Economics from the Top Down: Does Hierarchy Unify Economic Theory?

Fix, Blair Doctoral Thesis Economics from the Top Down: Does Hierarchy Unify Economic Theory? Provided in Cooperation with: The Bichler & Nitzan Archives Suggested Citation: Fix, Blair (2018) : Economics from the Top Down: Does Hierarchy Unify Economic Theory?, The Bichler and Nitzan Archives, Toronto, http://bnarchives.yorku.ca/548/ This Version is available at: https://hdl.handle.net/10419/180922 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-nd/4.0/ YORK UNIVERSITY Economics from the Top Down Does Hierarchy Unify Economic Theory? Blair Fix A Dissertation submitted to the Faculty of Graduate Studies in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Graduate Program in Environmental Studies York University Toronto, Ontario August 2018 © Blair Fix, 2018 II Abstract What is the unit of analysis in economics? The prevailing orthodoxy in mainstream economic theory is that the individual is the ‘ultimate’ unit of analysis. The implicit goal of mainstream economics is to root macro-level social structure in the micro-level actions of individuals. But there is a simple problem with this approach: our knowledge of human behavior is hopelessly inadequate for the task at hand. Faced with real-world complexities, economists are forced to make bold (and seldom tested) assumptions about human behavior in order to make models tractable. The result is theory that has little to do with the real world. This dissertation investigates an alternative approach to economics that I call ‘economics from the top down’. This approach begins with the following question: what happens when we take the analytical focus off of individuals and put it into social hierarchy? The effect of this analytical shift is that we are forced to deal with the realities of concentrated power. The focus on hierarchy leads to some surprising discoveries. First, I find evidence that hierarchical organization has a biophysical basis. I show that institution size (firms and governments) is strongly correlated with rates of energy consumption, and that the growth of institutions can be interpreted as the growth of social hierarchy. Second, I find that hierarchy plays an important role in shaping income and income distribution. I find that income scales strongly with hierarchical power (defined as the number of subordinates under one’s control), and that hierarchical power affects income more strongly than any other factor measured. Lastly, using an empirically informed model of the hierarchical structure of US firms, I find that hierarchy plays a dominant role in shaping the income distribution tail. These results hint that hierarchy can be used to unify the study of economic growth (understood in biophysical terms) and income distribution. I conclude by making the first prediction of how the concentration of hierarchical power should relate to the growth of energy consumption. This prediction sheds new light on the origin of inequality. While this ‘top down’ approach to economics is in its infancy, the results are encouraging. Focusing on hierarchy gives fresh insight into many of the important questions facing society — insight that cannot be obtained by focusing on individuals. III Acknowledgments It is not easy to forge your own scientific path, let alone to declare that much of what has been written in your field needs to be rethought. Contrarian thinking often leads to isolation. Thankfully, I have not been isolated during my time at York, and much of this has to do with the work of Jonathan Nitzan. Together with Shimshon Bichler, Jonathan has created a path-breaking approach to political economy that has strongly shaped my thinking. But more than this, Jonathan has provided many opportunities for me to share my research, and has offered extremely useful feedback. For this I am grateful. I would also like to thank the ‘capital as power’ community. The many discussions on the web forum have been intellectually invigorating. Parts of this dissertation have benefited from discussions with Shai Gorsky and James McMahon. I also owe a great debt to Charlie Hall, who took the risk of letting me (an unknown PhD student) write a book in his Energy Briefs series. The experience taught me a great deal about how to think like a scientist, and how to communicate effectively. I would also like to thank Ellie Perkins for advising me during my seven years at York, and for giving me the freedom to chart my own path. Thank-you also to Mark Thomas and Justin Podur for the many helpful comments. I am also grateful for financial support from the Social Sciences and Humanities Resource Council, the Ontario Graduate Scholarship Program, and the Lewis and Bennett Graduate Scholarship. Lastly, thank-you to Grace and Garry Fix for the many hours of proof reading, and to Emily and Petra for the support along this journey. Contents 1 Introduction: Economics from the Top Down 1 1.1 Summary of Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 A Glimpse of a Synthesis? . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2 Energy and Institution Size 20 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 Energy and Institution Size: Empirical Evidence . . . . . . . . . . . 23 2.3 The ‘How’ Question: Energy and Firm Dynamics . . . . . . . . . . . 27 2.4 The ‘Why’ Question: Energy, Technology and Hierarchy . . . . . . 32 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3 Evidence for a Power Theory of Personal Income Distribution 51 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 Theories of Personal Income Distribution . . . . . . . . . . . . . . . 52 3.3 A Hierarchical Power Theory of Personal Income Distribution . . . 58 3.4 Testing the Power-Income Hypothesis . . . . . . . . . . . . . . . . . . 62 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4 A Hierarchy Model of Income Distribution 92 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 CONTENTS V 4.2 A Hierarchy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.3 A Capitalist Gradient Hypothesis . . . . . . . . . . . . . . . . . . . . . 111 4.4 A Hierarchical Redistribution Hypothesis . . . . . . . . . . . . . . . . 125 4.5 Conclusions: Modeling from the Top Down . . . . . . . . . . . . . . 133 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5 Conclusion: A Glimpse of a Synthesis? 144 5.1 What is the Unit of Analysis in Economics? . . . . . . . . . . . . . . 144 5.2 The Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.3 A Synthesis of Growth and Distribution? . . . . . . . . . . . . . . . . 149 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 A Appendices for Energy and Institution Size 166 A.1 Sources and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 167 A.2 Assessing Size Bias within Firm Databases . . . . . . . . . . . . . . . 175 A.3 The Firm Size Distribution as a Variable Power Law . . . . . . . . . 185 A.4 Testing Gibrat’s Law Using the Compustat Database . . . . . . . . . 191 A.5 Instability of the Gibrat Model . . . . . . . . . . . . . . . . . . . . . . 195 A.6 Properties of Stochastic Models . . . . . . . . . . . . . . . . . . . . . 199 A.7 Bias and Error in the GDP Labor Time Method . . . . . . . . . . . . 202 A.8 A Hierarchical Model of the Firm . . . . . . . . . . . . . . . . . . . . 209 A.9 An Agrarian Model of Institution Size . . . . . . . . . . . . . . . . . . 213 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 B Appendices For Evidence for a Power Theory of Personal Income Distribution 221 B.1 Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 B.2 Hierarchical Structure and Pay within Case-Study Firms . . . . . . 235 B.3 A Hierarchical Model of the Firm . . . . . . . . . . . . . . . . . . . . 244 B.4 The Compustat Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 B.5 Estimating Compustat Model Parameters . . . . . . . . . . . . . . . . 257 B.6 Compustat Model Results . . . . . . . . . . . . . . . . . . . . . . . . . 263 CONTENTS VI B.7 A Sensitivity Analysis of the Compustat Model . . . . . . . . . . . . 271 B.8 The Between-Within Gini Metric and Effect Size . . . . . . . . . . . 273 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 C Appendices For A Hierarchy Model of Income Distribution 282 C.1 Sources and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 C.2 Hierarchical Structure and Pay within Case-Study Firms . . . . . . 290 C.3 Compustat Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 C.4 Hierarchy Model Equations . . . . . . . . . . . . . . . . . . . . . . . . 299 C.5 Restricting Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 C.6 The Adjusted Hierarchy Model . . . . . . . . . . . . . . . . . . . . . . 318 C.7 A Null Effect Model for Top Incomes and Firm Size . . . . . . . . . 323 C.8 How Hierarchy Generates the Power-Law Tail . . . . . . . . . . . . . 325 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 List of Figures 1.1 A Glimpse of a Synthesis? . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Different Forms of Networks . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Idealized Units of Social Interaction . . . . . . . . . . . . . . . . . . . 10 2.1 Institution Size vs. Energy Use per Capita at the International Level 24 2.2 Institution Size vs. Energy Use per Capita in the United States . . 25 2.3 Synthesizing Evidence — Firm Size vs. Energy Use per Person or Worker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4 Using Firm Age Data to Estimate International Firm Dynamics . . 29 2.5 Technological Scale and Social Coordination in Electricity Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.6 The Growth of Management as a Function of the Firm Size Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.7 Testing the Hierarchical Model of the Firm Using Managment Share of Total Employment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.8 A Case Study in Causality: The Collapse of the Soviet Union . . . 44 3.1 Labor Productivity Inequality vs. Income Inequality . . . . . . . . . 56 3.2 Calculating the Average Number of Subordinates . . . . . . . . . . 61 3.3 The Distribution of Power Within A Firm . . . . . . . . . . . . . . . . 61 3.4 Income Inequality vs. Power Inequality within Firms . . . . . . . . 63 3.5 Average Income vs. Hierarchical Power Within Case-Study Firms . 65 3.6 Changes in Hierarchical Power and Pay During Intra-Firm Promotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.7 Analysis of Variance Using the Gini Index . . . . . . . . . . . . . . . 68 3.8 Grouping Power By Hierarchical Level . . . . . . . . . . . . . . . . . 71 3.9 Visualizing the Compustat Model . . . . . . . . . . . . . . . . . . . . 74 LIST OF FIGURES VIII 3.10 The GBW Ratio for Different Income-Affecting Factors . . . . . . . . 77 3.11 The GBand GWIndex for Different Income-Effecting Factors . . . . 78 4.1 A Branching Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.2 A Tripartite Division of Income Distribution . . . . . . . . . . . . . . 99 4.3 A Landscape View of the Hierarchy Model . . . . . . . . . . . . . . . 102 4.4 Modeled Income Distribution vs. US Data . . . . . . . . . . . . . . . 104 4.5 Firm Size Distributions Associated With Top Incomes and Wealth . 107 4.6 A Counterfactual Analysis of Model Properties . . . . . . . . . . . . 109 4.7 The Neoclassical Vision of Capitalist Income . . . . . . . . . . . . . . 114 4.8 The Marxist Vision of Capitalist Income . . . . . . . . . . . . . . . . 114 4.9 A Hierarchical Power Vision of Capitalist Income . . . . . . . . . . . 116 4.10 A Gradient Vision of Capitalist Income . . . . . . . . . . . . . . . . . 116 4.11 Measuring Hierarchical Power . . . . . . . . . . . . . . . . . . . . . . 117 4.12 CEO Hierarchical Power . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.13 Capitalist Income Fraction of US CEOs . . . . . . . . . . . . . . . . . 119 4.14 A Landscape View of the Capitalist Gradient Model . . . . . . . . . 121 4.15 Comparing the Capitalist Gradient Model to US Data . . . . . . . . 122 4.16 Historical Income Distribution Trends in the United States . . . . . 126 4.17 Changing How Income Scales with Hierarchical Rank and Power . 128 4.18 The Hierarchical Redistribution Model vs. US Data . . . . . . . . . 130 4.19 A Visualization of US Hierarchical Income Redistribution . . . . . . 132 4.20 The Rich and Powerful — Hierarchical Power and Top Incomes . . 135 5.1 A ‘Reduction’ Looking for a Synthesis . . . . . . . . . . . . . . . . . . 148 5.2 Visualizing the Energy-Hierarchy Model . . . . . . . . . . . . . . . . 151 5.3 Hierarchical Power Concentration and Energy Use per Capita . . . 153 5.4 Inequality vs. Mode of Energy Capture . . . . . . . . . . . . . . . . . 155 A.1 The Effects of Truncating the GEM Database <1000 . . . . . . . . 168 A.2 Large Firms in Manufacturing Subsectors — Analyzing Bias Caused by Variations in the Number of Firms . . . . . . . . . . . . . . . . . . 171 A.3 Firm Size Distributions in Selected Micro Databases . . . . . . . . 176 Summary of Findings 3 hierarchy is ubiquitous. Hierarchy, I believe, is the basic building block for most (if not all) modern institutions, and it has deep roots that likely extend into prehistory [10,11]. And in evolutionary terms, humans are but one of a vast number of social animals, virtually all of which use hierarchy as a method of social organization [12–17]. A second reason to focus on hierarchy is that it offers a simple way of studying the class structure of society. While many social scientists have stressed a focus on class structure [18–28], there is no consensus on how classes should be defined and studied. Hierarchy is useful because it provides a mathematically-generalizable form for defining and studying social class. Lastly, I am interested in hierarchy because it is conspicuously absent from mainstream economic theory, and thus its role in shaping social structure is poorly understood. 1.1 Summary of Findings On the face of it, this dissertation is a sprawling journey through a wide variety of seemingly unrelated social phenomena. At various points, I investigate energy consumption, institution size, technological change, intra-firm income distribution, the different factors that affect income, personal income distribution, functional income distribution, and changes in income inequality over time. How are these things possibly related? The surprising finding in this dissertation is that all of these phenomena can be linked to social hierarchy. Let me explain how. In Chapter 2, I explore the relation between energy consumption and institution size. I find that as energy consumption increases (both across space and across time), there is a systematic increase in institution size. Specifically, as energy consumption increases, self-employment declines, employment in large firms increases, average firm size increases, and government employment increases. I find evidence that these trends are indicative of a general increase in social hierarchy with energy consumption. Why is hierarchy related to energy consumption? I hypothesize that increasing energy consumption requires increasing the scale and complexity of technology, which in turn, requires greater social coordination. But according to the work of anthropologist Robin Dunbar, brain size places a key limit on primate group size. [29–31]. Dunbar’s primate evidence predicts an average human group size of about 150 (Dunbar’s number). Building on the work of Turchin and Gavrilets [32], I propose that hierarchy allows humans to sidestep this group-size limitation. A hierarchy’s nested chain of command allows group size to grow without any corresponding increase in the Summary of Findings 4 number of required social relations. This suggests that increasing hierarchical organization plays a central role in increasing energy consumption. In Chapter 3, I turn the focus to personal income. The dominant paradigm in personal income distribution theory is that income stems from productivity. But this approach has a severe (but little discussed) problem: when differences in individual productivity are measured objectively (and not circularly), they are far too small to account for observed differentials in income. But if not productivity, then what explains differences in personal income? I propose that personal income is most strongly determined by hierarchical power. What is hierarchical power? I define it as the ability to influence subordinates within a hierarchical chain of command. I measure hierarchical power in terms of the number of subordinates under an individual’s control. Using this metric, I find that relative income within firms scales strongly with hierarchical power. Using data for intra-firm promotions/demotions, I also find that changes in relative income within firms scale strongly with changes in hierarchical power. Lastly, I find that grouping individuals by hierarchical level (across firms) affects income more strongly than any other factor measured. This evidence suggests that hierarchy plays a key role in shaping personal income. In Chapter 4, I keep the focus on income, but expand the scope of analysis. I conduct a general inquiry into how hierarchy affects income distribution. I build a hierarchical model that extrapolates the available firm-level data to create a large-scale simulation of the hierarchical structure of the United States economy over the last two decades. After showing that this model does a reasonably good job of reproducing the features of US income distribution, I use the model for a wide variety of analysis. This leads to three major findings. First, I find that hierarchy plays a dominant role in shaping the tail of US income distribution. This is important, because the power-law tail of income distribution is a celebrated empirical regularity that is usually explained in individualistic terms [33–49]. In contrast, I find that the power law scaling of top incomes is likely caused by hierarchical organization. The second major finding is that hierarchy can be used to relate personal and functional income distribution. Drawing on Nitzan and Bichler’s ‘capital as power’ hypothesis [8], I propose that earning capitalist income is a function of hierarchical power. I find that CEO pay evidence is consistent with this hypothesis. Moreover, a model that generalizes CEO pay trends accurately reproduces the distribution of capitalist income in the United States. Lastly, I investigate if the recent explosion in US top income shares can be understood in terms of a hierarchical redistribution of income. I find that a model implementing this hypothesis accurately reproduces several A Glimpse of a Synthesis? 5 Social Hierarchy Economic Growth Income Distribution Synthesis? Energy and Institution Size Evidence for a Power Theory of Personal Income Distribution A Hierarchy Model of Income Distribution Figure 1.1: A Glimpse of a Synthesis? This figure shows how I conceive the big-picture structure of this dissertation. Each of the three papers (Ch. 2-4) connects either biophysical economic growth or income distribution to social hierarchy. But this connection begs a question: are growth and income distribution also related? I explore this possibility in Chapter 5. key trends in US income distribution. To summarize, hierarchy seems to play a central role in shaping the size, composition, and dynamics of top incomes. 1.2 A Glimpse of a Synthesis? I have given this dissertation the inquisitive (and not declarative) subtitle “Does Hierarchy Unify Economic Theory?”. As I see it, each of the three papers in this dissertation connects either biophysical economic growth or income distribution to social hierarchy. This hints at a connection between growth and income distribution themselves (see Fig. 1.1). It is the possibility of unifying these two phenomena that informs the dissertation subtitle. The reader may be asking — what is biophysical economic growth? In short, it is the growth of the economy measured in biophysical rather than monetary terms. As discussed in the ‘Methods’ section below, I treat energy consumption as biophysical indicator of economic scale. Why? Energy is the life-blood of all non-equilibrium systems. The rate of energy flow limits the types of structure that a given system can achieve. As such, when I connect the growth of energy consumption to social hierarchy (Ch. 2), I view this as an implicit connection between biophysical economic scale and social hierarchy. Methods 6 In Chapter 5, I use the cumulative results in the dissertation to offer the glimpse of a synthesis between biophysical economic growth and income distribution. The basic thinking is as follows. If social hierarchy increases with energy consumption, and hierarchy is a mechanism for concentrating power, it follows that power should become more concentrated as energy consumption increases. Furthermore, if concentrations of hierarchical power lead to concentrations of income (as found in Ch. 3 and 4), the growth of energy consumption should be associated with an increase in income inequality. To make this prediction concrete, I use the results in Chapters 2-4 to build a model of how hierarchical power concentration might increase with energy consumption. If this model is correct (and there are many caveats), it indicates something surprising. It suggests that a society’s first order of magnitude increase in energy consumption — from subsistent metabolic levels to agrarian levels — should correspond with a massive increase in the concentration of hierarchical power. After this initial transition, the model suggests that further increases in energy consumption (to industrial levels) should have little effect on power concentrations. Given the connection between power inequalities and income inequalities, this suggests that the transition from hunter-gatherer societies to agrarian societies should be associated with a substantial increase in inequality. And counter-intuitively (to me at least), all subsequent changes to biophysical economic scale should have little effect on inequality. Interestingly, recent archaeological evidence suggests that this is what actually occurred [50]. Hunter-gatherer societies had very little inequality, but the transition to agriculture brought levels of inequality that were comparable to modern, industrial societies. My analysis suggests that this non-linear trend owes to the non-linear scaling behavior of hierarchy itself. To summarize, using hierarchy as the unit of analysis seems to be a fruitful way to do economic research. Hierarchy, it would seem, lies at the very heart of human social organization, and is related to many of the outstanding questions in economics (and social science in general). 1.3 Methods The methods used in this dissertation bear little resemblance to what most people would recognize as ‘economics’. Because my methods are so different, I want to make their intellectual origins explicitly clear. My approach has four main components, outlined below. Methods 7 A Biophysical Approach to Economics Put succinctly, a biophysical approach to economics means taking the laws of thermodynamics seriously. These laws outline the basic rules of energy transformation: (1) energy can neither be created nor destroyed; and (2) all energy transformation processes must incur losses. It is hard to overstate the scientific importance of these laws. Indeed, the physicist Arthur Eddington once remarked “if your theory is found to be against the [laws]of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation” [51]. The laws of thermodynamics imply that, without flows of energy, all roads lead to equilibrium. And thermodynamic equilibrium is a boring state. Most of the interesting things that scientists study are out of equilibrium, and are sustained by a constant flow of energy [52,53]. Life is perhaps the most compelling example. All life on earth is united by a common struggle — a “struggle for free energy available for work” [54]. The ability to harness energy places key constraints on the structure of life, from the level of the cell [55], to the organism [56,57], to the ecosystem [58]. Probably the first economist to take the laws of thermodynamics seriously was Nicholas Georgescu-Roegen [59]. Since Georgescu-Roegen’s work in the 1970s, there has been growing interest in reformulating economic theory to have a biophysical basis [60–64]. By far the most popular approach is to reform neoclassical growth theory by adding energy as a third factor of production, beside labor and capital. A non-exhaustive list of scholars who have pursued this approach would include [63,65–74]. I take the biophysical approach seriously. But unlike many other economists, my goal is not to use energy consumption to explain the growth of real GDP. In fact, I am not interested in economic output at all. Basic measurement issues (outlined below) conspire to make the objective measure of economic output impossible. Instead, I am interested in energy consumption in its own right. Because of its importance for sustaining non-equilibrium structure, I use energy consumption as a biophysical indicator of economic scale. (For more details about this approach, see [75]). Addressing The Measurement Problem Most of mainstream economic theory is prefaced on the idea that economic output is objectively measurable. This is true of neoclassical marginal productivity theory, which explains income distribution in terms of the output of labor and capital [76–81]. It is also true of neoclassical economic growth theory, Methods 8 which assumes that the economy has a measurable, aggregate output [82,83]. The curious thing, however, is that these theories are all derived using the assumption of a one-commodity economy. For instance, Giorgio Colacchio observes that “the only case consistent with the marginal productivity theory is that of a ‘one-commodity’ economy” [84]. Similarly, in formulating his canonical growth model, Robert Solow assumes: “There is only one commodity, output as a whole” [82]. Why do these theories begin with such a bizarre assumption? It is because this is the only condition under-which the comparison and aggregation of different outputs is possible. The central problem is this: if we want to add or compare two or more things that are qualitatively different, we need a common unit of measurement. However, for each different choice of unit, our comparison (or aggregation) will yield different results. Giampietro et al. call this the “epistemological predicament associated with purposive quantitative analysis ... the observer always affects what is observed when defining the descriptive domain” [85]. Economists make matters worse by choosing price as a unit of comparison. This does two things. Firstly, it makes marginal productivity theory circular. Why? Output is supposed to explain income, but by using prices to aggregate/compare output, we are actually measuring output in terms of income. Secondly, the fact that prices change over time causes a host of measurement problems. Francis Edgeworth observes: If one great group of commodities varies pretty uniformly in one direction, and another in a different direction (or even in the same direction but in a markedly different degree), then the task of restoring the level of prices can no longer be regarded as a purely objective ... problem. (cited in [86], emphasis added) Over the years, many authors have commented on one or more aspects of this measurement problem (a non-exhaustive list would include [87–93]). But while many critical economists are aware of the problem, few are willing to take the logical course of action. If heterogeneous output cannot be objectively compared or aggregated, then there is no sense in trying to measure it. As a result, we need to build theory that does not rely on the concept of economic output. As far as I know, Jonathan Nitzan and Shimshom Bichler [8]were the first to arrive at this conclusion. They propose an approach to political economy that focuses entirely on differential (price-ratio) quantities rather than on ‘real’ output. Inspired by Nitzan and Bichler, I have made the decision to abandon the measurement of economic output. Instead, I do one of two things. When I Methods 9 Figure 1.2: Different Forms of Networks This figure (taken from Barabasi and Otvai [94]) shows three different types of networks. On the left is a random network, generated by adding edges between nodes at random. In the middle is a scale-free network. This name owes to the fact that there is no typical scale for the number of connections between nodes. Some nodes have many connections, some have very few. Lastly, the right panel shows a hierarchical network, which is characterized by a nesting structure. want a measure of (biophysical) economic scale that is independent of monetary value, I use energy consumption per capita. Alternately, when I am interested in prices, I use differential ratios to allow comparisons. Recognizing Ultra-sociality There is a curious disconnect between how economists model humans, and how the more historical (and biological) oriented social sciences view our species. In economics, humans are treated as essentially asocial “globules of desire” [95]. Individuals exist purely to maximize their own utility. This asocial model is at odds with the rest of our scientific knowledge. Modern science recognizes that humans are but one form of primate, and all primates are social animals. Moreover, there is growing agreement that human sociality far surpasses our primate cousins. Rather than merely being social, humans are ultra-social [96– 102]. This means that we form very large groups and are capable of cooperating with non-kin in ways that other primates cannot. Taking ultra-sociality seriously means focusing on social connections between individuals. Network science offers a powerful way to do this [103]. We imagine individuals as ‘nodes’ in the network, and social relations as the ‘edges’. My Methods 10 Neoclassical Reciprical Exchange Marxist Capitalist Worker Surplus Capital as Power Power Figure 1.3: Idealized Units of Social Interaction This figure shows my understanding of the basic units of social interaction adopted by Neoclassical and Marxist theory. Neoclassical theory is predicated on reciprocal exchange between utility maximizing parties. Marxist theory is predicated on the production of surplus by workers and its appropriation by capitalists. I propose that social (branching) hierarchy should be used as the unit of interaction for a capital as power approach to political economy. The premise is that a superior wields power over one or more subordinates. focus on hierarchy is inspired by network science. As shown in Figure 1.2, a hierarchy is really just a particular type of network — one with a nested structure. A pure hierarchical network has a very important property. No matter where we begin, if we trace connections (going in only one direction) we will always end up in the same place [104]. To see how this works, think about a hierarchical chain of command. No matter which subordinate we begin on, if we move up the chain of command, we will always end at the same individual — the ‘ruler’. A hierarchy is special type of network that concentrates power in the hands of the few. This property, I believe, is extremely important for understanding human social structure. Capital as Power It is hard to overstate the importance of Nitzan and Bichler’s [8]‘capital as power’ framework to my approach. To begin with, Nitzan and Bichler offer a compelling critique of both the neoclassical and Marxist approaches to political economy. The problem, they argue, is that the prerequisite units simply do not exist. Neoclassical theory is based on the concept of reciprocal exchange in which individuals maximize utility. But utility is unobservable, even in principle. Marxist theory, on the other hand, is based on the concept of surplus value. Workers create value, which is then appropriated by capitalists. But like utility, Nitzan and Layout 11 Bichler convincingly argue that surplus value cannot (even in principle) be measured. Why? It is based on the non-existent unit of ‘socially-necessary abstract labor time’. Nitzan and Bichler argue that political economy needs a fresh start — a “ctrlalt-del” [105]. I find this boldness liberating — it unburdens us of centuries of dead-end theoretical baggage. So what is the way forward? Nitzan and Bichler argue that it involves focusing on the relation between power and monetary value. I agree. I take this focus on power (and value) and merge it with a focus on hierarchy. My contribution to capital as power is to add an idealized unit of social interaction — the power-relation between a superior and subordinates within a hierarchy (see Fig. 1.3). Of course, this is not the only type of social relation that humans engage in; rather, it is one that has received too little attention from political economists. 