Revenue
~30 min · WEC13 · 3.3.2
WEC13 · 3.3.2 · 30 min
Marginal revenue hits zero at exactly the output where demand stops being elastic — not by coincidence, but because they're the same fact seen from two different formulae.
Before you read on
Two or three questions on exactly what this lesson teaches. Being wrong here is fine — it's the fastest way to find out what to pay attention to next.
Three formulae, one underlying quantity
Total revenue is what a firm actually takes in: TR = P × Q, price times quantity sold. Average revenue is revenue per unit: AR = TR/Q. Substitute TR = P×Q into that and the Q cancels — AR = P, exactly. Average revenue and price are the same number for a firm charging one price to everyone; the AR curve and the demand curve are literally the same line, not two separate things that happen to coincide.
Marginal revenue is the revenue from selling one more unit: MR = ΔTR/ΔQ. For a firm with a downward-sloping demand curve, MR sits BELOW AR at every output beyond the first unit — because selling one more unit doesn't just add revenue from that unit, it also means every OTHER unit now sells for a slightly lower price too (assuming a single-price market, not price discrimination). That's the whole reason MR ≠ AR outside perfect competition: the price cut needed to sell the extra unit gets paid on the whole quantity, not just the marginal one.
This lesson's job is the relationship the spec asks for by name: price elasticity of demand's effect on total revenue, understood both numerically (with real calculations) and diagrammatically. It matters directly for the business-objectives lesson's revenue-maximisation rule (MR = 0): the worked-chain below derives exactly why MR hits zero at the unit-elastic point on the demand curve, rather than asking you to memorise that fact separately.
One more distinction sits underneath every revenue figure: nominal revenue is measured in the £ (or $) of the year it was earned; real revenue strips out the effect of general price-level (inflation) changes, so it measures whether a firm actually sold more, not just whether prices in the economy rose. Real revenue in a base year's prices = nominal revenue × (base-year price index ÷ current-year price index). A firm can report nominal revenue growth every year while its real revenue is flat or falling, if inflation over that period was high enough — the MCQ below works this exact calculation through.
Mechanism
Why a price change moves total revenue the way it does
A price change has two effects on total revenue that pull in opposite directions, and which one wins depends on elasticity. Cut the price, and every unit you were already selling now earns less — a revenue LOSS on the existing quantity. But the lower price also draws in extra buyers — a revenue GAIN on the extra quantity sold. If demand is elastic, quantity responds proportionally more than price moved, so the gain from extra units sold outweighs the loss from charging less on each — total revenue rises when price falls, and falls when price rises. If demand is inelastic, quantity barely responds, so the loss (or gain) from the price change on the existing quantity dominates — total revenue moves in the SAME direction as price. At exactly unit elasticity, the two effects are perfectly balanced and total revenue doesn't move at all.
Worked, in full
Deriving MR = 0 at the unit-elastic point
- 01
Start from TR = P×Q, where P and Q are related by the (downward-sloping) demand curve, so P is really a function of Q. The change in TR from a small change in Q is: ΔTR = P·ΔQ + Q·ΔP (the extra revenue from more units, plus the change in revenue from every unit now selling at a different price).
Earns: K — the product-rule structure of TR stated explicitly, not asserted.
- 02
Divide through by ΔQ: MR = ΔTR/ΔQ = P + Q·(ΔP/ΔQ).
Earns: An1 — the definition of MR applied directly to the expression from stage 1.
- 03
Price elasticity of demand is PED = (ΔQ/ΔP)·(P/Q). Rearranging for ΔP/ΔQ gives ΔP/ΔQ = P/(Q·PED). Substituting into stage 2: MR = P + Q·[P/(Q·PED)] = P + P/PED = P·(1 + 1/PED).
Earns: An2 — a genuine algebraic substitution, not a memorised formula presented from nowhere.
- 04
PED for a normal downward-sloping demand curve is negative, so it's conventional to write its magnitude |PED| = e and flip the sign: MR = P·(1 − 1/e). Check the three cases directly: at e = 1 (unit elastic), MR = P·(1−1) = 0 exactly. At e > 1 (elastic), 1/e < 1, so MR is positive. At e < 1 (inelastic), 1/e > 1, so MR is negative.
Earns: Eval — the revenue-maximising rule (MR=0) is shown to be identical to "demand is unit elastic," not two separate facts that happen to line up; and the sign of MR at every other elasticity level follows from the same one formula, not a separate rule per case.
x-axis: Quantity, Q · y-axis: Price / Revenue
- AR (=demand)
- A straight downward-sloping line from a high price-intercept to the quantity-intercept — AR = P at every output, for a single-price firm.
- MR
- A straight line starting at the SAME price-intercept as AR, falling at exactly twice AR's rate, reaching zero at HALF of AR's quantity-intercept — then continuing NEGATIVE beyond that, into the inelastic region the mechanism above derives (MR = P(1−1/e) < 0 when e < 1).
