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Cournot competition and spatial agglomeration revisited
Economics Letters 73 (2001) 175–177 www.elsevier.com / locate / econbase
Cournot competition and spatial agglomeration revisited Noriaki Matsushima* Faculty of Economics, Shinshu University, Asahi 3 -1 -1, Matsumoto, Nagano 390 -8621, Japan Received 16 October 2000; accepted 30 March 2001
Abstract In a letter previously published in this journal, Pal [Economics Letters 60(1998) 49] show that firms locate equidistant from each other in a circular city. However, we show that the following location pattern is also an equilibrium outcome: firms agglomerate at two points in a circular city. 2001 Elsevier Science B.V. All rights reserved. Keywords: Cournot; Spatial Competition; Agglomeration JEL classification: D43; L13
1. Introduction Hamilton et al. (1989) and Anderson and Neven (1991) analyze the two stage problem of location and quantity choice in Hotelling (1929)’s linear city model. In sharp contrast to the dispersion result under Bertrand competition, both papers conclude that under Cournot competition the firms agglomerate at the center of the market. Pal (1998) analyzes the two-stage problem of location and quantity choice in Salop (1979)’s Circular city model. In sharp contrast to the results obtained in a linear city model, he shows that in a circular city model, both Bertrand and Cournot competition yield identical locations in equilibrium: the firms locate equidistant from one another on the circle. We also analyze the two-stage problem of location and quantity choice in Salop (1979)’s Circular city model. We show that half of firms agglomerate at a point and the rest of the firms agglomerate at the side opposite of the point.
* Tel.: 181-263-37-2951; fax: 181-263-37-2344. E-mail address: [email protected] (N. Matsushima). 0165-1765 / 01 / $ – see front matter PII: S0165-1765( 01 )00481-5
2001 Elsevier Science B.V. All rights reserved.
N. Matsushima / Economics Letters 73 (2001) 175 – 177
176
2. The basic model The two-stage location–quantity game is modeled as follows. There are infinitely many consumers located uniformly on a circle with perimeter equal to 1. There are n private firms who produce and sell a homogenous output to the consumers. The firms have identical technology and constant marginal cost of production (zero without loss of generality). In the first stage of the game, the firms simultaneously choose locations on the circle. Let x i (i [ h1, 2, . . . , nj) be the locations of firm i (i [ h1, 2 . . . ,nj), respectively. Let p(x) denote the price of the product and Q(x) denote the total quantity supplied at x, where x is the point on the circle located at a distance from 0 (measured clockwise). Assume that the demand function at each point x is linear and is given by: p 5 a 2 bQ(x), where a . 0 and b . 0 are constants. The firms transport the good from the plant to the consumers. The good can be transported only along the circle. Each firm pays a transport cost tux 2 x i u (i [ h1 . . . ,nj), where t is constant and u u is the distance between x and x i to ship a unit of the product from its own location to a consumer point x. In the second stage of the game, each firm observes its competitors’ locations and simultaneously chooses its output to be offered at each point x so as to maximize its profit. To ensure that all firms always serve the whole market, we assume that a . nt /2.
3. Analysis We follow the Cournot assumption that firms compete in quantities at each point in the market. Since marginal production costs are constant, quantities set at different points by the same firm are strategically independent. Cournot equilibria can be characterized by a set of independent Cournot equilibria, one for each point x. Let p(i,x) denote firm i’s (i [ h1, . . . ,n) profit at x.
p (i, x) 5 (a 2 bQ(x) 2 tux 2 x i u)qi (x),
(1)
where qi (x) is firm i’s output offered for sale at x. From (1), the first-order condition of firm i is as follows.
O q (x) 2 2bq (x) 2 tux 2 x u 5 0. n
a2b
j
i
(2)
i
j ±i
The output of firm i at x and the total quantity at x are as follows.
