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Economic welfare in delivered pricing duopoly: Bertrand and Cournot
Economics Letters 89 (2005) 112 – 119 www.elsevier.com/locate/econbase
Economic welfare in delivered pricing duopoly: Bertrand and Cournot Toshihiro MatsumuraT, Daisuke Shimizu Institute of Social Science, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Received 19 August 2004; received in revised form 6 April 2005; accepted 16 May 2005
Abstract We investigate two delivered pricing models. We find that in the Bertrand (Cournot) model the equilibrium distance between two firms’ locations is too large (small) for maximum social welfare, while it is too small for maximum consumer surplus in both models. D 2005 Elsevier B.V. All rights reserved. Keywords: Spatial competition; Discriminatory pricing; Production substitution JEL classification: D43; R32
1. Introduction Since the seminal work of Hotelling (1929), a rich and diverse literature on spatial competition has emerged. Location models fall into two categories: shipping (or delivered pricing) models are those in which firms bear transport costs; shopping (or mill pricing) models are those in which consumers pay for transport. Each type allows for either Bertrand-type price-setting or Cournot-type quantity-setting. Lederer and Hurter (1986) and Hamilton et al. (1989) carried out pioneering works on delivered pricing models with Bertrand and Cournot competition, respectively. Hamilton et al. (1989) compared the equilibrium locations between two types of competition. Recently, the T Corresponding author. Tel.: +81 3 5841 4932; fax: +81 3 5841 4905. E-mail address: [email protected] (T. Matsumura). 0165-1765/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2005.05.019
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113
literature on location-quantity models has become richer and more diverse1, but few papers have investigated welfare implications. In this paper, we investigate welfare implications in shipping models using a general distribution of consumers. We analyzed two models: Bertrand and Cournot. We found that increasing the distance between firms raises producer surplus and reduces consumer surplus in both models. However, these two models yield opposite welfare implications. In the Bertrand model the equilibrium distance between the locations of two firms is too large, while in the Cournot it is too small for maximal social welfare.
2. Bertrand model We first present a two-stage location-price duopoly model in a linear city. The space is the interval [0,1]. Let f(x)(N 0) be a function denoting the density of consumers on [0,1]. The demand at point x a [0,1] is given by q(x) = (a p(x))f (x), where p(x) is the price of the homogeneous product at market x, q(x) is the total quantity supplied by the firms at x, and a is a positive constant. In the first stage, each firm i independently chooses its location x i (i = 1,2). Without loss of generality, we assume that x 1 V x 2. Each firm produces at a constant marginal cost, which is normalized to zero. To ship a unit of the product from its own location to a consumer at point x, each firm i pays transport cost, t i (|x x i |), where t i is increasing, differentiable, and t i (0) = 0. Let p i (x) be the price offered by firm i at each point x a [0,1]. In the second stage, after observing its competitor’s location, each firm i simultaneously chooses p i (x) a [0,l) for x a [0,1]. We assume that a N 2t i (1). This assumption ensures that the second-stage equilibrium price is lower than the monopoly price. Firms are able to discriminate among consumers since they control transportation. Consumer arbitrage is assumed to be prohibitively expensive. We analyze first the second-stage Bertrand competition, given x 1 and x 2(zx 1). Since the marginal costs are constant, prices set at different points by the same firm are strategically independent. The second-stage Bertrand equilibrium of this game can then be characterized by a set of independent Bertrand equilibria, one for each point x a [0,1]. As is well known, under the homogeneous goods Bertrand competition, the firm with a lower cost obtains the whole share of the market and the equilibrium price is equal to the rival’s cost. Note that there is a point x¯ a [x 1, x 2] such that t 1(x¯ x 1) = t 2(x 2 x¯ ), and firm 1 serves all markets x V x¯ and firm 2 serves markets x z x¯ . Thus, firm 1’s profit P 1B(x 1, x 2) and firm 2’s profit P 2B(x 2, x 1) are given respectively, Z x¯ Z x¯ B B P1 ðx1 ; x2 Þ ¼ p1 ð x; x1 ; x2 Þdx ¼ ðt2 ðx2 xÞ t1 ðjx1 xjÞÞða t2 ðx2 xÞÞf ð xÞdx 0
PB2 ðx2 ; x1 Þ ¼
Z
x¯
0
1
pB2 ð x; x2 ; x1 Þdx ¼
Z
1
ðt1 ð x x1 Þ t2 ðjx2 xjÞÞða t1 ð x x1 ÞÞf ð xÞdx;
ð1Þ
x¯
where pBi (x; x i , x j ) is firm i’s local profit at market x. In the first stage, each firm i chooses its location x i so as to maximize its profit PBi (x i , x j ), which is given above. Let (x 1B, x 2B) be the pair of the equilibrium locations in the Bertrand model. If x 1 = x 2, each 1
For the recent discussions, see Pal (1998), Matsushima (2001), and Shimizu (2002).
