Abstract
This paper investigates a new approach to understanding personal and functional income distribution. I propose that hierarchical power—the command of subordinates in a hierarchy—is what distinguishes the rich from the poor and capitalists from workers. Specifically, I hypothesize that individual income increases with hierarchical power, as does the share of individual income earned from capitalist sources. I test this idea using evidence from US CEOs, as well as a numerical model that extrapolates the CEO data. The results indicate that income tends to increase with hierarchical power, as does the capitalist composition of income. This suggests that hierarchical power may be a determinant of both personal and functional income.
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Notes
 1.
There is a subtle distinction between neoclassical and Marxist theory. Neoclassical theory attributes labor income directly to productivity. However, Marxist theory attributes income to the value of labor power. The latter is the labor time required to reproduce labor power. Since the labor power of more productive workers tends to take more to reproduce, more productive workers tend to earn higher wages.
 2.
In 2017, employee ownership plans had total assets of roughly $1.3 trillion [54], while total US market capitalization was roughly $30 trillion, according to the Russel 3000 index.
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Acknowledgements
I thank Jonathan Nitzan for comments on an earlier draft of this paper.
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Methods
Methods
Supplementary materials for this paper are available at the Open Science Framework [68]. The supplementary materials include source data, code for analysis, and model code.
US classbased income
Data for US classbased income come from the World Inequality Database (WID). My income measures are shown in Table 7. These are composed of the WID data series, as shown in Tables 8 and 9. I use two WID series to construct K1*, L1*, and T. This means that I merge statistics from both WID series.
The WID data come from Piketty et al. [56]. For the methods of this study, see their appendix [69]. This is the most detailed study to date of US classbased income. However, it comes with some caveats. Piketty et al. subdivide proprietor income into capitalist and labor components. The capitalist component is series fkbus. The labor component is series flmil (Table 9). I cannot find, in Piketty’s work, an explicit statement of the methods behind this split. However, according to Rognlie [70], Piketty assumes that proprietor income “has the same net capital share as the corporate sector”.
This leads to a difference between my definitions of classbased income (defined in the main paper) and the empirical data (Table 7). My twoclass definition of capitalist income (K1) includes all proprietor income. In contrast, the empirical measure K1* contains only a portion of proprietor income. My definition of labor income (L1) contains no proprietor income. In contrast, the empirical measure L1* contains a portion of proprietor income.
In addition to the capitalist income series provided by WID, I construct my own series K2 shown in Table 7. This includes equity and interest income (with capital gains).
Methods for estimating income distribution statistics
WID provides three types of data that I use to compute statistics:

1.
Income percentile (bin)

2.
Income share (by income percentile bin)

3.
Income threshold (by income percentile bin).
As an example, the WID data may indicate that percentiles P99–P100 have an income share of 15%. This means that the top 1% holds 15% of all income. The income threshold for this bin may be $200,000. This means that the lowest income of the top 1% is $200,000.
Gini index: I estimate the Gini index by constructing a Lorenz curve from WID data. The Gini index equals the area between the Lorenz curve and the line of perfect equality, divided by the total area under the line of perfect equality.
Top 1% share: This is provided directly by the WID data.
Powerlaw exponent: I estimate the powerlaw exponent of the top 1% of incomes using income percentile and threshold data. I create binned data where we know the proportion of people in each bin, and the income boundaries of each bin. I then use the method discussed in Ref. [71] to estimate the powerlaw exponent from this binned data.
Probability density: I estimate the probability density using income percentile and threshold data. I first normalize income threshold data, so that the median equals 1. I then construct a cumulative distribution. This is the fraction of individuals below a given income. I estimate the probability density function from the slope of the cumulative distribution.
Lorenz curve: The Lorenz curve is constructed from income percentile and income share data. It is the cumulative share of income vs. income percentile.
Cumulative distribution: I construct the cumulative distribution from income percentile and threshold data. I normalize income threshold data, so that the median equals 1.
Complementary cumulative distribution: I construct the complementary cumulative distribution (CCD) from the cumulative distribution (CD). The yvalue for the CCD is 1 minus the corresponding yvalue for the CD.
Income quantiles: These data are provided directly by WID (reported as income thresholds by income percentile bin).
