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Abstract

  1. Top of page
  2. Abstract
  3. 1. The Model
  4. 2. The Analysis
  5. 3. Long-run Analysis
  6. 4. Continuous Distribution of Downstream Costs
  7. 5. Demand-side Substitution
  8. 6. Conclusion
  9. Appendix A: Proofs of Propositions and Lemmas
  10. References
  11. Supporting Information

We consider a monopolistic supplier's optimal choice of wholesale tariffs when downstream firms are privately informed about their retail costs. Under discriminatory pricing, downstream firms that differ in their ex ante distribution of retail costs are offered different tariffs. Under uniform pricing, the same wholesale tariff is offered to all downstream firms. In contrast to the extant literature on price discrimination with non-linear wholesale tariffs, we find that banning discriminatory wholesale contracts often improves welfare. This also holds if the manufacturer is not an unconstrained monopolist. Moreover, uniform pricing increases downstream investments in cost reduction in the long run.

Third-degree price discrimination is a widely used business practice in intermediate-good markets, i.e. manufacturers often apply different conditions to identical transactions with different retailers.1 The pros and cons of this pricing practice have been discussed among legal and economic scholars since the 1930s and are still debatable. Whether price discrimination by a large manufacturer represents an abuse of its dominant position is a crucial question in many anti-trust decisions on both sides of the Atlantic Ocean.2 Most contributions to the economic literature on the welfare effects of price discrimination focus on linear wholesale tariffs (Katz, 1987; DeGraba, 1990; Yoshida, 2000; O'Brien, 2002; Valletti, 2003; Inderst and Valletti, 2009). A common pricing practice in business-to-business relations, however, is quantity rebate schemes, which is hardly surprising in the face of the well-known double marginalisation problem.3 Regarding the welfare effects of price discrimination, the extant literature that allows for non-linear wholesale tariffs – by and large – agrees upon a ban on price discrimination being detrimental for social welfare (O'Brien and Shaffer, 1992; Rey and Tirole, 2007; Inderst and Shaffer, 2009; Arya and Mittendorf, 2010).4

This clear-cut theoretical prediction is at odds with the legal practice in the EU as well as in the US, where anti-trust authorities regard quantity discounts as a justifiable pricing strategy of manufacturers as long as they are non-discriminatory. For instance, in the Michelin I judgement from 1981, the European Commission did not contest the quantity rebate scheme itself, but its alleged discriminatory nature with ‘comparable amounts purchased almost never result[ing] in the same or comparable discount being granted’ (Recital 42 of Commission decision 81/969/EEC).5 In contrast to the extant theoretical literature but in line with the usual legal practice, we derive conditions such that banning discriminatory non-linear wholesale tariffs is socially desirable. The novelty of our study is to allow for privately informed downstream firms.6

We investigate the welfare effects of banning discriminatory non-linear wholesale tariffs in a model with two downstream firms that have private information regarding their own retail cost, which is either high or low. Ex ante, downstream firms differ in the distribution of their retail cost and this is known by a monopolistic manufacturer. If third-degree price discrimination is permitted, the manufacturer offers a different menu of quantity–transfer pairs to downstream firms with different distributions of retail costs. Under uniform pricing, on the other hand, the same menu is offered to both downstream firms. When deciding whether to accept the manufacturer's offer, each downstream firm is privately informed about the realisation of its retail cost. Thus, the manufacturer offers non-linear tariffs not only to reduce double marginalisation but also to screen downstream firms according to retail efficiency.

We consider a model with two downstream firms, each of which serves an independent market. The quantities procured by low-cost retailers turn out to be independent of the pricing regime because under both regimes there is no distortion at the top. The quantities procured by high-cost retailers, on the other hand, are distorted downwards and the magnitude of this distortion depends on the pricing regime. Under price discrimination, the high-cost type of the ex ante more efficient firm – more likely to be a low-cost producer – procures a lower quantity than the high-cost type of the ex ante less efficient firm. The quantity procured by ex post high-cost retailers under uniform pricing is bracketed by the quantities assigned to high-cost retailers under price discrimination. Therefore, in expectation, banning price discrimination harms the market which is served by the ex ante less efficient firm, whereas the other market, which is served by the ex ante more efficient firm, benefits. Due to these opposing effects, general welfare results are hard to obtain. Nevertheless, we show that uniform pricing is optimal from a welfare point of view as long as price discrimination does not lead to an expansion of (expected) total output. For the case of linear demand and provided that all markets are being served under either pricing regime, price discrimination does not lead to an expansion of total output and thus is detrimental for welfare. This output criterion is known from the literature analysing third-degree price discrimination in final-good markets with linear tariffs and without asymmetric information (Schmalensee, 1981).7 For the case of homogeneous Cournot competition downstream, Schmalensee's observation, together with the fact that the less efficient downstream firm receives a discount under price discrimination, directly implies that banning price discrimination in input markets improves welfare if wholesale contracts are linear (Katz, 1987; DeGraba, 1990).8

In our model, it can be optimal for the manufacturer not to serve a high-cost retailer to cut back on information rents. In consequence, if the average probability of high-cost production is low but the ex ante less efficient firm is nevertheless quite likely to produce at high cost, then high-cost production takes place only under price discrimination and only in the ex ante less efficient market. Here, price discrimination leads to more markets being served in expectation and unambiguously improves welfare. This finding resembles the classic Chicago school argument in favour of price discrimination (Bork, 1978).9

By allowing downstream firms to invest in process innovation, DeGraba (1990) identifies another channel through which uniform pricing can improve welfare. By extending our model to a long-run analysis in the spirit of DeGraba (1990), we show that if downstream firms can invest in the (expected) efficiency of production, uniform pricing results in higher investment incentives, thereby potentially leading to overall higher welfare. This conjecture is confirmed for the case of linear demand downstream.

A widespread assumption in the literature on price discrimination in input markets is that the manufacturer is an unconstrained monopolist who can make take-it-or-leave-it offers to its retailers. Two exceptions are Inderst and Valletti (2009) and O'Brien (2002), both of whom focus on linear wholesale contracts. In Inderst and Valletti (2009), the manufacturer is constrained by the threat of demand-side substitution. O'Brien (2002) assumes that wholesale prices are determined by bilateral negotiations between the manufacturer and downstream firms. Both contributions show that relaxing the assumption of an unconstrained monopolistic manufacturer can give rise to circumstances where the welfare implications of price discrimination are reversed. To show robustness of our findings we introduce upstream competition in a similar vein to Inderst and Valletti (2009). Focusing on the cases where the retailers procure the inputs from the manufacturer in equilibrium, we show that banning price discrimination often improves welfare even in the presence of an alternative source of supply. Although outside the scope of our study, an important concern of anti-trust authorities is whether price discrimination is used by manufacturers to exclude potential rivals upstream, so called primary-line injuries. How a manufacturer can use discriminatory tariffs to exclude a rival is at the heart of the analysis of Giardino-Karlinger and Motta (2012), who consider a model with network effects and inelastic demand. In our model, in contrast, there are no network effects and demand is elastic, which implies that double marginalisation is an issue.

The article closest related to this study is Inderst and Shaffer (2009), who assume that the manufacturer offers observable two-part tariffs to the retailers. Focusing on asymmetric downstream firms, discriminatory contracts are shown to amplify differences in downstream firms' competitiveness. A ban on price discrimination tends to raise all final-good prices and thus to reduce total output. In consequence, banning price discrimination reduces consumer surplus and welfare. Similarly, Arya and Mittendorf (2010) show a ban on discriminatory two-part tariffs to be always welfare harming when downstream firms are asymmetric in the sense that one operates in multiple markets while the other downstream firm is active only in a single market. Whereas in Inderst and Shaffer (2009) as well as in Arya and Mittendorf (2010) the manufacturer is perfectly informed about the downstream firms' asymmetries, this is not the case in our model. We show that introducing asymmetric information can turn these previous welfare findings upside down.

