Question

Suppose a firm has a long run average total cost function given by: ATC= (3200/Q) + 10 + 8Q. The demand for this product is given by: QD= 2900-8P

Determine the optimal firm size. 20 units -answer

Calculate the long run number of firms.13 firms - answer

(answers are given, please show steps as to how to get there)

Answer 1

In the long-run a firm should produce at a point where ATC is the minimum.

ATC would be the minimum if its derivative is 0.

ATC = (3200/Q) + 10 + 8Q

Minimum of ATC = Derivative of ATC with respect to Q

0 = -3200/Q^2 + 0 + 8

3200/Q^2 = 8

3200 = 8Q^2

3200/8 = Q^2

400 = Q^2

Q = square root of 400

    = 20 units (Answer)

This is to be placed in the ATC function to get price.

ATC = (3200/Q) + 10 + 8Q

         = (3200/20) + 10 + 8 × 20

         = 160 + 10 + 160

         = 330

Therefore, this is the price. (P = 330)

Now,

QD = 2900 – 8 × 330

      = 2900 – 2640

      = 260

Required number of firms = QD / Q

                                           = 260 / 20

                                           = 13 (Answer)