Tying and entry deterrence in vertically differentiated markets∗ ´c ˇ† Eugen Kova (University of Bonn and CERGE-EI)

November 2007

Abstract This paper analyzes tying and bundling as entry deterrence tools. It shows that a multi-product firm can defend its monopoly position in one market via mixed bundling even without having a dominant position in another market. Such a strategy, however, leads to welfare losses and cannot be prevented by cooperation or a merger among rivals. This is shown in a model with two complementary goods. Each of the goods is vertically differentiated and consumers’ preferences for them are positively correlated. In addition, some implications for competition policy are discussed. Keywords: industrial organization, vertical differentiation, anti-trust policy, entry deterrence, foreclosure, tying, bundling JEL classification: L11, L12, L13, L41

∗

This paper is based on my dissertation submitted at CERGE-EI. I would like to thank Avner Shaked, Dirk Engelmann, and Kresimir Zigic for consultations and valuable suggestions and Anette Boom, Ekaterina Goldfain, Patrick Rey, Frank Riedel, Robert Schmidt, Jakub Steiner, and Jean Tirole for helpful comments. All errors are mine. † Address: Department of Economics, University of Bonn, Adenauerallee 24–26, 53113 Bonn, Germany; URL: www.uni-bonn.de/~kovac; e-mail: [email protected].

1

Introduction

Tying refers to the practice where a firm makes the purchase of one of its products conditional on the purchase of another of its products. According to leverage theory, tying “provides a mechanism whereby a firm with monopoly power in one market can use the leverage provided by this power to foreclose sales in, and thereby monopolize, a second market” (Whinston 1990). Therefore, tying is one of the key concepts in antitrust laws and policies dealing with monopolization.1 The most prominent example is probably Microsoft’s tying of Internet Explorer to the operating system Windows. In this case, by virtue of having a strongly dominant position in the operating system market,2 Microsoft has been found guilty of leveraging this market power to foreclose Netscape’s sales in the web browser market. Early anti-trust cases involving monopolization by tying required a proof of the monopoly power in the first market. However, following the argument by Posner (1976), “. . . how could a tie-in be imposed unless such power existed?” (p. 176), such proof was later omitted. Recent theoretical literature has already established that Posner’s argument does not hold in general and tying can indeed be profitable without monopoly power (Nalebuff 2000, Denicolo 2000, Kov´aˇc 2004).3 This paper takes a further step and provides a theoretical example in which a multi-product firm without being dominant in the first market uses tying to deter entry in the second market. I consider firms’ products to be differentiated vertically and assume that the multi-product firm has a weaker position (it produces a low-quality product) in the first market. The use of tying to deter entry in such setting contrasts strongly with the current understanding of tying as an entry deterrence tool. The theoretical literature on entry deterrence and foreclosure by tying falls in line with the argument by Posner (1976). In his seminal paper, Whinston (1990) introduces a theoretical model supporting the leverage hypothesis. The author examines the implications of tying in a model with independent goods, where a multi-product firm has monopoly power in one market and competes in price with a rival on another market. He argues that tying may lead to the foreclosure of the rival’s sales in the tied good market, provided the multi-product firm can credibly commit to the tying strategy. In that case tying is used as commitment device to fight entry. It is necessary to point out that both monopolistic position in the first market and the possibility of commitment are crucial assumptions for this result. Moreover, as Whinston (1990) concludes, tying is profitable for the monopolist precisely because of the “exclusionary effect on the market structure.” The main difference of the current paper from Whinston (1990) is that I do not require the multi-product firm to have 1

In the U.S. anti-trust laws, Section 2 of the Sherman Act (1890) declares monopolization as anti-competitive and Section 3 of the Clayton Act (1914) deals with tying contracts and exclusive dealing. 2 According to market researcher OneStat.com, in the year 2003, Windows operated more than 97% of personal computers. 3 In particular, in Kov´aˇc (2004), I consider two markets (for a horizontally differentiated and for a homogeneous product), and the multi-product firm is assumed to face an equal competitor on each market.

2

a monopoly on the first market. Moreover, I consider the case where it has a weaker position than its competitor and I show that commitment does not play an important role for my results. Protection of monopolized markets and entry deterrence are concepts closely related to foreclosure. Recently, they have been studied particularly with applications to the Microsoft case. Carlton and Waldman (2002) argue that a dominant firm can use bundling to remain dominant in an industry with rapid technological change.4 Applying the analysis to the Microsoft case, the authors claim that Microsoft’s tying and deterrence of Netscape’s entry into the market for Internet browsers could have increased social welfare. On the other hand, Choi and Stefanadis (2001) show that if an incumbent monopolist faces simultaneous entry in several markets, it may employ tying to discourage potential rivals from entry and innovation. Such action then has negative welfare effects. Nalebuff (2004) considers a monopolist in one market which also operates in another competitive market. Assuming that the consumption levels can be continuous, the author shows that the monopolist can extract an additional surplus via mixed bundling by lowering the monopoly price and increasing the unit price of the second good. Rey and Tirole (2005) provide an excellent survey of the literature on foreclosure and incorporate all surveyed models into one framework. However, they analyze foreclosure only in cases when a monopoly power in another market is present. The authors also discuss a critique of the leverage hypothesis made by the Chicago School (Director and Levi 1956, Posner 1976) who argue that for complementary products there is only one monopoly rent to extract, and hence, a monopolist has no incentives to monopolize a second market. Rey and Tirole (2005) conclude that the goods must be relatively independent so that the monopolist finds monopolization of the second market profitable. In a report for the Department of Trade and Industry of the UK, Nalebuff provides another survey of tying and bundling. The first part of the report (Nalebuff 2003a), surveys different motives for bundling and tying, analyzes their consequences, in particular, anti-competitive effects, and provides policy recommendations. The second part of the report (Nalebuff 2003b), applies the analysis to particular anti-trust cases. The whole report, however, considers monopolization and entry deterrence only in cases when the multi-product firm has monopoly power. Both surveys mentioned above suggest that very little is known about tying as an entry deterrence tool in cases, where the multi-product firm does not have monopoly power in the first market. The purpose of this paper is, therefore, to provide an example which shows that entry deterrence is also possible without having dominant position in the first market, when the goods are complements. In particular, I show that in this situation the multi-product firm can deter entry of inferior entrants in the second market via mixed bundling. In addition, when entry deterrence occurs in equilibrium, it leads to lower social welfare and consumer surplus. Moreover, entry 4

Bundling is a more general concept than tying and refers to a situation in which a package containing at least two different products is offered. The practice in which the firm offers only the bundle is called pure bundling. The practice in which the firm offers the bundle as well as some of the products separately is called mixed bundling. See also Table 2, p. 8.

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deterrence cannot be avoided by an entrant’s cooperation with the incumbent rival (in the first market). Besides the absence of dominant position, there are two other important aspects in which this paper differs from previous literature. First, the entry deterrence is achieved via mixed bundling, and not via pure bundling as in, for example, Whinston (1990). Second, the assumption of commitment is not crucial for my results, and entry deterrence can also occur when it is abandoned. The remainder of the paper is organized as follows. In Section 2, I describe the setup of the model and explain the basic intuition. In Sections 3, I analyze the multiproduct firm’s strategies when selling separate products, using pure bundling and mixed bundling. In Section 4, I investigate when the entry deterrence strategy is profitable for the multi-product firm, analyze its welfare consequences, show that entry deterrence cannot be prevented by cooperation among rivals, and discuss the commitment assumption. Section 5 concludes and discusses the relevance of my results for anti-trust policies. Appendix A contains proofs of all lemmas and propositions.

