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Consider two firms with the same constant average and marginal cost, AC = MC = 5...

Consider two firms with the same constant average and marginal cost, AC = MC = 5 (meaning the cost function is T C1 = 5q1 , T C2 = 5q2 ), facing the market demand curve q1 + q2 = 53 − P . We will use the Stackelberg model to analyze what will happen if one of the firms makes its output decision before the other.

  1. What is each firm’s equilibrium output and profit if they behave noncooperatively and move simultaneously?

  2. Now suppose Firm 1 is the Stackelberg leader (i.e., makes its output decisions before Firm 2). What is each firm’s equilibrium output and profit?

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Answer #1

Demand is P = 53 - q1 - q2. When they move simultaneously they face the demand function P = 53 - q1 - q2 and therefore, the profit functions are

π1 = (P - AC)q1 or π1 = (53 - q1 - q2 - 5)q1 or π1 = 48q1 - q1^2 - q1q2

and

π2 = (P - AC)q2 or π1 = (53 - q1 - q2 - 5)q2 or π2 = 48q2 - q2^2 - q1q2

Profit is maximum when  π1'(q1) = 0 and π2'(q2) = 0

48 - 2q1 - q2 = 0 and 48 - 2q2 - q1 = 0

q1 = 24 - 0.5q2 and q2 = 24 - 0.5q1

Solve this to get

q1 = 24 - 12 + 0.25q1

q1* = 12/0.75 = 16 units and q2* = 16 units

Price P = 53 - 32 = $21 per unit

Profits for each firm = (21 - 5)*16 = $256

Firm 1 is the Stackelberg leader. It produces (A - c)/4 = (53 - 5)/2 = 24 units and Firm 2 produces half of Firm 1 output which is 12 units. Price is 53 - 36 = $17.

Firm 1's profit = (17 - 5)*24 = $288 and firm 2's profit = (17 - 5)*12 = $144.

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