Question

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.

Answer #1

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 – 4q2^{2} – 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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