Backward Integration and Collusion in a Duopoly Model with Asymmetric Costs

Transcription

1 Backward Integration and Collusion in a Duopoly Model with Asymmetric Costs Pedro Mendi y Universidad de Navarra September 13, 2007 Abstract This paper formalyzes the idea that input transactions may be used to implement side payments among colluding rms. A model is proposed to analyze the e ect of backward integration on collusive outcomes in a downstream duopoly with asymmetric marginal costs. Vertical integration expands the set of collusive outcomes that are sustainable for a given realization of the discount factor. Side payments implemented by input sales are more important the larger the di erence in marginal costs, since they permit the shifting of production towards the relatively more e cient rms, while maintaining rms incentives to collude. A price of the input above that posted by an alternative source or sales of the input below cost may be observed, depending on the realization of downstream rms costs. JEL Codes: L42, L12. Keywords: Backward integration; Collusion; Duopoly. 1 Introduction Backward integration makes a rm present both in the input and in the output market, hence transforming it into a potential supplier of other nal good Department of Business Administration, Universidad de Navarra. Edi cio Bibliotecas- Este, Pamplona, Spain. pmendi@unav.es. y I thank Joel Sandonís, Rafa Moner and Pepe Sempere, as well as seminar participants at Universidad de Valencia and conference participants at EARIE-07 for their helpful comments on di erent versions of this paper. Financial support from PIUNA and Ministerio de Educación y Ciencia (SEJ /ECON) is gratefully acknowledged. All errors are my own. 1

2 producers. This could be of special interest if downstream rms are colluding, since the input price may be manipulated in order to implement side payments among them. In particular, a model is presented with a competitive upstream industry and a downstream duopoly, where the latter may have di erent marginal costs. Competition upstream would in principle ensure that the price of the input be equal to its marginal cost. However, it will be veri ed that, if one of the downstream rms is integrated backwards with one of the suppliers, this might not be the case, with the actual input price being above or below this level. The fundamental di erence between vertical separation and vertical integration is that this vertically integrated rm becomes a producer in the nal goods market and, simultaneously, a potential input supplier to the nonintegrated downstream rm. This presence in the input and in the nal good market becomes an additional instrument that will allow downstream rms to enlarge the set of sustainable collusive outcomes, and will also allow the rm that integrates to increase its pro ts relative to the vertical separation case. In order to isolate the impact of vertical integration, the model keeps bargaining power constant before and after integration takes place. In particular, it is assumed that the downstream rm that integrates makes the rm that does not integrate take-it-or-leave-it o ers, both in the case of vertical separation and in the case of vertical integration. The di erence between the two cases is in the contents of the o ers that this rm makes. In the case of vertical separation, the rm with bargaining power posts output levels to be produced by each of the two downstream rms. In the case of vertical integration, in addition to output levels, the vertically integrated rm posts a price at which it will sell the input. The model shows that, if the rm with bargaining power is less e cient, the vertically integrated rm sells the input above the price posted by competitive input suppliers, even if these alternative suppliers have the same cost as the 2

3 integrated upstream rm. Conversely, if the vertical structure is more e cient than the non-integrated rm, the price will be below upstream marginal cost. All these input transactions are made to compensate for reductions in market share, relative to the vertical separation case. These results may be used by antitrust authorities to evaluate a merger that has some vertical component. In particular, it highlights a potential role of input sales, namely to implement side payments among colluding rms. This has two potentially antagonistic e ects on welfare: a negative e ect, since it would facilitate collusion, and a positive e ect, since it would enhance production e ciency. Antitrust authorities should consider both e ects when deciding on a case that involves backward integration. The present paper combines the issues of collusion and vertical integration. Most papers that analyze the e ect of vertical integration use a static setting. For instance, Ordover, Saloner, and Salop (1990) propose a model where vertical integration can be pro table. Linnemer (2003) proposes a model to analyze the welfare e ects of backward integration where downstream rms have asymmetric costs. He nds that if the cost of the rm that integrated is low enough, backward integration may increase welfare. Riordan (1998) nds that a dominant rm with a cost advantage integrating backwards put upward pressure both on input and output prices. Chen (2001), using a model with two downstream rms that produce di erentiated products, identi es a collusive e ect of vertical integration. Speci cally, by integrating backward, a more aggressive pricing behavior downstream is detrimental for its input sales. On the other hand, while the issue of collusion has received extensive attention in the literature, the speci c case of cost asymmetries has been rarely considered (see Rothschild, 1999, or Vasconcelos, 2005, for two exceptions). Furthermore, it is observed that most colluding rms are also active in other stages of the value chain. In a paper that combines collusion with vertical -albeit 3

4 forward integration, Nocke and White (2007) study the e ect of vertical mergers on upstream collusion, considering several models where upstream rms o er two-part tari contracts to downstream rms, which compete either in quantities or in prices. They conclude that the net e ect of vertical integration is to facilitate collusion. The result of input sales at a price above that of a competing source contributes the literature on market foreclosure, summarized in Rey and Tirole (2003). For instance, Sandonís and Faulí-Oller (2005) present a model where vertical integration causes partial foreclosure, in the presence of a less e cient source. The novelty of the present paper is that now partial market foreclosure may be achieved in the absence of any cost advantage on the vertical structure s side, relative to the alternative supplier. What allows the vertical structure to sell at a price greater than that of the alternative source is fear of retaliation by the rm with bargaining power, which induces the non-integrated downstream rm to accept a marked-up input price. Finally, Normann (2004) proposes a model in a dynamic setting, where foreclosure may facilitate collusion. This paper is organized as follows: Section 2 describes the model to be used in the following two sections. Section 3 considers the case of vertical separation, while Section 4 considers the case of forward integration of one of the upstream rms with one of the downstream rms. Section 5 studies the particular case of a linear demand function and presents some numerical results, and Section 6 concludes. 2 The model Consider an upstream industry where a large number, N, of rms produce an input that is used by two downstream rms to produce a nal good to be sold to consumers. The marginal cost of production of upstream rms is c and these 4

