Question

What is the formula for the income elasticity of demand?

Short answer

The formula for income elasticity of demand (E) is: \[E = \frac{(Q_2 - Q_1)\cdot I_1}{(I_2 - I_1) \cdot Q_1}\] where, \(Q_1\) = initial quantity demanded, \(Q_2\) = final quantity demanded, \(I_1\) = initial income, and \(I_2\) = final income.

Step-by-step solution

Identify the variables involved in income elasticity of demand

The variables involved in income elasticity of demand are: 1. Quantity demanded (Q) 2. Income (I)

Define the income elasticity of demand

The income elasticity of demand (E) is defined as the percentage change in quantity demanded (Q) divided by the percentage change in income (I). Mathematically, it can be written as: \[E = \frac{\% \Delta Q}{\% \Delta I}\] where, Δ denotes the change in the respective variable, E = income elasticity of demand, Q = quantity demanded, and I = income

Express percentage change in terms of variables

To derive the formula for income elasticity of demand, we need to express the percentage change in quantity demanded and the percentage change in income in terms of their respective variables. The percentage change can be expressed as: \[\% \Delta Q = \frac{Q_2 - Q_1}{Q_1} \times 100\] \[\% \Delta I = \frac{I_2 - I_1}{I_1} \times 100\] where, Q1 and I1 represent the initial quantity demanded and income, respectively, Q2 and I2 represent the final quantity demanded and income, respectively

Substitute the percentage change expressions into the income elasticity of demand formula

Now, we'll substitute the expressions for the percentage change in quantity demanded and the percentage change in income into the income elasticity of demand formula: \[E = \frac{(Q_2 - Q_1) / Q_1}{(I_2 - I_1) / I_1} \]

Simplify the formula

We can simplify the formula by eliminating the common terms. Multiply the numerator and denominator by \(Q_1\) and \(I_1\), respectively: \[E = \frac{(Q_2 - Q_1) \cdot I_1}{(I_2 - I_1) \cdot Q_1} \] Now we have obtained the formula for income elasticity of demand: \[E = \frac{(Q_2 - Q_1)\cdot I_1}{(I_2 - I_1) \cdot Q_1}\]