Other elasticities identify relationships

The sign now carries economic meaning.

Cross-price elasticity divides the percentage change in demand for good X by the percentage change in price of good Y. A positive value indicates substitutes. A negative value indicates complements. Income elasticity is positive for normal goods and negative for inferior goods. Among normal goods, a value above one is commonly associated with a luxury and a value between zero and one with a necessity.

Here the sign carries relationship information because the denominator is not the focal good’s own price. If the price of tea rises and demand for coffee rises, both percentage changes have the same sign, producing positive cross-price elasticity and indicating substitutes. If the price of printers rises and demand for ink falls, the signs differ, producing a negative value and indicating complements. A value near zero suggests a weak relationship over the measured range.

Income elasticity uses income in the denominator. A 10 percent income rise that produces a 6 percent quantity increase gives E_I=0.6: the good is normal and income-inelastic, often described as a necessity in the basic classification. If quantity falls 4 percent, E_I=-0.4, indicating an inferior good. “Inferior” describes direction, not the size of the response or physical quality.

Price elasticity of supply is positive in the usual case. Supply becomes more elastic when firms have time, spare capacity, mobile inputs, inventories, or easy storage. A vertical supply curve is perfectly inelastic. A horizontal curve is perfectly elastic.

Supply elasticity is E_s=(%Δ Q_s)/(%Δ P). The sign is normally positive because quantity supplied moves with price. A fixed number of seats in a stadium tonight has nearly perfectly inelastic short-run supply. Over a longer horizon, organizers can schedule more events or expand capacity, making supply more elastic. Time does not guarantee a large response, but it opens adjustment margins.

Storage can shift response across time. A seller with inventories can release more units when price rises, increasing short-run supply elasticity. Fresh produce that spoils quickly offers less timing flexibility. Spare capacity, ease of hiring, and input mobility work similarly by determining how costly a quantity adjustment is.

Do not use the demand sign rule everywhere

When ride-share price rises 8 percent, demand for bus trips rises 4 percent. Cross-price elasticity is +0.5, so the services are substitutes. Taking an absolute value and calling the result “inelastic demand” would discard the sign’s relationship meaning and use an own-price classification for the wrong elasticity.

Always name the numerator and denominator in words before calculating. “Quantity of X over price of Y” is cross-price elasticity. “quantity of X over income” is income elasticity. The numbers alone cannot tell which interpretation belongs to the ratio.

When a question reports only a sign, do not invent a magnitude classification. A positive cross-price elasticity establishes substitutes but may be small or large. A negative income elasticity establishes an inferior good, but its absolute value still measures response strength. Direction and magnitude are separate pieces of evidence.