2

The model

2.1

Informal description and intuition

There are two markets for two indivisible complementary goods XD and XE , two incumbent firms M and D, and one potential entrant firm E. Firm M (for multiproduct) operates in both markets and firm D (for duopoly) operates only in the market for good XD . Firm E (for entrant) plans to enter the market for good XE . I assume that each of the goods is differentiated vertically and that consumers’ preferences for them are positively correlated. In this situation, there are four possible rankings of qualities. As the purpose of this paper is not to provide a general theory, but rather to present a possibility result, I will restrict my analysis to a specific ranking leading to entry deterrence. This ranking is specified by the following assumptions.5 Assumption 1. The multi-product firm M offers a low-quality product in the market for good XD and a high-quality product in the market for good XE (see Table 1). Assumption 2. Package containing multi-product firm’s goods XD and XE is superior to the package containing firm D’s good XD and firm E’s good XE . Under the above assumptions, I show that the multi-product firm M can protect its monopoly position in the second market (for good XE ) against low-quality entrants, provided the market is sufficiently narrow. By sufficiently narrow, I mean that in the absence of bundling, the market cannot accommodate more than two firms in equilibrium, as specified by Assumption 3 below.6 Entry deterrence is then achieved by mixed bundling when the multi-product firm offers the monopolistic good XE and the bundle. 5

I have also analyzed other ranking, but consistently with the intuition presented below, only this one leads to entry deterrence. Hence, the other cases are not analyzed in this paper. 6 Following Shaked and Sutton (1982), such markets are called natural oligopolies.

4

Assumption 3. The qualities on each market are such that the market is covered and both firms earn positive profits in the equilibrium of the no-bundling subgame.7 Note that the incentive for tying is different from what is known in the literature. Usually, tying is used either as a mechanism to leverage the market power (in order to deter entry) or as a tool for price discrimination between consumers with a high and low willingness to pay for the tied product. On the other hand, in this paper, although tying serves the purpose of deterring entry, it does not leverage any market power. Rather, it is used as a price discrimination tool between consumers with a high appreciation for quality and consumers with a low appreciation for quality. Quality First market (XD ) high incumbent firm (D) low multi-product firm (M )

Second market (XE ) multi-product firm (M ) potential entrant (E)

Table 1: Markets structure The intuition for entry deterrence is the following. By selling only its monopolistic good XE and the bundle consisting of both its goods, the multi-product firm M makes its low-quality good XD unavailable separately. When firm E enters the market, under Assumptions 1 and 2, the combinations available to consumers are: 1. firm D’s (incumbent specialist firm) good XD together with firm M ’s good XE , as the highest-quality combination; 2. the bundle by multi-product firm M ; 3. firm D’s good XD together with firm E’s (entrant) good XE , as the lowestquality combination. Thus, the entrant’s good, XE , can be purchased only in combination with the highquality good XD produced by firm D. This combination is inferior to the bundle (Assumption 2) and, hence, it is purchased only by consumers with a low appreciation for quality. However, if the market is sufficiently narrow (Assumption 3), there may be no consumers with sufficiently low appreciation for quality. In this case there is “not enough place” for all available combinations to have a positive market share. This will foreclose the entrant’s sales and will make entry unprofitable.

2.2

Formal description of the model

Consumers are characterized by their taste for quality (appreciation of quality) θ which is uniformly distributed on the interval [θ, θ]. I assume that consumers’ tastes for quality for both goods are positively correlated. In particular, I assume that each consumer has the same taste for quality for both goods. Moreover, consumers have 7

For better illustration, Assumption 3 is stated as a condition on equilibrium. Later it is formulated rigorously in terms of primitives of the model; see inequalities (4)–(6).

5

a positive marginal utility only from the first unit of both goods. Hence, they buy either one or zero units. Formally, I consider the following utility function (in reduced form):  θ(sd + se ) − p, if he buys goods XD , XE with qualities sd , se     by spending p, where d ∈ {D, M D}, e ∈ {E, M E}, Uθ = −p, if he buys only one good for price p,     0, if he does not buy.

(1)

The parameter θ can be interpreted as taste for quality for a package containing goods XD , XE with qualities sd , se . Both goods are assumed to have “equal weight” in the package and the quality of the package is simply the sum of qualities of the products it contains. The utility of −p when the consumer buys only one good means that the “direct” utility from consumption is zero. It reflects the complementarity of the goods.8 Note that whenever prices are positive, each consumer prefers not to buy at all to buying only one good. Remark 1. A more realistic approach would be, instead of characterizing consumers according to their taste for quality, to characterize them by their income, as in Shaked and Sutton (1982). Income is an economically measurable and clearly defined variable. However, Tirole (1992, pp. 96–97) shows that both approaches are equivalent. When consumers differ by income, the parameter 1/θ can be interpreted as the marginal rate of substitution between income and quality. Wealthier consumers correspond to higher values of θ, because they have a lower marginal utility of income. The assumption that the taste for quality is the same for both goods means that the marginal rate of substitution between income and quality is the same for both goods. Remark 2. The assumption of positive correlation is reasonable for complementary goods. I restrict the analysis to the case of perfect correlation, which significantly simplifies the model and makes it tractable, by allowing for one-dimensional characterization of consumers’ preferences. The same logic also applies when the taste for quality is not the same, but highly positively correlated. Carbajo, de Meza and Seidmann (1990) use a similar approach and assume that consumers’ valuations for two goods are the same. They argue that a high positive correlation is likely to occur when the goods are normal and when consumers are differentiated according to income. I assume that both production of each good and entry are costless.9 Goods (varieties) XD produced by firms M and D are differentiated vertically. Similarly, if entry occurs, goods XE produced by firms M and E will be differentiated vertically. The structure of the markets is shown in Table 1. All qualities are given exogenously and for i ∈ {D, E}, I denote sM i the quality of good Xi produced by firm M , and si the quality of the good produced by its rival i. With this notation, Assumptions 1 and 2 8

Alternatively, complementarity could be captured by using a non-additive form of utility from a package containing both goods. 9 Introduction of fixed costs of firm E’s entry would make entry deterrence even easier.

6

can be rewritten as: 0 < s M D < sD ,

0 < sE < sM E ,

sD + sE < sM D + sM E .

(2)

If follows then that the four possible combinations of goods XD and XE have ranking: 0 < sM D + sE < sD + sE < sM D + sM E < sD + sM E .

(3)

In addition, Assumption 3 can be rewritten in the following form: 2θ < θ, θ(sD − sM D ) ≤ θ(2sD + sM D ), θ(sM E − sE ) ≤ θ(2sM E + sE ).

(4) (5) (6)

Based on the analysis by Tirole (1992, p. 296), the first inequality means that the market should be wide enough to accommodate two firms. The other inequalities ensure that the markets are covered in equilibrium, when the goods are sold separately. I will call values of parameters admissible if they satisfy condition (2) and conditions (4)–(6). In order to simplify the analysis, I also denote τ = θ/θ,

∆i = sM i − si (i ∈ {D, E}),

ρ = ∆D /∆E .

(7)

Using this new notation, the above conditions can be rewritten as the following: ∆D < 0, ∆E > 0, ∆D + ∆E > 0, ½ ¾ sD − sM D sM E − sE 1 max , ≤τ ≤ . 2sD + sM D 2sM E + sE 2

(8) (9)

Parameter τ can be interpreted as a measure for narrowness of the market. The market is narrow when τ is high and is wide when τ is low. Hence, an upper bound on τ means that the market should not be too narrow (i.e., should be wide enough), whereas a lower bound on τ means that the market should not be too wide (i.e., should be narrow enough). Note that the maximum is achieved on the market with a higher quality ratio, which is sD /sM D on the market for good XD and sM E /sE on the market for good XE . Parameter ρ can be interpreted as a measure for toughness or softness of competition in one market relative to the other. The competition on one market is tougher when the difference in qualities (here denoted as ∆D and ∆E ) is lower (in absolute value), i.e., the goods are close to substitutes. Note that (8) implies ρ ∈ (−1, 0). Values of ρ close to 0 mean that competition on the market for XE is softer than competition on the market for XD , whereas values of ρ closer to −1 mean that competition on both markets is equally soft (or equally tough). The whole situation can be modelled as a three-stage game. In the first stage, firm M decides which combination of goods XD and XE it will sell — its strategies are listed in Table 2.10 In the second stage, firm E decides whether to enter the 10

Firm M engages in tying if it offers some product in the bundle, but not separately (see also Footnote 4). Here tying occurs in subgames Γpure , ΓmixD , and ΓmixE .

