Two goods have independent marginal utilities if \\[ \frac{d t U}{d Y d X} \quad \frac{d^{2} U}{d X d Y}=0 \\] Show that if we assume diminishing marginal utility for each good, then any utility function with independent marginal utilities will have a diminishing \(M R S\). Provide an example to show that the converse of this statement is not true. 3.10 a. Show that the CES function \\[ \begin{array}{c} x^{3} \\ 0 \end{array}+* \frac{Y}{T} \\] is homothetic. How does the \(M R S\) depend on the ratio \(\mathrm{F} / \mathrm{X}\) ? b. Show that your results from part (a) agree with Example 3.3 for the case \(6=1\) (perfect substitutes \()\) and \(5=0\) (Cobb-Douglas). c. Show that the \(M R S\) is strictly diminishing for all values of \(8<1\) d. Show that if \(X=Y\), the \(M R S\) for this function depends only on the relative sizes of a and j8. e. Calculate the \(M R S\) for this function when \(Y / X=.9\) and \(Y / X=1.1\) for the two cases \(8=5\) and \(8=-1 .\) What do you conclude about the extent to which the \(M R S\) changes in the vicinity of \(\mathrm{X}=\mathrm{F} ?\) How would you interpret this geometrically?

Starting from the given condition of independent marginal utilities: \[ \frac{d t U}{d Y d X} \frac{d^{2} U}{d X d Y}=0 \] Let's denote marginal utilities as: \[ MU_X = \frac{dU}{dX}, MU_Y = \frac{dU}{dY} \] Then, according to the condition above, we can rewrite it as: \[ \frac{dMU_Y}{dX} MU_X = 0 \] Diminishing marginal utility implies \( \frac{dMU_X}{dX} < 0 \) and \( \frac{dMU_Y}{dY} < 0 \) We can find the MRS as follows: \[ MRS = \frac{MU_X}{MU_Y} \] Now, we need to find the second derivative of MRS with respect to X and show that it is positive, which implies diminishing MRS: \[ \frac{d MRS}{dX} = \frac{\frac{dMU_X}{dX} \cdot MU_Y - MU_X\frac{dMU_Y}{dX}}{(MU_Y)^2} \] Given that independent marginal utilities condition: \(\frac{dMU_Y}{dX} = MU_X = 0\), we have: \[ \frac{d MRS}{dX} = \frac{\frac{dMU_X}{dX} \cdot MU_Y}{(MU_Y)^2} \] Now, we find the second derivative of MRS with respect to X: \[ \frac{d^2 MRS}{dX^2} = \frac{(\frac{d^2MU_X}{dX^2} \cdot MU_Y) \cdot (MU_Y)^2 - (\frac{dMU_X}{dX} \cdot MU_Y)^2}{(MU_Y)^4} \] Since diminishing marginal utility for X implies that \(\frac{dMU_X}{dX} < 0\) and \(\frac{d^2MU_X}{dX^2} \geq 0\), we can conclude that \(\frac{d^2 MRS}{dX^2} \geq 0\). Therefore, the MRS will be diminishing, proving the initial claim. 2. Converse of the statement not true (example)

Consider the following utility function: \[ U(X, Y) = X^2 + Y^2 \] Marginal utilities are: \[ MU_X = \frac{dU}{dX} = 2X, \ MU_Y = \frac{dU}{dY} = 2Y \] And MRS is: \[ MRS = \frac{MU_X}{MU_Y} = \frac{2X}{2Y} = \frac{X}{Y} \] Now we'll show that it has diminishing MRS, but doesn't have independent marginal utilities. First, let's find the second derivative of MRS with respect to X: \[ \frac{d^2 MRS}{dX^2} = \frac{d}{dX} \left( \frac{Y}{X^2} \right) = -\frac{2Y}{X^3} < 0 \] Since \(\frac{d^2 MRS}{dX^2} < 0\), the MRS is diminishing. However, the marginal utilities are not independent: \[ \frac{dMU_Y}{dX} = \frac{d(2Y)}{dX} = 0, \ \frac{dMU_X}{dY} = \frac{d(2X)}{dY} = 0 \] \[ \frac{dMU_Y}{dX} \cdot \frac{dMU_X}{dY} = 0 \cdot 0 \neq 0 \] This counterexample shows that the converse of the statement is not true, as even if MRS is diminishing, the marginal utilities might not be independent.