Question

1. Suppose, a perfectly competitive firm is trying to determine its profit-maximizing level of output. The product sells for $260 per unit. The total cost function is given by C = 1000 + 80Q – 6Q2 + .2Q3. Find the equilibrium price and maximum profits. Also, find the shutdown point for this firm.

2. You are the manager of a monopolistically competitive firm, and your demand and Cost functions are given by Q = 20 – 2P and C = 104 – 14Q + Q2. a) Find the inverse demand function for your firm’s product. b) Determine the profit-maximizing level of output and price. c) Calculate the firm’s maximum profits. d) What long-run adjustments should you expect?

Answer 1

1. A profit maximizing perfectly competitive firm produces at the point where market price = MC

TC = 1000 + 80Q - 6Q² + 0.2Q³

Or, MC = d(TC)/dQ = 80 - 12Q + 0.6Q²

If market price is $260, then

80 - 12Q + 0.6Q² = 260

Or, 0.6Q² - 12Q - 180 = 0

Or, Q² - 20Q - 300 = 0

Or, Q² -30Q + 10Q - 300 = 0

Or, Q(Q - 30) + 10(Q - 30) = 0

Or, (Q - 30) (Q + 10) = 0

Either, (Q -30) = 0 or, (Q + 10) = 0

Q can't be negative, so Q = 30

Therefore, profit maximizing quantity is 30 units. At this quantity, Total revenue = price * quantity = $(260 * 30) = $7800.

Total cost = 1000 + (80*30) - (6*30*30) + 0.2(30)³ = $3400

Therefore, equilibrium price is $260 per unit and maximum profit is (TR - TC) = $(7800 - 3400) = $4400.

A profit maximizing perfectly competitive firm's shut down point is price = minimum AVC.

From the cost function we can see that, TVC = 80Q - 6Q² + 0.2Q³

Or, AVC = (TVC/Q) = 80 - 6Q + 0.2Q²

When AVC is minimized, d(AVC)/dQ = 0

Or, -6 + 0.4Q = 0

Or, 0.4Q = 6

Or, Q = 6/0.4 = 15

At this quantity, AVC = 80 - (6*15) + (0.2*15*15) = $35

Therefore, shutdown point is, price = 35.