1
1
EXPLAINING COMPLEX DISTRIBUTIONS WITH SIMPLE
MODELS
A. Chatterjee and B. K. Chakrabarti
TOPICS TO BE COVERED IN THIS CHAPTER:
• Kinetic theory of gases: derivation of Maxwell-Boltzmann
distribution
• Bio-box on L Boltzmann and JW Gibbs
• Gibbs distribution in 1-D, 2-D etc.
• The Asset exchange model: the random exchange and
minimal exchange models - Gibbs distribution and condensate
• Bio-box on MN Saha and SN Bose
• Asset exchange model with savings: Gibbs bulk and power
law tail
• Box on contributions of: J Angle, BK Chakrabarti, A Chatterjee
et al
• Variations of the model giving Gibbs bulk and Pareto tail
Appropriate quotation here
1.1
Kinetic theory of gases
The ideal gas is a very important subject of study in statistical physics. It constitutes a system where the interaction between the particles or molecules is
very weak or negligible, Alternatively, the gas is particularly rarified so that
Econophysics. Sinha, Chatterjee, Chakraborti and Chakrabarti
c 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Copyright ISBN: 3-527-XXXXX-X
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1 EXPLAINING COMPLEX DISTRIBUTIONS WITH SIMPLE MODELS
the particles are typically large distance apart, such that the interaction is very
small. A typical molecule has its centre of mass at r, and let p denote the momentum of its centre of mass. In absence of external force fields, the energy E
of the molecule is given by
p2
E=
+ Eint
(1.1)
2m
where the first term on the right is the kinetic energy of the centre of mass,
while the second ther arises out of the asymmerty (polyatomic molecule) of
the molecule, and is simply the internal energy of rotation or vibration with
respect to the molecular centre of mass. The dilute gas assumption also allows us to neglect any potential energy of interaction with other molecules,
and thus, E is independent of r. The dilute gas approximation also allows
us to treat the translational degrees of freedom classically, while the internal
degrees of freedom are usually dealt using quantum mechanics.
1.1.1
Derivation of Maxwell-Boltzmann distribution
As for the statistical description, we restrict ourselves to a hard-sphere gas
with no intermolecular interactions. The easiest way to realise things is to
estimate the number of collisions which a sample of gas exerts on a planar
boundary surface which is orientated arbitrarily perpendicular to the x-axis.
Thus, the collisions per unit area, or pressure is
N
= ρ N ∆t ∑ v x f (v x ),
A
v x >0
(1.2)
ρ N being the number density of particles, ∆t is the collision time, v x is the
component of velocity along x axis, and m being the mass of each molecule.
Now, the change in momemtun suffered my the molecules on hitting the wall
is given by
(1.3)
∆p = N2mv x = 2mρ N A∆t ∑ v2x f (v x ),
v x >0
and hence the pressure is given by
P=
∆p
= 2mρ N ∑ v2x f (v x ).
A∆t
v x >0
(1.4)
For a stationary gas, we recognize that the sum of all velocities above zero is
equal to half the sum of all the velocities from negative to positive infinity:
P = 2mρ N
∑
v x >0
v2x f (v x ) = 2mρ N
1
2
∑ v2x f (v x )
vx
(1.5)
1.1 Kinetic theory of gases
The above can be rewritten in terms of macroscopic quantities using the ideal
gas law
nRT
= mρ N ∑ v2x f (v x ) = mρ N hv2x i
(1.6)
V
vx
which generalises into
RT
(1.7)
M
where R is the ideal gas constant and M is the molecular weight of the gas. For
an ideal gas in 3 dimensions, hv2 i = 3hv2x i = 3RT/M = 3k B T/m, k B being
the Boltzmann constant. Hence the expression for mean kinetic energy of the
ideal gas would look like
hv2x i =
h Ei =
1
3k B T
m h v2 i =
.
2
2
(1.8)
The distribution funchtion g(v) of velocities would be given by
g(v)dv = f (v x ) f (v y ) f (v z )dv x dvy dvz .
