Multlateral Lmt Prcng n Prce-Settng Games Eray Cumbul and Gábor Vrág, Aprl 18, 2018 Abstract In ths paper, we characterze the set of pure strategy undomnated equlbra n dfferentated Bertrand olgopoles wth lnear demand and constant unt costs when frms may prefer not to produce. When all frms are actve, there s a unque equlbrum. However, there s a contnuum of non-equvalent Bertrand equlbra on a wde range of parameter values when the number of frms (n) s more than two and n [2, n 1] frms are actve. In each such equlbrum, the frms that are relatvely more cost or qualty effcent lmt ther prces to nduce the ext of ther rval(s). When n 3, ths game does not need to satsfy supermodularty, the sngle-crossng property (SCP), or log-supermodularty (LS). Moreover, the best responses mght have negatve slopes. These results are very dfferent from those n the exstng lterature on Bertrand models wth dfferentated products, where unqueness, supermodularty, the SCP, and the LS usually hold under a lnear market demand assumpton, and best response functons slope upward. Our man results extend to a Stackelberg entry game where some establshed ncumbents frst set ther prces, and then a potental entrant sets ts prce. TOBB-ETU Unversty, ecumbul@mal.rochester.edu. I thank Tubtak for ther fnancal support. Unversty of Toronto, gabor.vrag@utoronto.ca Ths paper orgnates from Cumbul (2013). An earler verson was crculated under the ttle Non-supermodular Prce Settng Games. We would lke to thank semnar partcpants for valuable dscussons at the 2 nd Brazlan Game Theory Socety World Congress 2010, SED 2011 at the Unversty of Montréal, Mdwest Economcs Theory Meetngs 2011 at the Unversty of Notre Dame, Stony Brook Game Theory Festval 2011, the 4 th World Congress of Game Theory 2012, Blg Unversty, Istanbul, Unversty of Rochester 2010, 2011, and 2013, Unversty of Toronto 2013, Internatonal Industral Organzaton Conference 2014, EARIE 2014, Canadan Economc Theory Conference 2014, IESE-Barcelona 2014, Stony Brook Game Theory Festval 2014, and Koç Unversty Wnter Workshop 2017. We also thank Vctor Agurregabra, Alp Atakan, Paulo Barell, Erc Van Damme, Pradeep Dubey, Manuel Mueller Frank, Alberto Galasso, Srhar Govndan, Thomas D. Jetschko, Martn Osborne, Romans Pancs, Greg Shaffer, Ron Segel, Tayfun Sönmez, Adam Szedl, Wllam Thomson, Mhkel M. Tombak, Utku Ünver, and Xaver Vves for ther useful comments and suggestons.
1 Introducton In several markets, some frms may not be able to actvely partcpate, and many decde to shut down. A large amount of lterature has studed entry or ext decsons that are nduced by nformaton-based (.e., sgnalng-based) lmt prcng 1 and predatory prcng 2 practced by other frms. However, the entry and ext behavor of frms mght also be effcency-based n hghly compettve markets. Compettors cost-reducng nnovatons, the nablty to adapt to changng market condtons, a cost-effcent merger among rval frms, or frms strateges to rase rvals varable costs may nduce an exstng frm to ext or a potental entrant not to enter. Nevertheless, an nactve frm mght stll be effcent enough to lead the actve frm(s) to engage n effcency-based lmt prcng but not strong enough to enter the market. In ths paper, we study tradtonal statc prce-settng games among frms that have dfferent levels of qualty or cost effcences. The dfferences between these levels mght be due to one of the factors above. Our man am s to dentfy the set of actve and nactve frms n any pure strategy undomnated Bertrand equlbrum and to provde a full characterzaton of the equlbrum behavor of frms. 3 There are two types of equlbrum n ths game. An equlbrum s ether unconstraned or constraned (.e., lmt prcng/knked demand) f the prce decsons of the set of actve frms are unconstraned or constraned, respectvely, by the presence of nactve frm(s). We show that when all frms are actve, there s a unque equlbrum. However, f the margnal nactve frm s suffcently neffcent, then there s a contnuum of equvalent unconstraned equlbra when the number of frms (n) s greater or equal to two. If otherwse, there s a contnuum of non-equvalent constraned equlbra and the Bertrand best responses have negatve slopes n a regon for a wde range of parameter values when n 3 1 The earler lmt prcng lterature assumed that a low pre-entry prce mght deter entry because potental entrants would vew the prce as mplyng that low prces would be set postentry (e.g., Gaskns (1971), Kamen and Schwarts (1971), Baron (1973)). Mlgrom and Roberts (1982), Bagwell and Ramey (1988), Bagwell (2007), and Gedge et al. (2016) address ths ssue by ntroducng asymmetrc nformaton between the ncumbents and the potental entrant. 2 Predatory prcng means that a frm charges a prce that s below the frm s average costs wth the sole ntenton of drvng an exstng rval out of the market. 3 Such a characterzaton n statc quantty-settng games s trval. In partcular, standard exstence and unqueness results for the Cournot equlbrum extend to envronments where frms may prefer not to be actve (Novshek, 1985 and Gaudet and Salant, 1991) 1
and n [2, n 1] frms are actve. In each such equlbrum, the frms that are relatvely more cost or qualty effcent lmt ther prces to nduce the ext of ther rval(s). We also provde an teratve algorthm to fnd the set of actve frms n any equlbrum and show that ths set s the same n all equlbra. These results are very dfferent from the exstng lterature on Bertrand models wth dfferentated products, where unqueness hold under a lnear market demand assumpton and the best response functons slope upward. 