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Vertical mergers and downstream spatial competition with different product varieties
Economics Letters 101 (2008) 262–264
Contents lists available at ScienceDirect
Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o n b a s e
Vertical mergers and downstream spatial competition with different product varieties Hamid Beladi a,⁎, Avik Chakrabarti b,1, Sugata Marjit c,2 a b c
Department of Economics, College of Business, University of Texas at San Antonio, One UTSA Circle, San Antonio, Texas 78249-0633, United States Department of Economics, 816 Bolton Hall, College of Letters and Science, University of Wisconsin-Milwaukee, P.O. Box 413, Wisconsin 53201, United States Centre for Studies in Social Sciences, Calcutta 700-094, West Bengal, India
a r t i c l e
i n f o
Article history: Received 7 January 2008 Received in revised form 8 August 2008 Accepted 19 August 2008 Available online 24 August 2008
a b s t r a c t We show how, in an industry where no downstream firm can produce all varieties demanded, a vertical merger with a monopoly upstream will induce each downstream firm (inside and out of the merger) to deviate from the socially optimal location. © 2008 Elsevier B.V. All rights reserved.
Keywords: Product-differentiation Price-discrimination Spatial-competition Firm-location Merger JEL classification: L13 L42 D43 R32
1. Introduction
2. Model and propositions
Vertical merger activities have been drawing increasing attention of regulators, anti-trust authorities as well as those in the media and academics.3 In this paper, we extend Braid (2007)4 to demonstrate the effect of vertical mergers on the equilibrium locations of firms which sell different varieties of a product. In a vertically related industry, with an upstream monopoly, the downstream firms will not choose the socially optimal location. A vertical merger will exacerbate the distance of the downstream firms from the socially optimal location: the firm outside the merger will deviate more than the firm that is part of the merger.
Consider a vertically related industry with one upstream firm (M) producing an intermediate good and selling this good to 2 downstream firms (Rj:i = 1,2) who transform (on a one-to-one basis) the intermediate good into the differentiated final goods and sell to consumers distributed uniformly with unit density on a uni-dimensional (linear) market interval with support [0, 1]. The downstream firms R1 and R2 are located at x and y, respectively, on this interval. R1 sells products U and W, and R2 sells products V and W: a fraction c of consumers want to buy product U; a fraction c of consumers want to buy product V; and a fraction b of consumers want to buy product W.5 Transportation costs are td, where t
⁎ Corresponding author. Tel.: +1 210 458 7038; fax: +1 210 458 7040. E-mail addresses: [email protected] (H. Beladi), [email protected] (A. Chakrabarti), [email protected] (S. Marjit). 1 Tel.: +1 414 229 4680; fax: +1 414 229 3860. 2 Tel.: +91 852 2788 7745; fax: +91 852 2788 8806. 3 See Lafontaine and Slade (2007) for an insightful review of the contributions relevant to the related literature. 4 Braid (2007) showed that the equilibrium locations of two firms are partially centralized to the socially optimal extent if there is spatial price discrimination, and if each firm has two out of three products, or else one variety of a differentiated product with some consumers indifferent between varieties.
0165-1765/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2008.08.024
5 One can contemplate, following Braid (2007), an equivalent scenario where firm 1 sells product variety G, and firm 2 sells product variety H: a fraction c of consumers wants to buy only variety G; a fraction c of consumers wants to buy only variety H; and a fraction b of consumers is indifferent between varieties G and H. If each firm cannot price discriminate at each location between the different types of consumers who find its own variety desirable, then it might be possible to assume mixed price strategies, but unlike Dasgupta and Maskin (1986), who have a single mixed-strategy Nash equilibrium in mill prices for any given set of firm locations, there would be a different mixed-strategy Nash equilibrium in delivered prices at each consumer location for any given set of firm locations. We maintain that there is spatial price discrimination for product W of the sort first examined in Hoover (1937) and Lerner and Singer (1937), in which there is a Nash equilibrium in delivered price schedules.
