Assume there are two firms in a Stackelberg game. Firm 1 chooses its output first and Firm chooses its output after knowing what Firm 1’s output is. Market demand is given by P = 1,000 – 4Q. Each firm has identical marginal cost of 100. Determine the Nash equilibrium for this game.
In Stackelberg model where firm 1 is a first mover, it must take the reaction function of firm 2 in its computation of marginal revenue.
Derivation of firm 2’s reaction function
Total revenue of firm 2 = P*(q2) = (1000 – 4(q1 + q2))q2 = 1000q2 – 4q22 – 4q1q2
Marginal revenue = 1000 – 8q2 – 4q1
Marginal cost = 100
Solve for the reaction function
1000 – 8q2 – 4q1 = 100
8q2 = 900 – 4q1
q2 = (112.5 – 0.5q1)
Incorporate this in the reaction function of firm 1
Total revenue for firm 1 = P*(q1) = (1000 – 4(q1 + q2))q1
= (1000 – 4q1 – 4((112.5 – 0.5q1)))q1
= 1000q1 - 4q1^2 - 450q1 + 2q1^2
= 550q1 - 2q1^2
MR1 = 550 – 4q1
Equate MR = MC
550 – 4q1 = 100
q1 = 112.5
q2 = 112.5 – 0.5*112.5 = 56.25
This is the Nash equilibrium for this game.
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