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Wealth Condensation as a ZRP

Wealth Condensation as a ZRP. oops!. Z. Burda, J. Jurkiewicz, M. Kaminski, M.A. Nowak, G. Papp, I. Zahed Phys. Rev E65 026102. Wealth Condensation as a ZUM. Z. Burda, J. Jurkiewicz, M. Kaminski, M.A. Nowak, G. Papp, I. Zahed Phys. Rev E65 026102. The plan.

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Wealth Condensation as a ZRP

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  1. Wealth Condensation as a ZRP oops! • Z. Burda, J. Jurkiewicz, M. Kaminski, • M.A. Nowak, G. Papp, I. Zahed • Phys. Rev E65 026102

  2. Wealth Condensation as a ZUM • Z. Burda, J. Jurkiewicz, M. Kaminski, • M.A. Nowak, G. Papp, I. Zahed • Phys. Rev E65 026102

  3. 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...)

  4. The rich (really) are different • Distribution of wealth: • Majority - log normal: • Bill Gates and friends:

  5. Pareto (1897) • Pareto looked at personal income for the wealthy: • Pareto index, found to be between one and two

  6. Gibrat for the rest of us • Formulated in 1931 • is the Gibrat index

  7. The gentlemen in question ( +1) Pareto Gibrat Zipf

  8. A Wealth curve

  9. And another

  10. How might one arrive at such distributions? • Log normal from MSP

  11. Getting a power law • Poverty bound in MSP • Drift to is balanced by reflection at

  12. Other ways • Adding noise • Pareto Index

  13. Models with individual agents • Flow-like model - Bouchaud and Mézard • Generalized Lotka-Volterra, Solomon et.al.

  14. Mean-Field • Mean-field solution of Bouchaud, Mézard • where • Express in terms of normalized wealth

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

  16. Characterizing the distribution • Partial wealth • Inverse participation ratio

  17. 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

  18. What happens for finite total wealth? • The non-integrable tail gives the wealth condensation • So what happens in a finite economy?

  19. Balls in boxes (==ZUM) • Take Pareto distribution as given • Z(W,N) is an appropriate normalization • W balls in N boxes

  20. ZRP, ASEP..

  21. Steady state • Weights are determined by the jump rates • Or vice-versa

  22. Solving • Saddle point solution • where

  23. Solving II • Saddle point solution • where is a solution of • giving

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

  25. Solving IV • Effective Probabilities • Exact calculation

  26. What does this look like? Above threshold

  27. Below and through threshold

  28. Condensation • Above threshold the effective distribution is bare + delta

  29. Non-condensation • Below the threshold, damped power law

  30. Inverse Participation ratio • At threshold it changes • From zero to

  31. Tinkering with the “economy” • Suppose we started below threshold • Increasing decreases

  32. Why did I get interested?

  33. Effective polymer model

  34. Endpiece • What I haven’t discussed - dynamics i.e. ZRP, ZUM • Godreche - “zeta-urn”

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