Influence of saving propensity on the power law tail of wealth distribution

1
Influence of saving propensity on the power law
tail of wealth distribution

• Marco Patriarca(1,3), Anirban Chakraborti(2) and
  Guido Germano(1)
• (1) Fachbereich Chemie, Philipps-Universität
  Marburg
• (2) Brookhaven National Laboratory, Dep. of
  Physics, Upton, NY
• (3) Current address Theoretische Physik 1,
  Institut für Physik, Universität Augsburg

2
Motivations

• Kinetic multi-agents models make it possible to
  construct a simple quantitative microscopic
  models of a market economy which can reproduce
  some general relevant features of wealth
  distributions such as
• zero limit at small wealth
• a mode larger than zero
• an exponential tail at low and intermediate
  wealth
• a power law at large wealth
• a percentage of richest agents of a few
• Money here uses as a conserved, quantitatively
  well defined quantity which measures of all
  economic goods which contribute to wealth

3
Topics

• Generalities about kinetic multi-agent models
• I. The basic model
• II. The model with global saving propensity ?0 ?
  (0,1)
• III. The model with individual saving
  propensities ? i ? (0,1)
• The richest agents influence saving propensity
  distribution on wealth distribution
• Some examples of realistic distributions

4
3
1
2
j
4
?x
i
N-2
N-1
N
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I. Basic Model System 1,2

• N units (agents)
• Assign initial walth xi
• At every time step t two agents k and j are
  extracted at random
• x is re-distributed at random between k and j
• (r random number between 0 and 1)

• Time evolution is carried out until thermal
  equilibrium is reached

1 A. Dragulescu and V. M. Yakovenko,
Statistical mechanics of money, Eur. Phys. J. B
17 (2000) 723. 2 E. Bennati La simulazione
statistica nell'analisi della distribuzione del
reddito modelli realistici e metodo di
Montecarlo, ETS Editrice, Pisa,1988. Un metodo di
simulazione statistica nell'analisi della
distribuzione del reddito, Rivista Internazionale
di Scienze Economiche e commerciali, August
(1988) 735. Il metodo Montecarlo nell'analisi
economica, Rassegna di lavori dell'ISCO (4)
(1993) 31
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Equilibrium Distribution
i.e. the Boltzmann distribution, where ?x? is the
average value of x.

• The model is robust in that the corresponding
  Gibbs distribution is obtained in different
  conditions
• Different initial distributions of x
• Pairwise as well as multi-agent interactions
• Constant as well as random ?x
• Random, first neighbor, as well as consecutive
  selection of the interacting agents
• Various linear forms of ?x
• Rapid convergence to equilibrium distribution
  also for a very small number of agents

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II. Model with saving propensity ? (0 lt ? lt 1)
3,4,5

• N units (agents) with wealth xi
• At every time step t extract randomly two agents
  k and j
• x is then redistributed randomly between k and
  j , r random number in (0,1)

3 A. Chakraborti, PhD Thesis. 4 A.
Chakraborti and B. K. Chakrabarti, Eur. Phys. J.
B 17, 167 (2000). 5 A. Chakraborti, Int. J.
Mod. Phys. C 13, 1315 (2002)
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Equilibrium Distribution
The equilibrium distribution is a
gamma-distribution
where x is the average x and an the
normalization constant,
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(No Transcript)
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III. Model with individual saving propensities ?n
(0 lt ?n lt 1) 1,2

• N units (agents) with moneys xi
• At every time step t extract randomly two agents
  k and j .
• money is then redistributed randomly between k
  and j according to a dynamical money-conserving
  (stochastic) dynamics

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Equilibrium Distribution
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The Boltzmann equation approach 1,2,3,4

• The Boltzmann equation approach to multi-agent
  systems can predict many features of the wealth
  distribution.
• For instance, the power law for the
  distribution function is - 2 1,2,3,4 .
• The first 2 moments identical to those of the
  gamma-distribution 1.
• One can also predict the asymtotic time
  dependence 4.
• Also, pretious information about the role of
  agents with given (high) values of saving
  propensity can be obtained.

1 Das and Yarlagadda, A distribution function
analysis of wealth distribution, URL
arXivcond-mat/0310343. 2 P. Repetowicz, S.
Hutzler, and P. Richmond, Dynamics of Money and
Income Distributions, Phisica A 3 A. Chatterjee
and B. K. Chakrabarti and R. B. Stinchcombe,
Master equation for a kinetic model of trading
market and its analytic solution,
cond-mat/0501413 and Analyzing money
distributions in ideal gas models of markets
physics/0505047 4 S. Ispolatov, P. Krapivsky,
and S. Redner, Wealth distributions in asset
exchange models, Eur. Phys. J. B 2, 267 (1998).
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Correlation between money and saving propensity

• In models with individual saving propensity there
  is a correlation between
• the individual saving propensities ?n and the
  corresponding money xn .
• This happens quite generally, for random as well
  as deterministic assignement
• of ?n , for power or nonpower laws, for all
  distributions of ?n .
• dots are single agents,
• the continuous
• line is the average
• over x 1 / (1 ?).

.
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Richer agents remain richer(only poor agent
wealth varies appreciably during one trade)

• Notice that rich agents will never risk all
  their money they have a large saving propensity
  (? 1) and therefore a very low effective
  temperature
• T (1- ? )
• Thus they only invest a small amount of money in
  a trade
• This is shown by the small width of the
  x(t1)-x(t) map at large values of x.

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Thermodynamical modelling
T1
T2
N N1 N2 N3 N1 have saving propensity
?1, N2 have saving propensity ?2 N3 have
saving propensity ?3 ?x? ?x?i 1 / (1
?i) D 2 (1 2 ?i) / (1 ? ?i) T 2 ?x?i / D
?x?i (1 ? ?i) / (1 2 ?i)
D1
D2
T3
D3
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Influence of ?-cutoff
Example of a system with 100 000 agents and a
uniform ?-distribution from ? 0 up to ??M.
The wealth distribution ends at a cut-off
determined in turn by the cutoff of the saving
propensity distribution. The richest agent is
that with the highest saving propensity, as it
follows by the x-? correlation.
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Influence of ?-cutoff
The tail of the ?-distribution can influence in
turn the tail of the wealth distribution also
reavealing the single-agent structure at very
high ?.
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Toward a more realistic distribution
Following a suggestion by A. Chatterjee, B. K.
Chakrabarti, and S. S. Manna Physica Scripta
T106, 36 (2003) one can prepare a ?-distribution
with a small fraction of agents with uniformly
distributed ? and the rest with a fixed. Here is
an example with 99 of agents with ?0.2 and 1
with uniformly distributerd ?
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Conclusions

• Subsystems with a given ?i relax to
  exponential distributions
• They behave has open, coupled subsystems with
  their own temperature Ti and effective dimension
  Di .
• Power law are naturally obtained from the
  superpositions of such distributions
• Realistic shapes of wealth distributions can be
  obtained from suitable
• ?-distributions.

• Sponsors
• Chemistry Department, University of Marburg
• Numerical computations were partially carried
  out on the facilities of the Laboratory of
  Computational Engineering, Helsinki University of
  Technology, under support by the Academy of
  Finland, Research Centre for Computational
  Science and Engineering, project no. 44897
  (Finnish Centre for Excellence Program
  2000-2005).
• The work at Brookhaven National Laboratory was
  carried out under Contract No. DE-AC02-8CH10886,
  Division of Material Science, U.S. Department of
  Energy.