Equilibrium exclusive dealing in oligopoly. Steffen Ziss. Wilfrid Laurier University April, 2010

Transcription

1 Equlbrum exclusve dealng n olgopoly by Steffen Zss Wlfrd Laurer Unversty Aprl, 00 Abstract. Ths paper consders a settng n whch upstream olgopolsts delegate the retalng of ther dfferentated products to a set of undfferentated retalng agents. The downstream maret structure s assumed to consst of a set of ndependent agents that exclusvely sell the product of a sngle manufacturer and a common agent that sells the product of many manufacturers. A three-stage game s consdered. n the frst stage the manufacturers decde whether or not to maret ther products usng an ndependent agent or the common agent. n the second stage the manufacturers determne the terms of the two-part tarff contract offered to ther respectve agents. n the fnal stage the agents engage n ether output or prce competton wth other retalng agents. f the agents engage n output competton then the model shows that t wll ether be the case that all manufacturers use ndependent agents or that they all use the common agent. Whch of these two equlbra emerge depend on the degree of substtutablty between products and on the number of manufacturers. f the agents engage n prce competton then a thrd type of equlbrum also emerges n whch some frms adopt the common agent and others adopt ndependent agents. JEL lassfcaton Numbers. L0, L0, L40 Keywords. olgopoly, exclusve dealng Address. School of Busness & Economcs, Wlfrd Laurer Unversty, Waterloo, Ontaro, anada, NL 35 tel: (59) (ext. 776) fax: (59) emal: szss@wlu.ca

2 . ntroducton n a duopoly settng Ln (990) has argued that one of the reasons that manufacturers prefer exclusve dealng to non-exclusve dealng s that t serves to dampen prce competton. The purpose of ths paper s to show that the dampenng of competton argument does not hold n maret structure less concentrated than duopoly. Specfcally ths paper shows that frms that adopt exclusve dealng charge hgher wholesale and retal prces than frms that adopt non-exclusve dealng. Secondly, when a frm moves from non-exclusve dealng to exclusve dealng then frm rases both retal and wholesale prces but the mpact of the wholesale and retal prces of frm s rval s ambguous. Thrdly, non-exclusve dealng arses n equlbrum f marets are suffcently unconcentrated and/or goods are suffcently close substtutes, regardless of whether retalers engage n prce or output competton. Fnally f retalers engage n prce competton then the number of frms that adopt non-exclusve dealng rses as marets become less concentrated and/or goods become closer substtutes. A motvaton for ths paper s to explan recent ncreases n common agency n both the retal car and gasolne marets. n partcular the retal car ndustry has seen the emergence of common agency va the formaton of dealershp groups such as AutoNaton n the US and Pendragon Vardy n the UK. For example AutoNaton whch began n 996 had acqured 30 new vehcle franchses and 3 stores by the end of 008 and these stores sold 37 dfferent brands of new vehcles from manufacturers such as Toyota, Other motvatons for adoptng exclusve dealng are to () foreclosure rvals or rase rvals costs or () to prevent varous conflcts between manufacturers and retaler such free-rdng by manufacturers on the servces (advertsng, tranng, clent lsts) provded by a rval manufacturers (see Sass (005) for a bref survey.

3 Honda, Nssan, GM, Damler, BMW and hrysler. The UK dealer group Pendragon Vardy controls 350 dealers and the percentage of UK dealershps controlled by these dealershps s 9% for Jaguar, % for Land Rover, 8% for Ford and 7% for GM/Vauxhall 3. n the retal gasolne maret a smlar phenomenon has occurred va the emergence of chans of ndependent convenence store operators that sell gasolne. The most notable of these frms s ouche Tard whch s the largest convenence store operator n North Amerca and about 70% of ts stores sell gasolne. Many of ouche Tard stores are co branded wth rvng Ol. n the last decade ouche Tard has acqured convenence stores whch sell branded gasolne such as Exxon, Shell, BP and onoco Phllps 4. Ths paper consders a settng n whch upstream olgopolsts delegate the retalng of ther dfferentated products to a set of undfferentated retalng agents. The downstream maret structure s assumed to consst of a set of ndependent agents that exclusvely sell the product of one manufacturer and a common agent that sells the product of many manufacturers. Manufacturers are assumed to extract the full surplus from ther respectve agents usng a two-part tarff comprsng a wholesale prce and a fxed lcensng fee. A three-stage game s consdered. n the frst stage the upstream manufacturers determne whether to maret ther product usng and ndependent agent of the common agent. n the second stage the manufacturers determne the terms of the two-part tarff contract offered to ther respectve agents. The agents agree to maret a product provded they at least brea even at maretng that partcular product. n the fnal AutoNaton 008 Annual Report. 3 Feast (006) 4 and

