Question : A consumer's utility function is $U=X Y$. If the consumer is currently consuming $\mathrm{X}=4$ and $\mathrm{Y}=5$, what is the marginal rate of substitution (MRS) of $\mathrm{X}$ for $\mathrm{Y}$ ?

Option 1: 1/5

Option 2: 4/5

Option 3: 5/4

Option 4: 5/1

Correct Answer: 5/4

Solution : The correct answer is (c) $5 / 4$

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 $\mathrm{U}=\mathrm{XY}$, we can calculate the marginal utilities as follows:
$\mathrm{MUx}=\mathrm{dU} / \mathrm{dX}=\mathrm{Y}=5$ (since the derivative of $\mathrm{X}$ with respect to $\mathrm{X}$ is 1 , and the derivative of $\mathrm{Y}$ with respect to $\mathrm{X}$ is 0 )

$\mathrm{MUy}=\mathrm{dU} / \mathrm{dY}=\mathrm{X}=4$ (since the derivative of $\mathrm{X}$ with respect to $\mathrm{Y}$ is 0 , and the derivative of $\mathrm{Y}$ with respect to $\mathrm{Y}$ is 1 )

Now, we can substitute the values into the formula for MRS:
$
\operatorname{MRS}(\mathrm{X} \text { for } \mathrm{Y})=-\mathrm{MUx} / \mathrm{MUy}=-5 / 4=-5 / 4
$

Since MRS is typically represented as a positive value, we take the absolute value:
$
\mid \operatorname{MRS}(\mathrm{X} \text { for } \mathrm{Y}) \mid=5 / 4
$