Maria Letizia Bertotti – Faculty of Science and Technology
Mathematical Models for Socio-Economic Problems
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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- Wealth Inequality
- Income Distribution
- Tax Evasion
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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- Wealth Inequality
- Income Distribution
- Tax Evasion
in collaboration with Giovanni Modanese
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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“At this point we may allude to the broad view which is opened
by applying this science to the statistics of animated beings,
of human society, of sociology, etc., and not merely upon
mechanical particles.”
Ludwig Boltzmann
Über statistiche Mechanik (1904),
Conference at the scientific congress of St. Louis in 1904, first published in an english
translation with the title The relations of Applied Mathematics in Congress of Arts and
Sciences … St Louis 1904, Boston (1905)
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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Outline
 Introduction and perspective
 The subject at hand: taxation and redistribution process
 A mathematical framework
 A prototype model: analytical results and computational outputs
 Introducing tax evasion in the model
 Conclusions and further investigations
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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 Introduction and perspective
… a perspective which puts the interactions among the
heterogeneous individuals at the very heart of the question.
These interactions lead to self-organized aggregate patterns
and regularities, which emerge from the system as a whole.
No global controller.
Emphasis on the process through which structures and
patterns emerge from the “microscopic” interactions.
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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Economy as a Complex System
… a growing strand of economists …
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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Economy as a Complex System
… a growing strand of economists …
Thomas Shelling
,
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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Economy as a Complex System
… a growing strand of economists …
Thomas Shelling
,
Alan Kirman
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
,
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Economy as a Complex System
… a growing strand of economists …
Thomas Shelling
,
Alan Kirman
,
Brian W. Arthur
,
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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Economy as a Complex System
… a growing strand of economists …
Thomas Shelling
,
Alan Kirman
,
Brian W. Arthur
,
Mauro Gallegati
Maria Letizia Bertotti FaST – LUB
,…
Mathematical Models for Socio-Economic Problems
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Computational Agent-Based Models
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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Computational Agent-Based Models
Statistical Mechanics Tools
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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Computational Agent-Based Models
Statistical Mechanics Tools
Gas Kinetic Theory
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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Computational Agent-Based Models
Statistical Mechanics Tools
Kinetic Theory of Gases
Econophysics
(E.H. Stanley)
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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 The subject at hand: taxation and redistribution process
… monetary exchanges in a closed trading market society.
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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Divide a population of individuals in a finite number n of
classes, characterized by their average income. Let
r 1 < r2 < … < r n
τ1 ≤ τ2 ≤ … ≤ τn
the average incomes of the n classes
the tax rates
xi(t) the fraction at time t of individuals in the i-th class
S a fixed amount of money
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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government
S τk
h
S
Maria Letizia Bertotti FaST – LUB
k
Mathematical Models for Socio-Economic Problems
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government
S τk
h
S (1 - τk)
Maria Letizia Bertotti FaST – LUB
k
Mathematical Models for Socio-Economic Problems
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S τk
h
S (1 - τk)
Maria Letizia Bertotti FaST – LUB
k
Mathematical Models for Socio-Economic Problems
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indirect interaction
S τk
h
S (1 - τk)
k
direct interaction
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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 A mathematical framework
... a system of n ordinary di↵erential equations (a simplified version)
n X
n ⇣
⌘
X
dxi
i
i
=
Chk
+ T[hk]
(x) xh xk
dt
h=1 k=1
with
xi
n
X
xk ,
i = 1, ..., n,
k=1
• the direct transition probability densities
i
Chk
2 R+ ,
n
X
i
(
Chk
= 1 for any fixed h, k)
i=1
• the indirect transition variation densities
i
T[hk]
: Rn ! R,
n
X
i
(
T[hk]
(x) = 0 for any fixed h, k, for x 2 Rn )
i=1
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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 A prototype model
i
i
Take Chk
= Aihk + Bhk
, where the only nonzero elements Aihk are Aiij = 1 and
i
the only possibly nonzero elements Bhk
are of the form
i
Bi+1,k
= pi+1,k S
1
⌧k
,
ri
ri+1
1 ⌧i
i
Bi,k
= pk,i S
ri+1 ri
1 ⌧i
Bii 1,k = pk,i 1 S
ri ri
where
pi,k S
1
1
ri
⌧k
,
ri 1
,
1
ph,k = (1/4) min{rh , rk }/rn
with the exception of some terms (p1,k = 0, ph,n = 0, ...).
