Statistical Properties in Firms’ Large-scale Data (Evolutionary Economics and Social Complexity Science, 26) 9811622965, 9789811622960

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Evolutionary Economics and Social Complexity Science 26

Atushi Ishikawa

Statistical Properties in Firms’ Large-scale Data

Evolutionary Economics and Social Complexity Science Volume 26

Editors-in-Chief Takahiro Fujimoto, The University of Tokyo, Tokyo, Japan Yuji Aruka, Kyoto, Japan

The Japan Association for Evolutionary Economics (JAFEE) always has adhered to its original aim of taking an explicit “integrated” approach. This path has been followed steadfastly since the Association’s establishment in 1997 and, as well, since the inauguration of our international journal in 2004. We have deployed an agenda encompassing a contemporary array of subjects including but not limited to: foundations of institutional and evolutionary economics, criticism of mainstream views in the social sciences, knowledge and learning in socio-economic life, development and innovation of technologies, transformation of industrial organizations and economic systems, experimental studies in economics, agent-based modeling of socio-economic systems, evolution of the governance structure of firms and other organizations, comparison of dynamically changing institutions of the world, and policy proposals in the transformational process of economic life. In short, our starting point is an “integrative science” of evolutionary and institutional views. Furthermore, we always endeavor to stay abreast of newly established methods such as agent-based modeling, socio/econo-physics, and network analysis as part of our integrative links. More fundamentally, “evolution” in social science is interpreted as an essential key word, i.e., an integrative and /or communicative link to understand and re-domain various preceding dichotomies in the sciences: ontological or epistemological, subjective or objective, homogeneous or heterogeneous, natural or artificial, selfish or altruistic, individualistic or collective, rational or irrational, axiomatic or psychological-based, causal nexus or cyclic networked, optimal or adaptive, microor macroscopic, deterministic or stochastic, historical or theoretical, mathematical or computational, experimental or empirical, agent-based or socio/econo-physical, institutional or evolutionary, regional or global, and so on. The conventional meanings adhering to various traditional dichotomies may be more or less obsolete, to be replaced with more current ones vis-à-vis contemporary academic trends. Thus we are strongly encouraged to integrate some of the conventional dichotomies. These attempts are not limited to the field of economic sciences, including management sciences, but also include social science in general. In that way, understanding the social profiles of complex science may then be within our reach. In the meantime, contemporary society appears to be evolving into a newly emerging phase, chiefly characterized by an information and communication technology (ICT) mode of production and a service network system replacing the earlier established factory system with a new one that is suited to actual observations. In the face of these changes we are urgently compelled to explore a set of new properties for a new socio/economic system by implementing new ideas. We thus are keen to look for “integrated principles” common to the abovementioned dichotomies throughout our serial compilation of publications. We are also encouraged to create a new, broader spectrum for establishing a specific method positively integrated in our own original way. Editors-in-Chief Takahiro Fujimoto, Tokyo, Japan Yuji Aruka, Tokyo, Japan

Editorial Board Satoshi Sechiyama, Kyoto, Japan Yoshinori Shiozawa, Osaka, Japan Kiichiro Yagi, Neyagawa, Osaka, Japan Kazuo Yoshida, Kyoto, Japan Hideaki Aoyama, Kyoto, Japan Hiroshi Deguchi, Yokohama, Japan Makoto Nishibe, Sapporo, Japan Takashi Hashimoto, Nomi, Japan Masaaki Yoshida, Kawasaki, Japan Tamotsu Onozaki, Tokyo, Japan Shu-Heng Chen, Taipei, Taiwan Dirk Helbing, Zurich, Switzerland

More information about this series at http://www.springer.com/series/11930

Atushi Ishikawa

Statistical Properties in Firms’ Large-scale Data

Atushi Ishikawa Faculty of Economic Informatics Kanazawa Gakuin University Kanazawa, Ishikawa, Japan

ISSN 2198-4204 ISSN 2198-4212 (electronic) Evolutionary Economics and Social Complexity Science ISBN 978-981-16-2296-0 ISBN 978-981-16-2297-7 (eBook) https://doi.org/10.1007/978-981-16-2297-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

As I write this preface in April 2021, the world continues to suffer from the pandemic caused by the spread of Covid-19. Although experts from a variety of disciplines continue to persevere, an end to the pandemic does not seem imminent. Unfortunately, an economic crisis is choking the entire world. The last global economic crisis occurred about 90 years ago in the 1930s. What is very different today is that computers worldwide are collecting and analyzing big data for widespread use. Such technology has slowed the spread of Covid-19. In the same way, can the economic damage caused by a pandemic be ameliorated? Can economic recoveries be accelerated? I believe the answer to both questions lies in the handling of big data. Although big data is essentially just a collection of millions of bits of data, it doesn’t make any sense to analyze it blindly. To extract useful information from big data, we need a rough idea of the statistical properties of the data and their relationships. In this book, I introduce the statistical properties of firms’ financial big data and their relationships, which my collaborators and I have mainly studied in a young discipline called econophysics, which was born in the 1990s. The nature of the financial big data of firms during major economic changes is unknown, as we will describe. I believe that the knowledge gained from the analysis of big data from previous periods when economies did not change much will be a great guide. This book complies a series of my studies on statistical properties in firms’ largescale data, which were published in following fifteen academic journal articles: 1. Ishikawa A, Fujimoto S, Mizuno T (2019) Macroscopic properties in economic system and their relations. In: Chakrabarti A, Pichl L, Kaizoji T (eds) Network theory and agent-based modeling in economics and finance. Springer, Singapore, pp 133–157. 2. Ishikawa A (2006) Derivation of the distribution from extended Gibrat’s law. Physica A367:425–434 3. Ishikawa A (2007) The uniqueness of firm size distribution function from tentshaped growth rate distribution. Physica A383:79–84.

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4. Ishikawa A (2006) Annual change of Pareto index dynamically deduced from the law of detailed quasi-balance. Physica A371:525–535. 5. Ishikawa A (2009) Quasi-statically varying Pareto-law and log-normal distributions in the large and the middle scale regions of Japanese land prices. Prog Theor Phys Suppl 179:103–113. 6. Ishikawa A, Fujimoto S, Mizuno T (2011) Shape of growth rate distribution determines the type of non-Gibrat’s property. Physica A390:4273–4285. 7. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2016) Firm growth function and extended-Gibrat’s property. Adv Math Phys 2016:9303480. 8. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2016) Long-term firm growth properties derived from short-term laws and number of employees in Japan and France. Evolut Inst Econo Rev 13:409–422. 9. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2015) Firm age distributions and the decay rate of firm activities. In: Takayasu H, Ito N, Noda I, Takayasu M (eds) Proceedings of the international conference on social modeling and simulation, plus econophysics colloquium 2014. Springer, Tokyo, pp 187–194. 10. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2015) The relation between firm age distributions and the decay rate of firm activities in the United States and Japan. Big data, 2015 IEEE international conference on date of conference 2015, pp 2726–2731. 11. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2017) Dependence of the decay rate of firm activities on firm age. Evolut Inst Econo Rev 14:351–362. 12. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2017) Transition law of firms’ activity and the deficit aspect of non-Gibrat’s law. JPS Conf Proc 16:011005. 13. Ishikawa A, Fujimoto S, Mizuno T (2018) Statistical law observed in inactive rate of firms. Evol Inst Econ Rev 16:201–212. 14. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2014) Analytical derivation of power laws in firm size variables from Gibrat’s law and quasi-inversion symmetry: a geomorphological approach. J Phys Soc Jpn 83:034802. 15. Ishikawa A, Fujimoto S, Mizuno T (2020) Why does production function take the Cobb-Douglas form? Direct observation of production function using empirical data. Evol Inst Econ Rev. https://doi.org/10.1007/s40844020-00180-3. Kanazawa, Japan April 2021

Atushi Ishikawa

Acknowledgments

First, I express my deep gratitude to Dr. Yuji Aruka who gave me the opportunity to write this book. He recommended that I compile a series of my studies on the statistical properties of sets of firm-size variables and introduced me a publisher. I had never imagined writing a book, and I was surprised by his suggestion. However, my collaborators and researchers in the same field agreed that a book project was a good opportunity to organize my studies. I am grateful for this valuable opportunity. I also express my deep appreciation to the co-authors of the papers on which this book is based. Without their joint researches, this book would not exist. I continue to collaborate now with Dr. Takayuki Mizuno of the National Institute of Informatics and Dr. Shoji Fujimoto of Kanazawa Gakuin University. I previously collaborated with Dr. Tsutomu Watanabe of the University of Tokyo’s Graduate School and Dr. Masashi Tomoyose of the University of the Ryukyus. Collaboration with them was essential to the success of this book. I also thank Dr. Masahiro Anazawa of the Tohoku Institute of Technology and Dr. Tadao Suzuki of Fukushima Gakuin University, who co-authored papers that ignited this research series. In addition, I express my deep gratitude to Dr. Mieko Tanaka of Meiji University and Dr. Hideaki Aoyama of Kyoto University for hosting workshops where a series of research was presented for this book. I thank the many researchers who participated in workshops and discussed various issues. The researches presented in this book were fostered by such scientific presentations. Dr. Fujimoto carefully read the manuscript and pointed out a number of problems. It goes without saying that any problems are my responsibility. Finally, I thank Mr. Yutaka Hirachi and Ms. Vaishnavi Venkatesh of Springer who supported my book. I thank Kanazawa Gakuin University and my colleagues for providing me with a home for my research activities.

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Contents

1

Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 5

2 Non-Gibrat’s Property in the Mid-Scale Range . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Data .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Previous Research .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.1 Properties at Fixed Time . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.2 Properties in Short-Term Period . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.3 Derivation of Power Law from Short-Term Properties . . . . . . . 2.4 Non-Gibrat’s Property in the Mid-Scale Range of Current Profits . . . 2.5 Discussion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

7 7 9 10 10 12 14 16 19 20

3 Quasi-Statistically Varying Power-Law and Log-Normal Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Data .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Varying Power-Law and Log-Normal Distributions.. . . . . . . . . . . . . . . . . . 3.4 Quasi-Time-Reversal Symmetry . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4.1 Quasi-Statically Varying Power Law . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4.2 Quasi-Statically Varying Log-Normal Distribution .. . . . . . . . . . 3.4.3 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5 Discussion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

23 23 24 25 29 30 31 33 35 36

4 Extension of Non-Gibrat’s Property . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Data .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Properties at Fixed Time and in Short-Term Period .. . . . . . . . . . . . . . . . . . 4.3.1 Properties at Fixed Time . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.2 Properties in Short-Term Period . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

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4.4 Non-Gibrat’s Property of Operating Revenues and Total Assets . . . . . 4.5 Discussion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

47 52 53

5 Long-Term Firm Growth Derived from Non-Gibrat’s Property and Gibrat’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Data Analyses of Long-Term Growth of Firms. . . .. . . . . . . . . . . . . . . . . . . . 5.3 Data Analyses of Short-Term Growth of Firms . . .. . . . . . . . . . . . . . . . . . . . 5.4 Long-Term and Short-Term Firm Growth . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.5 Discussion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

55 55 58 62 64 66 68

6 Firm-Age Distribution and the Inactive Rate of Firms . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Data .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Data Analysis.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Inactive Rate of Firm and Firm-Age Distribution .. . . . . . . . . . . . . . . . . . . . 6.5 Discussion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

71 71 73 74 76 79 80

7 Statistical Properties in Inactive Rate of Firms. . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Activity Status and Financial Variables Dataset . . .. . . . . . . . . . . . . . . . . . . . 7.3 Dependence of Inactive Rate on Firm Size . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4 Relation Between Non-Gibrat’s Property and Inactive Rate .. . . . . . . . . 7.5 Discussion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

83 83 84 86 88 91 92

8 Power Laws with Different Exponents in Firm-Size Variables . . . . . . . . . 95 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 95 8.2 Data .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 96 8.3 Different Exponent Power Laws and Quasi-Inversion Symmetry . . . . 99 8.4 Identification of Ridge Using Surface Openness . .. . . . . . . . . . . . . . . . . . . . 103 8.5 Discussion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 106 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 109 9 Why Does Production Function Take the Cobb–Douglas Form? .. . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Data .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 Quasi-Inverse Symmetry in Two and Three Dimensions . . . . . . . . . . . . . 9.4 Modified Constant Returns to Scale. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.5 Analytical Preparation for Direct Observation of Cobb–Douglas Symmetric Plane . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

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9.6 Direct Observation of Cobb–Douglas Quasi-Inverse-Symmetric Plane . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 126 9.7 Conclusion and Discussion .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 129 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 134 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 137

About the Author

Atushi Ishikawa, is a member of the Faculty of Economic Informatics at Kanazawa Gakuin University. The author was originally a theoretical physicist of elementary particles. He now specializes in Econophysics and is primarily engaged in the study of the statistical properties of firms’ large-scale financial data. The study covers a wide range of other topics, including analyzing point-of-sale (POS) data, analyzing Twitter, and analyzing land prices.

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Chapter 1

Introduction

Abstract In this chapter, I present my views on the future direction of economics from the history of the development of physics. In physics, statistical rules found in macroscopic systems guide the theoretical framework of microscopic systems. In both physics and economics, macroscopic systems are aggregates of microscopic components. Therefore, the methodology of physics should be an important guide to the development of economics. From this perspective, this chapter describes the structure of this book.

In a system composed of many firms, various statistical properties are observed in terms of the quantities that represent the size of firms, such as operating revenues, assets, profits, and number of employees (herein referred to as firm-size variables). Interestingly, the universal structure observed in natural science also exists in economic science. In recent years, econophysics has recaptured economic phenomena from the viewpoint and method of physics. Many stylized facts about firms and their relationships are also being clarified (see [1–5], for example). In the history of physics, we extracted various universal structures from nature and described them logically and mathematically, including Newtonian mechanics in the seventeenth century, electromagnetism, relativity, and thermodynamics in the nineteenth century, and quantum mechanics in the twentieth century. Here we focus on thermodynamics, a system that describes the relationship between macroscopic matter and heat based on various observations and experiments codified in the mid-nineteenth century. Next came the kinetic theory of gas and statistical mechanics, and it became clear that matter’s microstructure is dominated by quantum mechanics. Surprisingly, despite the development of statistical and quantum mechanics, thermodynamics did not require any corrections, suggesting that its universal structure is not influenced by the unfocused details of microscopic structure and unknown elements. In other words, a universal structure exists that independently describes the macroscopic world of microscopic theory [6]. In addition to thermodynamics, the motion of rigid bodies, critical phenomena in magnetic materials and fluids, etc. are universal structures of the macroscopic world that are also independent of system details. Furthermore, statistical mechanics © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Ishikawa, Statistical Properties in Firms’ Large-scale Data, Evolutionary Economics and Social Complexity Science 26, https://doi.org/10.1007/978-981-16-2297-7_1

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1 Introduction

that discusses microscopic multi-body systems is a theoretical system that is designed to be consistent with thermodynamics. The above physical viewpoint is a guide that clarifies the structure of economic science. Using an analogy to physics, the macroscopic statistical properties of economic systems, obtained as a result of the actions of individual firms, may constitute a theoretical system independent of microscopic systems. The history of the development of physics suggests that the theoretical framework of the decisionmaking mechanisms of micro-individual firms should be structured with guidelines that are consistent with macro-statistical properties. Of course, it goes without saying that the theoretical framework of the decision-making mechanism of microindividual firms (or economic agents) is important. This is evident from the role that statistical mechanics plays a role in physics. Physics, in particular, is the field of natural science in which it is relatively easy to carry out experiments or construct theories while keeping research subjects in an ideal state. Because of these characteristics, physical methods have played a leading role in understanding nature through experiments and theories. However, even in relatively simple physics, combining the microscopic world described by statistical mechanics with the macroscopic world described by thermodynamics has been successful only in a limited system called the “equilibrium state.” On the contrary, the statistical mechanics of the “non-equilibrium state” remains only partially complete at a stage before being connected to macroscopic systems. Economics deals with fiscal phenomena that are much more complex than those studied in physics. In contrast to physics, a system in an equilibrium state can bridge the gap between microscopic and macroscopic theories. The research by Dr. Aoki and Dr. Yoshikawa was pioneering [7], and the work by Dr. Aoyama and his group is considered the fruits of such a bridge [5, 8–14]. On the other hand, however, whether in physics or in economics I believe that linking the microscopic and macroscopic worlds of “non-equilibrium states” is impossible with the mathematical tools presently possessed by humankind. In this context, I first discuss the macronature of financial data from the number of firms and their relationships. Note that in this book, the distribution formed by a set of data is viewed as a macroscopic object in the sense that it completely ignores the behavior of individual firms. A phenomenon in which distribution does not change is regarded as a phenomenon in which macroscopic statistics do not change. This approach is similar to that used in economics, where the decision-making mechanism of a microeconomic entity is regarded as a micro-object, and GDP, CPI, or the unemployment rate obtained by totaling individual data are used as a macroeconomic index. In the above sense, the corporate financial data used in this book are in equilibrium because there is no significant change over time in the macroscopic statistical properties. This may be because the data for the period being analyzed happened to be in equilibrium. Alternatively, some studies suggest that firm-size data may essentially be in equilibrium (for example, see [15]). Although in both economics and physics, state change is really quite interesting. For this reason, I also discuss systems that change over time. However, as noted earlier, since the corporate

1 Introduction

3

financial data presented in this book appear to be in equilibrium, my discussion focuses on Japanese land prices. In thermodynamics, a theory is constructed by requiring that the beginning and the end of a system of changing states be in equilibrium. We then consider the changes due to various processes during that state. As one example of them, the following changes are addressed. Because the time evolution of a system is so slow, we sometimes assume an extreme change that allows the system to be considered in equilibrium at any time during the course of the change. Such a slow change is called quasi-static. In this book, we follow this approach and treat fluctuations in land prices in Japan as quasi-static and construct a theory. This approach is consistent with observations of land prices, except when the system changes excessively. In this book from these points of view, I discuss various statistical properties and their relationships that are identified in a number of firm-size variables. It is organized as follows. First in Chap. 2, we consider the properties observed in a certain amount of firms at a certain time, such as a particular year. We introduce the idea that a powerlaw distribution is observed in the large-scale range [16–18] and a log-normal distribution is observed in the mid-scale range [19–21]. Next we introduce a study by Aoyama et al. that derives a power distribution from time-reversal symmetry and Gibrat’s law observed in the large-scale range [22, 23]. A time-reversal symmetry indicates that the system is in equilibrium. Gibrat’s law states that the growth-rate distribution, which is conditioned by the initial value of the variable, does not depend on the initial value. This book is based on these critical starting points. Finally, we show that a log-normal distribution is derived from the time-reversal symmetry observed in the large-scale range and the non-Gibrat’s property in the mid-scale range, which is not found in the large-scale range. We then prove that in the limited case of a firm’s positive current profits, the non-Gibrat’s property consistent with time-reversal symmetry is uniquely determined. In Chap. 3, as a time-varying system, we consider the properties of land prices in Japan. We describe how power-law distribution in the large-scale range and lognormal distribution in the mid-scale range are observed in land prices and argue that they change in time. By combining quasi-time-reversal symmetry, which denotes a quasi-static change in the system, and Gibrat’s law in the large-scale range or the non-Gibrat’s property in the mid-scale range, a quasi-static power-law distribution or a log-normal distribution is derived. Our analytical discussion is consistent with the land price data. In Chap. 4, the non-Gibrat’s property from Chap. 2 is extended. The special case discussed in Chap. 2 is that the conditional growth-rate distribution is linear on the log–log axis. Although this is a property of positive current profits, the conditional growth-rate distribution often has a convex downward curvature in many firm-size variables, such as operating revenues and total assets. Even in such cases, we can extend the discussion in Chap. 2 to describe the mid-scale non-Gibrat’s property and show that it can be combined with time-reversal symmetry to derive a log-normal distribution. This analytical discussion is supported by a firm’s operating revenue and total asset data.

4

1 Introduction

So far, we have discussed the relationships between the properties observed in time-constant firm-size variables, such as the power-law and log-normal distributions, as well as the properties observed in the short-term state changes of firm-size variables, such as time-reversal symmetry, quasi-time-reversal symmetry, Gibrat’s law, and non-Gibrat’s property. In Chap. 5, we combine short-term and long-term statistical properties. When considering a group of firms, the geometric mean of the firm-size variables is observed as follows. When the firm is young, elements of its firm size such as operating revenues grow rapidly under a power law depending on its age and shift to a gradual exponential growth. We then use simulations to explain that these two long-term growth properties are derived from the non-Gibrat’s property in the midscale range and Gibrat’s law in the large-scale range. So far, we have discussed the relationships between the properties of firm-size variables observed at a certain time in short- and long-term properties. Note that all of these properties are limited to firms that continue to operate. In the next two chapters, we discuss the statistical properties of those firms that ceased to operate. In Chap. 6, we discuss the short-term properties of firms’ inactivity and such long-term properties as firm-age distribution. We derived firm-age distribution from the firm-age dependency of the rate at which active firms cease operating (their inactivity rate). This derivation uses the identical approach as the theory of nuclear decay [24]. In Chap. 7, we discuss the relationships between the inactive rate of firms introduced in Chap. 6 and firm size. We show that when firm size exceeds a certain threshold, the inactive rate of firms does not depend on firm size, and when it is smaller than a threshold, the inactive rate decreases as firm size increases. These properties, which are Gibrat’s law and non-Gibrat’s property for firms that have ceased their activities, are paired with Gibrat’s law and non-Gibrat’s property observed in firms that continue their activity. Although we broached the temporal, short-term, and long-term properties of various firm-size variables as well as the short- and long-term statistical properties of the inactive rate of firms, we have only dealt with one type of firm-size variables. The last two chapters discuss the observed properties of different types of firm-size variables, such as operating revenues, tangible fixed assets, number of employees, and their relationships. In Chap. 8, we argue that the ratio of the exponents of the power-law distributions observed in the large-scale ranges of a firm’s operating revenues, its tangible fixed assets, and its number of employees equals the slope of the symmetry axis of the quasi-inversion symmetry of their combination. This idea simultaneously applies the quasi-time-reversal symmetry introduced in Chap. 2 to different firmsize variables. Then we confirm the result of this analysis framework by empirical data. To identify the axis of symmetry of quasi-inversion symmetry, we borrow an index called surface openness, which is used in geomorphology [25–27]. In Chap. 9, we extend Chap. 8’s discussion to quasi-inversion symmetry among three variables of firm size and combine the power-law distributions of three kinds of firm-size variables. Furthermore, the quasi-inversion symmetry of the three

References

5

variables is viewed in a three-variable space as symmetry about a two-dimensional plane. If the three firm-size variables are operating revenues, tangible fixed assets, and number of employees, the quasi-inversion symmetric plane can be expressed as a Cobb–Douglas type of production function [28], which is frequently used in economics. Using the surface openness introduced in Chap. 8, we identify a Cobb– Douglas type quasi-inversion symmetric plane, which is quantitatively consistent with that used in economics. This book is composed of the above nine Chapters. Although Chap. 1 begins with a discussion of the unique studies of econophysics, Chap. 9 introduces an econophysical approach to the origin of an important component of economics, the Cobb–Douglas production function. Except for Chaps. 7 and 9, all of the data analysis for this book was rewritten or revised. The analysis in Chap. 8 is new for this book, and Chap. 4’s discussion is an extension of an earlier paper. Other details were added to this book. Empirical studies using large-scale data provide clues to the macroscopic nature of economic phenomena and their relationships, such as thermodynamics in physics. The discussion in this chapter and book is based primarily on our previous works [29]. Most texts are reproduced under the Creative Commons Attribution License or with permission from publishers. The main text has been edited or rewritten to fit the context of this book.

References 1. Mantegna RN, Stanley HE (2000) Introduction to econophysics: correlations and complexity in finance. Cambridge University Press, Cambridge 2. Saichev AI, Malevergne Y, Sornette D (2010) Theory of Zip’s Law and beyond. Springer, Berlin 3. Garibaldi U, Scalas E (2010) Finitary probabilistic methods in econophysics. Cambridge University Press, Cambridge 4. Aoyama H, Fujiwara Y, Ikeda Y, Iyetomi H, Souma W (2010) Econophysics and companies: statistical life and death in complex business networks. Cambridge University Press, Cambridge 5. Aoyama H, Fujiwara Y, Ikeda Y, Iyetomi H, Souma W, Yoshikawa H (2017) Macroeconophysics: new studies on economic networks and synchronization (physics of society: econophysics and sociophysics). Cambridge University Press, Cambridge 6. Tazaki H (2000) Thermodynamics. Baifukan, Tokyo (in Japanese) 7. Aoki M, Yoshikawa H (2007) Reconstructing macroeconomics – a perspecrive from statistical physics and conbinatiorial stochastic processes. Cambridge University Press, Cambridge 8. Aoyama H, Yoshikawa H, Iyetomi H, Fujiwara Y (2009) Labour productivity superstatistics. Prog Theor Phys Suppl 179:80–92 9. Aoyama H, Fujiwara Y, Ikeda Y, Iyetomi H, Souma W (2009) Superstatistics of labour productivity in manufacturing and nonmanufacturing sectors. Eco Open Acces Open Assess E J 3:2009–2022 10. Souma W, Ikeda Y, Iyetomi H, Fujiwara Y (2009) Distribution of labour productivity in Japan over the period 1996–2006. Eco Open Acces Open-Asses E J 3:2009–2014 11. Aoyama H, Yoshikawa H, Iyetomi H, Fujiwara Y (2010) Productivity dispersion: facts, theory, and implications. J Econo Interact Coord 5:27–54

6

1 Introduction

12. Iyetomi H (2012) Labor productivity distributions with negative tempreature. Prog Theor Phys Suppl 194:135–143 13. Aoyama H, Iyetomi H, Yoshikawa H (2015) Equilibrium distribution of labor productivity. J Econo Interact Coord 10:57–66 14. Yoshikawa H (2020) Reconstructing macroeconomics – Keynes and Schumpeter. Iwanami Shoten, Tokyo (in Japanese) 15. Mizuno T, Katori M, Takayasu H, Takayasu M (2002) Statistical and stochastic laws in the income of Japanese companies. In: Takayasu H (ed) Empirical science of financial fluctuations – the advent of econophysics. Springer, Tokyo, p 321–330 16. Pareto V (1897) Cours d’Économie politique. Macmillan, London 17. Newman MEJ (2005) Power laws, Pareto distributions and Zipf’s law. Contemp Phys 46:323– 351 18. Clauset A, Shalizi CR, Newman MEJ (2009) Power-law distributions in empirical data. SIAM Rev 51:661–703 19. Gibra R (1932) Les Inégalités Économique. Sirey, Paris 20. Badger WW (1980) An entropy-utility model for the size distribution of income. In: West BJ (ed) Mathematical models as a tool for the social science. Gordon and Breach, New York, p 87–120 21. Montroll EW, Shlesinger MF (1983) Maximum entropy formalism, fractals, scaling phenomena, and 1/f noise: a tale of tails. J Stat Phys 32:209–230 22. Fujiwara Y, Souma W, Aoyama H, Kaizoji T, Aoki M (2003) Growth and fluctuations of personal income. Physica A321:598–604 23. Fujiwara Y, Guilmi CD, Aoyama H, Gallegati M, Souma W (2004) Do Pareto-Zipf and Gibrat laws hold true? An analysis with European firms. Physica A335:197–216 24. Rutherford E, Soddy F (1903) Radioactive change. Phil Mag 6:576–591 25. Yokoyama R, Sirasawa M, Kikuchi Y (1999) Representation of topographical features by opennesses. J Jpn Soc Photogramm Remote Sens 38(4):26–34 (in Japanese with English abstract) 26. Yokoyama R, Sirasawa M, Pike R (2002) Visualizing topography by openness: a new application of image processing to digital elevation models. Photogramm Eng Remote Sens 68(3):257–265 27. Prima ODA, Echigo A, Yokoyama R, Yoshida T (2006) Supervised landform classification of Northeast Honshu from DEM-derived thematic maps. Geomorphology 78:373–386 28. Cobb CW, Douglass PH (1928) A theory of production. Am Eco Rev 18:139–165 29. Ishikawa A, Fujimoto S, Mizuno T (2019) Macroscopic properties in economic system and their relations. In: Chakrabarti A, Pichl L, Kaizoji T (eds) Network theory and agent-based modeling in economics and finance. Springer, Singapore, p 133–157

Chapter 2

Non-Gibrat’s Property in the Mid-Scale Range

Abstract The distribution of the growth rate does not depend on the initial value in a large-scale range of firm-size variables, such as operating revenues, assets, current profits, and the number of employees. This is called Gibrat’s law. On the other hand, in the mid-scale range of firm-size variables, the growth-rate distribution changes regularly based on the initial value. This idea is referred to as the nonGibrat’s property in this book. In this chapter we show that when the system is in equilibrium, a log-normal distribution of firm-size variables is derived from the non-Gibrat’s property in the mid-scale range. When the growth-rate distribution is linear on the log–log axis, it is analytically proven that the form of the system’s nonGibrat’s property at equilibrium is uniquely determined. We then confirm that these properties can be observed with high accuracy in empirical data using the positive current profit data for Japanese firms in 1998 and 1999.

2.1 Introduction In this chapter, we focus on the distribution of firm-size variables observed in the mid-scale range at a given time, such as a “certain year.” The discussion is based primarily on my previous works [1, 2]. Most texts are reproduced under the Creative Commons Attribution License or with permission from the publisher. The main text is modified to fit the context of this book. In this book, we refer to the variables that represent the size of a firm, such as its operating revenues, number of employees, assets, profits, etc., as firm-size variables. The size of each firm changes over time due to various external and internal factors. Describing these microscopic changes is extremely difficult. However, when dealing with a very large group of such firms, macroeconomic properties emerge that differ from those of individual firms. In this book, the distribution formed by a set of data is viewed as a macroscopic object in the sense that it completely ignores the behavior or changes of individual firms. The study of this macroscopic property began with the discovery in the nineteenth century by Italian economist Vilfredo Pareto that the distribution of personal income in England follows a power law [3]. Since then, it © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Ishikawa, Statistical Properties in Firms’ Large-scale Data, Evolutionary Economics and Social Complexity Science 26, https://doi.org/10.1007/978-981-16-2297-7_2

7

8

2 Non-Gibrat’s Property in the Mid-Scale Range

has been clarified that various quantities follow the power-law distribution in fields such as economics, sociology, and financial engineering [4–18]. At the same time, as the amount of data to be analyzed increases, the power-law distribution holds only in the large-scale range. In this book, a range smaller than the large-scale range is called the mid-scale range. The distribution followed by mid-scale variables is considered the log-normal distribution at its simplest. Although such log-normal distribution is often described as the result of the simplest Gibrat’s process in a multiplicative stochastic process [19–21], the derived log-normal distribution is not stable and increases with time. This procedure is inappropriate for the stable firm-size variables discussed in this book. To obtain a stable distribution from a stochastic process, we can add additive noise to a multiplication stochastic process or introduce a reset event or a reflection wall [22–26]. In these cases, however, the stable distribution is a power law, and the log-normal distribution appears to violate the power-law distribution. Instead of using a specific model as a starting point like a stochastic process, Aoyama et al. focused on two properties observed in short-term changes in a very large set of personal income, firm-size variables, and so on and showed that a power-law distribution was automatically derived from them [27, 28]. One of these is Gibrat’s law, a concept that resembles Gibrat’s process. The other is the detailed balance (time-reversal symmetry) that is observed when the system is in equilibrium. Aoyama et al. first confirmed the existence of Gibrat’s law and the time-reversal symmetry in personal income for two consecutive years based on empirical data. They showed that a power-law distribution was derived by analytical discussion and its consistency was confirmed by empirical data. They validated this argument by confirming with empirical data that the same properties were observed for firm-size variables. Their method is different from the conventional method because they replaced the conventional Gibrat’s process with Gibrat’s law, which can be confirmed in empirical data, and introduced time-reversal symmetry, which can also be confirmed in empirical data to incorporate the system’s stability into the theory. Two possible directions exist for developing their argument and exploring such macroproperties as firm-size variables from a broader perspective. First, we extend the time-reversal symmetry observed when the system is in equilibrium to the quasi-time-reversal symmetry observed when the system changes quasi-statically and construct a theory that can explain the quasi-static change of the power-law distribution. This is introduced in the next chapter. The other direction focuses on a property that holds in the mid-scale range where Gibrat’s law does not hold (in this book, we refer to it as the non-Gibrat’s property) and combines it with the timereversal symmetry to identify the distribution that holds in the mid-scale range. This extension is the purpose of this chapter. This chapter is organized as follows. Section 2.2 describes the data used in it. In Sect. 2.3, we explain the above time-reversal symmetry and Gibrat’s law and introduce a study by Aoyama et al. who argued that the power law can be derived from the data. We discuss a large-scale range in the firm-size variables. Then in Sect. 2.4, as a counterpart to Gibrat’s law, we describe the non-Gibrat’s property

2.2 Data

9

observed in the mid-scale range and combine it with time-reversal symmetry to derive a log-normal distribution. When the growth-rate distribution is linear in a log–log axis, we also analytically prove that there is only one expression of nonGibrat’s property. We then confirm the consistency between the conclusions of these analytical discussions and the empirical positive current profit data. Finally, Sect. 2.5 summarizes this chapter.

2.2 Data This book deals with three types of databases. The first is Orbis, one of the world’s largest corporate finance databases provided by Bureau van Dijk [29]. The second is CD Eyes 50, a financial database of Japanese firms, provided by Tokyo Shoko Research [30]. The last is the official land price data provided by Japan’s Ministry of Land, Infrastructure and Transport [31]. The last data source is suitable for observing the temporal changes in the power-law and log-normal distributions. The details are given in Chap. 3. The methods used to collect data from the first two corporate financial databases share the following similarities. The first database contains financial data for up to ten consecutive years from the latest available year for a firm. The second database contains such data for up to three consecutive years. Therefore, these databases include a large amount of data near the year of purchase, although the amount of old data is small. We have three editions of Orbis. On the other hand, since we have the 1999 to 2009 editions of CD Eyes, we can analyze the financial data of about 500,000 firms in Japan from 1998 to 2008. Here we consider the positive current profit data as the firm-size variables with the simplest form of the nonGibrat’s property in Sect. 2.1. Table 2.1 shows the amount of annual current profit data presented on CD Eyes analyzed in this chapter. The years of the edition and the data differ by 1 year because the data of the year of the edition are scarce in the data Table 2.1 Amount of Japanese firms’ current profit data in CD Eyes

Year 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

Profit data 466,422 450,972 429,266 477,222 493,308 494,917 499,742 499,084 496,037 482,829 436,570

10

2 Non-Gibrat’s Property in the Mid-Scale Range

collection process. We observe the positive current profits of Japanese firms in 1998 and 1999 because they are the oldest and most abundant data we have. Similar analyses are possible for other years, and the results are summarized and tabulated at the end of this chapter. The same is confirmed in the current profit data of Japanese firms, which have the most available data in our most recent Orbis 2016 edition.

2.3 Previous Research In this section, as a preliminary study, we first introduce two properties that can be observed in the set of firm-size variables at a given time, such as a certain year. One is a power-law distribution [3, 17, 18] in the large-scale range, and the other is a log-normal distribution [19–21] in the mid-scale range. Next we introduce two properties observed in short-term changes in the firm-size variables, such as a year and the next year. One is the time-reversal symmetry, which can be observed in the joint distribution of firm-size variables over two consecutive years. The other is Gibrat’s law, which can be seen in the growth-rate distribution of firm-size variables over two consecutive years. Finally, we introduce a study by Aoyama et al. that shows that the power-law distribution in the large-scale range at a fixed time can be derived from the above two properties found in the short-term changes of the firm-size variables [27, 28].

2.3.1 Properties at Fixed Time As mentioned in the introduction, in the large amount of operating revenues, profits, assets, and number of employees that represent the size of firms (firm-size variables), properties are universally observed, which are unaffected by individual microscopic structures and firm behaviors. Here we first describe the power-law and log-normal distributions because they are the oldest known properties of firms. Both distributions are observed in the large- and mid-scale ranges of the firm-size variables at fixed times. The properties of the small-scale range have not been established because the amount of data is too scarce. In the low-scale range, several distributions have been proposed [32–35]. The power-law distribution has been observed in the large-scale range of various firm-size variables [3, 17, 18]: P (x) ∝ x −μ−1

for x0 < x.

(2.1)

Here x is firm-size variables such as operating revenues, number of employees, assets, and profits, P (x) is the probability density function (PDF), μ is a parameter that represents the spread of the power-law distribution, and x0 is the lower limit

2.3 Previous Research

 Log-normal Pareto

 

Px

Fig. 2.1 Positive current profit distribution of Japanese firms in 1998 as described in CD Eyes. In Eq. (2.1), Pareto index is measured by regression analysis as μ = 1.00 ± 0.01 in range x0 (= 104.6 ) < x < 108.0 . In addition, when Eq. (2.2) is applied to mid-scale range of xmin (= 102.7 ) < x < x0 (= 104.6 ), αLN = 0.17 ± 0.04 is estimated by regression analysis

11

    











Profits x in thousands ot yen

of the large-scale range. This is also called Pareto’s law, which was discovered in the UK’s personal income distribution by Italian economist Vilfredo Pareto [3]. Power-law index μ is also called the Pareto index. Although individual firm-size variables change annually, this property is universally observed in the large-scale range regardless of the type of firm-size variables, the country to which the firms belong, or the year that was measured. Pareto index μ is generally around 1, although it varies depending on the observation variables. Figure 2.1 shows the distribution of positive current profits of Japanese firms for 1998, as described in CD Eyes. The horizontal axis represents current profits (unit: 1000 Japanese yen) and the vertical axis represents PDF. Setting x0 = 104.6, the Pareto index can be estimated to be μ = 1.00 ± 0.01 by regression analysis. To eliminate the fluctuation of several points at the right end, the evaluation is made in the range of x0 (= 104.6 ) < x < 108.0. From Fig. 2.1, it can be confirmed that Eq (2.1) holds with high accuracy. Similar to the power law in the large-scale range of the firm-size variables, a log-normal distribution is also found in the mid-scale range [19–21]:   x P (x) ∝ x −μ−1 exp −α ln2 x0

for xmin < x < x0 .

