Nonlinear Contracts and Vertical Restraints in Bilateral Duopoly

Paolo Ramezzana∗ September 2016

Abstract This paper studies the competitive effects of a variety of publicly observable nonlinear contracts and vertical restraints in bilateral duopoly. When suppliers offer menus of contracts and inputs are sufficiently differentiated, there exist equilibria in which both retailers purchase from both suppliers at wholesale prices above marginal cost to soften downstream competition. In these common agency equilibria, vertical restraints such as all-units discounts, market-share requirements and no-steering rules affect upstream competition for marginal sales and lead to higher prices and lower welfare than twopart tariffs. Whereas with sequential contracting the industry monopoly outcome is the unique equilibrium, with simultaneous contracting coordination failures may lead to less profitable equilibria. Keywords: Bilateral oligopoly, nonlinear pricing, vertical restraints, facilitating practices. JEL Classification: D43, L13, L42.

∗

U.S. Federal Trade Commission. E-mail: [email protected]. I would like to thank Patrick Rey, David Schmidt, Marius Schwartz, Yossi Spiegel, Alison Weingarden and participants in the 2014 International Industrial Organization Conference (Chicago), the 2014 EARIE Conference (Milan), the 2014 Washington, DC IO Conference and in seminars at the U.S. Federal Trade Commission and Department of Justice for helpful comments and discussions. The views expressed in this paper are those of the author and do not necessarily represent those of the Federal Trade Commission or any of its Commissioners.

1

Introduction

In many industries competing downstream firms (“retailers”) each distribute multiple substitutable products or inputs supplied by differentiated upstream firms (“suppliers”). Examples include retail distribution, in which competing retailers often offer consumers a choice among different brands of a product and/or different payment methods; computer hardware or other durable goods, in which competing OEMs often offer their customers a choice among different brands of components; and the health care industry, in which competing insurers typically offer their subscribers a choice among different in-network health care providers. In these and many other industries, the supply relationships between suppliers and retailers are governed by vertical contracts that often include nonlinear pricing schedules and vertical restraints. Notwithstanding the fact that markets with these features are the norm rather than the exception, the theory of vertical contracting in such settings is still largely unexplored. The majority of the literature has studied “triangular” market structures in which two or more suppliers compete for the business of a single retailer (e.g., O’Brien and Shaffer, 1997; Bernheim and Whinston, 1998; and Calzolari and Denicol`o, 2013) or a single supplier sells an input to two or more competing retailers (e.g., Shaffer, 1991; O’Brien and Shaffer, 1992; McAfee and Schwartz, 1994; Segal, 1999; Segal and Whinston, 2003; Rey and Verg´e, 2004; Mikl´os-Thal, Rey and Verg´e, 2011; and Rey and Whinston, 2013). Although a few articles, discussed in further detail towards the end of this introduction, have explored settings with multiple products and multiple competing retailers, these articles do not focus on upstream competition for marginal sales between suppliers with market power (e.g., Inderst and Shaffer, 2010, and Nocke and Rey, 2014) or cannot reconcile this competition with the existence of delegated common agency equilibria in which all retailers enter nonlinear pricing contracts with all suppliers (e.g., Dobson and Waterson, 2007, and Rey and Verg´e, 2010). This paper attempts to fill this void by studying a model in which two differentiated suppliers offer menus of publicly observable bilateral contracts to two differentiated retailers. The analysis focuses on double common agency equilibria in which both retailers represent both suppliers. A key property of these equilibria is that suppliers charge wholesale prices above marginal cost to prevent excessive dissipation of industry profits through downstream

1

competition.1 The resulting positive upstream unit margins give suppliers incentives to compete for marginal sales, thus introducing competitive externalities in the upstream market in addition to the competitive externalities in the downstream market. The main objective of the paper is to analyze the implications of these externalities for the the properties and existence of double common agency equilibria under different types of vertical contracts. After laying out the formal model and discussing its assumptions in Section 2, in Section 3 I set the stage for subsequent sections by analyzing the case in which vertical restraints are banned and only two-part tariffs are allowed. Although product differentiation allows suppliers to raise wholesale prices above marginal cost in order to soften downstream competition, upstream competition prevents them from supporting the (higher) level of wholesale prices that would maximize industry profits. In addition to providing a benchmark against which to evaluate the competitive effects of the vertical restraints studied in the remainder of the paper, this section also explores some aspects that may be of independent interest, such as the effects of changes in the intensity of downstream competition on the contracts offered by suppliers. Section 4 allows suppliers to add minimum volume requirements to the nonlinear tariffs discussed above. In practice, this type of minimum quantity forcing is often implemented through contracts that include all-units discounts, i.e. discounts that become available only when a retailer’s purchases exceed a given volume threshold and that are calculated on all the units purchased by the retailer, not only on the incremental units above the threshold. By triggering a large (often prohibitive) upward jump in the average price paid by the retailer when his purchases fall below a minimum volume, these discounts often have the effect of forcing the retailer to purchase that minimum volume or not to deal with the supplier at all.2 I show that, in the presence of upstream competition, minimum volume requirements 1

The assumption of publicly observable contracts is important, though not crucial, for the existence of positive unit margins in the upstream market. Section 2 explains in some detail how it can be justified on both practical and methodological grounds in this context. 2 For a discussion of the minimum quantity forcing effects of all-units discounts in a bilateral monopoly setting see Kolay, Ordover and Shaffer (2004) and O’Brien (2014). Examples of high-profile antitrust cases involving all-units discounts include, in the United States, the actions brought by Advanced Micro Devices and the U.S. Federal Trade Commission against Intel (Civil Action 05-441-JJF and Case 061 0247, Docket 9341, respectively) and by ZF Meritor against Eaton (Civil Action 06-623-SLR) and, in Europe, by the European Commission against Michelin and British Airways (Cases T-203/01, ECR II-04071, and T-219/99, ECR II-5917, respectively). Note that the threshold that triggers all-units discounts can also be expressed as a minimum share of the retailer’s total purchases, a case that will be discussed further below in the context of contracts that reference rivals.

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can be used defensively by suppliers to reduce their rivals’ incentives to compete for marginal sales and, therefore, effectively amount to facilitating practices that lead to higher prices, higher profits and lower welfare. Specifically, when suppliers offer contracts simultaneously, there exists a multiplicity of equilibria and output can take any value between the industry monopoly and two-part tariff levels. When, instead, suppliers offer contracts sequentially, as in Section 6, the coordination failures that lead to this multiplicity of equilibria are resolved and the unique equilibrium of the model implements the industry monopoly outcome. More generally, Section 4 highlights a theme that will figure prominently in different guises throughout the paper. In models of common agency like the one in this paper, the out-of-equilibrium portions of contracts can have important effects on equilibrium outcomes even though such portions are not selected on the equilibrium path (see, e.g., Martimort and Stole, 2002, 2003). In the specific case of the all-units discounts discussed above, although a supplier can achieve any desired volume of sales with appropriately chosen two-part tariffs without needing to punish retailers drastically in the out-of-equilibrium event that the retailers’ purchases fall short of that volume, such out-of-equilibrium punishments affect the level of equilibrium output by affecting the profitability of the deviations available to the other supplier. This logic also explains the difference between my findings and those of Inderst and Shaffer (2010), who analyze a model similar to mine but in which one of the products is supplied by a perfectly competitive fringe that cannot offer vertical contracts. Inderst and Shaffer find that there is no loss of generality from restricting the set of own-quantity contracts that the only supplier with market power can use to two-part tariffs (or to direct mechanisms), since in their setting that supplier’s upstream competitors cannot respond by offering their own contracts. This is not the case in my paper. Section 5 studies the effects of a number of non-exclusive contracts that reference rivals (CRRs henceforth), i.e., contracts that do not require a retailer to be exclusive with the supplier offering the contract but nevertheless condition the terms received by the retailer on his dealings with other suppliers (see, e.g., Scott Morton, 2012). Non-exclusive CRRs may take different forms, but generally contain provisions that discourage or prohibit downstream firms from steering a significant portion of their business to rival suppliers. An example of such provisions is provided by the no-steering restraints (also known as no-discrimination rules or NDRs) imposed until recently by credit card companies on merchants. These restraints

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prevented merchants from offering consumers better prices for purchases made with different means of payment or with credit cards issued by competitors. Recently, the U.S. Department of Justice has successfully challenged the use of these restraints by American Express and its competitors, arguing that they suppressed the incentives of credit card companies to compete for additional charge volume at the point of sale.3 Similar rules requiring uniformity of retail pricing (also known as price-coherence rules or retail MFNs) are common in a number of other sectors (see, e.g., the discussion in Chen and Schwartz, 2015). Another common type of CRRs is constituted by market-share discounts, which require a downstream firm to source a minimum share of its total purchases from a given supplier in order to obtain a favorable price from that supplier. Although market-share discounts do not constrain relative retail prices directly, they do so indirectly by reducing a downstream firm’s willingness to steer business towards rival suppliers and have therefore competitive effects analogous to the nosteering rules discussed above.4 I show that CRRs can be used by a supplier s with both “defensive” and “offensive” purposes at the same time. In their defensive role, they reduce the incentives of a rival supplier, s0 , to compete by cutting its wholesale prices, since such price cuts would induce retailers to lower also the retail prices of product s and would thus yield a smaller increase in the sales of product s0 than in the absence of the CRRs offered by supplier s. In their offensive role, the CRRs offered by supplier s give that supplier incentives to impose a negative externality on supplier s0 by increasing the wholesale prices of product s since, in the presence of CRRs, retailers must respond by also raising the retail prices and reducing the volume of sales of product s0 . The combination of these two effects always leads to higher equilibrium prices than with two-part tariffs. In particular, when suppliers offer contracts sequentially, CRRs completely dampen upstream competition and implement the industry monopoly outcome as the unique equilibrium of the model. When instead suppliers offer contracts simultaneously, coordination failures may cause them to charge wholesale prices above the level that would 3

The DOJ issued its complaint against American Express on October 4, 2010 and on February 19, 2015 Judge Garaufis of the U.S. District Court of Brooklyn, NY, found in its favor. Mastercard and Visa settled their cases with DOJ in 2010. For a discussion of the economics of merchant restraints in the credit card industry see Carlton and Winter (2015). Another example of an antitrust case involving similar issues is the action brought in 2008 by the U.K. OFT against two tobacco manufacturers and a number of retailers, in which the challenged practices involved the fixing of the relative retail prices of different brands of cigarettes at the point of sale (see CE/2596–03, ‘Imperial Tobacco’). 4 Market-share requirements featured prominently in some of the antitrust cases mentioned in footnote 2 above.

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maximize industry profits, although this problem can be mitigated by adopting minimum volume requirements that dissuade excessive pricing by upstream competitors. Sections 5 and 6 derive these results in detail and compare them with those obtained by Inderst and Shaffer (2010) in a model in which one of the two products is supplied at marginal cost by a competitive fringe. The discussion has so far focused on double common agency equilibria in which both retailers deal with both suppliers. In Section 7 I show that these equilibria exist only under certain conditions that can be briefly discussed as follows. First, when retailers compete in the downstream market, a double common agency market structure does not necessarily maximize industry profits, since market structures involving exclusivity at one or both retailers reduce the intensity of downstream competition, although at the cost of some loss in variety. When products and retailers are sufficiently differentiated, deviations to exclusive market structures would, however, entail substantial losses in variety without significantly reducing downstream competition, and would thus be unprofitable relative to double common agency. Second, the presence of externalities between suppliers creates incentives for a coalition formed by a supplier and one or both retailers to exclude the other supplier. When competition with menus of contracts between supplier allows retailers to appropriate a sufficiently large share of equilibrium industry profits, retailers become, however, more invested in maintaining double common agency whenever this maximizes industry profits and would thus require a higher compensation from a supplier that attempted to deviate to exclusivity. This tends to make deviations to exclusivity less profitable for suppliers and thus expands the region of parameters for which double common agency equilibria exist. Besides characterizing the conditions for existence that briefly summarized above, this section also addresses some technical difficulties in the modeling of vertical contracting in bilateral duopoly, such as the possibility of multiple continuation equilibria in the selection of contracts by retailers and the ability of suppliers to engage in multilateral deviations at both retailers. As mentioned at the beginning of this introduction, a few other articles have studied vertical contracting in settings with multiple competing suppliers and retailers. Nocke and Rey (2014) focus on incentives for exclusivity and vertical integration in a model with secret contracts and quantity competition, which rules out above-cost wholesale prices and thus upstream competition for marginal sales. Rey and Verg´e (2010) analyze a model with publicly

5

observable contracts and thus above-cost wholesale prices, but restrict attention to the case in which suppliers offer a single contract to each retailer, with the consequence that double common agency equilibria do not exist when retailers have market power and suppliers cannot use resale price maintenance. Moreover, these authors consider a different type of vertical restraint (RPM) from the ones considered in this paper. Finally, Dobson and Waterson (2007) limit their attention to (bilaterally inefficient) vertical contracts with linear wholesale prices. For a discussion of modeling issues in settings with multiple suppliers and retailers see also Inderst (2010) and Mikl´os-Thal, Rey and Verg´e (2010).

2

The model

Two suppliers, denoted by s, s0 ∈ {1, 2}, produce two symmetrically differentiated products with the same constant marginal cost c and no fixed costs. The two products are distributed by two symmetrically differentiated retailers, denoted by r, r0 ∈ {1, 2}, which do not have to bear any additional costs besides the payments to suppliers. When both retailers sell both products, consumers can effectively choose between four differentiated products, corresponding to the four different supplier-retailer combinations. Let qsr and psr denote, respectively, the quantity and retail price of product s sold by retailer r. The inverse demand system is symmetric, with psr = P (qsr , qs0 r , qsr0 , qs0 r0 ) denoting the inverse demand of product s at retailer r, where P (·) is continuous and twice differentiable in all its arguments. I also assume that the inverse demand system has the following properties.5 Assumption 1 (Demand) ∂i P < 0, i = 1, 2, 3, 4, with |∂2 P | > |∂4 P | and |∂3 P | > |∂4 P |. This assumption states that the four products are substitutes and that product differentiation and retailer differentiation are cumulative, so that products s0 r and sr0 are closer substitutes than product s0 r0 for product sr. For illustrative purposes, in Section 3.3 I introduce a linear demand example that satisfies Assumption 1 and allows me to parametrize the degrees of product and retailer differentiation. I study competition in this market by means of a three-stage model, with suppliers offering menus of contracts in the first stage, retailers deciding which contracts to accept in the second 5

In keeping with the notation in Rey and Verg´e (2010), I use ∂i f to denote the partial first derivative of 2 f with respect to its ith argument. Analogously ∂ik f denotes the partial second derivative of f with respect 2 to its ith and kth arguments. For example, ∂1 P = ∂psr /∂qsr and ∂12 P = ∂ 2 psr /∂qsr ∂qs0 r .