Capitalist income share vs. percentile: Capitalist income share \(\text {K}1^*_{\text {frac}}\) is calculated by merging two series:
Capitalist income share \(\text {K}2^*_{\text {frac}}\) is calculated as:
US CEO data: the Compustat firm sample
Data for US CEOs (and their firms) come from the Execucomp and Compustat databases. I will refer to these data as the ‘Compustat firm sample’. I use these data for the case study of CEO pay and as the basis for the US hierarchy model. Methods are discussed below.
Finding the CEO
I identify CEOs using titles in the Execucomp series TITLEANN. I use a threestep algorithm:

1.
Find all executives whose title contains one or more of the words in the ‘CEO Titles’ list in Table 10.

2.
Of these executives, take the subset whose title does not contain any of the words in the ‘Subordinate Titles’ list in Table 10.

3.
If this returns more than one executive per firm per year, chose the executive with the highest pay.
CEO pay and capitalist income fraction
Execucomp contains several different estimates of CEO pay. These differ primarily in the valuation of stock option compensation. [55] argue that we should use actual realized gains. This is the difference between the market value of the option and the exercise value at the time of exercise. Importantly, actual realized gains are the income recorded by the IRS for tax purposes. I measure CEO total pay and capitalist income fraction (\(K_{\text {frac}}\)) using the following series:
Series descriptions are shown in Table 11.
CEO pay ratio and firm employment
I calculate the CEO pay ratio as:
Firm mean income is calculated by dividing total staff expenses (Compustat Series XLR) by total employment (Compustat Series EMP):
CEO pay ratio and firm mean income data are available for roughly 3000 firmyear observations from 2006 to 2016. Figure 16 shows summary statistics of these data.
Hierarchy model equations
The hierarchy model assumes that firms are hierarchically structured, with a span of control that increases exponentially with hierarchical rank. The model simulates individual income as a function of hierarchical power. I discuss here the model’s main equations. See Table 12 for notation.
The employment hierarchy
For each firm, the model generates an employment hierarchy using the span of control (s). This is the ratio of employment (E) between two consecutive hierarchical ranks (h). We let \(h = 1\) be the bottom hierarchical rank. We define the span of control in rank 1 as \(s= 1\). This leads to a piecewise function for the span of control:
Based on evidence from firm case studies (Fig. 18), the model assumes that the span of control increases exponentially with hierarchical rank, with a and b as free parameters:
As one moves up the hierarchy, employment in each consecutive rank (\(E_h\)) decreases by a factor of \(1/s_h\). This yields a recursive formula for calculating \(E_h\):
The model assumes that employment is a whole number and so rounds down to the nearest integer (notated by \(\lfloor \rfloor\)). By repeatedly substituting Eq. 15 into itself, we obtain a nonrecursive formula for hierarchical employment:
In product notation, Eq. 16 becomes:
Total employment \(E_T\) in the whole firm is the sum of employment in all hierarchical ranks. Defining n as the total number of hierarchical ranks, total firm employment is:
Because the model builds the hierarchy from the bottom–up, n is not known beforehand. The model defines n using Eq. 17. The model calculates employment in every hierarchical rank until it reaches a rank with zero employment. The top rank n is the highest rank with nonzero employment:
To summarize, the employment hierarchy in each firm is determined by 3 free parameters: the span of control parameters a and b, and employment in the bottom rank, \(E_1\). Code for this algorithm is located in exponents.h and hierarchy.h in the Supplementary Material [68].
The pay hierarchy
The model assumes that individual income is a function of hierarchical power:
Here, \(I_{i,h,f}\) is the income of the ith person in hierarchical level h of firm f. \(\bar{I}_{1,f}\) is the average income in the bottom hierarchical level of the firm. \(\bar{P}_{h,f}\) is average hierarchical power in level h of the firm. \(\beta _f\) is the powerincome exponent of the given firm. Lastly, \(\epsilon _i\) is a stochastic noise factor that adds dispersion to individual income.