H. Beladi et al. / Economics Letters 101 (2008) 262–264
is a constant and d is distance shipped. There is a maximum reservation price, denoted by k, that consumers are willing to pay which is sufficiently large so that it comes into play only for product varieties where there is no competition between the two firms. The game is played in three stages, with perfect monitoring i.e. all past actions become common knowledge at the end of each stage. In the first stage, M makes a take-it-or-leave-it two-part tariff offer to R1 and R2: M's offer takes the form (wj,Fj) where wj is the marginal wholesale price and Fj is the fixed fee extracted by M from Rj. At the same time, R1 and R2 simultaneously choose their location in the retail market. In the second stage, R1 and R2 simultaneously decide whether or not to accept or decline a contract. If Rj decides to accept a contract offered, the fixed fee (Fj) is collected by M at this stage. In the third stage, R1 and R2 engage in spatial price discrimination. In the final stage, quantities are demanded by consumers from the upstream firms at which time the wholesale price has to be paid for each unit that is ordered and then sold in the retail market to consumers. As usual, the game is solved by backward induction. We can then make a comparison between the un-integrated and integrated (where one6 of the downstream firms is merged upstream) equilibria with b . spatial price discrimination. For expositional convenience, let R ¼ 4ðbþc Þ Absent the possibility of a merger, M charges a uniform wholesale price w and a fixed fee F that extracts all of the profits from each downstream firm. The profits of R1, from selling varieties U and W, are Π1 ¼
i ct h 2 cðk−wÞ− x þ ð1−xÞ2 2 x ðxþy 2 Þ þ ∫ 0 b½t ð y−xÞ−wdz þ ∫ x b½t ðx þ y−2zÞ−wdz −F
ð1Þ
The profits of R2, from selling varieties V and W, are Π2 ¼
i ct h 2 cðk−wÞ− y þ ð1−yÞ2 2 y 1 þ ∫ ðxþyÞ b½t ðx þ y−2zÞ−wdz þ ∫ y b½t ðx−yÞ−wdz −F
ð2Þ
2
Solving the first order conditions for profit-maximization, we obtain ðx; yÞ ¼
! 2 4c þ b2 þ 4bc −bwðb þ cÞ 4c2 þ 3b2 þ 8bc −bwðb þ cÞ ; 4ðb2 þ 2c2 þ 3bcÞ 4ðb2 þ 2c2 þ 3bcÞ ð3Þ
This leads to our first proposition. Proposition I. Absent a merger, the Nash equilibriumh locations of the i 1 bw two downstream firms, in a vertically related industry, are 2 F R þ 4ðbþ2cÞ . If there is a merger between M and R1, M sets (w1,F1) = (0,0) and charges a wholesale price w2 N w and a fixed fee F2 that extracts all of the profits from R2. The profits of R1, from selling varieties U and W, are Π1 ¼
i ct h 2 ck− x þ ð1−xÞ2 2 x ðxþy 2 Þ þ ∫ 0 b½t ð y−xÞdz þ ∫ x b½t ðx þ y−2zÞdz
ð4Þ
263
Solving the first order conditions for profit-maximization, we obtain ðx; yÞ ¼
! 16c2 þ 4b2 þ 16bc −b2 w2 16c2 þ 12b2 þ 32bc −bw2 ð3b þ 4cÞ ð6Þ ; 16ðb2 þ 2c2 þ 3bcÞ 16ðb2 þ 2c2 þ 3bcÞ
This leads to our next proposition. Proposition II. When one of the downstream firms merges upstream, the Nash equilibrium locations downstream firms, in a h of the two i b2 w2 1 − R þ 16ðbþc for the integrated firm vertically Þðbþ2cÞ h related 2industry, are 2i b w2 2 for the un-integrated firm. and 12 þ R þ 16ðbþc þ 4ðbw Þðbþ2cÞ bþ2cÞ It may be noted that, irrespective of the vertical structure (integrated or not) of the market, the downstream firms gravitate toward the center if b = 0 (i.e. when, in the absence of any demand for W, the downstream firms are reduced to spatial price-discriminating monopolists choosing uniform delivered prices). This replicates the equilibrium one would expect, á la Hotelling (1929), in the market for a homogeneous product where the (mill) prices of the two firms are exogenous and equal, and consumers pay travel costs. Finally, if the upstream monopoly is broken up then each manufacturer of the intermediate good can only make zero profits on its contract notwithstanding the vertical structure: (wij,Fij) = (0,0) where wij is the marginal wholesale price and Fij is the fixed fee extracted by Mi from Rj. In effect, the equilibrium locations of the downstream firms will replicate, á la Braid (2007), the socially optimal outcome: 12 FR . The intuition behind this result lies in the classic Bertrand model: essentially, each upstream firm can always slightly undercut any contract which allows its rival to make a positive profit since upstream firms produce a homogeneous intermediate good and compete in (nonlinear) prices. 3. Conclusion This paper is a natural follow up of continued efforts to capture the consequences of vertical mergers. We view our analysis as one step forward along this road. We show that in a vertically related industry with an upstream monopoly, the downstream firms will not choose the socially optimal location under spatial price discrimination when none of the downstream firms produce all the varieties that consumers demand. A vertical merger will cause the distance of the downstream firms from the socially optimal location to increase. In particular, the firm outside the merger will move farther away from the socially optimal location than will the firm that is part of the merger. A couple of interesting extensions, we are currently working on, include a) integrating a dynamic version our model with Nocke and White (2007)7 to assess the impact of vertical mergers on firms' ability to collude and b) merging an open-economy version of our model with Neary (2007)8 to analyze the role that cross-border mergers play in comparative advantage, under conditions of spatial price discrimination in an industry where a retailer cannot deliver all the varieties that the market demands. We hope our efforts will open up areas of potential cross-fertilization for future work. References
ð5Þ
Braid, R.M., 2008. Spatial price discrimination and the locations of firms with different product selections or product varieties. Economics Letters 98, 342–347. Dasgupta, P., Maskin, E., 1986. The existence of equilibrium in discontinuous economic games, II: Applications. Review of Economic Studies 53, 27–41. Hoover, E.M., 1937. Spatial price discrimination. Review of Economic Studies 4, 182–191.
