Question : A consumer's utility function is $U=X^{\wedge} 0.5 \mathrm{Y}^{\wedge} 0.5$. If the consumer is currently consuming $\mathrm{X}=16$ and $\mathrm{Y}=9$, 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) $3 / 2$

The utility function is $\mathrm{U}=\mathrm{X}^{\wedge} 0.5 * \mathrm{Y}^{\wedge} 0.5$
The marginal utility of $\mathrm{X}$ (MUx) can be found by taking the partial derivative of the utility function with respect to $\mathrm{X}$ and multiplying it by 1 (since the exponent of $\mathrm{X}$ is 0.5 ):
$
\mathrm{MUx}=0.5 * \mathrm{X}^{\wedge}(-0.5) * \mathrm{Y}^{\wedge} 0.5
$

Similarly, the marginal utility of Y (MUy) can be found by taking the partial derivative of the utility function with respect to $\mathrm{Y}$ and multiplying it by 1 (since the exponent of $\mathrm{Y}$ is 0.5 ):
$
\mathrm{MUy}=0.5 * \mathrm{X}^{\wedge} 0.5 * \mathrm{Y}^{\wedge}(-0.5)
$

Now, let's substitute the given values $\mathrm{X}=16$ and $Y=9$ into the equations:
$
\begin{aligned}
& \text { MUx }=0.5 * 16^{\wedge}(-0.5) * 9^{\wedge} 0.5 \\
& \quad=0.5 *(1 / 4) * 3 \\
& =3 / 8
\end{aligned}
$
$
\mathrm{MUy}=0.5 * 16^{\wedge} 0.5^* 9^{\wedge}(-0.5)
$

$
\begin{aligned}
& =0.5 * 4 *(1 / 3) \\
& =2 / 3
\end{aligned}
$

Finally, we can calculate the MRS:
$
\text { MRS }=\text { MUx } / \text { MUy }
$

$
\begin{aligned}
& =(3 / 8) /(2 / 3) \\
& =(3 / 8) *(3 / 2) \\
& =9 / 16
\end{aligned}
$

Therefore, answer is $3 / 2$