5 rms are capacity unconstrained. As in Salinger (1988), the nal good is produced using a xed-coe cients technology. Speci cally, downstream rms use one unit of the input produced by upstream rms per unit of output. Their transformation costs are c 1 and c 2, respectively. Thus, if the input price is w, downstream rms marginal costs are w + c 1 and w + c 2, respectively. Given these marginal costs, downstream rms compete in quantities. Let p(q) be the inverse demand for the nal product. This demand is twice-continuously di erentiable, with p 0 (q) < 0, p 0 (q) + q i p 00 (q) 0. This demand function gives rise to continuous, downward-sloping reaction functions R i (q j ; c i ; c j ), where these functions are de ned as R i (q j ; c i ; c j ) arg max(p(q + q j ) c i )q (1) q and the intersection of these reaction functions constitutes a unique stable equilibrium. The game has an in nite number of periods. In period 0, downstream rm 1 may integrate backwards with one of the rms in the upstream industry. Starting in period 1, each period has three stages, namely: 1. Upstream rms post input prices. 2. Each downstream rm decides the amount of input to purchase and from what upstream rm to procure the input. Payments for input procurement activities are made. 3. Production takes place, and downstream rms collect revenues from sales of the nal product. Each upstream rm observes the amount of input that each downstream rm purchased from it, but not other input procurement activities by downstream 5

6 rms. Downstream rms do not observe other downstream rms input procurement activities, only their output levels. If input procurement was observable, then downstream rms would be able to punish deviations in this stage; with the assumption of unobservable input procurement, punishment is only triggered by a deviation in output. Downstream rms may tacitly collude in the production stage. Collusive outcomes will be sustained by reverting to single-period Nash reversion outcomes, as rst suggested in Friedman (1971). In the Nash equilibrium, all rms produce the Cournot output, which may be symmetric or asymmetric, depending on the realization of rms costs. The next two sections examine the cases of vertical separation and vertical integration, respectively. In the case of vertical integration, the vertical structure, i.e. downstream rm 1 integrated backward with one of the upstream rms, may sell the input to the non-integrated downstream rm in addition to producing the nal good. In both cases, bargaining power will be assigned to downstream rm 1, which will allow it to make take-it-or-leave-it (TIOLI) o ers to the other downstream rm. The reason to assign bargaining power to one of the two rms is to make the comparison between the vertical separation and vertical integration outcomes easier. 3 Vertical separation Suppose initially that both downstream rms purchase the input from the competitive industry at marginal cost. First, the price w at which downstream rms purchase the input is the result of price competition among the N rms in the upstream industry, which simultaneously post input prices. If upstream rms were to equally split upstream industry pro ts > 0, then a input price w > c 6

7 will be sustainable as long as N (1 ) + 0 (2) since Nash reversion involves pricing at marginal cost. Hence, the input price will be equal to marginal cost as long as < N 1 N. In order to simplify the analysis, it will be assumed that the number of upstream rms is N > 1 1, which ensures that the price at which downstream rms purchase the input is w = c. If rm 1 makes TIOLI o ers (q 1 ; q 2 ), given the discount factor and given the equilibrium input price in the upstream market, w = c, rm 1 will: max q 1(p(q 1 + q 2 ) (c 1 + c)) (3) q 1;q 20 8 >< 1 (q 1 ; q 2 ; c 1 + c; c 2 + c; ) 0 s.t. >: 2 (q 2 ; q 1 ; c 2 + c; c 1 + c; ) 0 where i (q i ; q j ; c i + c; c j + c; ) = q i (p(q i + q j ) (c i + c)) (1 )R i (q j ; c i + c; c j + c) (p(r i (q j ; c i + c; c j + c) + q j ) (c i + c)) qi C(c i + c; c j + c) p(qi C(c i + c; c j + c) + qj C(c j + c; c i + c)) (c i + c) (4) is rm i s incentive constraint. Whenever this function is non-negative, its incentive constraint will be satis ed, and rm i will not deviate from the collusive outcome. In this expression, R i (q j ; c i + c; c j + c) is rm i s reaction function, as de ned above, and qi C(c i + c; c j + c) is rm i s production in the Cournot-Nash equilibrium where rms have costs c i +c and c j +c. Of course, qi C(c i+c; c j +c) = R i (qj C(c j + c; c i + c); c i + c; c j + c). The constraints in this problem determine the set of feasible values of q 1 7

