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Let's work through this step by step.

### Given Information
- **Utility function**: \( U(x, y) = x y^3 \)
- **Budget constraint**: \( P_x x + P_y y = M \)
  
Let's solve each question based on this information.

---

### 1) Demand Function for Good \( X \)
To maximize utility, Frank should set the **marginal rate of substitution (MRS)** equal to the **price ratio**. For a utility function \( U(x, y) = x y^3 \), the MRS is given by the ratio of the marginal utilities:

1. **Marginal Utility of \( X \)**: \( MU_x = \frac{\partial U}{\partial x} = y^3 \)
2. **Marginal Utility of \( Y \)**: \( MU_y = \frac{\partial U}{\partial y} = 3x y^2 \)
3. **MRS**: \( \frac{MU_x}{MU_y} = \frac{y^3}{3x y^2} = \frac{y}{3x} \)

To find the demand function for \( X \), set the MRS equal to the price ratio:

\[
\frac{y}{3x} = \frac{P_x}{P_y}
\]

Solving for \( y \) in terms of \( x \):

\[
y = 3x \frac{P_x}{P_y}
\]

Now, substitute \( y \) back into the budget constraint:

\[
P_x x + P_y \left(3x \frac{P_x}{P_y}\right) = M
\]

Simplify this to solve for \( x \):

\[
P_x x + 3 P_x x = M
\]
\[
4 P_x x = M
\]
\[
x = \frac{M}{4 P_x}
\]

So, the demand function for \( X \) is:

\[
X = \frac{M}{4 P_x}
\]

---

### 2) Demand Function for Good \( Y \)
Using the relationship \( y = 3x \frac{P_x}{P_y} \) from above, substitute \( x = \frac{M}{4 P_x} \):

\[
y = 3 \left(\frac{M}{4 P_x}\right) \frac{P_x}{P_y}
\]

Simplify to get \( y \) in terms of \( P_x \), \( P_y \), and \( M \):

\[
y = \frac{3M}{4 P_y}
\]

Thus, the demand function for \( Y \) is:

\[
Y = \frac{3M}{4 P_y}
\]

---

### 3) Optimal Amount of \( X \) Given Specific Prices
Given:
- \( M = 100 \)
- \( P_x = 2 \)
- \( P_y = 6 \)

Using the demand function for \( X \):

\[
X = \frac{M}{4 P_x} = \frac{100}{4 \times 2} = \frac{100}{8} = 12.5
\]

So, the optimal amount of \( X \) Frank should consume is:

\[
X^* = 12.5
\]

---

### 4) Optimal Amount of \( Y \) Given Specific Prices
Using the demand function for \( Y \):

\[
Y = \frac{3M}{4 P_y} = \frac{3 \times 100}{4 \times 6} = \frac{300}{24} = 12.5
\]

So, the optimal amount of \( Y \) Frank should consume is:

\[
Y^* = 12.5
\]

---

### 5) Optimal Amount of \( X \) After Price Change
With the new prices:
- \( P_x = 5 \)
- \( P_y = 6 \)

Using the demand function for \( X \):

\[
X = \frac{M}{4 P_x} = \frac{100}{4 \times 5} = \frac{100}{20} = 5
\]

Thus, the optimal amount of \( X \) after the price change is:

\[
X^{**} = 5
\]

---

### 6) Optimal Amount of \( Y \) After Price Change
Using the demand function for \( Y \) with the updated prices:

\[
Y = \frac{3M}{4 P_y} = \frac{3 \times 100}{4 \times 6} = 12.5
\]

So, the optimal amount of \( Y \) remains:

\[
Y^{**} = 12.5
\]

---

Would you like further details or explanations on any step?

### Related Questions
1. How would the demand functions change if Frank’s utility function were different?
2. How does the MRS help determine the optimal consumption bundle?
3. What would happen if the budget \( M \) were doubled?
4. How would the demand functions adjust if both prices increased proportionally?
5. What if \( P_y \) changed instead of \( P_x \)?

#### Tip:
Remember, the MRS (marginal rate of substitution) reflects Frank's willingness to trade between goods \( X \) and \( Y \) at any consumption point.

Frank's utility function is given as U(x, y) = x * y^3. Find Frank's demand functions for goods X and Y and determine the optimal amounts of X and Y Frank should consume given his budget, prices, and after a price change.

Learn how to derive Frank's demand functions for goods X and Y, based on his utility function U(x, y) = x * y^3 and a given budget constraint. This problem involves utility maximization, marginal rate of substitution, and optimization, with step-by-step solutions for optimal consumption quantities before and after a price change.

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