On Tacit Collusion among Asymmetric Firms in Bertrand Competition

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1 On Tact Colluson among Asymmetrc Frms n Bertrand Competton Ichro Obara Department of Economcs UCLA Federco Zncenko Department of Economcs UCLA November 11, 2011 Abstract Ths paper studes a model of repeated Bertrand competton among asymmetrc frms that produce a homogeneous product. The dscountng rates and margnal costs may vary across frms. We dentfy the crtcal level of dscount factor such that a collusve outcome can be sustaned f and only f the average dscount factor wthn the lowest cost frms s above the crtcal level. We also characterze the set of all effcent collusve equlbra when frms dffer only n ther dscountng rates. Due to dfferental dscountng, mpatent frms gan a larger share of the market at an earler stage of the game and patent frms gan a larger share at a later stage n effcent equlbrum. Although there are many effcent collusve equlbra, our model provdes a unque predcton n the long run n the sense that every effcent collusve equlbrum converges to the unque effcent statonary collusve equlbrum wthn fnte tme. JEL Classfcaton: C72, C73, D43. Keywords: Bertrand Competton, Colluson, Dfferental Dscountng, Repeated Game, Subgame Perfect Equlbrum. 1 Introducton The model of repeated Bertrand competton explans how frms may be able to collude and sustan a hgh prce even when they produce dentcal goods. Thus t resolves so called Bertrand paradox, whch would arse n one-shot nteracton, that frms lose any monopoly power and make no proft as soon as two frms are present n the market. (Trole [7]). 1 Snce t s a smple and very convenent model, t has been used n numerous appled works. However, we stll do not fully understand when and how colluson can be sustaned except for the very specal case where frms are symmetrc. Ths assumpton of symmetrc frms s of course very strong and unrealstc; frms n general dffer n varous dmensons. What 1 There are many other ways to resolve Bertrand paradox such as ntroducng capacty constrants or dfferentated demands etc. 1

2 we thnk s partcularly strong s the assumpton of equal dscountng. There are at least two reason to beleve that future proft s dscounted dfferently by dfferent frms. Frst, some frms may be subject to a less favorable nterest rate than others due to some knd of credt market mperfecton. Second, even f the tme preference s the same across frms, the tme preferences of the managers who run those frms can be dfferent. Some manager may dscount future heavly f she expects to retre or be fred soon. Some manager s preference may be more n lne wth the preference of the frm f she may own more stocks (and stock optons) of the frm. The goal of ths paper s to understand the nature of colluson n the repeated Bertrand competton model when frms are asymmetrc, especally when dfferent frms dscount future profts n dfferent way. 2 We have two man results. Frst we dentfy the crtcal level of dscount factor such that a collusve outcome can be sustaned f and only f the average dscount factor wthn the lowest cost frms s above the crtcal level. More generally, we show that the necessary and suffcent condton for sustanng a colluson at a certan prce (or more) s that the average dscount factor of all the frms whose margnal cost s below the prce must be larger than n 1 n, where n s the number of such frms. A more patent frm s wllng to gve up more market shares to more mpatent frms, whose ncentve constrants are then relaxed. So the dstrbuton of dscountng rates matters n general. In our smple settng wth homogeneous good, the mean of dscountng rates among colludng frms determnes the possblty of colluson. Our second result s a characterzaton of all effcent (proft-maxmzng) collusve equlbra when frms dffer only n ther dscountng rates. In effcent equlbra, more mpatent frms gan a larger share of the market at an earler stage and more patent frms gan a larger share at a later stage. Such an ntertemporal substtuton of the market share s subjectve to the ncentve constrant: we cannot assgn 0% share forever even to the most mpatent frm. Hence the equlbrum outcome s not the frst best. Our characterzaton provdes a totally new pcture of colluson, whch s radcally dfferent from the one among symmetrc frms. Frst, the equlbrum market share n any effcent collusve equlbrum changes over tme. More specfcally, the market share dynamcs of each frm can be descrbed by three phases. In the frst phase, a frm has no share of the market, leavng the market to more mpatent frms. In the second phase, the frm enters 2 We assume that heterogeneous dscountng rates are gven exogenously. Of course, t would be nterestng to thnk about a model n whch they are endogenously determned for a varety of reasons. We thnk that our model wth fxed heterogeneous dscountng rates would open a possblty of buldng such a model. 2

