Collusion and Heterogeneity of Firms

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1 Collusio ad Heterogeeity of Firms Ichiro Obara Departmet of Ecoomics Uiversity of Califoria, Los Ageles Federico Ziceko Departmet of Ecoomics Uiversity of Pittsburgh November 5, 2015 Abstract We examie the impact of heterogeeous discoutig o collusio. Our aalysis clarifies exactly whe collusio ca be sustaied ad how collusio would be orgaized efficietly with heterogeeous discoutig. First we show that collusio is possible if ad oly if the average discout factor exceeds a certai threshold. Our characterizatio ca be exteded to the case where firms face capacity costraits. The we characterize the dyamics of market shares i all efficiet collusive equilibria. I every efficiet collusive equilibrium with the moopoly price, each firm s market share icreases first the decreases to the same stable level i order of icreasig patiece. Oe implicatio of our characterizatio is that every efficiet collusive equilibrium must coverge to the same uique statioary oe. Thus we ca make a uique predictio of the equilibrium market share i the log ru despite the severe multiplicity of efficiet collusive equilibria. I the log ru, the most impatiet firm gais a large market share for the sake of icetive ad the most patiet firm teds to gai a large market share for the sake of efficiecy. JEL Classificatio: C72, C73, D43. Keywords: Bertrad Competitio, Capacity Costrait, Collusio, Differetial Discoutig, Repeated Game, Subgame Perfect Equilibrium. 1 Itroductio We study a dyamic Bertrad competitio model where firms discout future profits at differet discout rates ad examie the impact of heterogeeous discoutig o collusio. The goal of this paper is to uderstad whe ad how collusio arises with heterogeeous discoutig. The model of dyamic Bertrad/Courot competitio is the stadard framework to aalyze ad uderstad collusio (Tirole [30]). The basic logic behid these models is We would like to thak a editor ad aoymous referees for their useful commets. We also like to thak the audiece at Cal Poly, Corell, Kyoto Uiversity, UCLA, Uiversity of Pittsburgh, USC, Yale, 2011 Asia Meetig of the Ecoometric Society, ad 2012 Southwest Ecoomic Theory Coferece. All remaiig errors are ours. 1

2 very simple: firms are willig to collude because they fear a price war i the future. Clearly the discoutig rate is the most critical parameter to determie the effectiveess of such itertemporal icetive scheme. However almost all the models of this kid assume symmetric discoutig. I fact, it is ofte the case to assume that firms are completely symmetric i every aspect. Symmetric models are very useful to uderstad a variety of issues associated with collusio because of its tractability. But the symmetry assumptio is urealistic, thus limits the scope of applicatios of such models, especially whe symmetry is imposed o such a crucial parameter of the model. 1 heterogeeity of discoutig rates to the dyamic model of collusio. This motivates us to itroduce There are at least two reasos to believe that future profit is discouted differetly by differet firms. The first oe is the cost of capital. If a firm faces a higher iterest rate tha the others for ay reaso (ex. asymmetric iformatio), the the firm would value profits i the short ru more tha the others would do. The secod reaso is heterogeeity of maagers discoutig rates. Eve if the cost of capital is the same across firms, the maagers who ru those firms may discout future profits differetly. Accordig to Stei [28], maagers focus o the short term whe their salaries deped heavily o curret stock prices. Shleifer ad Vishy [26] argues that short horizos of ivestors lead to short horizos of maagers who are averse to uderpricig of their equity. This type of aversio may occur whe low equity prices icreases the probability of replacemet. Narayaa [23] emphasizes that maagerial career cocers may result i a focus o short-term horizos. Specifically, Narayaa [23] states that if the maager possesses private iformatio uavailable to ivestors ad his ability is ukow, the he might choose quicker-retur projects i order to icrease his wage. To the extet that these cosideratios (compesatio scheme, ivestors horizo, maagerial career cocers, maager s private iformatio) vary across firms, future profit should be discouted differetly by differet firms. Formally, our model is a ifiitely repeated Bertrad (price-settig) game by firms with heterogeeous discoutig rates ad costat margial costs. We assume that all the firms produce the same product, ad the firm that charges the lowest price must serve the etire market as i the stadard Bertrad game. Whe two or more firms charge the same (lowest) price, we allow those firms to divide the market i a flexible way through commuicatio. 2 Thus firms ca collude i two dimesios: price ad market share. 3 1 Fershtma ad Pakes [12] emphasizes the importace of heterogeeity amog firms i the same market. 2 Geesove ad Mulli [14] documets actual examples of commuicatio for the sugar refiig cartel, although they do ot seem to be related directly to the way i which commuicatio works i our model. 3 Some papers study a dyamic oligopoly model where firms choose their capacities, the prices. See 2

3 Our mai fidigs cosist of two parts. The first part characterizes exactly whe collusio is possible with heterogeeous discoutig. We show that the average discout factor is the key variable to determie the possibility of collusio. Specifically, we show that the moopoly price (ideed ay price strictly above the margial cost) ca be sustaied with firms if ad oly if the average discout factor exceeds 1. If the average discout factor falls strictly below this critical threshold, the the competitive outcome prevails i every period i ay equilibrium. Thus heterogeeity i discoutig does ot discourage collusio per se. Sice the allocatio of market shares is flexible, a more patiet firm is willig to give up more market shares to more impatiet firms, whose icetive costraits are the relaxed. So the distributio of discoutig rates matters i geeral. I our Bertrad settig, it turs out that the first momet of the distributio determies the possibility of collusio completely. We ca exted this result to the case with productio capacity. I this case, the critical level of the average discout factor depeds o the price to be supported. The average discout factor eeds to be larger to support a collusive price that is closer to the moopoly price. It ca be the case that firms caot collude at the moopoly price, but ca collude at a price lower tha the moopoly price. Sice we focus o asymmetric discoutig ad keep every other part of the model symmetric itetioally, we assume symmetric capacity costraits for the most part. But we discuss asymmetric capacity costraits ad illustrate a possibility that asymmetry i capacity costraits facilitates collusio i the presece of asymmetric discoutig. The secod part examies how collusio should be orgaized. We provide a almost complete characterizatio of all collusive equilibria that are Pareto-efficiet from the firms perspective. First we show that the equilibrium prices are always set at the moopoly price i most of efficiet collusive equilibria. If this is ot the case, the the equilibrium prices must be icreasig mootoically ad quickly util it reaches the moopoly price ad stays there forever. Thus it is almost without loss of geerality to focus o efficiet collusive equilibria with the moopoly price, possibly except for the first few periods. The we proceed to provide a complete characterizatio of the dyamics of market shares i all efficiet collusive equilibria with the moopoly price. Our result paits a very differet picture of collusio, relative to the oe with symmetric discoutig. The stadard model with symmetric firms usually focuses o statioary equilibria where the prices ad the market shares are costat over time. I our settig, almost all statioary equilibria are iefficiet due Beoit ad Krisha [4] ad Staiger ad Wolak [27]. 3

