Lagrange multiplier problem
Hello I'm stuck on this problem.
P(L,K) = 100L^0.25 * K^0.75
P = production output
L = labour
K = capital
Suppose that labour costs $40 per unit and capital costs are $60 per unit. Units lagrange multipliers, find the least expensive way of producing 10 000 units of output
This is what i've done so far.
C(L,K) = 40L + 60K P(L,K) = 100L^0.25 x K^0.75 = 10 000
∆C = λ∆P
∆C = 40i + 60j ∆P = 25λ(K/L)^0.75 i + 75λ(L/K)^0.25 j
40 = 25λ(K/L)^0.75 60 = 75λ(L/K)^0.25
2(L/K)^0.25 = (L/K)^0.75
2L = K
P(L,2L) = 100L^0.25 x 2L^0.75 = 10 000
10 000 = 200L
L = 50
C(L,2L) = 40(50) + 60(100) = 8000
The examples I did in class were much different, they had values +/- for x, and you just use the - for the min.
I'm lost, I'm not sure if this is a min or a max.
Anyhelp will be greatful. =)