Cross-Price Elasticity

How cross-price elasticity works

Suppose margarine prices rise 10%. Some shoppers stop buying margarine and switch to butter; butter sales jump 7%. The cross-price elasticity of butter demand with respect to margarine price is +7% ÷ +10% = +0.7. The positive sign says the two goods are substitutes — they compete for the same need, and a higher price on one drives buyers to the other.

Now suppose gasoline prices jump 30%. Drivers cut their miles and put off buying new SUVs; SUV sales fall 9%. Cross-price elasticity of SUVs with respect to gasoline is −9% ÷ +30% = −0.3. The negative sign says the two are complements — a tank of gas and a vehicle that drinks it are consumed together, and pricing one out of reach drags the other down too.

The general formula is:

XED of X with respect to Y  =  (%ΔQx) ÷ (%ΔPy)
                            =  (ΔQx / Qx) ÷ (ΔPy / Py)

Three regions classify the relationship:

     XED          Relationship                        Real example
   ─────────  ──────────────────────  ─────────────────────────────────────
   XED > 0    Substitutes              Coke ↔ Pepsi, butter ↔ margarine
   XED ≈ 0    Independent / unrelated  Bread ↔ smartphones
   XED < 0    Complements              Cars ↔ gasoline, printers ↔ ink

Unlike own-price elasticity, cross-price elasticity is not always negative — and the sign is the whole point.

Worked example: butter and margarine

USDA spreads-tracking shows that in a typical year, U.S. consumers buy about 1.05 billion pounds of butter at $4.20/lb and 1.40 billion pounds of margarine at $3.10/lb. A poor canola harvest pushes margarine to $3.41/lb (a 10% rise). Suppose the empirical cross-price elasticity of butter with respect to margarine is +0.7 — a value squarely inside the range reported by Tonsor and Mintert's USDA-funded panel studies.

Predicted butter response:

%ΔQ_butter = XED × %ΔP_margarine
            = (+0.7) × (+10%)
            = +7%

New butter quantity = 1.05 billion × 1.07 = 1.124 billion lbs

Now suppose butter's own-price elasticity is −0.65. The first-order partial-equilibrium calculation: 75 million extra pounds of butter demanded at $4.20/lb is $315 million in extra revenue, before any butter-price adjustment. Margarine producers, meanwhile, lose roughly 10% × |PED_margarine| ≈ 14% of their volume — about 196 million lbs. The total spread market shrinks slightly while butter takes share.

The same calculation runs in reverse for complements. Suppose Apple cuts iPhone prices 5%. With a cross-price elasticity of AirPods to iPhone price of −0.6, AirPod sales rise about 3%. A pricing decision on the platform reverberates through every accessory.

Real-world cross-price elasticities

The numbers below come from a mix of supermarket scanner data (Nielsen, IRI), the U.S. Department of Justice's antitrust filings, and academic surveys. Magnitudes depend heavily on how narrowly each good is defined.

Good X	Good Y	XED of X w.r.t. P_Y	Type
Coca-Cola	Pepsi	+0.61	Strong substitutes
Butter	Margarine	+0.70	Substitutes
Beef	Pork	+0.40	Substitutes
Chicken	Beef	+0.16	Mild substitutes
Bus fares	Gasoline	+0.34	Modal substitutes
Bread	Smartphones	≈ 0	Unrelated
SUVs	Gasoline	−0.28	Complements
AirPods	iPhone	−0.60 (estimate)	Strong complements
Movie tickets	Popcorn	−0.45	Complements
Tennis racquets	Tennis balls	−1.20	Strong complements

Two patterns: brand-level substitution (Coke vs Pepsi) is far stronger than category-level substitution (chicken vs beef), and platform-accessory pairs (consoles, phones, racquets) consistently produce the largest negative numbers.

Graphical intuition: shifting demand curves

Cross-price elasticity is captured by a shift of one good's demand curve when the other good's price changes. For substitutes, the shift is outward; for complements, inward.

