Network Externality, Quality Asymmetry, and. Product-Line Design

Transcription

1 Network Externality, Quality Asymmetry, and Product-Line Design Bing Jing 44 West 4th Street, KMC 8-79 Stern School of Business, New York University New York, NY Tel: September 28,

2 Network Externality, Quality Asymmetry, and Product-Line Design Abstract: Inamodelofverticaldifferentiation, we examine how network externalities affect a firm s product line design under welfare maximization and duopoly. A social planner will offer the highest and lowest feasible qualities and subsidize an equal amount on their adoption. When the firms in a duopoly produce compatible goods, the high-end firm offers two distinct quality levels but the low-end firm offers a single quality level. When the firms in a duopoly produce incompatible goods, they each offer two distinct quality levels. The results contrast starkly with the case without network effects. Keywords: Networkeffects, product line, vertical differentiation. JEL Classification: D43,D62,L15. 2

3 1 Introduction In markets with network externalities, we frequently observe that many firms offer qualitydifferentiated products, even when the marginal costs remain by and large constant across qualities. For example, providers such as Hotmail and Yahoo! offer a free service with limited storage space, and also a priced service with a much larger storage space. At many dating or social websites (e.g., Match.com), people can register and post their personal profiles for free. A non-subscribing user can only be passively contacted by others, while a subscriber can also initiate a dialogue with other users. Clearly, second-degree price discrimination in these markets is not driven by cost concerns, but instead driven by network effects, because the unit variable costs remain about the same as quality increases. Product differentiation by the same vendor in such settings can not be explained with extant theory of price discrimination (e.g., Stokey 1979 and Salant 1989), which concludes that each firm should offer only one quality in the absence of network externalities. Compared with conventional markets, in markets with network effects price discrimination has the additional benefit that it expands the installed base and elevates consumers valuation for the products offered, allowing the firms to extract greater surplus. Such an intuition has been formally developed by Csoba and Hahn (2006) and Jing (2006) in monopoly settings. In a model of vertical differentiation, the present paper extends Jing (2006) to a duopolistic setting, and examines how network effects and a firm s relative quality (dis)advantage may jointly drive its product line structure. Csoba and Hahn (2006) present a model of asymmetric network effects, where the product can be decomposed into two separate functions. The full version of the product contains both functions, but the degraded version contains only the basic function. Further, each function has its own separate user group. 1 Asymmetry arises because the marginal network value of each additional user joining one group may differ from that of the other group. The 1 An example they give is Adobe Acrobat, which supports both reading and creating PDF documents, versus the Acrobat Reader, which only supports reading PDF documents. 3

4 spirit of their model is thus more similar to two-sided markets (e.g., Caillaud and Jullien 2003, Armstrong 2005, and Rochet and Tirole 2003) than to vertical differentiation. Under the same market conditions for market segmentation to be sub-optimal to the monopolist (as in Stokey 1979 and Salant 1989), Jing (2006) shows that introducing network externalities restores the optimality of price discrimination. The monopolist is shown to offer two distinct quality levels: the highest and the lowest qualities in its admissible quality set. In the language of marketing professionals, therefore, network externalities are a factor that favors extending a product line. The current paper generalizes this previous research into other market structures. We first analyze a social planner s product line design as a benchmark, and find that it will offer two distinct quality levels and will subsidize the same amount at each quality level. Mainly for tractability, our duopolistic analysis focuses on an asymmetric setting where one firm s quality domain is strictly higher than that of the other (as in Champsaur and Rochet 1989), i.e., we exclude the possibility that the firms offer interluded product lines 2. In a compatible duopoly, the firms share the same installed base, and the ability to internalize the benefits from network expansion depends on a firm s quality position: the first-order benefits of network expansion accrue to the high-end firm while the low-end firm only captures the second-order or indirect benefits. Therefore, the low-end firm will offer only a single quality variety, but the high-end firm will offer two distinct quality levels. The lower quality offered by the high-end firm reduces the degree of product differentiation between the firms and thus intensifies price competition. The suppressed price of the low-end firm then effectively expands the common installed base, which increases consumers valuations for the high-end firm s products. The high-end firm is then able to make higher profits. In contrast, in an incompatible duopoly, each firm operates its own proprietary installed base, which it has an incentive to promote through product line extension. Consequently, each firm will offer two distinct quality levels, independent of its relative quality position. 2 Please see the first two paragraphs of section 4 for more detailed justification. 4

