A producer is a price-taker with cost function $C(y)=4y^3-32y^2+96y$. Determine the supply function.
Antwoord 1 correct
Correct
Antwoord 2 optie
\[
y(p)=\left \{
\begin{array}{ll}
0 & {\rm if } \ p<4\\
2\frac{2}{3}+\frac{1}{24}\sqrt{48p-512} & {\rm if } \ p\geq 4
\end{array}
\right.
\]
y(p)=\left \{
\begin{array}{ll}
0 & {\rm if } \ p<4\\
2\frac{2}{3}+\frac{1}{24}\sqrt{48p-512} & {\rm if } \ p\geq 4
\end{array}
\right.
\]
Antwoord 2 correct
Fout
Antwoord 3 optie
\[
y(p)=\left \{
\begin{array}{ll}
0 & {\rm if } \ p<4\\
8+\frac{1}{8}\sqrt{16p-512} & {\rm if } \ p\geq 4
\end{array}
\right.
\]
y(p)=\left \{
\begin{array}{ll}
0 & {\rm if } \ p<4\\
8+\frac{1}{8}\sqrt{16p-512} & {\rm if } \ p\geq 4
\end{array}
\right.
\]
Antwoord 3 correct
Fout
Antwoord 4 optie
\[
y(p)=\left \{
\begin{array}{ll}
0 & {\rm if } \ p<32\\
4+\frac{1}{8}\sqrt{16p-512} & {\rm if } \ p\geq 32
\end{array}
\right.
\]
y(p)=\left \{
\begin{array}{ll}
0 & {\rm if } \ p<32\\
4+\frac{1}{8}\sqrt{16p-512} & {\rm if } \ p\geq 32
\end{array}
\right.
\]
Antwoord 4 correct
Fout
Antwoord 1 optie
\[
y(p)=\left \{
\begin{array}{ll}
0 & {\rm if } \ p<32\\
2\frac{2}{3}+\frac{1}{24}\sqrt{48p-512} & {\rm if } \ p\geq 32
\end{array}
\right.
\]
y(p)=\left \{
\begin{array}{ll}
0 & {\rm if } \ p<32\\
2\frac{2}{3}+\frac{1}{24}\sqrt{48p-512} & {\rm if } \ p\geq 32
\end{array}
\right.
\]
Antwoord 1 feedback
Correct: $AC(y)=4y^2-32y+96$ Then $AC'(y)=8y-32$ gives $y=4$. Since $AC''(y)=8$ it holds that $AC''(4)=8>0.$ Hence, according to the second-order condition for an extremum $AC(4)=32$ is the minimum of $AC(y)$.
The profit function of the producer is given by $\pi(y)=py-(4y^3-32y^2+96y)$. which gives
$$\begin{align*}
\pi'(y)=0
&\Leftrightarrow&
p-(12y^2-64y+96)=0\\
&\Leftrightarrow&
-12y^2+64y-96+p = 0.
\end{align*}$$
Hence,
\[
y=\frac{-64 - \sqrt{64^2 -4\cdot(-12)\cdot(-96+p)}}{-24} =2\tfrac{2}{3}+\tfrac{1}{24}\sqrt{48p-512}
\]
and
\[
y=\frac{-64 + \sqrt{64^2 -4\cdot(-12)\cdot(-96+p)}}{-24} =2\tfrac{2}{3}-\tfrac{1}{24}\sqrt{48p-512}.
\]
Since $\pi'(y)$ is a quadratic function with $a<0$ we find the maximum profit for an output quantity of $y=2\frac{2}{3}+\frac{1}{24}\sqrt{48p-512}$, whenever $p\geq 32$. We conclude that the supply function is defined by \[
y(p)=\left \{
\begin{array}{ll}
0 & {\rm if } \ p<32\\
2\tfrac{2}{3}+\tfrac{1}{24}\sqrt{48p-512} & {\rm if } \ p\geq 32
\end{array}
\right.
\]
Go on.
The profit function of the producer is given by $\pi(y)=py-(4y^3-32y^2+96y)$. which gives
$$\begin{align*}
\pi'(y)=0
&\Leftrightarrow&
p-(12y^2-64y+96)=0\\
&\Leftrightarrow&
-12y^2+64y-96+p = 0.
\end{align*}$$
Hence,
\[
y=\frac{-64 - \sqrt{64^2 -4\cdot(-12)\cdot(-96+p)}}{-24} =2\tfrac{2}{3}+\tfrac{1}{24}\sqrt{48p-512}
\]
and
\[
y=\frac{-64 + \sqrt{64^2 -4\cdot(-12)\cdot(-96+p)}}{-24} =2\tfrac{2}{3}-\tfrac{1}{24}\sqrt{48p-512}.
\]
Since $\pi'(y)$ is a quadratic function with $a<0$ we find the maximum profit for an output quantity of $y=2\frac{2}{3}+\frac{1}{24}\sqrt{48p-512}$, whenever $p\geq 32$. We conclude that the supply function is defined by \[
y(p)=\left \{
\begin{array}{ll}
0 & {\rm if } \ p<32\\
2\tfrac{2}{3}+\tfrac{1}{24}\sqrt{48p-512} & {\rm if } \ p\geq 32
\end{array}
\right.
\]
Go on.
Antwoord 2 feedback
Wrong: $y=4$ is the minimum location of the average cost function, not the minimum itself.
See Minimum/maximum.
See Minimum/maximum.
Antwoord 3 feedback
Wrong: $y=4$ is the minimum location of the average cost function, not the minimum itself.
See Minimum/maximum.
See Minimum/maximum.
Antwoord 4 feedback