A competitive firm has the short-run cost function c(y)= 2y3

Question

A competitive firm has the short-run cost function c(y)= 2y3 - 16y2 + 64y + 50. The firm will produce a positive amount in the short run if and only if the price is greater than
A) $16.
B) $64.
C) $32.
D) $35.
E) $31.

Quizplus · Accepted Answer

In the short run, a competitive firm will produce if its price is greater than or equal to its minimum average variable cost (AVC) because if the price is less than or equal to AVC, the firm will incur losses in the short run and will shut down. The minimum AVC is found where the marginal cost (MC) curve intersects the AVC curve from below. Taking the derivative of the cost function with respect to y, we have:

MC = 6y^2 - 32y + 64

Setting MC equal to AVC and solving for y, we have:

AVC = c(y)/y = 2y^2 - 16y + 64 + 50/y

6y^2 - 32y + 64 = 2y^2 - 16y + 64 + 50/y

4y^2 - 16y - 50/y = 0

Multiplying both sides by y, we have:

4y^3 - 16y^2 - 50 = 0

Dividing both sides by 2, we have:

2y^3 - 8y^2 - 25 = 0

Using the rational root theorem, we can see that y = 5 is a root. Dividing by (y-5), we get:

2y^2 + 2y + 5 = 0

which has no real roots, so y = 5 is the only critical point. We can confirm that this is a minimum by computing the second derivative:

d2c(y)/dy2 = 12y - 32

d2c(5)/dy2 = -8 < 0

Therefore, y = 5 is the global minimum of AVC, and AVC(5) = $32. Therefore, the firm will produce a positive amount in the short run if and only if the price is greater than $32.