- Email: [email protected]
On the rationale of spatial discrimination with quantity-setting firms
Research in Economics 65 (2011) 254–258
Contents lists available at ScienceDirect
Research in Economics journal homepage: www.elsevier.com/locate/rie
On the rationale of spatial discrimination with quantity-setting firms Stefano Colombo Largo A. Gemelli 1, I-20123, Catholic University of Milan, Milan, Italy
article
info
Article history: Received 16 April 2010 Accepted 16 November 2010 Keywords: Cournot Spatial discrimination Hotelling
abstract We show in a game-theoretic model that when quantity-setting firms first choose whether to discriminate or not and then set quantities, the unique equilibrium consists in all firms selling a uniform quantity to all consumers. This sharply contrasts with the case of pricesetting firms. © 2011 University of Venice. Published by Elsevier Ltd. All rights reserved.
1. Introduction A consistent body of literature originated by the pioneering work of Hamilton et al. (1989) assumes that firms are able to sell differentiated quantities to consumers which are located at different points in the space.1 That is, spatial discrimination is assumed to be a feasible option for firms. Moreover, Hamilton et al. (1989) and the literature thereafter implicitly assume that when spatial discrimination is a feasible option, firms actually discriminate across consumers by selling location-specific quantities.2 In other words, it is assumed that when discrimination is possible, discrimination occurs. However, this is far from being obvious. In fact, when a firm can sell location-specific quantities (i.e. discrimination is feasible), the firm can also sell the same quantity to every consumer: the possibility to discriminate implies the possibility not to discriminate (while the reverse is not always true). As a consequence, one should address the question whether discrimination is actually the equilibrium of a game in which firms choose whether to discriminate or not. The purpose of this note is to investigate whether quantity-setting firms choose to sell discriminatory quantities when they are allowed (but they are not constrained) to do so. The analysis is performed within the Hotelling spatial framework.3 We obtain that the unique sub-game perfect equilibrium consists in both firms selling the same quantity to all consumers: that is, discrimination does not emerge in equilibrium. As a consequence, the assumption that firms can sell location-specific quantities (‘‘discrimination is feasible’’) may be not a sufficient rationale for firms actually selling location-specific quantities (‘‘discrimination occurs’’).
E-mail address: [email protected]. 1 As Hamilton et al. (1989) argue, a situation where firms set location-specific quantities corresponds to the case of spatial price discrimination. In fact, when each firm chooses a location-specific quantity, the market-clearing condition determines the price at each location: therefore, prices are not uniform along the space, i.e. they are discriminatory. To avoid confusion, we shall call the situation where firms set location-specific quantities with the name of Cournot spatial discrimination. Instead, a situation where firms set location-specific prices shall be called Bertrand spatial discrimination. 2 A non-exhaustive list of articles which make this assumption comprehends Anderson and Neven (1991), Pal (1998), Mayer (2000), Chamorro-Rivas (2000), Matsushima (2001), Shimizu (2002), Yu and Lai (2003), Gross and Holahan (2003), Matsumura (2003), Gupta (2004), Gupta et al. (2004), Berenguer Maldonado et al. (2005), Matsumura and Schimizu (2005a,b), Gupta et al. (2006), Li (2006), Matsumura and Okamura (2006), Matsumura and Shimizu (2006), Pal and Sarkar (2006), Benassi et al. (2007), Matsumura and Schimizu (2008) and Ebina et al. (2009). In these papers, one can frequently read sentences like: ‘‘The firms deliver the product to the consumers and arbitrage is assumed to be infeasible. Thus the firms can discriminate across consumers’’ (Gupta et al., 2004, p. 763). The assumption of feasibility of spatial discrimination is (implicitly) supposed to be a sufficient condition also for the fact that firms actually discriminate across consumers. 3 The analysis within the Salop model yields identical results. 1090-9443/$ – see front matter © 2011 University of Venice. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.rie.2010.11.002
S. Colombo / Research in Economics 65 (2011) 254–258
255
More generally, this note can be seen as the Cournot version of the article by Thisse and Vives (1988). In a two-stage game where firms first decide whether to price discriminate or not and then set prices, Thisse and Vives (1988) show that the unique sub-game perfect equilibrium is characterized by price discrimination and that firms are trapped into a prisoner dilemma, because profits are lower under discrimination than under uniform pricing. We differentiate from Thisse and Vives (1988) by assuming that firms set quantities instead of prices. We show that, in a sharp contrast with Thisse and Vives (1988), the unique sub-game perfect equilibrium is characterized by both firms selling uniform quantities to all consumers (no discrimination occurs). However, as in Thisse and Vives (1988), this situation is a prisoner dilemma: the reason is that in the Cournot framework, price discrimination yields higher profits than uniform pricing, while the reverse holds in the Bertrand framework. This note is structured as follows. In Section 2 the model is introduced. In Section 3 we study the equilibrium. 