8 and q 2. Notice that when = 0, the two constraints become rm 1 and rm 2 s reaction functions, respectively, while when = 1, it is the intersection of the areas below rms 1 and 2 s isopro t curves that correspond to the Nash equilibrium levels. Firm 1 s problem is to choose the combination q 1 ; q 2 in this set that belongs to the lowest possible isopro t curve (that which is associated with the highest pro ts). The following lemma shows that in the optimal solution q1; q2, rm 2 s incentive constraint must be binding: Lemma 1 The solution q 1; q 2 to rm 1 s problem satis es 2 (q 2; q 1; c 2 +c; c 1 + c; ) = 0 Proof. See Appendix 1. Under vertical separation, both q1 and q2 have to be positive, since otherwise their incentive constraints would not be satis ed. Notice that this fact implies that in the case of di erent production costs, part of the output is produced by a relatively ine cient rm, which is clearly suboptimal from the point of view of the industry as a whole. Firm 2 s constraint equated to zero allows us to obtain an expression q 2 (q 1 ). In principle, the equality 2 = 0 does not guarantee the existence of a function q 2 (q 1 ). For instance, in the case of a linear demand, 2 is a quadratic polynomial of q 1 and q 2, and there may be two values of q 2 that satisfy 2 = 0 for the same value of q 1. Of course, rm 1 will choose the solution where it obtains greater pro ts, and that is precisely the one where q 2 is lowest. Inserting this particular relationship q 2 (q 1 ) into the objective function, the problem may be rewritten as: max q 1(p(q 1 + q 2 (q 1 )) (c 1 + c)) (5) q 10 s.t. 1 (q 1 ; q 2 ; c 1 + c; c 2 + c; ) 0 8

9 whose rst-order condition is: p(q 1 + q 2 ) dp (c 1 + c) + q dq 2 = 0 (6) dq dq 1 where dq2 dq 1 is obtained using the Implicit Function Theorem from 2 = 0, evaluated at the optimum: dq 2 = q 2 dq 1 p(q 1 + q 2 ) dp dp(r dq (1 )R 2+q 1) 2 dq (c 2 + c) + q 2 dp dq (7) Notice that for a high enough value of, dq 2 dq 1 > 0, whereas the opposite occurs for lower values of the discount factor. If the optimal values of q 1 and q 2 satisfy rm 1 s incentive constraint, this is indeed the solution. However, if rm 1 s incentive constraint is not satis ed, the solution is the intersection of the two constraints, i.e. the solution of the system 1 = 0, 2 = 0. This occurs for low values of. As approaches one, rm 1 is able to reduce rm 2 s pro ts towards the Nash equilibrium level. Indeed, when = 1, rm 2 s pro ts are exactly equal to Nash equilibrium pro ts. In this case of vertical separation, for any industry output, rm 1 tries to maximize its own market share, subject to acceptance by rm 2. Hence, the problem of production ine ciency is aggravated if the vertical structure s cost is greater than that of the non-integrated rm. Recall that, for any industry output, production costs are minimized by concentrating production in the relatively more e cient rm. The possibility of side payments between the two rms, which is possible under vertical integration, but not under vertical separation, will permit the implementation of more e cient distributions of industry output. In particular, with such side payments it will be possible to shift production from the ine cient towards the e cient rm, since the latter could compensate the former for the loss in market share. In other words, the 9

10 possibility of side payments provides the rm with bargaining power with an additional instrument that can be used to construct collusive outcomes that satisfy both rms incentive constraints. Input sales below or above marginal cost will be the mechanism by which these transfers are implemented. Next section precisely analyzes this case. 4 Vertical integration Assume now that rm 1 is integrated backwards with one of the upstream rms. This section characterizes the best outcome from rm 1 s perspective, provided that it is now able to supply the input that rm 2 uses, in addition to producing the nal good. Now an o er by the integrated rm consists on (q 1 ; q 2 ; w I ), i.e. output levels plus a price at which it supplies the input to the other rm. As in the case of vertical separation, collusive outcomes will be sustained by means of Nash reversion. Since now rm 1 may act as a supplier of the input as well as a nal good producer, reversion to the Nash equilibrium outcome occurs either if the non-integrated downstream rm fails to procure to input at the posted price or after a deviation in output. Notice that rm 1 now has the ability to punish deviations both in procurement and in nal product market behavior. In the case of vertical separation, reversion to the Nash equilibrium could only be triggered by a deviation in output. When choosing the triple (q 1 ; q 2 ; w I ), rm 1 has to keep in mind two possible deviations on rm 2 s side. First, rm 2 may fail to purchase the input from the vertical structure. This prompts retaliation in that same period s production stage, and both rms would earn Nash equilibrium pro ts from that same period on. This possibility implies that in any o er made by the vertical structure, rm 2 must make at least Nash equilibrium pro ts to accept it. On the other hand, 10

11 rm 2 may procure part of the input from the vertical structure, in particular those units that will allow it to produce its share of the nal output, at the posted input price, and some additional units from some other upstream rm, at marginal cost. Recall that the vertical structure does not observe whether rm 2 procures any input from other upstream rms. This deviation strategy prompts reversion to the Nash equilibrium outcome starting in the period after deviation takes place, allowing rm 2 to earn pro ts above the Nash equilibrium level for one period. Lemma 2 Firm 2 is better o secretly procuring additional input from the alternative source than failing to purchase from the integrated rm at the posted input price. Furthermore, as approaches 1, rm 2 s pro ts are lowered to the Cournot level. Proof. See Appendix 2. Lemma 2 states that the constraint that is to be taken into account by the vertical structure is that determined by a deviation in output. This is because rm 2 s pro ts in any collusive outcome will always exceed Nash equilibrium pro ts, and thus the acceptance constraint will never be binding. The optimal collusive outcome will now be analyzed. First, it will be established that the input price w I posted by the integrated rm will di er from the equilibrium input price c posted by the remaining upstream producers. In particular, w I > c will implement a transfer from rm 2 to rm 1, whereas the opposite occurs when w I < c. The case of no side payments is precisely w I = c. Call this side payment, or transfer t = (w I c)q 2, which could be negative, zero, or positive, depending on the relationship between w I and c. This way, rm 1 s 11