1.4 Layout This dissertation consists of the three self-contained papers: 1. Energy and Institution Size 2. Evidence for a Power Theory of Personal Income Distribution 3. A Hierarchy Model of Income Distribution ‘Energy and Institution Size’ has been published in PLOS ONE [106], and ‘Evidence for a Power Theory of Personal Income Distribution’ is currently under review at the Journal of Economic Issues. A note to the reader. The writing of these three papers spans a significant period of time, while the overarching theme that unifies them has only recently become clear to me. As such, each paper makes little reference to others. I leave the discussion of connections for the conclusion in Chapter 5. References 12 References 1. Brennan G, Tullock G. An economic theory of military tactics: Methodological individualism at war. Journal of Economic Behavior & Organization. 1982;3(2-3):225–242. 2. Leontief W. Theoretical assumptions and nonobserved facts. American Economic Review. 1971;61(1):1–7. 3. Wilson EO. Consilience: The unity of knowledge. New York: Random House; 1999. 4. Epstein JM, Axtell R. Growing artificial societies: social science from the bottom up. Washington, D.C.: Brookings Institution Press; 1996. 5. Hodgson G. Behind methodological individualism. Cambridge Journal of Economics. 1986;10(3):211–224. 6. Hodgson GM. Meanings of methodological individualism. Journal of Economic Methodology. 2007;14(2):211–226. 7. Brown C. Is there an institutional theory of distribution? Journal of Economic Issues. 2005;39(4):915–931. 8. Nitzan J, Bichler S. Capital as Power: A Study of Order and Creorder. New York: Routledge; 2009. 9. Peach JT. Distribution and economic progress. Journal of Economic Issues. 1987;21(4):1495–1529. 10. Price TD, Feinman GM. Foundations of social inequality. vol. 1. New York: Springer Science & Business Media; 1995. 11. Price TD, Feinman GM, editors. Pathways to power: New Perspectives on the Emergence of Social Inequality. New York: Springer; 2010. 12. Barroso FG, Alados CL, Boza J. Social hierarchy in the domestic goat: effect on food habits and production. Applied Animal Behaviour Science. 2000;69(1):35–53. 13. Guhl AM, Collias NE, Allee WC. Mating behavior and the social hierarchy in small flocks of white leghorns. Physiological Zoology. 1945;18(4):365– 390. References 19 99. Richerson PJ, Boyd R. The evolution of human ultra-sociality. In: Indoctrinability, ideology, and warfare: Evolutionary perspectives. New York: Berghahn Books; 1998. p. 71–95. 100. Tomasello M. The ultra-social animal. European journal of social psychology. 2014;44(3):187–194. 101. Turchin P. The Puzzle of Human Ultrasociality: How Did Large-Scale Complex Societies Evolve? In: Cultural Evolution: Society, Technology, Language, and Religion. Cambridge, MA: MIT Press; 2013. p. 61–74. 102. Williams LA, Bliss-Moreau E. Humans are ultrasocial and emotional. Behavioral and Brain Sciences. 2016;39. 103. Barabási AL. Network science. Cambridge: Cambridge university press; 2016. 104. Mones E, Vicsek L, Vicsek T. Hierarchy measure for complex networks. PloS one. 2012;7(3):e33799. 105. Bichler S, Nitzan J. Capital as power: Toward a new cosmology of capitalism. Real-World Economics Review. 2012;(61):65–84. 106. Fix B. Energy and Institution Size. PLOS ONE. 2017;12(2):e0171823. doi:doi:10.1371/journal. pone.0171823. Chapter 2 Energy and Institution Size Abstract Why do institutions grow? Despite nearly a century of scientific effort, there remains little consensus on this topic. This paper offers a new approach that focuses on energy consumption. A systematic relation exists between institution size and energy consumption per capita: as energy consumption increases, institutions become larger. I hypothesize that this relation results from the interplay between technological complexity and human biological limitations. I also show how a simple stochastic model can be used to link energy consumption with firm dynamics. 2.1 Introduction Throughout the last century, there has been a recurrent desire to connect human social evolution to changes in energy consumption [1–4]. The motivation is simple: the laws of thermodynamics dictate that any system that exists far from equilibrium must be supported by a flow of energy [5]. Since human societies are non-equilibrium systems, it follows that energy flows ought play an important part in social evolution. However, it has proved difficult to move from grand pronouncements based on the laws of thermodynamics to a quantitative understanding of the relation between energy use and social evolution [6]. This paper offers a contribution to such a quantitative understanding. This paper is concerned with one particular aspect of social change: the growth in size of the institutions that control human labor. While such institutions have taken many forms throughout history, in the modern era, the control of human labor is dominated by two institutions: the business firm and government. In this paper, institution size refers to the amount of human labor (i.e employment) controlled by an organization. Under this definition/metric of in- Introduction 21 stitution size, I demonstrate that a pervasive, positive correlation exists between institution size and energy use per capita. I pursue two avenues for understanding the relation between energy and institution size. The first approach draws on the rich history of stochastic modelling within firm size theory. Stochastic (random) models have been successfully used to link firm dynamics to the overall firm size distribution. Yet there is little understanding of what drives variations in firm dynamics. Using data on firm age and firm size to constrain a stochastic model, I demonstrate that firm dynamics are likely related to rates of energy consumption, and I offer a prediction of what this relation should look like. The second approach is more speculative, and aims to offer a general explanation of why rates of energy consumption are related to institution size. I propose two factors that mediate this relation: technological scale and social hierarchy. I hypothesize that increases in energy consumption involve a trend towards the use of technologies that are larger and more complex. These increasingly large technologies require the coordination of greater numbers of people. Given the limitations of the human brain [7], I argue that large-scale social coordination is most easily achieved through social hierarchy [8]and that firms and government are specific manifestations of this hierarchy. This paper is organized as follows. After a brief review of the strengths and weaknesses of various theories of institutional size (Sec. 2.1.1), Section 2.2 discusses the empirical evidence connecting energy consumption with institution size. Section 2.3 then uses a stochastic model to further illuminate the relation between energy use and firm dynamics. Finally, Section 2.4 presents and tests a series of hypotheses linking institution size to technological scale and social hierarchy. 2.1.1 Theories of Institutional Size Theories of institution size can be divided into two classes: those that concern themselves with the causes of institutional growth (‘why’ theories) and those that do not (‘how’ theories). ‘How’ theories have met with great empirical success, while ‘why’ theories have struggled to offer explanations that are testable. All ‘how’ theories of institutional size can be traced back to the work of the French economist Robert Gibrat, who discovered that the rate of growth of business firms seemed to be independent of their size [9]. While later investigation found this ‘law of proportional effect’ to be only approximately true — growth rate variance tends to decline with size [10–12]— it has led to a rich history of Introduction 22 stochastic firm growth models [13,14]. The basic principle is that firm growth is treated probabilistically. Each firm is submitted to a series of random shocks that make it grow (or shrink) over time. When applied to large numbers of firms, the result is a firm size distribution. The surprising finding is that these purely random models can very accurately predict the functional form of real-world firm size distributions (see Appendix A.6). Despite their success, ‘how’ theories are not particularly satisfying because they do not explain why institutions grow. Unfortunately, theories that do attempt to explain the cause of institution growth often rely on unmeasurable variables, and as a result, are untestable. The theory of the firm has been dominated by Ronald Coase’s transaction cost approach. According to Coase, “... a firm will tend to expand until the costs of organizing an extra transaction within the firm become equal to the costs of carrying out the same transaction by means of an exchange on the open market or the costs of organizing in another firm” [15]. Unfortunately, transaction costs have been notoriously difficult to define (let alone measure), rendering Coasian theory untestable [16,17] Other theories propose that management talent is the driver of firm growth. For instance, Robert Lucas assumes that the firm size distribution results from “allocat(ing) productive factors over managers of different ability so as to maximize output” [18]. Yet Lucas concedes that the causal factor in this model — the talent of managers — is “probably unobservable”. Despite this problem, Lucas’s theory remains popular [19,20]. Still other theories propose that firm growth is the result of a resource-driven competitive advantage [21,22]. Unfortunately, this approach has struggled to stipulate exactly how a particular resource is transformed into a value-creating competitive advantage. Priem and Butler argue that the ‘resource-based view’ advances a theory of value that is tautological — resources create value because they are (among other things) valuable [23]. In terms of measurability, theories of government size have faired no better than theories of firm size. One approach is to apply the rational-choice model to the behavior of voters. Government size is treated as a reflection of the preferences of utility maximizing voters [24,25]. However, without an objective measure of individuals’ internal preferences, this theory is untestable. Another approach is to assume that government bureaucracies (or government as a whole) are self-serving entities that attempt to maximize their budgets, but are restrained by voters and/or an institutional framework such as the constitution [26,27]. While maximizing behavior is one of the fundamental postu- Energy and Institution Size: Empirical Evidence 23 lates of neoclassical economics, the hypothesis that humans maximize external pay-offs has been falsified [28]. The lack of measurable variables has consistently plagued ‘why’ theories of institution size. If a new theory is to be successful, it must demonstrate a connection between institution size and some universally measurable quantity. Energy consumption is just such a quantity. 2.2 Energy and Institution Size: Empirical Evidence To study the relation between energy and institution size, I compare variations in energy use per capita to variations in the size of firms and government over both space and time. For firms, I investigate how changes in the base, tail and mean of the firm size distribution are related to changes in energy use per capita. I use self-employment data to investigate the base of the firm size distribution (relying on the assumption that self-employer firms are very small). To investigate the tail of the firm size distribution, I look at the employment share of the largest firms. To quantify the relative size of government, I measure the government share of total employment. Comparison of these institution size metrics with energy use per capita are shown in Figures 2.1-2.3. Figure 2.1 shows international trends (each colored line represents the path through time of a specific country), while Figure 2.2 shows time-series data for United States. Figure 2.3 (which focuses only on firms) merges data from Figures 2.1-2.2 and adds US sectoral and subsectoral level data. Although this synthesis merges data that are not identically defined (see Fig. 2.3 caption), the result is clear: the inclusion of sectoral data serves to extend (by two orders of magnitude) the trends found at the national level. In the case of small firms and mean firm size, the inclusion of sectoral data also increases the regression strength. To summarize our findings, the evidence in Figures 2.1-2.3 suggests the following ‘stylized’ facts. As energy use per capita increases: 1. The small firm employment share declines; 2. The large firm employment share increases; 3. The mean firm size increases; 4. The government employment share increases. Findings 1-3 suggest that increases in energy consumption are associated with a shift in employment from small to large firms. This indicates that the firm Energy and Institution Size: Empirical Evidence 24 ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● R2=0.66 1 2 5 10 20 50 100 5 10 20 50 100 200 500 1000 Energy Use per Capita (GJ) % of Total Employment A. Self−Employment R2=0.57 0.1 1 10 100 5 10 20 50 100 200 500 1000 Energy Use per Capita (GJ) % of Total Employment B. 25 Largest Corporations ● ●●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● R2=0.49 1 2 5 10 20 50 5 10 20 50 100 200 500 1000 Energy Use per Capita (GJ) Average Firm Size (Employees) C. Average Firm Size ● ● ● ● ● ● ● R2=0.38 1 2 5 10 20 50 100 5 10 20 50 100 200 500 1000 Energy Use per Capita (GJ) % of Total Employment D. Government Figure 2.1: Institution Size vs. Energy Use per Capita at the International Level This figure shows how different metrics of institution size vary with energy consumption per capita. Panels A-C analyze variations in firm size by looking at the base, tail, and estimated mean of the firm size distribution. Panel D analyzes variations in government size. In order to show as much evidence as possible, panels A, B and D are a mix of time series and scatter plot. Lines represent the path through time of individual countries while points represent a country with a single observation. Error bars in panel C represent the 95% confidence interval of mean firm size estimates. Variations in selfemployment, large-firm, and government employment share vs. energy are modelled with log-normal cumulative distribution functions. Mean firm size vs. energy is modelled with a power law. Grey regions indicate the 99% confidence region of each model. For sources and methodology, see Appendix A.1. Energy and Institution Size: Empirical Evidence 25 150 250 350 450 1880 1900 1920 1940 1960 1980 2000 2020 30 20 10 0 % of Total Employment Self−Employment (left) Energy use per capita (GJ, right) A. Self−Employment R2=0.86 150 250 350 450 1880 1900 1920 1940 1960 1980 2000 2020 0 5 10 15 20 25 % of Total Employment 200 Largest Firms (left) Energy use per capita (GJ, right) B. 200 Largest Firms R2=0.73 150 250 350 450 1880 1900 1920 1940 1960 1980 2000 2020 3 4 5 6 7 8 9 10 Number of Employees Average Firm Size (left) Energy use per capita (GJ, right) C. Average Firm Size R2=0.9 150 250 350 450 1880 1900 1920 1940 1960 1980 2000 2020 5 10 15 20 % of Total Employment Government (left) Energy use per capita (GJ, right) D. Government R2=0.6 Figure 2.2: Institution Size vs. Energy Use per Capita in the United States This figure shows the trends for various measures of institution size in the United States over the last century. Trends mirror those found at the global level. As energy consumption per capita increases, self-employment rates decline (panel A, note reverse scale), the large firm employment share increases (panel B), mean firm size increases (panel C), and the government employment share increases (panel D). Note that government regressions exclude World War II (dotted line). For sources and methodology, see Appendix A.1. Energy and Institution Size: Empirical Evidence 26 ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● R2=0.71 0.1 1.0 10.0 100.0 101102103104105 Energy Use per Person or Worker (GJ) % of Total Employment ● ● International United States US Industry US Manufacturing Subsectors A. Small Firms and Self−Employment ● ● ● ● ● ● ● ● ● ● ● ●●●● ●●● ● ● ● R2=0.57 0.1 1.0 10.0 100.0 101102103104105 Energy Use per Person orWorker (GJ) % of Total Employment ● International United States US Industry US Manufacturing Subsectors B. Large Firms ●●●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● R2=0.63 1 10 100 1000 101102103104105 Energy Use per Person or Worker (GJ) Average Firm Size ●● ●● International United States US Industry US Manufacturing Subsectors C. Average Firm Size Figure 2.3: Synthesizing Evidence — Firm Size vs. Energy Use per Person or Worker This figure combines data from 3 different units of analysis (nations, sectors, and subsectors) to offer a comprehensive picture of the relation between firm size and energy use per capita (or per worker). ‘US Industry’ consists of construction and manufacturing sectors, while ‘US Manufacturing Subsectors’ are the smallest subdivisions of the manufacturing sector. At the national level, energy use is measured per person, while at the sectoral level, it is measured per worker. In panel A, self-employment data (for nations and US Industry) is merged with the data for the employment share of firms with 0-4 employees in US manufacturing subsectors. In panel B, data for the employment share of the largest 25 firms (for nations and US Industry) is merged with data for the employment share of firms with more than 5000 employees in US manufacturing subsectors. Panel C shows mean firm size data at the national and sectoral level. Grey regions indicate the 99% confidence region of each model. For sources and methodology, see Appendix A.1. The ‘How’ Question: Energy and Firm Dynamics 27 size distribution becomes more skewed as energy consumption increases. In Appendix A.3, I demonstrate that this shift (at the national level) can be accurately modelled in terms of the changing exponent of a power law distribution. Assuming a correlation between energy use and GDP, then the evidence presented here is consistent with previous research that has focused on the relation between firm size and GDP per capita [18,20,29–31]. However, my focus here on energy use (rather than GDP) is intentional: it is part of a larger effort to ground economic theory in the laws of thermodynamics [32], and to root empirical analysis in biophysical (rather than monetary) phenomena [33–36]. Following the long-standing division in institution size theory between ‘how’ and ‘why’ theories, I adopt two separate approaches for understanding the relation between institution size and energy consumption. The first approach deals with the ‘how’ question: how exactly do changes in firm size occur? To answer this question, I use a stochastic model to illuminate the relation between energy use and firm dynamics. The second approach deals with the more difficult ‘why’ question: why is institution size related to energy consumption. To answer this question, I investigate the relation between energy, technological change, and social coordination. 2.3 The ‘How’ Question: Energy and Firm Dynamics Beginning with the work of Gibrat [9]and later Simon and Bonini [37], stochastic models have been successfully used to explain the functional form of the firm size distribution in terms of firm dynamics. The implication of these models is that changes in average firm size occur through changes in firm dynamics. Given the connection between energy consumption and firm size, it follows that firm dynamics ought to vary with changes in energy consumption. Ideally, we would look at this relation directly by investigating international variations in the firm growth rate distribution and comparing them to variations in energy consumption. Unfortunately, data constraints make such a comparison difficult. Calculating international firm growth rate distributions would require longitudinal data for a large, representative sample of firms in many countries. I am not aware of the existence of any such data at the present time. However, we can use what little data is available to make inferences about the relation between energy and firm dynamics. Firm age data provides an indirect window into firm dynamics. If we assume that new firms start at a small size, then we can infer the historic rate of growth of any firm, given its current age and size (i.e. a new, large firm likely grew The ‘How’ Question: Energy and Firm Dynamics 28 rapidly, while an old, small firm likely grew slowly). Figure 2.4A shows how firm age is related to rates of energy consumption per capita. The dataset used here (the GEM database) does not report firm age directly. Instead, it reports whether or not a firm is under 42 months of age. I use this data in Figure 2.4A to calculate the fraction of firms that are under 42 months of age. This fraction tends to decline as energy use per capita increases. This data clearly hints that a systemic relation exists between energy consumption and firm dynamics. In the following section, I use a stochastic model to make specific predictions about the form of this relation. 2.3.1 A Stochastic Model The essence of all stochastic firm models is that growth is treated probabilistically. Each firm begins with some arbitrary initial size L0. After every discrete time interval, the firm is subjected to a series of random ‘shocks’ (xi) that perturb it from its initial size. In our model, these shocks are drawn randomly from a Laplace distribution. At any point in time, each firm’s size L(t)is equal to the initial size times the product of all shocks (Eq. 2.1). If the time interval is years, then each shock can be interpreted as the annual growth rate (in fractional form). L(t) = L0·x1·x2·... ·xt(2.1) This basic Gibrat model is unstable unless additional stipulations are added (see Appendix A.5). I add a reflective lower bound that disallows firms from shrinking below the size L=1 (this is sometimes called the Keston process [38–40]). As long as firm growth rates have a downward drift, the model will produce a stable firm size distribution. Using this model requires the following assumptions: 1. The firm size distribution is a power law. 2. Firm growth rates are independent of size. 3. New firms are all born at size L=1. 4. The firm birth rate is equal to the firm death rate. 5. Firm growth rates come from a Laplace distribution. 6. The firm size distribution exists in an equilibrium. Assumption 1 is necessary because the model produces a power law distribution (see Appendix A.6). Recent studies have found that firm size distributions The ‘Why’ Question: Energy, Technology and Hierarchy 35 Table 2.1: Scale Increase of Various Industrial Technologies Type Early Prototype Largest Today Unit Scaling Factor Electric Power Plant 0.0125 2 2500 megawatts 1.80 ×106 Oil Refinery 5.5 1 240 000 barrels per day 2.24 ×105 Aluminium Smelter 5.7 1 060 000 tonnes per year 1.86 ×105 Internal Combustion Engine 0.75 107 390 horsepower 1.43 ×105 Mining Excavator 380 2 324 0000 cubic meters per day 6.12 ×104 Blast Furnace 0.3 5 500 cubic meters 1.83 ×104 Tanker Ship 1809 260 859 gross tonnage 1.44 ×102 This table shows the size of 7 selected industrial technologies at their earliest stage of development (‘Early Prototype’) and at the largest scale existing today. Column 5 shows the scaling factor between the largest and early technologies (largest/early). Technologies are ranked in descending order of scaling factor. For data sources, see Appendix A.1. changes in technological scale necessarily involve the increasing coordination of human labor. For instance, the largest oil refinery in the world, located in Jamnagar, India, employs 2500 people on site [50]. Rather than acting autonomously (like the users of consumer electronics), these individuals must coordinate their actions over a wide range of different tasks. This suggests that increases in technological scale require an increase in social coordination. But to what degree are increases in energy use per capita actually achieved through increases in technological scale? Given the complexity of technological change, this question is difficult to answer at a general level (for all technologies). Instead of a general test of hypothesis A, I present here a case study of electricity production and consumption in the United States (Fig. 2.5A-B). The results of this case study indicate that increases in technological scale have played an important role in meeting increases in per capita electricity use over the last century. Figure 2.5A shows how the indexed change in US electricity use per capita relates to the indexed change in mean power plant size (as measured by nameplate capacity). Over the last 100 years, the two series tracked together quite closely, with both electricity use and power plant size increasing rapidly between 1920 and 1980 and plateauing thereafter. How important was this change in technological scale for meeting per capita demand? To answer this question, Figure 2.5B plots the indexed ratio of mean power plant size to electricity use per capita. This ratio indicates the fraction of electricity use per capita growth The ‘Why’ Question: Energy, Technology and Hierarchy 36 Mean Capacity US Power Plants US Electricity Use per Capita 1 2 5 10 20 1920 1940 1960 1980 2000 2020 Year Indexed Growth (1920 = 1) A. Indexed Growth US Power Plant Mean Capacity US Electricity Use per Capita mean =0.56 0.25 0.50 0.75 1.00 1920 1940 1960 1980 2000 2020 Year 1920 = 1 B. 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Construction Time of Power Plants Figure 2.5: Technological Scale and Social Coordination in Electricity Generation Panel A shows the time-series relation between the mean capacity of US power plants and US electricity use per capita. Both series are indexed to 1 in the year 1920 in order to show relative growth. Power plants tend to get larger as electricity use per capita increases increases. Panel B shows the fraction of US per capita electricity use growth (since 1920) that was met by increases in mean plant size. The dashed line indicates the mean over the period 1920-2015, while the shaded region shows the standard deviation. Panel C shows the relation between power plant capacity and the estimated construction labor time. The entire range of electricity generation technology is included in this plot — from the smallest gasoline generators to the largest hydroelectric power plants. Different primary energy sources are indicated by color. Data is modelled with a power law. Grey regions indicate the 99% confidence region of the regression. For sources and methodology, see Appendix A.1. The ‘Why’ Question: Energy, Technology and Hierarchy 37 that was met by increases in power plant capacity. Between 1920 and 2015, increases in power plant capacity accounted for roughly half of the total increase in electricity use per capita. In the US electricity generation sector, increases in technological scale obviously played a major role in meeting increases in per capita electricity consumption. Was this increase in scale accompanied by a corresponding increases in the scale of social coordination (hypothesis B)? Answering this questions requires that we first define what we mean by the ‘scale’ of social coordination, and specify how this relates to a given technology. I define the ‘scale’ of social coordination as the number of people required to construct, maintain, and operate a specific technology. For measurement purposes, however, I limit my analysis only to construction labor time. This decision is driven primarily by data availability (and lack thereof). For the most part, published power plant data focuses almost exclusively on costs, and primarily on the cost of construction. Fortunately, with a few simplifying assumptions, construction cost data can be used to estimate construction labor time. I use this latter metric to quantify the scale of social coordination associated with a given power plant. To estimate construction labor time from costs, I first note that by the rules of double-entry accounting, all costs eventually become someone’s income. If we assume that all income accrues to labor (i.e. we neglect capitalist income) then we can divide the total cost of a project by an estimate of the average wage to obtain a rough estimate of the total labor time involved. I use GDP per capita as a measure of average income, giving equation 2.5 as my method for estimating labor time. Labor Time ≈Total Cost GDP per capita (2.5) Although this method contains some implicit bias/error, I show in Appendix A.7 that it is unlikely that this bias/error affects the integrity of the results (largely due to the vast size range of power plant studied here). Figure 2.5C applies this method to estimate the construction labor time of approximately 500 different power plants and generators. The capacity of these plants/generators ranges over 7 orders of magnitude — from the smallest gaspowered generator (1000 watts) to the largest hydroelectric dams (the 22.5 gigawatt Three Gorges Dam). Different energy sources are indicated by color. The results show a strong scaling relation between plant capacity and construc- The ‘Why’ Question: Energy, Technology and Hierarchy 38 tion labor time. This indicates that the scale of social coordination necessary to build a power plant is strongly related to the plant’s energy conversion capacity. To summarize, our case study of the electricity generation sector is consistent with both hypothesis A and B. We find that increases in power plant scale have played an important role in meeting increases in US per capita electricity consumption (hypothesis A). Furthermore, we find that power plant size is strongly related to construction labor time — our measure of the scale of social coordination (hypothesis B). Admittedly, a case study of a single technology represents limited evidence. However, the vast scaling of the other technologies shown in Table 2.1 indicates that this line of reasoning has promise. To continue my arguments, I will assume that the findings of this case study can be generalized to many other technologies. The result (we assume) is a that increases in energy consumption require a generalized increase in the scale of human social coordination. The question, then, is how is this coordination accomplished? 