A firm exist as a monopoly in the industry and has the following functions of revenue and cost.
TR = Total revenue and TC = Total cost
TR = 300q - 2q²
TC = 2q³ - 20q² + 60q + 60
Find the level of output at which the firm will maximize profits.
To find the level of output at which the firm will maximize profits, we need to calculate the firm's profit function.
Profit (π) is equal to total revenue (TR) minus total cost (TC):
π = TR - TC
π = (300q - 2q²) - (2q³ - 20q² + 60q + 60)
π = 300q - 2q² - 2q³ + 20q² - 60q - 60
π = -2q³ + 18q² + 240q - 60
To maximize profits, we need to find the level of output q at which the derivative of the profit function with respect to q is equal to 0:
dπ/dq = -6q² + 36q + 240
Setting the derivative equal to 0 and solving for q:
-6q² + 36q + 240 = 0
-6(q² - 6q - 40) = 0
q² - 6q - 40 = 0
Now, we can solve for q using the quadratic formula:
q = (6 ± √(6² - 4*(-40)) / 2
q = (6 ± √(36 + 160)) / 2
q = (6 ± √196) / 2
q = (6 ± 14) / 2
Either q = 10 or q = -4
Since q must be positive (output cannot be negative), the level of output at which the firm will maximize profits is q = 10.