Question

A monopolist faces demand curve p = 320 – 12Q. Production cost equals 1000 + cQ. What is
the maximum value of c for which the firm is profitable?

Answer 1

A monopolist faces demand curve p = 320 – 12Q. Production cost equals 1000 + cQ. The maximum value of c for which the firm is profitable is:
TP = 0 and MR = MC,
so TR - TC = 0,
(320 - 12Q)*Q - (1000 - cQ) = 0,
-12Q^2 + (320 + c)Q - 1000 = 0,
12Q^2 - (320 + c)Q + 1000 = 0,
MR = MC,
320 - 24Q = c, then:
12Q^2 - (640 - 24Q)Q + 1000 = 0,
36Q^2 - 640Q + 1000 = 0,
9Q^2 - 160Q + 250 = 0,
Q1 = (160 + (160^2 - 4*9*250)^0.5)/18 = 16 units,
Q2 = (160 - (160^2 - 4*9*250)^0.5)/18 = 1.73 units.
So, c = 320 - 24*1.73 = 278.5.