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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.Q,

in which P equals the output price. Assume that
P = 12 and the total cost function is

TC(Q) = Q3 – 4.5Q2 + 18Q – 7.

a) 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.

b) 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.