Oligopoly Intermediation, Relative Rivalry, and the Mode of Competition

Transcription

1 IDEI 818 October 2013 Olgopoly Intermedaton, Relatve Rvalry, and the Mode of Competton Stephen F. Hamlton, Phlppe Bontems and Jason Lepore 7

2 Olgopoly Intermedaton, Relatve Rvalry, and the Mode of Competton Stephen F. Hamlton Calforna Polytechnc State Unversty San Lus Obspo Phlppe Bontems Toulouse School of Economcs (GREMAQ, INRA, IDEI) Jason Lepore Calforna Polytechnc State Unversty San Lus Obspo October 15, 2013 Abstract Polcy desgn n olgopolstc settngs depends crtcally on the mode of competton between frms. We develop a model of olgopoly ntermedaton that reveals the mode of competton to be an equlbrum outcome that depends ontherelatvedegreeofrvalrybetweenfrms n the upstream and downstream markets. We examne two forms of sequental prcng games: Purchasng to stock (PTS), n whch frms select nput prces pror to settng consumer prces; and purchasng to order (PTO), n whch frms sell forward contracts to consumers pror to selectng nput prces. The equlbrum outcomes of the model range between Bertrand and Cournot dependng on the relatve degree of rvarly between frms n the upstream and downstream markets. Prces are strategc complements and the equlbrum prces concde wth the Bertrand outcome when the markets are equally rvalrous, whle prces are strategc substtutes when the degree of rvalry s suffcently hgh n one market relatve to the other. Cournot outcomes emerge under crcumstances n whch prces are strategcally ndependent n ether the upstream or downstream market. We derve testable mplcatons for the mode of competton that depend only on prmtve condtons of supply and demand functons. JEL Classfcaton: L13, L22, F13 Keywords: Olgopoly; Intermedaton; Strategc Pre-commtment; Polcy. Correspondence to: S. Hamlton, Department of Economcs, Orfalea College of Busness, Calforna Polytechnc State Unversty, San Lus Obspo CA Voce: , FAX: , emal: shamlto@calpoly.edu. Part of ths paper was wrtten whle Stephen F. Hamlton was vstng the Toulouse School of Economcs and the author thanks members of that department for ther hosptalty. We are ndebted to Robert Innes and Rchard Sexton for nvaluable advse on an earler draft of the paper, and thank semnar partcpants at UC Berkeley, Toulouse School of Economcs, the Unversty of Nebraska and the Unversty of Kel for helpful comments. 1

3 1 Introducton An obstacle to dervng polcy mplcatons from the olgopoly model s the senstvty of strategc pre-commtment devces to the mode of competton. As observed by Fudenberg and Trole (1984) and Bulow, Geanakoplos, and Klemperer (1985), the strategc underpnnngs of the olgopoly model depend fundamentally on the manner n whch frms choce varables alter the margnal proft expressons of rvals, a feature that has essental mplcatons for the desgn of effectve polcy. For strategc trade polcy, export subsdes are optmal when frms choce varables are strategc substtutes (Brander and Spencer 1985); however, export taxes are optmal when frms choce varables are strategc complements (Eaton and Grossman 1986). For strategc delegaton between the owner and manager of a frm, the optmal contract overcompensates sales when frms compete n strategc substtutes, but overemphaszes cost when agents compete n strategc complements (Fershtman and Judd 1987, Sklvas 1987). For contracts between vertcally-algned supplers, the optmal contract nvolves lump sum transfers from manufacturers to retalers when retalers compete n strategc complements (Shaffer 1991), but nvolves negatve lump sum payments when retalers compete n strategc substtutes (Vckers 1985). 1 In ths paper, we develop a model of olgopoly ntermedaton that generates testable hypotheses on the mode of competton. We frame our model around duopoly ntermedares who purchase an nput from a common upstream market and sell fnshed goods n a common downstream market. Both the nput and output prces selected by olgopoly frms are strategcally nterdependent, so that a prcng strategy pursued by a frm n one market, for nstance a prce promoton on the fnshed good desgned to ncrease market share, has mplcatons for both the supply and demand condtons facng the rval. The ntermedated olgopoly model provdes a convenent way to represent strategc nteracton between frms: It reduces to the 1 The mode of competton of the olgopoly model also has mportant mplcatons for frst-mover advantage n sequental games between frms. It has been recognzed snce von Stackelberg (1934) that advantage goes to the frst-mover when choce varables are strategc substtutes; however, beng the frst-mover s dsadvantageous when frms compete n strategc complements (Gal-Or 1985). 2

4 usual olgopoly (olgopsony) model when the supply (demand) functon s nfntely elastc, yet encompasses general forms of strategc nteracton when prces are nterdependent n both supply and demand functons. We characterze the strength of the olgopoly prcng nteracton between frms n the upstream and downstream markets n terms of the degree of market rvalry. The products procured and produced n each market are dfferentated, as would be the case when manufacturers rely on specalzed nputs to produce branded consumer goods, and a greater extent of product dfferentaton softens the degree of market rvalry. Our settng thus extends Stahl s (1988) analyss of the ntermedated olgopoly model to dfferentated product markets. We follow Stahl (1988) n examnng two forms of sequental prce competton: () purchasng to stock (PTS), n whch the frms select nput prces pror to settng output prces; and () purchasngtoorder (PTO), n whch the frms sell forward contracts to consumers pror to selectng nput prces. Our observatons on the mode of competton depend crtcally on the relatve degree of market rvalry n the upstream and downstream markets. We defne the relatve rvalry of the upstream market to be the dfference (n absolute terms) between the rato of the cross-prce to own-prce elastcty of supply and the rato of cross-prce to own-prce elastcty of demand. It s a measure of the relatve strength of the olgopoly nteracton at each pont of market contact between frms. If the cross-prce elastcty s zero n a market (the monopoly case), the market s non-rval, whereas f the rato of cross-prce elastcty to own-prce elastcty s approxmately one n a market (the case of commodtzed products), the market s hghly rvalrous. Markets are equally rvalrous when there s no dfference n the ntensty of the olgopoly nteracton n the upstream and downstream markets, as would be the case n the homogeneous product settng consdered by Stahl (1988). Our man results can be summarzed as follows. Frst, we fnd that an outcome of Bertrand merchants emerges only under crcumstances n whch the upstream and downstream markets are equally rvalrous. Such an outcome occurs rrespectve of 3

