Price elasticity of demand: practice problems with step-by-step solutions

Also called: PED · elasticity of demand

Free Price elasticity of demand practice problems with diagrams, hints and full worked solutions. New values every time, at three levels: percentage changes, midpoint method, linear demand.

Free practice problemEasy
ticket price−12%tickets per week?400 → QE = −0.8

A cinema sells 400 tickets per week. The price elasticity of demand is −0.8. If the ticket price is cut by 12%, how many tickets Q will it sell per week?

Q1 = 400E = −0.8%ΔP = −12%
tickets
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What you practise

  1. Easypercentage changes
  2. Mediummidpoint method
  3. Hardlinear demand

Key formulas

Price elasticity (midpoint)
E = (ΔQ/Q̄)/(ΔP/P̄)
Elastic / inelastic
|E| > 1 elastic, |E| < 1 inelastic
Revenue
TR = P·Q

Learn Price elasticity of demand from scratch

1How strongly buyers react to price

When a price goes up, people buy less. That is the law of demand. But how much less depends on the product.

If the price of salt doubles, you still buy about the same amount. If one brand of cola gets 20% dearer, many buyers switch to another brand.

The price elasticity of demand E puts a number on this reaction:

E = (% change in quantity) / (% change in price)

Example: the price rises by 10% and the quantity sold falls by 15%. Then E = −15% / 10% = −1.5.

E is negative, because price and quantity move in opposite directions. Keep the minus sign in your answer.

Self-check The price rises by 10% and the quantity demanded falls by 20%. What is the price elasticity of demand?
  1. +2
  2. −0.5
  3. −20
  4. −2

Answer: −2. E = −20% / 10% = −2. Dividing the other way round gives −0.5, and dropping the sign gives +2. The minus sign shows that quantity falls when price rises.

2Elastic, inelastic and revenue

Look at the size of E without its sign, written |E|:

  • Elastic, |E| > 1: quantity changes by a larger percentage than price. Buyers react strongly.
  • Inelastic, |E| < 1: quantity changes by a smaller percentage than price. Buyers hardly react.
  • Unit elastic, |E| = 1: both change by the same percentage.

Elasticity tells a firm what happens to its total revenue, TR = P · Q, when it changes the price.

  • Inelastic demand: a price rise increases revenue, because sales fall only a little.
  • Elastic demand: a price rise decreases revenue, because sales fall a lot.

Products with few substitutes and necessities, like petrol or medicine, tend to be inelastic. Products with many close substitutes tend to be elastic.

Self-check Demand for a product has E = −0.4. The firm raises its price. What happens to total revenue?
  1. It falls to zero
  2. It falls
  3. It rises
  4. It stays the same

Answer: It rises. |E| = 0.4 is less than 1, so demand is inelastic. Sales fall by a smaller percentage than the price rises, so P · Q goes up.

3The midpoint method

A change from €10 to €14 is +40%, but a change from €14 back to €10 is −28.6%. The answer would depend on the direction.

The midpoint (or arc) method avoids this. It divides each change by the average of the two values:

%ΔQ = (Q₂ − Q₁) / ((Q₁ + Q₂)/2)
%ΔP = (P₂ − P₁) / ((P₁ + P₂)/2)
E = %ΔQ / %ΔP
  1. Write down P₁, Q₁ (before) and P₂, Q₂ (after).
  2. Find the change in quantity and divide it by the average quantity.
  3. Find the change in price and divide it by the average price.
  4. Divide the first result by the second. Keep the sign.
  5. Classify: elastic if |E| > 1, inelastic if |E| < 1.

4A worked example

Follow every step. Try to predict the next one before you open it.

Worked example

When the price of a ticket changes from €10.00 to €14.00, weekly sales change from 500 to 300. Use the midpoint (arc) method to find the price elasticity of demand E.

1

Mark the two points

QPD500€10300€14

Before: P₁ = €10, Q₁ = 500. After: P₂ = €14, Q₂ = 300.

2

Relative change in quantity

The average quantity is (500 + 300)/2 = 400.

%ΔQ = (300 − 500) / 400 = −0.5
3

Relative change in price

The average price is (10 + 14)/2 = 12.

%ΔP = (14 − 10) / 12 = 0.3333
4

Divide

E = −0.5 / 0.3333 = −1.5
5

Interpret

|E| = 1.5 is more than 1, so demand is elastic. Check with revenue: 10 · 500 = €5000 before and 14 · 300 = €4200 after. The price rise lowered revenue, as elastic demand predicts.

Answer

E = −1.5: demand is elastic.

5Elasticity on a straight demand line

Some questions give a demand equation such as Q = 200 − 2P. Each €1 increase in price lowers Q by 2. This slope is dQ/dP = −2.

The point elasticity at one price is:

E = (dQ/dP) · P / Q

At P = €40: Q = 200 − 2 · 40 = 120, so E = −2 · 40 / 120 = −0.67. Demand is inelastic at that price.

QPD120€40100€50

Along a straight line the elasticity changes: demand is inelastic at low prices and elastic at high prices.

Revenue is largest where E = −1. For Q = a − bP that is at P = a/(2b). Here P = 200/4 = €50, with Q = 100 and revenue €5000.