2.4.2 Social Coordination and Human Biology Social coordination can conceivably be achieved in many different ways (customs, markets, institutions, etc.). Thus, an increase in social coordination does not necessarily imply an increase in firm and government size. Why, then, have these institutions increased in size as energy consumption increases? Hypotheses C-E propose a chain of reasoning explaining why institutions are the most effective way of organizing large groups of people. The key to this reasoning is hypothesis C: humans have a limited ability to maintain social relations. The evidence for this hypothesis comes primarily from the work of anthropologist Robin Dunbar, who has uncovered a startling relation between primate brain size and mean group size [7]: primate species with larger brains (as measured by the relative size of the neocortex) tend to live in larger groups. Dunbar has developed this finding into what he calls the social brain hypothesis: “primates evolved large brains to manage their unusually complex social system[s]”[49]. The implication of Dunbar’s findings is that the size of the human brain places limitations on the number of social relations that an individual is able to maintain. Dunbar uses his primate data to predict a mean human group size of about 150. While this number should be considered exploratory, Dunbar notes that early egalitarian societies had group sizes around this order of magnitude [51]. A key feature of egalitarian organization is that any member of a group may The ‘Why’ Question: Energy, Technology and Hierarchy 39 maintain relations with any other member of the group. Thus, the number of possible social relations increases linearly with group size. Given the hypothesized limitations in the human ability to maintain social relations, it follows that egalitarian social organization is not an effective method for coordinating large numbers of people. One way of increasing group size beyond Dunbar’s number is to organize groups in a way that limits human interaction. Turchin and Gavrilets note that this is a key feature of social hierarchies, which are characterized by a treelike chain of command [8]. Within a hierarchy an individual must maintain social relations only with his direct superior and direct inferiors. Thus, hierarchy allows group size to grow without any corresponding increase in the number of human relations (hypothesis D). As evidence for this line of reasoning, Turchin and Gavrilets demonstrate that a strong correlation exists between the population of historical agrarian empires and the number of administrative (hierarchical) levels within their respective governments. Similarly, Hamilton et al. find a strong relation between population size and the number of hierarchical levels with various hunter-gatherer societies [52]. This evidence suggests that social hierarchy is a common tool used for increasing the scale of social coordination. 2.4.3 Hierarchy and Institution Size Social hierarchies have taken many different forms at different points in human history. For instance, in many pre-state societies, social hierarchy took the form of the chiefdom. In middle-ages Europe, the feudal manor was the principle unit of hierarchy. In the modern era, I argue that business firms and governments are the principle unit of social hierarchy (hypothesis E). To test this hypothesis, I focus only on firms. The implication of hypothesis E is that increasing firm size constitutes an investment in social hierarchy. If this reasoning is correct, then mean firm size should be an indicator of the relative ‘top heaviness’ of a society. Why? Hierarchies tend to become more top heavy as they become larger — the fraction of individuals in the upper echelons tends to grow as the size of the hierarchy increases. Thus, if firms are the modern embodiment of social hierarchy, then mean firm size should be related to the relative size of the upper social echelon. Since the upper echelons of a hierarchy are almost exclusively involved in managing the activities of other people, it seems sensible to use the management profession as a metric for the size of this top cohort. Thus, if hypothesis E is The ‘Why’ Question: Energy, Technology and Hierarchy 40 Mean Firm Size Managers 5.0 12.5% 2.9 7.5% 1.7 0% Firm Size Distribution = Manager = Non-Manager Figure 2.6: The Growth of Management as a Function of the Firm Size Distribution This figure graphically demonstrates how the management fraction increases with firm size (assuming firms are ‘ideal hierarchies’). Firms are indicated by boxes (with the exception of single-person firms) with a worker’s hierarchical position shown vertically. The span of control — defined as the size ratio between adjacent hierarchical levels — is constant for all firms. In this picture, the span of control is 2. Managers (red) are assumed to be all individuals in and above the third hierarchical level. To maintain simplicity, this graphic does not use a power law firm size distribution. correct, we expect that increases in mean firm size should be associated with an increase in the employment share of managers. To refine this prediction, I develop a hierarchical firm model of society (Fig. 2.6) based on the following assumptions: 1. All firms are ‘ideal’ hierarchies with a single span of control. 2. All individuals in and above the third hierarchical level are considered ‘managers’. 3. The firm size distribution is a power law. Why assume that management begins at the third hierarchical level? Obviously, individuals within the lowest hierarchical level have no management responsibilities. Those in the second hierarchical level can be thought of as The ‘Why’ Question: Energy, Technology and Hierarchy 41 Figure 2.7: Testing the Hierarchical Model of the Firm Using Managment Share of Total Employment Panels A and B plot the country-level relation between the management fraction and mean firm size. Modelled data is also shown in the background, with the span of control indicated by color. Panels A and B use different (incommensurable) classification methodologies for ‘management’. Panel A uses ISCO-88 (which includes legislators, senior officials and managers) while panel B uses ISCO-1968 (which includes administrative and managerial workers). Error bars indicate the 95% confidence intervals for mean firm size. Panel C compares the span of control range from the model to the span distribution found by 12 different empirical studies. Red boxplots indicate case studies, and show the span of control distribution within a single firm. Blue boxplots indicate aggregate studies and show the span of control distribution across many different firms. The span of control distribution across all 12 studies is shown on the right. For sources and methodology, see Appendix A.1. The ‘Why’ Question: Energy, Technology and Hierarchy 42 ‘working supervisors’ — individuals who have some supervisory responsibilities but who spend a majority of their time engaged in ‘production’ [53]. I assume that individuals in and above the third hierarchical level are devoted mostly to managing the work of others. This model predicts that the management fraction of employment should grow non-linearly with firm size, eventually approaching an asymptote defined only by the span of control. If the span of control is s, then the asymptote occurs at 1/s2(see Appendix A.8 for the details of this calculation). In Figure 2.7 I test this model at the international level. Figure 2.7A and 2.7B plot the country-level relation between the management fraction of employment versus mean firm size (the two plots show different occupation classification regimes). Empirical data is shown in black, while model predictions are shown in the background with the span of control indicated by color. Different mean firm sizes are produced by varying the exponent of the firm size power law distribution (for a technical discussion of this model, see Appendix A.8). The model nicely reproduces the observed relation between mean firm size and the management fraction of employment. However, this fit is achieved by freely manipulating the span of control parameter. Thus, it is important to check that the modelled span of control range is consistent with the span range for real firms. Ideally we would be able compare the span range of the model to the span distribution of a large, global sample of firms. Unfortunately, data constraints make this impossible. Due to the proprietary nature of firm personnel data, only a handful of studies have analyzed firm hierarchies. Figure 2.7C shows data from 12 such studies that together sample firms from 7 different nations (Denmark, Japan, Netherlands, Portugal, the United Kingdom, the United States, and Sweden). The resulting firm sample gives relatively good coverage of wealthy nations, but unfortunately does not include any firms from developing countries (due to the lack of available studies). For a summary of the data sources, see Appendix A.1. Boxplots in Figure 2.7C correspond to the span of control range found by each study. Note that the data is a mixture of case studies of single firms and aggregate studies that analyze the structure of many different firms. While these aggregate studies give better scope than the case studies, many focus only on the upper levels of the hierarchy (where data is more easily obtained). The important finding in Fig. 2.7C is that the model’s fitted span of control range is consistent with the available empirical data. To summarize these findings, a simple hierarchical firm model of society is The ‘Why’ Question: Energy, Technology and Hierarchy 43 able to replicate the observed relation between mean firm size and the management share of employment. The changes in mean firm size are achieved by varying the exponent of a firm size power law distribution, while the management fraction of employment is fitted by ‘tuning’ the span of control range (assumed to be the same both within and between all modelled firms). Importantly, the resulting fitted span range is consistent with the existing empirical data on the internal structure of the firm. The success of this model gives support to hypothesis E, and suggests that increases in mean firm size are characteristic of a generalized increase in social hierarchy. 2.4.4 Causality I have proposed hypotheses A-E as a chain of reasoning connecting energy consumption to institution size. But which way does causation run? Do increases in energy consumption cause institutions to become larger, or is the reverse true? As I discuss below, it seems likely that causation runs in both directions. Although hypotheses A-E are framed in terms of increases in energy use (and institution size), I think that a discussion of causation is clearer when framed in terms of constraints and decline. For instance, I think it must be the case that energy constraints place limits on institution size. This is for the simple reason that energy conversion technology is useless without an energy input. I have proposed that large institutions provide the social coordination necessary to build and operate large technologies. But without sufficient energy input, these technologies cannot be operated, and the institution’s raison d’être ceases to exist. Imagine how long a large steel firm would stay in business if there was not enough coke to fuel its large blast furnaces. This line of thinking implies that a decline in energy consumption (due to scarcity) can cause a decline in institution size. However, recent history (the collapse of the Soviet Union) suggests that causality can operate in the reverse direction. Figure 2.8 shows energy and government employment share trends in six nation-states that emerged after the dissolution of the USSR. In the aftermath of the Soviet collapse, these six countries experienced drastic reductions in both government size and energy use. During this period, there was no global energy shortage, meaning biophysical energy constraints can likely be ruled out as a causal factor. Instead, it seems likely that institutional collapse is the driving factor here. This case is illustrative because the Soviet economy relied on an unusually high degree of government control of production, placing an enormous amount The ‘Why’ Question: Energy, Technology and Hierarchy 44 1990 2010 1990 2010 1990 2007 1990 2010 1990 2010 1990 2003 Armenia Azerbaijan Belarus Estonia Moldova Ukraine 0 25 50 75 0 25 50 75 0 100 200 300 0 100 200 300 0 100 200 300 Energy Use per Capita (GJ) Government % of Total Employment Figure 2.8: A Case Study in Causality: The Collapse of the Soviet Union This figure tracks the path through time of six nations that emerged after the collapse of the Soviet Union (in 1990-91). As the collapse unfolded, the fraction of people employed by the government shrank rapidly, as did energy use per capita. Since the USSR collapse was an institutional crisis (not an energy crisis), this suggests that at least in this case, causality runs from institution size to energy consumption. of power in the hands of a single institution. Not surprisingly, the collapse of this institution led to social chaos and widespread economic decline. I think this shows quite clearly that institutional collapse can cause a decline in energy consumption. The argument that causation can operate in both directions suggests that energy use and institution size exhibit a feedback relation (rather than linear causality). One possible avenue for furthering this research is to use systems modelling. Ugo Bardi has shown that a simple adaptation of the Lotka–Volterra equations can be used to model the relation between energy extraction and a technological stock [54]. A plausible line of future research would be to add institution size to this type of model. It is also important to note that changes in energy use and institution size occur alongside other social changes, the two most obvious being urbanization and changes in sector composition [35]. It seems likely that these phenomena Chapter 3 Evidence for a Power Theory of Personal Income Distribution Abstract This paper proposes a new ‘power theory’ of personal income distribution. I hypothesize that income is most strongly determined by hierarchical power — which I define as the number of subordinates under one’s control. Using this definition, I find that relative income within firms scales strongly with hierarchical power. I also find that hierarchical power has a stronger effect on income than any other factor for which data is available. I conclude that this is evidence for a power theory of personal income distribution. 3.1 Introduction Over the last decade, concerns about income inequality have risen to the forefront of public attention. As testament to this interest, Thomas Piketty’s expansive treatise on inequality, Capital in the Twenty-First Century, became an unlikely best seller when it was published in 2014. Due in no small part to the work of Piketty and colleagues [1–5],empirical study of income inequality has flourished. But this plethora of new data has not led to a corresponding theoretical revolution. The problem, I believe, is an unwillingness to question and test the basic assumptions on which current theory rests. Most theories of personal income distribution are deeply wedded to the assumption that income is proportional to productivity. However, this approach has a simple, but little discussed problem: income is distributed far more unequally than documented differentials in human labor productivity. But if not productivity, then what explains differentials in income? I hypothesize that personal income is explained most strongly by hierarchical Theories of Personal Income Distribution 52 power, as manifested by one’s rank in an institutional hierarchy. Using the common definition of power as the ‘ability to influence or control others’, I measure hierarchical power in terms of the number of subordinates under an individual’s control. From this definition, it follows that power, unlike productivity, tends to be very unequally distributed within hierarchies — a natural consequence of the tree-like chain of command that concentrates control at the top. I test the power-income hypothesis in two ways. First, using the available firm case study data, I look for correlation between income and my metric for hierarchical power. I find that relative income within firms is strongly correlated with hierarchical power. I also find a strong correlation between changes in income and changes in hierarchical power. Second, I test the strength of the power-income effect against a wide range of other income-affecting factors. I find that grouping individuals by hierarchical rank has the strongest effect on income. I conclude that this is evidence for a power theory of personal income distribution. The paper is organized into the following parts. In section 3.2, I review and critique existing theories of personal income distribution, and summarize the key failings of the dominant ‘productivist’ approach. In section 3.3, I outline the principles and motivations behind my proposed power theory of income distribution. In section 3.4, I test the power-income hypothesis against empirical evidence. All methods and sources are documented in the Appendix. 3.2 Theories of Personal Income Distribution My reading of the history of personal income distribution theory is that the field has struggled to meet the following two mutually contradictory goals: 1. Address and explain the ‘Galton-Pareto’ paradox; 2. Maintain consistency with prevailing theories of functional income distribution. The ‘Galton-Pareto paradox’ refers to the large discrepancy between the observed distribution of human abilities and the observed distribution of income. The former was first documented by Francis Galton [6], who found that human abilities were normally distributed, and hence quite equal. The latter was first documented by Vilfredo Pareto [7], who found that income distributions were highly skewed and unequal. Following the findings of Galton and Pareto, political economists have spent a century struggling to reconcile these two facts [8]. The process has been Theories of Personal Income Distribution 53 made difficult primarily because the two dominant theories of functional (classbased) income distribution assume a connection between individual productivity (hence ability) and income. At the present time, two main approaches to personal income distribution theory exist: the stochastic and the productivist approach. The stochastic school solves the Galton-Pareto paradox by ignoring prevailing theories of function income distribution. In contrast, the productivist school purports to both resolve the Galton-Pareto paradox and maintain consistency with the rest of economic theory. However, a closer look reveals that this ‘success’ relies on untestable assumptions and circular logic. I review both theories below. 3.2.1 Stochastic Theories The discrepancy between Galton and Pareto’s findings is a paradox only if one expects that income should be somehow related to ability. Clearly the simplest resolution is to assume that ability plays a negligible role in determining income. This is precisely the road taken by stochastic models, which explain income distribution in terms of random events that have little (if anything) to do with the characteristics of individuals. In 1953, David Champernowne demonstrated that a simple statistical process could be used to explain the ‘Pareto’ (or power law) distribution [9]. In this model, individuals are subjected to a series of random, exogenous ‘shocks’ that perturb their income. Over time, this process leads to an equilibrium power law distribution. Champernowne’s model was later recognized to be part of a general class of interrelated models in which ‘multiplicative’ randomness is the generative mechanism for a skewed distribution [10–14]. More recently, econophysicists have used this stochastic line of thinking to draw explicit parallels between the distribution of income and the distribution of kinetic energy in gases. These kinetic exchange models explain income distributions in terms of the random exchange of money between individuals [15–18]. Under the assumption that money is conserved, kinetic exchange models generate distributions of income that closely resemble those in the real world. Despite their successes, stochastic models have been mostly ignored by the economics profession. One reason is that the assumptions underlying this type of theory (especially kinetic exchange models) are often unrealistic [19]. Kinetic exchange models imply a world in which money is conserved for all time, nothing is ever produced, there are no groups, institutions or classes of people, and the world exists in static equilibrium. Theories of Personal Income Distribution 54 However, a more insidious reason that stochastic models have been ignored is that they are inconsistent with the prevailing theories of functional income distribution, and the latter form the ‘hard core’ of political economic theory. 3.2.2 Productivist Theories The discipline of political economy essentially arose in response to questions about class-based (or functional) income distribution. As David Ricardo saw it, the role of political economy was to “determine the laws” that regulate the distribution of income between the “classes of the community” [20]. Out of the 19th century debate over these laws, two great schools of thought merged — Marxist and neoclassical. Over the following century, virtually all economic theory was built on top of either Marxist or neoclassical assumptions about income distribution. The result is that if a new theory of personal income distribution contradicts these prevailing theories of functional income distribution, accepting the new theory logically requires discarding not only the functional income distribution theory, but a large part of political economic theory as well. Perhaps understandably, economists have hesitated to take this road. Instead, they have largely opted for personal income distribution theories that prioritize consistency with the rest of economic thought. Although Marxist and neoclassical schools are usually positioned in opposition to one another, they both posit a similar link between productivity and income [21]. In neoclassical theory, income is attributed to marginal productivity — the incremental increase in output caused by the incremental increase in inputs of capital/labor [22,23]. Thus, if a capitalist makes more than a worker, it is because an additional unit of his ‘capital’ adds more to output than an additional unit of the worker’s labor. The logical implication of this theory is that income differences between workers — who all earn labor income — must be due to differences in individual productivity. Out of this line of reasoning came human capital theory, which attributes workers’ productivity to some internal stock of ‘human capital’ [24–26]. Unlike neoclassical theory, Marxist theory posits that labor is the sole producer of value [27]. Therefore, both labor and capitalist income ultimately stem from workers’ productivity. The Marxist twist is to treat capitalist income as parasitic – the result of the expropriation of surplus value created by workers. The relative balance between labor and capitalist income is then a function of the ‘degree of exploitation’ of workers. But when it comes to income distribution among workers, Marxists come to conclusions that are very similar to their neo- Theories of Personal Income Distribution 55 classical counterparts. Since labor is the sole source of value, skilled workers who earn more than unskilled workers must somehow be more productive [28]. This productivity-income hypothesis has made it difficult for neoclassical and Marxist theories to address the Galton-Pareto paradox. Since individual productivity is presumably related to ability, one cannot take the easy road and simply negate any relation between ability and income. Instead, one must explain why productivity is as unequally distributed as income, but ability is not. The most common resolution to the Galton-Pareto paradox is to assume that different abilities, each normally distributed, somehow interact to have a multiplicative effect on productivity [29,30]. This multiplicative effect can be expressed as a production function in which a worker’s output (Y) is an exponential function of the sum of different abilities (ai): Y=ea1+a2+...+ai. This hypothesis is central to human capital theory, which proposes that investments in human capital yield multiplicative returns to productivity [24,25]. But is this actually the case? Is productivity as unequally distributed as income? Unfortunately, this question is not as easily answered as it might seem. The problem is this: how do we compare the productivity of different workers who have qualitatively different outputs? For instance, how can we determine if a farmer, who produces potatoes, is more productive than a composer, who produces music? Any such comparison of qualitatively different outputs inevitably requires choosing a common unit of analysis. But the choice of this unit is subjective, and different units will lead to different results. The logical implication is that there are no objective grounds for comparing the productivity of workers with qualitatively different types of output. The same problem occurs when attempting to measure the productivity of capital: one can only compare capitalists with exactly the same output. There are other measurement problems inherent in marginal productivity theory. These include the inability to objectively measure capital [21,32,33], as well as the inability to isolate the effect on output caused by changes in capital versus changes in labor (see Pullen [34]for a good review). Taking these measurement problems seriously means that one can compare productivity only between workers who have exactly the same output.