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![Page 1: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/1.jpg)
Statistical Physics and the “Problem of Firm Growth”
Dongfeng FuAdvisor: H. E. Stanley
K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia, M. Riccaboni, H. E. Stanley, 74, PRE 035103 (2006).
DF Fu, F. Pammolli, S. V. Buldyrev, K. Matia, M. Riccaboni, K. Yamasaki, H. E. Stanley 102, PNAS 18801 (2005) .
DF. Fu, S. V. Buldyrev, M. A. Salinger, and H. E. Stanley, PRE 74, 036118 (2006).
Collaborators:
![Page 2: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/2.jpg)
Motivation
Firm growth problem quantifying size changes of firms.
1) Firm growth problem is an unsolved problem in economics.
2) Statistical physics may help us to develop better strategies to improve economy.
3) Help people to invest by quantifying risk.
![Page 3: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/3.jpg)
Outline
1) Introduction of “classical firm growth problem”.
2) The empirical results of the probability density function of growth rate.
3) A generalized preferential-attachment model.
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Classical Problem of Firm Growth
t/year1 2 10
5
12log
)(
)1(log
tS
tSg
Firm growth rate:
Firm at time = 1
S = 5
Firm at time = 2
S = 12
Firm at time = 10
S = 33
Question: What is probability density function of growth rate P(g)?
![Page 5: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/5.jpg)
Classic Gibrat Law & Its Implication
Traditional View: Gibrat law of “Proportionate Effect” (1930)
S(t+1) = S(t) * t ( t is noise).
Growth rate g in t years
=
logS(t)
= logS(0) + log(t’ )t’=1
M
S(0)
S(t)log
M
t’=1= log(t’)
Gibrat: pdf of g is Gaussian.
Growth rate, g
Pro
babi
lity
den
sity
pdf(g) 2
2g
e
Gaussian
P(g) really Gaussian ?
![Page 6: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/6.jpg)
Databases Analyzed for P(g)
1. Country GDP: yearly GDP of 195 countries, 1960-2004.
2. American Manufacturing Companies: yearly sales of 23,896 U.S. publicly traded firms, based on Security Exchange Commission filings 1973-2004.
3. Pharmaceutical Industry: quarterly sales figures of 7184 firms in 21 countries (mainly in north America and European Union) covering 189,303 products in 1994-2004.
![Page 7: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/7.jpg)
Empirical Results for P(g) (all 3 databases)
Growth rate, g
PD
F,
P(g
)
Not Gaussian !
i.e. Not parabola
Traditional Gibrat view is NOT able to accurately predict P(g)!
![Page 8: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/8.jpg)
The New Model: Entry & Exit of Products and firms
Preferential attachment to add new product or delete old product
Rules:
b: birth prob. of a firm.
birth prob. of a prod.
death prob. of a prod.
( > )
New:
New: 1. Number n of products in a firm 2. size of product
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1. At time t, each firm has n(t) products of size i(t), i=1,2,…n(t),
where n and >0 are independent random variables that follow
the distributions P(n) and P(), respectively.
2. At time t+1, the size of each product increases or decreases by a
random factor i(t+1) = (t)i * i.
n
ii
n
ii
t
t
tS
tSg
1
1
)(
)1(log
)(
)1(log
Assume P() = LN(m,V), and P() = LN(m,V). LN Log-Normal.
“Multiplicative” Growth of Products
Hence:
![Page 10: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/10.jpg)
for large n. Vg = f(V, V)
= Variance
P(g|n) ~ Gaussian(m+V/2, Vg/n)
The shape of P(g) comes from the fact that P(g|n) is Gaussian but the convolution with P(n).
Growth rate, g
P(g
| n)
1
)|()()(n
ngPnPgPIdea:
How to understand the shape of P(g)
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Distribution of the Number of ProductsP
roba
bili
ty d
istr
ibut
ion,
P(n
)
Number of products in a firm, n
Pharmaceutical Industry Database
1.14
![Page 12: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/12.jpg)
1. for small g, P(g) exp[- |g| (2 / Vg)1/2].
2. for large g, P(g) ~ g-3 .
Characteristics of P(g)
Growth rate, g
P(g
)
222 )2|(|2
2)(
gg
g
VggVg
VgP
Our Fitting Function
P(g) has a crossover from exponential to power-law
![Page 13: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/13.jpg)
Our Prediction vs Empirical Data I
Scaled growth rate, (g – g) / Vg1/2
Sca
led
PD
F,
P(g
) V
g1/2
GDPPhar. Firm / 102
Manuf. Firm / 104
One Parameter: Vg
![Page 14: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/14.jpg)
Our Prediction vs Empirical Data IICentral & Tail Parts of P(g)
Central part is Laplace.
Scaled growth rate, (g – g) / Vg1/2
Sca
led
PD
F, P
(g)
Vg1/2
Tail part is power-law with exponent -3.
