Spatial Cournot competition in a linear–circular market

Abstract

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.

Appendix

1.1 Proof of Proposition 1

Solving $\partial R_1/\partial y_1=0$ and $\partial R_2/\partial y_2=0$ yields two equilibrium location pairs. The first one is the agglomeration solution at the center of the main street such that $(y_1^*,y_2^*)=(\frac{1}{2\pi },\frac{1}{2\pi })$. The second one is a separated solution such that $(y_1^*,y_2^*)=(\frac{2\pi \theta a-4\pi a-2\pi t-t\theta +6t}{4(2+\theta )t\pi },\frac{4\pi a-2\pi \theta a+2\pi t+5t\theta +2t}{4(2+\theta )t\pi })$. We divide our proof into three cases for the validity of solutions.

• 1. Since the first-order conditions $\partial R_1/\partial y_1=\partial R_2/\partial y_2=0$ at $y_1=y_2=L/2$, we should check the second-order conditions for the agglomeration solution:

  $$\begin{aligned} \frac{\partial ^2 R_1}{\partial y_1^2}=\frac{\partial ^2 R_2}{\partial y_2^2}=\frac{-4t(2t-4\pi a-\pi t-2t\theta +2\pi \theta a)}{9\pi b}<0. \end{aligned}$$

  The above condition implies that $\theta >\frac{4\pi a+\pi t-2t}{2(\pi a-t)}\equiv \theta _A^{{\textit{SOC}}}$ is required for $\partial ^2 R_1/\partial y_1^2<0$ and $\partial ^2 R_2/\partial y_2^2<0$. Moreover, the stability condition in location space $\frac{\partial ^2 R_1}{\partial y_1^2}\cdot \frac{\partial ^2 R_2}{\partial y_2^2} -\frac{\partial ^2 R_1}{\partial y_1\partial y_2}\cdot \frac{\partial ^2 R_2}{\partial y_1\partial y_2} =\frac{16t^2(\theta -2)(2\pi a-t)((\theta -2)2\pi a-t(2\pi +3\theta -2))}{81b^2\pi ^2}\ge 0$, which implies $\theta \ge \theta _A^*$, because $\theta _A^{SOC}>2$ and $\theta _A^*>2$.

• 2. From the second solution, $(y_1^*=0,y_2^*=L)$ at the separate solution requires a boundary condition $\theta \le \theta _\mathrm{Pal}^*=\frac{2(2\pi a+\pi t-3t)}{2\pi a-t}$.

• 3. Plugging the separate locations into the second-order conditions yields $\partial ^2R_1/\partial y_1^2=\partial ^2R^2/\partial y_2^2$ $=-\frac{4(\pi +\theta )t^2}{9b\pi }<0$. In other words, the separate equilibrium locations always satisfy the second-order condition. However, $y_1^*<L/2$ (or the same condition $y_1^*<y_2^*$) is required for this solution, which implies $\theta <\theta _A^*=\frac{2(2\pi a+\pi t-t)}{2\pi a-3t}$. Moreover, the stability condition $\frac{\partial ^2 R_1}{\partial y_1^2}\cdot \frac{\partial ^2 R_2}{\partial y_2^2} -\frac{\partial ^2 R_1}{\partial y_1\partial y_2}\cdot \frac{\partial ^2 R_2}{\partial y_1\partial y_2} =\frac{-16t^2(\theta -2)(2\pi a-t)((\theta -2)2\pi a-t(2\pi +3\theta -2))}{81b^2\pi ^2}\ge 0$ is satisfied here for $\theta \le \theta _A^*$.

Therefore, we have proved Proposition 1. $\square$

1.2 Proof of Proposition 2

The interior solutions are valid when $\theta _\mathrm{Pal}^*<\theta <\theta _A^*$. Differentiating $y_1^*$ with respect to $\theta$, a, and t yields

$$\begin{aligned}&\frac{\partial y_1^*}{\partial \theta }=\frac{4\pi a-4t+\pi t}{2t\pi (\theta +2)^2}>0, \\&\frac{\partial y_1^*}{\partial a} =\frac{\theta -2}{2t(\theta +2)}>0, \quad \text{ since }\; \theta _\mathrm{Pal}^*>2, \\&\frac{\partial y_1^*}{\partial t} =\frac{a(2-\theta )}{2t^2(\theta +2)}<0, \quad \text{ since }\; \theta _\mathrm{Pal}^*>2, \end{aligned}$$

which proves Proposition 2. $\square$

1.3 Proof of Proposition 3

• 1. Solving $y_1^S=\frac{1}{2\pi }$ for $\theta$ yields $\theta =\frac{8\pi a+7\pi t-4t}{4\pi a-9t}=\theta _A^\mathrm{SW}$.

• 2. Solving $y_1^o=0$ for $\theta$ yields $\theta =\frac{8\pi a+7\pi t-18t}{2(2\pi a-t)}$.

• 3. From the condition $y_1^o>0$, the interior solution is valid when $\theta _\mathrm{Pal}^\mathrm{SW}<\theta <\theta _A^\mathrm{SW}$.

$\square$

1.4 Proof of Proposition 4

• 1. Solving $\partial R_1/\partial y_1=0$ and $\partial R_2/\partial y_2=0$ simultaneously yields two solutions: $(y_1^*,y_2^*)=(\frac{1}{2\pi },\frac{1}{2\pi })$ and $(y_1^*,y_2^*)=(\frac{2\theta \pi a-\theta t-3\pi t}{4\theta \pi t},L-y_1^*)$. The agglomeration solution is valid only when the second-order conditions $\partial ^2 R_1/\partial y_1^2=\partial ^2 R_2/\partial y_2^2=\frac{-8t(\theta (\pi a-t)-\pi t)}{9\pi b}<0$, which leads to the condition $\theta >\frac{\pi t}{\pi a-t}$. Furthermore, the stability condition implies $\theta \ge \frac{3\pi t}{2\pi a-3t}$, which is greater than $\frac{\pi t}{\pi a-t}$. Therefore, $\theta \ge \frac{3\pi t}{2\pi a-3t}$ leads to the agglomeration solution.

• 2. The dispersed solution is valid by the assumption $a\ge 2t(1+\frac{1}{\pi })$. The second-order conditions for this solution are always valid since $\partial ^2 R_1/\partial y_1^2=\partial ^2 R_2/\partial y_2^2=-\frac{4t^2(\pi +\theta )}{9\pi b}<0$. The endpoint solution $(y_1^*,y_2^*)=(0,L)$ is valid when this dispersed solution satisfies the corner solution, which implies $\theta \le \frac{3\pi t}{2\pi a-t}$.

• 3. The interior solution $0<y_1^*<\frac{L}{2}$ implies the condition $\frac{3\pi t}{2\pi a-t}<\theta <\frac{3\pi t}{2\pi a-3t}$. Finally, $\partial y_1^*/\partial \theta =\frac{3}{4\theta ^2}>0$, $\partial y_1^*/\partial a=\frac{1}{2t}>0$, $\partial y_1^*/\partial t=-\frac{a}{2t^2}<0$ for the interior solution. $\square$