Sorin Solomon, Hebrew University of Jerusalem

1. Sorin Solomon, Hebrew University of Jerusalem Physics, Economics and Ecology Boltzmann, Pareto and Volterra

2. Sorin Solomon, Hebrew University of Jerusalem Physics, Economics and Ecology Boltzmann, Pareto and Volterra

3. Lotka Volterra + + ci (X.,t))Xi +j aij Xj dXi=( ai

4. Lotka Volterra Boltzmann ( ) + x + ci (X.,t))Xi +j aij Xj dXi=(randi

5. Lotka Volterra Boltzmann ( ) + x + ci (X.,t))Xi +j aij Xj dXi=(randi Efficient market hypothesis +

6. Lotka Volterra Boltzmann ( ) + x + c (X.,t))Xi +j aij Xj dXi=(randi Efficient market hypothesis = + P(Xi) ~ Xi–1-adXi Pareto

7. P(X) ~ X–1-adX

8. Davis [1941] No. 6 of the Cowles Commission for Research in Economics, 1941. 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 thegreat generalizations of human knowledge

9. dx=h (t)x + r => P(x) dx ~ x–1-ad x fixedhdistribution withnegative drift< ln1+h > < 0 Not good for economy !

10. dx=h (t)x + r => P(x) dx ~ x–1-ad x fixedhdistribution withnegative drift< ln1+h > < 0 Not good for economy ! Herbert Simon; intuitive explanation d ln x (t) = h (t) + lower bound = diffusion + down drift + reflecting barrier

11. dx=h (t)x + r => P(x) dx ~ x–1-ad x fixedhdistribution withnegative drift< ln(1+h) > < 0 Not good for economy ! Herbert Simon; intuitive explanation d ln x (t) = h (t) + lower bound = diffusion + down drift + reflecting barrier • Boltzmann (/ barometric) distribution for ln x P(ln x ) d ln x ~ exp(- aln x) d ln x

12. dx=h (t)x + r => P(x) dx ~ x–1-ad x fixedhdistribution withnegative drift< ln1+h > < 0 Not good for economy ! Herbert Simon; intuitive explanation d ln x (t) = h (t) + lower bound = diffusion + down drift + reflecting barrier • Boltzmann (/ barometric) distribution for ln x P(ln x ) d ln x ~ exp(- aln x) d ln x ~ x-1-a d x Pareto

13. Can one obtain stable power laws in systems with variable growth rates (economies with both recessions and growth periods) ? Yes! in fact the solution is suggested by the fact that: The list of systems with Power laws The list of systems described traditionally by the logistic Lotka-Volterra equations all one has to do is to recognize the statistical character of the Logistic Equation and introduce the noise term representing it.

14. 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) 'I would urge that people be introduced to the logistic equation early in their education… Not only in research but also in the everyday world of politics and economics … Sir Robert May Nature Volterra Montroll

15. 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) dX=(a-c X) X Volterra Montroll

16. 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) Lotka Volterra Montroll Eigen dX=(a-c X) X dXi =(ai + c (X.,t))Xi +j aij Xj

17. Stochastic Generalized Lotka-Volterra dXi =(rand i (t)+ c (X.,t))Xi +j aij Xj for clarity take jaij Xj = a / Nj Xj = a X Assume Efficient market: P(rand i (t) )= P(rand j (t) ) => THENthe Pareto power law P(Xi ) ~ X i–1-a holds with aindependent on c(X.,t)

18. Stochastic Generalized Lotka-Volterra dXi =(rand i (t)+ c (X.,t))Xi +j aij Xj for clarity take jaij Xj = a / Nj Xj = a X Assume Efficient market: P(rand i (t) )= P(rand j (t) ) => THENthe Pareto power law P(Xi ) ~ X i–1-a holds with aindependent on c(X.,t) Proof:

19. dXi =(rand i (t)+ c (X.,t) )Xi + a X

20. For #i >> e <rand2>/a: dXi =(rand i (t)+ c (X.,t) )Xi + a X dX=c (X.,t) )X+ a X

21. dXi =(rand i (t)+ c (X.,t) )Xi + a X Logistic eq. if c(X.,t)= -c X; dX=c (X.,t) )X+ a X Else=> chaos, etc; In any case: following analysis holds:

22. dXi =(rand i (t)+ c (X.,t) )Xi + a X dX=c (X.,t) )X+ a X Denote x i (t) = Xi (t) / X(t) Then dxi (t) = dXi (t)/ X(t) + X i (t) d (1/X)

23. dXi =(rand i (t)+ c (X.,t) )Xi + a X dX=c (X.,t) )X+ a X Denote x i (t) = Xi (t) / X(t) Then dxi (t) = dXi (t)/ X(t) + X i (t) d (1/X) =dXi (t) / X(t) - X i (t) d X(t)/X2

24. dXi =(rand i (t)+ c (X.,t) )Xi + a X dX=c (X.,t) )X+ a X Denote x i (t) = Xi (t) / X(t) Then dxi (t) = dXi (t)/ X(t) + X i (t) d (1/X) =dXi (t) / X(t) - X i (t) d X(t)/X2 = [randi (t) Xi +c(X.,t) Xi+ aX ]/ X

25. dXi =(rand i (t)+ c (X.,t) )Xi + a X dX=c (X.,t) )X+ a X Denote x i (t) = Xi (t) / X(t) Then dxi (t) = dXi (t)/ X(t) + X i (t) d (1/X) =dXi (t) / X(t) - X i (t) d X(t)/X2 = [randi (t) Xi +c(X.,t) Xi+ aX ]/ X -Xi/X [c(X.,t) X + a X ]/X

