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Let's work through each part of this problem in detail:

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

Let's solve each part.

---

### Part (a): Find Devin's Demand Functions for $ x $ and $ y $

To maximize utility given the budget constraint, we'll use the **Lagrange multiplier** method.

1. **Objective function**:
   \[
   U(x, y) = x^3 y
   \]

2. **Budget constraint**:
   \[
   10x + 1y = 600
   \]

3. **Lagrangian function**:
   \[
   \mathcal{L} = x^3 y + \lambda (600 - 10x - y)
   \]

4. **Set up partial derivatives** and solve for the demand functions $ x $ and $ y $.

   \[
   \frac{\partial \mathcal{L}}{\partial x} = 3x^2 y - 10\lambda = 0
   \]
   \[
   \frac{\partial \mathcal{L}}{\partial y} = x^3 - \lambda = 0
   \]
   \[
   \frac{\partial \mathcal{L}}{\partial \lambda} = 600 - 10x - y = 0
   \]

From the second equation, $ \lambda = x^3 $. Substitute this into the first equation:

\[
3x^2 y = 10x^3
\]

Dividing both sides by $ x^2 $ (assuming $ x \neq 0 $):

\[
3y = 10x
\]
\[
y = \frac{10}{3} x
\]

Now, substitute $ y = \frac{10}{3} x $ into the budget constraint:

\[
10x + \frac{10}{3} x = 600
\]

Combine terms:

\[
\frac{30x + 10x}{3} = 600
\]
\[
\frac{40x}{3} = 600
\]
\[
40x = 1800
\]
\[
x = 45
\]

Now, substitute $ x = 45 $ back into $ y = \frac{10}{3} x $:

\[
y = \frac{10}{3} \times 45 = 150
\]

So, the demand functions (values) are:
- $ x = 45 $
- $ y = 150 $

---

### Part (b): Devin's Optimal Consumption Bundle and Total Utility

Devin's optimal consumption bundle, based on the demand functions we found, is:
- $ x = 45 $
- $ y = 150 $

**Total Utility**:

\[
U = x^3 y = 45^3 \times 150
\]

Calculate $ 45^3 $:

\[
45^3 = 91125
\]

Now, multiply by 150:

\[
U = 91125 \times 150 = 13668750
\]

So, Devin's **total utility** at this bundle is **13,668,750**.

---

### Part (c): New Consumption Bundle and Total Utility if Price of $ x $ Decreases to $5

1. **Updated price of $ x $**: $ P_x = 5 $
   
2. **New budget constraint**:
   \[
   5x + y = 600
   \]

Using the same method as above, set up the new system to find the new demand functions:

With the utility maximization process similar to Part (a):

- The relationship $ y = \frac{10}{3} x $ still holds.

Substitute $ y = \frac{10}{3} x $ into the new budget constraint:

\[
5x + \frac{10}{3} x = 600
\]

Combine terms:

\[
\frac{15x + 10x}{3} = 600
\]
\[
\frac{25x

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 economic problem involving utility maximization, demand functions, and price change effects. Devin's demand function is U = x^3 y with a budget of $600 and prices of x and y at $10 and $1, respectively. Find the demand functions, optimal bundle, and total utility, and analyze the effect of a price drop for x from $10 to $5.

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