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: 4/3
Solution : The correct answer is (b) $4 / 3$
To find the marginal rate of substitution (MRS) of $\mathrm{X}$ for $\mathrm{Y}$, we need to calculate the ratio of the marginal utilities of $\mathrm{X}$ and $\mathrm{Y}$.
The marginal utility of $\mathrm{X}$ (MUx) can be obtained by taking the partial derivative of the utility function with respect to $\mathrm{X}$ :
$
\mathrm{MUx}=\partial \mathrm{U} / \partial \mathrm{X}=2 \mathrm{X}
$
Similarly, the marginal utility of Y (MUy) can be obtained by taking the partial derivative of the utility function with respect to $\mathrm{Y}$ :
$
\mathrm{MUy}=\partial \mathrm{U} / \partial \mathrm{Y}=2 \mathrm{Y}
$
Now we can calculate the MRS:
$
\operatorname{MRS}=\mathrm{MUx} / \mathrm{MUy}=(2 \mathrm{X}) /(2 \mathrm{Y})=\mathrm{X} / \mathrm{Y}
$
Substituting $X=4$ and $Y=3$ into the equation, we get:
$
\operatorname{MRS}=4 / 3
$
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