Publication: RAND Journal of Economics
Publication Date: 22-DEC-06
Delivery: Immediate Online Access
Author: Fang, Hanming ; Norman, Peter

Article Excerpt
Comparing monopoly bundling with separate sales is relatively straightforward in an environment with a large number of goods. We show that results similar to those for the asymptotic case can be obtained in the more realistic case with a given finite number of goods, provided that the distributions of valuations are symmetric and log-concave.

When I go to the grocery store to buy a quart of milk, I don't have to buy a package of celery and a bunch of broccoli.... I don't like broccoli. (U.S. Senator John McCain, in an interview on cable TV rates published in the Washington Post, CI, March 24, 2004)

1. Introduction

* Bundling, the practice of selling two or more products as a package deal, is a common phenomenon in markets where sellers have market power. It is sometimes possible to rationalize bundling by complementarities in technologies or in preferences. However, it has long been understood that bundling may be a profitable device for price discrimination, even when the willingness to pay for one good is unaffected by whether or not other goods in the bundle are consumed, and when no costs are saved through bundling (Adams and Yellen, 1976; Schmalensee, 1982). While the earliest literature of bundling typically understood it as a way to exploit negative correlation between valuations for different goods, McAfee, McMillan, and Whinston (1989) show that mixed bundling, which refers to a selling strategy where each good can be purchased either as a separate good or as part of a bundle, leads to a strict increase in profits relative to fully separate sales, provided that a condition on the joint distribution of valuations is satisfied. Importantly, the distributional condition holds generically and is implied by stochastic independence, so the profit-improving role of mixed bundling has nothing do with exploiting negative correlations of valuation distributions.

In this article we rule out mixed bundling by assumption and focus on the comparison of pure bundling and separate sales. By pure bundling, we refer to the case that any good IS sold either as an item in a larger bundle or as a separate item, but not both. Of course, mixed bundling does occur in the real world. For example, in many markets it is possible to buy access to cable TV at one price, high-speed Internet access at one price, and a bundle consisting of both cable TV and high-speed Internet access at a price that is lower than the sum of the component prices. (1) We offer three reasons for our focus on pure bundling.

First, McAfee, McMillan, and Whinston (1989) showed that, generically, any multiproduct monopolist should offer to sell all of the goods in mixed bundles. This powerful result does make some of the crude bundling schemes that we observe in the real world rather puzzling. For example, the question of why ESPN is available as a component of a bundle while championship boxing matches tend to be available only on a pay-per-view basis cannot be answered.

Second, in some cases technological reasons may make mixed bundling infeasible or too costly to implement. For example, in the context of bundling computer programs it does not seem farfetched to assume that selling components separately would require substantial extra programming costs in order to guarantee compatibility of the components with older softwares, costs that could be avoided if the new programs are bundled.

Third, in some cases the practice of mixed bundling is more likely than pure bundling to get in trouble with antitrust laws, which is explicitly expressed in terms of "anticompetitive mixed bundling." (2) Of course in general, the legal interpretation of "mixed" is unclear. But in a recent case in the United Kingdom, the decision by the Office of Fair Trading (2002) on the alleged anticompetitive mixed bundling by the British Sky Broadcasting Limited explicitly stated that "[m]ixed bundling refers to a situation where two or more products are offered together at a price less than the sum of the individual product prices--i.e., there are discounts for the purchase of additional products." This test, which compares marginal prices, requires that a product can be bought both as a bundle and as a separate good. Thus it has no bite at all when the monopolist uses pure bundling. (3)

In this article we obtain a rather intuitive characterization for when a multiproduct monopolist should bundle and when it should sell the goods separately in order to maximize its profits. To some extent our characterization confirms (mainly) numerical results in Schmalensee (1984), namely, the higher the marginal cost and the lower the mean valuation, the less likely that bundling dominates separate sales. When limiting our comparison to pure bundling and separate sales, we are able to highlight a clear intuition for what happens when two or more goods are sold as a bundle. The key effect driving all the results is that the variance in the relevant willingness to pay is reduced when goods are bundled. We shall provide a partial characterization for when this reduction in variance is beneficial for the monopolist and when it is not.

