Price Discrimination in Input Markets

Transcription

1 Price Discrimination in Input Markets Roman Inderst Tommaso M. Valletti First version March 2006; This version January 2007 Abstract We analyze the short- and long-run implications of third-degree price discrimination in input markets. In contrast to the extant literature, where the supplier is typically an unconstrained monopolist, in our model input prices are constrained by the potential for demand-side substitution. This modification has profound implications as, in contrast to thecasewherethesupplierisunconstrained,moreefficient and thus ultimately larger firms receive lower input prices. Also, it is now more efficient and no longer less efficient firms that are hurt by a ban on price discrimination. Finally, introducing the potential for demand-side substitution also reverses the findings for how a ban on price discrimination would affect consumer surplus and welfare, both in the short and in the long run. Keywords: Price Discrimination; Uniform Pricing; Input Market. We thank seminar participants at Bocconi, EARIE (Amsterdam), the Office of Fair Trading, and Southampton for helpful comments. University of Frankfurt, London School of Economics and CEPR. r.inderst@lse.ac.uk. Corresponding author. Imperial College London, University of Rome Tor Vergata and CEPR. t.valletti@imperial.ac.uk. 1

2 1 Introduction According to the extant theory on price discrimination in input markets, more efficient firms pay a higher price, implying that it should be them and not their smaller competitors that lobbyfortheimpositionofabanonpricediscrimination. By the same token, the imposition of uniform pricing would then increase investment incentives by protecting the profits of more efficient intermediary firms. 1 These implications follow from the assumption that the price discriminating firm is an unconstrained monopolist, which optimally extracts higher profits from more efficient firms by charging them a higher input price. If there is the potential for demand-side substitution, however, more efficient and thus ultimately larger firms should also have more attractive alternative options. In a simple model, where intermediary firms must incur some costs of investment or of search and switching when moving away from the incumbent supplier, we show that this leads to a reversal of the results from the existing literature. More efficient and thus ultimately larger firms receive now lower input prices. Moreover, the imposition of a ban on price discrimination hurts more efficient and benefits less efficient firms, while stifling incentives to improve efficiency. In fact, we find that as one firm becomes more efficient, then under discriminatory pricing this not only reduces its own input price but it also leads to an increase of the input price of its competitors. These predictions of our model seem to accord well, in particular, with the objectives that were typically pursued when passing bans on price discrimination, such as the famous Robinson- Patman Amendments to Section 2 of the Clayton Act, namely to protect smaller or otherwise weaker competitors. 2 Our findings also support the common judgement that by protecting weak competitors, the imposition of uniform pricing would reduce efficiency in the long run. The case where a supplier can act like an unconstrained monopolist may be relevant in 1 See DeGraba (1990) and Yoshida (2000). For recent overviews on the theory of price discrimination, though dealing largely with discrimination among final consumers, see Armstrong (2006) and Stole (2005). 2 As Wright Patman, the Texas Democrat who was the main force behind the bill that took is name, wrote on the first page of a book he published to explain the law (Patman (1938), p. 3): The expressed purpose of the Act is to protect the independent merchant, the public whom he serves, and the manufacturer from whom he buys, from exploitation by unfair competitors. See Ross (1984) and Schwartz (1986) for accounts of the roots and immediate consequences of the Robinson Patman Act. Gerardin and Petit (2006) provide an account of European law, Article 82(c) in particular. They also give a recent overview of the large literature, both economic and legal, on the Robinson Patman Act. 2

3 industries where, given their choice of technology, intermediary firms are highly locked into a relationship with a particular supplier. Likewise it may be more applicable to natural monopolies. Note, however, that such a monopoly position is not necessary in order for the firm to fall under the respective antitrust provisions that prohibit or restrict discriminatory pricing. All that is needed is that the respective supplier is to a sufficient extent shielded from effective competition. Likewise, mandatory provisions in regulated industries, in particular in network industries, often apply to firms that do not enjoy (or do no longer enjoy) a fully monopolistic position. 3 Our analysis proceeds as follows. In Section 2 we introduce the key aspects of our model with price discrimination in input markets. Section 3 provides an analysis for the case where intermediary firms operate in separate markets. Besides providing a natural firststepinthe analysis, the case where the markets of intermediary firms are geographically segmented has particular importance under European law, which intends to eliminate market segmentation along the boundaries of member states for the purpose of creating a single European market. 4 In Section 4 we introduce the case with competition between intermediary firms. Given that only in this case claims of secondary line injury could play a role, for antitrust purposes this case has particular importance. 5 In fact, the view that non cost-related discounts under unrestricted price discrimination may harm smaller rivals and thereby distort competition seems to have recently gained renewed importance, in particular in the area of retailing with the rise of powerful big box retailers that are also increasingly active across borders. 6 One of our key 3 A case in point is telecommunications. The EU s new regulatory framework for electronic communications lists nondiscrimination requirements as one of the key options available to National Regulatory Authorities to regulate the wholesale market (see Valletti (2003)). Comparable provisions can be found in the US Telecommunications Act of To sanction geographic price discrimination, Community courts have relied both on a wider interpretation of Article 82(c) and, in particular, on prohibiting restrictions on parallel trade (on the grounds of Article 81), which are often a prerequisite to sustain meaningful price differentials in input markets. See Malueg and Schwartz (1994) for an economic analysis. 5 First line injury would arise, for instance, if the respective discriminatory practice had the objective or the effect of putting the supplier s own competitors at a disadvantage. This could happen in the presence of a vertically-integrated firm (see, e.g., Biglaiser and Degraba (2001)). Secondary line price discrimination happens when price discrimination practices put a dominant firm s customers at a competitive disadvantage vis-à-vis other customers. 6 Though there is the perception that prosecutions under the Robinson Patman act have become less important 3

