Monopoly Profit Maximization

A streaming service launches a hit show. No rival has anything close. The firm faces the market demand curve. A high price means few subscribers. A low price means many subscribers but less revenue per head. Which price brings in the most money?

Start with the definition. Profit is total revenue minus total cost: $π = T R - T C$ . The firm picks $Q$ to make $π$ as large as possible. To find the top of a curve, take its slope (the derivative with respect to $Q$ ) and set the slope to zero. A zero slope means the curve has stopped rising.

Differentiate $π = T R - T C$ and set the slope to zero. $M R$ is the slope of $T R$ , $M C$ is the slope of $T C$ .

Rearranging gives the rule: $M R = M C$ . It is not an axiom. It falls out of one definition and one piece of calculus. The intuition is plain. Keep producing while the next unit’s revenue covers its cost. Stop the moment the two meet.

That fixes the quantity. For the price, go up to the demand curve at $Q_{m}$ and read it off. For a monopolist, $M R$ is below $P$ . So $P_{m}$ sits above $M C$ . The gap is the monopoly markup.

Key Term

Two-step recipe

Step 1. Find $Q_{m}$ where $M R = M C$ .

Step 2. Read $P_{m} = D (Q_{m})$ off the demand curve.

Profit: $π = (P_{m} - A T C) \times Q_{m}$ .

Profit is a rectangle. $P_{m}$ on top. $A T C$ on the bottom. Width $Q_{m}$ . In a competitive market, entry shrinks this rectangle to zero. For a monopolist, nothing does. The rights to the show keep rivals out. The rectangle stays.

Worked Example

Same streaming service. Demand: $P = 100 - 2 Q$ ( $Q$ in millions of subscribers, $P$ in dollars per month). Costs: $M C = 20 + Q$ , $A T C = 20 + 0.5 Q + 50/ Q$ . Find $Q_{m}$ , $P_{m}$ , and profit.

Interactive — Drag the quantity, watch profit rise, peak, then fall

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Demand, MR, MC, and ATC on one chart. Slide Q. The profit rectangle is largest at MR = MC.

Check yourself · no marks

A monopolist stops at $M R = M C$ . Why not keep going to $P = M C$ , the way a competitive firm does?

Commit to one first. Guessing and being wrong beats reading the answer cold.

Interactive — Drag the line to the profit-maximising quantity

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Demand, MR, and MC. Drop the vertical line where a monopolist would produce.

Practice · 1 / 4

No penalty for a wrong answer

Monopoly profit maximization:

A monopolist faces demand $P = 80 - Q$ with $M C = 20$ . The profit-maximizing quantity is:

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