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 $r=4$, and in which the relation between labor, capital and output quantity is described by the production function $P(L,K)=\sqrt{LK}$. Determine the minimum cost function.
$C(y)=5y$
$C(y)=2y$
$C(y)=16y$
$C(y)=4y$
Correct: We solve the following cost minimization problem:
$\begin{array}{ll}
\mbox{minimize} &C(L,K)=L+4K\\
\mbox{subject to}&\sqrt{LK}=y,\\
\mbox{where} & L>0 \ \mbox{and} \ K>0.
\end{array}
$
Since
\[
P_L'(L,K)=\frac{1}{2}L^{-\frac{1}{2}}K^{\frac{1}{2}}
\]
and
\[
P_K'(L,K)=\frac{1}{2}L^{\frac{1}{2}}K^{-\frac{1}{2}}
\]
it follows that
\[
MRTS(L,K)=\dfrac{\frac{1}{2}L^{-\frac{1}{2}}K^{\frac{1}{2}}}{\frac{1}{2}L^{\frac{1}{2}}K^{-\frac{1}{2}}}
=
\dfrac{K}{L}
\]
Subsequently, we determine the solutions of the system of equations corresponding to the first-order condition for cost minimization:
$$\left\{ \begin{array}{lll}
MRTS(L,K)&=&\dfrac{1}{4}\\
\sqrt{LK}&=&y
\end{array}\right.$$
From $\dfrac{K}{L}=\dfrac{1}{4}$ it follows that $L=4K$. Therefore, $y=\sqrt{4K\cdot K}=2K$, which gives $K=\frac{1}{2}y$. Consequently, $L=2y$.
Hence, $C(L,K)=C(2y,\frac{1}{2}y)=2y+4\cdot \frac{1}{2}y=4y$.
We have to verify whether this is indeed the minimum cost function. If $L=y$, then $y=\sqrt{yK}$, which gives that $K=y$. This would result in $C(L,K)=5y$.
If $L=4y$, then $y=\sqrt{4yK}$, which gives that $K=\frac{1}{4}y$. This would result in $C(L,K)=5y$.
Consequently, the minimum cost function is $C(y)=4y$.
Go on.
Wrong: Production can be cheaper.
See Example.
Wrong: $\sqrt{4K^2}\neq 4K$.
See Properties power functions.
Wrong: $2K=y$ and not $K=2y$.
See Example.