12 problem may be written as: max q 1(p(q 1 + q 2 ) (c + c 1 )) + t (8) q 1;q 2;t 1 (q 1 ; q 2 ; c + c 1 ; c + c 2 ; ) + t 0 s.t. 2 (q 2 ; q 1 ; c + c 2 ; c + c 1 ; ) t 0 w I = c + t q 2 0 where the t term appears with a factor in the inequality because the transfer has to be taken into account both in the computation of collusive pro ts as well as in the computation of deviation pro ts, since the transfer is paid even if deviation takes place. Notice also that w I does not a ect rm 2 s optimal deviation: this optimal deviation is computed taking into account that the marginal cost is c + c 2, not w I + c 2. Hence, w I acts as a mechanism to transfer pro ts between rms, and does not actually a ect marginal cost in the case of deviation or Nash equilibrium. Therefore, given any transfer of pro ts t, this could be implemented by means of an input price w I = c + t q 2 or with another two-part tari f; w such that f + wq 2 = t, i.e. such that the total payment is the same. This problem includes an additional constraint: w I 0. Thus, even though the integrated rm may sell the input below cost if it nds it optimal to do so, it can not actually sell the input at a negative price. This constraint may become binding if the solution requires a transfer from rm 1 to rm 2, which is implemented by means of sales of the input below cost. More precisely, for any output levels q 1 ; q 2, the restriction w I 0 imposes an upper bound on the transfer that can actually take place from rm 1 to rm 2. If rm 1 s cost is lower than rm 2 s, the constraint w I 0 will prevent reaching production e ciency, since the required input price would have to be minus in nity. Notice that the constraint w I 0 is never binding in the case of transfers from rm 2 12

13 to rm 1. As in the previous case, rm 1 chooses (q 1 ; q 2 ; w I ) to make rm 2 s incentive constraint binding. Hence, we may rewrite the problem as: max q 1(p(q 1 + q 2 ) (c + c 1 )) + 2 (q 2 ; q 1 ; c + c 2 ; c + c 1 ; ) (9) q 1;q 20 s.t. 1 (q 1 ; q 2 ; c + c 1 ; c + c 2 ; ) + 2 (q 2 ; q 1 ; c + c 2 ; c + c 1 ; ) 0 w I = c + t q 2 0 Of course, any combination q 1 ; q 2 that satis ed both rms incentive constraints in the case of vertical separation, also satis es the constraint of the problem in the case of vertical separation. Indeed, the set of feasible q 1 ; q 2 has been expanded. This is because under vertical integration, both rms incentive constraints had to hold simultaneously, which implied that the most restrictive constraint determined feasibility of output levels. By contrast, in the problem with transfers, both constraints are pooled, hence any output levels that were feasible in the previous problem continue to be feasible, and some previously unfeasible output levels now become feasible. Also notice that the problem in the case of vertical separation was the same as that under vertical integration, but imposing the additional constraint t = 0. Therefore, given the vertical separation optimum q1; q2, in the case of vertical integration the vertical structure may increase its pro ts choosing some other output levels, together with a non-zero transfer. In particular, it is typically the case (for su ciently high values of the discount factor), that the vertical structure shifts production towards the e cient rm, and away from the inef- cient rm, compensating the variation in pro ts with appropriate transfers, implemented by means of input sales. This highlights the importance of the constraint of zero transfers that is present in the case of vertical separation: relaxing that constraint makes the vertical structure choose a di erent output 13

14 allocation than in the case of vertical separation. The remainder of this section will discuss the di erent types of solution to the vertical structure s problem, distinguishing between corner and interior solutions. In our setting, a corner solution is that where one of the two downstream units does not produce any nal output at all. Ignoring for now the constraints w I 0 and , the rst-order conditions of this problem read: p(q 1 + q 2 ) (c 1 + c) + dp dq q 1 + dp dq q dp(r 2 + q 1 ) 2 (1 )R 2 dq q 1 dp dq + p(q 1 + q 2 ) 0; = 0 if q 1 > 0 (c 2 + c) + dp dq q 2 0; = 0 if q 2 > 0 (10) Consider rst the case q 1 = 0; q 2 > 0, i.e. the rm without bargaining power producing the whole product. Since rm 1 does not produce at all, the required transfer will be positive, with the input being sold above marginal cost: w I > c. In this case, the rst-order conditions become: (p(q 2 ) (c 1 + c)) + dp dq q dp(r 2 ) 2 (1 )R 2 dq p(q 2 ) 0 (c 2 + c) + dp dq q 2 = 0 (11) and therefore, from the bottom condition, we obtain that the optimal q 2 is q m 2, i.e. precisely the monopoly output with cost c 2. Thus, returning to the top condition, p(q2 m ) (c 1 + c) + dp dq qm 2 0 (12) which holds as long as c 1 c 2. Hence, when rm 2 is at least as e cient as rm 1, rm 1 is best o letting rm 2 produce the whole industry output. Notice that this also applies to the case of rms having symmetric costs. In order for 14

