Price, Income and Cross-Elasticities of Demand

Why a YED-driven income change shifts the whole demand curve, not just quantity along it

1. 01

   A demand curve has only two axes, price and quantity — every other influence on demand, income included, is held fixed ('ceteris paribus') at whatever level it happened to be when that particular curve was drawn. YED asks what happens when income, one of those held-fixed variables, is the thing that actually changes. Because income sits outside both axes, a change in income can never be shown by moving to a different point on the SAME curve — the curve was only ever valid at the old income level, so a genuine change in income makes it the wrong curve, and an entirely new one has to be drawn instead.

   Earns: K — the movement-vs-shift distinction re-derived from what a demand curve's two axes actually hold fixed, rather than asserted as a rule to remember.

2. 02

   YED = %ΔQd ÷ %ΔY rearranges, by multiplying both sides by %ΔY, to %ΔQd = YED × %ΔY — the demand response is the elasticity's sign multiplied by the income change's sign, exactly like any other product of two signed numbers. Four cases follow directly: YED positive (normal/luxury) with income RISING gives a positive %ΔQd — demand shifts RIGHT. YED positive with income FALLING gives a negative %ΔQd — demand shifts LEFT. YED negative (inferior) with income RISING gives a negative %ΔQd — demand shifts LEFT. YED negative with income FALLING gives a positive %ΔQd — demand shifts RIGHT. The sign of YED on its own never fixes the shift direction — it only fixes the direction once it's multiplied by whichever way the income change actually went.

   Earns: An1a — the general shift-direction rule derived algebraically as a sign-of-a-product argument, all four sign combinations stated explicitly rather than only the one the anchor question needs.

3. 03

   Applying this to the anchor's own numbers: real income in Laos fell by 12.32% (%ΔY = −12.32), and the income elasticity of demand for a domestic holiday was YED = +1.36 — positive, so domestic holidays are a normal good there, and specifically a luxury (|YED| > 1). From stage 2's identity, %ΔQd = 1.36 × (−12.32%) = −16.7552%, computed exactly (1.36 × 12.32 = 16.7552). A negative %ΔQd at every price level is precisely what a LEFTWARD shift of the whole demand curve means — less is now wanted at every price, not just at whatever the old equilibrium price happened to be.

   Earns: An1b — the Laos figures substituted into the general identity as a numeric check, with the exact arithmetic shown rather than the direction merely asserted.

4. 04

   Nothing about the fall in real income changes any determinant of SUPPLY for domestic holidays — income affects demand only, so the supply curve stays exactly where it was. At the OLD equilibrium price, the new, lower demand curve now wants less than the unchanged supply curve is offering — an excess supply appears at that price — so price is bid down; as price falls, quantity supplied contracts along the still-unmoved supply curve until it meets the new demand curve at a new equilibrium. That new equilibrium is necessarily at both a LOWER price and a LOWER quantity than before: pulling one curve left while holding the other fixed can only drag their intersection down and to the left along the fixed curve, never up or to the right.

   Earns: An2 — the equilibrium consequence of a leftward demand shift against a fixed supply curve derived from first principles (excess supply forcing price down), rather than quoted as a memorised shift-direction rule.

5. 05

   The chain generalises in both directions: whenever YED's sign and the income change's direction combine, via %ΔQd = YED×%ΔY, to give a negative %ΔQd, demand shifts left and — against an unchanged supply curve — equilibrium price AND quantity both fall; whenever the combination gives a positive %ΔQd, demand shifts right and both rise instead. Reading the chain through to this final equilibrium effect, rather than stopping at 'demand shifts left,' is what actually answers a question asking to illustrate the impact on 'the equilibrium price and quantity' — the exact phrase this anchor question uses, and exactly the three Application marks (original equilibrium; demand shifted left; new equilibrium) a diagram like this one is marked against.

   Earns: Eval — the direction rule generalised beyond the one anchor case and explicitly tied back to what the mark scheme actually rewards, rather than left as a fact true only for this specific question.

Why PED changes continuously along one straight-line demand curve

1. 01

   Take a firm facing the general straight-line demand curve P = a − bQ (a, b > 0) — the shape of every downward-sloping demand line on this paper. Concretely, let a = £20 and b = £2 per thousand units, so P = 20 − 2Q, with Q measured in thousands of units per week.

   Earns: K — both the general and a concrete form stated up front, so every later step can be checked against real numbers.

