Circle Model (Salop 1979) Consider firms locating on a circle of unit circumference, say, one mile. Consumers are distributed uniformly around the circle, and each buys one unit of the good from the firm charging the lowest delivered price. There are 200 consumers. Costs are made up of
(i) a fixed cost per firm of F = 4.
(ii) a marginal cost of production equal to 1 per unit.
(iii) transport cost (delivery costs) of 2 per unit distance per unit of the good.
In the market, firms set a mill price (the price at the factory). Delivered prices are equal to the mill price plus transport costs. (Problem requires calculus.)
(a) What is the density of consumers around the circle? How many consumers would reside in a circle segment with a length of 1/(2N) miles?
(b) Find the socially optimal number of firms (firms can also be interpreted as number of products).
(c) What would be the socially optimal number of firms if F = 0? What does this suggest about the likelihood of F = 0? What is the role of F in determining the optimum number of firms? The delivery cost?
(d) Find the Nash equilibrium number of firms/products if they are distributed at equal intervals around the circle and the Nash equilibrium price in the market if there is simultaneous free entry.
(e) Find the subgame perfect number of firms and the price charged in the market if firms enter sequentially and location costs are sunk.
(f) Suppose location costs are not sunk. Redo (e).