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market for good XE . In the third stage, all firms compete in prices. I analyze the pure-strategy equilibria of each subgame and look for a subgame perfect equilibrium in pure strategies of the whole game. To simplify the analysis, I consider equilibria where combinations of goods with a higher quality also have a higher price (otherwise nobody purchases the lower quality combination). I am particularly interested, whether entry deterrence can occur in subgame-perfect equilibrium.

Strategy

Products offered by M

no bundling

XD and XE

ΓnoB

X

X

X

X

pure bundling

bundle M = {XD , XE }   M and XD M and XE  M, XD and XE

Γpure

×

X

X

×

ΓmixD ΓmixE ΓmixDE

X × X

X X X

X X X

× X X

mixed bundling

Subgame

Available combinations sM D sD sM D sD + + + + sE sE sM E sM E

Table 2: Strategies of firm M in the first stage with available combinations (packages) In addition, I will assume that unbundling of goods in the bundle is impossible (or excessively costly) so that consumers cannot buy the bundle, abandon one product, and buy a higher quality variety from another firm. As has already been argued in the literature (Matutes and Regibeau 1992), pure bundling is then equivalent to making the products incompatible with rival ones (printers and cartridges can serve as a typical example). The unbundling assumption then means that use of incompatible products is impossible or excessively costly. Following Whinston (1990), I also assume that f irm M can precommit itself to its bundling strategy (e.g., it will not sell one of the goods separately if it previously decided otherwise). However, as shown in Subsection 4.4, this assumption is not necessary and results presented in this paper can also be obtained when the possibility of commitment is abandoned. However, for the ease of presentation, I start by assuming that such precommitment is possible.

3 3.1

Bundling strategies No bundling

Consider first the benchmark case where firm M decides to sell its products separately, and firm E enters the market (subgame ΓnoB ). Let pM D , pM E , pD , and pE be the prices of the respective goods offered by the firms (this notation will also be used in the following sections). In the absence of bundling, each consumer has four choices available (two for each good). If all customers are served with both goods, the two markets can be considered to be independent. On the market for good XE , a consumer with taste for quality 8

θ buys product XE from firm M , if and only if θ > θ∗ , where θ∗ = (pM E − pE )/∆E represents the indifferent consumer.11 In the opposite case, he buys product XE from firm E. Firms’ profits then are ΠM E = pM E (θ − θ∗ ) and ΠE = pE (θ∗ − θ). Their maximization yields the following equilibrium prices and profits pM E = 31 ∆E (2θ − θ),

pE = 31 ∆E (θ − 2θ),

ΠM E = 91 ∆E (2θ − θ)2 ,

ΠnoB = 19 ∆E (θ − 2θ)2 . E

(10)

The indifferent consumer is then indexed by θ∗ = 13 (θ + θ). Obviously, pM E > pE ∗ and ΠM E > ΠnoB E . Furthermore, condition (4) implies that θ < θ , and (6) assures that the θ consumer has a non-negative utility. Hence, the market is in equilibrium covered by two firms. Moreover, firm E’s profit is clearly positive for all admissible values of parameters, which means that it will always enter the market. The situation on the market for good XD is analogous, with the exception that now firm M has a low-quality good. In this case, the equilibrium prices and profits are (recall that ∆D < 0) pM D = − 13 ∆D (θ − 2θ),

pD = − 31 ∆D (2θ − θ),

ΠM D = − 91 ∆D (θ − 2θ)2 ,

2 1 ΠnoB D = − 9 ∆D (2θ − θ) .

(11)

Observe that all profits are homogeneous of degree 1 in (∆D , ∆E ) and homogeneous of degree 2 in (θ, θ). Summing up, I obtain firm M ’s equilibrium profit 2

2 2 1 ΠnoB M = ΠM D + ΠM E = 9 [(2 − τ ) − ρ(1 − 2τ ) ] · ∆E θ .

3.2

(12)

Pure bundling

In the case of pure bundling (subgame Γpure ), each consumer is allowed to buy one of two packages: products from firms D and E purchased separately (with qualities sD and sE ) and the bundle M from firm M . Let pM denote the price of the bundle M offered by firm M . Since unbundling is impossible, the current situation can be described as one vertically differentiated market with two products: the bundle M offered by firm M (with quality sM D + sM E and price pM as a high-quality combination) and the combination of products XD and XE (with quality sD + sE and price pD + pE as a low-quality combination).12 However, there is a difference from a single vertically differentiated market with qualities sM D + sM E and sD + sE , since here the price of the latter is not set by a single firm. A consumer with taste for quality θ buys the bundle M if and only if θ > θ∗ , where θ∗ = (pM −pD −pE )/(∆D +∆E ) represents the marginal (indifferent) consumer. The maximization of profits ΠM = pM (θ − θ∗ ),

ΠD = pD (θ∗ − θ),

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ΠE = pE (θ∗ − θ)

I ignore the case of equality since it corresponds to a set of consumers with measure zero. In the traditional economic literature a bundle means in general a combination of goods. To avoid misunderstandings, I will refer to the bundle only as the result of bundling, i.e., a package of goods XD and XE sold together by firm M . 12

9

yields the following equilibrium prices and profits pD = pE = 41 (∆D + ∆E )(θ − 2θ), Πpure = Πpure = D E Πpure = M

1 (∆D 16

1 (∆D 16

pM = 41 (∆D + ∆E )(3θ − 2θ),

+ ∆E )(θ − 2θ)2 =

+ ∆E )(3θ − 2θ)2 =

1 (1 16

1 (1 16

(13)

2

+ ρ)(1 − 2τ )2 · ∆E θ , 2

+ ρ)(3 − 2τ )2 · ∆E θ .

(14)

The indifferent consumer is then characterized by θ∗ = 14 (2θ + θ). Just as for separate markets, it is necessary to check whether the conditions for market coverage are satisfied in equilibrium. It is easy to verify that (4) implies θ < θ∗ and (5) and (6) imply that the θ consumer has a non-negative utility. Therefore, the market is covered in equilibrium. Again, entry will always occur, since firm E’s profit is positive for all admissible values of parameters. However, the profit is smaller than in the case of no bundling.

3.3

Mixed bundling with good XE — case of entry

I start the analysis of mixed bundling with the most interesting case, where firm M offers good XE and the bundle M (subgame ΓmixE ). This way, it makes the combination with the lowest quality (sM D + sE ) unavailable. Hence, if firm E enters the market, consumers are left with three possible packages with qualities as indicated in Table 2. The marginal consumer who is indifferent between buying the goods from firms D and E, and buying the bundle is characterized by θ2∗ = (pM −pD −pE )/(∆D + ∆E ). The marginal consumer who is indifferent between buying the bundle and the highest-quality combination is characterized by θ3∗ = (pM − pD − pM E )/∆D . All available combinations have a positive market share, if and only if θ < θ2∗ < θ3∗ < θ. In this case, firms’ profits are: ΠD = pD (θ2∗ − θ + θ − θ3∗ ),

ΠE = pE (θ2∗ − θ),

ΠM = pM (θ3∗ − θ2∗ ) + pM E (θ − θ3∗ ), The following proposition and corollary show that under certain conditions, there may be no place for firm E in the market. In order to specify that conditions, let τˆ(ρ) =

3 + 2ρ , 2(3 + ρ)

(∗)

for all ρ ∈ (−1, 0). Then τˆ(ρ) is increasing in ρ on interval (−1, 0) and its range is interval ( 14 , 12 ); see Figure 1 for illustration. Proposition 1. In the subgame ΓmixE after entry, there exists an equilibrium (in pure strategies) where firm E obtains a positive market share, if and only if τ < τˆ(ρ). Corollary 1. For any ρ ∈ (−1, 0), expression τˆ(ρ) represents a critical value such that firm M can deter entry of firm E by choosing mixed bundling with good XE , if and only if τ ≥ τˆ(ρ). 10

