How to Solve Elasticity of Demand: Worked Examples
Price elasticity of demand sits near the center of every introductory microeconomics course, and the calculation itself is not long. What trips students up is not the arithmetic but two subtler problems: using the wrong formula variant and mishandling the sign. Tracing through the OpenStax Principles of Economicsworked examples while building Classeva's economics problem bank, I found those two errors account for the majority of wrong answers on well-understood material. This walkthrough covers the midpoint formula from first principles and applies it to two fully worked problems, then isolates the sign error so you can stop losing marks on a concept you actually understand.
What Is Price Elasticity of Demand?
Price elasticity of demand (PED) measures how responsive quantity demanded is to a change in price. When a coffee shop raises its prices by 10% and sales fall by 20%, the demand for that coffee is elastic: buyers pulled back more than proportionally. When a utility company raises electricity prices by 10% and consumption falls by only 3%, demand is inelastic: buyers had nowhere else to go.
The ratio captures that responsiveness as a single number. A PED of 2 means every 1% price rise causes a 2% quantity drop. A PED of 0.3 means every 1% price rise causes only a 0.3% quantity drop. The number tells you not just the direction but the magnitude, which is what makes it useful for pricing decisions and exam questions alike.
The Midpoint Formula
The standard formula for price elasticity of demand using the midpoint method, as presented in OpenStax Principles of Economics (3rd edition), reads:
The key difference from the basic percentage change formula is the denominator. Instead of dividing by the starting value (Q1 or P1), both percentage changes use the average of the two values: (Q1 + Q2) / 2 and (P1 + P2) / 2. That symmetry means you get the same elasticity number whether you calculate the move from A to B or from B to A.
Elastic, Inelastic, and Unitary Elastic
The absolute value of PED places demand in one of three broad categories, with two extreme cases at either end.
| Category | Condition | What it means | Example goods |
|---|---|---|---|
| Perfectly elastic | |PED| = infinity | Any price rise sends quantity to zero | Commodity markets with perfect substitutes |
| Elastic | |PED| > 1 | Quantity changes more than proportionally | Luxury goods, brand-specific items with close rivals |
| Unitary elastic | |PED| = 1 | Quantity changes exactly proportionally | Theoretical midpoint; total revenue unchanged |
| Inelastic | |PED| < 1 | Quantity changes less than proportionally | Petrol, electricity, prescription medicines |
| Perfectly inelastic | |PED| = 0 | Quantity does not change at all | Approximated by critical medications with no alternatives |
The five elasticity categories with conditions and typical goods. Most real markets fall between perfectly inelastic and unitary elastic.
Worked Example 1: Coffee Shop Price Rise
A coffee shop raises the price of a flat white from $3.00 to $3.60. At $3.00, daily sales average 500 cups. At $3.60, daily sales drop to 380 cups. Calculate the price elasticity of demand using the midpoint formula and classify the result.
Step-by-Step Calculation
Assign your variables
P1 = $3.00, Q1 = 500 (original). P2 = $3.60, Q2 = 380 (new). Write these out before touching the formula to avoid swapping values mid-calculation.
Compute the midpoint price
Midpoint P = (3.00 + 3.60) / 2 = 6.60 / 2 = $3.30.
Compute the midpoint quantity
Midpoint Q = (500 + 380) / 2 = 880 / 2 = 440 cups.
Calculate percentage change in quantity
% change Q = (380 - 500) / 440 x 100 = (-120 / 440) x 100 = -27.27%.
Calculate percentage change in price
% change P = (3.60 - 3.00) / 3.30 x 100 = (0.60 / 3.30) x 100 = +18.18%.
Divide and apply absolute value
PED = |(-27.27%) / (+18.18%)| = |-1.50| = 1.50.
Classify the result
|PED| = 1.50 > 1, so demand is elastic. A 1% price rise at this coffee shop produces a 1.5% fall in quantity demanded.
Interpreting the Result
An elasticity of 1.50 signals that coffee buyers at this shop respond strongly to price. This might reflect nearby competitor cafes selling similar products. The revenue implication runs against the owner's intuition: raising price by 20% ($3.00 to $3.60) causes sales to fall by 27%, so total revenue actually drops. Before the rise: $3.00 × 500 = $1,500 per day. After: $3.60 × 380 = $1,368 per day. The price hike costs $132 per day.
When |PED| exceeds 1, a price rise reduces total revenue because the volume loss outweighs the per-unit gain. A price cut increases total revenue. The coffee shop owner would recover revenue by cutting prices if the elasticity estimate is reliable. This revenue-elasticity connection appears on virtually every microeconomics exam.
