Price elasticity of demand — PED — Economics

Before this

This leads to

PED and total revenue
Tax incidence

The law of demand tells you the direction, not the size

Suppose the price of insulin doubles. People with diabetes who need insulin will still buy almost the same amount, because their lives depend on it. Now suppose the price of a holiday abroad doubles. Many families will simply stay at home this year.

Both goods obey the law of demand: a higher price means a lower quantity demanded. But the law says nothing about how much lower. For insulin, the fall is tiny. For holidays, it is large. A business deciding its prices, and a government deciding a tax, both need to know which kind of good they are dealing with.

Economists measure this sensitivity with one number, the price elasticity of demand. This lecture shows how to calculate it, what its value means, and what makes it large or small.

Measuring sensitivity with percentages

Price elasticity of demand (PED)

$PED = \frac{% change in quantity demanded}{% change in price}$

It measures the percentage change in quantity demanded for each 1% change in price, other things being equal.

Why percentages, and not simple changes? Because simple changes depend on the units. A $1 rise means a lot for a $2 bus ticket and almost nothing for a $20 000 car. A fall of 100 units means a lot for a small shop and almost nothing for a national market. Percentages remove the units, so the PED of rice in Chennai can be compared with the PED of fuel in Barbados.

A percentage change is always measured from the starting value:

% change = \frac{new value - old value}{old value} \times 100

By the law of demand, price and quantity demanded move in opposite directions. When one percentage change is positive, the other is negative. So PED is a negative number for almost every good. Economists often compare the size of PED and ignore the minus sign. We write that size as $∣ PED ∣$ . For example, a PED of −2 has a size of 2.

What the number means

The key comparison is with 1, because 1 means "the same percentage".

$∣ PED ∣ > 1$ : elastic demand. Quantity demanded changes by a larger percentage than price. A 10% price rise causes a fall of more than 10%.
$∣ PED ∣ < 1$ : inelastic demand. Quantity demanded changes by a smaller percentage than price. A 10% price rise causes a fall of less than 10%.
$∣ PED ∣ = 1$ : unit elastic demand. Both change by the same percentage.

There are also two extreme cases. If quantity demanded does not change at all, PED = 0. This is perfectly inelastic demand, drawn as a vertical demand curve. If buyers will buy any amount at one price but nothing at all at a slightly higher price, demand is perfectly elastic, drawn as a horizontal demand curve. These two are useful as limits. Real goods sit somewhere between them.

Stop and think

The PED for a good is −0.5. Its price rises by 10%. Does the quantity demanded fall by 0.5%, by 5%, or by 50%?

Show the answer

By 5%. PED is the percentage change in quantity for each 1% change in price. So a 10% rise gives $- 0.5 \times 10% = - 5%$ . The answer 0.5% forgets to multiply by the 10%. The answer 50% divides instead of multiplying. Since 5% is less than 10%, demand is inelastic, as $∣ PED ∣ = 0.5 < 1$ says.

One straight line, every kind of elasticity

Here is a surprise. Take the straight-line demand curve $Q_{d} = 200 - 5 P$ . Every $1 rise in price lowers the quantity by exactly 5 units, everywhere on the line. So you might expect the PED to be the same everywhere. It is not.

At a price of $30, the quantity is 50. A $1 rise is only about a 3.3% rise in price, but losing 5 units is a 10% fall in quantity. Quantity responds far more than price: PED = −3, elastic.

At a price of $10, the quantity is 150. A $1 rise is a 10% rise in price, but losing 5 units is only about a 3.3% fall in quantity. Now quantity responds far less: PED ≈ −0.33, inelastic.

That is it. That is the whole idea. The change is always 5 units for $1, but the percentages depend on where you start. High up the line, the price is large and the quantity is small, so a fixed change in quantity is a big percentage. Low down, the opposite is true.

In general, each $1 changes the quantity on a straight line by the same amount. So PED at any point is that amount, with its minus sign, multiplied by $P \div Q$ at the point:

PED = \frac{change in Q}{change in P} \times \frac{P}{Q} = - 5 \times \frac{P}{Q}

The demand curve

Q_{d} = 200 - 5 P

. The top half of the line is elastic, the bottom half is inelastic, and the exact middle ($20, 100 units) is unit elastic.

Explore it below. Move along the line and watch PED change.

Explore — PED along a straight-line demand curve

Move the price. The point moves along the demand curve. Read what a $1 rise in price would do from that point, in units and in percentages.

Price: $30

PED at several points on $Q_{d} = 200 - 5 P$ , using PED = −5 × P ÷ Q:

Price ($)	Quantity	PED	Demand here is
36	20	−9	elastic
30	50	−3	elastic
20	100	−1	unit elastic
10	150	≈ −0.33	inelastic
4	180	≈ −0.11	inelastic

Did you see it? The line has the same slope everywhere, but PED runs from very elastic at the top to very inelastic at the bottom, passing −1 exactly at the middle. So never judge elasticity by how steep a curve looks. Steepness depends on the units chosen for the axes. Elasticity does not.