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firm’s profit is zero, and given the rival’s location each firm can increase its profit by any relocation. Thus, we have x 1B p x 2B. It is also well known that an interior solution is achieved in this setting, i.e., 0 b x 1B b x 2B b 1.
3. Welfare We now discuss the welfare implication of the locations. Let csB(x; x 1, x 2) be the consumer surplus at x and CSB(x 1, x 2) be the total consumer surplus. Routine manipulation yields
f ð xÞða t2 ðx2 xÞÞ2 =2 if x V x¯ f ð xÞða t1 ð x x1 ÞÞ2 =2 if x N x¯ 2 3 Z 1 Z 1 Z x¯ 1 CSB ðx1 ; x2 Þu csB ð x; x1 ; x2 Þdx ¼ 4 f ð xÞða t2 ðx2 xÞÞ2 dx þ f ð xÞða t1 ð x x1 ÞÞ2 dx5: 2 0 0 x¯ ð2Þ B
cs ð x; x1 ; x2 Þ ¼
Let P B(x 1, x 2) u P 1B + P 2B be the two firm’s joint profits. Let W B(x 1,x 2) u CSB + P B be the total social surplus. We now present our main results. Proposition 1. ðiÞ
BPB Bx1
j
ðx1 ;x2 Þ¼ðxB ;xB Þ b 0; 1
Proof. See Appendix A.
2
ðiiÞ
BCS B Bx1
j
ðx1 ;x2 Þ¼ðxB1 ;xB2 Þ
N 0;
and
ðiiiÞ
BW B Bx1
j
ðx1 ;x2 Þ¼ðxB1 ;xB2 Þ
N 0:
5
Proposition 1 implies that a slight decrease in the distance between the two firms’ locations from their equilibrium locations increases consumer surplus, reduces the two firm’s joint profits, and increases the total social surplus. This result indicates that the distance between the two firms’ locations is too large from the viewpoint of maximum welfare. We offer an intuitive explanation on why an increase in x 1 from x 1B improves welfare. A slight increase in x 1 does not affect firm 1’s profit because firm 1 chooses x 1 so as to maximize its profit (the envelope theorem). In the markets served by firm 1, the prices depend on firm 2’s cost, which is independent of x 1. Thus, the change in x 1 does not affect consumer surplus in these markets. In the markets served by firm 2, an increase in x 1 decreases the prices. It increases consumer surplus and reduces firm 2’s profit. Since the price is higher than firm 2’s marginal cost, a reduction in the difference between price and marginal cost improves welfare. An increase in x 1 accelerates competition. Some readers might think that a more aggressive competition obviously improves both consumer surplus and welfare. In the next section we present a result of the Cournot counterpart as a benchmark, where a more aggressive competition improves consumer surplus but does not improve welfare.
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4. Cournot model In this section we consider a Cournot model. The basic structure is mostly the same as the Bertrand counterpart. We consider a linear transport cost, t|x x i |, where t is a positive constant. Let q i (x) be the output offered by firm i at each point x a [0,1]. In the second stage, after observing its competitor’s location, each firm i simultaneously chooses q i (x) a [0,l) for x a [0,1]: Hence, we have q(x) u q 1(x) + q 2(x). We assume that a N 2t. This assumption ensures that the whole market is served by both firms. We analyze Cournot competition. Firms compete using quantities at each point. Since the marginal costs are constant, quantities set at different points by the same firm are strategically independent. The second-stage Cournot equilibrium of this game can then be characterized by a set of independent Cournot equilibria, one for each point x a [0,1]. The profit earned at each point x by firm i is given by p i (x) = f (x)(a q(x) / f (x) t|x x i |)q i (x). Routine manipulation yields the following Cournot equilibrium at each x a [0,1]: qCi ð xÞ ¼ f ð xÞ
a 2tjx xi j þ tjx xj j ; 3
qC ð x Þ ¼ f ð x Þ
2a tjx x1 j tjx x2 j ; 3
ð3Þ
where the superscript C signifies Cournot equilibrium. The resulting profit at any point x a [0,1] is given by 2 f ð xÞ pCi ð xÞ ¼ a 2tjx xi j þ tjx xj j ; 9
i; j ¼ 1; 2:
ð4Þ
Let PCi (x 1, x 2) be the total profit of firm i in the Cournot game. Let csC(x; x 1, x 2) be the consumer surplus at x. Routine manipulation yields csC(x; x 1, x 2) = f(x)(2a t|x x 1| t|x x 2|)2 / 18. Let CSC(x 1,x 2) be the total consumer surplus. This is given by "Z # Z 1 1 1 2 C CS ðx1 ; x2 Þ u csð x; x1 ; x2 Þdx ¼ f ð xÞð2a tjx x1 j tjx x2 jÞ dx : ð5Þ 18 0
0
Let P C(x 1, x 2) u P 1C + P 2C be the two firms’ joint profits and W C(x 1, x 2) u CSC + P C be the total social surplus. Proposition 2. ðiÞ
BPC Bx1
j
ðx1 ; x2 Þ¼ðxC1 ; xC2 Þ V 0;
ðiiÞ
BCS C Bx1
j
BW C z 0; and ð iii Þ ðx1 ; x2 Þ¼ðxC1 ; xC2 Þ Bx1
j
ðx1 ; x2 Þ¼ðxC1 ; xC2 Þ
V 0;
and (iv) the equalities are satisfied if and only if x 1C = x 2C .2 Proof. See Appendix A.