4 results is that a ban on price discrimination would, in particular, reduce consumer surplus and welfare in the long run by stifling retailers incentives to invest in becoming more efficient. The paper closes in Sections 5 and 6 with a discussion of several of our assumptions and with some concluding remarks. All proofs as well as detailed calculations for the case with linear demand can be found in the Appendix. 2 The Model The basic set-up follows much of the literature comparing discriminatory and uniform pricing in an intermediary industry. We consider a single supplier that provides an input to the intermediary industry. Firms in the intermediary industry use the input to produce a homogeneous final good. Firms transform one unit of the input into one unit of the output. 7 The supplier produces at constant marginal costs, which we take to be zero to simplify our expressions. Also, without much loss in generality we restrict attention to two downstream firms, i =1, 2. We denote firm i s own constant marginal costs of production by k i 0. Our analysis distinguishes between the following two cases. In the first case, the two downstream firms serve two identical but independent markets. For most of our analysis we assume that both markets are symmetric and characterized by the same inverse demand function P (q). In the second case that we analyze, both firms i =1, 2 are active in the same market, offer homogeneous goods, and compete in quantities. There, P (q) characterizes the single market. In both cases, we assume that P (q) is strictly decreasing and twice continuously differentiable where P (q) > 0. Moreover, we employ throughout this paper the standard (Cournot) assumption that P 0 < min{0, qp 00 } where P>0. 8 In specifying next the contractual game, we follow again closely the received literature. The supplier can make take-it-or-leave-it offers to the downstream firms. 9 Under price discrimination, in the US, Dukes et al. (2006) show that there are still approximately 14 private party suits pursued each year. 7 This specification is not important for our results. Retailing is a natural example where this specification is reasonable. 8 See, for instance, Vives (1999). This assumption ensures that under competition the firms maximization problem is strictly concave. If the two firms serve independent markets, the weaker condition 2P 0 < min{0, qp 00 } would suffice. 9 That the supplier can make a take-it-or-leave-it offer is not as restrictive as it seems. It is well-known that under the so-called outside-option principle the outcome from bilateral Nash bargaining is already pinned down 4

5 the supplier thus offers each firm a constant input price w i, while under uniform pricing the same input price w i = w applies to both firms. Consequently, upon accepting the supplier s offer, a firm s total marginal cost equals c i := w i + k i. Our restriction to linear contracts, which we discuss in detail after presenting our results, is shared with much of the literature on third-degree price discrimination. Our main deviation from much of the literature is that the supplier, despite having the ability to make take-it-or-leave-it offers, is no longer an unconstrained monopolist. Here, we follow Katz (1987) and suppose that instead of seizing operations following rejection of the supplier s offer, a downstream firm can turn to an alternative source of supply. As in Katz (1987), this comes at costs F > 0 and allows the respective firm to obtain the input at constant marginal costs bw 0. Below we also discuss the possibility that the alternative input is only an imperfect substitute. Katz (1987) refers to the buyers alternative as the option to integrate backwards, which given the associated high fixed costs only represents a viable option for the large buyer in his analysis. In contrast, we focus our analysis on the case where the supplier can not act as an unconstrained monopolist vis-à-vis any of the downstream firms. An alternative interpretation could be that the costs F are incurred to adopt another technology under which a differentinputthatissoldatthe(competitive)marketpriceof bw can be used instead. Section 3.4 discusses also the possibility that, after rejecting the supplier s offer, a buyer can reduce bw by expending more resources, e.g., by searching for a lower offer or, in the case of vertical integration, by making higher investments. 10 The final part of our model is an initial investment stage. Following DeGraba (1990), each downstream firm can initially spend resources to reduce its own marginal cost k i. 11 For our general analysis, we specify that initially a firm s own marginal cost is equal to k i > 0, which the firm can reduce by k when incurring the expenditure e( k ). We suppose that e is strictly increasing and continuously differentiable with e 0 (0) = 0 and e 0 ( k ) as k k i,which allows us to focus on interior solutions. The chosen investment levels and, consequently, the by the binding outside option of one party if this is sufficiently attractive. In our case, this would be the option of each intermediary firm to switch to another source of supply. 10 Note that we abstract from the possibility that a firm may organically procure from multiple suppliers so as to reduce F in case one contract has to be replaced. Such strategic considerations are part of the analysis in Biglaiser and Vettas (2005). 11 See Choi (1995) for a similar model in the context of government tariffs in international trade. 5

