Equilibrium Vertical Foreclosure in the Repeated Game

Transcription

1 Equilibrium Vertical Foreclosure in the Repeated Game Hans-Theo Normann y Royal HollowayCollege, University of London January 14, 005 Abstract This paper analyzes whether vertical foreclosure can emerge as an equilibrium in an in nitely repeated game. In the stage game, a vertically integrated rm gains from foreclosure due to a \raising rival's costs" e ect but foreclosure is not a static Nash equilibrium because of a commitment problem. This paper shows that, in the repeated game, foreclosure is a subgame perfect equilibrium. Regarding the price charged to the nonintegrated rm, equilibrium re nements predict that this price will not be lower than the monopoly price. Further, foreclosure as a collusive equilibrium may be easierto sustain than collusionin the nonintegrated industry. Keywords: collusion, foreclosure, vertical integration JEL classi cation numbers: D43, L13, L3, L40 I am grateful to DirkEngelmann for helpful comments. y Department of Economics, Egham, Surrey TW0 0EX, UK, fax: , hans.normann@rhul.ac.uk

2 1 Introduction One of the main arguments in the literature on vertical integration is that integration may cause foreclosure because this \raises rival's costs". The argument was rst put forward by Ordover, Saloner and Salop (1990), henceforth OSS, and is as follows. When a vertically integrated rm forecloses nonintegrated downstream rms, that is, when it withdraws from the input good market, upstream competition becomes weaker. This reduction of competition implies higher input cost for the nonintegrated downstream rms. Since the downstream unit of the integrated rm bene ts when rivals' costs are raised, the integrated rm is better o pursuing such a foreclosure strategy compared to the case where it actively competes. While OSS has been rather in uential in the theoretical industrial organization literature 1, it has also been criticized. Hart and Tirole (1990) and Rei en (199) pointed out that the result depends on the assumption that integrated rms can commit not to supply nonintegrated downstream rivals. Without commitment, vertically integrated rms will compete just like the other upstream rms. In the static Nash equilibrium without commitment, nonintegrated downstream rms are not foreclosed and equilibrium prices are the same with and without vertical merger. This paper investigates whether foreclosure can be an equilibrium of the in nitely repeated game. This possibility has been suggested by Riordan and Salop (1995) and the intuition is straightforward. Repeated interaction (Macauley, 1963) can serve as a commitment device in that it helps the integrated rm establishing a reputation for staying out of nonintegrated markets. Since all upstream rms bene t from vertical foreclosure in the long run, it could be an equilibrium of the repeated game. The oligopoly model of the stage game in this paper is very similar to the OSS's model. There is a duopoly both upstream and downstream. Upstream rms charge linear prices to the downstream rms and engage in perfect price competition. At the downstream level, there is di erentiated price competition. Whereas OSS consider a static game, we analyze the in nitely repeated game. The results are that foreclosure is indeed a subgame perfect Nash equilibrium of the in nitely repeated game, provided rms' discount factor is su±ciently high. In a general model, we show that in fact any individually rational price charged to the nonintegrated downstream rm can be part of a collusive equilibrium. Equilibrium re nements help reducing the set of plausible prices, and they suggest that the nonintegrated rm will be charged a price at least as high as the monopoly price. Using a parametrized 1 Salinger (1988), Hart and Tirole (1990) and OSS are seminal among the new foreclosure theories. Martin, Normann and Snyder (001) and Rey and Tirole (003) contain reviewsof foreclosure theories.

3 model, wefurther show that foreclosure as acollusiveequilibrium under verticalintegration often requires a lower minimum discount factor than collusion under vertical separation. In other words, vertical integration may facilitate collusion. Finally, we discuss whether downstream collusion can be a more attractive collusive strategy. Following the critique by Hart and Tirole (1990) and Rei en (199), various papers have shown that OSS's foreclosure result can be rigorously derived from game-theoretic models. Ordover, Saloner and Salop (199) re-establish their result in a descending-price auction. Riordan (1998) analyses backward integration by a dominant rm with a cost advantage. Choi and Yi (000) and Church and Gandal (000) show the result if upstream rms can commit to a technology which makes the input incompatible to nonintegrated downstream rms. In Chen (001) downstream rms strategically choose upstream suppliers, and Riordan and Chen (003) investigate the connection between vertical integration and exclusive dealing contracts. This paper contributes to this literature by showing equilibrium vertical foreclosure in the in nitely repeated game of the original OSS model. The next section introduces the general model and section 3 reports the results derived from that model. Section 4 introduces a parametrized model. Whether or not foreclosure facilitates collusion in the parametrized model is analyzed in the section 5 and downstream collusion is considered in section 6. The nal section is the conclusion. The Model and Static Nash Equilibrium Apart from minor di erences, the stage game market model is as in OSS. There are two upstream rms and two downstream rms. Call the two upstream rms U 1 and U; and the two downstream rms D1 and D. The integrated rm will be called U1-D1. We start by analyzing the downstream level. Downstream rms pay linear prices for the input which constitute their only cost. De ne c i as the price per unit rm Di pays. There is di erentiated price competition and Q i (p i ;p j ); i;j =1;;i 6= j; denotes the demand function of Di: Accordingly, Di's pro ts are ¼ Di = (p i c i )Q i (p i ;p j ); i;j = 1; ;i 6= j: (1) We impose the following assumptions on demand. Demand functions Q i (p i ;p j ) are twice continuously di erentiable i =@p i < i =@p j > 0; i =@p i =@p j < 0; i;j = 1; ; i 6= j: These This can be generalized to more complex downstream cost functions. SeeOSS. 3