5 the degree of product dfferentaton n each market. Thus, our analyss reveals the underpnnng of the Stahl (1988) outcome of Bertrand merchants to be determned by the relatve degree of market rvalry rather than the absolute degree of rvalry n the upstream and downstream markets. Second, we show frm profts to be greater under PTS than under PTO when the downstream market s relatvely rvalrous, whereas the PTO outcome Pareto domnates the PTS outcome n terms of frm and ndustry profts when the upstream market s relatvely rvalrous. In ether case, settng prces n the less rvalrous market serves as a pre-commtment devce for prces n the more rvalrous market, provdng frms wth the ablty to soften prce competton n the market where the degree of olgopoly nteracton s most ntense. Thrd, we demonstrate that when one market s more rvalrous than the other, prces can be ether strategc complements or strategc substtutes dependng on the relatve degree of market rvalry. As the relatve degree of market rvalry ncreases, the equlbrum outcome under PTS and PTO converges to Cournot as prces become strategcally ndependent n the less rvalrous market. For the PTS game, ths result relates to the fndng of Kreps and Schenkman (1983) that the Cournot outcome emerges n a two-stage game where frms frst choose capacty and then select prces. In general, the PTS game dffers from the settng consdered by Kreps and Schenkman (1983) n the sense that frms supply functon n the upstream nput (capacty) market have rval s nput prces as arguments; however, the models are somorphc when each frm has monopsony control of an ndependent upstream market. Fourth, we provde condtons under whch our observatons on olgopoly ntermedaton are robust to nventory-holdng behavor n symmetrc markets wth lnear supply and demand functons. Our analyss thus formally extends the fndng of Kreps and Schenkman (1983) to markets wth dfferentated products and nterdependent upstream markets. Fnally, we expand our analyss of sequental prcng outcomes to an extended 4

6 game n whch frms frst choose the tmng of prcng decsons pror to selectng prces. For the case of symmetrc markets wth lnear supply and demand functons, we verfy that the Pareto domnant outcomes n the prcng sub-game represent equlbrum strateges n the extended game. Namely, under crcumstances n whch ndustry profts are largest under PTS (PTO), we show that such a prcng strategy s also the equlbrum outcome of the extended game. Our observatons are related to prevous research by Magg (1996) that endogenzes the mode of competton n the olgopoly model. 2 In Magg s (1996) model, frms make capacty commtments before tradng occurs n a downstream nternatonal market, but can subsequently relax ther pror capacty commtments subject to an ex post adjustment cost parameter. Ths sequental capacty adjustment process produces a contnuum of outcomes that spans the modes of competton between Bertrand and Cournot accordng to the cost of capacty adjustment. Our model departs from ths framework by specfyng olgopsony nteracton n the nput (capacty) market n place of ex post adjustment cost. An advantage of our approach s that t results n testable hypotheses on the mode of competton that are readly estmable from market data. We llustrate the polcy mplcatons of the model for the case of a contract between a prncple (a domestc trade authorty, a frm, or a controllng shareholder) and an agent(a domestc frm, a suppler/consumer, or a manager) that mposes a unt tax or subsdy on the nput procured from the upstream market. We show that the optmal contract to maxmze olgopoly profts nvolves taxng the nput when the upstream and downstream markets are relatvely equal n terms of rvalry, but subsdzng the nput when the degree of relatve rvalry s suffcently large. numercally characterze these polcy outcomes for perturbatons n the relatve degree of market rvalry under lnear supply and demand condtons and show that the optmal value of the polcy varable follows an nverted u-shaped pattern: As the 2 We follow the conventon of Magg (1996) n characterzng the mode of competton accordng to whether prces are strategc substtutes or strategc complements n the olgopoly model. We 5

7 upstream market becomes ncreasngly rvalrous, the optmal polcy swtches from a subsdy to a tax under PTS before revertng back to a subsdy under PTO. The remander of the paper s structured as follows. In the next Secton we present the model and characterze equlbrum prces under PTS and PTO. In Secton 3, we compare the outcomes for frm and ndustry profts and classfy the Pareto domnant Nash equlbrum accordng to the relatve degree of rvalry n the upstream and downstream markets. In Secton 4, we extend these outcomes to consder nventoryholdng behavor and games wth endogenous tmng under symmetrc market condtons wth lnear supply and demand functons. In Secton 5, we derve the mode of competton n the lnear model and characterze condtons for prces to be strategc substtutes that depend only on observed market prces and estmable supply and demand elastctes n a gven ndustry. In Secton 6, we numercally llustrate the mplcaton of our fndngs for the case of vertcal contracts between frms and ther upstream supplers, and n Secton 7, we conclude. The proofs of all Propostons appear n the Appendx. 2 The Model We consder duopoly ntermedares who compete aganst each other n prces. The frms purchase dfferentated stocks ( nputs ) from supplers n an upstream market, and sell fnshed products ( outputs ) derved from the nputs to consumers n a downstream market. Both nput supplers and consumers are prce-takng agents n ther respectve markets. To clarfy the mplcatons of the model for olgopoly prcng outcomes, we consder fxed proportons technology. Specfcally, lettng x denote the quantty of the nput purchased n the upstream market by frm {1, 2}, we scale unts such that y = x denotes the quantty of the output sold by the frm n the downstream market under crcumstances where nventory s not held. Products n each market are dfferentated, and the degree of rvalry between frms potentally dffers at ther ponts of contact n the upstream and downstream markets accordng to the ntensty of cross-prce 6