4 stage the agents engage n ether dfferentated ournot or dfferentated Bertrand competton wth other agents. f the agents engage n dfferentated ournot competton then the model shows that t wll never be an equlbrum for some frms to use the common agent when other frms use ndependent agents. n other words t wll ether be the case that all manufacturers use ndependent agents or they all use the common agent. Whch of these two equlbra emerge depend on the degree f substtutablty between products and on the number of manufacturers. f there are two frms then the model yelds the results prevously obtaned by Ln (990) namely that exclusve dealng (.e. ndependent agents) dampens competton as compared wth non-exclusve dealng (.e. common agency) and s thus preferred by upstream manufacturers 5. f there are between 3 and 7 manufacturers then the exclusve dealng regme contnues to be an equlbrum for all parameter values but common agency also arses as an equlbrum provded products are suffcently close substtutes. As the number of manufacturers ncreases beyond 7 then the set of parameter values for whch the exclusve dealng (resp. common agency) regme emerges s reduced (resp. ncreased) and arses f products are suffcently poor (resp. good) substtutes. The retal prce competton results dffer from the retal output competton results n that the equlbrum number of frms adoptng common agency s unque rather than non unque and can tae on values between and rather than or n. The second 5 The result that exclusve dealng dampens competton and s thus preferred by upstream manufacturers holds for both lnear and two-part tarffs. n the two-part tarff case a crtcal assumpton s that the common agent requres brea even on each product. f the two-part tarff s set so as to observe the overall brea even constran of the common agent then Bernhem and Whnston (985) and O Bren and Shaffer (993) show that the dampenng of competton s acheved by non-exclusve dealng (.e. common agency) regmes and not by exclusve dealng regmes. As a result manufacturers prefer non-exclusve dealng provded ths regme doe not bestow any countervalng power on the common agent. 3

5 result mples that a mxed equlbrum arses n whch some frms adopt common agency and other adopt exclusve dealng. One smlarty between the prce and output competton results s that the extent of common agency ncreases as marets become less concentrated. The ntuton for these results s as follows. An ncrease n the sze of the common agency serves to ntensfy wholesale prce competton between products controlled by the common agent but serves to soften wholesale prce competton from products not controlled by the common agent. As a result an ncrease n the common agency serves to dsspate olgopoly rents but serves to shft the reduced olgopoly rents away from the manufacturers who use ndependent agents and toward the manufacturers who employ the common agent. The extent to whch rents shftng occurs depends on the number of remanng frms who contnue to use ndependent agents. For example f there are only two frms and they decde to use a common agent there are no remanng frms who use ndependent agents from who rents can be shfted. As a result the common agency regme wll not be adopted as t only results n rent dsspaton wthout resultng n rent shftng. As the number of manufacturers ncrease the scope for rent shftng s ncreased and common agency thus emerges as an equlbrum phenomenon. The paper s organzed as follows. Secton ntroduces the model and derves the retal output competton results. Secton 3 consders retal prce competton. Secton 4 offers some concludng remars. 4

6 . The Model The model conssts of n manufacturers. Each manufacturer produces a dfferentated product at constant margnal cost c and zero fxed cost. f the output of product s denoted q and the output of all products except frm s denoted Q - then the nverted demand functon for product s now assumed to be gven by (a) p ( q, Q ) a q Q =,,n (b) where 0 < s the substtutablty parameter. n partcular f = 0 then the products are completely unrelated whereas f = then the products are perfect substtutes. Each manufacturer delegates the retalng of ther product to a downstream agent. The downstream agent s ether an ndependent agent that sells the product of a sngle manufacturer of a common agent that sells the products of many manufacturers. The manufacturer s contract wth the agent specfes a wholesale prce w and a fxed fee F. f the frst products are sold by the common agent and the remanng n products are sold by ndependent agents. The manufacturer s pay-off s gven by () M ( w c) q F =,,n The retaler s pay-off s gven by (3) where R R, =,,n (4) 0,...,,..., n 5