5
Proposition 2 implies that a slight decrease in the distance between the two firms’ locations from their equilibrium locations increases consumer surplus and reduces social welfare and the two firms’ joint If the consumer distribution is uniform (i.e., f(x) = 1 for all x a [0,1]), x 1C = x 2C. Otherwise, x 1C p x 2C may hold. See Gupta et al. (1997). For the welfare implications of agglomeration in the Cournot model, see Matsumura and Shimizu (in press). 2
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profits. Proposition 2(i) and (ii) have the same inequality signs as Proposition 1(i) and (ii), respectively. However, unlike in the Bertrand counterpart, in the Cournot model the equilibrium distance between two firms is too small from the viewpoint of maximum social welfare. Suppose that x 1C b x 2C. We offer an intuitive explanation on why an increase in x 1 from x 1C reduces welfare in the Cournot game and on why Cournot and Bertrand results yield the contrasting welfare implications. The underlying factor is strategic substitutability vs. strategic complementarity. A slight increase in x 1 decreases firm 1’s costs for markets close to 1 (the right edge of the linear city). It affects firm 2’s output for these markets through strategic interaction between the two firms. In the Cournot (Bertrand) model it reduces (increases) firm 2’s output because of the strategic substitutability (complementarity). In these markets firm 2’s transport cost is lower than firm 1’s, so the decrease (increase) in firm 2’s output results in the loss (gain) of social welfare.3 Firm 1 chooses its location without considering this effect, and it yields Propositions 1(iii) and 2(iii).
5. Concluding remarks In this paper, we have analyzed welfare implications in two delivered pricing duopoly models. We found that, in both Bertrand and Cournot models, firms’ private incentives to locate closely to each other are insufficient from the viewpoint of maximum consumer surplus, but are excessive from the viewpoint of maximum producer surplus. From the viewpoint of total social surplus, the two models yield quite different implications. In the Bertrand model a decrease in the distance between the two firms improves welfare, while the opposite is true in the Cournot model. Until recently, plant locations were severely regulated in Japan. Firms could not build new plants in Tokyo or Osaka. Regulations of plant location to avoid excessive concentration are widespread. Typical theoretical support for such regulations is based on technological externalities such as congestion effects. Our paper might provide another rationale for these regulations without assuming congestion effects in the Cournot model. However, this result depends on the model specification. In the Bertrand model the opposite result is derived, and in this circumstance the government should take this pro-concentration effect into account.
Acknowledgement We are grateful to participants in the seminars at Kyoto University, the University of Tokyo, and Japan Economic Association Annual Meeting 2003 for their helpful comments and suggestions. We are also indebted to an anonymous referee for his/her valuable and constructive suggestions. Needless to say, we are responsible for any remaining errors. Financial supports of the Grant-in-Aid from Zengin Foundation for Studies on Economics and Finance and from the Japanese Ministry of Education, Science and Culture, and of the ISS project bThe Lost Decade? Re-appraising Contemporary JapanQ are greatly appreciated.
3
For discussion of the similar production substitution effect in the Cournot models, see Lahiri and Ono (1988), Matsumura (1998, 2003) and Matsushima and Matsumura (2003).
