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11 Socio-Economic Modeling
191
Fig. 11.3. Hongkong: in the thin w = O(1)-part of the Pareto distribution

tial differential equation of Fokker–Planck type for the distribution of money
among individuals. This diffusion-convection equation reads:
2
∂f λ ∂ ∂
= v f + (v − m)f . (11.8)
2
∂t 2 ∂v 2 ∂v
In (11.8) m is the mean wealth,

m = vf (v, t) dv ,
+
which is time-conserved assuming that f hasbeenscaledtobeaprobability
density. The same Fokker–Planck equation was obtained in [2] as the mean field
limit of a stochastic equation, as well as in [9,14] in the context of generalized
Lotka–Volterra dynamics.
The equilibrium state of the Fokker–Planck equation can be computed ex-
plicitly and is of Pareto type, namely it is characterized by a power-law tail for
the richest individuals. By assuming for simplicity m = 1wefind:

μ−1
μ exp −
(μ −1) v
f ∞ (v) = (11.9)
Γ(μ) v 1+μ
where
2
μ = 1+ > 1.
λ
We remark that the tails of the Pareto steady state of the Fokker–Planck
2
equation are related to the coefficients s and σ which appear in the collision
2
rule (11.6), with σ /s = λ!
Another important field in which microscopic kinetic models describing
the collective behavior and self-organization in a society [16] can be fruitfully
employed is the modeling of opinion formation (cfr. [1,13,15] and the references
therein).
In these studies, formation of opinion is described by mean field model equa-
tions. They are in general systems of ordinary differential equations or partial
differential equations of diffusive type. In [1], attention is focused on two aspects
of opinion formation, which in principle could be responsible for the formation
of coherent structures. The first one is the remarkably simple compromise pro-
cess, in which pairs of agents reach a fair compromise after exchanging opinions.
The second one is a diffusion process, which allows individual agents to change
their opinions in a random diffusive fashion. While the compromise process has
its basis in the human tendency to settle conflicts, diffusion accounts for the pos-
sibilitythatpeoplemaychangeopinionthroughaccesstoinformation.Atpresent