Cross-Price Elasticity

The calculation always takes the same shape: the percentage change in the quantity demanded of good X divided by the percentage change in the price of good Y, holding the price of X, income and other determinants constant. Over a discrete interval, candidates normally use the midpoint convention, dividing each change by the average of its beginning and ending values so the result does not depend on the direction of travel between the two observations.

When demand is expressed as an equation, the work is faster. In a linear demand function where quantity demanded of X depends on its own price, income and the price of a related good, the coefficient on the related good's price is multiplied by the ratio of that related price to the quantity demanded at the point of interest. The coefficient alone carries the sign and therefore the classification; the ratio scales it into an elasticity.

The sign is not an accounting artefact, it describes what buyers do when a relative price moves. If a price rise on good Y pushes consumers toward good X, the two compete for the same need and the ratio is positive; the goods are substitutes. If the price rise on good Y suppresses purchases of X because the two are consumed jointly, the ratio is negative and the goods are complements.

Magnitude then grades the strength of that link. Values far from zero indicate goods consumers treat as close alternatives or tightly bundled pairs; values close to zero indicate the pair sits in effectively separate markets. Note that the measure is not symmetric: the elasticity of X with respect to the price of Y need not equal the elasticity of Y with respect to the price of X, because the base quantities and the relative importance of each good in the budget differ.

Cross-price elasticity is the empirical backbone of market definition. Competition authorities assessing a merger ask whether a small but significant price increase by one firm would divert enough demand to a rival to make the increase unprofitable; that diversion is a cross-elasticity estimate. Weak substitution supports a narrow market definition and a stronger market-power finding.

It also governs razor-and-blade pricing, where a printer or console is sold thin because complementary cartridges or software carry the margin, and multi-product retail pricing, where a promotion on one line is judged on total basket revenue rather than the discounted item. In equity research it appears when analysts model demand spillovers between a company's own product lines, or the exposure of an airline's traffic to fuel-driven fare changes and to rail or coach alternatives on short-haul routes.

At Level I the required competencies are narrow but tested precisely. Calculate the elasticity from two price-quantity observations or from a stated demand function, classify the pair from the resulting sign, and rank the strength of relationships from competing values. Keep the three elasticities distinct: own-price elasticity relates quantity to the good's own price and is negative for a normal downward-sloping demand curve; income elasticity relates quantity to income and separates normal from inferior goods; cross-price elasticity relates quantity to another good's price.

Questions frequently supply a demand equation with three coefficients and ask which one answers the question posed. Identify the variable named in the question before touching a calculator, then substitute the given price and quantity levels rather than assuming the coefficient is itself the elasticity.