Complementary Goods, Monopoly vs. Monopoly Power:Evaluating Received Policy Wisdoms on Complementary

Goods Mergers*

By

Robert T. Masson,Serdar Dalkir,Ari Gerstle,

andDavid Eisenstadt**

* The authors wish to express their thanks to Dr. Abigail Ferguson (MiCRA, Inc.);Professor Michael Waldman (Cornell University); organizers and participants of: ERC/METUVI. International Conference in Economics - Ankara, September 2002; 29th EARIE AnnualConference - Madrid, September 2002; U.S. Federal Trade Commission and Department ofJustice Antitrust Division’s Joint Hearings on Health Care and Competition Law and Policy -Washington, DC, June 2003; Third Annual International Industrial Organization Conference -Atlanta, Georgia, April 2005; Professor Francesco Parisi (University of Minnesota, University ofBologna), and anonymous referees. All remaining errors are the authors’.

** Cornell University, MiCRA, U.S. Department of Justice, and MiCRA, respectively. Theviews expressed herein are the authors’ own and are not purported to reflect those of theemployers.

1 Introduction

In the last few years there has been a surge of intellectual interest and debate about “non-

horizontal mergers” on the part of the Antitrust agencies in the U.S. and the EU. This interest

and debate have primarily focused on the issue of “double marginalization” in vertically related

or complementary goods markets. The EU commissioned major economic reports on the state of

the literature: the Church Report (2004) and Bishop et al. (2005) (at 382 and 176 pages,

respectively) do not capture the issues we raise herein. It is true that both from a legal and an

economics point of view much of this recent focus has been on vertical mergers, but the policy

analyses address complementary product mergers both indirectly by analogy to vertical product

mergers and directly. Vertical and complementary product markets may involve “double

marginalization,” referred to as “Cournot effects.” We demonstrate, though, a large class of

complementary products cases for which there are no “Cournot effects.” In the explicit

modeling of complementary products we additionally show the role of potential commodity

bundling.

Clearly for policy analysis one needs to know the positive economics of “What will

happen?” before one can analyze the normative economics of “What should policy be?” We will

look at three key policy statement which purport to represent the positive economics of what will

happen with complementary goods mergers. We show that, as a matter of economic theory, all

three of these are incorrect and misleading – the antitrust view of complementary product

mergers is based on incorrect theoretical perceptions.

We address only the positive economics. We prove these policy statements to be false by

counter example. We use three mathematically simple and highly stylized demand structures for

our proofs. We do calculate consumer and producer surplus for each case, but the normative

issues of what to do about complementary products mergers more generally should depend not

only on correcting these misperceptions about complementary products mergers, but knowing

about more general demand structures, or the demand structures in specific cases. It should also

depend upon factors beyond our scope in this paper, such as the relative role of consumer and

producer surplus and, for that matter, whether the agencies in one jurisdiction should weigh their

own consumers or producers differently from foreign consumers or producers. With the

exception of a short aside, we do not address these issues herein.

One of the ideas in the policy statements is that in positive economic terms

complementary product mergers will have “Cournot effects” which will lower prices. So we

start with a demand structure which yields exactly the Cournot complementary product

monopoly results with his proven “double marginalization” which leads to prices that would be

reduced by the merger of the two firms. Note that following the Cournot analysis, the goods are

like components used in fixed factor proportions, e.g., one needs one of each of two components

to have a viable consumption good. We then add competitors for each product. We assume

there is only “vertical product differentiation” (quality differentiation) and assume that buyers

can costlessly “mix and match,” buy high quality of each, low of each, or high of one and low of

the other. The result? There is no double marginalization without merger – there are no Cournot

effect “benefits” from complementary product mergers!

Once this is established we vary from exactly reproducing the Cournot demand structure

and let consumers have preferences which value quality more highly for one good than for the

other good. We first analyze a limiting case in which, contrary to the policy statements, merger

leads to higher prices from (pure) bundling. Then with a related demand structure we analyze

another of the policy statements, which is that bundling increases consumer choice and hence

will benefit consumers as long as the bundling is “mixed.” That is, following the merger of the

high quality firms in each component market, the merged firm charges a price for each

component and has another price (at a discount) for buying the bundle of the two high quality

components. We show that contrary to the policy statement, offering this “discount” for the

bundle does not lower prices, instead it forces up the individual component prices and permits

the bundle price to exceed the sum of the premerger prices.

The policy statements about the pricing effects of complementary products mergers are

simply incorrect and do not provide a solid foundation for the development of coherent policies

for addressing the effects of complementarity for any proposed relevant welfare criteria.

2 Background

In 1838 Cournot examined the price and welfare implications of a merger between

monopoly producers of complementary goods. He assumed firms set prices, and found that such

a merger could allow for the elimination of a bi-lateral pricing externality (often called double-

marginalization) causing the merged firm to profitably increase output and lower price to the

benefit of consumers. The inescapable logic of Cournot’s result continues to inform the

decisions of regulators of competition policy today.

In its discussion of the mathematically related problems of successive (vertical) and

complementary monopoly, Tirole’s (1997 p. 175) text states, “What is worse than a monopoly?

A chain of monopolies.” While this adage refers specifically to a vertical arrangement where

one monopolist controls an input to be sold to another monopolist for use in production of a final

good, as Tirole points out (also on p. 175) the statement is just as true for two monopolists

producing complementary goods to be consumed in fixed proportions. This is because formally

the models are identical except for the vertical problem having a first mover (the input price is

set before the final good price) and the complementary monopoly has simultaneous moves (both

input prices are set before the buyers decide how much of each to purchase).

Standard analyses of these market structures conclude that a single integrated monopolist

is preferred to either a chain of monopolists or to two or more complementary product

monopolists. But does the perceived wisdom that complementary monopolies should be

replaced by a single integrated monopoly stand up if one replaces the term “monopoly” with

“oligopoly”?

Papers by Salinger (1989) and Economides and Salop (1992) analyze the problem of

complementary oligopoly in quantity setting games and find welfare enhancing results which are

1 See M. Whinston (1990) for an analysis of market foreclosure due to tying ofcomplementary goods and Bernheim and Whinston (1990) for an analysis of how multi-marketcontact may lead to small presence in markets of competitors providing convenient triggerstrategy replies to cheating on collusion in markets where the relative shares are reversed (e.g.,share less than normally perceived as significant horizontal overlap may lead to more stablecollusion).

2 Church (2008) on vertical mergers “...[there is] a presumption that vertical mergers arewelfare enhancing and good for consumers... However, vertical mergers can be anticompetitiveif they result in either foreclosure or enhanced coordination.”

qualitatively similar to the complementary monopoly and successive (vertical) oligopoly models.

In contrast, we use a price setting game, as did Cournot, and show how tenuous these welfare

results can be in an oligopoly setting.

3 Policy Statements we Contradict

The 1984 “U.S. Department of Justice Merger Guidelines” consisted of two separate

texts which respectively addressed horizontal and non-horizontal mergers. (These guidelines

were based on the theoretical knowledge of the early 1980s – the Horizontal part of the

Guidelines was substantially revised in 1992 and issued jointly with the Federal Trade

Commission, with minor revisions in 1997). The 1984 non-horizontal merger guidelines, which

have not been revised, addressed vertical and conglomerate mergers, including complementary

product mergers. Their only mention of a possibility of anticompetitive harm is if there is

“market foreclosure” or “an enhanced facilitation of collusion.”1,2

Now we turn to the current state of perceived knowledge. Proposed mergers and a

variety of policy statements from the U.S. and EU antitrust agencies led the EU to commission

two separate studies of the current state of knowledge about non-horizontal mergers (Church

2004, Bishop et al. 2005). Neither the policy statements discussed below nor the two EU reports

seem to be informed of the issues we raise.

We start with the logic of double marginalization, which is also referred to as “Cournot

effects” in the literature. This logic, from at least one perspective, is captured in the following:

3 This proposition (as well as other similar propositions we state below) may beoverstating the views of the original statements which they aim to formalize. We justify ourapproach by noting that the authors of the original statements do not appear to recognize that theeffects they discuss are not inherent and that other possibilities may exist.