Due to our focus on separate markets, it is of no relevance for an individual retailer whether he knows the supply conditions of the other retailers. If retailers operate in the same market, on the other hand, a crucial modelling assumption is whether retailers can observe their rivals' contracts. If contracts are unobservable, the manufacturer faces a commitment problem when price discrimination is permitted and the outcome optimal from the integrated structure's point of view cannot be obtained. Uniform pricing restores the manufacturer's commitment power which implies that banning price discrimination reduces welfare (see Rey and Tirole, 2007). This commitment issue – which first was raised by Hart and Tirole (1990), O'Brien and Shaffer (1992, 1994) and McAfee and Schwartz (1994) – is completely absent in our analysis.

With downstream firms operating in separate markets the question of the policy relevance of our findings is immediate, because both Art. 102(c) TFEU as well as the Robinson–Patman Act consider discriminatory pricing as an abuse of a dominant position only if a downstream firm is placed at a competitive disadvantage.10 Nevertheless, the European Commission and the Community courts have largely applied Art. 102(c) TFEU to manufacturers' pricing practices which have little to do with putting their retailers at a competitive disadvantage – most notably to ban geographic price discrimination across member states (Geradin and Petit, 2005). Fighting geographic price discrimination is of high relevance for competition policy in the EU. As stated in Art. 18 TFEU, one of the most basic principles of the Treaty of the EU is the avoidance of discrimination based on national grounds. The landmark case on geographic price discrimination is United Brands (Commission decision 76/353/EEC). United Brands Company (UBC) sold bananas to distributors/ripeners from various Member States at significantly different prices, with the prices charged from Danish distributors exceeding the prices charged from Irish distributors by 138%. According to the Commission, UBC's pricing practice constituted an abuse of its dominant position. Similarly, Tetra Pak, one of the world's leading companies for the packaging of liquids in cartons, charged considerably different prices across Member States (Tetra Pak II, Commission decision 92/163/EEC). The Commission and the CFI concluded that Tetra Pak's business strategy was an abuse of its dominant position and an infringement of Art. 102(c) TFEU.11

The rest of the article is organised as follows. In Section 'The Model', we introduce our basic model with a monopolistic input supplier. This model is analysed in Section 'The Analysis'. In Section 'Long-run Analysis', we conduct a long-run analysis by allowing downstream firms to invest in a reduction in production cost before contracting takes place. After considering a continuous-type distribution for downstream firms in Section 'Continuous Distribution of Downstream Costs', Section 'Demand-side Substitution' augments the basic model by assuming that the manufacturer is constrained by the threat of demand-side substitution. We conclude in Section 'Conclusion'. All proofs of Sections 'The Analysis', 'Long-run Analysis', 'Continuous Distribution of Downstream Costs' are relegated to the Appendix A, while the online Appendix provides additional material to Section 'Demand-side Substitution'.

1. The Model

  1. Top of page
  2. Abstract
  3. 1. The Model
  4. 2. The Analysis
  5. 3. Long-run Analysis
  6. 4. Continuous Distribution of Downstream Costs
  7. 5. Demand-side Substitution
  8. 6. Conclusion
  9. Appendix A: Proofs of Propositions and Lemmas
  10. References
  11. Supporting Information

Consider a vertically related industry where the upstream market is monopolised by manufacturer M. The manufacturer produces an essential input that is supplied to the downstream sector. For simplicity, we assume that the manufacturer produces quantity q at constant marginal cost, K > 0. There are two downstream firms, i ∊ {1,2}, that can transform one unit of the input into one unit of the final good.

We assume that downstream firms operate in separate and independent markets, i.e. each downstream firm is a local monopolist.12 Downstream markets are identical in size and characterised by the inverse demand function P(q), which is strictly decreasing, twice differentiable where P > 0, and satisfies the assumption inline image where P > 0.13

Downstream firm i produces at constant marginal cost and without fixed costs. The marginal cost of production is either high or low, inline image with inline image. The last inequality guarantees that the joint-surplus maximising quantity of a vertically integrated firm is strictly positive.

A downstream firm's type, i.e. its marginal cost of production, is private information. The manufacturer only knows the probability inline image with which downstream firm i produces at low marginal cost. Ex ante firm 1 is more likely to produce at low marginal cost than firm 2, i.e. inline image.14

The manufacturer can make take-it-or-leave-it offers to the downstream firms, where the wholesale tariff offered to downstream firm i takes the form of a list of quantity–transfer combinations. With only two ex post types, the manufacturer cannot benefit from specifying more than two items per list. So the tariff offered to firm i is inline image, specifying a quantity inline image and a transfer from firm i to the manufacturer, inline image, for each feasible cost type.15 We assume free disposal, i.e. when having purchased quantity inline image of the input, downstream firm i can produce any quantity inline image of the final output at cost inline image.

The sequence of events is as follows: first, nature draws the cost type for each downstream firm i ∊ {1,2}, which thereafter is privately observed by the respective downstream firm. Next, the manufacturer makes a take-it-or-leave-it offer to each downstream firm. Under price discrimination the manufacturer offers each downstream firm a possibly different tariff, whereas under uniform pricing one and the same tariff applies to both firms. A downstream firm either chooses one of the two offered quantity–transfer pairs or it rejects the manufacturer's offer. In case of rejection, the downstream firm obtains its reservation profit, which is normalised to zero. If the downstream firm accepts a quantity–transfer pair (qt), it decides how much of this acquired input to transform into the final good, and sells the produced output to consumers.

With the set of ex post types (potential ex post profits) being identical for both downstream firms in our model, the quantity discounts offered by the manufacturer under uniform pricing are available to both firms ex ante and in this sense practically available. If the firms' type spaces are different, say because one firm operates in a significantly larger market than the other firm, then the quantities assigned to the firm operating in the large market might never be attractive to the firm operating in the small market. If this is the case, the manufacturer can use a quantity rebate scheme to implement third-degree price discrimination indirectly. This is not feasible in our model, which therefore provides a clean comparison of a situation where price discrimination is permitted to a situation where price discrimination is not only forbidden but also not feasible indirectly via sophisticated quantity rebate schemes.

2. The Analysis

  1. Top of page
  2. Abstract
  3. 1. The Model
  4. 2. The Analysis
  5. 3. Long-run Analysis
  6. 4. Continuous Distribution of Downstream Costs
  7. 5. Demand-side Substitution
  8. 6. Conclusion
  9. Appendix A: Proofs of Propositions and Lemmas
  10. References
  11. Supporting Information

Let inline image denote the quantity optimally produced by a downstream firm that operates at marginal cost c. It is readily verified that inline image is strictly decreasing in c. Due to free disposal, downstream firm i's maximum profit when faced with quantity–transfer tuple (q,t) is inline image, where

  • display math(1)

Thus, downstream firm i's gross profit inline image is strictly increasing and strictly concave in q on inline image and constant for inline image. Moreover, inline image satisfies the following single-crossing property:

Lemma 1. A low-cost downstream firm benefits more from an increase in the quantity of the input than a high-cost downstream firm: for all inline image it holds that

  • display math

Furthermore, let inline image denote the optimal quantity produced by a vertically integrated structure comprising of the manufacturer and a downstream firm with marginal cost c.

2.1. Optimal Wholesale Tariffs

2.1.1. Discriminatory offers

If not restricted to offering the same wholesale tariffs to both downstream firms, the manufacturer solves two independent maximisation problems. When contracting with a downstream firm that produces at low costs with probability α, the manufacturer offers a wholesale tariff inline image to maximise expected upstream profits,

  • display math(2)

subject to Γ being incentive compatible and individually rational,

  • display math

As usual, incentive compatibility requires that the low-cost type obtains a higher quantity than the high-cost type, inline image. Moreover, due to free disposal in the optimum we must have inline image and inline image. Thus, the optimal contract satisfies the following monotonicity constraint:

  • display math(MON)

By standard arguments, the transfers inline image and inline image are uniquely determined by the two binding constraints, (IRinline image) and (ICinline image). Hence, the manufacturer chooses quantities inline image and inline image to maximise

  • display math(3)

subject to the monotonicity requirement (MON). Define

  • display math(4)

Proposition 1. Under discriminatory wholesale tariffs, a low-cost firm obtains the joint-surplus maximising quantity, inline image, whereas the quantity assigned to a high-cost firm is distorted downwards. If a firm is very likely to produce at low costs, inline image, then the quantity assigned to its high-cost type equals zero. If inline image, then inline image is defined by

  • display math(5)

Intuitively, as the probability of dealing with a low-cost downstream firm becomes smaller, the manufacturer chooses the quantity offered to the high-cost type closer to the joint-surplus maximising quantity inline image. If, on the other hand, the probability of contracting with a low-cost downstream firm is sufficiently high, the manufacturer prefers to offer a zero quantity to the high-cost type. This eliminates information rents and in turn allows the manufacturer to extract all the surpluses from the interaction with a low-cost type. A low-cost firm is always assigned the joint-surplus maximising quantity, which is the well-known no distortion at the top result. It is worthwhile pointing out that if the difference in possible retail costs is not too high, both cost types are rather likely to be served by the manufacturer.

