Total cost function of a firm is TC= 200+4Q+2Q squared If the firm is perfectly competitive and the price of its product is $24, what is its optimal output rate?

24

To find the optimal output rate for a perfectly competitive firm, we need to determine the quantity at which the firm's marginal cost equals the price of the product.

The marginal cost (MC) is the derivative of the total cost (TC) function with respect to quantity (Q). In this case, the TC function is given as TC = 200 + 4Q + 2Q^2.

Taking the derivative of the TC function with respect to Q, we can find the MC function:

MC = d(TC)/dQ = d(200 + 4Q + 2Q^2)/dQ

Differentiating each term with respect to Q, we get:

MC = 4 + 4Q

Now, since the firm is perfectly competitive, the price of its product is $24. In a perfectly competitive market, the price is equal to the marginal revenue (MR). Therefore, we set MR equal to MC to find the optimal quantity:

MR = MC

$24 = 4 + 4Q

Subtracting 4 from both sides, we get:

$20 = 4Q

Divide both sides by 4 to solve for Q:

Q = $20 / 4 = 5

Therefore, the optimal output rate for the firm is 5 units.