# Income elasticity

The income elasticity ( ) of the demand indicates how much the demand for a good changes relatively when the income of a household changes (relatively). ${\ displaystyle \ eta _ {i}}$ ${\ displaystyle i}$ It is defined as the percentage change in the amount demanded as a percentage change in income.

${\ displaystyle {\ frac {\ text {Percentage change in demand}} {\ text {Percentage change in income}}}}$ Example: If the demand for a certain good increases by 12% as a result of an increase in income of 10%, the income elasticity of this good is 1.2.

## Formal definition and classification

Formal: Be the Marshallian demand for a commodity as a function of the prices of all goods and household income . Then the following applies to the income elasticity : ${\ displaystyle x_ {i} (\ mathbf {p}, y) = x_ {i} (p_ {1}, \ ldots, p_ {n}; y)}$ ${\ displaystyle i}$ ${\ displaystyle n}$ ${\ displaystyle y}$ ${\ displaystyle \ eta _ {i}}$ ${\ displaystyle \ eta _ {i} \ equiv {\ frac {\ partial x_ {i} (\ mathbf {p}, y)} {\ partial y}} {\ frac {y} {x_ {i} (\ mathbf {p}, y)}}}$ If the income elasticities for a good are weighted with the share that consumption of the good has in total household income, and these weighted income elasticities are added up over all goods, they must add up to 1. Hence: ${\ displaystyle i}$ ${\ displaystyle \ sum _ {i = 1} ^ {n} \ left [{\ frac {p_ {i} x_ {i} (\ mathbf {p}, y)} {y}} \ cdot \ eta _ { i} \ right] = 1}$ ## properties

If the income elasticity is positive, one speaks of a normal good ; if it is negative, of an inferior good . On the basis of this, a distinction is sometimes made between different types of normal goods: A good whose demand increases disproportionately to income with an increase in income and which thus has an income elasticity of more than 1 is called a luxury good ; If demand increases disproportionately to income and if the income elasticity is above 0 and below 1, one speaks of a necessary good (see figure).

Other terms are also used for the different elasticity ranges; for this, reference is made to the article Inferiores Gut # Different Definitions .

## literature

• Geoffrey A. Jehle and Philip J. Reny: Advanced Microeconomic Theory. 3rd ed. Financial Times / Prentice Hall, Harlow 2011, ISBN 978-0-273-73191-7 .