Condition τ ≥ τˆ(ρ) requires the market to be narrow enough in order to leave no place for the entrant. This means it is not possible that all three available combinations have a positive market share. In such case, the lowest-quality combination will be the one which cannot have a positive market share. Several points are worth noting. First, for any ρ ∈ (−1, 0), I obtain τˆ(ρ) ∈ ( 41 , 12 ). Hence, for any ρ ∈ (−1, 0) there exists an open set of values of τ such that τ ≥ τˆ(ρ). Second, in order to analyze the effect of firm E’s quality, observe that τˆ(ρ) is decreasing in sE (as ∆E is decreasing in sE and ρ is increasing in ∆E ). This means that the lower the quality with which firm E enters, the less likely entry deterrence is (as the inequality τ ≥ τˆ(ρ) holds for a smaller set of values of τ ). Note that condition (6) imposes a lower bound on firm E’s quality. Otherwise, it would not be able to compete with firm M in the no-bundling case. Third, in the proof of this proposition, I show that for any non-negative price pE , firm cannot obtain a positive profit whenever firms D and M play a best response to both rivals’ prices. Note that introduction of fixed costs of entry would amplify this result even more. Fourth, this form of entry deterrence can occur only if the packages including firm E’s product XE have the lowest and the second-lowest quality. An alternative ordering of qualities with such property would be obtained when sD < sM D , sE < sM E , and sM D + sE < sD + sM E . However, it is possible to show that in this case firm E can enter the market. The main reason for this difference is the behavior of firm D. In the original ordering, firm D’s market share has two margins. As its product XD is part of the highest-quality combination, firm D sets a high price in order to earn a high profit from the highest-quality combination. In this way it sacrifices market share from the low-quality combination with firm E’s product. On the other hand, in the ordering introduced in this paragraph, firm D’s product has only one margin and is not part of the highest-quality combination. Hence, its price is lower, allowing all three available combinations to have a positive market share. In addition, as indicated earlier, it is possible to show that among all possible orderings, only (3) allows for entry deterrence.

3.4

Mixed bundling with good XE — case of no entry

In this subsection, I analyze the subgame ΓmixE , after firm E decides not to enter. In order to distinguish from the case of entry, I will denote this subgame Γdeter . As mentioned in in Subection 2.2, firm M commits to its bundling strategy in the first stage. Hence, when firm E does not enter, firm M is not allowed to change its strategy ex-post and has to sell the bundle and good XE .13 In this case, the consumer has only two packages available: the bundle and good XD from firm D together with good XE from firm M . Denote θ3∗ = (pM − pD − pM E )/∆D the consumer who is indifferent between them, and θ0∗ = pM /(sM D + sM E ) the consumer who is indifferent between buying the bundle and not buying at all. The following proposition specifies the equilibrium in this subgame. 13

However, it may charge such a high price for some of its products that nobody will buy them.

11

Proposition 2. There exists a unique equilibrium of the subgame Γdeter . In this equilibrium, the market is undercovered and: pD = − 13 ∆D θ,

pM = 12 (sM D + sM E )θ, 2

Πdeter = − 19 ∆D θ , D

2

pM E = pD + pM E , 2

Πdeter = − 91 ∆D θ + 14 (sM D + sM E )θ . M

(15)

Interestingly, the price of the bundle is lower than the price of good XE sold separately. The crucial assumption for this is the impossibility of unbundling. Although it is surprising, this form of pricing indeed occurs in reality (however, the motives may be different). Nalebuff (2003a, pp. 31–32) provides an example of cars and radios, where cars with radios are cheaper than cars without radios. If unbundling is possible, the same outcome can be achieved via exclusive dealing. Firm M can offer the consumers an exclusive contract to supply good XE more cheaply, if they do not buy the good XD from firm D. This requires that firm M is able to monitor consumers’ purchases. Following Nalebuff (2004), such monitoring is a reasonable assumption in business-to-business transactions. As an example, MasterCard and Visa “required their member banks to issue only MasterCard and Visa and not American Express” (Nalebuff 2004, p. 3). Although the above result may appear counterintuitive, it only means that consumers with a high appreciation for quality are charged a higher price. This can be interpreted as a form of price discrimination between consumers with a high appreciation for quality (buying good XD from firm D) and consumers with a low appreciation for quality (buying it from firm M ). 2 Note also that pM = 12 (sM D + sM E )θ is the monopoly price and 41 (sM D + sM E )θ is the monopoly profit, if firm M would be alone in the market offering a single product (the bundle) with quality sM D + sM E . The presence of firm D then allows firm M to extract additional surplus by offering its product XE separately, consistently with the price discrimination argument. This may also appear counterintuitive, since firm M is better off in a duopoly than in a monopoly. Note, however, that this result relies on complementarity. Similarly, as a monopolist producing only one complement is better off when another firm produces a second complement, here a firm M producing a high quality XE and only a low quality XD is better off when firm D produces a high quality XD . As an interesting consequence, firm M does not wish to foreclose the market for firm D. An undercovered market indicates a possibility of entry. As was shown in the previous section, there can be no low-quality entrants in the market for good XE . However, there could still be a potential entrant in the market for good XD . In this case, it would be necessary to analyze this firm’s entry decision in the second stage simultaneously with firm E’s decision. However, the narrowness of the market, according to (5), implies that such a firm would not be active in the no-bundling subgame (and likewise for the pure-bundling subgame). Hence, I will omit the possibility of additional entry in the market for good XD .

12

3.5

Mixed bundling with good XD

By offering good XD and the bundle (subgame ΓmixD ), firm M makes the combination with the highest quality (sD + sM E ) unavailable. Consumers are left with three available packages with qualities specified in Table 2. The marginal consumers are characterized by θ1∗ = (pM D − pD )/∆D and θ2∗ = (pM − pD − pE )/(∆D + ∆E ), and firms’ profits are ΠE = pE (θ2∗ − θ),

ΠD = pD (θ2∗ − θ1∗ ),

ΠM = pM (θ − θ2∗ ) + pM D (θ1∗ − θ). Proposition 3. In the subgame ΓmixD , there exists an equilibrium where the lowestquality combination has a positive market share, if and only if τ<

1+ρ . 5+ρ

(16)

The above proposition implies that although firm M decides to offer (in addition to the bundle) product XD , it prefers that nobody buys it, provided condition (16) does not hold. This may be achieved by offering good XD for sufficiently high price. Then any equilibrium in this subgame is outcome equivalent 14 to an equilibrium of the pure-bundling subgame. The equilibrium prices and profits are then given by (13)–(14) with pM D high enough so that nobody buys firm M ’s good XD together with firm E’s good XE .

3.6

Mixed bundling with both goods XD and XE

In the subgame ΓmixDE , each consumer has all combinations of goods XD and XE available, with ranking of qualities given by (3). Moreover, he can buy the products from firm M either in the bundle (for price pM ) or separately (for price pM D + pM E ). Offering a bundle makes sense only if pM < pM D + pM E . Otherwise nobody buys it, and the situation is equivalent to selling separate products. The marginal consumers are characterized by θ1∗ = (pM D − pD )/∆D , θ2∗ = (pM − pD − pE )/(∆D + ∆E ), and θ3∗ = (pM − pD − pM E )/∆D . When θ < θ1∗ < θ2∗ < θ3∗ < θ, firms’ profits then can be written as follows: ΠD = pD (θ2∗ − θ1∗ + θ − θ3∗ ),

ΠE = pE (θ2∗ − θ),

ΠM = pM (θ3∗ − θ2∗ ) + pM D (θ1∗ − θ) + pM E (θ − θ3∗ ). Proposition 4. In the subgame ΓmixD , there exists an equilibrium where the lowestquality combination has a positive market share, if and only if τ<

2(1 + ρ) , 7 + 4ρ

14

(17)

Two equilibria are outcome equivalent if they yield the same profits to each firm and the same utility to each consumer.

13

The above proposition implies that although firm M offers product XD separately, it prefers that nobody buys it, provided condition (17) does not hold. Similarly as in the previous subsection, this may be achieved by choosing sufficiently high price pM D . Then any equilibrium in this subgame is outcome equivalent to an equilibrium of the subgame where the multi-product firm offers good the bundle and good XE , causing no entry by firm E. The equilibrium outcome is then specified by Proposition 2.

