PARETO POWER LAWS Bridges between microscopic and macroscopic scales

1. PARETO POWER LAWS Bridges between microscopic and macroscopic scales

4. 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 the great generalizations of human knowledge

5. Alfred Lotka the number of authors with n publications in a bibliography is a power law of the form C/n1+aThe exponent  a is often close to 1.

6. LOTKA-VOLTERRA LOGISTIC EQUATIONS History, Applications

7. Franco Scudo, "The 'Golden Age' of theoretical ecology. A conceptual appraisal", Rev.Europ.Etud.Social., 22, 11-64 (1984) Franco Scudo e J.R. Ziegler, "The Golden Age of theoretical ecology: 1923-1949", Berlin, Springer, 1978

8. La ricerca ecologica ha avuto, nel periodo 1920-1940, alcuni "anni d'oro", come li ha definiti Franco Scudo (1), con importanti contributi anche italiani (per esempio di V. Volterra e U. D'Ancona).

9. Malthus : autocatalitic proliferation: dX/dt = a X with a=birth rate - death rate exponential solution: X(t) = X(0)ea t contemporary estimations= doubling of the population every 30yrs

10. Verhulst way out of it: dX/dt = a X – c X2 Solution: exponential==========saturation at X= a / c

11. – c X2= competition for resources and other the adverse feedback effects saturation of the population to the value X= a / c For humans data at the time could not discriminate between exponential growth of Malthus and logistic growth of Verhulst But data fit on animal population: sheep in Tasmania: exponential in the first 20 years after their introduction and saturated completely after about half a century.

12. Confirmations of Logistic Dynamics pheasants turtle dove humans world population for the last 2000 yrs and US population for the last 200 yrs, bees colony growth escheria coli cultures, drossofilla in bottles, water flea at various temperatures, lemmings etc.

13. Elliot W Montroll: Social dynamics and quantifying of social forces“almost all the social phenomena, except in their relatively brief abnormal timesobey the logistic growth''.

14. - default universal logistic behavior generic to all social systems - concept of sociological force which induces deviations from it Social Applications of the Logistic curve: technological change; innovations diffusion (Rogers) new product diffusion / market penetration (Bass) social change diffusion dX/dt ~ X(N – X ) X = number of people that have already adopted the change and N = the total population

15. Sir Ronald Ross  Lotka: generalized the logistic equation to a system of coupled differential equations for malaria in humans and mosquitoes a11 = spread of the disease from humans to humans minus the percentages of deceased and healed humans a12 = rate of humans infected by mosquitoes a112 = saturation (number of humans already infected becomes large one cannot count them among the new infected). The second equation = same effects for the mosquitoes infection

16. Volterra: dXi= Xi (ai- ciF(X1 … , X n) ) Xi = the population of species iai= growth rate of population i in the absence of competition and other species F = interaction with other species: predation competition symbiosis Volterra assumed F =a1 X1+ ……+ an Xn more rigorous  Kolmogorov. MPeshel and W Mende The Predator-Prey Model; Do we live in a Volterra World? Springer Verlag, Wien , NY 1986

17. Mikhailov Eigen equations relevant to market economics i = agents that produce a certain kind of commodity Xi = amount of commodity the agent i produces per unit time The net cost to an individual agent of the produced commodity is Vi = ai Xi ai = specific cost which includes expenditures for raw materials machine depreciation labor payments research etc Price of the commodity on the market is c c (X.,t) = i ai Xi / i Xi

18. The profits of the various agents will then be ri = c (X.,t)Xi - ai Xi Fraction k of it is invested to expand production at rate dXi=k (c (X.,t)Xi - ai Xi ) These equations describe the competition between agents in the free market This ecology market analogy was postulated already in Schumpeter and Alchian See also Nelson and Winter Jimenez and Ebeling Silverberg Ebeling and Feistel Jerne Aoki etc account for cooperation: exchange between the agents dXi=k (c (X.,t)Xi - ai Xi ) +j aij Xj-j aij Xi

19. GLV and interpretations

20. wi (t+t) – wi (t) = ri (t) wi (t) + a w(t) – c(w.,t) wi (t) w(t) is the average of wi (t) over all i ’s at time t a and c(w.,t) are of order t c(w.,t) means c(w1,. . ., wN,t) ri (t) = random numbers distributed with the same probability distribution independent of i with a square standard deviation <ri (t) 2> =Dof ordert One can absorb the average ri (t) in c(w.,t) so <ri (t) > =0

21. wi (t+t) – wi (t) = ri (t) wi (t) + a w(t) – c(w.,t) wi (t) admits a few practical interpretations wi (t) = the individual wealth of the agent i then ri (t) = the random part of the returns that its capital wi (t) produces during the time between t and t+t a = the autocatalytic property of wealth at the social level = the wealth that individuals receive as members of the society in subsidies, services and social benefits. This is the reason it is proportional to the average wealth This term prevents the individual wealth falling below a certain minimum fraction of the average. c(w.,t) parametrizes the general state of the economy: large and positive correspond = boom periods negative =recessions

