Wealth Condensation as a ZRP

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Wealth Condensation as a ZRP
oops!

• Z. Burda, J. Jurkiewicz, M. Kaminski,
• M.A. Nowak, G. Papp, I. Zahed
• Phys. Rev E65 026102

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Wealth Condensation as a ZUM

• Z. Burda, J. Jurkiewicz, M. Kaminski,
• M.A. Nowak, G. Papp, I. Zahed
• Phys. Rev E65 026102

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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...)

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The rich (really) are different

• Distribution of wealth
• Majority - log normal
• Bill Gates and friends

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Pareto (1897)

• Pareto looked at personal income for the wealthy
• Pareto index, found to be between one and two

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Gibrat for the rest of us

• Formulated in 1931
• is the Gibrat index

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The gentlemen in question ( 1)
Pareto
Gibrat
Zipf
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A Wealth curve
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And another
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How might one arrive at such distributions?

• Log normal from MSP

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Getting a power law

• Poverty bound in MSP
• Drift to

is balanced by reflection at
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Other ways

• Adding noise
• Pareto Index

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Models with individual agents

• Flow-like model - Bouchaud and Mezard
• Generalized Lotka-Volterra, Solomon et.al.

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Mean-Field

• Mean-field solution of Bouchaud, Mezard
• where
• Express in terms of normalized wealth

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Pareto Like distribution

• The steady state distribution
• Pareto exponent greater than one, but can be
  twiddled

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Characterizing the distribution

• Partial wealth
• Inverse participation ratio

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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

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What happens for finite total wealth?

• The non-integrable tail gives the wealth
  condensation
• So what happens in a finite economy?

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Balls in boxes (ZUM)

• Take Pareto distribution as given
• Z(W,N) is an appropriate normalization
• W balls in N boxes

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ZRP, ASEP..
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Steady state

• Weights are determined by the jump rates
• Or vice-versa

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Solving

• Saddle point solution
• where

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Solving II

• Saddle point solution
• where is a solution of
• giving

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Solving III

• Nature of saddle point solution
• As is increased decreases
• At some critical density, saddle point fails

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Solving IV

• Effective Probabilities
• Exact calculation

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What does this look like? Above threshold
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Below and through threshold
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Condensation

• Above threshold the effective distribution is
  bare delta

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Non-condensation

• Below the threshold, damped power law

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Inverse Participation ratio

• At threshold it changes
• From zero to

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Tinkering with the economy

• Suppose we started below threshold
• Increasing decreases

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Why did I get interested?
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Effective polymer model
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Endpiece

• What I havent discussed - dynamics i.e. ZRP, ZUM
• Godreche - zeta-urn