(2.2)

Here α is a parameter that represents the spread of the log-normal distribution, and x0 is the upper limit of the mid-scale range, which is a boundary with a large-scale range. Equation (2.2) can be connected smoothly with Eq. (2.1) in the large-scale range by setting α = 0. Therefore, we rewrite the description of the log-normal distribution from the standard one and express it as Eq. (2.2). Similar to the power law in the large-scale range, although the firm-size variables change from year to year, the log-normal distribution is universally found in the mid-scale range regardless of the type of firm-size variables, the country to which the firms belong, or the year that was measured.

12

2 Non-Gibrat’s Property in the Mid-Scale Range

On the log–log axis, the log-normal distribution (2.2) is drawn as a quadratic function. In Fig. 2.1, omitting the point under 102.7 where the fluctuation is large, in the mid-scale range of 102.7 < x < x0 (= 104.6 ), the log-normal distribution (2.2), which is a quadratic function, precisely fits the data. At this time, the regression analysis estimates αLN = 0.17 ± 0.04.

2.3.2 Properties in Short-Term Period Next we introduce two types of properties that are identified in the amount of firmsize variables in a short-term period for two consecutive years. The first is timereversal symmetry, which indicates that a firm-size variable system is in equilibrium for two consecutive years. The second is Gibrat’s law in the large-scale range that is found in the growth of firm-size variables from 1 year to the next. In this book, we use T as a symbol for the calendar year. Denote (xT , xT +1 ) as firm-size variables for two consecutive years and focus on joint PDF P (xT , xT +1 ). This is observed as a scatter plot of (xT , xT +1 ) in our data analysis. Symmetry is found in the joint PDF when the firm-size variables are in an equilibrium state. Even though individual firm-size variables vary, when their aggregates are in equilibrium, we studied a system’s symmetry with respect to a time reversal: xT ↔ xT +1 . This is called time-reversal symmetry, which is expressed using joint PDF [27, 28]: PJ (xT , xT +1 ) = PJ (xT +1 , xT ).

(2.3)

Note that its function form is identical on both sides of Eq. (5.3). The system is symmetrical with respect to a straight line: log10 xT +1 = log10 xT in both logarithmic axes. In Eq. (2.3), measure dxT dxT +1 is omitted because it is shared by both sides. Time-reversal symmetry, which is a concept that describes the equilibrium state of a system in thermodynamics, is also called a detailed balance. In this subsection, we first consider the system’s equilibrium and address the short-term growth rate of firm-size variables. We define the growth rate of the firmsize variables for two consecutive years as R = xT /xT +1 . At this time, conditional PDF Q(R|xT ) of growth rate R, which is conditioned by the firm-size variable of first year xT , has different characteristics in the large- and mid-scale ranges [36, 37]. Interestingly, Q(R|xT ) does not depend on the firm-size variable in first year xT in large-scale range xT > x0 [19, 38]: Q(R|xT ) = Q(R) for x0 < xT .

(2.4)

This is called Gibrat’s law, and lower bound x0 corresponds to the lower limit of the large-scale range in which the power law (2.1) holds. As will be described in this book, there are two types of growth-rate distributions, depending on the type of firm-size variables. One is linear in both logarithmic axes, and the other has a

2.3 Previous Research

13

Fig. 2.2 Log–log plot of positive current profits of Japanese firms in 1998 x1998 (in thousands of yen) and in 1999 x1999 (in thousands of yen)

n=1 n=2 n=3 n=4 n=5

 

qr|n

Fig. 2.3 Conditional PDFs of log growth rate r = log10 x1999 /x1998 , calculated using data plotted in Fig. 2.2: Initial value xT is divided into five logarithmically equal-sized bins: x1998 ∈ [101+0.4(n−1) , 101+0.4n ) (n = 1, 2, · · · , 5). Data range shown here is 10 ≤ xT < 103 (thousands of yen)

   -4

-3

-2

-1

0

1

2

3

4

r

downward convex curvature. In both cases, Q(R|xT ) does not depend on xT in the large-scale range. This property is quite interesting where the growth of the largescale, firm-size variables beyond threshold x0 is not dependent on its initial value. This universal structure has been identified in different firm-size variables, different countries, and different years. Figure 2.2 is a log–log plot of the positive current profits of Japanese firms in 1998 (x1998) used in Fig. 2.1 and the positive current profits of those firms in 1999 (x1999). Note that the figure does not include firms that had positive profits in 1998 and negative profits in 1999. From Fig. 2.2, the figure is symmetrical even if the vertical and horizontal axes are interchanged. This is time-reversal symmetry (2.3). Figures 2.3, 2.4, and 2.5 are conditional PDFs of the logarithmic growth rate of positive current profits R = log x1999/x1998 at T = 1998, calculated using the data plotted in Fig. 2.2. In their growth-rate distributions, the initial conditions are set such that the positive current profits for 1998 are divided into the following 15 logarithmically equal-sized bins: x1998 ∈ [101+0.4(n−1), 101+0.4n ) (n = 1, 2, · · · , 15). The xT dependence of the respective positive and negative standard deviations σ± of the 15 conditional PDFs in Figs. 2.3, 2.4, and 2.5 is

14

n=6 n=7 n=8 n=9 n=10

 

qr|n

Fig. 2.4 Conditional PDFs of positive log growth rate r = log10 x1999 /x1998 , calculated using data plotted in Fig. 2.2. Initial value xT is divided into five logarithmically equal-sized bins: x1998 ∈ [101+0.4(n−1) , 101+0.4n ) (n = 6, 7, · · · , 10). Data range shown here is 103 ≤ xT < 105 (thousands of yen)

2 Non-Gibrat’s Property in the Mid-Scale Range

   -4

-3

-2

-1

0

1

2

3

4

r n=11 n=12 n=13 n=14 n=15

 

qr|n

Fig. 2.5 Conditional PDFs of log growth rate r = log10 x1999 /x1998 , calculated using data plotted in Fig. 2.2: Initial value xT is divided into five logarithmically equal-sized bins: x1998 ∈ [101+0.4(n−1) , 101+0.4n ) (n = 11, 12, · · · , 15). Data range shown here is 105 ≤ xT < 107 (thousands of yen)

   -4

-3

-2

-1

0

1

2

3

4

r illustrated in Fig. 2.6. For xT greater than x0 ∼ 104.6, there is no significant change in σ± , depending on xT . This corresponds to the fact that in Fig. 2.5, the growth-rate distributions hardly changed at n > 10. That is, in the range where x > 104.6 (= x0 ), q(r|xT ) hardly depends on xT , confirming that Gibrat’s law (2.4) was established.

2.3.3 Derivation of Power Law from Short-Term Properties Here we derive a static power law (2.1) at a fixed time from Gibrat’s law (2.4) where the conditional growth-rate distribution does not depend on the value of the first year in the large-scale range under the equilibrium state of firm-size variables (time-reversal symmetry (2.3)) following pioneering studies [27, 28].

2.3 Previous Research

15

2

V+ V

sigma

1.5

1

0.5

0

1

2

4

3

5

6

7

logxT Fig. 2.6 xT dependence of positive and negative standard deviations σ± for 15 conditional PDFs in Figs. 2.3, 2.4, and 2.5

When the system is in equilibrium, the time-reversal symmetry (2.3) holds. Using growth rate R = xT +1 /xT , the time-reversal symmetry is rewritten by variables xT , R as follows: PJ (xT , R) = R −1 PJ (RxT , R −1 ).

(2.5)

Using conditional PDF Q(R|xT ) = PJ (xT , R)/P (xT ) and Gibrat’s law (2.4), this can be reduced: 1 Q(R −1 |RxT ) 1 Q(R −1 ) P (xT ) = = . P (RxT ) R Q(R|xT ) R Q(R)

(2.6)

Since the system has time-reversal symmetry, Gibrat’s law (2.4) is also established under the transformation of xT ↔ RxT (= xT +1 ). Since the right side of Eq. (2.6) is only a function of R, we signify it by G(R) and expand Eq. (2.6) by R = 1+ (  1) in . The 0-th order term of  is a trivial expression, and the 1-st order term yields the following differential equation: G (1)P (xT ) + xT

d P (xT ) = 0. dxT

(2.7)

16

2 Non-Gibrat’s Property in the Mid-Scale Range

Here G (·) denotes the R differentiation of G(·). No more useful information can be obtained from the second and higher order terms of . The solution to this differential equation is uniquely given: 

P (xT ) ∝ xT −G (1).

(2.8)

This solution satisfies Eq. (2.6), even if R is not near R = 1, when Q(R) =  R −G (1)−1 Q(R −1 ) holds (this is called a reflection law). Reflection laws have been confirmed with various actual data [28]. Finally, if we set G (1) = μ + 1, Eq. (2.8), derived from Gibrat’s law under time-reversal symmetry, matches the power law (2.1) observed at time T . Since this system has symmetry xT ↔ xT +1 , the power law holds for the same Pareto index μ even at time T + 1.

2.4 Non-Gibrat’s Property in the Mid-Scale Range of Current Profits In Sect. 2.3, a power law is derived from Gibrat’s law that holds in the large-scale range of firm-size variables under time-reversal symmetry. Here we show that a static log-normal distribution is derived from a non-Gibrat’s property that holds in the mid-scale range under time-reversal symmetry. Here we discuss the case where the logarithm of the conditional growth-rate distribution is linear with respect to the logarithmic growth rate: r = log10 R. When the logarithm of PDF q(r|xT ) of logarithmic growth rate r (which is conditioned on initial firm size xT ) is linear with respect to r, it is described as follows [1, 2]: log10 q(r|xT ) = c − t+ (xT ) r for r > 0,

(2.9)

log10 q(r|xT ) = c + t− (xT ) r for r < 0.

(2.10)

Using the relation between q(r|xT ) and Q(R|xT ), log10 q(r|xT ) = log10 Q(R|xT ) + r + log10 (ln 10),

(2.11)

Q(R|xT ) is described as follows: Q(R|xT ) = d R −1−t+ (xT ) for R > 1,

(2.12)

Q(R|xT ) = d R −1+t− (xT ) for R < 1.

(2.13)

2.4 Non-Gibrat’s Property in the Mid-Scale Range of Current Profits

17

Although c and d are the functions of xT in general, here we concentrate on the case where the xT dependency can be ignored. In this case, the expression up to the second equation of Eq. (2.6) becomes the following for R > 1: P (xT ) = R 1+t+ (xT )−t− (RxT ) . P (RxT )

(2.14)

Expanding Eq. (2.14) as R = 1 +  (  1) yields the following expressions from the 1st, 2nd, and 3rd terms of :   dP (x) = 0, 1 + t+ (x) − t− (x) P (x) + x dx dt+ (x) dt− (x) + = 0, dx dx dt+ (x) d 2 t+ (x) +x = 0. dx d 2x

(2.15) (2.16) (2.17)

The zeroth term in  is a self-explanatory expression, and no useful information can be obtained from the fourth or higher terms. Here xT is expressed as x for simplicity. The same differential equations are obtained for R < 1 whose solutions are generally given: x , x0 x t− (x) = C2 − α ln , x0  x . P (x) ∝ x −μ−1 exp −α ln2 x0 t+ (x) = C1 + α ln

(2.18) (2.19) (2.20)

Here C1 , C2 , and α are integration constants, μ = C1 − C2 , and x0 is a parameter that is introduced to connect with the power law (2.1) in the large-scale range. These solutions satisfy Eq. (2.14), except in the neighborhood of R = 1. Therefore, they are both necessary and sufficient conditions. Combining Eqs. (2.18), (2.19) and Eqs. (2.9), (2.10), when the logarithm of conditional growth-rate distribution q(r|xT ) is linear with respect to r, only the following non-Gibrat’s property,  xT r for r > 0, log10 q(r|xT ) = c − C1 + α ln x0  xT log10 q(r|xT ) = c + C2 − α ln r for r < 0, x0

(2.21) (2.22)

18

2 Non-Gibrat’s Property in the Mid-Scale Range

is permitted to be consistent with the time-reversal symmetry. In this book, we call this property a non-Gibrat’s property. We will investigate in Chap. 4 the extension of this property when the conditional growth-rate distribution has a downward convex curvature. Equation (2.20) is also the best log-normal distribution (2.2). Since this system has xT ↔ xT +1 symmetry, the log-normal distribution has identical parameters α and μ at time T + 1. We confirm the consistency of this analytical discussion with empirical data. In Fig. 2.1, the data follow a power-law distribution in the range of xT > 104.6 (= x0 ) and a log-normal distribution (2.2) in the range of 102.7 < xT < 104.6 (= x0 ). These data ranges correspond to Figs. 2.4 and 2.5, which show that it is a reasonable approximation to describe the conditional PDF of the logarithmic growth rate as a linear expression of r, such as Eqs. (2.9) and (2.10). Equations (2.9) and (2.10) are applied to the conditional PDFs in Figs. 2.3, 2.4, and 2.5 to obtain t± and c, and their dependence on xT is shown in Fig. 2.7. In the previous discussion in Sect. 2.4, the dependence of c on xT was ignored in Eqs. (2.9) and (2.10). Figure 2.7 confirms that the assumption is a good approximation in the range of xT > 102.7. When α is evaluated by the regression analysis of t+ (xT ) and t− (xT ) in the range of 102.7 < xT < 104.6 (= x0 ), α+ = 0.15±0.01 is obtained from t+ (xT ) and α− = 0.13±0.02 is obtained from t− (xT ). That is, α+ = α− is true within the error. This indicates that Eqs. (2.18) and (2.19) hold within the error range. In addition, these values coincide within the error range with αLN = 0.17 ± 0.04 obtained by assuming that the data in the range of 102.7 < xT < 104.6 (= x0 ) in Fig. 2.1 follow the log-normal distribution. Thus, the following argument is consistent with the empirical data: 4

c t+ t

c xT , t xT

3

2

1

0 1

2

3

4

5

6

7

logxT Fig. 2.7 xT dependencies of t± and c evaluated by applying Eqs. (2.9) and (2.10) to conditional growth-rate distributions in Figs. 2.3, 2.4, and 2.5

2.5 Discussion

19

Table 2.2 Consistency between αLN obtained by regression analysis of log-normal distribution and α± obtained by regression analysis of t+ (xT ) and t− (xT ) T 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

T +1 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

Amount of data 271,488 249,762 268,567 292,315 292,013 297,444 300,348 297,159 288,709 260,686

αLN 0.17 ± 0.04 0.17 ± 0.05 0.14 ± 0.03 0.15 ± 0.04 0.17 ± 0.05 0.16 ± 0.04 0.17 ± 0.04 0.17 ± 0.04 0.17 ±0.04 0.17 ± 0.04

α+ 0.15 ± 0.01 0.11 ± 0.03 0.20 ± 0.03 0.16 ± 0.01 0.16 ± 0.03 0.15 ± 0.04 0.22 ± 0.01 0.09 ± 0.02 0.12 ± 0.05 0.18 ± 0.01

α− 0.13 ± 0.02 0.10 ± 0.01 0.14 ± 0.03 0.14± 0.04 0.11 ± 0.05 0.12 ± 0.01 0.10 ± 0.02 0.06 ± 0.02 0.12 ± 0.02 0.12 ± 0.03

the log-normal distribution, which is seen in the mid-scale positive current profits range, is derived from the time-reversal symmetry and the non-Gibrat’s property. This consistency can be confirmed not only when T = 1998 but also in almost all the years of data we have (Table 2.2).

2.5 Discussion In this chapter, we focused on the distribution of firm-size variables in the mid-scale range, as observed at such fixed times as “a certain year.” Over 100 years ago it has been confirmed that the distribution in a large-scale range follows the power law for various quantities in fields such as economics, sociology, and financial engineering. Power-law distributions can be derived from various models, including stochastic processes. On the other hand, the distribution in the mid-scale range follows a log-normal distribution. However, various distributions other than the log-normal distribution have been proposed, and in many cases, the distribution’s superiority or inferiority has been discussed by comparing its applicability with the data. In this context, Aoyama et al. derived the power-law distribution not from a specific model as a starting point but from Gibrat’s law and the time-reversal symmetry observed in the short-term changes of a very large set, such as firm-size variables. This epoch-making study showed that when dealing with a huge group of firms, its macroscopic properties can be linked without assuming a specific model at the microlevel. Aoyama et al. argued that when the system is in equilibrium, the power-law distribution is derived from Gibrat’s law observed in the large-scale range with high accuracy in the large-scale range of empirical data. In this chapter, the property that the growth-rate distribution changes regularly based on the initial values in the mid-scale range is expressed as a non-Gibrat’s property. When the system is in equilibrium, the log-normal distribution is derived from the non-Gibrat’s property. Note that we can analytically show that when

20

2 Non-Gibrat’s Property in the Mid-Scale Range

the growth-rate distribution is linear on a log–log axis, such as positive current profits, the expressions of a non-Gibrat’s property that are allowed for a system in equilibrium are uniquely defined. We numerically confirmed these analytical arguments by positive current profit data for Japanese firms in 1989 and 1999. These properties are confirmed by the positive current profit data of around 500,000 Japanese firms from 1998 to 2008, which are listed in the CD Eyes database in our possession. Similarly, we can check the positive current profit data of Japanese firms in the Orbis database. Thus, this chapter extended Aoyama et al.’s discussion in the large-scale range to the mid-scale range, where the log-normal distribution is derived. As discussed in this chapter’s introduction, another direction to extend Aoyama et al.’s work is to replace the arguments that arise when the system is in equilibrium with those in quasi-static changes. This can be done by replacing the time-reversal symmetry with quasi-time-reversal symmetry. In the next chapter, we use this approach to derive quasi-statically varying power and log-normal distributions in terms of Gibrat’s law and the non-Gibrat’s property under quasi-time-reversal symmetry. Such analytical discussion is supported by the official land prices in Japan.

References 1. Ishikawa A (2006) Derivation of the distribution from extended Gibrat’s law. Physica A367:425–434 2. Ishikawa A (2007) The uniqueness of firm size distribution function from tent-shaped growth rate distribution. Physica A383:79–84 3. Pareto V (1897) Cours d’Économie Politique. Macmillan, London 4. Bak P, Tang C, Wiesenfeld K (1987) Self-organized criticality: an explanation of the 1/f noise. Phys Rev Lett 59:381–384 5. Bak P, Tang C, Wiesenfeld K (1988) Self-organized criticality. Phys Rev A38:364–374 6. Peng CK, Mietus J, Hausdorff JM, Havlin S, Stanley HE, Goldberger AL (1993) Long-range anticorrelations and non-Gaussian behavior of the heartbeat. Phys Rev Lett 70:1343–1346 7. Bonabeau E, Dagorn L (1995) Possible universality in the size distribution of fish schools. Phys Rev E51:R5220–R5223 8. Render S (1998) How popular is your paper? An empirical study of the citation distribution. Eur Phys J B4:131–134 9. Takayasu M, Takayasu H, Sato T (1996) Critical behaviors and 1/ noise in information traffic. Physica A233:824–834 10. Saichev AI, Malevergne Y, Sornette D (2010) Theory of Zip’s Law and beyond. Springer, Berlin 11. Kaizoji T (2003) Scaling behavior in land markets. Physica A326:256–264 12. Yamano T (2004) Distribution of the Japanese posted land price and the generalized entropy. Eur Phys J B38:665–669 13. Mantegna RN, Stanley HE (1995) Scaling behaviour in the dynamics of an economic index. Nature 376:46–49 14. Podobnik B, Horvatic D, Petersen AM, Uro˘sevi´c B, Stanley HE (2010) Bankruptcy risk model and empirical tests. Proc Natl Acad Sci USA 107:18325–18330

References

21

15. Fu D, Pammolli F, Buldyrev SV, Riccaboni M, Matia K, Yamasaki K, Stanley HE (2005) The growth of business firms: theoretical framework and empirical evidence. Proc Natl Acad Sci 102:18801–18806 16. Podobnik B, Horvatic D, Pammolli F, Wang F, Stanley H E, Grosse I (2008) Size-dependent standard deviation for growth rates: empirical results and theoretical modeling. Phys Rev E77:056102 17. Newman MEJ (2005) Power laws, Pareto distributions and Zipf’s law. Contem Phys 46:323– 351 18. Clauset A, Shalizi CR, Newman MEJ (2009) Power-law distributions in empirical data. SIAM Rev 51:661–703 19. Gibra R (1932) Les Inégalités Économique. Sirey, Paris 20. Badger WW (1980) An entropy-utility model for the size distribution of income. In: West BJ (ed) Mathematical models as a tool for the social science. Gordon and Breach, New York, pp 87–120 21. Montroll EW, Shlesinger MF (1983) Maximum entropy formalism, fractals, scaling phenomena, and 1/f noise: a tale of tails. J Stat Phys 32:209–230 22. Kesten H (1973) Random difference equations and Renewal theory for products of random matrices. Acta Math 131:207–248 23. Levy M, Solomon S (1996) Power laws are logarithmic Boltzmann laws. Int J Mod Phys C7:595–601 24. Sornette D, Cont R (1997) Convergent multiplicative processes repelled from zero: power laws and truncated power laws. J Phys I7:431–444 25. Takayasu H, Sato A, Takayasu M (1997) Stable infinite variance fluctuations in randomly amplified langevin systems. Phys Rev Lett 79:966–969 26. Sato A, Takayasu H, Sawada Y (2000) Invariant power law distribution of langevin systems with colored multiplicative noise. Phys Rev E61:1081–1097 27. Fujiwara Y, Souma W, Aoyama H, Kaizoji T, Aoki M (2003) Growth and fluctuations of personal income. Physica A321:598–604 28. Fujiwara Y, Guilmi CD, Aoyama H, Gallegati M, Souma W (2004) Do Pareto-Zipf and Gibrat laws hold true? An analysis with European firms. Physica A335:197–216 29. Bureau van Dijk Electronic Publishing KK. https://www.bvdinfo.com/en-gb 30. Tokyo Shoko Research, Ltd. http://www.tsr-net.co.jp/en/outline.html 31. National Land Numerical Information download sorvice. http://nlftp.mlit.go.jp/ksj-e/index. html 32. Dragulescu ˇ A, Yakovenko VM (2001) Exponential and power-law probability distributions of wealth and income in the United Kingdom and the United States. Physica A299:213–221 33. Silva AC, Yakovenko VM (2005) Temporal evolution of the “thermal” and “superthermal” income classes in the USA during 1983–2001. Europhys Lett 69:304–310 34. Anazawa M, Ishikawa A, Suzuki T, Tomoyose M (2004) Fractal structure with a typical scale. Physica A335:616–628 35. Ishikawa A, Suzuki T (2004) Relations between a typical scale and averages in the breaking of fractal distribution. Physica A343:376–392 36. Hart PE, Oulton N (1996) Growth and size of firms. Eco J 106:1242–1252 37. Lotti F, Santarelli E, Vivarelli M (2009) Defending Gibrat’s Law as a long-run regularity. Small Bus Eco 32:31–44 38. Sutton J (1997) Gibrat’s Legacy. J Econ Lit 35:40–59

Chapter 3

Quasi-Statistically Varying Power-Law and Log-Normal Distributions

Abstract In this chapter, we show that when a system changes quasi-statically, the power-law and log-normal distributions, which change quasi-statically, are derived from Gibrat’s law in the large-scale range and from the non-Gibrat’s property in the mid-scale range. Then using Japan’s publicly announced land price data from 1974 to 2020, analytical discussions can be confirmed accurately.

3.1 Introduction Much data in the fields of economics, sociology, and financial engineering follow a power-law distribution in the large-scale range [1–3] and a log-normal distribution in the mid-scale range [4–6] at a given time. As noted in Chap. 2, this book examines the macroproperties of firm-size variables, including operating revenues, number of employees, and assets. Although the size of individual firms changes from time to time, the distribution of all firms is stable over time (for example, see [7]). In other words, it is difficult to observe large temporal changes in the exponents of the power-law distribution or in the logarithmic standard deviation of the log-normal distribution. However, when we look at elements other than firm-size variables, we find that the power-law distribution and the log-normal distribution change over time, such as the land price distribution in Japan [8–10]. We describe such a system by quasi-statically extending the time-reversal symmetry discussed in the previous chapter. Based on the quasi-time-reversal symmetry and Gibrat’s law or nonGibrat’s property, we derive a quasi-statically varying power-law distribution and a log-normal distribution. This derivation is an extension of Aoyama et al.’s work on equilibrium states [11, 12] that described states that change quasi-statically in the time direction. In Chap. 8, this approach leads to a relationship between the power indices of different types of firm-size variables at the same time. The discussion here is based primarily on my previous work [13, 14]. Most texts are reproduced under the Creative Commons Attribution License or with permission from the publisher. The main text is modified to fit the context of this book. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Ishikawa, Statistical Properties in Firms’ Large-scale Data, Evolutionary Economics and Social Complexity Science 26, https://doi.org/10.1007/978-981-16-2297-7_3

23

24

3 Quasi-Statistically Varying Power-Law and Log-Normal Distributions

This chapter is organized as follows. The subject of its analysis is the data of a system in a quasi-statically changing state. It is also important to provide data for a system in which the axis of the quasi-time-reversal symmetry introduced here can be easily identified. Section 3.2 explains why the handling of publicly announced land price data in Japan satisfies the above two characteristics. In Sect. 3.3, we observe a power-law distribution in the large-scale range and a log-normal distribution in the mid-scale range in the posted land price data and their quasi-static changes. In Sect. 3.4, we derive a quasi-statically varying power-law distribution or a log-normal distribution from Gibrat’s law or non-Gibrat’s property under quasi-time-reversal symmetry. Then the correspondence between the parameter of the quasi-timereversal symmetry and the changes of the power index or the logarithmic standard deviation, derived from the above analytical discussion, can be confirmed by empirical data. Finally, Sect. 3.5 summarizes this Chap. 3.

3.2 Data Since Japan’s livable population density is very high, land is a very valuable asset, distinct from buildings. Based on the Public Notice of Land Prices Act, since 1970, the price of land (called standard land) is annually announced on January 1. Real estate appraisers evaluate the prices of about 20,000 reference sites in Japan (in recent years), and the Land Appraisal Committee of the Ministry of Land, Infrastructure, Transport and Tourism determines the prices and publicly announces them every March. The official land price is used as a standard for calculating land’s acquisition price for public works projects and as an indicator of land transaction prices. These publicly available, structured data can be easily analyzed by researchers [15]. During Japan’s bubble economy from 1986 to the early 1990s, the price difference of land rapidly increased and its land price system also changed greatly. Since Japan’s land prices are closely related to its economy, its land price system has continued to change even after the bubble economy burst. As noted earlier, posted land prices are surveyed and published for use as indicators of land acquisition and transactions. Therefore, the places where land prices are surveyed (standard land) gradually change year by year at a rate of around 1% annually, and they have become standard indicators of land prices throughout Japan. Therefore, the analysis results of posted land prices are expected to resemble those of the actual land prices throughout Japan. The annual change in land prices is very small compared with the change in the firm-size variables. For example, in the current profits covered in the previous chapter, many firms changed their value significantly the following year. Figure 2.2 in Chap. 2 shows the positive current profit data for Japanese firms for 1998 and 1999; the variance is stark. On the other hand, as we will see in the next section, the variance in land prices is very small. This is because it has been impossible, at least in recent years, for land prices to increase 10, 100, or even 1000 times the following year. Thus, for data with very

3.3 Varying Power-Law and Log-Normal Distributions Table 3.1 Amount of data on posted land prices

Year 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986

25

Data 970 1350 2800 5490 14,570 15,010 15,010 15,010 15,580 16,480 17,030 17,380 17,600 16,975 16,975 16,975 16,635

Year 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

Data 16,635 16,820 16,840 16,865 16,892 17,115 20,555 26,000 30,000 30,000 30,300 30,600 30,800 31,000 31,000 31,520 31,866

Year 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020

Data 31,866 31,230 31,230 30,000 29,100 28,227 27,804 26,000 25,983 25,984 23,363 23,363 25,255 25,988 25,988 25,993 25,993

small variance, the axis of symmetry introduced in this chapter is nearly identical to the regression line. The fact that this property cannot be achieved with highly dispersed data is the main reason why land price data are studied in this chapter. In the analysis in the next section, we excluded data from 1970 to 1973 when the official announcement of land prices was first made because the amount was insufficient. Our analysis was comprised of data from 1974 to 2020. The amount of data for each year is summarized in Table 3.1.

3.3 Varying Power-Law and Log-Normal Distributions Typical examples of the probability distribution functions (PDFs) of land prices in 2020 and 2007 are shown in Figs. 3.1 and 3.2. In each figure, the power-law distribution [8–10] P (x) = Cx −μ−1

for

x0 < x

(3.1)

is observed in the large-scale range. Here exponent μ is a parameter that represents the spread of the power-law distribution, and x0 is the lower limit of the large-scale range. At the same time, the log-normal distribution   x1 P (x) = Cx −(μ+1) exp −α ln2 x0

for xmin < x < x0

(3.2)

26

3 Quasi-Statistically Varying Power-Law and Log-Normal Distributions



Px

    

Log-normal Pareto 





Land Price x in yen/m Fig. 3.1 Official land price (x) distribution in Japan in 2020: Units are JP Yen/m2 . Power law (3.1) is applied to top 10% of data among large-scale data, and power index is μ = 1.16 ± 0.0. However, top 0.1% of fluctuating data was excluded. Logarithmic standard deviation, calculated by fitting mid-scale range to log-normal distribution (3.2) with xmin = 5000 and x0 = 501,000, is σ = 1.07 ± 0.01



Px

    

Log-normal Pareto 





Land Price x in yen/m Fig. 3.2 Official land price (x) distribution in Japan in 2007: Units are JP Yen/m2 . Power law (3.1) is applied to top 10% of data in large-scale data, and power index is μ = 1.27 ± 0.0. However, top 0.1% of fluctuating data was excluded. Logarithmic standard deviation, calculated by fitting mid-scale range to log-normal distribution (3.2) with xmin = 5000 and x0 = 501,000, is σ = 0.96 ± 0.01

is also observed in each mid-scale range. Here α is a parameter that represents the spread of the log-normal distribution, and x0 is the upper limit of the midscale range. xmin is the lower limit of the mid-scale range that probably reflects the insufficiency of data in the small-scale range. In the low-scale range, several distributions have been proposed [16–19]. α is related to logarithmic standard

3.3 Varying Power-Law and Log-Normal Distributions

27

deviation σ of the log-normal distribution described in the standard way: α=

1 . 2σ 2

(3.3)

The power exponents in Figs. 3.1 and 3.2 are μ = 1.16 ± 0.00 and 1.27 ± 0.00. These are the regression values of the cumulative expression (3.1) for the top 0.1% to 10% of the cumulative data. The logarithmic standard deviations of Figs. 3.1 and 3.2 are σ = 1.07 ± 0.01 and 0.96 ± 0.01, which are obtained by setting xmin = 5000, x0 = 501,000 and performing regression analysis using Eq. (3.2). In this method, the lower limit x0 of the large-scale range and the upper limit x0 of the mid-scale range do not always match. This is because the power index is obtained from the cumulative distribution, and the logarithmic standard deviation is obtained from the PDF. This inconsistency is not a major problem in this analysis. As a typical example, we showed the distribution of land prices in 2007 and 2020. Power index μ of the large-scale range and log-normal standard deviation σ of the mid-scale range in both years are different. This is not a special case; both constantly changed between 1974 and 2020, as illustrated in Fig. 3.3. Power exponent μ varies from approximately 1 to 1.75, where the smallest value is μ = 1.01 ± 0.00 in 1987 and the largest is μ = 1.75 ± 0.00 in 2001. The extent of the large-scale range following the power-law distribution is characterized by 1/μ. Figure 3.3 shows that the spread of land prices in the large-scale range was greatest in 1987, which is the year immediately after the Japanese economy entered its bubble period. Figure 3.3 shows that after the bubble burst in the early 1990s, its spread gradually dwindled to a minimum in 2001. The figure also shows that logarithmic standard deviation σ varies from about 0.9 to 1.3, with the largest value in 1991 at σ = 1.29 ± 0.03 and the smallest in 2004 and 2005 at σ = 0.93 ± 0.00. The spread of land prices in the mid-scale range was greatest in 1991 when Japan’s bubble economy collapsed. Thereafter, the extent gradually decreased, with the greatest decreases in 2004 and 2005 (Fig. 3.3). This figure suggests the following two main points. First, the power index and logarithmic standard deviation are constantly changing in conjunction

 

1.5

sigma

2 Pareto Index mu

Fig. 3.3 Temporal changes in Pareto index μ and logarithmic standard deviation σ of posted land prices in Japan from 1974 to 2020

1.5 1 1 75 80 85 90 95 00 05 10 15 20 Year T

28

3 Quasi-Statistically Varying Power-Law and Log-Normal Distributions 

Land Price xT+in yen/m













2019 - 2020 log10 xT+1 = log10 xT





   Land Price xT in yen/m





Fig. 3.4 Scatter plots of posted land prices for 2019 (= T ) and 2020 (= T + 1)

with the Japanese economy. Although the movements of the large-scale range following the power-law distribution and the mid-scale range following the lognormal distribution are linked, a time lag is seen between the movements of both. The time lag between this large- and mid-scale range change is important when measuring the quasi-time-reversal symmetry parameters described in the next section. At the end of this section, as an introduction to the next section, scatter plots of land prices for 2019 and 2020 are illustrated in Fig. 3.4, and those for 2006 and 2007 are illustrated in Fig. 3.5. As noted in the previous section, since land prices do not change significantly from year to year, the dispersion of the scatter plots is very small relative to the firm-size variables. This can be seen in Figs. 3.4 and 3.5. Also shown in Fig. 3.3, the change was small in both the power index and logarithmic standard deviation between 2019 and 2020. In such cases, a time-reversal symmetry can be approximately confirmed for the joint PDF. In fact, the data in Fig. 3.4 are distributed symmetrically with respect to the time inversion. On the other hand, as shown in Fig. 3.3, the change in the power index from 2006 to 2007 is particularly large compared with that from 2019 to 2020. This corresponds to the fact that the data shown in Fig. 3.5 are not symmetric with respect to the time inversion, especially in the large-scale range. In the next section, we show that by extending the symmetry for time reversal to a quasi-static one, we can describe a system in which the power exponent changes quasi-statically. This extension also facilitates the description of quasi-static changes in the log-normal distribution.

3.4 Quasi-Time-Reversal Symmetry

29



Land Price xT+in yen/m











  

2006 - 2007 log10 xT+1 = log10 xT

















Land Price xT in yen/m

Fig. 3.5 Scatter plots of posted land prices for 2006 (= T ) and 2007 (= T + 1)

3.4 Quasi-Time-Reversal Symmetry In this section we introduce quasi-time-reversal symmetry and derive a quasistatically varying power-law distribution and a log-normal distribution. In this book, T symbolizes the calendar year. Here we denote (xT , xT +1 ) as the land prices for two consecutive years and focus on joint PDF P (xT , xT +1 ). This is observed as a scatter plot of (xT , xT +1 ) in the data analysis. Two types of symmetries are observed in the joint PDF when the variables are in an equilibrium state and change quasi-statically. As mentioned in Chap. 2, even though individual variables vary, when their aggregates are in equilibrium, we study the system’s symmetry with respect to a time reversal: xT ↔ xT +1 . This is called time-reversal symmetry, which is expressed using the joint PDF [11, 12]: PJ (xT , xT +1 )dxT dxT +1 = PJ (xT +1 , xT )dxT dxT +1 .

(3.4)

Note that its function form is identical on both sides of Eq. (3.4). The system is symmetrical with respect to a straight line, log10 xT +1 = log10 xT , in both logarithmic axes. Time-reversal symmetry, a concept that describes the equilibrium state of a system in thermodynamics, is also called a detailed balance. When a system composed of a large number of variables changes quasi-statically, it is symmetric with respect to time inversion axT θ ↔ xT +1 , which is a quasi-static

30

3 Quasi-Statistically Varying Power-Law and Log-Normal Distributions

extension of time-reversal symmetry. We call this quasi-time-reversal symmetry, which is expressed using the following joint PDF [13, 14]: PJ (xT , xT +1 ) dxT dxT +1 





xT +1 1/θ xT +1 1/θ θ = PJ , a xT d d a xT θ . a a

(3.5)

Here θ and a are the parameters of this symmetry, and the system is symmetrical with respect to a straight line, log10 xT +1 = θ log10 xT + log10 a, on a logarithmic axis. That is, parameter θ is the slope of the symmetry axis, and log10 a is its intercept.

3.4.1 Quasi-Statically Varying Power Law First, we show that in a large-scale range where Gibrat’s law (2.4) holds, a power index change can be derived from the quasi-time-reversal symmetry where variables change quasi-statically in two consecutive years [13, 14]. When a system changes quasi-statically, quasi-time-reversal symmetry (3.5) is found. Using extended growth rate R = xT +1 /axT θ in the quasi-time-reversal system, this is rewritten by variables xT , R:

 PJ (xT , R) = R 1/θ−2 PJ R 1/θ xT , R −1 .

