Question

The weekly sales of Honolulu Red Oranges is given by $q = 1116 - 18p$.How do you calculate the price elasticity of demand when the price is $\$ 31$ per orange? (Yes, $\$ 31$per orange)

Answer 1

Price elasticity of demand:
In simple words it can be explained as the percentage change in the quantity demanded of a good or service divided by the percentage change in the price. It can be calculated by using the formula:
${\text{Price Elasticity of Demand}} = \dfrac{{\% {\text{change in quantity}}}}{{\% {\text{change in price}}}}$
It can also be written as:

$${\text{Price Elasticity of Demand}} = \dfrac{{\dfrac{{dq}}{q}}}{{\dfrac{{dp}}{p}}} \\\ = \dfrac{{dp}}{{dq}} \times \left( {\dfrac{p}{q}} \right) \\\$$

Now by using the above information we can solve the given question.

Complete step by step solution:
Given
$q = 1116 - 18p........................................\left( i \right)$
Here we know that:
$q:{\text{the demand}} \\\ p:{\text{the price}} \\\$
So here we have to find the price elasticity of demand when the price is $\$ 31$ per orange.
Now it can be found by using the equation:

$${\text{Price Elasticity of Demand}} = \dfrac{{\dfrac{{dq}}{q}}}{{\dfrac{{dp}}{p}}} \\\ = \dfrac{{dp}}{{dq}} \times \left( {\dfrac{p}{q}} \right) \\\$$

So to find $\dfrac{{dp}}{{dq}}$lets differentiate (i) with $p$, such that we can write:

$$q = 1116 - 18p \\\ \dfrac{{dq}}{{dp}} = - 18................................\left( {ii} \right) \\\$$

Now we know that it’s given the price is $\$ 31$ per orange.
So $p = 31$
Also from (i) we can find the value of demand $q$.
Such that:

$$q = 1116 - 18p \\\ q = 1116 - 18 \times 31 \\\ q = 1116 - 558 \\\ q = 558..........................................\left( {iii} \right) \\\$$

Now we have$\dfrac{{dp}}{{dq}},p{\text{ and }}q$. So let’s substitute the values in the specified equation.
Such that we can write:

$${\text{Price Elasticity of Demand}} = \dfrac{{dp}}{{dq}} \times \left( {\dfrac{p}{q}} \right) \\\ = 18 \times \left( {\dfrac{{31}}{{558}}} \right) \\\ = \dfrac{{18}}{{18}} \\\ = 1....................................\left( {iv} \right) \\\$$

Therefore the price elasticity of demand when the price is $\$ 31$ per orange would be $1$.

Note: If the price elasticity of demand is equal to 0 then we can say that the demand is perfectly inelastic which means that the demand will not change when the price changes. Also when the price elasticity of demand equals 1, demand would be unit elastic and if the value of price elasticity is greater than 1 then we can say that the demand is perfectly elastic.