In each firm, we define the average hierarchical power in level h as:
Here \(\bar{S}_h\) is the average number of subordinates per member of rank h:
Statistics
Mean income in a firm. Mean income in a firm (\(\bar{I}_T\)) is the average of mean income in each hierarchical rank (\(\bar{I}_h\)), weighted by the employment in each rank (\(E_h\)):
CEO pay ratio. The model defines the ‘CEO’ as the person in the top hierarchical rank, n. CEO pay is thus \(\bar{I}_n\), average income in the top hierarchical rank. The CEO pay ratio (C) is defined as CEO income divided by average income in the firm:
The United States hierarchy model
The US hierarchy model uses the equations from “Hierarchy model equations” to simulate the hierarchical structure of the US private sector. The model’s parameters are summarized in Table 13. I detail here how I restrict these parameters.
Simulating the size distribution of US firms
Evidence suggests that the size distribution of firms in the US (and other G7 countries) roughly follows a power law [72, 73]. In a powerlaw distribution, the probability of finding a firm of size x is:
Figure 17 compares the size distribution of US firms to a discrete power law. The inset plot shows the bestfit values for the powerlaw exponent \(\alpha\), fitted using the method described in Ref. [71].
To simulate the size distribution of US firms, I use a discrete powerlaw distribution of 1 million firms. In each iteration, the model sets the powerlaw exponent \(\alpha\) by sampling from the set of fitted US values (Fig. 17, inset). To ensure that the simulation produces realistically sized firms, I truncate the powerlaw distribution at a maximum firm size of 2.3 million. This is the present size of Walmart, the largest US firm that has ever existed.
Code for the random number generator for the discrete powerlaw distribution can be found in rpld.h, located in the Supplementary Material [68]. This code is an adaptation of Collin Gillespie’s [74] discrete powerlaw generator found in the R poweRlaw package. Gillespie’s generator is, in turn, an adaptation of the algorithm outlined by Clauset et al. [75].
Spanofcontrol parameters a and b
The shape of the employment hierarchy in simulated firms is determined by the spanofcontrol parameters a and b. To set these parameters, I regress Eq. 14 onto spanofcontrol data from casestudy firms (Fig. 18a). I then use Eqs. 14, 17, and 18 to create the employment hierarchy in each simulated firm. Note that all firms are assigned the same values for a and b.
The model incorporates uncertainty in a and b using the bootstrap method [76]. I run the model many times, with each iteration regressing a and b on a bootstrapped sample of the firm casestudy data. Figure 18b shows the shape of the modeled employment hierarchy for a generic large firm. Code implementing the bootstrap is located in boot_span.h in the Supplementary Material [68].
Employment in the base hierarchical level (\(E_T\))
Given span of control parameters a and b, each simulated firm hierarchy is constructed from the bottom hierarchical level up. To do this, we must estimate \(E_1\), the employment in the base hierarchical level.
To estimate \(E_1\) in each firm, I use the model to create a numerical function relating base level employment \(E_1\) to total firm employment \(E_T\). I input a range of different base employment values into Eqs. 14, 17, and 18, and calculate total employment for each value. The result is a discrete mapping relating baselevel employment to total employment. I then use the C++ Armadillo interpolation function to linearly interpolate between these discrete values. This creates a numerical function that returns \(E_1\) when given total firm employment \(E_T\) and spanofcontrol parameters a and b.
Code implementing this method is located in base_fit.h in the Supplementary Material [68].
Powerincome exponent \(\beta\)
The powerincome exponent (\(\beta\)) determines the rate that income increases with hierarchical power in simulated firms (see Eq. 20). Unlike the span of control parameters, I allow \(\beta\) to vary between firms.
I restrict the variation of \(\beta\) using a twostep process. I first ‘tune’ the model to data from Compustat firms. This returns a distribution of \(\beta\) that is specific to Compustat firms. I then fit these data with a parameterized distribution, from which simulated values for \(\beta\) are randomly chosen.
Fitting powerincome exponent \(\beta\) to Compustat firms
I fit \(\beta\) to Compustat firms using the CEO pay ratio (C). The first step of this process is to simulate the employment hierarchy for each Compustat firm using parameters a, b, and \(E_1\) (the latter is determined from total employment). Given the simulated employment hierarchy, the CEO pay ratio in the modeled firm is uniquely determined by the parameter \(\beta\). I choose \(\beta\), so that the model produces a CEO pay ratio that is equivalent to the Compustat data.
