The demand D of passenger automobiles is given by D = 0.90 I^1.1 p^-0.7 where I is the income and p is the price per car. Find the income elasticity of demand.

Steps to Solve

1. Identify the demand function Given the demand function for passenger automobiles: $$ D = 0.90 I^{1.1} p^{-0.7} $$ where $I$ is the income and $p$ is the price.

2. Apply the formula for income elasticity of demand The formula for income elasticity of demand (E) is: $$ E = \frac{\partial D}{\partial I} \cdot \frac{I}{D} $$

3. Calculate the partial derivative of demand with respect to income We first need to differentiate $D$ with respect to $I$: $$ \frac{\partial D}{\partial I} = 0.90 \cdot 1.1 I^{1.1 - 1} p^{-0.7} = 0.99 I^{0.1} p^{-0.7} $$

4. Substitute the derivative into the elasticity formula Now, substitute $\frac{\partial D}{\partial I}$ back into the elasticity formula: $$ E = \left(0.99 I^{0.1} p^{-0.7}\right) \cdot \frac{I}{0.90 I^{1.1} p^{-0.7}} $$

5. Simplify the expression This simplifies to: $$ E = \frac{0.99 I^{0.1} p^{-0.7} \cdot I}{0.90 \cdot 0.90 I^{1.1} p^{-0.7}} = \frac{0.99 I^{1.1}}{0.90 \cdot 0.90} $$

6. Final expression for income elasticity Thus, the income elasticity of demand is: $$ E = \frac{0.99}{0.90} = 1.1 $$