Question

Below is cost information on a firm in a perfectly competitive market, where q denotes the quantity produced by the firm. (Note that this firm's MC curve is a line rather than a check mark. That's okay.) TVC =1/2 q²= 8q; TFC=20;MC=q+8

a) Find a formula for AVC.
b) Graph the AVC and MC curves on the same diagram. Be sure to graph the formula for this AVC curve, which may be slightly different from the generic graph.
c) If one exists, find the shut-down price for this firm.
d) For each of the following prices, find the level of output for which the firm maximizes its profit in the short run. Compute, also, the amount of profit.
Show your reasoning. i) P=6 ii) P=10 iii) P=16
e) Sketch the firm's short-run supply curve.

Answer 1

The formula for AVC is 1/2q + 8. The graph shows AVC initially decreasing but at a slower rate and then increasing. The shut-down price is where AVC intersects TFC. The output level for maximum profit is 2 units at P = 10, and the short-run supply curve is the portion of MC above the shut-down price.

Step-by-step explanation:

In a perfect competition, the cost information of a firm is given as:

• TVC = 1/2q² = 8q
• TFC = 20
• MC = q + 8

a) Formula for AVC:

AVC is calculated by dividing total variable cost (TVC) by the quantity produced (q):

AVC = TVC / q = (1/2q² + 8q) / q = 1/2q + 8

b) Graph of AVC and MC:

Both AVC and MC are linear functions since MC = q + 8. The graph will have AVC decreasing initially but at a decreasing rate, and then increasing after a point. MC will be a straight upward-sloping line.

c) Shut-down price:

The shut-down price is the price at which the firm covers its variable costs but not its fixed costs. Since TFC is given as 20, the shut-down price is the price at which AVC (1/2q + 8) intersects with TFC (20) curve.

d) Output level and profit:

i) P = 6:

The profit-maximizing output level is found by equating MR = MC = P. Substituting P = 6 and MC = q + 8, we get q = 6 - 8 = -2. Since we can’t produce negative quantities, the firm will shut down.

ii) P = 10:

MR = 10 and MC = q + 8, equating these we get q = 10 - 8 = 2. The firm will produce 2 units of output to maximize profit.

iii) P = 16:

MR = 16, equating this with MC = q + 8 gives q = 16 - 8 = 8. The firm will produce 8 units of output to maximize profit.

e) Firm's short-run supply curve:

The firm's short-run supply curve is the portion of MC curve above the shut-down price, as the firm will produce only when the price is above the shut-down price.