Question : A consumer's utility function is $U=X+2 Y$. If the consumer is currently consuming $\mathrm{X}=4$ and $\mathrm{Y}=6$, 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: 6/4
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 calculate the ratio of the marginal utility of $\mathrm{X}$ to the marginal utility of $\mathrm{Y}$.
The marginal utility of $\mathrm{X}$ can be calculated as the partial derivative of the utility function with respect to $\mathrm{X}$, holding $\mathrm{Y}$ constant:
$
\mathrm{MUx}=\partial \mathrm{U} / \partial \mathrm{X}=1
$
The marginal utility of $\mathrm{Y}$ can be calculated as the partial derivative of the utility function with respect to $\mathrm{Y}$, holding $\mathrm{X}$ constant:
$
\mathrm{MUy}=\partial \mathrm{U} / \partial \mathrm{Y}=2
$
The MRS of $\mathrm{X}$ for $\mathrm{Y}$ is given by the ratio of the marginal utilities:
$
\operatorname{MRS}=\mathrm{MUx} / \mathrm{MUy}=1 / 2=1 / 2
$
