1. Firm Size Mobility and Validity of the Gibrat Model Yougui Wang Department of Systems Science School of Management, Beijing Normal University A talk @ the 4 th China-Europe Summer School on Complexity Science, Shanghai, August 14, 2010

2. Acknowledgments The works referred in this talk were carried out under collaborations with Jinzhong Guo, Beishan Xu, Jianhua Zhang and Qinghua Chen. Thanks to all organizers of this summer school. The invitation from Professor Yi-Cheng Zhang and Professor Binghong Wang is appreciated. I am grateful to Professor Shlomo Havlin and Paul Ormerod for his valuable suggestions and comments.

3. Outline of This Talk Background and Motivations Firm size distribution Firm Size Mobility Rank clock of firm size The validity of the Gibrat Model Conclusions

4. Firm Size distribution Firm size distribution has long been a topic in economics. Robert Gibrat carried out the pioneer work in this issue. In recent years, many works have been done on the firm size distribution such as Simon 1977,B.H.Hall 1987, Michael H. R. Stanley 1995, Robert L. Axtell 2001,J,Zhang 2009.

5. M.H.Stanley. Zipf plots and the size distribution of firms, Economics Letters 49 (1995) 453-457 Robert L. Axtell. Zipf Distribution of U.S. Firm Sizes, 1818 (2001); 293 Science

6. Firm Size Distribution in China J. Zhang, Q. Chen, Y. Wang, “Zipf distribution in top Chinese firms and an economic explanation”, Physica A, vol. 388, pp. 2020-2024. 2009.

7. Firm Size Distribution in USA

8. Firm Size Distribution in World

9. Changes under Stable Distributions Time

10. Changes under Stable Distributions Firm Size Distribution in China

11. Individual Evolution Variations in rank indicate it seems move randomly over time.

12. All Individuals’ Evolutions Aggregate variation in rank can not be simply figure out by putting all individuals’ evolutions together.

13. G. S. Fields and E. A. Ok, “The meaning and measurement of income mobility”, Journal of Economic Theory, vol. 71, pp. 349–377. 1996. Mobility of Firm Size where, N is the number of firms, x k0 and x k1 are the sizes of firm i at time 0 and 1 respectively.

14. Absolute Mobility Absolute mobility covers changes of all firm sizes and is somewhat independent of scale.

15. Relative Mobility The ranks of individuals can be derived from attributes of them.

16. Zipf Plot

17. Shlomo Havlin, “The distance between Zipf plots”, Physica A, vol. 216, pp. 148-150 ， 1995. Relative Mobility where, N is the number of firms, r k0 and r k1 are the rank of firm i at time 0 and 1 respectively.

18. Rank Clock of Top 100 Cities in USA Michael Batty, “Rank clocks”, Nature, vol. 444, pp. 592–596, 2006.

19. Seven firms ranked in top 500 of the world during the thirteen years Rank Clocks of Firms

20. Many firms ranked in 500 firms in USA Eight firms ranked in top 500 in USA Rank Clocks of Firms

21. Rank-shift Clocks of Firm Size Rank Shift Clocks of Firms

22. The Measurements of Mobility ChinaU.SGlobal revenue (million ￥ ) rankshare revenue (million $) rankshare revenue (million $) rankshare 19961105.67824.839480.000183021.6730.904980.0003 19971356.17326.288790.000232240.68730.583890.0005 19981708.31827.423840.000283106.87134.739730.0005 19991889.53725.740490.000253344.64534.248860.0002 20002342.27427.51810.000284084.4534.364440.0003 20012112.44825.672530.00034206.82630.723360.0003 200229247934.768230.00032264.78729.81360.000354031.17632.991150.0003 2003403837.330.751940.00031662.14618.867370.000195115.70129.317290.0003 2004620305.236.425640.00042085.19319.149470.000174585.30824.889870.0002 2005591840.631.951340.00022383.52220.732330.00025109.58528.687220.0002 2006720703.826.634620.00022171.82417.627450.000164958.13724.126910.0002 2007101225126.183570.00022217.62819.350650.000166428.09825.962390.0002 2008106494430.093530.00023219.16525.574840.00037953.17534.296380.0003

23. The Law of Proportional Effect The rate of growth of size X between period (t-1) and period t is a random variable, so that Then after a series of periods, the evolution of size can be integrated as Taking log and making some approximations, we thus obtain The distribution can be approximated by a normal distribution.

24. The Gibrat Model The change in size for firm i follows the proportional effect plus a steady growth, so that There is a minimum value of firm size and the firms whose size is less than it will be replaced by the current average one.

25. Simulation Results on the Evolution of US Firm Size Distribution

26. Validation of the Gibrat Model

27. Validity of Gibrat’s model Validation of the Gibrat Model

28. Growth Rate of Firm Size

29. Comparisons Between Simulations and Measurements component USChinaGlobal RealSimulateErrorRealSimulateErrorRealSimulateError 20.616 21.50701.2612 31.776 9 31.794 3 3.8860 31.197 7 26.739 7 4.8423 1.0741 1.11660.0422 1.13751.29070.04231.06091.06270.0802 1.0889 1.09440.0297 1.23201.24960.04341.06501.10940.0206 0.9953 1.02010.0177 0.92401.03310.01440.99771.04330.0611 0.0465 0.05160.0153 0.10200.11910.01450.05000.04710.0195 0.0350 0.03900.0118 0.09020.09650.01530.02680.02630.0084 0.0114 0.01260.0063 0.01180.02260.00740.02330.02070.0138

30. Conclusions The firm size distribution has a unified pattern over time and regions. Behind the stable Zipf distributions, the firm size mobility exhibit various characteristics. Gibrat model can reproduce both the distribution and mobility of firm size.

31. Many Thanks!