Question

The prices of inputs K and L are given as $12 per unit and $3 per unit respectively, and a firm operates with the production function

Q=25 K ^0.5 L^0.5

Q=25K0.5L0.5

.

(i) What is the minimum cost of producing 1,250 units of output?

Answer 1

$Q= 25K^{0.5}L^{0.5}$

$MPK= 25\times 0.5K^{-0.5}L^{0.5}$

$= 12.5K^{-0.5}L^{0.5}$

$MPL= 25\times 0.5K^{0.5}L^{-0.5}$

$= 12.5K^{0.5}L^{-0.5}$

$\frac{MPL}{MPK}=\frac{w}{r}$

$\frac{12.5K^{0.5}L^{-0.5} }{12.5K^{-0.5}L^{0.5}}=\frac{3}{12}$

$\frac{K}{L}=\frac{3}{12}$

L= 4K

K= $\frac{1}{4}L$ = 0.25L

Plug values the above in the production function

$1250= 25(0.25L^{0.5})L^{0.25}$

1250= 31.25L

L= 40

1250= 25K$^{0.5}4K^{0.5}$

1250= 100K

K= 12.5

TC= wL+rK

= $(40\times3)+(12\times12.5)= 270$