[31]have compiled data that does exactly that — they report differences in productivity among workers doing the same task. In Figure 3.1, I take this data and convert it into a Gini index of ‘productivity inequality’ so that it is directly comparable to income inequality within nation states. This evidence indicates that differences in productivity are systematically too small to account for observed levels of inequality. Theories of Personal Income Distribution 56 Mean = 0.40 Income Inequality Within Nation−States Mean = 0.10 Productivity Inequality Among Workers Doing the Same Task 0 2 4 6 8 10 12 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Gini Index Density Figure 3.1: Labor Productivity Inequality vs. Income Inequality Using a Gini index, this figure compares the inequality of worker productivity to income inequality within nation-states. Data for the former comes from Hunter et al. [31], who report the coefficient of variation of productivity among workers conducting the same task. Data plotted here shows the distribution of productivity inequality for 55 different tasks. I convert Hunter’s data to a Gini index by assuming that worker productivity is lognormally distributed. The Gini index (G) of a lognormal distribution with a coefficient of variation cvis G=erf(1 2Ælog(c2 v+1)). I plot the resulting distribution against the distribution of Gini indexes of income inequality for all country-year observations in the World Bank database (series SI.POV.GINI). Theories of Personal Income Distribution 57 However, the link between productivity and income is not typically measured in such restrictive terms. Instead, the standard practice is to adopt monetary value as a common unit of comparison for measuring different outputs. Thus, labor productivity is generally measured in terms of sales or value-added per worker [35–41]. The problem with this approach is that it relies on circular logic. According to theory, income is explained by productivity. But when the theory is tested, productivity is measured in terms of income. And based purely on accounting principles, we expect wages to be correlated with sales/valueadded per worker. Double entry accounting principles dictate that the value-added (Y) of a firm is equivalent to the sum of all wages/salaries (W) and capitalist income (K). If we divide by the number of workers (L), we find that value-added per worker is equivalent to the average wage (w=W/L) plus K/L: Y L=W+K L=w+K L(3.1) Sales (S) are similar, but include an additional non-labor cost term (C): S L=W+K+C L=w+K+C L(3.2) Thus, if we look for correlation between average wage (w) and value-added/sales per worker (Y/Lor S/L), we will surely find it, since simple accounting definitions dictate that the former is a major component of the later. To summarize, existing theories of personal income distribution are plagued by fundamental problems. The two main schools reviewed here — stochastic and productivists — both have major shortcomings. The stochastic approach, while interesting from a mathematical standpoint, makes assumptions that are unrealistic and have little to do with the real world. The productivist school, on the other hand, has waged an uphill battle with empirical evidence. Its successes have been achieved by basing empirical tests on circular logic. I argue that a new approach is needed. Rather than focus on productivity (or stochastic interactions) I propose that personal income is best explained by the hierarchical power structure of institutions. A Hierarchical Power Theory of Personal Income Distribution 58 3.3 A Hierarchical Power Theory of Personal Income Distribution The premise of this paper is that income distribution can be explained primarily in terms of differentials in hierarchical power. But before diving into the specifics of this theory, I want to provide a rationale based on the big picture of human history. Why? There is nothing like looking at the past to gain fresh insight into the present. Let’s ask a simple question: what aspects of human history suggest that hierarchical power might affect how we distribute resources (which is what income distribution is all about). Let’s begin with our deep history — the evolutionary backdrop of the human species. Humans are but one of a wide variety of social mammals, virtually all of which form dominance hierarchies, or ‘pecking orders’ [42–47]. A key characteristic of these dominance hierarchies is that high social rank is associated with preferential access to resources, particularly sexual mates [48–52]. Of course, human behavior is far more complex than even the most intelligent (non-human) primates. Just because we evolved from hierarchy-forming animals does not necessarily mean that hierarchical rank still plays a role in how we divide up the pie. However, there is good evidence that humans do have an instinctual behavior towards hierarchy formation. Several studies have shown that children and adolescents spontaneously form dominance hierarchies when placed into small groups [53–55]. Other studies have shown that, like other social mammals, human reproductive success increases with social status [56,57]. There is even evidence that social status at birth is epigenetically imprinted on human DNA [58]— something that also occurs in Rhesus monkeys [59]. Given our evolutionary heritage, it seems plausible that hierarchy plays a role in the way humans distribute resources. Another reason to suspect that resource distribution has to do with hierarchy and power is the ubiquity of inherited status in human history. It is hard to justify the wealth of a hereditary aristocracy as stemming from anything but power and privilege. Interestingly, inherited status has surprisingly deep historical roots. There is tentative archaeological evidence for inherited status beginning in the neolithic era [60–62], and widespread evidence beginning in the bronze age around 5000 years ago [63–67]. It is around this time that the first Egyptian dynasty formed [68], followed later by dynasties in Mesopotamia [69]and China [70]. Since then, as Gaetano Mosca observes, the existence of a hereditary ruling class has been the norm: A Hierarchical Power Theory of Personal Income Distribution 59 There is practically no country of longstanding civilization that has not had a hereditary aristocracy at one period or another in its history. We find hereditary nobilities during certain periods in China and ancient Egypt, in India, in Greece before the wars with the Medes, in ancient Rome, among the Slays, among the Latins and Germans of the Middle Ages, in Mexico at the time of the Discovery and in Japan down to a few years ago. [71] But while history may be sordid, there is always the possibility that modern societies have made a clean break with the past. Power may have played a central role in the distribution of resources in past societies, but in modern societies reciprocal exchange is what matters most. This is the story that emerged in the writings of Adam Smith [72]and was codified into neoclassical theory by Jevons, Menger and Walras [73–75]. To paraphrase George Orwell [76], this is now the prevailing orthodoxy that most right-thinking economists accept without question. But what if there has not been a clean break with the past? What if power still plays an important role in shaping resource distribution? A wide variety of scholars have argued that this is the case. A non-exhaustive list would include [21,77–89]. These scholars argue that power plays a central role in shaping income distribution. If there is to be a power-based theory of income distribution, what should it look like? According to Christopher Brown: ... [A]theory of distribution should be indistinguishable from a theory of power. A satisfactory theory of power would, beyond defining what power is, elucidate principles to explain how power is established, enlarged or diminished, protected and perpetuated, redistributed, exercised, and rendered legitimate or illegitimate. [90] A full-fledged theory of power is a tall order. In this paper, I narrow the focus to look only at hierarchical power in the context of personal income distribution. My ideas stem from the work of Simon [91]and Lydall [92], who independently proposed income distribution models based on the hierarchical structure of firms. The focus of Simon and Lydall’s work is the branching nature of institutional hierarchies, in which each superior controls multiple subordinates. This structure is unique to humans. All other animals form linear hierarchies — an ordinal ranking from top to bottom. The most important feature of a branching hierarchy is that it concentrates power in the hands of the few. I propose that this concentration of hierarchical power can be used to explain income inequality. The main theoretical contribution of this paper is to offer a quantifiable def- A Hierarchical Power Theory of Personal Income Distribution 60 inition of hierarchical power that allows power differentials to be directly compared to income differentials. I test the following hypothesis: Hypothesis: Income is most strongly determined by hierarchical power, as measured by the number of subordinates under one’s control. 3.3.1 Measuring Hierarchical Power What is hierarchical power? I define it as the ability to control subordinates within a hierarchical chain of command. The link between hierarchy and power is implicit in the etymology of the word ‘hierarchy’ itself, which derives from the Greek term hierarkhes, meaning ‘sacred ruler’ [93]. In essence, an institutional hierarchy is a nested set of power relations between a superior (a ruler) and subordinates (the ruled). It is a control structure that concentrates power at the top [94]. I propose that one’s power within a social hierarchy is proportional to the number of subordinates under one’s control. I put this in formula form as: hierarchical power =number of subordinates +1 (3.3) The logic of this equation is that all individuals start at a baseline power of 1, indicating that they have control over themselves. Power then increases linearly with the number of subordinates. If we had access to the exact chain of command structure of an institution, we could use this definition to measure the power of each individual within a hierarchy. Unfortunately, chain of command information is rarely available. Instead, existing case studies report aggregate hierarchical structure only — total employment by hierarchical level. While we cannot calculate the power of specific individuals, we can use this data to calculate the average power of all individuals in a specific hierarchical level: ¯ Ph=¯ Sh+1 (3.4) Here ¯ Phis the average power of individuals in hierarchical level h, and ¯ Sh is the average number of subordinates below these individuals. 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. Figure 3.2 shows a sample calculation, where red individuals occupy the level in question, and blue individuals are subordinates. Each red individual has 2 direct subordinates, and 4 indirect subordinates, for a total of 6 subordinates. The average hierarchical power in level three is therefore 7. Testing the Power-Income Hypothesis 67 the ratio of power after versus power before the promotion/demotion (Eq. 3.6). An individual’s power is defined by Eq. 3.4. Since we do not know the exact chain of command, I assign all individuals the average power of their respective hierarchical level. ∆¯ P=¯ Pafter ¯ Pbefore (3.6) For each promotion/demotion, I define the fractional change in income (∆I) as the ratio of income after versus income before the event (Eq. 3.7). In order to isolate the effect of the promotion from the exogenous effects of inflation and/or general wage increases, I measure all incomes relative to the firm mean income (¯ I) in the appropriate year. ∆I=Iafter/¯ Iafter Ibefore/¯ Ibefore (3.7) Figure 3.6 show the results of this dynamic analysis. Here each plotted point represents the fractional change in pay and power for the promotion/demotion of a single individual. For the over 16,000 promotions/demotion events analyzed here, a highly significant correlation exists between changes in power and changes in individual income. Interestingly, the correlation holds both for promotions and for demotions, the latter occurring when an individual drops hierarchical levels. The relative pay reductions accompanying these demotions are difficult to understand from a productivist approach. Do these individuals suddenly experience a drastic reduction in ability/productivity? The evidence in Figure 3.6 suggests a better explanation: within the BGH firm, pay is largely a function of the power of a specific hierarchical position, irrespective of the person holding this position. To conclude, the available evidence is consistent with hypothesis A. Relative income within firms is both statically and dynamically correlated with hierarchal power. Having survived this first hurdle, we now move on to test the powerincome effect in the more stringent form of hypothesis B. 3.4.2 The Strength of the Power-Income Effect Hypothesis B states that hierarchical power affects income more strongly than any other factor. To test this hypothesis, I use an analysis of variance method to quantify the income effect of from wide variety of different factors. In order Testing the Power-Income Hypothesis 68 Between −Group : GB=0.07 Within −Group : GW=0.45 Indicator : GBW =0.17 Income Density Group 1 Group 2 A. Small Effect on Income Between −Group : GB=0.57 Within −Group : GW=0.14 Indicator : GBW =4.07 Income Density Group 1 Group 2 B. Large Effect on Income Figure 3.7: Analysis of Variance Using the Gini Index This figure shows an example of the analysis of variance method that uses the Gini index. A hypothetical 2-group variable (like ‘sex’) is illustrated to have a small effect on income in panel A and a large effect in panel B. The means of each distribution are indicated by a dashed line. We use equation 3.8 to define the between-within Gini indicator, GBW . A small effect on income is indicated by a GBW that is close to zero, while a large effect is indicated by a GBW greater than one. to make the test as thorough as possible (given data constraints), some data is model dependent (see the Appendix for a detailed discussion). Method for Measuring Effect Size While there are many conceivable ways that hypothesis B could be tested, the format of available data makes the analysis of variance method the most appropriate. This is because many factors that affect income (such as ‘sex’ or ‘race’) are qualitative variables. Even factors like ‘education’ and ‘age’ that could conceivably be quantitatively measured (in units of time) are typically reported in qualitative groups such as ‘college graduate’ or ages ‘50-59’. The analysis of variance (ANOVA) method provides a simple way of determining how strongly qualitative variables affect income. The essence of this approach is to compare Testing the Power-Income Hypothesis 69 between-group income dispersion to within-group income dispersion for a given factor. The larger the between-group dispersion is relative to the within-group dispersion, the larger the effect on income. This approach is most easily understood by way of an example. Figure 3.7 shows a hypothetical example of how a two-group variable like ‘sex’ might affect income. When the separate income distributions of the two groups are plotted together, we can clearly see a small effect in Fig. 3.7A and a large effect in Fig. 3.7B. How do we quantify the size of this effect? Most people likely judge the difference in group means against the dispersion within each group. We might call this a signal-to-noise ratio, where the ‘signal’ is the difference in group means and the ‘noise’ is the within-group dispersion. The larger the signal is relative to the noise, the larger the effect. The ANOVA method allows us to generalize this concept of effect to more than two groups. The corresponding signal-to-noise ratio is often called Cohen’s f2. For this metric, the ‘signal’ is the dispersion between group means, while the ‘noise’ is the dispersion within groups, where dispersion is measured as the sum of squared differences from the mean [102,103]. While Cohen’s f2is a common measure of effect size, its calculation requires either raw data on individual income, or data for within-group variance (or standard deviation). Unfortunately, this type of data is difficult to obtain. Instead, what is readily available are aggregate statistics reporting within-group Gini indexes. Because of the ubiquity of the Gini index, I use it to measure effect size. Similar to Cohen’s f2, my effect size metric is a signal-to-noise ratio (Eq. 3.8). However, rather than the sum of squares, I use the Gini index to measure both within-group and between-group dispersion. I call this metric the betweenwithin Gini ratio (GBW ). GBW =GB GW (3.8) Here GBis the between-group Gini index (the Gini index of group mean incomes), while (GW) is the average of all within-group Gini indexes. For a detailed discussion of the relation between GBW and f2see Appendix B.8. The value of GBW can range from 0 to infinity, with larger values indicating a larger effect on income (see the example in Fig. 3.7). Of particular interest is the value GBW =1, which occurs when between-group dispersion is equal to within-group dispersion. Any factor that produces GBW >1 can be considered to have a significant impact on income, since inequality between groups is larger than inequality within groups. However the primary use of the GBW metric is not Testing the Power-Income Hypothesis 70 its absolute value, but its relative value when different income-affecting factors are compared. A well-known shortcoming of the Gini index is that it has a downward bias for small sample sizes. If the sample size is n, the maximum possible Gini index is: Gmax n=n−1 n(3.9) Thus a sample size of n=2 has a maximum Gini index of Gmax 2=0.5. This bias presents a problem for the calculation of the between-group Gini index GB because the number of groups (n) is often extremely small (i.e. n=2 for the factor ‘sex’). While this small nis not really a sample (it is the actual number of groups), it still causes a bias in the Gini index. The result is that we cannot safely compare GBbetween two income-affecting factors with different numbers of internal groups. To correct for this bias, I use the method proposed by George Deltas [104]. The bias-adjusted Gini index (Gad j) is defined by dividing the unadjusted Gini (G) by the maximum possible Gini (Gmax n), given the number of internal groups n: Gad j =G Gmax n (3.10) All between-group Gini calculations in this paper use the adjusted Gini index, Gad j. However, for notational simplicity I refer to this adjusted between-group Gini as GBfor the remainder of the paper. Some Clarifications on ‘Effect Size’ It is important to clarify that my empirical method measures the effect on income, not the effect on inequality. There is a subtle, but important difference. In measuring the effect on income, group size is irrelevant. Any factor that has a strong effect on income (like education) necessarily involves zeroing in on a small, elite group of people (for instance, a small fraction of the population has a graduate degree). But the effect on inequality takes group size into consideration. Consider the scenario where all people with a graduate degree are millionaires, but there are only 10 such people (out of millions). Having/lacking a graduate degree will have a strong effect on income, but not on inequality. I hope this gives an intuitive understanding of the difference between the two types of effect size. For a technical discussion, see Appendix B.8. Testing the Power-Income Hypothesis 71 1 2 4 3 5 h Figure 3.8: Grouping Power By Hierarchical Level This figure shows my method for grouping individuals by their power. In this figure, each hierarchy represents a different firm. My proposed groups consist of all individuals (regardless of firm) that share the same hierarchical level. Groups are indicated by color. On a different note, readers trained in econometrics will (correctly) observe that my method for measuring effect-size does not isolate the income-effects of a given factor. It does not show that, when all other factors are held constant, a change in factor Aby amount xaffects income by amount y. I make no attempt to do this because I think it is the wrong approach. As Keynes long ago argued, the only conceivable way that an econometric model can isolate an effect is if the model includes a complete list of causal factors [105]. But since we can never be sure that our causal list is complete, we can never know if our econometric model is wrong [106]. My thinking is more pragmatic. Given the complexities of human behavior, we can likely never isolate a factor to find its ‘true’ effect on income. But we can rank effect-size with the full understanding that when we measure one factor’s effect on income, enumerable other factors are included in this measurement. In the face of enumerable confounding variables, Occam’s razor would suggest that we simply chose the factor with the largest effect on income and use it to build a theory. Grouping Individuals By Hierarchical Level To test hypothesis B (that hierarchical power affects income more strongly than any other factor) using an analysis of variance method, we must group individuals into different categories/classes of social power. My method is to group individuals by hierarchical level across all firms, as illustrated in Figure 3.8. This method is theoretically attractive because hierarchical level is the principle determinant of power. If a firm has a constant ‘span of control’ (the number of Testing the Power-Income Hypothesis 72 subordinates below each superior), then power will increase exponentially with hierarchical level. In Figure 3.8, the span of control is constant both within and between firms. The result is that all individuals in each hierarchical level have the same power. In the real-world, we would expect this not to be the case. Evidence from firm case-study data suggests that the span of control varies both within and between firms (see Fig. 4 and 5 in Appendix B.2). As a result, we still expect that average power will increase exponentially with hierarchical level, but each hierarchical level will contain individuals with a range of different power. While there are other conceivable ways of grouping individuals by power, this method is both theoretically attractive and practical for empirical analysis. The available data on firm hierarchies is limited, and the most commonly reported metric is the distribution of income by hierarchical level. The Data To test hypothesis B, I use the 19 different income-affecting factors shown in Table 3.1. With two exceptions (discussed below), data comes from the United States. Data sources as well as details about each category are discussed in Appendix B.1. Before proceeding with a discussion of the data sources used for income by hierarchical level, it is worth reviewing why I do not use the same case study data that was used to test hypothesis A. Testing hypothesis B requires grouping Table 3.1: Income-Affecting Factors Used to Test Hypothesis B Geographic Physical Attribute Socioeconomic Census Block Group Age Education Census Tract Cognitive Score* Employee vs. Self-Employed County Race Firm* Urban vs. Rural Sex Full vs. Part Time Hierarchical Level* Home Owner vs. Renter Occupation Parents’ Income Percentile Public vs. Private Sector Religion Type of Income (Labor/Property) * Indicates variables that use model-dependent data (at least in part) Testing the Power-Income Hypothesis 73 individuals by hierarchical level across a large number of firms. To be consistent, the firms should all be in the same country (ideally the United States), and the observations (that are compared) should be in the same year. The case study data does not meet these requirements: it is a small sample, with firms from many different countries with many non-overlapping years. As a result, the case study data is not useful for testing hypothesis B. Instead, I use three different sources for estimating income distribution by hierarchical level. The first source is a seminal study by Mueller, Ouimet, and Simintzi [107]that reports income distribution by hierarchical level for 880 United Kingdom firms over the period 2004-2013. The second source is a study by Fredrik Heyman [108]that analyzes the pay distribution of the top 4 levels of management in 560 Swedish firms in the year 1995. Heyman’s data comes with the caveat that it does not represent all hierarchical levels — just the top four. For this reason, I mark Heyman’s results with an asterisk. I use this non-US data because I am not aware of any equivalent US study that reports income distribution by hierarchical level over a large number of firms. While comparing US to UK/Swedish studies is not ideal, I proceed because of the lack of alternative data. If anything, the UK and Swedish data should lead to an under-estimate of the power-income effect in the United States. Why? Both the UK and Sweden have significantly less income inequality than the US (according to the World Bank, the most recent UK and Swedish Gini indexs are 0.33 and 0.27, while the most recent US Gini index is 0.46). If there is less total inequality, the potential for between-group inequality is diminished, resulting in a lower GBW metric (see Eq. 3.8). My third source for hierarchical level data is a model that uses the insights from firm case study data to estimate the hierarchical pay structure of 713 US firms in the Compustat database (covering the years 1992-2015). This ‘Compustat Model’ is discussed in detail in the Appendix, but I review its core components here. The idea of the Compustat Model is that firm case-study data can be used to make generalizations about the hierarchical employment and pay structure of firms. Although different firms have differently shaped hierarchies (see Fig. 4 in Appendix B.2), there are underlying regularities shared by all firms. The following regularities are shown in Fig. 5 in Appendix B.2: 1. The span of control tends to increase with hierarchical level. 2. The ratio of average pay between adjacent hierarchical levels increases by level. Testing the Power-Income Hypothesis 74 Figure 3.9: Visualizing the Compustat Model This figure visualizes the results of the Compustat Model for selected US firms in the year 2010. The data and method underlying this model are discussed in detail in the Appendix. Each pyramid represents a separate firm with volume proportional to total employment. The vertical axis corresponds to hierarchical level. Income is indicated by color. Testing the Power-Income Hypothesis 75 3. Intra-level inequality tends to be constant across all hierarchical levels. I use these regularities to construct a hierarchical model of the firm (see Appendix B.3). Given appropriate input data, this model can be used to estimate income inequality by hierarchical level (across firms). To make this estimate, I use the Compustat database, which provides the following data for 713 US firms over the period 1992–2015: 1. Number of Employees 2. Total Staff Expenses 3. CEO Pay In conjunction with case-study regressions, this Compustat data can be used to estimate the hierarchical pay structure of individual US firms (see Appendix B.5 and F). While the details of the model are complex, the core idea is simple: since the CEO sits at the top of the corporate hierarchy, his/her relative pay (when compared to the average pay of all employees) gives an indication of the rate at which income increases by hierarchical level. When paired with assumptions about the ‘shape’ of the firm (derived from case-study regressions), the model gives an unambiguous prediction about firm internal pay structure. Results of the model are visualized in Figure 3.9 for selected firms in 2010. The skeptical reader may be wondering why, after dismissing the case study data as not useful for testing hypothesis B, I nonetheless construct a model that hinges on this very data. The model is useful because the Compustat data (to which the model is fitted) adds a great deal of new information that is not contained within the case study data itself. The Compustat data adds a large number of US firms that exist over a continuous time-series, each having a different size, different mean pay, and different CEO pay ratio. While the case study data determines the hierarchical shape of all firms, the Compustat data determines everything else. In Appendix B.7 I analyze the sensitivity of this model to the case study data. I find that the key metric — the GBW metric for income grouped by hierarchical level — is relatively robust to changes in case study data. In addition to income distribution by hierarchical level, I also use the Compustat model to estimate the strength of the firm-income effect (how much working for different firms affects income). In this case, the Compustat database can be used to directly measure income inequality between firms, and the model is used to estimate inequality within each firm. I use this model-dependent data because I am not aware of any studies that directly measure internal income distributions of a large sample of firms. Testing the Power-Income Hypothesis 76 Results The results of the analysis of variance test of hypothesis B are shown in Figures 3.10 and 3.11. Figure 3.10 shows the between-within Gini ratio (GBW ) for our 19 different income-affecting factors. For all factors except religion and cognitive score, the boxplots indicate the variation of GBW over time (typically the last 20 years). For religion, the boxplot range indicates uncertainty in the GBW estimate, while for cognitive score, it indicates variation between different studies. Figure 3.11 shows the same data, but in a slightly different format. The GBW metric consists of a ratio of between-to-within group income dispersion (Eq. 3.8). Figure 3.11 decomposes this ratio and shows the individual components of the metric — between-group inequality (GB) and within-group inequality ( ¯ GW). Aside from religion, density plots indicate the distribution of these values over time (for religion, density plots indicate uncertainty). The important information here is the relative position of between-group inequality relative to withingroup inequality. This test of hypothesis B yields conclusive results: of the 19 different incomeaffecting factors tested, hierarchical level has the strongest effect on income. We can conclude that the available evidence supports hypothesis B: hierarchical power appears to affect income more strongly than any other factor. Interestingly, the Compustat model and the data from Mueller et al. and Heyman give GBW ratios that are similar (although the underlying values of GBand ¯ GWare quite different). This may indicate that the strength of the hierarchy-income effect is consistent across countries that have different levels of inequality. In addition to the support for the power-income hypothesis, Figures 3.10 and 3.11 reveal a few other notable findings. Firstly, physical attributes (age, cognitive score, race, and sex) have a relatively insignificant effect on income. Geographic effects are also quite small, although they become larger as the geographic area decreases (geographic factors ranked from largest to smallest area are: county, tract, block group). 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Mind. 1950;59(236):433. 118. Boehm C, Barclay HB, Dentan RK, Dupre MC, Hill JD, Kent S, et al. Egalitarian behavior and reverse dominance hierarchy [and comments and reply]. Current Anthropology. 1993;34(3):227–254. Chapter 4 A Hierarchy Model of Income Distribution Abstract Based on worldly experience, most people would agree that firms are hierarchically organized, and that pay tends to increase as one moves up the hierarchy. But how this hierarchical structure affects income distribution has not been widely studied. To remedy this situation, this paper presents a new model of income distribution that explores the effects of social hierarchy. This ‘hierarchy model’ takes the limited available evidence on the structure of firm hierarchies, and generalizes it to create a large-scale simulation of the hierarchical structure of the United States economy. Using this model, I conduct the first quantitative investigation of hierarchy’s effect on income distribution. I find that hierarchy plays a dominant role in shaping the tail of US income distribution. The model suggests that hierarchy is responsible for generating the power-law scaling of top incomes. Moreover, I find that hierarchy can be used to unify the study of personal and functional income distribution, as well as to understand historical trends in income inequality. 