Abstract
Purpose
On the one hand, everyone agrees that economics should be fair, that workers should get equal pay for equal work. Any instance of unfairness causes a strong disagreement. On the other hand, in many companies, advanced workers – who produce more than others – get paid disproportionally more for their work, and this does not seem to cause any negative feelings. In this paper, the authors analyze this situation from the economic viewpoint.
Design/methodology/approach
To analyze the problem, the authors use general mathematical models of how utility – and hence, decisions – depends on the pay-per-unit.
Findings
The authors show that from the economic viewpoint, additional payments for advanced workers indeed make economic sense, benefit everyone, and thus – in contrast to the naive literal interpretation of fairness – are not unfair. As a consequence of this analysis, the authors also explain why the labor share of the companies' income is, on average, close to 50% – an empirical fact that, to the best of the authors’ knowledge, was never previously explained.
Originality/value
To the best of the authors’ knowledge, this is the first paper that explains the empirical fact – that the labor share of the income is close to 50%.
Keywords
Citation
Kosheleva, O. and Aguilar, S.R. (2022), "Is it fair that advanced workers get paid disproportionally more: economic analysis", Asian Journal of Economics and Banking, Vol. 6 No. 1, pp. 97-102. https://doi.org/10.1108/AJEB-06-2021-0079
Publisher
:Emerald Publishing Limited
Copyright © 2021, Olga Kosheleva and Sean R. Aguilar
License
Published in Asian Journal of Economics and Banking. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode.
1. Formulation of the problem
1.1 Fairness is important
We all want fairness. One of the important aspects of fairness is that all the workers should get equal pay for equal labor.
In the past, this was not the case in many countries: for example, in the 19th century, in the USA and in other industrialized countries, women were routinely paid less for the same result.
Good news is that, as time goes, more and more companies are providing such equal pay; more and more countries have legislation in place that prohibit discrimination of all kinds and require equal pay for equal work.
1.2 Disproportional payment for advanced workers – case of seeming unfairness
However, there is an aspect of the current pay system that seems to contradict this fairness idea: the fact that in many companies, the best workers get additional bonuses and other rewards – and in this sense, get paid disproportionally more.
Let us explain this using a simple example. Suppose that a company provides a 10% annual bonus to the best performer. Suppose that the two best workers perform at almost the same level, with a difference of 1%. The first worker's productivity was only 1% higher, but this worker gets the 10% bonus in addition to his/her pay, while naively understood fairness would mean that this worker should be paid only 1% more.
1.3 But is this really unfair?
From the purely mathematical viewpoint, if we view the idea of equal pay for equal labor literally, this arrangement – when advanced workers get paid disproportionally more – is clearly not fair. However, in contrast to other cases of unfairness, no one seems to protest against this practice as an unfair one. But why is this so?
In this paper, we analyze this situation from the economic viewpoint. We explain why the naive approach to fairness is not applicable to this situation, and why disproportionally higher pay to advanced workers makes perfect economic sense.
2. Analysis of the problem and the resulting explanation
2.1 Let us formulate the problem in precise terms
Suppose that the workers are paid amount s for each produced unit, and that, on average, the workers produce x0 units in a certain period of time: day, week, month, whatever. Then, the average worker's salary during this time period is s ⋅ x0. How can we encourage the workers to be more productive?
According to decision theory, when a rational person makes a decision, he/she maximizes a quantity called utility; see, for example, Fishburn (1969), Luce and Raiffa (1989), Raiffa (1997), Nguyen et al. (2009), and Kreinovich (2014). In these terms, the incentive works if the positive utility provided by this incentive is larger than the decrease in utility caused by the need to spend time and effort on producing additional units.
Let us denote by e the decrease in a person's utility caused by producing one unit. Then, when the worker produces a additional units, the time and effort needed for producing this unit decrease his/her utility by the amount a ⋅ e.
We need to compensate this loss of utility by an additional monetary reward m. It is known that the utility u(M) caused by the overall monetary gain M is proportional to the square root of the monetary amount:
in utility caused by an extra dollar decreases with M. This makes perfect sense:
If a person has no money, getting a dollar is a big deal, but
If a person already earned $1,000, having an extra dollar is practically unnoticeable.
The additional reward m increases the worker's salary from the original value s ⋅ x0 to the new value s ⋅ x0 + m. So, the additional reward m increases the worker's utility from the original value
2.2 What if we provide equal pay for equal work?
If the follow the idea of equal pay for equal work and pay the same amount s for each additional unit, then the additional pay should be equal to m = s ⋅ a. In this case, the above condition takes the following form:
This condition is equivalent to
By squaring both side of this inequality, we get an equivalent inequality
If we open parentheses, subtract c2 ⋅ s ⋅ x0 from both sides, and move the term
Dividing both sides by a ⋅ e2, we conclude that
So, if we literally offer equal pay for equal work, we get a limitation on the amount of extra work that the workers will be willing to do under this incentive.
2.3 How can we incentivize workers to be more productive?
In view of the above analysis, the only way to incentivize workers to be even more productive is to offer them a disproportionally larger pay for extra work. This is exactly what happens when advanced workers get additional bonuses. From this viewpoint, the disproportional pay to advanced workers makes perfect economic sense and is, thus, perfectly fair:
The whole society benefits from the increased productivity, and
This disproportionally larger pay to advanced workers is the only way to increase productivity further.
