Question : A consumer's utility function is $U=X Y$. If the consumer is currently consuming $\mathrm{X}=5$ and $\mathrm{Y}=4$, what is the marginal rate of substitution (MRS) of $\mathrm{X}$ for $\mathrm{Y}$ ?
Option 1: 4/5
Option 2: 5/4
Option 3: 20/16
Option 4: 16/20
Correct Answer: 4/5
Solution : The correct answer is (a) $4 / 5$
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 $\mathrm{U}=\mathrm{XY}$.
To find the marginal utility of $\mathrm{X}$, we differentiate the utility function with respect to $\mathrm{X}$ :
$
\partial \mathrm{U} / \partial \mathrm{X}=\mathrm{Y}
$
To find the marginal utility of $\mathrm{Y}$, we differentiate the utility function with respect to $\mathrm{Y}$ :
$
\partial \mathrm{U} / \partial \mathrm{Y}=\mathrm{X}
$
Now we can calculate the MRS:
$
\begin{gathered}
\mathrm{MRS}=(\partial \mathrm{U} / \partial \mathrm{X}) /(\partial \mathrm{U} / \partial \mathrm{Y}) \\
\quad=\mathrm{Y} / \mathrm{X}
\end{gathered}
$
Substituting $\mathrm{X}=5$ and $\mathrm{Y}=4$ :
$
\operatorname{MRS}=4 / 5
$
Therefore, the marginal rate of substitution (MRS) of $\mathrm{X}$ for $\mathrm{Y}$ is $4 / 5$.
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