4.1 Introduction The field of income distribution modeling is in need of new ideas. Ever since Pareto [1]discovered the power law scaling of top incomes and wealth, theorists have sought generative models for creating income distributions. In this regard, the field has been wildly successful. An impressive array of models now exist that can generate, from simple principles, observed distributions of income [2–21]. The problem is that this outward empirical success masks underlying assumptions that often have little to do with reality. To echo Leontief, “what is really needed, in most cases, is a very difficult and seldom very neat assessment and verification of these assumptions in terms of observed facts” [22]. A Hierarchy Model 99 Income Distribution Inter-Firm Intra-Firm Inter-Hierarchical Level Intra-Hierarchical Level Inter-Firm Inter-Hierarchical Level Intra-Hierarchical Level Figure 4.2: A Tripartite Division of Income Distribution This figure illustrates the income distribution grouping scheme used by the hierarchy model. The model allows for three sources of income dispersion. Inter-firm dispersion consists of differences in (average) pay between firms. Within each firm, there are two further sources of dispersion. Inter-hierarchical level dispersion consists of differences in (average) pay between hierarchical levels, while intra-hierarchical level dispersion consists of differences in pay within each hierarchical level. from regressions on case study data, in conjunction with firm-level data from the Compustat and Execucomp databases. Modeling the United States The model is designed to study the hierarchical structure of the US economy as it was (on average) over the years 1992-2015. At the highest level of abstraction, the model has three parts. First, the model creates a firm size distribution A Hierarchy Model 100 that dictates how many firms of a given size will exist. Second, for each firm in this distribution, the model creates a hierarchical structure. This means the model determines how many ranks will exist, and how many individuals will occupy each hierarchical rank. Lastly, the model uses each of the three dispersion sources (outlined above) to stochastically generate an income for every individual in every firm. In a sense, everything else amounts to details about how each of these steps is carried out. I review here the most important elements of each step. A technical discussion can be found in the Appendix. Step 1: Create a Firm Size Distribution. The first step of the model is to generate a distribution of firm sizes. The available evidence suggests that national firm size distributions can be modeled by a power law [68–70]. Under this assumption, the probability of finding a firm of size xis proportional to x−α, where αis a constant. I model the United States firm size distribution with 1 million firms distributed according to a discrete power law distribution with exponent α=2.01 (see Appendix C.5). Step 2: Endow Firms with Hierarchical Structure. The hierarchy model captures only the aggregate hierarchical structure of firms. That is, I model the number of employees in each hierarchical level, not the exact chain of command. I base the model on a number of recent case studies that have documented the aggregate hierarchical structure of firms in various developed countries (see Appendix C.2). From this data, I make generalizations about the hierarchical structure of firms. The evidence suggests that the span of control (the ratio between adjacent hierarchical levels) increases with rank. I model this increase with an exponential function. For simplicity, all firms in the model have the same hierarchical structure — that is, they are governed by the same span of control function. However, since there is a great deal of uncertainty in this function, I run the model many times. Each different model run uses a slightly different span of control function, determined by resampling from case study data. The result is that the hierarchical structure of firms varies stochastically between different model runs, allowing us to capture uncertainty in the underlying empirical data. For more details, see Appendix C.4 and C.5. Step 3: Endow Individuals with Income After each firm has a hierarchical structure, we begin the most important part of the model, which is to assign ev- A Hierarchy Model 101 ery individual an income. Because the model has three dispersion mechanisms, this last step has three components, outlined below. Step 3A: Generate Inter-Hierarchical Level Dispersion. In the model, hierarchical pay is constructed from the bottom up. Starting from the bottom rank, I define a function that determines the rate at which pay increases by hierarchical rank. This function is informed by case study data (see Appendix C.2). Unlike hierarchical employment structure, each modeled firm is given a different hierarchical pay structure. The process of assigning different hierarchical pay structure to each firm is heavily informed by firm-level data in the Compustat database. (See Appendix C.3 for a detailed discussion of the Compustat data). The basic idea is this: before running the full simulation, I fit the hierarchy model to Compustat data for real-world American firms. Compustat (in conjunction with Execucomp) provides data on CEO pay, average pay, and firm employment. Assuming the CEO occupies the top hierarchical level, we can use this information to model the hierarchical pay structure of each Compustat firm. Once this is complete, we have an indication of how hierarchical pay should vary across firms. The model’s main simulation is then informed by this variation. The result is a unique hierarchical pay structure for each firm. For more details, see Appendix C.4 and C.5. Step 3B: Generate Inter-Firm Dispersion. I create inter-firm income dispersion by varying (average) pay in the bottom hierarchical level of each firm. This variation is informed by firm-level data in the Compustat database. As discussed in Step 3A, prior to running a full-scale simulation, I fit the model to firms in the Compustat database. After having fit hierarchical pay, I use this information to estimate how base-level pay varies across these firms. This variation then informs the model’s main simulation. For more details, see Appendix C.4 and C.5. Step 3C: Generate Intra-Hierarchical Level Dispersion. The last step is to model the income dispersion within the hierarchical levels of each firm. The available case study evidence suggests that income dispersion within hierarchical levels is roughly constant across all hierarchical levels (see Appendix C.2). To simplify the model, I further assume that intra-hierarchical level dispersion is constant across all firms. Informed by case study data, I use a single parameterized distribution to randomly generate income dispersion within all hierarchical levels of every firm. For more details, see Appendix C.4 and C.5. A Hierarchy Model 102 Figure 4.3: A Landscape View of the Hierarchy Model This figure visualizes the US hierarchy model as a landscape of three dimensional firms. Each pyramid represents a single firm, with size indicating the number of employees and height corresponding to the number of hierarchical levels. If you look closely, you will see vertical lines corresponding to individuals. Income (relative to the median) is indicated by color. This visualization has 20,000 firms — a small sample of the actual model, which uses 1 million firms. Visualizing the US Hierarchy Model From the brief discussion here (or even from the technical discussion in the Appendix), it is not easy to gain an intuitive understanding of what the model ‘looks like’. To aid with such an intuitive understanding, Figure 4.3 shows a ‘landscape’ view of the model’s structure. Each pyramid represents a different hierarchically organized firm. The size of each pyramid corresponds to the number of employees, height represents hierarchical level, and color represents relative income. Figure 4.3 nicely highlights the main characteristics of the model. The firm power law distribution is clearly visible. The vast majority of firms are small, but there are a few behemoths. Inter-firm income dispersion and inter-hierarchical level income dispersion are also visible, while intra-hierarchical level income A Hierarchy Model 103 dispersion appears negligible. Lastly, top incomes are concentrated in upper hierarchical levels, and consequently occur mostly in larger firms. These facts, which are qualitatively visible here, become even more clear as we analyze the model results in quantitative terms. 4.2.2 Testing the Hierarchy Model (Part 1) The purpose of the hierarchy model is to study the hierarchical structure of the United States economy. The first step, then, is to make sure that the model produces realistic results. To that end, Figure 4.4 compares the model’s aggregate structure to US empirical data. Even though the model is an extrapolation from a limited set of data, it does a reasonably accurate job of reproducing US distribution of income. A few things are obvious from this comparison. Firstly, the model underestimates US income inequality, both in terms of the Gini index (Fig. 4.4A) and the income share of the top 1% (Fig. 4.4B). What is the source of this discrepancy? Looking at the income probability density in Figure 4.4D, it appears that the US income distribution is more ‘bottom heavy’ than the model. That is, the model produces too few extremely small incomes, relative to the US. This tendency is also evident in the cumulative distribution (Fig. 4.4F). Why does this discrepancy occur? I demonstrate in Appendix C.6 that the discrepancy can be removed by increasing the model’s inter-firm income dispersion. This suggests that the model’s under-estimate of US inequality is due to an under-estimate of inter-firm income dispersion. My guess is that this occurs because the model is based on Compustat firm data, which is not a representative sample of the US firm population. Compustat contains data for public firms only, and as a result, is biased towards large firms. I suspect that a more representative firm sample would give greater inter-firm income dispersion. (It is also possible that the model-empirical discrepancy results from some factor that is not included in the model. The most plausible would be unemployment, but many others are possible). I include adjusted results in the Appendix to show that the model is capable of closely reproducing the important features of US income distribution (as any well-parameterized model should be). I do not, however, use this adjusted data for any of the proceeding analysis. The purpose of the model is to extrapolate empirical data, warts and all. While the model slightly misrepresents the ‘body’ of US income distribution, it accurately reproduces the tail. This is evident in the complementary cumula- A Hierarchy Model 104 0.40 0.45 0.50 0.55 0.60 Model US (CPS) US (IRS) Gini Index A. Gini Index 0.10 0.15 0.20 0.25 Model US Fraction of Total Income B. Top 1% Income Share 2.25 2.50 2.75 3.00 Model US α C. Power Law Exponent 0.0 0.5 1.0 1.5 0.00 0.25 0.50 0.75 1.00 0 1 2 3 4 0.00 0.25 0.50 0.75 1.00 Normalized Income (mean = 1) Income Percentile Density Cumulative Fraction of Income D. Probability Density E. Lorenz Curve 0.00 0.25 0.50 0.75 1.00 10−4 10−3 10−2 10−1 100 0.01 0.1 1 10 100 0.01 0.1 1 10 100 Normalized Income (mean = 1) Normalized Income (mean = 1) Cumulative Proportion Cumulative Proportion F. Cumulative Distribution G. Complementary Cumulative Distribution Model Median United States Median Model 95% Range United States Range (1994−2015) Figure 4.4: Modeled Income Distribution vs. US Data This figure compares various aspects of the model’s income distribution to US data over the years 1992-2015. Panel A shows the Gini index, with two different US sources — the Current Population Survey (CPS) and the Internal Revenue Service (IRS). Panel B shows the top 1% income share, using data from 17 different time series. Panel C shows the results of fitting a power law distribution to the top 1% of incomes (where αis the scaling exponent). Panel D plots the income density curve with mean income normalized to 1 (using data from the CPS). Panels E, F, and G use IRS data to construct the Lorenz curve, cumulative distribution, and complementary cumulative distribution (respectively). The cumulative distribution shows the proportion of individuals with income less than the given xvalue. The complementary cumulative distribution shows the proportion of individuals with income greater than the given xvalue. Note the log scale on the x-axis for these last two plots. For sources and methods, see Appendix C.1. A Hierarchy Model 105 tive distribution (Fig. 4.4F) in the form of virtually identical model and empirical slopes in the right tail. How can these slopes be quantified? One way is to fit the tail of the income distribution to a power law — a method that dates back to the work of Pareto [1]. This approach provides a way of analyzing the tail of the income distribution independently from the body. Under a power law distribution, the probability of finding someone with income xis proportional to x−α, where αis a constant (the power law exponent). The approximate power law scaling of top incomes is visible as the straight line in the tail of the complementary cumulative distribution (when plotted on a loglog scale). The choice of where the distribution ‘tail’ begins is arbitrary. I define the tail as the top 1% of incomes — a threshold that has been popularized by Piketty [71]. Figure 4.4C shows the results of fitting a power law to the top 1% of incomes (for methods, see Appendix C.1). The model produces power law exponents that are statistically indistinguishable from those found in the US data. Both a Kolmogorov–Smirnov test and a t-test indicate no significant differences (at the 5% level) between the model and empirical results. To conclude, the model produces an income distribution that is roughly consistent with the US distribution of income. In particular the model closely reproduces the tail of the US distribution. 4.2.3 Testing the Hierarchy Model (Part 2) When discussing the model visualization shown in Figure 4.3, I noted that large incomes appear to be clustered at the tops of large firms. This is a defining feature of the hierarchy model. It occurs because income scales strongly with hierarchical rank. As a result, top earners are found at the tops of large firms, because these firms have the most hierarchical levels. This prediction is not made by any other model of income distribution (to my knowledge). It is important, therefore, that we put it to the test. To test this prediction, I look at the distribution of firm sizes associated with top earning individuals. What does this mean? I take a sample of Americans with top incomes, and then record the firms with which these individuals are associated. I then look at the size distribution of these firms. I do the same with the model, and compare the results. I conduct this test using data from the Forbes 400 and Execucomp. The Forbes 400 list is useful because it is a definitive ranking of the 400 richest Americans, and it provides the institutional source of each individual’s wealth. The caveat is that this list is a ranking by wealth, not income. I use the Forbes 400 as A Hierarchy Model 106 a proxy for top US incomes, under the assumption that wealth and income are strongly related. I supplement the Forbes 400 data with the ‘Execucomp 500’. The latter is composed of the 500 top paid US executives (in each year between 1992-2015) in the Execucomp database. The advantage of the Execucomp 500 is that it is a ranking explicitly by income. The disadvantage is that we do not know if these 500 executives are actually the top paid US individuals. Before discussing the results of this test, it is instructive to know what a nulleffect would look like. If there is absolutely no relation between income and firm membership, what sort of firm size distribution should be associated with top incomes? It turns out that for the United States, we should expect a null-effect to return a roughly log-uniform distribution (see Appendix C.7 for a derivation). Results for the Fortune 400 and Execucomp 500 firm size distributions are shown in the main panel of Figure 4.5. To be clear, these density plots represent the size distribution of firms associated with the richest 400 Americans and the 500 top paid executives in the Execucomp database (respectively). To better visualize the distribution, I plot the density of the logarithm of firm size. Under this transformation, the null-effect result will appear as a uniform distribution. From the evidence shown in Figure 4.5, we can immediately conclude that the null-effect is false. There is definitely a relation between top incomes (wealth) and firm size. But is it the relation that is predicted by the hierarchy model? To find out, I conduct the same analysis on the model. I select the model’s 500 top paid individuals and record the size distribution of associated firms. The results are shown in Figure 4.5 as the ‘Model 500’. The model predicts a relation between top incomes and firm size that is very similar to the US empirical data. To be sure, the model results are not identical to either the Forbes 400 or the Execucomp 500 distributions. But, given the paucity of data on which the model is based (as well as the general uncertainty in the empirical analysis of top incomes), I count this result as a success. The model produces results that are roughly consistent with the US data. Since the model has three sources of income dispersion, we naturally want to know which of these sources is responsible for producing the results in Figure 4.5. To answer this question, I use a counterfactual analysis. I create three different counterfactual models to supplement the original (Model A). Each counterfactual model isolates a single source of dispersion as it appears in the original model. Model B has intra-hierarchical dispersion only, Model C has inter-firm dispersion only, and Model D has intra-hierarchical level dispersion only. The results of this counterfactual analysis are shown in the right-hand panels in Figure 4.5. This analysis indicates that it is exclusively inter-hierarchical in- A Hierarchy Model 107 Log−Uniform (Null Effect) 0.0 0.1 0.2 0.3 0.4 0.5 0123456 log(firm size) Density Model 500 Forbes 400 Execucomp 500 Model A vs. Empirical Data Model D Model C Model B 0246 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 log(firm size) Counterfactual Models Model A =original model Model C =inter−firm dispersion only Model B =inter−hierarchical dispersion only Model D =intra−hierarchical dispersion only Figure 4.5: Firm Size Distributions Associated With Top Incomes and Wealth This figure shows the size distribution of firms associated with top earning individuals in the US and in the hierarchy model (of the US). The ‘Forbes 400’ represents the size distribution of firms associated with (owned by) the wealthiest 400 Americans in the year 2014. The ‘Execucomp 500’ represents the size distribution of firms associated with the 500 top earning American executives (in each year from 1992-2015) in the Execucomp database. The ‘Model 500’ represents the size distribution of firms associated with the 500 top earning individuals in the hierarchy model. Results for counterfactual models are shown on the right. Each counterfactual model isolates a single source of income dispersion. Model B shows inter-hierarchical dispersion only, Model C shows inter-firm dispersion only, and Model D shows intra-hierarchical level dispersion only. In all plots, I also show the log-uniform distribution (dotted line), which is predicted if there is no relation between firm membership and income. For sources and methods, see Appendix C.1. A Hierarchy Model 108 come dispersion (Model B) that is responsible for associating top incomes with large institutions. How do we know this? The inter-hierarchical dispersion model (B) produces results that are virtually identical to the original model. At the same time, inter-firm dispersion only (Model C) and intra-hierarchical level dispersion only (Model D) produce drastically different results. Note that with intra-hierarchical dispersion only (Model D), we recover the null-effect (a log-uniform distribution). Why? In this model, firms play no part in determining income. (Income for all individuals is determined by a single stochastic function). Interestingly, this is a world that is implied by many models of income that focus solely on interactions between individuals [2,3,5,6,10–13, 15,16,20]. In these models, there are no firms. The implicit assumption must be that firms play no role in the distribution of income. Given the evidence in Figure 4.5, it would seem that these models need rethinking. To conclude, the hierarchy model correctly predicts that top paid individuals should be associated with firms that are far larger than those of the general population. Moreover, the model indicates that this effect is purely a result of inter-hierarchical pay dispersion. 4.2.4 Quantifying Hierarchy’s Effect on Income Distribution Having established that the hierarchy model gives credible results, I now use it to investigate how hierarchy affects US income distribution. I isolate the effects of hierarchy by creating three different counterfactual version of the United States. Each version contains only one of the three sources of income dispersion used in the original model. By comparing these counterfactual models to the original model, we can determine how each dispersion source affects income distribution. Let’s begin with a seemingly simple question: how does hierarchy affect income inequality? The results in Figure 4.6 indicate that this question is not so simple. The affect seems to depend on how we measure inequality. Let’s begin by using the the Gini index (Figure 4.6A). Here we see that the model with interfirm dispersion has a Gini index that is closest to the original model. (The model with inter-hierarchical dispersion comes a distant second). This result suggests that hierarchy does not have a particularly strong effect on inequality. However, things change drastically when we switch to measuring inequality in terms of the income share of the top 1% (Fig. 4.6B). Now we find that the model with inter-hierarchical dispersion has inequality that is nearly identical A Capitalist Gradient Hypothesis 115 For the present argument regarding the basis of capitalist income, I set aside the question of how the firm’s income stream is derived. Instead, I am interested in how an owner wields power to partition a firm’s income stream. The central hypothesis in this paper is that firms are hierarchically organized. This hypothesis implies that ownership confers the right to sit at the top of the firm hierarchy. From this position of hierarchical power (as owner), the capitalist has the authority to divide up the firm’s income stream. This suggests that capitalist income stems from hierarchical power. This vision is illustrated in Figure 4.9. While this vision is intuitive (at least to me), it is almost certainly too simplistic. The problem is that it is based on a 19th century, all-or-nothing concept of ownership. In this vision, a capitalist is the owner of a firm. Unfortunately, the rise of joint-stock companies muddies this tidy theory. Joint-stock companies allow ownership to by divided among many people. In the modern world, partial ownership is the rule. This realization led to the famous ‘separation thesis’ posited by Berle and Means [80]. The idea is that ownership has become so diffuse that capitalists no longer control the corporate hierarchy. Instead, control is ceded to managers, who are employees. The problem with the separation thesis is that it acknowledges the rise of partial ownership, but insists on a traditional dichotomy between capitalists and laborers. The truth is that the line between being a capitalist and being a laborer has been blurred. Top managers often earn a large portion of their income from stock options. Conversely, owners of firms often pay themselves some form of salary. Instead of a capitalist-laborer dichotomy, what we need is a capitalistlaborer gradient. This implies that there is a steady range between being purely a capitalist and being purely a laborer. Figure 4.10 shows what this might look like when applied to a hierarchy. As one moves up the hierarchy, individuals become increasingly more capitalistic. This capitalist gradient hypothesis can be interpreted a number of ways. The simplest interpretation is to assume a gradient of ownership within a single firm. However, this is realistic only for firms that are 100% employee owned. While such firms do exist (and can become quite large), they are not the norm. It is more common for a firm to have partial employee ownership via an employee stock ownership plan. In 2017, about 14 million Americans were enrolled in employee stock ownership plans (ESOP) [81]. This represents about 9% of the workforce. It is quite plausible that these employee stock options are preferentially rewarded to the top tiers of the hierarchy. However, ESOP assets constitute a small minority (roughly 4%) of total US market capitalization.4This means A Capitalist Gradient Hypothesis 116 Capitalist Capitalist Income Labor Income Capital Figure 4.9: A Hierarchical Power Vision of Capitalist Income This figure shows my interpretation of the capital as power framework, when applied to a hierarchically organized firm. Unlike in neoclassical and Marxist visions of capital (Fig. 4.7 and 4.8, respectively) I do not show physical capital. This is not to say that physical capital does not exist — we simply do not focus on it. Rather, we focus on ownership of institutions. Capital is conceived solely in terms of property rights. By purchasing a firm, a capitalist purchases the legal right to sit at the top of the firm hierarchy. From this position of power, the capitalist has the right to divide up the firm’s income stream as he sees fit. Under this vision, hierarchical power is the source of capitalist income. Capitalist Laborer Figure 4.10: A Gradient Vision of Capitalist Income This figure shows a more nuanced (than Fig. 4.9) interpretation of the relation between capitalist income and firm hierarchy. In this model, there is a smooth gradient between being 100% capitalist (earning all your income from capitalist sources) and being 100% laborer (earning all your income from labor sources). I hypothesize that the capitalist share of individual income tends to increase with hierarchical power. A Capitalist Gradient Hypothesis 117 they are probably not the main source of capitalist income. Therefore, it is most realistic to interpret the gradient model as a statistical phenomenon that occurs at the societal level. We admit that the ownership structure of any given firm is likely complex. Similarly, we admit that individuals who earn capitalist income may receive it from a variety of firms. But at the aggregate level, we hypothesize that earning capitalist income is related to hierarchical class structure. This is the hypothesis that I test. 4.3.3 Measuring Hierarchical Power To test the capitalist gradient hypothesis, we need to measure hierarchical power. What is hierarchical power? I define it as the ability to control subordinates within a hierarchical chain of command. Unlike the more general concept of ‘social power’, hierarchical power is easier to pin down and quantify. This is because the chain of command structure of a hierarchy clearly delineates who has control over whom. A hierarchy is nothing but a nested set of power relations between superior and subordinates (ruler and ruled). It is a control structure that concentrates power at the top [48]. I propose that one’s power within a social hierarchy is proportional to the number of subordinates under one’s control. I put this in formula form as: hierarchical power =number of subordinates +1 (4.1) The logic of this equation is that all individuals start at a baseline power of 1, indicating that they have control over themselves. Power then increases linearly with the number of subordinates. Figure 4.11: Measuring Hierarchical Power 4In 2017, ESOPs had total assets of roughly $1.3 trillion [81], while total US market capitalization was roughly $30 trillion, according to the Russel 3000 index. A Capitalist Gradient Hypothesis 118 As an example, suppose we want to find the hierarchical power of the red individual in Figure 4.11. This person has two direct subordinates, each of whom have 2 subordinates. Thus the red individual has control over 6 subordinates in total, mean his/her hierarchical power is 7. The general form of a branching hierarchy means that hierarchical power increases exponentially with rank. 4.3.4 Testing the Capitalist Gradient Hypothesis (Part 1) If the capitalist gradient hypothesis is correct, we should be able to find evidence that capitalist income fraction increases with hierarchical power. I test the gradient hypothesis using CEO income data. This data is convenient for two reasons. First, CEO income data is easy to obtain. US regulation requires that public companies disclose CEO compensation. Second, we can estimate a CEO’s hierarchical power without any knowledge of the firm’s hierarchical structure. Under the assumption that the CEO holds the top hierarchical position in a firm, it follows that their hierarchal power is equivalent to the number of employees in the firm. This thinking is visualized in Figure 4.12. If a firm has xemployees, x−1 of them will be subordinate to the CEO. Since hierarchal power is defined as the number of subordinates plus one, the CEO’s hierarchical power is simply firm size x. Thus, if we have data for firm size, we automatically have data for CEO hierarchical power. So how do we calculate the ‘capitalist’ component of CEO income? I define the CEO capitalist income fraction as the portion of total income received from stock options: CEO Capitalist Income Fraction =Income from Stock Options Total Compensation (4.2) Unlike cash compensation, there are many different ways to value stock options [82–84]. This means that CEO capitalist income fraction has some inherent ambiguity. However, the nuances of stock option valuation do not concern me here. Instead, I am interested in general trends in CEO compensation. For this task, the standard methods for stock option valuation will do just fine. I use CEO income data from the Execucomp database. The data series and their underlying methods are discussed in Appendix C.3. Figure 4.13 shows the resulting relation between capitalist income fraction and firm size for roughly 40,000 American CEOs over the years 1992-2015. Two important findings emerge. Firstly, the capitalist fraction of CEO income tends A Capitalist Gradient Hypothesis 119 31 15 7 3 1 Hierarchical Power 13115 7 3 Firm Size = CEO Figure 4.12: CEO Hierarchical Power This figure shows the relation between firm size and CEO hierarchical power. Each hierarchy represents a different firm, with the CEO at the top (red). If hierarchical power is defined as the number of subordinates +1 (Eq. 4.1), CEOs have hierarchical power equal to firm size. Modeled Trend 0 10 20 30 40 50 60 70 80 90 100 100101102103104105106 Firm Size (Employees) = CEO Hierarchical Power Capitalist Income (% of Compensation) P25−P75 P50 (Median) Figure 4.13: Capitalist Income Fraction of US CEOs This figure plots the relation between capitalist income fraction and firm size for roughly 40,000 American CEOs over the years 1992-2015. Assuming that CEOs sit at the top of the corporate hierarchy, firm size is a direct indicator of CEO hierarchical power. The median (P50) and interquartile range (P25-P50) for capitalist income fraction are calculated using logarithmically spaced firm-size bins. The dashed line indicates the linear regression used for modeling purposes. Data comes from Execucomp and Compustat. For methods, see Appendix C.3. A Capitalist Gradient Hypothesis 120 to increase with firm size (and hence hierarchical power). Secondly, capitalist income fraction tends towards zero for CEOs in very small firms (fewer than 10 employees). These results are consistent with the capitalist gradient hypothesis — they support the idea that earning capitalist income is a gradient function of hierarchical power. 