Let's say an organization has 100 employees in the beginning of 2013 and grows to 110 employees by the beginning of 2014. This implies a growth rate of 10% for 2013.
Now, let's say a hiring manager wants to reach a size of 200 employees at the end of 2016 (aka beginning of 2017), 3 years later.
What I'm looking for is a formula to determine a growth rate for each intervening year (2014, 2015, 2016). But, I'm not looking for a growth rate that's the same every year, which is what a typical compound interest calculator assumes. Rather, I'm looking for a solution where the rate increases annually at a constant increment. It would take into account the 10% growth rate of 2013 and linearly increase it towards the final growth rate of 2016, arriving at an organization with 200 members.
This example uses 3 years and a concrete organization size, but I'm looking for a formulaic solution, where the number of years is variable (as well as the starting and ending number of employees and the starting growth rate). For simplification, fractional numbers of people is not a problem.
Thanks in advance.
Best Answer
To model your situation (in a ${continuous}$ fashion) you can use $y'=(a+bt)y$ which has a solution of $y(t)=y(0)e^{at+bt^2/2}$. To adjust $a$ and $b$ use
$\displaystyle \ln{y(t)\over y(0)}=at+bt^2/2$.
In your case $a=0.1$ and $t=3$ and the targeted growth is $y(3)/y(0)=2$. So $\ln(2)=0.1*3+b*3^2/2$ giving $b=(\ln(2)-0.1*3)*(2/3^2)=0.087$.
This again is for a continuous model. Companies act in step-wise/discrete fashion. In that case you have to use $y'=ay$ for one year, then $y'=(a+b)y$ for the next year, and then $y'=(a+2b)y$ in the next year, etc.