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 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 $\mathrm{U}=\mathrm{X}+2 \mathrm{Y}$, we can calculate the marginal utilities as follows:
$
\begin{aligned}
& \operatorname{MUx}=\mathrm{dU} / \mathrm{dX}=1 \\
& \mathrm{MUy}=\mathrm{dU} / \mathrm{dY}=2
\end{aligned}
$
Now, we can substitute the values into the formula for MRS:
$\operatorname{MRS}(\mathrm{X}$ for $\mathrm{Y})=-\mathrm{MUx} / \mathrm{MUy}=-1 / 2=-1 / 2$
Since MRS is typically represented as a positive value, we take the absolute value:
$\mid \operatorname{MRS}(\mathrm{X}$ for $\mathrm{Y}) \mid=1 / 2$
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