2.1.2. Uniform pricing

The requirement that both downstream firms have to be offered the same tariff, i.e. inline image, leaves the incentive compatibility and individual rationality constraints unchanged. The manufacturer chooses quantities inline image and inline image to maximise

  • display math(6)

where inline image. To characterise the optimal tariff under uniform pricing, let

  • display math(7)

denote the value of inline image that, for a given value of inline image, results in an average probability of contracting with a low-cost firm equal to inline image.

Proposition 2. Under a uniform wholesale tariff, a low-cost firm obtains the joint-surplus maximising quantity, inline image, whereas the quantity assigned to a high-cost firm is distorted downwards. If the average probability of both downstream firms producing at low costs is relatively high, inline image, then the quantity assigned to a high-cost type equals zero. If inline image, then inline image is defined by

  • display math(8)
2.1.3. Comparison of pricing regimes

The degree of the downward distortion in a high-cost type's quantity is determined by the respective firm's individual probability of producing at low cost under price discrimination and by the average probability of contracting with a low-cost firm under uniform pricing. The following Lemma orders the different quantities assigned to high-cost types.

Lemma 2. The quantity offered to high-cost firms under uniform pricing is bracketed by the quantities offered to the high-cost firms under price discrimination:

  • display math(9)

2.2. Welfare

We now turn to the welfare implications of banning price discrimination. Welfare under pricing regime r ∊ {D,U}, which is stochastic ex ante, is defined as the sum of consumer and producer surplus, inline image. Let the difference in expected welfare between the discriminatory pricing regime and the uniform pricing regime be inline image.

To delineate the importance of asymmetric information, we first consider a benchmark case with symmetric information.

Proposition 3. Suppose downstream firm 1 produces at low costs with certainty, inline image, and downstream firm 2 produces at high cost with certainty, inline image. Then, permitting price discrimination improves welfare, i.e. ΔW > 0.

With symmetric information, if price discrimination is allowed, it is optimal for the manufacturer to offer each downstream firm the joint-surplus maximising quantity — irrespective of its cost type – and to extract downstream profits fully via the transfer. Under uniform pricing this is not optimal and the manufacturer faces a metering problem similar to the screening problem under asymmetric information. The efficient firm 1 obtains the joint-surplus maximising quantity, but the quantity assigned to the less efficient firm 2 is distorted downwards. With even the joint-surplus maximising quantities being too low from a welfare perspective it is readily obtained that banning price discrimination is strictly welfare harming.16

Under asymmetric information ΔW depends only on the quantities produced by high-cost retailers because there is no distortion at the top under either regime. Formally,

  • display math(10)

Thus, we can distinguish the following four cases, as depicted in Figure 1:

  1. If both downstream firms are relatively unlikely to produce at low costs, inline image, then high-cost production takes place in both markets under either pricing regime, i.e. inline image;
  2. If firm 1 is relatively likely and firm 2 is relatively unlikely to produce at low costs, inline image, then high-cost production takes place in both markets under uniform pricing, whereas under price discrimination high-cost production takes place only in market 2, i.e. inline image;
  3. If firm 1 is very likely and firm 2 is relatively unlikely (but not too unlikely) to produce at low costs, inline image, then high-cost production takes place only under price discrimination and only in market 2, i.e. inline image;
  4. If both firms are relatively likely to produce at low costs, inline image, then high-cost production does not take place under either pricing regime, i.e. inline image.

In case IV, with the manufacturer never serving a high-cost downstream firm irrespective of the pricing regime, we have ΔW = 0. Therefore, in what follows, we focus on the interesting cases I–III.

Before characterising the welfare consequences of banning price discrimination, it is worthwhile pointing out how the different parties are affected by this policy. Clearly, the manufacturer is harmed if price discrimination is banned. Moreover, in expectation, consumers in market 1 benefit while consumers in market 2 are harmed. Finally, note that a downstream firm's ex ante expected profit is equal to its expected information rent, which is increasing in the quantity assigned to its high-cost type. Therefore, from an ex ante perspective, downstream firm 1 benefits while downstream firm 2 is harmed by a ban on price discrimination.

To state the main finding of this Section, define the expected change in quantity as inline image, with inline image denoting the aggregate quantity of the final good.

Proposition 4. Suppose that high-cost production takes place at least in market 2 under price discrimination, i.e. inline image.

  1. If high-cost production takes place under uniform pricing, inline image, then permitting price discrimination harms welfare whenever it does not lead to a strict expansion in expected total output, i.e. ΔQ ≤ 0[RIGHTWARDS DOUBLE ARROW]ΔW < 0;
  2. If high-cost production takes place only under price discrimination, inline image, then permitting price discrimination improves welfare, i.e. ΔW > 0.

In case III (part (ii) of Proposition 4) to cut back on information rents, M assigns a zero quantity to the high-cost type of firm 1 under price discrimination and to high-cost downstream firms in general under uniform pricing. Thus, price discrimination leads to more markets being served (in expectation), thereby benefiting welfare in the spirit of the classic Chicago school argument against non-discrimination clauses.

In cases I and II (part (i) of Proposition 4), it is not clear which pricing regime results in higher expected welfare due to opposing effects. In the case of high-cost production, the quantity sold in market 2 is lower, whereas the quantity sold in market 1 is higher under uniform pricing than under price discrimination – market 1 is not even served under price discrimination in case II. Even though a general welfare result cannot be derived in these cases, we can establish a sufficient condition – resembling Schmalensee's (1981) output test – for uniform pricing to improve welfare: if price discrimination does not lead to an expansion of expected total output, expected welfare decreases if price discrimination is permitted.17

The above discussion leads us to conjecture that banning price discrimination can be welfare enhancing, which is further supported by analysing case II in more detail.

Corollary 1. If high-cost production takes place under uniform pricing but only in market 2 under price discrimination, inline image, then permitting price discrimination becomes more likely to be welfare harming as firm 1 becomes more likely to be a high-cost type, i.e. inline image.

Corollary 1 suggests that banning price discrimination can switch from being welfare harming to being welfare enhancing as the probability of firm 1 to be the low-cost type decreases. We show below that this conjecture holds true for a linear demand function.

Even if price discrimination is legally banned, one might argue that, in practice, the manufacturer might get away with discriminatory wholesale tariffs as long as none of the downstream firms files a complaint. First of all, note that high-cost types obtain their reservation profits independent of the pricing regime. Moreover, as we have argued above, in the case of low-cost production, downstream firm 1 would benefit from the enforcement of the non-discrimination clause, whereas downstream firm 2 would suffer. Therefore, if a downstream firm brings a case, then it must be the low-cost type of firm 1. In the case of a complaint, however, the enforcement of the non-discrimination clause is welfare neutral at best: whereas the quantity in low-cost markets is unaffected, the quantity in market 2 in the case of high-cost production is reduced, which harms welfare. As we argue in Section 'Continuous Distribution of Downstream Costs', this rather negative observation is an artefact of the two-type case.

2.3. An Application with Linear Demand

Suppose demand is linear, P(q) = max{0,1−q}, and assume that inline image. In this case, inline image, inline image, and inline image. Letting inline image be implicitly defined by inline image, tedious but straightforward calculations yield the following result.

Proposition 5. Suppose that high-cost production takes place at least in market 2 under price discrimination, i.e. inline image. Then, permitting price discrimination harms welfare if and only if firm 1 is likely to produce at high cost, i.e. inline image.

