Cross Price Elasticity Calculator

Cross-price elasticity = [(Q₂ − Q₁) ÷ Q₁] ÷ [(P₂ − P₁) ÷ P₁] (P = the other product's price, Q = this product's quantity)

Q₁

The quantity of this product demanded before the other product's price changed

Q₂

The quantity demanded after — it can be lower, and it can be zero if demand disappears entirely

P₁

The other product's price before, which is also the base both percentage changes are measured against

P₂

The other product's price after — what changed, driving the whole calculation

Cross-price elasticity

The answer: signed, unit-free, and read by its sign first — positive for substitutes, negative for complements, zero for unrelated

Use it when you need to know whether two products compete, and by how much — pricing a second product line, working out what a rival's discount will cost you, or deciding whether a bundle makes sense. It is the tool for a question that a plain price elasticity cannot answer, because a product's own elasticity tells you what happens when its own price moves and says nothing about anybody else's. Read the sign first: negative means the two move together in demand, so a price cut by the other product hurts you; positive means demand shifts between them, so a price cut by the other product also hurts you, but for the opposite reason and with a different remedy. Then read the magnitude with the two caveats that make this number easy to misuse. It is measured at one point on one pair of prices, so it is not a constant — the same two products can be substitutes at high prices and complements at low ones. And it is a correlation in demand, not proof of cause: two products can show a strong relationship because a third factor moves both, so treat a surprising result as a question about the market rather than an answer about it.

1. Quantity rises 3% when the other price falls 5% — complements

   1. Quantity change: (206 − 200) ÷ 200 = 3 percent
   2. Other price change: (9.5 − 10) ÷ 10 = −5 percent
   3. Elasticity: 3 ÷ −5 = −0.6
   4. The sign is negative, so the two products are complements: the cheaper one gets, the more of the other people buy

   Both denominators here are the before values, 200 and 10, and that choice is deliberate — the page uses the initial value rather than the midpoint, so the two percentages can be reproduced by hand. It matters: the midpoint version of this same case gives −0.5882 rather than −0.6, and neither is wrong, they are different definitions. The initial-value form is the one that stays finite when a price falls all the way to zero.

2. Quantity rises 5% when the other price rises 20% — substitutes

   1. Quantity change: (210 − 200) ÷ 200 = 5 percent
   2. Other price change: (12 − 10) ÷ 10 = 20 percent
   3. Elasticity: 5 ÷ 20 = 0.25
   4. Positive, so the products are substitutes: people leave the one that got more expensive and buy this one instead

   The sign flipped relative to the first example and that is the entire finding — the number 0.25 is small, but the direction is what tells you these two products compete. Comparing the two examples is the point: the same quantity response, 3 percent against 5 percent, produces a negative answer in one and a positive one in the other purely because of which way the other price moved.

3. A 10% quantity fall on a 10% price rise — unit elasticity, −1

   1. Quantity change: (450 − 500) ÷ 500 = −10 percent
   2. Other price change: (22 − 20) ÷ 20 = 10 percent
   3. Elasticity: −10 ÷ 10 = −1
   4. Exactly −1 means the two percentage changes are the same size, so demand for this product falls in proportion to the other product's price rise

   A result of exactly −1 is a useful landmark because both changes are visible in the answer at once: a 1 percent move in the other price produces a 1 percent move in this quantity. Note how easily this could have been misread — a quantity that falls while the other price rises looks like a substitute relationship if you only watch the two directions, and the sign is what corrects you.

4. The other price collapses to zero — elasticity −0.5

   1. Quantity change: (300 − 200) ÷ 200 = 50 percent
   2. Other price change: (0 − 10) ÷ 10 = −100 percent
   3. Elasticity: 50 ÷ −100 = −0.5

   The other product becomes free or is withdrawn, a −100 percent move, and the measure still returns a usable −0.5 because the base is the price before. This is the case the initial-value definition handles and a percentage measured against the after value would not: the after price is zero, so there would be nothing to divide by. It is also the reason a free giveaway is such a strong test of whether two products are complements.

