Question

A firm has a long-run total cost function of C(Q) = 180,000 + 30Q + 2Q^2 for Q > 0 and C(0) = 0. With MC(Q) = 30 + 4Q:

a. What is the average total cost function?

b. At what quantity does the average total cost reach a minimum?

Answer 1

The average total cost function is
`(180,000 + 30Q + 2Q^2) / Q`. The average total cost reaches a minimum at a quantity of 3000.

Step-by-step explanation:

The average total cost (ATC) function can be calculated by dividing the total cost (TC) by the quantity (Q). In this case, the total cost function is given as
`C(Q) = 180,000 + 30Q + 2Q^2`, so the average total cost function is
`ATC(Q) = (180,000 + 30Q + 2Q^2) / Q`. Simplifying this equation gives
`ATC(Q) = 180,000/Q + 30 + 2Q`. The average total cost reaches a minimum where the marginal cost (MC) is equal to the average total cost (ATC). Set MC equal to ATC and solve for Q. In this case,
`30 + 4Q = 180,000/Q + 30 + 2Q.` Simplifying this equation gives
`6Q^2 - 180,000 = 0.` Solving this quadratic equation results in Q = 3000. Therefore, the average total cost reaches a minimum at a quantity of 3000.