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The problem provided involves analyzing Denise's demand functions and the effects of changes in the price of good $ x $ on her consumption choices. Let's break down each part of the problem as seen in the images.

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### Part (a)
**Problem**: Determine Denise's demand functions for $ x $ and $ y $.

The demand functions are derived from Denise's utility function, $ U = x^3 y^2 $, and given budget constraint with prices for $ x $ and $ y $ as $ p_x = 6 $ and $ p_y = 8 $ respectively, and income $ M = 60 $.

The demand functions are given by:
\[
x^* = \frac{3M}{5p_x} \quad \text{and} \quad y^* = \frac{2M}{5p_y}
\]

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### Part (b)
**Problem**: Calculate Denise's optimal consumption bundle and total utility given the prices and income.

1. **Calculate $ x^* $ and $ y^* $**:
   - For $ x $:
     \[
     x^* = \frac{3 \cdot 60}{5 \cdot 6} = \frac{180}{30} = 6
     \]
   - For $ y $:
     \[
     y^* = \frac{2 \cdot 60}{5 \cdot 8} = \frac{120}{40} = 3
     \]

2. **Calculate Total Utility**:
   - Substitute $ x = 6 $ and $ y = 3 $ into the utility function $ U = x^3 y^2 $:
     \[
     U = 6^3 \cdot 3^2 = 216 \cdot 9 = 1944
     \]

Thus, Denise's optimal consumption bundle is $ (6, 3) $ with a total utility of 1944.

---

### Part (c)
**Problem**: Suppose the price of $ x $ decreases from $6 to $4. Calculate Denise's new consumption bundle and total utility.

1. **New Optimal Quantities $ x^* $ and $ y^* $**:
   - For $ x $:
     \[
     x^* = \frac{3 \cdot 60}{5 \cdot 4} = \frac{180}{20} = 9
     \]
   - For $ y $ (unchanged as the price of $ y $ remains the same):
     \[
     y^* = \frac{2 \cdot 60}{5 \cdot 8} = 3
     \]

2. **New Total Utility**:
   - Substitute $ x = 9 $ and $ y = 3 $ into the utility function:
     \[
     U = 9^3 \cdot 3^2 = 729 \cdot 9 = 6561
     \]

Denise’s new consumption bundle is $ (9, 3) $ with a total utility of 6561.

---

### Part (d)
**Problem**: Solve for the income and substitution effects on $ x $ due to the price decrease using the Hicksian approach.

To solve for income and substitution effects:

1. **Calculate the Hicksian Demand for $ x $ at the new price $ p_x = 4 $** while keeping utility constant at 1944 (from the original bundle).
  
2. **Substitution Effect**:
   - Find the consumption of $ x $ that maintains utility at 1944 with the new price, holding purchasing power constant (this requires further calculations).
   
3. **Income Effect**:
   - Calculate the difference between the total effect (from $ x^* = 6 $ to $ x^* = 9 $) and the substitution effect.

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### Part (e)
**Problem**: Solve for the income and substitution effects on $ y $ due to the price decrease using the Hicksian approach.

Since the price of $ y $ hasn't changed, the substitution effect for $ y $ should be zero. However, there may be an **income effect** on $ y $ because the decrease in $ p_x $ changes real income, potentially leading to an increased quantity demanded of $ y $ due to increased purchasing power.

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Would you like a detailed breakdown of each effect calculation for parts (d) and (e)? 

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#### Related Questions to Explore Further:
1. How would the solution change if Denise's income were doubled?
2. What would be the effects on $ x $ and $ y $ if both prices increased by 50%?
3. How would Denise’s consumption choices change if her utility function had different exponents?
4. Can we generalize the Hicksian approach to non-linear demand functions?
5. How would a Cobb-Douglas utility function alter the substitution and income effects?

#### Tip:
When calculating the Hicksian demand, use the compensated budget approach to isolate substitution effects accurately.

Income & Substitution Effects (Hicksian) Practice Problem: Denise's demand function is U = u(x,y) = x^3 y^2. Denise’s income is $60, the price of x is $6, and the price of y is $8. Find demand functions for x and y, optimal consumption bundle, total utility, new bundle with reduced price for x, and income and substitution effects on x and y.

Analyze income and substitution effects for Denise's consumption of goods x and y, given her utility function U = x^3 y^2, income of $60, and initial prices of $6 and $8. Determine her demand functions, optimal consumption bundle, and changes in utility and bundle after a price decrease in x.

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