variation in price elasticity of demand along the length of a straight-line demand curve

Cambridge A‑Level Economics 9708 – Elasticities of Demand and Supply

1. Definitions, Formulae & Sign Conventions (ceteris paribus)

• Price elasticity of demand (PED)

  $$E_P=\frac{\%\Delta Q}{\%\Delta P}= \frac{dQ}{dP}\frac{P}{Q}$$

  Negative because of the law of demand (price ↑ → quantity ↓).

• Price elasticity of supply (PES)

  $$E_S=\frac{\%\Delta Q_s}{\%\Delta P}= \frac{dQ_s}{dP}\frac{P}{Q_s}$$

  Positive because of the law of supply (price ↑ → quantity supplied ↑).

• Income elasticity of demand (YED)

  $$E_I=\frac{\%\Delta Q}{\%\Delta I}$$

• Cross‑price elasticity of demand (XED)

  $$E_{XY}= \frac{\%\Delta Q_X}{\%\Delta P_Y}$$

All definitions assume ceteris paribus – all other relevant factors are held constant.

2. Why Elasticities Matter (AO2)

Elasticities enable us to predict the effect of changes in price, income or related‑good prices on quantities demanded or supplied. They are essential for:

1. Pricing and output decisions (business & government).
2. Revenue and profit forecasting (total‑revenue analysis).
3. Tax incidence and welfare analysis (who bears a tax, dead‑weight loss).
4. Understanding consumer/producer behaviour (normal vs. inferior goods, substitutes vs. complements, elastic vs. inelastic supply).

3. Determinants of Elasticity (ceteris paribus)

3.1 Determinants of PED

• Availability of close substitutes – more substitutes → higher |E_P| (more elastic).
• Proportion of income spent on the good – larger share → higher |E_P|.
• Time horizon – longer period gives consumers time to adjust → higher |E_P|.
• Nature of the good – luxuries are more elastic; necessities, addictive or habit‑forming goods are less elastic.

3.2 Determinants of PES

• Time to adjust production – longer period → higher |E_S|.
• Flexibility of inputs – mobile, easily stored or substitutable inputs increase |E_S|.
• Spare production capacity – excess capacity → higher |E_S|.
• Complexity of the production process – complex, specialised processes lower |E_S| (more inelastic).

3.3 Determinants of YED

• Nature of the good – necessities: 0 < E_I < 1; luxuries: E_I > 1.
• Income level – at low incomes many goods behave as necessities; as income rises, some become luxuries.
• Consumer preferences – strong preferences can raise the magnitude of YED.

3.4 Determinants of XED

• Degree of substitutability or complementarity – close substitutes or strong complements give larger |E_{XY}|.
• Proportion of expenditure on the related good – larger share → larger |E_{XY}|.
• Time horizon – more time to find alternatives → higher |E_{XY}|.

4. Interpreting Elasticity Values (AO2 / AO3)

5. Variation of PED Along a Straight‑Line (Linear) Demand Curve

5.1 The linear demand equation

A straight‑line demand curve can be written as:

$$Q = a - bP$$

where a and b are positive constants. The slope $\displaystyle\frac{dQ}{dP}= -b$ is constant, but PED varies because the term $\displaystyle\frac{P}{Q}$ changes at each point.

5.2 Algebraic location of the unit‑elastic point

Setting $|E_P|=1$ gives:

$$|E_P| = b\frac{P}{Q}=1 \;\;\Longrightarrow\;\; P = \frac{a}{2b},\qquad Q = \frac{a}{2}$$

Thus the **mid‑point** of the line (halfway between the price‑intercept and the quantity‑intercept) is the unit‑elastic point.

5.3 Elasticity in the three sections of a linear demand curve

• Upper‑right segment (high P, low Q): $P/Q$ is large → $|E_P|>1$ (elastic).
• Mid‑point ($P=a/2b$, $Q=a/2$): $|E_P|=1$ (unit‑elastic).
• Lower‑left segment (low P, high Q): $P/Q$ is small → $|E_P|<1$ (inelastic).