5 There exists a large body of literature on network externalities, addressing a broad range of issues such as firms compatibility choices (e.g., Baake and Boom 2001, Economides and White 1994, Katz and Shapiro 1985), technology adoption (e.g., Belleflamme 1998, Choi 1994, Church and Gandal 1993, Katz and Shapiro 1986, Shy 1996), penetration pricing (e.g., Dhebar and Oren 1985, Farrell and Saloner 1986), entry deterrence (e.g., Fudenberg and Tirole 2000), network interconnection (e.g., Laffont et al. 2003, Wright 2004), and protection of property rights (Conner and Rumelt 1991, Takeyama 1994). Complementary to these studies, the present paper investigates how network externalities affect the vertical product line decision of a firm. In Bental and Spiegel (1995), a firm s network size is its only quality-like dimension and the firms products are not differentiated otherwise. Baake and Boom (2001), Conner (1995), Esser and Leruth (1988) distinguish between a product s value derived from its inherent quality (the so-called "intrinsic value") and the value derived from its installed base (the so-called "network value"). Our treatment on a product s network and intrinsic values follows their approach. Among these papers, the present article is more closely related to Baake and Boom (2001). They examine a three-stage game in a duopoly of vertical differentiation. The firms first choose their inherent quality levels, then decide whether to make their products compatible, and finally, they compete in prices. There are two key differences between their model and ours. First, their model focuses on single-product firms, while in ours a firm s number of qualities is its endogenous variable. That is, we address the firms equilibrium product-line structures. Second, compatibility decisions are endogenous in their model, but are exogenous in ours. The current paper is also related to the literature on vertical differentiation. Mussa and Rosen (1978) and Moorthy (1984) examine how a monopolist should design its product line for a continuous and discrete distribution of consumer types, respectively. Gabszewicz and Thisse (1979) and Shaked and Sutton (1982) are Bertrand duopoly models of singleproduct firms. Champsaur and Rochet (1989) extend Mussa and Rosen (1978) to a duopoly, 5

6 allowing the firms to first commit to producing non-overlapping quality spectrums and then engage in price competition. Following the lead of Champsaur and Rochet (1989), our analysis also assumes that the firms have non-overlapping quality domains in order to make the analysis more tractable. Gal-Or (1983) and Johnson and Myatt (2005) are Cournot models of vertical differentiation, where in equilibrium the firms produce overlapping (or even identical) quality varieties. Many of these models center around two (often countervailing) forces: discriminating among one s own customers to better extract consumer surplus versus differentiating one s own varieties from the competitor s to mitigate competition. This basic trade-off also appears in our paper. However, a distinguishing aspect of this study is that we focus on how network effects and the firms relative quality positions drive their product line decisions. The paper is organized as follows. Section 2 presents the model. Sections 3 looks at the product-line structure of a social planner as a benchmark. Section 4 characterizes the equilibrium product-line structures in a compatible duopoly, where the firms share a common installed base. In section 5, we examine an incompatible duopoly, where the firms have their proprietary installed bases. Section 6 concludes. An Appendix contains the proofs. 2 Model In this market, the lowest (highest) feasible quality level is s L (s H ), where s L may be interpreted as some minimum quality standard due to regulation, or some cutoff quality level below which no consumer would buy, and s H the state-of-the-art quality level. In section 3, the social planner s quality domain is taken as the interval [s L,s H ]. In a duopoly (see sections 4 and 5), the low-end firm s quality domain is [s L,s M ] and the high-end firm s quality domain is (s M,s H ],wheres L <s M <s H (see the discussion in the beginning of section 4). There are no fixed costs of production. Each firm has to choose its number of product qualities as well as their specific levels. We assume that the unit variable cost is constant 6