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stage, and retailers competing in the downstream market in the third stage. I first give a concise description of these three stages and then discuss the reasons for and implications of the main assumptions of the game in some detail further below. Stage 1: Contract offers by suppliers – Each supplier s offers each retailer r a menu ne with two publicly observable contracts: a nonexclusive contract Csr and an exclusive e with which the retailer commits to selling only product s. Each type contract Csr

of contract j ∈ {ne, e} specifies a number of different tariffs, one for each possible contract acceptance decision made by the other retailer. Since, as I explain below, each combination of contract acceptance decisions by the two retailers results in a unique market structure, one can also think of the tariffs in each type of contract as being contingent on market structure. The following market structures are possible. • Double common agency (dca): Both retailers accept nonexclusive contracts. • Pairwise exclusivity (pe): One retailer accepts an exclusive contract from one supplier and the other retailer accepts an exclusive contract from the other supplier. • Upstream monopoly (um): Both retailers accept exclusive contracts from the same supplier. • Mixed outcome (mix): One retailer accepts an exclusive contract from one supplier, while the other retailer accepts nonexclusive contracts from both suppliers. • Bilateral monopoly (bm): One retailer accepts an exclusive contract from one supplier, while the other retailer does not accept any contract. Denote by tj,k sr (qsr ) the tariff charged by supplier s to retailer r for a contract of type j ∈ {ne, e} when the market structure is k ∈ {dca, pe, um, mix, bm}. To see how the applicable contingent tariff is determined consider, for example, the case in which e retailer r accepts the exclusive contract Csr from supplier s. If retailer r0 accepts

the exclusive contract Cse0 r0 from supplier s0 , the resulting market structure is pairwise e 0 exclusivity and the tariff that applies in the contract Csr is te,pe sr (qsr ); if r accepts e the exclusive contract Csr 0 from supplier s, the resulting market structure is upstream ne ne monopoly and te,um (qsr ); if r0 accepts the nonexclusive contracts Csr 0 and Cs0 r 0 from sr

both suppliers, the resulting market structure is mixed and te,mix (qsr ); finally, if r0 sr 7

does not accept any contract, the resulting market structure is bilateral monopoly and te,bm (qsr ).6 The tariffs that apply to the case in which r accepts nonexclusive contracts sr from both suppliers can be determined with an analogous reasoning. I assume that the j,k j,k , and unit wholesale price wsr tariffs take the form of two-part tariffs, with fixed fee Fsr j,k with which the retailer must comply and possibly one or more vertical restraints Rsr

to qualify for the terms of the contract.7 Sections 3 through 5 study the case in which the suppliers offer contracts simultaneously and Section 6 the case in which they do so sequentially. Stage 2: Contract acceptance by retailers – Each retailer observes all the contracts that have been offered to both retailers and decides which of its contracts to accept, taking as given the contract acceptance decision of the other retailer. The retailers’ decisions are simultaneous. If this contract acceptance subgame has multiple equilibria, I focus on coalition-proof equilibria, which in this two-player subgame correspond to equilibria that are Pareto-undominated from the point of view of retailers (see Bernheim, Peleg and Whinston, 1987). As discussed in Section 7, this will yield a unique continuation equilibrium in the application of interest. Stage 3: Downstream competition – Given the contracts accepted in stage 2, retailers compete in the downstream market. Since, under appropriate regularity conditions, the results do not depend on whether downstream competition is in quantities or prices, in the main text of the paper I study the (simpler) case of quantity competition and discuss the case of price competition in Appendix B. In both cases, I assume that for each profile of wholesale prices and each set of vertical restraints there always exists a unique downstream equilibrium. Discussion of assumptions – Before proceeding, it might be helpful to discuss briefly the reasons for and implications of some of the assumptions introduced above. First, the assumption that contracts are publicly observable and binding is important, though not crucial, in Although it is in principle also possible for r0 to accept a nonexclusive contract from one and only one of the suppliers, for the purpose of my analysis this case is equivalent to r0 accepting an exclusive contract from that supplier and will thus not be considered separately. 7 As discussed in further detail in subsequent sections, I assume that the wholesale price faced by a retailer that fails to comply with the vertical restraints included in a contract is prohibitive. 6

8

this context. As shown by O’Brien and Shaffer (1992), McAfee and Schwartz (1994), and Rey and Verg´e (2004), secret contracts would reduce the ability of a monopolistic supplier to soften downstream competition by charging positive upstream margins. In the multi-supplier context of this paper, this would translate into reduced incentives for suppliers to steal sales from each other, which would in turn reduce the importance and consequences of upstream competition. In the extreme, if suppliers were completely unable to raise wholesale prices above marginal cost, their business stealing incentives would be eliminated and the results obtained in the paper would lose much of their interest. The assumption of publicly observable contracts can, however, be justified along a number of dimensions. First, from a practical point of view, in a number of important sectors of the economy downstream firms know the terms of the contracts received by their rivals. Examples include the retail distribution of many consumer products, where it is common for retailers to obtain verifiable MFN commitments from suppliers; the auto distribution industry, where it is common knowledge that car manufacturers charge the same wholesale price to all dealers; and some health care markets, where insurance companies may have enough information to reconstruct the reimbursement rates negotiated with hospitals by their rivals.8 Second, from a methodological point of view, the qualitative consequences of assuming publicly observable contracts are less extreme than it might at first appear since, as shown by Rey and Verg´e (2004) for the case of a monopolistic supplier, upstream margins are positive also when contracts are secret, retailers hold wary beliefs about the offers received by their rivals, and downstream competition is in prices (as in Appendix B). Under those assumptions, the main insights of this paper, which depend on the existence of positive upstream unit margins, should therefore apply also to environments with secret contracts.9 Finally, from the point of view of the contribution of this paper to the literature, the assumption of public contracts allows me to study how the results obtained 8

In February 2007 the state of New Hampshire launched a website (http://nhhealthcost.nh.gov/) with detailed public information on the bilateral rates negotiated by individual insurers and hospitals for a large number of medical procedures. As of January 2016, 18 other states have adopted all-payer claims databases (APCDs) that collect prices by hospital and medical procedure from insurers, and in some cases publicly release average prices by hospital and procedure. Although these states, contrary to New Hampshire, do not yet release the bilateral prices negotiated by individual insurers and hospitals, some of them are planning to do so in the future. Moreover, knowledge of the average negotiated rates in the market is still an (imperfect) tool for an insurer to gauge whether a hospital has behaved opportunistically in setting such rates. 9 As explained in the concluding remarks of Section 8, I do not solve the model for the case of secret contracts since the analysis of wary beliefs, already very complex in the single-supplier setting considered by Rey and Verg´e (2004), would be virtually intractable in the multi-supplier setting of this paper.

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by Inderst and Shaffer (2010) under the same assumption but without upstream competition between suppliers with market power are affected by the introduction of such competition. Second, the assumption that suppliers offer menus of contracts, as opposed to a single contract, and that the tariffs in these contracts can be made contingent on market structure is meant to capture in a simple framework the essence of more complex negotiations, in which suppliers and retailers would find it optimal to agree on different terms if unexpected contract rejections gave rise to different market structures (see, e.g., Inderst and Wey, 2003, and Mikl´os-Thal, Rey and Verg´e, 2011). In the model of this paper, this assumption has two consequences. First, as explained in further detail in Section 3, it implies that the optimal contract offered by s to r for a given market structure is designed to maximize the joint profits of these two firms under that market structure, not to shift rents from the other supplier, r0 , by affecting that supplier’s disagreement payoff (see Segal, 1999). Second, when suppliers and retailers contract with one another using menus, retailers might be able to retain higher profits than if they received take-it-or-leave it offers involving single contracts. As discussed in Section 7, this has important implications for the existence of common agency equilibria in which both retailers represent both suppliers. Third and last, the contracting game presented above allows suppliers to engage in multilateral deviations in stage 1. For example, a supplier can organize a deviation to upstream monopoly by inducing both retailers to accept exclusive contracts. Allowing for this possibility constitutes an innovation relative to most existing models of contracting in bilateral oligopoly. These models assume that each firm sends a distinct representative to each bilateral negotiation, with each representative being unable to observe the contracts negotiated by the other representatives of the same firm or to coordinate his actions with those representatives (see, e.g., de Fontenay and Gans, 2014).10 Such an assumption, which is quite ad-hoc even in models with secret contracting, would be particularly unrealistic in the public contracting model of this paper, since it would amount to assuming that a representative of retailer r can observe the contracts negotiated by the representatives of another retailer r0 but not the contracts negotiated by the other representatives of retailer r herself. 10 For an exception, see Collard-Wexler, Gowrisankaran and Lee (2015), who also allow for multilateral deviations. The questions addressed by these authors, as well as many of their other assumptions, are however different from those in this paper.

10

3

Two-part tariffs

In this section I study the properties of DCA equilibria in which suppliers can only use twopart tariffs without vertical restraints, assuming for the time being that such equilibria exist (i.e., that no supplier finds it profitable to deviate to exclusivity). A full-fledged analysis of the conditions for the existence of DCA equilibria is presented in Section 7. The analysis in this section, besides possibly being of independent interest, provides a benchmark for subsequent sections. Since the focus of the analysis is on double common agency (DCA) equilibria, for ease of notation in the remainder of the paper I omit the superscripts ne and dca when referring to nonexclusive DCA tariffs and reserve the use of the superscripts j ∈ {e, ne} and k ∈ {pe, um, mix, bm} for tariffs that apply to other market structures. As shown in Section 7, these other tariffs, although not charged on the equilibrium path, still matter for the existence and properties of DCA equilibria.

3.1

Downstream competition and derived demand

If, in stage 2, both retailers have accepted common agency contracts from both suppliers, in stage 3 each retailer r solves the following profit-maximization problem, given the wholesale prices wsr , s = 1, 2, in its contracts and the quantities qsr0 , s = 1, 2, chosen by its downstream rival r0 , "

maximize πr = {q1r ,q2r }

# X

(psr − wsr ) qsr − Fsr .

(1)

s=1,2

The first-order condition of this problem with respect to qsr is (psr − wsr ) +

∂psr ∂ps0 r qsr + qs0 r = 0. ∂qsr ∂qsr

(2)

Given the four first-order conditions implied by (2) for s = 1, 2 and r = 1, 2, a change in wsr affects (directly or indirectly) the equilibrium value of all four retail quantities by inducing a different equilibrium in the downstream market. For future reference, I assume that, for any set of contracts accepted by retailer r, the equilibrium value of πr is increasing in the wholesale prices faced by the other retailer, i.e. dπr /dwsr0 > 0, s = 1, 2. To ensure that equilibrium quantities respond in sensible ways to changes in wholesale prices around a symmetric equilibrium, I also impose the following regularity conditions on the primitives of 11

the inverse demand system. Assumption 2 (Inverse demand) For any symmetric level of quantities qsr = q, s, r = 1, 2, the inverse demand system satisfies the following conditions A, B < 0 and |A| > |B|

(3)


2 2 2 2 A ≡ 2∂1 P + ∂3 P + ∂11 P + ∂13 P + ∂22 P + ∂24 P q

(4)


2 2 2 2 B ≡ 2∂2 P + ∂4 P + ∂12 P + ∂14 P + ∂21 P + ∂23 P q

(5)

where

These regularity conditions, that are satisfied by the linear demand system introduced for illustrative purposes in Section 3.3 below, have the following implications for a downstream equilibrium in which wsr = wsr0 = ws and qsr = qsr0 = qs , s = 1, 2. Lemma 1 (Downstream competition and derived demand) If Assumption 2 holds, around a symmetric equilibrium with two-part tariffs and for wsr = wsr0 = ws , dqs0 dqs <0< , dws dws

dqs dqs0 + < 0. dws dws

(6)

For a proof of Lemma 1 and of all other lemmas and propositions in the paper, see Appendix A. Lemma 1 establishes the intuitive properties that the derived demand for a product decreases with the wholesale price of that product and increases with the wholesale price of the rival product, and that the former effect is stronger than the latter.11

3.2

Equilibrium wholesale prices

Denote the total vertical profits generated by retailer r by Πr =

X

(psr − c) qsr .

(7)

s=1,2 11

These properties of derived demand are analogous to those assumed by Rey and Verg´e (2010) in a model with downstream price competition (see Appendix B of this paper for an analysis of downstream price competition in my model). However, whereas Rey and Verg´e can simply assume these properties to hold without loss of generality for their purposes, for the purposes of this paper I need to derive them from the restrictions on the primitives of final demand imposed in Assumption 2. The reason for this is that in Section 5 I will show that, under the same restrictions on the primitives of final demand as in Assumption 2, CRRs give rise to properties of derived demand that are different from those established in Lemma 1 above for the case of two-part tariffs.

12

These total vertical profits are split between the two suppliers and retailer r, with the two suppliers earning π1r and π2r and retailer r earning πr , so that π1r + π2r + πr = Πr . For future reference, I assume that total industry profits,

(8) P

r=1,2

Πr , reach a unique maximum

at the symmetric quantities qsr = q m > 0 in correspondence of the symmetric wholesale prices wsr = wm and are strictly concave in qsr and wsr , for all s and r. In the remainder of the paper I refer to q m and wm as the industry monopoly quantities and wholesale prices, respectively. A profit-maximizing supplier s that intends to implement a DCA market structure charges each retailer r a fixed fee for that market structure that extracts all the incremental profits generated by s in excess of the maximum profits that r could earn by dropping product s and representing supplier s0 exclusively. Let πr\s denote these maximum profits (throughout the paper I use the notation \s to indicate that a retailer does not sell product s). By rearranging the identity in (8), one can therefore write the profits earned by supplier s at retailer r in a DCA outcome in which both retailers accept the contracts of both suppliers as πsr (w) = Πr (w) − πs0 r (w) − πr\s ,

(9)

where w = (wsr , ws0 r , wsr0 , ws0 r0 ). The factors determining the level of πr\s will be discussed in Section 7 when studying the existence of DCA equilibria. For the time being it is sufficient to note that πr\s does not depend on the wholesale prices, wsr and wsr0 , offered by supplier s to the two retailers in his DCA contracts since, if one or both of the retailers rejected the DCA contract offered by s, contracts meant for market structures other than DCA would apply. P Using (9) and noting that πs0 r (w) = Fs0 r +(ws0 r − c) qs0 r (w), the total profits πs = r=1,2 πsr earned by supplier s at both retailers can be written as πs (w) =

X  Πr (w) − (ws0 r − c) qs0 r (w) − Fs0 r − πr\s ,

(10)

r=1,2

Given that πr\s and Fs0 r do not depend on w, the profit-maximization problem of supplier s is max

ws1 ,ws2

X

[Πr (w) − (ws0 r − c) qs0 r (w)] .

r=1,2

13

(11)

Denoting with the superscript t the values of variables in a DCA equilibrium with two-part tariffs, the first order conditions of problem (11) with respect to wsr are dπs (wt ) = dwsr

12

X X dΠj (wt )
dqs0 j − wst 0 j − c = 0. dwsr dwsr j=1,2 j=1,2 {z } {z } | | Competitive Effect on industry externality profits

(12)

The two effects shown in (12) lead to the following proposition. Proposition 1 In any subgame-perfect symmetric DCA equilibrium of the simultaneous contracting game with two-part tariffs it must be c < wt < wm , where wm is the wholesale price that implements the industry monopoly outcome. This result, which has already been derived by Rey and Verg`e (2010) for a model with downstream price competition, will be used throughout the paper as a benchmark against which to compare the effects of different vertical restraints. The intuition behind it is straightforward. If it were wt ≤ c, a small increase in wsr above wt would always be profitable for supplier s because it would strictly increase industry profits (by softening downstream competition without causing double marginalization) and reduce or leave unchanged the profits of supplier s0 , since for wt ≤ c the latter would earn a non-positive unit margin. If instead it were wt ≥ wm a small reduction in wsr below wt would not reduce total industry profits, which are maximized at wm , and would steal profitable sales from supplier s0 , thus increasing the profits of supplier s. The fact that wt > c leaves room for the vertical restraints discussed in subsequent sections to affect upstream competition between suppliers. Before discussing these restraints it is, however, helpful to introduce a linear demand example, which will be used throughout the paper to illustrate and verify a number of results.

