Calculate the income elasticity of demand for the above.

Understand the Problem

The question is asking to calculate the income elasticity of demand based on the information provided about the consumer's income and the quantity of the product they buy before and after the income change.

Answer

The income elasticity of demand is $E_i = 2.67$.

Steps to Solve

Identify the initial and new income and quantity demanded
Begin with the initial income ($I_1$) and quantity demanded ($Q_1$) as follows:

Initial income, $I_1 = R4,000$
Initial quantity demanded, $Q_1 = 120$ units
Next, identify the new income ($I_2$) and new quantity demanded ($Q_2$):
New income, $I_2 = R5,000$
New quantity demanded, $Q_2 = 200$ units

Calculate the change in income and quantity demanded
Determine the changes in income ($\Delta I$) and quantity demanded ($\Delta Q$):

Change in income:
$$ \Delta I = I_2 - I_1 = R5,000 - R4,000 = R1,000 $$
Change in quantity demanded:
$$ \Delta Q = Q_2 - Q_1 = 200 - 120 = 80 $$

Calculate the percentage change in income and quantity demanded
Calculate the percentage changes:

Percentage change in income:
$$ \text{Percentage change in income} = \frac{\Delta I}{I_1} \times 100 = \frac{1,000}{4,000} \times 100 = 25% $$
Percentage change in quantity demanded:
$$ \text{Percentage change in quantity demanded} = \frac{\Delta Q}{Q_1} \times 100 = \frac{80}{120} \times 100 \approx 66.67% $$

Calculate the income elasticity of demand
Use the formula for income elasticity of demand ($E_i$):
$$ E_i = \frac{\text{Percentage change in quantity demanded}}{\text{Percentage change in income}} $$
Substituting the values:
$$ E_i = \frac{66.67%}{25%} = 2.67 $$

The income elasticity of demand is $E_i = 2.67$.

More Information

Income elasticity of demand measures how much the quantity demanded of a good responds to a change in consumer income. An elasticity greater than 1 indicates that the good is a luxury, meaning as income rises, quantity demanded increases significantly.

Tips

Confusing the changes in income and quantity with their absolute values instead of their differences.
Forgetting to convert the changes into percentages before calculating elasticity.

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