4

Entry deterrence

4.1

Entry deterrence as a subgame perfect equilibrium

In this section, I compare entry-deterrence equilibrium with other equilibrium outcomes. It can be easily established that if τ ≥ τˆ(ρ), then conditions (16) and (17) do not hold (see Figure 1 for graphical illustration). Therefore, when τ ≥ τˆ(ρ), any equilibrium of the third stage after entry is outcome equivalent either to the equilibrium of the pure-bundling subgame, or to the equilibrium of the no-bundling subgame. Hence, entry deterrence is a subgame perfect equilibrium, if and only if Πdeter > Πpure M M and Πdeter > ΠnoB M M . The following proposition specifies, when these inequalities hold. τ 1 2

τˆ(ρ) =

3+2ρ 2(3+ρ)

1 4 2(1+ρ) 7+4ρ 1+ρ 5+ρ

0

ρ −1

Figure 1: Illustration of τˆ and conditions (16) and (17) Proposition 5. For all admissible values of parameters, the following statements hold: (i) Πdeter > ΠnoB M M . pure (ii) If, in addition, τ ≥ τˆ(ρ), then ΠnoB M > ΠM .

Corollary 2. If τ ≥ τˆ(ρ), then entry deterrence occurs in subgame-perfect equilibrium.

14

Observe that condition τ ≥ τˆ(ρ) is sufficient but not necessary for part (ii) of the above proposition.15 However, according to Proposition 1 and Corollary 1, it is sufficient and necessary for entry deterrence to occur. If, on the other hand, τ < τˆ(ρ), then firm E can enter the market and obtain a positive market share. In order to find the subgame-perfect equilibrium when τ < τˆ(ρ), it is necessary to compare firm M ’s equilibrium profits from the last stage subgames. However, a detailed examination of all cases would significantly extend the analysis and is, therefore, omitted. Remark 3. In the proof of Proposition 5, I show that condition (20) is necessary and > Πpure to hold. Its right-hand side is decreasing sufficient for the inequality ΠnoB M M 1 in τ on [0, 2 ] and attains value of 0 when τ = 12 (see Figure 2). Thus, leaving entry deterrence aside, for any fixed ρ, pure bundling is preferred by firm M to no bundling, if the market is sufficiently wide (i.e., τ is low). This result conforms to the results in Kov´aˇc (2004) in the sense that a multi-product firm without monopoly power can find tying profitable also without entry deterrence motives. There, tying yields to softer competition, which allows the firms to relax prices.

4.2

Welfare effects

In this subsection, I analyze the welfare consequences of entry deterrence. I compare social welfare and consumer surplus in equilibrium of the no-bundling subgame, purebundling subgame, and the subgame after entry deterrence. Recall that since firms’ revenues are equal to consumer spending, social welfare depends only on qualities of goods purchased by consumers and not on prices paid. The prices only determine indifferent consumers, and hence, the sizes of the segments where consumers buy the same products. I start by comparing the consumer surplus and social welfare in the pure-bundling and no-bundling subgame. Figure 2 and Table 3 illustrate the comparison (see Appendix C for details). The analysis shows that when firm M prefers no bundling to pure bundling, its choice leads also to a lower consumer surplus, but to a higher social welfare. On the other hand, pure bundling, when preferred by firm M , may enhance consumer surplus or social welfare, but never both simultaneously. Region

pure ΠnoB M − ΠM

CS noB − CS pure

SW noB − SW pure

A B C D

+ − − −

− − + +

+ + + −

Table 3: Comparison of equilibria in no-bundling and pure-bundling subgame (regions in Figure 2) 15

Consider, for example, the values sD = 0.2, sM D = 0.1, sE = 0.1, sM E = 0.9, θ = 1, and θ = 0.45, which are admissible and yield τ = 0.45, ρ = −0.125, and τˆ(ρ) = 0.4783. The equilibrium pure profits then are ΠnoB = 0.1929, and Πdeter = 0.2611. M = 0.2137, ΠM M

15

−ρ τˆ−1 (τ ) =

3(1−2τ ) 2(1−τ )

0.2

(1−2τ )(17−10τ ) 97−172τ +100τ 2

A

(1−2τ )(31−38τ ) 239−548τ +284τ 2

B

0.1

C

(1−2τ )(7+10τ ) 119−68τ +92τ 2

D 0

1 2

τ

Figure 2: Comparison of equilibria in no-bundling and pure-bundling subgame In order to understand these results intuitively, observe first that in both cases (nobundling and pure-bundling), the market is covered and split into two segments. In the no-bundling case, the high segment is served by the highest quality combination (sD + sM E ), whereas the low segment is served by the lowest quality combination (sM D + sE ). In the pure-bundling case, the segments are served by combinations with middle qualities (sM D + sM E for the high segment and sD + sE for the low segment). As the high segment is larger, one should expect that the effect on the high segment will prevail over the effect on the low segment.16 Therefore, pure bundling should lead to a decrease in (average) quality. In addition, there is a second effect that pure bundling leads to lower differentiation and tougher competition. When ρ is sufficiently low (close to −1), then the available combinations (with qualities sM D + sM E and sD + sE ) are close to perfect substitutes and the second effect dominates. Thus, the equilibrium prices in pure-bundling should be lower than in no-bundling, leading to an increase in consumer surplus. On the other hand, in this case, the decrease in quality in the high segment is more significant,17 which yields a lower social welfare. Let me proceed now by comparing the social welfare after no entry (entry deterrence) with social welfare in the no-bundling equilibrium and pure equilibrium. Proposition 6. For all admissible values of parameters the following statements hold: (i) SW noB > SW deter . (ii) SW pure < SW deter , when the market is sufficiently narrow, i.e., when τ0 < τ < for some τ0 ∈ (0, 12 ).

1 2

In all considered equilibria the size of the high segment is at least 12 θ. Recall that the indifferent consumer has taste for quality 13 (θ + θ) in no-bundling equilibrium; 14 (2θ + θ) in pure-bundling equilibrium; and 12 θ in entry-deterrence equilibrium. 17 This holds since sM E − sE > sD − sM D . Note that there are also consumers of small measure 1 (θ−2θ), who are in the low segment in the no-bundling equilibrium but switch to the high segment 12 in the pure-bundling equilibrium. 16

16

The proof of Proposition 6 can be found in Appendix C. The proposition shows that entry deterrence decreases social welfare compared to no bundling, but may increase social welfare compared to pure bundling, when the market is narrow enough. There are two effects driving these results. First, entry deterrence leads to higher qualities (two combinations with qualities sD + sM E and sM D + sM E are available), which has a positive impact on social welfare. Second, entry deterrence leads to undercovered market and therefore (given my assumptions) less consumers are served compared to pure bundling and no bundling. This has a negative impact on social welfare. Obviously, the second effect is more significant when the market is wide, since more consumers are not served. Therefore, only for narrow markets could the first effect dominate, and social welfare in entry deterrence may potentially exceed social welfare in no-bundling and pure-bundling equilibria. However, the analysis shows that this is never the case for the no-bundling equilibrium.18 In order to better illustrate the comparison of social welfare in entry-deterrence equilibrium and pure-bundling equilibrium, I used numerical simulations. Since all values (profits, social welfare, and consumer surplus) are homogeneous of degree 1 in (sD , sE , sM D , sM E ) and all conditions are homogeneous in (sD , sE , sM D , sM E ),19 I can normalize the total quality of the bundle to 1, i.e., set sM D + sM E = 1. Note that under this normalization, the set of all admissible values of parameters is bounded (by 0 from below and by 1 from above). The simulations show that the inequality SW pure < SW deter holds in approximately 48% of cases from all admissible values of parameters, but in 87% of cases from all admissible values of parameters where entry deterrence occurs, i.e., condition τ ≥ τˆ(ρ) is satisfied.20, 21 These percentages can also be interpreted as probabilities that the inequality occurs, provided that the joint distribution of (sD , sE , sM D , τ ) is roughly uniform. Now, I turn to the comparison of consumer surplus after entry deterrence with consumer surplus in the no-bundling equilibrium and pure-bundling equilibrium. Recall that consumer surplus can be obtained by subtracting firms’ profits from the social welfare. As opposed to social welfare, prices do play an important role here. As the prices for different segments of the market may change in different ways, intuition behind the effects on consumer surplus is much more complicated. However, one effect from the above analysis is still at work. Entry deterrence yields an undercovered market where the consumers who are not served have zero utility, as opposed to no-bundling and pure-bundling equilibria where all consumers are served and obtain a positive utility. This has a negative impact on consumer surplus. A detailed comparison of consumer surplus would require a discussion of many 18