![Page 15: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/15.jpg)
Universality w.r.t Different Countries
Scaled growth rate, (g – g) / Vg1/2
Sca
led
PD
F, P
g(g)
Vg1/
2
Growth rate, g
PD
F, P
g(g)
Original pharmaceutical data Scaled data
Take-home-message: China/India same as developed countries.
![Page 16: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/16.jpg)
Conclusions
1. P(g) is tent-shaped (exponential) in the central part and power-law with exponent -3 in tails.
2. Our new preferential attachment model accurately reproduced the empirical behavior of P(g).
![Page 17: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/17.jpg)
Scaled growth rate, (g – g) / Vg1/2
Sca
led
PD
F, P
(g)
Vg1/2
Our Prediction vs Empirical Data III
![Page 18: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/18.jpg)
),()(
)(),1()(
1),1(
)(
1),(tnP
tn
ntnP
tn
ntnP
tn
n
t
tnP
Master equation:
Math for Entry & Exit
Case 1: entry/exit, but no growth of products.
n(t) = n(0) + (- + b) t
Initial conditions: n(0) 0, b 0.
![Page 19: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/19.jpg)
),()(
)(),1()(
1),1(
)(
1),(tnP
tn
ntnP
tn
ntnP
tn
n
t
tnP
Master equation:
Math for Entry & Exit
Solution:
Pold(n) exp(- A n)
Pnew(n)
)()0(
)()0(
)0()( nP
btn
btnP
btn
nnP newold
)()]/(2[ nfn b
Case 1: entry/exit, but no growth of units.
n(t) = n(0) + (- + b) t
Initial conditions: n(0) 0, b 0.
![Page 20: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/20.jpg)
![Page 21: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/21.jpg)
Different Levels
Class
A Country
A industry
A firm
Units
Industries
Firms
Products
is composed of
is composed of
is composed of
![Page 22: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/22.jpg)
The Shape of P(n)
Number of products in a firm, n
PD
F, P
(n)
b=0 P(n) is exponential.
b0, n(0)=0 P(n) is power law.
P(n) = Pold(n) + Pnew(n).
P(n) observed is due to initial condition: b0, n(0)0.
(b=0.1, n(0)=10000, t=0.4M)
Number of products in a firm, n
![Page 23: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/23.jpg)
P(g) from Pold(n) or Pnew(n) is same
222 )2|(|2
2)(
gg
g
VggVg
VgP
2/3
2
2
)(1
2
)(
2
1)(
g
V
tn
V
tngP
gg(1)
(2)
Based on Pold(n):
Based on Pnew(n):
Growth rate, g
P(g
)
![Page 24: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/24.jpg)
Statistical Growth of a Sample Firm
Firm size S = 5
Firm size S = 33
t/year1 2 10
Firm size S = 12
3=1
1=2
3 products:
2=2
n = 3
2=11=4
3=5
4=2
7=5
3=5
6=4
1=6
4=1
2=2
5=10
n = 4
n = 7
L.A.N. Amaral, et al, PRL, 80 (1998)
![Page 25: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/25.jpg)
Number and size of products in each firm change with time.
What we do
Pharmaceutical Industry Database
Pro
babi
lity
dis
trib
utio
n
The number of product in a firm, n
Traditional view is
To build a new model to reproduce empirical results of P(g).
![Page 26: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/26.jpg)
![Page 27: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/27.jpg)
Average Value of Growth Rate
S, Firm Size
Mea
n G
row
th R
ate
![Page 28: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/28.jpg)
Size-Variance Relationship
S, Firm Size
g|
S)
![Page 29: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/29.jpg)
Simulation on
S, Firm Size
(g|
S)
![Page 30: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/30.jpg)
Other Findings
S, firm sale
E(
|S),
exp
ecte
d
S, firm saleE
(N|S
), e
xpec
ted
N
![Page 31: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/31.jpg)
![Page 32: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/32.jpg)
Mean-field Solution
noldt0
t
nold nnew
nnew(t0, t)
![Page 33: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/33.jpg)
The Complete ModelRules:1. At t=0, there exist N classes with n units.2. At each step: a. with birth probability b, a new class is born b. with , a randomly selected class grows one unit in size based on “preferential attachment”. c. with a randomly selected class shrinks one unit in size based on “preferential dettachment”.
