Multiproduct Duopoly with Vertical Di erentiation Yi-Ling Cheng y Shin-Kun Peng z Takatoshi Tabuchi x 3 August 2011 Abstract: This paper investigates a two-stage competition in a vertically differentiated industry, where each rm produces an arbitrary number of similar qualities and sells them to heterogeneous consumers. The number of products, qualities, prices, and the extent of the market coverage are endogenously determined. We show that when unit costs of quality improvement are increasing and quadratic, each rm has an incentive to provide a disconnected set of similar qualities approximating a continuum. The nding contrasts sharply with the single-quality outcome when the market coverage is exogenously determined. We also show that allowing for multiple qualities intensi es the level of competition, lowers the pro t of each rm, and raises the consumer surplus and the social welfare in comparison to the single-quality duopoly. (JEL Classi cation: D43, L13). Keywords: multiproduct rms, market segmentation, quality competition, continuum of qualities, vertical product di erentiation. We thank Chien-Fu Chou, Ching-I Huang and Kuo-Chih Yuan for their helpful comments and suggestions. y Academia Sinica and National Science Council. 128 Academia Road, Section 2, Nankang, Taipei 115, Taiwan. E-mail: d94323011@ntu.edu.tw z Academia Sinica and National Taiwan University. 128 Academia Road, Section 2, Nankang, Taipei, 115 Taiwan. E-mail: speng@econ.sinica.edu.tw x Faculty of Economics, University of Tokyo. Hongo-7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan. E-mail: ttabuchi@e.u-tokyo.ac.jp
1. Introduction The literature on vertical product di erentiation in an oligopoly normally assumes that any one rm can produce only one product. Although this is analytically convenient, in reality, rms normally produce and sell multiple di erentiated products. It is known in the literature that a monopolist o ers multiple products or an interval of vertically di erentiated qualities under this setup: quasi-linear utility, uniform distribution of consumer taste, and quadratic cost of quality improvement (Mussa and Rosen, 1978; Gabszewicz et al., 1986). However, it is also known that under a similar setup each duopolist produces a single quality rather than a range of qualities (Champsaur and Rochet, 1989, 1990; Bonnisseau and Lahmandi- Ayed, 2006). 1 The main purpose of this paper is to answer the question: whereas a monopolist o ers multiple qualities, why does each duopolist o er a single quality in these studies? A causal survey of the evidence suggests that vertically di erentiated product markets are usually multiproduct oligopolies. For example, in the market of cellular baseband processors, Qualcomm faces sti competition from MediaTek on the strength of MediaTek s low-cost chipsets. 2 Their product lines are vertically segmented: Qualcomm is producing chipsets for high-end smartphones and 3G mobile phones o ering the Snapdragon series that includes models QSD 8250, 8650, MSM 7230, 8260, 8660, etc.; MediaTek is producing chipsets for low-end feature phones o ering the MT series that includes models MT 6225, 6253, 6573, etc. Similarly, in the automobile industry, BMW and Mercedes-Benz both sell three classes of sedans. The sedans they produce are of higher qualities and prices than those o ered by Ford or Honda. 1 Notable exceptions are Katz (1984), Gilbert and Matutes (1993) and Barron et al. (2000) under di erent setups. 2 Together, they had a 56% share of cellular baseband revenues in 2009. 1
In this paper, we show that duopolistic rms o er multiple qualities in order to be consistent with this reality on the basis of the common assumptions made in Champsaur and Rochet (1989, 1990). For this purpose, we pay attention to the assumed unit costs of quality improvement and market coverage in analyzing vertically di erentiated product markets. These two factors play a crucial role in whether rms will o er single or multiple products. 3 As to the rst factor, the literature on duopoly in a vertically di erentiated industry often assumes that unit costs of quality improvement are zero, linear, or quadratic. Our model identi es the factors that in uence whether rms will produce single or multiple products; we do this by looking into the second derivative of the cost function. We reveal that when the unit cost is concave (including zero and linear), each rm produces a single quality. Since our focus is on investigating multiproduct rms, we assume strictly convex cost functions; for analytical tractability, we specify that unit costs of quality improvement are quadratic, just as Champsaur and Rochet (1989) and Motta (1993), among others, do. As to the second factor of market coverage, all consumers are assumed to be served in the literature such as Champsaur and Rochet (1989, 1990). However, there is no obligation for rms to serve all consumers, especially poor people with low willingness to pay. We therefore consider a more general situation where rms do not have to cover the whole market. That is, unlike the literature on exogenous market coverage, in this paper, the extent of market coverage (i.e., the number of consumers served by rms) is endogenously determined by strategic competition between rms. We show that equilibrium outcomes under exogenous market coverage are very di erent from those under endogenous market coverage under a two-stage 3 Fixed costs of each quality are also important considerations in their decision making. Although the xed costs would reduce the number of qualities, rms often produce multiple qualities due to the segmentation e ect. We explore that in this paper by disregarding the xed costs. 2
