Energy, Hierarchy and the Origin of Inequality

Fix, Blair Article — Published Version Energy, Hierarchy and the Origin of Inequality PLoS ONE Provided in Cooperation with: The Bichler & Nitzan Archives Suggested Citation: Fix, Blair (2019) : Energy, Hierarchy and the Origin of Inequality, PLoS ONE, ISSN 1932-6203, PLOS, San Francisco, CA, Vol. 14, Iss. 4, April, pp. 1-32, https://doi.org/10.1371/journal.pone.0215692 , http://bnarchives.yorku.ca/597/ This Version is available at: https://hdl.handle.net/10419/195951 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ RESEARCH ARTICLE Energy, hierarchy and the origin of inequality Blair FixID* York University, Toronto, Ontario, Canada *[email protected] Abstract Where should we look to understand the origin of inequality? I propose an unusual window of evidence—modern societies. I hypothesize that evidence for the origin of inequality is encoded in the institutional structure of industrial societies. To test this idea, I use a model to project modern trends into the past. This model takes the modern relation between energy, hierarchy, and inequality and creates a hindcast of the origin of inequality. The results are broadly consistent with the available evidence. The model predicts an explosion of inequality with the transition from hunter-gathering to agriculture, followed by a plateau. This finding potentially opens a new window of evidence into the origin of inequality. 1 Introduction The origin of inequality is one of the great mysteries of human social evolution. For the vast majority of our history, we lived in small bands that were fiercely egalitarian [1]. But then around 10,000 years ago, something changed [2,3]. For reasons that remain poorly understood, we began to abandon our ancestral state, and started allowing some individuals to command vastly more resources than others. At first inequality was the exception, but it soon spread until it became the most common form of organization. This great transition has puzzled scientists for centuries [2–11]. But like the origin of life, the origin of inequality is frustratingly difficult to study. The problem is that origins remain locked in the past, meaning evidence is sparse. Still, we have made progress. With great effort, we have found three ‘windows’ of evidence into the origin of inequality: the archaeological record [10– 19], surviving traditional societies [20–25], and the written record of inequality [26–30]. These windows focus either on societies that are long gone, or societies whose form is archaic. This is perfectly reasonable, but it also limits the evidence we can uncover. The archaeological and written record of inequality will always be sparse. And traditional societies are rapidly disappearing from the world. Given the limits of these windows, where else might we look to study the origin of inequality? I suggest we draw inspiration from evolutionary biology. One of the great breakthroughs in studying the origin of life was the discovery that the DNA of living organisms contains a coded history of their evolution [31]. Might something similar be true of human societies? Might the social structure of modern societies contain a coded history of the origin of inequality? I test this possibility here. I use institutions as the social corollary of DNA. Institutions are systems of organizing that are passed between generations. I think we can use modern institutional trends to infer the origin of inequality. PLOS ONE | https://doi.org/10.1371/journal.pone.0215692 April 24, 2019 1 / 32 a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 OPEN ACCESS Citation: Fix B (2019) Energy, hierarchy and the origin of inequality. PLoS ONE 14(4): e0215692. https://doi.org/10.1371/journal.pone.0215692 Editor: Stefan Cristian Gherghina, The Bucharest University of Economic Studies, ROMANIA Received: January 24, 2019 Accepted: April 5, 2019 Published: April 24, 2019 Copyright: ©2019 Blair Fix. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: Supplementary Materials are available at https://osf.io/7b8tu/. Funding: The author received no specific funding for this work. Competing interests: The author has declared that no competing interests exist. Looking at modern societies, I find two important trends (Section 2). First, societies that use more energy tend to have larger institutions. Second, modern institutions are hierarchically organized and income increases rapidly with rank. How does this relate to the origin of inequality? The key is that the growth of institution size can be interpreted as the growth of hierarchy. The idea is that as hierarchy grows it concentrates resources at the top, potentially leading to greater inequality. The modern trend is towards greater energy use and greater hierarchy. To infer the origin of inequality, I propose that we reverse this trend and project it backwards in time. I call this the ‘energy-hierarchy-inequality’ (EHI) hypothesis: Energy-hierarchy-inequality hypothesis We can infer the origin of inequality from the modern relation between energy use, hierarchy, and inequality. Like with DNA, these institutional trends do not give direct evidence of our past. Instead, they must be interpreted with a model. To test the EHI hypothesis, I use a model to project modern trends into the past (Section 3). The model gives a