Fig 1.
How firm size changes with energy use per capita.
Panel A shows how average firm size within nations varies with energy use per capita. Firm size is measured using employment. Each data point represents a country. Error bars indicate the 95% confidence interval in the estimates of mean firm size. Grey regions indicate the 95% confidence region of the regression. Panel B shows how the entire firm size distribution within nations varies by energy use. I put countries into 5 groups, ranked by energy use. I then plot the aggregate firm size distribution within each group. The inset graph shows average energy use per capita within each quintile. Here α refers to the estimated power-law exponent of the firm size distribution. For sources and methods, see Section 7.
Fig 2.
Hierarchical employment structure of six case-study firms.
This figure shows the hierarchical employment structure of six different case-study firms, named after the study authors [57–62].
Fig 3.
The exponential growth of subordinates with rank.
In an idealized hierarchy, the total number of subordinates (blue) tends to grow exponentially with hierarchical rank (red). The exact relation will depend on the span of control—the number of subordinates directly below each superior.
Fig 4.
Average income vs. hierarchical power within case-study firms.
This figure shows data from six firm case studies [57–62]. The vertical axis shows average income within each hierarchical level of the firm (relative to the base level), while the horizontal axis shows my metric for average power, which is equal to one plus the average number of subordinates below a given hierarchical level. Each point represents a single firm-year observation, and color indicates the particular case study. Grey regions around the regression indicate the 95% prediction interval. For methods, see Section 7.
Fig 5.
The growth of hierarchy concentrates power.
This figure illustrates how the growth of hierarchy leads to the concentration of hierarchical power. Below each hierarchy, I show the distribution of hierarchical power. (hierarchical power = 1 + the total number of subordinates). I then calculate the Gini index of hierarchical power concentration (G). The initial growth of hierarchy rapidly concentrates power. But further growth of hierarchy leads to progressively slower growth of hierarchical-power concentration.
Fig 6.
Visualizing the energy-hierarchy-inequality model.
This figure shows the EHI model as a landscape. Hierarchies are visualized as pyramids. Height and color indicate hierarchical rank. The top panel shows a subsistence society that consumes hunter-gatherer levels of energy use. The model predicts little hierarchical organization, and little concentration of hierarchical power. The bottom panel shows an industrial society with energy use on par with modern Iceland or Qatar. The model predicts considerable hierarchical organization, and considerable concentration of hierarchical power.
Fig 7.
Extrapolating the origin of inequality with the EHI model.
This figure shows the results of the energy-hierarchy-inequality model. Panel A shows how the concentration of hierarchical power changes with energy use per capita. Panel B shows the evolution of income inequality. Color indicates the scaling exponent β between hierarchical power and income (see Eq 3). Shaded regions show the energy use range for various societies throughout history. For sources and methods, see Section 7.
Fig 8.
Testing the energy-hierarchy-inequality model.
This figure compares the EHI model to empirical data. Panel A shows archaeological data from ancient societies, measured using housing size and reported by ‘adaptation’. Horizontal lines indicate the plausible range of energy use for each adaptation. Panel B shows income inequality in pre-industrial societies. Energy use is estimated from per capita income data (horizontal lines show the uncertainty). Panel C shows data for modern nation-states, with vertical lines showing the range of inequality estimates for each country. Panel D also shows modern data, but measures inequality using the top 1% income share. For sources and methods, see Section 7.
Fig 9.
Is the Kuznets curve caused by declining hierarchical despotism?
Panel A plots all of the empirical data in Fig 8A–8C. The red line shows the smoothed trend. It has an inverted U shape, often called a ‘Kuznets curve’. Panel B shows inferred β for each society. This is the scaling of income with hierarchical power that is required if the EHI model is correct. I infer β by matching real-world societies to the EHI model. I interpret β as an index of ‘hierarchical despotism’—it measures elites’ ability to use their hierarchical power to concentrate resources.
Fig 10.
Energy use estimates by adaptation.
This figure shows the energy range for historical societies sorted by adaptation. Data sources are shown in Table 1.
Table 1.
Data sources for energy use by adaptation.
Table 2.
Hierarchy model notation.
Fig 11.
Calculating the average number of subordinates.
Table 3.
Model parameters.
Fig 12.
Idealized hierarchy implied by firm case studies.
Panel A shows how the span of control varies with hierarchical level in case-study firms [57–62]. 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. Horizontal ‘jitter’ has been introduced to better visualize the data. The line indicates an exponential regression, with the grey region indicating the regression 95% confidence interval. Panel B shows the idealized firm hierarchy that is implied by the regression in Panel A. Error bars show the uncertainty in the hierarchical shape, calculated using a bootstrap resample of case-study data.
Fig 13.
Probability distribution of β in case-study institutions.
This figure shows the probability distribution of the parameter β in different case-study institutions. This parameter indicates the scaling behavior between income and hierarchical power: income ∝ (hierarchical power)β. Probabilities are determined using the bootstrap method. Panel A shows the β probability distribution for case-study firms [57–62]. Panel B shows the β probability distribution for a US slave estate (Cannon’s Point Plantation [161]). I show results for measuring inequality in terms of both house size and income.
Fig 14.
Determining the power-income ‘Noise’ parameter.
This figure shows the distribution of income dispersion within hierarchical levels of case-study firms [57–62, 162], measured using the Gini index. The mean of this distribution (with associated uncertainty) is used to set the power-income noise parameter σ. When not reported directly (or calculable from raw data), the within-hierarchical level Gini index is estimated from reported summary statistics in case studies.