Recall that in most cases, except a small region D on Figure 2, social welfare in no-bundling equilibrium is larger than the one in pure-bundling equilibrium. 19 In general, I say that a condition of the form f (x, y) ≥ 0 is homogeneous in x, if for any λ > 0 it is equivalent to f (λx, y) ≥ 0. Analogously, I define homogeneity for a strict inequality. 20 The simulations were performed using the Mathematica 5.0 software (the source code can be obtained upon request). I used a grid of size 100×100×100×100 on the set [0, 1]×[0, 1]×[0, 1]×[0, 21 ] in the (sD , sE , sM D , τ )-space and at each admissible point from the grid, I compared social welfare in entry-deterrence equilibrium to social welfare in pure-bundling equilibrium. 21 The simulations also reveal that the inequality τ ≥ τˆ(ρ) holds in 45% of the domain of all admissible parameter values.

17

cases. Therefore, similarly as for social welfare, I again used numerical simulations (performed in the same way as described in Footnote 20) in order to compare consumer surplus in entry-deterrence equilibrium to consumer surplus in no-bundling and purebundling equilibria. The results of the simulations are summarized in the following conjecture. Conjecture 1. For all admissible values of parameters the following inequalities hold: CS deter < CS noB and CS deter < CS pure . In this sense, my analysis contributes to the current debate on the abuse of dominant position. According to the traditional “market oriented” competition policy, such act of entry deterrence (or foreclosure) is considered anti-competitive. However, in light of the new welfare oriented competition policy, foreclosure should be judged based on its welfare effects. The recent DG Competition Discussion paper on the enforcement of Article 82 specifies foreclosure as anti-competitive, if it excludes an “equally efficient rival.”22 Here, the rival (firm E) is less efficient on the market for good XE than firm M . However, foreclosure leads to welfare losses and should, therefore, be considered anti-competitive.

4.3

Cooperation as defence against entry deterrence

As a possible defence against entry deterrence firm D could cooperate with firm E in order make the entry possible. I assume that such a decision would be made after firm M ’s bundling decision but before firm E’s entry decision, since it simultaneously should be a response to M ’s bundling strategy, and it should enable E’s entry. In particular, I consider two ways of cooperation. However, both turn out to be unprofitable for firm D. First, firm D can agree to sell its product XD only in a bundle with firm E’s product. The agreement may even involve side payments from firm E to firm D to improve firm D’s incentives to cooperate with firm E. This is a form of inter-firm bundling where two products produced by different firms are offered in a bundle. Bundling of hardware and software provides a typical example for such practice (e.g., CD-writers are usually purchased jointly with the appropriate application software). In this case, there are only two packages available: the bundle by firm M and the package consisting of firm D’s good XD and firm E’s good XE . Since both firms D and E behave as profit-maximizing individuals, in the last stage, this form of defence is equivalent to the pure-bundling subgame. Second, firms D and E can merge and bundle their products together. In this case, the packages available in the market are the same as in the previous case. However, firms D and E do not behave as two profit-maximizing individual firms, but as one firm. Therefore, this situation is equivalent to a single vertically differentiated market with two products of qualities sM D + sM E (offered by firm M ) and sD + sE (offered by the merged firms D and E). 22 See “DG Competition discussion paper on the application of Article 82 of the Treaty to exclusionary abuses” by the European Commission (2005) and Rey, Gual, Hellwig, Perrot, Polo, Schmidt and Stenbacka (2005).

18

Remark 4. Note that the second case suggests an additional potential motive for mergers as found in the literature. Traditional economic literature mainly considers mergers among firms that produce substitutes, in order to increase market power. Nalebuff (2002) describes the possibility of bundling as a motive for the merger of firms producing complements. His analysis is mainly meant to explain the GE-Honeywell merger. In this paper, the motive would be similar to the one by Nalebuff (2002). However, here it would be used as a defence against entry deterrence. Unfortunately, it turns out not to be profitable for firm D in this model. In order for firm D to cooperate with firm E, the cooperation should make firm D better off by increasing its profit compared to the equilibrium in entry-deterrence 2 = − 91 ∆D θ . Firm D’s profit in subgame. As shown in Section 3.4, this profit is Πdeter D the first case (without side payments) is the same as in the pure-bundling subgame 1 equilibrium, which is Πpure = 16 (∆D + ∆E )(θ − 2θ)2 . However, as a side payment, D firm E may transfer part of its profit to firm D. Hence, firm D may earn up to Πpure + Πpure = 2Πpure D E D . On the other hand, the joint profit in the second case is 1 merge Π = 9 (∆D + ∆E )(θ − 2θ)2 . It follows that < Πmerge < 2Πpure Πpure D . D

(18)

Hence, a merger is not profitable compared to a separate profit maximization with side payments. Moreover, it can be easily shown that firm M ’s equilibrium profit when competing against two separate firms is higher than the profit when competing against the merged multi-product firm. The reason for this is that a merger leads to more aggressive behavior, and hence, all firms experience lower profits. Nalebuff (2000) obtains a similar result for horizontally differentiated complements. Now it remains to compare the above profits to firm D’s profit after entry deterrence. The following proposition shows that the suggested tools are not sufficient as a defence against entry deterrence, when τ ≥ τˆ(ρ). Proposition 7. If τ ≥ τˆ(ρ), then cooperation is not profitable for firm D. Intuitively, by cooperation with firm E, firm D loses high profits from selling its XD together with firm M ’s good XE as the highest-quality combination. Therefore, firm D’s cooperation with firm E is not profitable, and hence, entry deterrence cannot be prevented.

4.4

Commitment

This section discusses the assumption of commitment and analyzes a variation of my model with no commitment. Until now I assumed, following Whinston (1990), that firm M can precommit itself to not changing its bundling strategy in a later stage (e.g., not to sell one of the goods separately if it previously decided otherwise). This precommitment can be achieved, for example, by a technological setting, and may involve sunk costs, e.g., design, advertising, etc; see Whinston (1990) and Nalebuff (2003a) for a more extensive discussion. In such cases, it is reasonable to assume that 19

the bundling strategy is chosen before the pricing decisions. As I show, the announcement of tying in the first stage can make firm E’s second-stage entry unprofitable. However, in contrary to Whinston (1990), precommitment is not necessary here.23 To illustrate this point, consider the subgame where firm M offers the bundle as well as both goods XD and XE . As follows from Subsections 3.6 and 4.1, in this case firm E (if it enters) cannot obtain a positive market share in equilibrium. Therefore, if bundling is not prohibited, the same outcome will be achieved in the game with the following timing. In the first stage, firm E decides whether to enter the market or not, and in the second stage, firms compete in prices — firm M chooses prices pM D , pM E , and pM (price of the bundle); firm D price pD , and (after entry); and firm E price pE . This subgame is equivalent to mixed bundling with selling both products XD and XE separately, which can lead to entry deterrence. Therefore, under condition τ ≥ τˆ(ρ), firm E anticipates that it cannot obtain a positive market share. Thus, it decides not enter the market. The above discussion implies that precommitment is not necessary for entry deterrence. This result highlights another aspect in which my paper differs from Whinston (1990). Unlike here, in Whinston (1990), foreclosure occurs only when the multiproduct firm can credibly commit to bundling.

