Hierarchy and the power-law income distribution tail

Abstract

What explains the power-law distribution of top incomes? This paper tests the hypothesis that it is firm hierarchy that creates the power-law income distribution tail. Using the available case-study evidence on firm hierarchy, I create the first large-scale simulation of the hierarchical structure of the US private sector. Although not tuned to do so, this model reproduces the power-law scaling of top US incomes. I show that this is purely an effect of firm hierarchy. This raises the possibility that the ubiquity of power-law income distribution tails is due to the ubiquity of hierarchical organization in human societies.

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Notes

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    The constant c is equal to \( (\alpha - 1) / (x_{ \text {min} })^{1 - \alpha } \), where \(x_{ \text {min}}\) is the lower bound of the power law (i.e., the beginning of the income distribution tail).

References

  1. 1.

    Pareto, V. (1897). Cours d’economie politique (Vol. 1). Geneva: Librairie Droz.

    Google Scholar 

  2. 2.

    Atkinson, A. B. (2017). Pareto and the upper tail of the income distribution in the UK: 1799 to the present. Economica, 84(334), 129.

    Article  Google Scholar 

  3. 3.

    Aoyama, H., Souma, W., Nagahara, Y., Okazaki, M. P., Takayasu, H., & Takayasu, M. (2000). Pareto’s law for income of individuals and debt of bankrupt companies. Fractals, 8(03), 293.

    Article  Google Scholar 

  4. 4.

    Coelho, R., Richmond, P., Barry, J., & Hutzler, S. (2008). Double power laws in income and wealth distributions. Physica A: Statistical Mechanics and its Applications, 387(15), 3847.

    Article  Google Scholar 

  5. 5.

    Clementi, F., & Gallegati, M. (2005). Pareto’s law of income distribution: Evidence for Germany, the United Kingdom, and the United States. In A. Chatterjee, S. Yarlagadda, & B. K. Chakrabarti (Eds.), Econophysics of wealth distributions (pp. 3–14). Berlin: Springer.

    Google Scholar 

  6. 6.

    Clementi, F., & Gallegati, M. (2005). Power law tails in the Italian personal income distribution. Physica A: Statistical Mechanics and Its Applications, 350(2–4), 427.

    Article  Google Scholar 

  7. 7.

    Di Guilmi, C., Gaffeo, E., & Gallegati, M. (2003). Power law scaling in world income distribution. Economics Bulletin, 15, 1–7.

    Google Scholar 

  8. 8.

    Drgulescu, A., & Yakovenko, V. M. (2001). Exponential and power-law probability distributions of wealth and income in the United Kingdom and the United States. Physica A: Statistical Mechanics and its Applications, 299(1–2), 213.

    Article  Google Scholar 

  9. 9.

    Nirei, M. (2009). Pareto distributions in economic growth models. IIR working paper WP#09-05

  10. 10.

    Toda, A. A. (2012). The double power law in income distribution: Explanations and evidence. Journal of Economic Behavior & Organization, 84(1), 364.

    Article  Google Scholar 

  11. 11.

    Silva, A. C., & Yakovenko, V. M. (2004). Temporal evolution of the thermal and superthermal income classes in the USA during 19832001. EPL (Europhysics Letters), 69(2), 304.

    Article  Google Scholar 

  12. 12.

    Souma, W. (2001). Universal structure of the personal income distribution. Fractals, 9(04), 463.

    Article  Google Scholar 

  13. 13.

    Lydall, H. F. (1959). The distribution of employment incomes. Econometrica: Journal of the Econometric Society, 27(1), 110.

    Article  Google Scholar 

  14. 14.

    Audas, R., Barmby, T., & Treble, J. (2004). Luck, effort, and reward in an organizational hierarchy. Journal of Labor Economics, 22(2), 379.

    Article  Google Scholar 

  15. 15.

    Baker, G., Gibbs, M., & Holmstrom, B. (1993). Hierarchies and compensation: A case study. European Economic Review, 37(2–3), 366.

    Article  Google Scholar 

  16. 16.

    Dohmen, T. J., Kriechel, B., & Pfann, G. A. (2004). Monkey bars and ladders: The importance of lateral and vertical job mobility in internal labor market careers. Journal of Population Economics, 17(2), 193.

    Article  Google Scholar 

  17. 17.

    Grund, C. (2005). The wage policy of firms: Comparative evidence for the US and Germany from personnel data. The International Journal of Human Resource Management, 16(1), 104.

    Article  Google Scholar 

  18. 18.

    Lima, F. (2000). Internal labor markets: A case study. FEUNL working paper, vol. 378.

  19. 19.

    Morais, F., & Kakabadse, N. K. (2014). The corporate gini index (CGI) determinants and advantages: Lessons from a multinational retail company case study. International Journal of Disclosure and Governance, 11(4), 380.

    Article  Google Scholar 

  20. 20.

    Treble, J., Van Gameren, E., Bridges, S., & Barmby, T. (2001). The internal economics of the firm: Further evidence from personnel data. Labour Economics, 8(5), 531.

    Article  Google Scholar 

  21. 21.

