How the rich are different: hierarchical power as the basis of income size and class

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.

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 class-based income

Data for US class-based 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 class-based 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 class-based income (defined in the main paper) and the empirical data (Table 7). My two-class 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).

Table 7 Measures of US class-based income

Table 8 World inequality database main series

Table 9 World inequality database component series

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.

Power-law exponent: I estimate the power-law 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 power-law 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 y-value for the CCD is 1 minus the corresponding y-value 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:

$$\begin{aligned} \text {K}1^*_{\text {frac}} = {\left\{ \begin{array}{ll} \text {fkinc} ~/~ \text {fainc} \\ \text {pkinc} ~/~ \text {ptinc}. \\ \end{array}\right. } \end{aligned}$$

(6)

Capitalist income share $\text {K}2^*_{\text {frac}}$ is calculated as:

$$\begin{aligned} \text {K}2^*_{\text {frac}} = (\text {fkequ} + \text {fkfix} ) ~/~ \text {fainc}. \end{aligned}$$

(7)

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.

Table 10 Titles used to identify the ‘CEO’

Finding the CEO

I identify CEOs using titles in the Execucomp series TITLEANN. I use a three-step 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.

Table 11 Execucomp compensation series

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:

$$\begin{aligned} \text {Total pay}= \text {TOTAL}\_\text {ALT2} \end{aligned}$$

(8)

$$\begin{aligned} K_{\text {frac}}= \frac{ \text {Actual realized gains from stock options} }{\text {Total pay}} \end{aligned}$$

(9)

$$\begin{aligned} K_{\text {frac}}= \frac{ \text {SHRS}\_\text {VEST}\_\text {VAL} + \text {OPT}\_\text {EXER}\_\text {VAL} }{\text {TOTAL}\_\text {ALT2}}. \end{aligned}$$

(10)

Series descriptions are shown in Table 11.

CEO pay ratio and firm employment

I calculate the CEO pay ratio as:

$$\begin{aligned} \text {CEO pay ratio} = \frac{\text {CEO pay}}{\text {Firm mean income}}. \end{aligned}$$

(11)

Firm mean income is calculated by dividing total staff expenses (Compustat Series XLR) by total employment (Compustat Series EMP):

$$\begin{aligned} \text {Firm mean income} = \frac{\text {Total staff expenses}}{\text {Total employment}}. \end{aligned}$$

(12)

CEO pay ratio and firm mean income data are available for roughly 3000 firm-year observations from 2006 to 2016. Figure 16 shows summary statistics of these data.

Fig. 16

Statistics of the Compustat firm sample. This figure shows selected statistics of the Compustat firm sample. a shows the number of firms in the sample over time, b the average firm size, and c the share of US employment held by these firms. d shows the logarithmic distribution of firm size, and e shows the logarithmic distribution of the CEO pay ratio. f shows the mean CEO pay ratio of all firms over time. g shows the logarithmic distribution of normalized mean pay (mean pay divided by the average pay of the firm sample in each year). h shows the ratio of mean pay in the sample relative to the US average (calculated from BEA Table 1.12 by dividing the sum of employee and proprietor income by the number of workers in BEA Table 6.8C-D. i shows the Gini index of firm mean pay over time

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.

Table 12 Model 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:

$$\begin{aligned} s_h \equiv {\left\{ \begin{array}{ll} ~~1 &{} \text {if} ~~ h = 1 \\ ~~\dfrac{E_{h-1}}{E_{h}} &{} \text {if} ~~ h \ge 2. \\ \end{array}\right. } \end{aligned}$$

(13)

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:

$$\begin{aligned} s_h = {\left\{ \begin{array}{ll} ~~1 &{} \text {if} ~~ h = 1 \\ ~~a \cdot e^{bh} &{} \text {if} ~~ h \ge 2. \\ \end{array}\right. } \end{aligned}$$

(14)

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 $E_h$:

$$\begin{aligned} E_h = \left\lfloor \frac{E_{h-1}}{s_h} \right\rfloor ~~~~~~ \text {for} ~~~~~~ h > 1. \end{aligned}$$

(15)

