Question

There are two firms in an industry. The industry demand curve is given by p = 3,600 - 4q. Each firm has one manufacturing plant and each firm i has a cost function C(q_i) = q^2_i, where q_i is the output of firm i. The two firms form a cartel and arrange to split total industry profits equally.
Required:
a) Under this cartel arrangement, they will maximize joint profits if ___________.

Answer 1

Answer:

Under this cartel arrangement, they will maximize joint profits if each of the firms produces 257.14 units and sells at $1,542.88 per unit.

Explanation:

p = 3,600 - 4q ..................................................... (1)

C(q_i) = q_i^2 ………………………………… (2)

MC_i = dC(q_i)/dq = 2q_i ……………………. (3)

Since q = q_1 + q_2, we have:

p = 3,600 - 4(q_1 + q_2)

p = 3,600 - 4q_1 - 4q_2 .................................... (4)

For Firm 1:

TR_1 = p * q_1 = (3,600 - 4q_1 - 4q_2)q_1

TR_1 = 3,600q_1 - 4q_1^2 - 4q_2q_1

MR_1 = dTR_1/dq_1 = 3,600 - 8q_1 - 4q_2

From equation (3), MC_1 = 2q_1

Since at the optimum MC_1 = MR_1, we have:

2q_1 = 3,600 - 8q_1 - 4q_2

10q_1 = 3,600 - 4q_2

q_1 = (3,600 - 4q_2)/10

q_1 = 360 - 0.4q_2 .......... (5)

For Firm 2:

TR_2 = p * q_2 = (3,600 - 4q_1 - 4q_2)q_2

TR_2 = 3,600q_2 - 4q_2q_1 - 4q_2^2

MR_2 = dTR_2/dq_2 = 3,600 - 4q_1 - 8q_2

From equation (3), MC_2 = 2q_2

Since at the optimum MC_2 = MR_2, we have:

2q_2 = 3,600 - 4q_1 - 8q_2

10q_2 = 3,600 - 4q_1

q_2 = (3,600 - 4q_1)/10

q_2 = 360 - 0.4q_1 .......... (6)

a. Calculation of Cournot equilibrium quantities

Substituting equation (6) for q_2 into equation (5), we have:

q_1 = 360 - 0.4(360 - 0.4q_1)

q_1 = 360 – 144 + 0.16q_1

q_1(1 – 0.16) = 216

q_1 = 216 / 0.84

q_1 = 257.14 <------------- Cournot equilibrium quantity for firm 1

Substitute for q_1 in equation (6), we have:

q_2 = 360 - 0.4(257.14)

q_2 = 360 – 102.86

q_2 = 257.14 <------------- Cournot equilibrium quantity for firm 2

b. Calculation of Cournot equilibrium price

Substitute for q_1 and q_2 into equation (4), we have:

p = 3,600 – 4(257.14) – 4(257.14)

p = 1,542.88

Therefore, under this cartel arrangement, they will maximize joint profits if each of the firm produces 25.14 and sells at $1,542.88 per unit.