A Simple Analogy for Understanding the Added Cost of Steps

I sent this simple analysis to Harford County, Maryland budget authority officials in February 1979 seeking to have them correct their erroneous claim regarding the “new/additional money” needed to provide steps or increments for teachers and other personnel.

Because Harford County officials have continued to make similar false assertions ( circa 2014-2016) regarding the added cost, this retired mathematician has examined the issue anew (this time more formally), created a web site, and placed the analysis therein. Hopefully, the widespread and readily available analysis will work toward officials ceasing to make false claims and accordingly chose to treat its employees fairly by routinely providing steps or increments since the added cost is negligible-as proven by the author’s analysis.

An experience increment of $500 dollars paid to an employee earning $11,000 per year amounts to approximately a 4% increase in salary. Discussion of the Harford County budget for the coming year in the local press indicates that the county treasurer and chief of administration plus certain writers analyzing the budget believe that an additional 4% over the previous year must be budgeted for each employee receiving the increment. This I claimed in my letter and enclosure of January 30, 1979, is not
correct. The enclosure contained a mathematical analysis which showed that a payroll system was cyclic in that people in the high experience levels, whose salary contained many increments, retired and thus compensated for many employees moving up the increment scale. I observed that the experience “shift” from year to year could cause an increase (or equally a decrease) depending on the experience statistics of the workforce. I claimed that the budgeting of 4% (as in my example above) was an
absolute upper bound that applied only in the unlikely circumstance of no retirees in a given year. The analogy below helps to put the problem in perspective and aids in understanding the point of contention.


Consider a classroom of 30 students and a teacher. Assume the 30 children are sitting in a row of chairs facing the teacher as depicted below.

C1          C2          C3 . . . C30

The student sitting in C1 has one piece of candy, the student sitting in C2 has two pieces of candy, etc. and the student sitting in C30 has thirty pieces of candy. The candy is analogous to the experience increments. The teacher instructs the child sitting in C30 to leave the room (analogous to retiring) and to leave his/her 30 pieces of candy on the vacated chair. The teacher then instructs each of the remaining 29 students to stand, leave their candy on their vacated chair, and to be seated in the adjacent chair to their left. The teacher then goes into the hallway and returns with a child whom
she seats in C1.

Happy laughing pupils of primary school having fun during break with their teacher, playing musical chairs

Now each child has one piece of candy more than when this exercise of chair moving began. Yet the teacher has not had to produce a single piece of additional candy. The moving from one chair to another is equivalent to personnel moving up in salary due to an additional year of experience. The child brought in from the hall is analogous to a new employee entering at a base salary. The analogy illustrates the point that when a workforce is stable or in steady-state, as far as average experience, then the year to year cost increase due to the increment is expected to be zero. Alternatively, in the absence of this steady-state, an average taken over a suitable number of years will find the increases are balanced by decreases.