Question

Given that the total cost function is:
TC = 100Q – Q^2 + 1/3Q^3
Where Q = rate of output and TC = total cost
a. Determine the marginal and average cost functions
b. Calculate the output level that minimizes average cost
c. Calculate the output level that minimizes marginal cost

Answer 1

Answer on Question #38995 – Economics – Other

Given that the total cost function is

$$TC = 100Q - Q^2 + \frac{1}{3} Q^3$$

Where Q = rate of output and TC = Total Cost

A. Determine the marginal and average cost functions

B. Calculate the output level that minimizes average cost

C. Calculate the output level that minimizes marginal cost

Solution

$$TC = 100Q - Q^2 + \frac{1}{3} Q^3$$

A. Marginal function MC = TC' = 100 - 2Q + Q^2

Average cost function ATC = TC/Q = 100 - Q + \frac{1}{3} Q^2

B. The output level that minimizes average cost is in the point, where ATC' = 0 (ATC'' = 1/3 > 0),

so

-1 + \frac{2}{3}Q = 0,

-2 + \frac{2}{3}Q = 1,

Q = 1.5 units

C. The output level that minimizes marginal cost is in the point, where MC' = 0 (MC'' = 1 > 0),

so

-2 + 2Q = 0,

2Q = 2,

Q = 1 unit.