constrained optimisation: Production function F(L,K) = 2xLK....

[SB_]

Production function F(L,K) = 2xLK, where x is a positive parameter, L is the amount of labour, and K is the amount of capital employed.
Cost of capital = r and cost of labour = w
How much capital and labour will be used and what do these quantities depend on? (Hint: suppose q units of output should be produced at minimal costs.)
Here is what I did:
Minimise wL + rK subject to F(L,K) = Q
2xLK = Q is the same as K = Q / 2xL
Substituting this into the objective equation gives:
wL + rQ / 2xL
The first order condition is : w – (2xrQ / (2xL)^2 ) = 0
Rearranging wL + rK gives : w = rK / L
Substituting gives :
rK / L – (2xrQ / (2xL)^2 ) = 0
giving Q = 2KL
However I think I need the answer in terms of either K or L, not both. Anybody know how to solve / or what I’m doing wrong?

 

[Ishuda]

Production function F(L,K) = 2xLK, where x is a positive parameter, L is the amount of labour, and K is the amount of capital employed.
Cost of capital = r and cost of labour = w
How much capital and labour will be used and what do these quantities depend on? (Hint: suppose q units of output should be produced at minimal costs.)
Here is what I did:
Minimise wL + rK subject to F(L,K) = Q
2xLK = Q is the same as K = Q / 2xL
Substituting this into the objective equation gives:
wL + rQ / 2xL
The first order condition is : w – (2xrQ / (2xL)^2 ) = 0
Rearranging wL + rK gives : w = rK / L
Substituting gives :
rK / L – (2xrQ / (2xL)^2 ) = 0
giving Q = 2KL
However I think I need the answer in terms of either K or L, not both. Anybody know how to solve / or what I’m doing wrong?

I'm not sure where you got the w=rK/L and following but you are correct in that
The first order condition is : w – (2xrQ / (2xL)^2 ) = 0,
(although I have never heard that terminology, I assume that it means the equation must be satisfied to obtain a minimum in cost).

That is the derivative of the cost function
C(L,K;Q) = \(\displaystyle w\, L\, +\, r\, K\)
with a given production quantity Q
Q = 2 x L K
for some positive parameter x, is zero when
w – (2xrQ / (2xL)^2 ) = 0
Since Q, x, r, and w are the given quantities, L and K should be expressed in those terms. So what does that then say about L and K. For example, solving the first order condition we get
L = \(\displaystyle \sqrt{\frac{r\, Q}{2\, w\, x}}\)
and
K = Q / 2xL = ...

So what does L and K depend on and what is that dependence?

EDIT: Whoops, the question was "How much capital and labour ..." so what are K and C at the minimum of the cost function C and what do they depend on?