competition in a vertically di erentiated industry. More precisely, the singleproduct outcome of Champsaur and Rochet (1990, Proposition 2) is attributed to exogenously given full market coverage, whereas the multiproduct outcome in this paper is ascribed to endogenously determined partial market coverage. This suggests that market coverage plays an important role in the emergence of multiproduct rms that produce distinct qualities to segment the market. When the market coverage is endogenously determined, rms have incentives to produce multiple qualities because more consumers would make a purchase thus, the size of the market could be enlarged. In some industries, product qualities are entangled and rms that deal in similar qualities compete vigorously with each other. In other industries, product qualities are segmented. This is because rms may want to relax price competition by market segmentation, because they may bene t from economies of scope in production by producing similar qualities, and because by doing so, rms may build a consumer brand image based on these qualities. We focus on the latter industries, where market is segmented by high-quality and low-quality multiproduct rms. The rest of this paper is organized as follows. Section 2 presents a model of two-stage competition in a vertically di erentiated industry based on the two key assumptions. Firms simultaneously choose both the number of products and qualities of its products in the rst stage, and then compete in terms of price in the second stage. 4 Section 3 investigates the rst-stage quality equilibrium and characterizes the subgame perfect Nash equilibrium (SPNE) under the quadratic cost of quality improvement. We show that each rm produces a set of qualities rather than a single quality under endogenous market coverage. Section 4 considers the welfare implications by computing the consumer surplus and social 4 If the number of products, qualities, and prices are selected in one stage, the results di er. See Katz and Economides (1985) for a continuum of consumers; see Gayer (2007) for a nite number of consumers. 3
welfare. Section 5 outlines the conclusions that can be made based on this model. 2. The Model There are two multiproduct rms, A and B, which have identical production technology in terms of their quality improvement. Each rm produces a nite number of products with di erent qualities. The products of rm A are indexed as 1; 2; :::; n a, and their qualities are indexed as q 1 ; q 2 ; :::; q na ; the products of rm B are indexed as n a + 1; n a + 2; :::; n a + n b, and their qualities are indexed as q na+1; q na+2; :::; q na+nb. The associated prices of these quality-di erentiated goods are denoted by p 1 ; p 2 ; :::; p na+nb, respectively. The two rms play a two-stage game. In the rst stage, they simultaneously choose both the number of products and the qualities of their products. In the second stage, they simultaneously select the prices of their products, having observed the number of products and their qualities. In principle, the strategy spaces are unrestricted nite quality sets that can overlap or entangle. However, since this is clearly out of reach in search for equilibria, we con ne ourselves to the noteworthy subset of nonoverlapping segmented equilibria. That is, we assume q i q i+1 for i = 1; 2; :::; n a + n b 1 so that both rms o er a connected range of similar qualities. The reason we make this assumption is explained in the following two steps. First, suppose each rm can produce at most two distinct qualities: rm A produces q 1 and q 2 and rm B produces q 3 and q 4. We numerically checked for all possible SPNE con gurations of qualities using the Newton-Raphson method with the initial values of q i = k where i = 1, 2, 3, 4 and k = 0:2, 0:4, 0:6, 0:8, 1. We revealed that the segmented con guration q 1 > q 2 > q 3 > q 4 is the only candidate for an SPNE, and that all entangled con gurations such as q 1 > q 3 > q 2 > q 4 and q 1 > q 3 > q 4 > q 2 are not candidates. In fact, we show 4
in Appendix 1 that the segmented con guration with one rm choosing two high qualities and one rm choosing two low qualities is indeed an SPNE. Second, if each rm can produce an arbitrary number of qualities, then we expect, from Mussa and Rosen, that each rm would o er a connected range of similar qualities. In order to show that this is an SPNE, we take into account the possibility of deviations to more general con gurations. Although this is out of reach, we may exclude entangled con gurations as candidates for SPNE. This is because any deviation to from segmented to entangled con gurations intensi es competition, which reduces the pro t. We may also exclude overlapping con gurations as candidates for SPNE. This is because no rm can extract a positive pro t from consumers in the overlapping qualities as is explained in a simple undercutting argument in footnote 7 in Champsaur and Rochet (1989). Thus we may focus on the segmented con guration of quality di erentiation with rm A producing high qualities and rm B producing low qualities following Champsaur and Rochet (1989). We show in the next section that, as it turns out, each rm provides a set of qualities approaching a continuum in the sense that the maximal distance from one variety to the next converges to zero. There is a continuum of consumers, each of whom has di erent taste regarding quality. Their willingness to pay for quality is distributed uniformly over the interval [; ] and the density is normalized to 1. Each consumer purchases one unit of the product from either A or B, or does not purchase at all. Following Tirole (1988), we assume the utility function of consumer is given by 8 < q i p i if she purchases quality q i at price p i U( i ; q i ) = : 0 otherwise This utility implies that consumers unanimously prefer higher quality at a given price and that consumers with higher will pay more for a higher quality. Consumer demand is determined as follows. Marginal consumers, indexed by i, are indi erent as to whether they will purchase quality q i at price p i or quality 5