hindcast of the origin of inequality —a prediction that can be compared to empirical evidence. The results are promising (Section 4). Consistent with the available evidence, the model predicts an explosion of inequality during the energy transition from hunter-gathering to agriculture. As energy use increases beyond agrarian levels, the model predicts that inequality should plateau. Whether this plateau is consistent with evidence is less clear. Depending on the inequality metric used, there is evidence that inequality declines slightly with industrialization. This may be because hierarchies become less ‘despotic’ as energy use increases. Future research is needed to test this possibility. The results suggest that institutional trends in modern societies provide a plausible window into the origin of inequality. I speculate about causal mechanisms in Section 5, but for now the evidence is too sparse to draw many conclusions. More importantly, this finding opens new doors for future research. It implies that looking to the past may not be the only way to understand the origin of inequality. Signs of humanity’s deep history may be encoded in the institutional structure of our own societies. 2 Energy, hierarchy, and inequality: The evidence I review here the evidence linking energy, hierarchy, and inequality. The chain of reasoning (but not necessarily causation) is: energy  !institution size  !hierarchy  !power  !income I begin with energy because, like many scientists [32–41], I think social evolution is tied to energy use. The rationale is simple: according to the laws of thermodynamics, a non-equilib- rium system must be supported by a flow of energy [42]. Since human societies are non-equi- librium systems, energy should play an important role in social evolution. The link between energy and inequality has been proposed before [43–46], but this paper makes two new contributions. First, I explicitly link energy and inequality through social hierarchy. Second, I develop a formal model that hindcasts the origin of inequality. 2.1 Energy and institution size The energy-hierarchy-inequality hypothesis begins with a link between energy and institution size. In modern societies, institution size is strongly correlated with energy use per capita [47, 48]. Fig 1 illustrates this effect using business firms. Fig 1A plots average firm size within different nations against their energy use per capita. Each point represents a country, with error Energy, hierarchy and the origin of inequality PLOS ONE | https://doi.org/10.1371/journal.pone.0215692 April 24, 2019 2 / 32 bars indicating the uncertainty in average firm size. As energy use per capita increases, average firm size increases as well. The growth of average firm size is not caused by a horizontal shift in the distribution. Instead, it is caused by a fattening of the distribution tail. Fig 1B visualizes this behavior. Here I group the countries of the world into quintiles (5 groups) ranked by energy use per capita. For each quintile, I plot the aggregate firm size distribution. Note how the slope of the firm size distribution decreases with greater energy use. This indicates that large firms become more common. The firm size distribution can be modeled by a power law [48–51]. This means that the probability of finding a firm of size xis roughly proportional to x −α , where αis the power-law exponent. A smaller power-law exponent indicates a fatter tail. As shown in Fig 1B, greater energy use is associated with a smaller power-law exponent for the firm size distribution. This provides a simple way to model the relation between energy use and firm size. 2.2 Institution size and hierarchy The second step of the energy-hierarchy-inequality hypothesis is to connect institution size to hierarchy. I hypothesize that (virtually) all human institutions are hierarchically organized. This means they have a nested chain of command that grows with institution size. As the hierarchy grows, new ranks are added at a logarithmic rate [52,53]. This scaling behavior has been observed in business firms [54], historical empires [55], and hunter-gather societies [56]. Hierarchical organization also means that elite ranks should become more common as a hierarchy grows. Assuming that managers occupy top ranks, this implies that the management share of employment should increase with average firm size. This trend has been observed at the international level [48]. The most direct evidence for hierarchical organization comes from firm case studies [57– 62]. Fig 2 shows the hierarchical structure of six case-study firms (which come from Britain, the Netherlands, Portugal, and the United States). Although the specific structure varies, all six 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 αrefers to the estimated powerlaw exponent of the firm size distribution. For sources and methods, see Section 7. https://doi.org/10.1371/journal.pone.0215692.g001 Energy, hierarchy and the origin of inequality PLOS ONE | https://doi.org/10.1371/journal.pone.0215692 April 24, 2019 3 / 32 firms share the pyramid shape that we expect of a hierarchy. I use these case studies to inform the energy-hierarchy-inequality model (see Section 7 for details). To summarize, the evidence suggests that institutions tend to become larger as energy use increases. If institutions are hierarchically organized, this implies that the growth of energy is associated with the growth of hierarchy. 