The Monopolist's Demand

A competitive firm is a price-taker — it faces a horizontal demand line at the market price. A monopolist faces the whole market demand curve, which slopes downward. To sell more, the monopolist has to lower the price on every unit. That single fact reshapes every revenue concept that follows. The cleanest way to see why is to start from the demand curve itself and build up — first to total revenue, then to marginal revenue.

Key Term

Demand is the same thing as average revenue

At any quantity $Q$ , the demand curve says: “this is the price the market will pay for the $Q$ -th unit.” If the monopolist sells $Q$ units at that price, every customer pays $P$ , so the revenue per unit the firm collects is just $P$ itself. That’s what average revenue ( $A R$ ) means: $A R = T R / Q = P$ . The demand curve and the AR curve are the same curve.

Interactive — Demand is AR — TR is the rectangle

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Slide Q. The demand curve gives the price (= AR). The shaded rectangle is TR = P × Q.

So total revenue is a rectangle whose height and width trade off along the demand curve. As Q rises, P falls, so the rectangle’s area $T R = P \times Q$ isn’t a straight line — it rises, peaks, then falls. To make this concrete, plug in the demand curve.

Multiply average revenue by quantity to get total revenue. A linear demand makes TR a downward-opening parabola.

Marginal revenue is the rate at which $T R$ changes when $Q$ rises by one unit — in calculus terms, $M R = d T R / d Q$ . Take that derivative and the result falls out cleanly:

MR is the slope of TR. Same vertical intercept as demand ( $a$ ), but twice the slope.

Key Term

Same intercept, twice the slope

For linear demand $P = a - b Q$ , marginal revenue is $M R = a - 2 b Q$ . Three consequences:

MR starts at $a$ — same vertical intercept as demand.
MR hits zero at $Q = a / (2 b)$ — exactly half the quantity where demand hits zero.
MR sits below demand at every $Q > 0$ — the “price effect” (price cut applied to all previous units) drags MR down faster than the demand curve.

Interactive — TR rises, peaks, then falls — MR is its slope

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Three curves on one chart: demand (= AR), TR (the parabola), and MR (the slope of TR). Slide Q and watch how the three are connected.

Worked Example

A monopolist faces demand $P = 100 - 2 Q$ . Build TR, take the derivative to find MR, and find the quantity where TR peaks.

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Why does $M R$ fall faster than the demand curve for a monopolist?

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The monopolist’s demand and MR:

For a monopolist, $M R$ is less than $P$ because:

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