Question

11. Given that total cost can be expressed as TC(X) = x3-140x2 + 4200, find: a. the average cost function b. the level of output that minimizes AC (check 2nd order conditions also) c. the value of average cost at its minimum level

Answer 1

(a)

Average cost (AC) = TC(X)/x = x² - 140 + (4200/x)

(b)

Average cost is minimized when dAC/dx = 0 (FOC) and d²AC/dx² > 0 (SOC).

dAC/dx = 2x - (4200/x²) = 0

2x = 4200/x²

2x³ = 4200

x³ = 2100

Taking cube root,

x = 12.81

d²AC/dx² = [2 + (8400/x³)] > 0 for all positive values of x. Therefore, SOC is satisfied.

(c)

When x = 12.81,

Minimum AC = (12.81)² - 140 + (4200/12.81) = 164.10 - 140 + 327.87 = 351.97