4 to heterogeeous discoutig. Efficiecy would be improved by a itertemporal trasfer of market shares betwee firms with differet discoutig rates. More specifically, we show that, i ay efficiet collusive equilibrium with the moopoly price, each firm s equilibrium market share dyamics ca be described by three phases. I the first phase, a firm has o share of the market, leavig the market to more impatiet firms. I the secod phase, the firm eters the market ad gais all the residual market shares after leavig the miimum level of market share to more impatiet firms, which correspods to the smallest market share for those firms that ca be supported by a statioary collusive equilibrium with the moopoly price. The fial phase for the firm starts whe a more patiet firm eters the market. I this phase, the firm s market share drops to its miimum level ad stays there forever. Firms go through these three phases i icreasig order of patiece. Thus each firm s market share is hump-shaped over time, first goes up the goes dow. 4 Two exceptios are the most impatiet firm ad the most patiet firm. The market share of the most impatiet firm decreases mootoically over time (i.e. there is o first phase) ad the market share of the most patiet firm icreases mootoically over time (i.e. there is o fial phase). Oe importat implicatio of our result is that every efficiet collusive equilibrium coverges to the same (uique) statioary efficiet collusive equilibrium i the log ru. As described above, each firm s market share coverges to the market share i a worst statioary collusive equilibrium (with the moopoly price) i ay efficiet collusive equilibrium, except for the most patiet firm. 5 This limit market share correspods to the statioary market share of the uique efficiet collusive equilibrium that is statioary. Hece our model delivers the uique predictio regardig the equilibrium market share i the log ru without ivokig ay equilibrium selectio criterio. With symmetric firms, there are may efficiet statioary equilibria with differet market shares because how to share the market is irrelevat for efficiecy. With asymmetric discoutig, however, efficiecy imposes a sharp restrictio o how the market should be allocated itertemporally. As a cosequece, eve though there are may efficiet equilibria, the log ru market share must be the same across all efficiet collusive equilibria. 4 Some efficiet collusive equilibrium that are relatively more asymmetric may require the market price to be lower tha the moopoly price (See the example i Subsectio 4.1). However, sice the equilibrium price must coverge to the moopoly price quickly i ay efficiet collusive equilibrium, the above descriptio of collusive patter is valid i every efficiet collusive equilibrium possibly except for the first few periods (Propositio 4.1). 5 The time to reach the statioary efficiet collusive equilibrium is bouded across all efficiet collusive equilibria for a give profile of discoutig rates. 4

5 Aother importat implicatio of our result is that the distributio of the log-ru market shares across firms is U-shaped with respect to the order of patiece. The most impatiet firm ad the most patiet firm teds to occupy a larger share of the market evetually, but for very differet reaso. The most impatiet firm eeds a large market share for the sake of icetive. The most patiet firm gais a large market share evetually for the sake of efficiecy. I a sese, the degree of heterogeeity i discoutig rates is magified edogeously i the log ru if it is measured usig the stable market shares as weights. Thus if we focus o ay observable characteristics of firms that are correlated with their discoutig rates, the we would fid that very differet firms become evetually domiat i the same market eve though they produce exactly the same product. From a more theoretical perspective, our paper delivers ew isights ito the theory of repeated games with differetial discoutig. As reviewed briefly ext, most of the available results for repeated games with differetial discoutig focus o the limitig case where players are ifiitely patiet. behavior i all efficiet equilibria for a give discout factor. Related Literature I our settig, we ca characterize the equilibrium There is a good reaso for symmetric models to have bee so popular i the literature o dyamic oligopolistic competitio. First, there is the issue of equilibrium selectio. There are always may equilibria - hece there is always the issue of equilibrium selectio - i repeated games. The model of dyamic Bertrad competitio is o exceptio. With symmetric firms, it might make more sese to focus o the symmetric collusive equilibrium, possibly as a focal poit. However, it is ot clear at all which equilibrium would be selected whe firms are asymmetric. Secodly, the theory of repeated games with differetial discoutig is still at its developmet stage. For these reasos, there are ot may works that study collusio amog firms with heterogeeous discoutig. Oe otable exceptio is Harrigto [16]. It shows that a statioary collusive equilibrium with possibly asymmetric market share ca be sustaied with asymmetric discoutig if ad oly if the average discout factor exceeds the critical threshold 1. Our first result builds o ad develop this result further. We provide a complete characterizatio regardig the possibility of collusio by cosiderig all equilibria icludig ostatioary oes ad by itroducig capacity costraits. We like to emphasize that it is especially importat to study ostatioary equilibria with heterogeeous discoutig. Ipatiet firms ad patiet firms are willig to trade their profits over time. I fact, our efficiecy result shows that almost all statioary equilibria are iefficiet with asymmetric discoutig. Aother 5

