One price, another market
The last lecture ended with a question: how does the demand for one good respond to the price of a different good? Here it is in action.
A town has two bakeries that sell almost the same bread. One of them raises its price by 20%. What happens at the other bakery? Its price has not changed, yet it sells more bread, because some buyers walk across the road. Now a different case. The price of petrol rises sharply. Nothing has happened to the price of large cars, yet fewer people buy them, because a large car is expensive to run.
In the lecture on what shifts the demand curve, you met these two relationships: substitutes, which compete for the same buyer, and complements, which are bought as a team. That lecture gave the direction of the shift. This lecture measures its size, with the third member of the elasticity family.
Cross elasticity of demand
Here is the quantity demanded.
It measures how strongly the demand for one good responds to a change in the price of another good, other things being equal.
Read the formula carefully. The top is about good A. The bottom is about good B. If both were about the same good, it would be price elasticity of demand (PED) again.
As with income elasticity of demand (YED), the sign is the first thing to read:
- XED above 0: substitutes. When B's price rises, demand for A rises, because buyers switch from B to A. The two changes have the same sign, so XED is positive. Examples: two brands of bread, tea and coffee, bus and train journeys.
- XED below 0: complements. When B's price rises, demand for A falls, because fewer people use B, so fewer need A. The changes have opposite signs, so XED is negative. Examples: petrol and large cars, printers and ink cartridges, game consoles and games.
- XED close to 0: unrelated goods. The price of B makes almost no difference to demand for A. Example: the price of salt and the demand for bicycles.
The size matters too. The further XED is from 0, the stronger the relationship. Two nearly identical brands of bread might have an XED of +2 or more: a small price gap sends many buyers across the road. Tea and coffee might have a much smaller positive XED: some people switch, but many keep drinking what they like.
That is it. That is the whole idea. The sign of XED tells you how two goods are related, and its size tells you how closely.
A price in one market shifts a curve in another
When the price of B changes, buyers of A face no change in A's price. So the change in A's demand is not a movement along A's demand curve. It is a shift of the whole curve. XED measures how far the curve shifts, as a percentage, for each 1% change in B's price.
Try it below. Change the price of B and choose how the two goods are related.
Choose a relationship. Then move the price of good B away from $2.50. Watch the demand curve for good A, and read the XED at a price of A of $3.00.
At a price of A of $3.00, buyers want 600 units of A a week when B costs $2.50. The table shows the new quantity of A when B's price changes (an invented example).
| Price of B | Substitutes (XED +0.8) | Complements (XED −0.6) |
|---|---|---|
| $3.00 (+20%) | 696 (+16%) | 528 (−12%) |
| $2.00 (−20%) | 504 (−16%) | 672 (+12%) |
For substitutes, a rise in B's price shifts A's demand curve to the right. For complements, it shifts it to the left. For unrelated goods, it does not move.
Did you see the difference? For substitutes, a higher price of B pushes A's demand curve to the right. For complements, it pushes it to the left. In both cases, the price of A never moved: A's buyers simply wanted more, or less, at every price.
The price of tea rises, and coffee sellers sell more coffee at the same price as before. Is this a movement along the demand curve for coffee?
Show the answer
No. A movement along a demand curve is caused only by a change in that good's own price. Coffee's price did not change. A determinant of coffee demand changed: the price of a substitute. So the whole demand curve for coffee shifts to the right. In the tea market, the higher price of tea does cause a movement along tea's demand curve. One price change can cause a movement in one market and a shift in another.
XED works one way at a time
XED of A with respect to B's price need not equal XED of B with respect to A's price. Suppose a small bus company runs alongside a big national railway. If rail fares rise, many rail passengers might try the bus, and the small bus company's demand could jump. If bus fares rise, the few bus passengers who switch to rail make almost no difference to the railway's huge number of passengers. The same relationship, substitutes, gives a large XED one way and a tiny XED the other.
Why firms and regulators care
Firms want a low XED with their rivals. If buyers switch the moment a rival cuts its price, a firm has little control over its own price. So firms try to make their product seem different: through advertising, quality, design and a recognisable name. Each of these aims to make buyers less willing to switch, which lowers the XED with rival products.
Firms use complements on purpose. A firm that sells both printers and the cartridges that fit them can sell the printer cheaply. The cheaper printer raises demand for its cartridges, because the two are complements. If the cartridges carry a high profit, the firm can earn more overall. This only works if buyers cannot easily use another firm's cartridges in the printer.
Regulators use the idea to define a market. When two firms want to merge, competition authorities ask which products really compete with each other. The European Commission's notice on market definition, revised in February 2024, describes the "SSNIP test": would buyers switch to other products if the price rose by a small but significant amount, typically 5% to 10%? If enough buyers would switch, those products are close substitutes and belong in the same market. This is a question about cross elasticity.
Worked examples
The price of cola brand B rises from $1.00 to $1.20 a bottle. Weekly sales of cola brand A, whose price does not change, rise from 5000 to 5700 bottles. Find the XED and interpret it.
Price of B: . Quantity of A: .
XED = +14 ÷ +20 = +0.7. Positive, so the brands are substitutes. A 1% rise in B's price raises demand for A by about 0.7%: buyers do switch, but not all of them.
The price of petrol rises by 30%. Demand for large-engine cars falls by 9% (an invented figure). Find the XED.
XED = −9 ÷ +30 = −0.3. Negative, so petrol and large cars are complements: they are used together, and dearer petrol makes a large car more costly to run.
The size, 0.3, is small. Many people keep the car they have, at least at first, and the cost of fuel is only part of the cost of owning a car.
For a printer firm, the XED of its ink cartridges with respect to the price of its printers is −0.5 (an invented figure). It cuts the printer price by 20%. Predict the change in demand for cartridges.