(3.6)

Equation (3.6) is reduced to Eq. (2.5) at θ = 1. Using conditional PDF Q(R|xT ) and Gibrat’s law (2.4), this is reduced to −1 1/θ x ) −1 P (xT ) T 1/θ−2 Q(R |R 1/θ−2 Q(R ) = R = R . P (R 1/θ xT ) Q(R|xT ) Q(R)

(3.7)

Here we assume that Gibrat’s law (2.4) holds under a transformation: xT ↔ R 1/θ xT (= (xT +1 /a)1/θ ). This is valid in a system that has quasi-time-reversal symmetry. Since the last term in Eq. (3.7) is only a function of R, we signify it by Gθ (R) and expand Eq. (3.7) to R near 1 as R = 1 +  (  1). The 0-th order of  is a trivial expression, and the 1-st order term yields the following differential equation: Gθ  (1)P (xT ) +

xT d P (xT ) = 0. θ dxT

(3.8)

Here Gθ  (·) denotes the R differentiation of Gθ (·). No more useful information can be obtained from the second and higher order terms of . The solution to this

3.4 Quasi-Time-Reversal Symmetry

31

differential equation is uniquely given: P (xT ) ∝ xT −θGθ

 (1)

.

(3.9)

As in Chap. 2, this solution satisfies Eq. (3.7), even if R is not near R = 1, when  Q(R) = R −Gθ (1)−1 Q(R −1 ) holds. Next in quasi-static system (xT , xT +1 ), we identify distribution P (xT +1 ). Actually, we should write PxT (xT ), PxT +1 (xT +1 ); however, because function forms are complicated, they are collectively written as P . From Eq. (3.9) and P (xT )dxT = P (xT +1 )dxT +1 , P (xT +1 ) can be expressed: P (xT +1 ) = P (xT )

dxT  ∝ xT +1 −Gθ (1)+1/θ−1. dxT +1

(3.10)

Here we signify Pareto indices at T , T + 1 by μT , μT +1 and represent P (xT ), P (xT +1 ): P (xT ) ∝ xT −μT −1 , P (xT +1 ) ∝ xT +1 −μT +1 −1 .

(3.11)

Comparing Eqs. (3.9) and (3.10) to Eq. (3.11), we obtain θ Gθ  (1) = μT + 1, Gθ  (1) − 1/θ + 1 = μT +1 + 1 and conclude the following relation among μT , μT +1 , and θ : θ=

μT 1/μT +1 = . μT +1 1/μT

(3.12)

From this, we understand that parameter θ in the quasi-time-reversal symmetry represents the rate of the change of Pareto indices μT , μT +1 at time T , T + 1. This is geometrically consistent with the idea that the power law’s width at time T can be expressed as 1/μT and at time T + 1 as 1/μT +1 on the logarithmic axis.

3.4.2 Quasi-Statically Varying Log-Normal Distribution Next we show that in the mid-scale range where the non-Gibrat’s property is observed, the change in the parameters of the log-normal distribution is derived from the quasi-time-reversal symmetry where variables change quasi-statically in two consecutive years [14]. Here we assume that the growth-rate distribution can be expressed as Eqs. (2.9) and (2.10). In this case, the expression up to the second equation of the quasi-static time-reversal symmetry (3.7) can be transformed as follows for R > 1: P (xT ) 1/θ = R 1/θ+t+ (xT )−t− (R xT ) . 1/θ P (R xT )

(3.13)

32

3 Quasi-Statistically Varying Power-Law and Log-Normal Distributions

Expanding this equation around R = 1, the following differential equations are obtained: {1 + θ (t+ (x) − t− (x))} P (x) + x

dP (x) = 0, dx

(3.14)

dt+ (x) dt− (x) + = 0, dx dx

(3.15)

d 2 t+ (x) dt+ (x) +x = 0. dx d 2x

(3.16)

Here xT is expressed as x for simplicity. The same differential equations are obtained for R < 1. Equations (3.15) and (3.16) from which t± (x) is obtained are identical as the case when there is time-reversal symmetry. The solutions of these differential equations are given by Eqs. (2.18) and (2.19) and the following:  x , P (x) ∝ x −μ−1 exp −θ α ln2 x0

(3.17)

where we set θ (C1 − C2 ) = μ. Since these solutions satisfy Eq. (3.13), except in the neighborhood of R = 1, they are both necessary and sufficient conditions. From the above, when the logarithm of conditional growth-rate distribution q(r|xT ) is linear with respect to r, only the non-Gibrat’s property is permitted to be consistent with the quasi-timereversal symmetry. At the same time, P (xT ) also follows a log-normal distribution. Next in quasi-static system (xT , xT +1 ), we identify distribution P (xT +1 ) as in the case of the large-scale range. From Eq. (3.17) and P (xT )dxT = P (xT +1 )dxT +1 , P (xT +1 ) can be expressed: P (xT +1 ) = P (xT )

 α μ xT +1  dxT . ∝ xT +1 − θ −1 exp − ln2 dxT +1 θ ax0θ

(3.18)

Using Eq. (3.12) and signifying the parameters that change quasi-statically at time T and T + 1 by αT and αT +1 , P (xT ) and P (xT +1 ) can be written:  xT  P (xT ) ∝ xT −μT −1 exp −αT ln2 , x0  xT +1  . P (xT +1 ) ∝ xT +1 −μT +1 −1 exp −αT +1 ln2 ax0θ

(3.19) (3.20)

In this case, αT = θ α, αT +1 = α/θ , and therefore, the relation among αT , αT +1 , and θ is concluded: θ2 =

αT . αT +1

(3.21)

3.4 Quasi-Time-Reversal Symmetry

33

As we will introduce in the next chapter, when the growth-rate distribution is not linear on the logarithmic axis but has a downward curvature, a similar result can be obtained. Using Eqs. (3.3) and (3.21) can be rewritten: θ=

σT +1 . σT

(3.22)

This is geometrically consistent because the width of the log-normal distribution at time T can be expressed as σT , and the width of the log-normal distribution at time T + 1 can be expressed as σT +1 on the logarithmic axis.

3.4.3 Data Analysis In the previous section, we analytically showed that the quasi-statically varying power-law distribution and the log-normal distribution are derived from Gibrat’s law in the large-scale range and non-Gibrat’s property in the mid-scale range under the quasi-time-reversal symmetry. We infer that the slope of the symmetry axis of quasitime-reversal symmetry θ , observed in joint PDF P (xT , xT +1 ) for two consecutive years, equals the ratio of power exponents μT /μT +1 or the ratio of logarithmic standard deviations σT +1 /σT . In this section, the validity of the discussion in the previous section is confirmed by verifying whether these analytical conclusions are reached in the empirical data. As mentioned above, the data to be handled are Japan’s posted land prices from 1974 to 2020. First, slope θ of the symmetry axis, which is a parameter of the quasi-timereversal symmetry, is evaluated in the scatter plots of the land prices for 2006 and 2007 and for 2019 and 2020 (Figs. 3.5 and 3.4). As noted earlier, in these scatter plots with very small variances, we assume that the regression line can be identified with the axis of the quasi-time-reversal symmetry. In Figs. 3.2 and 3.1, we measured power index μ and logarithmic standard deviation σ in a large-scale range from the top 0.1% of the data to 10% and a mid-scale range from xmin = 5000 to x0 = 501,000. For consistent measurements, we evaluated the parameters of the quasi-time-reversal symmetry in large-scale region θL and medium-scale region θM by performing regression analysis on data where the land prices for two consecutive years (xT , xT +1 ) were both in these ranges. As a result, we obtained values of θL = 1.03 ± 0.00, θM = 1.01 ± 0.00 in Fig. 3.4 and values of θL = 1.06 ± 0.00, θM = 1.03 ± 0.00 in Fig. 3.5. Similarly, θL , θM were measured for all the other data from 1974 to 2020, and the changes are illustrated in Fig. 3.6. θL , θM vary from approximately 0.9 to 1.1, where the largest peak of θL was 1.09 ± 0.00 in 1986 immediately after the bubble burst, and the second peak was 1.06 ± 0.00 in 2007. The largest peak of θM was 1.11 ± 0.00 in 1988 during the bubble period, and the second peak was 1.04 ± 0.00 in 2008. Thus, the time lag of the quasi-time-reversal symmetry between the largeand mid-scale regions described in Sect. 3.3 appears in Fig. 3.6. Such a time lag between θL and θM is likely to occur when the system moves significantly, like

34

3 Quasi-Statistically Varying Power-Law and Log-Normal Distributions

1.2

L M

theta

1.1 1 0.9

75 80 85 90 95 00 05 10 15 20 Year T+ Fig. 3.6 Transition of parameters θL , θM for quasi-time-reversal symmetry in large- and mid-scale regions

thetaL and muT/muT+

1.4

L T/T+

1.2

1

0.8

75

80

85

90

95

00

05

10

15

20

Year T+ Fig. 3.7 Comparison of time course of slope θL of symmetry axis of quasi-time-reversal symmetry in large-scale region and ratio of power index μT /μT +1

during a bubble period, and the movement in the large-scale region was gradually transmitted to the mid-scale region. Figures 3.7 and 3.8 compare θL , θM measured in this way with the changes in μT and σT (Fig. 3.3) measured in the land price distribution in T years. In Fig. 3.7, we confirmed that Eq. (3.12): θL = μT /μT +1 accurately holds in most years. Here θ in Eq. (3.12) is written as θL in the sense of a quasi-time-reversal symmetry parameter in the large-scale range. In Fig. 3.8, we confirmed that Eq. (3.12): θM = σT +1 /σT accurately holds in most years. Again, θ in Eq. (3.12) is written as θM , which denotes a parameter of the quasi-time-reversal symmetry in the mid-scale range. In this way, we confirmed the consistency of the analytical discussion conducted in the previous section using the data of land prices in Japan, which change year by year in a quasi-static manner.

thetaM and sigma/sigma

3.5 Discussion

35

1.2

M T+/T

1

0.8 75

80

85

90

95 00 05 Year T+

10

15

20

Fig. 3.8 Comparison of time course of slope θM of symmetry axis of quasi-time-reversal symmetry in mid-scale region and ratio of logarithmic standard deviation σT +1 /σT

3.5 Discussion In the various data studied in social science, a power-law distribution is observed in the large-scale range, and a log-normal distribution is observed in the mid-scale range. In Chap. 2, we introduced a study by Aoyama et al. that derived a power-law distribution from Gibrat’s law in which the growth-rate distribution of data does not depend on the initial value when the system is in equilibrium. We extended the theory to derive the log-normal distribution from the non-Gibrat’s property, which holds in the mid-scale range. In this chapter, we showed that when the system is in a quasi-equilibrium state, the power-law and log-normal distributions, which change quasi-statically, are derived from Gibrat’s law and the non-Gibrat’s property. The analytical discussion was supported by data on land prices in Japan, which are generally considered to change quasi-statically. We addressed quasi-timereversal symmetry, which holds when the system is in a quasi-equilibrium state, and confirmed that the relationship between the slope of the symmetry axis and the changes in the power index and the logarithmic standard deviation holds with good accuracy with empirical data. In this chapter, we easily identified the axis of quasi-time-reversal symmetry, largely because of the nature of Japanese land price data. Although land prices fluctuate from year to year, the changes are very small compared to the changes in the firm-size variables. As a result, the scatter plots for two consecutive years of data were linear with very small variance. Therefore, the axis of the quasi-timereversal symmetry can be approximately measured as a regression line of the data’s scatter plot. This explains why the consistency of the theory was easily confirmed in the empirical data. Applying the same approach to firm-size variables is difficult. The reason is that the variation in firm-size variables is very large, the scatter plots of two successive years of data are not linear, and identifying the axis of the symmetry

36

3 Quasi-Statistically Varying Power-Law and Log-Normal Distributions

of quasi-time-reversal symmetry is difficult by regression analysis. Techniques for solving this problem are introduced in Chap. 8. In this chapter, we assumed a non-Gibrat’s property introduced in the previous chapter as the basis for log-normal distribution in the mid-scale range. In fact, we did not directly detect whether our assumption is correct. This is because the amount of the land price data used in this chapter was only around 20,000, which is insufficient for measuring the growth-rate distribution conditioned by the initial values. However, the same discussion as in this chapter can be done even if the growth-rate distribution is not a non-Gibrat’s property, which becomes linear on the log–log axis, but rather has the extended non-Gibrat’s property described in Chap. 4. We will again focus our analysis on firm-size variables and consider operating revenues and total assets as representative examples. A major difference from the current profits discussed in Chap. 2 is that the operating revenues and total assets are not negative. In such cases, the growth-rate distribution of the firm-size variables does not become linear on the logarithmic axis but has a downward curvature. Chapter 4 will discuss the extension of non-Gibrat’s property in such cases.

References 1. Pareto V (1897) Cours d’Économie Politique. Macmillan, London 2. Newman MEJ (2005) Power laws, Pareto distributions and Zipf’s law. Contem Phys 46:323– 351 3. Clauset A, Shalizi CR, Newman MEJ (2009) Power-law distributions in empirical data. SIAM Rev 51:661–703 4. Gibra R (1932) Les Inégalités Économique. Sirey, Paris 5. Badger WW (1980) An entropy-utility model for the size distribution of income. In: West BJ (ed) Mathematical models as a tool for the social science. Gordon and Breach, New York, pp. 87–120 6. Montroll EW, Shlesinger MF (1983) Maximum entropy formalism, fractals, scaling phenomena, and 1/f noise: a tale of tails. J Stat Phys 32:209–230 7. Mizuno T, Katori M, Takayasu H, Takayasu M (2002) Statistical and stochastic laws in the income of Japanese companies. In: Takayasu H (ed) Empirical science of financial fluctuations – the advent of econophysics. Springer, Tokyo, pp 321–330 8. Kaizoji T (2003) Scaling behavior in land markets. Physica A326:256–264 9. Kaizoji T, Kaizoji M (2004) A mechanism leading bubbles to crashes: the case of Japan’s land markets. Physica A344:138–141 10. Yamano T (2004) Distribution of the Japanese posted land price and the generalized entropy. Eur Phys J B38:665–669 11. Fujiwara Y, Souma W, Aoyama H, Kaizoji T, Aoki M (2003) Growth and fluctuations of personal income. Physica A321:598–604 12. Fujiwara Y, Guilmi CD, Aoyama H, Gallegati M, Souma W (2004) Do Pareto-Zipf and Gibrat laws hold true? An analysis with European firms. Physica A335:197–216 13. Ishikawa A (2006) Annual change of Pareto index dynamically deduced from the law of detailed quasi-balance. Physica A371:525–535 14. Ishikawa A (2009) Quasi-statically varying Pareto-law and log-normal distributions in the large and the middle scale regions of Japanese land prices. Prog Theor Phys Suppl 179:103–113 15. National Land Numerical Information download sorvice. http://nlftp.mlit.go.jp/ksj-e/index. html

References

37

16. Dragulescu ˇ A, Yakovenko VM (2001) Exponential and power-law probability distributions of wealth and income in the United Kingdom and the United States. Physica A299: 213–221 17. Silva AC, Yakovenko VM (2005) Temporal evolution of the “thermal” and “superthermal” income classes in the USA during 1983–2001. Europhys Lett 69:304–310 18. Anazawa M, Ishikawa A, Suzuki T, Tomoyose M (2004) Fractal Structure with a Typical Scale. Physica A335:616–628 19. Ishikawa A, Suzuki T (2004) Relations between a typical scale and averages in the breaking of fractal distribution. Physica A343:376–392

Chapter 4

Extension of Non-Gibrat’s Property

Abstract Similar to current profits discussed in Chap. 2, operating revenues and total assets also follow a power-law distribution in the large-scale range and a lognormal distribution in the mid-scale range in a certain year. Also, observing such short-term changes as two consecutive years, there is a time-reversal symmetry in the joint probability density function of such firm-size variables over “a certain year” and “next year.” Furthermore, Gibrat’s law also holds, which states that the conditional growth-rate distribution of firm-size variables does not depend on the initial values in a large-scale range. However, unlike the positive current profits discussed in Chap. 2, the distribution of such growth rates as operating revenues and total assets is not linear on the logarithmic axis; it has a downward curvature. Even in this case, the initial dependence of the conditional growth-rate distribution in the mid-scale range is regular. Here we extend our discussion in Chap. 2 to present a non-Gibrat’s property that describes its regularity. We show that log-normal distribution is derived from time-reversal symmetry and the extended non-Gibrat’s property and conclude that the results are consistent with the empirical data.

4.1 Introduction As described in Chap. 2, when we observe at a fixed time, such as “a certain year,” such firm-size variables as operating revenues, number of employees, assets, and profits, we observe a power-law distribution in the large-scale range [1–4] and a log-normal distribution in the mid-scale range [5–8]. In Chap. 2, we examined the positive current profit data of Japanese firms [9, 10]. We introduced studies by Aoyama et al. [11, 12], which showed that the powerlaw distribution in the large-scale range is derived from time-reversal symmetry and Gibrat’s law in the change of the short-term period. Time-reversal symmetry is a property in which the system is in equilibrium and the joint probability density distribution (PDF) of the firm-size variables over two consecutive years is symmetrical with respect to the time reversal. Gibrat’s law states that the growth-

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Ishikawa, Statistical Properties in Firms’ Large-scale Data, Evolutionary Economics and Social Complexity Science 26, https://doi.org/10.1007/978-981-16-2297-7_4

39

40

4 Extension of Non-Gibrat’s Property

rate distribution conditioned by the initial value of the firm size does not depend on the initial value. Chapter 2 argued that such a relationship between statistical properties in the large-scale range can be extended to the mid-scale range. We showed that the log-normal distribution in the mid-scale range is derived from the time-reversal symmetry and non-Gibrat’s property. Here non-Gibrat’s property means that the conditional growth-rate distribution of the firm-size variables changes regularly according to the initial value. Mostly importantly, Chap. 2 analytically argued that if the conditional growth-rate distribution is linear on the logarithmic axis [13, 14], where the horizontal axis is the logarithmic growth rate and the vertical axis is the logarithm of the PDF, then under time-reversal symmetry, a non-Gibrat’s property is uniquely defined [9, 10]. The non-Gibrat’s property denotes that the larger the initial firm size, the smaller the positive growth-rate distribution and the larger the negative growth-rate distribution. In Chap. 2, we confirmed that these properties are accurately established in the positive current profits of Japanese firms, where the growth-rate distribution can be linearly approximated on a logarithmic scale. Recall that a linear growth-rate distribution on the log–log axis is a very strong constraint. In fact, the distribution of the growth rates of many firm-size variables, including operating revenues (net sales) and total assets, has a downward curvature [15–17]. In this chapter, we extend Chap. 2’s discussion to consider the non-Gibrat’s property [18–21] associated with such a growth-rate distribution that is not linear on the log axis but that has curvature. Our discussion in this chapter is based primarily on our previous work [22]. Most texts are reproduced under the Creative Commons Attribution License or with permission from the publisher. The main text is modified to fit the context of this book. Chapter 4 is organized as follows. Section 4.2 begins with a firm’s operating revenues and the total assets data used in it. In Sect. 4.3, we first confirm that a power-law distribution is observed in the large-scale range and that a log-normal distribution is seen in the mid-scale range of these operating revenues and total asset data. Then we confirm that time-reversal symmetry is seen in the scatter plots of two consecutive years of the data and observe growth-rate distributions conditioned by the initial values. In these data, Gibrat’s law is confirmed in the large-scale range. Section 4.4 extends the non-Gibrat’s property when the growth-rate distribution has a downward curvature on the log–log axis. Furthermore, as in Chap. 2, we show that a log-normal distribution can be derived from the extended non-Gibrat’s property and confirm the consistency between the conclusions of these analytical discussions and the empirical data. Finally, Sect. 4.5 summarizes this chapter.

4.2 Data As the main financial data of firms, this chapter analyzes operating revenues (net sales) and total assets data from the 2016 edition of Orbis, one of the world’s largest databases, provided by Bureau van Dijk [23]. The database CD Eyes, provided by

4.2 Data

41

Tokyo Shoko Research [24] in Chap. 2, includes the top 500,000 Japanese firms in terms of operating revenues. Since most belong to the large-scale range that follows a power-law distribution, they are unsuitable for the analysis of mid-scale data. Orbis contains financial data for around 200 million firms worldwide. One of its features is that its data size is huge. For example, the operating revenue data of around one million Japanese firms are recorded. Compared to CD Eyes, these data are more comprehensive, covering the operating revenues of not only large-sized but also mid-sized firms. An equally important feature is that national data are recorded in the same format. Therefore, we can compare the comprehensive corporate financial data of each country. However, as mentioned in Chap. 2, the database is structured to record corporate financial data for up to ten consecutive years since the last available year. Therefore, in many cases, the amount of data close to the year when the database was published is large, although the amount of old data is small. This chapter analyzes data on operating revenues and total assets from Japan, France, and Spain between 2010 and 2014, all of which have a sufficient amount of information and are contained in the 2016 edition of the database Orbis, which we will now denote as Orbis 2016. The latest data in the 2016 database are from 2015, although the amount is small because it is still being collected. Therefore, the year 2015 was excluded from the analysis. The amount of data for each year is shown in Tables 4.1, 4.2, and 4.3. The tables also include data on the number of employees for reference. However, since the annual changes in the number of employees are very small compared with the changes in operating revenues and total assets, it is difficult to investigate the changes in the distribution of the growth rates of firm-size variables under different initial conditions. Therefore, we excluded the number of employees from this chapter’s analysis.

Table 4.1 Amount of data on operating revenues, total assets, and number of employees of Japanese firms in Orbis 2016 Year 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

Operating revenue 227,573 294,651 345,304 419,033 518,754 610,833 969,821 1,146,845 1,039,638 344,067

Total assets 130,987 174,646 223,200 285,033 363,584 420,773 436,445 415,077 341,759 93,907

Number of employees 129,523 172,175 217,230 269,891 339,315 392,174 411,327 394,358 325,818 89,549

42

4 Extension of Non-Gibrat’s Property

Table 4.2 Amount of data on operating revenues, total assets, and number of employees of French firms in Orbis 2016 Year 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

Operating revenue 832,716 884,347 933,198 978,076 1,045,513 1,110,364 1,139,674 1,077,633 764,633 80,334

Total assets 835,546 884,827 933,284 977,911 1,045,510 1,110,402 1,139,833 1,077,737 764,702 80,338

Number of employees 387,963 376,138 342,805 353,610 371,585 345,889 277,829 284,490 261,799 26,316

Table 4.3 Amount of data on operating revenues, total assets, and number of employees of Spanish firms in Orbis 2016 Year 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

Operating revenue 740,225 748,675 767,088 769,174 751,935 739,322 720,004 687,076 608,497 1079

Total assets 728,665 712,745 746,167 736,449 723,055 706,650 668,201 612,541 23,939 263

Number of employees 585,478 600,934 623,063 624,138 597,626 581,798 557,070 526,718 468,319 951

4.3 Properties at Fixed Time and in Short-Term Period In this section, as in Chap. 2, we first confirm that the distribution of firm-size variables (x) at a given time (T ) follows a power-law distribution in the large-scale range [1, 3, 4], P (x) ∝ x −μ−1

for x0 < x,

(4.1)

and a log-normal distribution [5, 25],   x P (x) ∝ x −μ−1 exp −αLN ln2 x0

for xmin < x < x0 ,

(4.2)

4.3 Properties at Fixed Time and in Short-Term Period

43

using operating revenue data from Japanese firms in 2013. Here x0 is the boundary between the large- and mid-scale ranges, P is the probability density function (PDF) of firm-size variable x, and μ is the Pareto index. xmin is the lower limit of the mid-scale range, and α is 1/2 of the reciprocal of the logarithmic variance, which indicates the extent of the log-normal distribution. Then, again as in Chap. 2, the time-reversal symmetry of firm-size variables over two consecutive years (xT , xT +1 ), PJ (xT , xT +1 ) = PJ (xT +1 , xT ),

(4.3)

can be confirmed using operating revenue data for Japanese firms in 2013 and 2014. Here PJ is a joint PDF of (xT , xT +1 ), and importantly, both sides of Eq. (4.3) must have the same functional form. Finally, we observe conditional PDF Q(R|xT ) of the growth rate of operating revenues R = xT +1 /xT for two consecutive years. In the large-scale range as well as the positive current profits of Chap. 2, Gibrat’s law is observed: Q(R|xT ) = Q(R) for x0 < xT .

(4.4)

However, non-Gibrat’s property in the mid-scale range is different from that discussed in Chap. 2. Chapter 4 will explain this property and extend the discussion in Chap. 2 to combine the extended non-Gibrat’s property with log-normal distribution (4.2).

4.3.1 Properties at Fixed Time First, we examine the power law (4.1) in the large-scale range and the log-normal distribution (4.2) in the mid-scale range. Figure 4.1 shows the distribution of the operating revenues of Japanese firms in Orbis 2016 for 2013. The horizontal axis shows the operating revenues (thousands of yen), and the vertical axis shows the PDF. Setting x0 = 106 , the Pareto index is estimated to be μ = 0.86 ± 0.01 by regression analysis. To eliminate the fluctuation of several points at the right end, the evaluation is made in the range of x0 (= 106 ) < x < 109.4 . Figure 4.1 confirms that Eq. (4.1) holds with high accuracy. On the log–log axis, the log-normal distribution (4.2) is drawn as a quadratic function. In Fig. 4.1, omitting the point of the large fluctuation of 104 or less, the log-normal distribution as a quadratic function accurately fits the data in the midscale range of xmin (= 104) < x < x0 (= 106 ). At this time, regression analysis estimates that αLN = 0.45 ± 0.03.

44

4 Extension of Non-Gibrat’s Property

 Log-normal Pareto



Px

     









Sales x in yen Fig. 4.1 Distribution of operating revenues for Japanese firms in 2013, as described in Orbis 2016. Pareto index, measured by regression using power-law distribution (2.1) in the range of x0 (= 106 ) < x < 109.4 , is evaluated to be μ = 0.86 ± 0.01. In addition, when applying log-normal distribution (2.2) to the mid-scale range of xmin (= 104 ) < x < x0 (= 106 ), αLN = 0.45 ± 0.03 is estimated by regression analysis

Fig. 4.2 Scatter plot of operating revenues of Japanese firms in 2013 (x2013 in thousands of yen), which are used in Fig. 4.1, versus the operating revenues of those companies in 2014 (x2014 in thousands of yen)

4.3.2 Properties in Short-Term Period Next we observe the time-reversal symmetry (4.3), Gibrat’s law (4.4), and the nonGibrat’s property of two consecutive years of operating revenue data. Figure 4.2 shows scatter plots of the operating revenues of Japanese firms in 2013 (x2013), which is used in Fig. 4.1, and the operating revenues of those firms (x2014) in 2014

4.3 Properties at Fixed Time and in Short-Term Period



n=1 n=2 n=3 n=4 n=5



qr|n

45

    -3

-2

-1

0

1

2

3

r Fig. 4.3 Conditional PDFs q(r|xT ) or q(r|n) of log growth rate of operating revenues r = log10 x2014 /x2013 , calculated using data plotted in Fig. 4.2. Initial value xT is contained in five logarithmically equal-sized bins: x2013 ∈ [102+0.5(n−1) , 102+0.5n ) (n = 1, 2, · · · , 5). Data range shown here is 102 ≤ xT < 104.5 (thousands of yen)

on a log scale. Note that only firms with data for both 2013 and 2014 are included in this figure. Firms with data in 2013 and without in 2014 will be discussed in Chap. 7. From Fig. 4.2, as in the case of the positive current profits of Chap. 2, the figure is symmetrical when the vertical and horizontal axes are interchanged. In other words, the time-reversal symmetry (4.3) also holds for operating revenues. The exact discussion is based on the Kolmogorov–Smirnov test. See our paper for details [22]. Figures 4.3, 4.4, and 4.5 are conditional PDFs of the 2013 logarithmic growth rate of operating revenues r = log10 x2014/x2013, calculated using the data plotted in Fig. 4.2. They set the initial conditions in which operating revenues for 2013 are divided into the following 15 logarithmically equal-sized bins: x2013 ∈ [102+0.5(n−1), 102+0.5n ) (n = 1, 2, · · · , 15). The xT dependencies of the positive and negative standard deviations of the 15 conditional PDFs of Figs. 4.3, 4.4, and 4.5 are illustrated in Fig. 4.6. From Fig. 4.6, we confirmed that in the range of xT larger than x0 ∼ 106, there is no large change in σ± , depending on xT . This corresponds to the fact that in Fig. 4.5, the growth-rate distribution hardly changed at n > 10. That is, in the range of x > x0 (= 106), q(r|xT ) hardly depends on xT , and Gibrat’s law (4.4) is established. On the other hand, the behavior of σ− in the mid-scale range is quite different between Figs. 4.6 and 2.6 for positive current profits, which was discussed in Chap. 2. In Fig. 2.6, σ− increases as xT increases, although in Fig. 4.6, it decreases. As discussed in Chap. 2, such behavior is not permitted if the growth-rate distribution of the firm-size variables is linear on the log–log axis under time-reversal symmetry. This suggests that the distribution of the growth rates of firm-size variables, such as operating revenues, is not linear on the log scale. This idea can be

46

4 Extension of Non-Gibrat’s Property



n=6 n=7 n=8 n=9 n=10

qr|n

     -3

-2

-1

0

1

2

3

r Fig. 4.4 Conditional PDFs q(r|xT ) or q(r|n) of log growth rate of operating revenues r = log10 x2014 /x2013 , calculated using data plotted in Fig. 4.2. Initial value xT is contained in five logarithmically equal-sized bins: x2013 ∈ [102+0.5(n−1) , 102+0.5n ) (n = 6, 7, · · · , 10). Data range here is 104.5 ≤ xT < 107 (thousands of yen)



n=11 n=12 n=13 n=14 n=15

qr|n

     -3

-2

-1

0

1

2

3

r Fig. 4.5 Conditional PDFs q(r|xT ) or q(r|n) of log growth rate of operating revenues r = log10 x2014 /x2013 , calculated using data plotted in Fig. 4.2. Initial value xT is contained in five logarithmically equal-sized bins: x2013 ∈ [102+0.5(n−1) , 102+0.5n ) (n = 11, 12, · · · , 15). Data range here is 107 ≤ xT < 109.5 (thousands of yen)

seen in Figs. 4.3, 4.4, and 4.5. On a logarithmic scale, the growth-rate distribution of operating revenues has a convex curvature at the bottom and is not linear. The following section extends the discussion in Chap. 2, allowing us to describe the nonGibrat’s property in such cases.

4.4 Non-Gibrat’s Property of Operating Revenues and Total Assets

47

V+ V

sigma

0.5

0 2

3

4

5

6

7

8

9

logxT Fig. 4.6 xT dependencies of positive and negative standard deviations σ± of 15 conditional PDFs of Figs. 4.3, 4.4, and 4.5

4.4 Non-Gibrat’s Property of Operating Revenues and Total Assets In Sect. 2.4, we show that a static log-normal distribution is derived from a nonGibrat’s property that holds in the mid-scale range under time-reversal symmetry. In the process, the expression of the non-Gibrat’s property is only permitted in one form Eqs. (2.21) and (2.22) when the logarithm of the conditional growth-rate distribution is linear with respect to logarithmic growth rate r. When the logarithm of PDF q(r|xT ) of logarithmic growth rate r (conditioned by initial firm size xT ) is not linear on the logarithmic axis but has curvature, the simplest way to take the effect into account is to add quadratic term r to Eqs. (2.9) and (2.10) [22, 26]: log10 q(r|xT ) = c − t+ (xT ) r + ln 10 u+ (xT )r 2 for r > 0,

(4.5)

log10 q(r|xT ) = c + t− (xT ) r + ln 10 u− (xT )r 2 for r < 0.

(4.6)

Here ln 10 is a coefficient that simplifies the notation later. It is also assumed, though not explicitly, that there is an appropriate cutoff rc in the r range so that the integral of PDF does not diverge. Q(R|xT ) is given by Q(R|xT ) = d R −1−t+ (xT )+u+ (xT ) ln R for R > 1, Q(R|xT ) = d R

−1+t− (xT )+u− (xT ) ln R

for R < 1.

(4.7) (4.8)

48

4 Extension of Non-Gibrat’s Property

The expression up to the second equation of the time-reversal symmetry (2.6) is rewritten as follows for R > 1:   P (xT ) 1+t+ (xT )−t− (RxT )− u+ (xT )−u− (RxT ) ln R =R . (4.9) P (RxT ) As before, expanding this around R = 1 gives the following differential equations. By expanding Eq. (4.9) as R = 1+ (  1), we can obtain the following equations from the first, second, third, fourth, and fifth terms of : 

 dP (x) 1 + t+ (x) − t− (x) P (x) + x = 0, dx     dt+ (x) dt− (x) + x + 2 u+ (x) − u− (x) = 0, dx dx  2  du+ (x) d t+ (x) d 2 t− (x) dt+ (x) dt− (x) + +6 +x 2 + = 0, 2 dx dx dx d 2x d 2x  2  d t+ (x) d 2 t− (x) dt+ (x) dt− (x) + + 3x + dx dx d 2x d 2x  3  d t+ (x) d 3 t− (x) + x2 + = 0, d 3x d 3x 3 4 d 2 t+ (x) dt+ (x) 2 d t+ (x) 3 d t+ (x) + 7x + 6x + x = 0. dx d 2x d 3x d 4x

(4.10) (4.11) (4.12)

(4.13) (4.14)

The zeroth term in  is a self-explanatory expression, and no useful information can be obtained from the sixth or higher terms. The identical differential equations are obtained for R < 1. As their solutions, t± (x) and u± (x) can be expressed as follows: t+ (x) =

D−3 3 x D−2 2 x x ln ln + + α ln + D1 , 3 x0 2 x0 x0

t− (x) = −

(4.15)

D−3 3 x D+2 − D−2 2 x x ln ln + + (D+1 − α) ln + D2 , 3 x0 2 x0 x0

(4.16)

x D−3 2 x D+2 + D−2 ln ln − + D3 , 6 x0 6 x0

(4.17)

u+ (x) = −

x D−3 2 x 2D+2 − D−2 D+1 ln ln + D3 , + + 6 x0 6 x0 2  D−3 4 x D+2 − 2D−2 3 x ln ln P (x) ∝ x −μ−1 exp − + 6 x0 6 x0  2α − D+1 2 x ln − . 2 x0

u− (x) = −

(4.18)

(4.19)

4.4 Non-Gibrat’s Property of Operating Revenues and Total Assets

49

Here D−3 , D±2 , D+1 , α, D1 , D2 , and D3 are the integration constants, μ = D1 − D2 , and x0 is a parameter introduced to connect with the power law (4.1) in the large-scale range. Since these solutions satisfy Eq. (4.9), except in the neighborhood of R = 1, they are necessary and sufficient conditions. To consider the simplest extension of Chap. 2, we assume that t± (x) can be approximated by a linear function of ln xx0 . Then D−3 = D−2 = D+2 = 0, and Eqs. (4.15)–(4.19) are simplified: x + D1 , x0 x + D2 , t− (x) = α− ln x0 t+ (x) = α+ ln

u+ (x) = D3 , α+ + α− u− (x) = + D3 , 2   α+ − α− 2 x −μ−1 P (x) ∝ x exp − ln . 2 x0

(4.20) (4.21) (4.22) (4.23) (4.24)

Here we changed the symbols: α = α+ , D+1 −α = α− . If no 2nd term is introduced in Eqs. (4.5) and (4.6), then u± = 0, and so D3 = 0, α− = −α+ from Eqs. (4.22) and (4.23). In this case, Eqs. (4.20), (4.21), and (4.24) result in Eqs. (2.18), (2.19), and (2.20). Combining Eqs. (4.20) through (4.23) and Eqs. (4.5) and (4.6), when the logarithm of conditional growth-rate distribution q(r|xT ) has a downward curvature, the simplest extension of the non-Gibrat’s property is represented as follows:  xT r + ln 10 D3 r 2 for r > 0, log10 q(r|xT ) = c − D1 + α+ ln x0  xT log10 q(r|xT ) = c + D2 + α− ln r x0  α+ + α− + D3 r 2 for r < 0. + ln 10 2

(4.25)

(4.26)

At this time, from Eq. (4.24), P (x) follows a log-normal distribution (4.2). Since this system has xT ↔ xT +1 symmetry, the log-normal distribution also remains for identical parameters μ and αLN at time xT +1 . From the above, in both cases where the growth-rate distribution of the firm-size variables is linear or convex downward on both logarithmic axes in two consecutive years, the distribution function in the mid-scale range at a fixed time is a log-normal distribution. We can confirm the consistency of the above analytical discussion with empirical data. In Fig. 4.1, the data in the range of 104 (= xmin ) < xT < 106 (= x0 ) follow a

50

4 Extension of Non-Gibrat’s Property 8

c t+ t u+ u

7

c xT , t xT , u xT

6 5 4 3 2 1 0 2

3

4

5

6

7

8

logxT Fig. 4.7 xT dependencies of t± , u± , and c evaluated by applying Eqs. (4.5) and (4.6) to conditional growth-rate distributions in Figs. 4.3, 4.4, and 4.5

log-normal distribution (4.2) and the data in the range of xT > 106 follow a powerlaw distribution (4.1). These data ranges correspond to n = 5, · · · in Figs. 4.3, 4.4, and 4.5. These figures show that it is reasonable to approximate the conditional PDF of the logarithmic growth rate to the 2nd order of r, as in Eqs. (4.5) and (4.6). The conditional PDFs of Figs. 4.3, 4.4, and 4.5 are applied with Eqs. (4.5) and (4.6) to obtain t± , u± , and c, and their xT dependencies are shown in Fig. 4.7. In this chapter, as in Chap. 2, Eqs. (4.5) and (4.6) are discussed on the assumption that the xT dependency of c is ignored, and from Fig. 4.7, such an assumption is a good approximation in the range of 104 < xT < 106. Similarly, in the range of 104 < xT < 106 in Fig. 4.7, since the xT dependency of u± is negligibly small, the validity of Eqs. (4.22) and (4.23) is confirmed. Most critically, the xT dependency of t± must be evaluated. Within the range of 104 (= xmin) < xT < 106 (= x0 ), the regression analysis of t± (xT ) to evaluate α± yields α+ = 0.51±0.04 from t+ (xT ) and α− = 0.04±0.03 from t− (xT ). As a result, α+ −α− = 0.47±0.05 is obtained, which agrees with 2αLN = 0.45±0.03, obtained in Fig. 4.1 within the range of error. Thus, the following argument is consistent with the empirical data. The log-normal distribution observed in the mid-scale range of the operating revenues is derived from the time-reversal symmetry and the extended non-Gibrat’s property. This consistency is confirmed not only in Japan’s operating revenue data with T = 2013 but also in other years and countries for which we have sufficient data. Tables 4.4, 4.5, and 4.6, respectively, show the analysis results of the operating revenue data for Japanese, French, and Spanish firms from 2010 to 2014. In most cases, consistency with empirical data is confirmed. Similarly, as shown in

4.4 Non-Gibrat’s Property of Operating Revenues and Total Assets

51

Table 4.4 Consistency of operating revenues of Japanese firms with difference between αLN , which is obtained by regression analysis of log-normal distribution, and α± , which is obtained by regression analysis of t+ (xT ) and t− (xT ) T 2010 2011 2012 2013

T +1 2011 2012 2013 2014

Amount of data 428,254 483,523 824,165 944,116

2αLN 0.41 ± 0.02 0.43 ± 0.03 0.38 ± 0.02 0.45 ± 0.03

α+ 0.53 ± 0.03 0.58 ± 0.08 0.41 ± 0.09 0.51 ± 0.04

α− 0.14± 0.01 0.17 ± 0.03 0.06 ± 0.04 0.04 ± 0.03

α+ − α− 0.39 ± 0.03 0.40 ± 0.08 0.35 ± 0.10 0.47 ± 0.05

Table 4.5 Consistency of operating revenues of French firms with difference between αLN , obtained by regression analysis of log-normal distribution, and α± , obtained by regression analysis of t+ (xT ) and t− (xT ) T 2010 2011 2012 2013

T +1 2011 2012 2013 2014

Amount of data 928,371 956,693 915,997 653,932

2αLN 0.52 ± 0.02 0.52 ± 0.02 0.51 ± 0.02 0.39 ± 0.03

α+ 0.46 ± 0.06 0.57 ± 0.08 0.54 ± 0.04 0.65 ± 0.01

α− 0.02 ± 0.03 0.03 ± 0.02 0.00 ± 0.04 0.09 ± 0.03

α+ − α− 0.43 ± 0.06 0.55 ± 0.08 0.55 ± 0.06 0.56 ± 0.03

Table 4.6 Consistency of operating revenues of Spanish firms with difference between αLN , obtained by regression analysis of log-normal distribution, and α± , obtained by regression analysis of t+ (xT ) and t− (xT ) T 2010 2011 2012 2013

T +1 2011 2012 2013 2014

Amount of data 676,064 658,157 620,949 573,158

2αLN 0.45 ± 0.02 0.44 ± 0.02 0.40 ± 0.02 0.39 ± 0.02

α+ 0.46 ± 0.03 0.51 ± 0.03 0.51 ± 0.04 0.59 ± 0.02

α− 0.05 ± 0.01 0.08 ± 0.01 0.03 ± 0.01 0.11 ± 0.01

α+ − α− 0.40 ± 0.03 0.44 ± 0.03 0.48 ± 0.04 0.49 ± 0.02

Table 4.7 Consistency of total assets of Japanese firms with difference between αLN , obtained by regression analysis of log-normal distribution, and α± , obtained by regression analysis of t+ (xT ) and t− (xT ) T 2010 2011 2012 2013

T +1 2011 2012 2013 2014

Amount of data 328,125 364,488 369,461 315,842

2αLN 0.33 ± 0.03 0.33 ± 0.03 0.33 ± 0.03 0.31 ± 0.03

α+ 0.60 ± 0.02 0.73 ± 0.01 0.62 ± 0.03 0.74 ± 0.03

α− 0.29 ± 0.04 0.39 ± 0.02 0.30 ± 0.01 0.35 ± 0.04

α+ − α− 0.32 ± 0.05 0.34 ± 0.02 0.33 ± 0.03 0.39 ± 0.05

Tables 4.7, 4.8, and 4.9, consistency with the empirical data is confirmed for the total assets of the firms in those countries. As shown in Table 4.1, since the amount of Spain’s total asset data for 2014 is extremely small, Table 4.9 shows the results for 2009 to 2013, excluding 2014.