2.3 Hierarchical power and income The last component of the energy-hierarchy-inequality hypothesis is a relation between hierarchical power and income. The idea is that elites use their power within a hierarchy to gain preferential access to resources. Why might this be the case? Our evolutionary background provides some hints. Virtually all social mammals form dominance hierarchies [63–68]. In these hierarchies, high social status allows greater access to resources, particularly sexual mates [69–74]. Given our evolutionary heritage, we expect that humans should exhibit similar behavior. Unsurprisingly, there is a strong link between human hierarchical status and reproductive success [75–79]. Is the same true for income? Evidence suggests so. But before looking at this evidence, I note a key difference between human and non-human hierarchies. All other animals form linear hierarchies—an ordinal ranking from top to bottom. But humans form branching hierarchies, in which each superior controls multiple subordinates. This has important consequences for income distribution. In a branching hierarchy, the number of subordinates grows exponentially with rank (Fig 3). If income stems from power over subordinates, than it too should increase exponentially with rank. This means that hierarchy can lead to vast inequalities. To make this relation quantitative, I define ‘hierarchical power’ as: hierarchical power ¼1þnumber of subordinates ð1Þ The idea is that control over subordinates is a form of power—it increases “the possibility of imposing one’s will upon the behavior of other persons” [80]. All individuals start with a 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]. https://doi.org/10.1371/journal.pone.0215692.g002 Energy, hierarchy and the origin of inequality PLOS ONE | https://doi.org/10.1371/journal.pone.0215692 April 24, 2019 4 / 32 baseline power of 1, indicating they have control over themselves. Hierarchical power then increases proportionally with the number of subordinates. Is income within hierarchies a function of hierarchical power? Evidence from case-study firms suggests so. Fig 4 plots average income (relative to the bottom hierarchical level) against average hierarchical power for each rank in our six case-study firms. There is a strong correlation. A similar correlation exists between changes in income and changes in hierarchical power [81]. 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. https://doi.org/10.1371/journal.pone.0215692.g003 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. https://doi.org/10.1371/journal.pone.0215692.g004 Energy, hierarchy and the origin of inequality PLOS ONE | https://doi.org/10.1371/journal.pone.0215692 April 24, 2019 5 / 32 The power-income relation implies that inequality should increase as a hierarchy grows. This is because hierarchical power gets concentrated as a hierarchy gets larger (Fig 5). Importantly, this relation is non-linear. The initial growth of hierarchy rapidly concentrates power. But further growth of hierarchy leads to progressively slower growth of hierarchical-power concentration. If income scales with hierarchical power, the same should be true of inequality. As a hierarchy grows, inequality should explode and then plateau. To summarize, modern evidence suggests a joint relation between energy use, hierarchy, and inequality. As energy use increases, societies become more hierarchical. If income is proportional to hierarchical power, this should cause an increase in income inequality. To investigate the origin of inequality, I propose that we extrapolate this relation back in time. 3 An energy-hierarchy-inequality model To extrapolate the energy-hierarchy-inequality evidence, I create a numerical model. This model simulates the empirical relation between energy, hierarchy, and income. I discuss the basic components of the model below. For a technical discussion, see Section 7. 3.1 Model assumptions The energy-hierarchy-inequality model extrapolates modern trends into the distant past. To do this, we assume the following: Assumption 1. Institutions have a power-law size distribution. The growth of institution size is synonymous with a decline in the power-law exponent. Assumption 2. Institutions are hierarchically organized with a structure equivalent to modern firm hierarchies. 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. https://doi.org/10.1371/journal.pone.0215692.g005 Energy, hierarchy and the origin of inequality PLOS ONE | https://doi.org/10.1371/journal.pone.0215692 April 24, 2019 6 / 32 Assumption 3. The modern trend between energy use per capita and institution size applies to all societies. Assumption 4. Income scales with hierarchical power in all societies. The rate of scaling may vary over time and space. Are these assumptions realistic? Regarding assumption 1, there is evidence that pre-capital- ist societies had a power-law distribution of institution size. For instance, feudal manor size was roughly power-law distributed [82,83]. Similarly, slave estate size in the antebellum American South was roughly power-law distributed (see S1 Fig). Evidence also suggests that huntergatherer settlement sizes had a power-law distribution tail [84]. The types