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Math Help - Microeconomics Help

  1. #1
    Newbie
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    Microeconomics Help

    Hey all, first post, hopefully someone can help me with this.


    "Suppose the production of airframes is characterized by a CES production function: Q=(K^0.5 + L^0.5)^2 (MPL = (K^1/2 + L^1/2) /L^1/2), MPK = (K^1/2 + L^1/2) /K^1/2)

    Suppose that the price of labor is $10 per unit and the price of capital is $1 per unit. Find the cost minimizing combination of labor and input for an airframe manufacturer that wants to produce 121,000 airframes."


    Thanks in advance guys.
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  2. #2
    MHF Contributor
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    You have 2 unknowns so you need to find 2 simultaneous equations and solve them.

    There are 2 ways to do this, the long way and the short way.
    The long way to do this is to set up a lagrangian. You want to minimise the total cost, subject to the constraint that you have produced 121000 airframes.

    Total cost: K + 10L
    Constraint: 121000 = (\sqrt{L} + \sqrt{K})^{2}

    Lagrangian
    Z = K + 10L -\lambda ((\sqrt{L} + \sqrt{K})^{2} -121000)

    If you solve this in the usual way, you will get 2 simultaneous equations:
    121000 = (\sqrt{L} + \sqrt{K})^{2}
    \frac{MPL}{10} = \frac{MPK}{1}

    Solve those to get your answers of L and K.


    short cut
    You only need the ratio L/K. You can get this from the second simultaneous equation on its own
    \frac{MPL}{10} = \frac{MPK}{1}

    \frac{MPL}{MPK} = \frac{10}{1}

    \frac{MPL}{MPK} = 10


    \frac{(K^{0.5} + L^{0.5})L^{-0.5}}{(K^{0.5} + L^{0.5})K^{-0.5}} = 10


    \frac{L^{-0.5}}{K^{-0.5}} = 10

    \frac{K^{0.5}}{L^{0.5}} = 10

    \frac{K}{L} = 100


    Interestingly, for this production function, the ratio of K/L is constant at all levels of output. So you didn't need to know that there were 121000 airframes after all.
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