Marginal Revenue Curve From Demand Curve

You stare at the graph. But two lines. The other slopes down faster, cutting the horizontal axis at exactly half the quantity. One slopes down — that's demand. That's marginal revenue.

Your professor drew it in three seconds. You've been staring at it for twenty minutes.

Here's the thing nobody says out loud: the relationship between the demand curve and the marginal revenue curve isn't a trick. But it's not a "gotcha" for exams. Plus, it's the entire reason monopolies don't produce where demand equals marginal cost. Which means it's why pricing power exists. And once you actually see why the math works — not just memorize the rule — the rest of industrial organization stops feeling like magic.

What Is the Marginal Revenue Curve from Demand Curve

Marginal revenue is the extra revenue you get from selling one more unit. Simple definition. But the curve — the shape* of it — comes from a tension that most textbooks gloss over.

When you lower price to sell an additional unit, two things happen at once. So you gain revenue on that new unit. But you also lose revenue on every previous unit* because they all now sell at the lower price. That second effect — the inframarginal loss — is why marginal revenue sits below demand for any firm with market power.

For a linear demand curve written as P = a - bQ, the marginal revenue curve is MR = a - 2bQ. Day to day, same vertical intercept. Twice the slope. The MR curve hits the quantity axis at exactly half the distance where demand hits it.

The Geometry Nobody Explains

Draw a linear demand curve from (0, a) to (a/b, 0). The midpoint of that line — at quantity a/2b — is where marginal revenue crosses zero. Think about it: at that exact point, demand is unit elastic. Also, to the left, demand is elastic and MR is positive. To the right, demand is inelastic and MR turns negative.

Negative marginal revenue. Think about that. Selling more* actually reduces* total revenue. The price cut on all previous units outweighs the revenue from the new one. A firm would never voluntarily operate there unless forced.

Nonlinear Demand Changes the Picture

The "twice the slope" rule only holds for linear demand. With constant elasticity demand — Q = AP^(-ε) — the relationship becomes MR = P(1 - 1/ε). Also, marginal revenue is a constant fraction of price. The curves don't have that clean geometric relationship anymore.

This matters. They're usually steeper at high prices, flatter at low prices. Real demand curves aren't linear. The MR curve inherits that curvature in ways that don't follow a simple doubling rule.

Why It Matters / Why People Care

If you're a monopolist — or any firm facing a downward-sloping demand curve — you maximize profit where MR = MC. Not where demand equals MC. Not where price equals MC. Where marginal revenue* equals marginal cost.

The gap between demand and marginal revenue is your markup. It's the measure of your market power. The wider the gap, the more you can push price above marginal cost.

The Lerner Index Lives Here

The Lerner Index — (P - MC)/P — equals 1/|ε| at the profit-maximizing quantity. In practice, that elasticity? It's determined by the shape of the demand curve at the quantity where MR = MC*. The marginal revenue curve is the bridge between demand elasticity and actual pricing behavior.

Firms don't think in elasticities. They think in marginal revenue. "If I sell one more unit, what happens to my total revenue?Plus, " That's the operational question. In real terms, the demand curve answers "what price can I charge? " The marginal revenue curve answers "what do I actually gain?

Policy and Regulation

Antitrust authorities look at the gap between price and marginal cost. But they can't observe marginal cost directly. They can estimate demand curves. From estimated demand, they derive implied marginal revenue. The difference between observed price and implied MR gives a structural measure of market power — no cost data required.

This is how the DOJ evaluates mergers. In real terms, this is how the EU fines dominant firms. But the marginal revenue curve isn't academic. It's enforcement infrastructure.

How It Works (or How to Derive It)

Start with total revenue. TR = P × Q. But P is a function of Q — that's the demand curve. So TR(Q) = P(Q) × Q.

Marginal revenue is the derivative: MR = d(TR)/dQ = P(Q) + Q × dP/dQ.

That second term — Q × dP/dQ — is the inframarginal loss. Day to day, mR < P. Practically speaking, it's negative because dP/dQ is negative (demand slopes down). So MR = P + (negative number). Always, for downward-sloping demand.

Step-by-Step for Linear Demand

1. Write demand: P = a - bQ
2. Total revenue: TR = (a - bQ)Q = aQ - bQ²
3. Differentiate: MR = a - 2bQ
4. Check: same intercept (a), twice the slope (-2b vs -b)

That's it. The algebra takes thirty seconds. The intuition takes longer.

