Question

Suppose that the market price is given by max{0,10 -Q} where Q is the total market quantity. Firms in this market choose quantity and then the price in the market is revealed. Suppose that there are two firms in the market, Firm A and Firm B. Each firm has a constant marginal cost of one and no fixed costs. Suppose that Firm A chooses output first, it is then observed by Firm B and then Firm B makes her choice of output. Find the equilibrium level of each firm’s output and profit in this market. Compare this outcome to the outcome when both firms choose the level of output simultaneously. (Stakelberg Duopoly)

Answer 1

In Stackelberg model where firm B is a first mover, it must take the reaction function of firm A in its computation of marginal revenue.

Derivation of firm A’s reaction function

Total revenue of firm A = P*(q1) = (10 – (q1 + q2))q1 = 10q1 – q1² – q1q2

Marginal revenue = 10 - 2q1 - q2

Marginal cost = 1

Solve for the reaction function

10 - 2q1 - q2 = 1

9 - q2 = 2q1

q1 = 4.5 - 0.5q2

Incorporate this in the reaction function of firm B

Total revenue for firm B = P*(q2) = (10 – (q1 + q2))q2

= 10q2 - q1q2 - q2^2

= 10q2 - q2^2 - q2*(4.5 - 0.5q2)

= 10q2 - q2^2 - 4.5q2 + 0.5q2^2

= 5.5q2 - 0.5q2^2

MR = 5.5 - q2

MC = 1

MR = MC

5.5 - q2 = 1

q2 (B) = 4.5

q1 (A) = 4.5 - 0.5*4.5 = 2.25

Price = 10 - (4.5 + 2.25) = 3.25

Profit for B = (3.25 - 1)*4.5. = 10.125

Profit for A = (3.25 - 1)*2.25 = 5.0625

This is the equilibrium level of each firm’s output and profit in this market

Cournot outcome has both firms producing (A - c)/3B = (10 - 1)/3 = 3 units each.