Consumers are uniformly distributed along a linear city of length 1 (assume, for simplicity, U[0, 1]), have transportation costs t per unit of distance, and have unit demands. Except for their location, they are identical and have surplus s for the good sold by the monopolist. The monopolist is considering establishing shops at the borders of the city or in the center (at location 0, 1, or 1/2). The monopolist may set up 1, 2, or 3 shops with the cost of each shop being fixed f. There is no marginal cost of production.
a. Show that the monopolist will always open a shop at location 1/2 if it opens any shops.
b. Derive the profits the monopolist will earn if it opens 1 or 3 shops (there are multiple cases for each number of shops demarcated by the relationship between s and t and whether the market is covered.)
c. Explain whether the monopolist should open 1 shop or 3 shops. In other words, what are the conditions on s, t, and f for the monopolist to choose 1 shop as compared to 3?
Use hotelling model for horizontal differentiation