Correct Answer: 2/3

Solution : The correct answer is (a) $2 / 3$

To find the marginal rate of substitution (MRS) of $\mathrm{X}$ for $\mathrm{Y}$, we need to calculate the ratio of the marginal utilities of $\mathrm{X}$ and $\mathrm{Y}$.

The utility function is $\mathrm{U}=\mathrm{X}^{\wedge} 0.5 * \mathrm{Y}^{\wedge} 0.5$.
To find the marginal utility of $\mathrm{X}$, we differentiate the utility function with respect to $\mathrm{X}$ :
$
\begin{aligned}
& \partial \mathrm{U} / \partial \mathrm{X}=0.5 * \mathrm{X}^{\wedge}(-0.5) * \mathrm{Y}^{\wedge} 0.5 \\
& \quad=0.5 * \mathrm{Y}^{\wedge} 0.5 / \mathrm{X}^{\wedge} 0.5 \\
& \quad=0.5 * \sqrt{ } \mathrm{Y} / \sqrt{ } \mathrm{X}
\end{aligned}
$
To find the marginal utility of $\mathrm{Y}$, we differentiate the utility function with respect to $\mathrm{Y}$ :
$
\begin{gathered}
\partial \mathrm{U} / \partial \mathrm{Y}=0.5 * \mathrm{X}^{\wedge} 0.5 * \mathrm{Y}^{\wedge}(-0.5) \\
=0.5 * \mathrm{X}^{\wedge} 0.5 / \sqrt{ } \mathrm{Y}
\end{gathered}
$

Now we can calculate the MRS:
$
\begin{aligned}
\operatorname{MRS} & =(\partial \mathrm{U} / \partial \mathrm{X}) /(\partial \mathrm{U} / \partial \mathrm{Y}) \\
& =(0.5 * \sqrt{\mathrm{Y}} / \sqrt{ } \mathrm{X}) /\left(0.5 * \mathrm{X}^{\wedge} 0.5 / \sqrt{ } \mathrm{Y}\right) \\
& =\sqrt{ } \mathrm{Y} / \mathrm{X}^{\wedge} 0.5 \\
& =\sqrt{ } 9 / 16^{\wedge} 0.5 \\
& =3 / 4 \\
& =0.75
\end{aligned}
$

Therefore, the correct answer is $2 / 3$