COURNOT COMPETITION YIELDS SPATIAL DISPERSION

Introduction. A lot of works suggests that Cournot oligopolists competing in a spatial model, with a uniform distribution of consumers, agglomerate in the center of the market. In this paper revisited some results from [1]. In the paper [1] showed that Cournot-type oligopolists which discriminate over space will tend to agglomerate. The paper [2] considers the spatial model used by [2] to study firms’ decisions on locations without restricting the consumers’ reservation price. Purpose. This paper extend the analysis of the standard model of spatial discrimination with Cournot competition along the linear city for a high enough transport tariff. Results. It was obtained that for a high enough transport tariff the firms have a decision which lies on the boundary of the feasible locations region. We show that a change in the central agglomeration strategy to the dispersion strategy occurs at the point of transcritical bifurcation. The different effects come into play. Before bifurcation point the effect of minimizing transport costs is dominate. Firms choose the central agglomeration strategy to minimize a total distance of transportation. The growth of the transport tariff leads to a decrease in the total profit. In the bifurcation point begins to dominate the effect of market segmentation. Firms choose a dispersed strategy to monopolize adjacent markets. The growth of the transport tariff leads to an increase in total profits. The growth of total profit with growth of the transport tariff is due to the fact that when dispersion strategy, the firms supply more to adjoining markets and less to distant markets. In the case of multiple equilibria is shown that exactly the stable solution provides a large profit. The conditions for full coverage of the markets for both strategies are defined. Conclusions. In this paper we show that firms under Cournot competition will tend to dispersion. Thus, the article extends the analysis of the standard Hotelling spatial competition model. The results allow a deeper look at the causes of agglomeration and dispersion of firms. The analysis of equilibrium stability showed that the transport tariff is a bifurcation parameter for firms when choosing a spatial strategy.


Introduction and problem statement
In search of a solution to the Bertrand paradox, Hotelling proposed to take into account the factor of space under the price competition of firms. In Hotelling's linear city model [3], two firms compete on a segment with a unit demand at each point. Firms optimize their prices and location on the segment. Transportation delivery costs of goods are borne by consumers. Hotelling found that in an equilibrium state, firms would be minimally spatially differentiated, since they would be located in the center. This conclusion of the model analysis subsequently became a famous "principle of minimal differentiation".
Anderson and Neven [1] restricted the analysis to t b < 2 . Rivas [2] extend the analysis to t b ≤ allowing for different market configurations. The paper identified market patterns where firms compete over the whole market as well as patterns where a firm behaves as a monopoly in a market segment.
Formulation of article goals. In this paper we are extending the analysis to t b d 2 and showing that firms have location decisions which provide a full markets cover.

The linear city model
Two firms sell homogeneous goods on the unit segment, at each point of which is the consumer market x , x > @ 0 1 , . The distance of the firms from zero point is equal x 1 and x 2 accordingly, and x 1 2 ≤ x . Each firm faces linear transportation costs of t to move one good unit per one unit of distance. Consumer arbitrage is assumed to be prohibitively costly.
The linear demand curve in the market x : where p x -the price in the market x , q x 1 , q x 2 -the quantities supplied of firms in the market x , b -a minimum price, at which there is no demand (reservation price). Let us assume that firms supply products to all markets (full coverage): q 1 1 0 x t , Thus, zero quantities supplied are possible only at the boundaries of a unit segment. The profits of firms in the market x : The competitive game consists of two stages. In the first stage, the firms simultaneously select their locations. In the second stage, at the given location decisions, the firms simultaneously choose their supplied quantities. The equilibrium of the model is solved by backward induction.

