A monopolist faces a demand curve Q = 120 – 2p and has costs given by

A monopolist faces a demand curve Q = 120 – 2p and has costs given by C(Q) = 20Q + 100.

a. Write the monopolist’s profits in terms of the price it charges.
b. Use the derivative (w.r.t. price) to determine the monopolist’s profit-maximizing price.
c. Now, derive the monopolist’s inverse demand based on the demand equation above. Write out the monopolist’s profits in terms of quantity.
d. Use the derivative w.r.t. Q to determine the monopolist’s optimal quantity. What price does the monopoly charge?

ANSWER

a. (p) = (120 – 2p)p – 20(120 – 2p) – 100
b. d/dp = 120 – 4p + 40 = 0 p* = 40
c. p = 60 – .5Q
Pi(Q) = (60 – .5Q)Q – 20Q – 100
d. dPi/dQ = 60 – Q – 20 = 0 Q = 40 and p = 120 – 2(40 ) = 40. The answer is the same whether deriving the optimality condition based on price or quantity.