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University of Minnesota Law School Legal Studies Research Paper Series Research Paper No. 09-43

Separating Complements: The Effects of Competition and Quality Leadership Matteo Alvisi, Emanuela Carbonara, Francesco Parisi This paper can be downloaded without charge from the Social Sciences Research Network Electronic Paper Collection

Electronic copy available at: http://ssrn.com/abstract=1502919

Separating Complements: the E¤ects of Competition and Quality Leadership Matteo Alvisi, Emanuela Carbonara, Francesco Parisi November 9, 2009

Abstract The law and economics literature on the tragedy of the anticommons suggests that producers of complementary goods should integrate themselves. Recent decisions by the antitrust authorities seem to indicate that there is tradeo¤ between the “tragedy” and the lack of competition which might exist in an integrated market structure. In this paper we analyze such tradeo¤ in oligopolistic complementary markets when products are vertically di¤erentiated. We show that quality leadership plays a crucial role. When there is a quality leader, forcing divestitures or prohibiting mergers, thus increasing competition, lowers prices and enhances consumer surplus. However, when quality leadership is shared, “disintegrating” …rms may lead to higher prices. Therefore, concerns about the tragedy of the anticommons are well posed in antitrust decisions. Keywords: complements, anticommons, competition, mergers, vertical di¤ erentiation. JEL Codes: C7, D42, D43, K21, L11, L12, L13, L40, M21

Matteo Alvisi, Department of Economics, University of Bologna and SAIS, Johns Hopkins University, e-mail:[email protected]. Emanuela Carbonara, Department of Economics, University of Bologna, e-mail:[email protected]. Francesco Parisi, School of Law, University of Minnesota and Department of Economics, University of Bologna, e-mail:[email protected]. We thank Luigi Alberto Franzoni, Andrea Mantovani, Emanuela Michetti, Vincenzo Denicolò, and participants to the 2009 EALE Conference in Rome and to seminars in Bologna and Minneapolis for useful comments.

1


1

Introduction

Recently, law and economics literature has devoted a considerable amount of attention to a speci…c class of market distortions, known as “the tragedy of the anticommons” (Buchanan and Yoon 2000, Parisi et a. 2005, Dari-Mattiacci and Parisi 2007). Based on Cournot (1838)’s “complementary oligopoly”(later “double marginalization”in the industrial organization jargon), such literature argues that social welfare might be better served by policies favoring integration. In fact, when complementary goods are sold by di¤erent …rms, prices are higher than those set by a monopoly selling all the complementary goods. A merger of all independent producers would then yield a higher consumer surplus. While the resulting social welfare may fall short of the perfectly competitive one, a merger might represent a second best solution. Strictly speaking, this literature is applicable only to situations in which the markets for all complementary goods are monopolies. However, there are few real world examples in which markets for all complements forming a system are monopolies. More often, we face situations in which each complement is produced in an oligopolistic setting. Consider, for instance, software markets, where each component of a system is produced by many competing …rms, such as Microsoft, Apple, Unix and Linux for operating systems; Microsoft, Google, Apple, Mozilla for Internet browsers, and so on. Similarly, consider the market for photographic equipment, in which both camera bodies and lenses are produced by many competing companies (Nikon, Canon, Olympus, Pentax, etc.), some of which are active only in the market for lenses (Tamron, Sigma, Vivitar). In such cases, integration would entail two di¤erent “market failures”: on the one hand integration may reduce the extent of the tragedy, on the other hand it may lower welfare because of reduced competition. In this paper we look at the interplay of competition and double marginalization and we identify the conditions under which either of them prevails. We aim to formulate policy recommendations to guide antitrust authorities in cases in which complementary goods are produced in oligopolistic markets. Dari-Mattiacci and Parisi (2007) show that the nature of the anticommons problem changes when there are multiple sellers providing perfectly-substitutable complementary goods. Speci…cally, under Bertrand competition, two (perfect) substitutes for all but one complement are su¢ cient to eliminate the tragedy. In such case, competing inde-

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pendent …rms set prices whose sum equals the price chosen by a monopolist selling all components. Thus, if the goods are perfect substitutes, the tragedy of the anticommons is not an issue in oligopolistic markets. Integration has no impact on social welfare and the prescription for competition policy would be inactivity (i.e. maintaining the existing market con…guration, be that integration or separated …rms). We prove that these conclusions change when the assumption of perfect substitutability is relaxed. In particular, we study a case in which components are vertically di¤erentiated with respect to their quality. Until recently, antitrust authorities appear to have disregarded the tragedy of the anticommons. For instance, in United States v. Microsoft Corp., Microsoft was required to divest branches of its business other than operating systems, creating a new company dedicated to application development. The break-up plan (later abandoned in the US but enforced in Europe) would have created two …rms selling complementary goods, possibly generating a tragedy of the anticommons. More recent decisions have made explicit reference to the “tragedy”. In the case General Electric-Honeywell (jet engines and avionics), the European Commission explicitly acknowledged that the merger between the two …rms would generate lower prices. Interestingly, however, the merger was prohibited, since, according to the Commission, post-merger prices would be so low as to injure other …rms, thus reducing competition.1 Antitrust authorities seem to believe that they are facing a trade o¤ between the tragedy of the anticommons and the lack of competition, and that they should allow integration only when the former becomes a more serious problem than the latter.2 We show that the type of quality leadership in the market is the driving force behind the determination of the relative strength of anticommons problems and lack of competition. To de…ne quality leadership, consider a setting in which competing integrated …rms produce all components of a system (e.g., operating system plus word processor or camera body plus lenses). We have a “quality leader”when a single …rm produces better quality components than all other competitors. For instance, the same …rm produces both the high-quality operating system and the high-quality word processor. We have 1

European Commission Decision of 03/07/2001, declaring a concentration to be incompatible with the common market and the EEA Agreement Case, No. COMP/M.2220 - General Electric/Honeywell. For a thorough analysis of past European Commission decisions, GE-Honeywell case included, see also Russo, Schinkel, Gunster, Carree (2009). 2 The "e¢ ciency o¤ence" argument used by the EC in the GE-Honeywell case is analyzed by Motta and Vasconcelos (2005), which considers the impact of such Antitrsut decision in a dynamic setting.

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“shared quality leadership” in the opposite situation, in which one …rm manufactures the best operating system and another …rm manufactures the best word processor. When the market is characterized by the presence of a quality leader, either “disintegrating” (i.e. “breaking up”) a …rm producing complementary goods or prohibiting a merger leads to lower prices, lower pro…ts and higher consumer surplus. On the contrary, if a market is characterized by shared leadership, integration is welfare superior to competition, since disintegrating (or not allowing mergers) would create an anticommons problem. In other words, while the negative e¤ects of lack of competition always overcome the anticommons problem in the presence of a quality leader, the tragedy of the anticommons prevails in case of shared leadership. One of the key insights of our analysis is that, with full-quality leadership, complements produced by the same …rm are in fact perceived as substitutes, so that an increase in the price of one good increases the demand for the complements produced by the same …rm. In the case of shared leadership, on the other hand, cross-price e¤ects among complements produced by the same …rm have the usual impact in that an increase in the price of one good decreases the demand for all complements. The cross-price e¤ect among complements produced by di¤erent …rms has instead an opposite impact, with an increase in the price of a good inducing an increase in the demand of complements produced by other …rms. These …ndings stress how the type of quality leadership, together with the number of …rms, is a crucial factor in assessing the e¤ects of antitrust policies in complementary markets.

1.1

Related Literature

Besides the law and economics literature on the tragedy of the anticommons cited above, this paper is related to the literature on "mix & match": …rms producing all or some components of a system might sell them as a bundle or separately, allowing consumers to fully "mix and match" across …rms (Matutes and Regibeau 1988, Einhorn 1992, Denicolò 2000). There are two main di¤erences between such literature and our paper. First of all, we assume that consumer tastes are distributed across systems and not across single components. In the latter case, in fact, the demand for each component would be independent of the prices of other components, i.e. there are no cross-price

4

e¤ects, at least as long as …rms are not allowed to engage in mixed bundling practices.3 It is therefore questionable whether such previous literature is fully capable of giving account of the complexities of complementary markets where substitutes exist for each complement. Second, our approach is more policy oriented. Rather than focusing on …rms’strategic decisions we analyze the impact of integration and/or mergers on social welfare. Our results also provide a contribution to the literature studying pricing decisions and welfare e¤ects of mergers in complementary system markets (Dalkir et al. 2002, Tan and Yuan 2003, Gans and King 2006, Choi 2008). Such literature has only focused on cases where the merged …rm engages in “mixed bundling”. We choose not to consider such practices, since our model already o¤ers novel insights on the e¤ects of antitrust policies in complementary markets.4 As for the analysis of the tradeo¤ between the tragedy of the anticommons and competition, this paper is related to the literature on vertical di¤erentiation and entry (Nalebu¤ 2004, Chen and Nalebu¤ 2006, Casadesus-Masanell et al. 2007, Alvisi et al. 2009). In particular, while Nalebu¤ (2004) argues that integration tends to generate barriers to entry, Alvisi et al. (2009) show that allowing …rms to sell all components of a system may be both welfare enhancing and pro-competitive. The paper is organized as follows. Section 2 introduces the model. Section 3 and 4 analyze the structure of market demand under the two alternative assumptions of full and shared quality leadership. Section 5 presents the main results of the paper, showing how the e¤ects of mergers and disintegration change with di¤erent quality leadership. Intermediate cases in which integrated …rms compete with independent producers of separate components are analyzed separately in Section 6. Welfare analysis is performed in Section 7. Section 8 concludes and presents some extensions. Appendix A contains the proofs of some Propositions in the text and Appendix B sketches the proof for the existence of the equilibria studied in the paper. 3

Under “mixed bundling”, the merged …rm sells the individual components both separately and as a bundle (and the bundle is o¤ered at a discount). See, for instance, Matutes and Regibeau (1992). For an extensive review of the economic literature on bundling practices see Kobayashi (2005). 4 The analysis of of the impact of mixed- bundling practices under di¤erent types of quality leadership is left for future research.

