A monopoly maximizes profit with the same marginal rule as any firm, using its own marginal revenue (Why Marginal Revenue Is Below Price) and measuring profit as in Profit and Loss on the Graph.
Key idea
(1) Find the quantity where $MR = MC$. (2) Go up to the demand curve at that quantity to read the price. (3) Profit $= (P - ATC) \times Q$.
Why it works
Each unit adds $MR - MC$ to profit, so the monopoly keeps going while MR is at least MC, just like a competitive firm. The difference is the price: MR and MC tell the monopoly how many to sell, but the price it can charge for that many is whatever buyers will pay, and that is the height of the demand curve. Because MR is below demand, the price ends up above MC.
Worked example: Constant marginal cost
A monopoly faces demand $P = 20 - Q$ and has a constant marginal cost of \$4, with no fixed cost. Find its quantity, price and profit.
- $MR = 20 - 2Q$. Set $MR = MC$: $20 - 2Q = 4$, so $Q = 8$.
- Price from demand: $P = 20 - 8 = \$12$.
- With constant MC and no fixed cost, $ATC = \$4$. Profit $= (\$12 - \$4) \times 8 = \$64$.
Answer: $Q = 8$, $P = \$12$, profit \$64.
Worked example: With a fixed cost
Same monopoly, but it also has a fixed cost of \$40. Do its quantity and price change? What is its profit?
- A fixed cost doesn't change MC or MR, so $Q = 8$ and $P = \$12$ stay the same.
- Profit: $\$64 - \$40 = \$24$.
Answer: Same quantity and price; profit \$24.
Because a monopoly sets its price from the demand curve, it has no supply curve of the usual kind: the quantity it offers at a given price depends on the whole shape of demand, not on the price alone.
Common mistake
Reading the price off the MC or MR curve at the MR = MC quantity. At 8 units, MR and MC are both \$4, but buyers will pay \$12 for 8 units, and the monopoly charges what buyers will pay. Check: the price must be a point on the demand curve.