4 See Gerstle (2004) for more general demand structures using copulas.

To the extent the merging parties enjoyed large market shares andmarket power in complementary goods, there will be a tendencyfor prices to decline post-merger because the effects of price dropsin one complement on profits in the other will be internalizedinstead of being ignored.

In short, fears that a conglomerate merger involvingportfolio effects would lead to a welfare reducing type of pricediscrimination involving tying or bundling could be a thin reed tolean on as the sole rationale for blocking the merger.”

– OECD Roundtable on Portfolio Effects inConglomerate Mergers, Range Effects: The UnitedStates Perspective, (Oct. 12, 2001), pp 30-31.

The following proposition formalizes the reliance on Cournot effects implicit in the previous

statement.

First Proposition: A characteristic of complementary goods oligopolies in which

different firms produce different products is double marginalization, which can be

reduced by merger of at least one set of complementary product producers.

We prove that if the First Proposition is an accurate description of Cournot effects in this

literature (double marginalization or internalization of profits) then the proposition as stated is

false.3

Our proofs are all by counter example; each proof starts with the assumptions underlying

a proposition, and proceeds to show that at least under some conditions, these assumptions lead

to a result which contradicts with (is ruled out by) the proposition. In our proofs we apply

Occam’s Razor; our tenet is that the simplest model structure required to show a contradiction is

the most elegant. We do not seek to establish generalized functional forms4 or to perform

comparative statics using parameter values calibrated to the “real world.” Our twin goals are (1)

5 Cooper, Froeb, O’Brien and Vita (2005) note that the EU may have changed its policytoward “... protecting ‘competition rather than competitors...,’” but also note that there is someevidence that this may not be the case after all. The EU’s concerns may have weightedEuropean competitors differently from foreign competitors.

to disprove each proposition with the most compact model possible and (2) to provide the

intuition for why the proposition is wrong.

We next provide the background for two other recent policy statements.

At least as of 2001 the EU and the U.S. competition authorities apparently had widely

divergent views of complementary product mergers. For example, the EU challenged the

complementary product merger between GE and Honeywell (jet engines and avionics), and

another between Guinness and Grand Metropolitan (e.g., Johnnie Walker and Bacardi). The

“theory” behind EU’s challenge of the GE - Honeywell merger partially relied on a Cournot

effects/double marginalization hypothesis: that the combined entity would set prices so low as to

sufficiently injure other competitors and thereby reduce competition.5 The U.S. authorities and

various others also applied the Cournot intuition, but they argued that lower prices are good for

the consumers and not destructive of competition. Hal Varian analyzes the EU decision in a

June 28, 2001, New York Times article titled “In Europe, GE and Honeywell ran afoul of

19th-century thinking.” He concludes:

[A]ntitrust authorities rightly frown on companies' coming togetherto set prices, since the effect is often anticompetitive. On the otherhand, if the products are highly complementary and are producedin highly concentrated industries, producers left to their owndevices may set prices too high because of the “Cournot effect.”

In a speech he delivered on November 9, 2001, William J. Kolasky (U.S. Assistant

Attorney General from 2001 to 2002) discussed “Conglomerate Mergers and Range Effects: It’s

a long way from Chicago to Brussels.” He stated:

We simply could not identify any conditions under which aconglomerate merger, unlike a horizontal or vertical merger, wouldlikely give the merged firm the ability and incentive to raise price

6 A strand of the literature addresses the question whether antitrust should be concernedwith solely consumer surplus or with total surplus (or a composite of the two). See Pittman(2007) for recent references; also see Posner and Easterbrook (1981, pp. 152-170) for additionalviews including those of legal scholars. A related issue is whether EU policy should be based onEuropean (consumer or producer) surplus or world (consumer or producer) surplus? Of coursethe U.S. policy faces a similar question.

7 “Portfolio Effects” refers to mergers that combine firms which are not competing in thesame product markets. A merger of complementary goods producers is a special case of amerger involving portfolio effects.

and restrict output. ... [A] fundamental problem with the EU’sconclusion [is] that the merger would strengthen GE’s(non-existent) dominant position in engines. That findingnecessarily rests entirely on "range effects" -- in this case, thetheory that GE and Honeywell would engage in "mixed bundling"by offering a package of GE engines and Honeywell avionics andnonavionics systems at discounted prices because of the so-calledCournot effect. Honeywell brings nothing else to the party.

This first part of this statement is rather strong, because it rules out any conditions which

would lead to higher prices. Formalized as a proposition:

Second Proposition: Complementary goods mergers will never lead to higher prices.

We demonstrate that this Second Proposition is incorrect. In many cases prices will rise

with such mergers and at least consumer surplus would correspondingly fall.6 Note that the

Second Proposition refers to the Cournot effect on prices, which is addressed by the First

Proposition. The Second Proposition also refers to mixed bundling, which we address below.

An aspect common to EU authorities’ challenge of both mergers (GE/Honeywell and

Guinness/Grand Metropolitan) was the possibility of commodity bundling (or tying in the latter

case because products are not used in fixed factor proportions). Among submissions to the

OECD Roundtable on Portfolio Effects7 in Conglomerate Mergers in 2001 we find the following

statement:

For bundling, a key issue will be the extent to which thecomponents can still be bought individually. If individualpurchase remains an option, bundling actually creates additional

choice for consumers. Only if the components can exclusively bebought as a bundle does a potential problem emerge.

- EU Competition and Trade Group, “Bundling, Tying,and Economic Theory”, European Developments in theCommunications Sector, (July, 2001), p 4.

We state this as a formal proposition:

Third Proposition: If a complementary goods merger would lead to mixed bundling

(offering the goods both as parts of a bundle and individually outside of the bundle) then

the merger would benefit the consumers.

We show that this Third Proposition is incorrect. Indeed, we demonstrate a case in

which, if bundling is allowed, consumers would benefit if mixed bundling were prohibited. In

the absence of such a prohibition the firms’ profit-maximizing strategy would be to mixed

bundle. But we show that the effect of this is that the firm raises the prices of components before

offering a “discount” for the bundle, which reduces consumer surplus to a larger extent.

Although there was elaboration on these points following these cases, the issues we raise

below have not been addressed to our knowledge. Church (2004) was commissioned to provide

an economic study of these issues for the EU (“Church Report”). Cooper, Froeb, O’Brien and

Vita (2005) critique the Church Report. The critique focuses mainly on vertical mergers; it

mentions the removal of “double-markups” or “double marginalization” sixteen times within

nine pages. The critique appears to treat complementary goods similar to vertically related firms

(an approach we show to be true for monopolies but potentially very misleading for price setting

oligopolies). In Church’s reply (2005) he explicitly addresses complementary products mergers

and Cournot effects leading to double marginalization. In another EU commissioned economic

report, Bishop (2005) states “... in complementary relationships, each firm would benefit if the

other lowered the price of its product” (i.e. double marginalization). We show this is not a

robust result in the case of complementary price setting oligopolies.

4 Literature Review

In this section we first explain the Cournot analysis of a merger of two monopoly

producers of complementary goods; we then discuss commodity bundling. Next, we examine

two models that generalize the Cournot result to complementary oligopoly and discuss the

mathematically similar problem of successive oligopoly.

Complementary products, monopoly and monopoly power

The complementary monopoly problem posited by Cournot (1838) assumes buyers who

purchase two inputs (zinc and copper) to produce a final good (brass). Each input is sold by a

monopolist. Moreover, the inputs are used in fixed proportions, that is, there is no input

substitutability, and the demand for each input is inseparable from the demand for the two inputs

combined. Therefore, each consumer’s purchasing decision is based on the total cost of the two

inputs. Cournot argues that for a complementary monopoly the strategic (control) variable is

price, not quantity. Cournot’s key intuition is that since the buyers’ purchasing decisions are

based on the total cost of the two inputs, each monopolist input supplier sets its unit price

according to a reaction function that accounts for the price of the other input supplier. (Stated

more technically, there is a pricing externality between the two input suppliers.) This intuition is

parsimoniously expressed by the phrase “double marginalization.” Assuming a linear demand

function, Cournot demonstrates that a merger between the two input suppliers would enable the

input suppliers to internalize this externality (eliminating double marginalization), and to set a

single, profit-maximizing price. He shows that the single post-merger price for the two inputs

combined is less than the sum of the pre-merger prices. The is commonly called the “Cournot

effect.”

