Consider a production process in which the cost of one unit of labor $L$ is $w=1$ and the cost of one unit of capital $K$ is $r=2$, and in which the relation between labor, capital and output quantity is described by the production function $P(L,K)=\dfrac{LK}{L+2K}$. Determine the minimum cost function.
$C(y)=9y$
$C(y)=6y$
$C(y)=10\frac{1}{2}y$
$C(y)=8y$
Correct: We solve the following cost minimization problem:
$\begin{array}{ll}
\mbox{minimize} &C(L,K)=L+2K\\
\mbox{subject to}&\dfrac{LK}{L+2K}=y,\\
\mbox{where} & L>0 \ \mbox{and} \ K>0.
\end{array}
$
Since
\[
P_L'(L,K)=\dfrac{K\cdot(L+2K)-LK\cdot 1}{(L+2K)^2}
\]
and
\[
P_K'(L,K)=\dfrac{L\cdot(L+2K)-LK\cdot 2}{(L+2K)^2}
\]
it follows that
\[
MRTS(L,K)=\dfrac{K\cdot(L+2K)-LK\cdot 1}{L\cdot(L+2K)-LK\cdot 2}
=
\dfrac{2K^2}{L^2}
\]
Subsequently, we determine the solutions of the system of equations corresponding to the first-order condition for cost minimization:
$$\left\{ \begin{array}{lll}
MRTS(L,K)&=&\dfrac{1}{2}\\
\dfrac{LK}{L+2K}&=&y
\end{array}\right.$$
From $\dfrac{2K^2}{L^2}=\dfrac{1}{2}$ it follows that $L^2=4K^2$ and hence, $L=2K$. Therefore, $y=\dfrac{2KK}{2K+2K}=\frac{1}{2}K$, which gives $K=2y$. Consequently, $L=4y$.
Hence, $C(L,K)=C(4y,2y)=4y+2\cdot 2y=8y$.
We have to verify whether this is indeed the minimum cost function. If $L=3y$, then $y=\dfrac{3yK}{3y+2K}$, which gives that $K=3y$. This would result in $C(L,K)=9y$.
If $L=8y$, then $y=\dfrac{8yK}{8y+2K}$, which gives that $K=1\frac{1}{3}y$. This would result in $C(L,K)=10\frac{2}{3}y$.
Consequently, the minimum cost function is $C(y)=8y$.
Go on.
Wrong: $L^2=4K^2$ does not imply $L=4K$.
See Properties power functions.
Wrong: Note that $r=2$.
Try again.
Wrong: $L=2K$ and not $K=2L$.
See Example.