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11 Socio-Economic Modeling
189
Fig. 11.1. Angra dos Reis, Brazil: on the w = o(1)-part of the Pareto distribution

Fig. 11.2. Salvador de Bahia, Brazil: on the w = o(1)-part of the Pareto distribution

same effect can also be obtained by simply allowing the random variables in the
trading laws (11.6) to assume values on the whole real axis, and at the same time
discarding those trades for which one of the post-trade wealths is non-positive.
A critical analysis of the discussed collision=trading rules reveals a deep
analogy between the economic models described above and the granular ma-
terial flows modeling framework of Chapter 3. They share the property that
the steady (or, more generally, the self-similar asymptotic) states are different
from the classical Maxwell distribution of the Boltzmann equation of gas dy-
namics presented in Chap. 1. Another analogy becomes evident when looking
at the non-conservative properties of the economic and granular Boltzmann
equations, resulting from inelastic binary collision models.
Conservativeexchangedynamicsbetweenindividualsredistributethewealth
among people. Without conservation, the best way to extract information on the
large-time behavior of the solution relies on scaling the solution itself to keep
the average wealth constant after scaling. Nevertheless, the explicit form of the
limit distribution of the kinetic equation remains extremely difficult to recover,
and often requires the use of suitable numerical methods.
A complementary method to extract information on the steady state distri-
bution was linked in [5] to the possibility of obtaining particular asymptotics,
which mimic the characteristics of the solution of the original problem for large
times. The main result in this direction was to show that the kinetic model
converges (under appropriate assumptions) in a suitable scaling limit to a par-