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2

The Model

Consider two complementary goods, 1 and 2; which are valuable only if purchased together. An example of such a case would be a software package run on a complementary hardware product. Consumers combine 1 and 2 on a one to one basis to form a system. Initially there are two competing …rms, A and B; each manufacturing both complements. Components are fully compatible, so that there are four ways to form a system, de…ned as AA = fA1 ; A2 g ; BB = fB1 ; B2 g ; AB = fA1 ; B2 g ; BA = fB1 ; A2 g : We consider two distinct producer relationships: “full quality leadership”, where …rm A manufactures a superior version of both components and “shared quality leadership”, where …rm A manufactures a superior version of component 1 (hardware) and …rm B a superior version of component 2 (software). In analogy with Einhorn (1992), we also assume that, for all consumers, the incremental value of the hardware to the system is higher than the one provided by the software. As such, a system with good quality hardware is valued by consumers more than a system with good quality software. Under full quality leadership, the qualities of the four available systems will then be ranked as follows: qAA > qAB > qBA > qBB . Analogously, under shared quality leadership, the quality ranking is qAB > qAA > qBB > qBA: Let the price of component i1 be pi1 and the price of component j2 be pj2 (i = A; B and j = A; B). Then the system ij is available at a total price of pij = pi1 + pj2 ; (i = A; B and j = A; B). All …rms set their prices simultaneously. Without loss of generality, we assume that all components are produced at zero costs. Each consumer has the same reservation price V for the worst available system. Let

represent the consumer taste parameter for the quality of the system, where

is

uniformly distributed in the interval [0; 1]. The (indirect) utility function of a consumer purchasing system ij is then Uij = V + qij

pij : This functional form is similar to that

used by Gabszewicz and Thisse (1979) and Economides (1989). However, with respect to their approach, we only consider cases in which all consumers purchase one of the four available systems and all systems have positive demand.5 We then consider the possibility of breaking up both A and B into two separate entities, each producing one of the two components, leading to four independent producers, 5 Gabszewicz and Thisse (1979) consider a spectrum of consumers with varying tastes for quality who choose between a low- and a high-quality product. These authors consider some cases that we do not, i.e. some consumers may not purchase any system or some system may not have positive demand.

6

A1 ; A2 ; B1 and B2 : In such a setting the tragedy of the anticommons may reappear and we could observe higher prices with respect to the integrated market case. This is because …rms now do not need to consider the impact of raising their price on the demand of the complementary component. However, the fact that each …rm is now able to control the price of one component only makes competition …ercer, possibly generating lower system prices than in the integrated market case. Depending on the e¤ect that dominates, disintegration may lead to either lower prices and higher consumer welfare or to the opposite result. In the next Sections we will analyze the conditions under which each e¤ect dominates. We will …nd that the form of the quality leadership will play a crucial role in this analysis. In particular, under full quality leadership, competition leads to lower prices and enhances consumer surplus. On the contrary, when quality leadership is shared, breaking up integrated …rms (or, equivalently, prohibiting a merger) may lead to higher prices so that concerns about the tragedy of the anticommons are well posed in antitrust policies.

3

Full-quality leadership: complements as substitutes

Under full-quality leadership, A is the high quality producer for both components. Quality takes values in the [0; 1] interval, with the least- and highest-quality systems at its boundaries, i.e. qBB = 0 and qAA = 1: Demand functions for the four systems are AA AB

obtained in the standard way. De…ne

=

pA2 pB2 1 qAB

the parameter value of the mar-

ginal consumer who is indi¤erent between systems AA and AB; and similarly de…ne AB BA

=

pA1 +pB2 pA2 pB1 qAB qBA

and

BA BB

=

pA2 pB2 qBA :

Given the quality ranking under full lead-

ership, a necessary condition to have a positive demand for all four systems is6 0
0;

indicating that A1 and A2 are perceived as substitutes notwithstanding their technical complementarity. In fact, as pA2 increases, the demand for A1 does not decrease in the upper part of the market. As indicated in Figure 2, some consumers (segment C) might shift from system AA to AB, but they do not vary their demand of A1 : In particular, the consumer who was previously indi¤erent between systems AA and AB now prefers AB (

AA AB

in Figure 2 has moved to the right) but still purchases A1 : On the other hand,

the increase in pA2 raises the demand for AB, which is now relatively cheaper than BA. In fact, the threshold

AB BA

depends negatively on pA2 and thus shifts to the left.

Then, overall, the demand for A1 becomes strictly larger, increasing by the segment D in Figure 2. This same analysis can be very easily applied to the demand of B1 with respect to F with respect to p F pB2 ; to DA2 A1 and to DB2 with respect to pB1 : This special charac-

teristic of the demand functions under full leadership, in which technical complements produced by the same …rm behave as substitutes and exhibit “inverse” cross-price effects, will play a crucial role in the upcoming results and is described more formally in the following proposition. Proposition 1 Under full-quality leadership, the cross-price elasticity of the demand of component i with respect to the price of the same-quality complement j,

z ij

(z =

A; B; i; j = 1; 2 and i 6= j), is always positive, i.e. each …rm produces goods that consumers perceive as gross substitutes. It is well-known that when a …rm produces substitute goods, breaking it up into independent production processes unambiguously increases the degree of competition in the market, so that even if technically speaking we are studying complementary products, there may be no tragedy of the anticommons. We postpone such analysis, however, to Section 5, following the description of the shared leadership case. Now,

8

we de…ne pro…t functions and equilibrium prices under full leadership. In pursuit of this purpose, in order to simplify algebra, the analysis will be performed assuming that qAB = 2qBA :7 When …rms A and B produce both components, their overall pro…ts amount to FI A

F + p D F and = pA1 DA1 A2 A2

FI B

F + p D F ; where “F I” stands for “inte= pB1 DB1 B2 B2

grated market with full leadership”. Di¤erentiating FI B

FI A

with respect to pA1 and pA2 and

with respect to pB1 and pB2 and solving the …rst-order conditions simultaneously

yields the following Bertrand equilibrium prices:

Equilibrium pro…ts are

FI A

=

pFA1I =

2qBA (3 5qBA ) 3(1 qBA )

(6)

pFA2I =

4qBA (1 2qBA ) 3(1 qBA )

(7)

pFB1I =

qBA (3 5qBA ) 3(1 qBA )

(8)

pFB2I =

2qBA (1 2qBA ) 3(1 qBA )

(9)

4qBA (5 9qBA ) 9(1 qBA ) ;

FI B

=

qBA (5 9qBA ) 9(1 qBA ) :

Hence, the quality

leader A earns higher pro…ts than B:8

4

Shared quality leadership: the tragedy strikes back

Under shared leadership, A manufactures the high-quality component 1, whereas B manufactures the high-quality component 2. As before, quality takes values in the [0; 1] interval, with the least- and highest-quality systems at its boundaries, i.e. qBA = 0 and qAB = 1: Demand functions for the four systems are obtained as usual. De…ne AB AA

=

pB2 pA2 1 qAA

the parameter value of the marginal consumer who is indi¤erent between

systems AB and BA; and similarly de…ne

AA BB

=

pA1 +pA2 pB2 pB1 qAA qBB

and

BB BA

=

pB2 pA2 qBB :

7 This assumption is with no loss of generality, since all results would also hold in the more general case qAB > qBA . Given that, as stated in footnote 5, we require qAB + qBA > 1, qAB = 2qBA implies qBA > 31 : Moreover, since qAB < 1; then it must be that qBA < 12 . General proofs for generic qAB and qBA are available upon request from the authors. 8 Notice that prices are all positive. The demands for the four systems are however all positive only 1 if V is su¢ ciently large. In particular, condition (1) is always satis…ed whenever V : Also, this 3 equilibrium holds only if none of the …rms has an incentive to unilaterally deviate to di¤erent market con…gurations involving fewer then four systems being purchased. Such possibility is excluded for a su¢ ciently large value of qBA : qBA > 0:41: This value is obtained through tedious algebra and the procedure can be found in Appendix B.

