Statistical Physics and the “Problem of Firm Growth”

1. 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).

2. 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.

3. 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.

4. 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)?

5. 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).

6. 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.

7. 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)!

8. 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

9. “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:

10. 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).

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

12. 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

13. 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

14. 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.

15. 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.

16. 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).

17. Our Prediction vs Empirical Data III Scaled PDF, P(g) Vg1/2 Scaled growth rate, (g – g) / Vg1/2

18. 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.

19. 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.

21. Different Levels Class Units is composed of A Country Industries is composed of A industry Firms is composed of A firm Products

22. 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.

23. 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

24. 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)

25. 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

27. Average Value of Growth Rate Mean Growth Rate S, Firm Size

28. Size-Variance Relationship (g|S) S, Firm Size

29. Simulation on  (g|S) S, Firm Size

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

32. Mean-field Solution t0 nold nnew(t0, t) t nold nnew

33. 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:

34. Effect of b on P(n) Simulation The distribution, P(n) The number of units, n

35. The Size-Variance Relation

36. Math for 1st Set of Assumption Master equation: Solution: Pold(n)  exp[- n / nold(t)] Pnew(n)  n -[2 +b/(1-b)] f(n)

37. 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

38. 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)

39. Math for 2nd Set of Assumption Idea: for large n. From Pold(n): (3) From Pnew(n): (4) (b 0) (5)

41. 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).

42. PHID

43. Current Status on the Models of Firm Growth

44. 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

45. Bouchaud's Model: Firm S evolves like this: assuming x follows power-law distribution: Conclusion: 1. 2. 3.

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

47. 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:

48. 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)