O
n
a 1 t j51 ux 2 x j u 2 (n 1 1)tux 2 x i u qi (x) 5 ]]]]]]]]]]] (n 1 1)b
O
(3)
n
na 2 t i51 ux 2 x i u Q(x) 5 ]]]]]]. (n 1 1)b
(4)
The profit of firm i at x is
Sa 1 t O
n
D
2
ux 2 x j u 2 (n 1 1)tux 2 x i u p (i, x) 5 ]]]]]]]]]]]]] . (n 1 1)2 b j51
(5)
N. Matsushima / Economics Letters 73 (2001) 175 – 177
177
We now show an equilibrium which is different from Pal (1998)’s equidistant result. Consider the case where n / 2 firms locate at x50 and n / 2 firms locate at x51 / 2 (n is even). If a firm who locates at x50 does not have an incentive to locate at the other points (x±0), then the case is an equilibrium outcome. When the firm changes its location and locates at x i #1 / 2, its profit at x is 2
(a 1 (n / 2 2 1)tux 2 0u 1 (n / 2)tux 2 1 / 2u 2 ntux 2 x i u) p (i, x) 5 ]]]]]]]]]]]]]]]] (n 1 1)2 b
(6)
Total profit is
pi
5
E
(a 1 (n / 2 2 1)t(x 2 0) 1 (n / 2)t(1 / 2 2 x) 2 nt(x i 2 x))2 ]]]]]]]]]]]]]]]]] dx 0 (n 1 1)2 b 1/2 (a 1 (n / 2 2 1)t(x 2 0) 1 (n / 2)t(1 / 2 2 x) 2 nt(x 2 x i ))2 ]]]]]]]]]]]]]]]]] 1 dx xi (n 1 1)2 b x i 11 / 2 (a 1 (n / 2 2 1)t(1 2 x) 1 (n / 2)t(x 2 1 / 2) 2 nt(x 2 x i ))2 ]]]]]]]]]]]]]]]]] 1 dx 1/2 (n 1 1)2 b 1 (a 1 (n / 2 2 1)t(1 2 x) 1 (n / 2)t(x 2 1 / 2) 2 nt(1 2 x 1 x i ))2 ]]]]]]]]]]]]]]]]]] 1 dx x i 11 / 2 (n 1 1)2 b 48a 2 2 24at 1 (4 1 2n 1 n 2 )t 2 2 48nt 2 x i2 1 64nt 2 x i3 ]]]]]]]]]]]]]]]] . 48(n 1 1)2 b xi
E E E
5
The first-order condition and the second-order condition are as follows: 2nt 2 x i (1 2 2x i ) ≠ 2 p (i, x) 2nt 2 (1 2 4x i ) ≠p (i, x) ]]] 5 2 ]]]]] , ]]] 5 2 ]]]] . 2 2 2 ≠x i (1 1 n) b ≠x i (1 1 n) b ≠p (i, x) / ≠x i 50, at x i 50 and at x i 51 / 2. For 0,x i ,1 / 2, ≠p (i, x) / ≠x i ,0. Also, ≠ 2 p (i, x) / ≠x i2 , 0, for x i ,1 / 4 and ≠ 2 p (i, x) / ≠x 2i . 0, for x i .1 / 4. Therefore, the profit is maximized at x i 50. We find that the following case is an equilibrium outcome: n / 2 firms locate at x50 and n / 2 firms locate at x51 / 2 (n is even). This is a counter example to Pal (1998)’s equidistant result.
References Anderson, S.P., Neven, D.J., 1991. Cournot competition yields spatial agglomeration. International Economic Review 32, 793–808. Hamilton, J.H., Thisse, J.-F., Weskamp, A., 1989. Spatial discrimination: Bertrand vs. Cournot in a model of location choice. Regional Science and Urban Economics 19, 87–102. Hotelling, H., 1929. Stability in competition. Economic Journal 39, 41–57. Pal, D., 1998. Does Cournot competition yield spatial agglomeration? Economics Letters 60, 49–53. Salop, S.C., 1979. Monopolistic competition with outside goods. Bell Journal of Economics 10, 141–156.