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Appendix A Proof of Proposition 1. From the first-order condition for firm 1’s equilibrium location we have Z 1 B BPB BPB2 B Bp2 ð xÞ V t ¼ ¼ x x f ð xÞdx B B B B 1 1 Bpð xÞ Bx1 ðx1 ; x2 Þ¼ðx1 ; x2 Þ Bx1 ðx1 ; x2 Þ¼ðx1 ; x2 Þ x¯ Z 1 t1V x xB1 a 2t1 x xB1 þ t2 jx xB2 j f ð xÞb0: ¼
ð6Þ
x¯
Note that Bp2B(x) / Bp(x) N 0 since we assume that firm 2’s equilibrium price ( p(x) = t 1) must be smaller than its monopoly price ((a t 2) / 2). Z 1 Z 1 B BCS B B Bcs ð xÞ B B V V x x x x t t x x ¼ a t f ð xÞdxN0; f ð x Þdx ¼ B B 1 1 1 1 1 1 Bpð xÞ Bx1 ðx1 ; x2 Þ¼ðx1 ; x2 Þ x¯ x¯ Z 1 B B BW B B Bp2 ð xÞ þ cs ð xÞ V f ð xÞdx x x t ¼ B B 1 1 Bpð xÞ Bx1 ðx1 ; x2 Þ¼ðx1 ; x2 Þ x¯ Z 1 ¼ t1V x xB1 t1 x xB1 t2 jx xB2 j f ð xÞdxN0; ð7Þ x¯ 5
since t 1 N t 2 for x N x¯.
Proof of Proposition 2. From the first-order conditions for the two firm’s equilibrium locations we have " Z C Z xC x1 2 BPC1 4t C C a 2tx þ tx þ tx f ð x Þdx þ ða þ 2txC1 þ txC2 3txÞf ð xÞdx ¼ 1 2 ðx1 ; x2 Þ¼ðxC1 ; xC2 Þ C 9 Bx1 0 x1 # Z 1 a þ 2txC1 txC2 tx f ð xÞdx ¼ 0; ð8Þ þ xC2
" Z C x1 BPC2 4t a þ txC1 2txC2 þ tx f ð xÞdx ¼ C C ðx1 ; x2 Þ¼ðx1 ; x2 Þ 9 Bx2 0
Z
xC2
xC1
þ
Z
a txC1 2txC2 þ 3tx f ð xÞdx
1
xC2
a
txC1
þ
2txC2
# tx f ð xÞdx ¼ 0:
ð9Þ
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T. Matsumura, D. Shimizu / Economics Letters 89 (2005) 112–119
(i) "Z C x1 BPC BPC2 2t a þ txC1 2txC2 þ tx f ð xÞdx ¼ ¼ C C C C ðx1 ; x2 Þ¼ðx1 ; x2 Þ ðx1 ; x2 Þ¼ðx1 ; x2 Þ 9 Bx1 Bx1 0 # Z xC Z 1 2 C C C C a tx1 2tx2 þ 3tx f ð xÞdx ða tx1 þ 2tx2 txÞf ð xÞdx xC1
¼
2t 9
xC2
Z
xC2 xC1
2 a txC1 2txC2 þ 3tx f ð xÞdx;
ð10Þ
where the last equality is derived from (9). Note that (a tx 1C 2tx 2C + 3tx) N (a tx 1C 2tx 2C + 3tx 1C) = (a 2t(x 2C x 1C)) N 0 for all x a [x 1C,x 2C], where we use x 2C x 1C b 1 and a N 2t. Since (10) integrates over negative values throughout, if x 1C b x 2C then (10) is negative, and if x 1C = x 2C then (10) = 0. (ii) From (5), we obtain " Z C Z xC x1 2 BCS C t C C 2a tx1 tx2 þ 2tx f ð xÞdx þ 2a þ txC1 txC2 f ð xÞdx ðx1 ; x2 Þ¼ðxC1 ; xC2 Þ ¼ 9 Bx1 0 xC1 # Z 1 þ 2a þ txC1 þ txC2 2tx f ð xÞdx xC2
¼
t 9
Z
xC2 xC1
2 a txC1 2txC2 þ 3tx f ð xÞdx;
ð11Þ
where the last equality is derived using (8) and (9). The integrand is the same as in case (i) except that there is no negative sign, so this expression is nonnegative. (iii) Since W C(x 1, x 2) = P C(x 1, x 2) + CSC(x 1, x 2), from (10) and (11), we have Z C BW C 2t x2 a txC1 2txC2 þ 3tx f ð xÞdx; ðx1 ; x2 Þ¼ðxC1 ; xC2 Þ ¼ 9 xC1 Bx1 which is negative when x 1C b x 2C, and equal to 0 when x 1C = x 2C.
5
References Gupta, B., Pal, D., Sarkar, J., 1997. Spatial Cournot competition and agglomeration in a model of location choice. Regional Science and Urban Economics 27, 261 – 282.
