Hotelling's model of competition on a linear beach is used widely in many
applications, but one application that is difficult to study in the model is
free entry. Free entry is easiest to study in a model with symmetric firms,
but more than two firms on a line cannot be symmetric because those located
nearest the endpoints will have only one neighboring rival, whereas those
located nearer the middle will have two.
To avoid this problem, Steven Salop introduced competition on a circle.
\(^{18}\) As in the Hotelling model, demanders are located at each point, and
each demands one unit of the good. A consumer's surplus equals \(v\) (the value
of consuming the good) minus the price paid for the good as well as the cost
of having to travel to buy from the firm. Let this travel cost be \(t d\), where
\(t\) is a parameter measuring how burdensome travel is and \(d\) is the distance
traveled (note that we are here assuming a linear rather than a quadratic
travel-cost function, in contrast to Example 15.5 .
Initially, we take as given that there are \(n\) firms in the market and that
each has the same cost function \(C_{i}=K+c q_{i}\) where \(K\) is the sunk cost
required to enter the market [this will come into play in part (e) of the
question, where we consider free entry] and \(c\) is the constant marginal cost
of production. For simplicity, assume that the circumference of the circle
equals 1 and that the \(n\) firms are located evenly around the circle at
intervals of \(1 / n\). The \(n\) firms choose prices \(p_{i}\) simultancously.
a. Each firm \(i\) is free to choose its own price \(\left(p_{i}\right)\) but is
constrained by the price charged by its nearest neighbor to either side. Let
\(p^{*}\) be the price these firms set in a symmetric equilibrium. Explain why
the extent of any firm's market on either side
\((x)\) is given by the equation
$$p+t x=p^{*}+t[(1 / n)-x]$$
b. Given the pricing decision analyzed in part (a), firm \(i\) sells \(q_{i}=2 x\)
because it has a market on both sides. Calculate the profit-maximizing price
for this firm as a function of \(p^{*}, c, t,\) and \(n\)
c. Noting that in a symmetric equilibrium all firms' prices will be equal to
\(p^{*},\) show that \(p_{i}=p^{*}=c+t / n .\) Explain this result intuitively.
d. Show that a firm's profits are \(t / n^{2}-K\) in equilibrium.
e. What will the number of firms \(n^{*}\) be in long-run equilibrium in which
firms can freely choose to enter?
f. Calculate the socially optimal level of differentiation in this model,
defined as the number of firms (and products) that minimizes the sum of
production costs plus demander travel costs. Show that this number is
precisely half the number calculated in part (e). Hence this model illustrates
the possibility of overdifferentiation.