6 This allows us room for analyzing the impact of a vertical merger on an existing firm that is not part of the merger.
7 Nocke and White (2007) are among the first to have investigated the impact of vertical mergers in a dynamic game of repeated interaction between upstream and downstream firms. 8 Neary (2007) has constructed the first analytically tractable general equilibrium model of cross-border mergers.
The profits of R2, from selling varieties V and W, are Π2 ¼
i ct h 2 cðk−w2 Þ− y þ ð1−yÞ2 2 y 1 þ ∫ ðxþyÞ b½t ðx þ y−2zÞ−w2 dz þ ∫ y b½t ðx−yÞ−w2 dz −F2 2
264
H. Beladi et al. / Economics Letters 101 (2008) 262–264
Hotelling, H., 1929. Stability in competition. Economic Journal 39, 41–57. Lafontaine, F., Slade, M., 2007. Vertical integration and firm boundaries: the evidence Lafontaine, Francine. Journal of Economic Literature 45, 629–685. Lerner, A., Singer, H., 1937. Some notes on duopoly and spatial competition. Journal of Political Economy 45, 145–186.
Neary, P., 2007. Cross-border mergers as instruments of comparative advantage. Review of Economic Studies 4, 182–191. Nocke, V., White, L., 2007. Do vertical mergers facilitate upstream collusion? American Economic Review 97, 1321–1339.
Contents lists available at ScienceDirect
Economics Letters j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o n b a s e
Vertical mergers and downstream spatial competition with different product varieties Hamid Beladi a,⁎, Avik Chakrabarti b,1, Sugata Marjit c,2 a b c
Department of Economics, College of Business, University of Texas at San Antonio, One UTSA Circle, San Antonio, Texas 78249-0633, United States Department of Economics, 816 Bolton Hall, College of Letters and Science, University of Wisconsin-Milwaukee, P.O. Box 413, Wisconsin 53201, United States Centre for Studies in Social Sciences, Calcutta 700-094, West Bengal, India
a r t i c l e
i n f o
Article history: Received 7 January 2008 Received in revised form 8 August 2008 Accepted 19 August 2008 Available online 24 August 2008
a b s t r a c t We show how, in an industry where no downstream firm can produce all varieties demanded, a vertical merger with a monopoly upstream will induce each downstream firm (inside and out of the merger) to deviate from the socially optimal location. © 2008 Elsevier B.V. All rights reserved.
Keywords: Product-differentiation Price-discrimination Spatial-competition Firm-location Merger JEL classification: L13 L42 D43 R32
1. Introduction
2. Model and propositions
Vertical merger activities have been drawing increasing attention of regulators, anti-trust authorities as well as those in the media and academics.3 In this paper, we extend Braid (2007)4 to demonstrate the effect of vertical mergers on the equilibrium locations of firms which sell different varieties of a product. In a vertically related industry, with an upstream monopoly, the downstream firms will not choose the socially optimal location. A vertical merger will exacerbate the distance of the downstream firms from the socially optimal location: the firm outside the merger will deviate more than the firm that is part of the merger.
