definition of price elasticity, income elasticity and cross elasticity of demand (PED, YED, XED)

Price Elasticity, Income Elasticity, Cross‑Elasticity & Price Elasticity of Supply

Learning Objective

By the end of this unit you should be able to:

• State the definition, formula and sign‑convention for each of the four main elasticities required for Cambridge IGCSE/A‑Level (2.1‑2.4).
• Identify and explain the key determinants of each elasticity, including short‑run vs. long‑run effects.
• Calculate elasticities from given data and interpret the result in economic terms.
• Use appropriate diagrams to show movements, shifts and the welfare consequences of a change in price, income or related‑good price.
• Link each elasticity to real‑world business decisions (pricing, product‑mix, mergers) and government policy (taxes, subsidies, regulation).
• Evaluate the usefulness and limitations of elasticity analysis (AO3).

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2.1 Demand, Supply & Equilibrium – a fuller refresher

2.1.1 Demand

• Definition: The relationship between the price of a good (P) and the quantity demanded (Q_d) ceteris paribus (all other determinants held constant).
• Determinants (shifters):
  • Consumer income (normal vs. inferior goods)
  • Tastes & preferences
  • Prices of related goods (substitutes & complements)
  • Expectations of future prices or income
  • Number of buyers (market size)
  • Demographic factors (age, population)
• Movement vs. shift: A change in the good’s own price causes a movement along the demand curve; any other determinant causes the whole curve to shift left (decrease) or right (increase).

2.1.2 Supply

• Definition: The relationship between the price of a good (P) and the quantity supplied (Q_s) ceteris paribus.
• Determinants (shifters):
  • Input prices (wages, raw materials)
  • Technology & productivity
  • Expectations of future prices
  • Number of sellers
  • Government policies (taxes, subsidies, regulation)
  • Natural conditions (weather, disasters)
• Movement vs. shift: A change in the good’s own price moves us along the supply curve; any other determinant shifts the whole curve.

2.1.3 Market equilibrium

• Equilibrium point: The intersection of demand and supply where Q_d=Q_s and the market price (P*) is stable.
• Effect of a shift: Use the “shift‑shift” diagram to show how a right‑ward demand shift raises both equilibrium price and quantity, whereas a left‑ward supply shift raises price but lowers quantity, etc.
• Welfare concepts: Consumer surplus (area above price and below demand) and producer surplus (area below price and above supply). Elasticities determine how large these surpluses are and how they change when price moves.

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2.2 Elasticities of Demand

2.2.1 Price Elasticity of Demand (PED)

Definition: The percentage change in quantity demanded of a good resulting from a 1 % change in its own price, holding all other determinants constant.

Formula (percentage‑change method)

\[ \varepsilon_{p}= \frac{\%\Delta Q_{d}}{\%\Delta P} = \frac{\Delta Q_{d}/Q_{d}}{\Delta P/P} \]

Sign convention: Because the law of demand gives a negative relationship, PED is normally negative. In practice we use the absolute value \(|\varepsilon_{p}|\) for classification.

Classification

• \(|\varepsilon_{p}|>1\) – elastic (quantity responds strongly).
• \(|\varepsilon_{p}|=1\) – unit‑elastic (proportional response).
• \(|\varepsilon_{p}|<1\) – inelastic (quantity responds weakly).

Key determinants (including time‑dimension)

• Availability of close substitutes – more substitutes → higher elasticity.
• Proportion of income spent on the good – larger share → higher elasticity.
• Nature of the good – luxuries > necessities.
• Definition of the market – narrow market (e.g., “Coca‑Cola”) → higher elasticity than a broad market (“soft drinks”).
• Time horizon – demand is more elastic in the long run because consumers can adjust habits, find alternatives or change consumption patterns.

Worked numerical example

Before	After
P = £10, Qd = 120 units	P = £8, Qd = 150 units

Calculate PED using the midpoint (arc) formula:

\[ \varepsilon_{p}= \frac{(150-120)/( (150+120)/2 )}{(8-10)/( (8+10)/2 )} = \frac{30/135}{-2/9} = \frac{0.222}{-0.222} = -1.00 \]

Interpretation: demand is unit‑elastic over this price range; a 20 % fall in price leaves total revenue unchanged.

Total‑Revenue (TR) test

• If \(|\varepsilon_{p}|>1\) (elastic) a price cut increases TR (TR = P × Q).
• If \(|\varepsilon_{p}|<1\) (inelastic) a price rise increases TR.
• If \(|\varepsilon_{p}|=1\) TR is unchanged.

Diagram suggestion: Plot a linear demand curve, mark two price‑quantity points (P₁,Q₁) and (P₂,Q₂), draw the two TR rectangles, and annotate the direction of change in TR.

Elasticity along a linear demand curve

• Upper (high‑price, low‑quantity) segment: relatively elastic.
• Lower (low‑price, high‑quantity) segment: relatively inelastic.
• Reason: the same absolute price change is a larger percentage change at the top of the curve.

