Firm Size Mobility and Validity of
the Gibrat Model
Yougui Wang
Department of Systems Science
School of Management, Beijing Normal University
A talk @ the 4th China-Europe Summer School on
Complexity Science, Shanghai, August 14, 2010
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.
Outline of This Talk
•
•
•
•
•
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Background and Motivations
Firm size distribution
Firm Size Mobility
Rank clock of firm size
The validity of the Gibrat Model
Conclusions
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.
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
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.
Firm Size Distribution in USA
Firm Size Distribution in World
Changes under Stable Distributions
Firms :{1, 2,
Sizes :{x1 , x2 ,
, k,
N}
, xk ,
Time
 x10 , x20 ,  , xk 0 ,  x N 0 
 x , x ,, x , x 
11
21
k1
N1 



 x12 , x22 ,  , xk 2 ,  x N 2 





 x1t , x2t ,  , xkt ,  x Nt 

xN }
Changes under Stable Distributions
 x10 , x20 ,  , xk 0 ,  x N 0

 x , x ,, x , x

k1
N1
 11 21



 x , x ,, x , x

kt
Nt
 1t 2t

 x1t 1 , x2t 1 ,  , xkt 1 , x Nt 1 
Firm Size Distribution in China
Individual Evolution
 x10 , x20 ,  , xk 0 ,  x N 0 
 x , x ,, x , x 
11
21
k1
N1 



 x12 , x22 ,  , xk 2 ,  x N 2 





 x1t , x2t ,  , xkt ,  x Nt 

Variations in rank indicate it seems
move randomly over time .
All Individuals’ Evolutions
 x10 , x20 ,  , xk 0 ,  x N 0 
 x , x ,, x , x 
11
21
k1
N1 



 x12 , x22 ,  , xk 2 ,  x N 2 





 x1t , x2t ,  , xkt ,  x Nt 

Aggregate variation in rank can not
be simply figure out by putting all
individuals’ evolutions together.
Mobility of Firm Size
N
1
M
N
| x
1
M
N
N
k 1
k0
 xk 1 |
 | log x
k 1
k0
 log xk1 |
where, N is the number of firms, xk0 and xk1 are the sizes of firm i at
time 0 and 1 respectively.
G. S. Fields and E. A. Ok, “The meaning and
measurement of income mobility”, Journal of Economic
Theory, vol. 71, pp. 349–377. 1996.
Absolute Mobility
 x10 , x20 ,  , xk 0 ,  x N 0 
1 N
 x , x ,  , x ,  x  M  N  | xk 0  xk 1 |
k 1
11
21
k1
N1 



 x12 , x22 ,  , xk 2 ,  x N 2  Absolute mobility covers

 changes of all firm sizes

 and is somewhat

 x1t , x2t ,  , xkt ,  x Nt 
 independent of scale.
Relative Mobility
 x10 , x20 ,  , xk 0 ,  x N 0 
 x , x ,, x , x 
k1
N1 
 11 21

 x12 , x22 ,  , xk 2 ,  x N 2 




 x1t , x2t ,  , xkt ,  x Nt 
r10 , r20 ,  , rk 0 ,  rN 0 
r , r ,  , r ,  r 
k1
N1 
 11 21

r12 , r22 ,  , rk 2 ,  rN 2 
 


r1t , r2t ,  , rkt ,  rNt 
The ranks of individuals can be derived from attributes of
them.
Zipf Plot
 x1t , x2t ,, xkt , xNt 


