5. (i) A competitive firm’s total cost function is given by TC = .25Q2 + 25 (wit

Question

5. (i) A competitive firm’s total cost function is given by

TC = .25Q2 + 25
(with MC = .5Q).
The firm faces a market price of $15. Algebraically calculate the profit maximizing output and the level of optimal profit for the firm.

(ii) Suppose that fixed costs increase by $50 but the prevailing market price remains unchanged. Using algebra determine the effects of this change in cost on the profit maximizing output and the optimal profit. Do you see any change from your answer to (i) above? Explain why or why not.

Explanation / Answer

Haha, no this is not basic econ. This type of material is usually covered in intermediate and advanced econ courses. :)

1. In order for firms to maximize profit, they need to make sure that their marginal revenue (MR) equals their marginal cost (MC).

You are given MC = 0.5Q

You are not given MR. However, you are given the fact that you are dealing with a perfectly competitive firm. A perfectly competitive firm is also known as a price taker, which means that it must "take" the market price for its marginal revenue.

Therefore, MR= MP= 15

Again, in order to maximize profit a firm must operate at MC = MR.

Set the equations equal to one another and solve.

0.5Q= 15

Q= 30

The profit maximizing output is 30. The level of optimal profit for the firm occurs when they produce 30 units of output.

2. Your original total cost curve is

TC= 0.25Q^2 + 25

Given the TC curve, you are able to find the MC curve by taking the derivative of TC. By doing so, you will get

MC= 0.5Q. (This was already given in the first part of the problem. I just wanted to show you how the MC is derived)

If fixed costs increased by $50, your new total cost curve will be

TC= 0.25Q^2 + 75

In order to determine the level of profit maximization, you must take the TC curve and turn it into a MC curve. After taking the derivative, you end up with

MC = 0.5Q

Notice how your MC curve is the same even if fixed costs are increased by $50. Why is this the case? It is because in the long run, firms view fixed costs as "sunk costs". Sunk costs are past costs that have already been incurred and cannot be recovered. Therefore, it doesn't have any affect on the firm's long run goal of profit maximization.

Still don't believe it? Set MR= MC:

15= 0.5Q

Q= 30

Again, you end up with 30 units of output as the profit maximizing units of production.

I hope this helps! :)

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