Is it fair that advanced workers get paid disproportionally more: economic analysis

Purpose – Ontheone hand,everyoneagreesthateconomicsshouldbefair,thatworkersshouldget equalpay 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%.


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 workerscase 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 rewardsand 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.

But is this really unfair?
From the purely mathematical viewpoint, if we view the idea of equal pay for equal labor literally, this arrangementwhen advanced workers get paid disproportionally moreis 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 x 0 units in a certain period of time: day, week, month, whatever. Then, the average worker's salary during this time period is s $ x 0 . 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), andKreinovich (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: uðM Þ ¼ c$ ffiffiffiffiffi M p ; see, for example, Kahneman (2011) and references therein (see also Kosheleva and Kreinovich, 2019). This formula may sound strange at first glance, but it is actually in good accordance with common sense. Indeed, according to this empirical formula, the increase in utility caused by an extra dollar decreases with M. This makes perfect sense: (1) If a person has no money, getting a dollar is a big deal, but (2) 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 $ x 0 to the new value s $ x 0 þ m. So, the additional reward m increases the worker's utility from the original value c$ ffiffiffiffiffiffiffiffi ffi . We need to make sure that the resulting increase in utility is greater than or equal to that of the decrease in utility a $ e. In this case, the overall balance for the worker will be positive, and the worker will be incentivized to be more productive.

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 5 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 c 2 $ s $ x 0 from both sides, and move the term 2a$e$c$ ffiffiffiffiffiffiffiffi ffi s$x 0 p to the left-hand side, we conclude that Dividing both sides by a $ e 2 , 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.

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: (1) The whole society benefits from the increased productivity, and (2) 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 senseand 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?
(1) 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 $ xwhich are also proportional to the overall production.
(2) 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 ðp À s À AÞ$x: 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.

How production depends on the pay-per-unit: analysis
The overall production x is the sum of the amounts x i produced by individual workers: x 5 P x i . In line with the above-described general idea, each worker selects the production level x i that maximizes his/her overall utility, that is, by using notations from the previous section, the level x i 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, c i , and e i . To find the value x i that maximizes the expression (7), we can differentiate this expression with respect to x i and equate the derivative to 0. As a result, we get the equation Thus, the overall production has the form that is, the form where we denoted 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: ðp À AÞ$K$s À K$s 2 : To find the pay-per-unit s for which this benefit is the largest, we differentiate this expression with respect to s and equate the derivative to 0. As a result, we get ðp À AÞ$K À 2K$s ¼ 0; 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 $ xthat 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.

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 workerswhich is based on exactly the same modelis correct.