Hi, I'm new here and I'm not sure if this is the right section or not if it is in the wrong section please let me know and I'll post it there instead.
I have been racking my brain with this for a couple days and am embarrassed that I can't figure it out (at one point I considered myself good at this kind of thing)... Just so you know this is NOT Homework, but a personal project of mine that I have been working on. The field is economics and expenditure models.
I am looking for a f(x) that can be determined given a set of knowns at two different points:
@ x_{1}, f(x_{1}) and f '(x_{1}) are know and
@ x_{2}, f(x_{2}) and f '(x_{2}) are also known
relationship between the two points -- x_{2}>x_{1}, f(x_{2})>f(x_{1}), f '(x_{2})<f '(x_{1})
My though was to use a function in the form of f(x) = A log_{b}(x) - C, but I could not get it to work.
I am trying to "smooth" out the corners on an expenditure model relating to family spending, and convert 3 flat percentages into a series of decaying rates with no abrupt change in slope.
I have three sections of the model where spending is assumed to be linear, but at different rates (or slopes) relating to income level (Low Income rate>Middle Income Rate>High Income rate). The model does not start at zero but at a point >0.
I hope that clarifies what I am trying to do.
I should add that a log function is appropriate as the over model trends that way. My first attempt was trying to set up a single equation to fit the entire system but I wasn't able to do that either.
So:
f '(x) ~ 1/(x*ln(b))
f ''(x) ~ -1/(x^{2}*ln(b))
First, you should realize that , so you really have only two independent parameters.
So you're given values for , , , , , and and you would like to fit a function of the form to them. It turns out that you really only need one set , , to determine A and C.
so
, so
So hopefully when you use those three to determine A and C, the other three also work. If not, you'll need a new model.
- Hollywood