What makes demand elastic or inelastic

Five factors decide whether buyers respond strongly to a price change. All five come back to one question: how easily can buyers avoid the higher price?

1. The number and closeness of substitutes

If there are close substitutes, buyers can switch when the price rises, so demand is elastic. Insulin has no real substitute. Coffee from one café, on a street with ten cafés, has nine close substitutes a few steps away.

2. Necessity or luxury

Buyers cannot easily cut back on necessities such as basic food, medicine and electricity for a home. Demand for them tends to be inelastic. Luxuries, such as jewellery or a holiday abroad, are easy to give up or postpone, so demand for them tends to be elastic.

3. The share of income spent on the good

A 20% rise in the price of salt costs a family very little, so few people notice or change what they buy. A 20% rise in the price of a car or a rent is a large amount of money, so buyers search hard for alternatives. Goods that take a large share of income tend to have more elastic demand.

4. Time

Right after a price rise, buyers may be stuck with their habits, their car or their heating system. Over months and years, they can change them. So demand is usually more elastic in the long run. A review of many studies of fuel for cars looked at a lasting 10% rise in its price (Goodwin, Dargay and Hanly, 2004). It found that fuel use fell by about 2.5% within a year, and by about 6% in the longer run. That is a PED of about −0.25 in the short run and about −0.6 in the long run: inelastic in both, but much less so over time.

5. How broadly the market is defined

Demand for "food" as a whole is very inelastic, because there is no substitute for eating. Demand for one brand of breakfast cereal is elastic, because other brands are on the same shelf. The narrower the definition, the more substitutes there are, and the more elastic the demand.

Practise using the five factors below.

Sorter — likely elastic or likely inelastic?

Each card names a good. Decide whether its demand is likely to be elastic or inelastic, then place the card. Read the reason on every card.

Each good, its likely elasticity, and the main reason:

Good	Likely demand	Main reason
Insulin, for a person with diabetes	inelastic	a necessity with no substitute
Coffee at one café, on a street with ten cafés	elastic	many close substitutes
Salt	inelastic	a tiny share of income
A holiday abroad	elastic	a luxury that can be postponed
Fuel for cars, in the first month after a price rise	inelastic	little time to adjust
One brand of bottled water	elastic	other brands are on the same shelf
Food as a whole	inelastic	a broad market with no substitute
One brand of breakfast cereal	elastic	a narrow market with many substitutes
Electricity for a home	inelastic	a necessity, with few substitutes in the short run

Worked examples

Worked Example 1 — Calculating PED from two prices

A cinema raises its ticket price from $10 to $12. The number of tickets sold falls from 500 to 400 a week. Calculate the PED.

% change in price = \frac{12 - 10}{10} \times 100 = + 20%

% change in quantity = \frac{400 - 500}{500} \times 100 = - 20%

PED = −20% ÷ 20% = −1. Demand over this range is unit elastic.

One caution. If the price had fallen from $12 to $10, the same two points would give −1.5, because the percentages start from different values (+25% in quantity, −16.7% in price). So always measure from the starting point of the change described. Some textbooks avoid this problem by measuring from the midpoint of the two values instead. Follow the method your exam uses.

Worked Example 2 — PED at a point on a line

Demand for a good is $Q_{d} = 200 - 5 P$ . Find the PED at a price of $20, and at a price of $10.

At $20, $Q_{d} = 200 - 100 = 100$ . Each $1 changes quantity by −5, so PED = $- 5 \times 20 \div 100 = - 1$ . Unit elastic.

At $10, $Q_{d} = 200 - 50 = 150$ . PED = $- 5 \times 10 \div 150 \approx - 0.33$ . Inelastic.

Check the first answer with percentages: from $20, a $1 rise is a 5% rise, and losing 5 units from 100 is a 5% fall. The same percentage, so PED = −1.

Worked Example 3 — Using PED to predict

Suppose the PED for bus travel in a city is −0.4 (an invented figure). The bus company raises fares by 15%. What change in the number of journeys should it expect?

Rearrange the definition: % change in quantity = PED × % change in price = $- 0.4 \times 15% = - 6%$ . The company should expect about 6% fewer journeys, other things being equal. Because demand is inelastic, the fall in journeys (6%) is much smaller than the rise in fares (15%). (The next lecture shows what this means for the company's revenue.)

Worked Example 4 — A real case: Mexico's sugary-drinks tax

The policy. You met this case in the lecture on the law of demand. On 1 January 2014, Mexico put a tax of 1 peso per litre on drinks with added sugar, about 10% of their 2013 price.

What happened to prices. A study of prices in Mexican cities found that the tax was passed on to buyers. For fizzy sugary drinks, prices rose by roughly 1 peso per litre, about 11% (Colchero and others, PLoS ONE, 2015).

