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Let's go through each part of the problem step-by-step.

### Given Information
- **Utility Function**: $ U(x, y) = x^3 y $
- **Income (M)**: $600
- **Price of $ x $ (P_x)**: $10
- **Price of $ y $ (P_y)**: $1

### Part (a): Demand Functions for $ x $ and $ y $
The demand functions provided are:
\[
x^* = \frac{3M}{4P_x} \quad \text{and} \quad y^* = \frac{M}{3P_y}
\]
where $ x^* $ and $ y^* $ represent the optimal quantities of goods $ x $ and $ y $, respectively.

### Part (b): Optimal Consumption Bundle and Total Utility
1. **Calculate $ x^* $:**
   \[
   x^* = \frac{3M}{4P_x} = \frac{3 \times 600}{4 \times 10} = \frac{1800}{40} = 45
   \]

2. **Calculate $ y^* $:**
   \[
   y^* = \frac{M}{3P_y} = \frac{600}{3 \times 1} = \frac{600}{3} = 200
   \]

3. **Total Utility**:
   Substitute $ x = 45 $ and $ y = 200 $ into the utility function:
   \[
   U(x, y) = x^3 y = 45^3 \times 200
   \]
   Calculate $ 45^3 $:
   \[
   45^3 = 45 \times 45 \times 45 = 91125
   \]
   Then,
   \[
   U(45, 200) = 91125 \times 200 = 18225000
   \]

   So, the total utility for this bundle is **18,225,000**.

### Part (c): New Consumption Bundle and Total Utility (When $ P_x = 5 $)
1. **Calculate the new $ x^* $:**
   With the new $ P_x = 5 $,
   \[
   x^* = \frac{3M}{4P_x} = \frac{3 \times 600}{4 \times 5} = \frac{1800}{20} = 90
   \]

2. **Calculate $ y^* $ (unchanged, as $ P_y $ and $ M $ remain the same):**
   \[
   y^* = \frac{M}{3P_y} = \frac{600}{3 \times 1} = 200
   \]

3. **New Total Utility**:
   Substitute $ x = 90 $ and $ y = 200 $ into the utility function:
   \[
   U(x, y) = x^3 y = 90^3 \times 200
   \]
   Calculate $ 90^3 $:
   \[
   90^3 = 90 \times 90 \times 90 = 729000
   \]
   Then,
   \[
   U(90, 200) = 729000 \times 200 = 145800000
   \]

   So, the total utility for the new bundle is **145,800,000**.

### Summary of Answers
- **Part (a)**: The demand functions are $ x^* = \frac{3M}{4P_x} $ and $ y^* = \frac{M}{3P_y} $.
- **Part (b)**: Optimal bundle $ (x^*, y^*) = (45, 200) $, Total Utility = 18,225,000.
- **Part (c)**: New bundle $ (x^*, y^*) = (90, 200) $, New Total Utility = 145,800,000.

Would you like any more details or have any questions about these calculations?

#### Related Questions
1. How would Devin's utility change if the price of $ y $ increased instead?
2. What is the income elasticity of demand for $ x $ in this case?
3. How does the substitution effect play a role in Devin's consumption bundle change?
4. If Devin's income doubled, what would be the new optimal consumption bundle?
5. What is the effect of a proportional increase in both prices on Devin’s utility?

#### Tip
To analyze how sensitive consumption is to changes in income or prices, calculating elasticities can provide valuable insights.

Devin’s demand function is U = u(x,y) = x^3 * y. Devin’s income is $600, the price of x is $10, and the price of y is $1. a) What are Devin’s demand functions for x and y? b) What is Devin’s optimal consumption bundle of these two goods given the prices and income listed above? What is his total utility? c) Suppose the price of x decreases from $10 to $5. What is Devin’s new consumption bundle? What is his total utility for this new bundle?

Solve an optimization problem involving Devin’s utility function U(x,y) = x^3 * y. Find the optimal consumption bundle, calculate total utility, and analyze the effects of a price decrease in x from $10 to $5. Explore demand functions, utility maximization, and the Slutsky equation in this undergraduate economics example.

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