7 s an ndcator functon whch s for the common agent (denoted ) and 0 for ndependent agents (denoted ) and where (5) R = (p (q, Q - ) w )q F =,,n represents the retal pay-off assocated wth product. Now consder the followng threestage game that nvolves smultaneous choce n each stage. n the frst stage each manufacturer chooses whether to use the common agent or an ndependent agent to maret ther product. n the second stage the manufacturer chooses the terms of the contract (w, F ) and n the thrd stage the retalng agents chooses output q. The game s solved usng bacward nducton. Output stage. Substtutng () and (5) nto (3) and then dfferentatng wth respect to q yelds the followng frst order condton (6) a q Q q w =,,n, whch ndcates that the maret power externalty (.e. q ) nternalsed by the common agent s negatve and proportonal to output. n other words an ncrease n q lowers the prce of product and thus lowers the revenues from product n proporton to q. Rearrange (7) to obtan (7) q a w Q q =,,n, whch mples that the seller of product s more prepared to cede maret share to product f they also sell product ( = ) than f a rval seller sold product ( = 0). The reason for ths result s that an ncrease n q nduces a common agent to lower q not only because the margnal revenue of q has fallen but also because q now mposes a greater 6

8 maret externalty on product because q s hgher. Ths result mples that a common agent wll be more responsve to wholesale prce competton than an ndependent agent. Output stage comparatve statcs. Substtutng (4) nto (7) and then solvng yelds solutons denoted q (w, ) where w = (w,,w n ). n Appendx A equaton (6) s used to derve that the comparatve statc effects of w are (8a) (8b) q w Q w ( )[( ( n ))( ) ( )( ( n ))] ( ) ( )[( n )( ) ( )( ( n ))] ( ) for those manufacturers that employ the common agent and by =,, =,, (9a) (9b) q ( ( n ))( ( )) ( ) = +,,n w Q w [( n )( ( )) ( ) ] = +,,n for those manufacturers that employ ndependent agents, where > 0 s the determnant of the Jacoban matrx. ontract stage. Let A represent the mnmum return that the agent requres n order to sell the product of a manufacturer. An ncrease n the agent s barganng power wll ncrease the sze of A. t s assumed that A s not affected by the number of products of other manufacturers sold by an agent. Furthermore the barganng between the manufacturer and agent 6 results n the fxed fee beng set so as to acheve a return of A per product sold by an agent,.e. (0) F ( p ( q ( w), Q ( w )) w ) q ( w A =,,n ) 6 Our barganng assumpton follows Ln (990) and mples that the choce of whether to employ a common or ndependent agent has no affect on barganng or on the ablty of manufacturers to collude. n contrast O Bren and Shaffer (993) assume that the common agent acheves countervalng power by beng able to reect the product of the manufacturer whereas Bernhem and Whnston (985) assume that the fxed fees are set so as to satsfy the ont proft, thereby facltatng colluson between manufacturers. 7

9 And thus that the wholesale prce s chosen so as to maxmze the resdual proft earned by the manufacturer. Substtutng (0) nto () yelds that the manufacturer s resdual proft s gven by () M ( w ) ( p ( q ( w), Q ( w) c) q ( w A =,,n ) Dfferentatng () wth respect to w yelds that the choce of w satsfes P ( ) q ( w) p ( ) Q ( w) () p ( ) q ( w ) c q ( w) 0 =,,n q w Q w Now let (3) Q q v =,,n w w denote the effectve conectural varaton (or EV) of frm, then dvde () by q w and substtute (3) nto () to obtan p ( ) p ( ) (4) p ( ) q v c 0 =,,n q Q Snce p Q s negatve then (4) ndcates that a frm becomes more aggressve (.e. the margnal proftablty of output expansons rse) as the EV becomes more negatve. Substtutng (8a) and (8b) or (9a) and (9b) nto (3) yelds an EV equal to (5a) v = v [( n )( ) ( )( ( ( n ))( ) ( )( ( n )] ( n )) =,, for manufacturers that employ the common agent and by (5b) v = v = ( [( n ( n )( ))( ( ( )) )) ( ( ) ] ) = +,...,n for manufacturers employng ndependent agents. omparng (5a) and (5b) reveals that 8

