The price elasticity of supply of good X is half the price elasticity of supply of good Y. A 10% rise in the price of good Y results in a rise in its supply from 400 units to 520 units. calculate the %age change in quantity supplied of good X when its price falls from Rs 10 to Rs 8 per unit.

Let initial quantity supplied (Q₁) be 400 units
Final quantity supplied (Q₂) be 520 units

Change in quantity supplied $\left(∆Q\right)$ = $Q_2-Q_1=520-400=80 \mathrm{units}$

$$\begin{gathered} \mathrm{Percentage} \mathrm{change} \mathrm{in} \mathrm{quantity} \mathrm{supplied} \mathrm{of} \mathrm{good}\mathit{ }Y =\frac{\mathrm{Change} \mathrm{in} \mathrm{quantity} \mathrm{supplied}}{\mathrm{Initial} \mathrm{quantity} \mathrm{supplied}}\times 100 \\ =\frac{80}{400}\times 100 \\ =30\% \end{gathered}$$

$$\begin{gathered} \mathrm{Percentage} \mathrm{change} \mathrm{in} \mathrm{price} \mathrm{of} \mathrm{good} Y\mathit{ }=10\% \left(\mathrm{given}\right) \\ \mathrm{Percentage} \mathrm{change} \mathrm{in} \mathrm{price} \mathrm{of} \mathrm{good} X=\frac{P_2-P_1}{P_1}\times 100 \\ =\frac{8-10}{10}\times 100 \\ =-20\% \end{gathered}$$

According to the question,

Price elasticity of supply of good X = $\frac{1}{2}\times$ ​Price elasticity of supply of good Y

$$\begin{gathered} \Rightarrow \frac{\% \mathrm{change} \mathrm{in} \mathrm{quantity} \mathrm{supplied} \mathrm{of} \mathrm{good} X}{\% \mathrm{change} \mathrm{in} \mathrm{price} \mathrm{of} \mathrm{good}\mathit{ }X}=\frac{1}{2}\times \frac{\% \mathrm{change} \mathrm{in} \mathrm{quantity} \mathrm{supplied} \mathrm{of} \mathrm{good} Y}{\% \mathrm{change} \mathrm{in} \mathrm{price} \mathrm{of} \mathrm{good} X} \\ \mathrm{or}, \frac{\% \mathrm{change} \mathrm{in} \mathrm{quantity} \mathrm{supplied} \mathrm{of} \mathrm{good} X}{-20}=\frac{1}{2}\times \frac{30}{10} \\ \therefore \% \mathrm{change} \mathrm{in} \mathrm{quantity} \mathrm{supplied} \mathrm{of} \mathrm{good} X=-30 \end{gathered}$$

The negative sign denotes the percentage fall in the supply of good X.