3.3

Linear demand example

The following linear inverse demand system, which satisfies Assumptions 1 and 2, is particularly convenient, as it allows one to parametrize the degrees of product and retailer 12

I assume that, for every ws0 1 and ws0 2 , πs is strictly concave in ws1 and ws2 and has thus a unique maximum. I also assume that the game played by the two suppliers in the setting of wholesale prices has a unique and stable equilibrium. These assumptions hold in the linear demand example that I study in Section 3.3 below.

14

differentiation psr = v − (qsr + aqs0 r ) − b (qsr0 + aqs0 r0 ) ,

(13)

where v > c and a, b ∈ [0, 1].13 Lower values of a and b indicate, respectively, higher supplier and retailer differentiation, with a = 0 and b = 0 corresponding to the extreme case in which, respectively, suppliers or retailers are completely independent in demand and a = 1 and b = 1 to the extreme case in which they are perfect substitutes. Note that, like in Assumption 1, supplier and retailer differentiation are “cumulative”, in that the coefficient ab on qs0 r0 in (13) is smaller than the coefficients a and b on qs0 r and qsr0 , respectively. For future reference, note also that this demand specification can be obtained from maximization of the following quadratic quasi-linear utility function  X X 1 U =m+ v − (qsr + aqs0 r + bqsr0 + abqs0 r0 ) qsr 2 s=1,2 r=1,2 subject to the budget constraint m +

P

s,r=1,2

(14)

psr qsr = I. Without loss of generality, I also

assume c = 0. Figure 1 shows wholesale and retail prices in a symmetric DCA equilibrium with two-part tariffs for different levels of a and b.14 The left panel shows that, when the products become 0.6

0.6

p

m

pm

0.5

0.5

pt

0.4

p ,w

p ,w

0.4

0.3

pt

0.3

wm wm

0.2

0.2

0.1

0.1

wt 0.0 0.0

0.2

wt

a Hproduct substitutability L 0.4

0.6

0.8

0.0 0.0

1.0

0.2

b Hretailer substitutability L 0.4

0.6

0.8

1.0

Figure 1: Two-part tariff equilibrium. 13

This linear inverse demand specification is the same as that adopted by Inderst and Shaffer (2010). The calculations for the linear demand example underlying the figures presented in this and subsequent sections are performed in a Mathematica code available from the author on request. 14

15

closer substitutes (i.e. when a increases from 0 to 1), more intense upstream competition causes wt to fall and to diverge further from its monopoly level wm .15 In the limit case in which a = 1 (perfectly substitutable products), wt = c = 0 but pt remains above c = 0 because of Cournot downstream competition and, additionally, retailer differentiation. The right panel shows that wt increases as the retailers become closer substitutes (i.e. when b increases from 0 to 1). This is due to the fact that, as downstream competition intensifies, the gains from softening such competition through higher wholesale prices increase, while the costs associated with double marginalization decrease. This result suggests that changes in the downstream competitive environment – such as entry, exit or mergers – may have subtle implications for equilibrium vertical contracts. For example, entry in the downstream market, by intensifying competition in that market and reducing double marginalization, may cause upstream suppliers to increase marginal wholesale prices in order to mitigate dissipation of industry profits (while a downstream merger would have the opposite effect). An increase in b causes, however, the equilibrium retail price, pt , to fall, although the latter remains bound away from marginal cost even at b = 1 because downstream competition is Cournot.

4

Minimum volume requirements (all-units discounts)

I now study the effects of allowing suppliers to add minimum volume requirements to the twopart tariff contracts studied above.16 Specifically, I consider contracts that require retailer r to purchase a minimum volume q sr from supplier s in order to qualify for a non-prohibitive wholesale price wsr . As explained in the introduction, a supplier can implement such requirements by adopting all-units discounts with prohibitive non-discounted wholesale prices. The following result establishes that such contracts, when offered simultaneously, can support a large number (a continuum, in fact) of additional DCA equilibria beyond the two-part tariff equilibrium. All of these additional equilibria are less competitive, and thus jointly more profitable, than the two-part tariff equilibrium. 15

wm is the level of wholesale prices that would yield the industry monopoly outcome conditional on retailers competing ` a la Cournot in stage 2; i.e., wm is the value of the wholesale prices that the two suppliers would choose if they could collude in Stage 1. 16 I continue to consider the case of simultaneous contract offers and to assume that DCA equilibria exist. Sequential contracting and the conditions for the existence of DCA equilibria are analyzed in Sections 6 and 7, respectively.

16

Proposition 2 If deviations to exclusivity are not profitable, any wv ∈ [wt , wm ] can be supported as a symmetric subgame perfect DCA equilibrium of a game in which suppliers simultaneously offer nonlinear contracts with minimum volume requirements. A good way of discussing the intuition for this result is to explain how it differs from the conclusions reached by Inderst and Shaffer (2010) in a model in which product s is supplied by a firm that has market power and offers vertical contracts, whereas product s0 is supplied at marginal cost by a perfectly competitive fringe. Inderst and Shaffer show that, with quantity competition, minimum volume requirements implemented through all-units discounts have no additional effect on equilibrium outcomes relative to two-part tariff contracts. The reason for this is that, given the wholesale prices ws0 1 = ws0 2 = c of the competitively supplied product s0 , supplier s can induce retailers to select any specific quantity profile consistent with the first-order conditions in (2) by using two-part tariffs with appropriately chosen wholesale prices ws1 and ws2 . Adopting more complex pricing schedules or forcing the quantities qs1 and qs2 directly on retailers would not improve on this outcome.17 If the minimum volume requirements imposed by supplier s did not affect ws0 1 and ws0 2 , this would be true also in my model, since two-part tariffs with appropriately chosen ws1 and ws2 would be sufficient to induce any desired profile of quantities. However, contrary to Inderst and Shaffer (2010), in my model the adoption of minimum volume requirements by supplier s does affect the equilibrium wholesale prices ws0 1 and ws0 2 offered by supplier s0 . Specifically supplier s0 knows that retailer r, when faced with the minimum volume requirement imposed by supplier s, cannot shift sales from s to s0 in response to small reductions in ws0 r . Starting from any ws0 r = wv ≤ wm , this eliminates any incentive for supplier s0 to reduce ws0 r , since such a price reduction would reduce total industry profits without shifting any profits away from supplier s. Therefore, when both suppliers offer minimum volume requirements, no supplier has an incentive to cut prices starting from any equilibrium with wv ∈ [wt , wm ].18 In other 17

Formally, given the first-order conditions for qs0 1 and qs0 2 , which do not depend directly on ws1 or ws2 , supplier s can induce any pair of quantities qs1 and qs2 by choosing ws1 and ws2 in the first-order conditions for these quantities. As briefly discussed in Appendix B, the result that, given the wholesale prices offered by its rival, a supplier cannot improve on two-part tariffs by using any other own-quantity contract holds also in the case of downstream price competition, although the formal argument is less straightforward in that case. 18 Moreover, as shown in the proof of Proposition 2, supplier s0 has no incentive to raise its wholesale prices above wv either, since doing so would make the minimum volume restraints imposed by the rival supplier, s, non-binding, thus making the problem equivalent to that with two-part tariffs in Section 3 (and in that case profits are maximized at wt < wv ).

17

words, minimum volume requirements can have facilitating effects by allowing each supplier to reduce the other supplier’s incentives to compete for marginal sales. The reason that this cannot happen in Inderst and Shaffer’s (2010) model is that in their model the perfectly competitive suppliers of product s0 have no power to offer contracts or set prices, which rules out any strategic role for the contracts offered by supplier s. The analysis in this section is an application of the more general principle that, in the presence of competing principals, the out-of-equilibrium portions of pricing schedules (or, more generally, of menus of contracts) matter since, by affecting directly the out-of-equilibrium behavior of agents, they affect indirectly the profitability of deviations by the other principals (see, e.g., Martimort and Stole 2002, 2003).

5

Contracts that reference rivals (CRRs)

Non-exclusive CRRs are supply contracts that do not require complete exclusivity but nevertheless condition explicitly the terms offered to a retailer on that retailer’s decisions regarding the volume of sales or retail price of rival products. As discussed in the introduction, nonexclusive CRRs can restrict a retailer’s ability to sell and price rival products in various ways. Two common types of restraints are minimum share requirements, that require a retailer r to purchase a minimum share σsr of its total volume from supplier s, i.e. qsr / (qsr + qs0 r ) ≥ σsr , and restraints on relative retail prices, such as retail MFNs and no-steering rules, that prohibit a retailer r from lowering the price of a rival product s0 below some multiple of the price of product s, i.e. ps0 r ≥ νsr psr . Both types of restraints constrain the relative quantities of the two products that retailers can sell (directly in the case of market-share requirements and indirectly in the case of restraints on relative prices) and have therefore very similar implications for the equilibrium of the model. In the interest of brevity, in what follows I focus on minimum share requirements, with the understanding that the same conclusions apply also to restraints on relative prices. I first analyze the case in which CRRs do not include minimum volume requirements (Subsection 5.1) and then the case in which they do (Subsection 5.2). As in Sections 3 and 4, also in this section I postpone the discussion of sequential contracting and existence of DCA equilibria to Sections 6 and 7, respectively.

18

5.1

CRRs without minimum volume requirements

I assume that retailer r can purchase product s at a non-prohibitive wholesale price wsr only if it complies with the minimum share requirement qsr / (qsr + qs0 r ) ≥ σsr imposed by supplier s. These contracts are the same as those analyzed by Inderst and Shaffer (2010) in a model without upstream competition. Since, as I show further below, both suppliers have incentives to impose minimum share requirements that are binding, in a symmetric DCA equilibrium it must be σsr = 1/2 for s, r = 1, 2. This implies that retailers cannot respond to changes in either of the wholesale prices by altering the relative quantities of the two products, since they must ensure that q1r = q2r . In particular, following an increase in wsr above ws0 r , it is the restraint qsr ≥ qs0 r imposed by supplier s that becomes binding and prevents the retailers from steering consumers away from product s and towards product s0 ; while following a reduction in wsr below ws0 r the binding restraint is that imposed by supplier s0 , i.e. qs0 r ≥ qsr . As shown in the following lemma, symmetric restraints that imply q1r = q2r affect the equilibrium by affecting the way in which quantities respond to changes in wholesale prices. Lemma 2 If Assumption 2 holds, around a symmetric equilibrium with CRRs and for wsr = wsr0 = ws dqs0 dqs = < 0. dws dws Whereas Lemma 1) implied that, in the absence of vertical restraints, the cross-price derivatives of the derived demand for the two products are positive i.e. dqs0 /dws > 0, Lemma 2 establishes that, in the presence of CRRs and under the same restrictions on the primitives of final demand, these cross-price derivatives become negative, i.e. dqs0 /dws < 0. The intuitive reason for this can be seen from the first-order condition of retailer r with respect to the quantity of one of the two products in the presence of CRRs, which around a symmetric equilibrium can be written as19   wsr + ws0 r + (∂1 P + ∂2 P ) qsr = 0. psr − 2 19

See the proof of Lemma 2 in Appendix A for a detailed derivation.

19

(15)

With CRRs, retailer r responds to an increase in ws0 r as if it were an undifferentiated increase in his average marginal cost, (wsr + ws0 r ) /2, and passes through such an increase by reducing the quantities of both products in the same proportion. In other words, as explained by Calzolari and Denicol`o (2013), with non-exclusive CRRs retailers behave as if the products were perfect complements, regardless of their actual degree of substitutability in demand. The implications of this for the setting of wholesale prices by suppliers in stage 1 is a variant of the well-known problem of Cournot complements: with CRRs, the fact that dqs0 /dws < 0 gives supplier s incentives to impose a negative externality on supplier s0 by raising ws , which, as established further below, leads to equilibrium wholesale prices that are higher than the levels that maximize total industry profits. Although, as I discuss in further detail below, with simultaneous contracting CRRs may yield equilibrium outcomes that are jointly less profitable than under two-part tariffs, each supplier has unilateral incentives to adopt binding CRRs, since, by doing so, it can increase its profits for any given type of contracts offered by its rival. Lemma 3 When non-exclusive CRRs are allowed, it is a dominant strategy for every supplier to adopt them and include restraints that are binding. The implications of Lemmas 2 and 3 for equilibrium prices and quantities in a game with simultaneous contract offers are characterized in the following proposition, in which the superscript crr stands for CRRs. Proposition 3 In any symmetric subgame-perfect DCA equilibrium of the simultaneous contracting game with CRRs and no minimum volume requirements, equilibrium wholesale prices wcrr are above the level that maximizes industry profits, i.e. wcrr > wm > wt . Proof: If wcrr ≤ wm , a small increase in ws above wcrr would not decrease total industry profits (since these are concave and maximized at wm ) and, as a consequence of the binding restraint that reference rivals imposed by supplier s, would strictly decrease the sales, qs0 , and profits, πs0 , of supplier s0 (see Lemma 2). In light of (12), such an increase is therefore always strictly profitable for supplier s. This proves that wcrr ≤ wm cannot be an equilibrium. The result obtained in Proposition 3 for the simultaneous contracting game of this section differs from that obtained by Inderst and Shaffer (2010) in a model in which one of the two 20

products is supplied at marginal cost by a perfectly competitive fringe. In Inderst and Shaffer (2010) the adoption of minimum share requirements without minimum volume requirements by supplier s leads to an equilibrium that implements the industry monopoly outcome. The reason for the difference between their result and the result in this section is that in their model there are no externalities in the upstream market. Specifically, in their model supplier s cannot impose any externalities on the perfectly competitive fringe producing product s0 , since that fringe earns zero margins. Conversely, the perfectly competitive fringe does not have the market power to impose any externalities on supplier s, even though the existence of positive profit margins on product s would make imposing such externalities profitable. This is not the case in my model when contracting takes place simultaneously, since both suppliers have the incentives and the ability to steal profitable sales from each other. As shown in Section 6, however, when contracting takes place sequentially the predictions of my model regarding the effects of CRRs coincide with those of Inderst and Shaffer. Proposition 3 has implications for welfare and industry profits. Consumer and overall welfare are always lower in a DCA equilibrium with CRRs than in a DCA equilibrium with two-part tariffs, since prices are always higher and quantities always lower in the former than in the latter.20 When contracting takes place simultaneously, as is the case in this section, industry profits can instead be higher or lower with CRRs than with two-part tariffs. This ambiguity results from the fact that both contracts yield equilibrium prices that depart from the prices that maximize joint industry profits, with two-part tariffs yielding prices that are too low and CRRs prices that are too high. More concrete conclusions regarding this point can be reached by considering the linear demand example introduced in (13), which I discuss next. Based on the inverse linear demand system in (13), Figure 2 shows the wholesale prices, wcrr , and retail prices, pcrr , in a symmetric DCA equilibrium with CRRs. As can be seen in the left-hand panel, changes in the degree of product substitutability, a, do not affect wcrr or pcrr , since with CRRs retailers behave as if the products were no longer substitutes. The right-hand panel confirms, instead, the tendency (already discussed in Section 3.2 for the case of two-part tariffs) of suppliers to respond to increases in the intensity of downstream 20

As discussed in Section 7, the adoption of CRRs can also affect welfare by affecting the existence of a DCA equilibrium. Specifically, if symmetric DCA equilibria exist with CRRs but not with two-part tariffs, banning CRRs may lead to exclusive equilibria and possibly reduce welfare by causing a loss of variety.