10 (6a) v 0 f 0 < θ < and < n v (6b) v v 0 f θ = and < n (6c) v 0 f = (6d) v 0 f = n (6a) and (6b) mply that the manufacturers who employ a common agent wll be more aggressve than those that employ ndependent agents. Let the symmetrc equlbrum output be denoted as q f sold by a common agent and by q f sold by an ndependent agent. Substtute (), = a c and v = v, q = q for =,, and v = v, q = q for = +,,n nto (4) and re-arrange to obtan (7a) q [( ) q ( n ) q ] v q 0 (7b) q [ q ( n ) q ] v q 0 whch can be solved to obtan [ ( v )] (8) q, {, }, [ ( v )][ ( n v )] ( n ) and whch mply that the equlbrum prce can be expressed as (9) p c q ( v ) {, } and thus that the manufacturers equlbrum profts are gven by (0) ( v )( q ) {, } Substtutng () nto (6) yelds w p q ( ( )) whch when combned wth (9) mples that the equlbrum wholesale prce s () w c q ( v ( )) {, } 9

11 Proposton : The equlbrum outputs and prces are as follows () () () (v) n n = = n 0 < θ < θ = (a) q > q > 0 q > 0, q = 0 q > 0 q > 0 (b) p > p > c p = p = c p > c p > c (c) w < w < c w < c, w = c w < c w < c PROOF: See Appendx B. Proposton shows that manufacturers wll set wholesale prces below margnal cost regardless of whether they employ an ndependent agent or a common agent. The reason for ths result s two-fold. Frstly, manufacturers can use the fxed fee to recoup any losses that they ncur as a result of sellng below margnal cost. Secondly, by sellng below margnal cost the manufactures nduce ther agents to ncreasng output whch then serves to deter rval output and thereby shft more of the olgopoly rents from ther rvals to themselves. Proposton also shows that snce common agents are more responsve to wholesale prce competton (see (7)) than are ndependent agents then manufacturers that employ common agents are more aggressve n settng wholesale prces whch then results n more output and lower retal prces for these manufacturers. hoce of agent stage. f < < n represents the equlbrum number of manufacturers that choose to maret ther product usng the common agent then must satsfy () ( ) ( ) to ensure that none of the manufacturers wshes to leave the common agency. Secondly must satsfy 0

12 (3) ( ) ( ) to ensure than none of the n manufacturers wsh to on the common agency. For = to be an equlbrum only (3) needs to be satsfed whereas f = n s to be an equlbrum only () needs to be satsfed. Substtutng (8), (0), (5a) and (5b) nto () and (3) and then carry out numercal smulatons yelds the results n Table whch are summarzed n Result. Result : Under retal output competton the equlbrum value of s as follows () f n = then = s unque. () f 3 n 7 then (a) = s unque f the products are suffcently poor substtutes (b) otherwse s non-unque and nvolves ether = or = n. () f n 8 then (a) = s unque f the products are suffcently poor substtutes, (b) = n s unque f the products are suffcently close but not perfect substtutes (c) otherwse s non-unque and nvolves ether = or = n. (v) An ncrease n n reduces the set of θ values for whch = occurs.

13 Table : Values of and n for whch = and = n are possble equlbra ( under retal output competton Number of = = or = n = n Manufacturers (n) All None None 3 <.83 >.83 None 4 <.64 >.64 None 5 <.50 >.50 None 6 <.4 >.4 None 7 <.34 >.34 None 8 < < <.8 and θ >.95 9 <.6.6 < <.57 and θ >.99 0 <.3.3 < <.4 and θ >.99 0 <.. < <.3 and θ > ~ 50 < < <.044 and θ > ~ 00 < < <.009 and θ > ~.8 < < < <.98.4 < <.99.3 < < ~.044 < < ~.009 < < ~ = mples that all frms adopt exclusve dealng. = n mples that all frms employ the common agent. f = 0 then products are unrelated. f = then products are perfect substtutes.

