A Note on Complementary Goods Mergers between Oligopolists with Market Power: Cournot Effects, Bundling and Antitrust

These include:

“…when there are no economies of scope, when two producers of complementary products merge they may offer a lower price for a bundle of those products because the merger solves a “double-marginalization problem… This is the so-called “Cournot effect” … is all the more likely in those instances where the merging firms had been exercising a degree of market power before the merger. U.S. Antitrust Division submission for OECD Roundtable on Portfolio Effects in Conglomerate Mergers, Range Effects: The United States Perspective (“OECD Roundtable”), October 12, 2001, p. 11. http://www.justice.gov/atr/public/international/9550.htm.

To the extent the merging parties enjoyed large market shares and market power in complementary goods, there will be a tendency for prices to decline post merger … fears that a conglomerate merger involving portfolio effects would lead to a welfare reducing type of price discrimination involving tying or bundling could be a thin reed to lean on as the sole rationale for blocking the merger. Ibid, pp. 30–31.

A firm may bundle its product with a complement in order to soften competition. Bundling in this case increases the profits of all participants in the market… An easy way to detect whether softening competition is the motivation for bundling is to look at competitors’ reactions to the bundle: If competitors are complaining about the possibility, we can be pretty sure that it is not serving to soften competition. Ibid, pp. 30–31.

To the extent a merger of complements gives the merged firm the incentive to lower prices because it causes the firm to internalize the negative externalities associated with higher prices (the so-called Cournot effect), it moves prices in the right direction – toward marginal costs – enhancing allocative efficiency through the elimination of double marginalization and benefitting consumers with lower prices and increased output.” “We simply could not identify any conditions under which a conglomerate merger, unlike a horizontal or vertical merger, would likely give the merged firm the ability and incentive to raise price and restrict output. William Kolasky, [then Deputy Assistant Attorney General U.S. Department of Justice], Conglomerate Mergers and Range Effects: It’s a Long Way from Chicago to Brussels, November 9, 2001.

Improved coordination between suppliers of complementary goods is an essential aspect of efficiency. Such improved coordination not only raises the parties’ joint profits, but tends to increase overall efficiency as well through lower prices or improved quality. This externality between the parties could be better internalized by their vertical [stet] merger… OECD Policy Roundtables, Vertical Mergers, 2007, United States submission, pp. 239–248.

when producers of complementary goods are pricing independently, they will not take into account the positive effect of a drop in the price of their product on the sales of the other product. Depending on the market conditions, a merged firm may internalize this effect and may have a certain incentive to lower margins if this leads to higher overall profits (this incentive is often referred to as the “Cournot effect”). Official Journal of the European Union, October 18, 2008, paragraph 117.

In criticizing the EU’s decision to challenge the GE-Honeywell merger Hal Varian concludes that “GE-Honeywell ran afoul of 19th-century thinking.” Specifically:

[A]ntitrust authorities rightly frown on companies’ coming together to set prices, since the effect is often anticompetitive. On the other hand, if the products are highly complementary and are produced in highly concentrated industries, producers left to their own devices may set prices too high because of the “Cournot effect.” [New York Times, June 28, 2001].

Supra note 1.

The absence of CEs under our demand specifications results from pre-merger price competition between a high-quality version of a component and homogenous low-quality versions sold by multiple firms that engage in pure Bertrand pricing.

AC&P also consider the case in which each integrated firm sells one high- and one low-quality component; in this case divestiture may lead to double marginalization. Under this setup, their model predicts that double marginalization (i.e., “tragedy of the anticommons” in their terminology) will more than offset the benefits from the increase in competition.

Vertical differentiation refers to a situation where all consumers are willing to pay a premium for a particular version of a component. Horizontal differentiation means some consumers would pay a premium for one version of a component while others would pay a premium for a different version.

