The market for energy foods is booming, and hot startup PowerButter™️ is in hyper-growth mode. The CEO wants to double the company’s size every 5 months. That means churning out more managers. Trouble is, it takes 11 months to train a manager and only managers can train managers. What is the average number of trainees a manager is training at any given time?
Bonus challenge: Training isn’t going smoothly. So newly trained managers will now have to wait 3 months before training others. What will the average number of trainees per manager at any given time be after this update?
The company wants to double in size every $t_\text{double} = 5$ months
This means that every piece of the company will need to double in that time, including the managers and their trainees. For each manager that exists at time zero, they must be able to offer up a newly graduated trainee at time $t_\text{double}$.
However, it takes longer to train a manager than the time the company aims to double in. How can this situation add up?
It can work, but only if the company maintains overlapping cohorts of trainees.
We can find the ratio of managers to trainees under one of two assumptions. We can assume that the training follows a discrete schedule, and doubles exactly at the five month mark. We can also assume that the company is big enough that its growth is continuous.
Since the company is in continuous growth, the number of managers at time $t$ is
$$ M(t) = M_0\ 2^{t/t_\text{double}}. $$
Since it takes time $t_\text{train}$ for a trainee to graduate, the number of trainees at time $t$ is just the number of managers at time $(t+t_\text{train})$ minus the number of managers at time $t$
$$ T(t) = M(t+t_\text{train}) - M(t). $$
This relationship is true because anyone in training at time $t$ (no matter how far along they are), must be graduated by the point a training period has passed, $(t+t_\text{train})$.
So, the ratio of trainees to managers at time $t$ is just
$$ \begin{align*} \frac{T(t)}{M(t)} &= \frac{M(t+t_\text{train}) - M(t)}{M(t)} \\ &= \frac{2^{(t+t_\text{train})/t_\text{double}} - 2^{t/t_\text{double}}}{2^{t/t_\text{double}}} \\ &= 2^{t_\text{train}/t_\text{double}} - 1. \end{align*} $$
Plugging in the given training and doubling periods, we get $T(t)/M(t) = 2^{11/5} - 1 \approx 3.595$ trainees to every manager, independent of time.