q i+1 at price p i+1. Solving U( i ; q i ) = U( i ; q i+1 ) yields 8 < p i p i+1 q i = i q i+1 for i = 1; : : : ; n a + n b 1 n o : pna+nb (1) max q na+nb ; for i = n a + n b Thus, any consumer with an index greater than i will prefer q i to q i+1 for all i = 1; :::; n a +n b 1, and any consumer with an index less than p na+n b =q na+n b will prefer not to buy at all rather than to buy q na+n b. The literature on exogenous full market coverage assumes na+n b =, whereas we consider endogenous market coverage so that na+n b second line of equation (1). is determined by the market fundamentals given by the The demand x i for quality q i is computed by equation (1) as follows: 8 < 1 for i = 1 x i = : i 1 i for i = 2; : : : ; n a + n b We assume the variable cost of quality improvements, i.e., each unit s production incurs a cost related to its quality level. We assume that the xed cost of product customization is zero because this will not a ect the overall results qualitatively. 5 Following Tirole, we also assume that the production activities are fully additive; this means that the unit variable cost of quality improvement is independent of its quantities and dependent on quality. The pro ts of rm A and B are then given by Xn a a = [p i c(q i )]x i ; b = i=1 nx a+n b i=n a+1 where c(q i ) is the unit cost of quality improvement of quality q i. [p i c(q i )]x i (2) Assumption 1 The unit cost of quality improvement is strictly convex. Assumption 1 indicates that the marginal quality improvements is increasingly costly. This is also assumed by Mussa and Rosen and Champsaur and Rochet 5 See Noh and Moschini (2006) for the e ect of xed costs on the entry deterrence. 6
(1989, 1990), among others, in their models, on the one hand. On the other hand, Choi and Shin (1992) and Bonnisseau and Lahmandi-Ayed assume that quality improvement incurs linear costs. However, because the utility is of the form u = q i, the willingness to pay for a unit quality is constant. When the unit costs of quality improvement are concave, including linear, the higher the quality is, the lower the marginal cost of improving a unit quality will be. Therefore, rms have incentives to sell qualities at as high a price as possible; this implies that they have no reason to o er goods of lower qualities and segment the market, as proven in Appendix 2. In addition, human technologies often display decreasing returns to scale, and thus increasing quality is increasingly expensive. Apart from a few cases, most industries display convex costs of increasing quality. This is the situation we focus on. As the unit costs of quality improvement are convex, the marginal cost of a lower quality will be lower. Such a technology may enable rms to o er multiple products and segment the market. Assumption 2 The lower bound of the consumer distribution,, is less than 0:262. The threshold 0:262 is the lowest value of served consumers given by p na+n lim b n a;n b!1 q na+nb which is computed in the next section. Under Assumption 2, the market is always partially covered, which implies endogenous market coverage. If this assumption is found not to be the case, then the whole market may be covered. There is another possible situation between the two, which is called the corner solution. In this situation, the price of the lowest quality is set such that the consumer with the lowest = will be indi erent as to whether or not they will buy the lowest quality. When is greater than 0:262, there will be three possible strategies that rm B, who produces the lowest quality q na+n b in the second line of (1), can adopt. These strategies are as follows: (i) to uncover the market na+n b = p na+n b =q na+n b 7
for large =; (ii) to use the corner solution na+n b = = p na+n b =q na+n b for intermediate =; and (iii) to cover the market na+n b = for small =. 6 Because investigating these cases is very complicated (Wauthy, 1996; Liao, 2008), we adopt Assumption 2 in order to focus on the partially covered case, where the extent of market coverage is endogenously determined by rms. In our model, the two rms play a two-stage game under these two assumptions. Using backward induction, we rst solve the rst-order conditions for prices in the second stage. From this, the following lemma is obtained; its proof is in Appendix 3. Lemma 1 For any given qualities with an arbitrary number, there exists a unique price equilibrium given by 8 h i 1 c(q 2 i ) + q i + q bc(q a)+2q ac(q b ) 3q aq b 4q a q b for i = 1; 2; : : : ; n a 1 >< q a[2c(q a)+c(q b )+2(q a q b )] p 4q i = a q b p a for i = n a q b [c(q a)+2c(q b )+(q a q b )] 4q a q b p b for i = n a + 1 h >: c(q i ) + q i[2c(q a)+c(q b )+2(q a q b )] for i = n a + 2; : : : ; n a + n b 1 2 4q a q b i (3) where brands q a q na and q b q na+1 are in direct competition. Lemma 1 implies that the rms make a two-step optimization: setting a middle of the road price on the interior subsegment and a Hotelling price on the competitive boundary. When n a and n b reach in nity, we have 8 < i = dp i dq i = 1 2 c 0 (q i ) + for i = 1; 2; : : : ; n a 1 (4) : for i = n a + 2; : : : ; n a + n b 1 2 h i c 0 (q i ) + p a qa 6 Such a corner solution does not arise in a monopoly. In fact, Gabszewicz et al. show that a multiproduct monopolist will segment the market by o ering the maximum number of qualities permitted when = is below a threshold, and that he will only o er the top quality product when = is above the threshold. The former corresponds to (i) partial market coverage and the latter implies (iii) full market coverage. 8
from equations (1) and (3). This corresponds to in Mussa and Rosen (equation 17), where a monopolist perfectly discriminates among consumers by equalizing the marginal revenue and cost of increments of quality improvement. Equation (4) also shows that the price increases as quality increases, and its gradient is steeper for higher quality. Since Lemma 1 ensures the uniqueness of the second-stage price subgame for any qualities with an arbitrary number, we can safely focus on the rst-stage quality competition in the next section. 3. Quadratic Cost of Quality Improvement In order to obtain the SPNE explicitly, we set a quadratic cost of quality improvement c(q i ) = qi 2 =2, which satis es Assumption 1, 7 and we examine the endogenous market coverage on the basis of Assumption 2. Substituting the second-stage equilibrium prices (3) into the pro ts (2), we have the equilibrium pro ts, a and b, as functions of q 1; q 2 ; : : : ; q na+nb. The 7 Under di erent convex costs of quality improvement, most of the results do not di er much qualitatively, according to our numerical analysis. 9