6 differece betwee our paper ad [16] is that we obtai a uique equilibrium predictio i the log ru. To cope with the issue of multiple statioary equilibria with differet market share cofiguratio, Harrigto [16] uses the Nash bargaiig solutio to select oe statioary equilibrium. 6 I this paper, we show that the log ru equilibrium behavior is the same across all efficiet collusive equilibria. Thus we do ot eed to apply ay equilibrium selectio criterio other tha efficiecy to select a equilibrium as log as the log-ru outcome is cocered. 7 May papers examie the effect of asymmetric productio techology o collusio. Compte, Jey, ad Rey [9] ad Lambso [19] itroduce asymmetric capacity costraits to the ifiitely repeated Bertrad game. 8 Vascocelos [31] itroduce asymmetric margial costs to the ifiitely repeated Courot game. Maso, Phillips, ad Nowell [21] coducted a experimet o dyamic duopoly games with asymmetric costs ad foud that the symmetry of margial costs facilitates collusio. 9 The semial cotributio i the theory of repeated games with differetial discoutig is Lehrer ad Pauzer [20], which studies a geeral two-player repeated game with differetial discoutig. They characterize the set of feasible payoffs ad show that it is larger tha the covex hull of the uderlyig stage game payoffs. 10 They also characterize the limit equilibrium payoff set as discout factors go to 1 while keepig their log ratio fixed. I particular, they show that there exists some idividually ratioal ad feasible payoff that caot be sustaied i equilibrium o matter how patiet the players are. There are some recet developmets i the theory of repeated games with differetial discoutig. For the case of perfect moitorig, Che [6] ad Guero et.al [15] examies a folk theorem for a class of examples where eve a weak form of full dimesioality is violated. Che ad Takahashi [8] proves a folk theorem for a class of games that satisfy a certai dyamic versio of full dimesioality. For the case of imperfect moitorig, Sugaya [29] proves a folk theorem with full dimesioal payoffs ad some coditios o the 6 Also see Harrigto [17]. 7 Adersso [1] itroduces a istataeous alteratig offer bargaiig game that precedes the ifiitely repeated Bertrad game i [16] ad selects the equilibrium price ad the equilibrium price market share edogeously. But it does ot cosider ay ostatioary equilibrium for the repeated Bertrad game. 8 Brock ad Sheikma [5] is the first paper to study the ifiitely repeated Bertrad game with symmetric capacity costraits. See also Lambso [18]. 9 May papers study the effect of asymmetry of firms o the divisio of surplus withi the framework of cooperative game. Osboure ad Pitchik [24] applies the Nash bargaiig solutio to duopoly firms with differet capacities. Schmalesee [25] applies a variety of cooperative solutio cocepts to a set of firms with differet margial costs. Sice these models are static, they do ot discuss asymmetry i discoutig rates. 10 Che ad Fujishige [7] provides a differet characterizatio of the set of feasible payoffs for two-player repeated games. 6

7 moitorig structure. Fog ad Surti [13] study repeated prisoer s dilemma games with differetial discoutig ad with side paymets. They provide a ecessary ad sufficiet coditio o the average discoutig rates to support a particular class of (almost) statioary equilibrium where the players cooperate i every period. However they do ot cosider all equilibria ad their aalysis is restricted to the two-player case. Miller ad Watso [22] studies repeated games i which the stage game is preceded by a bargaiig phase ad a trasfer phase i each period. They provide a recursive characterizatio of the equilibrium payoff set for some refiemet of subgame perfect equilibrium ad exted it to the case with heterogeeous discoutig factors. This paper is orgaized as follows. We describe the model i detail i the ext sectio. I sectio 3, we prove our first result regardig the critical average discout factor. sectio 4, we provide a characterizatio of all efficiet collusive equilibria. We itroduce capacity costraits to our model ad geeralize our collusio possibility result i Sectio 5. We discuss other possible extesios i Sectio 6. The Sectio 7 cocludes. All the proofs are relegated to the appedix. 2 Model of Repeated Bertrad Competitio with Heterogeeous Discoutig This sectio describes the basic structure of our model, a ifiitely repeated Bertrad game with differetial discoutig. I what follows, we first defie the stage game, the costruct the ifiitely repeated game. The mai features of the stage game are the followigs. The players are 2 firms represeted by the umbers I = {1, 2,..., }, who produce the same homogeeous product. The demad of the product is give by cotiuous fuctio D : R + R +. Each firm has a liear cost fuctio C i : R + R + give by C i (q i ) = cq i with costat margial cost c 0, where q i idicates the quatity produced by firm i. We assume that D is decreasig o R + ad there exists the uique moopoly price p m > c that maximizes π(p) = D(p) (p c), which is odecreasig o [c, p m ]. Let π m = D(p m )(p m c) be the moopoly profit. At the begiig of a stage game, firms choose prices ad make a suggestio about how to allocate the market i case of a draw i prices. If a firm charges a price that is higher tha a price charged by aother firm, the the firm s market share is 0. The firm that charges the lowest price, which we call the market price ad deote by p, must produce eough products to meet the market demad. I case more tha oe firm charges the lowest 7 I