8 effects between frms n each market. One nterpretaton of the nature of product dfferentaton n the model s that demand and supply functons depend on the spatal locaton of supplers, consumers, and frms. Another nterpretaton s that the frms requre specalzed nputs to produce dfferentated consumer goods. Let p =(p 1,p 2 ) denote the vector of output prces n R 2. Consumer demand for product s gven by D = D (p), (1) where D s n the class of smooth functons on R 2. We assume D D / p < 0 and Dj D / p j 0, where the latter condton confnes attenton to the case of substtute goods. We restrct output prces to P =[0, p] [0, p], a convex and compact subset R 2,wherep>0 s a symmetrc prce such that D (p, p) =0for all p p and D (p, p) > 0 for all p [0, p). Let w =(w 1,w 2 ) denote the vector of nput prces. The supply functon facng frm n the upstream market s S = S (w), (2) where S s n the class of smooth functons on R 2 and S S / w > 0 and Sj S / w j 0 (.e., products n the upstream market are substtutes). As no frm can earn postve rents wth an nput prce greater than ts output prce, we accordngly restrct nput prces to be n P. Throughout the paper, we assume that the drect effect of a prce change outweghs the ndrect effect n each market; that s, D Dj j D j Dj > 0 and Σ S Sj j S j Sj > 0. These condtons ensure that the system of demand and supply equatons s nvertble. In Secton 4.2, we derve suffcent condtons for no strategc nventory-holdng behavor. For now, we streamlne the exposton of equlbrum outcomes under PTS and PTO by suppressng nventory-holdng behavor and the destructon or removal of goods. Wthout the possblty of holdng nventory, the demand and supply functons 7

9 facng each frm are lnked by the materal balance equatons, D(p) =S(w). (3) There are two mportant subgames we defne below: the PTS game and the PTO game. In the PTS game, the two frms smultaneously and ndependently choose ther nput prces. Gven that the frms are not able to hold nventory, the unque set of output prces assocated wth the nput prce par w =(w 1,w 2 ) are determned by condton (3), whch we denote by p(w). Frm s proft ntheptsgames π,s (w) = p (w) w S (w) F, {1, 2}, where F denotes fxedcosts,aportonofwhchmaybesunk. 3 To guarantee exstence and unqueness of the PTS equlbrum, we mpose the followng regularty condtons on profts. Axom 1 π,s (w) < 0, andπ,s (w)+π,s j (w) < 0 for all w P and {1, 2}. In the PTO game,the two frms smultaneously and ndependently choose ther output prces. We assume that frms must fll all orders and cannot hold forward contracts as nventory. As a consequence, the unque set of nput prces are determned by condton (3) under PTO as w(p). Frm s proft n the PTO game s π,o (w) = p w (p) D (p) F, {1, 2}, We mpose the followng regularty condton on profts under PTO. Axom 2 π,o (p) < 0, andπ,o (p)+π,o j (p) < 0 for all p P and {1, 2}. At tmes, we wsh to compare equlbrum prces under PTS and PTO to those that emerge under the two-sded Cournot game n whch quantty choces determne both upstream and downstream market prces at once. To do so, we defne 3 It s straghtforward to show that contnuty and dfferentablty of the the supply and demand functons mply that π,s s also n the class of smooth functons on R 2. 8

10 nverse demand and nverse supply from equatons (1) and (2) as P (y) and W (y), respectvely. The proft of frm under Cournot competton s gven by π,c (y) = P (y) W (y) y F, {1, 2}, (4) where y = (y 1,y 2 ) denotes the vector of retal (and wholesale) quanttes under quantty competton. To facltate comparson of the PTS and PTO equlbrum wth the Cournot outcome, we make the followng assumptons about the Cournot proft functons. Axom 3 π,c and {1, 2}. (y) < 0, andπ,c (y)+π,c j (y) < 0 for all y [0,D(0, p)] [0,D(0, p)] Ths assumpton guarantees exstence and unqueness of equlbrum n the Cournot game. In addton, we assume that the collectve proft s strctly concave n symmetrc quanttes. Axom 4 π,c (y)+2π,c j (y)+π,c jj (y) < 0 for all symmetrc y (y 1 = y 2 ). As a convenent benchmark for ths analyss, we frst examne the case n whch frms smultaneously select prces n the upstream and downstream markets. Throughout the paper, we refer to ths object as the Bertrand outcome. 2.1 Bertrand Outcomes Consder the case n whch frms select prces smultaneously n the upstream and downstream markets. Frm seeks to maxmze profts π,b (p, w) =p D (p) w S (w) F, {1, 2} (5) subject to the nventory constrant (3). Evaluatng the frst-order condtons wth respect to w and p, we descrbe the Bertrand outcome n the sense of Stahl (1988) as the smultaneous soluton to p w = S (w) S (w) D (p) {1, 2}, (6) (p), D 9

11 where subscrpts refer to partal dervatves. Smultaneously solvng equatons (6) subject to the nventory constrant (3) yelds the Bertrand prces, whch we defne n the symmetrc case as (w B,p B ). For future reference, t s helpful to express the symmetrc prce-cost margn of Bertrand merchants as p B w B = pb ε d o + wb ε s. (7) o 2.2 Relatve Rvalry The concept of relatve rvalry s mportant for the analyss to follow. We defne the relatve rvalry of markets as a measure of the strength of the olgopoly nteracton n the upstream market relatve to the downstream market, where a more rvalrous market s one n whch a change n the nput (output) prce selected by a frm leads to a greater change n supply (demand) for the rval. Focusng on the symmetrc market equlbrum, we descrbe the relatve rvalry of markets n terms of supply and demand elastctes. Specfcally, let ² s = εs c ε s o rato of cross-prce elastcty of supply (ε s c = Sj w j S supply (ε s o = S w S > 0), and let ² d = εd c ε d o elastctes n the downstream market, where ε d c = D j < 1 denote the absolute value of the > 0) to own-prce elastcty of < 1 denote the correspondng rato of demand p j D p > 0 and ε d o = D > 0. D Defnton 1 The relatve rvalry of the upstream market s gven by Θ = ² s ² d. We measure the relatve rvalry of the upstream market accordng to equlbrum values for the prces n the Bertrand outcome descrbed above. We refer to the upstream market as beng more rvalrous than the downstream market when Θ > 0, the downstream market as beng more rvalrous than the upstream market when Θ < 0, and the markets as beng equally rvalrous when Θ =0. We are now ready to examne the equlbrum outcomes of the ntermedaton model under two forms of sequental prcng behavor: () PurchasngtoStock(PTS); and () PurchasngtoOrder(PTO). 10