4 assumptions ensure downward sloping demand with substitutes goods. Further, we assume that goods are strategic complements, that Q i =@p j > 0. A nal assumption is Q i =@p i +@ Q i =@p j < 0: These assumptions imply that the upward sloping best-reply functions have a slope of less than one and, hence, they are su±cient to ensure the existence of a unique Nash equilibrium of the stage game. 3 Let p i (c i;c j ); i;j =1; ; i 6= j; denote the static Nash equilibrium prices at the D level. In the static Nash equilibrium, the input prices (c i ;c j ) su±ciently describe downstream competition, and we will often use Q i (c i;c j ) as a shortcut for Q i (p i (c i;c j );p j (c j;c i )); and ¼ Di (c i;c j ) for ¼ Di = (p i (c i;c j ) c i )Q i (c i;c j ): Given the above assumptions, one can show that raising the cost of a downstream rival is pro table, that Di(c i ;c j j j > 0; i;j = 1; ;i 6= j: () Note Di =@p j > 0 follows i =@p j > 0; j=@c j > 0 follows from comparative statics of the rst order Dj =@p j = 0: We now turn to the upstream level. U1 and U have constant marginal cost which we normalize to zero. The upstream rms compete in prices and we assume perfect Bertrand competition between them. The U rm posting the lower price to Di supplies Q i (c i;c j ) i;j=1; ; i 6= j; to this downstream rm and obtains a pro t of c i Q i (c i ;c j ). The high price rm sells nothing to Di and, in the case of a tie, each rm supplies Q i (c i;c j )=: The static Nash equilibrium at the upstream level is as follows. Without vertical integration, competition µa la Bertrand implies prices equal to marginal cost for both Di; that is, c 1 = c = 0. This static Nash equilibrium is unique. When U1 and D1 are integrated, the downstream segment of U 1-D1 is delivered internally at c 1 = 0 and the two upstream rms compete for D only. As emphasized by Hart and Tirole (1990) and Rei en (199), the unique static Nash equilibrium has U1-D1 and U charging a price equal to marginal cost. That is, we have c 1 = c = 0, just as in the case without integration. U earns zero pro ts and U 1-D1 earns ¼ D1 (0; 0) in the static Nash equilibrium.4 3 Collusive Foreclosure in the Repeated Game In this section, we will analyze under which conditions foreclosure emerges as an equilibrium outcome in the above general model. Collusion at the upstream level with vertical integration will be analyzed 3 The assumptions to ensure a unique static Nash equilibrium are merely made to simplify the analysis. Under weaker assumptions,thestagegamehasmultipleequilibriaandonewouldneedtodistinguishbetweenstableandunstableequilibria (see OSS). Note also that somewhatweaker conditionsmightbe su±cient to guarantee existence and uniqueness (see Vives, 1999). 4 Also OSS acknowledge that this is the Nash equilibrium without commitment. 4

5 in this section this is the case where foreclosure may occur. Collusion under vertical separation and collusion at the downstream level will be considered below using a parametrized model. Consider the vertically integrated industry and recall c 1 = 0 because U1 and D1 are integrated. We suppose that rms try to implement the following collusive foreclosure strategy where c denotes the collusive input price charged to D. Collusive foreclosure involves U charging D an input price c 0 and U1-D1 not supplying D. (Alternatively, U 1-D1 could also post a price slightly larger than c as part of the collusion. This alternative implies minor modi cations of the results which we discuss at the end of this section.) At the downstream level, static Nash equilibrium prices, p 1 (0;c ) and p (c ; 0); and outputs, Q 1(0;c ) and Q (c ; 0); will be realized. This notion of foreclosure is just as in the static game but, as a collusive strategy, it is very di erent from normal oligopoly collusion. When adhering to collusion, U 1-D1 stays out of the market whereas it enters (at a price smaller than c ) when defecting. U is simply a monopolist when collusion is successful but c will generally not be U's preferred monopoly price. When defecting U ; is still a monopolist and then it will surely charge its monopoly price. Many values of c can potentially be part of an equilibrium in the in nitely repeated game, and only careful application of equilibrium selection criteria can help identifying more plausible values of c. Before solving the repeated game, it is useful to de ne three particular levels of c : The rst is the one just mentioned where U charges the price which maximizes it's own pro ts. Denote this price by c mon and de ne formally c mon := arg max c c Q (c ; 0): (3) The second benchmark is the one that maximizes U 1-D1 pro ts if it serves D in addition to D1 as a monopolist. De ne this level formally by c int := arg max c ¼ D1 (0;c ) + c Q (c ; 0): (4) Note that c int > c mon due D1 =@c > 0: This second benchmark will be important when we analyze defection by U 1-D1: Finally, it is useful to de ne the level of c where U 's output (and therefore pro t) becomes zero, denoted by c. Formally c := fq (c ; 0) = 0g: (5) Note that c > c mon but c int may be smaller or larger than c : All three benchmarks are unique. 5