Proposition 5 is illustrated in Figure 2. If the anti-trust authority need not be overly concerned about the possibility of one or the other market not being served under either pricing regime (for inline image large), then – at least for linear demand – banning price discrimination is socially desirable. The area where both markets are served irrespective of the pricing regime is quite large if the difference in retail costs between a high-cost and a low-cost firm is relatively low. Hence, if differences in ex post retail costs are not overly large and if the demand function is sufficiently linear in the relevant range of prices, banning price discrimination – the usual legal practice in the EU – improves welfare.

The finding that banning price discrimination often is welfare improving is in contrast to findings in the extant literature on third-degree price discrimination under non-linear wholesale tariffs. Inderst and Shaffer (2009), for instance, consider a manufacturer who is perfectly informed about the retail costs of two asymmetric downstream firms. For the case of separate markets – Proposition 6 of Inderst and Shaffer – they show that banning price discrimination unambiguously reduces welfare, which parallels our symmetric information benchmark (Proposition 3). With symmetric information price discrimination allows the manufacturer to achieve the vertically integrated outcome, which can be considered as a second best. This second-best outcome is not achieved under uniform pricing, where the quantity procured by the inefficient firm is distorted downwards due to the additional incentive constraint. With asymmetric information, in contrast, the manufacturer is constrained by incentive compatibility under either pricing regime. As a result, the quantities procured by high-cost downstream firms are distorted downwards under both pricing regimes. In general, it remains unclear which pricing regime leads to the larger average distortion. Proposition 5 shows that the average distortion is higher under price discrimination than under uniform pricing if demand is linear, leading to a ban on price discrimination being welfare improving. Put differently, our finding shows that the strong welfare result of Inderst and Shaffer does not carry over to the case of asymmetric information.

Finally, the welfare assessment in Proposition 5 allows us to address the usefulness of the output test proposed in Proposition 4(i). Remember that this output test embodies a sufficient condition for price discrimination to be welfare harming. The relevant question to ask therefore is for what part of the dark-grey shaded area in Figure 2 the expected overall quantity is weakly smaller under price discrimination than under uniform pricing. When all markets are always served, case I, then ΔQ = 0 and we can conclude that a ban on price discrimination is welfare improving. In case II, when the high-cost type of firm 1 is not served under price discrimination, price discrimination always leads to an expansion of expected aggregate output. Thus, in case II, the output test does not help to specify which pricing regime is superior from a welfare point of view.

image

Figure 1. Welfare Comparison

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image

Figure 2. Linear Demand

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3. Long-run Analysis

  1. Top of page
  2. Abstract
  3. 1. The Model
  4. 2. The Analysis
  5. 3. Long-run Analysis
  6. 4. Continuous Distribution of Downstream Costs
  7. 5. Demand-side Substitution
  8. 6. Conclusion
  9. Appendix A: Proofs of Propositions and Lemmas
  10. References
  11. Supporting Information

For linear wholesale prices, DeGraba (1990) pointed out a further channel through which differences in the pricing regimes can translate into differences in welfare: with the more efficient downstream firm being discriminated against, price discrimination leads to lower incentives for downstream firms to invest in a more efficient retail technology, thereby harming welfare not only in the short run but also in the long run. In this Section, we show that a ban on price discrimination increases downstream firms' incentives to invest in cost reduction also when non-linear wholesale contracts are in place.

Specifically, suppose that initially both downstream firms produce at high cost with certainty. At some preliminary stage 0, before the manufacturer makes its offers, both downstream firms can simultaneously invest in R&D. If the research of a downstream firm is successful, this downstream firm produces at low cost inline image; otherwise, this downstream firm continues to operate at high cost inline image. If a downstream firm incurs investment cost ψ(α), its research is successful with probability α ∊ [0,1], where inline image and inline image. Thus, in a sense, α reflects a downstream firm's research intensity. The investment in R&D is observed by the manufacturer. Whether the research was successful, however, is private information of each downstream firm. We focus on symmetric equilibria in pure strategies.18 To obtain a clear-cut finding with respect to the difference in investment incentives under the two pricing regimes, we impose the following:

Assumption 1. Downstream marginal revenue is concave: inline image, whenever P > 0.

3.1. Investment Incentives

Remember that transfers charged by the manufacturer are pinned down by (IRinline image) and (ICinline image). Given R&D intensities inline image and inline image, downstream firm i's expected profit at the investment stage under pricing regime r ∊ {D,U} is

  • display math(11)

where inline image denotes the quantity offered to firm i's high-cost type under pricing regime r. Taking the derivative of (11) with respect to inline image yields the following first-order condition:

  • display math(12)

On the one hand, a higher investment makes it more likely that the downstream firm produces at low costs and thus obtains a positive information rent. On the other hand, the information rent decreases in a downstream firm's investment because the quantity assigned to a high-cost firm is decreasing in the investment level. Under price discrimination the expected information rent of firm i depends only on its own investment level, whereas under uniform pricing it depends on the average investment level of both firms. Thus, the manufacturer reacts more strongly to an increased investment of firm i – and cuts back this firm's information rent more severely – under price discrimination than under uniform pricing. In consequence, permitting discriminatory wholesale contracts stifles downstream firms' incentives to invest in a reduction in their production costs.19

Proposition 6. Suppose that Assumption 1 holds, so that downstream marginal revenue is concave. Then, a downstream firm's investment in cost reduction is higher under uniform pricing than under price discrimination, i.e. inline image.

3.2. Welfare

With investment incentives being higher under uniform pricing than under price discrimination, it seems likely that in the long run banning price discrimination is socially beneficial. The next finding establishes this conjecture for a specification with linear demand.

Proposition 7. Suppose that demand is linear and there are no investment costs, i.e. P(q)=max{1−q,0} and ψ(α)≡0 for all α ∊ [0,1]. Then, in the long run, welfare is higher under uniform pricing than under price discrimination.

In the long run, with inline image, both cost types of both downstream firms are always served. Moreover, with investment incentives being higher under uniform pricing than under price discrimination, a firm is more likely to produce at low cost under uniform pricing. This effect supports welfare under uniform pricing compared to price discrimination. With higher investment incentives under uniform pricing, however, the downward distortion in quantity for a high-cost firm is stronger, which reduces welfare under uniform pricing compared to price discrimination. According to Proposition 7 the direct effect due to an increased probability of producing at low costs outweighs the indirect effect of a higher quantity distortion, thereby making a ban on price discrimination socially desirable in the long run.

4. Continuous Distribution of Downstream Costs

  1. Top of page
  2. Abstract
  3. 1. The Model
  4. 2. The Analysis
  5. 3. Long-run Analysis
  6. 4. Continuous Distribution of Downstream Costs
  7. 5. Demand-side Substitution
  8. 6. Conclusion
  9. Appendix A: Proofs of Propositions and Lemmas
  10. References
  11. Supporting Information

In this Section, we allow for the marginal cost of downstream firm i ∊ {1,2} being continuously distributed, i.e. inline image with inline image. Firm i's cost is ex ante distributed according to c.d.f. inline image with density inline image for all inline image. The cost distributions of the two firms are different in the sense that values of inline image exist such that inline image. The manufacturer, who knows the firms' ex ante cost distributions but not effective retail costs, offers downstream firm i a quantity–transfer list inline image, specifying a quantity inline image and a transfer inline image for each feasible cost type. For this continuous-type case, we focus on linear demand P(q) = max{1−q,0}. The manufacturer's expected profit is given by

  • display math(13)

As before the manufacturer has to satisfy the individual rationality and incentive constraints: for all i ∊ {1,2} and inline image,

  • display math(IR)
  • display math(IC)

If price discrimination is banned, the manufacturer has to comply with the non-discrimination constraint inline image. Note that for the manufacturer it is more profitable to contract with low-cost downstream firms. This implies that the usual monotonicity requirement, which is necessary to satisfy incentive compatibility, here requires that inline image and inline image are non-increasing. To avoid bunching, we impose the following assumption in the spirit of the monotone hazard rate property.

Assumption 2. For all inline image it holds that inline image, with i ∊ {1,2}, and inline image are non-decreasing.

Note that Assumption 2, which guarantees that optimal quantity schedules are strictly decreasing, is satisfied if both density functions are weakly decreasing.

