The problem presented is asking to determine whether the given statement is "True," "False," or "Uncertain," and to provide reasoning. The key point is the comparison of the marginal rate of substitution (MRS) for two utility functions and the behavior of their indifference curves.

Utility Functions Given:

1. $U(x, y) = xy$ (Eq. 1)
2. $U(x, y) = x^2 y^2$ (Eq. 2)

Statement:

"The MRS equation for both of these utility functions is the same, but at least one of these utility functions does not produce convex indifference curves."

Steps to Approach:

1. Marginal Rate of Substitution (MRS): The MRS is defined as the negative ratio of the marginal utility of good $x$ to the marginal utility of good $y$: $\text{MRS} = - \frac{MU_x}{MU_y}$ where $MU_x$ is the marginal utility with respect to $x$, and $MU_y$ is the marginal utility with respect to $y$.

2. Calculate MRS for each utility function:

   • For $U(x, y) = xy$:

     • $MU_x = \frac{\partial U(x, y)}{\partial x} = y$
     • $MU_y = \frac{\partial U(x, y)}{\partial y} = x$
     • Thus, $\text{MRS} = -\frac{y}{x}$.

   • For $U(x, y) = x^2 y^2$:

     • $MU_x = \frac{\partial U(x, y)}{\partial x} = 2x y^2$
     • $MU_y = \frac{\partial U(x, y)}{\partial y} = 2y x^2$
     • Thus, $\text{MRS} = -\frac{2xy^2}{2yx^2} = -\frac{y}{x}$.

   Both utility functions give the same MRS equation: $\text{MRS} = -\frac{y}{x}$.

3. Convexity of Indifference Curves:

   • A utility function produces convex indifference curves if the MRS decreases as we move along the curve, i.e., diminishing marginal rate of substitution.
   • For $U(x, y) = xy$, the indifference curves are convex, since it corresponds to a Cobb-Douglas utility function, which always produces convex indifference curves.
   • For $U(x, y) = x^2 y^2$, this function is a transformation of the original Cobb-Douglas form, but it does not necessarily guarantee convexity. In fact, higher powers in utility functions can lead to different shapes in the indifference curves, potentially resulting in non-convex indifference curves.

Conclusion:

The statement is True because:

• Both utility functions yield the same MRS equation.
• However, the second utility function $U(x, y) = x^2 y^2$ does not guarantee convex indifference curves, unlike the first utility function.

Would you like further clarification or more details on any part of the solution?

Expanding Questions:

1. What are the conditions for utility functions to produce convex indifference curves?
2. How would the shape of the indifference curves change if we modified the powers of the variables in the utility function?
3. Can two different utility functions always produce the same MRS? Under what conditions?
4. How does the concept of diminishing marginal returns relate to convex indifference curves?
5. How does the mathematical form of utility functions impact consumer preferences and choices?

Tip: Always check the convexity of indifference curves by analyzing how the MRS behaves as you vary the quantities of the goods.