Worked Example 2: Public Transport Fare Change
A city raises bus fares from $1.50 to $2.00 per trip. Weekly ridership falls from 200,000 to 190,000 trips. Calculate the price elasticity of demand using the midpoint formula and classify the result.
Step-by-Step Calculation
Assign variables
P1 = $1.50, Q1 = 200,000. P2 = $2.00, Q2 = 190,000. The units (trips) can be in thousands throughout since they cancel in the ratio.
Compute midpoint price
Midpoint P = (1.50 + 2.00) / 2 = $1.75.
Compute midpoint quantity
Midpoint Q = (200,000 + 190,000) / 2 = 195,000 trips.
Calculate percentage change in quantity
% change Q = (190,000 - 200,000) / 195,000 x 100 = (-10,000 / 195,000) x 100 = -5.13%.
Calculate percentage change in price
% change P = (2.00 - 1.50) / 1.75 x 100 = (0.50 / 1.75) x 100 = +28.57%.
Divide and apply absolute value
PED = |(-5.13%) / (+28.57%)| = |-0.18| = 0.18.
Classify
|PED| = 0.18 < 1, so demand is inelastic. Riders barely reduce trips even when the fare rises by nearly a third.
Reading the Inelastic Result
A PED of 0.18 places bus ridership firmly in inelastic territory. This reflects a real pattern in public transit economics: many riders depend on buses for commuting and have no convenient alternative. A 33% fare rise (from $1.50 to $2.00) reduces trips by only about 5%, so total revenue rises for the city. Before: $1.50 × 200,000 = $300,000 per week. After: $2.00 × 190,000 = $380,000 per week. Revenue climbs by $80,000 weekly.
When |PED| falls below 1, a price rise increases total revenue because the per-unit gain outweighs the modest volume loss. This explains why governments and regulated monopolies often raise fares and utility prices: the demand is sufficiently inelastic that revenue rises rather than falls. A price cut on an inelastic good reduces revenue.
The Sign-Convention Mistake Most Students Make
The raw price elasticity of demand is always negative. Price and quantity move in opposite directions along any normal demand curve: price rises, quantity falls. When you divide a negative percentage change in quantity by a positive percentage change in price, the raw result is a negative number. Students who report a negative PED have technically produced the correct raw calculation but then failed to apply the convention their course requires.
Most microeconomics courses, including the OpenStax Principles of Economics textbook used widely at the university level, report PED as its absolute value. A result of |PED| = 1.50 is elastic; a result of -1.50 left as-is confuses the classification because you might mistake it for a value between 0 and 1 by looking only at the digit before the decimal. Always take the absolute value before classifying.
Slope Is Not Elasticity
A steeper demand curve does not always mean more inelastic demand, and this confuses students who conflate the two. Slope measures the ratio of the price change to the quantity change in units: rise over run. Elasticity measures the ratio of percentage changes, which strips out the units entirely. Two demand curves can have identical slopes but sharply different elasticities if they sit at different positions on the price-quantity space.
On a straight-line (linear) demand curve, the slope stays constant, but the elasticity changes at every point. At the top of the curve (high price, low quantity), demand is elastic. At the bottom (low price, high quantity), demand is inelastic. Only at the exact midpoint does |PED| equal 1. The visual steepness of a curve gives only a rough directional signal. The calculation gives the answer.
Why the Midpoint Formula Beats the Basic Formula
The basic formula divides the change in quantity by the original quantity (Q1), and divides the change in price by the original price (P1). That produces a different elasticity depending on which end you start from. Moving from $3.00 to $3.60 gives one result; moving from $3.60 back to $3.00 gives a different result. That asymmetry is a problem because the underlying economic relationship is the same in both directions.
The midpoint formula resolves this by using the average of both values as the base. The result is identical regardless of which direction the price moves. This is why OpenStax recommends the midpoint method as the standard approach for introductory economics. Use it as your default unless your course explicitly specifies the basic formula.
First: forgetting to take the absolute value, leaving a negative PED that misclassifies as inelastic. Second: using Q1 or P1 as the denominator instead of the midpoint average, producing an asymmetric result. Third: rounding percentage changes too early, which can shift a result from above 1 to below 1. Keep fraction form throughout the calculation and round only at the final step.
What Determines Whether Demand Is Elastic?