12 2.3 Purchasng to Stock (PTS) In the purchasng to stock (PTS) game, frms frst select nput prces and acqure stocks n the upstream market, and then subsequently select output prces for fnshed goods n the downstream market. Let w =(w 1,w 2 ) denote the vector of nput prces selected by the frms and let p (w) and p j (w) denote the assocated output prces mplctly defned by equatons (3). Under PTS, frm selects w to maxmze profts, π,s (w,w j )= p (w) w S (w) F, {1, 2}, (8) where F denotes fxed costs, a porton of whch may be sunk. The frst-order necessary condton for a proft maxmum s π,s (p (w) w )S (w)+(p (w) 1)S (w) =0, {1, 2}. (9) The effectofannputprcechangebyfrm on output prces can be derved by totally dfferentatng equatons (3). Holdng dw j =0,thsyelds " # D Dj dp S = dw dp. j D j D j j S j It follows that p (w) p w = Dj j S D j Sj < 0 p j pj (w) w = D Sj Dj S. (10) Equaton (2.3) represents the own-prce effect of an ncrease n the nput prce on the output prce of frm. Ths term s negatve. When frm rases hs nput prce, the frm procures a greater quantty of the nput, and ths drves down the frm s output prce and narrows hs prce-cost margn. Equaton (10) s the cross-prce effect of a nput prce ncrease by frm on the output prce of the rval frm j. The sgn of ths term s mportant for the results to follow. Frm j responds to a hgher nput prce by frm (dw > 0) byncreasng 11

13 hs output prce (dp j > 0) whenever D Sj >Dj S, and otherwse holds constant or decreases hs output prce. Under PTS, a change n a frm s nput prce has two offsettng effects on the output prce of the rval. The frst effect s the supply effect, D Sj dw 0. Selectng a hgher nput prce bds stocks away from the rval n the upstream nput market, whch leads to an nward shft of the rval s supply functon. The supply effect reduces the rval s procurement of the nput, thereby reducng the quantty sold n the downstream market by frm j and rasng the rval s output prce. The second effect s the demand effect, D j S dw 0. A rse n the nput prce of frm rases nput procurement for frm n the upstream market, leadng to a commensurate ncrease n producton and sales for frm n the downstream market and a decrease n p. The demand effect results n an nward shft of the rval s demand functon, placng downward pressure on p j. The relatve magntude of the supply and demand effects depends on the relatve rvalry of the upstream market. When products n the downstream market are commodtzed, for example, the demand effect domnates the supply effect, as total output would rse followng a unlateral ncrease n the nput prce of one frm, floodng the downstream market wth fnshed goods. In the symmetrc market equlbrum (D = D j,s = S j,p 1 = p 2 = p, w 1 = w 2 = w), the cross-prce effect n equaton(10) can be expressed n terms of relatve rvalry as p j (w) s = Θ, where s = denotes equal n sgn. Under crcumstances n whch the upstream market s relatvely rvalrous (Θ > 0), an ncrease n the nput prce by frm ncreases the output prce of frm j. When the downstream market s relatvely rvalrous (Θ < 0), an ncrease n the nput prce by frm decreases the output prce of frm j, and for an equal degree of rvalry, Θ =0, the output prce of frm j s ndependent of the nput prce selecton of frm. Relatve rvalry has essental strategc mplcatons for the olgopoly model. To 12

14 see ths, note that S (w,w j )=D (p (w),p j (w)) under the nventory constrant (3). Dfferentatng ths expresson wth respect to w and droppng arguments for notatonal convenence, we have S = D p + D jp j. When frm ncreases hs nput prce n the upstream market, ths leads to a drect ncrease n the output sold by frm, D p > 0. But the nput prce ncrease also facltates a strategc response by the rval frm, Dj pj 0, the sgn of whch depends on p j (and thus on Θ). Specfcally, frm perceves a smaller supply response to a rse n w when Θ < 0, whch dampens the value of an nput prce ncrease. Thus, engagng n PTS behavor serves as a commtment devce to refran from ncreasng nput prces when Θ < 0. It s of course n the nterest of both frms to mantan lower nput prces n the upstream market, as ths supports correspondngly hgher output prces n the downstream market and hgher prce-cost margns, makng sequental prcng behavor under PTS a facltatng practce whenever Θ < 0. The equlbrum under PTS s determned by the smultaneous soluton of equatons (9). Let w s =(w1 s,ws 2 ) denote the equlbrum nput prce vector that solves these equatons and let p s =(p s 1,ps 2 ) denote the assocated vector of equlbrum output prces mpled by equatons (3). The equlbrum prce-cost margn for symmetrc frms can be wrtten p s w s = k s ps ε d o + ws ε s, (11) o where k s =1+ Dj Dj j Θ. Notce that the second term on the rght-hand sde of equaton (11) s dentcal to the second term on the rght hand-sde of equaton (7), but that the frst term dffers from the Bertrand margn. Playng PTS ntroduces a weght on the demand-sde porton of the equlbrum prce-cost margn that jontly accounts for the relatve rvalry of the upstream market. By nspecton, k s > 1 f and only f Θ < 0. That s, equlbrum prce-cost margns are hgher for frms n the PTS 13

15 game than n the Bertrand outcome when the upstream market s less rvalrous than the downstream market (Θ < 0). The ntuton for ths fndng s qute clear: When Θ < 0, a decrease n the nput prce by a frm facltates an ncrease n the rval s output prce, thereby softenng prce competton n the downstream market. In Secton 3, we provde a more complete comparson of PTS and the Bertrand outcomes. Before turnng to ths analyss, we derve the market equlbrum for the remanng case of PTO. 2.4 Purchasng to Order (PTO) Suppose the frms sell forward contracts for delvery of fnshed goods to consumers pror to procurng nputs n the upstream market. Forward contracts are wdely used n practce, ncludng mported goods sold to retal dstrbutors and a sgnfcant porton of wholesale trade (Stahl 1988). Let p =(p 1,p 2 ) denote the vector of output prces selected by the frms and let w (p) and w j (p) denote the assocated nput prces defned by equatons (3). In the PTO game, frm selects hs output prce to maxmze profts of π,o (p,p j )= p w (p) D (p) F, {1, 2}. The frst-order necessary condton for a proft maxmum s π,o (p w (p))d (p)+(1 w (p))d (p) =0, {1, 2}. (12) We evaluate the PTO equlbrum by proceedng as above. Makng use of the mplct functon theorem on equatons (3) gves the nput prce responses w (p) w p = D Sj j Dj S j Σ < 0 (13) w j wj (p) p = Dj S D Sj Σ. (14) Equaton (13) measures the own-prce effect of an ncrease n the output prce, and equaton (14) measures the cross-prce effect of a output prce change by frm on the nput prce selected by frm j. As n the case of PTS, the sgn of the cross-prce 14