The most important limitation is that this is not a constant. Elasticity is measured at one pair of prices on one demand curve, and both curves move: the same two products can be substitutes in one price range and complements in another, and a value taken at 10 will not hold at 25. Treat any single figure as local. The second is that it is a correlation between two quantities, not an identified cause — if a third factor such as income, a season, or an advertising campaign moved both at once, this page will report a relationship that is really the third factor, and nothing in the arithmetic can detect that. It also assumes nothing else changed between the two observations, which in a real market is never quite true. It takes how much you charge for the other product as given rather than deriving it from a cost, so it says nothing about whether the price change was profitable. It uses the initial value as the denominator rather than the midpoint, which is a choice and not the only convention: an arc elasticity measured against the midpoint of the two values gives a different number, and it is the more common one in some textbooks, so quote the definition whenever you quote a figure. Both percentage changes are measured over an unspecified period, so the result carries an implicit time horizon you have to state yourself. It covers two products only — a real substitution question usually has several alternatives, and pairwise elasticities can be entirely consistent with each other and still give a misleading picture of the whole market. Finally it is a demand relationship and says nothing about supply, capacity or inventory.

What is cross-price elasticity of demand?

It is the percentage change in the quantity demanded of one product divided by the percentage change in the price of a different product, and it is unit-free because both halves are percentages. On the default case a quantity that rises 3 percent while another product's price falls 5 percent gives 3 ÷ −5 = −0.6. The sign carries most of the information: negative means the two are complements, positive means substitutes, zero means unrelated.

How do I calculate cross-price elasticity?

Divide the two percentage changes, each measured against its own before value. On the default numbers: quantity goes from 200 to 206, which is (206 − 200) ÷ 200 = 3 percent, and the other price goes from 10 to 9.5, which is (9.5 − 10) ÷ 10 = −5 percent. Then 3 ÷ −5 = −0.6. Both changes must be measured over the same period and neither should be reversed — the quantity is always this product's and the price is always the other product's.

What does a negative cross-price elasticity mean?

That the two products are complements: they are used together, so when one gets cheaper people buy more of both and when it gets more expensive they buy less of both. The default case is exactly this — a 5 percent cut in the other product's price brings a 3 percent rise in this one's quantity, for −0.6. The more negative the figure, the tighter the pairing: something near zero is a weak complement, and a large negative number means you cannot really sell one without the other.

What does a positive cross-price elasticity mean?

That the two products are substitutes, so demand shifts from one to the other when prices diverge. A 20 percent rise in the other product's price that brings a 5 percent rise in this one's quantity gives +0.25. The larger the positive number, the closer the substitutes: near zero they are barely in competition at all, and a large positive figure means customers treat them as interchangeable.

Can cross-price elasticity be greater than 1?

Yes, and it is not bounded at 1 the way a product's own price elasticity often is described to be. It can be far larger: a product whose quantity halves when a complementary product's price rises 10 percent gives −5, and one whose demand disappears entirely on a 10 percent price rise gives −10. That is why this calculator reports it as a plain signed number rather than a percentage — there is no upper bound to express.

Why does this calculator use the initial value instead of the midpoint?

Because both conventions exist and they give different answers, so the one being used has to be stated. This page divides each change by the value before it changed: (206 − 200) ÷ 200, not ÷ 203. The midpoint or arc convention would divide by the average of the two values and give −0.5882 on the same inputs rather than −0.6. Neither is wrong; the initial-value form is used here because it stays well defined when a price falls all the way to zero, which the midpoint form also handles but the after-value form does not. When you quote a number, quote the definition with it.

Is cross-price elasticity the same at every price?

No, and this is the trap. It is measured at one pair of prices on one demand curve, and the relationship between two products changes as prices move — two goods that are substitutes at high prices can become complements at low ones, and the same pair measured in a different season or a different market will not give the same figure. A single value is a local reading, not a constant, so use it to answer the question at the prices you actually face rather than as a permanent property of the two products.

• Commodity and Food Elasticities — the cross-price elasticity tables and the sign convention this page follows — U.S. Department of Agriculture, Economic Research Service (United States)
• Food Demand Analysis — how cross-price elasticities are estimated and what the sign means for substitutes and complements — U.S. Department of Agriculture, Economic Research Service (United States)
• Commodity and Food Elasticities — Documentation: how own-price and cross-price elasticity estimates are recorded and queried — U.S. Department of Agriculture, Economic Research Service (United States)