Moreover, we focus on cases where – irrespective of the pricing regime – the manufacturer serves all types of downstream firms, which corresponds to case I in the previous analysis.

Assumption 3. inline image.

Lemma 3. Suppose that Assumptions 2 and 3 hold, so that the optimal quantity schedules are all strictly decreasing in the cost type and all cost types procure a strictly positive quantity. Then, cost realisations exist for which the quantities assigned to the two firms under discrimination differ, i.e. there are inline image such that inline image for i,j ∊ {1,2} and i ≠ j. For all these cost realisations the quantity offered under uniform pricing is strictly bracketed by the two discriminatory quantities, i.e. inline image.

According to Lemma 3, one market benefits from price discrimination, whereas the other market is harmed compared to uniform pricing for a given cost realisation. Nevertheless, we obtain a clear welfare result.

Proposition 8. Suppose that Assumptions 2 and 3 hold, so that the optimal quantity schedules are all strictly decreasing in the cost type and all cost types procure a strictly positive quantity. Then, permitting price discrimination harms welfare, i.e. ΔW < 0.

In the two-type case we have argued that if a downstream firm has an incentive to bring a case, the enforcement of a non-discrimination clause is welfare neutral at best. As the next finding shows, with a continuum of cost types, if a firm has an incentive to bring a case, the enforcement of a uniform tariff can improve welfare.

Observation 1. Suppose inline image crosses inline image once from above at inline image, so that firm 1 is more likely to produce at low-cost levels than firm 2. If the manufacturer offers different tariffs to the downstream firms, then the ex ante efficient firm 1 has an incentive to bring a case when its costs are relatively high – in particular for inline image. Moreover, if the lawsuit is successful, output in market 1 increases which increases welfare in market 1.

5. Demand-side Substitution

  1. Top of page
  2. Abstract
  3. 1. The Model
  4. 2. The Analysis
  5. 3. Long-run Analysis
  6. 4. Continuous Distribution of Downstream Costs
  7. 5. Demand-side Substitution
  8. 6. Conclusion
  9. Appendix A: Proofs of Propositions and Lemmas
  10. References
  11. Supporting Information

As was shown by Caprice (2006) and Inderst and Valletti (2009), the implications of price discrimination in input markets for pricing decisions and welfare may be reversed if the assumption of a monopolistic input supplier is relaxed. As we show next, the main effect of downstream firms having an outside option in our model is to shift rents from the manufacturer to the downstream firms. As a result, by and large, our findings are robust towards relaxing the assumption of a monopolistic input supplier.20

Consider the same model as before but suppose that a downstream firm, when rejecting the manufacturer's offer, can turn to an alternative source of input supply. If a firm with marginal cost inline image acquires its input from the alternative supply, then its profits are inline image, with inline image.21, 22 We assume that the alternative supply is not too attractive in the sense that the joint surplus generated by the manufacturer and either type of downstream firm exceeds that downstream firm's profit obtained under the alternative supply.

Assumption 4. For all inline image it holds that inline image.

Define

  • display math(14)

which sets out how much more a low-cost firm benefits from the alternative input supply than a high-cost firm, relative to the low-cost firm's cost advantage. To stick close to our basic model, we keep inline image and inline image fixed and assume that any variation in ϕ arises due to changes in inline image or inline image.

For reasons of tractability, we focus on situations where, under the optimal contract, it is never the upward incentive constraint that is binding, i.e. we do not consider countervailing incentives in the sense of Lewis and Sappington (1989).23 Moreover, we restrict attention to circumstances where the manufacturer serves both types of downstream firms, which allows to draw welfare implications irrespective of the particular form the alternative supply takes. A sufficient condition for the optimal contract to satisfy these properties is as follows:

Assumption 5. inline image, where inline image.

While the existence of an alternative supply leaves the incentive compatibility constraints unchanged, the individual rationality constraints now reflect type-dependent outside options:

  • display math

For pricing regime r ∊ {D,U}, define inline image as the critical probability of low-cost production at which the quantity assigned to a high-cost type in the case without type-dependent outside options equals ϕ. Moreover, let inline image and note that inline image. The following result shows that our previous findings are robust towards relaxing the assumption of an unconstrained manufacturer.

Proposition 9. Suppose that Assumptions 4 and 5 hold, so that under the optimal contract countervailing incentives do not prevail and neither downstream firm procures its input from the alternative supply. Moreover, suppose that firm 2 is likely to produce at high cost, inline image.

  1. If firm 1 is rather likely to produce at high cost, inline image, then permitting price discrimination harms welfare whenever it does not lead to a strict expansion in expected total output, i.e. ΔQ ≤ 0 [RIGHTWARDS DOUBLE ARROW] ΔW < 0;
  2. If firm 1 is rather likely to produce at low cost, inline image, then permitting price discrimination improves welfare, i.e. ΔW > 0.

In comparison to our baseline model, instead of high-cost firms not being served at all, now high-cost firms are offered a ‘rather low’ quantity equal to ϕ if contracting with a low-cost firm is sufficiently likely. Otherwise, however, the intuition behind the welfare result of Proposition 9 is basically the same as the one behind Proposition 4. In particular, banning price discrimination can be beneficial for welfare also in situations where the manufacturer is not an unconstrained monopolist.23 This becomes apparent when investigating the case of linear demand.

Corollary 2. Suppose that Assumptions 4 and 5 hold, so that under the optimal contract countervailing incentives do not prevail and neither downstream firm procures its input from the alternative supply. Furthermore, suppose demand is linear, P(q)=max{1−q,0}. Then, if firm 1 is likely to produce at high cost, inline image, permitting price discrimination harms welfare, i.e. ΔW < 0.

According to Corollary 2, if the potential differences in retail costs are low, a ban on price discrimination improves welfare at least for linear demand. To see this, notice that inline image approaches 1 as inline image tends to inline image.

6. Conclusion

  1. Top of page
  2. Abstract
  3. 1. The Model
  4. 2. The Analysis
  5. 3. Long-run Analysis
  6. 4. Continuous Distribution of Downstream Costs
  7. 5. Demand-side Substitution
  8. 6. Conclusion
  9. Appendix A: Proofs of Propositions and Lemmas
  10. References
  11. Supporting Information

In this article, we analyse a vertically related industry with asymmetric information between the upstream and the downstream sector. The main purpose is to enquire into the welfare effects of banning third-degree price discrimination in intermediate-good markets when non-linear pricing schemes are feasible. This question is of immediate practical interest because, from a legal perspective, quantity discounts are commonly regarded as a justifiable pricing strategy of manufacturers as long as they are not discriminatory in the sense of applying different conditions to identical transactions with different trading partners.

While there has been considerable back and forth in the academic literature regarding the question whether banning price discrimination in input markets constitutes a desirable course of policy when wholesale prices are linear, among the few exceptions which consider non-linear wholesale pricing schemes the predominant opinion is that banning price discrimination is detrimental for welfare. In contrast to these findings, we show that even if non-linear pricing schemes are feasible, the reservation towards discriminatory pricing practices embodied in legal enactments may well be warranted when downstream firms have private information.

A weak point of our analysis is the focus on separate markets, which rules out potential competitive effects. We believe, however, that our findings carry over to situations where downstream firms compete in differentiated products – at least if the degree of differentiation is sufficiently large. We refrain from analysing competition downstream because, with our focus on fairly general wholesale contracts, it is far from straightforward how to model competition in a tractable manner.

Many real-life trading relationships are long lived. If a downstream firm's retail costs vary from period to period but are correlated over time, the manufacturer's assessment of the distribution of retail costs can become more precise over time and as a result the asymmetric information problem is reduced. To achieve incentive compatibility in early periods, however, becomes more costly, which increases the asymmetric information problem in early periods of the business relationship. To explore the repeated interaction between a manufacturer and privately informed retailers is a fascinating topic for future research.

Appendix A: Proofs of Propositions and Lemmas

  1. Top of page
  2. Abstract
  3. 1. The Model
  4. 2. The Analysis
  5. 3. Long-run Analysis
  6. 4. Continuous Distribution of Downstream Costs
  7. 5. Demand-side Substitution
  8. 6. Conclusion
  9. Appendix A: Proofs of Propositions and Lemmas
  10. References
  11. Supporting Information

Proof of lemma 1. First, suppose inline image. Then,

  • display math(A.1)

Next, suppose inline image. Then,

  • display math(A.2)

where the last inequality holds by inline image and inline image being strictly increasing in q on inline image.

