Question

Alpha company is producing chocolate bars and its average variable cost is given by the following equation: ???=0.5?2−24?+432.

3.1) If total fixed cost for the firm is 100, compute the total cost of producing chocolate bars for the firm.

3.2) Compute the number of chocolate bars that minimizes the average variable cost for the firm. What is the value of marginal cost at this point?

3.3) Assume that chocolate market is a perfectly competitive market and market price for chocolate is P=150. What do you think about profit (or loss) of Alpha company in the short run when market price is P=150?

3.4) Calculate the price for the shut-down point of Alpha company in the short-run (i.e. below which price Alpha company may shut down its plant?) Do you think Alpha company may shut- down its plant if market price is P=100?

Answer 1

(3.1)

Total cost (TC) = Total variable cost + Fixed cost (FC) = (AVC x Q) + FC

TC = 0.5Q³ - 24Q² + 432Q + 100

(3.2)

AVC is minimized when dAVC/dQ = 0.

dAVC/dQ = Q - 24 = 0

Q = 24

At Q = 24, MC = dTC/dQ = 1.5Q² - 48Q + 432 = (1.5 x 24 x 24) - (48 x 24) + 432 = 864 - 1152 + 432 = 144

(3.3)

A perfect competitive firm maximizes profit (or minimizes loss) by equating P with MC. When P = 150,

1.5Q² - 48Q + 432 = 150

1.5Q² - 48Q + 282 = 0

Q² - 32Q + 188 = 0

Solving this equation by quadratic equation solver,

Q = 24 or Q = 8 (assuming output is integer only)

As MC curve is U-shaped, the more output is produced, the more the profit. So profit maximizing output is 24.

Profit = Revenue - TC = (P x Q) - TC = 150Q - (0.5Q³ - 24Q² + 432Q + 100) = 24Q² - 0.5Q³ - 582

When Q = 24,

Profit = (24 x 24 x 24) - (0.5 x 24 x 24 x 24) - 582 = 6912 - 582 = 6330

Therefore economic profit is positive in short run.

(3.4)

A perfect competitive firm will shut down if revenue is lower than variable costs, or when Price (MC) is less than AVC.

When P = 100, equating P with MC,

1.5Q² - 48Q + 432 = 100

1.5Q² - 48Q + 332 = 0

Solving this equation by quadratic equation solver,

Q = 22 or Q = 10 (assuming output is integer only)

As MC curve is U-shaped, the more output is produced, the more the profit. So profit maximizing output is 22.

When Q = 22,

AVC = (0.5 x 22 x 22) - (24 x 22) + 432 = 242 - 528 + 432 = 146

Since at this output level, Price < AVC, firm will shut down when price is 100.