Wealth Condensation as

a ZRPZ. Burda, J. Jurkiewicz, M. Kaminski,

M.A. Nowak, G. Papp, I. Zahed

Phys. Rev E65 026102

oops!

Wealth Condensation as

a ZUMZ. Burda, J. Jurkiewicz, M. Kaminski,

M.A. Nowak, G. Papp, I. Zahed

Phys. Rev E65 026102

The plan

•Apologies (this is an exercise in re-labelling)

•Context: Pareto, Gibrat

•Obtaining observed distributions

•Finite total wealth

•Balls in boxes (ZRP, ZUM, ASEP...)

The rich (really) are different•Distribution of wealth:

•Majority - log normal:

•Bill Gates and friends:

Pareto (1897)•Pareto looked at personal income for the wealthy:

• Pareto index, found to be between one and two

Gibrat for the rest of us

•Formulated in 1931

• is the Gibrat index

The gentlemen in question ( +1)

Pareto Gibrat Zipf

A Wealth curve

And another

How might one arrive at such

distributions?•Log normal from MSP

Getting a power law

•Poverty bound in MSP

•Drift to is balanced by reflection at

Other ways

•Adding noise

•Pareto Index

Models with individual agents•Flow-like model - Bouchaud

and Mezard

•Generalized Lotka-Volterra, Solomon et.al.

Mean-Field•Mean-field solution of Bouchaud, Mezard

•where

•Express in terms of normalized wealth

Pareto Like distribution

•The steady state distribution

•Pareto exponent greater than one, but can be twiddled

Characterizing the distribution

•Partial wealth

•Inverse participation ratio

Wealth Condensation• Acts as a order parameter

•If one or more is extensive then

• mean wealth is finite

• mean wealth is infinite

•some guy gets

What happens for finite total wealth?

•The non-integrable tail gives the wealth condensation

•So what happens in a finite economy?

Balls in boxes (==ZUM)

•Take Pareto distribution as given

•Z(W,N) is an appropriate normalization

•W balls in N boxes

ZRP, ASEP..

Steady state

•Weights are determined by the jump rates

•Or vice-versa

Solving •Saddle point solution

•where

Solving II

•Saddle point solution

•where is a solution of

•giving

Solving III•Nature of saddle point solution

•As is increased decreases

•At some critical density, saddle point fails

Solving IV

•Effective Probabilities

•Exact calculation

What does this look like? Above threshold

Below and through threshold

Condensation

•Above threshold the effective distribution is bare + delta

Non-condensation

•Below the threshold, damped power law

Inverse Participation ratio

•At threshold it changes

•From zero to

Tinkering with the “economy”

•Suppose we started below threshold

•Increasing decreases

Why did I get interested?

Effective polymer model

Endpiece

•What I haven’t discussed - dynamics i.e. ZRP, ZUM

•Godreche - “zeta-urn”