Question

A homogeneous product duopoly faces a market demand function given by p = 300 - 3Q,where Q = q1 + q2. Both firms have constant marginal cost MC = 100.

1a. Derive the equation of each firm's quantity reaction function.

b. What are the Cournot equilibrium quantity and price in this market? How much does each firm produce?

c. What would be the equilibrium price and quantity in this market if it were perfectly competitive?

d. What would the equilibrium price in this market be if the two firm colluded to set the monopoly price?

e. What is the Bertrand equilibrium price and quantity in this market?

f. Suppose Firm 1 is the Stackelberg leader, what is the equilibrium price in this market if Firm 2 plays the follower in this duopoly market? What is the equilibrium quantity? How much does each firm produce?

g. In the diagram below, illustrate the Cournot, Bertrand and Stackelberg equilibria in parts (b), (e) and (f). Calculate the consumer surplus in each equilibrium.

Graph:
Price for vertical axis, quantity for horizontal axis. Demand line with a MC horizontal line.

Answer 1

(a)

p = 300 - 3Q = 300 - 3q1 - 3q2

For firm 1,

TR1 = p x q1 = 300q1 - 3q1² - 3q1q2

MR1 = TR1/q1 = 300 - 6q1 - 3q2

Setting MR1 = MC,

300 - 6q1 - 3q2 = 100

6q1 + 3q2 = 200..........(1) [reaction function, firm 1]

For firm 2,

TR2 = p x q2 = 300q2 - 3q1q2 - 3q2²

MR2 = TR2/q2 = 300 - 3q1 - 6q2

Setting MR2 = MC,

300 - 3q1 - 63q2 = 100

3q1 + 6q2 = 200..........(2) [reaction function, firm 2]

(b)

Multiplying (2) by 2,

6q1 + 12q2 = 400.........(3)

6q1 + 3q2 = 200.........(1)

(3) - (1) yields:

9q2 = 200

q2 = 22.22

q1 = (200 - 6q2)/3 [from (1)] = [200 - (6 x 22.22)]/3 = (200 - 133.32)/3 = 66.68/3 = 22.22

Q = 22.22 + 22.22 = 44.44

p = 300 - (3 x 44.44) = 300 - 133.32 = 166.68

(c)

In perfect competition, p = MC.

300 - 3Q = 100

3Q = 200

Q = 66.67

p = MC = 100

(d)

In monopoly, MR = MC.

TR = p x Q = 300Q - 3Q²

MR = dTR/dQ = 300 - 6Q

300 - 6Q = 100

6Q = 200

Q = 33.33

p = 300 - (3 x 33.33) = 300 - 99.99 = 200.01

NOTE_ As per Chegg Answering Policy, 1st 4 parts are answered.