rce and uantty Competton Revsted X. Henry Wang Unversty of Mssour - Columba Abstract By enlargng the parameter space orgnally consdered by Sngh and Vves (984 to allow for a wder range of cost asymmetry, Zanchettn (006 fnds that the Sngh and Vves result that frms always make larger profts under uantty competton than under prce competton fals to hold. Ths paper shows that whle proft rankng between prce and uantty competton can be (partally reversed the celebrated result by Sngh and Vves that frms always choose a uantty contract n a two-stage game contnues to hold n the enlarged parameter space. Ctaton: Wang, X. Henry, (008 "rce and uantty Competton Revsted." Economcs Bulletn, Vol. 4, No. 8 pp. -7 Submtted: February 7, 008. Accepted: March 3, 008. URL: http://economcsbulletn.vanderblt.edu/008/volume4/eb-08d4000a.pdf
. Introducton In ther semnal paper on prce and uantty competton n a dfferentated duopoly, Sngh and Vves (984 present three mportant fndngs (stated here assumng that goods are substtutes: ( both consumer surplus and total surplus are larger under prce competton than under uantty competton ( both frms profts are hgher under uantty competton than under prce competton and ( both frms choosng the uantty contract s a domnant strategy eulbrum n the two-stage game n whch frms choose between a prce contract and a uantty contract n the frst stage and then compete accordngly n the second stage. Recently, Zanchettn (006 relaxes the parameter restrcton mposed mplctly by Sngh and Vves to allow for a wder range of cost asymmetry and fnds that whle concluson ( above contnues to hold concluson ( above does not hold n the larger parameter space. In partcular, Zanchettn (006 fnds that, wth hgh degrees of cost asymmetry and/or low degrees of product dfferentaton, both the effcent frm s profts and total profts can be hgher under prce competton than under uantty competton. Snce concluson ( above s based on the rankng of frms profts under dfferent modes of competton, the fndng of Zanchettn (006 of the possblty of partal reversal n proft rankngs calls nto ueston whether concluson ( holds n the larger parameter space. The purpose of ths paper s to nvestgate the two-stage game orgnally studed by Sngh and Vves by allowng for the larger parameter space consdered by Zanchettn (006. Our man concluson s that, n the larger parameter space, both frms choosng the uantty contract s the only Nash eulbrum n the two-stage game t s ether a domnant strategy eulbrum or a weakly domnant strategy eulbrum. Hence the possblty of reversal n proft relatonshps wll not alter the concluson that n the two-stage game n whch frms frst commt between a prce contract and a uantty contract and then compete accordngly they wll always choose the uantty contract.. Model Setup Our model setup s the same as n Zanchettn (006. Two goods, and, are produced by frm and frm, respectvely. Frm (, has a constant unt cost of producton c. It s assumed that c c so that frm s at least as effcent as frm. The nverse demand functons for the two goods are gven by: p α γ j,, j, j. ( In (, α represents consumers reservaton prce for ether good and γ s the substtuton parameter. As n Zanchettn (006, we focus on the case of substtutng goods (.e., 0< γ<. The drect demand euatons are gven by: [( γ α p +γp j],, j, j. ( γ Zanchettn ntroduced the parameter x ( αc /( α c to measure the degree of cost
asymmetry between the two frms. The range of ths parameter s (0, and t ncreases as the cost gap between the two frms decreases. As ponted out by Zanchettn, the monopoly outcome n whch frm becomes a monopoly under ether prce or uantty competton prevals f x γ /. Zanchettn s focus and also our focus s thus the space S r {0 < γ < γ /< x }. Compared to Sngh and Vves (984, the space S r allows for a larger range of cost asymmetres between the two frms. In the space S r, both frms produce a postve output n the Cournot eulbrum. More specfcally, the Cournot eulbrum values are gven by (euaton (7 n Zanchettn (006: ( αc (γx 4 γ C C p c ( αc (x γ 4 γ C C p c C ( αc (γx 4 γ C ( α c (x γ 4 γ. (3 Wthn the space S r, both frms produce a postve output n the Bertrand eulbrum f L γ x > x ( γ γ (4 and the Bertrand eulbrum values n ths case are gven by (euaton (8 n Zanchettn (006: B B p c ( αc (γ γx γ ( γ (4 γ B B p c ( αc [( γ x γ] γ ( γ (4 γ B ( αc (γ γx γ 4γ B ( α c [( γ x γ] γ 4γ. (5 Wthn the space S r, the lmt-prcng eulbrum n whch only frm produces a postve output prevals f condton (4 s not satsfed and the Bertrand eulbrum values n ths case are gven by (euaton (0 n Zanchettn (006: L L ( αc x p c ( αc ( γx γ γ L ( αc ( γxx γ L L L p c 0. (6 The space S r ncludes as a subset the space consdered by Sngh and Vves whch s {0 < γ < x γ < }. Here, the space S r s slghtly dfferent from that defned n Zanchettn (006 n that the lne n whch γ s excluded. Ths s only for convenence of dscusson n the Bertrand competton case.