What happened to quantity. In 2014, purchases of taxed drinks were on average 6% lower than they would have been without the tax. By December 2014, they were 12% lower (Colchero and others, BMJ, 2016).

The elasticity. Taking the price rise as about 10%, the PED over the whole year was roughly $- 6% \div 10% \approx - 0.6$ : inelastic. By December, with a fall of about 12%, it was closer to −1.2. These are rough figures, because different drinks rose by different amounts. But the pattern is clear.

The verdict. Demand for sugary drinks was inelastic at first: people bought less, but not much less. As the year went on, the response grew, exactly as the time factor predicts. Buyers had time to change habits and find substitutes, such as bottled water, whose purchases rose by about 4%.

Three classic mistakes

Trap 1 — Judging elasticity by steepness

A steep-looking curve is not always inelastic, and a flat-looking one is not always elastic. A straight line has every elasticity along it. The look of a curve also changes with the scale of the axes. Calculate PED from percentages.

Trap 2 — Using changes instead of percentage changes

For $Q_{d} = 200 - 5 P$ , the change in quantity for each $1 is −5 everywhere. That is the slope, not the PED. PED needs percentage changes, so it also depends on the price and quantity at the point.

Trap 3 — The formula upside down

PED is the percentage change in quantity divided by the percentage change in price. Quantity goes on top, because it is the response. Price is the cause, so it goes underneath.

Try it yourself

Work each one on paper first. Open a hint only when you are stuck.

The price of a magazine rises from $4.00 to $4.20, and weekly sales fall from 2000 to 1700. Calculate the PED and say whether demand is elastic or inelastic.

Hint 1
Find both percentage changes from the starting values.

Hint 2
The price change is $0.20 \div 4.00 \times 100$ . The quantity change is $- 300 \div 2000 \times 100$ .

Full answer
Price: +5%. Quantity: −15%. PED = −15 ÷ 5 = −3. Since $∣ PED ∣ = 3 > 1$ , demand is elastic. Check: a 5% price rise caused a fall three times as large in percentage terms.
Suppose the PED for a brand of rice is −1.5. Its price falls by 8%. By what percentage does the quantity demanded change?

Hint 1
% change in quantity = PED × % change in price.

Hint 2
A fall in price is a negative percentage: −8%.

Full answer
$- 1.5 \times (- 8%) = + 12%$ . The quantity demanded rises by 12%. Check: elastic demand, so the response (12%) is larger than the price change (8%), and it is in the opposite direction.
For $Q_{d} = 200 - 5 P$ , find the price at which PED = −3, and check that the point is on the elastic part of the line.

Hint 1
PED = $- 5 P \div Q$ , and $Q = 200 - 5 P$ .

Hint 2
Solve $5 P = 3 (200 - 5 P)$ .

Full answer
$5 P = 600 - 15 P$ , so $20 P = 600$ and $P = 30$ . Then $Q = 50$ , and PED = $- 5 \times 30 \div 50 = - 3$ . The price $30 is above the midpoint price of $20, so the point is on the elastic upper half, as expected.
In an invented city, the buses carry 1 000 000 journeys a month at a fare of $2. The short-run PED is −0.3, and the long-run PED is −0.7. The fare rises by 10%. Estimate the number of journeys a month in the short run and in the long run, and explain why they differ.

Hint 1
Find the percentage change in journeys for each PED.

Hint 2
Short run: $- 0.3 \times 10% = - 3%$ . Apply each percentage to 1 000 000.

Full answer
Short run: −3%, so about 970 000 journeys. Long run: $- 0.7 \times 10% = - 7%$ , so about 930 000 journeys. They differ because of time. At first, people still have the same jobs, homes and habits. Over months, some buy bicycles or cars, share rides, or move closer to work, so more of them avoid the higher fare. Both values show inelastic demand, but the response more than doubles with time.

The bigger idea

Price elasticity turns a direction into a size. The law of demand says that buyers respond to price. PED says how strongly, and the five factors say why. Everything comes back to one question: how easily can buyers avoid a higher price?

That number matters to anyone who sets a price. A firm with inelastic demand can raise its price and lose few customers. A government taxing a good with inelastic demand changes buying habits only a little, but it collects a lot of tax. The next lecture follows this idea to its most useful result: what happens to a seller's revenue when the price changes.

Key concepts in this lesson

Price elasticity of demand is the percentage change in quantity demanded divided by the percentage change in price, and it is negative for almost every good.
Demand is elastic when |PED| is greater than 1, inelastic when it is less than 1, and unit elastic when it equals 1.
On a straight-line demand curve, PED changes from point to point, so steepness alone does not show elasticity.
Demand is more elastic when there are close substitutes, the good is a luxury or a large share of income, buyers have time to adjust, or the market is narrowly defined.
Real estimates show demand becoming more elastic over time, as with fuel and with Mexico's sugary-drinks tax.