2. 02

   The PED formula is %ΔQ ÷ %ΔP, which is algebraically the same as (ΔQ/Q) ÷ (ΔP/P) — each percentage change written out as the raw change over the original value. Dividing by a fraction means multiplying by its reciprocal: (ΔQ/Q) ÷ (ΔP/P) = (ΔQ/Q) × (P/ΔP). Reordering the four terms (multiplication doesn't care about order) gives (ΔQ/ΔP) × (P/Q) — a raw slope-ratio multiplied by the price-to-quantity ratio at that point.

   Earns: An1a — the invert-and-multiply step shown as its own line, not skipped over on the way from a division-of-fractions to a product.

3. 03

   To evaluate that slope-ratio ΔQ/ΔP, rearrange the demand curve P = a − bQ to make Q the subject: add bQ to both sides to get P + bQ = a, then subtract P from both sides to get bQ = a − P, then divide both sides by b to get Q = (a−P)/b. Every one-unit change in P now changes Q by exactly −1/b (the coefficient in front of P), so ΔQ/ΔP = −1/b — a constant, the same everywhere on the line, because the line has one constant slope.

   Earns: An1b — the P=a−bQ → Q=(a−P)/b isolation shown as its own three-line rearrangement, not asserted as a one-step jump.

4. 04

   Substituting both results back into PED = (ΔQ/ΔP) × (P/Q): PED = (−1/b) × (a−bQ)/Q = 1 − a/(bQ), using P = a−bQ for the P in the numerator and simplifying −(a−bQ)/(bQ) into 1 − a/(bQ) by splitting the fraction into two terms.

   Earns: An1c — the two derived pieces (the constant slope-ratio and the P=a−bQ substitution) combined into the single closed-form PED(Q), the line-by-line combination made explicit rather than presented as a finished result.

5. 05

   Evaluate PED(Q) = 1 − a/(bQ) at three points on the £20/£2 curve, computed exactly: at Q=2 (P=£16, near the price axis), PED = 1 − 10/2 = −4 — elastic. At Q=5 (P=£10, the exact midpoint of the line), PED = 1 − 10/5 = −1 — unit elastic. At Q=8 (P=£4, near the quantity axis), PED = 1 − 10/8 = −0.25 — inelastic.

   Earns: An2 — three concrete, independently checkable values plotted on the same line, showing the classification changes even though nothing about the line's slope did.

6. 06

   None of this is specific to a=20, b=2. The same algebra gives PED → −∞ as Q → 0 (right at the price axis: an almost-zero quantity is being sold at a price near the maximum the market will bear, so a small price cut pulls in disproportionately many extra buyers) and PED → 0 as Q → a/b (right at the quantity axis: price is already near zero, so cutting it further barely changes anything). And exactly at the midpoint, Q = a/(2b), the algebra gives PED = 1 − a/(b·a/(2b)) = 1 − 2 = −1 every single time — unit-elastic-at-the-midpoint is a forced mathematical consequence of any straight-line demand curve, not a coincidence of these particular numbers.

   Earns: Eval — the pattern generalised beyond one worked example, with both limiting cases named explicitly, which is what turns 'I checked three points' into 'I understand why this is true for every straight-line demand curve.'

The PED–total-revenue link, worked from MR = −bQ(1+PED)

1. 01

   Total revenue is TR = P × Q. On the same £20/£2 demand curve, TR(Q) = (20 − 2Q) × Q = 20Q − 2Q². Evaluate it at the same three quantities used for PED above: TR(2) = £32, TR(5) = £50, TR(8) = £32.

   Earns: K — TR computed as a genuine function of Q on the same curve already in use, not a fresh, disconnected example.

2. 02

   TR = P×Q, and P is really a function of Q along the demand curve, so a change in Q changes TR two ways at once: ΔTR = P·ΔQ + Q·ΔP — the extra revenue from the extra units themselves, plus the change in revenue on every unit already being sold, now selling at a different price. Dividing through by ΔQ gives MR = ΔTR/ΔQ = P + Q·(ΔP/ΔQ). The slope ΔP/ΔQ on this line was already pinned down in the PED worked chain above — ΔQ/ΔP = −1/b there, so ΔP/ΔQ = −b = −2, the same constant £2-per-thousand-units slope every point on P = 20 − 2Q shares. Substituting P = 20 − 2Q and ΔP/ΔQ = −2: MR = (20 − 2Q) + Q·(−2) = 20 − 2Q − 2Q = 20 − 4Q (the discrete version — the actual cash difference from selling one whole extra unit — is checked directly below, together with the exact reason the two numbers don't quite match).

   Earns: An1a — MR derived algebraically from TR = P×Q, reusing the constant slope already pinned down in the earlier worked chain, rather than quoted as 'the standard result' or reached through calculus this paper doesn't require.

