Is the Retail Gasoline Market Local or National?

Abstract

This paper studies the effect of station density on prices in the retail gasoline market in the Czech Republic. We estimate the impact of the number of competing stations in various driving-distance ranges around each station on prices. We find that station density has a negative effect on prices; the effect decreases with distance and is statistically significant up to six kilometers. This suggests that the retail gasoline market is local rather than national.

Appendix A: Modeling the effects of station density

The model has a structure of M spokes. Each spoke has a length d _i . There are N+1 firms, where N≤M. Firms have a constant marginal cost c. Firm 0 is located at the center; N firms are located at the extremes of N spokes. Firms are indexed such that firm i is located at the extreme of spoke i. There is a continuum of consumers with mass 1. The mass of consumers which are uniformly distributed along each spoke is 1/M. So, the distribution function of consumers along spoke i is 1/(M d _i ).

The consumers behave in the same way as in the Hotelling model. Each consumer buys one unit of goods; the preferences can be described by the utility function U = R−p−T x, where R is the reservation price; p is the price to be paid; T are the transportation costs; and x is the distance to the firm. The demand for the production of firm 0 along the spoke is given as follows

• $D_{0}=\frac {R-p_{0}}{T}$ if there is no competitor.

• $D_{0}=\frac {1}{2}+\frac {p_{i}-p_{0}}{2d_{i}T}$ if there is a competitor.

The profit of firm 0 is

$${\Pi}_0=\frac{M-N}{M}(p_0-c)\frac{R-p_0}{T}+\frac{1}{M} \sum\limits_{i=1}^N (p_0-c) \left( \frac{1}{2}+\frac{p_i-p_0}{2d_iT} \right) $$

The profit of firm $i \in \{1,\dots ,N \}$ is

$${\Pi}_i=(p_i-c) \left( \frac{1}{2}+\frac{p_0-p_i}{2d_iT} \right)$$

We derive best-response functions and solve for the equilibrium price $p_{0}^{*}$. The equilibrium price $p_{0}^{*}$ is implicitly given by the following equation.

$$ (M-N)(R-2p_{0}^{*}+c)+ \sum\limits_{i=1}^{N} \left( \frac{3T}{4}-\frac{ p_{0}^{*}-c}{4d_{i}} \right)=0 $$

(2)

Suppose without loss of generality that there are N ₁ firms at a distance d ₁ from the center, N ₂ firms in distance d ₂ from the center, and so on. Of course, it holds that $N_{1}+N_{2}+\dots +N_{k}=N$. The distances are indexed such that $d_{1}<d_{2}<\dots <d_{k}$.

By differentiating the Eq. 2 according to N _i we can express how the equilibrium price depends on the number of firms at a distance d _i :

$$ -R+2p_{0}^{*}-c+\frac{3T}{4}-\frac{ p_{0}^{*}-c}{4d_{i}}=\frac{\partial p_{0}^{*}}{\partial N_{i}} \left( 2(M-N)+\frac{1}{4} \left( \frac{N_{1}}{d_{1}}+\dots+ \frac{N_{k}}{d_{k}} \right) \right) $$

(3)

It follows from the condition (2) that the left-hand side of the equation is negative. The second term on the right-hand side is positive. Hence, we may conclude that if you increase the number of firms at any distance level d _i , the price of firm 0 falls:

$$\frac{\partial p_0^*}{\partial N_i}<0$$

Suppose that d _i <d _j . We want to compare $\frac {\partial p_{0}^{*}}{\partial N_{i}}$ and $\frac {\partial p_{0}^{*}}{\partial N_{j}}$, i.e. what happens with the price if we add a competitor at a distance d _i or at a distance d _j . The comparison of these two effects depends only on the left-hand side of Eq. 3 because the second term on the right-hand side is the same for both. Obviously, the absolute value of the left-hand side is decreasing in d _i . Therefore it follows that if we increase the number of firms at a lower distance d _i <d _j , the price of firm 0 falls more:

$$\left| \frac{\partial p_0^*}{\partial N_i} \right| > \left| \frac{\partial p_0^*}{\partial N_j} \right|$$