Question

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

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.

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