52

4 Extension of Non-Gibrat’s Property

Table 4.8 Consistency of total assets of French firms with difference between αLN , obtained by regression analysis of log-normal distribution, and α± , obtained by regression analysis of t+ (xT ) and t− (xT ) T 2010 2011 2012 2013

T +1 2011 2012 2013 2014

Amount of data 987,602 1,018,043 975,234 701,318

2αLN 0.28 ± 0.04 0.29 ± 0.04 0.30 ± 0.04 0.25 ± 0.04

α+ 0.53 ± 0.04 0.58 ± 0.03 0.54 ± 0.04 0.57 ± 0.02

α− 0.26 ± 0.03 0.25 ± 0.03 0.25 ± 0.05 0.31 ± 0.02

α+ − α− 0.27 ± 0.05 0.33 ± 0.04 0.29 ± 0.06 0.26 ± 0.03

Table 4.9 Consistency of total assets of Spanish firms with difference between αLN , obtained by regression analysis of log-normal distribution, and α± , obtained by regression analysis of t+ (xT ) and t− (xT ) T 2009 2010 2011 2012

T +1 2010 2011 2012 2013

Amount of data 717,697 700,584 664,739 601,911

2αLN 0.37 ± 0.01 0.36 ± 0.01 0.36 ± 0.01 0.35 ± 0.01

α+ 0.58 ± 0.04 0.48 ± 0.06 0.52 ± 0.06 0.59 ± 0.04

α− 0.14 ± 0.02 0.14 ± 0.02 0.14 ± 0.03 0.13 ± 0.03

α+ − α− 0.44 ± 0.04 0.34 ± 0.06 0.38 ± 0.07 0.46 ± 0.05

4.5 Discussion Firm-size variables, such as operating revenues and total assets, follow a power-law distribution in the large-scale range and a log-normal distribution in the mid-scale range at a certain year, like current profits, as discussed in Chap. 2. Pondering such short-term changes as two consecutive years, there is a time-reversal symmetry in the joint PDF of firm-size variables in “a given year” and “next year.” Furthermore, Gibrat’s law holds that the conditional growth-rate distribution of firm-size variables does not change with the initial value in the large-scale range. The major difference from Chap. 2 is that the shape of the growth-rate distribution is not linear on the logarithmic axis; it has a downward curvature. Chapter 4 described how to extend the discussion in Chap. 2 in such cases. We introduce the difference between the changes in the positive and negative growth-rate distributions due to the change in the initial value by adding the second order term on the logarithmic growth rate to the growth-rate distribution. The difference is reflected in the constant time distribution as the reciprocal of the logarithmic variance of the log-normal distribution (the spread of the log-normal distribution). Our previous work proposed an approximation that ignores the change in the negative growth-rate distribution caused by the change in the initial value [22]. The discussion in Chap. 4 is more general, including our previous work. In this chapter, we confirmed that the results obtained through these discussions are accurate based on recent data on operating revenues and total assets in Japan, France, and Spain. Note that in the case of operating revenues, the change in the negative growth-rate distribution is often smaller than the change in the positive distribution in the mid-

References

53

scale range. On the other hand, for total assets, although the change in the negative growth-rate distribution is also smaller than the change in the positive distribution, the difference of the changes is small. This suggests that the probability that the operating revenues will decrease is unlikely to decrease even if the initial operating revenues increase in the mid-scale range. On the other hand, the larger the initial total assets, the smaller is the probability that they will decrease. These assertions are reasonable because operating revenue is the main financial variable representing performance in each firm’s period (flow data), and total assets are another primary financial variable that represents a firm’s cumulative performance (stock data). The properties observed in a time-constant distribution, such as firm-size distribution, are derived from the properties observed in the short term, such as two consecutive years. In Chap. 5 we show that these short-term properties lead to the observed long-term growth of firm size.

References 1. Pareto V (1897) Cours d’Économie Politique. Macmillan, London 2. Axtell RL (2001) Zipf distribution of U.S. firm sizes. Science 293:1818–1820 3. Newman MEJ (2005) Power laws, Pareto distributions and Zipf’s law. Contem Phys 46:323– 351 4. Clauset A, Shalizi CR, Newman MEJ (2009) Power-law distributions in empirical data. SIAM Rev 51:661–703 5. Gibra R (1932) Les Inégalités Économique. Sirey, Paris 6. Badger WW (1980) An entropy-utility model for the size distribution of income. In: West BJ (ed) Mathematical models as a tool for the social science. Gordon and Breach, New York, pp 87–120 7. Montroll EW, Shlesinger MF (1983) Maximum entropy formalism, fractals, scaling phenomena, and 1/f noise: a tale of tails. J Stat Phys 32:209–230 8. Stanley MHR, Buldyrev SV, Havlin S, Mantegna R, Salinger MA, Stanley HE (1995) Zipf plots and the size distribution of Firms. Eco Lett 49:453–457 9. Ishikawa A (2006) Derivation of the distribution from extended Gibrat’s law. Physica A367:425–434 10. Ishikawa A (2007) The uniqueness of firm size distribution function from tent-shaped growth rate distribution. Physica A383:79–84 11. Fujiwara Y, Souma W, Aoyama H, Kaizoji T, Aoki M (2003) Growth and fluctuations of personal income. Physica A321:598–604 12. Fujiwara Y, Guilmi CD, Aoyama H, Gallegati M, Souma W (2004) Do Pareto-Zipf and Gibrat laws hold true? An analysis with European firms. Physica A335:197–216 13. Okuyama K, Takayasu M, Takayasu H (1999) Zipf’s law in income distribution of companies. Physica A269:125–131 14. Ishikawa A (2009) Power-law and log-normal distributions in temporal changes of firm-size variables. Eco 3 Spec Issue Reconstruct Macroecon 3:2009–2011 15. Amaral LAN, Buldyrev SV, Havlin S, Leschhorn H, Maass P, Salinger MA, Stanley HE, Stanley MHR (1997) Scaling behavior in economics: I. Empirical results for company growth. J Phys I France 7:621–633 16. Matia K, Fu D, Buldyrev SV, Pammolli F, Riccaboni M, Stanley HE (2004) Statistical properties of business firms structure and growth. Europhys Lett 67:498–503

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17. Buldyrev SV, Growiec J, Pammolli F, Riccaboni M, Stanley HE (2007) The growth of business firms: facts and theory. J Eur Eco Assoc 5(2–3):574–584 18. Aoyama H (2004) Ninth Annual Workshop on Economic Heterogeneous Interacting Agents (WEHIA) 19. Aoyama H, Fujiwara Y, Souma W (2004) The Physical Society of Japan 2004 Autumn Meeting 20. Aoyama H, Iyetomi H, Ikeda Y, Souma W, Fujiwara Y (2007) Pareto firms. Nihon Keizai Hyouronsha, Tokyo (in Japanese) 21. Takayasu H (2009) New way of financing firms based on the fat-tailed distribution of growth rate. APFA7 & Tokyo Tech. Hitotsubashi Interdisciplinary Conference 22. Ishikawa A, Fujimoto S, Mizuno T (2011) Shape of growth rate distribution determines the type of non-Gibrat’s property. Physica A390:4273–4285 23. Bureau van Dijk Electronic Publishing KK. https://www.bvdinfo.com/en-gb 24. Tokyo Shoko Research, Ltd. http://www.tsr-net.co.jp/en/outline.html 25. Sutton J (1997) Gibrat’s legacy. J Econ Lit 35:40–59 26. Tomoyose M, Fujimono S, Ishikawa A (2009) Non-Gibrat’s law in the middle scale region. Prog Theor Phys Suppl 179:114–122

Chapter 5

Long-Term Firm Growth Derived from Non-Gibrat’s Property and Gibrat’s Law

Abstract With the Orbis database provided by Bureau van Dijk, we analyzed the dependence of firms’ operating revenues, total assets, and number of employees (firm-size variables) on firm age in Japan and France from 2010 to 2013. As a result, we confirmed that the geometric mean value of the firm-size variables obeys a power-law growth for its first 10 years and subsequently follows exponential growth. Using numerical simulations, these long-term properties of firm-size growth were derived from short-term growth law and properties that were observed in two successive years. First, early power-law growth under a size threshold comes from the extended non-Gibrat’s property. Second, subsequent exponential growth over the threshold is derived from Gibrat’s law.

5.1 Introduction In the earlier chapters, we previously explained that the distribution of firm-size variables at a fixed time is derived from the short-term law and firm-size properties. As a next step, we observe how the same short-term laws and properties of firm-size variables lead to long-term growth in firm size. This chapter’s discussion is based primarily on our previous work [1, 2]. Most texts are reproduced under the Creative Commons Attribution License or with permission from the publisher. The main text is modified to fit the context of this book. Economic growth is a main subject of macroeconomics. Using the long-term GDP data of each country around the world, economic growth has been discussed, and the Solow growth model and the Romer model have been proposed [3, 4]. Since the exhaustive financial data of worldwide firms have become available to researchers, the growth of firms can be analyzed whose financial data constitute the GDP of each country. In this chapter, we investigate economic growth with such large-scale data (big data). First, we summarize the discussions in Chaps. 2 and 4. In econophysics, economic data are investigated from a physics perspective [5, 6], and various statistical laws and universality in firm activities have been studied [7–39]. Among them, the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Ishikawa, Statistical Properties in Firms’ Large-scale Data, Evolutionary Economics and Social Complexity Science 26, https://doi.org/10.1007/978-981-16-2297-7_5

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5 Long-Term Firm Growth Derived from Non-Gibrat’s Property and Gibrat’s Law

distribution of social quantities at a point in time has been frequently investigated. In particular, power-law distribution, observed in the large-scale range, is well known [7–9]. If a firm’s operating revenues, total assets, and number of employees (firmsize variables) in calendar year T are signified by xT , their distribution obeys a power law over size threshold x0 in number of years and countries as follows: P (xT ) ∝ xT −μ−1 for x0 < xT .

(5.1)

Here P (xT ) is the probability density function (PDF) of xT , and exponent μ is called Pareto’s index. Although each firm size xT varies annually, power laws have been surprisingly confirmed in the large-scale range. At the same time, Pareto’s index of firm-size data in Japan has not fluctuated for a 30-year period [17]. This is another intriguing property. On the other hand, in the mid-scale range, firm-size variables xT under size threshold x0 obey the following log-normal distribution:   xT P (xT ) ∝ xT −μ−1 exp −α ln2 for xT < x0 . x0

(5.2)

Here α, which is 1/2 of the reciprocal of the logarithmic variance, is a parameter representing the extent of the log-normal distribution. As seen in previous chapters, in the firm-size data of two successive years (xT , xT +1 ), statistical laws and properties are also observed, such as time-reversal symmetry, Gibrat’s law, and non-Gibrat’s property. Time-reversal symmetry means that the system is symmetric under the time-reversal exchange of variables xT +1 ↔ xT and is represented using joint PDF PJ (xT , xT +1 ) as follows [21, 22]: PJ (xT , xT +1 ) = PJ (xT +1 , xT ).

(5.3)

Gibrat’s law [25, 26], which is observed in the large-scale range where the power law is observed, is represented: Q(R|xT ) = Q(R) for x0 < xT .

(5.4)

Here R ≡ xT +1 /xT is the growth rate of the firm-size variables, and Q(R|xT ) is the conditional PDF. Equation (5.4) denotes that Q(R|xT ) does not depend on initial value xT over size threshold x0 . In Chap. 2, we described a non-Gibrat’s property in the mid-scale range where a log-normal distribution is observed and showed that, in the case of firms’ positive current profits, the PDFs of the growth rate of firm-size variables are approximated as follows [27–29]:  xT log10 q(r|xT ) = c − C1 + α ln r for r > 0, x0  xT r for r < 0. log10 q(r|xT ) = c + C2 − α ln x0

(5.5) (5.6)

5.1 Introduction

57

Here c, C1 , C2 , and α are the parameters. r ≡ log10 R is a logarithmic growth rate and q(r|xT ) is the conditional PDF related to Q(R|xT ) by log10 q(r|xT ) = log10 Q(R|xT ) + r + log10 (ln 10). The non-Gibrat’s property shows the dependence of q(r|xT ) on xT under size threshold x0 . Note that when log10 q(r|xT ) is described by the linear functions of r, under time-reversal symmetry, the dependence of q(r|xT ) on xT is uniquely determined by Eqs. (5.5) and (5.6), which we call the non-Gibrat’s property. In the case of a firm’s operating revenues, total assets, and many other firm-size variables, q(r|xT ) can be approximated [29, 30]:  xT r + ln 10 D3 r 2 for rc > r > 0, (5.7) log10 q(r|xT ) = c − D1 + α+ ln x0  xT log10 q(r|xT ) = c + D2 + α− ln r x0  α+ + α− + D3 r 2 for − rc < r < 0. (5.8) + ln 10 2 Here c, D1 , D2 , D3 , and α+ , α− are the parameters, and rc is a cut-off parameter to avoid diverging the integral of the PDF. α in the log-normal distribution (5.2) is identical to (α+ + α− )/2. In this case, if q(r|xT ) is described by the quadratic functions of r under the time-reversal symmetry, the dependence of q(r|xT ) on xT (expressed by Eqs. (5.7) and (5.8)) is the simplest extension of the non-Gibrat’s property expressed by Eqs. (5.5) and (5.6). In Chap. 4, we called this extended nonGibrat’s property. Note that the laws at a point in time Eqs. (5.1) and (5.2) are derived from short-term laws and properties observed in two successive years. First, the power law (5.1) in the large-scale range is derived from Gibrat’s law (5.4) under timereversal symmetry (5.3) [21, 22]. Second, log-normal distribution (5.2) is inferred from the non-Gibrat’s property ((5.5), (5.6)) or the extended non-Gibrat’s property ((5.7), (5.8)) under time-reversal symmetry (5.3) [27–30]. These analytical derivations are confirmed using firms’ positive current profits, operating revenues, and total assets. The study of laws observed in firm-size distributions at a point in time has a long history, and such laws have recently been related to the short-term laws or properties observed in two successive years. Chapters 2 and 4 discussed these issues. In this chapter, as a next step we concentrate on long-term properties that are statistically observed. Very interestingly, observing the dependence of the geometric mean of firm-size variables on firm age confirms that it grows rapidly in the first 10 years according to a power-law function and shifts to a gradual exponential growth. Furthermore, numerical simulations can reproduce the initial rapid power growth that is derived from the short-term extended non-Gibrat’s property and the subsequent slow exponential growth that is derived from Gibrat’s law. The rest of this chapter is organized as follows. In Sect. 5.2, our database is described. By analyzing the exhaustive data of firms’ operating revenues, total assets, and the number of employees in Japan and France, we identified the growth

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5 Long-Term Firm Growth Derived from Non-Gibrat’s Property and Gibrat’s Law

properties of the geometric mean values of these firm-size variables. In Sect. 5.3, Gibrat’s law and the extended non-Gibrat’s property are confirmed as a shortterm law and property observed in two successive years that employ the operating revenue data of Japanese firms as one example of firm-size variables. In Sect. 5.4, we introduce a stochastic model based on a short-term law and property in Sect. 5.3 and show that it leads to power-law growth and the subsequent exponential growth shown in Sect. 5.2. The last section concludes this chapter.

5.2 Data Analyses of Long-Term Growth of Firms In this study, we employ the Orbis database provided by Bureau van Dijk [40] that contains information on around 200 million listed and private firms worldwide, as of 2016. Since firm-size variables with sufficient data volume and growth properties can be easily observed, we analyzed the data of operating revenues, total assets, and the number of employees in Japan and France from 2010 to 2013 from the database. See Tables 4.1 and 4.2 in Chap. 4 for the amount of data for each country. The maximum amount of Japanese or French firms’ financial data in the database is around one million. Since the number of active firms in Japan is reported to range from around one to two million [41], this database is considered exhaustive. Since the Orbis database contains 10-year information, it is impossible to directly trace the trajectory of firm size xT over a decade. However, from each firm’s foundation year in the database, firm age t can be calculated. In this book, t denotes firm age. For the statistical property of the firm-size growth, we observe the dependence of the geometric mean values of firm size xt  on firm age t. The foundation year is t = 1. Figures 5.1 and 5.2 show the dependence of geometric mean operating revenues

xt  of Japanese and French firms in 2013 on firm age t. In Figs. 5.1a and 5.2a, we confirm that xt  in the first 10 years depends on t under the power-law function:

xt  ∝ t γ .

(5.9)

In Figs. 5.1b and 5.2b, we also confirm that xt  after the first 10 years depends on t under an exponential function:

xt  ∝ eβt .

(5.10)

The values of γ , β, which were estimated by the least squares method in 2013, are included in the captions of Figs. 5.1 and 5.2, and those in the other years did not change significantly. The average values of γ , β from 2010 to 2013 are γ OR JP = OR

OR

0.56 ± 0.03, β JP = 0.016 ± 0.001 in Japan and γ OR FR = 0.55 ± 0.03, β FR = 0.050 ± 0.001 in France. The power-law growth indices in Japan and France have

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5.2 Data Analyses of Long-Term Growth of Firms

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Fig. 5.1 Geometric mean values of operating revenues xt  of Japanese firms in 2013 are plotted by circles (◦). Horizontal axis is firm age t, and xt is in thousands of yen adjusted to 2010 prices. (a) In double logarithmic plots, xt  in range 1 ≤ t ≤ 10 is approximated by dashed line as follows: xt  ∝ t γ , γ = 0.52 ± 0.01. (b) In single logarithmic plots, xt  in range 10 < t ≤ 40 is approximated by solid line as follows: xt  ∝ eβt , β = 0.014 ± 0.001

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Fig. 5.2 Geometric mean values of operating revenues xt  of French firms in 2013 are plotted by circles (◦). Horizontal axis is firm age t, and xt is in thousands of euros adjusted to 2005 prices. (a) In double logarithmic plots, xt  in range 1 ≤ t ≤ 10 is approximated by dashed line as follows: xt  ∝ t γ , γ = 0.57 ± 0.03. (b) In single logarithmic plots, xt  in range 10 < t ≤ 40 is approximated by solid line as follows: xt  ∝ eβt , β = 0.052 ± 0.001

OR almost the same values (γ OR FR ∼ γ JP ). The exponential growth index in France OR

OR

exceeds that in Japan (β FR > β JP ). Figures 5.3 and 5.4 show the dependence of geometric mean total assets xt  of Japanese and French firms in 2013 on firm age t. In Figs. 5.3a and 5.4a, we confirm that xt  in the first 10 years depends on t under the power-law function (5.9). In Figs. 5.3b and 5.4b, we confirm that after the first 10 years, xt  depends on t under an exponential function (5.10). The values of γ , β, which were estimated by the least squares method in 2013, are included in the captions of Figs. 5.3 and 5.4, and those in the other years did not change significantly. The average values of γ , β

5 Long-Term Firm Growth Derived from Non-Gibrat’s Property and Gibrat’s Law

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Fig. 5.3 Geometric mean values of total assets xt  of Japanese firms in 2013 are plotted by circles (◦). Horizontal axis is firm age t and xt is in thousands of yen adjusted to 2010 prices. (a) In double logarithmic plots, xt  in range 1 ≤ t ≤ 10 is approximated by dashed line as follows: xt  ∝ t γ , γ = 0.44 ± 0.02. (b) In single logarithmic plots, xt  in range 10 < t ≤ 40 is approximated by solid line as follows: xt  ∝ eβt , β = 0.045 ± 0.002

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Fig. 5.4 Geometric mean values of total assets xt  of French firms in 2013 are plotted by circles (◦). Horizontal axis is firm age t and xt is in thousands of euros adjusted to 2005 prices. (a) In double logarithmic plots, xt  in range 1 ≤ t ≤ 10 is approximated by dashed line as follows:

xt  ∝ t γ , γ = 0.56 ± 0.01. (b) In single logarithmic plots, xt  in range 10 < t ≤ 40 is approximated by solid line as follows: xt  ∝ eβt , β = 0.050 ± 0.001

TA

from 2010 to 2013 are γ TA JP = 0.42 ± 0.02, β JP = 0.046 ± 0.002 in Japan and TA

γ TA FR = 0.53 ± 0.02, β FR = 0.050 ± 0.001 in France. The power-law growth index TA in France exceeds that in Japan (γ TA FR > γ JP ). The exponential index in France also TA

TA

exceeds that in Japan (β FR > β JP ). Similar firm growth properties are found not only in operating revenues and total assets but also in the number of employees. Figures 5.5 and 5.6 depict the dependence of the geometric mean of the number of employees xt  of Japanese and French firms in 2013 on firm age t. In Figs. 5.5 and 5.6, we confirm the power-

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5.2 Data Analyses of Long-Term Growth of Firms

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Fig. 5.5 Geometric mean values of the number of employees xt  of Japanese firms in 2013 are plotted by circles (◦). Horizontal axis is firm age t, and the unit of xt is one person. (a) In double logarithmic plots, xt  in range 1 ≤ t ≤ 10 is approximated by dashed line as follows:

xt  ∝ t γ , γ = 0.21 ± 0.02. This power-law approximation is statistically superior to exponential approximation because R 2 values of power-law and exponential approximations in this range are 0.95 and 0.89. (b) In single logarithmic plots, xt  in range 10 < t ≤ 40 is approximated by solid line as follows: xt  ∝ eβt , β = 0.027 ± 0.001

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Fig. 5.6 Geometric mean values of the number of employees xt  of French firms in 2013 are plotted by circles (◦). Horizontal axis is firm age t, and the unit of xt is one person. (a) In double logarithmic plots, xt  in range 1 ≤ t ≤ 10 is approximated by dashed line as follows:

xt  ∝ t γ , γ = 0.26 ± 0.02. This power-law approximation is not statistically superior to exponential approximation because R 2 values of power-law and exponential approximations in this range are 0.95 and 0.98. However, since the difference is very small, the power-law approximation is acceptable. (b) In single logarithmic plots, xt  in range 10 < t ≤ 40 is approximated by solid line as follows: xt  ∝ eβt , β = 0.049 ± 0.001

law growth (5.9) in the first 10 years and the subsequent exponential growth (5.10) that resembles the growth of the firms’ operating revenues and total assets. The estimated values of γ , β in 2013 are shown in the captions of Figs. 5.5 and 5.6, and those in the other years did not change significantly. The average values of NE γ , β from 2010 to 2013 are γ NE JP = 0.18 ± 0.01, β JP = 0.027 ± 0.001 in Japan

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Table 5.1 Average power-law growth index γ and exponential growth index β of operating revenues, total assets, and number of employees of Japanese and French firms from 2010 to 2013 OR

TA

NE

– γ OR β γ TA β γ NE β JP 0.56 ± 0.03 0.016 ± 0.001 0.42 ± 0.02 0.046 ± 0.002 0.18 ± 0.01 0.027 ± 0.001 FR 0.55 ± 0.03 0.050 ± 0.001 0.53 ± 0.02 0.051 ± 0.001 0.25 ± 0.02 0.046 ± 0.001

NE

and γ NE FR = 0.25 ± 0.02, β FR = 0.046 ± 0.001 in France. For the number of employees, the power-law and exponential growth indices in France exceed those in NE NE NE Japan (γ NE FR > γ JP , β FR > β JP ). These values are summarized in Table 5.1, which shows the magnitude of the following four indices. For Japanese firms, the power-law growth in the first 10 years of life is operating revenues first, followed by the total assets and number of TA NE employees: γ OR JP > γ JP > γ JP . The slow exponential growth that follows the initial power-law growth is the fastest in total assets, followed by the number of employees TA NE OR and operating revenues: β JP > β JP > β JP . For French firms, on the other hand, the power-law growth in the first 10 years is nearly as fast as the total assets and operating revenues, followed by the number TA NE of employees: γ OR FR ∼ γ FR > γ FR . The slow exponential growth that follows the initial power-law growth is also as fast as the total assets and operating revenue, OR TA NE followed by the number of employees: β FR ∼ β FR > β FR .

5.3 Data Analyses of Short-Term Growth of Firms The long-term growth properties of firm-size variables xt  are derived from shortterm laws and properties that are observed in two successive years. In this section, we investigate the short-term laws and properties by employing the operating revenue data of Japanese firms in (xT , xT +1 ) at T = 2011, T + 1 = 2012 as one example of firm-size variables. Figure 5.7 shows conditional PDFs q(r|xT ) of logarithmic growth rate r = log10 xT +1 /xT of operating revenues xT at T = 2011 in Japan. Here xT s are adjusted to 2010 prices in thousands of yen. Initial values xT are divided into logarithmically equal-sized bins xT ∈ [102+0.5(n−1), 102+0.5n ) (n = 1, 2, · · · , 12). In Fig. 5.7b, Gibrat’s law (5.4) confirms that q(r|xT ) does not depend on initial value xT . Conditional PDFs q(r|xT ) in Fig. 5.7 can be approximated by quadratic functions. The dependence on initial values xT in Fig. 5.7a is described as the extended non-Gibrat’s property (5.7), (5.8), where q(r|xT ) decreases as xT becomes larger for r > 0 and q(r|xT ) also decreases for r < 0. However, the change is more gradual than that of r > 0. Fig. 5.8 shows that the average value of |r(xT )| of Fig. 5.7 for r > 0, denoted by |r+ (xT )|, decreases as xT becomes larger and that the average value of |r(xT )| of Fig. 5.7 for r < 0, denoted by |r− (xT )|, also decreases.

5.3 Data Analyses of Short-Term Growth of Firms 



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Fig. 5.7 Conditional PDFs q(r|xT ) of logarithmic growth rate r = log10 xT +1 /xT of operating revenues xT at T = 2011 in Japan. Initial values xT are divided into logarithmically equalsized bins as follows: xT ∈ [102+0.5(n−1) , 102+0.5n ). Here xT s are adjusted to 2010 prices in thousands of yen. (a) Conditional PDFs for case n = 1, 2, · · · , 6. (b) Conditional PDFs for case n = 7, 8, · · · , 12







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xT Fig. 5.8 Average values of |r(xT )| in Fig. 5.7 for r > 0 and r < 0 are denoted by |r+ (xT )| and |r− (xT )|. In range 102.5 ≤ xT < 105 , |r+ (xT )| decreases as xT becomes larger and is approximated by |r+ (xT )| ∝ xT −δ+ , δ+ = 0.29 ± 0.03, and |r− (xT )| ∝ xT −δ− , δ− = 0.13 ± 0.01

However, the change is more gradual than that of r > 0. Note that |r+ (xT )| obeys the following power law in range xT ≤ 105 , as shown in Fig. 5.8 [42]:

|r+ (xT )| ∝ xT −δ+ for r > 0,

|r− (xT )| ∝ xT

−δ−

for r < 0.

(5.11) (5.12)

In Fig. 5.8, although |r± (xT )| decreases even in range xT ≥ 105 , it is negligible compared to the decrease in range xT ≤ 105 because the vertical axis is in a logarithmic scale. This reflects Gibrat’s law (5.4).

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5.4 Long-Term and Short-Term Firm Growth In this section, the long-term firm growth properties identified in Sect. 5.2 are derived from the short-term growth law (Gibrat’s law) and property (non-Gibrat’s property) in Sect. 5.3. First, we define firm-size variables’ growth rate R(xt ) of a firm-size variable at age t: R(xt ) =

xt +1 . xt

(5.13)

We assume that the short-term properties, which are confirmed in two successive years, (T , T + 1), also hold in two successive ages: (t, t + 1). In other words, we postulate that calendar year T in the previous section can be replaced by firm age t. Under this assumption, we interpret Eq. (5.13) as the following stochastic process: xt +1 = R(xt ) xt .

(5.14)

Here R(xt ) is a random variable that depends on initial value xt , and we assume that R(xt ) follows the extended non-Gibrat’s property and Gibrat’s law for xt < x0 and x0 < xt . In this model, there is no additive noise because the system’s stability is not required [43]. Using this model, we stochastically develop the system and investigate the growth of firm-size variables xt . The aim of this simulation is the confirmation of the power-law growth and the subsequent exponential growth of geometric mean values xt  observed in Sect. 5.2. The following is the simulation procedure: 1. As initial values of the firm-size variables, one thousand x1 s were randomly extracted from the operating revenue data in 2011 of Japanese firms that were founded in 2011. In other words, the firm’s age is one in 2011. 2. Multiplicative noise R(xt ) was randomly sampled from twelve growth-rate distributions of operating revenues in Fig. 5.7 based on initial value xt . 3. Iterating Eq. (5.14) 49 times, we obtain one thousand xt s (t = 1, 2, · · · , 50) and their geometric mean values xt . Figure 5.9 depicts the simulation result. Geometric mean values xt  (t = 1, 2, · · · , 50) first follow the power-law growth (5.9) and subsequently the exponential growth (5.10). At the same time, in Fig. 5.9b, the slope of the exponential growth is a tangent line of the power-law growth at the transition point from powerlaw growth to exponential growth. The indices are estimated as γ = 0.53 ± 0.00 and β = 0.035 ± 0.00 by the least squares method. In this simulation, the system is merely developed by multiplying growth rates R(xt ) observed in (T , T + 1) at T = 2011. The estimated indices in the simulation are close to the empirical indices observed in 2013 Japan, especially for power-law index γ . The difference between β in the simulation (Fig. 5.9) and β in the empirical data analysis (Fig. 5.5) probably

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5.4 Long-Term and Short-Term Firm Growth

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Fig. 5.9 Geometric mean values of firm size xt  in simulation based on the extended nonGibrat’s property and Gibrat’s law are plotted by circles (◦). Horizontal axis is firm age t, and x1 s are randomly extracted from operating revenue data of firms founded in 2011. (a) In double logarithmic plots, xt  in range 1 < t ≤ 20 is approximated by dashed line as follows:

xt  ∝ t γ , γ = 0.53 ± 0.00. This power-law approximation is statistically superior to exponential approximation because R 2 s of power-law and exponential approximations in this range are 0.98 and 0.97. (b) In single logarithmic plots, xt  in range 10 ≤ t ≤ 40 is approximated by dotted line as follows: xt  ∝ eβt , β = 0.035 ± 0.000

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Fig. 5.10 Geometric mean values of firm size xt  in simulation based on Gibrat’s law are plotted by circles (◦). Horizontal axis is firm age t, and x1 s are randomly extracted from operating revenue data of firms founded in 2011. In double logarithmic plots (a) and in single logarithmic plot (b),

xt  in range 1 ≤ t ≤ 40 is approximated by solid line as follows: xt  ∝ eβt , β = 0.014 ± 0.000. This exponential approximation is statistically superior to power-law approximation because R 2 s of power-law and exponential approximations in range 1 ≤ t ≤ 20 are 0.87 and 0.99

corresponds to the difference between the transition points in the simulation (Fig. 5.9) and in the empirical data analysis (Fig. 5.5). Note that early power-law growth does not emerge when multiplicative noise R does not depend on initial value xt . Figure 5.10 shows that xt  grows exponentially from the first step. In this simulation, we did not divide the empirical growth rate

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5 Long-Term Firm Growth Derived from Non-Gibrat’s Property and Gibrat’s Law

into bins. Since the simulation’s result can be explained analytically, the validity of this simulation procedure is confirmed. From these results, we conclude that the early power-law growth is derived from the extended non-Gibrat’s property and the subsequent exponential growth is reduced from Gibrat’s law. We also confirmed similar simulation results with respect to operating revenues in France and total assets and number of employees in Japan and France.