15 this to be the solution, the constraint has to hold, hence, q2 m (p(q2 m ) (c 2 + c)) C 1 C 2 (1 )R 1 (q m 2 )(p(r 1 + q m 2 ) (c 1 + c)) 0 (13) which holds for large enough realization of the discount factor given a di erence in costs c 1 c 2. If the constraint is satis ed, then production is carried out in the most e cient way, since it is the rm with the lowest cost the one producing the whole industry output. Notice that the rm with bargaining power sets the transfer so as to reduce rm 2 s pro ts to the Cournot level. In this case, the required transfer is positive, and the input price is w I = c + p(q m 2 ) (c 2 + c) C 2 q2 m (14) which is independent on. When the constraint is not satis ed, this constraint made binding determines a relationship between q 1 and q 2 that may be inserted into the objective function to reduce the set of choice variables, and both rms produce positive amounts in the optimum. These results may be summarized in the following proposition: Proposition 3 If c 1 c 2 for a su ciently high realization of, rm 2 produces the whole output, which is set at the monopoly level with costs c 2. The input price is set so that rm 2 s pro ts are reduced to the Cournot level. This proposition states that in the case of the non-integrated rm being more e cient than the vertical structure, and provided that the discount factor is above some minimum level, the vertical structure is best o letting the nonintegrated downstream rm produce the whole output. Of course, the vertical structure will make pro ts from its input sales to the non-integrated rm at 15

16 a price greater than marginal cost. A similar result was found in Mendi et al (2006), for the particular case of a linear demand function and homogeneous marginal costs, in a context of forward integration by an upstream producer. The solution to the vertical structure s problem could also be such that rm 2 does not produce any output, hence q 1 > 0; q 2 = 0. In this case, the rst-order conditions would read p(q 1 ) dp (c 1 + c) + q 1 dq (1 )R 2 dp(r 2 + q 1 ) dq = 0 q 1 dp dq + p(q 1) (c 2 + c) 0 (15) From the top condition, it may be seen that rm 1 would produce q 1 > q m 1 whenever < 1. Plugging the top equality into the bottom inequality, dp(r 2 + q 1 ) (1 ) p(q 1 ) (c 2 + c) + R 2 (c 2 c 1 ) (16) dq Examining this inequality, a necessary condition for it to hold is that c 2 c 1 > 0. Hence, the vertical structure would be the sole producer of the nal product only if it is more e cient than the non-integrated downstream rm. Not only that, but given the value of the discount factor, the di erence in costs c 2 c 1 has to be su ciently high. However, since transfer payments are based on sales of the input, if rm 2 does not produce in the optimal outcome, the required transfer is not feasible, since the required input price would be minus in nity. In other words, the implementability of this corner solution is limited by the constraint w I 0. Hence, when the non-integrated rm is less e cient than the vertical structure, the ine cient rm always produces a positive amount. The next proposition summarizes these results: Proposition 4 If c 1 < c 2, the ine cient rm always produces a positive amount. For high values of the discount factor, the restriction w I 0 prevents production 16

17 e ciency. When the rst-order conditions would in principle suggest the existence of a corner solution of the type q 1 > 0; q 2 = 0, the actual optimum is conditioned by the constraint w I = 0. Again, this determines a relationship between q 2 and q 1 that can be inserted in the objective function, leaving the problem as a single variable choice. Finally, when c 1 < c 2 and is su ciently high, the third possibility is that of an interior solution, where both rms produce the nal product. As seen above, a necessary condition for this to be the case is that c 2 > c 1. Ignoring the constraints, the solution is characterized by both rst-order conditions holding with equality: p(q 1 + q 2 ) dp dp (c 1 + c) + q 1 + q 2 dq dq q 1 dp dq + p(q 1 + q 2 ) (1 )R 2 dp(r 2 + q 1 ) dq (c 2 + c) + q 2 dp dq = 0 = 0 (17) and combining both constraints, dp(r 2 + q 1 ) (c 2 c 1 ) = (1 ) p(q 1 + q 2 ) (c 2 + c) + R 2 dq (18) Notice that, since the right-hand side of this expression is positive, a necessary condition for an interior solution to exist is that the vertical structure be more e cient than the non-integrated rm. Furthermore, and related to the case where the non-integrated rm does not produce at all, this is the solution for relatively low realizations of the discount factor. Summarizing all these di erent cases, if the non-integrated rm is at least as e cient as the vertical structure in the production of the nal product, the 17

18 vertical structure will nd it optimal to let he non-integrated rm produce the whole output, as long as the pooled incentive constraint is satis ed. This occurs when the discount factor is high enough. In the case of the vertical structure being more e cient, non-negativity of the input price prevents the implementation of production e ciency, and both rms will always produce in equilibrium. Furthermore, for high realizations of the discount factor, the output levels chosen are constrained by the non-negativity of input prices. Next section will present some numerical examples, using a linear demand function, to illustrate these results. The focus will be on output levels produced by each rm, comparing the vertical separation and vertical integration cases, input price, and the e ect of vertical integration on the pro tability of rm 1 s pro ts and on total welfare. 5 The case of a linear demand This section presents some results for a speci c functional form of the demand function. In particular, consider a linear inverse demand function p = a bq. The choice of this particular functional form will allow for parametrization of the above expressions, and will permit the calculation of numerical solutions, including some welfare calculations, for selected parameter values. Given this particular functional form of demand, the problem under vertical separation becomes max q 1(a b(q 1 + q 2 ) (c 1 + c)) (19) q 1;q 20 8 >< (a bq2 (c1+c))2 (a 2(c1+c)+(c2+c))2 q 1 (a b(q 1 + q 2 ) (c 1 + c)) (1 ) 4b 9b 0 s.t. >: (a bq1 (c2+c))2 (a 2(c2+c)+(c1+c))2 q 2 (a b(q 1 + q 2 ) (c 2 + c)) (1 ) 4b 9b 0 From the previous discussion, the bottom constraint is binding. Since this 18

