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.
Re: Log linear maximisation problem
You need to differentiate:
+b\ln\left(\frac{1}{t+1}\right )-tE)
with respect to t. There are three terms. The third one is easy: =-E)
.
The first:
)
}\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} \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
