Question : A consumer's utility function is $\mathrm{U}=\mathrm{X}^{\wedge} 2+\mathrm{Y}^{\wedge} 2$. If the consumer is currently consuming $\mathrm{X}=4$ and $\mathrm{Y}=3$, 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) $3 / 4$
Let's calculate the marginal rate of substitution (MRS) of X for Y correctly.
The utility function is $\mathrm{U}=\mathrm{X}^{\wedge} 2+\mathrm{Y}^{\wedge} 2$.
To find the MRS, we need to calculate the ratio of the marginal utilities:
$\operatorname{MRS}=(\partial \mathrm{U} / \partial \mathrm{X}) /(\partial \mathrm{U} / \partial \mathrm{Y})$
Differentiating the utility function with respect to $\mathrm{X}$ :
$
\partial \mathrm{U} / \partial \mathrm{X}=2 \mathrm{X}
$
Differentiating the utility function with respect to $\mathrm{Y}$ :
$
\partial \mathrm{U} / \partial \mathrm{Y}=2 \mathrm{Y}
$
Substituting $\mathrm{X}=4$ and $\mathrm{Y}=3$ into the partial derivatives:
$
\begin{aligned}
& \partial \mathrm{U} / \partial \mathrm{X}=2 * 4=8 \\
& \partial \mathrm{U} / \partial \mathrm{Y}=2 * 3=6
\end{aligned}
$
Now, we can calculate the MRS:
$\mathrm{MRS}=(\partial \mathrm{U} / \partial \mathrm{X}) /(\partial \mathrm{U} / \partial \mathrm{Y})$
$\begin{aligned} & =8 / 6 \\ & =4 / 3\end{aligned}$
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