4 To explan our results, consder a symmetrc three-frm dfferentated product Bertrand olgopoly where the margnal cost levels are c = ξ for = 1, 2, 3. All frms are actve; that s, ther equlbrum producton levels are all strctly postve. Suppose that a process nnovaton s avalable for frms 1 and 2. Accordngly, ther cost levels reduce to ξ = ĉ 1 = ĉ 2 < ĉ 3 = ξ. If the ntal cost level ξ s hgh enough, then there are two cutoff levels for ξ, say ξ 1 and ξ 2 wth 0 < ξ 1 < ξ 2, such that the frms equlbrum strateges are qualtatvely dfferent when ξ les n the regon [0, ξ 1 ], ( ξ 1, ξ 2 ), or [ ξ 2, ξ). More specfcally, f ξ [ ξ 2, ξ), then the level of nnovaton s not too hgh, and all three frms contnue to be actve n the market. At the other extreme, f ξ [0, ξ 1 ], then frm 3 becomes very neffcent compared to frms 1 and 2 and leaves the market. Accordngly, frms 1 and 2 charge unconstraned duopoly prces. The most nterestng regon s the ntermedate regon here, ξ ( ξ 1, ξ 2 ). Ths regon nvolves effcency-based lmt prcng nduced by frms 1 and 2 to keep frm 3 out of the market. If they gnored frm 3 and charged unconstraned duopoly prces, then frm 3 would contnue to be actve n the market. In the case of lnear demand, lmt prcng takes a partcularly smple form. Consder any prce combnaton of frms 1 and 2 such that p 1 + p 2 = M where M s unquely determned by the parameters of the model. If ether frm 1 or frm 2 charges a hgher prce, then frm 3 would start to produce, and the market would become a tropoly market. On the other hand, when ether frm decreases ts prce, the market s a duopoly market. For ths reason, the proft functons of frms 1 and 2 exhbt knks at prce combnatons where p 1 + p 2 = M. Moreover, the fact that demand s more senstve to a change n the prce that a frm sets n the regon where all three frms are actve 5 mples that the rght-hand dervatve 4 For nstance, Fredman (1977) shows that when the best response functons are contractons, costs are nondecreasng, and all frms produce mperfectly substtutable products, then there s a unque Bertrand equlbrum. 5 The reason s that when frm 1 changes ts prce n the duopoly regon (.e., where p 1 + p 2 < 2
of the proft of frm 1 wth respect to p 1 s more negatve (or less postve) than the left-hand dervatve f p 1 + p 2 = M as the demand drop s accelerated for prces where the thrd frm s actve. At such prce combnatons, the optmalty condtons for frm 1 requre the left-hand dervatve of the proft functon to be postve and the rght-hand dervatve to be negatve, whch can be satsfed by multple combnatons of p 1 and p 2 satsfyng p 1 + p 2 = M. As a result, there s a host of equlbra n our prce-settng game. Relatedly, the knk mples that the best response for frm 1 when frm 2 sets p 2 satsfes p 1 = M p 2, so the prce choces of frms 1 and 2 are strategc substtutes at such a pont. Our model has been extensvely studed n a two-frm set-up. For example, Muto (1993), Erkal (2005), and Zanchettn (2006) show that when there are two frms, there s a unque lmt prcng equlbrum, n whch the effcency gap between the two frms s suffcently hgh to rule out an nteror equlbrum, where both frms are actve, but not hgh enough to allow the most effcent frm to engage n (unconstraned) monopoly equlbrum. Ths paper generalzes the Bertrand equlbrum characterzaton results to an n-frm set-up when frms have any degree of cost and qualty asymmetres. The generalzaton of the lmt prcng equlbrum unvels a set of novel results such as the multplcty of the lmt prcng equlbra. There are several applcatons of the fndngs n the contexts of market ext after a cost-reducng process nnovaton or a cost-effcent merger 6 and of the comparsons of Cournot and Bertrand equlbra. For example, Zanchettn (2006) shows that the effcent frm s and ndustry profts can be hgher under Bertrand competton than under Cournot competton n the lmt prcng equlbrum regon. Ths fndng reverses Sngh and Vves (1984) rankng. It s clear from these arguments that the possblty of lmt prcng and multple equlbra mght gve rse to unexpected results n varous contexts. games. Our paper contrbutes to the lterature on supermodularty n prce-settng We show that under standard assumptons for demand and cost, the Bertrand best responses mght have negatve slopes, and thus, a Bertrand game wth dfferentated substtutable products may not satsfy supermodularty (Top- M), the frm s quantty responds relatvely mldly because there s only one other frm (frm 2), to whch customers dvert. In the regon where p 1 +p 2 M, any ncrease n p 1 makes customers dvert to frms 2 and 3. 6 Motta (2007) consders the possblty of a market ext after a cost-effcent Bertrand merger. Although a lmt prcng regon exsts, t has not been ponted out (Cumbul and Vrág, 2018b). 3
ks, 1979), the sngle-crossng property (Mlgrom and Shannon, 1994), or logsupermodularty f some frms produce zero output n equlbrum. Ths s n contrast wth the prevous lterature that showed that f all frms are actve n equlbrum, then the Bertrand olgopoly wth dfferentated substtutable products satsfes these propertes for a wde varety of cost and demand functons. 7,8 In partcular, n Topks (1979), Vves (1990), and Mlgrom and Roberts (1990), demand functon s assumed to be twce contnuously dfferentable. We argue that for standard demand functons, ths assumpton s satsfed only when all frms have postve producton. Gven that our game s not supermodular, the queston of the exstence and unqueness of a pure strategy equlbrum arses naturally. Fortunately, a standard fxed-pont theorem shows that a pure strategy Bertrand equlbrum exsts n our game. However, when n 3, the unqueness of equlbrum fals, and there s a contnuum of lmt prcng equlbra for a large confguraton of parameters n our model. 9,10 Such an equlbrum multplcty provdes nsghts nto how several 7 Topks (1979) shows that f the goods are substtutes wth lnear demand and costs and f the players strateges are prces constraned to le n an nterval [0, p], then the game s supermodular. Later, Vves (1990) extends the result to the case of convex costs. Buldng on Topks (1979), Mlgrom and Roberts (1990) show that there s a unque pure strategy Bertrand equlbrum wth lnear, constant elastcty of substtuton (CES), logt, and translog demand functons and constant margnal costs. 8 However, we mght have Bertrand equlbrum multplcty n the case of homogeneous products. Dastdar (1995) shows that wth dentcal, contnuous, and convex cost functons, Bertrand competton typcally leads to multple pure strategy Nash equlbra. Hoerng (2002) also fnds there s a contnuum of mxed strategy equlbra wth contnuous support. Moreover, there exsts a unque and symmetrc coaltonal-proof Bertrand equlbrum f the frms possess an dentcal and ncreasng average cost (Chowdhury and Sengupta, 2004). 9 Ledvna and Srcar (2011, 2012) and Federgruen and Hu (FH) (2015-2017b) also study prcesettng games where some frms may not produce n equlbrum. Ther set-ups cover our model wth further restrctons of postve demand ntercepts and substtutable products. Ledvna and Srcar (2011, Theorem 2.1) clam that there s always a unque undomnated Bertrand equlbrum. However, ther constraned (type 3) equlbrum s not mmune to some devatons, and thus, t s not an equlbrum of ths game. On the contrary, FH (2015, Theorem 3) clam that there s always a unque equlbrum, whch they call the component-wse smallest prce equlbrum or the specal equlbrum (FH, 2018), n our set-up. Ths equlbrum s weakly domnated as all nactve frms (f they exst) charge a prce strctly below ther margnal costs. However, our result establshes that there are nfntely many undomnated equlbra of ths game that are dfferent than the specal equlbrum when all frms are not actve (Cumbul and Vrág, 2018a). Our results have further mplcatons n the dynamc Bertrand olgopoly games (Ledvna and Srcar, 2011) and mean feld games (Chan and Srcar, 2015). 10 Our multplcty of (knked demand) lmt prcng equlbra result shows that knked demand equlbra are general and ntutve, and ratonalze prevous fndngs orgnally attrbuted to pecular characterstcs of specfc models. In partcular, Economdes (1994), Yn (2004), Cowan 4
