Firm Size Mobility and Validity of the Gibrat Model
Yougui Wang
Department of Systems ScienceSchool 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
• 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
:{1,2, , , }Firms k N
1 2:{ , , , , }k NSizes x x x x
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Nk
Nk
Nk
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Time
Changes under Stable Distributions
111211
21
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Nk
Nk
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Firm Size Distribution in China
Individual Evolution
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Variations in rank indicate it seems move randomly over time .
All Individuals’ Evolutions
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Aggregate variation in rank can not be simply figure out by putting all individuals’ evolutions together.
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
N
kkk xx
NM
110 ||1
where, N is the number of firms, xk0 and xk1 are the sizes of firm i at time 0 and 1 respectively.
N
kkk xx
NM
110 |loglog|1
Absolute Mobility
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Absolute mobility covers changes of all firm sizes and is somewhat independent of scale.
Relative Mobility
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The ranks of individuals can be derived from attributes of them.
Zipf Plot
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Shlomo Havlin, “The distance between Zipf plots”, Physica A, vol. 216, pp. 148-150 ， 1995.
Relative Mobility
where, N is the number of firms, rk0 and rk1 are the rank of firm i at time 0 and 1 respectively.
21
1
210 ]))()((1[
N
kk xrxrN
D
N
kkk xrxr
ND
110 )()(1
Rank Clock of Top 100 Cities in USA
Michael Batty, “Rank clocks”, Nature, vol. 444, pp. 592–596, 2006.
Seven firms ranked in top 500 of the world during the thirteen years
100 200 300 400 500
1997
2003
1998
2004
19992005
2000
2006
2001
2007
2002
1996/2008
CIGNA
DEUTSCHEPOST
FORTIS
FUJITSU
HOMEDEPOT
INDIAN OIL
WAL-MARTSTORES
Rank Clocks of Firms
100 200 300 400 500
1959
1986
1964
1991
19681995
1973
2000
1977
2004
1982
2008/1955
Many firms ranked in 500 firms in USA
Eight firms ranked in top 500 in USA
100 200 300 400 500
1959
1986
1964
1991
19681995
1973
2000
1977
2004
1982
2008/1955
PitneyBowesAbbottLaboratories3MAlcoaAltriaGroupAmericanStandardAnheuserBuschArcherDanielsMidland
Rank Clocks of Firms
Rank-shift Clocks of Firm SizeRank Shift Clocks of Firms
1990
50 100
150
1959
1986
1964
1967
1971
1999
1975
2004
1982
2008/1955
1995
Mobility of 3 real system
U.S.
Global
China
The Measurements of MobilityChina U.S Global
revenue(million
￥ )rank share
revenue(million
$)rank share
revenue(million
$)rank share
1996 1105.678 24.83948 0.00018 3021.67 30.90498 0.0003
1997 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
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
• 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.
1
1
t tt
t
x xx
1 0 1 2(1 ) (1 )(1 ) (1 )t t t tx x x
0 1 2l g logt to x x
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.
1( ) [ ( )] ( )i i iP t t P t
min( ) ( ) ( ) /iP t P t P t n
Simulation Results on the Evolution of US Firm Size Distribution
100
101
102
103
101
102
103
104
105
106
107
108
Rank
Rev
enue
Validation of the Gibrat Model
1
( )( ) ( ) ( ) ( )( )
( )( ){ }{ ( ) }( 1) ( 1)
ii i i
i i i
ii
i i
R tt p t t p tR t
p tR t p tR t p t
1( ) ( ) / ( )i i it R t R t 1( ) ( ) ( )i i iR t t R t
( ) ( ) / ( )i ip t R t R t ( ) ( )iiR t R t
( )( )( 1)R ttR t
( )( ) ( )
( 1)i
ii i
p tt p t
p t
Validity of Gibrat’s model
1log ( ) log[ ( ) / ( )]i i it R t R t
1
( )[ ( )] ( ) log ( ) ( ) log( )
( )( )log ( ) log( 1) ( 1)
ii i i
i i i
ii
i i
R tI t p t t p tR t
p tR t p tR t p t
( )[ ( )] log( 1)R tI tR t
( )[ ( )] ( ) log( 1)i
ii i
p tI t p t
p t
Validation of the Gibrat Model
Growth Rate of Firm Size
1960
1987
1964
1991
19691995
1973
2000
1978
2004
1982
2008/1955
growth rate of 3 system
2 1.5 1 0.5
U.S.
Global
China
Comparisons Between Simulations and Measurements
componentUS China Global
Real Simulate Error Real Simulate Error Real Simulate Error
20.616 21.5070 1.261231.776
931.794
3 3.8860 31.1977
26.7397 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
( ) /ii
t T
( ) /ii
t T
( ) /ii
t T
[ ] [ ( )] /ii
I I t T
[ ] [ ( )] /ii
I I t T
[ ] [ ( )] /ii
I I t T
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!