Last, suppose inline image. Then,

  • display math(A.3)

holds because inline image is strictly increasing in q on inline image.

Proof of proposition 1. It is readily verified that inline image is strictly concave. From the first-order condition inline image we obtain that inline image. Moreover, a high-cost type is served if and only if

  • display math(A.4)

If inline image, then inline image is characterised by the first-order condition inline image, see (5), and we have inline image. Finally, note that inline image and inline image satisfy the constraint (MON).

Proof of proposition 2. The result follows directly from the proof of Proposition 1 if we replace α by inline image and recognise that inline image is equivalent to inline image.

Proof of lemma 2. The desired statement follows from Propositions 1 and 2 together with the definition of inline image and the fact that

  • display math(A.5)

Proof of proposition 3. With inline image and inline image the manufacturer de facto knows that inline image and inline image. First, suppose that price discrimination is feasible. If the manufacturer contracts with a downstream firm that operates at marginal cost c, then – with incentive compatibility not being an issue – the manufacturer chooses [q(c),t(c)] to maximise t(c)−Kq(c) subject to π[q(c),c]−t(c) ≥ 0. In the optimum, the participation constraint must be binding such that the manufacturer's quantity choice effectively maximises joint surplus. In consequence, inline image.

Next, under uniform pricing the manufacturer chooses inline image and inline image to

  • display math(A.6)

where the constraints are those introduced in subsection 'Optimal Wholesale Tariffs'. The transfers are pinned down by the two binding constraints, (IRinline image) and (ICinline image). Thus, the manufacturer chooses quantities inline image and inline image according to

  • display math(A.7)

Clearly, there is no distortion at the top, inline image. If inline image, then inline image is characterised by

  • display math(A.8)

and inline image. If inline image, then inline image.

Regarding welfare, note that the outcome in low-cost market 1 does not depend on the pricing regime. The quantity in high-cost market 2, on the other hand, is distorted below inline image under uniform pricing. With inline image being too low from a welfare perspective, banning price discrimination therefore is unambiguously detrimental for welfare, i.e. ΔW > 0.

Proof of proposition 4. We prove each part of the Proposition in turn. To cut back on notation, define inline image for i ∊ {1,2}, and inline image.

  1. Note that the expected total output under price discrimination and under uniform pricing is given by
    • display math(A.9)
    and
    • display math(A.10)
    respectively. Thus, inline image is given by
    • display math(A.11)
    The change in expected welfare can be re-written as
    • display math(A.12)
    As inline image, we can find an upper bound for the first term and a lower bound for the second term (Varian, 1985). Hence, the change in expected welfare is bounded from above by
    • display math(A.13)
    Rearranging the above inequality yields
    • display math
    We conclude by noting that inline image because inline image.
  2. With inline image, we have inline image. According to (10), the difference in expected welfare under the two pricing regimes is
    • display math(A.14)

From the first-order condition (5) together with inline image whenever P(·) > 0, it follows that

  • display math(A.15)

which establishes the desired result.

Proof of corollary 1. In case II, with inline image, we have inline image. Note that inline image. Differentiation of (10) w.r.t. inline image yields

  • display math(A.16)

With inline image being defined by (8), we have inline image. Moreover, with inline image whenever P(·) > 0, from (8) it follows that

  • display math(A.17)

Taken together, these observations allow us to conclude that inline image.

Proof of proposition 5. With ΔW being given by

  • display math(A.18)

we consider in turn each of the three relevant cases identified in the main text:

  1. inline image
  2. inline image and
  3. inline image

. To cut back on notation, define inline image for i ∊ {1,2}, inline image, inline image and inline image.

  1. With inline image, we have math image and math image. Noting that inline image, ΔW < 0 follows from Proposition 4(i);
  2. With inline image, we have inline image, math image and math image. The difference in expected welfare thus equals

    • display math(A.19)

    Let inline image be implicitly defined by

    • display math(A.20)

    Differentiation of (A.20) with respect to inline image reveals that

    • display math(A.21)

    Substituting for inline image and inline image, and noting that inline image and inline image yields

    • display math(A.22)

    A first important observation is that each term in square brackets is strictly positive, which implies that inline image. Moreover, all the terms with inline image on the RHS of (A.22) cancel out, which allows us to re-write (A.22) as follows:

    • display math(A.23)

    Straightforward manipulation of the RHS yields

    • display math(A.24)

    As inline image if and only if inline image, the RHS of (A.23) is strictly negative. Therefore, with the term in square brackets on the LHS of (A.23) being strictly positive, we must have inline image. Taken together, the above observations imply

    • display math(A.25)

    Last, note that inline image. To see this, note that for inline image we have inline image, and in consequence

    • display math(A.26)

    With inline image, for ΔW = 0 we must have inline image, which holds for inline image. Together with inline image this last observation implies inline image. The result then follows immediately from Corollary 1;

  3. ΔW > 0 follows from Proposition 4(ii).

    Taken together, the above observations establish the desired result.

Proof of proposition 6. First, we show that the equilibrium investment levels are indeed characterised by the respective first-order conditions. Thereafter, we show that inline image.

Consider price discrimination first. Firm i's expected profit at the contracting stage does not depend on firm j's investment intensity inline image such that inline image. The information rent left to a low-cost downstream firm is zero if its investment level is too high, i.e. inline image for inline image. Moreover, inline image. Thus, inline image. Finally, note that inline image is a continuously differentiable function and thus inline image is characterised by the first-order condition

  • display math(A.27)

Under uniform pricing the profit of downstream firm i depends also on the rival's investment level inline image. If inline image, then firm i chooses a strictly positive investment level because inline image. For r = U, implicitly differentiating (11) with respect to inline image reveals

  • display math(A.28)

Given inline image, from (8) in combination with inline image, it follows that

  • display math(A.29)

Moreover, by Assumption 1,

  • math image(A.30)

This allows us to conclude that firm i's best-response function is weakly decreasing – weakly decreasing because it might be the case that firm i chooses inline image for values of inline image sufficiently close to zero, or inline image for values of inline image sufficiently close to 1. With downstream firms being ex ante symmetric, their best-response functions are symmetric. Existence of a symmetric Nash equilibrium with equilibrium investment level inline image then follows from best-response functions being continuous and firm i choosing an investment level strictly less than 1 for inline image sufficiently high. To see the latter point, note the following:

  1. if inline image, then inline image is a best response to inline image because a higher investment by firm i does not change the quantity allocation, inline image, but comes at higher cost and
  2. if inline image, then inline image, such that firm i's best response is smaller than inline image, which itself is smaller than 1 because inline image for inline image.

Last, note that any symmetric equilibrium under uniform pricing must have inline image: if inline image, then firm i's best response to inline image is not inline image but inline image; if inline image, then firm i can profitably deviate to inline image slightly below inline image, which results in strictly positive expected profits because inline image. Hence, inline image is implicitly defined by

  • display math(A.31)

Comparing (A.27) and (A.31) immediately reveals that inline image. Note that for inline image, we would have inline image. Suppose, in contradiction, that inline image, which implies that inline image. Let inline image, so that MR(q) denotes the marginal revenue of a downstream firm. With inline image, by hypothesis, that

  • display math

has to hold or equivalently,

  • display math(A.32)

The above inequality is violated because

  1. by hypothesis inline image and inline image, and
  2. by Assumption 1inline image is non-increasing.

This completes the proof.

Proof of proposition 7. For ψ(α) ≡ 0 and P(q) = max{1−q, 0}, where the latter implies inline image, it is straightforward to show that the investment level under price discrimination is given by

  • display math(A.33)

with inline image. The symmetric investment level under uniform pricing amounts to

  • display math(A.34)

With our focus on equilibria in undominated strategies, there is a unique equilibrium, which is symmetric. To see this, let inline image be the reaction function of firm i. The slope of the reaction function

  • display math(A.35)

Thus, there is only a symmetric equilibrium because the absolute value of the slope of the reaction function is always less than one.

