Question

When the price of commodity X falls by 10 percent, its demand rises from 150 units to 180 units. Calculate its price elasticity of demand. How much should the percentage fall in its price so that its demand rises from 150 to 210 units?

Answer 1

Hint: Assume 150 units as \[{{D}_{1}}\] and 180 units as \[{{D}_{2}}\] and calculate the increase in demand by subtracting \[{{D}_{1}}\] from \[{{D}_{2}}\]. Now, calculate percentage increase in the demand using the relation: - \[\dfrac{{{D}_{2}}-{{D}_{1}}}{{{D}_{1}}}\times 100%\]. Substitute this value in the formula: - E = $\dfrac{\text{Percentage increase in demand}}{\text{Percentage fall in price}}$ to calculate price elasticity of demand (E). Now, consider 210 units as \[{{D}_{3}}\] and calculate the percentage increase in demand from 150 to 210 units. Use the same relation of E to calculate the percentage fall in price in the second case.

Complete step-by-step solution
Here, we have been given that the demand for commodity X is rising from 150 units to 180 units when its price is falling by 10 percent. We have to calculate its price elasticity of demand.
Now, price elasticity of demand is the ratio of percentage change in demand to the percentage change in price. This change may be increasing or decreasing. So, here we are assuming 150 units as \[{{D}_{1}}\] and 180 units as \[{{D}_{2}}\], therefore percentage increase in the demand will be given as: -
\[\Rightarrow \] Percentage increase in demand = \[\left( \dfrac{{{D}_{2}}-{{D}_{1}}}{{{D}_{1}}} \right)\times 100%\]
\[\Rightarrow \] Percentage increase in demand = \[\left( \dfrac{180-150}{150} \right)\times 100%\]
\[\Rightarrow \] Percentage increase in demand = 20%
Now, assuming the price elasticity of demand as ‘E’, we get,
\[\Rightarrow \] E = $\dfrac{\text{Percentage increase in demand}}{\text{Percentage fall in price}}$
\[\Rightarrow \] E = \[\dfrac{20%}{10%}\]
\[\Rightarrow \] E = 2
Now, in the second case we have to calculate the percentage fall in price for the corresponding demand rise from 150 to 210 units. So, assuming 210 units as \[{{D}_{3}}\], we have,
\[\Rightarrow \] Percentage increase in demand = \[\dfrac{{{D}_{3}}-{{D}_{1}}}{{{D}_{1}}}\times 100%\]
\[\Rightarrow \] Percentage increase in demand = \[\dfrac{210-150}{150}\times 100%\]
\[\Rightarrow \] Percentage increase in demand = 40%
Since, the value of E will always remain same, therefore we have,
\[\Rightarrow \] E = $\dfrac{\text{Percentage increase in demand}}{\text{Percentage fall in price}}$
\[\Rightarrow \] 2 = $\dfrac{40\%}{\text{Percentage fall in price}}$
\[\Rightarrow \] Percentage fall in price required = \[\dfrac{40%}{2}=20%\]
Therefore, for the rise of demand from 150 to 210 units, the percentage fall in price should be 20%.

Note: One may note that without remembering the definition of price elasticity of demand we cannot calculate its value. You may see that we have used the same value of E in the calculation of percentage fall in price required for the second case, this is because E is a ratio and its value will remain fixed for any increase or decrease in price and demand.