Evolutionary and Institutional Economics Review

, Volume 14, Issue 2, pp 351–362 | Cite as

Dependence of the decay rate of firm activities on firm age

  • Atushi IshikawaEmail author
  • Shouji Fujimoto
  • Takayuki Mizuno
  • Tsutomu Watanabe


Using the ORBIS database that contains around 150 million firm-size data from all over the world, we investigated firm activity data in Germany, Spain, France, the United Kingdom, Italy, Japan, Korea, and the Netherlands and found that the decay rate of firm activities does not depend on firm age in Spain as in Japan. But in Germany, France, the United Kingdom, Italy, Korea, and the Netherlands, the decay rate of young firms is high and becomes lower and settles at a constant value as firms age. By approximating the decay rate of firm activities by an exponential function, we analytically derived the firm age distribution under the assumptions that the number of firms that is established annually is nearly constant and that the decay rate of firm activities does not change annually. Using empirical data from eight countries, we compared the parameters estimated by the decay rate of the firm activities with those by firm age distribution. Except in Spain and Germany, the two kinds of parameters estimated in two different ways were close to each other.


Econophysics Firm age distribution Decay rate of firm activity Dr. Jun-ichi Inoue 

JEL Classification




This study was supported by JSPS KAKENHI Grants 24510212 and 24710156.

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this article.


