Question

Ahmad has a total cost function for his furniture factory as follows: TC = 100 + 8Q – 0.12Q2 + 0.004Q3 a. Show that the ATC, AVC, MC, and AFC equations b. At what level of output do diminishing returns set in? c. At what level is AVC minimum? d. At what output level are MC and AVC equal? e. Why does AVC reach a minimum before ATC reaches a minimum? f. At the point where the ATC is at minimum, the marginal product of labour is 12 and the marginal product of capital is 18. If the price of labour per hour is RM8, what is the price of capital per hour?

Answer 1

TC = 100 + 8Q – 0.12Q2 + 0.004Q3
a. ATC = TC/Q = 100/Q + 8 – 0.12Q + 0.004Q2,
AVC = VC/Q = 8 – 0.12Q + 0.004Q2,
MC = TC' = 8 - 0.24Q + 0.012Q2,
AFC = FC/Q = 100/Q
b. Diminishing returns set in at such level of output, for which MC is minimal or MC' = 0, so:
-0.24 + 0.024Q = 0,
Q = 10 units.
c. AVC is at its minimum, when AVC = MC or AVC' = 0, so:
-0.12 + 0.008Q = 0,
Q = 15 units.
d. MC and AVC equal at the point, where AVC is at its minimum, so Q = 15 units.
e. AVC reaches a minimum before ATC reaches a minimum, because ATC > AVC and MC is increasing.
f. If at the point where the ATC is at minimum, MPL = 12, MPK = 18, w = RM8, then the price of capital per hour k is:
MPL/w = MPK/k,
12/8 = 18/k,
k = 12.