Bertrand Competition under Three-part Tari s in Vertically Related Markets

Transcription

1 Bertrand Competton under Three-part Tar s n Vertcally Related Markets Mara Alprant Emmanuel Petraks March 2015 Abstract Ths paper shows that Bertrand competton emerges n equlbrum n the downstream market of a vertcally related ndustry wth upstream monopoly and tradng va threepart tar s. Ths s n sharp contrast wth the bulk of the lterature n whch Cournot competton s the equlbum mode of competton. Bertrand competton turns out to be bene cal for all rms, but not for consumers and the socety. Keywords: Bertrand competton; Cournot competton; vertcal relatons. JEL class caton: D43; L13; L14 Alprant: Düsseldorf Insttute for Competton Economcs (DICE), Henrch-Hene nversty of Düsseldorf, e-mal: alprant@dce.hhu.de; Petraks: Department of Economcs, nversty of Crete, nv. Campus at Gallos, Rethymnon 74100, Greece, e-mal: petraks@uoc.gr. Ths research has been co- nanced by the European non (European Socal Fund - ESF) and Greek natonal funds through the Operatonal Program "Educaton and Lfelong Learnng" of the Natonal Strategc Reference Framework (NSRF) - Research Fundng Program: Thals - Athens nversty of Economcs and Busness - "New Methods n the Analyss of Market Competton: Olgopoly, Networks and Regulaton". Full responsblty for all shortcomngs s ours.

2 1 Introducton Nowadays, contracts sgned among vertcally related rms (e.g., nput sulers and nal good manufacturers) have become more complex, ncludng often besdes a per-unt of nput prce, a varety of other terms, such as trade dscounts, slottng and lstng fees, and other lump-sum payments. Such contracts have receved lately a lot of attenton by researchers and polcy makers due to ther sgn cant mplcatons on rms behavor, market outcomes and socal welfare. In ths paper, we nvestgate the mode of competton that arses n equlbrum n a vertcally related market n whch an upstream rm trades wth two competng downstream rms usng three-part tar contracts. The latter nclude a wholesale prce and a xed fee to be pad by the downstream rm to the upstream suler plus a compensaton fee (.e., a dscount over the xed fee) to be transferred downstream n the end of the game. The sequence of the moves s as follows. Frst, the upstream monopolst makes smultaneous take-t-or-leave-t o ers to the downstream rms stpulatng the type of contract (prce or quantty) to be o ered by the downstream rm to customers 1 along wth a compensaton fee that the suler wll pay back to the downstream rm f the latter abdes by the agreed contract. Second, the upstream monopolst bargans smultaneously and separately wth the downstream rms about ther two-part tar (wholesale prce plus xed fee) tradng terms. Fnally, downstream rms compete n the market. It s well known, snce the semnal paper of Sngh and Vves (1984), that f rms are free to choose ther strategc varable to compete n the market, t s a domnant strategy for each of them to choose quantty, provded that rms goods are substtutes. Thus, Cournot competton arses n equlbrum. Over the last thrty years, the Sngh and Vves result has been generalzed and also extended n a varety of alternatve economc settngs (see e.g., Tanaka, 2001a&b; Tasnád, 2006; Resnger and Ressner, 2009; Matsumura and Ogawa, 2012; Scrmtore, 2013). For nstance, Tanaka (2001a&b) generalze t to a market wth n olgopolstc rms and a market wth vertcal product d erentaton, respectvely. Resnger and Ressner (2009) pont out the senstvty of the Sngh and Vves result n markets wth demand uncertanty, whle Matsumura and Ogawa (2012) and Scrmtore (2013) explore ts robustness n mxed markets consstng of welfare maxmzng and pro t maxmzng rms. 2 In the present paper, we show that, contrary to Sngh and Vves (1984) and the bulk of the lterature thereof, Bertrand competton arses n equlbrum n a vertcally related 1 Here we adopt the termnology used n recent papers (see e.g., Matsumura and Ogawa, 2012; Chrco and Scrmtore, 2013): a downstream rm o erng a prce contract to customers s another way to say that the downstream rm chooses prce as ts strategc varable. 2 Matsumura and Ogawa (2012) show that n a mxed duopoly, prce nstead of quantty competton arses n equlbrum, ndependently of whether products are substtutes or complements. Whle Scrmtore (2013) ndcates that the latter can be reversed n the presence of su cently hgh rm subsdzaton. 1