2. The model As in Thisse and Vives (1988), there is a linear market of length 1. The left corner is denoted by 0, while the right corner is denoted by 1. Consumers are uniformly distributed, and their location is identified by x ∈ [0, 1]. There are two firms, firm A and firm B, whose location is identified respectively by a and b. Without loss of generality, we assume 0 ≤ a ≤ b ≤ 1. Firms set quantities. The market-clearing condition determines the price at each location. Arbitrage between consumers is excluded. Denote by qA,x and qB,x the quantity produced by firm A and firm B respectively at location x. If firm J = A, B sells the same quantity to all consumers, then qJ ,x = qJ ,x′ , ∀x, x′ ∈ [0, 1]: in this case the subscript x is omitted for simplicity. At each location x, the (inverse) demand function is linear as in Hamilton et al. (1989) and others, and it is given by: px = 1 − (qA,x + qB,x ). Each firm produces at constant marginal costs, which are normalized to zero. Fixed costs are nil, but the firms pay the transportation costs to ship the goods from the plant to the consumers’ locations. We assume linear transportation costs as in Hamilton et al. (1989) and others. That is, to ship one unit of the product from its plant a (resp. b) to a consumer located at x, firm A (resp. B) pays a transport cost equal to: t |a − x| (resp. t |b − x|), where t is the (strictly positive) unit transport cost. We assume that t ≤ 5/17: this condition guarantees that there are no local monopolies. This assumption is standard in Cournot spatial price discrimination literature.4 The timing of the game is the same as in Thisse and Vives (1988), with the addiction of a previous stage for the choice of locations as in Eber (1997) and Colombo (forthcoming). That is, at time 1 firms decide where to locate in the market, at time 2 firms decide whether to discriminate or not; at time 3 firms set the quantities. This yields four possible situations at time 3: both firms discriminate (DD), no firm discriminates (UU), only firm A discriminates (DU) and only firm B discriminates (UD). The profits earned by firm A at point x are given by: πAi ,x = (1−qiA,x −qiB,x −t ai − x)qiA,x , the profits earned by firm B at point x are given by: πBi ,x = (1−qiA,x −qiB,x −t bi − x)qiB,x ,
1
1
and the overall profits of firm A and firm B are respectively: 5iA = 0 πAi ,x dx and 5iB = 0 πBi ,x dx, where i = DD, UU , DU , UD. To save notation, in the following the superscript to the locations is omitted. We solve the game by backward induction. 3. The equilibrium As usual, we start from the last stage of the game. First, we calculate the equilibrium quantity schedules when both firms discriminate (case DD). We directly refer to Proposition 1 in Hamilton et al. (1989). Therefore: ∗ qDD A,x = (1 − 2t |a − x| + t |b − x|) /3
(1)
∗ qDD B,x = (1 − 2t |b − x| + t |a − x|) /3.
(2)
DD Substituting (1) and (2) into 5DD A and 5B , we get:
3 − 3t 1 − 4a + 4a2 + 2b − 2b2 + t 2 1 + 4a3 + 12a2 (1 − b) + 3b + 3b2 − 4b3 − 6a 1 + 2b − 2b2
∗ 5DD A
=
27 3 − 3t 2a − 2a2 + (1 − 2b)2 + t 2 1 + 4a3 + a2 (3 − 12b) − 6b + 12b2 − 4b3 + 3a(1 − 2b)2
∗ 5DD = B
27
.
Now we consider the case where no firm discriminates across consumers (case UU). The profit functions of firm A and 1 1 UU UU UU UU UU UU UU firm B are: 5UU A = 0 (1 − qA − qB − t |a − x|)qA dx and 5B = 0 (1 − qA − qB − t |b − x|)qB dx, which are concave UU in the quantity. Therefore, the equilibrium quantities are the solution of the system composed by ∂ 5UU = 0 and A /∂ qA
4 See Hamilton et al. (1989) for a discussion of this assumption. Our assumption is more stringent than in Hamilton et al. (1989) and others, where t ≤ 1/2. The reason is that we do not limit the analysis to the case of both firms selling discriminatory quantities: when only one firm discriminates, local monopolies are more likely to arise and a more stringent assumption on t is needed.
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S. Colombo / Research in Economics 65 (2011) 254–258
UU ∂ 5UU B /∂ qB = 0. Straightforward calculations yield: ∗ qUU = 2 − t 1 − 4a + 4a2 + 2b − 2b2 /6 A ∗ qUU = 2 − t 1 + 2a − 2a2 − 4b + 4b2 /6. B
Substituting (3) and (4) into
5UU A
and
5UU B
∗ 5UU B
(4)
we get:
∗ 5UU = 2 − t 1 − 4a + 4a2 + 2b − 2b2 A
(3)
2
/36
= [2 − t (1 + 2a − 2a − 4b + 4b )] /36. 2
2
2
Now we consider the cases where only one firm sells location-specific quantities while the other sells a uniform quantity to all consumers. As in Thisse and Vives (1988), De Fraja and Norman (1993), Eber (1997) and Colombo (forthcoming), we assume that the non-discriminating firm sets the quantity first and the discriminating firm successively sets the quantity at each location. As pointed out by Thisse and Vives (1988), the rationale of this assumption is that the non-discriminating firm’s decision (setting the uniform quantity) is less flexible than the decision of the discriminating firm (setting the quantity schedule). Suppose that in the first stage of the game firm A chooses not to discriminate, while firm B chooses discrimination (case UD). Suppose also that firm A sells the uniform quantity qUD A . Consider firm B. Firm B maximizes its profits by selling UD UD UD different quantities at each location: the best-reply function (given qUD A ) of firm B at x is the solution of ∂πB,x (qA )/∂ qB,x = 0 with respect to qUD B,x . That is:
UD UD qˆ UD B,x (qA ) = 1 − qA − t |b − x| /2.