Consider a vertically related industry with one upstream firm (M) producing an intermediate good and selling this good to 2 downstream firms (Rj:i = 1,2) who transform (on a one-to-one basis) the intermediate good into the differentiated final goods and sell to consumers distributed uniformly with unit density on a uni-dimensional (linear) market interval with support [0, 1]. The downstream firms R1 and R2 are located at x and y, respectively, on this interval. R1 sells products U and W, and R2 sells products V and W: a fraction c of consumers want to buy product U; a fraction c of consumers want to buy product V; and a fraction b of consumers want to buy product W.5 Transportation costs are td, where t
⁎ Corresponding author. Tel.: +1 210 458 7038; fax: +1 210 458 7040. E-mail addresses: [email protected] (H. Beladi), [email protected] (A. Chakrabarti), [email protected] (S. Marjit). 1 Tel.: +1 414 229 4680; fax: +1 414 229 3860. 2 Tel.: +91 852 2788 7745; fax: +91 852 2788 8806. 3 See Lafontaine and Slade (2007) for an insightful review of the contributions relevant to the related literature. 4 Braid (2007) showed that the equilibrium locations of two firms are partially centralized to the socially optimal extent if there is spatial price discrimination, and if each firm has two out of three products, or else one variety of a differentiated product with some consumers indifferent between varieties.
0165-1765/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2008.08.024
5 One can contemplate, following Braid (2007), an equivalent scenario where firm 1 sells product variety G, and firm 2 sells product variety H: a fraction c of consumers wants to buy only variety G; a fraction c of consumers wants to buy only variety H; and a fraction b of consumers is indifferent between varieties G and H. If each firm cannot price discriminate at each location between the different types of consumers who find its own variety desirable, then it might be possible to assume mixed price strategies, but unlike Dasgupta and Maskin (1986), who have a single mixed-strategy Nash equilibrium in mill prices for any given set of firm locations, there would be a different mixed-strategy Nash equilibrium in delivered prices at each consumer location for any given set of firm locations. We maintain that there is spatial price discrimination for product W of the sort first examined in Hoover (1937) and Lerner and Singer (1937), in which there is a Nash equilibrium in delivered price schedules.
H. Beladi et al. / Economics Letters 101 (2008) 262–264
is a constant and d is distance shipped. There is a maximum reservation price, denoted by k, that consumers are willing to pay which is sufficiently large so that it comes into play only for product varieties where there is no competition between the two firms. The game is played in three stages, with perfect monitoring i.e. all past actions become common knowledge at the end of each stage. In the first stage, M makes a take-it-or-leave-it two-part tariff offer to R1 and R2: M's offer takes the form (wj,Fj) where wj is the marginal wholesale price and Fj is the fixed fee extracted by M from Rj. At the same time, R1 and R2 simultaneously choose their location in the retail market. In the second stage, R1 and R2 simultaneously decide whether or not to accept or decline a contract. If Rj decides to accept a contract offered, the fixed fee (Fj) is collected by M at this stage. In the third stage, R1 and R2 engage in spatial price discrimination. In the final stage, quantities are demanded by consumers from the upstream firms at which time the wholesale price has to be paid for each unit that is ordered and then sold in the retail market to consumers. As usual, the game is solved by backward induction. We can then make a comparison between the un-integrated and integrated (where one6 of the downstream firms is merged upstream) equilibria with b . spatial price discrimination. For expositional convenience, let R ¼ 4ðbþc Þ Absent the possibility of a merger, M charges a uniform wholesale price w and a fixed fee F that extracts all of the profits from each downstream firm. The profits of R1, from selling varieties U and W, are Π1 ¼
i ct h 2 cðk−wÞ− x þ ð1−xÞ2 2 x ðxþy 2 Þ þ ∫ 0 b½t ð y−xÞ−wdz þ ∫ x b½t ðx þ y−2zÞ−wdz −F
ð1Þ
The profits of R2, from selling varieties V and W, are Π2 ¼
i ct h 2 cðk−wÞ− y þ ð1−yÞ2 2 y 1 þ ∫ ðxþyÞ b½t ðx þ y−2zÞ−wdz þ ∫ y b½t ðx−yÞ−wdz −F
ð2Þ
2
Solving the first order conditions for profit-maximization, we obtain ðx; yÞ ¼
! 2 4c þ b2 þ 4bc −bwðb þ cÞ 4c2 þ 3b2 þ 8bc −bwðb þ cÞ ; 4ðb2 þ 2c2 þ 3bcÞ 4ðb2 þ 2c2 þ 3bcÞ ð3Þ
This leads to our first proposition. Proposition I. Absent a merger, the Nash equilibriumh locations of the i 1 bw two downstream firms, in a vertically related industry, are 2 F R þ 4ðbþ2cÞ . If there is a merger between M and R1, M sets (w1,F1) = (0,0) and charges a wholesale price w2 N w and a fixed fee F2 that extracts all of the profits from R2. The profits of R1, from selling varieties U and W, are Π1 ¼
