Profitable Competition: A Multi-Product Firm Case

Transcription

1 Profitable Competition: A Multi-Product Firm Case Kentaro Inomata March 16, 2015 Abstract In contrast with the traditional wisdom that intensifying competition reduces firms profits, recently some works indicate opposite results in the context of new entry and product differentiation. The purpose of this paper is to investigate whether symmetric firms can benefit from the commoditization (homogenization) of products. Using a multi-product firm (MPF) model, we find that both in Cournot and Bertrand duopoly if each downstream MPF procures each product from an individual seller, those firms may increase their profits by providing closer substitutes because of the alleviation of the double marginalization. Such a result never holds in the case of standard single-product duopoly. The result implies that the commoditization can be beneficial for firms nevertheless it is recognized harmful by managers. Further we extend the model to three upstream merger cases: i) firm-specific seller, ii) brand-specific seller, iii) monopolistic seller, and then examine the welfare comparisons. We also show that upstream mergers may be desirable when products of a MPF are complements in i) and iii) both in Cournot and Bertrand. JEL classification: L13; D43 Keywords: product differentiation, multi-product firm (MPF), Cournot and Bertrand Competition, double marginalization, commoditization (homogenization), upstream merger This research was supported by a Grant-in-Aid for Japan Society for the Promotion of Science (JSPS) Fellowship (Grant number ). I would like to thank Takanori Adachi, Shumei Hirai, Shingo Ishiguro, Akira Ishii, Keisuke Kawata, Motonari Kurasawa, Masayoshi Maruyama, Toshihiro Matsumura, Noriaki Matsushima, Akira Miyaoka, Masayuki Nakabayashi, Tatsuhiko Nariu, Takao Okawa, Hideo Owan, Dan Sasaki, Kei Tomita and the conference participants at JAEA (2014) and the seminar participants at Nanzan University, The University of Tokyo (OEIO) for helpful and suggestive comments. Graduate School of Economics, Osaka University. Research Fellow (DC2), Japan Society for the Promotion of Science. kge001ik@mail2.econ.osaka-u.ac.jp 1

2 1 Introduction Traditionally in economics and management, it is likely to be said that the commoditization (homogenization) of goods makes competing firms less profitable and therefore imply that such firms should product-differentiate their products (Shy 1995). However, in reality, we often find the reversal situation, that is, firms provide similar goods. For example, foods, cafes, and bookstores, services and goods we usually see are not always differentiated enough from each other (provided commodities in the extreme case). Such cases have been usually recognized not as positive but as negative situations by both theorists and managers. In fact, almost all the studies in economics state that product differentiation makes firms more profitable. In addition, also, there are many management books enlighten so. The aim of this paper is to examine whether such expectations are always true. Recently there are works implying that firms may obtain higher revenues from new entry. One of the most important studies, Chen and Riordan (2007) suggest that an increase in the number of firms may cause increases in the profits of firms because of the change in the price elasticity of demand. Their result implies new entry as a type of intensifying competition can be profitable for all the firms. Another paper, Mukherjee and Zhao (2009) indicate the possibility that the most efficient firm may obtain a higher profit from new entry if the cost differences among firms are large enough. Also Ishida et al. (2011) imply that if an efficient firm has a cost advantage and an option to invest in cost reducing R&D, new entry may encourage the firm to invest and therefore such a firm gains more profit from new entry. The insight of these works is important in that it has a reverse implication for entry regulation. In contrast, in the context of product differentiation, there are few works studying the profitability of intensifying competition. Zanchettin (2006) is the first work to investigate it. His study suggests that an efficient firm may increase its profit by providing a closer substitute in duopoly. So Zanchettin (2006) indicates that providing similar goods is not always bad for at least one of the firms. Also Inomata (2014) implies that even if firms have no asymmetry in advance, when each firm has an option to provide another product, there are equilibria that only one of the firms becomes a multi-product firm (MPF) and benefits from providing closer substitutes. 1 These studies require the resulting asymmetries in some form and therefore those still imply that the commoditization (homogenization) of products is harmful for at least one firm. Different from those, by incorporating a vertical relation between a firm and its labor union, Fanti and Meccheri (2014) show that both the firms can benefit from providing closer substitutes each other. But their result can be applicable 1 MPF is a type of firms providing multiple differentiated goods. Since goods a MPF provides are differentiated, those are cannibalized if such goods are substitutes with each other and helpful if complements. This is the only characteristic of MPF and sometimes it shows MPF-specific results. For instance Lin and Zhou (2013) consider the MPF-specific problems and obtain intriguing results in the R&D port folio. Yoshida (2015) studies the effect of vertical relations on the downstream. 2