26. dXi =(rand i (t)+ c (X.,t) )Xi + a X dX=c (X.,t) )X+ a X Denote x i (t) = Xi (t) / X(t) Then dxi (t) = dXi (t)/ X(t) + X i (t) d (1/X) =dXi (t) / X(t) - X i (t) d X(t)/X2 = [randi (t) Xi +c(X.,t) Xi+ aX ]/ X -Xi/X [c(X.,t) X + a X ]/X = randi (t) xi + c(X.,t) xi + a

27. dXi =(rand i (t)+ c (X.,t) )Xi + a X dX=c (X.,t) )X+ a X Denote x i (t) = Xi (t) / X(t) Then dxi (t) = dXi (t)/ X(t) + X i (t) d (1/X) =dXi (t) / X(t) - X i (t) d X(t)/X2 = [randi (t) Xi +c(X.,t) Xi+ aX ]/ X -Xi/X [c(X.,t) X + a X ]/X = randi (t) xi + c(X.,t) xi + a -x i (t) [c(X.,t) + a ]=

28. dXi =(rand i (t)+ c (X.,t) )Xi + a X dX=c (X.,t) )X+ a X Denote x i (t) = Xi (t) / X(t) Then dxi (t) = dXi (t)/ X(t) + X i (t) d (1/X) =dXi (t) / X(t) - X i (t) d X(t)/X2 = [randi (t) Xi +c(X.,t) Xi+ aX ]/ X -Xi/X [c(X.,t) X + a X ]/X = randi (t) xi + c(X.,t) xi + a -x i (t) [c(X.,t) + a ]=

29. dXi =(rand i (t)+ c (X.,t) )Xi + a X dX=c (X.,t) )X+ a X Denote x i (t) = Xi (t) / X(t) Then dxi (t) = dXi (t)/ X(t) + X i (t) d (1/X) =dXi (t) / X(t) - X i (t) d X(t)/X2 = [randi (t) Xi +c(X.,t) Xi+ aX ]/ X -Xi/X [c(X.,t) X + a X ]/X = randi (t) xi + + a -x i (t) [ + a ]=

30. dXi =(rand i (t)+ c (X.,t) )Xi + a X dX=c (X.,t) )X+ a X Denote x i (t) = Xi (t) / X(t) Then dxi (t) = dXi (t)/ X(t) + X i (t) d (1/X) =dXi (t) / X(t) - X i (t) d X(t)/X2 = [randi (t) Xi +c(X.,t) Xi+ aX ]/ X -Xi/X [c(X.,t) X + a X ]/X = randi (t) xi + + a -x i (t) [ + a ]= = (randi (t) –a ) xi (t) + a

31. dxi (t) = (randi (t) –a ) xi (t) + a of Kesten type: dx=h (t) x + r and has constant negative drift ! Power law for large enough xi : P(xi ) d xi ~ xi-1-2 a/D d xi In fact, the exact solution is: P(xi ) = exp[-2 a/(D xi )]xi-1-2 a/D

32. dxi (t) = (randi (t) –a ) xi (t) + a of Kesten type: dx=h (t) x + r and has constant negative drift ! Power law for large enough xi : P(xi ) d xi ~ xi-1-2 a/D d xi In fact, the exact solution is: P(xi ) = exp[-2 a/(D xi )]xi-1-2 a/D Even for very unsteady fluctuations of c; X Depending on details Robust

33. Prediction: a =(1/(1-minimal income /average income)

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

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

36. Prediction: a =(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)

37. Prediction: a =(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) => a ~ 1.3-1.5 ; Pareto measured a ~ 1.4

38. Green gain statistically more (by 1 percent or so) M.Levy

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

40. P(x) ~ exp (-E(x) /kT) P(x) ~ x –1-a d x 1886 1897 In Statistical Mechanics, Thermal Equilibrium Boltzmann In Financial Markets, Efficient Market Pareto

41. Market Fluctuations in the Lotka-Volterra-Boltzmann model

42. 1000 50 1 Rosario Mantegna and Gene Stanley The distribution of price variations as a function of the time interval t

43. -1/b Rosario Mantegna and Gene Stanley The distribution of price variations as a function of the time interval t The relative probability of the price being the same after t as a function of the time interval t

44. -1/b Rosario Mantegna and Gene Stanley The distribution of price variations as a function of the time interval t The relative probability of the price being the same after t as a function of the time interval t P(0,t) ~ t –1/b

45. -1/b Rosario Mantegna and Gene Stanley The distribution of price variations as a function of the time interval t The relative probability of the price being the same after t as a function of the time interval t P(0,t) ~ t –1/b

46. -1/b Prediction of The Lotka-Volterra-Boltzmann model: b = a

47. a b M. Levy S.S -1/b The relative probability of the price being the same as a function of the time interval Prediction of The Lotka-Volterra-Boltzmann model: b = a

48. One more puzzle: For very dense (e.g. trade-by-trade) measurements and /or very large volumes the tails go like2 a ~ 3

49. One more puzzle: For very dense (e.g. trade-by-trade) measurements and /or very large volumes the tails go like2 a ~ 3 Explanation: P(trade volume > v) = P(ofer > v) x P(ask >v) = v – ax v – a= v –2 aP(volume = v) d v = v–1-2 ad v as in measurement

50. 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