To calculate the income elasticity of demand for automobiles given the demand function \( Q = 50,000 + 5y \), where \( Q \) is the quantity of automobiles demanded and \( y \) is the income per capita, we can follow these steps:
-
Calculate the initial and new quantities demanded:
- When income \( y = 10,000 \): \[ Q_1 = 50,000 + 5(10,000) = 50,000 + 50,000 = 100,000 \]
- When income \( y = 11,000 \): \[ Q_2 = 50,000 + 5(11,000) = 50,000 + 55,000 = 105,000 \]
-
Calculate the change in quantity demanded and change in income:
- Change in quantity demanded \( \Delta Q = Q_2 - Q_1 = 105,000 - 100,000 = 5,000 \)
- Change in income \( \Delta y = 11,000 - 10,000 = 1,000 \)
-
Calculate the average quantity and average income:
- Average quantity \( Q_{avg} = \frac{Q_1 + Q_2}{2} = \frac{100,000 + 105,000}{2} = 102,500 \)
- Average income \( y_{avg} = \frac{y_1 + y_2}{2} = \frac{10,000 + 11,000}{2} = 10,500 \)
-
Calculate the income elasticity of demand (E_y): \[ E_y = \frac{\Delta Q / Q_{avg}}{\Delta y / y_{avg}} \] Now substituting the values: \[ E_y = \frac{5,000 / 102,500}{1,000 / 10,500} \]
Simplifying both fractions:
- \( \frac{5,000}{102,500} = \frac{5}{102.5} \approx 0.04878 \)
- \( \frac{1,000}{10,500} = \frac{1}{10.5} \approx 0.095238 \)
So, \[ E_y = \frac{0.04878}{0.095238} \approx 0.511 \]
Therefore, the income elasticity of demand for automobiles is approximately 0.511. This indicates that the demand for automobiles is inelastic with respect to income, meaning that demand increases less proportionately than the increase in income.
1 answer