3 only when the labor union has a special objective (utility) function. So there still exists a question whether we can obtain the same result in other cases like where downstream firms buy their products from the upstream firms. Because usually real firms procure their goods from sellers and such a procurement problem often affects the strategic interactions among firms, to analyze the cases is still important. 2 It is well-known that intensifying competition alleviates the double marginalization problem which emerges when buyers need to buy the products from their sellers (Naylor 2002; Matsushima 2006). However, because such a positive effect is usually dominated by the negative competition effect in standard settings, it tends to be thought that firms certainly lose their benefits as competition becomes fiercer. This is correct in usual settings like single-product firms (henceforth SPFs). The SPF is a type of firms which provides only a good or a category, brand. But many firms in the real world are not single-product firms but multi-product firms (MPFs). Each of which provides multiple differentiated goods like super market, car manufacturer, and mobile phone carrier. Such a MPF can have an additional power to lower the wholesale prices than a SPF because of the existence of the cannibalization effect. Usually the effect is negative for the firm (if substitutes each other), but it can be positive when buying from suppliers in the sense that the wholesale price down effect is strengthened as implied by Symeonidis (2010). A MPF can be interpreted as the post-merged firm in differentiated market. He studies a positive impact of a downstream merger on the welfare in differentiated duopoly and suggests the positive aspect of downstream merger (providing multiple goods) on the alleviation of the double marginalization (the decrease in the wholesale price). It is simply because a MPF s profit optimization problem is slightly different from a SPF s one in that a MPF provides multiple differentiated goods and can simultaneously choose all the quantities (prices) of its own goods to maximize the profit. 3 So in this paper we focus on the case in which firms provide multiple differentiated goods which are produced by suppliers as a result of downstream mergers (or brand proliferation decisions). Specifically, we construct a Dobson and Waterson (1996)-type MPF model with vertical relations and investigate the possibility that firms obtain more profits as competition intensifies in both Cournot and Bertrand cases. We assume that there are four goods, good 1A and 1B, 2A and 2B, which are provided by downstream firm D 1 (former two goods) and D 2 (latter two), respectively. Also, in the basic model, each of those are procured by exclusive suppliers, U 1A, U 1B, U 2A, U 2B. We can interpret this situation as the cases in which firms procure their goods from dealers which are in the long-run relations. In this 2 For instance, big companies procuring their products from OEMs like Google are very standing-out now a days. Also, private shops like bakery or coffee shops in a town are the good examples of small firms. 3 One of the most recent textbook of Industrial organization mentions that A multiproduct firm has monopoly power over several products sets lower prices than separate firms (each controlling a single product) when the products are complements...sets higher prices when the goods are substitutes... (Belleflamme and Peitz, 2010, p. 29). This characteristic can work positively for competing firms when those buy goods from sellers. 3

4 case goods are exclusively provided by particular dealers even if goods sold by different firms are perfectly homogeneous. We define the key parameters in this paper. The degree of competition between firm 1 and 2 as the degree of product substitutability between 1A and 2A (1B and 2B), which is described by a slope parameter of demand, γ 12 ( (0, 1)). This is the one well-known in the context of product differentiation (Dixit 1979; Singh and Vives 1984). Why we define this as the degree of competition is to compare the effect of intensifying competition in the SPF case with the one in the MPF case. Likewise, the degree of product differentiation between two goods provided by a MPF is defined by γ AB ( (0, 1)). 4 This is the difference between the SPF and the MPF. In addition, it is the point of DW model, the indicator of cross effect between 1A and 2B (1B and 2A) is described by the product of two parameters above (γ C γ 12 γ AB ). 5 The timing and structure are as follows. In the first stage, each upstream firm determines its wholesale price given the rival three firms setting prices. And then, downstream firms compete in quantity or price with each other in the second stage. After each analysis, using the equilibrium outcomes, we investigate whether competition really always reduces the profits of competing firms. We find that i) in Cournot competition, all the competing firms (D i ) can benefit from intensifying competition if the degree of product differentiation between two goods provided by a MPF (γ AB ) is non-zero and the one between firms (γ 12 ) is relatively high. Similarly, ii) in Bertrand competition, firms yield more profits if two products provided by a MPF are complements (γ AB < 0) and the competition effect (γ 12 ) is in the medium level (not too high and low). Common with two modes of competition, to provide multiple differentiated goods is essential for obtaining the results, which implies that the intra-firm interaction may work positively on the profitability of firms in competition. This is because MPFs gain more profit from a stronger level of alleviation of the double marginalization problem and it can dominate the negative businessstealing effect and the direct market-shrink effect. In this case by buying closer substitutes or providing similar services, both the downstream firms may gain more profits. These results are applicable to many cases, for example, the relation between a big famous company and its manufacturers (including OEM), the one between a small private shop and its exclusive manufacturers. Otherwise long-run interactions with particular dealers. We also find that by intensifying competition, upstream firms can obtain higher profits in the Bertrand if goods of one firm are complements. Further we examine some upstream merger cases: i) both the goods of D i are sold by U i (Case II), ii) both i s A and j s A are sold by U i (Case III), iii) all 4 Particularly, the effect measured by this parameter is called the cannibalization effect if the value is positive. 5 As mentioned by Dobson and Waterson, the merits of this model are on the points that i) it can easily analyze both the interaction between two goods provided by an MPF and the strategic interaction between a good provided by an MPF and the other good by the rival MPF, ii) it seems more plausible than the setting that all the substitutabilities are specified by the same parameters. 4

