Partial collusion with asymmetric cross-price effects Luca Savorelli Quaderni - Working Papers DSE N 715

Transcription

1 Partal colluson wth asymmetrc cross-prce effects Luca Savorell Quadern - Workng Papers DSE N 715

2 Partal colluson wth asymmetrc cross-prce e ects Luca Savorell y October 8, 2010 Abstract Asymmetres n cross-prce elastctes have been demonstrated by several emprcal studes. In ths paper we study from a theoretcal stance how ntroducng asymmetry n the substtuton e ects n uences the sustanablty of colluson. We characterze the equlbrum of a lnear Cournot duopoly wth substtute goods, and consder substtuton e ects whch are asymmetrc n magntude. Wthn ths framework, we study partal colluson usng Fredman (1971) soluton concept. Our man result shows that the nterval of quanttes supportng colluson n the asymmetrc settng s always smaller than the nterval n the symmetrc benchmark. Thus, the asymmetry n the substtuton e ects makes colluson more d cult to sustan. Ths mples that prevous Anttrust decsons could be reversed by consderng the role of ths knd of asymmetry. Keywords: asymmetry, substtutes, Cournot duopoly, colluson, folk theorem. JEL class caton: C72, D43, L13. 1 Introducton In ths paper we study the sustanablty of colluson n a Cournot duopoly where market demands are asymmetrc n the magntude of substtuton e ects. The evdence of asymmetrc cross-prce elastctes s shown n several emprcal studes (e.g. Berry et al., 1995; Serthuraman et al., 1999; Km and Cotterll, 2008; Roas and Peterson, 2008). For example, n Berry et al. (1995) the cross-prce sem-elastcty between Nssan Sentra and Ford The author s grateful to hs PhD supervsor Vncenzo Dencolò, Fabo Pnna, Phlpp Schmdt-Dengler, Pasquale Schrald, and the semnar s audence at the London School of Economcs for ther useful comments and suggestons. The author s also grateful to STICERD - London School of Economcs and Poltcal Scence, where part of ths work has been developed n hs year as vstng research student. y Dpartmento d Scenze Economche, Unverstà d Bologna, pazza Scaravll 1, Bologna, Itala; and Dpartmento d Economa Poltca, pazza S. Francesco 7, 53100, Sena, Itala; e-mal: luca.savorell@unbo.t. 1

3 Escort s 1.375, whle between Ford Escort and Nssan Sentra s 8.024; n the US market for processed cheese Km and Cotterll (2008) nd that the cross-prce elastcty between Weght Watchers and Kraft s 0.25, whle between Kraft and Weght Watchers s 0.04; the cross-prce elastcty between Bud and Old Style s 0.003, whle the cross-prce elastcty between Old Style and Bud s (Roas and Peterson, 2008); nally, n a meta-analyss of 1060 cross-prce e ects, 19 grocery product categores and 280 brands, Serthuraman et al. (1999) provde an emprcal generalzaton of ths asymmetrc prce e ect. The evdence supports the dea that n general cross-prce elastctes are not symmetrc, and ths fact s also compatble wth a theoretcal perspectve, snce aggregate market demands need not satsfy any symmetry condton (see e.g. Dewert, 1980; Bonfrer et al. 2006), and also at the ndvdual level the Slutsky matrx need not be symmetrc (because the ncome e ect need not be symmetrc). In ths paper we consder a lnear duopoly model wth Cournot competton and substtute products. We extend Sngh and Vves (1984) by allowng for asymmetry n the magntude of the substtuton e ects, dervng the equlbrum quanttes, prces and profts. We compare them to those n a symmetrc equvalent duopoly settng, namely a duopoly n whch the substtuton parameters are symmetrc and equal to the average of the parameters n the asymmetrc case. Then, usng the folk theorem soluton concept and the penal code accordng to Fredman (1971), we analyze partal colluson. We derve the range of collusve quanttes that makes the colluson sustanable for both rms, and compare t to the symmetrc equvalent case. Ths paper adds the two followng man contrbutons to the exstng lterature. Frst, even though an extensve lterature on colluson consders other knds of asymmetres n market demands and n the characterstcs of rms (for an overvew see e.g. Feuersten, 2005), to the best of our knowledge, the e ect of asymmetrc cross-prce e ects on the stablty of colluson has not yet been studed from a theoretcal stance. Our man result shows that, gven the dscount rate of each rms, the nterval of quanttes whch supports the colluson s always smaller than n the symmetrc equvalent case. Intutvely, f rms are asymmetrc the ntervals of the quanttes makng the colluson stable are no longer concdent, and the e ect of asymmetry s to shft each rm s nterval n opposte drectons. Ths happens snce a hgh colluson quantty rases the relatve value of the devaton strategy for the weak rm, whose producton decson s relatvely more n uenced by the other. Smlarly, a low level of the colluson quantty makes the devatng behavor more convenent for the strong rm. Only the ntermedate levels of collusve quantty are therefore supported by both rms. In ths stylzed settng, we thus conclude that the asymmetry n the substtuton e ect makes colluson more d cult to sustan. Ths 2