19 constraint de nes a quadratic expression on q 1 and q 2, for every value of q 1, there are two possible values of q 2 that will make the constraint equal to zero. Of course, rm 1 will always choose the smaller value of q 2, since its pro ts are greater, given the value of q 1. This permits the de nition of a function q 2 (q 1 ), whose derivative, using the Implicit Function Theorem, may be written as: dq 2 = (1 )(a bq 1 2bq 2 (c 2 + c)) 2bq 2 dq 1 2(a bq 1 2bq 2 (c 2 + c)) (20) For high enough values of the discount factor, the solution will come from the rst-order condition of the problem de ned as one of choice of q 1 using the function q 2 (q 1 ). This rst-order condition is: q bq 2 (q 1 ) 2 + dq 2 bq 1 (c 1 + c) = 0 (21) dq 1 If the values of q 1 and q 2 that satisfy this equality satisfy rm 1 s incentive constraint, this is indeed the solution. Otherwise, the solution is the intersection of the two incentive constraints. On the other hand, the problem under vertical integration is written as: max q 1(a b(q 1 + q 2 ) (c 1 + c)) (22) q 1;q 2;t0 8 q 1 (a b(q >< 1 +q 2 ) (c 1 +c)) + t (1 ) (a bq2 (c 1+c)) 2 4b + t (a 2(c1+c)+(c2+c))2 9b 0 s.t. q 2 (a b(q 1 +q 2 ) (c 2 +c)) t (1 ) (a bq1 (c 2+c)) 2 4b t (a 2(c2+c)+(c1+c))2 9b 0 >: w I = c + t q

20 and since rm 2 s incentive constraint will be binding, max q 1(a b(q q 1 +q 2 ) (c 1 +c)) + q 2 (a b(q 1 +q 2 ) (c 2 +c)) (23) 1;q 20 (1 ) (a bq 1 (c 2 + c)) 2 (a 2(c 2 + c) + (c 1 + c)) 2 8 4b 9b q 1 (a b(q >< 1 +q 2 ) (c 1 +c)) + q 2 (a b(q 1 +q 2 ) (c 2 +c)) s.t. (1 ) (a bq2 (c 1+c)) 2 (a bq1 (c2+c))2 4b + 4b (a 2(c1+c)+(c 2+c)) 2 (a 2(c2+c)+(c1+c))2 9b + 9b >: w I (a bq1 (c2+c))2 (a 2(c2+c)+(c1+c))2 = c + (a b(q 1 +q 2 ) (c 2 +c)) (1 ) 4bq 2 9bq 2 0 and in this case the solution may be a corner one (with q 1 = 0), or an interior one. In the case of q 1 = 0, the rst-order condition with respect to q 2 reads: 0 a 2bq 2 c 2 = 0 ) q 2 = a c 2 2b (24) which is the monopoly outcome. In this case, the rst-order condition with respect to q 1 becomes (c 2 c 1 ) 0 (25) and, provided that the constraint is satis ed, this is indeed the solution. This occurs for high enough realization of the discount factor. Thus, for such values of, the non-integrated rm produces the monopoly output, and the vertically integrated rm does not produce at all. This outcome satis es productive ef- ciency, since it is the rm with the lowest cost the one producing the whole output. The vertically integrated rm s revenues come from input sales above marginal cost. In this case the constraint w I 0 is non-binding, since the required transfer is positive, and the input price is above the upstream industry s marginal cost. For lower values of the discount factor, the solution will be interior, and both rms will be producing the nal product. Notice that this arrangement does not satisfy productive e ciency. Also notice that the other 20

21 corner solution q 2 = 0 is not feasible because it does not satisfy the constraint w I 0. The numerical results will allow us to illustrate the outcome in terms of output and input prices as a function of the discount factor, both for the cases of vertical separation and vertical integration, for di erent realizations of rm 1 and rm 2 s costs. In the numerical calculations, the demand parameters have been set to a = 100, b = 1. Regarding costs, c = 20, and the analysis considers the cases c 1 = c 2 = c; c 1 = c; c 2 = c + 10; c 1 = c + 10; c 2 = c, i.e. both rms being equally e cient, the vertical structure being more e cient, and the non-integrated rm being more e cient. The gures will show how individual rms output and total output change with vertical integration, the input price prevalent under vertical integration, rm 1 s increased pro tability after integration, and the e ect of integration on total welfare. [insert Figures 1, 2, and 3 here] Figure 1. Output levels, case c1=c2=c ,01 0,08 0,15 0,22 0,29 0,36 0,43 0,5 0,57 0,64 0,71 0,78 0,85 0,92 0,99 q1,q2 q1ni q2ni q1i q2i Delta 21

22 Figure 2. Output levels, case c1=c, c2=c q1,q q1ni q2ni q1i q2i 0,01 0,08 0,15 0,22 0,29 0,36 0,43 0,5 0,57 0,64 0,71 0,78 0,85 0,92 0,99 Delta Figure 3. Output levels, case c1=c+10,c2=c ,01 0,08 0,15 0,22 0,29 0,36 0,43 0,5 q1,q2 0,57 0,64 0,71 0,78 0,85 0,92 0,99 q1ni q2ni q1i q2i Delta First, regarding output levels, Figures 1, 2, and 3 plot q 1 and q 2 under vertical integration and vertical separation, for the cases c 1 = c 2 = c; c 1 = c; c 2 = c+10; c 1 = c + 10; c 2 = c, respectively. If both downstream rms are equally e cient, under vertical separation total output decreases with the discount factor and rm 1 s share is greater than rm 2 s for every value of the discount factor, which also occurs for the case when rm 1 is more e cient than rm 2 (Figure 2). However, if rm 1 is less e cient than rm 2 (Figure 3), while total output also decreases with, rm 2 s share is greater under vertical separation. Under vertical integration, if rm 2 is at least as e cient than rm 1, which occurs both in Figure 1 and in Figure 2, rm 2 s share is increasing in, until the discount factor is high enough so that the monopoly solution with low cost 22