6 We now analyze collusion in the in nitely repeated game. Time is indexed from t = 0;:::;1: Firms discount future pro ts with a factor ±; where 0 ± < 1. When analyzing the repeated game, denote by ¼ c 1 and ¼ c the pro t U1-D1 and U earn respectively when both rms adhere to collusion. Let ¼ d i denote the pro t when a rm defects, and ¼ p i is the average pro t per period when a punishment path is triggered. We look for collusive equilibria that are subgame perfect. In order the prevent defection in any period t; we need or 1X 1X ± t ¼ c i ¼ d i + ± t ¼ p i (6) t=0 t=1 ± ¼d i ¼ c i ¼ d i ¼ p i := ± i ; (7) where i = 1; ; and ± i denotes the minimum discount factor required for rm i to adhere to collusion. Consider the collusive pro ts, ¼ c i; rst. Since c 1 = 0; U 1-D1's collusive pro t is ¼ c 1 = ¼ D1 (0;c ); that is, U 1-D1 does not make any pro t at the upstream level in equilibrium. U makes a collusive pro t of ¼ c = c Q (c ; 0): Defection from collusive foreclosure yields the following payo s. If U defects, it charges c mon no matter which c is part of the collusion and earns ¼ d = cmon Q (cmon ; 0): If U1-D1 defects, it undercuts U 's price, c, in order to gain the D business. If c c int, U1-D1 will simply charge a price in nitesimally smaller than c and we get ¼ d 1 = ¼ D1 (0;c ) + c Q (c ; 0) = ¼ c 1 + ¼c. For c > c int, U1-D1 will defect by charging c int (as follows from the de nition of c int ) and we obtain ¼ d 1 = ¼ D1(0;c int ) + c int Q (c int ; 0) > ¼ c 1 + ¼ c. Consider now the punishment following a defection. In the unique static Nash equilibrium, c = 0 and upstream pro ts are zero as there is perfect Bertrand competition. These are also the maximin pro ts and so more severe punishment strategies do not exist. Hence, we adopt simple trigger strategies with Nash reversion here and have ¼ p 1 = ¼ D1(0; 0) and ¼ p = 0 for U1-D1 and U ; respectively. Nash reversions are credible threats, so, as long as (7) holds, we have subgame perfect equilibria. Finally, collusive equilibria must be individually rational, that is, rms must get at least their maximin pro ts in a collusive equilibrium of the repeated game. Maximin pro ts, ¼ D1 (0; 0) for U 1-D1 and zero for U, result when c = 0: U also gets zero pro ts with c = c : Hence, any positive c < c implies ¼ c 1 = ¼ D1(0;c ) > ¼ D1(0; 0) (from ()) and ¼ c = c Q (c ; 0) > 0 and therefore ful lls the individual rationality requirement. 6

7 We are now ready to prove Proposition 1 Vertical foreclosure is a subgame perfect Nash equilibrium of the repeated game, provided rms' discount factor ± is su±ciently high. Moreover, any individually rational collusive input price 0 < c < c can be sustained with a high enough discount factor. Proof. We prove the more general second part by showing that ± i < 1; i=1,, for all 0 < c < c. Now, ± i < 1 if ¼ d i ¼ c i > ¼ p i. First, recall ¼c 1 = ¼ D1 (0;c ) > ¼ D1 (0; 0) = ¼p 1 and ¼c = c Q (c ;0) > 0 = ¼ p. This establishes the strict inequalities. Second, we prove ¼d i ¼ c i: If c c int, U1-D1 gets ¼ d 1 = ¼ c 1 + ¼ c > ¼ c 1 when defecting. If c > c int ; U 1-D1's optimal defection is to charge c int and we have ¼ d 1 = ¼ D1 (0;c int ) + c int Q (c int ; 0) > ¼ c 1: To prove ¼ d ¼ c ; note that ¼ d = c mon Q (c mon ; 0) ¼ c by de nition of c mon. Proposition 1 shows that vertical foreclosure can indeed by sustained as a subgame perfect Nash equilibrium of the in nitely repeated game. The result counters the criticism the OSS result received (not being a Nash equilibrium) whenever there is repeated interaction and the discount factor is su±ciently high. Note that for all 0 < c < c a strictly positive minimum discount factor is required. De ne ± := min c maxf± 1 (c );± (c )g: (8) ± > 0 follows from ¼ d 1 > ¼ c 1 > ¼ p 1 > 0 and therefore ± 1 > 0 for all 0 < c < c. Similarly ± > 0 unless c = c mon : Hence maxf± 1 ;± g > 0 for all c : Proposition 1 contains a Folk Theorem-like message on the action domain. All individually rational collusive input prices can be an outcome of an equilibrium in the repeated game. This raises the question if equilibrium selection criteria help reducing the set of equilibria. Proposition Only collusive equilibria with c c mon are Pareto e±cient. Proof. By de nition of c c =@c > 0 if and only if c < c mon. From c 1=@c > 0 for all c : Hence, an input prices c < c mon are not Pareto e±cient but those in the interval [c mon ;c ) are. Pareto e±ciency (from the rms' point of view) suggests that the collusive prices we may expect to occur will be at least as high as c mon : The intuition is that, when c is increased beyond c mon ; U 1-D1 gains unambiguously while rm U loses for either c < c mon and c > c mon : Hence, the bargaining 7

