Exercise 2 | Wiskunde op Tilburg University

The demand function of a good is given by

$q_d(p)=\frac{22}{p^2}$ ,

$(p> 0)$ and the supply function by

$q_s(p)=p^2+9$ ,

$(p\geq 0)$ . Determine all the market equilibria.

Antwoord 1 correct

Correct

Antwoord 2 optie

$(q,p)=(5.5,2)$

Antwoord 2 correct

Fout

Antwoord 3 optie

$(q,p)=(5.5,2)$ and

$(q,p)=(-11, 130)$

Antwoord 3 correct

Fout

Antwoord 4 optie

$(q,p)=(11,-\sqrt{2})$ and

$(q,p)=(11,\sqrt{2})$

Antwoord 4 correct

Fout

Antwoord 1 optie

$(q,p)=(11,\sqrt{2})$

Antwoord 1 feedback

Correct:

$\begin{align*} q_d(p)=q_s(p)& \Leftrightarrow \frac{22}{p^2}=p^2+9\\ & \Leftrightarrow 22=p^4+9p^2\\ & \Leftrightarrow p^4+9p^2-22=0. \end{align*}$

Substitute

$x=p^2$ . Then

$\begin{align*} x^2+9x-22= 0 & \Leftrightarrow (x+11)(x-2)=0\\ & \Leftrightarrow x=-11 \mbox{ or } x=2. \end{align*}$

$p^2=-11$ gives no solutions.

$p^2=2$ gives

$p=-\sqrt{2}$ or

$p=\sqrt{2}$ . Due to non-negativity

$p=-\sqrt{2}$ is not possible.

Hence,

$(q,p)=(11,\sqrt{2})$ is the market equilbrium.

Go on.

Antwoord 2 feedback

Wrong: Note that

$p^2=2$ , not

$p=2$ .

See for instance Feature logarithmic functions: Example (film).

Antwoord 3 feedback

Wrong: A quantity in a market equilibrium cannot be negative.

See Market equilibrium.

Antwoord 4 feedback

Wrong: A price cannot be negative.

See Market equilibrium.