4 mples that prevous anttrust decsons could be reversed by consderng the role of ths knd of asymmetry. The second result s related to the characterzaton of the equlbrum n the asymmetrc Cournot duopoly. We nd that the rm whose prce s relatvely more n uenced by the other rm s producton decsons sets a lower quantty, and sells at a lower prce wth respect to the rval, gettng lower pro ts. An ntuton for ths result s that, snce the two goods are strategc substtutes (but asymmetrcally) the producton decsons are drven by the rm whch s relatvely less n uenced by the rval. The symmetrc equvalent rm equlbrum prces, pro ts and quanttes le between those of the strong and the weak rm. The paper s structured as follows. In the next secton we characterze the equlbrum condton for the asymmetrc Cournot duopoly and compare t to the symmetrc equvalent case. In the thrd secton we derve the soluton of the colluson supergame, and we state the results on the mplct colluson stablty. In the fourth secton we draw the nal remarks and drectons for future theoretcal and emprcal research. The proofs are relegated to the appendx. 2 A lnear Cournot duopoly wth asymmetrc substtuton e ects In ths secton we consder a Cournot duopoly whch extends Sngh and Vves (1984) allowng for asymmetry n the magntude of the substtuton e ects. Ths extenson does not need any partcular assumpton, snce market demands need not satsfy any symmetry condtons such as those requred by the Slutsky equaton (see e.g. Dewert, 1980; Bonfrer et al., 2006). Moreover, f the ncome e ect n the Slutsky equaton s not symmetrc, the ndvdual demands are not symmetrc n the cross-prce e ect as well. Mastroleo and Savorell (2010) show whch class of ndvdual utlty functons underles the asymmetry n cross substtuton e ects. Let us consder a duopoly where rm and rm face the followng nverse demand functon p = a q b q ; f; g = f1; 2g (1) where b 2 [0; 1]; a > 0 and q ; 0:We normalze the own-prce e ect to one, but ths s not gong to a ect the results qualtatvely. If b 6= b, we say that the demands are asymmetrc, and f b = b we say that they are symmetrc. In a Cournot competton 3

