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Math Help - Finding a tractable solution to an implicit function

  1. #1
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    Finding a tractable solution to an implicit function

    Hi, I am working on an economics paper, and I find that the following first order condition for my variable of interest ( \mu \in [0,1]):

    \frac{\lambda(1+\mu)}{(1-\mu)}=N-x\frac{(1+\mu)}{\sqrt{\mu}}

    I would ideally like to provide an explicit solution, but unfortunately, this would amount to solving a fourth order polynomial. Is there any transformation I can make to \mu. which would allow me to provide a tractable explicit solution for the tranformation of the variable... or am I out of luck?

    thanks so much in advance!
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  2. #2
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    Re: Finding a tractable solution to an implicit function

    Are there any more relationships between the variables? If not, it doesn't seem possible ... not algebraically anyway.
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  3. #3
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    Re: Finding a tractable solution to an implicit function

    Hi there,

    first of all thank you so much for your help. I really appreciate it!
    In the original problem I tried to make the implicit function simpler by coupling a few variables (which eliminates the computational burden). I should have given you the full problem, perhaps. So here's how it originates. I am trying to find the \mu \in (0,1) that locally maximizes W (i.e., found by the first order condition). The function is as follows:



    W(\mu)=\frac{\left( \left( C-\left( \lambda +1\right) \left( 1+\mu\right) \sqrt{\frac{d}{\mu }}\right) ^{2}+2\alpha p\gamma \sqrt{\frac{d}{\mu }}\left(<br />
\left( 1+\mu \right) +\lambda \left( 1-\mu \right) \right) \right) }{\gamma }



    where \gamma>0, d>0, \lambda \in (0,1), p\in (0,1),\alpha \in (0,1)



    this would lead us to the foc:


    \frac{\lambda \left( 1+\mu \right) +\left( 1-\mu \right) }{\left( 1-\mu<br />
\right) \left( 1+\lambda \right) }=\frac{C-\left( 1+\lambda \right) \left(<br />
1+\mu \right) \sqrt{\frac{d}{\mu }}}{p\alpha \gamma }

    Which has a few critical points. But one of the roots is necessarily a local maximum \mu<1. I am trying to provide a more tractable soluton to this value of \mu

    As a matter of fact, if you could help give me some characterization of the critical points in general, that would be even also perfect!

    This problem has a few different variables than my original post. But again, the original post tried to ease some of the notational burdensome. Any further help would be greatly greatly appreciated!
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  4. #4
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    Re: Finding a tractable solution to an implicit function

    Quote Originally Posted by jcsm View Post
    Hi, I am working on an economics paper, and I find that the following first order condition for my variable of interest ( \mu \in [0,1]):

    \frac{\lambda(1+\mu)}{(1-\mu)}=N-x\frac{(1+\mu)}{\sqrt{\mu}}

    I would ideally like to provide an explicit solution, but unfortunately, this would amount to solving a fourth order polynomial. Is there any transformation I can make to \mu. which would allow me to provide a tractable explicit solution for the tranformation of the variable... or am I out of luck?

    thanks so much in advance!
    I just applied wolfram alpha to your equation, and well, try not to strain your eyes ...

    I changed \lambda to l, \mu to x, and x to y so that alpha could interpret the result easily. I'm assuming the displayed behemoth is the quartic equation in action. My only other suggestion would be to approximate \sqrt x with a partial Taylor sum around \mu=0.5 or x=0.5 as I've entered it in wolfram alpha (the first two terms gives this result, which I'm sure was solved using the cubic equation ... still no simplification from what i can see). Finally only one term gives this, although this does not look much better and you're losing accuracy as we go.

    All in all, if you were looking for a "nice" analytical solution, you may be out of luck. Try mathematica or matlab if you are used to those to check critical and inflection points in your equation. I'm sure it'll get ugly, but I also think if there is supposed to be a "nice" expression for this, maybe you made an error somewhere?

    Hopefully this gave you something to chew on ... good luck.

    Marian.
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  5. #5
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    Re: Finding a tractable solution to an implicit function

    oh wow, that was super super helpful. You don't know how much I appreciate it, this really solves my doubts and gives a rest to some of my wonders. I'll probably have to look for another approach. Thanks very very much!
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