A consumer whose utility function is given by $U(x,y)=x^{\frac{1}{2}}y^{\frac{1}{4}}$ spends his income $I=12$ on the goods $x$ and $y$ with prices $p_1=4$ and $p_2=2$, respectively. Determine the bundle that results in maximum utility.
Antwoord 1 correct
Correct
Antwoord 2 optie
$2^{\frac{3}{4}}$
Antwoord 2 correct
Fout
Antwoord 3 optie
$(1,4)$
Antwoord 3 correct
Fout
Antwoord 4 optie
$4^{\frac{1}{4}}$
Antwoord 4 correct
Fout
Antwoord 1 optie
$(2,2)$
Antwoord 1 feedback
Correct: The information translates in the utility maximization problem
$\begin{array}{ll}
\mbox{maximize}&x^{\frac{1}{2}}y^{\frac{1}{4}}\\
\mbox{subject to}&4x+2y=12,\\
\mbox{where} & x\geq 0 \ \mbox{and} \ y \geq 0.
\end{array}
$
$MRS(x,y)={\dfrac{p_1}{p_2}}$ then results in the equation $\dfrac{\frac{1}{2}x^{-\frac{1}{2}}y^{\frac{1}{4}}}{\frac{1}{4}x^{\frac{1}{2}}y^{-\frac{3}{4}}}=\dfrac{4}{2}$, which gives $y=x$.
Then we use the budget equation: $4x+2x=12$ gives $x=2$. Therefore, $y=2$.
To verify that $(x,y)=(2,2)$ is indeed the bundle that gives maximum utility we observe that $U(0,6)=U(3,0)=0<2^{\frac{3}{4}}=U(2,2)$.
Hence, $(2,2)$ is the bundle that gives the maximum utility.
Go on.
$\begin{array}{ll}
\mbox{maximize}&x^{\frac{1}{2}}y^{\frac{1}{4}}\\
\mbox{subject to}&4x+2y=12,\\
\mbox{where} & x\geq 0 \ \mbox{and} \ y \geq 0.
\end{array}
$
$MRS(x,y)={\dfrac{p_1}{p_2}}$ then results in the equation $\dfrac{\frac{1}{2}x^{-\frac{1}{2}}y^{\frac{1}{4}}}{\frac{1}{4}x^{\frac{1}{2}}y^{-\frac{3}{4}}}=\dfrac{4}{2}$, which gives $y=x$.
Then we use the budget equation: $4x+2x=12$ gives $x=2$. Therefore, $y=2$.
To verify that $(x,y)=(2,2)$ is indeed the bundle that gives maximum utility we observe that $U(0,6)=U(3,0)=0<2^{\frac{3}{4}}=U(2,2)$.
Hence, $(2,2)$ is the bundle that gives the maximum utility.
Go on.
Antwoord 2 feedback
Wrong: The question is not to find the maximum utility, but to find the corresponding bundle.
Try again.
Try again.
Antwoord 3 feedback
Antwoord 4 feedback
Wrong: The question is not to find the maximum utility, but to find the corresponding bundle.
Try again.
Try again.