4.3.5 Testing the Capitalist Gradient Hypothesis (Part 2) The evidence from US CEOs begs a question: does the relation between CEO capitalist income fraction and hierarchical power generalize to the broader US population? While data constraints stop us from answering this question directly (which is why we turned to CEO data in the first place), we can answer it indirectly by using the hierarchy model. I do this by using the CEO data to create a simple function relating capitalist income fraction to hierarchical power. Once I have this function, I plug it into the hierarchy model and endow each individual with a capitalist income. I then check the model’s results against US data. If the model produces results that are way off the mark, we know that the CEO results do not generalize to the whole population. However, if the model produces results that are consistent with US data, this is indirect evidence that capitalist income fraction increases with hierarchical power in the wider US population. The first step is to idealize the Figure 4.13 trend between CEO capitalist income fraction and hierarchical power. The simplest interpretation of this trend is that CEO income fraction increases linearly with the logarithm of hierarchical power. I fit the CEO data with a one-parameter logarithmic function, resulting in the ‘Modeled Trend’ line shown in Figure 4.13. This gives the following function relating capitalist income fraction (Kfrac) to hierarchical power (P):5 Kfrac =0.05ln(P)(4.3) This function is naive in the sense that it implies a deterministic relation between hierarchical power and capitalist income fraction — something that certainly does not exist in the real world. However, models are always simplifications, and it is often useful to simplify a noisy (stochastic) trend with a 5The discerning reader may note that, since a logarithmic function is uniformly increasing, Eq. 4.3 permits capitalist income fraction greater than 1. In practice, such results do not occur because the model does not permit firm sizes greater than 2.3 million — the largest US firm that has ever existed (Walmart, circa 2015). For this maximum hierarchical power of 2.3 million, Eq. 4.3 yields a capitalist income fraction of about 0.7. A Capitalist Gradient Hypothesis 121 Figure 4.14: A Landscape View of the Capitalist Gradient Model This figure visualizes the capitalist gradient model as a landscape of firms. Each pyramid represents a firm, with size indicating the number of employees. Hierarchical rank is indicated by height, and capitalist income fraction by color. deterministic one. If the results are good, we can add more realism later. If the results are bad we throw away the model. The next step is to plug this equation into the hierarchy model. We calculate the hierarchical power of each individual in the model (see Appendix C.4) and then use Eq. 4.3 to calculate the capitalist fraction of their income. The resulting capitalist gradient model is visualized in landscape form in Figure 4.14. As expected, capitalist income fraction is tightly related to hierarchical rank. If the CEO capitalist income trend is generalizable, the capitalist gradient model should produces results that match US data. So does it? Figure 4.15 compares the model to the United States. Let’s begin with the relation between capitalist income and total income size. This is effectively the relation between personal and functional income distribution — something that I have proposed that hierarchy can unify. Figure 4.15A plots Thomas Piketty’s data showing how US capitalist income fraction increases with income percentile [71]. As illustrated by the inset plot (which uses a linear x-axis scale), there is an explosion A Capitalist Gradient Hypothesis 122 US 2007 Without Capital Gains US 2007 With Capital Gains Model 0 25 50 75 90 95 100 Income Percentile Linear Scale 0.0 0.5 1.0 0 50 100 150 200 Linear Scale 10 20 30 40 50 60 70 10−6 10−5 10−4 10−3 10−2 10−1 100 0.0010.010.1110 1 2 5 10 20 50 100 200 Top Income Percentile (%) Income (Thousands $) Capitalist Income Fraction (%) Fraction of Individuals Model (Median) United States (Median) Model 95% Range United States Range (1990−2014) A. Capitalist Income Fraction vs. Income Percentile B. Size Distribution of Capitalist Income Capitalist Income 0.6 0.7 0.8 0.9 1.0 Model US Gini Index C. Gini Index Capitalist Income 0.2 0.3 0.4 0.5 0.6 0.7 Model US Top 1% Share D. Top 1% US Data Covers Years 1992−2014 2 4 6 8 10 12 Model US Capitalist US Dividends US Interest % of Total Income E. Capitalist Share of Total Income Figure 4.15: Comparing the Capitalist Gradient Model to US Data This figure compares the income distribution generated by the capitalist gradient model to US data. Panel A shows how capitalist income fraction increases with income percentile (ranked by total income). The inset plot uses a linear x-axis scale, while the main plot uses an inverted logarithmic scale of top incomes. Note that US empirical data has ‘steps’ that correspond to the bins in the source data. The blue line and shaded regions indicate the model’s median and 95% range, respectively. For panels B, C and D, US capitalist income is defined as the sum of income from dividends and interest. Data covers the years 1990 - 2014. Panel B shows the size distribution of capitalist income. The model data is normalized to have mean income in the same range as the US data. Panel C shows the inequality of capitalist income, as measured by the Gini index, while Panel D shows capitalist income inequality as measured by the income share of the top 1%. Panel E shows the capitalist share of total (national) income. For comparison, I also show the dividend and net interest share of US income. For sources and methods, see Appendix C.1. A Capitalist Gradient Hypothesis 123 of capitalist income that occurs in the topmost income percentiles. Evidently, those who earn very large incomes are overwhelmingly capitalists (and vice versa). The main panel spreads out this explosion by using an inverted logarithmic x-axis scale. Two different US trend-lines are shown. The upper line includes capital gains in the calculation of capitalist income, while the lower line does not. (The step-wise nature of these curves reflects Piketty’s income bins.) Like the US data, the capitalist gradient model predicts an explosion in capitalist income amongst top earners. Moving on, Figure 4.15B shows the size distribution of US capitalist income. For this graph (as well as Fig. 4.15C, D and E ), I define capitalist income as the sum of income from dividends and interest. Although many people do earn some capitalist income, the amount is usually inconsequentially small. This fact is reflected in the inset panel, which plots the capitalist income distribution on alinear scale. Nearly all reported capitalist incomes are lower than $5000. In order to see the tail of the distribution, the main plot uses a log-log scale. Again, the model is consistent with US data. To get these results, I do nothing but index the model data so it has the same mean as US data. Without tuning it to do so, the model effectively reproduces the tail of US capitalist income distribution. How about capitalist income inequality? Figure 4.15C and D show the Gini index and top 1% share of capitalist income, respectively. Just to be clear, the latter metric captures the share of total capitalist income held by the top 1% of reported capitalist incomes. First off, note how unequal US capitalist income is. The Gini index hovers around 0.9 (the maximum is 1), while the top 1% of capitalists earn about 40% of total capitalist income. The model reproduces this staggering income share of the top 1%, but falls short with the Gini index. Why? Part of the problem can be seen in Figure 4.15B — the model produces slightly too many capitalist incomes between $2000 to $5000. However, the primary problem has to do with the function used to determine capitalist income (Eq. 4.3). Capitalist income is assumed to increase linearly with the logarithm of hierarchical power. Since log(1) = 0, all individuals with a hierarchical power of 1 (the lowest amount possible) will have exactly zero capitalist income. When calculating inequality, these null incomes are (by convention) excluded. If we adjust the model slightly so that instead of having no income, these individuals have a tiny capitalist income, we get Gini index results that match US data. See Appendix C.6 for more details of this adjustment. Lastly, Figure 4.15E shows the capitalist share of total (national) income. The model produces a capitalist income share that is slightly lower than (but in a similar range as) the US data (from 1992-2014). For future reference, I also A Capitalist Gradient Hypothesis 124 include the individual components of US capitalist income. (In section 4.4, I model historic trends in the dividend share of national income). To summarize, the capitalist gradient model produces results that closely match US empirical data. This is indirect evidence suggesting that capitalist income fraction scales with hierarchical power in the general US population. 4.3.6 Property, Power, and Income The results shown in Figure 4.13 and 4.15 are preliminary, and should be treated with appropriate uncertainty. That being said, I want to reflect on their potential significance. In effect, the capitalist gradient model connects three things. It suggests that hierarchical class structure, ownership class structure, and personal income distribution are all related. Put another way, hierarchical elites, capitalists, and top earners are all the same people. What are we to make of this hypothesized relation between authority, property rights, and income? One interpretation is that it is nothing new. Suppose, when speaking about a feudal society, I stated that hierarchical elites, aristocrats, and the very rich are all the same people. This would be nothing particularly controversial. We are quite comfortable concluding that historical societies had a ruling class [85]. But many would bristle at that thought in our own society. Yet consider Reinhard Bendix’s description of the relation between authority, property rights, and income in German feudal society. He writes: governmental functions were usable rights which could be sold or leased at will. For example, judicial authority was a type of property. The person who bought or leased that property was entitled to adjudicate disputes and receive the fees and penalties incident to such adjudication. [86](p. 149) If we paraphrase Bendix, we arrive at the same reasoning that I used to derive the capitalist gradient hypothesis. Building on the work of Nitzan and Bichler, I suggested that ‘capitalist authority’ is a ‘type of property’. The person who buys this property is ‘entitled’ to wield hierarchical power and ‘receive income’ in return. From this reasoning came the hypothesis that capitalist income should be related to hierarchical rank and power. From the perspective of mainstream economic theory, this hypothesis is quite radical. It undermines the ubiquitous assumption that capitalists earn income from a productive asset. But given Bendix’s comments on feudal society, the capitalist gradient hypothesis may be quite conservative. Why? Conservatism implies a lack of change — a maintenance of the same order. The capitalist gradient hypothesis may be conservative because it suggests that income distribution A Hierarchical Redistribution Hypothesis 131 is tremendous uncertainty in this relation. More empirical research is needed to understand the source of this model discrepancy. 4.4.4 Discussion While we should always be cautious about drawing conclusions from a model, I want to offer my thoughts on the significance of these results. There has been a tendency, in political economy, to explain human income distribution in terms of ‘natural law’. For instance, John Bates Clark began his foundational text on marginal productivity by declaring: “It is the purpose of this work to show that the distribution of the income of society is controlled by a natural law”[75]. This tendency was only strengthened when Pareto discovered the ubiquitous power law scaling of top incomes [1]. But what is curious about ‘natural law’ theories is that they are almost always atomistic. Thus, Clark showed that perfectly competitive markets distribute income according to ‘natural law’. But leviathan governments are mysteriously absent from this picture. In a sense, the term ‘natural law’ is used as a euphemism for ‘in the absence of concentrated power’. Thus, ‘natural law’ explanations of skewed income distribution tails are typically based on atomistic premises, in which there are isolated individuals but no institutions [5,18]. From this perspective, power is a distortion. But what if concentrated power is the reason that income distribution has a power law tail? This is the story told by the hierarchy model. This model suggests that hierarchy — a form of concentrated power — is responsible for producing the fat tail of US income distribution. The same model suggests that changes in the tail are a result of a hierarchical redistribution of pay. Thus, hierarchy provides a potentially potent tool for understanding both the regularities of income distribution over time and space, but also the variation. I propose that the regularity of power-law income distribution tails owes to the ubiquity of social hierarchy. Conversely, I propose that variation in the tail owes to hierarchical redistribution. I conclude by visualizing the hierarchical redistribution that has occurred in the United States (as suggested by the hierarchy model). Figure 4.19 shows two modeled versions of the United States. On top is the 1965 version. On the bottom is the 2015 version. The difference between the two is subtle — it is almost completely isolated to the tops of large firms. Here we see a massive, order of magnitude increase in relative pay — a clear redistribution of income to top-ranked individuals. If the model is correct, we can conclude that the US has A Hierarchical Redistribution Hypothesis 132 Figure 4.19: A Visualization of US Hierarchical Income Redistribution This figure shows the model’s representation of historical hierarchical income redistribution in the United States. The top model represents the US in 1965 while the bottom represent the US in 2015. I create these models by choosing the hierarchical pay-scaling parameter that best matches the US CEO pay ratio, top 1% and dividend share data in the year in question. The difference between the two model’s is mostly visible at the tops of large firms as an order of magnitude increase in the pay of top-ranked individuals. Conclusions: Modeling from the Top Down 133 undergone a massive hierarchical redistribution of income in the last 30 years. 4.5 Conclusions: Modeling from the Top Down Many economists have an understandable desire to model human society from from the ‘bottom up’ [7]. This means that they seek to explain complex social structures solely in terms of the interaction of individuals. The bottom up strategy is a noble one, in principle. It would be a triumph of science if we could explain macro-level income distribution based purely on the interactions of individuals. In the same way, it would be a triumph of science if we could understand the emergence of consciousness based purely on the interactions of atoms and molecules. This is a noble pursuit in principle. In practice, however, it is misguided. The problem is two-fold. The first problem is computational feasibility. Suppose we had a highly accurate model of the human psyche, comparable to the accuracy of quantum mechanics. If we did, it’s highly likely that meaningful questions would be computationally unfeasible. Even though it is the general scientific consensus that consciousness emerges from matter alone (i.e. there is no mind-body dualism) I know of no attempt to simulate consciousness using the laws of physics. The problem is simply too difficult. Quantum physics is so computationally complex that it is difficult to simulate large molecules, let alone brains. The second problem is that to build a model from the bottom up, we need a highly accurate model of the ‘fundamental particles’. We have a pretty good model of atoms. Do we have a good model of the human psyche? Hardly. I believe we should be humble and admit that we know very little about human behavior. As a consequence, when we model from the bottom up, we are essentially groping in the dark. We must make blind assumptions about how agents behave. The problem is that the entirety of the modeling effort depends on these assumptions. The model may very well give good results — it may seem to ‘explain’ the social phenomena in question. But if the underlying assumptions are incorrect, the entire model is wrong. The dream of explaining income distribution from the bottom up is a noble one. The problem is that we are hopelessly far from being able to do this the right way. The bottom up models that do exist make extremely naive assumptions about how humans behave. While these models give good results, it is a fallacy to think that this validates their underlying assumptions. The alternative to the bottom-up approach is to model from the top down. Conclusions: Modeling from the Top Down 134 What does this mean? Instead of having social structure emerge from the bottomup actions of individuals, we (the modelers) impose structure from the top down. In essence, we impose structure on society and then explore the consequences. The origin of this structure is left unexplored. The top-down approach is useful because it allows realism and ignorance to coexist. A realistic model of income distribution must have institutions — they are simply too important to ignore. But we know very little about how and why institutions form. The top-down approach allows us to model institutions without having any idea of why they exist. This is the philosophy that underlies the hierarchy model. The model is based on two observations of the real-world: (1) firms are the dominant institution for organizing paid human activity (in capitalist societies); and (2) firms are hierarchically organized. The model takes these facts as given, and explores their consequences. The central finding of the hierarchy model is that hierarchy shapes the tail of the income distribution. According to our model, it is hierarchy that causes the distinctive power-law scaling of top incomes. This is important because explaining the power-law distribution of top incomes has been one of the primary concerns of income distribution modelers. The over-whelming majority of power law generating models are based on atomistic premises. As far as I am aware, the hierarchy model is the only power law generating model that includes institutions. But this is not all. The hierarchy model is, to my knowledge, the only power law generating model that is completely empirically grounded. As I have stated many times, the hierarchy model amounts to an extrapolation of real-world evidence. The model takes the little information of firm hierarchy that does exist, and extrapolates it to create a large-scale simulation of the US economy. To risk overstating this, there is nothing in the model that is not implied by empirical data. The story that the hierarchy model tells is this: the power-law distribution of top incomes arises from concentrations of power. The model suggests that without large, hierarchically organized firms, there would be no power law distribution of top incomes. This finding is significant in its own right, but made more so by its stark contrast with mainstream, neoclassical economic theory. James T. Peach summarizes the neoclassical approach: “Individual productivity and exogenously determined shifts in supply and/or demand curves determine distributive shares. ... [T]here is no power and there is no income distribution problem” [57]. If the hierarchy model is correct, concentrated power is not an Conclusions: Modeling from the Top Down 135 0 250 500 750 1000 0 10 20 30 40 50 60 70 80 90 100 Income Percentile Ave. Hierarchical Power Linear Scale 1 10 100 1000 10 000 100 000 0 10 20 30 40 50 60 70 80 90 100 Income Percentile Average Hierarchical Power Figure 4.20: The Rich and Powerful — Hierarchical Power and Top Incomes This figure plots average hierarchical power (number of subordinates +1) against income percentile for individuals in the hierarchy model of the United States. The shaded regions indicates the 95% range, while the line indicates the median. In order to show the entire range of data, the main panel uses a logarithmic scale on the y-axis. The inset panel uses a linear y-axis to illustrate how rapidly hierarchical power increases in the top 1% of incomes. aberration — it is the norm. Based on the model results, I have suggested that power-law scaling of top incomes is ubiquitous because concentrated power (in the form of hierarchical institutions) is also ubiquitous. To put matters simply, the hierarchy model gives new meaning to the phrase ‘rich and powerful’. This is made clear by Figure 4.20. Here I plot average hierarchical power against income percentile for the hierarchy model of the United States. Two completely different populations emerge — those with power and those without. The vast majority of people have very little hierarchical power. But things change drastically for the small minority in the upper income percentiles. Here there is an explosion of hierarchical power. This power, I believe, is the origin of the great inequalities that plague human society (now and in the past). Hierarchical power gives preferential access to resources, plain and Conclusions: Modeling from the Top Down 136 simple. That being said, there is no fixed relation between income and hierarchical power. Gerhard Lenski [55]gives the curious example of Robert McNamara’s move from the Ford Motor Company to the position of US Secretary of Defense. McNamara’s new position had far more power, and yet his income did not increase. Instead, it decreased by an order of magnitude. Why? These are questions we must ask. Unlike Clark’s theory of marginal productivity, a theory of income distribution based on hierarchy and power has no ‘laws’. Things can and do change. To conclude, the hierarchy model is a first attempt at quantitatively studying the distributional consequences of hierarchical organization. If nothing else, the model suggest that hierarchy must be taken seriously — it is a grave mistake to ‘assume’ hierarchy away when building income distribution models. If we want to alleviate income inequality, we need to understand it. 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Hierarchical Structure and Pay within Case-Study Firms 243 to define the lognormal scale parameter σthat produces a distribution with an equivalent coefficient of variation, cv: σh=qln(c2 v+1)(B.18) Once we have σh, we use equation B.19 to calculate the lognormal location parameter µfor each hierarchical level . Here ¯ Ihis the mean pay in hierarchical level h(which Lima reports directly). µh=ln(¯ Ih)−1 2σ2 h(B.19) Once we have the appropriate lognormal parameters for each hierarchical level, we use these distributions to create a simulated payroll. To do this, we draw Ehnumbers (employment in level h) from each lognormal distribution lnN(µh,σh). I then calculate the Gini index from this simulated payroll. Treble et al. Treble et al. [12]report the following summary statistics, which I use to estimate a firm Gini index: 1. Employment within each hierarchical level (Fig. 2); 2. Mean pay within each hierarchical level (Fig. 3); 3. 5th and 95th wage percentile by hierarchical level (Fig. 4). Again, I use Engauge Digitizer to pull data from all graphs. To estimate the intra-level Gini index, I adapt code writtent by Andrie de Vries to fit a parameterized distribution to the mean and 5th/95th percentiles. A Hierarchical Model of the Firm 244 Table B.9: Notation Symbol Definition aspan of control parameter 1 bspan of control parameter 2 cvcoefficient of variation CCEO to average employee pay ratio Eemployment feither (1) a generic function; or (2) a probability density function Fcumulative distribution function GGini index of inequality hhierarchical level ¯ Iaverage income µlognormal location parameter nnumber of hierarchical levels in a firm ppay ratio between adjacent hierarchical levels rpay-scaling parameter sspan of control σlognormal scale parameter Ttotal for firm ↓round down to nearest integer Qproduct of a sequence of numbers Psum of a sequence of numbers B.3 A Hierarchical Model of the Firm In this section, I outline the mathematics underlying my hierarchical model of the firm. The model assumptions, outlined below, are based on the stylized facts gleaned from the real-world firm data in section B.2. Model Assumptions 1. Firms are hierarchically structured, with a span of control that increases exponentially with hierarchical level. 2. The ratio of mean pay between adjacent hierarchical levels increases exponentially with hierarchical level. A Hierarchical Model of the Firm 245 3. Intra-hierarchical-level income is lognormally distribute and constant across all levels. Using these assumptions, I first develop an algorithm that describes the hierarchical employment within a model firm, followed by an algorithm that describes the hierarchical pay structure. B.3.1 Generating the Employment Hierarchy To generate the hierarchical structure of a firm, 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≡   1 if h=1 Eh−1 Eh if h≥2(B.20) Based on our empirical findings in Section B.2, we assume 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 (sh) with a simple exponential function, where aand bare free parameters: sh=(1 if h=1 a·ebh if h≥2(B.21) As one moves up the hierarchy, employment in each consecutive level (Eh) decreases by 1/sh. This yields Eq. B.22, a recursive method for calculating Eh. In this model, we want employment to be whole numbers. To accomplish this I have included the ↓symbol to indicate that the last step is to round down to the nearest whole number. By repeatedly substituting Eq. B.22 into itself, we can obtain a non-recursive formula (Eq. B.23). In product notation, Eq. B.23 can be written as Eq. B.24. Eh=↓Eh−1 sh for h>1 (B.22) Eh=↓E1·1 s2·1 s3·... ·1 sh (B.23) A Hierarchical Model of the Firm 246 Eh=↓E1 h Y i=1 1 si (B.24) Total employment in the whole firm (ET)is the sum of employment in all hierarchical levels. Defining nas the total number of hierarchical levels, we get Eq. B.25, which in summation notation, becomes Eq. B.26. ET=E1+E2+... +En(B.25) ET= n X h=1 Eh(B.26) In practice, nis not known beforehand, so we define it using Eq. B.24. 