5 the goods are sold by an upstream monopolist U (Case IV). We show that all the cases hold the standard result that downstream firms certainly reduce their profits from an increase in the product substitutability between firms, although some extension cases indicate increases in upstream firms profits. At last, we compare the welfares between the benchmark case I and other cases, II, III, and IV. We find that both in Cournot and Bertrand, a) the total surplus of Case I (benchmark) is always greater than Case III. But the ones of Case II and IV can be greater than the most competitive case I if goods provided by a downstream firm are complements. These results can be interpreted as extended ones of Ziss (1995) s work in that allowing complement cases. In his work the upstream merger is always anti-competitive but our analysis implies the possibility of reverse consequence. It is simply because the wholesale prices and market prices can be more in I than in II and IV. In contrast, it is always less in III than in I because competition in upstream disappears. There is a related work showing the positive aspect of upstream merger. Milliou and Pavlou (2013) also indicate the possibility that in differentiated single-product duopoly, upstream mergers may be desirable when considering the upstream cost-reducing R&D investments. In the reminder of this paper, section 2 sets the model and section 3 examines a comparative static for investigating whether competition benefits all the competing firms or not. Section 4 extends the basic setting to some cases. Section 5 concludes. 2 The Basic Model 2.1 Setting In this section, using Dobson and Waterson (1996) type inverse demands, we construct a multi-product firm (MPF) duopoly model with vertical relations. Consider the case there are two multi-product downstream firms and each of which, D i ( {1, 2}), provides two differentiated goods named, A and B. The firms compete via goods their own A and B procured by its exclusive suppliers, U ia and U ib. First, D i buys the intermediate good for producing B from U i in advance. Then each firm produces B and competes in quantity for A and B with each other in the second stage. The equilibrium outcomes are derived by backward induction (subgame perfect equilibrium: SPE). 2.2 Demands Firm i s inverse demands for good A and B are p ia a q ia γ 12 q ja γ AB q ib γ c q jb, (1) p ib a q ib γ 12 q jb γ AB q ia γ c q ja (2) where a is the willingness to pay for firm i s A and B, γ 12 ( (0, 1)) and γ AB ( ( 1, 1)) are the degree of product differentiation between firm 1 s good A 5

6 and firm 2 s A, and the one between good A and B provided by a multi-product firm (MPF), respectively. The former indicator implies the usual competition effect between the competing firms. Also, in the context of industrial organization, the latter is often called the cannibalization effect. This effect is the only characteristic of MPF. If γ AB > 0, then the sign of the coefficient is negative. It implies those goods are cannibalized each other (each good has a negative impact on its own another good provided by an MPF). In contrast, if negative, the sign turns to be positive and indicates that those are complements each other. We assume that in this paper, γ c equals the product of γ 12 and γ AB (γ c γ 12 γ AB ). This assumption is according to Dobson and Waterson (1996). By substituting γ 12 γ AB into γ c, eq. (1) and (2) can be rewritten as p ia a q ia γ 12 q ja γ AB q ib γ c q jb, a q ia γ 12 q ja γ AB q ib γ 12 γ AB q jb a (q ia + γ AB q ib ) γ 12 (q ja + γ AB q jb ), (3) p ib a q ia γ 12 q ja γ AB q ib γ c q jb, a q ib γ 12 q jb γ AB q ia γ 12 γ AB q ja a (q ib + γ AB q ia ) γ 12 (q jb + γ AB q ja ). (4) This expression helps us to understand what this type of inverse demand indicates: for example, if γ 12 1, i s A (B) and j s A (B) are homogeneous, further the difference between i s A and j s B is the same as the one between i s A and B. We also obtain the demand functions by solving the above equations. q ia α βp ia + βγ 12 p ja + βγ AB p ib βγ 12 γ AB p jb (5) q ib α βp ib + βγ 12 p jb + βγ AB p ia βγ 12 γ AB p ja (6) where α (1 γ 12 γ AB +γ 12 γ AB )a (1 γ 2 12 )(1 γ2 AB ), β 1 (1 γ 2 12 )(1 γ2 AB ) 2.3 2nd stage: Competition In the second stage, under Cournot (Bertrand) competition, each firm competes in quantity (price) with its rival. Π CD i Π BD i πia CD + πib CD (p ia w ia )q ia + (p ib w i )q ib (a q ia γ 12 q ja γ AB q ib γ 12 γ AB q jb w ia )q ia + (a q ib γ 12 q jb γ AB q ia γ 12 γ AB q ja w ib )q ib (7) π BD ia + π BD ib (p ia w ia )q ia + (p ib w ib )q ib (p ia w ia )(α βp ia + βγ 12 p ja + βγ AB p ib βγ 12 γ AB p jb ) + (p ib w ib )(α βp ib + βγ 12 p jb + βγ AB p ia βγ 12 γ AB p ja ) (8) 6