5.5 Discussion In this chapter, with the Orbis database provided by Bureau van Dijk, we analyzed the dependence of firms’ operating revenues, total assets, and number of employees (firm-size variables) on firm age in Japan and France from 2010 to 2013. We confirmed that the geometric mean value of the firm-size variables obeys a powerlaw growth for its first 10 years and subsequently follows exponential growth. Using numerical simulations, these long-term properties of firm-size growth were derived from a short-term growth law and property that were observed in two successive years. First, early power-law growth under a size threshold comes from the extended non-Gibrat’s property. Second, subsequent exponential growth over the threshold is derived from Gibrat’s law. Specifically, we introduced the assumption that a short-term law and property confirmed in two successive calendar years were also observed in two successive firm ages. This is natural and can be directly confirmed when investigation is possible. Then we introduced a simple stochastic model based on short-term law and properties and described the early power-law growth of firm-size variables and subsequent exponential growth. By changing the multiplicative noise of the stochastic equation based on the size of the variables, we confirmed that the early power-law growth is derived from the extended non-Gibrat’s property and the subsequent exponential growth is reduced from Gibrat’s law. The simulation showed that the transition from power-law growth to exponential growth is smooth when the multiplicative noise is stable. In this context, smooth transitions are continuous and share tangents at transition points, as shown in Fig. 5.9a and b. In such a case, γ = t0 β is established from Eqs. (5.9) and (5.10). Here t0 is the age at which a switch occurs from power growth to exponential growth, which is ten in the analysis of this chapter. In many cases, Table 5.1 confirms that this relationship is generally established. At the same time, this relational expression suggests that if t0 does not change, the magnitude relations of γ and β are identical. The only exception is Japan’s growth indices of operating revenues. Its initial power-law growth index γ OR JP exceeds that of the assets and the number of employees. However, the following OR moderate exponential growth index β JP is smaller than that of the assets and the number of employees. Such moderate growth might reflect that the transition

5.5 Discussion

67

from initial power-law growth to exponential growth was not smooth, caused by the bursting of Japan’s economic bubble. From the analysis in this chapter, we conclude that the strongest impact of the bubble bursting was on the growth of a firm’s operating revenues, probably because the operating revenues are financial variables representing the firms’ current performance. However, total assets are financial variables representing the firms’ cumulative performance. We also suggest that the difference between the average change of positive and negative growth rates δ± determines index γ of the initial power-law growth. δ± should be directly related to parameter α± of the non-Gibrat’s property. In other words, exponent γ of the initial power-law growth is determined by the difference between the non-Gibrat’s parameters α± : that is, the log-normal distribution’s parameter α. Since α is 1/2 of the reciprocal of the logarithmic variance of the log-normal distribution, the smaller α is, the greater the extent of the log-normal distribution. In summary, the larger is the spread of the log-normal distribution of firmsize variables, the smaller is initial power-law growth index γ of the firm-size variables. Conversely, the smaller the spread of the log-normal distribution of firm-size variables, the larger is initial power-law growth index γ of the firm-size variables. In fact, we confirmed that the spread of the log-normal distribution of the number of employees is larger than that of the operating revenues [44], a result that is consistent with the above consideration. In countries other than Japan and France, if the long-term growth of firm-size variables can be clearly observed, this speculation can be confirmed numerically. In Chap. 5, we showed that firm-size variables rapidly grow under a powerlaw function and subsequently grow more slowly under an exponential function, assuming that the extended non-Gibrat’s property and Gibrat’s law do not change annually in the long term. In Chaps. 2 and 4, we showed that the power law in the large-scale range and the log-normal distribution in the middle-scale range of firmsize variables at a point in time are derived from Gibrat’s law and the extended non-Gibrat’s property under time-reversal symmetry. Taken together, these findings can be summarized as follows. For older and largesized firms, not only the power law but also a long-term exponential growth law is observed. By combining them, the exponential firm-age distribution must be derived approximately. For young and mid-sized firms, not only log-normal distribution but also power-law growth is seen. By combining them, the firm-age distribution, which deviates from the exponential function, must also be derived. These features emerged when the decay rate of firm activities decreased as firms age [35]. In Chap. 6, we will examine the relationship between the long-term property of the age distribution of firms and the short-term property of the inactive rate of firms. In Chaps. 1 through 5, we only considered the statistical laws and properties of firms that continue to exist. Next we will identify statistical laws and properties associated with the cessation of firms’ activities that we did not cover in previous chapters.

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References 1. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2016) Firm growth function and extendedGibrat’s property. Adv Math Phys 2016:9303480 2. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2016) Long-term firm growth properties derived from short-term laws and number of emoplyees in Japan and France. Evolut Inst Econo Rev 13:409–422 3. Solow R (1956) A contribution to the theory of economic growth. Q J Econ 70:65–94 4. Romer PM (2007) Economic growth. In: Henderson DR (ed) The concise encyclopedia of economics. Liberty Fund, Indianapolis, pp. 128–131 5. Mantegna RN, Stanley HE (2000) Introduction to econophysics: correlations and complexity in finance. Cambridge University Press, Cambridge 6. Saichev AI, Malevergne Y, Sornette D (2010) Theory of Zip’s law and beyond. Springer, Berlin 7. Pareto V (1897) Cours d’Économie Politique. Macmillan, London 8. Newman MEJ (2005) Power laws, Pareto distributions and Zipf’s law. Contemp Phys 46:323– 351 9. Clauset A, Shalizi CR, Newman MEJ (2009) Power-law distributions in empirical data. SIAM Rev 51:661–703 10. Bonabeau E, Dagorn L (1995) Possible universality in the size distribution of fish schools. Phys Rev E51:R5220–R5223 11. Render S (1998) How popular is your paper? An empirical study of the citation distribution. Eur Phys J B4:131–134 12. Takayasu M, Takayasu H, Sato T (1996) Critical behaviors and 1/ noise in information traffic. Physica A233:824–834 13. Kaizoji T (2003) Scaling behavior in land markets. Physica A326:256–264 14. Mantegna RN, Stanley HE (1995) Scaling behaviour in the dynamics of an economic index. Nature 376:46–49 15. Axtell RL (2001) Zipf distribution of US firm sizes. Science 293:1818–1820 16. Okuyama K, Takayasu M, Takayasu H (1999) Zipf’s law in income distribution of companies. Physica A269:125–131 17. Mizuno T, Katori M, Takayasu H, Takayasu M (2002) Statistical and stochastic laws in the income of Japanese companies. In: Takayasu H (ed) Empirical science of financial fluctuations – the advent of econophysics. Springer, Tokyo, pp 321–330 18. Iyetomi H, Aoyama H, Fujiwara Y, Ikeda Y, Souma W (2012) A paradigm shift from production function to production copula: statistical description of production activity of firms. Quant Finan 12:1453–1466 19. Levy M, Solomon S (1996) Power laws are logarithmic Boltzmann laws. Int J Mod Phys C7:595–601 20. Sornette D, Cont R (1997) Convergent multiplicative processes repelled from zero: power laws and truncated power laws. J Phys I7:431–444 21. Fujiwara Y, Souma W, Aoyama H, Kaizoji T, Aoki M (2003) Growth and fluctuations of personal income. Physica A321:598–604 22. Fujiwara Y, Guilmi CD, Aoyama H, Gallegati M, Souma W (2004) Do Pareto-Zipf and Gibrat laws hold true? An analysis with European firms. Physica A335:197–216 23. Ishikawa A (2006) Annual change of Pareto index dynamically deduced from the law of detailed quasi-balance. Physica A371:525–535 24. Ishikawa A (2009) Quasi-statically varying Pareto-law and log-normal distributions in the large and the middle scale regions of Japanese land prices. Prog Theor Phys Suppl. 179:103–113 25. Gibra R (1932) Les Inégalités Économique. Sirey, Paris 26. Sutton J (1997) Gibrat’s legacy. J Econ Lit 35:40–59 27. Ishikawa A (2006) Derivation of the distribution from extended Gibrat’s law. Physica A367:425–434

References

69

28. Ishikawa A (2007) The uniqueness of firm size distribution function from tent-shaped growth rate distribution. Physica A383:79–84 29. Tomoyose M, Fujimoto S, Ishikawa A (2009) Non-Gibrat’s law in the middle scale region. Prog Theor Phys Suppl 179:114–122 30. Ishikawa A, Fujimoto S, Mizuno T (2011) Shape of growth rate distribution determines the type of non-Gibrat’s property. Physica A390:4273–4285 31. Coad A (2010) The exponential age distribution and the pareto firm size distribution. J Ind Compet Trade 10:389–395 32. Coad A (2010) Investigating the exponential age distribution of firms. Economics 4:2010–2017 33. Fujiwara Y (2004) Zipf law in firms bankruptcy. Physica A337:219–230 34. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2015) Firm age distributions and the decay rate of firm activities. In: Takayasu H, Ito N, Noda I, Takayasu M (eds) Proceedings of the international conference on social modeling and simulation, plus econophysics colloquium 2014. Spinger, Tokyo, pp 187–194 35. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2015) The relation between firm age distributions and the decay rate of firm activities in the United States and Japan. Big data, 2015 IEEE international conference on date of conference, pp 2726–2731 36. Coad A (2009) The growth of firms. Edward Elgar Publishing, Cheltenham 37. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2016) Firm growth function and extendedGibrat’s property. Adv Math Phys 2016:9303480 38. Luttmer EGJ (2011) On the mechanics of firm growth. Rev Econ Stud 78:1042–1068 39. Petersen AM, Riccaboni M, Stanley HE, Pammolli F (2012) Persistence and uncertainty in the academic career. Proc Natl Acad Sci 109:5213–5218 40. Bureau van Dijk, http://www.bvdinfo.com/Home.aspx/ 41. Statistics Bureau, Ministry of Internal Affairs and Communications, http://www.stat.go.jp/ index.htm 42. Matia K, Fu D, Buldyrev SV, Pammolli F, Riccaboni M, Stanley HE (2004) Statistical properties of business firms structure and growth. Europhys Lett 67:498–503 43. Takayasu H, Sato A, Takayasu M (1997) Stable infinite variance fluctuations in randomly amplified langevin systems. Phys Rev Lett 79:966–969 44. Ishikawa A, Mizuno T, Fujimoto S, Watanabe T (2011) Firm size distribution and nonGibrat’s law observed in the mid-scale range. Econophys-Kolkata VI (International workshop on “Econophysics of systemic risk and network dynamics”), Kolkata

Chapter 6

Firm-Age Distribution and the Inactive Rate of Firms

Abstract It is as important to consider how a firm will cease its activities as how it will continue those same activities. The short-term inactive rate of firms is observed in the long term as their age distribution. In this chapter, we identify the age dependence of the inactive rate of firms and link it to the long-term property of firm-age distribution. Specifically, we investigated the inactive rates of firm activities by comparing their 2014 and 2015 statuses in Germany, Spain, France, the United Kingdom, Italy, Japan, Korea, and the Netherlands. We found that in Japan, the inactive rate of firm activities does not depend on a firm’s age. In other countries, however, the inactive rate of young firms, which was higher than that of more established firms, gradually fell and eventually became constant as firms aged. This inactive rate leads to the following conclusion. In Japan, firm-age distribution decreases exponentially; in other countries, it exponentially decreases asymptotically, but young firms shift slightly upward. Using empirical data, we compared the inactive rate of firms with parameters measured from firmage distribution and confirmed our analytical discussion.

6.1 Introduction Over the last few decades, researchers have studied economies in a different way from traditional economics by applying to economic data a method borrowed from physics. This approach is called econophysics [1–3]. In econophysics, large amounts of the financial data of firms (operating revenues, number of employees, assets, and so on) are statistically analyzed to identify a substantial contingent of universal laws [4–24]. One main aim of econophysics is to systematically achieve an economy based on such laws. In Chaps. 2 and 4, we focused on the properties of the financial data of firms by observations in a single year or two successive years [25–34]. In Chap. 5, we discussed the long-term growth of firm-size variables, which by age group

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Ishikawa, Statistical Properties in Firms’ Large-scale Data, Evolutionary Economics and Social Complexity Science 26, https://doi.org/10.1007/978-981-16-2297-7_6

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statistically grew rapidly at an early stage following a power-law function and shifted to a gradual exponential growth. We argued that these properties can be understood from the non-Gibrat property observed in the mid-scale range and Gibrat’s law in the large-scale range. This argument links short-term and long-term properties. All our discussions so far have concentrated on the properties of and the relationships among surviving firms. Chapter 6 continues Chap. 5’s investigation of the long-term properties of firms and focuses on firm-age distribution. Some previous research concluded that the firm-age distributions of several countries obeyed exponential functions [35–40]. Such exponential firm-age distribution suggests that the inactive rate of firm activities does not depend on firm age by an analogy to nuclear decay [41]. In empirical data, the decay rate of Japanese firms was identified to be nearly constant after comparing their activities in 2008 and 2013 [42]. However, the decay rate of firm activities in the United States does depend on firm age [43]. The decay rate of young firms, which was higher than that of more established firms, fell and eventually became constant as firms aged. We approximated the distribution of the decay rate by an exponential function and analytically derived a firm-age distribution, which deviates from the exponential function, and confirmed the result with American empirical firm data. Using the Compustat database [44], we showed that the distribution of firms’ life span in the database follows an exponential function [45]. To summarize these studies, in this chapter, we investigate the activity data of the firms in countries where the amount of data is statistically sufficient and identify the dependence of the inactive rate on firm age. Our database comprises the following countries: Germany, Spain, France, the United Kingdom, Italy, Japan, Korea, and the Netherlands. Since the United States has excessive NA data, we excluded it from our analysis. In Japan, the decay rate of firm activities does not depend on firm age. On the other hand, in the remaining seven countries, the inactive rate does depend on it. The discussion in this chapter is based primarily on our previous work [42, 43, 46]. Most texts are reproduced under the Creative Commons Attribution License or with permission from the publisher. The main text is modified to fit the context of this book. The rest of Chap. 6 is organized as follows. Section 6.2 describes our database and explains the status data of firm activities. In Sect. 6.3, we show the analysis results of the inactive rate of firm activities and firm-age distribution and describe their qualitative properties. In Sect. 6.4, we approximate the distribution of the inactive rate observed in Sect. 6.3 by an exponential function, analytically derive the firm-age distribution, and quantitatively compare the parameters estimated by the inactive rate with those estimated by firm-age distribution. The last section concludes this chapter.

6.2 Data

73

6.2 Data We employed the Orbis database (provided by Bureau van Dijk [47]) released in 2015 and 2016 that contains around 200 million pieces of firm-size data from all over the world from those years. The Orbis databases contain the activity data of firms in 2014 and 2015. The firm activities in the Orbis database are largely categorized into the following three states: • Active – Active, Active (payment default), Active (insolvency proceedings), Active (dormant), Active (branch) • Inactive – In liquidation, Bankruptcy, Dissolved (merger or take-over), Dissolved (demerger), Dissolved (liquidation), Dissolved (bankruptcy), Dissolved, Inactive (branch), Inactive (no precision) Unknown In the database, the following nine countries have a sufficient amount of data for statistically analysis: Germany, Spain, France, the United Kingdom, Italy, Japan, Korea, the Netherlands, and the United States. Table 6.1 shows the number of active firms in 2014 as well as the number of active, inactive, unknown, and NA firms in 2015 that were active in 2014. We excluded the United States from the analysis because the IDs of very few firms matched between Orbis 2015 and Orbis 2016. This discrepancy was probably caused by a change in how such IDs were assigned. In this study, we defined firm inactivities as a transition from active to inactive and excluded data from firms whose statuses were unknown or NA in 2015. Here, we analyzed the eight countries shown in Table 6.1. Note that in this definition, firm

Table 6.1 Active firms with data of incorporation year in 2014 and active, inactive, unknown, and NA firms in 2015 that were active in 2014 in database

Country Germany Spain France United Kingdom Italy Japan Korea Netherlands

Active firms in 2014 1,849,601 2,308,053 1,751,302 3,428,365 3,156,270 2,659,627 462,591 2,623,934

Active firms in 2015 1,565,182 2,143,098 1,656,234 2,941,370 2,929,598 2,588,452 426,931 2,417,561

Inactive firms in 2015 48,262 149,232 91,898 486,216 226,352 68,690 30,166 4,204,333

Unknown firms in 2015 236,155 15,686 3170 558 141 2483 5493 1554

NA data in 2015 2 37 0 221 179 2 1 486

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6 Firm-Age Distribution and the Inactive Rate of Firms

inactivity does not necessarily denote bankruptcy. In Japan, for instance, there were 2,659,627 active firms in 2014, among which 2,588,452 remained active in 2015; 68,690 firms became inactive in this 1-year period. Of the total, 30 were categorized as Dissolved (merger or take-over), 1 as Dissolved (liquidation), 5 as Dissolved, 68, 654 as Inactive (no precision), 2, 483 as Unknown, and 2 as NA. This database also provides a firm’s incorporation year, which we regard as its starting point. Since firms have financial statements from their year of foundation, we denote their age of foundation as t = 1. Using this procedure, we can calculate firm age t. Table 6.1 shows the number of firms from which this information is available. We should mention that the Orbis database did not necessarily collect information on every small new firm in these countries.

6.3 Data Analysis In this section, we analyze the dependence of the inactive rate of firm activities on firm age and its distribution by employing the firm data from a database that includes Germany, Spain, France, the United Kingdom, Italy, Japan, Korea, and the Netherlands. The inactive rates were estimated by classifying the firms into age-rank bins with 5-year period and comparing their 2014 and 2015 activities. We signify them as D, defined by D≡

#(Inactive firms in 2015 that were Active in 2014) . (6.1) #(Active firms in 2014 that were not Unknown and NA in 2015)

We depict D and the number of firms N(t) in 2014. In each figure, the left axis is the number of firms N(t) in a log scale, and the right axis is the inactive rate of the firm activities in a 1-year D. Figure 6.1 shows the D (×) and N(t) (◦) distributions in German (a), Spain (b), France (c), the United Kingdom (d), Italy (e), Japan (f), Korea (g), and the Netherlands (h). In these countries, we observed a similar dependence of the decay rate of firm activities on firm age. The inactive rate of young firms, which exceeds that of the older firms, dropped and eventually became constant as they aged. Such dependence is rarely seen in Japan (f). In Korea (g), the inactive rate became lower and finally fell to “zero” as firms aged. Figure 6.1 also shows that the age distributions in these countries follow a convex downward function in a single logarithmic plot. In Japan, however, the semilogarithmic plot is almost linear. In all countries, there is a N(t) (◦) gap after age 60, which is a remnant of the effects of World War II. Note that the inactive rate D (×) measures the transition from activity states in 2014 to 2015. On the other hand, N(t) (◦) denotes the number of firms by age in 2015. That is, D (×) and N(t) (◦) share a horizontal axis, but generally D (×) does not determine the immediately above N(t) (◦). This result happens only if the distribution of D and the number of firms born each year do not change.

6.3 Data Analysis

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Fig. 6.1 Inactive rate of firm activity in 1-year D and firm-age distribution N(t) classified into age-rank bins of 5-year period in Germany (a), Spain (b), France (c), the United Kingdom (d), Italy (e), Japan (f), Korea (g), and the Netherlands (h). D is depicted as crosses (×) on right axis. N(t) is depicted as circles (◦) on left axis

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6.4 Inactive Rate of Firm and Firm-Age Distribution From the analysis results in the previous section, we assume that the inactive rate of a firm in 1-year D can generally be approximated using the following three parameters, λ1 , λ2 , and κ: D = λ1 + λ2 e−κt .

(6.2)

Next, we describe the number of firms N(t) at time t. Using an approximation, the inactive rate obeys the exponentially decreasing property, Eq. (6.2), which leads to a survival rate where firms are active in times t and t + 1 as follows: N(t + 1) = 1 − λ1 − λ2 e−κt . N(t)

(6.3)

This can be rewritten as the following differential equation: dN(t ) dt

N(t)

= −λ1 − λ2 e−κt .

(6.4)

The solution is written by 



λ2 −κt e −1 . N(t) = N0 exp −λ1 t + κ

(6.5)

Here, the constant of integration N0 is the number of firms at time t = 0. If the number of firms established annually is almost constant, N(t) can be interpreted as the number of firms of age t. On the one hand, for κt  1, Eq. (6.5) is approximated by   λ2 κ 2 t . N(t) ∼ N0 exp −(λ1 + λ2 )t − 2

(6.6)

On the other hand, for κt  1, Eq. (6.5) is approximated by N(t) = N0 exp [−λ1 t] .

(6.7)

Here, N0 = N0 e−λ2 /κ . By applying Eq. (6.2) to the inactive rate of firm activities D in Fig. 6.1, parameters λ1 , λ2 , and κ can be estimated. By applying Eq. (6.5) to firm-age distributions N(T ) in Fig. 6.1, parameters λ1 , λ2 , and κ can also be evaluated in principle. However, since directly calculating these parameters with Eq. (6.5) is very difficult, we estimate parameters λ1 + λ2 and λ2 κ/2 by applying the approximation in Eq. (6.6) to N(T ).

6.4 Inactive Rate of Firm and Firm-Age Distribution

77

Note that these parameters, which are estimated in two different ways, do not necessarily agree for the following reasons. In this chapter, the inactive rate of firms D is observed in the transition from an active state in 2014 to an inactive state in 2015. However, firm-age distribution N(t) represents the number of firms that have been successful for t years, in which the inactive rate of the firm activities has probably continued to change. In addition, the number of firms established annually is not necessarily approximated to be nearly constant. In this manner, several restrictions exist for the agreement of parameters λ1 , λ2 , and κ that are estimated by two different methods. When the parameters obtained by two different methods do not agree, firm data are beyond these restrictions, and vice versa. Table 6.2 shows the parameters estimated by these two different ways in eight D D countries. The parameters calculated by D are denoted by λD 1 , λ2 , and κ , and N N N those calculated by N(t) are denoted by λ1 , λ2 , and κ . λD 1 in Eq. (6.2) is the minimum D in the range of 1 ≤ t ≤ 100 in each country in Fig. 6.1, and the standard deviation of D is the error. Since the D of the older firms is not always smooth, in principle we also measured λD 2 and κ by applying Eq. (6.2) to D − λ1 of the firms in the range of 1 ≤ t ≤ 25. However, the Ds on the left side of Germany and Korea are lower than the second value, so they are excluded from the measurement. In Japan, D tends to increase in the range of 1 ≤ t ≤ 10. Since D in older firms was relatively smooth compared to other countries, we measured λD 2 and κ in the range of 10 < t ≤ 100. N N N On the other hand, λN 1 + λ2 and λ2 κ /2 were measured by applying Eq. (6.6) to N(t) in the range of 1 ≤ t ≤ 145 in each country in Fig. 6.1. However, because in Japan, the gap in the firm-age distribution caused by World War II, which is over 60 years old, is larger than the gaps in the firm-age distributions in other countries, we measured them in the range of 1 ≤ t ≤ 60. These measurement ranges exceed the applicable range of the approximation (6.6) (κt  1). However, since the downward curvature of the age distribution may not be captured by the measurement of several points within the applicable range, measurements were intentionally extended beyond the approximation’s applicable range. We confirmed from numerical calculations that the value measured beyond this approximation is estimated to be smaller than the true value. In particular, the deviation of the N N N N measured value of λN 2 κ /2 exceeds that of λ1 + λ2 . We can also measure λ1 by applying Eq. (6.7) to the application range (κt  1). However, this approach’s applicable range is narrow, and N(t) is not smooth, and furthermore, the scope of N(t) was determined by D more than 100 years ago. For these reasons, this approach was not adopted. First, in Table 6.2, considering the error, we confirmed that κ D in Japan is indistinguishable from 0. This means that dependence in Eq. (6.2) on t is negligible. This result is consistent with the observation that the inactive rate of firm activities D barely depends on the firm age in Fig. 6.1f. This is a major characteristic of Japan that is not seen in other countries. In Japan, the discrepancy among the parameters estimated by D and those by N(t) falls within the measurement error range, suggesting that over the last decade in Japan, the inactive rate of firm activities

λD 2

0.039 ± 0.002 0.086 ± 0.046 0.082 ± 0.004 0.160 ± 0.015 0.083 ± 0.008 0.004 ± 0.003 0.139 ± 0.026 0.060 ± 0.013

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0.008 ± 0.009 0.012 ± 0.022 0.003 ± 0.021 0.041 ± 0.034 0.009 ± 0.019 0.019 ± 0.004 0.000 ± 0.024 0.043 ± 0.014

Country

Germany Spain France United Kingdom Italy Japan Korea Netherlands

0.027 ± 0.003 0.027 ± 0.037 0.046 ± 0.004 0.137 ± 0.007 0.037 ± 0.007 0.000 ± 0.011 0.078 ± 0.013 0.092 ± 0.016

κD 0.050 ± 0.010 0.097 ± 0.029 0.077 ± 0.021 0.165 ± 0.040 0.077 ± 0.019 0.027 ± 0.004 0.053 ± 0.024 0.113 ± 0.017

D λD 1 + λ2

0.0006 ± 0.0001 0.0008 ± 0.0003 0.0014 ± 0.0001 0.00692 ± 0.0012 0.0009 ± 0.0001 0.0000 ± 0.0000 0.0008 ± 0.0001 0.0041 ± 0.0007

D λD 2 κ /2

0.076 ± 0.000 0.126 ± 0.000 0.063 ± 0.000 0.092 ± 0.000 0.108 ± 0.000 0.027 ± 0.000 0.125 ± 0.000 0.074 ± 0.000

N λN 1 + λ2

0.0002 ± 0.0001 0.0003 ± 0.0001 −0.0002 ± 0.0001 0.0002 ± 0.0000 0.0002 ± 0.0000 −0.0001 ± 0.0002 0.0003 ± 0.0001 0.0002 ± 0.0000

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N N D D N Table 6.2 Parameters calculated by D are denoted by λD 1 , λ2 , and κ , and those by N(t) are denoted by λ1 , λ2 , and κ in eight countries

78 6 Firm-Age Distribution and the Inactive Rate of Firms

6.5 Discussion

79

has remained almost stagnant and the number of firms established annually is nearly constant. In Spain, the inactive rate of firms that range from 1 to 30 years old is not smooth. This means that the inactive rate of its young firms is not statistically or significantly dependent on firm age. In France, the firm-age distribution from 60 to 120 years ago, which dates back to before World War II, is not smooth. This result reflects the fact that the inactive rate of firms in France has not been stable since before World War I. Thus, the firm-age distribution is not always smooth, mainly because of war. However, in countries (other than Japan) where the gap is very large, N(t) was measured for the entire range of the figure without taking this idea into account. Next, we compare two measurements: λ1 + λ2 and λ2 κ/2. For the former, the values measured by D and N(t) approximately agree with each other within the error range. On the other hand, in the latter case, the value measured by N(t) is often smaller than the value measured by D because the measurement is conducted outside the application range of the approximation.

6.5 Discussion In this chapter, we investigated the inactive rate of firms by comparing their 2014 and 2015 statuses in Germany, Spain, France, the United Kingdom, Italy, Japan, Korea, and the Netherlands. We found that the inactive rate of firms depends on firm age. The inactive rate of young firms in these countries, which was higher than that of more established firms, gradually lowered and eventually became constant as firms aged. The only exception was Japan, where the inactive rate of firms is almost independent of company’s age; if any dependence existed, it was too small to be observed. This difference is probably related to the institution of a firm’s establishment. According to the Small and Medium Enterprise Agency of Japan, Japan’s entry rate is considerably lower than in Western countries [48]. Japan might have more restrictions on the establishment of firms than other countries. Its low entry rate is probably related to the low inactive rate of young firms in Japan [48]. We approximated the inactive rate of firm activities by an exponential function and analytically derived firm-age distributions under the assumptions that the number of firms that are established annually is nearly constant and that the inactive rate of firm activities does not change. Using empirical data from eight countries, we compared the parameters estimated by the inactive rate of firm activities with those by firm-age distribution. The two kinds of parameters estimated by two different ways were close to each other. In this chapter, we investigated the inactive rates of firms and linked them to the long-term property of firm-age distribution. Chapter 5 looked at how firms grow in the short and long terms. It is equally important to consider how firms stop their activity. As a continuation of this chapter, Chap. 7 examines how a firm’s inactive rate is related to its size.

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References 1. Mantegna RN, Stanley HE (2000) Introduction to econophysics: correlations and complexity in finance. Cambridge University Press, Cambridge 2. Saichev AI, Malevergne Y, Sornette D (2010) Theory of Zip’s Law and beyond. Springer, Berlin 3. Aoyama H, Fujiwara Y, Ikeda Y, Iyetomi H, Souma W (2010) Econophysics and companies: statistical life and death in complex business networks. Cambridge University Press, Cambridge 4. Pareto V (1897) Cours d’Économie Politique. Macmillan, London 5. Newman MEJ (2005) Power laws, Pareto distributions and Zipf’s law. Contemp Phys 46:323– 351 6. Clauset A, Shalizi CR, Newman MEJ (2009) Power-law distributions in empirical data. SIAM Rev 51:661–703 7. Bonabeau E, Dagorn L (1995) Possible universality in the size distribution of fish schools. Phys Rev E51:R5220–R5223 8. Render S (1998) How popular is your paper? An empirical study of the citation distribution. Eur Phys J B4:131–134 9. Takayasu M, Takayasu H, Sato T (1996) Critical behaviors and 1/ noise in information traffic. Physica A233:824–834 10. Kaizoji T (2003) Scaling behavior in land markets. Physica A326:256–264 11. Yamano T (2004) Distribution of the Japanese posted land price and the generalized entropy. Eur Phys J B38:665–669 12. Mantegna RN, Stanley HE (1995) Scaling behaviour in the dynamics of an economic index. Nature 376:46–49 13. Axtell RL (2001) Zipf distribution of US firm sizes. Science 293:1818–1820 14. Podobnik B, Horvatic D, Petersen AM, Uro˘sevi´c B, Stanley HE (2010) Bankruptcy risk model and empirical tests. Proc Natl Acad Sci USA 107:18325–18330 15. Fu D, Pammolli F, Buldyrev SV, Riccaboni M, Matia K, Yamasaki K, Stanley HE (2005) The growth of business firms: theoretical framework and empirical evidence. Proc Natl Acad Sci 102:18801–18806 16. Podobnik B, Horvatic D, Pammolli F, Wang F, Stanley H E, Grosse I (2008) Size-dependent standard deviation for growth rates: empirical results and theoretical modeling. Phys Rev E77:056102 17. Okuyama K, Takayasu M, Takayasu H (1999) Zipf’s law in income distribution of companies. Physica A269:125–131 18. Ramsden JJ, Kiss-Haypál G (2000) Company size distribution in different countries. Physica A277:220–227 19. Mizuno T, Katori M, Takayasu H, Takayasu M (2002) Statistical and stochastic laws in the income of Japanese companies. In: Takayasu H (ed) Empirical science of financial fluctuations – the advent of econophysics. Springer, Tokyo, pp 321–330 20. Gaffeo E, Gallegati M, Palestrinib A (2003) On the size distribution of firms: additional evidence from the G7 countries. Physica A324:117–123 21. Zhang J, Chen Q, Wang Y (2009) Zipf distribution in top Chinese firms and an economic explanation. Physica A388:2020–2024 22. Levy M, Solomon S (1996) Power laws are logarithmic Boltzmann laws. Int J Mod Phys C7:595–601 23. Sornette D, Cont R (1997) Convergent multiplicative processes repelled from zero: power laws and truncated power laws. J Phys I 7:431–444 24. Takayasu H, Sato A, Takayasu M (1997) Stable infinite variance fluctuations in randomly amplified Langevin system. Phys Rev Lett 79:966–969 25. Gibra R (1932) Les Inégalités Économique. Sirey, Paris 26. Sutton J (1997) Gibrat’s legacy. J Econ Lit 35:40–59

References

81

27. Fujiwara Y, Souma W, Aoyama H, Kaizoji T, Aoki M (2003) Growth and fluctuations of personal income. Physica A321:598–604 28. Fujiwara Y, Guilmi CD, Aoyama H, Gallegati M, Souma W (2004) Do Pareto-Zipf and Gibrat laws hold true? An analysis with European firms. Physica A335:197–216 29. Ishikawa A (2006) Derivation of the distribution from extended Gibrat’s law. Physica A367:425–434 30. Ishikawa A (2006) Annual change of Pareto index dynamically deduced from the law of detailed quasi-balance. Physica A371:525–535 31. Ishikawa A (2007) The uniqueness of firm size distribution function from tent-shaped growth rate distribution. Physica A383:79–84 32. Ishikawa A (2009) Quasistatically varying log-normal distribution in the middle scale region of Japanese land prices. Prog Theor Phys Suppl 179:103–113 33. Tomoyose M, Fujimoto S, Ishikawa A (2009) Non-Gibrat’s law in the middle scale region. Prog Theor Phys Suppl 179:114–122 34. Ishikawa A, Fujimoto S, Mizuno T (2011) Shape of growth rate distribution determines the type of non-Gibrat’s property. Physica A390:4273–4285 35. Fujiwara Y (2004) Zipf law in firms bankruptcy. Physica A337:219–230 36. Coad A, Tamvada JP (2008) The growth and decline of small firms in developing countries. Working Paper on Economics and Evolution (Max Plank Institute of Economics) 2008-08 37. Coad A (2010) The exponential age distribution and the pareto firm size distribution. J Ind Compet Trade 10:389–395 38. Coad A (2010) Investigating the exponential age distribution of firms. Economics 4:2010–2017 39. Bottazzi G, Secchi A, Tamagni F (2008) Productivity, profitability and financial performance. Ind Corp Chang 17:711–751 40. Miura W, Takayasu H, Takayasu M (2012) Effect of coagulation of nodes in an evolving complex network. Phys Rev Lett 108:168701 41. Rutherford E, Soddy F (1903) Radioactive change. Phil Mag 6:576–591 42. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2015) Firm age distributions and the decay rate of firm activities. In: Takayasu H, Ito N, Noda I, Takayasu M (eds). Proceedings of the international conference on social modeling and simulation, plus econophysics colloquium 2014. Spinger, Tokyo, pp 187–194 43. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2015) The relation between firm age distributions and the decay rate of firm activities in the United States and Japan. Big data, 2015 IEEE international conference on date of conference, pp 2726–2731 44. Standard & Poor’s. COMPUSTAT, http://www.compustat.com 45. Daepp MIG, Hamilton MJ, West GB, Bettencourt LMA (2015) The mortality of companies. J R Soc Interface 12:20150120 46. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2017) Depencence of the decay rate of firm activites on firm age. Evolut Inst Econo Rev 14:351–362 47. Bureau van Dijk, http://www.bvdinfo.com/Home.aspx 48. Small and Medium Enterprise Agency, http://www.chusho.meti.go.jp/sme_english/index.html

Chapter 7

Statistical Properties in Inactive Rate of Firms

Abstract In this chapter, we investigate the dependence of the inactive rate of firms on the following main financial variables: total revenue, net income, total assets, and net assets. We used worldwide information on German, Spanish, French, British, Italian, Japanese, Korean, and Dutch firms recorded in the 2015 and 2016 editions of the comprehensive Orbis database of listed and unlisted firms. We confirmed that the inactive rate of firms is constant regardless of the size of the financial variables in the large-scale data range. In the mid-scale data range, the inactive rate of firms increases under a power law as the financial variables decrease. The boundary between the large- and mid-scale data ranges corresponds to the boundary between the power-law and log-normal distributions of the financial data.

7.1 Introduction Through Chap. 5, we discussed the statistical properties of time evolution, such as firm-size variables. In natural science, theories are based on the distributions of variables in a particular system of interest. Various statistical laws have been observed, which are related. Statistical laws are also studied in sets of firm-size variables, such as operating revenues, profits, and the number of employees. For example, the firmsize distribution in a certain year follows a power law in the large-scale range of data values [1–3] and a log-normal distribution in the mid-scale range of data values (for example, see [4]). For instance, the value of the boundary between the large- and mid-scale data ranges is around 104 thousand dollars for Japanese firms’ operating revenues. These distributions are closely related to statistical laws observed over two consecutive years. Under the time-reversal symmetry of the firm-size variables, power-law and log-normal distributions are derived from Gibrat’s law and nonGibrat’s property [5–11]. Time-reversal symmetry is a property in which the scatter plot of the firm-size variables between a certain year and the next year is symmetrical with respect to time reversal. Gibrat’s law reflects a property in which © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Ishikawa, Statistical Properties in Firms’ Large-scale Data, Evolutionary Economics and Social Complexity Science 26, https://doi.org/10.1007/978-981-16-2297-7_7

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the growth-rate distribution conditioned by firm-size variables does not depend on firm-size variables. On the other hand, non-Gibrat’s property describes the dependence of the conditional growth-rate distribution on them. These statistical properties are found in firms that continue to be active. Do statistical properties also exist in firms that stop their activity? Here the inactivity of firms refers not only to bankruptcy but also to the following inactive states: “In liquidation,” “Merger,” “Handover,” “Division,” “Liquidation,” and “Dissolution” in the Orbis database. Over the last decade in Japan, the number of bankruptcies has been around 10,000 per year [12]. According to the database we use, around 150,000 firms are annually suspended. Among them, around 40,000 firms disclosed their main financial variables. This is the same number published by the Small and Medium Enterprise Agency of Japan [12]. Therefore, our database is highly comprehensive. Chapter 6 studied the statistical properties of the inactive rates of firms in relation to firm-age distributions approximated by an exponential distribution [13–18]. We defined the inactivity of firms as their transition from an active to an inactive state. We found that in Japan, the inactive rate of firm activities does not depend on firm age. On the other hand, in many other countries, the inactive rate of young firms exceeded that of more established firms, gradually became lower, and eventually became constant as firms aged [19–21]. In this chapter, we expanded our research and clarified the statistical property observed between the inactive rate of firms and firm-size variables and found that the inactive rate of firms depends on various firmsize variables: total revenue, net income, total assets, and net assets. We conducted this analysis on firms in multiple countries with high data coverage in our database. The discussion in this chapter is based primarily on our previous work [22, 23]. Most texts are reproduced under the Creative Commons Attribution License or with permission from the publisher. The main text is modified to fit the context of this book. Chapter 7 is structured as follows. Section 7.2 describes the data used in it. In Sect. 7.3, we estimate the dependence of the inactive rate of firms on the values of basic financial variables, including total revenues, net income, total assets, and net assets. Section 7.4 discusses the relationship between the properties observed in Sect. 7.3 and Gibrat’s law observed in the large-scale range of financial data values or non-Gibrat’s law in the mid-scale data range. Finally, in Sect. 7.5, we summarize this study and describe future prospects.