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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. Lahiri, S., Ono, Y., 1988. Helping minor firms reduces welfare. Economic Journal 98, 1199 – 1202. Lederer, P.J., Hurter Jr., A.P., 1986. Competition of firms: discriminatory pricing and location. Econometrica 54 (3), 623 – 640. Matsumura, T., 1998. Partial privatization in mixed duopoly. Journal of Public Economics 70, 473 – 483. Matsumura, T., 2003. Consumer-benefiting exclusive territories. Canadian Journal of Economics 36, 1007 – 1025. Matsumura, T., Shimizu, D., in press. Spatial Cournot competition and economic welfare: a note, Regional Science and Urban Economics. Matsushima, N., 2001. Cournot competition and spatial agglomeration revisited. Economics Letters 73, 175 – 177. Matsushima, N., Matsumura, T., 2003. Mixed oligopoly and spatial agglomeration. Canadian Journal of Economics 36 (1), 62 – 87. Pal, D., 1998. Does Cournot competition yield spatial agglomeration? Economics Letters 60, 49 – 53. Shimizu, D., 2002. Product differentiation in spatial Cournot markets. Economics Letters 76, 317 – 322.
Economic welfare in delivered pricing duopoly: Bertrand and Cournot Toshihiro MatsumuraT, Daisuke Shimizu Institute of Social Science, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Received 19 August 2004; received in revised form 6 April 2005; accepted 16 May 2005
Abstract We investigate two delivered pricing models. We find that in the Bertrand (Cournot) model the equilibrium distance between two firms’ locations is too large (small) for maximum social welfare, while it is too small for maximum consumer surplus in both models. D 2005 Elsevier B.V. All rights reserved. Keywords: Spatial competition; Discriminatory pricing; Production substitution JEL classification: D43; R32
1. Introduction Since the seminal work of Hotelling (1929), a rich and diverse literature on spatial competition has emerged. Location models fall into two categories: shipping (or delivered pricing) models are those in which firms bear transport costs; shopping (or mill pricing) models are those in which consumers pay for transport. Each type allows for either Bertrand-type price-setting or Cournot-type quantity-setting. Lederer and Hurter (1986) and Hamilton et al. (1989) carried out pioneering works on delivered pricing models with Bertrand and Cournot competition, respectively. Hamilton et al. (1989) compared the equilibrium locations between two types of competition. Recently, the T Corresponding author. Tel.: +81 3 5841 4932; fax: +81 3 5841 4905. E-mail address: [email protected] (T. Matsumura). 0165-1765/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2005.05.019
T. Matsumura, D. Shimizu / Economics Letters 89 (2005) 112–119
113
literature on location-quantity models has become richer and more diverse1, but few papers have investigated welfare implications. In this paper, we investigate welfare implications in shipping models using a general distribution of consumers. We analyzed two models: Bertrand and Cournot. We found that increasing the distance between firms raises producer surplus and reduces consumer surplus in both models. However, these two models yield opposite welfare implications. In the Bertrand model the equilibrium distance between the locations of two firms is too large, while in the Cournot it is too small for maximal social welfare.
2. Bertrand model We first present a two-stage location-price duopoly model in a linear city. The space is the interval [0,1]. Let f(x)(N 0) be a function denoting the density of consumers on [0,1]. The demand at point x a [0,1] is given by q(x) = (a p(x))f (x), where p(x) is the price of the homogeneous product at market x, q(x) is the total quantity supplied by the firms at x, and a is a positive constant. In the first stage, each firm i independently chooses its location x i (i = 1,2). Without loss of generality, we assume that x 1 V x 2. Each firm produces at a constant marginal cost, which is normalized to zero. To ship a unit of the product from its own location to a consumer at point x, each firm i pays transport cost, t i (|x x i |), where t i is increasing, differentiable, and t i (0) = 0. Let p i (x) be the price offered by firm i at each point x a [0,1]. In the second stage, after observing its competitor’s location, each firm i simultaneously chooses p i (x) a [0,l) for x a [0,1]. We assume that a N 2t i (1). This assumption ensures that the second-stage equilibrium price is lower than the monopoly price. Firms are able to discriminate among consumers since they control transportation. Consumer arbitrage is assumed to be prohibitively expensive. We analyze first the second-stage Bertrand competition, given x 1 and x 2(zx 1). Since the marginal costs are constant, prices set at different points by the same firm are strategically independent. The second-stage Bertrand equilibrium of this game can then be characterized by a set of independent Bertrand equilibria, one for each point x a [0,1]. As is well known, under the homogeneous goods Bertrand competition, the firm with a lower cost obtains the whole share of the market and the equilibrium price is equal to the rival’s cost. Note that there is a point x¯ a [x 1, x 2] such that t 1(x¯ x 1) = t 2(x 2 x¯ ), and firm 1 serves all markets x V x¯ and firm 2 serves markets x z x¯ . Thus, firm 1’s profit P 1B(x 1, x 2) and firm 2’s profit P 2B(x 2, x 1) are given respectively, Z x¯ Z x¯ B B P1 ðx1 ; x2 Þ ¼ p1 ð x; x1 ; x2 Þdx ¼ ðt2 ðx2 xÞ t1 ðjx1 xjÞÞða t2 ðx2 xÞÞf ð xÞdx 0
PB2 ðx2 ; x1 Þ ¼
Z
x¯
0
1
pB2 ð x; x2 ; x1 Þdx ¼
Z
1
ðt1 ð x x1 Þ t2 ðjx2 xjÞÞða t1 ð x x1 ÞÞf ð xÞdx;
ð1Þ
x¯
where pBi (x; x i , x j ) is firm i’s local profit at market x. In the first stage, each firm i chooses its location x i so as to maximize its profit PBi (x i , x j ), which is given above. Let (x 1B, x 2B) be the pair of the equilibrium locations in the Bertrand model. If x 1 = x 2, each 1
For the recent discussions, see Pal (1998), Matsushima (2001), and Shimizu (2002).