3 ndustry where the upstream suler and the downstream rms trade va three-part tar contracts. In partcular, we show that each suler-buyer par (.e., a par consstng of the upstream monopolst and one downstream rm) has ncentves to agree on a prce nstead of a quantty contract to be o ered by the downstream rm to customers. Ths s so because the (excess) jont pro ts of the suler-buyer par are hgher under a prce rather than under a quantty contract, ndependently of the contract choce of the rval par. Nevertheless, the downstream rms obtan hgher pro ts o erng quantty rather than prce contracts, whle the ooste s true for the upstream suler. Thus, the latter can motvate the downstream rms to swtch to prce contracts by o erng them aroprate compensaton fees - dscounts over the xed fee of the two-part tar. In equlbrum, the upstream suler has ncentve to do so and thus Bertrand competton arses n the downstream market. Our work contrbutes to the lterature that examnes the endogenous selecton of strategc varables n vertcally related markets (see e.g., Correa-López, 2007; Chrco and Scrmtore, 2013; Manasaks and Vlasss, 2014). In partcular, Correa-López (2007) and Manasaks and Vlasss (2014) recon rm Sngh and Vves s result, respectvely, n a unonzed market wth an ndustry-wde unon and n a vertcally related market n whch vertcal chans trade va wholesale prce contracts. 3 Chrco and Scrmtore (2013) ndcate that n a market wth network externaltes and manageral delegaton a reversal of Sngh and Vves s result occurs, but only f network externaltes are su cently strong. Our analyss extends ths lterature, showng that Bertrand competton consttutes the equlbrum mode of competton when tradng n a vertcally related market s conducted va more complex three-part tar s contracts. We also contrbute to the recently growng lterature on vertcal contractng va threepart tar contracts that consders a number of alternatve forms of payments made by the upstream to the downstream rms, such as slottng allowances, upfront payments, and dscount fees (de Fontenay and Gans, 2005; Marx and Sha er, 2007; Mklós-Thal et al., 2011; Rey and Whnston, 2013). In partcular, we hghlght how the presence of lumpsum payments, n the form of dscounts over the xed fee of the two-part tar contracts, made by the upstream suler to the downstream rms can a ect the equlbrum mode of downstream competton. 2 The Model Followng Alprant et al. (2014), we consder an upstream rm,, that produces, at zero margnal cost, an nput that two downstream rms, and D 2, use, n one-to-one-proporton, to produce ther nal goods. Downstream rms ncur no costs of producton other than what 3 However, Correa-López (2007) shows that a reversal of ths result can occur under decentralzed unons, dependng upon the barganng power dstrbuton and the unon s relatve preference over wages. 2