7 where w ia is the wholesale price for i s good A charged by the exclusive supplier U ia and w i the one for B charged by U ib, respectively. Without loss of generality, a is normalized to one, c ia and c ib are zero. Given the outputs of firm j, firm i chooses both q ia and q ib (p ia and p ib ) for maximizing its profit in the same time. 6 Firm i s best responses to the rival j are derived: q ia (q ja) 1 + γ AB γ 12 q ja γ ABw B 2(1 γ 2 AB ), (9) q ib (q jb) 1 + γ AB γ 12 γ AB q ja w B 2(1 γ 2 AB ), (10) p ia (p ja) 1 γ 12 (1 γ 12 )γab 2 + (1 γ2 AB )w ia 2(1 γab 2 ) + γ 12p ja, (11) 2 p ib (p jb) 1 γ 12 (1 γ 12 )γab 2 + (1 γ2 AB )w ib 2(1 γab 2 ) + γ 12p jb 2 (12) Solving the above equations, the equilibrium quantities and prices in the second stage are determined: q C ia (2 γ 12)(1 γ AB ) 2(w ia γ AB w ib ) + γ 12 (w ja γ AB w jb ) (4 γ 2 12 )(1 γ2 AB ), (13) q C ib (2 γ 12)(1 γ AB ) 2(w ib γ AB w ia ) + γ 12 (w jb γ AB w ja ) (4 γ 2 12 )(1 γ2 AB ), (14) p B ia 2 γ 12 γ w ia + γ 12 w ja 4 γ12 2, (15) p B ib 2 γ 12 γ w ib + γ 12 w jb 4 γ12 2, (16) qia B δ ε[(2 γ12)w 2 ia + γ AB (2 γ12)w 2 ib + γ 12 w ja γ 12 γ AB w jb ], (17) q B ib δ ε[(2 γ 2 12)w ib + γ AB (2 γ 2 12)w ia + γ 12 w jb γ 12 γ AB w ja ], (18) (19) where δ (1 γ AB)[2 γ 12 (1+γ 12 )] (4 γ12 2 ), ε 1 )(1 γ2 12 )(1 γ2 AB (4 γ12 2 )(1 γ2 12 )(1 γ2 AB ). By these, we derive the equilibrium profits in the second stage. We only describe the profits when wholesale prices are exogenous and identical below. Endogenous profits are derived in the next subsection. Π CD i Π BD i 2(1 w) 2 (2 + γ 12 ) 2 (1 + γ AB ), (20) 2(1 γ 12 )(1 w) 2 (2 γ 12 ) 2 (1 + γ 12 )(1 + γ AB ). (21) 6 As mentioned in Introduction, since a MPF can choose the quantities of its products simultaneously, the equilibrium outcomes of an MPF are different from when those are provided by two single-product firms (SPFs). This difference works a crucial role for our main result (Proposition 3). 7

8 We examine a comparative static about the effect of competition on the profits. The process is as follows: first i) we use the inverse demands and demands including γ C (eq.(1),(2)) to identify the pure effect of competition and solve the problem above again. Then ii) differentiate the equilibrium profit with respect to γ 12 at γ C γ 12 γ AB. Therefore, we obtain Proposition 1 and Proposition 2. Proposition 1 (Cournot competition with exogenous wholesale prices) If all the wholesale prices are exogenous and symmetric (w ia w ib w ja w jb w), the equilibrium quantity is q C 1 w (γ 12, γ AB, γ C ) (2+γ C +γ 12 +2γ AB ) and hence an increase in γ 12 always reduces all the profits of firms. Proof of Proposition 1 By the F.O.C.s, the equilibrium profit must be π CD 2(1 + γ AB )q C 2 and it is a monotonically increasing function in q C. Since q C is monotonically decreasing in γ 12, we obtain the result. Q.E.D. Proposition 2 (Bertrand competition with exogenous wholesale prices) If all the wholesale prices are exogenous and symmetric (w ia w ib w ja w jb w), the equilibrium quantity is q (1+γ AB )(1 w) (1+γ C +γ 12 +γ AB )[2(1+γ AB ) γ C γ 12 ] and hence an increase in γ 12 always reduces all the profits of firms. Proof of Proposition 2 ΠBD i γc γ 12 γ AB > 0 1 γ 12 + γ12 2 < 0. However, it never holds. Therefore, we obtain the result. Q.E.D. Proposition 1 and Proposition 2 confirms that even if firms are MPFs, downstream competition is always harmful for all the firms if the equilibrium outcomes are symmetric st stage: Determination of Wholesale Prices In the first stage, under Cournot (Bertrand) competition, each upstream firm U C ia, U C ib (U B ia, U B ib ) determines its wholesale price level wc ia, wc ib (wb ia, wb ib ), given the demand for ia, ib which will be determined in the second stage and the other firms optimal choices (w ib, w ja, w jb ). That is, for instance, for U ia, π U ia(w ia, w ib, w ja, w jb) w ia q ia (w ia, w ib, w ja, w jb). (22) 8

