Suppose j j 2(0;1=2]: A necessary and su cent condton for the socal optmalty of symmetrc Bertrand competton s(x PP x QQ ) 2 (0;1=36): Otherwse, symmetrc Cournot competton s socally optmal. Proof. If j j 2(0;1=2]; we know from the prevous secton that rms compete nhomogeneous products, rrespectve of the strategc varables chosen at the rst stage. Gven =1, ndustry output s equal to one n all cases except the symmetrc Cournot; as a consequence, the cost of the overall capacty used n the ndustry must be the same everywhere except n the Cournot settng. Hence, from the socal standpont, relevant welfare levels are representednmatrx 5. j P Q P 1=2 2x PP 1=2 2x PP Q 1=2 2x PP 4=9 2x QQ Matrx 5 15

16 To prove the above Proposton, t su ces to observe that 1 2 2xPP > 4 9 2xQQ x PP x QQ < 1 36 : (28) If the opposte obtans,.e., x PP x QQ > 1=36; the tradeo between consumer surplus and the cost of capacty favours Cournot aganst Bertrand competton. Our next step n the welfare analyss conssts n verfyng whether there exsts any subset of the space de ned by the cost of capacty, wheren the prvately optmal equlbrum chosen by rms s socally e cent. We prove the followng Proposton 2 Suppose j j 2(0;1=2] andx QQ <x PP : If ()x QQ >x PP 1=36; and ()x QQ >x PQ +1=9; the unque duopoly equlbrum(p;p) s also socally e cent. If () x QQ <x PP 1=36; and () x QQ >x PQ +1=9; the unque duopoly equlbrum(q;q) s socally ne cent. Ifx QQ <x PQ +1=9; rms play a coordnaton game wth two equlbra,(p;p) and(q;q). When x QQ > x PP 1=36; (P;P) s socally e cent; otherwse (Q;Q) s socally e cent. Proof. To prove the above Proposton, we resort to a graphcal exposton. Fgure 2 below s drawn n the space x PP ;x QQª ; for a gven level ofx PQ : ² In regon ABCD, the duopoly equlbrum fp;pg concdes wth the socal optmum. ² In regon BDE, the duopoly equlbrum s agan fp;pg; whle the socal optmum s fq;qg: 16

17 Fgure 2 : Equlbrum analyss x QQ 6 C A B 0-1/36 F? D E - G x PQ +1=9 x PP ² In regon 0ABF, the duopoly game has two equlbra, fp;pg and fq;qg; the formerbeng socally optmal. ² In regon BEFG, the duopoly game has two equlbra, fp;pg and fq;qg; the latterbeng socally optmal. Consder now the second game where the choce of capacty takes place at the second stage. On the bass of Lemma 1 and Lemma 3, t can be establshed that a con ct between socal optmalty and prvate optmalty exsts asfaras the market varable s concerned. 5 Conclusons Adoptng the same demand structure as n Sngh and Vves (1984), we have endogensed both the choce of capacty and the degree of product substtutablty by ntroducng two stages where rms noncooperatvely undertake technologcal commtments before competng on the market. The consequences are twofold. Frst, n both cases the nvestment behavour of any 17

18 gven rm depends upon whether the rval s a quantty or a prce setter. Spec cally, we have establshed that () gven the market varable selected by the rval, beng a quantty setter s costler than beng a prce setter; and () a quantty setter s ncentve to nnovate s hgher when the rval s a prce setter, whle the opposte holds for a prce setter. Second, a set of equlbra emergesfromthe reducedformof the game, where rmschoose betweenprce and quantty. We have shown that, when the productvty of nvestment s relatvely low, the game exhbts two asymmetrc equlbra n whch one rm s a prce setter and the other s a quantty setter, and only the latter nvests n d erentaton. In the remanng scenaros of nnovaton technology, the equlbra are determned by the nterplay between the level of nvestment and the resultng level of d erentaton, vs à vs the comparson between revenues andthe cost of capacty. Socal welfare analysssuggests two consderatons. Frst, symmetrc prce competton s more expensve n terms of nstalled capacty. Ths mples that Bertrand behavour may not be socally e cent. Second, there are condtons under whch prvate ncentves may lead rms to play a socally optmal equlbrum. 18

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