Question

For clarity assume that the MPL=60-2L and
MPK =200-5K
The budget still remains at sh. 390, with the factor prices as sh.4 and sh. 20 for labour and capital respectively

Answer 1

MPL=60-2L, MPK =200-5K, B = sh. 390, PL = sh.4, PK = sh. 20.
Profit is maximized when MPK/PK = MPL/PL and B = PK*K + PL*L, so:
(200 - 5K)/K = (60 - 2L)/L
20K + 4L = 390 -> L = 97.5 - 5K,
(200 - 5K)/K = (60 - 2*(97.5 - 5K))/(97.5 - 5K),
(10K - 135)*K = (200 - 5K)*(97.5 - 5K),
10K^2 - 135K = 19500 - 487.5 - 1000K + 25K^2,
-15K^2 + 865K - 19012.5 = 0,
15K^2 - 865K + 19012.5 = 0,
3K^2 - 173K + 3802.5 = 0,
D = 173^2 - 4*3*3802.5 < 0, so there is no roots.