14 3. Prce competton f p denotes the prce of product and P - = product s assumed to be gven by p then the demand functon for (4a) q (p, P - ) = a bp + dp - where b = ( n ) and d =,,n n and where 0 < < denotes the substtutablty parameter whch s zero f goods are unrelated and approaches nfnty when goods are perfect substtutes. The aforementoned restrcton can be expressed as d (4b) 0 n where b The profts generated by product are thus gven by (5) = (p c)q (p, P - ) =,,n The retaler s pay-off for sellng product s gven by (6) R = (p w )q (p, P - ) F =,,n Prce stage. A general expresson for the obectve functon of each retaler s gven by (3). Substtute (4) and (6) nto (3), dfferentate wth respect to p, dvde by b and let and let d b to obtan the followng frst order condton a (7) b p + θp - + w +, ( p w ) = 0 =,,n whch ndcates that the maret power externalty nternalsed by the common agent s postve and proportonal to the retal mar-up. n other words an ncrease p rases the demand for product and thus rases the profts for product n proporton to the latter s mar-up. Re-arrange (7) to obtan 3

15 (8) p a b w P p w ), ( =,,n whch mples that the seller of product s less responsve to wholesale prce competton from product f they also sell product ( = ) than f a rval seller sold product ( = 0). The reason for ths result s that an ncrease n w wll rase p but lower p w. The ncrease n p shfts out the demand curve for product and ncreases the margnal revenue assocated wth rasng prce for product. f the seller of product also sells product then there s an offsettng effect due to the reducton n p w whch reduces the sze of the maret power externalty nternalzed by the seller of products and and whch nduces that seller to lower p. The maret power externalty effect only partally offsets the margnal revenue effect and thus results n the seller of products and rasng p n response to an ncrease n w but by an amount less than they would f ths seller only sold product. ontract stage. Followng the output competton analyss yelds that the prce competton verson of (4) s gven by q ( ) q ( ) (9) q ( ) ( p c) v c 0 where p P P p v =,,n w w denotes the EV under prce competton. Snce q P s postve then (9) ndcates that a frm becomes less aggressve (.e. the margnal proftablty of prce ncreases) as the EV becomes more postve. n Appendx t s shown that the EV s equal to ( n )( ( )) (30a) v = v =,, ( ( n ))( ( )) ( ) for manufacturers that employ the common agent and by 4

16 (30b) v = v = ( [( n ( n )( ))( ( )) ( ) ] ( )) ( ) = +,,n for manufacturers employng ndependent agents. omparng (4a) and (5b) reveals that (3a) v v f < < n 0 (3b) 0 v f = (3c) 0 v f = n (3a) mples that the manufacturers who employ a common agent wll be more aggressve than those that employ ndependent agents. n Appendx D t s shown that the (9) can be solved to obtan [ ( v )] (3) p c, {, }, [ ( v )][ ( n v )] ( n ) (9) also mples that the equlbrum quantty can be expressed as (33) q b( p c)( v ) {, } and thus that the manufacturers equlbrum profts are gven by (34) b( v )( p c) {, } q Substtutng (4a) nto (7) yelds w c p c whch when combned wth b( ( )) (33) mples that the equlbrum wholesale prce s ( p c) ( v ( )) (35) w c {, } ( ) 5

17 Proposton : The equlbrum outputs and prces are as follows () () () n = = n (a) p > p > c p > c p > c (b) q > q > 0 q > 0 q > 0 (c) w > {w, c} w > c w < c PROOF: See Appendx E. hoce of agent stage. Substtutng (3), (34), (30a) and (30b) nto () and (3) and then carry out numercal smulatons yelds the results n Table whch are summarzed n Result. Result : Under retal prce competton the unque equlbrum value of s as follows () f n = or 3 then =. () f 4 n 7 then (a) = f the products are suffcently poor substtutes (b) otherwse n and rses as products become closer substtutes. () f n 8 then (a) = f the products are suffcently poor substtutes (b) otherwse n and rses as products become closer substtutes. (v) An ncrease n n reduces the set of θ values for whch = occurs. 6

