Elasticity Formulas

Why This Matters

Elasticity is the concept that separates students who memorize supply and demand curves from those who actually understand how markets work. You're being tested on your ability to measure responsiveness—how much quantity demanded or supplied changes when prices, income, or related goods' prices shift. This matters because elasticity determines everything from tax incidence (who really pays a tax) to whether a firm should raise or lower prices to increase revenue.

The formulas themselves are straightforward, but the AP exam tests whether you can apply them correctly and interpret the results. You'll need to calculate elasticity from tables and graphs, determine whether demand is elastic or inelastic, predict revenue changes, and classify goods as normal, inferior, substitutes, or complements. Don't just memorize formulas—know what each coefficient tells you about consumer and producer behavior.

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Measuring Demand Responsiveness

These formulas quantify how sensitive consumers are to changes in price, income, or the prices of related goods. The numerator always captures the quantity response; the denominator captures the cause of that response.

Price Elasticity of Demand (PED)

• Formula: $PED = \frac{\%\ \text{Change in Quantity Demanded}}{\%\ \text{Change in Price}}$—always negative due to the law of demand, but we typically use absolute value for interpretation
• Interpretation thresholds: $|PED| > 1$ means elastic demand, $|PED| < 1$ means inelastic demand, $|PED| = 1$ means unit elastic
• Exam application: PED determines tax incidence—the more inelastic side of the market bears more of the tax burden

Income Elasticity of Demand (YED)

• Formula: $YED = \frac{\%\ \text{Change in Quantity Demanded}}{\%\ \text{Change in Income}}$—the sign matters here, unlike PED
• Classification tool: Positive YED indicates a normal good; negative YED indicates an inferior good
• Luxury vs. necessity: Normal goods with $YED > 1$ are luxuries; those with $0 < YED < 1$ are necessities

Cross-Price Elasticity of Demand (XED)

• Formula: $XED = \frac{\%\ \text{Change in Quantity Demanded of Good A}}{\%\ \text{Change in Price of Good B}}$—measures relationships between goods
• Sign interpretation: Positive XED means the goods are substitutes; negative XED means they are complements
• Market analysis: Higher absolute values indicate stronger relationships between goods—useful for predicting competitive effects

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Measuring Supply Responsiveness

Supply elasticity captures how producers respond to price changes. The key driver is production flexibility—can firms easily adjust output?

Price Elasticity of Supply (PES)

• Formula: $PES = \frac{\%\ \text{Change in Quantity Supplied}}{\%\ \text{Change in Price}}$—typically positive due to the law of supply
• Interpretation thresholds: $PES > 1$ means elastic supply, $PES < 1$ means inelastic supply, $PES = 1$ means unit elastic
• Policy relevance: When supply is inelastic, taxes create larger deadweight losses and producers bear more of the burden

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Calculation Methods

The AP exam tests two distinct approaches to calculating elasticity. Use the midpoint method for changes between two points; use point elasticity for a specific location on the curve.

Midpoint (Arc) Elasticity Formula

• Formula: $E = \frac{(Q_2 - Q_1) / [(Q_2 + Q_1)/2]}{(P_2 - P_1) / [(P_2 + P_1)/2]}$—uses averages to avoid different answers depending on direction
• When to use: Apply this method when given two price-quantity combinations from a table or graph
• Why it matters: The midpoint method gives the same elasticity whether price rises or falls—standard approach for AP calculations

Point Elasticity Formula

• Formula: $E = \frac{dQ}{dP} \times \frac{P}{Q}$—where $\frac{dQ}{dP}$ is the slope of the demand or supply curve (or its reciprocal)
• Linear demand insight: Elasticity varies along a straight-line demand curve—elastic at high prices, inelastic at low prices, unit elastic at the midpoint
• Slope vs. elasticity: A common exam trap—slope is constant along a linear curve, but elasticity changes at every point

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Applying Elasticity: The Total Revenue Test

This shortcut lets you determine elasticity without calculating—just observe what happens to revenue. Total revenue (TR) = Price × Quantity, so elasticity determines which effect dominates.

Total Revenue Test

• The rule: If price and total revenue move in opposite directions, demand is elastic; if they move in the same direction, demand is inelastic
• Elastic demand ($|PED| > 1$): Price increase → TR decreases; price decrease → TR increases (quantity effect dominates)
• Inelastic demand ($|PED| < 1$): Price increase → TR increases; price decrease → TR decreases (price effect dominates)

Unit Elastic Benchmark

• Definition: $|PED| = 1$ means the percentage change in quantity exactly equals the percentage change in price
• Revenue implication: Total revenue is maximized at the unit elastic point—any price change leaves TR unchanged
• Graphical location: On a linear demand curve, unit elasticity occurs at the midpoint where TR peaks

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Determinants of Elasticity

Understanding why elasticity varies helps you predict values even without data. These factors explain the economic intuition behind elasticity coefficients.

Key Factors Affecting PED

• Availability of substitutes: More substitutes → more elastic demand (consumers can easily switch)
• Necessity vs. luxury: Necessities tend to be inelastic; luxuries are more elastic
• Time horizon: Demand becomes more elastic over time as consumers find alternatives and adjust behavior

Additional Determinants

• Share of income: Goods that consume a larger portion of the budget tend to have more elastic demand
• Market definition: Narrowly defined markets (Coca-Cola) are more elastic than broadly defined ones (soft drinks)
• Habit and addiction: Products with habitual consumption patterns tend toward inelastic demand

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Quick Reference Table

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Self-Check Questions

1. If a 10% price increase causes a 15% decrease in quantity demanded, is demand elastic or inelastic, and what happens to total revenue?

2. A good has an income elasticity of $-0.5$. Is this a normal or inferior good, and is it a necessity or luxury?

3. Compare the cross-price elasticity between Coke and Pepsi versus Coke and hot dogs. Which is positive, which is negative, and why?

4. Using the midpoint method, calculate PED if price rises from $8 to $12 and quantity falls from 100 to 60 units. Show your work.

5. A government imposes a per-unit tax on a market where demand is highly inelastic and supply is highly elastic. Which side bears more of the tax burden, and why does elasticity determine this outcome?