Question

Given Q=100K^0.5L^0.5 ,w= 30, r=40

i) Find the quantity of labour and capital that the firm should use in order to minimize the cost of

producing 1444 units of output

ii) What is this minimum cost?

Answer 1

Given:

$Q = 100K^{0.5}L^{0.5}\\w =30\\r =4$

$MP_L=\frac{dQ}{dL}\\MP_L=100K^{0.5} × 0.5L^{-0.5}\\MP_L=50K^{0.5}L^{-0.5}\\MP_K = \frac{dQ}{dK}\\MP_K=100 × 0.5 K^{-0.5} L^{0.5}\\MP_K = 50K^{-0.5}L^{0.5}$

Cost minimizes at, 

$\frac{w}{r} = \frac{MP_L}{MP_K}\\\frac{30}{40}=\frac{50K^{0.5}L^{-0.5}}{50K^{-0.5}L^{0.5}}\\\frac{3}{4}=\frac{K}{L}\\3L=4K\\or,\\ K = \frac{3}{4}L\\L=\frac{4}{3}K\\K=\frac{3}{4}L$

Insert values in:

$Q = 100K^{0.5}×\frac{4}{3}K^{0.5}\\Q = \frac{400}{3} K\\When\space Q = 1444\\1444 = \frac{400}{3}K\\K =10.83\\ for L,\\Q = 100L^{0.5}×\frac{3}{4}L^{0.5}\\Q = \frac{300}{4} L\\When\space Q = 1444\\ 1444=\frac{300}{4} L\\L =19.25$

b). 

$C = wL + rK\\ C =30×19.25+40×10.83\\ C = 1010.7$