A producer is a price-taker on both the market for input factors labor and capital, and the market for end products. The cost of one unit of labor equals $w=2$, the cost of one unit of capital equals $r=8$, while the selling price of the end products equals $p=64$. The production function of this producer is given by $Y(L,K)=L^{\frac{1}{2}}K^{\frac{1}{4}}$. Determine the maximum profit.
Antwoord 1 correct
Correct
Antwoord 2 optie
$0$
Antwoord 2 correct
Fout
Antwoord 3 optie
$1024$
Antwoord 3 correct
Fout
Antwoord 4 optie
$48-\sqrt{2}$
Antwoord 4 correct
Fout
Antwoord 1 optie
$8192$
Antwoord 1 feedback
Correct: The revenue function is given by $R(L,K)=pY(L,K)=64L^{\frac{1}{2}}K^{\frac{1}{4}}$ and the cost function by $C(L,K)=wL+rK=2L+8K$, which results in the profit function
\[
\pi (L,K)=64L^{\frac{1}{2}}K^{\frac{1}{4}}-2L-8K.
\]
Since the partial derivatives of $\pi(L,K)$ equal
$\pi'_{L}(L,K) =32L^{-\frac{1}{2}}K^{\frac{1}{4}}-2$ and
$\pi'_{K}(L,K) =16L^{\frac{1}{2}}K^{-\frac{3}{4}}-8$,
the stationary points of $\pi(L,K)$ are solutions of the following system.
$$\begin{align*}
32L^{-\frac{1}{2}}K^{\frac{1}{4}}-2&=0\\
16L^{\frac{1}{2}}K^{-\frac{3}{4}}-8&=0
\end{align*}$$
Hence, $L^{\frac{1}{2}}=\frac{1}{2}K^{\frac{3}{4}}$ and therefore, $L=\frac{1}{4}K^{\frac{3}{2}}$.
Consequently, $32(\frac{1}{4}K^{\frac{3}{2}})^{-\frac{1}{2}}K^{\frac{1}{4}}=2$, which gives $2K^{-\frac{1}{2}}=\frac{1}{16}$. Thus, $K=1024$, which gives $L=8192$.
Hence, $(L,K)=(8192,1024)$ is the only stationary point. By the use of the criterion function we investigate whether or not this point is a maximum location. It holds that $\pi''_{LL}(L,K)=-16L^{-\frac{3}{2}}K^{\frac{1}{4}}$, $\pi''_{KK}(L,K)=-12L^{\frac{1}{2}}K^{-\frac{7}{4}}$ and $\pi''_{LK}(L,K)=8L^{-\frac{1}{2}}K^{-\frac{3}{4}}$, which implies that the criterion function is given by
$$\begin{align*}
C(L,K) &= \pi''_{LL}(L,K)\pi''_{KK}(L,K)-(\pi''_{LK}(L,K))^{2}\\
&=(-16L^{-\frac{3}{2}}K^{\frac{1}{4}}) \cdot (-12L^{\frac{1}{2}}K^{-\frac{7}{4}})-(8L^{-\frac{1}{2}}K^{-\frac{3}{4}})^2\\
&=192L^{-1}K^{-\frac{3}{2}}-64L^{-1}K^{-\frac{3}{2}}\\
&=128L^{-1}K^{-\frac{3}{2}}.\\
\end{align*}$$
Hence, as $C(1024,8192)>0$ and $\pi''_{LL}(1024,8192)<0$ it follows that $\pi (L,K)$ has a maximum at $(L,K)=(1024,8192)$, with value $\pi(1024,8192)=8192.$
Go on.
\[
\pi (L,K)=64L^{\frac{1}{2}}K^{\frac{1}{4}}-2L-8K.
\]
Since the partial derivatives of $\pi(L,K)$ equal
$\pi'_{L}(L,K) =32L^{-\frac{1}{2}}K^{\frac{1}{4}}-2$ and
$\pi'_{K}(L,K) =16L^{\frac{1}{2}}K^{-\frac{3}{4}}-8$,
the stationary points of $\pi(L,K)$ are solutions of the following system.
$$\begin{align*}
32L^{-\frac{1}{2}}K^{\frac{1}{4}}-2&=0\\
16L^{\frac{1}{2}}K^{-\frac{3}{4}}-8&=0
\end{align*}$$
Hence, $L^{\frac{1}{2}}=\frac{1}{2}K^{\frac{3}{4}}$ and therefore, $L=\frac{1}{4}K^{\frac{3}{2}}$.
Consequently, $32(\frac{1}{4}K^{\frac{3}{2}})^{-\frac{1}{2}}K^{\frac{1}{4}}=2$, which gives $2K^{-\frac{1}{2}}=\frac{1}{16}$. Thus, $K=1024$, which gives $L=8192$.
Hence, $(L,K)=(8192,1024)$ is the only stationary point. By the use of the criterion function we investigate whether or not this point is a maximum location. It holds that $\pi''_{LL}(L,K)=-16L^{-\frac{3}{2}}K^{\frac{1}{4}}$, $\pi''_{KK}(L,K)=-12L^{\frac{1}{2}}K^{-\frac{7}{4}}$ and $\pi''_{LK}(L,K)=8L^{-\frac{1}{2}}K^{-\frac{3}{4}}$, which implies that the criterion function is given by
$$\begin{align*}
C(L,K) &= \pi''_{LL}(L,K)\pi''_{KK}(L,K)-(\pi''_{LK}(L,K))^{2}\\
&=(-16L^{-\frac{3}{2}}K^{\frac{1}{4}}) \cdot (-12L^{\frac{1}{2}}K^{-\frac{7}{4}})-(8L^{-\frac{1}{2}}K^{-\frac{3}{4}})^2\\
&=192L^{-1}K^{-\frac{3}{2}}-64L^{-1}K^{-\frac{3}{2}}\\
&=128L^{-1}K^{-\frac{3}{2}}.\\
\end{align*}$$
Hence, as $C(1024,8192)>0$ and $\pi''_{LL}(1024,8192)<0$ it follows that $\pi (L,K)$ has a maximum at $(L,K)=(1024,8192)$, with value $\pi(1024,8192)=8192.$
Go on.
Antwoord 2 feedback
Antwoord 3 feedback
Wrong: The maximum profit is not equal to the value of $L$ in the maximum location.
See Minimum/maximum.
See Minimum/maximum.
Antwoord 4 feedback
Wrong: $(\frac{1}{4}K^{\frac{3}{2}})^{-\frac{1}{2}}\neq \frac{1}{4}K^{-\frac{3}{4}}$.
See Properties power functions.
See Properties power functions.