A firm uses a single plant with cost C=160+16Q+0.1Q^2 and faces the price equation P=96-0.4Q.
A) Find the firm's profit-maximizing price and quantity. What is its profit?
B) The firm's production manager claims that the firm's average cost of production is minimized at an output of 40 units. Furthermore, she claims that 40 units is the firm's profit=maximizing level of output. Explain whether these claims are correct.
C) Could the firm increase its profit by using a second plant (with cost identical to the first) to product the output in part (a)? Explain.
The firm's profit-maximizing output (Q) is where Marginal Revenue (MR) = Marginal Cost (MC)
Total Revenue (TR) = PQ
TR = (96 - 0.4Q)Q
TR = 96Q - 0.4Q^2
MR is the derivative of TR
MR = 96 - 2(0.4Q)
MR = 96 - 0.8Q
Total Cost (TC) = 160 + 16Q + 0.1Q^2
MC is the derivative of TC
MC = 16 + 2(0.1Q)
MC = 16 + 0.2Q
To maximize profit, let MR = MC
96 - 0.8Q = 16 + ...
A calculation of a firm's profit-maximizing price and output, followed by mathematical analysis of other possible production strategies.