7.2 Activity Status and Financial Variables Dataset In this chapter, we used the Orbis database published by Bureau van Dijk in 2015 and 2016 [24]. At that time, Orbis contained such information as the activity statuses and financial values of around 200 million listed and unlisted firms worldwide. One of its key features is that it not only contains a huge amount of information on firms

7.2 Activity Status and Financial Variables Dataset

85

around the world but also adopts a unified format that greatly simplifies analysis and comparison of data across national borders. We investigated firms in Germany, France, Spain, the United Kingdom, Italy, Japan, Korea, and the Netherlands with high data coverage of activity data and basic financial variables in Orbis published in 2015 and 2016. The activity status of firms was roughly divided into active and inactive states, where active states include “Payment default,” “Insolvency proceedings,” “Dormant,” and “Branch,” and inactive states include “In liquidation,” “Bankruptcy,” “Dissolved (merger or take-over),” “Dissolved (demerger),” “Dissolved (liquidation),” “Dissolved (bankruptcy),” “Dissolved,” “Inactive (branch),” “Inactive (no precision),” and “Unknown situation.” For example, the 2015 version of Orbis (Orbis 2015) contains 5,520,090 Japanese firms: 4,038,389 active and 1,481,701 inactive. In addition, the inactive firms are broken down as follows: 11 in “In liquidation,” 71 in “Bankruptcy,” 271 in “Dissolved (merger or take-over),” 1 in “Dissolved (liquidation),” 2 in “Dissolved (bankruptcy),” 117 in “Dissolved,” 464,790 in “Dissolved inactive (branch),” 1,016,168 in “Inactive (no precision),” and 270 in “Unknown situation.” Note that not all of these firms provided establishment-year data. Chapter 6 analyzed 2,659,627 firms described in Orbis 2015 that had both an active status in 2014 and establishment-year data. For this reason, the values in Table 6.1 differ from those above. In Orbis 2015 and the 2016 version (Orbis 2016), firm activity statuses in 2014 and 2015 are recorded. When the two versions are combined, a total of 5,517,259 Japanese firms are recorded with their activity statuses. Among them, 3,606,111 firms with establishment-year data are listed in Orbis 2015. Since the number of Japanese firms in 2015 is reported to be around 3.8 million [25], Orbis is a highly comprehensive database for firm activity status and establishment years. Among them, “1,789,348,” “1,137,483,” “626,197,” and “625,182” firms, respectively, have the latest total revenues, net income, total assets, and net assets data in Orbis 2015. These are not necessarily 2014 data; many are for 2014. To gain data volume, this chapter approximately treats them as 2014 data and maps them to the 2014 firm activity status. Total revenues and net income are the main financial variables representing performance in each firm’s period. Total assets and net assets are also leading financial variables representing a firm’s cumulative performance. In this way, a specific objective of this chapter is exploring the statistical properties among four different main financial variables thought to affect the activity of firms and their inactive rates. Of the 1,789,348 firms with total revenue data listed in Orbis 2015, 43,308 were active in Orbis 2015 and became inactive in Orbis 2016. All of the firms in this state identified as “Inactive (no precision).” The numbers of main financial variables in Germany, France, Spain, the United Kingdom, Italy, Japan, Korea, and the Netherlands are shown in Table 7.1. For the remainder of this book, status in Orbis 2015 and Orbis 2016 will be referred to as 2014 status and 2015 status.

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Table 7.1 Amounts of main financial variables of eight countries included in Orbis 2015: In Spain, France, Italy, and Japan, the amount of total revenue data is the largest, while in Germany, the United Kingdom, Korea, and the Netherlands, the amount of total assets data is the largest Country Germany Spain France United Kingdom Italy Japan Korea Netherlands

Total revenue 398,241 1,271,086 1,738,600 555,098 2,696,413 1,789,348 336,209 32,931

Positive/negative net income 89,882/28,334 25,554/12,651 845,068/434,987 373,824/231,711 648,923/532,715 816,709/285,006 268,984/73,904 37,244/24,817

Total assets 1,290,045 37,864 1,309,967 4,562,844 1,252,585 626,197 347,500 952,216

Positive/negative net assets 1,018,824/253,541 34,608/3191 1,054,073/246,782 2,214,280/967,161 1,042,996/200,878 465,177/158,692 309,544/32,148 705,001/236,798

7.3 Dependence of Inactive Rate on Firm Size In this chapter, as in the previous one, we define the inactive rate of firms D as follows: D≡

#(Inactive Firms in 2015 that were Active in 2014) . (7.1) #(Active Firms in 2014 that were not Unknown or NA in 2015)

In this section, we investigate the dependence of the inactive rate of firms on the main financial variables discussed in the previous section. First, we analyzed the total revenues of active Japanese firms in Orbis 2015. Figure 7.1a depicts the distribution of total revenue x (thousands of dollars) divided into logarithmically equal-sized bins as x ∈ [2n , 2n+1 ) (n = 1, 2, . . .). This distribution is represented by circles (◦). In Fig. 7.1a, the distribution of the total revenue of active firms in 2014 that became inactive in 2015 is shown by black circles (•) on the left vertical axis. In Fig. 7.1, power-law and log-normal distributions are confirmed in the largeand mid-scale data value ranges: N(x) ∝ x −μ−1 N(x) ∼ x −μ−1

  x exp −α ln2 x0

for x0 < x, for x < x0 .

(7.2) (7.3)

Here, N(x) is the number of firms, μ is Pareto’s index [1], α is 1/2 of the reciprocal of the logarithmic variance, and x0 is a size threshold around 104 . By definition (7.1), the ratio of these two distributions (◦, •) is the inactive rate of firms D in each total revenue class, and this is also shown in Fig. 7.1a by crosses (×). The scale is the right vertical axis. From Fig. 7.1a, the dependence of the inactive

7.3 Dependence of Inactive Rate on Firm Size 10

Active in 2014 Inactive in 2015 Inactive Rate

104

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10

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2

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6

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102 10

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87

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(f) Negative Net Assets

Fig. 7.1 Japanese active firms’ distributions in 2014 of total revenues (a), positive net income (b), negative net income (c), total assets (d), positive net assets (e), and negative net assets (f) are represented by circles (◦). Distributions of financial data of firms active in 2014 that became inactive in 2015 are shown by black circles (•). These scales are on left vertical axis. Inactive rate of firms D calculated by the ratio of the two distributions is shown by crosses (×) on right axis

rate of firms D on total revenue x is expressed as follows: D(x) ∼ Const.

for x0 < x,

(7.4)

D(x) ∝ x −ν

for x < x0 .

(7.5)

Here, the size threshold is x0 ∼ 104 , and ν for total revenues is a constant estimated as νTR = 0.28 ± 0.01 by applying a least squares method to the range of 101
x0 exists based on the logarithmic scale of the vertical axis of D. Figure 7.1b, c shows the results of the same analyses for positive and negative net income. For positive net income (x > 0), we observed identical dependence (7.4) and (7.5) with a threshold of x0 ∼ 102 and estimated the index as ν+NI = 0.39±0.01 in the range of 100.7 < x < x0 . On the other hand, for the negative net income (x < 0), the dependence of D on x is expressed as D(x) ∝ (−x)+ν

for − x < x0 .

(7.6)

Here, x0 ∼ 103.7, and ν−NI is estimated as ν−NI = 0.12 ± 0.02 in the range of 101 < −x < x0 . Figure 7.1d shows the results for total assets. The dependence of D on x is identical to Eqs. (7.4) and (7.5) with a threshold of x0 ∼ 103.1 , and the index is estimated as νTA = 0.22 ± 0.01 in the range of 101 < x < x0 . Figure 7.1e, f shows the results for positive and negative net assets. For the former (x > 0), the dependence of D on x is identical to Eqs. (7.4) and (7.5) with a threshold of x0 ∼ 103.7 , and the index is estimated as ν+NA = 0.20 ± 0.01 in the range of 101 < x < x0 . On the other hand, for the negative net assets (x < 0), the dependence of D on x is identical to Eq. (7.6) with a threshold of x0 ∼ 103.7 , and the index is estimated as ν−NA = 0.31 ± 0.03 in the range of 101 < −x < x0 . The property of such an inactive rate of firms can also be observed in countries other than Japan. For example, Fig. 7.2a–d, respectively, shows similar analyses for the total revenues of Korean firms where the index is estimated as νTR = 0.11±0.01 (x0 ∼ 104.2), for the positive net income of French firms (ν+NI = 0.35 ± 0.02, x0 ∼ 101.9 ), for the total assets of firms in the United Kingdom (νTA = 0.28 ± 0.01, x0 ∼ 102.8 ), and for the negative assets in Italy (ν−NA = 0.25 ± 0.01, x0 ∼ 103.1). Table 7.2 summarizes the results of these analyses.

7.4 Relation Between Non-Gibrat’s Property and Inactive Rate In this section, we consider the property of the decrease in the inactive rate of firms D under a power-law function as an increase in a main financial variable (7.5) as well as the property where D takes a constant value regardless of the main financial variables (7.4). As is shown in the previous section, the inactive rate of firms does not depend on the main financial variables in the large-scale data value range. This property corresponds to Gibrat’s law for continuing active firms where the growth-rate distribution does not depend on the firm size in the large-scale data range. This relationship is described as follows. In Figs. 7.1 and 7.2, both the distributions of

7.4 Relation Between Non-Gibrat’s Property and Inactive Rate 10

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-2

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(d) Negative Net Assets in Italy

Fig. 7.2 Examples of distributions of firms active in 2014 represented by circles (◦): total revenues of firms in Korea (a), positive net income of firms in France (b), total assets of firms in the United Kingdom (c), and negative net assets of firms in Italy (d). Distributions of financial items of firms active in 2014 that became inactive in 2015 are shown by black circles (•) on left vertical axis. Inactive rate of firms D calculated by the ratio of two distributions is shown by crosses (×) on right axis

main financial variables x of firms active in 2014 (◦) and those that became inactive in 2015 among the firms (•) follow a power law (7.4) in the large-scale data range. In Japan, for the total revenues of active firms in 2014, the index is estimated as μ = 0.87±0.01 in the range of x ≥ 104 by cumulative distribution N(x). The index is also estimated as μ = 0.86 ± 0.02 for firms that became inactive in 2015 among the firms. In the large-scale data range, these two distributions are identical. This value corresponds to the constant ratio of the number of active firms in 2014 and that became inactive in 2015. Therefore, the inactive ratio of firms D is constant in the large-scale data range. As mentioned in Chaps. 2 and 4, the power-law distribution is derived from Gibrat’s law. In this sense, the property in which D takes a constant value is related to Gibrat’s law. In the large-scale range, Gibrat’s law, which states that the growth-rate distribution of firms that continue to exist does not depend on the firm size, becomes independency of the inactive rate of firms on the firm size when they stop operating. On the other hand, in the mid-scale data range, as the financial variables become smaller, the proportion of firms that became inactive in 2015 among all active firms in 2014 (inactive rate of firms) increases under the power-law function.

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Table 7.2 ν (and x0 ) values for main financial variables of eight countries. TR, +NI, −NI, TA, +NA, and −NA, respectively, signify total revenues, positive net income, negative net income, total assets, positive net assets, and negative net assets. For items with less than 50,000 bits of data, no measured values are stated due to insufficient statistics Country Germany

νTR (x0 ) 0.32 ± 0.01 (103.7 ) Spain 0.26 ± 0.01 (102.2 ) France 0.27 ± 0.01 (103.4 ) United 0.24 ± 0.01 Kingdom (104.0 ) Italy 0.14 ± 0.01 (104.0 ) Japan 0.28 ± 0.01 (104.0 ) Korea 0.11 ± 0.01 (104.2 ) Netherlands –

ν+NI (x0 ) ν−NI (x0 ) 0.21 ± 0.01 – (102.5 ) – –

νTA (x0 ) 0.37 ± 0.02 (103.1 ) –

ν+NA (x0 ) 0.33 ± 0.01 (102.8 ) –

ν−NA (x0 ) 0.13 ± 0.01 (103.4 ) –

0.35 ± 0.02 (101.9 ) 0.34 ± 0.02 (102.8 ) 0.16 ± 0.01 (102.8 ) 0.39 ± 0.01 (102.0 ) 0.13 ± 0.01 (103.1 ) –

0.37 ± 0.01 (103.1 ) 0.28 ± 0.01 (102.8 ) 0.15 ± 0.01 (103.4 ) 0.22 ± 0.01 (103.1 ) 0.13 ± 0.01 (103.4 ) 0.43 ± 0.02 (103.1 )

0.35 ± 0.01 (102.8 ) 0.26 ± 0.01 (102.8 ) 0.20 ± 0.00 (103.4 ) 0.20 ± 0.01 (103.7 ) 0.19 ± 0.01 (103.7 ) 0.41 ± 0.01 (102.8 )

0.05 ± 0.01 (101.3 ) 0.10 ± 0.01 (104.3 ) 0.25 ± 0.01 (103.1 ) 0.31 ± 0.03 (103.7 ) –

0.015 ± 0.004 (101.9 ) −0.11 ± 0.01 (103.7 ) 0.16 ± 0.01 (103.1 ) 0.12 ± 0.02 (103.7 ) 0.09 ± 0.01 (103.1 ) –

0.09 ± 0.01 (103.1 )

This property is closely related to non-Gibrat’s property of firms that continue to exist. Here, non-Gibrat’s property describes the dependence of the growth-rate distribution of firm-size variables on the size of firms that continue to be active. In the growth-rate distribution in the mid-scale range, we observed the following interesting phenomenon. The value of the growth-rate distribution that became smaller is systematically lost in the data. In fact, this phenomenon was already identified in Fig. 4.1 in Chap. 4, which discusses non-Gibrat’s property of operating revenues. We explain this phenomenon with the analysis results in Chap. 4. Figure 4.2 is a scatter plot of the operating revenue data of Japanese firms for 2013 and 2014 (x2013, x2014). Figure 4.3 shows PDFs q(r|x2013) or q(r|n) of logarithmic growth rate r = log10 x2014/x2013 conditioned by initial values x2013 divided into five bins: x ∈ [102+0.5(n−1), 102+0.5n ) (n = 1, 2, . . . , 5). By scrutinizing the figure, we see that q(r|n) does not exist in the range of r < −0.4, r < −1.0, r < −1.4, r < −2.0, and r < −2.4 for n = 1, 2, . . . , 5. On average, there is no q(r|n) with r < −0.5n + 0.06. This range corresponds to the region below x2014 = 102 in Fig. 4.2. In fact, the upper limit of r in this area can be expressed as r = log10 x2014/x2013 = log10 102/102+0.5n = −0.5n, which is very close to the upper limit of r mentioned above. If there were still firms that were inactive in the second year and actually lost their data, they would fill this region. As a rough approximation, if a conditional PDF with r < 0 is approximated by q(r|xT ) ∝ 10νr , ignoring the r 2 term in Eq. (4.6), from the area of this region, the inactive rate can

7.5 Discussion

91

be expressed as  D∝

log10 102 /xT − inf

q(r|xT )dr ∝ xT −ν .

(7.7)

This is exactly Eq. (7.5). The values of ν do not match quantitatively within the error limits for a variety of reasons, including the use of a very rough approximation that ignores the r 2 term in Eq. (4.6) and with different target years. However, a previous work concluded that the values are close [22]. This data loss phenomenon can be understood as follows. There is a lower limit to the size at which firms maintain their activity. A negative growth below this limit rarely occurs because firms facing such situations stop their activities, and thus the financial data in the next year do not exist in the database. To summarize the above discussions, we interpreted the property where the inactive rate of firms is constant in the large-scale data range of financial variables as Gibrat’s law of firms that stop their activity. The property in which the inactive rate of firms decreases under the power law as financial variables increase in the midscale data range is related to non-Gibrat’s property of firms that stop their activity. This discussion necessarily leads to the conclusion that the boundary between these two properties should be identical as the boundary between the large-scale data range, in which the financial data distribution follows a power law, and the midscale data range, in which the financial variables follow a log-normal distribution. This assumption is also confirmed by the results in Figs. 7.1 and 7.2, where the boundaries are represented by dashed lines. The right and left sides of the lines are large- and mid-scale data ranges.

7.5 Discussion In Chap. 7, we investigated the dependence of the inactive rate of firms on the main financial variables: total revenues, net income, total assets, and net assets. We used the information on German, Spanish, French, British, Italian, Japanese, Korean, and Dutch firms recorded in the 2015 and 2016 editions of the comprehensive Orbis database of worldwide listed and unlisted firms. We confirmed that the inactive rate of firms is constant regardless of the size of the financial variables in the large-scale data range. In the mid-scale data range, the inactive rate of firms increases under a power law as the financial variables decrease. The boundary between the large- and mid-scale data ranges corresponds to the boundary between the power-law and log-normal distributions of the financial data. The power-law distribution is derived from Gibrat’s law in that the growthrate distribution does not depend on the firm-size variables, and the log-normal distribution is derived from non-Gibrat’s property that describes the dependence of

92

7 Statistical Properties in Inactive Rate of Firms

the growth-rate distribution on firm size. These laws and properties are observed in firms that continue to be active. We also observed properties for firms that stop their activity and explained how they are closely related to Gibrat’s law and non-Gibrat’s property. In this chapter, we analyzed the total revenues and net income as flow data and the total assets and net assets as stock data. Among them, net income and net assets also take negative values. In the Italian data, the inactive rate of firms takes a constant value regardless of net income or net assets in the large-scale data range of the negative value.1 This idea agrees with the intuition that the inactive ratio of firms becomes constant in the range of negative values larger than a certain threshold. In this chapter, we described that the same properties of the inactive rates of firms were observed in the main financial variables of the eight countries. We identified no big difference in the indices of the power-law functions in the midscale range. However, in total revenues, the power-law indices of Italy and Korea are slightly smaller than those of the other countries, suggesting that compared to the other countries, Italy and Korea have inactive rates of firms that become constant with greater total revenues. This difference is probably caused by each country’s particular economic system, including its legal structure. Another possible origin of such differences among nations is the structure of the industrial sector. In the future, we plan to analyze the inactive rate of firms based on industry type. We also discussed how the inactive rate depends on four types of main firm-size variables and compared the independently observed properties of different types of firm-size variables in previous chapters. However, since we have not yet directly discussed the relationships among different types of firm-size variables, we address this discussion in our last two chapters.

References 1. Pareto V (1897) Cours d’Économie Politique. Macmillan, London 2. Newman MEJ (2005) Power laws, Pareto distributions and Zipf’s law. Contemp Phys 46:323– 351 3. Clauset A, Shalizi CR, Newman MEJ (2009) Power-law distributions in empirical data. SIAM Rev 51:661–703 4. Stanley MHR, Buldyrev SV, Havlin S, Mantegna R, Salinger MA, Stanley HE (1995) Zipf plots and the size distribution of Firms. Econom Lett 49:453–457 5. Gibra R (1932) Les Inégalités Économique. Sirey, Paris 6. Sutton J (1997) Gibrat’s legacy. J Econ Lit 35:40–59

1 In Japan and the other countries studied here, the inactive rate of firms did not take a constant value regardless of the net income or the net assets in the large-scale data range of negative values. This is probably because the data of the negative values are not necessarily exhaustive. In Italy, compared to the other countries studied in this chapter, perhaps the data coverage of the negative values was too high.

References

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7. Fujiwara Y, Souma W, Aoyama H, Kaizoji T, Aoki M (2003) Growth and fluctuations of personal income. Physica A321:598–604 8. Fujiwara Y, Guilmi CD, Aoyama H, Gallegati M, Souma W (2004) Do Pareto-Zipf and Gibrat laws hold true? An analysis with European firms. Physica A335:197–216 9. Ishikawa A (2006) Derivation of the distribution from extended Gibrat’s law. Physica A367:425–434 10. Ishikawa A (2007) The uniqueness of firm size distribution function from tent-shaped growth rate distribution. Physica A383:79–84 11. Tomoyose M, Fujimoto S, Ishikawa A (2009) Non-Gibrat’s law in the middle scale region. Prog Theor Phys Suppl 179:114–122 12. The Small and Medium Enterprise Agency, http://www.chusho.meti.go.jp/sme_english/ 13. Fujiwara Y (2004) Zipf law in firms bankruptcy. Physica A337:219–230 14. Coad A, Tamvada JP (2008) The growth and decline of small firms in developing countries. Working Paper on Economics and Evolution (Max Plank Institute of Economics) 2008-08 15. Coad A (2010) The exponential age distribution and the pareto firm size distribution. J Ind Compet Trade 10:389–395 16. Coad A (2010) Investigating the exponential age distribution of firms. Economics 4:2010–2017 17. Bottazzi G, Secchi A, Tamagni F (2008) Productivity, profitability and financial performance. Ind Corp Change 17:711–751 18. Miura W, Takayasu H, Takayasu M (2012) Effect of coagulation of nodes in an evolving complex network. Phys Rev Lett 108:168701 19. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2015) Firm age distributions and the decay rate of firm activities. In: Takayasu H, Ito N, Noda I, Takayasu M (eds) Proceedings of the international conference on social modeling and simulation, plus econophysics colloquium 2014. Spinger, Tokyo, pp 187–194 20. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2015) The relation between firm age distributions and the decay rate of firm activities in the United States and Japan. Big data, 2015 IEEE international conference on date of conference, pp 2726–2731 21. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2017) Depencence of the decay rate of firm activites on firm age. Evolut Inst Econo Rev 14:351–362 22. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2017) Transition law of firms’ activity and the deficit aspect of non-Gibrat’s law. JPS Conf Proc 16:011005 23. Ishikawa A, Fujimoto S, Mizuno T (2018) Statistical law observed in inactive rate of firms. Evolu Inst Econ Rev 16:201–212 24. Bureau van Dijk, http://www.bvdinfo.com/Home.aspx 25. Statistics Bureau, Ministry of Internal Affairs and Communications, http://www.stat.go.jp/ english/index.htm.

Chapter 8

Power Laws with Different Exponents in Firm-Size Variables

Abstract In Chap. 8, we discuss the relationship between the power-law distributions observed in the large-scale range of different types of firm-size variables at the same time. We focused on operating revenues, tangible fixed assets, and the number of employees as firm-size variables and measured their power indices by country and year. We confirmed that the power indices of the three types of firm-size variables barely changed over time and that they differ depending on the type of variable. In Chap. 3, using time-reversal symmetry, we connected the power-law distributions in which the exponent changes due to the quasi-static time evolution of a system composed of single variables. After applying this argument to a system consisting of two sets of variables, we conclude that the ratio of the two power indices and the slope of the symmetry axis coincide by simultaneously considering the quasi-inversion symmetry from one firm-size variable to another. Our analytical conclusions are based on empirical data from firms in Japan, Spain, France, the United Kingdom, and Italy because they have sufficient data and their axis of symmetry can be easily observed. We confirmed that our analytical result is accurately established within the error range.

8.1 Introduction Including the previous chapter, we observed the statistical properties of firm-size variables at a given time as well as the short- and long-term statistical properties and developed discussions that link them [1–28]. The analyzed data included the current profits of firms (Chap. 2) [29, 30], posted land prices (Chap. 3) [31, 32], firm operating revenues and total assets (Chap. 4) [33], firm operating revenues, total assets, and the number of employees (Chap. 5) [34, 35], status of firms (Chap. 6) [36–38], and the total revenues, net income, total assets, and net assets of firms (Chap. 7) [39, 40]. What we have dealt with so far is the statistical properties of the time evolution of such a set of variables of one kind. This chapter, on the other hand, discusses the properties seen between sets of different types of firm-size variables at the same time. We performed the following analysis. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Ishikawa, Statistical Properties in Firms’ Large-scale Data, Evolutionary Economics and Social Complexity Science 26, https://doi.org/10.1007/978-981-16-2297-7_8

95

96

8 Power Laws with Different Exponents in Firm-Size Variables

As seen in various forms in previous chapters, a power-law distribution is observed in a large-scale range of firm’s operating revenues, tangible fixed assets, and the number of employees in a given year [1–3]. Such power indices barely change over time [4]. On the other hand, the power index of firm-size variables varies depending on the type of firm size (for example, see [41–43]). In Chap. 8, we focus first on the power index of operating revenues, tangible fixed assets, and the number of employees of Japanese firms between 2010 and 2014 because we have sufficient data. We then argue that the ratio of the three sets of power indices of these three types of firm-size variables is consistent with the slope of the quasi-inversion symmetry axis observed in their joint probability density functions (PDFs). This is an application of the quasi-time-reversal symmetry of the quasi-statically timevarying one-variable system discussed in Chap. 3 for a simultaneous two-variable system. Our discussion here is based primarily on our previous work [44, 45]. Most texts are reproduced under the Creative Commons Attribution License or with permission from the publisher. However, much of the discussion in this chapter is original. Chapter 8 is organized as follows. Section 8.2 describes the data analyzed in it. Section 8.3 describes the power-law distribution of the operating revenues, the tangible fixed assets, and the number of employees of Japanese firms in the largescale range and shows the measurement results of each power index from 2010 to 2014. Similar measurements are described for firms in Spain, France, the United Kingdom, and Italy. Using the discussion in Chap. 3, we analytically show that the ratio of the power indices of the three different types of firm-size variables in each year in each country corresponds to the slope of the quasi-inversion symmetry axis of the joint PDFs of the three sets of firm-size variables at the same time. Section 8.4 introduces the geographic surface openness index for identifying the axis of the quasi-inversion symmetry seen in the joint PDFs of different types of firm-size variables. Then, we described how the slopes of the identified axis of symmetry coincide with the ratio of the Pareto indices with high accuracy. The analysis was performed on firms in Spain, France, the United Kingdom, and Italy, and similar consistent results are obtained. Section 8.5 concludes with a summary.

8.2 Data This chapter analyzes the operating revenues (net sales), the tangible fixed assets, and the number of employees data from the 2016 edition of Orbis, one of the largest databases in the world, which is provided by Bureau van Dijk [46]. As discussed in Chap. 2, Orbis contains financial data for about 200 million firms worldwide at the time of its publication. Its main feature is its huge amount of data. For example, the operating revenue data of over one million Japanese firms are recorded. An equally critical feature is that the data of the countries are recorded

8.2 Data

97

in the same format, facilitating comparison of comprehensive corporate financial data by country. However, as mentioned in Chap. 2, the database is structured to record corporate financial data for up to ten consecutive years since the last available year. Therefore, in many cases, although the amount of data near the year when the database was published is large, the amount of older data is small. This chapter analyzes the data on operating revenues, tangible fixed assets, and the number of employees in Japan, Spain, France, the United Kingdom, and Italy from 2010 to 2014, all of which are statistically well informed and included in the 2016 edition of the database Orbis. The latest data in the 2016 database are from 2015, but the data volume is often small because they are still being collected. Therefore, data for the year 2015 are excluded from our analysis. The amounts of data for each country are shown in Tables 8.1, 8.2, 8.3, 8.4, and 8.5.

Table 8.1 Amount of data on operating revenues, tangible fixed assets, and number of employees of Japanese firms in Orbis 2016 Year 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

Operating revenue 227,573 294,651 345,304 419,033 518,754 610,833 969,821 1,146,845 1,039,638 344,067

Tangible fixed assets 130,930 174,565 223,081 284,905 363,447 420,545 436,210 414,826 341,580 93,905

Number of employees 129,523 172,175 217,230 269,891 339,315 392,174 411,327 394,358 325,818 89,549

Table 8.2 Amount of data on operating revenues, tangible fixed assets, and number of employees of Spanish firms in Orbis 2016 Year 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

Operating revenue 740,225 748,675 767,088 769,174 751,935 739,322 720,004 687,076 608,497 1079

Tangible fixed assets 780,028 781,877 800,910 802,805 782,655 769,862 750,188 716,635 631,258 1102

Number of employees 585,478 600,934 623,063 624,138 597,626 581,798 557,070 526,718 468,319 951

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8 Power Laws with Different Exponents in Firm-Size Variables

Table 8.3 Amount of data on operating revenues, tangible fixed assets, and number of employees of French firms in Orbis 2016 Year 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

Operating revenue 832,716 884,347 933,198 978,076 1,045,513 1,110,364 1,139,674 1,077,633 764,633 80,334

Tangible fixed assets 835,305 884,604 933,070 977,894 1,045,281 1,110,100 1,139,427 1,077,209 764,242 80,334

Number of employees 387,963 376,138 342,805 353,610 371,585 345,889 277,829 284,490 261,799 26,316

Table 8.4 Amount of data on operating revenues, tangible fixed assets, and number of employees of UK firms in Orbis 2016 Year 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

Operating revenue 348,888 320,314 279,729 279,077 269,502 241,829 233,488 224,054 216,225 898,227

Tangible fixed assets 1,112,401 1,117,705 1,142,252 1,172,291 1,215,972 1,270,818 1,340,275 1,432,269 1,597,245 405,742

Number of employees 88,758 90,335 89,522 92,128 97,590 98,221 99,181 100,947 100,601 900,843

Table 8.5 Amount of data on operating revenues, tangible fixed assets, and number of employees of Italian firms in Orbis 2016 Year 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

Operating revenue 620,064 890,924 936,083 949,894 971,713 982,850 970,982 2,806,513 2,381,964 11,736

Tangible fixed assets 620,024 893,325 939,461 959,097 978,858 983,489 970,965 953,292 876,153 11,736

Number of employees 216,424 255,519 374,496 316,745 244,717 500,167 545,291 2,448,516 2,474,746 6398

8.3 Different Exponent Power Laws and Quasi-Inversion Symmetry

99

8.3 Different Exponent Power Laws and Quasi-Inversion Symmetry As seen in previous chapters, the following power distributions are observed in the large-scale ranges of firms’ operating revenues (Y ), their tangible fixed assets (K), and the number of employees (L) [1–3]: P> (Y ) ∝ Y −μY P> (K) ∝ K

−μK

P> (L) ∝ L−μL

for Y > Y0 .

(8.1)

for K > K0 ,

(8.2)

for L > L0 ,

(8.3)

where μY , μK , and μL are the Pareto indices of Y , K, and L. Y0 , K0 , and L0 are the lower bounds of the power-law distributions of Y , K, and L. For example, power-law distributions are observed in the cumulative distributions of the operating revenues (Y ), the tangible fixed assets (K), and the number of employees (L) of Japanese firms in 2014 described in Orbis 2016 (Fig. 8.1). In each cumulative distribution in Fig. 8.1, the upper 10−3 of the total amount of data was set as an upper limit, and the lower limit was determined by a test in a previous work [47, 48] for discriminating the power-law distribution from the log-normal distribution. The power index was evaluated by the least squares method in the range. As a result, the Pareto indices in Fig. 8.1 were evaluated as μY = 0.84±0.00, μK = 0.82 ± 0.00, and μL = 0.91 ± 0.00. Tables 8.6, 8.7, 8.8, 8.9, and 8.10 summarize the Pareto indices for each firmsize variable by year and country calculated using the same method for the data described in the previous section. The first aspect identified from these tables is that the Pareto index for each firm-size variable in each country does not change over time [4]. In addition, the Pareto indices for the three types of firm-size variables are different in each country. μY ∼ μK < μL is observed in Japan and Italy. In Spain and the United Kingdom, μY < μK < μL is observed, and μK < μY < μL is observed in France. In this chapter, we simultaneously combined the power-law distributions of different indices. Our approach uses the quasi-inversion symmetry introduced in Chap. 3 [31, 32]. In Chap. 3, our argument linked the power-law distributions of different Pareto indices using a quasi-static time change. First, we introduced quasitime-reversal symmetry (3.5) in a system (xT , xT +1 ) where a set of single variables x, land prices, changes with time in a quasi-static manner. For extended growth rate R = axTx+1 corresponding to a quasi-statically changing system, we assumed Tθ that Gibrat’s law (2.4) holds. As a result, we analytically showed a relation (3.12) between the change of the power exponent and the parameter of the quasi-timereversal symmetry. We confirmed that the relationship was observed with high accuracy in the empirical data. In this chapter, we apply this quasi-static discussion of time evolution to the relationship between different types of firm-size variables. We define the

8 Power Laws with Different Exponents in Firm-Size Variables

Plndex= 0.836276285309598, Err= 0.00369494833786925

100

10 2

10 4

106

10 8

10 –6

10 –6

10 –4

10 –4

10 –2

10 –2

10 0

10 0

100

Plndex= 0.817885021672355, Err= 0.00389527066690891

10 –2

Y JP 2014

10 2 K JP 2014

10 4

10 6

10 8

(b)

10–6

10 –4

10 –2

100

(a)

100

Plndex= 0.905537044369221, Err= 0.00427185278133529

100

10 2

10 4

10 6

L JP 2014

(c) Fig. 8.1 Cumulative distributions of firms in 2014 Japan: (a) operating revenues (Y ), (b) tangible fixed assets (K), and (c) the number of employees (L). In each cumulative distribution, the upper 10−3 of total amount of data is set as an upper limit, and a lower limit is determined by a test for discriminating a power-law distribution and a log-normal distribution, and a power index is evaluated between them. Pareto indices of Y , K, and L are evaluated as μY = 0.84 ± 0.00, μK = 0.82 ± 0.00, and μL = 0.91 ± 0.00. Y and K units are 1000 US dollars, and L units are people. K starts at 10−2 due to the way the data are described and is not essential

relationship between the time-varying power index and the parameters of the quasitime-reversal symmetry (3.12) as the relationship between the power indices and the parameters of the quasi-inverse symmetry for different types of firm-size variables at the same time. Formally describing this step as an analytic argument is easy, where expression (3.12) for the variable (xT , xT +1 ) at time T and time T + 1 is replaced

8.3 Different Exponent Power Laws and Quasi-Inversion Symmetry

101

Table 8.6 Pareto indices for Japanese firms between 2014 and 2010, consisting of operating revenues (Y ), tangible fixed assets (K), and number of employees (L)

Year 2014 2013 2012 2011 2010

μY 0.84 ± 0.00 0.83 ± 0.00 0.85 ± 0.00 0.84 ± 0.00 0.84 ± 0.00

μK 0.82 ± 0.00 0.83 ± 0.00 0.85 ± 0.00 0.84 ± 0.00 0.83 ± 0.00

μL 0.91 ± 0.00 0.91 ± 0.00 0.92 ± 0.00 0.92 ± 0.00 0.92 ± 0.00

Table 8.7 Pareto indices for Spanish firms between 2014 and 2010, consisting of operating revenues (Y ), tangible fixed assets (K), and number of employees (L)

Year 2014 2013 2012 2011 2010

μY 0.96 ± 0.00 0.87 ± 0.00 0.89 ± 0.00 0.91 ± 0.00 0.92 ± 0.00

μK 0.87 ± 0.00 0.92 ± 0.00 0.95 ± 0.00 0.96 ± 0.00 0.95 ± 0.00

μL 1.00 ± 0.00 0.97 ± 0.00 1.01 ± 0.00 1.03 ± 0.00 1.06 ± 0.00

Table 8.8 Pareto indices for French firms between 2014 and 2010, consisting of operating revenues (Y ), tangible fixed assets (K), and number of employees (L)

Year 2014 2013 2012 2011 2010

μY 0.88 ± 0.00 0.86 ± 0.00 0.87 ± 0.00 0.88 ± 0.00 0.91 ± 0.00

μK 0.69 ± 0.00 0.67 ± 0.00 0.69 ± 0.00 0.72 ± 0.00 0.73 ± 0.00

μL 0.90 ± 0.00 0.89 ± 0.00 0.91 ± 0.00 0.96 ± 0.00 0.96 ± 0.00

Table 8.9 Pareto indices for UK firms between 2014 and 2010, consisting of operating revenues (Y ), tangible fixed assets (K), and number of employees (L)

Year 2014 2013 2012 2011 2010

μY 0.89 ± 0.00 0.87 ± 0.00 0.87 ± 0.00 0.89 ± 0.00 0.89 ± 0.00

μK 0.76 ± 0.00 0.74 ± 0.00 0.74 ± 0.00 0.75 ± 0.00 0.74 ± 0.00

μL 0.95 ± 0.00 0.93 ± 0.00 0.94 ± 0.00 0.94 ± 0.00 0.94 ± 0.00

Table 8.10 Pareto indices for Italian firms between 2014 and 2010, consisting of operating revenues (Y ), tangible fixed assets (K), and number of employees (L)

Year 2014 2013 2012 2011 2010

μY 0.92 ± 0.00 0.92 ± 0.00 0.93 ± 0.00 0.95 ± 0.00 0.98 ± 0.00

μK 0.95 ± 0.00 0.94 ± 0.00 0.95 ± 0.00 0.95 ± 0.00 0.94 ± 0.00

μL 1.03 ± 0.00 1.04 ± 0.00 1.05 ± 0.00 1.06 ± 0.00 0.98 ± 0.00

by variables (K, L), (L, Y ), and (K, Y ) [44]: θKL =

μK μL μK , θLY = , θKY = . μL μY μY

(8.4)

Here, θKL , θLY , and θKY are, respectively, the slopes of the axis of the quasiinversion symmetry of the joint PDFs of variables (K, L), (L, Y ), and (K, Y ). The question is whether the symmetry axes of the quasi-inversion symmetries of the joint PDFs of variables (K, L), (L, Y ), and (K, Y ) can be identified in the

102

8 Power Laws with Different Exponents in Firm-Size Variables

empirical data. Chapter 3 dealt with the quasi-time-reversal symmetry of land prices over a one-year period. In a single year, land prices do not usually increase by a factor of 10 or 100, let alone by a factor of 1000, and the scatter plots were linear with very small variances (see Figs. 3.4 and 3.5). Therefore, we specified the symmetry axis of the quasi-inversion symmetry by regression analysis. On the other hand, after one year, it is not unusual for firm-size variables to increase by 10, 100, or even 1000 times, and the scatter plots are highly dispersed (see Figs. 2.2 and 4.2). As mentioned earlier, since the Pareto index of a single firm-size variable does not change with time, it is sufficient to consider only the time-reversal symmetry. However, when we simultaneously combine the power-law distributions of different types of firm-size variables, as in this case, we need to find the axis of the quasiinversion symmetry in the scatter plots with a large variance, as shown in Fig. 8.2.