),,()(
)(),,1()(
1),,1(
)(
1),,(tnnP
tn
nttnP
tn
nttnP
tn
n
t
ttnPiii
i
Master equation:
Solution:
21),( IItnP )exp(1 nI
)]/(2[2
bnI
![Page 34: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/34.jpg)
Effect of b on P(n)
Simulation
The number of units, n
The
dis
trib
utio
n,
P(n
)
![Page 35: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/35.jpg)
The Size-Variance Relation
![Page 36: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/36.jpg)
),,()(
)(),,1()(
1),,1(
)(
1),,(tnnP
tn
nttnP
tn
nttnP
tn
n
t
ttnPiii
i
Master equation:
Solution:
)]/(2[2
bnI
Math for 1st Set of Assumption
)()0(
)()0(
)0()( nP
btN
btnP
btN
NnP newold
Pold(n) exp[- n / nold(t)]
Pnew(n) n -[2 + b/(1-b)] f(n)
![Page 37: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/37.jpg)
Math for 1st Set of Assumption
tn
tnbb
dt
tdn newnew
)0(
)()1(
)(
tn
tnb
dt
tdn oldold
)0(
)()1(
)(
nold(t) = [n(0)+t]1-b n(0)b
(1)
(2)
Initial condition:
nold(0)=n(0)
Solution:
nnew(t0, t) = [n(0)+t]1-b[n(0)+t0]b
)0(/)()( Ntntn oldold bb
new tntnttn 10
10 ])0(/[])0([),(
![Page 38: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/38.jpg)
Math Continued
)()0(
)()0(
)0()( nP
btN
btnP
btN
NnP newold
))(
exp()(
1)(
tn
n
tnnP
Pold(n) exp[- n / nold(t)]
Pnew(n) n -[2 + b/(1-b)] f(n)
Solution:
When t is large, Pold(n) converges to exponential distribution
![Page 39: Statistical Physics and the “Problem of Firm Growth” Dongfeng Fu Advisor: H. E. Stanley K. Yamasaki, K. Matia, S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649e835503460f94b84e3c/html5/thumbnails/39.jpg)
1
)|()()(n
gg ngPnPgP
]2/)(exp[2
)|( 2g
g
g VnggV
nngP
y
bg
bg dnnVngdyyygP .)2/exp()exp()(
)1
1
2
1(
2
0
1
1
Math for 2nd Set of Assumption
Idea:
222 )2|(|2
2)(
gg
gg
VggVg
VgP
2/3
2
2
)(1
2
)(
2
1)(
g
V
tn
V
tngP
ggg (3)
(4)
(5)(b 0)
for large n.
From Pold(n):
From Pnew(n):
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Empirical Observations (before 1999)
g(S) ~ S- , 0.2
S, Firm size
Sta
ndar
d de
viat
ion
of g
Small Medium Large
g, growth rate
Small firms Medium firms Large firms
Reality: it is “tent-shaped”! P
roba
bili
ty d
ensi
ty Empirical
pdf(g|S) ~)(
||
S
g
e
Michael H. Stanley, et.al. Nature 379, 804-806 (1996). V. Plerou, et.al. Nature 400, 433-437 (1999).
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PHID
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Current Status on the Models of Firm Growth
Models
IssuesGibrat Simon Sutton Bouchaud Amaral
p(N) is power law
p(S) is log-normal
p() is log-normal
(S) ~ S- = 0.5 = 0.22 depends 0.17
p(g|S) is “tent”
p(|S) & scaling
p(N|S) & scaling
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The Models to Explain Some Empirical Findings
Sutton’s Model
Simon's Model explains the distribution of the division number is power law.
Based on partition theory
2(S) =1/3(12 +12+ 12) + 1/3(12 + 22) + 1/3(32) = 17/3
1 1 1
1 2
3
S = 32(g) = 2(S/S) = 2(S)/S2 = 0.63 ~ S-2
= -ln(0.63)/2ln(3) = 0.21
The probability of growing by a new division is proportional to the division number in the firm. Preferential attachment.
The distribution of division number is power law.
3 firms industry
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Bouchaud's Model:
)()())()(1
(1
K
j
iiiji
ttttKdt
d
assuming follows power-law distribution:
1)(
op
Firm S evolves like this:
Conclusion:
2
1
21 1.
2.
3.
25.0
0 1
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The Distribution of Division Number N
N, Division Number
p(N
) , P
roba
bili
ty D
ensi
ty
PHID
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Example Data (3 years time series)
A1 0 3 4
A2 2 1 2
B1 0 10 1
B2 11 4 7
B3 5 6 7
Firm S N
A 2 1 2
B 16 2 11, 5
In the 1st year:
S g log(S(t+1)/S(t))
2 log(4/2)
4 log(6/4)
16 log(20/16)
20 log(15/20)
S N
2 1
16 2
S
2 2
16 11
16 5
A, B are firms. A1, A2 are divisions of firm A; B1, B2, B3 are divisions of firm B.
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Predictions of Amaral at al model
Scaled division size , /S
Sca
led
pdf()
, p(
)*S
1(|S) ~ S- f1(/S) 2(N|S) ~ S
- f2(N/S)
Scaled division number , N/S
Sca
led
pdf(
N),
p(N
)*S
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