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Math Help - Difficult maximisation problem

  1. #1
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    Difficult maximisation problem

    We need to find the value of X that maximizes U,

    U= [(w*X)^(1-a)-1]/(1-a)*[(s*(1-X))^(1-b)-1]/(1-b)

    where 0<X<1 w>1 s>1

    For special cases of a=1, we have as the first term Ln(wX); for b=1 we have Ln(s(1-X)
    We are allowed to assume w and s are large such that both terms are positive

    Taking the first differential using the product rule I get:

    dU/dX= 0= w(w*X)^-a * [(s*(1-X))^(1-b)-1]/(1-b) +
    s(s*(1-X))^-b *[(w*X)^(1-a)-1]/(1-a)


    Which doesn't really seem to get me very far!

    Any idea of alternative approaches or next steps ?

    We can get a neater solution under special conditions w=S -> infinity; a>1; b>1, which are acceptable approximations

    0 = [X^-a / (b-1)] + [(1-X)^-b / (a-1)]

    X^-a / (1-b) = -(1-X)^-b / (1-a)


    X^-a * (a-1) = (1-X)^-b *(b-1)


    But even here I am still stuck on how to proceed.
    Last edited by frustrated; April 6th 2013 at 06:09 AM.
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  2. #2
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    Re: Difficult maximisation problem

    Update

    The following seems to give a good approximation for the last problem but I don't know why:

    X= 1-1/[(a-b)(a-1)+2] as a-b -> 0 for small values of a-b the solution is almost perfect
    Last edited by frustrated; April 7th 2013 at 09:50 PM.
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  3. #3
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    Re: Difficult maximisation problem

    It appears that there is actually no analytic solution to this problem.
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