• Yonah Wilamowsky 


The Board of Education in New York City typically negotiated three-year contracts with their staff. It was found that costs over the three-year period invariably exceeded projections by 10%–20%. An analysis showed that although the number of employees was relatively constant, the distribution of employees with respect to salary levels had changed significantly. As a result, projections of salary costs were substantially understated. Over the course of the contract term, salaries increased due to longevity and additional employee education. Classic methods for forecasting costs include regression analyses, time series analysis, and simulation methods. Markov chains have also been used to project population changes over time. In this paper, we present a Markov chain model that was successfully used to forecast teacher populations as well as costs with much greater precision than had been possible previously. The model incorporated personnel as well as cost projections. For a given total salary budget, management was thus able to place salary increases in levels so as to keep costs to a minimum. As a result, management was able to obtain significant reductions in labor costs. In addition, the model can help by incorporating issues of diversity and inclusion in the workplace. By being able to track where employees would likely be down the line, a fairer distribution could be achieved.


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Forecasting personnel costs is a crucial component of any company’s plans. When negotiating a long-term contract with employees, a company must have a good understanding of what the contract will cost. Typically, salary ranges are fixed for each job category in the organization. It would then seem a simple matter to multiply the number of people in each job category by the proposed salary for that category, including whatever proposed increases are anticipated over the course of the contract. The Board of Education of New York City did exactly that when estimating the cost of a new three-year contract for its employees. They knew that the total number of employees in recent years had been relatively constant and anticipated that that would continue to be the case. Even so, they found that over the course of the last few contract negotiations, their estimates were consistently low, to the extent of 10% to 20%.

The Problem

An analysis of the problem showed that while, indeed, the total number of employees was relatively constant, the distribution of levels within this population was changing significantly. There were two types of employees: teacher and non-teacher positions. Over the course of three years, the non-teacher group was relatively stable, and the forecast was on target. However, overall, the teachers had been going up in levels by virtue of longevity and further education. As a result, their overall salary projections were therefore understated. The teachers were the overwhelming majority of employees, and therefore, the forecasts were very flawed.

Difficulty with Cost Projections

Additionally, it was discovered that cost projections had been even lower than they otherwise might have been due to the way management divided the salary adjustments. As is often the case, management was most interested in the total salary costs and paid less attention to the way that the sum was divided amongst levels. That was, after all, it thought, the determining factor in what its salary budget would be. In quoting the average salary of an employee, the figure they used was the mean salary. By multiplying the mean salary by the number of employees, they would have the total salary budget. Negotiators for labor, on the other hand, were more interested in the median or mode salary of employees. They knew that with a salary structure that was skewed to the right due to a number of highly paid principals and other specialists, the mean salary was not really representative of what the typical teacher was earning. They were, therefore, very interested in which categories the greater increases went to. By virtue of who was involved in the negotiations, they often preferred the greater increases to go to the more senior categories. For this reason, since the year-to-year movement tended to make the total distribution of personnel more senior, the total costs increased even faster.

The seniority system with respect to salaries was quite complex. A teacher would go up one step after his/her initial hiring twice per year. This would continue until after eight years of service so that there would be 16 possible levels. In addition, a teacher would receive a raise when completing a certain number of graduate credits or receiving a graduate degree. This created six different differential levels. In all, there were 96 (16 × 6) different salary levels.



Manpower projections have been approached using various quantitative techniques. Simulation (Geerlingset al., 2001; Hanna & Ruwanpura, 2007) and time-series forecasts (Feiring, 2006) are two of the more common methods.

One technique for forecasting population changes that has been popular for a long time is that of using Markov processes. Markov chains have emerged as a powerful tool in population forecasting due to their ability to model and predict the future states of a system based on its current state and transition probabilities. This mathematical framework is particularly well-suited for understanding and predicting the dynamics of populations in various fields such as demography, epidemiology, economics, and more.