16 effect depends on the relatve magntude of the supply effect, D Sj dp 0, andthe demand effect, D j S dp 0. Notce that the cross-prce effect n equaton (14) always takes the opposte sgn of the cross-prce effect n equaton (10); that s, p j (w) = s w j (p). Under condtons nwhchanncreasenafrm s nput prce ncreases the output prce of hs rval n the PTS game, an ncrease n the output prce decreases the nput prce of hs rval n the PTO game. In the symmetrc market equlbrum (D = D j,s = S j,p 1 = p 2 = p, w 1 = w 2 = w), w j (p) s = Θ. The equlbrum under PTO s determned by the smultaneous soluton of equatons (12). Defne the equlbrum output prce vector that solves these equatons as p o =(p o 1,po 2 ) and the assocated vector of equlbrum nput prces as wo =(w o 1,wo 2 ). The equlbrum prce-cost margn for symmetrc frms can be wrtten p o w o = po ε d o + k o wo ε s. o where k o =1 Sj Sj j Σ Θ. Inspecton of ths term reveals that ko > 1 f and only f Θ > 0. Equlbrum prce-cost margns are hgher n the PTO game than n the Bertrand outcome when the upstream market s relatvely rvalrous (Θ > 0). The reason s that a rse n the output prce by a frm decreases the nput prce set by hs rval when Θ > 0, whch facltates hgher prce-cost margns. 3 Equlbrum Outcomes In ths Secton we compare frm profts n the PTS and PTO games to the Bertrand outcome and dentfy Pareto domnant strateges n the symmetrc market equlbrum. To do so, we consder the symmetrc Bertrand prces (w B,p B )andexamne multlateral defectons from the Bertrand outcome that ncrease the profts of frms. 4 4 Wth slght abuse of notaton, we wrte demand, supply, and proft as functons of the scalar values of nput and output prces n the symmetrc equlbrum. 15

17 3.1 PTS Versus PTO Consder frst the PTS game. Evaluatng the nput prce condton (9) at the symmetrc Bertrand values (w B,p B )gves π,s (w B,p B ) Ã D j j S D j Sj 1! µ S S D D. Makng use of the market-clearng condton (3) and factorng terms yelds π,s (w B,p B ) S Dj S Θ = s Θ, where all terms are evaluated at (w B,p B ). In the PTS game, frms select hgher nput prces than the symmetrc Bertrand level when Θ > 0 and select lower nput prces when Θ < 0. Proceedng smlarly n the case of PTO, evaluatng the output prce condton (12) at the symmetrc Bertrand poston (w B,p B )gves π,o (w B,p B ) D D S j Σ Θ = s Θ. In the PTO game, frms select hgher output prces than the symmetrc Bertrand level when Θ > 0 and set lower output prces when Θ < 0. Proposton 1 For the ntermedated olgopoly model: () If Θ =0the equlbrum market outcome under ether PTO or PTS s Bertrand () If Θ > 0 (< 0) the Pareto domnant equlbrum s PTO (PTS). In Stahl s (1988) model, ntermedares compete n homogeneous product markets (Θ =0), and an outcome wth Bertrand merchants emerges under both PTO and PTS. Proposton 1 extends ths outcome to encompass any market that satsfes Θ =0. Thus, the Bertrand outcome represents an envelope of olgopoly equlbra characterzed by equal degrees of rvalry n the upstream market and downstream market. The essental underpnnng of the Bertrand outcome s that frms nput (output) prce choces are ndependent of the resultng output (nput) prces selected 16

18 by the rval. Ths outcome depends on the relatve degree (rather than the absolute degree) of market rvalry. When Θ 6= 0, frm profts are larger n the symmetrc equlbrum n cases where the frms enjoy wder prce-cost margns. Under crcumstances where the downstream market s relatvely rvalrous (Θ < 0), the PTS game facltates ths outcome, because the rval responds to a lower nput prce by selectng a hgher output prce n equaton (10). Equlbrum prce-cost margns and profts are accordngly hgher under PTS than under Bertrand. Under crcumstances where the upstream market s relatvely rvalrous (Θ > 0), profts are hgher n the PTO game, as the rval frm responds to a hgher output prce n ths case by selectng a lower nput prce n equaton (14). In ether case, the Pareto domnant equlbrum nvolves selectng prces n the relatvely less rvalrous market as a facltatng practce to soften prce competton n the remanng market. The anatomy of the ntermedated olgopoly model can be llustrated by descrbng crcumstances n whch playng the PTS (PTO) game produces Cournot outcomes. In the followng secton we characterze the Cournot equlbrum and derve formal condtons under whch the PTS and PTO games produce Cournot outcomes. 3.2 Cournot Outcomes The Cournot outcome s characterzed by maxmzng expresson (4) wth respect to y.thefrst-order necessary condton for a maxmum s µ π,c P (y) =P W + W y =0, y y {1, 2}. (15) Smultaneously solvng equatons (15) n the symmetrc equlbrum gves the equlbrum quantty, y C = y C 1 = yc 2, whch can be used to recover the symmetrc Cournot equlbrum prces (w C,p C ). Proposton 2 The Cournot outcome emerges n the PTS game when ² s =0and n the PTO game when ² d =0. 17

