Advanced Microeconomics/Monopoly Pricing

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A monopolist produces a good with the demand function q(p) which (by assumption) has an inverse  p(q) and costs given by c(q). Since it supplies the entire market, the monopolist simultaneously chooses output and price. Thus, the profit maximization problem may be written two ways;
 \max_{q} q\cdot p(q) - c(q)
 \max_{p} p\cdot q(p) - c(q(p))