Question

 A monopolistic producer of two goods, G1 and G2, has a total cost function

TC = 5Q(1) +10Q(2)

where Q(1) and Q(2) denote the quantities of G1 and G2 respectively. If P1 and P2 denote the corresponding prices then the demand equations are

P(1) = 50 - Q(1) - Q(2)

P(2) = 100 - Q(1) - 4Q(2)

Find the maximum profit if the firm's total costs are fixed at $100. Estimate the new optimal profit if total costs rise to $101.

Answer 1

The profit-maximizing quantity is produced at MR = MC.

$MR1 = TR'(Q1) = 50 - 2Q1 - Q2,$

$MC1 = TC'(Q1) = 5,$

50 - 2Q1 - Q2 = 5,

2Q1 + Q2 = 45.

$MR2 = TR'(Q2) = 100 - Q1 - 8Q2,$

$MC2 = TC'(Q2) = 10,$

100 - Q1 - 8Q2 = 10,

Q1 + 8Q2 = 90,

Q1 = 90 - 8Q2.

If we substitute the above equation into the previous one, then:

$2(90 - 8Q2) + Q2 = 45,$

15Q2 = 135,

Q2 = 9 units,

Q1 = 90 - 8×9 = 18 units.

P1 = 50 - 18 - 9 = 23,

P2 = 100 - 18 - 4×9 = 46.

Total profit is:

$TP = TR - TC = (23×18 + 46×9) - (5×18 + 10×9) = 648.$

If FC rise to $101, then the total profit will decrease.