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Physics, Economics and Ecology

Sorin Solomon, Hebrew University of Jerusalem. Physics, Economics and Ecology. Boltzmann, Pareto and Volterra. Pavia Sept 8 , 2003. Franco M.Scudo (1935-1998). Lotka. Volterra. +. + c i. ( X.,t ) ) X i +  j a ij X j. d X i = ( a i. Lotka. Volterra. Boltzmann. (. ). +. x.

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Physics, Economics and Ecology

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

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

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

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

  5. Alfred Lotka the number P(n) of authors with n publicationsis a power law P(n) ~ n-1-a with a ~ 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 thegreat generalizations of human knowledge

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

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

  9. dx=h (t)x + r => P(x) dx ~ x–1-ad x forfixedhdistribution withnegative drift< ln 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

  10. dx=h (t)x + r => P(x) dx ~ x–1-ad x forfixedhdistribution withnegative drift< ln 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

  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. 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) Montroll dXi =(ai + c (X.,t))Xi +j aij Xj

  13. 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) Volterra Lotka Montroll Scudo Eigen dXi =(ai + c (X.,t))Xi +j aij Xj

  14. 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

  15. 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) )

  16. 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(w.,t)

  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(w.,t) Proof:

  18. 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(w.,t) Xi+ aX ]/ X -Xi/X [c(w.,t) X + a X ]/X = randi (t) xi + c(w.,t) xi + a -x i (t) [c(w.,t) + a ]= = (randi (t) –a ) xi (t) + a

  19. dxi (t) = (ri (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 Even for very unsteady fluctuations of c; X

  20. dxi (t) = (ri (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

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

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

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

  24. 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)

  25. 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

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

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

  28. Market Fluctuations

  29. Paul Levy

  30. Paul Levy Gene Stanley

  31. Paul Levy Gene 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 a ~ 3

  34. One more puzzle: For very dense (trade-by-trade) measurements and very large volumes the tails go like 2 a ~ 3 Explanation: Volume of trade = minimum of ofer size and ask size P(volume > v) = P(ofer > v) x P(ask >v) = v –2 aP(volume = v) d v = v –1-2 a 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

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