Multiproduct Cournot oligopoly.

We study a Cournot industry in which each firm sells multiple quality-differentiated products. We use an upgrades approach, working not with the actual products but instead with upgrades from one quality to the next. The properties of single-product models carry over to the supply of upgrades, but not necessarily to the supply of complete products. Product line determinants and welfare results are presented. Strategic commitment to product lines is considered; firms may well choose to compete head-to-head.

1. Introduction

* Competition between multiproduct firms abounds. While understanding such competition is important, only limited progress has been made. We enhance understanding by presenting a general analysis of oligopolistic competition in quantities between firms offering multiple quality-differentiated products. We address three broad questions. First, when do the insights of single-product Cournot models carry over to a multiproduct world? Second, what factors determine firms' product lines (that is, the qualities that they offer), and how do qualities differ from the socially efficient ones? Third, to what extent does the opportunity to precommit strategically to a product line influence our results?

The setting is a market populated by consumers with general preferences for quality, and an arbitrary number of firms. We follow earlier work (Johnson and Myatt, 2003) by taking an "upgrades approach" This approach recasts competition in terms of conceptual upgrades from one quality level to the next, instead of in terms of actual products. (1)

A benefit of the upgrades approach is that it helps answer our first question. The insights of single-product Cournot models carry over to a multiproduct world when we think in terms of upgrades, but may fail when we think of the actual products themselves. For instance, when firms are symmetric, an increase in the number of competitors will yield an expansion in the supply of each upgrade. The supply of a particular quality, however, might well decline. Similarly, in an asymmetric setting, firms with lower costs produce more of every upgrade, echoing results from single-product industries. Once again, however, this does not necessarily apply to the supply of a specific complete product; it is possible that lower-cost firms produce zero units of some qualities while higher-cost firms offer positive supplies.

To answer our second question, we show that the key determinants of product lines include the returns to quality (that is, the change in the ratio of cost to willingness to pay as quality increases) and the changes in demand elasticity as quality increases. In contrast, the qualities offered by a social planner are determined solely by returns to quality. The quality of products consumed is distorted downward from the first-best. Relative to the second-best, however, quality is too high.

In answering our third question, we first show that firms competing in quantities never precommit not to sell the highest-quality good, since with strategic substitutes this makes them soft and leads to an expansion of their rivals' output. Interestingly, a decision instead not to sell a lower-quality product has conflicting effects. It toughens the committing firm's stance in the higher-quality upgrade market but softens it in the lower-quality upgrade market. While the net effect on profits is indeterminate, there is a bias toward not committing so that multiproduct firms may compete head-to-head (that is, with the same qualities) even given the opportunity to avoid doing so. These results stand in stark contrast to what prevails in price-setting models of competition in quality-differentiated markets such as that of Champsaur and Rochet (1989), who showed that duopolists precommit to producing qualities ranges that do not intersect. (2)

The upgrades technique was used in earlier work (Johnson and Myatt, 2003) in which we considered a multiproduct incumbent's response to entry. (3) Multiproduct quantity competition was also considered by Gal-Or (1983) and De Fraja (1996). Others considered competition in prices when goods are horizontally differentiated, (4) and monopoly price discrimination (Mussa and Rosen, 1978; Maskin and Riley, 1984).

Section 2 lays out our model, Sections 3 through 5 contain our analysis, and in Section 6 we conclude. All proofs are found in the Appendix.

2. Supply and demand in a multiproduct world

* Here we describe supply and demand in a market for quality-differentiated products. We look for pure-strategy Nash equilibria in which each multiproduct firm simultaneously chooses its outputs given those of other firms, and refer to such an equilibrium as a "multiproduct Cournot equilibrium." Our specification generalizes that in Johnson and Myatt (2003).

Formally, M distinct product qualities [q.sub.M] > ... > [q.sub.1] > 0 are supplied by N firms, where Zir [greater than or equal to] 0 is firm r's output of quality [q.sub.i], and [z.sub.i] = [[summation].sup.N.sub.r=1] [Z.sub.ir] is the total industry supply of [q.sub.i]. (5) We will make extensive use of the cumulative variables [Z.sub.ir] [equivalent to] [[summation].sup.M.sub.j=i] [Z.sub.jr] (firm r's supply at quality [q.sub.i] and above) and [Z.sub.i] = [[summation].sup.N.sub.r=1] [Z.sub.ir] (the corresponding industry supply). By construction, such cumulative variables satisfy the monotonicity constraints [Z.sub.1r] [greater than or equal to] [Z.sub.jr] [greater than or equal to] [Z.sub.Mr] [greater than or equal to] 0.

