Mathematical Problem: Competitive Firm Decisions. Suppose a perfectly competitiv
ID: 1223515 • Letter: M
Question
Mathematical Problem: Competitive Firm Decisions. Suppose a perfectly competitive firm experiences total cost (TC) and marginal cost (MC) according to the following equations, where Q is the quantity produced (output). The cost equations include both explicit and implicit costs.
TC = $4,000 + $5Q + $0.1Q2
MC = TC/Q = $5 + $0.2Q
4a. Calculate the profit-maximizing short run output and economic profit if the price the firm takes is $55.
4b. Calculate the profit-maximizing short run output and economic profit if the taken price rises to $65.
4c. In the long run, calculate the price, output, and economic profit. Explain what this level of profit means economically.
Explanation / Answer
ATC = TC / Q = (4,000 / Q) + 5 + 0.1Q
Profit = Q x (P - ATC)
A perfectly competitive firm maximizes profit by equating price with MC.
(4a). When P = MC = 55,
5 + 0.2Q = 55
0.2Q = 50
Q = 50 / 0.2 = 250
ATC = (4,000 / 250) + 5 + (0.1 x 250) = 16 + 5 + 25 = 46
Profit = 250 x $(55 - 46) = 250 x $9 = $2,250
(4b) P = MC = 65
5 + 0.2Q = 65
0.2Q = 60
Q = 300
ATC = (4,000 / 300) + 5 + (0.1 x 300) = 13.33 + 5 + 30 = 48.33
Profit = 300 x $(65 - 48.33) = 300 x $16.67 = $5,000
(4c) In long run, ATC = MC
(4,000 / Q) + 5 + 0.1Q = 5 + 0.2Q
4,000 / Q = 0.1Q
0.1Q2 = 4,000
Q2 = 40,000
Q = 200
P = MC = ATC = 5 + (0.2 x 200) = 5 + 40 = 45
Profit = Q x (P - ATC) = Q x 0 = 0
This means that firms are earning only a normal profit and zero excess (economic) profit.