(1.9)
and the partial derivative of g with respective to one of the axes looks like
dg v x
dg ∂v
dg ∂ q 2
∂g
v x + v2y + v2z =
=
=
,
∂v x
dv ∂v x
dv ∂v x
dv v
(1.10)
f (vy ) f (vz ) ∂ f (v x )
1 ∂g
1 dg
=
,
=
v dv
v x ∂v x
vx
∂v x
(1.11)
rearranging,
and what follows from the equivalence of the directions is
f (v x ) f (vy ) ∂ f (vz )
f (vy ) f (vz ) ∂ f (v x )
f (vz ) f (v x ) ∂ f (vy )
=
=
,
vx
∂v x
vy
∂vy
vz
∂v z
(1.12)
and dividing all by g one gets
∂ f (vy )
∂ f (v x )
1
1
∂ f (vz )
1
=
=
= −k
f (v x )v x ∂v x
f (vy )vy ∂vy
f (v z )vz ∂vz
(1.13)
k being some constant indep of the coordinates. Now we have a first order
differential equation for the velocity coordinates, integrating which one gets,
f (v x ) = B exp[−kv2x /2]. To determine value of the integrating constant B, we
note that for any probability distribution, the integral of the distribution over
the entire region of space it encompasses must be unity. Thus,
1=
Z ∞
−∞
f (v x )dv x = B
Z ∞
−∞
exp[−kv2x /2]dv x ,
(1.14)
3
4
1 EXPLAINING COMPLEX DISTRIBUTIONS WITH SIMPLE MODELS
q
q
k
k
, and thus f (v x ) = 2π
exp[−kv2x /2]. Thus, Eqn.(1.9)
which yields, B = 2π
can be rewritten as:
3/2
k
g(v) =
exp[−kv2x /2] exp[−kv2y /2] exp[−kv2z /2]
(1.15)
2π
The mean kinetic energy is thus
h Ei =
3m
2
k
2π
3/2 Z
∞
−∞
v2x exp[−kv2x /2]dv x
Z ∞
−∞
exp[−kv2y /2]dvy
Z ∞
−∞
exp[−kv2z /2]dvz
(1.16)
which simplifies to
h Ei =
3m
2
k
2π
3/2
2π
k
Z ∞
−∞
v2x exp[−kv2x /2]dv x =
Equating with the mean value of kinetic energy, we get k =
f (v x ) =
m
2πk B T
1/2
mv2x
.
exp −
2k B T
3m
.
2k
m
kB T .
(1.17)
Thus,
(1.18)
The distribution of molecular speeds is then given by
f (v) = 4π
m
2πk B T
3/2
mv2
,
v2 exp −
2k B T
(1.19)
where v2 = v2x + v2y + v2z . Subsequently, one can also find the distribution
function for energy E = 21 mv2 as:
g( E) = 2π
1
πk B T
3/2
E
1/2
E
exp −
kB T
(1.20)
1.1.2
Maxwell-Boltzmann distribution in D dimensions
Here we show that for integer or half-integer values of the parameter n the
Gamma distribution
γn (ξ ) = Γ(n)−1 ξ n−1 exp(−ξ ) .
(1.21)
Γ(n) is the Gamma function which represents the distribution of the rescaled
kinetic energy ξ (ξ = E/k B T) for a classical mechanical system in D = 2n
dimensions. T represents the absolute temperature of the system multiplied
by the Boltzmann constant k B .
The normalized probability distribution in momentum space is simply f (P) =
∏i (2πmk B T )− D/2 exp(−pi 2 /2mk B T ), where P = {p1 , . . . , p N } represents the
1.1 Kinetic theory of gases
5
momentum vectors of the N particles. Thus, since the kinetic energy distribution factorizes as a sum of single particle contributions, the probability density
factorizes as a product of single particle densities, each one of the form
1
p2
f (p) =
exp −
,
2mk B T
(2πmk B T ) D/2
(1.22)
where p = ( p1 , . . . , p D ) is the momentum of a generic particle. It is convenient
to introduce the momentum modulus p of a particle in D dimensions,
p2 ≡ p2 =
D
∑ p2k ,
(1.23)
k =1
where the pk ’s are thepCartesian components, since the distribution (1.22) depends only on p ≡
p2 . One can then integrate the distribution over the
D − 1 angular variables to obtain the momentum modulus distribution function, with the help of the formula for the surface of a hypersphere of radius p
in D dimensions,
2π D/2 D −1
SD ( p ) =
p
.
(1.24)
Γ( D/2)
One obtains
f ( p ) = SD ( p ) f (p) =
p2
2
D −1
p
exp
−
.
2mk B T
Γ( D/2)(2mk B T ) D/2
(1.25)
The corresponding distribution for the kinetic energy E = p2 /2m is therefore
D/2−1
E
exp
−
.
