Question 18 Suppose that the market for auto detailing in a city is perfectly co

Question

Question 18 Suppose that the market for auto detailing in a city is perfectly competitive. The auto detailing firms are identical and have long-run cost functions given by TCCQ) 1003-10002300 Q. Market demand is Q-5,000 90P Answer the following questions a. Derive the marginal and average cost curves for a firm in this industry. b. Find the quantity at which average total cost is minimized for each firm. c. Find the long-run equilibrium price in this industry. d. Use market demand to find the equilibrium total industry output. e. Find the equilibrium number of firms.

Explanation / Answer

TC = 10Q^3-100Q^2+300Q

MC = dTC/dQ = 3*10Q^2 - 2*100Q + 300

MC = 30Q^2-200Q+300

AC=TC/Q = 10Q^2 -100Q + 300

For min AC we take first order derivative of AC

=20Q - 100 = 0

20Q = 100

Q = 100/20 = 5

At Q= 5 the ATC is at its minimum or is minimised.

ATC at Q = 5,

ATC= 10*(5)^(2)-100*5+300 = 50

In the long run equilibrium the price equals the average total costs of production as the firm earns only normal profits.

So P=ATC = 50 this gives us P = 50

Now that the price is set to be 50

Industry Q = 5000-90*50 = 500

number of firms = Q/q = 500/5 = 100

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