1. Suppose there are 8 firms located equidistantly around a circle whose circumference is equal to 1. There are 400 consumers distributed uniformly around the circle. Each consumer gets utility U = 3.5 from buying a unit of the good from a firm which sells their ideal brand (a firm located at the consumer's location).
Also, the consumer's utility decreases by 4 times the distance that the consumer is located from the firm. So a consumer who buys from firm i gets utility given by Ui = 3.5 - 4xi where xi is the distance from the consumer's location to firm i's location. The consumer's surplus from purchasing a unit of the good from firm i is the difference between their utility from purchasing a unit of the good from firm i and the price that firm i charges.
So CS = Ui - Pi. Also, the consumer has no alternatives, so the consumer will buy a unit of the good if maximum surplus gained from purchasing from any firm is greater than 0.
a) Consider a particular firm denoted Firm 2. Let Firm 1 be the firm located on one side of Firm 2, and let Firm 3 be the firm located on the other side of Firm 2. Suppose Firm 1 charges a price P1 = 2, and Firm 3 charges a price P3 = 2. Write a general expression for Firm 2's demand.
b) Suppose Firm 2 can produce its good at constant marginal cost, MC = 1.5. What price should firm 2 charge if it wants to maximize its profit?
c) Suppose the firm has no close neighbors so that it acts as a monopolist. Therefore it captures all of the consumers whose surplus is greater than or equal to 0. Write a general expression for the firm's demand.
d) Suppose the monopoly firm can produce its good at constant marginal cost, MC = 1.5. What price should it charge if it wants to maximize its profit?