Cournot’s example is based on four specific assumptions. First, each complementary

input is sold by a monopolist. Second, the production technology is fixed proportions. Third, all

buyers use the same production technology. Fourth is the assumption of a linear, downward-

sloping demand curve.

In the first part of our analysis we retain Cournot’s second through fourth assumptions

but modify the first, changing monopoly to oligopoly. (Later we also modify his fourth

assumption.) We demonstrate that when the assumption of price-setting monopolies is replaced

by one of price-setting oligopolies, Cournot’s result does not obtain.

Complementary goods and commodity bundling

When manufacturers of two complementary goods merge, their strategy sets are enlarged.

Most commonly, pure bundling, mixed bundling, and tying are assumed to become available to

the merged firm as additional strategies. Of these, we consider only pure bundling and mixed

bundling (fixed factor proportions rules out tying).

The modern analysis of bundling can be traced to Stigler (1963) but most of the more

recent models are based on Adams and Yellen (1979), who assume that consumers have

reservation values for each of two complementary (or even unrelated) products. Using a number

of examples, Adams and Yellen demonstrate that any of the three strategies of independent

pricing, pure bundling, or mixed bundling may prevail depending on the distribution of

consumer valuations. Schmalensee (1984) adapts the Adams and Yellen model to consumer

valuations that are distributed bivariate Gaussian; he finds that (some type of) bundling will be

the preferred strategy as long as the correlation between valuations of the two products (across

consumers) does not equal +1. McAfee, McMillan, and Whinston (1989) assume a continuous

bivariate distribution of consumer valuations and characterize a sufficient condition for mixed

bundling by a monopolist to dominate unbundled sales.

Schmalensee also shows that bundling reduces “consumer heterogeneity” as perceived by

the seller. For example, suppose marginal costs are zero and that there are two consumers; the

first consumer’s valuations of the two complementary goods are {1, 5}, and the second

consumer’s valuations are {4, 2}. Without bundling, to sell both goods to both consumers a

monopolist would set individual product prices to 1 and 2 respectively, leading to a profit of 6.

Alternatively, to sell one unit of each good, the monopolist would set prices of 4 and 5

8 Under the assumption of a sequence of play, he finds that vertical integration can eitherincrease or decrease prices.

respectively, which would lead to a profit of 9. But the profit-maximizing strategy would be to

set a single bundled price of 6 and earn a profit of 12. Although this example assumes

negatively correlated preferences, negatively correlated preferences is not a necessary

assumption for bundling to be the profit-maximizing strategy. Gerstle (2004) assumes a demand

structure based on uniform marginal preferences on the unit square, which generates a linear

demand for each product. Using these “copulas,” he shows that bundling reduces consumer

surplus when the correlation between preferences for the two products is sufficiently strong,

regardless of its sign (negative or positive).

Complementary oligopoly

Salinger (1989) analyzes a homogeneous goods model where firms in each component

market compete with each other by setting individual quantities. He then considers a merger

between two firms producing different components to form a system. With fixed factor

proportions and constant returns to scale, the merged firm is assumed to “pure bundle” – that is,

sell the two components only as parts of a bundle (justified in Salinger 1988). In order to nest

vertical integration and complementary product integration into a single model, he analyzes

several alternative assumptions regarding sequence of play and about firms’ conjectures of

complementary good prices. In the case of simultaneous play by all firms, and assuming fixed

prices for complementary goods, he obtains the “Cournot effect.”8

Economides and Salop (1992) present a model with price competition and horizontal

product differentiation. There are initially two firms in each of two complementary good

markets. Their key assumption is that a price increase by any of the four firms results in an

increase in the demand for any of the remaining firms (this assumption does not apply to our

vertical product differentiation model). They analyze various equilibria. Of interest to us are

“composite goods competition” (pure bundling system competition), and “parallel vertical

9 Tirole refers to the merger of complementary goods producers as “horizontal”integration, the “...” in the quote is “(horizontally)” in the original. That use of this term ismisleading. In the legal nomenclature, horizontal integration refers to the merger of competitorsproducing the same good for sale in the same market. Complementary goods mergers in thelegal literature would be a subset of conglomerate mergers.

integration.” In the former equilibrium firms are integrated across markets (i.e. two separate

mergers between complementary goods), and each firm offers only a pure bundle of its system

(e.g., engines and cars). In the latter equilibrium each integrated firm sells a bundled system and

separate individual components, i.e. mixed bundling (for example, tuner amplifiers and

speakers). Their mixed bundling equilibrium, however, is restricted to setting the sum of the

component prices equal to the bundle price. (In our model, we show that relaxing this

assumption drives important additional results.)

Complementary and successive monopoly (oligopoly)

The complementary monopoly problem is related to the mathematically similar problem

of successive monopoly. The difference between the two models is that whereas the

intermediate product seller in the vertical model makes the first play, input producers in the

complement goods model make simultaneous plays. Tirole (1988) states that

The double marginalization (or chain-of-monopolies) problem isvery similar to that of two monopoly producers of complementarygoods... monopoly producers of complementary goods have anincentive to integrate ... in order to avoid double marginalizationand an excessive demand contraction (p. 175).9

The close relationship between the two cases is apparent in the way that they are handled in the

literature. For example, Tirole calls production and retailing “complements”; Economides and

Salop refer to one of their complementary products equilibria as “parallel vertical integration”;

Salinger (1988, 1989) nests the two models.

Mergers of complementary and successive oligopolists

The classical models of vertical integration and of mergers of complementary

10 Existing literature includes models which show that with variable factor proportions,vertical mergers can lead to declines in consumer surplus or total welfare; examples are Hay(1973), Warren-Boulton (1974), and Schmalensee (1973). Analogously, in complementaryproducts, removal of the assumption of fixed proportions demand (common across buyers) leadsto the potential for a tying rather than bundling equilibrium (Whinston 1990). These are notrelevant for our narrower purposes.

monopolists assume fixed factor proportions (common factor proportions across buyers) when

they derive the double-marginalization result. See Tirole (1988). The removal of double

marginalization through vertical merger leads to increases in both consumer surplus and

producer surplus.10

Greenhut and Ohta (1979) is the general starting point for analysis of successive

oligopolies. They model successive oligopolies with inputs that are used in fixed proportions to

produce a final good. They begin by stating, “Vertical integration of successive monopolists

(with fixed production coefficients) has long been known to provide merging monopolists with

greater profits and their customers with greater outputs at lower prices.” They show that when

the markets in two vertically related industries are both characterized by firms that compete by

setting quantities, the vertical integration of some firms causes industry output to increase and

subsequently market price to decrease. Ross (1992) commenting on this result states that,

“Because of these results, mergers to remove successive monopolies are widely considered

beneficial and therefore of no concern to antitrust officials” (p. 375).

Waterson (1982) extends Greenhut and Ohta by dropping the fixed factor proportions

assumption; he finds that there can be consumer surplus, and sometimes total welfare losses.

In the classical fixed factor proportions demand model, vertical mergers improve welfare,

with few exceptions. Salinger (1988) looks at a vertical merger when the pre merger market

structure consists of both integrated and unintegrated firms and the final good has fixed

proportions. Firms compete in quantities in both the intermediate input and final goods markets.

He justifies an assumption that vertically integrated firms do not sell their intermediate inputs to

any other firm given homogeneous products, constant returns and fixed proportions. When a

vertical merger occurs, the integrated firm expands output in the final product market, which

(ceteris paribus) decreases the price. But a vertical merger also reduces the number of the

intermediate good sellers, which (ceteris paribus) increases the price. For some parameter

values, the second effect dominates the first so the net effect is a higher price for the final

product.