rst-order conditions to be solved simultaneously are @ a @q 1 = 0 ) 3q 1 q 2 2 = 0 (5) @ a @q i = 0 ) q i 1 2q i + q i+1 = 0 for i = 2; : : : ; n a 1 (6) @ a @q a = 0 (7) @ b @q b = 0 (8) @ b @q i = 0 ) q i 1 2q i + q i+1 = 0 for i = n a + 2; : : : ; n a + n b 1 (9) @ b @q na+nb = 0 ) q na+nb 1 2q na+nb = 0 (10) a a(n a + 1; n b ; q(n a + 1); q(n b )) a(n a ; n b ; q(n a ); q(n b )) 0(11) b b(n a ; n b + 1; q(n a ); q(n b + 1)) b(n a ; n b ; q(n a ); q(n b )) 0(12) where q(n a ) (q 1 (n a ); :::; q na 1(n a ); q a (n a )) is the optimal quality line produced by rm A given n a, n b and q b (n b ), and q(n b ) (q b (n b ); q na+2(n b ); :::; q na+n b (n b )) is the optimal quality line by rm B given n a, n b and q a (n a ). Inequalities (11) and (12) imply that it is not pro table for each rm to increase the number of qualities. In order to compute the equilibrium values, we solve the simultaneous equations (5)-(12) in three steps. Step 1. Solving (5), (6), (9) and (10) leads to 8 < (2i 1)q a+2(n a i) qi 2n = a for i = 1; : : : ; n 1 a 1 (13) : n a+n b +1 i n b q b for i = n a + 2; : : : ; n a + n b From equation (13), we can readily show that q i q i+1 is constant within the same rm. This indicates that taking its ghting/boundary quality as a parameter, each rm picks evenly distributed qualities in order to minimize the cost for consumers of not getting their ideal choices. This way the rm maximizes the price it can get from consumers. Conditional on the boundary, this is in fact a local monopoly analysis. Consequently, duopoly yields the same results as 10
monopoly, as in Mussa and Rosen, for all brands except for ghting brands q a and qb. Also, we know from equation (13) that the non ghting qualities depend only on their own ghting quality (and are independent of the rival s ghting quality). Step 2. Inserting equation (13) into the pro ts, we obtain a(n a + 1; n b ; q a (n a ); q b (n b )) 2na( qa(na))3 a(n a ; n b ; q a (n a ); q b (n b )) = > 0 3(4n 2 a 1)2 b (n a; n b + 1; q a (n a ); q b (n b )) b (n a; n b ; q a (n a ); q b (n b )) = (2n b+1)q b (n b ) 3 > 0 48n 2 b (n b+1) 2 (14) The positive increments of pro ts in equation (14) indicate that each rm has an incentive to increase the number of qualities holding the quality of its ghting brand unchanged. Furthermore, q a (n a + 1) is the optimal quality of A s ghting brand given n a + 1, n b and q b (n b ), and q b (n b + 1) is the optimal quality of B s ghting brand given n a, n b + 1 and q a (n a ) by de nition. Because each optimal pro t is the largest, we necessarily have a(n a + 1; n b ; q a (n a + 1); q b (n b )) a(n a + 1; n b ; q a (n a ); q b (n b )) b (n a; n b + 1; q a (n a ); q b (n b + 1)) b (n a; n b + 1; q a (n a ); q b (n b )) (15) Putting equations (14) and (15) together, we get a > 0 and b implies that rms always bene t from adding more qualities. > 0, which Therefore, the admissible number of qualities, n a and n b, becomes large, and the equilibrium set of qualities approximates a continuum in the sense that the maximal distance from one variety to the next converges to zero. Step 3. Substituting equation (13) into equations (7) and (8), and taking the limit of n a ; n b! 1, we can reduce the FOCs to 8 f A 2q a (q a q b ) 8q 3 a 6q 2 aq b + 6q a q 2 b + q 3 b + 18qa q 3 b (20q a + q b ) q 3 b 2 = 0 8 The derivations are available from the authors upon request 11
and f B 16q 5 a 92q 4 aq b + 208q 3 aq 2 b 124q 2 aq 3 b + 20q a q 4 b q 5 b +16q a (q a q b ) 4q 2 a 11q a q b + q 2 b + 16q 2 a (4q a 7q b ) 2 = 0 In order to solve the equations f A = f B = 0 simultaneously, by using the Buchberger s algorithm (Cox et al., 1997), we compute the Gröbner bases, one of which is the 12th-order polynomial of q b. We can then readily verify that there exists a unique solution of q b in the interval of (0; ). Inserting it into f A = 0 gives us qa = 0:779 ; qb = 0:418 The second-order conditions for pro t maximization are shown to be satis ed. Taking the limit of (13), we get lim n a;n b!1 q 1 = ; lim n a;n b!1 q n a+n b = 0 Accordingly, we have obtained a unique candidate for SPNE. The fact that it is a unique SPNE is clearly demonstrated in the two-stage game as follows; the proof is in Appendix 4. Proposition 1 There exists a unique SPNE, where rms A and B o er the sets of qualities approximating continua 0:779; and 0; 0:418, respectively. Proposition 1 presents the main result of our model: there is a multiquality equilibrium where each duopolist produces an interval of qualities. This result contrasts sharply with the single-quality outcome outlined in Champsaur and Rochet (1989, Proposition 3; 1990, Proposition 2), where the market is assumed to be covered exogenously. Given the xed market size, the negative e ect of cannibalization dominates the positive e ect of segmentation. This functions in much the same way as Hotelling s (1929) spatial competition; hence, each rm has no incentive to provide multiple products. 12
On the contrary, the demand is more elastic in the partially covered market because consumers with lower willingness to pay for quality tend not to purchase any products if their prices are too high. In this case, rms are likely to segment and enlarge the market by o ering multiple products. Thus, in the partially covered market, the positive e ect of segmentation outweighs the negative e ect of cannibalization. According to Proposition 1, no rm produces intermediate qualities between 0:418 and 0:779, despite the fact that this interval is at the center of quality distribution. This is illustrated in the lower panel of Figure 1, where the dotted lines indicate the allocation of the qualities provided by the two rms. It is therefore in the interest of each rm to leave a gap between two product lines in order to relax the price competition at an SPNE. This is clearly an example of Champsaur and Rochet s (1989) Proposition 5. Figure 1: Intervals of willingness to pay and product quality From Proposition 1 and the equilibrium prices (3), the intervals of served 13
