What do monopolists do? If the monopolist is a private firm, generally we assume that it chooses the level of output to maximize profits. As an accounting matter, profits (PR) are equal to total revenue (TR) minus total costs (TC):
PR = TR - TC
For any firm, not just a monopolist, it can be shown that profit will be maximized at the quantity where marginal revenue is equal to marginal cost:
MR = MC
What makes a monopolist different from a competitive firm is that it faces a downward sloping demand curve: it knows that it will have to cut its price in order to sell more output. As a result, its MR curve will slope down rather than being flat like a competitive firm's.