√
kB T
p = 2mE
(1.26)
Comparison with the Gamma distribution, 1.21, shows that the Maxwell-Boltzmann
kinetic energy distribution in D dimensions can be expressed as
f ( E) =
dp
f ( p)
dE
=
1
Γ( D/2)k B T
E
kB T
f ( E) = (k B T )−1 γD/2 ( E/k B T ) .
(1.27)
The distribution for the rescaled kinetic energy,
ξ = E/k B T ,
is just the Gamma distribution of order D/2,
dE
1
f (ξ ) =
f ( E)
ξ D/2−1 exp(−ξ ) ≡ γD/2 (ξ ) .
=
dξ
Γ
(
D/2
)
E =ξk B T
(1.28)
(1.29)
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1 EXPLAINING COMPLEX DISTRIBUTIONS WITH SIMPLE MODELS
1.2
The Asset exchange model
In principle, any model of distribution of wealth defined by a process of exchange of asset could be called an ‘asset exchange model’. Generally, the economic activity in these models involve interaction between two individuals
resulting in redistribution of their assets. Of course, ‘individual’ here could
mean a single person or a comglomerate such as a company. ‘Asset’ generally
refers to anything that contributes to the overall wealth of the economy, and
could be cash or any other physical asset.
A model for evolution of wealth distribution in an economically interacting
population was introduced by Ispolatov et al [2], where specific amounts of
assets are exchanged between two individuals. Two different cases were studied: a ‘random’ exchange where either of the individuals are likely to gain,
and a ‘greedy’ exchange, where the richer individual always gain. Again, two
types of exchanges were considered: the ‘additive’ exchange where a fixed
amount of asset transfers from an individual to the other, and again, a ‘multiplicative’ exchange, where the amount of asset exchanged is a finite fraction
of the wealth of one of the individuals. This work reports a variety of distribution for the asset, depending on the exact trading rules. For the additive type,
a random exchange produces Gaussian distribution of wealth, while greedy
exchange produces a Fermi-like scaled wealth distribution. For the multiplicative exchanges, the random exchange produces a steady state, while the
greedy exchange produces a continuously evolving power law wealth distribution. Studies [7] have also shown that if the amount of money transfered is
a small fixed quantity, or a fixed fraction of the trading pair’s average wealth,
or a random fraction of the average wealth of the entire population, and even
in cases of debt or bankrupcy, the equilibrium distribution is exponential.
Hayes [3] proposed two models of asset exchange in a closed, non-evolving
economy based on simple exchange rules: yard-sale (YS) and theft-and-fraud
(TF). In the YS model, the amount of wealth exchanged is a finite fraction
of that of the poorer trader, and the resultant distribution corresponds to a
monopoly, where all the wealth accumulates with one trader, as reported in
an earlier study [4]. In the TF model, the trading pair randomly chooses the
loser, and the amount of wealth exchanged is a random fraction of the donor.
Thus the rich has more to lose while the poor has more to gain. The resultant
equilibrium distribution is exponential.
Some extensions of these models [6, 5] produce distributions which are of
considerable interest. In an asymmetric exchange model [6], the wealth dynamics is defined by:
h
i
(
w (t)
wi (t) + ǫ 1 − τ 1 − w i (t) w j (t), if wi (t) ≤ w j (t)
j
w i ( t + 1) =
(1.30)
wi (t) + ǫw j (t),
otherwise,
1.3 Gas-like models
ǫ is a random number between 0 and 1. Here, τ = 0 corresponds to random
exchange model [7] while the τ = 1 corresponds to the minimum exchange
model [4, 3]. In general, the relation between agents is asymmetric and the
richer agent dictates the terms of trade. τ is known as the ‘thrift’ parameter,
and it measures the degree to which the richer agent is able to use its power.
If one considers an uniform distribution of τ among agents between 0 and 1,
one observes a power law distribution for larger wealth, with Pareto exponent
1.5 [6].
1.3
Gas-like models
In 1960, Mandelbrot wrote “There is a great temptation to consider the exchanges of money which occur in economic interaction as analogous to the
exchanges of energy which occur in physical shocks between molecules. In the
loosest possible terms, both kinds of interactions should lead to similar states
of equilibrium. That is, one should be able to explain the law of income distribution by a model similar to that used in statistical thermodynamics: many
authors have done so explicitly, and all the others of whom we know have
done so implicitly.” [8]. Unfortunately Mandelbrot did not provide any reference to this material!