(5)
Using (5), overall firm A’s profits can be written as:
5UD A
∫ =
1
1−
qUD A
0
∫
b
[
+
1−
UD − qˆ UD B,x − t |a − x| qA dx =
qUD A
−
1 − qUD A − t (b − x) 2
a
∫
[ 1
1−
qUD A
a
∫
[ 1−
qUD A
−
1 − qUD A − t ( b − x)
0
2
]
− t (a − x) qUD A dx
] − t (x − a) qUD A dx
− t ( x − b)
]
− t (x − a) qUD A dx 2 2 = qUD 2 − qUD /4. A A − t 1 − 4a + 4a + 2b − 2b +
1−
b
qUD A
−
2
UD UD UD UD It easy to check that 5UD A is concave in qA . Solving ∂πA /∂ qA = 0 with respect to qA , we get the equilibrium uniform quantity sold by firm A. That is:
∗ qUD = 2 − t 1 − 4a + 4a2 + 2b − 2b2 A
/4.
(6)
Substituting (6) into (5) we get the quantity sold by firm B at x in equilibrium. That is: ∗ 2 2 qUD − 4 |x − b| /4. B,x = 2 + t 1 − 4a + 4a + 2b − 2b
(7)
Using (6) and (7) we can write the equilibrium profits of both firms. Therefore:
2 ∗ 5UD = 2 − t 1 − 4a + 4a2 + 2b − 2b2 /32 A ] [ 12 − 12t (1 + 4a − 4a2 − 6b + 6b2 ) + t 2 (7 − 96a3 + 48a4 − 36b + 96b2 − 120b3 + 60b4 ∗ 192. 5UD = B + 24a2 (1 + 6b − 6b2 ) + 24a(1 − 6b + 6b2 )) Finally, consider the case where firm A discriminates while firm B sells a uniform quantity (case DU). Since case DU is symmetric to case UD, we omit passages and only report the equilibrium profits, which are: ∗ 5DU A
12 − 12t (1 − 6a + 6a2 + 4b − 4b2 ) + t 2 (7 − 120a3 + 60a4 + 24b + 24b2 − 96b3 + 48b4 = + 48a2 (2 + 3b − 3b2 ) − 36a(1 + 4b − 4b2 ))
[
] 192
∗ 5DU = [2 − t (1 + 2a − 2a2 − 2b + 4b2 )]2 /32. B ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Simple comparisons show that 5UU ≥ 5DU and 5UD ≥ 5DD (similarly, 5UU ≥ 5UD and 5DU ≥ 5DD ). Moreover, A A A A B B B B UU ∗ DD ∗ DD ∗ ∗ we observe: 5UU ≤ 5 and 5 ≤ 5 . Therefore, we can state the following proposition, which represents the main A A B B contribution of this article:
Proposition 1. The unique equilibrium is characterized by both firms selling uniform quantities to all consumers. Moreover, a prisoner dilemma arises.
S. Colombo / Research in Economics 65 (2011) 254–258
257
Proposition 1 shows that a prisoner dilemma exists in Cournot spatial discrimination as in Bertrand spatial discrimination (Thisse and Vives, 1988), but the characteristic of the prisoner dilemma is the opposite in the two models: while in Bertrand discriminatory profits are lower than uniform profits and discrimination emerges in equilibrium, in Cournot uniform profits are lower than discriminatory profits (Proposition 1, second part) and uniformity emerges in equilibrium (Proposition 1, first part). Proposition 1 has a relevant implication: it shows that firms are not expected to sell discriminatory quantities even if this is technologically feasible. The intuition of Proposition 1 is the following. First, Proposition 1 shows that firms are better off when both discriminate with respect to the case when both set the same quantity. This is due to the fact that when both firms discriminate, each firm sets higher quantities in the proximity of its own location and lower quantities in the proximity of the rival’s location. Therefore, each firm serves its own market (those points which are near to it) with lower competition from the rival. It follows that profits are higher. In contrast, in Bertrand spatial discrimination, discrimination increases competition everywhere, thus determining lower profits with respect to uniform pricing (Thisse and Vives, 1988). Moreover, Proposition 1 shows that the dominant strategy of each firm consists in selling the same quantity to all consumers. The reason is the following. Suppose that firm A sells a uniform quantity while firm B discriminates across consumers. Firm A sells a high uniform quantity anticipating the fact that firm B (which is more flexible due to discrimination) will sell lower quantities in order to keep high prices. In other words, the non-discriminating firm adopts a ‘‘parasitic’’ strategy: it exploits the fact that the discriminating firm adapts its quantity schedules to maintain high prices when the non-discriminating firm sells a high quantity. In this way, the non-discriminating firm benefits from its own high quantity and the equilibrium high prices. Therefore, when firm B discriminates, firm A prefers not to discriminate. Similarly, if firm B does not discriminate, firm A chooses not to discriminate too, because it does not want to be parasitized by firm B. It follows that the dominant strategy consists in selling the same quantity to all consumers. The difference with Bertrand spatial discrimination is striking: while in Bertrand spatial discrimination each firm wants to maintain flexibility in setting prices and therefore chooses discrimination (Thisse and Vives, 1988), in Cournot spatial discrimination no firm wants to maintain flexibility in setting quantities, because this allows the rival to act as a parasite. Let us consider now stage 1 of the game. As in the second stage of the game the unique equilibrium is given by both firms choosing not to discriminate, we have to consider case UU. We state the following proposition: Proposition 2. The unique locational equilibrium in case UU is given by: aUU ∗ = bUU ∗ = 1/2. ∗ /∂ a = 2t (1 − 2a)[2 − t (1 − 4a + 4a2 + 2b − 2b2 )]/9. The term in the square brackets is strictly Proof. We have: ∂ 5UU A ∗ positive: then, the unique solution of ∂ 5UU /∂ a = 0 is aUU ∗ = 1/2, which is a global maximum. By symmetry, it follows A UU ∗ that b = 1/2.