Real‑world examples

• Petrol – short‑run inelastic (few substitutes), long‑run more elastic as consumers switch to public transport or electric cars.
• Smartphones – relatively elastic; a 5 % price cut can generate a >5 % rise in sales.
• Luxury handbags – highly elastic; a modest price rise leads to a sharp fall in quantity demanded.

2.2.2 Income Elasticity of Demand (YED)

Definition: The percentage change in quantity demanded of a good resulting from a 1 % change in consumer income, holding other determinants constant.

Formula

\[ \varepsilon_{y}= \frac{\%\Delta Q_{d}}{\%\Delta Y} = \frac{\Delta Q_{d}/Q_{d}}{\Delta Y/Y} \]

Interpretation

• \(\varepsilon_{y}>0\) – normal good (demand rises with income).
• \(\varepsilon_{y}<0\) – inferior good (demand falls as income rises).
• For normal goods:
  • \(0<\varepsilon_{y}<1\) – necessity (quantity rises, but less than proportionally).
  • \(\varepsilon_{y}>1\) – luxury (quantity rises more than proportionally).

Key determinants

• Nature of the good (necessity vs. luxury).
• Availability of higher‑quality substitutes (e.g., cheap rice vs. premium rice).
• Cultural and demographic preferences.
• Time horizon – in the long run consumers may re‑allocate spending, altering YED.

Worked numerical example

Income level	Quantity demanded
£20 000	200 units
£22 000 (10 % rise)	240 units (20 % rise)

\[ \varepsilon_{y}= \frac{20\%}{10\%}=2.0 \] Interpretation: the good is a luxury (YED > 1); demand grows faster than income.

Use in forecasting

• Predict market size under different economic scenarios (recession vs. boom).
• Guide product‑mix decisions – expand luxury lines when YED is high, protect inferior‑good lines during downturns.

2.2.3 Cross‑Elasticity of Demand (XED)

Definition: The percentage change in quantity demanded of good X resulting from a 1 % change in the price of a different good Y (X ≠ Y), ceteris paribus.

Formula

\[ \varepsilon_{xy}= \frac{\%\Delta Q_{d}^{x}}{\%\Delta P^{y}} = \frac{\Delta Q_{d}^{x}/Q_{d}^{x}}{\Delta P^{y}/P^{y}} \]

Interpretation

• \(\varepsilon_{xy}>0\) – goods are substitutes (price rise in Y raises demand for X).
• \(\varepsilon_{xy}<0\) – goods are complements (price rise in Y lowers demand for X).
• Magnitude indicates strength: \(|\varepsilon_{xy}|=2\) = strong relationship; \(|\varepsilon_{xy}|=0.2\) = weak.

Key determinants

• Degree of similarity (perfect substitutes vs. weak substitutes).
• Share of total expenditure on the two goods.
• Availability of other alternatives.
• Time horizon – relationships can strengthen as consumers adjust habits.

Worked numerical example (substitutes)

Good X (butter)	Good Y (margarine)
Price of butter unchanged (£2.00)	Price of margarine rises from £1.00 to £1.20 (20 % rise)
Quantity of butter rises from 500 kg to 560 kg (12 % rise)

\[ \varepsilon_{xy}= \frac{12\%}{20\%}=0.6>0 \] Interpretation: butter and margarine are substitutes, but the relationship is moderately weak.

Diagram suggestion

• Draw two demand curves for good X. Show a right‑ward shift when the price of a substitute Y rises, and a left‑ward shift when the price of a complement Y rises.
• Label the shift direction and annotate the sign of XED.

Business & policy applications

• Pricing strategy – a firm with a low positive XED (weak substitutes) can raise price without losing many customers.
• Merger analysis – high positive XED between two firms’ products signals strong competition concerns.
• Tax incidence – a tax on a complement (e.g., cigarettes & lighters) reduces demand for both goods.

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2.3 Price Elasticity of Supply (PES)

Definition: The percentage change in quantity supplied of a good resulting from a 1 % change in its own price, holding all other factors constant.

Formula

\[ \varepsilon_{s}= \frac{\%\Delta Q_{s}}{\%\Delta P} = \frac{\Delta Q_{s}/Q_{s}}{\Delta P/P} \]

Sign convention: PES is normally positive because a higher price encourages producers to supply more.

Determinants

• Availability of inputs – abundant inputs → higher PES.
• Time horizon – in the short run many factors (plant size, labour) are fixed, giving a low PES; in the long run firms can adjust capacity, making PES higher.
• Mobility of factors of production – easy entry/exit → high PES.
• Storage possibilities – goods that can be stored (e.g., wheat) have higher PES because producers can respond to price changes by releasing stocks.
• Complexity of production – highly specialised or capital‑intensive industries have low PES in the short run.

Worked numerical example

Before	After
P = £5, Qs = 200 units	P = £6 (20 % rise), Qs = 240 units (20 % rise)

\[ \varepsilon_{s}= \frac{20\%}{20\%}=1.0 \] Interpretation: supply is unit‑elastic at this point; a 20 % price rise leads to a proportionate 20 % increase in quantity supplied.