5 settng, each rm chooses the producton quantty to maxmze ts pro ts as follows: max q = (p c)q (2) where c s a constant margnal cost. The soluton of the model leads to the equlbrum quanttes and pro ts q C = (b 2)(a c) b b 4 C = (b 2) 2 (a c) 2 (b b 4) 2 (4) C = + = [(b 2) 2 + (b 2) 2 ](a c) 2 (b b 4) 2 : (5) We assume henceforth that a > c to guarantee that the Cournot equlbrum quanttes are postve. To have an ntuton about the strategc nteracton drvng the results, let us consder the followng best response functons: (3) q C = (a c b q C ) 2 (6) q C = (a c b q C) : (7) 2 By smple nspecton t s mmedately apparent that the two goods are strategc substtutes. followng Remark. Then, takng the cross dervatve of each best response functon leads to the Remark 1 Consder dqc dq C = b 2 and dqc dq C = b 2, then: 1. f b > b, dqc dq C 2. f b < b, dqc dq C < dqc dq C > dqc dq C ; : The above Remark states that, f e.g. b > b ; an expanson n rm producton has an mpact on s quantty choce greater than the mpact that an equvalent expanson by has on the quantty choce of : For ths reason, we call the weak substtute and the strong substtute (and vce versa f b < b ). Consstently, the strong substtute produces a hgher quantty and gets hgher pro ts wth respect to the weak. Henceforth, the equlbrum results assocated wth the asymmetrc demands wll be denoted by the superscrpt ASY: 4

6 To allow for a comparson wth the asymmetrc settng, we explot as benchmark a specal case of the symmetrc settng, such that b = b = b +b 2 : The substtuton parameters are chosen to be the mean of the parameters n the asymmetrc case. We call the benchmark settng a symmetrc equvalent, and we denote the assocated equlbrum values by the superscrpt SE: In the symmetrc equvalent case the soluton of problem 2 s q SE = q SE = SE = 2(a c) 4 + b + b (8) 4(a c) 2 (4 + b + b ) 2 : (9) To provde an ntutve comparson, n Fgure 1 we represent the asymmetrc Cournot equlbrum and ts symmetrc equvalent n the space of quanttes when b > b : q a c b 2( a c) b + b a c 2 SE C 0 a c 2 a c b 2( a c) b + b q Fgure 1 The best response functons are represented by the thck lnes n the asymmetrc case, and by the thn straght lnes n the symmetrc equvalent. The SE equlbrum s North 5

7 West of the asymmetrc Cournot equlbrum, thus the weak substtute produces a lower quantty wth respect to the SE case, and the strong substtute a hgher quantty. By consderng the two parabolas, representng the contour lnes of the pro t functons at the Cournot equlbrum, t s straghtforward to notce the asymmetry n the regon of the possble equlbra whch are Pareto superor to the Cournot-Nash. The followng Proposton characterzes the soluton to problem 2 wth respect to the equlbrum prces and the quantty levels, and compares them to the symmetrc equvalent case. Proposton 1 Let q SE = q SE = q SE and p SE = p SE = p SE : Then the followng holds: 1. b ASY > b ASY () q ASY < q SE < q ASY ; 2. b ASY > b ASY () p ASY < p SE < p ASY ; 3. b ASY > b ASY () ASY < SE < ASY : The above Proposton states that n the symmetrc equvalent case quanttes, prces, and pro ts le between the weak substtute s and the strong substtute s. The strategc weakness thus leads to lower quanttes, prces and pro ts wth respect to both the symmetrc equvalent and the strong substtute. 3 Partal colluson In ths secton we wll nvestgate how asymmetry n substtutablty a ects the stablty of colluson. As n the standard symmetrc case, the mplct level of colluson should not necessarly be the monopoly quantty. The reason s that there are n nte quanttes hgher than the monopoly one that stll provde pro ts greater than n the Cournot game. Moreover, as we wll show n Remark 3, when the dscount rate s su cently hgh the monopoly quantty does not make the colluson stable. Gven a common dscount factor ; we study the mnmum ndvdual quantty produced by a rm that allows colluson to be stable, and to what extent asymmetry n uences the stablty of colluson. We wll proceed through the followng steps. Frst, we wll set up the Cournot supergame and study the colluson and devaton strateges, ndng the nterval of quanttes for each rm that allows colluson to be sustanable. Second, we wll state n Proposton 2 whch s the nterval of collusve quanttes that makes colluson sustanable for both. Fnally, n Proposton 3 we wll state our man result, showng whether asymmetry n 6