Knowing that a PED is 1.50 or 0.18 is useful. Knowing why it takes that value helps you predict elasticity for unfamiliar goods on exam questions without doing the calculation. Four factors drive most of the variation.
| Factor | Makes demand more elastic | Makes demand more inelastic |
|---|---|---|
| Substitutes | Many close alternatives available | Few or no substitutes exist |
| Necessity vs luxury | Discretionary or luxury good | Necessity (food, medicine, utilities) |
| Share of budget | Large share of consumer spending | Small share (e.g., a box of matches) |
| Time horizon | Long run (buyers find alternatives) | Short run (consumers cannot adjust quickly) |
The four main determinants of price elasticity. Each acts on the buyer's ability or willingness to substitute away from the good.
The time horizon deserves emphasis because it catches students out. Petrol (gasoline) demand is inelastic in the short run: if prices spike this week, you still need to drive to work. Over two or three years, buyers buy more fuel-efficient cars, move closer to work, or switch to public transport. Long-run demand for petrol is considerably more elastic than short-run demand for the same good. The LibreTexts OpenStax Microeconomics edition uses fuel markets to illustrate exactly this short-run versus long-run elasticity shift. The time frame of the question changes the classification.
How Elasticity Connects to Total Revenue
The revenue-elasticity relationship follows directly from the definition, and it appears on exams frequently enough to deserve its own section. Total revenue equals price multiplied by quantity: TR = P × Q. When price rises, P goes up but Q goes down. Whether TR rises or falls depends entirely on which effect dominates, and that depends on elasticity.
| Demand type | Price rises | Price falls |
|---|---|---|
| Elastic (|PED| > 1) | TR falls (large Q drop) | TR rises (large Q gain) |
| Unitary elastic (|PED| = 1) | TR unchanged | TR unchanged |
| Inelastic (|PED| < 1) | TR rises (small Q drop) | TR falls (small Q loss) |
The revenue-elasticity rule. Unitary elastic demand is the only case where total revenue stays constant when price changes.
This relationship matters far beyond exam technique. Firms in markets with elastic demand cut prices to grow revenue. Governments tax goods with inelastic demand (tobacco, alcohol, petrol) precisely because the tax-driven price rise does not devastate quantity, so revenue climbs with minimal market disruption. The OpenStax economics textbook treats this revenue connection as the central applied implication of the elasticity concept. The Lumen Learning Microeconomics module extends the analysis to show how the same revenue rule applies to cross-price and income elasticity measures.
University Subject Calculators
Access economics calculators and tools to practice elasticity problems, supply and demand models, and other quantitative microeconomics exercises.
For a deeper treatment of how these concepts connect across your microeconomics module, the University resources hub collects study tools organized by subject. The GDP calculation walkthrough applies the same step-by-step method to macroeconomics data problems, and the hypothesis testing guide covers a comparable worked-example format for statistics questions. You can also explore the matrix multiplication guide for quantitative methods courses that overlap with economics programs.
For conceptual subjects like economics, pair these worked examples with active recall practice: work through each step without looking at the solution, then verify. When exams combine multiple topics, the quantitative exam revision guide shows how to build a problem bank and run timed practice that matches the pressure of a real paper. Browse all guides at the University blog for additional subject walkthroughs.
If you want to practice elasticity and related microeconomics problems with immediate worked feedback, the AI tutor below walks through questions at your level and flags the exact calculation step where errors appear.
Key Takeaways
- Price elasticity of demand measures the percentage change in quantity demanded divided by the percentage change in price. Use the midpoint formula: divide each change by the average of the two values, not the starting value.
- In the coffee shop example (price $3.00 to $3.60, quantity 500 to 380), PED = 1.50 (elastic). Revenue fell from $1,500 to $1,368 per day because the quantity drop outweighed the per-unit price gain.
- In the bus fare example (fare $1.50 to $2.00, trips 200,000 to 190,000), PED = 0.18 (inelastic). Revenue rose from $300,000 to $380,000 per week because the fare rise outweighed the small ridership loss.
- The most costly exam error is failing to take the absolute value of the raw (negative) PED. Always report |PED| before classifying as elastic or inelastic.
- Slope and elasticity are not the same thing. A linear demand curve has constant slope but changing elasticity along its length: elastic at the top, inelastic at the bottom, unitary at the midpoint.
- The four main determinants of elasticity are: availability of substitutes, whether the good is a necessity or luxury, the share of the budget the good consumes, and the time horizon available for adjustment.
- The revenue rule follows directly: a price rise on elastic demand reduces total revenue; a price rise on inelastic demand increases total revenue; unitary elastic demand keeps revenue constant.
