To derive the firm's average total cost, we first calculate its total cost function:
TC(q) = C(q) * q = (10 + 10q + q²) * q = q² + 10q + 10q²
The average total cost (ATC) is then given by:
ATC(q) = TC(q) / q = q + 10 + 10q
The average variable cost (AVC) is given by the variable costs per unit of output, which in this case is the sum of the variable cost and the marginal cost:
AVC(q) = (10 + 2q) / q
To determine the profit-maximizing level of output, the firm needs to equate marginal cost (MC) to market price (p), since it is a price taker in a perfectly competitive market. The marginal cost function is the derivative of the total cost function with respect to q:
MC(q) = dTC(q) / dq = 2q + 10
Setting MC(q) = p, we get:
2q + 10 = p
Solving for q, we get:
q = (p - 10) / 2
If the market price is 50, the firm should produce:
q = (50 - 10) / 2 = 20
To calculate the profit at this level of output, we need to subtract the total cost from the total revenue:
TR(q) = p * q = 50 * 20 = 1000
TC(q) = 20² + 10(20) + 10 = 530
Profit = TR(q) - TC(q) = 1000 - 530 = 470
if the market price is 50, the firm should produce 20 tons of laundry and will earn a profit of 470.