22. c(w.,t) limits the growth of w(t) to values sustainable for the current conditions and resources external limiting factors: finite amount of resources and money in the economy technological inventions wars , disasters etc internal market effects: competition between investors adverse influence of self bids on prices

23. A different interpretation: a set of companies i = 1, … , N wi (t)= shares prices ~ capitalization of the company i ~ total wealth of all the market shares of the company ri (t) = fluctuations in the market worth of the company ~ relative changes in individual share prices (typically fractions of the nominal share price) aw = correlation between wi and the market index w c(w.,t) usually of the form c w  represents competition Time variations in global resources may lead to lower or higher values of cincreases or decreases in the total w

24. Yet another interpretation: investors herding behavior wi (t)= number of traders adopting a similar investment policy or position. they comprise herd i one assumes that the sizes of these sets vary autocatalytically according to the random factor ri (t) This can be justied by the fact that the visibility and social connections of a herd are proportional to its size aw represents the diffusion of traders between the herds c(w.,t) = popularity of the stock market as a whole competition between various herds in attracting individuals

25. BOLTZMANN POWER LAWS IN GLV

26. Crucial surprising fact as long as the term c(w.,t) and the distribution of the ri (t) ‘s are common for all the i ‘s the Pareto power law P(wi) ~ wi–1-a holds and its exponent a is independent on c(w.,t) This an important finding since the i-independence corresponds to the famous market efficiency property in financial markets

27. take the average in both members of wi (t+t) – wi (t) = ri (t) wi (t) + a w(t) – c(w.,t) wi (t) assuming that in the N = limit the random fluctuations cancel: w(t+t) – w(t) = a w(t) – c(w.,t) w (t) It is of a generalized Lotka-Volterra type with quite chaotic behavior x i (t) = w i (t) / w(t)

28. and applying the chain rule for differentials d xi (t): dxi (t) =dwi (t) / w(t) - w i (t) d (1/w) =dwi (t) / w(t) - w i (t) d w(t)/w2 =[ri (t) wi (t) + a w(t) – c(w.,t) wi (t)]/ w(t) -w i (t)/w [a w(t) – c(w.,t) w (t)]/w = ri (t) xi (t) + a – c(w.,t) xi (t) -x i (t) [a – c(w.,t)]= crucial cancellation : the system splits into a set of independent linear stochastic differential equations with constant coefficients = (ri (t) –a ) xi (t) + a

29. dxi (t) = (ri (t) –a ) xi (t) + a Rescaling in t means rescaling by the same factor in <ri (t) 2> =D and a therefore the stationary asymptotic time distribution P(xi ) depends only on the ratio a/D Moreover, for large enough xithe additive term + a is negligible and the equation reduces formally to the Langevin equation for ln xi (t) d ln xi (t) = (ri (t) – a ) Where temperature = D/2 and force = -a => Boltzmann distribution P(ln xi ) d ln xi ~ exp(-2a/D ln xi ) d ln xi ~ xi-1-2 a/D d xiIn fact, the exact solution isP(xi ) = exp[-2 a/(D xi )] xi-1-2 a/D

30. P(w) 10-1 t=0 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-4 10-3 10-2 10-1 100 w

31. P(w) 10-1 10-2 10-3 10-4 t=10 000 10-5 10-6 10-7 10-8 10-9 10-4 10-3 10-2 10-1 100 w

32. P(w) 10-1 10-2 10-3 10-4 10-5 10-6 10-7 t=100 000 10-8 10-9 10-4 10-3 10-2 10-1 100 w

33. P(w) 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 t=1 000 000 10-9 10-4 10-3 10-2 10-1 100 w

34. P(w) 10-1 10-2 10-3 10-4 10-5 10-6 t=30 000 000 10-7 10-8 10-9 10-4 10-3 10-2 10-1 100 w

35. P(w) 10-1 t=0 10-2 10-3 10-4 10-5 10-6 t=10 000 t=30 000 000 10-7 t=100 000 10-8 t=1 000 000 10-9 10-4 10-3 10-2 10-1 100 w

39. K= amount of wealth necessary to keep 1 alive If wmin < K => revolts L = average number of dependents per average income Their consuming drive the food, lodging, transportation and services prices to values that insure that at each time wmean > KL Yet if wmean < KL they strike and overthrow governments. So c=x min = 1/L Therefore a ~ 1/(1-1/L) ~ L/(L-1) For L = 3 - 4 , a ~ 3/2 – 4/3; for internet L~ average nr of links/ site

40. In Statistical Mechanics, if not detailed balanceno Boltzmann In Financial Markets, if no efficient marketno Pareto

41. Further Analogies Thermal Equilibrium Efficient Market Pareto Law Boltzmann law One cannot extract energy from systems in thermal equilibrium One cannot gain systematically wealth from efficent markets Except for “Maxwell Demons” with microscopic information Except if one has access to detailed private information By extracting energy from non-equilibrium systems , one brings them closer to equilibrium By exploiting arbitrage opportunities, one eliminates them (makes market efficient) Irreversibility II Law of Theromdynamics Entropy Irreversibility ? ?

42. Paul Lévy Drawing by Mendes-France

45. Market Fluctuations Scaling