We consider a production process in which the cost of one unit of labor $L$ is $w=4$ and the cost of one unit of capital $K$ is $r=1$, and in which the relation between labor, capital and output quantity is described by the production function $P(L,K)=\dfrac{2LK}{4L+K}$.
We determine the cost function of the production process, which can be found by solving the following cost minimization problem:
$$\begin{array}{ll}
\text{ minimize } &C(L,K)=4L+K\\
\text{ subject to }&\dfrac{2LK}{4L+K}=y,\\
\text{ where } & L>0 \ \text{ and } \ K>0.
\end{array}$$
Since
\[
P_L'(L,K)=\dfrac{2K\cdot(4L+K)-2LK\cdot 4}{(4L+K)^2}
\]
and
\[
P_K'(L,K)=\dfrac{2L\cdot(4L+K)-2LK\cdot 1}{(4L+K)^2}
\]
it follows that
\[
MRTS(L,K)=\dfrac{2K\cdot(4L+K)-2LK\cdot 4}{2L\cdot(4L+K)-2LK\cdot 1}
=
\dfrac{2K^2}{8L^2}
\]
Subsequently, we determine the solutions of the system of equations corresponding to the first-order condition for cost minimization:
$$\left\{ \begin{array}{lll}
MRTS(L,K)&=&\dfrac{4}{1}\\
\left\dfrac{2LK}{4L+K}&=&y
\end{array}\right.$$
From $\dfrac{2K^2}{8L^2}=\dfrac{4}{1}$ it follows that $K^2=16L^2$, or $K=4L$. Therefore, $y=\dfrac{2L\cdot 4L}{4L+4L}=\dfrac{8L^2}{8L}=L$. Since $L=y$ it follows that $K=4y$.
Hence, $C(L,K)=C(y,4y)=4y+4y=8y$.
We have to verify whether this is indeed the minimum cost function. If $L=2y$, then $y=\dfrac{4yK}{8y+K}$, which gives that $K=\frac{8}{3}y$. This would results in $C(L,K)=10\frac{2}{3}y$.
If $L=\frac{3}{4}y$, then $y=\dfrac{1\frac{1}{2}yK}{3y+K}$, which gives that $K=6y$. This would result in $C(L,K)=9y$.
Consequently, the minimum cost function is $C(y)=8y$.