To find the bestfit value for \(\beta\), I use numerical optimization (the bisection method) to minimize the following error function:
Here, \(C_{\text {model}}\) is the modeled CEO pay ratio, and \(C_{\text {Compustat}}\) is the Compustat CEO pay ratio.
To ensure that there are no large errors, the model discards Compustat firms for which the bestfit \(\beta\) parameter produces an error larger than \(\epsilon = 0.01\). Figure 19 shows an example of the fitted \(\beta\) values for all Compustat firmyear observations. Code implementing this fitting method is located in fit_beta.h in the Supplementary Material [68].
Simulating the distribution of \(\beta\) for all US firms
Once we have estimated \(\beta\) for every Compustat firm, the next step is to fit a parameterized distribution to these data. For Compustat firms, the distribution of \(\beta\) is roughly lognormal, with dispersion that tends to decline with firm size
I model the distribution of \(\beta\) using a lognormal distribution with a constant location parameter \(\mu\) and a scale parameter \(\sigma _E\) that varies with firm size:
The location parameter \(\mu\) is constant for all firms and is given by:
To estimate the scale parameter \(\sigma\), I calculate the standard deviation of \(\ln (\beta _{\text {~Compustat}})\) within groups of firms binned by firm size E:
Figure 20A shows how \(\sigma _E\) varies with firm size. Each dot indicates \(\sigma _E\) calculated on a logspaced firmsize bin. I model \(\sigma _E\) as a loglinear function of firm size:
Once we have estimated the parameters \(\mu\) and \(\sigma _E\), we use Eq. 27 to generate \(\beta\) values for each simulated firm. Figure 20b shows how the modeled dispersion of \(\beta\) declines with firm size.
To test the above algorithm, I apply it back to Compustat firms. I ‘inject’ Compustatsized firms into the model, and test if the properties of these simulated firms match the properties of the realworld Compustat firms. Figures 20c and d show the results. Figure 20c compares the simulated distribution of \(\beta\) to the values fitted to Compustat firms (\(\beta _{Compustat}\)). Figure 20d shows how the CEO pay ratio grows with hierarchical power. Instead of plotting raw data, this figure shows a contour of the data for various density thresholds. In both cases, the model reasonably approximates Compustat values.
Mean pay in the base hierarchical level (\(\bar{I}_1\))
The baselevel pay parameter (\(\bar{I}_1\)) determines average pay in simulated firms. As with \(\beta\), I allow \(\bar{I}_1\) to vary across firms. I restrict this variation using a twostep process. I first ‘tune’ the model to data from Compustat firms. This creates a distribution of base pay specific to Compustat firms. I then fit these data with a parameterized distribution, from which simulation parameters are randomly chosen.
Estimating base pay \(\bar{I}_1\) in Compustat firms
Having already fitted a hierarchical pay structure to each Compustat firm (in the process of estimating \(\beta\)), we can use these data to estimate base pay for each firm. To do this, we set up a ratio between base level pay (\(\bar{I}_1\)) and firm mean pay (\(\bar{I}_T\)) for both the model and Compustat data:
Because the Compustat data cover multiple years, I first adjust firm mean pay (\(\bar{I}_T^{\text {Compustat}}\)) for inflation. I normalize \(\bar{I}_T^{\text {Compustat}}\) by dividing it by the average of firm mean pay for all firms in the given year.
The modeled ratio between base pay and firm mean pay (\(\bar{I}_1^{\text {~model}} / \bar{I}_T^{\text {~model}}\)) is independent of the choice of base pay. This is because the modeled firm mean pay is actually a function of base pay (see Eq. 23). If we run the model with \(\bar{I}_1^{\text {~model}} = 1\), then Eq. 31 reduces to:
To solve for \(\bar{I}_1^{\text {~Compustat}}\), we rearrange Eq. 32 to get :
The model uses Eq. 33 to estimate base pay \(\bar{I}_1^{\text {~Compustat}}\) for each Compustat firm. Code implementing this method is located in fit_beta.h in the Supplementary Material [68].