6 respective values of k i are common knowledge. 3 Separate Markets 3.1 The Short Run In this Section, we analyze the case where firms serve separate markets. We stipulate that they are both monopolists in their respective markets. 12 For the short-run analysis, we take the firms own marginal costs k i as given and solve for the respective price equilibria, both under price discrimination and under uniform pricing. Section 3.2 analyzes the long run, solving for the first stage of our model, where downstream firms can invest in a reduction of own marginal costs. Given total marginal costs of c i = k i + w i under discriminatory pricing, firm i optimally chooses the unique quantity and realizes the profits q(c i ):=argmax{q[p(q) c i ]} q π(c i ):=q(c i )[P (q(c i )) c i ], where both q and π are strictly decreasing in c i as long as P (0) >c i. The supplier has zero marginal costs and its objective is to choose w i to maximize q(c i )w i. Take first the benchmark case where the supplier is an unconstrained monopolist. Assuming that q(c i )w i is strictly quasiconcave where q>0, an unconstrained supplier would optimally set w i equal to w UC i := arg max w i {q(c i )w i }. (1) The distinctive feature of our model is that buyers alternative supply options put a constraint on the supplier s pricing power. For firm i the value of the alternative supply option equals V A i := π(bc i ) F, (2) where bc i := bw + k i denotes the total marginal costs under the alternative supply option. Consequently, the supplier s offer to each firm i must satisfy the respective participation constraint π(c i ) V A i. (3) 12 It is, however, straightforward to extend the analysis to the case where both firms face the same level of competition, that is as long as their competitors are not themselves purchasing from the same supplier. 6

7 In what follows, we focus on the case where Vi A is sufficiently attractive such that for each firm i, the respective condition (3) constrains the supplier s optimal choice of w i. Thisisthecase if the monopolistic input price w UC i is high enough, i.e., if π(c i ) <V A i holds for c i = k i + w UC i. It is in turn straightforward to show that this holds if both F and bw are not too large. 13 obtain the following result. Proposition 1. In the case of separate markets and with price discrimination, a downstream firm s input price w i is strictly lower the more efficient the firm is, i.e., the lower its own marginal costs k i. Proof. See Appendix. As noted in the Introduction, Proposition 1 is orthogonal to the results that are obtained in case the supplier can set its unconstrained optimal price w i = wi UC. In the latter case, the strictly higher purchase volume of a more efficient downstream firm makes it optimal for the supplier to charge a strictly higher input price. We In standard terminology, going back to the seminal analysis of Robinson (1933), the more efficient downstream firm would represent for the supplier the strong market, where it is optimal to set a higher price than in the weak market. To see why the constrained supplier must grant the more efficient firm a discount, note first that a reduction of k i increases both the firm s profits under the supplier s offer π(c i ) and the value of its alternative supply option Vi A = π(bc i ) F. As we will argue subsequently, the effect on Vi A is strictly larger such that the participation constraint (3) becomes tighter. This in turn implies that following a reduction of k i, the supplier must reduce w i in order to still satisfy the firm s participation constraint. 14 It thus remains to argue why a reduction in k i has a larger effect on the off-equilibrium profits Vi A than on the on-equilibrium profits π(c i ). This results from the following two observations. First, note that the firm s profits π(c) are strictly convex in its total marginal 13 Below, we provide explicit conditions for the case with linear demand. 14 Incidentally, the argument for why there is a size discount in our model is thus differentfromthatinkatz (1987). There, for a large intermediary firm that operates in several downstream markets the alternative of backward integration is more attractive as the associated fixed costs F are incurred only once, irrespective of whether the firm operates in one or in two separate markets. Formally, if there was no downstream competition, then in Katz (1987) the participation constraint for a small buyer controlling only one outlet would become π( c i ) π(c) F, while that for a larger buyer controlling two outlets would become 2[π( c i ) π(c)] F. 7

8 costs. If c i is already low and the firm thus produces a higher quantity, then it benefits more from a further reduction in c i. The second observation is that in equilibrium it holds that bc i <c i as from F>0the supplier s input price will always include an additional margin with w i bw >0. Hence, after switching the source of supply the firm s total marginal costs are lower. 15 We feel that the analyzed model and Proposition 1 provide a very parsimonious way to capture the threat of demand-side substitution and how this should intuitively vary with a firm s size, which in our model depends on firms levels of operating efficiency. Clearly, if F =0 then there is simply no scope for price discrimination and any intermediary firm will obtain the same input price w i = bw. Differences in efficiency and thus size matter precisely because there is some degree of lock-in due to F>0. If on the other side, F wasverylarge,thenwewould be back to the case of an unconstrained monopolist, which charges more efficient firms a higher price as the outside option plays no role. We have next the following immediate comparative results for the equilibrium input prices. Corollary 1. In the case of separate markets and with price discrimination, input prices are strictly increasing in both F and bw. Moreover, if firm i isthemoreefficient firm, then the price differential w j w i > 0 is strictly increasing in F. Proof. See Appendix. Note finally that, as a direct implication of Proposition 1, downstream firms that end up purchasing larger volumes will only pay a discounted input price. We turn next to the case of uniform pricing. If the supplier still chooses to make an offer that is acceptable to both firms, which is what we assume for the moment, then the following result follows directly from Proposition 1. Proposition 2. In the case of separate markets and uniform pricing, the supplier offers both downstream firms the price that it offers the more efficient firm under price discrimination. Consequently, compared to the case of price discrimination, under uniform pricing the input price for the more efficient firm stays unchanged while that of the less efficient firm decreases. 15 The argument does not hinge on the assumption that inputs from both sources are homogeneous. Precisely, if γ>1units of the alternative input were required to produce one unit of the output, then the respective margin on which the preceding argument is based would now be γw i w >0. 8

