Question

3.5

5. A firm's total-revenue and total-cost functions are TR 40 TC 0.0403-0.90+ 100 +5 (a) Determine the best level of output. (b) Determine the total profit of the firm at its best level of output.

Answer 1

5. A monopoly firm maximizes profit by producing at the point where MR = MC.

For this firm, TR= 4Q

Therefore, MR = d(TR)/dQ = 4

And, TC = 0.04Q³ - 0.9Q² + 10Q + 5

Therefore, MC = d(TC)/dQ = 0.12Q^​​2 - 1.8Q+ 10

Setting MR = MC, we get

0.12Q^​​2 - 1.8Q + 10 = 4

Or, 0.12Q² - 1.8Q + 6 = 0

Multiplying both sides by 0.12 we get,

Q² - 15Q + 50 = 0

Or, Q² -(10+5)Q + 50 = 0

Or, Q² - 10Q - 5Q +50 = 0

Or, Q(Q - 10) -5(Q - 10) = 0

Or, Q = 10 ; Q = 5

Therefore, the firm will be maximizing profit by producing either 10 units or 5 units. So, the best level of output it will be producing is 10 units as it is greater than 5 units.

b) When Q = 10, TR = (4*10) = 40 and TC = 0.04(10)³ - 0.9(10)² +(10*10) + 5 = 40 - 90 + 100 + 5 = 55

Therefore, total profit at this best level of output is (TR - TC) = (40 - 55) = -15, that is a loss of $15.