5.4 Limitation of the linear‑demand assumption

Real‑world demand curves are rarely perfectly straight; they may be curved, kinked or shift with income, tastes, etc. The linear model is a useful *approximation* for exam analysis, but students should recognise its simplification.

5.5 Numerical illustration (b = 1 for simplicity)

5.6 Sketch description (exam diagram)

• Draw a straight‑line demand curve sloping downwards from the price‑intercept (where $Q=0$) to the quantity‑intercept (where $P=0$).
• Mark the three points described above:
  • Upper‑right (elastic) – label $P_1$, $Q_1$.
  • Mid‑point (unit‑elastic) – label $P_u$, $Q_u$.
  • Lower‑left (inelastic) – label $P_2$, $Q_2$.
• Overlay a total‑revenue (TR) curve (TR = P·Q) on the same axes. The TR curve peaks exactly at the unit‑elastic point, illustrating the revenue‑elasticity relationship.
• Use arrows to show that moving left‑upwards along the demand curve makes PED become more elastic, while moving right‑downwards makes it more inelastic.

5.7 Implications for Total Revenue (TR)

Total revenue $TR = P \times Q$. The relationship with PED is:

• If $|E_P|>1$ (elastic) → a **price decrease** raises TR; a price increase lowers TR.
• If $|E_P|<1$ (inelastic) → a **price decrease** lowers TR; a price increase raises TR.
• If $|E_P|=1$ (unit‑elastic) → TR is unchanged by a price change.

These points should be linked to the TR curve drawn in the sketch.

6. Elasticity and Tax Incidence (AO2 / AO3)

6.1 Per‑unit excise tax example

Suppose the government imposes a £2 per‑unit tax on a good whose demand is relatively inelastic (|E_P| = 0.4) and supply is relatively elastic (|E_S| = 1.5).

• Because demand is inelastic, consumers bear a larger share of the tax burden – the price paid by consumers rises by more than £2, while the price received by producers falls.
• Numerical illustration (simplified):
  • Original equilibrium: $P=10$, $Q=100$.
  • After tax: consumer price $P_c = 11.6$, producer price $P_p = 9.6$ (tax wedge = £2).
• Dead‑weight loss (DWL) arises because the higher price reduces quantity demanded from 100 to, say, 90. The DWL is the area of the triangle between the old and new quantities under the demand and supply curves.

6.2 General rule for tax incidence

• If PED > PES (demand more elastic than supply) → producers bear a larger share of the tax.
• If PES > PED (supply more elastic) → consumers bear a larger share.
• When either curve is perfectly inelastic, the whole burden falls on the opposite side.

Link this analysis to welfare: the larger the elasticities, the larger the DWL (AO3 evaluation).

7. Summary Checklist (Cambridge 9708 – AS 2.2 & 2.3, A‑Level 7.1‑7.3)

1. State the four key elasticities, their formulas and the sign convention (PED < 0, PES > 0) – remember ceteris paribus.
2. Identify and explain the determinants that make each elasticity high or low; attach a brief “ceteris paribus” reminder to each.
3. Interpret the sign and magnitude of each elasticity, using the “elastic > 1 = responsive” shorthand and link to AO2/AO3 tasks (revenue, tax burden, welfare).
4. For a linear demand curve, locate the unit‑elastic point algebraically ($P=a/2b$, $Q=a/2$) and describe the three regions (elastic, unit‑elastic, inelastic). Mention the limitation of assuming a straight line.
5. Draw the required diagram: straight‑line demand, three elasticity points, and the total‑revenue curve showing the peak at the unit‑elastic point.
6. Apply elasticity to total‑revenue analysis, tax incidence (per‑unit tax example) and dead‑weight loss – evaluate the policy relevance.
7. Use the above knowledge to answer exam questions that require explanation (AO2) and evaluation (AO3) of pricing, tax, and welfare outcomes.