Why the supply curve under monopoly is indeterminate

The statement

A supply curve shows a unique quantity that sellers will offer at each price. The statement says that a monopolist has no such curve: there is no one-to-one relationship between the price it charges and the quantity it supplies, so its supply cannot be drawn as a curve.

Why a supply curve exists under perfect competition

A competitive firm takes the price as given, so P = MR, and it produces where P = MC. For each price there is exactly one output, read off the MC curve. The MC curve above minimum AVC is therefore the firm's supply curve, and summing these gives the industry supply curve.

Why there is no unique price-quantity relationship under monopoly

A monopolist is a price maker. It faces the downward-sloping market demand curve, so its MR lies below its demand (AR) curve, and it chooses output where MR = MC and then charges the highest price the demand curve allows for that output. The price it charges depends not only on marginal cost but also on the shape and elasticity of demand. Two cases show that price and quantity are not uniquely linked.

1. Same price, different quantities. Diagram: output on the horizontal axis, price and cost on the vertical axis, one MC curve. With demand curves D₁ and D₃ positioned so that their MR curves cut MC at different outputs OQ₁ and OQ₃, the price read off the respective demand curves can be the same, OP₁. One price is associated with two different quantities.
2. Same quantity, different prices. Diagram: output on the horizontal axis, price and cost on the vertical axis, one MC curve. Two demand curves D₁ and D₂ with different elasticities have marginal revenue curves MR₁ and MR₂ that both cut MC at the same point, so the monopolist produces the same output OQ₁ in both cases. Reading up to the demand curves, it charges OP₁ when demand is D₁ and the higher OP₂ when demand is the less elastic D₂. One quantity is supplied at two different prices.

Because the monopolist's output decision rests on MR, and MR depends on the elasticity of demand as well as on price (MR = P(1 − 1/e)), knowing the price alone does not tell us the quantity supplied. Shifts and rotations of demand can change price without changing quantity, or quantity without changing price. There is thus no unique relationship between price and quantity supplied: the monopoly supply curve is indeterminate, or, more simply, a monopolist has no supply curve.

Example (our own). Suppose a town's only cable operator has the same costs in two years and, by MR = MC, serves 10,000 homes each year. If in the second year viewers have fewer alternatives, its demand is less elastic, and it can charge a higher monthly fee for the same 10,000 connections. The same quantity goes with two prices, so no single fee schedule describes its supply.

Conclusion

Under monopoly, price and output are decided together by demand and cost through MR = MC. Since different demand conditions can give the same output at different prices, or different outputs at the same price, there is no unique price-quantity relationship, and so no supply curve. Only the equilibrium point for a given demand curve can be found.