of institution certainly vary across time and space. But regardless of type, the power-law distribution of institution size seems common. Assumptions 2, 3 and 4 are speculative. But given empirical evidence, why not extrapolate it and see where it takes us? 3.2 Model structure The energy-hierarchy-inequality model has four main steps, discussed below. For technical details, see Section 7. Step 1: Generate the institution-size distribution. The model generates an institution size distribution using a discrete power law. The power-law exponent varies stochastically over different model iterations. This simulates changes in institution size. Step 2: Estimate energy use from institution size. Energy use per capita (E pc ) is modeled as a function of average institution size � I: Epc ¼c1� Ic2ð2Þ The parameters c 1 and c 2 are determined from a regression on the international energy and firm data shown in Fig 1A. Step 3: Create hierarchical structure. The model uses firm case-study data (Fig 2) to determine the hierarchical structure of institutions. All modeled institutions have the same ‘shape’, but the number of ranks varies with institution size. Step 4: Endow individuals with income Individual income Iscales with hierarchical power P as I/Pb��ð3Þ where βdetermines the rate of scaling and �is a noise factor. To simulate variation between societies, βvaries stochastically between model iterations. I use case studies of modern firms, as well as an antebellum US slave estate, to determine a plausible range for this variation. The noise factor �adds a small amount of dispersion to the power-income relation. This is determined by income dispersion within hierarchical levels of the case-study firms. On its own, the noise factor corresponds to a Gini index of about 0.1. Between-Institution Income Dispersion. The model excludes income dispersion between institutions. US evidence suggests that between-institution income dispersion accounts for a minority of total income dispersion (about 30%) [85]. I assume that the growth of between-institution dispersion is not important for the emergence of inequality. Future research can determine if this is an appropriate assumption. Energy, hierarchy and the origin of inequality PLOS ONE | https://doi.org/10.1371/journal.pone.0215692 April 24, 2019 7 / 32 3.3 Visualizing the energy-hierarchy-inequality model Fig 6 visualizes the energy-hierarchy-inequality model as a landscape. Hierarchies appear as pyramids, with hierarchical rank indicated by height and color. On top is a subsistence society that consumes 5GJ of energy per capita per year. This is 3200 Kcal per day—not much above the metabolic needs of an average person. Hierarchical organization is negligible. Consequently, hierarchical power is very equally distributed, with a Gini index of 0.13. We expect very little inequality in this society. On the bottom is an industrial society that consumes 500GJ of energy per capita per year— similar to modern Iceland or Qatar. Hierarchical organization is ubiquitous. Consequently, 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. https://doi.org/10.1371/journal.pone.0215692.g006 Energy, hierarchy and the origin of inequality PLOS ONE | https://doi.org/10.1371/journal.pone.0215692 April 24, 2019 8 / 32 hierarchy needs to interact only with his direct superior and direct subordinates. This allows group size to grow without the need for more social interactions. If hierarchy confers energetic benefits (via coordination), we can imagine a feedback loop emerging: Hierarchical organization enables large-scale coordination that then enables greater energy use, that then enables more hierarchy (and so on). This explains why energy and hierarchy go together. But it leads to a problem. For the vast majority of human history, hierarchical organization was negligible. Clearly there was no energy-hierarchy feedback loop. What are we missing? The missing ingredient is resource distribution within the hierarchy. The problem is that hierarchy is a double-edged sword. It allows greater coordination, but it also leads to despotism. The nested chain of command gives enormous power to top-ranked individuals. When this power is (predictably) used for personal gain, it leads to vast inequalities. This would explain why income scales with hierarchical power. The resulting inequality means that hierarchy may not benefit low-ranking individuals. If the material gains from coordination are monopolized by elites, low-ranking individuals may be better off leaving the hierarchy. The stability of a hierarchy thus depends on the net advantage for low-ranking individuals [125]. If there is no advantage, the hierarchy will be unstable. For the majority of human history, the costs of hierarchical despotism likely outweighed any coordination benefits from hierarchy. We know that modern hunter-gatherers (and presumably ancient ones as well) aggressively suppress individuals with power-seeking tendencies [137,138]. Without a concentrated energy source (such as agriculture) the benefits to largescale coordination were likely marginal. Therefore, hierarchy was not tolerated because it conferred no advantage. This likely changed during the Neolithic revolution. The details remain poorly understood, but we can guess that the benefits of large-scale coordination increased. This