Question : A consumer's utility function is $\mathrm{U}=\mathrm{X}^{\wedge} 0.5 \mathrm{Y}^{\wedge} 0.5$. If the consumer is currently consuming $\mathrm{X}=9$ and $\mathrm{Y}=16$, what is the marginal rate of substitution (MRS) of $\mathrm{X}$ for $\mathrm{Y}$ ?
Option 1: 2/3
Option 2: 3/2
Option 3: 4/9
Option 4: 9/4
Correct Answer: 3/2
Solution :
The correct answer is (b) $3 / 2$
Let's calculate the marginal rate of substitution (MRS) of X for Y correctly. The utility function is $\mathrm{U}=\mathrm{X}^{\wedge} 0.5 * \mathrm{Y}^{\wedge} 0.5$.
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}=0.5 * \mathrm{X}^{\wedge}(-0.5) * \mathrm{Y}^{\wedge} 0.5
$
Differentiating the utility function with respect to $\mathrm{Y}$ :
$
\partial \mathrm{U} / \partial \mathrm{Y}=0.5 * \mathrm{X}^{\wedge} 0.5 * \mathrm{Y}^{\wedge}(-0.5)
$
Substituting $\mathrm{X}=9$ and $\mathrm{Y}=16$ into the partial derivatives:
$
\begin{aligned}
& \partial \mathrm{U} / \partial \mathrm{X}=0.5 * 9^{\wedge}(-0.5) * 16^{\wedge} 0.5=0.5 *(1 / 3) * 4=2 / 3 \\
& \partial \mathrm{U} / \partial \mathrm{Y}=0.5 * 9^{\wedge} 0.5 * 16^{\wedge}(-0.5)=0.5 * 3 *(1 / 4)=3 / 4
\end{aligned}
$
Now, we can calculate the MRS:
$
\operatorname{MRS}=(2 / 3) /(3 / 4)=(2 / 3) *(4 / 3)=8 / 9
$
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