PED and total revenue — the connection — Economics

Before this

This leads to

Tax incidence
Income elasticity of demand

Raise the price: more money, or less?

A seller's total revenue (TR) is the money it receives from sales: the price of each unit times the number of units sold.

TR = P \times Q

Now suppose the seller raises the price. The price goes up, which pushes revenue up. But by the law of demand, the quantity sold goes down, which pushes revenue down. Two forces pull in opposite directions. Which one wins?

In the last lecture, a bus company raised its fares by 15% when the price elasticity of demand (PED) for its journeys was −0.4. It lost about 6% of its journeys. The lecture promised to show what that meant for its revenue. This lecture keeps that promise, and the answer turns out to depend on one thing only: whether demand is elastic or inelastic.

Revenue is a rectangle

Here is a picture that makes the whole lecture clear. On a demand diagram, price is the height and quantity is the width. So $P \times Q$ is the area of a rectangle under the demand curve, from the origin to the point where the market is.

Take the demand curve from the last lecture, $Q_{d} = 200 - 5 P$ . At $30, 50 units are sold, so TR $= 30 \times 50 = $1500$ . Now raise the price to $32. The quantity falls to 40, and TR becomes $32 \times 40 = $1280$ . The rectangle has changed shape in two ways:

Raising the price from $30 to $32 on

Q_{d} = 200 - 5 P

. The rectangle gains a thin strip on top, because the 40 units still sold each bring $2 more. It loses a strip at the side: the 10 units no longer sold, each worth $30. The loss is bigger, so TR falls from $1500 to $1280.

A gain on top: the 40 units that are still sold each bring in $2 more. That adds $40 \times 2 = $80$ .
A loss at the side: the 10 units no longer sold each brought in $30. That removes $10 \times 30 = $300$ .

The loss is larger, so revenue falls by $300 - 80 = $220$ . Check: $1500 - 1280 = 220$ .

Elasticity decides which strip is bigger

The gain strip depends on the percentage rise in price. The loss strip depends on the percentage fall in quantity. And the comparison between those two percentages is exactly what PED measures.

Elastic demand ( $∣ PED ∣ > 1$ ): quantity falls by a larger percentage than price rises. The loss strip wins. A price rise lowers TR; a price cut raises TR.
Inelastic demand ( $∣ PED ∣ < 1$ ): quantity falls by a smaller percentage than price rises. The gain strip wins. A price rise raises TR; a price cut lowers TR.
Unit elastic demand ( $∣ PED ∣ = 1$ ): the two strips are about equal, so a small price change leaves TR almost unchanged.

That is it. That is the whole idea. With inelastic demand, price and revenue move in the same direction. With elastic demand, they move in opposite directions. The seller's question "will raising my price bring in more money?" is really the question "is my demand elastic or inelastic?"

PED and total revenue

If demand is inelastic, total revenue moves in the same direction as price. If demand is elastic, total revenue moves in the opposite direction to price. Where demand is unit elastic, total revenue is at its highest.

Now try it yourself. Move along the demand curve and watch the rectangle and the revenue together.

Graph — the revenue rectangle and the revenue arch

Move the price. The top graph shows TR as a rectangle under the demand curve. The bottom graph plots TR against price. Watch where TR is highest.

Price: $30

Total revenue at several prices on $Q_{d} = 200 - 5 P$ . TR is highest at $20, where PED = −1.

TR against price: an arch. On the left (low prices, inelastic demand), raising the price raises TR. On the right (high prices, elastic demand), raising the price lowers TR.

Price ($)	Quantity	TR ($)	PED
8	160	1280	−0.25
16	120	1920	≈ −0.67
20	100	2000	−1
24	80	1920	−1.5
32	40	1280	−4

Did you find the top of the arch? Revenue is highest at $20, the exact middle of the line, where PED = −1. Above $20, demand is elastic, so a price cut raises revenue. Below $20, demand is inelastic, so a price rise raises revenue. Both moves push the seller towards the middle.

A careful point about "unit elastic"

Many books say that when demand is unit elastic, a price change leaves revenue unchanged. That is true for small changes, but not exactly for large ones. Remember the cinema from the last lecture. Its price rose from $10 to $12 (+20%), and tickets fell from 500 to 400 (−20%), so PED = −1. Yet revenue fell from $10 \times 500 = $5000$ to $12 \times 400 = $4800$ .