To complete the two-stage game, we next provde eulbrum values for the cases n whch frm and frm choose dfferent contracts n the frst stage of the game. Consder frst the (, case n whch frm chooses the uantty contract whle frm chooses the prce contract. Maxmzng frm s proft takng frm s prce as gven gves rse to frm s best response functon n uantty as gven by ( γ α c +γp ( γ. (7 Maxmzng frm s proft takng frm s uantty as gven gves rse to frm s best response functon n prce as gven by p α+ c γ. (8 Solvng the system of euatons comprsng (7 and (8 yelds the eulbrum values for the (, case as gven by p c ( αc (γx γ p c ( αc [( γ x γ] ( αc (γx ( γ ( α c [( γ x γ]. (9 It s obvous to see that the soluton n (9 s vald for all parameters n the space S r. Consder next the (, case n whch frm chooses the prce contract whle frm chooses the uantty contract n the frst stage of the game. Maxmzng frm s proft takng frm s uantty as gven gves rse to frm s best response functon n prce as gven by p α+ c γ. (0 Maxmzng frm s proft takng frm s prce as gven gves rse to frm s best response functon n uantty as gven by ( γ α c +γp ( γ. ( Solvng the system of euatons comprsng (0 and ( yelds the eulbrum values for the (, case as gven by 3
α p c ( c (γ γx p c ( αc (x γ γ ( αc (γ γx ( α c (x γ ( γ. ( From (, both frms produce a postve uantty as long as condton (4 s satsfed. Hence, ( gves the soluton to the (, case when condton (4 holds. If condton (4 does not hold then t s straghtforward to verfy that the lmt-prcng soluton gven by (6 s the corner soluton for the (, case. 3. The Two-Stage Game We now study the two-stage game n whch the two frms each choose between a prce contract and a uantty contract n the frst stage and then n the second stage they compete accordng to ther frst-stage choce of contract. The reduced frst-stage game matrx may take one of the followng two forms. If condton (4 s satsfed then the game matrx s gven by Frm rce uantty Frm rce uantty (, (, B B (, ( C, C In ths game matrx, B B and are gven by (5, C C and are gven by (3, and are gven by (, and and are gven by (9. If condton (4 s not satsfed then the reduced frst-stage game matrx s gven by Frm rce uantty Frm rce uantty (, 0 (, L L (, 0 ( C, C 4
L In ths matrx, the second column s the same as n the frst matrx above and s gven by (6. The followng lemma wll help us fnd the Nash eulbrum n each of the above two game matrces. (The proof of ths lemma nvolves straghtforward algebra and s omtted. Lemma : C ( > > ( >. L C B >,, art ( of Lemma confrms the reversal n proft relatonshp between prce and uantty competton. Namely, f condton (4 s not satsfed then the more effcent frm s proft s hgher when the frms compete n prces than when they compete n uanttes. However, parts ( and ( of Lemma together mply that each frm s proft from choosng the uantty contract s never less than that from choosng the prce contract holdng the other frm s choce of prce or uantty contract fxed. The followng proposton follows mmedately from the relatonshps n Lemma. roposton : If condton (4 holds then the two-stage game has a domnant strategy eulbrum n whch both frms choose the uantty contract n the frst stage. If condton (4 does not hold then the two-stage game has a weakly domnant strategy eulbrum n whch both frms choose the uantty contract n the frst stage. roposton mples that the two-stage game n whch frms choose between a prce contract and a uantty contract n the frst stage and then compete accordngly n the second stage has a unue Nash eulbrum n that both frms choose the uantty contract n the frst stage of the game. The frst part of ths proposton s smply an extenson of smlar result by Sngh and Vves (984 except here the parameter space s larger than that consdered by Sngh and Vves. The second part of ths proposton confrms that the Sngh and Vves result essentally holds for the entre parameter space S r. We have thus shown that the possblty of (partal reversal n proft relatonshps between prce competton and uantty competton wll not alter the concluson that n the two-stage game they wll always choose the uantty contract n eulbrum. Ths s unfortunate on welfare grounds snce ths outcome s welfare nferor to the outcome when both frms choose the prce contract. 5
References Sngh, N., and X. Vves (984 rce and uantty Competton n a Dfferentated Duopoly Rand Journal of Economcs 5, 546-554. Zanchettn,. (006 Dfferentated Duopoly wth Asymmetrc Costs: New Results from a Semnal Model Journal of Economcs and Management Strategy 5, 999-05. 6