8 situation implicit in the repeated game can plausibly lead to equilibrium outcomes with c c mon. By contrast, in the one-shot game analyzed in OSS (assuming U 1-D1 can commit not to deliver D), U is a monopolist in the D market and would therefore never charge a price other than c mon. The characterization of Pareto e±cient equilibria does not indicate whether equilibria with c c mon are likely to meet the incentive constraint (7). To answer this question, the following lemma is helpful. It states how minimum discount factors ± 1 and ± as in (7) respond to changes of c. Lemma Let ± 1 (c ) and ± (c ) denote the minimum discount factors required by U 1-D1 and U respectively as functions of the collusive input price c : (i) If c c int 1(c )=@c R 0 if and only if (@¼ c =@c )(¼ c 1 ¼p 1 ) (@¼c 1 =@c )(¼ c ) R 0: If c > c int 1 (c )=@c < 0: (c )=@c R 0 if and only if c R c mon. Proof. Consider part (i). If c c int ; ¼ d 1 = ¼ c 1 + ¼ c ; we get ± 1 (c ) = ¼ c =(¼ c + ¼ c 1 ¼ p 1 ) and, 1 = (@¼c =@c )(¼ c 1 ¼ p 1 ) (@¼c 1=@c )(¼ c (¼ c 1 + ¼c : (9) ¼p 1 ) This yields the condition in part (i) of the lemma. If c > c int ; ¼ d 1 = ¼ D1(c int ; 0) + c int Q (c int ; 0), ± 1 = ¼ D1(c int ; 0) + c int Q ¼ c 1(c; 0) ¼ D1 (cint ; 0) + c int Q ¼ p 1 (10) and c 1 =@c ¼ D1 (cint ; 0) + c int Q ¼ p < 0: (11) 1 Then consider part (ii). Note that ¼ d = ¼c (cmon ; 0) is a constant, hence, we =@c = (@¼ c =@c )=(¼ d ) : =@c R 0 if and only if c R c mon : The lemma states that ± 1 decreases in c if either a regularity condition is met or if c > c int whereas ± has got a global minimum in c = c mon : The next proposition puts the lemma and proposition together by characterizing extremal equilibria. Extremal equilibria are the Pareto undominated equilibria in the set of subgame perfect Nash equilibria that are feasible given the incentive constraint (7). Proposition 3 Extremal subgame perfect Nash equilibria involve c c mon ; provided (@¼ c =@c )(¼ c 1 ¼ p 1 ) (@¼c 1 =@c )(¼ c ) < 0: 8

9 Proof. Consider a subgame perfect Nash equilibrium with c 0 < c mon and assume the condition in the proposition is met. One 1 =@c < 0 =@c < 0 from the lemma. From c =@c > 0 c 1=@c > 0: This implies that selecting an equilibrium with c > c 0 improves both rms pro ts and requires a lower minimum discount factor, as long as c < c mon. Therefore, no equilibrium with c 0 < c mon can be an extremal equilibrium, provided (@¼ c =@c )(¼ c 1 ¼ p 1 ) (@¼c 1=@c )(¼ c ) < 0. The subgame perfect Nash equilibria with c 0 c mon are not Pareto ranked. Hence, all subgame perfect Nash equilibria with c c mon are extremal equilibria. The proposition implies that rms are likely to collude on a price c c mon because any c below c mon reduces both rms' pro ts and makes collusion more di±cult to sustain for both rms. The condition (@¼ c =@c )(¼ c 1 ¼ p 1) (@¼ c 1=@c)(¼ c ) < 0 does not appear to be particularly restrictive. It holds, for example, in the model with linear demand (see below). Note 1 =@c < 0 if c c mon anyway; only if c < c mon is the sign 1 =@c ambiguous. c 1=@c > 0; so, the condition will be met c =@c or ¼ c 1 ¼ p 1 are small. One of the key results in OSS is to show that U 1-D1 has an incentive to commit to a certain price c to prevent U and D from merging. The logic is that, for some values of c ; the joint pro ts of U and D can be increased by vertically integrating and thus supplying D internally at marginal cost. For a parametrized model, OSS show that this price is lower than c mon. When U and D can merge, this upper bound on c is also relevant in the repeated game. Then the possibility of a U -D merger forces rms to collude on a price c < c mon ; that is, a Pareto inferior outcome. This is in contrast to proposition. Collusion is generally less likely as a higher ± is required, and less damaging for welfare as a lower c will occur. So, the threat of a U -D counter merger is bene cial from a policy perspective. Figure 1 summarizes the discussion of the general model. It shows minimum discount factors ± 1 (c ) and ± (c ) for values of c between zero and c : Figure 1 is drawn with the help of the lemma and the following properties of ± i : It is straightforward to verify that lim c!0 ± (c ) = 1: Further, ± (c mon ) = 0 and lim c!c ± (c ) = 1 as follows from the de nitions of c mon and c : Regarding ± 1 ; note 0 < ± 1 (c ) < 1 strictly for all c < c due to ¼ d 1 > ¼ c 1 > ¼ p 1 > 0: 5 Further lim c!c ± 1 (c ) = 0 if, as assumed in the gure, c int > c : 6 Finally, the gure is based on the condition in proposition, (@¼ c =@c )(¼ c 1 ¼p 1 ) 5 We cannot directly determine±1(0) since±1(c) =¼ c =(¼c +¼c 1 ¼p 1 ) but¼c (0) = 0 and¼c 1 (0;0) ¼p 1 = 0: However, l'h^opital'srule yields lim ± 1(c ) = lim c!0 c!0 ¼c (c ) 0 =(¼ c (c ) 0 +¼ c 1 (c ) 0 )<1 from¼ c (0)0 =Q and¼ c 1 (0)0 > 0: 6 This follows from¼ c (c )= 0 and so¼ d 1 =¼c 1 >¼p 1. Ifcint <c ;± 1 (c )>0 follows from¼ d 1 =¼c 1 +cint Q (cint ;0) so 9

