Answer:
A. Price Elasticity of Demand of Good X:
Using the formula for price elasticity of demand, we have:
\[ E_p = \frac{\frac{\delta Q_x}{Q_x}}{\frac{\delta P_x}{P_x}} = \frac{Q_x}{P_x} \cdot \frac{dQ_x}{dP_x} \]
Plugging the demand function, we get:
\[ E_p = \frac{Q_x}{P_x} \cdot \frac{d(-aP_x + bY)}{dP_x} = \frac{Q_x}{P_x} \cdot -a \]
\[ E_p = -a \]
Interpretation: The price elasticity of demand for good X is equal to -a. A negative value indicates that good X is an inelastic good, meaning that a change in price will result in a proportionally smaller change in quantity demanded.
B. Income Elasticity of Demand of Good X:
Using the formula for income elasticity of demand, we have:
\[ E_Y = \frac{\frac{\delta Q_x}{Q_x}}{\frac{\delta Y}{Y}} = \frac{Q_x}{Y} \cdot \frac{dQ_x}{dY} \]
Plugging the demand function, we get:
\[ E_Y = \frac{Q_x}{Y} \cdot b \]
Interpretation: The income elasticity of demand for good X is equal to b. The positive value of b indicates that good X is a normal good; as income increases, the quantity demanded of good X also increases.
C. Cross-Price Elasticity of Good X for Good Y:
Using the formula for cross-price elasticity of demand, we have:
\[ E_{py} = \frac{\frac{\delta Q_x}{Q_x}}{\frac{\delta P_y}{P_y}} = \frac{Q_x}{P_y} \cdot \frac{dQ_x}{dP_y} \]
Plugging the demand function, we get:
\[ E_{py} = \frac{Q_x}{P_y} \cdot 0 = 0 \]
Interpretation: The cross-price elasticity of demand for good X with respect to the price of good Y is equal to 0. This implies that there is no relationship between the price of good Y and the quantity demanded of good X, indicating that the two goods are not substitutes or complements in consumption.