frms may keep ther compettors out together. 11 Last, Topks (1998) argues that the log-supermodularty of demand s a crtcal suffcent condton for monotone best responses n Bertrand prce-settng games f one takes a frm to be specfed by ts unt cost (, ). However, the exstence result of non-monotonc best responses n the case of lnear demand shows that ths s not the case f a frm has a unt cost n [0, ). Ths fndng has smlartes to the fndngs of Amr and Grlo (2003). We also study varous extensons of the fndngs n the context of market entry. We consder an entry game wth establshed ncumbents and a potental entrant. In the frst stage, the ncumbents smultaneously choose ther prces. In the second stage, the entrant chooses ts prce. The lmt prcng equlbra of the assocated smultaneous move Bertrand game nclude the entry-deterrng lmt prcng equlbra of ths sequental move game. Thus, our man fndngs are robust n Stackelberg prce-settng games. Last, we show that each actve frm s proft decreases n own prce wthn the lmt prcng equlbrum set n our Bertrand and Stackelberg models. In the Stackelberg game, ths result contrbutes to the ongong debate about whether entry preventon can be seen as a publc good or not. Each frm could free-rde on the entry-preventng actvtes of ts compettors wth the potental mplcaton that there would be lttle entry deterrence. However, our fndng shows that every ncumbent would lke to contrbute to entry deterrence as much as the frm can f the entry s prevented. 12 We also show that a consumer surplus-maxmzng equlbrum prce vector can mnmze the total surplus (or the total producer and Yn (2010), and Merel and Sexton (2010) characterze the set of knked demand Bertrand equlbra between two actve frms, whch dffer only n ther locatons, n a Hotellng model of horzontal dfferentaton. Dfferent from ths lterature, for our multplcty of equlbra result, we need at least three frms, where there s at least one relatvely neffcent nactve frm. In each such equlbrum, the frms that are relatvely more cost or qualty effcent lmt ther prces to nduce the ext of ther rval(s). 11 The exstence of a contnuum of non-equvalent lmt output (.e., over-nvestment) strateges was found by Glbert and Vves (1986) n a Stackelberg quantty-settng entry game. Ths multplcty s due to the presence of the entry costs of the entrant when there are dscontnutes n the best reply and the proft functons of the ncumbents. See also Iacobucc and Wnter (2012) for a colluson-based analyss of jont excluson. 12 Ths has smlartes to the fndngs of Glbert and Vves (1986), where each ncumbent would lke to over-nvest to deter entry. However, some prevous authors, who have hghlghted the publc good aspect of noncooperatve entry preventon, would nclude Bernhem (1984), Waldman (1987,1991), Appelbaum and Weber (1992), and Kovenock and Roy (2005). 5
surplus) among the set of lmt prcng equlbrum prce vectors. In Secton 2, we descrbe the model. In Secton 3, we provde the Bertrand equlbrum analyss and our man results. In Secton 4, we provde the connecton between our results and the concepts of supermodularty, the sngle-crossng property, and log-supermodularty. In Secton 5, we dscuss the robustness of our results n dfferent models of horzontal dfferentaton. We also provde an extenson of the results n the case of sequental moves. In Secton 6, we study the welfare propertes of the Bertrand and Stackelberg equlbra. 2 The Bertrand Model Let N = {1, 2,..., n} be a fnte set of frms. Each frm N produces a sngle product (or provdes such a servce) at constant margnal cost c wthout ncurrng fxed costs. 13 Next, we descrbe the demand sde of the economy. The representatve consumer has an exogenous ncome I and maxmzes the utlty: U = A q λ q 2 λθ q q j (1) 2 N N N j> subject to y + N p q I, where the prce of the numerare good, y, s normalzed to 1, A s the exogenously gven measure of the qualty of varety n a vertcal sense, 14 θ ( 1, 1) s an nverse measure of product dfferentaton, n 1 q s the producton level of frm, and λ > 0 s the slope of the demand curve. Note that U(.) s strctly concave at θ ( 1, 1) and λ > 0 (Amr et al., 2017). n 1 Gven a prce vector p N = (p ) N = (p 1, p 2,..., p n ), the consumer wll consume a strctly postve amount of some s(p N ) products, whch are offered by the frms n set S(p N ) N. 15 The frst-order condton (FOC) of the consumer s problem 13 Our man results wll contnue to hold f we allow for avodable fxed costs. A formal analyss can be provded to the reader upon request. 14 For the nterpretaton of ths parameter, we follow Häckner (2000) and Martn (2009). Other thngs beng equal, an ncrease n A ncreases the margnal utlty of consumng good. 15 Throughout the paper, bold letters show that the consdered varable s wrtten n the vector form. 6