5 Analysis

A. Basic Model

In this section we first model complementary monopolies for specific demand structures,

a special case of which is the Cournot demand structure for complementary monopoly. Then we

introduce competing goods.

For example, consider the following problem. Along a river there may be two successive

tolling stations. The river may be the only way to deliver goods to market (complementary

monopoly). We assume that tolls can be credibly fixed before the trip starts (e.g., travel agents,

hence avoiding the “holdout problem” addressed by Feinberg and Kamien (2001). It is also

possible that there is a land route in competition with these two tolling stations. Yet another

possibility is that there is a land route around the first tolling station and another around the

second allowing either or both tolling stations to be bypassed at some cost. It turns out that

which form of competition prevails (none, competition for both goods as a package, competition

for each good individually) has significant implications for the welfare effects of a merger

between complementary product suppliers.

Suppose that consumers purchase one unit of each of two different components in order

to form a system (e.g., stereos and speakers, computers and monitors). If neither component

supplier has any competition, then this is a model of complementary monopoly. For

complementary oligopoly, we assume that for each component there are multiple firms.

Moreover, we assume there are two quality levels for each component: superior (high quality)

and inferior (low quality). For each component, one firm supplies the superior quality product

while two (or more) identical firms supply the inferior quality product. Firms engage in (quality-

differentiated) Bertrand price competition. We assume constant marginal costs, indexed to zero

(and no fixed costs). Initially there are no integrated firms, so that each firm produces only one

of the two components. We address the effects of a merger between the two superior quality

firms (across the two components). To do so we need to be more explicit about defining quality.

For each of the two components, we model each buyer’s valuation as an additional

willingness to pay (or a premium) for high quality over low quality. Initially, each buyer is

assumed to have an incremental valuation for each high quality good in the interval [0,1]

where i = 1, 2 indexes the component and j indexes the buyer. Given our assumptions, each low

quality firm will set a price that equals marginal cost, which is zero.

The valuations ( , ) can have an arbitrary joint distribution on the unit square. We

start with an extreme case of the joint distribution of valuations. We consider buyer (consumer)

valuations which are distributed uniformly on the ray from the origin to the point (1,1), which is

a linear subspace of the space spanned by the valuations (the unit square); these preferences lead

to the demand curves in Cournot’s 1838 treatise. This assumption implies that each consumer

values each high quality component equally. While certainly an extreme assumption, it is not

unbelievable. Consider individuals wishing to transport goods down a river for sale at a market.

This is actually one of the scenarios hypothesized by Cournot (1838) and also used in Gardner,

Gaston and Masson (2004) as well as Feinberg and Kamien (2001). Suppose that for river

transportation two tolls must be paid prior to arriving at the destination, and that each toll is set

independently. Suppose also that individuals have cargos of varying bulk where bulk affects the

costs of land transportation. The valuation that is placed on the ability to use the river for

transportation past either tolling station may be where cj is the individual’s cost of

11 As another example, one may consider stereo music players. Urban dwellers withexcellent reception may not care too much about the quality of their receivers, yet if they areinterested in music, they may care about the quality of their speakers. A rural dweller may careabout the sensitivity of the receiver, and if that listener is interested in news, not music, theremay not be a preference for high quality speakers. Still as another example, consider the factthat (although bundling engines and cars is mostly caused by production technology) somebuyers want a high power, light weight sports car, whereas others want a heavier car with asmoother ride, a full sized back seat, adequate horse power and better mileage.

avoiding one or the other toll station by shipping over land instead, and cj is uniformly

distributed over the interval [0,1]. The order of the tolls is inconsequential, and we have

assumed that the cost of avoiding each toll is the same (e.g., does not depend upon i, only

upon j). This results in consumer valuations having a perfect positive correlation and being

represented along the ray from (0,0) to (1,1). An important consequence of this assumption is

that consumer demand for each high quality good is represented by a linear, downward-sloping

demand curve. In the case where the high quality producers are monopolists in their respective

component markets, this leads to the system demand structure in Cournot’s (1838) example of

complementary monopoly.

We will later consider another extreme example which has consumers uniformly

distributed along the diagonal line from (0,1) to (1,0), the case of perfect negative correlation of

valuations. For example, consider health insurance plans. Young workers may want a health

plan to include the best pediatric care hospital but may not care about cardiology, whereas older

ones may care mostly about the quality of cardiology units but not care about pediatrics.11

Mathematically, these two distributions (the perfect positive correlation and the perfect

negative correlation) are referred to the Frechet-Hoeffding bounds, which are the limiting

distributions as the correlation between two jointly distributed random variables (distributed

according to a copula) approaches +1 and -1 respectively.

The qualitative results of the models we examine herein do not require these limiting

distributions. These two extreme cases were selected for two reasons. First, they nest the

demand structure in Cournot’s 1838 model. Second, they provide simple counterexamples. In

proof by counterexample, the simplest possible model structure that can establish the result is

usually the most elegant. These cases permit us to prove that complementary products mergers

may not lead to a lower price even under the assumption of consumer preferences consistent with

Cournot’s demand structure, which disproves the First Proposition above. Furthermore, using

the second set of extreme preferences we show that a complementary goods merger will lead to

higher prices through bundling, which disproves the Second Proposition above. Then, we

introduce a third set of preferences; this is the case in which consumer valuations are distributed

along the northeast arc of the unit circle centered on the origin. Through this third example we

establish that mixed bundling may lead to lower consumer surplus relative to a pure bundling or

the pre-merger equilibrium, which disproves the Third Proposition above.

Consumers are assumed to have independent valuations (independent across consumers)

for each of the higher quality components, giving each consumer a utility of the following form:

(1)

where pi is the price of the ith high quality component (i = 1, 2), and (vij - pi) is the net utility

(consumer surplus) from the ith component conditional upon purchase of the higher quality

product, indicated by δi / 1. (δi / 0 corresponds to purchase of a lower quality product for the ith

component.)

Consumers are assumed to purchase one unit of each component in order to form a

system. They can buy a lower quality component at a price indexed to zero or buy a higher

quality component at a positive price. All component producers, both high and low quality, are

assumed to have zero marginal cost. These assumptions insure that all consumers purchase a

system, although it may consist of one or even two low quality components.

B. The First Three Cases: Consumers distributed along (0,0) to (1,1)

Case 1: Monopoly with consumers distributed along (0,0) to (1,1)

Suppose that the two component producers are monopolists in their respective component

markets. In this case, each consumer’s choices are either to purchase a system consisting of both

components or to purchase nothing at all. Total valuations over such a system are distributed

uniformly on the interval [0,2] and so imply the following demand function for the system:

(2)

where P = p1 + p2 is the total system cost and Q = q1 = q2 is the system quantity demanded.

Figure1 shows the distribution of consumers and also the implied demand function for

the system under complementary monopoly.

Since consumers base their purchases on total system cost, each firm must take account

of the other firm’s price when setting its own. The profit maximization problem for firm i

becomes:

(3)

After differentiating over pi and simplifying we get the reaction function:

(4)

where k is the complementary product to i. Since this problem is symmetric we get an identical

expression for firm k. Substituting for pk from firm k’s response function and solving for pi we

get = 2/3 (one third of the vertical intercept if MC = 0). Therefore, P = 4/3 and 1/3 of all

consumers purchase the system while the remaining consumers purchasing nothing. This

replicates the classical complementary monopoly example solved by Cournot in 1838 (system

price equals two thirds of the demand intercept). Total firm profits are 4/9 = 0.4444. Consumer

surplus (CS) is calculated as the area under the component demand curve above the horizontal

price line for each component, yielding CS = 1/9 = 0.1111. Total surplus, the sum of consumer

and producer surpluses, is 5/9 = 0.5556.