9

Given the quality ranking under shared leadership, a necessary condition to have a positive demand for all four systems is9 BB BA

0
0 (i; j = 1; 2

and i 6= j). According to this result, we expect that under shared leadership the tragedy of the anticommons will play an important role in assessing the implications of breaking integrated producers into independent …rms, and that policy recommendations will be 9

By direct comparison of

AB AA ;

BB BA ;

condition (10) requires qAA + qBB > 1:

10

di¤erent from those indicated under full leadership. Again, we postpone such analysis to Section 5. Now, we de…ne pro…t functions and equilibrium prices under shared leadership. In doing this, as with the full leadership case, we simplify the algebra by assuming qAA = 2qBB : When …rms A and B produce both components, their overall pro…ts amount to SI A

S + p D S and to = pA1 DA1 A2 A2

SI B

S + p D S ; where the superscript SI = pB1 DB1 B2 B2

stands for “integrated market with shared leadership”. Di¤erentiating to pA1 and pA2 and

SI B

SI A

with respect

with respect to pB1 and pB2 and solving the …rst-order conditions

simultaneously, we obtain the Bertrand equilibrium prices pSI A1 =

pSI A2 =

qBB (3 4qBB ) 3(1 qBB ) qBB (1 2qBB ) 3(1 qBB )

Equilibrium pro…ts are

SI A

=

(16)

2 qBB 3(1 qBB )

(17)

qBB (1 2qBB ) 3(1 qBB )

(18)

pSI B1 = pSI B2 =

(15)

4qBB (5 6qBB ) 9(1 qBB ) ;

SI B

=

qBB (2 3qBB ) 9(1 qBB ) :

Hence, the producer of

the …rst component (A) earns higher pro…ts than the producer of the second component (B). This is reasonable, considering the assumption that component 1 provides the system with a higher incremental value than component 2 does. SI SI SI It should be noted that pSI A2 < 0, whereas pA1 ; pB1 and pB2 are all positive in the

relevant parameters range.10 Then, in equilibrium, …rm A would actually …nd it optimal to subsidize the consumption of A2 . As indicated in the previous proposition, while A1 and A2 are perceived as complements, B1 and A2 are perceived as substitutes. Thus, a decrease in pA2 actually increases the demand for component B1 to the advantage of A1 : This can be seen in Figure 4; where, as pA2 decreases, the demand for systems AB and BB decreases, enlarging that of AA and BA: Overall, the demands of A2 and, especially, of A1 increase, with a positive total e¤ect on pro…ts. Thus, it is perfectly reasonable that 10

In analogy with the full quality leadership case, V and qBB have to be su¢ ciently large to guarantee the existence of this equilibrium (see footnote 7). Also, in analogy with footnote 6, the assumption qAA = 2qBB implies qBB 2 13 ; 21 : Note that the negative sign of pSI A2 does not depend on the restriction we imposed on parameters, rather on the complementarity relationship between A1 and A2 and on the assumption of shared leadership. We have in fact obtained the same result also with general quality levels qAA > qBB :

11

…rm A …nds it pro…t-maximizing to sell one of its components below marginal cost in order to increase the consumption of the other complement.11 In the software industry, for instance, Adobe widely distributes its portable document reader for free. Similarly, in the past, Microsoft and Netscape (now part of AOL) have competed by creating new ways to freely distribute their Internet browsers. Finally, Sun Microsystems gives away both its Java virtual machine and Staro¢ ce, the most successful open source o¢ ce suite.12

5

The e¤ects of disintegration

Assume now that a decision of the antitrust authority is passed, requiring the breakup of previously integrated …rms. We then go from a market con…guration where only …rms A and B operate to one where four …rms, A1 ; A2 ; B1 and B2 are active. Component 1 is manufactured by …rms A1 and B1 ; whereas component 2 is manufactured by …rms A2 and B2 . We distinguish the two cases of full and shared quality leadership.

5.1

Full quality leadership

In this case …rms A1 and A2 are still "quality leaders" and produce higher quality goods than …rms B1 and B2 : Pro…ts for each …rm amount to

FD iz

F ; where i = A; B; = piz Diz

z = 1; 2 and the superscript F D stands for "disintegration of an integrated market with full leadership". Di¤erentiating

FD iz

with respect to piz and solving the …rst-order

conditions simultaneously yields the following Bertrand equilibrium prices: qBA (11 16qBA ) 14 19qBA

(19)

2 ) qBA (12 41qBA + 34qBA 2 28 80qBA + 57qBA

(20)

3(1 qBA )qBA 14 19qBA

(21)

pFA1D = pFA2D =

pFB1D = 11

If, for some reason, subsidization were not possible, …rm A would …x pA2 equal to its (zero) marginal cost as a corner solution. We have analyzed such a case but we have found that the main conclusions of the paper remain exactly the same, hence we decided not to include it in this paper. Calculations are available upon request. 12 Clearly, these are examples of cross-market subsidies that have been proved to be sustainable by previous literature in the presence of network externalities in oligopolies (see Shapiro and Varian 1999, Parker and Van Alstyne 2005). Here we …nd that simple complementarity is su¢ cient to justify such cross-subsidization.

12

pFB2D =

2 ) 2qBA (1 3qBA + 2qBA 2 28 80qBA + 57qBA

(22)

2 2 F D = (11 16qBA ) qBA F D = (11 16qBA ) qBA ; FD = A1 A1 A2 (14 19qBA )2 (14 19qBA )2 2q 2q 9(1 q ) 4(1 q ) (1 2q ) BA BA BA BA BA FD = ; FB2D = (14 19q : Again, it can 2 B1 (14 19qBA )2 BA ) (2 3qBA )

Equilibrium pro…ts are (12 17qBA )2 qBA (1 2qBA ) ; (14 19qBA )2 (2 3qBA )

be noticed that …rms producing high-quality components earn higher pro…ts ( F D; Bj

FD Aj

>

j = 1; 2), so that aggregate pro…ts in the high-quality sector are also higher.

The new equilibrium prices can be easily compared to the ones set by two integrated …rms, given by equations (6) to (9). In particular, the following result holds:13 Proposition 3 Under full leadership, breaking up integrated …rms involves lower prices for all components and then for all systems (pFizD < pFizI ; i = A; B; z = 1; 2, so that pFijD < pFijI ; i = A; B; j = A; B). In other terms, breaking up integrated …rms or prohibiting mergers when markets are characterized by full quality leadership would be bene…cial for each consumer, because all available systems can be purchased at a lower price. Thus, as expected, breaking an integrated duopoly in full leadership does not produce any tragedy of the anticommons. As indicated above, in this setting same-quality technical complements actually behave as substitutes. With full leadership, such complements are produced by the same …rm, which then internalizes the negative externality that a decrease in the price of one complement would have on the demand of the other (exactly as it would happen in the case of two substitute goods produced by the same …rm). That’s why integrated …rms set higher prices than independent producers.

5.2

Shared quality leadership

Under shared leadership, …rms A1 and B2 are "quality leaders" and produce higher quality goods than …rms B1 and A2 : Pro…ts for each …rm amount to

SD iz

S ; where = piz Diz

i = A; B; z = 1; 2; where the superscript SD stands for "disintegration of an integrated market with shared leadership". Di¤erentiating

SD iz

with respect to piz and solving the

…rst-order conditions simultaneously yields the following Bertrand equilibrium prices:

pSD A1 =

qBB (11 16qBB ) 14 19qBB

(23)

13 Proposition 3 is proved by direct comparison of prices in expressions (6) to (9) and (19) to (22). The same procedure is adopted to prove also Propositions 4 through 7.

13

2 ) 2qBB (1 3qBB + 2qBB 2 28 80qBB + 57qBB

(24)

3(1 qBB )qBB 14 19qBB

(25)

2 ) qBB (12 41qBB + 34qBB 2 28 80qBB + 57qBB

(26)

pSD A2 =

pSD B1 = pSD B2 =

2 2 SD = (11 16qBB ) qBB ; SD = 4(1 qBB ) qBB (1 2qBB ) ; A1 A2 (14 19qBB )2 (14 19qBB )2 (2 3qBB ) (12 17qBB )2 qBB (1 2qBB ) : Again, it can be noticed that …rms (14 19qBB )2 (2 3qBB )

SD B1

Equilibrium pro…ts are )2 q

9(1 qBB BB ; (14 19qBB )2

SD B2

=

facturing high-quality components obtain higher pro…ts (

SD A1

>

SD B1

and

SD B2

=

manu>

SD A2

i = A; B). Also in this case, aggregate pro…ts tend to be higher in sector A; which is the sector producing component 1; the one with the highest incremental value. The new equilibrium prices are therefore those listed in equations (23) to (26) and can be easily compared to the ones in the previous integrated con…guration, given by equations (15) to (18). In particular, the following result holds Proposition 4 Under shared leadership, breaking up integrated …rms involves higher SD SI SD prices for all components with the exception of pA1 (pSI A2 < pA2 ; pBz < pBz ; z = 1; 2; SD SI SD but pSI A1 > pA1 ). All systems’ prices are higher (pij < pij ; i = A; B; j = A; B).