8 the substtuton e ects makes colluson easer wth respect to the symmetrc equvalent benchmark. Over an n nte horzon, the two rms play grm trgger strateges n a Cournot supergame (1): We use the folk theorem soluton concept accordng to Fredman (1971). In the rst stage of the game, t = 0; the rms follow a collusve strategy and maxmze the ont pro ts. In general, pro ts dvson s not equal. In the remanng tme horzon, f both the rms played the collusve strategy n the prevous perod, the rms contnue to play the collusve strategy : Otherwse, f at least one rm devates from the collusve strategy playng D, the rms play the Cournot-Nash strategy C. We rst consder the collusve strategy. Snce there are n nte potental collusve outcomes, we use ont-pro t maxmzaton as the selecton crteron for the collusve focal pont (as n e.g. d Aspremont et al., 1983; and n asymmetrc envronments, e.g. Rothschld, 1999). Accordngly, we solve the followng problem: whch leads to the equlbrum quanttes and pro ts: max + (10) q ;q q = q a c = 2 + b + b (11) = (b + 1)(a c) 2 (2 + b + b ) 2 : (12) We can then state the followng remark, whch characterzes the ont-pro t maxmzaton equlbrum. Remark 2 When rm and rm maxmze the ont pro t, they produce the same quantty, and the weak substtute obtans a level of pro t lower than the strong. The ntuton for ths Remark s that, snce the two rms are technologcally symmetrc, when maxmzng the ont pro t they take nto account the strategc externalty dervng from the asymmetry n the strategc substtutablty, and nternalze t by playng lke symmetrc rms. Thus, t seems reasonable for ths equlbrum to be a focal pont also for lower levels of ont pro ts. When the rms play the strategy; each rm gets the pro ts = q (a c q b q ): Takng nto account the characterzaton n Remark 2, the rms set q therefore rewrte the above expresson as = q : We can 7

9 = q (a c (1 + b )q ): Notce agan that wth asymmetrc demands the collusve quanttes are stll symmetrc, but the pro ts are not. We now derve the equlbrum values for rm playng D, denotng the solutons wth the superscrpt D. In the perod of devaton, rm solves the followng maxmzaton problem: max = (p c)q = q D fq D q =q g (a c q D b q ) and obtans the followng per-perod quanttes and pro ts: q D = (a c b q ) 2 Thus, the ow of pro ts for the devatng rm s (13) D = (a c b q )2 : (14) 4 D = D (q ) + 1 ASY (15) where 2 [0; 1] s the dscount rate common to and. In what follows we wll study the problem of colluson stablty. We wll rst derve the nterval of quanttes of the rval rm that makes the colluson sustanable, and then the nterval of quanttes on whch both the rms agree to collude. For rm ; the collusve strategy s sustanable only f D (q )=(1 ):Then, the followng lemma states for whch values q; the colluson s stable for each sngle rms. Lemma 1 The colluson s stable for rm f and only f f and only f < q < qcol ; where: < q < qcol ;and for rm = (a c) (b b 4)[b ( 1) 2] + A (b b 4)[b 2 (2 + b ) 2 ] = (a c) (b b 4)[b ( 1) 2] A (b b 4)[b 2 (2 + b ) 2 ] = (a c) (b b 4)[b ( 1) 2] + A (b b 4)[b 2 (2 + b ) 2 ] = (a c) (b b 4)[b ( 1) 2] A (b b 4)[b 2 (2 + b ) 2 ] 8