The Cournot competition
According to the backward induction method we begin with the second stage. Let us assume that firms optimize supply volumes under Cournot competition. Solving the first-order conditions yields the reaction curves of the firms: The equilibrium supply volumes of firms to the market x : Let us define the feasible locations region of firms.
1. From previous studies [2,11,13,15] we know that the equilibrium in this model is symmetrical about the center: 2. In the center of line segment the firms minimize a total distance of traffic, therefore full markets coverage is possible with a highest transport tariff. Substituting into (1) the values x 1 1 2 = , x 2 1 2 = , x = 1 or into (2) the values x 1 1 2 = , x 2 1 2 = , x = 0 , we find that at any locations of firms the coverage of all markets is possible only at t d 2 b . 3. From (1) it follows that for firm 1 the minimum volume of deliveries is reaching in the market x = 1 . Therefore a condition of markets coverage for firm 1: For firm 2 the minimum volume of deliveries is reaching in the market x = 0 . Therefore a condition of markets coverage for firm 2: Solving the system of equations (4)-(5) yields РОЗВИТОК ТРАНСПОРТУ № 1(4), 2019 Thus, the feasible locations region are ( Fig.1):

Figure 1. The feasible locations region. Source: Own elaboration
The equilibrium profits of firms in the market x : In the first stage each firm selects a profit-maximizing location at a given location of the rival.
So, let us start with firm 1. The total profit of firm 1 in all markets: After integrating and identical transformations (10), we obtain: The optimal location is defined by the necessary condition: The sufficient condition for the existence of profit maximum for the firm 1: The necessary condition for the existence of the equilibrium location for firm 1 is the nonnegativity of the discriminant of the square equation (11): It is easy to make sure that D 1 0 > at x 2 1 2 ≥ . Therefore, due to condition (3), in the equilibrium state the discriminant (12) is always nonnegative.
The roots of the square equation (11) are: The root x 1 2 * does not satisfy the basic conditions of the model and therefore is not further analyzed. The total profit of firm 2 in all markets: After integrating and identical transformations (13), we obtain:

РОЗВИТОК ТРАНСПОРТУ № 1(4), 2019
The optimal location is defined by the necessary condition: The sufficient condition for the existence of profit maximum for the firm 2: The necessary condition for the existence of the equilibrium location for firm 2 is the nonnegativity of the discriminant of the square equation (14): It is easy to make sure that D 2 0 > at x 1 1 2 ≤ . Therefore, due to condition (3), in the equilibrium state, the discriminant (15) is always nonnegative. The roots of the square equation (14) are: The root x 2 2 * does not satisfy the basic conditions of the model and therefore is not further analyzed. Thus, we received the reaction curves of firms: x t 1 2 x t 2 1 2 1 2 12 Substituting (15) into (16), we are obtaining the symmetry condition (3). Using the symmetry condition (3), we find solutions of the system (15)-(16):

The analysis of the stability of equilibrium
Let us analyze a stability of the solutions (17)-(19). For this we consider a twodimensional map: x t  (17): From (21) we obtain two real multipliers: For P 1 2 1 , the fixed point is stable, for P 1 2 1 , ! the fixed point is unstable, for P 1 2 1 , the bifurcation occurs. From (22) it follows that the fixed point (17) is stable when ( From (23) we obtain two real multipliers: occurs a transcritical bifurcation, in which the spatial strategies exchange of stabilities.
The transcritical bifurcation diagram for b = 1 depicted in Fig. 2. The dynamics of the total profit of firm 1 at crossing of the bifurcation point for b = 1 depicted in Figure 3.  Fig. 3 we see that in the case of multiple equilibria (18)-(20), exactly the stable solution provides a large profit (Fig. 3). The Fig. 3 illustrates the effects that affect spatial strategies of firms. Before bifurcation point the effect of minimizing transport costs is dominate [14]. Firms choose the central agglomeration strategy to minimize a total distance of transportation. The growth of the transport tariff leads to a decrease in the total profit. In the bifurcation point begins to dominate the effect of market segmentation. Firms choose a dispersed strategy to monopolize adjacent markets. The growth of the transport tariff leads to an increase in total profits. The growth of total profit with growth of the transport tariff is due to the fact that when dispersion strategy, the firms supply more to adjoining markets and less to distant markets.
Note that the equilibrium profits of firms (4) are squares of supply volumes and, thus, "ignore" their negative values. For this reason, dispersion strategies (18) not take into account restrictions on full market coverage (4)- (5). Solving the systems of equations (4) and (18), (5) and (19), we find that dispersion strategies (18)