7 andremainsthesameacrossdifferent qualities. Without loss of generality, we normalize the unit cost to zero. A multi-product firm naturally makes compatible products in this static setting, i.e., all products made by the same firm will have the same installed base. We separately consider duopolies where the firms products are compatible and incompatible. 3 In a compatible duopoly, the firms share a common installed base, which is simply the sum of their unit sales. In an incompatible duopoly, each firm has its own proprietary installed base (measured by its own total unit sales). Each consumer demands zero or one unit of the product. A product of quality s has an intrinsic value of θs for a consumer of type θ. The intrinsic value is the utility a consumer obtains from the product s inherent features, regardless of the consumption by the other consumers. Consumer type θ stands for the consumer s constant marginal valuation for quality. We assume that θ is uniformly distributed on [ A, 1] with density 1, where A>0. Therefore, the total consumer population is A +1. Here A is sufficiently large so that the market is never entirely covered. Such a consumer distribution is reminiscent of that in Katz and Shapiro (1985). Because we normalize the marginal costs to zero, a negative consumer type means that the consumer holds a below-cost intrinsic value for the product. The firm only knows the overall consumer distribution but can not discern the type of each individual. The product has network externalities. Besides its intrinsic value, the product also has a network value, which depends on its network size or installed base. Following Farrell and Saloner (1986) and Katz and Shapiro (1985), we assume that network value is independent of consumer type. Consumers purchasing a product of quality s each obtain a network value of tsq, wheret 0 measures the strength of network effects and Q is the size of the product s network. Therefore, when purchasing a product of quality s at price p, consumer θ receives a net utility of θs + tsq p. That a higher-quality product generates higher network value 3 The firms endogenous compatibility choices are beyond the scope of the current paper. Here we are mainly concerned with how network effects drive the firms product line decisions, for given compatibility choices. 7

8 is supported by empirical evidence. For instance, a telephone with better voice clarity and advanced features such as call waiting and voice mail obviously creates higher network value than one without these features. An advanced piece of software (e.g., a word processor) supports creation and exchange of better quality documents, and thus generates greater network value than a primitive piece. Such a network value that is multiplicative in product quality and network size has previously appeared in Baake and Boom (2001) and Conner (1995). Afulfilled expectations equilibrium seems a natural solution concept to our static model. The firms first announce their product qualities and prices. Rational consumers then form identical expectations regarding the ultimate installed base(s), and make corresponding purchase decisions. In the fulfilled expectations equilibrium, the realized network size equals the consumers expected network size. 3 Welfare Maximization: A Benchmark Suppose a social planner s quality domain is [s L,s H ],where0 <s L <s H. To maximize welfare, the social planner will clearly produce compatible products. Consider K (an arbitrary number) compatible products of qualities s 1,..., s K (s L s 1 <...<s K s H ) with corresponding prices (or subsidies) p 1,..., p K, respectively. Because the unit cost is zero, a negative p k stands for a subsidy on product s k, and a positive p k its price. For 1 k K, letθ k denote the consumer indifferent between adopting s k and s k 1.Wethus have θ k s k + ts k Q p k = θ k s k 1 + ts k 1 Q p k 1,wheres 0 = p 0 =0. The expected network size, Q, must equal the realized network size in the fulfilled expectations equilibrium, i.e., Q =1 θ 1. Note that a consumer choosing s 0 does not purchase. From these two equations, we obtain Q = s 1 p 1 (1 t)s 1, (1) 8

9 and θ k = p k p k 1 s k s k 1 tq, for1 k K. (2) According to equation (1), the installed base increases linearly in the amount of subsidy on thelowestqualityproducts 1. We can verify that consumers in [θ k,θ k+1 ) will adopt product s k,whereθ K+1 =1. The social planner s problem is therefore Max p 1,...,p K KX 1 Z θk+1 θ k (θs k + ts k Q)dθ. (3) The social planner s first-order conditions (FOCs) w.r.t. p 2,..., p K lead to θ k+1 θ k =0,for2 k<k, (4) and p K p K 1 =0,fork = K. (5) Thelasttwoequationsjointlyimplythatp 1 =... = p K, i.e, the social planner will subsidize the same amount on the adoption of each quality variety. Equation (4) further implies that the intermediate qualities s 2,..., s K 1 are not adopted by any consumer. Endogenizing the social planner s quality choice over its quality domain, we arrive at the following Proposition. Proposition 1 Suppose 0 <t s L sl + s H. The social planner will offer two distinct qualities, s L and s H, with the same subsidy ts L s H (1 t) 2 s L t 2 s H. Without network effects (t =0), we can easily verify that the social planner will offer only the highest feasible quality s H at cost. With network effects, however, the social planner has the incentive to expand the installed base by also offering a lower-quality variety with a subsidy. The subsidized lower-quality variety effectively attracts additional low-valuation consumers who would not adopt this product otherwise. An enlarged installed base raises 9

