Question

2. The (total) cost function is given by C = 50 + 60Q – 18Q2 + 2Q3
a. Write down the fixed cost function FC(Q).
b. Write down the variable cost function VC(Q).
c. Write down the marginal cost function MC(Q).
d. Write down the average fixed cost function AFC(Q).
e. Write down the average variable cost function AVC(Q).
f. Write down the average total cost function AC(Q).
g. Find the break-even point (Q and AC).
h. Find the shut-down point (Q and AVC).
i. Draw a graph to illustrate AC, AVC, and MC functions for quantities Q on the interval between 1 and 10. Make sure you show (put the numbers there) where exactly the MC curve intercepts AVC and AC curves.
j. If the price is P = 60, calculate the profit-maximizing firm’s profit.

Answer 1

a. The fixed cost function is:

FC(Q) = 50.

b. The variable cost function is:

$VC(Q) = 60Q – 18Q^2 + 2Q^3.$

c. The marginal cost function is:

$MC(Q) = C'(Q) = 60 - 36Q + 6Q^2.$

d. The average fixed cost function is:

AFC(Q) = FC/Q = 50/Q.

e. The average variable cost function is:

$AVC(Q) = VC/Q = 60 - 18Q + 2Q^2.$

f. The average total cost function

$AC(Q) = C/Q = 50/Q + 60 - 18Q + 2Q^2.$

g. The break-even point (Q and AC) is:

$P = AC = 50/Q + 60 - 18Q + 2Q^2.$

h. The shut-down point (Q and AVC) is:

$P = AVC = 60 - 18Q + 2Q^2.$

i. AC = C/Q, AVC = (C - FC)/Q, MC = ∆C/∆Q.

j. If the price is P = 60, then:

P = MC,

$MC = C' = 60 - 36Q + 6Q^2 = 60,$

6Q(Q - 6) = 0,

Q1 = 0 or Q2 = 6 units.

TP = TR - C = 60*6 - (50 + 60*6 - 18*36 + 2*216) = 166 is the profit-maximizing firm’s profit.