# Income elasticity of demand

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In economics, the income elasticity of demand is the responsivenesses of the quantity demanded for a good to a change in consumer income. It is measured as the ratio of the percentage change in quantity demanded to the percentage change in income. If a 10% increase in Mr. Ruskin Smith's income causes him to buy 20% more bacon, Smith's income elasticity of demand for bacon is 20%/10% = 2.

## Interpretation

• A negative income elasticity of demand is associated with inferior goods; an increase in income will lead to a fall in the demand and may lead to changes to more luxurious substitutes.
• A positive income elasticity of demand is associated with normal goods; an increase in income will lead to a rise in demand. If income elasticity of demand of a commodity is less than 1, it is a necessity good. If the elasticity of demand is greater than 1, it is a luxury good or a superior good.
• A zero income elasticity of demand occurs when an increase in income is not associated with a change in the demand of a good.

Income elasticity of demand can be used as an indicator of future consumption patterns and as a guide to firms' investment decisions. For example, the "selected income elasticities" below suggest that an increasing portion of consumers' budgets will be devoted to purchasing automobiles and restaurant meals and a smaller share to tobacco and margarine.

Income elasticities are closely related to the population income distribution and the fraction of the product's sales attributable to buyers from different income brackets. Specifically, when a buyer in a certain income bracket experiences an income increase, their purchase of a product changes to match that of individuals in their new income bracket. If the income share elasticity is defined as the negative percentage change in individuals given a percentage increase in income bracken the income-elasticity, after some computation, becomes the expected value of the income-share elasticity with respect to the income distribution of purchasers of the product. When the income distribution is described by a gamma distribution, the income elasticity is proportional to the percentage difference between the average income of the product's buyers and the average income of the population.

## Mathematical definition

$\epsilon _{d}={\frac {\%\ {\mbox{change in quantity demanded}}}{\%\ {\mbox{change in income}}}}$ More formally, the income elasticity of demand, $\ \epsilon _{d}$ , for a given Marshallian demand function $Q(I,{\vec {P}}),$ with arguments income and a vector of prices of all goods, for arbitrarily small changes in price and quantity of a good is

$\epsilon _{d}={\frac {\partial Q}{\partial I}}{\frac {I}{Q}}$ This can be rewritten in the form

$\epsilon _{d}={\frac {d\ln Q}{d\ln I}}$ For discrete changes the elasticity is

$\epsilon _{d}={\Delta Q \over \Delta I}\times {(I_{1}+I_{2})/2 \over (Q_{1}+Q_{2})/2}={\Delta Q \over \Delta I}\times {I_{1}+I_{2} \over Q_{1}+Q_{2}},$ where subscripts 1 and 2 refer respectively to values before and after the change.

Necessity goods have an income elasticity of demand between zero and one: expenditure on these goods increases with income, but not as fast as income does, so the proportion of expenditure on these goods falls as income rises. This observation for food is known as Engel's law.

## Types of income elasticity of demand

There are five possible income demand curves:[citation needed]

• High-income elasticity of demand:

In this case, an increase in income is accompanied by a relatively larger increase in quantity demanded.

• Unitary income elasticity of demand:

In this case, an increase in income is accompanied by the same proportionate increase in quantity demanded.

• Low-income elasticity of demand:

In this case, an increase in income is accompanied by a less than proportionate increase in quantity demanded.

• Zero income elasticity of demand:

In this case, the quantity bought is constant regardless of changes in income.

• Negative income elasticity of demand:

In this case, an increase in income is accompanied by a decrease in quantity demanded.

## Selected income elasticities

• Automobiles 2.98
• Books 1.44
• A person's own life (also called "value of statistical life") 0.50 to 0.60
• Restaurant meals 1.40
• Tobacco 0.42
• Margarine −0.20
• Public transportation −0.36
• Water demand 0.15 

Income elasticities are notably stable over time and across countries due to the law of one price.