8 price, the market is allocated amog those lowest price firms accordig to their suggestios. Formally, firm i s pure actio is give by a 2-tuple a i = (p i, r i ) A i, where p i is the price choice, r i reflects firm i s request of market share i case of tie. Hece A i = R + [0, 1] is the set of pure actios available for firm i. The set of pure actio profiles is A = i I A i. Firm i s profit fuctio π i : A R ca be writte as D(p i )(p i c) if p i < p i, r i R D(p π i (a) = i )(p i c) if p i = p i ad R 0, 1 Î D(p i)(p i c) if p i = p i ad R = 0, 0 if p i > p i, where p i = mi j i p j, Î = {i I : p i = mi j I p j }, ad R = j Î r j. We emphasize that this is just oe way to model flexible market sharig i ocooperative way. The detail of this particular mechaism is ot importat. Essetially, what we eed is just that the firms agree o how to share the market o the equilibrium path. There are may other ways to model flexible market sharig without affectig ay of our results. 11 Give the stage game described above, we ow defie the ifiitely repeated game. We adopt the stadard discrete time model i which the above stage game is played i each of the periods t N. The distiguishig feature of our dyamic Bertrad competitio model is that firms have differet discout factors give by δ i [0, 1), i I. The set of possible histories i period t is give by H t = A t 1, where A 0 idicates the iitial history, ad A t deotes the t-fold product of A. A period t-history is thus a list of t 1 actio profiles. We suppose perfect moitorig throughout, i.e., all firms observe every actio profile chose i the past. Settig H = t N H t, a pure strategy for firm i is defied as a mappig σ i : H A i, ad cosequetly, a strategy profile is give by σ = (σ i ) i I. We say that a firm eters the market i period t whe the firm s market share is 0 util period t 1 ad becomes strictly positive for the first time i period t. Each strategy profile σ iduces a ifiite sequece of actio profiles a(σ) = (a t (σ)) t N A, where a t (σ) A deotes the actio profile iduced by σ i period t. We call the sequece a(σ) outcome path (or more simply, outcome) geerated by a strategy profile σ. Fially, for a give strategy profile σ, ad its correspodig outcome path a(σ) = (a t (σ)) t N, the discouted payoff for firm i at time t is give by U i,t (a(σ)) = δi τ t π i (a τ (σ)). τ=t 11 Athey ad Bagwell [2] adopts a similar mechaism where market shares are allocated flexibly via commuicatio. 8

9 I the followig sectios, we will focus o subgame perfect equilibrium, ad we will limit our attetio to pure strategy equilibria. We ofte suppress actios ad state the equilibrium coditios i terms of the firms profits. For simplicity, we deote firm i s profit ad the joit profit i period t o the equilibrium path by π i,t ad π t = π i,t respectively. If a sequece of payoffs π i,t, i I, t N is geerated by a equilibrium, the they must satisfy the followig icetive costraits i every period: π t U i,t = π i,t + δ i U i,t+1, t N O the other had, it is clear that ay sequece of profit profiles that is feasible ad satisfies this coditio ca be geerated by some equilibrium. Note that we ca use the worst equilibrium with 0 profit after ay uilateral deviatio without loss of geerality. Thus we ca use the above coditio as the equilibrium coditio. 3 Critical Average Discout Factor for Collusio I this sectio, we derive a ecessary ad sufficiet coditio to sustai a collusive equilibrium outcome. We say that the firms are colludig whe there is at least oe period i which the equilibrium outcome is ot a competitive oe, i.e. whe there is at least oe firm that makes positive profit i some period. We formalize this as follows. Defiitio 1. A outcome a = (a t ) t N is cosidered a collusive outcome if ad oly if there exists t N such that π i (a t ) > 0 for some i I. A collusive equilibrium is a subgame perfect equilibrium that geerates a collusive outcome. A p-collusive equilibrium for p > c is a collusive equilibrium i which every firm chooses p every period o the equilibrium path. A p-collusive equilibrium is statioary if the equilibrium market share of each firm does ot chage over time. The we ca obtai the followig sharp characterizatio regardig collusive equilibria, which says that there exists a p-collusive equilibrium if ad oly if the average discout factor exceeds some critical threshold. there exists a collusive equilibrium. Furthermore, p m -collusive equilibrium exists wheever If the average discout factor is strictly below this threshold, the the competitive outcome (price equals margial cost) prevails i every period i ay equilibrium. Theorem 3.1. There exists a collusive equilibrium if ad oly if i I δ i 1. 9

10 Wheever there exists a collusive equilibrium, there exists a statioary p-collusive equilibrium for ay p (c, p m ]. Whe the firms are symmetric, there exists a collusive equilibrium if ad oly if δ 1. Our result is a substatial geeralizatio of this well-kow result to the case with heterogeeous discoutig. The ituitio about why the average discoutig matters is as follows. A impatiet firm has a stroger icetive to break collusio. Thus a impatiet firm eeds to be assiged a larger market share to stay i collusio. O the other had, a patiet firm has a stroger icetive to keep collusio, thus is willig to give up its market share to more impatiet firms i order to sustai collusio. So the distributio of discoutig matters for the success of collusio. I our settig, it turs out the first momet of the distributio determies the possibility of collusio completely. The first part of this theorem is straightforward as show i Harrigto [16]. Suppose that every firm chooses p ad firm i s market share is α i every period. Such a statioary outcome ca be sustaied i equilibrium if ad oly if the followig icetive costrait is satisfied: π α iπ 1 δ i. By dividig both sides by π, multiplyig both sides by 1 δ i, ad summig up these iequalities across the firms, we obtai the above iequality o the average discout factor. Coversely it is clear that such α i ca be foud whe the average discout factor satisfies the above iequality. A much more difficult part of the proof, which is a more substatial cotributio of this paper, is to show that o collusive equilibrium exists whe the average discout factor is less tha 1, eve if ostatioary equilibria are cosidered. I ostatioary equilibrium, it is possible to trasfer market shares over time. Such itertemporal trasfer could be a Paretoimprovemet for the firms due to asymmetric discoutig, hece may facilitate collusio eve if collusio caot be sustaied i a statioary equilibrium. It turs out that such trasfer does ot work. To improve efficiecy, it is ecessary to let less patiet firms gai more market shares first ad let more patiet firms gai more shares later. Ituitively, such a arragemet is i coflict with less patiet firms icetive costraits i later periods. Here is a sketch of our formal proof of the above impossibility result. Firm i s icetive costrait i period t is give by the equality U i,t = π i,t + δ i U i,t+1 = π t + η i,t 10

