Spatial Cournot competition among multi-plant firms in a circular city.

• Author: Pal, Debashis

1. Introduction

   There is a long history of research on spatial competition that allows firms to choose their locations; most papers on this topic assume that each firm sets up only one plant. This is rather surprising because firms in most industries do set up multiple facilities. In industries where the cost of transportation is significant, such as cement, natural gas liquids, and ready-mixed concrete, producers usually manufacture a homogenous product at several production facilities. (1) There is empirical evidence of an abundance of spatial competition among multi-plant firms, and the model was introduced a long time ago. The paucity of literature on the subject, however, motivates the present work.

   The study of location decisions by multi-plant firms originated with Teitz (1968). In the context of Hotelling's (1929) linear city model with fixed prices and linear transport cost, Teitz (1968) pointed out that a Nash location equilibrium does not exist for firms having multiple plants. Subsequently, Martinez-Giralt and Neven (1988) considered both linear city and circular city models and assumed a quadratic transport cost together with mill pricing. They demonstrated that if each of two firms is allowed to open up to two plants, its optimal choice in equilibrium is not to open a second store. These results are intriguing given the prevalence of spatial competition involving firms with multiple facilities. They also highlight the challenges associated with the multi-plant oligopoly location problem.

   In recent years, Chamorro-Rivas (2000), Norman and Pepall (2000), and Pal and Sarkar (2002) undertake a different approach to study spatial competition among multi-plant firms. In contrast to assuming an exogenously fixed price or price competition, these authors assume that the firms compete in quantities (a la Cournot). (2)

   Norman and Pepall (2000) analyze the welfare and profitability effects of a horizontal merger between two firms in a spatial Cournot oligopoly serving a linear market. Their analysis involves spatial Cournot competition among many single-plant firms and a two-plant firm; and it shows that a horizontal merger of two firms improves production efficiency, increases profits for each firm, and may enhance welfare.

   Pal and Sarkar (2002) consider a multi-plant spatial duopoly in a linear market. They demonstrate that both spatial agglomeration and dispersion may arise as a solution to the equilibrium location problem. In particular, when the two firms have an equal number of plants, the plants belonging to competing firms agglomerate at discrete points that coincide with each firm's monopoly plant locations. The results also hold when there are more than two multi-plant firms.

   Chamorro-Rivas (2000), on the other hand, considers Salop's (1979) circular city model for a spatial duopoly with two plants each. He shows that in equilibrium all four plants are equally spaced on the circle, with each firm locating its two plants at opposite ends of a diameter.

   Research indicates that for a multi-plant oligopoly in a linear city model, the equilibrium location patterns are completely characterized. However, in a circular city model, it is not yet known whether the results for a two-plant duopoly extend to a multi-plant oligopoly. Nonetheless, the literature recognizes scenarios in which the circular city model is more appropriate than the linear city model. For example, when each plant faces competition from both sides, the circular city model is more appropriate than the linear city model. Therefore, a study of location choice for multi-plant oligopolists in a circular city model will fill a vacuum in the literature.

   We should point out that the circular city model is by no means applicable only to geographical markets that are shaped like a geometrical circle. If we assume that the goods can be transported only along the perimeter, the results obtained in a circular city model qualitatively extend to any market that is topologically equivalent to a non-intersecting closed curve. Furthermore, the circle may also represent the space of preferences. For example, in studying the competition among television networks choosing time slots for their various shows, or airlines choosing the arrival and departure times of their multiple flights, the twenty-four-hour time cycle represents the circular market in which competitors must choose locations for their multiple plants.

   In this paper, we analyze a three-stage plant, location, and quantity choice problem for multi-plant Cournot oligopolists in a circular city model. Chamorro-Rivas (2000) found that for two-plant duopolists, the equidistant location pattern, with plants of the same firm diametrically opposite each other, is the only subgame perfect Nash equilibrium (SPNE) outcome. We demonstrate that a straightforward generalization of this result to multi-plant oligopolists, though intuitively pleasing, is too simplistic.

