Suppose two firms in an industry face linear inverse demand curves P_i (q_i, q_j) = 7- q_i -q_j , i = 1; 2, i ≠ j. Firms compete in a two-stage game; first they set capacity and then they set price or output. At the first stage firms set capacities, at this stage the marginal costs of capacity is 6. Suppose that firms have zero marginal costs of production up until installed capacity and that production above capacity is not feasible. In case of rationing, rationing is assumed to be efficient.

1. Suppose each firm has a capacity of 7. Analyze competition at stage 2. Determine the Nash equilibrium if both firms set prices.

2. Consider the same situation as in (1) but suppose that firms choose quantities, not prices at stage 2. Determine the Nash equilibrium.

3. Consider the same situation as in (1) but suppose that consumers do not observe price and incur a cost of 1/2 if, after visiting one firm, they decide to visit the other firm. [You can think of identical consumers with the demand function as given above]. Characterize the equilibrium if both firms set prices. (What is the appropriate equilibrium concept here?). Give an explanation (at most 2 sentences).

4. Suppose that firms have given capacities q̅₁ and q̅₂, respectively. If firm 1 is the high-price firm, what is its demand function? Determine the Nash equilibrium in prices (provided that qi49=24, i = 1; 2). Show that equilibrium prices satisfy p₁ = p2 = a – q̅₁ – q̅₂.

5. Determine the subgame perfect equilibrium of the two-stage game in which firms first set capacities and then prices. Give an explanation (at most 3 sentences).

6. Suppose that firms collude at the stage at which they set capacity. What should they do?

7. Suppose that firms are able to use a less costly technology (e.g., the marginal cost of capacity falls from 6 to 11 = 2). What are the competitive effects of this reduction in capacity costs? What would happen if those costs fell to zero? Discuss your results.