This is also what happens when the company sometimes wants the workers to work overtime: they are paid disproportionally more per unit (or, alternatively, per hour). In the USA, the economic aspect of this extra pay for overtime is not evident, since the larger payment for overtime is required by the labor laws, but our arguments show that it also makes perfect economic sense – and indeed, many companies had the same practice before it was required by law.
3. Further economic analysis leads to a (somewhat unexpected) additional consequence
3.1 How can we determine the salary: formulation of the problem
In the above analysis, we assumed that the pay-per-unit amount s is fixed. A natural question is: how can we determine this amount?
A natural idea is to select the value s for which the overall positive effect on the company is the best. What does “the best” mean?
If it is a private company, it needs to maximize its profit. Each produced unit can be sold for p monetary units. If overall, the company produces x units, then its profit P can be obtained if we subtract, from the overall revenue p ⋅ x, the salary of the workers s ⋅ x and additional expenses A ⋅ x – which are also proportional to the overall production.
If it is a state-owned company, then its goal is to maximize the benefit for society as a whole. If the benefit from each unit is p, then to get the resulting overall gain in benefit, we also need to subtract from the overall benefit p ⋅ x the salary s ⋅ x and the additional expenses A ⋅ x.
In both cases, we need to select the pay-per-unit s so as to maximize the overall benefit
In this formula, the values p and A are known, but the overall production x depends on the pay-per-unit s: the more we pay per unit, the more incentivized will the workers be, and the larger will be their production.
So, to find the optimal pay-per-unit s, let us analyze how the overall production depends on the pay-per-unit.
3.2 How production depends on the pay-per-unit: analysis
The overall production x is the sum of the amounts xi produced by individual workers: x = ∑xi. In line with the above-described general idea, each worker selects the production level xi that maximizes his/her overall utility, that is, by using notations from the previous section, the level xi maximizes the difference:
In this formula, we took into account that both the value of money (as described by the coefficient c) and the effort needed to produce one unit (as described by the value e) are, in general, different for different workers. So, we denoted the values of c and e corresponding to the i-th worker by, correspondingly, ci, and ei.
To find the value xi that maximizes the expression (7), we can differentiate this expression with respect to xi and equate the derivative to 0. As a result, we get the equation
Thus, the overall production has the form
that is, the form
3.3 So what is the optimal pay-per-unit?
Substituting the expression (11) for the dependence of production x on the pay-per-unit s into formula (6) that describes the overall benefit (that we want to maximize), we end up with the following expression for the overall benefit:
To find the pay-per-unit s for which this benefit is the largest, we differentiate this expression with respect to s and equate the derivative to 0. As a result, we get
that is,
3.4 This is exactly what is happening
Formula (14) describes the optimal value. How does this formula compare with the economic reality? As we will show, it compares perfectly.
Indeed, if we multiply both sides of formula (14) by the overall production size x, we conclude that
In other words, the overall salary s ⋅ x of all the workers is equal exactly to one-half of the overall difference between the revenue p ⋅ x and the needed expenses A ⋅ x – that is, exactly to one-half of the company's income.
In other words, in this arrangement, the labor share of the company's income should be equal to (or at least close to) 50%. And this is exactly what is observed in advanced economics; see, for example, Manyika et al. (2019). This empirical confirmation shows that our approximate model (of course, all economic models are approximate) adequately captures the phenomenon.
This empirical confirmation should be expected: we deduced the 50% value exactly by taking into account that companies want to maximize their overall profit.
It should be mentioned that, to the best of our knowledge, ours is the first explanation of this empirical proportion.
3.5 This empirical confirmation strengthens our arguments about advanced workers' pay
This perfect fit with empirical data makes us confident that our explanation of disproportional pay for advanced workers – which is based on exactly the same model – is correct.
References
Fishburn, P.C. (1969), Utility Theory for Decision Making, John Wiley & Sons, New York, NY.
Kahneman, D. (2011), Thinking, Fast and Slow, Farrar, Straus, and Giroux, New York, NY.
Kosheleva, O. and Kreinovich, V. (2019), “How to assign points for chores”, Russian Digital Libraries Journal, Vol. 22 No. 6, pp. 759-761.
Kreinovich, V. (2014), “Decision making under interval uncertainty (and beyond)”, in Guo, P. and Pedrycz, W. (Eds), Human-Centric Decision-Making Models for Social Sciences, Springer-Verlag, Cham, pp. 163-193.
Luce, R.D. and Raiffa, R. (1989), Games and Decisions: Introduction and Critical Survey, Dover, New York, NY.
Manyika, J., Mischke, J., Bughin, J., Woetzel, J., Krishnan, M. and Cudre, S. (2019), “A new look at the declining labor share of income in the United States”, McKinsey Global Institute, San Francisco, CA, available at: https://www.mckinsey.com/featured-insights/employment-and-growth/a-new-look-at-the-declining-labor-share-of-income-in-the-united-states.
Nguyen, H.T., Kosheleva, O. and Kreinovich, V. (2009), “Decision making beyond arrow's ‘impossibility theorem', with the analysis of effects of collusion and mutual attraction”, International Journal of Intelligent Systems, Vol. 24 No. 1, pp. 27-47.
Raiffa, H. (1997), Decision Analysis, McGraw-Hill, Columbus, OH.