Welcome to Elasticities of Demand
Have you ever wondered why a cinema ticket discount suddenly fills every seat, but an increase in the price of tap water barely changes how much water people use? Or why sales of budget baked beans rise during economic downturns while luxury holidays boom when incomes grow?
The answer lies in elasticity. In economics, the word elasticity simply means responsiveness or sensitivity. It measures how much buyers change their behaviour when price, income, or the price of another good changes.
Don't worry if calculations and graphs in economics have felt intimidating in the past. We will break down each formula step-by-step with clear real-world examples and helpful memory tricks.
The Foundation: Calculating Percentage Change
Before looking at elasticity formulas, remember that all elasticity calculations rely on percentage change (\(\% \Delta\)), not absolute changes.
Percentage Change Formula:
\(\% \Delta = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\)
A Golden Rule to Remember: In every single elasticity equation, Quantity is on Top. You always put the percentage change in quantity demanded in the numerator (the top of the fraction), and the percentage change in the driving factor (price or income) in the denominator (the bottom).
1. Price Elasticity of Demand (PED)
Definition: Price Elasticity of Demand (PED) measures the responsiveness of the quantity demanded of a good to a change in its own price, ceteris paribus (all other factors remaining equal).
The Formula
\(\text{PED} = \frac{\% \Delta \text{ Quantity Demanded}}{\% \Delta \text{ Price}}\)
Understanding the Sign of PED
Due to the basic law of demand, price and quantity demanded move in opposite directions. If price rises, quantity demanded falls. Therefore, PED is mathematically negative (or zero). In economics, when we classify whether demand is elastic or inelastic, we look at the magnitude (the absolute value, written as \(|\text{PED}|\)).
Numerical Classifications of PED
1. Perfectly Inelastic Demand (\(\text{PED} = 0\)): A change in price causes no change at all in quantity demanded. The demand curve is vertical.
2. Relatively Price Inelastic (\(0 < |\text{PED}| < 1\)): The percentage change in quantity demanded is smaller than the percentage change in price. For example, if price rises by \(10\%\) and quantity demanded falls by only \(3\%\), \(|\text{PED}| = 0.3\). Consumers are relatively insensitive to price changes.
3. Unitary Elastic Demand (\(|\text{PED}| = 1\)): The percentage change in quantity demanded is exactly equal to the percentage change in price. If price rises by \(5\%\), quantity demanded falls by \(5\%\).
4. Relatively Price Elastic (\(|\text{PED}| > 1\)): The percentage change in quantity demanded is greater than the percentage change in price. For example, if price rises by \(10\%\) and quantity demanded falls by \(25\%\), \(|\text{PED}| = 2.5\). Consumers are very sensitive to price changes.
5. Perfectly Elastic Demand (\(|\text{PED}| = \infty\)): Any price increase causes quantity demanded to drop immediately to zero. The demand curve is horizontal.
Key Factors Influencing PED
Why are some goods elastic and others inelastic? Think about these key factors:
• Number and Closeness of Substitutes: Goods with many close substitutes (e.g., different brands of chocolate bars) have highly elastic demand because consumers can easily switch if the price rises. Goods with few or no substitutes (e.g., prescription medicines or water supply) are highly inelastic.
• Proportion of Income Spent: Inexpensive goods that take up a tiny fraction of consumer income (e.g., a box of matches or salt) tend to be price inelastic. Big-ticket purchases (e.g., cars or holidays) take a large share of income, making buyers far more price sensitive.
• Nature of the Good (Necessity vs. Luxury / Habit-Forming): Basic necessities (e.g., bread, electricity) and addictive/habit-forming goods (e.g., cigarettes, alcohol) have price inelastic demand. Luxury goods have price elastic demand.
• Time Period: In the short run, demand is typically more inelastic because consumers need time to adjust their habits or find alternatives. In the long run, demand becomes more elastic as substitutes are developed or discovered.
• Brand Loyalty and Consumer Lock-in: Strong brand loyalty makes demand more inelastic because consumers refuse to switch even if price increases.
PED and Total Revenue (TR)
Total Revenue is calculated as:
\(\text{Total Revenue (TR)} = \text{Price } (P) \times \text{Quantity } (Q)\)
Understanding PED helps businesses predict what will happen to their revenue when they adjust prices:
• When Demand is Price Elastic (\(|\text{PED}| > 1\)): Price and Total Revenue move in opposite directions.
- Price cut \(\rightarrow\) TR rises (the percentage gain in quantity outweighs the percentage cut in price).
- Price rise \(\rightarrow\) TR falls.
• When Demand is Price Inelastic (\(|\text{PED}| < 1\)): Price and Total Revenue move in the same direction.
- Price rise \(\rightarrow\) TR rises (the percentage gain from higher price outweighs the small percentage loss in quantity).
- Price cut \(\rightarrow\) TR falls.
• When Demand is Unitary Elastic (\(|\text{PED}| = 1\)): Total revenue is maximised. Small price changes leave total revenue unchanged.
Elasticity Along a Linear Demand Curve
Common Exam Trap: Do not confuse the slope of a demand curve with its elasticity! Along a downward-sloping linear demand curve:
• The top half is price elastic (\(|\text{PED}| > 1\)).
• The exact midpoint is unitary elastic (\(|\text{PED}| = 1\)), which is where Total Revenue reaches its peak.