11 where η i,t 0 is a slack variable (firm i s icetive costrait is bidig i period t if ad oly if η i,t = 0). Sice this equality holds i every period, we ca replace U i,t+1 with π t+1 + η i,t+1 to obtai π i,t + δ i π t+1 = π t + η i,t δ i η i,t+1. Summig up these equalities across the firms, we obtai the followig equatio regardig the sequece of joit profits: π t+1 = 1 i I δ π t + i where u i,t = η i,t δ i η i,t+1. 1 i I δ i u i,t, i I The coefficiet of π t is larger tha 1 if ad oly if the average discout factor is less tha 1. I fact, we ca show that, whe the joit profit is strictly positive i some period, the sequece {π t : t N} must diverge to ifiity, which is a cotradictio. To prove this formally, however, we eed to examie the behavior of the term i I u i,t carefully. This result will be geeralized to the case with capacity costraits i Sectio 5. 4 Characterizatio of Efficiet Collusive Equilibria Oe implicatio of the previous result is that we ca focus o statioary collusive equilibria without loss of geerality to examie the possibility of collusio. However it does ot tell us what is the best way to collude for the firms. I fact, almost all statioary collusive equilibria are iefficiet from the firms perspective, sice the firms would beefit from tradig market shares over time due to the heterogeeity of their patiece. I this sectio, we characterize the structure of efficiet collusive equilibria with heterogeeous discoutig. 12 I this sectio, we assume that 0 < δ 1 < δ 2 <... < δ 1 < δ < 1 for the sake of simplicity. But our results ca be exteded aturally to the case with equal discoutig for some of the firms (See footote 17). We also assume i=1 δ i > 1 throughout this sectio, which guaratees the existece of a statioary p m -collusive equilibrium by Theorem 3.1. I fact, there is a cotiuum of market shares that ca be supported by statioary p m - collusive equilibria i this case. 13 Let π i be firm i s per period profit i the statioary p m -collusive equilibrium that is worst from firm i s viewpoit. Specifically, π i is defied by 12 A collusive equilibrium is efficiet if there is o other collusive equilibrium that is Pareto-improvig for the firms. 13 Almost all such statioary p m -collusive equilibria are iefficiet because o firm s icetive costrait is bidig. 11

12 the followig bidig icetive compatibility coditio π i 1 δ i = π m. This implies that firm i s market share is 1 δ i i this statioary equilibrium. Note that the market share/profit per period is larger for more impatiet firms ad that the total discouted payoff i the worst statioary p m -collusive equilibrium is exactly π m for ay firm. 4.1 Efficiet Collusive Equilibrium with No-Moopoly Price With symmetric discoutig, every efficiet collusive equilibrium must be a p m -collusive equilibrium. Eve with asymmetric discoutig, if we restrict attetio to equilibria with statioary market share, the it is without loss of geerality to focus o p m -collusive equilibrium as Theorem 3.1 shows. However, this is ot the case i geeral. The price sometimes eeds to be below the moopoly price for some efficiet collusive equilibrium. This is because each firm s icetive costrait ca be relaxed by reducig the total profit. For example, if the total profit is reduced by ɛ by lowerig the price, the each firm s profit i the same period ca be reduced by ɛ without violatig its icetive costrait because the gai from a deviatio is the same. If we take ɛ-profit away from 1 firms i this way ad trasfer them to the remaiig firm, the this firm s et gai would be ( 2)ɛ, which is positive with three firms or more. The followig example shows that we ca apply this argumet to the best statioary p m -collusive equilibrium for firm to costruct a equilibrium with a o-moopoly price where firm s profit is higher tha its profit i ay p m -collusive equilibrium. Example with = 3 Set = 3. We maitai the assumptio that 0 < δ 1 < δ 2 < δ 3 < 1 ad δ 1 + δ 2 + δ 3 > 2. Cosider the followig sequece of payoff profiles. I period 1, π i,1 = π 1 δ i π m for i = 1, 2 ad π 3,1 = (δ 1 + δ 2 )π m π 1 for some joit profit π 1 [δ 2 π m, π m ]. From the secod period o, the best statioary p m -collusive equilibrium for firm 3 is played: π 1,t = π 1, π 2,t = π 2, ad π 3,t = (δ 1 + δ 2 1)π m for t = 2, 3,.... The icetive costraits of firm 1 ad 2 are bidig by costructio. Note that firm 3 s deviatio gai i period 1 is 2π 1 (δ 1 + δ 2 )π m, which is icreasig i π 1. Thus firm 3 s icetive costrait is satisfied as well. We will kow from Theorem 4.1 that firm 3 gais (δ 1 +δ 2 1)π m i every period i the p m -collusive equilibrium that is best for firm 3. So this equilibrium is better tha ay p m -collusive equilibrium for firm 3 if π 1 is set smaller tha π m (with a price lower tha p m ). It ca be show that ideed the best efficiet collusive equilibrium for firm 3 is obtaied whe π 1 = δ 2 π m A complete characterizatio of efficiet collusive efficiet equilibrium for the case of = 3 is available 12

