Tony's utility function is \( U(x, y) = x^7 y^3 \). This is a Cobb-Douglas utility function, where \( x \) and \( y \) represent quantities of two goods, and the exponents (7 and 3) reflect the relative importance of each good in Tony's preferences. In general, Cobb-Douglas utility functions have the form \( U(x, y) = x^a y^b \), where \( a \) and \( b \) are positive constants that sum to 1 in cases of constant returns to scale, but they can vary in different settings.

Key points to analyze:
1. **Marginal Utility of Good x**: This shows how Tony’s utility changes with small changes in the amount of good \( x \).
   \[
   MU_x = \frac{\partial U(x, y)}{\partial x} = 7x^6 y^3
   \]
   
2. **Marginal Utility of Good y**: Similarly, this shows how Tony’s utility changes with small changes in the amount of good \( y \).
   \[
   MU_y = \frac{\partial U(x, y)}{\partial y} = 3x^7 y^2
   \]

3. **Marginal Rate of Substitution (MRS)**: The MRS represents how much of good \( y \) Tony is willing to give up to get one more unit of good \( x \) while keeping utility constant.
   \[
   MRS = \frac{MU_x}{MU_y} = \frac{7x^6 y^3}{3x^7 y^2} = \frac{7y}{3x}
   \]

Would you like me to go into more detail about these concepts or help with another specific part of the utility function?

Here are some further questions related to this:
1. How does Tony's utility change if the quantity of good \( y \) doubles?
2. What happens to the MRS when the quantity of \( x \) increases?
3. How would you find the optimal bundle of goods given a budget constraint?
4. What is the elasticity of substitution between \( x \) and \( y \)?
5. What would happen to Tony's utility if both \( x \) and \( y \) increased by 10%?

**Tip**: When solving utility maximization problems, it's useful to equate the MRS with the price ratio of the goods to find the optimal consumption bundle.

Understand how to solve Tony's Cobb-Douglas utility function U(x, y) = x^7 y^3, including finding the marginal utility of goods x and y and calculating the Marginal Rate of Substitution (MRS). Explore advanced microeconomic concepts related to utility maximization.

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