Why? A 20% rise followed by a 20% fall does not return you to the start: $1.2 \times 0.8 = 0.96$ , which is 4% less. For small changes, the error is tiny. For example, on our demand curve, moving from $20 to $21 changes revenue only from $2000 to $1995. So the rule is a good guide for small changes, and the arch explains why: near its top, the arch is almost flat.

Stop and think

A shop's demand is inelastic. To bring in more customers and more money, the owner cuts prices. What happens to the shop's revenue?

Show the answer

It falls. With inelastic demand, quantity rises by a smaller percentage than the price falls. The shop does gain customers, but each one pays less, and the extra customers do not make up for it. More customers is not the same as more revenue. If the owner wants more revenue, inelastic demand points the other way: a price rise.

Test your intuition once more with a prediction.

Predict — a fare cut with inelastic demand

The bus company's demand has PED = −0.4. Instead of raising fares, it cuts them by 10% to win more passengers. What happens to its fare revenue? Choose one, then read the explanation.

The answer, worked out:

Journeys rise by $0.4 \times 10% = 4%$ . Revenue becomes $0.90 \times 1.04 = 0.936$ of its old value, a fall of about 6.4%. With inelastic demand, a price cut lowers revenue, even though the number of passengers rises.

Worked examples

Worked Example 1 — An elastic case: a restaurant

A restaurant raises the price of its main dish from $15 to $18. The number sold each week falls from 200 to 140. Find the PED and the change in revenue.

Price: $(18 - 15) \div 15 = + 20%$ . Quantity: $(140 - 200) \div 200 = - 30%$ . PED = −30 ÷ 20 = −1.5: elastic.

TR before: $15 \times 200 = $3000$ . TR after: $18 \times 140 = $2520$ . Revenue falls by $480, as the rule predicts for elastic demand. The diners have many other places to eat, so many of them leave.

Worked Example 2 — An inelastic case: a fuel station

A fuel station raises its price from $1.50 to $1.65 a litre. Sales fall from 1000 to 950 litres a day. Find the PED and the change in revenue.

Price: +10%. Quantity: −5%. PED = −5 ÷ 10 = −0.5: inelastic.

TR before: $1.50 \times 1000 = $1500$ . TR after: $1.65 \times 950 = $1567.50$ . Revenue rises by $67.50, as the rule predicts for inelastic demand. In the short run, drivers still need fuel.

Worked Example 3 — The bus company, finished

In the last lecture, fares rose by 15% with PED = −0.4, so journeys fell by 6%. What happens to fare revenue?

New revenue = new fare × new journeys = $1.15 \times 0.94 = 1.081$ times the old revenue. Revenue rises by about 8.1%.

A quick shortcut, "15% − 6% = 9%", gives about the right size but is not exact, because the 6% fall applies to the new, higher fare. Multiply the two factors instead.

There is a second lesson here. Inelastic demand often means that people have few alternatives. Many bus users cannot easily switch to a car. So a fare rise brings in more money, but much of it comes from people who have little choice. This is one reason why fares for public transport are often decided by governments, not only by the bus companies.

Worked Example 4 — A real case: the Philippines' tobacco taxes

The policy. In 2012, the Philippines passed a law that sharply raised taxes on tobacco and alcohol, often called the Sin Tax Reform. For the cheapest cigarette brands, the tax rose from 2.72 pesos per pack in 2012 to 30 pesos per pack by 2017. The price of one popular local brand rose by 167% over the same period.

What PED predicts. A government's tax revenue works like a seller's revenue: tax per pack × packs sold. Demand for cigarettes is widely found to be inelastic, because many smokers are addicted and there are few substitutes. So the prediction is that sales fall, but by a much smaller percentage than the price rises, and that tax revenue rises strongly.

What happened. Total cigarette sales fell by nearly 23% from 2012 to 2016. Tobacco tax revenue rose from 32.9 billion pesos in 2012 to 99.5 billion pesos in 2015, about three times as much. The share of adults who smoked fell from 29.7% in 2009 to 23.8% in 2015. (Figures from Health Taxes to Save Lives: Case Studies, Task Force on Fiscal Policy for Health, citing the Philippine Department of Finance, Euromonitor and the Global Adult Tobacco Survey.)