4 they pay to the upstream suler to obtan the nput. Consumers nverse and drect demands for D s nal good are: p a q q j and q (a p ) (a p j ) 1 2, ; j 1; 2, 6 j, where p and q are respectvely D s prce and quantty, and, wth 2 (0; 1), s a measure of the product substtutablty. The rms play a three stage game wth observable actons. In stage one, makes smultaneous take-t-or-leave-t o ers to the downstream rms regardng the type of contract (prce, p, or quantty, q) that each D wll o er to customers and a compensaton fee, G, to be pad by to D. Each downstream rm then, smultaneously and separately, accepts or rejects the o er. 4 We envsage ths compensaton fee as a transfer that wll make to D n the end of the game, provded that D abdes by the agreement. In stage two, bargans smultaneously and separately wth each D over the terms of a two-part tar contract,.e., over a wholesale prce, w, and a xed fee, F. Fnally, and D 2 choose each ts prce or ts quantty, dependng on the outcome of the rst stage. 5 Note that tradng between and D takes the form of a three-part tar contract (G ; F ; w ), where G s a back payment from to D n case that D comples wth the rst-stage agreement and (w ; F ) are the two-part tar terms. In ths sense, G can also be thought as a dscount over the xed fee F the downstream rm D transfers upstream. The barganng power of and D s gven, respectvely, by and 1, wth 2 (0; 1]: We solve the game by backwards nducton and by nvokng the Nash equlbrum of smultaneous generalzed Nash barganng problems n Stage 2. That s, the (; D ) par takes as gven the outcome of the smultaneously-run negotatons of the (; D j ) par. 6 Moreover, as s standard n the lterature, to obtan a unque equlbrum we assume parwse proofness on the equlbrum contracts,.e., we requre that the contract between and s mmune to a blateral devaton of wth D 2 and vse versa. 7 of all the (canddate) equlbra we assume throughout: 8 Assumpton 1: > () 3 (2 )(2 2 ) and < (p 17 1)4 that To guarantee exstence and stablty 4 Our results stll hold f we assume nstead that n the rst stage bargans, smultaneously and separately, wth each D over the type of contract (p or q) to be o ered from D to customers and a compensaton fee G : See also Footnote Accordng to Rey and Vergé s (2004) termnology, we assume that contracts are nterm observable. Interm observablty of contracts has been also used by, among others, Horn and Wolnsky (1988), Gal-Or (1991), McAfee and Schwartz (1995), and Mllou and Petraks (2007). 6 Ths s a common assumpton n the lterature on multlateral contractng used by, among others, Cremer and Rordan (1987), Horn and Wolnsky (1988), O Bren and Sha er (1992), and Mllou and Petraks (2007). 7 A more detaled analyss regardng the possble non-exstence of a pure strategy parwse proof equlbrum can be found n McAfee and Schwartz (1995), and Rey and Vergé (2004). 8 See Tremblay and Tremblay (2011) for detals regardng the stablty condtons under the mxed Cournot- Betrand model. 3

5 For notatonal reasons, we use superscrpt m, qq, denotng respectvely the equlbrum values under prce, quantty and mxed contracts o ered to customers n the downstream market. 3 Equlbrum Analyss Condtonal on the rst stage outcome, there are three possble subgames n the last stage of the game: Both downstream rms o er quantty contracts (qq), both downstream rms o er prce contracts (), and one downstream rm o ers a prce contract wth the other o erng a quantty contract (). Next, we analyze each of these subgames. () qq subgame: Each D chooses q to maxmze ts gross pro ts: max q (q ; q j ) (a q q j )q w q : (1) The reacton functons are: R qq (q j ) (a w q j )2: Thus, a reducton n w makes D a more aggressve compettor. Solvng the system of reacton functons, we obtan the quanttes and the upstream and downstream gross pro ts n terms of the wholesale prces: q qq (w ; w j ) a(2 ) 2w + w j 4 2 ; qq D (w ; w j ) [q qq (w ; w j )] 2 ; qq () subgame: Each D chooses p to maxmze ts gross pro ts: 2X 1 w q qq (w ; w j ): (2) max p (p ; p j ) (p w ) (a p ) (a p j ) 1 2 : (3) The reacton functons are: R (p j ) [a(1 ) + p j + w ]2: Thus, a reducton n w makes agan D a more aggressve compettor. Solvng the system of reacton functons, we obtan the quanttes and the upstream and downstream gross pro ts n terms of the wholesale prces: p (w ; w j ) a(2 2 ) + 2w + w j 4 2 ; q (w ; w j ) [p (w ; w j ) w ] D (w ; w j ) (1 2 )[q (w ; w j )] 2 ; 2X 1 w q (w ; w j ): (4) () subgame: Wthout loss of generalty, assume that o ers the prce contract and D 2 the quantty contract. Thus, chooses p 1, whle D 2 chooses q 2, n order each to maxmze ts gross pro ts: max p 1 1 (p 1 ; q 2 ) (a p 1 q 2 )(p 1 w 1 ): (5) max q 2 2 (p 1 ; q 2 ) [(1 )a (1 2 )q 2 + p 1 ]q 2 w 2 q 2 : (6) 4

