Question

1. Consider the following short- run variable cost function for a firm. VC = q3 -...

1. Consider the following short- run variable cost function for a firm.
VC = q3 - 10q2 + 100q
a. Find the equation for the short-run average variable cost and marginal cost.
b. Using your answer to part a, show that the output label at which short-run average variable cost is minimized is greater than the output level at which short-run marginal cost is minimized.
c. Using your answers to part (a) and (b), draw a rough sketch of the short run marginal and average cost curves.
d. Using your answers to part c, show the price label at which the firm would shut down and how for any one price label at which the firm earns a profit.

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Answer #1

VC = q3 - 10q2 + 100q

(a)

AVC = VC/q = q2 - 10q + 100

MC = dVC/dq = 3q2 - 20q + 100

(b)

AVC is minimized when dAVC/dq = 0

dAVC/dq = 2q - 10 = 0

2q = 10

q = 5

MC is minimized when dMC/dq = 0

dMC/dq = 6q - 20 = 0

6q = 20

q = 3.33 < 5 (Proved)

(c)

(d)

Firm will shut down when Price < Minimum AVC. So, shut-down price = Minimum AVC

AVC is minimum when q = 5. At this level,

Shut-down price = Minimum AVC = (5 x 5) - (10 x 5) + 100 = 25 - 50 + 100 = 75

In above graph, P* is the shut-down point. When Price is higher than P*, Price > AVC and therefore firm makes a profit.

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