Cross Elasticity of Demand

Cross elasticity of demand (also called cross-price elasticity) measures how the quantity demanded of one good (X) responds to a change in the price of another good (Y). It helps identify whether goods are substitutes, complements, or unrelated and by how much consumers switch between them.

Key takeaways

• Positive cross elasticity: goods are substitutes (demand for X rises when price of Y rises).
• Negative cross elasticity: goods are complements (demand for X falls when price of Y rises).
• Near zero: goods are largely unrelated.
• Firms use cross elasticity to set prices, design bundles, and plan loss-leader strategies (e.g., cheap printers to sell ink).

Definition and formula

Exy = (% change in quantity demanded of X) / (% change in price of Y)

In algebraic form:
Exy = (ΔQx / Qx) ÷ (ΔPy / Py) = (ΔQx / ΔPy) × (Py / Qx)

A common practical way to compute percentage changes that avoids asymmetry is the midpoint (arc) method:
%ΔX = (X2 − X1) / ((X1 + X2) / 2)

How to calculate (step-by-step)

1. Choose the initial and final values for Qx (quantity of X) and Py (price of Y).
2. Compute percentage change in Qx using the midpoint formula: (Qx2 − Qx1) / ((Qx1 + Qx2)/2).
3. Compute percentage change in Py using the midpoint formula: (Py2 − Py1) / ((Py1 + Py2)/2).
4. Divide %ΔQx by %ΔPy to get Exy.

Example:
– Coffee price rises from $4 to $5; tea quantity rises from 100 to 120.
– %ΔPy = (5 − 4) / ((5 + 4)/2) = 1 / 4.5 = 0.2222 (22.22%)
– %ΔQx = (120 − 100) / ((120 + 100)/2) = 20 / 110 = 0.1818 (18.18%)
– Exy = 0.1818 / 0.2222 ≈ 0.82 → positive, so tea and coffee are substitutes (moderately strong).

Interpretation

• Exy > 0: substitutes. Larger positive values mean stronger substitution (e.g., different brands of the same product).
• Exy < 0: complements. More negative values indicate stronger complementarity (e.g., printers and ink).
• Exy ≈ 0: unrelated goods; price changes in Y have little effect on demand for X.

Magnitude matters: values near zero indicate weak relationships; large absolute values indicate strong substitution or complementarity.

Examples

• Substitutes: If one restaurant raises burrito prices, demand at a competing restaurant typically increases (Exy > 0). Different brands of toothpaste or types of milk often show positive cross elasticity.
• Complements: If burger prices rise and fewer burgers are sold, demand for fries may fall (Exy < 0). Printers and ink cartridges are classic complements—selling printers cheaply can boost ink sales.

Uses for businesses and policymakers

• Pricing strategy: assess how a price change will shift demand between products.
• Product positioning: determine whether to differentiate or align products with competitors.
• Bundling and promotions: identify complementary products for packages or loss-leader tactics.
• Antitrust and market analysis: evaluate competitive relationships and market power.

How this differs from related elasticities

• Price elasticity of demand: measures how quantity demanded of a single good responds to changes in its own price.
• Cross elasticity of supply: measures how quantity supplied of one good responds to price changes of another good (supply-side analogue).

Simple explanation

If changing the price of item Y makes people buy more of item X, X and Y are substitutes. If changing the price of Y makes people buy less of X, they are complements. If nothing changes, they’re unrelated.

Conclusion

Cross elasticity of demand quantifies the interdependence between goods. By calculating Exy and interpreting its sign and magnitude, businesses and analysts can make informed decisions about pricing, product strategy, and competitive responses.