6 The reacton functons are: R 1 (q 2) (a q 2 + w 1 )2 and R 2 (p 1) [(1 )a w 2 + p 1 ]2(1 2 ): As above, a decrease n w makes D a more aggressve compettor. Solvng the system of reacton functons, we obtan the quanttes and the upstream and downstream gross pro ts n terms of the wholesale prces: q 1 (w 1; w 2 ) a(2 2 ) (2 2 )w 1 + w ; q 2 (w 1; w 2 ) a(2 ) 2w 2 + w ; (w 1 ; w 2 ) [q 1 (w 1; w 2 )] 2 ; D 2 (w 1 ; w 2 ) (1 2 )[q 2 (w 1; w 2 )] 2 ; (7) (w 1; w 2 ) w 1 q 1 (w 1; w 2 )] + w 2 q 2 (w 1; w 2 ): In the second stage, bargans wth D over (w, F ), takng as gven the equlbrum tradng terms o ered to D j, (w j, F j ). followng generalzed Nash product, where m qq; ;. In partcular, w and F are chosen to solve the max [ m (w ; wj ) + F + Fj d(wj ; Fj )] [ m D w ;F (w ; wj ) F ] 1 ; (8) where d(wj ; F j ) w j qmon j (wj )+F j ; wth qmon j (wj ) (a w j )2; s s dsagreement payo,.e., s pro ts when negotatons wth D break down and D j acts as a monopolst n the downstream market facng w j.9 Maxmzng (8) wth respect to F, we obtan F m D (w ; w j ) (1 )[m (w ; w j ) wj qmon j (wj )]: Substtutng ths nto (8), t follows that w s chosen n order to maxmze the excess jont surplus of and D,.e., the pro ts of and D mnus s dsagreement payo : max w [ m (w ; w j ) + m D (w ; w j ) w j q mon j (w j )]: (9) As s well-know n the lterature, the upstream monopolst faces the so-called commtment problem when t deals wth several downstream rms and the contracts are ether nterm observable or secret. 10 The unobservablty of the contracts gves room for oortunstc behavor by the upstream monopolst. Thus, snce s unable to commt not to act oortunstcally, t cannot fully explot ts monopoly power. As a consequence, the overall ndustry pro ts are not maxmzed. 11 Substtutng successvely (2), (4) and (7) nto (9), takng the rst order condtons and solvng the system of equatons, we obtan the respectve equlbrum wholesale prces n the 9 Ths s standard n the lterature. Note that our results stll hold f we assume nstead that D j acts as a duopolst n the downstream market facng w j (as n Horn and Wolnsky, 1988). 10 For more detals over the commtment problem see, among others, Hart and Trole (1990), O Bren and Sha er (1992), McAfee and Schwartz (1994, 1995), Rey and Vergé (2004) and de Fontenay and Gans (2005). A detaled dscusson of the nature of the commtment problem n our settng can be found n Alprant et al. (2014). 11 In contrast, under publc contracts n whch each downstream rm observes all rms contract terms before decdng to accept or not ts own terms, can publcly commt to all terms of trade, and therefore the overall ndustry pro ts are maxmzed. 5

7 qq; and subgames: w qq a 2 2(2 2 < 0: (10) ) w a2 4 > 0: (11) w 1 a 2 2(1 + )(2 2 < 0; w 2 ) a2 > 0: (12) 4(1 + ) Observe that n the qq subgame, the equlbrum wholesale prces are below s margnal cost,.e., subsdzes va the wholesale prces the downstream producton. Intutvely, a reducton n the wholesale prce charged by to D, ncreases D s output and decreases D j s output (quanttes are strategc substtutes) that results nto hgher gross pro ts for D that can transfer upstream va a hgher F. In contrast, n the subgame, there s no subsdzaton. Ths s because a reducton n the nput prce charged to D, leads to a decrease n both D s and D j s prces (prces are strategc complements), that causes a relatvely smaller ncrease n D s pro ts that can transfer upstream va F. Interestngly, n the subgame, subsdzes the prce settng downstream rm, whle there s no subsdzaton for the quantty settng one. In ths case, a reducton n the wholesale prce charged to - the prce settng downstream rm- leads to a reducton n s prce, and thus to a reducton n D 2 s output leadng to hgher s pro ts that can transfer upstream va F. The ooste s true for the quantty settng downstream rm D 2 ; hence t s charged a wholesale prce above s margnal cost. The above dscusson ndcates that has stronger ncentves to ncrease the aggressveness of the downstream rms when the latter o er quantty contracts to customers nstead of prce or mxed contracts. In other words, has stronger ncentves to behave oortunstcally, and thus t faces a more severe commtment problem n the qq subgame than n ether the or the subgame. The followng Lemma summarzes, Lemma 1 The equlbrum wholesale prces satsfy: w > w 2 > 0 > w 1 > wqq. After proper substtutons, the equlbrum market outcomes n each subgame are reported n Table 1. 6

