Exercise 3 | Wiskunde op Tilburg University

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.

Antwoord 2 feedback

Wrong: Note that

$\dfrac{\frac{1}{2\sqrt{x}}}{\frac{1}{2\sqrt{y}}}\neq \dfrac{\sqrt{x}}{\sqrt{y}}$ .

Try again.

Antwoord 3 feedback

Wrong:

$\dfrac{\sqrt{y}}{\sqrt{x}}\neq \dfrac{y}{x}$ .

Try again.

Antwoord 4 feedback

Wrong:

$U(1,64)\neq 216$ .

Try again.