If the firms were to merge and internalize the profit externality the integrated monopolist

would face the problem:

(5)

which results in an optimal system price P* = 1. Again, this replicates the Cournot result from

1838 (merger leading to system price of one half the vertical intercept). Compared to the case of

independent monopolists, component prices have decreased, total consumer welfare has

increased, and total profits have increased as well. Total consumer welfare has increased to 1/4

from 1/9 and profits have increased to ½ from 4/9. Total surplus has therefore increased from

0.5556 to 0.75, a change of 0.1944. In this example, a merger which eliminates “double

marginalization” and lowers prices increases both the profits and consumer welfare. This

example is substantively identical to Cournot’s original complementary monopoly problem.

Case 2: Consumers along (0,0) to (1,1) with an alternative package

This case is trivially the same as the previous case in terms of the high quality products;

the only difference is in terms of the consumption patterns of those who do not buy the high

quality products. Consumers in this case choose between a high quality system (composed

exclusively of the high quality components), and a low quality system. Systems that mix

different quality components are not permitted (although lower quality systems that mix

components sold by different producers are permitted). Compatibility is one reason that this may

occur. For example, an advanced (e.g., 64-bit) computer operating system may require a

powerful computer, and a powerful computer may not confer any gains for a less advanced

operating system. Another reason that only systems of uniform (both high or both low) quality

components are observed is technology (e.g., the bundling of engines and cars).

Returning to our river example, suppose that there is an alternative route to the two toll

stations. For example, a land route goes directly from the shipper’s point of origin to the

destination, and completely bypasses the river. Suppose that this alternative route is available to

the shipper at zero cost (in terms of tolls). But, suppose that the costs of land transportation

(e.g., fuel costs) are greater than those for river transportation and they are increasing in the

weight of the product to be shipped. In this setup, land transportation provides competition for

the river tolling stations.

Now, assume that the value of going down the river instead of over land is given by the

sum of the values on the ray from (0,0) to (1,1) and that shippers are distributed uniformly along

this ray. Then pre-merger and post merger outcomes are essentially the same as those described

for the previous example, with the exception that in the current example consumers who do not

purchase tolls purchase transportation via the alternative route, which corresponds to the “lower

quality” system. Unlike Cournot’s original example, this example involves the assumption of

system competition but its outcomes are identical to those of the classical Cournot

complementary monopoly problem.

Case 3: Consumers along (0,0) to (1,1) with alternative components

Now we return to the oligopoly setup that was initially described above where consumers

are free to purchase any combination of components when forming a system. In this example,

we assume away any incompatibility issues (e.g., we assume both fast and slow hard drives are

compatible with both fast and slow processors). For intuition, consider the tolling scenario.

There are two tolling stations, each circumvented by a separate land route. A shipper could

decide to pay one toll, and bypass the other tolling station. Since consumers are able to mix and

match “high quality” and “low quality” components, consumers are able to choose their systems

using the principle that they will not pay more for a component than their individual valuation of

that component. Consumers therefore evaluate 4 different systems defined as combinations of

high and low quality components, and choose the system that offers the highest net utility (net of

cost). This results in each consumer deciding whether or not to purchase each high quality

component based solely on its value and price. Consumer j purchases component i from the high

quality firm if and only if:

(6)

Since component valuations are distributed uniformly in the interval [0,1], the implied demand

for a given high quality component is:

(7)

which is independent of the price of the other high quality component. Firm i then solves the

profit maximization problem:

(8)

Solving this leads to an optimal component price p* = ½ for each component. The price of a

system with the two high quality components is hence 2p* = 1, which is identical to the system

price which would be charged by a firm formed by a merger between the two firms.

This example does not lead to the “Cournot effect”: a merger between the two high

quality complementary product producers does not change the price of the high quality

system; nor does it affect producer or consumer surplus.

By counter example we have demonstrated that the First Proposition (complementary

goods oligopoly in which different firms produce different products leads to double

marginalization, the extent of which will be reduced by a merger between at least one pair of

complementary product producers) is incorrect. In this context, the minimum necessary

condition for the Cournot effects to disappear is compatibility between high and low quality

components. Stated differently, when all components are compatible, there is no a priori reason

to expect a Cournot effect in an oligopoly! For a price setting oligopoly with vertical product

differentiation, only “pure” system competition (incompatibility) leads to the Cournot effect.

Notably, the demand structure arising out of the consumer preferences in our Cases 1-3 is

identical to the demand structure assumed by Cournot in his 1838 example of complementary

monopoly.

C. The Next Three Cases: Consumers distributed along (0,1) to (1,0)

12 Recall that Schmalensee (1984) looks at bivariate Gaussian distributions and varies thecorrelation from -1 to +1, finding some bundling in all cases other than the +1 case. Gerstle(2004) looks at the full range of correlations using copulas (uniform marginal distributions ofpreferences on the unit square) which yield general linear demand systems. Barring a correlationof +1 he finds mixed bundling is ubiquitous. The two cases we have illustrated so far are simplythe Frechet-Hoeffding bounds of the class of copula distributions he is uses.

Case 4: Monopoly with Consumers distributed on (0,1) to (1,0)

Our assumption about consumer preferences consistent with Cournot’s demand structure

was extreme, in that each individual values the two high quality components identically. These

preferences imply a correlation of +1 between component valuations each of which is distributed

uniformly across individuals. The usefulness of that assumption is it generates the Cournot

demand structure and its simplicity for demonstrating that the First Proposition is false. We

now turn to the opposite extreme: component valuations with a correlation of !1, and each with

a uniform distribution across individuals. The usefulness of this extreme assumption is its

simplicity for demonstrating that the Second Proposition is false.12

Again, we begin by examining the equilibrium under complementary monopoly and then

state the results under price-setting (Bertrand) oligopoly scenarios. When each of the two

quality leaders is a monopolist in their respective markets, we are back in a situation where

consumers either purchase a system (e.g., both components) or nothing at all and so are

concerned only with the system price (or sum of component prices), as in Case 1 above. It is

readily seen that consumer valuations distributed along the diagonal from (0,1) to (1,0) (see

figure 2a) implies a system demand that is perfectly inelastic at the unit quantity (Q = 1) for

system prices less than or equal to 1 and Q = 0 for all prices higher than 1 (see figure 2b).

Given any competitor’s price pk, the optimal response is to set price pi = 1 ! pk. There

are an infinite number of Nash equilibria that satisfy this. Here, we consider the focal-symmetric

equilibrium of component prices (p1, p2) = (½, ½). Note that all consumer surplus is fully

extracted in this equilibrium (neither product has an independent value outside of the system).

13 In our example, there are no such consumers, they all buy the high quality system.

The total of the two firms’ profits, which is also the total surplus in this case, is 1. Clearly the

optimal price for an integrated monopolist is also to set system price P* = 1. Thus the component

prices, total profits, and consumer welfare are unaltered by merger. Note that consumer surplus

is fully extracted under complementary monopoly as well as by the integrated monopolist.

In this case, even the monopoly case for each component does not lead to the Cournot

effect, so the First Proposition is false in this case even if we don’t introduce oligopoly issues.

Case 5: Consumers on (0,1) to (1,0) with an alternative system

The high and low quality components being incompatible with each other, as in Case 2

above, leads to outcomes that are identical to the complementary monopoly solution: all pricing

and welfare results are identical to Case 4. The only difference is that in this case, consumers

who would not purchase the high quality system are assumed to purchase the low quality

system.13

In this case, too, there is no Cournot effect from a complementary merger, and the First

Proposition is false.

Case 6: Consumers on (0,1) to (1,0): alternative components

This case has remarkably different results since each high quality component faces

compatible low quality alternatives. Each high quality firm faces an implied demand function:

(9)

Note that this demand structure is identical to the one in Case 3 even though consumer behavior

is much different in terms of system purchases. The profit maximizing component price is again

equal to ½. In contrast to the previous case, though, all consumers purchase mixed systems

composed of one high quality and one low quality component. (Consumers located at (½, ½)

have zero measure.) This is depicted graphically in Figure 3. Profit for each firm is equal to 1/4

for total profits of ½. Consumer welfare is given by the sum of the areas of the two triangles

directly beneath the diagonal (locus of consumer valuations). Total consumer welfare is 1/4.

