The market inverse demand curve is P = 90 – Q, where Q is the total market output consisting of Firm 1's output, q1, and Firm 2's output, q2. Both firms have a constant marginal cost of $10. If Firm 1 selects its output level first, how much output does each firm produce?
Question
The market inverse demand curve is P = 90 – Q, where Q is the total market output consisting of Firm 1's output, q1, and Firm 2's output, q2. Both firms have a constant marginal cost of $10. If Firm 1 selects its output level first, how much output does each firm produce?
Solution
To solve this problem, we need to use the concept of Stackelberg competition, where one firm (the leader, Firm 1 in this case) chooses its output level first and the other firm (the follower, Firm 2) then chooses its output level.
Step 1: Firm 2's Reaction Function First, we need to find Firm 2's reaction function. This is the output level q2 that maximizes Firm 2's profit given the output level of Firm 1. Firm 2's profit is its revenue minus its cost, which is (Pq2 - 10q2). Substituting P = 90 - Q and Q = q1 + q2, we get Firm 2's profit as ((90 - q1 - q2)q2 - 10q2). Taking the derivative of this with respect to q2 and setting it equal to zero, we get the reaction function q2 = 40 - 0.5*q1.
Step 2: Firm 1's Best Response Next, we need to find the output level q1 that maximizes Firm 1's profit given Firm 2's reaction function. Firm 1's profit is (Pq1 - 10q1). Substituting P = 90 - Q, Q = q1 + q2, and q2 = 40 - 0.5q1, we get Firm 1's profit as ((90 - q1 - (40 - 0.5q1))q1 - 10q1). Taking the derivative of this with respect to q1 and setting it equal to zero, we get q1 = 40.
Step 3: Firm 2's Output Finally, we substitute q1 = 40 into Firm 2's reaction function to get q2 = 40 - 0.5*40 = 20.
So, Firm 1 produces 40 units and Firm 2 produces 20 units.
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