Therefore, in equilibrium agglomeration arises, as both firms locate in the middle of the market in order to minimize the transportation costs. Until now we have assumed that the choice of the policy (whether to discriminate or not) is chosen after the choice of the location. However, one may argue that the policy may be chosen taking into account the optimal choice of locations. This situation amounts to a reversion of the first two stages of the game: firms first choose whether to discriminate or not, and then choose where to locate.5 Therefore, we have to calculate the equilibrium locations for each sub-game at stage 2. In the sub-game UU, the unique equilibrium location is characterized by agglomeration of firms (Proposition 2). The same occurs in the sub-game DD (see Hamilton et al., 1989), and in the sub-games UD and DU, as we show in the next proposition: Proposition 3. The unique locational equilibrium in the sub-games UD and DU is: aUD ∗ = bUD ∗ = aDU ∗ = bDU ∗ = 1/2. ∗ ∗ Proof. We only consider sub-game UD, as sub-game DU proceeds in the same way. As 5UU = 5UD , it must be that A A UD ∗ UD ∗ UD ∗ UD ∗ a = 1/2. Substituting a into 5B and taking the derivative with respect to b, we obtain: ∂ 5B /∂ b = t (1 − 2b)[3 − ∗ t (3 − 5b + 5b2 )]/8. As the term in the square brackets is strictly positive, the unique solution of ∂ 5UD /∂ b = 0 is b∗ = 1/2, B which is also a global maximum.
As agglomeration occurs in each sub-game, it follows that in the first stage of the game the unique equilibrium is still UU. References Anderson, S., Neven, D., 1991. Cournot competition yields spatial agglomeration. International Economic Review 32, 793–807. Benassi, C., Chirco, A., Scrimitore, M., 2007. Spatial discrimination and quantity competition and high transportation costs: a note. Economics Bulletin 12, 1–7. Berenguer Maldonado, M.I., Carbò Valverde, S., Fortes Escalona, M.A., 2005. Cournot competition in a two-dimensional circular city. Manchester School 73, 40–49.
