Page 190 -

11 Socio-Economic Modeling
186

∼ 1.6, Japan ∼ 1.8–2.2, [6]). It is also known from statistical studies that typi-
cally less than 20% of the population of any country own about 80% of the total
wealth of that country. The top income group obeys the above Pareto law while
the remaining low income population, in fact the majority (80% or more), follow
a different distribution, which is typically Gibbs [6] or log-normal.
Kinetic models of the time evolution of wealth distributions can be described
in terms of a Boltzmann-like equation which reads

∂f
= Q( f , f ) , (11.1)
∂t
where f = f (v, t) is the probability density of agents of wealth v ∈ R + at time
t ≥ 0, and Q is a bilinear operator which describes the change of f due to binary
trading events among agents. We shall refer to this equation in the sequel as
kinetic Pareto–Boltzmann equation.
Theinvolvedbinarytradingsare described by therules

v = p 1 v + q 1 w ; w = p 2 v + q 2 w , (11.2)
∗
∗
where (v, w) denote the (positive) moneys of two arbitrary individuals before the
∗
trading and (v , w ) the moneys after the trading. The transaction coefficients
∗
p i , q i , i = 1, 2 are either given constants or random variables, with the obvious
constraint of non-negativity. Also, they have to be such that the transformation
from the money states before trading and after trading is non-singular. Among
all possible kinetic models of type (11.1), (11.2) the conservative models are
characterized by the property

p 1 + p 2 = 1, q 1 + q 2 = 1,

where · denotes the probabilistic expectation. This guarantees conservation of
the total expected wealth of the market (which is the first order moment of the
distribution function, multiplied by the total number of individuals).
In weak form the collision operator Q( f , f )isdefined by

Q( f , f )(v)φ(v) dv

+

= 1 φ(v )+ φ(w )− φ(v)− φ(w) f (v)f (w)dv dw . (11.3)

∗
∗
2
+ +
Hereφ is a smooth test function with compact support in the non-negative reals.
Note that the collision operator is assumed to be ofso-called Maxwelliantype,
i.e. thescatteringkerneldoesnot depend on therelativewealthofcollisions and
can therefore be accounted for in the computation of the statistical expectation
by choosing the probability space appropriately.