• The bottom half is price inelastic (\(|\text{PED}| < 1\)).
Key Takeaway for PED: If demand is inelastic, raise the price to raise revenue. If demand is elastic, lower the price to raise revenue!
2. Income Elasticity of Demand (YED)
Definition: Income Elasticity of Demand (YED) measures the responsiveness of the quantity demanded of a good to a change in consumer income, ceteris paribus.
The Formula
\(\text{YED} = \frac{\% \Delta \text{ Quantity Demanded}}{\% \Delta \text{ Income}}\)
Interpreting the Sign and Magnitude of YED
Unlike PED, the plus or minus sign matters enormously in YED because it tells us the nature of the product:
• Inferior Goods (\(\text{YED} < 0\)): An increase in real income leads to a fall in quantity demanded. As consumers get richer, they abandon low-cost alternatives (e.g., supermarket value brands or budget bus travel) for higher-quality options.
• Normal Goods (\(\text{YED} > 0\)): An increase in real income leads to a rise in quantity demanded. Normal goods are subdivided into:
- Normal Necessities / Income Inelastic (\(0 < \text{YED} < 1\)): Demand rises at a slower rate than the increase in income (e.g., basic groceries, toothpaste).
- Luxury Goods / Income Elastic (\(\text{YED} > 1\)): Demand rises at a faster rate than the proportionate increase in income (e.g., fine dining, high-end electronics, overseas vacations).
Key Takeaway for YED: A positive sign (\(+\)) means a normal good; a negative sign (\(-\)) means an inferior good. Values above \(1\) represent luxuries.
3. Cross Elasticity of Demand (XED)
Definition: Cross Elasticity of Demand (XED) measures the responsiveness of the quantity demanded of Good A to a change in the price of Good B, ceteris paribus.
The Formula
\(\text{XED} = \frac{\% \Delta \text{ Quantity Demanded of Good A}}{\% \Delta \text{ Price of Good B}}\)
Interpreting the Sign and Magnitude of XED
The mathematical sign reveals the economic relationship between the two goods:
• Substitutes (\(\text{XED} > 0\)): A price increase for Good B causes consumers to switch, increasing demand for Good A.
- Close substitutes have a high positive XED (e.g., two rival smartphone brands).
- Weak substitutes have a low positive XED (e.g., tea and coffee for some consumers).
• Complements (\(\text{XED} < 0\)): Goods in joint demand. A price increase for Good B causes a decrease in demand for Good A.
- Close/strong complements have a high negative magnitude (e.g., games consoles and console video games).
- Weak complements have a low negative magnitude (e.g., cinema tickets and popcorn).
• Unrelated Goods (\(\text{XED} = 0\)): A price change in Good B has zero impact on demand for Good A (e.g., the price of trainers and demand for pineapples).
Memory Trick for Signs:
• Substitutes = Same direction (\(+\))
• Complements = Contrary direction (\(-\))
4. Significance to Businesses and Governments
Why Elasticity Matters to Firms
• Pricing Strategies: Firms use PED to choose pricing tactics. Knowing demand is price elastic encourages discounting, while price inelastic demand allows price increases without losing substantial sales.
• Price Discrimination: Firms can charge different prices to different customer groups based on differing PEDs (e.g., peak rail commuters have inelastic demand and pay higher fares than off-peak leisure travellers with elastic demand).
• Branding and Differentiation: Businesses invest heavily in advertising and quality improvements to build brand loyalty, reducing the closeness of substitutes and making their PED more inelastic.
• Stock and Production Planning (YED): Firms use YED to plan ahead for economic cycles. During economic booms (rising incomes), luxury producers expand capacity. During recessions (falling real incomes), discount retailers and inferior good producers prepare for higher demand.
• Competitive Reactions (XED): Knowing XED helps a firm anticipate how rival pricing decisions will impact their own sales and design defensive marketing campaigns.
Why Elasticity Matters to Governments
• Indirect Taxation: Governments place excise duties (indirect taxes) on goods with highly inelastic demand (e.g., tobacco, alcohol, and fuel). Because demand is inelastic, consumption falls very little, providing stable and massive tax revenue for the state.
• Tax Incidence (Tax Burden): When demand is relatively price inelastic compared to supply, consumers bear most of the tax burden in the form of higher prices. When demand is elastic, producers must absorb more of the tax themselves.
• Subsidies: If the government provides a subsidy to encourage consumption of a beneficial good, the expansion in quantity demanded will be substantially larger if the good is price elastic.
Quick Review: Common Pitfalls to Avoid in the Exam
• Pitfall 1: Confusing PED magnitude and sign. If a calculation gives \(\text{PED} = -2.5\), do not say "it is less than 1, so it is inelastic". The magnitude is \(|-2.5| = 2.5\), which is greater than 1, meaning demand is price elastic.
• Pitfall 2: Confusing YED and XED signs. For YED, a minus sign means an inferior good. For XED, a minus sign means complements.
• Pitfall 3: Flipping the fraction. Always put \(\% \Delta \text{ Quantity}\) on top and \(\% \Delta \text{ Price or Income}\) on the bottom.
• Pitfall 4: Using absolute changes instead of percentages. Never divide the raw change in units by the raw change in pounds (\(\frac{\Delta Q}{\Delta P}\)). You must calculate percentage changes first!