5 For a similar analysis in the case of a Bertrand model, see Eber (1997) and Colombo (forthcoming).
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Chamorro-Rivas, J.M., 2000. Plant proliferation in a spatial model of Cournot competition. Regional Science and Urban Economics 30, 507–518. Colombo, S., 2010, Discriminatory prices and the Prisoner Dilemma problem, The Annals of Regional Science, forthcoming (doi:10.1007/s00168-009-0335-2). De Fraja, G., Norman, G., 1993. Product differentiation, pricing policy and equilibrium. Journal of Regional Science 33, 343–363. Eber, N., 1997. A note on the strategic choice of spatial price discrimination. Economics Letters 55, 419–423. Ebina, T., Matsumura, T., Shimizu, D., 2009. Mixed oligopoly and spatial agglomeration in a quasi-linear city. Economics Bulletin 29, 2730–2737. Gross, J., Holahan, W., 2003. Credible collusion in spatially separated markets. International Economic Review 44, 299–312. Gupta, B., 2004. Spatial Cournot competition in a circular city with transport cost differentials. Economics Bulletin 4, 1–6. Gupta, B., Lai, F.-C., Pal, D., Sarkar, J., Yu, C.-M., 2004. Where to locate in a circular city? International Journal of Industrial Organization 22, 759–782. Gupta, B., Pal, D., Sarkar, J., 2006. Product differentiation and location choice in a circular city. Journal of Regional Science 46, 313–331. 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. Li, C., 2006. Location choice in a mixed oligopoly. Economic Modelling 23, 131–141. Matsumura, T., 2003. Consumer-benefiting exclusive territories. Canadian Journal of Economics 37, 1007–1025. Matsumura, T., Schimizu, D., 2005a. Economic welfare in delivered pricing duopoly: Bertrand and Cournot. Economics Letters 89, 112–119. Matsumura, T., Schimizu, D., 2005b. Spatial Cournot competition and economic welfare: a note. Regional Science and Urban Economics 35, 658–670. Matsumura, T., Okamura, M., 2006. Equilibrium number of firms and economic welfare in a spatial price discrimination model. Economics Letters 90, 341–396. Matsumura, T., Shimizu, D., 2006. Cournot and Bertrand competition in shipping models with circular markets. Papers in Regional Science 85, 585–598. Matsumura, T., Schimizu, D., 2008. A non-cooperative shipping Cournot duopoly with linear-quadratic transport costs and circular space. Japanese Economic Review 59, 498–518. Matsushima, N., 2001. Cournot competition and spatial agglomeration revisited. Economics Letters 73, 175–177. Mayer, T., 2000. Spatial Cournot competition and heterogeneous production costs across locations. Regional Science and Urban Economics 30, 325–352. Pal, D., 1998. Does Cournot competition yield spatial agglomeration?. Economics Letters 60, 49–53. Pal, D., Sarkar, J., 2006. Spatial competition among multi-plant firms in a circular city. Southern Economic Journal 73, 246–258. Shimizu, D., 2002. Product differentiation in spatial Cournot markets. Economics Letters 76, 317–322. Thisse, J.-F., Vives, X., 1988. On the strategic choice of spatial pricing policy. American Economic Review 78, 122–137. Yu, C.-M., Lai, F.-C., 2003. Cournot competition in spatial markets: some further results. Papers in Regional Science 82, 569–580.
Contents lists available at ScienceDirect
Research in Economics journal homepage: www.elsevier.com/locate/rie
On the rationale of spatial discrimination with quantity-setting firms Stefano Colombo Largo A. Gemelli 1, I-20123, Catholic University of Milan, Milan, Italy
article
info
Article history: Received 16 April 2010 Accepted 16 November 2010 Keywords: Cournot Spatial discrimination Hotelling
abstract We show in a game-theoretic model that when quantity-setting firms first choose whether to discriminate or not and then set quantities, the unique equilibrium consists in all firms selling a uniform quantity to all consumers. This sharply contrasts with the case of pricesetting firms. © 2011 University of Venice. Published by Elsevier Ltd. All rights reserved.
1. Introduction A consistent body of literature originated by the pioneering work of Hamilton et al. (1989) assumes that firms are able to sell differentiated quantities to consumers which are located at different points in the space.1 That is, spatial discrimination is assumed to be a feasible option for firms. Moreover, Hamilton et al. (1989) and the literature thereafter implicitly assume that when spatial discrimination is a feasible option, firms actually discriminate across consumers by selling location-specific quantities.2 In other words, it is assumed that when discrimination is possible, discrimination occurs. However, this is far from being obvious. In fact, when a firm can sell location-specific quantities (i.e. discrimination is feasible), the firm can also sell the same quantity to every consumer: the possibility to discriminate implies the possibility not to discriminate (while the reverse is not always true). As a consequence, one should address the question whether discrimination is actually the equilibrium of a game in which firms choose whether to discriminate or not. The purpose of this note is to investigate whether quantity-setting firms choose to sell discriminatory quantities when they are allowed (but they are not constrained) to do so. The analysis is performed within the Hotelling spatial framework.3 We obtain that the unique sub-game perfect equilibrium consists in both firms selling the same quantity to all consumers: that is, discrimination does not emerge in equilibrium. As a consequence, the assumption that firms can sell location-specific quantities (‘‘discrimination is feasible’’) may be not a sufficient rationale for firms actually selling location-specific quantities (‘‘discrimination occurs’’).