r1t , r2t ,, rkt , rNt 
Relative Mobility
1
2 2
N
1
D  [  (rk 0 ( x)  rk1 ( x)) ]
N  1
1
D
N
N
r
k 1
k0
( x)  rk1 ( x)
where, N is the number of firms, rk0 and rk1 are the rank of firm i at
time 0 and 1 respectively.
Shlomo Havlin, “The distance between Zipf plots”,
Physica A, vol. 216, pp. 148-150，1995.
Rank Clock of Top 100 Cities in USA
Michael Batty, “Rank clocks”, Nature, vol. 444,
pp. 592–596, 2006.
Rank Clocks of Firms
1996/2008
2007
1997
CIGNA
DEUTSCHE
POST
FORTIS
2006
1998
FUJITSU
HOME
DEPOT
INDIAN OIL
100
200
300
400
2005
500
1999
2004
2000
2003
2001
2002
Seven firms ranked in top 500 of the
world during the thirteen years
WAL-MART
STORES
Rank Clocks of Firms
2008/1955
2004
2008/1955
1959
2004
1959
2000
1964
100
1995
200
300
400
2000
1964
500
100
1968
1991
1973
300
400
1995
1991
1973
1977
1977
1982
Many firms ranked in
500 firms in USA
500
1968
1986
1986
200
1982
Eight firms ranked
in top 500 in USA
Pitney
Bowes
Abbott
Laboratories
3M
Alcoa
Altria
Group
American
Standard
Anheuser
Busch
Archer
Daniels
Midland
Rank-shift
Clocks
of
Firm
Size
Rank Shift Clocks of Firms
Mobility of 3 real system
2008/1955
1959
2004
1964
1999
China
Global
150
100
50
1967
1995
U.S.
1971
1990
1975
1986
1982
The Measurements of Mobility
China
revenue
(million
￥)
U.S
revenue
(million
$)
rank
1996
1105.678
1997
Global
share
revenue
(million
$)
rank
share
24.83948
0.00018
3021.67
30.90498
0.0003
1356.173
26.28879
0.00023
2240.687
30.58389
0.0005
1998
1708.318
27.42384
0.00028
3106.871
34.73973
0.0005
1999
1889.537
25.74049
0.00025
3344.645
34.24886
0.0002
2000
2342.274
27.5181
0.00028
4084.45
34.36444
0.0003
2001
2112.448
25.67253
0.0003
4206.826
30.72336
0.0003
rank
share
2002
292479
34.76823
0.0003
2264.787
29.8136
0.00035
4031.176
32.99115
0.0003
2003
403837.3
30.75194
0.0003
1662.146
18.86737
0.00019
5115.701
29.31729
0.0003
2004
620305.2
36.42564
0.0004
2085.193
19.14947
0.00017
4585.308
24.88987
0.0002
2005
591840.6
31.95134
0.0002
2383.522
20.73233
0.0002
5109.585
28.68722
0.0002
2006
720703.8
26.63462
0.0002
2171.824
17.62745
0.00016
4958.137
24.12691
0.0002
2007
1012251
26.18357
0.0002
2217.628
19.35065
0.00016
6428.098
25.96239
0.0002
2008
1064944
30.09353
0.0002
3219.165
25.57484
0.0003
7953.175
34.29638
0.0003
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
xt  xt 1
 t
xt 1
• Then after a series of periods, the evolution of size can be
integrated as
xt  (1   t ) xt 1  x0 (1  1 )(1   2 )
(1   t )
• Taking log and making some approximations, we thus obtain
l o g xt  log x0  1   2
 t
• The distribution can be approximated by a normal distribution.
The Gibrat Model
• The change in size for firm i follows the
proportional effect plus a steady growth, so
that
Pi (t )  [   i (t )]Pi 1 (t )
• 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.
Pi (t )  Pmin (t )  P(t ) / n
Simulation Results on the Evolution
of US Firm Size Distribution
8
10
7
10
6
Revenue
10
5
10
4
10
3
10
2
10
1
10
0
10
1
10
2
Rank
10
3
10
Validation of the Gibrat Model
i (t )  Ri (t ) / Ri 1 (t )
Ri (t )  i (t ) Ri 1 (t )
pi (t )  Ri (t ) / R(t )
R(t )   i Ri (t )
 (t )   pi (t )i (t )  
i
i
Ri (t )
pi (t )
Ri 1 (t )
pi (t )
R (t )
{
}{ pi (t )
}
R (t  1)
pi (t  1)
i
R(t )
(t ) 
R(t  1)
pi (t )
 (t )   pi (t )
pi (t  1)
i
Validity
of Gibrat’s
model Model
Validation
of the Gibrat
log i (t )  log[ Ri (t ) / Ri 1 (t )]
Ri (t )
I [ (t )]   pi (t ) logi (t )   pi (t ) log
Ri 1 (t )
i
i
pi (t )
R(t )
 log
  pi (t ) log
R(t  1) i
pi (t  1)
R(t )
I [(t )]  log
R(t  1)
pi (t )
I [ (t )]   pi (t )log
pi (t  1)
i
Growth Rate of Firm Size
growth rate of 3 system
2008/1955
1960
2004
China
1964
2000
Global
0.5
1
1.5
2
1969
1995
U.S.
1973
1991
1978
1987
1982
Comparisons Between Simulations
and Measurements
US
China
Global
component
Real
Simulate
Error
Real
Simulate
Error
Real
Simulate
Error
20.616
21.5070
1.2612
31.776
9
31.794
3
3.8860
31.197
7
26.739
7
4.8423
1.0741
1.1166
0.0422
1.1375
1.2907
0.0423
1.0609
1.0627
0.0802
1.0889
1.0944
0.0297
1.2320
1.2496
0.0434
1.0650
1.1094
0.0206
0.9953
1.0201
0.0177
0.9240
1.0331
0.0144
0.9977
1.0433
0.0611
0.0465
0.0516
0.0153
0.1020
0.1191
0.0145
0.0500
0.0471
0.0195
0.0350
0.0390
0.0118
0.0902
0.0965
0.0153
0.0268
0.0263
0.0084
0.0114
0.0126
0.0063
0.0118
0.0226
0.0074
0.0233
0.0207
0.0138
d
   i (t ) / T
i
   i (t ) / T
i
  i (t ) / T
i
I [ ]   I [i (t )] / T
i
I []   I [i (t )] / T
i
I [ ]   I [i (t )] / T
i
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.
Many Thanks!