Two-dimensional spatial Cournot competition with hyperbolic demand

1. IntroductionSince 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.1Most 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 model3 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 ). 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).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. …

This chapter reviews the literature on spatial Cournot competition with endogenous firms’ locations in the past 30 years, which started from Hamilton et al. (Spatial Discrimination: Bertrand vs. Cournot in a Model of Location Choice. Regional Science and Urban Economics, 19, 87–102, 1989) and Anderson and Neven (Cournot Competition Yields Spatial Agglomeration. International Economic Review, 32, 793–808, 1991). Linear markets and circular markets are two main streams in the spatial Cournot models. Overall speaking, spatial Cournot models can capture the real-world regularity (agglomeration at the market center) observed by Harold Hotelling and escape from the undercutting trap in Hotelling (Stability in Competition. Economic Journal, 39, 41–57, 1929). Moreover, diverse location patterns are shown in circular markets.

We investigate in a spatial Cournot competition setting how much product research and development (R&D) investment firms make in order to differentiate their goods. The model features a location competition in the first stage, R&D competition in the second stage, and Cournot competition in the third stage. There are two differentiation methods in the model—spatial differentiation affecting the firm costs, and differentiation in the demand function affecting the demand levels. We find that comparing to the setting with no spatial connotation, the firms tend to invest less in product R&D. We also consider welfare implications. Although the firms employ maximum differentiation in equilibrium, there are cases in which welfare can be improved if the firms are to locate closer to each other.

We reconsider a Cournot spatial competition in a circular city. We discuss an oligopoly model. We find that two equilibria exist if the transport cost function is nonlinear in distance, while a continuum of equilibria exists if it is linear. Thus, the result of the real indeterminacy of equilibria in the linear transport cost case is knife edge.

Spatial competition among multi-store firms

This paper explores a linear and circular model with spatial Cournot competition. It is shown that agglomeration of firms at the center of the main street is the equilibrium when the demand density on the main street (linear market) is high, and there exists a unique separated location equilibrium when the density is moderate. Moreover, the socially desirable interior locations are more dispersed than the equilibrium ones. Finally, extensions on tax policies, nonlinear demand, non-uniform distribution, and others are also discussed.

We characterize equilibrium plant locations for a spatial multi-plant Cournot oligopoly in a circular city. Previous work demonstrates that for a two-plant duopoly, all four plants are equally spaced. We establish that the equilibrium location pattern is unique if there are two firms with an equal number of plants. In most other scenarios, multiple equilibria arise. Next, we endogenously determine both the number of plants per firm and plant locations for a duopoly. It is shown that the subgame perfect Nash equilibrium may not be unique, and for identical set-up costs, the firms may choose different numbers of plants.

This paper analyzes spatial Cournot competition in a circular city with a directional delivery constraint, which means that a firm can only deliver its product in one direction. It reveals that, contrary to the standard result, the unique location equilibrium involves duopoly firms agglomerating at the same location when they deliver products in different directions or when the direction decisions are made endogenously. We point out that spatial agglomeration emerges from a central point of view for a firm in a circular city, showing relaxing quantity competition through cost differentiation.

We consider the standard model of spatial Cournot competition and show that for dispersion equilibria to exist, (a) a necessary condition is that the distribution be not unimodal, and (b) a sufficient condition is that the distribution be convex with a unique antimode and that asymmetry is not too strong.

This paper investigates spatial Cournot competition in a circular city, where the maximal service range of a vehicle is less than half of the perimeter, and a firm needs to initiate more than two dispatches to serve the whole market. We examine a multi-stage game of location and transportation mode choices, and the subsequent quantity competition between duopoly firms. The findings reveal that non-maximum dispersion is the unique location equilibrium when duopoly firms deliver products in different transportation modes or when the transportation mode decisions are made endogenously and the fixed cost of a transportation instrument is relatively high.

This paper analyzes the location equilibrium in two intersecting circular markets where two identical firms engage in Cournot competition. It is shown that each of the two intersection points occupied by one of the firms is the unique fully symmetric location equilibrium and also the social optimum, which is neither maximal nor minimal differentiation. The intuition of our result is that by each firm locating at each of the intersection points, firms can minimize their transport costs and partially avoid competition, compared with both locating at one of the intersection points. Our finding coincides with the real-world phenomenon that transport hubs may attract the presence of more firms.

The model developed in this paper extends the strategic location framework under Cournot competition in order to allow for different production costs across locations. The subgame perfect equilibria where two firms choose first a location and then quantities is analyzed under general production cost distributions. It appears that central agglomeration (the equilibrium under the uniform production costs distribution) only arises in the particular case where the center of the segment yields the minimal production cost. If the production cost distribution is globally convex, an agglomerated equilibrium exists at an intermediary point between the locations minimizing production cost and transport cost, respectively. The conditions are also derived for the existence of symmetric dispersed location equilibria. Two specific production cost distributions are analyzed: the linear and the inverted U one. It is demonstrated that the unique equilibrium in the linear distribution case is an agglomerated equilibrium and that the inverted U distribution yields a symmetric location of firms in equilibrium.

Spatial Cournot competition and agglomeration in a model of location choice

This paper models the distribution system as vertically related activities under spatial successive oligopoly.The retail market is characterized by a uniform and discrete distribution of intralocal concentrations of retailers along a linear market. Manufacturers are considered as wholesalers located arbitrarily along the linear (or over a plain) market and they can make sales to all local market points. The retailers are subject to competition at its own site as well as at a location some distance away while wholesalers are spatial Cournot oligopolists.Profit maximization condition for the upstreamers is based on the profit maximizing equilibrium conditions for the downstreamers since the marginal revenue function of the ratailer constitutes the average revenue function of the wholesaler. Therefore, resulting equilibrium conditions for the two vertically related markets determine the equilibrium output and retail price. Spatial competition in the retail market causes a change in equilibrium conditions for the downstreamers, which in turn induces a change in equilibrium conditions for the upstreamers while spatial competition and cost coditions in the wholesale market affect the latter equilibrium conditions directly.If the basic demands yield monotonically increasing (decreasing) elasticities with distance, smaller market area yields lesser (greater) elasticity of aggregate demand. Despite the same curvature of the basic demand, elasticity of the first derivative of aggregate demand does not necessarily decrease (increase) as market size decreases. Therefore, greater interlocal competition in the retail market may raise or lower the retail price and consumer surplus per unit area depending on the directions of changes in two types of elasticities. In contrast, greater intralocal competition yields lower retail price and greater consumer surplus per unit area due to the second curvature condition. Likewise, factors such as greater spatial competition in the wholesale market, lower marginal cost of production, lower transportation cost per unit of quantity and smaller total market area tend to lower retail price, thereby increasing consumer surplus per unit area.We expand Greenhut-Ohta [5] model to ponder some cases of vertical integration by spatial successive oligopolists when one upstreamer integrates with one or more downstreamers and moreover integrated enterprises can continue to offer independent downstreamers wholesale goods. We demonstrate that industry output is greater while both retail price and wholesale price are lower when partial vertical integration is contemplated than when independent retailers are allowed to purchase from independent manufacturers only.

The circular city model and the linear city model are extended to allow for asymmetric directional transportation costs. A two-stage location-then-quantity model is proposed. We show that in the circular city model, maximal dispersion arises in equilibrium, while in the linear city model, the unique equilibrium is represented by both firms agglomerating in a non-central point of the segment.