    Ariga, K., Brunello, G., Ohkusa, Y., & Nishiyama, Y. (1992). Corporate hierarchy, promotion, and firm growth: Japanese internal labor market in transition. Journal of the Japanese and International Economies, 6(4), 440.

    Article  Google Scholar 

  22. 22.

    Bell, B., & Van Reenen, J. (2012). Firm performance and wages: Evidence from across the corporate hierarchy. CEP discussion paper, vol. 1088.

  23. 23.

    Eriksson, T. (1999). Executive compensation and tournament theory: Empirical tests on Danish data. Journal of Labor Economics, 17(2), 262.

    Article  Google Scholar 

  24. 24.

    Heyman, F. (2005). Pay inequality and firm performance: Evidence from matched employer–employee data. Applied Economics, 37(11), 1313.

    Article  Google Scholar 

  25. 25.

    Leonard, J. S. (1990). Executive pay and firm performance. Industrial and Labor Relations Review, 43(3), 13.

    Article  Google Scholar 

  26. 26.

    Main, B. G., O’Reilly, C. A, I. I. I., & Wade, J. (1993). Top executive pay: Tournament or teamwork? Journal of Labor Economics, 11(4), 606.

    Article  Google Scholar 

  27. 27.

    Mueller, H. M., Ouimet, P. P., & Simintzi, E. (2016). Within-firm pay inequality. SSRN working paper.

  28. 28.

    Rajan, R. G., & Wulf, J. (2006). The flattening firm: Evidence from panel data on the changing nature of corporate hierarchies. The Review of Economics and Statistics, 88(4), 759.

    Article  Google Scholar 

  29. 29.

    Tao, H. L., & Chen, I. T. (2009). The level of technology employed and the internal hierarchical wage structure. Applied Economics Letters, 16(7), 739.

    Article  Google Scholar 

  30. 30.

    Kumamoto, S. I., & Kamihigashi, T. (2018). Power laws in stochastic processes for social phenomena: An introductory review. Frontiers in Physics, 6, 20.

    Article  Google Scholar 

  31. 31.

    Mitzenmacher, M. (2004). A brief history of generative models for power law and lognormal distributions. Internet mathematics, 1(2), 226.

    Article  Google Scholar 

  32. 32.

    Newman, M. E. (2005). Power laws, Pareto distributions and Zipf’s law. Contemporary Physics, 46(5), 323.

    Article  Google Scholar 

  33. 33.

    Gibrat, R. (1931). Les inegalites economiques. Paris: Recueil Sirey.

    Google Scholar 

  34. 34.

    Champernowne, D. G. (1953). A model of income distribution. The Economic Journal, 63(250), 318.

    Article  Google Scholar 

  35. 35.

    Gabaix, X., Lasry, J. M., Lions, P. L., & Moll, B. (2016). The dynamics of inequality. Econometrica, 84(6), 2071.

    Article  Google Scholar 

  36. 36.

    Nirei, M., & Souma, W. (2007). A two factor model of income distribution dynamics. Review of Income and Wealth, 53(3), 440.

    Article  Google Scholar 

  37. 37.

    Rutherford, R. (1955). Income distributions: A new model. Econometrica: Journal of the Econometric Society, 23(3), 277.

    Article  Google Scholar 

  38. 38.

    Wold, H. O., & Whittle, P. (1957). A model explaining the Pareto distribution of wealth. Econometrica, Journal of the Econometric Society, 25(4), 591.

    Article  Google Scholar 

  39. 39.

    Yule, G. U, I. I. (1925). A mathematical theory of evolution, based on the conclusions of Dr. JC Willis, FR S. Philosophical Transactions of the Royal Society B, 213(402–410), 21.

    Article  Google Scholar 

  40. 40.

    Simon, H. A. (1955). On a class of skew distribution functions. Biometrika, 42(3/4), 425.

    Article  Google Scholar 

  41. 41.

    Price, D. D. S. (1976). A general theory of bibliometric and other cumulative advantage processes. Journal of the Association for Information Science and Technology, 27(5), 292.

    Google Scholar 

  42. 42.

    Barabasi, A., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286(5439), 509.

    Article  Google Scholar 

  43. 43.

    Angle, J. (1986). The surplus theory of social stratification and the size distribution of personal wealth. Social Forces, 65(2), 293.

    Article  Google Scholar 

  44. 44.

    Angle, J. (2006). The inequality process as a wealth maximizing process. Physica A: Statistical Mechanics and Its Applications, 367(15), 388.

    Article  Google Scholar 

  45. 45.

    Chatterjee, A., Chakrabarti, B. K., & Chakraborti, A. (2007). Econophysics and sociophysics: Trends and perspectives. Milan: Wiley.

    Google Scholar 

  46. 46.

    Chatterjee, A., & Chakrabarti, B. K. (2007). Kinetic exchange models for income and wealth distributions. The European Physical Journal B, 60(2), 135.

    Article  Google Scholar 

  47. 47.