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T. Matsumura, D. Shimizu / Economics Letters 89 (2005) 112–119
firm’s profit is zero, and given the rival’s location each firm can increase its profit by any relocation. Thus, we have x 1B p x 2B. It is also well known that an interior solution is achieved in this setting, i.e., 0 b x 1B b x 2B b 1.
3. Welfare We now discuss the welfare implication of the locations. Let csB(x; x 1, x 2) be the consumer surplus at x and CSB(x 1, x 2) be the total consumer surplus. Routine manipulation yields
f ð xÞða t2 ðx2 xÞÞ2 =2 if x V x¯ f ð xÞða t1 ð x x1 ÞÞ2 =2 if x N x¯ 2 3 Z 1 Z 1 Z x¯ 1 CSB ðx1 ; x2 Þu csB ð x; x1 ; x2 Þdx ¼ 4 f ð xÞða t2 ðx2 xÞÞ2 dx þ f ð xÞða t1 ð x x1 ÞÞ2 dx5: 2 0 0 x¯ ð2Þ B
cs ð x; x1 ; x2 Þ ¼
Let P B(x 1, x 2) u P 1B + P 2B be the two firm’s joint profits. Let W B(x 1,x 2) u CSB + P B be the total social surplus. We now present our main results. Proposition 1. ðiÞ
BPB Bx1
j
ðx1 ;x2 Þ¼ðxB ;xB Þ b 0; 1
Proof. See Appendix A.
2
ðiiÞ
BCS B Bx1
j
ðx1 ;x2 Þ¼ðxB1 ;xB2 Þ
N 0;
and
ðiiiÞ
BW B Bx1
j
ðx1 ;x2 Þ¼ðxB1 ;xB2 Þ
N 0:
5
Proposition 1 implies that a slight decrease in the distance between the two firms’ locations from their equilibrium locations increases consumer surplus, reduces the two firm’s joint profits, and increases the total social surplus. This result indicates that the distance between the two firms’ locations is too large from the viewpoint of maximum welfare. We offer an intuitive explanation on why an increase in x 1 from x 1B improves welfare. A slight increase in x 1 does not affect firm 1’s profit because firm 1 chooses x 1 so as to maximize its profit (the envelope theorem). In the markets served by firm 1, the prices depend on firm 2’s cost, which is independent of x 1. Thus, the change in x 1 does not affect consumer surplus in these markets. In the markets served by firm 2, an increase in x 1 decreases the prices. It increases consumer surplus and reduces firm 2’s profit. Since the price is higher than firm 2’s marginal cost, a reduction in the difference between price and marginal cost improves welfare. An increase in x 1 accelerates competition. Some readers might think that a more aggressive competition obviously improves both consumer surplus and welfare. In the next section we present a result of the Cournot counterpart as a benchmark, where a more aggressive competition improves consumer surplus but does not improve welfare.
