Question : A consumer's utility function is $\mathrm{U}=\mathrm{X}^{\wedge} 2+\mathrm{Y}^{\wedge} 2$. If the consumer is currently consuming $X=3$ and $Y=4$, 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) $3 / 4$ To find the marginal rate of substitution (MRS) of X for Y, we need to calculate the ratio of the marginal utilities of $\mathrm{X}$ and $\mathrm{Y}$.
The utility function is $\mathrm{U}=\mathrm{X}^{\wedge} 2+\mathrm{Y}^{\wedge} 2$. To find the marginal utility of $\mathrm{X}$, we differentiate the utility function with respect to $\mathrm{X}$ :
$ \partial \mathrm{U} / \partial \mathrm{X}=2 \mathrm{X} $
To find the marginal utility of $\mathrm{Y}$, we differentiate the utility function with respect to $\mathrm{Y}$ : $ \partial \mathrm{U} / \partial \mathrm{Y}=2 \mathrm{Y} $
Now we can calculate the MRS: $ \begin{aligned} \operatorname{MRS} & =(\partial \mathrm{U} / \partial \mathrm{X}) /(\partial \mathrm{U} / \partial \mathrm{Y}) \\ & =(2 \mathrm{X}) /(2 \mathrm{Y}) \\ & =\mathrm{X} / \mathrm{Y} \\ & =3 / 4 \end{aligned} $
Question : A consumer's utility function is $\mathrm{U}=\mathrm{X}^{\wedge} 2+\mathrm{Y}^{\wedge} 2$. If the consumer is currently consuming $\mathrm{X}=3$ and $\mathrm{Y}=4$, what is the marginal rate of substitution (MRS) of $\mathrm{X}$ for $\mathrm{Y}$ ?
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 $\mathrm{Y}$ ?
Question : A consumer's utility function is $U=X^{\wedge} 0.5 \mathrm{Y}^{\wedge} 0.5$. If the consumer is currently consuming $\mathrm{X}=16$ and $\mathrm{Y}=9$, what is the marginal rate of substitution (MRS) of $\mathrm{X}$ for $\mathrm{Y}$ ?
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