A Markov chain is a stochastic process that undergoes transitions from one state to another in a probabilistic manner. The future behavior of the system is assumed to depend solely on its current state and not on its previous history. In population forecasting, a Markov chain can be applied by considering the population as a collection of discrete states, often defined by age, gender, health status, location, or other relevant attributes. The transitions between these states are determined by transition probabilities, which represent the likelihood of moving from one state to another within a given time frame. These probabilities can be derived from historical data, expert opinions, or statistical analyses.

Applications of Markov Chains in Forecasting

For example, consider an aging population in a certain region. By constructing a Markov chain model, policymakers and researchers can estimate the future distribution of age groups. The initial state of the system would be the current age distribution, and the transition probabilities would be based on observed patterns of age progression and mortality rates. By iteratively applying these transition probabilities, the model can project how the population’s age distribution will evolve over time, offering insights into potential challenges such as healthcare demands, pension requirements, and workforce trends.

Markov chains have similarly been used in economic forecasting, especially when analyzing consumer behavior and market trends. Businesses can employ these models to anticipate how customers move through different purchasing states or loyalty levels. By understanding these transitions, companies can tailor marketing strategies and optimize resource allocation to maximize profitability.

There are numerous applications of Markov chains to population forecasting in various fields (see Kingman, 1969, for an early review of such applications). To cite just a few: Lefevre (1988) discusses an application dealing with the spread of epidemics. Zhanget al. (2019) discuss traffic flow, Saadiet al. (2016) forecast travel behavior, and Chan (2015) deals with modeling market share.

Personnel Projections Using Markov Analysis

Markov analyses have been used to forecast personnel mobility rates (Valliant & Milkovich, 1977). They have been used in the prediction of military personnel planning (Setyaningrumet al., 2022). Levels of education in the workforce have been incorporated (Ledwith, 2019; Zatonatskaet al., 2022). Similarly, Markov processes have been used to forecast internal manpower supply based on job classifications (Rowland & Sovereign, 1969). In this paper, we combine education levels and job classification to project future manpower supply as well as costs.


It was determined that a population forecast using Markov analysis would be suitable for forecasting salary-level populations within the organization. For our personnel population change problem, we thus developed a Markov Chain model and set up a transition matrix giving the probabilities of going from one salary state to another each year. Grimshaw and Alexander (2011) present a discussion of transition matrix estimation. In our case, past employee-level history was used to determine these probabilities. This matrix would, therefore, need to be 96 × 96 to reflect the 96 different possible states. Complicating matters further is the fact that there are also retirements and new hires. Since the total population was thought to be constant, the new hires were assumed to equal the number of retirements. The percentage of retirements each year could also be determined based on historical data. The new hires could then be apportioned into each of the sixteen categories in whatever manner management desired to place new hires. Typically, new hires could start out at level 1, but this is not necessary, depending on management’s plans. The data files to set up the model were quite large, but the principles can be shown using a small example.

A Simplifying Example

In order to simplify the explanation of the process and the development of the Markov transition matrices, we will consider a case with three seniority levels and two differential states. Suppose the three seniority levels are 1, 2, and 3, and the three differential states are A and B. We then have, instead of 96 total states, only six and a 6 × 6 transition matrix. We must also incorporate retirements and new hires into this matrix. For illustrative purposes, consider the example given in Table I.

1A 1B 2A 2B 3A 3B Retire
1A 0 0 0.2 0.2 0.3 0.2 0.1
1B 0 0 0 0.4 0 0.4 0.2
2A 0 0 0 0.25 0.25 0.3 0.2
2B 0 0 0 0.2 0.2 0.3 0.3
3A 0 0 0 0 0.4 0.4 0.2
3B 0 0 0 0 0.3 0.3 0.4
Table I. Transition Matrix from Level to Level Each Year

Each cell represents the probability of somebody in the row category moving to the column category in the next year. For example, 30% of people in 1A will be in 3A the following year. For the people who retire, we must replace them with new hires. We will place them in whatever starting positions are appropriate to the management’s desires. Suppose the following represents the percentages in each category for new hires:

1A%–40% 1B%–20% 2A%–15% 2B%–10% 3A%–10% 3B%–5%

Looking for example at Row A, the 10% of people who retire will be apportioned as follows:

1A − 0.04 (0.10 * 0.40) 1B − 0.02 (0.10 × 0.20) 2A − 0.015 (0.10 × 0.15) 2B − 0.01 (0.10 * 0.10) 3A − 0.01 (0.10 × 0.10) 3B − 0.005 (0.10 × 0.05).

Thus, the percentage of 1A transitioning to each state in the following year is:

1A − 0.04 (0 + 0.04) 1B − 0.02 (0 + 0.02) 2A − 0.215 (0.20 + 0.015) 2B − 0.21 (0.20 + 0.01) 3A − 0.31 (0.30 + 0.01) 3B − 0.205 (0.20 + 0.005).

The entire matrix can be seen in Table II.

1A 1B 2A 2B 3A 3B
1A 0.04 0.02 0.215 0.21 0.31 0.20
1B 0.08 0.04 0.03 0.42 0.02 0.41
2A 0.08 0.04 0.03 0.27 0.27 0.31
2B 0.12 0.06 0.045 0.23 0.23 0.31
3A 0.08 0.04 0.03 0.02 0.42 0.41
3B 0.16 0.08 0.06 0.04 0.34 0.32
Table II. Transition Matrix from Level to Level Including New Hire

Similarly, we can complete the transition matrix (Call it T).

Let Yn = the starting distribution in year n. Suppose:

Y1 = 1A 1B 2A 2B 3A 3B.

(2000 1000 1500 1000 1000 500).

Then Y2 = Y1 × T.

Y2 = (560 280 610 1515 1865 2170).

For the labor negotiation done by management, the transition matrix was assumed to be constant over the three-year contract (thus Yk = Y1 × Tk−1). If desired, then T can be reevaluated as needed and substituted for succeeding time periods.

Results and Discussion

Implementation and Cost Savings

This forecasting system was implemented by the Board of Education in the succeeding contract negotiations. As expected, it showed that teachers were shifting toward higher levels. In about ten years, approximately 72% of all teachers would be at the highest level. This, of course, has important implications for the negotiations since the greatest increases had traditionally been placed with the most senior employees. As a result, management became more cognizant of how the total funds were distributed amongst the levels. They realized that placing most of the allocated funds in the higher categories would lead to higher costs in succeeding years and were able to make salary allocations accordingly. This led to significant savings over the course of the contract.


As mentioned previously, one of the assumptions in the model was that the total workforce would remain relatively constant. In our case, this assumption turned out to be flawed. In periods of expansion, it would certainly be necessary to increase the workforce. In the case of our model for the Board of Education, retrenchment and layoffs became an issue. Especially with such recent factors as changes brought about by COVID-19, layoffs and rehiring have become important factors. The model can easily be adapted to incorporate these issues. Other than the retirement category, the model can include a layoff category. Alternatively, new hires can be based not only on retirements but also on projections for staff needed.

Conclusions, Generalization, and Further Work

This model can be adapted to any industry. Such a system would be invaluable to managers, negotiators, and corporate planners. Using a comparable system of salary range levels, similar transition matrices would be developed. It would be important to allow for increases or decreases in staff size as needed, as well as keep track of possible changes in transitions over time. Wilinski (2019) deals with changing transition matrices in Markov chains. The value of such a model would not only be for negotiations and forecasting salary costs. It would also enable managers to track the effects of their hiring practices and maintain a balance of experience in their workforce. Goals could be set for experience levels needed over time, and hiring practices adjusted accordingly. It could also add great value by incorporating issues of diversity, equity, and inclusion in the workplace. By being able to track where employees would likely be down the line, a fairer distribution could be achieved.


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