21

competition (approximated by an increase in b in the figure) by raising wholesale prices. By increasing equilibrium upstream margins and thus the suppliers’ incentives to impose negative externalities on each other, increases in b also amplify the divergence between wcrr and wm . 0.7

0.7

pcrr

0.6

0.5

pm

0.4

p ,w

p ,w

0.5

wcrr

0.3

pcrr

0.6

pm

0.4

0.3

0.2

wcrr

0.2

w

0.1

0.0 0.0

0.2

a Hproduct substitutability L 0.4

0.6

wm

m

0.8

0.1

0.0 0.0

1.0

0.2

b Hretailer substitutability L 0.4

0.6

0.8

1.0

Figure 2: Equilibrium prices with simultaneous CRRs.

Figure 3 provides a comparison of industry profits under CRRs and two-part tariffs. As one would expect, industry profits with CRRs, Πcrr , are higher than industry profits with two-part tariffs, Πt , when product substitutability, a, is high, since Πcrr does not depend on a, while upstream competition causes Πt to fall when a increases.

5.2

CRRs with minimum volume requirements

Consider now the case in which suppliers require retailers to comply also with minimum volume requirements qsr ≥ q sr , s, r = 1, 2, in addition to the market-share requirements discussed above. By a logic analogous to that discussed in Section 4, suppliers can use minimum volume requirements to defend themselves from attempts by their competitor to impose negative externalities on them. While in the absence of CRRs minimum volume requirements discourage wholesale price cutting by rival suppliers, with simultaneous CRRs such requirements discourage the excessive wholesale price hiking by rival suppliers discussed in subsection 5.1 above. In both cases, minimum volume requirements move the equilibrium closer to the outcome that maximizes total industry profits. 22

1.0

b HRetailer substitutability L

0.8

P t >P crr

0.6

P crr >P t

0.4

0.2

0.0 0.0

0.2

a HProduct substitutability L 0.4

0.6

0.8

1.0

Figure 3: Industry profits with simultaneous CRRs (Πcrr ) and two-part tariffs (Πt ).

Proposition 4 If deviations to exclusivity are not profitable, any wcrr,v ∈ [wm , wcrr ] can be supported as a symmetric subgame perfect DCA equilibrium of a game in which suppliers simultaneously offer CRRs with minimum volume requirements. In the presence of CRRs, the requirement qs0 r ≥ q s0 r imposed by supplier s0 discourages small increases in wsr by supplier s (and vice versa). In particular, when qs0 r ≥ q s0 r is binding, starting from any wcrr,v ≥ wm (where the superscript crr, v denotes variables in an equilibrium with non-exclusive CRRs and minimum volume requirements) a small increase in wsr above wcrr,v would be unprofitable because it would reduce (or, at best, leave unchanged) industry profits without having any effect on πs0 , since the minimum volume requirement imposed by supplier s0 prevents retailer r from reducing qs0 r . Proposition 4 implies that, when adopted in a second-best world in which (simultaneous) CRRs are allowed, minimum volume requirements enhance both industry profits and consumer welfare, since all the additional DCA equilibria that they support have wholesale prices that are lower and closer to wm than in the DCA equilibrium with only CRRs.21 21

As already mentioned in footnote 20 and discussed in further detail in Section 7, minimum volume requirements may also affect welfare by affecting the existence of DCA equilibria.

23

6

Sequential contract offers

Although the fully symmetric model with simultaneous offers studied in Sections 3 through 5 may describe well some real-world situations – such as the antitrust case brought by the U.S. DOJ against the major credit card companies, in which all three companies were major and long-established market participants (see the introduction for a discussion) – in many other real-world situations one of the suppliers may have an advantage over its rivals. To investigate how the analysis above extends to such asymmetric cases, in this section I assume that one of the two suppliers, s, has a first-mover advantage. Specifically, I assume that stage 1 of the game unfolds in two sub-stages. In stage 1(a) supplier s offers public and binding menus of contracts to both retailers. After having observed the contracts offered by s, in stage 1(b) supplier s0 offers its menus of contracts to both retailers. Stages 2 and 3 of the game remain the same as in Section 2. Given the DCA contracts {wsr , Fsr , Rsr }, r = 1, 2, offered by supplier s in stage 1(a), where Rsr denotes any restraints in such contracts, in stage 1(b) supplier s0 chooses its DCA contracts to solve the following profit maximization problem. X maximize πs 0 = [Πr (w, R) − (wsr − c) qsr ] . {ws0 1 ,Rs0 1 },{ws0 2 ,Rs0 2 }

s. t.

r=1,2

(16)

the restraints Rs1 and Rs2 .

Note that, by a logic analogous to that discussed in Section 3, if wsr > c and the corresponding restraint Rsr permits it, supplier s0 has incentives to behave opportunistically and impose a negative externality on supplier s by inducing retailer r to reduce qsr . Consider next the profit-maximization problem of supplier s in stage 1(a). When offering its DCA contracts, supplier s takes into account that wsr and Rsr will affect the individual components {ws0 r , Fs0 r , Rs0 r }, r = 1, 2, of the contracts offered by supplier s0 , but not the level πs0 of the profits earned by that supplier in a DCA equilibrium, since the latter is determined independently by the out-of-equilibrium exclusive contracts offered by supplier s. In particular, since the fixed fees charged by supplier s0 are not yet determined in stage 1(a), they will adjust in response to changes in the DCA contracts offered by supplier s to ensure that a constant level of πs0 is maintained. This implies that supplier s cannot profit from behaving opportunistically and inducing retailers to lower qs0 1 and qs0 2 , as was instead the

24

case with the simultaneous contract offers. Given this, supplier s maximizes total industry P profits, r=1,2 Πr (w, R), subject to the reaction functions ws∗0 r (·) and Rs∗0 r (·) of supplier s0 determined by the solution to (16) above. X maximize πs = Πr (w, R) {ws1 ,Rs1 },{ws2 ,Rs2 }

s. t.

r=1,2

ws0 r = ws∗0 r (ws1 , ws2 , Rs1 , Rs2 ) ,

r = 1, 2,

Rs0 r = Rs∗0 r (ws1 , ws2 , Rs1 , Rs2 ) ,

r = 1, 2.

(17)

Given the framework above, one can proceed to study the properties of the subgame-perfect equilibria of this sequential game under two-part tariffs, CRRs and minimum volume requirements.

6.1

Two-part tariffs

It is straightforward to show that, by a logic analogous to that discussed in Section 3 for the case of simultaneous contracting, two-part tariff contracts are not sufficient to implement the industry monopoly outcome in the case of sequential contracting either. The reason for this is that if ws1 = ws2 = wm > c, as is necessary for a monopoly outcome without vertical restraints, then supplier s0 has an incentive to behave opportunistically and charge ws0 1 and ws0 2 strictly below wm in order to steal profitable sales from supplier s.

6.2

Minimum volume requirements

When suppliers can impose minimum volume requirements qsr ≥ q sr , s, r = 1, 2, allowing them to move sequentially resolves the indeterminacy that arose in the simultaneous game of Section 4 and yields the industry monopoly outcome as the unique equilibrium. Proposition 5 The sequential contracting game with minimum volume requirements has a unique subgame-perfect DCA equilibrium that implements the industry monopoly outcome. In this equilibrium supplier s imposes minimum volume requirements q s1 = q s2 = q m and offers ws1 = ws2 = wm , and supplier s0 responds by offering ws0 1 = ws0 2 = wm . Intuitively, in stage 1(a) supplier s sets its wholesale prices at the level wm that induces the industry monopoly outcome and imposes minimum volume requirements qsr ≥ q m to prevent supplier s0 from behaving opportunistically in stage 1(b) and lowering its wholesale prices 25

below wm . The logic of this result is very similar to that of the simultaneous contracting game with minimum volume requirements of Section 4, with only two differences. First, in the sequential game of this section supplier s0 does not need minimum volume requirements to prevent supplier s from behaving opportunistically, since the fact that s0 has not yet committed to any contracts when s offers its contracts leaves no room for such opportunistic behavior by s. Second, the fact that one of the two suppliers moves first eliminates the coordination failures that characterized the simultaneous game in Section 4 and helps the suppliers select the most profitable equilibrium among the many possible equilibria in that section.

6.3

CRRs without minimum volume requirements

Consider now the case in which suppliers can offer CRRs sequentially.22 Also in this case sequential contracting resolves any coordination failures between suppliers and yields the industry monopoly outcome, though with slightly different contracts and mechanisms from Subsection 6.2 above. Proposition 6 The sequential contracting game with CRRs and no minimum volume requirements has a unique subgame-perfect equilibrium that implements the industry monopoly outcome. In this equilibrium supplier s offers ws1 = ws2 = c and supplier s0 responds by imposing share requirements σs0 1 = σs0 2 = 1/2 and offering ws0 1 = ws0 2 = wm + (wm − c) > wm . Intuitively, after having eliminated any incentive for supplier s0 to behave opportunistically by setting ws1 = ws2 = c, supplier s delegates the task of monopolizing the market to supplier s0 . Supplier s0 can do so because it can raise wholesale prices sufficiently, while at the same time using binding minimum share requirements to induce retailers to reduce the quantities of both products. Specifically, given that with symmetric demand and costs industry profits are maximized at a symmetric outcome, supplier s0 imposes binding share requirements σs0 r = 1/2. As explained in Section 5, this effectively transforms the products into perfect complements in the eyes of retailers, which care only about the sum (or, equivalently, the average) of the wholesale prices of the two products. To induce a monopoly outcome supplier 22

As will become clear below, whether minimum volume requirements are also permitted in addition to CRRs is irrelevant in this sequential game. For simplicity, therefore, I consider the case in which such additional restraints are not adopted.

26

s0 must therefore raise the average wholesale price paid by the retailers to wm . In order to do so, supplier s0 must compensate for the lower levels of the wholesale prices ws1 = ws2 = c charged by s by charging ws0 1 = ws0 2 == wm + (wm − c) > wm .

7

Existence of double common agency equilibria

Having characterized the properties of DCA equilibria with different types of vertical contracts, I now turn to studying their existence. In this model DCA equilibria are not guaranteed to exist for two reasons. First, in the presence of downstream competition a DCA market structure does not necessarily maximize industry profits. Under certain conditions market structures such as pairwise exclusivity or upstream monopoly may yield higher industry profits than DCA by reducing the intensity of downstream competition at the cost of some loss in variety. Second, the presence of externalities between suppliers implies that a coalition of one of the two suppliers and one or both retailers may find it profitable to exclude the other supplier. The rest of this section derives conditions under which these two effects are not too strong and, thus, under which the DCA equilibria studied in previous section exist.23 Specifically, I first characterize the conditions for the existence of a DCA equilibrium for any type of DCA contracts and then apply these conditions to the specific types of DCA contracts (two-part tariffs, minimum volume requirements, and CRRs) studied in the previous sections. The following intermediate result is helpful for the characterization of equilibrium contracts. Lemma 4 In any equilibrium or deviation play in which supplier s sells to both retailers, the profits earned by s are decreasing in the wholesale price wsr0 \sr that s would make available to retailer r0 if retailer r rejected his contracts. If wholesale prices are exogenously constrained to be no lower than wmin , then in any equilibrium wsr0 \sr = wmin . Proof: To induce retailer r to accept its DCA contract, supplier s must guarantee r a profit at least as high as the profit πr\s that r could earn by rejecting the contract offered by s and selling product s0 exclusively. The maximum profits that s can earn from r are therefore decreasing in πr\s . Since, as assumed on page 11, dπr\s /dwsr0 > 0, s finds it optimal to lower 23

Note that the model may also have other equilibria, such as equilibria with upstream monopoly, pairwise exclusive or mixed market structures. An analysis of the conditions for the existence of these other equilibria is beyond the scope of this paper and is addressed in related but separate ongoing work.

27

wsr0 \sr as much as possible. Consequently, given an exogenous lower bound wmin on wholesale prices, wsr0 \sr = wmin . 

Intuitively, Lemma 4 says that profit maximization has implications also for contracts that are not meant to be taken in a continuation equilibrium. In particular, in order to maximize the profits that it can extract from retailer r in the continuation equilibrium that it intends to implement, supplier s commits to making retailer r0 as strong a competitor as it can in the out-of-equilibrium case in which r rejected s’s contract offer. Supplier s does so by offering as low a wholesale price wsr0 \sr as it can. Define the wholesale prices that maximize the joint profits of supplier s and the retailer(s) with which s deals in a DCA outcome and, respectively, in deviations to upstream monopoly, pairwise exclusivity, a mixed outcome in which s deals with both retailers, and a mixed outcome in which s deals only with retailer 1 as follows. dca dca es1 es2 w ,w




= arg

um um es1 es2 (w ,w ) = arg

max

ne,dca ne,dca ws1 ,ws2 r=1,2

we,ums1 ,ws2

= arg

ne,dca ne,dca esdca esdca wsr ,w ,w 0 r , wsr 0 0 r0



(18) (19)

r=1,2

e,pe espe0 r0 ) Πr (wsr , ∞, ∞, w

wsr




Πr



 i
ne,dca ne,dca esdca esdca esdca − w w ,w 0 r 0 , wsr 0 0 r − c qs0 r 0 r , wsr X e,um e,um Πr (wsr , ∞, wsr , ∞) max e,um 0

pe esr w = arg max e,pe mix mix es1,d2 es2,d2 w ,w

Xh

(20)

h   e,mix ne,mix max Π w , ∞, w , w 1 min s1 s2 e,mix ne,mix

ws1

,ws2

   i ne,mix e,mix ne,mix e,mix +Π2 ws2 , wmin , ws1 , ∞ − (wmin − c) qs0 2 wmin , ws2 , ∞, ws1 (21)   ne,mix mix es1,d1 esmix esmix w = arg max Π2 ws1 ,w 0 1,d2 , ∞, w 0 2,d2 ne,mix ws1

 
ne,mix esmix esmix esmix − w w ,w (22) 0 1,d2 − c qs0 1 0 1,d2 , ws1 0 2,d2 , ∞

Expressions (18) through (20) are self explanatory. Expressions (21) and (22) define, respectively, the most profitable wholesale prices offered by supplier s in a deviation to a mixed outcome in which s continues to sell to both retailers (hence the subscript d2) and in a deviation to a mixed outcome in which it sells only to retailer 1 (hence the subscript d1). For notational convenience one can furthermore define the value of the total vertical profits

28

generated by the two retailers in the outcomes above as follows

dca dca dca dca es2 esdca es1 esdca es1 esdca es2 esdca Πdca ≡ Π1 w ,w ,w = Π2 w ,w ,w 01 , w 02 02 , w 01

(23)

um um um um es1 es2 es2 es1 Πum ≡ Π1 (w , ∞, w , ∞) = Π2 (w , ∞, w , ∞)

(24)

pe pe esr espe0 r0 ) = Πr0 (∞, w espe0 r0 , w esr Πpe ≡ Πr (w , ∞, ∞, w , ∞)

(25)

mix mix es2,d2 es1,d2 , wmin ≡ Π1 w , ∞, w Πmix1 d2




(26)


mix mix es1,d2 es2,d2 ,∞ ≡ Π2 w , wmin , w Πmix2 d2

(27)

mix esmix esmix es1,d1 ≡ Π1 w ,w Πmix1 0 2,d2 0 1,d2 , ∞, w d1




(28)

mix esmix es1,d1 esmix ,w ≡ Π2 ∞, w Πmix2 0 1,d2 0 2,d2 , w d1




(29)

where the symmetry in (23) through (25) results from the assumed symmetry of demand and costs, and the notation for the profitability of deviations to mixed market structures in (26) through (29) should be interpreted as follows. The superscripts mix1 and mix2 denote, respectively, the retailer that sells only one product and the retailer that sells two products. As discussed above, the subscripts d1 and d2 indicate, respectively, whether the mixed market structure has arisen from a deviation in which the deviating supplier sells to only one retailer or from a deviation in which the deviating supplier sells to both retailers.24 For example, the total vertical profits generated by the retailer that sells only one product in a deviation in which a supplier sells to both retailers but excludes his rival from one of the two retailers are denoted by Πmix1 d2 . Before stating the main result of this section in Proposition 7 below, it is helpful to clarify an aspect that plays an important role in that proposition. The levels of the profits that e,mix suppliers and retailers earn in a DCA equilibrium depend on the profits Πmix d2 − ts0 r that

retailer r could earn in the (out-of-equilibrium) event that it rejected the DCA offer of supplier s and accepted the exclusive contract offered by supplier s0 , taking as given that retailer r0 accepts contracts from both suppliers. Since te,mix is not accepted by r on the equilibrium s0 r path and does not affect the equilibrium payoff of supplier s0 in any other way, it can take a large number of equilibrium values, which I denote by α. Note that α determines the split of 24 These two types of deviations may give rise to different industry profits because, as can be seen by comparing (21) and (22), they may entail different wholesale prices, although this is not the case in the linear demand example introduced on page 15 and discussed again further below.