E-mail address: [email protected]. 1 As Hamilton et al. (1989) argue, a situation where firms set location-specific quantities corresponds to the case of spatial price discrimination. In fact, when each firm chooses a location-specific quantity, the market-clearing condition determines the price at each location: therefore, prices are not uniform along the space, i.e. they are discriminatory. To avoid confusion, we shall call the situation where firms set location-specific quantities with the name of Cournot spatial discrimination. Instead, a situation where firms set location-specific prices shall be called Bertrand spatial discrimination. 2 A non-exhaustive list of articles which make this assumption comprehends Anderson and Neven (1991), Pal (1998), Mayer (2000), Chamorro-Rivas (2000), Matsushima (2001), Shimizu (2002), Yu and Lai (2003), Gross and Holahan (2003), Matsumura (2003), Gupta (2004), Gupta et al. (2004), Berenguer Maldonado et al. (2005), Matsumura and Schimizu (2005a,b), Gupta et al. (2006), Li (2006), Matsumura and Okamura (2006), Matsumura and Shimizu (2006), Pal and Sarkar (2006), Benassi et al. (2007), Matsumura and Schimizu (2008) and Ebina et al. (2009). In these papers, one can frequently read sentences like: ‘‘The firms deliver the product to the consumers and arbitrage is assumed to be infeasible. Thus the firms can discriminate across consumers’’ (Gupta et al., 2004, p. 763). The assumption of feasibility of spatial discrimination is (implicitly) supposed to be a sufficient condition also for the fact that firms actually discriminate across consumers. 3 The analysis within the Salop model yields identical results. 1090-9443/$ – see front matter © 2011 University of Venice. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.rie.2010.11.002
S. Colombo / Research in Economics 65 (2011) 254–258
255
More generally, this note can be seen as the Cournot version of the article by Thisse and Vives (1988). In a two-stage game where firms first decide whether to price discriminate or not and then set prices, Thisse and Vives (1988) show that the unique sub-game perfect equilibrium is characterized by price discrimination and that firms are trapped into a prisoner dilemma, because profits are lower under discrimination than under uniform pricing. We differentiate from Thisse and Vives (1988) by assuming that firms set quantities instead of prices. We show that, in a sharp contrast with Thisse and Vives (1988), the unique sub-game perfect equilibrium is characterized by both firms selling uniform quantities to all consumers (no discrimination occurs). However, as in Thisse and Vives (1988), this situation is a prisoner dilemma: the reason is that in the Cournot framework, price discrimination yields higher profits than uniform pricing, while the reverse holds in the Bertrand framework. This note is structured as follows. In Section 2 the model is introduced. In Section 3 we study the equilibrium. 2. The model As in Thisse and Vives (1988), there is a linear market of length 1. The left corner is denoted by 0, while the right corner is denoted by 1. Consumers are uniformly distributed, and their location is identified by x ∈ [0, 1]. There are two firms, firm A and firm B, whose location is identified respectively by a and b. Without loss of generality, we assume 0 ≤ a ≤ b ≤ 1. Firms set quantities. The market-clearing condition determines the price at each location. Arbitrage between consumers is excluded. Denote by qA,x and qB,x the quantity produced by firm A and firm B respectively at location x. If firm J = A, B sells the same quantity to all consumers, then qJ ,x = qJ ,x′ , ∀x, x′ ∈ [0, 1]: in this case the subscript x is omitted for simplicity. At each location x, the (inverse) demand function is linear as in Hamilton et al. (1989) and others, and it is given by: px = 1 − (qA,x + qB,x ). Each firm produces at constant marginal costs, which are normalized to zero. Fixed costs are nil, but the firms pay the transportation costs to ship the goods from the plant to the consumers’ locations. We assume linear transportation costs as in Hamilton et al. (1989) and others. That is, to ship one unit of the product from its plant a (resp. b) to a consumer located at x, firm A (resp. B) pays a transport cost equal to: t |a − x| (resp. t |b − x|), where t is the (strictly positive) unit transport cost. We assume that t ≤ 5/17: this condition guarantees that there are no local monopolies. This assumption is standard in Cournot spatial price discrimination literature.4 The timing of the game is the same as in Thisse and Vives (1988), with the addiction of a previous stage for the choice of locations as in Eber (1997) and Colombo (forthcoming). That is, at time 1 firms decide where to locate in the market, at time 2 firms decide whether to discriminate or not; at time 3 firms set the quantities. This yields four possible situations at time 3: both firms discriminate (DD), no firm discriminates (UU), only firm A discriminates (DU) and only firm B discriminates (UD). The profits earned by firm A at point x are given by: πAi ,x = (1−qiA,x −qiB,x −t ai − x)qiA,x , the profits earned by firm B at point x are given by: πBi ,x = (1−qiA,x −qiB,x −t bi − x)qiB,x ,
1
1
and the overall profits of firm A and firm B are respectively: 5iA = 0 πAi ,x dx and 5iB = 0 πBi ,x dx, where i = DD, UU , DU , UD. To save notation, in the following the superscript to the locations is omitted. We solve the game by backward induction. 3. The equilibrium As usual, we start from the last stage of the game. First, we calculate the equilibrium quantity schedules when both firms discriminate (case DD). We directly refer to Proposition 1 in Hamilton et al. (1989). Therefore: ∗ qDD A,x = (1 − 2t |a − x| + t |b − x|) /3
(1)
∗ qDD B,x = (1 − 2t |b − x| + t |a − x|) /3.
(2)
DD Substituting (1) and (2) into 5DD A and 5B , we get:
3 − 3t 1 − 4a + 4a2 + 2b − 2b2 + t 2 1 + 4a3 + 12a2 (1 − b) + 3b + 3b2 − 4b3 − 6a 1 + 2b − 2b2
∗ 5DD A
=
27 3 − 3t 2a − 2a2 + (1 − 2b)2 + t 2 1 + 4a3 + a2 (3 − 12b) − 6b + 12b2 − 4b3 + 3a(1 − 2b)2
∗ 5DD = B
27
.