6Common mistakes

  • Dividing the wrong way round: it is quantity change over price change.
  • Dropping the minus sign. Price elasticity of demand is negative.
  • Dividing by the starting value when the midpoint method is asked. Use the average of the two values.
  • Using absolute changes (−200 tickets, +€4) instead of relative changes.
  • Linking elasticity to revenue the wrong way: with inelastic demand a price rise increases revenue.
Self-check The price changes from €4 to €6 and the quantity from 200 to 100. What is E with the midpoint method?
  1. +1.67
  2. −0.6
  3. −1.67
  4. −1

Answer: −1.67. %ΔQ = −100/150 = −0.667 and %ΔP = 2/5 = 0.4, so E = −1.67. −1 divides by the starting values (−50%/50%), −0.6 divides the wrong way round, and +1.67 drops the sign.

7How exam questions look

  • Easy: percentage changes. Both percentages are given: divide the quantity change by the price change.
  • Medium: midpoint method. Two prices and two quantities are given: use the averages.
  • Hard: linear demand. A demand equation Q = a − bP is given: find the point elasticity at a price, or the price that gives the largest revenue.

Give E with its minus sign and say whether demand is elastic or inelastic.

Learn it step by step in the appLearn it step by step in the app

Worked examples

Example 1Easy
QPDQ1​P1​Q2​P2​

When the price of a product rises by 25%, the quantity demanded falls by 5%. Find the price elasticity of demand E.

%ΔP = +25%%ΔQ = −5%

Find E

Show hints
  1. Elasticity is the percentage change in quantity divided by the percentage change in price.
  2. Price and quantity move in opposite directions, so the result is negative.
  3. E = −5/25.
Show the answer and the step-by-step solution

Price elasticity of demand measures how strongly the quantity demanded reacts to price: E = (% change in quantity)/(% change in price). It is negative for normal goods. |E| > 1 is elastic (revenue falls when price rises), |E| < 1 is inelastic.

Divide the percentage changes

E = %ΔQ/%ΔP = (−5%)/(25%) = −0.2

Answer

E = −0.2

|E| < 1: demand is inelastic.

Example 2Medium
QPD1200€25.001580€15.00

When the price of a ticket changes from €25.00 to €15.00, weekly sales change from 1200 to 1580. Use the midpoint (arc) method to find the price elasticity of demand E.

P1 = €25.00P2 = €15.00Q1 = 1200Q2 = 1580

Find E

Show hints
  1. The midpoint method divides each change by the average of the two values, so the answer is the same whichever way the price moves.
  2. %ΔQ = (Q₂ − Q₁)/((Q₁ + Q₂)/2) and %ΔP = (P₂ − P₁)/((P₁ + P₂)/2).
  3. %ΔQ = 0.2734, %ΔP = −0.5.
Show the answer and the step-by-step solution

Price elasticity of demand measures how strongly the quantity demanded reacts to price: E = (% change in quantity)/(% change in price). It is negative for normal goods. |E| > 1 is elastic (revenue falls when price rises), |E| < 1 is inelastic.

Relative change in quantity

(1580 − 1200)/((1200 + 1580)/2) = 380/1390 = 0.2734

Relative change in price

(15 − 25)/((25 + 15)/2) = −10/20 = −0.5

Elasticity

E = 0.2734/−0.5 = −0.5468

Answer

E = −0.5468
Example 3Hard
QPDQP = ?

The demand for a product is Q = 1500 − 50P, with P in euros. At which price P is total revenue as large as possible?

Q = 1500 − 50P

Find P (€)

Show hints
  1. Revenue is R = P·Q = P·(a − bP). Maximise it by setting dR/dP = 0.
  2. Revenue is largest where demand is unit elastic, E = −1.
  3. dR/dP = 1500 − 100P.
Show the answer and the step-by-step solution

Price elasticity of demand measures how strongly the quantity demanded reacts to price: E = (% change in quantity)/(% change in price). It is negative for normal goods. |E| > 1 is elastic (revenue falls when price rises), |E| < 1 is inelastic.

Revenue as a function of price

R = P·(1500 − 50P) = 1500P − 50P²

Set the derivative to zero

dR/dP = 1500 − 100P = 0
P = 1500/100 = €15.00

Answer

P = €15.00

At this price E = −1: demand is unit elastic.

Questions and answers

What is Price elasticity of demand?

When a price goes up, people buy less. That is the law of demand. But how much less depends on the product.

If the price of salt doubles, you still buy about the same amount. If one brand of cola gets 20% dearer, many buyers switch to another brand.

How do I solve Price elasticity of demand problems step by step?

Price elasticity of demand measures how strongly the quantity demanded reacts to price: E = (% change in quantity)/(% change in price). It is negative for normal goods. |E| > 1 is elastic (revenue falls when price rises), |E| < 1 is inelastic.

What do the Price elasticity of demand exercises cover?
  • Easy: percentage changes
  • Medium: midpoint method
  • Hard: linear demand

Every problem has new values, so you learn the method instead of an answer. Stuck? Open a small hint first, and the full explanation only when you need it.

How practising works

  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Your progress, mistakes and saved problems are kept, so you can review before the exam.

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Printable worksheet

Price elasticity of demand worksheet with answers
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