9 The best response of UiA C (U ia B ) to the rivals in Cournot (Bertrand) competition are wia(w C ib, C wja, C wjb) C (1 γ AB)(2 γ 12 ) 4 + 2γ ABwiB C + γ 12wjA C γ 12γ AB wjb C, (23) 4 wia(w B ib, B wja, B wjb) B (2 γ AB)(2 γ 12 γ12) 2 2(2 γ12 2 ) + γ 12w B ja + γ AB(2 γ 2 12)w B ib γ 12γ AB w B jb 2(2 γ 2 12 ). (24) Because of symmetry, the equilibrium wholesale prices charged are w C (2 γ 12 )(1 γ AB ) 2(2 γ AB ) γ 12 (1 γ AB ), (25) w B (2 γ 12 γ 2 12)(1 γ AB ) 4 γ12 2 (2 γ, (26) AB) γ 12 (1 γ AB ) 2γ AB and so we obtain q C 2 (2 + γ 12 )[4 γ 12 (1 γ AB ) 2γ AB ](1 + γ AB ), (27) p B (1 γ 12 )(6 γ 2 12(2 γ AB ) 4γ AB ) (2 γ 12 )[4 γ12 2 (2 γ AB) γ 12 (1 γ AB ) 2γ AB ], (28) q B 2 γ12 2 (2 + γ 12 γ12 2 )[4 γ2 12 (2 γ AB) γ 12 (1 γ AB ) 2γ AB ](1 + γ AB ). (29) By substituting the wholesale prices and the quantities into the profit functions, we obtain those equilibrium profits, Π CD 8 (2 + γ 12 ) 2 [4 γ 12 (1 γ AB ) 2γ AB ] 2 (1 + γ AB ), (30) Π BD 2(1 γ 12 )(2 γ 2 12) 2 (2 γ 12 ) 2 (1 + γ 12 )[4 γ12 2 (2 γ AB) γ 12 (1 γ AB ) 2γ AB ] 2 (1 + γ AB ), (31) π CU 2 (2 γ 12 ) (1 γ AB ) (2 + γ 12 ) (4 γ 12 (1 γ AB ) 2γ AB ) 2 (1 + γ AB ), (32) π BU (1 γ 12 ) (2 + γ 12 ) ( 2 γ 2 12) (1 γab ) (2 γ 12 ) (1 + γ 12 ) (4 γ12 2 (2 γ AB) γ 12 (1 γ AB ) 2γ AB ) 2 (1 + γ AB ). 3 Effect of Competition on Equilibrium Profits (33) In this section, we examine a comparative static for clarifying the effect of competition on the profits of downstream firms. That is, whether the sign of 9

10 the first derivative of the equilibrium profit with respect to γ 12 is positive or not. First of all, to identify the pure effect of competition and compare with SPF case, we use the inverse demands eq. (1) and (2) which include γ C again and solve the problems as those done in the previous section. By this process, we can separate the effect on the relation between i s A (B) and j s A (B) from the one between i s A (B) and j s B (A). After that, partially-differentiate with respect to γ 12 at γ C γ 12 γ AB : So we obtain the main results as below. Π D γc γ 12 γ AB > 0. (34) Proposition 3 (Cournot competition with endogenous wholesale prices) In Cournot competition, all the firms competing raise their profits as competition intensifies if γ 12 is relatively high and γ AB is non-zero and relatively low. Proof of Proposition 3 Since Π CD 2(1 + γ AB )q 2, Π CD γc γ 12 γ AB > 0 qc γc γ 12 γ AB > 0, 4( 1 + γ 12 (1 + γ 2 AB )) (2 + γ 12 ) 2 (4 γ 12 (1 γ AB ) 2γ AB ) 2 (1 + γ AB ) 2 > 0 ( 1 + γ 12 (1 + γ 2 AB)) > 0. Q.E.D. From the last line of this proof, we can easily know that if γ AB 0, that is, firms are single-product firms, Proposition 3 never holds. The intuition behind the result is simple. To show it, we rewrite the first derivative of the equilibrium profit with respect to γ 12 and specify the path of competition effect on the profit. The first derivative can be divided into two parts: i) positive wholesale price down effect, ii) negative demand shrink effect. That is, q(w(γ 12, γ AB, γ C ), γ 12, γ AB, γ C ) γc γ 12 γ AB q w w + q (35) where w is the symmetric equilibrium wholesale price as a function of parameters. The first terms in (35) indicate the positive direct effect of decreases in the wholesale prices. The second is the negative market shrink effect. Usually the total effect is negative but it is opposite in this case. The intuition behind this result is below. In the downstream market, an increase in γ 12 simply shrinks the market so is harmful all the firms. But the existence of non-zero γ AB relaxes it. Similarly, γ AB makes the interactions between ia and jb (for i, j {1, 2}) via ib, so an increase in γ 12 makes wholesale price competition severer compared with SPFs. We also find the fact about the upstream firms as follows. 10

