Question
3.What is the income elasticity of automobiles as per capital income increases from, $10,000 to $11,000? The demand for automobiles as a function of income per capital is given by the equation. Q= 50,000 +5(y).
Answers
GPT-4o mini
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:
1. **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 \( y = 11,000 \):
\[
Q_2 = 50,000 + 5(11,000) = 50,000 + 55,000 = 105,000
\]
2. **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 \( \Delta y = 11,000 - 10,000 = 1,000 \)
3. **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 \( y_{avg} = \frac{y_1 + y_2}{2} = \frac{10,000 + 11,000}{2} = 10,500 \)
4. **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. **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 \( y = 11,000 \):
\[
Q_2 = 50,000 + 5(11,000) = 50,000 + 55,000 = 105,000
\]
2. **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 \( \Delta y = 11,000 - 10,000 = 1,000 \)
3. **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 \( y_{avg} = \frac{y_1 + y_2}{2} = \frac{10,000 + 11,000}{2} = 10,500 \)
4. **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.