Now we consider the case where no firm discriminates across consumers (case UU). The profit functions of firm A and 1 1 UU UU UU UU UU UU UU firm B are: 5UU A = 0 (1 − qA − qB − t |a − x|)qA dx and 5B = 0 (1 − qA − qB − t |b − x|)qB dx, which are concave UU in the quantity. Therefore, the equilibrium quantities are the solution of the system composed by ∂ 5UU = 0 and A /∂ qA
4 See Hamilton et al. (1989) for a discussion of this assumption. Our assumption is more stringent than in Hamilton et al. (1989) and others, where t ≤ 1/2. The reason is that we do not limit the analysis to the case of both firms selling discriminatory quantities: when only one firm discriminates, local monopolies are more likely to arise and a more stringent assumption on t is needed.
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UU ∂ 5UU B /∂ qB = 0. Straightforward calculations yield: ∗ qUU = 2 − t 1 − 4a + 4a2 + 2b − 2b2 /6 A ∗ qUU = 2 − t 1 + 2a − 2a2 − 4b + 4b2 /6. B
Substituting (3) and (4) into
5UU A
and
5UU B
∗ 5UU B
(4)
we get:
∗ 5UU = 2 − t 1 − 4a + 4a2 + 2b − 2b2 A
(3)
2
/36
= [2 − t (1 + 2a − 2a − 4b + 4b )] /36. 2
2
2
Now we consider the cases where only one firm sells location-specific quantities while the other sells a uniform quantity to all consumers. As in Thisse and Vives (1988), De Fraja and Norman (1993), Eber (1997) and Colombo (forthcoming), we assume that the non-discriminating firm sets the quantity first and the discriminating firm successively sets the quantity at each location. As pointed out by Thisse and Vives (1988), the rationale of this assumption is that the non-discriminating firm’s decision (setting the uniform quantity) is less flexible than the decision of the discriminating firm (setting the quantity schedule). Suppose that in the first stage of the game firm A chooses not to discriminate, while firm B chooses discrimination (case UD). Suppose also that firm A sells the uniform quantity qUD A . Consider firm B. Firm B maximizes its profits by selling UD UD UD different quantities at each location: the best-reply function (given qUD A ) of firm B at x is the solution of ∂πB,x (qA )/∂ qB,x = 0 with respect to qUD B,x . That is:
UD UD qˆ UD B,x (qA ) = 1 − qA − t |b − x| /2.
(5)
Using (5), overall firm A’s profits can be written as:
5UD A
∫ =
1
1−
qUD A
0
∫
b
[
+
1−
UD − qˆ UD B,x − t |a − x| qA dx =
qUD A
−
1 − qUD A − t (b − x) 2
a
∫
[ 1
1−
qUD A
a
∫
[ 1−
qUD A
−
1 − qUD A − t ( b − x)
0
2
]
− t (a − x) qUD A dx
] − t (x − a) qUD A dx
− t ( x − b)
]
− t (x − a) qUD A dx 2 2 = qUD 2 − qUD /4. A A − t 1 − 4a + 4a + 2b − 2b +
1−
b
qUD A
−
2
UD UD UD UD It easy to check that 5UD A is concave in qA . Solving ∂πA /∂ qA = 0 with respect to qA , we get the equilibrium uniform quantity sold by firm A. That is:
∗ qUD = 2 − t 1 − 4a + 4a2 + 2b − 2b2 A
/4.
(6)
Substituting (6) into (5) we get the quantity sold by firm B at x in equilibrium. That is: ∗ 2 2 qUD − 4 |x − b| /4. B,x = 2 + t 1 − 4a + 4a + 2b − 2b
(7)
Using (6) and (7) we can write the equilibrium profits of both firms. Therefore:
2 ∗ 5UD = 2 − t 1 − 4a + 4a2 + 2b − 2b2 /32 A ] [ 12 − 12t (1 + 4a − 4a2 − 6b + 6b2 ) + t 2 (7 − 96a3 + 48a4 − 36b + 96b2 − 120b3 + 60b4 ∗ 192. 5UD = B + 24a2 (1 + 6b − 6b2 ) + 24a(1 − 6b + 6b2 )) Finally, consider the case where firm A discriminates while firm B sells a uniform quantity (case DU). Since case DU is symmetric to case UD, we omit passages and only report the equilibrium profits, which are: ∗ 5DU A
12 − 12t (1 − 6a + 6a2 + 4b − 4b2 ) + t 2 (7 − 120a3 + 60a4 + 24b + 24b2 − 96b3 + 48b4 = + 48a2 (2 + 3b − 3b2 ) − 36a(1 + 4b − 4b2 ))
[
] 192
∗ 5DU = [2 − t (1 + 2a − 2a2 − 2b + 4b2 )]2 /32. B ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Simple comparisons show that 5UU ≥ 5DU and 5UD ≥ 5DD (similarly, 5UU ≥ 5UD and 5DU ≥ 5DD ). Moreover, A A A A B B B B UU ∗ DD ∗ DD ∗ ∗ we observe: 5UU ≤ 5 and 5 ≤ 5 . Therefore, we can state the following proposition, which represents the main A A B B contribution of this article:
Proposition 1. The unique equilibrium is characterized by both firms selling uniform quantities to all consumers. Moreover, a prisoner dilemma arises.
