The Naïve Model
The model is a very simple "one-period" model, with the following assumptions:
At the beginning of the period, we have an organization in place that is well integrated, well trained, and firing on all cylinders. We take the productivity of its team members to be the baseline for comparison.
During the period, we add new team members of the same average quality as those already in place.
At the end of the period, we have a "new" organization with more people and a greater capacity to get work done, because we have more people with the same average productivity as when we started.
What we deal with here is what happens to productivity during the period of transition.
Note an important assumption: We assume that the overall per capita productivity does not decrease because the organization gets larger. While this is often not the case, we naïvely assume that once we are done with the transition, the organization has gained productivity in direct proportion to the growth.
For organizations that are constantly growing, we can still use this model. We just sequence a series of one-period models one after the other. By superposition, we can make predictions for this case. It is mathematically a little more complex, but nothing new needs to be added conceptually.
Effect of New Hires During Transition: Contribution
Even during transition, we should add useful productive hours to the current total. Even an employee who is only 10 percent effective during his ramp-up period is still adding, for his or her part, four hours of "useful" work per week. We must view the remaining 36 hours as "training," "ramp up," or "investment in the future." We pay for those hours, but they do not show up in the current product.
 I use product development as the main focus. However, the analysis applies to other types of organizations as well. For example, in a sales organization, substitute the words "hours spent in direct selling" for "hours worked on the product." Everything else follows in parallel fashion. When we add new salespeople, our revenues go up, but so does our cost of sales on a per employee basis.
So, in this most naïve model, we gain hours but lose overall productivity during the transition period. That's a reasonable tradeoff, and we should quantify it so that we know what we are trading off for what. However, there is a secondary effect that we should add to the model to increase its fidelity: drag.
Effect of the New Hires on the Existing Team: Drag
It turns out that while new employees add hours by being productive at some marginal rate, they actually detract from the total by being there as well. That is because others in the organization lose productive time in working with the new, less productive employees. This can be time lost in supervision, explanation, "showing them the ropes" and any other manner of bringing the new employee up to speed, or correcting their mistakes. These are very costly hours, because the in-place employees are, by definition relative to the newbies, 100 percent productive. Another way of saying this is that the new employees place a drag on the organization because otherwise fully productive, in-place employees must interact with them.
I should point out some obvious exceptions to this rule. If we recruit a new team member who has a skill set that is needed and otherwise absent, there can be a spontaneous increase in overall productivity. Similarly, if we recruit a superior manager, we may turn around an otherwise dysfunctional team. Both these instances are examples of almost immediate improvement and represent "negative drag," as it were. But, generally speaking, these are exceptional cases.
The Model and Its Assumptions
As with all models, we make some simplifying assumptions, which I collect and enumerate here for clarity. Note that we can make successively more complicated models to increase fidelity, but at some point we hit the law of diminishing returns. The best models are the simplest.
Our assumptions are as follows:
Existing in-place team members are "100 percent productive." We ignore any loss of productivity due to their training and overhead. Basically we are using the existing people as the productivity benchmark, and everything is relative to them. We assume that the organization has a total productivity equal to the product of the average productivity of the existing team members times the number of them in steady state. Thus we do not assume that all existing employees have the same productivity but rather have some distribution that we can characterize with an average productivity.
During the ramp-up period (one year), new employees are characterized by an average fractional productivity P, with 0 P 1. We make the simplifying assumption that they are hired "en masse" at the beginning of the year. For example, P = 0.6 indicates that, during the ramp-up year, the average productivity of the new team members will be 60 percent of that of the existing team members; 60 percent of their hours go to "useful" work on the product, and 40 percent of their hours go to "getting up to speed." You can, of course, choose the ramp-up period to be anything you like without loss of generality; my observation is that it is usually longer than managers think it is.
A simple extension of the model could allow for "learning curve" behavior. That is, we could model the typical S-Curve that characterizes new-hire learning ramp-up during the period using a spreadsheet. In the interest of simplicity, I choose instead to assume a constant average productivity during the entire period.
We measure the annual growth of the team by the fraction G of new employees, usually between 0 and 1. For example, G = 0.1 indicates a 10-percent growth rate, or the addition of one new team member for each existing 10. G = 1.0 would correspond to 100 percent growth or attempting to add one new person for each existing person. As we will see later, growth rates of this magnitude are highly risky. Once again, we make the simplifying assumption that new team members are hired "en masse" at the beginning of the year. More complex models would stage, or phase, the arrival of the new hires; once again, this leads to more complicated math, but nothing that is conceptually novel. Once again, this could be built into a spreadsheet model.
The effect of new hires is characterized by a drag ratio D, with 0 D 1. For example, a D of 1.0 means that for every hour spent in non-productive work by a new team member, there is an associated loss of an hour of an existing team member. The smaller the D, the better for the organization; for example, a D of 0.2 means that we lose only two hours of work by existing team members for every 10 hours of lost work by newbies.
New team members have the same average hourly pay rate as the existing team members. This is reasonable, because we are assuming that once the ramp-up period is complete, the new team members will be just as productive as the existing ones; that is, we assume they will attain "100 percent productivity." Another way of saying this is that we assume that the newly hired people are of the same average quality as those already in place; hence, they are paid at the same average rate.