The difference in expected welfare between price discrimination and uniform pricing is

  • display math(A.36)

Thus, ΔW < 0 if and only if

  • display math(A.37)

which holds because inline image.

Proof of lemma 3. First, we analyse the manufacturer's screening problem for the continuous distribution of downstream types. Noting that neither the individual rationality constraints nor the incentive compatibility constraints depend on the pricing regime, we begin by drawing out the implications of these constraints for the optimal wholesale tariff. To cut back on notation, we suppress the subscript i indicating the downstream firm.

Define

  • display math(A.38)

Using a revealed preference argument for types inline image and inline image, we obtain

  • display math(A.39)

The above chain of inequalities implies that inline image except for points of discontinuity. Moreover, from (A.39) we immediately obtain that the incentive compatible quantity and transfer schedules, q(c) and t(c), are non-increasing. Using the insights from above, the transfer t(c) can be stated as

  • display math(A.40)

because inline image and inline image in the optimum.

Discriminatory Offers. With downstream firms operating in separate markets, the manufacturer solves two isolated maximisation problems. After integrating by parts, the manufacturer's problem regarding firm i = 1,2 can be stated as follows:

Program D:

  • display math

Ignoring the monotonicity constraint for the moment, point-wise maximisation yields

  • display math(A.41)

By Assumptions 2 and 3, the quantity schedule inline image is strictly decreasing and assigns a positive quantity to all types.

Uniform Pricing. Being restricted to offer the same wholesale tariff to both downstream firms, the manufacturer maximises

  • display math(A.42)

subject to the (IC) and (IR) constraints. As the constraints are the same as under price discrimination, the incentive compatible transfer schedule is still characterised by (A.40). Integrating by parts yields

  • display math(A.43)

such that the manufacturer faces the following problem:

  • display math

Ignoring the monotonicity constraint for the moment, point-wise maximisation yields

  • display math(A.44)

By Assumptions 2 and 3, the quantity schedule inline image is strictly decreasing and assigns a positive quantity to all types.

Based on the above insights, we now can prove Lemma 3. According to (A.41), if inline image, then inline image. In combination with (A.44), inline image is equivalent to

  • display math(A.45)

which establishes the desired result.

Proof of proposition 8. Inserting (A.41) and (A.44) into

  • display math(A.46)

and

  • display math(A.47)

respectively, reveals

  • display math(A.48)

Simplifying the above expression yields

  • display math(A.49)

which establishes the desired result.

Proof of Observation 1. Note that given our assumption inline image for all cinline image. First, we characterise a firm's profit in equilibrium for a given pricing regime r ∊ {D,U}. This allows us to calculate a firm's change in profits for a given cost realisation if we move from one regime to the other. Finally, we derive bounds for these changes that establish the result.

Let inline image denote the profit of firm i ∊ {1, 2} under pricing regime r given its retail cost is equal to c. Under price discrimination, we have

  • display math(A.50)

Likewise, under uniform pricing,

  • display math(A.51)

Retailer i strictly prefers a ban on price discrimination if inline image, where

  • display math(A.52)

Consider the ex ante efficient firm 1. Suppose firm 1's cost is relatively high, inline image and thus inline image. Hence,

  • display math(A.53)

Finally, to prove the last statement of the result, note that inline image iff

  • display math(A.54)

As we have shown above, the inequality is satisfied for inline image.

Notes
  1. 1

    In the following, the term price discrimination is used exclusively to refer to third-degree price discrimination, i.e. to situations where the manufacturer offers different wholesale tariffs (which may be more complex than linear tariffs) to its retailers.

  2. 2

    For an overview of landmark anti-trust cases in the EU see Russo et al. (2010).

  3. 3

    Empirical evidence for non-linear contracts being employed in vertical relations is presented, for instance, by Slade (1998) for the Canadian market of gasoline retailing and by Ferrari and Verboven (2012) for magazine distribution in Belgium.

  4. 4

    Building on the Rey–Tirole model and assuming that the manufacturer competes against a competitive fringe, Caprice (2006) shows that a ban on price discrimination leads to an increase in welfare if the fringe is sufficiently efficient.

  5. 5

    Likewise, in the European sugar industry decision from 1973, the Commission ruled that ‘the granting of a rebate which does not depend on the amount bought […] is an unjustifiable discrimination […].’ (Recital II-E-1 of Commission decision 73/109/EC) Other decisions include the Eurofix – Bauco/Hilti case, where the commission objected that the reduction in discounts was not linked primarily to any objective criteria such as quantity (Commission decision 88/138/EEC).

  6. 6

    We are not the first to consider privately informed downstream firms in a model of vertical relations. While other models of vertical relations, e.g. Rey and Tirole (1986) or Majumdar and Shaffer (2009), allow for downstream firms having private information regarding their stochastic retail cost, these studies do not discuss third-degree price discrimination. Price discrimination with privately informed buyers is also analysed by Bang et al. (forthcoming). They, however, neither analyse the welfare effects of a ban on price discrimination nor do they consider non-linear tariffs.

  7. 7

    A series of articles elaborates on Schmalensee's insight, see Varian (1985), Schwartz (1990), Malueg (1993) and Aguirre et al. (2010).

  8. 8

    As shown by Yoshida (2000), if firms differ in their efficiency in transforming the input into the final good, an increase in total output is a sufficient condition for price discrimination to reduce welfare.

  9. 9

    That price discrimination can lead to more (input) markets being served is also shown by Herweg and Müller (2012) for linear wholesale tariffs.

  10. 10

    In the US, the standard of proof for competitive harm is relatively low. According to the commonly applied Morton Salt rule, the existence of a substantial price difference for a substantial period of time is sufficient; actual proof of retailers competing for the same customers is not required. (FTC versus Morton Salt Co., 334 U.S. 37) In the recent Volvo case, however, the Supreme Court for the first time required actual proof of retailers competing for the same customers to establish competitive harm, thereby overruling the decision of a lower court. (Volvo Trucks North America, Inc. versus Reeder-Simco GMC, Inc. (04-905), 546 U.S. 164, 2006). For a more elaborate discussion of this point, see Luchs et al. (2010).

  11. 11

    Sometimes cases of geographic price discrimination are also considered as an infringement of Art. 101.1 TFEU. Notable examples are Glaxo Wellcome (Commission decision 2001/791/EC) in the pharmaceutical industry and Souris/Topps (Commission decision COMP/C-3/37.980) in the toy industry.

  12. 12

    With downstream firms operating in independent markets our model is equivalent to a model where the manufacturer sells directly to final consumers. We frame it as an input market setting because, on the one hand, non-discrimination laws are typically applicable to contracts between firms at different nodes of a supply chain, and, on the other hand, non-linear contracts are common in business-to-business relations.

  13. 13

    This assumption is weaker than the standard assumption inline image, which is typically imposed for the case of Cournot competition downstream (Vives, 1999; Inderst and Valletti, 2009).

  14. 14

    Our model can also be interpreted as a model of demand uncertainty where each downstream firm produces with constant marginal cost inline image. With probability inline image downstream firm i faces high demand P(q), otherwise it faces low demand inline image.

  15. 15

    This simple form of tariffs is indeed optimal under price discrimination. Under uniform pricing, on the other hand, the manufacturer benefits from offering a direct mechanism, which specifies quantity–transfer pairs depending on both firms' type announcements. Such a direct mechanism would implement the same allocation as an indirect mechanism with pre-contracting communication. Regarding indirect mechanisms, qualitatively similar results can be obtained when the manufacturer offers a three-part tariff to each downstream firm:

    • display math

    As we show in the online Appendix, for linear demand the quantities procured under the optimal three-part tariffs are exactly the same as the quantities optimally specified in the quantity–transfer lists we consider, which implies that the welfare findings are also identical.

  16. 16

    An alternative benchmark with symmetric information is the following: downstream firms' costs are stochastic but contracting takes place ex ante before the cost types are realised. Under price discrimination both firms sell the joint-surplus maximising quantities irrespective of the realised cost type. Under uniform pricing the manufacturer again faces a metering problem. As a consequence, the quantities sold by high-cost downstream firms are distorted downwards and banning price discrimination, again, is detrimental for welfare.