10 the utilities of all adopting consumers, and hence social welfare. That the social planner equally subsidizes adoption of the high-quality variety is somewhat surprising at first. When the same subsidy is offered for both products, we can verify that the consumer indifferent between adopting the two varieties is tq. If a positive subsidy p 1 is applied only to s L, then this indifferent consumer s type is tq + p 1 s H s L > tq. Clearly, the only effect of offering the same amount of subsidy on both varieties is to induce p consumers in ( tq, 1 s H s L tq) to adopt the high-quality variety (instead of the low-quality variety). Such an upward adjustment in adoption turns out to enhance social welfare. 4 Product Line Equilibrium in a Compatible Duopoly We now examine the firms product line decisions in a compatible duopoly. The firms thus share a product network, whose size is just their total unit sales. We focus on the case where the two firms have asymmetric, non-overlapping quality spaces, possibly due to different R&D capabilities. Without further loss of generality, suppose firm1hasafeasiblequality domain [s L,s M ],andfirm 2 has a feasible quality domain (s M,s H ],wheres L <s M <s H, i.e., firm 2 is the one with a quality advantage. The maximum feasible quality of each firm (s M for firm 1 and s H for firm 2) are fixed exogenously. (The firms R&D processes that lead to these maximum quality levels are beyond the scope of the current paper.) Our goal is to examine how network effects affect the firms product-line strategies, given their relative quality positions. Here the assumption of non-overlapping quality domains helps exclude the possibility of two interleaving product lines. In location models (of which vertical differentiation is a particular type), competition between firms with interluded product lines has been notoriously difficult to analyze. The difficulty mainly arises from the large number of possible permutations between the firms product varieties. 4 Such a difficulty becomes more pronounced 4 As an illustrative example, we restrict each firm s number of quality levels to two. Suppose firm 1 offers qualities a 1 and a 2,witha 1 <a 2,andfirm 2 offers qualities b 1 and b 2,withb 1 <b 2. If interleaving product 10

11 if the variety and location decisions of each firm are endogenous, as the space of permuting the firms products is essentially unbounded. It is for this reason that Bertrand competition between firms with interluded product lines has not been adequately studied in the literature. Champsaur and Rochet (1989) study a duopoly where each firm produces a distinct continuum of qualities, i.e., they also ignore the case in which the firms offer interluding quality spectrums. Our specification of the firms quality domains may be viewed as an analogue of their model. Competition proceeds in two phases. In the first phase, the firms simultaneously choose the number and qualities of their products. After observing each other s quality choices, the firms simultaneously set prices in the second phase. We first analyze the second stage. 4.1 Stage II: Price Equilibrium for Fixed Quality Sets In the second stage, the firms product lines are already fixed. Without loss of generality, suppose firm 1 offers N products, s 1,...,s N,wheres L s 1 <... < s N s M,andfirm 2 offers K products, s N+1,..., s N+K,wheres M <s N+1 <... < s N+K s H.LetQ denote the common installed base. Denote p k as the price of product k (1 k N +K). Let θ k denote the consumer indifferent between purchasing products s k and s k 1. We still have Q = s 1 p 1 (1 t)s 1 and θ k = p k p k 1 s k s k 1 tq, for1 k N + K. The pattern of market segmentation remains that consumers in [θ k,θ k+1 ) will purchase s k,whereθ N+K+1 =1. The firms choose prices simultaneously to maximize their profits: Firm 1: π 1 = NX p k (θ k+1 θ k ), (6) 1 line were allowed, there would then be three distinct permutation patterns: (1) a 1 <a 2 <b 1 <b 2,(2) a 1 <b 1 <a 2 <b 2,and(3)a 1 <b 1 <b 2 <a 2. As each firm s number of varieties increases, the number of possible permutations simply explodes. In a circular-city model of horizontal differentiation, Schmalensee (1978) allows interlaced product lines, but to make the analysis tractable he assumes that all varieties have the same, exogenously-fixed price. 11

