Chapter 5, Problem 6PA

Suppose that your demand schedule for DVDs is as follows:

Price	Quantity Demanded (income = $10,000)	Quantity Demanded (income = $12,000)
$8	40 DVDs	50 DVDs
10	32	45
12	24	30
14	16	20
16	8	12

1. a. Use the midpoint method to calculate your price elasticity of demand as the price of DVDs increases from $8 to $10 if (i) your income is $10,000 and (ii) your income is $12,000.
2. b. Calculate your income elasticity of demand as your income increases from $10,000 to $12,000 if (i) the price is $12 and (ii) the price is $16.

To determine

Calculating the price elasticity of demand.

Explanation of Solution

1. (i) If the income is $10,000, then the price of pizza rises from $8 to $10, and the quantity demanded decreases from 40 to 32. By midpoint method, the price elasticity of demand is calculated as follows:

$\begin{array}{c}{\text{Price elasticity of demand}}_{\text{Income \$20,000}}=\frac{\frac{{\text{Quantity}}_{\text{Present}}-{\text{Quantity}}_{\text{Previous}}}{\frac{{\text{Quantity}}_{\text{Present}}+{\text{Quantity}}_{\text{Previous}}}{2}}}{\frac{{\text{Price}}_{\text{Present}}-{\text{Price}}_{\text{Previous}}}{\frac{{\text{Price}}_{\text{Present}}+{\text{Price}}_{\text{Previous}}}{2}}}\\ =\frac{\frac{32-40}{\left(\frac{32+40}{2}\right)}}{\frac{10-8}{\left(\frac{10+8}{2}\right)}}\\ =\frac{\frac{-8}{\left(36\right)}}{\frac{2}{\left(9\right)}}\\ =\frac{-0.2222}{0.2222}\\ =-1\end{array}$

The price elasticity of demand for pizza is -1.

1. (ii) If the income is $12,000, then the price of pizza rises from $8 to $10, and the quantity demanded decreases from 50 to 45. By midpoint method, the price elasticity of demand is calculated as follows:

$\begin{array}{c}{\text{Price elasticity of demand}}_{\text{Income \$24,000}}=\frac{\frac{{\text{Quantity}}_{\text{Present}}-{\text{Quantity}}_{\text{Previous}}}{\frac{{\text{Quantity}}_{\text{Present}}+{\text{Quantity}}_{\text{Previous}}}{2}}}{\frac{{\text{Price}}_{\text{Present}}-{\text{Price}}_{\text{Previous}}}{\frac{{\text{Price}}_{\text{Present}}+{\text{Price}}_{\text{Previous}}}{2}}}\\ =\frac{\frac{45-50}{\left(\frac{45+50}{2}\right)}}{\frac{(10-8)}{\left(\frac{(10+8)}{2}\right)}}\\ =\frac{\frac{-5}{47.5}}{\frac{2}{9}}\\ =\frac{-0.11}{0.22}\\ =-0.5\end{array}$

The price elasticity of demand for pizza is -0.5.

Economics Concept Introduction

Concept Introduction:

Price elasticity of demand: Price elasticity of demand refers to the percentage change in the demand for goods and services due to change occurred in the price level.

To determine

Calculating the income elasticity of demand.

Explanation of Solution

1. (i) If the price is $12 and an income increases from $10,000 to $24,000, then the quantity demanded increases from 24 to 30. By midpoint method, the income elasticity of demand is calculated as follows:

$\begin{array}{c}{\text{Income elasticity of demand}}_{\text{Price \$12}}=\frac{\frac{{\text{Quantity}}_{\text{Present}}-{\text{Quantity}}_{\text{Previous}}}{\frac{{\text{Quantity}}_{\text{Present}}+{\text{Quantity}}_{\text{Previous}}}{2}}}{\frac{{\text{Income}}_{\text{Present}}-{\text{Income}}_{\text{Previous}}}{\frac{{\text{Income}}_{\text{Present}}+{\text{Income}}_{\text{Previous}}}{2}}}\\ =\frac{\frac{30-24}{\left(\frac{30+24}{2}\right)}}{\frac{12,000-10,000}{\left(\frac{12,000+10,000}{2}\right)}}\\ =\frac{\frac{6}{27}}{\frac{2,000}{11,000}}\\ =\frac{0.2222}{0.1818}\\ =1.22\end{array}$

The income elasticity of demand for pizza is 1.22.

1. (ii) If the price is $12 and an income increases from $20,000 to $24,000, then the quantity demanded increases from 24 to 30. By midpoint method, the income elasticity of demand is calculated as follows:

$\begin{array}{c}{\text{Income elasticity of demand}}_{\text{Price \$16}}=\frac{\frac{{\text{Quantity}}_{\text{Present}}-{\text{Quantity}}_{\text{Previous}}}{\frac{{\text{Quantity}}_{\text{Present}}+{\text{Quantity}}_{\text{Previous}}}{2}}}{\frac{{\text{Income}}_{\text{Present}}-{\text{Income}}_{\text{Previous}}}{\frac{{\text{Income}}_{\text{Present}}+{\text{Income}}_{\text{Previous}}}{2}}}\\ =\frac{\frac{12-8}{\left(\frac{12+8}{2}\right)}}{\frac{12,000-10,000}{\left(\frac{12,000+10,000}{2}\right)}}\\ =\frac{\frac{4}{10}}{\frac{2,000}{11,000}}\\ =0.4\times \frac{11}{2}\\ =2.2\end{array}$

The income elasticity of demand for pizza is 2.22.

Economics Concept Introduction

Concept Introduction:

Income elasticity of demand: It measures how much quantity demanded of a good responds to the change in consumers’ income.