1. Physics, Economics and Ecology Boltzmann, Pareto and Volterra Pavia Sept 8, 2003 Franco M.Scudo (1935-1998) Sorin Solomon, Hebrew University of Jerusalem

2. + d X i = ( a i LotkaVolterra + c i (X.,t)) X i +  j a ij X j

3. x + () d X i = (rand i LotkaVolterraBoltzmann + c i (X.,t)) X i +  j a ij X j

4. x + () = P ( X i ) ~ X i –1-  d X i LotkaVolterraBoltzmann Pareto d X i = (rand i + c (X.,t)) X i +  j a ij X j

5. Alfred Lotka the number P(n) of authors with n publications is a power law P(n) ~ n  with  ~ 1.

6. No. 6 of the Cowles Commission for Research in Economics, 1941. HAROLD T. DAVIS No one however, has yet exhibited a stable social order, ancient or modern, which has not followed the Pareto pattern at least approximately. (p. 395) Snyder [1939]: Pareto’s curve is destined to take its place as one of the great generalizations of human knowledge

7. d x =  (t) x +  P(x) dx ~ x –1-  d x for fixed  distribution with negative drift < 0 Not good for economy !

8. d x =  (t) x +  P(x) dx ~ x –1-  d x for fixed  distribution with negative drift < 0 Herbert Simon; intuitive explanation Not good for economy ! d ln x (t) =  (t) + lower bound = diffusion + down drift + reflecting barrier

9.  Boltzmann (/ barometric) distribution for ln x P(ln x ) d ln x ~ exp(-  ln x ) d ln x d x =  (t) x +  P(x) dx ~ x –1-  d x for fixed  distribution with negative drift < 0 Herbert Simon; intuitive explanation Not good for economy ! d ln x (t) =  (t) + lower bound = diffusion + down drift + reflecting barrier

10.  Boltzmann (/ barometric) distribution for ln x P(ln x ) d ln x ~ exp(-  ln x ) d ln x ~ x -1-  d x d x =  (t) x +  P(x) dx ~ x –1-  d x for fixed  distribution with negative drift < 0 Herbert Simon; intuitive explanation Not good for economy ! d ln x (t) =  (t) + lower bound = diffusion + down drift + reflecting barrier

11. Can one obtain stable power laws in systems with variable growth rates (economies with both recessions and growth periods) ? Yes! in fact all one has to do is to recognize the statistical character of the Logistic Equation

12. d X i = (a i + c (X.,t)) X i +  j a ij X j Montroll almost all the social phenomena, except in their relatively brief abnormal times obey the logistic growth. Elliot W Montroll: Social dynamics and quantifying of social forces (1978 or so)

13. d X i = (a i + c (X.,t)) X i +  j a ij X j Volterra Scudo Lotka Montroll Eigen almost all the social phenomena, except in their relatively brief abnormal times obey the logistic growth. Elliot W Montroll: Social dynamics and quantifying of social forces (1978 or so)

14. Stochastic Generalized Lotka-Volterra d X i = (rand i (t)+ c (X.,t) ) X i +  j a ij X j for clarity take  j a ij X j = a / N  j X j = a X

15. Stochastic Generalized Lotka-Volterra d X i = (rand i (t)+ c (X.,t) ) X i +  j a ij X j Assume Efficient market: P(rand i (t) )= P(rand j (t) ) for clarity take  j a ij X j = a / N  j X j = a X

16. Stochastic Generalized Lotka-Volterra d X i = (rand i (t)+ c (X.,t) ) X i +  j a ij X j Assume Efficient market: P(rand i (t) )= P(rand j (t) ) => THEN the Pareto power law P(X i ) ~ X i –1-  holds with  independent on c(w.,t) for clarity take  j a ij X j = a / N  j X j = a X

17. Stochastic Generalized Lotka-Volterra d X i = (rand i (t)+ c (X.,t) ) X i +  j a ij X j Assume Efficient market: P(rand i (t) )= P(rand j (t) ) => THEN the Pareto power law P(X i ) ~ X i –1-  holds with  independent on c(w.,t) for clarity take  j a ij X j = a / N  j X j = a X Proof:

18. d X i = (rand i (t)+ c (X.,t) ) X i + a X d X = c (X.,t) ) X + a X Denote x i (t) = X i (t) / X(t) Then dx i (t) = dX i (t)/ X(t) + X i (t) d (1/X) =dX i (t) / X(t) - X i (t) d X(t)/X 2 = ( rand i (t) –a ) x i (t) + a = [ rand i (t) X i +c(w.,t) X i + aX ]/ X -X i /X [ c(w.,t) X + a X ]/X = rand i (t) x i + c(w.,t) x i + a -x i (t) [ c(w.,t) + a ] =

19. dx i (t) = ( r i (t) –a ) x i (t) + a of Kesten type: d x =  (t) x +  and has constant negative drift ! Power law for large enough x i : P( x i ) d x i ~ x i -1-2 a/D d x i Even for very unsteady fluctuations of c; X

20. dx i (t) = ( r i (t) –a ) x i (t) + a of Kesten type: d x =  (t) x +  and has constant negative drift ! Power law for large enough x i : P( x i ) d x i ~ x i -1-2 a/D d x i In fact, the exact solution is : P( x i ) = exp[-2 a/(D x i )] x i -1-2 a/D Even for very unsteady fluctuations of c; X

21. Prediction:  =(1/(1-minimal income /average income)

22. Prediction:  =(1/(1-minimal income /average income) = 1/(1- 1/average number of dependents on one income)

23. Prediction:  =(1/(1-minimal income /average income) = 1/(1- 1/dependents on one income) = 1/(1- generation span/ population growth)

24. Prediction:  =(1/(1-minimal income /average income) = 1/(1- 1/dependents on one income) = 1/(1- generation span/ population growth) 3-4 dependents at each moment Doubling of population every 30 years minimal/ average ~ 0.25-0.33 (ok US, Isr)

25. Prediction:  =(1/(1-minimal income /average income) = 1/(1- 1/dependents on one income) = 1/(1- generation span/ population growth) 3-4 dependents at each moment Doubling of population every 30 years minimal/ average ~ 0.25-0.33 (ok US, Isr) =>  ~ 1.3-1.5 ; Pareto measured  ~ 1.4

26. Inefficient Market: Green gain statistically more (by 1 percent or so) No Pareto straight line

27. In Statistical Mechanics, Thermal Equilibrium  Boltzmann In Financial Markets, Efficient Market  no Pareto P(x) ~ exp (-E(x) /kT) 1886 P(x) ~ x –1-  d x 1897 Inefficient Market: Green gain statistically more (by 1 percent or so) No Pareto straight line M.Levy

28. Market Fluctuations

29. Paul Levy

30. Gene Stanley

31. Paul LevyGene Stanley (see him here in person)

32. M. Levy

33. One more puzzle: For very dense (trade-by-trade) measurements and very large volumes the tails go like 2 

34. One more puzzle: For very dense (trade-by-trade) measurements and very large volumes the tails go like 2  Explanation: Volume of trade = minimum of ofer size and ask size P(volume > v) = P(ofer > v) x P(ask >v) = v –2  P(volume = v) d v = v –1-2  d v as in measurement

35. Conclusion The 100 year Pareto puzzle Is solved by combining The 100 year Logistic Equation of Lotka and Volterra With the 100 year old statistical mechanics of Boltzmann