    Hegyi, G., Neda, Z., & Santos, M. A. (2007). Wealth distribution and Pareto’s law in the Hungarian medieval society. Physica A: Statistical Mechanics and its Applications, 380(1), 271.

    Article  Google Scholar 

  48. 48.

    Kitov, I. O. (2009). Mechanical model of personal income distribution (pp. 1–220). arXiv preprint arXiv:0903.0203

  49. 49.

    Pianegonda, S., Iglesias, J. R., Abramson, G., & Vega, J. L. (2003). Wealth redistribution with conservative exchanges. Physica A: Statistical Mechanics and its Applications, 322, 667.

    Article  Google Scholar 

  50. 50.

    Scheffer, M., Bavel, B. V., Leemput, I. A. V. D., Nes, E. H. V. (2017) Inequality in nature and society. Proceedings of the National Academy of Sciences, 114, 201706412. https://doi.org/10.1073/pnas.1706412114

    Article  Google Scholar 

  51. 51.

    Yakovenko, V. M., & Rosser, J. B, Jr. (2009). Colloquium: Statistical mechanics of money, wealth, and income. Reviews of Modern Physics, 81(4), 1703.

    Article  Google Scholar 

  52. 52.

    Simon, H. A. (1957). The compensation of executives. Sociometry, 20(1), 32.

    Article  Google Scholar 

  53. 53.

    Roberts, D. R. (1956). A general theory of executive compensation based on statistically tested propositions. The Quarterly Journal of Economics, 70(2), 270.

    Article  Google Scholar 

  54. 54.

    Axtell, R. L. (2001). Zipf distribution of US firm sizes. Science, 293, 1818.

    Article  Google Scholar 

  55. 55.

    Fix, B. (2017). Energy and institution size. PLoS ONE, 12(2), e0171823. https://doi.org/10.1371/journal.pone.0171823.

    Article  Google Scholar 

  56. 56.

    Gaffeo, E., Gallegati, M., & Palestrini, A. (2003). On the size distribution of firms: Additional evidence from the G7 countries. Physica A: Statistical Mechanics and its Applications, 324(12), 117. https://doi.org/10.1016/S0378-4371(02)01890-3.

    Article  Google Scholar 

  57. 57.

    Clauset, A., Shalizi, C. R., & Newman, M. E. (2009). Power-law distributions in empirical data. SIAM Review, 51(4), 661.

    Article  Google Scholar 

  58. 58.

    Piketty, T. (2014). Capital in the twenty-first century. Cambridge: Harvard University Press.

    Book  Google Scholar 

  59. 59.

    Brown, C. (1988). In Mangum, G. & P. Philips (Eds), Three worlds of labor economics (Vol. 51 pp. 515–530)

  60. 60.

    Brown, C. (2005). Is there an institutional theory of distribution? Journal of Economic Issues, 39(4), 915.

    Article  Google Scholar 

  61. 61.

    Dahrendorf, R. (1959). Class and class conflict in industrial society. Stanford: Stanford University Press.

    Google Scholar 

  62. 62.

    Huber, E., Huo, J., & Stephens, J. D. (2017). Power, policy, and top income shares. Socio-Economic Review. https://doi.org/10.1093/ser/mwx027.

    Google Scholar 

  63. 63.

    Kalecki, M. (1971). Selected essays on the dynamics of the capitalist economy 1933–1970. Cambridge: Cambridge University Press.

    Google Scholar 

  64. 64.

    Lenski, G. E. (1966). Power and privilege: A theory of social stratification. Chapel Hill: UNC Press Books.

    Google Scholar 

  65. 65.

    Marx, K. (1867). Capital (Vol. I). Penguin/New Left Review: Harmondsworth.

    Google Scholar 

  66. 66.

    Mills, C. W. (1956). The power elite. Oxford: Oxford University Press.

    Google Scholar 

  67. 67.

    Nitzan, J., & Bichler, S. (2009). Capital as power: A study of order and creorder. New York: Routledge.

    Book  Google Scholar 

  68. 68.

    Peach, J. T. (1987). Distribution and economic progress. Journal of Economic Issues, 21(4), 1495.

    Article  Google Scholar 

  69. 69.

    Sidanius, J., & Pratto, F. (2001). Social dominance: An intergroup theory of social hierarchy and oppression. Cambridge: Cambridge University Press.

    Google Scholar 

  70. 70.

    Weber, M. (1978). Economy and society: An outline of interpretive sociology. Berkeley: University of California Press.

    Google Scholar 

  71. 71.

    Wright, E. O. (1979). Class structure and income determination (Vol. 2). New York: Academic Press.

    Google Scholar 

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Acknowledgements

I would like to thank the Social Sciences and Humanities Resource Council of Canada (Grant no. 767-2015-1015) for its support. I thank also Jonathan Nitzan, who has offered feedback on aspects of this paper.

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Correspondence to Blair Fix.

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Fix, B. Hierarchy and the power-law income distribution tail. J Comput Soc Sc 1, 471–491 (2018). https://doi.org/10.1007/s42001-018-0019-8

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Keywords

  • Power law
  • Income distribution
  • Firm hierarchy
  • Economic modeling