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 non-recursive formula for hierarchical employment:

$$\begin{aligned} E_h = \left\lfloor E_1 \cdot \frac{1}{s_2} \cdot \frac{1}{s_3} \cdot ... \cdot \frac{1}{s_h} \right\rfloor . \end{aligned}$$

(16)

In product notation, Eq. 16 becomes:

$$\begin{aligned} E_h = \left\lfloor E_1 \prod _{i=1}^h \frac{1}{s_i} \right\rfloor . \end{aligned}$$

(17)

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:

$$\begin{aligned} E_T = \sum _{h=1}^{n} E_h. \end{aligned}$$

(18)

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 non-zero employment:

$$\begin{aligned} n = \lbrace h ~~~ | ~~~ E_{h} \ge 1 ~~~ \text{ and } ~~~ E_{h+1} = 0 \rbrace . \end{aligned}$$

(19)

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:

$$\begin{aligned} I_{i,h,f} = \bar{I}_{1,f} \cdot ( \bar{P}_{h,f}) ^ {\beta _f} \cdot \epsilon _i. \end{aligned}$$

(20)

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 power-income 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:

$$\begin{aligned} P_{h} = \bar{S}_h + 1. \end{aligned}$$

(21)

Here $\bar{S}_h$ is the average number of subordinates per member of rank h:

$$\begin{aligned} \bar{S}_h ~ = \sum _{i = 1}^{h -1} \frac{E_i}{E_h}. \end{aligned}$$

(22)

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 ($E_h$):

$$\begin{aligned} \bar{I}_T = \sum _{h = 1}^n \bar{I}_h \cdot \frac{E_h }{E_T}. \end{aligned}$$

(23)

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:

$$\begin{aligned} C = \frac{\bar{I}_n}{\bar{I}_T}. \end{aligned}$$

(24)

Table 13 Parameters in the US hierarchy model

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.

Fig. 17

The size distribution of firms in the United States. This figure compares the firm-size distribution in the United States to a discrete power-law distribution. The ‘steps’ indicate the firm-size bins. The inset plot shows the best-fit power-law exponent ($\alpha$) in each year. The US data combine ‘employer’ firms and unincorporated self-employed workers. Data for ‘employer’ firms are from the US Census Bureau, Business Dynamics Statistics. I augment these data with Bureau of Labor Statistics data for unincorporated self-employed workers (series LNU02032185 and LNU02032192). The histogram preserves firm-size bins used by the Census. I add self-employed individuals to the first bin. The last histogram bin contains all firms with more than 10,000 employees

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 power-law distribution, the probability of finding a firm of size x is:

$$\begin{aligned} p(x) \propto \frac{1}{x^\alpha }. \end{aligned}$$

(25)

Figure 17 compares the size distribution of US firms to a discrete power law. The inset plot shows the best-fit values for the power-law exponent $\alpha$, fitted using the method described in Ref. [71].

To simulate the size distribution of US firms, I use a discrete power-law distribution of 1 million firms. In each iteration, the model sets the power-law 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 power-law 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 power-law distribution can be found in rpld.h, located in the Supplementary Material [68]. This code is an adaptation of Collin Gillespie’s [74] discrete power-law generator found in the R poweRlaw package. Gillespie’s generator is, in turn, an adaptation of the algorithm outlined by Clauset et al. [75].

Fig. 18

Idealized hierarchy implied by firm case studies. Panel A shows how the span of control varies with hierarchical level in case-study firms [48,49,50,51,52,53]. The span of control is the subordinate-to-superior ratio between adjacent hierarchical levels. The x-axis corresponds to the upper hierarchical level in each corresponding ratio. Case-study firms are indicated by color. I have added horizontal ‘jitter’ to better visualize the data. The line indicates an exponential regression, with the grey region indicating the regression 95% confidence interval. b Shows the idealized firm hierarchy that is implied by the regression in a. Error bars show the uncertainty in the hierarchical shape, calculated using a bootstrap resample of case-study data

Span-of-control parameters a and b

The shape of the employment hierarchy in simulated firms is determined by the span-of-control parameters a and b. To set these parameters, I regress Eq. 14 onto span-of-control data from case-study 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 case-study 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 base-level 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 span-of-control parameters a and b.