Want to learn more? We recommend is the empty set a subset of all sets and what is the probability of an event that is impossible for further reading.

Step-by-Step for Constant Elasticity

1. Demand: Q = AP^(-ε) → P = (Q/A)^(-1/ε)
2. Total revenue: TR = P × Q = (Q/A)^(-1/ε) × Q = A^(1/ε) Q^(1 - 1/ε)
3. Differentiate: MR = (1 - 1/ε) A^(1/ε) Q^(-1/ε) = (1 - 1/ε) P
4. Result: MR = P(1 - 1/ε)

Notice the elasticity ε appears directly. Because of that, when ε = 1 (unit elastic), MR = 0. When ε < 1 (inelastic), MR < 0. When ε > 1 (elastic), MR > 0 but less than P.

Graphical Derivation Without Calculus

You can derive the linear MR curve geometrically. Worth adding: pick any quantity Q₀ on the demand curve. The price is P₀. Total revenue is the rectangle under (Q₀, P₀).

Now increase quantity by a tiny amount ΔQ. The new price is P₀ + ΔP (where ΔP is negative). The change in total revenue has two parts:

• Gain: ΔQ × (P₀ + ΔP) ≈ ΔQ × P

₀ (revenue from the extra unit sold at the new price)

• Loss: Q₀ × |ΔP| (revenue lost on all previous units because the price had to drop)

Marginal revenue is the net: Gain minus Loss. Since dP/dQ = -b (constant slope), the loss per unit of quantity is bQ₀. So as ΔQ → 0, the gain approaches P₀ΔQ. So the loss approaches Q₀|dP|. So MR = P₀ - bQ₀. Since P₀ = a - bQ₀, MR = a - 2bQ₀.

Geometrically, the MR curve bisects the horizontal distance between the demand curve and the vertical axis. On the flip side, at any price, the quantity where MR = 0 is exactly halfway to the quantity where P = 0. Total revenue is maximized where MR crosses zero — the midpoint of the linear demand curve.

Where the Standard Model Bends

Price Discrimination

The MR < P rule assumes a single price for all units. If a firm can charge different prices to different consumers — perfect price discrimination — the inframarginal loss disappears. The firm sells each unit at the buyer's willingness to pay. Marginal revenue equals* the demand curve. The firm produces the competitive quantity (where P = MC) but captures all surplus. The deadweight loss vanishes; the distribution problem remains.

Kinked Demand and Oligopoly

Sweezy’s kinked demand curve posits asymmetric reactions: rivals match price cuts but ignore price increases. This creates a discontinuity in the MR curve — a vertical gap at the kink. Within that gap, marginal cost can shift without changing the profit-maximizing price. Price rigidity emerges not from menu costs, but from the strategic shape of marginal revenue.

Dynamic Pricing and Capacity Constraints

When capacity is fixed (airlines, hotels, ride-share at peak), the marginal cost of the last seat is near zero — but the opportunity cost* is the expected value of selling that seat to a later, higher-paying customer. Marginal revenue becomes the derivative of the expected* revenue function under uncertainty. The firm doesn't just ask "what is MR today?" It asks "what is the marginal value of inventory?" That’s revenue management: MR as a shadow price.

Behavioral Kinks

Consumers reference prices. A price increase above a "fair" reference point may trigger disproportionate demand loss — a kink in demand created by psychology, not technology. The MR curve inherits that kink. Firms optimize around behavioral thresholds, not just cost curves.

The Bottom Line

Marginal revenue is the translation layer between the market’s willingness to pay and the firm’s production decision. Which means it internalizes the tension between volume and margin. Every pricing strategy — bundling, versioning, peak-load pricing, two-part tariffs — is an attempt to manipulate the marginal revenue curve, to shrink the gap between MR and P, to extract more of the area under the demand curve.

You don’t need to derive MR every day. The marginal revenue curve is the skeleton key of industrial organization. But if you don’t understand why it lies below demand, you don’t understand why monopolies restrict output, why price discrimination increases efficiency, why mergers raise prices, or why the Lerner Index measures market power. Master it, and the rest of the structure falls into place.