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 $\mathrm{Y}$ ?
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) $3 / 2$
The utility function is $\mathrm{U}=\mathrm{X}^{\wedge} 0.5 * \mathrm{Y}^{\wedge} 0.5$ The marginal utility of $\mathrm{X}$ (MUx) can be found by taking the partial derivative of the utility function with respect to $\mathrm{X}$ and multiplying it by 1 (since the exponent of $\mathrm{X}$ is 0.5 ): $ \mathrm{MUx}=0.5 * \mathrm{X}^{\wedge}(-0.5) * \mathrm{Y}^{\wedge} 0.5 $
Similarly, the marginal utility of Y (MUy) can be found by taking the partial derivative of the utility function with respect to $\mathrm{Y}$ and multiplying it by 1 (since the exponent of $\mathrm{Y}$ is 0.5 ): $ \mathrm{MUy}=0.5 * \mathrm{X}^{\wedge} 0.5 * \mathrm{Y}^{\wedge}(-0.5) $
Now, let's substitute the given values $\mathrm{X}=16$ and $Y=9$ into the equations: $ \begin{aligned} & \text { MUx }=0.5 * 16^{\wedge}(-0.5) * 9^{\wedge} 0.5 \\ & \quad=0.5 *(1 / 4) * 3 \\ & =3 / 8 \end{aligned} $ $ \mathrm{MUy}=0.5 * 16^{\wedge} 0.5^* 9^{\wedge}(-0.5) $
$ \begin{aligned} & =0.5 * 4 *(1 / 3) \\ & =2 / 3 \end{aligned} $
Finally, we can calculate the MRS: $ \text { MRS }=\text { MUx } / \text { MUy } $
$ \begin{aligned} & =(3 / 8) /(2 / 3) \\ & =(3 / 8) *(3 / 2) \\ & =9 / 16 \end{aligned} $
Therefore, answer is $3 / 2$
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 $\mathrm{U}=\mathrm{X}^{\wedge} 0.5 \mathrm{Y}^{\wedge} 0.5$. 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 $\mathrm{U}=\mathrm{X}^{\wedge} 2+\mathrm{Y}^{\wedge} 2$. 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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