Question

In a Stackelberg duopoly, each firm has the total cost function C=40q_i, where q_i is the quantity supplied by an individual firm (i=1,2). The total market demand is given by Q = 100 - 0.5p. Firm 1 is the leader and Firm 2 is the follower.

What is the Nash-Stackelberg output level for each? What is the Nash-Stackelberg market price and quantity? What is each firm's profit?

Answer 1

Marginal cost for both firms is $40. Inverse demand is P = 200 - 2Q

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) = (200 – 2(q1 + q2))q2 = 200q2 – 2q2² – 2q1q2

Marginal revenue = 200 – 4q2 – 2q1

Marginal cost = 40

Solve for the reaction function

200 – 4q2 – 2q1 = 40

160 - 2q1 = 4q2

This gives q2 = 40 - 0.5q1

Incorporate this in the reaction function of firm 1

Total revenue for firm 1 = P*(q1) = (200 – 2(q1 + q2))q1

TR = 200q1 - 2q1^2 - 2q1q2

= 200q1 - 2q1^2 - 2q1*(40 - 0.5q1)

= 200q1 - 2q1^2 - 80q1 + q1^2

= 120q1 - q1^2

MR = MC

120 - 2q1 = 40

q1 = 40 and so q2 = 40 - 0.5*40 = 20 units.

Nash-Stackelberg output level for leader is 40 units and for follower it is 20 units

Nash-Stackelberg market price is $80

Market quantity is 60 units

Leader's profit = (80 - 40)*40 = $1600

Follower's profit = (80 - 40)*20 = $800