consumers are computed as follows: Firm A serves consumers in the range of p lim a (q ) p b (q ) ; = [0:708; ] n a;n b!1 q a and rm B serves consumers in the range of p na+n lim b (q ) ; p a(q ) p b (q ) = [0:262; 0:708] n a;n b!1 qn b +n a qa qb q b Both p n b +n a (q ) and q n b +n a approach zero, but their ratio converges to 0:262 rather than zero. Because consumers in the range of [; 0:262] are not served, this market is not fully covered. While the non ghting qualities discriminate among their consumers perfectly, the two ghting brands q a and q b imperfectly discriminate among their consumers by bunching consumers in the range of p lim a (q ) p b (q ) ; p n a 1(q ) p a(q ) = [0:708; 0:890] and n a;n b!1 qa qb qn a 1 qa p lim b (q ) p n a+2(q ) ; p a(q ) p b (q ) = [0:472; 0:708] n a;n b!1 qb qn a+2 qa qb respectively. The upper panel of Figure 1 displays these intervals. Observe that the low-quality rm B provides a wider quality range and serves more consumers than does the high-quality rm A. Because the market is not fully covered, the low-quality rm wants to attract consumers who originally did not make a purchase. This is done by expanding the product line to include lower qualities and selling them for lower prices. As a result, the low-quality rm o ers a wider quality interval and serves more customers. In the market of home video game consoles, Nintendo introduced Wii and Sony introduced PlayStation 3 in 2006. 9 Compared to Wii, PlayStation has robust multimedia capabilities and a high-de nition optical disc format, Blu-ray Disc, which is targeted to professional players. PlayStation is sold at higher 9 Note that although Nintendo and Sony provide one model of game consoles rather than multiple models in each period, their retailers often bundle the console with its accessories in several combinations. 14
prices than Wii, and its market share is smaller than Wii (the market shares of PlayStation and Wii in 2009 are 26% and 50%, respectively). The uncovered range [; 0:262] here is narrower than that in a single-product duopoly ([; 0:376] as in Motta) or a monopoly ([; 0:5] as in Mussa and Rosen). This is because the multiquality competition is more intense than the others. The pro ts of rms can be decomposed into the two terms as follows: " n lim n a;n b!1 a = lim [p n a;n b!1 a (q ) c(q a )] X a 1 n a 1 n a + [p i (q ) c(q i )] # i 1 i = 0:0192 3 + 0:0125 3 = 0:0317 3 (16) " n lim n a;n b!1 b = lim [p n a;n b!1 b (q ) c(q b )] X a+n b n a n a+1 + [p i (q ) c(q i )] # i 1 i where 0 = and i i=1 i=n a+2 = 0:0156 3 + 0:0084 3 = 0:0240 3 (17) is given by equation (1) for i = 1; 2; :::; n a + n b. The rst terms in equations (16) and (17) are the pro ts from ghting brands q a qb, respectively, which are called pure di erentiation pro ts. The ghting brands imperfectly discriminate among their consumers by bunching consumers having di erent tastes for the same quality. On the other hand, the second terms are the pro ts made by o ering intervals of qualities (qa; q1] and [qn a+n b ; qb ), respectively, which are called pure segmentation pro ts. Because one rm s strategy on its non ghting products is independent of the other rm s strategy, each rm acts as a local monopolist and extracts consumers surplus by o ering a range of qualities with perfect discrimination. This is done by equalizing the marginal revenue and cost of increments of quality improvement given by (4). Further points are worth noting. First, the pure di erentiation pro t from the ghting quality is higher than the pure segmentation pro t from all non ghting qualities. The ghting brands make a larger pro t by bunching a large enough number of consumers in spite of facing intense competition. Second, the higherquality rm A is more pro table than the lower-quality rm B although the 15 and
former provides a narrower quality range and serves fewer consumers. Intuitively, this is because higher-quality products are purchased by consumers with higher willingness to pay. According to Motta, the pro ts are ( a; b ) = (0:03283 ; 0:0243 3 ) in the case of single-product duopoly with a partially covered market. Hence, we establish the following: Proposition 2 The pro t of each rm in a multiproduct duopoly is lower than that in a single-product duopoly. As the above breakdown shows, when rms are allowed to produce as many qualities as they like, they tend to o er an interval of qualities rather than a nite number of products. However, Proposition 2 shows that each rm s pro t becomes less due to price competition. This exempli es the prisoner s dilemma. However, note that Proposition 2 does not hold for monopoly. A monopolist who is allowed to produce an arbitrary number of qualities o ers a quality range of q 2 [0; ] and serves consumers only in the range of 2 [=2; ] (see Mussa and Rosen). On the other hand, if is not small and exceeds =2, then the optimal policy for the monopolist is to provide a single quality q1 = 2=3 and serve consumers with 2 [2=3; ]. The former pro t of multiproduct monopoly is calculated as 0:0833 3, which is higher than that of single-product monopoly given by 0:0741 3. This is because the monopolist does not fall into the prisoner s dilemma in the absence of inter rm competition, but exploits the consumer surplus by o ering a range of qualities. Although the high-quality rm A produces a narrower quality range and serves fewer consumers, the aggregate pro t of each quality of rm A is always higher than that of rm B according to (16) and (17). This implies that the negative e ect of the narrower range is overwhelmed by the positive e ect of the higher willingness to pay for higher qualities. 16
4. Social Welfare In this section, we look into the socially optimal production of qualities. The utility function U (; q) = maxfq p; 0g is quasi-linear and transferable. Therefore, the social welfare is de ned by W S + a + b where the consumer surplus is de ned by S Z = lim n a;n b!1 U (; q) d " Z 1 (q 1 p 1 ) d + nx a+n b i=2 Z i 1 i (q i p i ) d As a benchmark, let us rst consider the social optimum, that is, the rstbest assignment. As shown by Moorthy (1984), the social planner sets each price equal to each marginal cost p o i = c(q i ), assigns the quality range q o 2 lim na;n b!1[q 1 ; q na+n b ] = [0; ], and serves all consumers o 2 (; ). Since the pro ts are zero, the social welfare is equivalent to the consumer surplus. This is computed by W o = S o = 3 =6 = 0:1667 3 Next, consider the multiquality competition in a duopoly. Straightforward calculations yield S = 0:1004 3. Using (16) and (17), the social welfare in the multiquality competition is calculated by W = (0:1004 + 0:0317 + 0:0240) 3 = 0:1561 3 On the other hand, in the single-product competition, the social welfare as the sum of the consumer surplus and the pro ts of rms A and B is calculated by W = (0:0940 + 0:0328 + 0:0243) 3 = 0:1511 3 17 #