We skip the case of zero correlation, usually modeled as preferences for the two high-quality components distributed uniformly on a unit square. This preference distribution also leads to no CEs after a merger of the two high-quality component producers, and like case B below generates a loss of consumer surplus. Pre-merger, optimal high-quality component prices are ½, as a result, one-quarter of the population purchases each of the four system types. While a merger between the two high-quality producers followed by pure bundling does not change individual component prices, post-merger one-half of the population purchases the high-quality system while the other half purchases a system comprised of only low-quality components. The merged firm captures one-half of the consumer surplus that was earned pre-merger by the consumers that purchased a hybrid system. Both Einhorn (1992) and Matutes and Regibeau (1988) model the zero-correlation case; however, their models include at most four independent producers, each with some market power.

Although we report pre- and post-merger producer surplus, both the U.S. Horizontal Merger Guidelines (2010) and EU Non-Horizontal Merger Guidelines (2008) endorse a consumer welfare standard for evaluating transactions. (“Mergers should not be permitted to create, enhance, or entrench market power or to facilitate its exercise. A merger enhances market power if it is likely to encourage one or more firms to raise price, reduce output, diminish innovation, or otherwise harm customers as a result of diminished competitive constraints or incentives.” – DOJ, FTC Horizontal Merger Guidelines, August 2010, p. 2; “Effective competition brings benefits to consumers, such as low prices, high quality products, a wide selection of goods and services, and innovation. Through its control of mergers, the Commission prevents mergers that would be likely to deprive customers of these benefits by significantly increasing the market power of firms.” – Official Journal of the European Union, 2008/C 265/07, October 2008, paragraph 10.)

Presumably such a merger would be motivated by objectives outside of those addressed by this note.

Without suppliers of low-quality components, pre-merger each component monopolist accounts for the other’s price when setting its own. Profits for each monopoly producer would equal (1 – P/2)p_i, for i=1, 2, leading to identical reaction functions p_i=1 – p_j/2, optimal pre-merger prices for each component of 2/3, and a system price equal to 4/3. Their combination results in CEs because the merged firm maximizes total system profit of (1 – P/2)P leading to an optimal system price of 1. This same result is obtained when low-quality component producers compete pre-merger but high- and low-quality components are incompatible.

This amount is the sum of the valuations for that consumer who values the two high-quality components equally (and maximally across all consumers) and is derived from the formula for the circle, x² + y² = r² where x=y and r=1, i.e. 2x² = 1. Solving for x results in x=0.71 and 2x=1.42.

The Appendix also shows the merged firm would not choose to pure bundle. While its profits from pure bundling exceed the sum of the two firms’ pre-merger profits and consumer welfare increases, mixed bundling generates even greater profits because it allows the firm to price-discriminate and charge a higher price to those consumers who place a large value on only one of the high-quality components.

Even though the price of the bundle under either pure or mixed bundling is less than the sum of the pre-merger component prices, this result does not reflect the presence of CEs because no pricing externalities are internalized by the merger. A necessary and sufficient condition for the presence of CEs if all components are compatible is lower post-merger prices of the two high-quality components sold only separately. In case C, separate components pricing post-merger results in prices for the two high-quality components which equal their pre-merger prices. See Appendix, footnote 16.

That is, consumers with component valuations of at least (0.5834, 0.8122) or (0.8122, 0.5834).

At component price p∗=0.6522, total demand for either high-quality component is q∗=(2/π)cos−1(p∗)=0.5477. Inverse demand is Demand−1(q)=cosqπ2. For any one component total consumer surplus (CS) is ∫0q∗Demand−1(q)dq−p∗q∗=0.1254. Twice this amount, 0.2508, is total CS. On Ω, consumers who value the high-quality component more than v∗=1−p∗2=0.7583 do not purchase the other high-quality component. Their mass is q0=Demand(v∗)=0.4521. Consumers who purchase both high-quality components measure q1=q∗−q0=0.0956 and earn CS equal to ∫0q∗Demand−1(q)dq−p∗q1=0.0052. By symmetry, their CS on both components is 0.0104. CS earned by consumers who purchase only one high-quality component is 0.2508−0.0104=0.2404. See also Table 2.

Recall the demand for high-quality component j is given by (2/π)cos−1pjH for j=1,2. Thus the demand for 2H is independent of the price of 1H and vice versa. Given independent demands and absent bundling, the post-merger first-order conditions for profit maximization are identical to the pre-merger first-order conditions, implying identical pre- and post-merger prices.