29

profits between suppliers and retailers, with higher values yielding lower equilibrium profits for the retailers and higher equilibrium profits for the suppliers. Given this, the main result of this section can be stated in the following proposition. To simplify notation and without loss of generality, I assume that c = 0.25 Moreover, I denote the tariff that r must pay to s in a mixed outcome in which r is nonexclusive and r0 does not sell product i by tne,mix sr\ir0 . Proposition 7 If dca

Π



> max 2Π

um

−

pe Πmix1 d2 , Π ,

Πmix1 + Πmix2 d2 d1 2



(30)

there exist symmetric, weakly-truthful, subgame-perfect DCA equilibria. In these symmetric equilibria each supplier s offers the following tariffs = tne,dca = tne,dca sr sr0 = α, te,mix sr

Πdca − Πmix1 + min {α, Πmix1 d2 d2 } , 2

ne,dca dca dca ne,dca esr esr , wsr =w =w with wsr 0 0 ne,mix mix2 mix1 e,mix esr esr,d2 , wsr =w with wsr 0 ,d2 0 \s0 r = w

dca + ε, tne,mix − Πmix1 d2 sr0 \s0 r = Π

dca + min {α, Πmix1 tne,mix − Πmix1 d2 } , d2 sr\sr0 = Π

ne,mix with wsr\sr 0 = wmin

te,um = te,um = Πum − Πmix1 + min {α, Πmix1 sr d2 d2 } + ε, sr0

e,um e,um um um esr esr with wsr =w , wsr =w 0 0

e,pe pe mix1 te,pe + min {α, Πmix1 sr = tsr0 = Π − Πd2 d2 } + ε,

e,pe pe e,pe pe esr esr with wsr =w , wsr 0 = w 0

− 2Πum , and ε > 0 is arbitrarily small. Paretowhere α ∈ [0, α ¯ ], with α ¯ ≡ Πdca + Πmix1 d2 undominated equilibria from the point of view of suppliers are achieved at α=α ¯

if 2Πum ≥ Πdca

α ∈ [Πmix1 ¯] d2 , α

otherwise

A few aspects of Proposition 7 are worth discussing. First, the selection of contracts by competing retailers in stage 2 is a proper subgame in which the payoff to r from a given choice of contracts depends on the contracts that r expects r0 to choose.26 This subgame may have multiple equilibria. For example, r may select nonexclusive contracts if he expects r0 also to do so (and vice versa), or r may instead select an exclusive contract if he expects r0 also to do so (and vice versa). However, given the assumptions that retailers play Pareto j,k j,k ˜j,k Alternatively, one could assume c > 0 and replace tj,k sr with tsr = tsr + cqsr . All the results below j,k would then continue to hold for t˜sr . 26 This is different from models without externalities between agents, such as Prat and Rustichini (2003). 25

30

undominated equilibria in stage 2, the menus of tariffs specified in Proposition 7 ensure the existence of a unique coalition-proof continuation equilibrium. To see this note that in a DCA outcome both retailers earn exactly max {Πmix1 − α, 0}, while in any other outcome d2 − α, 0} − ε. Therefore, for ε > 0, all outcomes other at least one retailer earns max {Πmix1 d2 than DCA are weakly Pareto-dominated by DCA from the point of view of retailers and can therefore not be coalition-proof equilibria of this two-player game (see Bernheim, Peleg and Whinston, 1987). Note that the necessity to ensure the uniqueness of the continuation equilibrium imposes restrictions on the menus of contracts that suppliers can offer in stage 1, and thus on the region of parameter values for which a DCA equilibrium exists. For example, and te,pe in the absence of these restrictions, low levels of the out-of-equilibrium tariffs te,um sr sr would expand the range of parameters for which DCA equilibria exist by making deviations to market structures pe and um less profitable for supplier s0 . However, excessively low levels of these tariffs cannot be used by the suppliers to induce coalition-proof continuation equilibria with a DCA market structure, because they could be accepted by retailers on the equilibrium path and result in equilibria with market structures um and pe. Second, the subgame-perfect equilibrium of the overall game is truthful in the sense of Bernheim and Whinston (1986). This is important for the following reason. Models of common agency may have implausible equilibria that are supported by arbitrary outof-equilibrium portions of the menus offered by principals. This is the case because, in the absence of any restrictions on the menus offered by supplier s, deviations from these equilibria by supplier s0 may be prevented by unreasonable refusals to cooperate by supplier s. For example, in Bernheim and Whinston’s (1998) model of common agency with a single retailer, exclusive equilibria may exist even when they are significantly less profitable than common agency for the suppliers if supplier s prevents a deviation to common agency by supplier s0 by refusing to make a reasonable offer for that outcome. Analogously, in the multi-retailer model of this paper, deviations from a DCA equilibrium to a mixed market structure outcome in which supplier s0 intends to induce common agency only at retailer r0 and to seek exclusivity at or withdraw from retailer r are possible only if the tariffs tne,mix sr0 \sr charged, respectively, by supplier s in these two outcomes are not prohibitive. and te,mix sr0 Prohibitive tariffs would increase the likelihood that a DCA equilibrium exists by ruling out these deviations, but would do so arbitrarily, casting down on the plausibility of the

31

resulting equilibrium. This issue can be addressed by requiring the contracts offered by any supplier s for common agency at any retailer r0 to be truthful relative to a DCA outcome, i.e. to yield the same total profits for supplier s as in the candidate DCA equilibrium. It is + straightforward to check that the tariffs in Proposition 7 have this property, since tne,dca sr ne,mix + tne,mix tne,dca = te,mix sr sr0 sr0 \s0 r = tsr\sr0 .

Third and last, the proposition characterizes a multiplicity (a continuum, in fact) of symmetric DCA equilibria, corresponding to the different values of α ∈ [0, α ¯ ] for which a DCA equilibrium exists. As explained above, this multiplicity arises from the fact that, in a subgame-perfect DCA equilibrium, te,mix affects only the profits earned by supplier s0 , not sr = α ∈ [0, α ¯ ] constitutes the profits earned by supplier s itself, and therefore any value te,mix sr a (weak) best response by supplier s. Since the value of α has consequences for the the profits earned by suppliers and retailers in a DCA equilibrium and for the existence of such an equilibrium, a discussion of possible ways to impose some structure on this indeterminacy in Proposition 7, by reducing the may be helpful. As shown by the expression for tne,dca sr retailers’ outside options, higher values of α yield higher equilibrium profits for the suppliers. Therefore, if suppliers were able to engage in non-binding communication before stage 1 or to find some other way to avoid coordination failures, they would have incentives to raise α (see, e.g., Bernheim and Whinston (1998) and Calzolari and Denicol`o (2013) for discussions of these issues in models with a single retailer). These incentives must, however, take into account the fact that a DCA equilibrium no longer exists for α > α ¯ . The intuition for this is as follows. Whereas in a symmetric DCA equilibrium the two suppliers share the burden of ensuring participation by a given retailer, in a deviation to exclusivity the deviating supplier must bear this burden on his own.27 In light of this, the lower the outside option of the retailer, and thus the lower the profits required by the retailer for participation, the lower the costs of implementing a deviation to exclusivity and thus, all else equal, the more profitable such a deviation. Since the outside option of each retailer vis-`a-vis any (DCA or exclusive) contract offered by supplier s is max {Πmix1 − α, 0}, a higher α reduces the costs and increases d2 27

Games of simultaneous common agency like the present one pin down only the sum of the profits earned by each supplier in a common agency equilibrium, not the profits of each individual supplier and thus the contribution of that supplier to ensuring participation by the retailer. To resolve this indeterminacy, I focus on the symmetric case in which the two suppliers earn the same profits in a common agency equilibrium, and thus share the burden of ensuring retailer participation equally.

32

the profitability of deviations from a candidate DCA equilibrium.28 These observations imply that the most profitable DCA equilibrium for suppliers is the one with α = α ¯ . To the extent that suppliers move first, as is the case in this paper, it is reasonable to expect this equilibrium to be the focal one among DCA equilibria. The general results derived above can be applied to the specific types of contracts discussed in Sections 3 through 6. To be concrete, I use the linear demand example introduced in Section 3.3 and characterize the range of values of a (product substitutability) and b (retailer substitutability) for which DCA equilibria exist, as shown in Figure 4 for different types of contracts. Figure 4 shows that, generally, a DCA equilibrium exists when products and retailers are sufficiently differentiated (i.e. when a and b are sufficiently low). With high degrees of product differentiation deviations to market structures in which one of the products is not available from at least one retailer entail significant losses in variety and are thus less profitable. The effects of high degrees of retailer differentiation are, instead, more subtle. Deviations to market structures other than DCA tend to reduce the intensity of downstream competition. Specifically, deviations to pairwise exclusivity or mixed market structures cause the retailers to carry different mixes of products, thus increasing their effective degree of differentiation, while deviations to upstream monopoly give the upstream monopolist unconstrained ability to use (publicly observable) wholesale prices to soften downstream competition. When b is low and retailers are intrinsically very differentiated to start with, any incremental reduction in competition arising for exclusivity is small and dominated by the loss of variety that exclusivity entails. A comparison of the three panels of Figure 4 shows that the range of parameters for which there exists a DCA equilibrium varies across different types of contracts. Specifically, the conditions for the existence of a DCA equilibrium are more stringent with simultaneous CRRs without minimum volume requirements (top right panel) than with simultaneous two-part tariffs (top left panel). This is due to the fact that, as can be seen in Figure 3, CRRs yield lower equilibrium profits than two-part tariffs in the relevant range of parameter values. 28

For a discussion of a similar logic in a different model with two competing principals (retailers) and a single agent (supplier) see Miklos-Thal, Rey and Verg´e (2011) and Rey and Whinston (2013). For a model in which restricting the retailers to offering a single contract causes non-exclusive equilibria not to exist see instead Marx and Shaffer (2007).

33

1.0

0.8

0.8

b HRetailer substitutability L

b HRetailer substitutability L

1.0

0.6

0.4

DCA equilibrium with two-part tariffs exists

0.2

0.0 0.0

0.6

0.4

DCA equilibrium with simultaneous CRRs and no min. vol. req. exists

0.2

0.2

a HProduct substitutability L 0.4

0.6

0.8

0.0 0.0

1.0

0.2

a HProduct substitutability L 0.4

0.6

0.8

1.0

1.0

b HRetailer substitutability L

0.8

0.6

0.4

DCA equilibrium implementing industry monopoly exists

0.2

0.0 0.0

0.2

a HProduct substitutability L 0.4

0.6

0.8

1.0

Figure 4: Existence of DCA equilibrium with simultaneous two-part tariffs (top left), simultaneous CRRs (top right), and industry monopoly outcome (bottom).

One should also note that, as shown in the bottom panel, equilibria that implement the industry monopoly outcome (through simultaneous contracting with CRRs and minimum volume requirements or through sequential contacting with CRRs and/or minimum volume requirements) exist for a broader range of parameter values than other equilibria but are nevertheless not guaranteed to exist for all parameter values. The reason for this is the need for suppliers to ensure that DCA is the unique outcome of the continuation game in stage 2, as discussed above. This imposes constraints on the amount of profits that the suppliers can extract in a DCA outcome and may thus make such an outcome less profitable for suppliers than other outcomes, even when a DCA outcome maximizes total industry profits.

34

8

Concluding remarks

When differentiated suppliers offer publicly observable bilateral contracts to competing retailers, equilibrium wholesale prices are above the marginal cost of suppliers but below equilibrium retail prices. This gives rise to positive unit margins – and thus to competitive externalities between firms – in both the upstream and downstream markets. A first implication of these externalities is that common agency is not guaranteed to arise in equilibrium. It does so only when it yields sufficiently high industry profits, which is the case when products and/or retailers are highly differentiated, and when retailers retain a sufficiently large share of these profits. The derivation of these intuitive conditions within a fairly general framework, which allows for menus of contracts and multilateral deviations by suppliers, constitutes one of the main methodological contributions of the paper. A second, more applied, contribution of the paper is an analysis of the effects of various vertical contracts on the incentives of suppliers and retailers to compete for incremental volumes of sales in common agency equilibria. I have shown that minimum volume requirements and/or CRRs generally dampen the incentives of suppliers to compete relative to a benchmark in which only two-part tariffs are allowed. Specifically, when suppliers offer contracts sequentially, both types of restraints allow suppliers to maximize industry profits even in the absence of any (explicit or tacit) collusion. When instead suppliers offer contracts simultaneously, coordination failures may prevent the maximization of industry profits and lead to prices below the industry monopoly level with minimum volume requirements and above the industry monopoly level with CRRs. In terms of its policy implications, the paper has shed some light on a source of competitive harm that has received relatively little attention in the theoretical and applied antitrust literature. Historically, at least in the United States, all-units discounts and market-share rebates have been investigated in the context of private antitrust litigation initiated by suppliers complaining about having been partially or fully excluded from access to downstream distribution by a competitor.29 This paper has shown that, by acting as facilitating practices that benefit all suppliers (and, possibly, also all retailers if anticompetitive profits are shared through bargaining) but harm consumers, these restraints can raise competitive concerns also in situations in which no competitor has incentives to complain. In such cases, government 29

See, e.g., the antitrust cases discussed in footnote 2 of the introduction.

35

intervention, such as the actions brought by the U.S. DOJ against credit card companies and by the U.K. OFT against tobacco companies, may be necessary to preserve competition.30 The insights of the paper are also relevant for a growing body of, mostly empirical, research on bargaining between upstream and downstream firms in healthcare, cable television, and other markets (see, e.g., Gowrisankaran, Nevo and Town, 2015, and Crawford and Yurukoglu, 2012). This literature has so far focused to a large extent on the division of profits between upstream and downstream firms, and only to a lesser extent on the structure of the vertical contracts and restraints adopted by these firms and on their implications for marginal input, and thus retail, prices. The analysis in this paper can be seen as a first step towards understanding these implications. As an example, Ho and Lee (2015) find that an increase in competition in Californian health insurance markets has led to an increase in the prices charged by hospitals to health insurance companies and ascribe their findings to a reduction in the bargaining power of the latter. However, as discussed in the commentary to Figure 1 above, Ho and Lee’s findings are also consistent with (upstream) hospitals raising their marginal (input) prices to counter the dissipation of total profits that would otherwise result from an increase in (downstream) competition between insurance companies, regardless of the effects of the latter on the distribution of bargaining power. As for possible ways to address some of the limitations of the paper in future work, two aspects come to mind. First, my findings follow from the fact that suppliers earn positive unit margins in equilibrium. In the specific context of this model, this is a consequence of assuming that contracts are publicly observable. Although positive unit margins are the norm in most real-world upstream markets, in some of these markets they may not be explained by publicly observable contracts. One way to reconcile these two observations is to recognize that, as shown by Rey and Verg´e (2004) for the case of a monopolistic supplier, upstream margins would be positive also with secret contracts, wary beliefs and price competition. However, given that a formal derivation of Rey and Verg´e’s result in a setting with competing suppliers would be virtually intractable and that there is no a-priori reason to believe that their result would not carry over to such a setting, pursuing such a formal derivation 30

It should also be noted that my findings regarding the facilitating effects of these restraints on the marginal prices charged by all suppliers, including the complainant, still apply even when the main complaint revolves around the partial exclusionary effects of the restraints. A self-interested complainant is, however, unlikely to highlight this aspect of his theory of competitive harm.