Comparing the two welfare values, we can state the following: Proposition 3 When rms are allowed to o er an arbitrary number of qualities, both the consumer surplus and the social welfare are high in comparison to the single-quality duopoly. It is clear that the consumer surplus is higher in the multiquality competition although the pro t of each multiproduct rm is lower because of the prisoner s dilemma shown in Proposition 2. Proposition 3 indicates that the positive e ect on the consumer surplus is greater than the negative e ect on pro ts. Finally, we have somewhat similar results in the case of monopoly. A monopolist who is allowed to produce a range of qualities o ers a quality range of q 2 [0; ] and serves consumers in the range of 2 [=2; ]. The social welfare as the sum of the consumer surplus and the pro t is calculated by W = (0:0417 + 0:0833) 3 = 0:1250 3 In comparison with a duopoly situation, consumers are less well-o whereas the rm is better o. That is because multiproduct monopolists are able to exploit the consumer surplus more e ectively than the multiproduct duopolists. However, for society as a whole, a monopoly is worse than a duopoly due to lack of inter rm competition. In the case of single-product monopoly under > =2, the social welfare as the sum of the consumer surplus and pro t is W = (0:0370 + 0:0741) 3 = 0:1111 3 Thus, the single-product monopoly is shown to be in the worst interests of both consumers and the monopolist. 18
5. Conclusion In this paper, we have analyzed a two-stage competition between multiproduct duopolists in which the rms compete in terms of the number of similar products and their qualities and then compete in terms of prices. We have clari ed that the emergence of a multiproduct equilibrium crucially depends on the nature of quality improvement costs and on the nature of the endogenous market coverage. First, we showed that each rm must produce a single quality for any concave cost of quality improvement. This nding may justify the assumption of single-product rms under linear or zero costs of quality improvement. However, in reality, rms often o er multiple products in many industries, thus the linear or zero cost assumptions may be inappropriate in examining oligopolies in vertical product di erentiation. Second, we focused on the strictly convex, that is, quadratic cost of quality improvement and showed that each rm chooses to o er a disconnected continuum of qualities when the market coverage is endogenously determined. This outcome is in striking contrast with that in an exogenously covered market, as obtained in Champsaur and Rochet (1989, 1990). Third, we showed that the low-quality rm produces a wider range of qualities and serves more consumers than the high-quality rm, although the low-quality rm generates less pro t. Finally, we veri ed the prisoner s dilemma: the pro ts of multiproduct duopolists are smaller than those of single-product duopolists. However, the former yields higher consumer surplus and higher social welfare as a result of inter rm competition. These ndings may explain the characteristics of segmented market structures in the real world. Our results were obtained using a speci c model that made several assumptions on consumer preference and production technology; these assumptions are common in the literature on vertical product di erentiation. One drawback of our model is that each rm is assumed to produce segmented qualities. However, qualities are not necessarily segmented; indeed, they are often entangled between 19
rms in several industries. Appendix 1: The case that both rms can produce at most two qualities Proposition A1 When rms A and B can produce at most two qualities, there exists an SPNE, where high-quality rm A o ers two qualities f0:926; 0:779g and low-quality rm B o ers two qualities f0:407; 0:204g. This can be proven as follows. Let the qualities of rms A and B be fq a1 ; q a2 g and fq b1 ; q b2 g, respectively. Without loss of generality, assume q a1 > q b1, q a1 q a2, and q b1 q b2. (i) Benchmark case of q a1 q a2 q b1 q b2. The equilibrium prices in the second-stage subgame are given by (3) in (3) with n a = n b = 2. In the rst-stage quality game, the two rst-order conditions are given by q a1 = q a2 + 2 ; q b2 = q b1 3 2 Substituting (18) into the other two rst-order conditions, we have (18) f A 80qa2 5 + 204qa2q 4 b1 204qa2q 3 b1 2 + 89qa2q 2 b1 3 + 18q a2 qb1 4 8(16qa2 4 12qa2q 3 b1 + 3qa2q 2 b1 2 + 20q a2 qb1) 3 + 8(8qa2 3 6qa2q 2 b1 + 24q a2 qb1 2 + qb1) 3 2 = 0 and f B 64qa2 5 368qa2q 4 b1 + 768qa2q 3 b1 448qa2q 2 b1 + 68q a2 q b1 3q b1 +64q a2 (q a2 q b1 )(4qa2 2 11q a2 q b1 + qb1) 2 + 64(4qa2 3 7qa2q 2 b1 ) 2 = 0 In order to solve the equations f A = f B = 0 simultaneously, by using the Buchberger s algorithm, we compute the Gröbner bases, one of which is the 18th-order 20
polynomial of q b1. We can readily veri ed that there exists a unique solution of q b1 in the interval of (0; ). Plugging it into f A = 0 and (18) gives us a candidate for SPNE: fqa1; qa2g = f0:926; 0:779g; fqb1; qb2g = f0:407; 0:204g (19) The corresponding pro ts are A = 0:0326 3 ; B = 0:0236 3 (20) Next, we show that any deviation is unpro table for rm A given fq b1 ; q b2 g = f0:407; 0:204g. Five cases (ii)-(vi) may arise. (ii) A s deviation to q a1 q b1 q a2 q b2. The pro ts are given by qa1 2 A = p a1 p a1 p b1 + p 2 q a1 qb1 a2! pa1 (qb1 B = p )2 p b1 p b1 b1 2 where q b1 and q b2 q a1 q b1 q b1 p a2 q a2 qa2 2 pb1 p a2 p a2 p b2 2 qb1 q a2 q a2 qb2! pa2 (qb2 + p )2 p b2 b2 are given by (19). Pro t maximization with respect to prices yields a unique price vector. Plugging it into A in the above and maximizing it with respect to q a1 and q a2 yield two rst-order conditions. Again, using the Buchberger s algorithm, we compute the Gröbner bases, one of which is the 36thorder polynomial of q a2. There is a unique solution that satis es the constraints of q a1 qb1 q a2 qb2. It is given by fq a1; q a2g = f0:806; 0:238g 2 q a2 q b2 p b2 qb2 However, the corresponding pro t A = 0:0239 3 is strictly less than A Hence, rm A does not deviate to q a1 q b1 q a2 q b2. (iii) A s deviation to q b1 > q b2 q a1 q a2. in (20). 21