Notice …rst that “disintegrating the market”generates a tragedy of the anticommons for all prices but pA1 : This asymmetric e¤ect is the result of the interplay of the complex cross-price e¤ects among components, so that, while breaking up …rm A would certainly generate the standard tragedy of the anticommons and raise both pA1 and pA2 ; this would not be the whole story. In fact, we notice immediately that the break-up would certainly lead the new producer of component A2 (previously priced below marginal cost) to …x a higher (positive) price. Such substantial increase in pA2 brings about three di¤erent e¤ects. First, it generates an increase in pB2 ; the price of the direct competitor (and substitute) of A2 and this increases the price of A1 (perceived as substitute of B2 ), thus reinforcing the “tragedy e¤ect”. Second, the increase in pA2 raises pB2 ; and this would tend to reduce pB1 ; the complement of B2 : Because of this, the competition in the market for component 1 gets …ercer and pA1 tends to decrease. Finally, an increase in pA2 brings about a decrease in the price of its complement A1 . The latter two e¤ects more than compensate the increase in pA1 due to the tragedy and to the higher pB2 .14 14 SI Both these e¤ects are also at play with pB1 ; with the di¤erence that pSI B2 ; contrarily to pA2 ; is SI positive, so that the last two e¤ects are weaker and dominated by the …rst. Overall then, pSD > p B1 B1 :

14

SD In any event, even if pSI A1 > pA1 ; the total prices of all systems are larger with

independent producers. Then, breaking up integrated …rms or prohibiting mergers when markets are characterized by shared quality leadership would be detrimental for each consumer because all available systems can be purchased only at a higher price. Here, an integrated duopoly is welfare superior to competition, since breaking …rms would create an anticommons problem. It is …nally apparent how the type of quality leadership, together with the number of …rms (Parisi and Dari-Mattiacci 2007) and the correlation among the consumers’taste parameters for complements (Dalkir et al., 2002) should be one of the key factors in judging antitrust policies in complementary markets. In fact, not in all cases the tragedy of the anticommons drives the result in favor of an integrated market structure.

6

The case for “partial disintegration”

In this section we consider the intermediate cases in which only one …rm in the market is integrated. Again, we distinguish between the two cases of full and shared quality leadership. In this way, we can assess whether the conclusions obtained in Section 5 apply also to market structures in which integrated …rms compete with independent …rms that specialize in the production of a single component. In particular, we will be able to establish the implications for prices and pro…ts when an antitrust authority requires only one …rm (and not both) to divest in the market.

6.1

Full leadership

Under full leadership, we may have two cases: one in which there is an integrated …rm producing two high-quality complements A1 ; A2 ; competing with two independent lowquality …rms producing B1 and B2 (from now on, we label this case F A) and one in which the opposite holds and the integrated …rm produces two low-quality components (B1 and B2 ), henceforth labeled case F B. In case F A pro…ts for each …rm amount to (z = 1; 2). Di¤erentiating

FA A

FA A

F +p D F ; = pA1 DA1 A2 A2

with respect to pA1 and pA2 and

FA Bz

FA Bz

F ; = pBz DBz

with respect to

pFBzA and solving the …rst-order conditions simultaneously we obtain pFA1A =

2 qBA 29 89qBA + 68qBA 2 17 42qBA + 25qBA

15

(27)

pFA2A =

Equilibrium pro…ts are FA B2

=

16qBA (2qBA 1)(3qBA (25qBA 17)2

FA A 2)

2 qBA 19 65qBA + 54qBA 2 17 42qBA + 25qBA

(28)

pFB1A =

qBA (11qBA 7) 25qBA 17

(29)

pFB2A =

4qBA (2qBA 1) 25qBA 17

(30)

=

2 3 qBA (461 2128qBA +3257qBA 1654qBA ) ; (25qBA 17)2 (1 qBA )

FA B1

=

qBA (11qBA 7)2 , (25qBA 17)2

:

In case F B; pro…ts for each …rm amount to

FB Az

F ; (z = 1; 2), = pAz DAz

FB B

=

F + p D F . Equilibrium prices are pB1 DB1 B2 B2

pFA1B =

2qBA (11qBA 7) 25qBA 17

(31)

pFA2B =

8qBA (2qBA 1) 25qBA 17

(32)

pFB1B =

pF B qBA (11qBA 7) = A1 25qBA 17 2

(33)

pFB2B =

pF B 4qBA (2qBA 1) = A2 25qBA 17 2

(34)

Equilibrium pro…ts are 2 F B = qBA (41qBA 66qBA +25) : B (25qBA 17)2

FB A1

=

4qBA (11qBA 7)2 ; (25qBA 17)2

FB A2

=

64qBA (2qBA 1)(3qBA 2) (25qBA 17)2

and

We notice immediately that starting from a fully integrated market structure, breaking either …rm A or …rm B up (or similarly, prohibiting the merger of two independent …rms producing same-quality complements) results in lower prices. Moreover, prices would further be reduced by also breaking up the remaining integrated …rm (moving then from cases F A or F B towards case F D). The following Proposition summarizes these results. Proposition 5 Under full leadership, increasing the degree of market competition (i.e. breaking up integrated …rms or prohibiting a merger) reduces all components’ and systems’ prices (pFizD < pFizA < pFizI and pFizD < pFizB < pFizI ; i = A; B; z = 1; 2). In conclusion, with full leadership, increasing the degree of market competition (i.e. breaking up …rms or prohibiting mergers) is always bene…cial for consumers. In other words, starting from a fully integrated market, there is a clear direct relationship between 16

the equilibrium prices and the degree of integration. In fact, breaking one of the two …rms, reaching either case F A or F B; involves lower prices. Moreover, prices are further lowered moving from F A or F B; and breaking up the remaining integrated …rm. Figure 5 illustrates this relationship for pA1 and qBA = 0:4: These results suggest that the intervention of antitrust authorities trying to enhance competition should be supported, notwithstanding the presence of complementary goods.

6.2

Shared leadership

In the intermediate cases in which only one …rm in the market is integrated, break-up generates di¤erent e¤ects on prices depending on whether either A or B is integrated (cases SA and SB respectively, in analogy with subsection 5.1). In case SA; pro…ts for each …rm amount to

SA A

S +p D S ; = pA1 DA1 A2 A2

SA Bz

S ; = pBz DBz

(z = 1; 2). Equilibrium prices are 2 qB 40qB 55qB + 19 2 25qB 42qB + 17

pSA A1 =

2 2qB 10qB 11qB + 3 2 25qB 42qB + 17

(36)

pSA B1 =

qB (5qB 4) 25qB 17

(37)

pSA B2 =

5qB (2qB 1) 25qB 17

(38)

pSA A2 =

Equilibrium pro…ts are SA B2

=

SA A

(35)

=

2 3 qB (205 881qB +1260qB 600qBA ) ; (25qB 17)2 (1 qB )

SA B1

=

qB (5qB 4)2 (25qBA 17)2

and

25qB (2qB 1)(3qB 2) : (25qBA 17)2

In case SB; pro…ts for each …rm amount to

SB Az

S ; (z = 1; 2), = pAz DAz

SB B

=

S + p D S . Equilibrium prices are pB1 DB1 B2 B2

pSB A1 =

qB (16qB 11) 25qB 17

(39)

qB (2qB 1) 25qB 17

(40)

pSB A2 = pSB B1 =

2 qB 16qB 17qB 3 2 25qB 42qB + 17

17

(41)

pSB B2 = Equilibrium pro…ts are 2 3 qB (117 540qB +826qB 419qB ) : (25qB 17)2 (1 qB )

SB A1

=

2 qB 26qB 31qB + 9 2 25qB 42qB + 17 qB (16qB 11)2 ; (25qBA 17)2

SB A2

=

(42) qB (2qB 1)(3qB 2) (25qBA 17)2

and

SB B

=

We can now state the following results. Proposition 6 Starting from a fully integrated market with shared-quality leadership: a) Breaking up Firm A (case SB) implies lower prices for components A1 and B1 and higher prices for A2 and B2 : Also, prices of systems AA and BB increase, whereas prices of systems AB and BA decrease; b) breaking up Firm B (case SA) implies higher prices for all components and therefore a higher price for all systems. In part a) of Proposition 6 we notice that breaking up …rm A produces the standard tragedy of the anticommons for component 2 (prices for A2 and B2 increase) but increases competition in the market for component 1 (both pA1 and pB1 decrease). The intuition for lower prices for A1 and B1 is similar to the one provided to explain the decrease of pA1 in Proposition 4. Also notice that the decrease in pA1 more than counterbalances the increase in pB2 , so that the price of system AB is actually lower. Then, breaking up …rm A is not necessarily detrimental to all consumers. In particular, while the lower portion of the demand (i.e., consumers who are only willing to pay relatively lower prices) always su¤ers from this policy, consumers with high valuation for quality might actually take advantage of the lower price for component 1; obtaining now system AB for less. In conclusion, the welfare e¤ects of such a policy are less obvious and need a separate investigation, performed in Section 7. In part b) of the proposition, the tragedy e¤ect dominates and all prices increase. This happens because, under SB; the increase in pB2 after the break up is less substantial and does not generate the very strong negative e¤ect on both pB1 and pA1 as the increase in pA2 after the breakup of …rm A: Since it is a high-quality product, B2 is priced above marginal cost also with an integrated …rm B so that, ceteris paribus, the cross- price e¤ect between complements is smaller. Moreover, as assumed, the second component contributes less to the system’s value than the …rst, so that its price cannot increase too much after disintegration. 18

The analysis can be completed by studying policies of “sequential disintegration” starting from either SA or SB towards SD. The same intuitions provided for Proposition 6 explain the following results. Proposition 7 Under shared-quality leadership: a) starting from SA and breaking up Firm A implies lower prices for components A1 and B1 and to higher prices for A2 and B2 : All systems’ prices increase with the SD exception of pAB (pSA AB < pAB );

b) starting from SB and breaking up …rm B produces the standard tragedy of the anticommons, that is all systems’ prices increase. Propositions 4, 6 and 7 together o¤er a clear vision of how di¤erent the relationship is between the degree of integration and price levels in shared leadership.