12 N+K X Firm 2: π 2 = p k (θ k+1 θ k ). (7) N+1 Solving this sub-game leads to the next Proposition. Proposition 2 In the price equilibrium: (1) The high-end firm (firm 2) offers two quality levels (s N+1 and s N+K ), and the low-end firm (firm 1) offers only one quality level (s N ). (2) The low-end firm s profit equals the profit generated by the high-end firm s lower quality product. When the firms produce compatible goods, Proposition 2 reveals how their product line length may depend on their quality (dis)advantage. For fixed sets of non-interluding qualities, in the price equilibrium the high-end firm actively sells two quality varieties, while the lowend firm actively sells only a single quality variety. The reason for such a stark contrast is as follows. The demand of each product s k of firm 1 (the low-end firm) is θ k+1 θ k = p k+1 p k s k+1 s k p k p k 1 s k s k 1,for1 k N. The demand of firm 2 s top quality is 1 θ N+K = 1 p N+K p N+K 1 s N+K s N+K 1 + tq, wherethelastterm(tq) is the demand contribution due to network externalities. This indicates that network effects do not directly affect firm 1 s demand but directly expands firm 2 s demand. 5 Because firm 1, the low-end firm, only enjoys the indirect or second-order benefits of network expansion, it lacks the incentive to segment its market and instead sells only one variety. On the other hand, at a quality advantage firm 2 is able to harvest the first-order benefits of network expansion and thus will offer a lower-quality variety to expand the total product network. The top quality s N+K is firm 2 s primary revenue source. Offering the lower quality s N+1 has two effects. First, offering s N+1 helps achieve the strategic effect of expanding the total product network. After firm 2 inserts the lower quality s N+1, product differentiation between the firms decreases and hence price rivalry intensifies. This suppresses firm 1 s price and induces more consumers to join the installed base. A larger installed base 5 From firm 1 s profit expression in (12) below, we see that the low-end firm does indirectly benefit from network effects. 12

13 then raises consumers valuation of all products, including s N+K. Second, offering s N+1 allows firm 2 to price discriminate and extract more surplus from consumers purchasing its top quality s N+K. The equilibrium prices are and The equilibrium network size is thus p N = s N(s N+1 s N ) (4 3t)s N+1 s N, (8) p N+1 = 2s N+1(s N+1 s N ) (4 3t)s N+1 s N, (9) p N+K = 4s N+1s N+K s N s N+K 3s N s N+1. (10) 2[(4 3t)s N+1 s N ] Q = s N p N (1 t)s N = 3s N+1 (4 3t)s N+1 s N. (11) When t increases, i.e., when network effects get stronger, we can verify that all prices and the network size will increase. Therefore, in the compatible duopoly, stronger network effects relax price competition and lead to a larger installed base. It is also easy to verify that stronger network effectsincreaseeachproduct s unitsales. 4.2 Stage I: Product Line Equilibrium In the first stage, the firms choose qualities to maximize their profits. Substituting the equilibrium prices into (6) and (7), we obtain the firms profits as functions of qualities: π 1 = s Ns N+1 (s N+1 s N ) [(4 3t)s N+1 s N ] 2 (12) and π 2 = s N+K(4s N+1 s N ) 2 s N s N+1 (s N +8s N+1 ) 4[(4 3t)s N+1 s N ] 2. (13) 13

