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Description
We develop a model in which a main product (called product A) provides a performance quality z by itself, whereas a complementary product (called product B) is useless by itself but enhances the main product's performance quality to q > z. This asymmetric complementarity gives rise to the following results. First, if z is relatively small, then firms A and B behave as if the products are symmetrically complementary with the usual double marginalization problem. Second, if z is sufficiently large, then firms A and B price their products as if they are independent. Third, over a certain range of intermediate z, no pure-strategy Nash equilibrium exists.
1. Introduction
* In the computing industry, since 1990, there has been no single dominant vertically integrated firm. Instead, the industry is characterized by vertical disintegration, i.e., computer systems or platforms consist of many vertically related layers of components. Firms in different layers rely on one another, but at the same time they compete against each other for a bigger share of the industry profits. It is important to understand complementarity among different components.
In 1838, Cournot analyzed the pricing of symmetrically complementary products, like left and right shoes, and identified the well-known "double mark-up problem" i.e., when the two complementary products are supplied by two independent monopolies, the prices are higher than those set by an integrated monopoly. However, the complementarity relationship in the computing industry is quite different from that analyzed by Cournot and others. For instance, an advanced application program enhances the value of an operating system (OS), but it is useless without the OS. In contrast, the OS provides its basic functions without the advanced application program.
Furthermore, as Bresnahan (1999) and Bresnahan and Greenstein (1999) point out, in order to obtain a larger share of industry profits, a firm producing one product has an incentive to enter the others' "turf" by incorporating functions provided by the other firms. For example, in its early days, MS Windows did not include program functions such as WordPad, Internet Explorer (IE), and Windows Media, but over time, it has included these and other programs that were previously supplied by independent firms. Another example is secondary cache. Once a separate piece of hardware, secondary cache is now integrated into the Intel central processing unit (CPU). As firms constantly try to expand their product boundaries, the boundaries between adjacent layers and the relationships among those products change continuously as a consequence of both vertical competition and technological innovation.
This paper analyzes the strategic interactions between two firms whose products are asymmetrically complementary and attempts to shed light on vertical competition among different, layers of the computing industry by exploring the effects of changes in their product boundaries.
To model asymmetric complementarity, we assume that the "main product" A, produced by firm A, by itself provides a performance quality of z, but consumers may derive a higher performance quality of q (i.e., q > z) by combining it with an "enhancer" product B, produced independently by firm B. Unlike the main product A, product B does not provide any function by itself.
To explore the implications of asymmetric complementarities between products A and B, we first analyze a simultaneous pricing game between firms A and B given z, < z < q. It turns out that asymmetric complementarity combined with heterogeneous consumer preference over performance gives rise to the following three unexpected results. First, if z is relatively small, then products A and B are as if they are symmetrically complementary with z = and are always sold as a bundle. Second, if z is sufficiently large, then firms A and B price their products as if they are independent, in which case some consumers buy A alone while others buy both products. This result has an implication on the "double mark-up" problem: Even though products A and B are asymmetrically complementary, the firms set their prices independently, and the "double mark-up" problem vanishes. Third, over a certain range of intermediate value z, no pure-strategy Nash equilibrium exists. However, we can construct a mixed strategy equilibrium over the range.
Also, we examine the effects of increasing z, which can be interpreted as an expansion of firm A's product boundary. We analyze how an increase in z affects social welfare, industry profits, and consumer welfare.
There are several recent related studies on complementary technologies and patents (e.g., Farrell and Katz, 2000; Lerner and Tirole, 2004) and tying/bundling (e.g., Whinston, 1990; Choi and Stefanadis, 2001; Carlton and Waldman, 2002; Nalebuff, 2004).
Farrell and Katz (2000) analyze the incentive of a monopolist in product A to enter complementary product B's market in order to force independent suppliers of B to charge lower prices, which increases its own profits made from product A. If consumers in our model were homogeneous, then our results would become very similar to those of Farrell and Katz (2000): an increase in z "price squeezes" product B and always has a positive effects on firm A's profits. With heterogeneous consumer preference, however, we show that an increase in z does not have monotonic effects on firms' pricing and profits.
Our model is also closely related to Lerner and Tirole's (2004) model of patent portfolios, which allows a full range of complementarity and substitutability. There are several major differences between our model and theirs. First, their focus is on factors that encourage or hinder the formation of patent pools and the welfare effect of these pools, whereas our focus is on the firms' switching pricing behavior and the welfare effects of changes in z. Second, in their model, all users or licensees derive the same amount of marginal benefits from an additional patent, but in our model, different consumer types derive different marginal benefits from the basic product A and the bundle (A + B). Because of these differences, we obtain the result that the demand for A and B is independent of each other if z is sufficiently large and that no pure-strategy equilibrium exists for intermediate values of z.
Our paper is related to the literature on tying/bundling because product A in our model can be regarded as a bundle of two complementary products, [A.sub.1] and [B.sub.1] (i.e., [A.sub.1] and [B.sub.1] combine to yield a performance quality z, whereas [A.sub.1] and B combine to yield a performance quality q.) However, this literature either focuses on the entry deterrence role of tying or assumes that tying with a firm's own product excludes consumption of competing products. (1) However, when Microsoft ties its Windows OS and its applications such as IE, it still leaves room for consumers to add a rival product to its OS. We capture this product relationship by assuming that product B as an enhancer of the basic product A.
Nalebuff (2004) shows that, when consumers are heterogenous in their valuations of products A and B, an incumbent, by bundling A and B, can significantly lower the profits of a single-product entrant and that bundling could be quite an effective entry deterrence strategy. (2) However, our paper looks at the case in which one firm produces only a base product and the other firm produces a complementary product.