yelds that for all products that are consumed n a non-negatve quantty, p = A λq λθ q j. (2) j S\ When θ = 1 (θ = 1 resp.) and A s 1 = A j, j, all products are perfect substtutes (complements resp.). When θ = 0, each frm s a monopoly for the good the frm produces. To obtan non-trval results, we assume that A > c. Let also the qualty-cost dfferental be defned as δ = A c. Wthout loss of generalty, assume that δ 1 > δ 2 >... > δ n. 16 It s not easy to see at ths step that the s products, whch wll be consumed by the consumer, have the hghest qualty-cost dfferentals,.e., S = {1, 2,..., s}, n any equlbrum of ths game. 17 Solvng (2) for the quanttes yelds q = D S (p S ) = a,s b s p + d s p j, (3) j S\ where p S = (p ) S, a,s = (b s + d s )A d s S A 1+θ(s 2), b s =, and λ(1 θ)(1+θ(s 1)) θ d s =. Note that the demand ntercept a λ(1 θ)(1+θ(s 1)),s can be negatve and the products can be complements unlke Federgruen and Hu (2015-2018). Gven a prce vector p N, one can calculate the proft of each frm N as follows. The proft of frm, π (p N ) s equal to 0 f N\S(p N ). The proft of S(p N ) can be wrtten as π S (p N ) = (p c )(a,s(pn ) b s(pn )p + d s(pn ) 3 Equlbrum Analyss j S(p N )\ p j ). (4) Each frm sets ts prce p smultaneously, knowng all the cost and demand parameters of the game. A pure strategy equlbrum of the Bertrand game requres 16 All our results are vald for the case where some of the qualty-cost dfferentals are equal, as we assume n some examples, but the notaton becomes much more burdensome; therefore, we do not cover ths case formally. 17 A full characterzaton of S for any prce vector s not necessary at ths pont. We use the relevant propertes of S when we proceed wth our analyss. 7
that for all N t holds that p arg max x π (x, p ) where we let p be the vector of the prces set by all frms other than. Followng the lterature, we can argue that weakly domnated strateges are not credble n a one-shot Bertrand game. Thus, we assume p [c, A ] to characterze undomnated Bertrand equlbra and gnore the actons below the margnal cost levels unlke Federgruen and Hu (2015-2018). Let ( p, q ) denote an equlbrum prce-quantty par of frm. Let S N be the set of actve frms wth the cardnalty of S beng s at a gven prce vector p N = (p ) N, where p [c, A ], based on (3). Let h = arg max N\S δ and S = S {h}. In the Appendx, we show that the proft of frm s quas-concave wth respect to the prce of frm, and thus, a pure strategy equlbrum exsts. In partcular, when frm s actve, the frst dervatves of q and π wth respect to p ether exst and are non-ncreasng n p or the rght-hand dervatves are less than the left-hand dervatves. Moreover, the proft and the demand become zero when p s suffcently large. Therefore, the proft and demand functons are quas-concave and sngle-peaked n p. Ths argument shows that both functons exhbt a knk at the pont when a new frm becomes actve because at such a pont the demand of becomes more senstve to changes n s prce (that s, b s +1 > b s ) because the consumer may dvert to more frms than before (see Fgures 1a and 1b). Lemma 1. ) The demand and proft functons of each actve frm are snglepeaked n own prce, and the functons are knked when ther prces ht the crtcal set where a new frm starts postve demand by prcng above the margnal cost. ) The proft functon π s contnuous and quas-concave n p when p [c, A ] and q 0. Consequently, there exsts a pure strategy Bertrand-Nash equlbrum. Proof: All proofs are provded n the Appendx unless otherwse stated. Our analyss s conducted n the followng steps. Frst, we study a smpler game and gnore the non-negatvty constrant for the output levels. In effect, we use (3) to calculate the demand even f q < 0 for some S. In Lemma 2, we fnd the equlbrum of ths modfed game, whch we call a relaxed equlbrum. In the next step, we mpose the non-negatvty constrants to fnd the necessary condtons for the equlbra of the orgnal game (see Lemma 3). Then, n Lemma 4, we propose an teratve algorthm to fnd the frms that are actve n the equlbrum of the 8
orgnal game. Fnally, we characterze the equlbrum prces and quanttes n Proposton 1. To provde a defnton of a relaxed equlbrum, we use (3). In the S-frm market, a prce vector p S (S) = (p (S)) S 0 s a relaxed Bertrand-Nash equlbrum f for each S t holds that p (S) = arg max (x c )(a,s b s x + d s p j ). (5) x j S\ Gven our lnearty assumptons, there s a unque relaxed equlbrum, whch can be found from the FOCs by dfferentatng (5) wth respect to x and settng the dervatve to zero. The best response of frm S s then gven as BR S : R s 1 R s.t. BR S (p S\ ) = a,s + d s j S\ p j + b s c 2b s, (6) where p S\ = (p j ) j S\{} s the prce vector that does not contan the th dmenson. Assumng that all frms best respond, we obtan the relaxed equlbrum prce and the quantty levels as stated n the followng lemma. Lemma 2. Let S N. ) The unque relaxed equlbrum prce and the quantty strateges of frm S are gven by p (S) = δ (1 + θ(s 1))(2 + θ(s 3)) θ(1 + θ(s 2)) S δ (2 + θ(s 3))(2 + θ(2s 3)) + c (7) and q (S) = b s (p (S) c ). (8) ) q (S) > q j (S) f and only f δ > δ j. If all frms are actve, then Lemma 2-) unquely characterzes the prce and quantty strateges of frms for S = N. Ths fndng s consstent wth the unqueness of equlbrum characterzatons of Farahat and Peraks (2010) and Federgruen and Hu (2015-2018) when all products receve postve demand n a mult-product settng wth substtutable products. In the case of complementary or monopoly products,.e., θ ( 1, 0], all frms are actve and the equlbrum n 1 prce and quantty strateges of frms are unquely gven by (7) and (8). Therefore, 9
we assume θ (0, 1) from now on. An mmedate concluson from Lemma 2-) s that frms that have hgher qualty-cost dfferences produce more than frms that have relatvely lower qualty-cost dfferences n the relaxed equlbrum. We next derve the equlbrum strateges of frms when there s at least one nactve frm. We now mpose the constrant that the output of each frm s non-negatve. Frst, we derve a condton that ensures that f the set of actve frms n the market s S, then frm h does not want to enter. Our startng pont s that when frm h s nactve, any frm N\S that s less effcent than frm h can be gnored for the analyss as those frms are also nactve. Consequently, the demand that frm h faces when t sets p h = c h and takes p S as gven follows from (3): D S h (p S, p h = c h ) = b s +1δ h + d s +1( S p S A ). (9) It s clear that frm h can be actve (produce q h > 0) f and only f D S h (p S, p h = c h ) > 0 because otherwse even f frm h charges ts break-even prce c h, the frm faces a non-postve demand. Let us derve the necessary condtons for an equlbrum where only frms n S are actve. Lemma 3. If the set of actve frms s S (that s, q > 0 f and only f S ) n an equlbrum, then one of ) or ) holds 18 : ) (unconstraned equlbrum) If Dh S (p S (S )) < 0 and Dh S (p S (S ), p h = c h ) 0, then for all S, p = p (S ), q = q (S ); ) (lmt prcng equlbrum) If Dh S (p S (S )) < 0 and Dh S (p S (S ), p h = c h ) > 0, then Dh S ( p S (S ), p h = c h ) = 0. Lemma 3 shows that there are two possble types of equlbra of whch exactly one type occurs for any parameter values. In an unconstraned equlbrum, the actve frms, S, charge the prces they would f no frms other than the actve frms exsted n the market. If the most effcent nactve frm (frm h) receves a non-postve demand, then the actve frms are unconstraned, and they 18 A knfe-edge case may also occur f frm h produces exactly zero when t nteracts wth frms n S n the relaxed equlbrum (.e., Dh S (p S (S )) = 0). In ths case, all frms n S are actve when they charge ther relaxed equlbrum prces n the market of frms n S and h; that s, for all S, p = p (S ) and q = q (S ). It can be shown from (8) that for each S, q (S h) q (S ), whch explans why we consder ths knfe-edge case as a possblty. 10