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115
4. Cournot model In this section we consider a Cournot model. The basic structure is mostly the same as the Bertrand counterpart. We consider a linear transport cost, t|x x i |, where t is a positive constant. Let q i (x) be the output offered by firm i at each point x a [0,1]. In the second stage, after observing its competitor’s location, each firm i simultaneously chooses q i (x) a [0,l) for x a [0,1]: Hence, we have q(x) u q 1(x) + q 2(x). We assume that a N 2t. This assumption ensures that the whole market is served by both firms. We analyze Cournot competition. Firms compete using quantities at each point. Since the marginal costs are constant, quantities set at different points by the same firm are strategically independent. The second-stage Cournot equilibrium of this game can then be characterized by a set of independent Cournot equilibria, one for each point x a [0,1]. The profit earned at each point x by firm i is given by p i (x) = f (x)(a q(x) / f (x) t|x x i |)q i (x). Routine manipulation yields the following Cournot equilibrium at each x a [0,1]: qCi ð xÞ ¼ f ð xÞ
a 2tjx xi j þ tjx xj j ; 3
qC ð x Þ ¼ f ð x Þ
2a tjx x1 j tjx x2 j ; 3
ð3Þ
where the superscript C signifies Cournot equilibrium. The resulting profit at any point x a [0,1] is given by 2 f ð xÞ pCi ð xÞ ¼ a 2tjx xi j þ tjx xj j ; 9
i; j ¼ 1; 2:
ð4Þ
Let PCi (x 1, x 2) be the total profit of firm i in the Cournot game. Let csC(x; x 1, x 2) be the consumer surplus at x. Routine manipulation yields csC(x; x 1, x 2) = f(x)(2a t|x x 1| t|x x 2|)2 / 18. Let CSC(x 1,x 2) be the total consumer surplus. This is given by "Z # Z 1 1 1 2 C CS ðx1 ; x2 Þ u csð x; x1 ; x2 Þdx ¼ f ð xÞð2a tjx x1 j tjx x2 jÞ dx : ð5Þ 18 0
0
Let P C(x 1, x 2) u P 1C + P 2C be the two firms’ joint profits and W C(x 1, x 2) u CSC + P C be the total social surplus. Proposition 2. ðiÞ
BPC Bx1
j
ðx1 ; x2 Þ¼ðxC1 ; xC2 Þ V 0;
ðiiÞ
BCS C Bx1
j
BW C z 0; and ð iii Þ ðx1 ; x2 Þ¼ðxC1 ; xC2 Þ Bx1
j
ðx1 ; x2 Þ¼ðxC1 ; xC2 Þ
V 0;
and (iv) the equalities are satisfied if and only if x 1C = x 2C .2 Proof. See Appendix A.
5
Proposition 2 implies that a slight decrease in the distance between the two firms’ locations from their equilibrium locations increases consumer surplus and reduces social welfare and the two firms’ joint If the consumer distribution is uniform (i.e., f(x) = 1 for all x a [0,1]), x 1C = x 2C. Otherwise, x 1C p x 2C may hold. See Gupta et al. (1997). For the welfare implications of agglomeration in the Cournot model, see Matsumura and Shimizu (in press). 2
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T. Matsumura, D. Shimizu / Economics Letters 89 (2005) 112–119
profits. Proposition 2(i) and (ii) have the same inequality signs as Proposition 1(i) and (ii), respectively. However, unlike in the Bertrand counterpart, in the Cournot model the equilibrium distance between two firms is too small from the viewpoint of maximum social welfare. Suppose that x 1C b x 2C. We offer an intuitive explanation on why an increase in x 1 from x 1C reduces welfare in the Cournot game and on why Cournot and Bertrand results yield the contrasting welfare implications. The underlying factor is strategic substitutability vs. strategic complementarity. A slight increase in x 1 decreases firm 1’s costs for markets close to 1 (the right edge of the linear city). It affects firm 2’s output for these markets through strategic interaction between the two firms. In the Cournot (Bertrand) model it reduces (increases) firm 2’s output because of the strategic substitutability (complementarity). In these markets firm 2’s transport cost is lower than firm 1’s, so the decrease (increase) in firm 2’s output results in the loss (gain) of social welfare.3 Firm 1 chooses its location without considering this effect, and it yields Propositions 1(iii) and 2(iii).
5. Concluding remarks In this paper, we have analyzed welfare implications in two delivered pricing duopoly models. We found that, in both Bertrand and Cournot models, firms’ private incentives to locate closely to each other are insufficient from the viewpoint of maximum consumer surplus, but are excessive from the viewpoint of maximum producer surplus. From the viewpoint of total social surplus, the two models yield quite different implications. In the Bertrand model a decrease in the distance between the two firms improves welfare, while the opposite is true in the Cournot model. Until recently, plant locations were severely regulated in Japan. Firms could not build new plants in Tokyo or Osaka. Regulations of plant location to avoid excessive concentration are widespread. Typical theoretical support for such regulations is based on technological externalities such as congestion effects. Our paper might provide another rationale for these regulations without assuming congestion effects in the Cournot model. However, this result depends on the model specification. In the Bertrand model the opposite result is derived, and in this circumstance the government should take this pro-concentration effect into account.
Acknowledgement We are grateful to participants in the seminars at Kyoto University, the University of Tokyo, and Japan Economic Association Annual Meeting 2003 for their helpful comments and suggestions. We are also indebted to an anonymous referee for his/her valuable and constructive suggestions. Needless to say, we are responsible for any remaining errors. Financial supports of the Grant-in-Aid from Zengin Foundation for Studies on Economics and Finance and from the Japanese Ministry of Education, Science and Culture, and of the ISS project bThe Lost Decade? Re-appraising Contemporary JapanQ are greatly appreciated.
3
For discussion of the similar production substitution effect in the Cournot models, see Lahiri and Ono (1988), Matsumura (1998, 2003) and Matsushima and Matsumura (2003).
