Question

12. Suppose a firm’s cost function is C(q) = 3q² + 10. (Show work)

a. Find variable cost, fixed cost, average cost, average variable cost, and average fixed cost.(Hint: Marginal cost is given by MC = 6q.)

b. Find the output that minimizes average cost.

Answer 1

Given the total cost function, C, as below:

C = 3q^2 + 10

a.

Hence,

Variable cost (V) = 3q^2 [Answer]

Fixed cost (F) = 10 [Answer]

Average cost (AC) = C / q

                                = (3q^2 + 10) / q

                                = 3q + (10/q) [Answer]

Average variable cost (AVC) = V / q

                                                = (3q^2) / q

                                                = 3q [Answer]

Average fixed cost (AFC) = F / q

                                           = 10 / q

                                           = 10/q [Answer]

b.

AC would be the minimum if the derivative of it is equal to 0.

AC = 3q + (10/q)

Derivative of AC = (d/dq) [3q + (10/q)]

            0               = 3 + (10 × (-1))/q^2

0 = 3 – 10/q^2

3 = 10/q^2

3q^2 = 10

q^2 = 10/3

q = (10/3)^(1/2)

q = (10/3)^0.5 [Answer]

Alternative approach for the checking of part-b:

AC = MC

3q + (10/q) = 6q

10/q = 6q – 3q

10/q = 3q

10/3 = q^2

q = (10/3)^(1/2)

q = (10/3)^0.5 [Answer checked]