10 and A = [b (b (b 2) 2 + b (b 2 b 2 ) + 8(b b ))] 1 2, A = [b (b (b 2) 2 + b (b 2 b 2 ) + 8(b b ))] 1 2 : Henceforth we wll consder only real-valued boundares quanttes. 1 We call Q = f < q < qcol g and Q = f < q < qcol g the sets of sustanable colluson quanttes for rm and : As t s evdent by nspecton, the two nterval do not concde as long b 6= b : Some collusve quanttes are thus sustanable for rm ; but not for rm. Snce t could also be the case that the two ntervals do not overlap, n the followng Lemma we state when there s room for colluson among rms f the two rms are asymmetrc. Lemma 2 If b 6= b ; then Q 6= Q and Q \ Q always exsts. The above Lemma means that there s always an nterval of quanttes on whch the two rms can agree to collude, and whch s sustanable for both. Let us call Q ASY = Q \ Q the set of collusve quanttes whch are sustanable for both rm and rm : By contrast, when b = b ; = =, and the two sets concde. In ths case, we call the set of collusve quanttes Q SY M : The followng proposton characterzes = and = the set of collusve quanttes that makes the colluson stable for each case. Proposton 2 When: 1. b < b, Q ASY = f < q ; < qcol g; 2. b = b, Q SY M = f < q < g; 3. b > b, Q ASY = f < q ; < qcol g: Frst, notce that settng b 1 = b 2 = 1; the boundary values n the above Lemma reduce to the well known symmetrc case (9 5)(a c) 3(9 ) < q ; < a c 3 : Second, settng = 0; the boundares of the stable colluson quanttes are [ a c 2+b ; a c 2+b ] n case 1, and the reverse n case 2. The lower boundary s thus greater than the collusve monopoly quantty, a c 2+b +b. Ths leads to the followng remark. 1 When b > b ; A s always postve and A s postve when (8b (8+b 2 )b +b 3 ) ((b 2) 2 b ) < d and, n addton, to be real valued, f b < 13 p 41 ' 0; 825; t must be b 8 < 2(2 [(b 2) 2 (b +1)] b postve and analogous condtons can be obtaned by nvertng ndexes. 1 2 ) : When b < b ; A s always 9

11 Remark 3 There always exst a value 0 1 such that the collusve monopoly equlbrum s not stable. The above remark ponts out that, as n the symmetrc case, the monopoly quantty s not always a feasble quantty, and when ths s the case the analyss of partal colluson thus becomes more relevant. We proceed to answer the key research queston of ths paper,.e. whether asymmetry n the substtuton e ects makes colluson more d cult to sustan. Wthout qualtatvely a ectng our results, we normalze the parameter space by settng b = 1 b :We de ne the ndex of asymmetry as = b b = 1 2b :When the two values are symmetrc, b = b = 1 2 and = 0; whle = 1 when one parameter s 0 and the other 1, the maxmum level of asymmetry. We then use the average value of the parameter as benchmark, and we state the man result of ths paper n the followng proposton. Proposton 3 The nterval of collusve quanttes n the symmetrc equvalent case s always larger than the one n the asymmetrc case. The ntuton behnd ths proposton can be understood by hghlghtng the strategc nteracton between the two rms. Each rm determnes what s the level of producton of the other rm that makes the colluson stable for tself. When rms collude, t s optmal for both to produce the same quantty. Then, t can mmedately be seen from (13) that, f the weak substtute wants to devate from the agreement, he s gong to produce a lower devaton quantty wth respect to the devaton quantty of the strong substtute. The Cournot quantty n the followng perod s lower as well. The reason s that a hgh colluson quantty rases the relatve value of the devaton strategy for the weak substtute. Analogously, a low level of the colluson quantty makes the devatng behavor more convenent for the strong substtute. The asymmetry n the strategc substtuton e ect translates nto asymmetrc partal colluson strateges, whch stll overlap, but are no longer concdent. Each rm s collusve nterval s shfted n opposte drectons, and only the ntermedate levels of collusve quantty are supported by both rms. We can therefore conclude that the asymmetry n the substtuton e ect makes colluson more d cult to sustan wth respect to the symmetrc benchmark case. 4 Conclusons In ths paper we generalzed Sngh and Vves (1984) to account for asymmetry n the substtuton e ects, and to study ts mplcatons for mplct colluson. The rst result 10