Diagram suggestion

• Draw an upward‑sloping supply curve, mark two points (P₁,Q₁) and (P₂,Q₂). Show the percentage changes and label the calculated PES.
• Indicate the short‑run (steeper) and long‑run (flatter) supply curves to illustrate how elasticity changes over time.

Policy relevance

• Tax incidence – when PES is inelastic relative to PED, producers bear most of a tax burden.
• Subsidy design – a highly elastic supply means a subsidy quickly expands output, useful for encouraging green technology.
• Market‑entry decisions – industries with high long‑run PES attract new entrants after a price rise.

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2.4 Interaction of Elasticities with Market Equilibrium & Welfare

2.4.1 Effect on Consumer and Producer Surplus

• When demand is elastic, a price fall creates a large increase in quantity, expanding consumer surplus considerably while reducing producer surplus.
• When demand is inelastic, a price rise yields a small quantity change; producer surplus rises sharply and consumer surplus falls only a little.
• Similarly, a highly elastic supply curve means a price change causes a large quantity response, altering producer surplus more than consumer surplus.

2.4.2 Tax and Subsidy Incidence

• Tax on a good with \(|\varepsilon_{p}| > |\varepsilon_{s}|\) (demand more elastic than supply) → consumers bear a larger share of the tax burden.
• Subsidy on a good with highly elastic supply → output expands greatly, reducing the per‑unit cost to producers and increasing total welfare.

2.4.3 Policy & Business Decision‑Making

• Governments use PED to decide whether a tax on a harmful good (e.g., cigarettes) will raise revenue without causing a huge drop in consumption.
• Firms use YED to forecast demand growth in emerging markets where incomes are rising rapidly.
• Cross‑elasticity helps firms anticipate the impact of a competitor’s price change on their own sales.

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Summary Table of Elasticities

Elasticity	Symbol	Formula	Typical sign	Economic meaning
Price Elasticity of Demand	\(\varepsilon_{p}\)	\(\displaystyle \varepsilon_{p}= \frac{\%\Delta Q_{d}}{\%\Delta P}\)	Negative (use absolute value for classification)	Elastic, unit‑elastic, inelastic demand
Income Elasticity of Demand	\(\varepsilon_{y}\)	\(\displaystyle \varepsilon_{y}= \frac{\%\Delta Q_{d}}{\%\Delta Y}\)	Positive for normal goods, negative for inferior goods	Necessity, luxury, inferior good
Cross‑Elasticity of Demand	\(\varepsilon_{xy}\)	\(\displaystyle \varepsilon_{xy}= \frac{\%\Delta Q_{d}^{x}}{\%\Delta P^{y}}\)	Positive for substitutes, negative for complements	Strength of inter‑good relationship
Price Elasticity of Supply	\(\varepsilon_{s}\)	\(\displaystyle \varepsilon_{s}= \frac{\%\Delta Q_{s}}{\%\Delta P}\)	Positive (usually < 1 in short run, > 1 in long run)	Responsiveness of producers to price changes

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Key Points for Exam Answers (AO1‑AO3)

1. State the definition and write the correct formula. Use the appropriate symbols and show the percentage‑change notation.
2. Explain the sign and what it tells you about the good. Relate to normal/inferior, substitute/complement, elastic/inelastic.
3. Identify at least two determinants. Discuss how each determinant would move the elasticity up or down, and note any short‑run/long‑run differences.
4. Provide a worked numerical calculation. Show step‑by‑step use of the midpoint (arc) formula and interpret the result.
5. Draw a relevant diagram. Label axes, the original curve, the movement/shift, percentage changes and any welfare areas (consumer/producer surplus, TR rectangles).
6. Link to policy or business implications. Discuss tax incidence, pricing strategy, product‑mix decisions, merger concerns, or forecasting.
7. Evaluation (AO3). Consider:
   • Data reliability – elasticities are often based on short‑run observations.
   • The ceteris paribus assumption – real markets rarely hold all other factors constant.
   • Short‑run vs. long‑run differences – elasticities can change dramatically over time.
   • Behavioural factors – habits, brand loyalty or expectations may weaken the predictive power of simple elasticity analysis.

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Suggested Diagram Checklist (for exam practice)

• PED – two points on a linear demand curve; annotate ΔP, ΔQ, calculate elasticity, add TR rectangles to illustrate the total‑revenue test.
• YED – bar chart or two demand curves showing quantity at two income levels; label the percentage changes.
• XED – two demand curves for good X (original and shifted) showing the effect of a price change in good Y (substitutes vs. complements).
• Elasticity along a straight line – colour‑code the upper (elastic) and lower (inelastic) segments of a linear demand curve.
• PES – upward‑sloping supply curve with two points; show short‑run (steeper) vs. long‑run (flatter) supply curves.
• Welfare diagram – equilibrium with consumer and producer surplus; illustrate how a price change moves the surplus when demand or supply is elastic/inelastic.