In analogy to two-particle collisions with a resulting change in their individual kinetic energy (or momenta), income exchange models may be based on
two-agent interactions. Here two randomly selected agents exchange money
by some pre-defined mechanism. Assuming the exchange process does not
depend on previous exchanges, the dynamics follows a Markovian process:
mi (t)
m i ( t + 1)
(1.31)
=M
m j (t)
m j ( t + 1)
where mi (t) is the income of agent i at time t and the collision matrix M
defines the exchange mechanism.
In this class of models, one considers a closed economic system where total
money M and total number of agents N is fixed. This corresponds to a situation where no production or migration occurs and the only economic activity
is confined to trading. Each agent i, individual or corporate, possess money
mi (t) at time t. In any trading, a pair of traders i and j exchange their money
[9, 2, 7, 10], such that their total money is (locally) conserved and none end up
with negative money (mi (t) ≥ 0, i.e, debt not allowed):
mi (t + 1) = mi (t) + ∆m; m j (t + 1) = m j (t) − ∆m
(1.32)
following local conservation:
m i ( t ) + m j ( t ) = m i ( t + 1) + m j ( t + 1);
(1.33)
7
8
1 EXPLAINING COMPLEX DISTRIBUTIONS WITH SIMPLE MODELS
m (t)
i
m (t+1)
i
trading
m (t)
j
m (t+1)
j
Table 1.1 Schematic diagram of the trading process. Agents i and j redistribute their money in
the market: m i ( t) and m j ( t), their respective money before trading, changes over to m i ( t + 1)
and m j ( t + 1) after trading.
time (t) changes by one unit after each trading. The simplest model considers
a random fraction of total money to be shared [7]:
∆m = ǫij [mi (t) + m j (t)] − mi (t),
(1.34)
where ǫij is a random fraction ( 0 ≤ ǫij ≤ 1) changing with time or trading.
The steady-state (t → ∞) distribution of money is Gibbs one:
P(m) = (1/T ) exp(−m/T ); T = M/N.
(1.35)
Hence, no matter how uniform or justified the initial distribution is, the eventual steady state corresponds to Gibbs distribution where most of the people
have got very little money. This follows from the conservation of money and
additivity of entropy:
P ( m1 ) P ( m2 ) = P ( m1 + m2 ).
(1.36)
This steady state result is quite robust and realistic too! In fact, several variations of the trading, and of the ‘lattice’ (on which the agents can be put and
each agent trade with its ‘lattice neighbors’ only), whether compact, fractal
or small-world like [11], leaves the distribution unchanged. Some other variations like random sharing of an amount 2m2 only (not of m1 + m2 ) when
m1 > m2 (trading at the level of lower economic class in the trade), lead to
1.3 Gas-like models
even drastic situation: all the money in the market drifts to one agent and the
rest become truely pauper [3, 4].
1.3.1
Model with uniform savings
In any trading, savings come naturally [12]. A saving propensity factor λ was
therefore introduced in the random exchange model [10] (see [7] for model
without savings), where each trader at time t saves a fraction λ of its money
mi (t) and trades randomly with the rest:
(1.37)
mi (t + 1) = λmi (t) + ǫij (1 − λ)(mi (t) + m j (t)) ,
m j (t + 1) = λm j (t) + (1 − ǫij ) (1 − λ)(mi (t) + m j (t)) ,
(1.38)
where
∆m = (1 − λ)[ǫij {mi (t) + m j (t)} − mi (t)],
(1.39)
ǫij being a random fraction, coming from the stochastic nature of the trading.
The market (non-interacting at λ = 0 and 1) becomes ‘interacting’ for any
non-vanishing λ(< 1): For fixed λ (same for all agents), the steady state distribution P(m) of money is exponentially decaying on both sides with the mostprobable money per agent shifting away from m = 0 (for λ = 0) to M/N as
λ → 1 (Fig. 1.2). This self-organizing feature of the market, induced by sheer
self-interest of saving by each agent without any global perspective, is quite
significant as the fraction of paupers decrease with saving fraction λ and most
people end up with some finite fraction of the average money in the market
(for λ → 1, the socialists’ dream is achieved with just people’s self-interest
of saving!). Interestingly, self-organisation also occurs in such market models
when there is restriction in the commodity market [20]. Although this fixed
saving propensity does not give yet the Pareto-like power-law distribution,
the Markovian nature of the scattering or trading processes (eqn. (1.36)) is
effectively lost. Indirectly through λ, the agents get to know (start interacting with) each other and the system co-operatively self-organises towards a
most-probable distribution (m p 6= 0) (see Fig. 1.2).