  17. 17

    In analogy to Varian (1985), we can establish also a lower bound regarding the change in expected welfare: inline image. An immediate implication is that price discrimination improves welfare if (from the perspective of a vertically integrated firm) the profitability of the output under price discrimination exceeds the profitability of the output under uniform pricing valued at the discriminatory prices.

  18. 18

    For the case of zero investment cost, inline image, we focus on symmetric equilibria in undominated strategies.

  19. 19

    Following DeGraba (1990) we consider symmetric firms, which implies that no discrimination takes place on the equilibrium path. Nevertheless, when discrimination is possible (off equilibrium) this has an effect on the downstream firms' investment incentives.

  20. 20

    A detailed derivation of the results presented in this Section is to be found in the online Appendix.

  21. 21

    Here, the manufacturer faces a screening problem with a type-dependent outside option. This class of problems is thoroughly analysed, for instance, by Jullien (1996, 2000).

  22. 22

    For instance, if a competitive fringe supplies the input at per-unit wholesale price inline image, then inline image.

  23. 23

    See also Tirole (1988, p. 154).

  24. 24

    Moreover, just as in the case without alternative supply, if high-cost production takes place in both markets only under uniform pricing, inline image, then ΔW is strictly increasing in inline image.

References

  1. Top of page
  2. Abstract
  3. 1. The Model
  4. 2. The Analysis
  5. 3. Long-run Analysis
  6. 4. Continuous Distribution of Downstream Costs
  7. 5. Demand-side Substitution
  8. 6. Conclusion
  9. Appendix A: Proofs of Propositions and Lemmas
  10. References
  11. Supporting Information
  • Aguirre, I., Cowan, S. and Vickers, J. (2010). ‘Monopoly price discrimination and demand curvature’, American Economic Review, vol. 100(4), pp. 160115.
  • Arya, A. and Mittendorf, B. (2010). ‘Input price discrimination when buyers operate in multiple markets’, Journal of Industrial Economics, vol. 58(4), pp. 84667.
  • Bang, S.H., Kim, J. and Yoon, Y.R. (forthcoming.). ‘Reverse price discrimination with bayesian buyers’, Journal of Industrial Economics.
  • Bork, R. (1978). The Antitrust Paradox, New York: Basic Books.
  • Caprice, S. (2006). ‘Multilateral vertical contracting with an alternative supply: the welfare effects of a ban on price discrimination’, Review of Industrial Organization, vol. 28(1), pp. 6380.
  • DeGraba, P. (1990). ‘Input market price discrimination and the choice of technology’, American Economic Review, vol. 80(5), pp. 124653.
  • Ferrari, S. and Verboven, F. (2012). ‘Vertical control of a distribution network – an empirical analysis of magazines’, RAND Journal of Economics, vol. 43(1), pp. 2650.
  • Geradin, D. and Petit, N. (2005). ‘Price discrimination under EC competition law’, in (M. Bergman, ed.), The Pros and Cons of Price Discrimination, pp. 2163, Stockholm: Konkurrensverket (Swedish Competition Authority).
  • Giardino-Karlinger, L. and Motta, M. (2012). ‘Exclusionary pricing when scale matters’, Journal of Industrial Economics, vol. 60(1), pp. 75103.
  • Hart, O. and Tirole, J. (1990). ‘Vertical integration and market foreclosure’, Brookings Papers on Economic Activity. Microeconomics, vol. 1990, pp. 20586.
  • Herweg, F. and Müller, D. (2012). ‘Price discrimination in input markets: downstream entry and efficiency’, Journal of Economics and Management Strategy, vol. 21(3) , pp. 77399.
  • Inderst, R. and Shaffer, G. (2009). ‘Market power, price discrimination, and allocative efficiency in intermediate-good markets’, RAND Journal of Economics, vol. 40(4), pp. 65872.
  • Inderst, R. and Valletti, T. (2009). ‘Price discrimination in input markets’, RAND Journal of Economics, vol. 40 (1), pp. 119.
  • Jullien, B. (1996). ‘L'impact des options extérieures sur les échanges en information asymétrique’, Revue économique, vol. 47(3), pp. 43746.
  • Jullien, B. (2000). ‘Participation constraints in adverse selection models’, Journal of Economic Theory, vol. 93 (1), pp. 147.
  • Katz, M.L. (1987). ‘The welfare effects of third-degree price discrimination in intermediate good markets’, American Economic Review, vol. 77(1), pp. 15467.
  • Lewis, T. and Sappington, D.E.M. (1989). ‘Countervailing incentives in agency problems’, Journal of Economic Theory, vol. 49(2), pp. 294313.
  • Luchs, R., Geylani, T., Dukes, A. and Srinivasan, K. (2010). ‘The end of the Robinson-Patman act? Evidence from legal case data’, Management Science, vol. 56(12), pp. 212333.
  • Majumdar, A. and Shaffer, G. (2009). ‘Market-share contracts with asymmetric information’, Journal of Economics and Management Strategy, vol. 18(2), pp. 393421.
  • Malueg, D. (1993). ‘Bounding the welfare effects of third-degree price discrimination’, American Economic Review, vol. 83(4), pp. 101121.
  • McAfee, R.P. and Schwartz, M. (1994). ‘Opportunism in multilateral vertical contracting: nondiscrimination, exclusivity, and uniformity’, The American Economic Review, vol. 84(1), pp. 21030.
  • O'Brien, D.P. (2002). ‘The welfare effects of third-degree price discrimination in intermediate good markets: the case of bargaining’, Working Paper No. 245, Federal Trade Commission.
  • O'Brien, D.P. and Shaffer, G. (1992). ‘Vertical control with bilateral contracts’, The RAND Journal of Economics, vol. 23(3), pp. 299308.
  • O'Brien, D.P. and Shaffer, G. (1994). ‘The welfare effects of forbidding discriminatory discounts: a secondary-line analysis of Robinson-Patman’, Journal of Law, Economics, and Organization, vol. 10(2), pp. 296318.
  • Rey, P. and Tirole, J. (1986). ‘The logic of vertical restraints’, American Economic Review, vol. 76(5), pp. 92139.
  • Rey, P. and Tirole, J. (2007). ‘A primer on foreclosure’, in (M. Armstrong and R. Porter, eds.), Handbook of Industrial Organization 3, pp. 214520, Amsterdam: Elsevier.
  • Russo, F., Schinkel, M.P., Günster, A. and Carree, M. (2010). European Commission Decisions on Competition, Economic Perspectives on Landmark Antitrust and Merger Cases, Cambridge: Cambridge University Press.
  • Schmalensee, R. (1981). ‘Output and welfare implications of monopolistic third-degree price discrimination’, American Economic Review, vol. 71(1), pp. 2427.
  • Schwartz, M. (1990). ‘Third-degree price discrimination and output: generalising a welfare result’, American Economic Review, vol. 80(5), pp. 125962.
  • Slade, M.E. (1998). ‘Strategic motives for vertical separation: evidence from retail gasoline markets’, Journal of Economics and Management Strategy, vol. 14(1), pp. 84113.
  • Tirole, J. (1988). The Theory of Industrial Organization, Cambridge, MA: MIT Press.
  • Valletti, T. (2003). ‘Input price discrimination with downstream cournot competitors’, International Journal of Industrial Organization, vol. 21(7), pp. 96988.
  • Varian, H. (1985). ‘Price discrimination and social welfare’, American Economic Review, vol. 75(4), pp. 8705.
  • Vives, X. (1999). Oligopoly Pricing: Old Ideas and New Tools, Cambridge, MA: MIT Press.
  • Yoshida, Y. (2000). ‘Third-degree price discrimination in input market: output and welfare’, American Economic Reviews, vol. 90(1), pp. 2406.

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. The Model
  4. 2. The Analysis
  5. 3. Long-run Analysis
  6. 4. Continuous Distribution of Downstream Costs
  7. 5. Demand-side Substitution
  8. 6. Conclusion
  9. Appendix A: Proofs of Propositions and Lemmas
  10. References
  11. Supporting Information
FilenameFormatSizeDescription
ecoj12061-sup-0001-AppendixB-C.pdfapplication/PDF152K

Appendix B. Proofs and Supplementary Material to Section 6.

Appendix C. Three-Part Wholesale Tariffs.

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