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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i
i
i
And take T[hk]
(x) = U[hk]
(x) + V[hk]
(x), where
and
ph,k S ⌧k
i
P
U[hk] (x) = n
j=1 xj
i
V[hk]
(x) = ph,k S ⌧k
✓
✓
xi
ri
xi
1
ri
1
h,i+1
rh
Maria Letizia Bertotti FaST – LUB
ri
rh
ri+1
ri
◆
◆ Pn 1
h,i
j=1 xj
Pn
.
ri 1
j=1 xj
Mathematical Models for Socio-Economic Problems
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 Analytical results (for the prototype model)
Theorem In correspondence to any initial
Pncondition x0 = (x01 , . . . , x0n ),
for which x0i
0 for all i = 1, . . . , n and
i=1 x0i = 1, a unique solution
x(t) = (x1 (t), . . . , xn (t)) of the evolution equation exists, which is defined for
all t 2 [0, +1), satisfies x(0) = x0 and also
xi (t)
0 for all i = 1, . . . , n and
n
X
xi (t) = 1 for all t
0.
i=1
i
i
(The expressions of the U[hk]
(x) and V[hk]
(x) become linear in the variables xj
and, accordingly, the right hand sides of the equations are polynomials containing cubic terms as the highest degree ones.)
Pn
Theorem The scalar function µ(x) = i=1 ri xi , expressing the global income and, due to the population normalization, also the mean income, is a first
integral for the evolution equations.
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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 Computational outputs (for the prototype model)
we fix the following parameter values:
we take n = 25,
and we choose the rj and the ⌧j for j = 1, . . . , n.
For example, we take rj = 10j and
j
⌧j = ⌧min +
n
where
⌧min = 30/100,
Maria Letizia Bertotti FaST – LUB
1
(⌧max
1
⌧min )
⌧max = 60/100.
Mathematical Models for Socio-Economic Problems
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we stress that we could choose other parameter values;
for instance, the incomes rj can be taken
to increase in a nonlinear way as j increases.
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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 Computational outputs (for the prototype model)
• Uniqueness of the asymptotic stationary distribution
for any fixed value of µ
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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Final 1
Initial 1
class
class
population
population
Initial 2
Final 2
class
Maria Letizia Bertotti FaST – LUB
class
Mathematical Models for Socio-Economic Problems
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 Computational outputs (for the prototype model)
• Uniqueness of the asymptotic stationary distribution
for any fixed value of µ
• Dependence of the asymptotic stationary distribution
on ⌧max
⌧min
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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0.14
0.08
⌧min = 40%, ⌧max = 50%.
0.12
0.10
0.06
0.08
0.06
0.04
0.04
0.02
0.02
0.00
0.00
0.14
0.08
⌧min = 30%, ⌧max = 60%.
0.12
0.10
0.06
0.08
0.06
0.04
0.04
0.02
0.02
0.00
0.00
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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The di↵erence in each class of the fraction of individuals
of the asymptotic distribution relative to the model with
⌧min = 40%, ⌧max = 50%
and the asymptotic distribution relative to the model with
⌧min = 30%, ⌧max = 60%.
0.006
0.004
0.002
0.000
!0.002
!0.004
!0.006
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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 Computational outputs (for the prototype model)
• Uniqueness of the asymptotic stationary distribution
for any fixed value of µ
• Dependence of the asymptotic stationary distribution
on ⌧max
⌧min
• Emergence of distribution tails exhibiting
a power-law behaviour
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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For suitable values of µ,
the asymptotic stationary distributions
exhibit power-law decreasing behaviour in tails.
... V. Pareto
more than a century ago ...
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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The power-law behaviour of a function f (x) = cx ↵
is equivalently expressed by the equation: ln f (x) = ln c ↵ ln x.
Therefore, if the distribution tail exhibits such a behaviour,
in the log-log plot in the variables x and f a straight line must be seen.
!
0.0150
!
0.08
0.0100
!
!
0.06
0.0070
0.04
!
!
0.0050
!
0.02
0.0030
0.00
!
180.
Maria Letizia Bertotti FaST – LUB
200.
220.
Mathematical Models for Socio-Economic Problems
!
240.
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To check the power law character of the tails we may take e.g. incomes
which increase exponentially as ri = 3 (1, 67)i . The basis 1.67 is chosen in such
a way as to obtain an income increase by a factor 100 when the class index i
increases by 9 (the tail of the distribution is typically fitted to look for a power
law in the range from i = 16 to i = 24).
The tail of the distribution (for i=16, …24)
The log-log plot (for i=16, …24)
!
0.030
!
10"6
0.025
!
!
0.020
!
10"7
0.015
!
0.010
!
"8
10
0.005
0.000
!
2 ! 104
Maria Letizia Bertotti FaST – LUB
5 ! 104
1 ! 105
2 ! 105
Mathematical Models for Socio-Economic Problems
5 ! 105
!
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From a Report of Banca d’Italia
(Supplementi
al Bollettino Statistico …
I redditi delle famiglie italiane
2008)
Outputs of our simulations
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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 Introducing tax evasion in the model
Consider the case in which the tax evasion
provides an advantage not only to the individual
who on the occasion of a trade is earning money,
but also to that one who is paying.
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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tax compliance case
S τk
h
S (1 - τk)
Maria Letizia Bertotti FaST – LUB
k
Mathematical Models for Socio-Economic Problems
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θk< τk
tax evasion case
S θk
h
S (1 – (τk + θk)/2)
Maria Letizia Bertotti FaST – LUB
k
Mathematical Models for Socio-Economic Problems
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θk = 4/5 τk
tax evasion case
S θk
h
S (1 – (τk + θk)/2)
Maria Letizia Bertotti FaST – LUB
k
Mathematical Models for Socio-Economic Problems
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0.10
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0.00
0.00
asymptotic distribution with tax compliance
⌧min = 30%, ⌧max = 60%.
asymptotic distribution with tax evasion
0.4
0.3
0.010
0.2
0.005
0.1
0.000
0.0
!0.005
difference: (tax evasion) – (tax compliance)
Maria Letizia Bertotti FaST – LUB
percentage variation
Mathematical Models for Socio-Economic Problems
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0.12
0.10
⌧min = ⌧max = 45%.
0.10
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0.00
0.00
asymptotic distribution with tax compliance
asymptotic distribution with tax evasion
0.8
0.04
0.6
0.03
0.4
0.02
0.01
0.2
0.00
0.0
!0.01
!0.2
difference: (tax evasion) – (tax compliance)
Maria Letizia Bertotti FaST – LUB
percentage variation
Mathematical Models for Socio-Economic Problems
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The simulations systematically show that
the e↵ect of the tax evasion is that
there is an increment in the number of individuals
belonging to the poorest classes and the richest ones
at the detriment of the middle classes.
...
Also, the Gini index turns out to be
larger in the presence of tax evasion.
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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The Gini index is defined based on the Lorenz curve,
which plots the proportion of the total income of the population
(y axis)
that is cumulatively earned by the bottom x% of the population.
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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A
Gini index =
half square
% of total income 100
line of perfect equality
society with some inequality
A
society with more inequality
100
Maria Letizia Bertotti FaST – LUB
% of population
Mathematical Models for Socio-Economic Problems
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tax compliance
tax evasion
120
120
100
100
80
80
100sx
60
100sx
60
100srx/mu
100srx/mu
40
40
20
20
0
0
0 20 40 60 80 100 120 Gini index 0.321
0 20 40 60 80 100 120 Gini index 0.337
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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 Conclusions and further investigations
Summarizing:
this is a rudimentary model, but it seems to capture the main ingredients of the problem
our interest is not so much on a particular model as on the general explorative strategy
Remarks:
finite number of classes
the choice of the phk … a network structure? …
For future research:
… a number of aspects and directions towards which address investigation;
in particular:
a calibration of parameters closer to real world ones
analytical proofs of some computationally evident facts
…
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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In conclusion:
mathematics suggests that
paying taxes and use them correctly is
a good and civilized practice, a way to contribute
together to a reduction of social inequalites
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
Page 49
References
• Bertotti, M.L.: Modelling taxation and redistribution: a discrete active
particle kinetic approach. Appl. Math. Comput. 217, 752–762 (2010)
• Bertotti, M.L., Modanese G.: From microscopic taxation and redistribution
models to macroscopic income distributions. Physica A 390, 3782–3793 (2011)
• Bertotti, M.L., Modanese G.: Exploiting the flexibility of a family of models
for taxation and redistribution. Eur. Phys. J.B 85, 261 (2012)
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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Thank you for your attention!
Maria Letizia Bertotti FaST – LUB
Mathematical Models for Socio-Economic Problems
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