9 If the uniform offer w is acceptable to both firms, then by Proposition 2 output is unaffected in the market served by the more efficient firm while it strictly increases in the market served by the less efficient firm. If P (q) denotes the marginal valuation of a representative consumer, we then have the following result. Corollary 2. In the case of separate markets, uniform pricing strictly increases consumer surplus and welfare in the short run if k 1 6= k 2. Note that if the supplier is an unconstrained monopolist, the short-run implications of uniform pricing are typically ambiguous. If the supplier still sells to both firms, then the optimal uniform price lies strictly between the higher discriminatory price of the more efficient firm and the lower discriminatory price of the less efficient firm. (Inthecaseoflineardemand,which is analyzed in Section 3.3, consumer surplus remains unchanged, though.) Finally, there is yet another difference between the two cases if the supplier ends up selling only to one firm under uniform pricing. If this is the case, then the unconstrained monopolist will only sell to the more efficient firm, while in our model it will be the larger and more efficient firm that will switch. 16 Interestingly, even in this case consumer surplus is strictly higher under uniform pricing, which now follows, however, as from bw <w i theoutputofthemoreefficient firm will increase. 3.2 The Long Run At the first stage of our model, downstream firms can invest in a reduction of their own marginal costs. As a benchmark, suppose first that w i was exogenously fixed. (For instance, it could be the price prevailing in a perfectly competitive upstream market.) In this case, firm i optimally chooses the reduction k in own marginal costs k i = k i k to trade off the resulting marginal increase in profits π 0 (c i ),wherec i = k i + w i, with the marginal investment costs e 0 ( k ). If w i is, instead, strategically chosen by the supplier and thus varies with k i, firm i s marginal benefits from reducing k i are strictly higher. This follows as by Proposition 1 a reduction in k i also leads to a lower input price w i. Formally, the marginal benefits from a reduction of w i in case of discriminatory pricing are given by dπ(c µ i) = π 0 (c i ) dk i 1+ dw i dk i, (4) 16 As this is an immediate consequence of Propositions 1 and 2, we omit a formal statement of this result. 9

10 where dw i /dk i > 0. If now k i k j holds, then the marginal benefits from decreasing k i are unchanged if there is uniform pricing, which follows immediately from our previous observation that the uniform input price is the same as the discriminatory price for the more efficient firm. In contrast, if it holds that k i >k j, then the incentives for firm i are strictly lower under uniform pricing. This follows in turn as given k i >k j, even after a marginal reduction of k i firm i will still be the less efficient firm, implying that k i will not affect the prevailing uniform input price. Turning to the determination of the equilibrium at the investment stage, we assume that the expenditure function e( k ) is sufficientlyconvexsuchthatthefirms optimal investments are unique under discriminatory pricing. Suppose also for the moment that firms start out with the same initial marginal costs k i. While under discriminatory pricing they will also end up being symmetric with k 1 = k 2, under uniform pricing one firm will always end up being more efficient than the other. This follows as by our previous observations, the marginal benefits from increasing k under uniform pricing have a discontinuity at k 1 = k 2. Recall that this is in turn due to the fact that the uniform price only depends on the marginal costs of the ex-post more efficient firm.wethushavethefollowingresults. Proposition 3. If markets are separate, then in the long run, one firm will choose the same and the other firm a strictly lower investment than under price discrimination. This also implies that even if firms are initially symmetric, in the long run one firm will be more efficient than the other. Proof. See Appendix. Another immediate implication of Proposition 3 is that if firms are initially equally efficient, then both discriminatory and uniform pricing will lead to the same input prices. This holds as without discrimination, the input price is determined by the ex-post more efficient firm, which makes the same investment as under discriminatory pricing. However, under uniform pricing the other firm invests strictly less such that its total marginal costs are strictly higher under uniform pricing. We thus have the following result. Corollary 3. In the case with separate markets, if downstream firms are initially equally efficient and can invest in a reduction of own marginal costs, then total consumer surplus is strictly higher under price discrimination than under uniform pricing. In the linear example analyzed further below, we show that uniform pricing also leads to lower total welfare. Furthermore, while Corollary 3 only covers the case where firms are initially 10

