Chapter 20, Problem 2E

Plot the price and quantity data given in the demand schedule of exercise 1. Put price on the vertical axis and quantity on the horizontal axis. Indicate the price elasticity value at each quantity demanded. Explain why the elasticity value gets smaller as you move down the demand curve.

To determine

Reason behind the elasticity value getting smaller with the downward movement on demand curve.

Explanation of Solution

The required graph is below:

1. The arc or mid- point elasticity is calculated as follows:

Elasticity of Demand (E_d)

  $E_d=\frac{(Q_2-Q_1)}{(P_2-P_1)}\times \frac{\left(P_2+P_1\right)}{(Q_2+Q_1)}$

Where E_d = coefficient of elasticity

P₁ = Initial Price

P₂ = New Price

Q₁ = Initial Quantity

Q₂ = New Quantity

At price $5 per unit, quantity demanded = 100 units

At price $10 per unit, quantity demanded = 80 units

Thus, at the first point where price = $5 per unit, quantity demanded = 100 units

  $E_d=\frac{(Q_2-Q_1)}{(P_2-P_1)}\times \frac{\left(P_2+P_1\right)}{(Q_2+Q_1)}$

P₁= $10 P₂ = $5

Q₁ = 80 Q₂ = 100

  $\begin{array}{l}{E}_d=\frac{(80-100)}{(10-5)}\times \frac{\left(10+5\right)}{(80+100)}\\ E_d=\frac{-20}{5}\times \frac{15}{180}\\ =-0.33\end{array}$

Price elasticity of demand = -0.33

At price $10 per unit, the quantity demanded is 80 units, elasticity of demand is calculated as follows:

  $E_d=\frac{(Q_2-Q_1)}{(P_2-P_1)}\times \frac{\left(P_2+P_1\right)}{(Q_2+Q_1)}$

P₁= $10 P₂ = $15

Q₁ = 80 Q₂ = 60

  $\begin{array}{l}{E}_d=\frac{(60-80)}{(15-10)}\times \frac{\left(15+10\right)}{(80+60)}\\ E_d=\frac{-20}{5}\times \frac{25}{140}\\ =-0.71\end{array}$

At price $10 per unit, price elasticity of demand is -0.71.

At price $15 per unit, quantity demanded is 60 units, elasticity is calculated as follows:

  $E_d=\frac{(Q_2-Q_1)}{(P_2-P_1)}\times \frac{\left(P_2+P_1\right)}{(Q_2+Q_1)}$

P₁ = $15    P₂= $20

Q₁ = 60   Q₂= 40

  $\begin{array}{l}{E}_d=\frac{(40-60)}{(20-15)}\times \frac{\left(15+20\right)}{(60+40)}\\ E_d=\frac{-20}{5}\times \frac{35}{100}\\ =-1.4\end{array}$

Thus, at price $15 per unit, price elasticity of demand is -1.4

At price $20 per unit, quantity demanded is 40 units, sp, elasticity is calculated as follows:

  $E_d=\frac{(Q_2-Q_1)}{(P_2-P_1)}\times \frac{\left(P_2+P_1\right)}{(Q_2+Q_1)}$

P₁ = $25   P₂= $25

Q₁ = 40   Q₂= 20

  $\begin{array}{l}{E}_d=\frac{(20-40)}{(25-20)}\times \frac{\left(20+25\right)}{(40+20)}\\ E_d=\frac{-20}{5}\times \frac{45}{60}\\ =-3\end{array}$

Thus, at price $20 per unit, price elasticity of demand is -3.

At price $25 per unit, quantity demanded is 20 units, so, elasticity is calculated as follows:

  $E_d=\frac{(Q_2-Q_1)}{(P_2-P_1)}\times \frac{\left(P_2+P_1\right)}{(Q_2+Q_1)}$

P₁ = $25   P₂= $30

Q₁ = 20   Q₂= 10

  $\begin{array}{l}{E}_d=\frac{(10-20)}{(30-25)}\times \frac{\left(25+30\right)}{(20+10)}\\ E_d=\frac{-10}{5}\times \frac{55}{30}\\ =-3.67\end{array}$

Thus, at price $25 per unit, price elasticity of demand is -3.67

b)

Using price changes of $10, we compare the price and quantity changes with $10 increments. We will still follow the same mid point formula.

At price $5 per unit, quantity demanded is 100 units and at price $15 per unit, the quantity demanded is 60 units, thus elasticity at $5 per unit is:

  $E_d=\frac{(Q_2-Q_1)}{(P_2-P_1)}\times \frac{\left(P_2+P_1\right)}{(Q_2+Q_1)}$

P₁ = $5   P₂= $15

Q₁ = 100   Q₂= 60

  $\begin{array}{l}{E}_d=\frac{(60-100)}{(15-5)}\times \frac{\left(5+15\right)}{(100+60)}\\ E_d=\frac{-40}{10}\times \frac{20}{160}\\ =-0.5\end{array}$

Thus, at price $5 with price changes of $10, price elasticity of demand is -0.5

At price $10 per unit, the quantity demanded is 80 units and at price $20 per unit, quantity demanded is 40 units, thus elasticity at $10 per unit is:

  $E_d=\frac{(Q_2-Q_1)}{(P_2-P_1)}\times \frac{\left(P_2+P_1\right)}{(Q_2+Q_1)}$

  $\begin{array}{l}{E}_d=\frac{(40-80)}{(20-10)}\times \frac{\left(10+20\right)}{(80+40)}\\ E_d=\frac{-40}{10}\times \frac{30}{120}\\ =-1\end{array}$

Thus, at price $10 with price changes of $10, price elasticity of demand is -1

At price $15 per unit, the quantity demanded is 60 units and at price $25 per unit, quantity demanded is 20 units, thus elasticity at $15 per unit is:

  $E_d=\frac{(Q_2-Q_1)}{(P_2-P_1)}\times \frac{\left(P_2+P_1\right)}{(Q_2+Q_1)}$

  $\begin{array}{l}{E}_d=\frac{(20-60)}{(25-15)}\times \frac{\left(15+25\right)}{(60+20)}\\ E_d=\frac{-40}{10}\times \frac{40}{80}\\ =-2\end{array}$

Thus, at price $15 with price changes of $10, price elasticity of demand is -2

At price $20 per unit, the quantity demanded is 40 units and at price $30 per unit, quantity demanded is 10 units, thus elasticity at $25 per unit is:

  $E_d=\frac{(Q_2-Q_1)}{(P_2-P_1)}\times \frac{\left(P_2+P_1\right)}{(Q_2+Q_1)}$

  $\begin{array}{l}{E}_d=\frac{(10-40)}{(30-20)}\times \frac{\left(20+30\right)}{(40+10)}\\ E_d=\frac{-30}{10}\times \frac{50}{50}\\ =-3\end{array}$

Economics Concept Introduction

Price elasticity (E_d) is the ratio of percentage change in quantity demanded upon percentage change in price.

  ${\text{E}}_{\text{d}}\text{=}\frac{\text{Percentage}\text{change}\text{in}\text{quantity}\text{demanded}}{\text{Percemtage}\text{change}\text{in}\text{Price}}$

  ${\text{E}}_{\text{d}}\text{=}\frac{\%\Delta Qd}{\%\Delta P}$