13 As the above example shows, equilibrium price may eed to be strictly below the moopoly price i some efficiet collusive equilibrium. However the followig two propositios show that we do ot lose much from focusig o efficiet collusive equilibrium with moopoly price every period, which we call p m -efficiet collusive equilibrium. The first propositio shows that every efficiet collusive equilibrium must become a p m -efficiet collusive equilibrium withi a fiite umber of periods. Furthermore, the equilibrium prices must be strictly icreasig at a certai rate util they reach ad stay at the moopoly price, hece the time to reach p m -efficiet collusive equilibrium is bouded uiformly across all efficiet collusive equilibria. Therefore it is without loss of geerality to focus o p m -efficiet collusive equilibrium as log as we are ot so cocered with the iitial phase of efficiet collusive equilibria. Propositio 4.1. For ay efficiet collusive equilibrium, there exists T 1 such that the joit profit π t > 0 is mootoically icreasig over time ad satisfies 1 δ π t+1 π t for t = 1,..., T 2 ad π t = π m for ay t T. Furthermore, this T is bouded across all efficiet collusive equilibria. The ext propositio shows that every efficiet collusive equilibrium with o-moopoly price must be very asymmetric. More specifically, some firm s total discouted profit must be lower tha the profit i the worst statioary p m -collusive equilibrium, which is π m = π i 1 δ i. Propositio 4.2. Every efficiet collusive equilibrium where firm i s total discouted payoff is at least as large as π m = π i 1 δ i for every i I must be a p m -efficiet collusive equilibrium. Fially we observe that every efficiet collusive equilibrium must be a p m -efficiet collusive equilibrium for the special case of = 2. By Propositio 4.1, if the market price is ot p m i some period, the the first period price must be strictly below p m. I the first period of such a equilibrium, the icetive costraits must be bidig for both firms. Otherwise, it would be possible to geerate a Pareto-improvig outcome i the first period without violatig ay icetive costrait by icreasig the joit profit ad sharig the margial gai equally. The each firm s total discouted profit must be exactly π 1, which is smaller tha π m. But this outcome is Pareto domiated by ay statioary p m -collusive equilibrium outcome as π m correspods to the worst statioary p m -collusive equilibrium profit for both firm. upo request. 13

14 4.2 Market Share Dyamics i p m -Efficiet Collusive Equilibrium Now we provide a complete characterizatio of all p m -efficiet collusive equilibrium. Let s first cosider the simplest case with oly two firms to illustrate our mai result. With asymmetric discoutig, efficiecy ca be improved by havig more patiet firms to led some market share iitially to less patiet firms. It is ot difficult to see that the first best allocatios for the firms are characterized as follows: The market price is always the moopoly price p m. Firm 1 (more impatiet firm) gais the whole market share up to some period t 1. Firm 1 ad firm 2 share the market i some way i period t. Firm 2 gais the whole market share from period t + 1 o. Clearly this is ot a equilibrium outcome as firm 1 s icetive costrait is violated after period t + 1. Hece o first best allocatio ca be achieved by ay equilibrium. To keep firm 1 s icetive to collude, some market share must be left for firm 1. Our result shows that every efficiet collusive equilibrium takes the followig slightly differet form with two firms to resolve this trade-off betwee efficiecy ad icetive. Note that we ca focus o p m -efficiet collusive equilibrium without loss of geerality i this case as discussed i the ed of the previous subsectio. The market price is always the moopoly price p m. Firm 1 gais the whole market share up to some (ot too late) period t 1. Firm 1 ad 2 share the market i period t i such a way that firm 1 s profit is at least as large as π 1. Firm 1 gais π 1 forever ad firm 2 gais all the rest from period t + 1 o. This is very ituitive. A first best outcome is beig approximated as closely as possible subject to the icetive costrait of firm 1, who eeds to pay back i a later stage of the game. 15 The followig theorem provides a geeral versio of this result for the case with firms. Note that our assumptio implies πm π i > 0. i=1 δ i > 1 i=1 15 Hece firm 2 s icetive costrait is ot bidig i the log ru. 14

15 Theorem 4.1. Every p m -efficiet collusive equilibrium has the followig structure: there exists 1 = t 1 t 2 t such that, for every i, a sequece of equilibrium profit profiles {π i,t } i,t satisfy the followig properties (whe t i = t i+1, the descriptio at t i+1 applies to firm i s market share). 1. Firm i eters the market i period t i with π i,ti [0, π m i 1 2. π i,t = π m i 1 h=1 3. π i,t [ π i, π m i 1 π h for t = t i + 1,..., t i+1 1 h=1 4. π i,t = π i for t > t i+1 5. U i,1 π m for all i π h ] for t = t i+1 Coversely, if there exist (t 1, t 2,..., t ) ad a sequece of profit profiles π i,t such that the above properties are satisfied ad i I π i,t = π m for ay t, the there exists a p m -efficiet collusive equilibrium that geerates them. The equilibrium dyamics of each firm s market share is roughly divided ito three phases, which ca be described i words as follows. Firm i s market share is iitially 0 while more impatiet firms are gaiig profits.the firm i eters the market ad ejoys the maximum market share subject to the costrait that all less patiet firms gai their worst statioary p m -collusive equilibrium profit π h, h = 1,..., i 1 every period. As soo as a more patiet firm eters the market, firm i s per-period profit is slashed dow to π i with market share 1 δ i ad stays there forever. So each firm s market share/profit first goes up, the goes dow. Two exceptios are the most impatiet firm (firm 1) whose market share is mootoically decreasig over time ad the most patiet firm (firm ) whose marker share is mootoically icreasig over time. h=1 π h ] I the log ru, firm i s market share coverges to 1 δ i for i = 1,..., 1 ad firm s market share coverges to 1 1 i=1 (1 δ i) = 1 i=1 δ i ( 2) > 1 δ > 0 i every p m -efficiet collusive equilibrium Firm i s cotiuatio payoff is exactly π m i the log ru for i = 1,..., 1 as the icetive costrait is bidig. The most patiet firm s cotiuatio payoff is larger tha π m. 17 This theorem ca be exteded to the case where ot all discout factors are distict. For example, suppose that δ i = δ i+1 = δ ad all other discout factors are strictly ordered. The we ca treat these two firms as oe firm with discout factor δ ad their joit profit/market share behaves exactly as characterized by Theorem 4.1 except that their joit profit coverges to π i + π i+1 = 2(1 δ )π m whe i+1 <. Each firm s per-period profits ca be ostatioary i the secod phase as log as the joit profit satisfies the property i the theorem, as their icetive costraits are slack. But each firm s log-ru profit must coverge to (1 δ ) i the third phase whe i + 1 <. 15