11 Proposition 4 An increase in γ 12 always reduces Us profits. Proof of Proposition 4 π1cu γc γ 12 γ AB < 0 4(1 γ AB + 2γ 12 (1 2γ AB γab 3 ) (1 γ AB + γab 2 γ3 AB ) < 0. This inequality holds for any γ 12 (0, 1) and γ AB (0, 1). Proposition 4 is reasonable since the wholesale prices decrease and therefore downstream firms obtain higher profits. Next, we consider the case of Bertrand competition. As in the case of Cournot Competition, we reuse the demands including γ C and solve the problems. Differentiate the profit with respect to γ 12 at γ C γ 12 γ AB and we obtain Proposition 5. Proposition 5 (Bertrand competition with endogenous wholesale prices) In Bertrand competition, downstream firms may benefit from intensifying competition if γ 12 is the medium level and γ AB is positive and relatively high. Proof of Proposition 5 ΠBD γc γ 12 γ AB > 0 [4 4γ 12(2+2γ AB +3γ 2 AB )+4γ2 12 (1+2γ AB+3γ 2 AB )+γ3 12 (6+6γ AB+8γ 2 AB ) γ4 12 (5+6γ AB+8γ 8 AB ) γ5 12 (1 γ AB)(2+γ AB +2γ 2 AB )] (4 γ 12 2γ12 2 2γ AB+γ AB γ 12 +γ12 2 γ > AB) 0. We obtain the parameter domains which satisfy the inequality: approximately < γ 12 < and f(γ 12 ) < γ AB < 1 where f(γ 12 ) 8+6γ2 12 γ4 12 4(6 4γ γ4 12 ) γ γ γ γ γ γ γ γ γ γ 12 (1 γ 12. )(6 4γ12 2 +γ4 12 )2 We also obtain the result about Us profits. Q.E.D. Proposition 6 When goods provided by a MPF are complements (γ AB < 0) and the complementarity is moderate, intensifying competition may increase Us profits. Proof of Proposition 6 Mathematica file used to solve the inequality is available on request. By Proposition 5 and 6, we find that the substitutability between intra-brands yields the downstream firm a higher profit but makes the upstream firm less profitable. 4 Some Extensions We extend the basic model (henceforth Case I) to some cases, i) an upstream firm U i sells two goods, A and B to D i (Case II: firm-specific sellers), ii) U A sells A to both firm i and j (Case III: brand-specific seller), iii) a monopolistic upstream firm U m sells all the products to all the buyers (Case IV: monopolistic seller). Those cases can be also interpreted as the post-merged cases of the benchmark case I considered in the previous sections. 11

12 4.1 Case II In this case, seller i sets its wholesale prices of good A and B to maximize its profit in the same time. The equilibrium outcomes are π U i w ia q ia + w ib q ib (36) q 2C 8 (γ 1,2 + 1) (γ 1,2 4) 2 (γ 1,2 + 2) 2 (γ A,B + 1), (37) p 2B (γ 1,2 1) ( γ1,2 2 (γ A,B 2) 4γ A,B + 6 ) (γ 1,2 2) ( γ1,2 2 (γ ), (38) A,B 2) + γ 1,2 (γ A,B 1) 2γ A,B + 4 ( γ 2 w 2B 1,2 + γ 1,2 2 ) (γ A,B 1) γ1,2 2 (γ A,B 2) + γ 1,2 (γ A,B 1) 2γ A,B + 4. (39) As a result, we obtain Proposition 7. Proposition 7 Both in Cournot and Bertrand, all the sellers and buyers decrease their profit by intensifying competition. Proof of Proposition 7 For any γ 12 (0, 1), γ AB ( 1, 1), π 2CD 32(1 γ γc γ 12 γ AB 1,2) (γ 1,2 4) 3 (γ 1,2 +2) 3 (γ A,B +1) < 0, 2 π 2CU 8(γ γc γ 12γ AB 2 2γ1,2+4) 1,2 (γ 1,2 4) 3 (γ 1,2+2) 2 (γ A,B +1) < 0, 2 π 2BD γc γ 12 γ AB 4(2γ8 12 2γ7 12 9γ γ γ γ3 12 4γ γ 12 8) (γ 12 2) 3 (γ 12+1) 2 (2γ γ12 4)3 (γ AB +1) 2 < 0, π 2BU γc γ 12γ AB 4(2γ7 1,2 +2γ6 1,2 13γ5 1,2 6γ4 1,2 +28γ3 1,2 +2γ2 1,2 20γ1,2+8) (γ 1,2 2) 2 (γ 1,2 +1) 2 (2γ 2 1,2 +γ 1,2 4) 3 (γ A,B +1) 2 < 0. Q.E.D. This case helps us to understand the mechanism in Proposition 3 and 5. That is, Case II means each upstream firm, U i, is the monopolistic supplier for D i. In this case, as implied by Ziss (1995), the upstream merger is negative on the wholesale down effect. 4.2 Case III In Case III, each upstream firm sells A(B) to both D i and D j. π U A w ia q ia + w ja q ja, (40) π U B w ib q ib + w jb q jb. (41) 12