Code implementing this method is located in base_fit.h in the Supplementary Material [68].

Fig. 19

Estimating the power-income exponent. ($\beta$) Inside compustat firms. This figure shows the $\beta$ values fitted to Compustat firms. a shows how the fitted values of $\beta$ relate to firm size and the CEO pay ratio. The discrete changes in color (evident as vertical lines) correspond to changes in the number of hierarchical levels within simulated firms. b shows the distribution of the fitted values of $\beta$. Note that fitted values for $\beta$ vary between model iterations

Power-income exponent $\beta$

The power-income 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 two-step 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 power-income 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 best-fit value for $\beta$, I use numerical optimization (the bisection method) to minimize the following error function:

$$\begin{aligned} \epsilon (\beta ) = \left|~ C_{\text {model}} - C_{\text {Compustat}} \right|. \end{aligned}$$

(26)

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 best-fit $\beta$ parameter produces an error larger than $\epsilon = 0.01$. Figure 19 shows an example of the fitted $\beta$ values for all Compustat firm-year observations. Code implementing this fitting method is located in fit_beta.h in the Supplementary Material [68].

Fig. 20

Modeling the distribution of the parameter. $\beta$ This figure visualizes the algorithm used to simulate the distribution of the parameter $\beta$. This parameter determines how rapidly income increases with hierarchical power in a given firm. a shows binned data for $\sigma _E$ (the lognormal scale parameter) for Compustat firms. Each dot indicates $\sigma _E$ for the given firm-size bin. The straight line indicates the modeled relation (Eq. 30). b shows how the modeled dispersion of $\beta$ decreases with firm size. c compares the distribution of $\beta$ for Compustat firms to the simulated distribution, created by injecting Compustat-sized firms into the model. d uses the same method to compare the CEO pay ratio in Compustat firms to that produced by injecting Compustat-sized firms into the model. Contour P10 contains 10% of the data, contour P50 contains 50%, and contour P90 contains 90% of the data

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:

$$\begin{aligned} \beta (E) = \ln {\mathcal {N}} ( \beta ; \mu ,\sigma _{E} ). \end{aligned}$$

(27)

The location parameter $\mu$ is constant for all firms and is given by:

$$\begin{aligned} \mu = \overline{ \ln (\beta _{\text {~Compustat}}) }. \end{aligned}$$

(28)

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:

$$\begin{aligned} \sigma _{E} = \text {SD} \left[ ~ \ln ( \beta _{\text {~Compustat}}) ~\right] _{E}. \end{aligned}$$

(29)

Figure 20A shows how $\sigma _E$ varies with firm size. Each dot indicates $\sigma _E$ calculated on a log-spaced firm-size bin. I model $\sigma _E$ as a log-linear function of firm size:

$$\begin{aligned} \sigma _{E} = c_{1} \ln (E) + c_{2}. \end{aligned}$$

(30)

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’ Compustat-sized firms into the model, and test if the properties of these simulated firms match the properties of the real-world 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 base-level 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 two-step 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:

$$\begin{aligned} \dfrac{\bar{I}_1^{\text {~Compustat}}}{\bar{I}_T^{\text {~Compustat}}} = \dfrac{\bar{I}_1^{\text {~model}}}{\bar{I}_T^{\text {~model}}}. \end{aligned}$$

(31)

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:

$$\begin{aligned} \dfrac{\bar{I}_1^{\text {~Compustat}}}{\bar{I}_T^{\text {~Compustat}}} = \dfrac{1}{\bar{I}_T^{\text {~model}}}. \end{aligned}$$

(32)

To solve for $\bar{I}_1^{\text {~Compustat}}$, we rearrange Eq. 32 to get :

$$\begin{aligned} \bar{I}_1^{\text {~Compustat}} = \frac{\bar{I}_T^{\text {~Compustat}}}{\bar{I}_T^{\text {~model}}}. \end{aligned}$$

(33)

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].

