Consider a horizontally differentiated product market in which two firms are located at points l₁ = 0 and l₂ = 1 on the line. Firms produce at marginal costs c. There is a continuum of consumers of mass 1 who 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 = r – τ (x)²- p₁ if she buys one unit from firm 1 and v₂ = r -τ (1- x)² -p₂ if she buys from firm 2. Firms have marginal costs equal to c.

1. Suppose that firms simultaneously set a uniform price for all consumers. Characterize the equilibrium of the game. Determine equilibrium profits.

2. Suppose now that firms can price-discriminate between consumers located on [0, 1/2] (segment A) and [1/2, 1] (segment B). Determine the profit function of each firm. Characterize the pure-strategy Nash equilibrium of the game in which firms simultaneously set prices. Note: You are allowed to restrict attention to the part of the demand function, which is relevant for the equilibrium analysis.

3. Compare your result in (2) to the game in which firms cannot discriminate. In which environment obtain firms larger profits? Explain your findings.

4. Suppose now that firm 1 can discriminate between the two consumer segments and that firm 2 cannot. Characterize the Nash equilibrium of the price game in which firm 1 sets a possibly different price for each consumer segment, while firm 2 sets the same price to all consumers.

5. Compare the firm’s profits in situation (4) to those in situation (2).

6. Consider the possibility for firms to “invest”” (non-negative number) into the possibility to price discriminate between consumers in segments A and B at investment cost I. Characterize the equilibrium of the two-stage game in which firms simultaneously decide whether to invest in stage 1 and simultaneously set prices in stage 2 depending on the level of the investment cost I. Comment on your result.