16 Note that every cotiuatio equilibrium of p m -efficiet collusive equilibrium is a p m -efficiet collusive equilibrium. So p m -efficiet collusive equilibrium exhibits a ice reegotiatio-proof property o the equilibrium path. 18 The proof of Theorem 4.1 is based o the followig two ituitive equilibrium properties, which we establish i Appedix A.4. If a firm has ot etered the market (i.e. ever gaied a positive market share), the every firm that is more patiet should t have etered the market. Oce a firm eters the market, every firm that is more impatiet will receive its worst statioary p m -collusive equilibrium profit every period from the ext period o. All the properties of p m -efficiet collusive equilibria ca be deduced from these two properties. We would like to emphasize the followig properties of the equilibrium market share dyamics. Uique Equilibrium Selectio i the Log Ru Every efficiet collusive equilibrium evetually becomes a p m -efficiet collusive equilibrium by Propositio 4.1. Hece Theorem 4.1 implies that every firm s market share must coverge to the same level i the log ru for ay efficiet collusive equilibrium. More specifically, every efficiet collusive equilibrium coverges to the uique statioary efficiet collusive equilibrium outcome, which is the worst statioary p m -collusive equilibrium for firm i = 1,..., 1 ad the best oe for firm. 19 Therefore our model delivers a uique predictio i the log ru without ivokig ay equilibrium selectio criterio other tha efficiecy. 20 This is i cotrast with other papers that rely o statioary equilibria. As we have observed, there are usually cotiuum of statioary collusive equilibria, thus oe eeds to apply some equilibrium selectio rule to choose a equilibrium as i Harrigto [16]. 18 Of course it is ot fully reegotiatio-proof due to the perfect competitio equilibrium off the equilibrium path. 19 All the other efficiet collusive equilibria must be ostatioary. 20 Eve if ot all firms discout factors are distict, we have the same uique log-ru predictio for all firms except for the most patiet oes (see footote 17). If there are more tha oe most patiet firms, there ca be multiple log-ru statioary outcomes. For example, if δ 1 = δ = δ > δ 2, the ay (π 1, π ) such that 2 i=1 πi + π 1 + π = π m ad mi{π 1, π } (1 δ)π m ca be sustaied i the log ru i a statioary efficiet collusive equilibrium. Note that the log-ru market share ca be very sesitive for a small chage of discout factor aroud δ. If δ 1 chages from δ ɛ to δ + ɛ, the firm 1 s log-ru market share would jump up. To be precise, firm 1 s (possible) log-ru market share is upper hemicotiuous ad becomes a odegeerate closed iterval at (ad oly at) δ 1 = δ. 16

17 Heterogeeity of Firms i the Log Ru There are two firms that would potetially domiate the market i the log ru. Sice firm i(< ) s market share is 1 δ i, which is decreasig i patiece, the market share of the most impatiet firm teds to be large i the log ru. O the other had, the market share of the most patiet firm ca be quite large because it gais all the residual market share i the ed. 21 Hece the distributio of log-ru market shares ca be U-shaped with respect to the patiece of the firms. If we evaluate the degree of heterogeeity by usig the market shares as weights, the the heterogeeity of patiece i the market would be magified i the log ru. The reaso why those two firms would occupy a larger market share tha others is very differet. The most impatiet firm requires a large market share for the sake of icetive. The most patiet firm gais a large market share for the sake of efficiecy. Mootoicity of Market Share Dyamics Our result sheds light o the structure of the dyamics of efficiet equilibrium i repeated games with differetial discoutig, especially the behavior of middle-patiet players. Lehrer ad Pauzer [20] shows that every efficiet equilibrium for two player repeated games with differetial discoutig exhibits the followig type of mootoicity: the more impatiet player s cotiuatio payoff decreases over time ad the more patiet player s payoff icreases over time. Not much has bee kow for the structure of equilibrium whe there are more tha two players. I our model, the least patiet firm s payoff/market share ad the most patiet firm s payoff/market share exhibits the same mootoicity. 22 Furthermore, the middle-patiet firms payoffs/market shares are hump-shaped. They go up first, the go dow ad stay costat evetually. It is useful to compare our log-ru market share to the oe i Harrigto [16], which is derived by applyig the Nash bargaiig solutio to the set of statioary subgame perfect equilibrium payoffs. I Harrigto [16], k most impatiet firms receive the worst statioary market share 1 δ i for i = 1,..., k ad other k firms share the remaiig market equally, where k is the miimum umber such that the icetive costraits of these k firms are satisfied. Our log-ru outcome is quite differet from this allocatio as the most patiet firm plays a very special role as the residual claimat. The differece is most stark whe every firm, eve the most impatiet oe, is very patiet. I this case, a completely equal 21 The most patiet firm s log ru market share is larger tha the most impatiet firm s log ru market share if ad oly if 1 i=1 δi + δ1 > However, observe that firm 3 s cotiuatio profit decreases over time i our example i Subsectio 4.1. Hece this mootoicity does ot hold for efficiet collusive equilibrium with o-moopoly price. 17