Fig. 21

Modeling the distribution of base pay in Compustat firms. This figure shows the distribution of (fitted) mean pay in the base level of Compustat firms. Pay is normalized, so that the average income in the Compustat sample (in each year) is 1. I model these data with a gamma distribution

Simulating the distribution of base pay $\bar{I}_1$ for all US firms

Once each Compustat firm has a fitted value for base-level mean pay, we fit these data with a parametric distribution. I use a gamma distribution to model the distribution of base-level 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 over-represented 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 base-level pay in modeled firms. Code implementing this method is located in base_pay_sim.h in the Supplementary Material [68].

Power-income noise factor

I model noise ($\epsilon$) in the power-income relation using a lognormal random variate:

$$\begin{aligned} \epsilon \sim \ln {\mathcal {N}}(\mu , \sigma ). \end{aligned}$$

(34)

The noise factor is designed to reproduce the average income dispersion within hierarchical ranks of case-study firms. I set the lognormal scale parameter ($\mu$), so that the distribution of $\epsilon$ has a mean of 1:

$$\begin{aligned} \mu = \ln (1) - \frac{1}{2} \sigma ^2. \end{aligned}$$

(35)

To determine $\sigma$, we first calculate the mean Gini index ($\bar{G}$) of inequality within hierarchical ranks of case-study firms (Fig. 22). We then calculate $\sigma$ using:

$$\begin{aligned} \sigma = 2 \cdot \hbox {erf}^{-1} (\bar{G}). \end{aligned}$$

(36)

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 case-study 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].

Fig. 22

Determining the power-income ‘noise’ parameter. This figure shows the distribution of income dispersion (measured using the Gini index) within hierarchical ranks of case-study firms [48,49,50,51,52,53, 77]. I use the mean of this distribution (with associated uncertainty) to set the power-income noise parameter $\epsilon$. For methods used to calculate within-rank income dispersion in the case-study data, see the Appendix in [47]

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 ($\epsilon$):

$$\begin{aligned} K_{\text {frac}} \propto \ln (P) \cdot \epsilon . \end{aligned}$$

(37)

The labor fraction of income ($L_{\text {frac}}$) is then the complement of the capitalist fraction:

$$\begin{aligned} L_{\text {frac}} = 1 - K_{\text {frac}}. \end{aligned}$$

(38)

Conceptually, then, class-based 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:

$$\begin{aligned} K_{\text {frac}} \sim \mathcal {N}( K_{\text {frac}}; \mu _K, \sigma _K ) ; ~~~~ K_{\text {frac}} \le 1. ~~ \end{aligned}$$

(39)

I then create a lower bound by setting to zero all randomly drawn values that are less than zero:

$$\begin{aligned} \text {if}(K_{\text {frac}} ) < 0 ~~ \text {then} ~~ K_{\text {frac}} = 0. \end{aligned}$$

(40)

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 firm-size 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 log-linear function of firm size (Fig. 18a):

$$\begin{aligned} \mu _K = c_1 \ln (P) + c_2 \end{aligned}$$

(41)

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):

$$\begin{aligned} RSD = \left| \frac{\sigma _K}{\mu _K} \right| = c_1 P^{c_2}. \end{aligned}$$

(42)

I then define $\sigma _K$ as:

$$\begin{aligned} \sigma _K = RSD \cdot | \mu _K |. \end{aligned}$$

(43)

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 firm-size bins to estimate $\mu _K$ and $\sigma _K$. Code implementing this method is located in k_func.h in the Supplementary Material [68].

Fig. 23

The capitalist-gradient model. This figure shows the steps used to simulate the relation between the capitalist fraction of income and hierarchical power. Using Eqs. 39 and 40, a and b show how the parameters $\mu _K$ and $\sigma _K$ vary with firm size. Each point represents the value fitted to binned CEO data. The line indicates the modeled relation. c and d compare the CEO data to simulated data, created by injecting individuals with the same hierarchical power as our CEOs into the model. c shows the distribution of the capitalist fraction of income. Using data binned by firm size, d shows how the capitalist fraction of income changes with hierarchical power

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.

Table 14 Structure of the hierarchy model