23 is sustainable, with all production being carried out by rm 2. In Figure 1, this occurs for 0:7, whereas in Figure 3, this occurs for 0:43. For values of the discount factor greater than these threshold levels, rm 2 s pro ts are lowered to the Cournot level. By contrast, in the case of rm 1 being more e cient than rm 2, it is impossible to implement the solution where production is e ciently carried out, i.e. where rm 1 is the sole producer of the nal product. This is because the required transfer would be minus in nity, which would clearly violate the constraint w I 0. This is the reason why, for any value of the discount factor, rm 2 s share in total output is positive. [insert Figure 4 here] Figure 4. Input price, vertical integration ,01 0,09 0,17 0,25 0,33 0,41 0,49 0,57 0,65 0,73 0,81 0,89 0,97 w c1=c2=c c1=c,c2=c+10 c1=c+10,c2=c Delta Regarding the input price, Figure 4 shows that precisely when rm 1 is more e cient than rm 2, the constraint w I 0 becomes binding for 0:49. For these values of the discount factor, the vertically integrated rm would like to pay rm 2 to use its input, so as to increase its own market share. In the other two cases, this is not an issue, since the required transfer is positive and there is no upper bound on w I (although obviously, it will never exceed the nal good price). The input price is in all cases above the marginal cost, which is the 23

24 price posted by the other rms in the upstream industry, and is increasing in the discount factor. For high enough values of the discount factor, in particular 0:7 and 0:43 in gures 1 and 3, respectively, the input price is constant, and rm 2 s pro ts are equal to the Cournot level. [insert Figure 5 here] Figure 5. Firm 1's profit ratio, vertical integration to vertical separation 1,6 1,5 1,4 1,3 1,2 1,1 1 0,01 0,09 0,17 0,25 0,33 0,41 0,49 0,57 0,65 Ratio 0,73 0,81 0,89 0,97 c1=c2=c c1=c,c2=c+10 c1=c+10,c2=c Delta Vertical integration will never make rm 1 worse o, since the vertically integrated rm may always implement the optimal outcome under vertical separation, setting w I = c. For this reason, the ratio of rm 1 s pro ts under vertical integration to rm 1 s pro ts under vertical separation will always be at least one. Figure 5 plots this ratio for the three cases considered. It may be seen that the greatest gain occurs in the case when rm 1 is less e cient than rm 2. This is because under vertical integration rm 1 may shift production towards the e cient rm, and extract rents by means of marked-up input sales. By contrast, when rm 1 is more e cient than rm 2, the condition w I 0 constrains this shifting of production towards the e cient rm, implying that the gains to rm 1 will be more modest. In the case c 1 = c 2 = c, when = 1 24

25 the ratio is exactly one, since both rms are equally e cient. [insert Figure 6 here] Figure 6. Welfare ratio, vertical integration to vertical separation 1,15 1,1 1,05 1 0,95 0,9 0,01 0,09 0,17 0,25 0,33 0,41 0,49 0,57 0,65 0,73 Ratio 0,81 0,89 0,97 c1=c2=c c1=c,c2=c+10 c1=c+10,c2=c Delta Finally, in terms of welfare, there are two e ects that are to be taken into account. One is the impact of vertical integration on total output, and the second one is the e ect of vertical integration on the distribution of production between the two rms. Figure 6 plots the ratio of total welfare under vertical integration to total welfare under vertical separation. It may be observed that for most values of, in the case of both rms being equally e cient, vertical integration decreases total welfare. This is because vertical integration has the e ect of reducing total output in this case and, since both rms are equally e cient, there are no gains from transferring production across rms. By contrast, when rms di er in their production costs, the e ect of moving towards the e cient distribution of production dominates any output-reducing e ects of vertical integration. For this reason, vertical integration increases welfare in these two cases. Notice that the highest welfare gains occur when rm 1 is less e cient than rm 2: in this case the constraint w I 0 is non-binding and the e cient allocation of production can be achieved. 25

26 6 Conclusions This paper analyzes the e ect of vertical integration on competition in a downstream industry. A vertical structure, i.e. the result of integrating a downstream rm with an input supplier, might sell the input to the non-integrated downstream rm with the purpose of transferring pro ts between them. This enlarges the set of collusive outcomes that may be sustainable for a given realization of the discount factor. This paper explores this e ect for the particular case of one of the two downstream rms being able to specify collusive outcomes by means of take-it-or-leave-it o ers. The model shows that if the non-integrated rm is at least as e cient as the vertical structure, for a su ciently high value of the discount factor the vertical structure will let the non-integrated rm to produce the whole output, which is set at the monopoly level, and extract pro ts from the non-integrated rm by means of a input price above marginal cost. By contrast, if the vertical structure is more e cient than the non-integrated downstream rm, e cient production will not be accomplished, since this would require a negative input price to be implemented. In this case, production is carried out both by the e cient and the ine cient rm. Moreover, with c 1 < c 2, if both rms were vertically integrated and if rm 1 was able to demand an ask price for the input, for a su ciently high value of the discount factor, rm 1 would want to produce the whole output, while procuring the input from rm 2 at a price greater than marginal cost. In the model proposed in this paper, input sales play the role of transfers of pro ts between rms. This introduces a crucial di erence between the cases of vertical separation and vertical integration. While in the former rms may manipulate total output and market shares only, in the latter, rms may manipulate total output, market shares, and side payments between rms. These side 26

