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

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

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

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

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

No. 6 of the Cowles Commission for Research in Economics, 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

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

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

 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

 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

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

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)

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)

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

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

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

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:

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 ] =

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

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

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

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

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

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 ~ (ok US, Isr)

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 ~ (ok US, Isr) =>  ~ ; Pareto measured  ~ 1.4

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

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

Market Fluctuations

Paul Levy

Gene Stanley

Paul LevyGene Stanley (see him here in person)

M. Levy

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

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

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