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Math Help - Constrained optimization problem : Cost minimisation

  1. #1
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    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.?

    Thanks

    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??
    Last edited by Val92; March 29th 2011 at 04:46 AM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Val92 View Post
    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.?

    Thanks

    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??
    This is a linear program, it has a solution at a vertex of the feasible region. This means either L1=0 or L2=0 (which is also obvious for other reasons).

    CB
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