Modelos de Distribuição de
Riqueza: Desigualdade e
Criminalidade
José Roberto Iglesias
I.F. e F.C.E. - U.F.R.G.S., Porto Alegre, Brazil
Rio de Janeiro, 9 de novembro 2007
Porto Alegre
(30o S)
Econofis 2007
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Colaboradores
Porto Alegre (Brasil):
• Gaspar Machado Caon
• Sebastián Gonçalves
• Tiago Bonacina
• Vanessa Hoffmann
• Viktoriya Semeshenko
S. C. de Bariloche
(Argentina)
• Miguel Fuentes
• Marcelo Kuperman
• Guillermo Abramson
• Sebastián Risau-Gusman
• M. Fabiana Laguna
França:
• Mirta Gordon
• J.-Pierre Nadal
Mérida (México)
• Cristian Moukarzel
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Motivation...
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Pareto´s law
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Pareto’s law
N ( x) 
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A
( x  a)α
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The exponential + power law behavior
(Dragulescu & Yakovenko, 2001)
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Brazilian income distribution
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Inequality (Gini coefficient)
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Gini coefficient Map
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Models of Wealth Distribution
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Non-interacting agents
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A model with fluctuations and interactions
Other non-conservative
model: Richmond –
Solomon (LotkaVolterra)
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Interacting conservative models
 Agents are molecules of an ideal gas,
that exchange money as molecules
exchange energy.
wi (t  t )  wi (t )  w
 The simplest version (D-Y 2000)
delivers a Boltzmann – Gibbs
(exponential) distribution
w j (t  t )  w j (t )  w
 Many authors (Iglesias et. al.; A.
Chatterjee et. al.) introduced a kind of
multiplicative noise: “saving propensity”
and are able to obtain power laws
distribution
 Effect of size of market and taxes
(Ausloss, Pekalski, 2006)
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A model with Risk Aversion
A random (or not) fraction,
, of the agent´s wealth is
saved
 An agent site with wealth w1
exchanges with another of
wealth w2 a quantity (fair
rule):

dw  min[(1  1 )w1; (1   2 )w2 ]
Or: The winner takes all,
(winner rule) he receives dw
from his partner.

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What happens?
Condensation (or a frozen
society, where just one agent
concentrates all the wealth)
This transaction occurs with a
given probability of favoring
the poorer agent p, being either
p fixed for all the agents or p
given by:
p
1
w  w1
f 2
2
w2  w1
 being f : 0  f  0.5
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Effect of Risk aversion and pexch
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About wealth condensation
The European Physical Journal – Special Topics, 143, 75-79 (2007)
p*  
log 
2
log

Critical line for condensation
(Moukarzel et al, 2007)
Before arriving to condensation the system exhibits power law
With growing Pareto exponent, leading to a “thermal death of
trade”.
But the final state is one of equilibrium but with no equipartition
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About exchange models:
“Man is an animal that makes bargains: no
other animal does this - no dog exchanges
bones with another”
Adam Smith
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Comparing exchange rules
“Fair” Rule
(equal opportunities)
dw  min[(1  1 )w1; (1  2 )w2 ]
A
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B
Rule the winner takes all
dwwinner  (1  loser )wloser
A
B
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Hierarchy evolution
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Interactions within the same “economic” class
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Wealth
depending
interactions
Agents only
interact when
their wealth is
within a
threshold u
|wi-wk| < u
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Wealth distribution vs. time
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And if correlations are included
between risk-aversion and expected profits?
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Wealth distribution of “rational” agents
The poorer agent changes strategy
N=100.000 agents
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initial
wealth uniformly distributed {0,1000}
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Gini, red points
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Wealth distribution of
“irrational” agents
The richer agent changes
strategy
Econofislaw
2007 exponent –1.125
Power
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Gini: green points,
(blue points, poorer agent
Change strategy)
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Economic effects of criminality
M.B. Gordon, J.R.I., J.P. Nadal, V. Semeshenko (to appear)
Agents are characerized by its wealth wi and
a honesty coefficient Hi (changing)
Each “month” agents perceive a wage
Number of crime attemps each month
depends on <H>
p0 is probability of punishment of small
felonies and p1 of big ones. It varies as a
function of booty
Punishment consist in prison + fines
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Criminality
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Economic indicators
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Time evolution
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Wealth distribution
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Honesty distribution
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Conclusions
 Probability of punishment of light and
heavy felonies are correlated.
 Inequality increases and Economic
growth decreases with high criminality
 Time evolution is in avalanches
 Future: what treatment to relapses
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It is only by not paying one's bills that one can
hope to live in the memory of the commercial
classes.
Oscar Wilde
Quino
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Publicações: www.if.ufrgs.br/~iglesias
MUITO OBRIGADO!
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