Size and Sign
Definition
Coefficient of elasticity: the single number produced by an elasticity formula, whose size measures the strength of the response and whose sign shows its direction.
Elasticity=% ΔQd% ΔX \text{Elasticity} = \dfrac{\%\,\Delta Q_d}{\%\,\Delta X} Elasticity=%ΔX%ΔQd- An elasticity compares the relative change in quantity demanded with the relative change in its cause, both expressed as percentages.
- The sign of the coefficient shows the direction of the relationship.
- The size of the coefficient, read as an absolute value, shows the strength of the response.
Key Idea
- The sign identifies the type of relationship or good, while the size shows how strong that relationship is.
- Because both parts of the ratio are percentages, elasticity has no units and can be compared across different goods.
Relative Percentage Changes
Elasticity=% ΔQd% ΔX \text{Elasticity} = \dfrac{\%\,\Delta Q_d}{\%\,\Delta X} Elasticity=%ΔX%ΔQd- Elasticity uses percentage changes so that goods measured in different units, such as litres of petrol and tonnes of steel, can still be compared on the same scale.
- A large absolute change can be a small percentage change of a large base, and a small absolute change can be a large percentage change of a small base, so only percentages give a fair comparison.
- Comparing the two percentage changes shows whether quantity demanded responds more or less than the change that caused it, which is exactly what the coefficient captures.
The Sign
- PED is normally negative, for example −0.5 for petrol, because a price rise causes quantity demanded to fall, so the two percentage changes carry opposite signs.
- YED is positive for a normal good, such as +1.5 for restaurant meals, because higher income raises demand, and negative for an inferior good, such as −0.3 for supermarket value bread, because higher income leads buyers to switch away.
- XED is positive for substitutes, such as +0.5 between Coca-Cola and Pepsi, because a rise in one price sends buyers to the other, and negative for complements, such as −0.8 between consoles and games, because a rise in one price cuts demand for the other.
Example
- An XED of +0.5 between two coffee brands has a positive sign marking substitutes, and a small size marking a weak link, so a price cut by one only mildly dents the other's sales.
- An XED of −0.8 between consoles and games has a negative sign marking complements, and a larger size marking a stronger link, so dearer consoles noticeably reduce game sales.
The Size
- A size of 0 means no response at all, so quantity demanded is perfectly unresponsive.
- A size between 0 and 1, such as 0.5, means quantity demanded responds less than proportionately (inelastic).
- A size of exactly 1 means quantity demanded responds in exact proportion to the change (unit elastic).
- A size above 1, such as 2, means quantity demanded responds more than proportionately (elastic).
- A size approaching infinity means quantity demanded responds without limit at a single price (perfectly elastic).
Why Both Matter
- The sign alone cannot tell you how big the effect is, so it is not enough on its own.
- The size alone cannot tell you the direction of the effect, so it too is incomplete.
- Read together, they let firms and governments predict both the scale and the direction of a change in demand: a supermarket setting a price, or a chancellor taxing petrol, needs both to forecast the effect on sales and revenue.
Exam technique
- State the sign first to fix the type of relationship or good.
- State the size next to judge how strong the response is.
- Apply both to the specific good named in the question rather than in the abstract.
Common Mistake
- Do not judge strength from the sign, because a negative sign can accompany either a strong or a weak response.
- Do not ignore the size once the sign is known, because a small coefficient means a weak link even when the sign is correct.
Self review
- Why does elasticity use percentage changes rather than absolute amounts?
- What does the sign of PED normally show?
- What does a positive XED indicate?
- What does a coefficient greater than one in size tell you?
- Why do you need both the sign and the size to interpret a coefficient?
