Question

Suppose production function is
Q=k²L²-0.12k²-3.2L²
Then find the level of output where
APL=MPL,if k=15
MPK=0,if L=10

Answer 1

1.

$APL=\frac{Q}{L}$

$MPL=\frac{\Delta Q}{\Delta L}$

$Q=k^2L^2-0.12k^2-3.2L^2=15^2L^2-0.12\times15^2-3.2L^2=225L^2-27-3.2L^2=221.8L^2-27$

put Q into the formula:

$L=\sqrt{\frac{Q+0.12k^2}{k^2-3.2}}=\sqrt{\frac{Q+27}{221.8}}=\sqrt{L^2-\frac{27}{221.8}}$

APL=MPL

$Q=APL\times L$

put the found expressions in the formula and get Q

L cut on each other

Q=221.8L^2-27

$\Delta Q=APL\times\Delta L$

$\Delta Q=\frac{Q}{L}\times\Delta L$

$\Delta Q=\frac{221.8L^2-27}{\sqrt{L^2-\frac{27}{221.8}}}\times(\frac{1}{2\times \sqrt{L^2-\frac{27}{221.8}}})=110.9$

2.

$MPK=\frac{\Delta Q}{\Delta k}$

MPK=0 L=10

$\Delta Q=MPK\times\Delta k$

$\Delta Q=0$

Find the derivatives Q

Q'=2k2L-0.24k-6.4L=0

$2k2\times10-0.24k-6.4\times10=0$

40k-0.24k-64=0

39.76k=64

$k=\frac{64}{39.76}$

$Q'=2\frac{64}{39.76}2\times10-0.24\frac{64}{39.76}-6.4\times10=0$