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: n={h|Eh≥1 and Eh+1=0}(B.27) To summarize, the hierarchical employment structure of our model firm is determined by 3 free parameters: the span of control parameters aand b, and base-level employment E1. B.3.2 Generating Hierarchical Pay To model the hierarchical pay structure of a firm, we begin by defining the interhierarchical pay-ratio (ph) as the ratio of mean income (¯ I)between adjacent hierarchical levels. Again, it is helpful to use a piecewise function so that we can define a pay-ratio for hierarchical level 1: ph≡   1 if h=1 ¯ Ih ¯ Ih−1 if h≥2(B.28) Based on our empirical findings in Section B.2, we assume that the pay ratio A Hierarchical Model of the Firm 247 increases exponentially with hierarchical level. I model this relation with the following function, where ris a free parameter: ph=(1 if h=1 rhif h≥2(B.29) Using the same logic as with employment (shown above), the mean income Ihin any hierarchical level is defined recursively by Eq. B.30 and non-recursively by Eq. B.31. ¯ Ih=¯ Ih−1 ph (B.30) ¯ Ih=¯ I1 h Y i=1 pi(B.31) Mean income for all employees (¯ IT) is then the weighted average of hierarchical level mean income (¯ Ih) and hierarchical level employment (Eh): ¯ IT= n X h=1 ¯ Ih·Eh ET (B.32) We define the CEO as the person(s) in the top hierarchical level. Therefore, CEO pay is simply ¯ In, average income in the top hierarchical level. The CEO-toaverage-employee pay ratio is given by the equation below. For succinctness, I refer to this ratio as the ‘CEO pay ratio’ C: C=¯ In ¯ IT (B.33) To summarize, the hierarchical pay structure of our model firm is determined by 2 free parameters: the pay-scaling parameter r, and mean pay in the base level (¯ I1) B.3.3 Adding Intra-Level Pay Dispersion Up to this point, we have modelled only the mean income within each hierarchical level of a firm. The last step in the modelling process is to make the firm more realistic by adding pay dispersion within each hierarchical level. A Hierarchical Model of the Firm 248 Level Mean Income 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 012345678910 012345678910 Income Income DensityDensity Hierarchical Level (h) 1 2 3 4 5 Hierarchical Level (h) 1 2 3 4 5 A. Adding Pay Dispersion Within Each Hierarchical Level B. Relative Contribution to Intra−Firm Income Distribution Figure B.7: Adding Intra-Level Pay Dispersion to a Model Firm This illustrates a model firm with lognormal pay dispersion in each hierarchical level. The model firm has a pay-scaling parameter of r=1.2 and an intra-level Gini index of 0.13. Panel A shows the separate distributions for each level, with mean income indicated by a dashed vertical line. Panel B shows contribution of each hierarchical level to the resulting income distribution for the whole firm (income density functions are summed while weighting for their respective employment. Span of control parameters are identical to those used Table B.10. A Hierarchical Model of the Firm 249 For this model, I assume that pay dispersion within hierarchical levels is lognormally distributed. This means that income (I) in hierarchical level his described by the probability density function lnN(Ih;µ,σ), where µand σare the location and scale parameters, respectively: lnN(Ih;µ,σ) = 1 Ih·σp2πexp−(ln Ih−µ)2 2σ2(B.34) Our empirical investigation of firm case studies indicated that pay dispersion with hierarchical levels is relatively constant (see Fig. B.5C). Given this finding, I assume identical inequality within all hierarchical levels. This means that the lognormal scale parameter σis the same for all hierarchical levels. To define µ, I use Eq. B.35, the formula for the mean income (¯ Ih) of our lognormal distribution. Solving for µgives Eq. B.36. ¯ Ih=eµ+1 2σ2(B.35) µ=ln(¯ Ih)−1 2σ2(B.36) Given a value for σ(which is a free parameter), we can define the pay distribution within any hierarchical level of a firm. This process is shown graphically in Figure B.7. Figure B.7A shows the lognormal income distributions for each hierarchical level of a 5-level firm with pay-scaling parameter r=1.2. Figure B.7B shows the size-adjusted contribution of each hierarchical level to the overall intra-firm income distribution. Lower levels have more members, and thus dominate the overall distribution. Once we have defined the probability distributions governing income in each hierarchical level, the last step is to simulate individual pay, and ultimately construct a firm payroll. We do this by defining income as a random lognormal variable: Ih∼lnN(µh,σ)(B.37) We construct a completed firm payroll by drawing Ehrandom numbers for each level h, and combining them all in the payroll vector I. Using subscripts to denote the hierarchical level and superscripts to denote the individual in that level (ranging from 1 to Eh) we get: I={I1 1,I2 1,..., IE1 1,I1 2,I2 2,..., IE2 2,..., ¯ In}(B.38) A Hierarchical Model of the Firm 250 Note that the last entry, the CEO pay ¯ In, is not a random variable. In order to preserve the CEO pay ratio (dictated by the Compustat dataset) I do not allow this value to vary stochastically. B.3.4 Example of the Model Algorithm We begin by choosing the arbitrary values of a,b,E1,r, and ¯ I1shown in Table B.10. We then input these values into the hierarchy-building algorithm. Column A shows the hierarchical levels of the firm (h), where h=1 is the base level. Using parameters aand b, we first calculate the span of control (column B), which defines the employment-ratio between adjacent hierarchical levels. In column C, we begin with the base level and use Eq. B.22 to calculate employment in each hierarchical level. In column D, we calculate the pay ratio phusing the pay-scaling parameter r. Finally, in column E, we calculate mean income ¯ Ihin each hierarchical level Once we have this table of values, we can calculate aggregate statics like total employment (the sum of column C) and mean pay (the mean of column E, weighted by column C). We can also calculate the CEO pay ratio. These results are shown at the bottom of Table B.10 The last step of the model is to generate a simulated payroll by adding lognormal dispersion to each hierarchical level. For large firms, this involves drawing many random numbers from a lognormal distribution. For example purposes, it is convenient to choose a small firm. Table B.11 shows a firm with the same span of control parameters as in Table B.10, but with a base size of 10. As before, we use the model algorithm to calculate mean pay in each level. We then use Eq. B.36 to calculate the lognormal location parameter in each level. The last step is to create the simulated payroll. For each hierarchical level h, we draw Ehrandom numbers from a lognormal distribution with parameters σand µh. Note that we do not let income in the top hierarchical level vary stochastically — this preserves the CEO pay ratio on which the model is based. Once we have the simulated payroll, we can calculate the firm’s income inequality. The resulting Gini index will vary randomly, due to the stochastic nature of the model. For large firms (more than 1000 employees) this variation is negligible. For small firms, if we wish to know the ‘true’ Gini index that is predicted from the sum of the lognormal density functions, we can do two things: 1. Run the model many times and take the mean of resulting sample of Gini indexes; 2. Multiply all hierarchical employment Ehby a large, constant factor. A Hierarchical Model of the Firm 251 Table B.10: Example of the Model Algorithm Parameters a b E1r¯ I1 1 0.2 10 000 1.15 1 A B C D E Hierarchical Level Span of Control Employment Pay Ratio Mean Income h sh=e0.2hEh=↓Eh−1 shph=1.15h¯ Ih=¯ Ih−1·ph 10 7.39 0 – – 9 6.05 1 3.52 468.5 8 4.95 8 3.06 133.2 7 4.06 44 2.66 43.5 6 3.32 182 2.31 16.4 5 2.72 607 2.01 7.1 4 2.23 1652 1.75 3.5 3 1.82 3678 1.52 2.0 2 1.49 6703 1.32 1.3 1 – 10000 – 1 Results Total Employment Mean Pay CEO Pay CEO Pay Ratio ET= n X h=1 Eh¯ IT= n X h=1 ¯ Ih·Eh ET ¯ InC=¯ In/¯ IT 22 875 1.87 468.5 250 A Hierarchical Model of the Firm 252 Table B.11: Adding Intra-Level Pay Dispersion to a Firm Parameters a b E1r¯ I1σ 1 0.2 10 1.2 1 0.24 Hierarchical Level Pay Ratio Mean Pay Scale Parameter Location Paramter hph=1.2h¯ Ih=¯ Ih−1·phσ µh=ln(¯ Ih)−1 2σ2 41.73 2.99 0.24 1.07 31.44 1.73 0.24 0.52 21.20 1.20 0.24 0.15 1– 1.00 0.24 -0.03 Generating a Simulated Payroll Hierarchical Level Employment Simulated Payroll hEhIh∼lnN(µh,σ) 41{2.99} 33{1.62,1.88,1.16} 26{1.11,0.94,1.08, 1.15,1.07,2.13} 110 {0.75,0.65,1.04, 1.09,0.96,0.95, 0.97,1.09,0.75, 0.87} Estimating Compustat Model Parameters 259 ● ●●●●●●●●●●●● ● ● ● ● ●● ●● ●● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 100 101 102 103 104 105 100101102103104105106 Total Employment (ET) Base Employment (E1) ●Model Polynomial Fit Figure B.10: Finding Base Level Employment From Total Employment This figure shows the modeled relation between total employment in a firm (ET) and base-level employment (E1). This relation, defined by Eq. B.21,B.24, and B.26, depends on the span of control parameters aand b(here a=1.05 and b=0.13). I fit this numerical mapping with a high-order polynomial to allow fast (but accurate) estimation of E1from ET. The R code for this procedure is available in the Supplementary Material. this resampled data. Figure B.9 shows the probability density distributions resulting from this bootstrap analysis. To incorporate this uncertainty into the model, I run the model many times — once for each bootstrapped estimate of a,b, and σ. In each iteration, we first resample the case-study data and calculate values of a,b, and σ. We then use these values (particularly aand b) to calculate all other model parameters. The results shown in this paper are based on 5000 bootstrap runs of the model. B.5.2 Base Level Employment Having estimated the span of control parameters aand b, the next step is to calculate base-level employment E1for each Compustat firm. We do this by using data for total employment ET. The modeled-relation between total employment ETand base-level employment E1is determined by equations B.21,B.24, and B.26 (see Appendix B.3). Estimating Compustat Model Parameters 260 Given values for aand b, these equations produce a unique relation between E1 and ET. ET=fa,b(E1)(B.42) What we want is an inverse function that gives E1from ET: E1=f−1 a,b(ET)(B.43) Although there may be a way to define this inverse function analytically, it is beyond my mathematical abilities. Instead, I use the model to reverse engineer the problem. I define fa,bnumerically by inputting a range of different values for E1into equations B.21,B.24, and B.26 and calculating ETfor each value. The result is a discrete mapping relating base-level employment to total employment (see Fig. B.10). I then fit this mapping with a high-order polynomial, which then serves as an approximation to the inverse function f−1. This polynomial can then be used to quickly and accurately calculate E1from ETfor every Compustat firm. B.5.3 Pay-Scaling Parameter Once we have calculated base-level employment (E1) for all Compustat firms, we can estimate their respective pay-scaling ratios (r) using the CEO-to-averageemployee pay ratio (C). The pay-scaling ratio rdetermines the rate at which mean pay increases by hierarchical level. Having estimated a,b, and E1for each Compustat firm, the model (specifically equations B.24,B.29,B.31,B.32,and B.33) produces a CEO pay ratio (Cmodel) that is a unique function of the pay-scaling parameter r: Cmodel =fa,b,E1(r)(B.44) As with base-employment, I am not aware of an analytical method for defining the inverse function f−1 a,b,E1. Instead I use a numerical optimization method to solve for r. I define an error function ε(r)that quantifies the error between the actual value of a firm’s CEO pay ratio (Cempirical )and the value predicted by the model (Cmodel) for a given value of r: ε(r) = Cmodel −Cempirical(B.45) For each firm, the correct value of ris that which minimizes this error function. I use the R non-linear optimization function ‘nlminb’ to solve this mini- Estimating Compustat Model Parameters 261 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● 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12 14 16 18 1011021031041051061.0 1.1 1.2 1.3 1.4 1.5 Firm Size (Employees) Pay Scaling Parameter (r) CEO Pay Ratio (Empirical) Density 1.1 1.2 1.3 1.4 1.5 Pay Scaling Parameter (r) B. Pay−Scaling Parameter DistributionA. Fitted Pay−Scale Parameters Figure B.11: Fitting Compustat Firms with a Pay-Scaling Parameter This figure shows the fitted pay-scaling parameters (r) for all Compustat firms. Panel A shows the relation between the CEO pay ratio and firm size, with the fitted pay-scaling parameter indicated by color. The pay-scaling parameter distribution for all firms (and years) is shown in panel B. These results show the average of 5000 model runs, each with different bootstrapped parameters a,b, and σ. mization problem. To ensure that there are no large errors, I discard Compustat firms for which the best-fit rparameter produces an error that is larger than 5% of Cempirical. Fitted results for rare shown in Figure B.11. B.5.4 Base-Level Pay Once we have the pay-scaling parameter r, we can estimate base-level pay for each Compustat firm. To do this, we set up a ratio between base level pay (¯ I1) and firm mean pay (¯ IT) for both the model and Compustat data: ¯ ICompustat 1 ¯ ICompustat T = ¯ Imodel 1 ¯ Imodel T (B.46) The modeled ratio between base pay and firm mean pay (¯ Imodel 1/¯ Imodel T) is independent of the choice of base pay. This is because the modeled firm mean Estimating Compustat Model Parameters 262 pay is actually a function of base pay (see Eq. B.31 and B.32). If we run the model with ¯ Imodel 1=1, then Eq. B.46 reduces to: ¯ ICompustat 1 ¯ ICompustat T =1 ¯ Imodel T (B.47) We can then rearrange Eq. B.47 to solve for an estimated base pay for each Compustat firm (¯ ICompustat 1): ¯ ICompustat 1=¯ ICompustat T ¯ Imodel T (B.48) Compustat Model Results 263 B.6 Compustat Model Results I review here the results of the Compustat model that are not discussed in the main paper. All results are generated using 5000 bootstrap model runs over different values for the parameters a,b, and σ. From the data generated by the model, many different calculations are possible. I review here the following: (1) estimates for income inequality within Compustat firms; (2) estimates for income by hierarchical level; and (3) aggregate inequality of all firms in the model. B.6.1 Inequality Within Compustat Firms Figure B.12 shows estimate of income inequality within Compustat firms. In Figure B.12A, I illustrate how firm Gini indexes are related to both the CEO Pay ratio and firm size. Note that the CEO pay ratio is a reliable indicator of firm inequality only for firms of the same size. A general feature of a hierarchical firm model is that when internal inequality is held constant, the CEO pay ratio nonetheless tends to increase with firm size (a feature first demonstrated by Herbert Simon [22]). In Figure B.12A, this feature is evident as color contours of constant firm inequality that scale with both firm size and the CEO pay ratio. Figure B.12B shows the overall distribution of all firm Gini indexes. According to our model, 90% of Compustat firms have internal Gini indexes between 0.2 and 0.5. Note that the distribution is right-skewed — a small minority of firms have extremely unequal pay. In Figure B.12C I compare firm inequality in the Compustat model to inequality within the case-study firms discussed in Appendix B.2. The results indicate that Compustat firms are slightly more unequal than the case study firms. However, because the case-study sample size is small, this difference is not statistically significant. A Kolmogorov-Smirnov test gives a p-value of 0.20, indicating that there is a reasonable (20%) probability that the two firm samples (model and case study) come from the same distribution. Thus, under the conventional 5% significance level, we cannot reject the null-hypothesis that these samples come from the same distribution. But if the case study data and Compustat model produce firm internal Gini distributions that are statistically indistinguishable, why not simply use case study data for the test of hypothesis B (hierarchical power has the strongest effect on income)? There are several reasons the case study data cannot be used. Firstly, the case study sample size is extremely small. 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0.6 0.7 0.8 0.9 1.0 Model Case Studies 1995 2000 2005 2010 2015 Firm Size (Employees) Firm Internal Gini Index Year CEO Pay Ratio Density Firm Internal Gini Index Mean Gini Index 0.25 0.50 0.75 Gini Index B. Firm Gini Distribution C. Model vs. Case Studies D. Mean Firm Gini Over Time A. Modelled Gini Index Figure B.12: Compustat Model Results for Intra-Firm Inequality This figure shows the firm internal Gini index results of the Compustat model. Panel A shows how firm internal inequality (indicated by color) is related to the CEO pay ratio and firm size. Panel B shows the distribution of modeled Gini indexes for all firms. Panel C compares model results to the Gini index of case study firms (see section B.2.1 for case study methods). Panel D shows time evolution of the average Gini index of all modeled firms. The shaded region indicates the 95% confidence interval. All results are computed from 5000 model runs, each with different bootstrapped parameters a, b, and σ. Compustat Model Results 265 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● CNO FINANCIAL GROUP INC FIRST NBC BANK HOLDING CO STONE ENERGY CORP NATIONAL COMMERCE FINANCIAL CATHAY GENERAL BANCORP NUVEEN INVESTMENTS INC FUTUREFUEL CORP HCC INSURANCE HOLDINGS INC PHH CORP JACK IN THE BOX INC NATIONAL DISC BROKERS INC TRUSTCO BANK CORP/NY CEC ENTERTAINMENT INC BANK OF THE OZARKS INC DELPHI FINANCIAL GROUP INC OAK INDUSTRIES INC EAST WEST BANCORP INC LEUCADIA NATIONAL CORP WATERHOUSE INVESTORS SVCS NYFIX INC FIRST USA INC CHIPOTLE MEXICAN GRILL INC COLONIAL PROPERTIES TRUST ONBANCORP INC TPI ENTERPRISES INC REWARDS NETWORKS INC ROSLYN BANCORP INC TORCHMARK CORP NORTH FORK BANCORPORATION APOLLO EDUCATION GROUP INC FOOTHILL GROUP INC −CL A BOFI HOLDING INC ORITANI FINANCIAL CORP PHARMERICA CORP TALMER BANCORP INC HUDSON CITY BANCORP INC PROVIDIAN FINANCIAL CORP MONACO COACH CORP INTEGRATED HEALTH SVCS INC GBC BANCORP/CA FINANCIAL FEDERAL CORP HCI GROUP INC BLOCKBUSTER ENMNT CORP POLYMEDICA CORP CUSTOMERS BANCORP INC APPROACH RESOURCES INC SCORE BOARD INC CONSOL ENERGY INC GRAND CASINOS INC MICREL INC CALIFORNIA FED BANCORP INC PPL CORP BANGOR HYDRO−ELECTRIC CO CASCADE NATURAL GAS CORP DTE ENERGY CO ENERGY EAST CORP CH ENERGY GROUP INC NIAGARA MOHAWK HOLDINGS INC COMMONWEALTH ENERGY SYSTEM UNICOM CORP CONECTIV INC KEYSPAN ENERGY CORP ORANGE & ROCKLAND UTILITIES COAST SAVINGS FINANCIAL INC INTERSTATE POWER CO EVERSOURCE ENERGY UIL HOLDINGS CORP PG&E CORP CONSOLIDATED EDISON INC TNP ENTERPRISES INC FIRSTENERGY CORP RGS ENERGY GROUP INC BETHLEHEM STEEL CORP DAMES & MOORE GROUP NEW CENTURY ENERGIES INC AMEREN CORP IOWA−ILLINOIS GAS & ELEC IDACORP INC CALIBER SYSTEMS INC CONSTELLATION ENERGY GRP INC GPU INC ILLINOVA CORP KENNAMETAL INC MONTANA POWER CO LONGVIEW FIBRE CO CMP GROUP INC AMERICAN ELECTRIC POWER CO ARCBEST CORP NV ENERGY INC ALLEGHENY ENERGY INC XCEL ENERGY INC NEW ENGLAND ELECTRIC SYSTEM CLECO CORP CONSOLIDATED PAPERS INC GREAT PLAINS ENERGY INC PUGET ENERGY INC SOUTHERN CO EXELON CORP APOGEE ENTERPRISES INC DUKE ENERGY OHIO INC 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.14 0.16 0.18 0.20 0.22 Gini Index Estimate Gini Index Estimate A. 50 Most Unequal Firms B. 50 Most Equal Firms Figure B.13: The Most Equal and Unequal Compustat Firms This figure shows the 50 most unequal (panel A) and 50 most equal firms (panel B). Points indicate the mean Gini index for each firm, while the error bars show the 95% confidence interval calculated from 5000 bootstrap model runs. Compustat Model Results 266 cover many different countries (not just the US, the desired country). Thirdly, the observation years often do not overlap. To test hypothesis B, we need a large firm sample from a single country in a single year. While model dependent, the results inferred from Compustat data satisfy these conditions, while case study data does not. Figure B.12D shows the time-evolution of average inequality within Compustat firms. During the late 1990s inequality rapidly increased, followed by relative stability from 2000 onward. While the trend is clear, there is significant uncertainty in the absolute level of inequality (as indicated by the shaded region). This uncertainty is due to the small case-study sample size on which key model parameters are based (see Appendix B.5). Finally, Figure B.13 shows Gini index estimates for the 50 most equal and 50 most unequal firms. What is most interesting about these results is the sectoral composition of the 50 most equal firms. The vast majority (80%) are energy/utility companies. In the United States, firms in the utility sector are highly regulated, which leads to far more scrutiny over executive pay. Previous studies have found similar results — executives in regulated firms earn far less than those in unregulated firms [23]. This finding has important implications for a power theory of income distribution. It suggests that government regulation serves as a check on power, limiting the degree to which elites are able to use their status to amass wealth. B.6.2 Income By Hierarchical Level Besides estimating firm internal inequality, I use the Compustat model to estimate income and inequality by hierarchical level. To do this, we group all individuals by their hierarchical level, regardless of firm membership (see Fig. 8 of main paper). Model results are shown in Figure B.12, and are compared to the UK data documented by Mueller et al. [4]. Figure B.12A shows how mean income changes by hierarchical level. In both the Compustat model and Mueller’s data, mean income increases super-exponentially with hierarchical level — that is, it increases faster than an exponential function, which would appear as a straight line on the log-linear scale. Figure B.12B shows how intra-level income inequality changes by hierarchical level. For hierarchical levels 1-10, both the Compustat model and Mueller’s data show similar trends. The similarities between the model and Mueller’s data lend credence to the model. However, what explains the differences? One key factor is that the Compustat Model Results 267 ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●●●●●●● ● ● ●● ● ● ● ●●● ●●● ● ● ● 1 2 5 10 20 50 100 200 0.0 0.1 0.2 0.3 0.4 0.5 0.6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Hierarchical Level Hierarchical Level Mean Income Gini Index ●● ●● Model Mueller et al. ●● ●● Model Mueller et al. A. Mean Income By Hierarchical Level B. Gini Index By Hierarchical Level Figure B.14: Compustat Model Results for Income by Hierarchical Level This figure compares the results of the Compustat model to the UK data from Mueller et al. [4]. Panel A shows average income by hierarchical level (across all firms) indexed to pay in level 1. Panel B shows how intra-level inequality changes by hierarchical level. Shaded regions indicate the 95% confidence region of the model, estimated from 5000 bootstrap runs (see Appendix B.5 ). United States has much greater income inequality than the United Kingdom, and the Compustat firm sample comes from the former and Mueller’s sample the latter. As it turns out, both the Compustat model and Mueller’s data imply aggregate levels of inequality that are consistent with their respective national Gini indexes (see Fig. B.2 and B.15). In this light, the results in Figure B.12A make sense — in the more unequal United States, income scales more rapidly with hierarchical level than in the United Kingdom. The results in Figure B.12B can be similarly explained — in the more unequal United States, intra-hierarchical level income dispersion is greater than in the UK. Another interesting result in Figure B.12 is the conspicuous change in model trends for hierarchical levels above 11. Above this level, mean income no longer increases with hierarchical level, and intra-level inequality declines precipitously. The former result may simply be an artifact of the particular firm sample. Going back to Figure B.12A, note that the four largest firms have particularly low Compustat Model Results 268 CEO pay ratios. Given the model’s assumptions, only the very largest firms will have more than 11 hierarchical levels. Since the 4 largest firms have particularly low CEO pay ratios, resulting mean income in hierarchical levels 12-14 will be relatively low. The precipitous drop in intra-level inequality for hierarchical levels 12-14 is likely due to the convergence to a size of one. This is because there is often only one firm with 12 or more hierarchical levels, and the top level of this firm will contain only one individual. By definition, there is zero inequality in a sample size of one. B.6.3 Aggregate Inequality An important test of the Compustat model is to see if it produces aggregate levels of inequality that are comparable to US empirical data. Figure B.15 shows the results of such a test. Here I plot the time-series trends in both US historical inequality and aggregate inequality in the Compustat model. This latter metric is calculated by aggregating (by year) all individuals in the model into a single sample, and then calculating the inequality of the resulting income distribution. Figure B.15A compares the model’s aggregate Gini index against three different types of data published by the US Census: Gini by individual,family, and household. Two findings are evident. Firstly, the model is roughly consistent with the US empirical data over the period 2000-2015. However, the model produces too little inequality during the 1990s. Secondly, the US empirical data shows contradictory trends — roughly constant inequality among individuals, but secularly increasing inequality among families and households. The model reproduces the secular trend. But which empirical data should we believe? My vote is that the secular increase is the correct trend. Largely in response to his dissatisfaction with official inequality statistics, Thomas Piketty [24]has focused on measuring inequality in the tail of the income distribution. Figure B.15B and C show Piketty’s series for the top 10% and 1% income share in the United States. Both series show secularly increasing inequality over the period in question. The model reproduces these trends quite accurately, but at a lower absolute level of inequality. How can it be that the model more or less matches US Gini index data, but gives much less inequality than Piketty’s metrics? A plausible explanation is that official data simply underestimates inequality. However, the validity (or lack their of) of official inequality statistics is not something that this paper is concerned with. Rather, I simply take official data as a given, and use it to test my power- The Between-Within Gini Metric and Effect Size 275 dispersion within groups, as measured by the standard deviation. In the case of income, the size of the signal-to-noise ratio indicates how accurately we can predict someone’s income based only on knowledge of their group membership (either A or B). The larger the signal-to-noise ratio, the more accurate the prediction. In the example shown in Figure B.17, group A and B have the same difference in means and the same within-group standard deviation in both the left and right panel. Therefore Cohen’s dwould measure an identical effect size. To be clear, this is the effect on individual income (definition B), not the effect on inequality (definition A). B.8.2 Measuring Effect Size In this paper, I am concerned only with effect size definition B — the effect on individual income. I have proposed the between-within Gini metric (GBW ) as a measure of this type of effect size. This metric is defined by equation B.50, where GBis the Gini index of group means and ¯ GWis the mean of all within-group Gini indexes: GBW =GB ¯ GW (B.50) How does this metric relate to more standard measures of effect size? It amounts to a signal-to-noise ratio that is similar to Cohen’s f2measure, the latter of which is a generalization of Cohen’s dto many different groups. Cohen’s d uses the difference between means in the numerator. In order to generalize to many groups, f2uses the sum of squared differences (SS). To obtain f2(Eq. B.51), we divide the sum of squares between-groups (SSB) by the sum of squares within groups (SSW). See Fleishman [25]and Steiger [26]for a more detailed discussion of the f2metric.1 f2=SSB SSW (B.51) To be clear, SSBis the sum of squared differences between each group mean (¯ xi) and the grand mean (¯ xGM ), multiplied by group size n. Similarly, SSWis the sum of squared differences between each observation (xi j) and its group mean (¯ xi). This double sum operates over each of the kgroups and nobservations The Between-Within Gini Metric and Effect Size 276 within each group. Lastly, iindexes groups, and jindexes observations within each group. SSB=n k X i=1 (¯ xi−¯ xGM )2(B.52) SSW= k X i=1 n X j=1 (xi j −¯ xi)2(B.53) Like Cohen’s d, the f2metric is a signal-to-noise ratio. The ‘signal’ is the sum of squares between groups, while the ‘noise’ is the sum of squares within groups. When applied to income, the size of f2indicates the accuracy with which we can predict individual income from group membership. Comparing the form of f2and GBW , we see that the two measures of effect size are very similar. Both are signal-to-noise ratios, consisting of a ratio of between-group dispersion to within-group dispersion. The difference is that f2 uses the sum of squares to measure dispersion, while GBW uses the Gini index. Given the similarity between GBW and f2, there should be some relation between the two measures. Rather than attempt to show this similarity analytically, I use simulated data. I build a model based on the following assumptions: 1. Income within groups is lognormally distributed. 2. Within-group income dispersion is the same for all groups (but can vary over different model iterations). 3. Mean income between groups is lognormally distributed (and can vary between iterations). 4. Total inequality is (roughly) constant for all iterations. 5. The size of each group is constant. 6. The number of groups varies (between iterations) from 2 to 100. For each iteration of the model, we define the mean income of each group by drawing randomly from a lognormal distribution. We then simulate individuals within each group by drawing randomly from (a different) lognormal distribution. The model has 2 key parameters: the lognormal scale parameter that defines the dispersion between groups, and the lognormal scale parameter that 1A more common formula for this metric is f2=η2/(1−η2), where η=SSB/SST, the sum of squares between groups divided by the total sum of squares. Since SST=SSB+SSW, simple algebraic substitution can prove that the two definitions of f2are equivalent. The Between-Within Gini Metric and Effect Size 277 determines dispersion within groups. Varying these parameters changes the size of the group-income effect. For consistency, I use only parameter combinations that produce roughly the same level of total inequality (a Gini index of 0.5). For each set of simulated data, I calculate both GBW and f2. Because analysis of variance typically assumes that within-group data is normally distributed, I calculate f2using the logarithm of income. The results are shown in Figure B.18. As expected, there is an extremely strong relation between the two effectsize measures. This indicates that an f2test of the power-income effect would likely give very similar results to the GBW findings shown in Figure 10 of the main paper. To reiterate, I do not conduct such an f2test in this paper because the relevant data is not available. B.8.3 A Note on Group Size The simulation shown in Figure B.18 shows a special case where all groups have the same size. However, for the vast majority of the income-affecting factors studied in this paper, the groups do not have the same size. For instance, there are vastly more people in lower hierarchical levels than in upper hierarchical levels. How do we deal with this situation? The key ingredient of effect-size definition B is that it weights each different group equally (rather than weighting a group by its size). One way to achieve this equal weighting is to draw equal-sized samples from each group and calculate Cohen’s f2using equations B.51-B.53. But what if we do not have raw data? What if we have only summary statistics such as the mean and standard deviation of each group? We can proceed by noting that an alternative way to define Cohen’s f2is as the ratio of between-group variance σ2 Band mean within-group variance ¯ σ2 W: f2=σ2 B ¯ σ2 W (B.54) σ2 B= k X i=1 (¯ xi−¯ xGM )2 k(B.55) ¯ σ2 W= k X i=1 n X j=1 (xi j −¯ xi)2 nk = k X i=1 σ2 i k(B.56) Here, σ2 Band ¯ σ2 Ware derived by dividing SSBand SSW(respectively) by nk. 