Mathematically, it is expressed as

$PES=\frac{\%∆Q_s}{\%P}$

Where $\%Q_s$ is the percentage change in quantity supplied

$\%P$ is the percentage change in price

The accompanying examples will illustrate the steps in calculating price elasticity of supply

**Example 1**

Determine the price elasticity of supply if a $10\%$ increase in price results in an $18\%$ increase in quantity supplied.

$PES=\frac{\%∆Q_s}{\%P}$

$PES=\frac{18}{10}$

$PES=1.8$

The numerical value of price elasticity of supply is greater than $1$, hence supply is relatively elastic.

**Example 2**

Calculate the price elasticity of supply if a change in the price of books from €4 to €2 causes the quantity supplied to decrease from 10 pens to 5 pens.

$PES=\frac{\%∆Q_s}{\%P}$

Unlike the first example, we were not given the percentage change in price and quantity, hence we solve for percentage changes using the midpoint method.

For percentage change in quantity supplied,

$Q_s=\frac{Q_1-Q_0}{\frac{Q_1+Q_0}{2}}\times100$

$Q_1=10$, $Q_0=5$

$Q_s=\frac{5-10}{\frac{5+10}{2}}\times100$

$Q_s=\frac{-5}{7.5}\times100$

$Q_s=-66.7\%$

For percentage change in price,

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

$P_1=2$, $P_0=4$

$P=\frac{2-4}{\frac{2+4}{2}}\times100$

$P=\frac{-2}{3}\times100$

$P=-66.7\%$

$PES=\frac{-66.7\%}{-66.7\%}$

$PES=1$

The numerical value of price elasticity of supply is one, hence, supply is said to be unitary elastic.

**Example 3**

Determine the price elasticity of supply if a rise in the price of books from €4 to €8 causes the number of books supplies to rise from 8 books to 20 books.

Percentage change in quantity supplied is

$Q_s=\frac{Q_1-Q_0}{\frac{Q_1+Q_0}{2}}\times100$

$Q_1=20$, $Q_0=8$

$Q_s=\frac{20-8}{\frac{20+8}{2}}\times100$

$Q_s=\frac{12}{14}\times100$

$Q_s=85.7\%$

percentage change in price,

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

$P_1=8$, $P_0=4$

$P=\frac{8-4}{\frac{8+4}{2}}\times100$

$P=\frac{4}{6}\times100$

$P=66.7\%$

$PES=\frac{88.7\%}{66.7\%}$

$PES=1.3$

The coefficient of price elasticity of supply is greater than one, hence supply is relatively elastic.

**Example 4**

Determine the price elasticity of supply if the quantity of bags supplied increases from 4500 to 5500 as a result of a rise in the price of bags from €14 to €18 per bag.

Percentage change in quantity supplied is

$Q_s=\frac{Q_1-Q_0}{\frac{Q_1+Q_0}{2}}\times100$

$Q_1=5500$, $Q_0=4500$

$Q_s=\frac{5000-4500}{\frac{5500+4500}{2}}\times100$

$Q_s=\frac{1000}{5000}\times100$

$Q_s=20\%$

percentage change in price,

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

$P_1=18$, $P_0=14$

$P=\frac{18-14}{\frac{18+14}{2}}\times100$

$P=\frac{4}{16}\times100$

$P=25\%$

$PES=\frac{20\%}{25\%}$

$PES=0.8$

As you can see, the numerical value of the Price elasticity of supply is 0.8, which is lesser than one but greater than 0. Therefore, supply is relatively inelastic.

**Example 5**

A rise in the price of shoes from €14 to €18 per shoe causes the quantity supplied to rise from 4000 to 6000 shoes. Describe the price elasticity of supply

$Q_1=6000$, $Q_0=4000$

$Q_s=\frac{6000-4000}{\frac{6000+4000}{2}}\times100$

$Q_s=\frac{2000}{5000}\times100$

$Q_s=40\%$

$P_1=18$, $P_0=14$

$P=\frac{18-14}{\frac{18+14}{2}}\times100$

$P=\frac{4}{16}\times100$

$P=25\%$

$PES=\frac{40\%}{25\%}$

$PES=1.6$

The supply can be described as relatively elastic since the numerical value of PES is greater than one.

**Related post**

**Example 6**

The price of new cars increases by $5\%$. If the price elasticity of supply for new cars is 0.6, the quantity supplied of new cars will rise by how many percent?

$PES=\frac{\%∆Q_s}{\%P}$

Here percentage change in price{5\%} is given and the numerical value of Price elasticity of supply{0.6) is given, so, we would solve it as a mathematician does.

$0.6=\frac{\%∆Q_s}{5}$

Cross multiply

$0.6\times5=\%∆Q_s$

$3=\%∆Q_s$

$\%∆Q_s=3$

So, the quantity supplied will rise by $3\%$

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