Question

When the price of commodity A falls from Rs. 10 to Rs. 5 per unit, its quantity demanded doubles. Calculate its elasticity of demand. At what price will its quantity demanded fall by 50 per cent?

Answer 1

Hint: Calculate the elasticity of demand $ \left( {{E}_{d}} \right) $ using the definition:
 $ {{E}_{d}}=\dfrac{\dfrac{\Delta Q}{Q}}{\dfrac{\Delta P}{P}} $ , where $ \Delta Q $ is the change in the quantity demanded (Q), and $ \Delta P $ is the change in the price (P).
If the value of a quantity x changes by $ \Delta x $ , then the percentage change in its value is $ \dfrac{\Delta x}{x}\times 100 $ .
Mark that the change $ \Delta x $ is positive for a rise/increase in the value of x and negative for a fall/decrease in the value of x.
  $ \Delta x={{x}_{new}}-{{x}_{old}} $

Complete step by step solution:
Using the definition of change $ \Delta x={{x}_{new}}-{{x}_{old}} $ , the change in the price of the commodity A, when the price falls from P = Rs. 10 to Rs. 5, will be:
  $ \Delta P={{P}_{new}}-{{P}_{old}}=5-10=-5 $
It is also given that the quantity demanded (Q) doubles in value.
  $ \Delta Q={{Q}_{new}}-{{Q}_{old}}=2Q-Q=Q $
Using the formula for elasticity:
  $ {{E}_{d}}=\dfrac{\dfrac{\Delta Q}{Q}}{\dfrac{\Delta P}{P}} $
$\Rightarrow {{E}_{d}}=\dfrac{\dfrac{Q}{Q}}{\dfrac{-5}{10}} $
$\Rightarrow {{E}_{d}}=\dfrac{1}{1}\times \dfrac{2}{-1} $
$\Rightarrow {{E}_{d}}=-2 $
Therefore, the elasticity of demand is $ {{E}_{d}}=-2 $ .
Calculation of price:
If the quantity demanded falls by 50%, then:
  $ \dfrac{\Delta Q}{Q}\times 100=-50 $
$\Rightarrow \dfrac{\Delta Q}{Q}=\dfrac{-50}{100}=\dfrac{-1}{2} $
Using the value $ {{E}_{d}}=-2 $ , we have:
  $ {{E}_{d}}=\dfrac{\dfrac{\Delta Q}{Q}}{\dfrac{\Delta P}{P}} $
$\Rightarrow -2=\dfrac{\dfrac{-1}{2}}{\dfrac{\Delta P}{10}} $
$\Rightarrow -2=\dfrac{-1}{2}\times \dfrac{10}{\Delta P} $
$\Rightarrow \Delta P=\dfrac{5}{2} $
$\Rightarrow \Delta P=2.5 $
Therefore, the price will be $ 10+2.5 $ = Rs. 12.5.

Note: The value of elasticity of demand is always negative. This is because, of the two changes, the percentage change in the quantity demanded and the percentage change in the price, one change will be positive, the other negative.
The percentage change in revenue can be calculated by knowing the elasticity and the percentage change in price alone.
Perfectly inelastic $ \left( {{E}_{d}}=0 \right) $ : Changes in the price do not affect the quantity demanded; raising prices will always cause total revenue to increase.
Relatively inelastic $ \left( -1<{{E}_{d}}<0 \right) $ : The percentage change in the quantity demanded is smaller than that in its price. Hence, when the price is raised, the total revenue increases, and vice versa.
Unit (or unitary) elastic $ \left( {{E}_{d}}=-1 \right) $ : The percentage change in the quantity demanded is equal to that in the price, so a change in price will not affect total revenue.
Relatively elastic $ \left( -\infty <{{E}_{d}}<-1 \right) $ : The percentage change in the quantity demanded is greater than that in the price. Hence, when the price is raised, the total revenue falls, and vice versa.
Perfectly elastic $ \left( {{E}_{d}}=-\infty \right) $ : Any increase in the price, no matter how small, will cause the quantity demanded for the good to drop to zero. Hence, when the price is raised, the total revenue falls to zero.
Total revenue is maximized at the combination of price and quantity demanded where the elasticity of demand is unitary.