There have been a few attempts to analytically formulate this problem [13]
but no analytic expression has been arrived at. It has also been claimed through
heuristic arguments (based on numerical results) that the distribution is a
close approximate form of the Gamma distribution [14]:
P(m) =
nn
mn−1 exp(−nm)
Γ(n)
(1.40)
where Γ(n) is the Gamma function whose argument n is related to the savings
factor λ as:
3λ
n = 1+
.
(1.41)
1−λ
9
1 EXPLAINING COMPLEX DISTRIBUTIONS WITH SIMPLE MODELS
2.5
λ0 = 0
λ0 = 0.1
λ0 = 0.6
λ0 = 0.9
2
P(m)
10
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
4
m
Table 1.2 Steady state money distribution P ( m ) for the model with uniform savings. The data
shown are for different values of λ: 0, 0.1, 0.6, 0.9 for a system size N = 100. All data sets shown
are for average money per agent M/N = 1.
This result has also been supported by numerical results in Ref. [15]. However,
a later study [16] analyzed the moments, and found that moments upto the
third order agree with those obtained from the form of the eqn. (1.41), and
discrepancies start from fourth order onwards. Hence, the actual form of the
distribution for this model still remains to be found out.
It seems that a very similar model was proposed by Angle [17, 18] several
years back in sociology journals. Angle’s ‘one parameter inequality process’
model (OPIP) is described by the equations:
m i ( t + 1)
m j ( t + 1)
= mi (t) + Dt wm j (t) − (1 − Dt )wmi (t)
= mi (t) + (1 − Dt )wmi (t) − Dt wm j (t)
(1.42)
where w is a fixed fraction and Dt takes value 0 or 1 randomly. The numerical
simulation results of OPIP fit well to Gamma distributions.
In the gas like models with uniform savings, the distribution of wealth
shows a self organizing feature. A peaked distribution with a most-probable
value indicates an economic scale. Empirical observations in homogeneous
groups of individuals as in waged income of factory labourers in UK and
USA [19] and data from population survey in USA among students of different school and colleges produce similar distributions [18]. This is a simple
1.3 Gas-like models
case where a homogeneous population (say, characterised by a unique value
of λ) has been identified.
1.3.2
Model with distributed savings
In a real society or economy, the interest of saving varies from person to person, which implies that λ is a very inhomogeneous parameter. To move a step
closer to this real situation, one considers the saving factor λ to be widely distributed within the population [21, 22, 23]. The evolution of money in such a
trading can be written as:
(1.43)
mi (t + 1) = λi mi (t) + ǫij (1 − λi )mi (t) + (1 − λ j )m j (t) ,
m j (t + 1) = λ j m j (t) + (1 − ǫij ) (1 − λi )mi (t) + (1 − λ j )m j (t)
(1.44)
The trading rules are similar as before, except that
∆m = ǫij (1 − λ j )m j (t) − (1 − λi )(1 − ǫij )mi (t)
(1.45)
here; where λi and λ j are the saving propensities of agents i and j. The agents
have fixed (over time) saving propensities, distributed independently, randomly and uniformly (white) within an interval 0 to 1: agent i saves a random
fraction λi (0 ≤ λi < 1) and this λi value is quenched for each agent, i.e., λi
are independent of trading or t.
The distribution P(m) of money M is found to follow a strict power-law
decay, which fits to Pareto law (??) with ν = 1.01 ± 0.02 ( Fig. 1.3). This power
law is extremely robust. For a distribution
ρ ( λ ) ∼ | λ0 − λ | α ,
λ0 6= 1,
0 < λ < 1,
(1.46)
of quenched λ values among the agents, the Pareto law with ν = 1 is universal
for all α. The data in Fig. 1.3 corresponds to λ0 = 0, α = 0.
The role of the agents with high saving propensity (λ → 1) is crucial: the
power law behavior is truely valid upto the asymptotic limit if 1 is included.
Indeed, had we assumed λ0 = 1 in Eqn. (1.46), the Pareto exponent ν immediately switches over to ν = 1 + α. Of course, λ0 6= 1 in Eqn. (1.46) leads to
the universality of the Pareto distribution with ν = 1 (irrespective of λ0 and
R1
α). Obviously, P(m) ∼ 0 Pλ (m)ρ(λ)dλ ∼ m−2 for ρ(λ) given by Eqn. (1.46)
and P(m) ∼ m−(2+α) if λ0 = 1 in Eqn. (1.46) (for large m values).