T. Matsumura, D. Shimizu / Economics Letters 89 (2005) 112–119
117
Appendix A Proof of Proposition 1. From the first-order condition for firm 1’s equilibrium location we have Z 1 B BPB BPB2 B Bp2 ð xÞ V t ¼ ¼ x x f ð xÞdx B B B B 1 1 Bpð xÞ Bx1 ðx1 ; x2 Þ¼ðx1 ; x2 Þ Bx1 ðx1 ; x2 Þ¼ðx1 ; x2 Þ x¯ Z 1 t1V x xB1 a 2t1 x xB1 þ t2 jx xB2 j f ð xÞb0: ¼
ð6Þ
x¯
Note that Bp2B(x) / Bp(x) N 0 since we assume that firm 2’s equilibrium price ( p(x) = t 1) must be smaller than its monopoly price ((a t 2) / 2). Z 1 Z 1 B BCS B B Bcs ð xÞ B B V V x x x x t t x x ¼ a t f ð xÞdxN0; f ð x Þdx ¼ B B 1 1 1 1 1 1 Bpð xÞ Bx1 ðx1 ; x2 Þ¼ðx1 ; x2 Þ x¯ x¯ Z 1 B B BW B B Bp2 ð xÞ þ cs ð xÞ V f ð xÞdx x x t ¼ B B 1 1 Bpð xÞ Bx1 ðx1 ; x2 Þ¼ðx1 ; x2 Þ x¯ Z 1 ¼ t1V x xB1 t1 x xB1 t2 jx xB2 j f ð xÞdxN0; ð7Þ x¯ 5
since t 1 N t 2 for x N x¯.
Proof of Proposition 2. From the first-order conditions for the two firm’s equilibrium locations we have " Z C Z xC x1 2 BPC1 4t C C a 2tx þ tx þ tx f ð x Þdx þ ða þ 2txC1 þ txC2 3txÞf ð xÞdx ¼ 1 2 ðx1 ; x2 Þ¼ðxC1 ; xC2 Þ C 9 Bx1 0 x1 # Z 1 a þ 2txC1 txC2 tx f ð xÞdx ¼ 0; ð8Þ þ xC2
" Z C x1 BPC2 4t a þ txC1 2txC2 þ tx f ð xÞdx ¼ C C ðx1 ; x2 Þ¼ðx1 ; x2 Þ 9 Bx2 0
Z
xC2
xC1
þ
Z
a txC1 2txC2 þ 3tx f ð xÞdx
1
xC2
a
txC1
þ
2txC2
# tx f ð xÞdx ¼ 0:
ð9Þ
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T. Matsumura, D. Shimizu / Economics Letters 89 (2005) 112–119
(i) "Z C x1 BPC BPC2 2t a þ txC1 2txC2 þ tx f ð xÞdx ¼ ¼ C C C C ðx1 ; x2 Þ¼ðx1 ; x2 Þ ðx1 ; x2 Þ¼ðx1 ; x2 Þ 9 Bx1 Bx1 0 # Z xC Z 1 2 C C C C a tx1 2tx2 þ 3tx f ð xÞdx ða tx1 þ 2tx2 txÞf ð xÞdx xC1
¼
2t 9
xC2
Z
xC2 xC1
2 a txC1 2txC2 þ 3tx f ð xÞdx;
ð10Þ
where the last equality is derived from (9). Note that (a tx 1C 2tx 2C + 3tx) N (a tx 1C 2tx 2C + 3tx 1C) = (a 2t(x 2C x 1C)) N 0 for all x a [x 1C,x 2C], where we use x 2C x 1C b 1 and a N 2t. Since (10) integrates over negative values throughout, if x 1C b x 2C then (10) is negative, and if x 1C = x 2C then (10) = 0. (ii) From (5), we obtain " Z C Z xC x1 2 BCS C t C C 2a tx1 tx2 þ 2tx f ð xÞdx þ 2a þ txC1 txC2 f ð xÞdx ðx1 ; x2 Þ¼ðxC1 ; xC2 Þ ¼ 9 Bx1 0 xC1 # Z 1 þ 2a þ txC1 þ txC2 2tx f ð xÞdx xC2
¼
t 9
Z
xC2 xC1
2 a txC1 2txC2 þ 3tx f ð xÞdx;
ð11Þ
where the last equality is derived using (8) and (9). The integrand is the same as in case (i) except that there is no negative sign, so this expression is nonnegative. (iii) Since W C(x 1, x 2) = P C(x 1, x 2) + CSC(x 1, x 2), from (10) and (11), we have Z C BW C 2t x2 a txC1 2txC2 þ 3tx f ð xÞdx; ðx1 ; x2 Þ¼ðxC1 ; xC2 Þ ¼ 9 xC1 Bx1 which is negative when x 1C b x 2C, and equal to 0 when x 1C = x 2C.
5
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