[] Market demand. A unit mass of consumers is indexed by a type parameter [theta] [member of] [0, [bar.[theta]]]. We write H(z) for the type such that a mass z [member of] [0, 1] of consumers value quality more highly. H(z) is strictly decreasing and continuously differentiable, H(0) = [bar.[theta]] and H(z) = 0 for z [greater than or equal to] 1. (6) A consumer [theta] who pays a price [p.sub.i] for product i enjoys utility u([theta], [q.sub.i]) - [p.sub.i], where u([theta], q) is strictly increasing in [theta] and q, twice continuously differentiable, and exhibits increasing differences, so that u([theta], q) - u([theta], q') is strictly increasing in [theta] whenever q > q'. Furthermore, we assume that u([theta], 0) = u(0, q) = 0 for all [theta] and q. This equates a product of zero quality with no consumption, and ensures that the lowest type gains no value from a purchase. Each consumer purchases a single unit of the product that maximizes u([theta], [q.sub.i]) - [P.sub.i], unless doing so yields strictly negative utility, in which case she purchases nothing.

To derive an inverse demand system, we begin with the case where [Z.sub.1] = [[summation].sup.M.sub.i=1] [Z.sub.i] < 1, so that there is partial market coverage. We require a set of positive prices such that exactly [z.sub.i] consumers wish to purchase product i. Higher-quality products must carry a price premium and be purchased by consumers with higher types. Thus a consumer of type H([Z.sub.1]) must be indifferent between purchasing quality [q.sub.i] and not purchasing at all, and hence [p.sub.1] = u(H([Z.sub.1]), [q.sub.1]). Similarly, a type H([Z.sub.i]) with [Z.sub.i] others above her must be just indifferent between products i and i - 1, and so [p.sub.i] = [p.sub.i-1] + [u(H([Z.sub.i]), [q.sub.i]) - u(H([Z.sub.i]), [q.sub.i-1])]. We define [q.sub.0] = [p.sub.0] = 0 for convenience, and note that [p.sub.i] - [p.sub.i-1] is the price differential between products i and i - 1. This satisfies

[p.sub.i] - [p.sub.i-1] = [P.sub.i]([Z.sub.i]), where [P.sub.i](Z) [equivalent to] u(H(Z), [q.sub.i]) - u(H(Z), [q.sub.i-1]). (1)

It turns out that (1) also describes the system of inverse demand when there is complete market coverage, so that [Z.sub.1] [greater than or equal to] 1, and addresses the possibility that some products are in zero supply. In Johnson and Myatt (2003) we confirm that (1) is correct for all circumstances.

[P.sub.i]([Z.sub.i]) is the extra utility received by a type [theta] = H([Z.sub.i]) from increasing quality from [q.sub.i-1] to [q.sub.i], and is independent of [Z.sub.j] for j [not equal to] i; this is the strength of the upgrades approach. (7) Equivalently, it is the price of an "upgrade" in quality. We offer this (conceptual) interpretation: [Z.sub.1] units of a "baseline" product of quality [q.sub.1] are supplied, together with successive upgrades in order to achieve higher qualities.

We might consider [Z.sub.ir][P.sub.i]([Z.sub.i]) to be the revenue earned by firm r from providing quality increases. If upgrades corresponded to actual physical products, a familiar regularity condition would be that marginal revenue is decreasing.

Assumption. Marginal revenue is decreasing in each upgrade market, so that

[partial derivative][[P.sub.i]([Z.sub.i]) + [Z.sub.ir][P'.sub.i]([Z.sub.i])]/[partial derivative][Z.sub.i] < 0. (2)

We maintain this assumption throughout the entire article. (8) In a single-product world, (2) implies that a firm's best response is decreasing in the output of its competitors. In terms of complete physical products, a unit increase in [Z.sub.i] corresponds to a unit increase in [z.sub.i] and a unit decrease in [z.sub.i-1], so that (2) says that the marginal revenue of converting lower-quality units into higher-quality units is decreasing. It also implies that the marginal revenue of increasing the supply of a complete product i is decreasing.