The crucial idea is that bundling makes the tails of the distribution of willingness to pay thinner. However, what we need is a rather strong notion of what "thinner tails" mean. Specifically, we need to be able to conclude that for a given per-good price below (respectively, above) the mean, bundling increases (respectively, reduces) the probability of trade. This can be rephrased by saying that the average valuation is more peaked than the underlying distributions. Notice that the law of large numbers can be used to reach this conclusion if there are sufficiently many goods available, but for a given finite number of goods, counterexamples are easy to construct. We therefore need to make some distributional assumptions. Indeed, assuming that valuations are distributed in accordance to symmetric and log-concave densities, we can use a result from Proschan (1965) to unambiguously rank distributions in terms of relative peakedness.

Under these distributional assumptions, bundling reduces the effective dispersion in the buyers' valuations. This reduction of valuation dispersion is to the advantage of the monopolist when a good should be sold with high probability (either because costs are low or because valuations tend to be high). In such cases, we show that bundling increases the monopolist's profits. The reduction of taste dispersion may be to the disadvantage of the monopolist when the goods have only a thin market (either because the costs are high or because valuations tend to be low). Indeed in such cases the monopolist is better off relying on the fight tail of the distribution and selling all goods separately.

The idea that "bundling reduces dispersion" has been around for a long time, and there is even some emerging empirical evidence supporting it as a motivation to bundle (see Crawford, 2004). What is largely missing in the literature, however, are results that establish reasonably general conditions to explain bundling as a profit-maximizing selling strategy. The most related article is Schmalensee (1984), who considers the case with normally distributed distributions of valuations (which belongs to the class we consider). Relying mainly on numerical methods, he reaches a similar conclusion. Recently, Ibragimov (2005) has developed a related characterization relying on a generalization of the result in Proschan (1965).

In the context of "information goods" (goods with zero marginal costs), Bakos and Brynjolfsson (1999) and, more recently, Geng, Stinchcombe, and Whinston (2005), used a similar idea to argue that bundling is better than separate sales. While both sets of authors assume zero marginal costs, the main difference with our article is that they focus on results for large numbers of goods. Though we also prove some asymptotic results, our main contribution is to provide conditions under which we can obtain analogous results for the finite-good case.

The remainder of the article is structured as follows. Section 2 presents the model. Section 3 introduces the statistics notion of peakedness. Section 4 presents the asymptotic results in an environment with a large number of goods. Section 5 provides our main analysis for the finite-good case. Finally, Section 6 concludes. All proofs are relegated to Appendix A.

2. The model

* The underlying economic environment is the same as in McAfee, McMillan, and Whinston (1989), except that we allow for more than two goods. A profit-maximizing monopolist sells K indivisible products indexed by j = 1,..., K, and good j is produced at a constant unit cost [c.sub.j]. A representative consumer is interested in buying at most one unit of each good and is characterized by a vector of valuations [theta] = ([[theta].sub.1],..., [[theha].sub.K]), where [[theta].sub.j] is interpreted as the consumer's valuation of good j. The vector [theta] is private information to the consumer, and the utility of the consumer is given by

[K.summation over (j=1)] [I.sub.j][[theta].sub.j] - p,

where p is the transfer from the consumer to the seller and [I.sub.j] is a dummy taking on value one if good j is consumed and zero otherwise. Valuations are assumed stochastically independent, and we let [F.sub.j] denote the marginal distribution of [[theta].sub.j]. Hence, [[PI].sup.K.sub.j=i]/[F.sub.j]([[theta].sub.j]) is the cumulative distribution of [theta].

3. Peakedness of convolutions of log-concave densities

* A rough interpretation of the law of large numbers is that the distribution of the average of a random sample gets more and more concentrated around the population mean as the sample size grows. However, the law of large numbers does not imply that...



NOTE: All illustrations and photos have been removed from this article.



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