27 payments, implemented by means of sales of the input above or below marginal cost, may permit total output to be produced by the most e cient rms, while still satisfying rms incentive constraints. Thus, vertical integration expands the set of collusive outcomes that are sustainable. Furthermore, the di erence between the set of collusive outcomes that are sustainable with vertical integration and the set of collusive outcomes that are sustainable under vertical separation increases with the di erence in rms marginal costs. Finally notice that in some cases, the input is sold at a price greater than that posted by an alternative supplier. This is an interesting result, if related to the results obtained in the partial market foreclosure literature. In the case of the present model, the implementability of this greater input price depends crucially on the dynamic nature of the relationship between the rms that are active in the downstream industry. References [1] Chen, Y On Vertical Mergers and their Competitive E ects. Rand Journal of Economics, 32(4): [2] Friedman, J.W A Non-Cooperative Equilibrium for Supergames. Review of Economic Studies, 38: [3] Linnemer, L Backward Integration by a Dominant Firm. Journal of Economics and Management Strategy, 12(2): [4] Mendi, P., R. Moner-Colonques, and J. Sempere-Monerris Tari s, Collusion and Vertical Integration. Mimeo, Universidad de Navarra and Universidad de Valencia. [5] Nocke, V., and L. White Do Vertical Mergers Facilitate Upstream Collusion? Forthcoming, American Economic Review. 27

28 [6] Normann, H.T Equilibrium Vertical Foreclosure in the Repeated Game. Mimeo, University of London. [7] Ordover, J., G. Saloner, and S. Salop Equilibrium Vertical Foreclosure. American Economic Review, 80(1): [8] Perry, M., and R. Porter Oligopoly and the Incentive for Horizontal Merger. American Economic Review, 75: [9] Rey, P., and J. Tirole A Primer on Foreclosure. Forthcoming, Handbook of Industrial Economics. [10] Riordan, M Anticompetitive Vertical Integration by a Dominant Firm. American Economic Review, 88: [11] Salinger, M Vertical Mergers and Market Foreclosure. The Quarterly Journal of Economics, 103(2): [12] Sandonís, J., and R. Faulí-Oller On the Competitive E ects of Vertical Integration by a Research Laboratory. Forthcoming, International Journal of Industrial Organization, 24: [13] Vasconcelos, H Tacit Collusion, Cost Asymmetries, and Mergers. RAND Journal of Economics, 36(1): Appendix 1. Proof of Lemma 1 Assume that rm 1 s optimal o er is q1; q2 and that, given this o er, 1 (q1; q2; c 1 ; c 2 ; ) 0 and 2 (q2; q1; c 2 ; c 1 ; ) > 0. This means that rm 1 s incentive constraint may or may not be binding, whereas rm j s constraint is non-binding. If rm 1 s incentive constraint is non-binding, then reducing rm 2 s production increases rm 1 s pro ts. Given that rm 2 s constraint is assumed to be non-binding, a reduction in rm 2 s output may be found that still satis es both constraints. 28

29 This reduction in rm 2 s output increases rm 1 s pro ts, contradicting the optimality of q1; q2. If rm 1 s constraint is binding, while rm 2 s constraint is non-binding, consider the case of q1 > R 1 (q2), i.e. the solution being above rm 1 s reaction function. In this case, a reduction in rm 1 s output increases its pro ts and clearly satis es rm 1 s constraint, since rm 1 s deviation pro ts remain the same. Given that rm 2 s constraint was initially non-binding, there exists a reduction in rm 1 s output small enough that increases rm 1 s pro ts while satisfying rm 2 s constraint. A similar argument applies to the case q1 < R 1 (q 2), but with an increase rather than a decrease in rm 1 s output. In the case q 1 = R 1 (q 2), rm 1 increases its pro ts by moving along its own reaction function, increasing its own output and reducing rm 2 s output. Since rm 2 s constraint was initially non-binding, a rearrangement of output may be found that satis es rm 2 s constraint. Hence, if 1 (q 1; q 2; c 1 ; c 2 ; ) 0 and 2 (q 2; q 1; c 2 ; c 1 ; ) > 0, the proposed solution q 1; q 2 may be improved upon, and thus, in the optimal solution, it must be the case that 2 (q 2; q 1; c 2 ; c 1 ; ) = 0. Appendix 2. Proof of Lemma 2 First, if rm 2 fails to purchase from the integrated rm at the posted input price w I, its pro ts will be C 2 = q2 C (c 2 + w; c 1 + c) p(q2 C (c 2 + w; c 1 + c) + q1 C (c 1 + c; c 2 + w)) (c 2 + w) (26) i.e. Cournot pro ts with costs c 2 + w and c 1 + c, respectively. By contrast, if rm 2 decides to purchase q1 units of the input from the integrated rm at a price w I and R 2 (q 1; c 2 + w; c 1 + c) q 2 at price w, which is the price posted by the remaining N 1 upstream rms, its incentive constraint will read: 2 t (1 ) d 2 t + C 2 (27) 29

30 where t = w I q 2 and d 2 = R 2 (q 1; c 2 + w; c 1 + c)p(r 2 (q 1; c 2 + w; c 1 + c) + q 1) w(r 2 (q 1; c 2 + w; c 1 + c) q 2) 2. Given this, since 2 t C 2, because rm 2 may secure Cournot level of pro ts by failing to purchase from the integrated rm, it must be true that (1 ) d 2 t + C 2 C 2. On the other hand, notice that lim (1 ) d 2 t + C 2 = C 2 (28)!1 implying that, the larger, the lower rm 2 s pro ts. As approaches 1, rm 2 s pro ts converge to the Cournot level. 30