Starting

from SI; sequentially disintegrating the two …rms going …rst either to case SA and SB and then to case SD; generates a monotonic increase in both pA2 and pB2 , whereas the impact on pA1 and pB1 is more complex to analyze. Figure 6 provides a full illustration of the relationship between the degree of integration and the price of all products in the market for qB = 0:4: For instance, panel 6.2 in Figure 6 shows that, starting from SI; breaking up …rm A lowers pA1 , but then also breaking up …rm B and reaching case SD increases pA1 ; even if this second variation is of lower magnitude than the …rst. Similarly, pB1 decreases from SI to SB and then increases from SB to SD but, unlike the previous case, the second variation is larger in magnitude (see panel 6.4). The results illustrated so far have several interesting policy implications, summarized in the following Corollary: Corollary 1 With shared-quality leadership and a fully integrated market: a) breaking up both …rms or breaking up …rm B only (leading to cases SD or SA respectively) always generates the tragedy of the anticommons; b) breaking up …rm A (case SB) reduces the total prices of the highest and the lowest quality systems available; c) starting with full integration, disintegrating A decreases B 0 s pro…ts, whereas disintegrating B increases A0 s pro…ts. 19

7

Welfare analysis

We now proceed to analyze social welfare in the two cases of full and shared leadership.

7.1

Full leadership

By comparing pro…ts in cases F I; F A and F B, we …rst notice that breaking up one of the two …rms of a fully integrated market decreases pro…ts for all …rms, including the one that remains integrated.15 This implies that …rms should not be in favor of an authority prohibiting a merger of rival independent …rms or breaking up an integrated rival. In fact, with no tragedy of the anticommons at play, a lower degree of integration only increases the number of …rms in the market and makes the competition …ercer. Moreover, pro…ts for all …rms would further decrease when the market structure goes from being partially integrated to totally disintegrated, as we notice by comparing pro…ts in cases F A; F B and F D. In other terms, with full leadership there also exists a direct relationship between the degree of integration and individual pro…ts, so that each single …rm should always be against anti-mergers policies, no matter whether such policies hit them directly or their rivals only and whether they produce high- or low- quality components. As for aggregate producer surplus, if such direct relationship holds for each single …rm, it holds a fortiori for the whole industry. Thus, P S F I > P S F A > P S F D and P S F I > P S F B > P S F D , implying that with full leadership the industry as a whole would always be against policies of disintegration. The analysis of consumer surplus under full leadership is pretty straightforward and could be performed simply checking Proposition 3. The prices of all components are the highest with a fully integrated market and the lowest with four independent …rms, with the partially integrated market structures providing intermediate values. Thus, the rankings in terms of consumer surplus result CS F D > CS F A > CS F I and CS F D > CS F B > CS F I . As usual, policy recommendations so as to maximize social welfare depend on the relative weights attached to each social group. 15

When we lower the degree of integration, we break up at least one …rm into two separate entities. Thus, what happens more precisely is that the sum of the pro…ts of the newly independent …rms is always lower than the pro…t of the pre-break up integrated …rm(s).

20

7.2

Shared leadership

As suggested by Propositions 4 and 6 and 7, when quality leadership is shared the relationship between the degree of integration and …rms’ prices is not as direct as in the full leadership case. This implies that policy recommendations may not be clear cut and may require a very careful analysis of price levels under alternative market con…gurations. As for producer surplus (P S), our model suggests that the tragedy of the anticommons prevails, as indicated in the following proposition. Proposition 8 In cases of shared leadership, producer surplus decreases with the degree of integration. In particular, a) P S SD > P S SA > P S SI and b) P S SD > P S SB > P S SI : Proof. See Appendix A. The second part of inequality a) appears intuitive if analyzed together with Figures 6.1 and 6.3. Starting from an integrated market structure, breaking up …rm B increases all …rms’prices. Given that total market demand remains unchanged, all …rms’pro…ts and, consequently, producer surplus increase. Also, the …rst part of the inequality holds, notwithstanding the decrease in pA1 and pB1 obtained when shifting from case SA to case SD: The increase in pro…ts in selling A2 and B2 more than compensates for the lower pro…ts provided by components A1 and B1 ; and this happens even if three of the four available systems in the market cost more in case SA than in case SD; as stated in Proposition 6. Symmetrically, while the …rst part of inequality b) is a direct implication of the general price increase from case SB to case SD; the second part derives from the stronger e¤ects on pro…ts of components A2 and B2 :16 The comparison of equilibrium prices in Propositions 6 and 7 also indicates that system prices are never lower in case SA and SD than in cases SB and SI and this should be re‡ected in consumer surplus, as con…rmed by the following Proposition. Proposition 9 With shared leadership, CS SB > CS SI > CS SD and CS SI > CS SA > CS SD : Proof. See Appendix A. As for market con…gurations SB and SI; Proposition 7 indicates that starting from an integrated market and breaking up …rm A (that is, the …rm producing the high-quality 16

In such shift, two of the four available systems cost more and two cost less (see Proposition 7).

21

component that provides the higher incremental value to utility levels), increases the prices of the systems previously produced by the same …rm (AA and BB) and decreases the prices of “mixed systems” AB and BA: Notice that, under shared leadership, AA and BB are the “intermediate systems” in terms of quality, whereas AB and BA are the highest and lowest quality systems, respectively (see Figure 3). Then, breaking up …rm A especially bene…ts those consumers at the extremes of market demand (i.e., those purchasing system AB and BA). This e¤ect more than counterbalances the increases in pAA and pBB ; so that aggregate consumer surplus is actually higher in case SB than with fully integrated …rms. In an oligopolistic setting, then, while the tragedy makes full integration always better for consumers than the presence of four independent …rms, breaking up the …rm producing the component that at the same time is of high quality and provides the highest incremental value to a system (…rm A) increases consumer surplus even more. To fully understand this result we should recall that …rm A; when integrated, sets a price below marginal cost on the second (low-quality) component. Doing this allows A to set a very high price on A1 ; thus forcing consumers to buy A1 combined with A2 and reducing their possibility of combining it with B2 to build the highest-quality system. By breaking up …rm A; the new …rm producing A2 will have to set a price equal or above marginal cost and A1 will be forced to sell at a lower price. Consider, for example, the case of Adobe Writer and Adobe Reader. Adobe allows consumers to download its Reader for free from the Internet, pricing it at or below marginal cost. If we believe that the highest quality portable Writer in the market is indeed the Adobe one, our results would imply that breaking up Adobe in two …rms, one producing the Reader and the other the Writer, would decrease the price charged for the Writer and increase consumer surplus. Finally, starting from a disintegrated market structure, Proposition 9 indicates that sometimes mergers of independent …rms producing single components should be allowed, if not encouraged. In particular, consumer surplus may increase if the merged …rms produce goods of di¤erent quality levels (B1 and B2 in our case). In our initial example of software markets, the integrated production of a low-quality operating system and an high-quality internet browser should then be judged favorably from an antitrust perspective.

22

8

Conclusions

While in the past competition policy has disregarded the tragedy of the anticommons, more recent decisions have explicitly considered it when assessing the e¤ects of mergers and divestitures. The underlying belief shared by antitrust authorities is that complementary markets are characterized by a tradeo¤ between the tragedy of the anticommons and the lack of competition. In that sense, integration should be allowed only when double marginalization is a more prominent problem than the reduction in the number of active …rms in the market. In this paper we have analyzed such tradeo¤s in oligopolistic complementary markets, where each component is produced by more than one (vertically di¤erentiated) …rm. Previous literature (Dari-Mattiacci and Parisi 2007) argued that competition for all but one component solves the tragedy when goods are homogeneous. We prove that this may not be the case when goods are di¤erentiated. In particular, we have shown that the relative strength of anticommons problems and lack of competition is not simply related to the number of active …rms manufacturing each component of a system but also, and crucially, to the type of quality leadership characterizing the market. In the presence of a quality leader, forcing …rms to divest or prohibiting mergers leads to lower prices, lower pro…ts and higher consumer surplus. On the contrary, if a market is characterized by shared leadership, integration (at least "partial") is to be preferred to competition, since the tragedy of the anticommons prevails. With full quality leadership, complements produced by the same …rm appear as substitutes in the consumers’demand functions, while with shared leadership, they exhibit the standard cross-price e¤ect. It is the cross-price e¤ect among complements produced by di¤erent …rms that now has a reversed sign. Finally, the “mix & match”literature underlines how mergers and/or divestitures are not the only available strategy to …rms producing complementary goods. In fact, they can also increase pro…ts implementing pure or “mixed-bundling” practices. A possible extension would be to provide a full-‡edged analysis of mergers and divestitures, taking into account all possible selling strategies that …rms could adopt after merging. What might occur is that …rms might be able, through “mixed bundling”, to charge higher prices to consumers who buy components separately and do “mix & match”. In this case, allowing …rms to merge and then to make their products compatible can actually worsen

23

consumer surplus, contrary to common wisdom, even in comparison to the pre-merge oligopolistic price competition. While the antitrust authorities nowadays seem to have accepted the tragedy of the anticommons as a fundamental element of policies towards mergers, there seems to be no current clear recommendation to consider carefully the consequences of allowing compatibility after a merger.

