Question : A consumer's utility function is $U=X+2 Y$. If the consumer is currently consuming $\mathrm{X}=5$ and $\mathrm{Y}=3$, what is the marginal rate of substitution (MRS) of $\mathrm{X}$ for $\mathrm{Y}$ ?
Option 1: 1/2
Option 2: 2/3
Option 3: 3/2
Option 4: 5/3
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 calculate the ratio of the marginal utility of $\mathrm{X}$ to the marginal utility of $\mathrm{Y}$.
The utility function is $\mathrm{U}=\mathrm{X}+2 \mathrm{Y}$.
To find the marginal utility of $\mathrm{X}$, we differentiate the utility function with respect to $\mathrm{X}$ :
$
\partial \mathrm{U} / \partial \mathrm{X}=1
$
To find the marginal utility of $\mathrm{Y}$, we differentiate the utility function with respect to $\mathrm{Y}$ :
$
\partial \mathrm{U} / \partial \mathrm{Y}=2
$
Now we can calculate the MRS:
$
\begin{aligned}
\operatorname{MRS} & =(\partial \mathrm{U} / \partial \mathrm{X}) /(\partial \mathrm{U} / \partial \mathrm{Y}) \\
& =1 / 2 \\
& =1 / 2
\end{aligned}
$
Therefore, the marginal rate of substitution (MRS) of $\mathrm{X}$ for $\mathrm{Y}$ is $1 / 2$.