Simulating the distribution of base pay \(\bar{I}_1\) for all US firms
Once each Compustat firm has a fitted value for baselevel mean pay, we fit these data with a parametric distribution. I use a gamma distribution to model the distribution of baselevel pay (Fig. 21).
Note that the distribution of base pay in Compustat firms has a bimodal structure. I do not try to replicate this structure, because I feel that it is not representative of the US firm population. The lower mode in the Compustat data is composed mostly of chain restaurants, which seem to be overrepresented in the Compustat sample. While the gamma distribution fits the Compustat data quite roughly, it fits better than other parameterized distributions.
Once we have fitted the Compustat data with a gamma distribution, we then sample from this distribution to simulate baselevel pay in modeled firms. Code implementing this method is located in base_pay_sim.h in the Supplementary Material [68].
Powerincome noise factor
I model noise (\(\epsilon\)) in the powerincome relation using a lognormal random variate:
The noise factor is designed to reproduce the average income dispersion within hierarchical ranks of casestudy firms. I set the lognormal scale parameter (\(\mu\)), so that the distribution of \(\epsilon\) has a mean of 1:
To determine \(\sigma\), we first calculate the mean Gini index (\(\bar{G}\)) of inequality within hierarchical ranks of casestudy firms (Fig. 22). We then calculate \(\sigma\) using:
This equation is derived from the definition of the Gini index of a lognormal distribution: \(G=\hbox {erf}(\sigma / 2)\).
To incorporate uncertainty in the casestudy data, each model iteration uses a different bootstrap resample to calculate \(\bar{G}\). Code implementing this method is located in boot_sigma.h in the Supplementary Material [68].
Class composition of individual income
I model the class composition of individual income as a function of hierarchical power. The capitalist fraction of income (\(K_{\text {frac}}\)) increases with the logarithm of hierarchical power (P), with some associated noise (\(\epsilon\)):
The labor fraction of income (\(L_{\text {frac}}\)) is then the complement of the capitalist fraction:
Conceptually, then, classbased income is a simple function of hierarchical power. The complication, however, is that the dispersion \(\epsilon\) in capitalist income fraction is not simple. To replicate dispersion in the capitalist fraction of CEO income, I model \(K_{\text {frac}}\) as a partially truncated normal distribution. I draw values from a truncated normal distribution with an upper bound of 1:
I then create a lower bound by setting to zero all randomly drawn values that are less than zero:
The parameters \(\mu _K\) and \(\sigma _K\) are both functions of firm size. To model these parameters, I first fit a truncated normal distribution to CEO capitalist income fraction data, binned by firm size. Within each firmsize bin, I use numeric optimization to find the values for \(\mu _K\) and \(\sigma _K\) that best reproduce the distribution of the capitalist fraction of CEO income. I then model \(\mu _K\) as a loglinear function of firm size (Fig. 18a):
I model \(\sigma _K\) by first modeling the relative standard deviation \(\left \sigma _K / \mu _K \right\) as a power function of hierarchical power (Fig. 18a):
I then define \(\sigma _K\) as:
I test the above algorithm by applying it back to the CEO data. I ‘inject’ (into the model) individuals with the same hierarchical power as those in our CEO sample. I then simulate the capitalist component of their income. Figures 23c and d show how this simulation compares to the original data. Figure 23c compares the distribution of the capitalist fraction of income for all individuals. Figure 23d shows how the capitalist fraction of income grows with hierarchical power. In both cases, the model reproduces (with reasonable accuracy) the trends found in the empirical data.
To incorporate uncertainty, each model iteration uses different firmsize bins to estimate \(\mu _K\) and \(\sigma _K\). Code implementing this method is located in k_func.h in the Supplementary Material [68].
Summary of model structure
The model is implemented in C++ using a modular design. Each major task is carried out by a separate function that is defined in a corresponding header file. Table 14 summarizes the model’s structure in the order that functions are called. In each step, I briefly summarize the action that is performed, and reference the section where this action is described in detail.
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Fix, B. How the rich are different: hierarchical power as the basis of income size and class. J Comput Soc Sc (2020). https://doi.org/10.1007/s4200102000081w
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Keywords
 Hierarchy
 Power
 Functional income distribution
 Personal income distribution
 Inequality
 Capital as power
 Class