physics, economics and ecology boltzmann, pareto and volterra pavia sept 8, 2003 franco m.scudo...

Physics, Economics and Ecology

Boltzmann, Pareto and Volterra

Pavia Sept 8 , 2003

Franco M.Scudo (1935-1998)

Sorin Solomon, Hebrew University of Jerusalem

+

d Xi = ( ai

Lotka Volterra

+ ci (X.,t)) Xi +j aij Xj

x+( )d Xi = (randi

Lotka Volterra Boltzmann

+ ci (X.,t)) Xi +j aij Xj

x+( )

= P(Xi) ~ Xi –1- d Xi

Lotka Volterra Boltzmann

Pareto

d Xi = (randi + c (X.,t)) Xi +j aij Xj

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, 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 the great generalizations of human knowledge

d x= (t) x + P(x) dx ~ x –

1- d xfor fixed distribution with negative drift < ln > < 0

Not good for economy !

d x= (t) x + P(x) dx ~ x –


Herbert Simon; intuitive explanation


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 –






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 –






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

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

Volterra

Scudo

Lotka

MontrollEigen

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 Xi = (rand i (t)+ c (X.,t) ) Xi +j aij Xj

for clarity take j aij Xj = a / N j Xj = a X



Assume Efficient market: P(rand i (t) )= P(rand j (t) )





=> THEN the Pareto power law P(Xi ) ~ X i –1-

holds with independent on c(w.,t)





=> THEN the Pareto power law P(Xi ) ~ X i –1-

holds with independent on c(w.,t)


Proof:

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

d X = 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) –a ) xi (t) + a

= [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 ] =

dxi (t) = (ri (t) –a ) xi (t) + a of Kesten type: d x= (t) x +

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

dxi (t) = (ri (t) –a ) xi (t) + a of Kesten type: d x= (t) x +

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

Prediction:

=(1/(1-minimal income /average income)

Prediction:


= 1/(1- 1/average number of dependents on one income)

Prediction:


= 1/(1- 1/dependents on one income)

= 1/(1- generation span/ population growth)

Prediction:




3-4 dependents at each moment Doubling of population every 30 years minimal/ average ~ 0.25-0.33 (ok US, Isr)

Prediction:




3-4 dependents at each moment Doubling of population every 30 years minimal/ average ~ 0.25-0.33 (ok US, Isr) => ~ 1.3-1.5 ; 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

Paul Levy Gene Stanley

Paul Levy Gene 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

physics, economics and ecology boltzmann, pareto and volterra pavia sept 8, 2003 franco m.scudo...

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