Question

Management has estimated a competitive industry's demand function to be: Q= 4,500 - 25P

There are 90 identical profit-maximizing, price taker firms in the industry, each with an average variable cost curve of Ave = $8 +$0.75Q and fixed costs of $600 per hour. Q is the output per hour for the firm. Find the supply function, its market clearing price, the profit maximizing output of the typical firm, and its hourly profit.

Answer 1

Average Variable Cost, AVC = 8 + 0.75Q
So, Total Variable Cost, TVC = AVC*Q = (8+0.75Q)*Q = 8Q + 0.75Q²

Total Cost, TC = TVC + fixed cost = 8Q + 0.75Q² + 600

Supply function of a firm is given by P = MC
Marginal Cost, MC = d(TC)/dQ = 8 + 2(0.75Q) = 8 + 1.5Q

Now, P = MC gives,
P = 8 + 1.5Q
So, 1.5Q = P - 8
So, Q = (P/1.5) - (8/1.5)

Supply curve of market, Qs = 90Q = 90(P/1.5) - 90(8/1.5) = 60P - 480
Qs = 60P - 480 (supply function)

At equilibrium, demand = supply
So,  4,500 - 25P = 60P - 480
So, 60P + 25P = 4500 + 480
So, 85P = 4980
So, P = 4980/85 = 58.59 (market clearing price)

Q = (P/1.5) - (8/1.5) = (P-8)/1.5 = (58.59-8)/1.5 = 50.59/1.5 = 33.73 (profit maximizing output of the typical firm)

Hourly profit = Total revenue of a firm - TC = P*Q - (8Q + 0.75Q² + 600) = (58.59)(33.73) - 8(33.73) - 0.75(33.73)² - 600 = 1976.24 - 269.84 - 853.28 - 600 = 253.12