charge ther relaxed equlbrum quanttes n the S -frm market,.e., p = p (S ) (part )). However, t mght also be the case that frm h faces postve demand. In a lmt prcng (or constraned) equlbrum (LPE), the actve frms are constraned by the presence of frm h. Thus, they lmt ther unconstraned equlbrum prces to some p S such that frm h receves exactly zero demand (part )). Ths elmnates the producton ncentve of frm h. The result s ntutve because f frm h was not on the verge of enterng but was out of the market, then the actve frms would not be constraned by frms not n S when consderng small devatons. Next, we provde an algorthm that constructvely fnds the set of actve frms, namely, N, n any equlbrum of ths game. Lemma 4. Apart from the knfe-edge case, the set of actve frms s N n any equlbrum 19, where N s the set dentfed by the followng Bertrand teraton algorthm (BIA) for N = {1, 2,..., }. STEP 1: If q2(n 2 ) < 0, then N = N 1. Otherwse, proceed to the next step.. STEP : If q +1(N +1 ) < 0, then N = N. Otherwse, proceed to the next step.. STEP n 1: If qn(n) < 0, then N = N n 1. Otherwse, N = N. In the proof of ths lemma, we show that there cannot be an equlbrum when the set of actve frms s not N (apart from the knfe-edge case). But snce an equlbrum exsts by Lemma 1-), the set of actve frms should be N n any equlbrum (unconstraned or LPE). We provde an ntuton for why an equlbrum does not exst, n whch some frm s actve, but a more effcent frm j compared to frm (.e., δ j > δ ) s not. We show that f the proposed actve frms play ther unconstraned relaxed equlbrum strateges, then there s demand left for frm j. If they constrant ther prces so that frm j does not produce, then there always exsts a frm n the set of proposed actve frms such 19 qn (N ) = 0 s the knfe-edge case. In such a case, the set of actve frms s N \ n. 11
that t fnds a proftable prce devaton ncentve. Ths argument shows that only the most effcent frms can produce n an equlbrum. We next derve our set of equlbra when the set of actve frms s N. A necessary condton for an LPE to occur s that p n +1 = c n +1 and DÑn +1( p 1, p 2,..., p n, p n +1 = c n +1) = 0, or equvalently, by (9), the equlbrum prces of frms n N sum to a constant 20 : Condton 1: N p = M = A (1 + θ(n 1))δ n +1 θ N whch means that frm n + 1 s ndfferent about beng actve or not., (10) Condton 1 s not suffcent for an LPE to exst. For each N, the followng also needs to hold: Condton 2: π Ñ,R p pn, p n +1 =c n +1 0 and πn,l p pn 0, (11) where R and L denote the rght- and left-hand dervatves, respectvely, of the relevant functon. The frst dervatve n Condton 2 translates to p p B, whle the second translates to p p B, where and p B = (1 + θ(n 1))(δ δ n +1) 2 + θ(2n 3) p B = (1 + θn )(δ δ n +1) 2 + θ(2n 1) + c (12) + c (13) as we show n the proof. In Proposton 1, we also use the followng crtcal values for δ n +1: δ B n +1 = θ(1 + θ(n 2)) N δ (1 + θ(n 1))(2 + θ(n 3)) (14) and δ B n +1 = θ(1 + θ(n 1)) N δ θ 2 + (1 + θn )(2 + θ(n 3)). (15) 20 The equlbrum prces sum to a constant because each product s symmetrcally dfferentated from the rval products n our lnear model. In general, the condton s that the entrant s ndfferent between enterng or not, whch defnes a functonal on the prces of other frms. 12
We now state the two man propostons of our paper. The frst one characterzes all pure strategy equlbra of the game (both unconstraned and lmt prcng) when there s at least one nactve frm. Proposton 1. Let there be at least one nactve frm. ) A pure strategy unconstraned equlbrum exsts f and only f δ n +1 δ B n +1. In such an equlbrum, each frm N charges prce p = p (N ) and produces q = q (N ), whle each frm N \ N charges p c and produces q = 0. ) A prce vector ( p 1, p 2,..., p n ) s a pure strategy LPE prce vector for actve frms f and only f t satsfes (10) and p [p B, pb ] for all n. In each such equlbrum, p n +1 = c n +1 and q n +1 = 0; and for each > n + 1, p c and q = 0. ) A pure strategy LPE exsts f and only f δ n +1 I B = (δ B n +1, mn{δ B n +1, δ n }). In part ), we prove the condtons under whch an unconstraned equlbrum exsts, and we provde a full characterzaton when there s at least one nactve frm. When the margnal nactve frm s qualty-cost gap s suffcently low, the actve frms play the same equlbrum strateges that they would n a world n whch the nactve frms dd not exst and thus dsregard these strateges. Ths case has the same flavor as the occurrence of blockaded entry n standard entry deterrence models. The FOCs n the N -frm market hold wth equaltes, and the prce decson of each actve frm N s unquely determned as p (N ). As each nactve frm s ndfferent between chargng any prce above ts margnal cost level, there s a contnuum of equvalent pure strategy equlbra n the sense of generatng dentcal sales and proft volumes. In parts ) and ), we provde two characterzatons of the LPE prce vectors of any gven game. Condtons 1 and 2 are necessary and suffcent for an LPE to exst as they elmnate the devaton ncentves of all frms. Snce the frst condton holds, the proft functon of frm N exhbts a knk n p at the canddate equlbrum prce vector as we dscussed before Lemma 1. Moreover, whle rulng out devatons to a lower prce (to undercut actve frms) formally translates nto the condton that the left-hand dervatve of the proft functon of an actve frm s non-negatve, rulng out devatons to a hgher prce (allowng a new frm to enter) formally translates nto the condton that the rght-hand dervatve of the proft functon of an actve frm s non-postve by Condton 2. 13