5

Conclusion

This paper analyzes a situation where multi-product firm competes against a specialist firm in one market and faces a potential entrant in a second market. In particular, I consider the case of two markets for complementary products. Within each market, the goods are differentiated vertically. Unlike in the standard leverage argument, where the multi-product firm leverages its market power in the first market, I show that the multi-product firm can use tying to deter entry of a low-quality entrant to the second market, even without having a dominant position in the first market. Moreover, in contrast to the previous literature, the mechanism through which entry deterrence is achieved is mixed bundling. The presented argument does not rely on costs of entry, but rather on the market structure. I show that entry deterrence occurs when the market is narrow enough, in the sense of Shaked and Sutton (1982), i.e., when the consumers’ tastes are not too heterogeneous. The presence of fixed costs of entry would amplify this result even more. When entry deterrence occurs, it leads (on average) to higher qualities, but simultaneously makes the goods too expensive for consumers with low taste for quality. The welfare consequences of entry deterrence therefore depend on the balance between those two effects. In particular, I show that entry deterrence yields a lower social welfare compared to selling separate products. In addition, the entry deterrence cannot not be prevented by cooperation among the specialist firms. These results are relevant for anti-trust policies since they illustrate an anti-competitive practice which was until now not taken into account. In particular, in relation to the new welfare 23

I am grateful to Patrick Rey for this observation.

20

based approach to competition policy, my results indicate that even exclusion of an inefficient rival may lead to welfare losses. The understanding of this issue could be in the future extended in several directions. First, in the paper I consider the qualities of goods given exogenously. It would be interesting to analyze an extension of the present model, where qualities are determined endogenously at some preceding stage. Second, the markets considered in this paper are narrow in the sense that each of them cannot accommodate more than two firms in equilibrium in the no-bundling equilibrium. A relevant question is how my results extend to wider markets that are still natural oligopolies in the sense of Shaked and Sutton (1982) but allow for more firms, or even for different market structures. Third, I considered only cooperation among specialist firms as a way to prevent entry deterrence. Some other ways could be suggested and analyzed in the introduced framework. Fourth, it would be interesting to provide an example where the multi-product firm deters entry of a superior rival.

A

Appendix: Proofs

Proof of Proposition 1. Assume first that there is such equilibrium and that τ ≥ τˆ(ρ). The equilibrium prices satisfy first-order conditions for maximization of firm M ’s and D’s profits:24 2∆E pM − ∆E pD − 2(∆D + ∆E )pM E + ∆D pE = 0, 2pM − pD − 2pM E = ∆D θ, ∆E pM − 2∆E pD − (∆D + ∆E )pM E + ∆D pE = ∆D (∆D + ∆E )(θ − θ). Solution of this system (with pE as parameter) is ¤ 1 £ pM = (∆D + 3∆E )pE + 2∆D (∆D + ∆E )θ + (−2∆2D + ∆D ∆E + 3∆2E )θ , 6∆E 1 pM E = (pE + ∆E θ), 2 ∆D pD = [pE − 2(∆D + ∆E )(θ − θ)]. 3∆E Hence, θ2∗ − θ = −

∆D + 3∆E 1 pE + [(2∆D + 3∆E )θ − 2(∆D + 3∆E )θ]. 6∆E (∆D + ∆E ) 6∆E

It is easy to observe that the coefficient at pE is negative. Moreover, the last term is nonpositive, if and only if τ ≥ τˆ(ρ). In this case θ2∗ ≤ θ for any non-negative pE , which is a contradiction. Conversely, assume that τ < τˆ(ρ). Together with firm E’s first order condition pM − pD − 2pE = (∆D + ∆E )θ, I obtain the equilibrium prices as specified in Appendix B. These prices then establish an equilibrium with positive market shares and positive profits for all firms. 24

Because the profits are quadratic in prices, it is easy to check that second-order conditions also hold for any prices.

21

Proof of Proposition 2. If both combinations are purchased by a positive measure of consumers, firm M ’s profit in the subgame Γdeter is ΠM = pM E (θ − θ3∗ ) + pM (θ3∗ − max {θ0∗ , θ}).

(19)

First I analyze the case of overcovered marked, i.e., θ0∗ < θ < θ3∗ < θ. In this case, ∂ΠM 2pM − pD − 2pM E = − θ, ∂pM ∆D

∂ΠM 2pM − pD − 2pM E =θ− , ∂pM E ∆D

which implies ∂p∂M ΠM + ∂p∂M E ΠM = θ − θ > 0. Hence, at least one of the derivatives is positive. This means that ΠM has no interior maximum, i.e., such that θ0 < θ < θ3∗ < θ. If the market is exactly covered, then θ0∗ = θ < θ3∗ < θ. Hence, pM = (sM D + sM E )θ. If pM < (sM D + sM E )θ, the market is overcovered (i.e., θ0∗ < θ). On the other hand, if pM > (sM D + sM E )θ, the market is undercovered (i.e., θ0∗ > θ). Necessary conditions for maximization of firm M ’s profit are: ∂p∂M E ΠM = 0 and ∂p∂M ΠM |+ pM =(sM D +sM E )θ ≤ 0. Substitution yields (2pM −2pM E −pD )/∆D = θ and (2pM −2pM E −pD )/∆D ≤ 2pM /(sM D + sM E ). Hence, θ ≤ 2θ, which contradicts (4). It follows then that the market needs to be undercovered in equilibrium, i.e., θ < θ0 < θ3∗ < θ. In this case, firm M ’s profit is specified by (19) and firm D’s profit is ΠD = pD (θ − θ3∗ ). Their maximization yields the following first order conditions 2pM − 2pM E − pD = 2pM ∆D /(sM D + sM E ), 2pM − 2pM E − pD = ∆D θ, pM − pM E − 2pD = ∆D θ. Obviously, the prices specified in the proposition form the unique solution of the above system. The profits are obtained after substitution. Note that θ0∗ = pM /(sM D + sM E ) = 1 2 θ > θ, and hence, the market is indeed undercovered. Proof of Proposition 3. Proceeding similarly as in the proof of Proposition 1, assume first that there is such equilibrium and that condition (16) does not hold. If I fix pM E , then all other prices necessarily satisfy first-order conditions. These yield θ2∗ − θ =

1 1 pM E + [(∆D + ∆E )θ − (∆D + 5∆E )θ]. ∆D 2(∆D + 3∆E )

Again, the coefficient at pM E is negative and τ ≥ (1 + ρ)/(5 + ρ) implies that the last term is non-positive. Hence, θ2∗ ≤ θ for any positive pM E , which is a contradiction. Conversely, if τ < (1+ρ)/(5+ρ), then the first-order condition yield the following prices as Appendix B. These prices establish an equilibrium with θ < θ1∗ < θ2∗ < θ. Proof of Proposition 4. Proceeding again the same way as in the proof of Propositions 1 and 3, assume first that there is such equilibrium and that condition (17) does not hold. I will consider pE as parameter and show that condition (17) implies θ1∗ ≤ θ for any pE ≥ 0. First-order conditions for maximization of firm M ’s and firm D’s profits yield θ1∗ − θ = −

1 1 pE + [2(∆D + ∆E )θ − (4∆D + 7∆E )θ]. 6(∆D + 2∆E ) 6(∆D + 2∆E )

Again, if τ ≥ 2(1 + ρ)/(7 + 4ρ), then θ1∗ ≤ θ for all pE ≥ 0. Conversely, if τ < 2(1 + ρ)/(7 + 4ρ), then I obtain the following equilibrium prices as specified in Appendix B. These prices yield θ < θ1∗ < θ2∗ < θ3∗ < θ

22

Proof of Proposition 5. The inequality Πdeter > ΠnoB M M is equivalent to − 19 ∆D + 14 (sM D + sM E ) > − 19 ∆D (1 − 2τ )2 + 91 ∆E (2 − τ )2 . This can be obtained as sum of the following three inequalities: − 19 ∆D ≥ − 19 ∆D (1 − 2τ )2 ,

1 4 sM D

1 4 sM E

> 0,

≥ 19 ∆E (2 − τ )2 .

