Question

Consider an industry with the inverse demand function P(Q) = 12 − Q, where Q is the sum of the outputs q₁ and q₂ of the two firms in the industry. There is only fixed cost in production (e.g. investment cost for machine) but no variable cost for producing each output (e.g. zero cost for inputs). Both two firms have the same fixed cost at 4. No fixed cost incurs if a firm decides not to operate. Suppose that firm 1 is the leader in the industry. That is, first firm 1 chooses q₁; then firm 2, after seeing q₁, chooses q₂. Assume that a firm does not operate when profit is zero. (Hints: The fixed cost affects only the firm’s decision of whether or not to operate; it does not affect the output a firm wishes to produce if it wishes to operate.)

1. If firm 1 chooses to produce q₁ = 8, does firm 2 wish to operate or not? Explain briefly.
2. Describe firm 2's best reply to firm 1's choice q₁. Remember to distinguish the two cases in which firm 2 operates and does not operate. Show your calculations.
3. If firm 2 wishes to operate, what output (q₁) maximizes firm 1's profit? What is firm 1's maximum profit in this case? Show your calculations. (Note that the value of q₁ must be such that firm 2 wishes to operate.)
4. If firm 2 does NOT wish to operate, what output (q₁) maximizes firm 1's profit? What is firm 1's maximum profit in this case? Show your calculations. (Note that the value of q₁ must be such that firm 2 does NOT wish to operate.)
5. In the subgame perfect equilibrium of this game, what is firm 1's output (q₁) and what is firm 2's output (q₂)? Explain briefly. No need to draw any graph.

Answer 1