18 Table : Values of ˆ = (n )θ and n for whch varous values of are an equlbrum under retal prce competton (0 < ˆ < ) Number of = n = n Manufacturers (n) All None None 3 All None None 4 ˆ <.84 ˆ >.84 None 5 ˆ <.79 ˆ >.79 None 6 ˆ <.77 ˆ >.77 None 7 ˆ <.75 ˆ >.75 None 8 ˆ < < ˆ <.98 ˆ >.98 9 ˆ < < ˆ <.96 ˆ >.96 0 ˆ <.7.7 < ˆ <.95 ˆ >.95 0 ˆ < < ˆ <.9 ˆ >.9 50 ˆ < < ˆ <.88 ˆ > ˆ < < ˆ <.88 ˆ >.88 f ˆ = 0 then products are unrelated. f ˆ = then products are very close substtutes. n ths range the value of s unque and rses from to n as ˆ rses (.e. as the products become closer substtutes). 7

19 4. oncludng remars The maorty of papers that use the dampenng of competton framewor for analyzng the ncdence and effects of exclusve and non exclusve dealng assume duopoly 7. The current paper departs from the lterature n that t consders more than two frms and s thus able to show that the effects of exclusve dealng (.e. hgher wholesale and retal prces) are nvarant to maret structure but that the ncdence of exclusve dealng vares dramatcally wth maret structure. Specfcally the paper show that exclusve dealng becomes less prevalent (common agency becomes more prevalent) as marets become less concentrated. Our results thus provde alternate explanatons for some of the observatons made n the emprcal lterature on exclusve dealng 8. 7 hang (99), Dobson and Waterson (997), Ln (990), Mauleon, Sempere Monerrs and Vannetelbosch (005), Moner olonques, Sempere Monerrs and Urbano (004), Mycels, Ryanto and Wuyts (000), O Bren and Shaffer (993). 8 See results and a lterature revew n Sass (005). 8

20 REFERENES AUTONATON ANNUAL REPORT FOR 008 BERNHEM, B.D. AND WHNSTON, M.D., 985, ommon Maretng Agency as a Devce for Facltatng olluson, Rand Journal of Economcs, (6(), pp BONANNO, G. AND VKERS, J.S., 988, Vertcal Separaton, Journal of ndustral Economcs, 36(), pp hang, M.H. (99). Exclusve dealng contracts n a successve duopoly wth sde payments, Southern Economc Journal 59: ouche Tarde web ste Accessed Aprl 30, 00 Dobson, P.W. and Waterson, M. (997) Exclusve tradng contracts n successve dfferentated duopoly, Southern Economc Journal 63: LN, Y.J., 990, The Dampenng-of-ompetton Effect of Exclusve Dealng, Journal of ndustral Economcs, 39(), pp Mauleon, A., Sempere Monerrs, J.J. and Vannetelbosch, V.J. (005) Networs of Manufacturers and Retalers SSRN Worng Paper Seres, Rochester, July. Moner olonques, R., Sempere Monerrs, J.J., and Urbano, A. (004) The manufacturers choce of dstrbuton polcy under successve duopoly, Southern Economc Journal 70(3): Mycels, J., Ryanto, Y.E. and Wuyts, F., (000) nter and ntrabrand competton and the manufacturer retaler relatonshp, Journal of nsttutonal and Theoretcal Economcs 56:

21 O BREN, D.P. AND SHAFFER, G., 993, On the Dampenng-of-ompetton Effect of Exclusve dealng, Journal of ndustral Economcs, 4(), pp. 5-. Feast, R. (006) New group may domnate UK dealers. Automotve News Europe, (4), p.3 Sass, T.R. (005) The compettve effects of exclusve dealng: Evdence from the U.S. beer ndustry, nternatonal Journal of ndustral Organzaton 3, pp Wpeda, Accessed Aprl 30, 00 0

22 Appendx A: Dervaton of output stage comparatve statcs Let Q q and note that n Q ( n ) Q and, q = ( )Q. Substtute = 0 and Q - = Q q nto (6) to get (A). Substtute (4) nto (6), sum (6) over all =,,n to get (A), and over all =,, to get (A3). (A) a ( )q Q = w = +,,n (A) na ( + ( n )) Q ( n ) Q w (A3) a Q ( ( )) Q w Totally dfferentate any one of the n equatons gven n (A) to obtan (A4) and then totally dfferentate (A) and (A3) to obtan (A5) and (A6) respectvely (A4) (A5) (A6) ( n ) 0 ( ) ( ) 0 0 dq dq dq n dw dw dw {,..., n} Let be the determnant of the LHS matrx and then solve the above to get (9a) and (A7) Q w ( ) =,, (A8) Q w ( )( ( )) = +, n (A9) Q w ( )( ( n )) =,, (A8) mnus (9a) yelds (9b). Q - = Q q, n, q = Q q, = nto (6) yelds (A0) a ( )q Q Q w =,, mplct dfferentaton yelds q Q Q (A) w w w =,, Substtute (A7) and (A9) nto (A) to get (8a). Subtract (8a) from (A7) to get (8b).