Total surplus is then 3/4 = 0.75.

However, this case leads to a markedly different outcome than Case 3. A merger

between the two high quality component producers would enable them to bundle their goods,

forcing consumers to purchase both or none. This is readily seen to be the unique optimal

(linear) pricing strategy. A bundle price of P = 1 maximizes profits and extracts all consumer

surplus. Profits are equal to 1, up from the sum of their pre-merger profits of ½, and consumer

surplus is zero, down from 1/4 pre-merger. Although the merger does not affect the “nominal”

component prices, it is no longer possible to purchase only one high quality component. The

merger increases the implicit price (a concept that we formalize later) of each high quality

component. Total surplus actually increases from 0.75 to 1, but at the expense of consumers.

In this example, not only is there no pre-merger double marginalization, but the merger

leads to higher (implicit) prices and lower consumer surplus, rather than lower prices and higher

consumer surplus. So, in this case not only is the First Proposition false, so is the Second

Proposition that “Complementary goods mergers will never lead to higher prices” is

contradicted given these parameters.

D. The Final Case: consumers on the unit circle: alternative components

With this preference structure, the case of monopoly is one in which consumers buy a

system, or do not buy either good. The case of systems competition (e.g., due to

incompatibilities) which lead to high quality systems competing with lower quality systems as

the only form of competition is similar. In either case, each consumer consumes a bundle

(system) after the merger. The interesting case for examining the Third Proposition arises

when components are compatible and a merged seller has the option of selling a mixed bundle –

a system price and an individual price for each component. We directly proceed to discuss this

case.

Case 7: Consumers on the northeast arc of the unit circle: alternative components

Cases 1-6 have illustrated the crucial differences between complementary monopoly and

complementary oligopoly with vertically differentiated products when (1) competition is by

system, vs. (2) competition is by component. These extreme cases illustrated why results

relating to double marginalization under a complementary monopoly should not be presumed to

be the case for all complementary product mergers. This is because price competition by

component breaks the strict complementarity in consumption of the high quality goods, and

eliminates the pricing externality that is required for double marginalization to exist in the pre-

merger game. For the next illustration we simply offer a richer model of a complementary

oligopoly problem when there is competition by component. This model shows the Third

Proposition to be false; if firms select to use mixed bundling this does not imply that consumers

benefit from a merger.

In this model we posit a more complex purchasing behavior than in Cases 1-6, which

requires expanded notation. The first component has a high quality producer, firm 1H and at

least two low quality producers denoted generically as 1L. Similarly for the second component

there is a 2H and at least two 2L’s. Systems consisting of a high quality component and a low

quality component we call “mixed,” and write HL for the {1H,2L} system and LH for the

{1L,2H} system. A system comprised of both high quality components is referred to as a “high

quality system” and labeled HH for {1H,2H}.

Let ÷ denote the entire circumference of the unit circle centered on the origin of ú2.

Consider a unit mass of consumers distributed uniformly on the intersection of ÷ with the

positive quadrant of ú2. A subset of the arc length can be translated to a proportion of the

consumer population, as follows. The circumference of a circle with radius r is 2πr; the arc

length of the positive quadrant of the unit circle (with r = 1) is thus π/2. The arc length

associated with any particular set of preferences, when multiplied by 2/π, represents the

proportion of the total population associated with those preferences.

1. Equilibrium when components are priced independently

Consider the profit maximization problem of firms 1H and 2H. Each consumer has

valuations of the high quality components denoted v1H and v2H respectively (we suppress the

subscript j for the consumer’s identity). A consumer purchases a system that contains

component 1H (or 2H) when v1H > p1H (or v2H > p2H). Figure 4 illustrates the purchase decisions

of consumers for a given set of component prices (we show below that these are the equilibrium

prices). Clockwise from the vertical intercept consumers purchase the mixed system HL, the

high quality system HH, or the mixed system LH. The arc length of a portion of the unit circle

between points a and b on the circle is the angle (measured in radians) swept out by a ray from

the origin to point a as it moves to point b. The length of the arc is then either the inverse sine

(arcsin, or sin!1) of the vertical “v1H,p1H” coordinate or the inverse cosine (arccos, or cos!1) of the

horizontal “v2H,p2H” coordinate.

Since the total length of arc in the northeast quadrant equals (π/2), arc lengths are

normalized by multiplying them with (2/π). The total proportion of consumers to which each

firm sells are Market(Firm 1H) = 1 - (2/π) sin-1(p1H) = (2/π) cos-1(p1H) and Market(Firm 2H) =

(2/π) cos-1(p2H). Therefore, the profit maximization problem for each firm is to set a price piH to

solve:

(10)

Differentiating with respect to piH yields:

(11)

Solving (11) leads to prices p1H = p2H = 0.6522 and firm profits of 0.3572 for a combined profit

of 0.7144. Given these prices, only 9.54% of the consumers purchase the high quality system

while all other consumers purchase a mixed system. Under an assumption of independent

14 Elasticity of the system demand Q(P) is given by g = QN(P)×P/Q(P) (if price increases,there is a kink in demand for price decreasing). Since at P = 1 Q(P) = 1, the elasticity expression

(12)

valuations over the components, consumer surplus under this equilibrium is 0.2508. Total

surplus is thus 0.9652.

2. Post merger analysis: pure bundling

Consider the effects of a merger between firms 1H and 2H. Post merger, the combined

firm could continue to price components independently or it could engage in some form of

bundling. First, suppose that the merged firm chooses a pure bundling strategy. Suppose that

the bundle price is P. In this case, it sells the bundle to all consumers located on the arc to the

northeast of the line connecting the bundle price P on each axis – that is, only consumers who

fall outside of the line that slopes downward from (0, P) to (P, 0) purchase the bundle. Figure 5

illustrates purchase decisions of consumers given an arbitrary bundle price of 1.2. This is not

the equilibrium bundle price, as we show below. (We select a non-equilibrium price to illustrate

buyers who do not purchase the bundle, in the pure bundling equilibrium there are no such

buyers.)

Under pure bundling, system demand is given by

Solving for profits using this demand curve we can represent profits as a function of the bundle

price P in figure 6. (Note, the horizontal axis starts at the price of 1.) This is because it is trivial

to note that the firm will never set a bundle price less than 1, because at P=1 it sells the bundle to

all consumers; a lower price would not gain any customers and would only lower profits. Also,

at a price of , the bundle price line is tangent to the unit circle so profits are zero. The optimal

pure bundling price is equal to P* = 1, selling the bundle to the entire market.14

simplifies to where and .

Moreover, g2 simplifies to . Therefore at P = 1, g2 = 16/π2 which implies | g | = 4/π > 1,

i.e. the demand is elastic at P = 1. Since we are assuming a marginal cost of zero, this impliesthat the merged firm will not find it profitable to increase the price of its bundle from P = 1. Moreover, since Q(P) is concave, its elasticity increases with P. Therefore profits are actuallyeverywhere lower than profits at P = 1.

Profits at the equilibrium bundle price are equal to 1, greater than the combined firm

component pricing profits of 0.7144. Consumer surplus is calculated as CS =

, where Q* = Q(P*) = 1, Π* = Π(P*) = 1 and P(Q) is the inverse demand

function in Figure 6. This calculation yields CS = 0.2732. Both consumer and total surplus

actually increase with merger. We work this case because it is a stepping stone for evaluation of

the Third Proposition. In fact this equilibrium is moot in practice because the pure bundling

strategy is shown to be strictly dominated by mixed bundling.

3. Post merger analysis: mixed bundling

Finally, we analyze mixed bundling as an alternative post-merger strategy. Figure 5

demonstrated that a pure bundle price above 1 leaves two sections of the arc unserved by the

bundle. Our notation for the pure bundle price is P. Moving to mixed bundling, where some

buyers only buy one high quality component, it is useful to augment the notation. The bundle of

the two high quality components, HH, will be sold at price PHH. The two components, purchased

separately, will be sold at prices of p1H and p2H. Given prices for the components, a consumer

will purchase the bundle as long as (1) the value of the bundle exceeds its price

( ) and (2) the net value of buying only one component is less than the net value

of buying the bundle ( for i = 1, 2). This leads to the inequalities:

and . Otherwise the mixed system HL (LH) is purchased if

(13)

( ).