10 c 1=@c )(¼ c ) < 0. The gure i (c )=@c < 0; i=1,; when 0 < c < c mon ; suggesting that foreclosure as a collusive strategy will fail when ± < ± and will typically involve c c mon otherwise. [Figure 1 about here.] Finally, we discuss other forms of collusive foreclosure. First, U 1-D1 may not completely withdraw from the D market but instead post a price c + "; " being small, when colluding. The only thing that would change in this case is that U could not defect pro tably any more when c < c mon : To see this, note that U ideally wants to defect by charging the price c mon ; but since U1-D1 charges c +" < c mon ; this is not possible. This implies ¼ d = ¼ c and so ± = 0 if c < c mon : Everything else and in particular the results in this section remain unchanged. Second, U1-D1 and U could collude by both charging c to D; that is, they could each supply half of D's output, Q (c ; 0)=: In this case, ¼ c = c Q (c ; 0)= and so ± (c ) increases by c Q =(c mon Q ) if c > c mon : Even though ± 1 is lower in this case, ± as in (8) will often increase. Hence, while this is a subgame perfect Nash equilibrium of the repeated game, it will typically be more di±cult to sustain. 4 A ParametrizedModel In this section, we will develop a parametrized version of the model which is useful to derive further results. The market model is similar to the one in OSS's appendix and has linear demand. Demand is symmetric and the demand intercept is, without loss of generality, normalized to one Q i (p i ;p j ) = 1 kp i + dp j ; i;j; = 1; ;i 6= j; (1) where k > d 0. Products are entirely heterogenous if d = 0 while d! k would imply perfectly homogenous goods. Di's pro t is ¼ Di = (1 kp i + dp j )(p i c i ); i;j; = 1; ;i 6= j: (13) Myopic maximization at the downstream level yields Nash equilibrium prices that±1 =c int Q =(cint Q +¼c 1 ¼p 1 )>0: p i (c i ;c j ) = k + d + k c i + kdc j 4k d (14) 10

11 and equilibrium outputs Downstream pro ts are ¼ Di (c i;c j ) = (Q i ) =k: Q i (c i ;c j ) = k k + d (k d )c i + kdc j 4k d : (15) Consider now the in nitely repeated game with vertical integration. As above, U1-D1 forecloses D and U delivers D at a collusive price c 0. The integrated rm delivers its downstream unit at c 1 = 0. From (15), U 1-D1 and D will sell the following quantities U 1-D1's downstream pro t is whereas D's pro t is Q 1 (0;c ) = k k + d + kdc 4k d ; (16) Q (c ; 0) = k k + d (k d )c 4k d : (17) µ k + d + ¼ kdc D1(0;c ) = k 4k d (18) µ k + d (k ¼ d )c D(c ; 0) = k 4k d : (19) U 1-D1 does not make any pro t at the upstream level. U makes a pro t of c Q (c ; 0) or For U1-D1 punishment pro ts are ¼ U (c ; 0) = c k k + d (k d )c 4k d : (0) while ¼ p = 0: ¼ p 1 = k ¼ D1 (0; 0) = (k d) (1) and The benchmark prices can easily be obtained as The third benchmark is c int c mon = k + d (k d ) : () = (k + d) (4k + kd d ) (8k 4 7k d + d 4 : (3) ) c = k + d k d: (4) 11

12 Note c mon = c = but c int R c is ambiguous. First, consider U 1-D1's incentives to collude. If c < c int ; U1-D1's defection pro t is the sum of ¼ U as in (0) and ¼ D1; its own equilibrium pro t as in (18). If c c int ; U 1-D1's defection pro t is the sum of c int Q (cint ; 0) and ¼ D1 (0;cint ): The trigger strategy implies a punishment pro t as in (1). Plugging these values into (7), the minimum discount factor required for U 1-D1 to adhere to collusion is 8 >< ± 1 = >: (4k d )(k + d c (k d )) 8k (k + d) d 3 c (8k 4 7k d + d 4 ) if c <c int (8k (k + d) d 3 )c int (8k 4 7k d + d 4 )(c int ) c kd(k + d) c k d (8k (k + d) d 3 )c int (8k 4 7k d + d 4 )(c int if c ) c int where c int is de ned as in (3). In the general model, when c c 1 =@c < 0 if and only if (@¼ c =@c )(¼ c 1 ¼p 1 ) (@¼c 1 =@c)(¼ c ) < 0 from the lemma. For the parametrized model, this condition becomes (4k + kd d )c k 3 d (4k d ) (k d) When c > c int, we 1 =@c < 0 from the lemma. (5) < 0: (6) Now turn to U 's incentives to collude. U's collusive pro t is ¼ U(c ; 0); and U 's defection pro t is ¼ U (cmon ; 0): The punishment pro t is ¼ U (0; 0) = 0. Plugging these expressions into (7), one obtains µ d + c d + k 4c k ± = : (7) k + d We know the sign =@c from the lemma. 5 Does VerticalIntegrationFacilitateCollusion? The result that vertical foreclosure can emerge as an equilibrium outcome of the repeated game does not necessarily suggest a policy against vertical integration. The make a point against vertical integration, one needs to show that theindustry is more prone to collusion with integration than without, that is, one needs to show that vertical integration facilitates collusion. In an in nitely repeated game, an industrial policy can be said to facilitate collusion if the industry requires a lower minimum discount factor than without the policy. To investigate this, we now compare the required minimum discount factors with and without vertical integration. Without integration, it is straightforward to solve for the minimum discount factor. There are now two independent U rms competing for both downstream rms, D1 and D. Suppose rms collude on 1