3 the market and gans all the rest after leavng more mpatent frms the mnmum amount of statonary market share, whch correspond to the worst statonary collusve equlbrum market share for them. The fnal phase starts when a more patent frm enters the market. In the fnal phase, the frm s marker share drops to the level that corresponds to ts worst statonary collusve equlbrum market share and stays there forever. Secondly, our results delver the unque predcton n the long run. As descrbed above, the equlbrum market share for each frm, except for the most patent frm, converges to ts worst statonary collusve equlbrum market share n any effcent collusve equlbrum. More precsely, every effcent collusve equlbrum converges to the unque statonary collusve equlbrum wthn fnte tme. 3 We know that, wth symmetrc frms, there are many effcent statonary equlbra wth dfferent market shares because how to share the market s rrelevant for effcency. Wth asymmetrc dscountng, however, effcency mposes a sharp restrcton on how the market should be allocated ntertemporally. As a consequence, even though there are many effcent equlbra, the long run market share must be the same across all effcent equlbra. From a more theoretcal perspectve, our results delver new nsghts nto the theory of repeated games wth dfferental dscountng. As revewed brefly next, the major results for repeated games wth dfferental dscountng are restrcted to asymptotc results (.e. frms are nfntely patent) and the two-player case. In our settng, we characterze all the effcent equlbra wth n players for a fxed dscount factor, possbly due to some specal structure of Bertrand competton game. Related Lterature It s not wthout reason that prevous works have focused on the symmetrc model. Frst, there s the ssue of equlbrum selecton as mentoned. There are always many equlbra - hence there s always the ssue of equlbrum selecton - n repeated games. The model of dynamc Bertrand competton s no excepton. For symmetrc models, t mght make sense to focus on the symmetrc (and effcent) equlbrum, possbly as a focal pont. However, t s not clear whch equlbrum would be a focal pont when frms are asymmetrc. Secondly, the theory of repeated games wth dfferental dscountng s stll at ts development stage. For these reasons, there are not many works that study colluson among heterogeneous frms. In our vew, ths fact lmts the scope of applcatons of the repeated Bertrand competton model. 3 The tme to reach the effcent statonary collusve equlbrum s bounded across all effcent collusve equlbra for a gven profle of dscountng rates. 3

4 One notable excepton s Harrngton (1989) [4]. It shows that a statonary collusve equlbrum can be sustaned wth dfferental dscountng f and only f the average dscount factor exceeds some crtcal level. Our frst result bulds on and mproves on ths result. We provde a more complete characterzaton regardng the possblty of colluson by consderng all equlbra ncludng nonstatonary ones. 4 Clearly t s mportant to consder nonstatonary equlbra because almost all statonary equlbra are not effcent wth dfferental dscountng as our second result shows. Another dfference between our paper and [4] s that we obtan the unque equlbrum n the long run. To cope wth the ssue of multple statonary equlbra, Harrngton [4] uses a barganng soluton to select one equlbrum. On the other hand, we show that the long run equlbrum behavor s the same across all effcent equlbra. Thus we do not need to rely on any equlbrum selecton crteron other than effcency as long as we are concerned wth the long-run outcome. The semnal contrbuton n the theory of repeated game wth dfferental dscountng s Lehrer and Pauzner (1999) [5]. It studes a general two-player repeated game wth dfferental dscountng and shows that the set of feasble payoffs s larger than the convex hull of the underlyng stage game payoffs because players can mutually beneft from tradng payoffs across tme. They also characterze the lmt equlbrum payoff set as dscount factors go to 1 keepng ther rato fxed. In partcular, they show that there s some ndvdually ratonal and feasble payoff that cannot be sustaned n equlbrum no matter how patent the players are. There are some recent contrbutons n the theory of repeated games wth dfferental dscountng. Chen [1] and Gueron et. al [3] study stage games wth one dmensonal payoffs. Sugaya [6] proves a folk theorem for repeated games wth mperfect montorng and wth dfferental dscountng. Fong and Surt [2] study repeated prsoner s dlemma games wth dfferental dscountng and wth sde payments. Ths paper seems to be partcularly related to our paper because we use market share as a way to transfer utlty. Ths paper s organzed as follows. We descrbe the model n detal n the next secton. In secton 3, we prove our frst result regardng the crtcal average dscount factor. secton 4, we characterze effcent equlbra. We conclude and dscuss potental extensons of our results n the last secton. Most of the proofs are relegated to the appendx. 4 Snce a collusve outcome can be sustaned by a statonary equlbrum when the average dscount factor exceeds the crtcal level, the crucal step for our result s to show that no nonstatonary collusve equlbrum exsts when the average dscount factor s below the same crtcal level In 4

