Question

Consider a consumer with the utility function, \( U(x, y) = y + \sqrt{x} \), where \( x \) and \( y \) are quantities of two commodities consumed.
Which one of the following is TRUE?

• A. Income elasticity of demand for both goods is 1.
• B. Income elasticity of demand for good x is 0.
• C. Income elasticity of demand for good y is 0.
• D. Income elasticity of demand for good x is 0.5 and good y is 1.

Hint:

Income elasticity for goods with square root demand functions is typically low, reflecting inelastic demand with respect to income.

Answer 1

The Correct Option is B

Step 1: Utility function analysis.
The given utility function is \( U(x, y) = y + \sqrt{x} \). To determine income elasticity, we first need to find the demand functions for both goods using the utility maximization approach.
Step 2: Deriving income elasticity.
The income elasticity of demand measures the responsiveness of the quantity demanded of a good to changes in income. It is given by the formula: \[ \epsilon = \frac{dQ}{dI} \times \frac{I}{Q} \] where \( Q \) is the quantity demanded and \( I \) is the income.
For good \( x \), the utility function has a square root form, which leads to a demand that is less sensitive to income changes (i.e., an income elasticity of 0).
Step 3: Answer explanation.
- Option (A) is incorrect because the income elasticity of demand for both goods is not 1.
- Option (B) is correct because the income elasticity for \( x \) is 0, meaning that changes in income have no effect on the demand for good \( x \).
- Option (C) is incorrect because the income elasticity for \( y \) is not 0.
- Option (D) is incorrect because the values given for the elasticities do not match the utility function.
Final Answer: \[ \boxed{\text{Income elasticity of demand for good x is 0.}} \]