In the two-frm case, when n = 1, there s a unque LPE, whch s gven by ( p 1, p 2 ) = (A 1 δ 2 θ, c 2 ), by part ). Ths fndng n the duopoly market concdes wth the LPE characterzatons of Muto (1993) and Zanchettn (2006), respectvely, when there are only cost asymmetres (A 1 = A 2 = A); and δ 1 = 1, δ 2 (0, 1], and λ = 1. Our results extend the analyss to an n-frm framework by allowng cost and qualty asymmetres. In part ) of ths proposton, we fx the qualty-cost dfferences of frms apart from frm n + 1 as δ > δ j for < j. If δ n δ B n +1, then an LPE exsts f and only f δ n +1 (δ B n +1, δ B n +1). Ths characterzaton result proves that lmt prcng equlbra occur for a large set of parameter confguratons. In the Appendx (Proposton 7), we also provde varous comparatve statcs about the senstvty of lmt prcng strateges to the degree of substtutablty (θ). These results generalze and confrm known results of the duopoly case. For example, only frm 1 s actve n the case of homogeneous products wth asymmetrc costs. When A = A j for j, as θ 1, frm 1 s LPE prce converges to c 2 by Condton 1. Our second man result s the exstence of multple non-equvalent effcencybased lmt-prcng equlbra when there are at least three frms and n [2, n 1] of them are actve. Ths result follows from Proposton 1, but due to ts mportance, we state t as a separate result. Proposton 2. Assume that an LPE exsts. Each frm > n s nactve. There s a contnuum of non-equvalent LPE prce vectors ( p 1, p 2,..., p n ) for actve frms N when n 3 and n [2, n 1]. The LPE prce of frm 1 s unque when n = 1. When there are at least two actve frms and at least one nactve frm (n [2, n 1]), there s a contnuum of non-equvalent LPE prce vectors for actve frms N n the sense of generatng dfferent sales and proft volumes for them. The relatvely more effcent frms lmt ther prces n multple ways to nduce the ext of ther rval(s). Ths multplcty s drven by the fact that when more than one effcent frm engages n lmt prcng, the strategc nteracton among these actve frms becomes domnated by the ncentve of keepng the potental entrant out of the market whle stealng most of the potental entrant s demand. The knks n the proft functons (and thus, the multplcty of lmt prcng equlbra) arse from ths ncentve, whch s stronger than the usual ncentve of stealng the rvals demand. The exstence of multple equlbra n smple lnear Bertrand 14
models s n sharp contrast wth the prevous lterature, whch found a unque Bertrand equlbrum for a large class of demand functons. 21 Last, a smple numercal example helps fx these deas. Example: Let there be three frms, namely, N = {1, 2, 3}. Let (A 1, A 2 ) = (23, 23) and (c 1, c 2 ) = (2, 2). Thus, the qualty-cost dfferences of frms 1 and 2 are δ 1 = δ 2 = 21. Let the nverse demand be p = A q 0.8(q j + q l ), where, j, l = 1, 2, 3 and j l. The demand parameters are a 1,2 = a 2,2 = 115 b 2 = 25, d 9 2 = 20, b 9 3 = 45, and d 13 3 = 20. We dfferentate three cases: 13 Case 1: (Unconstraned tropoly) Let δ 3 (δ B 3, δ 1 ] = ( 756, 21]. Usng (7) 47 and (8), a three-frm relaxed equlbrum calls for q1(n) = q2(n) = 27(203 6δ 3), 286 q3(n) = 9(47δ 3 756), and p 286 (N) = c + 13q (N), = 1, 2, 3. By our ntal assumpton, 45 all frms are actve n the relaxed equlbrum, and thus, ths equlbrum s the unque equlbrum of ths case. Case 2: (Unconstraned duopoly) Let δ 3 [0, δ B 3 ] = [0, 140 ]. In ths case, there 9 s a contnuum of equvalent duopoly equlbra. In each such equlbrum, p 1 = p 2 = p 1({1, 2}) = p 2({1, 2}) = 5.5, and q 1 = q 2 = q 1({1, 2}) = q 2({1, 2}) = 175 18 by (7) and (8). Then D N 3 (p 1 = 5.5, p 2 = 5.5, p 3 c 3 ) 5(9δ 3 140) 13 0 by (3) and the startng assumpton of Case 2. Therefore, Lemma 3-) mples that unconstraned duopoly equlbra exst. Case 3: (Lmt prcng) Let δ 3 I B = (δ B 3, δ B 3 ) = ( 140 9, 756 47 ) and DN 3 (p 1 = 5.5, p 2 = 5.5, p 3 = c 3 ) = 5(9δ 3 140) 13 > 0. Therefore, substtutng from (3) and usng Lemma 3-), we have D 3 ( p 1, p 2, p 3 = c 3 ) = 0 or p 1 + p 2 = 184 9δ 3 4. For example, let A 3 = 25 and c 3 = 9 so that δ 3 = 16 I B. Thus, a 1,3 = a 2,3 = 75 13. Usng Condtons 1 and 2, there exsts a contnuum of non-equvalent lmt prcng equlbra n whch frms 1 and 2 are actve and the equlbra are of the form 22 109 22 p 1, p 2 111 22 and p 1 + p 2 = 10 and p 3 = c 3 = 9. (16) To explan equlbrum ncentves, take the trple (5, 5, 9), whch consttutes an LPE. For frm 1, t s not worth chargng a prce hgher than 5 because the rght- 21 Ledvna and Srcar (2011, 2012) clam the unqueness of the Bertrand equlbrum n our set-up. They argue that frm n + 1 charges at ts margnal cost n an LPE. However, one also needs to make sure that frm n + 1 produces a zero output. 22 There are only lmt prcng equlbra n ths case. 9, 15
hand dervatve of ts proft s negatve at (5, 5, 9) as Π R 1 p 1 = D N 1 (p 1 = 5, p 2 = 5, p 3 = 9) 3b 3 = 10 3 45 13 = 5 13 < 0. (17) It s not worth chargng a lower prce ether because the left-hand dervatve of ts proft s postve at (5, 5, 9) as Π L 1 p 1 = D {1,2} 1 (p 1 = 5, p 2 = 5) 3b 2 = 10 3 25 9 = 5 3 A symmetrc argument holds for frm 2. > 0. (18) The knks n the demand and proft functons of frms 1 and 2 are the key propertes that make multple equlbra possble (see Fgure 1b). We wll derve the assocated best responses of frms 1 and 2 n the next secton. Dscusson: The specal equlbrum of Federgruen and Hu (2015-2018) s the prce vector ( 461 94, 461 94, A 3 756 47 ) when δ 3 [0, 756 47 ).23 Clearly, p 3 = A 3 756 47 < c 3, and therefore, ths equlbrum s a weakly domnated LPE n Cases 2 and 3 (Cumbul and Vrág, 2018a). It exsts only f p 3 0 or A 3 756. However, n Case 1, the 47 specal equlbrum concdes wth our equlbrum when all frms are actve. 4 Supermodularty In ths secton, we show that (log-)supermodularty and the sngle-crossng property may not hold f some frms are not actve n equlbrum, and the best responses may have negatve slopes n a regon. 24 Proposton 3. Let n 3. A lnear Bertrand model wth contnuous best responses may not be supermodular or log-supermodular or satsfy the sngle-crossng property. Moreover, the best responses may be non-monotone. We prove ths result n the Appendx buldng on Case 3 of the example n 23 To fnd the specal equlbrum n Cases 2 and 3, we only decrease the margnal cost level of frm 3 to a hypothetcal level so that t produces exactly 0 quantty n the relaxed equlbrum wth three frms. 24 Roberts and Sonnenschen (1977) and Fredman (1983) provde examples of nonsupermodular dfferentated Bertrand duopoles where an equlbrum does not exst n pure strateges. Unlke our case, the non-supermodular examples of these artcles feature dscontnutes n the best responses. 16