i ct h 2 ck− x þ ð1−xÞ2 2 x ðxþy 2 Þ þ ∫ 0 b½t ð y−xÞdz þ ∫ x b½t ðx þ y−2zÞdz
ð4Þ
263
Solving the first order conditions for profit-maximization, we obtain ðx; yÞ ¼
! 16c2 þ 4b2 þ 16bc −b2 w2 16c2 þ 12b2 þ 32bc −bw2 ð3b þ 4cÞ ð6Þ ; 16ðb2 þ 2c2 þ 3bcÞ 16ðb2 þ 2c2 þ 3bcÞ
This leads to our next proposition. Proposition II. When one of the downstream firms merges upstream, the Nash equilibrium locations downstream firms, in a h of the two i b2 w2 1 − R þ 16ðbþc for the integrated firm vertically Þðbþ2cÞ h related 2industry, are 2i b w2 2 for the un-integrated firm. and 12 þ R þ 16ðbþc þ 4ðbw Þðbþ2cÞ bþ2cÞ It may be noted that, irrespective of the vertical structure (integrated or not) of the market, the downstream firms gravitate toward the center if b = 0 (i.e. when, in the absence of any demand for W, the downstream firms are reduced to spatial price-discriminating monopolists choosing uniform delivered prices). This replicates the equilibrium one would expect, á la Hotelling (1929), in the market for a homogeneous product where the (mill) prices of the two firms are exogenous and equal, and consumers pay travel costs. Finally, if the upstream monopoly is broken up then each manufacturer of the intermediate good can only make zero profits on its contract notwithstanding the vertical structure: (wij,Fij) = (0,0) where wij is the marginal wholesale price and Fij is the fixed fee extracted by Mi from Rj. In effect, the equilibrium locations of the downstream firms will replicate, á la Braid (2007), the socially optimal outcome: 12 FR . The intuition behind this result lies in the classic Bertrand model: essentially, each upstream firm can always slightly undercut any contract which allows its rival to make a positive profit since upstream firms produce a homogeneous intermediate good and compete in (nonlinear) prices. 3. Conclusion This paper is a natural follow up of continued efforts to capture the consequences of vertical mergers. We view our analysis as one step forward along this road. We show that in a vertically related industry with an upstream monopoly, the downstream firms will not choose the socially optimal location under spatial price discrimination when none of the downstream firms produce all the varieties that consumers demand. A vertical merger will cause the distance of the downstream firms from the socially optimal location to increase. In particular, the firm outside the merger will move farther away from the socially optimal location than will the firm that is part of the merger. A couple of interesting extensions, we are currently working on, include a) integrating a dynamic version our model with Nocke and White (2007)7 to assess the impact of vertical mergers on firms' ability to collude and b) merging an open-economy version of our model with Neary (2007)8 to analyze the role that cross-border mergers play in comparative advantage, under conditions of spatial price discrimination in an industry where a retailer cannot deliver all the varieties that the market demands. We hope our efforts will open up areas of potential cross-fertilization for future work. References
ð5Þ
Braid, R.M., 2008. Spatial price discrimination and the locations of firms with different product selections or product varieties. Economics Letters 98, 342–347. Dasgupta, P., Maskin, E., 1986. The existence of equilibrium in discontinuous economic games, II: Applications. Review of Economic Studies 53, 27–41. Hoover, E.M., 1937. Spatial price discrimination. Review of Economic Studies 4, 182–191.
6 This allows us room for analyzing the impact of a vertical merger on an existing firm that is not part of the merger.
7 Nocke and White (2007) are among the first to have investigated the impact of vertical mergers in a dynamic game of repeated interaction between upstream and downstream firms. 8 Neary (2007) has constructed the first analytically tractable general equilibrium model of cross-border mergers.
The profits of R2, from selling varieties V and W, are Π2 ¼
i ct h 2 cðk−w2 Þ− y þ ð1−yÞ2 2 y 1 þ ∫ ðxþyÞ b½t ðx þ y−2zÞ−w2 dz þ ∫ y b½t ðx−yÞ−w2 dz −F2 2
264
H. Beladi et al. / Economics Letters 101 (2008) 262–264
Hotelling, H., 1929. Stability in competition. Economic Journal 39, 41–57. Lafontaine, F., Slade, M., 2007. Vertical integration and firm boundaries: the evidence Lafontaine, Francine. Journal of Economic Literature 45, 629–685. Lerner, A., Singer, H., 1937. Some notes on duopoly and spatial competition. Journal of Political Economy 45, 145–186.
Neary, P., 2007. Cross-border mergers as instruments of comparative advantage. Review of Economic Studies 4, 182–191. Nocke, V., White, L., 2007. Do vertical mergers facilitate upstream collusion? American Economic Review 97, 1321–1339.