  1. Aoyama H, Fujiwara Y, Ikeda Y, Iyetomi H, Souma W (2011) Econophysics and companies: statistical life and death in complex business networks. Cambridge University Press, CambridgeGoogle Scholar
  2. Axtell RL (2001) Zipf distribution of US firm sizes. Science 293:1818CrossRefGoogle Scholar
  3. Bonabeau E, Dagorn L (1995) Possible universality in the size distribution of fish schools. Phys Rev E 51:R5220CrossRefGoogle Scholar
  4. Bottazzi G, Secchi A, Tamagni F (2008) Productivity, profitability and financial performance. Ind Corp Change 17:711CrossRefGoogle Scholar
  5. Clauset A, Shalizi CR, Newman MEJ (2009) Power-law distributions in empirical data. SIAM Rev 51:661CrossRefGoogle Scholar
  6. Coad A (2010a) The exponential age distribution and the Pareto firm size distribution. J Ind Compet Trade 10:389CrossRefGoogle Scholar
  7. Coad A (2010) Investigating the exponential age distribution of firms. Economics 4:2010–17Google Scholar
  8. Coad A, Tamvada JP (2008) The growth and decline of small firms in developing countries. Papers on economics and evolution 2008-08 (2008) Max Planck Institute of EconomicsGoogle Scholar
  9. Daepp MIG, Hamilton MJ, West GB, Bettencourt LMA (2015) The mortality of companies. J R Soc Interface 12:20150120CrossRefGoogle Scholar
  10. 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 102:18801CrossRefGoogle Scholar
  11. Fujiwara Y (2004) Zipf law in firms bankruptcy. Phys A 337:219CrossRefGoogle Scholar
  12. Fujiwara Y, Souma W, Aoyama H, Kaizoji T, Aoki M (2003) Growth and fluctuations of personal income. Phys A 321:598CrossRefGoogle Scholar
  13. Fujiwara Y, Guilmi CD, Aoyama H, Gallegati M, Souma W (2004) Do Pareto-Zipf and Gibrat laws hold true? An analysis with European firms. Phys A 335:197CrossRefGoogle Scholar
  14. Gaffeo E, Gallegati M, Palestrinib A (2003) On the size distribution of firms: additional evidence from the G7 countries. Phys A 324:117CrossRefGoogle Scholar
  15. Gibrat R (1932) Les Inégalités Économiques. Sirey, ParisGoogle Scholar
  16. Ishikawa A (2006) Derivation of the distribution from extended Gibrat’s law. Phys A 367:425CrossRefGoogle Scholar
  17. Ishikawa A (2006) Annual change of Pareto index dynamically deduced from the law of detailed quasi-balance. Phys A 371:525CrossRefGoogle Scholar
  18. Ishikawa A (2007) The uniqueness of firm size distribution function from tent-shaped growth rate distribution. Phys A 383:79CrossRefGoogle Scholar
  19. Ishikawa A (2009) Quasistatically varying log-normal distribution in the middle scale region of Japanese land prices. Prog Theor Phys Suppl 179:103CrossRefGoogle Scholar
  20. Ishikawa A, Fujimoto S, Mizuno T (2011) Shape of growth rate distribution determines the type of non-Gibrat’s property. Phys A 390:4273CrossRefGoogle Scholar
  21. Ishikawa A, Fujimoto S, Mizuno T and Watanabe T (2015) Firm age distributions and the decay rate of firm activities. In: Proceedings of the International Conference on Social Modeling and Simulation, plus Econophysics Colloquium 2014 (Springer Proceedings in Complexity, 2015), pp 187–194Google Scholar
  22. Ishikawa A, Fujimoto S, Mizuno T and Watanabe T (2015) The relation between firm age distributions and the decay rate of firm activities in the United States and Japan. In: Proceedings of Workshop in 2015 IEEE International Conference on Big Data (2015), pp 2726–2731Google Scholar
  23. Kaizoji T (2004) Scaling behavior in land markets. Phys A 326:256CrossRefGoogle Scholar
  24. Levy M, Solomon S (1996) Power laws are logarithmic Boltzmann laws. Int J Mod Phys C 7:595CrossRefGoogle Scholar
  25. Mantegna RN, Stanley HE (1995) Scaling Behavior in the dynamics of an economic index. Nature 376:46CrossRefGoogle Scholar
  26. Mantegna N, Stanley HE (2007) Introduction to econophysics: correlations and complexity in finance. Cambridge University Press, CambridgeGoogle Scholar
  27. Miura W, Takayasu H, Takayasu M (2012) Effect of coagulation of nodes in an evolving complex network. Phys Rev Lett 108:168701CrossRefGoogle Scholar
  28. Mizuno T, Katori M, Takayasu H, Takayasu M (2002) Statistical laws in the income of Japanese companies. In: Takayasu H (ed) Empirical science of financial fluctuations. Springer, Tokyo, pp 321–330CrossRefGoogle Scholar
  29. Newman MEJ (2005) Power-law, Pareto distributions and Zipf’s law. Contemp Phys 46:323CrossRefGoogle Scholar
  30. Okuyama K, Takayasu M, Takayasu H (1999) Zipf’s law in income distribution of companies. Phys A 269:125CrossRefGoogle Scholar
  31. Pareto V (1897) Cours d’Économie Politique. Macmillan, LondonGoogle Scholar
  32. Podobnik B, Horvatic D, Pammolli F, Wang F, Stanley HE, Grosse I (2008) Size-dependent standard deviation for growth rates: empirical results and theoretical modeling. Phys Rev E 77:056102CrossRefGoogle Scholar
  33. Podobnik B, Horvatic D, Petersen AM, Urošević B (2010) Bankruptcy risk model and empirical tests. Proc Natl Acad Sci 107:18325CrossRefGoogle Scholar
  34. Ramsden JJ, Kiss-Haypál G (2000) Company size distribution in different countries. Phys A 277:220CrossRefGoogle Scholar
  35. Render S (1998) How popular is your paper? An empirical study of the citation distribution. Eur Phys J B 4:131CrossRefGoogle Scholar
  36. Rutherford E, Soddy F (1903) Radioactive change. Philos Mag 6:576CrossRefGoogle Scholar
  37. Saichev A, Malevergne Y, Sornette D (2009) Theory of Zipf’s law and beyond. Springer, BerlinGoogle Scholar
  38. Sornette D, Cont R (1997) Convergent multiplicative processes repelled from zero: power laws and truncated power laws. J Phys I(7):431Google Scholar
  39. Sutton J (1997) Gibrat’s legency. J Econ Lit 35:40Google Scholar
  40. Takayasu M, Takayasu H, Sato T (1996) Critical behavior and \(1/f\) noise in information traffic. Phys A 233:824CrossRefGoogle Scholar
  41. Takayasu H, Sato A, Takayasu M (1997) Stable infinite variance fluctuations in randomly amplified Langevin system. Phys Rev Lett 79:966CrossRefGoogle Scholar
  42. Tomoyose M, Fujimoto S, Ishikawa A (2009) Non-Gibrat’s law in the middle scale region. Prog Theor Phys Suppl 179:114CrossRefGoogle Scholar
  43. Yamano T (2004) Distribution of the Japanese posted land price and the generalized entropy. Eur Phys J B 38:665CrossRefGoogle Scholar
  44. Zhang J, Chen Q, Wang Y (2009) Zipf distribution in top Chinese firms and an economic explanation. Phys A 388:2020CrossRefGoogle Scholar

Copyright information

© Japan Association for Evolutionary Economics 2017

Authors and Affiliations

  1. 1.Kanazawa Gakuin UniversityKanazawaJapan
  2. 2.National Institute of InformaticsTokyoJapan
  3. 3.The Graduate University for Advanced Studies [SOKENDAI]TokyoJapan
  4. 4.JST PRESTOTokyoJapan
  5. 5.University of TokyoTokyoJapan

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