8 qq subgame q qq p qq a(2 ) 2(2 2 ) a(1 )(2+) 2(2 2 ) qq D a2 (1 )(2 ) 2 8(2 2 ) subgame q a(2+) 4(1+) a(2 ) 4 p D a2 (1 )(2+)[ ] 32(1+) qq a2 (2 )[(2 )(2 2 ) 3 ] 4(2 2 ) 2 a2 (2+)[2(2 )+(1 ) 3 (1 )] 16(1+) q 1 a(2+) 4(1+) subgame q 2 a(2 ) 2(2 2 ) p 1 a( ) p 4(1+)(2 2 ) 2 a( ) 4(1+)(2 2 ) a2 (1 )[16(1+) 2 ( )] 32(1+) 2 (2 2 ) D 2 a2 (1 )(2 ) 2 8(2 2 ) a2 [(1+)+2 6 (2 3)+ 7 (1 )(4 )] 32(1+) 2 (2 2 ) 2 ; 16(4 5 2 ) 8 4 (1 + 4) Table 1: Equlbrum quanttes, prces, upstream and downstream gross (from compensaton fees) pro ts. Lemma 2 ) The equlbrum outputs and prces satsfy: q qq q 2 > q 1 q p 1 > p 2 > q. ) The equlbrum downstream pro ts satsfy: qq D D 2 qp > D >. ) The equlbrum upstream pro ts satsfy: > > qq. and p > v) The jont pro ts of and D, m ;D m + m D, satsfy: ;D > ;D 2 qp ; > ; > qq ;D. As wholesale prces are substantally lower when the downstream rms o er quantty contracts than prce contracts to customers (Lemma 1), quanttes are hgher and prces are lower n the qq than n the subgame. Further, n the subgame, the quanttes and prces le n between the latter two subgames. Yet, although the quantty settng downstream rm faces a hgher wholesale prce (Lemma 1), t produces more (and sets a lower prce) than the prce settng one. In addton, we observe that a downstream rm obtans hgher pro ts when t o ers a quantty rather than a prce contract. Intutvely, downstream rms are better-o n the qq than n the subgame, snce they enjoy hgher e cency under the former due to the substantally lower margnal producton costs. Further, n the subgame, the prce settng rm obtans the lowest pro ts snce ts quantty s lower and ts prce hgher than those of the quantty settng rm. More mportantly, the upstream monopolst and the downstream rms have completely oosng nterests: The upstream monopolst obtans the hghest pro ts n the subgame n whch case the downstream rms pro ts are the lowest ones. The ooste holds n the qq subgame. Ths s so because the upstream monopolst su ers more from the commtment problem n the latter case. For a smlar reason, the upstream rm s pro ts n the subgame le n between the qq and subgames. 7