5 2 Model of Repeated Bertrand Competton wth Dfferental Dscountng Ths secton descrbes the basc structure of our model, an nfntely repeated Bertrand game. In what follows, we frst defne the stage game, then construct the nfntely repeated game. The man features of the stage game are the followngs. The players are n 2 frms represented by the numbers I = {1, 2,..., n}. They offer a homogeneous product whose market demand s characterzed by contnuous functon D : R + R +. Each frm has a lnear cost functon C : R + R + gven by C (q ) = c q, where I, c 0 s the margnal cost, and q ndcates the quantty produced by frm. We suppose that c 1 c 2... c n wthout loss of generalty and denote I = { I : c = c 1 } and n = #(I ). We assume that n 2. Hence, n one-shot Bertrand competton, the market prce would be c 1 and no frm would make any proft. It s assumed that the demand functon satsfes the followng regularty condtons: D s decreasng on (0, ); there exsts the monopoly prce for each frm: p m > c for frm that maxmzes p (D(p) c ). We assume that the margnal costs are not very dfferent: even the hghest cost c n s less than p m 1. Ths mples that pm for any, j I. 5 At the begnnng of a stage game, frms make prce decsons and suggest how to allocate output quotas n case of a draw n prces. If a frm charges a prce that s hgher than a prce charged by another frm, then the frm s market share s 0. The frm that charges the lowest prce must produce enough output to satsfy the market demand. In case there are more than one frm that charges the lowest prce, the market s allocated among those frms accordng to ther suggestons. Formally, frm s pure acton s gven by a 2-tuple a = (p, r ) A, where p s the prce choce, r reflects frm s request of market share n case of te. Hence A = R + [0, 1] s the set of pure actons avalable for player. The set of pure acton profles s A = I A. Frm s proft functon π : A R can be wrtten as π [a] = D(p )(p c ) f p < p, r R D(p )(p c ) f p = p and R 0, 1 Î D(p )(p c ) f p = p and R = 0, 0 f p > p, where p = mn j p j, Î = { I : p = mn j I p j }, and R = j Î r j. 5 If the margnal of some frm s too hgh, t s lkely that the presence of such a frm s rrelevant for our analyss. > c j 5

6 Gven the stage game descrbed above, we now defne the nfntely repeated game. Bascally, we adopt a dscrete tme model n whch the prevous stage game s played n each of the perods t N. The dstngushng feature of our dynamc Bertrand competton model s that the players have dfferent dscount factors gven by δ (0, 1), I. The set of possble hstores n perod t s gven by H t = A t 1, where A 0 ndcates the empty set, and A t denotes the t-fold product of A. A perod t-hstory s thus a lst of t 1 acton profles. We suppose perfect montorng throughout,.e., at the end of each perod, all players observe the acton profle chosen n all the prevous perods. Settng H = t N H t, a pure strategy for frm s defned as a mappng σ : H A, and consequently, a strategy profle s gven by σ = (σ ) I. Each strategy profle σ nduces an nfnte sequence of acton profles a(σ) = (a t (σ)) t N A, where a t (σ) A denotes the acton profle nduced by σ n perod t. We call the sequence a(σ) outcome path (or more smply, outcome) generated by a strategy profle σ. Fnally, for a gven strategy profle σ, and ts correspondng outcome path a(σ) = (a t (σ)) t N, the tme-average repeated game payoff for frm at tme t s U,t [a(σ)] = (1 δ ) τ=t δ τ t π [a τ (σ)]. In the followng sectons, we wll just focus on subgame perfect equlbrum solutons, and we wll lmt our attenton to pure strategy equlbra. 3 Crtcal Average Dscount Factor for Colluson In ths secton, we derve a necessary and suffcent condton to sustan a collusve equlbrum outcome. We say that the frms are colludng when there s at least one perod n whch the equlbrum outcome s not a compettve one,.e. when there s at least one frm that makes postve proft n some perod. We formalze ths as follows. Defnton 1. An outcome a = (a t ) t N s consdered a collusve outcome f and only f there exsts t N such that π (a t ) > 0 for some I. A collusve equlbrum s a subgame perfect equlbrum that generates a collusve outcome. Then we can obtan the followng sharp characterzaton, whch says that a collusve outcome can be sustaned f and only f the average dscount factor among the lowest cost frms s above some threshold. 6

7 Theorem 3.1. There exsts a collusve equlbrum f and only f I δ n 1. n n Proof. See the appendx. When the frms are symmetrc, there exsts a collusve equlbrum f and only f δ n 1 n. Thus our result s a substantal generalzaton of ths well-known result to the case wth heterogeneous dscountng and costs. It follows from the result n [4] that n 1 n s the crtcal threshold to support a collusve outcome by a statonary collusve equlbrum,.e. an equlbrum n whch each frm keeps a certan level of market share every perod and the prce s always the same. Take any prce p strctly between the mnmum cost c = mn I c and the next smallest cost. There exsts a statonary collusve equlbrum by the lowest cost frms n whch the market prce s always p and frm ( I ) gans share α [0, 1] of the jont proft π n every perod f the followng nequaltes are satsfed for all I. (1 δ )π α π By dvdng both sdes by π and summng up these nequaltes across the frms, t can be shown that such α, I exsts f and only f the average dscount factor among the lowest cost frms s larger than or equal to n 1 n. A more dffcult part of the proof s to show that colluson s mpossble when the average dscount factor s less than n 1 n, even f nonstatonary equlbra are consdered. In nonstatonary equlbrum, t s possble to transfer market shares over tme to generate larger contnuaton profts n the future, whch may enable the frms to sustan colluson. It turns out that such transfer does not work. To mprove effcency, t s necessary to let less patent frms to gan more market shares frst and let more patent frms to gan more shares later. Intutvely, such an arrangement s n conflct wth less patent frms ncentve constrants n later perods. Here s a sketch of our formal proof. We assume that the margnal cost s the same across all frms to smplfy our exposton. Frm s ncentve constrant n perod t s gven by the equalty U,t = (1 δ ) π ( a t ) + δ U,t+1 = (1 δ ) π ( a t) + η,t 7

