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.
