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Statistical Physics and the “Problem of Firm Growth”. Collaborators:. Dongfeng Fu Advisor: H. E. Stanley. DF Fu, F. Pammolli, S. V. Buldyrev, K. Matia, M. Riccaboni, K. Yamasaki, H. E. Stanley 102 , PNAS 18801 (2005). - PowerPoint PPT Presentation
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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:
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.
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.
Classical Problem of Firm Growth
t/year1 2 10
512
log
)()1(
logtS
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)?
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
Prob
abili
ty d
ensi
ty
pdf(g) 2
2g
e
Gaussian
P(g) really Gaussian ?
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.
Empirical Results for P(g) (all 3 databases)
Growth rate, g
PDF,
P(g
)
Not Gaussian !
i.e. Not parabola
Traditional Gibrat view is NOT able to accurately predict P(g)!
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
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
tStSg
1
1
)(
)1(log
)()1(log
Assume P() = LN(m,V), and P() = LN(m,V). LN Log-Normal.
“Multiplicative” Growth of Products
Hence:
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)
Distribution of the Number of ProductsPr
obab
ility
dis
tribu
tion,
P(n
)
Number of products in a firm, n
Pharmaceutical Industry Database
1.14
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
Our Prediction vs Empirical Data I
Scaled growth rate, (g – g) / Vg1/2
Scal
ed P
DF,
P(g
) Vg1/2
GDPPhar. Firm / 102
Manuf. Firm / 104
One Parameter: Vg
Our Prediction vs Empirical Data IICentral & Tail Parts of P(g)
Central part is Laplace.
Scaled growth rate, (g – g) / Vg1/2
Scal
ed P
DF,
P(g
) V
g1/2
Tail part is power-law with exponent -3.
Universality w.r.t Different Countries
Scaled growth rate, (g – g) / Vg1/2
Scal
ed P
DF,
Pg(g
) Vg1/
2
Growth rate, g
PDF,
Pg(g
)
Original pharmaceutical data Scaled data
Take-home-message: China/India same as developed countries.
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).
Scaled growth rate, (g – g) / Vg1/2
Scal
ed P
DF,
P(g
) V
g1/2
Our Prediction vs Empirical Data III
),()(
)(),1()(1),1(
)(1),( tnP
tnntnP
tnntnP
tnn
ttnP
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.
),()(
)(),1()(1),1(
)(1),( tnP
tnntnP
tnntnP
tnn
ttnP
Master equation:
Math for Entry & Exit
Solution:
Pold(n) exp(- A n)
Pnew(n)
)()0(
)()0(
)0()( nPbtn
btnPbtn
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.
Different Levels
Class
A Country
A industry
A firm
Units
Industries
Firms
Products
is composed of
is composed of
is composed of
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
P(g) from Pold(n) or Pnew(n) is same
222 )2|(|2
2)(
gg
g
VggVg
VgP
2/3
2
2)(1
2)(
21)(
g
Vtn
VtngP
gg(1)
(2)
Based on Pold(n):
Based on Pnew(n):
Growth rate, g
P(g)
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)
Number and size of products in each firm change with time.
What we do
Pharmaceutical Industry Database
Prob
abili
ty d
istri
butio
n
The number of product in a firm, n
Traditional view is
To build a new model to reproduce empirical results of P(g).
Average Value of Growth Rate
S, Firm Size
Mea
n G
row
th R
ate
Size-Variance Relationship
S, Firm Size
g|
S)
Simulation on
S, Firm Size
(g|
S)
Other Findings
S, firm sale
E(|
S), e
xpec
ted
S, firm saleE(N
|S),
expe
cted
N
Mean-field Solution
noldt0
t
nold nnew
nnew(t0, t)
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
tnnttnP
tnnttnP
tnn
tttnP
iiii
Master equation:
Solution:
21),( IItnP )exp(1 nI
)]/(2[2
bnI
Effect of b on P(n)Simulation
The number of units, n
The
dist
ribut
ion,
P(
n)
The Size-Variance Relation
),,()(
)(),,1()(1),,1(
)(1),,( tnnP
tnnttnP
tnnttnP
tnn
tttnP
iiii
Master equation:
Solution:
)]/(2[2
bnI
Math for 1st Set of Assumption
)()0(
)()0(
)0()( nPbtN
btnPbtN
NnP newold
Pold(n) exp[- n / nold(t)]
Pnew(n) n -[2 + b/(1-b)] f(n)
Math for 1st Set of Assumption
tntnbb
dttdn newnew
)0()()1()(
tntnb
dttdn 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([),(
Math Continued
)()0(
)()0(
)0()( nPbtN
btnPbtN
NnP newold
))(
exp()(
1)(tn
ntn
nP
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
1
)|()()(n
gg ngPnPgP
]2/)(exp[2
)|( 2g
gg Vngg
VnngP
y
bg
bg dnnVngdyyygP .)2/exp()exp()(
)1
121(2
01
1
Math for 2nd Set of Assumption
Idea:
222 )2|(|2
2)(
gg
gg
VggVg
VgP
2/3
2
2)(1
2)(
21)(
g
Vtn
VtngP
ggg (3)
(4)
(5)(b 0)
for large n.
From Pold(n):
From Pnew(n):
Empirical Observations (before 1999)
g(S) ~ S- , 0.2
S, Firm size
Stan
dard
dev
iatio
n of
g
Small Medium Large
g, growth rate
Small firms Medium firms Large firms
Reality: it is “tent-shaped”! Pr
obab
ility
den
sity 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).
PHID
Current Status on the Models of Firm Growth Models Issues
Gibrat 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
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 23
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
Bouchaud's Model:
)()())()(1
(1
K
jiiij
itttt
Kdtd
assuming follows power-law distribution:
1)( op
Firm S evolves like this:
Conclusion:
21
21 1.
2.
3.
25.0
0 1
The Distribution of Division Number N
N, Division Number
p(N
) , P
roba
bilit
y D
ensi
ty
PHID
Example Data (3 years time series)
A1 0 3 4A2 2 1 2B1 0 10 1B2 11 4 7B3 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 N2 116 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.
Predictions of Amaral at al model
Scaled division size , /S
Scal
ed p
df(
), p(
)*S
1(|S) ~ S- f1(/S) 2(N|S) ~ S- f2(N/S)
Scaled division number , N/S
Scal
ed p
df(N
), p(
N)*
S