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 calculate the marginal rate of substitution (MRS) of X for Y, we need to find the ratio of the marginal utility of $\mathrm{X}$ to the marginal utility of $\mathrm{Y}$.
In this case, the utility function is $\mathrm{U}=\mathrm{XY}$. To find the marginal utility of $\mathrm{X}$ (MUx), we differentiate the utility function with respect to $\mathrm{X}$, holding $\mathrm{Y}$ constant:
$
\mathrm{MUx}=\partial \mathrm{U} / \partial \mathrm{X}=\mathrm{Y}
$
To find the marginal utility of $\mathrm{Y}$ (MUy), we differentiate the utility function with respect to $\mathrm{Y}$, holding $\mathrm{X}$ constant:
$
\mathrm{MUy}=\partial \mathrm{U} / \partial \mathrm{Y}=\mathrm{X}
$
Substituting the given values $\mathrm{X}=5$ and $\mathrm{Y}=4$, we get:
$
\begin{aligned}
& \operatorname{MUx}=\mathrm{Y}=4 \\
& \mathrm{MUy}=\mathrm{X}=5
\end{aligned}
$
The MRS of $\mathrm{X}$ for $\mathrm{Y}$ is the ratio of MUx to MUy:
$
\operatorname{MRS}=\mathrm{MUx} / \mathrm{MUy}=4 / 5
$
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