Statistical physics and the problem of firm growth
This presentation is the property of its rightful owner.
Sponsored Links
1 / 49

Statistical Physics and the “Problem of Firm Growth” PowerPoint PPT Presentation


  • 64 Views
  • Uploaded on
  • Presentation posted in: General

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).

Download Presentation

Statistical Physics and the “Problem of Firm Growth”

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Statistical physics and the problem of firm growth

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. Stanley102,

PNAS 18801 (2005) .

K. Yamasaki,K. Matia,S. V. Buldyrev, DF Fu, F. Pammolli, K. Matia,M. Riccaboni, H. E. Stanley, 74, PRE 035103 (2006).

DF. Fu, S. V. Buldyrev, M. A. Salinger, and H. E. Stanley, PRE 74, 036118 (2006).


Motivation

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

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

Classical Problem of Firm Growth

Firm at time = 2

S = 12

Firm at time = 10

S = 33

Firm at time = 1

S = 5

Firm growth rate:

t/year

1

2

10

Question: What is probability density function of growth rate P(g)?


Classic gibrat law its implication

Gibrat: pdf of g is Gaussian.

logS(t)

= logS(0) + ålog(t’ )

M

Gaussian

t’=1

Growth rate g in t years

=

Probability density

pdf(g) 

M

=  log(t’)

t’=1

Growth rate, g

P(g) really Gaussian ?

Classic Gibrat Law & Its Implication

Traditional View: Gibrat law of “Proportionate Effect” (1930)

S(t+1) = S(t) * t ( t is noise).


Databases analyzed for p g

Databases Analyzed for P(g)

  • Country GDP: yearly GDP of 195 countries, 1960-2004.

  • American Manufacturing Companies: yearly sales of 23,896 U.S. publicly traded firms, based on Security Exchange Commission filings 1973-2004.

  • 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

Empirical Results for P(g) (all 3 databases)

Not Gaussian !

i.e. Not parabola

PDF, P(g)

Growth rate, g

Traditional Gibrat view is NOT able to accurately predict P(g)!


The new model entry exit of products and firms

The New Model: Entry & Exit of Products and firms

New:

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: 1. Number n of products in a firm 2. size  of product


Statistical physics and the problem of firm growth

“Multiplicative” Growth of Products

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.

Assume P() = LN(m,V), and P() = LN(m,V). LN  Log-Normal.

Hence:


Statistical physics and the problem of firm growth

How to understand the shape of P(g)

Idea:

P(g|n) ~ Gaussian(m+V/2, Vg/n)

for large n.

Vg = f(V, V)

= Variance

P(g | n)

Growth rate, g

The shape of P(g) comes from the fact that P(g|n) is Gaussian but the convolution with P(n).


Distribution of the number of products

Distribution of the Number of Products

Pharmaceutical Industry Database

Probability distribution, P(n)

1.14

Number of products in a firm, n


Statistical physics and the problem of firm growth

Characteristics of P(g)

Our Fitting Function

1. for small g,

P(g)  exp[- |g| (2 / Vg)1/2].

P(g)

2. for large g, P(g) ~ g-3 .

Growth rate, g

P(g) has a crossover from exponential to power-law


Our prediction vs empirical data i

Our Prediction vs Empirical Data I

One Parameter: Vg

Scaled PDF, P(g) Vg1/2

GDP

Phar. Firm / 102

Manuf. Firm / 104

Scaled growth rate, (g – g) / Vg1/2


Our prediction vs empirical data ii central tail parts of p g

Our Prediction vs Empirical Data IICentral & Tail Parts of P(g)

Scaled PDF, P(g) Vg1/2

Scaled growth rate, (g – g) / Vg1/2

Tail part is power-law

with exponent -3.

Central part is Laplace.


Universality w r t different countries

Universality w.r.t Different Countries

Original pharmaceutical data

Scaled data

PDF, Pg(g)

Scaled PDF, Pg(g) Vg1/2

Growth rate, g

Scaled growth rate, (g – g) / Vg1/2

Take-home-message: China/India same as developed countries.


Conclusions

Conclusions

  • P(g) is tent-shaped (exponential) in the central part and power-law with exponent -3 in tails.

  • Our new preferential attachment model accurately reproduced the empirical behavior of P(g).


Statistical physics and the problem of firm growth

Our Prediction vs Empirical Data III

Scaled PDF, P(g) Vg1/2

Scaled growth rate, (g – g) / Vg1/2


Math for entry exit

Case 1: entry/exit, but no growth of products.

Math for Entry & Exit

Master equation:

n(t) = n(0) + ( -  + b) t

Initial conditions: n(0)  0, b  0.


Math for entry exit1

Case 1: entry/exit, but no growth of units.

Solution:

Pold(n)  exp(- A n)

Pnew(n) 

Math for Entry & Exit

Master equation:

n(t) = n(0) + ( -  + b) t

Initial conditions: n(0)  0, b  0.


