Correct Answer: 1/2

Solution : The correct answer is (a) $1 / 2$

To find the marginal rate of substitution (MRS) of $\mathrm{X}$ for $\mathrm{Y}$, we need to determine the ratio at which the consumer is willing to trade off $\mathrm{X}$ for $\mathrm{Y}$ while keeping the utility constant. Mathematically, the MRS is given by the negative ratio of the marginal utilities of $\mathrm{X}$ and $\mathrm{Y}$ :
$\operatorname{MRS}(\mathrm{X}$ for $\mathrm{Y})=-\mathrm{MUx} / \mathrm{MUy}$
Given the utility function $\mathrm{U}=\mathrm{X}+2 \mathrm{Y}$, we can calculate the marginal utilities as follows:
$
\begin{aligned}
& \operatorname{MUx}=\mathrm{dU} / \mathrm{dX}=1 \\
& \mathrm{MUy}=\mathrm{dU} / \mathrm{dY}=2
\end{aligned}
$

Now, we can substitute the values into the formula for MRS:
$\operatorname{MRS}(\mathrm{X}$ for $\mathrm{Y})=-\mathrm{MUx} / \mathrm{MUy}=-1 / 2=-1 / 2$
Since MRS is typically represented as a positive value, we take the absolute value:
$\mid \operatorname{MRS}(\mathrm{X}$ for $\mathrm{Y}) \mid=1 / 2$