Spatial Cournot competition in two intersecting circular markets

Transportation hubs have also received much attention in the transport geography literature. For instance, Hesse and Rodrigue (2004) mentioned that transport hubs are becoming more and more important and are attracting various distribution centers to locate nearby in the USA and Europe. Such transportation hubs are emerging along the Ohio River Valley in the USA and Benelux and the Netherlands in Europe.

There are few economic papers discussing competition along intersecting roads (markets). If we also consider some atypical market configurations in the literature, then there are more studies. For example, Braid (1989) analyzed spatial retail competition along intersecting roadways (two, three, or four ways). Retailers choose their locations in the first stage and engage in price competition in the second stage. He showed that there is no subgame perfect equilibrium in this model. Chen and Riordan (2007) developed the spokes model for nonlocalized spatial competition among oligopoly firms. Their spokes model was employed later by Caminal and Granero (2012), Germano and Meier (2013), and Buechel and Reggiani (2014). Moreover, a network structure with many nodes has been analyzed in Pinto et al. (2016), where they showed that firms prefer to be located at those network nodes with multiple exits. Other network structures for oligopoly competition can be found in Buechel and Roehl (2015), Hoernig et al. (2012), and Wang and Wang (2018).

Our two intersecting circular markets can also be applied to the case of conurbation, where two or more nearby cities may come to overlap as cities grow, such as Dallas–Fort Worth in the USA.

The intuition of the result in Pal (1998) has been further explained in Yu and Lai (2003, p. 573), where they described the “natural location effect” as a firm choosing the best location without considering the existence of its rival. Obviously, this effect is nil in a circular market. They also described the “strategic location effect” as a firm moving as far as possible from the rival’s location, which we identify here to avoid competition.

Other related circular Cournot models are: Shimizu and Matsumura (2003), Sun (2010), Pal and Sarkar (2006), Matsumura and Matsushima (2012), Matsushima and Matsumura (2003), Matsumura and Shimizu (2006), and Yu (2007).

For the analysis of spatial location choices, we believe that a setting of Cournot competition is better than a setting of Bertrand competition. This is because the competition in a Bertrand setting might be too strong and may encounter an undercutting problem leading to no location equilibrium, while Cournot competition is much milder and can escape from the undercutting trap.

For example, in the steamboat age, St. Louis was developed as an important inland city, because it was located at the intersection of the Mississippi River and the Missouri River.

For example, in Taipei metropolitan area, the government is now constructing a three-(intersecting)-circular and three-linear subway system with many intersections to make it easy for travelers to transfer.

If an \([x_1,y_2]\) pair is symmetric on one of the central east–west and the central north–south lines, then we defined it as semi-symmetric locations. For example, \([x_1,y_2]=[\frac{1}{4},\frac{1}{4}]\) is a semi-symmetric location pair, because it is only symmetric on the central north–south line; \([x_1,y_2]=[\frac{s}{2},1-\frac{s}{2}]\) is also a semi-symmetric location pair, because it is only symmetric on the central east–west line.

One of the referees asked whether \(x_1\in [0,\frac{1}{2}]\) and \(y_2=1-x_1\in [\frac{1}{2},1]\) (say, \([x_1,y_2]=[\frac{1}{4},\frac{3}{4}]\)) could be a equilibrium location pair. First, the pair \([x_1,y_2]=[\frac{1}{4},\frac{3}{4}]\) is neither fully symmetric nor semi-symmetric by our definition but symmetric on the linkage line of \(x=\frac{1}{4}\) and \(y=\frac{3}{4}\), while the line’s exact direction depends on the value of s. Second, we have proved that \([x_1,y_2]=[\frac{1}{4},\frac{3}{4}]\) is not an equilibrium location pair. Please see the proof in Appendix 3.

The detailed proofs that \([x_1,y_2]=[0,0]\) and \([x_1,y_2]=[1/2,1/2]\) are not location equilibria are provided in Appendices 1 and 2, respectively.

Although (\(x_1=0\), \(y_2=0\)) are fully symmetric locations, it is not an equilibrium location pair. Detailed proof is listed in Appendix 1.

More related explanation can be found in Yu and Lai (2003, p. 573). In contrast, the middle point of a linear market has a cost advantage for firms to serve the whole market, and this advantage is strong enough to dominate the effect of competition avoidance mentioned above, and thus we can see that firms agglomerate at the market center in Anderson and Neven (1991).

It is obvious that \([x_1,y_2]=[1/2,1/2]\) can avoid competition between firms. However, \([x_1,y_2]=[1/2,1/2]\) cannot be a location equilibrium, because the transportation cost is not minimized (see Appendix 2 for details).

The result that transportation hubs reduce competition comes from our setting that there exist two intersection points between two circular markets, and firms strategically choose to locate on separated intersection points to reduce competition. If there is only one transportation hub (\(s=0\)), then there exists no incentive to choose separated locations to avoid competition. In this case, the competition at the transportation hub is greater than at other locations.

One of the referees asked whether \([x_1,y_2]=[s,s]\) (or \([1-s,1-s]\)) might (also) be an equilibrium location. First, this location pattern is not fully symmetric in our definition and is excluded from our previous analysis. Second, we can show that \([x_1,y_2]=[s,s]\) (or \([1-s,1-s]\)) is not an equilibrium (see Appendix 4), because it will be dominated by \([x_1,y_2]=[s,1-s]\). Therefore, Proposition 1 is confirmed again.

Many thanks to one of the referees for pointing out this situation.

For example, given parameters \(a=2\), \(b=1\), \(t=1\), and \(s=1/32\), our equilibrium locations are valid when \(\lambda <32\). In other words, our equilibrium locations are quite robust. In fact, when \(a=2\), \(b=1\) and \(t=1\), all \(\lambda \le 14\) yield the same equilibrium as ours, no matter the value of s.

Similar calculations can prove that \([x_1,y_2]=[0,0]\) and \([x_1,y_2]=[1/2,1/2]\) are both local minimums in social welfare. Detailed calculations are available upon request to the authors. In fact, by considerable numerical calculation, this local maximum is also the unique global maximum.

Colombo (2016) showed that when \(c=0\), the equilibrium quantity is not well defined. Therefore, \(c>0\) is necessary.

Note that the consumer surplus under a hyperbolic demand function has no closed-form solution, so the corresponding welfare analysis cannot proceed.