A symmetric Circle City Hotelling model

05 Oct 2015

This model is very similar to Triangle City.

Circle City

Let the circle be of circumference \(d\pi\). There are \(n\) symmetric firms facing marginal cost \(c\) located equidistant from each other on the circle. Firm \(i\) faces two marginal consumers: \(x_{i-1}\) (the firm to its left) and \(x_i\) (the firm to its right). The marginal consumers are identical, and firm \(i\) and firm \(i+1\) compete over consumer \(x_i\). Consumers face a travel cost of t. \(x_i\)’s utility is given by:

\[\begin{align} v - p_i - tx_i &= v - p_{i+1} - t(\frac{td\pi}{n}-x_i) \cr \implies x_i &= \frac{p_{i+1}-p_i+(td\pi/n)}{2t} \cr \end{align}\]

Since the consumers are identical, this holds for every marginal consumer. Firm \(i\) faces \(x_i + x_{i-1}\), and solves

\[max_p (p_i-c)\left[\frac{p_{i+1}-p_i+(td\pi/n)}{2t} + \frac{p_{i-1}-p_i+(td\pi/n)}{2t} \right]\] \[\begin{align} \text{FOC:} &~~~ \left[\frac{p_{i+1}-p_i+(td\pi/n)}{2t} + \frac{p_{i-1}-p_i+(td\pi/n)}{2t} \right] - (p_i-c)\frac{2}{2t} \cr \implies & ~~~ \frac{td\pi}{n} - (p_i-c)\frac{1}{t} = 0 ~~~~~~ (\pi_i=\pi_j ~ \forall~i,j~\because \text{firms are symmetric})\cr \implies & p_i^* = c + \frac{td\pi}{n} \end{align}\]

So the markup is increasing in the diameter of the circle (\(\frac{\partial p}{\partial d} \gt 0\)) and the travel cost (\(\frac{\partial p}{\partial } \gt 0\)), and decreasing in the number of firms (\(n \to \infty\), \(p \to c\)).

The fact that firm \(i\) faces 2 (or \(k\), I suppose) symmetric marginal customers doesn’t affect the equilibrium prices \(p_i^*\), only equilibrium profits \(\pi_i^*\) (scaled by \(k\)).