3. 03

   Checking this against the discrete version directly: TR(3) − TR(2) = £42 − £32 = £10, the actual extra cash from selling a third unit — smaller than MR(2) = 20 − 4(2) = £12 from the formula just derived. The £2 gap has an exact source, not a vague 'rounding' one: fully expanding (P+ΔP)(Q+ΔQ) − PQ gives ΔTR = P·ΔQ + Q·ΔP + ΔP·ΔQ, three terms, but stage 2's ΔTR = P·ΔQ + Q·ΔP kept only the first two. The missing third term, evaluated over the Q=2-to-3 step (ΔQ=1, ΔP=−£2, the curve's constant slope), is ΔP·ΔQ = −£2 × 1 = −£2 — exactly the £12-vs-£10 shortfall. Since Q is measured in thousands of units, a 'step' of 1 in Q is really 1,000 physical units — at that scale the £2 cross-term is trivial next to the other two terms, which is why MR = 20 − 4Q is still an accurate per-unit figure even though the discrete and formula-derived readings of 'marginal revenue' are not identical in principle, as the exact £12-vs-£10 gap above shows.

   Earns: An1b — the discrete and formula-derived readings of 'marginal revenue' reconciled against a real numeric example, with the exact size of the gap traced to the specific term dropped in the derivation, rather than left implicitly conflated or blamed vaguely on 'approximation.'

4. 04

   Evaluate MR = 20 − 4Q at the same three quantities used for PED above: at Q=2, MR = +£12 — the revenue curve is still rising at this point. At Q=5, MR = £0 — the revenue curve is momentarily flat, its peak. At Q=8, MR = −£12 — the revenue curve is now falling.

   Earns: An2 — MR computed exactly at the same three points already used for PED, so the two derivations can be checked against each other directly.

5. 05

   Compare the two results at each point: Q=2 gave PED=−4 (elastic) and MR=+£12 (positive). Q=5 gave PED=−1 (unit elastic) and MR=£0. Q=8 gave PED=−0.25 (inelastic) and MR=−£12 (negative). The sign of MR matches the elastic/unit/inelastic classification exactly at every point checked — and this isn't a coincidence of these three points: algebraically, MR = −bQ(1+PED) for any straight-line demand curve, which is negative whenever PED is between −1 and 0 (inelastic), positive whenever PED is below −1 (elastic), and exactly zero at PED=−1. Checked against all nine integer quantities on this curve (Q=1 through Q=9), the identity holds exactly every time, not just at the three points quoted here.

   Earns: An3 — the exact algebraic identity MR = −bQ(1+PED) stated and cross-checked against every numeric point already computed, not just observed as a pattern in three cases.

6. 06

   Since Q rises as price falls along a demand curve, 'MR positive as Q rises' means the same thing as 'TR rises as price falls' — so a price CUT raises revenue exactly where demand is elastic (PED < −1), and a price cut LOWERS revenue exactly where demand is inelastic (−1 < PED < 0). Equivalently: a price RISE raises revenue where demand is inelastic, and lowers it where demand is elastic. This is not two rules to memorise — it is one fact (the sign of 1+PED) read in two directions depending on which way the price is moving.

   Earns: Eval — the classic PED-TR relationship stated as the forced consequence of the algebra just derived, in both directions, rather than as two separate memorised facts.

Complete the chain, step 1 of 2 — deriving why a POSITIVE XED means substitutes (heavily scaffolded)

1. 01

   XED = %ΔQd of good B ÷ %ΔP of good A.

2. 02

   Suppose the price of good A rises. A consumer who used to buy some of good A now finds it relatively more expensive than before, and has to decide what to do about the rest of their spending.

3. 03

   If B is a substitute for A — it fulfils a similar need — some consumers switch away from the now-pricier A and buy B instead, so Qd of B rises. %ΔP(A) is positive and %ΔQd(B) is also positive: the same sign, so XED is positive.

Complete the chain, step 2 of 2 — now derive both cases yourself, with no worked stages given

1. 01

   XED = %ΔQd of good B ÷ %ΔP of good A.

Evaluate the view that a government should always target an indirect tax at goods with the most price-inelastic demand in order to raise the maximum tax revenue. (VERIDIAN-original question, written in the pattern of WEC11's Section D essay format — 20 marks total, split 12 for Knowledge/Application/Analysis and 8 for Evaluation, confirmed directly against the Pearson IAL Economics Unit 1 Exemplars, Feb 2020, pp.53-55 — not a reproduction of any real past-paper question.)

20 marks available