I'm figuring out some kind of money saving function where the amount, in the end, should be 100.000 value with a time period of 10 years.
I thought of an exponential growth function since people start with jobs and with work experience, you will earn more with results into a higher amount that you can save.
With some search, I found this formula: $P(t) = P0 e^r*e^t$ (I can't format this propper) Where $P(t) = 100.000$ , $r=0.5298$, $P0=500$ and $t=10$
This will lead to an insane amount of money in year 9 and 10.
I'm looking for a more reasonable function where you start with the least amount in year 0 and the most amount in year 10 so the amount that you saved is $100.000$. But it should be reasonable like you can't save 20k in year 10. Also, $P0$ may differ for each person, but the time and the end goal is fixed. I do not take interest into account.
Best Answer
You can impose the following model:
Save $P(0)$ in year 0, $P(1) = \alpha P(0)$ in year 1, $P(2) = \alpha P(1) = \alpha^2 P(0)$ in year 2, etc. for some $\alpha>1$. So you are geometrically/exponentially increasing the amount you save each year. After 10 years you will have $P(0) + P(1) + \; ... \; + P(10)$ = $P(0)[1+ \alpha + \alpha^2 + \; ... \; + \alpha^{10}] = P(0) \frac{\alpha^{10}-1}{\alpha-1}$.
Say $\alpha=1.5$. At the end you will have $113.33 P(0)$, which means $P(0) = 882.38$ if you want the final amount to be 100,000. With this model, you will have to save $\$882.38$ in year 0, $\$1323.60$ in year 1, ..., and $\$5088.2$ in year 10 (which is still reasonable).