References [1] Alvisi, M., Carbonara E., Dari-Mattiacci, G., and Parisi, F. (2009), ‘Complementing Substitutes: Bundling, Compatibility and Entry’, Amsterdam Center for Law & Economics Working Paper No. 2009-10 [2] Buchanan J., and Yoon, Y.J. (2000), ‘Symmetric Tragedies: Commons and Anticommons’, 43, Journal of Law and Economics, 1-13. [3] Casadesus-Masanell, R., Nalebu¤, B., and Yo¢ e D., (2007), ‘Competing Complements’, Harvard Business School Working Paper, n.09-009. [4] Chen, M.K. and Nalebu¤, B. (2006), ‘One-Way Essential Complements’, Cowles Foundation Discussion Paper No.1588. [5] Choi, J.P. (2008), Mergers with bundling in complementary markets, The Journal of Industrial Economics, 56(3): 553-577. [6] Cournot, Augustin. Recherches sur les Principes Mathematiques de la Theorie des Richesses, Paris: Hachette, 1838. (English translation by N. T. Bacon published in Economic Classics [Macmillan, 1897] and reprinted in 1960 by Augustus M. Kelly.) [7] Dalkir, S.; Eisenstadt, D.; Gerstle, A. and Masson, R. T., 2002, ’Complementary Goods, Monopoly vs. Monopoly Power: A Reassessment of Merger E¤ects,’unpublished manuscript, Cornell University. [8] Dari-Mattiacci, G. and Parisi, F. (2007), ‘Substituting Complements’, Journal of Competition Law and Economics, 2 (3), 333-347. [9] Denicolò, V. (2000), ‘Compatibility and Bundling with Generalist And Specialist Firms’, The Journal of Industrial Economics, 48(2), 177-187.

24

[10] Economides, N. (1989), Desirability of Compatibility in the Absence of Network Externalities, American Economic Review, 79: 1165-1181. [11] Einhorn, M.A. (1992), ‘Mix and Match Compatibility with Vertical Product Dimension’, RAND Journal of Economics, 23 (4), 535-547. [12] Gabszewicz, J.J. and J.F. Thisse (1979), Price Competition, Quality and Income Disparities, Journal of Economic Theory, 20: 340-359. [13] Gans, J. S. and S. P. King (2006), ‘Paying for Loyalty: Product Bundling in Oligopoly’, The Journal of Industrial Economics, 54(1): 43-62. [14] Kobayashi, B. (2005), ‘Does Economics Provide a Reliable Guide to Regulating Commodity Bundling by Firms? A Survey of the Economic Literature’, Journal of Competition Law & Economics, 1(4): 707-746. [15] Motta, M. and Vasconcelos, H., (2005), ‘E¢ ciency Gains and Myopic Antitrust Authority in a Dynamic Merger Game’, International Journal of Industrial Organization, 23(9-10), 777-801. [16] Matutes, C. and Regibeau, P. (1988), “‘Mix and Match”: Product Compatibility without Network Externalities’, RAND Journal of Economics, 19 (2), 221-233. [17] Matutes, C. and P. Regibeau (1992), ‘Compatibility and Bundling of Complementary Goods in a Duopoly’, Journal of Industrial Economics, 40:37-53. [18] Nalebu¤, B. (2004), ‘Bundling as an Entry Barrier’, Quarterly Journal of Economics, 1:159-187. [19] Parisi, F., Schulz, N., and Depoorter, B. (2005), ‘Duality in Property: Commons and Anticommons’, International Review of Law and Economics, 25 (4), 578-591. [20] Parker, G.G. and Van Alstyne, M.W. (2005), ‘Two-Sided Network E¤ects: A Theory of Information Product Design’, Management Science, 51(10), 1494-1504. [21] Russo, F., Schinkel, M.P., Gunster, A.M., and Carree, M., (2009), ‘European Commission Decisions on Competition: Economic Analysis in Antitrust and Merger’, Cambridge University Press, forthcoming.

25

[22] Shapiro, C. and H. Varian (1999), Information Rules. A Strategic Guide to the Network Economy, Harvard Business School Press, Boston, Massachusetts. [23] Tan, G. and Yuan, L. (2003), ‘Strategic Incentives of Divestitures of Competing Conglomerates’, International Journal of Industrial Organization, 21, 673-697.

A

Appendix A

A.1

Proof of Proposition 8 P

De…ne P S SD P S SI

SI + A

P S SB

SB B

P

SD iz

i=A;B z=1;2 (7 9qBB ) = qBB 9(1 qBB ) ;

SI B

+

P S SI and either

P

z=1;2 P S SA

SB Az

=

2 +255403q 3 4 5 qBB (62319qBB 178642qBB BB 182111qBB +51805qB 8674) ; 2 +75q 3 (14 19qBB )2 (135qBB 176qBB 34) BB

=

2 3 ) P SA qBB (271 1162qBB +1650qBB 775qBB SA + ; A Bz = (17 25qb)^2(1 qb) z=1;2 2 3 ) qBB (240 1022qBB +1447qBB 681qBB : By comparing P S SD ; (17 25qBB )2 (1 qBB )

P S SA

or P S SB for qBB 2 ( 31 ; 12 ); it is possible to verify that P S SD >

P S SA > P S SI and P S SD > P S SB > P S SI in the relevant range of the parameters.

A.2

Proof of Proposition 9

Proof. De…ne

CS St

Z1

AB AA

AB

(V +

pSt A1

ZAA St (V + qAA pB2 )d +

pSt A1

pSt A2 )d +

AA BB

AA

ZBB + (V + qBB BB BA

BB

pSt B1

ZBA St pB2 )d + (V

pSt B1

pSt A2 )d

0

(t = I; A; B; D). Substituting the expressions for equilibrium prices relative to each market con…guration and rearranging: CS SI =

CS SD =

2 +18V (1 q 9 28qBB +27qBB BB ) 18(1 qBB )

4 +392(1+2V ) q 3 (7010+2166V )+2q 2 (3145+2318V ) 8q (2939qBB BB (317+413V )) BB BB 2(14 19qBB )2 (2 3qBB )

26

(43)

(44)

CS SA =

4 3 (39+10V )+q 2 (4427+2950V ) 2q ((2025qBB 125qBB BB (909+1139V )+289(1+2V )) BB 2(17 25qBB )2 (1 qBB )

(45)

CS SB =

4 3 (2118+625V )+q 2 (3983+2950V ) q (1731qBB 2qBB BB (1719+2278V )+289(1+2V )) BB 2(17 25qBB )2 (1 qBB )

(46)

By comparing CS SD ; CS SI and either CS SA or CS SB for qBB 2 ( 13 ; 12 ); it is possible to verify that CS SD > CS SA > CS SI and CS SD > CS SB > CS SI in the relevant range of the parameters.

B

Appendix B: Existence of equilibria with full leadership.

In this Appendix, we are going to ascertain the conditions under which all four systems AA; AB; BA and BB are sold in the Bertrand equilibrium with full quality leadership. From Section 3, we know that this happens only if equilibrium prices satisfy (1), which guarantees positive demand for all systems. However, this represents only a necessary condition for such market con…guration to emerge in equilibrium. In fact, by de…nition, in an equilibrium market con…guration neither …rm can …nd it pro…table to deviate from the equilibrium condition. Thus, we need to check that …rms are not willing to set prices that generate the following market con…gurations:17 M1) fAA; AB; BBg (meaning, a market con…guration where only these three bundles are sold); M2) fAA; BA; BBg ; M3) fAA; BBg : In what follows we will analyze the cases of integrated and disintegrated markets. The procedure for partial disintegration is similar and is therefore omitted. We will treat the two cases separately. 17

It is worth noting that deviation to market con…gurations where AA and BB are not sold is not possible. Intuitively, in order to kick the least-quality system BB out of a market where all consumers buy one system (covered-market con…guration), …rms should price higher-quality systems below BB; which is not feasible. Similarly, pushing AA out of the market involves pricing other systems low enough to compensate consumers with a high taste parameter for the reduction in quality, which would be unpro…table for all …rms.

