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.