Partial agglomeration or dispersion in spatial Cournot competition.

• Author: Matsumura, Toshihiro

1. Introduction

   Since the seminal work of Hotelling (1929), a rich and diverse literature on spatial competition has emerged. Location models can be classified into two types. One type is shipping or spatial price-discrimination model, where sellers bear transport costs. The other is shopping or mill-pricing model, where buyers pay for transport. For each type, one can have either Bertrand-type price setting or Cournot-type quantity setting. (1)

   Most studies on location theory use shopping (mill pricing) models with Bertrand competition. Although Cournot and Bertrand-types of non spatial models are equally popular, the body of literature on spatial competition that uses Cournot-type models is relatively small. Economists have recently considered shipping models with Cournot competition. Hamilton, Thisse, and Weskamp (1989) and Anderson and Neven (1991) carry out pioneering works on location models. (2) They use linear city models and show that all firms agglomerate at the central point. Pal (1998) shows that their result is crucially dependent on the assumption of a linear city. He investigates a circular city duopoly model (3) and finds that an equidistant location pattern appears in equilibrium; that is, locational dispersion appears. (4) Matsushima (2001a) extends Pal's model to an n-firm oligopoly and shows another equilibrium where half of the firms locate at one point and the other half locate at the opposite point (partial agglomeration). These results indicate the multiplicity of equilibria in spatial Cournot models with a circular city. The multiplicity of equilibria restricts the applicability of the model because the model does not give us a distinct prediction. In this article, we take a close look at Pal (1998) and Matsushima (2001a). We try to solve this problem by extending their linear transport cost model to one with nonlinear transport cost. (5) We consider a simplified model. The numbers of firms and markets (and so possible locations) are four. Four firms choose location A, B, C, or D (see Figure 1). If each of the four firms chooses a different location, we call this outcome Pal type. If two firms locate at A and the other two locate at C, we call this outcome Matsushima type. We find that the Pal-type equilibrium is much more robust because (i) Pal-type equilibrium always exists as long as transport cost is increasing in distance, while Matsushima-type equilibrium fails to exist when the transport cost function is significantly convex or concave (Proposition 1); (ii) if firms choose their locations sequentially, the unique equilibrium outcome is Pal type under nonlinear transport cost functions (Proposition 2); and (iii) the profit of each firm in Pal type is never smaller than that in Matsushima type, and the former is strictly larger if the transport cost function is nonlinear (Proposition 3).

   [FIGURE 1 OMITTED]

   We also compare the welfare implications of Pal-type and Matsushima-type models. If the transport cost function is linear, the two outcomes yield exactly the same profits, consumer surplus, and total social surplus. However, this equivalence does not hold if the transport cost function is nonlinear, either convex or concave. In this case, Pal type yields greater total social surplus and profit of each firm, while Matsushima type yields greater consumer surplus. These results indicate that the welfare implications are sensitive to whether or not the transport cost function is linear.

   Anderson and Neven (1991) show that, in the linear city model, a strong concavity in transport cost changes the equilibrium outcome. This result is related to our results, but we emphasize that our results are different from theirs. First, we show that both concavity and convexity of the transport cost function affect the results, while in Anderson and Neven (1991), only strong convexity changes the result. Second, in our Propositions 2 and 3, even a slight nonlinearity changes the results.

   We now remark on the applicability of the shipping spatial model. The most natural interpretation of the model is that each firm chooses where it builds a plant. There is another important interpretation. We can interpret space as product varieties. Each firm's location indicates the product or sector in which it has an advantage. Distant locations are the products for which the firm is at a disadvantage and to produce them it incurs additional costs. In short, location choice corresponds to technology choice and transportation costs correspond to the additional production costs. Hence, a shipping model is a suitable analytical tool for both spatial and nonspatial competition. (6) When we use a circular city shipping model in this context, multiplicity of equilibria might make analysis difficult. Our result indicates that we should focus on the Pal type in such a case because the Pal-type equilibrium is quite robust. Our result also indicates that, when we discuss the welfare implication of this model, the results can be crucially dependent on the shape of the transport cost function.

   The paper is organized as follows. Section 2 formulates the model. Section 3 investigates equilibrium outcomes and presents our main results. Section 4 discusses welfare. Section 5 concludes the paper. All proofs are presented in the Appendix.

2. The Model

   There are four markets, A, B, C, and D, which are shown in Figure 1. Each market generates a demand, p = a - Q, where p is the price of the homogeneous products and Q is the total quantity supplied by four firms. We assume that a is sufficiently large so as to ensure that all markets are served by all four firms. When shipping their products, the firms transfer them along the perimeter of the circle. The unit transport cost of shipping between neighboring markets (such as between A and B) is t and between the opposite markets (such as between A and C) is T, where t and T are positive constants. We assume that 0 < t < T (transport cost is increasing in distance). We normalize transport cost within the market to zero. We can measure the degree of concavity (or convexity) by T/t. If transport cost is proportional to distance (i.e., the transport cost function is linear in distance), T/t = 2. If the transport cost function is convex (concave) with respect to distance, T/t > (

   Each firm i chooses its location [x.sub.i] [member of] {A, B, C, D}. After observing the rivals' locations, firms compete a la Cournot. Each firm incurs a symmetric constant marginal cost of production, which we normalize to zero. The consumers are assumed to have a prohibitively costly transport cost, preventing arbitrage. (7) These assumptions are standard in the literature.

3. Equilibrium Location Pattern

   Cournot Competition

   The equilibrium concept used is subgame perfect Nash equilibrium. We solve the game by backward induction. Thus, the local competition in the final stage subgames is examined first.

   Because we assume constant marginal cost, each local market can be analyzed independently. Let [q.sub.i](J) and [[pi].sub.i](j) (i [member of] {1, 2, 3, 4}, j [member of] {A, B, C, D}) denote the equilibrium output and the equilibrium profit of firm i at market j in the second stage, respectively. The standard analysis yields

   [q.sub.i](j) = 1/5(a + [c.sub.1](j) + [c.sub.2](j) + [c.sub.3](j) + [c.sub.4](j) - 5[c.sub.i](j)), (1)

   [[pi].sub.i](j) = 1/25 [(a + [c.sub.1](j) + [c.sub.2](j) + [c.sub.3](j) + [c.sub.4](j) - 5[c.sub.i](j)).sup.2] = [([q.sub.i](j)).sup.2], (2)

   where [c.sub.i](j) denotes firm i's unit transport cost between its location and market j. The total profit of firm i is

   [[PI].sub.i] = [summation over (j [member of] {A,B,C,D})] [[pi].sub.i](j). (3)

   Simultaneous Location Choice

   First we discuss a model with simultaneous location choice. The game runs as follows. In the first stage, all firms simultaneously and independently choose their locations. After observing each other's locations, the four firms face Cournot competition described in section 3.1.

   There are eight possible location patterns, as shown in Table 1. For example, pattern 5 has two firms located at A, one at B, and one at C. These eight cover all possible patterns, as others differ only by symmetry from one of the other patterns. Note that 7 is the Matsushima-type outcome (two firms locate at A and the others locate at C) and 8 is the Pal-type outcome (each firm chooses a different location).

   As noted above, we use subgame perfection as our equilibrium concept. A location pattern is an equilibrium outcome if no firm is willing to deviate from its current location, considering how its location affects the second stage...