27

B.1

Integration

In the presence of two integrated …rms (A and B), the market con…guration in which four systems are sold is an equilibrium if neither …rm is willing to deviate from the prices set in equations (6) to (9). To prove that this is indeed the case for some parameters’ values, we need …rst the following result for V: Lemma 1 V =

1 3

is the minimum value of V that guarantees market coverage with

integration. Proof. Given that BB is the least-quality bundle, in order for the consumer with the lowest valuation for taste to buy it, it must be UBB = V pFB1I and pFB2I ; pFB1I + pFB2I =

qBA (5 9qBA ) 3(1 qBA ) ;

pFB1I

pFB2I

0: Summing up

which reaches is highest value for qBA = 31 : In

fact, when qBA = 31 ; pFB1I + pFB2I = 0:33; so that V = 0:33 is the minimum value to have market coverage. B.1.1

Deviation to market con…guration M1

We prove that neither A nor B is ever able or willing to deviate to M1. Firm A

In order to deviate to M1 …rm A would need to set pA1 and pA2 such that

demand for system BA; DBA =

AB BA

BA BB

0: Based on market con…guration M1 and

given pFB1I and pFB2I , A would set its prices to maximize I pA2 pF B2 1 2qBA

pA2 1

; obtaining pDEV = A1

qBA (9 11qBA ) 6(1 qBA ) ;

DEV A

pDEV = A2

ing pFB1I ; pFB2I ; pDEV and pDEV into DBA ; we obtain DBA = A1 A2

= pA1 1 3 (6

I pA1 pF B1 2qBA

2 7qBA +2qBA 6(1 qBA ) 2 ) 25qBA +21qBA 6(1+qBA )qBA

+

: Substitut: We can

immediately verify that DBA > 0 always for qBA > 31 ; hence, given that our restriction qAB = 2qBA implies qBA >

1 3

such deviation is never feasible for A:18

Firm B Based on market con…guration M1 and given pFA1I and pFA2I , B would set its I 1 pF pF DEV = p DEV = qBA (3 5qBA ) ; A1 pB1 A2 pB2 B1 2qBA + pB2 1 2qBA ; obtaining pB1 B 3(1 qBA ) 2qBA (1 2qBA ) F I F I DEV DEV DEV and pB2 into B ; we obtain 3(1 qBA ) : Substituting pA1 ; pA2 ; pB1 2 qBA (9 22qBA +9qBA ) : Comparing DEV to FBI in Section 3, we obtain DEV B B 18(1 qBA )2 2 qBA (1 3qBA ) < 0; which is always negative. Hence, such deviation is never 18(1 qBA )2

prices to maximize pDEV = B2 DEV B FI B

=

=

pro…table for B: 18

It is also trivial to verify that setting either pA1 or pA2 so that DAB = 0 (corner solution) is never a pro…table solution for A compared to the initial situation A1 A2 ; A1 B2 ; B1 A2 B1 B2 : In general, we will omit analysis of corner solutions, since it can be proven that they are never pro…table deviations for either …rm.

28

B.1.2


The same is true for deviations to market con…guration M2: deviation is never feasible for A; whereas B never …nds it pro…table. Firm A

In order to deviate to M2 …rm A would need to set pA1 and pA2 such that

demand for system AB; DAB =

AA AB

AB BA

0: Based on market con…guration M2 and

I pA1 pF DEV = p B1 + 1 A1 A 1 qBA I 2 pA2 pF 3 3q 2q q (5 7q ) BA B2 pA2 1 = BA ; obtaining pDEV = 6(1BAqBA )BA ; pDEV A2 A1 qBA 6(1 qBA ) : Substituting 2 +15q 3 16qBA 26qBA BA 3 pFB1I ; pFB2I ; pDEV : Again, it and pDEV into DAB ; we obtain DAB = 6(1 2 )q A1 A2 3qBA +2qBA BA can be veri…ed that DAB > 0 always for qBA > 13 : Hence, such deviation is never feasible

given pFB1I and pFB2I , A would set its prices to maximize

for A: Firm B Based on market con…guration M1 and given pFA1I and pFA2I , B would set its I I pF pF DEV = p DEV = qBA (3 5qBA ) ; A1 pB1 A2 pB2 B1 1 qBA + pB2 qBA ; obtaining pB1 B 3(1 qBA ) 2qBA (1 2qBA ) F I F I DEV DEV DEV DEV pB2 = 3(1 qBA ) : Substituting pA1 ; pA2 ; pB1 and pB2 into B ; we obtain 2 3 DEV = qBA (4 11qBA +2qBA +9qBA ) : Comparing DEV to F I , we obtain DEV FI = B B B B B 9(1 qBA )3 qBA (1 3qBA )2 (1 2qBA ) ; which is always negative. Hence, such deviation is never prof18(1 qBA )3

prices to maximize

itable for B: B.1.3


In this case, only two systems AA and BB are sold. This implies that …rms, although setting the prices for their components separately, have the possibility of recurring to cross-subsidization in order to make sure that demands for the other systems AB and BA are non-positive. We are going to prove that, while deviation from AA; AB; BA BB to M3 is never feasible for A (notwithstanding the afore mentioned price ‡exibility), it is never pro…table for B: Firm A

In order to deviate to M3, …rm A would need to set pA1 and pA2 such that

demand for system AB and BA are non positive. Given pFB1I and pFB2I , A would set its prices to maximize

DEV A

= (pA1 + pA2 ) 1

(pA1 + pA2

pFB1I

pFB2I ) : From

DEV A

it

is trivial to see that pA1 and pA2 are set as a sum, so that, de…ned (pA1 +pA2 )DEV = pDEV A as the system price that maximizes

DEV ; A

any combination pA1 and pA2 such that

pA1 + pA2 = pDEV can be set by …rm A: We obtain pDEV = A A 29

2 3+2qBA 9qBA 6(1 qBA ) :

Substituting

2

it into FI A

=

2 DEV ; DEV = (3+2qBA 9qBA ) A A 36(1 qBA )2 2 (1 3qBA )2 (9 14qBA +9qBA )

: Comparing it with

> 0 always for qBA 2

36(1 qBA )2

1 1 3; 2

FI A

in Section 3,

DEV A

: Hence, if deviation were

feasible, A would always deviate. Feasibility requires that both DAB and DBA are non positive: DAB

0 () pA1

p~A1 1 =

3pA2 (1

DBA

0 () pA1

p~A1 2 =

6pA2 (1

2 ) qBA ) + qBA (1 5qBA + 6qBA 2 ) 3(1 3qBA + 2qBA qBA ) qBA (1 3qBA ) 3(1 qBA )

(47) (48)

This is possible if and only if p~A1 ~A1 1
p A1 if

is the minimum value that guarantees market coverage. Also, p~A1 is increasing 1

30

and only if pA2 > pA2 =

1

2 4qBA + 13qBA 3

3 18qBA

(55)

It is possible to show that p A2 > pA2 always. Then, if p~A1 1 > p A1 ; satisfaction of all feasibility conditions requires p~A1 ~A1 1 < pA1 < p 2 ; p A2 < pA2 < requires p A2
p A1 : Consider then the case where p~A1 1 < p A1 : This occurs if and only if (55) is true, which requires p A2 < pA2 < pA2 : However, we veri…ed that p A2 > pA2 , so that all feasibility conditions are never satis…ed simultaneously and A is never able to deviate to AA BB: Firm B To deviate to M3, given pFA1I and pFA2I , B would set its prices to maximize DEV B

= (pB1 + pB2 )(pFA1I + pFA1I

pB1 + pB2 ): Again, B optimizes with respect to qBA (5 9qBA ) 3(1 qBA ) : Substituting qBA (5 9qBA )2 : Comparing DEV B 3(1 qBA )2

the sum (pB1 + pB2 )DEV = pDEV ; and pDEV = B B into and pDEV B DEV B

FI B

=

DEV = DEV ; we obtain B B qBA (1 3qBA )2 (5 9qBA ) ; which is 9(1 qBA )2

always negative for qBA 2

1 1 3; 2

pFA1I ; pFA2I to

FI, B

. Hence,

such deviation is never pro…table for B: This concludes the proof for the existence of the equilibrium market con…guration fAA; AB; BA; BBg when …rms A and B are integrated and A is the quality leader. It is worth noting that, while it would be always pro…table to deviate for the quality leader (although never feasible), the …rm producing lower quality is never willing to deviate.