12 we found characterzes Cournot equlbrum: The symmetrc benchmark rm equlbrum prces, pro ts and quanttes are lower than those of the strong, and hgher than those of the weak. The second result states that the asymmetry n the substtuton e ects makes partal colluson more d cult to support wth respect to ts symmetrc benchmark. Future extensons of ths model could explore the robustness of the results under d erent settngs and knds of competton (e.g. Bertrand competton, sem-colluson, R&D). Our feelng s that, n frameworks other than colluson, ntroducng asymmetres n the cross-prce e ect could lead to new theoretcal nsghts partcularly useful to emprcal analyss. Fnally, ths paper suggests that the asymmetry n the substtuton e ects s a relevant ssue when evaluatng the possblty of mplct colluson among rms. If the emprcal estmatons of market demands do not take nto account ths knd of asymmetry, t s lkely that the extent to whch the rms can collude s overestmated. We thus thnk that there s room for an emprcal re-assessment of prevous estmatons, and that n ths lght perhaps some ant-trust decsons could be reversed. Moreover, the theory proposed n ths paper could be usefully tested by usng expermental economcs methodology. 5 Appendx 5.1 Proof of Proposton 1 Part 1. By smple nspecton, as long as b ASY > b ASY we know that q ASY < q ASY : Then for value of the para- t s su cent to solve the nequaltes q ASY < q SE and q SE < q ASY meters of the problem, that s b ; 2 [0; 1] and a > c > 0: Part 2. From the solutons of the symmetrc and asymmetrc problem t s smple to get p ASY (b b 4) ; p SE = 2a+(2+b +b )c 4+b +b ; p ASY = a(b 2)+[b (b 1)]c 2 (b b 4) and analogously to Part 1 solvng the correspondng nequaltes. Part 3. The soluton to the nequalty s analogous and straghtforward by consderng 8 and Proof of Lemma 1 Let us consder the nequaltes D (q )=(1 ); that s: D (q ) + 1 ASY (q ) 1 = a(b 2)+[b (b 1)]c 2 (a c b q )2 4 + (b 2) 2 (a c) 2 1 (b b 4) 2 q (a 2q c) 1 11

13 The soluton of the assocated equaton gves the two values = (a c) (b b 4)[b ( 1) 2] + A (b b 4)[b 2 (2 + b ) 2 ] = (a c) (b b 4)[b ( 1) 2] A (b b 4)[b 2 (2 + b ) 2 ] where A = [b (b (b 2) 2 + b (b 2 b 2 ) + 8(b b ))] 1 2 : Invertng the ndexes we obtan analogous values for rm ; that s = (a c) (b b 4)[b ( 1) 2] + A (b b 4)[b 2 (2 + b ) 2 ] = (a c) (b b 4)[b ( 1) 2] A (b b 4)[b 2 (2 + b ) 2 ] where A = [b (b (b 2) 2 + b (b 2 b 2 ) + 8(b b ))] 1 2 : We want the nterval of collusve quanttes for both rm and to be real valued. When b > b ; A s radcand s always postve and A s s postve when (8b (8+b 2 )b +b 3 ) ((b 2) 2 b ) < d and, n addton, f b < 13 p 41 8 ' 0; 825; t must be b < 2(2 [(b 2) 2 (b +1)] 2 1 ) b : When b < b ; A s always postve and analogous condtons can be obtaned by nvertng ndexes. 5.3 Proof of Lemma 2 Lemma 3 If b 6= b ; then Q 6= Q and Q \ Q always exsts. We wll rst show that, when b > b () of the proof showng b < b () obtaned by nvertng the ndexes. < < < Let us consder b > b. We rst prove that nequalty < < < ; the second part s analogous, and can be < : Ths can be done solvng the (a c) (b b 4)[b ( 1) 2] A (b b 4)[b 2 (2 + b ) 2 ] < (a c) (b b 4)[b ( 1) 2] + A (b b 4)[b 2 (2 + b ) 2 ] 12