- MR = 0
- Occurs at exactly half of AR's quantity-intercept — the revenue-maximising output, and the unit-elastic point on the demand curve.
- The shaded rectangle IS total revenue — and nothing else
- Its area is exactly P×Q (30×£70 = £2,100 at Q=30 here): the full box from both axes out to the AR curve. This is the precise area a real Jan 2020 examiner report found candidates confusing with supernormal profit — see the trap-taxonomy below. Profit is a DIFFERENT, smaller rectangle, (price − average cost) × quantity, that cannot even be drawn on THIS diagram, because there is no AC curve here to measure it against.
Common error: Drawing MR falling at the same rate as AR (i.e. drawing MR and AR as parallel lines).
Correct: For a straight-line demand curve, MR must fall at exactly TWICE the rate of AR and hit zero at HALF of AR's quantity-intercept — a direct algebraic consequence of AR = a − bQ giving TR = aQ − bQ² and therefore MR = a − 2bQ, not a drawing convention to memorise separately.
x-axis: Quantity, Q · y-axis: Total revenue
- TR
- Rises from zero, peaks exactly at the output where MR = 0 (half of AR's quantity-intercept, from the diagram above), then falls back to zero — a single hump, not a straight line.
- TR peaks exactly where MR = 0
- Forced by definition: MR is the slope of TR (MR = ΔTR/ΔQ), so TR's turning point must be where its own slope is zero — the same Q=50 point as the MR diagram above.
- Left of the peak: demand is elastic
- MR is positive here — to the left of its zero-crossing in the diagram above — so TR is still climbing: the extra revenue from the units sold outweighs the lower price now charged on every existing unit, and MR's sign here is exactly PED's elasticity classification, per the worked-chain above.
- Right of the peak: demand is inelastic
- MR is negative here — past its zero-crossing in the diagram above — so TR is now falling: the loss from charging less on every existing unit outweighs the revenue gained from the few extra units sold. Same mechanism as the left side, just past the point where the two effects flip which one dominates.
Common error: Drawing TR as a straight line, or peaking it at a different output than where the paired AR/MR diagram's MR curve actually crosses zero.
Correct: TR is a curve (a parabola for a straight-line demand curve), and its peak must line up EXACTLY with MR's zero-crossing in the diagram above.
In your own words
In one sentence: why must marginal revenue fall below average revenue at every output beyond the very first unit, for any firm facing a downward-sloping demand curve?
Complete it yourself
Complete the chain — showing MR = P·(1 − 1/e) two other ways
- 01
At e = 2 (elastic), a firm charging P = £20 has MR = 20·(1 − 1/2) = £10.
- 02
At e = 4 (very elastic — close to a highly competitive market), the same firm's MR = 20·(1 − 1/4) = £15 — closer to price, the more elastic demand is.
Named traps
- confusing-a-fall-in-revenue-with-a-fall-in-price
- A price cut does not automatically mean lower total revenue, and a price rise does not automatically mean higher total revenue — the direction depends entirely on elasticity. This is a pure algebra trap, not a memorised exception: work out (or be told) the elasticity first, then apply the mechanism above, rather than assuming price and revenue always move together.
- treating-ar-and-mr-as-the-same-line
- AR = P always, for a single-price firm — but MR = AR only in the special, limiting case of perfectly elastic demand (perfect competition). Everywhere else, MR sits strictly below AR. Drawing them as the same line outside perfect competition is one of the most common diagram errors on this topic.
- unit-elastic-means-revenue-cant-change-not-wont-change-much
- "Unit elastic" is an exact boundary (e = 1, MR = 0 exactly), not an approximate description of "roughly proportional" responses. A question describing demand as unit elastic is telling you total revenue is UNCHANGED by the price change — not merely that it changes by a small amount.
- confusing-revenue-and-profit-areas-on-the-diagram
- Confirmed word-for-word in the real Jan 2020 examiner report, on the exact PED-and-total-revenue question this lesson is built around: "Fewer candidates were able to show the relationship in diagrammatic form. For example, confusing revenue and profit areas below the demand curve." Total revenue is the FULL rectangle from both axes out to the AR curve — price × quantity, nothing subtracted. Supernormal profit is a smaller rectangle nested inside it, (price − average cost) × quantity, and it cannot be shown at all without an AC curve on the diagram. A diagram with only AR and MR curves (like the one above) can only ever show revenue; don't shade an area as "profit" unless an AC curve is actually drawn and the shaded box sits between AR and AC, not between AR and the origin.
Beyond the spec
The spec asks you to apply the PED-revenue relationship but doesn't ask why a firm would ever want to know it. Knowing the actual business use closes that gap — without it, the relationship reads as a pure maths exercise rather than something a real pricing manager uses.