B.2

Disintegration

In this case we have four separate …rms A1 ; A2 ; B1 and B2 : Then, fAA; AB; BA; BBg is an equilibrium market con…guration if no …rm is willing to deviate from the prices set in equations (19) to (22) in Section 5.1. In analogy with section B.1. we …rst state the following result for V Lemma 2 V = 0:17 is the minimum value of V that guarantees market coverage in a disintegrated con…guration. in V; whereas p A1 is invariant with respect to V: Hence, if p~A1 > p A1 at the minimum admissible V; 1 p~A1 1 > p A1 for all V > 0:17:

31

B.2.1


We prove that …rm A2 has the incentive to deviate for su¢ ciently low levels of qBA : For all other …rms it is neither feasible nor pro…table to deviate to M1. Firm A1 In order to deviate to M1, …rm A1 would have to set pA1 such that DBA SD Given pFA2D ; pSD B1 and pB2 , A would set its prices to maximize

obtaining pDEV = A1 we obtain DBA = qBA 2

1 1 3; 2

qBA (31 41qBA ) : 28 38qBA 5(1+qBA ) 28 38qBA : We

DEV A1

= pA1 1

0:

D pA1 pF B1 2qBA

;

Substituting pFB1D ; pFB2D ; pFA2D and pDEV into DBA ; A1 can immediately verify that DBA > 0 always for

; hence such deviation is never feasible for A1 :

Firm A2 In order to deviate to M1, …rm A2 would need to set pA2 such that DBA Given pFA1D ; pFB1D and pFB2D , A2 would set its prices to maximize

DEV A2

= pA2 1

0:

D pA2 pF B2 1 2qBA

2 ) (1 2qBA )(28 78qBA +55qBA into = obtaining pDEV : Substituting pFB1D ; pFB2D ; pFA1D and pDEV 2 ) A2 A2 2(28 80qBA +57qBA 2 +157q 3 (154qBA 273qBA 28) BA DBA ; we obtain DBA = : It can be veri…ed that DAB < 0 for 2 ) qBA (28 80qBA +57qBA qBA < 52 : Hence, for qBA < 25 deviation to M1 is feasible for A2 : We need to check

whether it is also pro…table. Substituting pFB1D ; pFB2D ; pFA1D and pDEV into A2 Section 5.1,

DEV A2

FD A2

=

DEV and comparing it with F D in A2 A2 2 3 +38069q 4 5 ) (784 7088qBA +25196qBA 44100qBA 12986q BA BA ; which 2 )2 4(28 80qBA +57qBA

is positive for qBA < 0:38 and negative elsewhere. We can therefore conclude that A2 would never deviate for qBA > 0:38: Firm B1 Given pFA1D ; pFA2D and pFB2D , B1 would set its price to maximize pB1

D pF A1

pB1 2qBA ;

DEV B1

=

qBA (11 16qBA ) : Substituting pFA1D ; pFA2D ; pDEV and pFB2D B1 28 38qBA qBA (11 16qBA ) : Comparing DEV to FB1D , obtain DEV B1 B1 8(14 19qBA )2

obtaining pDEV = B1

DEV ; we obtain DEV = B1 B1 2 q (49 208q +184q BA BA FD = BA ) ; 2 B1 8(14 19qBA )

into

which is negative for qBA > 13 . Hence, such deviation is

never pro…table for B1 : Firm B2 Given pFA1D ; pFA2D and pFB1D , B2 would set its price to maximize pB2

D pF A2

pB2

1 2qBA

pDEV into B2

(1 2qBA )(12 17qBA ) ; obtaining pDEV = qBA : 2 ) B2 2(28 80qBA +57qBA 2 ) (4 9qBA +5qBA DBA ; we obtain DBA = (28 80q +57q2 ) ; BA BA

=

Substituting pFA1I ; pFA2I ; pFB1D and which is always positive in the

admissible range of qBA . Hence, such deviation is never pro…table for B:

32

DEV B2

;

B.2.2


In this case, two …rms may be able to deviate for su¢ ciently low levels of qBA : A1 and B2 : For the other two …rms, deviation is never feasible. Firm A1 In order to deviate to M2, …rm A1 would have to set pA1 such that DAB Given pFA2D ; pFB1D and pFB2D , A1 would set pA1 to maximize 2 7 15qBA +8qBA DEV A1 6(1 qBA ) 2 7 23qBA +16qBA AB (19qBA 14)qBA

obtaining p obtain D

=

DEV A1

0:

D pA1 pF B1 1 qBA

= pA1 1

;

: Substituting pFB1D ; pFB2D ; pDEV and pFA2D into DAB ; we A1

; which is negative for qBA < 0:44: Hence, such deviation

=

is feasible for qBA < 0:44: We need to check pro…tability. Substitute pFB1D ; pFB2D ; pDEV A1 and pFA2D into

DEV : Comparing DEV with FD A1 A1 A1 2 3 ) (49 282qBA +528qBA 320qBA ; which is positive for qBA (19qBA 14)2

in Section 5.1,

DEV A1

FD A1

=

< 0:42: Hence, A1 would never

deviate for qBA > 0:42: Firm A2 In order to deviate to M2, …rm A2 would have to set pA2 such that DAB Given pFA2D ; pFB1D and pFB2D , A1 would set pA2 to maximize 2 ) qBA (30 86qBA +61qBA 2 ) 2(28 80qBA +57qBA 2 3 ) (6 64qBA +151qBA 103qBA AB 2 +228q 3 272qBA 434qBA BA 56

obtaining pDEV = A2 DAB ; we obtain D

DEV A2

0:

D pA2 pF B2 qBA

= pA2 1

;

: Substituting pFB1D ; pFB2D ; pFA1D and pDEV into A2

=

> 0 in the admissible range for qBA .

Hence, such deviation is never feasible for A2 : Firm B1 Given pFA1D ; pFA2D ; pFB2D , …rm B1 would set its price to maximize pB1

D pF A1

pB1 1 qBA ;

DEV B1

qBA (11 16qBA ) DEV and FD FD (28 38qBA ) : Substituting pA1 ; pA2 ; pB1 1 6qBA 38 28qBA > 0 in the admissible range for qBA . Hence,

= obtaining pDEV B1

into DAB ; we obtain DAB =

=

pFB2D such

deviation is never feasible for B1 : Firm B2 In order to deviate to M2, …rm B2 would have to set pB2 such that DAB Given pFA1D ; pFA2D ; pFB1D and pDEV B2 , B2 would set pB2 to maximize 2 ) qBA (12 41qBA +34qBA 2 2(28 80qBA +57qBA ) 2 20 71qBA +61qBA AB 2 160qBA 114qBA 56

obtaining pDEV = B2 DAB ; we obtain D

DEV B2

=p

0:

D p pF B2 A2 B2 qBA

;

: Substituting pFA1D ; pFA2D ; pFB1D and pDEV into A2

=

; which is negative for qBA < 0:48: Hence, such

deviation is feasible for qBA < 0:48: We need to check pro…tability. Substitute pFA1D ; pFA2D ; pFB1D and pDEV into B2

DEV : Comparing B2 2 3 +1060q 4 ) qBA (112 808qBA +2145qBA 2484qBA BA ; 2 )2 4(28 80qBA +57qBA

DEV B2

with

F D; B2

we obtain

DEV B

FI B

=

which is negative for qBA > 0:41: Hence, B2

will never deviate to M2 if qBA > 0:41:

33

B.2.3


In this case we argue that, when the market is disintegrated, each single …rm is not able, with its control of a single price, to eliminate demands for two systems (AB and BA). We prove this for …rms A1 and A2 : The proof for the other two …rms follows the same methodology. Firm A1 In order to deviate to M3, …rm A1 would need to set pA1 such that demands DAB and DBA are non positive. Given pFA2D ; pFB1D and pFB2I , A1 would maximize pA1 1

(pA1 + pFA2D

into DBA ; DBA =

pFB1D

4+7qBA 28 38qBA

pFB2D ) : We obtain pDEV = A1

2 ) 7(2 3qBA +qBA : 28 38qBA

DEV A1

=

Substituting it

> 0 always in the admissible range for qBA . Hence, deviation

is never feasible for A1 . Firm A2 In order to deviate to M3, …rm A2 would need to set pA2 such that demands DAB and DBA are non positive. Given pFA1D ; pFB1D and pFB2I , A2 would maximize pA2 1

(pA1 + pFA2D

pFB1D

stituting it into DAB ; DAB =

pFB2D ) : We obtain pDEV = A2

28

2 3 +199q 4 28 158qBA +375qBA 434qBA BA 2 +434q 3 4 56qBA 272qBA BA 228qBA

2 94qBA +101qBA

56

DEV A2

3 35qBA 2 160qBA +114qBA

=

: Sub-

> 0 always in the admis-

sible range for qBA . Hence, deviation is never feasible for A2 . This concludes our proof of the existence of the equilibrium in market con…guration fAA; AB; BA; BBg: existence requires V

0:17 and qBA > 0:41:

34

DA1

DB1

DBB

DBA BA θ BB

0

DAB

DAA

AB θ BA

AA θ AB

1

DA2

DB2

Figure 1: Demands for systems and single components under full-quality leadership.

E=∆DBA0 DBA

AB θ BA

C=∆DAB>0 DAB

AA θ AB

DAA

1

Figure 2: Effect of an increase in pA2 on demands under full leadership. Demand for A1 increases: the two complements behave as (gross) substitutes. Demand for B1 decreases.

DB1

DA1

DBA

DBB

DAA

BB θ BA

0

DAB

AA θ BB

AB θ AA

1

DB2

DA2

Figure 3: Demands for systems and single components under shared-quality leadership.

H=∆DBA>0 0

DBA

BB θ BA

G=∆DAA>0 DBB

AA θ BB

F=∆DAB