14 For convenience, we only consider interior solutions to firm 1 s quality choice and firm 2 s choice of its lower quality. Let s N = t(8 7t)s H 8(1 t)+t 2. Proposition 3 Suppose s L <s N <s M < (7 6t)s N <s 4 3t H and 0 <t< The unique sub-game perfect equilibrium is firm 1 offering quality s N and firm 2 offering qualities s N+1 = (7 6t)s N 4 3t and s N+K = s H. Under the conditions of Proposition 3, the low-end firm (firm 1) positions its only product quality at s N = t(8 7t)s H. The high-end firm (firm 2) always offers its highest feasible quality 8(1 t)+t 2 s H, regardless of the extent of network effects. Besides, it also offers a lower quality (7 6t)s N. 4 3t Proposition 3 has two corollaries. Corollary 1 states that without network externalities each firm offers only a single quality level. Corollary 1. Suppose s L 4s H 7 s M. In the absence of network effects (i.e., when t =0), the unique sub-game perfect equilibrium involves firm 1 offering quality 4s H 7 and firm 2offering quality s H. Corollary 1 is thus consistent with the result of Choi and Shin (1992) on the equilibrium quality locations in a duopoly. Corollary 2. Suppose s L < t(8 7t)s H 8(1 t)+t 2 <s M and 0 <t< Ast increases, s N, s N+1 and s N+1 s N will all increase. Corollary 2 is an interesting result. Stronger network effects not only cause both firms to raise their qualities but also lead to greater product differentiation between the firms. The reason for the firms to raise their qualities is as follows. From the price equilibrium in the second-stage, we see that stronger network effects increase the demand of each firm (for any fixed quality levels). Both firms will then raise their quality levels in response to the increased demand for their products (except that firm 2 s top quality is bounded by s H ). However, it is unintuitive that stronger network externalities lead to a greater product differentiation between the firms. 14

15 5 An Incompatible Duopoly We now turn to the duopoly in which the firms produce incompatible products. With incompatible products, the firms operate their respective proprietary networks rather than share a common network. For the same reason as laid out in the previous section, we continue to consider two firms with non-overlapping quality spaces, i.e., firm1hasafeasiblequality domain [s L,s M ],andfirm 2 has a quality domain (s M,s H ],with0 <s L <s M <s H. Suppose firm 1 offers N products, s 1,..., s N,wheres L s 1 <... < s N s M,andfirm 2offers K products, s N+1,..., s N+K,wheres M <s N+1 <...<s N+K s H. Denote p k as the price of product k (1 k N + K), Q i thesizeoffirm i s (where i =1, 2) proprietary network. Since the products of firm i are all compatible with each other, Q i equals the total unit sales of firm i. The indifferent consumer θ k between two adjacent quality levels s k and s k 1 satisfies: for 1 k N, θ k s k + ts k Q 1 p k = θ k s k 1 + ts k 1 Q 1 p k 1, (14) where s 0 = p 0 =0,fork = N +1, θ N+1 s N+1 + ts N+1 Q 2 p N+1 = θ N+1 s N + ts N Q 1 p N, (15) and for N +2 k N + K, θ k s k + ts k Q 2 p k = θ k s k 1 + ts k 1 Q 2 p k 1. (16) Note that θ N+1 is the consumer indifferent between purchasing firm 1 s s N and firm 2 s s N+1, i.e., the consumer who separates the two firms installed bases. In the fulfilled expectations equilibrium, Q 1 = θ N+1 θ 1,andQ 2 =1 θ N+1. From these two equations and (14) through (16), we can obtain Q 1 and Q 2 as expressions of the prices and qualities (for the exact expressions, see the proof of Proposition 4 in the Appendix). From (14)-(16), we then 15