Secton 3 for A 3 = 25 and c 3 = 9. Snce supermodularty mples monotone best responses, t s suffcent to show that the best reples of frms 1 and 2 may be non-monotone n each others prce choces for our non-supermodularty result. Smlarly, log-supermodularty and the SSC property fal to hold n ths example as we show n the Appendx. Upon fxng p 3 = c 3 = 9, for each, j = 1, 2, j, the best response of frm s pece-wse lnear, non-monotone, and contnuous and gven by 3.3 + 0.4p j f p j < 67 14 p = BR (p j ) = 10 p j f 67 p 14 j 111 22 69+4p j otherwse. 18 The frst and thrd formulas n BR (.) correspond to the duopoly and tropoly best responses of frm, respectvely, by (6). Note that both best responses are ncreasng n the other actve frm s prce. The mddle case corresponds to the lmt prcng regon dscussed. In that regon, the best response of each actve frm s decreasng n the other actve frm s prce, whch mples the non-monotoncty result. Observe n Fgure 2 that the best responses ntersect along the segment seg[cd] = {(p 1, p 2 ) R 2 : 109 22 p 1 111 22 and p 1 + p 2 = 10}. That s, each ( p 1, p 2, p 3 ) R 3 + such that ( p 1, p 2 ) seg[cd] and p 3 = c 3 = 9 s a pure-strategy equlbrum of ths game as already derved n the prevous secton. 5 Dscussons and Extensons 5.1 Robustness of results Our multplcty of (knked demand) lmt prcng equlbra and the strategc substtutes mode of competton results are drven by the fact that the strategc nteracton among the actve frms becomes domnated by the ncentve of keepng the potental entrant out of the market. The knk n the demand and proft functons (thus, the multplcty of lmt prcng equlbra) and the nonsupermodularty results derve from ths ncentve. We would expect a smlar ncentve effect to produce a multplcty of equlbra and competton n strategc modes of competton n any model where, n some parameter regon, prce competton between actve frms (producng substtutable products) may cause an 17
outflow of the total market demand served by the competng frms. Ths suggests that most of the results should (qualtatvely) generalze to other specfcatons of the model, e.g., non-lnear and less symmetrc (.e., θ rather than θ n (2)) specfcatons of demand, non-lnear or avodable fxed costs, non-strctly rankable qualty-cost gaps, dfferent order of players moves (see the next secton), or even to dfferent models of horzontal dfferentaton (.e., Hotellng model). For nstance, Glbert and Vves (1986) found smlar lmt output equlbra and the strategc substtutes mode of competton results when the fxed cost dsadvantage of the potental entrant s not too drastc compared to the ncumbent frms n a Stackelberg quantty-settng game. In general, the presence of an nactve product s necessary for our multplcty of lmt prcng equlbra and the nonsupermodularty results. In our companon paper, we show that all of our man results are robust n a mult-product settng, where all frms are actve but some products do not receve postve demand even n a duopoly market (Cumbul and Vrág, 2018b). It s useful to pont out the smlartes between our results and the early results n the Hotellng lterature. For example, Merel and Sexton (2010) characterze the knked demand and proft Bertrand equlbra wthn the lnear Hotellng duopoly model wth fxed (extreme) frm locatons. They also show that prce competton turns nto competton n strategc substtutes under certan parameter condtons. The outflow n the overall demand comes from uncoverng the market nstead of a relatvely less effcent frm, but the deep ntuton of ths and our results are qute smlar. Both results are two applcatons of knked demand theory n dfferent models of horzontal dfferentaton. Our results, therefore, are general and ntutve and ratonalze prevous fndngs attrbuted to pecular characterstcs of specal models. 5.2 Sequental market entry In ths secton, we test the robustness of our results to the order of moves. We also follow our orgnal model s prelmnary assumptons and notaton. We consder the followng sequental move ncumbents and entrant game wth complete nformaton. Let N = {1, 2,..., n } denote the set of actvely partcpatng ncumbents n an establshed Bertrand olgopoly; that s, we assume 18
δ n > θ(1+θ(n 2)) N δ by Lemma 2-). Consder now that the threat of entry (1+θ(n 1))(2+θ(n 3)) by frm n + 1 appears. The Stackelberg game has two stages. 25,26 levels. Stage 1: The ncumbents smultaneously and ndependently set ther prce Stage 2: The potental entrant chooses whether or not to enter. If the frm decdes to enter the market, then the frm sets ts prce, takng the ncumbents prces as gven. For smplcty, assume there are no fxed costs of entry. We search for subgame perfect equlbra. Let us defne two crtcal cutoff values for the qualty-cost dfference of the entrant as δ SP n +1 and δ SP n +1, where δ SP n +1 = δ B n +1 and δ SP n +1 = θ(2 θ 2 + 2θ(n 1)(2 + θ(n 1))) N δ (1 + θ(n 1))(4 3θ 2 + 2θ(n 1)(3 + θ(n 2))). (19) In the Appendx, we show that there are three crtcal regons to consder assumng that δ SP n +1 < δ n : ) entry s blocked f δ n +1 < δ SP n +1, or ) entry s prevented through effcency-based lmt prcng f δ n +1 I SP = (δ SP n +1, δ SP n +1), or ) entry s allowed f δ n +1 > δ SP n +1. We further show that an LPE exsts f and only f Condton 1 and a strcter condton than Condton 2 holds. Thus, we obtan the followng proposton. Proposton 4. Let δ n +1 I SP = (δ SP n +1, mn{δ SP n +1, δ n }). Each LPE of the sequental move Stackelberg prce-settng game above s also an LPE of the smultaneous move Bertrand game among N {n + 1} players. In lght of Propostons 1 and 4, there s a contnuum of Stackelberg lmt prcng equlbra, and the Stackelberg prce-settng game may not be supermodular when n 2 and the entrant s nactve by Proposton 6 of the Appendx. 25 See Glbert and Vves (1986) for a smlar entry deterrence game, where the ex-ante symmetrc ncumbents choose outputs rather than prces n a homogeneous good set-up. 26 Kübler and Müller (2002) argue that prce leadershp s common n the U.S. cgarette ndustry durng the late 1920s and early 1930s, the German dye ndustry, the U.S. automoble ndustry after 1950, and the U.S. market for ready-to-eat cereals. Ther expermental results suggest that the Stackelberg equlbrum predctons are approprate under prce competton. 19