8 where a t s the acton profle n perod t, π ( a t) = π ( a t ) s the jont proft n perod t, U,t+1 s frm s contnuaton proft from perod t + 1 on, and η,t 0 s a slack varable (frm s ncentve constrant s bndng n perod t f and only f η,t = 0). Note that each frm gans the same equlbrum jon proft by prce-cuttng because the cost s assumed to be the same. Snce ths equalty holds n every perod, we can replace U,t+1 wth (1 δ ) π ( a t+1) + η,t+1 and dvde both sdes by 1 δ to obtan π ( a t ) + δ π ( a t+1) = π ( a t) + η,t δ η,t+1 1 δ. Summng up these equaltes across the frms, we have the followng equaton regardng π ( a t) : π ( a t+1) = n 1 I δ π ( a t) + 1 I δ u,t, I where u,t = η,t δ η,t+1 1 δ. The coeffcent of π ( a t) s larger than 1 f and only f the average dscount factor s less than n 1 n. In fact, we can show that, when the jont proft s strctly postve n some perod, the sequence π ( a t), t = 1, 2,.. must dverge to nfnty, whch s a contradcton. To prove ths formally, however, we need to examne carefully the behavor of I u,t, t = 1, 2, 3,... A collusve equlbrum we construct uses a prce between the lowest cost and the second lowest cost, so t s not very proftable when ths dfference between them s small. such a case, the lowest cost frms would prefer to nclude the second lowest cost frm(s) n ther coalton to rase the equlbrum prce. In Our result can be easly generalzed to accommodate such possblty. Let p be any prce. Let I(p) be the set of frms such that c p f and only f I(p) and I(p) = n (p). Call a subgame perfect equlbrum p-collusve equlbrum f the equlbrum prce s always at least as large as p. We can prove the followng generalzaton of the above result. Theorem 3.2. For any 0 < p p m, there exsts a p-collusve equlbrum f and only f I(p) δ n (p) n (p) 1. n (p) The proof s almost the same, hence omtted. Comment 8

9 When I = 1,.e. there s the unque lowest cost frm, Theorem 2 stll holds. But we need to rely on a less natural punshment. The assumpton I 2 guarantees that any devaton from a collusve outcome s punshed by Nash reverson wth 0 proft forever. If I = 1, the 0 proft equlbrum requres that there are at least two frms chargng c 1, but frm 1 serves the whole market (r 1 = 1, r = 0 for all 1). 4 Characterzaton of Effcent Collusve Equlbra In ths secton, we characterze effcent collusve equlbra wth dfferental dscountng rates. We assume that the margnal cost s the same across frms and normalze t to 0. Then the monopoly prce can be determned wthout any ambguty. Let p m be the monopoly prce and π m be the monopoly proft. We also assume that 0 < δ 1 < δ 2 <... < δ n 1 < δ n < 1 for the sake of smplcty. The result can be easly extended to the case where the dscountng factors of some frms are the same. Let π,t, = 1,..., n, t N be a sequence of profts assocated wth any collusve equlbrum. By defnton, they satsfy the followng ncentve compatblty condton n every perod: (1 δ ) π t U,t where U,t s frm s equlbrum contnuaton proft n the begnnng of perod t and π t = π,t. On the other hand, t s clear that any sequence of proft profles that satsfy those condtons are generated by a collusve equlbrum. Hence we use such a sequence of proft profles to descrbe any collusve equlbrum. A collusve equlbrum s effcent f there s no subgame perfect equlbrum that makes every frm better off weakly and some strctly. Observe that π t s always n (0, π m ] for any effcent collusve equlbrum. π t cannot exceed the monopoly proft by defnton. If π t < 0, then we can construct a more effcent equlbrum by just droppng perod t. We know that there exsts a statonary collusve equlbrum wth monopoly prce f and only f n =1 δ n n 1 n. When the average dscount factor s strctly larger than n 1 there s a range of market shares that can be supported by statonary collusve equlbrum. Let π be frm s per perod proft n the worst statonary collusve equlbrum proft for frm. Note that π = (1 δ ) π m by the ncentve compatblty condton. We assume n =1 δ n > n 1 n for the rest of ths secton. We frst prove that, n any effcent collusve equlbrum, the jont proft must be strctly ncreasng untl t reaches the monopoly proft and stays there forever. 9 n,

