Question

given the production Q=L^1/2 K^1/2 price per unit of K and L is birr 2 and birr 4 respectively and total cost is birr 80 a) determine the maximum output subject to the cost constraint b)caculate the MRTS_l,k at equilibrium point

Answer 1

The optimal L and K will be at the point where

$MRTS=\dfrac{MPL}{MPK}=\dfrac{w}{r}$

Therefore

$MPL=\dfrac{1}{2}L^{-1/2}K^{1/2}\\[0.4cm] MPK=\dfrac{1}{2}L^{1/2}K^{-1/2}\\[0.4cm]$

If $w=4 \text{ and } r=2$ , then

$\dfrac{\frac{1}{2}L^{-1/2}K^{1/2}}{\frac{1}{2}L^{1/2}K^{-1/2}}=\dfrac{4}{2}\\[0.4cm] \dfrac{K}{L}=2\\[0.4cm] K=2L$

The total cost is equal to

$80=4L+2K$

Therefore

$80=4L+2(2L)\\[0.4cm] 80=4L+4L\\[0.3cm] 80=8L\\[0.3cm] L^*=\boxed{10}$

The optimal capital is equal to

$K=2\times 10\\[0.3cm] K^*=\boxed{20}$

At the optimal L and K, the MRTS is equal to

$MRTS=\dfrac{10}{5}\\[0.3cm] MRTS=\boxed{2}$