11 symmetric, one can apply continuity arguments to show that the implications are the same as long as firms are initially not too different. Otherwise, i.e., if some firm j is initially sufficiently less efficient, then its total marginal costs may still go down under uniform pricing as the lower input price may more than compensate for its lower investment. To conclude this Section, we can compare our results for the long run with the outcome if the supplier is an unconstrained monopolist. There, the imposition of uniform pricing reduces a hold-up problem, which arises as the supplier skims off some of the gains if a firm becomes more efficient. 17 Unambiguous results for consumer surplus and welfare changes can be obtained for the case of linear demand, to which we turn next. 3.3 The Linear Case We take first the case of separate markets, where for linear inverse demand P (x) = 1 x we obtain the quadratic profits π(c i )= 1 c i Substitution into the binding participation constraint (3) yields the discriminatory prices w i =1 k i p (1 k i bw) 2 4F. (5) which immediately confirms that w i > bw is strictly increasing in both bw, F,andk i.usingthat the supplier s unconstrained optimum is now wi UC =(1 k i )/2, we can also make more explicit the condition for when the firm s participation constraint indeed binds, which is the case for all µ 1 ki bw 2 µ 1 2 ki F. 2 4 Under uniform pricing, it is next still optimal for the supplier to serve the more efficient firm, say i =1as k 1 <k 2,if w 1 ( 1 w 1 k w 1 k 2 2 ) w 2 ( 1 w 2 k 2 ), 2 where we can substitute w 1 and w 2 from (5) and which surely holds if either F is sufficiently low or, holding F fixed, if k 1 and k 2 are close enough. 17 Interestingly, a recent paper by Hermalin and Katz (2006) finds that improving the seller s information can actually decrease the buyer s hold-up problem and lead to more downstream investment. In their asymmetric information context, pricing conditional on a signal of investment can be rephrased as discriminatory pricing. 18 Linear demand is a frequent assumption in the literature on third-degree price discrimination (see, e.g., DeGraba (1990) and Yoshida (2000)). 11

12 For the long-run analysis, we follow DeGraba (1990) and take a quadratic investment cost function e( k )=t( k ) 2 /2, wherewithoutaffecting results we set now t =1. 19 We restrict the analysis to the case where firms are initially symmetric. As shown in the proof of Proposition 4, under price discrimination firms choose to reduce own marginal costs by k = PD := 1 k bw. Under uniform pricing, the ex-post less efficient firm chooses instead k = U := 2 p ( PD ) 2 F PD,whichgiven2 p ( PD ) 2 F<2 PD is indeed strictly lower than PD. Finally, we can also show that total welfare is strictly quasiconcave in k andthateventhe optimal choice under price discrimination, PD,isinefficiently low from the perspective of total welfare. Consequently, with linear demand uniform pricing reduces not only consumer surplus but also total welfare. Proposition 4. Suppose that firms are initially symmetric and that demand is linear and investment costs are quadratic. Then with separate markets consumer surplus and welfare are strictly higher in the short run but strictly lower in the long run. Proof. See Appendix. To conclude the discussion of linear demand and separate markets, we consider briefly the case with asymmetric markets. Clearly, all our key results, namely that a reduction of k i pushes down w i under price discrimination and that uniform pricing lowers incentives, depend on our assumption of symmetric markets. If markets are asymmetric, however, there is additional scope for price discrimination. Once again, the results from our model are markedly different from those that are obtained if the supplier is an unconstrained monopolist. Suppose therefore that inverse demand is now given by P i (q) =a i q, where a higher value of a i corresponds to a larger market. A discriminating unconstrained supplier would choose w i =(a i k i )/2, setting again a higher price in the stronger market. This is again the opposite to what we find if the supplier is constrained. To see this, observe that for a given input price w, which equals w i under the supplier s offer and bw under the alternative supply option, the downstream firm s profits are now equal to π =(a i k i w) 2 /4. The cross derivative of profits with respect to market size a i and the input price w is strictly negative, which is just a 19 Admittedly, as the derivative remains bounded as k k i, this does not satisfy one of the stipulated requirements for the general case. In what follows, we will, however, impose sufficient conditions to ensure that the equilibrium is still interior. (See the proofs of Propositions 4 and 8 for details.) 12

13 formal restatement of the intuitive result that the firm s gains from a market expansion are larger the lower its input price. Given an input price w i > bw, this in turn implies that a (marginal) increase in a i pushes up the value of the firm s alternative supply option Vi A by strictly more than the firm s profits under the supplier s offer. To still satisfy the firm s participation constraint, the supplier is then forced to reduce w i. 4 Competition 4.1 The Short Run Price Discrimination With competition, we have to be more specific about what is publicly observable. We assume that individual offers w i are not observable, while it can be observed whether some firm i has changed its source of supply. As will become clear below, these specifications are, however, not crucial for our results, given that input prices are linear. It is now convenient to first review the benchmark where the supplier is an unconstrained monopolist. In this case, the supplier optimally chooses input prices to maximize its total profits w 1 q(c 1,c 2 )+w 2 q(c 1,c 2 ),wherec i = w i +k i. At an interior solution, each input price w i solves the respective first-order condition, implying again that the supplier charges the more efficient firm a strictly higher price. Interestingly, while under discriminatory pricing the effect of a reduction of k i on the input price w j of its competitor is in general ambiguous, with linear demand w j will not be affected. To solve for the equilibrium input prices in our model, we specify that downstream firms have passive beliefs, i.e., that after receiving a deviating offer firm i does not change its beliefs about the offer that is simultaneously made to the other firm. If in addition F and bw are again sufficiently small, then by optimality for the supplier the participation constraints for both firms will be binding in equilibrium. We now denote the Cournot quantities if firms have known total marginal costs c i and c j by q(c i,c j ) and the respective profits by π(c i,c j ). Under competition, the binding participation constraint for firm i is then given by 20 π(c i,c j )=V A i := π(bc i,c j ) F (6) 20 Note that in equilibrium, each firm will form rational expectations about the offers made to other firms. By the preceding argument, we can thus for the sake of brevity omit any additional notation for firms beliefs. 13