19 Proposton 2 clarfes the essental fndng of Kreps and Schenkman (1983) as an outcome that depends on the strategc ndependence of prces n the upstream nput (capacty) market n the PTS game. The Cournot outcome also emerges n the PTO game when the output prces of frms are strategcally ndependent. In the followng secton, we llustrate the robustness of these outcomes by consderng crcumstances n whch frms can hold nventores. To see the ntuton for Proposton 2, consder the prce-cost margns under Cournot n the symmetrc market equlbrum. Makng use of equatons (2.3) and (13), the symmetrc equlbrum prce-cost margn can be wrtten as p C w C = ε d o p C 1 ² 2 d + w C ε s o (1 ² 2 s). (16) The essental dfference between ths outcome and the outcome under PTS and PTO s that quantty-settng frms jontly consder the strategc effect of a quantty ncrease on rasng ther procurement cost n the upstream market and on reducng ther sales revenue n the downstream market. In contrast, frms n the PTS equlbrum, who set prces sequentally n the upstream market pror to selectng prces n the downstream market, can consder only the mplcaton of ther frst-stage nput prce choce on ther subsequent level of sales. When prces n the upstream market are strategcally ndependent, ² s =0, however; the only nteracton that remans wth the rval n ths case occurs through the nterdependence of prces n the downstream olgopoly market, so that the nablty of the frm to account for the strategc nterdependence of nput prces n the upstream market no longer has any consequence. When ² s =0, the second term on the rght-hand sde of equaton (16) reduces to wc ε s o and the weght on the demand-sde porton of the equlbrum prce-cost margn n the PTS game reduces to k s = 1, resultng n the Cournot outcome. For a smlar 1 ² 2 d reason, the PTO game results n the Cournot olgopsony outcome when prces are strategcally ndependent n the downstream market, ² d =0. 18

20 4 Model Extensons In ths Secton we consder two extensons of the model. We frst extend the game structure to a settng n whch frms endogenously select the tmng of ther prcng decsons from the choce set {PTS,PTO} pror to choosng prces. We then extend the model to consder nventory-holdng behavor by allowng the supply procured by each frm to exceed demand n equatons (3). For each extenson, we examne the symmetrc market equlbrum under condtons of lnear supply and demand. Specfcally, we consder the lnear specalzaton, D (p) =max{a bp + cp j, 0}, (17) S (w) =max{βw γw j, 0}, (18) where =1, 2, 6= j, andwherea, b, β > 0 and c, γ 0 are postve constants. We restrct the each frm s prces to the nterval [0,P], wherep 2a/(b c). Demand (supply) condtons reduce to local monopoly (monopsony) markets when c =0( γ = 0) and products n the downstream (upstream) market become ncreasngly commodtzed as c b (γ β). 4.1 Endogenous Tmng Our observatons above on the proft motveoffrms to select prces sequentally n ntermedated olgopoly markets rases the queston of whether such behavor s also an equlbrum strategy n settngs where coordnaton on tmng s not possble pror to selectng prces. Here, we extend the game structure to a settng n whch frms frst choose whether to engage n PTS or PTO pror to selectng prces n each market. Consder the followng three stage game. In stage 1, frms select the tmng of ther prcng decsons from the choce set {PTS,PTO}. In stage 2, frms select prces ntherelevantsubgameaccordngtothetmng of prcng decsons determned n stage 1. The frms face symmetrc market condtons n the second stage accordng to the demand and supply functons n equatons (17) and (18). 19

21 PTS 1 PTO 1 S j S j j S j S j j PTS 2 PTO 2 Dj k s Sj k o Dj, Sj D j j S j j D j j S j j Dj k s, Sj D j j S j j Dj D j j k s, Sj k o Dj S j j D j j S j S j j k o Dj D j j Dj k s D j j, Sj k o Dj S j j D j j Table 1: Prce-Cost Margns for Frms n Response to Tmng Choces Our comparson of stage 2 outcomes nvolves examnng equlbrum prces under several possble alternatves: () frm plays PTS whle frm j plays PTO; () both frms play PTS; and () bothfrms play PTO. We have completely characterzed the equlbrum outcomes of the latter two subgames and we turn here to cases n whch frms select asychronous prces (PTS, PTO). Our analyss of the extended game s more transparent when makng use of the followng, general result. Proposton 3 The selecton of a gven tmng strategy {PTS,PTO} by frm provdes the rval frm j wth an equlbrum prce-cost margn that corresponds wth the strategy selected by frm. Table 1 descrbes the dfferent possbltes for the equlbrum prce cost margn for frm j dependng on the frm s strategy. The entres n Table 1 have mportant strategc mplcatons n the lnear case. Recall that k o = 1 S j Sj j Σ Θ and ks = 1+ Dj Dj j Θ. These terms are constant under lnear demand and supply functons n equatons (17) and (18). If frm defects from the PTO game to play PTS, the defecton has no mplcatons for frm s prce-cost margn, whch remans at the PTO level; however, rval frm j responds to the ntal nput prce selected by frm by adjustng hs prce-cost margn. In equlbrum, frm j selects the wder prce-cost margn of the PTS game when Θ < 0. The wder equlbrum prce-cost margn that emerges for the PTO player n the (PTS, PTO) subgame subsequently ncreases the volume of sales for the PTS player at the PTO equlbrum prce-cost margn, makng the choce of PTS a facltatng practce. The opposte s true defectng from PTS to the PTO subgame s a facltatng practce when Θ > 0. 20

22 Now consder the outcome for profts under the system of equatons (17) and (18). Evaluatng the symmetrc pay-off matrx n the frst stage of the game yelds: Proposton 4 Under lnear supply and demand condtons: () (Bertrand,Bertrand) s an equlbrum f and only f Θ =0; () (PTS,PTS) s the unque equlbrum of the game when Θ < 0; and () (PTO,PTO) s the unque equlbrum of the game when Θ > 0. The ntuton for ths result s that the choce of tmng allows frms to employ prces n the relatvely less rvalrous market to serve as a facltatng practce to soften prce competton n the more rvalrous market. When Θ < 0, afrm that chooses to play PTS causes hs rval to adjust hs prce-cost margn upward n equlbrum, because k s > 1. Moreover, as demonstrated by the entres n Table 1, ths adjustment occurs rrespectve of the tmng of prce-settng behavor chosen by the rval frm. The larger prce-cost margn selected by the rval frm benefts the frm playng PTS, and as a result, both frms select PTS n the frst stage of the game. A comparable outcome occurs when Θ > 0, allowng both frms to coordnate on PTO. Before examnng outcomes wth nventory-holdng behavor, t s worthwhle to consder the case n whch nput prces are strategcally ndependent n the upstream market, ² s =0. When ² s =0, (PTS, PTS) s the unque equlbrum outcome n the lnear case and the margn adjustment by each frm s k s =1+(D j )2 /. The unque equlbrum outcome of the game s Cournot. If the frms nstead engaged n PTO, k o reduces n ths case to k o =1, and the PTO equlbrum would produce the Bertrand outcome. As n Kreps and Schenkman (1983), the frms would wsh to defect from selectng output prces that result n Bertrand profts by frst commttng to the purchase of stocks ( capacty ), whch results n the Cournot outcome. 4.2 Inventory-Holdng Now consder a settng n whch nventory-holdng behavor s possble. For expostonal clarty, we lmt our attenton to market condtons that satsfy Θ < 0, whch 21