% change in demand for cartridges = XED × % change in printer price = .
A fall in the price of a complement raises demand: the two negatives give a positive.
The policy. As you saw in the lecture on price elasticity of demand, on 1 January 2014 Mexico put a tax of 1 peso per litre on drinks with added sugar. That was about 10% of their price in 2013.
The prediction. If water and other untaxed drinks are substitutes for sugary drinks, buyers should switch towards them. Their demand should rise even though their own price did not change: a positive XED.
What happened. A study of household purchases from shops (Colchero and others, BMJ, 2016) compared 2014 with the trend before the tax. Purchases of taxed drinks were about 6% lower. Purchases of untaxed drinks were about 4% higher, mainly because people bought more bottled plain water.
A rough XED. Untaxed drinks rose by about 4% when the price of taxed drinks rose by about 10%, so XED ≈ +4 ÷ +10 = +0.4.
The verdict. The sign matches the prediction: bottled water behaved as a substitute for sugary drinks. Treat the size with care. "Untaxed drinks" is a group of many drinks, the price rise was not exactly 10% for every drink, and other things may have changed in 2014. The direction is clear; +0.4 is only a rough estimate.
One common way of reading an XED calculation contains a single mistake. Find it.
A student works out an XED and explains it. Exactly one line is wrong. Tap the line you think is wrong.
A student's XED calculation and explanation. Line 4 is wrong.
- Line 1. The price of tea rises from $4.00 to $4.40 a packet: a 10% rise. Correct.
- Line 2. Demand for coffee, whose price does not change, rises from 2000 to 2080 jars a week: a 4% rise. Correct.
- Line 3. XED of coffee with respect to the price of tea = 4% ÷ 10% = +0.4. Correct.
- Line 4. "XED is less than 1, so tea and coffee are complements." Wrong: the sign decides. +0.4 is positive, so they are substitutes.
- Line 5. A 10% rise in the price of tea raises demand for coffee by only 4%, so some tea drinkers switch, but many do not. Correct.
Three classic mistakes
XED divides the change in demand for A by the change in the price of B. If you divide by the change in A's own price, you have worked out PED instead.
The XED of A with respect to B's price can be very different from the XED of B with respect to A's price, as the bus and railway example showed. Always say which good's demand and which good's price you mean.
If the price of B falls by 10% and demand for A falls by 5%, XED = −5 ÷ −10 = +0.5. The goods are substitutes, even though demand for A fell. Divide the signed changes, and read the sign of the answer.
Work each one on paper first. Open a hint only when you are stuck.
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Bus fares in a city rise by 8%. The number of train journeys rises by 6%. Find the XED of train journeys with respect to bus fares, and say how the two are related.
Hint 1
XED = % change in demand for train journeys ÷ % change in bus fares.
Hint 2
6 ÷ 8.
Full answer
XED = +6 ÷ +8 = +0.75. Positive, so buses and trains are substitutes. Check: dearer buses send some passengers to the train.
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The price of a games console falls by 15%. Sales of games for that console rise by 12%. Find the XED and classify the goods.
Hint 1
Keep the signs: the price change is −15%.
Hint 2
+12 ÷ −15.
Full answer
XED = +12 ÷ −15 = −0.8. Negative, so consoles and games are complements. Check: a cheaper console means more console owners, who buy more games.
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The XED of margarine with respect to the price of butter is +0.6. A shop sells 2000 tubs of margarine a week. The price of butter rises by 5%. Predict the new weekly sales of margarine, if nothing else changes.
Hint 1
% change in margarine demand = XED × % change in butter price.
Hint 2
.
Full answer
Demand rises by . tubs a week. Check: butter and margarine are substitutes, so dearer butter should raise margarine sales, and it does.
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A firm sells 2000 printers a month at $100 each (cost $70 each) and 10 000 ink cartridges a month at $25 each (cost $5 each). The PED for its printers is −1.5, and the XED of its cartridges with respect to the printer price is −0.5. (These are invented figures.) It cuts the printer price to $80. Find the change in monthly profit from printers, from cartridges, and overall. Should it cut the price?
Hint 1
The price cut is −20%. Use PED for printer sales and XED for cartridge sales.
Hint 2
Printers: , so 2600 sold at a profit of $10 each. Cartridges: , so 11 000 sold at a profit of $20 each.
Full answer
Printers: profit falls from to , a fall of $34 000. Cartridges: profit rises from to , a rise of $20 000. Overall, profit falls by $14 000 a month. On these figures, the cut does not pay. It would pay only if cartridges were more closely tied to printers (a larger negative XED) or carried a bigger profit. That is exactly the trade-off a firm using this strategy has to judge.
With XED, the elasticity family is complete on the demand side. PED asks how buyers respond to a good's own price. YED asks how they respond to their income. XED asks how they respond to the price of something else. Each one is a percentage change divided by a percentage change, and each one turns a direction from the demand-shift lecture into a number.
XED also shows that no market stands alone. A tax on sugary drinks changes the market for water. A change in fuel prices changes the market for cars. Because firms and regulators care about these links, XED helps decide which products count as competitors at all. The next lecture turns to the other side of the market: how strongly sellers respond to price.
- Cross elasticity of demand is the percentage change in quantity demanded of one good divided by the percentage change in the price of another good.
- A positive XED shows substitutes, a negative XED shows complements, and an XED near zero shows unrelated goods.
- The further XED is from zero, the closer the substitutes or the stronger the complements.
- A change in the price of B shifts the whole demand curve for A, because A's own price has not changed.
- XED need not be the same in both directions, and firms, and the authorities that judge mergers, use it to decide which products compete.