13 The wholesale price and market price are as follows. Then we obtain Proposition 8. q 3C 1 (γ 1,2 + 2) (2 γ A,B ) (γ A,B + 1), (42) p 3B γ 1,2 (γ A,B 2) 2γ A,B + 3, (γ 1,2 2) (γ A,B 2) (43) w 3B γ A,B 1 γ A,B 2. (44) Proposition 8 Both in Cournot and Bertrand, for any parameter values, all the buyers decrease their profit by intensifying competition. However sellers can benefit from intensifying competition. Proof of Proposition 8 Prove only the latter statement. π3cu γc γ 12γ AB 2(2 γ AB +γ 3 AB ) > 0 2 γ AB + γab 3 < 0. So for γ 12 (0, 1) and (2+γ AB ) 2 (2 γ AB ) 3 (1+γ AB ) 2 1 (1+ 2) 1 3 π 3BU + (1 + 2) 1 3 < γ AB < 1, the inequality holds. Similarly, in Bertrand, 2(1 2γ γc γ 12γ AB 12 )( 2+3γ AB +γ 3 AB ) (2 γ AB ) 3 (2 γ 12) 2 (1+γ 12) 2 (1+γ AB ) > 0 2+3γ 2 AB +γab 3 < 0 if 0 < γ 12 < 1 2, or 2 + 3γ AB + γab 3 > 0 if 1 2 < γ 12 < 1. Hence for γ 12 (0, 1 2 ) 1 and (1+ + (1 + 2) 1 2) 1 3 < γ AB < 1, or for γ 12 ( 1 3 2, 1) and 1 < γ AB < 1 (1+ 2) (1 + 2) Case IV Q.E.D. In the last case, the upstream monopolist sets all the goods selling the downstream firms. π 4U m w ia q ia + w ja q ja + w ib q ib + w jb q jb. (45) In both Cournot and Bertrand cases, the equilibrium wholesale prices are just w 4C w 4B 1 2. This is the monopolistic wholesale price. The quantity in Cournot and market price in Bertrand are By this, we finally find the results in Case IV. q 4C 1 2 (γ 1,2 + 2) (γ A,B + 1), (46) p 4B 3 2γ 1,2 4 2γ 1,2. (47) Proposition 9 Both in Cournot and Bertrand, the equilibrium wholesale price is just w 1 2. For any parameter values, downstream firms decrease their profits 13

14 both in Cournot and Bertrand. The monopolistic supplier decreases its profit by intensifying competition in Cournot. However in contrast, buyers in Bertrand can benefit from intensifying competition if γ 12 > 1 2 because of the increase in quantity. 5 Welfare Comparisons: vs Upstream Mergers In this section we also examine the welfare comparisons for studying whether the upstream merger cases that we analyzed in the previous section are always anti-competitive. To do this, we compare the surpluses in Case I with the ones in Case II, III, and IV respectively. The results of these analyses are summarized in the following proposition. Proposition 10 In both Cournot and Bertrand, the consumer s surplus and the total surplus can be greater under merged cases (II and IV) than in Case I if the products provided by a downstream firm are complements. In contrast, the ones in Case III are always lower than in Case I. This proposition implies that by allowing the consideration of complement cases (γ AB < 0) we obtain the intriguing results. That is, the welfare can be improved by the upstream mergers. It is of course consistent with Ziss (1995) s general analysis. But by focusing on the linear demand and extending the insight to complement cases our analysis shows somewhat important results. This type of consideration is meaningful when firms competing in downstream are MPFs which is very common in the real world. In SPF competition, such a case cannot be considered. 6 Conclusion This paper examined the positive impact of intensifying competition on the profits of competing firms which procure their products from suppliers. By using the MPF model first proposed by Dobson and Waterson (1996), we obtain the results that all the competing firms may yield higher profits by intensifying competition in both Cournot and Bertrand competition. Such results are opposite to the traditional ones of product differentiation. This result can be applicable to many cases where retailers need to procure from their exclusive suppliers. Also we investigated the welfare comparisons between the benchmark case I and others. By allowing complements cases, we find that the upstream mergers may be desirable in that the consumer s surplus and the total surplus can be greater in cases II and IV than in I. This is because the wholesale prices and the resulting market prices are higher in I than II and IV when goods provided by a downstream firm are complements. This result is contrast with Ziss (1995) s seminal work only considering substitute cases. 14