13 arbitrary collusive prices c 1 and c they charge D1 and D respectively. Let ¼ c denote the sum of pro ts a rm makes in the two markets and denote the defection pro t by ¼ d : If a rm defects, it will do so in both markets (Bernheim and Whinston, 1990). Hence, ¼ d = ¼ c : Finally, already simple Nash reversions yield ¼ p = 0. From symmetry, ¼ c ; ¼ d and ¼ p are the same for both rms. It follows that the collusive prices c 1 and c can be supported as a subgame perfect Nash equilibrium if and only if ± 1=: This minimum discount factor does not depend on c 1 and c or any functional forms. 7 Further, the fact that rms collude in two markets here does not a ect to propensity to collude (Bernheim and Whinston, 1990). Using the parametrized model, we now compare the threshold of 1= to the minimum discount factor required for collusion under vertical integration. Proposition 4 In the parametrized model, collusive foreclosure with vertical integration requires a lower discount factor than upstream collusion without integration if and only if d=k > 0:380; that is, when products are not too heterogenous. Proof. We need to show that maxf± 1 (c );± (c )g < 1= for some c : To do this, we solve for ± i (c ) 1=; i=1,, as in (5) and (7), and then search for c values such that both ± i (c ) are below the threshold. First, we look for solutions to ± 1 (c ) 1=. As above, we need to distinguish c R c int : Consider c < c int and de ne bc := f± 1 (c ) = 1=jc < c int g: We obtain a unique solution, bc = 8k3 4kd d 3 8k 4 5k d + d 4 : (8) 1 =@c < 0; it follows that ± 1 (c ) < 1= for all c > cb ; provided c < c int : For such c to exist necessarily requires bc < c int : There are two unknown variables, d and k; in this inequality but we can express the solution in terms of d=k; the ratio of the slope parameters indicating the degree of product di erentiation (note that 0 d=k < 1). It turns out that bc < c int if and only if d=k > 0: 541: Hence, c satisfying bc < c < c int exist if and only if d=k > 0: 541, and for these c we have ± 1 (c ) < 1=: Then consider c c int and de ne ec := f± 1 (c ) = 1=jc c int g. Solving ± 1 (c ) = 1= for c yields two solutions. Only the positive root is plausible ce = (k + d) + p m ; (9) kd 7 This is true only ifc 1 andc are not higher than the monopoly price. But we can discard prices higher than the monopoly price as they both reduce pro ts and require a higher discount factor. 13

14 where m = (8k + 8dk + d + c int ((8k (k + d) d 3 ) c int (8k 4 7k d + d 4 ))): 1 =@c < 0; we get ± 1 (c ) < 1= for all c > ec ; provided c c int : Note that c > ec is su±cient to get ± 1 (c ) < 1= if ec c int. Now, ec c int if and only if d=k 0:541: Hence, c satisfying c > ec c int exist if and only if d=k 0: 541 and for these c we again have ± 1 (c ) < 1=. (There may also be c c int > ec solving ± 1 (c ) < 1=:) Second, we look for solutions to ± (c ) = 1=: It is straightforward to verify that ± (c ) 1= if and only if c mon (1 p 1=) c c mon (1 + p 1=): We nally establish values of c such that both ± 1 (c ) < 1= and ± (c ) < 1=. To do this, it is useful to express bc and ce in terms of c mon and to make the distinction d=k Q 0:541 since solutions to ± 1 (c ) < 1= depend on the degree of product di erentiation. First, assume 0: 541 < d=k < 1 and consider bc. If d=k = 0:541; we get bc = 1: 354c mon : If d! k; bc = c mon =: Further bc =c mon monotonically decreases in d: This implies 1:354c mon > bc c mon = and therefore c mon (1 p 1=) < bc < c mon (1 + p 1=): Hence, there exist c satisfying bc < c < c int < c mon (1 + p 1=) such that maxf± 1 (c );± (c )g < 1=: Second, assume 0:541 d=k 0 and consider ec : If d=k = 0:541; then ec = 1: 354c mon, but lim d!0 ec = +1: Because ec =c mon monotonically decreases in d; we need to nd the minimum d=k such that ec < c mon (1+ p 1=): This threshold is d=k > 0:380: If this condition is met, c mon (1 p 1=) < ec < c mon (1+ p 1=): Hence, if 0:541 d=k > 0:380; there exist c satisfying c int < ec < c < c mon (1+ p 1=) such that maxf± 1 (c );± (c )g < 1=: Since c satisfying cb < c < c int do not exist if d=k < 0: 541, the condition stated in the proposition, d=k > 0: 380; is necessary to nd c which solve maxf± 1 (c );± (c )g < 1=: The proposition shows that the vertical integration facilitates collusion in a probabilistic sense, provided the condition on product di erentiation is met. Assume d=k > 0:380: For ± < ±; neither the integrated nor the separated industry are collusive and market outcomes are the same. For discount factors between ± and 1=; only the integrated industry is collusive and this is the case an active policy would want to prevent. Finally, for ± > 1=; both the integrated and nonintegrated structure are collusive. 8 If d=k < 0:380, collusive foreclosure requires a higher discount factor than 1= and the result is reversed. In that case, vertical foreclosure is an obstacle to collusion in a probabilistic sense. 8 Note that if±>0:5, both downstream markets are supplied atc > 0 under separation, whilethis is thecaseonly for the D market with integration (D1 is delivered at marginal cost). This suggests the possibility that vertical integration is preferablefrom a policy point of view if±> 0:5, although this depends on the speci ccollusive prices charged by rms. 14