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B.51) to my between-within Gini metric, GBW (Eq. B.50). The comparison uses simulated data, and each data point represents different parameter combinations (see model assumptions above). Color indicates the number of groups used in each iteration. R code for the model is available in the Supplementary Material. The Between-Within Gini Metric and Effect Size 279 Given group means (¯ xi) and within-group standard deviations( σi), we can use this alternative formula to calculate f2. What equation B.54 does is give identical weight to each group’s summary statistics. This accomplishes the same thing as if we took equal sized samples from raw data and used Eq. B.51-B.53 to calculate f2. This same logic applies to my construction of the GBW metric: it calculates a signal-to-noise ratio from summary statistics by giving equal weight to each group. References 280 References 1. Bowles S, Gintis H, Osborne M. The determinants of earnings: A behavioral approach. 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Journal of Population Economics. 2004;17(2):193–228. 9. Grund C. The wage policy of firms: comparative evidence for the US and Germany from personnel data. The International Journal of Human Resource Management. 2005;16(1):104–119. 10. Lima F. Internal Labor Markets: A Case Study. FEUNL Working Paper. 2000;378. 11. Morais F, Kakabadse NK. The Corporate Gini Index (CGI) determinants and advantages: Lessons from a multinational retail company case study. International Journal of Disclosure and Governance. 2014;11(4):380–397. 12. Treble J, Van Gameren E, Bridges S, Barmby T. The internal economics of the firm: further evidence from personnel data. Labour Economics. 2001;8(5):531–552. References 281 13. Ariga K, Brunello G, Ohkusa Y, Nishiyama Y. Corporate hierarchy, promotion, and firm growth: Japanese internal labor market in transition. Journal of the Japanese and International Economies. 1992;6(4):440–471. 14. Bell B, Van Reenen J. Firm performance and wages: evidence from across the corporate hierarchy. CEP Discussion paper. 2012;1088. 15. Eriksson T. Executive compensation and tournament theory: Empirical tests on Danish data. Journal of Labor Economics. 1999;17(2):262–280. 16. Leonard JS. Executive pay and firm performance. Industrial and Labor Relations Review. 1990;43(3):13–28. 17. Main BG, O’Reilly III CA, Wade J. Top executive pay: Tournament or teamwork? Journal of Labor Economics. 1993;11(4):606–628. 18. Rajan RG, Wulf J. The flattening firm: Evidence from panel data on the changing nature of corporate hierarchies. The Review of Economics and Statistics. 2006;88(4):759–773. 19. Tao HL, Chen IT. The level of technology employed and the internal hierarchical wage structure. Applied Economics Letters. 2009;16(7):739–744. 20. Axtell RL. Zipf distribution of US firm sizes. Science. 2001;293:1818–1820. 21. Brown C, Medoff J. The employer size-wage effect. Journal of political Economy. 1989;97(5):1027–1059. 22. Simon HA. The compensation of executives. Sociometry. 1957;20(1):32– 35. 23. Joskow P, Rose N, Shepard A, Meyer JR, Peltzman S. Regulatory constraints on CEO compensation. Brookings Papers on Economic Activity Microeconomics. 1993;1993(1):1–72. 24. Piketty T. Capital in the Twenty-first Century. Cambridge: Harvard University Press; 2014. 25. Fleishman AI. Confidence intervals for correlation ratios. Educational and Psychological Measurement. 1980;40(3):659–670. 26. Steiger JH. Beyond the F test: effect size confidence intervals and tests of close fit in the analysis of variance and contrast analysis. Psychological methods. 2004;9(2):164–182. Appendix C Appendices For A Hierarchy Model of Income Distribution Supplementary materials for this paper are available at the Open Science Framework repository: https://osf.io/3bsvt/ The supplementary materials include: 1. Data for all figures appearing in the paper; 2. Raw source data; 3. R code for all analysis; 4. Hierarchy model code. Sources and Methods 283 C.1 Sources and Methods Fig. 4.4: Modeled Income Distribution vs. US Data Complementary Cumulative Distribution The US complementary cumulative distribution is calculated from data in the IRS Individual Complete Report (Publication 1304), Table 1.1, from 1996 to 2015. Cumulative Distribution The US cumulative distribution is calculated from data in the IRS Individual Complete Report (Publication 1304), Table 1.1, from 1996 to 2015. Gini Index I use two sources for the US Gini index. The first source is the US Current Population Survey, Table PINC-08 (available from the US Census) over the years 1994 to 2015. The second source is the IRS Individual Complete Report (Publication 1304), Table 1.1, from 1996 to 2015. I estimate the Gini index by constructing a Lorenz curve from the reported cumulative frequency data. R code implementing this method is available in the Supplementary Material. The Census and IRS data are not mutually consistent. IRS data is based on tax units, not individuals. The advantage of the IRS data is that it is an administrative record. Current Population Survey (CPS) data, on the other hand, is obtained by interview. The advantage of the CPS data is that it explicitly counts individuals. The disadvantage is that “there is a tendency in household surveys for respondents to under report their income” [1]. Lorenz Curve The US Lorenz curve is calculated from data in the IRS Individual Complete Report (Publication 1304), Table 1.1, from 1996 to 2015. Power Law Exponents I estimate the power law exponent of the income distribution tail using the maximum likelihood method. US empirical data comes from the IRS Individual Complete Report (Publication 1304), Table 1.1. Since this data is reported in Sources and Methods 284 Table C.1: Power Law Cutoff Boundaries in US Data Year Percentile α 1996 0.987 2.92 1997 0.985 2.89 1998 0.996 2.58 1999 0.996 2.58 2000 0.995 2.54 2001 0.996 2.63 2002 0.996 2.67 2003 0.996 2.65 2004 0.995 2.59 2005 0.994 2.54 2006 0.993 2.54 2007 0.993 2.54 2008 0.994 2.66 2009 0.995 2.78 2010 0.994 2.73 2011 0.994 2.74 2012 0.992 2.64 2013 0.993 2.74 2014 0.992 2.70 2015 0.991 2.72 binned form, I use the binned log-likelihood equation developed by Virkar and Clauset [2]: L=n(α−1)·ln bmin + k X i=min hilnbi (1−α)−bi+1 (1−α)(C.1) Here αis the power law exponent, biand bi+1are consecutive bin boundaries, hiand hi+1are consecutive bin counts, kis the number of bins, and n is the sum of bin counts above bmin (the cutoff point for the power law). The best-fit exponent αis the value that maximizes the log-likelihood function (L). Since there is no closed-form solution to this maximization problem, I solve for αnumerically. To determine the power law exponent for the top 1% of incomes in each year, I set the power law cutoff boundary (bmin) to the empirical bin that is closest to the 99th percentile. Results are shown in Table C.1. Hierarchical Structure and Pay within Case-Study Firms 291 Figure C.3 shows data for these metrics for the 6 case study firms. Figure C.3A shows how the span of control changes as a function of hierarchical level. The data shows unambiguously that the span of control tends to increase as one moves up the hierarchy. Figure C.3B shows how the inter-level pay ratio changes as a function of hierarchical level. Again, this ratio tends to increase as one moves up the hierarchy. Figure C.3C shows the intra-level Gini index as a function of hierarchical level. Unlike the other two quantities, intra-level income inequality seems to be more-or-less constant across all hierarchical levels (a linear regression reveals no significant trend). This case study data plays a central role in the hierarchical model developed in this paper. From the case study evidence, I propose the following ‘stylized’ facts about firm employment and pay structure: 1. The span of control tends to increase with hierarchical level. 2. The inter-level pay ratio tends to increase with hierarchical level. 3. Intra-level income inequality is approximately constant across all hierarchical levels. The case-study evidence informs the basic structure of the model, and also some of its key parameters. Parameters for span of control are determined from regressions on data in Figure C.3A, while parameters for intra-level income dispersion are determined from the mean of data in Figure C.3C. For a detailed discussion of the model algorithm and parameter fitting procedure, see Sections C.4 and C.5. Hierarchical Structure and Pay within Case-Study Firms 292 Limi Morais & Kakabadse Treble et al. Audas et al. Baker et al.. Dohmen et al. Limi Morais & Kakabadse Treble et al. Audas et al. Baker et al.. Dohmen et al. 75 50 25 0 25 50 75 75 50 25 0 25 50 75 75 50 25 0 25 50 75 1 3 5 7 9 11 1 3 5 7 9 11 30 20 10 0 10 20 30 30 20 10 0 10 20 30 30 20 10 0 10 20 30 1 3 5 7 9 11 1 3 5 7 9 11 Percent of Employment Average Pay (Base = 1) Hierarchical LevelHierarchical Level A. Firm Hierarchical Employment Structure B. Firm Hierarchical Pay Structure Figure C.2: The Hierarchical Employment and Pay Structure of Six Different Firms This figure shows the pyramid structure of six different case study firms. Panel A shows the hierarchical structure of employment, while panel B shows the hierarchical pay structure. Hierarchical Structure and Pay within Case-Study Firms 293 ●● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●●● ● ●● ● ●●● ● ●● ● ●●● ● ●● ● ●● ● ● ● ● ● ●● ● ● ● ●● ●● ● ● ●● ● ●●● ● ● ● ● ●●● ● ●● ● ●●● ● ●● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ●●● ●● ● ● 0.1 0.2 0.5 1.0 2.0 5.0 2 3 4 5 6 7 8 9 10 11 12 13 Hierarchical Level Span of Control A. Span of Control ●●● ● ● ●● ● ●● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ●●● ● ● ●● ●●● ● ● ●● ●●●● ● ●● ●●●● ● ●● ●●●● ● ●● ●●●● ●●● ●●●● ●●● ●●●● ●●●●●●● ●● ● ●●●● ● ●●● ● ● ● ●● ● ●● ● ●● ● ● ● ● ●● ●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ● ● ● ● 0.5 1.0 1.5 2.0 2.5 3.0 2 3 4 5 6 7 8 9 10 11 12 13 Hierachical Level Pay Ratio B. Pay Ratio ●●● ● ● ● ● ● ●●● ● ● ● ● ● ●●● ● ● ● ● ● ●●● ● ● ● ●● ●●● ● ● ● ● ● ●●● ● ● ● ● ● ●●● ●● ● ● ● ●●● ●● ● ● ● ●●● ●● ● ● ● ●●● ●● ● ● ● ●●● ● ● ● ● ● ●●● ●● ● ● ● ●●● ●● ● ● ● ●●● ●● ● ● ● ●●● ●●● ●● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ●●● ●●●●● ● ●● ● ●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●●●●● ● ● 0.0 0.1 0.2 0.3 0.4 1 2 3 4 5 6 7 8 9 10 11 12 13 Hierarchical Level Gini Index C. Intra−Level Pay Inequality Sources: ● ● ● ● ● ● ● Audas et al. Baker et al.. Dohmen et al. Grund Limi Morais & Kakabadse Treble et al. Figure C.3: Case Studies of Firm Hierarchical Structure This figure shows data from 7 different single-firm case studies. Panel A shows how the span of control (the employment ratio between adjacent levels) relates to hierarchical level. Panel B shows how the ratio of mean pay between adjacent levels varies with hierarchical level. In these two panels, the x-axis corresponds to the upper hierarchical level in the ratio. Panel C shows levels of income inequality within individual hierarchical levels of each firm. Note that horizontal ‘jitter’ has been introduced in all three plots in order to better visualize the data (hierarchical level is a discrete variable). Grey regions correspond to the 95% confidence interval for regressions (or in panel C, the mean). Compustat Data 294 C.3 Compustat Data This paper makes extensive use of the Compustat and Execucomp databases. Compustat contains data for most publicly traded US companies, while Execucomp contains data for executive compensation. Three key statistics used throughout this paper are calculated from this data: firm mean income, the CEOto-average-employee pay ratio, and the capitalist income fraction of executives. I discuss the data and methods used for these calculations in the following sections. C.3.1 Firm Mean Income Firm mean income is calculated by dividing total staff expenses (Compustat Series XLR) by total employment (Compustat Series EMP): Firm Mean Income =Total Staff Expenses Total Employment (C.4) C.3.2 CEO Pay Ratio Throughout this paper, I use the term ‘CEO’ to refer to the executive at the top of the corporate hierarchy. I identify CEOs using the titles contained in the Execucomp series TITLEANN. Because titles vary greatly by company, identifying the top executive is not always a simple task. While a manual search would be most accurate, this is unrealistic given that the Execucomp database contains over 275 000 entries. Instead, I use the following three-step algorithm to identify the ‘CEO’: 1. Find all executives whose title contains one or more of the words in the ‘CEO Titles’ list (Table C.5). 2. Of these executives, take the subset whose title does not contain any of the words in the ‘Subordinate Titles’ list (Table C.5). 3. If this search returns more than one executive per firm per year, chose the executive with the highest pay. After identifying the CEO (and matching CEO pay data with firm data contained in the Compustat database), I calculate the CEO pay ratio using the following equation: CEO Pay Ratio =CEO Pay Firm Mean Income (C.5) Compustat Data 295 Table C.5: Titles Used to Identify the ‘CEO’ CEO Titles: Subordinate Titles president vp chairman v-p CEO cfo Chief Executive Officer vice chmn chief finance officer president of coo division div presidentgroup president chairmainco-president deputy chairman pres.- Chief Financial Officer Notes: This table shows the Execucomp titles used to identify the CEO of each company. CEOs are deemed to be those whose title contains words in the left column, but not those in the right column. Titles such as ‘president-’ and ‘president of’ are included in the subordinate list because they typically refer to a president of a division with the company: i.e. ‘president of western division’ or ‘president-western hemisphere’. CEO pay ratio and firm mean income data are collectively available for roughly 6000 firm-year observations over the period 1992-2016. I use this data to ‘tune’ my hierarchical model of the firm (see Section C.5) . Figure C.4 shows selected summary statistics of this dataset. C.3.3 Capitalist Income Share of Executives I define the capitalist income share of executives (Kfrac) as the ratio of stockoptions income to total income: Kfrac =Stock Options Total Income (C.6) Compustat Data 296 100 150 200 250 300 350 400 10 20 30 40 0.0 2.5 5.0 7.5 10.0 0 100 200 300 400 500 0 200 400 0 50 100 150 200 0 200 400 600 0.75 1.00 1.25 1.50 0.20 0.25 0.30 0.35 0.40 0.45 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015 2 3 4 5 6 0 1 2 3 4 1995 2000 2005 2010 2015 −3 −2 −1 0 1 2 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015 Year Year Year log10(Employees)log10(CEO Pay Ratio)Year log10(Normalized Mean Pay)Year Year Number Employees (Thousands) % of US Employment Number of Firms Number of Firms Number of Firms Sample Mean / US Mean Mean Pay Gini Index A. Number of Firms B. Mean Firm Size C. Employment Share D. Firm Size Distribution E. CEO Pay Ratio F. Mean CEO Pay Ratio G. Normalized Mean Pay H. Mean Pay Ratio I. Inter−Firm Inequality Figure C.4: Selected Statistics from the Firm Sample Used for Model Tuning This figure shows statistics for the Compustat firm sample used to tune my hierarchical model. Panel A shows the number of firms in the sample over time, Panel B the average firm size, and Panel C the share of US employment held by these firms. Panel D shows the logarithmic distribution of firm size, and Panel E shows the logarithmic distribution of the CEO pay ratio. Panel F shows the mean CEO pay ratio of all firms over time. Panel G shows the logarithmic distribution of normalized mean pay (mean pay divided by the average pay of the firm sample in each year). Panel H shows the ratio of mean pay in the Compustat 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. Panel I shows the Gini index of firm mean pay over time. Compustat Data 297 Table C.6: Data Used to Calculate Executive Capitalist Income Fraction Series Description Reporting Format RSTKGRNT The value of restricted stock granted during the year (determined as of the date of the grant). 1992 OPTION_AWARDS_BLK_VALUE The aggregate value of stock options granted to the executive during the year as valued using Standard & Poor’s Black-Scholes methodology. 1992 TDC1 Salary, Bonus, Other Annual, Total Value of Restricted Stock Granted, Total Value of Stock Options Granted (using BlackScholes), Long-Term Incentive Payouts, and All Other 1992 STOCK_AWARDS_FV Fair value of all stock awards during the year as detailed in the Plan Based Awards table. Valuation is based upon the grant-date fair value as detailed in FAS 123R. 2006 OPTION_AWARDS_FV Fair value of all options awarded during the year as detailed in the Plan Based Awards table. Valuation is based upon the grant-date fair value as detailed in FAS 123R. 2006 TDC1 Salary, Bonus, Non-Equity Incentive Plan Compensation, Grant-Date Fair Value of Option Awards, Grant-Date Fair Value of Stock Awards, Deferred Compensation Earnings Reported as Compensation, and Other Compensation. 2006 The Execucomp database contains two main accounting methods for valuing stock options: a ‘1992’ reporting format that applies from 1992 to 2005’, and a ‘2006’ reporting format that applies from 2006 onward. These series are summarized in Table C.6. For both reporting formats, the relevant total income series (TDC1) remains the same. I calculate the capitalist income fraction of executives using the following two formulas for 1992 format and 2006 format, respectively: Kfrac_1992 =RSTKGRNT +OPTION_AWARDS_BLK_VALUE TDC1 (C.7) Kfrac_2006 =STOCK_AWARDS_FV +OPTION_AWARDS_FV TDC1 (C.8) Compustat Data 298 Figure C.5: Firm Sales vs. Payroll in the Compustat US Database This figure plots normalized firm sales against normalized firm payroll for every firmyear observation in the Compustat US database from 1950 to 2015. Each dot is a specific firm in a specific year. To adjust for inflation, I divide sales and payroll by the database averages in the respective year. C.3.4 Firm Sales vs. Firm Payroll In section 4.4, I use the hierarchy model to reproduce historical trends in the CEO pay ratio. The empirical data from Mishel and Schieder [9]uses the CEOs in the top 350 US firms ranked by sales. Since the hierarchy model does not have sales, I calculate the CEO pay ratio by ranking firms by total payroll. Since payroll is highly correlated with firm sales (Fig. C.5), the former is a good proxy for the latter. Hierarchy Model Equations 299 C.4 Hierarchy Model Equations In this section, I outline the mathematics underlying my hierarchical model of the firm. The model assumptions, outlined below, are based on the stylized facts gleaned from the real-world firm data in section C.2. 1. Firms are hierarchically structured, with a span of control that increases exponentially with hierarchical level. 2. The ratio of mean pay between adjacent hierarchical levels increases exponentially with hierarchical level. 3. Intra-hierarchical-level income is lognormally distributed and constant across all levels. Using these assumptions, I first develop an algorithm that describes the hierarchical employment within a model firm, followed by an algorithm that describes the hierarchical pay structure. Table C.7: Notation Symbol Definition aspan of control parameter 1 bspan of control parameter 2 CCEO to average employee pay ratio Eemployment Fcumulative distribution function GGini index of inequality hhierarchical level ¯ Iaverage income µlognormal location parameter nnumber of hierarchical levels in a firm ppay ratio between adjacent hierarchical levels rpay-scaling parameter sspan of control σlognormal scale parameter Ttotal for firm ↓round down to nearest integer Qproduct of a sequence of numbers Psum of a sequence of numbers Hierarchy Model Equations 300 C.4.1 Generating the Employment Hierarchy To generate the hierarchical structure of a firm, 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≡   1 if h=1 Eh−1 Eh if h≥2(C.9) Based on our empirical findings in Section C.2, we assume 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 (sh) with a simple exponential function, where aand bare free parameters: sh=(1 if h=1 a·ebh if h≥2(C.10) As one moves up the hierarchy, employment in each consecutive level (Eh) decreases by 1/sh. This yields Eq. C.11, a recursive method for calculating Eh. Since we want employment to be whole numbers, we round down to the nearest integer (notated by ↓). By repeatedly substituting Eq. C.11 into itself, we can obtain a non-recursive formula (Eq. C.12). In product notation, Eq. C.12 can be written as Eq. C.13. Eh=↓Eh−1 sh for h>1 (C.11) Eh=↓E1·1 s2·1 s3·... ·1 sh (C.12) Eh=↓E1 h Y i=1 1 si (C.13) Total employment in the whole firm (ET)is the sum of employment in all hierarchical levels. Defining nas the total number of hierarchical levels, we get Eq. C.14, which in summation notation, becomes Eq. C.15. ET=E1+E2+... +En(C.14) Restricting Parameters 307 10−4 10−3 10−2 10−1 100 101102103104 Firm size Fraction of Firms Census Data Power Law Figure C.8: The United States Firm Size Distribution This figure shows the US firm size distribution compared to a power law distribution with exponent α=2.01 (a simulation with 15 million firms) . The US histogram combines data for ‘employer’ firms with data for unincorporated self-employed workers. Data for ‘employer’ firms is from the US Census Bureau, Statistics of U.S. Businesses (using data for 2013). This data is augmented with Bureau of Labor Statistics data for unincorporated self-employed workers (series LNU02032185 and LNU02032192). The histogram preserves Census firm-size bins, with self-employed data added to the first bin. The last point on the histogram consists of all firms with more than 10,000 employees. ponent α=2.01. Although not perfect, the fit is good enough for modeling purposes. I assume that the firm sizes can be modeled with a discrete power law random variate. I model the US firm size distribution with α=2.01. 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 firm 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 firm size of 2.3 million. This happens to be the present size of Walmart, the largest US firm in existence. Code for the discrete power law random number generator can be found in the C++ header file rpld.h, located in the Supplementary Material. This code Restricting Parameters 308 0 1 2 3 0.8 1.0 1.2 1.4 1.6 a Density 0 5 10 15 20 25 0.075 0.100 0.125 0.150 0.175 b Density Figure C.9: Density Estimates for Span of Control Parameters This figure shows density estimates for the parameters aand b, which together determine the ‘shape’ of the firm hierarchy. These parameters are determined from regressions on firm case-study data (Fig. C.3). The density functions are estimated using a bootstrap analysis, which involves resampling (with replacement) the case study data many times, and calculating the parameters aand bfor each resample. is an adaption of Collin Gillespie’s discrete power law generator found in the R poweRlaw package [19](which is, in turn, an adaption of the algorithm outline by Clauset [20]). C.5.2 Span of Control Parameters The parameters aand btogether determine the shape of firm employment hierarchy. These parameters are estimated from an exponential regression on case study data (Fig. C.3A). The model proceeds on the assumption that these parameters are constant across all firms. Because the case-study sample size is small, there is considerable uncertainty in these values. I incorporate this uncertainty into the model using the bootstrap method [21], which involves repeatedly resampling the case-study data (with replacement) and then estimating the parameters aand bfrom this resample. Figure C.9 shows the probability density distribution resulting from this bootstrap analysis. I run the model many times, each time with aand bdetermined by a bootstrap resample of case-study data. Restricting Parameters 309 Code implementing this bootstrap can be found in the C++ header file boot_span.h. C.5.3 Base Level Employment Given span of control parameters aand b, each firm hierarchy is constructed from the bottom hierarchical level up. Thus, we must know base level employment. In practice, however, we don’t know this value — instead we are given total employment for a particular firm. While it may be possible to use the equations in section C.4 to define an analytic function relating total employment to base level employment, this is beyond my mathematical abilities. Instead, I use the model to reverse engineer the problem. I input a range of different base employment values into equations C.10,C.13, and C.15 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 E1, given total employment ET. Code implementing this method can be found in the C++ header file base_fit.h, located in the Supplementary Material. C.5.4 Pay-Scaling Parameter The pay-scaling ratio rdetermines the rate at which mean pay increases by hierarchical level. Unlike the span of control parameters, the pay-scaling parameter is allowed to vary between firms. But how should it vary? I restrict the variation of this parameter in a two-step process. I first ‘tune’ the model to Compustat data. This results in a distribution of pay-scaling parameters specific to Compustat firms. I then fit this data with a parameterized distribution, from which simulation parameters are randomly chosen. Fitting Compustat Pay-Scaling Parameters I fit the pay-scaling parameter rto Compustat firms using the CEO-to-averageemployee pay ratio (C). The first step of this process is to build the employment hierarchy for each Compustat firm using parameters a,b, and E1(the latter is determined from total employment). Given this hierarchical employment structure, the CEO pay ratio in the modeled firm is uniquely determined by the parameter r. Thus, we simply choose rsuch that the model produces a CEO pay ratio that is equivalent to the empirical ratio. Restricting Parameters 310 ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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Pay−Scaling Parameter DistributionA. Fitted Pay−Scale Parameters Figure C.10: Fitting Compustat Firms with a Pay-Scaling Parameter This figure shows the fitted pay-scaling parameters (r) for all Compustat firms. Panel A shows the relation between the CEO pay ratio and firm size, with the fitted pay-scaling parameter indicated by color. The discrete changes in color (evident as vertical lines) correspond to changes in the number of hierarchical levels within firms. The pay-scaling parameter distribution for all firms (and years) is shown in panel B. To solve for this rvalue, I use numerical optimization (the bisection method) to minimize the error function shown in Eq. C.30. Here CCompustat and Cmodel are Compustat and modeled CEO pay ratios, respectively. ε(r) = Cmodel −CCompustat(C.30) For each firm, the fitted value of rminimizes this error function. To ensure that there are no large errors, I discard Compustat firms for which the best-fit r parameter produces an error that is larger than ε=0.01). Fitted results for r are shown in Figure C.10. Code implementing this method can be found in the C++ header file fit_model.h, located in the Supplementary Material. Generating a Pay Scaling Distribution Once we have generated rparameters for every Compustat firm, the next step is to fit a parameterized distribution to this data. 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● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ● ●●●● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Compustat Firms 1.0 1.1 1.2 1.3 1.4 102103104105106 Firm Size (Employment) r A. Pay−Scaling Parameter (r) ● ● ● ●● ● ● ● ● ● Model Compustat (log−spaced employment bins) 0.1 0.2 0.3 0.4 0.5 100101102103104105106 Firm Size (Employment) σE B. Modeling σE ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● ●● ●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ●● ● ●● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ●● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●●● ●●●● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Model 2 σ Range Compustat Firms −5 −4 −3 −2 −1 100101102103104105106 Firm Size (Employment) ln(r0) C. Model of r0 Simulation Compustat Firms 0 5 10 15 1.00 1.05 1.10 1.15 1.20 1.25 1.30 r Density D. Simulated r for Compustat Firms Figure C.11: Modeling the Firm Pay Scaling Distribution This figure visualizes the model used to simulate firm pay-scaling parameters (r). Panel A shows the relation between rand firm employment for Compustat firms. For the simulation, the distribution of ris modeled with the lognormal variate r0. Panel B shows how the lognormal scale parameter σE(defined by Eq. C.35) changes with firm size. The straight line indicates the modeled relation. Panel C shows how the modeled dispersion of ln(r0)declines with firm size, and how this relates to Compustat r0data. The 2σrange indicates 2 standard deviations from the mean (on log-transformed data). Panel D shows how the distribution of rfor Compustat firms compares to the simulated distribution achieved by applying the model to the same Compustat firms. Restricting Parameters 312 persion of ris approximately lognormal, and tends to decline with firm size (see Figure C.11A). I model ras a shifted function of the lognormal variate r0: r=1+lnN(r0)(C.31) The lognormal variate r0is defined by location parameter µand scale parameter σ. While µis assumed to be constant for all firms, σis a function of firm size E: r0(E) = lnN(r0;µ,σE)(C.32) I use the tuned Compustat data to solve for the parameters µand σ. We first transform Compustat rvalues using Eq. C.33 to get the Compustat distribution of r0: r0=r−1 (C.33) The best-fit value for µis defined by taking the mean of ln(r0): µ=ln(r0)(C.34) Similarly, we can solve for the best-fit value for σby taking the standard deviation of ln(r0). However, unlike µ, the value σwill depend on the size range of firms (E): σE=SD[ln(r0) ]E(C.35) Figure C.11B plots σEvs. Efor logarithmically spaced size groupings of Compustat firms. I model this relation using a log-linear regression. Figure C.11C shows how the modeled dispersion in r0varies with firm size, and how this compares to Compustat data. Once we have fitted the parameters µand σto the tuned Compustat data, we can generate rvalues for simulated firms using equations C.31 and C.32. Although the model is simple, it produces reasonably accurate results. To test this accuracy, we can apply the model to the same Compustat firms for which it is ‘tuned’. For each Compustat firm, we use the method outlined above to stochastically generate a pay-scaling value r. As Figure C.11D shows, the resulting simulated distribution of rfairly accurately reproduces the original data. When we move from simulating Compustat firms to a real-world distribution of firms, this model involves significant extrapolations for small firms. Why? [Document text truncated for crawler view.]