is likely related to sedentism and the development of agriculture [139,140]. Irrigation likely also played an important role [141,142]. I argue that during the Neolithic revolution, the energy-hierarchy feedback loop took hold. As a result, hierarchical power became more concentrated. Elites predictably used their power for personal gain, resulting in the emergence of inequality. I have so far treated inequality as an effect of hierarchy. But it may actually play a role in the growth of hierarchy. I have argued that the growth of hierarchy depends on the net advantage to low-ranking individuals. One way to increase this advantage is to increase the returns to hierarchical coordination (through environmental or technological change). But another way to increase the net advantage is to decrease hierarchical despotism. If the gains of hierarchy are more equally distributed, the net benefit to low-ranking members is greater. This reasoning means that inequality may play a causal role in the growth of hierarchy and the growth of energy use. This is speculation, but it fits with the inference that hierarchical despotism declines with energy use (Fig 9B). Perhaps limiting hierarchical despotism is a prerequisite for industrialization? Or put another way, is it possible to have an industrial economy built on slavery—the most despotic mode of human organization? These are open questions worth investigating. To summarize, I think that understanding the energy-hierarchy-inequality relation requires merging both functional and conflict theories of social stratification. It requires understanding what Wilson calls the “fundamental problem of social life” [134]. The idea is that cooperative groups beat uncooperative groups. But selfish individuals beat unselfish individuals within groups. Hierarchy nicely highlights both aspects of this problem. It is a powerful tool for coordination, and thus has potential group benefits. But it is also predictably used for selfish gain, thus resulting in great inequality. Thinking in this way may provide an important tool for understanding the origin of inequality. Energy, hierarchy and the origin of inequality PLOS ONE | https://doi.org/10.1371/journal.pone.0215692 April 24, 2019 15 / 32 6 Conclusions Origin questions are some of the most seductive in science. At the same time, they are among the most difficult questions to answer. The problem is that origins are always locked in the past, meaning evidence is frustratingly sparse. Scientific progress on origin questions happens when we find reliable windows into the past. It is instructive to see how new windows of evidence have advanced other fields. In modern cosmology, the breakthrough came when Edwin Hubble discovered that galaxies are receding from us. Reversing this trend implied that the universe had once been smaller—perhaps infinitely so. And so the big bang theory was born [143]. In biology, the breakthrough came with the discovery of DNA. By comparing the DNA of different organisms, we can infer the history of evolution. It suggests that all life has a single origin [31]. What about the origin of inequality? Obviously we should continue to gather historical and archaeological evidence. But this evidence will always remain limited. We should also continue studying traditional societies. But these societies are rapidly disappearing from the world. That leaves modern societies as a source of evidence. I have proposed that the institutional structure of modern societies contains a coded history of the origin of inequality. To test this idea, I used a model to project into the past the modern relation between energy use, hierarchy, and inequality. The model predictions are generally consistent with the evidence. This suggests we may have found a new window into the origin of inequality. 7 Methods 7.1 Data sources and methods Sources for Fig 1.Data for firm size comes from the Global Entrepreneurship Monitor (GEM), series ‘omnowjob’. To calculate firm size, I merge all data over the years 2001-2014. Because the GEM data over-represents large firms, I use only firms with 1000 or fewer employees. For method details, see the Appendix in Ref. [48]. Uncertainty in average firm size is estimated using the bootstrap method. Firm size distribution power-law exponents are estimated using the R PoweRlaw package [144]. Energy data comes from the World Bank, series EG. USE.PCAP.KG.OE. Sources for Figs 2and 4.Firm case-study data comes from [57–62]. For a description of this data, see the Appendix in Ref. [145]. Hierarchical power is defined as P=1+S, where Pis hierarchical power and Sis the number of subordinates. Because the case studies provide data for aggregate hierarchical structure only (not the chain of command), I calculate average hierarchical power, � Ph¼1þ� Sh. Here S h is the average number of subordinates below level h. It is defined as the sum of employment (E) in all subordinate levels, divided by employment in the level in question: � Sh¼Ph1 i¼1Ei=Eh. Income is normalized relative to the average income in the base hierarchical level (in the year in question). Sources for Fig 7.I assume that human metabolic needs range from 2000 Kcal to 2500 Kcal per day. Western and Eastern Eurasia energy use data comes from Morris [86]. US total energy consumption is from Historical Statistics of the