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Statistical Physics and the “Problem of Firm Growth”

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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. 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).

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.

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.

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)?

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 ?

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

S(t+1) = S(t) * t ( t is noise).

- 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.

Not Gaussian !

i.e. Not parabola

PDF, P(g)

Growth rate, g

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

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

“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:

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).

Pharmaceutical Industry Database

Probability distribution, P(n)

1.14

Number of products in a firm, n

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

One Parameter: Vg

Scaled PDF, P(g) Vg1/2

GDP

Phar. Firm / 102

Manuf. Firm / 104

Scaled growth rate, (g – g) / Vg1/2

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.

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.

- 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).

Our Prediction vs Empirical Data III

Scaled PDF, P(g) Vg1/2

Scaled growth rate, (g – g) / Vg1/2

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

Master equation:

n(t) = n(0) + ( - + b) t

Initial conditions: n(0) 0, b 0.

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

Solution:

Pold(n) exp(- A n)

Pnew(n)

Master equation:

n(t) = n(0) + ( - + b) t

Initial conditions: n(0) 0, b 0.

Class

Units

is composed of

A Country

Industries

is composed of

A industry

Firms

is composed of

A firm

Products

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: b0, n(0)0.

b=0 P(n) is exponential.

b0, n(0)=0 P(n) is power law.

Based on Pold(n):

(1)

Based on Pnew(n):

(2)

P(g)

Growth rate, g

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

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)

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

Mean Growth Rate

S, Firm Size

Size-Variance Relationship

(g|S)

S, Firm Size

(g|S)

S, Firm Size

E(N|S), expected N

E(|S), expected

S, firm sale

S, firm sale

t0

nold

nnew(t0, t)

t

nold

nnew

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:

Simulation

The distribution, P(n)

The number of units, n

Master equation:

Solution:

Pold(n) exp[- n / nold(t)]

Pnew(n) n -[2 +b/(1-b)] f(n)

(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

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)

Math for 2nd Set of Assumption

Idea:

for large n.

From Pold(n):

(3)

From Pnew(n):

(4)

(b 0)

(5)

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).

PHID

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

Firm S evolves like this:

assuming x follows power-law distribution:

Conclusion:

1.

2.

3.

PHID

p(N) , Probability Density

N, Division Number

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

In the 1st year:

Scaled pdf(N), p(N)*S

Scaled pdf(), p()*S

Scaled division size , /S

Scaled division number , N/S

1(|S) ~ S- f1(/S)

2(N|S) ~ S- f2(N/S)