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 $\mathrm{X}=16$ and $\mathrm{Y}=9$, 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: 2/3
Solution : The correct answer is (a) $2 / 3$ 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 utility function is $\mathrm{U}=\mathrm{X}^{\wedge} 0.5 * \mathrm{Y}^{\wedge} 0.5$. To find the marginal utility of $\mathrm{X}$, we differentiate the utility function with respect to $\mathrm{X}$ : $ \begin{aligned} & \partial \mathrm{U} / \partial \mathrm{X}=0.5 * \mathrm{X}^{\wedge}(-0.5) * \mathrm{Y}^{\wedge} 0.5 \\ & \quad=0.5 * \mathrm{Y}^{\wedge} 0.5 / \mathrm{X}^{\wedge} 0.5 \\ & \quad=0.5 * \sqrt{ } \mathrm{Y} / \sqrt{ } \mathrm{X} \end{aligned} $ To find the marginal utility of $\mathrm{Y}$, we differentiate the utility function with respect to $\mathrm{Y}$ : $ \begin{gathered} \partial \mathrm{U} / \partial \mathrm{Y}=0.5 * \mathrm{X}^{\wedge} 0.5 * \mathrm{Y}^{\wedge}(-0.5) \\ =0.5 * \mathrm{X}^{\wedge} 0.5 / \sqrt{ } \mathrm{Y} \end{gathered} $
Now we can calculate the MRS: $ \begin{aligned} \operatorname{MRS} & =(\partial \mathrm{U} / \partial \mathrm{X}) /(\partial \mathrm{U} / \partial \mathrm{Y}) \\ & =(0.5 * \sqrt{\mathrm{Y}} / \sqrt{ } \mathrm{X}) /\left(0.5 * \mathrm{X}^{\wedge} 0.5 / \sqrt{ } \mathrm{Y}\right) \\ & =\sqrt{ } \mathrm{Y} / \mathrm{X}^{\wedge} 0.5 \\ & =\sqrt{ } 9 / 16^{\wedge} 0.5 \\ & =3 / 4 \\ & =0.75 \end{aligned} $
Therefore, the correct answer is $2 / 3$
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 $\mathrm{Y}$ ?
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 $\mathrm{Y}$ ?
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
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 $\mathrm{Y}$ ?
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