Cournot competition and spatial agglomeration revisited Noriaki Matsushima* Faculty of Economics, Shinshu University, Asahi 3 -1 -1, Matsumoto, Nagano 390 -8621, Japan Received 16 October 2000; accepted 30 March 2001
Abstract In a letter previously published in this journal, Pal [Economics Letters 60(1998) 49] show that firms locate equidistant from each other in a circular city. However, we show that the following location pattern is also an equilibrium outcome: firms agglomerate at two points in a circular city. 2001 Elsevier Science B.V. All rights reserved. Keywords: Cournot; Spatial Competition; Agglomeration JEL classification: D43; L13
1. Introduction Hamilton et al. (1989) and Anderson and Neven (1991) analyze the two stage problem of location and quantity choice in Hotelling (1929)’s linear city model. In sharp contrast to the dispersion result under Bertrand competition, both papers conclude that under Cournot competition the firms agglomerate at the center of the market. Pal (1998) analyzes the two-stage problem of location and quantity choice in Salop (1979)’s Circular city model. In sharp contrast to the results obtained in a linear city model, he shows that in a circular city model, both Bertrand and Cournot competition yield identical locations in equilibrium: the firms locate equidistant from one another on the circle. We also analyze the two-stage problem of location and quantity choice in Salop (1979)’s Circular city model. We show that half of firms agglomerate at a point and the rest of the firms agglomerate at the side opposite of the point.
* Tel.: 181-263-37-2951; fax: 181-263-37-2344. E-mail address: [email protected] (N. Matsushima). 0165-1765 / 01 / $ – see front matter PII: S0165-1765( 01 )00481-5
2001 Elsevier Science B.V. All rights reserved.
N. Matsushima / Economics Letters 73 (2001) 175 – 177
176
2. The basic model The two-stage location–quantity game is modeled as follows. There are infinitely many consumers located uniformly on a circle with perimeter equal to 1. There are n private firms who produce and sell a homogenous output to the consumers. The firms have identical technology and constant marginal cost of production (zero without loss of generality). In the first stage of the game, the firms simultaneously choose locations on the circle. Let x i (i [ h1, 2, . . . , nj) be the locations of firm i (i [ h1, 2 . . . ,nj), respectively. Let p(x) denote the price of the product and Q(x) denote the total quantity supplied at x, where x is the point on the circle located at a distance from 0 (measured clockwise). Assume that the demand function at each point x is linear and is given by: p 5 a 2 bQ(x), where a . 0 and b . 0 are constants. The firms transport the good from the plant to the consumers. The good can be transported only along the circle. Each firm pays a transport cost tux 2 x i u (i [ h1 . . . ,nj), where t is constant and u u is the distance between x and x i to ship a unit of the product from its own location to a consumer point x. In the second stage of the game, each firm observes its competitors’ locations and simultaneously chooses its output to be offered at each point x so as to maximize its profit. To ensure that all firms always serve the whole market, we assume that a . nt /2.
3. Analysis We follow the Cournot assumption that firms compete in quantities at each point in the market. Since marginal production costs are constant, quantities set at different points by the same firm are strategically independent. Cournot equilibria can be characterized by a set of independent Cournot equilibria, one for each point x. Let p(i,x) denote firm i’s (i [ h1, . . . ,n) profit at x.
p (i, x) 5 (a 2 bQ(x) 2 tux 2 x i u)qi (x),
(1)
where qi (x) is firm i’s output offered for sale at x. From (1), the first-order condition of firm i is as follows.
O q (x) 2 2bq (x) 2 tux 2 x u 5 0. n
a2b
j
i
(2)
i
j ±i
The output of firm i at x and the total quantity at x are as follows.