10 We start wth the followng lemma. Lemma 4.1. Consder any effcent collusve equlbrum where, for some t 1, π t+1 < π m and there s a frm such that U,t+1 > (1 δ )π t+1 and π,t+1 > 0. Then π t+1 πt δ n. Proof. Defne Ĩt+1 = { I : U,t+1 = (1 δ )π t+1 }, whch s not empty (otherwse the jont proft can be ncreased to mprove effcency). Suppose that π t+1 < πt δ n. Then π,t π t δ π t+1 π t δ n π t+1 > 0 for all Ĩt+1. Consequently, the profts can be perturbed as follows: π,t = π,t δ ε and π,t+1 = π,t+1 + ε, for Ĩt+1; whereas π,t = π,t+ Ĩt+1 δ ε and π,t+1 = π,t+1 ( Ĩt+1 1)ε. Snce π t+1 < π m and π,t+1 > 0, ths new allocaton s feasble and ncentve compatble for ε > 0 small enough. Moreover, as Ĩt+1 δ > Ĩt+1 1, t also Pareto-domnates the ntal one. Ths s a contradcton. The next theorem proves a strong monotoncty property for effcent collusve equlbra. Theorem 4.1. For any effcent collusve equlbrum, there exsts T such that π t < π t+1 for t = 1,..., T 1 and π t = π m for any t T. Furthermore, ths T s bounded across all effcent collusve equlbra. Proof. Take any effcent collusve equlbrum. Let π t (0, π m ] be a jont proft n any perod t. We assume that π t > δ n π t+1 and π t+1 < π m, and derve a contradcton. If those two condtons are satsfed, then t must be the case that π t+1 = π,t+1 by Lemma Ĩt Therefore, there s j Ĩt+1 such that π j,t+1 > (1 δ j )π t+1, otherwse, π t+1 = π,t+1 π t+1 (1 δ ) = π t+1 ( Ĩt+1 δ ), Ĩt+1 Ĩt+1 Ĩt+1 but Ĩt+1 δ > Ĩt+1 1. As a result, (1 δ j )π t+2 U j,t+2 < (1 δ j )π t+1. The frst nequalty s derved from the ncentve constrant n perod t + 2, whereas the second one from the fact that π j,t+1 > (1 δ j )π t+1 and U j,t+1 = (1 δ j )π t+1. Then, π t+1 > π t+2. We can proceed n a smlar manner to obtan π t+k > π t+k+1 for every k 1, whch contradcts the effcency assumpton. Hence t must be the case that ether π t δ n π t+1 or and π t+1 = π m. Clearly ths mples that there s T such that π t < π t+1 for t = 1,..., T 1 and π t = π m for any t T. Fnally we prove that ths T s bounded across all effcent equlbra. For any gven T, each frm s proft per perod s at most δ T 1 n π m for the frst T T perods for any T T. If T s large, then frm s payoff s less than π. Such payoff profle s Pareto-domnated by any statonary collusve equlbrum. 10

11 Next we provde an (almost) complete characterzaton of effcent collusve equlbra. Consder any effcent collusve equlbrum where frm s equlbrum proft exceeds π. Then every frm s ncentve constrant s not bndng n the frst perod, hence the equlbrum jont proft must be π m n the frst perod. Gven our monotoncty result, ths mples that the equlbrum prce s always p m for ths class of effcent collusve equlbra. We call such collusve equlbrum p m -effcent collusve equlbrum. The next theorem characterzes the structure of p m -effcent collusve equlbrum. Observe that ths characterzaton s a complete characterzaton of the asymptotc behavor of all effcent collusve equlbra, because every effcent collusve equlbrum converges to some p m -effcent collusve equlbrum eventually wthn fnte tme by our prevous result. In p m -effcent collusve equlbrum, more patent frms lend the market share ntally to more mpatent frms. However, the ablty of mpatent frms to pay back the market share s lmted by the requrement that each frm s proft cannot be lower than ts worst statonary equlbrum proft π. Theorem 4.2. Every p m -effcent collusve equlbrum has the followng structure: there exsts t 1 t 2..., t n 1 such that, for every, 1. π,t = 0 for every t < t 1 2. π,t [0, π m 1 3. π,t = π m 1 4. π,t [ π, π m 1 π h ] for t = t 1 π h for t = t 1 + 1,..., t 1 5. π,t = π for t > t π h ] for t = t 6. Incentve Constrants n the frst perod [ ( ) { }] δ t 1 1 (1 δ ) π,t 1 + δ δ t 1 t 1 π m π h [ ] +δ t 1 1 (1 δ ) δ t t 1 π,t + δ t t 1 +1 π (1 δ ) π m Furthermore, f there exst (t 1, t 2,..., t n 1 ) and a sequence of proft profles π,t that satsfy the above condtons, then there exsts a correspondng p m -effcent collusve equlbrum that generates them. 11