Substitutes (butter when margarine ↑)         Complements (SUVs when gas ↑)

P ↑                                            P ↑
  | D_old   D_new                                | D_new   D_old
  |  \       \                                    |  \       \
  |   \       \  ← shift right                    |   \       \  ← shift left
  |    \       \                                  |    \       \
  +─────\───────\──→ Q                            +─────\───────\──→ Q

Quantity rises at every price             Quantity falls at every price

The size of the shift, holding price constant, is exactly the cross-price elasticity multiplied by the percent change in the related good's price. A 10% jump in margarine price shifts butter's demand curve out by about 7% if XED = +0.7. A 30% rise in gasoline shifts SUV demand inward by about 9% if XED = −0.3.

Variants and decompositions

	Marshallian (uncompensated)	Hicksian (compensated)
What's held constant?	Income	Utility
Captures	Substitution + income effects	Substitution effect only
Sign	Can flip for inferior goods	Always non-negative for net substitutes
Used for	Empirical demand systems	Welfare analysis
Symmetry	Generally not symmetric	Symmetric (Slutsky)

The Marshallian (uncompensated) cross-price elasticity is what scanner-data regressions actually recover; it lumps the substitution effect and the income effect together. The Hicksian (compensated) version isolates the substitution effect — what happens when relative prices move but the consumer is given enough income to stay on the same indifference curve. The Slutsky decomposition makes the link explicit:

XED_Marshallian  =  XED_Hicksian  −  (share of income spent on Y) × YED

For small-share goods (salt, postage), the income-effect correction is negligible and the two definitions nearly coincide. For housing or food in poor countries, the two can differ substantially — and welfare calculations need the Hicksian version.

A second variant is the SSNIP test, the workhorse of antitrust market definition: a candidate market is "small" if a hypothetical 5% non-transitory price increase by a single firm would lose enough customers to neighboring goods to be unprofitable. High cross-price elasticities to those neighbors mean the market is too narrow and must be expanded. Low cross-elasticities mean the market is well-defined and the firm has potential power.

What makes cross-elasticity large

• Functional similarity. Goods that satisfy the same need (Coke and Pepsi, beef and chicken, bus and rideshare) substitute strongly.
• Tight bundling. Goods that are consumed together in roughly fixed proportions (printers and ink, razors and blades, consoles and games) complement strongly.
• How narrowly each good is defined. The narrower the definition, the larger |XED|. "Tropicana orange juice" has higher cross-elasticity to "Florida's Natural" than "orange juice" has to "apple juice".
• Income share of the related good. If good Y takes a tiny share of income, even a big move in P_Y barely matters for X.
• Time horizon. Like own-price elasticity, cross-elasticities tend to grow over time as consumers and firms adjust.

Common pitfalls

• Assuming symmetry. XED of butter w.r.t. margarine price is not generally the same as XED of margarine w.r.t. butter price. The income share of each good and the slopes of each demand curve differ. Only Hicksian elasticities are guaranteed symmetric.
• Using a single number for all price ranges. Cross-elasticity, like own-price elasticity, is local. The substitutability of butter for margarine when both cost $3-4/lb may not hold when margarine collapses to $1/lb.
• Confusing "complement" with "frequently bought together". Two products that often appear in the same shopping cart (eggs and milk) are not necessarily economic complements. The test is whether a price change in one moves the quantity of the other in the opposite direction.
• Forgetting the income effect. If a price change is large and the good is a big share of income, the Marshallian XED can mask substitution behavior. Welfare analyses require the Hicksian decomposition.
• Picking the wrong sign for inferior goods. When the related good is inferior (margarine for low-income shoppers, say) and the price change is large, the income and substitution effects can pull in opposite directions and produce a counter-intuitive sign on the Marshallian XED.
• Confusing cross-elasticity with the cross-derivative ∂Q/∂P alone. Cross-elasticity is unitless because it scales by the levels Q and P. The raw partial derivative carries units (e.g., gallons per dollar) and is not comparable across markets.