The verdict. The prediction held. Sales fell by a much smaller percentage than prices rose, so revenue rose sharply. Notice that both things happened at once: the government collected far more money, and people smoked less. That combination is possible only because demand is inelastic. These figures cover several years, in which incomes and population also changed, so they show the pattern rather than an exact PED.

Three classic mistakes

Trap 1 — "More customers means more revenue"

A price cut always brings more customers, by the law of demand. It brings more revenue only if demand is elastic. With inelastic demand, the extra customers do not make up for the lower price.

Trap 2 — Subtracting percentages

A 15% price rise and a 6% fall in quantity do not give a 9% rise in revenue. Multiply the factors: $1.15 \times 0.94 = 1.081$ , an 8.1% rise. The shortcut is close only when both changes are small.

Trap 3 — One answer for the whole demand curve

On a straight-line demand curve, PED changes from point to point. So "should we raise the price?" has different answers at different prices: yes below the midpoint, no above it. Always ask where on the curve the seller is.

Try it yourself

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

A concert's ticket price rises from $40 to $50, and sales fall from 1000 to 700. Find the PED and the change in total revenue.

Hint 1
Find both percentage changes from the starting values.

Hint 2
Then find TR before and after: price × quantity.

Full answer
Price: +25%. Quantity: −30%. PED = −30 ÷ 25 = −1.2: elastic. TR falls from $40 \times 1000 = $40 000$ to $50 \times 700 = $35 000$ . As expected, the price rise lowered revenue.
The PED for household electricity is −0.3 (an invented figure). The price rises by 20%. By what percentage does revenue change?

Hint 1
Quantity changes by PED × 20%.

Hint 2
Revenue factor = 1.20 × (1 − 0.06).

Full answer
Quantity falls by $0.3 \times 20% = 6%$ . Revenue factor $= 1.20 \times 0.94 = 1.128$ , a rise of about 12.8%. Inelastic demand, so price and revenue move in the same direction.
On $Q_{d} = 200 - 5 P$ , the price falls from $30 to $24. Find TR before and after, and explain the change using PED.

Hint 1
At $30, Q = 50. At $24, Q = 80.

Hint 2
Both prices are above the midpoint price of $20.

Full answer
TR before: $30 \times 50 = $1500$ . TR after: $24 \times 80 = $1920$ . Revenue rises. Both prices are on the elastic upper half of the line (PED = −3 at $30 and −1.5 at $24), so a price cut raises revenue.
A seller on $Q_{d} = 200 - 5 P$ charges $10. Its manager says: "Our demand is inelastic, so we should keep raising our price. Every rise will bring in more revenue." Explain what is right and what is wrong in this, and find the best price for revenue.

Hint 1
Find PED at $10, and then at higher prices. Does demand stay inelastic?

Hint 2
Revenue is highest where PED = −1. Where is that on this line?

Full answer
At $10, PED ≈ −0.33, so demand is inelastic, and a price rise does raise revenue: TR goes from $10 \times 150 = $1500$ towards higher values. That part is right. But PED is not fixed. As the price rises, demand becomes less inelastic, reaches PED = −1 at $20 (TR = $2000), and then becomes elastic. Above $20, every further rise lowers revenue: at $30, TR is back down to $1500. So the best price for revenue is $20. Keep raising the price "for ever", and revenue eventually falls to zero at $40.

The bigger idea

Elasticity turns a sales question into a revenue question. Any seller who sets a price, from a street stall to a government setting a tax, is choosing a rectangle under a demand curve. Inelastic demand gives sellers the power to raise prices and earn more. Elastic demand takes that power away, because buyers can walk to a substitute.

Revenue is not the whole story. A firm also cares about its costs, and so about profit, and a government also cares about who pays. The lecture on tax incidence, and later lectures on firms, build on exactly this link between elasticity and revenue.

Key concepts in this lesson

Total revenue is price times quantity, which on a demand diagram is the area of a rectangle under the demand curve.
When demand is inelastic, total revenue moves in the same direction as price, so a price rise raises revenue.
When demand is elastic, total revenue moves in the opposite direction to price, so a price cut raises revenue.
On a straight-line demand curve, total revenue is highest at the midpoint, where PED = −1.
To find the change in revenue, multiply the price factor by the quantity factor rather than adding or subtracting percentages.