The city is of length 1 unit, firms have constant marginal costs of c per unit and no fixed costs, and each consumer buys one unit of the good. Consumer i's utility derived from buying product j is given by uij = u t(xi yj ) 2 pj , for j = 1, 2. where t is a taste cost for the consumer. Suppose that consumers are uniformly distributed across the whole city (between 0 and 1) and that firm 1 is at one end of the city (y1 = 0) while firm 2 is at the other end (y2 = 1).

(a) What is the demand for each firm's product?

(b) Find the Nash equilibrium prices and profits.

(c) Consider the same set up as above but now assume that at the start of the game both firms can invest in advertising that affects the taste parameter t. In particular, let t = a1 + a2 where ai is the level of advertising for firm i, i = 1, 2. The cost of advertising for each firm is a 2 i . Find the optimal advertising choices in a subgame perfect equilibrium.

(d) Find the level of advertising that maximizes the joint profit of both firms. How is this different from part (c)? Why?