Question

If the duopolists in question 24 behave according to the Stackelberg Leader-Follower model, determine the (1) equilibrium price, (2) quantity, and (3) economic profits for the total market and (4) the consumer surplus, and (5) dead weight loss.

24:Cournot duopolists face a market demand curve given by P = 90 - Q where Q is total market demand. Each firm can produce output at a constant marginal cost of 30 per unit. There are no fixed costs. Determine the (1) equilibrium price, (2) quantity, and (3) economic profits for the total market, (4) the consumer surplus, and (5) dead weight loss.

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) = (90 – (q1 + q2))q2 = 90q2 – q2² – q1q2

Marginal revenue = 90 – 2q2 – q1

Marginal cost = 30

Solve for the reaction function

90 – 2q2 – q1 = 30

60 - q1 = 2q2

This gives q2 = 30 - 0.5q1

Incorporate this in the reaction function of firm 1

Total revenue for firm 1 = P*(q1) = (90 – (q1 + q2))q1

TR = 90q1 - q1^2 - q1q2

= 90q1 - q1^2 - q1*(30 - 0.5q1)

= 90q1 - q1^2 - 30q1 + 0.5q1^2

= 60q1 - 0.5q1^2

MR = MC

60 - q1 = 30

q1 = 30 and so q2 =  30 - 0.5*30 = 15 units.

(1) equilibrium price,= 90 - (30 + 15) = $45

(2) quantity, = 45 units

(3) economic profits for the total market = (P - MC)*total quantity = (45 - 30)*45 = $675

(4) the consumer surplus, = 0.5*(Max price - current price)*current qty = 0.5*(90 - 45)*45 = 1012.50

and (5) dead weight loss = 0.5*(45 - 30)*(60 - 45) = 112.50.