Similar to case (ii), we rst solve the price subgame, and then solve the quality subgame given fqb1 ; q b2g = f0:407; 0:204g. Using the Buchberger s algorithm, one of the Gröbner bases is shown to be the 7th-order polynomial of q a2. There is a unique solution that satis es the constraints of q b1 q b2 q a1 q a2 > 0. It is given by fq a1; q a2g = f0:112; 0:056g However, the corresponding pro t A = 0:0047 3 is less than A in (20). Hence, rm A does not deviate to q b1 > q b2 q a1 q a2. (iv) A s deviation to q a1 q b1 > q b2 q a2. Likewise, we rst solve for prices, and then solve for qualities given fq b1 ; q b2 g = f0:407; 0:204g. Using the Buchberger s algorithm, one of the Gröbner bases is shown to be the 24th-order polynomial of q a2. However, there is no interior solution that satis es the constraints of q a1 qb1 q a2 qb2, which suggests that rm A does not provide two qualities. That is, we should consider the case that rm A o ers one quality q a1, given that rm B o ers two qualities fqb1 ; q b2 g = f0:407; 0:204g. The case of q a qb1 > q b2 is included in case (i) by setting q a1 = q a2, and the case of q b1 > q b2 q a is included in case (iii) by setting q a1 = q a2. Hence, rm A does not deviate to q a1 q b1 > q b2 q a2. (v) A s deviation to q b1 q a1 q b2 q a2. Likewise, we rst solve the price subgame, and then solve the quality subgame given fqb1 ; q b2g = f0:407; 0:204g. One of the Gröbner bases is shown to be the 28th-order polynomial of q a2. There is a unique solution that satis es the constraints of q b1 q a1 q b2 q a2 > 0. This is given by fq a1; q a2g = f0:302; 0:034g However, the corresponding pro t A = 0:0046 3 is less than A rm A does not deviate to q b1 q a1 q b2 q a2. (vi) A s deviation to q b1 q a1 q a2 q b2. in (20). Hence, 22
Likewise, we rst solve the price subgame, and then solve the quality subgame given fqb1 ; q b2g = f0:407; 0:204g. One of the Gröbner bases is shown to be the 23rd-order polynomial of q a2. However, there is no interior solution that satis es the constraints of qb1 q a1 q a2 qb2, which suggests that rm A provides one quality q a1, given that rm B o ers two qualities fqb1 ; q b2g = f0:407; 0:204g. This is the case of qb1 q a qb2. The rst-order condition is shown to be the 7th-order polynomial of q a. There is a unique solution q a = 0:302 that satis es the constraint of q b1 q a q b2. However, the corresponding pro t A = 0:0046 3 is less than A in (20). Hence, rm A does not deviate to q b1 q a1 q a2 q b2. Finally, we show that any deviation is unpro table for rm B given fqa1; qa2g = f0:926; 0:779g. Five cases (vii)-(xi) may arise. (vii) B s deviation to qa1 q b1 qa2 q b2. Similar to case (ii), one of the Gröbner bases is shown to be the 28th-order polynomial of q b2. There is a unique solution that satis es the constraints of qa1 q b1 qa2 q b2 > 0. It is given by fq b1; q b2g = f0:877; 0:401g However, the corresponding pro t B = 0:010 3 is strictly less than B Hence, rm B does not deviate to q a1 q b1 q a2 q b2. in (20). (viii) B s deviation to q b1 q b2 qa1 qa2. Likewise, one of the Gröbner bases is shown to be the 10th-order polynomial of q b2. However, there is no interior solution that satis es q b1 q b2 qa1 qa2, which suggests that rm B provides one quality q b, given that rm A provides two qualities fqa1; qa2g = f0:926; 0:779g. This is the case of q b qa1 qa2. The rst-order condition is shown to be the 5th-order polynomial of q b. There is a unique solution qb = 1:117 that satis es q b qa1. However, the corresponding pro t B = 0:0033 3 is less than B in (20). Hence, rm A does not deviate to q b1 q b2 q a1 q a2. 23
(ix) B s deviation to q b1 qa1 qa2 q b2. Likewise, one of the Gröbner bases is shown to be the 24th-order polynomial of q b2. However, there is no interior solution that satis es q b1 qa1 qa2 q b2 > 0, implying that rm B does not provide two qualities. That is, we consider the case that rm B o ers one quality q b, given that rm A o ers two qualities fqa1; qa2g = f0:926; 0:779g. The case of q b qa1 qa2 is included in case (viii) by setting q b1 = q b2, and the case of qa1 qa2 q b is included in case (i) by setting q b1 = q b2. Hence, rm B does not deviate to q b1 qa1 qa2 q b2. (x) B s deviation to qa1 q b1 q b2 qa2. Likewise, one of the Gröbner bases is shown to be the 23rd-order polynomial of q b2. However, there is no interior solution that satis es qa1 q b1 q b2 qa2, implying that rm B does not provide two qualities. That is, we consider the case that rm B o ers one quality q b, given that rm A o ers two qualities fqa1; qa2g = f0:926; 0:779g. This is the case of qa1 q b qa2. The rst-order condition is the 7th-order polynomial of q b. There is a unique solution qb = 0:848 that satis es qa1 q b qa2. However, the corresponding pro t B = 0:0017 3 is less than B in (20). Hence, rm B does not deviate to q a1 q b1 q b2 qa2. (xi) B s deviation to q b1 qa1 q b2 qa2. Likewise, one of the Gröbner bases is shown to be the 36th-order polynomial of q b2. However, there is no interior solution that satis es q b1 qa1 q b2 qa2, implying that rm B does not provide two qualities. That is, we consider the case that rm B o ers one quality q b, given that rm A o ers two qualities fqa1; qa2g = f0:926; 0:779g. The case of q b qa1 qa2 is included in case (viii), and the case of qa1 q b qa2 is included in case (x) by setting q b1 = q b2. Hence, rm B does not deviate to q b1 qa1 q b2 qa2. Appendix 2: Concave Cost of Quality Improvement Proposition A2 If the unit cost of quality improvement is concave, each rm 24