14 for i =1, 2 and j 6= i. We thus obtain two equations that characterize equilibrium input prices. As we show in the proof of Proposition 5 below, for all sufficiently low F the solution is also unique. We will focus on this case. Proposition 5. Suppose for the case with competition that the partial derivatives of equilibrium profits satisfy π 11 > 0 and π 12 < 0. Then under discrimination, equilibrium input prices are uniquely determined and a more efficient firm obtains a strictly lower price. Moreover, if some firm i becomes more efficient as k i decreases, then this both lowers its own input price and raises that of its competitor j. Proof. See Appendix. The joint assumption on the profit function, namely that π 11 > 0 and π 12 < 0, iscommonly made in the literature and holds, in particular, under linear demand. For instance, these assumptions are made in the literature on R&D races and strategic cost reduction such as in Katz (1986). They are also satisfied by most common functional specifications for demand. 21 In our model, these assumptions allow to make precise predictions on how a change in a firm s marginal costs affects both its own participation constraint and that of its competitor and thereby the respective input prices. We next provide more details on this. As we have already argued in the case of separate markets, a reduction in a firm s own marginal costs k i increases both the value of its alternative supply option Vi A and the ( onequilibrium ) profits from the supplier s offer. To sign the effect on the equilibrium input price w i, however, it was important that the marginal effect on Vi A was stronger, which formally held as the monopoly profits π(c) were strictly convex and as w i > bw. If the Cournot profits π(c i,c j ) are still convex in own costs as π 11 > 0, then this result still holds. That is, a reduction in k i still makes the participation constraint of firm i tighter, which in turn forces the supplier to lower w i. Turning to the impact on the firm s competitor j, a reduction in the marginal costs of firm i reduces both its competitor s off-equilibrium profits Vj A and its on-equilibrium profits π(c i,c j ). If π 12 < 0 holds, then the effect on Vj A is again stronger, implying that the participation constraint for firm j is now relaxed. This allows the supplier to raise the respective input price w j. In what follows, we will always invoke the assumption that π 11 > 0 and π 12 < 0. If two competing firms purchase inputs from the same supplier, then our model suggests that discriminatory input prices amplify efficiency differences in two ways, namely via a reduction in 21 See Athey and Schmutzler (2001) for a detailed discussion. 14

15 the more efficient firm s input price and via an increase in the less efficient firm s input price. One reading of Proposition 5 is that the exercise of buyer power on the procurement market may lead to higher input prices for other, possibly less powerful, buyers, which then exacerbates their competitive disadvantage in the downstream market. In the European Union, this has been recognized as a potential harm in antitrust decisions (see, for example, Faull and Nikpay (1999), para ). To rationalize such a waterbed effect, it is sometimes argued that suppliers will try to recoup the margin they lose with the more powerful buyer. In contrast to this argument, which leaves open the question of why the supplier did not try to charge a higher price earlier on, Proposition 5 formalizes an alternative channel by which such a waterbed effect could arise. Uniform Pricing Recall that with separate markets, the imposition of a nondiscrimination requirement lowered the input price of the less efficient firm while leaving that of the more efficient firm unchanged. Consequently, the more efficient firm was not affected. With competition, however, the more efficient firm is adversely affected both as its competitor s input price decreases and as its own input price increases. To see why this is the case, suppose that k i <k j and that, following the imposition of uniform pricing, the supplier would offer both firms the discriminatory price of the more efficient firm. As we know from Proposition 5, however, the reduction in the less efficient firm s input price relaxes the participation constraint of the more efficient firm, which in turn allows the supplier to increase the uniform price w i = w j = w. Proposition 6. In case firms compete and have different own marginal costs, the imposition of a nondiscrimination requirement leads to a unique uniform input price. The uniform price lies strictly above the discriminatory price of the more efficient firm and strictly below that of the less efficient firm. Proof. See Appendix. For general demand functions, we can not unambiguously determine how output and thus consumer surplus change under uniform pricing, though with linear demand we show below that both increase, which mirrors our results under separate markets. Finally, we have the following additional comparative result, which will prove useful for the subsequent analysis of the long run. 15