United States, Tables Db164-171 (1900-1948) and Energy Information Agency Table 1.3 (1949-2000). US population is from Maddison [146]. Qatar data comes from the World Bank (series EG.USE. PCAP.KG.OE). Sources for Fig 8.Fig 8A. Archaeological inequality data is from Kohler et al. [18] and is measured using house size. I estimate the energy use range for each adaptation using the data in Table 1. Results for this energy range are shown in Fig 10.Fig 8B. Pre-industrial inequality data is from Milanovic [29]. I estimate energy use from reported values of GDP per capita. To Energy, hierarchy and the origin of inequality PLOS ONE | https://doi.org/10.1371/journal.pone.0215692 April 24, 2019 16 / 32 do this, I extrapolate the modern international relation between real GDP per capita and energy use per capita. Data for this regression comes from the World Bank (series EG.USE. PCAP.KG.OE and NY.GDP.PCAP.PP.KD). Fig 8C. Inequality data comes from three sources: the World Inequality Database (Gini index calculated from Lorenz curves), the United Nations World Income Inequality Database, and the OECD. I merge all data into a single database and estimate the range of inequality from this data. Energy use data comes from the World Bank, series EG.USE.PCAP.KG.OE. Fig 8D. Top 1% income share data is from the World Inequality Database. Energy use data is from the World Bank, series EG.USE.PCAP.KG.OE. Sources for Fig 9.Fig 9A. Merges all sources from Fig 8.Fig 9B. The hierarchical despotism index βis estimated by matching empirical data to the best fit model iteration. βis chosen by minimizing the following error function: �i¼ jlog Erlog Em;ijþjGrGm;ij ð4Þ E r and G r are energy use per capita and the Gini index of inequality (respectively) in the realworld society. E m,i and G m,i are energy use per capita and the Gini index of inequality (respectively) in the model iteration i. I assign real-world societies the model parameter β i associated with the best-fit model iteration i. 7.2 Hierarchy model equations This section provides technical details for the algorithm used to generate institutional hierarchies. Notation is shown in Table 2. Table 1. Data sources for energy use by adaptation. Society Energy (GJ/capita) Adaptation Source Agrarian max 38 agriculture [86] Bangladesh circa 1979 11.4 agriculture [147] Catalonia 1860 34.6 agriculture [148] Classical Greek 30.5 agriculture [86] Classical Greek 38 agriculture [86] Czechia 1850 39 agriculture [149] England Wales 1560 20 agriculture [150] England Wales 1600 17.4 agriculture [150] Europe 1500 CE 30 agriculture [151] Generic 26 agriculture [152] Han China 41 agriculture [86] Rome 9.2 agriculture [153] Rome 16.8 agriculture [153] Rome 38 agriculture [86] Sang Saeng 48 agriculture [154] Song China 45 agriculture [86] Trinket Island 39 agriculture [155] World 1820 19.2 agriculture [156] Generic 12 horticulture [152] Human-powered agriculture 9.5 horticulture [157] Generic 3.8 hunting-gathering [157] Generic 5 hunting-gathering [152] Western Eurasia 10,000 BCE 7.6 hunting-gathering [86] Western Eurasia 14,000 BCE 6.1 hunting-gathering [86] https://doi.org/10.1371/journal.pone.0215692.t001 Energy, hierarchy and the origin of inequality PLOS ONE | https://doi.org/10.1371/journal.pone.0215692 April 24, 2019 17 / 32 7.2.1 Generating the employment hierarchy. To generate the hierarchical structure of an institution, we begin by defining the span of control (s) as the ratio of employment (E) between two consecutive hierarchical levels (h), where h= 1 is the bottom hierarchical level. It simplifies later calculations if we define the span of control in level 1 as s= 1. This leads to the following piecewise function: sh� 1if h¼1 Eh Eh1 if h�2 8 < :ð5Þ The model assumes that the span of control is not constant; rather it increases exponentially with hierarchical level. I model the span of control as a function of hierarchical level (s h ) with 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. https://doi.org/10.1371/journal.pone.0215692.g010 Table 2. Hierarchy model notation. Symbol Definition aspan of control parameter 1 bspan of control parameter 2 Eemployment hhierarchical level nnumber of hierarchical levels in an institution sspan of control Ttotal for institution #round down to nearest integer ∏product of a sequence of numbers ∑sum of a sequence of numbers https://doi.org/10.1371/journal.pone.0215692.t002 Energy, hierarchy and the origin of inequality PLOS ONE | https://doi.org/10.1371/journal.pone.0215692 April 24, 2019 18 / 32 an exponential function, where aand bare free parameters: sh¼1if h¼1 a�ebh if h�2 (ð6Þ As one moves up the hierarchy, employment in each consecutive level (E h )decreases by 1/s h . This yields Eq 7, a recursive method for calculating E h . Since we want employment to be whole numbers, we round down to the nearest integer (notated by #). By repeatedly substituting Eq 7 into itself, we can obtain a non-recursive formula (Eq 8). In product notation, Eq 8 can be written as Eq 9. Eh¼# Eh1 sh for h>1ð7Þ Eh¼# E1�1 s2�1 s3�. . . �1 shð8Þ Eh¼# E1Y h i¼1 1 sið9Þ Total employment in the whole institution (E T ) is the sum of employment in all hierarchical levels. Defining nas the total number of hierarchical levels, we get Eq 10, which in summation notation, becomes Eq 11. ET¼E1þE2þ. . . þEnð10Þ ET¼X n h¼1 Ehð11Þ In practice, nis not known beforehand, so we define it using Eq 9. We progressively increase huntil we reach a level of zero employment. The highest level nwill be the hierarchical level directly below the first hierarchical level with zero employment: n¼ fhjEh�1and Ehþ1¼0g ð12Þ To summarize, the hierarchical employment structure of our model institution is determined by 3 free parameters: the span of control parameters aand b, and base-level employment E 1 . Code for this hierarchy generation algorithm can be found in the C++ header files hierarchy.h and exponents.h, located in the Supplementary Material [158]. 