Question

A firm faces the following Average Cost function

AC = 1/3Q^2- 18Q + 120 + 15/Q
  

Calculate the output level that minimizes
   (i) Marginal Cost
   (ii) Average Variable Cost.

Answer 1

Answer : Given,

AC = 1/3Q^2- 18Q + 120 + 15/Q

AC = TC / Q

=> TC = AC * Q

So, TC = (1/3Q^2- 18Q + 120 + 15/Q) * Q

=> TC = 1/3 Q^3 - 18Q^2 + 120Q + 15

(i) MC (Marginal Cost) = TC / Q

=> MC = Q^2 - 36Q + 120

To find the output level where MC is minimum we have to take derivative of MC and then have to take equal to zero for the outcome.

dMC / dQ = 2Q - 36 = 0

=> 2Q = 36

=> Q = 36 / 2

=> Q = 18

Therefore, the marginal cost is minimum when output level is Q = 18.

(ii) TC = 1/3 Q^3 - 18Q^2 + 120Q + 15

Here VC (Variable Cost) = 1/3 Q^3 - 18Q^2 + 120Q

AVC (Average Variable Cost) = VC / Q

=> AVC = (1/3 Q^3 - 18Q^2 + 120Q) / Q

=> AVC = 1/3 Q^2 - 18Q + 120

To find the output level where MC is minimum we have to take derivative of MC and then have to take equal to zero for the outcome.

dAVC / dQ = (1/3 * 2Q) - 18 = 0

=> 2Q / 3 = 18

=> 2Q = 18 * 3 = 54

=> Q = 54 / 2

=> Q = 27

Therefore, the average variable cost is minimum when output level is Q = 27.