The utility function given is $U(x, y) = x^{\frac{2}{5}} y^{\frac{3}{5}}$, which suggests that this is a Cobb-Douglas utility function. In Cobb-Douglas utility functions, the share of income spent on each good depends only on the exponents of the goods in the utility function, not on their prices.

In this case, the exponents are:

• $\frac{2}{5}$ for $x$
• $\frac{3}{5}$ for $y$

For a Cobb-Douglas utility function, the share of income spent on each good is constant and equal to the exponent on that good. Therefore:

• The share of income spent on $x$ is $\frac{2}{5}$,
• The share of income spent on $y$ is $\frac{3}{5}$.

Since the income shares depend on the exponents and not on the prices, the shares of income spent on each good remain the same, even if the price of good $x$ increases.

Thus, the correct answer is:

(c) Both shares stay the same.

Would you like further details on the Cobb-Douglas utility function? Here are some related questions to consider:

1. How is the optimal consumption bundle determined from a Cobb-Douglas utility function?
2. What is the effect of an income increase on the consumption of goods in a Cobb-Douglas utility model?
3. How do price changes affect the demand for goods in non-Cobb-Douglas utility functions?
4. What is the significance of the exponents in Cobb-Douglas utility functions?
5. How does the Cobb-Douglas utility function compare to other types of utility functions, like Leontief or CES?

Tip: In Cobb-Douglas utility functions, income shares remain constant because the budget is allocated proportionally to the exponents on the goods in the utility function.