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To find Frank's demand functions for goods \( X \) and \( Y \), given the utility function:

\[
U(x, y) = x^7 y^3
\]

we'll use the Cobb-Douglas utility maximization process.

### 1. Demand Function for Good \( X \)

Let:
- \( p_x \): Price of good \( X \)
- \( p_y \): Price of good \( Y \)
- \( M \): Income

In Cobb-Douglas utility functions, the demand functions for \( X \) and \( Y \) can be found by allocating the income in proportion to the exponents on \( x \) and \( y \). Here, the exponents on \( x \) and \( y \) are \( 7 \) and \( 3 \) respectively, giving a total weight of \( 7 + 3 = 10 \).

The demand function for \( X \) is derived by spending a fraction of income \( \frac{7}{10} \) on \( X \):

\[
X^* = \frac{\frac{7}{10} M}{p_x} = \frac{7M}{10 p_x}
\]

### 2. Demand Function for Good \( Y \)

Similarly, the demand function for \( Y \) is derived by spending a fraction of income \( \frac{3}{10} \) on \( Y \):

\[
Y^* = \frac{\frac{3}{10} M}{p_y} = \frac{3M}{10 p_y}
\]

### Final Answers

1. **Demand function for good \( X \):**
   \[
   X^* = \frac{7M}{10 p_x}
   \]

2. **Demand function for good \( Y \):**
   \[
   Y^* = \frac{3M}{10 p_y}
   \]

Would you like more details on any part of this solution?

### Follow-Up Questions

1. How would these demand functions change if the exponents on \( x \) and \( y \) in the utility function were different?
2. Can you derive the indirect utility function using these demand functions?
3. How does a change in income \( M \) affect Frank's demand for goods \( X \) and \( Y \)?
4. What would happen to the demand functions if the prices \( p_x \) or \( p_y \) increased?
5. How would the demand functions change if we introduced a budget constraint that allowed borrowing?

### Tip

In Cobb-Douglas utility functions, the exponents represent the share of income allocated to each good. This simplifies the process of finding demand functions.

Frank's utility function is given by U(x, y) = x^7 y^3. What are Frank's demand functions for goods X and Y in terms of Px, Py, and M?

Find Frank's demand functions for goods X and Y using his utility function U(x, y) = x^7 y^3, with prices Px, Py, and income M. Learn how to apply Cobb-Douglas utility maximization to derive demand in terms of income and prices.

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