15 The proof of the proposition gives a characterization of the range of c values for which the result holds. The interval includes c > c mon ; that is, some of the Pareto e±cient extremal equilibria prices. 6 Extending the Model to Downstream Collusion So far, downstream rms were assumed to charge static equilibrium Nash prices. Using the parametrized model again, we now analyze the integrated industry when downstream collusion is possible and check whether downstream collusion is a more pro table strategy than upstream collusive foreclosure. If rms' discount factor is such that both upstream and downstream collusion are feasible, it is the integrated rm that will determine at which level collusion will occur. Since U1-D1 operates at both levels, it will price competitively at one level of the industry and collude at the other. The reason is that the integrated rm's pro t is ultimately a ected only by the prices at the downstream level (as it still delivers D1 at marginal cost), and U1-D1 does not care whether downstream prices are higher because of collusive foreclosure or because of downstream collusion. In any event, U 1-D1 will prevent collusion at both levels in order to avoid double marginalization. Will U1-D1 prefer collusive foreclosure or downstream collusion? A general answer to this question is di±cult as the outcomes of collusive foreclosure and downstream collusion are usually rather di erent in nature and often impossible to compare. We can, however, compare discount factors when the outcomes of upstream and downstream collusion coincide. Assume there is downstream collusion by U1-D1 and D and upstream competition, that is, c 1 = c = 0. Foreclosure-type collusion yields a unique outcome, determined by c 1 = 0 and c 0, and implies downstream prices of p 1 (0;c ) and p (c ; 0) respectively. Now assume that this very outcome occurs as a result of downstream collusion. Downstream rms implement collusive prices, denoted by p c i; i=1,, identical to those which would occur with foreclosure, that is, p c 1 = p 1 (0;c ) and p c = p (c ; 0); even though c 1 = c = 0. In other words, downstream rms charge collusive prices as if input costs were c 1 = 0 and c 0. We obtain p c 1 = p 1 (0;c ) = k + d + kdc 4k d ; (30) p c = p (c ; 0) = k + d + k c 4k d : (31) 15

16 Collusive quantities and pro ts are immediate ¼ c D1 = k (k + d + kdc ) (4k d ) ; (3) ¼ c D = k k + d k c + d c k + d + k c (4k d ) : (33) Note that ¼ c D = pc Q and ¼c D1 = ¼ D1 as in (18). U 1-D1 bene ts from this collusion, exactly as much as with collusive foreclosure. Also D bene ts since ¼ c D = pc Q = p (c ;0)Q > (p (c ; 0) c )Q = ¼ D(c ; 0). When analyzing defection, we need to solve for best replies. Now, p c 1 = p 1(0;c ) is a already best reply to p c = p (c ; 0): This implies ¼ d D1 = ¼ c D1 and hence ± D1 = 0 (where ± Di denotes Di's minimum discount factor) and we can henceforth focus on D: D's best reply is p d = 1 + dp 1 k = 4k + d + c d (4k d ; (34) ) which implies a defection pro t of µ 4k + d + c d : (35) ¼ d = k 4 4k d We know the static Nash equilibrium pro t is ¼ D (0; 0) = ¼ p D = k= (k d) as above. Plugging ¼ c D; ¼ d D and ¼p D into (7), we obtain ± D (c) = c 4k d (8k + c d + 4d)d : (36) Proposition 5 Consider the parametrized model and downstream collusive prices of p c 1 = p 1(0;c ) and p c = p (c ; 0): For a given discount factor, collusive foreclosure at the upstream level yields higher pro ts if ± maxf± 1 (c mon d =k );± (c mon d =k )g whereas downstream collusion yields higher pro ts if ± < ±. Proof. Regarding the rst part, note ± D (c ) 1 if and only if c c mon d =k : That is, collusive prices c c mon d =k cannot be sustained with downstream collusion. By contrast, we know that these prices were an equilibrium of the repeated game with collusive foreclosure. Since higher c raise U 1-D1's pro t, given any ± maxf± 1 (c mon d =k );± (c mon d =k )g; collusive foreclosure at the upstream level yields higher pro ts. Regarding the second part, note ± D (0) = ± D1 = 0 D (c )=@c > 0. That is, by choosing a su±ciently low c ; D1 and D can lower the minimum discount factor arbitrarily close to zero. Above, 16