36

does not appear to be a fruitful extension. Studying settings in which there are other reasons for the existence of positive upstream margins, such as the need to provide adequate risk-insurance to retailers (see, e.g., Bernheim and Whinston 1998, Section 5) or adequate incentives for suppliers to make non-contractible investments that increase sales of their products, is instead likely to be a more productive research avenue. This research may also yield helpful insights into the possible efficiency-enhancing effects of vertical restraints in common agency settings. Second, although for the purpose of the analysis in this paper I have focused on common agency equilibria, under different parameter values the model may also have equilibria involving exclusivity. For example, market structures with upstream monopoly or pairwise exclusivity may increase industry profits by softening downstream competition and thus emerge as equilibria under certain conditions. An exploration of these conditions might be a worthwhile topic for future research.

37

References Bernheim, Douglas, Bezalel Peleg and Michael Whinston (1987), “Coalition-Proof Nash Equilibria I: Concepts,” Journal of Economic Theory, Vol. 42, pp. 1-12. Bernheim, Douglas, and Michael Whinston (1986), “Menu Auctions, Resource Allocations, and Economic Influence,” Quarterly Journal of Economics, Vol. 101, pp. 1-31. Bernheim, Douglas and Michael Whinston (1998), “Exclusive Dealing,” Journal of Political Economy, Vol. 106 No. 1, pp. 64-103. Calzolari, Giacomo and Vincenzo Denicol` o (2013), “Competition with Exclusive Contracts and Market-share Discounts,”American Economic Review, Vol. 103 No. Carlton, Dennis and Ralph Winter (2014), “Competition Policy and Regulation in Credit Card Markets: Insights from Single-sided Market Analysis,” Competition Policy International, Vol. 10 No. 2, pp. 53-69. Chen, Yongmin and Marius Schwartz (2015), “Differential Pricing When Costs Differ: A Welfare Analysis,” RAND Journal of Economics, Vol. 46 No. 2, pp. 442-460. Collard-Wexler, Allan, Gautam Gowrisankaran and Robin Lee (2015), “‘Nash-inNash’ Bargaining: A Microfoundation for Applied Work,” working paper, August. Crawford, Gregory and Ali Yurukoglu (2012), “The Welfare Effects of Bundling in Multichannel Television Markets,” American Economic Review, Vol. 102 No. 2, pp. 643685. de Fontenay, Catherine and Joshua Gans (2014), “Bilateral Bargaining with Externalities,” Journal of Industrial Economics, Vol. LXII No. 4, pp. 756-788. Dobson, Paul and Michael Waterson (2007), “The Competitive Effects of Industry-wide Vertical Price Fixing in Bilateral Oligopoly,” International Journal of Industrial Organization, Vol. 25, pp. 935-962. Gowrisankaran, Gautam, Aviv Nevo, and Robert Town (2015), “Mergers When Prices Are Negotiated: Evidence from the Hospital Industry,” American Economic Review, Vol. 105 38

No. 1, pp. 172-203. Ho, Kate and Robin Lee (2015), “Insurer Competition in Health Care Markets,” working paper, June. Inderst, Roman (2010), “Models of Vertical Market Relations,” International Journal of Industrial Organization, Vol. 28, pp. 341-344. Inderst, Roman and Greg Shaffer (2010), “Market-share Contracts as Facilitating Practices,” RAND Journal of Economics, Vol. 41 No. 4, Winter, pp. 709-729. Inderst, Roman and Christian Wey (2003), “Bargaining, Mergers, and Technology Choice in Bilaterally Oligopolistic Industries,” RAND Journal of Economics, Vol. 34, No. 1, Spring, pp. 1-19 . Martimort, David and Lars Stole (2002), “The Revelation and Delegation Principles in Common Agency Games,” Econometrica, Vol. 70, No. 4, pp. 1659-1673. Martimort, David and Lars Stole (2003), “Contractual Externalities and Common Agency Equilibria,” Advances in Theoretical Economics, Vol. 3: Article 4. McAfee, Preston and Marius Schwartz (1994), “Opportunism in Multilateral Vertical Contracting: Nondiscrimination, Exclusivity, and Uniformity,” American Economic Review, Vol. 84, pp. 210-230. Marx, Leslie and Greg Shaffer (2007), “Upfront Payments and Exclusion in Downstream Markets,” RAND Journal of Economics, Vol. 38, pp. 823843. Mikl´ os-Thal, Jeanine, Rey, Patrick and Thibaud Verg´ e (2010), “Vertical Relations,” International Journal of Industrial Organization, Vol. 28, pp. 345-349. Mikl´ os-Thal, Jeanine, Rey, Patrick and Thibaud Verg´ e (2011), “Buyer Power and Intrabrand Coordination,” Journal of the European Economic Association, Vol. 9 No. 4, pp721-741. Nocke, Volker and Patrick Rey (2014), “Exclusive Dealing and Vertical Integration in Interlocking Relationships,” working paper, available at http://idei.fr/vitae.php?i=50.

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O’Brien, Daniel (2014), “All-units Discounts and Double Moral Hazard,” FTC Working Paper No. 316. O’Brien, Daniel and Greg Shaffer (1992), “Vertical Control with Bilateral Contracts,” RAND Journal of Economics, Vol. 23 (Autumn), pp. 299-308. O’Brien, Daniel and Greg Shaffer (1997), “Nonlinear Supply Contracts, Exclusive Dealing, and Equilibrium Market Foreclosure,” Journal of Economics and Management Strategy, Vol. 6 No. 4, pp. 755-785. Prat, Andrea and Aldo Rustichini (2003), “Games Played Through Agents,” Econometrica, Vol. 71 No. 4, pp. 989-1026. Rey, Patrick and Thibaud Verg´ e (2004), “Bilateral Control with Vertical Contracts, ” RAND Journal of Economics, Vol. 35, pp. 728-746. Rey, Patrick and Thibaud Verg´ e (2010), “Resale Price Maintenance and Interlocking Relationships,” Journal of Industrial Economics, 58(4) December, pp. 928-961. Rey, Patrick and Michael Whinston (2013), “Does Retailer Power Lead to Exclusion?” RAND Journal of Economics, Vol. 44, No. 1, pp 75–81. Scott-Morton, Fiona (2012), “Contracts that Reference Rivals,” Remarks of the DAAG for Economic Analysis, U.S. DOJ; http://www.justice.gov/atr/public/speeches/281965.pdf. Segal, Ilya (1999), “Contracting with Externalities,” Quarterly Journal of Economics, Vol. CXIV No. 2, pp.337-388. Segal, Ilya and Michael Whinston (2003), “Robust Predictions for Bilateral Contracting with Externalities,” Econometrica, Vol. 71, No. 3, May, pp. 757-791. Shaffer, Greg (1991), “Slotting Allowances and Resale Price Maintenance: A Comparison of Facilitating Practices,” RAND Journal of Economics, Vol. 22 (Spring), pp. 120-135.

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APPENDIX A Omitted proofs Proof of Lemma 1: Consider the first-order conditions of retailer r with respect to q1r and q2r as given in (2) and totally differentiate both conditions with respect to wsr and the four quantities q1r , q2r , q1r0 , and q2r0 . Since in the symmetric equilibria studied in the paper each supplier s finds it optimal to offer the same wholesale prices wsr = wsr0 = ws to both retailers, when studying the responses of the downstream equilibrium quantities to changes in wholesale prices one can limit attention to dwsr = dwsr0 = dws and thus dqsr = dqsr0 = dqs for s = 1, 2. Evaluating the two equations obtained from total differentiation of the first-order conditions around qs = qs0 = q, one obtains −dws + Adqs + Bdqs0 = 0 (A-1) Bdqs + Adqs0 = 0 where A and B are given in Assumption (2). Solving the system in (A-1) yields the following comparative statics dqs A = 2 , dws A − B2

dqs0 B =− 2 dws A − B2

(A-2)

Assumption (2) then implies the result.  Proof of Proposition 1: The proof can be simplified by noting that, when evaluated at ws0 r = ws0 r0 = wt , the profit function of suppliers s, πs (wsr , wsr0 , wt , wt ), is symmetric in wsr and wsr0 . This implies that supplier s maximizes its profits by charging wsr = wsr0 = ws and that the solution of the problem in (11) can be found by solving a single first-order condition with respect to ws . To see why this is the case, set wsr = wsr0 = ws in πs (wsr , wsr0 , wt , wt ) and take the first-order condition of πs (ws , ws , wt , wt ) with respect to ws , which yields ∂1 πs (ws , ws , wt , wt )+∂2 πs (ws , ws , wt , wt ) = 0. Since symmetry of πs implies that ∂1 πs (ws , ws , wt , wt ) = ∂2 πs (ws , ws , wt , wt ) at wsr = wsr0 = ws , the first-order condition corresponds to 2∂1 πs (ws , ws , wt , wt ) = 2∂2 πs (ws , ws , wt , wt ) = 0 . The latter is equivalent to the first-order conditions ∂1 πs (ws , ws , wt , wt ) = ∂2 πs (ws , ws , wt , wt ) = 0 obtained by first partially differentiating πs (wsr , wsr0 , wt , wt ) with respect to wsr or wsr0 and then setting wsr = wsr0 = ws . 41

With this observation in mind, for ws0 r = ws0 r0 = wt , one can write the first-order condition of supplier s in (12) as X dΠr (wt )
dqs0 dπs (wt ) = − 2 wt − c = 0. dws dws dws r=1,2

(A-3)

Given the expression for Πr resulting from (7), the first part of (A-3) can be written as X dΠr (wt ) dqs = 2 [(ps − c) + ∂1 P qs + ∂2 P qs0 + ∂3 P qs + ∂4 P qs0 ] + dws dws r=1,2

(A-4)

dqs0 . 2 [(ps0 − c) + ∂1 P qs0 + ∂2 P qs + ∂3 P qs0 + ∂4 P qs ] dws Noting that (ps − c) = (ps − ws ) + (ws − c) and (ps0 − c) = (ps0 − ws0 ) + (ws0 − c) and that the first order conditions of retailers with respect to qs and qs0 imply, respectively, (ps − ws ) + ∂1 P qs + ∂2 P qs0 = 0 and (ps0 − ws0 ) + ∂1 P qs0 + ∂2 P qs = 0, (A-4) can be written as X dΠr (wt ) dqs = [(ws − c) + ∂3 P qs + ∂4 P qs0 ] + dw dw s s r=1,2

[(ws0 − c) + ∂3 P qs0 + ∂4 P qs ]

(A-5) dq . dws s0

Around a symmetric equilibrium with ws = ws0 = wt and qs = qs0 = q t condition (A-5) can be further simplified to read   X dΠr (wt ) 
 dqs dqs0 t t = w − c + (∂3 P + ∂4 P ) q + , dws dws dws r=1,2

(A-6)

which allows one to re-write (A-3) as  
 dqs
dqs0 dπs (wt )  t dqs0 t = w − c + (∂3 P + ∂4 P ) q + = 0, − 2 wt − c dws dws dws dws

(A-30 )

Since, from Lemma 1, (dqs /dws + dqs0 /dws ) < 0 and dqs0 /dws > 0, one has dπs (wt ) /dws > 0 for wt ≤ c. This implies that in any equilibrium it must be wt > c. Consider next the case in P which wt ≥ wm . Since r=1,2 Πr is single-peaked in ws and reaches a maximum at wm , the first term in (A-3) (effect on industry profits) is non-positive for wt ≥ wm , whereas the second term (competitive externality) is strictly positive, which implies that dπs (wt ) /dws < 0 for wt ≥ wm . These two facts imply c < wt < wm .  Proof of Proposition 2: I first establish that each supplier is indifferent between implementing a given profile of

42

downstream quantities by using a two-part tariff contract or a contract with minimum volume requirements. I then show that, starting from any candidate symmetric DCA equilibrium with minimum volume requirements and wv ∈ [wt , wm ], there exists no profitable deviation to alternative contracts that induce DCA. For given ws0 r and ws0 r0 , consider the first-order conditions for qs0 r and qs0 r0 implied by (2). These first-order conditions depend on qsr and qsr0 but not on wsr and wsr0 . Therefore, the retailers’ choices of qs0 r and qs0 r0 are independent of whether supplier s induces retailers to choose the quantities qsr and qsr0 by offering two-part tariff contracts with appropriately chosen wsr and wsr0 or minimum quantity forcing contracts. Moreover, for given ws0 r and ws0 r0 , supplier s is also indifferent between inducing the quantities qsr and qsr0 with two-part tariffs or by forcing such quantities on the retailer through minimum volume requirements. These two facts imply that, for given ws0 r and ws0 r0 , supplier s obtains the same profits by implementing the four quantities qsr , qsr0 , qs0 r and qs0 r0 through two-part tariffs or minimum volume requirements. Given a profit-maximizing choice of wsr and wsr0 , offering contracts with minimum quantity forcing restraints that are just binding leaves, nevertheless, the supplier’s profits unchanged and is thus a (weak) best response. In what follows I characterize the wholesale prices that can be supported as equilibria when suppliers adopt such (weak) best responses. Consider a candidate symmetric DCA equilibrium with minimum volume requirements specifying ws = wv for qs ≥ q v (and a prohibitive ws for the out-of-equilibrium case in which ≥ q v ), where wv ∈ [wt , wm ] and q v ∈ [q m , q t ]. Starting from this candidate equilibrium, consider first a deviation to a contract specifying ws < wv and q s > q v . By Lemma 1, absent the minimum quantity forcing restraint qs0 ≥ q v imposed by supplier s0 , such a deviation would give retailers incentives to reduce qs0 . Given the minimum volume requirement qs0 ≥ q v , howP ever, qs0 remains unaffected, which, by (A-3), implies that dπs /dws |ws =wv− = r=1,2 dΠr /dws . P Since r=1,2 Πr is maximized at wm and everywhere concave, dπs /dws |ws =wv− > 0 at any ws < wv ≤ wm . This establishes that, starting from any wv ≤ wm , lowering ws below wv is not a profitable deviation for s. Consider next a deviation to a contract with ws > wv and qs ≥ q s , where q s < q v . By Lemma 1, this deviation gives retailers incentives to increase qs0 , and thus makes the minimum volume requirements imposed by supplier s0 non-binding, so that dπs (wv ) /dws to the right of wv is the same as in (A-3). Given that πs is concave and 43

maximized at wt ≤ wv , this implies dπs (wv ) /dws < 0 in an interval around wv , and thus that raising ws above wv is not a profitable deviation. Taken together, these results imply that, when a deviation to exclusivity is not profitable, any wv ∈ [wt , wm ] can be supported as a symmetric DCA equilibrium with minimum volume requirements. To complete the proof, one needs to show that neither wv < wt nor wv > wm can be symmetric DCA equilibria. When wv < wt , there always exists a profitable deviation to contracts with ws > wv which would preserve DCA and increase πs (since πs is maximized at wt > wv ). When instead wv > wm , there always exists a profitable deviation to a ws < wv , since such a deviation P would not affect qs0 whenever qs0 > 0 and would increase total industry profits, r=1,2 Πr , thus increasing πs .  Proof of Lemma 2: Given wholesale prices wsr and ws0 r and the quantities qsr0 and qs0 r0 sold by retailer r0 , when retailer r faces the constraint qsr = qs0 r it chooses qsr according to the following first order condition (with an analogous first order condition applying to the choice of qs0 r ) 

(psr − wsr ) + (ps0 r − ws0 r ) +

∂psr ∂psr + ∂qsr ∂qs0 r





qsr +

∂ps0 r ∂ps0 r + ∂qsr ∂qs0 r



qs0 r = 0.