23 provdes strategc ncentves for PTS to emerge as the unque equlbrum outcome n the case where holdng nventory s not possble. Analogous condtons apply for PTO n settngs where forward contracts do not represent bndng commtments to delver fnshed goods to consumers n the downstream market. 5 Suppose stocks procured from the upstream nput market n the PTS game can be freely dsposed. The possblty of free dsposal of nputs relaxes the nventory constrant n equaton (3), whch becomes D (p) S (w) for =1, 2. To study the prcng subgame wth fxed supples and potentally slack nventory constrants, we follow Kreps and Schenkman (1983) and assume that resdual demand s effcently ratoned. We then characterze the proftablty of defecton strateges from the no-nventory equlbrum. Ths analyss results n the followng: Proposton 5 The nventory constrant n equaton (3) s always bndng when 6 P j (y)s j (w) 1+εs o. (19) The mplcaton of Proposton 5 s that the PTS equlbrum descrbed above under the assumpton of no nventory-holdng concdes wth the equlbrum outcome n a general settng wth nventory-holdng behavor provded that condton (19) holds. The nterpretaton of ths condton s as follows. After nput procurement has taken place, the procurement cost of the frm s sunk, leavng the frm wth the potental to sell less than the procured quantty wth no addtonal cost under the assumpton of free dsposal. Proposton 5 descrbes market condtons under whch a frm selectng quanttes n a settng wth no producton cost would wsh to select an output level that s (at least weakly) greater than the output level mpled by PTS, thereby resultng n a bndng nventory constrant. 5 The possblty of holdng negatve nventory under PTO mplctly assumes that forward contracts wth consumers can be renegotated. In the event that forward contracts are non-renegotable, free dsposal of forward contracts would not be possble, and an addtonal deterrent would exst to holdng nventory under PTO. 6 In the lnear case, all terms n condton (19) are constant and the nequalty can be wrtten, γc(β γ) (2β γ), whch holds for suffcently small c, γ. 22

24 5 Mode of Competton Characterzng and measurng the mode of competton n the olgopoly model s essental for dervng polcy prescrptons n settngs wth strategc pre-commtment. It s also mportant for dervng nferences on the type of market condtons that warrant anttrust scrutny, for nstance market features that favor the use of slottng allowances as a practce to soften prce competton. 7 Our goal n ths secton s to develop testable hypotheses on the mode of competton n ndustral settngs. To characterze the mode of competton n ntermedated olgopoly settngs, consder the second partals of proft under PTS. Droppng arguments for notatonal convenence, the effect of an nput prce change by frm j on the margnal proft of frm under PTS s π,s j (p w )S j +(p 1)S j + p js + p js, {1, 2}. (20) Prces may be strategc complements (π,s j the sense of Bulow, Geanakoplos, and Klemperer (1985). > 0) or strategc substtutes (π,s j < 0) n To provde an ntutve characterzaton of these outcomes, consder the frst-order effects n expresson (20). Classfyng the mode of competton n the lnear model s mportant for dervng testable hypotheses for emprcal work that reles on lnear estmaton technques. On substtuton of equatons (2.3) and (10), the mode of competton n the PTS game can be expressed as à D j π,s j = j S! à D j Sj D j 1 Sj j + S j! D j Sj j S. In the symmetrc market equlbrum, ths condton becomes π s j s = (p w) ε d p o² s 1 ² 2 d + Θ, (21) where the frst term on the rght hand sde of equaton (21) s postve and Θ < 0 s negatve n the PTS game by Proposton 1. It can be seen mmedately upon nspecton of terms n equaton (21) that prces are strategc complements when markets 7 The Federal Trade Commsson (FTC), whch regulates the grocery ndustry, refused to provde gudelnes for slottng allowances, ctng the need for further nvestgaton on the effcency effects of the practce (FTC 2001). 23

25 are equally rvalrous, Θ =0(Bertrand), whereas prces are strategc substtutes n the case where prces are strategcally ndependent n the upstream market, ² s =0 (Cournot). In general, Θ < 0 s a necessary but not a suffcent condton for prces to be strategc substtutes n the PTS game. In the PTO game, the effect of an output prce change by frm j on the margnal proft offrm s π,o j =(p w t )D j +(1 w )D j w jd w jd, {1, 2}. (22) Confnng attenton to frst-order effects n expresson (22) and makng substtutons from (13) and (14) n the symmetrc market equlbrum yelds π o j s (p w) = w εs o² d 1 ² 2 s Θ. (23) As n the PTS game, output prces may be strategc complements (π,o j > 0) or strategc substtutes (π,o j < 0) dependng on the magntude of Θ > 0. WhenΘ > 0, the frst term on the rght-hand sde of the expresson s postve and the second term s negatve. The relatve rvalry of the upstream market nfluences the mode of competton under PTO n the opposte manner as under PTS: The model reduces to Bertrand when Θ =0and to Cournot olgopsony when ² d =0. A relatvely rvalrous upstream market (Θ > 0) s necessary but not suffcent for output prces to be strategc substtutes n the PTO game. Proposton 6 Under condtons of lnear supply and demand, prces are strategc substtutes n the ntermedated olgopoly model when: () ² s h 1+ (p w) p ² d ε d o(1 ² 2 d) ;or () ² d h ² s. 1+ (p w) w εs o(1 ² 2 s) Under crcumstances n whch Θ < 0 (² s <² d ), the upstream market s relatvely less rvalrous than the downstream market. Frms are able to soften downstream prce competton by selectng nput prces pror to output prces n the PTS game. Part 24