17 we saw that foreclosure-type collusion requires ± > 0: Here, for ± < ±, collusion at the D level is feasible and therefore downstream collusion yields higher pro ts. The proposition shows that upstream collusive foreclosure can be both advantageous and disadvantageous compared to downstream collusion. The advantage is that prices higher than c mon d =k can only be implemented with upstream foreclosure-type collusion. The disadvantage is that any collusive foreclosure requires ±> 0 and so it may not be feasible when rms' discount factor is low. With downstream collusion, the minimum discount factor required for collusion can be arbitrarily close to zero since colluding on a low c yields prices close to the static Nash equilibrium where incentives to deviate are small for D rms. The conclusion is that upstream collusive foreclosure can be more pro table than downstream collusion and the possibility of downstream collusion does not make collusive foreclosure redundant. The proposition only states su±cient conditions. One can show that there exists some ± (±; maxf± 1 (c mon d =k );± (c mon d =k )g) such that collusive foreclosure at the upstream level yields higher pro ts if and only if ± > ± : However, the computation of ± is very cumbersome and the functional forms are not particularly informative. There are, of course, plausible forms of collusion at the D level other than p c 1(0; 0) = p 1 (0;c ) and p c (0; 0) = p (c ;0). Note, however, that U1-D1 can always switch to upstream (foreclosure-type) collusion as an outside option. That is, collusion at the D level should give U1-D1 no less than it would get in the foreclosure outcome and so collusion with p c 1(0; 0) = p 1(0;c ) and p c (0; 0) = p (c ; 0) is focal. 7 Conclusions This paper shows that vertical foreclosure can by sustained as an equilibrium of an in nitely repeated duopoly game with a raising rival's cost e ect. The result counters the criticism that foreclosure is not a Nash equilibrium of the static game analyzed by Ordover, Saloner, and Salop (1990). The results also indicate that collusive foreclosure, if successful, can be damaging from a policy perspective as equilibrium selection criteria suggest that non-integrated rms will be charged at least the monopoly price for the input good. The paper further shows that vertical foreclosure can be preferable to downstream collusion. Finally, comparing the industry with and without vertical integration, the paper shows that the minimum discount factor required for collusion is lower with vertical integration unless products are very di erentiated. In other words, vertical integration often facilitates collusion. The result that vertical integration may facilitates collusion has rst been obtained by Nocke and 17

18 White (003). Using a model with two-part tari s, they analyze collusion in vertically related oligopolies. For a variety of settings, they show that vertical integration facilitates collusion at the upstream level compared to the nonintegrated industry. In their model, a vertically integrated rm still competes for nonintegrated downstream rms, so foreclosure does not occur in their paper. References [1] Bernheim, D.C., and Whinston, M.D. (1990): Multimarket Contact and Collusive Behavior, RAND Journal of Economics 39, [] Chen, Y. (001): On Vertical Mergers and Their Competitive E ects, RAND Journal of Economics 3, [3] Chen, Y. and Riordan, M. (00): Vertical Integration, Exclusive Dealing and Ex-Post Cartelization, mimeo, Columbia University. [4] Choi, J.P. and Yi, S.-S. (000): Vertical Foreclosure and the Choice of Input Speci cations, RAND Journal of Economics 31, [5] Church, J. and Gandal, N. (000): Systems Competition, Vertical Merger, and Foreclosure, Journal of Economics and Management Strategy 9, [6] Hart, O. and Tirole, J. (1990): Vertical Integration and Market Foreclosure, Brookings Papers on Economic Activity: Microeconomics, 05{76. [7] Macauley, S.: Non-Contractual Relations in Business, American Sociological Review 8, [8] Martin, S., Normann, H.T. and Snyder, C.M. (001): Vertical Foreclosure in Experimental Markets, RAND Journal of Economics, 3, [9] Nocke, V. and White, L. (003): Do Vertical Mergers Facilitate Upstream Collusion? Penn Institute for Economic Research Paper Series, [10] Ordover, J.A., Saloner, G., and Salop, S.C. (1990): Equilibrium Vertical Foreclosure, American Economic Review 80, [11] Ordover, J.A., Saloner, G., and Salop, S.C. (199): Equilibrium Vertical Foreclosure: Reply, American Economic Review 8,

19 [1] Rei en, D. (199): Equilibrium Vertical Foreclosure: Comment, American Economic Review 8, [13] Rey, P. and Tirole, J. (003): A Primer on Foreclosure, Handbook of Industrial Organization, Vol. 3 (forthcoming). [14] Riordan, M. (1998): Anticompetitive Vertical Integration by a Dominant rm. American Economic Review 88, [15] Riordan, M. and Salop, G. (1995): Evaluating Vertical Mergers{A Post Chicago Approach, Antitrust Law Journal 63, [16] Salinger, M. (1988): Vertical Mergers and Market Foreclosure, The Quarterly Journal of Economics, 103, [17] Vives, X. (1999): Oligopoly Pricing{Old Concepts and New Ideas, MIT Press. 19

20 Figure 1. Critical discount factors as functions of c.