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Math Help - Log linear maximisation problem

  1. #1
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    Log linear maximisation problem

    We are given to solve for t to maximise W,

    W=(1-(1/(t+1)))^a*(1/(t+1))^b*e^-tE

    where t>0 or t=0

    Which can be log linearised to obtain

    lnW=aLn(1-(1/(t+1)))+bLn(1/(t+1))-tE

    Obviously to obtain the maximimum we need to differentiate the above with respect to t and find the zero value(s) for the differential, (and then locate which is the maximum) but I am stuck on how to proceed with the differentiation.

    Any help would be most appreciated.
    Last edited by frustrated; November 25th 2012 at 02:49 AM.
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  2. #2
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    Re: Log linear maximisation problem

    You need to differentiate:

    a\ln\left(1-\frac{1}{t+1}\right)+b\ln\left(\frac{1}{t+1}\right  )-tE

    with respect to t. There are three terms. The third one is easy: \frac{d}{dt}(-tE)=-E.

    The first:

    \frac{d}{dt}a\ln\left(1-\frac{1}{t+1}\right) = a\frac{1}{\left(1-\frac{1}{t+1}\right)}\frac{d}{dt}\left(\frac{-1}{t+1}\right) = a\frac{1}{\left(1-\frac{1}{t+1}\right)}\frac{1}{(t+1)^2}

    The second:

    \frac{d}{dt}b\ln\left(\frac{1}{t+1}\right) = b\frac{1}{\left(\frac{1}{t+1}\right)}\frac{d}{dt} \left(\frac{1}{t+1}\right)  = -b\frac{1}{\left(\frac{1}{t+1}\right)}\frac{1}{(t+1  )^2}

    And I'll leave it to you to simplify the result.

    You might be able to differentiate the original function, too, but this seems to give a simpler answer.

    - Hollywood
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