16 Corollary 4. A marginal reduction of the more efficient firm s own marginal costs strictly reduces the uniform input price w, while a marginal reduction of the less efficient firm s own marginal costs increases w. Proof. See Appendix. Both results in Corollary 4 follow intuitively from Proposition 6. In particular, if firm j is the less efficient firm, a (marginal) reduction of k j relaxes the participation constraint for the more efficient firm i. As the participation constraint for the more efficient firm i determines the uniform price w, this implies a higher input price for both firms. 4.2 The Long Run We consider again first the case of discriminatory pricing. A marginal reduction in k i increases the firm s profits by dπ(c i,c j ) ci = dk i µ = q i k i π 1 (c i,c j )+ c j 1+ w i k i π 2 (c i,c j ) k i µ 1 q j c i P 0 w j k i q j c j P 0, (7) where we abbreviate q i = q(c i,c j ) and q j = q(c j,c i ). Recall from (4) that with separate markets, the marginal benefits from a reduction in k i were given by q i ³1+ dw i dk i, reflecting the direct effect on the firm s own production costs and the reduction in its input price. With competition, incentives to reduce own marginal costs arise now from two additional channels. First, firm i s lower own marginal costs imply that its competitor reduces its output q j, which in turn increases the price on the downstream market. Second, we know from Proposition 5 that a reduction of k i leads to an increase in the competitor s input price w j, which in turn leads to a further reduction of q j and thus again to a higher price on the downstream market. In expression (7), the latter effect is accounted for by the final terms, where we can substitute that w j / k i < 0 and q j / c j < 0. Under uniform pricing, where the same price w applies to both firms, we have that dπ(c µ i,c j ) = q i 1+ w µ 1 q j P 0 w q j P 0. (8) dk i k i c i k i c j We take first the case where k i k j. An important difference compared to the incentives under price discrimination, as given by (7), is that a lower input price is now shared by the firm s competitor, which reduces incentives. Formally, this is captured by the last term in (8), 16

17 wherewenowhave w/ k i > 0 instead of w j / k i < 0 as in (7). In addition, for given levels of efficiency we know that under uniform pricing the quantity that is sold by the more efficient firm i is strictly lower, which further dampens incentives to reduce own marginal costs. We have the following result. Lemma 1. In case of competition, the marginal benefits from reducing own costs are strictly lower for the more efficient firm under uniform pricing than under price discrimination. Proof. See Appendix. It is useful to recall at this point that with separate markets, the more efficient firm s incentives were unaffected by the imposition of uniform pricing. Lemma 1 provides thus a stronger result if there is competition. Despite this observation, Lemma 1 does not imply that in equilibrium the ex-post more efficient firm will always make a lower investment under uniform pricing, though we show that this is the case under linear pricing (and for low values of F ). This holds for the following reason. If the ex-post less efficient firm invests sufficiently less than under discriminatory pricing, then even though both firms have the same input price under uniform pricing the ex-post more efficient may end up with a much larger market share of the market. Its output may thus be higher than under discriminatory pricing, which provides a countervailing force to Lemma 1. We already used in our previous arguments that under uniform pricing firms will always have different levels of efficiency in the long run. This was also the case with separate markets, where incentives for the ex-post less efficient firm were lower as its own marginal costs did not affect the prevailing uniform price. Incentives are further muted with competition where, as we know from Corollary 4, a reduction in the less efficient firm s own marginal costs even increases the uniform input price. 22 Proposition 7. Under competition, if there is uniform pricing then in the long run firms always end up having different levels of efficiency and thus different market shares. This holds, in particular, also for the case where firms are initially symmetric, in which case the ex-post less efficient firm will always invest strictly less than under price discrimination. Proof. See Appendix. 22 We make here use of the common assumption that a reduction in w would increase both firms profits (see, for instance Vives (1999, p. 105)). 17

18 4.3 The Linear Case With linear demand we can again obtain, also in the long run, some unambiguous results on consumer surplus. In fact, with competition the extant literature often considers exclusively the case of linear demand, as in DeGraba (1990) and Yoshida (2000). Here, we obtain equilibrium quantities q(c i,c j )=(1 2c i + c j )/3 and profits π(c i,c j )=(1 2c i + c j ) 2 /9. As the existing literature shows, with linear demand the optimal discriminatory prices of an unconstrained supplier are identical to those with separate markets: w i =(1 k i )/2. Note next that π satisfies π 11 =8/9 > 0 and π 12 = 4/9 < 0. With price discrimination, the binding participation constraint (6) changes to (w i bw)[1 2k i +2k j bw (w i w j )] = 9 F. (9) 4 After some transformation, we obtain from (9) that the average discriminatory price W := (w i + w j )/2 must solve 2 k i k j +4bw 2W A i A j =0, (10) where A i = p (1 2k i + k j 3 bw +2W ) 2 18F and A j is defined symmetrically. Under uniform pricing, if k i k j we have from the participation constraint for firm i that w = bw + 9F 4(1 2k i + k j bw). (11) What facilitates now the comparison of the two cases is the general insight that under homogenous Cournot competition total output only depends on the average marginal costs. Hence, to compare consumer surplus under the two regimes it is thus sufficient to compare w from (11) with W, which solves equation (10). This allows us to obtain, at least for low F, the result that with linear demand consumer surplus is again higher in the short run if price discrimination is banned. However, in the long run the opposite result holds, which again perfectly mirrors the findings for separate markets. Proposition 8. As in Proposition 7, suppose again that firms are initially symmetric and that demand is linear and investments costs are quadratic. Then, at least for low F, under uniform pricing consumer surplus is again strictly higher in the short run but strictly lower in the long run. Proof. See Appendix. 18

