Consider the utility function $U(x,y)=(\sqrt{x}+\sqrt{y})^3$, $(x>0, y>0)$. Determine the point $(x,y)$ where the utility is equal to 216 and the marginal rate of substitution equals $\dfrac{1}{8}$.
Antwoord 1 correct
Correct
Antwoord 2 optie
$(x,y)=(\frac{4}{9},28\frac{4}{9})$.
Antwoord 2 correct
Fout
Antwoord 3 optie
$(x,y)=((\frac{6}{1+\sqrt{\frac{1}{8}}})^2,\frac{1}{8}(\frac{6}{1+\sqrt{\frac{1}{8}}})^2)$
Antwoord 3 correct
Fout
Antwoord 4 optie
$(x,y)=(1,64)$
Antwoord 4 correct
Fout
Antwoord 1 optie
$(x,y)=(28\frac{4}{9},\frac{4}{9})$
Antwoord 1 feedback
Correct: $MRS(x,y)=\dfrac{\sqrt{y}}{\sqrt{x}}=\dfrac{1}{8}$ gives $y=\frac{1}{64}x$.
Hence, $U(x,y)=(\sqrt{x}+\sqrt{\frac{1}{64}x})^3=(\frac{9}{8}\sqrt{x})^3=216$ gives $\frac{9}{8}\sqrt{x}=6$ and hence, $\sqrt{x}=\frac{16}{3}$ and therefore, $x=28\frac{4}{9}$. Consequently, $y=\frac{4}{9}$.
Go on.
Hence, $U(x,y)=(\sqrt{x}+\sqrt{\frac{1}{64}x})^3=(\frac{9}{8}\sqrt{x})^3=216$ gives $\frac{9}{8}\sqrt{x}=6$ and hence, $\sqrt{x}=\frac{16}{3}$ and therefore, $x=28\frac{4}{9}$. Consequently, $y=\frac{4}{9}$.
Go on.
Antwoord 2 feedback
Wrong: Note that $\dfrac{\frac{1}{2\sqrt{x}}}{\frac{1}{2\sqrt{y}}}\neq \dfrac{\sqrt{x}}{\sqrt{y}}$.
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Try again.
Antwoord 3 feedback
Wrong: $\dfrac{\sqrt{y}}{\sqrt{x}}\neq \dfrac{y}{x}$.
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Try again.
Antwoord 4 feedback
Wrong: $U(1,64)\neq 216$.
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Try again.