Chapter 20, Problem 1E

To determine

(a)

Price elasticity when price changes by $5.

Answer to Problem 1E

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

At price $15, price elasticity of demand = -1.4.

At price $20 per unit, the price elasticity of demand = -3.

At price $25 per unit, price elasticity of demand = -3.67

Explanation of Solution

The arc or mid-point elasticity is calculated in the following way:

Elasticity of Demand (E_d) as per mid-point method:

$E_d=\frac{(Q_2-Q_1)}{(P_2-P_1)}\times \frac{\left(P_2+P_1\right)/2}{(Q_2+Q_1)/2}$

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

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 is 100 units, and at price $10 per unit quantity demanded = 80 unit, thus, the elasticity at first point = $5 per unit where price = $5 per unit and 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= $5, Q₁ = 100

P₂ = $10, Q₂= 80

$E_d=\frac{(80-100)}{(10-5)}\times \frac{\left(10+5\right)}{(80+100)}$

$\begin{array}{l}{E}_d=\frac{-20}{5}\times \frac{15}{180}\\ =-0.33\end{array}$

Thus, E_d= -0.33

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

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

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

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

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

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

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

At price $15, price elasticity of demand = -1.4.

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

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

At price $20 per unit, the price elasticity of demand = -3.

At price $25 per unit, the quantity of demand = 20 units, so, elasticity is calculated as:

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

At price $25 per unit, price elasticity of demand = -3.67

Economics Concept Introduction

Introduction:

The arc or mid-point elasticity is calculated in the following way:

Elasticity of Demand (E_d) as per mid-point method:

$E_d=\frac{(Q_2-Q_1)}{(P_2-P_1)}\times \frac{\left(P_2+P_1\right)/2}{(Q_2+Q_1)/2}$

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

Where E_d = coefficient of elasticity

P₁ = Initial Price

P₂ = New Price

Q₁ = Initial Quantity

Q₂ = New Quantity

To determine

(b)

Price elasticity when price changes by $10.

Answer to Problem 1E

At price $5 with price changes of $10, price elasticity of demand is -0.5.

At price $10 with price changes of $10, price elasticity of demand = -1

At price $15 with price changes of $10, price elasticity of demand is -2.

At price changes $10, the price elasticity of demand is -3.

Explanation of Solution

Using price changes of $10, we compare the price and quantity changes with $10 increments.

E_d as per mid-point method.

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

Where E_d = coefficient of elasticity

P₁ = Initial Price

P₂ = New Price

Q₁ = Initial Quantity

Q₂ = New Quantity

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

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

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

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

At price of $10 per unit. Quantity demanded = 80 unit and at price = $20 per unit, quantity demanded= 40 units, thus, elasticity at $10 per unit is:

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

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

At price $10 with price changes of $10, price elasticity of demand = -1

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

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

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

At price $20 per unit, quantity demanded = 40 units and at price = $30 per unit, quantity demanded = 10 units, thus elasticity at $20 per unit.

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

Thus, at price changes $10, the price elasticity of demand is -3.