7.2.2 Calculating hierarchical power in the hierarchy model. I define an individual’s hierarchical power as one plus the number of subordinates (S) under their control: P¼1þSð13Þ Because the hierarchy model simulates only the aggregate structure of institutions (employment by hierarchical level), hierarchical power is calculated as an average per rank. For hierarchical rank h, the average hierarchical power (� Ph) is defined as the average number of subordinates (� Sh) plus 1: � Ph¼1þ� Shð14Þ Energy, hierarchy and the origin of inequality PLOS ONE | https://doi.org/10.1371/journal.pone.0215692 April 24, 2019 19 / 32 Each individual with rank his assigned the average power � Ph. The average number of subordinates � Shis equal to the sum of employment (E) in all subordinate levels, divided by employment in the level in question: � Sh¼X h1 i¼1 Ei Ehð15Þ As an example, consider the hierarchy in Fig 11. The average number of subordinates below each individual in hierarchical level 3 (red) would be: � S3¼E1þE2 E3¼16 þ8 4¼6ð16Þ Therefore, these individuals would all be assigned a hierarchical power of 7. 7.3 Restricting model parameters The model’s parameters are summarized in Table 3. My method for restricting these parameters is detailed below. Fig 11. Calculating the average number of subordinates. https://doi.org/10.1371/journal.pone.0215692.g011 Table 3. Model parameters. Parameter Definition Action Scope αInstitution size distribution exponent Determines the skewness of the institution size distribution — a,bSpan of control parameters Determines the shape of the institution hierarchy. Identical for all institutions. E 1 Employment in base hierarchical level Used to build the employment hierarchy from the bottom up. Determines total employment. Specific to each firm. βPower-income exponent Determines scaling relation between income and hierarchical power. Identical for all institutions. σNoise parameter Used to add noise to the power-income relation. It is the scale parameter of a lognormal distribution Identical for all institutions. https://doi.org/10.1371/journal.pone.0215692.t003 Energy, hierarchy and the origin of inequality PLOS ONE | https://doi.org/10.1371/journal.pone.0215692 April 24, 2019 20 / 32 7.3.1 Institution size distribution power-law exponent. Recent studies have found that firm size distributions in the United States [49] and other G7 countries [51] can be modeled accurately with a power law. A power law has the simple form shown in Eq 17, where the probability of observation xis inversely proportional to xraised to the exponent α: pðxÞ / 1 xað17Þ The hierarchy model assumes that all human societies have power-law institution size distributions. The model simulates different societies by allowing the power-law exponent αto vary stochastically between different model iterations. A characteristic property of power-law distributions is that as αapproaches 2, the mean becomes undefined. In the present context, this means that the model can produce institution sizes that are extremely large—far beyond anything that exists in the real world. To deal with this difficulty, I truncate the power-law distribution at a maximum institution size of 2.3 million. This is the present size of Walmart, the largest firm that has ever existed. Code for the discrete power-law random number generator can be found in the C++ header file rpld.h, located in the Supplementary Material [158]. This code is an adaptation of Collin Gillespie’s [144] discrete power-law generator found in the R poweRlaw package (which is, in turn, an adaptation of the algorithm outline by Clauset [159]). 7.3.2 Span of Control Parameters. The parameters aand btogether determine the shape of the model’s institutional hierarchies. These parameters are estimated from an exponential regression on firm case-study data (Fig 12A). The model assumes that these parameters are constant across all institutions. The resulting modeled hierarchy shape is shown in Fig 12B. Because the case-study sample size is small, there is considerable uncertainty in the span of control parameters. I incorporate this uncertainty into the model using the bootstrap method [160], which involves repeatedly resampling the case-study data (with replacement) and then estimating the parameters aand bfrom this resample. I run the model many times, each time with aand bdetermined by a bootstrap resample of case-study data. The resulting variation in the shape of the model’s hierarchies is indicated by the error bars in Fig 12B. Code implementing this bootstrap can be found in the C++ header file boot_span.h located in the Supplementary Material [158]. 