15.11 Competition on a circle 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 while 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 (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 id, 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 a firms in the market and that each has the same cost function C, K+ cq,, 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 I and that the # firms. are located evenly around the circle at intervals of 1/n. The firms choose prices p, simultaneously.

a. Each firm i is free to choose its own price (p) 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+tx=p+t(1/n)-x.

b. Given the pricing decision analyzed in part (a), firm i sells q, 2x because it has a market on both sides. Calculate the profit-maximizing price for this firm as a function of p",

c, t, and .

c. Noting that in a symmetric equilibrium all firms' prices will be equal to p*, show that p = p+t/n. Explain this result intuitively.

d. Show that a firm's profits are t/m - K in equilibrium.

e. What will the number of firms * 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.