Let Q(P) denote the mass of consumers who purchase the bundle at price P. For these consumers v1H+v2H>P. Let F(P) be the mass of consumers for whom v1H+v2H<P. Thus Q(P)+F(P)=1, and bundle demand is Q(P)=1−F(P). The bundle price is represented as a line with a slope of –1 along which x+y=P. (The x and y respectively correspond to v2H and v1H.) The unit circle is defined by the equality x2+y2=1. For any value of P strictly between 1 and 2, the bundle price line and the unit circle intersect (and the bundle price line bisects the unit circle) at two distinct points (x∗,y∗) and (x∗∗,y∗∗). Let y∗∗>y∗. Half of the consumers for whom v1H+v2H<P are located on the lowermost part of Ω below the lower intersection point (x∗,y∗) and the other half are located to the left of (x∗∗,y∗∗). Then, (x∗,y∗) can be derived as follows: The price line is y=P−x and the unit circle is y=1−x2. Equating these, P−x=1−x2 leads to the quadratic expression 2x2−2Px−1−P2=0, which has a root at x∗=P+2−P2/2 and y∗=1−x∗. By the trigonometric discussion immediately preceding equation [5], the length of the arc from the “base” of Ω to the point of intersection (x∗,y∗) equals cos−1x∗. Half of the mass of consumers who do not buy the bundle equals this arc length normalized by 2/π. This mass is doubled to account for an equal mass of consumers at the “apex” of Ω who do not buy the bundle. Thus F(P)=2(2/π)cos−1P+2−P2/2. Since Q(P)=1−F(P), the demand for the bundle at price P equals 1−2(2/π)cos−1P+2−P2/2.

The formal argument as to why the post-merger pure bundling equilibrium occurs at (P∗,Q∗)=(1,1) is as follows. (There is a kink in the demand curve at Q=1, so the following calculus defines “partial derivative with respect to P at P=1” as P approaches 1 from above.) Elasticity of the system demand Q(P) is given by ε=Q′(P)P/Q(P). At P=1, Q=1, this elasticity simplifies to ɛ=Q′(P)=num(P)/den(P) where num(P)≡22P1−P2−P2 and den(P)≡1−P2−P2π. Moreover, ɛ2 simplifies to 162−P2π2. Therefore at P=1, ɛ2=16/π2 which implies |ɛ|=4/π>1, i.e. the demand is elastic at P=1. We will first argue that the merged firm does not have an incentive to raise price above P=1. At P=1, elasticity exceeds 1 and marginal cost is constant at zero. Therefore, if the merged firm priced at P>1 it would lose quantity (and revenue) without avoiding cost. Since Q(P) is concave, demand elasticity increases with P meaning any P>1 is dominated by P=1. Next, we argue that the merged firm does not have an incentive to price below P=1. Since all consumers have a reservation price of at least one, any P<1 is dominated by P=1. Stated technically, if elasticity >1, and MC = 0, there is an incentive to lower price and raise quantity because MR>MC. At the kink, however, lowering price does not lead to higher quantity, causing the same quantity to be sold for less. Thus, the merged firm will maximize profit at P∗=1.

The profit function [8] follows from the bundle demand expression [7] and the individual expressions for the high-quality components [5]. The first term captures profits from bundle sales, and equals the product of the bundle price PHH with the bundle demand expression [7], given that p=P+2−P2/2 and P=PHH. The second term captures profits from individual component sales and is identical to the profit expression [5], with p substituted for piH. The expression p=piH=PHH+2−PHH2/2 is identically derived as, and identical to, the expression for x∗ in footnote 17 above, with PHH replacing P.

Pre-merger utility is v1−p1H0; post-merger it equals v1−(p1H,2H′−v2) when written to illustrate the change in the implicit price of 1H. Hence, ΔU=v2−PHH′+p1H0.

The function ψ is the inverse of θ and Ψ is the inverse of Θ.

Calculated as the solution to α in α−0.8122=0.5834 where (0.5834, 0.8122) are the coordinates of point K on Ω, and QKκ=Θ(PKκ).