14 consderng the relevant range of the parameters a; b ; b ; c;, and the condtons stated n Lemma 1 for the quanttes to be real-valued, ths nequalty always holds. Then, we prove that < : The followng nequalty (a c) (b b 4)[b ( 1) 2] A (b b 4)[b 2 (2 + b ) 2 ] < (a c) (b b 4)[b ( 1) 2] + A (b b 4)[b 2 (2 + b ) 2 ] always holds, consderng as above the relevant range of the parameters and the condton for real-values quanttes. Fnally, to ensure that the two ntervals always overlap, we prove that < ; that s (a c) (b b 4)[b ( 1) 2] + A (b b 4)[b 2 (2 + b ) 2 ] < (a c) (b b 4)[b ( 1) 2] A (b b 4)[b 2 (2 + b ) 2 ] always holds. The remanng nequaltes can be derved by transtvty. 5.4 Proof of Proposton 2 Proposton 2 follows drectly by Lemma 1 and 2. The two asymmetrc rms can agree on the collusve quantty belongng to the nterval where the two ndvdual partal colluson strateges overlap. 5.5 Proof of Proposton 3 Let us normalze the parameter space by settng b = 1 Proposton 2, l ASY = : b :Then, consder b < b : By < q; < qcol and, snce assumng postve real-valued outputs, we call the length of the set of collusve quanttes n the asymmetrc case,.e. l ASY = (a c) (4+b (b 1)) 2 [2( 15)+b ( 5+2b 2 ( 1)+(12+)+b (5 3(2+))]+2(4 b +b )[B 1 +B 2 ] [4+b (b 1)] 2 [(b 3) 2 +(b 1) 2 ][b 2 (2+b ) 2 +b 2 ] where: B 1 = [b (b (17 2b + (b 2) 2 8))] 1 2, and B 2 = [(7 + + b (b (11 + 2b + (b 2)) 20)] 1 2 : In the standard equvalent case, when b = b = 1 2 ; the nterval lse = q l SE 16 = (a c) 5( 25) : q s gven by Then, consderng the nequalty l ASY > l SE, (a c) cancel out and the resultng nequalty s ver ed for all the values of the parameters of the problem (b ; b ; ) and t s ndependent also from the calbraton of a and c: 13

15 The case n whch b < b can be proved analogously by consderng the length l ASY = and comparng t to l SE : The results and the proof are analogous, and can be obtaned smply by nvertng ndexes. References Berry, S., J. Levnsohn, and A. Pakes (1995). Automoble prces n market equlbrum. Econometrca: Journal of the Econometrc Socety 63 (4), Bonfrer, A., E. Berndt, and A. Slk (2006). Anomales n Estmates of Cross-Prce Elastctes for Marketng Mx Models: Theory and Emprcal Test. NBER Workng Paper. D Aspremont, C., A. Jacquemn, J. Gabszewcz, and J. Weymark (1983). On the stablty of collusve prce leadershp. The Canadan Journal of Economcs / Revue canadenne d Economque 16 (1), Dewert, W. (1980). Symmetry condtons for market demand functons. The Revew of Economc Studes 47 (3), Feuersten, S. (2005). Colluson n ndustral economcs - A survey. Journal of Industry, Competton and Trade 5 (3), Fredman, J. (1971). A non-cooperatve equlbrum for supergames. The Revew of Economc Studes 38 (1), Km, D. and R. Cotterll (2008). Cost pass-through n d erentated product markets: the case of us processed cheese. Journal of Industral Economcs 56 (1), Mastroleo, M. and L. Savorell (2009). Asymmetrc substtutablty and complementarty n ndvdual demand. MIMEO. Roas, C. and E. Peterson (2008). Demand for d erentated products: Prce and advertsng evdence from the US beer market. Internatonal Journal of Industral Organzaton 26 (1), Rothschld, R. (1999). Cartel stablty when costs are heterogeneous. Internatonal Journal of Industral Organzaton 17 (5), Sethuraman, R., V. Srnvasan, and D. Km (1999). Asymmetrc and neghborhood crossprce e ects: some emprcal generalzatons. Marketng Scence 18 (1),

16 Sngh, N. and X. Vves (1984). Prce and quantty competton n a d erentated duopoly. The RAND Journal of Economcs 15 (4),

17