Consider a rm in a competitive market in the long run The av

Consider a .rm in a competitive market in the long run. The average cost is given by AC (q) (q-10)^2 + 12. Assume the firm is in a constant cost industry. The market demand is given by D(p) = 5000-300p. What is the marginal cost of the firm? What is the minimum average cost and the corresponding output the firm produces? With free entry and exit, what is the equilibrium price? How many firms are there in the industry? What is the profit of each firm?

Solution

Answer (a)

AC = (q-10)² + 12 and Demand D(p) = 5000 – 300p

Total cost             = AC * q

                                = ((q-10)² + 12)q

                                = (q²+ 100 – 20q +12)q

                                = q³+ 100q – 20q² +12q

                                = q³– 20q² + 88q

Marginal Cost = Change in TC / Change in Q

Differentiating total cost of q Marginal cost = 3q²– 40q + 88

Answer (b)

Minimum average cost and corresponding output the firm produces

Find the critical point of the average cost function: To take the derivative of the average cost function, rewrite it as

AC = (q-10)² + 12 or AC = q²+100- 20q +12

AC = q²+100- 20q +12

AC = q²- 20q +88

To take the derivative of the average cost function, rewrite it as

2q – 20, This function is undefined at x = 0, and is equal to zero when

2q -20 = 0

2q = 20 or q = 10

So we have critical points at q = 0 and q = 10.

Average cost at critical point =

AC = q²- 20q +88 (substituting q = 10)

AC = 100 – 200 +88

AC = 22

Or

If Marginal Cost = Average Cost, average cost is considered at minimum

q²- 20q +88 = 3q²– 40q + 88

2q²-20q=0

2q²=20q

q²= 10q

q = 10

Answer (c)

With free entry and exit, what is the equilibrium price? How many firms are therein in the industry? What is the profit for each firm

The equilibrium quantity = 10 and Price as 22

Demand D(p) = 5000 – 300p substituting Price

5000 – 300 * 22

= 5000 – 6600

Quantity = 1600

Since each firm makes 10 units of the product the number of firm

= 1600/ 10 = 160

Profit for each firm

At equilibrium the profit for each firm is zero.