Lemma: Under mixed bundling, the optimal choice of bundle and component prices

involves the component price lines intersecting the circle at the points where the bundle

price line intersects the circle. That is, for a given bundle price, the integrated firm

chooses component prices so that all consumers not served by the bundle will purchase

one of the high quality components.

Proof :(see appendix)

The Lemma implies that, given a bundled system price, the component prices are

determined by the intersections of the bundle price line with the circle. Therefore, the firm’s

profit function can be expressed as a function of a single parameter, the bundle price. The

intuition for this is straightforward. If profits are higher when an individual purchases the bundle

rather than a component, then the bundle price and the component prices will be set so that this

individual has no incentive to buy only one component. Once the bundle price is determined, the

component prices are set so that all buyers purchase either the bundle or one component.

Given the lemma above, profits under mixed bundling can be expressed as the following

equation:

PH H denotes the bundle price and and are determined by the intersections of the bundle

price line with the circle. The first term in Π gives profits due to sales of the bundled good. The

second term gives profits due to sales of components 1H and 2H. Figure 7 shows mixed

bundling profits as a function of the bundle price.

15 Because mixed bundling dominates pure bundling, the integrated monopolist gainsfrom assuring that its products are compatible with the low quality products.

16 Recall Economides and Salop (1992) analyze mixed bundling in a quantity settingmodel. They, however, constrain component prices to add up to the bundle price. In our casethe sum of component prices of 1.92 far exceeds the bundle price of 1.24.

The maximum of the profit function is found by differentiating the profit function with

respect to the bundle price and solving for the price at which the derivative is zero. The unique

solution yields a bundle price of PHH = 1.2356. From Equation (13), symmetric component

prices are p1H = p2H = 0.9618. All of the market is covered by either bundle or component sales.

Profits under mixed bundling are 1.1389; this is 13.89% higher than pure bundling profits

(and 59.4% higher than pre merger profits). The implication is that mixed bundling strictly

dominates pure bundling as a post merger strategy. Consumer welfare in this equilibrium is

0.0855 which is markedly lower than the pre-merger consumer welfare of 0.2508.15

Recall the statement formalized in the Third Proposition: “If individual purchase

remains an option, bundling actually creates additional choice for consumers.” The choice,

however, is at component prices of 0.9618 and a bundle price of 1.2356,16 compared to the pre-

merger component prices of 0.6522 each and also compared to the pure bundling price of 1,

which is an inferior strategy relative to mixed bundling. The “additional choice” comes at a very

high cost in terms of consumer welfare. We illustrate this in Figure 8. Pre-merger prices and

consumption patterns have a superscript 0 and post-merger prices and consumption patterns are

marked with a prime symbol (N).

With the help of Figure 8, we analyze the distribution of consumer gains and losses from

the merger. Consumers who purchase mixed systems both before and after merger account for

35.31% of the population. An additional 55.15% of consumers purchase a mixed system pre

merger but then purchase the high quality bundle under mixed bundling. The remaining

consumers purchase the high quality system under either scenario.

17 Pre-merger utility is , post-merger it is when written to

illustrate the implicit price. So .

We can now contrast the pre and the post merger equilibria, keeping in mind that the pure

bundling solution can be ignored, as pure bundling is a strictly dominated strategy. For those

consumers who purchase a mixed system both before and after the merger, the effect is drastic:

these consumers suffer a 47.5% increase in price and a commensurate decrease in surplus. On

the other hand those consumers purchasing a high quality system both before and after enjoy a

5.27% decrease in system price.

The majority of consumers, however, are those that chose to purchase a mixed system

under independent pricing, but opt to purchase the high quality system in the mixed bundling

equilibrium. Since the purchase decision of these consumers changes along with the prices, it is

desirable to calculate the implicit price change of the component purchased under both regimes.

Let be the price of component i purchased by one such consumer pre-merger and let

be the system price in the mixed bundling equilibrium. In addition, let the component

valuations of this consumer be and . Suppose that the consumer purchased component

1H pre-merger, then the implicit price that is paid for component 1H under mixed bundling is

which is the total price paid net of consumer surplus associated with the value of

having good 2H as part of the system. The change in consumer surplus is then given by

.17 Note that this implicit price change is also equal to the change in

consumer surplus for each of these individuals. It is important to notice that not all consumers

who purchased components pre-merger and the bundled system post-merger suffer an increase in

the implicit component price.

In Figure 9, point K shows the consumer who purchases a mixed system with firm 1H's

18 This person is to the left of those who would buy both high quality goods pre-merger.

19 Gerstle (2004) examines bundling by a monopolist facing a distribution of preferenceswhich has positive density on the unit square. The distribution is defined by a copula (which isspecified by uniform marginal distributions) leading to linear demands. He shows that mixedbundling always dominates pure bundling and that for a wide class of parameters consumersurplus is lower with mixed bundling than with component pricing alone. For a subset of theseparameters total surplus is lower. Finally, that there is a wide class of parameters for whichconsumer surplus under mixed bundling is lower than that under pure bundling (which would not

component pre-merger and purchases the high quality bundle post-merger and find that his or her

implicit price of purchasing the component 1H has not changed. (Point K is defined by ΔU = 0).

The point K then is the valuation of a person who is indifferent between the pre and post merger

outcome.18 Consumers to the immediate right of K are better off, those to the left are worse off.

In the figure, the pre-merger component price of each component is 0.65 and the bundle price is

the optimal post merger bundle price of 1.24.

Geometrically, P'HH ! p1H is the distance 1.24 ! .65 on the vertical axis, which is equal to

v2 = .59 (= 1.24 ! .65) on the horizontal axis. Point κ is analogously constructed for the value of

good 1.

The set of consumers who are better off with the merger is given by the arc length

between points K and κ in the diagram. Those to the left of K (and those below κ) are worse off

post-merger. Table 1 details consumers’ purchase behavior pre and post merger and whether

their consumer surplus increased or decreased.

Most consumers suffer a loss of surplus due to the merger. The magnitude of the gain in

surplus to some consumers is also much less than the magnitude of others’ losses, the net

change in consumer surplus is therefore negative. Recall that the decrease in consumer surplus

is less with merger if firms select pure bundling (which they would not do unless mixed bundling

were prohibited). This demonstrates the serious consequences of a merger analysis that restricts

the set of price discrimination strategies that accrue to firms through non-horizontal merger as is

sometimes assumed in the literature.19

be chosen in any case).

In Case 7 there is no Cournot effect, so the First Proposition is false. Prices

(appropriately calculated) increase for a significant fraction of consumers, showing the Second

Proposition to be false. As for the Third Proposition, “If a complementary goods merger

would lead to mixed bundling (offering goods both as a bundle and separately not in a bundle)

then the merger would be to the benefit of consumers.” Not only is this false, consumer surplus

declines, but the entire intuition of the Third Proposition is also incorrect: It is the mixed

bundling itself which is responsible for the decline in consumer surplus, component prices are

considerably enhanced before offering the bundle “at a discount” from the sum of component

prices. If mixed bundling were prohibited, but pure bundling allowed, the merger would benefit

consumers with this demand structure.

E. Summary and Assessment of Results

We analyzed mergers between producers of complementary products under three sets of

market conditions: 1) complementary monopoly; 2) oligopoly with system competition; 3)

oligopoly with component competition.

We assumed that there was solely vertical product differentiation and that firms compete

by setting prices. We examined three preference patterns: 1) Consumers uniformly distributed

along (0,0) to (1,1); 2) Consumers uniformly distributed along (0,1) to (1,0); 3) Consumers

uniformly distributed along the positive arc of the unit circle. We found that the Cournot effect

is not ubiquitous, and that under several of our cases a merger will lower consumer surplus.

These results are summarized in Table 2.

