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.