A monopolist faces the (inverse) demand for its product: p = a – bQ. T

A monopolist faces the (inverse) demand for its product: p = a – bQ. The monopolist has a marginal cost given by c and a fixed cost given by F.

a. Assume that F is sufficiently small such that the monopolist produces a strictly positive level of output. What is the profit-maximizing price and quantity?
b. Compute the maximum profit for the monopolist.
c. For what values of F will the monopolist earn negative profit?

ANSWER

a. The monopolist will choose p = MR (or derive from first order condition of profit function).
a – 2bQ = c
Solving for Q
Q* = (a – c)/2b
The price follows from plugging the optimal output into the demand:
p* = a – b(a – c)/2b = a – (a – c)/2 = (a + c)/2
b. The profit comes from plugging the price and quantity into the profit equation:
Pi* =[(a + c)/2 – c](a – c)/2b – F = (a – c)2/4b – F
c. Find F* such that Pi* = 0:
F* = (a – c)2/4b
For F > F*, profits will be negative.