6 Lmt Prcng and Market Performance For the welfare analyss, t s mportant to know how equlbrum multplcty affects consumer welfare, producer surplus, and total welfare n the descrbed Bertrand and Stackelberg prce-settng games. Based on Propostons 1 and 6, an equvalent way of wrtng the set of LPE prces of frm N s provded n the followng corollary. Corollary 1. Let n 2 and w = B, SP represent the Bertrand or Stackelberg prce-settng game played among N {n + 1} players. Let M be gven by (10). The set of the LPE prces of frm N n the w game s gven by K w = { p such that p [max{p w, M p w j }, mn{p w, M p w }]}. j j N \ j N \ The queston, then, s whch prces n the set of the LPE prce vectors of K w maxmze the well-beng of dfferent agents of the economy. 6.1 The surplus of ndvdual producers Our thrd man result shows that n the set of LPE prces, each actve frm prefers the equlbrum where the frm s prce s the lowest (and the other frms total prce s the hghest). Proposton 5. Let w = B, SP. Frm prefers to charge ts lowest LPE prce,.e., max{p w, M j N \ pw j }, among the set of the LPE prces of frm (K w ) n the w game. Ths result s ntutve as t states that each frm s proft s ncreasng n the other frms total prce (ther products beng substtutes). Therefore, n the set of equlbrum prce vectors K w, each frm naturally prefers the other frms to charge as hgh a prce as s consstent wth equlbrum. Ths observaton mples that each frm has a strong nterest n keepng out weaker rvals by chargng a low prce tself, and thus, a free-rder problem s not assocated wth ths jont preemptve behavor. Ths result n the Stackelberg game also contrbutes to the ongong debate of whether entry preventon can be seen as a publc good or not. If any frm sets 20
ts prces suffcently low to prevent entry, then all frms are protected from competton. Thus, each frm could free-rde on the entry-preventng actvtes of ts compettors wth the potental mplcaton that there would be lttle entry deterrence. Our result shows that ths s not the case n our model. Every ncumbent would lke to contrbute to entry deterrence as much as the frm can gven that the entry wll be prevented. 27 Ths s n contrast to typcal publc good provson problems. 6.2 Consumer and total surpluses For smplcty, let n = 2 from now on. The results for the consumer surplus (CS), the total producer surplus (TPS), and the total surplus (TS=the sum of the CS and the TPS) are less straghtforward. By Proposton 8 of the Appendx, we show that extreme prces maxmze the CS because the CS s convex n the prces. Moreover, dependng on the parameter values, the TPS and the TS may be ether a corner soluton or an nteror prce vector where the two actve frms charge prces that are more symmetrc by Proposton 9 of the Appendx. Wthout gong nto detals here, we would lke to provde an ntuton about why the TPS or the TS may be maxmzed at an nteror or at a corner soluton n the set K w, where w = B, SP. We also dscuss how the TS and the CS may be maxmzed at smlar or dfferent prces n K w. Case 1. When the two actve frms are symmetrc (A 1 = A 2 and c 1 = c 2 ), then the TPS and the TS are maxmzed when the two frms charge an equal prce (p, p ). As we already argued, the CS s always maxmzed at a corner prce vector of K w. Moreover, we show n the Appendx that the CS s mnmzed at the symmetrc prce vector (p, p ) (see Fgure 3a). Consequently, we have an nterestng case where maxmzng the CS and the TS (or the TPS) yelds polar opposte recommendatons. In partcular, f the actve frms are able to coordnate how they wsh to keep out potental rvals, then they would choose the equlbrum wth equal prces, whch yelds the lowest CS and the hghest TS. We provde a geometrc nterpretaton of ths argument n Fgure 3a wth two addtonal remarks. Frst, f there are multple equlbra when the actve frms keep a rval out, then t s better for the CS f they choose 27 Ths fndng resembles the result of Glbert and Vves (1986) n a homogenous good Stackelberg quantty-settng game. 21
dfferent prces. For example, f the actve frms alternate over tme n terms of choosng prce combnatons so that the rval s kept out, then ths behavor enhances the CS. Second, as δ 3 changes n the range (δ 3, mn{δ 3, δ 2 }) where the equlbra are constraned, there are equlbra for a hgher value of δ 3, whch make the consumers better off than some equlbra that occur when δ 3 s lower. Case 2. If the two actve frms are asymmetrc enough (.e., δ 1 ( (2 θ)(1+2θ)δ 2 (2 θ 2 )δ 2 θ 2+θ, )), 28 then the above dchotomy dsappears. All welfare measures are maxmzed at extreme prces as the frms are now dfferent enough that t s better for all groups to let the more effcent (attractve) frm produce as much as possble (see Fgure 3b). 7 Concluson In ths paper, we characterzed the set of equlbra n a statc olgopolstc model of prce competton wth dfferentated products and constant margnal costs. We dd not mpose any restrcton on the strctly rankable qualty-cost effcency gaps among frms. We showed that there are essentally two cases, the unconstraned case and the lmt prcng (constraned) case. In the former case, the actve frms play the same equlbrum strateges they would n a world n whch nactve frms dd not exst. In the latter case, the frms prce low enough to ensure that the margnal frm does not become actve. Thus, n ths case the presence of nactve frms stll nfluences the behavor of the actve frms and cannot be gnored by the actve frms. We frst showed that for an exogenously gven set of parameters of the model, the set of actve frms s unquely determned; thus, the product assortments are dentcal n any equlbrum of ths game. In the unconstraned case, each equlbrum of the game s equvalent n the sense of generatng dentcal sales and proft volumes. However, n the lmt prcng case, the proft and demand functons of the actve frms have knks when evaluated at an equlbrum. We show that f there s more than one actve frm, there s a contnuum of nonequvalent lmt prcng equlbra n the sense of generatng dfferent sales and proft volumes for the actve frms. In any such equlbrum, a devaton to a lower prce would attract lmted shfts n quanttes from the competng frms 28 See Proposton 9 of the Appendx for more dscusson for the Bertrand game. 22