# Chapter – 2: Elasticity of Demand – I

## (b) Explain the relationship between price elasticity of demand, income elasticity of demand and elasticity of substitution.

Ans. (a) Price Elasticity of Demand. Price elasticity of demand measures the degree of responsiveness of quantity demanded to change in the price of a commodity. It is denoted as $$e$$.

Elasticity of demand $$\left( e \right)$$ is defined as the ratio of percentage change in quantity demanded (Q) to percentage change $$\left( \Delta \right)$$ in price of the commodity $$\left( P \right)$$.

$e = \frac{\frac{\Delta Q}{Q}\times 100}{\frac{\Delta P}{P}\times 100}$

So, $e = \frac{\Delta Q}{\Delta P}\times \frac{P}{Q}$

where P and Q refer to original price and quantity respectively

$\Delta Q = Q_{n} – Q_{n-1}$

$\Delta P_{n} = P_{n-1} – P_{n}$

The value of $$e$$ varies from 0 to $$\infty$$. Normally $$e$$ is – ve due to the inverse relationship between price and quantity.

Since $$\frac{\Delta P}{\Delta Q}$$ is slop of demand curve, then given P and Q price elasticity is inversely related to slope of the demand curve $$\left ( \frac{\Delta Q}{\Delta P}=\frac{1}{\frac{\Delta P}{\Delta Q}}\right ).$$

So, for different values of $$e$$ the slope/shape of the demand curve will be different as given in the diagrams below.

When $$e = 0$$, demand is called perfectly inelastic. The demand curve is a vertical line with slope = $$\infty$$ as shown in Diagram 1.

When $$e < 1$$ and greater than 0, demand is inelastic and the demand curve is steeper as shown in Diagram 2.

When $$e = 1$$, demand is unitary elastic and the demand curve is a rectangular hyperbola as shown in Diagram 3.

When $$e > 1$$, demand elastic and demand curve is flatter as shown in Diagram 4.

When $$e = \infty ,$$ demand is perfectly elastic and demand curve is horizontal with zero slope as shown in Diagram 5.

Income Elasticity of Demand: Income elasticity of demand measures the degree of responsiveness of quantity demanded to change in income of the consumer. It is denoted as $$e_{I}$$.

$$e_{I}$$ is defined as the ratio of percentage change in quantity demanded (Q) to percentage change in income (I).

$e_{I} =\frac{\frac{\Delta Q}{Q}\times 100}{\frac{\Delta I}{I}\times 100}$

$e_{I} = \frac{\Delta Q}{\Delta I}\times \frac{I}{Q}$

Where, $$\Delta$$ = Change in , Q = Quantity (original) and I = Income (Original)

The value of $$e_{1}$$ differs for different types of goods consumed.

For inferior goods $$e_{1} < 0.$$

For necessities $$e_{1} = 0$$

For normal goods $$0 < e_{1} < 1$$

For luxury goods $$e_{1} > 1$$

The relationship between income and quantity demanded is given by the Engle curve (E). The slopes/shapes of the Engel curve as per the different values of income elasticities are given in the diagram.

In the diagram,  $$E_{1}$$ is the Engel curve for inferior goods because inferior goods have negative income elasticities so the Engel curve is downward sloping.  $$E_{2}$$ is the vertical Engel curve for necessities for whom $$e_{1} = 0. E_{3}$$ is the upward sloping steeper Engel curve for normal goods having $$e_{1} < 1. E_{4}$$ is the upward sloping but flatter Engel curve for luxury goods having $$e_{1} < 1.$$

Cross Elasticity of Demand: It is the degree of responsiveness of quantity demanded of one product with respect to change in the price of another relative product ceteris paribus. Hence, the concept of cross elasticity of demand explains the relationship between the price of one commodity and the quantity demanded of another commodity.

The cross elasticity of demand is measured by:

$$\,\,\,\,\,\,{e_{XY}} = \frac{{{\rm{Percentage}}{\mkern 1mu} {\rm{Change}}{\mkern 1mu} {\rm{in}}{\mkern 1mu} {\rm{Quantity}}{\mkern 1mu} {\rm{Demanded}}{\mkern 1mu} {\rm{of}}{\mkern 1mu} {\rm{Commodity}}{\mkern 1mu} {\rm{X}}}}{{{\rm{Percentage}}{\mkern 1mu} {\rm{Change}}{\mkern 1mu} {\rm{in}}{\mkern 1mu} {\rm{Price}}{\mkern 1mu} {\mkern 1mu} {\rm{of}}{\mkern 1mu} {\rm{related}}{\mkern 1mu} {\rm{Commodity}}{\mkern 1mu} Y}}$$

$$or,\,{\mkern 1mu} {e_{XY}} = \frac{{\Delta {Q_X}}}{{{Q_X}}} \div \frac{{\Delta {P_Y}}}{{{P_Y}}}$$

$$or,\,{\mkern 1mu} {e_{XY}} = \frac{{\Delta {Q_X}}}{{\Delta {P_Y}}} \times \frac{{{P_Y}}}{{{Q_X}}}$$

$$or,\,{\mkern 1mu} {e_{XY}} = \frac{{\partial {Q_X}}}{{\partial {P_Y}}} \times \frac{{{P_Y}}}{{{Q_X}}}$$

where,

$${e_{XY}}$$ = Cross elasticity of demand of two related commodities X & Y

$${{Q_X}}$$ = Quantity demanded of commodity X

$${{P_Y}}$$ = Price of another related commodity Y

$$\Delta$$ =  “delta” means “change in”

Cross elasticity of demand between two commodities helps us to specify the nature of the relation between two goods. As two commodities are termed as substitutes if a fall in the price of one (say Y) causes a decrease in quantity demanded of other (say X) or vice versa.

Two commodities are complementary (jointly demanded) if a fall in the price of one (say Y) results in an increase in the demand of another (say X). Two commodities will be independent when a change in the price of one (say Y) does not affect the quantity demanded of the other (say X).

Utility of Cross Elasticity of Demand: Measure of cross elasticity of demand prove useful in defining whether firms producing similar or different products are in competition with each other. For example, men’s wear and women’s wear have low cross elasticity. A manufacturer of men’s wear is not in competition with a manufacturer of women’s wear, whereas Coke and Pepsin have a high cross elasticity of demand and hence products of Coke and Pepsi are in strict competition with each other.

(b) The relationship between three elasticities can be mathematically shown. The relationship between their elasticities can be given for a single consumer faced with two goods – X and Y in a single equation. where, for good X

$$e_{p}$$ = Price elasticity of demand

$$e_{I}$$ = Income elasticity of demand

$$e_{s}$$ = Elasticity of substitution

and kX = The proportion of the income of the consumer spent on good X Thun

${e_p} = kX \times {e_I} + \left( {1 – kX} \right){e_{es}}$

This equation holds in all situations. Hence, with this equation, if one knows any two elasticities, the third one can be found out.