Question

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 cost. (Just need B through C answer please)

a. Find the equilibrium price, quantity and economic profit for the total market, consumer surplus and Dead weight loss

b. If the duopolists in question above behave, instead, according to the Bertrand model, what will be the equilibrium price, quantity and economic profit for the total market, consumer surplus and Dead weight loss

c. If the duopolists in question above behave according to the Stackelberg Leader-Follower model, what will be the equilibrium price, quantity and economic profit for the total market, consumer surplus and Dead weight loss (dupolists is the leader)

d. If the duopolists in question above behave as a shared monopoly what will be the equilibrium price, quantity and economic profit for the total market, consumer surplus and Dead weight loss

Answer 1

b. If the duopolists in question above behave, instead, according to the Bertrand model, what will be the equilibrium price, quantity and economic profit for the total market, consumer surplus and Dead weight loss

Under Bertrand model with identical good and same marginal cost the outcome is determined by P = MC because undercutting results in pricing reaching the MC. Here MC is 30 so each firm produces Q = 90 - 30 = 60/2 units = 30 units and total market quantity in equilibrium is 60 units. Price = MC = $30. Economic profit is zero because MC = AC = P all are equal to $30. CS = 0.5*(max price - current price)*qty = 0.5*(90 - 30)*60 = $1800. There is no deadweight loss because P = MC results in no efficiency loss.

c. If the duopolists in question above behave according to the Stackelberg Leader-Follower model, what will be the equilibrium price, quantity and economic profit for the total market, consumer surplus and Dead weight loss (dupolists is the leader)

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*(current price - MC)*(competitive quantity - current quantity) = 0.5*(45 - 30)*(60 - 45) = 112.50.