Patriarca et al. [27] studied the correlation between the saving factor λ and
the average money held by an agent whose savings factor is λ. This numerical study revealed that the product of this average money and the unsaved
fraction remains constant, or in other words, the quantity hm(λ)i(1 − λ) is a
11
1 EXPLAINING COMPLEX DISTRIBUTIONS WITH SIMPLE MODELS
10
x-2
0
10-2
P(m)
12
10-4
10
-6
10-8
10-3
N = 1000
M/N = 1
10-2
10-1
100
101
102
103
m
Table 1.3 Steady state money distribution P ( m ) for the distributed λ model with 0 ≤ λ < 1 for a
system of N = 1000 agents. The x −2 is a guide to the observed power-law, with 1 + ν = 2. Here,
the average money per agent M/N = 1.
constant. This result turns out to be the key to the formulation of a mean field
analysis to the model [28].
In a recent mean field approach, Mohanty [28] calculated the distribution
for the ensemble average of money for the model with distributed savings.
It is assumed that the distribution of money of a single agent over time is
stationary, which means that the time averaged value of money of any agent
remains unchanged independent of the initial value of money. Taking the
ensemble average of all terms on both sides of Eqn. 1.43, one can write
"
#
1 N
hmi i = λi hmi i + hǫi (1 − λi )hmi i + h ∑ (1 − λ j )m j i .
(1.47)
N j =1
The last term on the right is replaced by the average over the agents, where
it is assumed that any agent (ith agent here) on the average, interacts with all
others in the system, which is the mean field approach.
Writing
*
+
1 N
h(1 − λ)mi ≡
(1.48)
(1 − λ j ) m j
N j∑
=1
1.3 Gas-like models
101
10
δ = -0.8
δ = -0.4
δ = +0.4
δ = +0.8
1 + ν =2
0
10-1
P(m)
10-2
10-3
10-4
10-5
N = 200, M/N = 1
10-6
10-7
10-2
10-1
100
101
102
m
Table 1.4 Steady state money distribution P ( m ) in the model for N = 200 agents with λ distributed as ρ( λ) ∝ λα with different values of α. For all cases, the average money per agent
M/N = 1.
and since ǫ is assumed to be distributed randomly and uniformly in [0, 1], so
that hǫi = 1/2, Eqn. 1.47 reduces to
(1 − λi )hmi i = h(1 − λ)mi.
(1.49)
Since the right side is free of any agent index, it seems that this relation is true
for any arbitrary agent, i.e., hmi i(1 − λ) = constant, where λ is the saving
factor of the ith agent. What follows is, dλ = const. dm
. An agent with a
m2
(characteristic) saving propensity factor (λ) ends up with wealth (m) such that
one can in general relate the distributions of the two:
P(m) dm = ρ(λ) dλ.
(1.50)
Therefore, the distribution in m is bound to be of the form:
P(m) ∝
1
m2
for uniform distribution of savings factor λ, i.e, ν = 1. This analysis can also
explain the non-universal behavior of the Pareto exponent ν, i.e, ν = 1 + α for
13
14
References
ρ(λ) = (1 − λ)α . Thus, this mean field study explains the origin of the universal (ν = 1) and the non-universal (ν 6= 1) Pareto exponents in the distributed
savings model.
These model income distributions P(m) compare very well with the wealth
distributions of various countries: data suggests Gibbs like distribution in the
low-income range (more than 90% of the population) and Pareto-like in the
high-income range [24, 25, 26] (less than 10% of the population) of various
countries. In fact, we compared one model simulation of the market with saving propensity of the agents distributed following Eqn. (1.46), with λ0 = 0 and
α = −0.7 [21]. The qualitative resemblance of the model income distribution
with the real data for Japan and USA in recent years is quite intriguing. In
fact, for negative α values in Eqn. (1.46), the density of traders with low saving propensity is higher and since λ = 0 ensemble yields Gibbs-like income
distribution (1.35), we see an initial Gibbs-like distribution which crosses over
to Pareto distribution (??) with ν = 1.0 for large m values. The position of the
crossover point depends on the value of α. It is important to note that any
distribution of λ near λ = 1, of finite width, eventually gives Pareto law for
large m limit. The same kind of crossover behavior (from Gibbs to Pareto) can
also be reproduced in a model market of mixed agents where λ = 0 for a finite
fraction p of population and λ is distributed uniformly over a finite range near
λ = 1 for the rest 1 − p fraction of the population.
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