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

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

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

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 Efficient market hypothesis +

P ( X ) ~ X –1-  d X

Davis [1941] No. 6 of the Cowles Commission for Research in Economics, 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 Not good for economy ! fixed  distribution with negative drift < 0

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

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

Can one obtain stable power laws in systems with variable growth rates (economies with both recessions and growth periods) ? 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. Yes! in fact the solution is suggested by the fact that:

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 Montroll '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

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 Montroll d X = (a - c X) X

d X i = (a i + c (X.,t)) X i +  j a ij X j Volterra 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) d X = (a - c X) 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(X.,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(X.,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 For #i >> e /a :

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

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)

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

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) = [ rand i (t) X i +c(X.,t) X i + aX ]/ X =dX i (t) / X(t) - X i (t) d X(t)/X 2

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) = [ rand i (t) X i +c(X.,t) X i + aX ]/ X =dX i (t) / X(t) - X i (t) d X(t)/X 2 -X i /X [ c(X.,t) X + a X ]/X

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) = [ rand i (t) X i +c(X.,t) X i + aX ]/ X = rand i (t) x i + c(X.,t) x i + a =dX i (t) / X(t) - X i (t) d X(t)/X 2 -X i /X [ c(X.,t) X + a X ]/X

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) = [ rand i (t) X i +c(X.,t) X i + aX ]/ X = rand i (t) x i + c(X.,t) x i + a =dX i (t) / X(t) - X i (t) d X(t)/X 2 -X i /X [ c(X.,t) X + a X ]/X -x i (t) [ c(X.,t) + a ] =

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) = [ rand i (t) X i +c(X.,t) X i + aX ]/ X = rand i (t) x i + c(X.,t) x i + a =dX i (t) / X(t) - X i (t) d X(t)/X 2 -X i /X [ c(X.,t) X + a X ]/X -x i (t) [ c(X.,t) + a ] =

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) = [ rand i (t) X i +c(X.,t) X i + aX ]/ X = rand i (t) x i + + a =dX i (t) / X(t) - X i (t) d X(t)/X 2 -X i /X [ c(X.,t) X + a X ]/X -x i (t) [ + a ] =

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) = [ rand i (t) X i +c(X.,t) X i + aX ]/ X = rand i (t) x i + + a =dX i (t) / X(t) - X i (t) d X(t)/X 2 -X i /X [ c(X.,t) X + a X ]/X -x i (t) [ + a ] = = ( rand i (t) –a ) x i (t) + a

dx i (t) = ( rand 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

dx i (t) = ( rand 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 Depending on details Robust

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

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

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

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

Market Fluctuations in the Lotka-Volterra-Boltzmann model

Rosario Mantegna and Gene Stanley The distribution of price variations as a function of the time interval 

Rosario Mantegna and Gene Stanley The distribution of price variations as a function of the time interval   The relative probability of the price being the same after  as a function of the time interval 

Rosario Mantegna and Gene Stanley The distribution of price variations as a function of the time interval  The relative probability of the price being the same after  as a function of the time interval  P(0,  –  

Rosario Mantegna and Gene Stanley The distribution of price variations as a function of the time interval  The relative probability of the price being the same after  as a function of the time interval  P(0,  –  

Prediction of The Lotka- Volterra- Boltzmann model:  

The relative probability of the price being the same as a function of the time interval M. Levy S.S   Prediction of The Lotka- Volterra- Boltzmann model:  

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

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

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

PS NOTE: The main point is NOT the detailed form of the L-V Equation BUT the coupling between the dynamics of the individuals and the average of the entire ensemble. As such, considering the autocatalytic (multiplicative) dynamics of a single element (as often suggested) does not solve the problem of systems with variable growth (/recession) rate (I.e does not get us past the Kesten result).