Calculate the income elasticity of demand for the above.

Steps to Solve

1. Identify Initial and New Values

Let the initial income be $Y_1 = R20,000$ and the initial quantity demanded be $Q_1 = 100$ units.

Let the new income be $Y_2 = R25,000$ and the new quantity demanded be $Q_2 = 150$ units.

2. Calculate the Change in Income and Quantity Demanded

Next, we calculate the change in income ($\Delta Y$) and the change in quantity demanded ($\Delta Q$):

$$ \Delta Y = Y_2 - Y_1 = R25,000 - R20,000 = R5,000 $$

$$ \Delta Q = Q_2 - Q_1 = 150 - 100 = 50 $$

3. Calculate the Average Values

Now, compute the average income and average quantity demanded:

$$ \bar{Y} = \frac{Y_1 + Y_2}{2} = \frac{R20,000 + R25,000}{2} = R22,500 $$

$$ \bar{Q} = \frac{Q_1 + Q_2}{2} = \frac{100 + 150}{2} = 125 $$

4. Use the Income Elasticity of Demand Formula

The formula for income elasticity of demand ($E_d$) is:

$$ E_d = \frac{\frac{\Delta Q}{\bar{Q}}}{\frac{\Delta Y}{\bar{Y}}} $$

Substituting the values we calculated:

$$ E_d = \frac{\frac{50}{125}}{\frac{R5,000}{R22,500}} $$

5. Calculate Each Component

Calculate the numerator and denominator separately:

Numerator:

$$ \frac{50}{125} = 0.4 $$

Denominator:

$$ \frac{R5,000}{R22,500} = \frac{5,000}{22,500} = \frac{2}{9} \approx 0.2222 $$

6. Divide to Find Elasticity

Finally, substitute the values into the formula:

$$ E_d = \frac{0.4}{0.2222} \approx 1.8 \text{ (rounded)} $$