POWER LAWS Bridges between microscopic and macroscopic scales.

POWER LAWS

Bridges between microscopic and macroscopic scales

Wolff's [1996] findings regarding the holdings of the top 1%, top 5%, and top 10% of the U.S. population in 1992 are reported in Table

Table 2k Pk

1% 37.2%

5% 60.0%

10% 71.8%

Pk denotes the percentage of total wealth

held by the top k percent

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

LOGISTIC EQUATIONS

History, Applications

contemporary estimations= doubling of the population every 30yrs

Malthus : autocatalitic proliferation:

dX/dt = a X with a =birth rate - death rateexponential solution: X(t) = X(0)ea t

Verhulst way out of it: dX/dt = a X – c X2

Solution: exponential ==========saturation at X= a / c

– 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.

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.

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

- 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

X = number of people that have already adopted the change and

N = the total population

dX/dt ~ X(N – X )

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

Xi = the population of species i

ai = growth rate of population i in the

absence of competition and other species

F = interaction with other species: predation competition symbiosis

Volterra assumed F =1 X1 + ……+ n Xn more rigorous Kolmogorov.

Volterra:

MPeshel and W Mende The Predator-Prey Model;

Do we live in a Volterra World? Springer Verlag, Wien , NY 1986

d Xi = Xi (ai - ci F ( X1 … , X n ) )

Eigen Darwinian selection and evolution in prebiotic environments.

Autocatalyticity = The fundamental property of:

life ; capital ; ideas ; institutions

each type i of DNA and RNA molecules

replicate in the presence of proteins at rate = ai

Typos: instead of generating an identical molecule of type i

probability rate aij for a mutated molecule of type j

d Xi = Xi (ai - cii Xi - j cij Xj) +j aij Xj -j aij Xi

cij = competition of the replicators for various resources

saturation

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

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

d Xi = 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

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

Eigen: aij variability of the population

adaptability and survivaleconomic system: social security or

some form of mutual help

conglomerates with aij

more stable in a stochastically changing environment

agents that are not fit now might become later the fittest

GLV and interpretations

wi (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

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> =D of order

One can absorb the average ri (t) in c(w.,t) so

< ri (t) > =0

wi (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+

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

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

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

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

POWER LAWS IN GLV

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-

holds and its exponent is independent on c(w.,t)

This an important finding since the i-independence corresponds

to the famous market efficiency property in financial markets

take the average in both members of

wi (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+) – 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)

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

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 xi the 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(-2 a/D ln xi ) d ln 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

10010-3 10-2 10-110-4

10-9

10-4

10-5

10-6

10-7

10-8

10-1

10-2

10-3

t=0P(w)

w

10010-3 10-2 10-110-4

10-9

10-4

10-5

10-6

10-7

10-8

10-1

10-2

10-3

t=10 000

P(w)

w

10010-3 10-2 10-110-4

10-9

10-4

10-5

10-6

10-7

10-8

10-1

10-2

10-3

t=100 000

P(w)

w

10010-3 10-2 10-110-4

10-9

10-4

10-5

10-6

10-7

10-8

10-1

10-2

10-3

t=1 000 000

P(w)

w

10010-3 10-2 10-110-4

10-9

10-4

10-5

10-6

10-7

10-8

10-1

10-2

10-3

t=30 000 000

P(w)

w

10010-3 10-2 10-110-4

10-9

10-4

10-5

10-6

10-7

10-8

10-1

10-2

10-3

t=0

t=10 000

t=100 000

t=1 000 000

t=30 000 000

P(w)

w

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 ~ 1/(1-1/L) ~ L/(L-1)

For L = 3 - 4 , ~ 3/2 – 4/3; for internet L average nr of links/ site

In Statistical Mechanics,

if not detailed balance no BoltzmannIn Financial Markets,

if no efficient market no Pareto

Thermal Equilibrium Efficient Market

Further Analogies

Boltzmann law

One cannot extract energy from systems in thermal equilibrium

Except for “Maxwell Demons” with microscopic information

By extracting energy from non-equilibrium systems , one brings them closer to equilibrium

Irreversibility

II Law of Theromdynamics

Entropy

Pareto Law

One cannot gain systematically wealth from efficent markets

Except if one has access to detailed private information

By exploiting arbitrage opportunities, one eliminates them (makes market efficient)

Irreversibility

?

?

Market Fluctuations Scaling

Feedback

Volatility Returns

=> Long range Volatility correlations