In the model of perfect competition, all firms are price-takers since they treat
ID: 1199886 • Letter: I
Question
In the model of perfect competition, all firms are price-takers since they treat price as a market-determined constant. Firm Perfcomp's total revenue function is TR(Q) = P middot Q, in which P equals the output price. Assume that P = 12 and the total cost function is TC(Q) = Q^3 - 4.5Q^2 + 18Q - 7. Determine the firm's profit function and the level of output at which Firm Perfcomp should produce in order to maximize profits. Confirm that this quantity represents maximum profits for the firm by using the second-order condition. According to microeconomic theory, perfectly competitive firms will maximize profits by producing at the quantity where price equals marginal cost and where the slope of the marginal revenue curve is less than that of the marginal cost curve. Show that the theory holds in this example.Explanation / Answer
(a)TR = P*Q
=12Q
TC = Q^3 – 4.5Q^2 + 18Q – 7
Profits = TR-TC
Profits = 12Q – (Q^3 – 4.5Q^2 + 18Q – 7)
To know the maximum profits we take first derivative of the profit function with respect to output Q
D(Profits)/dQ = 12 – 3Q^2 + 9Q – 18 = 0
3Q^2 - 9Q + 6 = 0
3Q^2 – (6+3)Q + 6 =0
3Q^2 – 6Q - 3Q + 6 =0
3Q (Q – 2) -3(Q-2) = 0
3(Q-2)(Q-1) = 0
Q = 2 or 1
Second order derivative
-6Q + 9 =0
Q = -9/6 or -3/2
The negative sign implies that the profit is maximized.
(b) TR = P*Q
TR=12Q
MR = 12
TC = Q^3 – 4.5Q^2 + 18Q – 7
MC = 3Q^2 – 9Q + 18
MR=MC
12 = 3Q^2 – 9Q + 18
3Q^2 - 9Q + 6 = 0
3Q^2 – (6+3)Q + 6 =0
3Q^2 – 6Q - 3Q + 6 =0
3Q (Q – 2) -3(Q-2) = 0
3(Q-2)(Q-1) = 0
Q = 2 or 1
Which implies that MR=MC condition is satisfied only at the maximum profit.
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