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

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)?

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 ?

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

PD

F,

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

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:

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 ProductsP

roba

bili

ty d

istr

ibut

ion,

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

Sca

led

PD

F,

P(g

) V

g1/2

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

Sca

led

PD

F, P

(g)

Vg1/2

Tail part is power-law with exponent -3.

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.

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

Sca

led

PD

F, P

(g)

Vg1/2

Our Prediction vs Empirical Data III

),()(

)(),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.

),()(

)(),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.

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

)(

2

1)(

g

V

tn

V

tngP

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

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).

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),

exp

ecte

d

S, firm saleE

(N|S

), e

xpec

ted

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

tn

nttnP

tn

nttnP

tn

n

t

ttnPiii

i

Master equation:

Solution:

21),( IItnP )exp(1 nI

)]/(2[2

bnI

Effect of b on P(n)

Simulation

The number of units, n

The

dis

trib

utio

n,

P(n

)

The Size-Variance Relation

),,()(

)(),,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)

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([),(

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

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):

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).

PHID

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

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

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

The Distribution of Division Number N

N, Division Number

p(N

) , P

roba

bili

ty D

ensi

ty

PHID

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.

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