7.3.3 Base-level employment. Given span of control parameters aand b, each hierarchy is constructed from the bottom hierarchical level up. Thus, we must know base level employment. To get this value, I input a range of different base employment values into Eqs 6,9, and 11 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 allows us to predict base level E 1 , given total employment E T . Code implementing this method can be found in the C++ header file base_fit.h, located in the Supplementary Material [158]. 7.3.4 Power-income exponent. The model assumes that income scales with hierarchical power as Ih I1¼ ðPhÞb��ð18Þ where I h is income in hierarchical level h,I 1 is income in the base hierarchical level, Pis hierarchical power, and �is the stochastic noise factor. To simulate variation between societies, I allow βto vary over different model iterations. I use two different data sources to determine a plausible range for this variation. The first is case-study data from modern firms [57–62]. I determine βfrom regressions on the data shown Energy, hierarchy and the origin of inequality PLOS ONE | https://doi.org/10.1371/journal.pone.0215692 April 24, 2019 21 / 32 in Fig 4. For each case-study firm, I regress log(I h /I 1 ) onto logP h . The slope of the relation is the estimate for β. I estimate the uncertainty in βusing the bootstrap method [160]. I repeatedly resample case-study data and re-run the regression to estimate β. The resulting probability distribution of βis shown in Fig 13A for each case-study firm. 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. https://doi.org/10.1371/journal.pone.0215692.g012 Energy, hierarchy and the origin of inequality PLOS ONE | https://doi.org/10.1371/journal.pone.0215692 April 24, 2019 22 / 32 The second data source is a case study of a US slave estate—Cannon’s Point Plantation [161]. I estimate βfrom the living standard of the plantation owner relative to his slaves. For this estimate, we solve the power-income relation for β: b¼logðIh=I1Þ logðPhÞð19Þ Although we do not know the hierarchical structure of the slave estate, we know that the owner sits on top of the hierarchy. All of the slaves are his subordinates. Therefore the number of slaves (n slave ) gives us a rough estimate for the owner’s hierarchical power: Powner �1þnslave ð20Þ If we know the living standard of the owner (I owner ) and slaves (I slave ), we can combine Eqs 19 and 20 to get a rough estimate for β: b�logðIowner=IslaveÞ logð1þnslaveÞð21Þ The living standard of the owner is equal to his income. But slaves have no income, so we must use another method to estimate their living standards. One way is to use the slave expenses paid by the owner. Another method is to compare the owner and slaves in terms of house size. The results for both methods are shown in Fig 13B. Again, I use the bootstrap technique to investigate the plausible range of βthat is implied by the Cannon’s Point data. I sample different values for the owner’s income, the slaves’ income (living standard), and the number of slaves and put them repeatedly into Eq 21. As we would expect, the resulting βfor our slave estate is far higher than in our case-study firms. In a slave regime, the evidence suggests that βcould approach 1. To put this in perspective, this means income scales linearly with hierarchical power. If this were the case in industrial societies, the CEO of Walmart would earn 2 million times that of an entry-level worker. Nothing like this exists in industrial societies—for good reason. They are not based on slavery. But slavery was ubiquitous in human history, so we need to allow for its existence in our model. Based on the case-study data in Fig 13, I allow βto vary over the range 0.2 �β�1. 7.3.5 Power-income noise factor. Noise (�) in the power-income relation is modeled with a lognormal random variate with dispersion determined by the parameter σ: ��ln NðsÞ ð22Þ The noise factor reproduces the average within-hierarchical level income dispersion in casestudy firms [57–62]. The distribution of within-hierarchical level income dispersion is shown in Fig 14. To determine σ, we first calculate the mean Gini index (� G) of the case-study data shown in Fig 14. We then calculate σusing: s¼2�erf1ð� GÞ ð23Þ This equation is derived from the definition of the Gini index of a lognormal distribution: G= erf(σ/2). To incorporate uncertainty in the case-study data, each model iteration uses a different bootstrap resample to calculate � G. Code implementing this method can Energy, hierarchy and the origin of inequality PLOS ONE | https://doi.org/10.1371/journal.pone.0215692 April 24, 2019 23 / 32 be found in the C++ header file boot_sigma.h, located in the Supplementary Material [158]. 7.3.6 Estimating energy use from average institution size. The energy-hierarchy- inequality model assumes that energy use E pc is proportional to average institution size I: Epc ¼c1� Ic2ð24Þ The parameters c 1 and c 2 are determined from regressions on the international firm data shown in Fig 1A. Fig 13. Probability distribution of βin case-study institutions. This figure shows the probability distribution of the parameter βin 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 βprobability distribution for casestudy firms [57–62]. Panel B shows the βprobability distribution for a US slave estate (Cannon’s Point Plantation [161]). 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