The MR = P·(1 − 1/e) relationship derived above is the mathematical foundation of a real pricing technique: a firm that has separately estimated the elasticity of demand it faces (e.g. from past sales data at different prices) can set its profit-maximising price directly from its marginal cost, using the rearranged formula P = MC / (1 − 1/e) — sometimes called inverse-elasticity pricing. A firm facing more elastic demand (e large) sets a price close to marginal cost, because a big proportional demand response punishes any markup heavily; a firm facing less elastic demand (e small, closer to 1) can sustain a much bigger markup over cost before losing enough customers to make it unprofitable. This is the same underlying logic that shows up in real-world third-degree price discrimination — charging a higher markup to the group with less elastic demand — but applied to a single market rather than splitting one market into several.
Same question, every level
Discuss the likely impact of a price increase on a firm's total revenue, given that its product faces price-inelastic demand. (VERIDIAN-original question, an 8-mark item of the type this paper's own verified pattern marks as one blended Knowledge/Application/Analysis/Evaluation band across L1-L4 — see How This Paper Is Structured — not a reproduction of any past-paper question.)
8 marks available
If the firm puts its price up, it will earn more money because it is charging more for each unit.
States the conclusion without reference to elasticity or quantity at all — asserts price and revenue always move together, exactly the trap named above. No diagram, no use of the given information that demand is inelastic.
Retrieval — with feedback on every choice
A firm sells 100 units at $10 each. It cuts its price to $9 and quantity demanded rises to 130 units (midpoint elasticity = −2.48, i.e. elastic).
What happens to total revenue? (VERIDIAN-original — a calculation item of the type this spec point is actually assessed by.)
A firm sells 100 units at $10 each. It raises its price to $12 and quantity demanded falls to 92 units (midpoint elasticity = −0.46, i.e. inelastic).
What happens to total revenue?
A firm's marginal revenue is currently negative. What must be true of demand at this output?
A firm's demand curve is a straight line, AR = 50 − 2Q. Using the fact that MR falls at twice the rate of AR for a straight-line demand curve, what is MR when Q = 10?
A firm's nominal revenue rises from $100,000 in Year 1 (price index = 100) to $115,000 in Year 2 (price index = 110).
What is the firm's real revenue growth, in Year 1 prices? (Tests the money-to-real-terms conversion skill listed as a quantitative skill for this qualification.)
- TR = P×Q. AR = TR/Q = P (for a single-price firm). MR = ΔTR/ΔQ.
- MR = P·(1 − 1/e), where e = |PED|. MR = 0 exactly at e = 1 (unit elastic) — this is why MR=0 is the revenue-maximising rule.
- Elastic demand (e>1): price and TR move in OPPOSITE directions. Inelastic (e<1): price and TR move in the SAME direction. Unit elastic (e=1): TR unchanged.
- Straight-line demand curve: MR falls at exactly TWICE AR's rate, crossing zero at HALF of AR's quantity-intercept.
- MR = AR = P only in the limiting case of perfectly elastic demand (perfect competition) — everywhere else, MR < AR.
- Real revenue (base-year prices) = nominal revenue × (base-year price index ÷ current-year price index). Nominal growth ≠ real growth whenever the price index has moved.
- Total revenue on a diagram is the FULL rectangle P×Q out to the AR curve — profit is a smaller, different rectangle (P − AC) × Q that needs an AC curve to draw at all. A real examiner report confirms candidates confuse the two.
Not affiliated with or endorsed by Pearson Edexcel. Every quotation and figure attributed to a mark scheme or examiner report in this lesson was independently verified against the primary Pearson document, not carried over from prior course material.
A firm sells 100 units at $10 each. It cuts its price to $9 and quantity demanded rises to 130 units (midpoint elasticity = −2.48, i.e. elastic).
What happens to total revenue? (VERIDIAN-original — a calculation item of the type this spec point is actually assessed by.)
- AFalls from $1,000 to $900
This only accounts for the lower price on the ORIGINAL 100 units — it ignores that 30 extra units are now being sold too, which is exactly the trap of assuming price and revenue always move together.
- Rises from $1,000 to $1,170
Correct. TR before = $10×100 = $1,000. TR after = $9×130 = $1,170. Elastic demand means the 30% rise in quantity outweighs the 10.5% fall in price, so total revenue rises despite the lower price — check it yourself: 9×130=1,170.
- CStays exactly the same at $1,000
That would only be true at UNIT elastic demand (e = 1) — this firm's demand is elastic (e ≈ 2.48), well past the point where revenue stays constant.
- DCannot be calculated from the information given
It can — TR = P×Q is calculable directly from the price and quantity given, both before and after the price change. No further information is needed.
