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### Course: Microeconomics>Unit 3

Lesson 3: Income elasticity of demand and cross-price elasticity of demand

# Lesson Overview - Cross Price Elasticity and Income Elasticity of Demand

## Summary

In a previous lesson we learned about price elasticity of demand, but there are many other types of elasticity that measure how agents respond to variables other than the change in a good's price. Two of these are Cross Price Elasticity of Demand and Income Elasticity of Demand. The sign of each of these conveys important information about the good.
As we learned previously, inferior goods have an inverse relationship between income and demand, which results in a negative income elasticity of demand. On the other hand, normal goods have a positive relationship between income and demand which is reflected in a positive income elasticity of demand.
We determine whether goods are complements or substitutes based on cross price elasticity - if the cross price elasticity is positive the goods are substitutes, and if the cross price elasticity are negative the goods are complements.

## Cross price elasticity of demand

Cross price elasticity of demand $\left(XED\right)$ measures the how a change in the price of one good will affect the quantity demanded of another good. The formula for XED is:
$XED=\frac{\mathrm{%}\mathrm{\Delta }{Q}_{D}\phantom{\rule{0.5em}{0ex}}of\phantom{\rule{0.5em}{0ex}}Good\phantom{\rule{0.5em}{0ex}}A}{\mathrm{%}\mathrm{\Delta }P\phantom{\rule{0.5em}{0ex}}of\phantom{\rule{0.5em}{0ex}}Good\phantom{\rule{0.5em}{0ex}}B}$
Unlike the always negative price elasticity of demand, the value of the cross price elasticity can be either negative or positive, and the sign provides important information about whether the goods are complements and substitutes. The magnitude of the elasticity tells the degree to which the goods are complementary or substitutable.
We can interpret the cross-price elasticity of demand as summarized in the table below:
If the sign of $XED$ is...and the elasticity isthe goods are
negativeperfectly elastic $-\mathrm{\infty }$perfect complements that must be consumed in fixed proportions
negativeelastichighly complementary goods
negativeinelasticsomewhat complementary goods
$0$$0$unrelated goods (neither complements nor substitutes)
positiveinelasticsomewhat substitutable
positiveelasticvery substitutable
positiveperfectly elastic $\left(\mathrm{\infty }\right)$perfect substitutes
We can visualize these along a number line:

# Income elasticity of demand

Income Elasticity of Demand $\left(YED\right)$ measures how a change in buyers income will lead to a change in the demand for a good. The formula for $YED$ is:
$YED=\frac{\mathrm{%}\mathrm{\Delta }{Q}_{D}}{\mathrm{%}\mathrm{\Delta }Y}$
Where $Y$ is the income consumers of a good.
We can interpret the income elasticity of demand as summarized in the table below:
If the sign of $YED$ is...and the elasticity isthe goods are
negativeelastic or inelasticinferior good
$0$perfectly inelasaticabsolute necessity
positiveinelasticnormal necessity
positiveelasticnormal luxury
We can visualize these along a number line:

## Key Terms

TermDefinition
Cross price elasticity of demandAlso written as $XED$, measures the responsiveness of consumers purchases of one good to a change in the price of a different good (a substitute or a complement).
SubstitutesGoods that can be consumed instead of one another. The $XED$ coefficient for substitutes is always positive
ComplementsGoods that are usually consumed together. The $XED$ coefficient for two complements will always be negative
Income elasticity of demandAlso written as $YED$, measures the responsiveness of consumers to a change in their incomes
Normal goodSometimes called a superior good, A good with a direct relationship between income and demand. $YED$ is always positive for a normal good
Inferior goodA good with an inverse relationship between income and demand. $YED$ is always negative for an inferior good
LuxuryA normal good with a relatively elastic $YED$
NecessityA normal good with a relatively inelastic $YED$

## Key equations - calculating $XED$‍  and $YED$‍

The formula for calculating both XED and YED is essentially the same as that for calculating the price elasticity of demand. The only difference is what goes on the bottom of the equation.
As with price elasticity of demand, if percentage changes in income, the price of related goods and quantity of the good in question are not given, and we know the initial prices, they can be calculated using the formulas below:
$XED=\frac{\left(\frac{{Q}_{2A}-{Q}_{1A}}{{Q}_{1A}}\right)}{\left(\frac{{P}_{2B}-{P}_{1B}}{{P}_{1B}}\right)}$
Where people buy ${Q}_{1A}$ units of good A when the price of good B is ${P}_{1B}$, and people buy ${Q}_{2A}$ units of good A when the price of good B is ${P}_{2B}$.
Income elasticity can be calculated as follows:
$YED=\frac{\left(\frac{{Q}_{2}-{Q}_{1}}{{Q}_{1}}\right)}{\left(\frac{{Y}_{2}-{Y}_{1}}{{Y}_{1}}\right)}$
Where ${Y}_{1}$ is the consumer's original income and ${Y}_{2}$ is the consumers new income.

## Things to consider and common mistakes

• The signs in your calculations of $XED$ and $YED$ are particularly important. If the price and quantity change in opposite directions when calculating $XED$ then the goods must be complements and the coefficient will be negative. If income and quantity change in opposite directions when calculating $YED$ then the good must be inferior and the coefficient will be negative. A positive $XED$ coefficient means goods are substitutes and a positive $YED$ coefficient means the good is normal.
• The absolute value of $YED$ and $XED$ tell you about the elasticity.
• A note on terminology: When describing the price elasticity of demand for a good it is simple enough to say "demand is elastic" or "demand is inelastic". But when describing the cross and income elasticities of demand special attention should be paid to your use of the terminology. For $XED$ you must specify that demand is cross-price elastic or inelastic with respect to another good. For $YED$ you must specify that demand for a good is either income elastic or income inelastic.
• It can be tempting to make normative judgments about the qualities of a good, but it is important to remember that these are objective measures. What makes a good normal or inferior, or two goods complements or substitutes, depends on how we respond to these conditions changing, not any assumption we make about the good beforehand.

## Want to join the conversation?

• Hello, could you tell me when I should use the midpoint method to calculate the percentage change? I am a little confused, because in some videos you use the midpoint method, and in others the regular one.