(A-7)

Around a symmetric equilibrium in which wsr = wsr0 = ws and ws0 r = ws0 r0 = ws0 both retailers choose the same quantity of each product, i.e. qsr = qsr0 = qs and qs0 r = qs0 r0 = qs0 . Moreover, the constraints that reference rivals imply that qs = qs0 , and thus that the two terms with partial derivatives within parentheses in (A-7) are the same and equal to (∂1 P + ∂2 P ). The first order condition in (A-7) can thus be written as (ps − ws ) + (ps0 − ws0 ) + (qs + qs0 ) (∂1 P + ∂2 P ) = 0.

(A-8)

Totally differentiating (A-8) with respect to ws and the four quantities, evaluating the result at the symmetric quantities q, and rearranging one obtains dqs dqs0 1 = = <0 dws dws 2 (A + B)

(A-9)

where A and B are defined in (4) and (5) and the sign follows from Assumption 6. Proof of Lemma 3: Consider first a DCA contract Cbsdca offered by supplier s to both retailers. Assume that this contract does not include any binding restraints that reference rivals and specifies a wholesale 44

bs that maximizes πs , given the contract Csdca price w offered by supplier s0 to both retailers, 0 bs , ws0 ) /∂ws = 0. Such a contract is always dominated by a nonexclusive contract i.e. ∂πs (w

that references rivals, Cesdca , with market-share restraints that are just binding (i.e. marketshare restraints that induce the same relative quantities qbs0 /qbs as those that prevail under the es > w bs . To see this note that a restraint that is just contract Cbsdca ) and a wholesale price w bs , ws0 ) unchanged but implies ∂πs (w bs , ws0 ) /∂ws |ws =wbs+ > 0 binding leaves the value of πs (w bs . This follows from the fact that CRRs cause the sign of for small increases in ws above w

dqs0 /dws in (A-30 ) to switch from positive to negative, as can be seen by comparing Lemmas es > w bs such that πs (w es , ws0 ) > πs (w bs , ws0 ) 1 and 2, and implies that there always exists a w

and thus that the adoption of non-exclusive CRRs is a dominant strategy for both suppliers. 

Proof of Proposition 4: Starting from any candidate symmetric DCA equilibrium with wcrr,v ∈ [wm , wcrr ], the share requirement qs ≥ qs0 , and the minimum volume requirement qs ≥ q crr,v , where q crr,v ∈ [q crr , q m ], consider first a deviation by supplier s to a different DCA contract with ws > wcrr,v , the same share requirement qs ≥ qs0 , and a lower minimum volume requirement q˜crr,v < q crr,v . Given the minimum volume requirements imposed by supplier s0 such a deviation is unprofitable, as it does not affect qs0 r or qs0 r0 , and reduces total industry profits, since wcrr,v > wm and industry profits are concave. Consider next a deviation by supplier s to a contract with ws < wcrr,v , the same share requirement qs ≥ qs0 , and a higher minimum volume requirement q˜crr,v > q crr,v . In the presence of the share requirement imposed by supplier s0 , such a deviation would induce the retailer to increase sales of both products, rendering the minimum volume requirement of supplier s0 non-binding. This makes the problem equivalent to that with CRRs but without minimum volume requirements studied in Proposition 3. Since πs is maximized at wcrr > wcrr,v , lowering ws below wcrr,v would reduce πs and thus be unprofitable. This establishes that any symmetric DCA equilibrium with wcrr,v ∈ [wm , wcrr ] is immune to deviations to other contracts that would still induce DCA. One still needs to prove that neither wcrr,v < wm nor wcrr,v > wcrr can be symmetric DCA equilibria. If wcrr,v < wm , there would always exist a profitable deviation to contracts with ws > wcrr,v . Such a deviation would make the minimum volume requirement imposed by supplier s0 nonbinding and increase πs since, with CRRs, πs is maximized at wcrr > wm > wcrr,v . If instead 45

wcrr,v > wcrr > wm , there would always exist a profitable deviation to some ws < wcrr,v , since such a deviation would not affect qs0 whenever qs0 > 0 and would increase total industry profits thus increasing πs .  Proof of Proposition 5 If supplier s offers wholesale prices ws1 = ws2 = wm and imposes minimum volume requirements qs1 ≥ q m and qs2 ≥ q m in stage 1(a), it is a best response for supplier s0 to also offer ws0 1 = ws0 2 = wm and implement qs0 1 = qs0 2 = q m in stage 1(b). This can be proven as follows. For ws1 = ws2 = wm and symmetric demand, the profit-maximization problem of supplier s0 in (16) is symmetric, implying that a best response must have ws0 1 = ws0 2 . Assume then, by contradiction, that the best response of supplier s0 is ws0 1 = ws0 2 < wm . This implies that the minimum volume requirements imposed by s are binding, which makes the second term in P the objective function in (16) equal to the constant r=1,2 (wm − c) q m and induces supplier P P s0 to acts as if maximizing total industry profits r=1,2 Π (w, R). Since r=1,2 Π (w, R) is maximized when all wholesale prices are equal to wm , this contradicts the assumption that ws0 1 = ws0 2 < wm is a best response. Assume next, always by contradiction, that supplier s0 chooses ws0 1 = ws0 2 > wm . This implies that the minimum volume requirements imposed by s are not binding. A reduction in ws0 1 and/or ws0 2 towards wm increases the profits of supplier P P s0 in (16) by increasing r=1,2 Π (w, R) and reducing r=1,2 (wm − c) qsr , which contradicts the assumption that ws0 1 = ws0 2 > wm is a best response. Finally, if supplier s deviates from offering wholesale prices equal to wm or from imposing minimum volume requirements equal to q m in stage 1(a), supplier s0 will not act to maximize industry profits in stage 1(b). Since the objective function of supplier s is increasing in industry profits, such deviations in stage 1(a) would not be profitable for supplier s. Proof of Proposition 6 Assume that in stage 1(a) supplier s offers DCA contracts with ws1 and ws2 and does not impose strictly binding vertical restraints (I show that the latter is optimal further below). In stage 1(b) supplier s0 can implement any profile of quantities it desires by imposing binding share requirements and adjusting its wholesale prices. To see this, note that when the share requirements imposed by s0 are binding (which I will show below to be optimal for s0 ) one

46

has qsr = φs0 r qs0 r with φs0 r ≡ (1 − σs0 r ) /σs0 r and the profits of retailer r can be written as πr = [φs0 r (psr − wsr ) + (ps0 r − ws0 r )] qs0 r

(A-10)

Given φs0 r and wholesale prices, each retailer r solves the following single first-order condition with respect to qs0 r 

 ∂psr ∂psr + φ s0 r φs0 r qs0 r + φs0 r (psr − wsr ) + ∂qs0 r ∂qsr   ∂ps0 r ∂ps0 r (ps0 r − ws0 r ) + + φs0 r qs0 r = 0. ∂qs0 r ∂qsr

(A-11)

Given the two first-order conditions in (A-11) for r = 1, 2, supplier s0 can induce any value of qs0 1 and qs0 2 by appropriately choosing ws0 1 and ws0 2 , and any value of qs1 and qs2 by appropriately choosing φs0 1 and φs0 2 . Anticipating this, in stage 1(a) supplier s sets ws1 = ws2 = c and no binding share requirements, so that the problem (16) faced by s0 in stage 1(b) corresponds to the maximization of industry profits, which is the same problem in (17) faced by s in stage 1(a). In other words, s delegates to s0 the maximization of industry profits. Given that, with symmetric demand and costs, maximization of industry profits requires all quantities to be at the same level q m and that s0 can only induce this outcome by raising ws0 1 and ws0 2 above ws1 = ws2 = c while preserving the symmetry of quantities, s0 finds it optimal to impose binding and symmetric share requirements φs0 1 = φs0 2 = 1. This causes qsr = q and psr = p for all s and r and (A-10) becomes   c + w s0 r + (∂1 P + ∂2 P ) q = 0. p− 2

(A-12)

Condition (A-12) corresponds to the first-order condition with respect to the quantity of either product of a retailer facing no vertical restraints and two-part tariffs with wholesale price (c + ws0 r )/2 in a fully symmetric outcome (see (2)). By definition, the wholesale price that implements q = q m in such a setting is wm . Therefore, in order to induce q = q m supplier s0 must charge ws0 r such that (c+ws0 r )/2 = wm , which yields ws0 r = wm +(wm − c) > wm , r = 1, 2. Uniqueness follows from the fact that any wsr 6= c would make the profit-maximization problem (16) solved by supplier s0 in stage 1(b) different from the profit maximization problem (17) solved by s and would thus yield lower profits for supplier s. Proof of Proposition 7: Given the equilibrium menus of contracts offered by supplier s0 , one can derive the conditions 47

under which supplier s finds it optimal to induce a DCA continuation equilibrium over other possible continuation equilibria in the following two steps: 1) calculate the highest profit that s can obtain in each of the different continuation equilibria that it can induce, and 2) derive the conditions under which a DCA continuation equilibrium yields the highest profits for s among all possible continuation equilibria. In relation to step 1), if supplier s wants to implement a given outcome k it must ensure that each retailer r earns at least as much in outcome k as in other outcomes when retailer r takes as given that retailer r0 accepts contracts also designed for outcome k. Supplier s thus maximizes the profits it obtains from implementing outcome k by charging each retailer i) wholesale prices that maximize the joint profits of the supplier and the two retailers under outcome k, and ii) fixed fees that leave each retailer indifferent between accepting the contract that implements outcome k or rejecting it and resorting to its best alternative. Part i) follows from the fact that supplier s is the residual claimant to joint profits, since the amount of profits that it must guarantee to the retailers to induce them to accept the contracts that implement outcome k, determined in ii), does not depend on the wholesale prices for that outcome. As for part ii), for the time being I ignore the issue of multiple continuation equilibria, which I address towards the end of the proof. One can calculate the maximum profits that supplier s can obtain under different outcomes as follows. ne DCA continuation equilibrium Assume that r takes as given that r0 accepts Csr 0 and   dca ne ne = Πdca − + tne,dca = Πdca − tne,dca Csne 0 r 0 . If r also accepts Csr and Cs0 r it earns πr sr s0 r
ne + min {α, Πmix1 tne,dca − Πdca − Πmix1 sr d2 } /2. If instead r rejected Csr its best alternative d2

would be to accept only Cse0 r , with which it would earn πrmix1 = Πmix1 − te,mix1 = d2 s0 r Πmix1 − min {α, Πmix1 d2 d2 }. The highest tariff supplier s can therefore charge while still
ne is tne,dca = Πdca − Πmix1 + min {α, Πmix1 inducing r to accept Csr sr d2 d2 } /2. This implies that the maximum profits that s can earn from both retailers in a DCA continuation equilibrium are  tne,dca + tne,dca = Πdca − Πmix1 + min α, Πmix1 . sr d2 d2 sr0

(A-13)

e Upstream monopoly continuation equilibrium Consider the deviation contracts Cˆsr e and Cˆsr 0 that implement a continuation equilibrium with upstream monopoly by s. e 0 Assume that r takes as given that r0 accepts Cˆsr 0 and rejects all contracts offered by s .

48

e If r accepts Cˆsr and rejects all contracts offered by s0 it earns πrum = Πum − tˆe,um sr . If ine stead r rejected Cˆsr its best alternative would be to accept Cse0 r , with which it would earn mix1 πrpe = Πpe −te,pe −min {α, Πmix1 d2 }−ε. The highest tariff supplier s can therefore s0 r = Πd2 e + min {α, Πmix1 = Πum − Πmix1 is tˆe,um charge while still inducing r to accept Cˆsr sr d2 d2 } + ε.

This implies that the maximum profits that s can earn from both retailers in an upstream monopoly continuation equilibrium are 
+ min α, Πmix1 +ε . + tˆe,um = 2 Πum − Πmix1 tˆe,um d2 d2 sr sr0

For ε → 0, inducing a DCA continuation equilibrium is a better response for s than inducing an upstream monopoly continuation equilibrium if and only if  mix1 Πdca ≥ 2Πum − Πmix1 + min α, Π . d2 d2

(A-14)

Note that (A-14) is more likely to hold when α is low. Since I assume that α ≥ 0, a necessary condition for (A-14) to hold, and for DCA to be an equilibrium, is therefore Πdca ≥ 2Πum − Πmix1 d2 , as reflected in condition (30). e and Pairwise exclusivity continuation equilibrium Consider deviation contracts Cˆsr j Cˆsr 0 that implement a continuation equilibrium with pairwise exclusivity with pairs j 0 sr and s0 r0 (Cˆsr 0 has prohibitive terms and is meant not to be taken by retailer r ).

Assume that r takes as given that r0 accepts Cse0 r0 and rejects all contracts offered by e s. If r accepts Cˆsr and rejects all contracts offered by s0 it earns πrpe = Πpe − tˆe,pe sr . If e instead r rejected Csr its best alternative would be to accept Cse0 r , with which it would

− min {α, Πmix1 earn πrum = Πum − te,um = Πmix1 d2 d2 } − ε. The highest tariff supplier s can s0 r e therefore charge while still inducing r to accept the deviation contract Cˆsr is

 pe mix1 tˆe,pe + min α, Πmix1 + ε, sr = Π − Πd2 d2

For ε → 0, inducing a DCA continuation equilibrium is a better response for s than inducing a pairwise exclusive continuation equilibrium if and only if Πdca ≥ Πpe .

(A-15)

Mixed continuation equilibrium in which s sells only to one retailer Consider devij ne ation contracts Cˆsr and Cˆsr 0 that implement a continuation equilibrium in which sup-

49

j plier s sells only to retailer r, while supplier s0 sells to both retailers (Cˆsr 0 has prohibitive

terms and is meant not to be taken by retailer r0 ). Assume that r takes as given that ne r0 accepts Cse0 r0 and rejects the contracts offered by s. If r accepts Cˆsr and Csne 0 r it earns  
ne,mix dca − tˆne,mix = Πmix2 − tˆne,mix − Πmix1 + ε . If instead πrmix2 = Πmix2 d1 d1 d2 sr\sr0 + ts0 r\sr0 sr\sr0 − Π ne its best alternative would be to accept Cse0 r , with which it would earn r rejected Cˆsr

− min {α, Πmix1 = Πmix1 πrum = Πum − te,um d2 } − ε. The highest tariff supplier s can d2 s0 r ne is therefore charge r while still inducing r to accept Csr,d1

 dca mix1 mix2 − Π + min α, Π . tˆne,mix = Π 0 d2 d1 sr\sr

(A-16)

Comparing (A-16) to (A-13) one can see that inducing a DCA continuation equilibrium is a better response for s than inducing a mixed continuation equilibrium in which it sells only to one retailer if and only if Πdca ≥

+ Πmix2 Πmix1 d1 d2 . 2

(A-17)

Mixed continuation equilibrium in which s sells to both retailers Consider deviation ne e and Cˆsr contracts Cˆsr 0 that implement a continuation equilibrium in which supplier s

induces retailer r to exclude supplier s0 , while retailer r0 sells the products of both suppliers.