12 Proof. See the appendx. In words, every p m -effcent collusve equlbrum has the followng propertes. From perod 1 to perod t 1 1, frm 1 gets the whole share. In perod t 1, frm 1 and 2 shares the market where π,t1 π 1. After ths perod, frm 1 s share s gong to be always π 1. From perod t to perod t 2 1, frm 2 gets π m π 1. In perod t 2, frm 2 and 3 shares the market where π,t2 π 2. After ths perod, frm 2 s share s gong to be always π 2. From perod t to perod t 3 1, frm 3 gets π m π 1 π After perod t n 1, frm n gets π m n 1 π h and frm h < n gets π h forever. There are two crtcal perods for frm : t 1 and t. Up to t 1, frm s market share s 0. The perods between t 1 and t s the pay back tme when frm gets all the market share subject to the constrant that each less patent frm h < gans π h. After t, frm s proft s reduced to π and stay there forever. It may be the case that there s some overlap: t k = t k+1 =,..., = t m = t for some m > k. Note that π,t such a case. π for = t k, t k+1,..., t m 1 n Comment One mplcaton of our theorem s that there exsts the unque effcent statonary collusve equlbrum, to whch every effcent collusve equlbrum converges. Ths s the statonary collusve equlbrum where the prce s p m, frm s market share s π for = 1,..., n 1 and frm n s market share s π m =1,...,n 1 π n every perod, whch corresponds to the worst statonary collusve equlbrum for frm = 1,..., n 1 (and the best one for frm n). All the other effcent collusve equlbra must be nonstatonary. Our result delvers the unque predcton n the long run wthout any equlbrum selecton. Ths s not the case f we focus on statonary collusve equlbra. 12

13 When δ = δ +1 for some, ther market share s characterzed by smlar condtons: ther market share s 0 before t 1, π,t = π 1,t = π (= π +1 ) after t, and can be somewhat arbtrary between t and t 1 (but we can assume that ther market shares are constant wthn these perods wthout loss of generalty). 5 Concluson and Dscusson In the context of Bertrand prce competton n an nfntely repeated game, ths paper studes collusve behavor among n frms wth asymmetrc dscount factors and asymmetrc margnal costs. We fnd that t s possble to sustan a colluson f and only f the average dscount factor of the lowest cost frm s at least as large as (n 1)/n, where n s the number of the lowest cost frms. Ths paper also studes effcent collusve equlbra among n frms wth dfferental dscountng when the margnal cost s the same across frms. We succeed n characterzng the structure of effcent collusve equlbra. More specfcally, we show the followngs. In any effcent collusve equlbrum, the jont proft must be strctly ncreasng over tme untl t reaches the monopoly proft level wthn fnte tme and stay there forever. Every effcent collusve equlbrum where no frm s payoff s not too low must be a collusve equlbrum wth the monopoly prce n every perod ( p m -effcent collusve equlbrum ). In every p m -effcent collusve equlbrum, a frm s market share s 0 ntally, reaches a peak, then converges to the market share that corresponds to the worst statonary collusve equlbrum wth the monopoly prce (except for the most patent frm). Every effcent collusve equlbrum converges to the unque effcent statonary collusve equlbrum n the long run, where the equlbrum prce s p m, frm s proft per perod s π for = 1,..., n 1 and 1 =1,...,n 1 π for = n n every perod. 13

14 References [1] B. Chen (2008). On Effectve Mnmax Payoffs and Unequal Dscountng, Economcs Letters, 100, 1: [2] Y-F. Fong and J. Surt (2009). The Optmal Degree of Cooperaton n the Repeated Prsoners Dlemma wth Sde Payments, Games and Economc Behavor, 67,1: [3] Y. Guéron, T. Lamadon and C. Thomas (2011). On the Folk Theorem wth Onedmensonal Payoffs and Dfferent Dscount Factors, Games and Economc Behavor, 73, 1: [4] J. Harrngton (1989). Colluson among Asymmetrc Frms: The case of Dfferent Dscount Factors, Internatonal Journal of Industral Organzaton, 7: [5] Lehrer, E. and A. Pauzner (1999). Repeated Games wth Dfferental Tme Preferences, Econometrca, 67, 2: [6] T. Sugaya (2010), Characterzng the Lmt Set of PPE Payoffs wth Unequal Dscountng, mmeo. [7] J. Trole (1988). The Theory of Industral Organzaton, MIT press. 14