O
n
a 1 t j51 ux 2 x j u 2 (n 1 1)tux 2 x i u qi (x) 5 ]]]]]]]]]]] (n 1 1)b
O
(3)
n
na 2 t i51 ux 2 x i u Q(x) 5 ]]]]]]. (n 1 1)b
(4)
The profit of firm i at x is
Sa 1 t O
n
D
2
ux 2 x j u 2 (n 1 1)tux 2 x i u p (i, x) 5 ]]]]]]]]]]]]] . (n 1 1)2 b j51
(5)
N. Matsushima / Economics Letters 73 (2001) 175 – 177
177
We now show an equilibrium which is different from Pal (1998)’s equidistant result. Consider the case where n / 2 firms locate at x50 and n / 2 firms locate at x51 / 2 (n is even). If a firm who locates at x50 does not have an incentive to locate at the other points (x±0), then the case is an equilibrium outcome. When the firm changes its location and locates at x i #1 / 2, its profit at x is 2
(a 1 (n / 2 2 1)tux 2 0u 1 (n / 2)tux 2 1 / 2u 2 ntux 2 x i u) p (i, x) 5 ]]]]]]]]]]]]]]]] (n 1 1)2 b
(6)
Total profit is
pi
5
E
(a 1 (n / 2 2 1)t(x 2 0) 1 (n / 2)t(1 / 2 2 x) 2 nt(x i 2 x))2 ]]]]]]]]]]]]]]]]] dx 0 (n 1 1)2 b 1/2 (a 1 (n / 2 2 1)t(x 2 0) 1 (n / 2)t(1 / 2 2 x) 2 nt(x 2 x i ))2 ]]]]]]]]]]]]]]]]] 1 dx xi (n 1 1)2 b x i 11 / 2 (a 1 (n / 2 2 1)t(1 2 x) 1 (n / 2)t(x 2 1 / 2) 2 nt(x 2 x i ))2 ]]]]]]]]]]]]]]]]] 1 dx 1/2 (n 1 1)2 b 1 (a 1 (n / 2 2 1)t(1 2 x) 1 (n / 2)t(x 2 1 / 2) 2 nt(1 2 x 1 x i ))2 ]]]]]]]]]]]]]]]]]] 1 dx x i 11 / 2 (n 1 1)2 b 48a 2 2 24at 1 (4 1 2n 1 n 2 )t 2 2 48nt 2 x i2 1 64nt 2 x i3 ]]]]]]]]]]]]]]]] . 48(n 1 1)2 b xi
E E E
5
The first-order condition and the second-order condition are as follows: 2nt 2 x i (1 2 2x i ) ≠ 2 p (i, x) 2nt 2 (1 2 4x i ) ≠p (i, x) ]]] 5 2 ]]]]] , ]]] 5 2 ]]]] . 2 2 2 ≠x i (1 1 n) b ≠x i (1 1 n) b ≠p (i, x) / ≠x i 50, at x i 50 and at x i 51 / 2. For 0,x i ,1 / 2, ≠p (i, x) / ≠x i ,0. Also, ≠ 2 p (i, x) / ≠x i2 , 0, for x i ,1 / 4 and ≠ 2 p (i, x) / ≠x 2i . 0, for x i .1 / 4. Therefore, the profit is maximized at x i 50. We find that the following case is an equilibrium outcome: n / 2 firms locate at x50 and n / 2 firms locate at x51 / 2 (n is even). This is a counter example to Pal (1998)’s equidistant result.
References Anderson, S.P., Neven, D.J., 1991. Cournot competition yields spatial agglomeration. International Economic Review 32, 793–808. Hamilton, J.H., Thisse, J.-F., Weskamp, A., 1989. Spatial discrimination: Bertrand vs. Cournot in a model of location choice. Regional Science and Urban Economics 19, 87–102. Hotelling, H., 1929. Stability in competition. Economic Journal 39, 41–57. Pal, D., 1998. Does Cournot competition yield spatial agglomeration? Economics Letters 60, 49–53. Salop, S.C., 1979. Monopolistic competition with outside goods. Bell Journal of Economics 10, 141–156.