18 market sharig would be obtaied with k = 0 i [16]. Our log-ru statioary market share would be very asymmetric i this case. Sice 1 δ i is small for every firm, firm s market share would be close to 1 ad every other firm s market share is very small. O the other had, our solutio would coicide with the oe i [16] whe k = 1. This is the case if ad oly if the remaiig log-ru market share for firm i our model (i.e. 1 1 i=1 (1 δ i)) is strictly smaller tha the log-ru market share for firm 1, which traslates to the followig coditio: 1 i=1 δ i + δ 1 < 1. This coditio is satisfied whe the average discout factor is close to the threshold for Theorem Capacity Costrait ad Heterogeeous Discoutig We ca exted our collusio possibility result i Sectio 3 to the case where firms face capacity costraits. Suppose that each firm ca produce at most K uits at costat margial cost c. To avoid complicatios, we assume that D(c) ( 1)K, which meas 1 firms ca meet the whole demad at the competitive price. Uder this assumptio, 0 profit outcome ca be still sustaied as a equilibrium outcome, hece used as a puishmet as before. 23 We restrict attetio to collusive equilibrium with uiform price, where every firm charges the same price withi each period. 24 Whe there is o capacity costrait (i.e. K = ), every collusive equilibrium is essetially a collusive equilibrium with uiform price. So this otio geeralizes collusive equilibrium to the case where firms are capacitycostraied. Clearly p-collusive equilibrium is a example of a uiform price equilibrium, but uiform-price equilibrium is more geeral as the equilibrium prices ca chage over time. Let R(p, K) = mi{d(p),k} D(p) be the ratio of the maximum market demad oe firm ca steal by chargig a price slightly below the market price p. Sice D(p) is decreasig, R(p, K) is odecreasig i p. R(p, K) = 1 whe K is large eough. The followig result is a geeralizatio of our collusio possibility result with capacity costraits. 23 I geeral, capacity costrait has a ambiguous effect o collusio. O oe had, a smaller productio capacity makes price-cuttig less profitable. O the other had, it may make puishmet less harsh. This assumptio meas that the secod effect is abset. Hece the itroductio of capacity costrait makes collusio uambiguously easier. 24 Whe there are capacity costraits, i geeral we eed a ratioig rule to determie who would be able to purchase the product whe the demad exceeds the total supply give ay realized price (Davidso ad Deeckere [11]). However, sice we focus o uiform-price equilibria ad assume D(c) ( 1)K, the choice of ratioig rule is irrelevat i our paper. A firm would make o sale whe chargig a higher price tha the equilibrium price idepedet of the choice of ratioig rule. 18

19 Theorem 5.1. There exists a collusive equilibrium with uiform price if ad oly if i I δ i R(p,K) 1. for some p (c, p m ]. Whe this iequality is satisfied for p (c, p m ], there exists a statioary p-collusive equilibrium. Note that the set of prices that ca be sustaied i collusive equilibrium depeds o the size of the capacity costrait. Give K ad other parameters, the set of prices that ca be supported is give by some iterval (c, p), where p may be strictly below the moopoly price p m. Without capacity costrait, p m -collusive equilibrium exists wheever there exists a collusive equilibrium. With capacity costrait, eve if the moopoly price caot be supported, a lower price may be supported i collusive equilibrium. This is because the proportio of market share a firm ca steal by price-cuttig decreases as the market price decreases. 6 Discussio ad Extesio Asymmetric Capacity Costraits This paper examies the impact of asymmetric discoutig o collusio. Several papers such as Compte, Jey ad Rey [9] ad Lambso [19] examie the impact of asymmetric capacity costraits o collusio. A geeral message that emerges from those papers is that asymmetry i capacity costraits makes it harder to sustai collusio. To isolate the effect of asymmetric discoutig, we itetioally kept every other part of our model symmetric. But this does ot mea that we caot allow for asymmetric techology. We ca itroduce asymmetric capacity costraits to our model ad replicate some of our results i a straightforward way. Let K i be firm i s capacity costrait ad R i (p, K i ) = mi{d(p),k i} D(p). The defie firm i s effective discout factor by δ i (p, K i ) = 1 (1 δ i )R i (p, K i ). Note that a smaller capacity costrait makes the effective discout factor higher. The it is straightforward to prove the followig theorem. Theorem 6.1. There exists a p-collusive equilibrium with p (c, p m ] if ad oly if i I δ i(p, K i ) 1. Wheever a p-collusive equilibrium exists, there exists a statioary oe. 19

20 This theorem is ot as geeral as our previous theorems such as Theorem 3.1 ad Theorem 5.1 as we restrict attetio to p-collusive equilibrium. 25 Noetheless, this result already has a iterestig implicatio because of the way i which asymmetric discoutig ad asymmetric capacity costraits iteract i our model: asymmetric capacity costraits may facilitate collusio whe firms differ i their patiece. Ituitively, this is because a smaller capacity would mitigate the icetive problem of a impatiet firm. The above theorem shows that collusio is more easily sustaied whe i I (1 δ i)r i (p, K i ) is smaller. We ca make this umber smaller by icreasig oe uit of capacity of a patiet firm ad decreasig oe uit of capacity of a impatiet firm. Thus a certai kid of asymmetry i capacity costraits may facilitate collusio with asymmetric discoutig. More Geeral Model of Discoutig We assume that discout factors are fixed throughout the game. Aother way to model asymmetric discoutig would be to assume that discout factors are radom ad may chage over time. For example, Dal Bo [10] cosiders a repeated Bertrad model where discout factors are i.i.d. radom variables across time, but are commo to all firms. It would be iterestig to examie whether ad how our results could be exteded a model i which differet discout factors are allowed together with some flexible market sharig rule. 26 We expect that it is possible to obtai similar results to ours for the i.i.d. case. Bagwell ad Staiger [3] cosiders a model i which demad growth alterates betwee fast ad slow growth phases accordig to a Markov process ad proves that collusive prices are weakly procyclical whe growth rates are positively correlated through time. A reiterpretatio of this growth dyamics is that the discout factor moves betwee high ad low values. It would be iterestig to cosider a model where growth phases ca differ across firms. 27 We ca go further to explicitly model the factors (ex. imperfectio i credit market) that geerate asymmetric discoutig edogeously. I such a model, the dyamics of discoutig factors may be either i.i.d. or completely persistet. This would be a very iterestig ad challegig problem to study, but we expect that our ituitio about the trade-off betwee icetive ad efficiecy is still robust ad valid eve i such a geeral model. This topic is left for future research. 25 Due to the depedece of δ i(p, K i) o p, it is ot straightforward to exted the statemet to all collusive equilibria. 26 Dal Bo [10] assumes a equal sharig rule. 27 To model this formally, we eed to itroduce differetiated products so that each firm has its ow market. 20