16 have θ k = p k p k 1 s k s k 1 tq 1,for1 k N, (17) θ N+1 = p N+1 p N s N+1 s N t(s N+1Q 2 s N Q 1 ) s N+1 s N,for1 k N, (18) θ k = p k p k 1 s k s k 1 tq 2,forN +2 k N + K, (19) The firms objective functions are the same as those given in (6) and (7). They choose prices simultaneously to maximize their profits. Proposition 4 Suppose (1 t) 2 s N+1 s N > 0. In the incompatible duopoly, in the unique price equilibrium each firm sells its highest and lowest quality levels, i.e., firm 1 sells s 1 and s N and firm 2 sells s N+1 and s N+K. For any given qualities s N and s N+1, the condition (1 t) 2 s N+1 s N > 0 holds when network effects are not too strong (t <1 q s N s N+1 ). Just as in the compatible duopoly, here the high-end firm still offers its top and bottom qualities. The key difference made by incompatibility is reflected in the product line decision of the low-end firm, firm 1, which also offers two distinct quality levels. When the firms produce incompatible products, consumers valuation for a firm s products depends only on its exclusive installed base. Therefore, the high-end firm no longer directly benefits from the market segmentation conducted by the low-end firm. Because the low-end firm can fully capture the first-order benefits of expanding its exclusive installed base, it now has the incentive to cultivate its network by extending its product line. So far, we have only analyzed the outcome in the price equilibrium for given quality sets. In the incompatible duopoly, we are unable to obtain the sub-game perfect product line equilibrium in closed form, because the algebra is no longer tractable. However, we can conjecture that when the firms endogenize their quality locations, the high-end firm will continue to set its top quality at s H,itshighestfeasiblequality;thereisnoanticipatedfactor that would force it to choose its top quality otherwise. 16

17 6 Conclusion In a framework of vertical differentiation, this article has examined how network externalities and asymmetric quality domains jointly affect the equilibrium product-line structures in a duopoly. We first analyze a social planner s optimal product line as a benchmark. Without network effects, it offers only the highest feasible quality, because any lower-quality product costs no less to produce but is valued strictly less by all consumers under our utility and cost functions. With network effects, however, the social planner will offer the highest and lowest feasible qualities and subsidize an equal amount on the adoption of these two products. The duopolistic analysis forms the core of this paper. Here we find that the firm with a higher quality domain always offers two distinct quality levels, whether its low-end competitor produces compatible or incompatible goods. On the other hand, the firm with a lower quality domain will offer only one quality level in a compatible duopoly, and will offer two distinct quality levels in an incompatible duopoly. The underlying rationale is as follows. In an incompatible duopoly, each firm has its own installed base, which determines the network values of its products. Therefore, each firm has the incentive to offer two distinct quality levels to segment its market and expand its installed base, regardless of its relative quality position. In a compatible duopoly, the firms share one installed base. In this case, the firms relative quality positions matter. The firm with a higher (lower) quality domain can capture the first-order (second-order) benefits of network expansion. The high-end firm thus has the incentive to offer two quality levels to expand the installed base via reducing product differentiation with the low-end rival and thus intensifying price competition. The low-end firm, however, lacks such an incentive. We conclude by discussing the implications of some key assumptions of this model. To simplify exposition, we have assumed that consumer types are uniformly distributed. Note that the equilibrium product line structures are primarily driven by network effects, and our key results will continue to hold valid under more general consumer distributions (except that the exact quality locations in Proposition 3 may change). Confining the duopolistic 17

18 analysis to firms with asymmetric quality domains is arguably a limitation of this research. However, as pointed out in the second paragraph of section 4, such a treatment ensures a more tractable analysis. Attempts in lifting such a restraint is left for future research. 7 Appendix Proof of Proposition 1: For K given quality levels, we have shown above that the social planner offers only the two extreme qualities, s 1 and s K. Solving the two FOCs w.r.t. p 1 and p K gives p 1 = p K = ts 1 s K (1 t) 2 s 1 t 2 s K. Substituting the optimal subsidies into (3) yields social welfare as a function of the quality choices W = t 2 (s K ) 2 2[(1 t) 2 s 1 t 2 s K ] 2 < 0 and W s K = (1 t)2 (s 1 ) 2 2[(1 t) 2 s 1 t 2 s K ] 2 > 0. Q.E.D. s 1 s K W 2[(1 t) 2 s 1 t 2 s K.Wethenhave ] s 1 = ProofofProposition2: Rewriting the first order conditions of each firm with θ k = p k p k 1 s k s k 1 tq, we obtain the unit sales of its products. For firm 1, θ N+1 θ N = p N+1, for k = N, (20) 2(s N+1 s N ) θ k+1 θ k =0, for 1 k N 1, (21) and for firm 2, 1 θ N+K = 1 (1 + tq) > 0, for k = N + K, (22) 2 θ k+1 θ k =0, for N +2 k N + K 1, (23) θ N+2 θ N+1 = p N, for k = N +1, (24) 2(s N+1 s N ) where p N and p N+1 are the equilibrium prices of products s N and s N+1. The first statement of the Proposition follows directly from (20) through (24). By (20), firm 1 s revenue is p N (θ N+1 θ N ) = p N p N+1. By (24), the revenue of s 2(s N+1 s N ) N+1 is also p N+1 (θ N+2 θ N+1 )= p N p N+1 2(s N+1 s N. From these, the second statement immediately follows. ) Q.E.D. 18