15 Appendx Proof of Theorem 3.1 Proof. We already dscussed that there exsts a collusve statonary subgame perfect equlbrum when the nequalty s satsfed. Thus we just need to show that there s no collusve I δ n subgame perfect equlbrum when < n 1 n. By contradcton, begn by assumng that ã = (ã t ) t N s a collusve equlbrum outcome, and wthout loss of generalty, assume that π (ã 1 ) = I π (ã 1 ) > 0. Note frst that for each I, there exsts a bounded nonnegatve sequence {η (t) : t N} defned by η (t) = U,t (ã) (1 δ )π (ã t ). Moreover, snce U,t (ã) = (1 δ )π (ã t ) + δ U,t (ã), we have that (1 δ )π (ã t ) + δ U,t (ã) = (1 δ )π (ã t ) + η (t), and therefore (1 δ )π (ã t ) + δ [(1 δ )π (ã t+1 ) + η (t+1) ] = (1 δ )π (ã t ) + η (t), or equvalently, [ ] [ ] π (ã t ) = π (ã t ) + η(t) δ π (ã t+1 ) + η(t+1). (1 δ ) (1 δ ) Addng up ths nequalty across I and denotng s = I δ, we obtan or more shortly, π (ã t ) = n π (ã t ) s π (ã t+1 ) + η (t) δ η (t+1), (1 δ ) I π (ã t+1 ) = γπ (ã t ) + 1 s where γ = (n 1)/s and u,t = (η (t) I u,t, δ η (t+1) )/(1 δ ). Before proceedng, t s useful to note that γ > 1 and therefore π (ã t+1 ) π (ã t ) + 1 s I u,t for every t N. π (ã 1 ) + 1 s I j=1 t u,j, 15

16 Now consder the seres t j=1 u,j for I, and observe that t maybe wrtten as t j=1 t u,j = η(1) (1 δ ) + j=2 η (j) δ η (t+1) (1 δ ). Snce the equlbrum proft s bounded above for each frm by assumpton, the equlbrum proft s bounded below as well for each frm; otherwse the average dscounted proft s negatve. Ths mples that there exsts M such that 0 η (j) M for all j N and I. Observe that ths mples that the seres t j=2 η(j) must be ether unbounded above or convergng to a fnte (nonnegatve) real number. Suppose frst that j=2 η(j) s unbounded above for some I. On the one hand, we know that I ( t j=1 u,t) s unbounded above, too. On the other hand, we have that π (ã t+1 ) π (ã 1 ) + (1/s ) t I j=1 u,t, whch s a contradcton because the sequence {π (ã t ) : N} s bounded above. Suppose now that j=2 η(j) s fnte for all I. Then we have that η (t) as well as u,t converge to zero for all I. If η (j) = 0 for all I and j N, t follows mmedately that I ( t j=1 u,t) 0. On the other hand, f η (t ) = c > 0 for some I and t N, there exsts T > t such that η (t) < c (1 δ )/(2δ ) for all t T. As a result, we have that t u,j j=1 η(1) (1 δ ) + c c 2, when t > T. As I s a fnte set, there s T N (ndependent of ) such that I ( t j=1 u,t) 0 for all t T, and consequently, π (ã t ) π (ã 1 ) as long as t T. Before proceedng, observe frst that there s t N such that γ t π (a 1 ) > 2M. Secondly, snce I s a fnte set and u,t converge to zero for each I, there exsts T N (ndependent of ) such that T > T and u,t < (s M )/(n tγ t ) for all I and t T. The followng nequalty s a straghtforward mplcaton: π (ã T +t ) = γπ (ã T +t 1 ) + 1 s I u, T +t 1 γπ (ã T +t 1 ) M tγ t, and by nducton, we can prove that π (ã T +t ) γ t π (ã T t 1 M ), tγ t j j=0 16

17 for every t N. Fnally, after replacng t = t n the prevous nequalty, we obtan the desred result: t 1 π (ã T + t ) γ t π (ã T ) > γ t π (ã 1 ) > 2M M. M tγ t j j=0 t 1 j=0 M t The second nequalty follows by π (ã T ) π (ã 1 ) and γ > 1, whereas the last one by γ t π (ã 1 ) > 2M. And obvously, ths s a contradcton because π (ã T + t ) M. Proof of Theorem 4.2 We prove the theorem through a seres of lemmata. Lemma 5.1. For any effcent subgame perfect equlbrum, f frm s ncentve constrant s not bndng n perod t > 1, then π j,t 1 = 0 for every j >. Proof. Suppose not,.e. there exsts some monopoly-prce effcent SPE where frm s ncentve constrant s not bndng n perod t > 1 and π j,t 1 > 0 for some j >. Then there s a perod t > t such that frm s ncentve constrant s not bndng for t, t + 1,..., t and π,t > 0. We can fnd such t, otherwse π,t+1 = π,t+2 =... = 0 (f π,t+1 = 0, then U,t+2 > π hence s ncentve constrant n perod t + 2 s not bndng. If π,t+2 = 0, then U,t+2...). Such a path s not sustanable. Now perturb the proft of frm and j as follows. π,t = π,t + ε, π j,t = π j,t ε, π,t = π,t ε, π j,t = π j,t + ε, We are bascally exchangng frm j s market share n perod t wth frm s market share n perod t,keepng every other frm s proft at the same level. Snce δ < δ j, π j,t > 0 and π,t > 0, we can pck ε, ε > 0 so that frm j s contnuaton payoff n every perod from t to t ncreases and frm s contnuaton payoff n perod t ncreases. So ths allocaton Paretodomnates the orgnal SPE allocaton. Frm j s ncentve constrants are not affected at all. Frm s ncentve constrants n perod t s satsfed by constructon. Fnally, we can 17

