HOW TO SOLVE PRICE ELASTICITY OF DEMAND

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Price elasticity is the degree of responsiveness of quantity demanded to price changes.

It is calculated as:

$$PED=\frac{\%∆QD}{\%∆P}$$

Where $\%∆QD$ is the percentage change in quantity demanded.

             $\%∆P$ is the percentage change in price.

Ok, Let's try these seven examples

Example 1

Determine the price elasticity of demand if a $10\%$ increase in the price of rice leads to a $20\%$ reduction in the quantity demanded.

Solution:

Recalled that $PED=\frac{\%∆QD}{\%∆P}$

Hence, $PED=\frac{-20\%}{10\%}$

$PED=-2$

Note: the negative sign is usually ignored

The numerical value of PED is 2, thus, demand is elastic. 

Wondering why elastic demand, see my post on types of the price elasticity of demand here.

In some cases (as is the case below), you have to determine the percentage change in prices and quantity.

Example 2

Determine the price elasticity of demand if a rise in the price of oil from €11 to €13 per barrel decreases the quantity demanded from 25,000 to 23,000 barrels.

Solution:

You know $PED=\frac{\%∆QD}{\%∆P}$

Where $\%∆QD=\frac{Q_1-Q_0}{Q_0}$

             $\%∆P=\frac{P_1-P_0}{P_0}$

 

Using the simple percentage method

If $Q_0=25,000$, $Q_1=23,000$, then

$\%∆QD=\frac{23000-25000}{25000}\times100$

This translates to:

$\%∆QD=-8\%$

If $P_0=11$, $P_1=13$, then

$\%∆P=\frac{13-11}{11}\times100$

$\%∆P=18.2\%$

Having gotten the percentage in price and quantity, let's calculate the PED

$PED=\frac{-8}{18.2}$

$PED=-0.44$

The numerical value(0.44 is lesser than 1) hence the oil demand is inelastic.

Example 3

Determine the elasticity of demand if a rise in the price of phones from €28 to €36 per phone causes a fall in the quantity demanded from 6000 to 4000 pens.

Solution:

$P_0=28$, $P_1=36$

$\%∆P=\frac{36 -28}{28}\times100$

$\%∆P=28.6\%$

$Q_0=6000$, $Q_1=4000$

$\%∆QD=\frac{4000-6000}{6000}\times100$

$\%∆QD=-33.3\%$

$PED=\frac{-33.3}{28.6}=-1.2$

The numerical value of PED is greater than 1, hence, demand is elastic

So far, We have been calculating using the simple method. 

But this method has one disadvantage. One may obtain different elasticity between two price points, 

Hence, the need for a more accurate method–The midpoint method.

Since the midpoint method uses the same bases for the price increase and decrease, one will almost always obtain the same elasticity between two price points regardless of price increase or decrease.

But, what is the midpoint formula?

It is expressed as: 

$PED=\frac{\%∆QD}{\%∆P}$

Where $\%∆QD=\frac{Q_1-Q_0}{\frac{Q_1+Q_0}{2}}\times100$

           $\%∆P=\frac{P_1-P_0}{\frac{P_1+P_0}{2}}\times100$

Ok, let's try solving with the midpoint method.

Example 4

A fall in the price of petrol from €42 to €38 per barrel increases the quantity demand from 38400 to 41600. Calculate the numerical value of PED.

Solution:

$P_1=€38$, $P_0=€42$

$\%∆P=\frac{38-42}{\frac{38+42}{2}}\times100$

$\%∆P=-10\%$

$Q_1=€41,600$, $Q_0=38,400$

Where $\%∆QD=\frac{41600-38400}{\frac{41600+38400}{2}}\times100$

$\%∆QD=8\%$

Recalled that:

$PED=\frac{\%∆QD}{\%∆P}$

$PED=\frac{8}{-10}$

$PED=-0.8$

Example 5

Determine the price elasticity of demand assuming a rise in the price of rice from €5.5 to €6.5 per bag decreases the quantity demanded from 25,000 to 23,000 bags of prices.

Solution

$Q_1=€23,000$, $Q_0=25,000$

Where $\%∆QD=\frac{23000-25000}{\frac{23000+25000}{2}}\times100$

$\%∆QD=-8.3\%$

$P_1=€6.5$, $P_0=€5.5$

$\%∆P=\frac{6.5-5.5}{\frac{6.5+5.5}{2}}\times100$

$\%∆P=16.7\%$

Recalled that:

$PED=\frac{\%∆QD}{\%∆P}$

$PED=\frac{-8.3}{-16.7}$

$PED=-0.5$

The numerical value of PED(0.5) is lesser than 1. Hence, demand is inelastic

Look easy, try this!!!

Example 6

Amazon.com, an online bookseller, wants to increase its total revenue. They decided to offer a 20% discount on all books. This resulted in the quantity demanded Increasing from 10,000 to 11,000 books. Using the midpoint method, will total revenue increased?

Solution:

We have already been given the percentage change in price: which is a 20% reduction.

Let's compute the percentage increase in quantity demanded.

$Q0=10,000$, $Q1=10,500$

$\%∆QD=\frac{11000-10000}{\frac{11000+10000}{2}}\times100$

$\%∆QD=9.5\%$

$PED=\frac{5}{-10}$

$PED=-0.5$

The numerical value of PED is 0.5(signifying that demand is inelastic)

hence, the total revenue will not increase rather it will decrease as a result of the discount.

And now, the last question.....

Related post

Example 7

The quantity of new cars increases by $10\%$. If the price elasticity of demand for new cars is 1.25, the price of new cars will fall by how many percent?

Solution: 

You know that $PED=\frac{\%∆QD}{\%∆P}$

If $%∆QD=10$, $PED=1.25$, Then

$1.25=\frac{10}{\%∆P}$

Cross multiply

$1.25(\%∆P)=10$

Divide through by $1.25$

$\frac{\require{cancel}\bcancel{1.25}\%∆P}{\require{cancel}\bcancel{1.25}}=\frac{10}{1.25}$

${\%∆P}=8$

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