Consider a horizontally differentiated product market in which firms are located at the extreme points of the unit interval. Firms produce at marginal costs equal to zero. A continuum of consumers of mass 1 are uniformly distributed on the unit interval. They have unit demand and have an outside utility of -∞. A consumer located at x ∈ [0, 1] obtains indirect utility v₁ = r₁ - tx - p₁ if she buys one unit from firm 1 and v₂ = r₂ - t(1 - x) - p₂ if she buys from firm 2. Firms have marginal costs equal to zero.

1. Suppose that firms have set prices at p₁ and p₂ respectively. Determine the demand function for each firm for each admissible price pair (p₁, p2).

2. Suppose that the social planner chooses first-best optimal prices. Which price pairs would be socially optimal.

3. Suppose that the two firms simultaneously prices. Determine the market equilibrium for all possible combinations of (r₁, r₂).

4. From now on consider the special case that t = 1. Suppose that each firm i can use advertising to increase the willingness to pay from r_i = 1 to r_i = 2. Consider the two-stage game in which firms choose advertising at the first stage and price at the second stage. Characterize the subgame- perfect Nash equilibrium of the game depending on the advertising cost A. Consider the cases A = 2/9, A = 3/9, and A = 4/9. What is the welfare ranking?

5. What are the equilibria for A = 5/18 and A = 7/18?

6. What are the welfare consequences of a reduction in advertising the advertising cost from A = 5/18 + ɛ to A = 5/18 - ɛ for the limit where ɛ → 0 (determine whether total surplus increases or decreases and by how much)? Comment on your result in one sentence.

7. What are the welfare consequences of a reduction in advertising the advertising cost from A = 7/18 + ɛ to A = 7/18 - ɛ for the limit where ɛ → 0 (determine whether total surplus increases or decreases and by how much)? Comment on your result in one sentence.