A foundations note — keeping the building blocks sharp.
A monopolist is the sole seller and therefore faces the whole market’s downward-sloping demand curve. That single fact drives everything. To sell one more unit it must cut the price — and the cut applies to every unit sold, not just the marginal one — so marginal revenue falls below price:
MR = P·(1 + 1/ε) < Pwhere ε < 0 is the price elasticity of demand. (With linear demand, the MR curve has twice the slope of demand.)
The optimum is two steps. Profit is still maximised where MR = MC — that pins down the quantity Q_m. Then you read the price off the demand curve at that quantity: P_m = P(Q_m) > MC. The contrast with perfect competition is the whole point: competition drives P = MC; monopoly sustains a markup, P > MC.
Measuring the markup. Combine MR = MC with MR = P(1 + 1/ε) and rearrange:
(P − MC)/P = −1/εThe left side is the Lerner index, a measure of market power. The less elastic the demand (|ε| small), the bigger the markup; as |ε| → ∞ — the competitive limit — the markup vanishes. A consequence: a monopolist only ever operates on the elastic part of demand (|ε| > 1), because where demand is inelastic MR < 0 and cutting output would raise revenue.
The welfare cost. Because Q_m is below the competitive quantity, there is a range of units on which buyers’ valuation (the demand price) exceeds MC but which go unproduced. That forgone surplus is the deadweight loss — the monopoly’s efficiency cost against the competitive benchmark — on top of a transfer of surplus from consumers to the monopolist as profit.
Natural monopoly. When strong economies of scale make average cost fall across the whole relevant range, one firm supplies the market most cheaply, and competition would wastefully duplicate fixed costs. Regulators then step in — typically with average-cost pricing (P = AC, zero profit) or marginal-cost pricing (P = MC, efficient but needing a subsidy).
Common slips: thinking a monopolist can “charge anything” (the demand curve binds — too high a price collapses quantity); reading the price off the MR = MC height instead of off the demand curve; and imagining production on the inelastic part of demand, where MR < 0 is never optimal.