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Math Help - Optimization: transformation of target function and conditions

  1. #1
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    Optimization: transformation of target function and conditions

    In a recent microeconomics lecture I was confronted with the following problem:
    maxx1, x2, x3 2*sqrt(x1)
    s.t.
    2*sqrt(x2y)=64
    y=2*sqrt(2x3)
    x1+x2+x3=112

    the professor reformulated the problem without explanation to
    maxx1, x2, x3 4x1
    s.t.
    16x22y22=644
    y2=8x3
    x1+x2+x3=112
    and then solved by plugging the conditions into the target function. How can such a transformation be determined without changing the problem at hand? what is the logic or the reason behind this?

    thanks for any clue
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  2. #2
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    Re: Optimization: transformation of target function and conditions

    If 2\sqrt{x_1} is at a maximum, then so is 4x_1=(2\sqrt{x_1})^2. The changes made to the conditions are similar - taking the 4th power of both sides and squaring both sides.

    - Hollywood
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