9 Last, but not least, the jont pro ts of each (; D ) par n the subgame always exceed those n the qq subgame, whle the respectve pro ts n the subgame le n between. In addton, the jont pro ts of the quantty settng par exceeds those of the prce settng par n the subgame. We now turn to the rst stage. makes smultaneous take-t-or-leave-t o ers to the downstream rms regardng the contract type (p or q) that each D wll o er to customers along wth a compensaton fee, G, that wll pay back to D n the end of the game provded that D abdes by the agreed contract. Each downstream rm then, smultaneously and separately, accepts or rejects the o er. 12 Lemma 2 nforms us that prefers both downstream rms to o er p contracts to customers; moreover, that the jont pro ts of each (; D ) par are hgher under a p than under a q contract, rrespectvely of whether the rval downstream rm o ers a p or a q contract. Therefore, can make an o er (p; G ), wth G qp D, to each downstream rm such that D wll accept. Ths s so because G compensates D for ts losses when swtchng from a q to a p contract o ered to customers (gven that D j o ers a p contract). But does have ncentves to make such o ers? Ths wll be the case f 2G > qq, or else, 2( qp D ) > qq. It can be checked that the latter nequalty always holds. It remans to be checked whether has ncentves to make an o er only to one of the downstream rms, say, to. By Lemma 2 we know that wll accept an o er (p; G 1 ) only f G 1 > G b qq D 2, qq D qp and < ( qq D > qq, so n prncple t could be so. However, qp > G (snce by Lemma D ). But f makes such an o er (p; G) b to, then ts pro ts ), whch are lower than those obtaned when makes (p; G ) o ers wll be to both downstream rms. Hence, wll not make a p o er to just one downstream rm. Proposton 1 summarzes our ndngs. Proposton 1 In equlbrum, the upstream monopolst o ers (p; G ), wth G qp D ; to each downstream rm and these o ers are accepted by both downstream rms. Hence, Bertrand competton s establshed n the downstream market. Fnally, the welfare consequences of alternatve contract con guratons o ered to customers by downstream rms are summarzed n the next Proposton. 12 If nstead bargans, smultaneously and separately, wth each D over p or q and G the analyss s as follows. Barganng over a dscrete varable (p or q) n case that transfers are allowed reduces to the choce of the contract type that maxmzes (; D ) s jont pro ts, gven the contract choce of the other par (; D j). Moreover, the ncremental (; D ) s jont pro ts stemmng from the best choce over the alternatve one are dstrbuted among the two partes accordng to ther respectve barganng powers, and 1 : Accordng to Lemma 2, ndependently of the contract type chosen by (; D j), the jont pro ts of the (; D ) par are always hgher when selectng a p than a q contract. Hence, t s a domnant strategy for each (; D ) par to choose a contract type p nstead of q and as a result, Bertrand competton arses n equlbrum. 8

10 Proposton 2 Consumers surplus and total welfare satsfy: CS qq > CS > CS and T W qq > T W > T W : Proposton 2 nform us that there s a msalgnment of socetal and market ncentves. Cournot downstream competton s socally preferable than ether mxed (.e. Bertrand- Cournot) or Bertrand downstream competton, but Bertrand competton prevals n the market equlbrum (Proposton 1). Consumers are better o when both downstream rms o er quantty contracts, snce n the qq subgame the prces are lower and the output s hgher than under any other subgame. Total welfare s hgher too n ths case, due to the hgher consumer surplus and downstream rms pro ts (Lemma 1). However, the exstence of back payments that can make to the downstream rms leads to socally ne cent contracts o ered to customers. 4 Concluson We have shown that n a vertcally related market n whch an upstream monopolst trades wth two competng downstream rms va three-part tar contracts, Bertrand competton arses as the equlbrum mode of competton n the downstream market. In the presence of compensaton fees that the upstream monopolst can pay back to the downstream rms, all rms are better-o when prce contracts are o ered n the downstream market. The explanaton behnd ths result comes from the fact that a prce contract yelds hgher jont pro ts for each suler-buyer par than a quantty contract, rrespectvely of the contract o ered by the rval downstream rm. As the losses of the downstream rms when selectng prce nstead of quantty contracts are outweghed by the overall gan of the upstream monopolst, the latter o ers them aroprate compensaton fees, or else dscounts over the x fee of the two-part tar, n order to motvate them to swtch to prce contracts. Fnally, note that our results have been obtaned under a monopolstc upstream market structure and nterm observable two-part tar contracts employed n the nput tradng between the vertcally related rms. It would be nterestng to check whether our results stll hold under alternatve assumptons about the upstream market structure and the nature of nput tradng terms between the upstream and the downstream rms. 5 Aendx Proof of Lemma 1: w qq a 3 2(1+)(2 2 ) Frst, w > 0. Hence, w w 2 a3 4+4 Proof of Lemma 2: ) Frst, note that q qq 0; hence, q qq q 2 > q > 0; second, w 2 > 0 > w 1 > w 2 > 0 > w 1 > wqq : q 2 and q q 1 : Further, p q 1 p 1 ; also, qqq q ; nally, w 1 a 3 4(1+)(2 2 ) > a 4 > 0; p 4(1+)(2 2 ) 1 9