o ers a single quality. This can be proven as follows. Substituting the prices (3) into (1), we have 8 >< i = >: h i 1 + c(q i) c(q i+1 ) 2 q i q i+1 q a[c(q a) c(q b )] 4qh a 2 5qaq b+qb 2 i 1 c(qi ) c(q i+1 ) 2 q i q i+1 + 2c(qa)+c(q b)+2(q a q b ) 4q a q b c(q a)+2(q a q b ) 4q a q b for i = n a for i = 1; 2; :::; n a 1 for i = n a + 1; : : : ; n a + n b 1 Then, in the second stage, the rst-order condition of the top quality is @ a = 1 c (q 1 ) c (q 2 ) 2c 0 (q 1 ) + c (q 1) c (q 2 ) @q 1 4 q 1 q 2 q 1 q 2 > 1 c (q 1 ) c (q 2 ) c (q1 ) c (q 2 ) 2c 0 (q 1 ) + c (q 1) c (q 2 ) 4 q 1 q 2 q 1 q 2 q 1 q 2 = 1 c (q 1 ) c (q 2 ) c (q1 ) c (q 2 ) c 0 (q 1 ) 2 q 1 q 2 q 1 q 2 0 (21) where the rst inequality is due to x 1 = 1 c (q 1 ) c (q 2 ) > 0 2 q 1 q 2 () c (q 1 ) c (q 2 ) q 1 q 2 and the second inequality is due to concavity of c(q): c (q 1 ) c (q 2 ) q 1 q 2 c 0 (q 1 ) That is, the top quality goes to in nity. 10 10 In order to avoid in nite quality, Choi and Shin and Bonnisseau and Lahmandi-Ayed assume an upper bound of q i. However, assuming an upper bound should alter the assumption on the unit cost of quality improvement as follows c 0 (q i ) 0, c 00 (q i ) 0 for q i q c (q i ) is suddenly very large for q i > q This means that c (q i ) is no more concave. 25
From (21), the demand for intermediate quality i (= 2; 3; :::; n a 1; n a + 2; :::; n a + n b 1) is given by x i = i 1 i = p i 1 p i p i p i 1 qi 1 qi qi qi 1 = 1 c (qi 1 ) c (q i ) c (q i ) c (q i+1 ) 2 q i 1 q i q i q i+1 (22) Because the rst term is the slope between points (q i 1 ; c (q i 1 )) and (q i ; c (q i )), and the second term is that between (q i ; c (q i )) and (q i+1 ; c (q i+1 )), their di erence is non-positive for any concave c (q i ) with q i 1 > q i > q i+1. In addition, the demand for quality n a + n b is written as i x na+n b = q na+n b 1 h c(qna+nb 1) q na+nb 1 2(q na+n b 1 q na+n b ) c(q na+nb ) q na+nb 0: for any concave c (q i ). Hence, we have shown that rm A produces at most two qualities and rm B at most one quality q b. It remains to show that (n a ; n b ) = (2; 1) is not an SPNE. In this case, rm A optimizes q 1 and q 2 (= q a ). Since @ a=@q 1 > 0, q 1 is the maximum quality, which is denoted by q. Let q 2 = rq and q 3 = sq with 0 < s < r < 1. Then, the other rst-order condition is given by where @ a @q 2 = s2 (20r + s) 2 q 2 + C(q) 4 (4r s) 3 q 2 C(q) 1 (r; s)c (q) 2 + 2 (r; s)c (q) q + 3 (r; s)c (q) c 0 (q) q + 4 (r; s)c 0 (q) q 2 and i (r; s) are constants. From concavity of c (q), the degree of C(q) is does not exceed two. When the degree is less than two, we have @ a lim = s2 (20r + s) 2 q!1 @q 2 4 (4r s) 3 > 0 26
Therefore, we have q 2! q, which implies single-product outcome. When the degree is equal to two, the unit cost of quality improvement is linear or zero c (q) = q ( 0). Then, we get 11 @ a @q 2 = s2 (20r + s)( ) 2 4 (4r s) 3 > 0 Hence, we have shown that the multiproduct duopoly setting ends up with the single-product duopoly. Appendix 3: Proof of Lemma 1 The rst-order conditions are readily computed as 12 @ a 2p 1 2p 2 c(q 1 ) + c(q 2 ) = = 0 for i = 1 @p 1 q 1 q 2 @ a = 2p i 1 2p i c(q i 1 ) + c(q i ) 2p i 2p i+1 c(q i ) + c(q i+1 ) = 0 @p i q i 1 q i q i q i+1 for i = 2; : : : ; n a 1 @ a = 2p n a 1 2p na c(q na 1) + c(q na ) 2p na p na+1 c(q na ) = 0 for i = n a @p na q na 1 q na q na q na+1 @ b = p n a 2p na+1 + c(q na+1) 2p na+1 2p na+2 c(q na+1) + c(q na+2) = 0 @p na+1 q na q na+1 q na+1 q na+2 for i = n a + 1 @ b = 2p i 1 2p i c(q i 1 ) + c(q i ) 2p i 2p i+1 c(q i ) + c(q i+1 ) = 0 @p i q i 1 q i q i q i+1 for i = n a +2; : : : ; n a +n b 1 @ b = 2p n a+n b 1 2p na+nb c(q na+nb 1) + c(q na+nb ) 2p na+nb c(q na+nb ) = 0 @p na+nb q na+nb 1 q na+nb q na+nb for i = n a + n b where the second and the fth equations are ignored when n a = 2 and n b = 2. 11 Furthermore, because @ b = s2 (4r 7s)( ) 2 @q 3 4 (4r s) 3 = 0 we obtain q 1; q 2 = q and q 3 = 4q=7. This is the result by Choi and Shin assuming single-product duopoly. 12 If n a = 2 and n b = 2, the second and the fth equations in the following are ignored. 27
Using @ a =@p i = 0 (i = 1; : : : ; n a as a function of p 1 : 1), p 2 ; : : : ; p na can be successively expressed 1 p i = p 1 c(q1 ) c(q na ) + (q 1 q na ) for i = 2; : : : ; n a (23) 2 From @ b =@p i = 0 (i = n a + 2; : : : ; n a + n b ), p na+1; : : : ; p na+n b solved as a function of p na+nb : p i = 1 c(q i ) + q i (2p na+n 2 q b c(q na+nb )) na+nb are successively for i = n a +1; : : : ; n a +n b 1 (24) Plugging (23) and (24) into @ a =@p na = @ b =@p na+1 = 0, we obtain p 1 and p n a+n b as functions of q i, n a and n b, respectively. Substituting them into (23) and (24) yields the equilibrium prices (3). Because the pro t functions are quadratic and concave in p i, the second-order conditions are satis ed. Appendix 4: Proof of Proposition 1 Because the unique solution given by (13) and Proposition 1 satis es the n a + n b rst-order conditions, it is su cient to show that both pro t functions are concave in the neighborhood of q i = q i for i = 1; 2; : : : ; n a + n b. This can be con rmed by computing the Jacobian matrix of each pro t function. For rm A, by using (13), we have 8 3( q a) 4(2n a for i = j = 1 1) @ 2 a @q i @q j = >< >: q a 2(2n a 1) for i = j = 2; : : : ; n a 1 q a 4(2n a 1) for i = j 1 and i = 1; 2; : : : ; n a 0 for i = j 2 @f A @q a for i = j = n a (25) Let J i be the determinant of the ith principal minor of the Jacobian matrix given by (25). Suppose " # i q a jj i j = (2i + 1) (26) 4 (2n a 1) 28
holds for i = k. The following shows that it is true for k = 1; 2. jj 1 j = 3 q a 4 (2n a 1) ; jj 2j = 5 q 2 a 16 (2n a 1) 2 In addition, we can show that (26) also holds for i = k + 1 as follows: jj k+1 j = @2 a @ 2 2 a jj @qk 2 k j jj k 1j @q k @q k 1 " # k " # " # k 1 " q a q a q a qa = (2k + 1) (2k 1) 4 (2n a 1) 2(2n a 1) 4 (2n a 1) 4 (2n a 1) " # k+1 q a = [2 (k + 1) + 1] 4 (2n a 1) Hence, (26) holds for all i = 1; 2; : : : ; n a lim jj @f A n a j = lim jj na n a!1 n a!1 @q a @f A = lim jj na n a!1 @q a 1. For i = n a! 1, we have @ 2 2 a 1j jj na 2j @q na @q na 1 " qa 1j 0:147 lim n a!1 jj n a Therefore, the signs of jj na j and jj na 1j 4 (2n a 1) # 2 jj na 2j 1j are opposite, and hence, jj i j is negative for all odd i and positive for all even i, verifying concavity near q i = q i for i = 1; 2; : : : ; n a. The similar proof is applied for rm B. # 2 References Barron, J. M., B. A. Taylor, J. R. Umbeck. 2000. A Theory of Quality-Related Di erences in Retail Margins: Why There Is a Premium on Premium Gasoline. Economic Inquiry. 38(4) 550 569. Bonnisseau, J.-M., R. Lahmandi-Ayed. 2006. Vertical Di erentiation: Multiproduct Strategy to Face Entry? Berkeley Electronic Journal of Theoretical Economics. 6(1) 1 14. 29