Consider first retailer r and assume that it takes as given that r0 ac-

0 ne ne ˆe cepts Cˆsr 0 and Cs0 r 0 . If r accepts Csr and rejects the contracts offered by s it earns e its best alternative would be to accept . If instead r rejected Cˆsr − tˆe,mix πrmix1 = Πmix1 sr d2

− min {α, Πmix1 = Πmix1 − te,mix Cse0 r , with which it would earn πrmix1 = Πmix1 d2 }. The d2 d2 s0 r e highest tariff supplier s can therefore charge r while still inducing r to accept Cˆsr is 0 tˆe,mix = min {α, Πmix1 sr d2 }. Consider next retailer r and assume that it takes as given that e ne ne r accepts Cˆsr and rejects the contracts offered by s0 . If r0 accepts Cˆsr 0 and Cs0 r 0 it earns  
ne,mix mix2 ˆne,mix πrmix2 = Πmix2 − tˆne,mix − tsr0 \s0 r − Πdca − Πmix1 + min {α, Πmix1 0 d2 d2 d2 } . sr0 \s0 r + ts0 r0 \s0 r = Πd2 ne e If instead r0 rejected Cˆsr 0 its best alternative would be to accept Cs0 r 0 , with which it mix1 would earn πrpe0 = Πpe − te,pe − min {α, Πmix1 d2 } − ε. The highest tariff supplier s s0 r0 = Πd2 ne mix2 can therefore charge r0 while still inducing r0 to accept Cˆsr is tˆne,mix − Πdca + ε. sr0 \s0 r = Πd2

By implementing a deviation to a mixed continuation equilibrium in which it sells to both retailers and excludes supplier s0 from one retailer, supplier s can therefore earn

50

at most  ne,mix mix1 ˆ + t = min α, Π tˆe,mix + Πmix2 − Πdca + ε. 0 0 sr d2 d2 sr \s r

For ε → 0, inducing a DCA continuation equilibrium is a better response for s than inducing a mixed continuation equilibrium in which it sells to both retailers if and only if Π

dca

Πmix1 + Πmix2 d2 d2 ≥ . 2

(A-18)

mix2 Since Πmix2 ≥ Πmix2 ≥ Πmix2 d1 d2 , (A-18) always holds when (A-17) does. To see why Πd1 d2 , mix es1,d1 compare (27) and (29), taking into account that w ≥ wmin and that Πmix2 is mix mix es1,d1 es1,d1 = wmin . at w symmetric and increasing in w

For α ∈ [0, α ¯ ] and ε → 0, all of the above implies that offering contracts that implement a DCA outcome is a best response for supplier s if and only if (30) holds. Regarding the multiplicity of continuation equilibria, note that, given the tariffs in Proposition 7, any of the possible market structures (dca, um, pe, mix) is a Nash equilibrium of the contract acceptance game played by retailers in stage 2. However, it is straightforward to check that, for ε > 0, the DCA equilibrium is the only coalition-proof Nash equilibrium of that game. It is also straightforward to check that, for α ∈ [0, α ¯ ] and ε → 0, the equilibrium tariffs offered by suppliers for outcomes that involve common agency for at least one retailer ne,mix are truthful relative to the DCA outcome, since tne,dca + tne,dca = te,mix + tne,mix sr sr sr0 sr0 \s0 r = tsr\sr0 in

Proposition 7. Finally, the level of tariffs in Pareto-undominated equilibria from the point of view of suppliers can be found as follows. Since the profits of each supplier s in a DCA equilibrium are mix1 πsdca = Πdca −Πmix1 α, Πmix1 d2 +min {α, Πd2 }, any DCA equilibrium with α < min {¯ d2 } is Pareto

dominated by another DCA equilibrium with α0 ∈ (α, α ¯ ]. If 2Πum ≥ Πdca one has α ¯ ≤ Πmix1 d2 , and thus the unique Pareto-undominated equilibrium has α = α ¯ . If instead 2Πum < Πdca one mix1 has α ¯ > Πmix1 ¯ ] is Pareto undominated.  d2 , thus any equilibrium with α ∈ [Πd2 , α

51

APPENDIX B (Not necessarily for publication or for online publication)

Price competition in the downstream market In this appendix I show that the results obtained in the main text for the case of quantity competition hold also in the case of price competition. Let qsr = D(psr , ps0 r , psr0 , ps0 r0 ) denote the direct demand for product s at retailer r obtained by inverting the inverse demand system in (13) and assume that D (·) satisfies the following regularity conditions, which parallel the conditions presented in Assumption 2 for the case of quantity competition. Assumption 3 (Direct demand) An increase in psr reduces total demand for product s and increases total demand for product s0 (i.e., ∂1 D + ∂3 D < 0 and ∂2 D + ∂4 D > 0); an increase in psr reduces total demand at retailer r and increases total demand at retailer r0 (i.e., ∂1 D + ∂2 D < 0 and ∂3 D + ∂4 D > 0); and an increase in all prices reduces the demand P4 for any product at any retailer (i.e. i=1 ∂i D < 0). Moreover, for any symmetric level of quantities qsr = q, s, r = 1, 2, the direct demand system satisfies the following conditions |X| > |Y |

X < 0,

X∂2 D − Y ∂1 D ∂2 D + ∂4 D < − , X∂1 D − Y ∂2 D ∂1 D + ∂3 D

(B-1) (B-2)

where X ≡ 2∂1 D + ∂3 D −

2 2 2 2 ∂11 D + ∂13 D + ∂22 D + ∂24 D q ∂1 D + ∂2 D

2 2 2 ∂ 2 D + ∂14 D + ∂21 D + ∂23 D Y ≡ 2∂2 D + ∂4 D − 12 q ∂1 D + ∂2 D

(B-3)

When these conditions hold, one has that dqs /dws < 0 and dqs0 /dws > 0, as in Lemma 1, and wt > c, as in Proposition 1, also in the case of downstream price competition. Since these properties of the equilibrium constitute the cornerstones of all the results in the paper, it is straightforward to establish the following conclusion. Proposition B.1 If Assumption 3 holds, the results obtained in Sections 3 through 7 for the case of downstream quantity competition hold also for the case of downstream price competition. B-1

Proof: See the end of this appendix. 

For a concrete comparison of the properties of equilibria under price and quantity competition, one can resort again to the linear demand system introduced in Section 3.3. Given the consumer preferences represented by the quadratic utility function in (14), one can invert the inverse demand system in (13) and obtain the following direct demand function for product s at retailer r. qsr = ρ [(1 − a)(1 − b)v − psr + aps0 r + bpsr0 − abps0 r0 ] ,

(B-4)

where ρ ≡ 1/ [(1 − a2 )(1 − b2 )] ≥ 1.31 It can be verified that this direct demand specification satisfies the regularity conditions in Assumption 3.32 Figure 1 shows equilibrium prices for the case of two-part tariffs contracts. Just as in the case of quantity competition, wholesale and retail prices fall as product substitutability, a, increase; while wholesale prices increase and retail prices decrease as retailer substitutability, b, increases. Contrary to the case of quantity competition mark ups vanish, i.e. pt,b = wt,b , when retailers are perfect substitutes, i.e. b = 1. Note that, consistent with price competition being generally more intense than quantity competition, pt,b is lower and falls more rapidly than pt,c as a and b increase. Figure 2 shows the equilibrium value of the same variables for the case of CRRs. Note that, contrary to the case of two-part tariffs discussed above, with CRRs equilibrium retail prices are higher under price than under quantity competition. This is due to the fact 31

When one of the products, say product ij, is not available or not consumed in positive quantities, the demand for the other three products must be adjusted by setting pij at that virtual level peij for which qij = 0. 32 Note that in a DCA outcome in which both products are offered by both retailers qsr depends negatively on ps0 r0 . This is due to significant diversion from product s0 r0 towards products s0 r and sr0 in response to an increase in ps0 r0 , which crowds out sales of product sr. This notwithstanding, it is still the case that P ∂Πr /∂ps0 r0 > 0 and ∂Πs /∂ps0 r0 > 0, where Πs ≡ r=1,2 (psr − c) qsr and the game analyzed in this paper is thus still well defined. Other authors, such as Rey and Verg´e (2010), specify direct demand as qsr = v −psr +aps0 r +bpsr0 +abps0 r0 . Such an ad-hoc demand specification poses, however, a number of issues. First, it is not clear whether this direct demand specification has been derived from a utility function representing reasonable consumer preferences. In fact, by inverting it, one can see that the resulting inverse demand, and thus consumer marginal utility, for product sr is increasing qs0 r0 , which suggests complementarity, not substitutability, between the two products in the underlying consumer preferences. Second, and related to the previous issue, in order for the demand specification in Rey and Verg´e (2010) to be well behaved one needs to impose ad-hoc restrictions on the parameters a and b (e.g., a + b + ab < 1) to ensure that demand for a given product decreases when all prices increase. These restrictions reduce the range of a and b over which one can conduct comparative statics and make it, for example, impossible to analyze the limit case of perfectly substitutable suppliers (a = 1) and/or retailers (b = 1). This is instead possible with a specification that is explicitly derived from a utility function, like the one that I adopt in this paper.

B-2

0.6

0.6

pm

pm

0.5

0.5

pt,c pt,b

0.3

0.2

pt,b

0.3

0.2

wt,b

0.1

0.0 0.0

pt,c

0.4

p ,w

p ,w

0.4

0.2

a Hproduct substitutability L 0.4

0.6

0.8

wt,b

0.1

0.0 0.0

1.0

0.2

b Hretailer substitutability L 0.4

0.6

0.8

1.0

Figure 5: Equilibrium with two-part tariffs (price competition).

that, even absent CRRs, wholesale prices are higher under price competition than under quantity competition, because the need to soften downstream competition is greater, and the resulting double marginalization costs lower, under the former than under the latter. With CRRs, higher upstream margins imply greater incentives for suppliers to impose negative externalities on each other by increasing wholesale prices further, ultimately leading to higher retail prices. 0.7

0.7

p

pcrr,b

crr,b

0.6

0.6

p

pcrr,c 0.5

pm

0.4

w

p ,w

p ,w

0.5

crr,c

crr,b

0.3

0.4

wcrr,b 0.3

wm,b

0.2

wm,b

0.2

0.1

0.0 0.0

pm

0.1

0.2

a Hproduct substitutability L 0.4

0.6

0.8

0.0 0.0

1.0

0.2

b Hretailer substitutability L 0.4

0.6

0.8

1.0

Figure 6: Equilibrium with simultaneous CRRs (price competition).

Figure 7 shows that, under price competition, an equilibrium with CRRs yields higher industry profits than one with two-part tariffs for a broader range of values of a and b than B-3

under quantity competition. 1.0

Quantity competition

b HRetailer substitutability L

0.8

P t >P crr

0.6

P crr >P t

0.4

Price competition 0.2

0.0 0.0

0.2

a HProduct substitutability L 0.4

0.6

0.8

1.0

Figure 7: Industry profits with simultaneous CRRs (Πcrr ) and two-part tariffs (Πt ) (price competition).

The reason for this is that higher wholesale prices increase industry profits more under price competition than under quantity competition, because the former is more intense than the latter and double marginalization less of a problem in the former than in the latter. Since an equilibrium with CRRs has higher wholesale prices than one with two-part tariffs, the former tends to be relatively more profitable than the latter when downstream competition is in prices instead of quantities. 1.0

1.0

Quantity competition

Quantity competition 0.8

b HRetailer substitutability L

b HRetailer substitutability L

0.8

Price competition 0.6

0.4

DCA equilibrium with two-part tariffs exists

0.2

0.0 0.0

0.6

Price competition 0.4

DCA equilibrium with simultaneous CRRs and no min. vol. req. exists 0.2

0.2

a HProduct substitutability L 0.4

0.6

0.8

0.0 0.0

1.0

0.2

a HProduct substitutability L 0.4

0.6

0.8

Figure 8: Existence of DCA equilibria with two-part tariffs and CRRs (price competition).

B-4

1.0

Regarding the existence of DCA equilibria, as shown in Figure 8, such equilibria exist for a narrower range of values of a and b under price competition than under quantity competition for both the case of two-part tariffs and CRRs. This is so because DCA equilibria are less profitable in the former than in the latter case.

Proof of Proposition B.1 Since all the results in the main text of the paper depend on the mode of downstream competition only through the properties of derived demand and upstream margins implied by Lemma 1 and Proposition 1, it is sufficient to prove that these properties hold also in the case of downstream price competition. Lemma 1 with downstream price competition Consider the first-order condition of a price-setting retailer r with respect to psr when suppliers use two-part tariff contracts without vertical restraints. qsr + (psr − wsr )

∂qsr ∂qs0 r + (ps0 r − ws0 r ) = 0. ∂psr ∂ps0 r

(B-5)

Totally differentiate the two first-order conditions for psr and ps0 r implied by (B-5) with respect to wsr , ws0 r , psr , ps0 r , psr0 and ps0 r0 and, for a reasoning analogous to that explained in the proof of Lemma 1, evaluate the two resulting equations at wsr = wsr0 = ws , psr = psr0 = ps , dwsr = dwsr0 = dws and dpsr = dpsr0 = dps , s = 1, 2, to obtain −∂1 Ddws − ∂2 Ddws0 + Xdps + Y dps0 = 0 (B-6) −∂1 Ddws0 − ∂2 Ddws + Y dps + Xdps0 = 0 where X and Y are defined in (B-3). Solving (B-6) yields X∂1 D − Y ∂2 D dps = , dws X2 − Y 2

dps0 X∂2 D − Y ∂1 D = dws X2 − Y 2

(B-7)

Given the conditions imposed on X and Y by (B-1) it is straightforward to prove that dps dps0 dps dps0 > and + >0 dws dws dws dws

(B-8)

Moreover, differentiating the demand functions for qsr and qs0 r with respect to all four retail prices, imposing the same symmetry assumptions as above, dividing throughout for dws , and

B-5

using the results in (B-7) one obtains dqs (∂1 D + ∂3 D) (X∂1 D − Y ∂2 D) + (∂2 D + ∂4 D) (X∂2 D − Y ∂1 D) = dws X2 − Y 2 (∂1 D + ∂3 D) (X∂2 D − Y ∂1 D) + (∂2 D + ∂4 D) (X∂1 D − Y ∂2 D) dqs0 = dws X2 − Y 2

(B-9)

Using (B-9), it is straightforward to verify that the results in Lemma 1 hold if the conditions in Assumption 3 hold. Proposition 1 with downstream price competition The proof follows the same logic as the proof of Proposition 1. In particular, dπs (wt ) /dws is still given by (A-3) and, following steps that are similar to (A-4) and (A-5), one can prove that  i  dp X Πr (wt ) h

X4 dps0 s t t t = p − w (∂3 D + ∂4 D) + w − c ∂i D + i=1 dws dws dws r=1,2

Given (B-8), and since (∂3 D + ∂4 D) > 0 and

P4

i=1

(B-10)

∂i D < 0, (B-10) is strictly positive for

wt ≤ c. Having established this fact, the rest of the proof proceeds as the proof of Proposition 1. 

B-6