The Burr Corporation’s total cost function (where TC is the total cost in dollars and Q is quantity) is TC = 200 + 4Q + 2Q^2
a. If the fi rm is perfectly competitive and the price of its product is $24, what is its optimal output rate?
b. At this output rate, what is its profitt?
To find the optimal output rate and profit for the Burr Corporation, we need to use the concepts of marginal cost and marginal revenue. Here are the steps to find the answers to both parts of the question:
a. To find the optimal output rate, we need to equate marginal cost (MC) with the price (P) of the product. In a perfectly competitive market, this will maximize the firm's profit.
1. Calculate the marginal cost (MC) function by taking the derivative of the total cost function with respect to quantity (Q): MC = dTC/dQ.
Given TC = 200 + 4Q + 2Q^2, differentiate the total cost function with respect to Q:
dTC/dQ = 0 + 4 + 4Q.
The marginal cost function is MC = 4 + 4Q.
2. Set the marginal cost (MC) equal to the price (P) of the product:
MC = P.
Since the price is given as $24, we have:
4 + 4Q = 24.
3. Solve for Q to find the optimal output rate:
4Q = 20,
Q = 5.
Therefore, the optimal output rate for the Burr Corporation is 5 units.
b. To find the profit at this output rate, we need to calculate the total revenue (TR) and subtract total cost (TC):
1. Calculate the total revenue (TR) function by multiplying the price (P) with the quantity (Q):
TR = P * Q,
TR = 24 * 5,
TR = 120.
2. Calculate the total cost (TC) at the optimal output rate (Q = 5):
TC = 200 + 4Q + 2Q^2,
TC = 200 + 4(5) + 2(5)^2,
TC = 200 + 20 + 2(25),
TC = 200 + 20 + 50,
TC = 270.
3. Calculate the profit by subtracting total cost (TC) from total revenue (TR):
Profit = TR - TC,
Profit = 120 - 270,
Profit = -$150.
Therefore, at the optimal output rate of 5 units, the Burr Corporation would have a loss (negative profit) of $150.