19 π Proof of Proposition 3: We can verify that 2 s N+K = (4s N+1 s N ) 2 4[(4 3t)s N+1 s N )] 2 s π N+K = s H.That 2 s N+1 =0is equivalent to > 0. Therefore, s 2 N 24ts N+1 s H + s N [(20 3t)s N+1 +6ts H ]=0, and that π 1 s N =0is equivalent to (7 6t)s N (4 3t)s N+1 =0. Solving these two equations simultaneously leads to the optimal quality choices s N and s N+1. When t < 4 45 (10 10) = 0.608, we can verify that s N+1 <s N+K = s H and s L <s N <s M, i.e., both are interior solutions in the respective firms quality domains. Q.E.D. ProofofCorollary1: The proof is by setting t =0in the proof of Proposition 3. When t =0, π 2 s N+K = 1 4 > 0 and π 2 s N+1 = s2 N (s N +20s N+1 ) 4(4s N+1 s N ) 3 > 0. Therefore, s N+1 = s N+K = s H. That π 1 s N =0leads to 7s N 4s N+1 =0,andthuss N = 4s H 7. Q.E.D. ProofofCorollary2: The proof is straightforward by verifying the respective first derivatives. Q.E.D. ProofofProposition4: The proof is done by obtaining each product s unit sales in the price equilibrium by transforming the FOCs. However, we first need to derive the expressions of Q 1 and Q 2. From (14)-(16), we have the network sizes as functions of equilibrium prices: Q 1 = s 1(p N+1 p N + ts N+1 ) p 1 ((1 t)s N+1 s N ), 2s 1 [(1 t) 2 s N+1 s N ] Q 2 = t(p N+1s 1 + p 1 s N s 1 s N+1 )+(1 t)s 1 p N + s 1 (s N+1 s N p N+1 ). 2s 1 [(1 t) 2 s N+1 s N ] 19

20 Firm2 sfocsare π 2 p N+K =1 2(p N+K p N+K 1 ) s N+K s N+K 1 + tq 2 =1 2θ N+K tq 2 =0, and π 2 p k =2(θ k+1 θ k )=0, π 2 p N+1 =2(θ N+2 θ N+1 )+tq 2 + θ N+1 + (1 t)2 p N+1 s N (1 t) 2 s N+1 =0. π That 2 p N+K =0yields 1 θ N+K =(1+tQ 2 )/2 > 0, indicating s N+K attracts strictly positive sales in the price equilibrium. Clearly, the unit sales of each s k,forn +2 k N + K 1, areθ k+1 θ k =0. The unit sales of s N+1 are θ N+2 θ N+1 = 1 2 µ (1 t) 2 p N+1 [1 (1 t)q (1 t) 2 2 ] s N+1 s N = (1 t)2 s 1 p N + ts N [(1 t)p 1 + s 1 ] 2s 1 [(1 t) 2 s N+1 s N ] > 0. By inspecting firm 1 s FOCs, we obtain the unit sales of each of its products in equilibrium: θ N+1 θ N = p N p N 1 p N + > 0, for k = N, s N s N 1 (1 t) 2 s N+1 s N θ 2 θ 1 = p N 2 θ k+1 θ k =0,for2 k N 1, µ θn+1 + t Q 1 = p 1 p 1 t(1 t)p N s N+1 > 0, for k =1. 2s 1 [(1 t) 2 s N+1 s N ] Therefore, in the price equilibrium the top and bottom qualities of firm 1 both attract strictly positive sales. This shows that the low-end firm, firm 1, will offer two distinct qualities when its products are incompatible with those of the competitor s. Q.E.D. 20

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