108

104

L

Y

106

104

102 102

100

100 10–2

101

104

107

100

102

K

104 L

(b)

(a) 108

Y

106

104

102

100 10–2

101

104

(c)

107

K

Fig. 8.2 Scatter plots among firms in 2014 Japan: (a) tangible fixed assets (K) and number of employees (L), (b) number of employees (L) and operating revenues (Y ), and (c) tangible fixed assets (K) and operating revenues (Y ). Y and K units are 1000 US dollars, and L units are people. K starts at 10−2 due to the way the data are described and is not essential

8.4 Identification of Ridge Using Surface Openness

103

The important point here is that identifying the symmetry axes by regression analysis is difficult using the least squares method. For example, with a large distribution of variance (Fig. 8.2), the regression lines obtained using the horizontal axis as the explanatory variable and the vertical axis as the objective variable and using the vertical axis as the explanatory variable and the horizontal axis as the objective variable are significantly different. This means that the regression line is not the axis of symmetry. In the next section, for identifying the axis of symmetry independent of the way in which explanatory and objective variables are taken, we introduce a method that identifies the ridge of the data density using an index called surface openness, which is used in geomorphology [49–51].

8.4 Identification of Ridge Using Surface Openness In this section, the axis of the quasi-inversion symmetry in Fig. 8.2 is identified as the ridge of the data by an index used in geomorphology called surface openness. To clearly comprehend the density, we divided it into logarithmically equal-sized cells and expressed the amount of data points in them by different shades. Figure 8.3 is a diagram of Japanese firms in 2014 created in this way. The logarithm of a cell’s density is its height. As the steepest-ascent line in the profit space, a ridge was previously discussed [52]. Here, we determined the cells that constitute the ridge using the surface openness defined as follows [49–51]. Figure 8.4 depicts a grid linked by the center points of the cells. From grid point A in Fig. 8.4, we counterclockwisely represent each azimuth as D = 1, 2, . . . , 8. As shown in Fig. 8.5, the minimum zenith and nadir angles at grid point A within distance L in azimuth D are represented by D φL and D ψL . Positive openness L is defined by the mean value of D φL along the eight azimuths, and negative openness L is the corresponding mean of D ψL . The surface openness is defined by the following difference: L −  L =

8 8 1 1 φ − D L D ψL . 8 8 D=1

(8.5)

D=1

The surface openness takes a negative value at the depressions and the valleys, zero at the level surface, the saddle point, and the uniform slope, and positive values at the ridge (Fig. 8.6) and the summit (Fig. 8.7). The shading in Fig. 8.3 shows only the absolutely high and low points. However, using surface openness, the points that are relatively higher than the circumference (a ridge or a summit) or relatively lower than it (a valley or a depression) can be easily extracted. Unlike regression analysis, this technique does not change the analysis results depending on how the objective and explanatory variables are taken. This is the advantage of surface openness. In this analysis, by setting L = 5, we estimate the surface openness for each cell and extract the cells of the openness that exceed 0.99 for Fig. 8.3a, 0.8 for Fig. 8.3b,

104

8 Power Laws with Different Exponents in Firm-Size Variables Slope:0.930894308943089, Err:0.0459216599620264

Slope:0.957077971233914, Err:0.0475666791627787 108

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L

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JP 2014

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(b)

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Y

106

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K JP 2014

(c)

Fig. 8.3 Data points in scatter plot of Fig. 8.2 are placed in cells separated at equal logarithmic intervals, and the amount of data in each cell is represented by shading. Cell’s center, judged to form a ridge by surface openness, is indicated by black circles. (a) KL plane data count topographic map and ridge. (b) LY plane data count topographic map and ridge. (c) KY plane data count topographic map and ridge

and 0.95 for Fig. 8.3c, the values of which appropriately determine the ridge in this dataset. In Fig. 8.3, the cells are expressed by black dots. The ridge’s inclination was measured by the least squares method from the cells constituting the ridge thus specified. Since the variance of the cells constituting the ridge is small, we can ignore the differences due to the exchange of the explanatory and objective variables. The ridge’s inclination thus identified is, respectively, measured as θKL = 0.93 ± 0.05, θLY = 0.96 ± 0.05, and θKY = 0.83 ± 0.16 in Fig. 8.3a–c. Figure 8.8 shows scatter plots in which the slope of quasi-inversion

8.4 Identification of Ridge Using Surface Openness Fig. 8.4 Grid linked by center points of cells. From a grid point, we counterclockwisely represent each azimuth as D = 1, 2, . . . , 8

Fig. 8.5 Dots represent the height of cells within distance L in azimuth D. From grid point A within distance L, we estimate zenith angles and denote minimum as D φL . Similarly, the minimum nadir angle is expressed by D ψL

Fig. 8.6 Schematic illustration of a ridge. In ridge indicated by thick lines, surface openness is positive because the surrounding area is often lower than ridge’s height

105

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8 Power Laws with Different Exponents in Firm-Size Variables

Fig. 8.7 Schematic illustration of a summit. At summit, surface openness is positive because all surroundings are lower than the summit

symmetry θ , measured in the same manner, is the vertical axis, and the ratio of Pareto index μ, in Table 8.1, is the horizontal axis. Since the error of the Pareto index is very small, the error bar on the horizontal axis is not written. Similar measurements were carried out outside Japan, and the results from Spain, France, the United Kingdom, and Italy, corresponding to Fig. 8.8b, which simplified the extraction of the cells constituting the ridge, are shown in Fig. 8.9. From Figs. 8.8 and 8.9, we confirmed that the ratio of the Pareto indices and the inclination of the quasi-inverse symmetry axis match within the range of error in most cases. Although not shown in the figure, this result is also often observed for the ratio of the power indices and the slope of the quasi-inversion symmetry of other firm sizes.

8.5 Discussion In this chapter, we discussed the relationship between the power-law distributions seen in the large-scale range of different types of firm-size variables at the same time. We focused on operating revenues, tangible fixed assets, and the number of employees as firm-size variables and measured their power indices by country and year. We confirmed that the power index of the three types of firm-size variables barely changed over time, although the value of the power index differs depending on the type of firm-size variables. In Chap. 3, we connected the power-law distributions, in which the exponent changes due to the quasi-static time evolution of a system composed of a set of variables, by the quasi-time-inversion symmetry. In this chapter, we applied this discussion to a system consisting of two sets of simultaneous variables. We concluded that the ratio of the two power indices coincides with the slope of the

107

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1

1

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thetaKL

8.5 Discussion

0.6

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0.6

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2014 2013 2012 2011 2010

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1.2 1

thetaKY

0.8 0.6 0.4 0.2 0

2014 2013 2012 0

0.2

0.4 0.6 0.8 muK / muY

1

1.2

Fig. 8.8 Between 2010 and 2014 in Japanese firms: (a) comparison of ratio (μK /μL ) of Pareto index of tangible fixed assets (K) and number of employees (L) and inclination (θKL ) of quasiinversion symmetry axis of their joint PDF, (b) comparison of ratio (μL /μY ) of Pareto index of number of employees (L) and operating revenues (Y ) and inclination (θLY ) of quasi-inversion symmetry axis of their joint PDF, and (c) comparison of ratio (μK /μY ) of Pareto index of tangible fixed assets (K) and operating revenues (Y ) and inclination (θKY ) of quasi-inversion symmetry axis of their joint PDF. Data were excluded for the years in which the cells comprising the ridge could not be identified using surface openness

quasi-inversion symmetry axis by considering the quasi-inversion symmetry from one firm-size variable to another. Then, we observed the above analytical conclusion on the empirical data of firms in Japan, Spain, France, the United Kingdom, and Italy, which have sufficient data quantity and whose symmetry axes are easy to observe. A good accuracy was established within the error range. However, since the Pareto indices of the firm-size variables barely change over time, we found no significant change in the ratios from year to year. Therefore, no appreciable

8 Power Laws with Different Exponents in Firm-Size Variables 1.2

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thetaLY

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2014 2013 2012 2011 2010

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1

2014 2013 2012 2011 2010

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Fig. 8.9 Comparison between ratio (μL /μY ) of Pareto index of employees (L) to that of operating revenue (Y ) and slope (θLY ) of quasi-inverted symmetry axis of their PDF for Spanish (a), French (b), British (c), and Italian (d) firms from 2010 to 2014

correlation exists between the ratio of the power exponents and the slope of the quasi-inversion symmetry axis. In all cases, however, they agree with each other, almost within allowable error. What does the difference in Pareto index by firm-size variables mean? First, the firm-size variables that follow the power-law distribution are in the large-scale range. The distribution of firm-size variables with a small power index is wider than those with a large power index. Since the power index for the number of employees was the highest in all five countries that we analyzed, the extent of the distribution of the number of employees was the smallest among the other firm-size variables. In Spain and the United Kingdom, the distribution of operating revenues is wider than the tangible fixed assets, although in France, their distribution is wider than the

References

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operating revenues. The distribution of operating revenues and tangible fixed assets is similar in Japan and Italy. In this chapter, we discussed three sets of three firm-size variables that follow a power-law distribution with different indices and introduced a two-dimensional quasi-inversion symmetry. In our book’s final chapter, we extend this discussion to contemporaneous three-dimensional quasi-inversion symmetry. This leads to a discussion of the production function [53], which is a basic component of economics.

References 1. Pareto V (1897) Cours d’Économie Politique. Macmillan, London 2. Newman MEJ (2005) Power laws, Pareto distributions and Zipf’s law. Contemp Phys 46:323– 351 3. Clauset A, Shalizi CR, Newman MEJ (2009) Power-law distributions in empirical data. SIAM Rev 51:661–703 4. Mizuno T, Katori M, Takayasu H, Takayasu M (2002) Statistical and stochastic laws in the income of Japanese companies. In: Takayasu H (ed) Empirical science of financial fluctuations -the advent of econophysics. Springer, Tokyo, pp 321–330 5. Bonabeau E, Dagorn L (1995) Possible universality in the size distribution of fish schools. Phys Rev E51:R5220–R5223 6. Render S (1998) How popular is your paper? An empirical study of the citation distribution. Eur Phys J B4:131–134 7. Takayasu M, Takayasu H, Sato T (1996) Critical behaviors and 1/ noise in information traffic. Physica A233:824–834 8. Saichev AI, Malevergne Y, Sornette D (2010) Theory of Zip’s law and beyond. Springer, Berlin Heidelberg 9. Kaizoji T (2003) Scaling behavior in land markets. Physica A326:256–264 10. Yamano T (2004) Distribution of the Japanese posted land price and the generalized entropy. Eur Phys J B38:665–669 11. Mantegna RN, Stanley HE (1995) Scaling behaviour in the dynamics of an economic index. Nature 376:46–49 12. Axtell RL (2001) Zipf distribution of US firm sizes. Science 293:1818–1820 13. Podobnik B, Horvatic D, Petersen A M, Uro˘sevi´c B, Stanley HE (2010) Bankruptcy risk model and empirical tests. Proc Natl Acad Sci U S A 107:18325–18330 14. Fu D, Pammolli F, Buldyrev SV, Riccaboni M, Matia K, Yamasaki K, Stanley HE (2005) The growth of business firms: theoretical framework and empirical evidence. Proc Natl Acad Sci 102:18801–18806 15. Podobnik B, Horvatic D, Pammolli F, Wang F, Stanley H E, Grosse I (2008) Size-dependent standard deviation for growth rates: empirical results and theoretical modeling. Phys Rev E77:056102 16. Okuyama K, Takayasu M, Takayasu H (1999) Zipf’s law in income distribution of companies. Physica A269:125–131 17. Ramsden JJ, Kiss-Haypál G (2000) Company size distribution in different countries. Physica A277:220–227 18. Gaffeo E, Gallegati M, Palestrinib A (2003) On the size distribution of firms: additional evidence from the G7 countries. Physica A324:117–123 19. Zhang J, Chen Q, Wang Y (2009) Zipf distribution in top Chinese firms and an economic explanation. Physica A388:2020–2024

110

8 Power Laws with Different Exponents in Firm-Size Variables

20. Kesten H (1973) Random difference equations and renewal theory for products of random matrices. Acta Math 131:207 21. Levy M, Solomon S (1996) Power laws are logarithmic Boltzmann laws. Int J Mod Phys C7:595–601 22. Sornette D, Cont R (1997) Convergent multiplicative processes repelled from zero: power laws and truncated power laws. J Phys I7:431–444 23. Takayasu H, Sato A, Takayasu M (1997) Stable infinite variance fluctuations in randomly amplified Langevin system. Phys Rev Lett 79:966–969 24. Stanley HE (1971) Introduction to phase transitions and critical phenomena. Clarendon Press, Oxford 25. Gibra R (1952) Les Inégalités Économique. Sirey, Paris 26. Sutton J (1997) Gibrat’s legacy. J Econ Lit 35:40–59 27. Fujiwara Y, Souma W, Aoyama H, Kaizoji T, Aoki M (2003) Growth and fluctuations of personal income. Physica A321:598–604 28. Fujiwara Y, Guilmi CD, Aoyama H, Gallegati M, Souma W (2004) Do Pareto-Zipf and Gibrat laws hold true? An analysis with European firms. Physica A335:197–216 29. Ishikawa A (2006) Derivation of the distribution from extended Gibrat’s law. Physica A367:425–434 30. Ishikawa A (2007) The uniqueness of firm size distribution function from tent-shaped growth rate distribution. Physica A383:79–84 31. Ishikawa A (2006) Annual change of Pareto index dynamically deduced from the law of detailed quasi-balance. Physica A371:525–535 32. Ishikawa A (2009) Quasistatically varying log-normal distribution in the middle scale region of Japanese land prices. Prog Theor Phys Suppl 179:103–113 33. Ishikawa A, Fujimoto S, Mizuno T (2011) Shape of growth rate distribution determines the type of non-Gibrat’s property. Physica A390:4273–4285 34. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2016) Firm growth function and extendedGibrat’s property. Adv Math Phys 2016:9303480 35. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2016) Long-term firm growth properties derived from short-term laws and number of employees in Japan and France. Evol Inst Econ Rev 13:409–422 36. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2015) Firm age distributions and the decay rate of firm activities. In: Takayasu H, Ito N, Noda I, Takayasu M (eds) Proceedings of the international conference on social modeling and simulation, plus econophysics colloquium 2014. Springer, Tokyo, pp 187–194 37. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2015) The relation between firm age distributions and the decay rate of firm activities in the United States and Japan. In: Big Data, 2015 IEEE international conference on date of conference 2015, pp 2726–2731 38. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2017) Dependence of the decay rate of firm activities on firm age. Evol Inst Econ Rev 14:351–362 39. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2017) Transition law of firms’ activity and the deficit aspect of non-Gibrat’s law. JPS Conf Proc 16:011005 40. Ishikawa A, Fujimoto S, Mizuno T (2018) Statistical law observed in inactive rate of firms. Evol Inst Econ Rev 16:201–212 41. Watanabe T, Mizuno T, Ishikawa A, Fujimoto S (2011) Shape of production function of mega firms: analytical methods and applications. Econ Rev (Keizai Kenkyuu) 62:193 (in Japanese) 42. Mizuno T, Watanabe T, Ishikawa A, Fujimoto S (2012) Power laws in firm productivity. Prog Theor Phys Suppl 194:122–134 43. Ishikawa A, Fujimoto S, Watanabe T, Mizuno T (2013) The emergence of different tail exponents in the distributions of firm size variables. Physica A392:2104–2113 44. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2014) Analytical derivation of power laws in firm size variables from Gibrat’s law and quasi-inversion symmetry: A geomorphological approach. J Phys Soc Jpn 83:034802

References

111

45. Ishikawa A, Fujimoto S, Mizuno T (2020) Why does production function take the CobbDouglas form? Direct observation of production function using empirical data. Evol Inst Econ Rev. https://doi.org/10.1007/s40844-020-00180-3 46. Bureau van Dijk, http://www.bvdinfo.com/Home.aspx 47. Malevergne Y, Pisarenko V, Sornette D (2011) Testing the Pareto against the lognormal distributions with the uniformly most powerful unbiased test applied to the distribution of cities. Phys Rev E 83:036111 48. Fujimoto S, Ishikawa A, Mizuno T, Watanabe T (2011) A new method for measuring tail exponents of firm size distributions. Economics E-J 5:2011–2020 49. Yokoyama R, Sirasawa M, Kikuchi Y (1999) Representation of topographical features by opennesses. J Jpn Soc Photogramm Remote Sens 38(4):26–34 (in Japanese with English abstract) 50. Yokoyama R, Sirasawa M, Pike R (2002) Visualizing topography by openness: a new application of image processing to digital elevation models. Photogramm Eng Remote Sens 68(3):257–265 51. Prima ODA, Echigo A, Yokoyama R, Yoshida T (2006) Supervised landform classification of Northeast Honshu from DEM-derived thematic maps. Geomorphology 78:373–386 52. Souma W (2007) Networks of firms and the ridge in the production space. In: Chatterjee A, Chakrabarti BK (eds) Econophysics of markets and business networks. Springer, Italia, pp 149–158 53. Cobb CW, Douglass PH (1928) A theory of production. Am Econ Rev 18:139–165

Chapter 9

Why Does Production Function Take the Cobb–Douglas Form?

Abstract We directly observed a Cobb–Douglas symmetric plane using the index of surface openness, which is used in geography, and successfully identified it. Based on this observation, we measured the capital shares (capital elasticity) and labor shares (labor elasticity) and compared them with the results of multiple regression analysis used in economics. We confirmed consistent agreement in seven countries: Japan, Germany, France, Spain, Italy, the United Kingdom, and the Netherlands. Thus, we show that the Cobb–Douglas production function can be clearly captured in empirical data as a geometric entity with a quasi-inverse symmetry of variables. Based on the above discussion, we theoretically clarified why the Cobb–Douglas production function is better fit to empirical data in economics, because it uniquely derives the fact that their variables follow a powerlaw distribution.

9.1 Introduction In the previous chapter, we focused on three types of firm-size variables: operating revenues, tangible fixed assets, and the number of employees, all of which follow power-law distributions with different Pareto indices. The quasi-inversion symmetry of the three sets of firm-size variables can connect their power-law distributions. By expanding this discussion of the quasi-inversion symmetry of two variables to three variables, we develop a discussion of the production function, which is a major foundation of economics. In economics, firms are regarded as economic entities that produce goods (Y ) using capital, labor, and other resources, which are called production factors (x1 , x2 , . . . , xn ). The production activity of firms is modeled as a function that inputs production factors and outputs a total production: Y = F (x1 , x2 , . . . , xn ). This is called a production function. As a simplified model, economists have proposed two-variable production functions in which the production factors are capital (K) and labor (L).

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Ishikawa, Statistical Properties in Firms’ Large-scale Data, Evolutionary Economics and Social Complexity Science 26, https://doi.org/10.1007/978-981-16-2297-7_9

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9 Why Does Production Function Take the Cobb–Douglas Form?

A typical example is the Cobb–Douglas production function [1]: Y (K, L) = AK α Lβ .

(9.1)

Here, A is the total factor productivity, which is interpreted as a firm’s production efficiency or technological capability that cannot be measured by capital (K) or labor (L). α and β are the capital share (capital elasticity) and labor share (labor elasticity) and represent production’s capital or labor dependency. The constant elasticity of substitution (CES)-type production function, Y (K, L) =  − 1 A δK −ρ + (1 − δ)L−ρ ρ , is an extension of the Cobb–Douglas production function [2, 3]. Here, δ and ρ are distribution and substitution parameters. The CES production function matches the Cobb–Douglas production function (9.1) at α + β = 1 in the ρ → 0 limit as well as the Leontief production function, Y = A min{K, L}, in the limit of ρ → +∞. Furthermore, economists proposed a translog-type production function, log Y (K, L) = log A + α log K + β log L + γ1 log2 K+γ2 log2 L+γ3 log K log L, as an extended version that explicitly includes the Cobb–Douglas production function [4]. In economics, various forms of production functions have been proposed and widely used by researchers in microeconomics and macroeconomics. Total factor productivity has been applied to industries around the world since the late 1950s as an indicator of productivity, and many empirical studies have been addressed in terms of production functions. However, conventional research is limited to comparisons of the superiority of the production function based on the regression’s good fit to the shape of several production functions described above, for example. Note that there is only limited argument concerning why a production function takes a functional form, such as the Cobb–Douglas type. Among these studies, Houthakker focused on the similarity between the Cobb–Douglas production function and the power-law distribution, followed by its variables from a microeconomic approach [5]. Unfortunately, his argument fails to reach the constant returns to scale discussed below. In this context, as in Houthakker’s work, where the Cobb–Douglas production function is a power-law function of variables (K, L, Y ), we show that the Cobb– Douglas production function can be interpreted as an inverse-symmetric plane and its residual in the 3D space of variables (K, L, Y ) [6–8]. Here, we observed quasiinverse symmetry in the joint probability density function (PDF) PKLY (K, L, Y ) under variable exchange, Y ↔ aK α Lβ , which is deeply related to the power-law distribution in the large-scale range of three variables (K, L, Y ) and log-normal distribution in the mid-scale range [9, 10]. Importantly, the observed fact that each variable (K, L, Y ) of the production function follows a power-law distribution is uniquely concluded from the form of the Cobb–Douglas production function. In this chapter, we directly observed a Cobb–Douglas symmetric plane using the index of surface openness (introduced in Chap. 8), which is used in geography, and successfully identified it. Based on this observation, we measured the capital share (capital elasticity) and labor share (labor elasticity), compared our results

9.2 Data

115

with the results of multiple regression analysis used in economics, and confirmed consistent agreement in seven countries: Japan, Germany, France, Spain, Italy, the United Kingdom, and the Netherlands. Thus, we showed that the Cobb–Douglas production function can be clearly captured in empirical data as a geometric entity with a quasi-inverse symmetry of variables. Based on the above discussion, we theoretically clarified why the Cobb–Douglas production function is better fit to empirical data in economics. The rest of this chapter is structured as follows. Section 9.2 describes the data we used in our analysis. In Sect. 9.3, we show analytically that a power law of (K, L, Y ) is derived from 3D quasi-inversion symmetry and Gibrat’s law. In Sect. 9.4, we describe how the constant returns to scale, which are often required in economics for Cobb–Douglas production functions, are naturally derived in a form standardized by the power index. Section 9.5 makes analytical preparations to facilitate the observations in Sect. 9.6. In Sect. 9.6, we address the Cobb–Douglas symmetric plane by identifying a ridge in Sect. 9.5’s analytical preparation and measuring the capital share (capital elasticity) and labor share (labor elasticity). We measure both kinds of shares by multiple regression analysis and confirm that they are consistent in seven countries: Japan, Germany, France, Spain, Italy, the United Kingdom, and the Netherlands. Finally, in Sect. 9.7, we summarize the main points of this chapter and present future perspectives.

9.2 Data We used the 2016 edition of the Orbis database provided by Bureau van Dijk [11]. It is the world’s largest corporate financial database with approximately 200 million listed and unlisted firms (at that time) collected worldwide from more than 120 local credit bureaus and information vendors, including from Asia, the Americas, Europe, the Middle East, and Africa. This database is characterized not only by its huge amount of data but also by the fact that its data are arranged in a standardized format and can be compared internationally. Its total includes the corporate financial data for 1,831,481 firms in Japan (JP), 3,458,922 firms in Italy (IT), 1,953,140 firms in France (FR), 1,204,584 firms in Germany (DE), 5,070,698 firms in the United Kingdom (GB), 1,604,553 firms in Spain (ES), and 3,213,808 firms in the Netherlands (NL). This study focuses on the firms in these countries that have sufficient data for statistical analysis. Although the number of firms collected in the United States is sufficient, we did not include them because of an unnatural bias that may be attributable to the method of data collection by local credit bureaus or information vendors. Since Japan has about 1–2 million active firms [12], the data analyzed here are highly comprehensive. We consider that this situation is the same in other countries. To most effectively exploit the features of this database, we use operating revenues (Y ) (total sales), tangible fixed assets (K), and the number of employees (L). When economists consider a production function, they generally use Y for

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9 Why Does Production Function Take the Cobb–Douglas Form?

added value, K for fixed assets in manufacturing, K for liquid assets in nonmanufacturing, and L for total labor hours or wages. On average, however, only around 1/10 of the firms in our database provided detailed data to calculate added value, etc. If we use the same amount as economists, we lose the data’s completeness and have to filter out firms for which we do have detailed data to calculate added value, etc. Most crucially in this study, K, L, and Y follow power-law distributions in the large-scale ranges. This property is common to both the (K, L, Y ) used in economics and the simple (K, L, Y ) we use. If the argument of directly observing a quasi-inversion symmetric plane is self-consistent, both quantities should allow for identical discussion. Therefore, to ensure a statistical advantage with a sufficient amount of data, this study proceeds with its analysis using the quantities mentioned above. The amount of data for Japan, Spain, France, the United Kingdom, and Italy is shown in Tables 8.3, 9.2, and 8.5 in the previous chapter. The amount of data for Germany and the Netherlands is shown in Tables 9.1 and 9.2.

Table 9.1 Amount of data on operating revenues, tangible fixed assets, and number of employees of German firms in Orbis 2016 Year 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

Operating revenue 125,444 124,269 147,664 242,788 293,058 335,285 346,633 344,803 272,842 26,261

Tangible fixed assets 604,286 656,045 693,913 708,504 698,961 710,170 562,307 461,275 337,954 3459

Number of employees 148,715 155,754 152,998 155,584 154,722 151,118 136,829 157,440 194,158 348,483

Table 9.2 Amount of data on operating revenues, tangible fixed assets, and number of employees of Dutch firms in Orbis 2016 Year 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015

Operating revenue 13,492 14,905 17,443 19,464 18,968 19,262 19,064 18,260 14,751 456

Tangible fixed assets 335,719 366,812 441,304 531,864 553,601 579,396 572,656 572,762 490,545 8509

Number of employees 384,706 511,742 695,108 1,038,610 1,299,558 1,533,303 1,333,659 1,208,607 1,128,279 528,017

9.3 Quasi-Inverse Symmetry in Two and Three Dimensions

117

9.3 Quasi-Inverse Symmetry in Two and Three Dimensions As discussed in Chap. 8, such variables as operating revenues (Y ), tangible fixed assets (K), and the number of employees (L), all of which represent a firm’s size, follow a power-law distribution in the large-scale range [13–15]: PY (Y ) ∝ Y −μY −1 .

(9.2)

PK (K) ∝ K −μK −1 ,

(9.3)

−μL −1

PL (L) ∝ L

(9.4)

.

Here, P is a probability density function (PDF), and power-law exponent μ is also called Pareto’s index. As repeatedly stated in this book, Fujiwara et al. [16, 17] argued that these power distributions (9.2)–(9.4) can be derived from the time-reversal symmetry observed between the variables of one year (xT ) and those of the next (xT +1 ), PJ (xT , xT +1 ) dxT dxT +1 = PJ (xT +1 , xT ) dxT +1 dxT ,

(9.5)

and Gibrat’s law, where growth-rate distribution Q (R|xT ) conditioned by first-year size (xT ), does not depend on first-year variables [9, 10]: Q (R|x1 ) = Q (R) .

(9.6)

Here, xT represents (K, L, Y ) in one year, and xT +1 represents it in the next. PJ is the joint PDF, and R = xT +1 /xT is the growth rate. In time-reversal symmetry, note that joint PDF PJ has the same shape on both sides of Eq. (9.5) under the permutation of variables xT ↔ xT +1 . The derivation by Fujiwara et al. is effective when the power-law index μ does not change. This derivation was described in Chap. 2. In Chap. 3, we extended the argument by introducing an extended growth rate of R = xT +1 /axT θ in quasi-time-reversal symmetry under the permutation of axT θ ↔ xT +1 variables: PJ (xT , xT +1 ) dxT dxT +1 = PJ



x

T +1

a

1 θ

, axT

θ





xT +1  θ1 d d axT θ . a (9.7)

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9 Why Does Production Function Take the Cobb–Douglas Form?

We also showed that a power-law distribution can be derived from quasi-timereversal symmetry (9.7) and Gibrat’s law (9.6), even when the power-law index varies quasi-statically. Here, a and θ , which are parameters of quasi-time-reversal symmetry, are related to power exponents (μT , μT +1 ) of variable (xT , xT +1 ): θ=

μT . μT +1

(9.8)

This derivation was described in Chap. 3. We confirmed this relation by the temporal change of power-law index μ, which can be observed in Japanese land prices [18, 19]. In Chap. 8, we showed that this quasi-inverse symmetry can be similarly observed among such different firm-size variables as tangible fixed assets, the number of employees, and operating revenues (K, L, Y ) at the same time and confirmed a relation (9.8) between the rate of their power-law indices and the slopes of the quasi-inverse symmetry axes [6–8]. In this case, we consider (xT , xT +1 ) to be (K, L) , (K, Y ), and (L, Y ). We can extend the concept of quasi-inverse symmetry and the growth rate to three simultaneous variables (K, L, Y ) by introducing growth rate R = Y/aK α Lβ into a system with quasi-inverse symmetry under the variable substitution of aK α Lβ ↔ Y : PKLY (K, L, Y ) dKdLdY   1  1 α β Y Y α β , , aK L = PKLY aLβ aK β  1   1  α β

Y Y d d aK α Lβ . d β α aL aK

(9.9)

The quasi-inverse symmetry (9.9) and Gibrat’s law of the three variables, Q (R|K, L) = Q (R) ,

(9.10)

lead to power-law distributions of K, L, and Y , as in the case of two variables. The derivation of this conclusion is described below. Three-variable quasi-inverse symmetry (9.9) is rewritten in three variables (K, L, R): PKLR (K, L, R) dKdLdR

1  1  1 

 1 = PKLR R α K, R β L, R −1 d R α K d R β L d R −1 .

(9.11)

9.3 Quasi-Inverse Symmetry in Two and Three Dimensions

119

This idea can be expressed as follows: 1

PKLR (K, L, R) = R α

+ β1 −2

1  1 PKLR R α K, R β L, R −1 .

(9.12)

From the definition of a conditional PDF, Q (R|K, L) = PKLR (K, L, R) /PKL (K, L), this equation can be modified: 1 PKL (K, L) + 1 −2 = Rα β

1 1  PKL R α K, R β L

=R

1 1 α + β −2

 1 1 Q R −1 |R α K, R β L Q (R|K, L)

Q R −1 . Q (R)

(9.13)

In the modification from the second to the third equation, we used a three-variable Gibrat’s law (9.10). Since the right-hand side of Eq. (9.13) is only a function of R, if we express it as Gαβ (R), it can be written as

1 1  PKL (K, L) = Gαβ (R) PKL R α K, R β L .

(9.14)

By expanding the right-hand side of Eq. (9.14) by R = 1 +  (0 <   1), the zeroth order of  becomes an obvious expression, and the first order becomes the following differential equation:  Gαβ



 L ∂ K ∂ + PKL (K, L) = 0. (1) + α ∂K β ∂L

(9.15)



Here, Gαβ (·) represents the differential of Gαβ (·) by R. No more useful information is available except the first order of . As shown in Fig. 8.2a, the two variables (K, L) are strongly correlated and are not independent. Therefore, the differential equation (9.15) cannot be solved as it is. We can convert variables (K, L) into normalized variables (k, l), log k =

log K − mK , σK

log l =

log L − mL , σL

(9.16)

and rotate them as −π/4 to finally obtain orthogonal independent variables (Z1 , Z2 ): 1 1 log Z1 = √ (log k + log l) , log Z2 = √ (− log k + log l) . 2 2

(9.17)

Here, (mK , mL ) and (σK , σL ) are the mean and standard deviation of (log K, log L) in the power-law region of variables (K, L). In Eq. (9.16), the distribution’s width is

120

9 Why Does Production Function Take the Cobb–Douglas Form?

normalized by dividing it by (σK , σL ) after subtracting the average (mK , mL ) from (log K, log L). As shown in Ref. [8], we can numerically show that this operation yields orthogonal independent variables (Z1 , Z2 ). We can rewrite the differential equation (9.15) with independent variables (Z1 , Z2 ) and solve it to analytically show that variable (Z1 , Z2 ) obeys the power-law distribution to numerically confirm the result in empirical data. No correlation between Z1 and Z2 does not necessarily mean that they are independent variables. However, in the power-law region of K and L, Z1 and Z2 are not only uncorrelated but also independent. This is numerically confirmed using empirical data (see [8] for details). The independence of variables Z1 and Z2 can be analytically explained. Two variables (K, L), whose coordinate system has quasi-inverse symmetry aKL K θKL ↔ L, are transformed into two variables, (k, l), whose coordinate system has inverse symmetry k ↔ l. Note that k and l are defined in Eq. (9.16) and that aKL and θKL are constant parameters. Inverse symmetry is invariance under the exchange of variables with respect to a line whose slope is π/4. Since Eq. (9.17) rotates the coordinate system by −π/4, the new variables (Z1 , Z2 ) must be independent. The system (Z1 , Z2 ) also has Z2 ↔ 1/Z2 symmetry. By setting 1 (σK α + σL β) , 2

(9.18)

1 (−σK α + σL β) , 2

(9.19)

θ1 = θ2 =

K α Lβ is reduced to Z1 θ1 Z2 θ2 , and the quasi-inverse symmetry is rewritten as Y ↔ a  Z1 θ1 Z2 θ2 . Here, a  (= 10αmK +βmL a) is a constant parameter. As a result, quasi-inverse symmetry and Gibrat’s law are similarly observed in the (Z1 , Z2 , Y ) coordinates. Therefore, using a similar discussion, the following partial differential equations are obtained: G+  (1) PZ1 Z2 (Z1 , Z2 ) +

Z2 ∂ Z1 ∂ PZ1 Z2 (Z1 , Z2 ) + PZ Z (Z1 , Z2 ) = 0, θ1 ∂Z1 θ2 ∂Z2 1 2

(9.20)

G−  (1) PZ1 Z2 (Z1 , Z2 −1 ) +

Z2 −1 ∂ Z1 ∂ PZ1 Z2 (Z1 , Z2 −1 ) − PZ Z (Z1 , Z2 −1 ) = 0, θ1 ∂Z1 θ2 ∂Z2−1 1 2 (9.21)

where G± (R) = R 1/θ1 ±1/θ2 −2 Q(R −1 )/Q(R) and R = Y/a  Z1 θ1 Z2 θ2 . Equation (9.21) is obtained from Z2 ↔ 1/Z2 symmetry.

9.3 Quasi-Inverse Symmetry in Two and Three Dimensions

121

As mentioned above, Z1 and Z2 are independent variables. Therefore, with the variable separation method, the solution PZ1 Z2 (Z1 , Z2 ) of Eqs. (9.20) and (9.21) is uniquely determined to be the product of the power-law functions of Z1 and Z2 . Note that the power-law index of Z2 for Z2 > 1 is different from the index of Z2 for Z2 < 1. Therefore, the analytical solution is expressed as follows: PZ1 Z2 (Z1 , Z2 ) = C Z1 −μ1 −1 Z2 −μ2 −1 for log10 Z2 > 0, PZ1 Z2 (Z1 , Z2 ) = C Z1

−μ1 −1

Z2

+μ2 −1

for log10 Z2 < 0,

(9.22) (9.23)

where the conditions of variable separation μ1θ+1 + μ2θ±1 = G±  (1). 1 2 This solution satisfies Z2 ↔ 1/Z2 symmetry: PZ1 Z2 (Z1 , Z2 )dZ1 dZ2 = PZ1 Z2 (Z1 , Z2 −1 )dZ1 d(Z2 −1 ). Note that Eqs. (9.22) and (9.23) continue to be a general solution to PZ1 Z2 (Z1 , Z2 ) = G± (R)PZ1 Z2 (R 1/θ1 Z1 , R 1/θ2 Z2 ), even if R deviates substantially  from R = 1, as long as Q(R) satisfies Q(R) = R −G± (1)+1/θ1±1/θ2 −2 Q(R −1 ) = R −μ1 /θ1 −μ2 /θ2 −2 Q(R −1 ). Using the transformation of integration measure dZ1 dZ2 =| as

∂(Z1 ,Z2 ) ∂(K,L)

| dKdL =

2

2 −1 σL −1 dKdL, σK σL K L +

PKL (K, L) is expressed

+

PKL (K, L) = C+ K θK −1 L−θL −1 for l > k, PKL (K, L) = C− K

− −θK −1

θL− −1

L

for l < k.

(9.24) (9.25)

Here, μ2 − μ1 μ2 + μ1 , θL+ = , σK σL μ2 + μ1 μ2 − μ1 = , θL− = . σK σL

+ = θK

(9.26)

− θK

(9.27)

Note that variables K and L are not independent because the power-law indices for l > k are different from the indices for l < k, although PKL (K, L) is also the product of the power-law functions of K and L. By integrating PKL (K, L) by L or K, from the leading order terms, power-law functions K and L are obtained as   + −  ∞ μ C+ aKL −θL C− aKL θL −2 1 −1 dLPKL (K, L) ∼ + K σK , (9.28) P (K) = + − θL θL 0   + −  ∞ μ C+ aLK θK C− aLK −θK −2 σ 1 −1 L L P (L) = dKPKL (K, L) ∼ + , (9.29) + − θK θK 0 where log10 aKL = mL − mK σL /σK and log10 aLK = mK − mL σK /σL . By comparing Eqs. (9.3) and (9.28) and Eqs. (9.4) and (9.29), the relations among

122

9 Why Does Production Function Take the Cobb–Douglas Form?

power-law indices are found as follows: μK = 2

μ1 μ1 , μL = 2 . σK σL

(9.30)

At the same time, from the result in Refs. [20, 21], we conclude the following. Under quasi-inverse symmetry Y ↔ a  Z1 θ1 Z2 θ2 , Eqs. (9.22) and (9.23) imply that Y obeys a power law (9.2) and index μY is identified [7]:  μ1 μ2 . , μY = min θ1 θ2 

(9.31)

From the data analyses of various countries [8], the relations among power-law indices are observed as μ1 /θ1 < μ2 /θ2 , and Eq. (9.31) is reduced to μY =

μ1 . θ1

(9.32)

The accuracy of Eqs. (9.30) and (9.32) with empirical data confirms the validity of the analytical argument that three-dimensional quasi-inversion symmetry and Gibrat’s law lead to a power law of three variables [8]. Furthermore, by writing A = R a, the definition of R becomes the Cobb– Douglas production function: log Y = α log K + β log L + log A.