Different levels

Different Levels

Class

Units

is composed of

A Country

Industries

is composed of

A industry

Firms

is composed of

A firm

Products


The shape of p n

The Shape of P(n)

PDF, P(n)

Number of products in a firm, n

Number of products in a firm, n

(b=0.1, n(0)=10000, t=0.4M)

P(n) = Pold(n) + Pnew(n).

P(n) observed is due to initial condition: b0, n(0)0.

b=0 P(n) is exponential.

b0, n(0)=0 P(n) is power law.


P g from p old n or p new n is same

P(g) from Pold(n) or Pnew(n) is same

Based on Pold(n):

(1)

Based on Pnew(n):

(2)

P(g)

Growth rate, g


Statistical growth of a sample firm

1=4

2=1

1=2

1=6

2=2

2=2

3=5

3=1

3=5

4=2

5=10

4=1

n = 3

3 products:

7=5

n = 4

6=4

n = 7

Statistical Growth of a Sample Firm

Firm size S = 5

Firm size S = 12

Firm size S = 33

t/year

1

2

10

L.A.N. Amaral, et al, PRL, 80 (1998)


Statistical physics and the problem of firm growth

Pharmaceutical Industry Database

Probability distribution

The number of product in a firm, n

What we do

To build a new model to reproduce empirical results of P(g).

  • Number and size of products in each firm change with time.

Traditional view is


Average value of growth rate

Average Value of Growth Rate

Mean Growth Rate

S, Firm Size


Statistical physics and the problem of firm growth

Size-Variance Relationship

(g|S)

S, Firm Size


Simulation on

Simulation on 

(g|S)

S, Firm Size


Other findings

Other Findings

E(N|S), expected N

E(|S), expected 

S, firm sale

S, firm sale


Mean field solution

Mean-field Solution

t0

nold

nnew(t0, t)

t

nold

nnew


The complete model

The Complete Model

Rules:

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”.

Master equation:

Solution:


Effect of b on p n

Effect of b on P(n)

Simulation

The distribution, P(n)

The number of units, n


The size variance relation

The Size-Variance Relation


Math for 1st set of assumption

Math for 1st Set of Assumption

Master equation:

Solution:

Pold(n)  exp[- n / nold(t)]

Pnew(n)  n -[2 +b/(1-b)] f(n)


Math for 1 st set of assumption

Math for 1st Set of Assumption

(1)

Initial condition:

nold(0)=n(0)

(2)

Solution:

nold(t) = [n(0)+t]1-b n(0)b

nnew(t0, t) = [n(0)+t]1-b[n(0)+t0]b


Math continued

Math Continued

When t is large, Pold(n) converges to exponential distribution

Solution:

Pold(n)  exp[- n / nold(t)]

Pnew(n)  n -[2 +b/(1-b)] f(n)


Statistical physics and the problem of firm growth

Math for 2nd Set of Assumption

Idea:

for large n.

From Pold(n):

(3)

From Pnew(n):

(4)

(b 0)

(5)


Empirical observations before 1999

Empirical Observations (before 1999)

Empirical

Small firms Medium firms Large firms

Reality: it is “tent-shaped”!

Probability density

pdf(g|S) ~

g, growth rate

Small

Medium

Large

Standard deviation of g

g(S) ~ S- ,   0.2

Michael H. Stanley, et.al. Nature 379, 804-806 (1996).

S, Firm size

V. Plerou, et.al. Nature 400, 433-437 (1999).


Statistical physics and the problem of firm growth

PHID


Current status on the models of firm growth

Current Status on the Models of Firm Growth


Statistical physics and the problem of firm growth

The Models to Explain Some Empirical Findings

Simon's Model explains the distribution of the division number is power law.

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

Sutton’s Model

Based on partition theory

2(S) =1/3(12 +12+12) + 1/3(12 +22) + 1/3(32) = 17/3

1 1 1

S = 3

1 2

2(g) = 2(S/S) = 2(S)/S2 = 0.63 ~ S-2

3

 = -ln(0.63)/2ln(3) =0.21


Bouchaud s model

Bouchaud's Model:

Firm S evolves like this:

assuming x follows power-law distribution:

Conclusion:

1.

2.

3.


The distribution of division number n

The Distribution of Division Number N

PHID

p(N) , Probability Density

N, Division Number


Example data 3 years time series

Example Data (3 years time series)

A, B are firms. A1, A2 are divisions of firm A; B1, B2, B3 are divisions of firm B.

In the 1st year:


Predictions of amaral at al model

Scaled pdf(N), p(N)*S

Scaled pdf(), p()*S

Scaled division size , /S

Predictions of Amaral at al model

Scaled division number , N/S

1(|S) ~ S- f1(/S)

2(N|S) ~ S- f2(N/S)


  • Login