23 Appendx B: Proof of Proposton 0 < θ, n, (6a), (6b) and (6c) mply (B) ( + θ( + v ))( + θ(n + v )) (n )θ > 0 a. From (8) f follows that (B) q q ( v [ ( v )][ ( n v )] ( n ) v ) () & () Follow from (B), (B), (6a) or (6b), (8), (b) or θ =. () & (v) Follow from (8), (B), (6c) or (6d), (b). b. Substtute Q - = Q q nto (a) to obtan that the dfference n retal prces s (B3) p p = ( θ)(q q ) () Follows from (B3), (b), q > q > 0, (9), (6a). () Follows from (9), q = 0 or (6b). () & (v) Follow from (9), (6c) or (6d), q > 0 or q > 0. c. Substtute (4) and Q - = Q q nto (6) to get that the dfference n wholesale prces s (B4) w w = [( θ)(q q ) + ( )θq ] () & () Follow from (B4), q > q > 0 or q > q = 0, (b) or θ =, (), (4), (6a) or (6b). () & (v) Follow from (), q > 0 or q > 0, (4), (6c) or (6d).

24 Appendx : Dervaton of comparatve statcs and EV under prce competton Let P p, P n p. Use (7) and follow the analyss n Appendx A to obtan () () (3) Let 0 ( n ) ( ) 0 ( ) 0 dp dp dp ( ( ) dw dw n ( ( ) dw m be the determnant of the Jacoban matrx and then solve the above system to get m dw {,..., n} (4) P w ( ( ))( ) =,, (5) P w ( )( ( )) = +,,n (6) P w ( ( ))( )( ( n )) =,, (7) p ( ( n ))( ( )) ( ) = +,,n w (5) mnus (7) yelds (8). (8) dvded by (7) yelds (30b). (8) P w [( n )( ( )) ( ) ] = +,,n Substtute P - = P p, n, p = P p, = nto (7) and mplctly dfferentate to get p P P (9) w w w =,, Substtute (4) and (6) nto (9) to get (0) p w ( )[( ( n ))( ( )) ( ) Subtract (0) from (4) to get (). () dvded by (0) yelds (30a). () P w ] =,, ( )( n )( ( )) =,, 3

25 Appendx D: Dervaton of prce competton solutons Let the symmetrc equlbrum prces from common and ndependent agents be denoted p and p respectvely. Substtute (4a), v = v, p = p for =,, and v = v, p = p for = +,,n nto (9). Now dvde by b, add and subtract c, add and subtract θ(n )c and then let c c (n ) a b, d b, pˆ p c and pˆ p c to get (D) p ˆ [( ) pˆ ( n ) pˆ ] v pˆ 0 (D) p ˆ [ pˆ ( n ) pˆ ] v pˆ 0 whch can be solved to obtan (3). Appendx E: Proof of Proposton 0 < θ < n, n, (3a) and (3b) mply (E) ( θ( + v ))( θ(n + v )) (n )θ > 0 a. From (3) f follows that (E) p p ( v [ ( v )][ ( n v )] ( n ) v ) () & () Follow from (E), (E), (3a) or (3b), (3), (4b). () Follows from (3), (3c), (4b). b. Substtute P - = P p nto (4a) to obtan that the dfference n output s (E3) q q = b( + θ)(p p ) () Follows from (E3), p > p > c, (4a), (33), (3a). () & () Follow from (33), p > c or p > c, (3b) or (3c). c. Substtute λ =, p w = p w for =,, and (4a) nto (7) to obtan (E4) p w = p w = q =,, b ( ) Substtute (4), (E4) and P - = P p nto (7) to get that the wholesale prce dfference s 4

26 (E5) w w = [( + θ)(p p ) + ( ) b ( ) θq ] () Follows from (E4), (4b), p > p > c, q > 0. () & () Follow from (35), (4), (4b), p > c or p > c, (3b) or (3c). 5