26 () of Proposton 6 s the relevant crtera for the mode of competton, and prces are strategc substtutes when ² s s suffcently small relatve to ² d. The opposte s true when Θ > 0 (² d <² s ). Part () of Proposton 6 s the relevant crtera for the mode of competton, and prces are strategc substtutes when ² d s suffcently small relatve to ² s. Inethercase,themodeofcompettonsdetermnedbytherelatve rvalry of the upstream market. 6 Polcy Implcatons The mode of competton n the olgopoly model has mportant mplcatons n a number of polcy settngs. In ths secton, we hghlght polcy mplcatons of the model for the case of contract desgn between a prncpal (a domestc trade authorty, a frm, or a controllng shareholder) and an agent (a domestc frm, a suppler/consumer, or a manager). For expostonal clarty, we consder strategc polces by the prncpal that tax or subsdze nput procurement by the agent n the upstream market under condtons of lnear supply and demand. Ths allows us to numercally compute the optmal tax level for varaton n the relatve rvalry of the upstream market by perturbng c and γ n the system of equatons (17) and (18). We consder the followng three-stage game. In stage 1, the prncpal of frm mposes a unt tax t on nputs procured n the upstream market by agent. In stage 2, agents take prncpals polcy decsons parametrcally and select the tmng of prcng decsons, and n stage 3, agents select prces n the relevant subgame. Lettng π (p, w) denote the proft ofprncpal, we denote agent s proft Π (p, w) =π (p, w) t S (w)+ω, {1, 2}, (24) where Ω s a lump-sum transfer that subsumes fxed costs. proft gves the frst-order condton Maxmzng the agent s Π,s p (w) w t S (w)+(p (w) 1)S (w) =0, {1, 2}, (25) for the PTS game and Π,o (p w (p) t )D (p)+(1 w (p))d (p) =0, {1, 2}, (26) 25

27 for the PTO game. Proposton 7 The choce of PTO or PTS as an equlbrum strategy of agent s nvarant to the prncpal s choce of polcy varable t. Thus, the market equlbrum s Bertrand f Θ =0,PTOfΘ > 0, andptsfθ < 0. The magntudes of the polcy varables, t, =1, 2 nfluence nether the sgn nor the value of Θ. An mplcaton of Proposton 7 s that our observatons on strategc pre-commtment devces are robust to dfferent polcy structures that alter the margnal returns facng olgopoly agents. Confnng attenton to frst-order effects, the second partal of Π wth respect to w and w j s Π,s j = p j(w)s(w)+(p (w) 1)Sj(w) =π,s j, {1, 2}, for the PTS game and the second partal of Π wth respect to p and p j s Π,o j =(1 w (p))dj(p) wj(p)d (p) =π,o j, {1, 2}. In the symmetrc market equlbrum, t s straghtforward to show that the optmalvaluesofthepolcyvarablessatsfy t s = π,s s j = π,o j, {1, 2}.8 (27) The optmal value of the polcy varable for each agent depends on the mode of competton between olgopoly frms. Taxes (t > 0) are optmal when prces are strategc complements, whereas subsdes (t < 0) are optmal when prces are strategc substtutes. The ntermedated olgopoly model allows the optmal value of the strategc precommtment polcy to be explctly computed accordng to supply and demand condtons facng frms. We llustrate ths outcome for the case of lnear supply and demand functons by calculatng the (unque) Nash equlbrum n polcy varables 8 Full dervaton of the optmal polcy varbles s provded n the Web Appendx. 26

28 for varaton n the degree of prcng nterdependence n each market, c and γ. For our numercal analyss, we select parameters a = b = β =1anddentfytheoptmal tax polcy n the symmetrc pre-commtment equlbrum, t = t = t j for varatons n c (0, 1] and γ (0, 1]. Fgure 1 depcts the contour lnes of the symmetrc trade polcy equlbrum n the space (c, γ). Note that PTS s the unque equlbrum of the game when γ >c, whleptostheunqueequlbrumofthegamewhenγ <c. The optmal trade polcy s a tax (t > 0) n the shaded regon of the fgure and a subsdy (t < 0) n the non-shaded area of the fgure. The contour lne where the lassez-fare outcome s optmal (t =0) s represented by the contour lne that separates these two areas. For a fxed γ (respectvely c) the optmal polcy reveals an nverted u-shaped pattern as c (respectvely γ) ncreases from 0 to 1. 7 Concludng Remarks The olgopoly ntermedaton model results n testable hypotheses on the mode of competton that can be used to derve polcy mplcatons n ndustral settngs. Our analyss reveals the prevalng mode of competton n a partcular ndustry to be an emprcal queston that depends on estmable parameters of supply and demand functons n the upstream and downstream markets where frms nteract. We have demonstrated that olgopoly frms have an ncentve to select nput prces pror to choosng output prces n the PTS game when the downstream market s relatvely rvalrous, but to select output prces pror to choosng nput prces n the PTO game when the upstream market s relatvely rvalrous. Under reasonably mld condtons, we have shown these outcomes to be robust to the possblty of strategc nventory-holdng behavor of frms. For both the PTS and PTO games, prces are strategc complements when the upstream market and the downstream market are relatvely equal n terms of rvalry, whereas prces are strategc substtutes under crcumstances where the upstream (downstream) market s suffcently rvalrous relatve 27

29 g c Fgure 1: 28

30 to the remanng market. In the case where the markets are equally rvalrous, the equlbrum outcome of the olgopoly model s Bertrand and n cases where prces are strategcally ndependent n ether the upstream market or the downstream market, the equlbrum outcome of the olgopoly model s Cournot. Our model provdes emprcal drecton for examnng the mode of competton n ndustral settngs that can be used to talor pre-commtment polces to the partcular condtons facng frms. Ths feature of the model can provde a useful tool for polcy formulaton as well as for the enforcement of ant-trust regulatons n a gven ndustry. An nterestng avenue for future research s to consder an actve role for frms n contrbutng to market rvalry through product dfferentaton. To the extent that olgopoly frms engage n actvtes to dfferentate themselves from rvals, whether by creatng dfferentated products from common nputs or by dscoverng new technques for producng exstng products from specalzed nputs, dfferentaton can provde a tool to soften prce competton wth rvals. But, perhaps surprsngly, our analyss clarfes that relaxng market rvalry s not necessarly a globally desrable endeavor. The reason s that t s the relatve degree of market rvalry not the absolute degree that s the source of strategc advantage. Indeed, a frm that seeks to specalze nput requrements would have a greater ablty to soften prce competton wth rvals when downstream products are commodtzed than when downstream products are hghly dfferentated. Our model suggests the potental for strategc standardzaton to occur under crcumstances n whch nputs (outputs) n the upstream (downstream) market are hghly specalzed. 29