Section 2 develops a simple model and Section 3 analyzes the game and demonstrates the possible nonexistence of pure-strategy Nash equilibrium. Section 4 analyzes the effect of z on firms' profits, consumer surplus, and social welfare. In Section 5, we check the robustness of the main results when consumers' preferences vary along two dimensions. Concluding remarks follow in the final section.
2. A model of product boundary
* There are two firms, A and B, that provide complementary products A and B, respectively. Product A provides some basic functions, and its performance level is measured by a parameter z. Product B by itself does not provide any function, but enhances product A's performance. The combination of products A and B (denoted by (A + B) hereafter) provides a higher performance level q [greater than or equal to] z. Let product i's (i = A, B) price and unit production cost be denoted by [p.sub.i] and [c.sub.i], respectively. We assume that the two firms set their prices simultaneously.
Given [p.sub.A] and [p.sub.B], consumers make their purchase decisions. Consumers differ in their valuation of product quality. The utility function of a type-[theta] consumer, [theta] [member of] [0, 1], is given by [theta] Q + I, where I is her income spent on numeraire goods and Q is a quality index of a product. Let the cumulative distribution function and continuous density functions be given by G([theta]) and g([theta]), respectively. Define F([theta]) as the proportion of consumers whose type is higher than [theta] and f([theta]) as F's density function, i.e., F([theta]) = 1 - G([theta]) and f([theta]) = -g([theta]) < 0. We make the standard assumption that the distribution of [theta] satisfies the increasing hazard-rate condition: namely, -f([theta])/F([theta]) is increasing in [theta]. (3) This increasing hazard rate condition yields strictly quasi-concave profit functions for firms A and B.
We impose the following restrictions on the model's key parameters throughout our analysis.
Assumption 1 . [c.sub.A] + [c.sub.B] [less than or equal to] q , [less than or equal to] z [less than or equal to] [bar.z] = q - [c.sub.B].
The first restriction implies that the maximum willingness to pay for product (A + B) is larger than or equal to its unit production cost. Without this restriction, (A + B) will never be supplied. The second restriction implies that the quality enhancement brought about by product B (i.e., q - z) is larger than or equal to [c.sub.B]. Without the second restriction, there will be no supply of product B. Under Assumption 1, both firms A and B are active and the classic double mark-up problem may arise.
[] Demand functions for products A and B. Consumer [theta] has three options: (i) to buy product A alone and gain net utility [v.sub.A]([theta]) = z[theta] - [p.sub.A]; (ii) to buy (A + B) and gain net utility [v.sub.A+B] ([theta]) = q[theta] - [p.sub.A] - [p.sub.B]; (iii) and to buy neither and gain zero net utility. A necessary condition for the consumer to buy A alone is [theta] [greater than or equal to] [theta]A = [p.sub.A]/z. Similarly, a necessary condition for a consumer to buy (A + B) is [theta] [greater than or equal to] [[theta].sub.A+B] = ([p.sub.A] + [p.sub.B])/q. Consumers get additional benefits of (q - z)[theta] by purchasing product B in addition to product A. Thus, a necessary condition for a consumer to buy B in addition to A is that [theta] [greater than or equal to] [[theta].sub.B] = [p.sub.B]/(q - z).
[FIGURE 1 OMITTED]
Because [v.sub.A]([theta]) intersects the steeper function [V.sub.A+B]([theta]) at one point, [[theta].sub.B], there are three possible cases.[c.sub.B]
Case 1. Virtually independent products: [[theta].sub.A] < [[theta].sub.A+B] < [[theta].sub.B].
This case is illustrated in Figure 1. Consumer types between [[theta].sub.A] and [[theta].sub.B] will buy product A alone, whereas consumer types [theta] [greater than or equal to] [[theta].sub.B] will buy (A + B). That is, consumers with [theta] [greater than or equal to] [[theta].sub.A] will buy product A and consumers with [theta] [greater than or equal to] [[theta].sub.B] will additionally buy product B. Substituting the definition of [[theta].sub.A] and [[theta].sub.B], the demand functions for A and B become
[D.sub.A]([p.sub.A], [p.sub.B]) = F([[theta].sub.A]) = F ([p.sub.A]/z) [D.sub.B]([p.sub.A], [p.sub.B]) = F([[theta].sub.B]) = F ([p.sub.B]/q - z]). (1)
As long as their prices satisfy [[theta].sub.A] < [[theta].sub.A+B] < [[theta].sub.B], the demand for A depends only on [p.sub.A] and the demand for B depends only on [p.sub.B]. Firms A and B act as independent firms, and we call this case "virtually independent products" and refer to the firms' pricing as "independent pricing" in the rest of this paper. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], denote the Nash equilibrium prices under independent pricing. For example, when F([theta]) = 1 - [theta] (i.e., [theta] is uniformly distributed), the Nash equilibrium prices are (z + [c.sub.A])/2 and (q - z + [c.sub.B])/2, respectively.
Case 2. Virtually strict complements: [[theta].sub.B] < [[theta].sub.A+B] < [[theta].sub.A].
Figure 2 illustrates this case. Consumers with [theta] < [[theta].sub.A+B] will buy neither product, but consumers with [theta] [greater than or equal to] [[theta].sub.A+B] will buy (A + B). None will buy product A alone. Substituting the definition of [[theta].sub.A+B], the demand functions for A and B become
[D.sub.A]([p.sub.A], [p.sub.B]) = [D.sub.B]([p.sub.A], [p.sub.B]) = F([p.sub.A] + [p.sub.B]/q). (2)
[FIGURE 2 OMITTED]
As long as [[theta].sub.B] [absolute value of [R'.sub.B]] so that there is a unique interaction of the two best response functions. The Nash... |
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