wealth condensation as a zrp z. burda, j. jurkiewicz, m. kaminski, m.a. nowak, g. papp, i. zahed...
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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”