A monopolist has demand and cost curves given by: Quantity demand = 10000 - 20P TC = 1000 + 10Q +

a. Find the monopolist's profit-maximizing quantity and price.

We write the demand into:
P = 500-.05Q
Then TR = P*Q = 500Q - .05Q^2
It should be noticed that the firm is a monopolist and is facing a downward sloping demand curve.
Then the firm's marginal revenue is the first derivative of the total revenue w. r. t. quantity, i.e.:
MR = dTR / dQ = 500 - .1 Q
MC = dTC / dQ = 10 + .1 Q
We know that the first order condition is MR = MC for profit maximization problem. (Because if MC < MR, it is more profitable to produce and sell one more unit of product; and if MC > MR, the firm will lose money on the extra unit produced.)
Therefore, 500 - .1 Q = 10 + .1 Q
.2Q = 490
Q* = 2450
Sub Q into demand curve, we get P* = 500-.05*2450

b. Find the monopolist's profit.
Profit = TR - TC = P*Q* - (1000 + 10Q* + .05Q*^2)
= 377.5*2450 - (1000 + 10*2450 + .05*2450^2)