Continuously%20Tunable%20Pareto%20Exponent%20in%20a

1
Econophys Kolkata I
Continuously Tunable Pareto Exponent in a Random
Shuffling Money Exchange Model
K. Bhattacharya, G. Mukherjee and S. S. Manna
Satyendra Nath Bose National Centre for Basic
Sciences manna_at_bose.res.in
2
Random pair wise conservative money
shuffling A.A. Dragulescu and V. M. Yakovenko,
Eur. Phys. J. B. 17 (2000) 723.
? N traders, each has money mi (i1,N),
?Ni1miN, ltmgt1 ? Time t number of pair wise
money exchanges ? A pair i and j are selected 1
i,j N, i ? j with uniform probability who
reshuffle their total money
? Result Wealth Distribution in the
stationary state
3
? Fixed Saving Propensity (?) A. Chakraborti and
B. K. Chakrabarti, Eur. Phys. J. B 17 (2000) 167.
? Result Wealth Distribution in the
stationary state
Gamma distribution P(m) ma exp(-bm) Most
probable value mpa/b
4
(No Transcript)
5
? Quenched Saving Propensities (?i, i1,N) A.
Chatterjee, B. K. Chakrabarti, and S. S. Manna,
Physica A, 335, 155 (2004)
? Result Wealth Distribution in the stationary
state
Pareto distribution P(m) m-(1?)
6
(No Transcript)
7

? Dynamics with a tagged trader
? N-th trader is assigned ?max and others 0? lt
?max for 1 i N-1 ? ?max is tuned and
ltm(?max)gt are calculated for different ? ?
ltm(?max)gt diverges like
8
ltm(?max)gt/NN-0.125 G(1-?max)N1.5 where
Gx?x-d as x?0 with d 0.725 ltm(?max)gtN-9/8
(1-?max)-3/4N-9/8 assuming 0.725 ¾ ltm(?max)gt
(1-?max)-3/4 For a system of N traders (1-?max)
1/N. Therefore ltm(?max)gt N3/4
9
? Approaching the Stationary State
? As ?max?1, the time tx required for the N-th
trader to reach the stationary state diverges.
? Scaling shows that tx (1-?max)-1
10
PRESENT WORK
Weighted selection of traders
Rule 1 Probability of selecting the i-th
trader is pi mia where a is a continuously
varying tuning parameter Rule 2 Trading is
done by random pair wise conservative money
exchange as before
11
Money Distribution in the Stationary State
P(m,N) follows a scaling form
Where G(x) ?x-(1?(a)) as x?0 G(x)
?const. as x?1
Results for a2
?(2)1 and ?(2)2 giving ?(2)1
Height of hor. part 1/N2 Length of hor. part
N Area under hor. part 1/N
12
Results for a3/2
?(3/2)3/2 and ?(3/2)1 giving ?(3/2)1/2
13
Results for a1
?(1)0
14
Conclusion ? There are complex inherent
structures in the model with quenched random
saving propensities which are disturbing.
More detailed and extensive study are
required. ? Model with weighted selection of
traders seems to be free from these
problems. Thank you.