Table 3 contrasts our results with the propositions we derived from the policy literature.

We demonstrated via counter examples that the sweeping Antitrust–Competition Policy

assessments of a priori lower prices for consumers from complementary good mergers are

simply unsupported in economic theory. (They may be supported empirically on a case by case

20 Suppose that a seller of X, which must be used with Y, merges with a seller of Y. Themerged firm may be able to produce an X-Y bundle less expensively than the sum of the costs ofa unit of X and a unit of Y produced independently.

basis, which we do not address.)

Notably with our stylized demand structures, when there is component competition, there

is no Cournot effect, even with the Cournot demand specification. This is due in part to

assuming that firms compete by setting prices and that there is no horizontal product

differentiation (cf., Salinger 1989 and Economides and Salop 1992). For two of our demand

specifications, mergers lead to bundling (pure and mixed) and higher prices when both pure and

mixed bundling are available as post-merger strategies. Even if there would be Cournot effects

for some demand specifications and assumptions about post merger strategy sets (e.g. no

bundling or no pure bundling), the potential for bundling may lead to price increases.

We should note some other topics that we have left out which might also affect the

pricing outcomes of complementary products mergers. First, complementary mergers and the

following firm behavior may raise entry barriers (cf. Nalebuff 2004). Second, synergies may be

created due to technical complementarities.20 Our analysis does not create a presumption about

whether the results of complementary goods mergers are inherently good or bad, but it

demonstrates that the a priori assumption accepted by policy makers that they are essentially

always good is not supported by economic theory.

A third issue that we do not address is the divergence in policy making in the U.S. versus

the EU. In, for example, GE-Honeywell, it appears that the U.S. felt that Cournot effects would

lower prices and benefit consumers. The EU decided that they would lower prices and harm

21 The EU did not articulate a clear economic rationale for its decision. Presumably thisis one reason why it commissioned economic reports on these issues after their decision. SeeKühn, Stillman and Caffarrra (2004) for a similar conclusion.

22 We do not address whether the goals would relate to European competitors andconsumers, or world competitors and consumers.

competitors (with some possible longer term consequences harmful to consumers).21,22 We have

only focused on the presumption that complementary goods mergers will lower prices, and

demonstrated that this proposition is not robust to market structure, consumer preferences, and

the set of strategies available to post-merger firms.

6 Conclusion

There is ongoing policy analysis of the potential effects of conglomerate mergers, in

particular, mergers of complementary goods producers. Both U.S. and EU antitrust authorities

have been involved in related inquiries, and the EU commissioned two economic reports, over

500 pages of work summarizing the state of economic knowledge about non-horizontal mergers.

In all of this it appears that antitrust authorities accept the Cournot complementary monopoly

double marginalization model as an underpinning of appropriate policy towards such mergers.

What we demonstrate herein is that Cournot effects are not ubiquitous, indeed for a large

class of complementary products oligopolies there are simply no Cournot effects!

We do not do a full policy analysis in this paper. A full policy analysis would need to

address the role of producer surplus and the role of consumer surplus, and whether, for example,

the EU antitrust authorities should be equally weighting consumer surplus and producer surplus

or whether they should be looking at surplus in only the EU or also in non-EU countries.

Similarly, should U.S. antitrust authorities be equally weighting American and foreign gains or

losses from mergers.

Before addressing normative policy analysis, however, one needs to know the positive

economic answers to questions such as “What will happen to prices” if there is a merger of

complementary oligopolists.

We briefly look at Merger Guidelines, to verify that they do not address the issues we

address in this paper. We then take three antitrust statements about what will happen with

mergers of complementary product firms. These statements are rephrased as theoretical

propositions about the effects of complementary products mergers. One statement is that prices

will fall due to Cournot effects. We show that even with Cournot’s demand specification, this

presumption is false in the case of vertical product differentiation (quality differentiation) for

price setting firms selling compatible complementary products. A second statement is that prices

will never rise with complementary goods mergers (barring foreclosure or entry barriers). By

varying from Cournot’s demand specification, we show this to be false, firms may bundle and

raise prices. A third policy statement is that as long as bundling is “mixed” (e.g., one can buy

either component separately or the bundle of the two components) that consumers will be better

off. In demonstrating that this is false, we even demonstrate that pure bundling (buy both or

neither) will lead to better merger outcomes for consumers than they would have if mixed

bundling were allowed.

Our proofs are by counter example, using the simplest parameters and functional forms to

disprove the propositions stated in the policy literature. We cannot say for certain that the

Cournot intuition is robust in the sense that “most” complementary products mergers will lead to

lower consumer prices. On the other hand, we do demonstrate strong reasons to believe that

when there is component competition, and products can be mixed and matched (i.e. are

compatible), the Cournot effect disappears. More generally, relying solely on “19th-century

thinking” (Varian, New York Times, June 28, 2001) appears to have led to naive policy making.

Appendix: Proof of Lemma, component price lines intersect bundle price line on the circle

To prove that the equilibrium component price lines intersect the circle at exactly the same

points as the bundle price line, we first show that it would not be profit maximizing for the firm

to choose lower component prices. This is clear from the diagram in Figure A1. Given a bundle

price PHH, suppose the firm chooses component prices p1H and p2H as shown so that each

component price is less than the point of intersection of the bundle price line and circle. Those

consumers on the arc between the vertical axis and point K purchase the mixed quality system

with component 1H, those on the arc between K and κ purchase the high quality bundle, and

those on the arc from κ to the horizontal axis purchase the mixed quality system with component

2H. Note that the firm could increase the bundle price from PHH to P'HH without altering the

purchase decisions of consumers. That is, it can continue to sell the high quality system to the

same set of consumers at the higher price P'HH, but at this higher bundle price, the component and

bundle price lines intersect the circle at the same points.

Now suppose that the component prices are greater than the point of intersection of the

bundle price line and the circle. We know from the derivation of the pure bundling equilibrium

that profits from bundling monotonically decrease as price increases away from 1; profits

therefore monotonically increase as price decreases toward 1. Therefore as long as there are

consumers not served by either high quality component or the bundle, it will be profitable for the

firm to reduce the bundle price in order to serve those consumers who otherwise would not

purchase any high quality components.

Whether it is more profitable to lower or raise the bundle price to meet the component

prices, or else lower or raise the component prices to meet the intersection points of the bundle

price line and circle is therefore the question that the integrated firm must answer. Given the

optimality of setting the component prices to intersect the circle at the same point as the bundle

price line, we are able to reduce the dimensionality of the firm’s profit maximization problem.

Profit maximization is simply a choice of a bundled system price since component prices are

determined by the intersection points with the circle.

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Figure 8

Figure 9

Table 1. Consumer choice and the effect on consumersurplus, proportions of consumers

Post merger choicemixed quality

systemhigh quality system

CS decrease CS increase CS decrease

Pre

mer

ger c

hoic

e mixed qualitysystem 35.31% 11.15% 44.00%

high qualitysystem n.a. 9.54% n.a.

Table 2. Merger Effects: Complementary Goods given Price Competition and Vertical Differentiation

preferencescompetition

{0,0},{1,1} {0,1},{1,0} unit arc

monopoly Cournot effectΔCS>0

No Cournot effectΔCS=0

n.a.

system competition Cournot effectΔCS>0

No Cournot effectΔCS=0

n.a.

componentcompetition

No Cournot effectΔCS=0

No Cournot effectPure bundlingΔCS<0

No Cournot effectMixed bundlingdominates pureΔCS<0

n.a. not analyzed, but there clearly is no Cournot effect for the reasons which generate theconclusions in the prior column.

Table 3. Merger Effects: Complementary Goods and the Rejection of Propositions Derived from the Literature

preferencescompetition

along (0,0) to (1,1) along (0,1) to (1,0) positive unit arc

monopoly Consistent withliterature

Rejects First andSecond Proposition

Not Analyzed

system competition Consistent withliterature

Rejects First andSecond Proposition

Not Analyzed

componentcompetition

Rejects FirstProposition

Rejects First andSecond Proposition

Rejects All ThreePropositions