Question : A consumer's utility function is $\mathrm{U}=\mathrm{X}^{\wedge} 0.5 \mathrm{Y}^{\wedge} 0.5$. If the consumer is currently consuming $\mathrm{X}=9$ and $\mathrm{Y}=16$, what is the marginal rate of substitution (MRS) of $\mathrm{X}$ for $\mathrm{Y}$ ?

Option 1: 2/3

Option 2: 3/2

Option 3: 4/9

Option 4: 9/4

Correct Answer: 3/2

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

To find the marginal rate of substitution (MRS) of $\mathrm{X}$ for $\mathrm{Y}$, we need to take the partial derivative of the utility function with respect to $\mathrm{X}$ (holding $\mathrm{Y}$ constant) and divide it by the partial derivative of the utility function with respect to $\mathrm{Y}$ (holding X constant).

The utility function is $\mathrm{U}=\mathrm{X}^{\wedge} 0.5 * \mathrm{Y}^{\wedge} 0.5$
Taking the partial derivative of $\mathrm{U}$ with respect to $\mathrm{X}$ :
$
\partial \mathrm{U} / \partial \mathrm{X}=0.5 * \mathrm{X}^{\wedge}(-0.5) * \mathrm{Y}^{\wedge} 0.5
$

Taking the partial derivative of $\mathrm{U}$ with respect to $\mathrm{Y}$ :
$
\partial \mathrm{U} / \partial \mathrm{Y}=0.5 * \mathrm{X}^{\wedge} 0.5 * \mathrm{Y}^{\wedge}(-0.5)
$

Now, we can calculate the MRS of $\mathrm{X}$ for $\mathrm{Y}$ :
$
\begin{aligned}
\mathrm{MRS} & =(\partial \mathrm{U} / \partial \mathrm{X}) /(\partial \mathrm{U} / \partial \mathrm{Y}) \\
& =\left(0.5 * \mathrm{X}^{\wedge}(-0.5) * \mathrm{Y}^{\wedge} 0.5\right) /\left(0.5 * \mathrm{X}^{\wedge} 0.5 * \mathrm{Y}^{\wedge}(-0.5)\right) \\
& =\mathrm{Y}^{\wedge} 0.5 / \mathrm{X}^{\wedge} 0.5 \\
& =\sqrt{ } \mathrm{Y} / \sqrt{ } \mathrm{X}
\end{aligned}
$

Substituting $\mathrm{X}=9$ and $\mathrm{Y}=16$ into the MRS equation:
$
\begin{gathered}
\text { MRS }=\sqrt{ } 16 / \sqrt{ } 9 \\
=4 / 3
\end{gathered}
$

Therefore, the correct answer is $3 / 2$