Let’s solve this step by step based on the information given:

Problem Statement

• Denise’s utility function is $U(x, y) = x^3 y^2$.
• Denise’s income $M = 60$.
• Price of $x$ (denoted as $p_x$) is $6.
• Price of $y$ (denoted as $p_y$) is $8.

Part (a): Demand Functions for $x$ and $y$

The demand functions provided in the image are:

• $x^* = \frac{3M}{5p_x}$
• $y^* = \frac{2M}{5p_y}$

Let’s move on to part (b).

Part (b): Optimal Consumption Bundle and Total Utility

1. Find the Optimal Consumption Bundle (Values of $x$ and $y$): Substitute the values of $M = 60$, $p_x = 6$, and $p_y = 8$ into the demand functions:

   $x^* = \frac{3 \times 60}{5 \times 6} = \frac{180}{30} = 6$

   $y^* = \frac{2 \times 60}{5 \times 8} = \frac{120}{40} = 3$

   So, Denise’s optimal consumption bundle is $(x^*, y^*) = (6, 3)$.

2. Calculate the Total Utility at This Bundle: Substitute $x = 6$ and $y = 3$ into the utility function $U(x, y) = x^3 y^2$:

   $U(6, 3) = 6^3 \times 3^2 = 216 \times 9 = 1944$

   Thus, Denise’s total utility for this bundle is 1944.

Part (c): New Bundle and Total Utility with a Price Change for $x$

Now, the price of $x$ decreases from $6 to $4. Let’s find the new values.

1. New Consumption Bundle: Substitute the new price $p_x = 4$ (keeping $M = 60$ and $p_y = 8$ unchanged):

   $x^* = \frac{3 \times 60}{5 \times 4} = \frac{180}{20} = 9$

   $y^* = \frac{2 \times 60}{5 \times 8} = \frac{120}{40} = 3$

   With the price change, Denise’s new consumption bundle is $(x^*, y^*) = (9, 3)$.

2. Total Utility for the New Bundle: Substitute $x = 9$ and $y = 3$ into the utility function:

   $U(9, 3) = 9^3 \times 3^2 = 729 \times 9 = 6561$

   Thus, Denise’s total utility for the new bundle is 6561.

Summary of Answers:

• (a) Demand functions for $x$ and $y$ are $x^* = \frac{3M}{5p_x}$ and $y^* = \frac{2M}{5p_y}$.
• (b) Optimal bundle: $(6, 3)$, Total utility: 1944.
• (c) New bundle after price decrease: $(9, 3)$, Total utility: 6561.

Would you like a more detailed explanation for any part, or do you have further questions?

Additional Questions:

1. How does a change in income affect Denise’s optimal bundle and total utility?
2. What would happen to the demand functions if the utility function had different exponents?
3. Can we find Denise’s optimal bundle if only her total utility is given?
4. How would the substitution effect alone change if we used a different utility function form?
5. What does the increase in utility in part (c) indicate about Denise’s preferences?

Tip:

When utility functions involve exponents, the demand functions often reflect how sensitive each good is to income and price changes, giving insight into consumption patterns.