11 p 2 a(1 )3 > 0; and 4(1+)(2 2 p ) 2 p qq ) Frst, note that qq D D D 2. Then D 2 a 4 4(1+)(2 > 0; hence, 2 p ) > p 1 > p 2 > q. D a2 (1 ) 3 [8 (1+)(4 2 )] > 0 and 32(1+)(2 2 ) a2 (1 ) 5 [4(1+) 2 (2+)] 32(1+)(2 2 ) > 0; hence, qq D D 2 > D >. ) Frst, a 3 [ ], where 16( ) + 5 ( ) , 32(1+) 2 (2 2 ) ( ) (8(2 + ) 2 ( ) and > 0. Thus > 0; second, > qq : v) Frst, ;D ;D 2 a2 4 [8 2 2 ( ) 6 +(8(1++ 2 ) 2 3 (2 )+ 5 )] 32(1+) 2 (2 2 ) 2 > 0; second, ;D 2 qq a3 [16(1+)+(8+( 2) 2 )((1 ) 2 2))] 32(1+) 2 (2 2 ) 2 > 0; hence, > ; a2 (1 ) 3 (8+( 2) 2 ) 32(1+) 2 (2 2 ) > 0; thrd, ; hence, ;D > ;D 2 > ; > qq ;D : Proof of Proposton 1: Proof of Proposton 1: Frst, G qp 0; second, 2G qq 2 4 (4 ) 2 > 0; hence 8(1+)(2 2 ) 2 ) ( qq D hence qq ;D a2 4 [4(1+) 2 ] 16(1+) 2 (2 2 ) 2 > 0; D a2 (1 ) 3 [8 (2 )(1+)(2+)] 32(1+)(2 2 ) > 2G > qq ; thrd, a2 3 [(1+)(2 2 )((2 )(1+)(2+) 8)+(2+)(8+( ))] < 0; 16(1+) 2 (2 2 ) 2 ( qq D ) < : Proof of Proposton 2: Consumers surplus s gven by CS m 1 2 (qm2 1 + q2 m2 + 2q1 mqm 2 ). Substtutng the respectve equlbrum quanttes (Table 1), t follows that, CS qq a 2 (2 ) 2 (1+), CS a2 [ (1+ 3 ) (4 2 )], CS a2 (2+) 2 4(2 2 ) 2 32(1+) 2 (2 2 ) 2 16(1+). Takng the d erences, CS qq CS a2 3 ( ) > 0 and CS CS a2 3 ( ) > 32(1+) 2 (2 2 ) 2 32(1+) 2 (2 2 ) 2 0; hence, CS qq > CS > CS. Total welfare s gven by T W m CS m + m + m + m D 2. From Table 1 t follows that, T W qq a2 (2 )(6 3 2 ), T W a2 [96+128(1 2 ) (4+3 2 ) 72 2 ] and T W 4( 2 2) 2 32(1+) 2 (2 2 ) 2 a 2 (2+)(6 ) 16(1+). Takng the d erences, T W qq T W a2 3 [8+(4 (8+3)] > 0 and T W 32(1+) 2 (2 2 ) 2 T W a2 3 [8(1 2 ) ] > 0;hence, T W qq > T W > T W. 32(1+) 2 (2 2 ) 2 References [1] Alprant, M., Mllou, C., Petraks, E., Prce vs. quantty competton n a vertcally related market. Economcs Letters 124, [2] Chrco, A., Scrmtore, M., Choosng prce or quantty? The role of delegaton and network externaltes. Economcs Letters 121, [3] Correa-López, M., Prce and quantty Competton n a d erentated duopoly wth upstream sulers. Journal of Economcs and Management Strategy 16, [4] Cremer, J., Rordan, M.H., On governng multlateral transactons wth blateral contracts. Rand Journal of Economcs 18,

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