S. Colombo / Research in Economics 65 (2011) 254–258
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Proposition 1 shows that a prisoner dilemma exists in Cournot spatial discrimination as in Bertrand spatial discrimination (Thisse and Vives, 1988), but the characteristic of the prisoner dilemma is the opposite in the two models: while in Bertrand discriminatory profits are lower than uniform profits and discrimination emerges in equilibrium, in Cournot uniform profits are lower than discriminatory profits (Proposition 1, second part) and uniformity emerges in equilibrium (Proposition 1, first part). Proposition 1 has a relevant implication: it shows that firms are not expected to sell discriminatory quantities even if this is technologically feasible. The intuition of Proposition 1 is the following. First, Proposition 1 shows that firms are better off when both discriminate with respect to the case when both set the same quantity. This is due to the fact that when both firms discriminate, each firm sets higher quantities in the proximity of its own location and lower quantities in the proximity of the rival’s location. Therefore, each firm serves its own market (those points which are near to it) with lower competition from the rival. It follows that profits are higher. In contrast, in Bertrand spatial discrimination, discrimination increases competition everywhere, thus determining lower profits with respect to uniform pricing (Thisse and Vives, 1988). Moreover, Proposition 1 shows that the dominant strategy of each firm consists in selling the same quantity to all consumers. The reason is the following. Suppose that firm A sells a uniform quantity while firm B discriminates across consumers. Firm A sells a high uniform quantity anticipating the fact that firm B (which is more flexible due to discrimination) will sell lower quantities in order to keep high prices. In other words, the non-discriminating firm adopts a ‘‘parasitic’’ strategy: it exploits the fact that the discriminating firm adapts its quantity schedules to maintain high prices when the non-discriminating firm sells a high quantity. In this way, the non-discriminating firm benefits from its own high quantity and the equilibrium high prices. Therefore, when firm B discriminates, firm A prefers not to discriminate. Similarly, if firm B does not discriminate, firm A chooses not to discriminate too, because it does not want to be parasitized by firm B. It follows that the dominant strategy consists in selling the same quantity to all consumers. The difference with Bertrand spatial discrimination is striking: while in Bertrand spatial discrimination each firm wants to maintain flexibility in setting prices and therefore chooses discrimination (Thisse and Vives, 1988), in Cournot spatial discrimination no firm wants to maintain flexibility in setting quantities, because this allows the rival to act as a parasite. Let us consider now stage 1 of the game. As in the second stage of the game the unique equilibrium is given by both firms choosing not to discriminate, we have to consider case UU. We state the following proposition: Proposition 2. The unique locational equilibrium in case UU is given by: aUU ∗ = bUU ∗ = 1/2. ∗ /∂ a = 2t (1 − 2a)[2 − t (1 − 4a + 4a2 + 2b − 2b2 )]/9. The term in the square brackets is strictly Proof. We have: ∂ 5UU A ∗ positive: then, the unique solution of ∂ 5UU /∂ a = 0 is aUU ∗ = 1/2, which is a global maximum. By symmetry, it follows A UU ∗ that b = 1/2.
Therefore, in equilibrium agglomeration arises, as both firms locate in the middle of the market in order to minimize the transportation costs. Until now we have assumed that the choice of the policy (whether to discriminate or not) is chosen after the choice of the location. However, one may argue that the policy may be chosen taking into account the optimal choice of locations. This situation amounts to a reversion of the first two stages of the game: firms first choose whether to discriminate or not, and then choose where to locate.5 Therefore, we have to calculate the equilibrium locations for each sub-game at stage 2. In the sub-game UU, the unique equilibrium location is characterized by agglomeration of firms (Proposition 2). The same occurs in the sub-game DD (see Hamilton et al., 1989), and in the sub-games UD and DU, as we show in the next proposition: Proposition 3. The unique locational equilibrium in the sub-games UD and DU is: aUD ∗ = bUD ∗ = aDU ∗ = bDU ∗ = 1/2. ∗ ∗ Proof. We only consider sub-game UD, as sub-game DU proceeds in the same way. As 5UU = 5UD , it must be that A A UD ∗ UD ∗ UD ∗ UD ∗ a = 1/2. Substituting a into 5B and taking the derivative with respect to b, we obtain: ∂ 5B /∂ b = t (1 − 2b)[3 − ∗ t (3 − 5b + 5b2 )]/8. As the term in the square brackets is strictly positive, the unique solution of ∂ 5UD /∂ b = 0 is b∗ = 1/2, B which is also a global maximum.
As agglomeration occurs in each sub-game, it follows that in the first stage of the game the unique equilibrium is still UU. References Anderson, S., Neven, D., 1991. Cournot competition yields spatial agglomeration. International Economic Review 32, 793–807. Benassi, C., Chirco, A., Scrimitore, M., 2007. Spatial discrimination and quantity competition and high transportation costs: a note. Economics Bulletin 12, 1–7. Berenguer Maldonado, M.I., Carbò Valverde, S., Fortes Escalona, M.A., 2005. Cournot competition in a two-dimensional circular city. Manchester School 73, 40–49.
5 For a similar analysis in the case of a Bertrand model, see Eber (1997) and Colombo (forthcoming).
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