   For a duopoly, we endogenously determine the number of plants for each firm. Previous research in this area assumes the number of plants per firm to be exogenous (Chamorro-Rivas 2000, Pal and Sarkar 2002). Since a significant cost typically is associated with setting up a plant, the setup cost should be an important element in determining the number of plants each firm would like to establish.

   The paper is organized as follows. Section 2 describes the model. For the exogenously given number of plants per firm, section 3 characterizes various properties of the location and quantity equilibria, and sections 4, 5, and 6 identify the equilibrium locations for firms with identical and nonidentical numbers of plants. Section 7 endogenously determines the number of plants per firm for a duopoly. Section 8 concludes the paper. All proofs are placed in the Appendix.

2. Model

   We consider a spatial multi-plant Cournot oligopoly serving a circular market with perimeter of length one. The points on the circle are identified with numbers in [0, 1], with the southernmost point being 1/2 and the values increasing in a clockwise direction. Thus, the northern most point is considered both 0 and 1. We assume that consumers are distributed uniformly on the circle. The market inverse demand at each point x on the circle is given by p(x) = max[0, a - bQ(x)], where a > 0, b > 0 are constants. Here, Q(x) is the aggregate quantity supplied at x and p(x) is the market price at x.

   There are n multi-plant firms that choose their plant locations on the perimeter of the circle. Each firm faces a positive cost to set up a plant. Let the setup cost per plant for firm i (1 [less than or equal to] i [less than or equal to] n) be [F.sub.i] > O. The setup costs [F.sub.1], [F.sub.2] ..., [F.sub.n], market demand, production, and transportation costs determine the number of plants per firm. In sections 3-6, we first solve for optimal plant locations, treating the number of plants per firm as exogenously given. In section 7, we endogenously determine the number of plants per firm for a duopoly.

   Let firm i (1 [less than or equal to] i [less than or equal to] n) set up [m.sub.i] [greater than or equal to] 0 plants. The vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the plant locations of firm i (1 [less than or equal to] i [less than or equal to] n). By convention, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The vector [??] = ([[[xi].bar].sub.1], [[[xi].bar].sub.2], ... [[[xi].bar].sub.n]) denotes the plant locations of all n firms and the vector [[??].sup.-i] = ([[[xi].bar].sub.1], ..., [[[xi].bar].sub.i-1], [[[xi].bar].sub.i+1], ..., [[[xi].bar].sub.n]) denotes the plant locations of all other firms except firm i (1 [less than or equal to] i [less than or equal to] n).

   The firms produce and sell a homogeneous output. The firms deliver the product to the consumers and arbitrage among consumers is assumed to be infeasible. (3) Therefore, the firms can discriminate across consumers. The firms have identical production and transportation technologies. Each firm produces at a constant marginal and average variable cost (both normalized to zero) and transports the goods along the perimeter, paying a linear transport cost of t > 0 per unit distance. Without loss of generality, we assume t = 1.

   Each firm serves a market point x incurring the lowest possible transport cost. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denote firm i's delivered marginal cost at x, where [absolute value of y - x] is the length of the shorter arc from y to x. Consequently, the market points served by plant j of firm i, {x : [c.sub.i](x) = [absolute value of [[xi].sub.ij] - x]}, turn out to be contiguous. We also assume a [greater than or equal to] n/2. This condition ensures that each firm will always serve the entire market.

3. Properties of Location Equilibrium

   First, for exogenously given ([m.sub.1], [m.sub.2], ..., [m.sub.n]), we study the SPNE of a two-stage game in which the firms simultaneously choose their plant locations in stage one; in stage two, after observing their competitors' locations, the firms simultaneously compete in quantities. We proceed by backward induction and first characterize the quantity equilibrium in the second stage for given locations. Since marginal production costs are constant and arbitrage among the consumers is infeasible, quantities supplied to different markets by the same firm are strategically independent. Therefore, the second-stage equilibrium can be characterized by a set of independent Cournot equilibria, one for each market point x.

   At each market point x, firm i (i = 1, 2, ..., n) chooses [q.sub.i](x) to maximize its profit [p(x) - [c.sub.i](x)][q.sub.i](x). By simultaneously solving the first-order conditions for profit maximization of the firms, we obtain the following equilibrium outcomes at each x:

   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN...