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evasion models may be categorized according to the features of this individual interac-
tion process. In fact, in econophysics models this process is driven by statistical me-
chanics using the Ising model (Ising, 1925) that is well known in physics and describes
objects which can be in one of two states and interact on a given lattice structure. Ex-
amples include Zaklan et al. (2008, 2009), Lima and Zaklan (2008), and Lima (2010)
which have identified the Ising states with compliant and non-compliant tax payers. In
contrast, if the interacting process is driven by parameter changes that induce behavioral
changes via a utility function and (or) by stochastic processes that do not have physi-
cal roots, these models belong to the economics domain. Examples include Mittone and
Patelli (2000); Davis et al. (2003); Bloomquist (2004, 2008); Korobow et al. (2007); An-
tunes et al. (2007); Szab o et al. (2009, 2010); Hokamp and Pickhardt (2010); Meder et
al. (2012); Nordblom and Zamac (2012); Andrei et al. (2013); Hokamp (2013); Pellizari
and Rizzi (2013) of which some are summarized by Bloomquist (2006) and Pickhardt
and Seibold (2013).
In agent-based tax evasion models of the econophysics type the Ising model is used
to mimic conditional cooperation among agents (Zaklan et al., 2009). Yet, the actual
patterns and levels of tax evasion in these models depend on two additional factors: the
network structure of society and the tax enforcement mechanism. The network structure
is implemented by alternative lattice types and tax enforcement consists of the two
economic standard parameters audit probability and penalty rate. To this extent, rational
behavior patterns are essentially reconstructed by means of statistical mechanics.
In previous work Pickhardt and Seibold (2013) and more recently Seibold and Pick-
hardt (2013) have extended the Ising-based econophysics approach to tax evasion to-
ward the implementation of different agent types. This theory is able to reproduce
results from agent-based economics models (Hokamp and Pickhardt, 2010) so that it
should be also appropriate for a quantitative analyis. Following this idea we aim in
the present contribution to apply the model to an analysis of the shadow economy in
France and Germany with respect to the percentage of rational agents in both countries.
Note that in the present paper we use the term 'shadow economy' synonymous with the
participation in black market services.
In Sec. 2 we outline the basic ingredients of our econophysics model and exem-
plify the approach for a black market with homogeneous agents. We apply our model
to the analysis of the shadow economies in Germany and France in Sec. 3 where we
deduce the essential parameters entering the simulations from previous experimental
and agent-based investigations in the literature. Finally, we discuss our results for dif-
ferent enforcement schemes and give policy recommendations to combat the shadow
economy in Sec. 4.
2
The Agent-Based Econophysics Approach
Our considerations are based on the Ising model hamiltonian
∑
ij
−
∑
i
H
=
−
J
S
i
S
j
B
i
S
i
(1)
where
J
describes the coupling of Ising variables (spins)
S
i
=+1,
−
1 between adjacent
lattice sites denoted by
ij
. In the present context
S
i
=+1(
−
1) is interpreted as a
