Question 12.19: If a firm faces the marginal cost schedule MC = 180 + 0.3q² ......
If a firm faces the marginal cost schedule MC = 180 + 0.3q²
and the marginal revenue schedule MR = 540 − 0.6q²
and total fixed costs are £65, what is the maximum profit it can make? (Assume that the second-order condition for a maximum is met.)
Profit is maximized when MC = MR. Therefore,
180 + 0.3q² = 540 − 0.6q²
0.9q² = 360
q² = 400
q = 20
To find the actual profit (π), we now integrate to get TR and TC and then subtract TC from TR.
TR = \int MR dq = \int (540 − 0.6q²)dq = 540q − 0.2q³
TC = \int MC dq + TFC = \int (180 + 0.3q²)dq + 65 = 180q + 0.1q³ + 65
π = TR − TC
= 540q − 0.2q³ − (180q + 0.1q³ + 65)
= 540q − 0.2q³ − 180q − 0.1q³ − 65
= 360q − 0.3q³ − 65
Thus when q = 20 the maximum profit level is
π = 360(20) − 0.3(20)³ − 65 = £4,735