The demand function for a firm’s product is Q(P) = 50-P/10. The firm’s cost of production is C(Q) = Q^3-20Q^2+125Q. The firm’s problem is to choose the value of Q≥0 that maximizes its profit. You may occasionally find an irrational number and in those cases simplify your answer as much as possible.
(1) Find all values of Q (if any) that satisfy the first-order condition for the firm’s problem.
(2) Calculate the firm’s profit-maximizing price and quantity. Justify your answer carefully.
(3) Calculate the firm’s maximized profit and the revenue and cost that produce that profit.
(4) Calculate the elasticity of demand at the profit-maximizing point.
(5) What is the firm’s markup at the profit-maximizing point? Confirm that this markup has the expected relationship to the elasticity of demand calculated in part (4).