Question

Suppose a firm has two plants with the following costs: Plant 1 C1 q-3003-2,400q2+1,000,000q and Plant 2. C2 = q4-45q3-3,000q2 + 2,000 000q. At what output is the average cost minimized in each plant? Plant 1 average costs are minimized at output q= (Enter a numeric response using an integer.)

Need to know the same thing for Plant 2 too.

Answer 1

Plant 1

Average cost = C/q = (q4 – 30q3 – 2400q2 + 1,000,000q)/q = q3 – 30q2 – 2400q + 1,000,000

AC is minimized when marginal average cost is 0

dAC/dq = 0

3q2 – 60q – 2400 = 0

q2 – 20q – 800 = 0

It is a quadratic equation with roots q = (20 +/- (400 + 4*800)^0.5)/2 = 40 or -20. The correct value is 40. Hence Plant 1 average cost is minimized at q = 40.

Plant 2

AC = q3 – 45q2 – 3000q + 2,000,000

dAC/dq = 0

3q2 – 90q – 3000 = 0

q2 – 30q – 1000 = 0

It is a quadratic equation with roots q = (30 +/- (900 + 4*1000)^0.5)/2 = 50 or -20. The correct value is 50. Hence Plant 2 average cost is minimized at q = 50.