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Finding an explicit solution for this variable

jcsm

New member
Mar 8, 2013
3
Hi, I am working on an economics paper, and I find that the following first order condition for my variable of interest ([TEX]\mu \in [0,1][/TEX]):

[TEX]\frac{\lambda(1+\mu)}{(1-\mu)}=N-x\frac{(1+\mu)}{\sqrt{\mu}}[/TEX]

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 [TEX]\mu[/TEX]. 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!
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,192
Well, I have several comments.

1. If you have exact values for $n, \lambda,$ and $x$, then you could find a numerical approximation for $\mu$.

2. $\mu\in(0,1)$, not $[0,1]$, because in the original equation as you've given it, neither endpoint is allowed.

3. There is an explicit solution available for the general quartic. I'd recommend using Mathematica if you want to find it. WolframAlpha will find it, though. It's messy.

Might I ask from where this equation originated? That is, could you give us a bit more context for it? Maybe there's a simplification earlier in the process that we could employ.
 

chisigma

Well-known member
Feb 13, 2012
1,704
Hi, I am working on an economics paper, and I find that the following first order condition for my variable of interest ([TEX]\mu \in [0,1][/TEX]):

[TEX]\frac{\lambda(1+\mu)}{(1-\mu)}=N-x\frac{(1+\mu)}{\sqrt{\mu}}[/TEX]

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 [TEX]\mu[/TEX]. 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!
The implicit function is fourth order in $\mu$ but first order in x, so that You can easily find...


$\displaystyle x= \frac{\sqrt{\mu}}{1-\mu^{2}}\ \{N\ (1+\mu) - \lambda\ (1-\mu) \}$ (1)

May be that the (1) is itself useful... if not a single value inverse function of (1) around the point $x=0$ and $\mu=0$ can be found...


Kind regards


$\chi$ $\sigma$
 

jcsm

New member
Mar 8, 2013
3
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 \(\displaystyle \mu \in (0,1)\) that locally maximizes \(\displaystyle W\) (i.e., found by the first order condition). The function is as follows:



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



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



this would lead us to the foc:


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

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

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!
 
Last edited:

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,192
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 \(\displaystyle \mu \in (0,1)\) that locally maximizes \(\displaystyle W\) (i.e., found by the first order condition). The function is as follows:



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



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



this would lead us to the foc:


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

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

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!
Thank you for this info, but I was thinking even more fundamental. What is $W$? What are all the other variables, including units if there are any? Why are you trying to maximize $W$? Does this problem come from a physical situation? If so, what is it?
 

jcsm

New member
Mar 8, 2013
3
Than you for your reply!

This is a game theoretical game, in which the variables have no fundamental relationship (I have already included all of these relationships in W, where W is the social welfare).

I was mainly looking for some properties or nice characterizations of the explicit solution. However, I guess that this will not be possible.

Thanks anyhow!
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,192
You can try taking the second derivative of $W(\mu)$ to find out whether certain critical points that you find are local mins or local maxes. Second derivative positive at a critical point means you found a local min, and second derivative negative at a critical point means you found a local max. If the second derivative is zero at a critical point, then you must revert back to the First Derivative test.