Question

Given the two-factor product in, Q = 150L0.5K

0.5, wage rate of labour equals to 50 and rental 

cost of capital equals to 40.

Determine the amounts of labour and capital that will minimize the cost of producing 1118 

units of output.

Answer 1

Minimize Cost of Production.

$\frac{MP_K}{r}=\frac{MP_L}{w}$

$Q=150L0.5K$

$Q=1118$

$MP_k=0.5(150L)$

$MP_k=75L$

$MP_l=150(0.5K)$

$MP_l=75K$

$w=50$

$r=40$

$\therefore \frac{75L}{40}=\frac{75K}{50}$

$3750L=3000K$

$K=1.25L$

but;

$1118=150L0.5K$

$1118=150L(0.5\times1.25L)$

$1118=93.75L^2$

thus;

$\bold{L=3.45=4}$

$\bold{K=1.25(4)=5}$