The second inequality is trivial. The first inequality holds since ∆D < 0 and τ ∈ [0, 12 ]. The third inequality is equivalent to 4(2 − τ )2 sE ≥ (1 − 2τ )(7 − 2τ )sM E . Now note that (6) can be rewritten as (1 + τ )sE ≥ (1 − 2τ )sM E . Hence, in order to prove the third inequality, it is sufficient to establish that 4(2 − τ )2 ≥ (7 − 2τ )(1 + τ ), or 6(τ − 12 )(τ − 3) ≥ 0, which obviously holds for τ ≤ 12 . This completes the proof of part (i). Using (12) and (14), inequality in part (ii) can be rewritten as −ρ(97 − 172τ + 100τ 2 ) > (1 − 2τ )(17 − 10τ ).

(20)

At this point I will use inequality τ ≥ τˆ(ρ), which can be rewritten in the “inverse form” as −ρ ≥

3(1 − 2τ ) = τˆ−1 (τ ). 2(1 − τ )

(21)

Since 97 − 172τ + 100τ 2 > 0 for all τ ∈ (0, 12 ), it is sufficient to show that (1 − 2τ )(17 − 10τ ) 3(1 − 2τ ) > , 2(1 − τ ) 97 − 172τ + 100τ 2

or equivalently

(1 − 2τ )(257 − 462τ + 280τ 2 ) > 0. 2(1 − τ )(97 − 172τ + 100τ 2 )

The last inequality obviously holds for all τ ∈ (0, 21 ). Hence, τ ≥ τˆ(ρ) implies (20), which completes the proof. Proof of Proposition 7. According to (18), it is sufficient to show that Πdeter > 2Πpure D D , 1 whenever τ ≥ τˆ(ρ) holds. In terms of ρ and τ , this inequality is for all τ ∈ (0, 2 ) equivalent to 9(1 − 2τ )2 − 19 ρ > 18 (1 + ρ)(1 − 2τ )2 , or −ρ> . 17 − 36τ + 36τ 2 Now it remains to show that condition (21) implies the above condition, i.e., that 3(1 − 2τ ) 9(1 − 2τ )2 > , 2 17 − 36τ + 36τ 2(1 − τ )

or equivalently

3(1 − 2τ )(11 − 18τ + 24τ 2 ) > 0. 2(1 − τ )(17 − 36τ + 36τ 2 )

This obviously holds for all τ ∈ (0, 12 ), which completes the proof.

B

Appendix: Mixed bundling equilibrium prices

This appendix provides the equilibrium prices in mixed-bundling equilibria, which were omitted in the proofs. Mixed bundling with good XE — case of entry:

23

pM =

∆D +∆E ∆D +9∆E

[(−∆D + 6∆E )θ + (∆D − 3∆E )θ],

pM E =

1 ∆D +9∆E

[(∆2D + 3∆D ∆E + 6∆2E )θ − (∆D + ∆E )(∆D + 3∆E )θ],

pD = − pE =

∆D (∆D +∆E ) ∆D +9∆E

∆D +∆E ∆D +9∆E

(5θ − 4θ),

[(2∆D + 3∆E )θ − 2(∆D + 3∆E )θ].

Mixed bundling with good XD : pM =

3∆E (∆D +∆E ) ∆D +9∆E

pM D = −

∆D ∆D +9∆E

(2θ − θ),

[(∆D + ∆E )θ − (∆D + 5∆E )θ],

pD = −

∆D (∆D +∆E ) ∆D +9∆E

pE = −

∆D +∆E ∆D +9∆E

(2θ − θ),

[(∆D + 3∆E )θ − (∆D + 6∆E )θ].

Mixed bundling with both goods XD and XE : pM =

∆D +∆E 2(5∆D +9∆E )

pM D = − pM E =

pE =

C

∆D 4(5∆D +9∆E )

1 4(5∆D +9∆E )

pD = −

[(5∆D + 6∆E )θ − 3(∆D + 2∆E )θ],

[(5∆2D + 21∆D ∆E + 24∆2E )θ − (∆D + ∆E )(7∆D + 12∆E )θ],

∆D (∆D +∆E ) 2(5∆D +9∆E )

∆D +∆E 2(5∆D +9∆E )

[5(∆D + ∆E )θ − (11∆D + 19∆E )θ],

(5θ − θ),

[(5∆D + 6∆E )θ − (7∆D + 12∆E )θ].

Appendix: Welfare comparison

Symbolically consumer surplus and social welfare can be written as Z θ Z Z θ ˜ ˜ = [˜ s(θ)θ − p˜(θ)] dθ = s˜(θ)θ dθ − Π, SW = CS + Π CS = θ

θ

θ

s˜(θ)θ dθ.

θ

˜ is total industrial profit (i.e., sum of profits of all firms), s˜(θ) stands for total where Π quality of the package consumer θ purchases, and p˜(θ) stands for the price he pays.25 A direct computation yields the following social welfare: SW noB = 12 (sM D + sM E )(1 − τ )(1 + τ ) − pure

=

deter

=

SW SW

1 2 (sM D 3 8 (sM D

+ sM E )(1 − τ )(1 + τ ) − + sM E ) −

1 18 [∆D (2 − τ )(4 + τ ) + ∆E (1 − 1 32 (∆D + ∆E )(1 − 2τ )(1 + 6τ ),

2τ )(1 + 4τ )],

5 18 ∆D .

Consumer surplus can be then obtained by subtracting firms’ equilibrium profits. Proceeding analogously as in part (ii) of Proposition 5, it can be verified that: SW noB > SW pure ,

if and only if

CS noB < CS pure ,

if and only if

25

(1 − 2τ )(7 + 10τ ) , 119 − 68τ + 92τ 2 (1 − 2τ )(31 − 38τ ) −ρ > . 239 − 548τ + 284τ 2 −ρ >

When the market is undercovered, the lower bound of the integrals needs to be changed respectively. For example, in the equilibrium of the subgame Γdeter it would be pM /(sM D + sM E ) = 12 θ.

24

Note that both these conditions are fulfilled when τ ≥ τˆ(ρ); see Figure 2 and Table 3 for illustration. Furthermore, observe that SW deter does not depend on τ and that both SW noB and SW pure are decreasing in τ . This can be obtained directly by taking derivatives: noB d dτ SW pure d dτ SW

= 91 [∆D − ∆E − (sD + 8sE + 8sM D + sM E )τ ], = 18 [−(∆D + ∆E ) − 2(3sD + 3sE + sM D + sM E )τ ],

which are obviously negative (recall that ∆D < 0 < ∆E and ∆D + ∆E > 0). Therefore, both SW noB and SW pure achieve their minimum at τ = 12 . For this value I get 7 SW noB − SW deter = − 12 ∆D > 0,

SW pure − SW deter = 85 ∆D < 0,

which proves Proposition 6. It is important to note that condition τ ≥ τˆ(ρ) is not sufficient for inequality SW pure < SW deter to hold. Consider, for example, sD = 0.42, sE = 0.48, sM D = 0.36, sM E = 0.64. 1 1 Then (9) reads as max{ 20 , 11 } ≤ τ ≤ 12 , ρ = − 38 , and τˆ(ρ) = 73 . However, τ = 0.44 > 37 noB yields SW = 0.4233, SW pure = 0.4018, and SW deter = 0.3917. The comparison of consumer surplus is much more cumbersome than the comparison of social welfare. A direct computation reveals that the consumer surplus in no-bundling and pure-bundling equilibria are concave, but may not be monotone in τ . Therefore, conditions (9) and τ ≥ τˆ(ρ) come into play (note that they were not required for part (i) of Proposition 6). A partial result can be obtained by comparing the consumer surplus in the boundary case τ = 12 , where I get CS noB = 18 (sD + 2sE + 2sM D + sM E ),

CS pure = 81 (2sD + 2sE + sM D + sM E ), CS deter =

1 72 (4sD

+ 5sM D + 9sM E ).

Thus, CS pure > CS deter and CS noB > CS deter . Numerical simulations then suggest that these relations hold for all admissible values of parameters (see Conjecture 1).

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