Exercise 3 | Wiskunde op Tilburg University

Which of the following graphs depicts the indifference curves of the utility function

$U(x,y)=4x^{\tfrac{3}{4}}y^{\tfrac{1}{4}}$ ,

$(x>0, y>0)$ with values

$k=2$ ,

$k=4$ and

$k=8$ ?

Antwoord 1 correct

Correct

Antwoord 2 optie

Antwoord 2 correct

Fout

Antwoord 3 optie

Antwoord 3 correct

Fout

Antwoord 4 optie

Antwoord 4 correct

Fout

Antwoord 1 optie

Antwoord 1 feedback

Correct: We are dealing with a utility function that is only defined for

$x>0$ and

$y>0$ . The rewriting of

$4x^{\tfrac{3}{4}}y^{\tfrac{1}{4}} = k$ for

$k=2$ gives:

$4x^{\tfrac{3}{4}}y^{\tfrac{1}{4}}= 2 \iff y^{\tfrac{1}{4}} = \dfrac{1}{2x^{\tfrac{3}{4}}} \iff y = \pm\dfrac{1}{16x^3} \stackrel{y\geq0}{=}\dfrac{1}{16x^3}.$
The graph of this is the left curve. The rewriting of

$4x^{\tfrac{3}{4}}y^{\tfrac{1}{4}} = k$ for

$k=4$ gives:

$4x^{\tfrac{3}{4}}y^{\tfrac{1}{4}}= 4 \iff y^{\tfrac{1}{4}} = \dfrac{1}{x^{\tfrac{3}{4}}} \iff y = \pm\dfrac{1}{x^3} \stackrel{y\geq0}{=}\dfrac{1}{x^3}.$
The graph of this is the curve in the middle. The rewriting of

$4x^{\tfrac{3}{4}}y^{\tfrac{1}{4}} = k$ for

$k=8$ gives:

$4x^{\tfrac{3}{4}}y^{\tfrac{1}{4}}= 8 \iff y^{\tfrac{1}{4}} = \dfrac{2}{x^{\tfrac{3}{4}}} \iff y = \pm\dfrac{16}{x^3} \stackrel{x\geq0}{=}\dfrac{16}{x^3}.$
The graph of this is the curve is the right curve.

Go on.

Antwoord 2 feedback

Wrong: How many indifference curves are asked for?

Try again.

Antwoord 3 feedback

Wrong: Level curves, and hence also indifference curves, do not intersect.

See Level curves, Example 1 and Example 2.

Antwoord 4 feedback

Wrong: What are the domains of

$x$ and

$y$ ?

Try again.