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, M. Riccaboni, H. E. Stanley, 74, PRE (2006). DF Fu, F. Pammolli, S. V. Buldyrev, K. Matia, M. Riccaboni, K. Yamasaki, H. E. Stanley 102, PNAS (2005). DF. Fu, S. V. Buldyrev, M. A. Salinger, and H. E. Stanley, PRE 74, (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/year 1210 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 M =  log (  t’ ) Gibrat: pdf of g is Gaussian. Growth rate, g Probability density pdf(g)  Gaussian P(g) really Gaussian ?

Databases Analyzed for P(g) 1.Country GDP: yearly GDP of 195 countries, American Manufacturing Companies: yearly sales of 23,896 U.S. publicly traded firms, based on Security Exchange Commission filings Pharmaceutical Industry: quarterly sales figures of 7184 firms in 21 countries (mainly in north America and European Union) covering 189,303 products in

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. Assume P(  ) = LN(m ,V  ), and P(  ) = LN(m ,V  ). LN  Log-Normal. “Multiplicative” Growth of Products Hence:

for large n.V g = f(V , V  ) = Variance P(g|n) ~ Gaussian(m  +V  /2, V g /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) Idea: How to understand the shape of P(g)

Distribution of the Number of Products Probability distribution, P(n) Number of products in a firm, n Pharmaceutical Industry Database 1.14

1. for small g, P(g)  exp[- |g| (2 / V g ) 1/2 ]. 2. for large g, P(g) ~ g -3. Characteristics of P(g) Growth rate, g P(g) Our Fitting Function P(g) has a crossover from exponential to power-law

Our Prediction vs Empirical Data I Scaled growth rate, (g – g) / V g 1/2 Scaled PDF, P(g) V g 1/2 GDP Phar. Firm / 10 2 Manuf. Firm / 10 4 One Parameter: V g

Our Prediction vs Empirical Data II Central & Tail Parts of P(g) Central part is Laplace. Scaled growth rate, (g – g) / V g 1/2 Scaled PDF, P(g) V g 1/2 Tail part is power-law with exponent -3.

Universality w.r.t Different Countries Scaled growth rate, (g – g) / V g 1/2 Scaled PDF, P g (g) V g 1/2 Growth rate, g PDF, P g (g) Original pharmaceutical dataScaled 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) / V g 1/2 Scaled PDF, P(g) V g 1/2 Our Prediction vs Empirical Data III

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.

Master equation: Math for Entry & Exit Solution: P old (n)  exp(- A n) P new (n)  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

The Shape of P(n) Number of products in a firm, n PDF, P(n) b=0 P(n) is exponential. b  0, n(0)=0 P(n) is power law. P(n) = P old (n) + P new (n). P(n) observed is due to initial condition: b  0, n(0)  0. (b=0.1, n(0)=10000, t=0.4M) Number of products in a firm, n

P(g) from P old (n) or P new (n) is same (1) (2) Based on P old (n): Based on P new (n): Growth rate, g P(g)

Statistical Growth of a Sample Firm Firm size S = 5 Firm size S = 33 t/year 1210 Firm size S = 12  3= 1  1=2 3 products:  2= 2 n = 3  2= 1  1=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 Probability distribution 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 Mean Growth Rate

Size-Variance Relationship S, Firm Size  g|S)

Simulation on  S, Firm Size  (g|S)

Other Findings S, firm sale E(  |S), expected  S, firm sale E(N|S), expected N

Mean-field Solution n old t0t0 t n new n new (t 0, t)

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) Simulation The number of units, n The distribution, P(n)

The Size-Variance Relation

Master equation: Solution: Math for 1st Set of Assumption P old (n)  exp[- n / n old (t)] P new (n)  n -[2 + b/(1-b)] f(n)

Math for 1 st Set of Assumption n old (t) = [n(0)+t] 1-b n(0) b (1) (2) Initial condition: n old (0)=n(0) Solution: n new (t 0, t) = [n(0)+t] 1-b [n(0)+t 0 ] b

Math Continued P old (n)  exp[- n / n old (t)] P new (n)  n -[2 + b/(1-b)] f(n) Solution: When t is large, P old (n) converges to exponential distribution

Math for 2 nd Set of Assumption Idea: (3) (4) (5) (b 0) for large n. From P old (n): From P new (n):

Empirical Observations (before 1999)  g (S) ~ S - ,   0.2 S, Firm size Standard deviation of g SmallMediumLarge g, growth rate Small firms Medium firms Large firms Reality: it is “tent-shaped”! Probability density Empirical pdf(g|S) ~ Michael H. Stanley, et.al. Nature 379, (1996). V. Plerou, et.al. Nature 400, (1999).

PHID

Current Status on the Models of Firm Growth Models Issues GibratSimon Sutton BouchaudAmaral 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( ) + 1/3( ) + 1/3(3 2 ) = 17/ S = 3  2 (g) =  2 (  S/S) =  2 (  S)/S 2 = 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: assuming  follows power-law distribution: Firm S evolves like this: Conclusion:

The Distribution of Division Number N N, Division Number p(N), Probability Density PHID

Example Data (3 years time series) A1034 A2212 B10101 B21147 B3567 FirmSN  A212 B16211, 5 In the 1 st year: S g  log(S(t+1)/S(t)) 2log(4/2) 4log(6/4) 16log(20/16) 20 log(15/20) SN S  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  Scaled pdf(  ), p(  )*S   1 (  |S) ~ S -  f 1 (  /S  )  2 (N|S) ~ S -  f 2 (N/S  ) Scaled division number , N/S  Scaled pdf(N), p(N)*S 