Traps tested: Ignores quantity response · Confuses elastic with unit elastic · Overclaims uncertainty
A firm sells 100 units at $10 each. It raises its price to $12 and quantity demanded falls to 92 units (midpoint elasticity = −0.46, i.e. inelastic).
What happens to total revenue?
- Rises from $1,000 to $1,104
Correct. TR before = $10×100 = $1,000. TR after = $12×92 = $1,104. Inelastic demand means the 8.3% fall in quantity is smaller than the 18.2% rise in price, so revenue rises along with price — check it yourself: 12×92=1,104.
- BFalls from $1,000 to $920
This applies the new quantity to the OLD price ($10×92) — it never actually applies the new $12 price to the calculation at all.
- CFalls, because raising price always reduces the number of units sold
Fewer units ARE sold (92 instead of 100) — but that alone doesn't determine what happens to total revenue. Revenue depends on both price and quantity together, and here the higher price per unit more than compensates for the small quantity fall.
- DStays exactly the same at $1,000
That would only hold at exactly unit elastic demand — this firm's demand is clearly inelastic (e ≈ 0.46), which is precisely why revenue moves WITH the price change rather than staying fixed.
Traps tested: Uses wrong price · Conflates quantity fall with revenue fall · Confuses inelastic with unit elastic
A firm's marginal revenue is currently negative. What must be true of demand at this output?
- ADemand is price elastic (e > 1)
Elastic demand gives POSITIVE marginal revenue (from MR = P(1−1/e), e>1 means 1/e<1, so MR>0) — the opposite of what's described here.
- Demand is price inelastic (e < 1)
Correct. MR = P·(1−1/e). When e < 1, 1/e > 1, making (1−1/e) negative, so MR is negative. A firm here is selling into the inelastic region of its demand curve, where an extra unit sold actually reduces total revenue.
- CDemand is unit elastic (e = 1)
Unit elastic demand gives MR = P·(1−1) = 0 exactly — zero, not negative.
- DThe firm is making a loss
Marginal revenue is about the effect of one more unit on REVENUE, not about profit or cost at all — a firm can have negative MR while still being profitable overall, or positive MR while loss-making. The two are independent.
Traps tested: Direction reversed · Confuses zero with negative · Conflates revenue and profit
A firm's demand curve is a straight line, AR = 50 − 2Q. Using the fact that MR falls at twice the rate of AR for a straight-line demand curve, what is MR when Q = 10?
- AMR = 30
This computes 50 − 2Q rather than 50 − 4Q — using AR's own formula rather than doubling its slope to get MR's.
- MR = 10
Correct. MR = 50 − 4Q, so at Q=10: MR = 50 − 40 = 10. AR at the same output is 50 − 2(10) = 30, confirming MR (10) sits below AR (30), as it always must for a downward-sloping demand curve.
- CMR = 40
This computes 50 − Q rather than 50 − 4Q — using AR's own coefficient (2) instead of doubling it to get MR's coefficient (4).
- DMR = 50
This is just the price-intercept, ignoring the effect of quantity entirely — MR falls as Q rises exactly like AR does, just twice as steeply.
Traps tested: Used ar slope not doubled · Wrong coefficient · Ignores slope
A firm's nominal revenue rises from $100,000 in Year 1 (price index = 100) to $115,000 in Year 2 (price index = 110).
What is the firm's real revenue growth, in Year 1 prices? (Tests the money-to-real-terms conversion skill listed as a quantitative skill for this qualification.)
- A15%, the same as nominal growth
This ignores the price index entirely and reads nominal growth straight off — but 10 percentage points of that 15% rise is just economy-wide inflation, not the firm selling more.
- Approximately 4.5%
Correct. Real Year 2 revenue in Year 1 prices = $115,000 × (100/110) = $104,545. Real growth = ($104,545 − $100,000)/$100,000 = 4.55%. Most of the firm's headline 15% nominal growth is inflation; only about a third of it reflects a genuine increase in what the firm actually sold.
- C10%, the size of the inflation adjustment
This confuses the inflation rate itself (10%, from the price index rising 100→110) with the REMAINING real growth after removing inflation's effect — those aren't the same number; you have to actually divide nominal revenue by the price ratio, not just subtract the inflation rate from nominal growth.
- D0%, because the price index rose
A rising price index means SOME of the nominal growth is inflation, not that ALL of it is. Here nominal growth (15%) is bigger than the price rise (10%), leaving genuine real growth of about 4.5% — the calculation has to be done, not assumed to cancel to zero.
Traps tested: Ignores price index · Subtracts instead of deflating · Assumes full offset
Practice this for real
This site teaches the mechanism; the exam is sat on Pearson's own real questions. Go find and attempt these yourself — nothing here substitutes for actually sitting a timed paper.
Pearson's official past-papers portalSelect International Advanced Level → Economics → any series, then look for WEC13.
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