19 input prices w W w w k 1 Figure 1: Input prices in the short run Our findings in Proposition 4 are, however, stronger as we could show that with linear demand and separate markets, also welfare and thus not only consumer surplus increased in the short run but decreased in the long run after banning price discrimination. With competition, matters are complicated by the fact that a change in relative input and thus also final prices causes a shift of market shares between the intermediary firms. Whilewewerenolongerable to prove a general result, for all examples that we studied we found, however, that also total welfare increased under uniform pricing in the short run. In Figure 1 we report an example where bw =0.3, F =0.01, andk 2 =0.15. Thefigure shows how a change in k 1 k 2 affects the uniform price w (the solid line) and the average price under discrimination W (the dotted line). While the two prices clearly coincide when firms are symmetric, k 1 = k 2, the uniform price lies otherwise always strictly below the average price under price discrimination. For this example, we verified that total welfare is strictly higher under uniform pricing for all k 1 <k 2, and that the difference is strictly increasing in the difference of marginal costs k 2 k 1. It is also interesting to note that the average price W in Figure 1 is relatively flat. As k 1 decreases, the reduction of w 1 is almost completely compensated by an increase in the competitor s input price w 2. For the long-run analysis, recall that in the general case we could only show that the expost less efficient firm (under uniform pricing) will always invest strictly less than under price discrimination (Proposition 7). At least for very small F, where we can use the derivatives around F =0, we can now show that both firms will have strictly higher marginal costs in the long run if price discrimination is banned. While this implies that consumer surplus must be 19

20 lower, as claimed in Proposition 7, we can also show that, again for low F,welfaremustbe lower in the long run. (Note that we use here again a quadratic investment technology with e( ki )=t( k ) 2 /2.) Proposition 9. For low F and linear demand, both firms invest less under uniform pricing, which reduces not only consumer surplus but also total welfare in the long run. Proof. See Appendix. We conclude with an example in Figure 2. Setting t =2, bw =0.3, and k i = k =0.4, the left panel plots as a function of F the prevailing equilibrium investment levels under price discrimination (the symmetric equilibrium described by the dotted line) and under uniform pricing (the asymmetrci equilibrium described by the solid lines). Uniform pricing raises long-run marginal costs for both firms. The right panel plots the resulting welfare under price discrimination (the dotted line) and under uniform pricing (the solid line). Intuitively, welfare decreases with F in both cases. It is always strictly higher in the long run under price discrimination than under uniform pricing. k F Welfare F Figure 2: Investments (left panel) and welfare (right panel) in the long run 4.4 Discussion Firms Alternative Supply Option We stipulated that all firms have the same alternative supply option. A different and equally plausible scenario could be that a more efficient, and thus ultimately larger, firm would spend more resources to reduce marginal costs under its alternative supply option, e.g., by investing more when integrating backwards or by searching more intensively for a more suitable substitute. Formally, we could suppose that after rejecting the supplier s offer a firm can invest h( w ) to reduce its marginal costs under the alternative supply option from w>0down to bw i = w w. 20

21 We have analyzed this modification and found that all our results continue to hold. addition, making bw i endogenous further increases the differential in input prices between more and less efficient firms. To be more precise, take first the case of separate markets, where the valueoftheoutsideoptionisnowvi A := max wi [π(k i + bw i ) h(w bw i )] F. That the optimal choice of bw i is strictly decreasing in k i follows immediately from strict convexity of π. Moreover, we have that dvi A /dk i = bq i,wherebq i is higher the lower bc i = k i + bw i, implying that a lower value of k i has now both a direct effect and an indirect effect via a reduction of bw i. 23 With competition, we have in addition that if firm i takes up its alternative supply option, then the competing firm j anticipates that firm i will end up with a lower bw i the lower k i,implyingthat it will choose a lower quantity bq i. This further increases the value Vi A the equilibrium input price w i. In and thus pushes down Linear Contracts We conclude this Section by discussing our assumption of linear supply contracts. In the case of separate markets, though under additional assumptions also with competition, the marginal input price for each firm would be the same if we allowed for more complex contracts such as two-part tariffs. 24 While supply contracts are indeed often much more complex than the simple linear contracts that we study, e.g., as they specify quantity discounts or slotting fees, the notion that the distribution of profits between up- and downstream firms has no impact on marginal input prices and thus on both quantities and prices on the final market seems equally extreme. Both casual evidence and more systematic data collection suggest that discounts given to particular retailers are at least partially passed on to consumers Here we assume that h( w ) is sufficiently convex and smooth to make the problem interior and allow to apply the envelope theorem. 24 Note that here and in what follows we continue to assume that contracts are only bilateral and non-observable to other parties. 25 An example is provided by the banana wars between the UK s main retailing chains in (as documented, for instance, in Concentration in Food Supply and Retail Chains, August 2004, WP Department for International Development, UK). Following a huge volume discount negotiated by Asda, a fully owned subsidiary of Wal-Mart, with DelMonte, Asda started a prolonged price war by cutting the price of loose bananas from 1.08 to 0.94 pounds per kilo. More systematic evidence for the UK was gathered in the Competition Commission s 2000 Supermarket report, documenting how large discounters received substantial discounts, which ultimately showed up in lower shelf prices (see Competition Commission, Supermarkets: A Report on the Supply of Groceries from Multiple Stores in the United Kingdom, Report Cm-4842, 2000). See also Milliou et al. (2004) for a recent 21