Elasticity

Elasticity

Price Elasticity of Demand (PED)

PED measures the responsiveness of quantity demanded to a change in price:

$\mathrm{PED} = \frac{\% \Delta Q_d}{\% \Delta P}$

Since the demand curve slopes downward, PED is negative, but it is conventionally expressed as an Absolute value.

For precise calculations, the midpoint (arc) formula avoids the ambiguity of which price and Quantity to use as the base:

$\mathrm{PED} = \frac{\frac{Q_2 - Q_1}{(Q_1 + Q_2)/2}}{\frac{P_2 - P_1}{(P_1 + P_2)/2}}$

Determinants of PED:

• Availability of substitutes: more substitutes = more elastic. A good with many close substitutes (e.g., Coca-Cola vs. Pepsi) has highly elastic demand
• Proportion of income: goods that take a larger share of income (e.g., cars, holidays) tend to be more elastic than inexpensive goods (e.g., salt, matches)
• Necessity vs. Luxury: necessities (food, medicine) tend to be inelastic; luxuries (designer clothing) tend to be elastic
• Time horizon: demand becomes more elastic over time as consumers adjust behaviour and find alternatives. Short-run PED for petrol is low; long-run PED is higher as consumers switch to electric vehicles or public transport
• Addictiveness: addictive goods (cigarettes, alcohol) tend to be inelastic in the short run, though demand may become more elastic in the long run as addiction is broken

Relationship with total revenue:

$\mathrm{TR} = P \times Q$

• If demand is elastic ($\|\mathrm{PED}\| > 1$), a price decrease increases total revenue
• If demand is inelastic ($\|\mathrm{PED}\| < 1$), a price decrease decreases total revenue
• If demand is unit elastic ($\|\mathrm{PED}\| = 1$), total revenue is unchanged (TR is maximised where PED $= -1$)

Income Elasticity of Demand (YED)

$\mathrm{YED} = \frac{\% \Delta Q_d}{\% \Delta Y}$

YED is important for understanding how demand changes as an economy grows. In a booming economy, Demand for luxuries rises disproportionately, while demand for inferior goods falls.

Price Elasticity of Supply (PES)

$\mathrm{PES} = \frac{\% \Delta Q_s}{\% \Delta P}$

Determinants of PES:

• Spare production capacity: firms with unused capacity can respond more quickly to price increases
• Mobility of factors of production: labour and capital that can be reallocated increase PES
• Ability to store stocks: goods that are non-perishable and inexpensive to store tend to have higher PES
• Time period: supply is more elastic in the long run than in the short run. In the momentary run, supply is perfectly inelastic (output cannot change). In the short run, supply is somewhat elastic. In the long run, all factors are variable, and supply is highly elastic

Cross-Price Elasticity of Demand (XED)

$\mathrm{XED} = \frac{\% \Delta Q_{d,A}}{\% \Delta P_B}$

• XED $> 0$: goods are substitutes (the higher the positive value, the closer the substitute)
• XED $< 0$: goods are complements (the more negative, the stronger the complement relationship)
• XED $= 0$: goods are unrelated

XED is used by firms to assess competitive threats. A high positive XED between two products Indicates they are close competitors.

Common Pitfalls

• Confusing PED with slope. Elasticity changes along a linear demand curve even though the slope is constant
• Forgetting that PED is expressed as an absolute value. A PED of $-0.5$ means demand is inelastic ($|-0.5| < 1$)
• Stating “demand is elastic” when you mean “demand is price elastic.” The unqualified word “elastic” is ambiguous
• Confusing income elasticity sign with elasticity magnitude. Normal goods have $YED > 0$; inferior goods have $YED < 0$. The magnitude still matters for determining necessity ($0 < YED < 1$) vs luxury ($YED > 1$)
• Using the wrong formula for cross-price elasticity. XED measures the responsiveness of quantity demanded of good X to a change in the price of good Y — not the other way around
• Calculating percentage change incorrectly when there is no base value given. Always use the midpoint (arc elasticity) formula when the base is ambiguous: $\frac{\Delta Q}{(Q_1 + Q_2)/2} \div \frac{\Delta P}{(P_1 + P_2)/2}$
• Confusing total revenue with total expenditure. They are the same number ($P \times Q$) but represent different perspectives (seller vs buyer)

Worked Examples

Example 1: Calculating PED

A firm raises the price of its product from $10 to$12. Quantity demanded falls from 500 to 400 units.

$\text{PED} = \frac{\%\Delta Q_d}{\%\Delta P} = \frac{\frac{400-500}{500} \times 100}{\frac{12-10}{10} \times 100} = \frac{-20\%}{+20\%} = -1.0$

Demand is unit elastic ($|\text{PED}| = 1$). Total revenue remains unchanged: 10 \times 500 = \5,000$and$12 \times 400 = $4,800$.

Wait — actually \5,000 \neq $4,800$, which means the PED calculated using the initial-point method gives$-1.0$ but the revenue changed. Using the midpoint method:

$\text{PED} = \frac{\frac{400-500}{(400+500)/2}}{\frac{12-10}{(12+10)/2}} = \frac{-100/450}{2/11} = \frac{-0.222}{0.182} \approx -1.22$

Demand is price elastic, consistent with the fall in total revenue.

Example 2: Income Elasticity

When income rises from $30,000 to$35,000, a household”s purchases of fresh vegetables increase from 200 to 230 kg/year, and purchases of instant noodles fall from 150 to 120 packets/year.

$\text{YED}_{\text{vegetables}} = \frac{\frac{30}{200}}{\frac{5000}{30000}} = \frac{0.15}{0.167} \approx 0.90$

Fresh vegetables are a normal good ($YED > 0$) and a necessity ($0 < YED < 1$).

$\text{YED}_{\text{noodles}} = \frac{\frac{-30}{150}}{\frac{5000}{30000}} = \frac{-0.20}{0.167} \approx -1.20$

Instant noodles are an inferior good ($YED < 0$).

Summary

• Price elasticity of demand (PED) measures how responsive quantity demanded is to a price change; $|PED| < 1$ = inelastic, $|PED| = 1$ = unit elastic, $|PED| > 1$ = elastic
• Price elasticity of supply (PES) depends on time horizon, spare capacity, and factor mobility
• Income elasticity (YED) distinguishes normal goods ($YED > 0$) from inferior goods ($YED < 0$); luxuries have $YED > 1$
• Cross-price elasticity (XED) identifies substitutes ($XED > 0$) and complements ($XED < 0$)
• PED and total revenue are linked: when demand is inelastic, a price rise increases total revenue; when elastic, a price rise decreases total revenue
• Key formula: $\text{PED} = \frac{\%\Delta Q_d}{\%\Delta P}$