Question : A consumer's utility function is $\mathrm{U}=\mathrm{X}^{\wedge} 2+\mathrm{Y}^{\wedge} 2$. If the consumer is currently consuming $\mathrm{X}=3$ and $\mathrm{Y}=4$, what is the marginal rate of substitution (MRS) of $\mathrm{X}$ for $\mathrm{Y}$ ?
Option 1: 3/4
Option 2: 4/3
Option 3: 9/16
Option 4: 16/9
Correct Answer: 3/4
Solution : The correct answer is (a) $3 / 4$
To find the marginal rate of substitution (MRS) of $\mathrm{X}$ for $\mathrm{Y}$, we need to determine the rate 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} \text { for } \mathrm{Y})=-\mathrm{MUx} / \mathrm{MUy}
$
Given the utility function $\mathrm{U}=\mathrm{X}^{\wedge} 2+\mathrm{Y}^{\wedge} 2$, we can calculate the marginal utilities as follows:
$
\begin{aligned}
& \mathrm{MUx}=\mathrm{dU} / \mathrm{dX}=2 \mathrm{X}=2 * 3=6 \\
& \mathrm{MUy}=\mathrm{dU} / \mathrm{dY}=2 \mathrm{Y}=2 * 4=8
\end{aligned}
$
Now, we can substitute the values into the formula for MRS:
$
\operatorname{MRS}(\mathrm{X} \text { for } \mathrm{Y})=-\mathrm{MUx} / \mathrm{MUy}=-6 / 8=-3 / 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=3 / 4
$
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