(9.33)

In this case, the three-dimensional Gibrat’s law (9.10) guarantees that the distribution of total factor productivity A does not depend on K and L.

9.4 Modified Constant Returns to Scale In the previous section, variables (K, L) were normalized by dividing them by logarithmic standard deviations, which are indicators of the width of the distribution in the large-scale range where the distribution follows a power law. On the other hand, the width of the distribution of the variables following the powerlaw distribution of exponent μ is also given as 1/μ on a logarithmic scale. Thus, the normalization of (log K, log L), divided by distribution width (σK , σL ) in Eq. (9.16), is equivalent to dividing by (1/μK , 1/μL ) or multiplying by (μK , μL ):



log k = μK log K − log Kup , log l = μL log L − log Lup .

(9.34)

Here, the origin is shifted by subtracting the logarithm of the power-law ranges’ upper limits (log Kup , log Lup ).

9.4 Modified Constant Returns to Scale

123

We first consider the quasi-inverse symmetry of two variables, (K, L). Because the normalized widths of the k and l distributions are equal, a two-variable quasiinverse-symmetric line, log L = θ log K + log a, is converted to a slope 1 inversesymmetric line: log l = log k + log a  . This is consistent with Eq. (9.8) where the relationship between slope θ of the quasi-inverse-symmetric axis and exponents μK and μL is given as 1 1 =θ . μL μK

(9.35)

Here, we have substitutions: μT = μK and μT +1 = μL . Extending this concept to three variables (K, L, Y ), the following relationship is established between the capital and labor shares (α, β) of the three-variable quasiinverse-symmetric plane, log Y = α log K + β log L + log a,

(9.36)

and the power-law exponents (μK , μL , μY ): 1 1 1 =α +β . μY μK μL

(9.37)

This is an extension of the constant returns to scale: 1 = α + β.

(9.38)

Using the following normalized notation, α = α

μY μY , β = β , μK μL

(9.39)

Eq. (9.37) is simply expressed as 1 = α + β  .

(9.40)

In Fig. 9.1, we compared the accuracy of Eqs. (9.38) and (9.40) using αM , βM , α  M , and β  M , which are evaluated by multiple regression analysis used in economics in Japanese firms from 2010 to 2014. The database we used was created by a collection method that reduced the number of firms in the past. Therefore, Fig. 9.1 shows data from 2014 to the previous 5 years. From Fig. 9.1, we confirmed that the accuracy of the constant returns to scale, normalized by the power-law index (9.40), becomes higher than the conventional one (9.38). Results similar to those of Japanese firms can be confirmed for firms in the countries that we analyzed [22].

124

9 Why Does Production Function Take the Cobb–Douglas Form?

1.3

D+E D’+E’

1.2

1.1

1.0

0.9 2014

2013

2012

2011

2010

Year

Fig. 9.1 Comparison of constant returns to scale (α +β = 1) of Japanese firms from 2014 to 2000 and a standardized one (α  + β  = 1)

More interestingly, normalized capital share α  (capital elasticity) and labor share β  (labor elasticity) can be directly estimated by observing the Cobb–Douglas symmetric plane, as described in the next section.

9.5 Analytical Preparation for Direct Observation of Cobb–Douglas Symmetric Plane In Sects. 9.3 and 9.4, we introduced the Cobb–Douglas type symmetric plane (9.36) and showed that the power-law distributions of K, L, and Y are derived from it. We have not yet confirmed, however, whether our quasi-inverse-symmetric plane is consistent with the Cobb–Douglas production function (9.33) or (9.1), which is treated in economics. We cannot deny the possibility that capital share (α) and labor share (β) are different between the quasi-inverse-symmetric plane (9.36) (that we introduced) and the multiple regression plane. This section provides an analytical preparation to confirm this possibility. In Sect. 9.3, we rewrote the quasi-inverse symmetry of the (K, L, Y ) space to the quasi-inverse one of the (K, L, R) space and limited the discussion to the powerlaw region of (K, L), where Gibrat’s law holds, and derived a differential equation for (K, L) by expanding growth rate R around “1.” In this discussion, we only investigated the neighborhood of the quasi-inverse-symmetric plane. To directly observe the Cobb–Douglas symmetric plane (9.36), the entire space of the three variables must be observed without adopting this method.

9.5 Analytical Preparation for Direct Observation of Cobb–Douglas. . .

125

Thus, in addition to Eq. (9.34), we should consider the standardization of Y :

log y = μY log Y − log Yup . (9.41) For two variables (K, L), the direction vector of the quasi-inverse-symmetric axis is (1/μK , 1/μL ), and the normalized direction vector is (1, 1) in two-variable √ space (k, l). Using Eq. (9.17), we rotated it to be 2, 0 , resulting in independent variables (Z1 , Z2 ). For three variables (K, L, Y ), the densest direction vector in the quasi-inverse-symmetric plane (9.36) is (1/μK , 1/μL , 1/μY ). The rotation of  √ 3, 0, 0 normalized direction vector (1, 1, 1) in a three-variable space (k, l, y) to is given by 1 log z1 = √ (log k + log l + log y) , 3 1 log z2 = √ (− log k + log l) , 2 1 log z3 = √ (− log k − log l + 2 log y) . 6

(9.42) (9.43) (9.44)

This rotation caused the densest directional vector in the Cobb–Douglas quasiinverse-symmetric plane (9.36) to overlap the z1 axis (Fig. 9.2). To align a Cobb–Douglas quasi-inverse-symmetric plane with the w3 = Const. plane, we need to rotate it further about the z1 axis. The rotation, expressed using parameter ψ, is as follows (see Fig. 9.3): 1 log w2 =  (log z2 + ψ log z3 ) , 2 ψ +1

(9.45)

1 log w3 =  (−ψ log z2 + log z3 ) . 2 ψ +1

(9.46)

Fig. 9.2 Rotation (9.42)–(9.44) causes densest directional vector in Cobb–Douglas quasi-inverse-symmetric plane (1, 1, 1) to overlap z1 axis

126

9 Why Does Production Function Take the Cobb–Douglas Form?

Fig. 9.3 Rotation (9.45), (9.46) aligns a Cobb–Douglas quasi-inverse-symmetric plane with w3 = Const. plane. For clarity, the figure is written as Const. = 0

Due to the above rotations in Eqs. (9.42) through (9.46), the Cobb–Douglas quasiinverse-symmetric plane overlaps the w3 = Const. plane. If the w3 = Const. plane is expressed by the quasi-inverse-symmetric plane in the (k, l, y) space by inverse transformation, it can be expressed as log y =

1−

√ √ 1 + 3ψ 3ψ log k + log l + Const. 2 2

(9.47)

From the above equation, using parameter ψ, which represents the slope of the Cobb–Douglas quasi-inverse-symmetric plane, normalized capital share α  (capital elasticity) and labor share β  (labor elasticity) can be expressed as 

α =

1−

√ 2

3ψ



, β =

1+

√ 3ψ . 2

(9.48)

9.6 Direct Observation of Cobb–Douglas Quasi-Inverse-Symmetric Plane In the previous section, we analytically discussed the following procedures to simplify observing the Cobb–Douglas quasi-inverse-symmetric plane (9.36) in a three-variable space (K, L, Y ). First, we converted three variables (K, L, Y ) into variables (k, l, y), which are normalized, such as the values of the power-law exponents in the power-law region to be “1.” Next, we considered a rotation (9.42)–(9.44) that converts the densest direction vector (1, 1, 1) in a Cobb–Douglas quasi-inverse-symmetric plane (9.36) into a vector of only the first component and obtains variables (z1 , z2 , z3 ).

9.6 Direct Observation of Cobb–Douglas Quasi-Inverse-Symmetric Plane

127

If 3D data are projected onto the z1 z2 plane in the z3 direction, then the densest direction vector should overlap the z1 axis (Fig. 9.3). If 3D data are projected onto the z1 z3 plane in the z2 direction, the densest direction vector should still overlap the z1 axis (Fig. 9.3). When 3D data are then projected onto the z2 z3 plane in the z1 direction, the densest direction vector is observed, and its slope is given as ψ (Fig. 9.3). In this section, we first confirm the above analytical discussions by directly observing the empirical data. In such direct observation, identifying the densest directional vector in a 2D plane is critical. We observed a dense directional vector as a ridge using the geographical index of surface openness, as in Chap. 8. The method is briefly described below. z1 and z2 data points are scattered in the z1 z2 plane. For example, Fig. 9.4 is a scatter plot of Japanese firms in 2014. To clearly comprehend the density, we divided it into logarithmically equal-sized cells and expressed the amount of data points in the cells by different shades. For example, Fig. 9.5 is a diagram of Japanese firms in 2014 created in this way. The logarithm of a cell’s density is its height. Then, as stated above, the ridge must be observed horizontally in the z1 z2 plane. As the steepest-ascent line in the profit space, a ridge was previously discussed [23]. In Chap. 8, we determined the cells that constitute the ridge using surface openness [24–26]. In this analysis, by setting L = 5, we estimate the surface openness for each cell and extract the cells of the openness that exceed 0.9, the value of which appropriately determines the ridge in this dataset. In Fig. 9.5, the cells are expressed by black dots. As expected, the series of densest cells observed as a ridge overlap the z1 axis. Fig. 9.4 Scatter plot of (z1 , z2 ) with Japanese firm data for 2014 projected on z1 z2 plane in z3 direction

Fig. 9.5 Data points in scatter plot of Fig. 9.4 are placed in cells separated at equal logarithmic intervals, and the amount of data in each cell is represented by shading. Cell’s center, judged to form a ridge by surface openness, is indicated by black circles

9 Why Does Production Function Take the Cobb–Douglas Form? 104

102

z2

128

100

102

104 108

106

104

102

100

102

z1 JAPAN 2014

A scatter plot in which 3D data are projected onto the z1 z3 plane in the z2 direction is similarly processed to obtain Fig. 9.6, which also shows that (as expected) the series of densest cells observed as a ridge overlap the z1 axis. A scatter plot in which 3D data are projected onto the z2 z3 plane in the z1 direction is similarly processed to obtain Fig. 9.7, which shows that the ridge, a series of the densest cells, is located from the upper right to the lower left. We measured the angle between this ridge and the z2 axis as ψ, assuming that a ridge is a straight line that regresses the center of cells with a surface openness greater than 0.9, which was appropriately determined to be a ridge cell. For example, for Japanese firms in 2014, ψ = 0.39 ± 0.04. In this case, using Eq. (9.48), we can estimate α  R = 0.16 ± 0.03 and β  R = 0.84 ± 0.16. On the other hand, with the same data and multiple regression analysis, we also estimate α  M = 0.16 ± 0.00 and β  M = 0.84 ± 0.00; both values are in good agreement. Figure 9.8 shows measurements taken of Japanese firms from 2014 to 2010 and firms in Germany, France, Spain, Italy, and the Netherlands during the 5-year period for which data were available (from 2009 to 2013). Among the measurements, those in the year with the largest number of firms in each country are plotted in Fig. 9.9. From Figs. 9.8 and 9.9, we confirmed that capital elasticity αR  , measured by the ridge specified by the surface openness, and capital elasticity αM  , measured by the multiple regression analysis used in economics, are in good agreement with the 5-year firm data of the seven countries.

9.7 Conclusion and Discussion

129

104

z3

102

100

102

104 108

106

102

104

100

102

z1 JAPAN 2014

Fig. 9.6 Japanese firms’ data for 2014 in database are projected on z1 z3 plane in z2 direction. (z1 , z3 ) data points are placed in logarithmically evenly spaced cells, and the amount of data is represented by shading. Cell’s center, judged to form a ridge by surface openness, is indicated by black circles

9.7 Conclusion and Discussion In this chapter, we showed that the Cobb–Douglas production function (9.1) or (9.33) can be directly observed using empirical data. If firms’ assets, labor, and production are expressed as a set of points in 3D space, the Cobb–Douglas production function can be interpreted as a quasi-inverse-symmetric plane in 3D space (9.36) and a residual R from the plane. From this viewpoint, we rotated the Cobb–Douglas quasi-inverse-symmetric plane in 3D space (9.36) to simplify the observations and projected the data in 3D space onto the 2D plane. With the geographic index of surface openness, we identified the densest axis in the plane as the ridge. Identifying this axis is difficult when the dispersion is large in the 2D plane. For example, by dividing the data on a 2D plane into equally spaced bins in a certain direction and calculating the average value or the logarithmic average value in the vertical direction for each bin, we can combine them and find the axis. However, since the average value or the logarithmic average value varies, depending on the selection method in a certain direction, the axis obtained by combining the average value or the logarithmic average value is not fixed to one axis. What is not affected by such arbitrariness is the advantage of the index of surface openness adopted

Fig. 9.7 Japanese firms’ data for 2014 in database are projected on z2 z3 plane in z1 direction. (z2 , z3 ) data points are placed in logarithmically evenly spaced cells, and the data amount is represented by shading. In addition, cell’s center, which is judged to form a ridge by surface openness, is indicated by black circles

9 Why Does Production Function Take the Cobb–Douglas Form? 104

102

z3

130

100

102

104

104

102

100 z2

102

104

JAPAN 2014 1.0

0.8

0.6 DR’

Fig. 9.8 Normalized capital elasticity (αM  ), calculated by multiple regression analysis for firms of each country in database of Japan, Italy, the United Kingdom, and the Netherlands from 2010 to 2014 and France, Germany, and Spain from 2009 to 2013, is compared with normalized capital elasticity (αR  ) calculated from ridge judged by surface openness

0.4

DE ES FR GB IT JP NL

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

DM’

in Chap. 8. By using it, we successfully and directly observed the Cobb–Douglas production function as a quasi-inverse-symmetric plane and a residual from the plane. The approach proposed by Hildenbrand [27] and developed by Dosi et al. [28] shares our arguments and perspectives on firms’ productivity, although it measures

9.7 Conclusion and Discussion Fig. 9.9 Comparison between αM  and αR  in Fig. 9.8 for Japan, Italy, the United Kingdom, and the Netherlands in 2014 and France, Germany, and Spain in 2013

131 1.0

0.8

DR’

0.6

0.4

DE ES FR GB IT JP NL

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

DM’

it with a very different approach. They theoretically showed that measuring productivity is possible without depending on the shape of the production function by the volume of a polyhedron composed of points in 3D space and the angle formed with the axis by the main diagonal. They confirmed their approach by empirical data. Their approach to measuring productivity is very innovative. The advantage of our study, on the other hand, is that it geometrically identifies the form of the production function that they avoided, making it possible to discuss specifically how production is related to assets and labors. Interestingly in our procedure, the capital and labor elasticities of the Cobb– Douglas production function can be expressed using the slope of a quasi-inversesymmetric plane in 3D space (9.36). By comparing the measured values of the capital elasticity observed in this way with those by the multiple regression analysis conventionally used in economics, we showed that the accuracy of these measured values agrees with the accuracy found using the data for 5 years from seven countries (including Japan) from which we obtained a sufficient quantity of data [22]. It is not obvious that the planes of multiple regression analysis and quasiinverse symmetry become identical. If all the data in the 3D space exist on a plane, they coincide. In practice, however, the data are widely dispersed. If they are spread vertically about the plane based on a log-normal distribution and are not concentrated on the symmetry plane, clearly identifying that the ridge is difficult even with quasi-inverse symmetry. This chapter clarified that such a ridge actually exists in empirical data using the index of surface openness in 2D distribution in which a 3D distribution is projected. This very interesting feature was only discovered by direct observation of the actual distribution of three different types of firm-size variables.

132

9 Why Does Production Function Take the Cobb–Douglas Form?

The fact that these observations are consistent in the seven countries with sufficient data means that a 3D space composed of capital, labor, and production has a ridge where the data are concentrated as a distinct entity. This idea can probably be applied to countries other than the seven surveyed in this chapter. If sufficient data exist, our results in Chap. 9 can be reproduced. As a result, we clarified from empirical data that the Cobb–Douglas production function, which has been discussed in economics, can be interpreted as the quasi-inverse symmetric plane of three variables and the residual from the plane. This study theoretically clarified why the Cobb–Douglas production function fits the empirical data better than a simple comparison of candidate production functions due to the goodness of the fit of the data. In the process, we also carried out the following new consideration on the constant returns to scale, which is a feature of the Cobb–Douglas production function. In economics, when the variables of the production function (K and L in this case) are multiplied by λ and production (Y ) is also multiplied by λ such as α + β = 1, the production function is called the constant returns to scale. The increasing returns if it exceeds λ times, such as α + β > 1, and the diminishing returns if it is less than λ times, such as α + β < 1. These three cases of returns to scale are easy to understand by considering the following simple examples. When the number of factories is multiplied by λ, the assets and labor of a firm are each multiplied by λ if the additional factories are the same size. A case where there is no change in the production efficiency and the production quantity becomes λ times is called a constant returns to scale; a case where the production efficiency improves and the production quantity becomes larger than λ times is called increasing returns; and a case where the production efficiency deteriorates and the production quantity becomes smaller than λ times is called diminishing returns. Economic arguments often assume constant returns to scale in the production function, and in fact, this is often observed approximately in various empirical data analyses (for example, see [29]). These observations indicate that on average, a constant return to scale is achieved in firm production. We showed by direct observation that the Cobb– Douglas production function can be understood as a quasi-inverse-symmetric plane in the space created by three variables (K, L, Y ) and the residual from the plane. We concluded that a constant return to scale is strictly satisfied with α  and β  normalized by the power-law indices. Since the values of power-law exponents μK , μL , and μY are all close to “1”, constant returns to scale were approximately observed even in the non-standardized capital share (capital elasticity) α and labor share (labor elasticity) β. In this chapter, we did not limit the data area in which the Cobb–Douglas production function was observed to the large-scale region where the power law holds. As our previous study showed [6–8, 22, 29], there is also quasi-inverse symmetry in the mid-scale region. The collapse of the power-law distribution observed in the large-scale region reflects that Gibrat’s law does not hold in the mid-scale region. This concept is independent of quasi-inverse symmetry. On the other hand, serious questions remain about the completeness of the data in the

9.7 Conclusion and Discussion

133

small scale. The current study did not exclude small data. If such a study could be implemented in a reliable way, the numerical consistency between our proposed method and multiple regression analysis might increase. This is future work. In our method, a firm’s assets (K), the number of employees (L), and production (Y ) are regarded as one point in a three-variable space (K, L, Y ), and we focus on how those several million groups are distributed. This approach to micro foundations attempts to derive the macroscopic properties of their distribution from the microscopic objects of individual points of view of the relationships among different types of firm-size variables rather than the distribution of individual variables. This critical issue has been studied in economics for many years. Finally, we describe the relationship between our discussion and the pioneering study by Aoki et al. who chose a physical approach to this problem [30–38]. Aoki and Yoshikawa hypothesized a multisystem in which the total number of employees and the total production amount of firms are fixed and found that randomness (entropy) is the greatest in a society in equilibrium. For such cases, they used physics methodologies to claim that the average number of employees in firms follows a Boltzmann distribution [30]. Aoyama et al. actually noticed a large fluctuation in the total amount of production and incorporated this effect by superimposing the Boltzmann distribution. They theoretically argued that the labor productivity of (Y/L) follows a power-law distribution in the large-scale range and the Boltzmann distribution with negative temperature in the middle- and low-scale ranges. This result was confirmed by empirical data [30–38]. These discussions can be described geometrically. First, consider the 2D plane of the logarithmic axes of labor productivity (Y/L) on the horizontal axis and the number of employees (L) on the vertical axis. The horizontal axis is divided into logarithmically equal-sized bins, and the vertical average of the data in the bins is plotted. The distribution followed by these points can be described by the superposition of the Boltzmann distribution. In this chapter, we showed that the total factor productivity (Y/K α Lβ ) is divided into logarithmically equal-sized bins and observed the distribution of the amount of data in the bins. We can determine (α, β, a) so that the distribution is symmetric with respect to constant a. Thus, our observations differ from those of Aoki et al. To unite the two, we need to extend their discussion to total factor productivity, including capital (K). If we can do this, we can discuss the distribution of the productivity of firms’ capital as well as labor and analyze firms in various industries in terms of both labor and capital. This is a major future challenge. In this book, we started our discussion based on the observation that the growthrate distribution of firm-size variables is tent-shaped, although why is a critically unsolved problem. It would be very interesting to derive the labor productivity distribution from the above statistical mechanics. This discussion can be extended to the distribution of capital productivity and total factor productivity. Furthermore, the growth-rate distribution of firm-size variables can probably be discussed in the same way. These considerations may provide a basis for discussing the micro foundations of economic phenomena in a statistical manner. Another important issue is addressing how the data resulting from the major changes in economic

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9 Why Does Production Function Take the Cobb–Douglas Form?

conditions will be incorporated into these discussions. If the time variation is quasistatic, I believe the discussions and methods in this book will improve the economic situation that may occur in the near future.

References 1. Cobb CW, Douglass PH (1928) A theory of production. Am Econ Rev 18:139–165 2. Solow RM (1956) A contribution to the theory of economic growth. Q J Econ 70:65–94 3. Arrow KJ, Chenery HB, Minhas BS, Solow RM (1961) Capital-labor substitution and economic efficiency. Rev Econ Stat 43:225–250 4. Christensen LR, Jorgenson DW, Lau LJ (1973) Transcendental logarithmic production function. Rev Econ Stat 55:28–45 5. Houthakker HS (1955) The Pareto distribution and the Cobb-Douglas production function in activity analysis. Rev Econ Stud 23:27–31 6. Mizuno T, Ishikawa A, Fujimoto S, Watanabe T (2012) Power laws in firm productivity. Prog Theor Phys Suppl 194:122–134 7. Ishikawa A, Fujimoto S, Watanabe T, Mizuno T (2013) The emergence of different tail exponents in the distributions of firm size variables. Physica A392:2104–2113 8. Ishikawa A, Fujimoto S, Mizuno T, Watanabe T (2014) Analytical derivation of power laws in firm size variables from Gibrat’s law and quasi-inversion symmetry: A geomorphological approach. J Phys Soc Jpn 83:034802 9. Gibra R (1932) Les Inégalités Économique. Sirey, Paris 10. Sutton J (1997) Gibrat’s legacy. J Econ Lit 35:40–59 11. Bureau van Dijk, http://www.bvdinfo.com/Home.aspx/ 12. Statistics Bureau, Ministry of Internal Affairs and Communications, https://www.stat.go.jp/ english/index.html 13. Pareto V (1897) Cours d’Économie Politique. Macmillan, London 14. Newman MEJ (2005) Power laws, Pareto distributions and Zipf’s law. Contemp Phys 46:323– 351 15. Clauset A, Shalizi CR, Newman MEJ (2009) Power-law distributions in empirical data. SIAM Rev 51:661–703 16. Fujiwara Y, Souma W, Aoyama H, Kaizoji T, Aoki M (2003) Growth and fluctuations of personal income. Physica A321:598–604 17. Fujiwara Y, Guilmi CD, Aoyama H, Gallegati M, Souma W (2004) Do Pareto-Zipf and Gibrat laws hold true? An analysis with European firms. Physica A335:197–216 18. Ishikawa A (2006) Annual change of Pareto index dynamically deduced from the law of detailed quasi-balance. Physica A371:525–535 19. Ishikawa A (2009) Quasistatically varying log-normal distribution in the middle scale region of Japanese land prices. Prog Theor Phys Suppl 179:103–113 20. Jessen AH, Mikosch T (2006) Regularly varying functions. Publ Inst Math 80:171–192 21. Gabaix X (2009) Power laws in economics and finance. Annu Rev Econ 1:255–294 22. Ishikawa A, Fujimoto S, Mizuno T (2019) Normalized constant returns of scale in CobbDouglas production function. J Kanazawa Gakuin Univ 17:152–163 (in Japanese) 23. Souma W (2007) Networks of firms and the ridge in the production space. In: Chatterjee A, Chakrabarti BK (eds) Econophysics of markets and business networks. Springer, Italia, pp 149–158 24. Yokoyama R, Sirasawa M, Kikuchi Y (1999) Representation of topographical features by opennesses. J Jpn Soc Photogramm Remote Sens 38(4):26–34 (in Japanese with English abstract)

References

135

25. Yokoyama R, Sirasawa M, Pike R (2002) Visualizing topography by openness: a new application of image processing to digital elevation models. Photogramm Eng Remote Sens 68(3):257–265 26. Prima ODA, Echigo A, Yokoyama R, Yoshida T (2006) Supervised landform classification of Northeast Honshu from DEM-derived thematic maps. Geomorphology 78:373–386 27. Hildenbrand W (1981) Short-run production functions based on microdata. Econometrica 49(5):1095–1125 28. Dosi G, Grazzi M, Marrengo L, Settepanella S (2016) Production theory: accounting for firm heterogeneity and technical change. J Ind Econ 64(4):875–907 29. Watanabe T, Mizuno T, Ishikawa A, Fujimoto S (2011) Shape of production function of mega firms: analytical methods and applications. Econ Rev (Keizai Kenkyuu) 62:193 (in Japanese) 30. Aoki M, Yoshikawa H (2007) Reconstructing macroeconomics - a perspective from statistical physics and combinatorial stochastic processes. Cambridge University Press, Cambridge 31. Aoyama H, Yoshikawa H, Iyetomi H, Fujiwara Y (2009) Labour productivity superstatistics. Prog Theor Phys Suppl 179:80–92 32. Aoyama H, Fujiwara Y, Ikeda Y, Iyetomi H, Souma W (2009) Superstatistics of labour productivity in manufacturing and nonmanufacturing sectors. Economics Open-Access OpenAssess E-J 3:2009–2022 33. Souma W, Ikeda Y, Iyetomi H, Fujiwara Y (2009) Distribution of labour productivity in Japan over the period 1996–2006. Economics Open-Access Open-Assess E-J 3:2009–2014 34. Aoyama H, Yoshikawa H, Iyetomi H, Fujiwara Y (2010) Productivity dispersion: facts, theory, and implications. J Econ Interact Coord 5:27–54 35. Iyetomi H (2012) Labor productivity distributions with negative temperature. Prog Theor Phys. Suppl 194:135–143 36. Aoyama H, Iyetomi H, Yoshikawa H (2015) Equilibrium distribution of labor productivity. J Econ Interact Coord 10:57–66 37. Aoyama H, Fujiwara Y, Ikeda Y, Iyetomi H, Souma W, Yoshikawa H (2017) Macroeconophysics: new studies on economic networks and synchronization (physics of society: econophysics and sociophysics). Cambridge University Press, Cambridge 38. Yoshikawa H (2020) Reconstructing macroeconomics - Keynes and Schumpeter. Iwanami Shoten, Tokyo (in Japanese)

Index

A A certain year, 7, 10, 19, 39, 52, 83 Activity statuses, 84–86 Added value, 116 Additive noise, 8, 64 Aoki, M., 2, 133 Aoyama, H., 2, 3, 8, 10, 19, 20, 23, 35, 39, 133 Assets, 1, 3–5, 7, 10, 23, 24, 36, 39–42, 47–53, 56–62, 66–67, 71, 84–92, 95–102, 106–109, 113, 115–118, 129, 131–133 Azimuth, 103, 105 B Bankruptcy, 73, 74, 84, 85 Boltzmann distribution, 133 Bubble burst, 27, 67 Bubble economy, 24, 27 Bubble period, 27, 33, 34 Bureau van Dijk, 9, 40, 58, 66, 73, 84, 96, 115 C Calendar year, 12, 29, 56, 64, 66 Capital, 113–115, 123, 124, 126, 128, 130–133 Capital elasticity, 114, 115, 124, 128, 130–132 Capital share, 114, 115, 124, 126, 132 CD Eyes, 9, 11, 20, 40, 41 Cobb-Douglas production function, 5, 114, 115, 122, 124, 129, 131, 132 Cobb-Douglas quasi-inverse-symmetric plane, 125–129 Cobb-Douglas symmetric plane, 114, 115, 124–126

Compustat database, 72 Conditional growth-rate distribution, 3, 14, 16–18, 32, 40, 47, 49, 50, 52, 84 Conditional PDF, 13–15, 18, 30, 43, 45–47, 50, 56, 57, 62, 63, 90, 119 Constant Elasticity of Substitution (CES) type production function, 114 Constant returns to scale, 114, 115, 122–124, 132 CPI, 2 Critical phenomena in magnetic materials and fluids, 1 Current profits, 3, 9–11, 13, 16–20, 24, 36, 39, 40, 43, 45, 52, 56, 57, 95

D Decay rate, viii, 67, 72 Decision-making mechanism, 2 Densest directional vector, 125, 127 Detailed balance, 8, 12, 29 Diminishing returns, 132 Dosi, 130 Downward convex curvature, 13, 18

E Economics, vii, 1, 2, 5, 8, 19, 23, 55, 67, 71, 92, 109, 113–116, 123, 124, 128, 131–134 agents, 2 growth, 55 phenomena, 1, 5, 133 science, 1

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Ishikawa, Statistical Properties in Firms’ Large-scale Data, Evolutionary Economics and Social Complexity Science 26, https://doi.org/10.1007/978-981-16-2297-7

137

138 Electromagnetism, 1 Entropy, 133 Equilibrium, 2, 3, 8, 12, 14, 15, 19, 20, 23, 29, 35, 39, 133 Equilibrium state, 2, 12, 14 Explanatory variable, 103 Exponential growth, 4, 57–59, 61, 62, 64, 66, 67, 72 Extended non-Gibrat’s property, 36, 40, 43, 57, 58, 62, 64–67 F Financial engineering, 8, 19, 23 Firm activities, viii, 55, 67, 72–77, 79, 84, 85 Firm age, viii, 4, 57–61, 64–67, 71–79, 84 Firm-age distribution, viii, 4, 67, 71–79, 84 Firm-size variables, viii, 3–5, 7–14, 16, 19, 23, 24, 28, 35, 36, 39–41, 43, 45, 52, 56–58, 62, 64, 66, 67, 71, 83, 84, 90–92, 95–109, 113, 131, 133 Flow data, 53, 92 Foundation year, 58 Fujiwara, Y., 117 G GDP, 2, 55 Geometric mean, 4, 58–61, 64–66 Geomorphology, 4, 103 Gibrat’s law, vii, viii, 3, 4, 8, 10, 12, 14–16, 19, 20, 23, 24, 30, 33, 35, 39, 40, 43, 44, 52, 55–67, 72, 83, 84, 88, 89, 91, 92, 99, 115, 117–120, 122, 124, 132 Gibrat’s process, 8

Index L Labor, 113–116, 123, 124, 126, 129, 131–133 elasticity, 114, 115, 124, 126, 131, 132 productivity, 133 share, 114, 115, 123, 126 Land prices, viii, 3, 9, 20, 23–29, 33–36, 95, 99, 102, 118 Leontief production function, 114 Logarithmic growth rate, 13, 16, 18, 40, 45, 47, 50, 52, 57, 62, 63, 90 Logarithmic standard deviation, 23, 24, 26, 27, 33, 35, 122 Logarithmic variance, 43, 52, 56, 67, 86 Log-normal distribution, 3, 8–12, 16, 18–20, 23–36, 39, 40, 42–44, 47, 49–52, 56, 57, 67, 83, 86, 91, 100, 131 Long-term growth, 4, 53, 55, 57–62, 71 Long-term property, 67, 79

M Macroeconomics, 2, 7, 55, 114 Macroscopic nature, 5 Macroscopic object, 2, 7 Micro foundations, 133 Microscopic multi-body systems, 2 Microscopic structure, 1, 10 Microstructure, 1 Multiple regression analysis, 115, 123, 128, 130, 131, 133 Multiplicative noise, 64–66 Multiplicative stochastic process, 8

J Japan’s Ministry of Land, Infrastructure and Transport, 9 Joint PDF, 12, 28–30, 33, 43, 52, 56, 96, 101, 107, 117

N Natural science, 1, 2, 83 Negative openness, 103 Net assets, 84–92, 95 Net income, 84–92, 95 Net sales, 40, 96 Newtonian mechanics, 1 Non-equilibrium state, 2 Non-Gibrat’s property, 3, 4, 7–20, 24, 31–33, 35, 36, 39–53, 55–67, 72, 84, 88–92 Nuclear decay, 72 Number of employees, 1, 4, 5, 7, 10, 39, 41, 42, 56–58, 60–62, 66, 67, 71, 83, 95–102, 107, 108, 113, 115–118, 133 Numerical simulations, 57, 66

K Kolmogorov-Smirnov test, 45

O Objective variable, 103, 104

H Hildenbrand, 130 Houthakker, 114 I Inactive rate, 4, 67, 71–80, 83–92 Increasing returns, 132

Index Operating revenues, 3–5, 7, 10, 23, 36, 39–53, 56–67, 71, 83, 90, 95–102, 106–109, 116–118 Orbis, 9, 10, 20, 40–44, 58, 66, 73, 74, 84–86, 91, 96–99, 115, 116

P Pareto, 7, 11, 16, 26, 27, 31, 43, 44, 56, 86, 96, 99–102, 106–108, 113, 117 Pareto’s index, 56, 86, 117 Personal income, 7, 8, 11 Physics, 1, 2, 5, 55, 71, 133 Polyhedron, 131 Positive current profits, 3, 9–11, 13, 19, 20, 24, 39, 40, 43, 45, 57 Positive openness, 103 Power index, 24, 26–28, 33–35, 96, 99, 100, 106, 108, 115 Power-law, 3, 7, 23, 39, 56, 72, 83, 96, 113 Power-law distribution, 3, 4, 8, 10, 18, 19, 23–25, 27, 35, 39–42, 44, 56, 89, 91, 96, 99, 100, 102, 106, 108, 109, 113, 114, 116–118, 122, 124, 132, 133 Power-law growth, 58, 60, 62, 64–67 Production factors, 113 Production function, 5, 109, 113–134 Profits, 1, 3, 7, 9–11, 13, 16–20, 24, 36, 39, 40, 43, 45, 52, 56, 57, 83, 95, 103, 127 Public Notice of Land Prices Act, 24

Q Quantum mechanics, 1 Quasi-inverse-symmetric axis, 106, 107, 118, 125 Quasi-inverse-symmetric plane, 114, 123–130, 132 Quasi-inversion symmetric plane, 5, 116 Quasi-inversion symmetry, 4, 96, 99–103, 106–109, 113, 115, 122 Quasi-static, 3, 8, 20, 24, 28, 29, 31, 32, 34, 35, 99, 106 Quasi-statically varying, 20, 23, 24, 30–33, 96, 118 Quasi-time-reversal symmetry, 3, 4, 8, 20, 23, 24, 28–36, 96, 102, 117, 118

R Reflection laws, 16 Reflection wall, 8

139 Regression analysis, 11, 18, 19, 27, 33, 36, 43, 44, 50–52, 102, 103, 115, 123, 128, 130, 131, 133 Regression lines, 25, 33, 35, 103 Relativity, 1 Reset event, 8 Residual, 114, 129, 130, 132 Ridge, 103–107, 115, 127–132 Romer, P.M., 55

S Short-term growth, 12, 62–66 Short-term properties, 4, 14–16, 53, 64, 67 Sociology, 8, 19, 23 Solow, R.M., 55 Standard deviations, 13, 15, 23, 26–28, 33, 35, 45, 47, 77, 119, 122 Statistical laws, 56, 67, 83 Statistical mechanics, 1, 2, 133 Status of firms, 85, 95 Stochastic equation, 66 Stochastic model, 58, 66 Stochastic process, 8 Stock data, 53, 92 Superposition, 133 Surface openness, 4, 5, 96, 103–107, 114, 127–131

T Tangible fixed assets, 4, 5, 96–102, 106–109, 113, 115–118 The kinetic theory of gas, 1 The motion of rigid bodies, 1 Thermodynamics, 1–3, 5, 12, 29 The Small and Medium Enterprise Agency of Japan, 79 The unemployment rate, 2 Three-dimensional Gibrat’s law, 122 Three-dimensional quasi-inversion symmetry, 109, 122 Three-variable space, 5, 125, 126, 133 Time-reversal symmetry, 3, 4, 8–10, 12–16, 18–20, 23, 24, 28–36, 39, 40, 43–45, 47, 48, 50, 52, 56, 57, 67, 83, 96, 99, 100, 102, 117 Time-varying system, 3 Tokyo Shoko Research, 9, 41 Topographic map, 104 Total assets, 3, 36, 40–42, 47–53, 56–62, 66, 67, 84–92, 95

140 Total factor productivity, 114, 122, 133 Total revenues, 84–92, 95 Total sales, 115 Trajectory, 58 Translog-type production function, 114 U Universality, 55 Universal structure, 1, 13

Index W World War II, 74, 77, 79 Y Yoshikawa, H., 2, 133

Z Zenith and nadir angles, 103