Constrained optimization problem : Cost minimisation
Hi, I have this problem:
To produce one unit of a good a manufacturer needs m units of materials that
cost r per unit. An unskilled worker can be hired at an hourly rate of w1 and
would need 10 hours to produce one unit of output. A skilled worker can be
hired for an hourly rate of w2 and is five times as productive as an unskilled
worker. Let Li, i = 1, 2 denote the number of hours worked by a low and
high skilled worker respectively, and let K denote the amount of material used.
Write down the production function that represents this technology and derive
the corresponding cost function.
I think that the production function should look like this
f(L1,L2,K) = (L1/10+L2/2)*K/m
(since both labour and materials are needed to produce one unit)
The minimisation problem is thus:
min C = w1*L1+w2*L2+r*K subject to f(L1,L2,K) = y
I used the Lagrange method but i can't manage to find the values for L1,L2 and K. Have i made a mistake in the above analysis or have i just made a mistake in the simultaneous equations.?
IMPORTANT EDIT: If we assume that an exact ratio of labour and capital is needed then the production function would take the form f(L1,L2,K) = min( L1/10 + L2/2 ; K/m) ... How can I differentiate this??