18 take ε, ε > 0 small enough so that frm s ncentve constrant from perod t + 1 to t s stll not bndng. So we can construct a more effcent SPE n ths case, a contradcton. Lemma 5.2. For any monopoly-prce effcent subgame perfect equlbrum, f π,t < π, then π j,t = 0 for every j >. Proof. If π,t < π, then U,t+1 > π. Hence frm s ncentve constrant s not bndng n perod t + 1. Then π j,t = 0 for every j > by Lemma 1. Lemma 5.3. For any effcent subgame perfect equlbrum wth π > π, (1)π 1,t π 1 for every t 1 and (2) π 1,t +k = π 1 for any k = 0, 1,... f frm 1 s ncentve constrant s bndng n perod t. Proof. If π 1,t < π 1 for any t, then π j,t = 0 for every j = 2, 3,..., n by Lemma 2. Ths contradcts to π,t = π m. Frm 1 s ncentve constrant s bndng n perod t f and only f U 1,t = π 1. Clearly ths holds f and only f π 1,t +k = π 1 for k = 0, 1, 2... By nducton, a smlar property holds for every frm. Lemma 5.4. For any effcent subgame perfect equlbrum wth π > π, suppose that π h,t+k = π h for every k = 0, 1, 2,... and every h = 1, 2,..., for some t and some I. Then (1) π +1,t+k π +1 for every k = 0, 1, 2,... and (2) π +1,t +k = π +1 for every k = 0, 1, 2... f frm + 1 s ncentve constrant s bndng n perod t t. Proof. Suppose that π +1,t+k < π +1 for any k. Then U +1,t+k+1 > π +1. Hence frm + 1 s ncentve constrant s not bndng n perod t + k + 1. Then π j,t+k+1 = 0 for every j > + 1 by Lemma 1. However, π h,t+k+1 = +1 π h,t+k+1 = π h + π +1,t+k+1 < +1 π h π, h whch s a contradcton. Ths proves (1). As for (2), frm + 1 s ncentve constrant s bndng n perod t t f and only f U,t = π. By Lemma 4, ths holds f and only f π +1,t +k = π +1 for every k = 0, 1, 2... Now we can prove Theorem

19 n Proof. In perod 1, we have π 1 such that (1) π h,1 = π m and (2) π h,1 π h for h = [ 1, 2,..., h 1 1, π h1,1 0, π m h ] 1 1 π h, and π h,1 = 0 for h > h 1 for some h 1 1 by.lemma 2. By Lemma 1, the ncentve constrant must be bndng for h = 1, 2,..., h 1 1 n perod 2. By Lemma 3 and Lemma 4, π h,1+k = π h for h = 1, 2,..., h 1 1 for the rest of the game (k = 1, 2,...). n In perod 2, we have π 2 such that (1) π h,2 = π m, (2) π h,2 = π h for h = 1, 2,..., h 1 1 [ (by the prevous step), (3) π h,2 π h for h = h 1, h 1 + 1,..., h 2 1, π h2,2 0, π m h ] 2 1 π h, and π h,2 = 0 for h > h 2 for h 2 for some h 2 h 1 by.lemma 4. By Lemma 1, the ncentve constrant must be bndng for h = h 1,..., h 2 1 n perod 3. By Lemma 3 and Lemma 4, π h,2+k = π h for h = h 1,..., h 2 1 for the rest of the game (k = 1, 2,...) and so on... Ths proves 1-6 n the statement of the theorem. On the other hand, suppose that there exst (t 1, t 2,..., t n 1 ) and a sequence of proft profles π,t that satsfy 1-6. It s clear that ths corresponds to some monopoly-prce SPE. So we just show that t s an effcent equlbrum. Suppose not. Then there exsts a Paretosuperor monopoly-prce effcent SPE, whch of course satsfes 1-6. Let ( t 1, t 2,..., t n 1 ) be the correspondng crtcal perods and π,t be the assocated sequence of proft profles. Snce ths equlbrum s more effcent than the former one, t must be the case that ether (1) t 1 < t 1 or (2) t 1 = t 1 and π 1,t1 π 1,t1. In ether case, t must be the case that, for frm 2, ether (1) t 2 < t 2 or (2) t 2 = t 2 and π 1,t2 π 1,t2. By nducton, ether (1) or (2) holds up to frm n 1. Then frm n s average proft gven π n,t s hgher than frm n s average proft gven π n,t. Ths s a contradcton to the assumpton that the latter equlbrum s more effcent. 19