15 7 Appendix: Proofs of Propositions in Section Case I vs Case II First we consider Case I and II. The consumer s surplus, producers surplus, and total surplus in Cournot are as follows. CS 1C 8 (γ 1,2 + 1) (γ 1,2 + 2) 2 (γ 1,2 (γ A,B 1) 2γ A,B + 4) 2 (γ A,B + 1), (48) T S 1C 8 ( γ1,2 2 (γ A,B 1) 4γ A,B + γ 1,2 + 7 ) (γ 1,2 + 2) 2 (γ 1,2 (γ A,B 1) 2γ A,B + 4) 2 (γ A,B + 1), (49) CS 2C 128 (γ 1,2 + 1) 3 (γ 1,2 4) 4 (γ 1,2 + 2) 4 (γ A,B + 1), (50) T S 2C 32 ( γ1,2 5 3γ1,2 4 20γ1, γ1, γ 1, ) (γ 1,2 4) 4 (γ 1,2 + 2) 4. (51) (γ A,B + 1) We note here that in Cournot all the equilibrium outcomes like the price and wholesale price and the resulting surplus can be rewritten as the function of the quantity because of the FOCs. That is, any differences of outcomes and surpluses between two cases like CS 1i ( CS i CS 1 ), T S 1i ( T S i T S 1 ), p 1i ( p i p 1 ), and w 1i ( w i w 1 ) for i {2, 3, 4}, can be simply explained by the quantity difference q 1i ( q i q 1 ). Since q 1i can be positive, p 1i and w 1i are negative in the case and therefore the surpluses may become positive if the goods provided by a firm in downstream are complements and the goods in the same brand but provided by different firms are relatively closer substitutes: < γ 1,2 < 1 and 1 < γ A,B < γ3 1,2 2γ2 1,2 12γ 1,2+16 4γ1,2 2 4γ (< 0). 1,2 8 Also the Bertrand ones are CS 1B 2 ( γ1,2 2 2 ) 2 (γ 1,2 2) 2 (γ 1,2 + 1) ( γ1,2 2 (γ A,B 2) + γ 1,2 (γ A,B 1) 2γ A,B + 4 ) 2 (γ A,B + 1), (53) T S 1B 2 ( γ1,2 2 2 ) ( 2γ1,2 3 (γ A,B 2) + γ1,2 2 (5 2γ A,B ) + γ 1,2 (12 8γ A,B ) + 8γ A,B 14 ) (γ 1,2 2) 2 (γ 1,2 + 1) ( γ1,2 2 (γ A,B 2) + γ 1,2 (γ A,B 1) 2γ A,B + 4 ) 2 (γ A,B + 1), (52) (54) CS 2B 2 ( γ1,2 2 2 ) 2 (γ 1,2 2) 2 (γ 1,2 + 1) ( 2γ1,2 2 + γ 1,2 4 ) 2 (γ A,B + 1), (55) T S 2B 2 ( γ1,2 2 2 ) ( 4γ1,2 3 5γ1,2 2 12γ 1, ) (γ 1,2 2) 2 (γ 1,2 + 1) ( 2γ1,2 2 + γ 1,2 4 ) 2 (γ A,B + 1). (56) As mentioned in the Cournot case, the Bertrand case can be also explained by the differences of the price and the wholesale price. Since the differences are 15

16 negative if complements, the surplus differences are positive in the case implying the upstream mergers are socially desirable. That is, if 0 < γ 1,2 < 3 1 and 1 < γ A,B < (< 0), the upstream mergers improve the welfares. γ 1,2 γ 2 1,2 +γ 1, Case I vs Case III Second is Case I and Case III. The surpluses in III are calculated as CS 3C 2 (γ 1,2 + 1) (γ 1,2 + 2) 2 (γ A,B 2) 2 (γ A,B + 1), (57) T S 3C γ 1,2 (6 4γ A,B ) 8γ A,B + 14 (γ 1,2 + 2) 2 (γ A,B 2) 2 (γ A,B + 1), (58) CS 3B 2 (γ 1,2 2) 2 (γ 1,2 + 1) (γ A,B 2) 2 (γ A,B + 1), (59) T S 3B 4γ 1,2 (γ A,B 2) 8γ A,B + 14 (γ 1,2 2) 2 (γ 1,2 + 1) (γ A,B 2) 2 (γ A,B + 1). (60) In this case, both the differences of wholesale price and the market price are always positive meaning the surplus is certainly less in III than in I. 7.3 Case I vs Case IV Finally we compare Case I with Case IV. The surpluses in IV are described as CS 4C γ 1, (γ 1,2 + 2) 2 (γ A,B + 1), (61) T S 4C 3γ 1, (γ 1,2 + 2) 2 (γ A,B + 1), (62) CS 4B 1 2 (γ 1,2 2) 2 (γ 1,2 + 1) (γ A,B + 1), (63) T S 4B 7 4γ 1,2 2 (γ 1,2 2) 2 (γ 1,2 + 1) (γ A,B + 1). (64) q 14 > 0 if 0 < γ 1,2 < 1 and 1 < γ A,B < γ 1,2 γ (< 0). Similarly 1,2 2 w4b w 1B > 0 if {0 < γ 1,2 γ 3 1 and 1,2 γ1,2 2 +γ1,2 2 < γ A,B < 1} or { 3 1 < γ 1,2 < 1 and 1 < γ A,B < 1}, we obtain the results. References Q.E.D. [1] Belleflamme, Paul and Peitz, Martin, Industrial Organization: Markets and Strategies Cambridge University Press (2010) [2] Chen, Yongmin, and Michael H. Riordan. Price and Variety in the Spokes Model. The Economic Journal (2007):

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